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Removing DESpecs directory which deserted to git
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0" encoding="UTF-8"?> <TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <teiHeader> <fileDesc> <titleStmt> <title xml:lang="la">Diversarum Speculationum mathematicum, & physicarum liber</title> <author>Benedetti, Giovanni Battista de</author> </titleStmt> <publicationStmt> <date>1585</date> <idno>/permanent/library/163127KK</idno> <availability> <p>open access</p> <p>http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration</p> <p>free</p> </availability> </publicationStmt> <sourceDesc> <p>Automatically created from the ECHO source; ECHO version 1.0; Script version 2011-06-03</p> </sourceDesc> </fileDesc> <profileDesc> <langUsage> <language ident="lat">lat</language> </langUsage> </profileDesc> </teiHeader> <text xml:lang="la" type="book"> <front> <div type="cover"> <pb facs="0001"/> <pb facs="0002"/> <pb facs="0003"/> <pb facs="0004"/> <pb facs="0005"/> </div> <div type="title"> <head xml:space="preserve">IO. BAPTISTAE <lb/>BENEDICTI <lb/>Patritij Veneti Philoſophi.</head> <head xml:space="preserve"><hi rend="italics">DIVERSARVM SPECVLATIONVM <lb/>Mathematicarum, & Phyſicarum</hi> <lb/>Liber.</head> <head xml:space="preserve">Quarum ſeriem ſequens pagina indicabit.</head> <head xml:space="preserve">AD SERENISSIMVM CAROLVM EMANVELEM <lb/>ALLOBROGVM, ET SVBALPINORVM <lb/>DVCEM INVICTISSIMVM.</head> <!-- <figure position="here"> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0005-01"/> </figure> --> <figure place="here"> <!-- <graphic url="0005-01"/> --> <graphic url="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0005-01"/> </figure> <head xml:space="preserve"><hi rend="small caps">Tavrini</hi>, Apud Hæredem Nicolai Beuilaquæ, <num value="1585" rend="small caps">mdlxxxv</num>. <lb/><hi rend="italics">Superioribus permiſſum</hi>.</head> <pb facs="0006"/> </div> <div type="toc"> <head rend="italics" xml:space="preserve">TRACTATVS QVI IN HOC <lb/>volumine continentur.</head> <head xml:space="preserve">Theoremata Arithmetica.</head> <head xml:space="preserve">Derationibus operationum perſpectiuæ.</head> <head xml:space="preserve">De Mechanicis.</head> <head xml:space="preserve">Diſputationes de quibuſdam placitis Ariſt.</head> <head xml:space="preserve">In quintum Euclidis librum.</head> <head xml:space="preserve">Phyſica, & Mathematica reſponſa per Epiſtolas.</head> <pb facs="0007"/> </div> <div type="dedication"> <head xml:space="preserve">SERENISSIMO <lb/>CAROLO EMANVELI <lb/>Sabaudiæ Duci, <choice><ex>&c.</ex><am>&c.</am></choice></head> <p rend="italics"> <s xml:space="preserve"><hi rend="small caps italics">AGitvr</hi> nonusdecimus annus ex quo litte-<lb/>ris Serenißimi patris tuæ Celſitudinis, ac-<lb/>cerſitus ex vrbe Parmenſi in banc me ciui-<lb/>tatem contuli. </s> <s xml:space="preserve">Is aduenientem tam bumanè <lb/>excepit, tanta deinde liberalitate fuit com-<lb/>plexus ego vicißim ei deſeruiendi, tam vebe-<lb/>menti cupiditate fui accenſus, vt ſub eius ditione quodſuper-<lb/>eßet vitæ agere conſtituerem. </s> <s xml:space="preserve">Cuius in me benignitas, mea <lb/>in illum obſeruantia mirum in modum mutuo vſu, & conſue-<lb/>tudine eſt adaucta, vt idem Dux me ſecum dum ruſticaretur <lb/>eße vellet, ſæpè etiam ſecum pernoctare; </s> <s xml:space="preserve">quo quidem tempo-<lb/>re de Matbematicis ſcientijs mecum agebat, in quibus perdi-<lb/>ſcendis mea opera vtebatur, quæſtiones, Arithmeticam, Geo<lb/>metriam, Opticen, Muſicam, aut Astrologiam ſpectantes <lb/>proponens. </s> <s xml:space="preserve">Cui vt quod in me eßet ſatisfacerem, acrius <lb/>quàm anteainea studia (adquætamen ſemper fui propenſißi-<lb/>mus) incubui. </s> <s xml:space="preserve"><choice><ex>Illiusque</ex><am>Illiusq́ꝫ</am></choice> imitatione (vt ferècæteri Principum <lb/>studiaimitantur) non pauci aut præſentes, aut per litter as me <lb/>de his, atque illis Mathematicis quæstionibus conſuluerunt. <lb/></s> <s xml:space="preserve">Cùmque ego nunquam laborem amicorum cauſa defugerim, <lb/>euenit vt post tot annorum curricula, mea ſcrinia ſcrutatus, <lb/>inuenerim tot abſolutas quæſtiones, vt ex eis corpus mediocre <lb/>effici poſſe videretur. </s> <s xml:space="preserve">Quas, cùm rationibus in epiſtola ſub-<lb/>ſequenti allatis edere constituiſſem, non ſub cuiuſque alte-<lb/>rius nomine, & auſpicijs quam tuæ Celſitudinis volui apparere; <lb/></s> <s xml:space="preserve">tum quòd patri debitum libellum filio reddere par erat, tum <pb facs="0008"/> quòd in tuæ Celſitudine paternam in me fouendo, & augendo<unclear reason="illegible"/> <lb/>benignit atem ineße ſemper ſum expertus, tum quòd tuæ Celſi-<lb/>tudinis interrog ationibus excitatus non pauca quæ hoc volu-<lb/>mine continentur, elucubraui. </s> <s xml:space="preserve">Acceßit, quod ego ſemper in <lb/>his dedic ationibus ſpectandum put aui, tuam Celſitudinem t<unclear reason="illegible"/>an-<lb/>tos progreßus in Mathematicis feciſſe, vt vel idonea æſtima-<lb/>trix mearum vigiliarum eſſe poßit. </s> <s xml:space="preserve">Quare, & veterum Per-<lb/>ſarum Regum gloriam æquauit, & nos veluti in ſpem certam <lb/>fælicitatis buius ſæculi induxit, ſi verum eſt Platonis va-<lb/>ticinium, beat am eam futuram Rempublic am in qua <lb/>Principes Philoſophentur. </s> <s xml:space="preserve">Tua igitur celſi-<lb/>tudo libellum tot ei nominibus debitum, <lb/>ea qua ſolet bumanitate accipe-<lb/>re nè grauetur. </s> <s xml:space="preserve">Deus tuas <lb/>omnes cogitationes, <lb/>& conatus ad <lb/>fœlicißi-<lb/>mos <lb/>ſemper exitus perducat, <lb/><choice><ex>teque</ex><am>teq́ꝫ</am></choice> diutißimè ſer-<lb/>uet incolu-<lb/>mem.</s> </p> <pb facs="0009"/> </div> <div type="preface"> <head xml:space="preserve">AD LECTOREM</head> <p> <s xml:space="preserve"><hi rend="small caps">CVm</hi> Varijs temporibus permulta in diuerſis <lb/>diſciplinis contemplatus ſim, partim à præ-<lb/>ſtantibus viris patronis ac amicis meis exci-<lb/>tatus, quiſuper eis ſententiam meam exquire-<lb/>bant, partim, abingenito mihi deſiderio, ra-<lb/>tionem, & cauſam eorum percipiendi, com-<lb/>mittendum non putaui, quin qualiacunque <lb/>meaſcripta in illis ſcientijs, ſtudioſis impartirer, <lb/>non dubitans quin illis aliquid commodi atque vtilitatis allatura ſint, prę <lb/>ſertim cum in eiuſmodi quæſtionibus inueſtigandis atque perpendendis, <lb/>nemo ( quod ſciam ) hactenus elaborauerit. </s> <s xml:space="preserve">Nihil enim his libris à me <lb/>traditum eſt, quod aut legiſſe, aut ab alijs audiuiſſe meminerim, nam ſi <lb/>aliena attigi, ea, aut cum aliqua differentia demonſtrationis, aut diluci-<lb/>dius ſcripſi, quod ſi forte alius eadem tradidit, aut eius lucubrationes ad <lb/>me non peruenerunt, aut earum perlectionis memoria excidit. </s> <s xml:space="preserve">Vtenim <lb/>etiam Ariſtoteles ipſe ſenſit facilè fieri poteſt, vt pluribus, eædem opinio-<lb/>nes in mentem veniant. </s> <s xml:space="preserve">Immo multa ſcribenti euenire poteſt, vt cum <lb/>iamdiu aliquid ſcripſerit, iam oblitus, idem repetat, quod mihi etiam <lb/>nonnunquam accidit. </s> <s xml:space="preserve">In his autemlibris non ſuſcepi munus integræ ali <lb/>cuius ſcientiæ tradendæ, ne, quæ abalijs iam tradita ſunt, ipſe inutiliter re <lb/>peterem, mihiq́ue viderer exalienis laboribus laudem voluiſſe comparare. <lb/></s> <s xml:space="preserve">Singularum enim ſcientiarum volumina, iam ab alijs collecta, at-<lb/>que in ordinemſunt digeſta, & ſi pauciſſimi ſint libri quorum omnes <lb/>ſententiæ, omniaq́ue inuenta vnius ſint authoris, excipio Archime-<lb/>dis volumina. </s> <s xml:space="preserve">Cumque multi ſint, qui vel vnam rem à ſe inuentam <lb/>in publicum proferre non dubitent, multo magis mihi qui multa ex-<lb/>cogitaui, & ſi inter ſe hætereogenea, atque vtcunque expreſſa, idem <lb/>licere ſum arbitratus. </s> <s xml:space="preserve">In his autem meditandis, ex Arithmeticis autho-<lb/>ribus quos inſpexi, præcipuus fuit Nicolaus Tartalea, quippe quem fe-<lb/>rè omnia ab alijs ſcripta collegiſſe conſtat, nec alios ex præcipuis, quos le-<lb/>gere potui omittendos duxi, inter quos ſunt Hieronymus Cardanus, Mi-<lb/>chael Stifelius, Gemma Friſius, Ioannes Nouiomagus, Cuthebertus <lb/>Tonſtallus, <choice><ex>cæterique</ex><am>cæteriq́;</am></choice> huiuſinodi. </s> <s xml:space="preserve">Quorundam tamen volumina illorum <lb/>qui à Tartalea citantur, vt Leonardi Piſani, Proſdocimi, Ioannis Infor-<lb/>tunati, Fratris Lucæ, Petri Borgi, aliorumq́ue aliquot inſpiciendorum, <pb facs="0010"/> facultas mihi non fuit. </s> <s xml:space="preserve">Præterea, licet in his libris nonnullę inueniantur <lb/>propoſitiones, quæ diſiunctam ab alijs habeant rationem, eæ non ſper-<lb/>nendæ tamen ſunt, viam fortaſſe alicui aperient vlterius progrediendi. <lb/></s> <s xml:space="preserve">Quemadmodum enim, exempli gratia, ex ſub contraria coni ſectione, <lb/>ſumpta poſtea fuit diuina illa Planisferijdelineation, quæ ſub Ptolomæi no-<lb/>mine legitur, & ſicuti ex <ref>penultima primi Euclidis</ref>, quam Pythagoras <lb/>excogitauit propè innumeræ pulchræ conſequentiæ in Aſtronomia, in <lb/>Architectura, in <choice><ex>multisque</ex><am>multisq́;</am></choice> alijs ſcientijs deſumptæ ſunt, immo quemad-<lb/>modum ex ſingulis propoſitionibus à noſtris maioribus excogitatis mul-<lb/>ta egregia ſunt deducta, ita fortaſſe continget, vt ex mearum muentio-<lb/>num aliqua, <choice><ex>nonnihil</ex><am>nõnihil</am></choice> in poſterum vtilitatis deſumatur. </s> <s xml:space="preserve">Si quid verò, hic in-<lb/>ueneris, quod tuo genio non arrideat, illa prudentiſſimi hominis ſen-<lb/>tentia in mentem veniat. </s> <s xml:space="preserve"><hi rend="italics">Quot capita, tot ſententiæ</hi>, ac per raro con-<lb/>tingere, vt idem omnibus probari, atque placere queat, & perdifficulter <lb/>inueniri hominem cui placeant omnia quæ alteri ſatisfaciunt. </s> <s xml:space="preserve">Nec te mo <lb/>ueat, quodhęc Theoremata ſiue excogitationes non videas ordine illo di-<lb/>ſpoſitas, quo collocari debere exiſtimaueris, tum in Arithmeticis, tum in <lb/>cæteris. </s> <s xml:space="preserve">Cum enim in huiuſmodi rebus ordo non ſit neceſſarr<unclear reason="illegible"/>us, vi-<lb/>ſum eſt mihi poſſe me, ſine repræhenſione, illum negligere, cum ſpe-<lb/>culationi, ſiue inuentioni preęcipuè adeo mihi incumbendum decreuerim <lb/>vtin collocatione operam ponere, & tempus abſumere operæpretium <lb/>non duxerim, quod idem in epiſtolarum collocatione feci, in quibus per-<lb/>ſonarum ad quas ſcribo nullus ferè graduum ordo ſeruatus eſt, nec tem-<lb/>poris, quo ſunt ſcriptæ, quæſitorum tantummodo ratione habita. </s> <s xml:space="preserve">Nec <lb/>admirari quenquam velim, quod in ſpeculandis numerorum paſſioni-<lb/>bus, figuris vtar geometricis, ita enim in <ref>.2. libr.</ref> fecit Euclides, qui mo-<lb/>dus, eo magis mihi arridet, quo minus eſt abſtractus, <hi rend="italics">quoniam oportet in-<lb/>telligentem phantaſmata ſpeculari</hi>, cum pręterea perſpicuum ſit, diſcretum <lb/>omne, ex continui diuiſione aliquo modo oriri, ſiue actu, ſiue potentia. <lb/></s> <s xml:space="preserve">Deinde ſi forte meis in deinonſtrationibus tibi videbor aliquando bre-<lb/>uior, illud in cauſa fuiſſe ſcias, quod ibi ad viros ſcribebam in his diſcipli-<lb/>nis exercitatos, quibus ſatis fuit rem ſignificare. </s> <s xml:space="preserve">Libuit autem mihi om-<lb/>nes voluminis Arithmetici propoſitiones potius vocabulo theorema-<lb/>tum appellare, quam problematum, quia pars earum ſpeculatiua tan-<lb/>tum mea eſt, & ſi ex varijs eiuſmodi propoſitionibus etiam operatiuam <lb/>adinuenerim. </s> <s xml:space="preserve">Quoniam verò multis in locis accidit, vt veritatis iudi-<lb/>candæ cauſa neceſſe mihi fuerit quorundam ſententijs aduerſari nolim te <pb facs="0011"/> hoc mihi vitio tribuere, <choice><ex>meque</ex><am>meq́;</am></choice> hoc nomine carptorem <choice><ex>maledicumque</ex><am>maledicumq́;</am></choice> ha-<lb/>bere quod alienos errores aperiam, cum potius habenda ſit mihi gratia, <lb/>quod in ijs interdum laborans (quę Antiſthenes in diſciplinis magis ne-<lb/>ceſſaria eſſe dixit, <hi rend="italics">vt mala ſcilicet prius dediſcantur<sic>)</sic></hi> falſas opiniones euel-<lb/>lere ſtudeam, <choice><ex>veritatemque</ex><am>veritatemq́;</am></choice> oſtendere, quam omnis philoſophus, Ariſto-<lb/>telis exemplo, pluris quam cuiuſuis hominis authoritatem, aut gratiam <lb/>facere debet. </s> <s xml:space="preserve">Cumq́ue in hoc volumine aliquid eiuſmodi legeris <lb/>te oratum volo, vt in iudicando, affectum omnem exuas, <lb/>Salluſtianum illud præ oculis habens. </s> <s rend="italics" xml:space="preserve">Omnes qui dere-<lb/>bus <choice><ex>dubijs</ex><am>dubijs</am></choice> conſultant, ab odio amicitia, ira, atque <lb/>miſericordia vacuos eſſe decet. </s> <s xml:space="preserve">Hinc fiet, vt <lb/>non perſonæ (vt multiſolent) ſed <lb/>veritati, quę ſummo ſtudio di-<lb/>gniſſima eſt, ſemper po <lb/>tius faueas. </s> <s xml:space="preserve">Vale <lb/>noſtrisq́ue <lb/>labo-<lb/>ribus vtere, ſi quem inde fructum, <lb/>ſicuti ſpero tuleris, illi præ-<lb/>cipuè habeas gratiam à <lb/>quo omnes fluunt <lb/>ſcientiæ.</s> </p> <pb facs="0012"/> <pb facs="0013"/> </div> </front> <body> <div type="chapter"> <head xml:space="preserve">IO. BAPTISTAE <lb/>BENEDICTI <lb/>PATRITII VENETI <lb/>SERENISS. CAR. EM. <lb/>ALLOBROGVM DVCIS <lb/>PHILOSOPHI.</head> <head rend="italics" xml:space="preserve">Theoremata Arithmetica.</head> <p> <s xml:space="preserve"><hi rend="small caps">PRaeclare</hi> multa veteres mathematici philoſophi de nu<lb/>meris eorumq́ue effectibus excogitata poſteris tradide-<lb/>runt, quorum cum vix vllam rationem reddiderint, aut <lb/>certè per exiguam, occaſione diuerſorum problematum <lb/>mihi à Sereniſſimo Sabaudiæ Duce propoſitorum præbi-<lb/>ta, de ijs quæ ab antiquis propoſita fuerunt contemplanda <lb/>nonnulla occurrerunt, quæ poſteritati comendare non <lb/>inutile arbitratus fum, ne hæ meæ cogitationes intercide-<lb/>rent, & occaſionem præberem quamplurimis abſtruſa hęc <lb/>indagandi, quæ problematibus & thæorematibus inuoluta, vix aliquem qui euol-<lb/>ueret nacta funt.</s> </p> <p> <s xml:space="preserve">Inter cætera vero à me queſita, hoc fuit theorema.</s> </p> <div type="math:theorem"> <head xml:space="preserve">THEOREMA PRIMVM.</head> <p> <s xml:space="preserve"><hi rend="small caps">INterrogavit</hi> me Sereniſſimus Dux Sabaudiæ, qua ratione cognoſci poſ-<lb/>ſet ſcientificè & ſpeculatiue (vt dicitur) productum ex duobus fractis numeris, <lb/>quolibet producentium minus eſſe. </s> <s xml:space="preserve">Cui reſpondi, mente & cogitatione conci-<lb/>piendum eſſe fractos producentes cum fractis productis, non vnius eiuſdemq́ue na-<lb/>turæ eſſe, imò longè diuerfæ.</s> </p> <p> <s xml:space="preserve">Exempli gratia, fractis numeris propofitis <seg type="var">.a.i.</seg> et <seg type="var">.a.c.</seg> quorum integri ſint <seg type="var">.a.<lb/>b.</seg> et <seg type="var">.a.d.</seg> qui tanquam lineæ cogitentur, apertum fanè eſſet productum <seg type="var">.c.i.</seg> fu-<lb/>perficiale futurum, quod nomen caperet à producto ſuperficiali <seg type="var">.d.b.</seg> generato ex <lb/>vno in aliud totorum linearium, nam ſi conſtitueretur <seg type="var">.a.i.</seg> octauum ipſius <seg type="var">.a.b.</seg> et <seg type="var">.a.<lb/>c.</seg> dimidium <seg type="var">.a.d.</seg> multiplicato <seg type="var">.a.i.</seg> cum <seg type="var">.a.c.</seg> produceretur fextumdecimum ipſius <seg type="var">.<lb/>d.b</seg>. </s> <s xml:space="preserve">Quare <seg type="var">.d.b.</seg> eſſet totum <choice><ex>relatiuum</ex><am>relatiuũ</am></choice> ipſius <seg type="var">.c.i.</seg> non aliquod totum producentium. <lb/></s> <s xml:space="preserve">Mirum itaque non eſt ſi productum <seg type="var">.c.i.</seg> minus videatur fuis producentibus, cum <lb/>toto, diuerſæ naturæ à primis conferatur, fractum fiquidem ab integro eiuſdem <lb/>naturæ, linearis, ſuperficialis, aut corporeæ denominatur.</s> </p> <p> <s xml:space="preserve">Quòd ſi amplioris cognitionis gratia ex ſcientiæ præceptis ſpeculari voluerit a@ <pb facs="0014" n="2"/><fw type="head">IO. BAPT. BENED.</fw> quis, qua ratione fractus numerus <seg type="var">.c.i.</seg> minor ſit in ſuo integro <seg type="var">.d.b.</seg> fracto <seg type="var">.a.i.</seg> in <lb/>ſuo integro <seg type="var">.a.b.</seg> aut fracto <seg type="var">.a.c.</seg> in ſuo integro <seg type="var">.a.d.</seg> conſideret is quo pacto pro-<lb/>portio <seg type="var">.c.i.</seg> ad <seg type="var">.d.b.</seg> minor ſit proportione <seg type="var">.a.i.</seg> ad <seg type="var">.a.b.</seg> et <seg type="var">.a.c.</seg> ad <seg type="var">.a.d.</seg> hac ratione. </s> <s xml:space="preserve">Ma-<lb/>nifeſtum eſt ex <ref>prima ſexti</ref> de quantitate <lb/>continua, aut <ref>.18. ſeptimi Euclidis</ref> de diſcre <lb/> <ptr xml:id="fig-0014-01a" corresp="fig-0014-01" type="figureAnchor"/> ta, proportionem ipſius <seg type="var">.d.i.</seg> ad <seg type="var">.d.b.</seg> eſſe ſi-<lb/>cut <seg type="var">.a.i.</seg> ad <seg type="var">.a.b.</seg> & cum <seg type="var">.c.i.</seg> minor ſit <seg type="var">.d.i.</seg> <lb/>velut pars ſuo toto, proportio, <seg type="var">c.i.</seg> ad <seg type="var">.d.b.</seg> <lb/>minor erit proportione <seg type="var">.d.i.</seg> ad <seg type="var">.d.b.</seg> ex .8. <lb/>quinti, </s> <s xml:space="preserve">quare minor erit pariter proportio-<lb/>ne <seg type="var">.a.i.</seg> ad <seg type="var">.a.b.</seg> ex <ref>.12. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice></ref> vnà etiam pro-<lb/>portio <seg type="var">.c.i.</seg> ad <seg type="var">.d.b.</seg> minor erit <seg type="var">.a.c.</seg> ad <seg type="var">.a.d.</seg> <lb/>ex eiſdem cauſis, medio <seg type="var">.c.b</seg>. </s> <s xml:space="preserve">Ex quibus pa-<lb/>tet ratio, cur fracti diuerſarum denomina-<lb/>tionum ad vnicam reducantur. </s> <s xml:space="preserve">Cur etiam <lb/>numeros integros in partes fractis ſimiles <lb/>frangere liceat, quæ omnia ex ſubſequenti <lb/>figura facilè cognoſci poſſunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0014-01" corresp="fig-0014-01a"> <graphic url="0014-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="2">II</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVae</hi> ſit ratio, cur hi, qui numeros, fractos diuerſarum denominationum col-<lb/>ligere volunt, & in ſummam redigere, multiplicent vnum ex numerantibus <lb/>per denominatorem alterius, & poſtmodum denominatores adinuicem, quorum <lb/>vltimum productum, commune eſt denominans duorum priorum productorum, <lb/>quæ collecta in ſummam efficiunt quod quærebatur.</s> </p> <p> <s xml:space="preserve">Qua in re ſciendum eſt, denominantes conſiderari tanquam partes vnius <choice><ex>eiuſdem- q́ue</ex><am>eiuſdẽ-q́ue</am></choice> magnitudinis quantitatis continuæ, linearum (verbigratia) <seg type="var">a.b.</seg> et <seg type="var">.a.d.</seg> <choice><ex>æqualium</ex><am>æqualiũ</am></choice> <lb/>in longitudine, <choice><ex>quarum</ex><am>quarũ</am></choice> <seg type="var">.a.b.</seg> in quatuor partes diuidatur, et <seg type="var">.a.d.</seg> in tres. </s> <s xml:space="preserve">Quare ſi colli-<lb/>gere voluerimus duo tertia cum tribus quartis, multiplicabimus <seg type="var">.a.c.</seg> duo tertia, <lb/>cum <seg type="var">.a.b.</seg> diuiſa in 4. partes, produceturq́ue <seg type="var">.c.b.</seg> octo partium ſuperficialium, de-<lb/>hinc multiplicando <seg type="var">.a.i.</seg> tres quartas cum <seg type="var">.a.d.</seg> diuiſa in .3. partes producetur <seg type="var">.i.d.</seg> pri <lb/>mis ſingulis æqualis, nouem partium ſuper <lb/>ficialium, multiplicata deinde <seg type="var">a.b.</seg> diui-<lb/> <ptr xml:id="fig-0014-02a" corresp="fig-0014-02" type="figureAnchor"/> ſa in .4. partes per <seg type="var">.a.d.</seg> in .3. diuiſa, produ-<lb/>cetur quadratum <seg type="var">.d.b.</seg> in continuo, in 12. <lb/>partes diuiſum, quod erit totum commune <lb/>ſingulis productis, quorum primum erat <seg type="var">.c.<lb/>b</seg>. </s> <s xml:space="preserve">Quare <seg type="var">.c.b.</seg> ita ſe habet ad totum <seg type="var">.d.b.</seg> ſi-<lb/>cut <seg type="var">.a.c.</seg> ad <seg type="var">.a.d.</seg> ex prima ſexti in continuis, <lb/>aut .18. ſeptimi in diſcretis quantitatibus, <lb/>et <seg type="var">.d.i.</seg> ad <seg type="var">.d.b.</seg> ſicut <seg type="var">.a.i.</seg> ad <seg type="var">.a.b.</seg> ex eiſdem <lb/>propoſitionibus. </s> <s xml:space="preserve">Collectis deinde parti-<lb/>bus producti <seg type="var">.c.b.</seg> cum partibus producti <seg type="var">.<lb/>d.i.</seg> manifeſtè depræhendetur eiuſmodi <lb/>ſummam componi ex partibus vnius totius <lb/>communis ſingulis earum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0014-02" corresp="fig-0014-02a"> <graphic url="0014-02"/> </figure> </div> </body> </floatingText> <pb facs="0015" n="3"/> <fw type="head">THEOR. ARITH.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="3">III</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> reperturi qualis ſit fractus aliquis numerus reſpectu alterius; </s> <s xml:space="preserve">multiplicare <lb/>debeant numeratores adinuicem & ita etiam denominatores, ex quo produ-<lb/>ctum ex numeratoribus nomen capiat à producto denominatorum.</s> </p> <p> <s xml:space="preserve">Huius ſi cauſam noſce vis, ſume <seg type="var">.o.i.</seg> & <seg type="var">.o.u.</seg> pro totis denominatoribus, tum <seg type="var">.o.e.</seg> <lb/>& <seg type="var">.o.a.</seg> pro numeratoribus (exempli cauſa) ſit <seg type="var">.o.i.</seg> ſenarius <seg type="var">.o.u.</seg> quaternarius <seg type="var">.o.e.</seg> <lb/>quinarius <seg type="var">.o.a.</seg> ternarius. </s> <s xml:space="preserve">Si noſce vis quæ ſint tres quartę partes quinque ſextarum, <lb/>patet ex regulis practicis oriri quindecim vigeſimaſquartas. </s> <s xml:space="preserve">Id quomodo fiat, ex <lb/>ſubſcripta ſigura depræhendetur, memores tamen eſſe oportet, quodlibet <choice><ex>productum</ex><am>productũ</am></choice> <lb/>conſiderari <choice><ex>tanquam</ex><am>tanquã</am></choice> ſuperficiem, producentia <choice><ex>autem</ex><am>autẽ</am></choice> tan-<lb/>quam lineas. </s> <s xml:space="preserve">In hac igitur ſigura productum ex totis <lb/> <ptr xml:id="fig-0015-01a" corresp="fig-0015-01" type="figureAnchor"/> linearibus eſt <seg type="var">.u.i.</seg> aggregatum ex .24. partibus, & <seg type="var">.u.e.</seg> <lb/>productum aggregatum ex .20. </s> <s xml:space="preserve">Quodita ſe habebit <lb/>ad productum totale <seg type="var">.u.i.</seg> ſicut <seg type="var">.o.e.</seg> ad <seg type="var">o.i.</seg> ex prima <lb/>ſexti aut .18. ſeptimi, ita <seg type="var">.u.e.</seg> erunt quinque ſextæ par <lb/>tes <seg type="var">.u.i.</seg> quarum in propoſito exemplo, tres quartæ <lb/><choice><ex>quæruntur</ex><am>quærũtur</am></choice>. </s> <s xml:space="preserve">Si <choice><ex>itaque</ex><am>itaq;</am></choice> multiplicabitur <seg type="var">.o.e.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.o.a.</seg> orietur <lb/>productum <seg type="var">.a.e.</seg> ita <choice><ex>proportionatum</ex><am>proportionatũ</am></choice> ad <seg type="var">.u.e.</seg> ſicut <seg type="var">.o.a.</seg> ad <lb/><seg type="var">o.u.</seg> reperitur, ex prædictis rationibus. </s> <s xml:space="preserve">Quòd ſi <choice><ex>ſtatutum</ex><am>ſtatutũ</am></choice> <lb/>eſt <seg type="var">.o.a.</seg> tres quartas partes eſſe ipſius <seg type="var">.u.o.</seg> <choice><ex>etiam</ex><am>etiã</am></choice> <seg type="var">.a.e.</seg> tres <lb/>quartæ partes <choice><ex>erunt</ex><am>erũt</am></choice> <seg type="var">.u.e.</seg> ſed <seg type="var">.u.e.</seg> quinque ſextæ ſunt ip-<lb/>ſius <seg type="var">.u.i.</seg> ex quo ſequitur bonum eſſe huiuſmodi opus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0015-01" corresp="fig-0015-01a"> <graphic url="0015-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="4">IIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> multiplicaturi fractos cum integris, rectè multiplicent numerantem fra-<lb/>cti per numerum integrorum, partianturq́ue productum per <choice><ex>denominantem</ex><am>denominantẽ</am></choice> <lb/>fracti, ex quo numerus quæſitus colligitur.</s> </p> <p> <s xml:space="preserve">Propter quod mente concipiamus in ſubſequenti figura, numerum integrorum <lb/>tanquam lineam <seg type="var">.a.e.</seg> qui, verbigratia, ſit denarius, quorum vnuſquiſque ſit æqualis <lb/><seg type="var">a.i.</seg> cogiteturq́ue productum ipſius <seg type="var">.a.e.</seg> in <seg type="var">.a.i.</seg> ſitq́ue <seg type="var">.u.e.</seg> quod quidem erit dena-<lb/>rius ſuperficialis, conſtituta prius <seg type="var">.a.u.</seg> æqualis <seg type="var">.a.i.</seg> & <seg type="var">.a.o.</seg> ſint duæ tertiæ <seg type="var">.a.u.</seg> <choice><ex>quarum</ex><am>quarũ</am></choice> <lb/>duarum tertiarum productum in numerum <seg type="var">.a.e.</seg> ſit <seg type="var">.o.e.</seg> pariter <seg type="var">.u.i.</seg> vnitas ſit ſuper-<lb/>ficialis prout <seg type="var">.a.i.</seg> vnitas eſt linearis, quam <seg type="var">.u.i.</seg> reſpicere debet productum <seg type="var">.o.e.</seg> ex <lb/>quo integer ſuperficialis <seg type="var">.u.i.</seg> erit tanquam ternarius, & productum <seg type="var">.o.i.</seg> tanquam bi <lb/>narius, & quia quælibet pars è viginti ipſius <seg type="var">.o.e.</seg> æqualis eſt tertiæ parti <seg type="var">.u.i.</seg> vnita-<lb/>tis ſuperficialis; </s> <s xml:space="preserve">ſi cupiamus ſcire quot integræ vnitates ſint in partibus <seg type="var">.o.e.</seg> conſul-<lb/>tum eſt eaſdem diuidere per denominantem <seg type="var">.u.i.</seg> compoſitum ex tribus partibus ſu <lb/>perficialibus, & cum tam linea <seg type="var">u.a.</seg> quam ſuperficies <seg type="var">.u.i.</seg> diuidatur in 3. partes <choice><ex>aequa les</ex><am>ęquales</am></choice> noſce peroportunum eſt eiuſmodi partitionem numeri <seg type="var">.o.e.</seg> fieri per numerum <lb/>ipſius <seg type="var">.u.i.</seg> non <seg type="var">.u.a.</seg> ex prædictis cauſis.</s> </p> <figure place="here"> <graphic url="0015-02"/> </figure> <pb facs="0016" n="4"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="5">V</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">ALia</hi> quoque via prædicti effe <lb/> <ptr xml:id="fig-0016-01a" corresp="fig-0016-01" type="figureAnchor"/> ctus cauſa, ſpeculando inno-<lb/>teſcere poteſt, cuius rei gratia for-<lb/>metur ſequens figura <seg type="var">.e.o.a.u.n.</seg> <lb/>eiuſmodi, vt <seg type="var">a.e.</seg> ſit numerus li-<lb/>linearis integrorum, & <seg type="var">o.e.</seg> produ-<lb/>ctum numerantis ipſorum <choice><ex>fractorum</ex><am>fractorũ</am></choice> <lb/>in integris, ex quo <seg type="var">.a.o.</seg> erunt duæ <lb/>tertiæ, verbigratia, <seg type="var">a.i.</seg> aut <seg type="var">a.u.</seg> qua-<lb/>rum <choice><ex>linearum</ex><am>linearũ</am></choice> ſingulę ſtatuuntur æqua <lb/>les vnitati lineari, ſuperficies autem <lb/>parallelogramma <seg type="var">.u.n.</seg> conſtituatur <lb/>æqualis magnitudinis ſuperficiei <seg type="var">.o.<lb/>e.</seg> ex quo <seg type="var">.u.n.</seg> erit nobis cognita ſu-<lb/>perficies. </s> <s xml:space="preserve">Cognoſcetur pariter quan <lb/>titas partium <seg type="var">.a.u.</seg> quam in propoſi-<lb/>to exemplo diximus eſſe trium par-<lb/>tium. </s> <s xml:space="preserve">ex regula igitur de tribus, di-<lb/>cemus ſi <seg type="var">.u.a.</seg> dat <seg type="var">.a.e.</seg> ſine dubio <seg type="var">.o.<lb/>a.</seg> dabit <seg type="var">.a.n.</seg> numerum linearem. <lb/>quæ regula ex 15. ſexti in continuis, <lb/>& ex 20. ſeptimi in diſcretis, depro-<lb/>mitur. </s> <s xml:space="preserve">rectè igitur <choice><ex>multiplicantur</ex><am>multiplicãtur</am></choice> fra-<lb/>cti numerantes cum integris, & productum diuiditur per <choice><ex>denominantem</ex><am>denominantẽ</am></choice> fractorum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0016-01" corresp="fig-0016-01a"> <graphic url="0016-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="6">VI</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">ITem</hi> & alia ſpeculatione cognoſci poteſt hoc rectè fieri, mul-<lb/>tiplicantes enim has duas tertias per decem, debemus conſide-<lb/> <ptr xml:id="fig-0016-02a" corresp="fig-0016-02" type="figureAnchor"/> rare quantitatem duarum <choice><ex>tertiarum</ex><am>tertiarũ</am></choice> decies produci, ex quo oriuntur <lb/>20. tertia, quandoquidem ſingulæ vnitates, </s> <s xml:space="preserve">tunc pro duobus ter-<lb/>tijs ſumuntur, ſed c<unclear reason="illegible"/>um quilibet integer tria fragmenta contineat, <lb/>ideo ex ratione partiendi quoties ternarius ingrediatur viginti, <lb/>ſtatim cognoſcemus quod optabamus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0016-02" corresp="fig-0016-02a"> <graphic url="0016-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Id ipſum accideret ſi integri in eiuſmodi ſpecie fractorum diui-<lb/>derentur. quo facto hi multiplicandi eſſent cum numerante propo <lb/>ſito, & <choice><ex>partiendum</ex><am>partiendũ</am></choice> productum per quadratum denominantis.</s> </p> <p> <s xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s> <s xml:space="preserve">Sit linea <seg type="var">.a.e.</seg> conſtans ex <choice><ex>quinque</ex><am>quinq;</am></choice> <lb/>integris numeris, quorum <choice><ex>vnuſquiſque</ex><am>vnuſquiſq;</am></choice> æqualis ſit <seg type="var">.a.u.</seg> vel <seg type="var">.a.i.</seg> & <seg type="var">.a.o.</seg> <lb/>ſint duo tertia vnitatis integræ linearis. </s> <s xml:space="preserve">cogitemus nunc hos <choice><ex>quinque</ex><am>quinq;</am></choice> <lb/>integros diuidi in ſua <choice><ex>fragmenta</ex><am>fragmẽta</am></choice> linearia, quę in propoſito exemplo <lb/>erunt 15. multiplicatis iam 15. cum propoſitis, videlicet <seg type="var">a.o.</seg> orie-<lb/>tur productum <seg type="var">.o.e.</seg> triginta fragmentorum ſuperficialium, <choice><ex>quorum</ex><am>quorũ</am></choice> <lb/>in ſingulos integros ſuperficiales <choice><ex>cadunt</ex><am>cadũt</am></choice> <choice><ex>nouem</ex><am>nouẽ</am></choice> in hoc <choice><ex>exemplo</ex><am>exẽplo</am></choice>, & cum <lb/>notauerimus quoties <choice><ex>nouem</ex><am>nouẽ</am></choice> ingrediatur triginta, propoſitum con-<lb/>ſequemur.</s> </p> <pb facs="0017" n="5"/> <fw type="head">THEOR. ARITH.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="7">VII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> multiplicaturi integros numeros & fractos, cum integris & fractis, de-<lb/>beant integros reducere ad ſpecies fractorum, eos colligendo cum fractis: <lb/></s> <s xml:space="preserve">deinde multiplicare hos vltimos numerantes adinuicem & productum partiri <lb/>per productum denominantium.</s> </p> <p> <s xml:space="preserve">Vt (exempli cauſa) ſi volumus multiplicare vnum & duo tertia, per duo & tria <lb/>quarta, reducentur omnia in fractos, ex quo vna ex parte eſſent quinque ter-<lb/>tia, multiplicanda cum vndecim quartis ex altera, quo facto oriretur productum <lb/>quinquagintaquinque fractorum, quod diuiſum per <lb/>productum ternarijin quaternarium, videlicet per duode <lb/>cim, quatuor integri proferentur cum ſeptem duodeci-<lb/>mis fractis vnius integri.</s> </p> <figure place="here"> <graphic url="0017-01"/> </figure> <p> <s xml:space="preserve">Detur ſubſequens figura in qua linea <seg type="var">a.i.</seg> æqualis ſit li-<lb/>neæ <seg type="var">.u.a.</seg> quarum <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> <choice><ex>conſideretur</ex><am>cõſideretur</am></choice> pro integro nume <lb/>ro: </s> <s xml:space="preserve"><choice><ex>cogiteturque</ex><am>cogiteturq́;</am></choice> <seg type="var">.a.i.</seg> valere quatuor in pręſenti <choice><ex>exemplo</ex><am>exẽplo</am></choice>, & <seg type="var">.a.<lb/>u.</seg> tria: </s> <s xml:space="preserve">detur deinde linea <seg type="var">.a.o.</seg> æquipollens vni integro <choice><ex>cum</ex><am>cũ</am></choice> <lb/>duobus tertijs, & <seg type="var">a.e.</seg> æquipollens duobus integris & tri-<lb/>bus quartis. </s> <s xml:space="preserve">Iam ſi hæ duæ lineæ in ſuos fractos redu-<lb/>cantur, multiplicata (vt in ſequenti figura apparet.) <seg type="var">a.o.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <lb/><seg type="var">a.e.</seg> orietur productum <seg type="var">o.e.</seg> fractorum ſuperficialium <lb/><choice><ex>quinquagintaquinque</ex><am>quinquagintaquinq;</am></choice>, quorum integer ſuperficialis va-<lb/>let duodecim, ſcilicet <seg type="var">.u.i.</seg> vt cuique manifeſtum eſt, ex <lb/>quo, quærenti media partitione, quoties duodecim in-<lb/>grediatur quinquagintaquinque, citra errorem, quæſitum <lb/>occurret.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="8">VIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">ID ipsvm</hi> accideret ſi fractiad vnam eandemq́ue denominationem reduceren-<lb/>tur, qui poſtmodum ſimul multiplicarentur, productumq́ue partiremur per qua-<lb/>dratum denominantis communis.</s> </p> <p> <s xml:space="preserve">Exempli cauſa, ſint eadem quinque tertia, & vndecim quarta adinuicem multi-<lb/>plicanda, quæ ſi reducantur ad vnam & eandem denominationem quinarius <lb/>numerans vnius, multiplicabitur cum quaternario deno-<lb/>minante alterius, & vndenarius ſecundi cum ternario de-<lb/>nominante primi. ex quo vna ex parte eſſent viginti, ex <lb/> <ptr xml:id="fig-0017-02a" corresp="fig-0017-02" type="figureAnchor"/> altera 33. numerantia vnius <choice><ex>communis</ex><am>cõmunis</am></choice> denominantis, quod <lb/>eſſet productum ternarij in quaternarium, videlicet duo-<lb/>decim, vt ex veteri regula patet. </s> <s xml:space="preserve">Iam ſi multiplicentur vi <lb/>ginti cum trigintatribus, dabuntur 660. fracti, quorum in-<lb/>teger erit quadratum duodenarij, nempe 144. quibus qui-<lb/>dem 660. diuiſis per 144. proferentur quatuor integri & <lb/>ſeptem duodecimi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0017-02" corresp="fig-0017-02a"> <graphic url="0017-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Cuius rei gratia ſit in ſubſcripta figura linea <seg type="var">.a.i.</seg> & ei <lb/>æqualis <seg type="var">.a.u.</seg> pro integro lineari, quæ <seg type="var">.a.i.</seg> diuidatur in qua-<lb/>tuor partes, & <seg type="var">.a.u.</seg> in tres, & linea <seg type="var">.a.e.</seg> ſit vndecim <choice><ex>partium</ex><am>partiũ</am></choice> <lb/>talium qualium <seg type="var">.a.i.</seg> eſt quatuor, & <seg type="var">.a.o.</seg> ſit quinque pro-<lb/>ut <seg type="var">.a.u.</seg> eſt trium. </s> <s xml:space="preserve">nunc multiplicato <seg type="var">.a.o.</seg> & <seg type="var">.a.i.</seg> orietur pro-<lb/>ductum <seg type="var">.o.i.</seg> viginti partium ſuperficialium. </s> <s xml:space="preserve">tum multipli- <pb facs="0018" n="6"/><fw type="head">IO. BAPT. BENED.</fw> cato <seg type="var">.a.e.</seg> per <seg type="var">.a.u.</seg> dabitur productum <seg type="var">.u.e.</seg> <choice><ex>trigintatrium</ex><am>trigintatriũ</am></choice> <lb/> <ptr xml:id="fig-0018-01a" corresp="fig-0018-01" type="figureAnchor"/> partium. </s> <s xml:space="preserve">ad hæc quadratum <seg type="var">.u.i.</seg> conſtabit ex duode-<lb/>cim partibus eiuſdem rationis cum reliquis duobus <lb/>productis, quod quadratum <seg type="var">.u.i.</seg> vnitas eſt ſuperficia-<lb/>lis, & communis denominans duorum productorum. <lb/></s> <s xml:space="preserve">quod ſi in præſentiarum cogitabimus lineam <seg type="var">.c.d.</seg> tri-<lb/>gintatrium partium æqualium, et <seg type="var">.c.t.</seg> duodecim ſimi-<lb/>lium, et <seg type="var">.c.f.</seg> viginti <seg type="var">.c.n.</seg> duodecim, multiplicato <seg type="var">.c.<lb/>d.</seg> cum <seg type="var">.c.f.</seg> dabitur ſuperficies <seg type="var">.f.d.</seg> 660. fractorum <lb/>ſuperficialium, quorum vnitas integra ſuperficialis <lb/>erit quadratum <seg type="var">.n.t.</seg> 144. partium cuiuſmodi <seg type="var">.f.d.</seg> <lb/>partes habet .660. diuiſo itaque <seg type="var">.f.d.</seg> per <seg type="var">.n.t.</seg> pro-<lb/>poſitum conſequetur. </s> <s xml:space="preserve">eo quòd eadem proportio erit <lb/> <ptr xml:id="fig-0018-02a" corresp="fig-0018-02" type="figureAnchor"/> producti <seg type="var">.f.d.</seg> ad <seg type="var">.n.t.</seg> quæ producti eius quòd fit ex <seg type="var">.<lb/>a.e.</seg> in <seg type="var">.a.o.</seg> ad <seg type="var">.u.i.</seg> nam proportio <seg type="var">.c.d.</seg> ad <seg type="var">.c.t.</seg> ea-<lb/>dem eſt quæ <seg type="var">.a.e.</seg> ad <seg type="var">.a.i.</seg> & <seg type="var">c.f.</seg> ad <seg type="var">.c.n.</seg> vt <seg type="var">.a.o.</seg> ad <seg type="var">.a.<lb/>u.</seg> ex prima ſexti vel 18. ſeptimi, ſed vt <seg type="var">.f.d.</seg> ad id <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>fit ex <seg type="var">.f.c.</seg> in <seg type="var">.c.t.</seg> eſt vt <seg type="var">.c.d.</seg> ad <seg type="var">.c.t.</seg> & vt eius <choice><ex>quod</ex><am>ꝙ</am></choice> fit ex <lb/><seg type="var">f.c.</seg> in <seg type="var">.c.t.</seg> ad <seg type="var">.n.t.</seg> eſt vt <seg type="var">.f.c.</seg> ad <seg type="var">.c.n.</seg> ex dictis pro-<lb/>poſitionibus </s> <s xml:space="preserve">quare ex æqua proportionalitate, eodem <lb/>modo diſcurrendo in figura <seg type="var">.o.a.e.</seg> ita ſe habebit <seg type="var">.f.d.</seg> <lb/>ad <seg type="var">.n.t.</seg> vt <seg type="var">.o.e.</seg> ad <seg type="var">.u.i</seg>. </s> <s xml:space="preserve">Porrò ex ijs, quæ hactenus de <lb/>fractorum multiplicatione conſiderata fuerunt, apertè <lb/>ratio deprehenditur, cur productum, ſingulis producen <lb/>tibus ſemper minus ſit, cum producta ſint ſuperficialia <lb/>producentia verò ſemper linearia, omiſſis productis <lb/>corporeis, quæ omnia ad ſuperficialia reducuntur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0018-01" corresp="fig-0018-01a"> <graphic url="0018-01"/> </figure> <figure xml:id="fig-0018-02" corresp="fig-0018-02a"> <graphic url="0018-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="9">IX</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">IN Ipsa</hi> fractorum diuiſione, animaduertendum eſt, denominantes numeros <lb/>ſemper æquales inuicem eſſe debere, vnius ſcilicet ſpeciei, quòd ſi æquales non <lb/>fuerint, neceſſe eſt via multiplicationis ipſorum denominantium adinuicem effice-<lb/>re æquales vt ſint, ex quo productum oritur eiuſmodi, vt aptum ſit habere partes <lb/>fractorum, quæ deſiderabantur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponerentur diuidenda ſeptem octaua per tria quarta præ-<lb/>cipit antiquorum regula, vt ad vnam tantum denominationem reducantur. </s> <s xml:space="preserve">quare <lb/>multiplicant denominantes inuicem. </s> <s xml:space="preserve">ex quo productum in materia propoſita ori-<lb/>tur triginta duarum partium commune denominans, cuius duo numerantes ſunt vi-<lb/>gintiquatuor & vigintiocto, producti ex multiplicatione vnius numerantis in deno <lb/>minantem alterius, ex quo dantur vigintiquatuor tamquam tria quarta trigintaduo <lb/>rum, & vigintiocto tanquam ſeptem octaua particularum vniformium, prout ope <lb/>primæ ſexti aut decimæoctauæ ſeptimi in ſubſcripta figura cognoſci poteſt.</s> </p> <pb facs="0019" n="7"/> <fw type="head">THEOR. ARITH.</fw> <p> <s xml:space="preserve">Sit itaque linea <seg type="var">.a.i.</seg> diuifa in partes octo, & ei æqualis in longitudine <seg type="var">.a.u.</seg> in qua-<lb/>tuor, productum verò vnius in alteram <lb/> <ptr xml:id="fig-0019-01a" corresp="fig-0019-01" type="figureAnchor"/> ſit <seg type="var">.u.i.</seg> trigintaduarum particularum <lb/>fuperficialium fimilium & <choice><ex>æqualium</ex><am>æqualiũ</am></choice> ad-<lb/>inuicem. </s> <s xml:space="preserve">fit deinde <seg type="var">.a.e.</seg> ſeptem <choice><ex>partium</ex><am>partiũ</am></choice> <lb/>lineæ <seg type="var">.a.i.</seg> & <seg type="var">.a.o.</seg> trium partium <seg type="var">.a.u</seg>. <lb/></s> <s xml:space="preserve">tunc productum <seg type="var">.a.e.</seg> in <seg type="var">.a.u.</seg> erit <seg type="var">.u.e.</seg> <lb/>particularum ſuperficialium vigintiocto <lb/>& productum <seg type="var">.a.o.</seg> in <seg type="var">.a.i.</seg> erit <seg type="var">.o.i.</seg> par <lb/>ticularum <choice><ex>ſuperficialium</ex><am>ſuperficialiũ</am></choice> vigintiquatuor <lb/>eiuſdem naturæ cum partibus triginta-<lb/>duabus totius denominantis communis. <lb/></s> <s xml:space="preserve">vnde diuifo numerante vigintiocto per-<lb/>numerantem vigintiquatuor, dabitur <lb/>vnum cum fexta parte illius vnius.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0019-01" corresp="fig-0019-01a"> <graphic url="0019-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="10">X</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">PArtiri</hi> ſeu diuidere vno numero alium numerum, eſt etiam quodammodo <lb/>eiuſmodi partem numeri diuifibilis inuenire refpectu totius numeri diuifibilis, <lb/>cuiuſmodi eſt vnitas in diuidente refpectu totius diuidentis, partem inquam numeri <lb/>diuiſibilis ſic ſe habentem ad totum numerum diuiſibilem ſicut vnitas ad totum di-<lb/>uidentem, quod ſimiliter ex regula de tribus præſtamus dicentes, ſi tantus numerus <lb/>diuidens dat <choice><ex>vnitatem</ex><am>vnitatẽ</am></choice>, quid dabit numerus diuifibilis, quemadmodum ex <ref>.15. ſexti</ref> <lb/>ſeu <ref>.20. ſeptimi</ref> licet ſpeculari, Idcircò quotieſcunque minorem numerum per <lb/>maiorem diuidimus, ſemper qui prouenit fractus eſt.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi cogitaremus lineam <seg type="var">.a.e.</seg> diuiſam in octo partes æquales, qua <lb/>rum vna ſcilicet vnitas effet <seg type="var">.a.i.</seg> & cupere-<lb/>mus eam diuidere in nouem partes, ac ſcire <lb/> <ptr xml:id="fig-0019-02a" corresp="fig-0019-02" type="figureAnchor"/> quan a ſit nona illius pars; </s> <s xml:space="preserve">manifeſtum eſſet, <lb/>nonam partem ipſius <seg type="var">.a.e.</seg> minorem futuram <lb/>ipſa <seg type="var">.a.i.</seg> cum <seg type="var">.a.i.</seg> diminui debeat à ſua inte-<lb/>gritate eadem proportione, qua <seg type="var">.a.e.</seg> minor <lb/>reperitur vna linea nouem partium æqualium <lb/>fingularum <seg type="var">.a.i</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0019-02" corresp="fig-0019-02a"> <graphic url="0019-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quod vt dilucidè cuiuis innoteſcat, hoc <lb/>etiam modo licebit videre ſitlinea <seg type="var">.n.c.</seg> no-<lb/>nupla ad <seg type="var">.a.i.</seg> & parallela ad <seg type="var">.a.e.</seg> dubium non <lb/>eſt quin <seg type="var">.n.c.</seg> maior futura ſit ipſa <seg type="var">.a.e.</seg> iam ſi <lb/>earum extrema congiungantur medijs duabus <lb/>lineis <seg type="var">.n.a.</seg> et <seg type="var">.c.e.</seg> quæ ſimul concurrant in <lb/>puncto <seg type="var">.o.</seg> (quod eſt probatu facillimum) da-<lb/>buntur certe duo trianguli fimiles <seg type="var">.a.o.e.</seg> et <seg type="var">.n.o.c</seg>. </s> <s xml:space="preserve">Sit deinde <seg type="var">.n.t.</seg> vna è partibus <lb/>ipſius <seg type="var">.n.c.</seg> quæ <seg type="var">.n.t.</seg> æqualis erit <seg type="var">.a.i.</seg> ex præſuppoſito. </s> <s xml:space="preserve">ducatur deinde <seg type="var">.o.t.</seg> quę <lb/>interſecet <seg type="var">.a.e.</seg> in puncto <seg type="var">.x.</seg> dico <seg type="var">.a.x.</seg> tanto minorem futuram <seg type="var">.a.i.</seg> quanto <seg type="var">.a.e.</seg> <lb/>minor eſt <seg type="var">.n.c.</seg> neque enim dubium eſſe poteſt quin proportiones <seg type="var">.n.t.</seg> ad <seg type="var">.a.x.</seg> et <seg type="var">. <pb facs="0020" n="8"/><fw type="head">IO. BAPT. BENED.</fw> n.c.</seg> ad <seg type="var">.a.e.</seg> ſint æquales inuicem quandoqui-<lb/> <ptr xml:id="fig-0020-01a" corresp="fig-0020-01" type="figureAnchor"/> dem vnaquæque earum ex triangulorum ſimi <lb/>litudine æqualis eſt proportioni <seg type="var">.o.n.</seg> ad <seg type="var">.o.a</seg>. <lb/></s> <s xml:space="preserve">itaque <seg type="var">.n.t.</seg> hoc eſt <seg type="var">.a.i.</seg> tanto maior erit <seg type="var">.a.x.</seg> <lb/>quanto <seg type="var">.n.c.</seg> maior eſt <seg type="var">.a.e.</seg> vnde ficut <seg type="var">.a.e.</seg> con-<lb/>ſtat octo nonis ipſius <seg type="var">.n.c.</seg> ita pars <seg type="var">.a.x.</seg> ipſius <seg type="var">.<lb/>a.e.</seg> octo nonis conſtabit ipſius <seg type="var">.a.i</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0020-01" corresp="fig-0020-01a"> <graphic url="0020-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Hinc patet ratio cur partituri numerum mino <lb/>rem per maiorem collocent minorem fupra <lb/>virgulam & maiorem infra & zerum ad <choice><ex>læuam</ex><am>læuã</am></choice>.</s> </p> <p> <s xml:space="preserve">Sciendum eſt præterea diuidere numerum <lb/>per numerum: </s> <s xml:space="preserve">eſſe inuenire <choice><ex>alterum</ex><am>alterũ</am></choice> latus à quo <lb/>producitur, ſuppoſito ſemper quòd numerus <lb/>diuifibilis ſuperſicialis ſit, & rectangulus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponantur triginta diuidenda per quinarium, nihil aliud erit <lb/>hæc diuiſio, quam inuentio alterius numeri, qui multiplicatus per quinarium produ-<lb/>cat triginta ſuperficies rectangulas, huiuſmodi verò eſt ſenarius, cuius ſingulæ vnita-<lb/>tes ſuperficiales erunt.</s> </p> <p> <s xml:space="preserve">Cuius rei gratia ſit ſubſcriptum rectangulum <seg type="var">.a.e.</seg> triginta vnitatum <choice><ex>ſuperſicialium</ex><am>ſuperſicialiũ</am></choice>, <lb/>cuius latus <seg type="var">.e.n.</seg> ſit quinque vnitatum. </s> <s xml:space="preserve">hinc latus <seg type="var">.a.n.</seg> erit ſex vnitatum; </s> <s xml:space="preserve">ita diuiden-<lb/>tes rectangulum <seg type="var">.e.a.</seg> nihil a iud faciemus, quam vt inue-<lb/> <ptr xml:id="fig-0020-02a" corresp="fig-0020-02" type="figureAnchor"/> nia mus quantum valeat latus <seg type="var">.a.n.</seg> quod erit ſex vnitatum. <lb/></s> <s xml:space="preserve">Sin verò diuiſerimus per latus <seg type="var">.a.n.</seg> quæremus latus <seg type="var">.e.n.</seg> <lb/>quinque vnitatum. </s> <s xml:space="preserve">ex quo, proportio totius numeri diuifi-<lb/>bilis ad numerum qui oritur, erit ſicut diuidentis ad vnita-<lb/>tem, ex prima ſexti, aut .18. vel .19. ſeptimi, & permutatim <lb/>ita ſe habebit diuiſibile ad diuidentem, ſicut numerus qui <lb/>oritur ad vnitatem.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0020-02" corresp="fig-0020-02a"> <graphic url="0020-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Partiri igitur nihil aliud eſt, quam inuenire latus rectanguli, quod productum in <lb/>diuidente, numerum diuiſibilem compl at, ex quo numerus diuiſibilis ſuperficialis <lb/>eſt, diuidens autem, & qui oritur, numeri lineares & latera producentia huiuſcemodi <lb/>numerum diuiſibilem. </s> <s xml:space="preserve">nam multiplicare & diuidere opponuntur inuicem, cum au-<lb/>tem ex multiplicatione laterum ſiue linearum generatur ſuperficies, ex diuiſione po-<lb/>ſtea ipſius ſuperficiei inuenitur alterum latus. </s> <s xml:space="preserve">quare mirum non eſt, ſi proueniens ex <lb/>vna diuiſione (via fractorum) ſit ſemper maius numero diuiſibili.</s> </p> <p> <s xml:space="preserve">Exempli gratia, diuidendo dimidium per tertiam partem, reſultat vnus integer nu <lb/>merus cum dimidio pro numero qui oritur. </s> <s xml:space="preserve">Sit itaque dimidium ſuperſiciale diuiſi-<lb/>bile <seg type="var">.b.c.</seg> cuius totum ſit <seg type="var">.b.p.</seg> quadratum. </s> <s xml:space="preserve">tertium verò lineare diuidens, <seg type="var">b.n.</seg> cuius to-<lb/>tum lineare ſit <seg type="var">.b.d.</seg> quærendum nobis eſt latus <seg type="var">.b.s.</seg> quod cum latere <seg type="var">.b.n.</seg> producat re <lb/>ctangulum <seg type="var">.n.s.</seg> æquale dimidio ſuperſiciali propoſito <seg type="var">.b.c.</seg> quod ſi ſiat, ex .15. ſexti, <lb/>aut .20. ſeptimi. erit eadem proportio <seg type="var">.b.n.</seg> ad <seg type="var">.b.q.</seg> quæ eſt <seg type="var">.q.c.</seg> ad <seg type="var">.b.s.</seg> dicemus itaque <lb/>ſi <seg type="var">.n.b.</seg> dat <seg type="var">.b.q.</seg> quid dabit <seg type="var">.q.c</seg>? </s> <s xml:space="preserve">certè <seg type="var">.b.s.</seg> ſed <seg type="var">.n.b.</seg> eſt tertium lineare et <seg type="var">.b.q.</seg> lineare <choice><ex>in- tegrum</ex><am>in-tegrũ</am></choice>, & <seg type="var">b.s.</seg> proueniens lineare. </s> <s xml:space="preserve">& quia <seg type="var">.b.c.</seg> dimidium ſuperficiale, producitur à <seg type="var">.q.c.</seg> <lb/>dimidio lineari in <seg type="var">.q.b.</seg> integro lineari. </s> <s xml:space="preserve">quare cum <seg type="var">.n.s.</seg> ſit ęqualis <seg type="var">.b.c.</seg> & productum ex <seg type="var">.<lb/>b.n.</seg> minori <seg type="var">.q.c.</seg> neceſſe eſt, vt producatur in <seg type="var">.b.s.</seg> maiore <seg type="var">.q.b.</seg> quod <seg type="var">.q.b.</seg> maius eſt <seg type="var">.q.c.</seg> <lb/>quod quidem <seg type="var">.q.c.</seg> ita appellatur ſicut <seg type="var">.b.c</seg>. </s> <s xml:space="preserve">quare mirum non eſt ſi proueniens per fra-<lb/>ctos numeros ex diuiſione, maior ſit numero diuiſibili.</s> </p> <pb facs="0021" n="9"/> <fw type="head">THEOR. ARITH.</fw> <p> <s xml:space="preserve">Hinc manifeſte patet quamlibet <choice><ex>diuiſionem</ex><am>diuiſionẽ</am></choice> aut partitionem oriri ex regula de tri-<lb/>bus, quandoquidem ſinguli diuidentes æquipollent vni integro, & loco illius ſu-<lb/>muntur. </s> <s xml:space="preserve">Perinde enim eſt diuidere centum per viginti, ac <choice><ex>regulam</ex><am>regulã</am></choice> obſeruare de tri-<lb/>bus <choice><ex>dicentes</ex><am>dicẽtes</am></choice>, ſi viginti æquipollent vni, quibus <choice><ex>ęquiualebunt</ex><am>ęquiualebũt</am></choice> <choice><ex>centum</ex><am>cẽtum</am></choice>? </s> <s xml:space="preserve">Hoc autem ex ſub <lb/>ſequenti figura facile deprehendetur, in qua linea <seg type="var">.a.b.</seg> ſignificat viginti, et <seg type="var">.a.o.</seg> <choice><ex>vni- tatem</ex><am>vni-tatẽ</am></choice> <choice><ex>linearem</ex><am>linearẽ</am></choice>, et <seg type="var">.a.c.</seg> vnitates lineares <choice><ex>centum</ex><am>centũ</am></choice>: </s> <s xml:space="preserve">o.c. verò centum vnitates ſuperficiales, <lb/>et <seg type="var">.a.d.</seg> <choice><ex>quinque</ex><am>quinq;</am></choice> vnitates lineares, et <seg type="var">.d.b.</seg> centum vnitates ſuperficiales, ex quo manife-<lb/>ftè deprehenditur quòd quemadmodum multiplicare, nihil aliud eſt, quam inueni <lb/>re <choice><ex>productum</ex><am>productũ</am></choice> ex duobus lateribus propoſitis, it a partiri nihil aliud eſt, quam da-<lb/>to vno latere inuenire aliud latus producti propoſiti.</s> </p> <figure place="here"> <graphic url="0021-01"/> </figure> <figure place="here"> <graphic url="0021-02"/> </figure> <p> <s xml:space="preserve">Nam <choice><ex>quotieſcunque</ex><am>quotieſcunq;</am></choice> <choice><ex>ratiocinantes</ex><am>ratiocinãtes</am></choice> dicimus tantundem numeri, immediate produci <lb/>mus ſuperficiem, <choice><ex>mediante</ex><am>mediãte</am></choice> vnitate in huiuſmodi numero, qui numerus <choice><ex>antequam</ex><am>antequã</am></choice> pro-<lb/>ducatur in vnitatem, mente concipiendus eſt tanqua m linearis, tanquam linea in-<lb/>quam diuiſa in totidem particulas lineares, ſingulas continuas & æquales vnitati <lb/>propoſitæ. </s> <s xml:space="preserve"><choice><ex>Cum</ex><am>Cũ</am></choice> verò productus fuerit numerus in vnitate ſuperficialis, erit ac ſi tot eſ-<lb/>ſent vnitates quadratæ, quod ſi ita non eſſet, nulla mentio facienda eſſet quo-<lb/>rumuis <choice><ex>fractorum</ex><am>fractorũ</am></choice>. </s> <s xml:space="preserve">Ex <choice><ex>eadem</ex><am>eadẽ</am></choice> regula de tribus reduci poteſtad praxim <choice><ex>tertium</ex><am>tertiũ</am></choice> theorema.</s> </p> <p> <s xml:space="preserve">Quare cupientes ſcire quæ ſint illæ partes, quæ ſunt tres quartę, ipſarum quin-<lb/>que ſextarum, dicemus ſi quatuor dant tria, quid dabunt <choice><ex>quinque</ex><am>quinq;</am></choice> ſextæ? </s> <s xml:space="preserve">dabunt .15. <lb/>vigeſimas quartas, quæ quindecim ſunt tres quartæ ipſius .20. viginti <choice><ex>autem</ex><am>autẽ</am></choice> <choice><ex>quinque</ex><am>quinq;</am></choice> ſex <lb/>tæ vigintiquatuor, quandoquidem nos numerum quęrimus, cui ita proportionentur <lb/><choice><ex>quinque</ex><am>quinq;</am></choice> ſextæ alterius numeri, ſicut quatuor ad tria, vnde ſic ſe habent .20. ad .15. ſi-<lb/>cut .4. ad .3. ipſe autem .20. <choice><ex>quinque</ex><am>quinq;</am></choice> ſextę partes ſunt vigintiquatuor, vt per ſe <choice><ex>notum</ex><am>notũ</am></choice> eſt.</s> </p> <p> <s xml:space="preserve">Ex eadem regula de tribus, huiuſmodi quęſito reſponderi poteſt, ſi conſtituamus <lb/>prædictas <choice><ex>quinque</ex><am>quinq;</am></choice> ſextas eſſe numerum, cuius tres quartæ quęrantur, dicentes, ſi vnus <lb/>integer dat tres quartas, quid dabunt <choice><ex>quinque</ex><am>quinq;</am></choice> ſextæ? </s> <s xml:space="preserve">quare ſequentes regulam de <lb/>tribus, dabuntur quindecim vigeſimæ quartæ. </s> <s xml:space="preserve">Valet eadem regula de tribus; </s> <s xml:space="preserve">vt quis <lb/>ſcire poſſit, quæ pars aut partes numeri propoſiti ſit aliquis numerus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſcire cupienti, quæ pars aut partes ipſius vigintiquatuor ſint ſex-<lb/>decim, conſtituentur .24. tanquam vnum totum, cuius pars aut partes ſint ſexdecim, <lb/>dicemus igitur ſi .24. dant ſexdecim, quid dabit vnum? </s> <s xml:space="preserve">ſexdecim videlicet vigeſi-<lb/>masquartas, quæ cum ad primos numeros reductæ fuerint, erunt duæ tertiæ. <lb/></s> <s xml:space="preserve">Eadem ratione qui ſcire uellet, quæ partes aut pars eſſent tres quartæ, octo no-<lb/>narum, diceret, ſi octo nonæ danttres quartas, quid dabit vnum? </s> <s xml:space="preserve">prouenient .27. <lb/>trigeſimęſecundæ.</s> </p> <p> <s xml:space="preserve">Subſeruit pariter ad <choice><ex>ſciendum</ex><am>ſciendũ</am></choice> <choice><ex>naturam</ex><am>naturã</am></choice> <choice><ex>partium</ex><am>partiũ</am></choice> numeri propoſiti. </s> <s xml:space="preserve">Exempli cauſa, ſi quis <lb/>quærat, cuius numeri, duodecim ſint duæ tertiæ partes. </s> <s xml:space="preserve">Dicet ſi duo dant tria, quid <pb facs="0022" n="10"/><fw type="head">IO. BAPT. BENED.</fw> dabunt duodecim? </s> <s xml:space="preserve">nempe dabunt decemocto, numerum quæſitum ſcilicet, <lb/></s> <s xml:space="preserve">Tunc autem nil aliud pręſtamus quam quòd quærimus numerum ad quem ita ſe <lb/>habeant duodecim, ſicut duo ad tria. </s> <s xml:space="preserve">Ita etiam ſi quis quærat, cuius numeri duo <lb/>tertia ſint tres quintę, dicet, ſi tria dant <choice><ex>quinque</ex><am>quinq;</am></choice>, quid dabunt duo tertia? </s> <s xml:space="preserve">nempe da-<lb/>bunt integrum cum fracto nono. </s> <s xml:space="preserve">Hoc erit <choice><ex>itaque</ex><am>itaq;</am></choice> quęrere numerum ad quem ſic ſe <lb/>habeant duo tertia ſicut tria ad <choice><ex>quinque</ex><am>quinq;</am></choice>, quod manifeſtum eſt per ſe.</s> </p> <p> <s xml:space="preserve">Eadem ratione qui ſcire vellet, cuius numeri duæ ſeptimæ, eſſent octo integra-<lb/>rum cum duabus quintis, diceret, ſi duo dant ſeptem quid dabunt octo integra cum <lb/>duabus quintis? </s> <s xml:space="preserve">nempe dabunt .29. integra cum duabus quintis numerum quæſi-<lb/>tum. </s> <s xml:space="preserve">Sic etiam qui transferre uellet fractum numerum in fractum, id perficeret <lb/>ex regula de tribus.</s> </p> <p> <s xml:space="preserve">Exempli gratia ſi proponerentur vnde cim tertiædecimæ vnius totius, toto diui-<lb/>ſo in .13. partes, <choice><ex>deſideraremusque</ex><am>deſideraremusq́;</am></choice> ſcire, quot partes totius <choice><ex>eſsent</ex><am>eſsẽt</am></choice> vndecim <choice><ex>tertiaedeci- mæ</ex><am>tertiędeci-mæ</am></choice>, toto in .4. partes diuiſo, diceremus ſi .13. dant .11. quid dabunt quatuor? </s> <s xml:space="preserve">nem <lb/>pe <choice><ex>dabunt</ex><am>dabũt</am></choice> tres quartas <choice><ex>cum</ex><am>cũ</am></choice> <choice><ex>quinque</ex><am>quinq;</am></choice> tertijsdecimis unius quartæ, hoc verò nihil aliud eſt <lb/>quam querere numerum, ad quem ſic ſe habeat totum in 4. partes diuiſum, ſicut <lb/>idem totum diuiſum in tredecim ſe habet ad undecim tertiasdecimas, Porrò ad <lb/>alia etiam multa hæc regula accommodata eſt.</s> </p> <p> <s xml:space="preserve">Hæc enim <choice><ex>non</ex><am>nõ</am></choice> ſine propoſito dicta ſunt, ſed ut <choice><ex>quiſque</ex><am>quiſq;</am></choice> videat cauſam ſimilium ope-<lb/>rationum, quæ à practicis circa fractos numeros ſcriptæ ſunt, omnem à diuina illa <lb/>regula de tribus originem trahere ut etiam in ſequentibus videbimus.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="11">XI</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> productum ex eo quod oritur in diuidente, ſemper æquale eſt numero <lb/>diuiſibili ſi queras ita accipe.</s> </p> <p> <s xml:space="preserve">Sit numerus diuiſibilis <seg type="var">.b.</seg> quod oritur ſit <seg type="var">.c.</seg> diuidens <seg type="var">.d.</seg> & vnitas diuidentis <seg type="var">.t.</seg> cum <lb/>igitur, vt in præcedenti theoremate dictum <lb/>fuit, eadem ſit proportio <seg type="var">.b.</seg> ad <seg type="var">.c.</seg> quæ eſt <seg type="var">.d.</seg> <lb/> <ptr xml:id="fig-0022-01a" corresp="fig-0022-01" type="figureAnchor"/> ad <seg type="var">.t.</seg> manifeſte deprehenditur ex .20. ſepti <lb/>mi, productum ex <seg type="var">.b.</seg> in <seg type="var">.t.</seg> æquale eſſe pro-<lb/>ducto <seg type="var">.c.</seg> in <seg type="var">d</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0022-01" corresp="fig-0022-01a"> <graphic url="0022-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="12">XII</num>.</head> <p> <s xml:space="preserve">ID ipſum alia ratione contemplari licet.</s> </p> <p> <s xml:space="preserve">Numerus diuiſibilis ſignificetur per lineam <seg type="var">.n.e.</seg> diuidens verò per lineam <seg type="var">.a.e.</seg> <lb/>quod oritur linea <seg type="var">.u.e.</seg> vnitas diuidentis <seg type="var">.o.e.</seg> <choice><ex>quam</ex><am>quã</am></choice> cogitamus eſſe vnitatem linearem; <lb/></s> <s xml:space="preserve">ad hæc productum ex <seg type="var">.u.e.</seg> in <seg type="var">.a.e.</seg> ſit ſuperficies <seg type="var">.u.a</seg>. </s> <s xml:space="preserve">Dico ſuperficiem <seg type="var">.u.a.</seg> componi <lb/>ex tot vnitatibus ſuperficialibus quot linearibus conſtat linea <seg type="var">.n.e.</seg> nam ex ijs quæ <lb/>diuidendi ratione notauimus, <choice><ex>conſtituitur</ex><am>cõſtituitur</am></choice> <lb/>eandem proportionem eſſe <seg type="var">.n.e.</seg> ad <seg type="var">.u.e.</seg> <lb/> <ptr xml:id="fig-0022-02a" corresp="fig-0022-02" type="figureAnchor"/> quę eſt <seg type="var">.a.e.</seg> ad <seg type="var">.o.e</seg>. </s> <s xml:space="preserve">At ex prima ſexti aut <lb/>18. ſeptimi ſic ſe habet totale <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.<lb/>u.a.</seg> ad partiale <seg type="var">.u.o.</seg> ſicut <seg type="var">.a.e.</seg> ad <seg type="var">.o.e</seg>. <lb/></s> <s xml:space="preserve">quare ſic ſe habebit <seg type="var">.u.a.</seg> ad <seg type="var">.u.o.</seg> ſicut <seg type="var">.n.<lb/>e.</seg> ad <seg type="var">.u.e.</seg> ſed <seg type="var">.u.e.</seg> et <seg type="var">.u.o.</seg> numero non differunt, cum ſint vnius & eiuſdem ſpeciei, (ta-<lb/>met ſi numerus <seg type="var">.u.o.</seg> ſit ſuperficialis et <seg type="var">.u.e.</seg> linearis). </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex nona quinti numerus <seg type="var">.<lb/>u.a.</seg> æqualis erit numero <seg type="var">.n.e</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0022-02" corresp="fig-0022-02a"> <graphic url="0022-02"/> </figure> </div> </body> </floatingText> <pb facs="0023" n="11"/> <fw type="head">THEOREM. ARITH.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA. <num value="13">XIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> diuidentibus numerum diuiſibilem per proueniens, oritur numerus diui-<lb/>dens?</s> </p> <p> <s xml:space="preserve">Sit ſubſcriptus rectangulus <seg type="var">.o.e.</seg> numerus diuiſi <lb/> <ptr xml:id="fig-0023-01a" corresp="fig-0023-01" type="figureAnchor"/> bilis, qui producitur, tam ex <seg type="var">.a.o.</seg> in <seg type="var">.a.e.</seg> quám ex <seg type="var">.a.<lb/>e.</seg> in <seg type="var">.a.o</seg>. </s> <s xml:space="preserve">quare ſi <seg type="var">.a.o.</seg> diuidens fuerit <seg type="var">.a.e.</seg> proue-<lb/>niens erit, ſi veró <seg type="var">.a.e.</seg> diuidens extiterit, <seg type="var">a.o.</seg> pro-<lb/>ueniens erit futurum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0023-01" corresp="fig-0023-01a"> <graphic url="0023-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA. <num value="14">XIIII</num>.</head> <p> <s xml:space="preserve">HOcipſum, alia <choice><ex>quoque</ex><am>quoq;</am></choice> uia licebit ſpeculari.</s> </p> <p> <s xml:space="preserve">Sit linea <seg type="var">.a.</seg> <choice><ex>denotans</ex><am>denotãs</am></choice> numerum diuiſibilem, et <seg type="var">.o.</seg> primi prouenientis linea <seg type="var">.e.</seg> pri <lb/>mi diuidentis <seg type="var">.u.</seg> ſecundi prouenientis ideſt cum <seg type="var">.o.</seg> pro diuidente ſumetur. </s> <s xml:space="preserve">Iam ex <lb/>indicata definitione diuiſionis nono theoremate huius libri, dabitur proportio <seg type="var">.a.</seg> <lb/>ad <seg type="var">.o.</seg> prout datur <seg type="var">.e.</seg> ad vnitatem ſignificatam li-<lb/>nea <seg type="var">.i.</seg> & permutatim <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> ſicut <seg type="var">.o.</seg> ad <seg type="var">.i.</seg> ſed <seg type="var">.a.</seg> <lb/> <ptr xml:id="fig-0023-02a" corresp="fig-0023-02" type="figureAnchor"/> ad <seg type="var">.u.</seg> ſic ſe habet prout <seg type="var">.o.</seg> ad <seg type="var">.i.</seg> ex eadem definitio-<lb/>ne diuiſionis, <choice><ex>itaque</ex><am>itaq;</am></choice> ſic ſe habebit <seg type="var">.a.</seg> ad <seg type="var">.u.</seg> ſicut <seg type="var">.a.</seg> ad <seg type="var">.<lb/>e.</seg> vnde <seg type="var">.u.</seg> æqualis erit <seg type="var">.e.</seg> ex .9. quinti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0023-02" corresp="fig-0023-02a"> <graphic url="0023-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA. <num value="15">XV</num>.</head> <p> <s xml:space="preserve">VNde prouenit, vt qui velit cognoſcere cuius numeri quatuor quintæ par-<lb/>tes, ſint duæ tertię, aut quid ſimile, <choice><ex>conſultiſſime</ex><am>cõſultiſſime</am></choice> faciat, ſi ad unam <choice><ex>eandemque</ex><am>eandemq;</am></choice> <lb/>denominationem reduxerit.</s> </p> <p> <s xml:space="preserve">Prout in propoſito exemplo, <choice><ex>cum</ex><am>cũ</am></choice> <choice><ex>denominans</ex><am>denominãs</am></choice> <choice><ex>communis</ex><am>cõmunis</am></choice> ſit quindecim, cuius duæ ter <lb/>tiæ ſunt <choice><ex>decem</ex><am>decẽ</am></choice>, & quatuor quintæ duodecim, <choice><ex>communis</ex><am>cõmunis</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> denominans .15. multipli <lb/>candus ſit per quatuor quintas, ſcilicet duodecim, & productum diuidendum per <lb/>duas tertias, hoc eſt decem, ex quo oriantur decemocto quęſitus numerus?</s> </p> <p> <s xml:space="preserve">Quod ad <choice><ex>reductionem</ex><am>reductionẽ</am></choice> <choice><ex>numeratorum</ex><am>numeratorũ</am></choice> ad vnam & eandem denominationem attinet, <lb/>ea de cauſa fit quo uti poſſimus regula de tribus, quæ tribus tantummodo notis ter-<lb/>minis indiget, quo quartus à prędictis dependens, inueniri poſſit, quandoquidem <lb/>bini illi reſpectus, tribus terminis comprehendi <choice><ex>poſsunt</ex><am>poſsũt</am></choice>. </s> <s xml:space="preserve">At quod ad multiplicatio-<lb/>nem ſpectat denominantis <choice><ex>communis</ex><am>cõmunis</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> numerante denominantis in cogniti & diui-<lb/>ſionem producti per numerantem <choice><ex>cognitum</ex><am>cognitũ</am></choice> illę nihil aliud ſunt, quam <choice><ex>quartum</ex><am>quartũ</am></choice> <choice><ex>terminum</ex><am>terminũ</am></choice> <lb/>inuenire, ita proportionatum tertio, vt ſecundus primo.</s> </p> <p> <s xml:space="preserve">Excmpli gratia, ſit <seg type="var">.a.</seg> <choice><ex>denotans</ex><am>denotãs</am></choice> nume-<lb/>rantem denominantis cogniti, qui ſigni <lb/> <ptr xml:id="fig-0023-03a" corresp="fig-0023-03" type="figureAnchor"/> ficetur linea <seg type="var">.o.</seg> et <seg type="var">.e.</seg> ſit denominantis in-<lb/>cogniti numerans, denotati linea <seg type="var">.u.</seg> imò <lb/>verò & cogniti <seg type="var">.o.</seg> nempe quatuor <lb/>quintæ, Iam ſi <seg type="var">.o.</seg> cum <seg type="var">.e.</seg> multiplicemus, & productum per <seg type="var">.a.</seg> diuidemus dabitur <seg type="var">.u.</seg> <lb/>ſic ſe habens ad <seg type="var">.e.</seg> ſicut <seg type="var">.o.</seg> ad <seg type="var">.a.</seg> ex .20. ſeptimi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0023-03" corresp="fig-0023-03a"> <graphic url="0023-03"/> </figure> </div> </body> </floatingText> <pb facs="0024" n="12"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="16">XVI</num>.</head> <p> <s xml:space="preserve">INuenire autem cupienti cuius numeri, duæ tertiæ, ſint quatuor quintę partes, mul<lb/>tiplicandę eſſent duæ tertiæ per denominantem communem, & productum diui-<lb/>dendum per quatuor quintas ipſius de-<lb/>nominantis. </s> <s xml:space="preserve">Ac ſi quis diceret ſi <seg type="var">.e.</seg> dat <seg type="var">.<lb/> <ptr xml:id="fig-0024-01a" corresp="fig-0024-01" type="figureAnchor"/> o.</seg> quid dabit <seg type="var">.a</seg>? </s> <s xml:space="preserve">nempe dabit <seg type="var">.u.</seg> nam in <lb/>propoſito exemplo, terminus <seg type="var">.a.</seg> loco <seg type="var">.e.</seg> <lb/>duos ſortietur denominantes, cognitum <lb/>videlicet <seg type="var">.o.</seg> et <seg type="var">.u.</seg> incognitum quod po-<lb/>ſtea cognitum oritur ex regula de tribus, vt dictum eſt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0024-01" corresp="fig-0024-01a"> <graphic url="0024-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="17">XVII</num>.</head> <p> <s xml:space="preserve">QVA ratione cognoſci poterit proportionem quantitatis cenſicæ cenſicæ ad <lb/>ſimilem quantitatem quadruplam eſſe ad eam, quæ eſt ſuarum radicum; </s> <s xml:space="preserve">pro-<lb/>portionem <choice><ex>autem</ex><am>autẽ</am></choice> primarum relatarum eſſe quintuplam, <choice><ex>atque</ex><am>atq;</am></choice> ita deinceps?</s> </p> <p> <s xml:space="preserve">Cuiusrei gratia, <choice><ex>ſciendus</ex><am>ſciẽdus</am></choice> eſt modus <choice><ex>productionis</ex><am>ꝓductionis</am></choice> <choice><ex>harum</ex><am>harũ</am></choice> <choice><ex>dignitatum</ex><am>dignitatũ</am></choice> qui oritur ex produ-<lb/>ctione primæ radicis in ſeipſam, prout qui <choice><ex>cubum</ex><am>cubũ</am></choice> requirit, ducat radicé in ſuo quadra-<lb/>to, & orietur cubus, hæc poſtea ducta in cubum, <choice><ex>quantitatem</ex><am>quantitatẽ</am></choice> cenſicam <choice><ex>cenſicam</ex><am>cenſicã</am></choice>, et in <lb/>hanc, prædictam radicem, dabit quantitatem primam relatam. </s> <s xml:space="preserve">Quod vbi ſciueri-<lb/>mus, meminiſſe oportet Euclidem decimaoctaua ſexti aut .11. octaui docere, pro-<lb/>portionem quadrati ad <choice><ex>quadratum</ex><am>quadratũ</am></choice>, duplam eſſe proportioni ſuarum radicum, & .36. <lb/>vndecimi aut .11. octaui, cubi ad <choice><ex>cubum</ex><am>cubũ</am></choice> triplam eſſe, ego verò nunc aſſero, cenſici cen <lb/>ſici ad radicum proportionem quadruplam eſſe, primi verò relati ad primum re-<lb/>latum quintuplam <choice><ex>atque</ex><am>atq;</am></choice> ita gradatim.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, detur linea <seg type="var">.d.</seg> quæ cubum maiorem ſignificet. et <seg type="var">.b.</seg> <lb/>minorem <seg type="var">.c.</seg> verò ſit radixipſius <seg type="var">.d.</seg> et <seg type="var">.e.</seg> ipſius <seg type="var">.b.</seg> ita ordinate adinuicem, vt in ſub-<lb/>ſcripta figura cernitur. </s> <s xml:space="preserve">Iam <seg type="var">.c.</seg> cum <seg type="var">.d.</seg> producatur <choice><ex>proueniatque</ex><am>proueniatq́;</am></choice> <seg type="var">.q.</seg> cenſicum cenſi-<lb/>cum, tum producatur <seg type="var">.e.</seg> cum <seg type="var">.b.</seg> et dabitur <seg type="var">.p.</seg> alterum cenſicum cenſicum. </s> <s xml:space="preserve">Dico <lb/>igitur proportionem <seg type="var">.q.</seg> ad <seg type="var">.p.</seg> quadruplam eſſe proportioni <seg type="var">.c.</seg> ad <seg type="var">.e.</seg> hac de <lb/>cauſa quòd proportio <seg type="var">.q.</seg> ad <seg type="var">.p.</seg> compo-<lb/>natur ex proportione <seg type="var">.d.</seg> ad <seg type="var">.b.</seg> et <seg type="var">.c.</seg> ad <seg type="var">.e.</seg> <lb/> <ptr xml:id="fig-0024-02a" corresp="fig-0024-02" type="figureAnchor"/> prout facile ex .24. ſexti, aut quinta octaui <lb/>depręhenditur. </s> <s xml:space="preserve">Quare <choice><ex>cum</ex><am>cũ</am></choice> proportio <seg type="var">.d.</seg> ad <seg type="var">.<lb/>b.</seg> proportioni <seg type="var">.c.</seg> ad <seg type="var">.e.</seg> tripla ſit, patet pro-<lb/>portionem <seg type="var">.q.</seg> ad <seg type="var">.p.</seg> quadruplam eſſe pro-<lb/>portioni <seg type="var">.c.</seg> ad <seg type="var">.e</seg>. </s> <s xml:space="preserve">Idem de cæteris dignitati <lb/>bus dico, ſumptis ſemper <seg type="var">.d</seg> et <seg type="var">.b.</seg> pro duo-<lb/>bus cenſibus cenſuum, aut duobus primis relatis, aut alio quouis axiomate.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0024-02" corresp="fig-0024-02a"> <graphic url="0024-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA. <num value="18">XVIII</num>.</head> <p> <s xml:space="preserve">CVR diuidentibus nobis dignitatem, per dignitatem, radix prouenientis: </s> <s xml:space="preserve">pro <lb/>ueniens ſit diuiſionis vnius radicis per alteram?</s> </p> <p> <s xml:space="preserve">Sint exempli gratia duę lineæ <seg type="var">.b.q.</seg> et <seg type="var">.f.g.</seg> quæ ſignificent duas radices cuiuſuis <lb/>dignitatis; </s> <s xml:space="preserve"><choice><ex>demusque</ex><am>demusq́;</am></choice> eſſe radices duorum quadratorum, <choice><ex>quadratumque</ex><am>quadratumq́;</am></choice> ipſius <seg type="var">b.q.</seg> <lb/>per quadratum ipſius <seg type="var">.f.g.</seg> diuidatur; </s> <s xml:space="preserve">quadrataq́ue radix prouenientis ſit <seg type="var">.d.q.</seg> <lb/>vnitas verò linearis ſit <seg type="var">.i.g</seg>. </s> <s xml:space="preserve">Dico ipſam <seg type="var">.d.q.</seg> eſſe proueniens ex diuiſione <seg type="var">.b.q.</seg> <lb/>per <seg type="var">.f.g</seg>. </s> <s xml:space="preserve">Patet enim ex definitione diuiſionis nono theoremate tradita quadra- <pb facs="0025" n="13"/><fw type="head">THEOR. ARITH.</fw> tum ipſius <seg type="var">.d.q.</seg> talem eſſe partem quadrati ipſius <seg type="var">.b.q.</seg> qualis quadratum ipſius <seg type="var">.g.i.</seg> <lb/>eſt quadrati ipſius <seg type="var">.f.g</seg>. </s> <s xml:space="preserve">Scimus pręterea ex .19. ſexti, aut vndecima octaui, propor-<lb/>tioné quadrati ipſius <seg type="var">.b.q.</seg> ad <choice><ex>quadratum</ex><am>quadratũ</am></choice> ipſius <seg type="var">.d.q.</seg> duplam eſſe proportioni <seg type="var">.b.q.</seg> ad <seg type="var">.<lb/>d.q.</seg> ſuarum radicum (cuborum enim tripla eſſet & cenſuum cenſuum, quadrupla, <lb/><choice><ex>atque</ex><am>atq;</am></choice> ita deinceps ex præcedenti theoremate) Id ipſum dico de dignitatibus ipſius <seg type="var">.<lb/>f.g.</seg> et <seg type="var">.i.g.</seg> reſpectu radicum <seg type="var">.f.g.</seg> et <seg type="var">.i.g</seg>. </s> <s xml:space="preserve">Vnde <lb/>cum proportio dignitatis ipſius <seg type="var">.b.q.</seg> ad il-<lb/>lam <seg type="var">.d.q.</seg> ęqualis ſit proportioni dignitatis <lb/> <ptr xml:id="fig-0025-01a" corresp="fig-0025-01" type="figureAnchor"/> ipſius <seg type="var">.f.g.</seg> ad illam <seg type="var">.g.i.</seg> ex communi ſcien-<lb/>tia apertè cognoſcemus ſimplices propor-<lb/>tiones eſſe interſe æquales, nempe eam quę <lb/>eſt <seg type="var">.b.q.</seg> ad <seg type="var">.d.q.</seg> æqualem eſſe ei, quæ eſt <seg type="var">.f.<lb/>g.</seg> ad <seg type="var">.i.g.</seg> <choice><ex>itaque</ex><am>itaq;</am></choice> ſequitur ex definitione diuiſionis <seg type="var">.d.q.</seg> eſſe proueniens ex diuiſione <seg type="var">.<lb/>b.q.</seg> per <seg type="var">.f.g</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0025-01" corresp="fig-0025-01a"> <graphic url="0025-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="19">XVIIII</num>.</head> <p> <s xml:space="preserve">CVR productum ex duabus radicibus quadratis, eſt quadrata radix, producti <lb/>ſuorum quadratorum ſimul?</s> </p> <p> <s xml:space="preserve">In cuius rei gratiam, ſint duo quadrata <seg type="var">.d.a.</seg> et <seg type="var">n.o.</seg> coniuncta ſimul, prout in ſub-<lb/>ſcripta figura apparet, ita tamen vtangulus <seg type="var">.a.n.u.</seg> ſitre <lb/>ctus, </s> <s xml:space="preserve">quare ex quartadecima primi, duo latera <seg type="var">.n.c.</seg> et <seg type="var">.<lb/> <ptr xml:id="fig-0025-02a" corresp="fig-0025-02" type="figureAnchor"/> n.a.</seg> directe <choice><ex>coniungentur</ex><am>coniũgentur</am></choice> adinuicem, prout etiam reli-<lb/>qua duo latera <seg type="var">.n.u.</seg> et <seg type="var">.n.d</seg>. </s> <s xml:space="preserve">Cogitato deinde <seg type="var">.a.u.</seg> pro <lb/>ducto ipſius <seg type="var">.a.n.</seg> in <seg type="var">.n.u.</seg> duarum videlicet radicum <lb/>quadratarum ſimul, dabitur ex prima ſexti, aut de-<lb/>cimaottaua ſeptimi, productum <seg type="var">.a.u.</seg> medium propor <lb/>tionale inter quadratum <seg type="var">.a.d.</seg> et <seg type="var">.u.c.</seg> quod ſi cogi-<lb/>temus has tres ſuperficies, tres numeros eſſe, pate-<lb/>bit ex vigeſimaprima ſeptimi productum <seg type="var">.a.u.</seg> in ſe-<lb/>ipſum, quadratum ſcilicet <seg type="var">.a.u.</seg> æquale eſſe producto <seg type="var">.<lb/>a.d.</seg> in <seg type="var">.u.c.</seg> ex quo propoſiti euidentia conſequetur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0025-02" corresp="fig-0025-02a"> <graphic url="0025-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="20">XX</num>.</head> <p> <s xml:space="preserve">QVA ratione id ipſum in cubis cognoſci poterit. <lb/></s> <s xml:space="preserve">Sit cubus <seg type="var">.l.b.</seg> & cubus <seg type="var">.o.p.</seg> quorum productum ſit <seg type="var">.u.g.</seg> quod aſſero eſle <lb/> <ptr xml:id="fig-0025-03a" corresp="fig-0025-03" type="figureAnchor"/> cubum, quamuis Eucli. idem probet <lb/>in <ref>.4. noni.</ref> cuius radicem demonſtra-<lb/>bo eſſe numeri æqualis numero <seg type="var">.m.q.</seg> <lb/>qui <seg type="var">.m.q.</seg> productum eſt ipſius <seg type="var">.m.e.</seg> in <seg type="var">.e.<lb/>q.</seg> radicum propoſitorum cuborum. </s> <s xml:space="preserve">Pa-<lb/>tet enim ex præcedenti theoremate <seg type="var">.m.<lb/> <ptr xml:id="fig-0025-04a" corresp="fig-0025-04" type="figureAnchor"/> <pb facs="0026" n="14"/><fw type="head">IO. BAPT. BENED.</fw> q.</seg> radicem eſſe quadratam producti <seg type="var">.l.e.</seg> in <seg type="var">.e.p.</seg> quod <choice><ex>productum</ex><am>productũ</am></choice> ſit quadratuni<unclear reason="illegible"/> <lb/>corporeum <seg type="var">.c.g.</seg> cogitemus pariter duo quadrata <seg type="var">.l.e.</seg> et <seg type="var">.e.p.</seg> eſſe pariter corpo-<lb/>rea, tantę profunditatis, quantam, vnitas linearis radicum <seg type="var">.m.e.</seg> et <seg type="var">.e.q.</seg> requirit. <lb/></s> <s xml:space="preserve">Hæc duo corpora producentur à ſuperficie in vnitatem, <choice><ex>vocenturque</ex><am>vocenturq́;</am></choice> <seg type="var">.l.x.</seg> et <seg type="var">.x.p.</seg> quo <lb/>facto, cogitemus corpus <seg type="var">.a.g.</seg> tamquam productum cubi <seg type="var">.l.b.</seg> in quadratum <seg type="var">.e.p</seg>. </s> <s xml:space="preserve">Vn-<lb/>de ex decimaoctaua, aut decimanona ſeptimi, eadem erit proportio <seg type="var">.a.g.</seg> ad <seg type="var">.c.g.</seg> <lb/>quæ eſt <seg type="var">.l.b.</seg> ad <seg type="var">.l.x.</seg> corporeum, ſed ex .25. vndecimi & prima ſexti, ita ſe habet <seg type="var">.a.K.</seg> <lb/>ad <seg type="var">.K.c.</seg> vnitatem linearé ſicut <seg type="var">.a.g.</seg> ad <seg type="var">.c.g.</seg> & ex <choice><ex>eiſdem</ex><am>eiſdẽ</am></choice> ita ſe habebit <seg type="var">.b.e.</seg> ad <seg type="var">.e.x.</seg> vnita-<lb/>tem linearem, ſicut <seg type="var">.l.b.</seg> ad quadratum <seg type="var">.l.x.</seg> corporeum. </s> <s xml:space="preserve">Itaque ſic ſe habebit <seg type="var">.b.e.</seg> ad <lb/>vnitatem linearem <seg type="var">.e.x.</seg> videlicet <seg type="var">.K.c.</seg> ſicut <seg type="var">.a.K.</seg> ad ipſam <seg type="var">.K.c</seg>. </s> <s xml:space="preserve">Vnde ex nona quinti <seg type="var">.<lb/>a.K.</seg> æqualis erit <seg type="var">.e.b.</seg> & conſequenter æqualis <seg type="var">.m.e</seg>. </s> <s xml:space="preserve">Iam verò ſit <seg type="var">.u.g.</seg> productum <seg type="var">.l.b.</seg> <lb/>cubi, in cubum <seg type="var">.o.p.</seg> vt ſupra dictum eſt, Hinc patebit ex quauis duarum propoſitio-<lb/>num, decimaoctaua, aut decimanona ſeptimi, eandem futuram proportionem <seg type="var">.u.g.</seg> <lb/>ad <seg type="var">.a.g.</seg> quæ eſt <seg type="var">.o.p.</seg> ad <seg type="var">.x.p.</seg> quadratum corporeum. </s> <s xml:space="preserve">Quare ex poſtremis, dictis ratio-<lb/>nibus, eadem erit proportio <seg type="var">.u.K.</seg> ad <seg type="var">.a.K.</seg> quæ eſt <seg type="var">.o.e.</seg> ad vnitatem linearem <seg type="var">.e.x.</seg> at <lb/>ex dictis decimaoctaua & decimanona ſeptimi, ita ſe habet <choice><ex>numerus</ex><am>numerꝰ</am></choice> <seg type="var">.m.q.</seg> ad <choice><ex>numerum</ex><am>numerũ</am></choice> <lb/><choice><ex>ſuperficialem</ex><am>ſuperficialẽ</am></choice> <seg type="var">.m.e.</seg> qui <choice><ex>producitur</ex><am>ꝓducitur</am></choice> à lineari <seg type="var">.m.e.</seg> in vnitaté <choice><ex>linearem</ex><am>linearẽ</am></choice> ipſius <seg type="var">.e.q.</seg> ſicut nume <lb/>rus <seg type="var">.q.e.</seg> ad ſuam vnitaté, ſed <choice><ex>cum</ex><am>cũ</am></choice> numerus <seg type="var">.a.K.</seg> æqualis ſit numero <seg type="var">.m.e.</seg> vt <choice><ex>probatum</ex><am>probatũ</am></choice> eſt <lb/>erit ergo ex vndecima & nona quinti, numerus <seg type="var">.u.K.</seg> æqualis numero <seg type="var">.m.q</seg>. </s> <s xml:space="preserve">At <seg type="var">.f.g.</seg> <lb/>pariter æqualis eſt numero <seg type="var">.m.q.</seg> ex præcedenti theoremate, vnde <seg type="var">.K.u.</seg> pariter æqua <lb/>lis erit <seg type="var">.f.g</seg>. </s> <s xml:space="preserve">Itaque ſequitur <seg type="var">.u.g.</seg> cubum eſſe, & <seg type="var">f.g.</seg> radicem ipſius, æqualem numero <seg type="var">.<lb/>m.q.</seg> quod quærebatur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0025-03" corresp="fig-0025-03a"> <graphic url="0025-03"/> </figure> <figure xml:id="fig-0025-04" corresp="fig-0025-04a"> <graphic url="0025-04"/> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0026-01"/> </figure> <figure place="here"> <graphic url="0026-02"/> </figure> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="21">XXI</num>.</head> <p> <s xml:space="preserve">VT autem in uniuerſum ſciri poſſit totum <choice><ex>infinitum</ex><am>infinitũ</am></choice> dignitatum, hoc eſt radicem <lb/>producti duarum dignitatum ſimilium, productum eſſe duarum radicum ea-<lb/>rundem dignitatum.</s> </p> <p> <s xml:space="preserve">Ponamus, exempli gratia, duas radices quadratas <seg type="var">.q.p.</seg> et <seg type="var">.g.K.</seg> incognitas, quas <lb/>qui velit adinuicem multiplicare, cogatur earum quadrata cognita <seg type="var">.n.</seg> cum <seg type="var">.i.</seg> multi-<lb/>plicare, quorum productum ſit quadratum <seg type="var">.m.</seg> radix cuius ſit <seg type="var">.b.d.</seg> quam dico æqualé <pb facs="0027" n="15"/><fw type="head">THEOREM. ARIT.</fw> eſſe <choice><ex>producto</ex><am>ꝓducto</am></choice> <seg type="var">.q.p.</seg> in <seg type="var">.g.k.</seg> <choice><ex>quod</ex><am>qđ</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> ſit <seg type="var">.o</seg>. </s> <s xml:space="preserve">Patet enim <choice><ex>proportionem</ex><am>proportionẽ</am></choice> <seg type="var">.o.</seg> ad <seg type="var">.q.p.</seg> <choice><ex>eandem</ex><am>eandẽ</am></choice> eſſe <lb/>cum proportione <seg type="var">.g.k.</seg> ad ſuam vnitatem linearem, ex decimaoctaua, aut decima-<lb/>nona ſeptimi, hæc vero vnitas linearis ſit <seg type="var">.t.</seg> cuius ſuperficialis ſit <seg type="var">.u.</seg> vnitas ſcilicet to-<lb/>ties in ſeipſam multiplicata quoties propoſita dignitas patitur, tametſi in præſen <lb/>ti exemplo quadrata dignitas ſumatur. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex eiſdem propoſitionibus decimaocta <lb/>ua aut decimanona, ſic ſe habet <seg type="var">.m.</seg> ad <seg type="var">.n.</seg> ſicut <seg type="var">.i.</seg> ad <seg type="var">.u</seg>. </s> <s xml:space="preserve">Scimus pręterea <choice><ex>proportionem</ex><am>proportionẽ</am></choice> <seg type="var">.<lb/>m.</seg> ad <seg type="var">.n.</seg> (eo quod in propoſito exemplo ſint quadrata) duplam eſſe proportioni <seg type="var">.b.<lb/>d.</seg> ad <seg type="var">.q.p.</seg> et ipſius <seg type="var">.i.</seg> ad <seg type="var">.u.</seg> pariter duplam proportioni <seg type="var">.g.k.</seg> ad <seg type="var">.t.</seg> iam autem dictum <lb/>fuit ſic ſe habere <seg type="var">.m.</seg> ad <seg type="var">.n.</seg> ſicut <seg type="var">.i.</seg> ad <seg type="var">.u</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <seg type="var">.<lb/>b.d.</seg> ſic ſe habebit ad <seg type="var">.q.p.</seg> ſicut <seg type="var">.g.k.</seg> ad <seg type="var">.t.</seg> <lb/> <ptr xml:id="fig-0027-01a" corresp="fig-0027-01" type="figureAnchor"/> quandoquidem ſic ſe habeattotum ad <choice><ex>to- tum</ex><am>to-tũ</am></choice>, ſicut pars ad <choice><ex>partem</ex><am>partẽ</am></choice>, <choice><ex>dum</ex><am>dũ</am></choice> ſimiles ſint, proba <lb/><choice><ex>tum</ex><am>tũ</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> eſt ſuperius ita ſe habere <seg type="var">.o.</seg> ad <seg type="var">.q.p.</seg> <lb/>ſicut <seg type="var">.g.k.</seg> ad <seg type="var">.t.</seg> <choice><ex>itaque</ex><am>itaq;</am></choice> <seg type="var">.o.</seg> ſic ſe habebit ad <seg type="var">.q.p.</seg> <lb/>ſicut <seg type="var">.b.d.</seg> ad <seg type="var">.q.p.</seg> vnde <seg type="var">.o.</seg> æqualis erit <seg type="var">.b.d.</seg> <lb/>Hocipſum cęteris dignitatibus conueniet, <lb/>mutatis tantummodo proportionibus <seg type="var">.m.<lb/>n.</seg> ad proportionem <seg type="var">.b.d</seg>: <seg type="var">q.p.</seg> ſic propor-<lb/>tionibus duarum dignitatum <seg type="var">.i.u.</seg> ad pro-<lb/>portionem ſuarum radicum <seg type="var">.g.k.t</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0027-01" corresp="fig-0027-01a"> <graphic url="0027-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="22">XXII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">DOcent</hi> veteres, quòd ſi quilibet numerus in duas partes inæquales diuiſus <lb/>fuerit, <choice><ex>totumque</ex><am>totumq́</am></choice> diuiſum per <choice><ex>vnam</ex><am>vnã</am></choice> partium, & per eandem pars altera diuiſa fue-<lb/>rit: </s> <s xml:space="preserve">differentia prouenientium ſemper vnitas erit. </s> <s xml:space="preserve">quodquidem veriſſimum eſt.</s> </p> <p> <s xml:space="preserve">Detur enim <seg type="var">.b.d.</seg> propoſitus numerus in duas partes inæquales diuiſus <seg type="var">.b.c.</seg> et <seg type="var">.c.d.</seg> <lb/>& in primis <choice><ex>totum</ex><am>totũ</am></choice> <seg type="var">.b.d.</seg> per <seg type="var">.c.d.</seg> diuidatur, ex quo oriatur <seg type="var">e.o.</seg> vnitas autem <choice><ex>per</ex><am>.ꝑ</am></choice> <seg type="var">.i.o.</seg> ſigni-<lb/>ficetur, tum pars ipſa <seg type="var">.b.c.</seg> <choice><ex>per</ex><am>ꝑ.</am></choice> <choice><ex>eandem</ex><am>eãdem</am></choice> <seg type="var">.c.d.</seg> diuidatur, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <choice><ex>proueniens</ex><am>proueniẽs</am></choice> <seg type="var">.a</seg>. </s> <s xml:space="preserve">Sanè ex defini-<lb/>tione diuiſionis, eadem erit proportio <seg type="var">.b.d.</seg> ad <seg type="var">.e.o.</seg> quæ eſt <seg type="var">.c.d.</seg> ad <seg type="var">.i.o.</seg> et ita <seg type="var">.b.c.</seg> ad <seg type="var">.a.</seg> <lb/>ſicut <seg type="var">.c.d.</seg> ad <seg type="var">.i.o</seg>. </s> <s xml:space="preserve">Ex <ref>.19. autem quinti</ref>, ita ſe habet <seg type="var">.b.c.</seg> ad <seg type="var">.e.i.</seg> ſicut <seg type="var">.b.d.</seg> ad <seg type="var">.e.o.</seg> at <seg type="var">.b.d.</seg> <lb/>ad <seg type="var">.e.o.</seg> ſic ſe habet ſicut <seg type="var">.c.d.</seg> ad <seg type="var">.i.o.</seg> hoc eſt ſicut <seg type="var">.b.c.</seg> ad <seg type="var">.a</seg>. </s> <s xml:space="preserve">Quare ex .11. quinti ſic ſe <lb/>habebit <seg type="var">.b.c.</seg> ad <seg type="var">.e.i.</seg> ſicut .ad <seg type="var">.a.</seg> ex quo ex .9. <choice><ex>praedi cti</ex><am>prędicti</am></choice> <seg type="var">.a.</seg> æqualis erit <seg type="var">.e.i.</seg> ſed <seg type="var">.e.i.</seg> minor eſt <seg type="var">.e.o.</seg> <lb/> <ptr xml:id="fig-0027-02a" corresp="fig-0027-02" type="figureAnchor"/> per <seg type="var">.i.o</seg>. </s> <s xml:space="preserve">Quare ſequitur propoſitum verum eſ<lb/>ſe. </s> <s xml:space="preserve">Quod ipſum pauciſſimis verbis ſic definiri <lb/>poteſt, ſi dixerimus, eiuſmodi diuidens .in par-<lb/>te diuiſibili, <choice><ex>quam</ex><am>quã</am></choice> in toto, ſemel minus ingredi, <lb/>quandoquidem altera pars eſt, ex qua totum integrum perficitur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0027-02" corresp="fig-0027-02a"> <graphic url="0027-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="23">XXIII</num>.</head> <p> <s xml:space="preserve">HOcipſum alia ratione contemplari po<lb/> <ptr xml:id="fig-0027-03a" corresp="fig-0027-03" type="figureAnchor"/> terimus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0027-03" corresp="fig-0027-03a"> <graphic url="0027-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Significetur enim totalis numerus per <seg type="var">.a.e.</seg> <lb/>in duas partes diuiſus <seg type="var">.a.u.</seg> et <seg type="var">.u.e.</seg> totius autem diuidens ſit <seg type="var">.u.e.</seg> & partis alterius <seg type="var">.a.u.</seg> <lb/>totius verò <choice><ex>proueniens</ex><am>proueniẽs</am></choice> ſit <seg type="var">.a.c.</seg> partis <choice><ex>autem</ex><am>autẽ</am></choice>, ſit <choice><ex>proueniens</ex><am>proueniẽs</am></choice> <seg type="var">.a.n.</seg> tum differentia ſit <seg type="var">.n.c.</seg> vni <pb facs="0028" n="16"/><fw type="head">IO. BAPT. BENED.</fw> tas vero cui <choice><ex>differentiam</ex><am>differentiã</am></choice> <seg type="var">.n.c.</seg> æquari dico, ſit <seg type="var">.a.i</seg>. </s> <s xml:space="preserve">Patet enim in primis, eandem propor <lb/>tionem eſſe <seg type="var">.a.e.</seg> ad <seg type="var">.a.c.</seg> quæ eſt <seg type="var">.u.e.</seg> ad <seg type="var">.a.i.</seg> ex definitione diuiſionis, et eandem <lb/>eſſe <seg type="var">.a.u.</seg> ad <seg type="var">.a.n.</seg> quæ eſt <seg type="var">.u.e.</seg> ad <seg type="var">.a.i.</seg> vnde ex .<lb/>11. quinti ſic ſe habebit <seg type="var">.a.e.</seg> ad <seg type="var">.a.c.</seg> ſicut <seg type="var">.a.<lb/> <ptr xml:id="fig-0028-01a" corresp="fig-0028-01" type="figureAnchor"/> u.</seg> ad <seg type="var">.a.n.</seg> et ex .19. eiuſdem ſic ſe habe-<lb/>bit <seg type="var">.u.e.</seg> ad <seg type="var">.n.c.</seg> ſicut <seg type="var">.a.e.</seg> ad <seg type="var">.a.c.</seg> ſed. ſic ſe <lb/>habebat <seg type="var">.u.e.</seg> ad <seg type="var">.a.i</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex prædicta .11. quinti, ſic ſe habebit <seg type="var">.u.e.</seg> ad <seg type="var">.n.c.</seg> ſicut ad <seg type="var">.a.<lb/>i</seg>. </s> <s xml:space="preserve">Quare ex .9. eiuſdem <seg type="var">.n.c.</seg> æqualis erit <seg type="var">.a.i.</seg> etidcirco <seg type="var">.n.c.</seg> pariter vnitas erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0028-01" corresp="fig-0028-01a"> <graphic url="0028-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="24">XXIIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> quibuslibet duobus numeris diuiſis adinuicem, <choice><ex>multiplicatisque</ex><am>multiplicatisq́</am></choice> prouenien <lb/>tibus ſimul, productum, ſemper eſt vnitas ſuperficialis? </s> <s xml:space="preserve">Nempe ex .20. ſeptimi, <lb/>quoniam vnitas linearis ſemper media proportionalis eſt inter bina prouenientia. <lb/></s> <s xml:space="preserve">Quodita ſpecularilicet.</s> </p> <p> <s xml:space="preserve"><choice><ex>Significentur</ex><am>Significẽtur</am></choice> duo propoſiti numeri per <seg type="var">.b.p.</seg> et <seg type="var">.b.d.</seg> mutuo diuiſi, proueniens au-<lb/>tem <seg type="var">.b.p.</seg> per <seg type="var">.b.d.</seg> diuiſum ſit <seg type="var">.b.n.</seg> tum proueniens <seg type="var">.b.d.</seg> diuiſum per <seg type="var">.b.p.</seg> ſit <seg type="var">.b.a.</seg> <lb/>et <seg type="var">.b.t.</seg> ſit vnitas <seg type="var">.b.p.</seg> et <seg type="var">.b.e.</seg> vnitas <seg type="var">.b.d.</seg> ex quo <seg type="var">.b.t.</seg> æqualis erit <seg type="var">.b.e</seg>.</s> </p> <p> <s xml:space="preserve">Iam ex definitio ne diuiſionis, dabitur eadem proportio <seg type="var">.b.p.</seg> ad <seg type="var">.b.n.</seg> quæ eſt <seg type="var">.b.d.</seg> <lb/>ad <seg type="var">.b.e.</seg> et proportio <seg type="var">.b.d.</seg> ad <seg type="var">.b.a.</seg> quæ eſt <seg type="var">.b.p.</seg> ad <seg type="var">.b.t</seg>. </s> <s xml:space="preserve">Sed cum ſic ſe habeat <seg type="var">.b.<lb/>p.</seg> ad <seg type="var">.b.n.</seg> ſicut <seg type="var">.b.d.</seg> ad <seg type="var">.b.e.</seg> permutando ſic ſe habebit <seg type="var">.b.p.</seg> ad <seg type="var">.b.d.</seg> ſicut <seg type="var">.b.n.</seg> ad <seg type="var">.b.<lb/>e.</seg> hoc eſt ad <seg type="var">.b.t.</seg> et cum ſic ſe habeat <seg type="var">.b.d.</seg> ad <seg type="var">.b.a.</seg> ſicut <seg type="var">.b.p.</seg> ad <seg type="var">.b.t</seg>: permutando ſic ſe <lb/>habebit <seg type="var">.b.d.</seg> ad <seg type="var">.b.p.</seg> ſicut <seg type="var">.b.a.</seg> ad <seg type="var">.b.t</seg>. <lb/></s> <s xml:space="preserve">Quare euerſim ſic ſe habebit <seg type="var">.b.p.</seg> ad <seg type="var">.<lb/> <ptr xml:id="fig-0028-02a" corresp="fig-0028-02" type="figureAnchor"/> <lb/>b.d.</seg> ſicut <seg type="var">.b.t.</seg> ad <seg type="var">.b.a.</seg> ſed <seg type="var">.b.n.</seg> ad <seg type="var">.b.t.</seg> ſic <lb/>ſe habebat vt <seg type="var">.b.p.</seg> ad <seg type="var">.b.d</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex .11. <lb/>quintiſic ſe habebit <seg type="var">.b.n.</seg> ad <seg type="var">.b.t.</seg> ſicut <seg type="var">.b.<lb/> <ptr xml:id="fig-0028-03a" corresp="fig-0028-03" type="figureAnchor"/> e.</seg> ad <seg type="var">.b.a</seg>. </s> <s xml:space="preserve">Dictum autem eſt <seg type="var">.b.e.</seg> et <seg type="var">.b.t.</seg> idem omnino eſſe. </s> <s xml:space="preserve">Quare ex .20. ſeptimi pro-<lb/>poſiti veritas innoteſcet.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0028-02" corresp="fig-0028-02a"> <graphic url="0028-02"/> </figure> <figure xml:id="fig-0028-03" corresp="fig-0028-03a"> <graphic url="0028-03"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="25">XXV</num>.</head> <p> <s xml:space="preserve">IDipſum & hac altera uia patebit.</s> </p> <p> <s xml:space="preserve">Duo illi numeri per <seg type="var">.o.</seg> et <seg type="var">.u.</seg> ſignificentur mutuo diuiſi, proueniens <choice><ex>autem</ex><am>autẽ</am></choice> <seg type="var">.o.</seg> per <seg type="var">.<lb/>u.</seg> ſit <seg type="var">.e.</seg> et proueniens <seg type="var">.u.</seg> per <seg type="var">.o.</seg> ſit <seg type="var">.x.</seg> vnitas uerò per <seg type="var">.i.</seg> ſignificetur, quas tamen quanti-<lb/>tates ſubſcripto modo ad inuicem diſponi-<lb/>to. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex definitione diuiſionis, eadem erit <lb/> <ptr xml:id="fig-0028-04a" corresp="fig-0028-04" type="figureAnchor"/> proportio <seg type="var">.o.</seg> ad <seg type="var">.e.</seg> quę eſt <seg type="var">.u.</seg> ad <seg type="var">.i.</seg> et <seg type="var">.o.</seg> ad <seg type="var">.i.</seg> quę <lb/>eſt <seg type="var">.u.</seg> ad <seg type="var">.x</seg>. </s> <s xml:space="preserve">Quare ex æqualitate <choice><ex>proportionum</ex><am>proportionũ</am></choice> <seg type="var">.<lb/>c.</seg> ad <seg type="var">.i.</seg> ſic ſe habebit ſicut <seg type="var">.i.</seg> ad <seg type="var">.x.</seg> erit enim <seg type="var">.i.</seg> <lb/>media proportionalis inter <seg type="var">.e.</seg> et <seg type="var">.x.</seg> ex .20. <choice><ex>autem</ex><am>autẽ</am></choice> <lb/>ſeptimi propoſitum concludetur. </s> <s xml:space="preserve">Huiuſmodi rei cauſa etiam eſt, quod proueniens <lb/>diuiſionis vnius eſt numerator æqualis denominatori diuiſionis alterius.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0028-04" corresp="fig-0028-04a"> <graphic url="0028-04"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="26">XXVI</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> duobus numeris mutuo diuiſis, <choice><ex>sumptis</ex><am>sũptis</am></choice> deinde prouenientibus ſimul et adinui <lb/>cem, & per hanc ſummam, diuiſa ſumma quadratorum dictorum <choice><ex>propoſitorum</ex><am>propoſitorũ</am></choice> <pb facs="0029" n="17"/><fw type="head">THEOREM. ARITH.</fw> numerorum, proueniat numerus æqualis numero producti duorum primorum nu-<lb/>m erorum ſimul.</s> </p> <p> <s xml:space="preserve">Sint exempli gratia propoſiti numeri .2. et .8. qui mutuo diuiſi in primis dent pro <lb/>uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-<lb/>cta dabunt ſummam quatuor integrorum et quartæ partis vnius, ſumma autem qua <lb/>dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar <lb/>tam partem vnius diuiſus dabit .16. pro proueniente, quæ .16. æqualia erunt pro <lb/>ducto binarii in octonarium.</s> </p> <p> <s xml:space="preserve">Cuius rei hæc erit ſpeculatio, ſint duæ lineæ <seg type="var">.o.e.</seg> et <seg type="var">.o.n.</seg> quæ duos numeros pro-<lb/>poſitos ſignificent, inuicem ad angulum rectum <seg type="var">.o.</seg> coniunctæ, quarum quadrata <lb/>ſint <seg type="var">.o.a.</seg> et <seg type="var">.o.p.</seg> ipſorum productum ſit <seg type="var">.n.e.</seg> tum <seg type="var">.o.t.</seg> ſit proueniens ex diuiſione <seg type="var">.o.e.</seg> <lb/>per <seg type="var">.o.n</seg>. </s> <s xml:space="preserve">Hęc ſingulatim conſideremus (<choice><ex>nam</ex><am>nã</am></choice> ſi in partibus ſimplicibus quod dicimus ac <lb/>ciderit, id ipſum in compoſitis conſequenter eueniet) quamobrem ex definitione di <lb/>uiſionis dabitur eadem proportio <seg type="var">.o.e.</seg> ad <seg type="var">.o.t.</seg> quæ eft <seg type="var">.o.n.</seg> ad vnitatem, quæ ſit <seg type="var">.o.<lb/>x</seg>. </s> <s xml:space="preserve">Nunc cogitemus <choice><ex>ſuperficiem</ex><am>ſuperficiẽ</am></choice> <choice><ex>rectangulam</ex><am>rectangulã</am></choice> <seg type="var">.o.c.</seg> <choice><ex>æqualem</ex><am>æqualẽ</am></choice> quadrato <seg type="var">.o.a</seg>. </s> <s xml:space="preserve">tunc numerus <seg type="var">.<lb/>c.t.</seg> proueniens erit, ut patet, ex diuiſione numeri quadrati <seg type="var">.o.a.</seg> per <choice><ex>numerum</ex><am>numerũ</am></choice> <seg type="var">.o.t.</seg> <choice><ex>eritque</ex><am>eritq́</am></choice> <lb/><choice><ex>eadem</ex><am>eadẽ</am></choice> proportio <seg type="var">.c.t.</seg> ad <seg type="var">.o.e.</seg> quæ eſt <seg type="var">.o.e.</seg> ad <seg type="var">.o.t.</seg> ex ſecunda parte quintæ decimæ ſexti, <lb/>aut .20. ſeptimi. </s> <s xml:space="preserve"><choice><ex>Iam</ex><am>Iã</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> dictum eſt <seg type="var">.o.e.</seg> ad <seg type="var">.o.t.</seg> ſic ſe habere ſicut <seg type="var">.o.n.</seg> ad <seg type="var">.o.x</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex .<lb/>11. quinti ſic ſe habebit <seg type="var">.c.t.</seg> ad <seg type="var">.o.e.</seg> ſicut <seg type="var">.o.n.</seg> ad <seg type="var">.o.x</seg>. </s> <s xml:space="preserve">Sed ex prima ſexti, aut .18. vel .<lb/>19. ſeptimi, ſic ſe habet <choice><ex>productum</ex><am>ꝓductum</am></choice> <seg type="var">.n.e.</seg> ad <seg type="var">.e.x.</seg> ſicut <seg type="var">.o.n.</seg> ad <seg type="var">.o.x</seg>. </s> <s xml:space="preserve">quare denuo ſic ſe ha-<lb/>bebit numerus <seg type="var">.c.t.</seg> ad numerum <seg type="var">.o.e.</seg> ſicut nume-<lb/>rus <seg type="var">.n.e.</seg> ad numerum <seg type="var">.x.e</seg>. </s> <s xml:space="preserve">Sed numerus <seg type="var">.o.e.</seg> cum <lb/> <ptr xml:id="fig-0029-01a" corresp="fig-0029-01" type="figureAnchor"/> numero <seg type="var">.x.e.</seg> ſpecie idem eſt, igitur ex .9. quinti nu <lb/>merus <seg type="var">.c.t.</seg> numero <seg type="var">.n.e.</seg> æqualis erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0029-01" corresp="fig-0029-01a"> <graphic url="0029-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Id ipſum de quadrato ipſius <seg type="var">.o.n.</seg> videlicet <seg type="var">.p.o.</seg> <lb/>dico. </s> <s xml:space="preserve">Nam ſi proueniens <seg type="var">.o.n.</seg> diuiſo per <seg type="var">.o.e.</seg> ideſt <seg type="var">.<lb/>o.i.</seg> proportionale reſpondens ad <seg type="var">.o.t.</seg> cum <seg type="var">.o.t.</seg> <lb/><choice><ex>coniunctum</ex><am>coniunctũ</am></choice> fuerit, et per <choice><ex>hanc</ex><am>hãc</am></choice> ſummam diuiſa ſumma <lb/>quadratorum <seg type="var">.o.a.</seg> et <seg type="var">.o.p.</seg> patet per ſe proueniens <lb/>futurum eiuſdem numeri <seg type="var">.c.t.</seg> <choice><ex>ipſumque</ex><am>ipſumq́</am></choice> <seg type="var">.c.t.</seg> proue-<lb/>niens ſemper ſuturum.</s> </p> <p> <s xml:space="preserve">Quo autem lucidius res hæc innoteſcat. </s> <s xml:space="preserve">Cogi <lb/>temus proueniens quadrati <seg type="var">.o.p.</seg> diuiſi ab <seg type="var">.o.i.</seg> re-<lb/><choice><ex>ſpondentisque</ex><am>ſpondentisq;</am></choice> <seg type="var">.o.t.</seg> eſſe <seg type="var">.i.u.</seg> quod via prædicta inue-<lb/>nitur æqualis eſſe numero <seg type="var">.n.e.</seg> ex quo conſe-<lb/>quenter æquale <seg type="var">.c.t</seg>: cogitato deinde rectangu-<lb/>lo <seg type="var">.o.u.</seg> æquali <seg type="var">.o.p.</seg> coniuncto <seg type="var">.o.c</seg>:totum <seg type="var">.t.u.</seg> æqua-<lb/>le erit compoſito duorum quadratorum <seg type="var">.o.a.</seg> et <seg type="var">.o.<lb/>p.</seg> cum in nullo numerus <seg type="var">.c.t.</seg> mutetur, tam ex com-<lb/>poſito <seg type="var">.t.u.</seg> <choice><ex>quam</ex><am>quã</am></choice> ex ſimplici <seg type="var">.o.c.</seg> ex quo propoſiti ſe <lb/>ſe ueritas profert.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="27">XXVII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">PRoposvervnt</hi> veteres nobile quidem problema, ſed quod tamen citra al-<lb/>gebraticam effectionem, aut neſcierunt, aut noluerunt diſſoluere, quod nihi-<lb/>lominus facillimum eſt.</s> </p> <pb facs="0030" n="18"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Proponunt hi numerum in binas eiuſmodi partes diuidendum, vt ſumma qua-<lb/>dratorum dictarum partium, alteri numero poſsibili propoſito æqualis ſit, poſſi-<lb/>bili inquam, etenim ſi eiuſmodi numerus propoſitus, minor eſſet producto totius <lb/>primi in ſuum dimidium, eſſet huiuſmodi factum impoſſibile. </s> <s xml:space="preserve">Quod nos exequi <lb/>cupientes, ſumamus primum <choice><ex>numerum</ex><am>numerũ</am></choice> propoſitum, quem in ſe ipſum multiplice-<lb/>mus. </s> <s xml:space="preserve">ab hoc quadrato deducamus ſecundum numerum propoſitum, tum quod re-<lb/>manſerit duplicemus, quod duplum denuo iubeo ex eodem primo quadrato detra-<lb/>hi, accepta poſtea radice quadrata reſidui & dempta ex priori numero propoſito, <lb/></s> <s xml:space="preserve">tunc dimidium reſidui vna pars erit ex duabus primi numeri quæſita.</s> </p> <p> <s xml:space="preserve">Exempli gratia proponantur .20. diuidenda in duas eiuſmodi partes, vt ſumma <lb/>quadratorum ipſarum partium æqualis ſit .272. qui numerus maior eſt .200. maior <lb/>inquam dimidio quadrati .400. ipſorum .20. hic autem numerus .272. è quadra-<lb/>to .400. deducatur, <choice><ex>remanebunt</ex><am>remanebũt</am></choice> enim .128. quod duplicari iubeo, <choice><ex>producentur</ex><am>producẽtur</am></choice> <choice><ex>ſiquidem</ex><am>ſiquidẽ</am></choice> <num value="256">.<lb/>256.</num> quæ pariter deducta è quadrato totali, remanebunt .144. cuius radicem ſumi <lb/>volo, quæ erit .12. & dempta ex .20. priori numero dato remanebit .8. cuius di-<lb/>midium erit .4: pars vna ex quæſitis, quæ ex primo numero propoſito .20. detra-<lb/>hetur, <choice><ex>remanebitque</ex><am>remanebitq́</am></choice> .16. pro altera parte.</s> </p> <p> <s xml:space="preserve">Cuius demonſtrationis cauſa, in primis cogitemus quadratum <seg type="var">.a.c.</seg> cognitum nu-<lb/>meri <seg type="var">.a.b.</seg> primò propoſiti, qui cogitetur diuiſus in duo quadrata <seg type="var">.d.e.</seg> et <seg type="var">.e.b.</seg> <choice><ex>duo- que</ex><am>duo-q́</am></choice> ſupplementa <seg type="var">.a.e.</seg> et <seg type="var">.e.c.</seg> numerus autem ſummæ duorum quadratorum <seg type="var">.d.e.<lb/>b.</seg> pro ſecundo propoſito datur; </s> <s xml:space="preserve">ex quo, ſumma duorum ſupplementorum <seg type="var">.a.e.c.</seg> <lb/>conſequenter erit cognita, quę cum duplicata fuerit, & quatuor hæc ſupplementa<unclear reason="illegible"/> <lb/>cogitatione accommodata, prout in <lb/>quadrato <seg type="var">.f.g.</seg> apparet (<choice><ex>quanuis</ex><am>quãuis</am></choice> idipſum <lb/> <ptr xml:id="fig-0030-01a" corresp="fig-0030-01" type="figureAnchor"/> proueniret ſi modo Eucl. octaua <choice><ex>ſecundi</ex><am>ſecũdi</am></choice> <lb/>aptaretur) æquali quadrato <seg type="var">.a.c.</seg> ita vt <lb/>cogitatis quatuor ſupplementis numeri <lb/>cogniti in quadrato <seg type="var">.f.g.</seg> ex conſequen-<lb/>ti cognoſcetur numerus quadrati partia <lb/>lis <seg type="var">.h.i.</seg> & vna etiam eius radix qua de-<lb/>tracta ex numero <seg type="var">.a.b.</seg> aut <seg type="var">.f.n.</seg> (quod <lb/>idem eſt) primo propoſiti, relinquetur numerus cognitus duplum <seg type="var">.x.k.n.</seg> aut <seg type="var">.t.b.</seg> <lb/>pars vna totius <seg type="var">.a.b.</seg> ex quo uerum erit hoc meum problema.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0030-01" corresp="fig-0030-01a"> <graphic url="0030-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="28">XXVIII</num>.</head> <p> <s xml:space="preserve">SI quis & aliam rationem perficiendæ <lb/> <ptr xml:id="fig-0030-02a" corresp="fig-0030-02" type="figureAnchor"/> huius rei quærat, hoc præſtet inuen-<lb/>to numero huius ſupplementi, cum in <lb/>præcedenti theoremate dictum fuerit, <lb/>qua ratione manifeſtetur duplum ſupple-<lb/>menti ipſius.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0030-02" corresp="fig-0030-02a"> <graphic url="0030-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Cogitemus in ſubſcripta figura lineam <seg type="var">.<lb/>a.b.</seg> tanquam primum numerum propoſi-<lb/>tum, & productum <seg type="var">.a.e.</seg> ſupplemento <seg type="var">.a.e.</seg> primæ præcedentis figuræ æquale ſit, <lb/>ac deinde ordine ab antiquis tradito procedatur, ad quadratum reducto dimidio <seg type="var">.<lb/>a.b.</seg> videlicet <seg type="var">.b.c.</seg> quod erit <seg type="var">.b.d.</seg> ex quo detrahatur deinde <seg type="var">.a.e</seg>. </s> <s xml:space="preserve">quare remane- <pb facs="0031" n="19"/><fw type="head">THEOREM. ARIT.</fw> bit quadratum <seg type="var">.e.d.</seg> cognitum, cuius radix æqualis erit <seg type="var">.c.t.</seg> qua coniuncta dimi-<lb/>dio <seg type="var">.c.a.</seg> ex quinta ſecundi Eucli. dabit quod propoſitum erat.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="29">XXIX</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVid</hi> cauſæ eſt, cur ſubtracto duplo producti duorum numerorum ad inui-<lb/>cem <choice><ex>multiplicatorum</ex><am>multiplicatorũ</am></choice> ex ſumma ſuorum quadratorum, ſemper quod ſuper <lb/>eſt duorum numerorum quadratum differentiæ ſit?</s> </p> <p> <s xml:space="preserve">Exempli gratia ſi proponerentur duo numeri .16. et .4. duplum producti eorum <lb/>eſſet .128. quò detracto ex ſumma ſuorum quadratorum, nempè ex .272. rema-<lb/>neret .144. cuius quadrati radix eſſet .12. tanquam differentia inter .4. et .16.</s> </p> <p> <s xml:space="preserve">Id vtſciamus, duo numeri propoſiti, duabus lineis ſignificentur, maiore <seg type="var">.q.g.</seg> <lb/>et minore <seg type="var">.g.p.</seg> directè coniunctis, ſuper quas, totale quadratum extruatur <seg type="var">.a.p.</seg> <lb/>in quo cogitetur diameter <seg type="var">.a.p.</seg> et à puncto <seg type="var">.g.</seg> ducatur parallela <seg type="var">.g.n.c.</seg> et à pun-<lb/>cto <seg type="var">.n.</seg> parallela <seg type="var">.n.s.r.</seg> ex quo duo producta <choice><ex>dabuntur</ex><am>dabũtur</am></choice> <seg type="var">.q.n.</seg> et <seg type="var">.n.u.</seg> ſingula æqualia pro-<lb/>ducto <seg type="var">.q.g.</seg> in <seg type="var">g.p.</seg> et <seg type="var">.a.n.</seg> et <seg type="var">.n.p.</seg> duo quadrata dictorum numerorum propoſi-<lb/>torum, quod ſatis <choice><ex>ſuperque</ex><am>ſuperq́</am></choice> , probatur quarta ſecundi Eucli. </s> <s xml:space="preserve">Cogitemus deinde <seg type="var">.n.<lb/>o.</seg> æqualem <seg type="var">.n.p.</seg> et à puncto <seg type="var">.o.</seg> ducatur <seg type="var">.o.m.t.</seg> parallela <seg type="var">.r.s.</seg> et <seg type="var">.o.e.</seg> ad <seg type="var">.n.<lb/>c</seg>. </s> <s xml:space="preserve">quare ex allatis ab Eucli. octaua ſecundi, dabi-<lb/>tur quantitas <seg type="var">.m.n.</seg> æqualis <seg type="var">.q.n.</seg> producto <seg type="var">.q.g.</seg> in <lb/> <ptr xml:id="fig-0031-01a" corresp="fig-0031-01" type="figureAnchor"/> <seg type="var">g.p.</seg> et quantitas <seg type="var">.o.c.</seg> minor ipſo producto, ex <lb/>quantitate quadrati <seg type="var">.n.p.</seg> ex quo quantitas <seg type="var">.m.n.e.</seg> <lb/>vna cum quadrato <seg type="var">.n.p.</seg> æqualis erit duplo produ-<lb/>cti <seg type="var">.q.g.</seg> in <seg type="var">.g.p.</seg> ſed hæ duæ quantitates, ſunt par-<lb/>tes duorum quadratorum dictorum, & quæ ſuper <lb/>eſt <seg type="var">.m.e.</seg> quadratum differentiæ vnius numeri pro-<lb/>poſiti ab altero, prout in ſubſcripta figura licebit cui <lb/>libet conſiderare. </s> <s xml:space="preserve">Itaque veritas hæc manifeſta <lb/>erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0031-01" corresp="fig-0031-01a"> <graphic url="0031-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="30">XXX</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">CVr</hi> ij qui ex duobus numeris propoſitis maiorem per minorem diuidunt, ſi <lb/>proueniens per maiorem numerum multiplicauerint, productum æquale <lb/>erit prouenienti ex diuiſione quadrati maioris numeri per minorem?</s> </p> <p> <s xml:space="preserve">Exempli gratia ſi proponantur duo numeri .20. et .4. <choice><ex>ipſeque</ex><am>ipſeq́</am></choice> .20. per .4. diui-<lb/>datur, dabit quinque, tum .400. quadrato .20. diuiſo per prioré .4. dabit .100. <lb/>quod proueniens, producto ex .20. in .5. primo prouenienti adæquatur.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, ſint duo numeri, qui lineis <seg type="var">.x.u.</seg> et <seg type="var">.x.s.</seg> maiore <choice><ex>atque</ex><am>atq;</am></choice> mi-<lb/>nore ſignificétur, tum <seg type="var">.u.x.</seg> numerus per <seg type="var">.s.x.</seg> di-<lb/>uidatur, ſitq́ue proueniens <seg type="var">.x.n.</seg> poſtmodum qua-<lb/> <ptr xml:id="fig-0031-02a" corresp="fig-0031-02" type="figureAnchor"/> dratum <seg type="var">.u.x.</seg> ſit <seg type="var">.x.o.</seg> et productum ex <seg type="var">.n.x.</seg> in <seg type="var">.u.<lb/>x.</seg> ſit <seg type="var">.x.e.</seg> quod æquale eſſe dico prouenienti ex <lb/>diuiſione quadrati <seg type="var">.o.x.</seg> per <seg type="var">.s.x.</seg> quod ſit <seg type="var">.m</seg>. </s> <s xml:space="preserve">Patet <lb/>enim ex definitione diuiſionis, talem futuram pro-<lb/>portionem <seg type="var">.u.x.</seg> ad <seg type="var">.n.x.</seg> qualis eſt <seg type="var">.s.x.</seg> ad vnitatem, <lb/>& quadratum <seg type="var">.o.x.</seg> ad rectangulum <seg type="var">.e.x.</seg> ita ſe ha- <pb facs="0032" n="20"/><fw type="head">I O. BAPT. BENED.</fw> biturum, ſicut <seg type="var">.u.x.</seg> ad <seg type="var">.n.x.</seg> ex prima ſexti aut .18. vel .19. ſeptimi, </s> <s xml:space="preserve">quare ex 11. <lb/>quinti ita ſe habebit <seg type="var">.o.x.</seg> ad <seg type="var">.e.x.</seg> ſicut <seg type="var">.s.x.</seg> ad vnitatem; </s> <s xml:space="preserve">ſed ſicut ſe habet <seg type="var">.s.x.</seg> ad. <lb/>vnitatem, ita ſe habet pariter <seg type="var">.o.x.</seg> ad <seg type="var">.m</seg>. </s> <s xml:space="preserve">vnde ex .11. prædicta ita ſe habebit <seg type="var">.o.<lb/>x.</seg> ad <seg type="var">.m.</seg> ſicut idipſum <seg type="var">.o.x.</seg> ad <seg type="var">.e.x.</seg> itaq́ue ex .9. prædicti quinti <seg type="var">.m.</seg> æqualis erit <seg type="var">.o.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0031-02" corresp="fig-0031-02a"> <graphic url="0031-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="31">XXXI</num>.</head> <p> <s xml:space="preserve">CVR propoſito aliquo numero in duas partes inæquales diuiſo, ſi rurſus per <lb/>quamlibet ipſarum diuidatur, prouenientia tantumdem coniuncta quantum <lb/>multiplicata efficiant.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſit denarius prop oſitus numerus, per binarium & octonarium <lb/>diuiſus, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta <lb/>crunt .6. cum quarta parte lineari, quæ ſi mul multiplicata, pariter erunt .6. cum <lb/>quarta parte ſuperficiali.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, totalis numerns, linea <seg type="var">.q.p.</seg> ſignificetur, eius duæ <lb/>partes, per <seg type="var">.k.</seg> maiorem et <seg type="var">.u.</seg> minorem, ipſa vnitas per .t: proueniens ex diuiſio-<lb/>ne <seg type="var">.q.p.</seg> per <seg type="var">.k.</seg> ſit <seg type="var">.q.i.</seg> proueniens autem ipſius <seg type="var">.q.p.</seg> per <seg type="var">.u.</seg> ſit <seg type="var">.q.f.</seg> </s> <s xml:space="preserve">quare ex defini-<lb/>tione diuiſionis ita ſe habebit <seg type="var">.q.p.</seg> ad <seg type="var">.q.i.</seg> ſicut <seg type="var">.k.</seg> ad <seg type="var">.t.</seg> et <seg type="var">.q.p.</seg> ad <seg type="var">.q.f.</seg> ſicut <seg type="var">.u.</seg> ad <seg type="var">.t.</seg> <lb/>hoc eſt <seg type="var">.q.f.</seg> ad <seg type="var">.q.p.</seg> ſicut <seg type="var">.t.</seg> ad <seg type="var">.u.</seg> vnde ex æqualitate <choice><ex>proportionum</ex><am>proportionũ</am></choice> ſic ſe habebit <seg type="var">.q.f.</seg> <lb/>ad <seg type="var">.q.i.</seg> ſicut <seg type="var">.k.</seg> ad <seg type="var">.u.</seg> et conuerſim. </s> <s xml:space="preserve">Ad hæc in linea <seg type="var">.q.p.</seg> vnitas, per lineam <seg type="var">.q.o.</seg> ſigni-<lb/>ficetur, quo facto, dicamus, ſi <seg type="var">.q.p.</seg> ad <seg type="var">.q.i.</seg> ſic ſe habet vt <seg type="var">.k.</seg> ad <seg type="var">.q.o.</seg> itaque permu-<lb/>tando, ſic ſe habebit <seg type="var">.q.p.</seg> ad <seg type="var">.k.</seg> ſicut <seg type="var">.q.i.</seg> ad <seg type="var">.q.o.</seg> hoc eſt <seg type="var">.k.u.</seg> ad <seg type="var">.k.</seg> ſicut <seg type="var">.i.q.f.</seg> ad <seg type="var">.<lb/>q.f.</seg> (nam <seg type="var">.k.u.</seg> partes ſunt integrales totius <seg type="var">.q.p.</seg> et <seg type="var">.k.u.</seg> ad <seg type="var">.k.</seg> eſt ſicut <seg type="var">.i.q.f.</seg> ad <seg type="var">.q.f.</seg> <lb/>ex .18. quinti) </s> <s xml:space="preserve">Quare ita erit <seg type="var">.i.q.f.</seg> ad <seg type="var">.q.f.</seg> ſicut <seg type="var">.q.i.</seg> ad vnitatem <seg type="var">.q.o.</seg> ex .11. quinti <lb/>Addatur deinde <seg type="var">.q.i.</seg> ad <seg type="var">.q.f.</seg> et <seg type="var">.q.i.</seg> per <seg type="var">.<lb/>q.f.</seg> multiplicetur, cuius multiplicatio-<lb/> <ptr xml:id="fig-0032-01a" corresp="fig-0032-01" type="figureAnchor"/> nis productum, ſit <seg type="var">.x.f.</seg> quod probabo <lb/>æquale eſſe ſummæ <seg type="var">.f.q.</seg> cum <seg type="var">.q.i</seg>. </s> <s xml:space="preserve">Sece-<lb/>tur enim linea <seg type="var">.q.x.</seg> in puncto <seg type="var">.s.</seg> ita. vt <seg type="var">.<lb/>q.s.</seg> æqualis ſit <seg type="var">.q.o.</seg> ſigneturq́ue pro-<lb/>ductum <seg type="var">.s.f.</seg> </s> <s xml:space="preserve">quare <choice><ex>eadem</ex><am>eadẽ</am></choice> erit propor-<lb/>tio quantitatis <seg type="var">.x.f.</seg> ad <seg type="var">.s.f.</seg> quæ eſt <seg type="var">.q.x.</seg> <lb/>ad <seg type="var">.q.s.</seg> ex prima ſexti, aut .18. vel 19. <lb/>ſeptimi, hoc eſt, ſicut <seg type="var">.q.i.</seg> ad <seg type="var">.q.o.</seg> et <lb/>ex .11. quinti (vt dictum eſt) ſicut <seg type="var">.i.q.<lb/>f.</seg> ad <seg type="var">.q.f.</seg> ſed numerus <seg type="var">.s.f.</seg> fuperficia-<lb/>lis tantus eſt, quantus linearis <seg type="var">.q.f</seg>. <lb/></s> <s xml:space="preserve">quare ex .9. quinti tantus erit (ſu-<lb/>perficialiter) numerus <seg type="var">.x.f.</seg> quantus <lb/>(lineariter). <seg type="var">f.q.i.</seg> quod erat pro-<lb/>poſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0032-01" corresp="fig-0032-01a"> <graphic url="0032-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA. <num value="32">XXXII</num>.</head> <p> <s xml:space="preserve">CVR numero aliquo in duas partes inæquales diuiſo, ſi rurſus diuidatur per <lb/>ſingulas partes, ſumma duorum prouenientium per binarium, ſemper ma-<lb/>ior ſit ſumma prouenientium ex diuiſione vnius partis per alteram.</s> </p> <p> <s xml:space="preserve"><choice><ex>Exempli</ex><am>Exẽpli</am></choice> gratia, ſi proponeretur numerus .24. qui in duas partes inæquales diuide <pb facs="0033" n="21"/><fw type="head">THEOREM. ARIT.</fw> retur .20. ſcilicet et .4. certè .24. perſingulas partes diuiſo, daretur vnum proue-<lb/>niens ſex integra, & alterum vnum & quinta pars, quorum ſumma eſſet ſeptem in-<lb/>tegra cum quinta parte, tum altera parte per alteram diuiſa, daretur vnum proue-<lb/>niens quinque integrorum & alterum vnius quinti tantum, quorum ſumma eſſet <lb/>quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-<lb/>tium per binarium.</s> </p> <p> <s xml:space="preserve">Cuius conſiderationis cauſa, propoſitus numerus linea <seg type="var">.q.p.</seg> ſignificetur, eius duę <lb/>partes lineis <seg type="var">.q.x.</seg> et <seg type="var">.x.p.</seg> <choice><ex>tum</ex><am>tũ</am></choice> <seg type="var">.q.f.</seg> ſit proueniens ex diuiſione totius <seg type="var">.q.p.</seg> per <seg type="var">.x.p.</seg> et <seg type="var">.<lb/>q.i.</seg> ſit proueniens ex diuiſione eiuſdem <seg type="var">.q.p.</seg> per <seg type="var">.q.x.</seg> adhæc <seg type="var">.h.m.</seg> ſit proueniens, <lb/>ex diuiſione <seg type="var">.q.x.</seg> per <seg type="var">x.p.</seg> et <seg type="var">.h.k.</seg> proue-<lb/>niensex diuiſione <seg type="var">.p.x.</seg> per <seg type="var">.q.x.</seg> patet igi-<lb/> <ptr xml:id="fig-0033-01a" corresp="fig-0033-01" type="figureAnchor"/> tur ex .22. theoremate huiuslibri proue-<lb/>niés.h.m. minus eſſe proueniente <seg type="var">.q.f.</seg> per <lb/>vnitaté, & proueniens <seg type="var">.h.k.</seg> minus proue-<lb/>niente <seg type="var">.q.i.</seg> per alteram vnitatem. </s> <s xml:space="preserve">Itaque <seg type="var">.<lb/>f.q.i.</seg> maior erit <seg type="var">.m.h.k.</seg> per numerum binarium, quoderat propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0033-01" corresp="fig-0033-01a"> <graphic url="0033-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA. <num value="33">XXXIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVilibet</hi> numerus, medius eſt <lb/>proportionalis inter numerum <lb/> <ptr xml:id="fig-0033-02a" corresp="fig-0033-02" type="figureAnchor"/> ſui quadrati & vnitatem.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0033-02" corresp="fig-0033-02a"> <graphic url="0033-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Detur enim numerus propoſitus, <lb/>qui linea <seg type="var">.a.u.</seg> ſignificetur, cuiusqua-<lb/>dratum ſit <seg type="var">.u.n.</seg> vnitas linearis ſit <seg type="var">.i.a.</seg> <lb/>et ſuperficialis <seg type="var">.o.</seg> patebit ex .18. ſexti <lb/>aut 11. octaui proportionem <seg type="var">.u.n.</seg> ad <seg type="var">.<lb/>o.</seg> futuram duplam proportioni <seg type="var">.u.a.</seg> <lb/>ad <seg type="var">.i.a.</seg> ſed <seg type="var">.i.a.</seg> e<unclear reason="illegible"/>t.o. eadem (ſpecie) <lb/>res <choice><ex>sunt</ex><am>sũt</am></choice>, tanta ſcilicet <seg type="var">.a.i.</seg> quanta <seg type="var">.o.</seg> vni <lb/> <ptr xml:id="fig-0033-03a" corresp="fig-0033-03" type="figureAnchor"/> tas eſt, Itaque proportio numeri <seg type="var">.u.n.</seg> <lb/>ad <seg type="var">.u.a.</seg> æqualis erit proportioni <seg type="var">.u.a.</seg> <lb/>ad <seg type="var">.i.a</seg>. </s> <s xml:space="preserve">Quare numerus <seg type="var">.u.a.</seg> inter nu-<lb/>merum <seg type="var">.u.n.</seg> & vnitatem, medius erit <lb/>proportionalis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0033-03" corresp="fig-0033-03a"> <graphic url="0033-03"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="34">XXXIIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">HOc</hi> ipſum quod diximus & alia ratione ſpeculari licebit.</s> </p> <p> <s xml:space="preserve">Propoſitus numerus, nunc etiam per <seg type="var">.a.u.</seg> ſignificetur, eius quadratum per <seg type="var">.<lb/>u.n.</seg> vnitas linearis per <seg type="var">.a.i.</seg> <choice><ex>productumque</ex><am>productumq́;</am></choice> <seg type="var">.a.u.</seg> in <seg type="var">.a.i.</seg> terminetur, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.n.i</seg>. </s> <s xml:space="preserve">quare <lb/><seg type="var">n.i.</seg> conſtabit numero íuperficiali æquali numero lineari <seg type="var">.a.u.</seg> & ex prima fexti aut .<lb/>18. vel .19. ſeptimi, eadem erit proportio <seg type="var">.u.n.</seg> ad <seg type="var">.i.n.</seg> quæ eſt <seg type="var">.a.u.</seg> ad <seg type="var">.a.i.</seg> ſed nu-<lb/>merus <seg type="var">.a.u.</seg> cum numero <seg type="var">.n.i.</seg> idem ſpecie eſt. </s> <s xml:space="preserve">Itaque medius eſt proportiona-<lb/>lis inter <seg type="var">.u.n.</seg> & vnitatem.</s> </p> <pb facs="0034" n="22"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="35">XXXV</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVivis</hi> numerus per alterum multiplicatus, & diuiſus, medius eſt propor-<lb/>tionalis inter productum multiplicationis, & proueniens diaiſionis.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi .20. <choice><ex>multiplicentur</ex><am>multiplicẽtur</am></choice> per quinque & inde per quinque diuidantur <lb/>productum erit .100. proueniens .4. inter quos numeros .20. medius eſt propor-<lb/>tionalis.</s> </p> <p> <s xml:space="preserve">Hoc vt ſpeculemur, proponatur numerus multiplicandus & diuidendus, qui ſi-<lb/>gnificetur linea <seg type="var">.u.e.</seg> multiplicans autem & diuidens linea <seg type="var">.a.u.</seg> multiplicationis <lb/>productum ſit <seg type="var">.e.a.</seg> proueniens ex diuiſione ſit <seg type="var">.o.e</seg>. </s> <s xml:space="preserve">Nunc proueniens <seg type="var">.e.o.</seg> per <choice><ex>nu- merum</ex><am>nu-merũ</am></choice> <seg type="var">.a.u.</seg> diuidentem multiplicetur, cuius multiplicationis productum ſit <seg type="var">.e.i.</seg> <lb/>quare, eadem erit proportio numeri <seg type="var">.a.e.</seg> <lb/>ad numerum <seg type="var">.e.i.</seg> quæ eſt numeri <seg type="var">.u.e.</seg> ad <lb/> <ptr xml:id="fig-0034-01a" corresp="fig-0034-01" type="figureAnchor"/> numerum <seg type="var">.e.o.</seg> ex prima ſextiaut .18. vel <lb/>19. ſeptimi. </s> <s xml:space="preserve">Sed cum numerus <seg type="var">.u.e.</seg> ex <ref>.11. theoremate præſentis libri</ref>, numero <seg type="var">.e.<lb/>i.</seg> æqualis ſit. </s> <s xml:space="preserve">verum eſſe, quod propoſi-<lb/>tum fuit conſequetur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0034-01" corresp="fig-0034-01a"> <graphic url="0034-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="36">XXXVI</num>.</head> <p> <s xml:space="preserve">CVR ij, qui propoſitum numerum ita multiplicare & diuidere cupiunt, vt pro <lb/>ductum multiplicationis, tam ſit multiplex prouenienti ex diuiſione, quam <lb/>quæritur, rectè ſumant aliquem numerum pro multiplicante & diuidente, qui ſit ra <lb/>dix quadrata denominantis quęſitę multiplicitatis.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponuntur .20. multiplicanda atque diuidenda, ita vt pro-<lb/>ductum multiplicationis nonuplum ſit prouenienti ex diuiſione, nempè, vt pro-<lb/>ueniens, nona pars ſit eiuſmodi producti, </s> <s xml:space="preserve">quare quadratam radicem ipſorum no-<lb/>uem, ideſt denominantis ſumunt, tria ſcilicet, multiplicant igitur & diuidunt <lb/>data .20. ex quo productum erit .60. proueniens autem .6. cum duabus tertijs. </s> <s xml:space="preserve">& <lb/>propoſitum ſequitur.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, ſignificetur numerus propoſitus linea <seg type="var">.u.e.</seg> multipli-<lb/>cans autem & diuidens linea <seg type="var">.u.a.</seg> productum ſit <seg type="var">.e.a.</seg> proueniens <seg type="var">.e.o.</seg> quadratum <lb/>verò <seg type="var">.a.u.</seg> ſit <seg type="var">.x.a.</seg> erit igitur proportio <seg type="var">.a.e.</seg> ad <seg type="var">.e.o.</seg> dupla proportioni <seg type="var">.a.e.</seg> ad nume <lb/>rum <seg type="var">.u.e.</seg> ex præcedenti theoremate: </s> <s xml:space="preserve">Adhæc, cogitemus in linea <seg type="var">.u.a.</seg> vnitatem <seg type="var">.<lb/>u.i.</seg> <choice><ex>terminenturque</ex><am>terminenturq́;</am></choice> duo producta <seg type="var">.e.i.</seg> et <seg type="var">.x.i.</seg> </s> <s xml:space="preserve">quare eadem erit proportio <seg type="var">.a.e.</seg> ad <seg type="var">.e.i.</seg> <lb/>quæ eſt <seg type="var">.a.e.</seg> ad <seg type="var">.u.e.</seg> numerus enim <seg type="var">.e.i.</seg> (quamuis ſuperficialis) idem eſt cum nume-<lb/>ro lineari <seg type="var">.u.e.</seg> ſed <seg type="var">.a.e.</seg> ad <seg type="var">.e.i.</seg> ſic ſe habet ſicut <seg type="var">.a.u.</seg> ad <seg type="var">.u.i.</seg> ex prima ſexti aut .18. <lb/>vel .19. ſeptimi, (quod ipſum dico de <seg type="var">.a.x.</seg> ad <seg type="var">.x.i.</seg>) </s> <s xml:space="preserve">quare proportio <seg type="var">.a.x.</seg> ad <seg type="var">.x.i.</seg> hoc <lb/>eſt <seg type="var">.x.u.</seg> ęqualis erit <choice><ex>proportioni</ex><am>ꝓportioni</am></choice> <seg type="var">.a.e.</seg> ad <seg type="var">.u.e.</seg> at trigeſimotertio & trigeſimoquarto theo <lb/>remate probatum eſt proportionem numeri <seg type="var">.a.x.</seg> ad vnitatem, duplam eſſe propor-<lb/>tioni eiuſdem numeri <seg type="var">.a.x.</seg> ad <seg type="var">.u.x.</seg> ſequitur <lb/>igitur cum dimidia ſint æqualia, tota etiam <lb/>æqualia eſſe: </s> <s xml:space="preserve">hoc eſt proportionem numeri <seg type="var">.<lb/> <ptr xml:id="fig-0034-02a" corresp="fig-0034-02" type="figureAnchor"/> a.e.</seg> ad numerum <seg type="var">.e.o.</seg> æqualem eſſe propor <lb/>tioni numeri <seg type="var">.a.x.</seg> ad vnitatem. </s> <s xml:space="preserve">Itaque rectè <lb/>ſumitur numerus <seg type="var">.a.u.</seg> eiuſmodi vt <choice><ex>quadratum</ex><am>quadratũ</am></choice> <pb facs="0035" n="23"/><fw type="head">THEOR. ARITH.</fw> ipſius <seg type="var">.a.x.</seg> tam ſit multiplex ad vnitatem, quam cupimus numerum <seg type="var">.a.e.</seg> numero <seg type="var">.<lb/>e.o.</seg> multiplicem eſſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0034-02" corresp="fig-0034-02a"> <graphic url="0034-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="37">XXXVII</num>.</head> <p> <s xml:space="preserve">CVR inuenire cupientes duos numeros, quorum quadrata in ſummam colle-<lb/>cta, æqualia ſint numero propoſito, & ijſdem numeris multiplicatis ad-<lb/>inuicem, productum alteri numero propoſito ſit æquale, rectè ſumant dimidium <lb/>primi numeri propoſiti, cui ſumma quadratorum æquari debet, <choice><ex>hocque</ex><am>hocq́;</am></choice> dimidium <lb/>in ſeipſum multiplicent, vnà etiam alterum numerum propoſitum in ſeipſum <lb/>multiplicent, quod quadratum detrahunt de primo, & reſidui quadratam radicem, <lb/>dimidio primi numeri propoſiti coniungunt, ex qua ſumma, quadratam radicem <lb/><choice><ex>eruunt</ex><am>eruũt</am></choice>, quæ duobus quæſitis numeris maior erit, cuius quadrato de primo numero <lb/>detracto, & exreliquo erutaradice quadrata, detur minor numerus, duorum <choice><ex>quae- ſitorum</ex><am>quę-ſitorum</am></choice>.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponerentur .34. pro primo numero cui æquari de-<lb/>beret ſumma duorum quadratorum, quorum radicum productum æquale eſſe de-<lb/>beret alteri numero, verbi gratia .15. iubet antiquorum regula, dimidium primi <lb/>numeri in ſeipſum multiplicari, cuius dimidij quadratum erit .289. è quo ſi detra-<lb/>has quadratum ſecundi numeri, nempe .225. remanebit .64. <choice><ex>atque</ex><am>atq;</am></choice> huius ſi quadra-<lb/>tam radicem ſumas nempe .8. quam dimidio primi numeri, nempe .17. coniun-<lb/>gas, dabitur duorum quadratorum numerorum quęſitorum maior numerus .25. hac <lb/>deinde radice è dimidio detracta, minus quadratum dabitur .9. ſcilicet, quorum <lb/>radices .5. et .3. eſſent ij numeri, qui quæruntur.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, cogitemus primum numerum, cui quadratorum fum <lb/>ma æquari debet, ſignificari linea <seg type="var">.a.n.</seg> tum concipiamus quæſita quadrata ſignifi-<lb/>cari, <choice><ex>coniungique</ex><am>coniungiq́</am></choice> modo ſubſcripto <seg type="var">.t.b.k.</seg> ſecundum porrò numerum propoſitum, <lb/>ſignificari producto <seg type="var">.d.b</seg>. </s> <s xml:space="preserve">Iam nil ſupereſt aliud quam vt quantitates <seg type="var">.d.p.</seg> et <seg type="var">.b.p.</seg> <lb/>quæramus.</s> </p> <p> <s xml:space="preserve">Itaque cum in linea <seg type="var">.a.n.</seg> ſummæ quadratorum numerus detur, quadratum di-<lb/>midij <seg type="var">.o.a.</seg> ſit <seg type="var">.s.a.</seg> quod nobis erit cognitum; </s> <s xml:space="preserve">ſit etiam <seg type="var">.a.u.</seg> numerus quadrati ma <lb/>ioris, et <seg type="var">.u.n.</seg> minoris, et <seg type="var">.a.z.</seg> productum vnius in alterum; </s> <s xml:space="preserve">qui quidem numerus <seg type="var">.a.<lb/>z.</seg> æqualis erit <lb/>quadrato nume <lb/> <ptr xml:id="fig-0035-01a" corresp="fig-0035-01" type="figureAnchor"/> ri <seg type="var">.d.b.</seg> ex .19. <lb/>theoremate hu-<lb/>ius libri. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <lb/><seg type="var">a.z.</seg> cognitum <lb/>erit, cum eius <lb/>radix <seg type="var">.d.b.</seg> ſit <choice><ex>ſe- cundus</ex><am>ſe-cũdus</am></choice> numerus <lb/>propoſitus, quæ <lb/>minor erit <seg type="var">.a.s.</seg> ex quinta ſecundi, aut ſeptima conſequentia poſt .16. noni Eucli-<lb/>dis. </s> <s xml:space="preserve">Iam ſubtracta quantitate <seg type="var">.z.a.</seg> è quadrato <seg type="var">.a.s.</seg> cognoſcetur quadratum <seg type="var">.t.x.</seg> <lb/>cuius radix æqualis erit <seg type="var">.o.u.</seg> ex poſtremo adductis, Itaque cognoſcemus <seg type="var">.o.u.</seg> qui <lb/>numerus coniunctus dimidio <seg type="var">.o.a.</seg> cognito, dabit quadratum <seg type="var">.a.u.</seg> cognitum, at-<lb/>queita <seg type="var">.u.n.</seg> pariter cognoſcetur, & eorum radices conſequenter.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0035-01" corresp="fig-0035-01a"> <graphic url="0035-01"/> </figure> </div> </body> </floatingText> <pb facs="0036" n="24"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Hoc ipſum & alia ratione perfici poteſt, nempe, iuncta ſumma <seg type="var">.k.b</seg>: <seg type="var">b.d</seg>: ec<unclear reason="illegible"/> <seg type="var">.<lb/>b.t.</seg> alteri rectangulo æquali <seg type="var">.b.d.</seg> quod ſit <seg type="var">.b.c.</seg> ex quo totum quadratum lineæ <seg type="var">.d.k.</seg> <lb/>cognitum erit, <choice><ex>atque</ex><am>atq;</am></choice> ita etiam conſequenter eius radicem <seg type="var">.d.k.</seg> cognoſcemus, cuius <lb/>ope ac producti <seg type="var">.d.b.</seg> cognoſcemus <seg type="var">.d.p.</seg> et <seg type="var">.p.k.</seg> prout ex theoremate quadrageſi-<lb/>moquinto huius libri patebit.</s> </p> <p> <s xml:space="preserve">Michael Stifelius, vndecimo cap. tertij libri, problema eiuſmodi proponit, <lb/>quod tamen ipſe via algebræ diſsoluit.</s> </p> <figure place="here"> <graphic url="0036-01"/> </figure> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="38">XXXVIII</num>.</head> <p> <s xml:space="preserve">CVR ij, qui duos numeros inuenire volunt, quorum productum alicui nu-<lb/>mero propoſito æquetur, & quadratorum eorundem differentia alteri nu-<lb/>mero propoſito æqualis ſir. </s> <s xml:space="preserve">Rectè dimidium ſecundi numeri propoſiti in ſeipſum <lb/>multiplicent, cui quidem numero differentia quadratorum æquari debet; </s> <s xml:space="preserve">porrò <lb/>huic quadrato primi propoſiti numeri, cui æquandum eſt productum numerorum <lb/>quæſitorum, quadratum adiungant; </s> <s xml:space="preserve">tum radicem quadratam huius ſummæ co-<lb/>pulet dimidio ſecundi numeri propoſiti, ei inquam, cui differentia quadratorum <lb/>æqualis eſſe debet, ex quo quadratum maius conſurgit, à quo, detracto ſecundo <lb/>numero, ſupereſt quadratum minus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponeretur primo loco numerus .8. cui æquandum eſt <lb/>productum numerorum quæſitorum, tum proponeretur numerus .12. cui, detra-<lb/>cto minore à maiore, differentia quadratorum vtriuſque quæſiti numeri æqualis <lb/>eſſe debet, oportet huius vltimi numeri .12. dimidium in ſeipſum multiplicare, <choice><ex>fient- q́ue</ex><am>fiẽt-q́ue</am></choice> .36. quadratum dimidij, vnde in ſummam colligeremus quadratum primi <lb/>numeri .8. quod eſſet .64. quæ cum .36. efficerent .100. cuius centenarij radice, nem <lb/>pe .10. collecta in ſummam cum dimidio ſecundi numeri, nempe .6. daretur qua-<lb/>dratum maius, nempe .16. ex quo, detracto ſecundo numero, nempe .12. rema-<lb/>neret quadratum minus .4.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, maius quadratum <lb/> <ptr xml:id="fig-0036-02a" corresp="fig-0036-02" type="figureAnchor"/> incognitum ſignificetur linea <seg type="var">.q.g.</seg> minus verò <lb/>pariter incognitum linea <seg type="var">.g.i.</seg> </s> <s xml:space="preserve">quare <seg type="var">.q.i.</seg> eorum <lb/>differentia, tanquam data remanebit cognita, <lb/>vnà etiam <seg type="var">.b.i.</seg> et <seg type="var">.q.b.</seg> ſua dimidia; </s> <s xml:space="preserve">tunc cogite-<lb/>tur quadratum <seg type="var">.y.g.</seg> ſuper <seg type="var">.b.g.</seg> et <choice><ex>parallelogram- mum</ex><am>parallelogrã-mum</am></choice> rectangu<unclear reason="illegible"/>lum <seg type="var">.g.r.</seg> deſignatum, & ita etiam <lb/>gnomon <seg type="var">.u.g.t.</seg> prout ſexta ſecundi Euclidis pro <lb/>ponitur, ex quo quadratum <seg type="var">.b.i.</seg> nempe <seg type="var">.u.t.</seg> co-<lb/>gnitum erit, ſed gnomon æqualis eſt rectangulo <seg type="var">.g.r.</seg> ex prædicta, aut ex .8. poſt .16. <pb facs="0037" n="25"/><fw type="head">THEOREM. ARIT.</fw> noni, <choice><ex>hocque</ex><am>hocq́;</am></choice> rectangulum <seg type="var">.g.r.</seg> quadratum eſt primi numeri propoſiti ex .19. theo-<lb/>remate huius libri, <choice><ex>itaque</ex><am>itaq;</am></choice> cognitum erit. </s> <s xml:space="preserve">vnà etiam gnomon <seg type="var">.u.g.t.</seg> cognoſcetur, <lb/>quare totum quadratum <seg type="var">.g.y.</seg> <choice><ex>eiusque</ex><am>eiusq́;</am></choice> radix <seg type="var">.b.g.</seg> manifęſta erit, cui coniuncta <seg type="var">.q.b.</seg> <lb/>data, maius quadratum <seg type="var">.q.g.</seg> cognoſcetur, ex qua <seg type="var">.b.g.</seg> detracta <seg type="var">.b.i.</seg> data, cogno-<lb/>ſcetur <seg type="var">.i.g.</seg> quadratum minus conſequenter, etiam eorum radices notæ erunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0036-02" corresp="fig-0036-02a"> <graphic url="0036-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="39">XXXIX</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">ALia</hi> etiam ratione idipſum definiri poteſt, prætermiſſa antiquorum via, <lb/>nempe multiplicatis in ſemetipſis primo & ſecundo, numeris propoſitis, qua-<lb/><choice><ex>druplicatoque</ex><am>druplicatoq́;</am></choice> quadrato primi, qua ſumma coniuncta cum quadrato ſecundi nume-<lb/>ri, & ex hac altera ſumma eruta radice quadrata, ex qua detracto ſecundo nume-<lb/>ro, & è reliquo ſumpto dimidio, quod erit <choice><ex>quadratum</ex><am>quadratũ</am></choice> minus, quo detracto ex radi-<lb/>ce poſtremo iuncta, ſupererit quadrarum maius.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponeretur numerus .8. cui productum duorum numerorum <lb/>quæſitorum æquandum eſt, proponeretur idem .12. cui differentia quadratorum <lb/>duorum numerorum æqualis eſſe debet. </s> <s xml:space="preserve">Iubeo primum numerum, nempe .8. in ſe <lb/>ipſum multiplicari, ex quo exurget .64. pro numero ſui quadrati, quod quadru-<lb/>plicari volo, <choice><ex>eritque</ex><am>eritq́;</am></choice> productum .256. quod cenſeo <choice><ex>coniungendum</ex><am>coniũgendum</am></choice> cum quadrato ſe-<lb/>cundi numeri propoſiti, nempe .144. <choice><ex>eritque</ex><am>eritq́;</am></choice> ſumma .400. ex quaſumetur radix, ſci <lb/>licet .20. & ex hac detrahetur ſecundus numerus .12. <choice><ex>reſiduique</ex><am>reſiduiq́;</am></choice> dimidium, nempe <num value="4">.<lb/>4.</num> pro quadrato minore, quo in ſummam collecto cum, 12. dabit quadratum <lb/>maius .16.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, quadratum maius per lineam <seg type="var">.q.g.</seg> minus per <seg type="var">.g.p.</seg> ſi-<lb/>gnificetur: </s> <s xml:space="preserve">ſuper integram autem <seg type="var">.q.p.</seg> erigatur quadratum integrum <seg type="var">.d.p.</seg> diuiſum, <lb/>vt quadratum <seg type="var">.f.g.</seg> vigeſimiſeptimi theorematis huius libri, (idipſum accideret di-<lb/>uiſo quadrato modo octauæ ſecundi Euclidis) quæ quidem diuiſio, eſt via quatuor <lb/>productorum <seg type="var">.q.g.</seg> in <seg type="var">.g.p.</seg> è quibus vnum ſit <seg type="var">.g.r.</seg> quod erit cognitum ex .19. theore <lb/>mate cum ſit <choice><ex>quadratum</ex><am>quadratũ</am></choice> primi numeri ppoſiti, ex quo illa quatuor cognita <choice><ex>erunt</ex><am>erũt</am></choice>. </s> <s xml:space="preserve">Iam <lb/>verò ſi cogitemus <seg type="var">.q.p.</seg> ſectam in puncto <seg type="var">.t.</seg> ita vt <seg type="var">.q.t.</seg> æqualis ſit <seg type="var">.p.g.</seg> dabitur differen <lb/>tia <seg type="var">.t.g.</seg> cognita, vt radix quadrati <seg type="var">.e.o.</seg> cum ex præſup-<lb/>poſito <seg type="var">.r.n.</seg> æqualis ſit <seg type="var">.q.g.</seg> et <seg type="var">.r.e</seg>: <seg type="var">g.p.</seg> ex quo etiam <seg type="var">.q.t.</seg> <lb/> <ptr xml:id="fig-0037-01a" corresp="fig-0037-01" type="figureAnchor"/> ita pariter <seg type="var">.e.n.t.g.</seg> æqualis erit. </s> <s xml:space="preserve">Collecto <choice><ex>itaque</ex><am>itaq;</am></choice> quadra <lb/>to <seg type="var">.e.o.</seg> ipſius <seg type="var">.t.g.</seg> cum quadruplo <seg type="var">.g.r</seg>: cognitum erit <lb/>quadratum <seg type="var">.d.p.</seg> ipſius <seg type="var">.q.p.</seg> </s> <s xml:space="preserve">quare cognoſcetur <seg type="var">.q.p.</seg> de <lb/>quo numero detracta differétia quadratorum cognita <seg type="var">.<lb/>t.g.</seg> ſupererit aggregatum <seg type="var">.p.g.</seg> et <seg type="var">.q.t.</seg> cognitum. </s> <s xml:space="preserve">Qua-<lb/>re ex conſequenti, dimidium aggregati, nempe <seg type="var">.g.p.</seg> <lb/>cognoſcetur, tanquam minus duorum quadratorum. <lb/></s> <s xml:space="preserve">cui iuncta <seg type="var">.g.t.</seg> aut detracta <seg type="var">.p.g.</seg> ex <seg type="var">.p.q.</seg> quadratum <seg type="var">.q.<lb/>g.</seg> maius cognitum remanebit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0037-01" corresp="fig-0037-01a"> <graphic url="0037-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="40">XL</num>.</head> <p> <s xml:space="preserve">CVR ijs, qui volunt duos eiuſmodi numeros inuenire, vt eorum maior mi-<lb/>norem, numero propoſito ſuperet, & productum vnius in alterum, alteri nu-<lb/>mero propoſito adęquetur, conſultiſsimum ſit dimidium primi numeri propoſiti, <pb facs="0038" n="26"/><fw type="head">IO. BAPT. BENED.</fw> numerum inquam, cui differentia duorum quæſitorum æquanda eſt, in ſeipſum <lb/>multiplicare, atque huic quadrato, ſecundum numerum propoſitum iungere, cui, <lb/>productum numerorum quæſitorum æquale eſſe debet, & ex hac ſumma eruere qua <lb/>dratam radicem, quæ coniuncta dimidio primi numeri propoſiti, dabit maiorem <lb/>duorum numerorum & ex eadem radice detracto dimidio primi numeri, minorem <lb/>numerum duorum quæſitorum.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponeretur .12. cui differentia vnius numeri ab altero æqua-<lb/>ri deberet, tum proponeretur .64. cui productum multiplicationis duorum quæſi-<lb/>torum ſimul <choice><ex>æquandum</ex><am>æquãdum</am></choice> eſſet. </s> <s xml:space="preserve">Dimidium primi numeri in ſeipſum multiplicaremus, <lb/><choice><ex>proueniretque</ex><am>proueniretq́;</am></choice> <choice><ex>quadratum</ex><am>quadratũ</am></choice> .36. cui coniuncto ſecundo, nempe .64. totum eſſet .100. <lb/>ex quo detracta quadrata radice .10. etipſi coniuncto ſenario, dimidio primi nume <lb/>ri, & ex eadem detracto eodem dimidio .6. pro maiore numero proueniret .16. & <lb/>pro minore .4.</s> </p> <p> <s xml:space="preserve">Cuius rei ſpeculatio hæc eſt. </s> <s xml:space="preserve">Sit <seg type="var">.e.o.</seg> differentia cognita duorum incognitorum <lb/>numerorum <seg type="var">.a.o.</seg> et <seg type="var">.a.e.</seg> quorum productum datum ſiue cognitum ſit <seg type="var">.a.s</seg>: conſide-<lb/>remus nunc <seg type="var">.e.i.</seg> dimidium <seg type="var">.e.o.</seg> datæ differentiæ, & ex compoſito <seg type="var">.a.i.</seg> imaginetur <lb/>quadratum <seg type="var">.a.x.</seg> in quo protracta ſit <seg type="var">.t.u.</seg> æquidiſtans lateri <seg type="var">.a.i.</seg> & tam ab ipſa <seg type="var">.a.i.</seg> re <lb/>mota, quam <seg type="var">.x.i.</seg> ab <seg type="var">.s.e.</seg> vnde <seg type="var">.t.e.</seg> quadratum erit <seg type="var">.e.i.</seg> <lb/>dimidiæ ſcilicet differentiæ datæ <seg type="var">.e.o.</seg> et <seg type="var">.t.n.</seg> rectan-<lb/> <ptr xml:id="fig-0038-01a" corresp="fig-0038-01" type="figureAnchor"/> gulum æquale erit rectangulo <seg type="var">.n.c.</seg> vt cuilibet licet <lb/>per ſe conſiderare, vnde ſequitur gnomonem <seg type="var">.e.r.t.</seg> <lb/>æqualem eſſe producto <seg type="var">.a.s.</seg> ideo cognitus, qui <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/>gnomon, ſi coniunctus fuerit quadrato <seg type="var">.e.t.</seg> cognito <lb/>ex radice <seg type="var">.e.i.</seg> cognita (vt dimidia toralis differentię <seg type="var">.<lb/>e.o.</seg> datæ) habebimus quadratum totale <seg type="var">.a.x.</seg> cogni-<lb/>tum, & ita eius radicem <seg type="var">.a.i.</seg> cognitam & reliqua om <lb/>nia conſequenter quæ quidem ſpeculatio eadem eſt <lb/>quæ .6. ſecundi ſeu .8. noni Euclidis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0038-01" corresp="fig-0038-01a"> <graphic url="0038-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Poteris tamen ex modo & rationibus præceden-<lb/>ti theoremate allatis, hocipſum concludere.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="41">XLI</num>.</head> <p> <s xml:space="preserve">CVR ij, qui aliquo propoſito numero, inuenturi ſunt duos numeros inter ſe <lb/>differentes, quorum quadratorum ſumma altero numero propoſito æqualis <lb/>ſit, rectè primum numerum propoſitum in ſeipſum multiplicant, quod quadratum <lb/>exſecundo numero <choice><ex>detrahunt</ex><am>detrahũt</am></choice>, & dimidium reſidui ſumunt, quod productum erit <lb/>multiplicationis duorum numerorum interſe, in reliquis præcedentis theorematis <lb/>ordinem ſequuntur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponeretur .12. tanquam numerus, cui differentia duorum <lb/>numerorum quæſitorum æquanda eſt, proponerentur præterea .272. quibus ſum-<lb/>ma quadratorum duorum numerorum quæſitorum æquari deberet, oporteret ſanè <lb/>primum numerum, nempe .12. in ſeipſum multiplicare, cuius <choice><ex>quadratum</ex><am>quadratũ</am></choice> hoc loco <lb/>eſſet .144. atque hoc detrahere ex ſecundo numero, ſupereſſet .128. ſumpto <lb/>deinde dimidio huiuſce numeri, népe .64. producto in quam duorum numerorum <lb/><choice><ex>quæſitorum</ex><am>quæſitorũ</am></choice>. </s> <s xml:space="preserve">Cum hoc .64. proſtea et duodenario primo propoſito numero, præceden <lb/>tis theorematis ordinem ſequeremur.</s> </p> <pb facs="0039" n="27"/> <fw type="head">THEOREM. ARIT.</fw> <p> <s xml:space="preserve">Quod vt ſpeculemus, conſideremus ſubſcriptam figuram, vigefiminoni theore-<lb/>matis figuræ ſimilem, in qua numeri quæſiti duabus <lb/>lineis directè coniunctis <seg type="var">.q.g.</seg> et <seg type="var">.g.p.</seg> fignificentur, ho <lb/> <ptr xml:id="fig-0039-01a" corresp="fig-0039-01" type="figureAnchor"/> <choice><ex>rum</ex><am>rũ</am></choice> quadrata <choice><ex>erunt</ex><am>erũt</am></choice> <seg type="var">.r.c.</seg> et <seg type="var">.g.s.</seg> <choice><ex>quorum</ex><am>quorũ</am></choice> <choice><ex>summa</ex><am>sũma</am></choice> <choice><ex>iterum</ex><am>iterũ</am></choice> propo <lb/>nitur, quare etiam cognita. </s> <s xml:space="preserve"><choice><ex>Differentia</ex><am>Differẽtia</am></choice> autem <choice><ex>duorum</ex><am>duorũ</am></choice> <lb/>numerorum primo propofita fit <seg type="var">.q.i.</seg> eius verò qua-<lb/>dratum <seg type="var">.m.e.</seg> quod cognitum eſt ex ſua radice <seg type="var">.q.i</seg>. <lb/></s> <s xml:space="preserve">quare gnomon <seg type="var">.e.n.m.</seg> ſimul cum quadrato minori <seg type="var">.<lb/>g.s.</seg> cognitus erit, quæ ſumma æqualis eſt duplo <seg type="var">.g.r.</seg> <lb/>producto datorum numerorum. </s> <s xml:space="preserve">Itaque & ipſa <seg type="var">.g.<lb/>r.</seg> cognoſcetur, nunc ſi præcedentis theorematis ſpe-<lb/>culationem in reliquis conſuluerimus propoſitum <lb/>conſequemur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0039-01" corresp="fig-0039-01a"> <graphic url="0039-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="42">XLII</num>.</head> <p> <s xml:space="preserve">ADhuc etiam & alia ratione idipſum conſequi poſſemus, non conſulto qua-<lb/>drageſimo theoremate. </s> <s xml:space="preserve">Nam ſubtracto quadrato differentiæ, numeri primi <lb/>(<choice><ex>inquam</ex><am>inquã</am></choice>) propoſiti, ex <choice><ex>summa</ex><am>sũma</am></choice> duorum quadratorum, nempe ex ſecundo numero pro-<lb/>poſito colligendum eſſet reſiduum in ſummam cum prædicto ſecundo numero, & <lb/>ex ſumma hac deſumenda quadrata radix, quæ duorum numerorum ſumma erit, <lb/>de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius <lb/>dimidio addito primo numero propoſito, aut detracto minore inuento ex radice <lb/>poſtremo inuenta, dabitur numerus maior, qui quæritur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, cum ſuperfuerint .128. hæc ſi cum ſecundo numero <choice><ex>nempe</ex><am>nẽpe</am></choice> .272. <lb/>iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo <lb/>propoſito, nempe .12. ſupererunt .8. quorum <choice><ex>dimidium</ex><am>dimidiũ</am></choice> erit .4. quo ex .20. detracto <lb/>aut coniuncto .12. maior numerus orietur.</s> </p> <p> <s xml:space="preserve">Cuius rei contemplatio, præcedenti figura aperitur. </s> <s xml:space="preserve">Nam reſiduum detractionis <lb/>quadrati <seg type="var">.m.e.</seg> ex ſumma <choice><ex>duorum</ex><am>duorũ</am></choice> quadratorum <seg type="var">.r.c.</seg> et <seg type="var">.g.s.</seg> numerum præbet æqua-<lb/>lem duobus ſupplementis <seg type="var">.q.n.</seg> et <seg type="var">.n.u.</seg> ex .8. ſecundi Euclidis. qui coniunctus duo-<lb/>bus quadratis (quorum ſumma ſecundo propoſita fuit) cognitionem profert qua-<lb/>drati <seg type="var">.q.u.</seg> & eius radicis <seg type="var">.q.p.</seg> de qua, detracto primo dato numero, ſcilicet <seg type="var">.q.i.</seg> ſu-<lb/>pereſt <seg type="var">.i.p.</seg> cuius dimidium nempe <seg type="var">.g.p.</seg> minor eſt numerus qui quęritur; </s> <s xml:space="preserve">reſiduum <lb/>verò totius <seg type="var">.g.q.</seg> maior ſcilicet.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="43">XLIII</num>.</head> <p> <s xml:space="preserve">CVR ij, qui volunt duos numeros inuenire, quorum ſumma æqualis propo-<lb/>fito alicui numero futura ſit, & ſumma quadratorum maior eorum produ-<lb/>cto per quantitatem alterius propoſiti numeri, rectè dimidium primi dati numeri in <lb/>ſeipſum multiplicant, quod quadratum ex <choice><ex>ſecundo</ex><am>ſecũdo</am></choice> dato numero detrahunt, ſumunt<lb/>q́ue tertię partis refidui quadratam radicem, quam dimidio primi numeri coniun-<lb/>gunt, ex quo maior numerus <choice><ex>duorum</ex><am>duorũ</am></choice> <choice><ex>quæſitorum</ex><am>quæſitorũ</am></choice> datur, quo ex toto primo detracto, ſu-<lb/>pererit minor.</s> </p> <p> <s xml:space="preserve">Exempli gratia, propoſito numero .20. cui æquanda eſt ſumma duorum nume-<lb/>rorum quæſitorum, <choice><ex>datoque</ex><am>datoq́;</am></choice> ſecundo numero .208. qui ſemper maior eſſe debet <pb facs="0040" n="28"/><fw type="head">IO. BAPT. BENED.</fw> quadrato dimidij, prout ex ſpeculatione huiuſmodi operis cognoſcetur, <choice><ex>cuiæquanda</ex><am>cuiæquãda</am></choice> <lb/>eſt <choice><ex>differentia</ex><am>differẽtia</am></choice> inter <choice><ex>ſummam</ex><am>ſummã</am></choice> <choice><ex>quadratorum</ex><am>quadratorũ</am></choice> <choice><ex>duorum</ex><am>duorũ</am></choice> qui <choice><ex>quæruntur</ex><am>quærũtur</am></choice> <choice><ex>numerorum</ex><am>numerorũ</am></choice>, ſimul <choice><ex>cum</ex><am>cũ</am></choice> pro <lb/>ducto <choice><ex>eorum</ex><am>eorũ</am></choice> radicum. </s> <s xml:space="preserve">Dimidium numeri .20. in ſeipſum multiplicandum eſſet, qua-<lb/><choice><ex>dratumque</ex><am>dratumq́;</am></choice> detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua <lb/>drata radix eſſet .6. quæ ſi iuncta fuerit dimidio .20. nempe .10. daretur maior nu-<lb/>merus quæſitus .16. quo detracto è .20. darentur .4.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, datus primus numerus ſignificetur linea <seg type="var">.g.h.</seg> in qua <lb/>maior numerus incognitus ſit <seg type="var">.g.h.</seg> minor verò <seg type="var">.b.h.</seg> quorum quadrata ſint <seg type="var">.y.t.</seg> et <seg type="var">.<lb/>b.l.</seg> in quadrato maximo <seg type="var">.g.p.</seg> tum productum <seg type="var">.g.b.</seg> in <seg type="var">.b.h.</seg> ſit <seg type="var">.g.c.</seg> <choice><ex>cogitenturque</ex><am>cogitenturq́;</am></choice> duo <lb/>diametri <seg type="var">.q.h.</seg> et <seg type="var">.g.p.</seg> diuiſi per medium in puncto <seg type="var">.o.</seg> per quod duę lineæ ducan-<lb/>tur <seg type="var">.f.d.</seg> et <seg type="var">.k.m.</seg> parallelæ lateribus maximi quadrati. </s> <s xml:space="preserve">Hæ dictum quadratum in <lb/>quatuor quadrata æqualia diuident, quorum <choice><ex>vnumquodque</ex><am>vnumquodq́;</am></choice>, æquale erit quadrato <seg type="var">.<lb/>g.f.</seg> dimidij ipſius <seg type="var">.g.h.</seg> datę, </s> <s xml:space="preserve">quare eorum <choice><ex>vnumquodque</ex><am>vnumquodq́;</am></choice> cognitum erit. </s> <s xml:space="preserve">Iterum co <lb/>gitemus <seg type="var">.s.x.</seg> per <seg type="var">.e.</seg> <choice><ex>parallelam</ex><am>parallelã</am></choice> <seg type="var">.g.k.</seg> tantum diſtan-<lb/>tem à <seg type="var">.g.k.</seg> quantum <seg type="var">.y.l.</seg> ab <seg type="var">.g.h.</seg> diſtare inueni-<lb/> <ptr xml:id="fig-0040-01a" corresp="fig-0040-01" type="figureAnchor"/> tur. </s> <s xml:space="preserve">Cogitetur pariter <seg type="var">.z.i.a.</seg> per punctum <seg type="var">.i.</seg> <lb/>parallela <seg type="var">.d.p.</seg> </s> <s xml:space="preserve">quare <seg type="var">.a.t.</seg> æqualis erit <seg type="var">.f.c.</seg> et <seg type="var">.y.x.</seg> <lb/>æqualis <seg type="var">.f.e.</seg> et <seg type="var">.y.s</seg>: <seg type="var">b.l.</seg> æqualis. </s> <s xml:space="preserve">Ita ſubtractis è <lb/>duobus quadratis ſuperius dictis <seg type="var">.a.t.y.x.</seg> et <seg type="var">.b.l.</seg> <lb/>producto <seg type="var">.y.b.</seg> æqualibus, ſupererunt <seg type="var">.k.d.</seg> et <seg type="var">.a.c.<lb/>x.</seg> cognita, tanquam æqualia dato ſecundo nu-<lb/>mero, ſed <seg type="var">.k.d.</seg> quadratum eſt medietatis <seg type="var">.g.f.</seg> <lb/>cognitæ, cognoſcetur igitur reſiduum <seg type="var">.a.c.x.</seg> vnà <lb/>etiam ſingulæ tertiæ partes nempe quadrata <seg type="var">.o.<lb/>i.o.c.</seg> et <seg type="var">.o.e.</seg> & radix <seg type="var">.b.f.</seg> vel <seg type="var">.f.s.</seg> ſingularum, <lb/>qua coniuncta dimidio <seg type="var">.g.f.</seg> <choice><ex>rurfusque</ex><am>rurfusq́;</am></choice> ab <choice><ex>eodem</ex><am>eodẽ</am></choice> de-<lb/>tracta, propoſitum conſequemur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0040-01" corresp="fig-0040-01a"> <graphic url="0040-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="44">XLIIII</num>.</head> <p> <s xml:space="preserve">CVR ſi quis cupiat numerum propoſitum in duas eiuſmodi partes diuidere, vt <lb/>quadratum maioris, quadratum minoris ſuperet quantitate alterius numeri <lb/>propoſiti, rectè primum numerum in ſeipſum multiplicabit, & ab eodem ſecun-<lb/>dum numerum detrahet, reſiduum verò per duplum primi diuidet, ex quo proue-<lb/>niens primi pars minor erit, quæ ex illo primo detracta, partem maiorem <lb/>proferet.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponantur .20. diuiſa in duas eiuſmodi partes, vt <choice><ex>quadratum</ex><am>quadratũ</am></choice> <lb/>maioris ſuperet quadratum minoris numero æquali ipſi .240. oportebit primum <lb/>numerum, qui quadratus cum fuerit, erit .400. in ſeipſum multiplicare, & ex hoc <lb/>quadrato ſecundum numerum nempe .240. detrahere, </s> <s xml:space="preserve">tunc remanebunt .160. quę <lb/>diuiſa per .40. <choice><ex>numerum</ex><am>numerũ</am></choice> <choice><ex>duplum</ex><am>duplũ</am></choice> primo, dabuntur quatuor pro minori numero, à reſi-<lb/>duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</s> </p> <p> <s xml:space="preserve">Quod vt exactè conſideremus, primus numerus propoſitus ſignificetur linea <seg type="var">.q.<lb/>h.</seg> diuidendus in duas partes <seg type="var">.q.p.</seg> et <seg type="var">.p.h.</seg> tales quales quærimus. </s> <s xml:space="preserve">Poſtmodum eriga <lb/><gap reason="illegible" extent="2" unit="chars"/>r quadratum <seg type="var">.q.e.</seg> diuiſum diametro <seg type="var">.f.h.</seg> <choice><ex>ductisque</ex><am>ductisq́;</am></choice> <seg type="var">.p.o.t.</seg> et <seg type="var">.a.o.c.</seg> parallelis lateri-<lb/>bus quadrati, dabuntur imaginaria quadrata <seg type="var">.c.t.</seg> et <seg type="var">.p.a.</seg> duarum partium <seg type="var">.q.p.</seg> et <seg type="var">.p.<lb/>h.</seg> incognitarum. </s> <s xml:space="preserve">Ad hæc cogitemus quadratum <seg type="var">.u.n.</seg> æquale quadrato <seg type="var">.p.a.</seg> è quadra <pb facs="0041" n="29"/><fw type="head">THEOR. ARITH.</fw> to maiore <seg type="var">.c.t.</seg> extractum quare reſiduum qua-<lb/> <ptr xml:id="fig-0041-01a" corresp="fig-0041-01" type="figureAnchor"/> drati <seg type="var">.c.p.</seg> cognitum erit, quam quantitatem co-<lb/>gnitam, cum ſit ſecundo loco data, cogitemus <lb/>detrahi è toto quadrato cognito <seg type="var">.q.e.</seg> ex quo <lb/>ſumma duorum ſupplementorum <seg type="var">.q.o.</seg> et <seg type="var">.o.e.</seg> <lb/>cognoſcetur, vnà cum quadratis <seg type="var">.u.n.</seg> et <seg type="var">.p.a.</seg> du <lb/>plo ſcilicet <seg type="var">.q.a.</seg> quo diuiſo per duplum <seg type="var">.q.h.</seg> aut <lb/>ſimplex <seg type="var">.q.a.</seg> per <seg type="var">.q.h.</seg> ſimplicem, dabitur <seg type="var">.a.h.</seg> <lb/>nempe <seg type="var">.p.h.</seg> minor numerus quæſitus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0041-01" corresp="fig-0041-01a"> <graphic url="0041-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="45">XLV</num>.</head> <p> <s xml:space="preserve">CVR volentes diuidere numerum propoſitum in duas eiuſmodi partes, vt pro <lb/>ductum vnius in alteram, alteri numero propoſito æquetur, rectè dimidium <lb/>primi dati numeri in ſeipſum multiplicant, ex quo quadrato ſecundum datum nu-<lb/>merum detrahunt, <choice><ex>reſiduique</ex><am>reſiduiq́;</am></choice> radicem ſumunt, qua coniuncta vni dimidio primi nu-<lb/>meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeſtabit.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi numerus partiendus eſſet .34. alter verò numerus eſſet .64. cui <lb/>productum vnius partis in alteram æquale eſſe deberet. </s> <s xml:space="preserve">Dimidium primi numeri, in <lb/>ſeipſum multiplicaremus, cuius quadratum eſſet .289. de quo detracto ſecundo nu-<lb/>mero nempe .64. remaneret .225. cuius quadrata radix nempe .15. coniuncta .17. <lb/>dimidio .34. proferet .32. maiorem partem, <choice><ex>detractoque</ex><am>detractoq́;</am></choice> ex .17. ſupereſſet .2. pars <lb/>inquam minor.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus propoſitus ſignificetur linea <seg type="var">.a.d.</seg> cu-<lb/>ius dimidium <seg type="var">.c.d.</seg> cognitum erit, vnà etiam eius quadratum <seg type="var">.c.f.</seg> quo diuiſo per dia <lb/>metrum <seg type="var">.e.d.</seg> ſupponantur partes ignotæ <lb/> <ptr xml:id="fig-0041-02a" corresp="fig-0041-02" type="figureAnchor"/> ipſius <seg type="var">.a.d.</seg> eſſe <seg type="var">.a.b.</seg> et <seg type="var">.b.d.</seg> & à puncto <seg type="var">.b.</seg> <lb/>duci lineam <seg type="var">.b.h.g.</seg> parallelam <seg type="var">.d.f.</seg> et <seg type="var">.m.<lb/>h.k.</seg> parallelam <seg type="var">.d.a.</seg> extructa figura ſimi <lb/>li figuræ quintæ ſecundi Eucli. </s> <s xml:space="preserve">quare da <lb/>bitur <choice><ex>gnomon</ex><am>gnomõ</am></choice> <seg type="var">.l.d.g.</seg> æqualis producto <seg type="var">.b.<lb/>k.</seg> & proinde cognitus, quo detracto è <lb/>quadrato, <seg type="var">c.f.</seg> remanebit quadratum <seg type="var">.g.l.</seg> <lb/>cuius radice æquali <seg type="var">.c.b.</seg> coniuncta <seg type="var">.a.c.</seg> <lb/>& detracta ex <seg type="var">.c.d.</seg> partes <seg type="var">.a.b.</seg> et <seg type="var">.b.d.</seg> quæſitæ dabuntur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0041-02" corresp="fig-0041-02a"> <graphic url="0041-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="46">XLVI</num>.</head> <p> <s xml:space="preserve">CVR propoſitis tribus numeris, quorum prior in duas eiuſmodi partes diui-<lb/>dendus ſit, ut mutuò diuiſæ, & per ſummam prouenientium diuiſo ſecundo <lb/>numero, proueniens vltimum ſit æquale tertio numerorum propoſitorum. </s> <s xml:space="preserve">Conſul <lb/>tiſsimum ſit ſecundum numerum per tertium diuidere, ex quo proueniens ſit ſum-<lb/>ma prouenientium è duabus partibus mutuò diuiſis, quam ſummam ſi quis velit di-<lb/>ſtinguere, rectè poſſit medio operationis <choice><ex>pręcedentis</ex><am>pręcedẽtis</am></choice> theorematis <choice><ex>sumpta</ex><am>sũpta</am></choice> vnitate ſuper <lb/>ficiali pro ſecundo numero diſtinctis poſtmodum prouenientibus, rectè meo iudi-<lb/>cio operabimur per <choice><ex>regulam</ex><am>regulã</am></choice> de tribus (quod fuit ab antiquis prætermiſſum) Si dixe- <pb facs="0042" n="30"/><fw type="head">IO. BAPT. BENED.</fw> rimus, ſi ſumma vnius dictorum prouenientium cum vnitate dat primum numerum, <lb/>quid ipſa eadem vnitas dabit? </s> <s xml:space="preserve">ex quo propoſitum oriatur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponuntur tres numeri, primus .20. ſecundus .34. tertius .8. <lb/>Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prębeant duo <lb/>prouenientia, quorum ſumma tanta ſit vt per eam diuiſo .34. proueniat numerus <lb/>æqualis tertio numero .8. </s> <s xml:space="preserve">Quod vt præſtemus iubet regula ſecundum .34. per <choice><ex>tertium</ex><am>tertiũ</am></choice> <num value="8">.<lb/>8.</num> diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit ſumma pro <lb/>uenientium ex diuiſione duarum partium quæſitarum, quæ ſi diſtinguere volueri-<lb/>mus, præcedentis theorematis methodum ſequemur, vnitate ſuperficiali pro ſecun <lb/>do numero propoſito ſumpta, ac ſi diceremus, diuidatur .4. cum vna quarta parte <lb/>in duas eiuſmodi partes, vt productum vnius in alteram ſit vnitas ſuperficialis, cer-<lb/>tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemſeptem <lb/>quartæ lineares, verum cum neceſſe ſit, ex præcedenti theoremate, dimidium in <lb/>ſeipſum multiplicare, <choice><ex>eſſetque</ex><am>eſſetq́;</am></choice> dimidium .8. quartarum partium cum octaua, com-<lb/>modius totum conſtituetur .34. octauarum, quarum dimidium, nempe decemſep-<lb/>tem octauæ, in ſeipſum multiplicatum erunt .289. ſexageſimæ quartæ vnius integri <lb/>ſuperficialis, quandoquidem <choice><ex>integrum</ex><am>integrũ</am></choice> ſuperficiale, cuius vnitas linearis in .8. partes <lb/>diuiditur eſt .64. vt ex primo theoremate huius libri depræhendi poteſt. </s> <s xml:space="preserve">Nunc vni-<lb/>tate hac ſuperficiali, nempe .64. ex .289. detracta, ſupererit .225. cuius radix qua-<lb/>drata, ſcilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit <lb/>maius proueniens .32. <choice><ex>detractaque</ex><am>detractaq́;</am></choice> ex altero dimidio, dabit proueniens minus .2. hoc <lb/>eſt pro maiore proueniente .32. octauas, & pro minore duas, quatuor ſcilicet inte-<lb/>gros pro maiore, & quartam partem vnius integri pro minore. </s> <s xml:space="preserve">Nunc ſi ex regula <lb/>de tribus dixerimus, ſi .4. iuncta vni, nempe .5. dant .20. primum numerum, quid <lb/>dabunt .4. integra (proueniens inquam maius) <choice><ex>dabunt</ex><am>dabũt</am></choice> certè .16. partem maiorem. <lb/></s> <s xml:space="preserve">Tum ſi dixerimus, ſi quarta pars coniuncta vnitati dat .20: </s> <s xml:space="preserve">quid dabit quarta illa <lb/>pars (hoc eſt proueniens minus) dabit <choice><ex>profectò</ex><am>ꝓfectò</am></choice> quatuor ſcilicet <choice><ex>minorem</ex><am>minorẽ</am></choice> partem, quod <lb/>ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-<lb/>ſcientes ijs vti ad inueniendas duas primi numeri partes.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, demus primum numerum ſignificari linea <seg type="var">.e.u.</seg> cuius <lb/>partes <seg type="var">.e.a.</seg> & <seg type="var">a.u.</seg> ſint quæ quæruntur, alter verò numerus ſignificetur linea <seg type="var">.b.<lb/>d.</seg> tertius linea <seg type="var">.g.f.</seg> proueniens <choice><ex>autem</ex><am>aũt</am></choice> diuiſionis <seg type="var">.e.a.</seg> per <seg type="var">.a.u.</seg> ſit <seg type="var">.n.t.</seg> diuiſionis <choice><ex>autem</ex><am>aũt</am></choice> <seg type="var">.a.u.</seg> <lb/>per <seg type="var">.a.e.</seg> ſit <seg type="var">.t.o.</seg> ſumma erit <seg type="var">.n.t.o.</seg> vnitas verò <seg type="var">.n.i.</seg> et <seg type="var">.o.i</seg>. </s> <s xml:space="preserve">Iam ſi numerus <seg type="var">.f.g.</seg> tertiò <lb/>propoſitus ex diuiſione ſecundi per <seg type="var">.o.t.n.</seg> proferri debet. </s> <s xml:space="preserve">Ex .13. theoremate patet, <lb/>quòd ſi <seg type="var">.b.d.</seg> per <seg type="var">.g.f.</seg> diuiſerimus, proferetur <seg type="var">.o.t.n.</seg> qui cum fuerit inuentus, <choice><ex>ſummam</ex><am>ſummã</am></choice> <lb/>eſſe oportet <choice><ex>duorum</ex><am>duorũ</am></choice> <choice><ex>prouenientium</ex><am>prouenientiũ</am></choice>, ex diuiſione mutua <choice><ex>duorum</ex><am>duorũ</am></choice> numerorum, nempe <seg type="var">.<lb/>a.e.</seg> per <seg type="var">.a.u.</seg> et <seg type="var">.a.u.</seg> per <seg type="var">.a.e.</seg> deinde manifeſtum eſt ex .24. aut .25. theoremate <choice><ex>eorum</ex><am>eorũ</am></choice> <lb/>productum (multiplicatis prouenientibus adinuicem) vnitatem ſuperficialem futu <lb/>ram eſſe. </s> <s xml:space="preserve">Hactenus igitur, totum <seg type="var">.o.n.</seg> ex doctrina præcedentis theorematis diui-<lb/>ditur in puncto <seg type="var">.t.</seg> ita vt productum <seg type="var">.o.t.</seg> in <seg type="var">.t.n.</seg> <lb/>ſolam vnitatem ſuperficialem <choice><ex>contineat</ex><am>cõtineat</am></choice>, quo <lb/> <ptr xml:id="fig-0042-01a" corresp="fig-0042-01" type="figureAnchor"/> facto, ſi, vt antedictum eſt, cogitauerimus <seg type="var">.n.<lb/>t.</seg> <choice><ex>proueniens</ex><am>proueniẽs</am></choice> eſſe ex diuiſione <seg type="var">.e.a.</seg> per <seg type="var">.a.u.</seg> et <seg type="var">.<lb/>t.o.</seg> proueniens ex diuiſione <seg type="var">.a.u.</seg> per <seg type="var">.a.e.</seg> pa-<lb/>tebit ex definitione diuiſionis, quod eadem <lb/>erit proportio <seg type="var">.a.e.</seg> ad <seg type="var">.n.t.</seg> quæ eſt <seg type="var">.a.u.</seg> ad vni-<lb/>tatem <seg type="var">.n.i.</seg> et <seg type="var">.a.u.</seg> ad <seg type="var">.o.t.</seg> eadem quæ eſt <seg type="var">.e.a.</seg> <pb facs="0043" n="31"/><fw type="head">THEOREM. ARITH.</fw> ad vnitatem <seg type="var">.o.i.</seg> <choice><ex>permutandoque</ex><am>permutandoq́;</am></choice> <seg type="var">.e.a.</seg> ad <seg type="var">.a.u.</seg> ſicut <seg type="var">.t.n.</seg> ad <seg type="var">.n.i.</seg> & componendo <seg type="var">.e.a.u.</seg> <lb/>ad <seg type="var">a.u.</seg> ſicut <seg type="var">.t.n.i.</seg> ad <seg type="var">.n.i</seg>: & euerſim <seg type="var">.e.a.u.</seg> ad <seg type="var">.e.a.</seg> vt <seg type="var">.t.n.i.</seg> ad <seg type="var">.t.n</seg>. </s> <s xml:space="preserve">Quare, ex .20. ſepti <lb/>mi, recte vtimur regula de tribus. </s> <s xml:space="preserve">Idem & de altera parte dico, quamuis qui vnam <lb/>teneat, alteram quo que habiturus ſit. </s> <s xml:space="preserve">Non mirum tamen ſi huiuſmodi problema <lb/>ab antiquis definitum non fuerit, qui hanc vltimam partem non cognouerunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0042-01" corresp="fig-0042-01a"> <graphic url="0042-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="47">XLVII</num>.</head> <p> <s xml:space="preserve">CVR duobus numeris mutuó diuiſis, ſi per ſummam prouenientium, produ-<lb/>ctum vnius in alterum multiplicetur, vltimum productum, ſummæ quadra-<lb/>tn<gap reason="illegible" extent="2" unit="chars"/>m duorum numerorum æquale futurum ſit.</s> </p> <p> <s xml:space="preserve">Exempli gratia, propoſitis .16. et .4. mutuò diuiſis, ſumma prouenientium erit <num value="4">.<lb/>4.</num> integrorum cum quarta parte, qua ſumma multiplicata cum producto <choice><ex>primorum</ex><am>primorũ</am></choice> <lb/>numerorum, nempe .64. dabuntur .272. integri ſuperficiales, qui ſummæ quadra-<lb/>torum duorum numerorum æquantur.</s> </p> <p> <s xml:space="preserve">Hoc vt conſideremus, duo numeri partibus <seg type="var">.a.e.</seg> et <seg type="var">.e.i.</seg> in linea <seg type="var">.a.i.</seg> ſignificentur, <lb/>quorum productum ſit <seg type="var">.e.d.</seg> & <choice><ex>quadratum</ex><am>quadratũ</am></choice> ipſius <seg type="var">.a.e.</seg> ſit <seg type="var">.e.p</seg>: ipſius verò <seg type="var">.e.i.</seg> ſit <seg type="var">.e.q.</seg> pro-<lb/>ueniens <choice><ex>autem</ex><am>aũt</am></choice> ex diuiſione <seg type="var">.e.i.</seg> per <seg type="var">.a.e.</seg> ſit <seg type="var">.o.u.</seg> proueniens <choice><ex>autem</ex><am>aũt</am></choice> <seg type="var">.a.e.</seg> per <seg type="var">.e.i.</seg> ſit <seg type="var">.o.t.</seg> quo-<lb/>rum ſumma ſit <seg type="var">.o.u.t.</seg> tum productum <seg type="var">.e.d</seg>: linea <seg type="var">.u.n.</seg> ſignificetur ad angulum <choice><ex>rectum</ex><am>rectũ</am></choice> <lb/>coniuncta in puncto <seg type="var">.u.</seg> extremo ipſius <seg type="var">.o.u.t.</seg> productum <choice><ex>autem</ex><am>aũt</am></choice> <seg type="var">.u.o.t.</seg> in <seg type="var">.u.n.</seg> ſit <seg type="var">.n.t</seg>. </s> <s xml:space="preserve">Iam <lb/>probandum nobis eſt <seg type="var">.n.t.</seg> æqualem eſſe ſummæ duorum quadratorum <seg type="var">.q.e.p</seg>. </s> <s xml:space="preserve">Quod <lb/>ſingillatim probo, & aſſero productum <seg type="var">.o.n.</seg> æquale eſſe quadrato <seg type="var">.q.e.</seg> & <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.<lb/>s.t.</seg> quadrato <seg type="var">.e.p</seg>. </s> <s xml:space="preserve">Nam ex .35. theoremate patet numerum <seg type="var">.e.i.</seg> medium eſſe <choice><ex>pro- portionalem</ex><am>pro-portionalẽ</am></choice> inter <seg type="var">.e.d.</seg> et <seg type="var">.o.u</seg>: cum numerus <seg type="var">.e.i.</seg> ex præſuppoſito ab <seg type="var">.e.a.</seg> multiplicetur <lb/>& diuidatur, cuius multiplicationis produ-<lb/>ctum eſt <seg type="var">.d.e</seg>: nempe <seg type="var">.u.n.</seg> & proueniens ex <lb/> <ptr xml:id="fig-0043-01a" corresp="fig-0043-01" type="figureAnchor"/> diuiſione eſt <seg type="var">.o.u</seg>: </s> <s xml:space="preserve">quare ex dicto theorema-<lb/>te <seg type="var">.e.i.</seg> media proportionalis eſt inter <seg type="var">.u.n.</seg> et <seg type="var">.<lb/>u.o</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> productum <seg type="var">.o.n.</seg> æquale eſt qua-<lb/>drato <seg type="var">.e.q.</seg> ex .16. ſexti vel .20. ſeptimi. </s> <s xml:space="preserve">Idem <lb/>dico de producto <seg type="var">.s.t.</seg> <choice><ex>nempe</ex><am>nẽpe</am></choice> æquale eſſe qua-<lb/>drato <seg type="var">.e.p.</seg> quandoquidem numerus <seg type="var">.a.e.</seg> ab <lb/><seg type="var">e.i.</seg> multiplicatur ac diuiditur, cuius multi-<lb/>plicationis productum eſt <seg type="var">.d.e.</seg> nempe <seg type="var">o.s.</seg> & <lb/>proueniens ex diuiſione <seg type="var">.o.t</seg>: </s> <s xml:space="preserve">inter quæ ex .<lb/>35. theoremate <seg type="var">.a.e.</seg> media proportionalis <lb/>eſt. </s> <s xml:space="preserve">Quare ex allatis propoſitionibus <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.s.t.</seg> æquale eſt quadrato <seg type="var">.e.p.</seg> ſed <choice><ex>totum</ex><am>totũ</am></choice> <lb/>productum <seg type="var">.n.t.</seg> ſumma eſt duorum productorum <seg type="var">.o.n.</seg> et <seg type="var">.s.t.</seg> ex prima ſecundi Eucli. <lb/></s> <s xml:space="preserve">Itaque verum eſſe quod dictum eſt, conſequitur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0043-01" corresp="fig-0043-01a"> <graphic url="0043-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="48">XLVIII</num>.</head> <p> <s xml:space="preserve">CVR ſi quis maiorem duorum numerorum ſola vnitate inter ſe differentium, <lb/>per minorem diuidat, <choice><ex>maioremque</ex><am>maioremq́;</am></choice> per proueniens multiplicet, productum, <lb/><choice><ex>summæ</ex><am>sũmæ</am></choice> ipſius maioris cum eodem proueniente æquale erit.</s> </p> <p> <s xml:space="preserve">Exempli gratia .10 per .9. diuiſo, datur vnum cum nona parte, quo multiplica-<lb/>to per proueniens, ipſo nempe .10: </s> <s xml:space="preserve">datur productum .11. cum nona parte, tantum ſci <pb facs="0044" n="32"/><fw type="head">I.O. BAPT. BENED.</fw> licet quanta ſumma eſt maioris cum proueniente.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, maior numerus ſignificetur <seg type="var">.a.i.</seg> et minor linea <seg type="var">.a.o.</seg> ex <lb/>quo ex præſupoſito <seg type="var">.o.i.</seg> vnitas erit. </s> <s xml:space="preserve">Sit autem proueniens ex diuiſione <seg type="var">.a.i.</seg> per <seg type="var">.a.o.<lb/>a.e</seg>: </s> <s xml:space="preserve">quod <seg type="var">.e.a.</seg> directè coniungatur ipſi <seg type="var">.a.i.</seg> et productum <seg type="var">.a.i.</seg> in <seg type="var">.a.e.</seg> ſit <seg type="var">.u.i</seg>. </s> <s xml:space="preserve">Probabo <lb/>numerum ſuperficialem <seg type="var">.u.i.</seg> æqualem eſſe lineari <seg type="var">.i.a.e</seg>. </s> <s xml:space="preserve">quare meminiſſe oportet, <lb/>decimotertio theoremate probatum fuiſſe, quod ſi numerus diuiſibilis per pro-<lb/>ueniens diuidatur, proueniens futurus ſit numerus diuidens, </s> <s xml:space="preserve">quare <seg type="var">.a.o.</seg> erit pro-<lb/>ueniens ex diuiſione <seg type="var">.a.i.</seg> per <seg type="var">.a.e.</seg> & ex deſinitione diuiſionis ita ſe habebit <seg type="var">.e.a.</seg> ad <seg type="var">.<lb/>a.i.</seg> ſicut <seg type="var">.o.i.</seg> ad <seg type="var">.o.a.</seg> & componondo ita <seg type="var">.e.i.</seg> ad <seg type="var">.a.i.</seg> ſicut <seg type="var">.i.a.</seg> ad <seg type="var">.o.a.</seg> </s> <s xml:space="preserve">quare <seg type="var">.a.i.</seg> erit me-<lb/>dia pportionalis inter <seg type="var">.e.i.</seg> et <seg type="var">.a.o.</seg> ſed <seg type="var">.a.i.</seg> non modò diuiſa <choice><ex>nunc</ex><am>nũc</am></choice> cogitatur ab <seg type="var">.e.a.</seg> ex <lb/>quo ſit proueniens <seg type="var">.a.o.</seg> ſed etiam per eandem <seg type="var">.e.a.</seg> multiplicata, ex quo produ-<lb/>ctum oriatur <seg type="var">.u.i</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> ex .25. theobema-<lb/>te <seg type="var">.a.i.</seg> media eſt proportionalis inter <seg type="var">.u.</seg> <lb/> <ptr xml:id="fig-0044-01a" corresp="fig-0044-01" type="figureAnchor"/> i. et <seg type="var">.a.o</seg>. </s> <s xml:space="preserve">Quare. ex .11. quinti. eadem erit <lb/>proportio <seg type="var">.u.i.</seg> ad <seg type="var">.a.i.</seg> ſicut <seg type="var">.e.i.</seg> ad eandem <seg type="var">.<lb/>a.i</seg>. </s> <s xml:space="preserve">Igitur ex .9. prædicti numerus <seg type="var">.u.i.</seg> <lb/>æqualis erit numero <seg type="var">.e.i.</seg> quod erat propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0044-01" corresp="fig-0044-01a"> <graphic url="0044-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="49">XLIX</num>.</head> <p> <s xml:space="preserve">IDipſtim etiam alia ratione conſiderari poteſt.</s> </p> <p> <s xml:space="preserve">Linea <seg type="var">.u.a.</seg> ſecetur in puncto <seg type="var">.t.</seg> ita vt <seg type="var">.a.t.</seg> æqualis ſit vnitati <seg type="var">.o.i.</seg> & media paral <lb/>lela <seg type="var">.t.n.</seg> terminetur productum <seg type="var">.t.i.</seg> quod conſtabit æquali numero, quamuis ſuperfi-<lb/>ciali, numero <seg type="var">.a.i.</seg> tametſi lineari. </s> <s xml:space="preserve">Tumparallela ducatur à puncto <seg type="var">.o.</seg> ipſi <seg type="var">.a.u.</seg> termi <lb/><choice><ex>neturque</ex><am>neturq́;</am></choice> productum <seg type="var">.o.u.</seg> ex quo bina producta dabuntur <seg type="var">.u.o.</seg> et <seg type="var">.t.i.</seg> inter ſe æqualia <lb/>ex .15. ſexti aut .20. ſeptimi cum ita ſe habeat <seg type="var">.a.i.</seg> ad <seg type="var">.a.u.</seg> ſicut <seg type="var">.a.o.</seg> ad <seg type="var">.a.t.</seg> ſed <seg type="var">.a.i.</seg> ad <seg type="var">.<lb/>a.o.</seg> permutando ſic ſe habet ſicut <seg type="var">.a.u.</seg> ad <seg type="var">.a.t.</seg> & ex prima ſexti aut .18. vel .19. ſepti-<lb/>mi ſic ſe habet <seg type="var">.u.i.</seg> ad <seg type="var">.u.o.</seg> ſicut <seg type="var">.a.i.</seg> ad <seg type="var">.a.</seg> <lb/> <ptr xml:id="fig-0044-02a" corresp="fig-0044-02" type="figureAnchor"/> o. hoc eſt <seg type="var">.u.i.</seg> ad <seg type="var">.t.i.</seg> ope .11. quinti. </s> <s xml:space="preserve">Iam <lb/>ex definitione diuiſionis ita ſe habet <seg type="var">.a.e.</seg> <lb/>ad <seg type="var">.a.i.</seg> ſicut <seg type="var">.o.i.</seg> ad <seg type="var">.o.a.</seg> & componendo <seg type="var">.<lb/>e.i.</seg> ad <seg type="var">.a.i.</seg> ſicut <seg type="var">.i.a.</seg> ad <seg type="var">.o.a</seg>. </s> <s xml:space="preserve">Itaque ex præ-<lb/>dicta .11. ſic ſe habebit <seg type="var">.e.i.</seg> ad <seg type="var">.i.a.</seg> ſicut <seg type="var">.u.<lb/>i.</seg> ad <seg type="var">.t.i.</seg> ſed <seg type="var">.t.i.</seg> numero conſtat æquali <seg type="var">.a.<lb/>i</seg>. </s> <s xml:space="preserve">quare ex .9. quinti numerus <seg type="var">.u.i.</seg> numero <seg type="var">.e.i.</seg> æqualis erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0044-02" corresp="fig-0044-02a"> <graphic url="0044-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="50">L</num>.</head> <p> <s xml:space="preserve">CVR diuidentes numerum propoſitum in duas eiuſmodi partes, vt <choice><ex>productum</ex><am>productũ</am></choice> <lb/>vnius in alteram cum i pſarum differentia in ſummam collectum, æquale ſit <lb/>alicui alteri numero maiori primo. </s> <s xml:space="preserve">Rectè primum ex ſecundo detrahunt, reſiduum <lb/>verò conſeruant, tum ex primo ſemper binarium deſumunt, <choice><ex>dimidiumque</ex><am>dimidiumq́;</am></choice> conſer-<lb/>uant, alterum verò dimidium in ſeipſo multiplicant, & ex quadrato numerum con <lb/>ſeruatum eruunt, <choice><ex>reſiduique</ex><am>reſiduiq́;</am></choice> radicem ex dimidio conſeruato, quod vltimum reſi-<lb/>duum propoſiti numeri quæſita pars minor eſt.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponatur numerus .20. ita <choice><ex>diuidendus</ex><am>diuidẽdus</am></choice>, vt <choice><ex>productum</ex><am>productũ</am></choice> vnius partis <lb/>in alteram, cum partium differentia collectum in ſummam, æquale ſit propoſito <pb facs="0045" n="33"/><fw type="head">THEOR. ARITH.</fw> numero, verbi gratia .92. præcepit regula detrahi primum numerum ex ſecundo, <lb/>nempe .20. ex .92. cuius reſiduum, ſcilicet .72. conſeruetur, tum detrahi iubet bi <lb/>narium ex primo, ſic in propoſito exemplo remanebunt .18. huius autem .18. dimi <lb/>dium in ſeipſum multiplicari iubet, quod cum ſit .9. datur numerus .81. ex quo .81. <lb/>primum numerum conſeruatum, nempe .72. vult regula detrahi, ſic remanebit .9. <lb/>tum huius .9. quadrata radix detrahenda eſt ex dimidio ipſius .18. quod fuit ante qua <lb/>dratum, ſic ſupererit .6. hoc eſt .9. excepta radice quadrata, qui .6. erit minor pars <lb/>quæſita, maior verò .14. quarum productum .84. coniunctum cum partium differen <lb/>tia præbet exactè .92.</s> </p> <p> <s xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s> <s xml:space="preserve">Primus numerus minor, qui proponitur diuiſibilis <lb/>ſignificetur linea <seg type="var">.q.g.</seg> maior vero linea <seg type="var">.x.</seg> tum cogitemus <seg type="var">.q.g.</seg> diuiſam, cuius maior <lb/>pars ſit <seg type="var">.q.o.</seg> minor <seg type="var">.o.g.</seg> differentia <seg type="var">.q.p.</seg> ex quo <seg type="var">.p.o.</seg> æqualis erit <seg type="var">.o.g.</seg> ſit autem produ-<lb/>ctum <seg type="var">.b.o</seg>. </s> <s xml:space="preserve">Oportet igitur, ut <seg type="var">.b.o.</seg> ſimul cum differentia <seg type="var">.q.p.</seg> æquale ſit numero <seg type="var">.x.</seg> ſe-<lb/>cundò propoſito, qui notus eſt, </s> <s xml:space="preserve">quare etiam ſumma producti <seg type="var">.b.o.</seg> cum differentia <lb/><seg type="var">q.p.</seg> cognita erit, ex qua detracto primo numero <seg type="var">.q.g.</seg> reſiduum cognitum erit, nunc <lb/>igitur quodnam erit hoc reſiduum? </s> <s xml:space="preserve">attendamus qua ratione ex ſumma <seg type="var">.b.o.</seg> et <seg type="var">.q.p.</seg> <lb/>detrahenda ſit <seg type="var">.q.g</seg>. </s> <s xml:space="preserve">In primis ſi ſubtraxerimus ex dicta ſumma <seg type="var">.q.p.</seg> quę pars eſt <seg type="var">.q.g.</seg> <lb/>ſupererit detrahenda <seg type="var">.p.g.</seg> ex <seg type="var">.b.o.</seg> pars inquam ipſius <seg type="var">.q.g.</seg> quod fiet quotieſcunque <lb/>cogitauerimus <seg type="var">.q.o.</seg> duabus vnitatibus diminutam, et per <seg type="var">.o.g.</seg> multiplicatam, ſit au-<lb/>tem productum <seg type="var">.b.e.</seg> nam cum <seg type="var">.o.g.</seg> toties <seg type="var">.b.o.</seg> ingrediatur, quot ſunt in <seg type="var">.q.o.</seg> vnitates <lb/>ex prima ſexti aut .18. vel .19. ſeptimi, <choice><ex>detrahendaque</ex><am>detrahendaq́;</am></choice> ſit <seg type="var">.p.g.</seg> ex <seg type="var">.b.o.</seg> quæ <seg type="var">.p.g.</seg> dupla <lb/>eſt <seg type="var">.o.g.</seg> patebit <seg type="var">.o.c.</seg> æqualem eſſe <seg type="var">.p.g.</seg> fu-<lb/>pererit ita que <seg type="var">.b.e.</seg> productum <seg type="var">.q.e.</seg> in <seg type="var">.e.</seg> <lb/> <ptr xml:id="fig-0045-01a" corresp="fig-0045-01" type="figureAnchor"/> i. cognitum, erutis autem ex <seg type="var">.q.g.</seg> ijſdem <lb/>duabus vnitatibus, remanebit <seg type="var">.q.i.</seg> nobis <lb/>nota, ex quo <seg type="var">.e.i.</seg> æqualis erit <seg type="var">.e.c</seg>. </s> <s xml:space="preserve">Cum <lb/>igitur productum <seg type="var">.q.e.</seg> in <seg type="var">.e.i.</seg> cognoſcamus <lb/>ſimul cum <seg type="var">.q.i</seg>: Sivoluerimus partes <seg type="var">.q.e.</seg> <lb/>et <seg type="var">.e.i.</seg> cognoſcere, vtemur .45. theorema-<lb/>te huius libri, & propoſitum obtinebimus, nam cognoſcemus <seg type="var">.e.i.</seg> & ex conſequen-<lb/>ti <seg type="var">.o.g.</seg> eius æqualem.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0045-01" corresp="fig-0045-01a"> <graphic url="0045-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="51">LI</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">DIvidere</hi> numerum in duas eiuſmodi partes, quæ pro medio proportionali <lb/>alterum numerum propoſitum recipiant, primi dimidio minorem, aliud ni <lb/>hil eſt, quàm binas primi numeri partes inuenire, quæ inter ſe multiplicatæ quadra <lb/>to ſecundi numeri numerum æqualem proferant, ex .16. ſexti aut .20. ſeptimi, quod <lb/>tamen .45. theoremate fuit à nobis ſpeculatum.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="52">LII</num>.</head> <p> <s xml:space="preserve">CVR pro poſitis tribus numeris quibuſcunque, ſi productum primi in ſecun-<lb/>dum per tertium multiplicetur, atque ſecundum hoc productum <choice><ex>corporeum</ex><am>corporeũ</am></choice>, <lb/>per primum numerum diuidatur, proueniens erit numerus æqualis producto ſe-<lb/>cundi in tertium.</s> </p> <p> <s xml:space="preserve">Exempli cauſa, proponantur hi tres numeri .10. 11. 12. <choice><ex>multiplicenturque</ex><am>multiplicenturq́;</am></choice> .10. <choice><ex>cum</ex><am>cũ</am></choice>.</s> <pb facs="0046" n="34"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve">11. dabuntur .110. quo producto multiplicato cum .12. dabuntur .1320. hoc pro <lb/>ueniens per primum nempe .10. diuiſum dabit .132. numerum æqualem producto <lb/>ſecundi in tertium numerorum propoſitorum, ſcilicet .132.</s> </p> <p> <s xml:space="preserve">Hoc vt ſpeculemur, primus numerus ſignificetur line <seg type="var">a.o.u.</seg> ſecundus <seg type="var">.e.o.</seg> tertius <seg type="var">.<lb/>e.a.</seg> productum verò <seg type="var">.o.u.</seg> in <seg type="var">.o.e.</seg> ſit <seg type="var">.o.i.</seg> ipſius ve <lb/>rò <seg type="var">.o.i.</seg> per <seg type="var">.e.a.</seg> <choice><ex>productum</ex><am>productũ</am></choice> <choice><ex>corporeum</ex><am>corporeũ</am></choice> ſit <seg type="var">.i.c.</seg> tum <lb/> <ptr xml:id="fig-0046-01a" corresp="fig-0046-01" type="figureAnchor"/> <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.e.o.</seg> in <seg type="var">.e.a.</seg> ſit <seg type="var">.e.c</seg>. </s> <s xml:space="preserve">Dico <choice><ex>nunc</ex><am>nũc</am></choice> quod di-<lb/>uiſo numero corporeo <seg type="var">.i.c.</seg> per <choice><ex>primum</ex><am>primũ</am></choice> <seg type="var">.o.u.</seg> <choice><ex>proue</ex><am>ꝓue</am></choice> <lb/>niens æquale erit numero producti <seg type="var">.e.c</seg>. </s> <s xml:space="preserve">Qua-<lb/>re in primis cogitandum eſt, quod cum produ-<lb/>ctum <seg type="var">.i.c.</seg> ortum fuerit ex multiplicatione <seg type="var">.o.i.</seg> <lb/>in <seg type="var">.e.a</seg>: dictum <seg type="var">.o.i.</seg> toties ingredietur <seg type="var">.i.c.</seg> quo-<lb/>ties vnitas reperitur in <seg type="var">.e.a.</seg> eadem ratione, to-<lb/>ties <seg type="var">.e.c.</seg> in <seg type="var">.i.c.</seg> quot vnitates erunt in <seg type="var">.o.u</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <lb/>ſequitur quòd diuiſo <seg type="var">.i.c.</seg> per <seg type="var">o.u.</seg> proueniens ſit <lb/><seg type="var">e.c.</seg> corporeum, æquale nihilominus producto <seg type="var">.e.c.</seg> ſuperficiali.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0046-01" corresp="fig-0046-01a"> <graphic url="0046-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="53">LIII</num>.</head> <p> <s xml:space="preserve">CVR diuidens propoſitum numerum in tres partes ſic ſe habentes vt produ-<lb/>ctum primi in ſecundam, in tertia <choice><ex>multiplicatum</ex><am>multiplicatũ</am></choice>, præbeat numerum alteri nu-<lb/>mero propoſito æqualem. </s> <s xml:space="preserve">Rectè ſecundum numerum per quemcunque alium mino <lb/>rem primo diuidit, qui diuidens vna erit ex tribus partibus quæſitis, proueniens <lb/>autem erit productum vnius in alteram reliquarum duarum, quarum ſumma cogni <lb/>ta erit, detracto numero diuidente ex primo dato, quam quidem ſi diſtinguere <lb/>quis voluerit, vtetur theoremate .45.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponitur numerus .20. in tres partes diuidendus, quæ ſic ſe <lb/>habeant, ut productum primæ in ſecundam in tertia multiplicatum det .90. itaque <lb/>ſumenda erit pro prima vna pars ipſius .20. quæcunque illa ſit, verbi gratia .2. qua <lb/>ſecundus numerus, nempe .90. diuidatur, dabitur igitur .45. quod erit productum <lb/>cæterarum partium inter ſe, quarum ſumma eſt .18. quam ſummam ſi diſtinguere <lb/>volueris in cęteris duabus partibus ſeparatis, vteris .45. theoremate, vt quàm citiſ-<lb/>ſimè quod cupis exequaris, erunt autem partes .3. et .15.</s> </p> <p> <s xml:space="preserve">In cuius ſpeculationis gratiam nihil aliud occurrit, quàm quod præcedenti theo-<lb/>remate, & ſuperiore .45. allatum eſt.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="54">LIIII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">DIvidere</hi> numerum in .3. eiuſmodi partes, vt quadratum vnius ſit æquale <lb/>producto reliquarum duarum inter ſe, idem omnino eſt cum 51. theoremate. <lb/></s> <s xml:space="preserve">Nam qui ſumet quamlibet partem propoſiti numeri, quæ tertia parte maior tamen <lb/>non ſit, <choice><ex>reſiduumque</ex><am>reſiduumq́</am></choice> in duas tales partes diuiſerit, vt prima ſumpta, media proportio <lb/>nalis ſit ex probatione .51. theoremate allata, propoſitum conſequetur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="55">LV</num>.</head> <p> <s xml:space="preserve">ID ipſum alia ratione ab ea diuerſa <choice><ex>quam</ex><am>quã</am></choice> .51. theoremate adduximus, <choice><ex>profici</ex><am>ꝓfici</am></choice> poteſt.</s> </p> <pb facs="0047" n="35"/> <fw type="head">THEOREM. ARIT.</fw> <p> <s xml:space="preserve">Sumantur enimtres numeri continui proportionales, cuiuſcunque denique pro <lb/>portionalitatis, qui in ſummam colligantur, ac poſtmodum, regula de trib. dica-<lb/>mus. </s> <s xml:space="preserve">Si ſumma hæc primo numero propoſito in tres partes diuidendo reſpondet, <lb/>cuireſpondebit vna ex tribus partibus huiuſcę <choice><ex>summæ</ex><am>sũmæ</am></choice>? </s> <s xml:space="preserve">idem dereliquis duabus pa<unclear reason="illegible"/>rti <lb/>bus dico.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponatur numerus .57. diuidendus in tres continuas partes <lb/>proportionales proportione ſeſquialtera, tres numeros in eiuſmodi proportio-<lb/>nalitate diſtinctos ſumemus, vt potè .4. 6. 9. qui in ſummam collecti dabunt <choice><ex>ſum- mam</ex><am>ſum-mã</am></choice> .19. <choice><ex>dicemusque</ex><am>dicemusq́;</am></choice> ſi .19. dant .4. quid <choice><ex>dabunt</ex><am>dabũt</am></choice> .57? </s> <s xml:space="preserve">vnde proueniens vnius partis erit <num value="12">.<lb/>12</num>. </s> <s xml:space="preserve">Tum ſi dicamus, ſi .19. dat .6. quid dabit .57? </s> <s xml:space="preserve">nempe dabit .18. </s> <s xml:space="preserve">Poſtremò, ſi <num value="19">.<lb/>19.</num> dat .9. quid dabit .57? </s> <s xml:space="preserve">nempe .26. atque ita dabitur .18. cuius quadratum æqua-<lb/>bitur producto reliquarum duarum partium inter ſe.</s> </p> <p> <s xml:space="preserve">Quod vt ſciamus, numerus propoſitus in tres quaſlibet partes diuidendus ſi-<lb/>gnificetur linea <seg type="var">.a.d.</seg> tres autem numeri dictæ proportionalitatis, lineis <seg type="var">.e.f</seg>: <seg type="var">f.g.</seg> <lb/>et <seg type="var">.g.h.</seg> directè inter ſe coniunctis denotentur. </s> <s xml:space="preserve">Cogitemus pariter lineam <seg type="var">.d.a.</seg> in <lb/>tres partes diuiſam <seg type="var">.a.b</seg>: <seg type="var">b.c.</seg> et <seg type="var">.c.d.</seg> eadem cum cæteris proportionalitate, </s> <s xml:space="preserve">tunc ea-<lb/>dem erit proportio <seg type="var">.a.d.</seg> ad quamlibet ſuarum partium, quæ eſt <seg type="var">.e.h.</seg> ad reſponden <lb/>tem ipſius in <seg type="var">.a.d</seg>: Verbi gratia reſpondentem <seg type="var">.a.b.</seg> ipſi <seg type="var">.e.f.</seg> et <seg type="var">.b.c</seg>: <seg type="var">f.g.</seg> et <seg type="var">.c.d</seg>: <seg type="var">g.h</seg>. </s> <s xml:space="preserve">Di <lb/>co enim quòd ita ſe habebit <seg type="var">.a.d.</seg> ad <seg type="var">.c.d.</seg> ſicut <seg type="var">.e.h.</seg> ad <seg type="var">.g.h</seg>. </s> <s xml:space="preserve">Nam cum ſic ſe habeat <seg type="var">.a.<lb/>b.</seg> ad <seg type="var">.b.c.</seg> ſicut <seg type="var">.e.f.</seg> ad <seg type="var">.f.g.</seg> ex præſuppoſito, permutando ſic ſe habebit <seg type="var">.a.b.</seg> ad <seg type="var">.e.f.</seg> ſi-<lb/>cut <seg type="var">.b.c.</seg> ad <seg type="var">.f.g.</seg> & eadem ratione ſic ſe habe-<lb/>bit <seg type="var">.c.d.</seg> ad <seg type="var">.g.h.</seg> ſicut <seg type="var">.b.c.</seg> ad <seg type="var">.f.g.</seg> & <choice><ex>conſequen- ter</ex><am>cõſequen- ter</am></choice> ſicut <seg type="var">.a.b.</seg> ad <seg type="var">.e.f.</seg> ex quo ex .13. quinti ſic <lb/>ſe habebit tota <seg type="var">.a.d.</seg> ad totam <seg type="var">.e.h.</seg> ſicut <seg type="var">.c.d.</seg> <lb/>ad <seg type="var">.g.h.</seg> aut <seg type="var">.b.c.</seg> ad <seg type="var">.f.g.</seg> aut <seg type="var">.a.b.</seg> ad <seg type="var">.e.f.</seg> per-<lb/>mutando itaque propoſitum manifeſtum erit, ipſum autem productum <seg type="var">.a.b.</seg> in <seg type="var">.c.b.</seg> <lb/>æquale erit quadrato <seg type="var">.b.c.</seg> ex .15. fexti aut .20. ſeptimi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0047-01" corresp="fig-0047-01a"> <graphic url="0047-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="56">LVI</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">VEteres</hi> aliud quoque problema indeterminatum propoſuerunt, quod ex <lb/>more ratione à me definietur, eſt autem eiuſmodi.</s> </p> <p> <s xml:space="preserve">Quomodo propoſitus numerus in tres eiuſmodi partes diuidatur, vt <choice><ex>quadratum</ex><am>quadratũ</am></choice> <lb/>vnius æquale fit fummæ quadratorum reliquarum duarum partium.</s> </p> <p> <s xml:space="preserve">Hoc vt efficiamus tria quadrata ſeparata ſumamus, <choice><ex>quorum</ex><am>quorũ</am></choice> <choice><ex>vnum</ex><am>vnũ</am></choice> æquale ſit reliquis <lb/>duobus; </s> <s xml:space="preserve"><choice><ex>eorum</ex><am>eorũ</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> radices in ſummam ſimul colligantur, tum regulam de tribus ſe <lb/>quemur, ratione præcedenti theoremate demonſtrata, & rectè vt infra docebimus, <lb/>quod autem dico de quadratis, etiam de cubis, & quibuſuis dignitatibus aſſero.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi numerus diuiſibilis proponatur .30. in tres eiuſmodi partes di <lb/>uidendus, vt quadratum vnius æquale ſit ſummæ quadratorum reliquarum duarum <lb/>partium, in primis radices trium quadratorum ſumemus, ſic quomodocunque ſe <lb/>habentes, vt maius ipſorum æquale ſit ſummæ reliquorum duorum, verbi gratia .25. <lb/>16. et .9. nempe .5. 4. et .3. quæ ſi colligantur in ſummam efficiunt .12. </s> <s xml:space="preserve">Tum ex regu-<lb/>la de tribus dicemus, ſi .12. reſpondet .30: </s> <s xml:space="preserve">cui, 5. radix maior reſpondebit? </s> <s xml:space="preserve">nem-<lb/>pe .12. cum dimidio.</s> </p> <p> <s xml:space="preserve">Deinde ſi dixerimus ſi .12. valet .30. quid valebit .4. radix media? </s> <s xml:space="preserve">nempe vale-<lb/>bit .10. tertia autem minor .7. cum dimidio. </s> <s xml:space="preserve">Itaquetota ſumma erit .30. & quadra- <pb facs="0048" n="36"/><fw type="head">IO. BAPT. BENED.</fw> tum .12. cum dimidio erit .155. quod æquale erit ſummæ quadratorum duarum par <lb/>tium, nempe .100. cum .55.</s> </p> <p> <s xml:space="preserve">Hoc vt <choice><ex>demonſtremus</ex><am>demõſtremus</am></choice>, numerus diuiſibilis propoſitus ſignificetur linea <seg type="var">.a.d.</seg> & ſum <lb/>ma radicum, noſtro modo ſumptarum, linea <seg type="var">.e.h.</seg> quarum prima & maior ſit <seg type="var">.e.f.</seg> ſe-<lb/>cunda <seg type="var">.f.g.</seg> tertia <seg type="var">.g.h.</seg> cogitemus etiam lineam <seg type="var">.a.d.</seg> ea ratione diuiſam eſſe qua <seg type="var">.e.h.</seg> <lb/>patebit cnim ex modo præcedentis theorematis vnamquanque partium <seg type="var">.a.d.</seg> ita ſe <lb/>habituram ad ſuum totum ſicut ſe habent ſingulæ <seg type="var">.e.h.</seg> ad ſuum. </s> <s xml:space="preserve">Quod ideo dico, vt <lb/>intelligamus rectè nos dicere. </s> <s xml:space="preserve">Si <seg type="var">.e.h.</seg> dat <seg type="var">.a.d.</seg> ergo <seg type="var">.e.f.</seg> dabit <seg type="var">.a.b.</seg> <choice><ex>atque</ex><am>atq;</am></choice> ita de cæteris. <lb/></s> <s xml:space="preserve">Quare permutando ſic ſe habebit <seg type="var">.a.b.</seg> ad <seg type="var">.b.c.</seg> ſicut <seg type="var">.e.f.</seg> ad <seg type="var">.f.g.</seg> idem dico de reliquis. <lb/></s> <s xml:space="preserve">Igitur ex .18. ſexti aut .11. octaui, eadem erit proportio quadrati <seg type="var">.a.b.</seg> ad <choice><ex>quadratum</ex><am>quadratũ</am></choice> <seg type="var">.<lb/>b.c.</seg> quæ quadrati <seg type="var">.e.f.</seg> ad quadratum <seg type="var">.f.g.</seg> tota enim ſunt æqualia, cum eorum partes <lb/>ſimiles inter ſe ſunt æquales. </s> <s xml:space="preserve">Idem dico de proportione qu@drati <seg type="var">.a.b.</seg> nempe ita <lb/>ſe habere ad <seg type="var">.c.d.</seg> ſicut quadratum <seg type="var">.e.f.</seg> ad quadratum <seg type="var">.g.h.</seg> ex quo ex .24. quinti pro-<lb/>portio quadrati <seg type="var">.a.b.</seg> ad ſummam quadratorum duarum partium <seg type="var">.b.c.</seg> et <seg type="var">.c.d.</seg> ſic ſe ha <lb/>bebit ut quadrati <seg type="var">.e.f.</seg> ad ſummam quadra-<lb/>torum <seg type="var">.f.g.</seg> et <seg type="var">.g.h</seg>. </s> <s xml:space="preserve">At quadratum <seg type="var">.e.f.</seg> æquale <lb/> <ptr xml:id="fig-0048-01a" corresp="fig-0048-01" type="figureAnchor"/> eſt ſummæ quadratorum <seg type="var">.f.g.</seg> et <seg type="var">.g.h.</seg> igitur <lb/>ſic etiam ſe habebit quadratum <seg type="var">.a.b.</seg> nempe <lb/>æquale quadratis <seg type="var">.b.c.</seg> et <seg type="var">.c.g</seg>. </s> <s xml:space="preserve">Idipſum de cæ <lb/>teris dignitatibus dices, <choice><ex>vterisque</ex><am>vterisq́;</am></choice> .21. theoremate huius libri.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0048-01" corresp="fig-0048-01a"> <graphic url="0048-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="57">LVII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">SImile</hi> quoque problema ab antiquis indeterminatum proponitur, quod eiuſ-<lb/>modi eſt.</s> </p> <p> <s xml:space="preserve">An numerus aliquis in tres eiuſmodi partes di@idi poſſit, vt quadratum vnius æ-<lb/>quale ſit ſummæ quadratorum cæterarum duarum partium ſimul cum producto <lb/>vnius in alteram.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponatur numerus .50. vt iam dictum eſt diuidendus, repe <lb/>riendus erit alius quilibet numerus, qui tamen ſumma ſit trium radicum ſic ſe ha-<lb/>bentium, vt quadratum vnius æquale ſit ſummæ quadratorum duarum partium ſi-<lb/>mul cum producto vnius in alteram, eum autem qui primò occurrit ſumamus, utpo <lb/>tè .30. qui ſumma eſt numerorum .6. 10. 14. partium ſic ſe habentium, vt quadratum <lb/>ipſius .14. æquale ſit ſummæ quadratorum cæterarum partium ſimul cum produ-<lb/>cto vnius in alteram, agamusq́ue regula de tribus, ac dicamus, ſi .30. valet <num value="50">.<lb/>50.</num> quid valebit .14. nempe .23. cum tertia parte. </s> <s xml:space="preserve">Idem efficiemus in cæte-<lb/>ris partibus, quarum vna erit .16. cum duabus tertijs, altera verò .10. abſque @ractis, <lb/>ex quo quadratum primæ erit .544. cum .4. nonis, ſecundæ .277. cum ſeptem nonis, <lb/>tertiæ .100. & productum ſecundæ in tertiam .166. cum .6. nonis, quod productum, <lb/>cum quadratis ſecundæ & tertiæ collectum erit .544. cum .4. nonis.</s> </p> <p> <s xml:space="preserve">Huius rei ſpeculatio eadem eſt, quę fuit præcedentis theorematis vſquequo no-<lb/>ueris eandem proportionem eſſe quadrati <seg type="var">.a.b.</seg> ad ſummam quadratorum <seg type="var">.b.c.</seg> et <seg type="var">.c.<lb/>d.</seg> quæ quadrati <seg type="var">.e.f.</seg> ad ſummam quadratorum <seg type="var">.f.g.</seg> et <seg type="var">.g.h</seg>. </s> <s xml:space="preserve">Sed cum hic non demus <lb/>quadratum <seg type="var">.e.f.</seg> æquale ſummæ quadratorum <seg type="var">.f.g.</seg> et <seg type="var">.g.h.</seg> fed maius ex producto <seg type="var">.g.h.</seg> <lb/>in <seg type="var">.f.g.</seg> aut quod idem eſt, è contrario, ſubſequentes figuræ cogitandæ erunt, qua-<lb/>rum <seg type="var">.i.</seg> ſit quadratum <seg type="var">.a.b</seg>: l. ſit quadratum <seg type="var">.e.f</seg>: x. quadratum <seg type="var">.b.c</seg>: y. quadratum <seg type="var">.f.g</seg>: p. <lb/>quadratum <seg type="var">.c.d</seg>: q. quadratum <seg type="var">.g.h</seg>: k. ſit productum <seg type="var">.b.c.</seg> in <seg type="var">.c.d</seg>: m. ſit productum <seg type="var">.f.</seg> <pb facs="0049" n="37"/><fw type="head">THEOREM. ARITH.</fw> g. in <seg type="var">.g.h</seg>. </s> <s xml:space="preserve">Nunc ex ſpeculatione præcedentis theorematis, eadem erit proportio <seg type="var">.n.<lb/>t.</seg> ad <seg type="var">.o.u.</seg> quæ eſt <seg type="var">.n.s.</seg> ad <seg type="var">.o.r.</seg> </s> <s xml:space="preserve">quare pro-<lb/>ductum <seg type="var">.k.</seg> ex definitione ſimile erit <lb/> <ptr xml:id="fig-0049-01a" corresp="fig-0049-01" type="figureAnchor"/> producto <seg type="var">.m.</seg> cum vtraque ſint rectan-<lb/>gula, vnde proportio <seg type="var">.k.</seg> ad <seg type="var">.m.</seg> ad pro-<lb/>portionem <seg type="var">.n.t.</seg> ad <seg type="var">.o.u.</seg> ex .18. ſexti du-<lb/>pla erit. </s> <s xml:space="preserve">Igitur proportio <seg type="var">.k.</seg> ad <seg type="var">.m.</seg> æ-<lb/>qualis erit proportioni <seg type="var">.x.</seg> ad <seg type="var">.y.</seg> et <seg type="var">.p.</seg> <lb/>ad <seg type="var">.q.</seg> et <seg type="var">.i.</seg> ad <seg type="var">.l.</seg> & permutando ſic ſe ha-<lb/>bebit <seg type="var">.k.</seg> ad <seg type="var">.i.</seg> ſicut <seg type="var">.m.</seg> ad <seg type="var">.l.</seg> ſed <seg type="var">.x.p.</seg> ad <seg type="var">.i.</seg> <lb/>ſicſe habere probatum eſt vt <seg type="var">.y.q.</seg> ad <seg type="var">.l</seg>. <lb/></s> <s xml:space="preserve">Quare ex eadem .24. quinti ſic ſe habe <lb/>bit <seg type="var">.x.p.k.</seg> ad <seg type="var">.i.</seg> ſicut <seg type="var">.y.q.m.</seg> ad <seg type="var">.l.</seg> ſed <seg type="var">.y.q.<lb/>m.</seg> æqualis eſt <seg type="var">.l</seg>. </s> <s xml:space="preserve">Itaque <seg type="var">.x.p.k.</seg> pariter <seg type="var">.i.</seg> <lb/>æqualis erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0049-01" corresp="fig-0049-01a"> <graphic url="0049-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="58">LVIII</num>.</head> <p> <s xml:space="preserve">ALIVD quoque problema, nec tamen definitum, veteres propoſuerunt, <lb/>nempe an aliquis numerus in .4. eiuſmodi partes diuidi poſſit, vt ſumma qua-<lb/>dratorum duarum partium dupla ſit ſummæ quadratorum reliquarum duarum.</s> </p> <p> <s xml:space="preserve">Verum huius effectio & ſpeculatio non erit difficilis, <choice><ex>cum</ex><am>cũ</am></choice> ſit eadem quæ præmiſsis <lb/>proximè duobus theorematibus allata fuit, ſumpta nempe ſumma radicum quarun <lb/>cunque ſic ſe habentium, prout dictum fuit. </s> <s xml:space="preserve">Verbigratia .44. cuius partes erunt. <lb/>16. 12. 14. 2. <choice><ex>tunc</ex><am>tũc</am></choice> progrediemur regula de tribus dicentes. </s> <s xml:space="preserve">Si .44 numerum propoſi-<lb/>tum valet, quid .16. pars maior? </s> <s xml:space="preserve">nempe valebit partem maiorem numeri propoſi-<lb/>ti reſpondentem .16. idem de cæteris dico.</s> </p> <p> <s xml:space="preserve">Porrò ſpeculatio eadem eſt cum ſuperioribus.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="59">LIX</num>.</head> <p> <s xml:space="preserve">CVR diuidens propoſitum numerum in duas eiuſmodi partes, vt productum <lb/>radicum quadratarum ipſarum partium æquale ſit alteri numero propoſito, <lb/>cuius <choice><ex>tamen</ex><am>tamẽ</am></choice> quadratum maius <choice><ex>non</ex><am>nõ</am></choice> ſit quadrato dimidij primi numeri propoſiti. </s> <s xml:space="preserve">Rectè <lb/>ſecundum numerum propoſitum in ſeipſum multiplicat, & <choice><ex>eundem</ex><am>eundẽ</am></choice> ex quadrato di-<lb/>midij primi detrahit, <choice><ex>reſiduique</ex><am>reſiduiq́;</am></choice> quadratam radicem ſubtrahit ex dimidio ipſius pri-<lb/>mi, ex quo datur minor pars quæſita, quaipſi dimidio coniuncta, maior pars ha-<lb/>betur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponatur numerus, 20. propoſito modo, in duas partes <lb/>eiuſmodi diuidendus, vt productum radicum æquale ſit (verbigratia) 8. </s> <s xml:space="preserve">Dimi-<lb/>dium priminumeri in ſeipſum multiplicabimus, cuius quadratum erit .100. ex <lb/>quo quadratum ſecundi numeri, nempe .64. detrahemus, <choice><ex>remanebitque</ex><am>remanebitq́;</am></choice> .36. cuius radi <lb/>ce quadrata coniuncta .10. dimidio inquam primi numeri propoſiti, dabitur nume <lb/>rus .16. pars maior, & ſubtracta à dimidio, dabitur minor pars, nempe .4.</s> </p> <pb facs="0050" n="38"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Hoc vt demonſtremus, primus nu-<lb/> <ptr xml:id="fig-0050-01a" corresp="fig-0050-01" type="figureAnchor"/> merus linea <seg type="var">.a.b.</seg> ſignificetur, quam di-<lb/>uiſam cogitemus in puncto <seg type="var">.c.</seg> in partes <lb/>quæſitas, ex quo præſupponitur duas li-<lb/>neas <seg type="var">.a.c.</seg> et <seg type="var">.c.b.</seg> duo quadrata eſſe, quæ <lb/>in altera figura ſignificetur per <seg type="var">.d.</seg> et <seg type="var">.e.</seg> <lb/>productum autem radicum cognitum <seg type="var">.<lb/>f.</seg> quandoquidem datum eſt, cuius qua-<lb/>dratum æquale erit producto quadra-<lb/>torum <seg type="var">.d.e.</seg> adinuicem, nempe <seg type="var">.b.c.</seg> in <seg type="var">.a.c.</seg> ex .19. theoremate huius. </s> <s xml:space="preserve">Quod verbi <lb/>gratia ſit <seg type="var">.x.</seg> <choice><ex>itaque</ex><am>itaq;</am></choice> cognitum, quo facto, doctrinam .45. theorematis libri huius ſecuti, <lb/>propoſitum conſequemur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0050-01" corresp="fig-0050-01a"> <graphic url="0050-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="60">LX</num>.</head> <p> <s xml:space="preserve">CVR productum differentiæ duarum radicum in ſummam ipſarum, ſemper <lb/>differentia ſit quadratorum ipſarum radicum.</s> </p> <p> <s xml:space="preserve"><choice><ex>Exempli</ex><am>Exẽpli</am></choice> gratia, quoslibet duos numeros pro radicibus ſumpſerimus, vt potè .3. et <num value="5">.<lb/>5.</num> quorum differentia eſt .2. certè ſi differentiam hanc per ſummam radicum ſcili-<lb/>cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eſt <lb/>ſuorum quadratorum, nempeinter .9. et .25.</s> </p> <p> <s xml:space="preserve">Hoc vt ſpeculemur, duæ radices in linea <seg type="var">.n.i.</seg> ſignificentur, quarum vna ſit <seg type="var">.n.c.</seg> & <lb/>altera <seg type="var">.c.i.</seg> ipſarum autem differentia <seg type="var">.n.t.</seg> ex quo <seg type="var">.t.<lb/>c.</seg> æqualis erit <seg type="var">.c.i</seg>. </s> <s xml:space="preserve">Tum cogitato toto quadrato <seg type="var">.d.i.</seg> <lb/> <ptr xml:id="fig-0050-02a" corresp="fig-0050-02" type="figureAnchor"/> cum diametro <seg type="var">.d.i.</seg> <choice><ex>ductaque</ex><am>ductaq́</am></choice> parallela lateri <seg type="var">.n.d.</seg> à <lb/>puncto <seg type="var">.c.</seg> & altera à puncto <seg type="var">.t.</seg> & à puncto <seg type="var">.o.</seg> tertia <lb/>ipſi <seg type="var">.n.i.</seg> & à puncto <seg type="var">.a.</seg> quarta <seg type="var">.x.a.e.</seg> parallela ipſi <seg type="var">.<lb/>o.</seg> inueniemus <seg type="var">.b.n.</seg> productum eſſe differentiæ <seg type="var">.n.<lb/>t.</seg> in ſumma radicum <seg type="var">.n.i.</seg> & cum <seg type="var">.d.o.</seg> et <seg type="var">.a.o.</seg> ſint <lb/>quadrata radicum prædictarum: </s> <s xml:space="preserve">b.e. æquale erit <seg type="var">.<lb/>n.u.</seg> cum vtrunque horum productorum æquale ſit <seg type="var">.<lb/>x.u.</seg> ex quo gnomon <seg type="var">.e.d.u.</seg> æqualis erit producto <seg type="var">.<lb/>b.n.</seg> quod ſcire cupiebamus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0050-02" corresp="fig-0050-02a"> <graphic url="0050-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="61">LXI</num>.</head> <p> <s xml:space="preserve">CVR propoſitum aliquem numerum diuiſuri in duas eiuſmodi partes, vt diffe-<lb/>rentia radicum quadratarum æqualis ſit alteri numero propoſito, cuius ta-<lb/>men quadratum dimidij primi quadratum non excedat. </s> <s xml:space="preserve">Rectè ſecundum numerum <lb/>in ſeipſum multiplicant, productum verò ex primo numero detrahunt, <choice><ex>rurſusque</ex><am>rurſusq́;</am></choice> di <lb/>midium reſidui quadrant, & quadratum hoc ex quadrato dimidij primi ſubtrahunt, <lb/>atque ita radice quadrata reſidui, dimidio primi coniuncta, pars maior datur, qua <lb/>ex ipſo dimidio detracta, pars minor relinquitur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, propoſito numero .20. ita ut propoſitum eſt, diuidendo, nem-<lb/>pe vt differentia radicum quadratarum dictarum partium æqualis ſit binario, bina-<lb/>rium hocin ſeipſum multiplicabimus, cuius quadratum .4. è primo numero .20. de <pb facs="0051" n="39"/><fw type="head">THEOREM. AR IT.</fw> trahemus, <choice><ex>ſupereritque</ex><am>ſupereritq́;</am></choice> numerus .16. cuius dimidium ſcilicet .8. in ſeipſum multipli-<lb/>cabimus, <choice><ex>dabiturque</ex><am>dabiturq́;</am></choice> numerus .64. qui cum ex quadrato dimidij primi detractus fue-<lb/>rit, nempe ex .100. & reſiduo .36. radix quadrata nempe .6. coniuncta denario, di-<lb/>midio primi, dabit .16. partem maiorem, & ex denario detracta, partem minorem.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus <lb/>propoſitus ſigniſicetur linea <seg type="var">.x.y.</seg> pro voto diui-<lb/> <ptr xml:id="fig-0051-01a" corresp="fig-0051-01" type="figureAnchor"/> ſa in puncto <seg type="var">.c.</seg> et <seg type="var">.x.t.</seg> productum ſit ipſius <seg type="var">.x.<lb/>c.</seg> in <seg type="var">.c.y.</seg> pariter etiam <seg type="var">.q.p.</seg> ſit ſumma radicum <lb/>quadratarum, nempe <seg type="var">.q.g.</seg> ipſius <seg type="var">.t.c.</seg> et <seg type="var">.g.p.</seg> ip-<lb/>ſius <seg type="var">.c.y</seg>. </s> <s xml:space="preserve">Tum ſuper <seg type="var">.q.p.</seg> extruatur & diuidatur <lb/>quadratum <seg type="var">.q.u.</seg> ea ratione qua .41. theoremate <lb/>aut .29. diuiſimus, in quo ſanè quadrato, quadra <lb/>tum ipſius <seg type="var">.q.i.</seg> cernemus datæ differentiæ, & in <lb/>eo collocata quadrata <seg type="var">.x.c.</seg> et <seg type="var">.c.y.</seg> ita etiam & <lb/>rationem, qua cognoſcimus productum <seg type="var">.g.r.</seg> (vſi <lb/>modo .29. theorematis) cuius quidem <seg type="var">.g.r.</seg> qua-<lb/>dratum, ex .19. theoremate æquale erit produ-<lb/>cto <seg type="var">.x.t.</seg> ideo etiam <choice><ex>cognitum</ex><am>cognitũ</am></choice>, ac proinde cum no <lb/>uerimus <seg type="var">.x.y.</seg> ſi rationem ſequemur .45. theore <lb/>mate cognoſcemus non ſolum ratione .41. theoremate allata hocrectè perfici, ſed <lb/>hac etiam alia ratione.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0051-01" corresp="fig-0051-01a"> <graphic url="0051-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="62">LXII</num>.</head> <p> <s xml:space="preserve">CVR propoſitum numerum diuiſuri in duas eiuſmodi partes, vt differentia <lb/><choice><ex>ſuarum</ex><am>ſuarũ</am></choice> <choice><ex>radicum</ex><am>radicũ</am></choice> <choice><ex>quadratarum</ex><am>quadratarũ</am></choice> æqualis ſit alteri numero propoſito. </s> <s xml:space="preserve">Cuius <choice><ex>tamen</ex><am>tamẽ</am></choice> <choice><ex>qua- dratum</ex><am>qua-dratũ</am></choice> maius non ſit quadrato medietatis ipſius primi propoſiti numeri. </s> <s xml:space="preserve">Rectè <choice><ex>etiam</ex><am>etiã</am></choice> <lb/><choice><ex>quadratum</ex><am>quadratũ</am></choice> dimidij ſecundi numeri ex dimidio primi <choice><ex>detrahunt</ex><am>detrahũt</am></choice>, <choice><ex>reſiduique</ex><am>reſiduiq́;</am></choice> radicem per <lb/>ſecundum multiplicant, & productum ex dimidio primi detrahunt, vt reſiduum <lb/>pars quæſita minor ſit, & illud alterum totius reſiduum, pars maior.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi numerus .50. in <lb/>prædictas duas partes diuidendus pro-<lb/> <ptr xml:id="fig-0051-02a" corresp="fig-0051-02" type="figureAnchor"/> poneretur, & alter etiam .6. quadratum <lb/>dimidij ſecundi numeri eſſet .9. eo detra <lb/>cto ex dimidio primi, remaneret .16. cu <lb/>ius radix .4. ſcilicet per totum ſecundum <lb/>nempe .6. multiplicata, proferet .24. <lb/>quo producto ex dimidio primi detra-<lb/>cto, nempe .25. dabitur .1. pars minor, <lb/>maior <choice><ex>autem</ex><am>autẽ</am></choice> erit <choice><ex>reſidum</ex><am>reſidũ</am></choice> .50. hoc eſt .49. <lb/>radices autem erunt .1. et .7. differentes <lb/>inter ſe, numero ſenario.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0051-02" corresp="fig-0051-02a"> <graphic url="0051-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Hocvt ſciamus, duo numeri lineis <choice><ex>ſi- gnificentur</ex><am>ſi-gnificẽtur</am></choice>, primus linea .b: <choice><ex>ſecundus</ex><am>ſecũdus</am></choice> linea <seg type="var">.<lb/>c.</seg> duæ autem partes <seg type="var">.b.</seg> duobus quadra-<lb/>tis <seg type="var">.q.i.</seg> et <seg type="var">.i.d.</seg> notentur, eorum verò radi-<lb/>ces lineis <seg type="var">.a.g.</seg> et <seg type="var">.g.d.</seg> differentia porrò ip <lb/>ſi <seg type="var">.c.</seg> æqualis & co gnita ſit <seg type="var">.a.h.</seg> ex quo <seg type="var">.h.</seg> <pb facs="0052" n="40"/><fw type="head">IO. BAPT. BENED.</fw> g. æqualis erit <seg type="var">.g.d.</seg> tum productum <seg type="var">.a.g.</seg> in <seg type="var">.g.d.</seg> ſit <seg type="var">.a.i.</seg> et <seg type="var">.t.i.</seg> æqualis <seg type="var">.a.i.</seg> et <seg type="var">.l.i.</seg> pariter <lb/>ſecetur æqualis <seg type="var">.t.i.</seg> quæ omnia ex diametro <seg type="var">.q.d.</seg> cogitari poſſunt: </s> <s xml:space="preserve">erit igitur <seg type="var">.u.i.</seg> æ-<lb/>qualis <seg type="var">.i.d.</seg> <choice><ex>ſupereritque</ex><am>ſupereritq́;</am></choice> quadratum <seg type="var">.q.u.</seg> differentiæ <seg type="var">.a.h.</seg> cognitum, hoc verò cogi-<lb/>temus diuiſum eſſe in .4. partes æquales medijs diametris <seg type="var">.p.r.</seg> et <seg type="var">.n.e.</seg> </s> <s xml:space="preserve">quare <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> <lb/>partium cognoſcetur, & <choice><ex>quadratum</ex><am>quadratũ</am></choice> erit ipſius <seg type="var">.a.K.</seg> aut ipſius <seg type="var">.K.h.</seg> dimidij <seg type="var">.a.h</seg>. </s> <s xml:space="preserve">Quòd <lb/>ſi aliquod iſtorum quadratorum detrahere voluerimus, nempe <seg type="var">.n.r.</seg> ex dimidio ſum <lb/>mæ <seg type="var">.b.</seg> duorum quadratorum <seg type="var">.q.i.</seg> et <seg type="var">.i.d.</seg> cognitæ, hac via procedemus, primum con <lb/>ſiderabimus <seg type="var">.t.r.</seg> coniunctam <seg type="var">.t.i.</seg> quæ quantitates erunt ſumma dimidij <choice><ex>duorum</ex><am>duorũ</am></choice> qua-<lb/>dratorum <seg type="var">.q.i.</seg> et <seg type="var">.i.d.</seg> quando quidem <seg type="var">.t.r.</seg> <lb/><choice><ex>dimidium</ex><am>dimidiũ</am></choice> eſt quadrati <seg type="var">.t.l.</seg> et <seg type="var">.t.i.</seg> <choice><ex>dimidium</ex><am>dimidiũ</am></choice> <lb/> <ptr xml:id="fig-0052-01a" corresp="fig-0052-01" type="figureAnchor"/> gnomonis <seg type="var">.t.i.l.</seg> coniunctum dimidio <lb/>quadrati <seg type="var">.i.d.</seg> ex quo <seg type="var">.i.t.r.</seg> dimidium erit <seg type="var">.<lb/>b.</seg> ex qua quantitate <seg type="var">.i.t.r.</seg> cogitare debe <lb/>mus detrahi quadratum ipſius <seg type="var">.K.h.</seg> nem <lb/>pe <seg type="var">.n.r</seg>: </s> <s xml:space="preserve">quare quod ſupereſt cognitum <lb/>erit nempe <seg type="var">.y.s.</seg> cum <seg type="var">.n.i.</seg> ſed <seg type="var">.y.m.</seg> æqualis <lb/>eſt <seg type="var">.n.i.</seg> et <seg type="var">.y.m.</seg> cum <seg type="var">.y.s.</seg> conſtituunt qua-<lb/>dratum <seg type="var">.p.m</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <seg type="var">.p.m.</seg> quadratum & <lb/>conſequenter <seg type="var">.p.s.</seg> eius radix cognoſce-<lb/>tur, ita etiam & productum huius <seg type="var">.p.s.</seg> in <seg type="var">.<lb/>s.x.</seg> æqualis <seg type="var">.c.</seg> nempe <seg type="var">.p.x</seg>: <choice><ex>eſtque</ex><am>eſtq́;</am></choice> produ-<lb/>ctum huiuſmodi ſemper minus quantita <lb/>te <seg type="var">.r.t.i</seg>: per <seg type="var">.u.i.</seg> æquale quadrato minori <seg type="var">.<lb/>i.d</seg>. </s> <s xml:space="preserve">quare <seg type="var">.i.d.</seg> cognoſcetur, conſequen-<lb/>ter <seg type="var">.i.</seg> @q. tanquam reſiduum ex <seg type="var">.b.</seg> & eo-<lb/>rum radices quadratæ cognoſcentur <seg type="var">.a.<lb/>g.</seg> et <seg type="var">.g.d</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0052-01" corresp="fig-0052-01a"> <graphic url="0052-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="63">LXIII</num>.</head> <p> <s xml:space="preserve">IDEM præſtari hac alia via, meo iudicio poteſt. </s> <s xml:space="preserve">Secundus numerus in <choice><ex>ſuum</ex><am>ſuũ</am></choice> dimi <lb/><choice><ex>dium</ex><am>diũ</am></choice> multiplicetur, <choice><ex>productum</ex><am>productũ</am></choice> autem ex dimidio primi detrahatur, ex quo re-<lb/>manens erit productum vnius quadratæ radicis in alteram partium primi numeri <lb/>quæſitarum, deinde productum hoc duplicetur, & primo numero dato coniunga-<lb/>tur, <choice><ex>ſicque</ex><am>ſicq́;</am></choice> huius ſummæ quadrata radix erit ſumma radicum quadratarum dictarum <lb/>partium, cui iuncto producto ex quadrageſimoquinto theoremate ſingulæ radices <lb/>proferentur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, primus numerus diuiſibilis erat .50. alter verò .6. </s> <s xml:space="preserve">Iam ſi multi-<lb/>plicemus .6. per .3. nempe dimidium proferetur numerus .18. quo ex dimidio pri-<lb/>mi, nempe .25. detracto, ſupererit .7. productum vnius radicis in alteram, quod du <lb/>plicatum dabit .14. quo coniuncto cum primo numero .50. dabitur numerus .64. <lb/>cuius quadrata radix ſcilicet .8. erit ſumma radicum duarum partium quæſitarum, <lb/>qua & producto .7. ex quadrag eſimoquinto theoremate dictæ radices diſtinguen, <lb/>tur, quarum vna erit .7. & altera <seg type="var">.I</seg>.</s> </p> <p> <s xml:space="preserve">Vtautem hocſpeculemur, præcedenti figura vti poterimus, in qua patet <seg type="var">.t.r.</seg> pro <lb/>ductum eſſe ſecundi numeri <seg type="var">.c.</seg> nempe <seg type="var">.a.h.</seg> hoc eſt <seg type="var">.t.u.</seg> in dimidio <seg type="var">.a.e.</seg> ſcilicet <seg type="var">.p.t.</seg> re-<lb/>ſiduum autem dimidij primi <seg type="var">.b.</seg> eſſe <seg type="var">.t.i.</seg> nempe <seg type="var">.a.i.</seg> productum radicum, quod ſupple <pb facs="0053" n="41"/><fw type="head">THEOREM. ARITH.</fw> mentum eſt quadrati <seg type="var">.q.d.</seg> totalis. </s> <s xml:space="preserve">Quare duplicato <seg type="var">.a.i.</seg> & coniuncto <seg type="var">.b.</seg> cognoſci-<lb/>mustotum <seg type="var">.q.d.</seg> & conſequenter <seg type="var">.a.d.</seg> ſuam radicem, hoc eſt ſummam duarum radi <lb/>cum <seg type="var">.a.g.</seg> et <seg type="var">.g.d.</seg> quæ medio <seg type="var">.a.i.</seg> cognito, & quadrageſimoquinto theoremate ſingu-<lb/>læ cognoſcuntur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="64">LXIIII</num>.</head> <p> <s xml:space="preserve">CVR propoſitum aliquem num erum in duas eiuſmodi partes diuiſuri, vt ſum-<lb/>ma radicum dictarum partium æqualis ſit alteri numero propoſito. </s> <s xml:space="preserve">Rectè ſe-<lb/>cundum numerum in ſeipſum multiplicant, ex quo quadrato, primum datum nu-<lb/>merum detrahunt, <choice><ex>rurſusque</ex><am>rurſusq́;</am></choice> reſiduum in ſeipſum multiplicant, & ex eo quadrato <lb/>quartam partem deſumunt, <choice><ex>quam</ex><am>quã</am></choice> ex quadrato dimidij primi numeri detrahunt, radi-<lb/>cemq́ue qua dratam reſidui cum iunxerint, & ex dimidio primi numeri detraxerint, <lb/>partes quæſitæ proferuntur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponeretur primus numerus .20. diuidendus et .6. ſecundus <lb/>pro ſumma radicum, hunc ſecundum .6. in ſeipſum multiplicabimus, <choice><ex>dabiturque</ex><am>dabiturq́;</am></choice> nu-<lb/>merus .36. ex quo quadrato primus numerus detrahetur, <choice><ex>ſupereritque</ex><am>ſupereritq́;</am></choice> numerus .16. <lb/>qui quadratus dabit .256. cuius numeri quarta pars ſumetur, nempe .64. quæ ex qua <lb/>drato dimidij primi numeri detrahetur, nempe .100. <choice><ex>ſupereritque</ex><am>ſupereritq́;</am></choice> .36. cuius radix qua <lb/>drata .6. coniuncta & detracta ex .10. dabit .16. partem maiorem et .4. minorem.</s> </p> <p> <s xml:space="preserve">Cuius rei hæc ſpeculatio, primus numerus diuiſibilis ſignificetur linea <seg type="var">.a.b.</seg> diui-<lb/>ſa in puncto <seg type="var">.e.</seg> in partes adhuc incognitas, et <seg type="var">.a.c.</seg> ſit productum <seg type="var">.a.e.</seg> in <seg type="var">.e.b.</seg> item <seg type="var">.q.<lb/>p.</seg> ſecundum numerum ſignificet, æqualem ſummæ radicum, quæ puncto <seg type="var">.n.</seg> diſtin-<lb/>guantur. </s> <s xml:space="preserve">Poſtmodum totum quadratum <seg type="var">.p.d.</seg> erigatur (quod nobis eſt cognitum), <lb/>in duo quadrata diuiſum <seg type="var">.o.p.</seg> et <seg type="var">.o.d.</seg> quorum ſumma <seg type="var">.a.b.</seg> cum detur, cognita rema-<lb/>net ſumma <choice><ex>duorum</ex><am>duorũ</am></choice> <choice><ex>ſupplementorum</ex><am>ſupplementorũ</am></choice> <seg type="var">.o.u.</seg> et <seg type="var">.o.q.</seg> qua quadrata <choice><ex>cum</ex><am>cũ</am></choice> fuerit dabit quadru <lb/><choice><ex>plum</ex><am>plũ</am></choice> quadrati <choice><ex>ſupplementi</ex><am>ſupplemẽti</am></choice> <seg type="var">.o.q.</seg> <choice><ex>nempe</ex><am>nẽpe</am></choice> <choice><ex>quadruplum</ex><am>quadruplũ</am></choice> producti <seg type="var">.a.c.</seg> etenim <seg type="var">.a.c.</seg> ex .19. theo <lb/>remate huius libri quadratum eft ipſius <seg type="var">.q.o.</seg> <choice><ex>ſicque</ex><am>ſicq́;</am></choice> poterant etiam veteres quadrare <lb/>dimidium differentiæ <seg type="var">.a.b.</seg> ab <seg type="var">.p.d.</seg> nempe quadrato tantummodo ſupplemento <seg type="var">.q.<lb/>o</seg>. </s> <s xml:space="preserve">Tunc habito <seg type="var">.a.c.</seg> eius ope tanquam producti <seg type="var">.a.e.</seg> in <seg type="var">.e.b.</seg> ex .45. theoremate ſingu <lb/>læ partes cognoſcentur.</s> </p> <p> <s xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito ſupplemento <seg type="var">.<lb/>q.o.</seg> diſtinguendæ radices <seg type="var">q.n.</seg> et <seg type="var">.n.p.</seg> ex .45. theoremate, quibus cognitis, eorum <lb/>etiam quadrata cognoſcuntur.</s> </p> <figure place="here"> <graphic url="0053-01"/> </figure> <figure place="here"> <graphic url="0053-02"/> </figure> <pb facs="0054" n="42"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="65">LXV</num>.</head> <p> <s xml:space="preserve">CVR propoſito numero in tres qualeſcunque partes diuiſo, ſi prima in <lb/>tertiam multiplicetur, & huic producto, ſecundæ in primam productum <lb/>coniungatur, <choice><ex>itemque</ex><am>itemq́;</am></choice> ſecundæ in tertiam, hæc ſumma duplicata æqualis ſit ſummæ <lb/>productorum ſingularum in cæteras duas.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponatur .20. diuiſus in tres partes nempe .12. 5. 3. multipli-<lb/>cato primo .12. per .3. tertiam partem dabitur .36. ſecunda verò multiplicata per re <lb/>liquas duas, hoc eſt .5. per .12. et .3. in primis dabitur .60. poſtea .15. <choice><ex>quorum</ex><am>quorũ</am></choice> <choice><ex>trium</ex><am>triũ</am></choice> pro <lb/>ductorum ſumma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-<lb/>citur ſummæ productorum ſingularum partium in reliquas duas, nempe ſummæ .60. <lb/>36. 60. 15. 36. 15. hoc eſt ipſis .222.</s> </p> <p> <s xml:space="preserve">Cuius rei per ſe patet ſpeculatio, cum in his ſex vltimis productis, ſingula tria <lb/>prima duplicentur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="66">LXVI</num>.</head> <p> <s xml:space="preserve">CVR propoſito numero in .3. qualeſcunque partes diuiſo, ſi in reliquas duas ſin-<lb/>gulæ multiplicentur, & hæc producta cum ſumma ſuorum quadratorum con-<lb/>iungantur, tota ſumma hæc vltima æqualis erit quadrato totali propoſiti numeri.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi fuerit idem numerus .20. in .3. partes diuiſus .12. 5. 3. </s> <s xml:space="preserve">Si .12. in <lb/>5. et .3. producatur, ſumma productorum erit .96. at .5. in .12. et .3. erit .75. poſtmo-<lb/>dum .3. in .12. et .5. erit .51. nempe in vniuerſum .222. quadratorum porrò ſumma <lb/>erit .178 quæ coniuncta .222. dabit .400. quadratum ipſius .20.</s> </p> <p> <s xml:space="preserve">Erit autem huiuſce rei facillima ſpeculatio, ſi ſequentem figuram mente conce-<lb/>perimus, in qua <seg type="var">.a.b.</seg> propoſitum numerum ſignificet, cuius partes diſtinctæ ſint me-<lb/>dio <seg type="var">.e.</seg> et <seg type="var">.c</seg>. </s> <s xml:space="preserve">Ip ſum autem <seg type="var">.q.b.</seg> ſit quadratum <lb/>totale parallelis <seg type="var">.e.s.</seg> et <seg type="var">.c.x.</seg> diuiſum, quæ qua <lb/> <ptr xml:id="fig-0054-01a" corresp="fig-0054-01" type="figureAnchor"/> dratum in triarectangula diuident, quorum <lb/>primum erit <seg type="var">.q.e.</seg> compoſitum ex producto <seg type="var">.a.<lb/>e.</seg> in ſemetipſam, nempe quadratum <seg type="var">.o.e.</seg> & <lb/>ex producto eiuſdem <seg type="var">.a.e.</seg> in <seg type="var">.e.b.</seg> quod erit re <lb/>ctangulum <seg type="var">.o.s.</seg> ex quo tria rectangula <seg type="var">.o.s.</seg> et <seg type="var">.<lb/>n.x.</seg> et <seg type="var">.t.u.</seg> tria producta erunt ſingularum par <lb/>tium in cæteras duas, et <seg type="var">.e.o</seg>: <seg type="var">c.n</seg>: <seg type="var">b.t.</seg> tria qua-<lb/>drata erunt: </s> <s xml:space="preserve">quibus ſex quantitatibus quadra <lb/>tum totale <seg type="var">.q.b.</seg> completur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0054-01" corresp="fig-0054-01a"> <graphic url="0054-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="67">LXVII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">VEteres</hi> aliud quoque problema indefinitum propoſuerunt, quod tamen à <lb/>nobis determinabitur.</s> </p> <p> <s xml:space="preserve">Cur diuiſuri propoſitum numerum in duas eiuſmodi partes, vt mutuò diuiſis, & <lb/>per ſummam prouenientium diuiſa ſumma qua dratorum partium, oriatur proue-<lb/>niens alter numerus propoſitus.</s> </p> <p> <s xml:space="preserve">Propoſito deinde tertio quolibet numero diuidendo per ſingulas partes primi, <pb facs="0055" n="43"/><fw type="head">THEOREM. ARIT.</fw> ita vt ſimul prouenientibus in ſummam collectis huius fummæ ad primum nume-<lb/>rum propoſitum proportio futura ſit ea quæ eſt tertij ad ſecundum. </s> <s xml:space="preserve">Rectè dimidium <lb/>primi numeri in ſeipſum multiplicant, ex quo quadrato ſecundum numerum detra <lb/>hunt, tum reſidui radicem ſumunt, quam iungentes, & detrahentes ex dimidio <lb/>primi, partes quæſitas habent, cætera ex neceſsitate ſubſequuntur, prout nunc a <lb/>me docebitur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponitur numerus .20. in duas partes diuidendus, quibus po <lb/>ſtea mutuò diuiſis, & per ſummam prouenientium diuiſa ſumma quadratorum, <lb/>dent <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> numerum propoſitum .36. nam reliqua conſequuntur. </s> <s xml:space="preserve">Itaque .10. <lb/>dimidium primi in ſeipſum multiplicatur, & ex quadrato .100. eruitur numerus .36. <lb/>nempe ſecundus propoſitus reſidui porrò .64. quadrata radix .8. fumitur, quam con <lb/>iungimus & detrahimus ex dimidio primi ſcilicet .10. ex quo partes quæſitæ dabun <lb/>tur .18. et .2. quæ mutuo diuiſæ dabunt ſuorum prouenientium ſummam .9. cum no-<lb/>na parte, per quam diuidentes .328. ſummam quadratorum ipſarum partium, <lb/>exactè dabitur numerus .36. qui fuit ſecundò propoſitus. </s> <s xml:space="preserve">Tum ſi per ſingu-<lb/>las iam inuentas partes quilibet numerus diuiſus fuerit, verbi gratia .72. ſumma pro <lb/>uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. ſer <lb/>uabit, quam tertius propoſitus .72. cum ſecundo .36.</s> </p> <p> <s xml:space="preserve">Quod vt ſpeculemur, primus numerus ſignificetur linea <seg type="var">.n.e.</seg> ita diuidendus à <lb/>puncto <seg type="var">.o.</seg> vt diuiſa parte <seg type="var">.n.o.</seg> per <seg type="var">.o.e.</seg> et <seg type="var">.o.e.</seg> per <seg type="var">.n.o.</seg> & per ſummam prouenien-<lb/>tium diuiſa ſumma quadratorum <seg type="var">.n.o.</seg> et <seg type="var">.o.e.</seg> detur ſecundus numerus notatus linea <seg type="var">.<lb/>q.K</seg>. </s> <s xml:space="preserve">Porrò meminiſſe oportet quòd .26. theoremate probatum fuit vltimum hoc <lb/>proueniens æquale producto partium inter ſe futurum, nempe producto <seg type="var">.n.o.</seg> in <seg type="var">.o.<lb/>e.</seg> quod ſignificetur rectangulo <seg type="var">.n.e</seg>. </s> <s xml:space="preserve">Itaque datis <seg type="var">.n.e.</seg> et <seg type="var">.q.K.</seg> ſi .45. theorema conſu-<lb/>luerimus, partes <seg type="var">.n.o.</seg> et <seg type="var">.o.e.</seg> cognoſcemus.</s> </p> <p> <s xml:space="preserve">Proponitur deinde tertius quilibetnumerus, verbi gratia <seg type="var">.x.</seg> diuidendus per <seg type="var">.o.e.</seg> <lb/>et <seg type="var">.o.n.</seg> qui ſi diuidatur per <seg type="var">.o.e.</seg> dabit pro <lb/>ueniens <seg type="var">.b.o</seg>. </s> <s xml:space="preserve">Si verò per <seg type="var">.n.o.</seg> proueniens <lb/> <ptr xml:id="fig-0055-01a" corresp="fig-0055-01" type="figureAnchor"/> erit <seg type="var">.d.n.</seg> nunc aſſerimus <choice><ex>ſummam</ex><am>ſummã</am></choice> duorum <lb/>horum prouenientium, ſic primo nume-<lb/>ro <seg type="var">.n.e.</seg> dato proportionatam eſſe, ſicut <lb/>tertius <seg type="var">.x.</seg> <choice><ex>ſecundo</ex><am>ſecũdo</am></choice> <seg type="var">.q.K</seg>. </s> <s xml:space="preserve">Producatur enim li-<lb/>nea <seg type="var">.d.n.</seg> donec <seg type="var">.n.q.</seg> æqualis ſit <seg type="var">.o.b.</seg> ex <lb/>quo <seg type="var">.q.d.</seg> erit ſumma vltimò prouenien-<lb/>tium: </s> <s xml:space="preserve">item producatur <seg type="var">.e.n.</seg> donec <seg type="var">.n.u.</seg> æ-<lb/>qualis ſit <seg type="var">.o.e.</seg> <choice><ex>termineturque</ex><am>termineturq́</am></choice> rectangulum <seg type="var">.<lb/>q.u.</seg> quod tertio numero propoſito <seg type="var">.x.</seg> vt <lb/>patet, æquale erit, </s> <s xml:space="preserve">quare ex .15. ſexti aut .<lb/>20. ſeptimi eadem erit proportio <seg type="var">.d.n.</seg> ad <lb/><seg type="var">n.q.</seg> quæ <seg type="var">.u.n.</seg> nempe <seg type="var">.o.e.</seg> ad <seg type="var">.o.n.</seg> & com-<lb/>ponendo <seg type="var">.d.q.</seg> ad <seg type="var">.q.n.</seg> ſicut <seg type="var">.e.n.</seg> ad <seg type="var">.n.o.</seg> & <lb/>permutando <seg type="var">.d.q.</seg> ad <seg type="var">.e.n.</seg> quæ <seg type="var">.q.n.</seg> hoc eſt <seg type="var">.<lb/>b.o.</seg> ad <seg type="var">.o.n.</seg> nempe ſicut <seg type="var">.b.e.</seg> ad <seg type="var">.e.n.</seg> ſuperficialem, ex prima ſexti aut .18. vel .19. <lb/>ſeptimi, ſed rectangulum <seg type="var">.e.n.</seg> conſtitutum fuit æquale numero <seg type="var">.q.K</seg>. </s> <s xml:space="preserve">itaque verum <lb/>eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0055-01" corresp="fig-0055-01a"> <graphic url="0055-01"/> </figure> </div> </body> </floatingText> <pb facs="0056" n="44"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="68">LXVIII</num>.</head> <p> <s xml:space="preserve">CVR numero per numerum diuiſo, <choice><ex>productoque</ex><am>productoq́;</am></choice> duorum numerorum per pro-<lb/>ueniens multiplicato, quod vltimò productum eſt, diuiſi numeri ſemper qua <lb/>dratum exiſtat.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi diuidamus .10. per .2. proueniens erit .5. quo producto ex duo <lb/>bus numeris multiplicato, nempe .20. habe <lb/>bimus .100. quadratum numeri diuiſi.</s> </p> <figure place="here"> <graphic url="0056-01"/> </figure> <p> <s xml:space="preserve">Cuius gratia duo numeri ſint <seg type="var">.a.</seg> et <seg type="var">.e.</seg> por <lb/>rò <seg type="var">.a.</seg> per <seg type="var">.e.</seg> diuiſo detur <seg type="var">.u.</seg> tum <seg type="var">.o.</seg> produ-<lb/>ctum <seg type="var">.a.</seg> in <seg type="var">.e.</seg> eſſe conſtituatur, quo per <seg type="var">.u.</seg> <lb/>multiplicato dabitur <seg type="var">.x.</seg> quadratum <seg type="var">.a.</seg> pro-<lb/>ptereà quòd <seg type="var">.a.</seg> medium eſt proportionale <lb/>inter <seg type="var">.o.</seg> et <seg type="var">.u.</seg> ex .35. theoremate. </s> <s xml:space="preserve">itaque <lb/>ex .16. ſexti aut .20. ſeptimi, propoſiti veri-<lb/>tas eluceſcet.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="69">LXIX</num>.</head> <p> <s xml:space="preserve">CVR numero aliquo per duos alios multiplicato & diuiſo, ſi per horum duo-<lb/>rum productum, ſumma duorum primorum productorum diuiſa fuerit, vl-<lb/>timum proueniens, ſummæ duorum primorum prouenientium æquale ſit.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden <lb/>dus ſumma productorum crit .336. prouenientium autem .7. ſi igitur ſummam .336. <lb/>productorum per productum duorum ſecundorum numerorum nempe .48. diuiſe-<lb/>rimus, proueniens pariter erit .7.</s> </p> <p> <s xml:space="preserve">In cuius <choice><ex>gratiam</ex><am>gratiã</am></choice> primus numerus ſignificetur linea <seg type="var">.q.b.</seg> multiplicandus & diuiden-<lb/>dus numeris deſignatis per <seg type="var">.k.m.</seg> et <seg type="var">.y.m.</seg> productorum ſumma ſit <seg type="var">.k.z.</seg> prouenien-<lb/>tium autem <seg type="var">.a.e</seg>: et <seg type="var">.a.o.</seg> ex <seg type="var">.k.m.</seg> et <seg type="var">.o.e.</seg> ex <seg type="var">.y.m</seg>: tum productum <seg type="var">.k.m.</seg> in <seg type="var">.m.y.</seg> ſit <seg type="var">.f.<lb/>m</seg>. </s> <s xml:space="preserve">Dico quòd ſi <seg type="var">.k.z.</seg> per <seg type="var">.f.m.</seg> diuiſerimus proueni et <seg type="var">.a.e</seg>. </s> <s xml:space="preserve">Quod cum ſic fuerit, erit <lb/>quoque verum quòd diuiſa <seg type="var">.k.z.</seg> per <seg type="var">.a.e.</seg> proueniet <seg type="var">.f.m.</seg> numerus ſcilicet æqualis <lb/>numero <seg type="var">.f.m.</seg> ex .13. theoremate huius. </s> <s xml:space="preserve">Itaque quotieſcunque probauero quòd di-<lb/>uiſa <seg type="var">.k.z.</seg> per <seg type="var">.a.e.</seg> proueniat numerus æqualis ipſi <seg type="var">.f.m.</seg> propoſitum verum eſſe con <lb/>ſequetur. ex .13. theoremate. </s> <s xml:space="preserve">Quòd ſi proueniens ex diuiſione <seg type="var">.k.z.</seg> per <seg type="var">.a.e.</seg> æqua <lb/>le fuerit <seg type="var">.f.m.</seg> patet ex .7. quinti quòd <choice><ex>eadem</ex><am>eadẽ</am></choice> erit proportio numeri <seg type="var">.k.m.y.</seg> ad ipſum <lb/>proueniens, quæ ad numerum <seg type="var">.f.m</seg>. </s> <s xml:space="preserve">Cogitemus <choice><ex>itaque</ex><am>itaq;</am></choice> <seg type="var">.k.u.</seg> æqualem <seg type="var">.a.e.</seg> ſuper quam <lb/>mente concipiamus rectangulum <seg type="var">.u.p.</seg> æqualem <seg type="var">.k.z.</seg> ex quo eadem erit proportio <seg type="var">.<lb/>k.p.</seg> ad <seg type="var">.k.y.</seg> quæ <seg type="var">.g.k.</seg> ad <seg type="var">.k.u.</seg> ex .15. ſexti, aut, 20. ſeptimi, numerus autem <seg type="var">.k.p.</seg> erit <lb/>proueniens, quod probandum eſt æquale eſſe <seg type="var">.f.m</seg>.</s> </p> <p> <s xml:space="preserve">Probabitur autem ſic, ex .9. quinti, nempe demonſtrato quòd numerus <seg type="var">.k.p.</seg> ean <lb/>dem proportionem habeat ad numerum <seg type="var">.k.y.</seg> quam habet numerus <seg type="var">.f.m.</seg> ad eundem <lb/><seg type="var">k.y</seg>. </s> <s xml:space="preserve">Sed probatum eſt ſic ſe habere <seg type="var">.k.g.</seg> ad <seg type="var">.k.u.</seg> ſicut <seg type="var">.k.p.</seg> ad <seg type="var">.k.y.</seg> ſufficiet igitur pro-<lb/>bare ſic ſe habere <seg type="var">.k.g.</seg> ad <seg type="var">.k.u.</seg> ſicut <seg type="var">.f.m.</seg> ad <seg type="var">.k.y</seg>. </s> <s xml:space="preserve">Sed <seg type="var">.k.g.</seg> dicitur æqualis eſſe <seg type="var">.q.b</seg>: et <seg type="var">.k.</seg> <lb/>u; </s> <s xml:space="preserve">a.e. ſatis erit igitur probare ita ſe habere <seg type="var">.q.b.</seg> ad <seg type="var">.a.e.</seg> ſicut <seg type="var">.f.m.</seg> ad <seg type="var">.k.y</seg>. </s> <s xml:space="preserve">Scimus au-<lb/>tem quòd eadem eſt proportio <seg type="var">.q.b.</seg> ad <seg type="var">.a.o.</seg> quæ <seg type="var">.m.k.</seg> ad vnitatem, quæ ſit <seg type="var">.x.</seg> & quod <lb/>proportio <seg type="var">.o.e.</seg> ad <seg type="var">.q.b.</seg> eadem eſt, quæ <seg type="var">.x.</seg> ad <seg type="var">.m.y.</seg> ex definitione diuiſionis. </s> <s xml:space="preserve">Quare <lb/>ex æqualitate proportionum eadem erit proportio <seg type="var">.k.m.</seg> ad <seg type="var">.m.y.</seg> quæ <seg type="var">.e.o.</seg> ad <seg type="var">.o.a.</seg> & <pb facs="0057" n="45"/><fw type="head">THEOREM. ARIT.</fw> componendo ſic ſe habebit <seg type="var">.k.y.</seg> ad <seg type="var">.m.y.</seg> ſicut <seg type="var">.e.a.</seg> ad <seg type="var">.o.a.</seg> & permutando <seg type="var">.k.y.</seg> ad <seg type="var">.e.<lb/>a.</seg> ſicut <seg type="var">.m.y.</seg> ad <seg type="var">.o.a.</seg> & ex .19. quinti ita <seg type="var">.k.m.</seg> ad <seg type="var">.e.o.</seg> ſicut <seg type="var">.k.y.</seg> ad <seg type="var">.e.a.</seg> & permutando <seg type="var">.<lb/>k.m.</seg> ad <seg type="var">.k.y.</seg> ſicut <seg type="var">.e.o.</seg> ad <seg type="var">.e.a</seg>. </s> <s xml:space="preserve">Nunc producatur <seg type="var">.f.t.</seg> donec <seg type="var">.t.i.</seg> æqualis ſit <seg type="var">.k.y.</seg> <choice><ex>produ- ctaque</ex><am>produ-ctaq́;</am></choice> <seg type="var">.m.t.</seg> done <seg type="var">c.t.s.</seg> æqualis ſit vnitati <seg type="var">.x.</seg> <choice><ex>termineturque</ex><am>termineturq́;</am></choice> rectangulum <seg type="var">.s.i.</seg> ex quo da-<lb/>bitur proportio numeri <seg type="var">.f.m.</seg> ad numerum <seg type="var">.s.i.</seg> compoſita ex <seg type="var">.m.t.</seg> ad <seg type="var">.t.s.</seg> et <seg type="var">.f.t.</seg> ad <seg type="var">.t.i.</seg> <lb/>ex .24. ſexti, aut quinta octaui, ſed ita etiam proportio <seg type="var">.q.b.</seg> ad <seg type="var">.a.e.</seg> componitur ex <lb/>eiſdem proportionibus, nempe ex <seg type="var">.q.b.</seg> ad <seg type="var">.o.e.</seg> æquali <seg type="var">.m.t.</seg> ad <seg type="var">.t.s.</seg> & ex proportione <seg type="var">.<lb/>o.e.</seg> ad <seg type="var">.a.e.</seg> æquali <seg type="var">.f.t.</seg> ad <seg type="var">.t.i.</seg> ita que proportio numeri <seg type="var">.f.m.</seg> ad <seg type="var">.s.i.</seg> hoc eſt ad <choice><ex>numerum</ex><am>numerũ</am></choice> <lb/>ipſius <seg type="var">.k.y.</seg> ęqualis eſt proportioni numeri <seg type="var">.q.b.</seg> ad <seg type="var">.a.e.</seg> <choice><ex>nempe</ex><am>nẽpe</am></choice> <seg type="var">.k.g.</seg> ad <seg type="var">.k.u.</seg> hoc eſt <seg type="var">.k.p.</seg> ad <lb/><seg type="var">x.y.</seg> ex quo ſequitur <seg type="var">.k.p.</seg> conſtare numero ęquali <seg type="var">.f.m.</seg> proueniens igitur ex diuiſione <lb/>numeri <seg type="var">.k.z.</seg> per <seg type="var">.f.m.</seg> æquale eſt numero ipſius <seg type="var">.a.e</seg>.</s> </p> <figure place="here"> <graphic url="0057-01"/> </figure> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="70">LXX</num>.</head> <p> <s xml:space="preserve">HAEC porrò concluſio alia etiam via demonſtrari poteſt.</s> </p> <p> <s xml:space="preserve">Significetur numerus diuidendus atque multiplicandus linea <seg type="var">.b.a</seg>. </s> <s xml:space="preserve">Deinde <lb/>diuidentes & <choice><ex>multiplicantes</ex><am>multiplicãtes</am></choice> ſint <seg type="var">.k.m.</seg> et <seg type="var">.m.y.</seg> prouenientia ex diuiſione ſint <seg type="var">.a.o.</seg> et <seg type="var">.o.<lb/>e.</seg> atque <seg type="var">.a.o.</seg> ex <seg type="var">.m.y</seg>: <seg type="var">o.e.</seg> verò ex <seg type="var">.k.m.</seg> proueniat, quorum ſumma ſit <seg type="var">.a.e</seg>: productum <lb/>autem <seg type="var">.b.a.</seg> in <seg type="var">.k.m.</seg> ſit <seg type="var">.b.p.</seg> et <seg type="var">.p.s.</seg> productum <seg type="var">.b.a.</seg> in <seg type="var">.m.y.</seg> ad hæc rectangulum <seg type="var">.k.y.</seg> ſit <lb/>productum <seg type="var">.k.m.</seg> in <seg type="var">.m.y</seg>: quo to-<lb/>tum productum <seg type="var">.a.s.</seg> diuidatur, pro <lb/> <ptr xml:id="fig-0057-02a" corresp="fig-0057-02" type="figureAnchor"/> <choice><ex>ueniensque</ex><am>ueniensq́;</am></choice> ſit <seg type="var">.a.c.</seg> cui, <seg type="var">a.c</seg>: <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.<lb/>a.s.</seg> <choice><ex>eandem</ex><am>eãdẽ</am></choice> <choice><ex>proportionem</ex><am>proportionẽ</am></choice> ſeruabit, <choice><ex>quam</ex><am>quã</am></choice> <lb/><seg type="var">k.y.</seg> rectangulum ad vnitatem ex <lb/>definitione diuiſionis, hoc autem <lb/>proueniens <seg type="var">.a.c.</seg> <choice><ex>conſtare</ex><am>cõſtare</am></choice> numero æ-<lb/>quali aſſero ſummæ <seg type="var">.a.e</seg>. </s> <s xml:space="preserve">Primum <lb/>enim ex dicta definitione diuiſio-<lb/>nis habemus eandem eſſe propor-<lb/>tionem <seg type="var">.b.a.</seg> ad <seg type="var">.a.o.</seg> quæ <seg type="var">.m.y.</seg> ad <lb/>vnitatem, & quod ſic ſe habet <seg type="var">.b.a.</seg> <lb/>ad <seg type="var">.o.e.</seg> ſicut <seg type="var">.k.m.</seg> ad eandem vnita <lb/>tem. </s> <s xml:space="preserve">Itaque vnitas hæc linearis ſi-<lb/>gnificetur per <seg type="var">.m.x.</seg> in ſingulis late-<lb/>ribus <seg type="var">.k.m.</seg> et <seg type="var">.m.y.</seg> producentibus rectangulum <seg type="var">.k.y</seg>: ſuperficialis autem vnitas ſit. <pb facs="0058" n="46"/><fw type="head">IO. BAPT. BENED.</fw> <seg type="var">g.m.</seg> <choice><ex>cogiteturque</ex><am>cogiteturq́;</am></choice> rectangulum <seg type="var">.y.x.</seg> & rectangulum <seg type="var">.k.x</seg>. </s> <s xml:space="preserve">Itaque dabitur eadem pro <lb/>portio <seg type="var">.k.m.</seg> ad <seg type="var">.m.x.</seg> nempe <seg type="var">.k.x.</seg> rectanguli ad <seg type="var">.m.g.</seg> quæ eſt <seg type="var">.b.a.</seg> ad <seg type="var">.o.e.</seg> et <seg type="var">.y.x.</seg> ad <seg type="var">.m.<lb/>g.</seg> quæ <seg type="var">.b.a.</seg> ad <seg type="var">.a.o.</seg> ſed ex prima ſexti aut .18. vel .19. ſeptimi, ſic ſe habet rectangu-<lb/>lum <seg type="var">.k.y.</seg> ad <seg type="var">.x.y.</seg> ſicut <seg type="var">.k.m.</seg> ad <seg type="var">.m.x.</seg> </s> <s xml:space="preserve">quare ſicut <seg type="var">.b.a.</seg> ad <seg type="var">.o.e.</seg> ex .11. quinti, & eiuſdem <lb/>rectanguli <seg type="var">.k.y.</seg> ad rectangulum <seg type="var">.k.x.</seg> ſicut <seg type="var">.y.m.</seg> ad <seg type="var">.x.m.</seg> nempe <seg type="var">.b.a.</seg> ad <seg type="var">.a.o</seg>. </s> <s xml:space="preserve">Quare <lb/>ex communi ſcientia, ſic ſe habebit duplum rectanguli <seg type="var">.k.y.</seg> ad ſummam <seg type="var">.y.x.</seg> cum <seg type="var">.<lb/>k.x.</seg> rectangulorum, ſicut duplum <seg type="var">.b.a.</seg> ad ſummam <seg type="var">.a.o.e.</seg> et proportio ſummæ re-<lb/>ctangulorum <seg type="var">.y.x.</seg> et <seg type="var">.k.x.</seg> duplo <seg type="var">.g.m.</seg> ſicut duplum <seg type="var">.b.a.</seg> ad <seg type="var">.a.o.e</seg>. </s> <s xml:space="preserve">Igitur ſumma duo-<lb/>rum rectangulorum <seg type="var">.y.x.</seg> et <seg type="var">.x.k.</seg> media proportionalis erit inter duplum rectanguli <seg type="var">.<lb/>k.y.</seg> & duplum vnitatis ſuperſicialis <seg type="var">.g.m</seg>. </s> <s xml:space="preserve">Nunc terminetur rectangulum <seg type="var">.a.r.</seg> ex quo <lb/>dabitur eadem proportio dupli <seg type="var">.a.s.</seg> ad <seg type="var">.a.r.</seg> ſicut dupli <seg type="var">.b.a.</seg> ad <seg type="var">.a.e.</seg> ex propoſitioni-<lb/>bus notatis, ſexti aut ſeptimi. </s> <s xml:space="preserve">Quare etiam ſicut dupli rectanguli <seg type="var">.k.y.</seg> ad <choice><ex>ſummam</ex><am>ſummã</am></choice> <lb/>rectangulorum <seg type="var">.y.x.</seg> et <seg type="var">.k.x</seg>. </s> <s xml:space="preserve">Iam verò ſi conſtituatur <seg type="var">.e.c.</seg> pro vnitate lineari ipſius <seg type="var">.<lb/>e.r.</seg> certi erimus numerum <seg type="var">.a.c.</seg> æqualem eſſe <seg type="var">.a.e.</seg> & proportionem <seg type="var">.r.e.</seg> ad <seg type="var">.e.c.</seg> hoc <lb/>eſt <seg type="var">.a.r.</seg> ad <seg type="var">.a.c.</seg> eandem quæ <seg type="var">.y.x.</seg> et <seg type="var">.x.k.</seg> rectangulorum ad <seg type="var">.m.g.</seg> ex prædictis rationi-<lb/>bus, & ex hypotheſi, nempe quòd <seg type="var">.<lb/>e.r.</seg> æqualis ſit numero <seg type="var">.k.m.y.</seg> <lb/> <ptr xml:id="fig-0058-01a" corresp="fig-0058-01" type="figureAnchor"/> hoc eſt rectangulorum <seg type="var">.y.x.</seg> et <seg type="var">.x.<lb/>k</seg>. </s> <s xml:space="preserve">Quamobrem <seg type="var">.a.r.</seg> ex communi <lb/>ſcientia <choice><ex>medium</ex><am>mediũ</am></choice> proportionale erit <lb/>inter duplum <seg type="var">.a.s.</seg> & duplum <seg type="var">.a.c.</seg> <choice><ex>ea demque</ex><am>eadẽq́;</am></choice> <choice><ex>proportio</ex><am>ꝓportio</am></choice> dupli prędicti <seg type="var">.a.s.</seg> ad <lb/>duplum <seg type="var">.a.c.</seg> ex æqualitate propor-<lb/>tionum ſimul collectarum, eadem <lb/>erit qùæ proportio dupli rectangu-<lb/>li <seg type="var">.k.y.</seg> ad duplum <seg type="var">.m.g.</seg> hoc eſt <seg type="var">.a.s.</seg> <lb/>ſimplicis ad ſimplicem <seg type="var">.a.c.</seg> quæ ſim <lb/>plicis rectanguli <seg type="var">.k.y.</seg> ad ſimplicem <lb/>vnitatem <seg type="var">.g.m.</seg> ſic enim ſe habet ſim <lb/>plex ad ſimplex, ſicut duplum ad <lb/>duplum. </s> <s xml:space="preserve">Sed pariter ita ſe habet <seg type="var">.a.s.</seg> ad <seg type="var">.a.</seg> c<unclear reason="illegible"/>. cogitato <seg type="var">.a.c.</seg> tamquam proueniente <lb/>ex diuiſione <seg type="var">.a.s.</seg> per rectangulum <seg type="var">.k.y.</seg> vt conſtitutum eſt, ſicut <seg type="var">.k.y.</seg> ad <seg type="var">.m.g.</seg> ex defi-<lb/>nitione diuiſionis vt iam dictum eſt, </s> <s xml:space="preserve">quare numerus <seg type="var">.a.c.</seg> æqualis erit numero <seg type="var">.a.o.e</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0057-02" corresp="fig-0057-02a"> <graphic url="0057-02"/> </figure> <figure xml:id="fig-0058-01" corresp="fig-0058-01a"> <graphic url="0058-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="71">LXXI</num>.</head> <p> <s xml:space="preserve">CVR propoſitis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-<lb/>dis, ſi <choice><ex>adinuicem</ex><am>adinuicẽ</am></choice> diuiſi fuerint, <choice><ex>duoque</ex><am>duoq́;</am></choice> <choice><ex>prouenientia</ex><am>proueniẽtia</am></choice> <choice><ex>inuicem</ex><am>inuicẽ</am></choice> multiplicata <choice><ex>quenuis</ex><am>quẽuis</am></choice> nu <lb/>merum producant, qui ſeruetur, ſi deinde ijdem numeri verſa vice mutuo diuiſi fue <lb/>rint, & inter ſe multiplicata prouenientia, <choice><ex>productum</ex><am>productũ</am></choice> hoc, primo ſeruato numero <lb/>æquale erit.</s> </p> <p> <s xml:space="preserve">Exempli gratia propoſitis his .4. numeris .20. 30. 5. 10. duo autem .20. ſcilicet <lb/>et .30. ſint numeri diuidendi, porrò .5. et .10. numeri diuidentes, <choice><ex>nempe</ex><am>nẽpe</am></choice> vt primo .20 <lb/>per .5. diuidatur, tum .30. per .10. producetur .4. et .3. qui ſimul multiplicati <choice><ex>proferent</ex><am>proferẽt</am></choice> <num value="12">.<lb/>12.</num> tum .20. per .10. d iuiſo et .30. per .5. prouenientia erunt .2. 6. quæ inter ſe multi-<lb/>plicata producent etiam .12.</s> </p> <pb facs="0059" n="47"/> <fw type="head">THEOR. ARITH.</fw> <p> <s xml:space="preserve">Cuius rationem ſi quæris, ſignificentur .4. numeri lineis, <seg type="var">a.e.o.u.</seg> <choice><ex>diuidaturque</ex><am>diuidaturq́;</am></choice> .2. <lb/>per <seg type="var">.o.</seg> & <choice><ex>oriatur</ex><am>oriat̃</am></choice>. s. & per <seg type="var">.u.</seg> <choice><ex>oriatur</ex><am>oriat̃</am></choice> <seg type="var">.y.</seg> et <seg type="var">.<lb/> <ptr xml:id="fig-0059-01a" corresp="fig-0059-01" type="figureAnchor"/> e.</seg> diuiſo per <seg type="var">.o.</seg> oriatur <seg type="var">.z.</seg> & per <seg type="var">.u.</seg> <lb/>proueniat <seg type="var">.f.</seg> tum <seg type="var">.n.</seg> ſit productum <seg type="var">.z.</seg> <lb/>in <seg type="var">.y.</seg> et <seg type="var">.m.</seg> productum <seg type="var">.s.</seg> in <seg type="var">.f</seg>. </s> <s xml:space="preserve">Dico <lb/>n. futurum æquale <seg type="var">.m</seg>. </s> <s xml:space="preserve">Sit deinde <seg type="var">.<lb/>x.</seg> vnitas, quare ex definitione diui-<lb/>ſionis eadem erit proportio <seg type="var">.s.</seg> ad <seg type="var">.a.</seg> <lb/>et <seg type="var">.z.</seg> ad <seg type="var">.e.</seg> quæ <seg type="var">.x.</seg> ad <seg type="var">.o</seg>. </s> <s xml:space="preserve">Sed ita ſe ha-<lb/>bet <seg type="var">.a.</seg> ad <seg type="var">.y.</seg> et <seg type="var">.e.</seg> ad <seg type="var">.f.</seg> ſicut <seg type="var">.u.</seg> ad <seg type="var">.x.</seg> ex <lb/>quo ſic ſe habebit <seg type="var">.s.</seg> ad <seg type="var">.a.</seg> ſicut <seg type="var">.z.</seg> ad <lb/>e. et <seg type="var">.a.</seg> ad. y, ſicut <seg type="var">.e.</seg> ad <seg type="var">.f</seg>. </s> <s xml:space="preserve">Itaque ex <lb/>æqualitate proportionum ſic ſe ha-<lb/>bebit s. ad <seg type="var">.y.</seg> ſicut <seg type="var">.z.</seg> ad <seg type="var">.f</seg>. </s> <s xml:space="preserve">Igitur ex <lb/>15. ſexti aut .20. ſeptimi productum <seg type="var">.<lb/>n.</seg> producto <seg type="var">.m.</seg> æquale erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0059-01" corresp="fig-0059-01a"> <graphic url="0059-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="72">LXXII</num>.</head> <p> <s xml:space="preserve">ALIVD quoque problema à me inuentum eſt, nempe vt proponantur .4. <lb/>numeri qualeſcunque tandem, quorum duo diuiſibiles ſint, tertius diuiſor <lb/>vnius è duobus pro libito, <choice><ex>quæramusque</ex><am>quæramusq́;</am></choice> alterius diuidentem, qui ſic ſe habeat vt pro <lb/>ductum duorum prouenientium quarto numero propoſito ſit æquale.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponuntur .4. numeri .20. 48. 5. 12. porrò .20. et .48. numeri <lb/>ſint diuiſibiles et .5. <choice><ex>diuidens</ex><am>diuidẽs</am></choice> vnius, ut potè .20. </s> <s xml:space="preserve"><choice><ex>Quærendus</ex><am>Quærẽdus</am></choice> nunc erit diuidens alterius <lb/>nempe .48. eiuſmodi vt productum prouenientium æquale ſit .12. </s> <s xml:space="preserve">Diuidam itaque <num value="20">.<lb/>20.</num> per .5. <choice><ex>prouenietque</ex><am>prouenietq́;</am></choice> 4. quem per .48. multiplicabo, nempe per alterum diuiſibi-<lb/>lem, <choice><ex>ſicque</ex><am>ſicq́;</am></choice> proueniet .192. quod productum per quartum numerum nempe .12. diui-<lb/>fum dabit .16. qui erit diuidens quæſitus, quo diuiſo .48. proueniet .3. ſecundum ſci <lb/>licet proueniens, quo per alterum hoc eſt .4. multiplicato producetur quartus nu-<lb/>merus .12.</s> </p> <p> <s xml:space="preserve">Quod vt ſciamus, primus nume-<lb/>rus diuiſibilis ſignificetur <choice><ex>rectangulo</ex><am>rectãgulo</am></choice> <seg type="var">.<lb/> <ptr xml:id="fig-0059-02a" corresp="fig-0059-02" type="figureAnchor"/> a.i.</seg> ſecundus rectangulo <seg type="var">.o.u.</seg> primus <lb/>diuidens latere <seg type="var">.a.e.</seg> quartum nume-<lb/>rum rectangulo <seg type="var">.i.o.</seg> primum proue-<lb/>niens latere <seg type="var">.e.i.</seg> ſecundus diuidens la <lb/>tere <seg type="var">.e.u.</seg> (hic autem eſt quem quæri-<lb/>mus) tum alterum proueniens ſigni <lb/>ficetur latere <seg type="var">.e.o</seg>. </s> <s xml:space="preserve">Iam <choice><ex>eadem</ex><am>eadẽ</am></choice> erit pro-<lb/>portio <seg type="var">.e.i.</seg> ad <seg type="var">.e.u.</seg> quæ <seg type="var">.o.i.</seg> ad <seg type="var">.o.u.</seg> <lb/>Sed cum cognitæ ſint tres quantita-<lb/>tes <seg type="var">.e.i</seg>: <seg type="var">i.o</seg>: et <seg type="var">.o.u.</seg> quarta quoque. e<unclear reason="illegible"/> <seg type="var">.u.</seg> exregula de tribus immediatè cognoſcetur, <lb/>cætera in ſubſcripta figura facillimè patebunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0059-02" corresp="fig-0059-02a"> <graphic url="0059-02"/> </figure> </div> </body> </floatingText> <pb facs="0060" n="48"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="73">LXXIII</num>.</head> <p> <s xml:space="preserve">HOC etiam problema à me inuentum eſt, nempe ſi duæ radices quadratæ in <lb/>ſummam collectæ fuerint, & ex dimidio eiuſmodi ſummæ detracta fuerit mi <lb/>nor radix, <choice><ex>reſiduique</ex><am>reſiduiq́;</am></choice> quadratum duplicatum <choice><ex>eique</ex><am>eiq́;</am></choice> ſummæ coniungatur du-<lb/>plum producti ipſius reſidui in dimidium ſummæ radicum, atque huic ſummæ du-<lb/>plum producti eiuſdem reſidui in radicem minorem coniunctum fuerit; </s> <s xml:space="preserve">vltima hæc <lb/>ſumma differentia erit duorum quadratorum propoſitorum.</s> </p> <p> <s xml:space="preserve">Exempli gratia duæ radices quadraræ ſint .5. et .11. harum ſumma erit .16. & dimi <lb/>dium .8. differentia minoris ab ipſo dimidio erit .3: duplum quadrati huius differen <lb/>tiæ erit .18: </s> <s xml:space="preserve">duplum producti huius differentię in dimidium ſummę radicum erit .48. <lb/>item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium <lb/>ſumma erit .96. tantaq́ue erit differentia ſuorum quadratorum, quorum vnum <lb/>erit .25. alterum verò .121.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ſcientia, duæ quadratæ radices ſint <seg type="var">.h.o.</seg> et <seg type="var">.o.d.</seg> directæ inter ſe con-<lb/>iunctæ, quæ ſumma per medium in puncto <seg type="var">.e.</seg> diuidatur, tum cogitetur <seg type="var">.e.b.</seg> æqualis <lb/><seg type="var">o.e.</seg> perpendicularis <seg type="var">.h.d.</seg> <choice><ex>ducanturque</ex><am>ducanturq́;</am></choice> lineæ <seg type="var">.b.h</seg>: <seg type="var">b.o.</seg> et <seg type="var">.b.d</seg>. </s> <s xml:space="preserve">Iam ex .4. primi <seg type="var">.b.h.</seg> æqua <lb/>lis erit <seg type="var">.b.d.</seg> & quadratum <seg type="var">.b.h.</seg> æquale quadrato <seg type="var">.h.o.</seg> & quadrato <seg type="var">.o.b.</seg> ſimul cum du <lb/>plo producti <seg type="var">.o.e.</seg> in <seg type="var">.o.h.</seg> ex .12. ſecundi Eucli. </s> <s xml:space="preserve">Sed ex .13. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> quadratum <seg type="var">.b.d.</seg> <lb/>minus eſt quadrato <seg type="var">.o.d.</seg> cum quadrato <seg type="var">.o.b.</seg> ex duplo producti <seg type="var">.o.e.</seg> in <seg type="var">.o.d.</seg> at duplum <lb/>eiuſmodi producti æquale eſt duplo qua-<lb/>drati <seg type="var">.o.e.</seg> & duplo producti <seg type="var">.o.e.</seg> in <seg type="var">.e.d.</seg> ex <lb/> <ptr xml:id="fig-0060-01a" corresp="fig-0060-01" type="figureAnchor"/> tertia eiuſdem, itaque duo quadrata ſcili-<lb/>cet <seg type="var">.o.b.</seg> et <seg type="var">.o.d.</seg> maiora erunt duobus qua-<lb/>dratis, nempe <seg type="var">.o.b.</seg> et <seg type="var">.o.h.</seg> collectis cum du <lb/>plo producti <seg type="var">.o.e.</seg> in <seg type="var">.o.h.</seg> ex duplo quadrati <lb/><seg type="var">o.e.</seg> vna <choice><ex>cum</ex><am>cũ</am></choice> duplo producti <seg type="var">.o.e.</seg> in <seg type="var">.e.d</seg>. </s> <s xml:space="preserve">Qua <lb/>re <choice><ex>differentia</ex><am>differẽtia</am></choice> ſummæ duorum quadratorum <lb/><seg type="var">o.b.</seg> et <seg type="var">.o.d.</seg> à ſumma duorum <seg type="var">o.b.</seg> et <seg type="var">.o.h.</seg> du <lb/>plum erit quadrati <seg type="var">.o.e.</seg> cum duplo produ-<lb/>cti <seg type="var">.o.e.</seg> in <seg type="var">.e.d.</seg> & duplo producti <seg type="var">.o.e.</seg> in <seg type="var">.o.h.</seg> <lb/>Quòd ſi ex ſingulis duabus ſummis quadratorum demptum fuerit quadratum <seg type="var">.o.b.</seg> <lb/>eadem producta & quadrata ipſius <seg type="var">.o.e.</seg> remanebunt, tanquam differentia duorum <lb/>quadratorum <seg type="var">.o.u.</seg> et <seg type="var">.h.c</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0060-01" corresp="fig-0060-01a"> <graphic url="0060-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="74">LXXIIII</num>.</head> <p> <s xml:space="preserve">CVR ſumma duorum <choice><ex>extremorum</ex><am>extremorũ</am></choice> quatuor terminorum <choice><ex>proportionalium</ex><am>proportionaliũ</am></choice> arith-<lb/>meticè, æqualis eſt ſummæ duorum mediorum, vbi nota hac in re neceſſa-<lb/>rium non eſſe proportionalitatem continuam exiſtere.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea <lb/>dem eſſet primi ad ſecundum quæ tertij ad quartum, ſumma primi cum quarto eſſet <lb/>26. <choice><ex>tantaque</ex><am>tantaq́;</am></choice> ſecundi cum tertio.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis cauſa, primus <choice><ex>maiorque</ex><am>maiorq́;</am></choice> numerus ſignificetur linea <seg type="var">.e.o.</seg> ſecun-<lb/>dus <seg type="var">.s.q.</seg> tertius <seg type="var">.u.c.</seg> quartus <seg type="var">.g.t.</seg> differentia porrò inter <seg type="var">.e.o.</seg> et <seg type="var">.s.q.</seg> ſit <seg type="var">.i.o.</seg> quæ æqualis <lb/>erit differentiæ <seg type="var">.r.c.</seg> qua quartus à tertio ſuperatur ex hypotheſi. </s> <s xml:space="preserve">Itaque aſſero ſum <lb/>mam <seg type="var">.e.o.</seg> cum <seg type="var">.g.t.</seg> nempe <seg type="var">.a.o.</seg> æqualem eſſe ſummę <seg type="var">.q.s.</seg> et <seg type="var">.u.c.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.q.p</seg>. </s> <s xml:space="preserve">Nam in <seg type="var">.a.o.</seg> <pb facs="0061" n="49"/><fw type="head">THEOREM. ARIT.</fw> Secundus tertiusq́ue terminus reperiuntur, eſt <lb/> <ptr xml:id="fig-0061-01a" corresp="fig-0061-01" type="figureAnchor"/> enim ſecundus <seg type="var">.e.i.</seg> tertius <seg type="var">.i.o.</seg> et <seg type="var">.e.a.</seg> quando-<lb/>quidem ex præſuppoſito <seg type="var">.e.i.</seg> æqualis eſt <seg type="var">.s.q.</seg> et <lb/><seg type="var">i.o.</seg> æqualis <seg type="var">.r.c.</seg> et <seg type="var">.a.e.</seg> cum ſit æqualis <seg type="var">.g.t.</seg> cui <lb/>pariter æqualis eſt <seg type="var">.r.u.</seg> ex quo <seg type="var">.a.e.</seg> æqualis <lb/>eſt <seg type="var">.u.r</seg>. </s> <s xml:space="preserve">Itaque illud ſequitur <seg type="var">.a.o.</seg> ipſi <seg type="var">.q.p.</seg> <lb/>æqualem eſſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0061-01" corresp="fig-0061-01a"> <graphic url="0061-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="75">LXXV</num>.</head> <p> <s xml:space="preserve">CVR ſumma duorum terminorum extremorum imparium arithmeticæ pro-<lb/>portionalitatis ſemper duplo medij termini æqualis eſt.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſunt hitres termini proportionalitatis arithmeticæ .20. 15. 10 <lb/>ſumma duorum extremorum erit .30. quæ duplum eſt medij termini .15.</s> </p> <p> <s xml:space="preserve">Quod vt ſpeculemur, tres termini, tribus lineis <seg type="var">.b.d</seg>: <seg type="var">n.u.</seg> et <seg type="var">.q.p.</seg> <choice><ex>ſignificentur</ex><am>ſignificẽtur</am></choice>. </s> <s xml:space="preserve">Di-<lb/>co nunc quòd ſumma <seg type="var">.b.d.</seg> cum <seg type="var">.q.p.</seg> nempe <seg type="var">.<lb/>h.d.</seg> ſemper duplo <seg type="var">.n.u.</seg> ſcilicet <seg type="var">.g.u.</seg> æqualis <lb/> <ptr xml:id="fig-0061-02a" corresp="fig-0061-02" type="figureAnchor"/> erit. </s> <s xml:space="preserve">Tum differentia <seg type="var">.b.d.</seg> ad <seg type="var">.n.u.</seg> ſit <seg type="var">.c.d.</seg> quæ <lb/>æqualis erit <seg type="var">.e.u.</seg> differentiæ inter <seg type="var">n.u.</seg> et <seg type="var">.q.p.</seg> <lb/>patet enim in linea <seg type="var">.h.d</seg>: <seg type="var">b.c.</seg> æqualem eſſe <seg type="var">.n.<lb/>u.</seg> ſed <seg type="var">.n.u.</seg> ex <seg type="var">.n.e.</seg> componitur æquali <seg type="var">.q.p.</seg> et <lb/>ex <seg type="var">.e.u.</seg> æquali <seg type="var">.c.d.</seg> cum <choice><ex>itaque</ex><am>itaq;</am></choice> in <seg type="var">.h.d.</seg> partem <seg type="var">.<lb/>h.b.</seg> reperiamus æqualem <seg type="var">.n.e.</seg> gratia <seg type="var">.q.p.</seg> & <lb/>partem <seg type="var">.c.d.</seg> æquale <seg type="var">m.e.u.</seg> manifeſtum erit <lb/><seg type="var">h.d.</seg> æqualem eſſe <seg type="var">.g.u</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0061-02" corresp="fig-0061-02a"> <graphic url="0061-02"/> </figure> </div> </body> </floatingText> </div> <div type="unknown"> <head rend="italics" xml:space="preserve">BINA PROBLEMAT A EX DVOBVS PRAEDICTIS <lb/>THEOREMATIBVS DEPENDENTIA.</head> <p> <s xml:space="preserve">EX duobus prædictis theorematibus duo problemata oriuntur, <choice><ex>quorum</ex><am>quorũ</am></choice> primum <lb/>eſt. </s> <s xml:space="preserve">Datis tribus quantitatibus cognitis, ſi quis quartam inuenire voluerit, <lb/>quæ eiuſmodi ſit reſpectu tertiæ, qualis eſt ſecunda reſpectu primæ, ſecunda cum <lb/>tertia in ſummam colligenda erit, ex qua detracta prima, ſupererit quarta.</s> </p> <p> <s xml:space="preserve">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. ſi quartam inuenire vo <lb/>luerimus eiuſmodi proportionem cum tertia arithmeticè ſeruantem, quam ſecunda <lb/>cum prima, ſecundam cum tertia in ſummam colligemus, <choice><ex>dabiturque</ex><am>dabiturq́;</am></choice> ſumma .26. ex <lb/>qua detracta prima quantitate, quarta relinquetur nempe .6. quod ex .74. theore-<lb/>mate dependet.</s> </p> <p> <s xml:space="preserve">Idipſum tamen proueniret ſi quis ex tertio termino differentiam primi atque ſe-<lb/>cundi detraheret; </s> <s xml:space="preserve">hæc tamen via non tam vniuerſalis eſtqu àm illa. </s> <s xml:space="preserve">N ſi quartus ter <lb/>minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-<lb/>mino in ſummam colligere oporteret.</s> </p> <p> <s xml:space="preserve">Alterum problema eſt, quòd inuentis duobus terminis, ſi tertius requiratur, ſe-<lb/>cundus duplicandus erit, ex qua ſumma detracto primo, ſtatim tertius proferetur, <lb/>quod problema ex præcedenti theoremate dependet.</s> </p> <pb facs="0062" n="50"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Progredi nihilominus etiam hac in re poſſemus per differentiam primi & ſecun-<lb/>di termini, eam detrahendo aut in ſummam cum ſecunda colligendo, attamen prior <lb/>ratio magis latè patet, ideſt vniuerſalior eſt.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="76">LXXVI</num>.</head> <p> <s xml:space="preserve">CVR ſi quis cupiat ſecundum terminum inuenire, quatuor terminorum arith-<lb/>meticè proportionalis continuæ, quorum nobis duo extrema proponantur. <lb/></s> <s xml:space="preserve">Rectè primum duplicabit <choice><ex>coniungetque</ex><am>coniungetq́;</am></choice> vltimo termino, nempe quarto, ex qua ſum-<lb/>ma tertiam partem deſumet, quæ erit ſecundus terminus quęſitus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi horum quatuor terminorum .12. 9. 6. 3. duo nobis extrema <lb/>proponantur. </s> <s xml:space="preserve">nempe .12. et .3. quorum ſecundus inueniendus ſit, ſumpto quolibet <lb/>pro primo, ſit autem .3. primus numerus, quartus verò .12. </s> <s xml:space="preserve">quare duplicato 3. vtpo <lb/>tè primo, & coniuncto .12. quarto, ſumma erit .18. cuius eſt tertia pars .6. ſecundus <lb/>numerus ſcilicet ſumpto principio à minimo. </s> <s xml:space="preserve">Idipſum euenit ſumpto principio à <lb/>maximo. </s> <s xml:space="preserve">Nam ſi datur ſecundus à minimo aut à maximo, illico tertius datur diffe-<lb/>rentia inter hunc & primum, ſecundo coniuncta, aut ex eodem detracta.</s> </p> <p> <s xml:space="preserve">Cuius ratio ſic demonſtratur, quatuor termini quatuor lineis <seg type="var">.m.g</seg>: <seg type="var">q.p</seg>: <seg type="var">u.n</seg>: <seg type="var">c.t.</seg> <lb/>ſignificentur, quorum <seg type="var">.m.g.</seg> et <seg type="var">.c.t.</seg> tantummodo cognoſcantur. </s> <s xml:space="preserve"><choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.m.g.</seg> primus ac <lb/>maior terminus: </s> <s xml:space="preserve">k.g. verò ſit duplum primi <seg type="var">.m.g</seg>: cui coniungatur <seg type="var">.b.k.</seg> æqualis <seg type="var">.c.t.</seg> <lb/>Dico tertiam partem <seg type="var">.b.g.</seg> quæ ſumma totalis eſt, æqualem eſſe <seg type="var">.q.p</seg>. </s> <s xml:space="preserve">In primis enim <lb/>certi ſumus <seg type="var">.m.f.</seg> in <seg type="var">.m.g.</seg> reperiri æqualem <seg type="var">.q.p.</seg> <choice><ex>ſupereſtque</ex><am>ſupereſtq́;</am></choice> <seg type="var">.f.g.</seg> differentia inter <seg type="var">.m.g.</seg> <lb/>et <seg type="var">.q.p.</seg> æqualis <seg type="var">.e.p.</seg> differentiæ inter <seg type="var">.q.p.</seg> et <seg type="var">.u.n.</seg> & æqualis <seg type="var">.o.n.</seg> differen-<lb/>tiæ inter <seg type="var">.u.n.</seg> et <seg type="var">.c.t</seg>: ſimul etiam in <seg type="var">.k.m.</seg> habemus <seg type="var">.d.m.</seg> æqualem <seg type="var">.m.f.</seg> </s> <s xml:space="preserve">quare etiam <seg type="var">.q.<lb/>p.</seg> et <seg type="var">.k.d.</seg> æqualem <seg type="var">.f.g.</seg> nempe <seg type="var">.e.p.</seg> aut <seg type="var">.o.n</seg>: Hactenus in <seg type="var">.k.g.</seg> reperimus duplum <seg type="var">.q.<lb/>p.</seg> ſimul cum <seg type="var">.f.g.</seg> et <seg type="var">.k.d.</seg> æqualibus <seg type="var">.e.p.</seg> et <seg type="var">.o.n.</seg> & quia <seg type="var">.b.K.</seg> æqualis <seg type="var">.c.t.</seg> fuit coniuncta. <lb/></s> <s xml:space="preserve">conſiderandum eſt an hætres quantitates <seg type="var">.f.g</seg>: <seg type="var">K.d.</seg> et <seg type="var">.b.K.</seg> ſimul æquales ſint <seg type="var">.q.p.</seg> <lb/>quod tamen per ſe manifeſtum eſt. </s> <s xml:space="preserve">nam <seg type="var">.q.p.</seg> ſuperat <seg type="var">.u.n.</seg> per <seg type="var">.e.p.</seg> et <seg type="var">.u.n.</seg> ex-<lb/>cedit <seg type="var">.c.t.</seg> per <seg type="var">.o.n.</seg> æqualem <seg type="var">.e.p.</seg> </s> <s xml:space="preserve">quare <seg type="var">.q.p.</seg> per duplum differentię <seg type="var">.f.g.</seg> ſuperat <seg type="var">.c.t.</seg> ita <lb/>que <seg type="var">.f.g</seg>: <seg type="var">k.d.</seg> et <seg type="var">.K.b.</seg> ipſi <seg type="var">.q.p.</seg> ſunt <choice><ex>ae- quales</ex><am>ę-quales</am></choice>, ex quo ſequitur <seg type="var">.q.p.</seg> <choice><ex>tertiam</ex><am>tertiã</am></choice> <lb/> <ptr xml:id="fig-0062-01a" corresp="fig-0062-01" type="figureAnchor"/> partem eſſe <seg type="var">.b.g.</seg> Hæc quæ hacte-<lb/>nus dicta fuerunt, in genere maio-<lb/>ris inæqualitatis probata fuerunt. <lb/></s> <s xml:space="preserve">At in genere minoris, ſumpto or-<lb/>dinis principio à minimo termino <lb/>rum, duplicetur <seg type="var">.c.t.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> duplum <lb/>hoc <seg type="var">.K.t.</seg> cui <seg type="var">.k.b.</seg> æqualis <seg type="var">.m.g.</seg> con-<lb/>iungatur, quæſumma ſit <seg type="var">.b.t</seg>. </s> <s xml:space="preserve">Di-<lb/>co <seg type="var">.u.n.</seg> tertiam eſſe partem ipſius. <lb/></s> <s xml:space="preserve">Nam in primis in <seg type="var">.b.t.</seg> datur termi <lb/>nus <seg type="var">.b.K.</seg> æqualis vltimo <seg type="var">.m.g.</seg> in <lb/>quo ſemel reperitur <seg type="var">.u.n.</seg> vnà cum <lb/>duabus differentijs, nempe <seg type="var">.i.g.</seg> in <lb/>ipſa autem <seg type="var">.b.t</seg>: <seg type="var">u.n.</seg> ſignificetur pri <lb/>mo loco per <seg type="var">.r.K.</seg> ex quo ſupererit <seg type="var">.b.r.</seg> duabus differentijs prædictis æqualis, ſed ex <lb/>præſuppoſito <seg type="var">.u.n.</seg> componitur ex <seg type="var">.o.u.</seg> æquali <seg type="var">.c.t.</seg> et <seg type="var">.o.n.</seg> ęquali vni differentiæ. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <pb facs="0063" n="51"/><fw type="head">THEOREM. ARIT.</fw> cum in <seg type="var">.b.t.</seg> præter <seg type="var">.r.K.</seg> bis detur <seg type="var">.c.t.K.t.</seg> et <seg type="var">.b.r.</seg> duabus differentijs æquipol-<lb/>lens, illud efficitur <seg type="var">.u.n.</seg> pariter ipſius <seg type="var">.b.t.</seg> eſſe tertiam partem, quod erat <choice><ex>propoſitum</ex><am>propoſitũ</am></choice>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0062-01" corresp="fig-0062-01a"> <graphic url="0062-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="77">LXXVII</num>.</head> <p> <s xml:space="preserve">CVR ſi quis velit ſecundum quinque continuorum proportionalium termi-<lb/>num inuenire, ſolis extremis cognitis. </s> <s xml:space="preserve">Rectè <choice><ex>vltimum</ex><am>vltimũ</am></choice> triplo primi coniunget, <lb/>ex qua ſumma quartam partem detraher, quæ erit ſecundus terminus quæſitus. <lb/></s> <s xml:space="preserve">Quod ipſum faciet qui inuenire vult ſecundum terminum ſenarij ſeptenarij, octo-<lb/>narij aut alterius cuiuſcunque, creſcente tamen multiplicatione primi, <choice><ex>vltimoque</ex><am>vltimoq́;</am></choice> <lb/>coniuncto.</s> </p> <p> <s xml:space="preserve">Exempli gratia, dantur duo extremi termini, horum quinque numerorum .18. <lb/>16. 14. 12. 10. nempe .18. et .10. ſi .18. primus erit, hoc eſt, ſi à genere maioris inæ-<lb/>qualitatis progrediemur, triplicabimus terminum .18. <choice><ex>dabunturque</ex><am>dabunturq́;</am></choice> .54. cui numero <lb/>coniuncto quinto termino .10. dabitur numerus .64. cuius quarta pars erit .16. vtpo <lb/>tè ſecundus terminus gratia, aut ſecundi ſex terminorum, quadruplicandus eſſet pri <lb/>mus .18. deinde adiuncto vltimo, quinta pars ſummæ eſſet ſecundus terminus, <choice><ex>atque</ex><am>atq;</am></choice> <lb/>ita deinceps.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, dicti termini lineis <seg type="var">.z.h</seg>: <seg type="var">f.s</seg>: <seg type="var">u.p</seg>: <seg type="var">e.g.</seg> et <seg type="var">.r.x.</seg> <choice><ex>ſigniſicentur</ex><am>ſigniſicẽtur</am></choice>. <lb/></s> <s xml:space="preserve">In primis ex genere maioris inæqualitatis, triplicabimus <seg type="var">.z.h.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> triplum hoc <seg type="var">.k.<lb/>h.</seg> <choice><ex>cuiconiungatur</ex><am>cuicõiungatur</am></choice> <seg type="var">.b.k.</seg> ęqualis vltimo termino <seg type="var">.r.x</seg>. </s> <s xml:space="preserve">Dico <seg type="var">.f.s.</seg> <choice><ex>quartam</ex><am>quartã</am></choice> partem eſſe ſum-<lb/>mę <seg type="var">.b.h</seg>. </s> <s xml:space="preserve">Nam in <seg type="var">.k.h.</seg> ſecundus terminus <seg type="var">.f.s.</seg> ter cum tribus differentijs æqualibus <seg type="var">.n.h.</seg> <lb/>reperitur. </s> <s xml:space="preserve">Probandum nunc eſt tres has differentias <seg type="var">.n.h</seg>: <seg type="var">a.c.</seg> et <seg type="var">.d.k.</seg> ſimul cum <seg type="var">.b.<lb/>K.</seg> ęquales eſſe <seg type="var">.f.s.</seg> <lb/> <ptr xml:id="fig-0063-01a" corresp="fig-0063-01" type="figureAnchor"/> quod in <choice><ex>dubium</ex><am>dubiũ</am></choice> re <lb/>uocari <choice><ex>non</ex><am>nõ</am></choice> poteſt, <lb/>cum <seg type="var">.f.s.</seg> ſuperet <seg type="var">.<lb/>r.x.</seg> per <seg type="var">.o.s</seg>: <seg type="var">t.p.</seg> et <seg type="var">.<lb/>i.g</seg>. </s> <s xml:space="preserve">At in genere <lb/>minoris inæquali <lb/>tatis, triplum <seg type="var">.r.x.</seg> <lb/>ſit <seg type="var">.x.a.</seg> et <seg type="var">.a.b.</seg> ſit <lb/>æqualis <seg type="var">.z.h.</seg> & <choice><ex>cum</ex><am>cũ</am></choice> <lb/><seg type="var">z.h.</seg> tribus <choice><ex>differem</ex><am>differẽ</am></choice> <lb/>tijs <seg type="var">.n.h</seg>: <seg type="var">o.s</seg>: <seg type="var">t.p.</seg> ſu-<lb/>peret <seg type="var">.e.g.</seg> quæ in <seg type="var">.<lb/>a.b.</seg> ſint <seg type="var">.b.K</seg>: <seg type="var">K.d</seg>: <lb/><seg type="var">d.c.</seg> ex quo <seg type="var">.a.c.</seg> <lb/>æqualis erit <seg type="var">.e.g.</seg> <lb/>et <seg type="var">.a.x.</seg> cum <seg type="var">.b.c.</seg> tripla <seg type="var">.e.g</seg>. </s> <s xml:space="preserve">Itaque tota ſumma <seg type="var">.b.x.</seg> qua drupla erit <seg type="var">.e.g</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0063-01" corresp="fig-0063-01a"> <graphic url="0063-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="78">LXXVIII</num>.</head> <p> <s xml:space="preserve">QVantitates quæ fuerint inuicem in proportionalitate arithmetica proportio-<lb/>nales, permutan do quoque proportionales erunt.</s> </p> <pb facs="0064" n="52"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Sint exempli gratia .4. quantitates <seg type="var">.a.b</seg>: <seg type="var">c.d</seg>: <seg type="var">e.f</seg>: et <seg type="var">.g.h</seg>: inuicem proportionales in <lb/>proportionalitate arithmetica. </s> <s xml:space="preserve">Hoc eſt vt quæ proportio (licet impropriè dicta) <lb/>eſt ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.c.d.</seg> <choice><ex>eadem</ex><am>eadẽ</am></choice> ſit ipſius <seg type="var">.e.f.</seg> ad <seg type="var">.g.h</seg>. </s> <s xml:space="preserve">Tunc permutando dico eandem pro <lb/>portionem fore ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.e.f.</seg> quæ ipſius <seg type="var">.c.d.</seg> ad <seg type="var">.g.h</seg>.</s> </p> <p> <s xml:space="preserve">Nam, ex hypotheſi, differentia qua <seg type="var">.a.b.</seg> ſuperat <seg type="var">.c.d.</seg> (quæ ſit <seg type="var">.m.b.</seg>) æqualis eſt <lb/>differentiæ qua <seg type="var">.e.f.</seg> ſuperat <seg type="var">.g.h.</seg> (quæ ſit <seg type="var">.i.f.</seg>) vnde <seg type="var">.a.m.</seg> reſiduum ex <seg type="var">.a.b.</seg> æquale erit <lb/><seg type="var">c.d.</seg> & reſiduum <seg type="var">.e.i.</seg> æquale <seg type="var">.g.h</seg>. </s> <s xml:space="preserve">Sit igitur exempli gratia <seg type="var">.c.d.</seg> maior <seg type="var">.g.h.</seg> per <seg type="var">.c.n.</seg> <lb/>vnde <seg type="var">.n.d.</seg> æqualis erit <seg type="var">.g.h.</seg> </s> <s xml:space="preserve">quare <seg type="var">.a.m.</seg> maior erit <seg type="var">.e.i.</seg> per <seg type="var">.a.K.</seg> æqualem <seg type="var">.c.n.</seg> ex com-<lb/>muni ſcientia. </s> <s xml:space="preserve">Vnde <seg type="var">.K.m.</seg> æqualis erit <seg type="var">.n.d.</seg> hoc eſt ipſi <seg type="var">.g.h.</seg> hoc eſt ipſi <seg type="var">e.i</seg>. </s> <s xml:space="preserve">Quare ex <lb/>communi conceptu <seg type="var">.b.K.</seg> æqualis erit ipſi <seg type="var">.f.e.</seg> ſed <seg type="var">.n.d.</seg> æqualis eſt <seg type="var">.g.h.</seg> vt dictum eſt. <lb/></s> <s xml:space="preserve">Cum ergo <seg type="var">.b.K.</seg> æqualis ſit <seg type="var">.e.f.</seg> et <seg type="var">.d.n.</seg> ipſi <seg type="var">.g.h.</seg> et <seg type="var">.a.b.</seg> maior ſit ipſa <seg type="var">.K.b.</seg> per <seg type="var">.a.K.</seg> æqua-<lb/>lem ipſi <seg type="var">.c.n.</seg> per quam <seg type="var">c.n</seg>: <seg type="var">d.c.</seg> maior eſt ipſa <seg type="var">.d.n.</seg> ſequitur verum eſſe <choice><ex>propoſitum</ex><am>propoſitũ</am></choice> hoc <lb/>eſt, quod eadem proportio ſit ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.e.f.</seg> quæ <seg type="var">.c.d.</seg> ad <seg type="var">.g.h.</seg> arithmetice ſcilicet.</s> </p> <figure place="here"> <graphic url="0064-01"/> </figure> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="79">LXXIX</num>.</head> <p> <s xml:space="preserve">CVR prouenientia duorum numerorum diuidentium eiuſdem numeri diuiſi-<lb/>bilis, geometricè <choice><ex>eandem</ex><am>eandẽ</am></choice> inter ſe <choice><ex>proportionem</ex><am>proportionẽ</am></choice> ſeruant, <choice><ex>quam</ex><am>quã</am></choice> ipſimet <choice><ex>diuidentes</ex><am>diuidẽtes</am></choice>.</s> </p> <p> <s xml:space="preserve">Exempli gratia ſi per ſenarium & octonarium numerus vigintiquatuor diuida-<lb/>tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</s> </p> <p> <s xml:space="preserve">Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis <seg type="var">.u.x.</seg> et <seg type="var">.n.e.</seg> diuidentes <lb/>autem ſint <seg type="var">.u.o.</seg> et <seg type="var">.e.o.</seg> </s> <s xml:space="preserve">quare ex ijs, quæ .10. <lb/> <ptr xml:id="fig-0064-02a" corresp="fig-0064-02" type="figureAnchor"/> theoremate dicta fuerunt <seg type="var">.u.x.</seg> per <seg type="var">.u.o.</seg> diui-<lb/>ſo dabit <seg type="var">.x.o.</seg> & diuiſo <seg type="var">.n.e.</seg> per <seg type="var">.e.o.</seg> dabit <seg type="var">.o.<lb/>n</seg>. </s> <s xml:space="preserve">Dicimus itaque <choice><ex>eandem</ex><am>eandẽ</am></choice> eſſe <choice><ex>proportionem</ex><am>proportionẽ</am></choice> <lb/><seg type="var">o.x.</seg> ad <seg type="var">.o.n.</seg> quæ <seg type="var">.e.o.</seg> ad <seg type="var">.o.u.</seg> quod patet ſub <lb/>ſcriptam figuram conſiderantibus, in qua, <lb/>ex .15. ſexti aut .20. ſeptimi, eadem propor-<lb/>tio cernitur <seg type="var">.o.x.</seg> ad <seg type="var">.o.n.</seg> quæ <seg type="var">.o.e.</seg> ad <seg type="var">.o.u</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0064-02" corresp="fig-0064-02a"> <graphic url="0064-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="80">LXXX</num>.</head> <p> <s xml:space="preserve">CVR quauis quantitate, tribus <lb/> <ptr xml:id="fig-0064-03a" corresp="fig-0064-03" type="figureAnchor"/> aut quatuor aut etiam pro libi-<lb/>to pluribus diuidentibus numeris di-<lb/>uifa, prouenientia eandem prorſus <lb/>inter ſe proportionem ſeruabunt, <lb/>quam ipſi diuidentes habere compe <lb/>riuntur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0064-03" corresp="fig-0064-03a"> <graphic url="0064-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Exempli gratia, proponitur nu-<lb/>merus .60. quinque numeris diuiden <lb/>dus, vtpotè .30. 20. 15. 12. 10. pro-<lb/>uenientia erunt .2. 3. 4. 5. 6. eadem <pb facs="0065" n="53"/><fw type="head">THEOREM. ARITH.</fw> proportione diuidentium, quamuis ex aduerſo.</s> </p> <p> <s xml:space="preserve">Cuius ratio ex .15. ſexti aut .20. ſeptimi dependet. </s> <s xml:space="preserve">prout in ſubſcripto ordine fa-<lb/>cillimè deprehendi poteſt.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="81">LXXXI</num>.</head> <p> <s xml:space="preserve">CVR quantitate in tres continuas partes proportionales ſecta, & per ſingulas <lb/>ipſarum diuiſa, ſumma trium prouenientium quadrato medij prouenientis <lb/>æqualis eſt.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponitur .14. diuidendus in tres continuas partes proportio-<lb/>nales, nempe .8. 4. 2. <choice><ex>ipſeque</ex><am>ipſeq́;</am></choice> numerus .14. per ſingulas diuiditur, ex quo tria proue-<lb/>nientia oriuntur, nempe ex prima parte .8. <choice><ex>proueniens</ex><am>proueniẽs</am></choice> erit .1. cum tribus quartis par <lb/>tibus ex ſecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-<lb/>nient .7. integri, qui in ſummam collecti dant .12. integros & vnam quartam par-<lb/>tem tantumdem, videlicet quantum quadratum prouenientis medij, nempe .3. <lb/>cum dimidio.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, totalis numerus ſignificetur linea <seg type="var">.n.c.</seg> qui in tres par-<lb/>tes diuidatur <seg type="var">.n.a</seg>: <seg type="var">a.e.</seg> et <seg type="var">.e.c.</seg> quæ ſint continuæ proportionales, quarum ſingulis, <lb/>numerum <seg type="var">.n.c.</seg> diuiſum eſſe cogitemus, proueniens autem ex diuiſione <seg type="var">.n.c.</seg> per <seg type="var">.n.<lb/>a.</seg> ſit <seg type="var">.i.d.</seg> quod verò prouenit ex diuiſione <seg type="var">.n.c.</seg> per <seg type="var">.a.e.</seg> ſit <seg type="var">.d.u.</seg> proueniens quoque ex <lb/>diuiſione <seg type="var">.n.c.</seg> per <seg type="var">.e.c.</seg> ſit <seg type="var">.u.o.</seg> quorum ſumma ſit <seg type="var">.i.o.</seg> quæ aſſeritur eſſe numeri æqua-<lb/>lis numero quadrati <seg type="var">.d.u</seg>. </s> <s xml:space="preserve">Quod hac ratione probabo, producatur linea <seg type="var">.i.o.</seg> donec <seg type="var">.<lb/>o.p.</seg> æqualis ſit <seg type="var">.o.u.</seg> <choice><ex>erigaturque</ex><am>erigaturq́;</am></choice> <seg type="var">.o.m.</seg> æqualis <seg type="var">.d.i.</seg> perpendiculariter <seg type="var">.o.p.</seg> in puncto <seg type="var">.o.</seg> <lb/>quæ producatur donec <seg type="var">.o.q.</seg> vnitati ſit æqualis, <choice><ex>terminenturque</ex><am>terminenturq́;</am></choice> duo rectangula <seg type="var">.m.p.</seg> <lb/>et <seg type="var">.q.i.</seg> ex quo habebimus rectangulum, aut productum <seg type="var">.m.p.</seg> æquale quadrato <seg type="var">.d.u.</seg> <lb/>ex .16 ſexti aut .20. ſeptimi, quandoquidem tria prouenientia <seg type="var">.o.u</seg>: <seg type="var">u.d.</seg> et <seg type="var">.d.i.</seg> ex <lb/>pręcedenti theoremate ſunt inter ſe continua proportionalia, proportionalitate qua <lb/>partes <seg type="var">.n.c</seg>. </s> <s xml:space="preserve">Iam verò ſi probauero <seg type="var">.q.i.</seg> productum, producto <seg type="var">.m.p.</seg> æquale eſſe, pro-<lb/>poſitum quoque probatum erit. </s> <s xml:space="preserve">Numerus enim producti <seg type="var">.q.i.</seg> æqualis eſt numero. <lb/></s> <s xml:space="preserve">ſummæ <seg type="var">.i.o</seg>. </s> <s xml:space="preserve">Habemus autem ex definitione diuiſionis ita ſe habere <seg type="var">.n.c.</seg> ad <seg type="var">.i.d.</seg> ſicut <seg type="var">.<lb/>n.a.</seg> ad <seg type="var">.o.q</seg>. </s> <s xml:space="preserve">Itaque permutando ſic ſe habebit <seg type="var">.n.c.</seg> ad <seg type="var">.n.a.</seg> ſicut <seg type="var">.d.i.</seg> hoc eſt <seg type="var">.m.o.</seg> ad <seg type="var">.<lb/>o.q.</seg> ſed ſicut ſe habet <seg type="var">.n.c.</seg> ad <seg type="var">.n.a.</seg> ita pariter ſe habet <seg type="var">.i.o.</seg> ad <seg type="var">.o.u.</seg> hoc eſt ad <seg type="var">.o.p</seg>. </s> <s xml:space="preserve">Ita-<lb/>que <seg type="var">.i.o.</seg> ad <seg type="var">.o.p.</seg> ſic ſe habebit ſicut <seg type="var">.m.o.</seg> ad <seg type="var">.o.q.</seg> ex quo ex .15. ſexti aut .20. ſeptimi <seg type="var">.<lb/>q.i.</seg> æqualis erit <seg type="var">.m.p.</seg> & conſequenter quadrato <seg type="var">.d.u</seg>. </s> <s xml:space="preserve">Vt autem lector minori labo-<lb/>re cognoſcere queat <seg type="var">.i.o.</seg> ad <seg type="var">.o.u.</seg> ſic ſe habere, vt <seg type="var">.n.c.</seg> ad <seg type="var">.n.a.</seg> ſciendum eſt quòd, ſic <lb/>ſe habet <seg type="var">.i.d.</seg> ad <seg type="var">.d.u.</seg> ut <seg type="var">.c.e.</seg> ad <seg type="var">.e.a.</seg> ex quo componendo ſic ſe habebit <seg type="var">.i.u.</seg> ad <seg type="var">.d.u.</seg> ſi-<lb/>cut <seg type="var">.c.a.</seg> ad <seg type="var">.a.e.</seg> & permutando ita <seg type="var">.i.u.</seg> <lb/> <ptr xml:id="fig-0065-01a" corresp="fig-0065-01" type="figureAnchor"/> ad <seg type="var">.c.a.</seg> vt <seg type="var">.d.u.</seg> ad <seg type="var">.e.a.</seg> ſed cum ex <choice><ex>præ- cedenti</ex><am>præ-cedẽti</am></choice> theoremate ſic ſe habeat <seg type="var">.d.u.</seg> <lb/>ad <seg type="var">.u.o.</seg> ſicut <seg type="var">.e.a.</seg> ad <seg type="var">.a.n.</seg> permutando <lb/>ſic ſe habebit <seg type="var">.d.u.</seg> ad <seg type="var">.a.e.</seg> ſicut <seg type="var">.u.o.</seg> ad <lb/><seg type="var">a.n.</seg> ex quo ex .11. quinti ſic ſe habe-<lb/>bit <seg type="var">.i.u.</seg> ad <seg type="var">.c.a.</seg> prout <seg type="var">.o.u.</seg> ad <seg type="var">.a.n.</seg> per-<lb/>mutandoq́ue <seg type="var">.i.u.</seg> ad <seg type="var">.u.o.</seg> vt <seg type="var">.c.a.</seg> ad <seg type="var">.a.n.</seg> & componendo, ita <seg type="var">.i.o.</seg> ad <seg type="var">.u.o.</seg> ſicut <seg type="var">.c.n.</seg> <lb/>ad <seg type="var">.a.n</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0065-01" corresp="fig-0065-01a"> <graphic url="0065-01"/> </figure> </div> </body> </floatingText> <pb facs="0066" n="54"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="82">LXXXII</num>.</head> <p> <s xml:space="preserve">CVR quantitate aliqua in quatuor partes <choice><ex>continuas</ex><am>cõtinuas</am></choice> proportionales ſecta per-<lb/>q́ue ſingulas diuiſa, ſumma quatuor prouenientium æqualis ſit producto ſe-<lb/>cundi in tertium.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi triginta in quatuor partes proportionales ſecetur, hoc eſt. <lb/>16. 8. 4. 2. <choice><ex>perque</ex><am>perq́;</am></choice> harum ſingulas idem numerus .30. diuidatur, primum proueniens <lb/>erit .1. cum ſeptem octauis partibus. </s> <s xml:space="preserve">Secundum .3. cum tribus quartis, tertium .7. <lb/>cum dimidio, quartum .15. integri, quorum ſumma erit .28. cum octaua parte, tan <lb/><choice><ex>tumque</ex><am>tumq́;</am></choice> erit productum ſecundi prouenientis in tertium.</s> </p> <p> <s xml:space="preserve">Quod vt ſciamus, quantitas <seg type="var">.n.c.</seg> in partes continuas proportionales quatuor ſe-<lb/>cetur <seg type="var">.n.a</seg>: <seg type="var">a.t</seg>: <seg type="var">t.e.</seg> et <seg type="var">.e.c.</seg> <choice><ex>rurſusque</ex><am>rurſusq́;</am></choice> per ſingulas partes illa ipſa diuiſa, prouenientia <lb/>ſint <seg type="var">.i.d</seg>: <seg type="var">d.x</seg>: <seg type="var">x.u</seg>: <seg type="var">u.o.</seg> <choice><ex>quorum</ex><am>quorũ</am></choice> ſumma ſit <seg type="var">.i.o.</seg> hanc <choice><ex>ſummam</ex><am>ſummã</am></choice> dicimus æqualem eſſe nume-<lb/>ro producti <seg type="var">.d.x.</seg> in <seg type="var">.x.u</seg>.</s> </p> <p> <s xml:space="preserve">Quod hac ratione probo, cogito productam eſſe lineam <seg type="var">.i.o.</seg> <choice><ex>quousque</ex><am>quousq́;</am></choice> <seg type="var">.o.p.</seg> æqua <lb/>lis ſit <seg type="var">.o.u.</seg> <choice><ex>erectamque</ex><am>erectamq́;</am></choice> <seg type="var">.m.o.</seg> æqualem <seg type="var">.i.d.</seg> perpendiculariter <seg type="var">.o.p.</seg> & productam donec <seg type="var">.<lb/>o.q.</seg> vnitati ſit æqualis. </s> <s xml:space="preserve">Iam terminatis rectangulis <seg type="var">.m.p.</seg> et <seg type="var">.i.q.</seg> patebit ex .15. ſexti <lb/>aut .20. ſeptimi, productum <seg type="var">.m.p.</seg> producto <seg type="var">.d.x.</seg> in <seg type="var">.x.u.</seg> æquale eſſe. </s> <s xml:space="preserve">Ita quòd ſi pro-<lb/>bauero productum <seg type="var">.i.q.</seg> producto <seg type="var">.m.p.</seg> æquale eſſe, facile patebit propoſitum. </s> <s xml:space="preserve">Cuius <lb/>gratia, ſequuti præcedentis theorematis ordinem, primum ex <choice><ex>definitionem</ex><am>definitionẽ</am></choice> diuiſionis, <lb/>eadem proportio erit <seg type="var">.n.c.</seg> ad <seg type="var">.i.d.</seg> quæ <seg type="var">.n.a.</seg> ad <seg type="var">.o.q.</seg> ex quo permutando <seg type="var">.n.c.</seg> ad <seg type="var">.n.a.</seg> ſic <lb/>ſe habebit vt <seg type="var">.i.d.</seg> hoc eſt <seg type="var">.m.o.</seg> ad <seg type="var">.o.q.</seg> & ſi progrediamur eodem ordine, quo præ-<lb/>cedenti theoremate, ſumpto principio ab <seg type="var">.i.d.</seg> et <seg type="var">.e.c.</seg> verſus <seg type="var">.d.x.</seg> et <seg type="var">.e.t.</seg> gradatimq́ue <lb/>permutando ac coniungendo, inue-<lb/> <ptr xml:id="fig-0066-01a" corresp="fig-0066-01" type="figureAnchor"/> niemus eandem proportionem eſſe <lb/><seg type="var">c.n.</seg> ad <seg type="var">.n.a.</seg> quæ <seg type="var">.i.o.</seg> ad <seg type="var">.o.u.</seg> nempe <seg type="var">.<lb/>o.p.</seg> ex quo ex .11 quinti, ita ſe habe <lb/>bit <seg type="var">.i.o.</seg> ad <seg type="var">.o.p.</seg> vt <seg type="var">.m.o.</seg> ad <seg type="var">.o.q.</seg> </s> <s xml:space="preserve">quare <lb/>ex .15. ſextiaut .20. ſeptimi <choice><ex>produ- ctum</ex><am>produ-ctũ</am></choice> <seg type="var">.i.q.</seg> erit producto<unclear reason="illegible"/> <seg type="var">.m.p.</seg> æquale, <lb/>ex quo etiam æquale erit producto <seg type="var">.<lb/>d.x.</seg> in <seg type="var">.x.u</seg>. </s> <s xml:space="preserve">Idem ordo in qualibet <lb/>quantitate in quantaſuis partes diuiſa ſeruari poterit, cum huiuſmodi <choice><ex>ſcientia</ex><am>ſciẽtia</am></choice> in vni <lb/>uerſum pateat.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0066-01" corresp="fig-0066-01a"> <graphic url="0066-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="83">LXXXIII</num>.</head> <p> <s xml:space="preserve">CVR termini medij cubus, trium continuè proportionalium, ſemper producto <lb/>rectanguli compræhenſi à maximo & medio in minimo termino æqualis ſit.</s> </p> <p> <s xml:space="preserve">Exempli gratia, datis his tribus terminis continuis proportionalibus .9. 6. 4. ſi <lb/>ſumpſerimus productum maximi in medium nempe .54. quod per <choice><ex>minimum</ex><am>minimũ</am></choice> .4. multi-<lb/>plicemus, dabitur numerus .216. cubo medij .6. æqualis.</s> </p> <p> <s xml:space="preserve">In cuius gratiam tres numeri continui proportionales tribus lineis <seg type="var">.a.e.i.</seg> <choice><ex>ſignifi- centur</ex><am>ſignifi-cẽtur</am></choice>, cubus autem <seg type="var">.e.</seg> ſignificetur figura <seg type="var">.d.n.</seg> <choice><ex>productumque</ex><am>productumq́</am></choice> <seg type="var">.a.</seg> in <seg type="var">.e.</seg> ſit <seg type="var">.b.n.</seg> ipſius <choice><ex>au- temmet</ex><am>au-tẽmet</am></choice> in <seg type="var">.i.</seg> ſit <seg type="var">.p.o.</seg> ita quod <seg type="var">.q.p.</seg> aut <seg type="var">.b.o.</seg> cum ſint <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> ſpeciei, æqualis erit .a: et <seg type="var">.o.n.</seg> <pb facs="0067" n="55"/><fw type="head">THEOR. ARITH.</fw> æqualis .e: et <seg type="var">.q.n.</seg> æqualis <seg type="var">.i</seg>. </s> <s xml:space="preserve">Nunc co-<lb/> <ptr xml:id="fig-0067-01a" corresp="fig-0067-01" type="figureAnchor"/> gitemus abſolui corpus <seg type="var">.n.h.</seg> ita ut <seg type="var">.b.<lb/>o.c.</seg>ſit vnica recta linea, ex quo ex .25. <lb/>vndecimi proportio <seg type="var">.n.h.</seg> ad <seg type="var">.n.k.</seg> ea-<lb/>dem eſt quæ <seg type="var">.o.h.</seg> ad <seg type="var">o.k.</seg> ſed ſic ſe ha-<lb/>bet <seg type="var">.o.h.</seg> ad <seg type="var">.o.k.</seg> vt <seg type="var">.h.b.</seg> ad <seg type="var">.b.k.</seg> <lb/>ex prima ſexti aut .18. vel .19. ſe-<lb/>ptimi itaque <seg type="var">.n.h.</seg> ad <seg type="var">.n.k.</seg> ex .11. <lb/>quinti ſic ſe habebit. vt <seg type="var">.h.b.</seg> ad <seg type="var">.b.k.</seg> <lb/>ſed <seg type="var">.n.h.</seg> ad <seg type="var">.n.d.</seg> ex eiſdem ſic ſe habet <lb/>ut <seg type="var">.h.u.</seg> ad <seg type="var">.d.u.</seg> et <seg type="var">.h.u.</seg> ad <seg type="var">.u.d.</seg> ita ut <seg type="var">.h.<lb/>b.</seg> ad <seg type="var">.b.k.</seg> ex præſuppoſito. </s> <s xml:space="preserve">Itaque ex <lb/>11. prædicta <seg type="var">.n.h.</seg> ad <seg type="var">.n.k.</seg> eadem erit <lb/>proportio quæ <seg type="var">.n.h.</seg> ad <seg type="var">.n.d</seg>. </s> <s xml:space="preserve">Quare <lb/>ex .9. quinti <seg type="var">.n.k.</seg> æqualis erit <seg type="var">.n.d.</seg> <lb/>Quod erat propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0067-01" corresp="fig-0067-01a"> <graphic url="0067-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="84">LXXXIIII</num>.</head> <p> <s xml:space="preserve">CVR quadrato vnius quantitatis radice proportionalis, per ſingulos tres termi <lb/>nos diuiſo, prouenientia, ſingulis dictis terminis ſint æqualia.</s> </p> <p> <s xml:space="preserve"><choice><ex>Exempli</ex><am>Exẽpli</am></choice> gratia, datis tribus terminis continuis proportionalibus .9. 6. 4. qua <lb/>dratum medij erit .36. quod per .9. diuiſum dabit .4: per .6: 6. per .4: 9.</s> </p> <p> <s xml:space="preserve">Cuius gratia, ſint tres termini <choice><ex>continui</ex><am>cõtinui</am></choice> <choice><ex>proportionales</ex><am>ꝓportionales</am></choice> <seg type="var">.a.o</seg>: <seg type="var">o.c.</seg> et <seg type="var">.c.q.</seg> <choice><ex>quadratum</ex><am>quadratũ</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> <lb/>medij ſit <seg type="var">.e.c</seg>. </s> <s xml:space="preserve">Iam ſi applicetur <choice><ex>rectangulum</ex><am>rectangulũ</am></choice> <seg type="var">.a.d.</seg> æquale quadrato <seg type="var">.e.c.</seg> ipſi <seg type="var">.a.o.</seg> & re-<lb/>ctangulum <seg type="var">.q.p.</seg> æquale eidem quadrato <seg type="var">.e.c.</seg> ipſi <seg type="var">.c.q.</seg> ſi quadratum <seg type="var">.e.c.</seg> per <seg type="var">.a.o.</seg> diui <lb/>ſerimus, proueniens erit <seg type="var">.o.d.</seg> <choice><ex>diuiſoque</ex><am>diuiſoq́</am></choice> per <seg type="var">.c.q.</seg> proueniens erit <seg type="var">.c.p.</seg> quod ſi per ſuam <lb/>radicem <seg type="var">.o.c.</seg> diuidatur, proueniens erit <seg type="var">.o.</seg> <lb/> <ptr xml:id="fig-0067-02a" corresp="fig-0067-02" type="figureAnchor"/> e. quod ſine dubio æquale eſt <seg type="var">.o.c.</seg> ſed dico <seg type="var">.<lb/>o.d.</seg> æqualem eſſe <seg type="var">.c.q</seg>. </s> <s xml:space="preserve">Nam ex .16. ſexti aut <lb/>20. ſeptimi eadem eſt proportio <seg type="var">.a.o.</seg> ad <seg type="var">.o.<lb/>c.</seg> quę <seg type="var">.o.e.</seg> ad <seg type="var">.o.d.</seg> nempe <seg type="var">.o.c.</seg> ad <seg type="var">.o.d.</seg> itaque <lb/><seg type="var">o.d.</seg> ex .9. quinti æqualis eſt <seg type="var">.c.q.</seg> quandoqui <lb/>dem ex .11. ſic ſe habet <seg type="var">.o.c.</seg> ad <seg type="var">.o.d.</seg> ſicut <seg type="var">.o.<lb/>c.</seg> ad <seg type="var">.c.q</seg>. </s> <s xml:space="preserve">Applicatis ijſdem rationibus ipſi <seg type="var">.<lb/>p.c.</seg> probabimus <seg type="var">.c.p.</seg> æqualem eſſe <seg type="var">.a.o.</seg> cum <lb/><seg type="var">o.c.</seg> media ſit proportionalis, <choice><ex>tam</ex><am>tã</am></choice> inter <seg type="var">.c.p.</seg> et <lb/><seg type="var">c.q.</seg> quam inter <seg type="var">.a.o.</seg> et <seg type="var">.c.q.</seg> itaque <seg type="var">.c.p.</seg> æqua-<lb/>lis eſt <seg type="var">.a.o</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0067-02" corresp="fig-0067-02a"> <graphic url="0067-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="85">LXXXV</num>.</head> <p> <s xml:space="preserve">CVR propoſitis tribus quantitatibus continuis proportionalibus proportione <lb/>aliarum duarum nobis datarum, multiplicata maiori poſtremarum dua-<lb/>rum in ſummam mediæ cum minima trium primarum, productum æqua-<lb/>le ſit producto minoris duarum in ſummam maximæ cum media trium.</s> </p> <p> <s xml:space="preserve">Exempli gratia proponuntur quantitates .9. 6. 4. proportione numerorum pro- <pb facs="0068" n="56"/><fw type="head">IO. BAPT. BENED.</fw> poſitorum .3. et .2. multiplicato .3. per .10. <choice><ex>ſummam</ex><am>ſummã</am></choice> .6. cum .4. dantur .30. quod pro-<lb/>ductum æquale erit producto .2. per .15. nempe per ſummam 9. et .6.</s> </p> <p> <s xml:space="preserve">Quod vt cognoſcamus, tres quan <lb/> <ptr xml:id="fig-0068-01a" corresp="fig-0068-01" type="figureAnchor"/> titates continuæ proportionales ſint <lb/><seg type="var">b.a.p.</seg> proportione <seg type="var">.d.q.</seg> productum <lb/>autem <seg type="var">.d.</seg> in ſummam <seg type="var">.a.</seg> cum <seg type="var">.p.</seg> ſit <seg type="var">.f.t.</seg> <lb/>& productum <seg type="var">.q.</seg> in ſummam <seg type="var">.b.a.</seg> ſit <seg type="var">.<lb/>K.h.</seg> et <seg type="var">.K.n.</seg> ſit æqualis <seg type="var">.b.</seg> et <seg type="var">.n.o.</seg> æqua <lb/>lis <seg type="var">.a.</seg> & ita etiam <seg type="var">.o.u.</seg> eidem <seg type="var">.a.</seg> et <seg type="var">.u.t.</seg> <lb/>æqualis <seg type="var">.p.</seg> et <seg type="var">.h.o.</seg> ipſi <seg type="var">.q.</seg> et <seg type="var">.f.o.</seg> ipſi <seg type="var">.d</seg>. <lb/></s> <s xml:space="preserve">quare ita ſe habebit <seg type="var">.K.n.</seg> ad <seg type="var">.n.o.</seg> ſicut <lb/><seg type="var">o.u.</seg> ad <seg type="var">.u.t.</seg> & componendo <seg type="var">.K.o.</seg> ad <seg type="var">.<lb/>n.o.</seg> vt <seg type="var">.o.t.</seg> ad <seg type="var">.u.t.</seg> & permutando <seg type="var">.K.<lb/>o.</seg> ad <seg type="var">.o.t.</seg> vt <seg type="var">.n.o.</seg> hoc eſt <seg type="var">.o.u.</seg> ad <seg type="var">.u.t.</seg> & <lb/>pariter <seg type="var">.f.o.</seg> ad <seg type="var">.o.h.</seg> vt <seg type="var">.o.u.</seg> ad <seg type="var">.u.t</seg>. </s> <s xml:space="preserve">Ita-<lb/>que ſicut <seg type="var">.k.o.</seg> ad <seg type="var">.o.t.</seg> ex quo ex .15. ſexti aut .20. ſeptimi <seg type="var">.K.h.</seg> æqualis erit <seg type="var">.f.t</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0068-01" corresp="fig-0068-01a"> <graphic url="0068-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="86">LXXXVI</num>.</head> <p> <s xml:space="preserve">CVR multiplicatis ſingulis tribus quantitatibus continuis proportionalibus in <lb/>reliquas duas, ſex producta æqualia ſint producto dupli ſummæ ipſarum trium <lb/>in mediam proportionalem.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponuntur hitres termini continui proportionales .9. 6. 4. pro <lb/>ductum .9. in .6. erit .54. at .9. in .4. erit .36. et .6. in .9: 54. et .6. in .4: 24. et .4. in .9: 36. et <num value="4">.<lb/>4.</num> in .6: 24. quæ producta ſimul collecta efficiunt numerum .228 ſed <choice><ex>tantum</ex><am>tantũ</am></choice> eſt pro-<lb/>ductum dupli ſummæ trium terminorum in ſecundum nempe .38 in .6.</s> </p> <p> <s xml:space="preserve">Cuius <choice><ex>intelligentiæ</ex><am>intelligẽtiæ</am></choice> cauſa, tres termini <choice><ex>continui</ex><am>cõtinui</am></choice> proportionales ſignificentur linea <seg type="var">.<lb/>b.e.</seg> nempe <seg type="var">.b.d</seg>: <seg type="var">d.c</seg>: <seg type="var">c.e.</seg> cuius duplum ſit <seg type="var">.u.e.</seg> et <seg type="var">.b.f.</seg> æqualis ſit <seg type="var">.b.d.</seg> et <seg type="var">.f.n</seg>: <seg type="var">d.c.</seg> et <seg type="var">.n.u</seg>: <lb/>c. e productum verò <seg type="var">.u.e.</seg> in <seg type="var">.d.c.</seg>ſit <seg type="var">.u.s.</seg> cui dico æqualem eſſe ſummam productorum <lb/>ſingulorum trium terminorum in reliquos duos. </s> <s xml:space="preserve">Quamobrem ducantur perpendi-<lb/>culares <seg type="var">.c.g</seg>: <seg type="var">d.o</seg>: <seg type="var">b.i</seg>: <seg type="var">f.a.</seg> et <seg type="var">.n.p.</seg> inter <seg type="var">.u.e.</seg> et <seg type="var">.q.s.</seg> ex quo pro producto <seg type="var">.c.e.</seg> in <seg type="var">.c.d.</seg> ha-<lb/>bebimus rectangulum <seg type="var">.c.s.</seg> & rectan-<lb/> <ptr xml:id="fig-0068-02a" corresp="fig-0068-02" type="figureAnchor"/> gulum <seg type="var">.d.g.</seg> pro producto <seg type="var">.c.e.</seg> in <seg type="var">.d.b.</seg> <lb/>ex .16. ſexti aut .20. ſeptimi itemq́ue <lb/>rectangulum <seg type="var">.q.n.</seg> pro producto <seg type="var">.d.c.</seg> <lb/>in <seg type="var">.c.e.</seg> & rectangulum <seg type="var">.b.o.</seg> ex <seg type="var">.d.c.</seg> in <seg type="var">.<lb/>b.d.</seg> & rectangulum <seg type="var">.b.a.</seg> ex <seg type="var">.b.d.</seg> in <seg type="var">.d.<lb/>c.</seg> et <seg type="var">.p.f.</seg> ex <seg type="var">.d.b.</seg> in <seg type="var">.c.e.</seg> ex .16. aut .20. <lb/>prędictas. </s> <s xml:space="preserve">Quare ſex producta æquantur inter ſe, <choice><ex>replentque</ex><am>replentq́</am></choice> productum <seg type="var">.u.s.</seg> ex quo <lb/>verum eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0068-02" corresp="fig-0068-02a"> <graphic url="0068-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="87">LXXXVII</num>.</head> <p> <s xml:space="preserve">QVA ratione cognoſci poſſit <choice><ex>verum</ex><am>verũ</am></choice> eſſe proportionem ſummæ quatuor quan-<lb/>titatum continuarum proportionalium ad ſummam ſecundæ & tertiæ, ean-<lb/>dem eſſe, quæ ſummæ primæ & tertiæ ad ſecundam ſimplicem.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi inue nirentur hæ quatuor quantitates continuæ proportiona-<lb/>es .16. 8. 4. 2. earum ſumma erit .30. ſunima verò ſecundæ & tertiæ .12. tum ſumma <pb facs="0069" n="57"/><fw type="head">THEOREM. ARITH.</fw> primæ cum tertia .20. ex quo ſicſe habet .20. ad .8. nempe ad ſecundam, vt .30. <lb/>ad .12.</s> </p> <p> <s xml:space="preserve">Quod vt ſciamus, quatuor prædictæ quantitates ſignificentur linea <seg type="var">.a.e.i.o.</seg> pro-<lb/>babo ita ſe habere <seg type="var">.a.e.i.o.</seg> ad <seg type="var">.e.i.</seg> vt <seg type="var">.a.i.</seg> ad <seg type="var">.e</seg>. </s> <s xml:space="preserve">Nam cum ſic ſe habeat <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> ut <seg type="var">.e.</seg> <lb/>ad <seg type="var">.i.</seg> & vt <seg type="var">.i.</seg> ad .o: ex æqualitate proportionum vel permutando ita ſe habebit <seg type="var">.a.</seg> ad <seg type="var">.i.</seg> <lb/>vt <seg type="var">.e.</seg> ad <seg type="var">.o.</seg> & è conuerſo ita <seg type="var">.o.</seg> ad <seg type="var">.e.</seg> vt <seg type="var">.i.</seg> ad <seg type="var">.a.</seg> & <choice><ex>componendo</ex><am>cõponendo</am></choice> ita <seg type="var">.o.e.</seg> ad e. vt <seg type="var">.i.a.</seg> ad <seg type="var">.a.</seg> <lb/><choice><ex>permutandoque</ex><am>permutandoq́</am></choice> <seg type="var">.o.e.</seg> ad <seg type="var">.i.a.</seg> vt <seg type="var">.e.</seg> ad <seg type="var">.a.</seg> nempe <seg type="var">.i.</seg> ad <seg type="var">.e.</seg> & componendo ita <seg type="var">.o.i.e.a.</seg> ad <seg type="var">.<lb/>i.a.</seg> vt <seg type="var">.i.e.</seg> ad <seg type="var">.e.</seg> & permutando ita <seg type="var">.o.i.e.a.</seg> ad <seg type="var">.i.e.</seg> vt <seg type="var">.i.a.</seg> ad <seg type="var">.e.</seg> quod erat propoſitum. <lb/></s> <s xml:space="preserve">Ex quo patet error antiquorum quiidipſum, accidere arbitrati ſunt in quantitatibus <lb/>diſcretæ proportionalitatis, quod tamen falſum eſt.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eſt quæ .4. <lb/>ad .2. </s> <s xml:space="preserve">Sed à proportione .6. ad .4. frangitur, cum non ſit eadem quæ .12. ad .6. harum <lb/>autem ſumma erit .24. & ſumma ſecundæ cum tertia .10. ſed primæ cum tertia erit <lb/>16. ex quo .16. ad .6. non ſic ſe habebit vt .24. ad .10. <lb/>At in ſpeculatione quatuor quantitatum <seg type="var">.a.</seg> <lb/> <ptr xml:id="fig-0069-01a" corresp="fig-0069-01" type="figureAnchor"/> <seg type="var">e.i.o.</seg> ſi proportio <seg type="var">.e.</seg> ad <seg type="var">.i.</seg> non eſſet eadem <lb/>quæ <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> minimè licuiſſet dicere ita ſe <lb/>habere <seg type="var">.i.</seg> ad <seg type="var">.e.</seg> vt <seg type="var">.e.</seg> ad <seg type="var">.a</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0069-01" corresp="fig-0069-01a"> <graphic url="0069-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="88">LXXXVIII</num>.</head> <p> <s xml:space="preserve">CVR extribus quantitatibus quibuſlibet, productum duarum in tertiam, vna <lb/>ſemper <choice><ex>eademque</ex><am>eademq́;</am></choice> ſit quantitas.</s> </p> <p> <s xml:space="preserve">Exempli gratia, proponuntur .15. 8. 2. ſi multiplicauerimus .15. per .8. tum produ <lb/>ctum per .2. tantum erit quantum ſi quis multiplicaret .8. per .2. & hoc per .15. et .15. <lb/>per .2. <choice><ex>rurſusque</ex><am>rurſusq́;</am></choice> per .8.</s> </p> <p> <s xml:space="preserve">Quod ut pateat, tres quantitates tri-<lb/> <ptr xml:id="fig-0069-02a" corresp="fig-0069-02" type="figureAnchor"/> bus lineis ſignificentur <seg type="var">.m.f</seg>: a. et <seg type="var">.o</seg>. </s> <s xml:space="preserve">Dico <lb/>productum <seg type="var">.m.f.</seg> in <seg type="var">.a.</seg> multiplicatum. <lb/></s> <s xml:space="preserve">per <seg type="var">.o.</seg> æquale eſſe producto <seg type="var">.a.</seg> in <seg type="var">.o.</seg> mul-<lb/>tiplicato per <seg type="var">.m.f.</seg> aut producto <seg type="var">.m.f.</seg> in <seg type="var">.<lb/>o.</seg> multiplicato per <seg type="var">.a</seg>. </s> <s xml:space="preserve">Sit enim corpus <seg type="var">.d.<lb/>u.</seg> <choice><ex>rectangulum</ex><am>rectãgulum</am></choice>, cuius latus <seg type="var">.n.u.</seg> ſit æquale <lb/><seg type="var">m.f.</seg> et <seg type="var">.u.t</seg>: a: et <seg type="var">.u.c</seg>: o. patebit manifeſtè <lb/><seg type="var">n.t.</seg> eſſe productum <seg type="var">.m.f.</seg> in <seg type="var">.a.</seg> quod <seg type="var">.n.t.</seg> <lb/>multiplicatum in <seg type="var">.u.c.</seg> æquali <seg type="var">.o.</seg> producit <lb/>corpus <seg type="var">.d.u.</seg> ſed idipſum corpus <seg type="var">.d.u.</seg> ex <lb/>multiplicatione producti <seg type="var">.c.t.</seg> in latus <seg type="var">.n.<lb/>u.</seg> æquale <seg type="var">.m.f.</seg> oritur, & idipſum <seg type="var">.d.u.</seg> ex <lb/>multiplicatione <seg type="var">.n.c.</seg> in latus <seg type="var">.u.t.</seg> æquale <seg type="var">.a.</seg> profertur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0069-02" corresp="fig-0069-02a"> <graphic url="0069-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="89">LXXXIX</num>.</head> <p> <s xml:space="preserve">CVR quarumcunque quatuor quantitatum, ſi prima in ſecundam multiplice-<lb/>tur & hoc productum in tertiam, <choice><ex>rurſusque</ex><am>rurſusq́</am></choice> hoc alterum in quartam, vltimum <lb/>productum æquale ſit producto producti ſecundæ in tertiam, in productum primæ <lb/>in quartam.</s> </p> <pb facs="0070" n="58"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Exempli gratia, caſu ſeſe offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8. <lb/>per .5. & hoc .40. per .3. rurſus hoc .120. per .2. vltimum productum eſſet .240. æqua <lb/>le producto .15. (quod ex .5. in .3. oritur) in productum .16. quod ex .8. in .2. pro-<lb/>fertur.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, cogitemus quatuor numeros quatuor lineis <seg type="var">.a.e.i.o.</seg> <lb/>ſignifi cari, productum autem <seg type="var">.e.</seg> in <seg type="var">.i.</seg> eſſe <seg type="var">.m.f.</seg> et <seg type="var">.r.s.</seg> ſimiliter & productum <seg type="var">.a.</seg> in <seg type="var">.o.</seg> eſ-<lb/>ſe <seg type="var">.m.z</seg>: et <seg type="var">.z.f.</seg> productum eſſe <seg type="var">.m.f.</seg> in <seg type="var">.m.z.</seg> cui productum <seg type="var">.a.</seg> in <seg type="var">.e.</seg> multiplicatum per <lb/>i. & hoc tandem per <seg type="var">.o.</seg> æquari debet.</s> </p> <p> <s xml:space="preserve">Sit itaque <seg type="var">.u.y.</seg> productum <seg type="var">.a.</seg> in <seg type="var">.e.</seg> quod <seg type="var">.u.y.</seg> per <seg type="var">.i.</seg> multiplicatum proferat <seg type="var">.u.s.</seg> <lb/>hocq́ue <seg type="var">.u.s.</seg> multiplicatum per <seg type="var">.o</seg>. </s> <s xml:space="preserve">Dico quod dabit numerum æqualem numero <seg type="var">.f.z.</seg> <lb/>Quamobrem <seg type="var">.r.s.</seg> aut <seg type="var">.m.f.</seg> quod idem eſt, in figura præcedentis theore matis ſigni-<lb/>ficetur linea <seg type="var">.n.u.</seg> & linea <seg type="var">.r.u.</seg> hu-<lb/>ius, nempe <seg type="var">.a.</seg> ſignificetur per <seg type="var">.u.t.</seg> <lb/> <ptr xml:id="fig-0070-01a" corresp="fig-0070-01" type="figureAnchor"/> præcedentis, ex quo numerus pro <lb/>ducti <seg type="var">.u.s.</seg> præſentis, in præcedenti <lb/>ſignificabitur producto <seg type="var">.n.t.</seg> quod <lb/><choice><ex>productum</ex><am>ꝓductũ</am></choice> <seg type="var">.u.s.</seg> <choice><ex>pręsens</ex><am>pręsẽs</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> <choice><ex>præsens</ex><am>præsẽs</am></choice> <seg type="var">.o.</seg> mul<lb/>tiplicatum, quod erat in præceden <lb/>ti <seg type="var">.u.c.</seg> ſignificabitur per <seg type="var">.d.u.</seg> præce <lb/>dentis, quod non modo ex multi-<lb/>plicatione <seg type="var">.n.t.</seg> præcedentis, nempe <seg type="var">.u.s.</seg> præſentis. in <seg type="var">.u.c.</seg> præcedentis æquali <seg type="var">.o.</seg> præ-<lb/>ſentis oritur, ſed etiam ex <seg type="var">.c.t.</seg> præcedentis æquali <seg type="var">.m.z.</seg> præſentis in <seg type="var">.n.u.</seg> præceden <lb/>tis æquali <seg type="var">.m.f.</seg> præſentis. </s> <s xml:space="preserve">Itaque verum eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0070-01" corresp="fig-0070-01a"> <graphic url="0070-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="90">XC</num>.</head> <p> <s xml:space="preserve">CVR quibuſlibet & quantiſuis numeris in ſummam collectis, ſi ab vnitate in ſe-<lb/>cunda ſpecie progreſſionis arithmeticę imparium numerorum progreſſi fue-<lb/>rimus, eiuſmodi ſumma ſemper eſt quadratus numerus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi horum quatuor diſparium numerorum <choice><ex>ſummam</ex><am>ſummã</am></choice>, in dicta pro-<lb/>greſſione arithmetica quis ſumat, principio ab vnitate ſumpto, nempe .1. 3. 5. 7. ſum-<lb/>ma erit .16. numerus quadratus inquam. </s> <s xml:space="preserve">Idem de cæteris.</s> </p> <p> <s xml:space="preserve">Quamobrem animaduertendum eſt, vnitatem, tam ſumi pro ſui ipſius radicem, <lb/>quam pro quadrato, cubo, cenſo cenſi, primo relato, & alia quauis dignitate. <lb/></s> <s xml:space="preserve">Nunc autem pro quadrato ſumamus per <seg type="var">.o.</seg> ſignificato, <choice><ex>cogitemusque</ex><am>cogitemusq́</am></choice> quadratum <seg type="var">.o.</seg> <lb/>includi quadrato vnitatem ſequenti, quod, vt patet, eſt quatuor vnitatum, ac pro-<lb/>priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex ſimi-<lb/>litudine quam cætera quadrata cum hoc primo retinent, ex quaternario denomina-<lb/>tionem acceperunt. </s> <s xml:space="preserve"><choice><ex>Hocitaque</ex><am>Hocitaq;</am></choice> ſit <seg type="var">.o.u.c.e.</seg> ita ex communi ſcientia quadrato <seg type="var">.o.</seg> iun-<lb/>gitur gnomon <seg type="var">.e.c.u.</seg> conſtans tribus vnitatibus, quare primus gnomon, numero im-<lb/>pari conſtat. </s> <s xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-<lb/>ſparibus ſummam excreſcere, cum propius accedere <choice><ex>quam</ex><am>quã</am></choice> binario nequeant, ex quo <lb/>medio binario, ſibi inuicem ſuccedunt. </s> <s xml:space="preserve">Dico igitur quòd quinario ternarium ſub <lb/>ſequente, coniuncto quadrato <seg type="var">.o.u.c.e.</seg> profertur quadratum, quod in numeris, bi-<lb/>narij quadratum ſequitur, <choice><ex>eritque</ex><am>eritq́;</am></choice> ternarij, <choice><ex>quodque</ex><am>quodq́;</am></choice> ſignificetur per <seg type="var">.o.f.</seg> patet enim pri <lb/>mo non differre ab <seg type="var">.o.c.</seg> præter quam gnomone <seg type="var">.b.f.d.</seg> qui coniungitur quadrato <seg type="var">.o.<lb/>c.</seg> quique duabus vnitatibus maior eſt <seg type="var">.e.c.u</seg>. </s> <s xml:space="preserve"><choice><ex>Iam</ex><am>Iã</am></choice> ſcimus gnomonem <seg type="var">.e.o.u.</seg> æqualem <pb facs="0071" n="59"/><fw type="head">THEOREM. ARIT.</fw> eſſe gnomoni <seg type="var">.e.c.u.</seg> <choice><ex>itemque</ex><am>itemq́;</am></choice> gnomonem <seg type="var">.b.f.d.</seg> æqualem gnomoni <seg type="var">.b.o.d.</seg> at hic gno-<lb/>mon <seg type="var">.b.o.d.</seg> ex præſuppoſito, maior eſt gnomone <seg type="var">.e.o.u.</seg> duabus vnitatibus <seg type="var">.b.</seg> et <seg type="var">.d.</seg> <lb/>Itaque etiam gnomon <seg type="var">.b.f.d.</seg> duabus vnitatibus gnomonem <seg type="var">.e.c.u.</seg> ſuperabit. </s> <s xml:space="preserve">Qua-<lb/>re <seg type="var">.b.f.d.</seg> erit impar immediatè ſequens ternarium, qui coniunctus quadrato <seg type="var">.o.c.</seg> <lb/>quadratum ſubſequens componet. </s> <s xml:space="preserve">Eadem ratione probabitur de quadrato <seg type="var">.o.n.</seg> ſe <lb/>quenti <seg type="var">.o.f.</seg> & gnomone <seg type="var">.i.n.a.</seg> cum hic ordo ſpeculationis ſit vniuerſalis. </s> <s xml:space="preserve">In <lb/>quo cernitur quemlibet gnomonem ſibi <choice><ex>contiguum</ex><am>contiguũ</am></choice> inferiorem ſemper duabus vni-<lb/>tat ibus excedere, cumque quadrata non niſi gnomonibus ſibi inuicem ſuccedant. <lb/></s> <s xml:space="preserve">Sed <choice><ex>cum</ex><am>cũ</am></choice> primus <seg type="var">.e.c.u.</seg> diſpar fuerit, <choice><ex>proculdubio</ex><am>ꝓculdubio</am></choice> <choice><ex>etiam</ex><am>etiã</am></choice> <choice><ex>neceſſarioque</ex><am>neceſſarioq́;</am></choice> cæteri diſpares <choice><ex>erunt</ex><am>erũt</am></choice>. <lb/></s> <s xml:space="preserve">Ex qua ſpeculatione, oritur regula ab antiquis tradita <lb/>inueniendi vltimi numeri diſparis <choice><ex>concurrentis</ex><am>cõcurrentis</am></choice> ad <choice><ex>compo ſitionem</ex><am>cõpo ſitionem</am></choice> alicuius quadrati. </s> <s xml:space="preserve">Vt ſi quis ſeire deſideret nu-<lb/>merum vltimum diſparem, quo mediante quadratum <seg type="var">.<lb/>o.n.</seg> conſtitutum fuit, quod aliud non eſt quam ſcire <lb/>quantus ſit numerus vltimi gnomonis <seg type="var">.i.n.a.</seg> æqualis gno <lb/>moni <seg type="var">.i.o.a</seg>. </s> <s xml:space="preserve">Itaque vt ſciamus hunc gnomonem <seg type="var">.i.o.a.</seg> <lb/>patet duplicandam eſſe radicem <seg type="var">.o.e.b.i.</seg> <choice><ex>dabiturque</ex><am>dabiturq́,</am></choice> <seg type="var">.o.e.<lb/>b.i.</seg> et <seg type="var">.o.u.d.a.</seg> vbi bis reperitur <seg type="var">.o.</seg> nos autem tantummo <lb/>do quærimus ſcire gnomonem .i.b.e.o.u.d.a. </s> <s xml:space="preserve">Itaque <lb/>minor eſt vnitate duplo radicis, cum unitas <seg type="var">.o.</seg> bis repe-<lb/>tatur, quæ tamen in gnomone ſemel tantum ſumebatur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0071-01" corresp="fig-0071-01a"> <graphic url="0071-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="91">XCI</num>.</head> <p> <s xml:space="preserve">CVR ſumma quadratorum, quorum radices ſunt in proportione ſeſquitertia <lb/>nempe .4. ad .3. quadrata ſit.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſumemus quadratum .3. ſcilicet 9. quod in ſummam cum qua-<lb/>drato .4. colligemus, nempè .16. <choice><ex>eritque</ex><am>eritq́;</am></choice> quadratum .25. & ita quadratum .6. hoc eſt <num value="36">.<lb/>36.</num> collectum cum quadrato .8. nempè .64. efficiet quadratum .100. ita etiam qua-<lb/>dratum .9. hoceſt .81. coniunctum quadrato .12. nempè .144. producet quadra-<lb/>tum .225.</s> </p> <p> <s xml:space="preserve">In cuius gratiam ſint duo quadrata ſubſcripta <seg type="var">.q.o.</seg> et <seg type="var">.q.a.</seg> quorum radices ſint <seg type="var">.q.</seg> <lb/> <ptr xml:id="fig-0071-02a" corresp="fig-0071-02" type="figureAnchor"/> g. et <seg type="var">.q.p.</seg> hoc eſt <seg type="var">.q.g.</seg> quatuor vnitatum, et <seg type="var">.q.<lb/>p.</seg> trium, ex quo <seg type="var">.q.a.</seg> erit .16. vnitatum et <seg type="var">.q.o.</seg> <lb/>nouem. </s> <s xml:space="preserve">Ad hæc cogitemus applicari quadra-<lb/>to <seg type="var">.q.a.</seg> gnomonem <seg type="var">.f.s.h.</seg> tam amplum ſiue la-<lb/>tum <choice><ex>quam</ex><am>quã</am></choice> gnomon <seg type="var">.b.a.g.</seg> nempè vt <seg type="var">.h.</seg> ſit æqua <lb/>lis .g: g. verò differentia ſit qua <seg type="var">.q.g.</seg> maior eſt <seg type="var">.<lb/>q.p.</seg> <choice><ex>huncque</ex><am>huncq́;</am></choice> gnomonem <seg type="var">.f.s.h.</seg> dico ęqualem eſ <lb/>ſe quadrato <seg type="var">.q.o.</seg> nam ex preſuppoſito <seg type="var">.g.</seg> terra <lb/>dicem <seg type="var">.q.p.</seg> ingreditur, & quater <seg type="var">.q.g.</seg> ex quo, <lb/>tres partes <seg type="var">.q.k.p.</seg> inter ſe æquales ſunt vnde <lb/>etiam quadratum <seg type="var">.q.o.</seg> nouem partibus ſuper-<lb/>ficialibus quadratis conſtabit, quarum ſingula <lb/>rum radix æqualis erit <seg type="var">.g.</seg> cumque præcedenti <lb/>theoremate didicerimus quemlibet gnomo-<lb/>nem quadrati immediatè ſequentis æquę amplitudinis cum gnomone præcedentis, <pb facs="0072" n="60"/><fw type="head">IO. BAPT. BENED.</fw> per duab. vnitatibus ſuperficialibus creſcere, <choice><ex>quarum</ex><am>quarũ</am></choice> <choice><ex>ſingularum</ex><am>ſingularũ</am></choice> radix æqualis eſt <seg type="var">.g.</seg> ne <lb/>ceſſariò ſequitur gnomonem <seg type="var">.b.a.g.</seg> duabus partibus aut vnitatibus gnomonem <seg type="var">.d.<lb/>o.p.</seg> ſuperare, ita vt gnomon <seg type="var">.b.a.g.</seg> ſeptem vnitatibus, aut partibus ſuperficialibus <lb/>quadratis conſtet. </s> <s xml:space="preserve">Quare eadem ratione gnomon <seg type="var">.f.s.h.</seg> conſtabit nouem ſimilibus. <lb/></s> <s xml:space="preserve">Itaque æqualis erit quadrato <seg type="var">.q.o</seg>. </s> <s xml:space="preserve">Quamobrem verum eſt, quòd quadrato <seg type="var">.q.o.</seg> <lb/>coniuncto quadrato <seg type="var">.q.a.</seg> proueniet quadratum <seg type="var">.q.s.</seg> cuius radix ita differet à <seg type="var">.q.g.</seg> vt <seg type="var">.<lb/>q.g.</seg> à <seg type="var">.q.p</seg>: ex quo tres radices arithmeticè inter ſe continuæ proportionales erunt. <lb/></s> <s xml:space="preserve">Idipſum dico ſi <seg type="var">.q.p.</seg> fuerit .6. et <seg type="var">.q.g</seg>: 8: </s> <s xml:space="preserve">tunc enim ſingulæ partes <seg type="var">.q.k.p.g.h.</seg> æquipol <lb/>lebunt duabus vnitatibus, quæ cogitabuntur <lb/> <ptr xml:id="fig-0072-01a" corresp="fig-0072-01" type="figureAnchor"/> in ſummam collectæ, ut cum patribus <seg type="var">.q.k.p.<lb/>g.h.</seg> integris contemplari liceat. </s> <s xml:space="preserve">Idem acci-<lb/>det fi <seg type="var">.q.p.</seg> erit .9. et <seg type="var">.q.g.</seg> 12. fingulæ enim par-<lb/>tes <seg type="var">.q.K.p.g.h.</seg> tripartitæ erunt. </s> <s xml:space="preserve">Idcircò dixi <lb/>gnomonem <seg type="var">.f.s.h.</seg> tam amplum cogitari de-<lb/>bere, quam gnomon <seg type="var">.b.a.g.</seg> nempè ut <seg type="var">.h.</seg> æqua <lb/>lis ſit <seg type="var">.g</seg>. </s> <s xml:space="preserve">Idem occurret ſi <seg type="var">.q.g.</seg> erit .12. et <seg type="var">.q.p.</seg> <lb/>quinque, quod cum fuerit patebitex præce-<lb/>dentis theorematis ſpeculatione, gnomonem <lb/><seg type="var">f.s.h</seg>: 25. vnitatibus conſtare, cogitatum am-<lb/>plitudinis ſimplicis vnitatis denominatæ in <seg type="var">.q.<lb/>p.</seg> aut <seg type="var">.q.g.</seg> non amplitudinis gnomonis <seg type="var">.b.a.g.</seg> <lb/>qui ſeptem vnitatibus latus eſſet. </s> <s xml:space="preserve">Cum igitur <seg type="var">.<lb/>q.p.</seg> quinque vnitatibus linearibus conſtet ſcimus <seg type="var">.q.o</seg>: 25. ſuperficialibus conſtare, <lb/>collecto itaque in ſummam quadrato <seg type="var">.q.o.</seg> cum quadrato <seg type="var">.q.a.</seg> cognoſcetur quadra-<lb/>tum <seg type="var">.q.s.</seg> vnà etiam eius radix. </s> <s xml:space="preserve">Eadem ratione, alia multa quadrata ſimilia contem-<lb/>plari licebit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0071-02" corresp="fig-0071-02a"> <graphic url="0071-02"/> </figure> <figure xml:id="fig-0072-01" corresp="fig-0072-01a"> <graphic url="0072-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="92">XCII</num>.</head> <p> <s xml:space="preserve">CVR propoſito numero pari maiori binario, qui detrahi & in ſummam colli-<lb/>gi debeat ex altero numero quærendo, vt tam reſiduum quam ſumma ſint <lb/>quadrata numerorum integrornm. </s> <s xml:space="preserve">Rectè dimidium propoſiti numeri in ſeipſum <lb/>multiplicamus, & quadrato huic addimus vnitatem, <choice><ex>eritque</ex><am>eritq́;</am></choice> numerus quæfitus.</s> </p> <p> <s xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-<lb/>ro inueſtigando, ut reſiduum detractionis, & ſumma ſint quadrati numeri. </s> <s xml:space="preserve">Addi-<lb/>ta vnitate ipſi .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s> </p> <p> <s xml:space="preserve">Cuius ſpeculationis gratia, ſubſcripta quatuor quadrata cogitemus <seg type="var">.g.p</seg>: <seg type="var">u.i</seg>: <seg type="var">t.c</seg>: <seg type="var">n.<lb/>K.</seg> <choice><ex>cogitemusque</ex><am>cogitemusq́;</am></choice> quadratum <seg type="var">.g.p.</seg> eſſe quadratum ſummæ, <seg type="var">K.n.</seg> verò reſidui ſubtractio-<lb/>nis: <seg type="var">u.i.</seg> <choice><ex>autem</ex><am>aũt</am></choice> numerum <choice><ex>inueſtigandum</ex><am>inueſtigãdũ</am></choice>, ex quo gnomon <seg type="var">.u.d.i.</seg> cognoſcetur ita etiam et <seg type="var">.n.<lb/>o.K.</seg> qui inter ſe ſunt æquales. </s> <s xml:space="preserve">Iam certi erimus <seg type="var">.e.i.</seg> eſſe plus quam dimidium gno-<lb/>monis <seg type="var">.n.o.K</seg>. </s> <s xml:space="preserve">Itaque cogitemus rectangulum <seg type="var">.r.c.</seg> exactum <choice><ex>dimidium</ex><am>dimidiũ</am></choice> eſſe gnomonis <seg type="var">.<lb/>n.o.K.</seg> ex unitatibus ſuperficialibus quarum una erit <seg type="var">.m.a</seg>.</s> </p> <p> <s xml:space="preserve">Cuius numeri quadratum ſit <seg type="var">.t.c.</seg> vnde etiam cognitum & cum <seg type="var">.K.c.</seg> ex communi <lb/>ſcientia ſit vnitas linearis, </s> <s xml:space="preserve">propterea quod <seg type="var">.m.a.</seg> eſt ſuperficialis hoc eſt quadrata, <lb/>quæ detracta ex <seg type="var">.q.c.</seg> dimidio gnomonis <seg type="var">.n.o.K.</seg> (quamuis lineari) ſupererit <seg type="var">.K.q.</seg> co <lb/>gnita, numerorum integrorum (nota <seg type="var">q.K.i.</seg> ſemper minor erit duabus vnitatibus li-<lb/>nearibus & maior vna ex dictis vnitatibus, ut ex te ipſo contemplari potes) </s> <s xml:space="preserve">quare <seg type="var">. <pb facs="0073" n="61"/><fw type="head">THEOR. ARITH.</fw> n.k.</seg> ipſius quadratum numerorum integrorum cognoſcetur, cui addito gnomone <seg type="var">.<lb/>n.o.K.</seg> cognoſcemus numerum <seg type="var">.u.i.</seg> quæſitum.</s> </p> <p> <s xml:space="preserve">Sed cum nobis hæc via, tenenda propoſitum non fuit, hoc eſt primo loco inue <lb/>niendi quadrati minoris <seg type="var">.n.K.</seg> ideo ſupereſt probandum gnomonem <seg type="var">.t.o.c.</seg> vnitati <choice><ex>ae- qualem</ex><am>ę-qualem</am></choice> eſſe, nempe quadratulo <seg type="var">.m.a.</seg> quod patebit, ſi conſideremus nos ſumpſiſſe <lb/>rectangulum <seg type="var">.r.c.</seg> pro dimidio gnomonis <seg type="var">.n.o.K</seg>. </s> <s xml:space="preserve">etenim ſi ſupplemento etiam <seg type="var">.n.r.</seg> qua <lb/>dratulum æquale <seg type="var">.m.a.</seg> adderetur, pateret gnomonem <seg type="var">.n.a.K.</seg> cum dicto quadratulo <lb/>collectum, æqualem eſſe gnomoni <seg type="var">.n.o.K</seg>: cum duo ſupplementa <seg type="var">.m.t.</seg> et <seg type="var">.m.c.</seg> inter ſe <lb/>fint æqualia. </s> <s xml:space="preserve">Quamobrem inuento quadrato <seg type="var">.t.c.</seg> ex dimidio gnomonis cognito, <lb/>additur vnitas, gnomon ſcilicet <seg type="var">.t.o.c.</seg> ex quo cognoſcitur numerus <seg type="var">.u.i.</seg> quæſitus. <lb/></s> <s xml:space="preserve">Quod autem quadratum <seg type="var">.g.p.</seg> numeris integris conſtet, hac ratione probatur viſum <lb/>enim fuit ſupra quadratum <seg type="var">.n.K.</seg> verè quadratum eſſe, & numeris integris conſtare, <lb/>pariter etiam <seg type="var">.t.c.</seg> <choice><ex>ſeque</ex><am>ſeq́;</am></choice> mutuo conſequi (nam <seg type="var">.K.c.</seg> eſt vnitas linearis) ex quo gnomon <lb/><seg type="var">n.a.K.</seg> numero diſpari conſtabit, ex ijs quæ .90. theoremate probata fuerunt. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <lb/>ex eodem theoremate neceſſe eſt gnomonem <seg type="var">.t.d.c.</seg> etiam numero diſpari conſtare, <lb/>ita vt à numero <seg type="var">.n.a.K.</seg> non niſi duabus vnitatibus differat, nempe vt <seg type="var">.c.p.</seg> ſit vnitas li-<lb/>nearis, ſed ita reuera eſt, numerus enim <seg type="var">.u.d.i.</seg> ex præſuppoſito par eſt, </s> <s xml:space="preserve">quare nume <lb/>rus <seg type="var">.t.d.c.</seg> diſpar erit, cum alterum vnitate ſuperet, videlicet gnomone <seg type="var">.t.o.c.</seg> vnita <lb/>ri æquali, tum <seg type="var">.n.a.K.</seg> minor eſt <seg type="var">.n.o.K.</seg> ex eodem gnomone <seg type="var">.t.o.c.</seg> unitati æquali. </s> <s xml:space="preserve">Ita <lb/>que <seg type="var">.n.a.K.</seg> minor erit <seg type="var">.u.d.i.</seg> per vnitatem, & minor <seg type="var">.t.d.c.</seg> per duas unitates, ex quo ſe-<lb/>quitur <seg type="var">.g.p.</seg> eſſe quadratum <choice><ex>integrorum</ex><am>integrorũ</am></choice> ex dicto theoremate ac con ſequens quadrato <lb/><seg type="var">t.c</seg>. </s> <s xml:space="preserve">quare <seg type="var">.c.p.</seg> vnitas erit, & radices <seg type="var">.q.K.</seg> et <seg type="var">.q.p.</seg> horum quadratorum numero bina-<lb/>rio inter ſe different. </s> <s xml:space="preserve">Vnà etiam ſcienda eſt cauſa, cur numerus propoſitus neceſſa <lb/> <ptr xml:id="fig-0073-01a" corresp="fig-0073-01" type="figureAnchor"/> riò binario maior eſſe debeat. </s> <s xml:space="preserve">Etenim <choice><ex>cum</ex><am>cũ</am></choice> ipſe <lb/>ſit futurus gnomon <seg type="var">.n.o.K.</seg> nec poſſit minor eſſe <lb/>numero ternario, vt patet ex .90. theoremate, <lb/>idcirco ſequitur neceſſariò maiorem eſſe bina-<lb/>rio debere. </s> <s xml:space="preserve">Quòd ſi diſpar numerus propone-<lb/>retur, nec forma operis nec ſpeculationis <choice><ex>mutan- da</ex><am>mutã-da</am></choice> eſſet. </s> <s xml:space="preserve">Non erit tamen neceſſarium vt ipſa <lb/>quadrata <seg type="var">.n.K.</seg> et <seg type="var">.g.p.</seg> numeris integris conſta-<lb/>rent. </s> <s xml:space="preserve">Sæpius enim fractis <choice><ex>componerentur</ex><am>cõponerentur</am></choice>, quod <lb/>ex .90. theoremate facile erit ſpeculari nihilo-<lb/>minus fractis integris, <choice><ex>ipſisque</ex><am>ipſisq́;</am></choice> collectis cum ſuis <lb/>fractis ſummæ eſſent quadratæ.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0073-01" corresp="fig-0073-01a"> <graphic url="0073-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="93">XCIII</num>.</head> <p> <s xml:space="preserve">CVR propoſitis duobus numeris altero pari, altero verò diſpari, duplo primi <lb/>minore per vnitatem, ſi alium inuenire numerum voluerimus, cui alterum iſto <lb/>rum coniunctum proferat quadratum, & altero detracto, quadratum ſuperſit. </s> <s xml:space="preserve">Re-<lb/>ctè datos numeros in ſummam colligemus, quam ſummam in duas quam maximas <lb/>poterimus partes diuidemus, quarum vna pari, altera diſpari conſtet, tum vtran-<lb/>que in ſeipſam multiplicabimus, & quadrato minori, duorum numerorum propo-<lb/>ſitorum quemuis ademus, ex quo cupimus nobis quadratum minus ſupereſſe, & pro <lb/>ueniet nobis numerum quæſitum.</s> </p> <p> <s xml:space="preserve">Exempli gtatia, proponuntur numeri .11. et .6. quorum alter alicui numero ad- <pb facs="0074" n="62"/><fw type="head">IO. BAPT. BENED.</fw> dendus, alter ex eodem detrahendus ſit, ex quo proferri debeant bina qua-<lb/>drata. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> numeri illi in ſummam collecti dabunt .17. differentiam minoris quadra <lb/>ti & maioris. </s> <s xml:space="preserve">I am ſi ex hoc .17. binas partes fecerimus, altera erit .8. altera .9. qui <lb/>bus in ſeipſis multiplicatis alterum quadratum erit .64. alterum .81. addito <choice><ex>itaque</ex><am>itaq;</am></choice> ipſi. <lb/>64. 11. aut .6. pro libito, propoſitum numerum conſequemur. </s> <s xml:space="preserve">cui addito .6. vel .11. <lb/>dabit nobis .81. vel ex ipſo detracto .11. vel .6. relinquet nobis 64. in pręſenti autem <lb/>exemplo talis numerus erit, aut .70. vel .75. </s> <s xml:space="preserve">Huius autem theorematis ſpeculatio <lb/>ex .90. dependet, quo demonſtratum fuit gnomonem proximè quadratum ſequen <lb/>tem, vnitate duplo radicis minorem eſſe.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="94">XCIIII</num>.</head> <p> <s xml:space="preserve">CVR ſi quis cupiat ſummam progreſſionis arithmeticæ quam citiſſimè cogno <lb/>ſcere. </s> <s xml:space="preserve">Rectè coniunget vltimo termino vnitatem primum terminum, huius <lb/>poſtea vltimi termini dimidium cum numero terminorum multiplicabit, ex <lb/>quo multiplicationis productum, erit omnium propoſitorum terminorum ſumma, <lb/>aut eundem vltimum terminum iunctum primo, per dimidium numeri terminorum <lb/>multiplicabit. </s> <s xml:space="preserve">Nam idipſum eueniet.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proponerentur .17. termini in prima progreſſione arithmeti-<lb/>ca naturali, vltimus eſſet .17. cui coniuncta vnitate primo termino ſumma erit .18. <lb/>cuius dimidium cum numero terminorum, nempe .17. multiplicatum cum fuerit, <lb/>oritur productum .153. </s> <s xml:space="preserve">Idpſum eueniet, multiplicato dimidio numeri <choice><ex>terminorum</ex><am>terminorũ</am></choice> <lb/>per vltimum coniunctum vnitati primo termino.</s> </p> <p> <s xml:space="preserve">Quod vt ſciamus, cogitemus terminos progreſſionis collocari, vt in figura ſub-<lb/>ſcripta <seg type="var">.a.o.n.</seg> collocantur, <choice><ex>tanquam</ex><am>tanquã</am></choice> per gradus, ſumpto principio ab vnitate <seg type="var">.n.</seg> tum <seg type="var">.<lb/>u.t.</seg> atque ita gradatim. </s> <s xml:space="preserve">Sic cogitato abſoluto parallelogrammo <seg type="var">.q.o.</seg> ſciemus aper-<lb/>tè ſummam progreſſionis tanto maiorem eſſe dimidio totius <choice><ex>parallelogrammi</ex><am>parallelogrãmi</am></choice>, quan <lb/>tum dimidium numeri diametri <seg type="var">.a.e.i.c.u.n.</seg> requirit. </s> <s xml:space="preserve">Nam cum parallelogram-<lb/>mum diuidatur à dl<unclear reason="illegible"/>ametro in tres partes, diameter vnam occupat, reliquæ verò duę <lb/>ambientes diametrum inter ſe ſunt æquales. </s> <s xml:space="preserve">Sumpto <choice><ex>itaque</ex><am>itaq;</am></choice> diametro cum altera di <lb/>ctarum duarum partium, patet ſumi pluſquam <choice><ex>dimidium</ex><am>dimidiũ</am></choice> totius <choice><ex>parallelogrammi</ex><am>parallelogrãmi</am></choice>. </s> <s xml:space="preserve">pro <lb/>tanta portione, quantum eſt dimidiam occupatam à diametro, qui <choice><ex>cam</ex><am>cã</am></choice><unclear reason="illegible"/> ex diſcretis <lb/>reſpondentibus numero terminorum componatur, conſtat numero æquali eſſe di-<lb/>cto numero terminorum <seg type="var">.o.n</seg>. </s> <s xml:space="preserve">Iam ſi quis multiplicet <seg type="var">.a.o.</seg> per dimidium <seg type="var">.o.n.</seg> procul <lb/>dubio, ex prima ſexti aut .18. ſeptimi, orietur <choice><ex>dimidium</ex><am>dimidiũ</am></choice> numeri <choice><ex>parallelogrammi</ex><am>parallelogrãmi</am></choice> <seg type="var">.q.o.</seg> <lb/>quod minus erit ſumma progreſſionis dimidio numeri diametri, aut quod idem eſt <lb/>dimidio <seg type="var">.o.n.</seg> ſed hoc dimidium <seg type="var">.o.n.</seg> æquale eſt producto dimidij vnitatis <seg type="var">.n.</seg> in <seg type="var">.o.n.</seg> <lb/>ex .20. ſeptimi, cum dimidium <seg type="var">.o.n.</seg> ſit eius productum in <choice><ex>vnitatem</ex><am>vnitatẽ</am></choice>. </s> <s xml:space="preserve">Itaque multipli-<lb/>cato <seg type="var">.n.o.</seg> per dimidium <seg type="var">.o.a.</seg> coniunctum dimidio vnitatis <seg type="var">.n.</seg> oritur ſumma quæſita <lb/>propoſitæ progreſſionis. </s> <s xml:space="preserve">Idipſum accidet multiplicata ſumma <seg type="var">.o.a.</seg> & vnitate <seg type="var">.n.</seg> <choice><ex>per</ex><am>ꝑ</am></choice> <lb/>dimidium <seg type="var">.o.n.</seg> ex .20. ſeptimi, cum proportio totius ad totum eadem ſit, quæ dimi <lb/>dijad dimidium, ex cauſa permutationalitatis. </s> <s xml:space="preserve">Patet etiam in progreſſionibus, <lb/>quæ ab vnitate initium ducunt, ſi fiat aſcenſus per binarium ſumma vltimi termini <lb/>cum primo ſemper duplam futuram eſſe numero terminorum, quod ſequentes figu <pb facs="0075" n="63"/><fw type="head">THEOREM. ARIT.</fw> ras confideranti ſpeculari licebit, Diametros harum figurarum notaui literis ſiue <lb/>characteribus <seg type="var">.a.e.i.c.u.n</seg>.</s> </p> <figure place="here"> <graphic url="0075-01"/> </figure> <figure place="here"> <graphic url="0075-02"/> </figure> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="95">XCV</num>.</head> <p> <s xml:space="preserve">IN progreſſionibus, quæ ab alio termino quam vnitate incohantur, idipſum vt <lb/>monuimus accidit, hoc tamen notato, quòd ex conſequenti quælibet pars dia-<lb/>metri <choice><ex>parallelogrammi</ex><am>parallelogrãmi</am></choice>, minimo termino æqualis erit, prout in progreſſionibus quæ <lb/>ab vnitate originem ducunt, ſingulæ partes diametri, vnitati ſui primi termini æ-<lb/>quales ſunt. </s> <s xml:space="preserve">At in reliquis progreſſionibus, vt in figura patet, eadem eſt propor-<lb/>tio totius diametri ad <seg type="var">.o.n.</seg> quæ minimi termini ad vnitatem ex .13. quinti, nempe <seg type="var">.<lb/>a.o.</seg> ad <seg type="var">.o.n.</seg> vt <seg type="var">.n.n.n.n.</seg> ad <seg type="var">.n</seg>. </s> <s xml:space="preserve">In eiuſmodi progreſſionibus accidit quoque <choice><ex>parallelo- grammum</ex><am>parallelo-grãmum</am></choice> à diametro in tres partes diuidi, quarum vnam ipſe occupat, reliquæ ve-<lb/>ro inter ſe æquales ipſum ambiunt. </s> <s xml:space="preserve">Ex quo illud etiam ſequitur, productum <seg type="var">.a.o.</seg> in <lb/>dimidium <seg type="var">.o.n.</seg> æquale eſſe dimidio <choice><ex>parallelogrammi</ex><am>parallelogrãmi</am></choice>, quod minus eſt ſumma progreſ-<lb/>ſionis dimidio diametri, quod dimidum ſi inuenire voluerimus, minimum <choice><ex>terminum</ex><am>terminũ</am></choice> <seg type="var">.<lb/>n.n.n.n.</seg> per dimidium <seg type="var">.o.n.</seg> multiplicabimus, & ex .18. aut .19. ſeptimi ipſum habe-<lb/>bimus, <choice><ex>quandoquidem</ex><am>quandoquidẽ</am></choice> minimo termino per totum <seg type="var">.o.n.</seg> multiplicato profertur integer <lb/>diameter ex .20. prædicti. </s> <s xml:space="preserve">Etenim vt diximus, eadem eſt proportio totius diame-<lb/>tri ad <seg type="var">.o.n.</seg> quæ minimi termini ad vnitatem. </s> <s xml:space="preserve">Ita etiam dico ex dicta .20. ſeptimi. <lb/></s> <s xml:space="preserve">idem dimidium diametri oriri, ſi quis dimidium minimi termini nempè <seg type="var">.n.n.</seg> per to <lb/>tum <seg type="var">.o.n.</seg> multiplicauerit. </s> <s xml:space="preserve">Quamobrem qui ſtatim ſummam propoſitæ progreſſionis <lb/>cognoſcere voluerit, <lb/> <ptr xml:id="fig-0075-03a" corresp="fig-0075-03" type="figureAnchor"/> ſemper primum termi <lb/>num <seg type="var">.n.n.n.n.</seg> cum <seg type="var">.a.o.</seg> <lb/>coniunget, qua ſumma <lb/>per <choice><ex>dimidium</ex><am>dimidiũ</am></choice> <seg type="var">.o.n.</seg> mul-<lb/>tiplicata, aut <seg type="var">.o.n.</seg> per <lb/>dimidium dictæ ſum-<lb/>mæ, ex prædictis rationibus propofitum conſequemur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0075-03" corresp="fig-0075-03a"> <graphic url="0075-03"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="96">XCVI</num>.</head> <p> <s xml:space="preserve">CVR ſi quis numerum terminorum inuenire velit, cognitis tantummodo pri <lb/>mo atque vltimo, rectè vltimum per primum diuidet, ex quo proueniens <pb facs="0076" n="64"/><fw type="head">IO. BAPT. BENED.</fw> numerus quæſitus erit.</s> </p> <p> <s xml:space="preserve">Quod intelligendum eſttamen quoties primus terminus differentia <choice><ex>terminorum</ex><am>terminorũ</am></choice> <lb/>eſt, nempe aſcendens ipſorum ter minorum.</s> </p> <p> <s xml:space="preserve">Cuius ratio manifeſtè ſpeculari poteſt in figura præcedentis theorematis. </s> <s xml:space="preserve">Nam <lb/>diuiſa <seg type="var">.a.o.</seg> per <seg type="var">.n.n.n.n.</seg> eadem proportio erit <seg type="var">.a.o.</seg> ad proueniens, quæ. n <seg type="var">.n.n.<lb/>n.</seg> ad vnitatem <seg type="var">.n.</seg> ex definitione diuiſionis. </s> <s xml:space="preserve">At ſuperius dictum fuit ita ſe ha bere <seg type="var">.a.<lb/>o.</seg> ad <seg type="var">.o.n.</seg> vt <seg type="var">.n.n.n.n.</seg> ad <seg type="var">.n.</seg> ex quo ſequitur ex .11. et .9. quinti pr oueniens eſſe nume-<lb/>rum quæſitum <seg type="var">.o.n</seg>.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="97">XCVII</num>.</head> <p> <s xml:space="preserve">VBI verò primus terminus, reliquorum non erit differentia. </s> <s xml:space="preserve">Hac de caufa ne-<lb/>ceſſe eſt detrahere primum ex vltimo, <choice><ex>reſiduumque</ex><am>reſiduumq́;</am></choice> per numerum aſcenden-<lb/>tem differentiam ſcilicet, partiri, <choice><ex>proueniensque</ex><am>proueniensq́;</am></choice> vnitati coniungere, quò numerum <lb/>terminorum habere poſſimus. </s> <s xml:space="preserve">Scimus etenim tam multas vnitates eſſe in vltimo <lb/>terminorum quot in omnibus interuallis aut differentijs in ſummam collectis ſimul <lb/>cum vnitatibus primi termini, <choice><ex>totque</ex><am>totq́;</am></choice> funt termini, quot interualla ſimul cum pri-<lb/>motermino. </s> <s xml:space="preserve">Quare fi minimus terminus interuallo æqualis fuerit. </s> <s xml:space="preserve">Vltimo per pri-<lb/>mum diuiſo, ex a dductis præcedenti theoremate propofitum confequemur. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <lb/>primo termino ex vltimo detracto <choice><ex>refiduoque</ex><am>refiduoq́;</am></choice> per interuallum, hoc eft numerum dif-<lb/>ferentiæ diuifo, proueniens erit numerus terminorum abſque primo quod vnus eft, <lb/>coni uncto quoque dicto prouenienti propoſitum conſequemur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="98">XCVIII</num>.</head> <p> <s xml:space="preserve">CVR fi quis arithmeticæ progreſſionis dato primo & vltimo fimul cum nume <lb/>ro terminorum, afcendentem numerum cognofcere voluerit. </s> <s xml:space="preserve">Rectè primuin <lb/>ex vltimo detrahet, <choice><ex>refiduumque</ex><am>refiduumq́;</am></choice> per numerum terminorum excepto vno diuidet.</s> </p> <p> <s xml:space="preserve">Huius theorematis ſpeculatio ex .13. theoremate manifeſta crit, nam in præce-<lb/>denti cap. numerus terminorum erat proueniens diuiſionis reſidui ſubtractionis pri-<lb/>mi termini ex vltimo.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="99">XCIX</num>.</head> <p> <s xml:space="preserve">CVR ſi quis maximum omnium terminorum dictæ progreffionis cognofcere <lb/>voluerit, dato primo vnà cum numero aſcendenti, <choice><ex>numeroque</ex><am>numeroq́;</am></choice> terminorum. </s> <s xml:space="preserve">Re-<lb/>ctè numerum afcendentem cum numero terminorum excepto vno multiplicabit, <lb/><choice><ex>productoque</ex><am>productoq́;</am></choice> primum terminum coniunget.</s> </p> <p> <s xml:space="preserve">Cuius quidem theorematis tum ex vndecimo, tum ex ijs quæ præcedentibus ca-<lb/>pitibus dicta fuerunt, aperta eſt ratio.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="100">C</num>.</head> <p> <s xml:space="preserve">CVR veteres cupientes obtinere ſummam pr<unclear reason="illegible"/>ogreffionis continuæ naturalis, <lb/>quæab vnitate initium ducit, dato vltimo termino tantummodo. </s> <s xml:space="preserve">Dimidium <lb/>vltimi-termini <choice><ex>cum</ex><am>cũ</am></choice> toto fequente multiplicabant, <choice><ex>productumque</ex><am>productumq́;</am></choice> ſumma quæſita erat.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi vltimus terminus eiuſmodi progreſſionis fuerit .7. multiplica- <pb facs="0077" n="65"/><fw type="head">THEOR. ARITH.</fw> to dimidio ipſius nempe .3. & dimidio, cum numero ipſum terminum <choice><ex>ſequenti</ex><am>ſequẽti</am></choice>, nem <lb/>pè .8. ſumma dictorum terminorum erit .28.</s> </p> <p> <s xml:space="preserve">Huius autem ſpeculatio ex .94. theoremate dependet, in quo facilè depræhen-<lb/>dere licet ex figura continuæ progreſſionis naturalis, numerum terminorum maxi-<lb/>mo termino ſemper æqualem eſſe; </s> <s xml:space="preserve">ex quo <choice><ex>tantum</ex><am>tãtum</am></choice> eſt dimidium numeriterminorum, <lb/>quantum maximi dimidium, <choice><ex>tantusque</ex><am>tantusq́;</am></choice> eſt vltimus terminus vnitati coniunctus, quan <lb/>tus numerus is, qui vltimum terminum conſequitur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="101">CI</num>.</head> <p> <s xml:space="preserve">CVR antiqui idip fum, quod iam dictum eft, in ea progreſſione, cuius vltimus ter <lb/>minus diſpar eſt ſcire cupientes, numerum integrorum proximè dimidium <lb/>maximi ſequentem ſumebant, quem per maximum multiplicabant, ex quo <lb/>ſumma quæſita oriebatur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi dimidium maximi fuiſſet .3. cum dimidio, fumebant quatuor, <lb/>& per maximum .7. multiplicabant, ex quo pariter proferebatur ſumma .28.</s> </p> <p> <s xml:space="preserve">Cuius ratio ex .20. ſeptimi Euclidis oritur, cum eadem ſit proportio numeri fe-<lb/>quentis ma ximum ad numerum dimidium maximi ſequentem; </s> <s xml:space="preserve">quæ maximi ad <choice><ex>fuum</ex><am>fuũ</am></choice> <lb/>dimidium, eſt enim dupla.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="102">CII</num>.</head> <p> <s xml:space="preserve">TRaditum eſt à nonnullis, à veteribus obſeruatam fuiſſe hancregulam, qua ſci-<lb/>re poſſent ſummam alicuius progreſſionis arithmeticæ diſcontinuæ aut inter <lb/>cifæ, quæ numero pari terminetur. </s> <s xml:space="preserve"><choice><ex>Multiplicabant</ex><am>Multiplicabãt</am></choice> enim <choice><ex>dimidium</ex><am>dimidiũ</am></choice> vltimi termini per <lb/>pro ximum numerum dimidio dicto maiorem, ex quo <choice><ex>inquiebant</ex><am>inquiebãt</am></choice> ſemper productum <lb/>ſummæ quæſitæ æquale eſſe, <choice><ex>ſubijciuntque</ex><am>ſubijciuntq́;</am></choice> exemplum progreſſionis, quæ à binario in-<lb/>choata crefcit per binarium. </s> <s xml:space="preserve">In qua quidem progreſſione non per fe, fed per acci-<lb/>dens regula vera eft. </s> <s xml:space="preserve">Hoc eſt, non quia ex ſe vnus ex producentibus numeris dimi-<lb/>dium termini maioris futurus ſit, alter uerò proximè ſequens dimidium, fed quia <lb/>vt dictum eſt .95. theoremate, eadem eſt proportio maximi termini ad numerum <lb/>terminorum, quæ minimi ad vnitatem. </s> <s xml:space="preserve">Cumq́ue in præfenti exemplo minimum <lb/>ſit duplum vnitati in eiuſmodi caſu, numerus terminorum, dimidio maximi termini <lb/>æqualis eſt, qui terminorum numerus ex ſe, vt patet, vnus eſt ex producentibus, al-<lb/>ter verò producens numerus, eſt proximè dimidium ſequens, non exſe, fed quia nu <lb/>merus ſequens, dimidium eſt ſummæ maximi, & minimi, quæ per fe alter eſſe de-<lb/>bet producens numerus. </s> <s xml:space="preserve">In cæteris enim progreſſionibus, quæ binario non creſcút <lb/>regulafal<unclear reason="illegible"/>fa eſt, prout facilè patere poteſt ei, qui ex ſcientiæ legibus ope ſpeculatio-<lb/>nis .95. theorematis ſpeculatus fuerit.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="103">CIII</num>.</head> <p> <s xml:space="preserve">ALIAM quoque tradunt regulam, qua veteres vſos fuiſſe dicunt, quo ſum-<lb/>mam ſcire poſſent progreſſionis diſcontinuæ, quænumero diſpari abſolui-<lb/>tur. </s> <s xml:space="preserve">Ea autem eſt eiuſmodi. </s> <s xml:space="preserve">Vltimum terminum in duas quam maximè poterant ma-<lb/>ximas partes diuidebant, quarum vna ſemper altera maior erat, banc autem maio-<lb/>rem in ſeipſam multiplicabant, at que quadratum hoc, ſummam progreffionis effe <pb facs="0078" n="66"/><fw type="head">IO. BAPT. BENED.</fw> affirmabant. </s> <s xml:space="preserve">Quæ ſanè regula, non ſemper, etſi interdum vera ſit.</s> </p> <p> <s xml:space="preserve">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creſcit per bina <lb/>rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè ſe-<lb/>quens, nempe è duabus partibus vltimi termini maior, æqualis ſit numero termino <lb/>rum, qui per ſe vnus è producentibus, ex ijs que .94. theoremate diximus, eſſe debet; <lb/></s> <s xml:space="preserve">alter vero producens, qui per ſe dimidium ſummæ primi & vltimi eſſe debet, per <lb/>accidens pars maior eſt duarum vltimi termini, & alteri producenti æqualis.</s> </p> <p> <s xml:space="preserve">Aut alio modo ratiocinemur, dicentes, in huiuſmodi progreſſione dimidium <lb/>ſummæ vltimi termini cum primo, ſemper medium proportionale eſt inter eam <lb/>ſummam & dimidium numeri terminorum, etenim huiuſmodi ſumma numero ter-<lb/>minorum ſemper dupla eſt, prout .94. theoremate tradimus. </s> <s xml:space="preserve">Itaque ex .20. ſeptimi, <lb/>quadratum partis maioris, producto ſummæ dictæ in numerum dimidij <choice><ex>terminorum</ex><am>terminorũ</am></choice> <lb/>æquale erit, quod productum per ſe ſummæ progreſſionis eſt æquale. </s> <s xml:space="preserve">At in cæte-<lb/>ris eiuſmodi progreſſionibus fallit regula, vt ex ſupradictis facilè demonſtratur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="104">CIIII</num>.</head> <p> <s xml:space="preserve">PErmultis terminis ad libitum propoſitis, diſpoſitis nihilominus progreſſio-<lb/>ne, aut proportionalitate geometrica continua, ſi minimus ex maximo & exfe-<lb/>quenti minimum detrahatur, reſiduum maximi, eam proportionem ad fum-<lb/>mam reliquorum omnium terminorum retinebit, quam reſiduum ſecundi ad pri-<lb/>mum.</s> </p> <p> <s xml:space="preserve">Proponuntur, exempli gratia, quatuor termini .3. 12. 48. 192. continui geome-<lb/>tricè proportionales, ſi primum, hoc eſt minimum, ex ſecundo, & maximo detra <lb/>has, exſecundo ſupererit .9. ex maximo .189. quod ſi minimum per reſiduum maxi <lb/>mi multiplicaueris, hoc eſt .189. orietur .567. tum ſi huiuſmodi productum per .9. <lb/>( refiduum ſecundi ) diuiſeris, proueniet .63. quod proueniens æquale erit ſummæ <lb/>reliquorum omnium terminorum, maximo excepto. </s> <s xml:space="preserve">Ex quo inferre licet ex .20. ſe <lb/>ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut ſi reſiduum ſecundi <lb/>per ſummam dictorum terminorum multiplicaueris produceturidem .567. </s> <s xml:space="preserve">quare <lb/>ex .20. ſeptimi & cætera.</s> </p> <p> <s xml:space="preserve">Quod vt <choice><ex>ſcientificè</ex><am>ſciẽtificè</am></choice> poſſimus, & in vniuerſum ſpeculari. </s> <s xml:space="preserve">Quatuor termini propo-<lb/>ſiti, quatuor ſubſcriptis lineis <choice><ex>ſignificentur</ex><am>ſignificẽtur</am></choice> <seg type="var">.b.i</seg>: <seg type="var">c.a</seg>: <seg type="var">f.r</seg>: <seg type="var">m.s.</seg> (quod <choice><ex>autem</ex><am>aũt</am></choice> de his quatuor di <lb/>co de <choice><ex>centummillibus</ex><am>centũmillibus</am></choice>, & eo amplius dicere poſſum.) </s> <s xml:space="preserve">Nunc minimus terminus <seg type="var">.m.s.</seg> ex <lb/>maximo <seg type="var">.b.i.</seg> detrahatur, <choice><ex>ſuperſitque</ex><am>ſuperſitq́;</am></choice> <seg type="var">.n.i.</seg> <choice><ex>idemque</ex><am>idemq́;</am></choice> <seg type="var">.m.s.</seg> ex ſecundo termino <seg type="var">.f.r.</seg> ſubtra-<lb/>hatur, <choice><ex>ſuperſitque</ex><am>ſuperſitq́;</am></choice> <seg type="var">.o.r</seg>. </s> <s xml:space="preserve">Dico proportionem <seg type="var">.n.i.</seg> ad ſummam reliquorum omnium ter-<lb/>minorum <seg type="var">.c.a</seg>: <seg type="var">f.r</seg>: <seg type="var">m.s.</seg> eandem effe, quæ <seg type="var">.o.r.</seg> ad <seg type="var">.m.s</seg>. </s> <s xml:space="preserve">Quamobrem ex tertio & quar-<lb/>to ſecundus <seg type="var">.f.r.</seg> <choice><ex>detrahatur</ex><am>detrahat̃</am></choice>, ex <choice><ex>tertioque</ex><am>tertioq́;</am></choice> ſuperſit <seg type="var">.t.a.</seg> & ex quarto <seg type="var">.e.i.</seg> ita etiam tertius <seg type="var">.<lb/>c.a.</seg> ex quarto <seg type="var">.b.i.</seg> <choice><ex>ſuperſitque</ex><am>ſuperſitq́;</am></choice> <seg type="var">.d.i.</seg> ſanè <lb/> <ptr xml:id="fig-0078-01a" corresp="fig-0078-01" type="figureAnchor"/> ſic ſe habebit <seg type="var">.c.a.</seg> ad <seg type="var">.f.r.</seg> vt <seg type="var">.c.t.</seg> ad <seg type="var">.f.o.</seg> <lb/>vt <choice><ex>quisque</ex><am>quisq;</am></choice> per ſe ſcire poteſt. </s> <s xml:space="preserve">Quare ex <lb/>19. quinti ſic ſe habebit <seg type="var">.a.t.</seg> ad <seg type="var">.r.o.</seg> vt <seg type="var">.<lb/>c.a.</seg> ad <seg type="var">.f.r.</seg> & permutando ita <seg type="var">.a.t.</seg> ad <seg type="var">.a.<lb/>c.</seg> vt <seg type="var">.o.r.</seg> ad <seg type="var">.r.f.</seg> & ſeparando ſic <seg type="var">.a.t.</seg> ad <seg type="var">.<lb/>a.c.</seg> (hoc eſt <seg type="var">.f.r.</seg>) vt <seg type="var">.r.o.</seg> ad <seg type="var">.o.f.</seg> vide-<lb/>licet <seg type="var">.m.s</seg>. </s> <s xml:space="preserve"><choice><ex>Idem</ex><am>Idẽ</am></choice> dico de <seg type="var">.d.i.</seg> ad <seg type="var">.a.c.</seg> nem-<lb/>pe ſic ſe habebit <seg type="var">.d.i.</seg> ad <seg type="var">.a.c.</seg> vt <seg type="var">.a.t.</seg> ad <seg type="var">. <pb facs="0079" n="67"/><fw type="head">THEOREM. ARIT.</fw> r.f.</seg> hoc eſt <seg type="var">.o.r.</seg> ad <seg type="var">.m.s.</seg> ex .11. quinti. </s> <s xml:space="preserve">Itaque ex communi ſcientia ſic ſe habe-<lb/>bit <seg type="var">.d.i.</seg> ad <seg type="var">.d.b.</seg> vt <seg type="var">.e.d.</seg> ad <seg type="var">.e.b</seg>: cum <seg type="var">.e.d.</seg> æqualis ſit <seg type="var">.t.a</seg>. </s> <s xml:space="preserve">Ita etiam vt <seg type="var">.e.n.</seg> ad <seg type="var">.n.b</seg>: cum <seg type="var">.n.<lb/>e.</seg> æqualis ſit <seg type="var">.o.r</seg>. </s> <s xml:space="preserve">Iam ſi ſic ſe habeat <seg type="var">.d.i.</seg> ad <seg type="var">.d.b.</seg> vt <seg type="var">.d.e.</seg> ad <seg type="var">.e.b.</seg> permutando <choice><ex>quoque</ex><am>quoq;</am></choice> ſic <lb/>ſe habebit <seg type="var">.d.i.</seg> ad <seg type="var">.d.e.</seg> vt <seg type="var">.d.b.</seg> ad <seg type="var">.b.e.</seg> & compon endo ita <seg type="var">.i.d.e.</seg> ad <seg type="var">.e.d.</seg> vt <seg type="var">.d.b.e.</seg> ad <seg type="var">.e.<lb/>b.</seg> & permutando ſic <seg type="var">.i.d.e.</seg> ad <seg type="var">.d.b.e.</seg> vt. de <seg type="var">.a.d.e.b.</seg> nempe vt <seg type="var">.e.n.</seg> ad <seg type="var">.n.b.</seg> & permutan <lb/>do ita <seg type="var">.i.d.e.</seg> ad <seg type="var">.e.n.</seg> vt <seg type="var">.d.b.e.</seg> ad <seg type="var">.b.n.</seg> & componendo ita <seg type="var">.i.d.e.n.</seg> ad <seg type="var">.n.e.</seg> vt <seg type="var">.d.b.e.</seg> et <seg type="var">.b.<lb/>n.</seg> ad <seg type="var">.b.n.</seg> & permutando ſic <seg type="var">.i.d.e.n.</seg> ad <seg type="var">.d.b.e.</seg> et <seg type="var">.b.n.</seg> nempe ad <seg type="var">.a.c</seg>: <seg type="var">f.r</seg>: <seg type="var">m.s</seg>: vt <seg type="var">.e.n.</seg> ad <seg type="var">.<lb/>n.b.</seg> hoc eſt. ut <seg type="var">.o.r.</seg> ad <seg type="var">.m.s.</seg> quod erat propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0078-01" corresp="fig-0078-01a"> <graphic url="0078-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="105">CV</num>.</head> <p> <s xml:space="preserve">CVR deſideranti ſummam quorumcunque terminorum progreſſionis conti-<lb/>nuæ geometricæ cognoſcere. </s> <s xml:space="preserve">Rectè minimus terminus ex maximo detrahen <lb/>dus eſt, <choice><ex>reſiduumque</ex><am>reſiduumq́;</am></choice> per denominantem progreſſionis dempta vnitate diuidendum, <lb/><choice><ex>prouenientique</ex><am>prouenientiq́;</am></choice> maximum terminum addendum, ex quo oritur ſumma quæſita.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi darentur quatuor termini continui proportionales .8. 12. 18. <lb/>27. primum hoc eſt minimum .8. ex vltimo .27. detraheremus: </s> <s xml:space="preserve"><choice><ex>remaneretque</ex><am>remaneretq́;</am></choice> .19. qui <lb/>per denominantem progreſſionis, dempta vnitate, diuideretur. </s> <s xml:space="preserve">Quo loco animad <lb/>uertendum eſt, quamlibet <choice><ex>denominationem</ex><am>denominationẽ</am></choice> cuiuſcunque proportionis numerorum <lb/>ſupra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eſt, & <lb/>idipſum de ſuperparticularibus, & ſuperpartientibus eſt intelligendum, vt in præ-<lb/>ſenti proportio ſeſquialtera inter duos terminos cogitanda eſt, nempe inter vnum <lb/>& dimidium, atque vnum. </s> <s xml:space="preserve">Seſquitertia autem inter vnum & tertiam partem, <lb/>& vnum. </s> <s xml:space="preserve">Seſquiquinta inter vnum cum quinta parte, & vnum. </s> <s xml:space="preserve">De ſuperpartien <lb/>tibus idem aſſero quod de proportione <choice><ex>ſuperbipartiente</ex><am>ſuperbipartiẽte</am></choice> tertias appellata, vt .5. <lb/>ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, ſuperbipartiens quar-<lb/>tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans ſcilicet, ſem <lb/>per ſit vnitas, alter verò denominans. </s> <s xml:space="preserve">Idem de cæteris. </s> <s xml:space="preserve">Quare in præſenti exem <lb/>plo, detracta vnitate ex denominante progreſſionis, ſupererit tantummodo dimi-<lb/>dium, quo diuiſo .19. proueniet .38. qui numerus æqualis erit ſummæ <choice><ex>reliquorum</ex><am>reliquorũ</am></choice> <lb/>omnium terminorum, cui coniuncto vltimo termino .27. dabitur ſumma quæſita .65.</s> </p> <p> <s xml:space="preserve">Pro cuius ſpeculatione, quatuor termini ſignificentur, quatuor lineis <seg type="var">.m.s</seg>: <seg type="var">f.r</seg>: <seg type="var">c.a.<lb/>b.i.</seg> primus autem terminus <seg type="var">.m.s.</seg> ex vltimo <seg type="var">.b.i.</seg> detrahatur, <choice><ex>reſiduumque</ex><am>reſiduumq́;</am></choice> ſit <seg type="var">.n.i.</seg> & ex <lb/>ſecundo <seg type="var">.f.r.</seg> cuius reſiduum ſit <seg type="var">.o.r.</seg> proportio verò progreſſionis ea ſit, quæ <seg type="var">.g.h.</seg> ad <seg type="var">.<lb/>y.</seg> quo vnitas repræſentatur (ex quo ſic ſe habebit <seg type="var">.g.h.</seg> ad <seg type="var">.y.</seg> vt <seg type="var">.f.r.</seg> ad <seg type="var">.m.s.</seg>) qua <seg type="var">.y.</seg> de <lb/>tracta ex <seg type="var">.g.h.</seg> ſuperſit <seg type="var">.h</seg>. </s> <s xml:space="preserve">Tum erecta <lb/>cogitetur linea <seg type="var">.n.u.x.</seg> indefinita per <lb/> <ptr xml:id="fig-0079-01a" corresp="fig-0079-01" type="figureAnchor"/> pendicularis <seg type="var">.b.i.</seg> à puncto <seg type="var">.n.</seg> quę diui <lb/>datur in puncto <seg type="var">.x.</seg> ita vt <seg type="var">.n.x.</seg> æqualis <lb/>ſit vnitati <seg type="var">.y.</seg> & in puncto <seg type="var">.u.</seg> ita. vt <seg type="var">.n.<lb/>u.</seg> æqualis ſit <seg type="var">.h.</seg> ex quo eadem erit <lb/>proportio <seg type="var">.n.u.</seg> ad <seg type="var">.n.x.</seg> vt <seg type="var">.h.</seg> ad <seg type="var">.y.</seg> <choice><ex>nem- pe</ex><am>nẽ-pe</am></choice> <seg type="var">.o.r.</seg> ad <seg type="var">.m.s</seg>. </s> <s xml:space="preserve">Nam cú ſic ſe habeat <seg type="var">.<lb/>f.r.</seg> ad <seg type="var">.m.s.</seg> hoc eſt ad <seg type="var">.f.o.</seg> vt <seg type="var">.g.h.</seg> ad <seg type="var">.y</seg> <lb/>hoc eſt ad <seg type="var">.g.</seg> permutando <choice><ex>quoque</ex><am>quoq;</am></choice> ſic <lb/>ſe habebit <seg type="var">.f.r.</seg> ad <seg type="var">.g.h.</seg> vt <seg type="var">.f.o.</seg> ad <seg type="var">.g</seg>. </s> <s xml:space="preserve">Ita <lb/>que ex .19. quinti <seg type="var">.o.r.</seg> ad <seg type="var">.h.</seg> vt <seg type="var">.f.r.</seg> ad <seg type="var">.g.h.</seg> ex quo ex .11. eiuſdem <seg type="var">.o.r.</seg> ad <seg type="var">.h.</seg> vt <seg type="var">.f.o.</seg> ad <pb facs="0080" n="70"/><fw type="head">IO. BAPT. BENED.</fw> g. & permutando <seg type="var">.o.r.</seg> ad <seg type="var">.f.o.</seg> hoc eſt ad <seg type="var">.m.s.</seg> vt <seg type="var">.h.</seg> ad <seg type="var">.g.</seg> hoc eſt <seg type="var">.y</seg>. </s> <s xml:space="preserve">Quamobrem ea-<lb/>dem erit proportio <seg type="var">.o.r.</seg> ad <seg type="var">.m.s.</seg> quæ <seg type="var">.n.u.</seg> ad <seg type="var">.n.x</seg>. </s> <s xml:space="preserve">Abfoluantur itaque duo rectangu <lb/>la <seg type="var">.x.i.</seg> et <seg type="var">.u.z.</seg> ita tamen vt <choice><ex>rectangulum</ex><am>rectangulũ</am></choice> <seg type="var">.u.z.</seg> cogitetur ęquale rectangulo <seg type="var">.x.i.</seg> cuius <seg type="var">.x.i.</seg> <lb/>ſuperficialis numerus ex communi conceptione lineari <seg type="var">.n.i.</seg> æqualis erit, </s> <s xml:space="preserve">quare ex <lb/><choice><ex>eadem</ex><am>eadẽ</am></choice> communi conceptione, numerus ſuperficialis <seg type="var">.u.z.</seg> lineari <seg type="var">.n.i.</seg> æqualis erit, qui <lb/>quidem numerus in figura rectangu-<lb/>la ſuperficialis cognitandus erit, cum <lb/> <ptr xml:id="fig-0080-01a" corresp="fig-0080-01" type="figureAnchor"/> diuidendus ſit per <seg type="var">.h.</seg> hoc eſt per <seg type="var">.n.u.</seg> <lb/>ex quo proueniens ex huiuſmodi di <lb/>uiſione erit numerus <seg type="var">.n.z.</seg> ex ijs <lb/>quæ .10. theoremate dicta fuerunt. <lb/></s> <s xml:space="preserve">Sed ex .15. fexti aut .20. ſepti-<lb/>mi eadem eſt proportio <seg type="var">.n.i.</seg> ad <seg type="var">.n.z.</seg> <lb/>quæ <seg type="var">.n.u.</seg> ad <seg type="var">.n.x.</seg> hoc eſt <seg type="var">.o.r.</seg> ad <seg type="var">.m.s.</seg> <lb/>videlicet vt <seg type="var">.n.i.</seg> ad aggregatum reli-<lb/>quorum omnium terminorum <seg type="var">.c.a</seg>: <seg type="var">f.<lb/>r</seg>: </s> <s xml:space="preserve">m.s. ex præcedenti theoremate, & ex .11. quinti Euclidis. </s> <s xml:space="preserve">Itaque ex .9. eiuf-<lb/>dem numerus <seg type="var">.n.z.</seg> æqualis erit ſummæ trium terminorum <seg type="var">.c.a</seg>: f. num <seg type="var">.s.</seg> cui coniuncto <lb/>quarto termino <seg type="var">.b.i.</seg> propoſitum obtinetur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0079-01" corresp="fig-0079-01a"> <graphic url="0079-01"/> </figure> <figure xml:id="fig-0080-01" corresp="fig-0080-01a"> <graphic url="0080-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="106">CVI</num>.</head> <p> <s xml:space="preserve">PRopoſuere veteres quæſita nonnulla de itineribus <choice><ex>interque</ex><am>interq́;</am></choice> hoc vnum fuit. </s> <s xml:space="preserve">Po-<lb/>namus duos iter agere per eandem viam quorum alter quatuor milliaria ſin-<lb/>gulis diebus conficiat, alter verò prima die milliare vnum, ſecunda duo, tertia tria, <lb/>atque ita ſingulis diebus milliare addit; </s> <s xml:space="preserve">quærimus quot dierum ſpacio ſocium con <lb/>ſequetur.</s> </p> <p> <s xml:space="preserve">Quamobrem numerus milliarium primi viatoris duplicatur, ſic ſunt .8. milliaria. <lb/></s> <s xml:space="preserve">ex quo ſemper vnitas detrahicur, quæ in præſenti exemplo erit .7. <choice><ex>totque</ex><am>totq́;</am></choice> dies erunt <lb/>quibus ſocius ſocium conſequetur, & milliarium numerum æqualem abſoluerit.</s> </p> <p> <s xml:space="preserve">Cuius rei facilis erit ſpeculatio, ſi ſubſcripta figura diligenter conſideretur, in <lb/>qua primus viator, die prima, quatuor milliaria linea <seg type="var">.q.d.</seg> ſignificata conficit, at-<lb/>que illa ipſa die alter vnum tantum defignatum per <seg type="var">.d.</seg> perficit, ita vt primus via-<lb/>tor tribus milliaribus ſocium anteceſſerit, altera verò die fecundus uiator cum duo <lb/>milliaria <choice><ex>conficiat</ex><am>cõficiat</am></choice>, excedètur à primo duobus milliaribus tantummodo, quę cum tri-<lb/>bus primæ diei quinque erunt; </s> <s xml:space="preserve">tertia die ijſdem de cauſis primus ſex tantum millia-<lb/>ribus à ſecundo diſtabit, cum verò quarta die tot ſecundus quot primus milliaria <lb/>conficiat, primus à ſecundo amplius quam antea non diſtabit; </s> <s xml:space="preserve">quinta verò cum ſe <lb/>cundus vnum milliare amplius quam primus conficiat. </s> <s xml:space="preserve">propius accedit ad primum <lb/>vno ex ſex milliaribus, quibus anteà diſtabat, tum ſexta cum duobus primum ſupe-<lb/>ret, detrahet ex ſex milliaribus præteritæ diſtantiæ tria, ſeptima tandem illa ſex <lb/>detraxerit. </s> <s xml:space="preserve">In quo conſiderandum eſt ſecundum viatorem iter agere progreſſio-<lb/>ne arithmetica continua naturali <seg type="var">.d.c.f.</seg> primum autem per rectangulum <seg type="var">.q.f.</seg> <choice><ex>quarum</ex><am>quarũ</am></choice> <lb/>duarum figurarum <seg type="var">.d.o.p.f.</seg> pars <choice><ex>communis</ex><am>cõmunis</am></choice> eſſe reperitur, quæ quantitates ſi inuicem <lb/>æquales eſſe debent, neceſſe eſt ſeparatas partes <seg type="var">.u.q.n.</seg> et <seg type="var">.t.i.c.</seg> inter ſe æquales eſſe, <lb/>& quoniam quarta die (hoc eſt die ſic diſtante à primo, nempè numero milliarium <pb facs="0081" n="71"/><fw type="head">THEOREM. ARITH.</fw> primi viatoris) tot milliaria abſoluat vnus <lb/> <ptr xml:id="fig-0081-01a" corresp="fig-0081-01" type="figureAnchor"/> quot alter abſque vlla differentia, quæ ſigni-<lb/>ficetur per <seg type="var">.o.s.</seg> neceſſe eſtitaque ex communi <lb/>conceptione tot dies eſſe poſt <seg type="var">.o.s.</seg> quotante-<lb/>ceſſerant, vt exceſſus æqualis ſit defectui, qui <lb/>ſimul collecti, iuncta etiam <seg type="var">.o.s.</seg> duplum erunt <lb/><seg type="var">d.s.</seg> dempta vnitate, prout facilè in ſubſcripta <lb/>figura qui ſque per ſe ſcientificè poterit ſpecu <lb/>lari. </s> <s xml:space="preserve">Quamobrem conſultum erit duplicare <lb/>numerum <seg type="var">.o.s.</seg> & exduplo vnitatem detrahe-<lb/>re, quandoquidem dies ſupra <choice><ex>infraque</ex><am>infraq́;</am></choice> <seg type="var">.o.s.</seg> cum <lb/>die <seg type="var">.o.s.</seg> minores ſunt duplo numeri <seg type="var">.d.s.</seg> aut <seg type="var">.o.<lb/>s.</seg> (quodidem eſt) vnitate.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0081-01" corresp="fig-0081-01a"> <graphic url="0081-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="107">CVII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> <choice><ex>ſiſecundus</ex><am>ſiſecũdus</am></choice> viator <choice><ex>ordinem</ex><am>ordinẽ</am></choice> <choice><ex>ſecundæ</ex><am>ſecũdæ</am></choice> progreſſionis arithmeticæ <choice><ex>ſeruansiter</ex><am>ſeruãsiter</am></choice> <lb/>agat, nempe ea quæ ab vno per binarium aſcendit, ſemper numerus dierum <lb/>æqualis erit numero milliarium diurnorum primi viatoris.</s> </p> <p> <s xml:space="preserve">In cuius gratiam animaduertendum eſt numerus ne milliarium diurnorum pri-<lb/>mi viatoris par an impar ſit. </s> <s xml:space="preserve">Etenim ſi par eſt, primus viator in fine ſingulorum die-<lb/>rum primæ medietatis numeri omnium dierum ſecundum antecedet numero diſpa <lb/>ri milliarium; </s> <s xml:space="preserve">altero verò dimidio numero dierum, à ſecundo numero etiam diſpa <lb/>ri præteribitur, vt in ſequenti figura patet. </s> <s xml:space="preserve">Nam prima die, ſecundus ex primo <lb/>milliare vnum ex numero pari, qui à primo conficitur detrahit; </s> <s xml:space="preserve">ſecunda verò die <lb/>idem ſecundus, duo ſubtrahit milliaria exdiſpari, qui primo reliquus fuerat, <choice><ex>ſicque</ex><am>ſicq́;</am></choice> <lb/>perpetuò diſpar remanet vſque ad vnitatem, ad quam cum peruenerint, nempe ad <lb/>illius diei exitum, quo primus ſecundum vnitate tantummodò ſuperat, manifeſtè <lb/>depræhendetur ſubſequente dieſecundum vnitate primum ſuperaturum, altera ve <lb/>rò tribus vnitatibus, prout penultima die ſecundus à primo tribus vnitatibus ſupera <lb/>batur. </s> <s xml:space="preserve">Quare neceſſe erit, tot diebus ſecundum cum primo iter agere, inchoan-<lb/>do ab ea die, qua fecundus primam ſuperabit, quot egerat dum à primo ſuperare-<lb/>tur, vt ex communi conceptione, media figura <seg type="var">.A.</seg> depræhendi poteſt. </s> <s xml:space="preserve">Quod au-<lb/>tem ſingula dimidia dierum, dimi-<lb/> <ptr xml:id="fig-0081-02a" corresp="fig-0081-02" type="figureAnchor"/> dia ſint numeri milliarium diurno-<lb/>rum primi; </s> <s xml:space="preserve">patebit exſequenti fi-<lb/>gura, cogitato termino <seg type="var">.u.n.</seg> vlti-<lb/>mo progreſſionis ſuperatę à primo <lb/>vſque ad vnitatem <seg type="var">.e.</seg> quiterminus <lb/><seg type="var">u.n.</seg> coniunctus primo <seg type="var">.o.</seg> nempe <seg type="var">.e.</seg> <lb/>ſemper <choice><ex>duplum</ex><am>duplũ</am></choice><unclear reason="illegible"/> eſt numeritermino-<lb/>rum <seg type="var">.o.n.</seg> vt .94. theoremate circa <lb/>finem dictum fuit. </s> <s xml:space="preserve">Sed <seg type="var">.u.n.</seg> cum <seg type="var">.e.</seg> <lb/>numero æquali conſtat numero <lb/>milliarium diurnorum primi viatoris, ex quo ſequitur totum numerum dierum, quo <lb/>rum <seg type="var">.o.n.</seg> dimidium eſt, æqualem eſſe numero milliarium diuruorum primi via-<lb/>toris.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0081-02" corresp="fig-0081-02a"> <graphic url="0081-02"/> </figure> </div> </body> </floatingText> <pb facs="0082" n="70"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="108">CVIII</num>.</head> <p> <s xml:space="preserve">AT ſi numerus milliarium primi viatoris diſpar fuerit, ſecundum numero pari <lb/>ſemper ſuperabit, vt facile erit ſequentem figuram conſideranti intelligere, <lb/>ex quo illud ſequetur, futuram quandam diem, qua paria milliaria conficient. </s> <s xml:space="preserve">Sit-<lb/>q́ue illa dies <seg type="var">.u.n.</seg> ſequitur <choice><ex>etiam</ex><am>etiã</am></choice> tranſacta ea die, tot diebus vtrique ambulandum eſſe <lb/>quot iter egere anteaquam ad diem <seg type="var">.u.n.</seg> <lb/>peruenirent, vt tanto numero primus à ſe-<lb/> <ptr xml:id="fig-0082-01a" corresp="fig-0082-01" type="figureAnchor"/> cundo ſuperetur, <choice><ex>quanto</ex><am>quãto</am></choice> ſecundum primus <lb/>ſuperauerat, vnde totalis numerus <seg type="var">.o.f.</seg> mi <lb/>nor erit duplo <seg type="var">.o.n.</seg> vnitate ex communi <lb/>conceptione, ſed ita etiam ſe habet termi-<lb/>nus <seg type="var">.u.n.</seg> hoc eſt minor duplo <seg type="var">.o.n.</seg> per <seg type="var">.o.</seg> vt <lb/>94. theoremate dictum fuit, itaque <seg type="var">.o.f.</seg> æ-<lb/>qualis erit <seg type="var">.u.n.</seg> quoderat propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0082-01" corresp="fig-0082-01a"> <graphic url="0082-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="109">CIX</num>.</head> <p> <s xml:space="preserve">SIN verò progreſſio ſecundi viatoris, non ab vnitate ſed à binario inchoata, <lb/>per binarium quoque aſcenderet, <choice><ex>numerusque</ex><am>numerusq́;</am></choice> milliarium diurnorum primi via <lb/>toris par eſſet, abſque dubio quadam die paria milliaria <choice><ex>vterque</ex><am>vterq;</am></choice> conficeret, quę ſigni <lb/>ficetur <seg type="var">.u.n.</seg> qua tranſacta, tot diebus vtrique <choice><ex>ambulandum</ex><am>ambulãdum</am></choice> erit, quot <choice><ex>fuerunt</ex><am>fuerũt</am></choice> <choice><ex>dum</ex><am>dũ</am></choice> primus <lb/>ſecundum ſuperaret, vt totidem alijs pri-<lb/>mus à ſecundo ſuperetur, in qua tamen <lb/> <ptr xml:id="fig-0082-02a" corresp="fig-0082-02" type="figureAnchor"/> progreſſione terminus <seg type="var">.u.n.</seg> ſemper duplus <lb/>eſt numero terminorum <seg type="var">.o.n.</seg> ex .95. theo-<lb/>remate, <choice><ex>totque</ex><am>totq́;</am></choice> ſunt infra <seg type="var">.u.n.</seg> termini vſque <lb/>ad <seg type="var">.f.</seg> quot ſupra. </s> <s xml:space="preserve">ex quo illud ſequitur om <lb/>nesterminosaut dies <seg type="var">.o.n.f.</seg> pauciores eſſe <lb/><seg type="var">u.n.</seg> vnitate, atque ita præcipit, regula de-<lb/>trahendam eſſe vnitatem ex numero mil-<lb/>liarium diurnorum primi viatoris, ſi dierum numerum habere voluerimus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0082-02" corresp="fig-0082-02a"> <graphic url="0082-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="110">CX</num>.</head> <p> <s xml:space="preserve">SED ſi in eiuſmodi progreſſione numerus milliarium diurnorum primi viato-<lb/>ris diſpar fuerit, patet quòd primus ſecundum numero diſpari ſuperabit, do-<lb/>necad vnitatem perueniatur <choice><ex>vi- ciſſimque</ex><am>vi- ciſſimq́;</am></choice> primum ſecundus, in-<lb/>choando ab vnitate, </s> <s xml:space="preserve">quare nul-<lb/>la <choice><ex>vnquam</ex><am>vnquã</am></choice> die paria milliaria vter-<lb/>que conficiet, ſit itaque vltima <lb/>dies, qua primus ſecundum vnita <lb/>tem antecedit <seg type="var">.u.n.</seg> qui terminus <lb/>duplus eſt numero terminorum <lb/><seg type="var">o.n.</seg> & cum illa die primus ſecun-<lb/>dum milliario antecedat, ſequen <pb facs="0083" n="71"/><fw type="head">THEOREM. ARIT.</fw> te verò à ſecundo milliario vno primus antecedatur, ex communi ſcientia neceſſe <lb/>eſt ſecundum tot diebus <choice><ex>cum</ex><am>cũ</am></choice> primo iter agere quot ſunt <seg type="var">.o.n.</seg> qui ſimul æquales erunt <seg type="var">.<lb/>u.n.</seg> ſed <seg type="var">.u.n.</seg> minor eft numero milliarium diurnorum primi vnitate <seg type="var">.e</seg>. </s> <s xml:space="preserve">Itaque rectè <lb/>ſequemur regulam, quæ iubet ex numero milliarium vnitatem demere, quo nu <lb/>merum dierum habere poſſimus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0082-03" corresp="fig-0082-03a"> <graphic url="0082-03"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="111">CXI</num>.</head> <p> <s xml:space="preserve">SI verò ſecundi viatoris progreſſio per ternarium aſcenderet, ſumpto initio ab <lb/>ipſo ternario, animaduertendum eſt an numerus milliarium diurnorum primi, <lb/>ternario menſuretur necne, etenim ſi menſuretur, tandem aliquando paria millia-<lb/>ria conficient, quæ dies ſit <seg type="var">.u.n.</seg> </s> <s xml:space="preserve">quare ſub <lb/> <ptr xml:id="fig-0083-01a" corresp="fig-0083-01" type="figureAnchor"/> <seg type="var">u.n.</seg> totidem quot ſupra termini <choice><ex>erunt</ex><am>erũt</am></choice>, <lb/>& <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.o.n.</seg> tertia ſit pars <seg type="var">.u.n.</seg> ex .95. theo-<lb/>remate. </s> <s xml:space="preserve">Itaque tota <seg type="var">.o.f.</seg> minor erit <lb/>duabus tertijs <seg type="var">.u.n.</seg> vnitate, vtiam re-<lb/>ctè ſumendæ ſint duæ tertiæ partes <seg type="var">.u.n.</seg> <lb/>ex quibus vnitas detrahatur ſuperſitq́ue <lb/>numerus <seg type="var">.o.f.</seg> dierum quæſitorum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0083-01" corresp="fig-0083-01a"> <graphic url="0083-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="112">CXII</num>.</head> <p> <s xml:space="preserve">CVM verò milliarium numerus p rimi viatoris metirinon poterit à numero <lb/>aſcendente ſecundi, patet nullam futuram diem qua pari milliaria conficient, <lb/></s> <s xml:space="preserve">quare illa vltima qua primus ſecundum antecedet, vno aut duobus milliaribus an-<lb/>tecedet in præſenti caſu. </s> <s xml:space="preserve">Antecedat itaque duobus milliaribus, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> dies <seg type="var">.u.n.</seg> & alte <lb/>ra <seg type="var">.t.i.</seg> ſecundus primum vno milliari ſuperabit, ita quod ſub <seg type="var">.t.i.</seg> non poterunt plu-<lb/>res integros dies iter agere, quam ambulauerunt ante diem <seg type="var">.u.n.</seg> hoc eſt vſquequo <lb/>ſecundtis iunctus ſit primo, qui numerus dierum, tertia parte <seg type="var">.o.n.</seg> ipſius <seg type="var">.u.n.</seg> vnitate <lb/>minor erit, cum ex .95. theoremate <seg type="var">.o.n.</seg> ſit tertia pars <seg type="var">.u.n.</seg> ex quo numerus <seg type="var">.o.f.</seg> ter-<lb/>minorum aut dierum intergrorum cognitus erit, qui ſi cum numero alcendente <lb/>cognoſcetur, ſtatim ex .99. theoremate deueniemus in cognitionem vltimi diei in <lb/>tegri <seg type="var">.s.f.</seg> atque ita etiam totius ſummæ progreſſionis ex .95. theoremate. </s> <s xml:space="preserve">Iam verò <lb/>cognito numero milliarium diurnorum primi, ſimul cum numero terminorum, aut <lb/>dierum conſequenter nouerimus rectanguli ſummam, hoc eſt productum à primo <lb/>viatore formatum, quarum duarum ſummarum in præſenti caſu ſemper ea, quæ <lb/>huiuſmodi producti eſt, maior erit, cum conſtitutum fuerit ſecundum viatorem à <lb/>primo ſuperari ipſa die <seg type="var">.u.n.</seg> vno milliari amplius quam ſequente die <seg type="var">.t.i.</seg> primus à ſe <lb/>cundo ſuperatur, tum pari gradu iter egerunt ſub <seg type="var">.t.i.</seg> quo ſupra <seg type="var">.u.n.</seg> ambulauerant. <lb/></s> <s xml:space="preserve">Hoc animaduertendo, quòd ſi ſumma progreſſionis maior eſſet rectangulo, ex ea <lb/>ſumma neceſſe eſſet <choice><ex>numerum</ex><am>numerũ</am></choice> mil <lb/>liarium vltimi termini in ſumma <lb/> <ptr xml:id="fig-0083-02a" corresp="fig-0083-02" type="figureAnchor"/> incluſi detrahere, & reſiduo ope-<lb/>rari. </s> <s xml:space="preserve">Nunc verò ſummam pro-<lb/>greſſionis exſumma rectanguli à <lb/>primo viatore facti ſubtrahi de-<lb/>bet, <choice><ex>reſiduumque</ex><am>reſiduumq́;</am></choice> ſeruari <choice><ex>voceturque</ex><am>voceturq́;</am></choice> <pb facs="0084" n="72"/><fw type="head">IO. BAPT. BENED.</fw> <choice><ex>primum</ex><am>primũ</am></choice> <choice><ex>reſiduum</ex><am>reſiduũ</am></choice>. </s> <s xml:space="preserve">Ad hæc <choice><ex>numerum</ex><am>numerũ</am></choice> <choice><ex>milliariorum</ex><am>milliariorũ</am></choice>, quæ <choice><ex>ſecundus</ex><am>ſecũdus</am></choice> viator die <choice><ex>ſeqennti</ex><am>ſeqẽnti</am></choice> <seg type="var">.s.f.</seg> confi <lb/>ciet <choice><ex>ſumatur</ex><am>ſumat̃</am></choice>, ex quo numerus <choice><ex>milliarium</ex><am>milliariũ</am></choice> <choice><ex>diurnorum</ex><am>diurnorũ</am></choice> primi detrahatur, <choice><ex>residuumque</ex><am>reſiduũq́;</am></choice> pariter re <lb/>ſeruetur, <choice><ex>voceturque</ex><am>voceturq́;</am></choice> ſecundum reſiduum, poſtmodum numerum milliarium primi <lb/>vnius diei multiplicetur per primum reſiduum ſeruatum, <choice><ex>productumque</ex><am>productumq́;</am></choice> per <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> <lb/>reſiduum diuidatur <seg type="var">.a.c.</seg> <choice><ex>proueniens</ex><am>proueniẽs</am></choice> (<choice><ex>quod</ex><am>ꝙ</am></choice> erit iter primi in ſequenti die) iungatur reſi-<lb/>ſiduo primo, tot enim erunt milliaria conficienda a ſecundo ſequenti die, vt <lb/>ſeſe conſequantur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0083-02" corresp="fig-0083-02a"> <graphic url="0083-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Vtautem ſciamus quantam partem diei <choice><ex>ſeqenntis</ex><am>ſeqẽntis</am></choice>, ſingulos itinere agere oporteat, <lb/><choice><ex>proueniens</ex><am>proueniẽs</am></choice> per .24. horas multiplicetur (ſuppoſito quod <choice><ex>ambulantes</ex><am>ambulãtes</am></choice> <choice><ex>nullam</ex><am>nullã</am></choice> <choice><ex>requiem</ex><am>requiẽ</am></choice> nec <lb/>die nec nocte <choice><ex>capiant</ex><am>capiãt</am></choice>), <choice><ex>productumque</ex><am>ꝓductũq́;</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> <choice><ex>numerum</ex><am>numerũ</am></choice> milliariorum vuius diei primi viatoris di <lb/>uidatur, ex quo dabitur <choice><ex>quantitas</ex><am>quãtitas</am></choice> <choice><ex>horarum</ex><am>horarũ</am></choice>, & pars horæ, qua <choice><ex>cuique</ex><am>cuiq;</am></choice> illa die <choice><ex>ambulandum</ex><am>ambulandũ</am></choice> <lb/>eſt. </s> <s xml:space="preserve">Idem accideret ſi primum reſiduum reſeruatum cum proueniente in ſummam <lb/>colligeretur, <choice><ex>eaque</ex><am>eaq́;</am></choice> ſumma per .24. horas multiplicaretur, <choice><ex>productumque</ex><am>productumq́;</am></choice> per <choice><ex>nume- rum</ex><am>nume-rũ</am></choice> milliariorum ſequenti die à ſecundo conficiendorum diuideretur. </s> <s xml:space="preserve">I dipſum quo-<lb/>que eueniret multiplicato primo reſiduo per .24. & producto per ſecundum reſi-<lb/>duum diuiſo.</s> </p> <p> <s xml:space="preserve">Exempli gratia, primus viator diurna milliaria vndecim conficit, ſecundus, pri-<lb/>ma die tria, ſecunda .6. tertia .9. <choice><ex>atque</ex><am>atq;</am></choice> ita deinceps, diuidatur ergo .11. per .3. vnde <lb/>pro numero <seg type="var">.o.n.</seg> dabitur .3. <choice><ex>ſupereritque</ex><am>ſupereritq́;</am></choice> .2. </s> <s xml:space="preserve">quare: <seg type="var">u.n.</seg> ab <seg type="var">.e.n.</seg> duobus milliaribus ſu-<lb/>perabitur, et <seg type="var">.i.t.</seg> dictum <seg type="var">.e.n.</seg> vno milliario, ex quo ante diem <seg type="var">.e.u.n.</seg> duobus diebus <lb/>iter egerunt, <choice><ex>totque</ex><am>totq́;</am></choice> diebus ambulandum erit poſt <seg type="var">.t.i.</seg> hoceſt .6. in vniuerſum inte-<lb/>gris. </s> <s xml:space="preserve">Ad hęc multiplicato <seg type="var">.o.f.</seg> hoc eſt .6. per <seg type="var">.x.o.</seg> hoc eſt .3. habebimus <seg type="var">.s.f.</seg> <choice><ex>milliariorum</ex><am>milliariorũ</am></choice> <lb/>18. <choice><ex>tum</ex><am>tũ</am></choice> <choice><ex>coniuncto</ex><am>cõiũcto</am></choice> <seg type="var">.x.o.</seg> primo termino hoc eſt .3. <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.s.f.</seg> hoc eſt .18. vltimo termino, habe <lb/>bimus .21. quo multiplicato <choice><ex>cum</ex><am>cũ</am></choice> dimidio <seg type="var">.o.f.</seg> hoc eſt .3. habebimus totam ſummam <lb/>progreſſionis .63. ſex dierum integrorum ex .94. theoremate, tum multiplicato .11. <lb/>nempe numero <choice><ex>milliariorum</ex><am>milliariorũ</am></choice> diurnorum primi cum .6. hoc eſt cum <seg type="var">.o.f.</seg> habebimus pa <lb/>rallelogrammum à primo ſex diebus integris confectum milliariorum .66. ex quo <lb/>detracta .63. ſumma inquam progreſſionis, ſupererit pro primo reſiduo .3. ſumptis <lb/>poſtea milliaribus .21. pro itinere, quod ſecundus die ſequenti <seg type="var">.s.f.</seg> conficeret, & ex <lb/>ijs detracto numero <choice><ex>milliariorum</ex><am>milliariorũ</am></choice> diurnorum primi, nempe .11. ſecundum reſiduum <lb/>erit 10. quod pro diuidenti ſeruabitur. </s> <s xml:space="preserve">Iam multiplicato .11. cum primo reſiduo <num value="3">.<lb/>3.</num> dabitur .33. qui diuiſus per .10. ſecundum reſiduum profert .3. cum tribus decimis, <lb/><choice><ex>eritque</ex><am>eritq́</am></choice> iter à primo viatore ſequenti die conficiendum, hoc etiam ipſum proueniens <lb/>cum primo reſiduo .3. coniunctum, dat .6. cum tribus decimis, quod eſt iter ſecundi <lb/>viatoris illa ſequenti die. </s> <s xml:space="preserve">Ad inueniendam autem quantitatem diei, qua vtrique <lb/>ambulandum eſt, perinde erit multiplicare proueniens .3. & tres decimas per .24. ho <lb/>ras, & productum per .11. dimidium iter primi viatoris partiri, ac multiplicare ſum <lb/>mam .6. & tres decimas cum .24. horis, <choice><ex>productumque</ex><am>productumq́</am></choice> diuidere per .21. hoc eſt periter <lb/>ſecundi viatoris ſequentis diei, vtrinque enim ſemper ſeptem horæcum .12. minu <lb/>tis prouenient. </s> <s xml:space="preserve">Idipſum accidet multiplicato per .24. horas primo reſiduo .3. pro-<lb/><choice><ex>ductoque</ex><am>ductoq́;</am></choice> diuiſo per ſecundum reſiduum .10.</s> </p> <p> <s xml:space="preserve">Quarum ſpeculationum gratia, totum iter parallelogrammi primi viatoris die-<lb/>rum integrorum fignificetur linea <seg type="var">.n.e.</seg> ſumma verò progreſſionis ſecundi linea <seg type="var">.f.m.</seg> <lb/>parallela <seg type="var">.n.e.</seg> <choice><ex>eritque</ex><am>eritq́;</am></choice> <seg type="var">.f.m.</seg> minor <seg type="var">.n.e</seg>. </s> <s xml:space="preserve">Conſtituamus deinde à termino <seg type="var">.f.n.</seg> (majoris <lb/><choice><ex>intelligentię</ex><am>intelligẽtię</am></choice> gratia) vtranque <choice><ex>perpendiculariter</ex><am>perpẽdiculariter</am></choice> duci, <choice><ex>producatur</ex><am>ꝓducatur</am></choice> deinde <seg type="var">.n.e.</seg> donec <seg type="var">.e.<lb/>d.</seg> æqualis ſit itineri diurno primi viatoris, item etiam producatur <seg type="var">.f.m.</seg> donec <seg type="var">.m.K.</seg> <lb/>æqualis ſit itineri à ſecundo confecto ſequenti die vltimum integrum progreſſio- <pb facs="0085" n="73"/><fw type="head">THEOREM. ARITH.</fw> nis, ex quo <seg type="var">.m.k.</seg> prolixior erit <seg type="var">.e.d.</seg> ex præſup poſito. </s> <s xml:space="preserve">Poſtmodum <seg type="var">.m.e.</seg> et <seg type="var">.k.d.</seg> dua-<lb/>bus lineis rectis coniungantur, quæ productæ concurrentin puncto <seg type="var">.b.</seg> ducatur pari-<lb/>ter <seg type="var">.e.g.</seg> à puncto <seg type="var">.e.</seg> parallela <seg type="var">.b.k.</seg> et <seg type="var">.m.a</seg>: <seg type="var">e.h.</seg> et <seg type="var">.b.q.</seg> parallelæ <seg type="var">.f.n.</seg> ex quo <seg type="var">.f.m.</seg> æqua-<lb/>lis erit <seg type="var">.n.a.</seg> et <seg type="var">.m.h</seg>: <seg type="var">a.e.</seg> et <seg type="var">.h.q</seg>: <seg type="var">e.o.</seg> et <seg type="var">.g.k</seg>: <seg type="var">e.d.</seg> et <seg type="var">.f.q</seg>: <seg type="var">n.o.</seg> ex .34. primi Eucli. </s> <s xml:space="preserve">vnde pro <lb/>portio <seg type="var">.m.h.</seg> ad <seg type="var">.h.q.</seg> erit vt <seg type="var">.m.g.</seg> ad <seg type="var">.g.k.</seg> quandoquidem vtraque æqualis eſt propor-<lb/>tioni <seg type="var">.m.e.</seg> ad <seg type="var">.e.b.</seg> ex .2. ſexti, ſed cum <seg type="var">.m.k.</seg> et <seg type="var">.g.k.</seg> notæ ſint, pariter cognoſcetur <seg type="var">.m.<lb/>g.</seg> ſecundum reſiduum, cum etiam notæ ſint <seg type="var">.n.e.</seg> et <seg type="var">.n.a</seg>. </s> <s xml:space="preserve">Itaque cognoſcemus <seg type="var">.a.e.</seg> hoc <lb/>eſt <seg type="var">.m.h.</seg> cognitis verò <seg type="var">.m.g</seg>: <seg type="var">g.k.</seg> et <seg type="var">.m.h.</seg> ex .15. ſexti aut .20. ſeptimi cognoſcetur <seg type="var">.h.<lb/>q.</seg> erit igitur <seg type="var">.a.e.</seg> aut quod idem eſt <seg type="var">.m.</seg> hprimum reſiduum, et <seg type="var">.m.g.</seg> ſecundum, et <seg type="var">.h.<lb/>q.</seg> aut <seg type="var">.e.o.</seg> proueniens, et <seg type="var">.n.o.</seg> et <seg type="var">.f.q.</seg> itinera vtriuſque viatoris inter ſe æqualia. <lb/></s> <s xml:space="preserve">Nec verò prætermittenda eſt ſpeculatio vltimæ rationis inueniendæ quantitatis <lb/>diei, quæ conſtat ope diuiſionis producti <seg type="var">.m.h.</seg> in .24. per <seg type="var">.m.g</seg>. </s> <s xml:space="preserve">Ea autem eiuſmodi <lb/>eſt. </s> <s xml:space="preserve">Probatum fuit ſic ſe habere <seg type="var">.m.h.</seg> ad <seg type="var">.h.q.</seg> ut <seg type="var">.m.g.</seg> ad <seg type="var">.g.k</seg>. </s> <s xml:space="preserve">Itaque componendo <lb/>ſic ſe habebit <seg type="var">.m.q.</seg> ad <seg type="var">.h.q.</seg> vt <seg type="var">.m.k.</seg> ad <seg type="var">.g.k.</seg> & permutando <seg type="var">.m.q.</seg> ad <seg type="var">.m.k.</seg> vt <seg type="var">.h.q.</seg> ad <seg type="var">.g.<lb/>k</seg>. </s> <s xml:space="preserve">Sed cum ſic ſe habeat <seg type="var">.m.h.</seg> ad <seg type="var">.h.q.</seg> vt <seg type="var">.m.g.</seg> ad <seg type="var">.g.k.</seg> permutando ſic ſe habebit <seg type="var">.m.<lb/>h.</seg> ad <seg type="var">.m.g.</seg> vt <seg type="var">.h.q.</seg> ad <seg type="var">.g.k.</seg> itaque <lb/>ex .11. quinti ita <seg type="var">.m.h.</seg> ad <seg type="var">.m.g.</seg> vt <seg type="var">.<lb/> <ptr xml:id="fig-0085-01a" corresp="fig-0085-01" type="figureAnchor"/> m.q.</seg> ad <seg type="var">.m.k.</seg> ex quo permutando <lb/><seg type="var">m.h.</seg> ad <seg type="var">.m.q.</seg> vt <seg type="var">.m.g.</seg> ad <seg type="var">.m.k.</seg> ſed <lb/><choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.m.k.</seg> ſit motus toti diei reſpon <lb/>dens, ſecurè dicere poterimus, ſi <lb/><seg type="var">m.g.</seg> talis eſt reſpectu horarum <num value="24">.<lb/>24.</num> ſignificatarum per <seg type="var">.m.k.</seg> qualis <lb/> <ptr xml:id="fig-0085-02a" corresp="fig-0085-02" type="figureAnchor"/> erit <seg type="var">.m.h.</seg> & quo <lb/>tæ parti dieire-<lb/>ſpondens: </s> <s xml:space="preserve">quæ <lb/><choice><ex>poſtmodum</ex><am>poſtmodũ</am></choice> erit <seg type="var">.<lb/>m.q.</seg> quæ, vt <choice><ex>di- ctum</ex><am>di-ctũ</am></choice> fuit, talis eſt <lb/>reſpectu <seg type="var">.m.k.</seg> <lb/>qualis <seg type="var">.m.h.</seg> re-<lb/>ſpectu <seg type="var">.m.g</seg>. </s> <s xml:space="preserve">Reli <lb/>quę duæ ſpecula <lb/>tiones priorum <lb/><choice><ex>modorum</ex><am>modorũ</am></choice>, vna & <lb/>eadem eſt, <choice><ex>facilisque</ex><am>facilisq́;</am></choice> per ſe mediocriter intelligenti. </s> <s xml:space="preserve">Eodem modo reliquæ omnes <lb/>progreſſiones ſecundi viatoris <choice><ex>cum</ex><am>cũ</am></choice> rectangulo primi conferri ex hoc theoremate <lb/>poterunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0085-01" corresp="fig-0085-01a"> <graphic url="0085-01"/> </figure> <figure xml:id="fig-0085-02" corresp="fig-0085-02a"> <graphic url="0085-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="113">CXIII</num>.</head> <p> <s xml:space="preserve">PRoponitur & aliud, primum ſcilicet viatorem iter incipere diebus aliquot an-<lb/>tè ſecundum, primum tamen lentius, quàm ſecundum ambulare, & utrunque <lb/>eorum certa quædam milliaria conficere. </s> <s xml:space="preserve">Iam ſiſcire voluerimus in quot diebus <lb/>ſeſe conſequentur, uulgaris regula iubet, inſpici quot milliaria primus ſolus iter a-<lb/>gens confecerit, tum animaduerti differentiam diurnam motus vnius ab altero, <choice><ex>atque</ex><am>atq;</am></choice> <lb/>milliarium numerum primi viatoris ſoli abundantis per hanc <choice><ex>differentiam</ex><am>differentiã</am></choice> diuidi, pro <lb/>ueniens autem erit numerus dierum quæſitus.</s> </p> <pb facs="0086" n="74"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Exempli gratia, ſi primus octo diebus antequam ſecundus iter arripuiſſet, con-<lb/><choice><ex>feciſſetque</ex><am>feciſſetq́;</am></choice> fingulis diebus .20. milliaria, tum ſecundus .25. quotidie perfeciſſet, mul <lb/>tiplicandus eſſet numerus .8. cum .20. ex quo darentur .160. milliaria à primo ſolo <lb/><choice><ex>ambulante</ex><am>ambulãte</am></choice> confecta, quibus diuiſis per .5. differentiam motuum diurnorum, daretur <num value="32">.<lb/>32.</num> numerus quæſitus dierum.</s> </p> <p> <s xml:space="preserve">Cuius ratio apertiſſima eſt. </s> <s xml:space="preserve">Sint enim duo rectanguli <seg type="var">.a.n.</seg> et <seg type="var">.u.i.</seg> æquales inter <lb/>ſe, quibus motus itinerarium ſignificentur, quorum <seg type="var">.a.n.</seg> ſit primi, et <seg type="var">.u.i.</seg> ſecundi, præ <lb/>tereà. <seg type="var">a.c.</seg> numerum milliarium diurnorum primi .et <seg type="var">.u.e.</seg> ſecundi, ex quo <seg type="var">.a.c.</seg> minor <lb/>erit <seg type="var">.u.e.</seg> per <seg type="var">.o.e.</seg> atque ita <seg type="var">.o.e.</seg> co-<lb/>gnoſcetur. </s> <s xml:space="preserve">Tum <seg type="var">.c.o.</seg> numerum <choice><ex>dierum</ex><am>dierũ</am></choice> <lb/> <ptr xml:id="fig-0086-01a" corresp="fig-0086-01" type="figureAnchor"/> primi ſoli iter agentis denotet, <choice><ex>cumque</ex><am>cũq;</am></choice> <lb/>conſtituamus <seg type="var">.a.n.</seg> æqualem eſſe <seg type="var">.u.i.<lb/>o.i.</seg> ęqualis erit <seg type="var">.o.a.</seg> atque <seg type="var">.o.a.</seg> cogni <lb/>tus ex ſuis producentibus <seg type="var">.a.c.</seg> et <seg type="var">.c.o.</seg> <lb/>itaque <seg type="var">.o.i.</seg> etiam cognitus, qui diui-<lb/>ſus per latus cognitum <seg type="var">.o.e.</seg> dabit <seg type="var">.e.<lb/>i.</seg> cognitum numerum ſcilicet dierum, quibus ſecundo ambulandum eſt, vt primum <lb/>conſequatur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0086-01" corresp="fig-0086-01a"> <graphic url="0086-01"/> </figure> </div> </body> </floatingText> </div> <div type="unknown"> <head xml:space="preserve">APPENDIX THEOREM. CXIII.</head> <p> <s xml:space="preserve">AB hoc theoremate ſumpſi ordinem illius operationis, numeris mediantibus, ad <lb/>inueniendam exactam temporis quantitatem, ſeu interuallum, <choice><ex>quod</ex><am>ꝙ</am></choice> tranſit, vel in <lb/>tercedit inter vnam mediocrem coniunctionem & aliam proximam <choice><ex>ſequentem</ex><am>ſequentẽ</am></choice> duo <lb/>rum planetarum, vt patet in epiſtola noſtra ad Illuſtrem Bernardum Trottum con-<lb/>tra Benedictum Altauillam repræhenſorem Ephemeridum. </s> <s xml:space="preserve">Verum tamen eſt <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>cum praxis huiuſmodi theorematis ſit multiplex, viſum fuit vnam proponere, quę <lb/>non ita perſpicua ſit, ſed ſubobſcura, non quòd aliquid voluerim latere illum ami <lb/>cum mihi dilectiſſimum, cui priuatim omnes modos prius oſtenderam, ſed vt cere-<lb/>brum illius mei aduerſarij in laberintum conijcerem inextricabilem vt feci, quam-<lb/>uis modus ille egregius etiam ſit, vt nunc oſtendam.</s> </p> <p> <s xml:space="preserve">In dicta epiſtola igitur mente cogitaui medium motum tardioris planetæ, pu-<lb/>ta ſaturni, illius temporis quo velocior planeta, ſcilicet Iupiter, percurrit ſuo medio <lb/>motu totum zodiacum, incipiendo ambo eodem temporis puncto, nec non ab vna <lb/>eorum media coniunctione, hoc eſt ab eodem zodiaci puncto, in quo coniunctę fue <lb/>runt eorum lineæ mediorum motuum, vbi inueni vi regulæ de tribus, quòd Satur <lb/>nus ſpacio dierum vnius mediocris reuolutionis Iouis, qui ſunt .4328. progreditur <lb/>medio motu gra .145. min .4. hoc eſt min .8704. pofito quòd ipſe Saturnus perficiat <lb/>vnam mediam reuolutionem ſpacio dierum .10740. vt dixi. </s> <s xml:space="preserve">Incipiendo igitur ite <lb/>rum Iupiter aliam reuolutionem percurrere, reperto Saturno per min .8704. ante <lb/>ipſum ſpacio .4328. dierum, certus eram hos dies ſignificatos eſſe à linea <seg type="var">.a.u.</seg> vel <seg type="var">.c.<lb/>o.</seg> (æquales enim inuicem ſunt) in figura huiuſmodi theorematis, & quòd rectangu <lb/>lum <seg type="var">.a.o.</seg> præbebat ſummam graduum .145. min .4. hoc eſt min .8704. et quòd <seg type="var">.a.c.</seg> <lb/>vel <seg type="var">.o.u.</seg> ſignificabat iter vnius diei ipſius Saturni, et <seg type="var">.u.e.</seg> iter vnius diei Iouis. </s> <s xml:space="preserve">Cogi-<lb/>temus nunc <seg type="var">.u.x.</seg> ſignificari dies .30. & à puncto <seg type="var">.x.</seg> productam eſſe <seg type="var">.x.f.</seg> parallelam ipſi <lb/><seg type="var">u.o.e.</seg> vnde certi erimus rectangulum <seg type="var">.e.x.</seg> ſignificare iter Ionis ſpacio temporis die-<lb/>rum .30. rectangulum verò <seg type="var">.o.x.</seg> iter Saturni eodem temporis interuallo, vnde rectan <pb facs="0087" n="75"/><fw type="head">THEOREM. ARIT.</fw> gulum <seg type="var">.e.x.</seg> erit minutorum .149. & ſecundorum .43. et <seg type="var">.o.x.</seg> minutorum .60. & ſecun. <lb/>20. vt in dicta epiſtola, vnde rectangulum <seg type="var">.o.f.</seg> erit min .89. & ſecun .23. & quia re-<lb/>ctangulum <seg type="var">.o.i.</seg> æquale eſt rectangulo <seg type="var">.a.o.</seg> ergo <seg type="var">.o.i.</seg> ſimiliter continebit min .8704. <lb/>Nunc quia <seg type="var">.a.c.</seg> vel <seg type="var">.o.u.</seg> denotat iter vnius diei Saturni et <seg type="var">.u.e.</seg> vnius diei Iouis vt di-<lb/>ximus ergo <seg type="var">.u.o.</seg> erit minutorum .2. ſecun <seg type="var">.o.</seg> & tertiarum .40. videlicet tertiarum <num value="7240">.<lb/>7240.</num> ſuppoſito periodo totaliipſius Saturni dierum .10740. et <seg type="var">.u.e.</seg> erit <choice><ex>minutorum</ex><am>minutorũ</am></choice>. <lb/>4. ſecun .59. & ter .27. vel circa hoc eſt tertiarum .17967. vnde <seg type="var">.o.e.</seg> erit tertiarum <num value="10727">.<lb/>10727.</num> </s> <s xml:space="preserve">Nuncſi dixerimus cum <seg type="var">.o.e.</seg> tertiarum .10727. dat <seg type="var">.o.u.</seg> vel <seg type="var">.a.c.</seg> (nam tam <lb/>vna quam altera eſt tertiarum .7240.) quid dabit <seg type="var">.a.u.</seg> vel <seg type="var">.o.c.</seg> (quia tam vna quam <lb/>altera eſt partium .4328.) clarum erit quòd dabit <seg type="var">.o.n.</seg> vel <seg type="var">.u.t.</seg> uel <seg type="var">.e.i.</seg> quia tam vna <lb/>quam altera erit partium .2921. quæ partes coniunctæ cum fuerint cum partibus ip-<lb/>ſius <seg type="var">.a.u.</seg> dabunt totam <seg type="var">.a.t.</seg> <choice><ex>partium</ex><am>partiũ</am></choice> .7249. quæ erunt tot dies, hoc eſt periodus quæſita.</s> </p> <p> <s xml:space="preserve">Alia methodo ſimiliter poſſumus idem cognoſcere, ſcilicet dicendo ſi rectangu <lb/>lum <seg type="var">.f.o.</seg> quod eſt minutorum .89. & ſecun .23. hoc eſt ſecundorum .5363. dat rectan <lb/>gulum <seg type="var">.o.x.</seg> minutorum .60. & ſecun .20. hoc eſt ſecun .3620. quid dabit <seg type="var">.a.u.</seg> partium <lb/>4328. vnde veniet <seg type="var">.u.t.</seg> partium .2921. ſimiliter, eo quod eadem proportio eſt rectan <lb/>guli <seg type="var">.f.o.</seg> ad <seg type="var">.o.x.</seg> quæ <seg type="var">.e.o.</seg> ad <seg type="var">.o.u.</seg> ex prima ſexti, vel .18. 19. ſeptimi ſeu .15. quinti.</s> </p> <p> <s xml:space="preserve">Poſſet etiam aliquis dicere ſi <seg type="var">.f.o.</seg> dat <seg type="var">.o.x.</seg> quid dabit <seg type="var">.o.a.</seg> vnde veniet <seg type="var">.o.t.</seg> quo <lb/>diuiſo per <seg type="var">.o.u.</seg> daret <seg type="var">.u.<lb/>t.</seg> quia ita ſe habet <seg type="var">.a.o.</seg> <lb/> <ptr xml:id="fig-0087-01a" corresp="fig-0087-01" type="figureAnchor"/> ad <seg type="var">.o.t.</seg> vt <seg type="var">.a.u.</seg> ad <seg type="var">.u.t.</seg> ex <lb/>ſupra hic iam citatis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0087-01" corresp="fig-0087-01a"> <graphic url="0087-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ego, in dicta epi-<lb/>ſtola, aliam methodum <lb/>obſeruaui, quæ eſt multi <lb/>plicando minuta .8704. <lb/>per .30. <choice><ex>productumque</ex><am>productumq́;</am></choice> di <lb/>uiſi per min .5363. quaſi <lb/>dicens. </s> <s xml:space="preserve">Si <seg type="var">.o.f.</seg> dat <seg type="var">.o.i.</seg> <lb/>quid dabit <seg type="var">.e.f</seg>. </s> <s xml:space="preserve">Vnde exiam ſupradictis propoſitionibus veniet <seg type="var">.e.i.</seg> & quia permu-<lb/>tando ita ſe habet <seg type="var">.o.f.</seg> ad <seg type="var">.e.f.</seg> vt <seg type="var">.o.i.</seg> ad <seg type="var">.e.i.</seg> ideo dixi, ſi min .89. cum ſecun .23. dat .30 <lb/>quid dabit min .8704.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="114">CXIIII</num>.</head> <p> <s xml:space="preserve">PRoponunt veteres & quærunt aliud, nempe ſi duo iter agentes, eodem in-<lb/>ſtanti diuerſis è locis proficiſcantur, ita vt vnus locum vnde alter profectus <lb/>eſt petat, <choice><ex>alterque</ex><am>alterq́;</am></choice> altero velocior ſit, quo loco <choice><ex>quanue</ex><am>quãue</am></choice> die ſibi inuicem occurrent.</s> </p> <p> <s xml:space="preserve">Exempli gratia, Patauio profectus quidam Taurinum petit, eodem inſtanti al-<lb/>ter Taurino Patauium, <choice><ex>eſtque</ex><am>eſtq́;</am></choice> iter .400. milliarium, ille tamen vndecim diebus, hic <lb/>9. motu regulari & vniformi appellit. </s> <s xml:space="preserve">Quærimus quot milliaria quiſque confece-<lb/>rit, <choice><ex>quotque</ex><am>quotq́;</am></choice> diebus iter egerit, priuſquam ſibi occurrant.</s> </p> <p> <s xml:space="preserve">Iubent nos veteres dies vtriuſque inuicem inter ſe multiplicare, <choice><ex>eritque</ex><am>eritq́;</am></choice> produ-<lb/>ctum .99. item etiam in ſummam colligere, <choice><ex>eritque</ex><am>eritq́;</am></choice> ſumma .20. per quam <choice><ex>productum</ex><am>productũ</am></choice>. <lb/>99.@diuiſerimus dabuntur dies .4. cum .19. vigeſimis vnius diei. </s> <s xml:space="preserve">At pro milliaribus <lb/>vtriuſque, pro eo qui .11. diebus iter conficit, multiplicatis .400. per .4. et .19. vigeſi <lb/>mis, tum diuiſo per .11. dabitur numerus .180. à Patauio Taurinum & è contra, qui <pb facs="0088" n="76"/><fw type="head">IO. BAPT. BENED.</fw> Taurino Patauium .220. quæ quiſque confecerit.</s> </p> <p> <s xml:space="preserve">Dum autem hæc ſpecularer attentius, occurrit alius ſoluendi modus, quamuis pro <lb/>lixior. </s> <s xml:space="preserve">Is <choice><ex>autem</ex><am>aũt</am></choice> eſt eiuſmodi. </s> <s xml:space="preserve"><choice><ex>Accipiatur</ex><am>Accipiat̃</am></choice> medietas minoris numeri <choice><ex>dierum</ex><am>dierũ</am></choice>, <choice><ex>nempe</ex><am>nẽpe</am></choice> .4. <choice><ex>cum</ex><am>cũ</am></choice> dimi <lb/>dio, & per .400. multiplicetur, <choice><ex>productumque</ex><am>productũq́;</am></choice> per <choice><ex>maiorem</ex><am>maiorẽ</am></choice> numerum diuidemus ſcilicet <lb/>11. ex quo dabuntur .163. cum .7. vndecimis, quo proueniente è dimidio <choice><ex>millia- riorum</ex><am>millia-riorũ</am></choice> itineris .200. detracto, & <choice><ex>preſiduum</ex><am>preſiduũ</am></choice> <choice><ex>nempe</ex><am>nẽpe</am></choice> .36. <choice><ex>cum</ex><am>cũ</am></choice> .4. vndecimis multiplicato pro <lb/><choice><ex>ductoque</ex><am>ductoq́;</am></choice> diuiſo <choice><ex>per</ex><am>ꝑ</am></choice> <choice><ex>ſummam</ex><am>ſummã</am></choice> dimidij itineris .200. <choice><ex>cum</ex><am>cũ</am></choice> primo <choice><ex>prouentu</ex><am>prouẽtu</am></choice> .163. et .7. vndecimis <lb/><choice><ex>nempe</ex><am>nẽpe</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> .363. ct .7. vndecimas partes <choice><ex>proueniet</ex><am>ꝓueniet</am></choice> .16. <choice><ex>cum</ex><am>cũ</am></choice> .4. vndecimis, quo <choice><ex>coniuncto</ex><am>cõiuncto</am></choice> pri <lb/>mo <choice><ex>prouenienti</ex><am>ꝓueniẽti</am></choice>, primus .180. milliaria <choice><ex>confecerit</ex><am>cõfecerit</am></choice>, quæ è .400. detracta ſupererunt .220. <lb/>pro itinere ſecundi, qui .9. diebus iter abſoluit. </s> <s xml:space="preserve">Ad hæc ſi tempus ſcire velimus <lb/>eius, qui .11. diebus appellit, multiplicabimus .11. cum .180. <choice><ex>productumque</ex><am>productumq́;</am></choice> per .400. <lb/>partiemur, <choice><ex>prouenientque</ex><am>prouenientq́;</am></choice> paulominus, quam quinque dies, nempe .4. cum .22. horis <lb/>et .48. minutis, quod tempus vtrique viatori inſeruiet, quandoquidem idipſum pro <lb/>uenit multiplicato .220. per .9. <choice><ex>productoque</ex><am>productoq́;</am></choice> per .400. diuiſo.</s> </p> <p> <s xml:space="preserve">Huius autem, qui à me pręſcribitur modi, ſpeculatio talis eſt. </s> <s xml:space="preserve">Duo termini duabus <lb/>rectis lineis æqualibus, & parallelis inter ſe <seg type="var">.b.p.</seg> et <seg type="var">.d.q.</seg> ſignificentur, quæ alijs dua-<lb/>bus <seg type="var">.b.d.</seg> et <seg type="var">.q.p.</seg> <choice><ex>coniungantur</ex><am>coniungant̃</am></choice>, quę parallelæ & æquales erunt ex .33. primi, quibus ſigni <lb/>ficentur duo itinera. </s> <s xml:space="preserve">Viator primus quidem lentior à. b in <seg type="var">.d.</seg> velocior à <seg type="var">.q.</seg> in <seg type="var">.p</seg>. </s> <s xml:space="preserve">Iam <lb/>ſumatur <choice><ex>punctum</ex><am>punctũ</am></choice> medium <seg type="var">.q.p.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.k.</seg> & ab ipſo ad <seg type="var">.b.d.</seg> ducatur <seg type="var">.k.i.</seg> parallela <seg type="var">.d.q.</seg> aut <lb/><seg type="var">b.p.</seg> quod idem eſt, ex quo <seg type="var">.b.i.</seg> æqualis erit <seg type="var">.p.k.</seg> ex .34. primi, hoc eſt <seg type="var">.q.k.</seg> <choice><ex>certique</ex><am>certiq́;</am></choice> eri-<lb/>mus primum viatorem <seg type="var">.q.p.</seg> in dimidio itineris <seg type="var">.q.k.</seg> occurrere non potuiſſe viatori ip <lb/>ſius <seg type="var">.b.i.</seg> quandoquidem eo tempore, quo is, qui ipſius <seg type="var">.q.p.</seg> mouetur per <seg type="var">.q.k.</seg> (cum ſit <lb/>altero velocior) qui per <seg type="var">.b.d.</seg> nondum peruenerit ad .i: Sit itaque punctum <seg type="var">.c.</seg> in quo <lb/>lentior reperitur, dum velocior eſt in <seg type="var">.k.</seg> ex quo certi erimus eos inter <seg type="var">.c.</seg> et <seg type="var">.i.</seg> ſibi in-<lb/>uicem obuiaturos eſſe. </s> <s xml:space="preserve">Cogito deinde rectam lineam ductam <seg type="var">.k.c.</seg> & ut ſe habet <seg type="var">.i.<lb/>c.</seg> ad <seg type="var">.c.b.</seg> ita cogito ſe habere. u<unclear reason="illegible"/> <seg type="var">.k.</seg> ad <seg type="var">.k.q.</seg> & à puncto <seg type="var">.u.</seg> ad <seg type="var">.i.</seg> duco <seg type="var">.u.i.</seg> quæ, vt manife <lb/>ſtum eſt, lineam <seg type="var">.k.c.</seg> in puncto <seg type="var">.e.</seg> interſecabit, à quo cum fuerit ducta <seg type="var">.e.o.n.</seg> parallela <lb/><seg type="var">k.i.</seg> habebimus <seg type="var">.o.n.</seg> ea ſcilicet puncta, quibus occurrunt ſibijpſis, nam cum ſic ſe ha <lb/>beat <seg type="var">.q.k.</seg> ad <seg type="var">.k.u.</seg> vt <seg type="var">.b.c.</seg> ad <seg type="var">.c.i.</seg> et <seg type="var">.k.u.</seg> ad <seg type="var">.k.n.</seg> vt <seg type="var">.c.i.</seg> ad <seg type="var">.c.o.</seg> ex ſimilitudine manifeſta <lb/>triangulorum, ex æqualitate proportionum ſic ſe habebit <seg type="var">.q.k.</seg> ad <seg type="var">.k.n.</seg> vt <seg type="var">.b.c.</seg> ad <seg type="var">.c.o.</seg> <lb/>& permutando ita <seg type="var">.k.q.</seg> ad <seg type="var">.b.c.</seg> vt <seg type="var">.k.n.</seg> ad <seg type="var">.c.o.</seg> & cum <seg type="var">.q.k.</seg> et <seg type="var">.b.c.</seg> ſpatia ſint tempori-<lb/>bus æqualibus confecta, itaque ſpatia <seg type="var">.k.n.</seg> et <seg type="var">.c.o.</seg> ex communi ſcientia temporibus <lb/>æqualibus conficientur.</s> </p> <p> <s xml:space="preserve">Quare rectè dicimus, ſi tot diebus à <seg type="var">.b.</seg> in <seg type="var">.d.</seg> aliquis peruenit, quot milliaria in di <lb/>midio temporis alterius viatoris idem conficiet? </s> <s xml:space="preserve">ex quo ex regula de tribus quam <lb/>primum iter <seg type="var">.b.c.</seg> cognoſcitur, quo ex dimidio itineris detracto, remanet <seg type="var">.c.i.</seg> cogni <lb/>tus, ſed cum probauerimus <seg type="var">.q.k.</seg> ad <seg type="var">.k.n.</seg> hoc eſt <seg type="var">.i.o.</seg> (cum ſint æquales inter ſe, ex .34 <lb/>primi) ita ſe habere. vt <seg type="var">.b.c.</seg> ad <seg type="var">.c.o.</seg> permutando ſic ſe habebit <seg type="var">.q.k.</seg> ad <seg type="var">.b.c.</seg> vt <seg type="var">.i.o.</seg> ad <seg type="var">.<lb/>o.c.</seg> & <choice><ex>componendo</ex><am>cõponendo</am></choice> <seg type="var">.q.k.</seg> et <seg type="var">.b.c.</seg> ad <seg type="var">.b.c.</seg> vt <seg type="var">.i.c.</seg> ad <seg type="var">.c.o.</seg> </s> <s xml:space="preserve">quare rectè dicimus ſi ſumma <seg type="var">.q.<lb/>k.</seg> cum <seg type="var">.b.c.</seg> dat <seg type="var">.b.c.</seg> quid dabit <seg type="var">.i.c</seg>? </s> <s xml:space="preserve">nempe dabit <seg type="var">.c.o.</seg> quo coniuncto cum <seg type="var">.b.c.</seg> cogno-<lb/>ſcitur <seg type="var">.b.o.</seg> quo <seg type="var">.b.o.</seg> detracto ex <seg type="var">.b.d.</seg> remanet cognitus <seg type="var">.o.d.</seg> nempe <seg type="var">.q.n.</seg> illi æqualis <lb/>ex .34. prædicta. </s> <s xml:space="preserve">Gratia verò <choice><ex>temporis</ex><am>tẽporis</am></choice> patet nos rectè dicere ſi <seg type="var">.b.d.</seg> tot diebus abſolui <lb/>tur, aut etiam <seg type="var">.q.p</seg>: quo <seg type="var">.b.o.</seg> aut <seg type="var">.q.n.</seg> abſoluetur.</s> </p> <p> <s xml:space="preserve">Vt autem ad ſpeculationem regulæ antiquorum deueniamus, cogitemus pri-<lb/>mum viatorem ipſius <seg type="var">.q.p.</seg> velociorem eo, qui per <seg type="var">.b.d.</seg> iter agit, tanto tempore præ <lb/>tergredi <seg type="var">.p.</seg> quanto alter <seg type="var">.b.d.</seg> abſoluit. </s> <s xml:space="preserve">Is autem ad <seg type="var">.g.</seg> pertingat, ex quo eadem pro-<lb/>portio ſpacij <seg type="var">.q.g.</seg> ad <seg type="var">.q.p.</seg> hoc eſt <seg type="var">.b.d.</seg> dabitur, quæ temporis quo <seg type="var">.b.d.</seg> abſoluitur ab <pb facs="0089" n="77"/><fw type="head">THEOREM. ARIT.</fw> eo qui per <seg type="var">.b.d.</seg> ad tempus quo <seg type="var">.q.p.</seg> ſolum, qui per <seg type="var">.q.p.</seg> mouetur (mo-<lb/>tus enim continui regulares & vniformes conſtituuntur) eadem ratione ita-<lb/>que ea erit proportio <seg type="var">.q.k.</seg> ad <seg type="var">.b.c.</seg> quæ <seg type="var">.q.g.</seg> ad <seg type="var">.q.p.</seg> & cum probatum <lb/>fuerit ita ſe habere <seg type="var">.k.n.</seg> ad <seg type="var">.c.o.</seg> vt <seg type="var">.q.k.</seg> ad <seg type="var">.b.c.</seg> itaque ſic ſe habebit <seg type="var">.k.n.</seg> ad <seg type="var">.c.<lb/>o.</seg> ut <seg type="var">.q.g.</seg> ad <seg type="var">.q.p.</seg> probatum etiam fuit ita ſe habere <seg type="var">.q.k.</seg> ad <seg type="var">.k.n.</seg> vt <seg type="var">.b.c.</seg> ad <seg type="var">.c.o.</seg> ex quo <lb/>componendo ſic ſe habebit <seg type="var">.q.n.</seg> ad <seg type="var">.n.k.</seg> vt <seg type="var">.b.o.</seg> ad <seg type="var">.o.c.</seg> & permutando ita <seg type="var">.q.n.</seg> ad <seg type="var">.b.<lb/>o.</seg> vt <seg type="var">.k.n.</seg> ad <seg type="var">.c.o.</seg> hoc eſt <seg type="var">.q.g.</seg> ad <seg type="var">.q.p.</seg> nempe vt tempus lenti ad tempus velocis itine-<lb/>rantis, & componendo ita <seg type="var">.q.n.</seg> cum <seg type="var">.o.b.</seg> hoc eſt <seg type="var">.b.d.</seg> ad <seg type="var">.b.o.</seg> vt ſumma <choice><ex>dierum</ex><am>dierũ</am></choice> vnius & <lb/>alterius viatoris ad <choice><ex>minorem</ex><am>minorẽ</am></choice> <choice><ex>numerum</ex><am>numerũ</am></choice> <choice><ex>dierum</ex><am>dierũ</am></choice> velocioris. </s> <s xml:space="preserve">Breuiter <choice><ex>itaque</ex><am>itaq;</am></choice> obtineremus in <lb/><choice><ex>tentum</ex><am>tentũ</am></choice> <choice><ex>quando</ex><am>qñ</am></choice> diceremus ſi ſumma dierum, quibus iter agitur à viatoribus talis eſt (20) re-<lb/>ſpectu numeri dierum velocioris(9) qualis & cui <choice><ex>reſpondebit</ex><am>reſpõdebit</am></choice> totum ſpacium <seg type="var">.b.d</seg>? </s> <s xml:space="preserve">vn-<lb/>de dabitur ſpacium <seg type="var">.b.o.</seg> vnde reliqua omnia nobis cognita emergent.</s> </p> <p> <s xml:space="preserve">Cum autem antiquorum regula iubeat numerum dierum vnius, cum numero die-<lb/>rum alterius multiplicari, ac poſtmodum diuidi productum per ſummam omnium <lb/>dierum, rectèid quidem fit. </s> <s xml:space="preserve">Nam cum ſic ſe habeat <seg type="var">.b.d.</seg> ad <seg type="var">.b.o.</seg> vt ſumma omnium <lb/>dierum ad minorem quantitatem dierum velocioris ſcilicet. </s> <s xml:space="preserve">Ideo temporis propor <lb/>tio à mobili per <seg type="var">.b.d.</seg> abſumpti ad tempus mobilis per <seg type="var">.b.o.</seg> eadem erit, quæ ſummæ <lb/>omnium dierum ad namerum dierum velocioris. </s> <s xml:space="preserve">Quarerectè dicemus, ſi eiuſmodi <lb/>ſumma talem reſpectum habet ad minorem numerum dierum, quem numerum re-<lb/>ſpiciet dies ipſius <seg type="var">.b.d</seg>? </s> <s xml:space="preserve">ex quo proferentur dies reſpondentes ipſi <seg type="var">.b.o.</seg> cætera iam <lb/>dicta fuerunt.</s> </p> <p> <s xml:space="preserve">Huiuſmodi verò ſpeculationis am-<lb/> <ptr xml:id="fig-0089-01a" corresp="fig-0089-01" type="figureAnchor"/> plitudo ad pauciſſima verba reduci <lb/>poteft, in cuius <choice><ex>gratiam</ex><am>gratiã</am></choice> ſit ſubſcripta <lb/>figura pars <choice><ex>inquam</ex><am>inquã</am></choice> pręcedentis, in qua <lb/><choice><ex>conſtituamus</ex><am>cõſtituamꝰ</am></choice> <seg type="var">.o.n.</seg> <choice><ex>locum</ex><am>locũ</am></choice> <choice><ex>eum</ex><am>eũ</am></choice> eſſe quo ſibi <lb/>viatores obuient, ex quo ſpacium <seg type="var">.q.<lb/>n.</seg> à ſuo viatore conficietur, eo ipſo <lb/>tempore, quo à ſuo ſpacium <seg type="var">.b.o.</seg> ita <lb/>que eadem erit proportio <seg type="var">.q.n.</seg> ad <seg type="var">.b.</seg> <lb/> <ptr xml:id="fig-0089-02a" corresp="fig-0089-02" type="figureAnchor"/> o. quæ <seg type="var">.q.g.</seg> ad <seg type="var">.b.d.</seg> eadem erit <choice><ex>inquam</ex><am>inquã</am></choice> <lb/>proportio <seg type="var">.d.o.</seg> ad <seg type="var">.o.b.</seg> quæ numeri <lb/>dierum eius, qui à <seg type="var">.b.</seg> pergit in <seg type="var">.d.</seg> ad <lb/>numerum dierum alterius qui à <seg type="var">.q.</seg> in <lb/>p. proficiſcitur, & componendo <choice><ex>eadem</ex><am>eadẽ</am></choice> <lb/>erit proportio <seg type="var">.d.b.</seg> ad <seg type="var">.b.o.</seg> quæ ſum-<lb/>mæ dierum ad minorem numerum ipſorum, & eadem quæ dierum <seg type="var">.b.d.</seg> ad dies <lb/>ipſius <seg type="var">.b.o</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0089-01" corresp="fig-0089-01a"> <graphic url="0089-01"/> </figure> <figure xml:id="fig-0089-02" corresp="fig-0089-02a"> <graphic url="0089-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="115">CXV</num>.</head> <p> <s xml:space="preserve">CIRCA hæc ipſa itinera aliud quæritur peruenuſtè, in quo quæſito illud con<lb/>ſtituitur cognitum eſſe, nempe interuallum inter duo diuerſa loca, è quibus <lb/>duo viatores eodem inſtanti vt ſibi occurrant proficiſcuntur, <choice><ex>certaque</ex><am>certaq́;</am></choice> milliaria ſin-<lb/>gulis diebus conficiant, ita tamen, ut unus ordinatè plura altero ambulet, quæritur <lb/>deinde quoto die ſibi occurrent. </s> <s xml:space="preserve">Hoc autem fit diuiſo toto interuallo locorum per <lb/>ſummam milliariorum quam vterque quotidie abſoluit.</s> </p> <pb facs="0090" n="78"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Exempli gratia, diſtant loca .100. milliaribus à ſe inuicem; </s> <s xml:space="preserve">vnus autem viator <lb/>ſingulis diebus .15 milliaria, alter .10. conficit ſi ita que .15. cum .10. <choice><ex>coniungamus</ex><am>coniũgamus</am></choice>, <lb/>ſumma erit .25. per <choice><ex>quam</ex><am>quã</am></choice> diuiſis milliaribus .100. totius interualli proferetur .4. nume <lb/>rus quæſitus dierum quo viatoribus iter agendum erit prius quam ſibi obuient.</s> </p> <p> <s xml:space="preserve">In cuius ſpeculationis gratiam totum iter ſignificetur linea <seg type="var">.a.u</seg>: primi autem via-<lb/>toris iter diurnum fit <seg type="var">.a.e.</seg> & alterius <seg type="var">.u.o</seg>: terminus verò <seg type="var">.i.</seg> ſit occurſus ita vt eodem <lb/>tempore, alter ſpacium <seg type="var">.a.i.</seg> alter <seg type="var">.u.i.</seg> confecerit, ſpacij autem <seg type="var">.a.e.</seg> tempus <lb/>per <seg type="var">.b.</seg> ſignificetur & tempus ſpacij <seg type="var">.u.o.</seg> per <seg type="var">.c.</seg> quæ tempora erunt inter ſe <lb/>æqualia, porrò ſpacij <seg type="var">.a.i.</seg> tempus per <seg type="var">.d.</seg> & ſpacij <seg type="var">.u.i.</seg> per <seg type="var">.f.</seg> denotetur, æquali <lb/>bus inquam, ex quo eadem proportio erit <seg type="var">.a.e.</seg> ad <seg type="var">.a.i.</seg> quæ <seg type="var">.b.</seg> ad <seg type="var">.d.</seg> et <seg type="var">.o.u.</seg> ad <seg type="var">.u.i.</seg> quæ <lb/>c. ad <seg type="var">.f.</seg> vnde permutando eadem erit proportio itineris ipſius <seg type="var">.b.</seg> ad iter ipſius <seg type="var">.c.</seg> quæ <lb/>itineris <seg type="var">.d.</seg> ad iter ipſius <seg type="var">.f.</seg> & componendo itinerum ipſius <seg type="var">.b.c.</seg> ad iter <seg type="var">.c.</seg> vt itinerum <seg type="var">.<lb/>d.f.</seg> ad iter <seg type="var">.f.</seg> & permutando itinerum <lb/><seg type="var">b.c.</seg> ad itinera <seg type="var">.d.f.</seg> vt itineris <seg type="var">.c.</seg> ad. iter <lb/> <ptr xml:id="fig-0090-01a" corresp="fig-0090-01" type="figureAnchor"/> ipſius <seg type="var">.f.</seg> meritò itaque quęritur ſi itine <lb/>ra <seg type="var">.b.c.</seg> dat itinera <seg type="var">.d.f.</seg> quid dabit tem-<lb/>pus <seg type="var">.c.</seg> nempe dabit tempus <seg type="var">.f.</seg> ſed <seg type="var">.c.</seg> <lb/>ſignatum eſt pro vna die, </s> <s xml:space="preserve">quare in pro <lb/>poſito exemplo <seg type="var">.f.</seg> ſignificabit 4: dies.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0090-01" corresp="fig-0090-01a"> <graphic url="0090-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="116">CXVI</num>.</head> <p> <s xml:space="preserve">ANtiquorum monumentis traditum motum reperimus diuinandi numeri quem <lb/>quis mente conceperit, quo iubemus eum qui numerum cogitauerit, dimi-<lb/>dium cogitari numeri addere cogitato, atque huic ſummæ, rurſus eiuſdem ſummę <lb/>dimidium adiungere, tum quærimus, quoties noueratius totam eam ſummam ingre <lb/>diatur patefactis fractis ſi qui occurrant.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi quis cogitaſſet numerum .12. iubebant huic dimidium addi, <lb/>nempe .6. ex quo ſumma erat .18. iubebant, præterea dimidium huius ſummæ nem-<lb/>pe .9. toti ſummæ adiungi, quæ fuiſſet .27. adhæc quærebant ſibi patefieri quoties <num value="9">.<lb/>9.</num> ſummam prædictam ingrederetur, & ſi in prima aut ſecunda diuiſione aut <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>vtraque, fracti reperirentur, ac quoties nouem vltimam ſummam ingrediebatur, <lb/>toties .4. multiplicabant. </s> <s xml:space="preserve">Quod ſi in prima diuiſione fracti erant, vltimo produ-<lb/>cto addebant vnitatem; </s> <s xml:space="preserve">ſin verò in ſecunda, binarium adiungebant, ex quo exa-<lb/>ctus numerus quæſitus proferebatur.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ratione ſit <seg type="var">.a.</seg> numerus cogitatione compræhenſus et <seg type="var">.e.</seg> ipſius <seg type="var">.a.</seg> cum <lb/>eiuſdem medietate ſumma et <seg type="var">.i.</seg> ipſius <seg type="var">.e.</seg> cum eiuſdem medietate itidem ſumma, vn <lb/>de <seg type="var">.i.e.a.</seg> tres numeri continui proportionales, in ſeſquialtera proportione euadent. <lb/></s> <s xml:space="preserve">Sumantur nunc tres numeri .4. 6. 9. in eadem proportionalitate. </s> <s xml:space="preserve">Vnde ratione ęqua <lb/>litatis proportionum ita ſe habebit <seg type="var">.i.</seg> ad <seg type="var">.a.</seg> <choice><ex>quemadmodum</ex><am>quẽadmodum</am></choice> .9. ad .4. & permutando <seg type="var">.i.</seg> <lb/>ad .9. quemadmodum <seg type="var">.a.</seg> ad .4. & ob id .4. toties ingredietur <seg type="var">.a.</seg> quoties .9. ipſam <seg type="var">.i</seg>. </s> <s xml:space="preserve">Sed <lb/>quia ſępe contingit, vt in ſecunda diuiſione, aut in ambabus etiam diuiſionibus re <lb/>periantur numeri fracti, anima duertendum eſt numerum animo compræhenſum <seg type="var">.a.</seg> <lb/>ſcilicet aut parem aut imparem ſemper futurum. </s> <s xml:space="preserve">Si par eſt, aut multiplex erit ad <num value="4">.<lb/>4.</num> aut non. </s> <s xml:space="preserve">Si priori modo ſe habebit in duabus diuiſionibus, nullus numerus fra-<lb/>ctus admittetur, ſed ſi ad .4. multiplex non erit, à multiplicibus per duo ſemper dif <lb/>feret, & ſi per medium diuidatur, eiuſdem medietas impar ſemper erit, vnde prior <pb facs="0091" n="79"/><fw type="head">THEOR. ARITH.</fw> quoque ſumma par nunquam exiſtet, cuius medietatem aliquod medium ſemper <lb/>ingredietur, & hanc ob cauſam poſterior ſumma cum fracto ſemper erit, & nume-<lb/>rum deſumptum maiorem eſſe multiplici ad quatuor per duo ſignificabit.</s> </p> <p> <s xml:space="preserve">At verò ſi inter impares reponatur, aut eorum erit qui ſuperant multiplicem <lb/>ipſius quatuor per vnum, ſeu per tria, quod hinc innoteſcet, nempe, quia ſi eorum <lb/>erit qui dictum multiplicem per vnum tantum vincunt, ſua medietate ipſi numero <lb/>addita, & præter hanc medietatem medio etiam integro adiuncto, tota hæc prior <lb/>ſumma in numerum parem ſemper euadet, vnde in poſteriori ſumma nullus nume-<lb/>rus fractus conſpicietur, & hanc ob <choice><ex>causam</ex><am>causã</am></choice> multiplici ipſius .4. vnitas ſemper addetur.</s> </p> <p> <s xml:space="preserve">Sed ſi numerus deſumptus, in ſerie eorum, qui multiplicem ipſius .4. pertria ſu-<lb/>perant, collocabitur, hinc compræhendetur, quia primæ ſummæ numerus cum <lb/>media vnitate ſemper impar erit, vnde ſecunda ſumma præter integras cum me-<lb/>dia vnitate nobis ſemper occur ret.</s> </p> <p> <s xml:space="preserve">Quod autem nobis prodere faciamus an in prima diuiſione, & ſecunda numerus <lb/>aliquis fractus conſiſtat, eò tantum nobis inſeruit, quò deueniamus in cognitionem <lb/>an numerus animo conceptus multiplicem ipſius .4. per vnum, per duo, aut tria ſupe <lb/>ret. </s> <s xml:space="preserve">Quòd etiam medias eas vnitates ad integros reducere faciamus, eò tantum re <lb/>fertur, vt minori labore eum, qui numerum imaginatione compræhendit, onere-<lb/>mus, quia reuera numerus impar nunquam mente concipi poteſt, quin aliquis fra-<lb/>ctus in prima diuiſione, aut in ſecunda ſequatur: </s> <s xml:space="preserve">vnde à numeris imparibus, qui mul <lb/>tiplicem ipſius .4. unitatis tantum exceſſu <choice><ex>ſuperant</ex><am>ſuperãt</am></choice>, poſterior ſumma <choice><ex>cum</ex><am>cũ</am></choice> quarta parte <lb/>vnitatis, præter integros numeros, & ab imparibus qui dictum multiplicem ipſius <num value="4">.<lb/>4.</num> per tria vincunt, cum tribus quartis vnius integri præter integras vnitates ; </s> <s xml:space="preserve">& à <lb/>numeris paribus, qui multiplicem ipſius .4. per duo cum medietate vnitatis præter <lb/>integros ſemper procedit. </s> <s xml:space="preserve">Ita cum is qui numerum ſecum conſiderat, ſi in nume-<lb/> <ptr xml:id="fig-0091-01a" corresp="fig-0091-01" type="figureAnchor"/> ris fractis verſatus eſſet, qui eum in-<lb/>terrogat prudenter ſe gereret, ſi ſibi <lb/>declarari curaret, quis nam ex fractis <lb/>ſu per integros <choice><ex>ſecundæ</ex><am>ſecũdæ</am></choice> <choice><ex>summæ</ex><am>sũmæ</am></choice> remane <lb/>ret, quia <choice><ex>per</ex><am>ꝑ</am></choice> quot quarta integros <choice><ex>ſecun- dæ</ex><am>ſecũ-dæ</am></choice> ſummæ ſuperaret, per <choice><ex>totidem</ex><am>totidẽ</am></choice> inte <lb/>gros numerus mente conceptus multiplicem ipſius .4. ſuperaret.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0091-01" corresp="fig-0091-01a"> <graphic url="0091-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="117">CXVII</num>.</head> <p> <s xml:space="preserve">VNDE fiat, vt ſi ali quis quemuis numerum animo compræhendat, eique <lb/>numero alium etiam quemlibet numerum propoſitum addat, & à tertia par <lb/>te huius ſummæ tertiam partem numeri imaginati detrah et, reſiduum ſecundi nu-<lb/>meri adiuncti, ideſt propoſiti, tertia pars erit.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia, ſi aliquis de numero denario cogitaſſet, <choice><ex>huicque</ex><am>huicq́;</am></choice> .24. adderet, <lb/>vnde triginta quatuor efficerent, detra hendo nunc tertiam partem numeri de na-<lb/>rij cogitatione concepti, ideſt .3. cum tertia parte vnius, à tertia parte huius ſum mæ <lb/>ideſt ab vndecim & vna tertia parte remanerent .8. ideſt tertia pars numeri additi. <lb/></s> <s xml:space="preserve">Id quod mihi inter iocos in honeſtorum hominum cætu in mentem venit.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione, prior numerus ima <lb/> <ptr xml:id="fig-0091-02a" corresp="fig-0091-02" type="figureAnchor"/> ginatus mediante linea <seg type="var">.a.b.</seg> et is, qui ad-<lb/>ditus eſt <choice><ex>intercedente</ex><am>intercedẽte</am></choice> linea <seg type="var">.b.d.</seg> è directo <pb facs="0092" n="80"/><fw type="head">IO. BAPT. BENED.</fw> coniunctis denotetur, et <seg type="var">.b.e.</seg> ſit tertia pars ipſius <seg type="var">.a.b.</seg> prioris numeri im aginati, et. b <lb/>c. tertia pars ipſius, <seg type="var">b.d.</seg> ſecundi numeri propoſiti, vnde coniunctum vnius harum ter <lb/>tiarum <choice><ex>partium</ex><am>partiũ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> alia ſit <seg type="var">.e.c.</seg> quod quidem <seg type="var">.e.c.</seg> eſſe tertiam partem ſummæ duorum <lb/>primorum ideſt <seg type="var">.a.d.</seg> aſſero. </s> <s xml:space="preserve">Iam manifeſtum eſt ipſius <seg type="var">.d.b.</seg> ad <seg type="var">.b.c.</seg> eſſe quemadmo <lb/>dum ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.b.e.</seg> vnde viciſſim ipſius <seg type="var">.d.b.</seg> ad <seg type="var">.b.a.</seg> erit quemadmodum ipſius <seg type="var">.b.<lb/>c.</seg> ad <seg type="var">.b.e.</seg> & coniunctim ipſius <seg type="var">.d.a.</seg> ad <seg type="var">.a.b.</seg> quemadmodum ipſius <seg type="var">.c.e.</seg> ad <seg type="var">.e.b.</seg> & viciſ-<lb/>ſim ipſius <seg type="var">.d.a.</seg> ad <seg type="var">.c.e.</seg> quemadmodum ipſius <seg type="var">.b.a.</seg> ad <seg type="var">.b.e.</seg> ſed proportio ipſius <seg type="var">.b.a.</seg> ad <seg type="var">.<lb/>b.e.</seg> eſt tripla, ergo ea quæ eſt ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.e.c.</seg> erit quoque tripla; </s> <s xml:space="preserve">vnde ſumendo <seg type="var">.e.<lb/>c.</seg> pro tertia parte ipſius <seg type="var">.a.d.</seg> & ab ipſa <seg type="var">.e.c.</seg> ſubtrahendo tertiam partem ipſius <seg type="var">.a.b.</seg> <lb/>tertia pars ipſius <seg type="var">.b.d.</seg> remanebit <seg type="var">.b.c</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0091-02" corresp="fig-0091-02a"> <graphic url="0091-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Aut alio hoc modo, ſupponendo <seg type="var">.e.c.</seg> tertiam partem ipſius <seg type="var">.a.d.</seg> et <seg type="var">.e.b.</seg> ipſius <seg type="var">.a.<lb/>b.</seg> exiſter. </s> <s xml:space="preserve">Dico <seg type="var">.b.c.</seg> tertiam partem ipſius <seg type="var">.b.d.</seg> futuram:</s> <s xml:space="preserve">quia ſi totius <seg type="var">.a.d.</seg> ad totum <lb/><seg type="var">e.c.</seg> ita ſe habet, quemadmodum <seg type="var">.a.b.</seg> à toto <seg type="var">.a.d.</seg> diffecti atque diuulſi ad <seg type="var">.e.b.</seg> à toto <seg type="var">.<lb/>e.c.</seg> diſractum, ergo ex <ref>.19. lib. quinti Eu-<lb/> <ptr xml:id="fig-0092-01a" corresp="fig-0092-01" type="figureAnchor"/> clid.</ref> reſidui <seg type="var">.b.d.</seg> totius <seg type="var">.a.d.</seg> ad reſiduum <seg type="var">.b.c.</seg> <lb/>totius <seg type="var">.e.c.</seg> erit, vt totius <seg type="var">.a.d.</seg> ad <choice><ex>totum</ex><am>totũ</am></choice> <seg type="var">.e.c.</seg> at-<lb/>que hic quidem modus rem <choice><ex>propoſitam</ex><am>propoſitã</am></choice> ſpe-<lb/>culandi mihi aptior & commodior eſſe videtur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0092-01" corresp="fig-0092-01a"> <graphic url="0092-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="118">CXVIII</num>.</head> <p> <s xml:space="preserve">PErmulta ac varia problemata inuenerunt antiqui, longioribus verò vijs reſolu-<lb/>ta, proptereà quòd <choice><ex>non</ex><am>nõ</am></choice> ſemper nobis ſuccurrit breuiſſima in vnaquaque re ex-<lb/>plicatio. </s> <s xml:space="preserve">Vt exempli gratia, proponitur numerus .50. diuidendus in tres tales par-<lb/>tes, quod ſecunda dupla ſit primę, & adhuc eam ſuperet tribus vnitatibus, tertia ve <lb/>rò æqualis ſit aggregato primæ cum ſecunda, & amplius ipſum aggregatum ſuperet <lb/>quinque vnitatibus.</s> </p> <p> <s xml:space="preserve">Ad hoc autem quæſitum ſoluendum antiqui vtebantur regula falſi, quod reuera <lb/>breuiori modo poteſt ſolui, videlicet detra hendo illud ſecundum exceſſum, quin-<lb/>que ſcilicet ex .50. ita vt nobis .45. remaneret, cui medietati hoc eſt .22. cum dimidia <lb/>vnitate, ſi addiderimus illud quinque habebimus .27. cum dimidia vnitate pro ter-<lb/>tia parte quæſita ipſius numeri .50. deinde ſi ab eodem numero .22. cum dimidia <lb/>vnitate detractum fuerit illud .3. primus exceſſus datus, remanebit .19. cum dimi-<lb/>dia vnitate, cuius tertia pars, hoc eſt .6. cum dimidia vnitate, prima pars, ex tri-<lb/>bus quæſita erit, quæ quidem ſi detraxerimus ex .19. cum dimidia vnitate, reli-<lb/>quum erit .13. cui <choice><ex>cum</ex><am>cũ</am></choice> additus fuerit primus exceſſus ideſt .3. </s> <s xml:space="preserve">Iam propoſitum re-<lb/>ſultabit nobis .16. pro ſecunda parte quæſita.</s> </p> <p> <s xml:space="preserve">Ratio verò huiuſmodi operationis talis eſt, ſit verbi gratia totalis numerus pro-<lb/>poſitus ſignificatus per lineam <seg type="var">.a.b.</seg> cuius ſecundæ partis numerus datus ſignificetur <lb/>per lineam <seg type="var">.g.</seg> & numerus tertiæ partis propoſitus per lineam <seg type="var">.h</seg>. </s> <s xml:space="preserve">Nunc dempta <seg type="var">.h.</seg> ex <lb/><seg type="var">a.b.</seg> nobis cognita, remanebit <seg type="var">.f.a.</seg> qua <choice><ex>quidem</ex><am>quidẽ</am></choice> per æqualia imaginatione diuiſa in pun <lb/>cto <seg type="var">.e.</seg> & ipſi <seg type="var">.e.f.</seg> addita <seg type="var">.f.b.</seg> tota <seg type="var">.e.b.</seg> nobis cognita erit, quæ quidem tertia pars <lb/>quæſita ipſius <seg type="var">.a.b.</seg> erit, proptereà quòd <seg type="var">.a.e.</seg> (quæ æqualis eſt ipſi <seg type="var">.e.f.</seg>) erit ſumma <lb/>primæ, & ſecundæ partis. </s> <s xml:space="preserve">Detrahatur poſteà. g. ex <seg type="var">.e.a.</seg> & remanebit <seg type="var">.d.a.</seg> cuius ter <lb/>tia pars ſit <seg type="var">.a.c.</seg> quæ quidem prima pars quæſita erit, & nunc cognita, & ita <seg type="var">.c.d.</seg> <lb/>cognita, cui cum addita fuerit <seg type="var">.d.e.</seg> habebimus ſecundam partem quæſitam, quæ <lb/>compo- <pb facs="0093" n="81"/><fw type="head">THEOREM. ARIT.</fw> componitur ex <seg type="var">.d.c.</seg> dupla .ad <seg type="var">.a.c.</seg> pri-<lb/> <ptr xml:id="fig-0093-01a" corresp="fig-0093-01" type="figureAnchor"/> mam partem, & ex <seg type="var">.d.e.</seg> numero dato. <lb/></s> <s xml:space="preserve">tertia verò pars <seg type="var">.e.b.</seg> compoſita eſt ex <seg type="var">.<lb/>e.f.</seg> æquali <seg type="var">.a.e.</seg> hoc eſt æquali compoſi-<lb/>to ex prima, & ſe cunda parte, & ex <seg type="var">.f.<lb/>b.</seg> numero dato vt proponebatur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0093-01" corresp="fig-0093-01a"> <graphic url="0093-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="119">CXIX</num>.</head> <p> <s xml:space="preserve">INter alia problemata ab antiquis inuenta, hoc etiam ponitur. </s> <s xml:space="preserve">Aliquis inter-<lb/>rogat quot ſint horæ, alius verò reſpondit tot eſſe, quot duæ tertiæ præteriti <lb/>temporis ſimul iuncta cum tribus quintis futuri temporis totius dieri naturalis effi-<lb/>ciunt. </s> <s xml:space="preserve">Nunc quæritur quot ſint horę.</s> </p> <p> <s xml:space="preserve">Antiqui, hoc etiam problema ſoluebant mediante regula falſi, ſed mihi alio mo <lb/>do ſoluendum eſſe dictum problema videtur. </s> <s xml:space="preserve">Accipio enim ex quinque, tres vni-<lb/>tates, pro parte futuri temporis, quas quidem in tres vnitates præteriti temporis <lb/>duco, vnde proueniunt mihi nouem vnitates, quod productum coniungo <choice><ex>cum</ex><am>cũ</am></choice> quin-<lb/>que futuri temporis, vnde veniunt .14. vnitates, ex regula poftea de tribus ita dico <lb/>ſi ex .14. mihi prouenit .9. quid reſultabit ex .24. & prouenient mihi horæ .15. cum <lb/>tribus ſeptimis vnius horæ, hoc eſt minuta ferè .26.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione, quinque vnitates, feu partes temporis futuri ſignificentur à <lb/>linea <seg type="var">.e.u.</seg> quarum trium ſigniſicentur a linea <seg type="var">.e.i.</seg> ſumpta deinde ſit linea <seg type="var">.e.o.</seg> æqualis <lb/>lineæ <seg type="var">.e.i.</seg> et <seg type="var">.e.a.</seg> tripla ſit ad <seg type="var">.o.e.</seg> vel ad <seg type="var">.e.i.</seg> quod idem eſt, vnde <seg type="var">.a.e.</seg> compoſita erit <lb/>ex <seg type="var">.a.o.</seg> (hoc eſt ex duabus tertijs ip ſius <seg type="var">.a.e.</seg>) & ex <seg type="var">o.e.</seg> (hoc eſt ex. tribus quintis ip-<lb/>ſius <seg type="var">.e.u.</seg>) vnde <seg type="var">.a.u.</seg> ad <seg type="var">.a.e.</seg> eandem rationem obtinebit, quæ .14. ad .9. </s> <s xml:space="preserve">propterea igi <lb/>tur poſſumus recte ratiotinari <lb/> <ptr xml:id="fig-0093-02a" corresp="fig-0093-02" type="figureAnchor"/> fi .14. nobis dat .9. quid dabit .24. <lb/>qui quidem .24. nobis dabit .15. <lb/>cum min .26. quod rectè factum <lb/>erit ex <ref>.20. ſeptimi Euclidis</ref>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0093-02" corresp="fig-0093-02a"> <graphic url="0093-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="120">CXX</num>.</head> <p> <s xml:space="preserve">SVpponunt etiam antiqui tres ſocios nummos habere, quorum ſumma primi & <lb/>ſecundi cognita ſit, item ſumma primi & tertij cognita & ſumma ſecundi & <lb/>tertij item cognita, at que ex huiuſmodi tribus aggregatis veniunt in cognitionem <lb/>particularem vniuſcuiuſque illorum.</s> </p> <p> <s xml:space="preserve">Gemafriſius ſoluit hoc problema ex regula ſalſi. </s> <s xml:space="preserve">At ego tali ordine progredior. <lb/></s> <s xml:space="preserve">Sit verbi gratia, ſumma primi cum ſecundo .50. & ſecundi cum tertio .70. & primi <lb/>cum tertio .60. harum trium ſummarum accipiantur duæ quæuis, vt puta .50. & .70 <lb/>quæ coniunctæ ſimul dabunt .120. à qua ſumma detrahatur reliqua, ideſt .60. & <lb/>reſtabit nobis .60. cuius medietas erit .30. hoc eſt numerus nummorum ſecundi <lb/>ſocij quo numero detracto à .70. hoc eſt à ſumma ſecundi cum tertio remanebit .40. <lb/>hoc eſt numerus tertij ſocij, & hic numerus deſumptus à .60. reſiduus erit nume-<lb/>rus primi ſocij.</s> </p> <pb facs="0094" n="82"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Pro cuius ratione conſideremus triangulum hic ſubnotatum <seg type="var">.a.b.c.</seg> cuius <lb/>unumquodque latus ſignificet ſummam duorum ſociorum, vtputa latus <seg type="var">.a.b.</seg> ſignifi-<lb/>cet ſummam primi cum ſecundo, latus verò <seg type="var">.b.c.</seg> ſummam ſecundi cum tertio, la-<lb/>rus autem <seg type="var">.a.c.</seg> ſummam primi cum tertio, et <seg type="var">.a.e.</seg> ſeu <seg type="var">.a.o.</seg> ſit numerus primi ſocij, et <seg type="var">.<lb/>e.b.</seg> vel <seg type="var">.b.u.</seg> ſit ſecundi ſocij, et <seg type="var">.c.u.</seg> ſeu <seg type="var">.c.o.</seg> ſit tertij, cum autem <seg type="var">.a.e.</seg> æqualis ſit <seg type="var">.a.o.</seg> <lb/> <ptr xml:id="fig-0094-01a" corresp="fig-0094-01" type="figureAnchor"/> et <seg type="var">.b.e</seg>:æ qualis <seg type="var">.b.u.</seg> et <seg type="var">.c.u.</seg> æqualis <seg type="var">.c.o.</seg> <lb/>ex ſuppoſito ſi <choice><ex>dempta</ex><am>dẽpta</am></choice> fuerit ſumma ſeu <lb/>latus <seg type="var">.a.c.</seg> datum ex aggregato laterum <seg type="var">.<lb/>a.b.</seg> cum <seg type="var">.b.c.</seg> reliquarum ſummarum, re <lb/>linquet nobis cognitum aggregatum <lb/>ex <seg type="var">.b.e.</seg> cum <seg type="var">.b.u</seg>. </s> <s xml:space="preserve">Quare & eius medic-<lb/>tas <seg type="var">.b.e.</seg> ſiue <seg type="var">.b.u.</seg> nobis cognita erit, qua <lb/>detracta exſumma <seg type="var">.b.a.</seg> relinquetur no <lb/>bis cognitus numerus <seg type="var">.a.e.</seg> detracto ve-<lb/>ro numero <seg type="var">.a.e.</seg> hoc eſt <seg type="var">.a.o.</seg> ex <seg type="var">.a.c.</seg> ſum-<lb/>ma, ſeu latus, aut <seg type="var">.b.u.</seg> ex <seg type="var">.b.c.</seg> remanebit <lb/><seg type="var">o.c.</seg> ſeu <seg type="var">.c.u.</seg> cognitus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0094-01" corresp="fig-0094-01a"> <graphic url="0094-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="121">CXXI</num>.</head> <p> <s xml:space="preserve">HAC etiam methodo hoc facere poſſumus non <choice><ex>ſolum</ex><am>ſolũ</am></choice> de tribus ſocijs, ſed <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>de omnibus quotquot volueris, vt exempli gratia, <lb/> <ptr xml:id="fig-0094-02a" corresp="fig-0094-02" type="figureAnchor"/> ſint ſex ſocij <seg type="var">.a.b.c.d.e.f.</seg> quorum ſumma per binos co-<lb/>gnita, vtputà ſumma numeri <seg type="var">.a.</seg> cum <seg type="var">.b.</seg> cognita nobis ſit, <lb/>& ſumma numeri <seg type="var">.b.</seg> cum <seg type="var">.c.</seg> & ſumma <seg type="var">.c.</seg> cum <seg type="var">.d.</seg> & ſum-<lb/>ma <seg type="var">.d.</seg> cum <seg type="var">.e.</seg> & ſumma <seg type="var">.e.</seg> cum <seg type="var">.f.</seg> neceſle eft etiam ſcire <lb/>ſummam duorum vno relicto, vtputa ſummam <seg type="var">.a.</seg> cum <lb/>c. vt poſſimus triangulum <seg type="var">.a.b.c.</seg> conſtituere. </s> <s xml:space="preserve">Vnde ex <lb/>præmiffa, cognitus numerus nobis erit vniuſcuiuſque <seg type="var">.a.<lb/>b.c</seg>. </s> <s xml:space="preserve">Quapropter dempto numero <seg type="var">.c.</seg> ex ſumma <seg type="var">.c.</seg> cum <lb/>d. & numero <seg type="var">.d.</seg> ex ſumma <seg type="var">.d.</seg> cum <seg type="var">.e.</seg> & numero <seg type="var">.e.</seg> ex ſum <lb/>ma <seg type="var">.e.</seg> cum <seg type="var">.f.</seg> habebimus intentum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0094-02" corresp="fig-0094-02a"> <graphic url="0094-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="122">CXXII</num>,</head> <p> <s xml:space="preserve">CVM aliquando, illud quod Archimedes inuenit, vt furtum Regiab aurifa-<lb/>bro in regia corona factum, quemadmodum ſcribit Vitruuius, proderet, con-<lb/>templarer, mihi etiam viſum eſt, vt aliquem modum ſcientiſicum inueſtigarem, quo <lb/>proportio auri ad argentum, quod in aliquo propoſito corpore exipſis miſto cogni <lb/>ti ponderis cognoſci poſſet. </s> <s xml:space="preserve">Et cum multos diuerſis temporibus excogitarim offi-<lb/>cio meo deeſſe nolui in ijſdem literarum monumentis mandandis, quorum hic <lb/>vnus erit: </s> <s xml:space="preserve">propoſita nobis ſint tria corpora <seg type="var">.A.M.V.</seg> æqualia inter ſe, ſed diuer-<lb/>ſarum ſpecierum materiei, vtputa quod <seg type="var">.A.</seg> ſit argenteum, & omogeneum <seg type="var">.V.</seg> ve-<lb/>rò aureum omogeneum, & M. mixtum exauro, & argento, ideſt heterogeneum, <lb/>cupimusergo ſcire <choice><ex>iuſtam</ex><am>iuſtã</am></choice> quantitatem auri & argenti, quæ eſt in ipſo corpore <seg type="var">.M.</seg> <lb/>miſto. </s> <s xml:space="preserve">Ita igitur faciamus. </s> <s xml:space="preserve">Videamus primum quantum ſit pondus vniuſcuiuſque <lb/>ipſorum corporum, ponamus autem pondus corporis <seg type="var">.V.</seg> auri eſſe vt .234. pondus <pb facs="0095" n="89"/><fw type="head">THEOR. ARITH.</fw> autem corporis <seg type="var">.M.</seg> miſti <choice><ex>vt</ex><am>.vt</am></choice> .216. argentei verò <seg type="var">.A.</seg> vt .156. detrahatur nunc pon-<lb/>dus <seg type="var">.A.</seg> ex pondere <seg type="var">.V</seg>. </s> <s xml:space="preserve">Reliquum erit .78. quod vocetur prima differentia ſeruan-<lb/>da, dematur etiam pondus <seg type="var">.M.</seg> ex pondere <seg type="var">.V.</seg> reliquum erit .18. pro ſecunda diffe-<lb/>rentia, etiam ſeruanda, multiplicetur poſteà pondus <seg type="var">.A.</seg> per ſecundam differen-<lb/>tiam, productum verò diuidatur per primam differentiam. </s> <s xml:space="preserve">Vnde in præſenti exem <lb/>plo proueniet nobis .36. quiquidem prouentus erit quantitas argenti ipſius corpo-<lb/>ris miſti <seg type="var">.M.</seg> quo etiam detracto ex pondere totali ipſius <seg type="var">.M.</seg> reliquum erit quanti-<lb/>tas auri eius corporis, hoc eſt .180.</s> </p> <p> <s xml:space="preserve">In cuius operationis ſpeculatione, aliquid natura ſua prius cognitum præcedere <lb/>oportet hoc eſt, quod omnia corpora omogenea eandem proportionem obtinent <lb/>inter quantitates, quam inter pondera. </s> <s xml:space="preserve">Quo ſuppoſito denotetur corpus <seg type="var">.A.</seg> li-<lb/>nea <seg type="var">.o.a.</seg> corpus autem <seg type="var">.V.</seg> linea <seg type="var">.o.c.</seg> & corpus <seg type="var">.M.</seg> linea <seg type="var">.e.u</seg>: ſed <seg type="var">.e.o.</seg> ſignificet par-<lb/>tem argenti, et <seg type="var">.o.u.</seg> partem auri in corpore miſto <seg type="var">.M.</seg> vnde ex communi conceptu <lb/>habebimus <seg type="var">.o.e.</seg> æqualem <seg type="var">.u.c.</seg> cum ex hypotheſi <seg type="var">.e.u.</seg> æqualis ſit <seg type="var">.o.c.</seg> et <seg type="var">.a.o.</seg> ſimiliter. <lb/></s> <s xml:space="preserve">Significetur poſteà pondus <seg type="var">.a.o.</seg> ab <seg type="var">.f.</seg> & pondus <seg type="var">.e.u.</seg> ab <seg type="var">.b.x.</seg> & pondus <seg type="var">.o.c.</seg> ab <seg type="var">.f.g.</seg> pon <lb/>dus verò <seg type="var">.o.e.</seg> ab <seg type="var">.b.</seg> pondus autem <seg type="var">.o.u.</seg> ab <seg type="var">.x.</seg> pondus enim <seg type="var">.u.c.</seg> ab <seg type="var">.b.d.</seg> et <seg type="var">.g.</seg> ſit diffe-<lb/>rentia, qua <seg type="var">.f.g.</seg> maior eſt .f: et <seg type="var">.d.</seg> <lb/> <ptr xml:id="fig-0095-01a" corresp="fig-0095-01" type="figureAnchor"/> differentia qua <seg type="var">.b.d.</seg> maior eſt <seg type="var">.b</seg>. <lb/></s> <s xml:space="preserve">Vnde ex ratione omogeneitatis ea <lb/>dem proportio erit <seg type="var">.a.o.</seg> ad <seg type="var">.e.o.</seg> vt <seg type="var">.<lb/>f.</seg> ad <seg type="var">.b.</seg> et <seg type="var">.o.c.</seg> ad <seg type="var">.u.c.</seg> quæ <seg type="var">.x.b.d.</seg> ſeu <lb/><seg type="var">f.g.</seg> (quodidem eſt) ad <seg type="var">.b.d</seg>. </s> <s xml:space="preserve">Quare <lb/>ex <ref>.11. quinti <choice><ex>eadem</ex><am>eadẽ</am></choice></ref> erit proportio <seg type="var">.<lb/>f.</seg> ad <seg type="var">.b.</seg> vt <seg type="var">.f.g.</seg> ad <seg type="var">.b.d.</seg> & permutan-<lb/>do ita erit <seg type="var">.f.</seg> ad <seg type="var">.f.g.</seg> vt <seg type="var">.b.</seg> ad <seg type="var">.b.d.</seg> & <lb/>ſeparando ita <seg type="var">.f.</seg> ad <seg type="var">.g.</seg> vt <seg type="var">.b.</seg> ad <seg type="var">.d</seg>. </s> <s xml:space="preserve">Sed <seg type="var">.g.</seg> cognita nobis eſt, vt differentia in <lb/>ter <seg type="var">.f.</seg> g, et <seg type="var">.f</seg>: cognita nobis eſt etiam <seg type="var">.f</seg>: cognoſcimus itidem <seg type="var">.d.</seg> vt differentiam inter <seg type="var">.<lb/>x.b.d.</seg> et <seg type="var">.b.x.</seg> quapropter cognoſcemus <seg type="var">.b.</seg> ex .20. ſeptimi Eucli. & ſic <seg type="var">.x.</seg> reſiduum. <lb/></s> <s xml:space="preserve">ex <seg type="var">.b.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0095-01" corresp="fig-0095-01a"> <graphic url="0095-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="123">CXXIII</num>.</head> <p> <s xml:space="preserve">NVNC ex methodo præcedentis propoſiti deuenire poſſumus in cognitio-<lb/>nem veræ quantitatis auri, & argenti confuſi in corona Hieronis conſtituen-<lb/>do primum duo corpora ſimplicia æqualia inter ſe, & coronæ hoc modo videlicet, <lb/>immergendo coronam, ſeu corpus miſtum in aliquod vas aqua plenum, & diligen-<lb/>ter colligere aquam, quæ ex eo effundetur, poſteà verò oportet, aliud vas inuenire <lb/>præciſæ capax illius a quæ collectæ, in quod demum infundatur tantum auri, & po-<lb/>ſteà tantum argenti, quantum ſieri poteſt, vnde vnumquodque horum duorum cor <lb/>porum ſimplicium æquale erit mixto, ſeu coronæ, & ſic quod dictum eſt in præce-<lb/>cedenti theoremate exequemur.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="124">CXXIIII</num>.</head> <p> <s xml:space="preserve">SED vt breuiori methodo idem præſtemus, quod in antecedenti propoſito di-<lb/>ctum eſt, quædam theoremata præmittenda ſunt, videlicet quòd quotíeſcunque <lb/>fuerint tria corpora, quorum duo inuicem æqualia ſint in quantitate, ſed diuerſa- <pb facs="0096" n="84"/><fw type="head">IO. BAPT. BENED.</fw> rum ſpecierum materiæ, tertium verò corpus maius, vel minus ſit in quantitate vtro-<lb/>que illorum, ſed eiuſdem materiæ vnius quod vis illorum, ponderis verò alterius, <lb/><choice><ex>semper</ex><am>sẽper</am></choice> eadem proportio erit inter pondera æqualium corporum, quæ inter <choice><ex>quantita- tem</ex><am>quãtita-tem</am></choice> corporis inæqualis, & eam quæ vnius cuiuſuis æqualium.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſit <seg type="var">.b.</seg> corpus aliquod aureum æquale corpori <seg type="var">.u.</seg> argenteo, ſit <lb/>etiam corpus <seg type="var">.a.</seg> argenteum maius corpore <seg type="var">.b.</seg> vel <seg type="var">.u.</seg> ſed ponderis eiuſdem, quod au-<lb/>ri <seg type="var">.b</seg>. </s> <s xml:space="preserve">Tunc dico eandem eſſe proportionem ponde-<lb/> <ptr xml:id="fig-0096-01a" corresp="fig-0096-01" type="figureAnchor"/> ris <seg type="var">.b.</seg> ad pondus <seg type="var">.u.</seg> quæ eſt magnitudinis <seg type="var">.a.</seg> ad ma-<lb/>gnitudinem <seg type="var">.u</seg>. </s> <s xml:space="preserve">Quod ratiocinemur hoc modo, nam <lb/>cum proportio corporeitatis <seg type="var">.a.</seg> ad corporeitatem <seg type="var">.u.</seg> <lb/>eadem ſit, quæ ponderis <seg type="var">.a.</seg> ad pondus <seg type="var">.u.</seg> ex ratione <lb/>omogeneitatis, ponderis verò <seg type="var">.b.</seg> ad pondus <seg type="var">.u.</seg> ex .7. <lb/>quinti, eadem quæ ponderis <seg type="var">.a.</seg> ad pondus <seg type="var">.u.</seg> ideo ex <lb/>11. eiuſdem proportio ponderis <seg type="var">.b.</seg> ad pondus <seg type="var">.u.</seg> eadem erit, quæ corporeitatis <seg type="var">.a.</seg> <lb/>ad corporeitatem <seg type="var">.u.</seg> vel ad corporeitatem <seg type="var">.b.</seg> quæ æqualis eſt alteri.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0096-01" corresp="fig-0096-01a"> <graphic url="0096-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="125">CXXV</num>.</head> <p> <s xml:space="preserve">QVotieſcunque nobis propoſita fuerint duo corpora cuiuſuis magnitudinis æ-<lb/>que ponderantia, ſed diuerſarum ſpecierum materiæ, cum ſcire volueri-<lb/>mus proportionem ponderum illarum ſpecierum inter ipſas hoc modo faciemus.</s> </p> <p> <s xml:space="preserve">Sint exempli gratia, duo nobis propoſita corpora <seg type="var">.a.</seg> et <seg type="var">.b.</seg> (vt dictum eſt) quæ ſi <lb/>fuerint æqualium magnitudinum inter ſe, clarum erit quod quæritur, ſed inæqua-<lb/>lia erunt, immergatur <choice><ex>unumquodque</ex><am>unumquodq;</am></choice> eorum in vas aqua plenum, & collecta ſit aqua <lb/>effuſa ab vnoquoque illorum, </s> <s xml:space="preserve">tunc <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> iſtarum aquarum æqualis magnitudi-<lb/>nis erit ſui corporis impellentis, & proportio ponderoſitatis illarum eadem erit, <lb/>quæ earum magnitudinum ex omogeneitate, quapropter ſi vnamquamque illarum <lb/>ponderabimus, habebimus propoſitum ex præcedenti theoremate.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="126">CXXVI</num>.</head> <p> <s xml:space="preserve">SED cum ſcire voluerimus pondus alicuius magnitudinis aquæ æqualis alicui <lb/>corpori ponderoſo, breuiſſimus modus erit ponderando ipſum corpus tam in ae-<lb/>re, quàm in aqua, & quia ſemper leuius erit in aqua, </s> <s xml:space="preserve">tunc differentia ponderum ip-<lb/>ſius corporis, erit pondus quæſitum, hoc eſt vnius corporis aquei æqualis magnitu-<lb/>dinis magnitudini corporis propoſiti ex <ref>.7. propoſitione lib. Archimedis de inſi-<lb/>dentibus aquæ</ref>. </s> </p> <p> <s xml:space="preserve">Quare ex præmiſſis quotieſcunque immerſa fuerint in aquam dicti vaſis duo cor <lb/>pora æquè ponderantia, ſed diuerſarum ſpecierum, vt dictum eſt, proportio pon-<lb/>deris aquæ maioris ad pondus aquæ minoris magnitudinis eadem ſemper erit, quæ <lb/>ponderis minoris corporis ad pondus alicuius corporis eidem æqualis, ſpeciei verò <lb/>maioris, vel eadem proportio ponderis alicuius corporis æqualis maiori, ſpeciei ve <lb/>rò minoris ad pondus ipſius maioris.</s> </p> <p> <s xml:space="preserve">Vt puta ſit corpus <seg type="var">.a.</seg> argenteum æqualis ponderis corpori <seg type="var">.b.</seg> aurei, & corpus <seg type="var">.u.</seg> <lb/>argenteum æqualis magnitudinis corpori <seg type="var">.b.</seg> aurei, corpus verò <seg type="var">.n.</seg> aureum æqualis <lb/>magnitudinis corpori <seg type="var">.a.</seg> argentei, corpus verò <seg type="var">.f.</seg> aqueum æqualis magnitudinis cor- <pb facs="0097" n="85"/><fw type="head">THEOREM. ARIT.</fw> pori <seg type="var">.a.</seg> argentei, corpus autem <seg type="var">.e.</seg> <choice><ex>aqueum</ex><am>aqueũ</am></choice> æqualis ma-<lb/> <ptr xml:id="fig-0097-01a" corresp="fig-0097-01" type="figureAnchor"/> gnitudinis corpori <seg type="var">.b.</seg> aurei. </s> <s xml:space="preserve">Tunc dico proportio-<lb/>nem ponderis <seg type="var">.f.</seg> ad pondus <seg type="var">.e.</seg> eadem eſſe, quæ pon-<lb/>deris <seg type="var">.b.</seg> ad pondus <seg type="var">.u.</seg> vt in præcedenti theoremate <lb/>iam dictum eſt, vel quæ ponderis <seg type="var">.n.</seg> ad pondus <seg type="var">.a.</seg> ex <lb/>11. quinti Euclidis. </s> <s xml:space="preserve">Proptereà quòe ponderis <seg type="var">.<lb/>n.</seg> ad pondus <seg type="var">.a.</seg> eft vt poderis <seg type="var">.b.</seg> ad pondus <seg type="var">.u.</seg> eo <lb/>quòd permutando ponderis <seg type="var">.n.</seg> ad pondus <seg type="var">.b.</seg> eſt vt <lb/>ponderis <seg type="var">.a.</seg> ad pondus <seg type="var">.u.</seg> ex corporum omogenei-<lb/>tate, & ex æqualitate magnitudinum corporum antecedentium & conſequentium.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0097-01" corresp="fig-0097-01a"> <graphic url="0097-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="127">CXXVII</num>.</head> <p> <s xml:space="preserve">SCire etiam nos oportet, quòd quotieſcumque fuerint duo corpora aquea, quo-<lb/>rum vnum æqualis magnitudinis ſit alicui miſto, quod quidem miſtum graue <lb/>ſit tam in aere, quàm in aqua, alterum verò corpus aquem æqualis ſit magnitudi-<lb/>nis alicui corpoli ſimplici, quod quidem corpus ſimplex æqualis ponderis ſit dicto <lb/>corpori miſto. </s> <s xml:space="preserve">Tunc proportio ponderis aquei, cuius magnitudo æquatur magni <lb/>tudini corporis miſti, ad pondus corporis aquei, cuius magnitudo æqualis eſt ma-<lb/>gnitudini corporis ſimplicis, eadem erit, quæ proportio ponderis alicuius corpo-<lb/>ris ſimplicis, cuius magnitudo æqualis ſit magnitudini corporis miſti ſuperius dicti, <lb/>ſed ſpeciei corporis ſimplicis iam dicti, ad pondus dicti miſti.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſit corpus aqueum <seg type="var">.e.</seg> magnitudinis æqualis corpori <seg type="var">.m.</seg> mixto, <lb/>corpus verò aqueum <seg type="var">.i.</seg> æqualis magnitudinis ſit corpori ſimplici <seg type="var">.a.</seg> quod quidem <lb/>corpus <seg type="var">.a.</seg> æqualis ponderis ſit cum corpore <seg type="var">.m.</seg> & corpus <seg type="var">.u.</seg> ſit æqualis magnitudinis <lb/>cum corpore <seg type="var">.m.</seg> ſed ſpeciei corporis <seg type="var">.a</seg>. </s> <s xml:space="preserve">Tunc dico proportionem ponderis <seg type="var">.e.</seg> ad <lb/>pondus <seg type="var">.i.</seg> <choice><ex>eandem</ex><am>eãdem</am></choice> eſſe, quæ ponderis <seg type="var">.u.</seg> ad pondus <seg type="var">.m.</seg> primum nulli dubium eſt, quin <lb/>eadem proportio ſit magnitudinis <seg type="var">.e.</seg> ad magnitudinem <seg type="var">.i.</seg> quæ magnitudinis <seg type="var">.m.</seg> ad <lb/>a. ſed <seg type="var">.m.</seg> ad <seg type="var">.a.</seg> eſt vt <seg type="var">.u.</seg> ad <seg type="var">.a.</seg> ex .7. quinti </s> <s xml:space="preserve">quare <lb/>ex .11. eiuſdem proportio <seg type="var">.e.</seg> ad <seg type="var">.i.</seg> erit vt <seg type="var">.u.</seg> ad <seg type="var">.a.</seg> <lb/> <ptr xml:id="fig-0097-02a" corresp="fig-0097-02" type="figureAnchor"/> de ipſius magnitudinibus loquendo, ſed propor-<lb/>tio ponderis <seg type="var">.u.</seg> ad pondus <seg type="var">.a.</seg> eadem eſt, quæ ma <lb/>gnitudinis <seg type="var">.u.</seg> ad magnitudinem <seg type="var">.a.</seg> ex omogenei-<lb/>tate. </s> <s xml:space="preserve">Idem dico de pondere <seg type="var">.e.</seg> ad pondus .1. </s> <s xml:space="preserve">Qua-<lb/>re proportio ponderis <seg type="var">.e.</seg> ad pondus <seg type="var">.i.</seg> eadem erit <lb/>quæ ponderis <seg type="var">.u.</seg> ad pondus <seg type="var">.a</seg>. </s> <s xml:space="preserve">Sed ponderis <seg type="var">.u.</seg> <lb/>ad pondus <seg type="var">.m.</seg> eadem eſt quæ ponderis <seg type="var">.u.</seg> ad pondus <seg type="var">.a.</seg> ex .7. quinti, ergò ex .11. <lb/>eiuſdem proportio ponderis <seg type="var">.e.</seg> ad pondus <seg type="var">.i.</seg> eadem erit, quæ ponderis <seg type="var">.u.</seg> ad pon-<lb/>dus <seg type="var">.m.</seg> quod eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0097-02" corresp="fig-0097-02a"> <graphic url="0097-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="128">CXXVIII</num>.</head> <p> <s xml:space="preserve">NVNC ad cognoſcendam proportionem duarum diuerſarum ſpecierum in <lb/>corpore miſto propoſito, tribus corporibus aqueis mediantibus, quæ <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/>corpora æqualium magnitudinum ſint alijs tribus corporibus vnius & eiuſdem pon <lb/>deris, quorum vnum ſit mixtum, reliqua verò duo ſimplicia, ſed ſpecierum mixti, <lb/>hoc ordine procedemus.</s> </p> <pb facs="0098" n="96"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Sint exempli gratia, tria corpora æquè ponderantia, & vnumquodque illorum <lb/>ſitquinque librarum, quorum vnum ſit aureum, aliud argenteum, reliquum verò <lb/>mixtum ex ijs duobus metallis, vnde corpus aureum ſimplex minus erit, & argen <lb/>teum maius corpore mixto, quod nulli dubium eſt, ſit nunc pondus corporis aquei <lb/>ęqualis corpori aureo, <choice><ex>librarum</ex><am>librarũ</am></choice> .3. aquei verò ęqualis miſto, ſit <choice><ex>librarum</ex><am>librarũ</am></choice> 3. <choice><ex>cum</ex><am>cũ</am></choice> quarta par <lb/>te, aquei demum æqualisargenteo, librarum .4. cum dimidia, vnde exijs, quæ in præ <lb/>cedenti theoremate, & in .126. theoremate diximus, ſi imaginatione concipiemus <lb/>alia duo corpora ſimplicia, auri, & argenti, ſed æqualium magnitudinum mixto, <lb/>habebimus proportionem ponderis aurei ad pondus corporis mixti vt <choice><ex>trium</ex><am>triũ</am></choice> librarum <lb/>cum quarta vnius ad .3. libras, & proportio ponderis mixti ad pondus argentei erit, <lb/>vt proportio librarum .4. cum dimidia ad tres libras cum quarta parte vnius libræ, <lb/>& proportio ponderis aurei ad pondus argentei vt librarum .4. cum dimidia ad li-<lb/>bras .3: hoc eſt aurei ad mixtum, vt .13. ad .12. & mixti ad argenteum, vt .18. ad .13. <lb/>& aurei ad argenteum, vt .3. ad .2. ideſt, vt .18. ad .12.</s> </p> <p> <s xml:space="preserve">Nunc inueniantur duo numeri ita inter ſe proportionati, vt .3. ad .2. habentes ta-<lb/>men inter ipſos numerum ita proportionatum ad maximum, vt .12. ſe habet ad <num value="13">.<lb/>13.</num> & ita proportionatum ad minimum, vt ſe habet .18. ad .13. quod hoc modo in-<lb/>ueniemus, multiplicabimus .18. per .12. & proueniet nobis .216. pro numero me-<lb/>dio, poſteà multiplicabimus .18. per .13. & proueniet .234. pro maximo, <choice><ex>demum</ex><am>demũ</am></choice> multi <lb/>plicando .12. per .13. proueniet .156. pro minimo, ita quod .234. correſpondebit <lb/>ponderi corporis aurei: </s> <s xml:space="preserve">216. verò ponderi mixti, et .156. ponderi argentei æqua-<lb/>lium magnitudinum.</s> </p> <p> <s xml:space="preserve">Cum autem proportiones horum trium corporum inuenerimus, ſi ordinem theo <lb/>rematis .122. ſequemur, habebimus quod quærebamus, & inueniemus in præſenti <lb/>exemplo proportionem ponderis auri ad pondus argenti in corpore mixto eſſe, vt <num value="180">.<lb/>180.</num> ad .36. ſed quia ſuppoſitum fuit corpus mixtum eſſe quinque librarum, propte-<lb/>reà dicemus. </s> <s xml:space="preserve">Si .216. hoc eſt toti corpori mixto correſpondent quinque libræ tunc <lb/>parti .180. hoc eſt auro in ipſo corpore mixto, correſpondent libræ .4. cum duabus <lb/>vncijs, ex regula detribus, reſiduum verò quinque librarum, ideſt vnciæ decem, <lb/>correſpondent parti .36. hoc eſt argento in dicto corpore mixto.</s> </p> <p> <s xml:space="preserve">Sed ſi tria corpora dicta fuiſſent inuicem ita proportionata, vt .40. 47. 60. </s> <s xml:space="preserve">tunc <lb/>proportio auri ad argentum in corpore mixto eſſet vt .13. ad .7. quapropter <choice><ex>cum</ex><am>cũ</am></choice> pon <lb/>dus mixti fuiſſet .120. librarum, </s> <s xml:space="preserve">tunc aurum ipſius eſſet librarum .78. argentum ve-<lb/>rò librarum .42. ex eadem regula.</s> </p> <p> <s xml:space="preserve">Pro quarum rerum ſpeculatione nil aliud oportet nunc dicere cum ſatis dictum à no <lb/>bis ſuperius fuerit, vno excepto, hoc eſt rationem reddere, qua motus fui ad inue <lb/>niendos illos .3. numeros ita inter ſe diſpoſitos, vt dictum eſt, quæ quidem ratio fuit, <lb/>vt haberemus .3. numeros ita inter ipſos ordinatè diſpoſitos, vt ſunt pondera trium <lb/>illorum corporum æqualium magnitudinum. </s> <s xml:space="preserve">Proptereà quòd quamuis inter pri-<lb/>mos .3. numeros ponderum corporum aqueorum eædem fuerint proportiones pon <lb/>derum corporum metallicorum, nihilominus medius numerus extra proprium lo-<lb/>cum, & inordinatè inueniebatur, reſpectu extremorum, vnde medius numerus in <lb/>ſuo vero ſitu inter .18. et .12. fuiſſent .16. <choice><ex>cum</ex><am>cũ</am></choice> .8. tertijs decimis, ſed vt <choice><ex>fractorum</ex><am>fractorũ</am></choice> incom <lb/>moditatem euitemus, præcepi, vt multiplicarentur extrema per .13. vnde produ-<lb/>cti fuerunt numeri .234. et .156. in <choice><ex>eadem</ex><am>eadẽ</am></choice> proportione, quæ eſt .18. ad .12. ex .18. ſepti <lb/>mi, iuſſi etiam multiplicari .18. per .12. vt nobis prodiret .216. ad quem numerum, <lb/>numerus .234. ita ſe haberet, ut .13. ad .12. ex .19. ſeptimi, quod autem ita ſit propor <pb facs="0099" n="87"/><fw type="head">THEOREM. ARITH.</fw> tionatus .216. ad .156. vt .18. ad .13. maniteſtum eſt exijſdem, nam tam .18. quam <num value="13">.<lb/>13.</num> multiplicatus fuit per .12.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="129">CXXIX</num>.</head> <p> <s xml:space="preserve">ALIVD proponitur problema hoc modo: </s> <s xml:space="preserve">ſupponitur obſidio alicuius loci, vbi <lb/>alimento ad nutriendos .10000. homines ſufficiunt pro quinque menſibus tan-<lb/>tum, ſed quia eum locum obſidione non liberari putatur niſi .18. menſibus exactis, <lb/>quæritur, quot homines eo tempore illis alimentis nutriri poſſint, hoc eſt .18. <lb/>menſibus.</s> </p> <p> <s xml:space="preserve">Præcipitregula, vt multiplicetur primus numerus, hoc eſt hominum .10000. cum <lb/>ſecundo, hoc eſt menſium quinque, productum verò diuidatur per .18. hoc eſt men-<lb/>ſium, </s> <s xml:space="preserve">tunc proueniet .2777. cum .7. nonis.</s> </p> <p> <s xml:space="preserve">Cuius operationis ratio eſt hæc, ſint exempli gratia duo hic ſubſcripta producta <lb/>ſuperficialia <seg type="var">.a.n.</seg> et <seg type="var">.o.u.</seg> inuicem æqualia, ſed tal@ figura delineata, vt proportio <seg type="var">.u.<lb/>x.</seg> ad <seg type="var">.x.o.</seg> ſit, vt .10000. ad quinque, & proportio <seg type="var">a.x.</seg> ad <seg type="var">.x.o.</seg> ſit vt .18. ad quinque, <lb/>ct <seg type="var">.x.n.</seg> ſit nobis ignota, quæ quidem eſt illa, quæ indagatur, ita <choice><ex>quod</ex><am>ꝙ</am></choice> vnumquodque <lb/>iſtorum productorum ſignificabit alimentum, et <seg type="var">.u.x.</seg> ſignificabit numerum homi-<lb/>num .10000. qui quidem homines comederent totum alimentum <seg type="var">.u.o.</seg> ſpacio tem-<lb/>poris <seg type="var">.x.o.</seg> quinque menſium, proptereà quòd <seg type="var">u.o.</seg> ſupponitur productum eſſe ab <seg type="var">.<lb/>u.x.</seg> in <seg type="var">.x.o</seg>. </s> <s xml:space="preserve">Deinde <choice><ex>ſupponendo</ex><am>ſupponẽdo</am></choice> <seg type="var">.a.x.</seg> tem <lb/> <ptr xml:id="fig-0099-01a" corresp="fig-0099-01" type="figureAnchor"/> pus eſſe .18. menſium, ergo <seg type="var">.x.n.</seg> ſignifi-<lb/>cabit numerum hominum, qui eo tem-<lb/>poris ſpacio ali poſſunt, hoc eſt <seg type="var">.x.a.</seg> ali-<lb/>mento <seg type="var">.n.a.</seg> eo quòd <seg type="var">.a.n.</seg> producitur ex <seg type="var">.<lb/>n.x.</seg> in <seg type="var">.a.x.</seg> vnde ex .15. ſexti, ſeu ex, 20. <lb/>ſeptimi proportio <seg type="var">.x.u.</seg> ad <seg type="var">.x.n.</seg> <choice><ex>eadem</ex><am>eadẽ</am></choice> erit, <lb/>quę <seg type="var">.a.x.</seg> ad <seg type="var">.x.o.</seg> quapropter rectè factum <lb/>erit accipere <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.u.o.</seg> quodidem <lb/>eſt in quantitate, quod productum .2. n. & ipſum diuidere per <seg type="var">.a.x.</seg> vnde nobis <lb/>proueniat <seg type="var">.n.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0099-01" corresp="fig-0099-01a"> <graphic url="0099-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="130">CXXX</num>.</head> <p> <s xml:space="preserve">QVotieſcunque nobis propoſitum fuerit inuenire tertium terminum, trium ter <lb/>minorum continuè proportionalium armonicæ proportionalitatis, quo-<lb/>tum duo nobis cogniti ſint, ita agemus.</s> </p> <p> <s xml:space="preserve">Sint, exempli gratia, tres termini <seg type="var">.q.p</seg>: <seg type="var">a.g.</seg> et <seg type="var">.e.c.</seg> continuæ proportionalium at <lb/>monicæ proportionalitatis, quorum <seg type="var">.q.p.</seg> maior et <seg type="var">.a.g.</seg> medius ſint nobis cogniti, <lb/>cum ergo voluerimus tertium <seg type="var">.e.<lb/>c.</seg> cognitum nobis eſſe: </s> <s xml:space="preserve">a.g. detra-<lb/> <ptr xml:id="fig-0099-02a" corresp="fig-0099-02" type="figureAnchor"/> hatur ex <seg type="var">.q.p.</seg> differentia verò <seg type="var">.d.<lb/>p.</seg> addatur <seg type="var">.q.p.</seg> quorum ſumma <lb/>erit <seg type="var">.q.o.</seg> cognita, qua mediante <lb/>diuidatur productum, quod ex <seg type="var">.a.<lb/>g.</seg> in <seg type="var">.d.p.</seg> exurgit, & proueniet no <lb/>bis <seg type="var">.n.g.</seg> hoc e@t minor differentia, eo quòd productum <seg type="var">.q.o.</seg> in <seg type="var">.n.g.</seg> æquale eſt pro- <pb facs="0100" n="88"/><fw type="head">IO. BAPT. BENED.</fw> ducto .2. g. in <seg type="var">.d.p.</seg> ex .20. ſeptimi, proptereà quòd proportio <seg type="var">.q.o.</seg> ad <seg type="var">.o.p.</seg> hoc eſt ad <seg type="var">.<lb/>d.p.</seg> eſt vt <seg type="var">.a.g.</seg> ad <seg type="var">.g.n.</seg> coniunctim cum diſiunctim it a ſit <seg type="var">.q.p.</seg> ad <seg type="var">.p.o.</seg> vt <seg type="var">.a.n.</seg> ad <seg type="var">.n.g.</seg> <lb/><choice><ex>permutando</ex><am>permutãdo</am></choice> eo quòd <seg type="var">.q.p.</seg> ad <seg type="var">.a.n.</seg> (ideſt ad <seg type="var">.e.c.</seg>) ita ſe <choice><ex>hent</ex><am>hẽt</am></choice> ut <seg type="var">.p.o.</seg> (hoc eſt <seg type="var">.d.p.</seg>) ad <seg type="var">.n.g.</seg> <lb/>ex <choice><ex>conditionibus</ex><am>cõditionibus</am></choice> armonicæ proportio nalitatis. </s> <s xml:space="preserve">Deinde ſi detraxerimus <seg type="var">.n.g.</seg> ex <seg type="var">.a.g.</seg> <lb/>remanebit <seg type="var">.e.c.</seg> minor terminus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0099-02" corresp="fig-0099-02a"> <graphic url="0099-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi <seg type="var">.e.c.</seg> tertius terminus nobis propoſitus eſſet ſimul cum <seg type="var">.a.g.</seg> medio, & volue <lb/>rimus maiorem inuenire <seg type="var">.q.p.</seg> ſcilicet, oportebit <seg type="var">.e.c.</seg> ex <seg type="var">.a.g.</seg> detrahere, differentiam <lb/>verò <seg type="var">.n.g.</seg> ſimiliter demeremus <lb/>ex <seg type="var">.e.c.</seg> unde remaneret nobis <seg type="var">.e.t.</seg> <lb/> <ptr xml:id="fig-0100-01a" corresp="fig-0100-01" type="figureAnchor"/> cognitum, quo reſiduo <seg type="var">.c.t.</seg> me-<lb/>diante diuidemus productum, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>furgit ex <seg type="var">.a.g.</seg> in <seg type="var">.t.c.</seg> & prouentus <seg type="var">.<lb/>d.p.</seg> erit differentia maior, eo <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/><choice><ex>productum</ex><am>productũ</am></choice> quod ſit ex <seg type="var">.e.t.</seg> in <seg type="var">.d.p.</seg> <lb/>æquale eſt producto quòd fit ex <seg type="var">.a.g.</seg> in <seg type="var">.t.c.</seg> per 20. ſeptimi Eucli. eo quòd <seg type="var">.a.g.</seg> (id-<lb/>eſt <seg type="var">.q.d.</seg>) ad <seg type="var">.d.p.</seg> eſt ut <seg type="var">.e.t.</seg> ad <seg type="var">.t.c.</seg> diſiunctim, cum coniunctim ita ſit <seg type="var">.q.p.</seg> ad <seg type="var">.d.p.</seg> vt <seg type="var">.e.<lb/>c.</seg> ad <seg type="var">.t.c.</seg> permutando, quia <seg type="var">.q.p.</seg> ad <seg type="var">.e.c.</seg> eſt vt <seg type="var">.d.p.</seg> ad <seg type="var">.t.c.</seg> hoc eſt ad <seg type="var">.n.g.</seg> ex legibus <lb/>dictis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0100-01" corresp="fig-0100-01a"> <graphic url="0100-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="131">CXXXI</num>.</head> <p> <s xml:space="preserve">ALIA etiam methodo hoc perfici poſſe comperi. </s> <s xml:space="preserve">Propoſiti enim cum nobis fue <lb/>rint duo termini <seg type="var">.c.e.</seg> minimus et <seg type="var">.g.a.</seg> medius, maximus verò quærendus ſit, de <lb/>trahatur differentia <seg type="var">.g.n.</seg> ex <seg type="var">.e.c.</seg> & per reſiduum <seg type="var">.e.t.</seg> diuidatur productum <choice><ex>quod</ex><am>ꝙ</am></choice> fit ex <seg type="var">.a.<lb/>g.</seg> in <seg type="var">.e.c.</seg> prouentus quæ erit <seg type="var">.q.p.</seg> terminus quæſitus.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione, ponamus in eſſe terminum <seg type="var">.q.p.</seg> </s> <s xml:space="preserve">tunc ex forma huius proportio <lb/>nalitatis nulli dubium erit quin <seg type="var">.q.p.</seg> ad <seg type="var">.e.c.</seg> fit vt <seg type="var">.d.p.</seg> ad <seg type="var">.n.g.</seg> hoc eft ad <seg type="var">.t.c.</seg> vnde ex <lb/>19. quinti vel .12. ſeptimi ita eſſet <seg type="var">.q.d.</seg> ad <seg type="var">.e.t.</seg> vt <seg type="var">.q.p.</seg> ad <seg type="var">.e.c.</seg> </s> <s xml:space="preserve">quare ex .20. @cptimi pro <lb/>ductum <choice><ex>quod</ex><am>ꝙ</am></choice> naſcitur ex <seg type="var">.p.d.</seg> (hoc eſt <seg type="var">.a.g.</seg>) in <seg type="var">.e.c.</seg> æquale eric producto <seg type="var">.e.t.</seg> in <seg type="var">.q.p.</seg> qua-<lb/>propter ſi diuiſerimus id per <seg type="var">.e.t.</seg> proueniet nobis <seg type="var">.q.p</seg>.</s> </p> <p> <s xml:space="preserve">Sed <choice><ex>cum</ex><am>cũ</am></choice> nobis propoſiti fuerint duo termini <seg type="var">.q.p.</seg> maximus, et <seg type="var">.a.g.</seg> medius, ſi <choice><ex>mini- mum</ex><am>mini-mũ</am></choice> <seg type="var">.e.c.</seg> <choice><ex>voluerimus</ex><am>voluerimꝰ</am></choice> inuenire. </s> <s xml:space="preserve">Termino <seg type="var">.q.p.</seg> maximo, <choice><ex>iungatur</ex><am>iũgat̃</am></choice>. <seg type="var">p.o.</seg> ęqualis, <seg type="var">p.d.</seg> <choice><ex>differentię</ex><am>differẽtię</am></choice> <lb/>propoſitæ, diuidatur poſtea productum <choice><ex>quod</ex><am>ꝙ</am></choice> ex <seg type="var">.q.p.</seg> in <seg type="var">.a.g.</seg> generatur per <seg type="var">.q.o.</seg> prouen <lb/>tus autem ſit <seg type="var">.e.c.</seg> qui quidem erit terminus quæſitus.</s> </p> <p> <s xml:space="preserve">Cuius operationis ſpeculutio hæc erit, ſupponatur terminum <seg type="var">.e.c.</seg> inuentum eſſe <lb/>vnde <seg type="var">.n.g.</seg> differentia ſit inter <seg type="var">.e.c.</seg> <lb/>et <seg type="var">.a.g.</seg> ex forma igitur armonicæ <lb/> <ptr xml:id="fig-0100-02a" corresp="fig-0100-02" type="figureAnchor"/> proportionalitis ita erit <seg type="var">.q.p.</seg> ad <seg type="var">.a.<lb/>n.</seg> vt <seg type="var">.p.o.</seg> ad <seg type="var">.n.g.</seg> vnde ex .13. quin-<lb/>ti. </s> <s xml:space="preserve">Ita erit <seg type="var">.q.o.</seg> ad <seg type="var">.a.g.</seg> vt <seg type="var">.q.p.</seg> ad <seg type="var">.a.<lb/>n.</seg> ergo <choice><ex>productum</ex><am>productũ</am></choice> quòd fit ex <seg type="var">.a.g.</seg> <lb/>in <seg type="var">.q.p.</seg> (ex .20. ſeptimi) æquale erit <lb/>producto <seg type="var">.q.o.</seg> in <seg type="var">.a.n</seg>. </s> <s xml:space="preserve">Quare ſi diuiſum fuerit tale productum per <seg type="var">.q.o.</seg> proueniet no-<lb/>bis <seg type="var">.e.c.</seg> quòd querebamus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0100-02" corresp="fig-0100-02a"> <graphic url="0100-02"/> </figure> </div> </body> </floatingText> <pb facs="0101" n="89"/> <fw type="head">THEOR. ARITH.</fw> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="132">CXXXII</num>.</head> <p> <s xml:space="preserve">SED quia aliquis poſſet in dubium reuocare, an poſſibile ſit inuenire tertium <lb/>terminum rationalem, ſeu communicantem duobus datis terminis inter ſe com <lb/>municantibus in tali proportionalitate, hoc eſt harmonica. </s> <s xml:space="preserve">Vthoc oſtendatur.</s> </p> <p> <s xml:space="preserve">Sint duo termini dati <seg type="var">.a.o.</seg> et <seg type="var">.a.e.</seg> inter ſe communicantes, tertius verò inuentus <lb/>ſit <seg type="var">.a.c.</seg> qui maximus, primò, ſit in ea proportionalitate, quem dico communicantem <lb/>eſſe cum primis datis.</s> </p> <p> <s xml:space="preserve">Nam ex conditionibus huiuſmodi proportionalitatis, habebimus primum ean-<lb/>dem proportionem eſſe <seg type="var">.a.c.</seg> ad <seg type="var">.a.o.</seg> quæ eſt <seg type="var">.e.c.</seg> ad <seg type="var">.e.o.</seg> vnde permutando ita erit <seg type="var">.a.<lb/>c.</seg> ad <seg type="var">.e.c.</seg> vt <seg type="var">.a.o.</seg> ad <seg type="var">.o.e.</seg> & quia ex .9. decimi Euclid <seg type="var">.a.o.</seg> communicat cum <seg type="var">.o.e.</seg> </s> <s xml:space="preserve">quare <lb/>ex .10. eiuſdem <seg type="var">.a.c.</seg> communicabit cum <seg type="var">.e.c.</seg> & per .9. cum <seg type="var">.a.e.</seg> et per .8. cum <seg type="var">.a.o.</seg> <lb/>quod<unclear reason="illegible"/>eſt propoſitum.</s> </p> <p> <s xml:space="preserve">Sed ſi datus fuerit maximus <seg type="var">.a.c.</seg> cum medio <seg type="var">.a.e.</seg> interſe communicantes mini-<lb/>mum verò <seg type="var">.a.o.</seg> probabo <choice><ex>communicantem</ex><am>cõmunicantem</am></choice> cum illis eſſe. </s> <s xml:space="preserve">Cogitemus ergo <seg type="var">.c.f.</seg> æqua-<lb/>jem eſſe differentiæ <seg type="var">.c.e.</seg> cognitæ, vnde habebimus proportionem, <seg type="var">a.c.</seg> ad <seg type="var">.c.f.</seg> vt <seg type="var">.a.o.</seg> <lb/>ad <seg type="var">.o.e.</seg> & componendo <seg type="var">.a.f.</seg> ad <seg type="var">.f.c.</seg> vt <seg type="var">.a.e.</seg> ad <seg type="var">.e.o.</seg> & quia (ex ſuppoſito). <seg type="var">a.c.</seg> commu-<lb/>nicat cum <seg type="var">.e.c.</seg> hoc eſt cum <seg type="var">.c.f.</seg> </s> <s xml:space="preserve">quare <lb/>ex eadem .9. dicti decimi <seg type="var">.a.f.</seg> et <seg type="var">.f.c.</seg> <choice><ex>erunt</ex><am>erũt</am></choice> <lb/> <ptr xml:id="fig-0101-01a" corresp="fig-0101-01" type="figureAnchor"/> inter ſe communicantes, & per .10. <seg type="var">a.e.</seg> <lb/>communicabit cum <seg type="var">.o.e.</seg> & per .9. <seg type="var">a.e.</seg> cò <lb/>municabit cum <seg type="var">.a.o.</seg> vnde per .8. <seg type="var">a.o.</seg> communicabit cum <seg type="var">.a.c.</seg> ſimiliter.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0101-01" corresp="fig-0101-01a"> <graphic url="0101-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="133">CXXXIII</num>.</head> <p> <s xml:space="preserve">SED ſi nobis duo extremi termini propoſiti fuerint, & medium inuenire deſide <lb/>remus in dicta proportionalitate, ita faciendum erit.</s> </p> <p> <s xml:space="preserve">Sint, exempli gratia, duo termini dati <seg type="var">.q.b.</seg> et <seg type="var">.b.r.</seg> minor <seg type="var">.b.r.</seg> ex maiori <seg type="var">.b.q.</seg> de-<lb/>trahatur, reſiduum verò <seg type="var">.q.x.</seg> multiplicetur per <seg type="var">.b.r.</seg> productum poſteà diuidatur per <lb/><seg type="var">q.r.</seg> vnde proueniet nobis <seg type="var">.x.l.</seg> pro differentia minori, quæ addita cum <seg type="var">.b.x.</seg> minimo <lb/>termino, dabit nobis <seg type="var">.b.l.</seg> mcdium terminum harmonicum.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione cogitemus dictum medium terminum <seg type="var">.b.l.</seg> iam inuentum eſſe, <lb/>vnde ita erit proportio <seg type="var">.q.l.</seg> ad <seg type="var">.l.x.</seg> vt <seg type="var">.q.b.</seg> ad <seg type="var">.b.r.</seg> ex forma huius proportionalitatis, <lb/></s> <s xml:space="preserve">quare coniunctim ita erit <seg type="var">.q.r.</seg> ad <seg type="var">.r.b.</seg> vt <lb/><seg type="var">q.x.</seg> ad <seg type="var">.x.l.</seg> & proptereà ex .20. ſeptimi <lb/> <ptr xml:id="fig-0101-02a" corresp="fig-0101-02" type="figureAnchor"/> productum, quod fit ex <seg type="var">.q.r.</seg> in <seg type="var">.x.l.</seg> æqua-<lb/>le erit producto <seg type="var">.q.x.</seg> in <seg type="var">.b.r</seg>. </s> <s xml:space="preserve">Rectè igitur <lb/>fit cum diuiditur hoc productum per <seg type="var">.q.r.</seg> vt proueniat nobis <seg type="var">.x.l.</seg> differentia minor.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0101-02" corresp="fig-0101-02a"> <graphic url="0101-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="134">CXXXIIII</num>.</head> <p> <s xml:space="preserve">POſſumus etiam harmonicè diuidere vnam datam proportionem abſque aliqua <lb/>diuiſione productorum, ne nobis fractiones proueniant, hoc modo videlicet. <lb/></s> <s xml:space="preserve">Nobis propoſitum ſit diuidere harmonicè ſeſquialteram <choice><ex>proportionem</ex><am>proportionẽ</am></choice> inuenian-<lb/>tur primo minimi termini huius proportionis ut putà .3. et .2. quarum ſumma, hoc <lb/>eſt quinque, multiplicetur per minorem ideſt .2. vnde proueniet nobis .10. qui qui-<lb/>dem erit minor terminus trium quæſitorum, quorum maximus erit productum ſum <pb facs="0102" n="90"/><fw type="head">IO. BAPT. BENED.</fw> mæ iam dictæ in maiorem eorum, hoc eſt quod fit ex quinque in .3. quod erit .15. </s> <s xml:space="preserve">Vt <lb/>autem medium terminum harmonicum inter iſtos habeamus, accipiatur <choice><ex>duplum</ex><am>duplũ</am></choice> pro-<lb/>ducti, quod fit ex primis minimis terminis, quod erit .12.</s> </p> <p> <s xml:space="preserve">Cuius rei ſpeculatio eſt iſta: </s> <s xml:space="preserve">ſignificentur duo termini datæ proportionis ab <seg type="var">.q.b.</seg> <lb/>et <seg type="var">.b.r.</seg> quorum ſumma erit <seg type="var">.q.r.</seg> cuius quadratum ſit <seg type="var">.q.o.</seg> ſit etiam imaginata <seg type="var">.b.e.</seg> <lb/>parallela ad <seg type="var">.o.r</seg>. </s> <s xml:space="preserve"><choice><ex>Sitque</ex><am>Sitq́;</am></choice> <seg type="var">.b.x.</seg> æqualis <seg type="var">.b.r.</seg> et <seg type="var">.q.u.</seg> ſimiliter, & ducatur <seg type="var">.x.y.</seg> parallela ad <lb/><seg type="var">r.o.</seg> et <seg type="var">.u.l.</seg> ad <seg type="var">.q.x</seg>. </s> <s xml:space="preserve">Tunc habebimus <seg type="var">.b.o.</seg> æquale ei producto, quod fit ex <seg type="var">.q.r.</seg> in <seg type="var">.b.r.</seg> <lb/>et <seg type="var">.b.y.</seg> eidem etiam æquale, et <seg type="var">.q.e.</seg> pro producto, quod fit ex <seg type="var">.q.r.</seg> in <seg type="var">.q.b.</seg> et <seg type="var">.q.l.</seg> pro <lb/>eo, quod fit ex <seg type="var">.q.x.</seg> in <seg type="var">.b.r</seg>. </s> <s xml:space="preserve">Vnde <seg type="var">.q.l.</seg> cum <seg type="var">.b.y.</seg> æquale fiet duplo ei, quod fit ex <seg type="var">.q.b.</seg> <lb/>in <seg type="var">.b.r</seg>. </s> <s xml:space="preserve">Dico nunc <seg type="var">.b.o.</seg> eſſe minimum terminum eorum, quos quærimus, et <seg type="var">.y.b.</seg> cum <seg type="var">.<lb/>x.u.</seg> medium <seg type="var">.q.e.</seg> verò maximum huiuſmodi proportionalitatis.</s> </p> <p> <s xml:space="preserve">Primum ergo certi ſcimus ex prima ſexti vel .18. ſeptimi eandem exiſtere pro-<lb/>portionem <seg type="var">.q.e.</seg> ad <seg type="var">.b.o.</seg> ſeu ad <seg type="var">.b.y.</seg> quæ <seg type="var">.q.b.</seg> ad <seg type="var">.b.r</seg>: ſed <seg type="var">.u.y.</seg> ad <seg type="var">.u.x.</seg> eſt vt <seg type="var">.y.l.</seg> ad <seg type="var">.l.x.</seg> <lb/>hoc eſt vt <seg type="var">.q.b.</seg> ad <seg type="var">.b.r.</seg> ideſt vt <seg type="var">.q.e.</seg> ad <seg type="var">.b.o.</seg> & ſumma <seg type="var">.u.y.</seg> cum <seg type="var">.u.x.</seg> ideſt <seg type="var">.q.y.</seg> minor eſt <lb/>quam <seg type="var">.q.e.</seg> maximus terminus per <seg type="var">.b.y.</seg> minimum ter-<lb/>minum. </s> <s xml:space="preserve">& <choice><ex>coniunctim</ex><am>cõiunctim</am></choice> <seg type="var">.q.y.</seg> ad <seg type="var">.q.l.</seg> vt <seg type="var">.y.x.</seg> ad <seg type="var">.x.l.</seg> hoc eſt <lb/> <ptr xml:id="fig-0102-01a" corresp="fig-0102-01" type="figureAnchor"/> vt <seg type="var">.q.r.</seg> ad <seg type="var">.r.b</seg>. </s> <s xml:space="preserve">Vnde ex ſpeculatione <choice><ex>præcedentis</ex><am>præcedẽtis</am></choice> theo <lb/>rematis, ſequitur <seg type="var">.u.y.</seg> eſſe differentiam inter <choice><ex>maximum</ex><am>maximũ</am></choice> <lb/>& medium terminum, et <seg type="var">.u.x.</seg> eſſe differentiam inter <lb/>medium & minimum dictæ proportionalitatis. </s> <s xml:space="preserve">Nam <lb/>eadem proportio eſt <seg type="var">.q.e.</seg> maximi termini ad <seg type="var">.b.o.</seg> mi-<lb/>nimi. quæ <seg type="var">.u.y.</seg> (differentia inter <seg type="var">.q.e.</seg> & gnomonem <seg type="var">.<lb/>u.b.y.</seg>) ad <seg type="var">.u.x.</seg> (differentia inter dictum <seg type="var">.u.b.y.</seg> et <seg type="var">.b.y.</seg> <lb/>minimum terminum, quia ſunt ambæ ut <seg type="var">.q.b.</seg> ad <seg type="var">.b.r.</seg> <lb/>vt diximus. </s> <s xml:space="preserve">Quare <seg type="var">.b.y.</seg> <choice><ex>coniunctum</ex><am>coniunctũ</am></choice> cum <seg type="var">.x.u.</seg> medius <lb/>terminus erit, qui quidem (vt dictum eſt) duplus eſt ei <lb/>quod fit ex <seg type="var">.q.b.</seg> in <seg type="var">.b.r</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0102-01" corresp="fig-0102-01a"> <graphic url="0102-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="135">CXXXV</num>.</head> <p> <s xml:space="preserve">ALIVM etiam modum ab antiquis traditum ad hoc problema perficiendum <lb/>inueni, qui talis eſt. </s> <s xml:space="preserve">Inueniatur primo inter datos terminos extremos, me-<lb/>dius terminus in arithmetica proportione, per <choice><ex>quem</ex><am>quẽ</am></choice> <lb/> <ptr xml:id="fig-0102-02a" corresp="fig-0102-02" type="figureAnchor"/> multiplicetur vnuſquiſque dictorum extremorum, <lb/>deinde multiplicentur ipſi extremi interſe, vnde <lb/>habebimus tria producta eadem proportione inui <lb/>cem exiſtentia, vt quærebatur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0102-02" corresp="fig-0102-02a"> <graphic url="0102-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Exempli gratia, ponamus duos propoſitos ter-<lb/>minos eſſe .3. et .2. quorum medius arithmeticè <lb/>eſſet .2. cum dimidia vnitate, per quem cum vnum <lb/>quemque priorum multiplicauerimus, <choice><ex>emergent</ex><am>emergẽt</am></choice> no-<lb/>bis duo producta, quorum primum ideſt maius eſſet <lb/>7. cum dimidia vnitate, reliquum verò eſſet <lb/>quinque, productum poſteà quod ex ipſis extremis <lb/>prouenit, erit .6. quod quidem eſt harmonicè collo <lb/>catum inter .7. cum dimidia vnitate, & quinque.</s> </p> <p> <s xml:space="preserve">Cuius rei ſpeculatio omnis à præcedenti theore-<lb/>mate dependet. </s> <s xml:space="preserve">Sint exempli gratia, duo termini <pb facs="0103" n="91"/><fw type="head">THEOREM. ARIT.</fw> propoſiti <seg type="var">.a.e.</seg> maior, et <seg type="var">.e.o.</seg> minor, <choice><ex>Sitque</ex><am>Sitq́;</am></choice> <seg type="var">.o.k.</seg> medius arithmeticus inter dictos, vn-<lb/>de clarè patebit <seg type="var">.o.k.</seg> eſſe dimidium ſummæ dictorum terminorum ex .75. theorema <lb/>te huius libri. </s> <s xml:space="preserve">Sit ergo productum <seg type="var">a.t.</seg> id quod fit ex <seg type="var">.a.e.</seg> in <seg type="var">.o.k.</seg> et <seg type="var">.o.t.</seg> ſit <choice><ex>productum</ex><am>productũ</am></choice> <lb/>quod fit ex <seg type="var">.e.o.</seg> in <seg type="var">.o.k.</seg> et <seg type="var">.n.m.</seg> ſit productum quod ſit ex <seg type="var">.a.e.</seg> in <seg type="var">.e.o.</seg> quorum vnum-<lb/>quodque erit dimid ium vniuſcuiuſque producti præcedentis theorematis, <lb/>ex .18. et .19. ſeptimi Eucli. vnumquodque ſui relatiui. </s> <s xml:space="preserve">Quare argumentando per <lb/>mutando à concluſionibus præcedentis theorematis ad has præſentis, habebimus <lb/>productum.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="136">CXXXVI</num>.</head> <p> <s xml:space="preserve">MEDIVM autem contra <choice><ex>harmonicum</ex><am>harmonicũ</am></choice> inuenire cum quis voluesit inter duos <lb/>propoſitos terminos, ita faciendum erit, hoc eſt per ſummam datorum ex <lb/>tremorum diuidatur productum quod fit ex minimo termino in <choice><ex>differentiam</ex><am>differẽtiam</am></choice> dato-<lb/>rum, prouentus poſtea erit differentia inter maximum & med<unclear reason="illegible"/>um quæſitum.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia, ſi nobis propoſiti fuerint hi duo termini .3. et .2. ſumma eo-<lb/>rum erit quinque, per quam cum diuiſerimus productum, quod naſcitur ex mini-<lb/>mo .2. in differentiam eorum, quæ eſt vnum, quod quidem erit .2. </s> <s xml:space="preserve">tunc duæ quintæ <lb/>partes prouenient, quæ ſi demptæ fuerint ex maximo termino, reliquum erit .2. <choice><ex>cum</ex><am>cũ</am></choice> <lb/>3. quintis, hoc eſt medius terminus contta harmonicus.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione cogitemus <seg type="var">.u.d.</seg> et <seg type="var">.x.c.</seg> eſſe duosterminosnobis propoſitos, in-<lb/>ter quos deſideremus inuenire <seg type="var">.o.s.</seg> medium ita illis <choice><ex>relatum</ex><am>relatũ</am></choice>, vt proportio exceſſus ip-<lb/>ſius ſupra <seg type="var">.x.c.</seg> (qui ſit <seg type="var">.e.n.</seg>) ad exceſ-<lb/>ſum <seg type="var">.u.d.</seg> ſupra <seg type="var">.o.s.</seg> (qui ſit <seg type="var">.n.d.</seg>) ea-<lb/> <ptr xml:id="fig-0103-01a" corresp="fig-0103-01" type="figureAnchor"/> dem ſit quæ <seg type="var">.u.d.</seg> ad <seg type="var">.x.c</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0103-01" corresp="fig-0103-01a"> <graphic url="0103-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Cogitemus igitur <seg type="var">.x.c.</seg> coniunctum <lb/>eſſe cum <seg type="var">.u.d.</seg> & hæcſumma vocetur <seg type="var">.<lb/>b.d.</seg> vnde habebimus proportionem <seg type="var">.<lb/>u.d.</seg> ad <seg type="var">.u.b.</seg> vt <seg type="var">.e.n.</seg> ad <seg type="var">.n.d</seg>. </s> <s xml:space="preserve">Quare <choice><ex>com- ponendo</ex><am>cõ-ponendo</am></choice> ita erit <seg type="var">.d.b.</seg> ad <seg type="var">.u.b.</seg> ut <seg type="var">.e.d.</seg> 3d.n.d. ſed quia <seg type="var">.d.b</seg>: <seg type="var">u.b.</seg> et <seg type="var">.e.d.</seg> quantitates no-<lb/>bis cognitę ſunt, ideò <seg type="var">.d.n.</seg> ex .20. ſeptimi cognita nobis erit.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="137">CXXXVII</num>.</head> <p> <s xml:space="preserve">SVpponunt antiqui aliquot mercatores dantes pecunias lucro in diuerſis vnius <lb/>anni temporibus, </s> <s xml:space="preserve">tunc in fine anni ſumma torius lucri datur cognita, ſed quæ-<lb/>ritur quantuni<unclear reason="illegible"/> vnicuique illorum exipſa ſumma debeatur.</s> </p> <p> <s xml:space="preserve">Exempli gratia, primus in principio anni poſuit .100. aurcos, ſecundus verò .100 <lb/>diebus poſt primum poſuit .50. aureos tertius autem .200. diebus poſt primum po-<lb/>ſuit .25. aureos ſumma lucri poſtea in fine anni fuit aureorum .60.</s> </p> <p> <s xml:space="preserve">Nunc vt ſciamus quantum huius ſummæ vniduique illorum proueniat, præcipit <lb/>regula, vt faciamus tria producta, quorum primum ſit ex numero dierum totius an-<lb/>ni in numerum aureorum primi, vnde tale productum in præſenti caſu erit .36500. <lb/>ſecundum verò ſit ex numero dierum à primo die in quo ipſe ſecundus poſuit uſque <lb/>ad finem anni, in numerum ipſorum nummorum, quod erit .13250. tertium autem <lb/>productum ex diebus tertij in numerum ſuorum aureorum, quod <choice><ex>quidem</ex><am>quidẽ</am></choice> erit .4125. <lb/>quæ producta ſimul collecta faciunt .53875. deinde multiplicetur vnumquodque <pb facs="0104" n="92"/><fw type="head">IO. BAPT. BENED.</fw> ipſorum prochictorum per ſummam lucri hoc eſt per .60. vnde multiplicatio primi <lb/>producti erit .2190000. multiplicatio verò ſecundi producti erit .795000. tertij po <lb/>ſtca erit .247500. quarum multiplicationum vnaquæque diuidatur per ſummam <lb/>53875. productori<unclear reason="illegible"/>t, & proueniet ex prima diuiſione .40. <choice><ex>cum</ex><am>cũ</am></choice> fractis .35000. vnius in-<lb/>tegri diuiſi in partes .53875. quod erit lucrum primi, prouentus autem ſecundæ di-<lb/>uiſionis erit .14. cum fractis .41050. vnius integri diuiſi in partes .53875. lucrum <choice><ex>ſecu di.</ex><am>ſecũdi</am></choice> </s> <s xml:space="preserve">prouentus verò quartæ diuiſionis erit .4. cum fractis .32000. vnius integri, vt ſu <lb/>pra diuiſi in partes .53875. hoc eſt lucrum tertij.</s> </p> <p> <s xml:space="preserve">Cuius rei ſpeculatio ex ſe in ſub ſcripta figura patet, vbi <seg type="var">.a.q.</seg> ſignificat numerum <lb/>dierum totius anni pro primo mercatore <seg type="var">.q.n.</seg> autem ſignificat numerum dierum ſe <lb/>cundi mercatoris <seg type="var">.e.q.</seg> poſteà ſignificat numerum dierum tertij ſit etiam <seg type="var">.s.a.</seg> pro nu-<lb/>mero denariorum primi, et <seg type="var">.o.n.</seg> pro numero ſecundi, et <seg type="var">.e.t.</seg> pro numero <lb/>tertij, productum autem <seg type="var">.q.s.</seg> ſignificet valorem primi lucri, et <seg type="var">.q.o.</seg> ſecundi, <lb/>et <seg type="var">.q.t.</seg> tertij <seg type="var">.x.y.</seg> autem ſignificet ſummam lucri omnium, et <seg type="var">.x.i.</seg> ſignificet <lb/>partem primi, et <seg type="var">.i.p.</seg> ſecundi, et <seg type="var">.p.y.</seg> tertij. </s> <s xml:space="preserve">vnde clarè patebit ex communi <lb/>ſcientia quòd eadem proportio erit <seg type="var">.x.y.</seg> ad <seg type="var">.x.i.</seg> quæ aggregati omnium producto-<lb/>rum <seg type="var">.q.s</seg>: <seg type="var">q.o.</seg> et <seg type="var">.q.t.</seg> ad <seg type="var">.q.s.</seg> & ita <seg type="var">.x.y.</seg> ad <seg type="var">.i.p.</seg> vt aggregati dictiad <seg type="var">.q.o.</seg> et <seg type="var">.x.y.</seg> ad <seg type="var">.p.y.</seg> <lb/>vt dicti aggregati ad <seg type="var">.q.t</seg>. </s> <s xml:space="preserve">Rectè igitur ex regula de tribus multiplicatio <seg type="var">.q.s.</seg> in <seg type="var">.x.y.</seg> <lb/>diuiditur per aggregatum omnium <lb/> <ptr xml:id="fig-0104-01a" corresp="fig-0104-01" type="figureAnchor"/> productorum, ita vt ſi aliquis dice-<lb/>ret, ſi ex dicto aggregato, prouenit <lb/><seg type="var">x.y.</seg> quid proueniet vnicuique <choice><ex>illo- rum</ex><am>illo-rũ</am></choice> <choice><ex>productorum</ex><am>productorũ</am></choice>. </s> <s xml:space="preserve"><choice><ex>Nam</ex><am>Nã</am></choice> ſi numerus dena-<lb/>riorum <choice><ex>ſecundi</ex><am>ſecũdi</am></choice> æqualis eſſet numero <lb/><seg type="var">a.s.</seg> primi vt putà. <seg type="var">n.b</seg>. </s> <s xml:space="preserve">tunc eius <choice><ex>lucrum</ex><am>lucrũ</am></choice> <lb/>ſignificaretur à rectangulo <seg type="var">.q.b.</seg> & ita <lb/>de tertio dico <choice><ex>quod</ex><am>ꝙ</am></choice> ſignificaretur à <choice><ex>re- ctangulo</ex><am>re-ctãgulo</am></choice> <seg type="var">.q.c.</seg> vel ſi ſi<unclear reason="illegible"/>antibus <choice><ex>ijſdem</ex><am>ijſdẽ</am></choice> <choice><ex>denariorum</ex><am>denariorũ</am></choice> quantitatibus <seg type="var">.n.o.</seg> et <seg type="var">.e.t.</seg> omnes ſuas pe-<lb/>cunias eodem tempore poſuiſſent, </s> <s xml:space="preserve">tunc rectangula ſignificantia eorum lucra eſlent <lb/><seg type="var">q.s.q.d.</seg> et <seg type="var">.q.f.</seg> ſed cum nec eodem tempore, nec eandem quantitatem poſueruntr<unclear reason="illegible"/>e <lb/>ctè eorum lucra ſignificantur à rectangulis <seg type="var">.q.s.q.o.</seg> et <seg type="var">.q.t.</seg> <choice><ex>quod</ex><am>ꝙ</am></choice> ex prima .6. vel .18. aut <num value="19">.<lb/>19.</num> ſeptimi ratiocinando clarè patebit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0104-01" corresp="fig-0104-01a"> <graphic url="0104-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="138">CXXXVIII</num>.</head> <p> <s xml:space="preserve">NIcolaus Tartalea in primo libro vltimæ partis numerorum ad .35. quæſitum <lb/>docet inuenire quantitatem laterum vnius propoſiti trianguli, cuius la-<lb/>r<unclear reason="illegible"/>erum proportio nobis data ſit ſimul cum area ſuperſiciali ipſius trianguli, ſed quia <lb/>ipſe Tartalea vtiturregula algebræ, mihi viſum eſt breuiori methodo hoc idein fa <lb/>cere, & etiam vniuerſaliori via.</s> </p> <p> <s xml:space="preserve">Supp onamus igitur duo triangula, quorum vnum <seg type="var">.u.n.i.</seg> ſit nobis <choice><ex>propoſitum</ex><am>propoſitũ</am></choice>, & <lb/>cognitæ ſuperficiei, proportiones ſimiliter laterum <seg type="var">.i.n.</seg> ad <seg type="var">.n.u</seg>: et <seg type="var">.u.n.</seg> ad <seg type="var">.u.i.</seg> ſint no <lb/>bis datæ, <choice><ex>alterum</ex><am>alterũ</am></choice> verò <choice><ex>triangulum</ex><am>triangulũ</am></choice> ſit <seg type="var">.a.o.u.</seg> à nobis tamen ita <choice><ex>confectum</ex><am>confectũ</am></choice>, v<unclear reason="illegible"/>latera ſint in<lb/>er ſe proportionata eodem modo, quo latera prioris trianguli, ſed hæc nobis <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>cognita ſint, <choice><ex>quod</ex><am>ꝙ</am></choice> facillimum eſt. </s> <s xml:space="preserve">Nunc vero ſi <choice><ex>demptum</ex><am>demptũ</am></choice> fuerit <choice><ex>quadratum</ex><am>quadratũ</am></choice> <seg type="var">.a.o.</seg> minimi <lb/>lateris, ex quadrato <seg type="var">.o.u.</seg> maximi, relinquet nobis duplum producti <seg type="var">.o.u.</seg> in <seg type="var">.u.e.</seg> per <lb/><choice><ex>penultimam</ex><am>penultimã</am></choice> .2. Eucli. <choice><ex>ſupponendo</ex><am>ſupponẽdo</am></choice> <seg type="var">.a.e.</seg> perpendicularem ad <seg type="var">.o.u.</seg> vnde tale productum <lb/>quòd fit ex <seg type="var">.o.u.</seg> in <seg type="var">.u.e.</seg> conſequenter nobis cognitum erit, & quia <seg type="var">.o.u.</seg> nobis cogni- <pb facs="0105" n="93"/><fw type="head">THEOREM. ARIT.</fw> tum eſt, ideo cognoſcemus <seg type="var">.e.u.</seg> ſed <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.e.u.</seg> minor ſit <seg type="var">.a.u.</seg> ex .18. & penultima primi, <lb/>ſi <choice><ex>demptum</ex><am>demptũ</am></choice> fuerit quadratum <seg type="var">.e.u.</seg> ex quadrato <seg type="var">.a.u.</seg> remanebit nobis <choice><ex>cognitum</ex><am>cognitũ</am></choice> <choice><ex>quadra- tum</ex><am>quadra-tũ</am></choice> <seg type="var">.a.e.</seg> & ſic nota erit nobis perpendicularis <seg type="var">.a.e.</seg> ex penultima primi, quæ quidem <seg type="var">.<lb/>a.e.</seg> ſi multiplicata fuerit in dimidium <seg type="var">.o.u.</seg> dabit nobis <choice><ex>ſuperficiem</ex><am>ſuperficiẽ</am></choice> trianguli <seg type="var">.a.o.u.</seg> ex <lb/>41. dicti libri. </s> <s xml:space="preserve">Et quia proportio trianguli <seg type="var">.a.o.u.</seg> ad triangulum <seg type="var">.u.i.n.</seg> (propter ſimi <lb/>litudinem) eſt vt quadrati <seg type="var">.o.u.</seg> ad quadratum <seg type="var">.n.i.</seg> ex communi ſcientia cum vna-<lb/>quæque iſtarum proportionum dupla ſit proportioni <seg type="var">.o.u.</seg> ad <seg type="var">.n.i.</seg> ex .17. et .18. ſexti, <lb/>deinde cum nobis cognitæ ſint tres iſtarum quatuor quantitatum hoc eſt ſuperficies <lb/>trianguli <seg type="var">.a.o.u.</seg> ſuperficies trianguli <seg type="var">.u.n.i.</seg> & quadrati <seg type="var">.o.u.</seg> </s> <s xml:space="preserve">quare ex regula de tribus <lb/>cognoſcemus etiam quadratum <seg type="var">.n.i.</seg> & ſic <seg type="var">.n.i.</seg> latus primi trianguli, vnde reliqua la <lb/>tera illicò nobis innoteſcent exipſa regula de tribus, cum dixerimus, ſi <seg type="var">.o.u.</seg> dat nobis <lb/><seg type="var">u.a</seg>. </s> <s xml:space="preserve">tunc <seg type="var">.i.n.</seg> dabit <seg type="var">.u.n.</seg> quòd etiam infero de <seg type="var">.u.i</seg>.</s> </p> <p> <s xml:space="preserve">Poſſemus etiam ita hoc perficere, <lb/>ſcilicet inuenire <seg type="var">.x.</seg> quantitatem me-<lb/> <ptr xml:id="fig-0105-01a" corresp="fig-0105-01" type="figureAnchor"/> diam proportionalem inter duas ſu-<lb/>perficies triangulorum, vnde ſuper-<lb/>ficies trianguli <seg type="var">.i.a.u.o.</seg> ad <seg type="var">.x.</seg> ſe ha-<lb/>beret ut <seg type="var">.o.u.</seg> ad <seg type="var">.i.n.</seg> & ita ex regula <lb/>detribus cognoſcemus <seg type="var">.i.n</seg>. </s> <s xml:space="preserve">Multo <choice><ex>tem</ex><am>tẽ</am></choice> <lb/>pore poſtquàm hoc theorema conſtruxi, ipſum conſcriptum inueni in decimo <lb/>ſecundi libri Ioannis de monte Regio, ſatis tamen obſcurè expreſſum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0105-01" corresp="fig-0105-01a"> <graphic url="0105-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="139">CXXXIX</num>.</head> <p> <s xml:space="preserve">IN eodem primo libro vltimæ partis numerorum, Tartalea probat, via algebrę <lb/>quòd quælibet duo latera trianguli orthogonij, angulumrectum continentia, <lb/>ſint tertio longiora per diame-<lb/> <ptr xml:id="fig-0105-02a" corresp="fig-0105-02" type="figureAnchor"/> trum circuli inſcriptibilis in ip-<lb/>ſo triangulo. </s> <s xml:space="preserve">ſed hoc breuius <lb/>geometricè poteſt <choice><ex>demonſtrari</ex><am>demõſtrari</am></choice>, <lb/>quemadmodum in ſubſcripta <lb/>hic figura videre eſt, proptereà <lb/>quòd cum anguli <seg type="var">.A.o.u.</seg> et <seg type="var">.n.</seg> <lb/>omnes ſint recti et <seg type="var">.A.u.</seg> æqualis <lb/><seg type="var">o.n.</seg> et <seg type="var">.A.n.</seg> ęqualis <seg type="var">.u.o.</seg> ipſæ <seg type="var">.A.<lb/>u.</seg> et <seg type="var">.A.n.</seg> æquales erunt diame-<lb/>tro ipſius circuli. </s> <s xml:space="preserve">Sed eædem <seg type="var">.<lb/>A.u.</seg> et <seg type="var">.A.n.</seg> ſunt ſuperfluum, quo <seg type="var">.A.B.</seg> et <seg type="var">.A.C.</seg> ſunt maiores <seg type="var">.B.C.</seg> cum <seg type="var">.B.u.</seg> et <seg type="var">.C.n.</seg> <lb/>ſint æquales <seg type="var">.B.C.</seg> ex penultima tertij Eucli.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0105-02" corresp="fig-0105-02a"> <graphic url="0105-02"/> </figure> </div> </body> </floatingText> </div> <div type="unknown"> <head xml:space="preserve">THEO. SEQVENS THEO. CXXXIX.</head> <p> <s xml:space="preserve">SImiliter in nono capite ſecundi libri nouæ ſcientiæ poterat ipſe Tartalea breuio <lb/>ri methodo abſque vlla operatione ipſius Algebræ inuenire <seg type="var">.A.H.</seg> reſpectu <seg type="var">.A.<lb/>E.</seg> eſſe vt .4. <choice><ex>cum</ex><am>cũ</am></choice> vno ſeptimo ad <choice><ex>vnum</ex><am>vnũ</am></choice>. </s> <s xml:space="preserve"><choice><ex>Nam</ex><am>Nã</am></choice> ipſe ſupponit <seg type="var">.A.E.</seg> <choice><ex>decimam</ex><am>decimã</am></choice> <choice><ex>partem</ex><am>partẽ</am></choice> eſſe ipſius <pb facs="0106" n="94"/><fw type="head">IO. BAPT. BENED.</fw> <seg type="var">A.I.</seg> vnde quadratum lineæ <seg type="var">.A.I.</seg> erit .100. idem dico de quadrato lineæ <seg type="var">.I.L</seg>. </s> <s xml:space="preserve">quare <lb/>ex penultima primi <seg type="var">.A.L.</seg> erit radix quadrata quadrati .200. ideſt .14. cum vno ſepti-<lb/>mo ferè. </s> <s xml:space="preserve">quare <seg type="var">.A.L.</seg> iuncta <seg type="var">.A.O.</seg> erit .28. cum duobus ſeptimis. </s> <s xml:space="preserve">ſed <seg type="var">.L.O.</seg> ex ſuppoſi-<lb/>to erit .20. eo quòd <seg type="var">.L.I.</seg> ęquatur ipſi <seg type="var">.A.I.</seg> ſimiliter et <seg type="var">.I.O.</seg> vt ipſe etiam probauit. </s> <s xml:space="preserve">qua <lb/>dempta ex <seg type="var">.L.A.O.</seg> relinquetur <seg type="var">.H.A.M.</seg> (nam <seg type="var">.L.H.</seg> cum <seg type="var">.O.M.</seg> æquatur ipſi <seg type="var">.L.O.</seg> ex .<lb/>35. tertij ipſius Eucli. partium .8. <choice><ex>cum</ex><am>cũ</am></choice> duabus ſeptimis. cuius <choice><ex>dimidium</ex><am>dimidiũ</am></choice> hoc eſt <seg type="var">.A.H.</seg> erit <lb/>4. cum una ſeptima, quod eſt propoſitum. </s> <s xml:space="preserve">Reſpice figuram ipſius Tartaleæ.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="140">CXL</num>.</head> <p> <s xml:space="preserve">QVadrageſimum nonum quæſitum ſimiliter poſſumus alio modo ſoluere, vt <lb/>putà cum vnumquodque latus rhombi ſimul cum area cognitum, ſeu datum <lb/>nobis ſit <choice><ex>cognitum</ex><am>cognitũ</am></choice> ſimiliter nobis erit quadratum lateris <seg type="var">.a.d.</seg> hoc eſt ſumma duorum <lb/>quadratorum <seg type="var">.a.o.</seg> et <seg type="var">.o.d.</seg> ex penultima primi Euclid. </s> <s xml:space="preserve">cúmque nobis cognita etiam <lb/>ſit totalis ſuperficies rhombi, cognita etiam nobis erit eius medietas, hoc eſt produ-<lb/>ctum <seg type="var">.o.d.</seg> in <seg type="var">.o.a.</seg> vnde ex methodo .37. Theorematis cognoſcemus .a<unclear reason="illegible"/><seg type="var">.o.</seg> et <seg type="var">.o.d.</seg> & ſic <lb/>etiam eorum dupla, quod quærebatur.</s> </p> <figure place="here"> <graphic url="0106-01"/> </figure> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="141">CXLI</num>.</head> <p> <s xml:space="preserve">PVlchrum quæſitum fuit id, quod Tartalea ponit pro .18. noni libri in quarto fo-<lb/>lio, quod huiuſmodi eſt. </s> <s xml:space="preserve">Aliquis habet dolium mero plenum, ex quo <lb/>duas vrnas extrahit ipſius vini, ſed loco ipſius vini infundit duas vrnas aquæ. </s> <s xml:space="preserve">Dein <lb/>de poſt aliquot dies extrahit iterum alias duas vrnas illius miſti, & iterum infundit <lb/>duas vrnas aquæ, & poſt alios aliquot dies idem facit, & hac vltima tertia vice in-<lb/>uenit aquam tantam eſſe, quantum vinum. </s> <s xml:space="preserve">Quæritur nunc quot vrnas capiat il-<lb/>lud dolium.</s> </p> <p> <s xml:space="preserve">Solutio ipſius Tartaleæ bona eſt, cum ſupponat illas quatuor quantitates vini eſſe <lb/>inuicem continuas proportionales, vt putà primò totum vinum merum, poſteà re-<lb/>ſiduum pro ſecunda quantitate, deinde pro tertia in ſecunda, & pro quarta in ter-<lb/>tia extractione, hoc eſt quòd proportio totius vini meri ad vinum in prima ſit, vt hu <lb/>ius ad vinum in ſecunda, & vt huius ad vinum in tertia miſtione. </s> <s xml:space="preserve">Sed quia ipſe <lb/>non probat hanc continuam proportionalitatem ex methodo ſcientifica, mihi <choice><ex>visum</ex><am>visũ</am></choice> <lb/>eſt hoc loco illam deſcribere.</s> </p> <p> <s xml:space="preserve">Cogitemus igitur <seg type="var">a.u.</seg> pro capacitate dolij, et <seg type="var">.a.i.</seg> pro quantitate duarum vrna-<lb/>rum. </s> <s xml:space="preserve">Nunc uerò ſupponamus quamlibet partem huius miſti omogeneam eſſe ſuo <lb/>toto, quapropter ſequetur eandem proportionem eſſe vini ad aquam in qualibet <lb/>parte, quæ erit in toto, & ideò imaginemur <seg type="var">.e.o.</seg> æqualem <seg type="var">.a.i</seg>. </s> <s xml:space="preserve">Sed in puncto <seg type="var">.i.</seg> tali <lb/>modo diuiſam, vt proportio <seg type="var">.i.e.</seg> ad <seg type="var">.i.o.</seg> eadem ſit quæ <seg type="var">.i.a.</seg> ad <seg type="var">.i.u</seg>. </s> <s xml:space="preserve">Supponamus <choice><ex>etiam</ex><am>etiã</am></choice> <pb facs="0107" n="95"/><fw type="head">THEOREM. ARITH.</fw> <seg type="var">e.o.</seg> eſſe duas primas vrnas vini miſti hoc eſt primæ miſtionis, vnde cum eadem pro <lb/>portio ſit <seg type="var">.a.i.</seg> ad <seg type="var">.i.u.</seg> vt <seg type="var">.e.i.</seg> ad <seg type="var">.i.o.</seg> ita erit (ex .19. quinti). <seg type="var">a.e.</seg> ad <seg type="var">.o.u.</seg> ut <seg type="var">.a.i.</seg> ad <seg type="var">.i.u.</seg> & <lb/><choice><ex>componendo</ex><am>componẽdo</am></choice> ita erit <seg type="var">.a.e.</seg> cum <seg type="var">.o.u.</seg> hoc eſt <seg type="var">.i.o.u.</seg> (proptereà quòd <seg type="var">.i.o.</seg> æqualis eſt <seg type="var">.a.e.</seg> <lb/>vt reſidua totorum æqualium) ad <seg type="var">.o.u.</seg> quemadmodum <seg type="var">.a.i.u.</seg> ad <seg type="var">.i.u</seg>. </s> <s xml:space="preserve">Quare <seg type="var">.i.u.</seg> erit <lb/>media proportionalis inter <seg type="var">.a.u.</seg> et <seg type="var">.o.u.</seg> vnde proportio <seg type="var">.a.u.</seg> ad <seg type="var">.o.u.</seg> dupla erit pro <lb/>portioni <seg type="var">.i.u.</seg> ad <seg type="var">.o.u</seg>. </s> <s xml:space="preserve">Nunc autem cum extracta fuerit quantitas <seg type="var">.e.o.</seg> ex primo mi-<lb/>ſto, & poſteà infuſa aqua vſque ad plenitudinem dolij, proportio ingredientium <lb/>huius ſecundi miſti erit ea, quæ eſt inter <seg type="var">.o.u.</seg> et <seg type="var">.o.a.</seg> eo quòd in prima miſtione pro-<lb/>proportio ingredientium erat ea, quæ eſt inter <seg type="var">.o.u.</seg> et <seg type="var">.a.e.</seg> vel inter <seg type="var">.a.e.</seg> et <seg type="var">.o.u.</seg> <lb/>vt demonſtrauimus. </s> <s xml:space="preserve">Accipiamus ergo <seg type="var">.t.m.</seg> huiuſmodi ſecundi mifti, magnitudi-<lb/>nis <seg type="var">.a.i.</seg> vel <seg type="var">.e.o.</seg> ſignificantis duas vrnas, & permutemus eum in tantam aquam, <lb/><choice><ex>ſitque</ex><am>ſitq́;</am></choice> punctum <seg type="var">.o.</seg> quod nobis diuidat <seg type="var">t.m.</seg> in <seg type="var">.o.m.</seg> et, <seg type="var">o.t.</seg> partes ſimplices, tali propor <lb/>tione inuicem relatas, vt ſunt <seg type="var">.o.u.</seg> et <seg type="var">.o.a.</seg> vnde habebimus ex ſupradictis rationibus <lb/>eandem proportionem ipſius <seg type="var">.a.t.</seg> ad <seg type="var">.m.u.</seg> vt <seg type="var">.a.o.</seg> ad <seg type="var">.o.u.</seg> & componendo <seg type="var">.a.t.</seg> cum <seg type="var">.m.<lb/>u.</seg> hoc eſt <seg type="var">.i.m.u.</seg> (eo quod cum <seg type="var">.t.m.</seg> æqualis ſit <seg type="var">.a.i.</seg> per conſequens <seg type="var">.i.m.</seg> æqualis erit <seg type="var">.<lb/>a.t.</seg>) ad <seg type="var">.m.u.</seg> vt <seg type="var">.a.o.u.</seg> ad <seg type="var">.o.u.</seg> ſed proportio <seg type="var">.a.o.u.</seg> ad <seg type="var">.o.u.</seg> dupla erat proportioni <seg type="var">.i.o.<lb/>u.</seg> ad <seg type="var">.o.u.</seg> quemadmodum ſupra diximus. </s> <s xml:space="preserve">Ergo proportio <seg type="var">.i.m.u.</seg> ad <seg type="var">.m.u.</seg> erit dupla <lb/>ſimiliter proportioni <seg type="var">.i.o.u.</seg> ad <seg type="var">.o.<lb/>u.</seg> quapropter <seg type="var">.o.u.</seg> erit media pro<lb/> <ptr xml:id="fig-0107-01a" corresp="fig-0107-01" type="figureAnchor"/> portionalis inter <seg type="var">.i.u.</seg> et <seg type="var">.m.u</seg>. </s> <s xml:space="preserve">Ec-<lb/>ce igitur quomodo eadem eſt pro <lb/>portio <seg type="var">.a.u.</seg> ad <seg type="var">.i.u.</seg> quæ <seg type="var">.i.u.</seg> ad <seg type="var">.o.u.</seg> & quæ <seg type="var">.o.u.</seg> ad <seg type="var">.m.u.</seg> qui quidem modus neceſſarius <lb/>eſt vt intellectus acquieſcat, id quod experientia non facit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0107-01" corresp="fig-0107-01a"> <graphic url="0107-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="142">CXLII</num>.</head> <p> <s xml:space="preserve">PRæcedens Tartaleæ quæſitum elegans quidem eſt, ſed pulchrum etiam vide-<lb/>tur quærere proportionem ingredientium in ultima miſtione, cum cognita fue <lb/>rit nobis proportio continentiæ dolij ad capacitatis vrnæ ſimul <choice><ex>cum</ex><am>cũ</am></choice> numero vitium <lb/>extractionum & impletionum.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proportio <seg type="var">.a.u.</seg> ad <seg type="var">.a.i.</seg> cognita nobis fuerit, cognoſcemus etiam <lb/><seg type="var">e.i.</seg> ex regula de tribus & per conſequens etiam <seg type="var">.i.o.</seg> reſiduum ex <seg type="var">.e.o.</seg> & ſimiliter ag-<lb/>gregatum <seg type="var">.a.i.</seg> cum <seg type="var">.i.o.</seg> & ſic <seg type="var">.o.u.</seg> reſiduum totius, et <seg type="var">.o.t.</seg> ſimiliter, eo quòd <seg type="var">.a.u.</seg> ad <seg type="var">.a.<lb/>o.</seg> eſt ut <seg type="var">.t.m.</seg> ad <seg type="var">.o.t.</seg> vnde cognoſcemus etiam <seg type="var">.o.m.</seg> vt reſiduum <seg type="var">.t.m.</seg> & ſimiliter ag-<lb/>gregatum <seg type="var">.a.o.</seg> cum <seg type="var">.o.m.</seg> hoc eſt <seg type="var">.a.m.</seg> & etiam <seg type="var">.m.u.</seg> reſiduum totius.</s> </p> <p> <s xml:space="preserve">Cognoſcere autem proportionem totius dolij ad vrnam, vel ècontrà, cum cogni <lb/>ta nobis fuerit proportio ingredientium in vltima miſtione ſimul cum numero vi-<lb/>tium extractionum, & repletionum, quod ſcribit Tartalea, hoc etiam modo <lb/>poſſumus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proportio <seg type="var">.m.u.</seg> ad <seg type="var">.m.a.</seg> cognita nobis fuerit, illicò ſcie-<lb/>mus proportionem <seg type="var">.a.u.</seg> ad <seg type="var">.m.u.</seg> & cum ſciuerimus numerum vitium extractionum, <lb/>& impletionum illicò cognoſci-<lb/>mus multiplicitatem proportio-<lb/>nis <seg type="var">.a.u.</seg> ad <seg type="var">.m.u.</seg> ad proportionem <seg type="var">.<lb/> <ptr xml:id="fig-0107-02a" corresp="fig-0107-02" type="figureAnchor"/> o.u.</seg> ad <seg type="var">.m.u.</seg> quapropter propor-<lb/>tio <seg type="var">.o.u.</seg> ad <seg type="var">.m.u.</seg> nobis cognita erit <lb/>hoc eſt <seg type="var">.a.u.</seg> ad <seg type="var">.i.u.</seg> & ſimiliter ea, quæ eſt <seg type="var">.a.u.</seg> ad <seg type="var">.a.i.</seg> & è conuerſo ſimiliter.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0107-02" corresp="fig-0107-02a"> <graphic url="0107-02"/> </figure> </div> </body> </floatingText> <pb facs="0108" n="96"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Vnde cum aliquis diceret priori modo, dolium habeo vrnarum .400. vini, & per <lb/>vices .25. extraxi & impleui ipſum, vt dictum eſt. </s> <s xml:space="preserve">Nunc verò velim ſcire proportio-<lb/>nem vini ad a quam hac vltima vice. </s> <s xml:space="preserve">Nunc igitur ſi procedemus iuxta doctrinam <lb/>primi exempli huius theorematis, obtinebimus quod quærebamus.</s> </p> <p> <s xml:space="preserve">Sed ſi diceret iuxta Tartaleæ quæſitum, hoc eſt dolium habeo, quod ignoro quot <lb/><choice><ex>nam</ex><am>nã</am></choice> urnas contineat, volo tamen per .25. vices extrahere, & implere vt <choice><ex>ſupradictum</ex><am>ſupradictũ</am></choice> <lb/>eſt, ita vt vltima vice proportio vini ad aquam ſit ſeſquialtera. </s> <s xml:space="preserve">Tunc ſi iuxta mo-<lb/>dum ſecundi exempli huius theorematis procedemus habebimus quod cupimus.</s> </p> <p> <s xml:space="preserve">Alio etiam modo aliquis quærere poſſet, hoc eſt, habeo <choice><ex>dolium</ex><am>doliũ</am></choice> quod capit .400. <lb/>vrnas. </s> <s xml:space="preserve">Habeo etiam vas trium vrnarum, quo mediante me oportet extrahere, & <lb/>implere. </s> <s xml:space="preserve">Velim tamen ſcire quoties me hoc facere oporteat, ita vt poſtrema vi-<lb/>ce vinum ſe habeat ad aquam in proportione ſeſquialtera, vnde multoties accidet <lb/>vltimam extractionem, & impletionem mutilatam, ſeu imperfectam, euadere.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſi proportio vini ad aquam in vltima miſtione deberet eſſe vt <seg type="var">.n.<lb/>u.</seg> ad <seg type="var">.n.a.</seg> ita vt extrema vice fuiſſet <seg type="var">.t.m.</seg> quæ quidem <seg type="var">.t.m.</seg> excederet terminum per <seg type="var">.<lb/>n.m.</seg> quæ <seg type="var">.n.m.</seg> reuera eſſet nobis cognita, eò quòd ex priori modo hic ſupra dicto <lb/>proportio <seg type="var">.a.m.</seg> ad <seg type="var">.m.u.</seg> nobis in-<lb/>noteſceret, & proportio <seg type="var">.n.a.</seg> ad <seg type="var">.<lb/>n.u.</seg> nobis data eſt ſimul cum <choice><ex>quan titate</ex><am>quã titate</am></choice> <seg type="var">.a.u.</seg> </s> <s xml:space="preserve">quare quantitas <seg type="var">.n.u.</seg> & <lb/><seg type="var">m.u.</seg> nobis cognita, remanebit, et <lb/><seg type="var">n.m.</seg> eorum differentia ſimiliter, etiam, et <seg type="var">.t.n.</seg> reſiduum vaſis, quo metimur, vnde <lb/>neceſſe erit, quo<unclear reason="illegible"/>d vltima vice vas contineret ſolum <seg type="var">.t.n.</seg> reliqua uerò per ſe patent.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0108-01" corresp="fig-0108-01a"> <graphic url="0108-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="143">CXLIII</num>.</head> <p> <s xml:space="preserve">HIeronymus Cardanus in <ref>lib. ſuæ arithmeticæ cap .66. quæſtione .56.</ref> quam Car<lb/>danicam vocat, ita inquit.</s> </p> <p> <s xml:space="preserve">Quidam perambulauit prima die certam quantitatem ſpatij, & ſecunda die, <choice><ex>tan tò</ex><am>tãtò</am></choice> plus proportionaliter, quantò diameter eſt maior coſta, & tertia die tantò plus <lb/>ſecunda, quantò proportionaliter portio lineæ diuiſæ ſecundum proportionem ha <lb/>bentem medium, & duo extrema excedit minorem portionem, & quarta die in <lb/>proportione ad tertiam vt ſecunda ad primam, & quinta die proportionaliter tan-<lb/>tò plus quarta, quantò in tertia plus ſecunda, & ita alternatis vicibus in diebus no-<lb/>uem peregit nouem milliaria. </s> <s xml:space="preserve">Quæritur igitur quantum ambulauit die prima.</s> </p> <p> <s xml:space="preserve">Hoc autem nihil aliud eſt, quàm ſi aliquis diceret, propono tibi, exempli gratia, <lb/>lineam <seg type="var">.a.l.</seg> nouem partibus inuicem non æqualibus ita diuiſam <seg type="var">.a.c</seg>: <seg type="var">c.d</seg>: <seg type="var">d.e</seg>: & cæte-<lb/>ris, quarum partium proportiones tibi etiam do, vt putà. <seg type="var">a.c.</seg> ad <seg type="var">.c.d.</seg> et <seg type="var">.c.d.</seg> ad <seg type="var">.d.e.</seg> et <seg type="var">.<lb/>d.e.</seg> ad <seg type="var">.e.f.</seg> & ſic de cæteris vſque ad poſtremam <seg type="var">.k.l.</seg> quæ quidem proportiones ſint <lb/>etiam inuicem diſſimiles, ſeu inæquales, do tibi etiam <choice><ex>proportionem</ex><am>proportionẽ</am></choice> totius lineæ <seg type="var">.a.l.</seg> <lb/>ad <seg type="var">.a.b.</seg> ſuam partem, quæ vt in propoſito exemplo nonupla eſt.</s> </p> <p> <s xml:space="preserve">Quæro nunc quam proportionem habebit <seg type="var">.a.c.</seg> ad <seg type="var">.a.b.</seg> & ſic de cæteris partibus <lb/>eiuſdem ad eandem <seg type="var">.a.b</seg>.</s> </p> <p> <s xml:space="preserve">Quod quidem facillimum erit ſpeculari, nec non operari vnicuique, qui omnino <lb/>practicæ numerorum ignarus non fuerit, dum ab ordine ſcientifico non diſcedat.</s> </p> <p> <s xml:space="preserve">Cum enim cognoſcimus proportionem <seg type="var">.a.c.</seg> ad <seg type="var">.c.d.</seg> conſequenter cognoſcemus <lb/>ctiam proportion em aggregati <seg type="var">.a.c.d.</seg> ad <seg type="var">.c.d.</seg> cum autem cognouerimus proportio- <pb facs="0109" n="97"/><fw type="head">THEOREM. ARIT.</fw> nem <seg type="var">.c.d.</seg> ad <seg type="var">.d.e.</seg> ſi <seg type="var">.c.d.</seg> accipiemus, vt medium inter <seg type="var">.a.d.</seg> et <seg type="var">.d.e.</seg> cognoſcemus etiam <lb/>proportionem <seg type="var">.a.d.</seg> ad <seg type="var">.d.e.</seg> </s> <s xml:space="preserve">quare etiam eam quæ <seg type="var">.a.e.</seg> ad <seg type="var">.d.e.</seg> collocando poſteà. <lb/><seg type="var">d.e.</seg> inter <seg type="var">.e.f.</seg> et <seg type="var">.a.e.</seg> innoteſcet ea, quæ eſt <seg type="var">.a.e.</seg> ad <seg type="var">.e.f.</seg> & ita gradatim accedenrus ad <lb/>perfectam cognitionem proportionis totius <seg type="var">.a.l.</seg> ad <seg type="var">.k.l</seg>. </s> <s xml:space="preserve">Nunc autem mediante <seg type="var">.k.l.</seg> <lb/>cognoſcemus proportionem totius <seg type="var">.a.l.</seg> ad <seg type="var">.i.k.</seg> & hac mediante, cam cognoſcemus, <lb/>quæ totius <seg type="var">.a.l.</seg> ad <seg type="var">.g.h.</seg> & hac mediante eam quæ totius <seg type="var">.a.l.</seg> ad <seg type="var">.f.g.</seg> & ſic gradatim, co <lb/>gnita nobis erit proportio totius <lb/>lineæ <seg type="var">.a.l.</seg> ad ſuam partem <seg type="var">.a.c.</seg> be-<lb/> <ptr xml:id="fig-0109-01a" corresp="fig-0109-01" type="figureAnchor"/> neficio poſteà totius lineæ <seg type="var">.a.l.</seg> co <lb/>gnoſcemus proportionem <seg type="var">a.c.</seg> ad <lb/><seg type="var">a.b.</seg> & ſic aliarum reſpectu lineæ <seg type="var">.a.b.</seg> vt quærebatur, quæ quidem propoſitio, etſi car <lb/>danica uocetur leuiſſima tamen eſt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0109-01" corresp="fig-0109-01a"> <graphic url="0109-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="144">CXLIIII</num>.</head> <p> <s xml:space="preserve">QVamuis multi de modo in ſumma colligendi, ſubtrahendi, <choice><ex>multiplicandi</ex><am>multiplicãdi</am></choice>, & di <lb/>uidendi proportiones ſcripſerint, nullus tamen (quod ſciam) perfectè, ac <lb/>ſcientificè ſpeculatus eſt has operationes, quapropter hanc rem cum ſilentio tranſi <lb/>re nolui, quin aliquid de ipſa conſcribam à ſumma dictarum proportionum in-<lb/>cohando.</s> </p> <p> <s xml:space="preserve">Quotieſcunque igitur volunt duas proportiones inuicem aggregare, ſimul ea-<lb/>rum antecedentia multiplicant, & ſimiliter earum conſequentia. </s> <s xml:space="preserve">Tunc proportio <lb/>terminata ab illis productis euadit in ſummam illarum duarum propoſitarum <lb/>proportionum.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia, ſi voluerimus colligere proportionem ſeſquialteram cum ſeſ-<lb/>quitertia, multiplicando .3. cum .4. antecedentia ſcilicet, pro ductum erit .12. poſteà <lb/>multiplicando .2. cum .3. conſequentia, tunc productum erit .6. </s> <s xml:space="preserve">Proportio igitur, <lb/>quæ inter .12. et .6. reperitur. (quæ dupla eſt) eſt ſumma propoſitarum <choice><ex>proportionum</ex><am>proportionũ</am></choice>.</s> </p> <p> <s xml:space="preserve">Cuius rei ſpeculatio erit huiuſmodi ſint <seg type="var">.x.</seg> et <seg type="var">.u.</seg> <lb/>duo antecedentia quarunruis proportionum <seg type="var">.t.</seg> <lb/> <ptr xml:id="fig-0109-02a" corresp="fig-0109-02" type="figureAnchor"/> verò et. n ſint eorum conſequentia, productum <lb/>autem antecedentium ſit <seg type="var">.a.g.</seg> illud verò quod <choice><ex>con</ex><am>cõ</am></choice> <lb/>ſequentium ſit <seg type="var">.d.a.</seg> vnde proportio <seg type="var">.a.g.</seg> ad <seg type="var">.a.d.</seg> <lb/>compoſita erit ex proportione <seg type="var">.x.</seg> ad <seg type="var">.t.</seg> & ex ea, <lb/>quæ eſt <seg type="var">.u.</seg> ad <seg type="var">.n.</seg> per .24. ſexti vel quintam octaui. <lb/></s> <s xml:space="preserve">Patet igitur ratio rectè faciendi, vt ſuprà dictum <lb/>eſt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0109-02" corresp="fig-0109-02a"> <graphic url="0109-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="145">CXLV</num>.</head> <p> <s xml:space="preserve">QVotieſcunque deinde detrahere volunt vnam proportionem ex altera mul-<lb/>tiplicant antecedens vnius cum conſequenti alterius. </s> <s xml:space="preserve">Tunc proportio, quę <lb/>inter talia duo producta incluſa reperitur, eſt reſiduum, ſeu differentia illarum dua-<lb/>rum proportionum datarum.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia, ſi aliquis vellet ex proportione dupla detrahere ſeſquialte-<lb/>ram, multiplicaret .2. antecedens duplæ cum .2. conſequenti ſeſquialteræ, quorum <lb/>productum eſſet .4. pro antecedenti reſiduę proportionis. </s> <s xml:space="preserve">Deinde multiplicaret .3 <lb/>antecedens ſeſquialteræ cum .1. conſequenti duplæ, & productum eſſet .3. pro <choice><ex>con- ſequenti</ex><am>cõ- ſequenti</am></choice> reſiduę proportionis; </s> <s xml:space="preserve">quæ quidem reſidua proportio eſſet vt .4. ad .3. hoc <lb/>eſt ſeſquitertia, & ſic de cæteris.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione, ſit proportio <seg type="var">.x.</seg> ad <seg type="var">.n.</seg> ea quæ (exempli gratia) maior ſit, à <lb/>qua volumus demere proportionem <seg type="var">.t.</seg> ad <seg type="var">.u.</seg> minorem ſcilicet. </s> <s xml:space="preserve">Nunc autem <lb/>productum <seg type="var">.x.</seg> in <seg type="var">.u.</seg> ſit <seg type="var">.a.g.</seg> illud verò <seg type="var">.t.</seg> in <seg type="var">.<lb/>n.</seg> ſit <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Tunc dico proportionem <seg type="var">.a.g.</seg> ad <seg type="var">.a.</seg> <lb/> <ptr xml:id="fig-0110-01a" corresp="fig-0110-01" type="figureAnchor"/> d. eſſe reſiduam quæſitam. </s> <s xml:space="preserve">Sit <seg type="var">.b.a.</seg> productum <lb/>u. in <seg type="var">.n.</seg> vnde eadem proportio erit producti <seg type="var">.a.<lb/>g.</seg> ad productum <seg type="var">.a.b.</seg> quę <seg type="var">.x.</seg> ad <seg type="var">.n.</seg> et <seg type="var">.a.d.</seg> ad <seg type="var">a.b.</seg> <lb/>quæ <seg type="var">.t.</seg> ad <seg type="var">.u.</seg> ex prima ſexti, ſeu .18. vel .19. ſe-<lb/>ptimi, ſed proportio <seg type="var">.a.g.</seg> ad <seg type="var">.a.b.</seg> hoc eſt <seg type="var">.x.</seg> ad <seg type="var">.<lb/>n.</seg> componitur ex ea, quæ eſt <seg type="var">.a.g.</seg> ad <seg type="var">.a.d.</seg> & ea, <lb/>quæ eſt <seg type="var">.a.d.</seg> ad <seg type="var">.a.b.</seg> hoc eſt <seg type="var">.t.</seg> ad <seg type="var">.u.</seg> ergò ea, quę <lb/>eſt <seg type="var">.a.g.</seg> ad <seg type="var">.a.d.</seg> erit quàm quærebamus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0110-01" corresp="fig-0110-01a"> <graphic url="0110-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="146">CXLVI</num>.</head> <p> <s xml:space="preserve">RATIO verò, quòd rectè fiat, quotieſcunque aliquam proportionem dupli-<lb/>care volentes, quadramus terminos ipſius proportionis, vel ſi eam triplicare <lb/>voluerimus, cubamus ipſos terminos, vel ſi eam quadruplicare voluerimus <lb/>inuenimus cenſicos cenſicos terminorum ipſius proportionis, & ſic de ſingulis, in <ref>.17 <lb/>Theo. huiuſmodi tractatus</ref> manifeſta eſt.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="147">CXLVII</num>.</head> <p> <s xml:space="preserve">QVotieſcunque nobis propoſiti fuerint duo numeri ad libitum, deſideraremus<lb/>q́ue duas proportiones tali relatione inuicem refertas, quali ſunt hi duo pro <lb/>poſiti numeri inter ſe, ita faciendum erit.</s> </p> <p> <s xml:space="preserve">Sciendum primo eſt proportionem maioris numeri propoſiti ad minorem ſem-<lb/>per eſſe alicuius ex quinque generum, hoc eſt aut erit generis multiplicis, aut ſu-<lb/>perparticularis, aut multiplicis ſuperparticularis, aut ſuper partientis, aut multi-<lb/>plicis ſuperpartientis.</s> </p> <p> <s xml:space="preserve">Nunc autem ſi erit ex genere multiplici, iam ab antiquis traditus eſt modus, <choice><ex>quem</ex><am>quẽ</am></choice> <lb/>ſequi debemus. </s> <s xml:space="preserve">Cuius ſpeculatio à me inuenta patet .in .17. Theo. huius libri, vt <lb/>in præcedenti dixi.</s> </p> <p> <s xml:space="preserve">Sed ſi talis proportio datorum numerorum erit alicuius aliorum generum, ita <lb/>agemus, ſi fuerit ſuperparticularis.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia, ſeſquialtera, tunc ſumantur duo numeri inuicem inæquales, <lb/>quos à caſu volueris <seg type="var">.o.</seg> et <seg type="var">.c.</seg> qui quidem cubentur, & eorum cubi ſint <seg type="var">.a.</seg> et <seg type="var">.e</seg>. </s> <s xml:space="preserve">Inuenia <lb/>tur poſteà. u. ita proportionatus ad <seg type="var">.o.</seg> vt <seg type="var">.o.</seg> eſt ad <seg type="var">.c.</seg> ex regula de tribus, hoc eſt diui-<lb/>dendo quadratum ipſius <seg type="var">.o.</seg> per <seg type="var">.c.</seg> vnde nobis proueniat <seg type="var">.u.</seg> & quia proportio <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> <lb/>tripla eſt proportioni <seg type="var">.o.</seg> ad <seg type="var">.c.</seg> & proportio <seg type="var">.u.</seg> ad <seg type="var">.c.</seg> dupla eſt <choice><ex>eidem</ex><am>eidẽ</am></choice>, quæ <seg type="var">.o.</seg> ad <seg type="var">.c.</seg> ideo <lb/>proportio <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> ſeſquialtera erit proportioni <seg type="var">.u.</seg> ad <seg type="var">.c</seg>.</s> </p> <p> <s xml:space="preserve">Sed ſi proportio numerorum propoſitorum fuerit ſeſquitertia, faciemus <seg type="var">.a.</seg> et <seg type="var">.e.</seg> <lb/>eſſe cenſica cenſica ipſius <seg type="var">.o.</seg> et <seg type="var">.c</seg>. </s> <s xml:space="preserve">tunc ſumemus <seg type="var">.u.</seg> conſequentem ad <seg type="var">.o.</seg> vt dictum eſt, <lb/>deinde inueniremus <seg type="var">.i.</seg> conſequens ad <seg type="var">.u.</seg> ita ut <seg type="var">.u.</seg> conſequens ipſius <seg type="var">.o</seg>. </s> <s xml:space="preserve">tunc habebi-<lb/>mus proportionem <seg type="var">.i.</seg> ad <seg type="var">.c.</seg> triplam, & eam quæ eſt <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> quadruplam proportio- <pb facs="0111" n="99"/><fw type="head">THEOREM. ARIT.</fw> ni <seg type="var">.o.</seg> ad <seg type="var">.c</seg>. </s> <s xml:space="preserve">Idem dico de reliquis proportionibus ſuperparticularibus.</s> </p> <p> <s xml:space="preserve">Sed ſi data proportio numerorum fuerit ex ſuper partientibus, vt exempli gra-<lb/>tia de quinque ad tria, efficiemus, vt <seg type="var">.a.</seg> et <seg type="var">.e.</seg> ſint prima relata ipſius <seg type="var">.o.</seg> et <seg type="var">.c.</seg> vnde <lb/>proportio <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> ita ſe habe-<lb/>bit ad proportionem <seg type="var">.o.</seg> ad <seg type="var">.c.</seg> <lb/> <ptr xml:id="fig-0111-01a" corresp="fig-0111-01" type="figureAnchor"/> vt quinque ad <choice><ex>vnum</ex><am>vnũ</am></choice> & propor-<lb/>tio <seg type="var">.i.</seg> ad <seg type="var">.c.</seg> ut tria ad <choice><ex>vnum</ex><am>vnũ</am></choice>. </s> <s xml:space="preserve">Qua-<lb/>re proportio <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> ad pro-<lb/>portionem <seg type="var">.i.</seg> ad <seg type="var">.c.</seg> ſe habebit, <lb/>vt quinque ad tria, & ſic de reliquis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0111-01" corresp="fig-0111-01a"> <graphic url="0111-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Pro alijs, eundem ordinem ſeruando, obtinebimus quod volumus.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="148">CXLVIII</num>.</head> <p> <s xml:space="preserve">QVamuis in .16. ſexti et .20. ſeptimi manifeſtè pateat ratio, quare rectè fiatac <lb/>cipiendam radicem quadratam illius producti, quod fit ex duobus datis <lb/>terminis, vt medium proportionale geometricè inter ipſos habeamus: </s> <s xml:space="preserve">nihilomi-<lb/>nus, quia per aliam methodum hoc idem ſcire poſſumus, inconueniens non erit a-<lb/>liquid circa hoc dicere.</s> </p> <p> <s xml:space="preserve">Cogitemus igitur exempli gratia, tres numeros continuè proportionales geo-<lb/>metricè <seg type="var">.a.b</seg>: <seg type="var">c.d.</seg> et <seg type="var">.e.f.</seg> quorum <seg type="var">.a.b.</seg> et <seg type="var">.e.f.</seg> tantummodo nobis cogniti ſint, imagine-<lb/>mur etiam <seg type="var">.g.a.</seg> eſſe productum quod fit ex <seg type="var">.a.b.</seg> in <seg type="var">.e.f.</seg> et <seg type="var">.d.k.</seg> quadratum <seg type="var">.c.d.</seg> et <seg type="var">.a.h.</seg> <lb/>id quod fit ex <seg type="var">.a.b.</seg> vnde eandem proportionem habebimus <seg type="var">.a.h.</seg> ad <seg type="var">.a.g.</seg> quæ eſt <seg type="var">.h.b.</seg> <lb/>ad <seg type="var">.b.g.</seg> ex prima .6. aut .18. vel .19. ſepti-<lb/>mi, ſed per .11. octaui ita eſt quadrati <seg type="var">.a.</seg> <lb/> <ptr xml:id="fig-0111-02a" corresp="fig-0111-02" type="figureAnchor"/> h. ad quadratum <seg type="var">.k.d.</seg> vt <seg type="var">.a.b.</seg> ad <seg type="var">.e.f.</seg> hoc <lb/>eſt vt <seg type="var">.h.b.</seg> ad <seg type="var">.b.g.</seg> ergo per .11. quinti ita <lb/>erit <seg type="var">.a.h.</seg> ad <seg type="var">.a.g.</seg> vt ad <seg type="var">.k.d.</seg> vnde <seg type="var">.a.g.</seg> æqua <lb/>le erit <seg type="var">.k.d.</seg> per .9. quinti. </s> <s xml:space="preserve">Rectè ergo erit <lb/>accipere radicem quadratam <seg type="var">.a.g.</seg> pro <seg type="var">.c.<lb/>d.</seg> quod etiam eſt diuidere vnam datam <lb/><choice><ex>proportionem</ex><am>proportionẽ</am></choice> per æqualia, hoc eſt in duas <lb/>æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poſteriori-<lb/>bus Theorematibus ad demonſtrandum priora illis, ſed hoc .148. dictum ſit luden <lb/>di loco.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0111-02" corresp="fig-0111-02a"> <graphic url="0111-02"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="149">CXLIX</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">Vnde</hi> fiat <choice><ex>quod</ex><am>ꝙ</am></choice> ſi quis inuenire voluerit ſecundum terminum ex quatuor nume <lb/>ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-<lb/>modo nobis cogniti ſint, rectè factum ſit quadrare primum eorum, & hoc quadra-<lb/>tum poſteà per alium terminum cognitum multiplicare, cuius producti demum ac-<lb/>cipere radicem cubam pro ſecundo termino quæſito, hocloco videbimus.</s> </p> <p> <s xml:space="preserve">Imaginemur quatuor terminos continuè proportionales, vt dictum eſt, eſſe.</s> <pb facs="0112" n="100"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve">a.b: <seg type="var">c.d</seg>: <seg type="var">e.f.</seg> et <seg type="var">.g.h.</seg> quorum <seg type="var">.a.b.</seg> et <seg type="var">.g.h.</seg> nobis tantummodo cogniti ſint, <choice><ex>ſitque</ex><am>ſitq́</am></choice> imagina <lb/>tione deſcriptus cubus <seg type="var">.a.q.</seg> primi termini, <choice><ex>cubusque</ex><am>cubusq́</am></choice> <seg type="var">.d.k.</seg> ſecundi rermini, conſidere-<lb/>mus etiam baſim <seg type="var">.a.i.</seg> quadratam ipſius cubi <seg type="var">.a.q.</seg> hoc eſt præcedentem dignitatem ip <lb/>ſius cubi eiuſdem radicis, quæ quidem baſis <seg type="var">.a.i.</seg> multiplicetur per quartum <choice><ex>terminum</ex><am>terminũ</am></choice> <lb/><seg type="var">g.h.</seg> productum autem ſit <seg type="var">.g.a.</seg> vnde eadem proportio erit <seg type="var">.a.q.</seg> ad <seg type="var">.a.g.</seg> quæ <seg type="var">.b.q.</seg> ad <seg type="var">.b.<lb/>g.</seg> per .25. vndecimi, ſed per primam ſexti, vel .18. aut .19. ſeptimi ita eſt <seg type="var">.q.i.</seg> ad <seg type="var">.i.g.</seg> <lb/>vt <seg type="var">.b.q.</seg> ad <seg type="var">.b.g.</seg> </s> <s xml:space="preserve">quare per .11. quinti <lb/>ita erit <seg type="var">.a.q.</seg> ad <seg type="var">.a.g.</seg> vt <seg type="var">.q.i.</seg> ad <seg type="var">.i.g.</seg> ideſt <lb/> <ptr xml:id="fig-0112-01a" corresp="fig-0112-01" type="figureAnchor"/> vt <seg type="var">.a.b.</seg> ad <seg type="var">.g.h.</seg> ſed vt eſt <seg type="var">.a.b.</seg> ad <seg type="var">.g.h.</seg> <lb/>ſic eſt <seg type="var">.a.q.</seg> ad <seg type="var">.k.d.</seg> per .36. vndecimi, <lb/>ſeu per .11. octaui, vnde per .11. quin <lb/>ti ſic erit <seg type="var">.a.q.</seg> ad <seg type="var">.a.g.</seg> vt ad <seg type="var">.k.d</seg>. </s> <s xml:space="preserve">Qua-<lb/>re per .9. eiuſdem <seg type="var">.a.g.</seg> ęqualis erit <seg type="var">.k.<lb/>d</seg>. </s> <s xml:space="preserve">Vnde rectè erit accipere radicem <lb/>cubam <seg type="var">.a.g.</seg> pro <choice><ex>ſecundo</ex><am>ſecũdo</am></choice> termino <seg type="var">.c.d.</seg> <lb/>id, quod nobis inſeruit ad inueniendam tertiam partem vnius propoſitæ propor-<lb/>tionis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0112-01" corresp="fig-0112-01a"> <graphic url="0112-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="150">CL</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">Sed</hi> vt ſpeculatio iſta ita vniuerſalis fiat vt ad <choice><ex>oens</ex><am>oẽs</am></choice> dignitates applicari poſſit; <lb/></s> <s xml:space="preserve">Supponamus <seg type="var">.a.q.</seg> et <seg type="var">.k.d.</seg> eſſe duas dignitates quas volueris vnius, ſed eiuſdem <lb/>ſpeciei, et <seg type="var">.a.i.</seg> dignitas præcedens dignitatem <seg type="var">.a.q.a.</seg> cuius multiplicatione in <seg type="var">.a.b.</seg> <lb/>eius radix producitur dignitas <seg type="var">.a.q.</seg> & ab ipſius <seg type="var">.a.i.</seg> multiplicatione in <seg type="var">.g.h.</seg> reſultet <seg type="var">.a.<lb/>g.</seg> vnde ex .18. vel .19. ſeptimi eadem proportio erit <seg type="var">.a.q.</seg> ad <seg type="var">.a.g.</seg> quæ <seg type="var">.a.b.</seg> ad <seg type="var">.g.h.</seg> ſed <lb/>eadem etiam eſt <seg type="var">.a.q.</seg> ad <seg type="var">.k.d.</seg> ex ijs, quæ in .17. theoremare dixi, vnde ex .11. quinti, <lb/>ita erit <seg type="var">.a.q.</seg> ad <seg type="var">.a.g.</seg> vt ad <seg type="var">.k.d</seg>. </s> <s xml:space="preserve">Quapropter <seg type="var">.a.g.</seg> æqualis erit <seg type="var">.k.d.</seg> & ideo cum inuenta <lb/>fuerit radix huiuſmodi dignitatis ex quantitate <seg type="var">.a.g.</seg> habebimus <seg type="var">.c.d.</seg> ſecundum ter-<lb/>minum quæſitum.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="151">CLI</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">Vnde</hi> verò fiat, quòd cum quis voluerit dimidium alicuius datæ proportio-<lb/>nis inuenire, rectè faciat, ſi accipiat radices quadratas illorum datorum rer-<lb/>minorum, etſi voluerit tertiam partem, accipiat radices cubas: </s> <s xml:space="preserve">ſi autem quartam, <lb/>accipereradices cenſicas cenſicas ipſorum, & ſic de ſingulis in .17. </s> <s xml:space="preserve">Theoremate om-<lb/>nia patent.</s> </p> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="152">CLII</num>.</head> <p> <s xml:space="preserve"><hi rend="small caps">Vnde</hi> autem fiat, vt cum quis voluerit multiplicare aliquam proportionem <lb/>per fractos, rectè faciat prius multiplicando eam per numeratorem, dein-<lb/>de productum diuiſerit per denominationem ipſorum fractorum.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia, cum aliquis voluerit multiplicare proportionem ſeſquiquar-<lb/>tam per duo tertia, multiplicabit prius ipſam proportionem per numeratorem .2. <lb/>& productum, erit proportio .25. ad .16. qua poſtea diuiſa per .3. denominatorem, <lb/>prouentus erit proportio radicis cubæ .25. ad radicem cubam .16. vel vt proportio.</s> <pb facs="0113" n="101"/> <fw type="head">THEOREM. ARIT.</fw> <s xml:space="preserve">25. ad radicem cubam .10000. quæ quidem proportiones æquales inuicem ſunt, cu <lb/>tam vna, quàm alia, ſit tertia pars totius.</s> </p> <p> <s xml:space="preserve">Pro cuius ratione cogitem is <seg type="var">.a.b.</seg> eſſe aliquod totum, quod multiplicare cupimus <lb/>per duas tertias, quod <choice><ex>quidem</ex><am>quidẽ</am></choice> nihil aliud eſt, quàm accipere duas tertias partes vnius <lb/>totius ſuperficialis, imaginemur igitur hoc totum <seg type="var">.a.b.</seg> lineare diuiſum eſſe in tertias <lb/>partes mediantibus <seg type="var">.e.</seg> et <seg type="var">.d</seg>. </s> <s xml:space="preserve">& tunc multiplicando ipſum per 2. tertias lineares produ-<lb/>ctum erit <seg type="var">.a.c.</seg> ſex vnitatum ſuperficialium, quod quidem productum poſteà diuiſum <lb/>per .3. dabit <seg type="var">.d.c.</seg> hoc eſt duas tertias ſuperficiales (quæ eſt tertia pars ipſius <seg type="var">.a.c.</seg>) & <lb/>ęquales numero <seg type="var">.c.b.</seg> duabus vnitatibus linearibus, ideſt duabus tertijs ipſius <seg type="var">.a.b</seg>. </s> <s xml:space="preserve">No <lb/>tandum etiam eſt, quòd cum ferè omnia reducantur ad regulam de tribus, proptereà <lb/>etiam multiplicatio alicuius quantitatis per aliam quantitatem, nihil aliud eſt quàm <lb/>quædam operatio ipſius regulæ de tribus, vt eyempli gratia volo multiplicare .25. <lb/>per 20. hoc nihil aliud eſt niſi quærere alium numerum ita proportionatum ad .25. <lb/>vt 20. ſe habetad vnum, vnde multiplicando .25. cum .20. & productum diuidendo <lb/>per vnum exregula de tribus, prouentus eſt idem numerus ipſius producti, & propte <lb/>rea cum volumus multiplicare aliquem numerum per fractos hoc nihil aliud eſt <lb/>quàm quærere aliquem numerum ita proportionatum ad ipſum numerum datum, <lb/>vt ſe habet numerator ad denominatorem, exempli gratia ſi .24. aliquis voluerit mul <lb/>tiplicare per duo tertia hoc idem eſt vt ſi quæreret numerum ad quem .24. ita ſe <lb/>habeat, vt .3. ad .2. & idem dico de proportionibus, hoc eſt quod aliud non eſt mulri-<lb/>plicare aliquam proportionem per fractos, quàm aliam proportionem quærere ad <lb/><choice><ex>quam</ex><am>quã</am></choice> data ſe habeat, vt denominator ſe <choice><ex>hent</ex><am>hẽt</am></choice> ad <choice><ex>numeratorem</ex><am>numeratorẽ</am></choice>; </s> <s xml:space="preserve">& hoc exregula de tribus <lb/>perficitur, <choice><ex>conſtituendo</ex><am>cõſtituẽdo</am></choice> <choice><ex>denominatorem</ex><am>denominatorẽ</am></choice> in primo loco, quilocus eſt diuiſoris, numerato <lb/><choice><ex>rem</ex><am>rẽ</am></choice> verò in <choice><ex>ſecundo</ex><am>ſecũdo</am></choice> loco, <choice><ex>multiplicando</ex><am>multiplicãdo</am></choice> poſteà pro <lb/>portionem per <choice><ex>numeratorem</ex><am>numeratorẽ</am></choice>, & <choice><ex>productum</ex><am>productũ</am></choice> <choice><ex>diuidem</ex><am>diuidẽ</am></choice> <lb/> <ptr xml:id="fig-0113-01a" corresp="fig-0113-01" type="figureAnchor"/> do per denominatorem, prouentus demum erit <lb/>proportio, ad quam data ſe habebit, vt denomi-<lb/>nator ſe <choice><ex>hent</ex><am>hẽt</am></choice> ad numeratorem ex ratione ipſius re <lb/>gulę de tribus. </s> <s xml:space="preserve">Ratio verò methodi <choice><ex>diuidendi</ex><am>diuidẽdi</am></choice> <choice><ex>vnam</ex><am>vnã</am></choice> <lb/>datam <choice><ex>proportionem</ex><am>proportionẽ</am></choice> per fractos, ex ſe ſatis patet, <lb/>cum idem ſit modus diuidendi quemhbet nume <lb/>rum integrum per fractos. </s> <s xml:space="preserve">Quare, quæ vnius, <lb/>& alterius eſt ratio.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0113-01" corresp="fig-0113-01a"> <graphic url="0113-01"/> </figure> </div> </body> </floatingText> </div> <div type="math:theorem"> <head xml:space="preserve">THEOREMA <num value="153">CLIII</num>.</head> <p> <s xml:space="preserve">NIcolaus Tartalea in .3. lib. quintæ partis numerorum ſoluit .24. quæſitum ſi-<lb/>bi propoſitum à Hieronymo Cardano, via particulari & non generali. </s> <s xml:space="preserve">Quæ-<lb/>ſitum autem tale eſt quamlibet propoſitam rectam lineam in duas partes ita diuide <lb/>re via Euclidis, ut cubus totius lineæ ad cubos partium ſe habeat in proportione <lb/>tripla.</s> </p> <p> <s xml:space="preserve">Tartalea igitur inquit quòd vt ſatisfiat ſpeculatiuis ingenijs ſoluendum ſit huiuſ-<lb/>modi quæſitum, ſecando lineam propoſitam <seg type="var">.a.b.</seg> in tres æquales partes, quarum vna <lb/>fit <seg type="var">.c.b.</seg> vnde problema ſolutum erit.</s> </p> <p> <s xml:space="preserve">Verum dicit, ſed hæc non eſt methodus generalis, proptereà, quod cum tale <lb/>problema alterius fuiſlet proportionis quam triplæ, talis methodus nihil valeret.</s> <pb facs="0114" n="102"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve">Quapropter non tacebo quod mihi in mentem venit circa hoc problema.</s> </p> <p> <s xml:space="preserve">Sit ergo linea <seg type="var">.a.b.</seg> diuiſibilis in puncto <seg type="var">.c.</seg> ita vt cubum totius dictæ <seg type="var">.a.b.</seg> lineæ ad <lb/>ſummam cuborum <choice><ex>ſuarum</ex><am>ſuarũ</am></choice> partium <seg type="var">.a.c.</seg> et <seg type="var">.c.b.</seg> oporteat eam proportionem <choice><sic>habére</sic><corr>habere</corr></choice>, <lb/>exempli gratia, vt .125. ad .65. vt vitemus fracta pro nunc, notantes talem propor-<lb/>tionem quadrupla nunquam maiorem eſſe poſſe, vt quilibet ex ſe contemplari po-<lb/>teſt, conſtituendo punctum <seg type="var">.c.</seg> in medio loco inter <seg type="var">.a.</seg> et <seg type="var">.b.</seg> vnde proportio totalis <lb/>cubi ad ſummam partialium eſſet omnium maxima quæ poſſint eſſe, collocando <seg type="var">.c.</seg> <lb/>vbi volueris in dicta linea <seg type="var">.a.b.</seg> & hæc eſſet quadrupla.</s> </p> <p> <s xml:space="preserve">Sed vt ad propoſitum reuertamur, conſiderabimus cubum totalem ipſius <seg type="var">.a.b.</seg> <lb/>eſſe vt .125. & ſummam partialium vt .65. quam detrahemus ex cubo totali & nobis <lb/>remanebit .60. pro ſumma trium ſolidorum inuicem æqualium, quorum longitu-<lb/>do vniuſcuiuſque erit tota linea <seg type="var">.a.b.</seg> nobis cognita vt radix dati cubi totalis, quæ erit <lb/>in hoc exemplo quinque partium, latitudo verò vniuſcuiuſque dictorum <choice><ex>ſolidorum</ex><am>ſolidorũ</am></choice> <lb/>erit <seg type="var">.a.c.</seg> pars maior ipſius <seg type="var">.a.b.</seg> quæ quidem <seg type="var">.a.c.</seg> adhuc nobis ignota eſt, profunditas <lb/>ſeu altitudo vniuſcuiuſque illorum ſolidorum, erit <seg type="var">.c.b.</seg> pars reliqua ipſius <seg type="var">.a.b.</seg> & <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>nobis incognita, ſed quia ſumma horum trium ſolidorum nobis manifeſta ſuperius <lb/>fuit, quæ erat .60. propterà nobis cognita erit quantitas vniuſcuiuſque illorum ſoli-<lb/>dorum, vt tertia pars totius ſummæ ipſorum quæ erit .20. in propoſito <choice><ex>exemplo</ex><am>exẽplo</am></choice>, dein <lb/>de cum vnumquodque illorum ſolidorum producatur à ſuperficie contenta ſeu pro<lb/> ducta ab <seg type="var">.c.a.</seg> in <seg type="var">.c.b.</seg> in tota linea <seg type="var">.a.b.</seg> ſequitur quòd ſi diuiſerimus hoc ſolidum .20. <lb/>per lineam <seg type="var">.a.b.</seg> quinque partium proueniet nobis cognita ſuperficies producta ab <seg type="var">.<lb/>a.c.</seg> in <seg type="var">.c.b.</seg> quatuor partium, ſed cum quadratum totius <seg type="var">.a.b.</seg> nobis cognitum ſit, eo <lb/>quod <seg type="var">.a.b.</seg> vt eius latus etiam cognitum eſt. </s> <s xml:space="preserve">Tunc dictum quadratum erit .25. quod <lb/>quidem æquale eſt quadruplo illius quod fit ex <seg type="var">.a.c.</seg> in <seg type="var">.c.b.</seg> ſimul cum quadrato diffe <lb/>rentiæ inter <seg type="var">.a.c.</seg> et <seg type="var">.c.b.</seg> per .8. ſecundi Eucli. </s> <s xml:space="preserve">Vnde quia quadruplum illius quod fit <lb/>ex <seg type="var">.a.c.</seg> in <seg type="var">.c.b.</seg> nobis cognitum eſt, vt <lb/>16. eo quod ſimplum quod eſt .4. <choice><ex>iam</ex><am>iã</am></choice> <lb/> <ptr xml:id="fig-0114-01a" corresp="fig-0114-01" type="figureAnchor"/> inuentum fuit, ideo ſi hoc quadru-<lb/>plum .16. demptum fuerit ex totali <lb/>quadrato .25. reliquum erit .9. qua <lb/><choice><ex>dratum</ex><am>dratũ</am></choice> ſcilicet vnius partis <seg type="var">.a.c.</seg> ipſius <lb/>hoc eſt illius partis, quæ differentia <lb/>eſt inter <seg type="var">a.c.</seg> et <seg type="var">.c.b.</seg> quæ quidem erit <num value="3">.<lb/>3.</num> partium quæ differentia cum ſub-<lb/>tracta fuerit ex <seg type="var">.a.b.</seg> reliquum erit du <lb/>plum ipſius <seg type="var">.c.b.</seg> duo ſcilicet. </s> <s xml:space="preserve">Quare <seg type="var">.<lb/>c.b.</seg> erit vt <seg type="var">.I.</seg> et <seg type="var">.a.c.</seg> vt .4. & productum <seg type="var">.a.c.</seg> in <seg type="var">.c.b.</seg> erit .4. vnitatum ſuperficialium. <lb/></s> <s xml:space="preserve">& c.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0114-01" corresp="fig-0114-01a"> <graphic url="0114-01"/> </figure> </div> </body> </floatingText> <pb facs="0115" n="103"/> </div> <div type="appendix"> <head xml:space="preserve">APPENDIX</head> <head xml:space="preserve">DE SPECVLATIONE <lb/>REGVLAE FALSI.</head> <p> <s xml:space="preserve"><hi rend="small caps">Nvnc</hi> idem ferè mihi accidit, quod & Michaeli Stifelio, à quo <lb/>cum Petreius Tipographus nuper totam ſuam Arithmeticam re <lb/>cepiſſet, mox poſteà per literas petijt <choice><ex>explicationem</ex><am>explicationẽ</am></choice> regulæ falſi.</s> </p> <p> <s xml:space="preserve">Similiter poſt inciſas omnes ſuperiorum Theorematum figu-<lb/>ras, <choice><ex>opereque</ex><am>opereq́;</am></choice> Typographo commiſſo, amicus quidam omnium <lb/>ſcientiarum ornatiſſimus maxima neceſſitudine mecum coniun-<lb/>ctus monuit me, vt aliquid de regula falſi ſcribere vellem, cuius <lb/>ſuaſu hæc, quæ ſequuntur appendicis vice ponere libuit, nelector, quidpiam quod <lb/>ad hancrem pertinet iure merito à nobis deſiderare poſſet; </s> <s xml:space="preserve">vt autem ad ipſam <choice><ex>re- gulam</ex><am>re-gulã</am></choice> accedamus Ego ſicut, & in alijs multis, ita & in huiuſcæ regulę inuentione cum <lb/>ipſo Stifelio maximè conuenio, putans regulam falſi, ſeu falſarum poſitionum in-<lb/>uentam fuiſſe per paruos numeros in quæſtionibus facillimis & cognitis, eodem fer <lb/>mè modo, quo ipſe monſtrat illis duobus exemplis, quæ quamuis ipſe appellet theo <lb/>remata, nihilominus the oremata ego illa non vocarem, niſi adiuncta fuerit ſpecu-<lb/>latio ab ipſo præterita, & non experientia tantummodo, vt ipſe fecit. </s> <s xml:space="preserve">Primum eius <lb/>exemplum eſt, quòd.</s> </p> <p> <s xml:space="preserve">Quorumcumque <choice><ex>duorum</ex><am>duorũ</am></choice> numerorum differentia, ſi fuerit multiplicata in aggre <lb/>gatum eorum, producit ipſam differentiam, quæ eſt inter quadrata eorum.</s> </p> <p> <s xml:space="preserve">Secundum verò exemplum eſt, quod.</s> </p> <p> <s xml:space="preserve">Datis tribus numeris ſecundum progreſſionem arithmeticam diſpoſitis, facit mul <lb/>tiplicatio medij in ſe, <choice><ex>quantum</ex><am>quãtum</am></choice> multiplicatio extremorum inter ſe cum multiplicatio <lb/>ne differentiarum inter ſe.</s> </p> <p> <s xml:space="preserve">Talia enim exempla ipſe aliter non probat niſi experientia in aliquibus numeris, <lb/>arbitratus ex eo inuentam eſſe regulam falſi, experientia tantummodo confirma-<lb/>tam, quod quidem etiam & ego credo. </s> <s xml:space="preserve">At experientia in philoſophia mathema-<lb/>tica, aut <choice><ex>nullam</ex><am>nullã</am></choice> prorſus facit <choice><ex>ſcientiam</ex><am>ſcientiã</am></choice>, aut omnino ſuperfluus fuit Euclides in multis <lb/>ſuis propoſitionibus, & præcipuè in eius ſecundo libro, ſi ſufficeret experientia. </s> <s xml:space="preserve">Id-<lb/>circo quo magis ad euidentiam ipſius veritatis, quam profiteor, deuenire poſſim, <lb/><choice><ex>accipiam</ex><am>accipiã</am></choice> primò primum exemplum <lb/>ipſius Stifelij hic ſuperius citatum, <lb/>& pro numero maiori, in prima hic <lb/> <ptr xml:id="fig-0115-01a" corresp="fig-0115-01" type="figureAnchor"/> ſubſcripta figura .AE. accipio <seg type="var">.a.i.</seg> <lb/>cuius quadratum ſit <seg type="var">.a.c</seg>: pro minori <lb/>vero numero capio <seg type="var">.a.e.</seg> <choice><ex>partem</ex><am>partẽ</am></choice> ipſius <lb/><seg type="var">a.i.</seg> cuius quadratum fit <seg type="var">.a.t.</seg> differen <lb/>tia autem horum numerorum erit <seg type="var">.<lb/>e.i.</seg> reliqua pars ipſius <seg type="var">.a.i</seg>: & differen <lb/>tia ipſorum quadratorum erit gno-<lb/>mon <seg type="var">.e.c.o</seg>: Nunc autem protraho <seg type="var">.<lb/>i.c.</seg> latus quadrati maioris quouſque <lb/><seg type="var">c.n.</seg> æqualis ſit <seg type="var">.a.e.</seg> numero minori, <lb/><choice><ex>perficioque</ex><am>perficioq́;</am></choice> rectangulum <seg type="var">.e.n.</seg> quod <pb facs="0116" n="104"/><fw type="head">IO. BAPT. BENED.</fw> producitur ex <seg type="var">.i.e.</seg> differentia in <seg type="var">.i.n.</seg> aggregatum amborum numerorum, ſed hoc pro <lb/>ductum excedit productum <seg type="var">e.c</seg>: partem gnomonis dicti per <seg type="var">.u.n.</seg> quod quidem <seg type="var">.u.<lb/>n.</seg> æquatur ipſi <seg type="var">.u.o.</seg> reliquæ ſcilicet parti ipſius gnomonis, <choice><ex>nam</ex><am>nã</am></choice> <seg type="var">.e.u.</seg> æqualis eft <seg type="var">.i.c.</seg> qua <lb/>re et <seg type="var">.a.i.</seg> ſed <seg type="var">.e.t.</seg> ęquatur <seg type="var">.e.a.</seg> vnde <seg type="var">.t.u.</seg> æqualis erit <seg type="var">.e.i.</seg> </s> <s xml:space="preserve">quare et <seg type="var">.u.c</seg>: at cum <seg type="var">.c.n.</seg> æqua <lb/>lis ſit ipſi <seg type="var">.a.e.</seg> erit etiam æqualis ipſi <seg type="var">.<lb/>o.t</seg>. </s> <s xml:space="preserve">quare <seg type="var">.u.n.</seg> æqualis erit ipſi <seg type="var">.u.o.</seg> <lb/>& tunc intellectus quieſcit, & <choice><ex>abſque</ex><am>abſq;</am></choice> <lb/> <ptr xml:id="fig-0116-01a" corresp="fig-0116-01" type="figureAnchor"/> aliqua alia experientia verè ſcientifi <lb/><choice><ex>ceque</ex><am>ceq́;</am></choice> dicere poteft, quòd.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0115-01" corresp="fig-0115-01a"> <graphic url="0115-01"/> </figure> <figure xml:id="fig-0116-01" corresp="fig-0116-01a"> <graphic url="0116-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quorumcumque duorum nume-<lb/>rorum differentia, fi fuerit multipli-<lb/>cata in aggregatum eorum, producit <lb/>ipſam <choice><ex>differentiam</ex><am>differentiã</am></choice>, quæ eftinter qua-<lb/>drata eorum.</s> </p> <p> <s xml:space="preserve">Hæcautem propoſitio à me ipſo <lb/>etiam in .60. </s> <s xml:space="preserve">Theoremate huius libri <lb/>aliter demonftrata fuit.</s> </p> <p> <s xml:space="preserve">DE ſpeculatione autem, etſcientia ſecundi exempli, in ſecunda hic ſubſcripta <lb/>figura <seg type="var">.ω.</seg> cogitemus lineam <seg type="var">.u.a.</seg> tribusin partibus arithmeticè diuiſam, qua <lb/>rum maxima ſit <seg type="var">.u.o.</seg> media. ſit <seg type="var">.o.e.</seg> minima verò ſit <seg type="var">.e.a.</seg> multiplicatio autem mediæ <seg type="var">.<lb/>o.e.</seg> in ſe ſit quadratum <seg type="var">.o.t.</seg> abſcindatur deinde ex <seg type="var">.o.e</seg>: <seg type="var">e.i.</seg> æqualis <seg type="var">.e.a.</seg> </s> <s xml:space="preserve">tunc <seg type="var">.o.i.</seg> erit <lb/>differentia inter <seg type="var">.o.e.</seg> et <seg type="var">.e.a.</seg> & æqualis differentiæ inter <seg type="var">.o.e.</seg> et <seg type="var">.o.u.</seg> ex hypotefi, quæ <lb/>quidem <seg type="var">.o.i.</seg> in ſe ducta procreabit quadratum <seg type="var">.o.c.</seg> quod erit productum ex differen <lb/>tijs ipſarum partium, & erit pars quadrati <seg type="var">.o.t.</seg> ſuperius dicti, vt exſe patet. </s> <s xml:space="preserve">Nunc <lb/>autem dico gnomonem <seg type="var">.i.t.n.</seg> æqualem eſſe ei quod fit ex <seg type="var">.a.e.</seg> in <seg type="var">.o.u</seg>. </s> <s xml:space="preserve">Producatur igi <lb/>tur <seg type="var">.e.t.</seg> quouſque <seg type="var">.t.r.</seg> æqualis ſit ipſi <seg type="var">.o.i</seg>. </s> <s xml:space="preserve">tunc <seg type="var">.e.r.</seg> erit æqualis <seg type="var">.o.u.</seg> quod etiam clarum <lb/>eſt. </s> <s xml:space="preserve">Claudatur ergo rectangulum <seg type="var">.i.r.</seg> quod erit æquale producto ipſius <seg type="var">.e.a.</seg> in <seg type="var">.o.u.</seg> <lb/>Nam <seg type="var">.e.i.</seg> ſumpta fuit <lb/>æqualis <seg type="var">.e.a.</seg> ſed ex ra <lb/> <ptr xml:id="fig-0116-02a" corresp="fig-0116-02" type="figureAnchor"/> tionibus in priori <choice><ex>exem</ex><am>exẽ</am></choice> <lb/>plo allatis, <choice><ex>productum</ex><am>ꝓductum</am></choice> <seg type="var">.<lb/>i.r.</seg> æquale erit gno-<lb/>moni <seg type="var">.i.t.n</seg>. </s> <s xml:space="preserve">Nuncau <lb/>tem verè, ſcientifice-<lb/>q́ue poſſumus affirma <lb/>re, quòd. </s> <s xml:space="preserve">Datis tribus <lb/>numeris <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> pro <lb/>greffionem arithme-<lb/>ticam diſpofitis, fa-<lb/>cit multiplicatio me-<lb/>dij in ſe quantum mul <lb/>tiplicatio extremorum inter ſe, cum multiplicatione differentiarum inter ſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0116-02" corresp="fig-0116-02a"> <graphic url="0116-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Et ſic de alijs huiuſmodi inuentionibus infero.</s> </p> <p> <s xml:space="preserve">DIcturus igitur aliquid circa <choice><ex>regulam</ex><am>regulã</am></choice> falſi, videtur mihi nullam oportere facere <lb/>mentionem de origine huiuſcæ regulæ, cum in hoc Stifelius ſatisfecerit, ſed <pb facs="0117" n="105"/><fw type="head">THEOREM. ARITH.</fw> potius veras rationes <choice><ex>propriaque</ex><am>propriaq́;</am></choice> fundamenta huiuſmodi operationis oftendere, fu-<lb/>mendo eadem exempla propoſita abipſis practicis, & maximè à Nicolao Tartalea <lb/>viro accuratiffimo, qui vbicunque potuit ſpeculatus eſt cauiſas <choice><ex>ipſarum</ex><am>ipſarũ</am></choice> operationum, <lb/>etſi de huiuſmodi falſi regula circa finem cap .8. lib. 17. promittat poſtea loqui, nub-<lb/>libi tamen loquutus eft. </s> <s xml:space="preserve">Monendum etiam cenſeo, me nihil de rationibus regulæ <lb/>falſi ſimplicis dicturum, cum ex ſeipſis ſatis appareant, quod non ita eſt de poſitio-<lb/>nibus duplis. </s> <s xml:space="preserve">Incipiam ergo à primo problemate lib. 17. ipſius Tartaleæ, quo <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>ipſe vtitur pro exemplo docendi gratia, ipſam regulam duplæ poſitionis, quod qui <lb/>dem problema aliter à me <choice><ex>ſolutum</ex><am>ſolutũ</am></choice> fuit in .118. </s> <s xml:space="preserve">Theoremate huius mei lib. quod ſimi <lb/>liter ob hanc demum occaſionem mihi oblatam, alia etiam via, ſpeculatus ſumidem <lb/>poſſe fieri, quæ quidem via ſeu methodus generalis erit, & ita ſe habet.</s> </p> <p> <s xml:space="preserve">Accipio enim propoſitum numerum diuiſibilem, à quo detraho ſummam <lb/>datorum numerorum, primo duplicato, eo quòd tam in ſecunda quam in <lb/>tertia parte reperitur, vt in propofito exemplo, datus numerus eft, 50. à <lb/>quo detraho ſummam dictorum numerorum, quæ eſt .11. nam tres, & tres, & <lb/>quinque ſunt vndecim, eo quòd primus ingreditur in ſecunda, & in tertia parte, <lb/>dempto igitur hoc numero .11. ex .50. remanet .39. qui quidem numerus intelligen-<lb/>dus eſt pro ſumma trium partium ſimplicium adhuc incognitarum, à quo extrahen <lb/>da eſt prima, eo modo quo nunc proponam exregula de tribus, hoc eſt aggregan <lb/>do dictas partes ſimplices ſine aliqua additione vtcunque volueris (ſed commodius <lb/>erit in minimis numeris) iuxta propoſitum, quod quidem propoſitum eſt, vt ſecun <lb/>da pars dupla ſit primæ, tertia verò æqualis fit primæ & ſecundæ, quæ partes in di-<lb/>ctis minimis numeris, ita diſpoſitæ erunt .1. 2. 3. quarum ſumma erit .6. </s> <s xml:space="preserve">Nunc ſi ex <lb/>regula de tribus dixerimus, cum hæc ſumma proueniat nobis ab vno, à quo proue-<lb/>niet .39. et veniet nobis .6. cum dimidio pro prima parte quæfita in propoſito nume-<lb/>ro .39. cum ergo habuerimus primam <choice><ex>partem</ex><am>partẽ</am></choice>, reliquas poſteà illicò cognoſcemus.</s> </p> <p> <s xml:space="preserve">Huiuſmodi verò operationis ratio ex ſe manifeſta patet, eo quòd proportio ſum <lb/>mæ partium in minimis numeris ad primam eorum partem eadem eſſe debet, quæ <lb/>ipſius .39. ad primam partem quæſitam huiuſmodi aggregati partium <choice><ex>ſimplicium</ex><am>ſimpliciũ</am></choice>, ſed <lb/>quia nemo adhuc, quod ſciam, ſatis animaduertit rationem modorum, qui ab anti-<lb/>quis obſeruati ſunt, qui quidem modi duo ſunt circa hoc Helcataym duplæ falſæ <lb/>pofitionis, igitur non prætermittam aliquid de hacreſpeculari, & primo de pri-<lb/>mo modo.</s> </p> <p> <s xml:space="preserve">In primis igitur <choice><ex>ſciendum</ex><am>ſciendũ</am></choice> eft, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/> <ptr xml:id="fig-0117-01a" corresp="fig-0117-01" type="figureAnchor"/> veritas ita inueniri poterit eo-<lb/>rum modo, me diantibus ſimpli<lb/>cibus partibus, vt <choice><ex>etiam</ex><am>etiã</am></choice> <choice><ex>median- tibus</ex><am>median-tibꝰ</am></choice> <choice><ex>compoſitis</ex><am>cõpoſitis</am></choice>, ut in pręſenti <choice><ex>exem</ex><am>exẽ</am></choice> <lb/>plo pro primis pofitionibus ac-<lb/>ceperunt .10. et .8. pro ſecundis <lb/>verò compoſitis <choice><ex>cum</ex><am>cũ</am></choice> numero .3. <lb/><choice><ex>inuenerunt</ex><am>inuenerũt</am></choice> .23. et .19. pro tertijs <lb/><choice><ex>autem</ex><am>aũt</am></choice> <choice><ex>compoſitis</ex><am>cõpoſitis</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> <choice><ex>quinque</ex><am>quinq;</am></choice>, notaue <lb/>runt .38. et .32. vnde prima ſum <lb/>marefultauit .71. ſecunda verò <lb/>59. ita <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>primus</ex><am>primꝰ</am></choice> error remanebat <lb/>21. <choice><ex>ſecundus</ex><am>ſecũdꝰ</am></choice> <choice><ex>autem</ex><am>aũt</am></choice> .9. vt in figura <seg type="var">.A</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0117-01" corresp="fig-0117-01a"> <graphic url="0117-01"/> <head xml:space="preserve">Compositorum</head> </figure> </div> </body> </floatingText> <pb facs="0118" n="106"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">SEDijdem errores proueniunt exſummis partium ſimplicium.</s> </p> <p> <s xml:space="preserve">Vtexempli gratia, in figura <seg type="var">.B.</seg> ſumma propoſita partium ſimplicium eſt .39. <lb/>vt diximus, eo quòd ab ipſo .50. detraxerimus .11. ſumma ſcilicet numerorum adij <lb/>ciendorum ad efficiendas partes compofitas, ſumma poſteà fimplicium partium <lb/>primæ poſitionis, erit .60. eo quòd prima pars erat .10. ſecunda autem ſimplex 20. <lb/>tertia verò fimplex .30. iuxta ordinem propoſiti. </s> <s xml:space="preserve">Summa deinde ſimplicium <choice><ex>partium</ex><am>partiũ</am></choice> <lb/>fecundæ poſitionis effet .48. quia prima eius pars erat .8. ſecunda verò ſimplex .16. <lb/>tertia autem ſimplex .24. vnde prima ſumma excederet datam .39. per .21. differen-<lb/>tiæ, ſecunda verò per .9. vt ſupra vidimus de ſummis compoſitis à dato .50. compo-<lb/>fito, & hoc quidem mirandum non eft, quod ſcilicet tres ſummæ fimplicium par-<lb/>tium ſintinuicem inæqua-<lb/>les, ijſdem differentijs me-<lb/> <ptr xml:id="fig-0118-01a" corresp="fig-0118-01" type="figureAnchor"/> diantibus, quibus <choice><ex>differunt</ex><am>differũt</am></choice> <lb/>dictæ tres ſummæ compofi <lb/>tæ, cum ab vnaquaque <choice><ex>con</ex><am>cõ</am></choice> <lb/><choice><ex>poſitarum</ex><am>poſitarũ</am></choice> ablatus fit nume-<lb/>rus .11. æqualiter, vnde ex <lb/>neceſſitate, permutando, <lb/><choice><ex>earum</ex><am>earũ</am></choice> differentiæ <choice><ex>relinquem</ex><am>relinquẽ</am></choice> <lb/>dæ erant æquales inuicem <lb/>ex <ref>.78. theoremate hu-<lb/>ius noſtri lib.</ref> ſummæ enim <lb/>compofitæ erant .71. 59. et <lb/>50. fimplices verò .60. 48. <lb/>et .39. differentes à primis <lb/>per .11. vt dictum eft, qua <lb/>re veritas ita manabit à compofitis, quemadmodum à fimplicibus, ſed à fimplici-<lb/>bus per ſe, & a compofitis per accidens vtiam iam videbimus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0118-01" corresp="fig-0118-01a"> <graphic url="0118-01"/> <head xml:space="preserve">Simpricium</head> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">ANtiquorumigitur primus m odus vtitur regula detribus, hocordine, multi-<lb/>plicando ſcilicet ſecundum errorem, qui eft .9. cum differentia primarum par <lb/>tium pofitarum, quæ eft .2. & productum diuidendo per differentiam errorum, quæ <lb/>eft .12. proueniens poftea quod eft .1. cum dimidio additur hoc loco primæ parti ſe-<lb/>cundæ poſitionis. <choice><ex>&c.</ex><am>&c.</am></choice> quòd benè ſe habet. </s> <s xml:space="preserve">Vbi animaduertendum eſt, quod ille <lb/>numerus .12. non eft accipiendus per ſe vt differentia errorum hoc eft .21. et .9. nifi <lb/>peràccidens, fed benè perfe, vt <choice><ex>differentia</ex><am>differẽtia</am></choice> inter .60. er .48. ſimplices ſummas, quem <lb/>admodum .9. in hoc propoſito eft differentia per ſe inter .48. et .39 per accidens ve-<lb/>ro inter .59. et .50.</s> </p> <p> <s xml:space="preserve">Cognoſcendum igitur eft mediante .24. quinti Eucli. quod eadem proportio <lb/>eft primæ ſummæ (ſimplicium dico) ad ſuam primam partem, quæ ſecundæ ſum-<lb/>mæ ad ſuam, & tertiæ ſummæ ad fuam fimiliter (vbi rectè etiam feciffent hoc in lo-<lb/>co antiqui ſi multiplicauiffent tertiam ſummam fim plicem cum prima parte prioris <lb/>fummæ fimplicis, & productum diuififfent per primam ſummam, vnde prima pars <lb/>quæſita tertiæ ſummæ orta fuiffet, abſque ullo negotio ipfius plus velminus) </s> <s xml:space="preserve">Quare <lb/>habebimus tres terminos antecedentes ab vna parte, & tres terminos conſequen-<lb/>tesab alia parte continentes vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> proportionem, vnde ex .19. quinti, <lb/>vel .12. ſeptimi eorum differentiæ proportionales erunt, hoc eft, <choice><ex>quod</ex><am>ꝙ</am></choice> eadem propor <pb facs="0119" n="107"/><fw type="head">THEOREM. ARIT.</fw> tio erit eius differentiæ, quæ eſt inter primam & fecundam ſummam, ad differen-<lb/>tiam quæ eſt inter primas earum partes, quæ illius differentiæ, quæ eſt inter ſecun-<lb/>dam & tertiam ſummam, ad differentiam, quæ eft inter primas illarum partes, ſed <lb/>harum .4. differentiarum, tres nobis cognitæ ſunt, ideft .12. 2. et .9. ergo ex regula de <lb/>tribus ab Eucli. in .20. ſeptì<unclear reason="illegible"/>mi ſpeculata inueniebatur quarta differentia, quæ eft .1. <lb/>cum dimidio.</s> </p> <p> <s xml:space="preserve">A compofitis ſummis idem etiam proueniet, ſed non vt ex proprijs caufis, & per <lb/>ſe, ſedper accidens. </s> <s xml:space="preserve">Nam quamuis eadem differentia fit inter 71. et .59. quæ in-<lb/>ter .60. et .48. & <choice><ex>eadem</ex><am>eadẽ</am></choice> inter .59. et .50. quæ inter .48. et .39. </s> <s xml:space="preserve">Nihilominus non eft <choice><ex>eadem</ex><am>eadẽ</am></choice> <lb/>proportio (propriè) ipſius .71. ad .59. quæ ipſius .60. ad .48. nec ea quæ ipſius .59. ad <num value="50">.<lb/>50.</num> eft quæ ipſius .48. ad .39: </s> <s xml:space="preserve">Vnde non erit eadem proportio ipſius .71. ad .59. quæ <lb/>ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipſius .8. ad .6. cum dimidio. </s> <s xml:space="preserve">Sed <lb/>minores illis. </s> <s xml:space="preserve">Nam ex æqualibus additamentis diminuuntur proportiones maio-<lb/>ris inęqualitatis.</s> </p> <p> <s xml:space="preserve">A fimplicibus igitur ſummis pendet ratio huiuſmodi effectus.</s> </p> <p> <s xml:space="preserve">Si vero prima pars fecundæ poſitionis effet .4. tunc ſecunda eius pars effet .8. & ter-<lb/>tia .12. quarum ſumma effet .24. (harum fimplicium partium ſeilicet) & minor vera <lb/>(39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum <lb/>partium effet .6. differentia vero primæpartis ſecundæ poſitionis, a prima parte quę <lb/>fita effet .2. cum dimidio. </s> <s xml:space="preserve">Vnde in huiuſmodi exemplo videre eft quare colligan-<lb/>tur errores inuicem, quando alter eorum eccedit, reliquus vero deficit à numero pro <lb/>pofito. </s> <s xml:space="preserve">Quod quidem ob aliam caufam non fit, nifi vt cognoſcatur differentia .36. <lb/>differentia ſcilicet ſimplicium ſummarum ipſarum poſitionum.</s> </p> <p> <s xml:space="preserve">Secundus autem modus ab antiquis magis exercitatus eſt, quod multiplicabant <lb/>diametraliter errores cum primis partibus, hoc eſt primum errorem cum prima par <lb/>te, hoc eſt cum numero ſecundæ poſitionis, ſecundum vero errorem cum prima <lb/>parte, hoc eſt cum numero primæ poſitionis, differentiam poſteà vel aggregatum <lb/>horum duorum productorum diuidebant per differentiam vel aggregatum dicto-<lb/>rum errorum, proueniens poſteà erat prima pars quæſita numeri propoſiti. </s> <s xml:space="preserve">Vn-<lb/>de oriebantur tria producta, quorum <choice><ex>tertium</ex><am>tertiũ</am></choice>, hoc eſt differentia, ſeu aggregatum il-<lb/>lorum conſtituebatur ex differentia feuaggregato errorum, & ex numero quæ-<lb/>fito.</s> </p> <p> <s xml:space="preserve">Vtin præfenti exemplo, primus error eſt .21. qui multiplicatus cum prima par-<lb/>te ſecundæ poſitionis, quæ eſt .8. producit .168. <choice><ex>ſecundus</ex><am>ſecũdus</am></choice> verò error eſt .9. qui multi-<lb/>plicatus cum prima parte primę poſitionis producit .90. differentia autem horum <lb/>productorum eſt .78. quæ diuifa per differentiam errorum, quæ eſt 12. dabit .6. <choice><ex>cum</ex><am>cũ</am></choice> di <lb/>midio, pro prima parte quæſita dati numeri diuiſibilis, qui erat .50.</s> </p> <p> <s xml:space="preserve">Hæc omnia rectè ſe habent. </s> <s xml:space="preserve">Sed, vt ſupra dixi diuiſor non eft per ſe differentia <lb/>errorum, neque etiam differentia per ſe ſummarum compoſitarum, fed bene fim-<lb/>plicium.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ſpeculatione, accipiendæ ſunt ſummæ ſimplices, quarum differen-<lb/>tiæ per ſe vtiles ſunt in huiuſmodi operatione; </s> <s xml:space="preserve">& quia etiam rationes veritatis ex <lb/>iſtis, & non ex illis fluunt; </s> <s xml:space="preserve">quamuis tam vnæ, quam aliæ ſint eædem in quantitate, <lb/>ideſt æquales.</s> </p> <pb facs="0120" n="108"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Diſponantur igitur huiuſmo-<lb/> <ptr xml:id="fig-0120-01a" corresp="fig-0120-01" type="figureAnchor"/> di numeri tali ordine, vt fim-<lb/>plex ſumma, quæ ab vna reli-<lb/>quarum ſuperatur, & aliam ſupe-<lb/>rat, medium locum teneat; </s> <s xml:space="preserve">@t <lb/>in propoſito exemplo ſumma <lb/>mediocris eft .48. quę à ſumma <num value="60">.<lb/>60.</num> ſuperatur, & ſuperat ſum-<lb/>mam .39. locata igitur fit hęc .48. <lb/>inter illas, ſuæ verò primæ partes <lb/>fimiliter conftitutæ ſint ſupra di-<lb/>ctas ſummas, cum ſuis <choice><ex>differentijs</ex><am>differẽtijs</am></choice>, <lb/>& tria producta iam dicta, vt in fi <lb/>guris <seg type="var">.C.</seg> et <seg type="var">.D.</seg> arithmeticis <lb/>clarè patet: </s> <s xml:space="preserve">figura enim <seg type="var">.C.</seg> eft <lb/>pro exemplo ipſius plus ſimpli-<lb/>citer: </s> <s xml:space="preserve">figura verò <seg type="var">.D.</seg> pro exem-<lb/>plo ipſius plus, & minus. </s> <s xml:space="preserve">Et fic <lb/> <ptr xml:id="fig-0120-02a" corresp="fig-0120-02" type="figureAnchor"/> in figura <seg type="var">.C.</seg> habebimus tres <lb/>numeros confequentes .60. 48. <lb/>39. & tres antecedentes .10. 8. <lb/>6. cum dimidio, vnam, & ean-<lb/>dem proportionem terminantes, <lb/>ex .24. quinti, vt diximus; </s> <s xml:space="preserve">qua-<lb/>re eorum differentiæ fimiliter <lb/>proportionales erunt, quod etiam <lb/>vidimus. </s> <s xml:space="preserve">Supponamus nunc nos <lb/>ignorare æqualitatem maximi <lb/>producti cum reliquis duobus, <lb/>accipiendo ſolum pro hypoteſi, <lb/>quòd dicta producta oriantur <lb/>ex lateribus iam dictis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0120-01" corresp="fig-0120-01a"> <graphic url="0120-01"/> </figure> <figure xml:id="fig-0120-02" corresp="fig-0120-02a"> <graphic url="0120-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Demonſtrandum nobis nunc relinquetur, maximum productum æquale effere-<lb/>liquis duobus; </s> <s xml:space="preserve">hoc eſt productum .168. æquale effe productis .90. et .78. quorum <lb/>duorum productorum alterum .90. ſcilicet, generatur à differentia .9. quæ eft ſe-<lb/>cundę, & tertię ſummæ, in primum numerum antecedentem, qui eſt .10. alterum vc-<lb/>ro productum .78. ſcilicet, generatur à differentia .12. quę eſt primę, & ſecundę, ſum <lb/>mę in tertium numerum antecedentem, qui eſt .6. cum dimidio, maximum vero <lb/>productum .168. ſcilicet generatur à differentia maxima .21. quę eft primę, & tertię <lb/>ſummę (& ſemper ęqualis prioribus duabus differentijs .12. et .9.) in ſecundum nu-<lb/>merum antecedentem, qui eſt .8.</s> </p> <p> <s xml:space="preserve">Conſtituantur igitur duo producta fimul iuncta ęqualia duobus .90. et .78. <lb/>lateralibus ſupra vnam aliquam rectam lineam <seg type="var">.q.p.</seg> <choice><ex>fitque</ex><am>fitq́;</am></choice> productum <seg type="var">.f.g.</seg> ęquale <num value="90">.<lb/>90.</num> productum verò <seg type="var">.g.n.</seg> ęquale .78. fit etiam baſis <seg type="var">.g.p.</seg> vt .9. et <seg type="var">.g.q.</seg> vt .12. vnde <seg type="var">.g.i.</seg> <lb/>vel <seg type="var">.q.n.</seg> erit vt .6. cum dimidio .et <seg type="var">.g.d.</seg> vel <seg type="var">.p.f.</seg> vt .10. & ideo <seg type="var">.i.d.</seg> differentia erit .3. <pb facs="0121" n="109"/><fw type="head">THEOREM. ARIT.</fw> cum dimidio, ut in figura <seg type="var">.C.</seg> geometrica hic ſubſcripta videre licet, et <seg type="var">.q.p.</seg> erit .21. <lb/>Cogitemus nunc differentiam <seg type="var">.d.i.</seg> diuiſam eſſe in puncto <seg type="var">.e.</seg> ita vt eadem proportio <lb/>ſit ipſius <seg type="var">.d.e.</seg> ad <seg type="var">.e.i.</seg> quæ ipſius <seg type="var">.q.g.</seg> ad <seg type="var">.g.p.</seg> hoc eſt vt .1 2. ad .9. quapropter <seg type="var">.d.e.</seg> erit <num value="2">.<lb/>2.</num> et <seg type="var">.e.i.</seg> erit .1. cum dimidio, vt in dicta figura <seg type="var">.C.</seg> arithmetica reperiuntur eſſe dif-<lb/>ferentiæ ipſorum antecedentium numerorum, deinde à puncto <seg type="var">.e.</seg> ducatur imagina-<lb/>tione <seg type="var">.u.e.o.</seg> æ quidiſtans ipſi <seg type="var">.q.p.</seg> & producatur <seg type="var">.q.n.</seg> vſque ad <seg type="var">.u.</seg> vnde ita ſe habebit <lb/><seg type="var">u.e.</seg> ad <seg type="var">.e.o.</seg> ut <seg type="var">.q.g.</seg> ad <seg type="var">g.p</seg>. </s> <s xml:space="preserve">quare vt <seg type="var">.d.e.</seg> ad <seg type="var">.e.i.</seg> ideo ex .15. ſexti vel .20. ſeptimi <seg type="var">.n.e.</seg> <lb/>rectangulum æquale crit ipſi <seg type="var">.e.f.</seg> qua propter rectang ulum <seg type="var">.q.o.</seg> æquale erit duobus<lb/> rectangulis <seg type="var">.f.g.</seg> et <seg type="var">.g.n</seg>: ſed cum <seg type="var">.g.i.</seg> ſit vt .6. cum dimidio, et <seg type="var">.i.e.</seg> vt .1. cum dimidio, er <lb/>go <seg type="var">.g.e.</seg> erit ut .8. qui quidem numerus multiplicatus cum <seg type="var">.q.p.</seg> 21. producit .168. ve <lb/>rum eſt igitur quod dictum fuit, hoc eſt <choice><ex>quod</ex><am>ꝙ</am></choice> maximum productum ęquale ſit reliquis <lb/>duobus.</s> </p> <figure place="here"> <graphic url="0121-01"/> </figure> <figure place="here"> <graphic url="0121-02"/> </figure> <pb facs="0122" n="110"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">DEmpto poſteà quo volueris horum altero productorum ex maximo, <choice><ex>diuiſoque</ex><am>diuiſoq́;</am></choice> <lb/>reliquo per differentiam conſequentium, ipſi diametraliter oppoſitam, pro <lb/>ueniet tibi numerus antecedens <choice><ex>correſpondensque</ex><am>correſpondensq́;</am></choice> illi.</s> </p> <p> <s xml:space="preserve">Animaduertendum tamen eſt, quòd ſi in figura à me ita ordinata, ſumma ſim-<lb/>plex propoſita medium locum occuparet, vt in figura <seg type="var">.D.</seg> arithmetica videri poteſt; <lb/></s> <s xml:space="preserve">tunc vt habeatur eius productum, addenda ſimul erunt circunſtantia producta .eo <lb/><choice><ex>quod</ex><am>ꝙ</am></choice> eius ſecundum latus eſſet antecedens medio loco conſtitutum, & prima pars <choice><ex>quae- ſita</ex><am>quę-ſita</am></choice> numeri propoſiti: </s> <s xml:space="preserve">in qua figura <seg type="var">.D.</seg> manifeſtè patet ratio, quare colligendi ſint <lb/>tam errores, quam producta, dum eorum alterum eſt plus, reliquum verò minus.</s> </p> <p> <s xml:space="preserve">Speculatio figurę <seg type="var">.D.</seg> arithmeticę videbitur in figura <seg type="var">.D.</seg> geometrica, eodem fe <lb/>rè modo quo fecimus in figuris <seg type="var">.C.</seg> mutatis mutandis, reſpectu ipſius plus, & minus.</s> </p> <p> <s xml:space="preserve">Collectio namque <choice><ex>errorum</ex><am>errorũ</am></choice> ſimiliter accidentalis eſt, eo quod eſſentialis numerus <lb/>diuiſor per ſe, eſt maxima differentia ſummarum ſimplicium, vt in dicta figura <seg type="var">.D.</seg> <lb/>cerni poteſt.</s> </p> <p> <s xml:space="preserve">Sed vt ſuperius dixi, nunc etiam repeto, quòd rectè hoc loco multiplicabatur <lb/>ſumma ſimplex propoſita, cum prima par <lb/>te primę poſitionis, vt productum diuide <lb/>retur per primam ſimplicem ſummam, <lb/> <ptr xml:id="fig-0122-01a" corresp="fig-0122-01" type="figureAnchor"/> vnde proueniret nobis pars prima <choice><ex>quaeſi- ta</ex><am>quęſi-ta</am></choice> noſtri numeri propoſiti, ex regula de <lb/>tribus, vnica poſitione.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0122-01" corresp="fig-0122-01a"> <graphic url="0122-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Vt exempli gratia, datus numerus diui <lb/>dendus ſit .100. in quinque partes, tales <lb/>verò, <choice><ex>quod</ex><am>ꝙ</am></choice> ſecunda duplo maior ſit prima <lb/>cum .2. ſimul, tertia autem æqualis ſit pri-<lb/>mæ & ſecundæ cum .3. vnitatibus iunctis, <lb/>quarta poſteà maior ſit prima ſecunda, & <lb/>tertia per .4. vnitates, quinta demum ſu-<lb/>peret reliquas omnes per quinque vnita <lb/>tes, vt in figura <seg type="var">.E.</seg> videre eſt, quæ quidem <lb/>partes compoſitæ (ſumpta vnitate pro <lb/>prima) ita diſpoſitæ erunt .1. 4. 8. 17. 35. <lb/>quarum ſumma erit .65. ſimplices autem <lb/>cum diſpoſitæ fuerint erunt .1. 2. 3. 6. 12. <lb/>quarum ſumma erit .24. dempta igitur <lb/>cum fuerit hæc ſimplex ſumma .24. à com <lb/>poſita .65. reſiduum erit .41. hoc eſt ſum-<lb/>ma numerorum propoſitorum cum ſuis <lb/>iterationibus in ipſis partibus, quod cum <lb/>per ſe clariſſimum ſit, ſuperſluum eſt <choice><ex>ipsam</ex><am>ipsã</am></choice> <lb/>ſummam annatomizare per ſingulas par-<lb/>tes, niſi quis habuerit eius cerebrum à fi-<lb/>gura Omega <choice><ex>terminatum</ex><am>terminatũ</am></choice>, cui tamen poſ-<lb/>ſemus dicere dictam ſummam .41. in .4. <lb/>partes diuidi, cuius prima eſſet .2. pro ad <lb/>ditione ad <choice><ex>ſecundam</ex><am>ſecũdam</am></choice> partem ſimplicium, <pb facs="0123" n="111"/><fw type="head">THEOREM. ARIT.</fw> ſecunda verò eſſet .5. pro additione ad tertiam partem ſimplicium, tertia autem eſ-<lb/>ſet .11. pro additione ad quartam partem ſimplicium, quarta demum eſſet .23. pro <lb/>additione quintæ partis ſimplicium, quarum partium .2. 5. 11. 23. ſumma eſt .41. vt <lb/>diximus. </s> <s xml:space="preserve">Hæc igitur ſumma .41. ſubducenda eſt à numero .100. propoſito, vnde re-<lb/>linquetur .59. pro ſumma partium ſimplicium numeri propoſiti, quarum prima erit <lb/>2. cum vndecim vigeſimisquartis ex diuiſione huiuſmodi .59. per .24. ſummam par-<lb/>tium ſimplicium ex viregulæ de tribus, dicendo ſi .24. prouenit nobis ab .1. prima <lb/>partium ſimplicium, à quo proueniet nobis .59? </s> <s xml:space="preserve">vnde proueniet à .2. cum vndecim <lb/>vigeſimisquartis pro prima parte quæſita, ſecunda verò iuxta propoſitum, erit .6. <lb/>cum .22. vigeſimisquartis, tertia autem .12. cum nouem vigeſimisquartis, quarta po<lb/>ſteà .25. cum .18. vigeſimisquartis, quinta demum erit .52. cum .12. vigeſimisquartis, <lb/>quarum omnium ſumma erit .100.</s> </p> <p> <s xml:space="preserve">STifelius in primo exemplo regulæ falſi, ita inquit.</s> </p> <p> <s xml:space="preserve">Quæratur numerus, à cuius dimidio ſubtractæ partes tertia, & quarta relin-<lb/>quatur .300.</s> </p> <p> <s xml:space="preserve">Ipſe enim ſupponit .300. pro reſiduo cognito alterius numeri incogniti, deinde <lb/>accipit .24. pro prima poſitione numeri cogniti, à cuius medietate abſcindit tertiam <lb/>& quartam partem ipſius medietatis, vnde remanet .5. qui quidem numerus .5. ex <num value="22">.<lb/>22.</num> quinti vel .15. ſeptimiſe ha-<lb/>bebit ad .24. vt .300. ad <choice><ex>numerum</ex><am>numerũ</am></choice> <lb/> <ptr xml:id="fig-0123-01a" corresp="fig-0123-01" type="figureAnchor"/> quæſitum, </s> <s xml:space="preserve">quare cum quis multi <lb/>plicauerit .300. per .24. & produ-<lb/>ctum diuiſerit per .5. proueniet <num value="1440">.<lb/>1440.</num> numerus quæſitus, ex vi <lb/>regulæ de tribus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0123-01" corresp="fig-0123-01a"> <graphic url="0123-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Conſideremus igitur <choice><ex>meam</ex><am>meã</am></choice> di-<lb/>ſpoſitionem numerorum huiuſ-<lb/>modi exempli, in figura hic ſup-<lb/>poſita <seg type="var">.F.</seg> in qua videre licebit <lb/>quo pacto ipſe etiam Stifelius ac <lb/>cipiat diuiſorem .5. vt <choice><ex>differentiam</ex><am>differentiã</am></choice> <lb/>errorum & non ut differentiam <lb/>duorum conſequentium .5. et .10 <lb/>ſicuti eſt re uera, ut diuiſor dico, <lb/>ex rationibus à me hic ſupra ad-<lb/>ductis, quamuis vna & eadem ſit <lb/>quantitas neceſſariò ut patet.</s> </p> <p> <s xml:space="preserve">ACcipiamus adhuc aliud exemplum à Tartalea propoſitione .9. <choice><ex>datum</ex><am>datũ</am></choice>, & <choice><ex>oppoſitum</ex><am>oppoſitũ</am></choice> <lb/>priori; </s> <s xml:space="preserve">nam ſicut in illo numerus ſimplex habebatur per ſubtractionem ſum-<lb/>mæ numerorum adijciendorum, in hoc fitèconuerſo, hoc eſt per additionem nu-<lb/>merorum ſubtrahendorum.</s> </p> <p> <s xml:space="preserve">Problema igitur ita ſe habet. </s> <s xml:space="preserve">Fuit quidam mercator qui habebat aliquot au-<lb/>reos, cuius quantitas poſteà quærenda erit, hic enim fecit duo itinera, ut aliquod <lb/>dictis aureis mediantibus lucrum faceret, in primo autem itinere duplicauit nume-<lb/>rum ſuorum aureorum, ex quibus poſteà conſumpſit .4. pro aliquibus expenſis, in <pb facs="0124" n="112"/><fw type="head">IO. BAPT. BENED.</fw> ſecundo itinere iterum duplicauit ſuos aureos, ex quibus etiam poſtea conſumpſit <num value="8">.<lb/>8.</num> numeratis poſteà pecunijs reperit tantummodo .24. aureos in eius marſupio, <choice><ex>quae ritur</ex><am>quęritur</am></choice> nunc quot habebat aureos in principio primi itineris.</s> </p> <p> <s xml:space="preserve">Intali caſu, cum ipſe quolibet itinere duplicabat eius pecuniam, nulli dubium <lb/>eſt quòd in fine ſecundi itineris ipſe habuiſſet pecuniam ſuam quadruplicatam, ſi <lb/>ex ipſa nihil detractum fuiſſet, ſed quia in fine primi itineris conſumpſit .4. aureos, <lb/>quibus alios .4. lucratus eſſet in ſecundo itinere, poſteà conſumpſit iterum <num value="8">.<lb/>8.</num> aureos, ita <choice><ex>quod</ex><am>ꝙ</am></choice> ex quadruplo ſuæ primæ pecuniæ, rectè dici poteſt, quod conſum-<lb/>pſerit .16. aureos; </s> <s xml:space="preserve">qui quidem numerus ex communi conceptu erit differentia in-<lb/>ter .24. & quadruplum prioris pecuniæ, cum qua profectus fuit in principio eius iti-<lb/>neris; </s> <s xml:space="preserve">quapropter ſi addiderimus .16. ipſi .24. habebimus .40. pro quadruplo eius <lb/>prioris pecuniæ. </s> <s xml:space="preserve">Rectè igitur dici poteſt, ſi .4. prouenit ab vno, à quo numero pro <lb/>ueniet .40.</s> </p> <p> <s xml:space="preserve">Videamus igitur nunc quo pacto hoc reſpondeat cum methodo antiquorum. <lb/></s> <s xml:space="preserve">Ego enim inueni duas poſitiones ſcriptas à Tartalea pro prima pecunia hoc eſt .12. <lb/>et .14. ſed à .12. pro primo errore reperi .8. more antiquo à .14. verò pro ſecundo er-<lb/>rore proueniebat .16. producta autem horum numerorum diametraliter, ſunt .112. <lb/>et .192. quorum differentia eſt .80. pro tertio producto, quo diuiſo per differen-<lb/>tiam <choice><ex>errorum</ex><am>errorũ</am></choice> .8. ſcilicet, præbetnobis .10. pro pecunia quæſita, vt etiam ego inueni.</s> </p> <p> <s xml:space="preserve">Sed hoc mihi viſum eſt ſubtilius examinare mea methodo mediante, vtin figu-<lb/>ra <seg type="var">.G.</seg> videre eſt, prius enim ſuo loco poſuitria producta dicta, deinde duas poſitio <lb/>nes .12. et .14. & quia ſciebam productum .112. oriri à multiplicatione .14. cum .8. <lb/>ideo poſui talem numerum .8. ſuo loco diametraliter oppoſito ei producto .112. <lb/>& quia ſciebam etiam productum .192. naſci ex .12. et .16. ideo ſuo loco poſui hunc <lb/>numerum .16. qui eſt maxima differentia inter duos conſequentes ( ita à me ſupra <lb/>nominatos) à qua differentia dempta priori .8. iam inuenta, reliqua .8. mihi daba-<lb/>tur, <choice><ex>quam</ex><am>quã</am></choice> ſuo loco notaui, ſuo <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>loco ſcripſi .2. differentiam inter <lb/>12. et .14. antecedentium. </s> <s xml:space="preserve">ſed <lb/> <ptr xml:id="fig-0124-01a" corresp="fig-0124-01" type="figureAnchor"/> quia ſciebam eandem proportio <lb/>nem eſſe inter hanc differentiam <lb/>& <choice><ex>differentiam</ex><am>differentiã</am></choice> .8. huic <choice><ex>ſuppoſitam</ex><am>ſuppoſitã</am></choice>, <lb/>quæ reperitur inter .12. <choice><ex>antecedem</ex><am>antecedẽ</am></choice> <lb/>tem, & <choice><ex>ſuum</ex><am>ſuũ</am></choice> conſequentem; </s> <s xml:space="preserve">ideo <lb/>poſui .48. pro dicto conſequenti, <lb/>diuiſi poſtea productum .80. per <num value="8">.<lb/>8.</num> differentiam ei diametraliter <lb/>oppoſitam, vnde prouenit mihi <lb/>10. cui ita proportionatus eſt <choice><ex>ſuus</ex><am>ſuꝰ</am></choice> <lb/>numerus conſequens .40. vt .48. <lb/>ad .12. et .56. ad .14. exijſdem ra-<lb/>tionibus à meſupra dictis. <lb/></s> <s xml:space="preserve">In tali igitur figura videntur nu-<lb/>merinaturaliter <choice><ex>correſpondentes</ex><am>correſpondẽtes</am></choice> <lb/>ipſis poſitionibus, & hac metho-<lb/>do poſſumus inuenire tales numeros conſequentes in omnibus alijs exemplis à no-<lb/>ſtris maioribus ſcriptis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0124-01" corresp="fig-0124-01a"> <graphic url="0124-01"/> </figure> </div> </body> </floatingText> <pb facs="0125" n="113"/> <fw type="head">THEOREM. ARIT.</fw> <p> <s xml:space="preserve">PRoponitur etiam quoddam vas, cuius pes ſit quarta pars totius vaſis cum oper <lb/>culo, pars autem media ſine operculo, ſit quinta pars ipſius pedis, operculum <lb/>verò .18. libras pendeat. </s> <s xml:space="preserve">quæritur nunc quantitas dicti pedis.</s> </p> <p> <s xml:space="preserve">Ex methodo enim antiquorum inuentus eſt pes .4. cum .14. decimisnonis ta-<lb/>lium partium, ſeu librarum, qualium operculus eſt .18. </s> <s xml:space="preserve">Videamus igitur & nos ex <lb/>noſtra figura, quo pacto hoc reſpondeat veritati.</s> </p> <p> <s xml:space="preserve">Inuenta enim ſunt tria producta, <choice><ex>iam</ex><am>iã</am></choice> orta ex dicta methodo .10. 100. 90. quæ ſuis <lb/>locis notaui, vt in ſigura <seg type="var">.H.</seg> ſubſcripſi etiam duas illorum poſitiones .5. et .10. cum <lb/>ſua differentia .5. & cum productum .10. oriretur ab vno latere .10. reliquum erat <num value="1">.<lb/>1.</num> quod ſuo loco notaui, ſimiliter quia .100. productum, pro vno eius laterum erat <num value="5">.<lb/>5.</num> reliquum autem .20. ſuo loco poſui, & quia differentia inter .20. et .1. duo latera, <lb/>quę eſt .19. æqualis eſt ei, quæ inter duo conſequentia duarum poſitionum, etiam <lb/>ſuo loco ipſam conſtitui, ſed quia hæc differentia eſt vnum laterum producti .90. er <lb/>go reliquum latus <choice><ex>quæſitum</ex><am>quæſitũ</am></choice> erit .4. cum .14. decimisnonis, rectè igitur operatur. <lb/></s> <s xml:space="preserve">ſed cum eadem proportio ſit inter differentiam .5. ſuperiorem, et .19. inferiorem, <lb/>quæ eſt vnius <choice><ex>antecedentis</ex><am>antecedẽtis</am></choice> ad ſuum <lb/>conſequens, </s> <s xml:space="preserve">quare .10. <choice><ex>antecedens</ex><am>antecedẽs</am></choice> <lb/>habebit pro ſuo conſequenti .38. <lb/> <ptr xml:id="fig-0125-01a" corresp="fig-0125-01" type="figureAnchor"/> et .5. habebit .19. et .4. cum .14. de-<lb/>cimisnonis habebit .18. rectè <choice><ex>igitur</ex><am>igit̃</am></choice> <lb/>dictum fuiſſet ſi .19. prouenit .à .5. <lb/>à quo proueniet .18?</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0125-01" corresp="fig-0125-01a"> <graphic url="0125-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Huiuſmodi autem rei ratio ita <lb/>ſe <choice><ex>hent</ex><am>hẽt</am></choice>, eſto linea <seg type="var">.a.e.u.</seg> cuius pars <lb/>a. ſit quarta reliquarum <seg type="var">.e.u.</seg> iuncta <lb/>rum, ſed <seg type="var">.e.</seg> ſit quinta ipſius <seg type="var">.a</seg>. </s> <s xml:space="preserve">Tunc <lb/>clarum erit quod <seg type="var">.e.</seg> erit vigeſima <lb/>dictarum <seg type="var">.e.u.</seg> </s> <s xml:space="preserve">quare erit decima-<lb/>nona ipſius <seg type="var">.u.</seg> ſed <choice><ex>cum</ex><am>cũ</am></choice> u. <choice><ex>sumpta</ex><am>sũpta</am></choice> ſit vt <num value="18">.<lb/>18.</num> rectè igitur dici poteſt, ſi .u: ut <num value="19">.<lb/>19.</num> prouenit ab <seg type="var">.a.</seg> ut quinque, à <lb/>quot ipſius <seg type="var">.a.</seg> proueniet <seg type="var">.u.</seg> ut .18.</s> </p> <p> <s xml:space="preserve">Quis enim non uidet quod diui <lb/>ſa cum fuerit <seg type="var">.u.</seg> in partes .19. quod <lb/>quinque illarum æquabuntur ipſi <seg type="var">.<lb/>a.</seg> cum quælibet fuerit æqualis <seg type="var">.e.</seg> <lb/>quintæ parti ipſius <seg type="var">.a</seg>.</s> </p> <p> <s xml:space="preserve">HAc igitur mea numerorum diſpoſitione mediante reperiuntur ipſi numeri in <lb/>feriores naturaliter conſequentes, correſpondentesq́ue ipſis ſuperioribus an <lb/>tecedentibus; </s> <s xml:space="preserve">quamuis multoties <choice><ex>contingere</ex><am>cõtingere</am></choice> poſſit, ut generationes ſeu com-<lb/>poſitiones ipſorum ignorentur: </s> <s xml:space="preserve">& quia tam à differentijs errorum, quam ab illis, <lb/>quę ſunt inter ueros conſequentes numeros ( propter eorum æqualitatem ) elicitur <lb/>ipſa ueritas, proptereà rectè antiqui illis vſi ſunt, quamuis ſint potius ſenſum <lb/>ſequuti, uel experientiam, quam rationem: </s> <s xml:space="preserve">quæ quidem ratio pendet ab ipſis na-<lb/>turalibus numeris conſequentibus ( ut ſupra uidimus ) etſi incognitis ut plurimum, <lb/>quod ſi ipſos inuenire primò nobis datum fuiſſet, unica <choice><ex>tantummodo</ex><am>tantũmodo</am></choice> poſitio ſuffice- <pb facs="0126" n="114"/><fw type="head">IO. BAPT. BENED.</fw> ret, mediante ipſa regula de tribus, vt <choice><ex>iam</ex><am>iã</am></choice> ſępius <choice><ex>dictum</ex><am>dictũ</am></choice> eſt, quod <choice><ex>etiam</ex><am>etiã</am></choice> clarè patet ex di-<lb/>uerſis problematibus .17. lib. ipſius Tartaleæ, vt ex primo, quod aſſumpſimus pro <lb/>noſtro etiam primo exemplo, ex .9. 15. 16. 17. 18. 19. 20. 27. 28. 29. 30. 33. & ex <lb/>alijs multis, vbi facillimè inue nitur conſequens ipſius poſitionis, qui quidem nume-<lb/>rus eſt diuiſor producti ipſius numeri propoſiti in numerum poſitionis, vnde poſteà <lb/>prouenit <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> latus huiuſmodi producti, hoc eſt numerus quæſitus, per <choice><ex>regulam</ex><am>regulã</am></choice> <lb/>de tribus, vt dixi.</s> </p> <p> <s xml:space="preserve">Alia verò multa problemata inueniuntur, pro quorum re@olutione poſſumus ali <lb/>qua methodo vti, in qua manifeſtè pateant <choice><ex>eorum</ex><am>eorũ</am></choice> rationes abſque regula falſi, cuius <lb/>regulæ rationes non ita promptè ipſi intellectui ſe offerunt, vt ſupra vidimus.</s> </p> <p> <s xml:space="preserve">Accipiamus pro exemplo .21. problema ipſius Tartalæ in dicto .17. libr. vbi ſup-<lb/>ponit vnum hædum diuiſum in .4. partes, quarum quælibet vendebatur eodem pre <lb/>cio, interiora vero .6. denarijs minus quam quælibet dictarum partium, ſumma <lb/>autem omnium iſtorum denariorum fuit .127. quæritur nunc precium cuiuſque <lb/>partis.</s> </p> <p> <s xml:space="preserve">Tale enim problema hoc etiam alio breuiori modo poteſt ſolui, vt rationes ma-<lb/>gis pateant, quam ex regula falſi.</s> </p> <p> <s xml:space="preserve">Nam ſi illi numero .127. denariorum, additus fuerit numerus .6. ſumma erit .133. <lb/>qua diuiſa per quinque, illico proueniet .26. cum tribusquintis pro precio vniuſcu-<lb/>iuſque quatuor partium, à quo .26. cum tribusquintis dempto .6. remanebit .20. cum <lb/>tribusquintis pro precio interiorum.</s> </p> <p> <s xml:space="preserve">Simili modo in .24. problemate inquit.</s> </p> <p> <s xml:space="preserve">Duodecim pyra cum .28. pomis venduntur .36. denarijs, et .20. pyra. cum .200 po <lb/>mis <choice><ex>venduntur</ex><am>vẽduntur</am></choice> .44. denarijs, <choice><ex>quæritur</ex><am>quærit̃</am></choice> nunc, quod <choice><ex>nam</ex><am>nã</am></choice> fuerit <choice><ex>precium</ex><am>preciũ</am></choice> <choice><ex>vniuſcuiuſque</ex><am>vniuſcuiuſq;</am></choice> illorum.</s> </p> <p> <s xml:space="preserve">Hoc etiam problema, hac alia methodo ſolui poteſt, dicendo exregula de tribus, <lb/>ſi ex .20. vtrorunque qui ea vendit, vult .44. quid volet ex .12? </s> <s xml:space="preserve"><choice><ex>manifeſtum</ex><am>manifeſtũ</am></choice> erit quod <lb/>volet .26. cum duobus quintis, </s> <s xml:space="preserve">quare .12. pyra cum .12. pomis valebunt .26. cum duo <lb/>bus quintis, ſed 12. cum .28. pomis valebant .36. ergo .16. poma ſola valebunt .9. <lb/>cum tribus quintis, hoc enim clarè ex ſe patet; </s> <s xml:space="preserve">quare cum dixerimus, ſi .16. poma ſo <lb/>la valent .9. cum tribusquintis, vnum valebit <seg type="var">.o.</seg> cum tribusquintis, ſed quemadmo-<lb/>dum .20. pyra cum .20. pomis valent .44. vnum pyrum, cum vno pomo valebunt .2. <lb/>cum quinta parte, à quo numero detractus cum fuerit <seg type="var">.o.</seg> cum tribus quintis, precio <lb/>ſcilicet vnius pomi, reliquum .1. cum tribusquintis, erit precium vnius pyri.</s> </p> <p> <s xml:space="preserve">Idem etiam dico de .28. problemate, vbi ſupponit quod quidam comparaſſet <lb/>quatuor petias, vt vulgo dicitur, panni pro ducatis .96. quarum primæ precium ob-<lb/>litus ſit, ſed memoria tenet pro ſecunda ſoluiſſe .6. plusquam pro prima, & pro ter-<lb/>tia ſoluiſſe .8. plus quam pro ſecunda, & pro quarta ſoluiſſe .10. plus quam pro ter-<lb/>tia, quæritur nunc quantum fuerit precium vniuſcuiuſque illarum.</s> </p> <p> <s xml:space="preserve">Quod <choice><ex>quidem</ex><am>quidẽ</am></choice> problema <lb/> <ptr xml:id="fig-0126-01a" corresp="fig-0126-01" type="figureAnchor"/> breuius eſſetita ſolui, vt in <lb/>ſubſcripta figura <seg type="var">.I.</seg> videri <lb/>poteſt, <choice><ex>addendo</ex><am>addẽdo</am></choice> ſimul omnes <lb/>exceſſus. </s> <s xml:space="preserve">Nam exceſſus <choice><ex>ſecum</ex><am>ſecũ</am></choice> <lb/>dæ ſupra primam eſt .6. ſed <lb/>cum exceſſus tertiæ ſupra ſe <lb/>cundam ſit .8. ergo exceſſus <lb/>tertiæ ſupra primam erit .14 <pb facs="0127" n="115"/><fw type="head">THEOREM. ARIT.</fw> ſed exceſſus quartæ ſupra tertiam eſt .10. vnde ſupra ſecundam erit .18. & ſupra pri-<lb/>mam erit .24. quæ omnia ſimul addita erunt .44. & in qualibet harum trium remane-<lb/>bit una pars æqualis primæ quantitati, </s> <s xml:space="preserve">quare ſi ex .96. detractus fuerit numerus .44. <lb/>reliquus 52. erit quadruplus primæ, </s> <s xml:space="preserve">quare prima pars valebit .13. ſecunda .19. ter-<lb/>tia .27. & quarta .37. quarum omnium ſumma eſt .96.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0126-01" corresp="fig-0126-01a"> <graphic url="0126-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">EX poſitionibus autem Tartaleæ in noſtra figura <seg type="var">.K.</seg> digeſtis, videre poſſumus <lb/>quo pacto <choice><ex>colligantur</ex><am>colligãtur</am></choice> huiuſ <lb/>modi <choice><ex>conſequentes</ex><am>conſequẽtes</am></choice> numeri ſimpli-<lb/>ces .36. et .52. more figuræ <seg type="var">.E.</seg> quia <lb/> <ptr xml:id="fig-0127-01a" corresp="fig-0127-01" type="figureAnchor"/> colliguntur primò partes compoſi <lb/>tæ .9. 15. 23. 33. ex quarum ſumma <lb/>80. ſubtrahitur .36. ſumma ſim-<lb/>plex ex ſimplicibus partibus .9. 9. <lb/>9. 9. & <choice><ex>reſiduum</ex><am>reſiduũ</am></choice> quod eſt .44. ſubdu <lb/>citur ex .96. ſumma compoſita & <lb/>propoſita, vnde remanet .52. pro <lb/>ſumma ſimplici, ex numero dato, <lb/>cuius proportio ad .13. eadem eſt <lb/>quæ .36 ad .9. & proptereà ſuper-<lb/>flua eſt ſecunda poſitio, <choice><ex>quando</ex><am>quãdo</am></choice> ſci <lb/>mus inuenire tales duos numeros <lb/>conſequentes, vt in hoc exemplo <lb/>ſunt .36. et .52. quia ex regula de <lb/>tribus poſteà elicitur veritas quæ-<lb/>ſita. </s> <s xml:space="preserve"><choice><ex>Idem</ex><am>Idẽ</am></choice> dico de 33. problemate.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0127-01" corresp="fig-0127-01a"> <graphic url="0127-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">PRO quo .33. problemate acci <lb/>piantur poſitiones primi <choice><ex>exem</ex><am>exẽ</am></choice> <lb/>pli Tonſtalli hoc eſt .33. et .31. vt in figuris hic ſubiectis <seg type="var">.P.Q.</seg> facile quis poteſt vi-<lb/>dere, vbi in figura P. videbit nume-<lb/>ros compoſitos, in figura verò <seg type="var">.Q.</seg> cer <lb/> <ptr xml:id="fig-0127-02a" corresp="fig-0127-02" type="figureAnchor"/> net numeros ſimplices, à quibus pro <lb/>ueniunt rationes per ſe huiaſmodi <lb/>operationis, in figura autem <seg type="var">.R.</seg> vide <lb/>bitur meus ordo, & iſtæ tres figuræ ſi <lb/>miles <choice><ex>erunt</ex><am>erũt</am></choice> tribus illis primis <seg type="var">.A.B.C.</seg> <lb/>ita quòd cum quis illas intellexerit, il <lb/>lico etiam iſtas cognoſcet, vbi <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>videbit quam confusè <choice><ex>ratiocinentur</ex><am>ratiocinẽtur</am></choice> ij <lb/>qui ignorant hunc meum ordinem <lb/>ſimplicium <choice><ex>numerorum</ex><am>numerorũ</am></choice>, à quibus fluit <lb/>tota ratio (vt ſupra dixi) huiuſcemo <lb/>di operationis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0127-02" corresp="fig-0127-02a"> <graphic url="0127-02"/> </figure> </div> </body> </floatingText> <pb facs="0128" n="116"/> <fw type="head">IO. BAPT. BENED.</fw> <figure place="here"> <graphic url="0128-01"/> </figure> <figure place="here"> <graphic url="0128-02"/> </figure> <p> <s xml:space="preserve">I Dem etiam poteſt dici de .15. problemate (ſicut de alijs multis) vbi ponit tres <lb/>homines habentes .40. aureos quorum primus habet duas quintas partes ſecun-<lb/>di, <choice><ex>ſecundus</ex><am>ſecũdus</am></choice> verò <choice><ex>quinque</ex><am>quinq;</am></choice> octauas tertij, <choice><ex>quæritur</ex><am>quærit̃</am></choice> <choice><ex>nunc</ex><am>nũc</am></choice> quot ducatos habeat vnuſquiſque.</s> </p> <p> <s xml:space="preserve">Quis non videt quæſo, <choice><ex>quod</ex><am>ꝙ</am></choice> omnes partes erunt .15. quare cum dixerimus ſi .15. dat <lb/>nobis .2. (pro prima portione primi hominis) quid dabit .40? </s> <s xml:space="preserve">vnde nobis proueniet <lb/>5. cum tertia parte.</s> </p> <p> <s xml:space="preserve">Et de .29. ſimiliter aſſero, vbi ponit <choice><ex>alium</ex><am>aliũ</am></choice> emiſſe tria fruſta panni pro ducatis .48. <lb/>quarum ſecundam habuit pro dimidio precio primæ, tertiam autem pro quarta <lb/>parte ipſius ſecundæ, </s> <s xml:space="preserve">quare omnes partes erunt .13. quapropter precium tertiæ pe-<lb/>tiæ erit tertiadecima pars ipſius .48. hoc eſt .3. cum .9. tertijs decimis.</s> </p> <p> <s xml:space="preserve">Adhuc duo exempla videtur mihi proponere, quorum primum eſt .38. eiuſdem <lb/>lib. vbi ſupponitur operarium quendam velle perficere opus quoddam ſpacio die-<lb/>rum .36. tali pacto, quod qualibet die, in qua ipſe operaturus ſit lucretur ſolidos <num value="16">.<lb/>16.</num> qualibet verò die, in qua nihil agat perdat ſolidos .24. </s> <s xml:space="preserve">Tunc accidit, vt exacto <lb/>termino <choice><ex>perfectoque</ex><am>perfectoq́;</am></choice> opere, tantum lucratus ſit, quantum perdiderit. </s> <s xml:space="preserve">Quæritur <choice><ex>nunc</ex><am>nũc</am></choice> <lb/>quot fuerint dies lucri, quotúe perditionis.</s> </p> <p> <s xml:space="preserve">Huiuſmodi problematis operatio breuiſſima abſque vlla falſa poſitione ita erit, <lb/>hoc eſt diuidendo productum .36. in .24. per .40. ideſt per aggregatum ipſius .24. <choice><ex>cum</ex><am>cũ</am></choice> <lb/>16. & prouentus erit .21. cum tribus quintis pro diebus lucri, vnde reliquum ex .36. <lb/>erit .14. cum duabus quintis pro diebus perditionis.</s> </p> <p> <s xml:space="preserve">Cuius operationis ratio ex ſe ſatis patet, cum duo producta, vnum lucri, alterum <lb/>vero perditio-<lb/>nis æqualia eſſe <lb/> <ptr xml:id="fig-0128-03a" corresp="fig-0128-03" type="figureAnchor"/> debeant, vnde <lb/>ex duodecima, <lb/>& vigeſimaſepti <lb/>mi ex regula de <lb/>tribus reperiun <lb/>tur partes ipſius <lb/>36. eodem mo- <pb facs="0129" n="117"/><fw type="head">THEOR. ARITH.</fw> do ſe inuicem habentes, vt .24. et .16. quæ ſunt .21. cum tribus quintis, et .14. <choice><ex>cum</ex><am>cũ</am></choice> dua <lb/>bus quintis, ex quo ſequitur, vt quod fit ex .21. cum tribus quintis, in 16. ęquale ſit ei <lb/>quod fit ex .14 cum duabus quintis, in .24. & ita reperiuntur duo producta æqualia, <lb/>vnum lucri, reliquum vero perditionis, vt in figura <seg type="var">.M.</seg> clarè videtur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0128-03" corresp="fig-0128-03a"> <graphic url="0128-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">A Liud verò exemplum eſt .39. quod quidem à ſuperiori non differt, niſi quod <lb/>in fine operationis, operarius dictus lucratus eſt ſolidos .60: </s> <s xml:space="preserve">quęritur <choice><ex>nunc</ex><am>nũc</am></choice> vt ſu-<lb/>pra, quot fuerunt dies lucri, & quot perditionis.</s> </p> <p> <s xml:space="preserve">Hoc etiam abſque vlla falſa poſitione dicto citius poteſt ſolui, hoc modo, <choice><ex>diuidem</ex><am>diuidẽ</am></choice> <lb/>do ſcilicet illos .60 ſolidos per .40. ideſt per aggregatum .24. cum .16. proueniens <lb/>autem, quod erit .1. cum dimidio, adde ad latus ſuperius inuentum, hoc eſt .21. cum <lb/>tribus quintis, & ſunima erit .23. cum decima parte pro numero dierum lucri, dein-<lb/>de idem prouentum deme ex alio latere ſuperius reperto .14. cum duabus quintis, & <lb/>refiduum erit .12. cum nouem decimis, vnde habebis numerum dierum perdi-<lb/>tionis.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ſpeculatione cogitemus in figura <seg type="var">.N.</seg> duo dicta producta inuicem <lb/>æqualia <seg type="var">.o.b.</seg> et <seg type="var">.n.c.</seg> exiſtente latere <seg type="var">.u.c.</seg> vt .24. <seg type="var">u.o.</seg> ut .16: </s> <s xml:space="preserve">b.u. vt .21. cum tribus quin <lb/>tis, et <seg type="var">.u.n.</seg> vt .14. cum duabus quintis. </s> <s xml:space="preserve">Nunc verò ſi mente concepta fuerit recta <seg type="var">.e.<lb/>a.t.</seg> æquidiſtans <seg type="var">.o.c.</seg> ita vt rectangulum <seg type="var">.o.e.</seg> ſit .60. </s> <s xml:space="preserve">tunc rectangulum, ſeu productum <lb/><seg type="var">b.t.</seg> ſuperabit rectangulum ſeu productum <seg type="var">.n.e.</seg> per idem .60. ex communi conceptu, <lb/>eo quòd ex producto <seg type="var">.n.c.</seg> ſublatum eſt productum <seg type="var">.a.c.</seg> 24. & producto <seg type="var">.o.b.</seg> additum <lb/>eſt productum <seg type="var">e.a.</seg> 16. rectè igitur feci cum diuiſerim .60 per .40. vnde prouenit mi <lb/>hi <seg type="var">.u.a.</seg> ideſt .1. cum dimidio, quod additum ipſi <seg type="var">.b.u.</seg> compoſuit <seg type="var">.b.a.</seg> & dempto ex <seg type="var">.u.<lb/>n.</seg> relinquit <seg type="var">.a.n.</seg> pro lateribus duorum productorum <seg type="var">.b.t.</seg> et <seg type="var">.n.e</seg>.</s> </p> <p> <s xml:space="preserve">Sed ſi idem operator perdidiſſet .60. tunc cogitaremus parallelam dictam <seg type="var">.e.a.</seg> t <lb/>ſuperius ductam eſſe ita vt ſecaret <seg type="var">.b.u.</seg> & non <seg type="var">.u.n.</seg> vnde adderet .24. ipſi producto <seg type="var">.n.<lb/>c.</seg> & d@meret .16. à producto <seg type="var">.b.o</seg>.</s> </p> <figure place="here"> <graphic url="0129-01"/> </figure> <p> <s xml:space="preserve">CIRCA verò talia quæſita videtur mihi non inutile fore ſi aliquid notatu di-<lb/>gnum aduerterim, hoc eſt quod ſæpe accidere poterit ut caſus impoſſibiles <lb/>proponantur. </s> <s xml:space="preserve">Quemadmodum ſi aliquis diceret, cupio mihi ueſtimentum con-<lb/>ficere ex duobus pannis colore & pretio differentibus, quorum unus exempli gra- <pb facs="0130" n="118"/><fw type="head">IO. BAPT. BENED.</fw> tia ſit albus, rubeus uerò alter, deinde albus ſit pretij .40. ſolidorum uniuſcuiuſ-<lb/>que cubiti, rubeusuerò precij .50. uellemq́ue omnes cubitos eſſe .8. nec plus <lb/>nec minus. </s> <s xml:space="preserve">Vellem etiam ſoluere ſolidos 450. neque minus.</s> </p> <p> <s xml:space="preserve">Hic igitur caſus impoſſibilis eſt, eo quòd .8. cubiti totius rubei eſſent precij ſo-<lb/>lidorum .400. tantummodo, unde ex alio panno albo minoris precij ſumere ali-<lb/>quid non poſſumus.</s> </p> <p> <s xml:space="preserve">Idem etiam eueniret ſi uoluiſſet ſoluere ſolidos .320. neque plus, eo quòd .8. cu-<lb/>biti illius minoris precij, hoc eſt .40. ſolidorum, eſsent ualoris .320. ſolidorum tan <lb/>tummodo, quare pro alio panno nullus eſset locus. </s> <s xml:space="preserve">Animaduertendum igitur erit <lb/>quod numerus poſſibilis ad ſoluendum tale quæſitum erit inter .400. et .320. & non <lb/>extra iſtos terminos, vt vnicuique patere poteſt.</s> </p> <p> <s xml:space="preserve">Similiter idem in hoc alio caſu accidere poterit, ut ſi quis diceret.</s> </p> <p> <s xml:space="preserve">Emi quinque petias panni pro aureis .55. pretium tamen primæ oblitus ſum, ſed <lb/>memoria teneo, quòd ſecunda altioris pretij erat quam ipſa prima per .4. & ter-<lb/>tia precioſior ſecunda per .7. et quarta carior tertia per .9. quinta verò ſuperabat <lb/>quartam per .2.</s> </p> <p> <s xml:space="preserve">Hic etiam reperitur impoſſibilitas quædam, eo quòd aggregatum omnium ha-<lb/>rum rerum, dato etiam quòd pro prima nihil ſolutum eſſet, ſuperat aureos .55 quòd <lb/>quidem nullo pacto fieri poteſt, vt veri ſint ſupra dicti exceſsus, ſi verus eſt numerus <lb/>totalis aureorum .55. </s> <s xml:space="preserve">Nam .4. cum .7. faciunt .11. qui quidem .11. cum .9. efficiunt <num value="20">.<lb/>20.</num> & hic cum .2. facit .22. ſed .22. cum .20. et .11. et .4. dant .57. qui numerus maior <lb/>eſt quam .55.</s> </p> <fw type="footer">FINIS THEOREM. ARIT.</fw> <pb facs="0131" n="119"/> </div> </div> <div type="chapter"> <head xml:space="preserve">DE RATIONIBVS <lb/>OPERATIONVM <lb/>PERSPECTIVAE.</head> <div type="section"> <head xml:space="preserve">CAP.I.</head> <p> <s xml:space="preserve">CVM nullus adhuc (quod ſciam) veras <choice><ex>internasque</ex><am>internasq́</am></choice> cauſas <lb/>operationis perſpectiuæ perſectè docuerit, operæpre-<lb/>cium exiſtimaui <choice><ex>aliquam</ex><am>aliquã</am></choice> de ijs diſputationem ſuſcipere.</s> </p> <p> <s xml:space="preserve">Multi enim <choice><ex>eorum</ex><am>eorũ</am></choice>, qui huiuſmodi operationis regulas <lb/>præſcribunt, cum eius effectuum veras cauſas igno-<lb/>rent, varios <choice><ex>diuerſosque</ex><am>diuerſosq́;</am></choice> errores committunt, vt exempli <lb/>gratia in ſubſcripta figura ſuperficiali <seg type="var">.A.</seg> volentes degra <lb/>dare (vt dicunt) rectangulum <seg type="var">.q.a.</seg> in triangulo <seg type="var">.i.d.q.</seg> du-<lb/>cunt <choice><ex>parallelam</ex><am>parallelã</am></choice> ipſi <seg type="var">.q.d.</seg> à puncto <seg type="var">.B.</seg> interſecationis lineæ-<lb/>o.l. cum latere <seg type="var">.i.d.</seg> trianguli, & (idem) indifferenter, ean-<lb/>dem quoque à puncto <seg type="var">.Z.</seg> interſecationis ipſius <seg type="var">.o.l.</seg> cum perpendiculari <seg type="var">.x.i.</seg> ducunt. <lb/></s> <s xml:space="preserve">neſcientes hunc ſolum eſſe verum modum, n onitem alium, quia ſi alius, talis eſſet, <lb/>hic, verus non exiſteret, nam ſi vellent ſeſe excuſ are, quòd ducendo dictam paralle-<lb/>lam à puncto <seg type="var">.B.</seg> hoc fiat præſupponendo planum ipſius <seg type="var">.i.d.q.</seg> verſus rectangulum <seg type="var">.<lb/>q.a.</seg> orizontale inclinatum, ſecundum angulum <seg type="var">.i.d.q.</seg> hæc excuſatio accipien-<lb/>da non eſſet, quia horum conſenſu, præſupponendo planum <seg type="var">.i.d.q.</seg> inclinatum, <lb/>anguli inferiores rectanguli degradati, non tam acuti, quam ſunt duo <seg type="var">.i.d.q.</seg> et <seg type="var">.i.q.<lb/>d.</seg> eſſe deberent, quod facilè eorum ratione innoteſcet, quæ de figura corporea <seg type="var">.<lb/>A.</seg> hîc ſubſcripta mox proponam, præter id, quòd volentes deinde aſpicere qua-<lb/>dratum degradatum, oporteret huiuſmodi planum reſpectu oculi ita collocare, <lb/>quemadmodum ſe habet linea <seg type="var">.i.d.</seg> reſpectu <seg type="var">.o.</seg> quod factu nimis arduum eſſet.</s> </p> <p> <s xml:space="preserve">Vera igitur ratio erit ducere parallelam <seg type="var">.e.r.</seg> ad <seg type="var">.q.d.</seg> à puncto <seg type="var">.Z.</seg> communi ip-<lb/>ſis <seg type="var">.o.l.</seg> et <seg type="var">.x.i.</seg> perpendiculari ipſi <seg type="var">.l.p</seg>.</s> </p> <figure place="here"> <graphic url="0131-01"/> </figure> <pb facs="0132" n="120"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Pro cuius rei ſpeculatione imaginemur in figura corporea .A: <seg type="var">q.a.</seg> eſſe figuram re-<lb/>ctangulam <choice><ex>orizontalemque</ex><am>orizontalemq́;</am></choice> ad degradandam ſuper aliquod planum perpendiculare <lb/>orizonti, & cum eo primum coniunctam in linea <seg type="var">.q.d.</seg> cuius plani triangulum <seg type="var">.i.q.d.</seg> <lb/>pars erit, ſit autem oculus reſpicientis <seg type="var">.o.</seg> cuius altitudo <seg type="var">.o.p.</seg> ab orizonte, qui <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/>conſpicit rectangulum dictum orizontale <seg type="var">.q.a.</seg> in pyramide <seg type="var">.o.q</seg>: <seg type="var">o.u</seg>: <seg type="var">o.a.</seg> et <seg type="var">.o.d.</seg> <lb/>terminata quatuor triangulis <seg type="var">.o.q.u</seg>: <seg type="var">o.u.a</seg>: <seg type="var">o.a.d.</seg> et <seg type="var">.o.d.q.</seg> ſit verò primum ita <lb/>collocatus pes <seg type="var">.p.</seg> eius qui reſpicit, vt linea <seg type="var">.p.l.</seg> perpendicularis ipſi <seg type="var">.u.a.</seg> lateri re-<lb/>ctanguli, medio loco poſita ſit, inter <seg type="var">.a.n.</seg> et <seg type="var">.u.s</seg>. </s> <s xml:space="preserve"><choice><ex>Idque</ex><am>Idq́;</am></choice> primum nobis erit exem-<lb/>plum.</s> </p> <p> <s xml:space="preserve">Imaginemur nunc lineas <seg type="var">.u.q.</seg> et <seg type="var">.a.d.</seg> indefinitè productas eſſe, quæ in ſuperficie-<lb/>bus duorum triangulorum <seg type="var">.o.u.q.</seg> et <seg type="var">.o.a.d.</seg> & rectanguli orizontalis <seg type="var">.q.a.</seg> ex <ref>prima <lb/>vndecimi Euclid.</ref> poſitæ erunt. </s> <s xml:space="preserve">Imaginemur etiam lineam <seg type="var">.p.s.n.</seg> perpendicula-<lb/>rem ipſi <seg type="var">.p.l.</seg> quæ etiam cum duabus <seg type="var">.u.q.s.</seg> et <seg type="var">.a.d.n.</seg> ex .34. primi Euclid. angulos <lb/>rectos conſtituet, cum ex .28. duæ <seg type="var">.u.q.s.</seg> et <seg type="var">.a.d.n.</seg> ſint parallelæ ipſi <seg type="var">.p.l.</seg> et <seg type="var">.s.n.</seg> ipſi <seg type="var">.u.<lb/>a.</seg> & quia ſupponitur <seg type="var">.o.p.</seg> perpendicularis plano orizontali, Angulus ergò <seg type="var">.o.p.l.</seg> re-<lb/>ctus erit ex ſecunda definitione .11. Euclid. </s> <s xml:space="preserve">Imaginemur quoque ductas eſſe <lb/>duas <seg type="var">.o.s.</seg> et <seg type="var">.o.n.</seg> vnde <seg type="var">.l.p.</seg> ei ſuperficiei, in qua ſunt duæ lineæ <seg type="var">.o.p.</seg> et <seg type="var">.s.n.</seg> ex .4. <lb/>11. perpendicularis erit, & ſuperficies orizontalis <seg type="var">.a.s.</seg> perpendicularis erit cum dicta <lb/><seg type="var">o.s.n.</seg> ex .18. eiuſdem lib. vnde ex dicta definitione <seg type="var">.o.s.u.</seg> et <seg type="var">.o.n.a.</seg> erunt anguli recti <lb/>et <seg type="var">.o.s.</seg> et <seg type="var">.o.n.</seg> ex communi ſcientia, in ſuperficiebus duorum triangulorum <seg type="var">.o.u.q.</seg> et <seg type="var">.<lb/>o.a.d.</seg> erunt, ſi noluerimus cogere aduerſarium ad confitendum duas lineas rectas in-<lb/>cludere ſuperficiem, quemadmodum cogere-<lb/> <ptr xml:id="fig-0132-01a" corresp="fig-0132-01" type="figureAnchor"/> tur facere, ſi opinaretur duas alias rectas per <lb/>eadem puncta <seg type="var">.o.s.n.</seg> tranſire, quæſunt in di-<lb/>ctis ſuperficiebus. </s> <s xml:space="preserve">Vnde <seg type="var">.o.s.</seg> et <seg type="var">.o.n.</seg> communes <lb/>erunt ſectiones duarum dictarum <choice><ex>ſuperficierum</ex><am>ſuperficierũ</am></choice> <lb/>cum ſuperficie <seg type="var">.o.s.n</seg>. </s> <s xml:space="preserve">Imaginemur nunc has <lb/>duas ſuperficies <seg type="var">.o.u.</seg> et <seg type="var">.o.a.</seg> quarum commu-<lb/>nis ſectio ſit <seg type="var">.o.t.</seg> (quæ erit linea recta ex .3. lib. <lb/>II.) quæ erunt perpendiculares ſuperficiei <seg type="var">.o.s.<lb/>n.</seg> ex .4. et .14. iam dictis. </s> <s xml:space="preserve">& ex .19. eiuſdem <lb/><seg type="var">o.t.</seg> perpendicularis eidem ſuperficiei <seg type="var">.o.s.n.</seg> <lb/>erit, & ex .6. eiuſdem hæc linea <seg type="var">.o.t.</seg> duabus <seg type="var">.u.<lb/>q.s.</seg> et <seg type="var">.a.d.n.</seg> parallela exiſter, & ex .9. eiuſdem <lb/>hæc linea <seg type="var">.o.t.</seg> duabus <seg type="var">.u.q.s.</seg> et <seg type="var">.a.d.n.</seg> parallela <lb/>exiſtet, & ex eadem .9. erit parallela ipſi <seg type="var">.p.l.</seg> <lb/>Imaginemur nunc planum, ſuper quod deſide <lb/>remus videre quadrangulum orizontale, quod <lb/>planum, exempli gratia, ſit primo, vt iam dixi-<lb/>mus, locatum in linea <seg type="var">.q.d.</seg> ad angulos rectos <lb/>cum plano orizontali, cuius communes ſectio <lb/>nes cum ſuperficiebus <seg type="var">.s.t.</seg> et <seg type="var">.n.t.</seg> viſionis la-<lb/>terum <seg type="var">.u.q.</seg> et <seg type="var">.a.d.</seg> ſint <seg type="var">.i.q.</seg> et <seg type="var">.i.d.</seg> & com-<lb/>munis ſectio trianguli <seg type="var">.o.u.a.</seg> ideſt viſionis <lb/>lateris <seg type="var">.a.u.</seg> cum dicto plano, ſit <seg type="var">.r.e</seg>. </s> <s xml:space="preserve">Vnde ex <lb/>communi ſcientia rectangulum orizontale, <lb/>oculo <seg type="var">.o.</seg> ſeipſum patefaciet in plano <seg type="var">.i.q.d.</seg> ſe- <pb facs="0133" n="121"/><fw type="head">DE PERSPECT.</fw> cundum figuram quadrilateram <seg type="var">.q.d.r.e</seg>. </s> <s xml:space="preserve">Communis autem ſectio ſuperficiei <seg type="var">.p.t.</seg> <lb/>cum dicto plano, ſit <seg type="var">.i.x.</seg> quæ <seg type="var">.i.x.</seg> perpendicularis erit <seg type="var">.s.a.</seg> ſuperficiei orizontali ex <lb/>19. lib. 11. quia <seg type="var">.p.t.</seg> eſt etiam orizonti perpendicularis ex .18. eiuſdem, cum <seg type="var">.o.p.</seg> ei-<lb/>dem perpendicularis exiſtat. </s> <s xml:space="preserve">Vnde <seg type="var">.i.x.</seg> erit altitudo trianguli <seg type="var">.i.q.d.</seg> & æqualis ipſi <seg type="var">.<lb/>o.p.</seg> ex <ref>.34. primi</ref>. </s> <s xml:space="preserve">Sit deinde <seg type="var">.o.l.</seg> <choice><ex>communis</ex><am>cõmunis</am></choice> ſectio ſuperficiei triangularis <seg type="var">.o.a.u.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <lb/>ſuperficie <seg type="var">.p.t.</seg> quæ <seg type="var">.o.l.</seg> ſecando lineam <seg type="var">.e.r.</seg> in puncto <seg type="var">.Z.</seg> nobis oſtendet quantum di-<lb/>ſtare ſeu eminens eſſe debeat latus <seg type="var">.e.r.</seg> in plano ab <seg type="var">.q.d.</seg> medio ipſius <seg type="var">.z.x</seg>. </s> <s xml:space="preserve">Et quia <lb/>præſuppoſuimus <seg type="var">.p.l.</seg> in eodem medio, inter <seg type="var">.u.s.</seg> et <seg type="var">.a.n.</seg> ideo <seg type="var">.x.q.</seg> ęqualis erit <seg type="var">.x.d.</seg> <lb/>& ex .4. lib. primi <seg type="var">.i.q.</seg> ipſi <seg type="var">.i.d.</seg> et <seg type="var">.e.r.</seg> parallela ipſi <seg type="var">.q.d.</seg> ex .6. lib. 11. cum ipſa quoque <lb/>ſit perpendicularis ſuperficiei <seg type="var">.p.t.</seg> ex <ref>.19. eiuſdem</ref>. </s> <s xml:space="preserve">Hucuſque igitur in figura cor-<lb/>porea <seg type="var">.A.</seg> prodeunt in lucem omnes cauſæ effectuum figuræ ſuperficialis <seg type="var">.A.</seg> ideſt vn <lb/>de fiat, vt in ipſa figura ſuperficiali, triangulum <seg type="var">.o.p.l.</seg> tale conſurgat, & quid ſignifi-<lb/>cet <seg type="var">.o.</seg> et <seg type="var">.o.p.</seg> et <seg type="var">.p.l.</seg> et <seg type="var">.o.l.</seg> & quam ob cauſam tale quoque formetur triangulum <seg type="var">.i.<lb/>q.d.</seg> atque in tantam altitudinem, quantam obtinet <seg type="var">.o.p.</seg> & quid ſint latera <seg type="var">.i.q.</seg> et <seg type="var">.i.<lb/>d.</seg> & quare erigatur <seg type="var">.x.i.</seg> parallela ipſi <seg type="var">.p.o.</seg> ab eadem <seg type="var">.p.o.</seg> tanto ſpatio diſtans, & qua <lb/>ratione producatur à puncto <seg type="var">.Z.</seg> ipſa <seg type="var">.Z.r.e.</seg> parallela ipſi <seg type="var">.q.d</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0132-01" corresp="fig-0132-01a"> <graphic url="0132-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nunc obſeruandum eſt, quòd ſi planum ipſius <seg type="var">.i.q.d.</seg> in figura corporea aliquan-<lb/>tulum inclinatum eſſet orizontem verſus, anguli <seg type="var">.i.q.d.</seg> et <seg type="var">.i.d.q.</seg> maiores exiſterent, <lb/>quàm cum idem eſt ipſi orizonti perpendiculare, quemadmodum clarè demonſtra-<lb/>tum fuit in .39. primi Vitelionis.</s> </p> <p> <s xml:space="preserve">Non igitur rectè fit ſi in figura ſuperficiali ducatur à puncto <seg type="var">.B.</seg> parallela ipſi <seg type="var">.q.d.</seg> <lb/>abſque maiori apertura angulorum <seg type="var">.i.q.d.</seg> et <seg type="var">.i.d.q</seg>.</s> </p> <figure place="here"> <graphic url="0133-01"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> </div> <div type="section"> <head xml:space="preserve">CAP. II.</head> <p> <s xml:space="preserve">CVM verò duæ præcedentes figuræ intellectæ erunt, facilè quoque erit intel-<lb/>ligere duas ſubſequentes <seg type="var">.B.B.</seg> in corporea quarum <seg type="var">.p.l.</seg> extra lineas <seg type="var">.u.s.</seg> et <seg type="var">.a.n.</seg> <lb/>reperitur, vbi enim aduertendum erit oportere ſumere ſemper <seg type="var">.p.x.</seg> figuræ ſuperfi-<lb/>cialis æqualem ei, quæ eſt corporeæ, & eidem ſuperficiali, adiungere <seg type="var">.x.d.</seg> æqualem <lb/>ei, quæ eſt corporeæ, & compoſito <seg type="var">.p.d.</seg> ex dictis duabus lineis, in figura ſuperficiali, <lb/>addere <seg type="var">.d.q.</seg> æqualem ei, quæ eſt figuræ corporeæ, deinde accipere punctum <seg type="var">.l.</seg> in fu-<lb/>perficiali <pb facs="0134" n="122"/><fw type="head">IO. BAPT. BENED.</fw> perficiali, ita diſtans à <seg type="var">.p.</seg> vt in corporea reperitur. </s> <s xml:space="preserve">Po-<lb/> <ptr xml:id="fig-0134-01a" corresp="fig-0134-01" type="figureAnchor"/> ſteà. <seg type="var">x.i.</seg> erigetur æqualis lineæ <seg type="var">.p.o.</seg> & ſuis terminis <lb/>concludetur triangulum <seg type="var">.i.q.d.</seg> & id quod remanet. <lb/></s> <s xml:space="preserve">Vnde ſi longius diſerendo progrediare, patebit ex <num value="4">.<lb/>4.</num> primi <seg type="var">.i.d.</seg> ſuperficialem, futuram æqualem <seg type="var">.i.d.</seg> <lb/>corporeæ. </s> <s xml:space="preserve">Idem dico de <seg type="var">.i.q.</seg> & de reliquis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0134-01" corresp="fig-0134-01a"> <graphic url="0134-01"/> <head xml:space="preserve">CORPOREA.</head> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0134-02"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> </div> <div type="section"> <head xml:space="preserve">CAP. III.</head> <p> <s xml:space="preserve"><hi rend="small caps">MOdvs</hi> hic, proprius eſt, & vniuerſalis, licet in figura ſuperficiali <seg type="var">.A.</seg> ſupe-<lb/>rius poſita, ſecundum communem antiquorum conſuetudinem exemplum <lb/>dederim, effectus enim idem eſt. </s> <s xml:space="preserve">Sed ſi quis vellet conſiderare dictam figuram ſu-<lb/>perficialem <seg type="var">.A.</seg> ſecundum eum modum, quem de figura <seg type="var">.B.</seg> ſuperficiali pręſcripſi, id <lb/>poterit in ſubſcripta figura <seg type="var">.C.</seg> ſpeculari in hunc modum. </s> <s xml:space="preserve">Accipiet enim <seg type="var">.p.x.</seg> ſuper <lb/>ficialem, æqualem corporeæ, & quia in ipſa corporea <seg type="var">.A.</seg> ſuppoſita fuit linea <seg type="var">.p.l.</seg> <lb/>ideſt punctum <seg type="var">.x.</seg> inter duas <seg type="var">.u.s.</seg> et <seg type="var">.a.n.</seg> ſecabimus <seg type="var">.p.x.</seg> ſuperficialem in puncto <seg type="var">.d.</seg> ita, <lb/>vt <seg type="var">.d.x.</seg> ſuperficialis, æqualis ſit corporeæ, & ipſi ſuperficiali <seg type="var">.p.x.</seg> addetur <seg type="var">.x.q.</seg> æqualis <lb/>corporeæ. </s> <s xml:space="preserve">vnde <seg type="var">.q.d.</seg> ſuperfi-<lb/>cialis æqualis erit corporeę,<lb/> <ptr xml:id="fig-0134-03a" corresp="fig-0134-03" type="figureAnchor"/> et <seg type="var">.p.x.</seg> ſuperficiali addetur <seg type="var">.<lb/>x.l.</seg> æqualis corporeæ. </s> <s xml:space="preserve">De <lb/>ijs poſtea quæ dicenda ſuper-<lb/>ſunt, iam ſatis ſuperq́ue di-<lb/>ximus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0134-03" corresp="fig-0134-03a"> <graphic url="0134-03"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quamobrem, punctum <seg type="var">.<lb/>x.</seg> aut intra, aut extra lineam <seg type="var">.<lb/>q.d.</seg> veniat, hunc modum ſe-<lb/>quentes, in errorem non in-<lb/>cidemus, imò efficietur qua -<lb/>drilaterum <seg type="var">.q.r.</seg> ſuperficiale, <lb/>ſimile, & æquale corporeo.</s> </p> </div> <div type="section"> <head xml:space="preserve">CAP. IIII.</head> <p> <s xml:space="preserve"><hi rend="small caps">PVnctvm</hi> verò <seg type="var">.i.</seg> (quod verum eſt punctum perſpectiuæ, vt practici dicere ſo <lb/>lent) quid ſit, hac via & ratione ſub noſtram cognitionem cadit: </s> <s xml:space="preserve">quòd nihil <pb facs="0135" n="123"/><fw type="head">DE PERSPECT.</fw> aliud eſt, quàm punctum, varijs ſectionibus commune, & huiuſmodi punctum, ocu <lb/>lus non eſt, quemadmodum multi Pictores, Sculptores, Architecti, & Perſpectiui <lb/>ignari, ipſum punctum, oculum appellando, falsò crediderunt, quaſi <choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.i.</seg> per-<lb/>ſpectiuæ oculus eſſet.</s> </p> <p> <s xml:space="preserve">In ſupradictis igitur figuris manifeſte eluceſcit cauſa diminutionis obiectorum, <lb/>& altitudinis trianguli æqualis ei, quæ eſt oculi à plano orizontali, vt etiam diſtantię <seg type="var">.<lb/>p.l.p.x.</seg> & cuiuſuis tandem rei. </s> <s xml:space="preserve">Sed vt huius effectus ſcientia magis in vniuerſum pa-<lb/>retur. </s> <s xml:space="preserve">Volo duas hic ſubſcriptas figuras <seg type="var">.D.</seg> corpoream, & <seg type="var">.D.</seg> ſuperficialem à vo-<lb/>bis conſiderari, in quarum corporea, linea <seg type="var">.p.l.</seg> ſit extra duas <seg type="var">.u.s.</seg> et <seg type="var">.a.n.</seg> vt in figu-<lb/>ra <seg type="var">.B.</seg> locata, ita tamen vt planum trianguli <seg type="var">.i.q.d.</seg> diſiunctum ſit à rectangulo ſuper-<lb/>ficiali, ideſt, vt ſeparatum exiſtat à linea <seg type="var">.q.d.</seg> latere ipſius rectanguli, & ſit etiam obli <lb/>quum, reſpectu ipſius rectanguli, ideſt vt communis ſectio dicti plani cum ſuperficie <lb/><seg type="var">a.s.</seg> orizontalis ipſi <seg type="var">.u.a.</seg> parallela non ſit, ſed ſit obliqua, ſi tamen idem planuni per-<lb/>pendiculare dictæ ſuperficiei orizontali <seg type="var">.a.s.</seg> erit: </s> <s xml:space="preserve">& dicta communis ſectio exprima<lb/>tur characteribus .q.ω.α.d.x. nunc in figura corporea habebimus figuram <seg type="var">.e.r.c.m.</seg> <lb/>in plano, quod viſualem pyramidem ſecat, medio cuius figuræ <seg type="var">.e.r.c.m.</seg> oculus po-<lb/>ſitus in <seg type="var">.o.</seg> rectangulum orizontale conſpicit. </s> <s xml:space="preserve">Volentes vero nunc in figura <seg type="var">.D.</seg> ſuper-<lb/>ficiali eam deſcribere, faciem us <seg type="var">.p.x.</seg> ſuperficialem, æqualem corporeæ, eiq́ue <lb/>addemus <seg type="var">.x.l.</seg> æqualem corporeæ, aut ſumemus <seg type="var">.p.l.</seg> eidem corporeæ <choice><ex>aequa- lem</ex><am>ęqua-lem</am></choice>, quam ſecabimus in puncto <seg type="var">.x.</seg> eodem planè modo, quo corporea reperi-<lb/>tur diuiſa; </s> <s xml:space="preserve">erigemus deinde <seg type="var">.p.o.</seg> et <seg type="var">.x.i.</seg> æquales corporeis. </s> <s xml:space="preserve">Secabimus deinde <seg type="var">.x.q.</seg> <lb/>æqualem corporeæ, & ducemus <seg type="var">.q.i.</seg> et <seg type="var">.l.o.</seg> vnde habebimus triangulos <seg type="var">.o.p.l.</seg> et <seg type="var">.i.x.<lb/>q.</seg> ſimiles & æquales corporeis ex .4. primi Eucli. </s> <s xml:space="preserve">Secabimus deinde <seg type="var">.q.x.</seg> in pun-<lb/>cto <seg type="var">.d.</seg> eadem ratione, qua ſecta fuit corporea, & ducemus lineam <seg type="var">.d.i.</seg> vnde habebi-<lb/>mus triangulos <seg type="var">.i.d.q.</seg> et <seg type="var">.i.d.x.</seg> ſimiles corporeis. </s> <s xml:space="preserve">& mediante triangulo <seg type="var">.i.q.d.</seg> hu-<lb/> <ptr xml:id="fig-0135-01a" corresp="fig-0135-01" type="figureAnchor"/> cuſque habebimus ſitus duorum laterum figurę <lb/>rectanguli degradati, ideſt ſitus ipſius <seg type="var">.e.m.</seg> et <seg type="var">.r.<lb/>c.</seg> etiam ſi adhuc neſciatur in qua parte ipſius <seg type="var">.i.<lb/>q.</seg> & ipſius <seg type="var">.i.d.</seg> eſſe <choice><ex>debeant</ex><am>debeãt</am></choice>. </s> <s xml:space="preserve">Quod ſi ſcire volue <lb/>rimus <choice><ex>ſecabitur</ex><am>ſecabit̃</am></choice>. <seg type="var">p.l.</seg> in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.g.</seg> ſimilis corporeæ, <lb/>ſi in ipſa tamen corporea prius protraxerimus <lb/>lineam <seg type="var">.q.d.</seg> latus rectanguli vſque ad <seg type="var">.p.l.</seg> in pun <lb/>cto <seg type="var">.g</seg>. </s> <s xml:space="preserve">Ducetur deinde linea <seg type="var">.o.g.</seg> ſuperficialis, <lb/>quæ ſecabit lineam <seg type="var">.i.x.</seg> in puncto <seg type="var">.f.</seg> linea vero <seg type="var">.<lb/>o.l.</seg> in puncto <seg type="var">.z.</seg> punctis ſitis in <seg type="var">.i.x.</seg> ſuperficiali, <lb/>pręcisè vt in corporea, <choice><ex>quemadmodum</ex><am>quemadmodũ</am></choice> quilibet <lb/>ex ſe facilè cognoſcere poteſt. </s> <s xml:space="preserve">Deinde in cor <lb/>porea, in ſuperficie orizontali ducatur <seg type="var">.p.q.</seg> et <lb/> <ptr xml:id="fig-0135-02a" corresp="fig-0135-02" type="figureAnchor"/> <pb facs="0136" n="124"/><fw type="head">IO. BAPT. BENED.</fw> <seg type="var">.p.u.</seg> & imaginemur <seg type="var">.o.q.</seg> in ſuperficie <seg type="var">.t.s.</seg> vnde trianguli <seg type="var">.o.p.q.</seg> et <seg type="var">.o.p.u.</seg> erunt perpen <lb/>diculares orizonti ex <ref>.18. lib. 11.</ref> et <seg type="var">.ω.m.</seg> et. α. e. communes ſectiones dictorum <choice><ex>duorum</ex><am>duorũ</am></choice> <lb/>triangulorum cum plano trianguli <seg type="var">.i.q.x.</seg> ipſi quoque plano ex .19. eiuſdem lib. erunt <lb/>perpendiculares. </s> <s xml:space="preserve">Nuncautem ſecetur <seg type="var">.q.x.</seg> ſuperficialis in punctis <seg type="var">.ω.</seg> et .α. eadem ra-<lb/>tione; </s> <s xml:space="preserve">qua corporea ſecta ſuit à duabus <seg type="var">.p.q.</seg> et <seg type="var">.p.u.</seg> à quibus punctis <seg type="var">.ω.</seg> et. α. ſuperficia <lb/>libus ductæ ſint duæ <seg type="var">ω.m.</seg> et .α.e. perpendiculares vſque ad latus <seg type="var">.i.q.</seg> in punctis <seg type="var">.m.</seg> et <seg type="var">.<lb/>e.</seg> quę ſitum habebunt in <seg type="var">.i.q.</seg> ſuperficiali pręcisè, vt in corporea, ex .26. primi, du-<lb/>cendo deinde in ſuperficiali duas <seg type="var">.m.f.</seg> et <seg type="var">.e.Z.</seg> eæ æquales erunt corporeis ex .4. pri-<lb/>mi, & ſic anguli <seg type="var">.i.e.z.</seg> et <seg type="var">.i.m.f.</seg> & eę duę lineę <seg type="var">.e.z.</seg> et <seg type="var">.m.f.</seg> fectę erunt à linea <seg type="var">.i.d.</seg> in duo<lb/> <ptr xml:id="fig-0136-01a" corresp="fig-0136-01" type="figureAnchor"/> bus punctis <seg type="var">.r.</seg> et <seg type="var">.c.</seg> vnde <seg type="var">.e.r.</seg> et <seg type="var">.m.c.</seg> æquales <lb/>erunt corporeis ex .26. primi, ſed ita quo-<lb/>que ſe habent duę <seg type="var">.e.m.</seg> et <seg type="var">.r.c.</seg> ſi verum eſt <choice><ex>quod</ex><am>ꝙ</am></choice> dif <lb/>ferentię rerum æqualium ſint adinuicem etiam <lb/>æquales. </s> <s xml:space="preserve">Hac ratione igitur habebimus figu-<lb/>ram quadrilateram <seg type="var">.m.e.r.c.</seg> ſuperficialem om <lb/>ninò ſinlilem, & ęqualem corporeæ. </s> <s xml:space="preserve">Is tamen <lb/>modus prolixus eſt, & arduus, quam ob cau-<lb/>ſam neque ego vnquam <choice><ex>eum</ex><am>eũ</am></choice> viui accommo-<lb/>darem, neque alijs, vt eodem vterentur ſua-<lb/>derem.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0135-01" corresp="fig-0135-01a"> <graphic url="0135-01"/> <head xml:space="preserve">CORPOREA.</head> </figure> <figure xml:id="fig-0135-02" corresp="fig-0135-02a"> <graphic url="0135-02"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> <figure xml:id="fig-0136-01" corresp="fig-0136-01a"> <graphic url="0136-01"/> <head xml:space="preserve">CORPOREA.</head> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0136-02"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> </div> <div type="section"> <head xml:space="preserve">CAP.V.</head> <p> <s xml:space="preserve"><hi rend="small caps">ESt</hi> igitur ſciendum, quòd qui ſciuerit vnum ſolum punctum locare in perſpe <lb/>ctiua, eo modo quem nunc proponam, facilè quoque ſciet ſupra quoduis <choice><ex>planum</ex><am>planũ</am></choice> <lb/>(quod tamen ſit perpendiculare orizonti) quamlibet rem locare. </s> <s xml:space="preserve">Quam ob cauſam <lb/>imaginemur hic ſubſcriptas duas figuras <seg type="var">.E.</seg> <choice><ex>corpoream</ex><am>corporeã</am></choice>, & E. ſuperficialem, & in qua-<lb/>drilatero rectangulo orizontali <seg type="var">.a.u.q.d.</seg> imaginemur eſſe punctum <seg type="var">.b.</seg> quodlibet col-<lb/>locandum in aliquo plano perpendiculari orizonti locato, quemadmodum ſuppo-<lb/>nebatur in figura <seg type="var">.A.</seg> corporea. </s> <s xml:space="preserve">Imaginemur ergo in ipſa figura <seg type="var">.E.</seg> corporea radi-<lb/>um viſualem <seg type="var">.o.b.</seg> qui ſectus ſit à noſtro plano in <seg type="var">.k.</seg> quod quidem <seg type="var">.k.</seg> quærendum eſt <lb/>in triangulo <seg type="var">.i.q.d.</seg> ipſius plani. </s> <s xml:space="preserve">Volo ob hanc igitur rem, vt à puncto <seg type="var">.b.</seg> in figura <seg type="var">.E.</seg> <lb/>ſuperficiali ducatur <seg type="var">.b.c.</seg> ad rectos <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.q.d.</seg> & à puncto <seg type="var">.c.</seg> ad <seg type="var">.i.</seg> ducatur linea <seg type="var">.c.i.</seg> et <seg type="var">.b.m.</seg> <lb/>parallela ipſi <seg type="var">.q.d.</seg> quę ab ipſa <seg type="var">.x.l.</seg> in puncto <seg type="var">.m.</seg> erit diuiſa, & hęc <seg type="var">.x.m.</seg> è directo con-<lb/>iuncta cum <seg type="var">.p.x.</seg> ducatur <seg type="var">.o.m.</seg> quæ ab <seg type="var">.i.x.</seg> ſecta erit in puncto <seg type="var">.f.</seg> à quo ducendo dein-<lb/>de <seg type="var">.f.g.h.</seg> parallela <seg type="var">.q.d.</seg> ab <seg type="var">.i.c.</seg> in puncto <seg type="var">.K.</seg> erit diuiſa. </s> <s xml:space="preserve">Atque id erit quod nobis <lb/>inquirendum propoſueramus.</s> </p> <pb facs="0137" n="125"/> <fw type="head">DE PERSPECT.</fw> <p> <s xml:space="preserve">Ad cuius rei <choice><ex>ſpeculationem</ex><am>ſpeculationẽ</am></choice>, imaginatione con<lb/> <ptr xml:id="fig-0137-01a" corresp="fig-0137-01" type="figureAnchor"/> cipiamus lineam <seg type="var">.b.c.</seg> corpoream, protractam eſ <lb/>ſe vſque ad <seg type="var">.y.</seg> lineæ <seg type="var">.s.n.</seg> & imaginatione ſit com <lb/><choice><ex>præhenſa</ex><am>præhẽſa</am></choice> linea <seg type="var">.y.o.</seg> et <seg type="var">.b.</seg> R<unclear reason="illegible"/>. parallela eidem, ideo <lb/>ob rationes iam dictas de figura <seg type="var">.A.</seg> hæ tres li-<lb/>neæ <seg type="var">.o.y</seg>: <seg type="var">i.c</seg>: et. </s> <s xml:space="preserve">R<unclear reason="illegible"/> <seg type="var">.b.</seg> ſimul cum linea <seg type="var">.o.b.</seg> erunt <lb/>in vna eademq́ue ſuperficie plana, quam cha-<lb/>racteribus <seg type="var">.y.</seg> R<unclear reason="illegible"/>. notemus .et <seg type="var">.i.c.</seg> eius erit ſe-<lb/>ctio communis cum plano, in quo quæritur <choice><ex>pun- ctum</ex><am>pũ-ctum</am></choice>, et <seg type="var">.f.k.</seg> ipſius plani cum triangulo <seg type="var">.o.b.m.</seg> <lb/>erit ſectio communis, & parallela ipſi <seg type="var">.q.d.</seg> ex <ref>.<lb/>6. lib. 11.</ref> quia <seg type="var">.k.f.</seg> perpendicularis eſt ſuperfi-<lb/>ciei <seg type="var">.p.t.</seg> ex .19. eiuſdem cum triangulus <seg type="var">.o.<lb/>b.m.</seg> eidem ſuperficiei <seg type="var">.p.t.</seg> ex .18. eiuſdem <lb/>perpendicularis exiſtat. </s> <s xml:space="preserve">Vnde perſpicuè pa-<lb/>tet ratio quare protracta<unclear reason="illegible"/> ſit parallela <seg type="var">.b.c.</seg> et <lb/>quare ducta ſit <seg type="var">.i.c.</seg> et coniuncta <seg type="var">.x.m.</seg> cum <seg type="var">.x.<lb/>p.</seg> directè, & quare ducta ſit <seg type="var">.o.m.</seg> et <seg type="var">.f.k</seg>. </s> <s xml:space="preserve">Lau-<lb/>do igitur vt ſemper præſupponatur <seg type="var">.p.x.</seg> perpen <lb/>dicularis baſi ipſius plani & præſupponatur, (vt <lb/>rem totam vnò verbo complectar) ſuperficies <seg type="var">.<lb/>p.t.</seg> perpendicularis plano, & orizonti. </s> <s xml:space="preserve">Quod <lb/>reliquum eſt, neceſſariv<unclear reason="illegible"/>m non eſt, niſi ad ſpe-<lb/>culandum. </s> <s xml:space="preserve">Neceſſariæ ergo non ſunt aliæli-<lb/>neæ, quàm.p.x: <seg type="var">p.o.x.i</seg>: <seg type="var">b.c</seg>: et <seg type="var">.x.m.</seg> è dire-<lb/>cto coniuncta cum <seg type="var">.p.x.</seg> (quæ <seg type="var">.x.m.</seg> coniuncta <lb/> <ptr xml:id="fig-0137-02a" corresp="fig-0137-02" type="figureAnchor"/> <pb facs="0138" n="126"/><fw type="head">IO. BAPT. BENED.</fw> æqualis ſit ipſi <seg type="var">.b.c.</seg>) <lb/> <ptr xml:id="fig-0138-01a" corresp="fig-0138-01" type="figureAnchor"/> <seg type="var">o.m.</seg> etiam <seg type="var">.i.c</seg>: et <seg type="var">.f.<lb/>k.</seg> vt in figura <seg type="var">.F.</seg> cla <lb/>riſſimè patet. </s> <s xml:space="preserve">Alias <lb/><choice><ex>autem</ex><am>autẽ</am></choice> multas lineas in <lb/>alijs figuris non <choice><ex>aliam</ex><am>aliã</am></choice> <lb/>ob <choice><ex>camm</ex><am>cãm</am></choice> duxi, <choice><ex>quam</ex><am>quã</am></choice> ad <lb/>facilius <choice><ex>eruendas</ex><am>eruẽdas</am></choice> è te-<lb/>nebris ignorantiæ, & <lb/>in cognitionis lucem <lb/>proferendas horum <lb/>effectuum cauſas, vt <lb/>dixi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0137-01" corresp="fig-0137-01a"> <graphic url="0137-01"/> <head xml:space="preserve">CORPOREA.</head> </figure> <figure xml:id="fig-0137-02" corresp="fig-0137-02a"> <graphic url="0137-02"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> <figure xml:id="fig-0138-01" corresp="fig-0138-01a"> <graphic url="0138-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">CAP. VI.</head> <p> <s xml:space="preserve"><hi rend="small caps">SEd</hi> vtlocum altitudinis, in noſtro plano perpendiculari orizonti, & ita <choice><ex>locatum</ex><am>locatũ</am></choice><unclear reason="illegible"/>, <lb/>vt poſtremo diximus, inueniamus; </s> <s xml:space="preserve">duas hîc ſubſcriptas figuras conſiderabimus <seg type="var">.<lb/>G.</seg> corpoream, & G. ſuperficialem, ſimiles duabus <seg type="var">.E.E.</seg> proximè præcedentibus, <lb/>in quarum corporea ſit linea <seg type="var">.b.M.</seg> altitudinis perpendicularis orizonti. </s> <s xml:space="preserve">Quare ſi <lb/>deſiderabis inuenire in noſtro plano ſitum puncti <seg type="var">.M.</seg> ideſt punctum radij <seg type="var">.o.M.</seg> vi-<lb/>ſualis in quo ipſe radius à plano eſt diuiſus, quod ſit <seg type="var">.R.</seg> quamuis extra <choice><ex>triangulum</ex><am>triangulũ</am></choice> <lb/><seg type="var">i.q.d.</seg> tibi imaginatione confige ductam eſſe lineam <seg type="var">.p.b.</seg> quæ erit ſectio commu-<lb/>nis orizontis cum ſuperficie <seg type="var">.o.p.b.M.</seg> quæ ſuperficies erit perpendicularis ipſi ori-<lb/>zonti ex .18. lib 11. </s> <s xml:space="preserve">Quòd autemnon minus <seg type="var">.o.p.</seg> quàm.M.b. ſit in vna eademq́ue <lb/>ſuperficie dubitandum non eſt, quia ſi imaginabimur ductam eſſe lineam <seg type="var">.p.M.</seg> ha <lb/>bebimus triangulum <seg type="var">.o.p.b.</seg> cum triangulo <seg type="var">.M.b.p.</seg> communibus partibus in vna ea-<lb/>demq́ue ſuperficie conſtantem, vt triangulum quoque <seg type="var">.o.p.M.</seg> cum triangulo <seg type="var">M.b.</seg> <lb/>o & triangulum <seg type="var">.o.p.b.</seg> cum triangulo <seg type="var">.o.p.M.</seg> & triangulum <seg type="var">.M.b.p.</seg> cum triangulo <seg type="var">.<lb/>M.b.o</seg>. </s> <s xml:space="preserve">Vnde cum quilibet triangulus in vnica tantum ſuperficie ſit ex .2. lib. 11. ſe-<lb/>quetur ſuperficiem <seg type="var">.o.p.b.M.</seg> planam eſſe, & vnicam, cuius communis ſectio cum no-<lb/>ſtro plano ſit. θ.K.R. quæ perpendicularis orizonti exiſtet ex .19. lib. 11. eritq́ue pa-<lb/>rallela ipſi <seg type="var">.i.x.</seg> ex .6. eiuſdem. </s> <s xml:space="preserve">Imaginare nunc erectam eſſe <seg type="var">.m.T.</seg> æqualem ipſi <seg type="var">.<lb/>b.M.</seg> orizonti perpendicularem, quæ extenſa erit in ſuperficie <seg type="var">.p.t.</seg> quod ex ſe ad <lb/>conſiderandum admodum facilè, clarumq́ue exiſtit, reducendo ad impoſſibilia <lb/>quemlibet hæc negare volentem. </s> <s xml:space="preserve">Imaginemur quoque ductam eſſe lineam <seg type="var">.M.<lb/>T.</seg> quæ <seg type="var">.b.m.</seg> ex .33. primi erit parallela, quia <seg type="var">.m.T.</seg> ęqualis <seg type="var">.b.M.</seg> parallela eſt <lb/>ipſi <seg type="var">.b.M.</seg> ex .6. lib. 11. præter hæc <seg type="var">.b.m.</seg> parallela eſt ipſi <seg type="var">.q.d.</seg> quia ſic fuit ducta <lb/>ſuperius, vnde <seg type="var">.M.T.</seg> parallela erit ipſi <seg type="var">.q.d.</seg> ex .9. vndecimi, & obid perpendi-<lb/>cularis erit ſuperficiei <seg type="var">.b.t.</seg> ex .8. eiuſdem. </s> <s xml:space="preserve">Nunc ſit <seg type="var">.R.V.</seg> communis ſectio trian-<lb/>guli <seg type="var">.o.M.T.</seg> cum noſtro plano, vnde <seg type="var">.R.V.</seg> perpendicularis erit ſuperficiei <seg type="var">.p.t.</seg> <lb/>ex .19. lib. 11. quam ob cauſam parallela erit ipſi <seg type="var">.q.d.</seg> ex .6. aut ex .9. eiuſdem <lb/>quia ex .6. dicta, parallela eſt ipſi <seg type="var">.M.T</seg>. </s> <s xml:space="preserve">Atſi <seg type="var">.R.V.</seg> parallela eſt ipſi <seg type="var">.q.d.</seg> <lb/>etiam <seg type="var">.f.K.</seg> probatum iam fuit parallelam eſſe eidem, ergo <seg type="var">.R.V.</seg> parallela erit <lb/>ipſi <seg type="var">.K.f.</seg> ex .30. primi, </s> <s xml:space="preserve">Vnde ex .34. æqualis erit ipſi <seg type="var">.K.f</seg>. </s> <s xml:space="preserve">Accedamus nunc <lb/>ad <choice><ex>figuram</ex><am>figurã</am></choice> <seg type="var">.G.</seg> <choice><ex>extructam</ex><am>extructã</am></choice> ſupra figuram <seg type="var">.E.</seg> ſuperficialem, & erigamus <seg type="var">.m.T.</seg> perpendi-<lb/>cularem ipſi <seg type="var">.m.p.</seg> ſed æqualem perfectæ altitudini, & ducamus <seg type="var">.T.o.</seg> vt ſecet li-<lb/>neam <seg type="var">.i.x.</seg> in puncto <seg type="var">.V.</seg> ab ipſo ducentes <seg type="var">.V.R.</seg> parallelam ipſi <seg type="var">.q.d.</seg> ducendo de- <pb facs="0139" n="127"/><fw type="head">DE PERSPECT.</fw> inde <seg type="var">.k.R.</seg> parallelam ipſi <seg type="var">.i.a.</seg> habebimus altitudinem <seg type="var">.k.R.</seg> quam quærebamus in <lb/>noſtro plano. </s> <s xml:space="preserve">Quod cum ſui natura clarum euadat, laborem ratiocinandi de eo, <lb/> <ptr xml:id="fig-0139-01a" corresp="fig-0139-01" type="figureAnchor"/> cuilibet vel mediocriter in præclariſſima hac ſcien-<lb/>tia erudito relinquo. </s> <s xml:space="preserve">ideſt, vt probetur <seg type="var">.k.R.</seg> <choice><ex>ſu- perficialem</ex><am>ſu-perficialẽ</am></choice>, <choice><ex>æqualem</ex><am>æqualẽ</am></choice> eſſe corporeæ. </s> <s xml:space="preserve">Sed <choice><ex>tollendo</ex><am>tollẽdo</am></choice> <choice><ex>ſuper- fluitatem</ex><am>ſuper-fluitatẽ</am></choice> linearum, & hoc <choice><ex>accommodantes</ex><am>accõmodantes</am></choice> vt in figura <seg type="var">.<lb/>F.</seg> diligenter conſideretur figura <seg type="var">.H</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0139-01" corresp="fig-0139-01a"> <graphic url="0139-01"/> <head xml:space="preserve">CORPOREA.</head> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0139-02"/> <head xml:space="preserve">SVPERFICIALIS.</head> </figure> <figure place="here"> <graphic url="0139-03"/> <head xml:space="preserve">SVPERFICIALIS</head> </figure> <pb facs="0140" n="128"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head xml:space="preserve">CAP. VII.</head> <p> <s xml:space="preserve">ALiarn tamen inueni viam breuiorem vt in figura <seg type="var">.H.H.</seg> in qua ſit punctus <seg type="var">.<lb/>b.</seg> perfecti, & <seg type="var">.k.</seg> degradati plani. </s> <s xml:space="preserve">Nunc ducatur <seg type="var">.b.c.s.</seg> ad rectos cum <seg type="var">.<lb/>p.m.</seg> indefinitè, quæ quidem abſcindatur in puncto <seg type="var">.s.</seg> ita quòd <seg type="var">.c.s.</seg> æqualis ſit alti <lb/>tudini perfectæ, deinde coniungatur rectà. s. cum <seg type="var">.i</seg>. </s> <s xml:space="preserve">Tunc ſi ab <seg type="var">.k.</seg> vſque ad <choice><ex>protractam</ex><am>protractã</am></choice> <lb/><seg type="var">i.s.</seg> ducta fuerit <seg type="var">.k.R.</seg> parallela li-<lb/> <ptr xml:id="fig-0140-01a" corresp="fig-0140-01" type="figureAnchor"/> neę <seg type="var">.c.s.</seg> hæc <seg type="var">.R.k.</seg> erit altitudo <lb/>quæſita ſeu degradata.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0140-01" corresp="fig-0140-01a"> <graphic url="0140-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quod ita probo. </s> <s xml:space="preserve">Iam nulli du <lb/>bium eſt quin <seg type="var">.f.V.</seg> ſit æqualis alti-<lb/>tudini quęſitæ ſeu degradatę, quo <lb/><choice><ex>tieſcunque</ex><am>tieſcunq;</am></choice> ergo <choice><ex>probauerimus</ex><am>ꝓbauerimus</am></choice> <seg type="var">.k.R.</seg> <lb/>æqualem eſſe lineæ <seg type="var">.f.V.</seg> habebi-<lb/>mus propoſitum. </s> <s xml:space="preserve">Quare certum <lb/>nobis erit eandem proportionem <lb/>eſſe lineæ <seg type="var">.c.s.</seg> ad <seg type="var">.k.R.</seg> quam <seg type="var">.c.i.</seg> ad <lb/><seg type="var">k.i.</seg> et <seg type="var">.c.i.</seg> ad <seg type="var">.k.i.</seg> vt <seg type="var">.x.i.</seg> ad <seg type="var">.f.i.</seg> et <seg type="var">.x.<lb/>i.</seg> ad <seg type="var">.f.i.</seg> vt <seg type="var">.m.o.</seg> ad <seg type="var">.f.o.</seg> et <seg type="var">.m.o.</seg> ad <seg type="var">.<lb/>f.o.</seg> vt <seg type="var">.m.T.</seg> ad <seg type="var">.f.V.</seg> ex ſimilitudine <lb/>triangulorum. </s> <s xml:space="preserve">Ergo <seg type="var">.m.T.</seg> ad <seg type="var">.f.V.</seg> <lb/>erit vt <seg type="var">.c.s.</seg> ad <seg type="var">.k.R.</seg> ex .11. quinti, <lb/>ſed <seg type="var">.c.s.</seg> ſumpta fuit æqualis <seg type="var">.m.T</seg>. <lb/></s> <s xml:space="preserve">quare <seg type="var">.c.s.</seg> ad <seg type="var">.f.V.</seg> erit, vt <seg type="var">.m.T.</seg> ad <lb/><choice><ex>eandem</ex><am>eãdẽ</am></choice> <seg type="var">.f.V.</seg> ex .7. <choice><ex>quinti</ex><am>ꝗnti</am></choice>, & ex .11. eiuſ-<lb/>dem <seg type="var">.c.s.</seg> ad <seg type="var">.f.V.</seg> erit vt <seg type="var">.c.s.</seg> ad <seg type="var">.k.R.</seg> <lb/>quapropter ex .9. eiuſdem <seg type="var">.k.R.</seg> æqualis erit <seg type="var">.f.V</seg>.</s> </p> </div> <div type="section"> <head xml:space="preserve">CAP. VIII.</head> <p> <s xml:space="preserve">MOdus ab antiquis philoſophis obſeruatus, eſt etiam vtilis, <choice><ex>compendioſaque</ex><am>compendioſaq́;</am></choice> via <lb/>progreditur, cuius ſpeculationem, in ſubſcripta figura, quadam ex parte <choice><ex>ſecun- dum</ex><am>ſecũ-dum</am></choice> morem antiquum, quadam etiam ex parte ſecundum ingenij mei vires <choice><ex>conſtru- cta</ex><am>cõſtru-cta</am></choice>, cognoſcemus. </s> <s xml:space="preserve">In qua ego diuiſi <seg type="var">.x.i.</seg> in puncto <seg type="var">.s.</seg> ab <seg type="var">.x.</seg> ita eleuato, quanta eſt <lb/> <ptr xml:id="fig-0140-02a" corresp="fig-0140-02" type="figureAnchor"/> <pb facs="0141" n="129"/><fw type="head">DE PERSPECT.</fw> vera altitudo ipſius <seg type="var">.M.T.</seg> et <seg type="var">I.s.</seg> duxi ſupponendo eſſe <seg type="var">.I.</seg> <choice><ex>punctum</ex><am>punctũ</am></choice> pſpectiuæ <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> <lb/>antiquos, ideſt angulum ſupremum trianguli antiquorum à punctoq́ue <seg type="var">.k.</seg> meo duxi <lb/><seg type="var">k.f.</seg> parallelam ipſi <seg type="var">.c.m.p.</seg> vſque ad <seg type="var">.i.x.</seg> in puncto <seg type="var">.f.</seg> & à puncto à communi ipſis <seg type="var">.k.f.</seg> <lb/>et <seg type="var">.i.x.</seg> vſque ad <seg type="var">.I.s.</seg> duxi quoque <seg type="var">.A.B.</seg> parallelam ipſi <seg type="var">.i.x.</seg> atque hæc omnia ex more <lb/>antiquo præſtiti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0140-02" corresp="fig-0140-02a"> <graphic url="0140-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nunc verò eum conſiderans modum, quem ego de figuris <seg type="var">.G.H.</seg> antecedentibus <lb/>præſcripſi, videndum eſt, an punctum <seg type="var">.B.</seg> tribus lineis <seg type="var">.A.B.I.s.</seg> et <seg type="var">.R.V.</seg> quarum hęc vl <lb/>tima à me iam ducta fuit, commune exiſtat, ideſt vtrum <seg type="var">.A.B.</seg> æqualis exiſtat ipſi <seg type="var">.K.<lb/>R.</seg> quam ſecundum modum à me adinuentum, reuera ſcimus eſſe deſideratam altitu <lb/>dinem in perſpectiua. </s> <s xml:space="preserve">Quod tunc à nobis probatum erit, quando rationibus clarè <lb/>patebit ipſam <seg type="var">.A.B.</seg> æqualem eſſe ipſi <seg type="var">.f.V</seg>. </s> <s xml:space="preserve">Quamobrem ducamus <seg type="var">.I.f.</seg> vſque ad <seg type="var">.ω.</seg> <lb/>lineæ <seg type="var">.c.p.</seg> vnde ratione ſimilitudinis triangulorum manifeſtè intelligemus, eandem <lb/>proportionem eſſe ipſius <seg type="var">.m.T.</seg> ad <seg type="var">.f.V.</seg> quæ eſt <seg type="var">.m.o.</seg> ad <seg type="var">.f.o.</seg> & eius, quæ eſt <seg type="var">.m.o.</seg> ad <seg type="var">.f.<lb/>o.</seg> quæ eſt <seg type="var">.ω.I.</seg> ad <seg type="var">.f.I.</seg> & eius, quæ eſt <seg type="var">.ω.I.</seg> ad <seg type="var">.f.I.</seg> quæ eſt <seg type="var">.x.I.</seg> ad <seg type="var">.A.I.</seg> & eius, quæ eſt <seg type="var">.x.<lb/>I.</seg> ad <seg type="var">.A.I.</seg> quæ eſt <seg type="var">.x.s.</seg> ad <seg type="var">.A.B.</seg> ideſt vt eius, quæ eſt <seg type="var">.m.T.</seg> ad <seg type="var">.A.B.</seg> ſed idem <choice><ex>quoque</ex><am>quoq;</am></choice> erat <lb/>de <seg type="var">.m.T.</seg> ad <seg type="var">.f.V</seg>. </s> <s xml:space="preserve">Vnde ſequitur <seg type="var">.A.B.</seg> æqualem eſſe <seg type="var">.f.V.</seg> ex .9. quinti Eucli. <choice><ex>atque</ex><am>atq;</am></choice> etiam <lb/>ipſi <seg type="var">.k.R.</seg> quod à nobis propoſitum eſt inquirendum.</s> </p> </div> <div type="section"> <head xml:space="preserve">CAP. IX.</head> <figure place="here"> <graphic url="0141-01"/> </figure> <p> <s xml:space="preserve"><hi rend="small caps">INstitvens</hi> etiam ſermonem de figuris ſu-<lb/>perficialibus orizontalibus, ſeu de plantis, <lb/>pulcherrimum quendam modum, quem ego ad <lb/>locandum quodlibet punctum in perſpectiua, <lb/>(degradatum cum fuerit <choice><ex>parallelogrammum</ex><am>parallelogrãmum</am></choice> quod <lb/>dam rectangulum, in noſtro plano perpendicula <lb/>ri orizonti, quemadmodum in ſuperioribus figu-<lb/>ris <seg type="var">.A.</seg> demonſtrauimus) conſideraui, ſilentio <lb/>haud prætereundum eſſe.</s> </p> <p> <s xml:space="preserve">Sit igitur in ſubſcripta hîc figura <seg type="var">.K.</seg> in paralle <lb/><choice><ex>logrammo</ex><am>logrãmo</am></choice> perfecto <choice><ex>punctum</ex><am>pũctum</am></choice> <seg type="var">.b.</seg> quod locari debeat <lb/>in degradato <seg type="var">.e.q.d.r</seg>. </s> <s xml:space="preserve">Nunc à duobus quorumli-<lb/>bet quatuor angulorum <seg type="var">.q.u.a.d.</seg> ducuntur duæ li-<lb/>neæ occultæ <seg type="var">.q.g.</seg> et <seg type="var">.u.f.</seg> per punctum <seg type="var">.b.</seg> vſque ad <lb/>latera <seg type="var">.q.d.</seg> et <seg type="var">.u.a.</seg> ita tamen vt eorum extremita-<lb/>tes <seg type="var">.g.</seg> et <seg type="var">.f.</seg> intus cadant inter <seg type="var">.q.d.</seg> et <seg type="var">.u.a.</seg> ipſorum <lb/>laterum, ideſt vt non ſecent duo latera <seg type="var">.q.u.</seg> aut <seg type="var">.d.<lb/>a</seg>. </s> <s xml:space="preserve">Deinde punctum <seg type="var">.f.</seg> inter <seg type="var">.q.</seg> et <seg type="var">.d.</seg> <choice><ex>coniungatur</ex><am>cõiungatur</am></choice> oc-<lb/>cultè cum angulo degradato <seg type="var">.e.</seg> qui <choice><ex>correſpondet</ex><am>correſpõdet</am></choice> <seg type="var">.<lb/>u.</seg> perfecti, mediante linea <seg type="var">.e.f.</seg> quæ erit <seg type="var">.u.f.</seg> degra <lb/>dita in noſtro plano. </s> <s xml:space="preserve">Deinde ſumatur punctum <seg type="var">.<lb/>n.</seg> in linea <seg type="var">.q.d.</seg> tam diſtans à <seg type="var">.q.</seg> quàm.g. diſtat ab <seg type="var">.<lb/>u.</seg> ducaturq́ue linea <seg type="var">.i.n.</seg> quæ lineam <seg type="var">.e.r.</seg> in puncto <lb/>c. diuidet, quod exijs, quæ ſuperius iam diximus <lb/>ad ipſum <seg type="var">.g.</seg> referetur. </s> <s xml:space="preserve">Ducendo poſtea lineam oc <lb/>cultam <seg type="var">.q.c.</seg> patebit eam correſpondere lineæ <seg type="var">.q.g.</seg> quæ ſecans lineam <seg type="var">.e.f.</seg> in puncto <seg type="var">.<lb/>t.</seg> hoc, communi ſcientiæ ratione, reſpondebit ipſi <seg type="var">.b.</seg> vt omnes cognoſcent.</s> </p> <pb facs="0142" n="130"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Sed ſi fortè punctum <seg type="var">.b.</seg> eſſet in aliquo <lb/> <ptr xml:id="fig-0142-01a" corresp="fig-0142-01" type="figureAnchor"/> laterum, puta <seg type="var">.q.u.</seg> volo vt in rectangulo per-<lb/>fecto <seg type="var">.q.d.a.u.</seg> ducta ſit vna diagonalis quam <lb/>volueris puta <seg type="var">.q.a.</seg> deinde à puncto <seg type="var">.b.</seg> ad reli-<lb/>quum angulum oppoſiti lateris ducta ſit recta <seg type="var">.<lb/>b.d.</seg> ita quod à diagonali ſecetur in puncto <seg type="var">.ω.</seg> <lb/>per quod punctum demum à reliquo angulo la <lb/>teris <seg type="var">.q.u.</seg> ducta ſit <seg type="var">.u.ω.</seg> vſque ad latus <seg type="var">.q.d.</seg> in <choice><ex>pum</ex><am>pũ</am></choice> <lb/>cto <seg type="var">.f.</seg> quo facto, ita faciendum erit in rectangu <lb/>lo degradato, hoc eſt ducenda erit diagonalis <seg type="var">.<lb/>q.r.</seg> quę correſpondet diagonali <seg type="var">.q.a.</seg> perfecti <lb/>deinde <seg type="var">.f.e.</seg> quæ correſpondet rectæ <seg type="var">.f.u.</seg> perfe-<lb/>cti, quæ etiam interſecabitur à diagonali <seg type="var">.q.r.</seg> <lb/>in puncto <seg type="var">.o.</seg> correſpondens <seg type="var">.ω.</seg> perfecti, per <choice><ex>quem</ex><am>quẽ</am></choice> <seg type="var">.<lb/>o.</seg> à puncto <seg type="var">.d.</seg> cum ducta fuerit <seg type="var">.d.o.</seg> vſque ad <seg type="var">.t.</seg> <lb/>in latere <seg type="var">.q.e.</seg> hoc <choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.t.</seg> correſpondebit pun <lb/>cto <seg type="var">.b.</seg> perfecti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0142-01" corresp="fig-0142-01a"> <graphic url="0142-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Idem eueniet ſi loco diametri <seg type="var">.q.a.</seg> ſumpta <lb/>fuerit diameter <seg type="var">.u.d.</seg> & loco <seg type="var">.b.d.</seg> protracta fue <lb/>rit <seg type="var">.b.a.</seg> deinde loco <seg type="var">.u.ω.f.</seg> ducta fuerit <seg type="var">.q.ω.f.</seg> vn-<lb/>de punctum correſpondens ipſi <seg type="var">.f.</seg> in figura de-<lb/>gradata erit in latere ſupremo <seg type="var">.e.r.</seg> correſpon-<lb/>dens lateri <seg type="var">.u.a.</seg> & ita ducenda erit diameter <seg type="var">.d.<lb/>e.</seg> correſpondens diametro <seg type="var">.d.u.</seg> et <seg type="var">.q.f.</seg> ſurſum <lb/>verſus correſpondens <seg type="var">.q.f.</seg> imum verſus deinde <seg type="var">.<lb/>r.o.</seg> reſpondens <seg type="var">.a.ω.</seg> quæ terminabicur ab <choice><ex>eodem- met</ex><am>eodẽ-met</am></choice> puncto <seg type="var">.t.</seg> vt prius.</s> </p> </div> <div type="section"> <head xml:space="preserve">CAP.X.</head> <p> <s xml:space="preserve"><hi rend="small caps">Ex</hi> mea figura <seg type="var">.F.</seg> ſuperficiali perſpectiuæ facillimum modum locandi quoduis <lb/>punctum in perſpectiua elicui. </s> <s xml:space="preserve">Iuſſi enim vt aptaretur tabula quædam rectan-<lb/>gula exactè plana, triplo aut quadruplo, aut quanto volueris maioris longitu-<lb/>dinis, quàm latitudinis protenſa, quæ quidem latitudo erat ad duos circiter pedes <lb/>deſignata ab <seg type="var">.A.B.C.D.</seg> cuius duobus lateribus <seg type="var">.A.B.</seg> et <seg type="var">.B.C.</seg> iuſſi, vt duæ regulæ affi <lb/>gerentur, quæ ſuperficiem eiuſdem tabulæ excederent, vt <choice><ex>vnum</ex><am>vnũ</am></choice> ex lateribus alicuius <lb/>anguli recti materialis, qui <choice><ex>appellatur</ex><am>appellat̃</am></choice> norma (vt inferius dicam) ei adherere poſſit, cu-<lb/>raui poſtea, ut iuxta angulum <seg type="var">.D.</seg> in puncto <seg type="var">.o.</seg> fixo mobilis regula <seg type="var">.o.Q.</seg> affigeretur <lb/>tantæ longitudinis, aut paulò minoris, quantam occupabat latus <seg type="var">.D.A.</seg> quæ circum <seg type="var">.<lb/>o.</seg> volueretur, in rectitudine poſteà. <seg type="var">o.i.</seg> parallela ipſi <seg type="var">.D.A.</seg> in puncto <seg type="var">.i.</seg> duobus pedi-<lb/>bus longè à latere <seg type="var">.A.B.</seg> aliam quoque mobilem appendere feci <seg type="var">.i.M.</seg> in tantam ferè <lb/>longitudinem extenſam, quanta conſtat <seg type="var">.A.B.</seg> <choice><ex>conſtitui</ex><am>cõſtitui</am></choice> etiam, vt <choice><ex>quoddam</ex><am>quoddã</am></choice> angulum re <lb/>ctum materiale tantæ magnitudinis, quanta nobis vſui eſſe poterat ſuper eadem ta-<lb/>bula; </s> <s xml:space="preserve">necnon regula quædam materialis neceſſariæ longitudinis <choice><ex>ſtatuerentur</ex><am>ſtatuerent̃</am></choice>, <choice><ex>atque</ex><am>atq;</am></choice> hæc <lb/>omnia tenuiſſima, vt fierent curaui. </s> <s xml:space="preserve">Quandam deinde lineam ad <seg type="var">.o.i.</seg> parallelam, <lb/>ideſt <seg type="var">.p.E.</seg> ſuper eadem tabula adeò diſtantem ab <seg type="var">.o.i.</seg> vt inter <seg type="var">.E.p.</seg> et <seg type="var">.B.c.</seg> perfe-<lb/>ctæ res, quæ degradari debebant, locari poſſent, ſignaui. </s> <s xml:space="preserve">Hæc autem diſtantia, quæ <pb facs="0143" n="131"/><fw type="head">DE PERSPECT.</fw> inter <seg type="var">.o.i.</seg> et <seg type="var">.p.E.</seg> intercedebat, altitudinem oculi ab orizonte ſignificabat. </s> <s xml:space="preserve">Signa-<lb/>ui etiam lineam <seg type="var">.i.G.</seg> perpendicularem lineæ <seg type="var">.E.p.</seg> cui affigi poſſet non nihil chartæ <lb/>quotieſcunque volebam in perſpectiua aliquid delineare. </s> <s xml:space="preserve">Quod cum facere deſi-<lb/>derabam, ponebam perfectum optimè affixum in quadrangulo <seg type="var">.E.G</seg>: & in quadran <lb/>gulo <seg type="var">.E.i.</seg> aliquod folium papyri affigebam. </s> <s xml:space="preserve">Ponamus nunc, me voluiſſe conſtitaere <lb/><choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.b.</seg> <choice><ex>ſumebam</ex><am>ſumebã</am></choice> <choice><ex>angulum</ex><am>angulũ</am></choice> <choice><ex>rectum</ex><am>rectũ</am></choice> <choice><ex>materialem</ex><am>materialẽ</am></choice>, ſeu <choice><ex>normam</ex><am>normã</am></choice>, & eius vnum latus, iuxta latus <seg type="var">.<lb/>B.c.</seg> ponebam, atque aliud per punctum <seg type="var">.b.</seg> tranſire faciebam, & vbi hoc latus <choice><ex>lineam</ex><am>lineã</am></choice> <lb/><seg type="var">E.p.</seg> diuidebat, punctum .c ſignabam per quod efficiebatur, vt regula <seg type="var">.i.M.</seg> tranſiret, <lb/>quieſceretq́ue aliquantulum aliquo modo in huiuſmodi ſitu, opera deinde circini <lb/>interuallum <seg type="var">.b.c.</seg> ſumebam, & in <seg type="var">.p.E.</seg> à puncto <seg type="var">.x.</seg> verſus <seg type="var">.E.</seg> punctum <seg type="var">.m.</seg> <choice><ex>ſignabam</ex><am>ſignabã</am></choice>: </s> <s xml:space="preserve">per <lb/>quod faciebam, vt tranſiret regula <seg type="var">.o.Q.</seg> quæ lineam <seg type="var">.x.i.</seg> in puncto <seg type="var">.f.</seg> diuidebat. <lb/></s> <s xml:space="preserve">Angulum deinderectum materialem accipiebam, cuius vnum latus <seg type="var">.A.B.</seg> ponebam, <lb/>aliud verò per punctum <seg type="var">.f.</seg> tranſibat, quod quidem latus regulam <seg type="var">.i.M.</seg> in puncto <seg type="var">.k.</seg> <lb/>(quod ſtatim ſuper folio papyri ſignabatur) interſecabat, atque hoc erat <choice><ex>punctum</ex><am>punctũ</am></choice>, <lb/>quod quærebam, puncto <seg type="var">.b.</seg> correſpondens. </s> <s xml:space="preserve">Huiuſmodi effectus rationes ab ijs, quæ <lb/>ſuperius dixi eliciuntur. </s> <s xml:space="preserve">Atque hæc ad baſes rerum, vt in ſubſcripta figura eluceſcit, <lb/>ſpectabant.</s> </p> <figure place="here"> <graphic url="0143-01"/> </figure> </div> <div type="section"> <head xml:space="preserve">CAP. XI.</head> <p> <s xml:space="preserve"><hi rend="small caps">Ad</hi> degradandas deinde altitudines, vſus ſum mea figura tam <seg type="var">.H.</seg> <choice><ex>quam</ex><am>quã</am></choice> <choice><ex>ent</ex><am>ẽt</am></choice> <seg type="var">.H.H.</seg> <lb/>vt milii ſeſe offerebat occaſio. </s> <s xml:space="preserve">In primis ratione modi figuræ <seg type="var">.H.</seg> <choice><ex>curabam</ex><am>curabã</am></choice>, vt <pb facs="0144" n="132"/><fw type="head">IO. BAPT. BENED.</fw> vnum ex lateribus anguli recti, ſeu normæ regulæ <seg type="var">.B.C.</seg> anniteretur, aliud verò <lb/>per <seg type="var">.m.</seg> in rectitudine cuius ſignabam <seg type="var">m.T.</seg> interuallum æquale altitudini perfecti, <lb/>ideſt punctum <seg type="var">.T.</seg> æqualiter diſtans ab <seg type="var">.m.</seg> tranſire faciebam, deinde regulam <seg type="var">.o.Q.</seg> <lb/>per punctum <seg type="var">.T.</seg> tranſire quoque faciebam: </s> <s xml:space="preserve">& notabam interſectionem ipſius cum <lb/>linea <seg type="var">.i.x.</seg> in puncto <seg type="var">.V.</seg> efficiebam deinde vt vnum ex lateribus anguli recti, lateri ta-<lb/>bulæ <seg type="var">.B.C.</seg> anniteretur, aliudq́ue per punctum <seg type="var">.k.</seg> tranſire faciebam, & in huiuſmodi <lb/>rectitudine à puncto k. ſignabam quandam menſuram æ qualem lineæ <seg type="var">.f.V.</seg> quę erat <lb/><seg type="var">k.R.</seg> pro altitudine degradata.</s> </p> </div> <div type="unknown"> <head xml:space="preserve">ALITER IDEM.</head> <p> <s xml:space="preserve"><hi rend="small caps">MEdiante</hi> deinde figura <seg type="var">.H.H.</seg> vnum ex lateribus anguli recti, lateri tabu-<lb/>læ <seg type="var">.B.C.</seg> vt anniteretur faciebam; </s> <s xml:space="preserve">aliud verò per punctum <seg type="var">.b.</seg> perfecti, ideſt ba <lb/>ſis eiuſdem perfecti tranſire faciebam. </s> <s xml:space="preserve">Et in huiuſmodi rectitudine ſignabam <seg type="var">.c.s.</seg> <lb/>æquale interuallum altitudini perfecti, ideſt punctum <seg type="var">.s.</seg> ita diſtans à <seg type="var">.c.</seg> efficiendo de <lb/>inde, vt latus anguli recti, lateri <seg type="var">.B.C.</seg> tabulæ anniteretur, aliudq́ue per punctum <seg type="var">.k.</seg> <lb/>tranſire faciens ſignabam <seg type="var">.k.R.</seg> indeterminatam. </s> <s xml:space="preserve">Faciens deinde tranſire regulam <lb/><seg type="var">i.M.</seg> per punctum <seg type="var">.s.</seg> notabam punctum <seg type="var">.R.</seg> interſectionis eiuſdem cum linea <seg type="var">.k.R.</seg> <choice><ex>iam</ex><am>iã</am></choice> <lb/>ducta. </s> <s xml:space="preserve">Itaque altitudinem <seg type="var">.k.R.</seg> degradatam habebam. </s> <s xml:space="preserve">Hæc autem via aliquan-<lb/>tulum breuior, expeditiorq́ue altera.</s> </p> <figure place="here"> <graphic url="0144-01"/> </figure> <pb facs="0145" n="133"/> <fw type="head">DE PERSPECT.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">JACOBO SOLDATO MEDIOLANENSI <lb/>Serenißimi Ducis Sabaudiæ Architecto peritißimo.</head> <head xml:space="preserve">CAP. VII.</head> <p> <s xml:space="preserve"><hi rend="small caps">SVperioribvs</hi> diebus non diu poſtquam de perſpectiuis inter nos ſermonem <lb/>habuimus, dum animus totus adhuc in his eſſet. </s> <s xml:space="preserve">Illud in mentem venit quòd exi <lb/>mius ille vir, & profundiſſimæ doctrinæ, nec vnquam ſatis laudatus Daniel Barba-<lb/>rus ſe accepiſſe profitetur à Ioanne Zamberto patritio Veneto, qui ad verbum om <lb/>nia deſumpſerat a Ioanne Cuſino Pariſienſe. </s> <s xml:space="preserve">Nec parum mirabar peritiſſimum il-<lb/>lum Cuſinum, quod in capite quarto ſecundæ partis perſpectiuæ, vt quodpiam pla-<lb/>num quadrilaterum in quadratam figuram redigeret, ſuper vnam datam <choice><ex>lineam</ex><am>lineã</am></choice> qua-<lb/>dratam compoſuiſſe. </s> <s xml:space="preserve">Non animaduertens diſtantiam aut interuallum <seg type="var">.b.c.</seg> degra-<lb/>datum ín linea <seg type="var">.b.f.</seg> (quod eſt <seg type="var">.b.E.</seg>) ita eſſe poſſe latus parallelogrammi rectanguli <lb/>magis longi quam lati, aut magis lati quam longi, vt etiam latus quadrati, quod be-<lb/>neficio ſubſcriptæ hic figuræ facilè depræhendi poteſt. </s> <s xml:space="preserve">Vbi <seg type="var">.b.c.</seg> latitudo eſſe po-<lb/>teſt, tam perfecti degradati in triangulo <seg type="var">.b.n.m.</seg> aut in triangulo <seg type="var">.b.q.t.</seg> quam in trian <lb/>gulo <seg type="var">.b.a.c</seg>. </s> <s xml:space="preserve">Sed perfectum degradati in triangulo <seg type="var">.b.n.m.</seg> magis longum quam <choice><ex>latum</ex><am>latũ</am></choice> <lb/>& perfectum degradatum in triangulo <seg type="var">.b.q.t.</seg> magis latum quam longum, & perfe-<lb/>ctum degradati in triangulo <seg type="var">.b.a.c.</seg> quadratum erit quemadmodum à meis etiam fi-<lb/>guris <seg type="var">.A.</seg> ſcientificè intelligi poteſt. </s> <s xml:space="preserve">Hinc, ad inueniendum perfectum alicuius pla-<lb/>ni degradati, non ſufficere degradationem ſolum interualli inter duos terminos <lb/>ſolos, ideſt <seg type="var">.b.E.</seg> aſſignare, apertè patet, quia non omnia parallelogramma. perfecta <lb/>ab vno <choice><ex>tantum</ex><am>tm̃</am></choice> interuallo producuntur, eo <choice><ex>quod</ex><am>ꝙ</am></choice> non ſunt omnia quadrata. </s> <s xml:space="preserve">Ad <choice><ex>inquirendum</ex><am>inquirendũ</am></choice> <lb/>igitur perfectum alicuius plani <choice><ex>parallelogrammi</ex><am>parallelogrãmi</am></choice>, alicuius propoſiti degradati, oportet <lb/>vniuerſam degradationem tam latitudinis, <choice><ex>quam</ex><am>ꝗ̃</am></choice> longitudinis, & <choice><ex>non</ex><am>nõ</am></choice> folius longitudinis <lb/>aſſignare; </s> <s xml:space="preserve">Vt <choice><ex>exempli</ex><am>exẽpli</am></choice> gratia, in ſubſcripta hic figura, <choice><ex>volendo</ex><am>volẽdo</am></choice> inuenire perfectum paral <lb/>lelogrammum degradati <seg type="var">.b.h.l.m.</seg> dando diſtantiam orizontalem <seg type="var">.b.d.</seg> à pede <seg type="var">.d.</seg> ho-<lb/> <ptr xml:id="fig-0145-01a" corresp="fig-0145-01" type="figureAnchor"/> <pb facs="0146" n="134"/><fw type="head">IO. BAPT. BENED.</fw> minis vſque ad planiſitum in quo degradatio facta ſit: </s> <s xml:space="preserve">ſtatim altitudo <seg type="var">.A.</seg> oculi à pe <lb/>de, quæ tanta ſemper eſſe debet quanta eſt altitudo trianguli <seg type="var">.b.n.m.</seg> qui clauditur, <lb/>protrahendo <seg type="var">.m.l.</seg> et <seg type="var">.b.h.</seg> vſque ad concurſum in <seg type="var">.n.</seg> in lucem prodibit. </s> <s xml:space="preserve">Oporter <lb/>deinde erigere lineam <seg type="var">.b.o.</seg> perpendicularem lineæ <seg type="var">.d.b.m.</seg> & vſque ad eandem pro <lb/>ducere lineam <seg type="var">.l.h.</seg> in puncto <seg type="var">.E.</seg> et à puncto <seg type="var">.A.</seg> per <seg type="var">.E.</seg> vſque ad <seg type="var">.c.</seg> ipſius <seg type="var">.d.b.m.</seg> produ <lb/>ctæ ducere <seg type="var">.A.E.c.</seg> atque deinde protrahere lineam <seg type="var">.o.b.</seg> vſque ad <seg type="var">.T.</seg> ita vt <seg type="var">.b.T.</seg> ęqua <lb/>lis ſit ipſi <seg type="var">.b.c.</seg> & ad ipſam à puncto <seg type="var">.m.</seg> ducere parallelam <seg type="var">.m.R.</seg> & à puncto <seg type="var">.T.</seg> ducere <seg type="var">.<lb/>T.R.</seg> parallelam ipſi <seg type="var">.b.m</seg>. </s> <s xml:space="preserve">Vnde ex .34. primi Eucli <seg type="var">.m.R.</seg> æqualis erit ipſi <seg type="var">.b.T.</seg> et <seg type="var">.R.<lb/>T.</seg> ipſi <seg type="var">.m.b.</seg> & anguli in rectos euadent, atque hoc parallelo grammum rectangulum <lb/>erit verum perfectum degradati <seg type="var">.b.m.l.h.</seg> obrationes à me circa <choice><ex>figuram</ex><am>figurã</am></choice> <seg type="var">.A.</seg> adductas. <lb/> <ptr xml:id="fig-0146-01a" corresp="fig-0146-01" type="figureAnchor"/> </s> <s xml:space="preserve">Sed eſt hic quod magis nos commoueat, quia cum ex linea <seg type="var">.b.c.</seg> quadratum <seg type="var">.b.g.</seg> pro <lb/>duxerit, vult eum poſtea degradare. </s> <s xml:space="preserve">Quod vt faciat (hanc figuram videbis in cap <num value="4">.<lb/>4.</num> ſecundæ partis Danielis Barbari) oculum <seg type="var">.A.</seg> in eadem ſuperficie extenſa, <lb/>quadrati <seg type="var">.b.g.</seg> collocat. </s> <s xml:space="preserve">Quod rectè fieti non poteſt, quia oculum hoc mo-<lb/>do locantes, viſualesq́ue radios beneficio vnius plani ſituati in <seg type="var">.b.f.</seg> ſecantes in ipſo <lb/>plano, nihil aliud quam <choice><ex>dictam</ex><am>dictã</am></choice> <choice><ex>lineam</ex><am>lineã</am></choice> <seg type="var">.b.f.</seg> & <choice><ex>nullam</ex><am>nullã</am></choice> <choice><ex>degradationem</ex><am>degradationẽ</am></choice> <choice><ex>inuenient</ex><am>inueniẽt</am></choice>. </s> <s xml:space="preserve">Id quod, & ſi <lb/>natura ſua ſit omnibus notum, ponit <choice><ex>tantum</ex><am>tñ</am></choice> id ipſum Vitelio pro quinta propoſitione <lb/>quarti libri de perſpectiua. </s> <s xml:space="preserve">Præter hæc, credit latera <seg type="var">.b.d.</seg> et <seg type="var">.c.e.</seg> quadrati degra-<lb/>dati ſemper videri mediantibus angulis <seg type="var">.b.A.c.</seg> et <seg type="var">.f.A.g.</seg> quod fieri <choice><ex>non</ex><am>nõ</am></choice> poteſt, quem-<lb/>a dmodum ex mea figura corporea <seg type="var">.A.</seg> facilè cognoſcere poſſumus, </s> <s xml:space="preserve">propterea quòd <lb/>latera <seg type="var">.d.r.</seg> et <seg type="var">.q.e.</seg> meæ figuræ, mediantibus angulis <seg type="var">.d.o.r.</seg> et <seg type="var">.q.o.e.</seg> qui extra ſuperfi-<lb/>ciem <seg type="var">.s.a.</seg> exiſtunt videntur, vnde ſi quis imaginaretur in puncto <seg type="var">.p.</seg> oculum eſſe, & <lb/>ab ipſo ad <seg type="var">.u.</seg> et <seg type="var">.q.</seg> duas lineas duceret, angulus <seg type="var">.q.o.u.</seg> nunc maior, nunc minor eſſet <lb/>angulo <seg type="var">.q.p.u.</seg> aliquando etiam æqualis, quamuis rariſſime; </s> <s xml:space="preserve">Sub diuerſis igitur an-<lb/>gulis, pro maiori parte, deteguntur latera, à partibus quadrati tam degradati, quàm <lb/>perfecti, quæ non ſunt anguli <seg type="var">.b.A.c.</seg> et <seg type="var">.f.A.G</seg>. </s> <s xml:space="preserve">Quod vero idem poſtea dicat eam <lb/>proportionem eſſe ab <seg type="var">.b.E.</seg> ad <seg type="var">.f.h.</seg> & ſimul ad <seg type="var">.c.g.</seg> quæ eſt ab <seg type="var">.a.g.</seg> ad <seg type="var">.h.g.</seg> id tuo relin <pb facs="0147" n="135"/><fw type="head">DE PERSPECT.</fw> quam iudicio. </s> <s xml:space="preserve">Tibi quoque conſiderandum relinquo; </s> <s xml:space="preserve">cum rationabilis degrada-<lb/>tio eſſe debeat, qua ratione neceſſarium ſit, vt diſtantiæ reſq́ue, in vna & <choice><ex>eadem</ex><am>eadẽ</am></choice> pro-<lb/>portione cum altitudine oculi ad rem degradatam exiſtant? </s> <s xml:space="preserve">Cum poſtea degrada-<lb/>uerit <choice><ex>quadratum</ex><am>quadratũ</am></choice>, is ſcriptor, in figura <seg type="var">.d.b.c.e.</seg> eum bene & ex perſpectiuæ optimis <lb/>legibus degradatum fuiſſe probare nititur; </s> <s xml:space="preserve">ſolum probans <seg type="var">.d.e.</seg> æqualem eſſe ipſi <seg type="var">.<lb/>E.h.</seg> <choice><ex>qua</ex><am>q̃</am></choice> <seg type="var">.E.h.</seg> <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> ipſum eſt degra datio lateris <seg type="var">.c.g.</seg> & <choice><ex>cum</ex><am>cũ</am></choice> ſuperius dixerit, ſetria <lb/>quadrati plana degradauiſſe, quia <seg type="var">.b.E.</seg> degradat <seg type="var">.b.c.</seg> et <seg type="var">.E.h.</seg> degradat <seg type="var">.c.g.</seg> et <seg type="var">.f.<lb/>h.</seg> degradat <seg type="var">.f.g.</seg> nec quidem de lateribus <seg type="var">.b.d.</seg> et <seg type="var">.c.e.</seg> loquitur, quia ſi <seg type="var">.c.g.</seg> <lb/>perfecti, degradatum eſt in <seg type="var">.E.h</seg>: et <seg type="var">.d.e.</seg> rectè protracta exiſtit, cum ſit æqua-<lb/>lis ipſi <seg type="var">.E.h.</seg> cum etiam <seg type="var">.b.d.</seg> et <seg type="var">.c.e.</seg> rectè protractæ eſſe debeant: </s> <s xml:space="preserve">qua de cau-<lb/>ſa ipſis <seg type="var">.b.E.</seg> et <seg type="var">.f.h.</seg> quæ, ex ipſo, ſunt degradationes <seg type="var">.b.c.</seg> et <seg type="var">.f.g.</seg> æquales eſſe non de-<lb/>bent? </s> <s xml:space="preserve">Poſſet is mihi quidem reſpondere, <choice><ex>quod</ex><am>ꝙ</am></choice> hoc pacto nulla ſuperficies clauderetur. <lb/></s> <s xml:space="preserve">Ergo tria latera <seg type="var">.b.c</seg>: <seg type="var">c.g.</seg> et <seg type="var">.g.f.</seg> <choice><ex>non</ex><am>nõ</am></choice> benè ſunt degradata, <choice><ex>eiusque</ex><am>eiusq́;</am></choice> <choice><ex>proportionalitates</ex><am>ꝓportionalitates</am></choice> ma <lb/>lè intellectæ nil probant. </s> <s xml:space="preserve">quia ſi dictæ proportionalitates, nobis tutò promitterent <lb/>degradationes, ab eo primum effectas, in linea <seg type="var">.b.f.</seg> eſſe bonas, ergo duæ <seg type="var">.b.d.</seg> et <seg type="var">.e.c.</seg> <lb/>falſæ exiſterent, quarum quælibet maior eſt <seg type="var">.b.E.</seg> et <seg type="var">.f.h.</seg> ex .18. primi Eucli. </s> <s xml:space="preserve">Omitta-<lb/>mus etiam quod vbi is ſcribit eam eſſe rationem, aut comparationem ab <seg type="var">.A.d.</seg> ad <seg type="var">.b.<lb/>E.</seg> quæ eſt ab <seg type="var">.d.c.</seg> ad <seg type="var">.b.c.</seg> eandemq́ue eſſe ab <seg type="var">.E.h.</seg> ad <seg type="var">.c.g.</seg> quæ eſt ab <seg type="var">.A.E.</seg> ad <seg type="var">.A.c.</seg> nil <lb/>probet; </s> <s xml:space="preserve">nec ſimilitudinem triangulorum, nec aliquam propoſitionem Eucli. citans. <lb/></s> <s xml:space="preserve">In quo excuſari non poteſt, quòd non ſoleat Euclidem, aut alium quemuis autorem <lb/>citare, cum vel in ipſo operis principio capite .3. primæ partis, A pollonium <choice><ex>Pergeum</ex><am>Pergeũ</am></choice> <lb/><choice><ex>Euclidemque</ex><am>Euclidemq́;</am></choice>, & ſi etiam præter rem, citet. </s> <s xml:space="preserve">Deinde <choice><ex>quum</ex><am>quũ</am></choice> idem probare vult <seg type="var">.d.e.</seg> æqua <lb/>lem eſſe ipſi <seg type="var">.E.h.</seg> eandem inquit eſſe proportionem <seg type="var">.a.b.</seg> ad <seg type="var">.a.d.</seg> quæ eſt ipſius <seg type="var">.A.c.</seg> <lb/>ad <seg type="var">.A.E.</seg> quod & ſi verum ſit, hic tamen modus ratiocinandi nullo ordine nititur, <lb/>quia rectius dixiſſet pro clariori intelligentia ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.e.</seg> eandem proportio-<lb/>nem eſſe, quæ eſt <seg type="var">.A.c.</seg> ad <seg type="var">.A.E.</seg> propter ſimilitudinem, quæ inter duos triangulos <seg type="var">.A.<lb/>c.a.</seg> et <seg type="var">.E.c.e.</seg> intercedit, cum <seg type="var">.E.e.</seg> ſupponatur parallela ipſi <seg type="var">.A.a.</seg> quod etiam vt de-<lb/>monſtraretur longiori oratione ei opus fuiſſet ſi voluiſſet intellectum eorum, qui pa <lb/>rum ſunt exercitati, perduci ad <choice><ex>cognoſcendum</ex><am>cognoſcendũ</am></choice> idem planè futurum de <seg type="var">.a.c.</seg> ad <seg type="var">.a.e.</seg> vt <lb/>eſt ipſius <seg type="var">.A.c.</seg> ad <seg type="var">.A.E.</seg> in hunc modum, ideſt probando primùm duos triangulos <seg type="var">.A.<lb/>c.a.</seg> et <seg type="var">.E.c.e.</seg> æquiangulos eſſe, mediante .29. primi Eucli. cum <seg type="var">.A.a.</seg> et <seg type="var">.E.e.</seg> inuicem <lb/>ſint parallelæ. </s> <s xml:space="preserve">Vnde ex .4. ſexti. idem extitiſſet de <seg type="var">.A.c.</seg> ad <seg type="var">.E.c.</seg> vt <seg type="var">.a.c.</seg> ad <seg type="var">.e.c.</seg> et. ex <lb/>16. quinti idem de <seg type="var">.A.c.</seg> ad <seg type="var">.a.c.</seg> vt ipſius <seg type="var">.E.c.</seg> ad <seg type="var">.e.c.</seg> & ex .19. eiuſdem de <seg type="var">.A.E.</seg> ad <seg type="var">.a.<lb/>e.</seg> vt ipſius <seg type="var">.A.c.</seg> ad <seg type="var">.a.c.</seg> & ex .16. iam dicta de <seg type="var">.A.E.</seg> ad <seg type="var">.A.c.</seg> vt ipſius <seg type="var">.a.e.</seg> ad <seg type="var">.a.c.</seg> ideſt <lb/>ipſius <seg type="var">.A.c.</seg> ad <seg type="var">.A.E.</seg> vt eſt ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.e</seg>: Aut hoc alio modo, qui breuior eſt pro-<lb/>cedendum, incipiendo ſcilicet à ſecunda ſexti Eucli. dicendo <choice><ex>quod</ex><am>ꝙ</am></choice> exiſtente <seg type="var">.E.e.</seg> paral <lb/>lela ipſi <seg type="var">.A.a</seg>: ex dicta .2. lib. 6. erit idem de <seg type="var">.c.E.</seg> ad <seg type="var">.E.A.</seg> vt de <seg type="var">.c.e.</seg> ad <seg type="var">.e.a.</seg> vnde ex <num value="18">.<lb/>18.</num> quinti innotuiſſet ſtatim quod de <seg type="var">.c.A.</seg> ad <seg type="var">.E.A.</seg> vt de <seg type="var">.c.a.</seg> ad <seg type="var">.e.a.</seg> extitiſſet. </s> <s xml:space="preserve">Nunc <lb/>mediantibus ſupradictis duabus propoſitionibus ideſt .29. primi, & 4. ſexti, cogno-<lb/>ſcitur idem planè eſſe de <seg type="var">.b.c.</seg> ad <seg type="var">.d.e.</seg> quod ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.e.</seg> & ex eiſdem idem eſſe <lb/>de <seg type="var">.c.g.</seg> ad <seg type="var">.E.h.</seg> quod ipſius <seg type="var">.A.c.</seg> ad <seg type="var">.A.E.</seg> vnde ex .11. quinti bis repetita idem erit de <lb/><seg type="var">b.c.</seg> ad <seg type="var">.d.e.</seg> quod de <seg type="var">.c.g.</seg> ad <seg type="var">.E.h.</seg> ſed cum ex ſuppoſito <seg type="var">.c.g.</seg> ſit æqualis ipſi <seg type="var">.c.b.</seg> idem <lb/>erit de <seg type="var">.c.g.</seg> ad <seg type="var">.e.d.</seg> quod ipſius <seg type="var">.c.b.</seg> ad eandem ex .7. quinti, vnde ex .11. idem erit de <lb/><seg type="var">c.g.</seg> ad <seg type="var">.E.h.</seg> quod eiuſdem <seg type="var">.c.g.</seg> ad <seg type="var">.e.d.</seg> ex .9. igitur eiuſdem <seg type="var">.d.e.</seg> æqualis erit ipſi <seg type="var">.E.<lb/>h.</seg> atque hic verus eſt modus ducendi intellectum parum exercitatum in cognicio-<lb/>nis campum. </s> <s xml:space="preserve">quem quidem mihi obſeruandum proponerem ſi onus ſcribendi ſu-<lb/>ſciperem ijs, qui in ſcientijs parum verſati ſunt, quos tanquam puerulos manu du- <pb facs="0148" n="136"/><fw type="head">IO. BAPT. BENED.</fw> cere oportet. </s> <s xml:space="preserve">Ratio verò ab ipſo adducta propter quam <seg type="var">.E.</seg> repreſentatur oculo al-<lb/>tius quam <seg type="var">.b.</seg> nempe eo quod <seg type="var">.A.</seg> ſuperſtet ipſi <seg type="var">.E.</seg> nihil valet, quia ſi inferius eſſet, <lb/>idem contingeret, ſed hoc euenit eo quod <seg type="var">.E.</seg> altius eſt ipſo <seg type="var">.b</seg>. </s> <s xml:space="preserve">Idem dico de <seg type="var">.h.</seg> <lb/>vbi ſimiliter decipitur. </s> <s xml:space="preserve">Idem etiam in .7. cap. fallitur in ſecundo modo, quem oſten <lb/>dit pro ſecundo quadrato aliquo degradato à parallelogrammo degradato magis <lb/>longo quàm lato, cum ducat parallelam <seg type="var">.l.m.</seg> ad <seg type="var">.b.c.</seg> à puncto <seg type="var">.l.</seg> interſection is ipſius <seg type="var">.<lb/>o.c.</seg> id, quod non rectè efficitur quemadmodum ex rationibus à me allegatis circa <lb/>meas figuras <seg type="var">.A.A.</seg> facilè innoteſcit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0145-01" corresp="fig-0145-01a"> <graphic url="0145-01"/> </figure> <figure xml:id="fig-0146-01" corresp="fig-0146-01a"> <graphic url="0146-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nono deinde cap. contrario planè ordine, quam oporteret proceſsit, quia <choice><ex>cum</ex><am>cũ</am></choice> <lb/>angulus .2. trianguli perfecti magis diſtet à plano ſuper quod degradari debet <lb/>triangulum, quàm latus .1. 3. oppoſitum dicto angulo .2. & per confequens longère <lb/>motior ſit ab oculo, ipſe in degradato, <choice><ex>eum</ex><am>eũ</am></choice> magis propinquum eſſe facit, è con-<lb/>tra eap .10. rectè fecit contra id, quod capite .9. tradiderat.</s> </p> <p> <s xml:space="preserve">Quod autem deinceps in prima parte .11. & vltimi capitis aſſerit eſt, <choice><ex>admittendum</ex><am>admittendũ</am></choice>. <lb/></s> <s xml:space="preserve">Quod verò in ſecunda parte ab eo traditur, ideſt alius quidam modus quem de <choice><ex>tranſ</ex><am>trãſ</am></choice> <lb/>ferendis punctis à perfecto in degradato proponit, non eſt modus vniuerſalis; </s> <s xml:space="preserve">quia <lb/>ſi altitudo <seg type="var">.T.Q.</seg> oculi à plano orizontali, non eſſet æqualis medietati lateris <seg type="var">.B.D.</seg> <lb/>perfecti, interualla <seg type="var">.a.b.c.d.e.</seg> lateris <seg type="var">B.D.</seg> admittenda non eſſent.</s> </p> <p> <s xml:space="preserve">Pro cuius rei intelligentia ſit in ſubſcripta hic figura corporea <seg type="var">.ω.</seg> parallelogram-<lb/>mum rectangulum <seg type="var">A.B.C.D.</seg> in plano orizontali, & linea <seg type="var">.Q.H.</seg> illud per medium <lb/>diuidat, quæ ſit parallela duobus lateribus <seg type="var">.A.B.</seg> et <seg type="var">.C.D.</seg> in cuius quolibet puncto <seg type="var">.<lb/>Q.</seg> ſit infimus terminus altitudinis oculi, & in <seg type="var">.<lb/> <ptr xml:id="fig-0148-01a" corresp="fig-0148-01" type="figureAnchor"/> T.</seg> ad perpendiculum ipſius <seg type="var">.Q.</seg> ſit verus ſitus <lb/>eiuſdem, tantum eleuatus à <seg type="var">.Q.</seg> quanta eſt <lb/>medietas ipſius <seg type="var">.D.B.</seg> ſitq́ue figura corpo-<lb/>rea finita ſimilis meæ <seg type="var">.A.</seg> vnde <seg type="var">.Q.T.</seg> æqualis <lb/>erit ipſi <seg type="var">.Q.æ.</seg> & planum perpendiculare <choice><ex>orizon- ti</ex><am>orizõ-ti</am></choice>, ſuper quod punctum <seg type="var">.k.</seg> perfecti duci debet <lb/>ſit <seg type="var">.R.D.B.</seg> ſintq́ue ductæ per imaginationem <lb/>lineæ <seg type="var">.T.K</seg>: <seg type="var">Q.K.</seg> et ſit <seg type="var">.K.N.</seg> perpendicularis la-<lb/>teri <seg type="var">.C.D.</seg> à quo puncto <seg type="var">.N.</seg> imaginatione ſit <choice><ex>con</ex><am>cõ</am></choice> <lb/>præhenſa linea <seg type="var">.N.Q.</seg> at que hæ tres lineæ ſectæ <lb/>ſint à plano in punctis <seg type="var">.c.i.</seg> et .2. quorum <choice><ex>punctum</ex><am>punctũ</am></choice>. <lb/>2. erit quæſitum plani. </s> <s xml:space="preserve">Imaginemur nunc duos <lb/>triangulos <seg type="var">.K.T.Q.</seg> et <seg type="var">.N.Q.æ.</seg> qui ſecti <choice><ex>erunt</ex><am>erũt</am></choice> <lb/>à plano <seg type="var">.R.B.D.</seg> quorum communes ſectiones <lb/>erunt .1. 2. et <seg type="var">.D.c.</seg> & quia <seg type="var">.N.K.D.i.</seg> et <seg type="var">.æ.Q.</seg> <lb/>inuicem ſunt parallelæ, ſequitur eandem pro-<lb/>portionem futuram ipſius <seg type="var">.Q.K.</seg> ad <seg type="var">.K.i.</seg> quæ eſt <lb/>ipſius <seg type="var">.æ.N.</seg> ad <seg type="var">.N.D.</seg> imaginatione concipien <lb/>do a puncto <seg type="var">.K.</seg> vſque ad <seg type="var">.æ.Q.</seg> quandam paral-<lb/>lelam ipſi <seg type="var">.N.æ.</seg> quemadmo dum ex te ipſo intel <lb/>ligere potes. </s> <s xml:space="preserve">Sed ratione ſimilitudinis trian-<lb/>gulorum ita ſe res habet de <seg type="var">.æ.Q.</seg> ad <seg type="var">.D.c.</seg> vt de <seg type="var">.<lb/>æ.N.</seg> ad <seg type="var">.N.D.</seg> vt quoque de <seg type="var">.T.Q.</seg> ad .2. 1. quemadmodum ipſius <seg type="var">.Q.K.</seg> ad <seg type="var">.K.i.</seg> vn-<lb/>de ex .11. quinti, idem erit de <seg type="var">.Q.T.</seg> ad .1. 2. quod de <seg type="var">.Q.æ.</seg> ad <seg type="var">.c.D.</seg> & ex .16. eiuſdem <lb/>de <seg type="var">.Q.T.</seg> ad <seg type="var">.Q.æ.</seg> quod de .1. 2. ad <seg type="var">.c.D.</seg> & exiſtente <seg type="var">.æ.Q.</seg> ex ſuppoſito æquali ipſi.</s> <pb facs="0149" n="137"/> <fw type="head">DE PERSPECT.</fw> <s xml:space="preserve">1. 2. </s> <s xml:space="preserve">Vnde huiuſmodi regula tunc bona redditur, quando <seg type="var">T.Q.</seg> æqualis eſt ipſi .œ.<lb/>Q. ideſt medietati ipſius <seg type="var">.D.B.</seg> at verò ſi æqualis non eſſet hoc minime ſequeretur, <lb/>vt facilè patet. </s> <s xml:space="preserve">Quòd verò .2. <seg type="var">R.Z.</seg> &. ſint benè diſpoſita, dubitandum non eſt, quia <lb/>punctum <seg type="var">.i.</seg> meæ hic ſubſcriptæ figuræ, quod coreſpondet K. eius ſiguræ adeò diſtat <lb/>a medio <seg type="var">.R.X.</seg> trianguli <seg type="var">.R.B.D.</seg> vt .2. cum .1. 2. dicto medio <seg type="var">.R.X.</seg> ex .6. </s> <s xml:space="preserve">Vndecimi fit <lb/>parallela. </s> <s xml:space="preserve">Idem de reliquis dico. quod manifeſtè cognoſci poteſt, ab eo, quod in <lb/>ſuperius poſitis figuris corporeis dixi. </s> <s xml:space="preserve">Huiuſmodi modus ducendi res in perſpectiua, <lb/>non ſolum à Gallis, ſed à Germanis etiam in vſum reducitur. </s> <s xml:space="preserve">Sed quia ad hæc <choice><ex>vſque</ex><am>vſq;</am></choice> <lb/>tempora eiuſdem perfectionis ratio, quam ego ſuperius propoſui, <choice><ex>nondum</ex><am>nõdum</am></choice> in lucem <lb/>emerſit, factum fuit, vt <choice><ex>errorum</ex><am>errorũ</am></choice> laqueis irretirentur, ſumentes <seg type="var">.T.Q.</seg> modo maiorem, <lb/>modo minorem medietate lateris <seg type="var">.D.B</seg>. </s> <s xml:space="preserve">Cum hunc igitur modum hic Autor <lb/>vniuerſalem eſſe putet, labitur in errorem, cum debuiſſet longitudinem ipſius <seg type="var">.T.Q.</seg> <lb/>debere eſſe æqualem medietati ipſius <seg type="var">.D.B.</seg> proferre. </s> <s xml:space="preserve">Aſſerit deinde diſtantiam ip-<lb/>ſius <seg type="var">.T.Q.</seg> à latere <seg type="var">.B.D.</seg> æ qualem eſſe debere lateri <seg type="var">.C.D.</seg> quod neceſſarium non eſt, <lb/>quia in quibuslibet diſtantijs, iuſta operatio fieri poteſt, quemadmodum in ſubſcri-<lb/>pta hîc figura facile patet, ideſt, quòd quibuſcunque modis <seg type="var">.c.D.</seg> æqualis remaneat <lb/>ipſi .1. 2. & ſic interualla, quæ <choice><ex>per</ex><am>ꝑ</am></choice> tranſuerſum aguntur <choice><ex>vſque</ex><am>vſq;</am></choice> ad <choice><ex>medium</ex><am>mediũ</am></choice> trianguli <seg type="var">.D.R.B.</seg> <lb/>Neque etiam probandus eſt auctor ille, cum pro oculo, ſuum <seg type="var">.T.</seg> loco <seg type="var">.Q.</seg> à me poſi-<unclear reason="illegible"/><lb/>ti, ponit, cum is locus ſit verus ſitus pedis eius quireſpicit, & non oculi. </s> <s xml:space="preserve">Quòd <choice><ex>autem</ex><am>autẽ</am></choice> <lb/>Auctor iſte, modo vniuerſali intelligat, vt iam diximus, <choice><ex>conſideretur</ex><am>cõſideretur</am></choice> figura tertij mo <lb/>di primi cap. tertiæ partis, in qua ſuum oculum (vt ita dicam) ponit in <seg type="var">.o.</seg> altius ſeu <lb/>diſtans à rectitudine lateris <seg type="var">.c.d.</seg> plus quam ſit totum latus <seg type="var">.d.b</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0148-01" corresp="fig-0148-01a"> <graphic url="0148-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">AD EVNDEM IACOBVM. <lb/>CAP. XIII.</head> <p> <s xml:space="preserve"><hi rend="small caps">TVas</hi> accepiliteras omnis humanitatis & officij plenas, in quibus requiris cau-<lb/>ſam, quæ me in alijs meis literis impulit ad <choice><ex>dicendum</ex><am>dicendũ</am></choice>, <choice><ex>angulum</ex><am>angulũ</am></choice> <seg type="var">.q.o.u.</seg> modo ma-<lb/>iorem, modo verò minorem futurum angulo <seg type="var">.q.p.u.</seg> meæ figuræ corporeæ <seg type="var">.A.</seg> hanc <lb/>igitur ob cauſam imagineris in ſubſcripta hîc figura duo triangula <seg type="var">.q.o.u.</seg> et <seg type="var">.q.p.u.</seg> <lb/>quorum <seg type="var">.q.p.u.</seg> perpendiculariter ſit ſuper ſuperficie trianguli <seg type="var">.q.o.p.</seg> collocatum, <lb/>præcisè vt in mea figura corporea <seg type="var">.A.</seg> ſuperficies verò trianguli <seg type="var">.q.o.p.</seg> ſit exempli-<lb/>gratia <seg type="var">.V.M.</seg> & trian-<lb/>guli <seg type="var">.u.o.p.</seg> ſit <seg type="var">.V.D.</seg> <lb/> <ptr xml:id="fig-0149-01a" corresp="fig-0149-01" type="figureAnchor"/> quarum <choice><ex>communis</ex><am>cõmunis</am></choice> ſe-<lb/>ctio ſit <seg type="var">.V.p.o.x.</seg> non <lb/>eſt enim <choice><ex>dubitandum</ex><am>dubitãdum</am></choice> <lb/>quin triangulum <seg type="var">.q.<lb/>p.u.</seg> ſit perpendicula-<lb/>re triangulo <seg type="var">.q.o.p.</seg> <lb/><choice><ex>cum</ex><am>cũ</am></choice> hoc ex .18. lib. 11. <lb/>Eucli. perpendicula-<lb/>re ſit ſuperficiei <seg type="var">.a.s.</seg> <lb/>in qua reperitur <choice><ex>trian- gulum</ex><am>triã-gulum</am></choice> <seg type="var">.q.p.u.</seg> & hoc <lb/>ex linea <seg type="var">.o.p.</seg> perpendiculari dictæ ſuperficiei <seg type="var">.a.s</seg>. </s> <s xml:space="preserve">Nunc dico angulum <seg type="var">.q.o.u.</seg> modo <lb/>maiorem, modo minorem eſſe angulo <seg type="var">.q.p.u</seg>. </s> <s xml:space="preserve">Notiſſimum igitur primum nobis <pb facs="0150" n="138"/><fw type="head">IO. BAPT. BENED.</fw> eſt angulum <seg type="var">.p.q.u.</seg> obtuſum eſſe; </s> <s xml:space="preserve">Imaginemur ergo circa triang ulum <seg type="var">.p.q.u.</seg> circun-<lb/>ſcriptum eſſe circulum, cuius portio <seg type="var">.p.q.u.</seg> minor erit medietate eiuſdem medij cir-<lb/>culi, vt iam ex <ref>30. Eucli. lib. tertij</ref> nouiſti. </s> <s xml:space="preserve">nunc imaginemur dictum circulum circum <lb/>lineam <seg type="var">.q.u.</seg> loco axis verſus <seg type="var">.x.</seg> moueri, vnde girus eiuſdem, per quem tranſibat linea <lb/><seg type="var">V.x.</seg> remouebitur ab eadem linea non nihil cum motus erit à primo ſitu vſquequò <lb/>ad ſecandam dictam lineam <seg type="var">.V.x.</seg> in alio quodam puncto inter <seg type="var">.p.</seg> et <seg type="var">.x.</seg> redibit; </s> <s xml:space="preserve">quod <lb/>quidem punctum ſi <lb/>erit inter <seg type="var">.o.</seg> et <seg type="var">.x.</seg> angu <lb/> <ptr xml:id="fig-0150-01a" corresp="fig-0150-01" type="figureAnchor"/> lus <seg type="var">.q.o.u.</seg> maior erit <lb/>angulo <seg type="var">.q.p.u</seg>. </s> <s xml:space="preserve">Sed ſi <lb/>idem <choice><ex>punctum</ex><am>punctũ</am></choice> erit in-<lb/>ter <seg type="var">.p.</seg> et <seg type="var">.o.</seg> dictus an-<lb/>gulus <seg type="var">.q.o.u.</seg> minor <lb/>erit <seg type="var">.q.p.u.</seg> de qua <choice><ex>qui- dem</ex><am>ꝗ-dẽ</am></choice> re tu ipſe median-<lb/>te .20. lib. 3. et .16. lib. <lb/>primi certior fieri po-<lb/>tes. </s> <s xml:space="preserve">Valde miror <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>hæc Ioannis Cuſini di <lb/>cta ad hæc vſque tempora tanto in prætio ſint habita, vt ab excellentibus ſcriptori-<lb/>bus quaſi ſi proprij eorum ingenij partus eſſent, de verboad verbum vt theſauros, in <lb/>fuis <choice><ex>ipſorummet</ex><am>ipſorũmet</am></choice> libris reſcripta fuerint, quemadmodum iam omnes admonui in mea <lb/>gnomonica Orontium, Munſterum, <choice><ex>aliosque</ex><am>aliosq́;</am></choice> permultos feciſſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0149-01" corresp="fig-0149-01a"> <graphic url="0149-01"/> </figure> <figure xml:id="fig-0150-01" corresp="fig-0150-01a"> <graphic url="0150-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">CAP. XIIII.</head> <p> <s xml:space="preserve"><hi rend="small caps">Ex</hi> ijs, qu æ de nonnullis effectibus ducendo in perſpectiua tertíum corpus regu <lb/>lare, <choice><ex>quod</ex><am>ꝙ</am></choice> octo triangulis æquilateribus eſt term inatum, ſcire deſideras, hoc <choice><ex>vnum</ex><am>vnũ</am></choice> <lb/>eſt caput: </s> <s xml:space="preserve">vnde fiat, aut quomodo probetur quaſlibet duas facies oppoſitas eiuſ-<lb/>dem corporis octoaedri <choice><ex>inuicem</ex><am>inuicẽ</am></choice> æquidiſtantes eſſe. </s> <s xml:space="preserve">Quamobrem ſit hîc <choice><ex>ſubſcriptum</ex><am>ſubſcriptũ</am></choice> <lb/><choice><ex>octoaedrum</ex><am>octoaedrũ</am></choice>, cuius diameter vna ſit <seg type="var">.b.q.</seg> et <seg type="var">.b.p.<lb/>l.</seg> vna ex faciebus, cui opponatur facies <seg type="var">.q.k.</seg> <lb/> <ptr xml:id="fig-0150-02a" corresp="fig-0150-02" type="figureAnchor"/> d. quas <choice><ex>adinuicem</ex><am>adinuicẽ</am></choice> æquidiſtantes eſſe contendo <lb/>ſint aliæ duæ facies, quæ inter has ponuntur <seg type="var">.<lb/>b.d.k.</seg> et <seg type="var">.q.p.l.</seg> & à punctis extremis <seg type="var">.b.q.</seg> dia-<lb/>metri. ductæ ſint quatuor lineæ <seg type="var">.b.a</seg>: <seg type="var">b.u</seg>: <seg type="var">q.a</seg>: <seg type="var">q.<lb/>u.</seg> ad puncta <seg type="var">.a.</seg> et <seg type="var">.u.</seg> diuidentia <seg type="var">.k.d.</seg> et <seg type="var">.l.p.</seg> per <lb/>medium, vnde ex 4. primi Eucli. quatuor hæ <lb/>lineæ adinuicem ęquales erunt <choice><ex>ſumendo</ex><am>ſumẽdo</am></choice> eas vt <lb/>baſes <choice><ex>triangulorum</ex><am>triangulorũ</am></choice> <seg type="var">.a.d.b</seg>: <seg type="var">u.l.b</seg>: <seg type="var">a.d.q.</seg> et <seg type="var">.u.l.q.</seg> <lb/><choice><ex>adinuicem</ex><am>adinuicẽ</am></choice> <choice><ex>quoque</ex><am>quoq;</am></choice> <choice><ex>æquidiſtabunt</ex><am>æꝗdiſtabũt</am></choice> <seg type="var">.a.b.</seg> ab <seg type="var">.u.q.</seg> et <seg type="var">.b.<lb/>u.</seg> ab <seg type="var">.q.a.</seg> ex .27. primi; </s> <s xml:space="preserve"><choice><ex>quia</ex><am>ꝗa</am></choice> ſi imaginabimur dia <lb/>metrum <seg type="var">.b.q.</seg> tunc ex .4. aut ex .8. eiuſdem lib. <lb/>habebimus angulos <seg type="var">.a.b.q.</seg> et <seg type="var">.u.q.b.</seg> æquales <lb/>inuicem; </s> <s xml:space="preserve">ſed ob eaſdem rationes <seg type="var">.p.l.</seg> paralle-<lb/>la eſt ipſi <seg type="var">.d.k.</seg> vnde ex 15. lib. 11. facies <seg type="var">.b.p.l.</seg> <lb/>parallela fit, aut æquidiſtans ipſi <seg type="var">.q.d.k.</seg> ideſt <lb/>primum propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0150-02" corresp="fig-0150-02a"> <graphic url="0150-02"/> </figure> </div> </body> </floatingText> <pb facs="0151" n="139"/> <fw type="head">DE PERSPECT.</fw> <p> <s xml:space="preserve">Ad habendam deinde quantitatem diſtantiæ, aut interualli ſimul cum ſitu, in fa-<lb/>cie <seg type="var">.q.d.k.</seg> quem latus <seg type="var">.p.l.</seg> perpendiculariter reſpicit. </s> <s xml:space="preserve">Imaginemur à puncto <seg type="var">.u.</seg> ſuper <lb/><seg type="var">q.a.</seg> cad ere lineam perpendicularem <seg type="var">.u.o.</seg> quæ illico reperitur cum triangulum <seg type="var">.a.<lb/>u.q.</seg> ex lateribus datis & cognitis conſtet, <choice><ex>quodquidem</ex><am>quodquidẽ</am></choice> triangulum, medietas eſt qua-<lb/>drilateri, ſeu. rumbi <seg type="var">.q.a.b.u.</seg> cui vnaquæque dictarum quatuor facierum perpendi-<lb/>cularis exiſtit ex .4. ct .18. lib. 11. & ob id linea <seg type="var">.u.o.</seg> extenſa in ſuperficie dicti quadri-<lb/>lateri, & perpendicularis lineæ <seg type="var">.q.a.</seg> perpendicularis erit faciei <seg type="var">.q.d.k.</seg> & ex .29. <lb/>primi, angulus <seg type="var">.b.u.o.</seg> rectus erit, ut <choice><ex>etiam</ex><am>etiã</am></choice> angulus <seg type="var">.o.u.l.</seg> ex .2. definitione lib. 11. vnde <lb/>ex .4. eiuſdem lib <seg type="var">.o.u.</seg> perpendicularis erit faciei <seg type="var">.b.p.l</seg>. </s> <s xml:space="preserve">Ha bebimus ergo ſitum in fa-<lb/>cie <seg type="var">.q.d.k.</seg> qui reſpicietur ad angulos rectos à linea <seg type="var">.p.l.</seg> quiquidem erit in perpendi-<lb/>culari à puncto <seg type="var">.o.</seg> ad <seg type="var">.q.a.</seg> ducta.</s> </p> <p> <s xml:space="preserve">Quòd autem <seg type="var">.a.o.</seg> ſit latus exagoni æquilateris circumſcrip tibilis ab eodem circu <lb/>lo, qui vnam ex faciebus triangularibus æquilateribus propoſiti corporis circunſcri-<lb/>bere pot eſt, ita oſtenditur. ſit <choice><ex>comprehenſum</ex><am>cõprehenſum</am></choice> imaginatione, triangulum <seg type="var">.a.q.u.</seg> ſepara <lb/>tim, cuius latus <seg type="var">.a.u.</seg> æquale eſt vni ex lateribus <choice><ex>triangulorum</ex><am>triangulorũ</am></choice> eiuſdem corporis ex .33. <lb/>primi, quo dlibet verò aliorum duorum æquale perpendicularibus dictorum trian-<lb/>gulorum, in quo triangulo <seg type="var">.a.u.q.</seg> ducta ſit perpendicularis <seg type="var">.u.o.</seg> ab vna <choice><ex>extremitatum</ex><am>extremitatũ</am></choice> <lb/>lateris maioris, ad vnum ex minoribus lateribus, quę perpendicularis intra triangu-<lb/>lum cadet, quia dictum triangulum oxigonium eſt. </s> <s xml:space="preserve">quod autem attinet ad duos angu <lb/>los <seg type="var">.a.</seg> et <seg type="var">.u.</seg> cum æquales ſint ex quinta lib. primi; </s> <s xml:space="preserve">17. nos certiores facit; </s> <s xml:space="preserve">quod verò an<lb/>gulus <seg type="var">.q.</seg> ſit <choice><ex>etiam</ex><am>etiã</am></choice> acutus: </s> <s xml:space="preserve">30. lib. tertii nos cer-<lb/>tos reddit, <choice><ex>quia</ex><am>ꝗa</am></choice><seg type="var">.a.u.</seg> minor eſt diametro <choice><ex>ſphae ræ</ex><am>ſphę ræ</am></choice> datum corpus circumſcribentis, cum <seg type="var">.q.</seg> <lb/>dictæ ſphęrę ſuperficiem tangat.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0151-01" corresp="fig-0151-01a"> <graphic url="0151-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Ad probandum <seg type="var">.a.o.</seg> ęqualem eſſe lateri <lb/>exagoni dicti, ſatis erit probare <seg type="var">.a.q.</seg> ſeſqui <lb/>alteram eſſe ad <seg type="var">.a.o.</seg> quia ſi in ſubſcripto <lb/>hîc circulo ducemus duas ſemidiametros <seg type="var">.<lb/>n.p.</seg> et <seg type="var">.n.l.</seg> ad. angulos <choice><ex>trianguli</ex><am>triãguli</am></choice> ęquilateri <seg type="var">.p.</seg> <lb/>et <seg type="var">.l.</seg> & cum quodlibet laterum ipſius exago <lb/>ni, ęquale ſit ſemidiametro circuli ex .15. <lb/>lib. 4. habebimus ex .8. primi, angulum <seg type="var">.n.<lb/>p.l.</seg> æqualem angulo <seg type="var">.q.p.l</seg>. </s> <s xml:space="preserve">Vnde ex .4. eiuſ <lb/>dem <seg type="var">.o.n.</seg> ęqualiserit ipſi <seg type="var">.o.q.</seg> ideſt <seg type="var">.q.a.</seg> ſeſ <lb/>quialtera erit ad <seg type="var">.a.o</seg>.</s> </p> <p> <s xml:space="preserve">Ad probandum nunc in triangulo <seg type="var">.a.q.<lb/>u</seg>: <seg type="var">a.q.</seg> ſeſquialteram eſſe ad <seg type="var">.a.o.</seg> eſt <choice><ex>quoque</ex><am>quoq;</am></choice> <lb/>ſciendum primò omne latus trianguli ęquilateri in potentia ſeſquitertium eſſe ad <lb/>perpendicularem eiuſdem trianguli, quod vndecima lib. 14. Eucli. breuiter demon <lb/>ſtratum eſt.</s> </p> <pb facs="0152" n="140"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Ponamus nunc quadratum lateris <seg type="var">.a.u.</seg> eſſe .12. clarum erit quodlibet quadratum <lb/>aliorum duorum laterum <seg type="var">.a.q.</seg> et <seg type="var">.u.q.</seg> futurum nouem, ex ijs quæ poſteriore loco dixi <lb/>mus, & quia quadratum ipſius <seg type="var">.q.a.</seg> eſt tantò minus aliorum duorum quadratorum <lb/>ſumma, quantum eſt duplum producti ipſius <seg type="var">.q.a.</seg> in <seg type="var">.a.o.</seg> ex .13. ſecundi, ſed alia duo <lb/>quadrata ſimul collecta faciunt .21. à quo numero ſubtrahendo quadratum ipſius <seg type="var">.a.<lb/>q.</seg> ideſt nouem, remanebit numerus .12. pro duplo producti ipſius <seg type="var">.q.a.</seg> in <seg type="var">.a.o.</seg> cuius <lb/>dupli me-<lb/> <ptr xml:id="fig-0152-01a" corresp="fig-0152-01" type="figureAnchor"/> dia pars, id-<lb/>eſt ſimplex <lb/>productum <lb/>ipſius <seg type="var">.q.a.</seg> <lb/><choice><ex>in</ex><am>ĩ</am></choice> <seg type="var">a.o.</seg> erit 6. <lb/>Sed <choice><ex>quia</ex><am>ꝗa</am></choice> qua <lb/>dratum ip-<lb/>ſius <seg type="var">.q.a.</seg> eſt <lb/>nouem, <lb/>eius radix <seg type="var">.<lb/>q.a.</seg> crit .3. <lb/>per <choice><ex>quam</ex><am>quã</am></choice> di-<lb/>uidendo .6. <lb/>productum <lb/>ipſius <seg type="var">.q.a.</seg> <lb/>in <seg type="var">.a.o.</seg> pro <lb/>latere <seg type="var">.a.o.</seg> <lb/>conſurgent <lb/>duo, cum er <lb/>go <seg type="var">.a.o.</seg> ſint <lb/>duo tertia <lb/>ipſius <seg type="var">.a.q.</seg> <lb/>certi <choice><ex>erimus</ex><am>erimꝰ</am></choice> <lb/><seg type="var">a.o.</seg> eſſe latus dicti exagoni.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0152-01" corresp="fig-0152-01a"> <graphic url="0152-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">CAP. XV.</head> <p> <s xml:space="preserve"><choice><ex>DEſiderantes</ex><am>DEſiderãtes</am></choice> ſcire deinde <seg type="var">.l.k.</seg> in figura <seg type="var">.M.</seg> quar <lb/> <ptr xml:id="fig-0152-02a" corresp="fig-0152-02" type="figureAnchor"/> ti cap. tertiæ partis perſpectiuę Danielis <lb/>Barbari, ſeu Zamberti, eſſe veram <choice><ex>altitudinem</ex><am>altitudinẽ</am></choice> cor-<lb/>poris octoaedri, <choice><ex>primum</ex><am>primũ</am></choice> ſcire debemus <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>exiſtente</ex><am>exiſtẽte</am></choice> <seg type="var">.b.<lb/>h.</seg> vt <choice><ex>etiam</ex><am>etiã</am></choice> <seg type="var">.b.l.</seg> tripla ad <seg type="var">.b.k.</seg> vt ex ijs, quę ſuperius <choice><ex>iam</ex><am>iã</am></choice> <lb/>diximus, facile percipi poteſt; </s> <s xml:space="preserve">ex penultima primi <seg type="var">.<lb/>b.l.</seg> in potentia, ſeſquioctaua erit ad <seg type="var">.k.l.</seg> ipſa et <seg type="var">.k.<lb/>l.</seg> dupla <choice><ex>inpotentia</ex><am>inpotẽtia</am></choice> ad <seg type="var">.h.k.</seg> & ob id ducta <choice><ex>cum</ex><am>cũ</am></choice> eſſet <seg type="var">.h.<lb/>l.</seg> exiſteret in potentia tripla ad <seg type="var">.h.k.</seg> & ſeſquialtera <lb/>ad <seg type="var">.l.k.</seg> & ſeſquitertia ad <seg type="var">.l.b.</seg> & ſic ad <seg type="var">.h.b.</seg> vnde <seg type="var">.l.h.</seg> <lb/>æqualis eſſet vni ex lateribus <choice><ex>trianguli</ex><am>triãguli</am></choice> ęquilateri di-<lb/>cti corporis. </s> <s xml:space="preserve">Ex rationibus igitur ſuperius hîc poſi-<lb/>tis <seg type="var">.l.k.</seg> erit altitudo dicta, id eſt diſtantia inter duas <lb/>facies inuicem oppoſitas, octoaedri.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0152-02" corresp="fig-0152-02a"> <graphic url="0152-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve"><choice><ex>Neque</ex><am>Neq;</am></choice> volo te ignorare <choice><ex>alium</ex><am>aliũ</am></choice> <choice><ex>non</ex><am>nõ</am></choice> <choice><ex>paruum</ex><am>paruũ</am></choice> fuiſſe <choice><ex>errorem</ex><am>errorẽ</am></choice> <lb/>illius Zamberti: </s> <s xml:space="preserve">cum <choice><ex>eodem</ex><am>eodẽ</am></choice> capite affirmet angulos <lb/>octoacdri rectos eſſe <choice><ex>cum</ex><am>cũ</am></choice> ſint acuti, <choice><ex>nam</ex><am>nã</am></choice> <choice><ex>vnuſquiſque</ex><am>vnuſquiſq;</am></choice> minor eſt angulo cubi ſolido.</s> </p> <pb facs="0153" n="141"/> </div> </div> <div type="chapter"> <head xml:space="preserve">DE MECHANICIS.</head> <p rend="italics"> <s xml:space="preserve"><hi rend="small caps">SCripservnt</hi> multi multa, & quidem ſcitißimè, de mechn<unclear reason="illegible"/>-<lb/>nicis, at cum natura <choice><ex>vſusque</ex><am>vſusq;</am></choice> aliquid ſemper vel nouum, vel <lb/>Latens in apertum emittere ſoleant, nec ingenui aut grati ſit <lb/>animi, posteris inuidere, ſi quid ei contigerit comperuiße prius <lb/>tenebris inuolutum: </s> <s xml:space="preserve">cum tam multa ipſe ex aliorum diligentia <lb/>ſit conſequut us. </s> <s xml:space="preserve">Paucula <choice><ex>quædam</ex><am>quædã</am></choice> futùra, vt reor, non ingrata his <lb/>qui in biſce mechanicis verſantur, nuſquam ante bac tentata, <lb/>aut ſatis exastè explicata in medium proferre volui: </s> <s xml:space="preserve">quo vel iuuandi deſiderium, vel <lb/>ſaltem non ocioſi ingenioli argumentum aliquod exbiberem: </s> <s xml:space="preserve">at que vel boc vno modo me <lb/>inter bumanos vixiſſe testatum relinquerem.</s> </p> <div type="section"> <head rend="italics" xml:space="preserve">De differentia ſitus brachiorum libra.</head> <head xml:space="preserve">CAP.I.</head> <p> <s xml:space="preserve"><hi rend="small caps">OMne</hi> pondus poſitum in extremitate alicuius brachij libræ maiorem, aut mi-<lb/> <ptr xml:id="hd-0153-01a" corresp="hd-0153-01" type="handwrittenAnchor"/> norem grauitatem habet, pro diuerſa ratione ſitus ipſius brachij. </s> <s xml:space="preserve">ſit exe<unclear reason="illegible"/>mpli <lb/>gratia <seg type="var">.B.</seg> centrum, aut, quod diuidit brachia alicuius libræ, & <seg type="var">.A.B.Q.</seg> vertica-<lb/>lis linea, aut, vt rectius dicam, axis orizontis, & <seg type="var">.B.C.</seg> vnum brachium dictæ li-<lb/>bræ, & in <seg type="var">.C.</seg> ſit pondus, & <seg type="var">.C.O.</seg> linea inclinationis, ſeuicineris<unclear reason="illegible"/> <seg type="var">.C.</seg> verſus cen-<lb/>trum mundi, cum qua <seg type="var">.B.C.</seg> angulum rectum conſtituat in puncto <seg type="var">.C</seg>. </s> <s xml:space="preserve">Exiſtente <lb/>igitur in huiuſmodi ſitu brachio <seg type="var">.B.C.</seg> dico pondus <seg type="var">.C.</seg> grauius futurum, quam <lb/>in alio quolibet ſitu. </s> <s xml:space="preserve">quia ſupra centrum <seg type="var">.B.</seg> omninò non quieſcet, quemadmodum <lb/>in quouis alio ſitu faceret. </s> <s xml:space="preserve">Ad quod intelligendum, ſit dictum brachium, in ſitu <seg type="var">.B.<lb/>F.</seg> cum eodem pondere in puncto <seg type="var">.F.</seg> & linea itineris ſeu inclinationis dicti ponderis <lb/>ſit <seg type="var">.F.u.M.</seg> per quam lineam dictum pondus progredi non poteſt, niſi brachium <seg type="var">.B.F.</seg> <lb/>breuius redderetur. </s> <s xml:space="preserve">Vnde clarum erit <lb/> <ptr xml:id="fig-0153-01a" corresp="fig-0153-01" type="figureAnchor"/> quòd pondus <seg type="var">.F.</seg> aliquantulum ſupra cen <lb/>trum <seg type="var">.B.</seg> mediante brachio <seg type="var">.B.F.</seg> nititur. <lb/></s> <s xml:space="preserve">Eſt quidem verum, quòd pondus <seg type="var">.C.</seg> nec <lb/>ipſum etiam per lineam <seg type="var">.C.O.</seg> proficiſce-<lb/>tur, quia iter extremitatis brachij eſt cir-<lb/>cularis, & <seg type="var">.C.O.</seg> in vno <choice><ex>quodam</ex><am>quodã</am></choice> puncto eſt <lb/>contingens. </s> <s xml:space="preserve">Sit hociter <seg type="var">.A.C.Q</seg>. </s> <s xml:space="preserve">Opor-<lb/>tet nunc præſupponere pondus extremi-<lb/>tatis brachij deberetanto magis <choice><ex>centro</ex><am>cẽtro</am></choice> <seg type="var">.B.</seg> <lb/>inniti, quanto magis linea ſuæ inclinatio-<lb/>nis (ponamus <seg type="var">.F.u.M.</seg>) propinqua erit di <lb/>cto centro <seg type="var">.B.</seg> quod ſequenti cap. proba-<lb/>bo, vt exempli gratia, ſit <seg type="var">.F.</seg> ſuper <seg type="var">.u.</seg> pun-<lb/>ctum medij ex æquo inter <seg type="var">.C.</seg> et <seg type="var">.B.</seg> qua-<lb/>propter <seg type="var">.u.B.</seg> æqualis erit <seg type="var">.u.C.</seg> vndeſe- <pb facs="0154" n="142"/><fw type="head">IO. BAPT. BENED.</fw> quetur dictum pondus grauius futurum pro parte <seg type="var">.F.C.</seg> quam pro ea, quæ eſt <seg type="var">.A.F.</seg> & <lb/>minus ſupra centrum <seg type="var">.B.</seg> pro dicta parte <seg type="var">.F.C.</seg> quam pro parte <seg type="var">.A.F.</seg> quieturum; </s> <s xml:space="preserve">& <lb/>dictum brachium quanto magis orizontale erit à ſitu <seg type="var">.B.F.</seg> tantò minus-ſupra dictum <lb/>centrum <seg type="var">.B.</seg> quieſcet, & hac ratione grauius quoque erit, & quanto magis vicinum <lb/>erit ipſi <seg type="var">.A.</seg> à dicto <seg type="var">.F.</seg> tantò magis ſuper centrum <seg type="var">.B.</seg> quoque quieſcet, vnde <choice><ex>tantò</ex><am>tãtò</am></choice> quo-<lb/>que leuius exiſtet. </s> <s xml:space="preserve">Idem dico de omni ſitu brachij per girum inferiorem <seg type="var">.C.Q.</seg> vbi <lb/>pondus pendebit à centro <seg type="var">.B.</seg> dictum centrum attrahendo, quemadmodum ſuperius <lb/>illud impellebat. </s> <s xml:space="preserve">Hæc verò omnia cap. ſequenti melius percipientur.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0153-01" corresp="hd-0153-01a"/> <figure xml:id="fig-0153-01" corresp="fig-0153-01a"> <graphic url="0153-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">De proportione ponderis extremitatis brachij libr & <lb/>in diuerſo ſitu ab orizontali.</head> <head xml:space="preserve">CAP. II.</head> <p> <s xml:space="preserve"><hi rend="small caps">PRoportio</hi> ponderis in <seg type="var">.C.</seg> ad idem pondus in F. erit quemadmodum totius <lb/>brachij <seg type="var">.B.C.</seg> ad partem <seg type="var">.B.u.</seg> poſitam inter centrum & lineam <seg type="var">.F.u.M.</seg> inclinatio-<lb/>nis, quam pondus ab extremitate <seg type="var">.F.</seg> liberum verſus mundi <choice><ex>centrum</ex><am>centrũ</am></choice> conficeret. </s> <s xml:space="preserve">Quod <lb/>vt facilius intelligamus imaginemur <choice><ex>alterum</ex><am>alterũ</am></choice> brachium libræ <seg type="var">.B.D.</seg> & in extremo <seg type="var">.D.</seg> <lb/>locatum aliquod pondus minus pondere <seg type="var">.C.</seg> vt <seg type="var">.B.u.</seg> pars <seg type="var">.B.C.m.</seg> nor eſt <seg type="var">.B.D.</seg> cla-<lb/>rè cognoſcetur ex .6. lib. primi de ponderibus Archimedis, quòd ſi in puncto <seg type="var">.u.</seg> col-<lb/>locatum erit pondus ipſius <seg type="var">.C.</seg> libra nihil penitus à ſitu orizontali dimouebitur. </s> <s xml:space="preserve">Sed <lb/>perinde eſt quòd pondus <seg type="var">.F.</seg> æquale <seg type="var">.C.</seg> ſit in extremo <seg type="var">.F.</seg> in ſitu brachij <seg type="var">.B.F.</seg> <choice><ex>quam</ex><am>quã</am></choice> vt ſit <lb/>in puncto <seg type="var">.u.</seg> in ſitu ipſius <seg type="var">.B.u.</seg> orizontali. </s> <s xml:space="preserve">Ad cuius rei euidentiam imaginemur <choice><ex>filum</ex><am>filũ</am></choice> <seg type="var">.<lb/>F.u.</seg> perpendiculare, & in cuius extremo <seg type="var">.u.</seg> pendere pondus, quod erat in <seg type="var">.F.</seg> vnde cla <lb/>rum erit quòd eundem effectum gignet, ac ſi fuiſſet in <seg type="var">.F.</seg> quod, vt iam diximus re-<lb/>manens affixum puncto <seg type="var">.u.</seg> brachij <seg type="var">.B.u.</seg> tantò minus graue eſt ſitu ipſius <seg type="var">.C.</seg> quantò <seg type="var">.u.<lb/>B.</seg> minus eſt ipſo <seg type="var">.B.C</seg>. </s> <s xml:space="preserve">Idem aſſero ſi brachium eſſet in ſitu <seg type="var">.e.B.</seg> quod facilè cogno-<lb/>ſcere poterimus, ſi imaginemur filum appenſum ipſi <seg type="var">.u.</seg> brachij <seg type="var">.B.C.</seg> & vſque ad <seg type="var">.e.</seg> <lb/><choice><ex>perpendicularem</ex><am>perpendicularẽ</am></choice>, in quo extremo <choice><ex>appensum</ex><am>appensũ</am></choice> eſſet pondus æquale ponderi <seg type="var">.C.</seg> & <choice><ex>liberum</ex><am>liberũ</am></choice> <lb/>ab <seg type="var">.e.</seg> brachij <seg type="var">.B.e.</seg> vnde libra orizontalis manebit. </s> <s xml:space="preserve">Sed ſi brachium <seg type="var">.B.e.</seg> conſolida-<lb/>tum fuiſſet in tali ſitu cum orizontali <seg type="var">.B.D.</seg> <lb/> <ptr xml:id="fig-0154-01a" corresp="fig-0154-01" type="figureAnchor"/> & <choice><ex>appenſo</ex><am>appẽſo</am></choice> <choice><ex>pondere</ex><am>põdere</am></choice> <seg type="var">.C.</seg> in <seg type="var">.e.</seg> libero à filo, nec <lb/><choice><ex>aſcenderet</ex><am>aſcẽderet</am></choice>, <choice><ex>neque</ex><am>neq;</am></choice> deſcenderet. </s> <s xml:space="preserve">quia tantum <lb/>eſt quod ipſum ſit appenſum filo, <choice><ex>quod</ex><am>ꝙ</am></choice> pendet <lb/>ab <seg type="var">.u.</seg> quantum quòd ab ipſo liberum <choice><ex>appem</ex><am>appẽ</am></choice> <lb/>nſum fuiſſet <seg type="var">.e.</seg> brachij <seg type="var">.B.e.</seg> & hoc procede <lb/>ret ab eo quòd partim pendereta centro <seg type="var">.<lb/>B.</seg> & ſi <choice><ex>brachium</ex><am>brachiũ</am></choice> eſſet in ſitu <seg type="var">.B.Q.</seg> totum <choice><ex>pon</ex><am>põ</am></choice> <lb/>dus centro <seg type="var">.B.</seg> remaneret appenſum, <choice><ex>quem- admodum</ex><am>quem-admodũ</am></choice> in ſitu <seg type="var">.B.A.</seg> <choice><ex>totum</ex><am>totũ</am></choice> dicto centro an-<lb/>niteretur. </s> <s xml:space="preserve">vnde fit vt hoc modo pondus <lb/>magis aut minus ſit graue, quò magis <lb/>aut minus à centro pendet, aut eidem niti-<lb/>tur: </s> <s xml:space="preserve"><choice><ex>atque</ex><am>atq;</am></choice> hæc eſt cauſa proxima, & per ſe, <lb/> <ptr xml:id="hd-0154-01a" corresp="hd-0154-01" type="handwrittenAnchor"/> qua fit vt vnum <choice><ex>idemque</ex><am>idemq;</am></choice> pondus in vno eo-<lb/><choice><ex>demque</ex><am>demq́;</am></choice> medio magis aut minus graue exi- <pb facs="0155" n="143"/><fw type="head">DE MECHAN.</fw> ſtat. </s> <s xml:space="preserve">Et quamuis appellem latus <seg type="var">.B.C.</seg> orizontale, ſupponens illud angulum rectum <lb/>cum <seg type="var">.C.O.</seg> facere, vnde angulus <seg type="var">.C.B.Q.</seg> fit vt minor ſit recto, ob quantitatem vnius <lb/>anguli ęqualis ei, quem duæ <seg type="var">.C.O.</seg> et <seg type="var">.B.Q.</seg> in centro regionis <choice><ex>elementaris</ex><am>elemẽtaris</am></choice> <choice><ex>conſtituunt</ex><am>conſtituũt</am></choice>, <lb/>hoc tamen nihil refert, cum dictus angulus inſenſibilis ſit magnitudinis. </s> <s xml:space="preserve">Ab iſtis au-<lb/>tem rationibus elicere poſſumus, quod ſi punctus <seg type="var">.u.</seg> erit ex æquo medius inter cen-<lb/>trum <seg type="var">.B.</seg> & extremum <seg type="var">.C.</seg> pondus <seg type="var">.F.</seg> aut <seg type="var">.M.</seg> pendebit, aut nitetur pro medietate dicto <lb/>centro <seg type="var">.B.</seg> & ſi dictum <seg type="var">.u.</seg> erit propius <seg type="var">.B.</seg> quam puncto <seg type="var">.C.</seg> pendebit ab ipſo, aut nitetur <lb/>ipſi amplius <choice><ex>quam</ex><am>quã</am></choice> exmedietate, & ſi magis verſus <seg type="var">.C.</seg> minus <choice><ex>quam</ex><am>quã</am></choice> ex medietate <choice><ex>nitetur</ex><am>nitet̃</am></choice>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0154-01" corresp="fig-0154-01a"> <graphic url="0154-01"/> </figure> <note xml:id="hd-0154-01" corresp="hd-0154-01a"/> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quòd quantit as cuiuſlibet ponderis, aut uirtus mouens re-<lb/>ſpectu alterius quantitatis cognoſcatur beneficio <lb/>perpendicularium ductarum à centro <lb/>libr & ad line am inclinationis.</head> <head xml:space="preserve">CAP. III.</head> <p> <s xml:space="preserve">EX ijs, quæ à nobis hucuſque ſunt dicta, facilè intelligi poteſt, <choice><ex>quod</ex><am>ꝙ</am></choice> quantitas <seg type="var">.B.u.</seg> <lb/>quæ ferè perpendicularis eſt à centro <seg type="var">.B.</seg> ad lineam <seg type="var">.F.u.</seg> inclinationis, ea eſt, <lb/> <ptr xml:id="hd-0155-01a" corresp="hd-0155-01" type="handwrittenAnchor"/> quæ nos ducit in cognitionem quantitatis virtutis ipſius <seg type="var">.F.</seg> in huiuſmodi ſitu, conſti <lb/>tuens videlicet linea <seg type="var">.F.u.</seg> cum brachio <seg type="var">.F.B.</seg> angulum acutum <seg type="var">.B.F.u</seg>. </s> <s xml:space="preserve">Vt hoc tamen <lb/>melius intelligamus, imaginemur libram <seg type="var">.b.o.a.</seg> fixam in centro <seg type="var">.o.</seg> ad. cuius etrema <lb/>ſint appenſa duo pondera, aut duæ virtutes mouentes <seg type="var">.e.</seg> et <seg type="var">.c.</seg> ita tamen <choice><ex>quod</ex><am>ꝙ</am></choice> linea incli-<lb/>nationis <seg type="var">.e.</seg> ideſt <seg type="var">.b.e.</seg> faciat angulum rectum cum <seg type="var">.o.b.</seg> in puncto <seg type="var">.b.</seg> linea verò inclina <lb/>tionis <seg type="var">.c.</seg> ideſt <seg type="var">.a.c.</seg> faciat angulum acutum, aut obtuſum cum <seg type="var">.o.a.</seg> in puncto <seg type="var">.a</seg>. </s> <s xml:space="preserve">Imagi-<lb/>nemur ergo lineam <seg type="var">.o.t.</seg> perpendicularem lineæ <seg type="var">.c.a.</seg> inclinationis, vnde <seg type="var">.o.t.</seg> minor <lb/>erit <seg type="var">.o.a.</seg> ex .18. primi Euclidis. ſecetur deinde imaginatione <seg type="var">o.a.</seg> in puncto <seg type="var">.i.</seg> ita ut <lb/><seg type="var">o.i.</seg> æqualis. </s> <s xml:space="preserve">ſit <seg type="var">.o.t.</seg> & puncto <seg type="var">.i.</seg> appenſum ſit pondus æquale ipſi <seg type="var">.c.</seg> cuius inclinationis <lb/>linea parallela ſit lineæ inclinationis ponderis <seg type="var">.e.</seg> ſupponendo tamen pondus aut vir <lb/>tutem <seg type="var">.c.</seg> ea ratione maiorem eſſe ea, quæ eſt <seg type="var">.e.</seg> qua <seg type="var">.b.o.</seg> maior eſt <seg type="var">.o.t.</seg> abſque dubio <lb/>ex .6. lib. primi Archi. de ponderibus <seg type="var">.b.o.i.</seg> non mouebitur ſitu, ſed ſi loco <seg type="var">.o.i.</seg> imagi <lb/>nabimur <seg type="var">.o.t.</seg> conſolidatam cum <seg type="var">.o.b.</seg> & per lineam <seg type="var">.t.c.</seg> attractam virtute <seg type="var">.c.</seg> ſimiliter <lb/>quoque continget ut <seg type="var">b.o.</seg> t; </s> <s xml:space="preserve">communi quadam ſcientia, non moueatur ſi tu. </s> <s xml:space="preserve">Eſt ergo <lb/> <ptr xml:id="hd-0155-02a" corresp="hd-0155-02" type="handwrittenAnchor"/> quod propoſuimus verum quantitatem alicuius ponderis reſpectu ad eam, quæ eſt <lb/>alterius debere depræhendi à perpendicularibus, quæ à centro libræ ad lineas incli <lb/>nationis exiliunt. </s> <s xml:space="preserve">Hinc autem innoteſcit facillimè, quantum vigoris, & vis pondus, <lb/>aut virtus <seg type="var">.c.</seg> ad angulum rectum cum <seg type="var">.o.a.</seg> minimè trahens, amitttat. </s> <s xml:space="preserve">Hinc quoque co <lb/>rollarium quoddam ſequetur, quò d quantò propinquius erit centrum <seg type="var">.o.</seg> libræ cen-<lb/>tro regionis elementaris, tantò quo que minus erit graue.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0155-01" corresp="hd-0155-01a"/> <note xml:id="hd-0155-02" corresp="hd-0155-02a"/> </div> </body> </floatingText> <figure place="here"> <graphic url="0155-01"/> </figure> <pb facs="0156" n="144"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quemadmodum exſupradictis cauſis omnes staterarum & <lb/>uectium cauſæ dependeant.</head> <head xml:space="preserve">CAP. IIII.</head> <p> <s xml:space="preserve">VIs brachij longioris alicuius ſtateræ, aut vectis, maior breuioris, ab ijs, quæ in ſu <lb/>perioribus capitibus diximus, ideſt <choice><ex>quod</ex><am>ꝙ</am></choice> nitatur <choice><ex>pendeatuem</ex><am>pendeatuẽ</am></choice> magis aut minus à <lb/>centro pondus in extremitate brachij maioris poſitum, oboritur. </s> <s xml:space="preserve">Quamobrem illud <lb/>à nobis primò eſt cognoſcendum, ſtateras, aut vectes, puras mathematicas li-<lb/>neas non eſſe, ſed naturales, hincque exiſtere corpora cum materia coniuncta. </s> <s xml:space="preserve">Nunc <lb/>igitur imaginemur <seg type="var">.n.s.</seg> eam ſuperficiem eſſe, quæ ſecundum longitudinem axem ſta <lb/>teræ ſcindit. </s> <s xml:space="preserve">& ſupponamus ipſius centrum eſſe primum in <seg type="var">.i.</seg> & maius brachium eſſe <lb/> <seg type="var">.i.u</seg>: minus autem <seg type="var">.i.n.</seg> & lineam verticalem <seg type="var">.i.o.</seg> quæ tanta ſit, quanta eſt ſpiſſitu-<lb/>do, aut craſſities ipſius ſtateræ à ſuperiori latere ad inferius, ad faciliorem intelligen-<lb/>tiam, ſupponendo <seg type="var">.n.s.</seg> <choice><ex>parallelogrammam</ex><am>parallelogrãmam</am></choice>. </s> <s xml:space="preserve">Poſitis igitur duobus ponderibus æquali-<lb/> <ptr xml:id="hd-0156-01a" corresp="hd-0156-01" type="handwrittenAnchor"/> bus in extremitatibus brachiorum, experientia innoteſcit, <choice><ex>quod</ex><am>ꝙ</am></choice> pondus ad <seg type="var">.u.s.</seg> appen-<lb/>ſum, viol entiam faciet ponderi appenſo ad <seg type="var">.n.x.</seg> ſed nos volumus inueſtigare <choice><ex>causam</ex><am>causã</am></choice> <lb/>huius effectus, quæ à nemine vnquam literarum monumentis, <choice><ex>quod</ex><am>ꝙ</am></choice> ſciam, conſignata <lb/> <ptr xml:id="hd-0156-02a" corresp="hd-0156-02" type="handwrittenAnchor"/> fuit. </s> <s xml:space="preserve">Iam diximus ſtateram, aut vectem materialem eſſe & <seg type="var">.n.s.</seg> eius ſuperficiem me-<lb/>diam, ſupponendo <seg type="var">.i.</seg> eſſe centrum quo nititur dicta ſtatera aut vectis; </s> <s xml:space="preserve">Cum hocer-<lb/>go ita ſe habeat, ſint <seg type="var">.u.s.</seg> et <seg type="var">.n.x.</seg> lineæ inclinationum ponderum, & imaginemur, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>dicta pondera pendeant à punctis <seg type="var">.u.</seg> et <seg type="var">.n.</seg> vt reuera pendent, etiam ſi appenſa eſſent <lb/>ſub <seg type="var">.s.</seg> et <seg type="var">.x.</seg> quia punctum <seg type="var">.u.</seg> & punctum <seg type="var">.n.</seg> ita coniuncta ſunt cum <seg type="var">.s.</seg> et <seg type="var">.x.</seg> ut qui <choice><ex>vnum</ex><am>vnũ</am></choice> <lb/>trahit alterum quoque trahat. </s> <s xml:space="preserve">Imaginemur quoque duas lineas <seg type="var">.i.u</seg>: <seg type="var">i.n.</seg> et <seg type="var">.i.e.</seg> quę <lb/><seg type="var">i.e.</seg> faciat angulum <seg type="var">.o.i.e.</seg> æqualem angulo <seg type="var">.o.i.n</seg>. </s> <s xml:space="preserve">Hinc clarè nobis patebit, ſi quis ipſi <lb/>e. pondus ipſius <seg type="var">.u.</seg> (<choice><ex>quod</ex><am>ꝙ</am></choice> æquale eſt ponderi <seg type="var">.n.</seg>) appenderet, id eandem planè vim habe <lb/>ret, quam pondus ipſius <seg type="var">.n.</seg> habet, & ſtateram neque ſurſum, neque deorſum moue-<lb/>ret, quia ambo pondera ad centrum <seg type="var">.i.</seg> mediantibus lineis <seg type="var">.e.i.</seg> et <seg type="var">.n.i.</seg> exęquo annite-<lb/>rentur, ſed dicto pondere poſito in .u: linea <seg type="var">.u.i.</seg> per quam pondus centro annititur, <lb/>magis orizontalis quam <seg type="var">.e.i.</seg> fit, & linea <seg type="var">.u.s.</seg> inclinationis longius diſtans à centro <seg type="var">.i.</seg> <lb/> <ptr xml:id="hd-0156-03a" corresp="hd-0156-03" type="handwrittenAnchor"/> quàm linea <seg type="var">.e.t.</seg> vnde huiuſmodi pondus magis quoque liberum à centro <seg type="var">.i.</seg> reſultat. <lb/></s> <s xml:space="preserve">magisque ponderoſum, quam cum erat in <seg type="var">.e.</seg> ratione eorum, quæ primo & ſecundo <lb/>capitibus diximus, & ob hanc cauſam ſuperat pondus poſitum in <seg type="var">.n</seg>. </s> <s xml:space="preserve">Sed ſi centrum <lb/>fuerit .in <seg type="var">.o.</seg> imaginabimur duas lineas <seg type="var">.o.s.</seg> et <seg type="var">.o.x.</seg> & ſupponemus quòd pondera po-<lb/>ſita ſint in <seg type="var">.s.</seg> et <seg type="var">.x.</seg> vnde exiſtente magis orizontali linea <seg type="var">.o.s.</seg> quam erit <seg type="var">.o.x.</seg> & linea <lb/><seg type="var">u.s.</seg> inclinationis longius diſtante à centro <seg type="var">.o.</seg> quàm linea <seg type="var">.e.t.</seg> eius pondus erit <choice><ex>quoque</ex><am>quoq;</am></choice> <lb/> <ptr xml:id="fig-0156-01a" corresp="fig-0156-01" type="figureAnchor"/> <ptr xml:id="fig-0156-02a" corresp="fig-0156-02" type="figureAnchor"/> <pb facs="0157" n="145"/><fw type="head">DE MECHAN.</fw> grauius, quia tantò minus pendebit à centro <seg type="var">.o.</seg> & ratiocinando, vt ſuperius dixi-<lb/>mus, inueniemus eundem effectum verum eſſe. </s> <s xml:space="preserve">In ſtateris, rectè & propriè appella <lb/>ri poteſt <seg type="var">.x.i.s.</seg> aut <seg type="var">.n.o.u.</seg> orizontalis, ſed in omni vectium ſpecie, hoc <choice><ex>tantum</ex><am>tãtum</am></choice> per quan <lb/>dam ſimilitudinem dicetur. </s> <s xml:space="preserve">Idem contemplari licet ſupponendo centrum in medio <lb/>inter <seg type="var">.o.</seg> et <seg type="var">.i.</seg> quod vnuſquiſque ex ſe abſque alterius auxilio facile præſtare poterit.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0156-01" corresp="hd-0156-01a"/> <note xml:id="hd-0156-02" corresp="hd-0156-02a"/> <figure xml:id="fig-0156-01" corresp="fig-0156-01a"> <graphic url="0156-01"/> </figure> <note xml:id="hd-0156-03" corresp="hd-0156-03a"/> <figure xml:id="fig-0156-02" corresp="fig-0156-02a"> <graphic url="0156-02"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">De quibuſdam rebus animaduerſione dignis.</head> <head xml:space="preserve">CAP.V.</head> <p> <s xml:space="preserve">NOn omittenda mihi <choice><ex>videntur</ex><am>vidẽtur</am></choice> quædam, quæ ad <choice><ex>tractationem</ex><am>tractationẽ</am></choice> vectium admodum <lb/>ſunt neceſſaria. </s> <s xml:space="preserve">Quod autem quærimus, in eo conſiſtit, quòd aliqui vectes <lb/>adhibeantur ad opus, quorum centrum, quod Græci <choice><ex>hypomochlion</ex><am>hypomochliõ</am></choice> appellant vnum <lb/>eſt ex extremis ipſius vectis, & pondus, quod ſurſum eleuari debet, inter ipſa-<lb/>met extrema iacet, propinquum tamen hypomochlio, vt exempli gratia, ſi vectis <lb/>eſſet infraſcripta figura <seg type="var">.o.s.u.x.</seg> cuius hypomochlion eſſet in puncto <seg type="var">.o.</seg> & pondus in <lb/>puncto <seg type="var">.n.</seg> clarum erit, <choice><ex>quod</ex><am>ꝙ</am></choice> cum eleuari debeat <seg type="var">.n.</seg> oportebit quoque opera manus ele-<lb/>uari <seg type="var">.u</seg>. </s> <s xml:space="preserve">Nunc conſiderandum eſt quomodo pondus <seg type="var">.n.</seg> annitatur ad <seg type="var">.u</seg>. </s> <s xml:space="preserve">Hanc ob cau <lb/>ſam imaginabimur rectas lineas <seg type="var">.n.o</seg>: <seg type="var">n.i</seg>: <seg type="var">n.e</seg>: <seg type="var">n.t.</seg> et <seg type="var">.n.u.</seg> quarum <seg type="var">.n.i.</seg> verſus mundi cen <lb/>trum ſit poſita, et <seg type="var">.n.t.</seg> faciat angulum <seg type="var">.i.n.t.</seg> æqualem angulo <seg type="var">.i.n.o</seg>. </s> <s xml:space="preserve">Nunc ponendo ali <lb/>quam virtutem in <seg type="var">.i.</seg> æquali inclinatione ad ſuperius conſtante, vt <seg type="var">.n.</seg> ad inferius (re-<lb/>mota tamen grauitate materiæ vectis) huiuſmodi virtus, totum pondus ipſius <seg type="var">.n.</seg> com <lb/>muni quadam ſcientiæ notione ſuſtinebit. </s> <s xml:space="preserve">& ſi <choice><ex>pondus</ex><am>põdus</am></choice> ipſius <seg type="var">.n.</seg> eſſet in <seg type="var">.x.</seg> è directo ſu-<lb/>per <seg type="var">.o.</seg> totum pondus ſuper hypomochlio ſe haberet, & tanta virtus ipſius hypomo-<lb/>chlij ſufficeret ad reſiſtendum pro ſuſtinendo, quanta eſt grauitas ipſius ponderis, <lb/>ſed ipſum iterum ponamus in <seg type="var">.n.</seg> ibi clarum erit, quòd ſi alia virtus à parte inſeriori <lb/>ad ſuperiorem vectis non opponitur, excepto tamen hypomochlio, oportebit virtu <lb/>te cuiuſdam partis ponderis <seg type="var">.n.</seg> (abſque conſideratione tamen, vt iam dixi, ponderis <lb/>materiæ vectis) vt vectis à parte <seg type="var">.s.u.</seg> deprimatur, & dixi vnius cuiuſdam partis pon-<lb/>deris <seg type="var">.n.</seg> quia alia <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> ponderis pars annititur ipſi hypomochlio <seg type="var">.o.</seg> <choice><ex>mediante</ex><am>mediãte</am></choice> linea <lb/><seg type="var">o.n.</seg> quæ angulos rectos cum <seg type="var">.o.x.</seg> non facit. </s> <s xml:space="preserve">Si autem à puncto <seg type="var">.t.</seg> opponet ſeſe huiuſ-<lb/>modi reſiſtentia, vt vectis non deprimatur, clarum erit communi ſcientia, <choice><ex>quod</ex><am>ꝙ</am></choice> virtus <lb/>ponderis <seg type="var">.n.</seg> diuiſa erit per medium æqualiter, cuius vna medietas ſuper <seg type="var">.o.</seg> quieſcet, <lb/>& alia ſuper <seg type="var">.t.</seg> mediantibus duabus lineis <seg type="var">.n.o.</seg> et <seg type="var">.n.t</seg>. </s> <s xml:space="preserve">Imaginemur nunc reſiſtentiam <lb/>t. ablatam eſſe, <choice><ex>poſitamque</ex><am>poſitamq;</am></choice> in <seg type="var">.e.</seg> clarum quoque erit, <choice><ex>quod</ex><am>ꝙ</am></choice> maior pars ponderis <seg type="var">.n.</seg> ipſi <seg type="var">.e.</seg> <lb/>annitetur beneficio lineæ <seg type="var">.n.e.</seg> quàm ipſi <seg type="var">.o.</seg> cum linea <seg type="var">.n.i.</seg> inclinationis ipſi <seg type="var">.e.</seg> ſit pro <lb/>pinquior quam <seg type="var">.o.</seg> quia omnis reſiſtentia aut in <seg type="var">.i.</seg> aut in <seg type="var">.e.</seg> aut in <seg type="var">.t.</seg> aut in <seg type="var">.u.</seg> eſt loco <lb/>centri, quemadmodum eſt <seg type="var">.o.</seg> & alter alterius opera iuuatur. </s> <s xml:space="preserve">Si verò eadem reſiſten <lb/>tia poſita erit in <seg type="var">.u.</seg> clarum quoque erit, <choice><ex>quod</ex><am>ꝙ</am></choice> minor pars ponderis <seg type="var">.n.</seg> annitetur ipſi <seg type="var">.u.</seg> <choice><ex>quam</ex><am>quã</am></choice> <lb/>ipſi <seg type="var">.o.</seg> cum dicta <seg type="var">.n.i.</seg> à centro <seg type="var">.u.</seg> longius quam à centro <seg type="var">.o.</seg> diſter, & proportio partis <lb/>ponderis <seg type="var">.n.</seg> in <seg type="var">.o.</seg> ad propor-<lb/>tionem partis ponderis <seg type="var">.n.</seg> in <lb/> <ptr xml:id="fig-0157-01a" corresp="fig-0157-01" type="figureAnchor"/> u. non erit <choice><ex>ſecundum</ex><am>ſecũdum</am></choice> propor <lb/>tionem angulorum <seg type="var">.u.n.i.</seg> et <lb/><seg type="var">o.n.i.</seg> ſed ſecundum propor <lb/>tionem <seg type="var">.u.i.</seg> ad <seg type="var">.i.o.</seg> quod cla <lb/>rè compræhendi poteſt ab <pb facs="0158" n="146"/><fw type="head">IO. BAPT. BENED.</fw> huius effectus conuerſo, ideſt, vt quemadmodum nunc ſupponuntur <seg type="var">.o.</seg> et <seg type="var">.u.</seg> eſſe duo <lb/>centra quibus <choice><ex>ſuſtinetur</ex><am>ſuſtinet̃</am></choice> pondus <seg type="var">.e.</seg> ipſius <seg type="var">.n.</seg> imaginemur <seg type="var">.n.</seg> eſſe quoddam centrum à <lb/>quo pendeant duo pondera <seg type="var">.o.</seg> et <seg type="var">.u.</seg> ſic inuicem proportionata, ut ſunt <seg type="var">.u.i.</seg> et <seg type="var">.i.o.</seg> <lb/>certe horum ponderum cauſa ſtatera <seg type="var">.o.s.</seg> quam vectem appellabamus à nulla parte <lb/>inclinabitur. </s> <s xml:space="preserve">Redeuntes nunc ad propoſitum, dicemus <choice><ex>quod</ex><am>ꝙ</am></choice> annitente pondere ipſius <seg type="var">.<lb/>n.</seg> minus ad <seg type="var">.u.</seg> quam ad <seg type="var">.o.</seg> ideſt ad <seg type="var">.t.</seg> minori vi opus erit in <seg type="var">.u.</seg> quàm in <seg type="var">.t.</seg> ad attollen-<lb/>dum pondus ipſius <seg type="var">.n.</seg> & ſic per conſequens quantò longius erit punctum <seg type="var">.u.</seg> ab <seg type="var">.t.</seg> tan <lb/>tò minori quoque vi egebit, & conſequenter quando vis, aut reſiſtentia in <seg type="var">.u.</seg> ita pro <lb/>portionata erit illi, quæ eſt ipſius <seg type="var">.o.</seg> vt eſt <seg type="var">.o.i.</seg> ad <seg type="var">.i.u.</seg> vectis non mouebitur. </s> <s xml:space="preserve">Sed quan <lb/>do erit proportio maior, reſiſtentiæ ipſius <seg type="var">.u.</seg> ad eam, quæ eſt ipſius <seg type="var">.o.</seg> ea, quæ eſt <seg type="var">.o.<lb/>i.</seg> ad <seg type="var">.i.u.</seg> </s> <s xml:space="preserve">tunc vectis à par-<lb/>teipſius <seg type="var">.u.s.</seg> eleuabitur, ſi <lb/> <ptr xml:id="fig-0158-01a" corresp="fig-0158-01" type="figureAnchor"/> vero proportio minor eſſet <lb/>quàm.o.i. ad <seg type="var">.i.u.</seg> </s> <s xml:space="preserve">tunc ve-<lb/>ctis ab eadem parte depri-<lb/>metur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0157-01" corresp="fig-0157-01a"> <graphic url="0157-01"/> </figure> <figure xml:id="fig-0158-01" corresp="fig-0158-01a"> <graphic url="0158-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">De ratione cuiuſdam uis adauctæ.</head> <head xml:space="preserve">CAP. VI.</head> <p> <s xml:space="preserve">QVibuſdam in locis vtuntur <choice><ex>quidam</ex><am>quidã</am></choice> <choice><ex>quodam</ex><am>quodã</am></choice> <choice><ex>inſtrumento</ex><am>inſtrumẽto</am></choice> piſtorio ad <choice><ex>ſubigendam</ex><am>ſubigẽdã</am></choice> pa-<lb/>ſtam, vnius tantum hominis ui adhibita, quæ quidem machina cum mihi di-<lb/>gna contemplatione eſſe videatur, eius aliquam rationem proponere volui, pro cu-<lb/>ius deſcriptione imaginemur planum, in quo ſedet ille, qui voluit paſtam, & in quo <lb/>ipſa paſta eſt repoſita <seg type="var">.T.S.D.</seg> & triangulum <seg type="var">.T.A.S.</seg> immobile perpendiculare-<lb/>q́ue ſuperficiei dicti plani, angulo autem <seg type="var">.A.</seg> coniunctum lignum <seg type="var">.A.E.</seg> vt ſemidiame <lb/>trum mobilem, & æqualem perpendiculari ipſius trianguli, </s> <s xml:space="preserve">unde <seg type="var">.A.</seg> loco centri erit <lb/>et <seg type="var">.D.O.</seg> ſit ſemidiameter, qui paſtam contundit, & ab eius extremo <seg type="var">.O.</seg> (quod <seg type="var">.O.</seg> <lb/>quando <seg type="var">.D.O.</seg> orizontalis eſt, in baſi dicti trianguli reperitur) veniat lignum <seg type="var">.O.V.</seg> <lb/>quod cum <seg type="var">.A.V.</seg> ſit æquale perpendiculari imaginatæ ab angulo <seg type="var">.A.</seg> baſi <seg type="var">.T.S.</seg> <choice><ex>deno- datum</ex><am>deno-datũ</am></choice> <choice><ex>tantum</ex><am>tñ</am></choice> utvulgo <choice><ex>dicitur</ex><am>dicit̃</am></choice> ſeu flexile in <seg type="var">.O.</seg> & in <seg type="var">.V.</seg> vt elleuare <choice><ex>atque</ex><am>atq;</am></choice> deprimere ſemidiame <lb/>trum <seg type="var">.D.O.</seg> poſſit, et <seg type="var">.V.O.</seg> ſit æqualis <seg type="var">.A.V.</seg> et <seg type="var">.V.</seg> medium ſit inter <seg type="var">.A.</seg> et <seg type="var">.E.</seg> vnde <seg type="var">.A.V.</seg> <lb/>cum <seg type="var">.O.V.</seg> æquales erunt <seg type="var">.A.E.</seg> ſunt deinde duo ligna <choice><ex>perpendicularia</ex><am>perpẽdicularia</am></choice> ab <seg type="var">.A.</seg> ad baſim <lb/>fixa, & immobilia inter ſe adeò diſtantia, vt inter ipſa <choice><ex>pertranſeant</ex><am>pertrãſeãt</am></choice> <seg type="var">.O.V.</seg> et <seg type="var">.D.O.</seg> ſupra <lb/>& infra, ne deuiet ſemidiametrum <seg type="var">.D.O</seg>. </s> <s xml:space="preserve">In extremitate deinde ipſius <seg type="var">.E.</seg> ſit lignum <lb/>quoddam tenue, vt digitus polex, ad angulos rectos cum <seg type="var">.A.E.</seg> quod ab aliquo, qui <lb/>antedictam machinam ſtet, manibus teneatur, qui quidem homo idipſum lignum, <lb/>ideſt ſemidiametrum <seg type="var">.A.E.</seg> à ſuperficie trianguli dicti, ad ſe trahendo, & deinde ver <lb/>ſus eundem triangulum impellendo, vim quandam maximam mediante ſemidia <lb/>metro <seg type="var">.D.O.</seg> ſuper paſtam excitat.</s> </p> <p> <s xml:space="preserve">Pro cuius rei contemplatione volo vt ſecundam hanc ſubſcriptam figuram <seg type="var">.b.a.<lb/>u.x.</seg> imaginemur, in qua <seg type="var">.u.</seg> exprimat <seg type="var">.A.</seg> primæ figuræ, & <seg type="var">.a.</seg> denotet <seg type="var">.O.</seg> & <seg type="var">.o.V.</seg> & <seg type="var">.x.<lb/>E.</seg> imaginemur etiam <seg type="var">.u.a.</seg> baſem trianguli <seg type="var">.a.u.o.</seg> cui <seg type="var">.o.t.</seg> perpendicularis dictæ baſi <seg type="var">.<lb/>u.a.</seg> addatur. </s> <s xml:space="preserve"><choice><ex>Hucuſque</ex><am>Hucuſq;</am></choice> igitur <seg type="var">.u.o.</seg> æqualis erit <seg type="var">.o.x.</seg> & ipſi <seg type="var">.o.a.</seg> imaginemur etiam <seg type="var">.a.o.</seg> <lb/>vſque ad <seg type="var">.b.</seg> ita productam vt <seg type="var">.o.b.</seg> æqualis ſit <seg type="var">.o.a.</seg> ponamus etiam pondus in <seg type="var">.a.</seg> impel- <pb facs="0159" n="147"/><fw type="head">DE MECHAN.</fw> lere verſus <seg type="var">.u.</seg> vnde linea eius inclinationis ſit ſemper <seg type="var">.a.u.</seg> ſupponamus etiam <seg type="var">.a.o.b.</seg> <lb/>eſſe <choice><ex>libram</ex><am>librã</am></choice>, aut ſtateram, aut vectem, & <seg type="var">.o.</seg> eius centrum, vnde vis, aut virtus ipſius <seg type="var">.a.</seg> <lb/>proportionalis erit ipſi <seg type="var">.o.t.</seg> reſpectu virtutis, aut vis imaginatæ in <seg type="var">.b.</seg> inclinationis <lb/>perpendicularis ipſi <seg type="var">.b.a.</seg> quæ quidem virtus, aut vis in <seg type="var">.b.</seg> proportionalis erit ipſi <seg type="var">.b.<lb/>o.</seg> ex tertio capite huius tractatus; </s> <s xml:space="preserve">Si ergo fuiſſet poſita in <seg type="var">.b.</seg> virtus quædarn ad an-<lb/>gulum rectum, trahens lineam <seg type="var">.b.o.</seg> tam proportionatam virtuti perpendiculari ip-<lb/>ſius <seg type="var">.a.</seg> quam eſt <seg type="var">.o.t.</seg> proportionata ipſi <seg type="var">.o.b.</seg> ſtatera <seg type="var">.b.o.a.</seg> non moueretur, ſed quæuis <lb/>portio maior in <seg type="var">.b.</seg> ſuperaret <seg type="var">.a.</seg> cum autem fuerit <seg type="var">.o.x.</seg> æqualis ipſi <seg type="var">.o.b.</seg> <choice><ex>idem</ex><am>idẽ</am></choice> planè eue-<lb/> <ptr xml:id="fig-0159-01a" corresp="fig-0159-01" type="figureAnchor"/> <ptr xml:id="fig-0159-02a" corresp="fig-0159-02" type="figureAnchor"/> niet, communi quadam ſcientia, ponen-<lb/>do virtutem <seg type="var">.b.</seg> in <seg type="var">.x</seg>. </s> <s xml:space="preserve">Quantitas ergo virtu <lb/>tis in <seg type="var">.x.</seg> quæ ſuperare debet reſiſtentiam <lb/>in <seg type="var">.a.</seg> quæ ipſi <seg type="var">.u.</seg> contraponitur, debet ha-<lb/>bere aliquantulum maioris proportionis <lb/>ad reſiſtentiam, quæ in <seg type="var">.a.</seg> angulum re-<lb/>ctum efficeret cum <seg type="var">.a.o.</seg> ea, quæ eſt <seg type="var">.o.t.</seg> <lb/>ad <seg type="var">.o.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0159-01" corresp="fig-0159-01a"> <graphic url="0159-01"/> </figure> <figure xml:id="fig-0159-02" corresp="fig-0159-02a"> <graphic url="0159-02"/> </figure> </div> </body> </floatingText> <pb facs="0160" n="148"/> <fw type="head">IO. BABPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">De quibuſdam erroribus Nicolai Tartaleæ circa pondera <lb/>corporum & eorum motus, quorum aliqui deſumpti <lb/>fuerunt à fordano ſcriptore quodam antiquo.</head> <head xml:space="preserve">CAP. VII.</head> <p> <s xml:space="preserve">CVm magis amici veritatis eſſe debeamus quàm cuiuſquam hominis, quemad-<lb/>modum Ariſto. ſcribit, detegam hoc loco quoſdam errores Nicolai Tartaleę <lb/>de ponderibus corporum, & velocitatibus motuum localium. </s> <s xml:space="preserve">Et primum decipitur <lb/>is in .8. lib. ſuarum diuerſarum inuentionum in ſecunda propoſitione, cum non ani-<lb/>maduerterit quanti momenti ſint extrinſecæ reſiſtentiæ.</s> </p> <p> <s xml:space="preserve">Subiectum quoque tertiæ propoſitionis eſt malè demonſtratum, quia idem pla-<lb/>nè ex eius demonſtratione iam dicta corporibus hætereogeneis, aut figura diuerſis <lb/>contingeret, quod ad velocitates attinet.</s> </p> <p> <s xml:space="preserve">In quarta propoſitione, quod ad <choice><ex>diſputandum</ex><am>diſputãdũ</am></choice> proponit <choice><ex>non</ex><am>nõ</am></choice> concludit melius. </s> <s xml:space="preserve"><choice><ex>autem</ex><am>autẽ</am></choice> id <lb/>ab eo <choice><ex>ſequitur</ex><am>ſequit̃</am></choice>, quod Archimedes in .6. propoſitione lib. primi de <choice><ex>ponderibus</ex><am>põderibus</am></choice> <choice><ex>probauit</ex><am>ꝓbauit</am></choice>.</s> </p> <note/> <p> <s xml:space="preserve">Sed in ſecunda parte quintę propoſitionis non uidet <choice><ex>quod</ex><am>ꝙ</am></choice> uigore ſitus eo modo, quo <lb/>ipſe diſputat, nulla elicitur ponderis differentia. </s> <s xml:space="preserve">quia ſi corpus <seg type="var">.B.</seg> deſcendere debet <lb/>per arcum <seg type="var">.i.l.</seg> corpus <seg type="var">.A.</seg> aſcendere debet per arcum <seg type="var">.u.s.</seg> æqualem, & ſimilem. eadem <lb/>quoque ratione ſituatum, vt eſt arcus <seg type="var">.i.l.</seg> vnde vt eſt facilè corpori <seg type="var">.B.</seg> deſcendere <lb/>per arcum <seg type="var">.i.l.</seg> difficile ita erit corpori <seg type="var">.A.</seg> aſcendere per arcum <seg type="var">.u.s</seg>. </s> <s xml:space="preserve">Hęc autem qnin <lb/>ta propoſitio Tartaleæ eſt ſecuuda quæſtio à Iordano propoſita.</s> </p> <p> <s xml:space="preserve">Quòd autem ad primum corollarium dictæ propoſitionis attinet, verum ille qui <lb/>dem ſcribit, eius tamen effectus cauſa & à Iordano prius, & ab ipſo poſtea citata, na-<lb/>tura ſua vera non eſt. </s> <s xml:space="preserve">quia vera cauſa per ſe ab eo oritur, <choice><ex>quod</ex><am>ꝙ</am></choice> à centro libræ dependeat <lb/>vt primo cap. huius tractatus oſtendi. </s> <s xml:space="preserve">Secundum verò corollarium falſum eſſe, ijs ra <lb/>tionibus quas nunc ſubiungam, patebit. </s> <s xml:space="preserve">Imaginemur <seg type="var">.u.</seg> pro centro regionis ele-<lb/>mentaris, & libram <seg type="var">.b.o.a.</seg> obliquam reſpectu ad <seg type="var">.u.</seg> & brachijs æqualibus <choice><ex>conſtantem</ex><am>conſtãtem</am></choice>, <lb/>& pondera in <seg type="var">.a.</seg> et in <seg type="var">.b.</seg> etiam æqualia. </s> <s xml:space="preserve">lineæ autem inclinationum ſint <seg type="var">.a.u.</seg> et <seg type="var">.b.u.</seg> <lb/>imaginemur etiam lineam <seg type="var">.o.u.</seg> & à centro <seg type="var">.o.</seg> libræ duas <seg type="var">.o.t.</seg> et <seg type="var">.o.e.</seg> perpendiculares <lb/>inclinationum lineis; </s> <s xml:space="preserve">vnde pondus ipſius <seg type="var">.a.</seg> in huiuſmodi ſitu tam erit proportiona <lb/>tum ponderi <seg type="var">.b.</seg> quam proportionata erit linea <seg type="var">.o.t.</seg> lineæ <seg type="var">.o.e.</seg> ex eo <choice><ex>quod</ex><am>ꝙ</am></choice> tertio cap. hu-<lb/>iustractatus probaui, ſed linea <seg type="var">.o.t.</seg> maior eſt linea <seg type="var">.o.e.</seg> quod ſic probo. </s> <s xml:space="preserve">Imaginemur <lb/>triangulum <seg type="var">.u.a.b.</seg> circunſcriptum eſſe à circulo <seg type="var">.u.a.n.b.</seg> cuius <seg type="var">.c.</seg> ſit centrum, <choice><ex>quod</ex><am>ꝙ</am></choice> erit <lb/>extra lineam <seg type="var">.u.o.</seg> cum ſupponatur <seg type="var">.a.o.b.</seg> obliquam eſſe reſpectu ad <seg type="var">.u.o</seg>. </s> <s xml:space="preserve">Imagine-<lb/>mur deinde à centro <seg type="var">.c.</seg> lineam <seg type="var">.c.o.s.</seg> vſque ad circunferentiam, quæ perpendicula-<lb/>ris erit ipſi <seg type="var">.a.b.</seg> ex tertia lib. 3. Eucli. </s> <s xml:space="preserve">ſi poſteà imaginemur duas lineas <seg type="var">.c.a.</seg> et <seg type="var">.c.b.</seg> ha <lb/>bebimus ex .8. lib. primi, angulum <seg type="var">.a.c.o.</seg> æqualem angulo <seg type="var">.b.c.o</seg>. </s> <s xml:space="preserve">Vnde ex .25. lib. 3. <lb/>arcus <seg type="var">.a.s.</seg> æqualis erit arcui <seg type="var">.b.s.</seg> ſed ſi imaginabimur <seg type="var">.u.o.</seg> ad circunferentiam vſque <lb/>productam, clarum erit <choice><ex>quod</ex><am>ꝙ</am></choice> arcum <seg type="var">.s.b.</seg> ſecaret in puncto <seg type="var">.n.</seg> vnde arcus <seg type="var">.n.b.</seg> minor erit <lb/>arcu <seg type="var">.n.a.</seg> & ſic etiam angulus <seg type="var">.n.u.b.</seg> minor erit angulo <seg type="var">.n.u.a.</seg> ex <ref>ultima lib. 6.</ref> </s> <s xml:space="preserve">Imagi-<lb/>nemur nunc alium quendam circulum, cuius <seg type="var">.o.u.</seg> ſit diameter, cuius circunferentia <lb/>per duo puncta <seg type="var">.e.</seg> et <seg type="var">.t.</seg> <choice><ex>prætergradiatur</ex><am>prætergradiat̃</am></choice>, cum in ipſis ſint angulirecti, quod quilibet ex <lb/>ſeratio cinando colligere poteſt, ſi .30. lib. 3. in mentem reuocauerit. </s> <s xml:space="preserve">Sed cum angu-<lb/>lus <seg type="var">.o.u.t.</seg> ſit maior angulo <seg type="var">.o.u.e.</seg> arcus <seg type="var">.o.t.</seg> maior erit arcu <seg type="var">.o.e.</seg> ex vltima .6. vnde cor <lb/>da <seg type="var">.o.t.</seg> maior erit corda ipſius <seg type="var">.o.e.</seg> ex conuerſo .27. lib. 3. quod eſt propoſitum. </s> <s xml:space="preserve">Pon-<lb/> <ptr xml:id="hd-0160-01a" corresp="hd-0160-01" type="handwrittenAnchor"/> dusigitur ipſius <seg type="var">.a.</seg> in huiuſmodi ſitu, pondere ipſius <seg type="var">.b.</seg> grauius erit. </s> <s xml:space="preserve">Quod è directo ijs <lb/>repugnat quæ Tartalea in 2. parte quinræ propoſitionis ediſerit, & per conſequens <lb/>2. corollarij falſitatem oſtendit, vt eam quoque, quæ in 6. propoſitione latet. </s> <s xml:space="preserve">quia <choice><ex>cum</ex><am>cũ</am></choice> <pb facs="0161" n="149"/><fw type="head">DE MECHAN.</fw> proportio <choice><ex>ponderis</ex><am>põderis</am></choice> <seg type="var">.a.</seg> ad pon <lb/>dusipſius <seg type="var">.b.</seg> eadem ſit cum <lb/>ea quę eſt <seg type="var">.o.t.</seg> ad <seg type="var">.o.e.</seg> ſub co <lb/><choice><ex>gnitionem</ex><am>gnitionẽ</am></choice> noſtram cadere po <lb/>teſt, primum cognoſcendo <lb/>angulos obliquitatis librę, <lb/>ideſt angulos <seg type="var">.b.o.u.</seg> et <seg type="var">.a.o.<lb/>u.</seg> quia oportet ſemper ſup-<lb/> <ptr xml:id="fig-0161-01a" corresp="fig-0161-01" type="figureAnchor"/> ponere ſitum aliquem no-<lb/>tum. </s> <s xml:space="preserve">Si nobis deinde co-<lb/>gnita erit proportio ipſius <seg type="var">.<lb/>o.u.</seg> ad <seg type="var">.o.b.</seg> et. ad <seg type="var">.o.a.</seg> aſſe-<lb/>quemur cognitionem angu <lb/>li <seg type="var">.b.</seg> et <seg type="var">.o.a.u.</seg> & per conſe-<lb/>quens ipſius <seg type="var">.o.a.t.</seg> eius reſi-<lb/>dui, vnde poſtea beneficio <lb/>angulorum <seg type="var">.e.</seg> et <seg type="var">.t.</seg> rectorum <lb/>& laterum <seg type="var">.o.b.</seg> et <seg type="var">.o.a.</seg> cogni <lb/>torum in cognitionem <seg type="var">.o.t.</seg> <lb/>et <seg type="var">.o.e.</seg> facile deueniemus.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0160-01" corresp="hd-0160-01a"/> <figure xml:id="fig-0161-01" corresp="fig-0161-01a"> <graphic url="0161-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">CAP. VIII.</head> <p> <s xml:space="preserve">QVod autem idem Tartalea in .6. propoſitione, & Iordanus in ſecunda parte. <lb/></s> <s xml:space="preserve">ſecundæ propoſitionis ſcribunt, maximum quoque errorem inſe continet. <lb/></s> <s xml:space="preserve">Dicunt enim <choice><ex>angulum</ex><am>angulũ</am></choice> <lb/><seg type="var">h.a.f.</seg> differentem ab <lb/>angulo <seg type="var">.d.b.f.</seg> alia ra-<lb/>tione non eſſe quàm <lb/>per angulum conta-<lb/>ctus <choice><ex>duorum</ex><am>duorũ</am></choice> <choice><ex>circulorum</ex><am>circulorũ</am></choice>, <lb/>vt in ſua figura ſcribit <lb/>Tartalea; </s> <s xml:space="preserve">id quod fal-<lb/>ſiſſimum eſt. </s> <s xml:space="preserve"><choice><ex>Quam</ex><am>Quã</am></choice> ob <lb/>cauſam in ſubſcripta <lb/>figura ſit libra <seg type="var">.B.A.</seg> <lb/> <ptr xml:id="fig-0161-02a" corresp="fig-0161-02" type="figureAnchor"/> & eius centrum. </s> <s xml:space="preserve">C et <seg type="var">.<lb/>u.</seg> <choice><ex>centrum</ex><am>centrũ</am></choice> regionis ele <lb/>mentaris, et <seg type="var">.A.u.</seg> et <seg type="var">.B.<lb/>u.</seg> lineæ <choice><ex>inclinationum</ex><am>inclinationũ</am></choice>. <lb/></s> <s xml:space="preserve">Imaginemur deinde <lb/>lineam <seg type="var">.B.K.</seg> <choice><ex>parallelam</ex><am>parallelã</am></choice> <lb/>ipſi <seg type="var">.A.u.</seg> quæ gyrum <seg type="var">.<lb/>B.F.A.</seg> in puncto <seg type="var">.K.</seg> <lb/>communi ſcientiæ <choice><ex>prae cepto</ex><am>pręcepto</am></choice> ſcindet, & habe <lb/>bimus angulum <seg type="var">.K.B.<lb/>Z.</seg> æqualem angulo <seg type="var">.<lb/>H.A.F.</seg> ideſt <seg type="var">.u.A.F.</seg> <lb/>(quia <seg type="var">.H.u.</seg> et <seg type="var">.D.</seg> <choice><ex>unum</ex><am>unũ</am></choice> <lb/>ſunt) cum ex .29. libr. <lb/>primi Euclidis angu- <pb facs="0162" n="150"/><fw type="head">IO. BAPT. BENED.</fw> lus <seg type="var">u.A.C.</seg> æqualis ſit <lb/>angulo <seg type="var">.K.B.T.</seg> & an-<lb/>gulus <seg type="var">.C.A.F.</seg> æqua-<lb/>lis angulo <seg type="var">.T.B.Z.</seg> <choice><ex>nunc</ex><am>nũc</am></choice> <lb/>comparatio eſt inter <lb/>angulum <seg type="var">.D.B.F.</seg> & an <lb/>gulum <seg type="var">.K.B.Z.</seg> miſtili-<lb/>neos, qui quidem duo <lb/>anguli, <choice><ex>communem</ex><am>cõmunem</am></choice> ha-<lb/>bent angulum miſtili <lb/>neum <seg type="var">.K.B.F.</seg> quapro-<lb/> <ptr xml:id="fig-0162-01a" corresp="fig-0162-01" type="figureAnchor"/> pter ſi angulus <seg type="var">.K.B.Z.</seg> <lb/>miſtilineus maior eſt <lb/>angulo <seg type="var">.D.B.F.</seg> miſti-<lb/>lineo per angulum <seg type="var">.<lb/>K.B.Z.</seg> contingentiæ, <lb/>circulorum ergo angu <lb/>lus miſtilineus com-<lb/>munis <seg type="var">.K.B.F.</seg> æqualis <lb/>erit miſtilineo, angu-<lb/>lo <seg type="var">.D.B.F.</seg> pars vide-<lb/>licet ſui toto. </s> <s xml:space="preserve">Omnis <lb/>autem error in quem <lb/>Tartalea, <choice><ex>Iordanusque</ex><am>Iordanusq;</am></choice> <lb/>lapſi fuerunt ab eo, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/> <ptr xml:id="hd-0162-0a" corresp="hd-0162-01" type="handwrittenAnchor"/> lineas inclinationum <lb/>pro parallelis viciſſim <lb/>ſumpſerunt, emana-<lb/>uit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0161-02" corresp="fig-0161-02a"> <graphic url="0161-02"/> </figure> <figure xml:id="fig-0162-01" corresp="fig-0162-01a"> <graphic url="0162-01"/> </figure> <note xml:id="hd-0162-01" corresp="hd-0162-01a"/> </div> </body> </floatingText> <p> <s xml:space="preserve">Septima propoſitio Tartaleæ, quæ eſt <choice><ex>quinta</ex><am>ꝗnta</am></choice> quæſtio Iordani mihi <choice><ex>videtur</ex><am>videt̃</am></choice> excipien-<lb/>da riſu, cum pondus ipſius <seg type="var">.A.</seg> ponderi ipſius <seg type="var">.B.</seg> exiſtens æquale, grauius ſit pondere <lb/>eiuſdem <seg type="var">.B.</seg> ratione minoris aperturæ anguli contingentiæ in <seg type="var">.A.</seg> quam in <seg type="var">.B.</seg> in quo <lb/>idem error committitur, qui in præcedenti committebatur, cum ſcilicet ipſe putet <lb/>lineas <seg type="var">.A.E.</seg> et <seg type="var">.B.D.</seg> figuræ ab eo confictæ ſibi inuicem eſſe parallelas, quæ etiam ſi <lb/>æquidiſtantes eſſent (vnde angulus <seg type="var">.E.A.G.</seg> minor eſſet angulo <seg type="var">.D.B.F.</seg>) non eam ta <lb/>men ob cauſam huiuſmodiangulorum differentia cauſa eſſet differentiæ <choice><ex>grauitatum</ex><am>grauitatũ</am></choice> <lb/>ipſorum <seg type="var">.A.</seg> et <seg type="var">.B.</seg> ob ea quæ cap .4. huius tractatus poſui.</s> </p> <p> <s xml:space="preserve">Octaua autem propoſitio, quæ eſt .6. quæſtio Iordani Iongè melius demonſtratur <lb/>ab Archi. in .6. lib. primi de ponderibus, cum nec à Iordano, nec à Tartalæa probata <lb/>fuerit, cum ijdem non probauerint præcedentes, quas in dicta .8. </s> <s xml:space="preserve">Tartalęa citat, qui <lb/>neque etiam probat nonam .10. 11. 12. et .13. cum ad pręcedentes probandas mini <lb/>mè acceſſerit.</s> </p> <p> <s xml:space="preserve">Quartadecima verò, quæ eſt .10. quęſtio Iordani, duas ob cauſas eſt falſa, quarum <lb/>vna eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> (ſupponendo <seg type="var">.A.D.E.G.B.</seg> eſſe vnum brachium librę et <seg type="var">.A.</seg> punctum <choice><ex>centri</ex><am>cẽtri</am></choice> <lb/>eiuſdem, et <seg type="var">.D.</seg> pondus ęquale ponderi <seg type="var">.E.</seg> & lineas inclinationum <seg type="var">.D.K.</seg> et <seg type="var">.E.M.</seg>) an <lb/>guli <seg type="var">.K.D.E.</seg> et <seg type="var">.M.E.G.</seg> ſibi <choice><ex>inuicem</ex><am>inuicẽ</am></choice> <choice><ex>non</ex><am>nõ</am></choice> ſunt ęquales; </s> <s xml:space="preserve"><choice><ex>cum</ex><am>cũ</am></choice> ille angulus ſit intrinſecus, hic <lb/>verò extrinſecus & oppoſitus dicto intrinſeco <choice><ex>vnius</ex><am>vniꝰ</am></choice> <choice><ex>trianguli</ex><am>triãguli</am></choice> terminati à. <seg type="var">D.E.</seg> à <seg type="var">.D.K.</seg> <pb facs="0163" n="151"/><fw type="head">DE MECHAN.</fw> et <seg type="var">.E.M.</seg> lineis productis vſque ad centrum regionis elementaris, vnde dictus angu-<lb/>lus <seg type="var">.M.E.G.</seg> maior eſt alio, ex .16 lib. primi Eucli. </s> <s xml:space="preserve">Qua ratione fit, vt hanc ob cauſam <lb/>E. grauius ſit ipſo <seg type="var">.D.</seg> cum minus dependeat à centro <seg type="var">.A.</seg> vt primo cap. huius tractatus <lb/>iam dixi. </s> <s xml:space="preserve">Alia quoque eſtratio, qua dictum <seg type="var">.E.</seg> grauius fit ipſo <seg type="var">.D.</seg> quę quidem eſt <lb/>maior diſtantia à centro <seg type="var">.A.</seg> libræ, per ſimiles rationes capit .4. huius tractatus ci-<lb/>tatas.</s> </p> <p> <s xml:space="preserve">Decimaquinta <choice><ex>quoque</ex><am>quoq;</am></choice> nil penitus valet, quę eſt .11. quęſtio Iordani, cuius Autho-<lb/>ris opuſculum opera Traiani Bibliopolę Venetijs è tenebris in lucem emerſit.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quòdſummaratione ſtateræper æqualia interualla <lb/>ſint diuiſæ.</head> <head xml:space="preserve">CAP. IX.</head> <p> <s xml:space="preserve">MAgna cum ratione <choice><ex>diuiduntur</ex><am>diuidũtur</am></choice> ſtateræ per interualla ęqualia, in libras, aut in <lb/>vncias, aut quoquo alio modo. </s> <s xml:space="preserve">Nam ſit ſtatera exempli gratia <seg type="var">.a.b.</seg> <lb/>& punctum, <choice><ex>quod</ex><am>ꝙ</am></choice> eam ſuſtinet ſit <seg type="var">.c.</seg> & vas illud, <choice><ex>quod</ex><am>ꝙ</am></choice> continetid, quod ponderari debet <lb/>f. </s> <s xml:space="preserve">Imaginemur nunc quod pondus brachij <seg type="var">.c.b.</seg> ab una parte, & pondus brachij <seg type="var">.c.a.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <lb/>eo, <choice><ex>quod</ex><am>ꝙ</am></choice> eſt dicti vaſis <seg type="var">.f.</seg> ab altera parte, ſint cauſę, quibus ſtatera <seg type="var">.a.b.c.</seg> ſtet orizonta-<lb/>lis. cui ſic orizontali manenti imaginemur ad punctum <seg type="var">.a.</seg> adiunctum eſſe pondus, <lb/>veluti vnius librę. & ad punctum <seg type="var">.d.</seg> tam diſtanti à <seg type="var">.c.</seg> ut eſt <seg type="var">.a.</seg> ab ipſo <seg type="var">.c.</seg> aliud quoque <lb/>pondus vnius libræ <choice><ex>additum</ex><am>additũ</am></choice> eſſe, vnde <choice><ex>coni</ex><am>cõi</am></choice> <choice><ex>quadam</ex><am>quadã</am></choice> ſcientia ſtatera, non mouebitur ſitu. <lb/></s> <s xml:space="preserve"><choice><ex>quia</ex><am>ꝗa</am></choice> exiſtentibus duobus hiſce ponderibus æqualibus, altero in <seg type="var">.d.</seg> & altero in <seg type="var">.a.</seg> remo <lb/>ta cum eſſent <seg type="var">.d.b.</seg> et <seg type="var">.f.</seg> abſque dubio <seg type="var">.a.d.</seg> non mutaret ſitum, ſed <seg type="var">.d.b.</seg> et, f. in ſitu, in <lb/>quo reperiuntur, à centro paribus viribus prędita ſunt. </s> <s xml:space="preserve">Addendo igitur <seg type="var">.d.b.</seg> ipſi <seg type="var">.d.</seg> <lb/>et <seg type="var">.f.</seg> ipſi .a: ſumma earum, æqualibus quoque viribus conſtabunt. ex communi ſen-<lb/>tentia, quæ habet ſi ęqualibus addas ęqualia, tota quoque fient ęqualia. </s> <s xml:space="preserve">Si verò <lb/>ponderi ipſius <seg type="var">.a.</seg> aliud adderetur eidem ęquale, haberemus in <seg type="var">.a.</seg> duplum pon-<lb/>dus ei <choice><ex>quod</ex><am>ꝙ</am></choice> eſt ipſius <seg type="var">.d.</seg> ſed volentes vt ſolum cum pondere ipſius <seg type="var">.d.</seg> ſtatera ſtet orizon <lb/>talis, ſi dictum pondus ipſius <seg type="var">.d.</seg> longè diſtabit à centro <seg type="var">.c.</seg> per duplum ipſius <seg type="var">.c.a.</seg> ideſt <lb/>ipſius <seg type="var">.c.d.</seg> id <choice><ex>quod</ex><am>ꝙ</am></choice> volumus aſſeque-<lb/>mur, beneficio ſupradictarum ra <lb/> <ptr xml:id="fig-0163-01a" corresp="fig-0163-01" type="figureAnchor"/> tionum, adiuti opera ſextę lib. pri<lb/>mi de <choice><ex>ponderibus</ex><am>põderibus</am></choice> Archimedis. </s> <s xml:space="preserve">Et <lb/>ſi quis aliud <choice><ex>quoque</ex><am>quoq;</am></choice> pondus adiun <lb/>geret ipſi <seg type="var">.a.</seg> æquale illi priori, ad <lb/><choice><ex>efficiendum</ex><am>efficiẽdum</am></choice>, vt ſtatera ſemper ori <lb/>zontalis maneret, oporteret, vt <choice><ex>pondus</ex><am>põdus</am></choice> ipſius <seg type="var">.d.</seg> ab <seg type="var">.c.</seg> longè diſtaret, ita vt huiuſmodi <lb/>diſtantia tripla eſſet primæ, & ſic per quoſdam quaſi gradus interualla redderentur <lb/>æqualia.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0163-01" corresp="fig-0163-01a"> <graphic url="0163-01"/> </figure> </div> </body> </floatingText> <pb facs="0164" n="152"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quòd line a circularis non habe at concauum cum con-<lb/>uexo coniunctum, & quod Aristo. cir caproportio <lb/>nes motuum aberrauerit.</head> <head xml:space="preserve">CAP.X.</head> <p> <s xml:space="preserve">ARiſtoteles in principio quæſtionum Mechanicarum ait lineam, quæ terminat <lb/> <ptr xml:id="hd-0164-01a" corresp="hd-0164-01" type="handwrittenAnchor"/> circulum videtur conuexum habere coniunctum cum concauo, quod falſum <lb/>eſt: </s> <s xml:space="preserve">quia huiuſmodi linea partes nullas ſecundum latitudinem habet, (vt ipſe etiam <lb/>confirmat) ſed eſt idem conuexum circuli: </s> <s xml:space="preserve">linea verò quæ terminus eſt ſuperficiei <lb/>ambientis, & amplectentis circulum eſt eadem concauitas dictæ ſuperficiei eun-<lb/>dem circulum ambientis, quæ nullam conuexitatem habet. </s> <s xml:space="preserve">& hæ duæ ſunt lineæ, <lb/>quarum vna diuerſa eſt ab alia, neque altera alterius, quod ad conuexum, & ad con-<lb/>cauum attinet.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0164-01" corresp="hd-0164-01a"/> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed illud, quod Ariſtoteles ſcribit de duplici reſpectu motus vnius puncti ſecun <lb/>dum vnam datam pro portionem, non ſufficit, ille enim ſic ait.</s> </p> <p> <s xml:space="preserve">Sit proportio ſecundum quam latum fertur, quam habet <seg type="var">.A.B.</seg> ad <seg type="var">.A.C.</seg> et <seg type="var">.A.</seg> qui <lb/>dem feratur verſus .B: <seg type="var">A.B.</seg> verò ſubterferatur verſus <seg type="var">.M.C.</seg> latum autem ſit <seg type="var">.A.</seg> <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/>ad <seg type="var">.D.</seg> vbi autem eſt <seg type="var">.A.B.</seg> verſus <seg type="var">.E</seg>. </s> <s xml:space="preserve">Quoniam igitur lationis erat proportio, quam <seg type="var">.<lb/>A.B.</seg> habet ad <seg type="var">.A.C.</seg> neceſſe eſt & <seg type="var">.A.D.</seg> ad <seg type="var">.A.E.</seg> hanc habere rationem. </s> <s xml:space="preserve">Simile igi <lb/> <ptr xml:id="hd-0164-02a" corresp="hd-0164-02" type="handwrittenAnchor"/> tur eſt pro portione paruum quadr ilaterum maiori. </s> <s xml:space="preserve">Quamobrem etc.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0164-02" corresp="hd-0164-02a"/> </div> </body> </floatingText> <p> <s xml:space="preserve">Cui reſpondeo, punctum <seg type="var">.A.</seg> quod mouetur in linea <seg type="var">.A.M.</seg> ab <seg type="var">.A.</seg> verſus <seg type="var">.M.</seg> vſque <lb/>ad <seg type="var">.F.</seg> non moueriab aliqua proportione determinata magis quàm ab alia: </s> <s xml:space="preserve">vnde <choice><ex>non</ex><am>nõ</am></choice> <lb/>ſolum poſſumus imaginari dictum punctum <seg type="var">.A.</seg> moueri ab <seg type="var">.A.</seg> vſque ad <seg type="var">.F.</seg> eiuſdem <lb/>velocitatis ſub alia quadam proportione, ſed etiam ſub alia, quæ iam datæ contraria <lb/>ſit, vt eſt proportio ipſius <seg type="var">.A.C.</seg> ad <seg type="var">.A.B.</seg> <choice><ex>imaginantes</ex><am>imaginãtes</am></choice> moueri <seg type="var">.A.</seg> verſus <seg type="var">.C.</seg> et <seg type="var">.A.C.</seg> ver <lb/>ſus <seg type="var">.B.M.</seg> delatam. </s> <s xml:space="preserve">Dico etiam idem <seg type="var">.A.</seg> moueri vſque ad <seg type="var">.F.</seg> ſecundum proportio-<lb/>nem ipſius <seg type="var">.A.O.</seg> ad <seg type="var">.A.N</seg>. </s> <s xml:space="preserve">Quamobrem imaginemur à puncto <seg type="var">.F.</seg> lineam <seg type="var">.F.H.</seg> cum <lb/>linea <seg type="var">.F.A.</seg> efficere angu-<lb/>lum æqualem angulo <seg type="var">.O.<lb/>P.A.</seg> & à puncto <seg type="var">.A.</seg> <choice><ex>lineam</ex><am>lineã</am></choice> <lb/> <ptr xml:id="fig-0164-01a" corresp="fig-0164-01" type="figureAnchor"/> <seg type="var">A.H.</seg> <choice><ex>cum</ex><am>cũ</am></choice> linea <seg type="var">.A.F.</seg> face-<lb/>re <choice><ex>angulum</ex><am>angulũ</am></choice> <choice><ex>æqualem</ex><am>æqualẽ</am></choice> angulo <lb/><seg type="var">O.A.P.</seg> unde angulus <seg type="var">.H.</seg> <lb/>æqualis erit angulo <seg type="var">.O.</seg> <lb/>ex .32. libr. primi Eucl. <lb/></s> <s xml:space="preserve">& <choice><ex>triangulum</ex><am>triangulũ</am></choice> <seg type="var">.A.H.F.</seg> ęqui <lb/>angulum erit triangulo <seg type="var">.<lb/>A.O.P</seg>. </s> <s xml:space="preserve">Quam ob <choice><ex>causam</ex><am>causã</am></choice> <lb/><choice><ex>eadem</ex><am>eadẽ</am></choice> proportio erit <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <lb/><seg type="var">A.H.</seg> ad <seg type="var">.F.H.</seg> quę <choice><ex>enipſius</ex><am>ẽipſius</am></choice> <lb/><seg type="var">A.O.</seg> ad <seg type="var">.O.P.</seg> punctum <lb/>igitur <seg type="var">.A.</seg> vſque ad <seg type="var">.F.</seg> mouetur ſecundum proportionem etiam ipſius <seg type="var">.A.O.</seg> ad <seg type="var">.O.P.</seg> <lb/>Huiuſmodi igitur conſideratio, ab Ariſtotele facta, nullius eſt momenti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0164-01" corresp="fig-0164-01a"> <graphic url="0164-01"/> </figure> </div> </body> </floatingText> <pb facs="0165" n="153"/> <fw type="head">DE MECHAN.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quod Aristo. in prima mechanicarum quæstionum eius quod <lb/>inquir it, uer am cauſam non attulerit.</head> <head xml:space="preserve">CAP. XI.</head> <p> <s xml:space="preserve">QVærens Ariſtoteles vnde fiat, vt eæ libræ, quæ brachia habent alijs longiora, <lb/>ſint exactiores cæteris, ait hoc euenire ratione maioris velocitatis extremo <lb/>rum earundem. </s> <s xml:space="preserve">Quod verum non eſt; </s> <s xml:space="preserve">quia hîc effectus nil aliud eſt, quam clarius pro <lb/>ponere ob omnium oculos obliquitatem brachiorum à linea orizontali, & oſtende-<lb/>re etiam facilius à dicto orizontali ſitu exire brachia iam dicta. </s> <s xml:space="preserve">Quæ quidem per ſe <lb/>neque à velocitate, neque à tarditate motus, ſed à ratione vectis, & à ma-<lb/>iori interuallo inter ſecundum ſitum extremorum à primo proficiſcuntur. </s> <s xml:space="preserve">Vt exem-<lb/>pli gratia, imaginemur magnam libram <seg type="var">.A.B.</seg> orizontalem, cuius centrum ſit <seg type="var">.E.</seg> et <lb/>pondus <seg type="var">.B.</seg> maius ſit pondere ipſius <seg type="var">.A.</seg> vnde conceditur, quòd ob hanc rationem di-<lb/>cta libra ſitum mutabit, qui ſecundus ſitus ſit in <seg type="var">.H.F</seg>. </s> <s xml:space="preserve">Imaginemur etiam <choice><ex>paruam</ex><am>paruã</am></choice> <choice><ex>quan- dam</ex><am>quã-dam</am></choice> libram <seg type="var">.a.e.b.</seg> orizontalem, quæ pondera habeat <seg type="var">.a.</seg> et <seg type="var">.b.</seg> æqualia duobus ponde <lb/>ribus alterius libræ & ſecundus ſitus ſit in <seg type="var">.h.f.</seg> ita tamen vt anguli circa <seg type="var">.e.</seg> æquales <lb/>ſint ijs, qui ſunt circa <seg type="var">.E.</seg> ideſt <seg type="var">.b.e.f.</seg> ſit ęqualis <seg type="var">.B.E.F</seg>. </s> <s xml:space="preserve">Nunc dico ſitum <seg type="var">.H.F.</seg> <choice><ex>exa- ctiorem</ex><am>exa-ctiorẽ</am></choice> futurum & clariorem ſitu <seg type="var">.h.e.f.</seg> ratione interualli <seg type="var">.B.F.</seg> maioris, interuallo <seg type="var">.<lb/>b.f.</seg> quod <seg type="var">.B.F.</seg> in eadem proportione maior eſt ipſo <seg type="var">.b.f.</seg> in qua <seg type="var">.B.E.</seg> maius eſt <seg type="var">.b.e.</seg> <lb/>quod autem interuallum <seg type="var">.B.F.</seg> breuiori, aut longiori temporis ſpacio quam <seg type="var">.b.f.</seg> ſit fa <lb/>ctum, nil planè refert. </s> <s xml:space="preserve">Ratione vectis deinde, dico <choice><ex>quod</ex><am>ꝙ</am></choice> ſi ſupponemus duas libras pa-<lb/>res <choice><ex>æqualesque</ex><am>æqualesq́;</am></choice> in omni alio reſpectu, præter quàm in brachiorum longitudine, pon-<lb/>dus <seg type="var">.B.</seg> maiorem vim habebit ad deprimendum brachium <seg type="var">.E.B.</seg> quàm pondus <seg type="var">.b.</seg> quia <lb/>libræ materiales, cum ſuſtineantur ab <seg type="var">.E.e.</seg> & non à puncto mathematico, ſed <lb/>à linea, aut ſuperficie naturali in materia exiſtente. </s> <s xml:space="preserve">vnde aliqua reſiſtentia ipſi mo-<lb/>tui brachiorum oritur, & hanc ob cauſam, ſupponendo hanc reſiſtentiam æqualem <lb/>tam in <seg type="var">.E.</seg> quàm in <seg type="var">.e.</seg> clarum erit ob ea, quæ in cap .4. huius tractatus oſtendi <seg type="var">.B.</seg> cum <lb/>minus dependeat ab <seg type="var">.E.</seg> aut minus quoque eidem <seg type="var">.E.</seg> annitatur, ponderoſum magis <lb/>futurum, quam <seg type="var">.b.</seg> & hac de cauſa mouebit ad partem inferiorem, maiori cum agilita <lb/>te, brachium <seg type="var">.E.B.</seg> multo magis etiam illud ipſum deprimet, ideſt maiorem etiam an <lb/>gulum <seg type="var">.B.E.F.</seg> quàm erit angulus <seg type="var">.b.e.f.</seg> faciet.</s> </p> <figure place="here"> <graphic url="0165-01"/> </figure> <pb facs="0166" n="154"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">De uer a cauſa ſecundæ, & tertiæ quæstionis mechanicæ <lb/>ab Ariſtotele nonperſpecta.</head> <head xml:space="preserve">CAP. XII.</head> <p> <s xml:space="preserve">ARiſtoteles in ſecunda quæſtionum mechanicarum quærens illius rationem ſic <lb/>ſcribit.</s> </p> <p> <s xml:space="preserve">Cur ſiquidem ſurſum fuerit ſpartum quando deorſum lato pondere quiſpiam id <lb/>admouet rurſus aſcendit libra: </s> <s xml:space="preserve">ſi autem deorſum conſtitutum fuerit non aſcendit, <lb/>ſed manet? </s> <s xml:space="preserve">an quia ſurſum quidem ſparto exiſtente plus libræ extra perpendiculum <lb/>ſi (ſpartum enim eſt perpendiculum) quare neceſſe eſt deorſum ferriid, quod plus <lb/>eſt, quare & cætera.</s> </p> <p> <s xml:space="preserve">Sed vera cauſa, vnde fiat, vt ſi ſpartum fuerit ſurſum, & brachium vnum <lb/>ipfius libræ deprimendo, & idem liberum deinde permittendo, ad ſitum ori-<lb/>zontalem redeat, non ſolum eſt maior quantitas ponderis brachiorum quæ iam præ <lb/>tergreſſa eſt vltra verticalem lineam, ſed etiam eſt longitudo brachij eleuati, quæ vl <lb/>tra verticalem lineam reperitur, vnde eius extremi pondus redditur grauius in pro-<lb/>portione, quam in hoc exemplo proponam, ſit <seg type="var">.A.B.</seg> libra in ſitu orizontali, cuius <lb/>ſpartum ſit <seg type="var">.E.</seg> ſuper ipſam. </s> <s xml:space="preserve">& deprimentes brachium ipſius <seg type="var">.A.</seg> vſque ad <seg type="var">.F.</seg> eius ſitus <lb/>ſit in <seg type="var">.F.H.</seg> vnde medium <choice><ex>punctum</ex><am>pũctum</am></choice> <seg type="var">.G.</seg> prætergreſſum erit lineam verticalem <seg type="var">.V.Z.</seg> ver <lb/>ſus <seg type="var">.B.</seg> quæ <seg type="var">.V.Z.</seg> ſecabit brachium <seg type="var">.F.G.</seg> in puncto <seg type="var">.D.</seg> vnde <seg type="var">.D.H.</seg> longius erit ipſo <seg type="var">.<lb/>F.D</seg>. </s> <s xml:space="preserve">Nunc nobis ſupponendum eſt id, <lb/>quod veriſſimum exiſtit, dictam ſcilicet li <lb/> <ptr xml:id="fig-0166-01a" corresp="fig-0166-01" type="figureAnchor"/> bram in ſitu <seg type="var">.F.H.</seg> <choice><ex>etiam</ex><am>etiã</am></choice> ſi ſuſtineatur à pun-<lb/>cto <seg type="var">.E.</seg> idem tamen futurum ac ſi ſuſtenta-<lb/>retur in puncto <seg type="var">.D.</seg> vnde ſequitur, quod <lb/>pondus appenſum ex ipſa <seg type="var">.H.</seg> ita grauius <lb/>reddatur, ipſo <seg type="var">.F.</seg> in eadem propor-<lb/>tione, quæ maioreſt <seg type="var">.D.H.</seg> ipſo <seg type="var">.D.F.</seg> ob <lb/>rationes quas in primis huius tractatus ca-<lb/>pitibus poſui, vt etiam ſi <seg type="var">.D.H.</seg> quodmate <lb/>riale eſſe ſupponitur, nullam planſe4; </s> <s xml:space="preserve">graui-<lb/>tatem haberet, <choice><ex>ſolustamem</ex><am>ſolustamẽ</am></choice> exceſſus vis pon <lb/>deris in <seg type="var">.H.</seg> poſiti, longè maior pondere in <lb/>F. collocato pro maiorilongitudine ipſius <lb/><seg type="var">D.H.</seg> ſufficiat. ad præſtandum vt libra ad <lb/>ſitum orizontalem redeat.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0166-01" corresp="fig-0166-01a"> <graphic url="0166-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">In ſecunda deinde huius quęſtionis par <lb/> <ptr xml:id="hd-0166-01a" corresp="hd-0166-01" type="handwrittenAnchor"/> te, in qua ſcribit libram in ſitu, in quo poſi <lb/>ta eſt, firmam manere, toto cęlo aberrat, quia <choice><ex>neceſſarium</ex><am>neceſſariũ</am></choice> eſt, vt omninò cadat, eò <choice><ex>uſque</ex><am>uſq;</am></choice> <lb/>quò ſpartum ſurſum remaneat: </s> <s xml:space="preserve">ablato tamen omni impedimento, quod nulla eget <lb/>probatione, cum natura ſua clariſſimè pateat.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0166-01" corresp="hd-0166-01a"/> </div> </body> </floatingText> <p> <s xml:space="preserve">Cauſa, deinde, vera tertiæ quæſtionis non eſt ea, quam Ariſtoteles ponit, ſed hu-<lb/>iuſmodi effectus ab eo, quod capitibus .4. et .5. huius tractatus propoſui originem <lb/>habet.</s> </p> <pb facs="0167" n="155"/> <fw type="head">DE MECHAN.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quòd Ariſtotelisratio in 6. quæſtione poſit a non ſit admittenda.</head> <head xml:space="preserve">CAP. XIII.</head> <p> <s xml:space="preserve">VOlens Ariſtoteles rationem proponere, vnde fiat, vt nauis velocius moueatur <lb/>cum antennam altiorem quàm cum depræſſiorem habet, id ad vectis ratio-<lb/>nem refert, quod verum <choice><ex>non</ex><am>nõ</am></choice> eſt. </s> <s xml:space="preserve">Huiuſmodi enim ratione nauis tardius potius, quàm <lb/>velocius ferri deberet, quia quantò altius eſt velum, vi venti impulſum, <choice><ex>tantò</ex><am>tãtò</am></choice> magis <lb/>proram ipſius nauis in aquam demergit. </s> <s xml:space="preserve">Sed huiuſmodi effectus à maiori potius <lb/>quantitate venti quam recipit, quàm ab alia aliqua cauſa oritur, quia ventus liberius <lb/><choice><ex>vehementiusque</ex><am>vehementiusq́;</am></choice> in altiore parte, quàm in depræſſione vagatur & perflat.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quòdrationes ab Ariſtotele de octaua quæstione confictæ <lb/>ſufficient es non ſint.</head> <head xml:space="preserve">CAP. XIIII.</head> <p> <s xml:space="preserve">RAtiones etiam ab Ariſtotele propoſitæ pro indaganda octauæ quæſtionis ve-<lb/>ritate, in qua quærit vnde fiat, vt corpora rotundæ figuræ, ad <choice><ex>voluendum</ex><am>voluendũ</am></choice> ſint <lb/>faciliora reliquis, quarum reuolutionum corporum tres ſpecies aſſignat, <choice><ex>quarum</ex><am>quarũ</am></choice> vna <lb/>eſt, vt rotarum <choice><ex>curruum</ex><am>curruũ</am></choice>; </s> <s xml:space="preserve">altera vt rotarum puteorum, aut trochlearum, quibus hauri-<lb/>tur aqua; </s> <s xml:space="preserve">& tertia, vt paruorum vaſorum a figulis fabricatorum, <choice><ex>ſufficientes</ex><am>ſufficiẽtes</am></choice> <choice><ex>non</ex><am>nõ</am></choice> ſunt.</s> </p> <p> <s xml:space="preserve">Incipiens autem à prima dico dubium non eſſe, quin tangente corpore aliquo ro <lb/>tundo aliquod planum mediante ſolo quodam puncto contingat, quemadmodum <lb/>probat Theodoſius in .3. lib. primi & Vitellio in .71. lib. primi, & <choice><ex>ducendo</ex><am>ducẽdo</am></choice> per <choice><ex>centrum</ex><am>centrũ</am></choice> <lb/>ſphæræ lineam vſque ad punctum contactus, ipſa erit perpendicularis plano contin-<lb/>genti ſphęram dictam, vt probat <choice><ex>idem</ex><am>idẽ</am></choice> Thęodoſius in .4. lib. primi <choice><ex>Alhazem</ex><am>Alhazẽ</am></choice> in .25. quar-<lb/>ti, & Vitellio in .7. primi. </s> <s xml:space="preserve">Verum etiam eſt omnem inclinationem ponderoſam huiuſ <lb/>modi corporis homogęnei totam hanc lineam æqualiter omni ex parte circundare; <lb/></s> <s xml:space="preserve">cuius quidem rei exemplum in carta deſcribere poſſumus mediante figura circulari <lb/>hîc ſubſcripta <seg type="var">.a.n.e.u.</seg> contigua lineæ rectæ <seg type="var">.b.d.</seg> in puncto <seg type="var">.a.</seg> vnde <seg type="var">.e.o.a.</seg> perpendicu <lb/>laris erit ipſi <seg type="var">.b.d.</seg> ex .17. lib. 3. Eucli. & <choice><ex>tantum</ex><am>tantũ</am></choice> ponderis habebimus à parte <seg type="var">.a.u.e.</seg> quan <lb/>tum ab ipſa <seg type="var">.a.n.e</seg>. </s> <s xml:space="preserve">Nuncigitur ſi imaginabimur ductum eſſe centrum verſus <seg type="var">.u.</seg> per <lb/>lineam <seg type="var">.o.u.</seg> parallelam ipſi <seg type="var">.a.d.</seg> clarum nobis <lb/>erit, <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>abſque</ex><am>abſq;</am></choice> vlla difficultate aut reſiſtentia <choice><ex>idem</ex><am>idẽ</am></choice> <lb/> <ptr xml:id="fig-0167-01a" corresp="fig-0167-01" type="figureAnchor"/> ducemus, quia huiuſmodi centrum ab inferiori <lb/>parte ad ſuperiorem, nunquam mutabit ſitum <lb/>reſpectu <choice><ex>diſtantiæ</ex><am>diſtãtiæ</am></choice> ſeu interualli, quę inter ipſum <lb/>lineamq́ue <seg type="var">.a.d.</seg> intercedit, <choice><ex>quod</ex><am>ꝙ</am></choice> quidem centrum <lb/>in ſe colligittotum pondus figurę <seg type="var">.a.n.e.u.</seg> & be <lb/>neficio lineæ <seg type="var">.e.o.a.</seg> illud ipſum puncto <seg type="var">.a.</seg> in li-<lb/>nea <seg type="var">.b.a.d.</seg> committit, productum <seg type="var">.a.</seg> nil refert, <lb/>vt magis, aut minus verſus ipſum <seg type="var">.d.</seg> aut verſus <lb/>b. <choice><ex>dirigatur</ex><am>dirigat̃</am></choice>; </s> <s xml:space="preserve">ita vt <choice><ex>cum</ex><am>cũ</am></choice> non oporteat vt huius figuræ <lb/><choice><ex>pondus</ex><am>põdus</am></choice>, vna vice, magis eleuetur, quàm alia, ſed <lb/>ſemper ęqualiter ſuper lineam <seg type="var">.b.a.d.</seg> quieſcat.</s> <pb facs="0168" n="156"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve"><choice><ex>Sitque</ex><am>Sitq;</am></choice> ſemper diuiſum à linea <seg type="var">.a.o.e.</seg> per medium, ſequitur communi quodam con-<lb/>ceptu, nullam nobis difficultatem oborituram, dictum centrum ad quam volueri-<lb/>mus partem ducendo, quemadmodum à qualibet alia figura, quæ perfectè rotunda <lb/>non eſſet, emergeret; </s> <s xml:space="preserve">Vt <choice><ex>exempli</ex><am>exẽpli</am></choice> gratia, ſi imaginabimur pentagonum <seg type="var">.K.i.h.f.l.</seg> quie <lb/>ſcere <choice><ex>ſuper</ex><am>ſuꝑ</am></choice> <choice><ex>eandem</ex><am>eandẽ</am></choice> <choice><ex>lineam</ex><am>lineã</am></choice> <seg type="var">.a.b.K.</seg> ita ut <choice><ex>primum</ex><am>primũ</am></choice> <choice><ex>totum</ex><am>totũ</am></choice> latus <seg type="var">.i.K.</seg> in linea <seg type="var">.b.K.</seg> <choice><ex>extendatur</ex><am>extẽdat̃</am></choice>, <choice><ex>ducen- do</ex><am>ducẽ-do</am></choice> poſteà centrum <seg type="var">.o.</seg> (ponamus.) verſus <seg type="var">.l.</seg> dubium non eſt, quin oporteat, vt dictum <lb/>centrum <seg type="var">.o.</seg> à linea <seg type="var">.b.d.</seg> eleuetur, ab <choice><ex>eademque</ex><am>eademq;</am></choice> magis diſtet, voluens ſe per <choice><ex>arcum</ex><am>arcũ</am></choice> vnum <lb/>circuli, <choice><ex>qui</ex><am>ꝗ</am></choice> <choice><ex>pro</ex><am>ꝓ</am></choice> ſuo ſemidiametro habeat <seg type="var">.o.K.</seg> quę maior eſt ipſa <seg type="var">.o.a.</seg> ex .18. li. primi Eu <lb/>cli. vnde ſi à puncto <seg type="var">.K.</seg> imaginabimur lineam <seg type="var">.K.c.</seg> reſpicientem centrum regionis <lb/>elementaris, dubium non eſt, quin ſi velimus transferre <choice><ex>centrum</ex><am>cẽtrum</am></choice> hoc à priori ſitu <choice><ex>vſque</ex><am>vſq;</am></choice> <lb/>ad dictam lineam, oporteat addere pondus parti ipſius <seg type="var">.l.</seg> quæ à linea <seg type="var">.K.c.</seg> fuit ſecta, <lb/>aut aliquid de ipſo pondere partis centri detrahere. </s> <s xml:space="preserve">quod quibuſuis modis fiat, ar-<lb/>duum certè eſt ad efficiendum; </s> <s xml:space="preserve">neque hoc etiam accidit figuræ perfectè rotundæ, <lb/>cum <choice><ex>centrum</ex><am>cẽtrum</am></choice> <choice><ex>quod</ex><am>ꝙ</am></choice> perfectè in medio ipſius ponderis eſt, reperiatur ſemper in linea per-<lb/>pendiculari ipſi plano, in quo animaduertendum eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> etiam ſi ipſum planum ap-<lb/>pellem; </s> <s xml:space="preserve">pro plano tamen perfecto intelligi nolo, ſed pro ſuperficie perfectè <choice><ex>ſphaeri- ca</ex><am>ſphęri-ca</am></choice> circa centrum à corporibus grauibus expetitum; </s> <s xml:space="preserve">nam ratione magnæ amplitudi-<lb/>nis huiuſmodi ſuperficiei, nullam differentiam notatu dignam à perfecto aliquo pla <lb/>no exigui interualli ad curuitatem eiuſdem ſuperficiei imaginari poterimus. </s> <s xml:space="preserve">Sed ut <lb/>redeamus ad ſermonem de reuolutione figuræ rotundæ ſuſceptum, <choice><ex>clarum</ex><am>clarũ</am></choice> igitur erit <lb/>quamlibet minimam vim (vt ita dicam) quę trahat, aut impellat centrum <seg type="var">.o.</seg> verſus <seg type="var">.u.</seg> <lb/>huiuſmodi figuram reuoluturam, cuius media pars ad trahendum, aut impellendum <lb/>punctum <seg type="var">.e.</seg> ſufficiere; </s> <s xml:space="preserve">Imaginemur autem <choice><ex>quod</ex><am>ꝙ</am></choice> li <lb/>nea <seg type="var">.n.o.u.</seg> eſſet libra <choice><ex>quędam</ex><am>quędã</am></choice> in figura perfectè <lb/> <ptr xml:id="fig-0168-01a" corresp="fig-0168-01" type="figureAnchor"/> rotunda <seg type="var">.a.n.e.u.</seg> poſita, & vis, quę trahere cen <lb/>trum deberet, diuiſa eſſet per medium, cuius <lb/>medietas appenſa eſſet extremitati <seg type="var">.u.</seg> diame-<lb/>tri <seg type="var">.n.o.u.</seg> <choice><ex>clarum</ex><am>clarũ</am></choice> erit, <choice><ex>quod</ex><am>ꝙ</am></choice> abſque vlla difficultate <lb/>reuolueret figuram ſuper lineam <seg type="var">.b.a.d.</seg> verſus <seg type="var">.<lb/>d.</seg> quia huius vis, aut pondus <choice><ex>nullum</ex><am>nullũ</am></choice> contra pon <lb/>dus haberet vltra centrum <seg type="var">.o.</seg> uerſus <seg type="var">.n.</seg> <choice><ex>quod</ex><am>ꝙ</am></choice> cen-<lb/>trum <seg type="var">.o.</seg> perpetuo quieſcit <choice><ex>ſuper</ex><am>ſuꝑ</am></choice>. a. in linea <seg type="var">.e.o.<lb/>a.</seg> per medium diuidente ſemper totum pon-<lb/>dus figurę ſuppoſitę. </s> <s xml:space="preserve">Tantò facilius ergo tota <lb/>dicta vis ap <lb/> <ptr xml:id="fig-0168-02a" corresp="fig-0168-02" type="figureAnchor"/> <ptr xml:id="fig-0168-03a" corresp="fig-0168-03" type="figureAnchor"/> plicata cen <lb/>tro, <choice><ex>ipsum</ex><am>ipsũ</am></choice> ver <lb/>ſus <seg type="var">.u.</seg> <choice><ex>trahens</ex><am>trahẽs</am></choice> <lb/>per lineam <lb/><choice><ex>parallelam</ex><am>parallelã</am></choice> ip <lb/>ſi <seg type="var">.a.d.</seg> <choice><ex>dictam</ex><am>dictã</am></choice> <lb/>figuram re-<lb/>uolueret. </s> <s xml:space="preserve">Et <lb/>ſi linea qua <lb/>dictum cen <lb/>trum trahi-<lb/>tur ab ipſo <pb facs="0169" n="157"/><fw type="head">DE MECHAN.</fw> <seg type="var">b.a.d.</seg> non æquediſtaret, ſed ſurſum traheret ſuper <seg type="var">.u.</seg> aut ſubter, aliquid de ſua vi vir <lb/><choice><ex>tuteque</ex><am>tuteq́;</am></choice> amitteret, & tantò plus, quantò inclinata magis eſſet verſus <seg type="var">.a.o.e.</seg> & tandem <lb/>cum eſſet vnita cum <seg type="var">.a.o.e.</seg> aut ad ſuperius, aut ad inferius quantalibet ui, etiam ſi in-<lb/>finita, figuram extra ſitum primæ lineæ <seg type="var">.a.o.e.</seg> non moueret, ſed ſi ſurſum traheret ſe <lb/>iungeret eam à linea <seg type="var">.b.a.d.</seg> non ob id tamen efficeret, ut centrum <seg type="var">.o.</seg> exiret extra pri <lb/>mam lineam <seg type="var">.a.o.e</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0167-01" corresp="fig-0167-01a"> <graphic url="0167-01"/> </figure> <figure xml:id="fig-0168-01" corresp="fig-0168-01a"> <graphic url="0168-01"/> </figure> <figure xml:id="fig-0168-02" corresp="fig-0168-02a"> <graphic url="0168-02"/> </figure> <figure xml:id="fig-0168-03" corresp="fig-0168-03a"> <graphic url="0168-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Secunda verò ſpecies, tribus reuolutionum modis, abſque axis mutatione conſta <lb/>re poteſt, ideſt modo, quo reuoluuntur trochleæ mediante fune, & quo reuoluuntur <lb/>aliquæ rotæ, in quibus aliquod animal incedit; </s> <s xml:space="preserve">& quo reuoluuntur illæ, quæ in homi <lb/>nis manu circunuoluuntur medio alicuius manubrij inflexi. </s> <s xml:space="preserve">Hi omnes modi cum <lb/>circulari figura magis, <choice><ex>quam</ex><am>quã</am></choice> cum alia quauis, faciliores euadunt. </s> <s xml:space="preserve">Et primò ſi priorem <lb/>modum conſiderabimus, vt mediante fune quælibet figura, quæ circularis non ſit, <lb/>voluatur, ſupponamus exemplo debere reuolui pentagonum æquiangulum <seg type="var">.a.e.i.o.<lb/>u.</seg> circa centrum <seg type="var">.c.</seg> mediante fune <seg type="var">.q.u.a.e.i.p.</seg> neceſſariò occurrent (in hac figura an-<lb/>gulorum, <choice><ex>laterumque</ex><am>laterumq́;</am></choice> diſparium) plures inæqualitates, quæ reuolutionem eiuſdem fi-<lb/>guræ irregularem efficient; </s> <s xml:space="preserve">quarum vna erit, quod duæ partes funis, ideſt <seg type="var">.u.q.</seg> et <seg type="var">.i.p.</seg> <lb/>non erunt in vna <choice><ex>eademque</ex><am>eademq́;</am></choice> inter ſe diſtantia ſemper, quod facile intellectu erit, ſi ima <lb/>ginabimur ductas eſſe lineas <seg type="var">.a.i</seg>: <seg type="var">u.i</seg>: et <seg type="var">.i.c.t.</seg> ſi funis duo pondera habebit alterum <lb/>altero maius, ſuis extremis appenſa, vnde debeat figura virtute ponderis maioris cir <lb/>cunuolui: </s> <s xml:space="preserve">dictæ duæ partes <seg type="var">.u.q.</seg> et <seg type="var">.i.p.</seg> eiuſdem funis, <choice><ex>mundi</ex><am>mũdi</am></choice> centrum, dum firmæ ma <lb/>nebunt, reſpicient; </s> <s xml:space="preserve">ſed permittentes pondera libera; </s> <s xml:space="preserve">maius, efficiens vt circunuolua-<lb/>tur figura; </s> <s xml:space="preserve">efficiet, vt aliquando vnum exlateribus, eiuſdem figuræ mundi <choice><ex>quoque</ex><am>quoq;</am></choice> cen <lb/>trum reſpiciet, vt in <lb/> <ptr xml:id="fig-0169-01a" corresp="fig-0169-01" type="figureAnchor"/> figura <seg type="var">.A.</seg> <choice><ex>ſicque</ex><am>ſicq́;</am></choice> etiam <lb/>linea <seg type="var">.i.c.t.</seg> (pro <choice><ex>exem- plo</ex><am>exẽ-plo</am></choice>) erit menſura di-<lb/>ſtantiæ funium inter <lb/>ipſas, & deinde <choice><ex>circum</ex><am>circũ</am></choice> <lb/>uoluendo etiam di-<lb/>ſtabuntinter ſe per li <lb/><choice><ex>neam</ex><am>neã</am></choice> <seg type="var">.i.a.</seg> aut <seg type="var">.i.u.</seg> vt in <lb/>figura <seg type="var">.B.</seg> <choice><ex>innotuit</ex><am>ĩnotuit</am></choice> <choice><ex>exem</ex><am>exẽ</am></choice> <lb/>plo, & ſic etiam ali-<lb/>quando erunt magis <lb/><choice><ex>diſtantes</ex><am>diſtãtes</am></choice>, quàm linea <lb/><seg type="var">t.i.</seg> & minus quàm.i.<lb/>a: </s> <s xml:space="preserve">nunquam tamen minus quam <seg type="var">.t.i.</seg> neque magis <choice><ex>quam</ex><am>quã</am></choice> <lb/><seg type="var">i.a.</seg> aut <seg type="var">.i.u.</seg> quæ ſunt æquales; </s> <s xml:space="preserve">Quæ quidem varietas, <lb/> <ptr xml:id="fig-0169-02a" corresp="fig-0169-02" type="figureAnchor"/> in hanc, & in illam partem impellet partes penden-<lb/>tes funis, vnde æqualiter non trahent. </s> <s xml:space="preserve">Idem dico, ſi <lb/>extrema <seg type="var">.q.</seg> et <seg type="var">.p.</seg> eſſent quoque ſemper in vna <choice><ex>eademque</ex><am>eadẽq́;</am></choice> <lb/>diſtantia; </s> <s xml:space="preserve">neque à corpore <choice><ex>ponderoſo</ex><am>põderoſo</am></choice> eſſent attracta, <lb/>quia aliæ partes ipſius <seg type="var">.u.q.</seg> et <seg type="var">.i.p.</seg> ex ſupradictis ratio-<lb/>nibus vnam <choice><ex>eademque</ex><am>eademq́;</am></choice> diſtantiam <choice><ex>non</ex><am>nõ</am></choice> ſemper <choice><ex>ſeruarent</ex><am>ſeruarẽt</am></choice>. <lb/></s> <s xml:space="preserve">vnde fieret vt cum diuerſis angulis tam <seg type="var">.i.p.</seg> <choice><ex>quam</ex><am>quã</am></choice> <seg type="var">.u.q.</seg> <lb/><choice><ex>traherent</ex><am>traherẽt</am></choice> ſemidiametros <seg type="var">.c.i</seg>: <seg type="var">c.e</seg>: <seg type="var">c.a</seg>: <seg type="var">c.u.</seg> et <seg type="var">.c.o.</seg> quia <choice><ex>non</ex><am>nõ</am></choice> <lb/>ſemper traherent ope ſeu virtute anguli æqualis ipſi <seg type="var">.<lb/>c.i.p</seg>. </s> <s xml:space="preserve">Hæc autem inę qualitas communis eſt omnibus <pb facs="0170" n="158"/><fw type="head">IO. BAPT. BENED.</fw> figuris rectilineis tam paris, quàm diſparis numeri. </s> <s xml:space="preserve">Sed aliam quandam maiorem <lb/>inęqualitatem habent hæ figuræ numeri diſparis, quæ eſt, quòd <choice><ex>quando</ex><am>quãdo</am></choice> linea <seg type="var">.t.i.</seg> tam <lb/> <seg type="var">.u.q.</seg> quàm ipſi <seg type="var">.i.p.</seg> <lb/> <ptr xml:id="fig-0170-01a" corresp="fig-0170-01" type="figureAnchor"/> <choice><ex>perpendicularis</ex><am>ꝑpẽdicularis</am></choice> fuerit, <lb/>ideſt <choice><ex>quando</ex><am>quãdo</am></choice> <seg type="var">.t.i.</seg> cum <lb/>dictis partibus funis <lb/>angulos rectos con-<lb/>ſtituerit, <choice><ex>tunc</ex><am>tũc</am></choice> ratione <lb/><choice><ex>longitudinis</ex><am>lõgitudinis</am></choice> ipſius <seg type="var">.c.<lb/>i.</seg> maioris quam <seg type="var">.t.<lb/>c.</seg> (quia cum ſit <seg type="var">.c.i.</seg> <choice><ex>ae- qualis</ex><am>ę-qualis</am></choice> ipſi <seg type="var">.c.a.</seg> et <seg type="var">.c.a.</seg> <lb/>maior ipſa <seg type="var">.c.t</seg>: <seg type="var">c.i.</seg> <lb/>etiam maior ſit ipſa <seg type="var">.<lb/>c.t.</seg>) pondus aut vis <lb/>ipſius <seg type="var">.p.</seg> ſuperabit <choice><ex>eam</ex><am>eã</am></choice> <lb/>quæ eſt ipſius <seg type="var">.q.</seg> ſed <lb/>quando <seg type="var">.t.</seg> erit in oppoſita parte, et <seg type="var">.i.</seg> in ea, quæ eſt <lb/>ipſius .t: q. <choice><ex>eandem</ex><am>eãdem</am></choice> ob cauſam ſuperabit <seg type="var">.p.</seg> & ſic mo <lb/> <ptr xml:id="fig-0170-02a" corresp="fig-0170-02" type="figureAnchor"/> tum faciet irregularem, & <choice><ex>non</ex><am>nõ</am></choice> vniformem; </s> <s xml:space="preserve">& obid <lb/>etiam perarduum, præter ictus, quos infligunt an-<lb/>guli in partem pendentem <choice><ex>aſcendentem</ex><am>aſcendẽtem</am></choice> funis, <choice><ex>quan- do</ex><am>quã-do</am></choice> vnum exlateribus vnitur cum fune.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0169-01" corresp="fig-0169-01a"> <graphic url="0169-01"/> </figure> <figure xml:id="fig-0169-02" corresp="fig-0169-02a"> <graphic url="0169-02"/> </figure> <figure xml:id="fig-0170-01" corresp="fig-0170-01a"> <graphic url="0170-01"/> </figure> <figure xml:id="fig-0170-02" corresp="fig-0170-02a"> <graphic url="0170-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Aliam inęqualitatem habent figuræ pares, quæ <lb/>etiam in imparibus cernitur, etſi aliquantulum di-<lb/>uerſa; </s> <s xml:space="preserve">quæ ab eo oritur, quod funes ſit modò ma-<lb/>gis, modo minus propinquę centro; </s> <s xml:space="preserve">quæ inæqualis <lb/>diſtantia, maiorem <choice><ex>minoremque</ex><am>minoremq́;</am></choice> vim ſuper dictum <lb/>centrum ob rationes in ſecunda parte cap. decimi <lb/>huius tractatus propoſitas, gignit. </s> <s xml:space="preserve">Nulla autem <lb/>ex ijs inæqualitatibus circulari figuræ contingit. </s> <s xml:space="preserve">Illud verò, quod de pentagonis fi-<lb/>guris dixi, omnibus aliis figuris diſparibus accommodari poteſt.</s> </p> <p> <s xml:space="preserve">Secundus modus eſt earum rotarum, in quibus aliquod animal incedit, quæ ſi cir-<lb/>culares non eſſent, tantò difficilius voluerentur, quantò pauciores angulos haberent. <lb/></s> <s xml:space="preserve">quod cum per ſe pateat, non demonſtrabo. </s> <s xml:space="preserve">Si ergo quantò plures angulos habebit <lb/>dicta figura, tantò ad circunuoluendum hoc modo agilior erit. </s> <s xml:space="preserve">Circularis igitur fi-<lb/> <ptr xml:id="hd-0170-01a" corresp="hd-0170-01" type="handwrittenAnchor"/> gura, quæ ex infinitis angulis efficitur, omnium agillima erit.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0170-01" corresp="hd-0170-01a"/> </div> </body> </floatingText> <p> <s xml:space="preserve">Tertius modus eſt earum rotarum, quæ manubrium habent, quæ etiam quantò <lb/>pauciores angulos habebunt, tanto <choice><ex>quoque</ex><am>quoq;</am></choice> difficiliores reddentur, tam ratione inimi <lb/>citiæ: </s> <s xml:space="preserve">quam exercet cum vacuo natura, quàm <choice><ex>violentię</ex><am>violẽtię</am></choice>, quam anguli aeri faciunt, eum <lb/>expellendo, vt ipſi occupent locum, quem ipſe <choice><sic>aér</sic><corr>aer</corr></choice> implebat. </s> <s xml:space="preserve">Quod nullo modo po <lb/>teſt euenire circulari figuræ.</s> </p> <p> <s xml:space="preserve">Nunc nobis ad dicendum reſtat de ſpecie reuolutionis rotarum, quæ parallelæ <lb/>ſunt orizonri, quibus accidit poſſe volui primo <choice><ex>tertioque</ex><am>tertioq́;</am></choice> modo ſecundę ſpeciei, & ob <lb/>id ſi circulares non erunt, eadem ſubibunt incommoda, de quibus in ſecunda illa ſpe <lb/>cie loquuti ſumus. </s> <s xml:space="preserve">ſed circulares rotæ huius tertiæ ſpeciei ad reuoluendum erunt re-<lb/>liquis eò faciliores, <choice><ex>quod</ex><am>ꝙ</am></choice> vno <choice><ex>ſolum</ex><am>ſolũ</am></choice> polo nituntur; </s> <s xml:space="preserve">Quod alijs nequaquam conceditur.</s> </p> <pb facs="0171" n="159"/> <fw type="head">DE MECHAN.</fw> <p> <s xml:space="preserve">Super hac tertia ſpecie formari poteſt problema, vnde fiat, vt quieſcens huiuſ-<lb/>modi rota parallela orizonti ſuper vnum punctum, & quantò fieri poteſt exiſtens <choice><ex>ae- qualis</ex><am>ę-qualis</am></choice>, ſi eam circunuoluamus maiore qua poterimus ui, & <choice><ex>eandem</ex><am>eãdem</am></choice> poſtea dimitten-<lb/>tes non perpetuò circunuoluatur.</s> </p> <p> <s xml:space="preserve">Hoc quidem, quatuor fit ob cauſas. quarum prima eſt, quia huiuſmodi motus, eius <lb/>rotæ non ſit naturalis. </s> <s xml:space="preserve">ſecunda eſt, quia etiamſi rota ſuper punctum mathematicum <lb/>quieſceret, oporteret tamen vt ſuperius <choice><ex>alterum</ex><am>alterũ</am></choice> haberet polum, qui ipſam <choice><ex>orizontalem</ex><am>orizontalẽ</am></choice> <lb/>teneret, qui quidem munimento aliquo corporeo indigeret; </s> <s xml:space="preserve">vnde fricatio quędam <lb/>conſequeretur, ex qua reſiſtentia prodiret.</s> </p> <p> <s xml:space="preserve">Tertia eſt, quia aer contiguus eam perpetuò aſtringit, <choice><ex>hocque</ex><am>hocq́;</am></choice> modo eius motui <lb/>reſiſtit.</s> </p> <p> <s xml:space="preserve">Quarta eſt, quia quęlibet pars corporea, quę à ſe mouetur, impetu eidem à quali-<lb/>bet extrinſeca virtute mouente impręſſo, habet naturalem inclinationem ad rectum <lb/>iter, non autem curuum, vnde ſi à dicta rota particula aliqua ſuę circunferentiæ <choice><ex>diſium</ex><am>diſiũ</am></choice> <lb/>geretur, abſque dubio per aliquod temporis ſpatium pars ſeparata recto itinere fer <lb/>retur per aerem, vt exemplo à fundis, quibus iaciuntur lapides, ſumpto, cognoſce <lb/>re poſsumus, in quibus, impetus motus impręſſus naturali quadam propenſione <lb/>rectum iter peragit, cum euibratus lapis, per lineam rectam contiguam giro, quem <lb/>primo faciebat, in puncto, in quo dimiſſus fuit, rectum iter inſtituat, vt rationi con-<lb/>ſentaneum eſt.</s> </p> <p> <s xml:space="preserve">Eadem, quoque ratione fit, vt quantò maior eſt aliqua rota, tantò maiorem quo <lb/>que impetum, & impreſſionem motus eius circunferentiæ partesrecipiant, vnde <choice><ex>ſae pe</ex><am>ſępe</am></choice> euenit, vt dum eam ſiſtere volumus, id <choice><ex>cum</ex><am>cũ</am></choice> labore & cum diſſicultate agamus ; </s> <s xml:space="preserve">quia <lb/>quantò maior eſt diameter vnius circuli, tantò minus curua eſt eiuſdem circunferen <lb/>tia, & tantò propius accedit angulum eiuſdem circunferentiæ ad quantitatem duo-<lb/>rum angulorum rectorum rectilineorum, ideſt circunferentia ad rectitudinem linea <lb/>rem. </s> <s xml:space="preserve">Vnde earundem partium dictæ circunferentiæ motus ad inclinationem ſibi à <lb/>natura tributam, quæ eſt incedendi per lineam rectam, magis accedit.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quod Aristotelis ratio none queſtionis <lb/>admittendanon ſit.</head> <head xml:space="preserve">CAP. XV.</head> <p> <s xml:space="preserve">VEra ratio nonæ quęſtionis à ſecunda parte decimi cap. huius tractatus, & non <lb/>aliunde, accerſiri debet.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quod Aristotelis rationes de decima queſtione <lb/>ſint reijciende.</head> <head xml:space="preserve">CAP. XVI.</head> <p> <s xml:space="preserve">ARiſtotelis rationes, vnde fiat, vt facilius moueantur libræ vacuæ, quàm plenè <lb/>ad propoſitam diſputationem non pertinent; </s> <s xml:space="preserve">quia ſemper ineunda eſt ratio <lb/>proportionis virtutis mouentis ſuper mobile; </s> <s xml:space="preserve">quod ipſe non fecit.</s> </p> <pb facs="0172" n="160"/> <fw type="head">IO. BABPT. BENED.</fw> <p> <s xml:space="preserve">Sit exempli gratia libra <seg type="var">.a.i.e.</seg> quæ in vtraque extremitate vnciam vnam ſolum <lb/>ponderis obtineat, & ſit libra <seg type="var">.n.i.u.</seg> æqualis priori, quæ pro ſingula extremitate <choice><ex>vnam</ex><am>vnã</am></choice> <lb/> <ptr xml:id="hd-0172-01a" corresp="hd-0172-01" type="handwrittenAnchor"/> ponderis libram habeat. </s> <s xml:space="preserve">Ariſtoteles admiratur, quòd addendo ipſi <seg type="var">.e.</seg> mediam pon <lb/>deris vnciam, brachium <seg type="var">.i.e.</seg> velocius cadat, quàm <choice><ex>adijciendo</ex><am>adijciẽdo</am></choice> <choice><ex>ipsam</ex><am>ipsã</am></choice> <choice><ex>mediam</ex><am>mediã</am></choice> <choice><ex>vnciam</ex><am>vnciã</am></choice> ipſi <seg type="var">.u.</seg> <lb/>brachij <seg type="var">.i.u</seg>. </s> <s xml:space="preserve">Quod à duabus cauſis proficiſcitur, quarum prior eſt, magna differentia <lb/>proportionis vnius libræ ad medietatem vnius vnciæ, ad proportionem vnius vnciæ <lb/>ad ipſam medietatem, quia ſi pondus adiectum extremo <seg type="var">.u.</seg> dimidiæ eſſet libræ , & <lb/>cum eadem tarditate brachium moueret, optimo iure in admirationem poſſet Ari-<lb/>ſtoteles duci. </s> <s xml:space="preserve">Sed hoc fieri non poſſet, quia ipſum deprimeret cum eadem quaſi ve <lb/>locitate, qua media vncia brachium <seg type="var">.i.e</seg>. </s> <s xml:space="preserve">Dixi autem quaſi, quia nonnihil diſcrimi-<lb/>nis intercederet, quod proficiſcitur à ſecunda ratione. </s> <s xml:space="preserve">Et hæc, reſiſtentia eſt , quæ <lb/>oritur à ſparto, quia quantò maius pondus continet libra, tantò magis præmit ſpar <lb/>tum in loco, in quo ſuſtinetur; </s> <s xml:space="preserve">vnde maior reſiſtentia in circunuolutione <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> ſpar <lb/>ti, in loco, in quo quieſcit, exoritur, quia ipſum eſt corpus materiale. </s> <s xml:space="preserve">Si quis autem <lb/>vellet, vt brachium <seg type="var">.i.u.</seg> eadem agilitate, qua <seg type="var">.i.e.</seg> deſcenderet, oporteret, vt propor-<lb/>tio dimidiæ librę adiectæ ponderi ipſius <seg type="var">.u.</seg> <lb/> <ptr xml:id="fig-0172-01a" corresp="fig-0172-01" type="figureAnchor"/> quod eſt vnius libræ, vim ſuam haberet, <lb/>quæ excederet reſiſtentiam ſui ſparti (me-<lb/>dio brachiorum maiorum ijs qui ſunt <seg type="var">.a.i.<lb/>e.</seg>) ita proportionatam, vt proportionata <lb/>eſt vis dimidiæ vnciæ ipſi e. iunctæ, reſiſten <lb/>tiæ ſui ſparti. </s> <s xml:space="preserve">Huiuſmodi rationes cum ro-<lb/>tis grauioribus <choice><ex>leuioribusque</ex><am>leuioribusq;</am></choice>, & ijs, quę à cor <lb/>poribus quibuſlibet grauibus impelluntur, accommodatæ fuerint, titubantem intel <lb/>lectum confirmabunt.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0172-01" corresp="hd-0172-01a"/> <figure xml:id="fig-0172-01" corresp="fig-0172-01a"> <graphic url="0172-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">De uer a cauſa .12. questionis mechanice.</head> <head xml:space="preserve">CAP. XVII.</head> <p> <s xml:space="preserve">VEra ratio, cur multò longius corpus aliquod graue impellatur funda, quam <lb/>manu, inde oritur, quòd circunuoluendo fundam, maior impræſſio impetus <lb/>motus fit in corpore graui, quàm fieret manu, quod corpus liberatum deinde cum <lb/>fuerit à funda, natura duce, iter <choice><ex>fuum</ex><am>fuũ</am></choice> à puncto, à quo proſilijt, per lineam contiguam <lb/>giro, quem poſtremò faciebat, ſuſcipit. </s> <s xml:space="preserve"><choice><ex>Dubitandumque</ex><am>Dubitandumq́</am></choice> non eſt, quin dicta funda <lb/>maior impetus motus dicto corpori imprimi poſſit, <choice><ex>cum</ex><am>cũ</am></choice> ex multis circumactibus, ma-<lb/>ior ſemper impetus dicto corpori accedat. </s> <s xml:space="preserve">Manus autem eiuſdem corporis motus, <lb/>dum illud ipſum circunuoluitur (pace Ariſtotelis dixerim) centrum non eſt, neque <lb/>funis eſt ſemidiameter. </s> <s xml:space="preserve">Immo manus quam maximè fieri poteſt in orbem cietur; <lb/></s> <s xml:space="preserve">qui quidem motus in orbem, vt circumagatur etiam ipſum corpus, cogit, quod qui-<lb/>dem corpus, naturali quadam inclinatione, exiguo quodam impetu iam incępto, <lb/>vellet recta iter peragere, vt in ſubſcripta figura patet, in qua <seg type="var">.e.</seg> ſignificat manum <seg type="var">.a.</seg> <lb/>corpus <seg type="var">.a.b.</seg> lineam rectam tangentem girum <seg type="var">.a.a.a.a.</seg> quando corpus liberum rema-<lb/>net. </s> <s xml:space="preserve">Verum quidem eſt, impręſſum illum impetum, continuò paulatim decreſcere <lb/>vnde ſtatim inclinatio grauitatis eiuſdem corporis ſubingreditur, quæ ſeſe miſcens <lb/>cum impręſſione facta per vim, non permittit, vt linea <seg type="var">.a.b.</seg> longo tempore recta per <pb facs="0173" n="161"/><fw type="head">DE MECHAN.</fw> maneat, ſed citò fiat curua, cum dictum corpus <seg type="var">.a.</seg> duabus virtutibus moueatur, qua-<lb/>rum vna eſt, violentia impræſſa, & alia natura, contra opinionem Tartaleæ, qui ne-<lb/>gat corpus aliquod motibus violen <lb/> <ptr xml:id="fig-0173-01a" corresp="fig-0173-01" type="figureAnchor"/> to & naturali ſimul & ſemel moueri <lb/>poſſe. </s> <s xml:space="preserve"><choice><ex>Neque</ex><am>Neq;</am></choice> eſt <choice><ex>ſilentio</ex><am>ſilẽtio</am></choice> <choice><ex>prætereundus</ex><am>prætereũdus</am></choice> <lb/>hac in re <choice><ex>quidam</ex><am>ꝗdã</am></choice> notatu <choice><ex>dignus</ex><am>dignꝰ</am></choice> effectus, <lb/>qui eiuſmodi eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> quanto magis <lb/>creſcit impetus in corpore <seg type="var">.a.</seg> cauſa <lb/>tus ab augumento velocitatis giri <lb/>ipſius <seg type="var">.e.</seg> <choice><ex>tantò</ex><am>tãtò</am></choice> magis oportet, vt ſen-<lb/>tiat ſe trahi manus à dicto corpore <lb/>a. mediante fune, quia quantò ma-<lb/>ior impetus motus ipſi <seg type="var">.a.</seg> eſt impręſ <lb/>ſus, tantò magis dictum corpus <seg type="var">.a.</seg> <lb/>ad rectum iter peragendum incli-<lb/>natur, vnde vt recta incedat, tantò <lb/>maiore quoque vi trahit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0173-01" corresp="fig-0173-01a"> <graphic url="0173-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">De decimatertia questione.</head> <head xml:space="preserve">CAP. XVIII.</head> <p> <s xml:space="preserve">DEcimatertia quæſtio ad vectem omnino eſt referenda. </s> <s xml:space="preserve">Imaginari debemus <lb/>axem cylindrici iugi, hypomochlion eſſe. </s> <s xml:space="preserve">Quod reſtat, illud ipſum totum de <lb/>pendet à .4. <choice><ex>quintoque</ex><am>quintoq́;</am></choice> cap. huius tractatus. </s> <s xml:space="preserve">Vna tamen differentia inter hanc machi-<lb/>nam, <choice><ex>vectemque</ex><am>vectemq́;</am></choice> reperitur, quæ eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> iugum aliquam reſiſtentiam pro coniunctione <lb/>calcata in loco, in quo voluitur, magis quàm hypomochlion vecti efficiat.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">De decimaquart a queſtione.</head> <head xml:space="preserve">CAP. XIX.</head> <p> <s xml:space="preserve">RAtiones etiam decimæquartæ quæſtionis dependent ab ijs, quæ ſunt vectis, vt <lb/>exempli gratia ſit lignum <seg type="var">.a.b.c.d.</seg> frang endum in medio, annitendo genibus <lb/>in punctum <seg type="var">.o.</seg> clariſſimè tunc videbimus, <choice><ex>quod</ex><am>ꝙ</am></choice> tenentes marlus longè à medio, in locis <lb/>a. et <seg type="var">.c.</seg> facilius <choice><ex>minorique</ex><am>minoriq́;</am></choice> cum labore illum frangemus, quàm ſi eaſdem vicinas me-<lb/>dio eiuſdem ligni in locis <seg type="var">.e.</seg> et <seg type="var">.i.</seg> poneremus. </s> <s xml:space="preserve">Cuius rei rationes <choice><ex>eædem</ex><am>eædẽ</am></choice> ſunt <choice><ex>cum</ex><am>cũ</am></choice> ijs, quæ <lb/>primis huius tractatus capitibus propoſitæ fuerunt. </s> <s xml:space="preserve">Imaginemur lineas rectas ductas <lb/>à puncto <seg type="var">.o.</seg> ad loca <seg type="var">.a.e.i.</seg> et <seg type="var">.c.</seg> hinc manifeſtè perſpiciemus eorum, quæ iam diximus <lb/>ratione, <choice><ex>quod</ex><am>ꝙ</am></choice> loca <seg type="var">.e.</seg> et <seg type="var">.i.</seg> mediantibus duabus lineis <seg type="var">.e.o.</seg> et <seg type="var">.i.o.</seg> magis annitentur <seg type="var">.o.</seg> cen <lb/>tro, quàm loco <seg type="var">.a.</seg> et <seg type="var">.c.</seg> <choice><ex>duarum</ex><am>duarũ</am></choice> <choice><ex>linearum</ex><am>linearũ</am></choice> <seg type="var">.a.o.</seg> et <seg type="var">.c.o.</seg> beneficio; </s> <s xml:space="preserve">vnde vim <choice><ex>quoque</ex><am>quoq;</am></choice> maiorem <lb/><choice><ex>habebunt</ex><am>habebũt</am></choice> <lb/> <ptr xml:id="fig-0173-02a" corresp="fig-0173-02" type="figureAnchor"/> in <seg type="var">.a.</seg> et <seg type="var">.c.</seg> <lb/>quàm in <lb/>e. et <seg type="var">.i</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0173-02" corresp="fig-0173-02a"> <graphic url="0173-02"/> </figure> </div> </body> </floatingText> <pb facs="0174" n="162"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">De uer a r atione .17. queſtionis.</head> <head xml:space="preserve">CAP. XX.</head> <p> <s xml:space="preserve">DEcimaſeptima quæſtio ab Ariſtotele haud benè percepta fuit, quia is non ac-<lb/>commodat partes vectis ſuis locis. </s> <s xml:space="preserve">Quamobrem imaginemur duos vectes <seg type="var">.<lb/>a.o.n.</seg> et <seg type="var">.o.e.u.</seg> quorum centra, quæ hypomochlia appellantur ſint <seg type="var">.o.</seg> & pondera, <lb/>quæ ſunt attollenda ſint <seg type="var">.a.</seg> et <seg type="var">.e.</seg> inter ſe æqualia, & diſtantię ſint <seg type="var">.a.o.</seg> et <seg type="var">.e.o.</seg> ſibi <choice><ex>inuicem</ex><am>inuicẽ</am></choice> <lb/>æquales, ſed <seg type="var">.o.n.</seg> æqualis ſit ipſi <seg type="var">.o.u</seg>: clarum erit, <choice><ex>quod</ex><am>ꝙ</am></choice> ad eleuandum <seg type="var">.a.</seg> oportebit depri <lb/>mere <seg type="var">.n.</seg> & ad eleuandum <seg type="var">.e.</seg> oportebit attollere <seg type="var">.u</seg>. </s> <s xml:space="preserve">Et quia omnia ſupponuntur æ qua <lb/> <ptr xml:id="hd-0174-01a" corresp="hd-0174-01" type="handwrittenAnchor"/> lia, clarum quoque erit, commu-<lb/>ni ſcientia, tantam virtutem in <lb/> <ptr xml:id="fig-0174-01a" corresp="fig-0174-01" type="figureAnchor"/> n. quanta ſufficiet ad <choice><ex>attollendum</ex><am>attollendũ</am></choice> <lb/>a. in <seg type="var">.u.</seg> <choice><ex>quoque</ex><am>quoq;</am></choice> ſuffecturam ad ele-<lb/>uandum <seg type="var">.e.</seg> quia <choice><ex>cum</ex><am>cũ</am></choice> æqualibus an <lb/>gulis ijs, quibus duæ virtutes <seg type="var">.a.</seg> <lb/>et <seg type="var">.n.</seg> annituntur <seg type="var">.o.</seg> centro, ita <seg type="var">.e.</seg> <lb/>et <seg type="var">.u.</seg> è contrario ſuo centro <seg type="var">.o.</seg> an <lb/>nituntur. </s> <s xml:space="preserve">& omnes rationes pro <lb/>vecte <seg type="var">.a.o.n.</seg> quarto <choice><ex>quintoque</ex><am>quintoq́;</am></choice> huius tracta-<lb/> <ptr xml:id="fig-0174-02a" corresp="fig-0174-02" type="figureAnchor"/> tus capitibus citatæ, vecti <seg type="var">.o.e.u.</seg> vt ſatis ſu <lb/><choice><ex>perque</ex><am>perq́;</am></choice> dixi in dicto capit .5. conuenire poſ-<lb/>ſunt.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0174-01" corresp="hd-0174-01a"/> <figure xml:id="fig-0174-01" corresp="fig-0174-01a"> <graphic url="0174-01"/> </figure> <figure xml:id="fig-0174-02" corresp="fig-0174-02a"> <graphic url="0174-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nunc ſit aliqua pars ligni cindenda ſe-<lb/>cundum venulas ſuas <seg type="var">.d.e.f.g.</seg> & ſit cuneus <lb/><seg type="var">a.b.c.</seg> qui vi mallei <seg type="var">.P.</seg> vſque ad <seg type="var">.t.x.</seg> pene-<lb/>trarit. </s> <s xml:space="preserve">Hinc clarum erit, quòd apertura <lb/><seg type="var">i.m.r.</seg> ligni, poſt quam infigitur cuneus ſe <lb/>cundum venas, longíor erit parte <seg type="var">.x.b.t.</seg> cu <lb/>nei, quæ ingreſſa eſt. </s> <s xml:space="preserve">Oportet nunc ima-<lb/>ginari duos vectes ſimiles ſupradictæ <seg type="var">.u.e.<lb/>o.</seg> in hunc modum, vt puncta <seg type="var">i.r.</seg> lìni ſint <lb/>loco <seg type="var">.u.</seg> extremi <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> vectis, et <seg type="var">.t.x.</seg> loco vir <lb/>tutis applicatæ ipſi <seg type="var">.u.</seg> & reſiſtentia circa <lb/>punctum <seg type="var">.m.</seg> loco ponderis <seg type="var">.e.</seg> vectis <seg type="var">.o.e.u.</seg> <lb/>dicti, & pars <seg type="var">.K.</seg> quaſi immediata poſt <seg type="var">.m.</seg> <lb/>verſus extremitatem <seg type="var">.f.e.</seg> ligni, ſit loco hy-<lb/>pomochlij <seg type="var">.o</seg>. </s> <s xml:space="preserve">Hinc fiet vt quanto longio <lb/>res erunt lineæ <seg type="var">.i.m.K.</seg> et <seg type="var">.r.m.K.</seg> tantò quo <lb/>que facilius virtutes <seg type="var">.t.x.</seg> impellent <seg type="var">.i.r</seg>.</s> </p> <pb facs="0175" n="163"/> <fw type="head">DE MECHAN.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">De uera & intrinſeca cauſa trocble arum.</head> <head xml:space="preserve">CAP. XXI.</head> <p> <s xml:space="preserve">PRo intelligenda vera, & intrinſeca ratione, vnde fiat ut multitudo rotularum in <lb/>trochleis cauſa ſit, ut exigua vis ſurſum moueat, aut attollat <choice><ex>pondera</ex><am>põdera</am></choice> magna. </s> <s xml:space="preserve">Ima <lb/>ginemur duas hîc ſubſcriptas trochlæas explicatas tranſuerſaliter in hunc modum, <lb/>ideſt ſit <choice><ex>paruum</ex><am>paruũ</am></choice> <choice><ex>tignum</ex><am>tignũ</am></choice> <seg type="var">.a.b.</seg> fixum & <choice><ex>parallelum</ex><am>parallelũ</am></choice> orizonti. cui ſint rotulæ appenſe ab infe <lb/>riori parte ad ſuperiorem <choice><ex>huicque</ex><am>huicq́;</am></choice> è regione <choice><ex>oppoſitus</ex><am>oppoſitꝰ</am></choice> ſit aliud <choice><ex>tignum</ex><am>tignũ</am></choice> <seg type="var">.c.d.</seg> quod moueri <lb/>poſſit ab imo ad ſumum, ſuper quod totidem ſint rotulæ aut radij, <choice><ex>cum</ex><am>cũ</am></choice> annexa poſtea <lb/>fuerit funis puncto <seg type="var">.b.</seg> fixo, eam faciendo pertranſire per rotulas tam à parte ſupe-<lb/>riore, quam ab inferiore; </s> <s xml:space="preserve">& appenſum deinde cum erit paruo illi tigno <seg type="var">.c.d.</seg> mobili <lb/>pondus <seg type="var">.E.</seg> ducendo poſtmodum extremum <seg type="var">.f.</seg> funis tranſeuntis per rotulas, idem pla <lb/>nè fiet quod à trochlęis ſimul unitis fieri ſolet. </s> <s xml:space="preserve">Cuius quidem effectus ratio ſub no-<lb/>ſtram cognitionem cadet facilius in huiuſmodi figura. </s> <s xml:space="preserve">Imaginemur ſeparatim ſta-<lb/>teram <seg type="var">.g.h.</seg> cuius <choice><ex>centrum</ex><am>cẽtrum</am></choice> ſit <seg type="var">.K.</seg> ita ſitum, ut brachium <seg type="var">.g.k.</seg> ſit duplum ad brachium <seg type="var">.K.<lb/>h.</seg> ſupponendo igitur in puncto <seg type="var">.g.</seg> pondus, aut virtutem mouentem unius libræ, & in <lb/>h. duarum librarum, <choice><ex>abſque</ex><am>abſq;</am></choice> dubio hæ duæ uirtutes in huiuſmodi diſtantijs à centro <lb/> <ptr xml:id="hd-0175-01a" corresp="hd-0175-01" type="handwrittenAnchor"/> ęquales <choice><ex>inuicem</ex><am>inuicẽ</am></choice> <choice><ex>erunt</ex><am>erũt</am></choice>, ob rationes prioribus capitibus iam allatas, & ſtatera orizontalis <lb/>manebit. </s> <s xml:space="preserve">Vnde clarum erit, <choice><ex>quod</ex><am>ꝙ</am></choice> quæuis etiam exigua virtus adiuncta ipſi <seg type="var">.g.</seg> mouebit <lb/>ſtateram extra orizontalem ſitum. </s> <s xml:space="preserve">Nunc ſi puncto <seg type="var">.i.</seg> ex æquo medio inter <seg type="var">.g.</seg> et <seg type="var">.K.</seg> <lb/>applicata erit virtus ipſius <seg type="var">.h.</seg> non amplius conſiderato brachio <seg type="var">.K.h.</seg> inclinante uirtu-<lb/>te ipſius <seg type="var">.i.</seg> eandem partem verſus, in quam inclinabat, quando erat in <seg type="var">.h.</seg> ſed uirtus ip <lb/>ſius <seg type="var">.g.</seg> inclinet contrario modo, <choice><ex>diuerſoque</ex><am>diuerſoq́;</am></choice> ab eo, quo inclinabat prius; </s> <s xml:space="preserve">clarum <choice><ex>quoque</ex><am>quoq;</am></choice> <lb/>erit, communi conceptu, & ob ea, quæ cap .5. huius tractatus ſunt dicta <seg type="var">.g.h.</seg> ſemper <lb/>in eodem ſitu abſque motu manſuram, <choice><ex>hancque</ex><am>hancq́;</am></choice> ſtateram appellabimus mobilem, & <lb/>primam. </s> <s xml:space="preserve">Imaginemur nunc à puncto <seg type="var">.e.</seg> fixo deſcendere funem <seg type="var">.e.K.</seg> quæ fulciat pun <lb/>ctum <seg type="var">.K.</seg> extremum diametri <seg type="var">.g.K.</seg> quam intelligo pro diametro vnius ex rotulis infe <lb/>rioribus trochleæ; </s> <s xml:space="preserve">& ſit <seg type="var">.n.l.m.</seg> diameter vnius ex rotulis ſuperioribus alterius parui <lb/>tigni defixi à parte inclinationis ipſius <seg type="var">.g.</seg> & parallela diametro <seg type="var">.g.K.</seg> cuius diametri <lb/>centrum fixum ſit <seg type="var">.l.</seg> & ſit coniunctum <seg type="var">.g.</seg> punctum, à fune cum puncto <seg type="var">.m.</seg> quæ <choice><ex>tam</ex><am>tã</am></choice> per-<lb/>pendicularis ſit primo diametro <seg type="var">.g.i.K.</seg> quàm ſecundo <seg type="var">.n.m.</seg> ideſt ita vt anguli <seg type="var">.n.m.g.</seg> <lb/> <ptr xml:id="fig-0175-01a" corresp="fig-0175-01" type="figureAnchor"/> <pb facs="0176" n="164"/><fw type="head">IO. BAPT. BENED.</fw> et <seg type="var">.m.g.k.</seg> ſint recti. </s> <s xml:space="preserve">Imaginemur <choice><ex>quoque</ex><am>quoq;</am></choice> virtutem ipſius <seg type="var">.g.</seg> applicatam eſſe extremo <seg type="var">.<lb/>n.</seg> cum inclinatione tamen contraria, ideſt ad inferiorem partem, quæ quidem virtus <lb/>communi quodam conceptu eandem poſſidebit vim ſuſtentandi immobilem diame <lb/>trum <seg type="var">.g.i.k.</seg> quam habebat, <choice><ex>quando</ex><am>qñ</am></choice> erat in <seg type="var">.g.</seg> cum inclinatione ad ſuperiorem partem, <lb/>& ſic etiam diameter <seg type="var">.n.l.m.</seg> non magis ab una, quàm ab alia parte declinabit, quia <lb/>cum quædam virtus in <seg type="var">.n.</seg> reperiatur æqualis medietati uirtutis ipſius <seg type="var">.i.</seg> quæ uirtus ip <lb/>ſius <seg type="var">.i.</seg> uim habet deprimendi ipſum <seg type="var">.g.</seg> ideſt <seg type="var">.m.</seg> pro dimidia ſui ipſius parte, ſequitur <seg type="var">.<lb/>n.m.</seg> debere immobilem permanere. </s> <s xml:space="preserve">Nunc ſi alia diameter rotulæ mobilis erit de-<lb/>ſumpta, quæ ſit <seg type="var">.p.q.o.</seg> cuius centrum ſit <seg type="var">.q.</seg> in ſitu parallelo ipſi <seg type="var">.n.l.m.</seg> & ſic collocata, <lb/>vt coniungendo <seg type="var">.o.</seg> cum <seg type="var">.n.</seg> anguli <seg type="var">.m.n.o.</seg> et <seg type="var">.n.o.p.</seg> ſint recti: </s> <s xml:space="preserve">ſi imaginati fuerimus <choice><ex>tranſ</ex><am>trãſ</am></choice> <lb/>latum eſſe <choice><ex>pondusipſius</ex><am>pondusipſiꝰ</am></choice> <seg type="var">.n.</seg> in <seg type="var">.o.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <choice><ex>eadem</ex><am>eadẽ</am></choice> inclinatione ad depræſſiorem partem, illud ip <lb/>ſum, ac ſi eſſet in <seg type="var">.n.</seg> communi conceptu, ſine alicuius diametri mutatione præſtabit. <lb/></s> <s xml:space="preserve">Et ſi centrum <seg type="var">.q.</seg> fixum eſſet, & extremo <seg type="var">.p.</seg> appoſitum fuiſſet pondus ipſius <seg type="var">.o.</seg> cum in <lb/>clinatione ad ſuperiorem partem, idem etiam planè pręſtaret, etiam ſi nullum ullius <lb/>diametri ſitum, communi ſcientia, mutaret, cum extremum <seg type="var">.m.</seg> deorſum ſit ductum <lb/>à. g. uirtute dimidiæ partis ipſius <seg type="var">.i.</seg> & ab alia huic ſimili <seg type="var">.m.</seg> quoque deorſum ſit tra-<lb/>ctum ab .o: quod quidem <seg type="var">.o.</seg> deorſum eſt alteratum, ob inclinationem ad ſuperius <lb/>à uirtute poſita in <seg type="var">.p.</seg> ſupponendo centrum <seg type="var">.q.</seg> fixum. </s> <s xml:space="preserve">Sed ſi loco centri fixi, imagina <lb/>bimur in <seg type="var">.q.</seg> pondus aliquod æquale ipſi <seg type="var">.i.</seg> quod duplum erit in uirtute ad eam, quæ <lb/>eſt ipſius <seg type="var">.p.</seg> & ipſius quoque .g: ſequetur <choice><ex>etiam</ex><am>etiã</am></choice> eadem immobilitas horum trium dia-<lb/>metrorum. </s> <s xml:space="preserve">Quia cum ſit huiuſmodi pondus ſeu virtus in <seg type="var">.q.</seg> cum inclinatione con-<lb/>traria virtuti in <seg type="var">.p.</seg> quæ æquipollet dimidiæ parti ipſius <seg type="var">.q.</seg> & ſic ei quæ eſt ipſius <seg type="var">.o.</seg> ſi-<lb/>militer quia <seg type="var">.o.</seg> tractum eſt ſupra ab <seg type="var">.n.</seg> virtute ipſius <seg type="var">.g.</seg> quod <seg type="var">.m.</seg> deorſum trudit; </s> <s xml:space="preserve">idcir <lb/>co quanta erit vis quam habebit virtus in <seg type="var">.q.</seg> ferendi deorſum diametrum <seg type="var">.p.o.</seg> tanta <lb/>quoque virtutes ipſorum <seg type="var">.p.</seg> et <seg type="var">.o.</seg> æquales, & æqualiter diſtantes à <seg type="var">.q.</seg> ipſum ad ſupe-<lb/>riorem partem inclinabunt. </s> <s xml:space="preserve">Quamobrem nec aſcender, nec deſcendet, nec locum <lb/>mutabit. </s> <s xml:space="preserve">Supponamus nunc quartum diametrum rotulæ <seg type="var">.s.t.r.</seg> quæ ſit ſecunda rotu <lb/>larum fixarum, parallela ipſi <seg type="var">.p.o.</seg> & in eo ſitu, quo coniungendo extrema <seg type="var">.r.p.</seg> anguli <lb/><seg type="var">o.p.r.</seg> et <seg type="var">.p.r.s.</seg> ſint recti, & imaginemur virtutem ipſius <seg type="var">.p.</seg> reperiri in <seg type="var">.s.</seg> cum inclinatio <lb/>ne tamen contraria, ideſt deorſum verſus, ex his <choice><ex>idem</ex><am>idẽ</am></choice> quoque planè ſequetur, ideſt <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>nulla <choice><ex>harum</ex><am>harũ</am></choice> quatuor diametrorum mouebitur. </s> <s xml:space="preserve">quia eundem <choice><ex>effectum</ex><am>effectũ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> inclinatione <lb/>deorſum verſus efficeret dicta virtus in <seg type="var">.s.</seg> quem in <seg type="var">.p.</seg> cum inclinatione ſurſum verſus. <lb/></s> <s xml:space="preserve">et iam dictum eſt virtutem ipſius <seg type="var">.g.</seg> dimidium virtutis ipſius <seg type="var">.i.</seg> trahere <seg type="var">.m.</seg> quæ <choice><ex>mediam</ex><am>mediã</am></choice> <lb/> <ptr xml:id="hd-0176-01a" corresp="hd-0176-01" type="handwrittenAnchor"/> te <seg type="var">.n.</seg> attrahit <seg type="var">.o.</seg> eodem robore, et <seg type="var">.s.</seg> eadem vi trahit <seg type="var">.p.</seg> medio ipſius <seg type="var">.r</seg>. </s> <s xml:space="preserve">Hucuſque <choice><ex>ſcien- tificè</ex><am>ſciẽ-tificè</am></choice> nouimus pondus, aut virtutem ipſius <seg type="var">.s.</seg> quæ eſt dimidium <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.i.</seg> ſuſtinere uim <lb/>ipſorum <seg type="var">.i.</seg> et <seg type="var">.q.</seg> nam quater tantum, quanta ipſamet virtus ipſius <seg type="var">.s.</seg> eſſe conſpicitur. <lb/></s> <s xml:space="preserve">Et ſi adiunctę nobis eſſent duæ aliæ diametri cum ijſdem planè conditionibus <choice><ex>ijſdem</ex><am>ijſdẽ</am></choice> <lb/>rationibus vtentes, cognoſceremus quod eadem medietas ipſius <seg type="var">.i.</seg> ſexies tantum <choice><ex>pon</ex><am>põ</am></choice> <lb/>deris, quanta ipſa exiſteret, ſeſtineret. </s> <s xml:space="preserve">Vnde <choice><ex>manifeſtum</ex><am>manifeſtũ</am></choice> euadit, <choice><ex>quod</ex><am>ꝙ</am></choice> eidem medietati <lb/>ipſius <seg type="var">.i.</seg> in <seg type="var">.s.</seg> nonnihil virtutis addendo, dictæ diametri, illicò <choice><ex>mouerentur</ex><am>mouerẽtur</am></choice> ſitu. </s> <s xml:space="preserve">Et quia <lb/>rotulæ in quolibet puncto, aliquam diametrum habent, neceſſariò ſequitur <choice><ex>quod</ex><am>ꝙ</am></choice> infe-<lb/>riores ad ſuperiores accedere debeant. </s> <s xml:space="preserve">Attamen ſi forte extremum immobile ip-<lb/>ſius funis non pendet à puncto <seg type="var">.e.</seg> trochleæ ſuperioris, ſed alligatum fuerit ad <choice><ex>medium</ex><am>mediũ</am></choice> <lb/>inferioris trochleæ ut ad punctum <seg type="var">.i.</seg> ope unius trochleę ſuperioris immobilis vt in fi <lb/>gura <seg type="var">.A.</seg> videre licet, clarè patebit <choice><ex>quod</ex><am>ꝙ</am></choice> à tribus virtutibus æqualibus pondus in <seg type="var">.i.</seg> <choice><ex>poſitum</ex><am>poſitũ</am></choice> <lb/>ſuſtinebitur: </s> <s xml:space="preserve">hoc eſt à <seg type="var">.g.</seg> ab <seg type="var">.i.</seg> & ab <seg type="var">.k.</seg> <choice><ex>quarum</ex><am>quarũ</am></choice> vnaquęque tertia pars erit ipſius <seg type="var">.i.</seg> in con <lb/>c<unclear reason="illegible"/>ontrariam <choice><ex>partem</ex><am>partẽ</am></choice>, hoc eſt tertia pars reſiſtentiæ. </s> <s xml:space="preserve">propterea <choice><ex>quod</ex><am>ꝙ</am></choice> ex æquo inter ſe <choice><ex>diſtant</ex><am>diſtãt</am></choice>. <pb facs="0177" n="165"/><fw type="head">DE MECHAN.</fw> <seg type="var">g.i.</seg> et .K: </s> <s xml:space="preserve">Quà propter augebitur virtus per numeros impares, hoc modo; </s> <s xml:space="preserve">Nam <seg type="var">.g.</seg> <lb/>eſſet tertia pars reſiſtentię, quemadmodum prius media erat. </s> <s xml:space="preserve">Idem infero de <seg type="var">.m.n.<lb/>o.p.r.</seg> et <seg type="var">.s</seg>. </s> <s xml:space="preserve">Sed cum oporteat pondus <seg type="var">.q.</seg> tantum eſſe vt <choice><ex>ſuffieiant</ex><am>ſuffieiãt</am></choice> reſiſtentiæ in <seg type="var">.o.</seg> et <seg type="var">.p.</seg> <lb/>ipſum ſuſtinere, idcirco ipſum pondus <seg type="var">.q.</seg> ſubſeſquialter erit <choice><ex>ponderi</ex><am>põderi</am></choice> in <seg type="var">.i.</seg> poſiti. </s> <s xml:space="preserve">Qua-<lb/>propter <seg type="var">.s.</seg> quinta pars erit ponderum <seg type="var">.i.</seg> et <seg type="var">.q</seg>. </s> <s xml:space="preserve">Deinde ſi adhuc. duo diametri vnus in-<lb/>ferior, alter verò ſuperior additi fuerint cum pondere æquali <seg type="var">.q.</seg> ad medium diame-<lb/>tri inferioris, </s> <s xml:space="preserve">tunc pondus <seg type="var">.s.</seg> erit ſeptima pars trium ponderum <seg type="var">.i.q.</seg> & tertij additi, ex <lb/> <ptr xml:id="fig-0177-01a" corresp="fig-0177-01" type="figureAnchor"/> ſupradictis rationibus. </s> <s xml:space="preserve">Et quia virtus ſuſti <lb/> <ptr xml:id="fig-0177-02a" corresp="fig-0177-02" type="figureAnchor"/> nens totale pondus trochleæ inferiori ap-<lb/>penſum in tot diuiditur partes æquales, <lb/>quot ſunt diametri orbiculorum trochleæ <lb/>inferioris, quando extremum immobile fu <lb/>nis alligatum fuerit trochleę ſuperiori, vt <lb/>puta in puncto <seg type="var">.e.</seg> cum verò alligatum fue-<lb/>rit trochleæ inferiori, virtus primi diame-<lb/>tri <seg type="var">.g.i.K.</seg> trochleæ inferioris ſemper ſeſqui <lb/>altera erit vnicuique aliorum <choice><ex>diametrorum</ex><am>diametrorũ</am></choice> <lb/>ideò virtus reſiſtentię alterius extremi mo <lb/>bilis funis, puta <seg type="var">.s.</seg> ſubmultiplex erit totalis <lb/>ponderis, eo modo quo diximus, cuius vir <lb/>tus, ſeu grauitas diuiditur ſeu diſtrubuitur <lb/>diametris inferioris trochleæ vt dictum eſt.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0175-01" corresp="hd-0175-01a"/> <note xml:id="hd-0176-01" corresp="hd-0176-01a"/> <figure xml:id="fig-0175-01" corresp="fig-0175-01a"> <graphic url="0175-01"/> </figure> <figure xml:id="fig-0177-01" corresp="fig-0177-01a"> <graphic url="0177-01"/> </figure> <figure xml:id="fig-0177-02" corresp="fig-0177-02a"> <graphic url="0177-02"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Depropria cauſa .24. quæſtionis.</head> <head xml:space="preserve">CAP. XXII.</head> <p> <s xml:space="preserve">VEra cauſa effectus, qui vigeſimaquarta quæſtione exprimitur, adhuc à nemine <lb/>(quod ſciam) animaduerſa fuit, licet non ſit admodum ardua vel obſcura. </s> <s xml:space="preserve">Ima <lb/>ginemur ergo duos circulos <seg type="var">.c.f.</seg> et <seg type="var">.b.g.</seg> concentricos, <choice><ex>itaque</ex><am>itaq;</am></choice> ſimul coniunctos, vt ſi ip <lb/>ſorum vnus feratur in orbem, alius quoque circumagatur, eo modo, quo curruum ro <lb/>tæ voluuntur. </s> <s xml:space="preserve">Et imaginemur primò ſuper lineam <seg type="var">.f.i.</seg> reuolui maiorem, & quando <lb/>idem circulus erit in <seg type="var">.l.</seg> dictam lineam <seg type="var">.f.i.</seg> tangere circunferentiam eiuſdem in pun- <pb facs="0178" n="166"/><fw type="head">IO. BAPT. BENED.</fw> cto <seg type="var">.c.</seg> vnde linea <seg type="var">.g.m.</seg> mediante <seg type="var">.K.</seg> continget circunferentiam circuli minoris in pun <lb/>cto .b: et <seg type="var">.K.g.</seg> ex .34. primi Eucli. æqualis erit ipſi <seg type="var">.f.l.</seg> quia ex .17. tertii, anguli <seg type="var">.f.</seg> et <seg type="var">.g.</seg> <lb/>ſunt æquales, vnde ex .28. primi <seg type="var">.f.l.</seg> et <seg type="var">.g.K.</seg> ſunt parallelæ. </s> <s xml:space="preserve">& ſic erunt <seg type="var">.k.l.</seg> cum <seg type="var">.f.g.</seg> ex <lb/>eadem ſupradicta. </s> <s xml:space="preserve">Ratio autem, qua arcus <seg type="var">.g.b.</seg> tranſierit lineam <seg type="var">.g.K.</seg> maiorem ipſa, <lb/>eſt, quia dum mouetur, quodlibet punctum ipſius <seg type="var">.g.b.</seg> virtute reuolutionis ipſius <seg type="var">.f.c.</seg> <lb/>omne punctum eiuſdem arcus <seg type="var">.g.b.</seg> vlterius verſus <seg type="var">.K.</seg> quam ſi moueretur virtute re-<lb/>uolutionis ipſius <seg type="var">.g.b.</seg> ſuper lineam <seg type="var">.g.m.</seg> defertur. </s> <s xml:space="preserve">vt exempli gratia, quando virtute <lb/>reuolutionis maioris circuli, centrum <seg type="var">.a.</seg> reperitur in ſitu lineæ <seg type="var">.l.K.</seg> punctum <seg type="var">.g.</seg> confe <lb/>cerit iter <seg type="var">.g.u.</seg> & punctum <seg type="var">.b.</seg> iter <seg type="var">.b.K.</seg> etiam reliqua omnia puncta inter <seg type="var">.g.b.</seg> magna <lb/>itinera egerint, cum à magno circulo ſint ante delata. </s> <s xml:space="preserve">Imaginemur quoque hos cir <lb/>culos eſſe delatos virtute reuolutionis circuli minoris, & <choice><ex>partem</ex><am>partẽ</am></choice> <seg type="var">.g.t.</seg> rectè <seg type="var">.g.m.</seg> dimen-<lb/>ſam fuiſſe ab arcu <seg type="var">.g.b</seg>. </s> <s xml:space="preserve"><choice><ex>Quando</ex><am>Quãdo</am></choice> ergo <seg type="var">.b.</seg> erit in <seg type="var">.t.</seg> factum erit iter <seg type="var">.b.t.</seg> ab ipſo <seg type="var">.b.</seg> et <seg type="var">.g.</seg> fa-<lb/>ciet iter <seg type="var">.g.n.</seg> quę itinera alijs multò breuiora ſunt, quia breuioribus cruribus reuolu-<lb/>ta ſunt dicta puncta; </s> <s xml:space="preserve">& ſic dico de reliquis omnibus punctis inter <seg type="var">.g.</seg> et <seg type="var">.b.</seg> & in hoc ca <lb/>ſu punctum <seg type="var">.f.</seg> erit in <seg type="var">.q.</seg> & punctum <seg type="var">.c.</seg> erit in <seg type="var">.e</seg>. </s> <s xml:space="preserve">Quamobrem omnia puncta <choice><ex>contingen- tiæ</ex><am>cõtingen-tiæ</am></choice> inter <seg type="var">.f.</seg> et <seg type="var">.c.</seg> non ſolum non erunt delata anteà, ſed potius à primo ſitu retrorſum <lb/>erunt repulſa. </s> <s xml:space="preserve">Vnde non eſt, quòd in tantam admirationem ducamur ſi dum reuol <lb/>uitur circulus maior, arcus <seg type="var">.g.b.</seg> circuli minoris, totam lineam <seg type="var">.g.K.</seg> tranſire videtur, <lb/>& dum reuoluitur minor, apparet arcum <seg type="var">.f.c</seg>: maius iter quam ab <seg type="var">.f.</seg> ad <seg type="var">.e.</seg> non facere, <lb/>cum maiore ſeſe in orbem ferente, quodlibet punctum arcus <seg type="var">.g.b.</seg> ad vnam <choice><ex>eandemque</ex><am>eandẽq;</am></choice> <lb/>partem duos motus obtineat. </s> <s xml:space="preserve">vt exempli gratia punctum <seg type="var">.b.</seg> non ſolum mouetur ver <lb/>ſus <seg type="var">.m.</seg> quòd circa centrum <seg type="var">.a.</seg> feratur, cum ipſum etiam centrum moueatur verſus <seg type="var">.m.</seg> <lb/>ſed quia pręter hoc deferantur quoque à circulo maiori verſus <seg type="var">.m.</seg> vſque ad lineam <seg type="var">.<lb/>k.l</seg>. </s> <s xml:space="preserve">Dum verò minor circulus in girum ducitur, habet quodlibet punctum arcus <seg type="var">.f.c.</seg> <lb/>duos motus contrarios, quorum alter verſus <seg type="var">.i.</seg> virtute reuolutionis circuli minoris, <lb/>& alter ex eo, <choice><ex>quod</ex><am>ꝙ</am></choice> dictus circulus maior circa centrum <seg type="var">.a.</seg> voluatur, vnde omne <choice><ex>punctum</ex><am>punctũ</am></choice> <lb/>contactus circuli maioris cum recta <seg type="var">.f.i.</seg> tetrorſum pellitur verſus <seg type="var">.x</seg>.</s> </p> <figure place="here"> <graphic url="0178-01"/> </figure> <pb facs="0179" n="167"/> <fw type="head">DE MECHAN.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">De uer a cauſa .30. quæstionis.</head> <head xml:space="preserve">CAP. <sic>XXIIII.</sic></head> <p> <s xml:space="preserve">VEra ratio, cur homo dum ſedet ( non tamen Turcarum more ) ſi velit <lb/>ſeſe in pedes erigere, calcaneos retrahit, vt efficiat angulum acutum, cum <choice><ex>fae- moribus</ex><am>fę-moribus</am></choice> coxis à parte inferiori, & ventrem inclinat, ad conſtituendum etiam angu <lb/>lum acutum in ſuperiori parte, ea eſt; </s> <s xml:space="preserve">vt totius corporis pondus, ex ęquo, ideſt ab <lb/>oppoſitis partibus circundet lineam rectam, quæ tranſit per locum, in quo conquie <lb/>ſcunt pedes verſus mundi centrum. </s> <s xml:space="preserve">ideſt, ut edatur ęquilibrium ponderis ipſius cor-<lb/>poris circum lineam illam, quę ſub pedibus inſeruit pro ſparto. </s> <s xml:space="preserve">Vnde aperiendo, <lb/>deinde dictos duos angulos circa dictam <choice><ex>lineam</ex><am>lineã</am></choice>, abſque vlla difficultate erigitur cor-<lb/>pus, & abſque periculo in alterutram partem cadendi.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Deratione .35. & ultimæ quæstionis.</head> <head xml:space="preserve">CAP. <sic>XXV.</sic></head> <p> <s xml:space="preserve">VEra ratio, quare, quę reperiuntur in vorticibus aquarum, ſemper verſus <lb/>medium ipſarum vertiginum vniuntur, inde promanat, quod media <lb/>vertiginum ſemper depreſſiora ſunt. </s> <s xml:space="preserve">vnde quòd dicta corpora ad medium acce-<lb/>dant, nihil aliud eſt, quàm ipſa corpora ſuo pondere grauitateq́ue deſcendere, figu <lb/>ra enim vorticibus eſt quaſi conica, & concaua cum angulo deorſum, & gyro baſis <lb/>ſurſum. </s> <s xml:space="preserve">Atque hæc vera eſt huius effectus cauſa, & non ea quam Ariſtoteles ponit, <lb/>à<unclear reason="illegible"/> quo aliarum omnium quæſtionum, quas ego omiſi rationes ſunt benè propoſitæ.</s> </p> <pb facs="0180" n="168"/> </div> </div> <div type="chapter"> <head xml:space="preserve">DISPVTATIONES <lb/>DE QVIBVSDAM PLACITIS <lb/>ARISTOTELIS.</head> <p rend="italics"> <s xml:space="preserve">TANTA A eſt certè Ariſtotelis amplitudo at que authoritas, vt dif-<lb/>ficillimum ac periculoſum ſit quidpiam ſcribere contra quam <lb/>ipſe docuerit, & mihi præſertim, cui ſemper viſa est viri <lb/>illius ſapientia admirabilis. </s> <s xml:space="preserve">Veruntamen studio veritatis im-<lb/>pulſus, cuius ipſe amore in ſeipſum ſiviueret excitaretur, in me <lb/>dium <choice><ex>quædam</ex><am>quædã</am></choice> proferre non dubitaui, in quibus me inconcußa mathematicæ philoſophiæ <lb/>baſis, cui ſemper inſiſto, ab eo dißentire coegit.</s> </p> <div type="section"> <head rend="italics" xml:space="preserve">Qualiter & ubi Ariſtoteles de uelocitate motuum natura-<lb/>lium localium aliter tractauerit quam nos <lb/>ſentiamus.</head> <head xml:space="preserve">CAP.I.</head> <p> <s xml:space="preserve">VOlens Ariſtoreles probare vacuum non eſſe in rerum natura .8. cap. lib. 4. phy<lb/>ſicorum ait, idem corpus per varia <choice><ex>diuerſaque</ex><am>diuerſaq́;</am></choice> media, vt per <choice><ex>aerem</ex><am>aerẽ</am></choice>, & per <choice><ex>aquam</ex><am>aquã</am></choice> <lb/>ſi moueretur, proportionem velocitatis eiuſdem corporis per aerem, ei, quæ per <lb/>aquam fit, vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> futuram cum ea, quæ eſt ſubtilitatis aereę ad ſubtilitatem <lb/>aquæ. </s> <s xml:space="preserve">In poſtrema autem parte eiuſdem capitis ſic ſcribit: </s> <s xml:space="preserve">Nam cum ea quę ma-<lb/>iorem vel ponderis velleuitatis pręſtantiam habent, ſi ſimili ſigura ſint, <choice><ex>ſpacium</ex><am>ſpaciũ</am></choice> par, <lb/>& æquale, maiore celeritate conficere cernamus, ea quam magnitudines inter ſe ha <lb/>bent, proportione: </s> <s xml:space="preserve">profectò idem etiam perinane fieret. </s> <s xml:space="preserve">Aliam quoque rationem <lb/>proponit phyloſophus .2. cap. ſexti phyſicorum ſcribens eademmet proportione, <lb/>qua tempus diuiditur, magnitudinem etiam diuidi. </s> <s xml:space="preserve">Sexto autem cap. primi de cœ-<lb/>lo ſcribit, tempora eandem proportionem habere, quam habentè conuerſo ponde-<lb/>ra; </s> <s xml:space="preserve">vt ſi media pars vnius ponderis, vnius horæ ſpatio moueretur, vniuerſum pondus <lb/>in media hora moueretur. </s> <s xml:space="preserve">Secundo cap. lib. 3. de cœlo duobus in locis apertè com<lb/>monſtrat velocitatem corporis minoris, maiori corpori comparatam, in eadem exi-<lb/>ſtere proportione, in qua dicta corpora adinuicem relata exiſtunt. </s> <s xml:space="preserve">Quinto cap. eiuſ<lb/>dem lib. idem affirmat, exemplo ab igne deſumpto. </s> <s xml:space="preserve">Ex alijs etiam plurimis locis <lb/>cognoſci poteſt, ſenſiſſe Ariſtotelem duo corpora eadem ſpecie, & figura prædita <lb/>eandem planè proportionem in ſuorum motuum velocitatibus, quam in ſuis ma-<lb/>gnitudinibus habent, retinere. </s> <s xml:space="preserve">Alij quoque permulti eandem opinionem retinue <lb/>runt, & <choice><ex>omnium</ex><am>omniũ</am></choice> poſtremus Nicolaus Tartalea, ſecunda propoſitione vigeſiminoni <lb/>quæſiti octaui libri, vbi profitetur ſe demonſtratiuè probare hanc propoſitio-<lb/>nem veram exiſtere; </s> <s xml:space="preserve"><choice><ex>neque</ex><am>neq;</am></choice> videt quàm magna reſiſtentiarum ſit differentia, quætam <lb/>ex diuerſitate figurarum, quàm ex magnitudinum varietate exoriri poteſt; </s> <s xml:space="preserve">quas qui <lb/>dem diuerſitates ne conſiderat quidem.</s> </p> <pb facs="0181" n="169"/> <fw type="head">DISPVTATIONES.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quædam ſupponenda ut conſtet cur circa uelocit atem motuum <lb/>natur alium localium ab Ariſtotelis placitis <lb/>recedamus.</head> <head xml:space="preserve">CAP. II.</head> <p> <s xml:space="preserve">CVM ſuſceperimus prouinciam probandi quod Ariſtoteles circa motus <lb/>locales naturales deceptus fuerit, ſunt quædam primo veriſſima & obie-<lb/>cta intellectus perſe cognita pręſupponenda, ac primum quælibet duo corpora, <lb/>grauia, aut leuia, area æquali, <choice><ex>ſimilique</ex><am>ſimiliq́</am></choice> figura, ſed ex materia diuerſa conſtantia, <choice><ex>eodem q́ue</ex><am>eodẽq́ue</am></choice> modo ſitum habentia, eandem proportionem velocitatis inter ſuos motus loca <lb/>les naturales, ut inter ſuamet pondera aut leuitates in vno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> medio, ſeruatu-<lb/>ra. </s> <s xml:space="preserve">Quod quidem natura ſua notiſſimum eſt ſi conſiderabimus non aliunde maio-<lb/>rem tarditatem, aut velocitatem gigni, quàm à .4. cauſis (dummodo medium vnifor <lb/>mè ſit & quietum) ideſt à maiori aut minori pondere aut leuitate; </s> <s xml:space="preserve">à diuerſa figura; <lb/></s> <s xml:space="preserve">à ſitu ciuſdem figuræ diuerſo, reſpectu lineę directionis, quæ recta inter mundi cen-<lb/>trum, & circunferentiam extenditur; </s> <s xml:space="preserve">& ab inæquali magnitudine. </s> <s xml:space="preserve">Vnde patebit, <lb/>quòd figuram non variando, nec in qualitate nec in quantitate, neque eiuſdem figu-<lb/>ræ ſitum, motum fore proportionatum virtuti mouenti, quæ erit pondus aut leuitas. <lb/></s> <s xml:space="preserve">Quod autem de qualitate, de quantitate & ſitu eiuſdem figuræ dico, reſpectu reſi-<lb/>ftentiæ ipſius medii dico: </s> <s xml:space="preserve">Quia diffimilitudo aut inęqualitas figurarum, aut ſitus di-<lb/>uerſus non <choice><ex>parum</ex><am>parũ</am></choice> alterat dictorum corporum motus, cum figura parua facilius diui-<lb/>dat continuitatem medij, quam magna; </s> <s xml:space="preserve">vt etiam cęlerius idem facit acuta, quàm ob <lb/>tufa; </s> <s xml:space="preserve">& illa quæ cum angulo, qui antecedat mouebitur velocius quàm illa, quæ ſecus. <lb/></s> <s xml:space="preserve">Quotieſcunque igitur duo corpora vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> reſiſtentiam ipſorum ſuperfi-<lb/>ciebus, aut habebunt aut recipient, eorum motus inter ſeipſos eodem planè modo <lb/>proportionati conſurgent, quo erunt ipſorum virtutes mouentes: </s> <s xml:space="preserve">& è conuerſo, quo <lb/>tieſcunque duo corpora vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> grauitatem, aut leuitatem, & diuerſas reſi <lb/>ſtentias habebunt, eorum motus inter ſeipſos eandem <choice><ex>proportionem</ex><am>proportionẽ</am></choice> ſortientur, <choice><ex>quam</ex><am>quã</am></choice> <lb/>habebunteorum reſiſtentiæ conuerſo modo; </s> <s xml:space="preserve">quæ quidem reſiſtentiæ inter ſeipſas, <lb/>eandem proportionem quàm ipſarum ſuperficies habebunt, aut in qualitate ſola fi <lb/>guræ, aut in quantitate ſola, aut in ſitu, aut in aliquibus ex dictis rebus, eo tamen mo <lb/>do, quiſuperius poſitus fuit, vt ſcilicet corpus illud quod alteri comparatum, æqua-<lb/>lis erit ponderis, aut leuitatis, ſed minoris reſiſtentiæ, exiſtet velocius altero, in <choice><ex>eadem</ex><am>eadẽ</am></choice> <lb/>proportione, cuius ſuperficies reſiſtentiam ſuſcipit minorem ea quæ alterius eſt cor-<lb/>poris, ratione facilioris diuiſionis continuitatis aeris, aut aquæ; </s> <s xml:space="preserve">Vt exempli gratia, <lb/>ſi proportio ſuperficiei corporis maioris ſuperficiei minoris ſeſquitertia eſſet, pro-<lb/>portio velocitatis dicti corporis maioris, velocitati corporis minoris, eſſet ſubſeſqui <lb/>tertia; </s> <s xml:space="preserve">vnde velocitas minoris corporis, maior eſſet velocitate corporis maioris, <choice><ex>quem</ex><am>quẽ</am></choice> <lb/>admodum quaternarius numerus ternario maior exiſtit.</s> </p> <p> <s xml:space="preserve">Aliud quoque ſupponendum eſt, velocitatem ſcilicet motus naturalis alicuius <lb/>corporis grauis, in diuerſis medijs, propor-<lb/>tionatam eſſe ponderi eiuſdem corporis in <lb/>ijſdem medijs; </s> <s xml:space="preserve">Vt exempli gratia, ſi pondus <lb/> <ptr xml:id="fig-0181-01a" corresp="fig-0181-01" type="figureAnchor"/> totale alicuius corporis grauis ſignificatum <lb/>crit ab <seg type="var">.a.i.</seg> quo corpore poſito in aliquo me- <pb facs="0182" n="170"/><fw type="head">IO. BABPT. BENED.</fw> dio minus denſo, quàm ipſum ſit, (quia in medio ſe denſiore ſi poner etur, non graue <lb/>e<gap reason="illegible" extent="2" unit="chars"/>et, ſed leue, quemadmodum Archimedes oſtendit) illud medium ſubtrahat par-<lb/>tem <seg type="var">.e.i.</seg> vnde pars <seg type="var">.a.e.</seg> eiuſdem ponderis libera manear; </s> <s xml:space="preserve">& poſito deinde eodem cor <lb/>pore in aliquo alio medio denſiore, minus tamen denſo quam ipſum ſit corpus, <lb/>hoc medium ſubtrahat partem <seg type="var">.u.i.</seg> dicti ponderis, vnde pars <seg type="var">.a.u.</seg> ei uſdem <lb/>ponderis remanebit. </s> <s xml:space="preserve">Dico proportionem velocitatis eiuſdem corporis per <choice><ex>medium</ex><am>mediũ</am></choice> <lb/>minus denſum, ad velocitatem eiuſdem per medium magis denſum futuram vt <seg type="var">.a.e.</seg> <lb/>ad <seg type="var">.a.u.</seg> vt eſt etiam rationi conſonum, magis quàm ſi dicamus huiuſmodi velocitates <lb/>eſſe, vt <seg type="var">.u.i.</seg> ad <seg type="var">.e.i.</seg> cum velocitates à virtutibus mouentibus ſolum (cum figura vna, <lb/><choice><ex>eademque</ex><am>eademq́;</am></choice> in qualitate, quantitate <choice><ex>ſituque</ex><am>ſituq́</am></choice> erit) proportionentur. </s> <s xml:space="preserve">Quæne<unclear reason="illegible"/>nc diximus, <lb/>planè ſimilia ſuntijs, quæ ſupra ſcripſimus, quia idem eſt dicere, proportionem velo <lb/>citatum, duorum corporum hetereogeneorum, ſed ſimilium figura, & magnitudine <lb/>æ qualium, in vno ſolo medio, æqualem eſſe <lb/>proportioni ponderum ipſorum, vt ſi dicam?</s> <s xml:space="preserve"><unclear reason="illegible"/> <lb/>proportionem velocitatum vnius ſolum cor-<lb/> <ptr xml:id="fig-0182-01a" corresp="fig-0182-01" type="figureAnchor"/> poris per diuerſa media eandem eſſe cum ea. <lb/>quæ eſt <choice><ex>ponderum</ex><am>ponderũ</am></choice> dicti corporis in iſidem medijs.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0181-01" corresp="fig-0181-01a"> <graphic url="0181-01"/> </figure> <figure xml:id="fig-0182-01" corresp="fig-0182-01a"> <graphic url="0182-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Poſſe uelocitatem alicuius corporis proportionem contrariam <lb/>in diuerſis medijs habere cum denſitate eorum.</head> <head xml:space="preserve">CAP. III.</head> <p> <s xml:space="preserve">POſſibile eſt in rerum natura corpus aliquod huiuſmodi denſitate præditum re-<lb/>periri, vt velocitas eius motus naturalis per aerem, velocitati per aquamita pro <lb/>portionata exiſtat, vt eſt <choice><ex>denſitas</ex><am>dẽſitas</am></choice> aquæ denſitati aeris. </s> <s xml:space="preserve">Denſitas aquæ notetur (exem-<lb/>pli gratia) per <seg type="var">.u.i.</seg> & ea, quæaeris eſt per <seg type="var">.e.i.</seg> & pondus alicuius corporis in aere per <lb/><seg type="var">e.a.</seg> & pondus eiuſdem corporis in aqua per <seg type="var">.u.a.</seg> ita tamen, quod eadem proportio <lb/>ſit <seg type="var">.e.a.</seg> ad <seg type="var">.u.a.</seg> vt <seg type="var">.u.i.</seg> ad <seg type="var">.e.i.</seg> vnde per vltimam ſuppoſitionem <choice><ex>præcedentis</ex><am>præcedẽtis</am></choice> capitis, pro <lb/>portio velocitatis prædicti corporis per aerem, <lb/>proportioni eiuſdem corporis per aquam erit, vt <lb/> <ptr xml:id="fig-0182-02a" corresp="fig-0182-02" type="figureAnchor"/> <seg type="var">e.a.</seg> ad <seg type="var">.u.a.</seg> ergo per .11. quinti, vt <seg type="var">.u.i.</seg> ad <seg type="var">.e.i</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0182-02" corresp="fig-0182-02a"> <graphic url="0182-02"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Oſcitanter ab Ariſtotele nonnibil prolatum cap 8. <lb/>lib. 4 Phyſicorum.</head> <head xml:space="preserve">CAP. IIII.</head> <p> <s xml:space="preserve">EX ſupradictis patet in vniuerſum non eſſe verum quod Ariſto .8. cap .4. lib. phy<lb/>ſicorum ſcribit, velocitates ſcilicet motuum alicuius corporis per diuerſa me-<lb/>dia, proportionatas eſſe denſitatibus eorundem mediorum. </s> <s xml:space="preserve">Quocirca, ſit propor-<lb/>tio <seg type="var">.u.i.</seg> ad <seg type="var">.e.i.</seg> vt <choice><ex>denſitatis</ex><am>dẽſitatis</am></choice> aquę ad <choice><ex>aeream</ex><am>aereã</am></choice> <choice><ex>denſitatem</ex><am>dẽſitatem</am></choice> .et <seg type="var">.e.a.</seg> ad <seg type="var">.u.a.</seg> vt ponderis alicuius <lb/>corporis in aere ad pondus eiuſdem in aqua, ita tamen vt maior aut minor propor-<lb/>tio ſit <seg type="var">.e.a.</seg> ad <seg type="var">.u.a.</seg> quam <seg type="var">.u.i.</seg> ad <seg type="var">.e.i.</seg> vnde exiſtente proportione velocitatis per <choice><ex>aerem</ex><am>aerẽ</am></choice> <pb facs="0183" n="171"/><fw type="head">DISPVTATIONES.</fw> ad velocitatem per aquam vt <seg type="var">.e.a.</seg> ad <seg type="var">.a.u.</seg> non erit ergo vt <seg type="var">.u.i.</seg> ad <seg type="var">.e.i</seg>. </s> <s xml:space="preserve">Ob hanc igitur <lb/>cauſam nimis diſſentaneum eſt rationi, opi-<lb/>nari proportionem velocitatis omnium cor <lb/>porum grauium per aerem vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> <lb/> <ptr xml:id="fig-0183-01a" corresp="fig-0183-01" type="figureAnchor"/> eſſe cum velocitate eorundem per aquam, <lb/>quemadmodum Ariſtoteles ſenſit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0183-01" corresp="fig-0183-01a"> <graphic url="0183-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Exempla dictorum.</head> <head xml:space="preserve">CAP.V.</head> <p> <s xml:space="preserve">POnamus, exempli gratia, aquam eſſe in denſitate dupla ad aerem, & aliquod <lb/>graue corpus in aqua duplum ad denſitatem ipſius aquæ, vnde dictum corpus in <lb/>denſitate ad aerem quadruplum erit; </s> <s xml:space="preserve">quam ob cauſam, mediam ſui ponderis tota-<lb/>lis partem in aqua, & in aere quartam partem, ex .7. lib. de inſidentibus aquæ ab Ar-<lb/>chimede conſcripto, amitteret. </s> <s xml:space="preserve">Moueretur igitur in aqua virtute illius mediæ partis <lb/><choice><ex>ponderis</ex><am>põderis</am></choice> ſui, in aere <choice><ex>autem</ex><am>aũt</am></choice> uirtute <choice><ex>trium</ex><am>triũ</am></choice> <choice><ex>quartarum</ex><am>quartarũ</am></choice>; </s> <s xml:space="preserve">vnde proportio facultatis <choice><ex>mouentis</ex><am>mouẽtis</am></choice> dicti <lb/>corporis in aere ad facultatem mouentem eiuſde m in aqua ſeſquialtera erit. </s> <s xml:space="preserve"><choice><ex>hocque</ex><am>hocq́;</am></choice> <lb/>corpus appelletur <seg type="var">.A</seg>. </s> <s xml:space="preserve">Sit aliud quoque corpus, quod <seg type="var">.B.</seg> nominetur, ſimile figura, & <lb/>magnitudine corporea corpori <seg type="var">.A.</seg> ſed <choice><ex>denſitate</ex><am>dẽſitate</am></choice>, in proportione ſeſquialtera ad <choice><ex>aquam</ex><am>aquã</am></choice>, <lb/>& denſius erit aere in proportione tripla. </s> <s xml:space="preserve">quamobrem corpus <seg type="var">.A.</seg> grauius erit cor-<lb/>pore <seg type="var">.B.</seg> in aere in proportione ſeſquialtera, vnde etiam velocius erit ipſo <seg type="var">.B.</seg> in aere <lb/>in eadem proportione, ſed corpus <seg type="var">.B.</seg> in aere, duplo maius pondus habebit, <choice><ex>quam</ex><am>quã</am></choice> in <lb/>aqua, cum in aere remaneant ei duæ ponderis tertiæ partes, & in aqua vna tantum, <lb/>ita vt Ariſtoteli concedam corpus <seg type="var">.B.</seg> in aere, quam in aqua velocius futurum in ea-<lb/>dem proportione, in qua, aqua eſt <choice><ex>denſior</ex><am>dẽſior</am></choice> aere, ex Euclidis vndecima propoſitione <lb/>lib. quinti. </s> <s xml:space="preserve">Sed præter hæc omnia, ſi corpus <seg type="var">.A.</seg> eſſet etiam velocius in aere, <choice><ex>quam</ex><am>quã</am></choice> in <lb/>aqua, in eadem proportione, ſequeretur ex .16. dicti lib. quinti proportionem velo-<lb/>citatis <seg type="var">.A.</seg> in aqua ad <choice><ex>velocitatem</ex><am>velocitatẽ</am></choice> ipſius <seg type="var">.B.</seg> in aqua etiam ſeſquialteram eſſe. </s> <s xml:space="preserve">Sed cum <lb/>corpus <seg type="var">.A.</seg> in denſitate ad aquam <choice><ex>duplum</ex><am>duplũ</am></choice> ſit, & corpus <seg type="var">.B.</seg> <choice><ex>ſeſquialterum</ex><am>ſeſquialterũ</am></choice> ad ipſam <choice><ex>aquam</ex><am>aquã</am></choice>, <lb/>ſequetur <choice><ex>proportionem</ex><am>proportionẽ</am></choice> ponderis ipſius <seg type="var">.A.</seg> ad <choice><ex>pondus</ex><am>põdus</am></choice> ipſius <seg type="var">.B.</seg> in aqua eſſe in propor-<lb/>tione dupla; </s> <s xml:space="preserve">Vnde ex primo ſuppoſito capitis ſecundi proportio velocitatis <seg type="var">.A.</seg> ad <lb/>velocitatem <seg type="var">.B.</seg> in aqua dupla erit, non ſeſquialtera. </s> <s xml:space="preserve">Si ergo proportio velocitatis <seg type="var">.<lb/>A.</seg> ad eam quæ eſt <seg type="var">.B.</seg> in aqua dupla eſt, & ea, quæeſt <seg type="var">.B.</seg> in aere, ad eam, quæ eſt ipſius <lb/>per aquam eſt etiam dupla (vnde ea quę eſt <seg type="var">.A.</seg> per aquam ęqualis erit ei, quæ eſt <seg type="var">.B.</seg> <lb/>peraerem, ex .9. lib. quinti) & cum ea, quæ eſt <seg type="var">.A.</seg> ſit ei, quæ eſt <seg type="var">.B.</seg> per aerem ſeſqui-<lb/>altera, erit ergo ea, quæ eſt <seg type="var">.A.</seg> per aerem, ei, quæ eſt ſuimet ipſius per aquam ſeſqui <lb/>altera, non autem dupla, ex .7. eiuſdem libr. quinti. </s> <s xml:space="preserve">Hiſce rationibus accedimus ad <lb/>confirmandam veritatem vltimi ſuppoſiti cap .2. proportionem videlicet velocita <lb/>tis <choice><ex>motus</ex><am>motꝰ</am></choice> naturalis in diuerſis medijs <choice><ex>alicuius</ex><am>alicuiꝰ</am></choice> corporis <choice><ex>ponderoſi</ex><am>põderoſi</am></choice> in ipſis medijs eſſe ean <lb/>dem cum ea, quæeſt inter pondera <lb/> <ptr xml:id="fig-0183-02a" corresp="fig-0183-02" type="figureAnchor"/> dicti corporis in dictis medijs. de ijs <lb/>tamen medijs intelligendo, quæ un-<lb/>ctuoſa, aut pinguia non ſunt, ut ſunt <lb/>oleum, lac, aut huiuſmodialia, quæà <lb/>qualibet minima qualitate frigoris <lb/>aut caloris alterantur, & impermea-<lb/>biles fiunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0183-02" corresp="fig-0183-02a"> <graphic url="0183-02"/> </figure> </div> </body> </floatingText> <pb facs="0184" n="172"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quod proportiones ponderum eiuſdem corporis in diuerſis medijs pro <lb/>portiones eorum mediorum denſit atum non ſeruant. Unde ne-<lb/>ceßariò inæquales proportiones uelocitatum <lb/>producuntur.</head> <head xml:space="preserve">CAP. VI.</head> <p> <s xml:space="preserve">OMne corpus graue variat proportionem ponderis per diuerſa media, vnde <lb/>proportiones velocitatum inæquales exiſtunt. </s> <s xml:space="preserve">Vt exempli gratia, ſi fue-<lb/>rit corpus <seg type="var">.A.</seg> cuius pondus totale ſit <seg type="var">.o.a.</seg> quod in aqua diminutum ſit ratione partis <seg type="var">.<lb/>e.o.</seg> ita vt ei ſolum relinquatur pondus <seg type="var">.a.e.</seg> & in aeie adempta ſit ei pars <seg type="var">.i.o.</seg> vnde ſo <lb/>lum remaneat pondus <seg type="var">.a.i</seg>. </s> <s xml:space="preserve">Supponamus aliud <choice><ex>quoque</ex><am>quoq;</am></choice> medium in eadem proportio-<lb/>ne minus denſum, quàm aer, quemadmodum aer minus denſus eſt, aqua, in quo, cor <lb/>pus <seg type="var">.A.</seg> ammittat partem <seg type="var">.t.o.</seg> ponderis ſui, vnde ex .7. lib. de inſidentibus aquæ Ar-<lb/>chimedis, eadem proportio erit <seg type="var">.e.o.</seg> ad <seg type="var">.i.o.</seg> quæ eſt <seg type="var">.i.o.</seg> ad <seg type="var">.t.o</seg>. </s> <s xml:space="preserve">Supponamus <choice><ex>quoque</ex><am>quoq;</am></choice> <lb/>eandem proportionem eſſe <seg type="var">.a.i.</seg> ad <seg type="var">.a.e.</seg> eſt <seg type="var">.e.o.</seg> ad <seg type="var">.i.o.</seg> </s> <s xml:space="preserve">tunc dico non futuram ean-<lb/>dem proportionem <seg type="var">.t.a.</seg> ad <seg type="var">.a.i.</seg> quæ eſt <seg type="var">.i.o.</seg> ad <seg type="var">.t.o</seg>. </s> <s xml:space="preserve">Cum ſit ergo proportio <seg type="var">.a.i.</seg> <lb/>ad <seg type="var">.a.e.</seg> ut <seg type="var">.e.o.</seg> ad <seg type="var">.i.o.</seg> erit diſiunctim <seg type="var">.e.i.</seg> ad <seg type="var">.e.a.</seg> vt <seg type="var">.e.i.</seg> ad <seg type="var">.i.o</seg>. </s> <s xml:space="preserve">Quare ex .9. libr. quin<lb/>ti erit <seg type="var">.a.e.</seg> æqualis <seg type="var">.i.o.</seg> ſed cum ita ſehabeat <seg type="var">.e.o.</seg> ad <seg type="var">.i.o.</seg> vt <seg type="var">.i.o.</seg> ad <seg type="var">.t.o.</seg> ita quoque <lb/>ſe habebit, ex vndecima quinti <seg type="var">.a.i.</seg> ad <seg type="var">.e.a.</seg> ut <seg type="var">.i.o.</seg> ad <seg type="var">.t.o</seg>. </s> <s xml:space="preserve">Cum autem (vt vidimus). <seg type="var">a.e.</seg> <lb/>ęqualis ſit ipſi <seg type="var">.i.o.</seg> non poterit eſſe proportio <seg type="var">.t.a.</seg> ad <seg type="var">.i.a.</seg> vt eſt <seg type="var">.o.i.</seg> ad <seg type="var">.t.o.</seg> quia ſi <lb/>hoc eſſet, eſſet etiam diſiunctim proportio <seg type="var">.i.t.</seg> ad <seg type="var">.i.a.</seg> vt eſt <seg type="var">.i.t.</seg> ad <seg type="var">.t.o.</seg> & ex ſupradicta <lb/>9. lib. quinti <seg type="var">.a.i.</seg> æqualis eſſet <seg type="var">.t.o</seg>. </s> <s xml:space="preserve">Maximum autem inconueniens eſſet <seg type="var">.t.o.</seg> minorem <lb/><seg type="var">o.i.</seg> ideſt minorem <seg type="var">.a.e.</seg> æqualem eſſe <seg type="var">.a.i.</seg> quæ maior eſt <seg type="var">.a.e</seg>. </s> <s xml:space="preserve">Oſtenſiuè tamen idem <lb/>hoc modo probari poteſt, vt exiſtente <seg type="var">.i.o.</seg> ęquali ipſi <seg type="var">.a.e.</seg> per conſequens <choice><ex>quoque</ex><am>quoq;</am></choice> erit <lb/>minor ipſa <seg type="var">.a.i.</seg> cum <seg type="var">.a.e.</seg> pars ſit ipſius <seg type="var">a.i</seg>. </s> <s xml:space="preserve"><choice><ex>Pereandem</ex><am>Pereãdem</am></choice> tamen rationem <seg type="var">.o.t.</seg> minoreſt <seg type="var">.<lb/>o.i</seg>. </s> <s xml:space="preserve">Tanto magis igitur minor erit <seg type="var">.t.o.</seg> ipſa <seg type="var">.i.a</seg>. </s> <s xml:space="preserve">Vnde ex .8. libri quinti maiorem pro <lb/>portionem habebit <seg type="var">.i.t.</seg> <lb/>ad <seg type="var">.t.o.</seg> quam ad <seg type="var">.i.a.</seg> & <lb/>ex .28. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> lib <seg type="var">.i.o.</seg> ad <lb/><seg type="var">t.o.</seg> <choice><ex>maiorem</ex><am>maiorẽ</am></choice> proportio-<lb/> <ptr xml:id="fig-0184-01a" corresp="fig-0184-01" type="figureAnchor"/> <choice><ex>nem</ex><am>nẽ</am></choice> habebit, quàm.t.a. <lb/>ad <seg type="var">.i.a.</seg> ex .12. igitur di-<lb/>cti quinti maiorem pro <lb/>portionem habebit <seg type="var">.i.a.</seg> ad <seg type="var">.e.a.</seg> quàm.t.a. ad <seg type="var">.i.a.</seg> ita ergo ſe habebunt ipſorum velo-<lb/>citates.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0184-01" corresp="fig-0184-01a"> <graphic url="0184-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Corpora grauia aut leuia eiuſdem figur æ et materiæ ſed inæqualis <lb/>magnitudinis, in ſuis motibus natur alibus uelocit atis, in eo <lb/>dem medio, proportionem longè diuerſam ſeruatura <lb/>eße quam Aristoteliuiſum fuerit.</head> <head xml:space="preserve">CAP. VII.</head> <p> <s xml:space="preserve">ESt mihi nunc probandum <choice><ex>quod</ex><am>ꝙ</am></choice> in uno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> mcdio duo corpora inæqualia, ſed <lb/>ſimili figura & materia, mouebuntur naturali motu, diuerſa tamen ratione ab <pb facs="0185" n="173"/><fw type="head">DISPVTATIONES.</fw> ea, quàm Ariſtoteles præſcripſit.</s> </p> <p> <s xml:space="preserve">Sintigitur corpora <seg type="var">.a.</seg> et <seg type="var">.o.</seg> inæqualia, <choice><ex>eadentamen</ex><am>eadẽtamen</am></choice> figura & materia prædita, quo-<lb/>rum <seg type="var">.a.</seg> maius ſit, & per conſequens in eadem quoque proportione grauius ipſo <seg type="var">.o.</seg> in <lb/>qua eſt maius, communi omnium ſententia.</s> </p> <p> <s xml:space="preserve">Scribit ergo Ariſtoteles proportionem velocitatis corporis <seg type="var">.a.</seg> ad eam, quæ eſt <lb/>corporis <seg type="var">.o.</seg> (naturaliterſe vnoquoque mouente) eandem futuram, quæ eſt magnitu <lb/>dinis, aut grauitatis corporis <seg type="var">.a.</seg> ad magnitudinem, aut grauitatem corporis <seg type="var">.o</seg>. </s> <s xml:space="preserve">Ima-<lb/>ginemur igitur corpus u. eadem magnitudine & figura, qua corpus <seg type="var">.a.</seg> præditum eſt, <lb/>ſed eandem grauitatem obtinere, quæ communicata eſt corpori <seg type="var">.o.</seg> quod ex quauis <lb/>materia conſter. </s> <s xml:space="preserve">Hinc ex primo ſuppoſito ſecundi capitis certi erimus proportio-<lb/>nem velocitatis corporis <seg type="var">.a.</seg> ſi comparetur cum velocitate corporis <seg type="var">.u.</seg> futuram, vt <choice><ex>eam</ex><am>eã</am></choice>, <lb/>quæ eſt ponderis corporis <seg type="var">.a.</seg> ad pondus ipſius corporis <seg type="var">.u</seg>. </s> <s xml:space="preserve">Ex .9. igitur lib. quinti Eu-<lb/>cli. cogitur fateri Ariſtoteles velocitatem corporis <seg type="var">.o.</seg> eſſe vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> in ſpe-<lb/>cie, quæ eſt corporis <seg type="var">.u</seg>. </s> <s xml:space="preserve">Quod primo ſuppoſito cap. ſecundi huius lib. planè repugna<lb/>ret. </s> <s xml:space="preserve">Igitur hæc Ariſtotelis opinio falſa eſt. </s> <s xml:space="preserve">Idem quoque probaretur mediante cor <lb/>pore <seg type="var">.i.</seg> æquali magnitudine, <choice><ex>ſimilique</ex><am>ſimiliq́;</am></choice> figura cum corpore <seg type="var">.o.</seg> prædito, ſed, quòd ad <lb/>quantitatem attinet, æquali corpori <seg type="var">.a.</seg> vnde ex primo ſuppoſito cap. ſecundi huius li <lb/>bri in eadem pro <lb/>portione <choice><ex>velocius</ex><am>velociꝰ</am></choice> <lb/>eſſet corpore <seg type="var">.o.</seg> <lb/> <ptr xml:id="fig-0185-01a" corresp="fig-0185-01" type="figureAnchor"/> in qua grauius eſt. <lb/>ex .9. igitur quin-<lb/>ti cogitur Ariſto-<lb/>teles affirmare <choice><ex>tam</ex><am>tã</am></choice> <lb/>velox eſſe corpus <lb/>a. <choice><ex>quam</ex><am>quã</am></choice> eſt corpus <lb/>i. vnde idem pla-<lb/>nè inconueniens emergit ex ſecundo ſuppoſito cap. ſecundi huius lib..</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0185-01" corresp="fig-0185-01a"> <graphic url="0185-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quod duo corpor a in æqualia eiuſdem materia in diuerſis <lb/>medijs eandem uelocitatis proportionem <lb/>retinebunt.</head> <head xml:space="preserve">CAP. VIII.</head> <p> <s xml:space="preserve">QVælibet duo corpora inæqualia ſimili tamen figura & eadem materia con-<lb/>ſtantia, naturaliter ſe per diuerſa media mouentia, vnam <choice><ex>eandemque</ex><am>eandemq́;</am></choice> ſem-<lb/>per proportionem velocitatum ſeruant.</s> </p> <p> <s xml:space="preserve">Sint duo corpora <seg type="var">.A.</seg> et <seg type="var">.B.</seg> ſibi inuicem inæqualia quorum <seg type="var">.A.</seg> ſit maius, ſed ſimile <lb/>figura & idem materia, <lb/>cuius pondus totaleſit <seg type="var">.<lb/> <ptr xml:id="fig-0185-02a" corresp="fig-0185-02" type="figureAnchor"/> x.o.</seg> & pondus totaleip <lb/>ſius <seg type="var">.B.</seg> ſit <seg type="var">.u.s</seg>. </s> <s xml:space="preserve">Imagine-<lb/>mur quoque corpus <seg type="var">.A.</seg> <lb/>poſitum in aqua amitte <lb/>re <choice><ex>partem</ex><am>partẽ</am></choice> <seg type="var">.o.e.</seg> ponderis. <pb facs="0186" n="174"/><fw type="head">IO. BAPT. BENED.</fw> <seg type="var">o.x.</seg> et <seg type="var">.B.</seg> quoque in eodem loco amittere <seg type="var">.c.s.</seg> et <seg type="var">.A.</seg> in <choice><ex>aem</ex><am>aẽ</am></choice><unclear reason="illegible"/>re partem <seg type="var">.i.o.</seg> et <seg type="var">.B.</seg> partem. <lb/> <seg type="var">.t.s</seg>. </s> <s xml:space="preserve">Nunc quia corpus aqueum, cui correſpondet <seg type="var">.e.o.</seg> æquale eſt ipſi <seg type="var">.A.</seg> & corpus <lb/>aqueum, cui correſpondet <seg type="var">.c.s.</seg> æquale eſt i pſi <seg type="var">.B.</seg> vt eſt ab Archimede <choice><ex>probatum</ex><am>probatũ</am></choice>: </s> <s xml:space="preserve">com <lb/>muni quadam ſcientiæ ratione, ſequitur eandem proportionem futuram <seg type="var">.o.x.</seg> ad <seg type="var">.e.o.</seg> <lb/>quæ eſt <seg type="var">.u.s.</seg> ad <seg type="var">.c.s.</seg> ob <choice><ex>eaſdemque</ex><am>eaſdemq́;</am></choice> rationes idem erit de <seg type="var">.x.o.</seg> ad <seg type="var">.i.o.</seg> ut <seg type="var">.u.s.</seg> ad <seg type="var">.t.s.</seg> & <choice><ex>idem</ex><am>idẽ</am></choice> <lb/>etiam erit de <seg type="var">.o.x.</seg> ad <seg type="var">.s.u.</seg> vt de <seg type="var">.e.o.</seg> ad <seg type="var">.c.s.</seg> vt etiam de <seg type="var">.o.i.</seg> ad <seg type="var">.s.t</seg>. </s> <s xml:space="preserve">Vnde ex .19. lib. <lb/>quintí erit de <seg type="var">.x.i.</seg> ad <seg type="var">.u.t.</seg> quemadmodum de <seg type="var">.x.o.</seg> ad <seg type="var">.u.s.</seg> idem dico de <seg type="var">.x.e.</seg> ad <seg type="var">.u.c</seg>. </s> <s xml:space="preserve">Ex <lb/>11. igitur dicti lib. erit. de <seg type="var">.x.i.</seg> ad <seg type="var">.u.t.</seg> quemadmodum de <seg type="var">.x.e.</seg> ad <seg type="var">.u.c.</seg> ex quibus <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/>proportionibus, ſi ſubtra <lb/> <ptr xml:id="fig-0186-01a" corresp="fig-0186-01" type="figureAnchor"/> hantur proportiones @reſi <lb/><choice><ex>ſtentiarum</ex><am>ſtẽtiarum</am></choice> extrinſecus <choice><ex>ad- uenentium</ex><am>ad-uenẽtium</am></choice>, proportiones <lb/>quæ remanebunt, exter-<lb/>tio communi axiomate <lb/>ab Eucli. in principio pri<lb/>mi lib. poſito, ad inuicem <lb/>erunt æquales, ſecundum quas eorundem corporum ſunt velocitates.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0185-02" corresp="fig-0185-02a"> <graphic url="0185-02"/> </figure> <figure xml:id="fig-0186-01" corresp="fig-0186-01a"> <graphic url="0186-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Anrectè Aristoteles diſeruerit de proportionibus mo-<lb/>tuum in uacuo.</head> <head xml:space="preserve">CAP. IX.</head> <p> <s xml:space="preserve">CVm verò Ariſtoteles circa finem cap .8. lib. 4. phyſicorum ſubiungit quod ea-<lb/>dem proportione dicta corpora mouerentur in vacuo, vt in pleno, id pace <choice><ex>eius</ex><am>eiꝰ</am></choice> <lb/><choice><ex>dictum</ex><am>dictũ</am></choice> ſit planè <choice><ex>erroneum</ex><am>erroneũ</am></choice> eſt. </s> <s xml:space="preserve">quia in pleno dictis corporibus ſubtrahitur proportio reſi <lb/>ſtentiarum extrinſecarum à proportione ponderum, vt velocitatum proportio re-<lb/>maneat, quę nulla eſſet, ſi dictarum reſiſtentiarum proportio, ponderum propor-<lb/>tioni æqualis eſſet, & hanc ob cauſam diuerſam velocitatum proportionem in va-<lb/>cuo haberent ab ea, quæ eſt in pleno.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quòd in uacuo corpor a eiuſdem materiæ æquali uelocita-<lb/>te mouerentur.</head> <head xml:space="preserve">CAP.X.</head> <p> <s xml:space="preserve">QVòd ſupradicta corpora in vacuo naturaliter pari velocitate mouerentur, <lb/>hac ratione aſſero.</s> </p> <p> <s xml:space="preserve">Sint enim duo corpora <seg type="var">.o.</seg> et <seg type="var">.g.</seg> omogenea, et <seg type="var">.g.</seg> ſit dimidia pars ipſius <seg type="var">.o.</seg> ſint alia <lb/>quoque duo corpora <seg type="var">.a.</seg> et <seg type="var">.e.</seg> omogenea primis, quorum quodlibet æquale ſit ipſi <seg type="var">.g.</seg> <lb/>& imaginatione compręhendamus ambo poſita in extremitatibus alicuius lineæ, cu <lb/>ius medium ſit <seg type="var">.i.</seg> clarum erit, tantum pondus habiturum, punctum <seg type="var">.i.</seg> quantum <choice><ex>centrum</ex><am>centrũ</am></choice> <lb/>ipſius <seg type="var">.o.</seg> quod <seg type="var">.i.</seg> virtute corporis <seg type="var">.a.</seg> et <seg type="var">.e.</seg> in vacuo, <lb/> <ptr xml:id="fig-0186-02a" corresp="fig-0186-02" type="figureAnchor"/> eadem velocitate moueretur, quacentrum ipſius .<lb/>o: </s> <s xml:space="preserve">cum autem difiuncta eſſent dicta corpora <seg type="var">.a.</seg> et <seg type="var">.e.</seg> <lb/>à dicta linea, non ideo aliquo modo ſuam velocita <pb facs="0187" n="175"/><fw type="head">DISPVTATIONES.</fw> tem mutarent, quorum quodlibet eſſet quoque tam velox, quam eſt .g: igitur <seg type="var">.g.</seg> <lb/>tam velox eſſet quam <seg type="var">.o</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0186-02" corresp="fig-0186-02a"> <graphic url="0186-02"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Corpora licet inæqualia eiuſdem materiæ & figuræ, ſireſiſten-<lb/>tias habuerint ponderibus proportionales <lb/>æqualiter mouebuntur.</head> <head xml:space="preserve">CAP. XI.</head> <p> <s xml:space="preserve">EAdem ratione, quam cap. antecedente præſcripſimus, poſſet oſtendi, ſi duo cor-<lb/>pora <seg type="var">.o.</seg> et <seg type="var">.g.</seg> ſuas reſiſtentias, ita ad inuicem proportionatas haberent, utſunt <lb/>eorum pondera, in pleno pari velocitate prædita eſſe, quod in fine capitis noni leui <lb/>ter attigi, quia punctum <seg type="var">.i.</seg> tam velox eſſet, ut centrum ipſius <seg type="var">.o.</seg> cum à tanto pondere <lb/>i. motum eſſet; </s> <s xml:space="preserve">quanto centrum ipſius <seg type="var">.o.</seg> atquetan <lb/> <ptr xml:id="fig-0187-01a" corresp="fig-0187-01" type="figureAnchor"/> tam reſiſtentiam duo corpora <seg type="var">.a.</seg> et <seg type="var">.e.</seg> <choice><ex>quanta</ex><am>quãta</am></choice> ipſum <lb/>o. ſolum haberet ex hypotheſi, dicta tamen corpo <lb/>ra <seg type="var">.a.</seg> et <seg type="var">.e.</seg> tam ſeparata, quam coniuncta, eandem <lb/>velocitatem retinerent <seg type="var">.g.</seg> igitur tam velox eſſet, <lb/>quam <seg type="var">.o</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0187-01" corresp="fig-0187-01a"> <graphic url="0187-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Maior hic demonſir atur eſſe proportio ponder is corpor is den <lb/>ſioris ad pondus minus denſi in <choice><ex>medijs</ex><am>medijs</am></choice> <choice><ex>denſioribus</ex><am>dẽſioribus</am></choice>, quam <lb/>ſit eorundem corporum in medio minus denſo, nec <lb/>corporum ponder a ſeruare proportionem <lb/>denſitatis mediorum.</head> <head xml:space="preserve">CAP. XII.</head> <p> <s xml:space="preserve">PRopoſita nobis cum fuerint duo corpora <seg type="var">.A.</seg> et <seg type="var">.B.</seg> area corporea æqualia, quo-<lb/>rum <seg type="var">.A.</seg> denſius ſit ipſo <seg type="var">.B.</seg> probabo in medio magis denſo, maiorem proportio <lb/>nem futuram ponderis ipſius <seg type="var">.A.</seg> ad pondus <seg type="var">.B.</seg> quàm in medio minus denſo.</s> </p> <p> <s xml:space="preserve">Sit igitur <seg type="var">.p.g.</seg> pondus totale ipſius corporis <seg type="var">.A.</seg> et <seg type="var">.q.k.</seg> ipſius corporis <seg type="var">.B.</seg> vnde <seg type="var">.p.g.</seg> <lb/>maius erit ipſo <seg type="var">.q.k</seg>. </s> <s xml:space="preserve">Sit quoque <seg type="var">.o.g.</seg> pondus, quod medium magis denſum ſubtra-<lb/>hit à pondere <seg type="var">.p.g.</seg> et <seg type="var">.n.k.</seg> ſit pondus, quod idem medium ſubtrahit à pondere <seg type="var">.q.k.</seg> et <lb/><seg type="var">f.g.</seg> ſit pondus, quod medium minus denſum ſubtrahit à <seg type="var">.p.g.</seg> et <seg type="var">.i.k.</seg> illud, quodid@m <lb/><choice><ex>medium</ex><am>mediũ</am></choice> ſubtrahit ab <seg type="var">.q.k.</seg> vnde <seg type="var">.o.g.</seg> æquale erit <seg type="var">.n.k.</seg> et <seg type="var">.f.g.</seg> ipſi <seg type="var">.i.k.</seg> quia quod ad <choice><ex>aream</ex><am>areã</am></choice> <lb/>attinet, corpora ſupponuntur æqualia, vnde proportio <seg type="var">.p.f.</seg> ad <seg type="var">.q.i.</seg> maior erit ea, quæ <lb/>eſt <seg type="var">.o.f.</seg> ad <seg type="var">.n.i.</seg> communi <lb/> <ptr xml:id="fig-0187-02a" corresp="fig-0187-02" type="figureAnchor"/> ſcientiæ notione, quia ſi <lb/>ſcinderet <choice><ex>aliquis</ex><am>aliꝗs</am></choice>.p.f. in pun <lb/>cto <seg type="var">.c.</seg> ita. vt <seg type="var">.c.f.</seg> æquale eſ-<lb/>ſet ipſi <seg type="var">.q.i.</seg> proportio <seg type="var">.c.f.</seg> <lb/>ad <seg type="var">.q.i.</seg> eſſet vt ea, quæ eſt <seg type="var">.<lb/>o.f.</seg> ad <seg type="var">.n.i.</seg> (hoc eſt nulla) <pb facs="0188" n="176"/><fw type="head">IO. BAPT. BENED.</fw> ſed proportio <seg type="var">.p.f.</seg> ad <seg type="var">.q.i.</seg> maior eſſet ea, quæ eſt <seg type="var">.c.f.</seg> ad <seg type="var">.q.i.</seg> ex. octaua lib. quinti, vn-<lb/>de ex .12. eiuſdem lib. maior eſſet <seg type="var">.p.f.</seg> ad <seg type="var">.q.i.</seg> quàm.o.f. ad <seg type="var">.n.i.</seg> ex .33. igitur eiuſdem, <lb/>maior erit proportio <seg type="var">.p.o.</seg> ad <seg type="var">.q.n.</seg> quàm.p.f. ad <seg type="var">.q.i</seg>. </s> <s xml:space="preserve">Sic quoque ſe habebunt ad inui <lb/>cem velocitates, quod eſt propoſitum. </s> <s xml:space="preserve">Cum autem proportio <seg type="var">.p.o.</seg> ad <seg type="var">.q.n.</seg> maior ſit, <lb/>quàm.p.f. ad <seg type="var">.q.i.</seg> permurando igitur maior erit proportio <seg type="var">.p.o.</seg> ad <seg type="var">.p.f.</seg> quam <seg type="var">.q.n.</seg> ad <seg type="var">.<lb/>q.i.</seg> aut euerſim maior erit proportio <seg type="var">.q.i.</seg> ad <seg type="var">.q.n.</seg> quàm.p.f. ad <seg type="var">.p.o.</seg> vnde ſi proportio <lb/><seg type="var">p.f.</seg> ad <seg type="var">.p.o.</seg> eſſet ac ea, quæ eſt <seg type="var">.o.g.</seg> ad <seg type="var">.f.g.</seg> non eſſet <seg type="var">.q.i.</seg> ad <seg type="var">.q.n.</seg> ut eſt <seg type="var">.o.g.</seg> ad <seg type="var">.f.g.</seg> aut <lb/>vt <seg type="var">.n.k.</seg> ad <seg type="var">.i.k.</seg> quodidem <lb/>eſt, de quibus quidem re-<lb/> <ptr xml:id="fig-0188-01a" corresp="fig-0188-01" type="figureAnchor"/> bus, exemplis propoſitis <lb/>quinto capite <choice><ex>mentionem</ex><am>mẽtionem</am></choice> <lb/>feci.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0187-02" corresp="fig-0187-02a"> <graphic url="0187-02"/> </figure> <figure xml:id="fig-0188-01" corresp="fig-0188-01a"> <graphic url="0188-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Velocitatibus autem ſe-<lb/>quentibus pondera, ſequi <lb/>tur proportionem veloci-<lb/>citatum duorum corporum hetereogeneorum eandem non eſſe per diuerſa media, <lb/>contra id, quod ſequeretur ſi Ariſtotelis opinionem .8. cap. lib. 4. phyſicorum re-<lb/>ciperemus.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Longe aliter ueritatem ſe habere quam Aristoteles <lb/>doceat in fine libri ſeptimi phyſicorum.</head> <head xml:space="preserve">CAP. XIII.</head> <p> <s xml:space="preserve">NOn tam facile eſt aſſignare proportionem velocitatum duorum corporum na <lb/>turalium, quam Ariſtoteles vltimo cap. lib. 7. phyſicorum putauit.</s> </p> <p> <s xml:space="preserve">Quamobrem ſint duo corpora <seg type="var">.B.</seg> et <seg type="var">.D.</seg> materia <choice><ex>magnitudineque</ex><am>magnitudineq́;</am></choice> diuerſa, pondere <lb/>tamen, & figura ſimilia, & proportio reſiſtentiarum, quas recipiunt à medio <choice><ex>dum</ex><am>dũ</am></choice> mo-<lb/>uentur, ſit. ut <seg type="var">.o.i.</seg> ad <seg type="var">.a.e.</seg> denotentur deinde velocitates totales abſque vlla reſiſten-<lb/>tia ab <seg type="var">.a.u.</seg> et <seg type="var">.o.c.</seg> quæ æquales erunt ad inuicem per communem ſcientiam ex ſup-<lb/>poſito, ſint alia deinde duo corpora <seg type="var">.V.</seg> et <seg type="var">.M.</seg> eodem modo ſe habentia ut prima <seg type="var">.B.</seg> <lb/>et <seg type="var">.D.</seg> in eodem medio, ſed ex diuerſa materia ab ea, quæ eſt illorum duorum corpo <lb/>rum, magnitudine tamen & figura ijſdem ſimilia: </s> <s xml:space="preserve">ſignificentur quoque eo-<lb/>rundem reſiſtentiæ per <seg type="var">.t.s.</seg> et <seg type="var">.n.r.</seg> & eorundem velocitates à nulla ex reſiſtentijs di-<lb/>minutæ, per <seg type="var">.n.x.</seg> et <seg type="var">.t.g.</seg> vnde <seg type="var">.n.r.</seg> æqualis erit <seg type="var">.a.e.</seg> et <seg type="var">.t.s.</seg> ipſi <seg type="var">.o.i.</seg> et <seg type="var">.n.x.</seg> ipſi <seg type="var">.t.g</seg>: <seg type="var">n.x.</seg> ta-<lb/>men et <seg type="var">.t.g.</seg> non erunt ęqualia <seg type="var">.a.u.</seg> et <seg type="var">.o.c</seg>. </s> <s xml:space="preserve">Sed exempli gratia, ponamus ea eſſe mi-<lb/>nora. </s> <s xml:space="preserve">Supponamus nunc <seg type="var">.e.u.</seg> velocitatem eſſe quæ remanet ipſi <seg type="var">.B.</seg> cum applicata <lb/>erit reſiſtentia <seg type="var">.a.e.</seg> dicto corpori <seg type="var">.B.</seg> quæ diminutam facit totam <seg type="var">.a.u.</seg> per <seg type="var">.a.e.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.i.c.</seg> <lb/>ea, quę remanet ipſi <seg type="var">.o.c.</seg> corporis <seg type="var">.D.</seg> et <seg type="var">.r.x.</seg> ea, quæ remanet <seg type="var">.n.x.</seg> corporis <seg type="var">.V.</seg> et <seg type="var">.s.g.</seg> <lb/>ea, quæ eſt ex <seg type="var">.t.g.</seg> corporis <seg type="var">.M</seg>. </s> <s xml:space="preserve">Vnde communi omnium <choice><ex>conſenſu</ex><am>cõſenſu</am></choice> aſſequemur <seg type="var">.e.u.</seg> ma <lb/>iorem futuram <seg type="var">.r.x.</seg> et <seg type="var">.i.c.</seg> ipſa <seg type="var">.s.g</seg>. </s> <s xml:space="preserve">Scindatur deinde <seg type="var">.a.m.</seg> ad ęqualitatem <seg type="var">.n.x.</seg> et <seg type="var">.o.z.</seg> <lb/>ipſius <seg type="var">.t.g.</seg> vnde <seg type="var">.a.m.</seg> ad <seg type="var">.o.z.</seg> et <seg type="var">.m.u.</seg> ad <seg type="var">.z.c.</seg> æquales habebimus, ut quoque <seg type="var">.e.m.</seg> ad <seg type="var">.r.<lb/>x.</seg> et <seg type="var">.i.z.</seg> ad <seg type="var">.s.g.</seg> quamobrem <seg type="var">.e.m.</seg> maior erit ipſa <seg type="var">.z.i.</seg> maior igitur erit proportio <seg type="var">.z.c.</seg> <lb/>ad <seg type="var">.z.i.</seg> quàm.m.u. ad <seg type="var">.m.e.</seg> (quia <seg type="var">.z.c.</seg> ad <seg type="var">.z.i.</seg> ita ſe habet vt <seg type="var">.m.u.</seg> ad <seg type="var">.i.z.</seg> ex .7. lib. quin-<lb/>ti, ſed <seg type="var">.m.u.</seg> ad <seg type="var">.i.z.</seg> maior eſt quam ad <seg type="var">.m.e.</seg> ex .8. dicti lib. vnde ex .12. eiuſdem <seg type="var">.z.c.</seg> ad <lb/>ad <seg type="var">.z.i.</seg> maior erit, quàm.m.u. ad <seg type="var">.m.e</seg>. </s> <s xml:space="preserve">Ergo ex .28. maior proportio erit <seg type="var">.c.i.</seg> ad <seg type="var">.z.i.</seg> <pb facs="0189" n="177"/><fw type="head">DISPVTATIONES.</fw> quam <seg type="var">.u.</seg> ad <seg type="var">.m.e.</seg> & ex .27. maior erit proportio <seg type="var">.c.i.</seg> ad <seg type="var">.u.e.</seg> quam <seg type="var">.z.i.</seg> ad <seg type="var">.e.m.</seg> ideſt <seg type="var">.s.</seg> g <lb/>ad <seg type="var">.r.x.</seg> quod Ariſtoteli in mentem non venerat. </s> <s xml:space="preserve">Alijs quoque modis idem proba-<lb/>ri poteſt, vt ſi diceret aliquis, maiorem proportionem eſſe <seg type="var">.e.m.</seg> ad <seg type="var">.m.u.</seg> quam <seg type="var">.i.z.</seg> ad <lb/><seg type="var">z.c.</seg> (quia <seg type="var">.e.m.</seg> ad <seg type="var">.m.u.</seg> eadem eſt ratio vt ad <seg type="var">.z.c.</seg> ex .7. quinti, ſed proportio <seg type="var">.e.m.</seg> ad <seg type="var">.<lb/>z.c.</seg> maior eſt quam <seg type="var">.i.z.</seg> ad <seg type="var">.z.c.</seg> ex .8. eiuſdem, ergo ea, quæ eſt <seg type="var">.e.m.</seg> ad <seg type="var">.m.u.</seg> ex .12. ma <lb/>for erit, quam <seg type="var">.i.z.</seg> ad <seg type="var">.z.c.</seg>) vnde componendo, ea quæ eſt <seg type="var">.e.u.</seg> ad <seg type="var">.m.u.</seg> maior erit illa, <lb/>quæ eſt <seg type="var">.i.c.</seg> ad <seg type="var">.z.c.</seg> & <choice><ex>permutando</ex><am>permutãdo</am></choice>, quam ea, quæ eſt <seg type="var">.e.u.</seg> ad <seg type="var">.i.c.</seg> ea, quæ eſt <seg type="var">.m.u.</seg> ad <seg type="var">.z.c.</seg> <lb/>& ex .33. quinti, ea, quæ eſt <seg type="var">.e.m.</seg> ad <seg type="var">.i.z.</seg> maior erit ea, quæ eſt <seg type="var">.e.u.</seg> ad <seg type="var">.i.c</seg>.</s> </p> <figure place="here"> <graphic url="0189-01"/> </figure> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quid ſequatur ex ſupradistis.</head> <head xml:space="preserve">CAP. XIIII.</head> <p> <s xml:space="preserve">EX præcedenti capite manifeſtè depræhenditur, in vniuerſum Ariſtotelis opi-<lb/>nionem veram non eſſe in prima parte vltimi capitis. lib. 7. phyſicorum; </s> <s xml:space="preserve">quia <lb/>in eo loco ſupponens ipſe corpus <seg type="var">.B.</seg> pręcedentis capitis eſſe dimidiam partem ipſius <lb/>D. quantum ad aream corpoream ſpectat (ſunt tamen pondere ad inuicem æqualia) <lb/>ait <seg type="var">.B.</seg> futurum duplo velocius ipſo <seg type="var">.D</seg>. </s> <s xml:space="preserve">Ego verò præcedenti capite accepi <seg type="var">.e.u.</seg> pro <lb/>velocitate reſidua corporis <seg type="var">.B.</seg> (ſubtracta ea tamen parte, quam ei reſiſtentia adimit, <lb/>quæ erat <seg type="var">.e.a.</seg>) et <seg type="var">.i.c.</seg> pro ea, quæ eſt corporis <seg type="var">.D.</seg> et <seg type="var">.r.x.</seg> pro ea, quæ eſt corporis <seg type="var">.V.</seg> et <seg type="var">.<lb/>s.g.</seg> pro ea, quæ eſt corporis <seg type="var">.M</seg>. </s> <s xml:space="preserve">Dicat nunc Ariſtoteles, quę nam harum duarum pro <lb/>portionum dupla erit? </s> <s xml:space="preserve">quia ſi earum aliqua talis erit, alia nullo modo eſſe poterit, <lb/>vt iam oſtendi, etiamſi duo corpora <seg type="var">.V.</seg> et <seg type="var">.M.</seg> eaſdem conditiones habeant, quas <seg type="var">.B.</seg> <lb/>et <seg type="var">.D</seg>. </s> <s xml:space="preserve">Ratio autem, quæ Ariſtotelem induxerit ad illud credendum, nulla alia eſſe <lb/>potuit, quàm quod putarit reſiſtentias proportionatas eſſe magnitudinibus corpo-<lb/>reis, ideſt quemadmodum <seg type="var">.B.</seg> erat corporaliter dimidia pars ipſius <seg type="var">.D.</seg> ſic etiam habe <lb/>ret medietatem eius reſiſtentiæ, quam habuiſſet corpus <seg type="var">.D</seg>. </s> <s xml:space="preserve">Quod etſi verum eſſet, <lb/>non tamen ſequeretur neceſſariò in quibuſlibet corporibus futuram velocitatum <lb/>proportionem eandem, quæ reſiſtentiarum eſt, vt ſuperiore capite oſtendimus.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Numrestè ſenſerit Philoſophus reſistentias proportionales <lb/>eße cum corporibus mobilibus.</head> <head xml:space="preserve">CAP. XV.</head> <p> <s xml:space="preserve">QVòd Ariſtoteles crediderit reſiſtentias proportionatas eſſe corporibus, erra-<lb/>uit. </s> <s xml:space="preserve">Si ſuperficies ijſdem proportionatæ eſſent, dubium non eſt, quin <lb/>reſiſtentiæ quoque ipſæ, ijſdem proportionatæ exiſterent, ſupponendo eas ſimiles <lb/>ſitu, dum eadem corpora mouerentur. </s> <s xml:space="preserve">Sed eadem proportio non eſt inter ſuperfi- <pb facs="0190" n="178"/><fw type="head">IO. BAPT. BENED.</fw> cies, & quæ inter corpor a <choice><ex>reperitur</ex><am>reperit̃</am></choice>: </s> <s xml:space="preserve">Ariſtoteles igitur in eo defecit. </s> <s xml:space="preserve">Quòd <choice><ex>autem</ex><am>autẽ</am></choice> inter <lb/>ſuperficies non eadem ſit proportio, quæ inter corpora extat, ſi primo ad ſphęricas <lb/>mentem verterimus, intelligemus proportionem eam, quæ inter duas ſphæras repe <lb/>ritur triplam ſemper exiſtere ei, quæ eſt inter ipſarum diametros ex vltima .12. libr. <lb/>Euclid. </s> <s xml:space="preserve">Eſt autem proportio, quæ eſt inter ſuperficies ſphęricas ęqualis ei, quæ eſt <lb/>ipſorum circulorum maiorum ex .16. lib. quinti, cum ex .31. primi de ſphæra & cy-<lb/>lindro Archimedis, omnis ſphærica ſuperficies quadrupla, ſit maiori circulo ipſius <lb/>ſphęræ, ſed proportio, quæ eſt inter dictos circulos, eſt dupla ei, quæ eſt inter <choice><ex>eorun- dem</ex><am>eorũ-dẽ</am></choice> diametros ex .2. lib. 12. Euc. </s> <s xml:space="preserve">ergo <choice><ex>proportio</ex><am>ꝓportio</am></choice>, quæ eſt inter corpora, ſeſquialtera erit <lb/>ei, quæ eſt ſuperficierum, & non æqualis, ut Ariſtoteles putauit. </s> <s xml:space="preserve">Idem de corporibus <lb/>ſimilibus à ſuperficiebus planis terminatis dico, ratiocinando mediante .36. lib. 11. <lb/>et .18. ſexti, vnde cognoſcemus proportionem corporum, proportioni laterum, tri-<lb/>plam futuram, & ſuperficierum proportionem, laterum proportioni duplam. </s> <s xml:space="preserve">Quare <lb/>corporum proportio, ei, quæ ſuperficierum eſt, ſeſquialtera erit, ita ut ſi velocitates <lb/>extitiſſent ad inuicem proportionatæ, vt ſuperficies, proportio velocitatis corporis <seg type="var">.<lb/>B.</seg> ei, quæ eſt corporis <seg type="var">.C.</seg> fuiſſet ſubſeſquialtera proportioni corporum, & non æqua <lb/>lis eidem.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Fdipſum aliter demonſtr atur.</head> <head xml:space="preserve">CAP. XVI.</head> <p> <s xml:space="preserve">ALio quoque modo probari poteſt non eſſe in vniuerſum verum id, quod Ari-<lb/>ſtoteles in prima parte capitis vltimi lib. 7. phyſicorum ait, ſic ſcribens.</s> </p> <p> <s xml:space="preserve">Si <seg type="var">.A.</seg> quidem ſit id quod mouet <seg type="var">.B.</seg> verò id quod mouetur, et <seg type="var">.C.</seg> ſit longitudo per <lb/>quam, et <seg type="var">.D.</seg> tempus in quo eſt motum, in tempore nimirum ęquali, potentia æqua-<lb/>lis <seg type="var">.A.</seg> dimidium ipſius <seg type="var">.B.</seg> per duplum mouebit ipſius <seg type="var">.C.</seg> per ipſum autem <seg type="var">.C.</seg> in dimi <lb/>dio temporis <seg type="var">.D.</seg> ſic enim erit rationis ſimilitudo.</s> </p> <p> <s xml:space="preserve">Sit ergo corpus <seg type="var">.o.</seg> ſeptimi capitis pondere æquali corpori <seg type="var">.u.</seg> eiuſdem capitis, ſed <lb/>area corporea minusipſo <seg type="var">.u.</seg> pro medietate. </s> <s xml:space="preserve">Simile tamen figura. </s> <s xml:space="preserve">Imaginemur <choice><ex>nunc</ex><am>nũc</am></choice> <lb/>tertium aliud corpus omogeneum ipſi <seg type="var">.u.</seg> quod ſit <seg type="var">.i.</seg> magnitudine & figura ſimile ipſi <lb/>o. vnde minor erit ipſo <seg type="var">.u.</seg> pro media parte, & hanc ob cauſam ipſum <seg type="var">.u.</seg> erit duplo ma <lb/>gis graue, quàm ipſum <seg type="var">.i.</seg> & per conſequens ipſum quoque <seg type="var">.o.</seg> duplo grauius erit <choice><ex>quam</ex><am>quã</am></choice> <lb/>ſit ipſum <seg type="var">.i.</seg> ex .7. libr. quinti Euclidis. </s> <s xml:space="preserve">Ipſum ergo corpus <seg type="var">.o.</seg> duplo velocius erit, <lb/>quàm ipſum <seg type="var">.i.</seg> ex primo ſuppoſito cap .2. huius lib. </s> <s xml:space="preserve">Vnde ex .9. quinti, velocitas ipſius <lb/>i. æqualis eſſet ei, quæ eſt ipſius u. cum Ariſtoteles ſcribat <seg type="var">.o.</seg> quoque futurum duplo <lb/>velocius ipſo <seg type="var">.u.</seg> <choice><ex>quod</ex><am>ꝙ</am></choice> cap .7. huius lib. falſum eſſe demonſtraui.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">De alio Aristo. lapſu.</head> <head xml:space="preserve">CAP. XVII.</head> <p> <s xml:space="preserve">SCribit Ariſtoteles in ultimo cap. lib. 7. phyſicorum in hunc modum. <lb/></s> <s xml:space="preserve">Si duo quædam ſeorſum per tantum ſpatium tanto tempore duo ſeorſum pon <lb/>dera mouent, & compoſita per longitudinem æqualem, <choice><ex>ęqualiuem</ex><am>ęqualiuẽ</am></choice> in tempore, com-<lb/>poſitum ex ponderibus <choice><ex>vtriſque</ex><am>vtriſq;</am></choice> mouebunt, eſt enim in eis eadem ratio.</s> </p> <pb facs="0191" n="179"/> <fw type="head">DISPVTATIONES.</fw> <p> <s xml:space="preserve">Quod in vniuerſum nec etiam poteſt eſſe verum in pleno, quia cap .14. iam pro-<lb/>baui, non eandem proportionem eſſe inter ſuperſicies corporum, & ipſa corpora.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quomodo dignoſcatur proportio uelocitatis duorum ſimilium <lb/>corporum omogeniorum inaqualium.</head> <head xml:space="preserve">CAP. XVIII.</head> <p> <s xml:space="preserve">ETiam ſi reperire in qua proportione motus naturaliter moueantur duo corpo-<lb/>ra, figura & materia ſimilia, inęqualia tamen ad inuicem, non facile ſit, oſten-<lb/>dam tamen qua ratione id conſequi poſſimus.</s> </p> <p> <s xml:space="preserve">Proponantur nobis, exempli gratia, duo corpora <seg type="var">.a.</seg> et <seg type="var">.o.</seg> ſphęrica, inęqualia inui-<lb/>cem, omogenea tamen materia, quorum <seg type="var">.a.</seg> maius ſit; </s> <s xml:space="preserve">ſi voluerimus inuenire in qua <lb/>nam velocitatis proportione naturaliter mouerentur. </s> <s xml:space="preserve">Volo vt inquiratur corpus <seg type="var">.i.</seg> <lb/>ſphęricum, alia tamen & diuerſa materia conſtans, ſed pondere ęquale corpori <seg type="var">.o.</seg> & <lb/>ſuperſicie tam proportionata ſuperficiei corp oris <seg type="var">.a.</seg> quàm eſt ea, quæ eſt ſui ponde-<lb/>ris ad pondus ipſius <seg type="var">.a</seg>. </s> <s xml:space="preserve">Hoc facto, indagetur, quænam erit proportio inter ſu-<lb/>perficies corporum <seg type="var">.i.</seg> et <seg type="var">.o.</seg> quę ſemper dupla eſt, vel ſubdupla ei quæ eſt diametro-<lb/>rum; </s> <s xml:space="preserve">ut iam cap .15. dixi, & hęc proportio ſuperficierum ſphęricarum <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.o.</seg> et <seg type="var">.i.</seg> ſub <lb/>trahatur ab æqualitate, quod igitur remanebit, erit proportio <choice><ex>velocitatum</ex><am>velocitatũ</am></choice> inter duo <lb/>corpora <seg type="var">.o.</seg> et <seg type="var">.i.</seg> ideſt inter <seg type="var">.o.</seg> et <seg type="var">.a.</seg> vt exempli gratia, ſi proportio ſuperficiei <seg type="var">.o.</seg> ſuperfi <lb/>ciei ipſius <seg type="var">.i.</seg> ſeſquitertiα<unclear reason="illegible"/> eſſet, ſub <lb/>trahendo eam ab ęqualitate, rema-<lb/> <ptr xml:id="fig-0191-01a" corresp="fig-0191-01" type="figureAnchor"/> neret <choice><ex>proportio</ex><am>ꝓportio</am></choice> ſubſeſquitertia, vnde <lb/>velocitas corporis maioris ( quod in <lb/>pręſenti loco ſupponitur eſſe <seg type="var">.o.</seg>) ei, <lb/>quę eſt corporis minoris, quale eſt <lb/>corpus <seg type="var">.i.</seg> ſubſeſquitertia eſſet; </s> <s xml:space="preserve">aut <lb/>dicamus quòd <seg type="var">.i.</seg> eſſet velocius ipſo <lb/>o. in proportione ſeſquitertia ex ſe <lb/>cundo ſuppoſito ſecundi capitis huius libri. </s> <s xml:space="preserve">Sed <seg type="var">.i.</seg> tam velox eſt quam ipſum <seg type="var">.a.</seg> ex <num value="11">.<lb/>11.</num> cap. ergo proportio velocitatis ipſius <seg type="var">.a.</seg> ſeſquitertia erit ei. quæ eſt ipſius <seg type="var">.o</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0191-01" corresp="fig-0191-01a"> <graphic url="0191-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Quam ſit inanis ab Ariſtotele ſuſcepta demonſtratio quod <lb/>uacuum non detur.</head> <head xml:space="preserve">CAP. XIX.</head> <p> <s xml:space="preserve">EX ijs, quæ ſuperius <choice><ex>demonſtrauimus</ex><am>demõſtrauimus</am></choice> facilè cognoſci poteſt irritam eſſc eam ratio <lb/>nem, quam Ariſtoteles .8. cap. lib. 4. phyſicorum ad deſtruendum vacuum, <choice><ex>con</ex><am>cõ</am></choice> <lb/>finxit. </s> <s xml:space="preserve">Vtigitur idem facilius oſtendamus, compræhendamus imaginatione infini-<lb/>ta media corporea, quorum vnum altero rarius ſit, in qua placuerit nobis ex propor <lb/>tionibus, incipiendo ab uno, imaginemur etiam corpus <seg type="var">.Q.</seg> denſius primo medio, cu-<lb/>ius corporis, totalis grauitas ſit <seg type="var">.a.b.</seg> & poſitum in ipſo medio, amittat partem <seg type="var">.e.b.</seg> ip-<lb/>ſius grauitatis, & in ſecundo medio amittat <seg type="var">.i.b.</seg> & ſic per gradus vnde nobis patebie<unclear reason="illegible"/> <pb facs="0192" n="180"/><fw type="head">IO. BAPT. BENED.</fw> dicto corpori <seg type="var">.Q</seg>. </s> <s xml:space="preserve">Nunquam remanſuram ſuam totalem grauitatem <seg type="var">.a.b.</seg> in quolibet <lb/>ex-dictis medijs. </s> <s xml:space="preserve">Nunc ſi quærat à me Ariſtoteles proportionem velocitatis corpo-<lb/>ris <seg type="var">.Q.</seg> per vacuum ad velocitatem dicti corporis per plenum, ego ei proponam pro-<lb/>portionem ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.a.e.</seg> exempli gratia, dicens, <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>quemadmodum</ex><am>quẽadmodum</am></choice> <seg type="var">.a.b.</seg> maius eſt <lb/>ip ſo <seg type="var">.a.e.</seg> ſic etiam corpus <seg type="var">.Q.</seg> velocius erit in vacuo, quàm in pleno, dicti autem ple-<lb/>ni denſitatem appellabimus <seg type="var">.e.b</seg>. </s> <s xml:space="preserve">Ariſtoteles dicet nunc, <choice><ex>quod</ex><am>ꝙ</am></choice> aliud quoddam medium <lb/>in eadem proportione ſubtilius ipſo <seg type="var">.e.b.</seg> deſumatur; </s> <s xml:space="preserve">quemadmodum <seg type="var">.a.e.</seg> minus eſt <lb/>ipſo <seg type="var">.a.b.</seg> ſit ergo iſtud <seg type="var">.i.b.</seg> in quo Ariſtoteles credit corpus Q. futurum tam velox ut <lb/>in vacuo, in quo aberrat, <choice><ex>quia</ex><am>ꝗa</am></choice> proportio velocitatis corporis <seg type="var">.Q.</seg> in medio <seg type="var">.i.b.</seg> ad velo <lb/>citatem eiuſdem in medio <lb/><seg type="var">e.b.</seg> ita ſe hàbebit, ut <seg type="var">.i.a.</seg> ad <lb/> <ptr xml:id="fig-0192-01a" corresp="fig-0192-01" type="figureAnchor"/> <seg type="var">e.a.</seg> ex ultimo ſuppoſito ca <lb/>pit .2. huius libr. quæ minor <lb/>eſſet ea, quæ eſt <seg type="var">.a.b.</seg> ad <seg type="var">.a.e.</seg> ex .8. lib. quinti Eucli.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0192-01" corresp="fig-0192-01a"> <graphic url="0192-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Non ſatis dilucidè Ariſtotelem de loco ratiocinatum fuiße.</head> <head xml:space="preserve">CAP. XX.</head> <p> <s xml:space="preserve">QVæ Ariſtoteles de loco ſcribit multas in ſe continent difficultates. </s> <s xml:space="preserve">Primum, <lb/>cap .4. lib. 4. phyſicorum ait, omne corpus eſſe in ſuo proprio loco, ſupponen <lb/>do vnum centrum pro loco grauium, et unam circunferentiam pro loco leuium cor <lb/>porum. </s> <s xml:space="preserve">Sed quomodo punctum poteſt eſſe locus ipſius corporis, cum omni dimen <lb/>ſione <choice><ex>capacitateque</ex><am>capacitateq́;</am></choice> ſit denudatum? </s> <s xml:space="preserve">vnde ſi <choice><ex>centrum</ex><am>centrũ</am></choice> locus eſſet corporum grauium, om <lb/>nia dicta corpora grauia, extra proprium locum exiſterent, quia nullum ex iis eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>ſit in centro. </s> <s xml:space="preserve">Adde quod neque hoc cum loci definitione ab ipſo poſita conſentiret <lb/>cum ipſe dicat in eodem cap. locum eſſe ſuperſiciem quandam, & non interuallum, <lb/>licet huiuſmodi definitio falſa appareat primo ex <choice><ex>inconuenienti</ex><am>incõuenienti</am></choice> falſo, quod ipſe hinc <lb/>ſequuturum dicit, ideſt, quod ſi locus interuallum eſſet, infinita loca exiſterent, quod <lb/>reuera nec ob hanc cauſam inconueniens exiſtit, quia eodem planè modo quo ali-<lb/>quod corpus poteſt eſſe infinita corpora, (quod ipſe diceret in potentia) ſic etiam in <lb/>teruallum aliquod poſſet eſſe infinita interualla. </s> <s xml:space="preserve">Cum autem dicat ſuperficies cor-<lb/>poris ambientis eſſe locum eius corporis, quod continetur, cogitur dicere lineam, <lb/>quæ circundat ſuperficiem, ſuperficiei locum eſſe, & puncta ipſius lineæ, quod reue <lb/>ra abſurdum eſt. </s> <s xml:space="preserve">Locus corporis eſt interuallum illud eadem magnitudine & figu-<lb/>ra, qua corpus ipſum pręditum eſt, quod ſi non eſſet, ſed eſſet ſuperficies, quemad-<lb/>modum Ariſtoteles voluit, maximum inconueniens ſequeretur, ſcilicet æquales lo-<lb/>cos capere inęqualia corpora, aut corpora æqualia, locos inęquales occupare, quod <lb/>ſcitu facillimum eſt, cum Theon ſuper Ptolomęi Almageſtum iam probarit ſphæ-<lb/>ricam ſuperficiem maius interuallum corporeum continere, quàm aliam <choice><ex>quanuis</ex><am>quãuis</am></choice> ſu-<lb/>perficiem dictæ ſphęricæ æqualem, vnde poſſent facilè reperiri duo loci, quorum al-<lb/>ter millies altero maior eſſet, capaces tamen corporum æqualium, aut reperiri duo <lb/>corpora, quorum alterum millies maius eſſet altero, quę tamen corpora apta eſſent <lb/>ad occupandos locos ęquales, quamuis Ariſtoteles dicat, locum, neque maiorem ne <lb/>que minorem eſſe debere locato. </s> <s xml:space="preserve">Sed interualla corporea ęqualia à quauis figura <lb/>terminata, continebunt ſemper corpora ęqualia. </s> <s xml:space="preserve">Corporeum igitur interuallum eſt <pb facs="0193" n="181"/><fw type="head">DISPVTATIONES.</fw> reuera locus corpori adęquatus, cum corpus in interuallum ſuperſiciale non intret, <lb/>quam @is interuallum corporeum ingrediatur. </s> <s xml:space="preserve">Et hoc modo <choice><ex>nullum</ex><am>nullũ</am></choice> eſt corpus, quod <lb/>in m@ do aut extra mundum ( dicat autem Ariſtoteles quicquid voluerit ) locum <lb/>ſuum non habeat.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">V<unclear reason="illegible"/>trum bene Aristoteles ſenſerit de infinito.</head> <head xml:space="preserve">CAP. XXI.</head> <p> <s xml:space="preserve">TRactans Ariſtoteles in fine quinti cap. lib. 3. phyſicorum de infinito ait, impoſ<lb/>ſibile cum ſit inuenire locum infinitum, & omne corpus in loco cum ſit, impoſ <lb/>ſibile quoque eſſe in rerum natura aliquod: </s> <s xml:space="preserve">infinitum corpus reperiri. </s> <s xml:space="preserve">Omittamus <lb/>quòd cum Ariſtoteles debuerit beneficio loci deſtruere infinitum, ordine peruerſo <lb/>de infinito prius, quàm de loco diſputationem inſtituat; </s> <s xml:space="preserve">ſed dicamus ipſum intelli-<lb/>gere de infinito corporeo, & cum probauerimus corporis locum eſſe corporeum in <lb/>teruallum, non autem ſuperficiem, neque opus ſit in definitione interualli mentio <lb/>nem aliquam facere terminorum, vnde ipſum infinitum eſſe poteſt, neque aliqua ra <lb/>tione de hac re dubitari poteſt; </s> <s xml:space="preserve">hoc modo nullum inconueniens ſequeretur, quòd <lb/>extra cęlum reperiri poſſit corpus aliquod infinitum, quamuis, id ipſe nulla euiden-<lb/>ti ratione inductus perneget. </s> <s xml:space="preserve">Senſit quoque, abſque eo, <choice><ex>quod</ex><am>ꝙ</am></choice> aliquam rationem propo <lb/>nat, aliquid extra cœlum reperiri quemadmodum apparet ex fine cap .9. lib. primi <lb/>de cœlo, cum etiam ait cap .8. lib. 8. phyſicorum, infinitas partes alicuius continui eſ-<lb/>ſe ſolum in potentia, non item in actu, hoc non eſt illico concedendum, quia ſi omne <lb/>totum continuum, & re ipſa exiſtens, in actu eſt, omnis quoque eius pars erit in actu, <lb/>quia ſtultum eſſet credere, ea quæ actu ſunt, ex ijs, quæ potentia exiſtunt, componi. <lb/></s> <s xml:space="preserve">Neque etiam dicendum eſt continuationem earundem partium efficere, vt poten-<lb/>tia ſint ipſæ partes, & omni actu priuatæ; </s> <s xml:space="preserve">Sit exempli gratia linea recta <seg type="var">.a.u.</seg> continua <lb/>quæ deinde diuidatur in puncto <seg type="var">.e.</seg> per æqualia, dubium non eſt, quin ante <choice><ex>diuiſionem</ex><am>diuiſionẽ</am></choice>, <lb/>medietas <seg type="var">.a.e.</seg> tam in actu (licet coniuncta cum alia <seg type="var">.e.u.</seg>) reperiretur, quàm totum .2. <lb/>u. licet à ſenſu diſtincta non eſſet. </s> <s xml:space="preserve">Idem affirmo de medietate <seg type="var">.a.e.</seg> ideſt de quarta <lb/>parte totius <seg type="var">.a.u.</seg> & pariter de octaua, de milleſima, & de quauis, ita vt eſſentia actua <lb/>lis infiniti hoc modo tutò concedi poſſit, <choice><ex>cum</ex><am>cũ</am></choice> ita ſit in natura. </s> <s xml:space="preserve">Sed peius etiam ſenſit <lb/>Ariſtoteles eodem loco capitis quinti lib. 3. phyſicorum, negando infinitum poſſe <lb/>connumerari inter quantitates, dicens vnam aliquam quantitatem intelligi vt cubi <lb/>tum, tricubitum, & cætera; </s> <s xml:space="preserve">vbi non conſiderat eadem etiam ratione intelligi poſſe <lb/>aliquam quantitatem <choice><ex>infinitorum</ex><am>infinitorũ</am></choice> cubitorum, & in quantitatis definitione nullam eſ-<lb/>ſe neceſſitatem terminorum, vt exempli gratia in definitione numeri, non eſt neceſ <lb/>ſitas alicuius determinati numeri, quia multitudo, non minus infinita, quàm finita, <lb/>intelligi poteſt. </s> <s xml:space="preserve">Vbi poſteà cap .8. libr .4. phyſicorum ait nullam eſſe differentiam <lb/>inter infinitum, & vacuum, reuera nihil abſurdius hoc dicere fingereue poterat.</s> </p> <pb facs="0194" n="182"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Exagitatur ab Ariſtotele adductatemporis definitio.</head> <head xml:space="preserve">CAP. XXII.</head> <p> <s xml:space="preserve">CVM ſenſerit Ariſtoteles <choice><ex>tempus</ex><am>tẽpus</am></choice> abſque motu eſſe <choice><ex>non</ex><am>nõ</am></choice> poſſe, ea tamen ab <choice><ex>inui- cem</ex><am>inui-cẽ</am></choice> ſeparans, <choice><ex>volens</ex><am>volẽs</am></choice> definire <choice><ex>tempus</ex><am>tẽpus</am></choice> ait, <choice><ex>ipsum</ex><am>ipsũ</am></choice> eſſe <choice><ex>motus</ex><am>motꝰ</am></choice> <choice><ex>menſuram</ex><am>menſurã</am></choice> <choice><ex>numerumque</ex><am>numerũq́;</am></choice>. </s> <s xml:space="preserve">Quæ <lb/>quidem definitio, natura ſua non eſt bona, quia tempus, neque numerus eſt, neque <lb/><choice><ex>etiam</ex><am>etiã</am></choice> eſt <choice><ex>menſura</ex><am>mẽſura</am></choice> motus <choice><ex>per</ex><am>ꝑ</am></choice> ſe, ſed <choice><ex>tantum</ex><am>tm̃</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> <choice><ex>accidens</ex><am>accidẽs</am></choice>, quia nihil eſt, <choice><ex>quod</ex><am>qđ</am></choice> numeret aut menſuret <lb/>aliud, quod non ſit <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> ſpeciei <choice><ex>cum</ex><am>cũ</am></choice> illo quod <choice><ex>menſuratur</ex><am>mẽſuratur</am></choice>, aut numero <choice><ex>circunſcribitur</ex><am>circunſcribit̃</am></choice>, <lb/>vt <choice><ex>exempli</ex><am>exẽpli</am></choice> gratia, nulla <choice><ex>vnquam</ex><am>vnquã</am></choice> ſuperficies <choice><ex>per</ex><am>ꝑ</am></choice> ſe numerabit aut <choice><ex>menſurabit</ex><am>mẽſurabit</am></choice> <choice><ex>lineam</ex><am>lineã</am></choice>, aut cor-<lb/>pus; </s> <s xml:space="preserve">neclinea ſuperficiem <choice><ex>aliquam</ex><am>aliquã</am></choice>, aut corpus: </s> <s xml:space="preserve">nec corpus <choice><ex>lineam</ex><am>lineã</am></choice> <choice><ex>aliquam</ex><am>aliquã</am></choice> aut ſuperfi-<lb/>ciem; </s> <s xml:space="preserve">Sed linea lineam menſurabit; </s> <s xml:space="preserve">ſuperficies ſuperficiem; </s> <s xml:space="preserve">& corpus corpus; </s> <s xml:space="preserve"><choice><ex>etianſi</ex><am>etiãſi</am></choice> <lb/>tam vna ex iis quantitatibus quàm altera ſit continua. </s> <s xml:space="preserve">Cum verò motus non ſit tem <lb/>pus, neque tempus ſit motus, ſed inter ſe maximè differant, ſequetur ex iis, alterum <lb/>nullo modo per ſe eſſe menſuram alterius, niſi per accidens. </s> <s xml:space="preserve">Et ſi alicui videtur, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>ad ſignificandam aliquam quantitatem motus, dicere huiuſmodi operationem dua-<lb/>rum horarum, aut duorum dierum, aut duorum annorum ſpatio completam eſſe, ſit <lb/>ponere tantum tempus: </s> <s xml:space="preserve">animaduertere debet hoc ſimpliciter non eſſe verum, quia <lb/>horarum, dierum. </s> <s xml:space="preserve">& annorum interualla, imaginatione <choice><ex>concipiuntur</ex><am>concipiũtur</am></choice> vt motus corpo-<lb/>rum cęleſtium, ſine quibus, neque anni, neque dies, neque horę exiſterent, <choice><ex>etiam</ex><am>etiã</am></choice> ſi om <lb/>nis motus ſit (vt ita dicam) locatus in tempore, ut corpus in loco, vnde motus motu, <lb/>& tempus tempore, non autem aliud ab alio menſuratur. </s> <s xml:space="preserve">Tempus ex neceſſita-<lb/>te (phyloſophicè tamen loquendo) res eſt æterna, motus non item, quia diuerſis mo <lb/>dis terminari poteſt & ceſſare, & interim dum ceſſabit quieſcet corpus, quod <lb/>primo mouebatur. </s> <s xml:space="preserve">nihilominus tamen, tempus continuabit curſum ſuum. </s> <s xml:space="preserve">Tempus <lb/>igitur potius locus motus erit dicendum, quàm numerus aut menſura eius, & tale eſt, <lb/>vt conſumatum uideatur à continuò quodam fluxu vnius inſtantis, quemadmodum <lb/>iam dixi in .38. capite meę gnomonicæ, & cum dico ab vno inſtanti, vnum in ſpecie, <lb/>& non in numero intelligo, quod à ſenſibus noſtris percipi non poteſt, <choice><ex>neque</ex><am>neq;</am></choice> etiam <lb/>notari, quia nouum ſemper inſtans nobis occurrit. </s> <s xml:space="preserve">& ſi aliquis aliquod <choice><ex>exemplum</ex><am>exemplũ</am></choice> (lar <lb/>go modo) incompræhenſibilitatis ipſius inſtantis deſideraret, imaginetur rotam ali <lb/>quam albam, in qua ſit nigrum aliquod punctum ſenſibile, aut è contra rotam <choice><ex>nigram</ex><am>nigrã</am></choice> <lb/>imaginetur, in qua ſit punctum album, quæ rota velociſſimè moueatur; </s> <s xml:space="preserve">huiuſmodi <lb/>punctum, nullo modo aſſignari poterit, magis ab una parte quàm ab altera; </s> <s xml:space="preserve">immo ſe <lb/>ſe nobis offeret ſemper in forma lineæ circularis. </s> <s xml:space="preserve">poſſumus aliquo modo etiam ſu-<lb/>mere exemplum à ſono, quia omnis chorda cuiuſlibet inſtrumenti muſici, dum ſo-<lb/>nus editur, tremit, unde huiuſmodi ſonus, appellari poteſt aggregatum aliquod ex <lb/>innumerabilibus ſonis. </s> <s xml:space="preserve">eodem modo ſe habet ſonus, quem ędunt campanę, & omnia <lb/>inſtrumenta tam naturalia, quàm artificialia, quæ quantò velocius <choice><ex>tremunt</ex><am>tremũt</am></choice>, tanto acu <lb/><choice><ex>tiorem</ex><am>tiorẽ</am></choice> generant ſonum, & quantò tardius, tantò grauiorem. </s> <s xml:space="preserve">Neque eſt quòd in ad-<lb/>mirationem ducamur, quòd ſenſui unum aliquod continuum appareat id, quod di-<lb/>ſcretorum eſt multitudo ( non putet tamen aliquis me negare continuitatem ſucceſ <lb/>ſiuam ipſius temporis) quod clare cognoſci poteſt à niue, aut à chryſtallo, aut à vi-<lb/>tro, aut à ſaccaro in minutiſſimas partes redacto, quæ continuam aliquam <choice><ex>albedinem</ex><am>albedinẽ</am></choice> <lb/>nobis ad inſpiciendum offerunt, quod nihil aliud eſt, quàm innumerabilis quædam <lb/>multitudo minutorum reflexorum. </s> <s xml:space="preserve"><choice><ex>Idem</ex><am>Idẽ</am></choice> dico de ſputo, & qualibet ſpuma, & quan- <pb facs="0195" n="183"/><fw type="head">DISPVTATIONES.</fw> to minutiora ſunt corpuſcula à quibus vt à ſpeculis reflectitur lumen, tantò magis ag <lb/><choice><ex>gregatum</ex><am>gregatũ</am></choice> illud <choice><ex>album</ex><am>albũ</am></choice> apparet. </s> <s xml:space="preserve">Hæc <choice><ex>autem</ex><am>autẽ</am></choice> exempla <choice><ex>cum</ex><am>cũ</am></choice> ſint, nec non largo modo ſumpta, <lb/><choice><ex>mirum</ex><am>mirũ</am></choice> non erit ſi claudicare <choice><ex>videbuntur</ex><am>videbũt̃</am></choice>. </s> <s xml:space="preserve">Sed ut ad <choice><ex>motum</ex><am>motũ</am></choice>, & <choice><ex>tempus</ex><am>tẽpus</am></choice> reuertamur ( quæ ſunt <lb/><choice><ex>continua</ex><am>cõtinua</am></choice> ſucceſſiua) Ariſtoteles in definiendo tempore, non reduxit in mentem, quod <lb/>ſcribit decimo metaphyſicę et .4. cap. ſecundo. libr. de cęlo, omnia videlicet, ab eo, <lb/>quod minimum eſt in ſuo genere, menſurari, & ex ſeipſo in phyſicorum libris, tem-<lb/>pus non eſt de genere motus; </s> <s xml:space="preserve">ergo eius ipſius rationum ui, tempus non erit menſura <lb/>motus, ſed motus quidem poteſt menſurare motum, videlicet velocior minus velo-<lb/>cem, & breuior longiorem; </s> <s xml:space="preserve">& <choice><ex>numerus</ex><am>numerꝰ</am></choice> <choice><ex>menſuratur</ex><am>menſurat̃</am></choice> numero, & tempus tempore in quan <lb/>tum longum eſt, aut breue, non in quantum velox, aut tardum; </s> <s xml:space="preserve">Nullum autem in-<lb/>conueniens ſequetur ſumendo tempus tam ptoportionale motui, quam locus cor-<lb/>pori, quia motus decem milliarium, quæ aliquis vnius horæ ſpatio conficiat, erit pro <lb/>portionalis corpori denſo, & motus vnius milliaris eadem hora peracti, proportiona <lb/>lis erit corpori raro; </s> <s xml:space="preserve">& quemadmodum corpus denſum occupat minus interuallum <lb/>loci, contra quam fiat in corpore raro: </s> <s xml:space="preserve">ſic etiam motus velox breuiori temporis ſpa-<lb/>tio peragetur, quam tardus.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Motum rectum eſſe continuum, uel dißentiente <lb/>Ariſtotele.</head> <head xml:space="preserve">CAP. XXIII.</head> <p> <s xml:space="preserve">ARiſtoteles .8. capi .8. phyſicorum ait impoſſibile eſſe aliquid per <choice><ex>lineam</ex><am>lineã</am></choice> rectam <lb/>nunc vno modo, nunc altero, ideſt eundo, & redeundo per dictam lineam in <lb/>extremis abſque quiete moueri. </s> <s xml:space="preserve">Id quod contrà poſſibile eſſe dico. </s> <s xml:space="preserve">Pro ſpecula-<lb/>tione cuius rei imaginemur circulum <seg type="var">.u.a.n.</seg> motu continuo circa centrum <seg type="var">.o.</seg> in <choice><ex>quam</ex><am>quã</am></choice> <lb/>libet partem, aut <choice><ex>dextram</ex><am>dextrã</am></choice>, aut <choice><ex>ſiniſtram</ex><am>ſiniſtrã</am></choice> ferri; </s> <s xml:space="preserve">& imaginemur <choice><ex>punctum</ex><am>pũctum</am></choice> <seg type="var">.b.</seg> extra ipſum, ubi <lb/>magis nobis videbitur, à quo ducantur duæ lineæ recte <seg type="var">.b.u.</seg> et <seg type="var">.b.n.</seg> contiguæ ipſi cir-<lb/>culo in punctis <seg type="var">.u.</seg> et <seg type="var">.n</seg>. </s> <s xml:space="preserve">Imaginatione quoque inter has duas lineas, alteram quæ ſit <seg type="var">.<lb/>u.n.</seg> aut <seg type="var">.c.d.</seg> aut <seg type="var">.e.f.</seg> aut <seg type="var">.g.h.</seg> conſtituamus in quali <lb/> <ptr xml:id="fig-0195-01a" corresp="fig-0195-01" type="figureAnchor"/> bet parte, ſumemus etiam punctum <seg type="var">.a.</seg> circun-<lb/>ferentiæ dicti circuli, à quo vſque ad <seg type="var">.b.</seg> lineam <seg type="var">.<lb/>b.a.</seg> imaginemur <choice><ex>fixam</ex><am>fixã</am></choice> in <seg type="var">.b.</seg> ſed quod remanat mo <lb/>bile, ſecundum quod mouebitur punctum <seg type="var">.a.</seg> vn-<lb/>de <choice><ex>aliquando</ex><am>aliquãdo</am></choice> hæc linea erit eadem cum <seg type="var">.b.u.</seg> & ali <lb/>quando cum <seg type="var">.b.n.</seg> & aliquando ab <seg type="var">.b.u.</seg> verſus <seg type="var">.b.<lb/>n.</seg> proficiſcetur, & aliquando ab <seg type="var">.b.n.</seg> verſus <seg type="var">.b.u.</seg> <lb/>vt accidit lineæ directionis, & retrogradationis <lb/>planetarum, vnde circulus <seg type="var">.u.a.n.</seg> erit vt epiciclus <lb/>et <seg type="var">.b.</seg> vt terræ centrum. </s> <s xml:space="preserve">Clarum nunc erit, quòd <lb/>quando linea <seg type="var">.b.a.</seg> eadem erit cum <seg type="var">.b.u.</seg> aut cum <lb/><seg type="var">b.n.</seg> non quieſcet, quia in inſtanti reuertetur, quia <lb/><seg type="var">b.u.</seg> et <seg type="var">.b.n.</seg> in puncto, <choice><ex>dictum</ex><am>dictũ</am></choice> circulum tangunt, & <lb/>dicta <seg type="var">.b.a.</seg> interſecabit ſemper aliquam ex dictis <lb/><seg type="var">u.n.</seg> aut <seg type="var">.c.d.</seg> aut <seg type="var">.e.f.</seg> aut <seg type="var">.g.h.</seg> quod interſectionis <lb/>punctum ſit <seg type="var">.t</seg>. </s> <s xml:space="preserve">Imaginemur nunc quod <choice><ex>ſecundum</ex><am>ſecũdũ</am></choice> <lb/>punctum <seg type="var">.t.</seg> aliquid per aliquam ex dictis lineis <pb facs="0196" n="184"/><fw type="head">IO. BABPT. BENED.</fw> moueatur, clarum erit quod tale aliquid, nunquam quieſcet, etiam ſi ſit in quouis ex <lb/>tremo. </s> <s xml:space="preserve">Ariſtotelis igitur opinio, tuta non eſt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0195-01" corresp="fig-0195-01a"> <graphic url="0195-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Idem uir grauisſimus an bene ſenſerit de motibus corporum <lb/>uiolentis & natur alibus.</head> <head xml:space="preserve">CAP. XXIIII.</head> <p> <s xml:space="preserve">ARiſtoteles in fine .8. phyſicorum ſentit corpus per vim motum, & ſeparatum à <lb/>primo mouente, moueri, aut motum eſſe per aliquod tempus ab aere, aut ab <lb/>aqua, quæ ipſum <choice><ex>ſequuntur</ex><am>ſequũtur</am></choice>. quod fieri non poteſt; </s> <s xml:space="preserve">quia imo aer, qui in locum defer-<lb/>tum à corpore ſubintrat ad fugandum vacuum, non ſolum hoc corpus non impellit, <lb/>ſed potius id cohibet à motu, quia aer per vim à corpore ducitur retrò, & diuiſus à <lb/>parte anteriori à dicto corpore, reſiſtit ſimiliter, & quantum dictus aer in dicta parte <lb/>condenſatur, tantum in poſteriori rarefit, vnde per vim ſeſe rarefaciens non permit-<lb/>tit, vt dictum corpus cum ea velocitate fugiat, cum qua aufugeret, quia omne agens <lb/>in agendo patitur. </s> <s xml:space="preserve">Quamobrem cum aer à dicto corpore rapiatur, corpus quoque <lb/>ipſum ab aere rapitur. </s> <s xml:space="preserve">Huiuſmodi autem rarefactio aeris, naturalis non eſt, ſed vio <lb/>lenta; </s> <s xml:space="preserve">& hanc ob cauſam reſiſtit, & ad ſe trahit, ſed non ſufferente natura, vt inter <choice><ex>vnum</ex><am>vnũ</am></choice> <lb/>& aliud ex dictis corporibus reperiatur vacuum; </s> <s xml:space="preserve">iccirco funt hæc ſemper contigua, <lb/>& mobile corpus aerem deſerere cum nequeat, eius velocitas impeditur. </s> <s xml:space="preserve">Huiuſmo <lb/>di igitur corporis ſeparatim à primo mouente velo citas oritur à quadam naturali im <lb/>pręſſione, ex impetuofitate recepta à dicto mobili, quæ impręsſio & impetuoſitas, <lb/>in motibus rectis naturalibus continuò creſcit, cum perpetuò inſe cauſam <choice><ex>mouentem</ex><am>mouẽtẽ</am></choice>, <lb/>ideſt propenſionem eundi ad locum ei à natura asſignatum habeat. </s> <s xml:space="preserve">Ariſto .8. cap. <lb/>primi lib. de cœlo, dicere non deberet <choice><ex>qui</ex><am>ꝗ</am></choice> quantò propius accedit corpus ad <choice><ex>terminum</ex><am>terminũ</am></choice> <lb/>ad quem, tantò magis ſit velox; </s> <s xml:space="preserve">ſed potius, <choice><ex>qui</ex><am>ꝗ</am></choice> quantò longius diſtat à termino à quò <lb/>tantò velocius exiſtit. </s> <s xml:space="preserve">quia tantò maior fit femper impræsfio, quantò magis moue-<lb/>tur naturaliter corpus, & continuò nouum impetum recipit, cum in fe motus caufam <lb/>contineat, quæ eſt inclinatio ad locum ſuum eundi, extra quem per vim confiftit. <lb/></s> <s xml:space="preserve">Neque etiam rectè ſeripſit Ariſto .9. cap. lib. 8. phyficorum et .2. lib. primi de cœlo <lb/>eſſe aliquem motum ex recto & circulari mixtum, <choice><ex>qui</ex><am>ꝗ</am></choice> omninò imposſibile eſt.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Motum rectum & natur alem non eſſe primo & per ſe <lb/>quicquid Ariſtoteli uiſum ſit.</head> <head xml:space="preserve">CAP. XXV.</head> <p> <s xml:space="preserve">MOtus rectus corporum naturalium ſurſum, aut dcorfum, non eft naturalis pri <lb/>mò & per ſe, quia motus naturalis perpetuus eſt, aut vt melius dicam, inceſ-<lb/>fabilis, & alius eſſe non poteft quàm circularis, <choice><ex>nullaque</ex><am>nullaq́;</am></choice> pars cum fuo toto coniun-<lb/>cta, alium motum naturalem habere poteft, quàm eum, qui eft totius. </s> <s xml:space="preserve">fi autem à ſuo <lb/>toto diuulfa atque difiuncta fit, <choice><ex>libereque</ex><am>libereq́;</am></choice> vagetur, ſpontè, & quàm breuisſima poteft <lb/>via, ad locum, <choice><ex>ſuitotius</ex><am>ſuitotiꝰ</am></choice> à natura ſtatutum proficiſcitur. </s> <s xml:space="preserve">hic motus primò, & per fe di-<lb/>cti corporis, naturalis non eft, cum à caufa naturæ fuæ contraria fit generatus, ideft, <pb facs="0197" n="185"/><fw type="head">DISPVTATIONES.</fw> ab co quod fit extra ſuum locum, vbi contra naturam ſuam reperitur. </s> <s xml:space="preserve">Vnde hu-<lb/>iuſmodi motus, partim & non omninò, naturalis eft. </s> <s xml:space="preserve">Is autem proprius eſt & natura <lb/>lis motus, qui dicti corporis eſſentiam conſeruat. </s> <s xml:space="preserve">hoc autem non præſtat hic rectus, <lb/>cum deſtruat, ergò hic motus primò & per ſe naturalis non eft.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Omne corpus eſſe in loco proprio graue, ut Aristoteli placuit, <lb/>non eft admittendum.</head> <head xml:space="preserve">CAP. XXVI.</head> <p> <s xml:space="preserve">ARift .4. cap. lib. 4. de cęlo fic ſcribit.</s> </p> <p> <s xml:space="preserve">Suo enim in loco grauitatem habent omnia præter ignem, fignum cuius eft <lb/>vtrem inflatum plus ponderis, quam vacuum habere, & c.</s> </p> <p> <s xml:space="preserve">Quo in loco, manifeftè indicat ſe caufam nec grauitatis, nec leuitatis corporum <lb/>naturalium nofce, quæ eft denfitas auto raritas corporis grauis, aut leuis, maior denſi-<lb/>tate, aut raritate medij permeabilis, in quo reperitur.</s> </p> <p> <s xml:space="preserve">Exemplum <choice><ex>qui</ex><am>ꝗ</am></choice> ipſe de vtre inflato proponit, debuiſſet ſaltem ei oculos ad verita-<lb/>tem, quæ clarisſimè fulget, inſpiciendum aperire. </s> <s xml:space="preserve">Verisſimum eſt, vtrem inflatum <lb/>plus ponderis habere quàm vacuum, aut quando aer in eo non eft per vim inclufus.</s> </p> <p> <s xml:space="preserve">Ratio autem huius rei eft, quia quando inflatus eft, ea quantitas aeris, in eum <lb/>per vim iniecti, minorem occupat locum, quàm ſi eidem liberè vagari permit-<lb/>teretur, vnde violenter, quodam modo, con denfata eft, & quia corpus denfum in <lb/>minus denfo, femper deſcendit, & minus denſum in magis denſo aſcendit. </s> <s xml:space="preserve">Hanc ob <lb/>caufam vter inflatus plenus corpore magis denſo, quàm eft medium quod eum cir-<lb/>cundat, deſcendit, non quia aer inaere, aut aqua in aqua fit grauis.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Haud admittendam opinionem Principis Peripateticorum <lb/>de circulo, & ſpbæra.</head> <head xml:space="preserve">CAP. XXVII.</head> <p> <s xml:space="preserve">CVm Ariftoteles fenſerit circulum eſſe figurarum ſuperficialium <choice><ex>primam</ex><am>primã</am></choice>, & <choice><ex>ſphae ram</ex><am>ſphęrã</am></choice> eſſe <choice><ex>primam</ex><am>primã</am></choice> <choice><ex>corporearum</ex><am>corporearũ</am></choice> <choice><ex>prope</ex><am>ꝑꝑ</am></choice> <choice><ex>earum</ex><am>earũ</am></choice> periferias, decipitur. </s> <s xml:space="preserve">Sunt enim vltimæ, <lb/>non primæ. </s> <s xml:space="preserve">Sunt quidem (in quò rectè ſentit) perfectè, licet rationem huius rei non <lb/>nouerit. </s> <s xml:space="preserve">Nam <choice><ex>centrum</ex><am>centrũ</am></choice> cuiuſlibet rei, eiuſdem rei <choice><ex>principium</ex><am>principiũ</am></choice> eft, & eę figurę, quæ ipſum <lb/>æqualiter circundant, poſſunt appellari perſectæ, ſiue ſint ſuperficiales, ſiue corpo-<lb/>reæ, & ècontrà illæ, quæ contrario modo ſe habent, imperfectæ. </s> <s xml:space="preserve">Quòd autem per-<lb/>ſectum eſt, licet natura fit primum, eſt tamen vltimum generatione. </s> <s xml:space="preserve">Sed quando <lb/>Ariftoteles duas dictas figuras pronuntiauit primas, vt perfectas, prioritate ſcilicet <lb/>ea, quæ oritur à perfectione, verum dixit; </s> <s xml:space="preserve">fed quando de figuris ſuperficialibus lo-<lb/>quens, vult circulum effe primum, quia ab vna <choice><ex>tantum</ex><am>tãtum</am></choice> linea terminetur; </s> <s xml:space="preserve"><choice><ex>non</ex><am>nõ</am></choice> minus pro <lb/>circulo, quam pro oxigonia ſeu elipſi, aut cucurbitali, aut aliis multis figuris ab vna <lb/>tantum linea terminatis concludit. </s> <s xml:space="preserve">Neque etiam hæc ratio perfectionem circuli <choice><ex>mon</ex><am>mõ</am></choice> <lb/>ſtrat, quia aliæ figuræ, à lineis curuis terminatę, eandem conditionem fortiuntur. <lb/></s> <s xml:space="preserve">Circulus ſphę<choice><ex>raque</ex><am>raq́;</am></choice>, non ex vno ſolo angulo recto conſtant, vt idem Ariftoteles putat <pb facs="0198" n="186"/><fw type="head">IO. BAPT. BENED.</fw> cap .4. lib. 4. de cęlo, etiam fi triangulus ex duobus angulis rectis conſurgat, ſed ſunt <lb/>figurę infinitorum angulorum rectorum, & hanc ob cauſam à me dicuntur vltimæ & <lb/>perfectę, quia infinito nihil addi poteſt. </s> <s xml:space="preserve">Numerus angulorum rectorum circuli, eft <lb/>minor duplo infinito per duo infinita angulorum contingentiæ, quæ duo infinita mi <lb/>nora funt quouis angulo acuto rectilineo, & numerus angulorum rectorum <choice><ex>folidorum</ex><am>folidorũ</am></choice> <lb/>ſphęræ, minor eft quadruplo infinito per .4. infinita angulorum ſolidorum <choice><ex>contingen- tiæ</ex><am>cõtingen-tiæ</am></choice>, quæ .4. infinita, minora ſunt quouis angulo ſolido acuto terminato à tribus pla-<lb/>nis. </s> <s xml:space="preserve">Triangulus inter figuras planas ſuperſiciales eft primus, & circulus vltimus; </s> <s xml:space="preserve">& <lb/>pyramis quadrilatera, inter corpora eft prima, & ſphęra vltima.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Occultam fuiße grauisſimo Stagirit & canſam ſcintilla-<lb/>tionis ſtellarum.</head> <head xml:space="preserve">CAP. XXVIII.</head> <p> <s xml:space="preserve">VBi Ariſtoteles ait ſcintillationem ſtellarum ſieriratione aſpectus @oſtri ob, ma <lb/>ximam diſtantiam, maximum errorem committit, vt etiam facid quum putat <lb/>vifionem fieri extramittendo, contra id, quod alio loco, immo contra veritatem ip <lb/>ſam afferuit. </s> <s xml:space="preserve">Scintillatio ergo ſtellarum, neque aſpectus noſtri ratione, neque ali-<lb/>cuius mutationis earundem ſtellarum, ſed ab inæqualitate motus corporum diapha <lb/>norum mediorum naſcitur, <choice><ex>quemadmodum</ex><am>quẽadmodum</am></choice> clarè cernitur, quòd fi inter aliquod obie <lb/>ctum, & nos, aliquis ſumus, qui aſcendat, intercefferit, videbimus obiectum illud qua <lb/>ſi tremere. </s> <s xml:space="preserve">Hoc autem tantò magis fiet, quantò magis diſtabit obiectum ab ipſo fu <lb/>mo; </s> <s xml:space="preserve">vnde admirationi locus non erit, fi ftellas fixas magis ſcintillare, quam errantes <lb/>cernamus. </s> <s xml:space="preserve">Lumen ſtellæ ad oculum noſtrum accedens, perpetuò per diuerfas dia-<lb/>phaneitates penetrat, medio continuorum motuum corporum mediorum, vnde <lb/>continuò eorum lumen variatur, & hoc in <choice><ex>longinquis</ex><am>lõginquis</am></choice> magis, quàm in propinquis ſtel <lb/>lis apparet, quemadmodum ab exemplo de fumo allato, & etiam ab aliquibus vi-<lb/>tris ex ſuperficie non plana, ſed irregulari conſtantibus, quilibet cognoſcere poteft.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Daricontinuum infinitum motum ſuper rectam at que <lb/>finitam lineam.</head> <head xml:space="preserve">CAP. XXIX.</head> <p> <s xml:space="preserve">OMnes hactenus ſenſerunt imposfibile eſſe dari per <choice><ex>imaginationem</ex><am>imaginationẽ</am></choice> motum con-<lb/>tinuum & <choice><ex>perpetuum</ex><am>perpetuũ</am></choice> <lb/> <ptr xml:id="fig-0198-01a" corresp="fig-0198-01" type="figureAnchor"/> ſuper vnam lineam rectam <lb/>finit: </s> <s xml:space="preserve">in quo <choice><ex>tantum</ex><am>tñ</am></choice> decipiuntur. <lb/></s> <s xml:space="preserve">Imaginemur <choice><ex>ion</ex><am>iõ</am></choice> duas lineas <lb/>parallelas <seg type="var">.a.b.</seg> et <seg type="var">.t.x.</seg> <choice><ex>quarum</ex><am>quarũ</am></choice> <lb/><seg type="var">b.a.</seg> fit <choice><ex>infinita</ex><am>ĩfinita</am></choice> à qualibet par <lb/>te, & in ea imaginemur pun <lb/>ctum <seg type="var">.a.</seg> moueri continuò ad <lb/>quam voluerimus partem, <lb/>& <pb facs="0199" n="187"/><fw type="head">DISPVTATIONES.</fw> & in linea <seg type="var">.t.x.</seg> imaginemur punctum fixu@, quod fit <seg type="var">.c.</seg> imaginemur etiam inter <seg type="var">.c.</seg> @. <lb/>a. vnam lineam rectam <seg type="var">.c.a.</seg> & inter duas parallelas dictas <seg type="var">.r.x.</seg> fixam, & motus punct@i <lb/>fit ab <seg type="var">.b.</seg> verfus <seg type="var">.a.</seg> ita ut <seg type="var">.c.a.</seg> fecet <seg type="var">.r.x.</seg> in puncto <seg type="var">.i.</seg> quod interfectionis punctum mo-<lb/>uebitur ab <seg type="var">.r.</seg> verfus <seg type="var">.x.</seg> continuò, in tempore infinito, neque vnquam idem erit cum <lb/>puncto <seg type="var">.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0198-01" corresp="fig-0198-01a"> <graphic url="0198-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head rend="italics" xml:space="preserve">Non eſſe ſolis calorem à motu localι ipſius corporis ſolaris, <lb/>ut Ariſtoteli placuit.</head> <head xml:space="preserve">CAP. XXX.</head> <p> <s xml:space="preserve">JD nullo planè modo eſt admittendum quod Ariftoteles credidit calorem folis à <lb/>motu locali ipſiuſmet corporis folaris, & non à lumine, prouenire, quemadmo-<lb/>dum manifeftè aſſerit primo metheororum cap .3. circa finem fic fcribens.</s> </p> <p> <s xml:space="preserve">Vtigitur repor gignatur atque calor, folis latio duntaxat, ſatis eſt eſſicere, & c. ſed <lb/>cap .7. lib. 2. de cælo fic ſeribit, Caliditas autem ab ipſis, <choice><ex>lumenque</ex><am>lumenq́;</am></choice> ideo fit, quia aer <lb/>ab illorum motione fricatur.</s> </p> <p> <s xml:space="preserve">Vbi non folum oftendit fe opinari, quòd motus corporum cœleſtium fit caufa ca <lb/>loris, ſed eriam luminis, paulò autem poſt dicit, ſuperiorum autem corporum vnum <lb/>quodque fertur in ſphæra, vt ipſa quidem non igniantur. </s> <s xml:space="preserve">Opinio profecto abfur-<lb/>da. </s> <s xml:space="preserve">Nam cùm corpus ſolate fixum fit in ſpisfitudine ſui orbis deferentis, fe-<lb/>cundum communem opinionem, non mouetur per fe, ſed accidentaliter, cum ſei-<lb/>licet fertur à dicto ſuo orbe, vnde fieri poteſt, vt in motu fui orbis, nullum ex <lb/>orbibus fuorum deferentium augis fricet, fed fi fricaret, id faceret mediante vno fo <lb/>lo puncto, vt cuilibet, aliquantulum in mathematicis verfato patet. </s> <s xml:space="preserve">Quam ob cau <lb/>ſam, rationi <choice><ex>confentaneum</ex><am>cõfentaneum</am></choice> non eſſet credere, quòd tantùm caloris gigneretur. </s> <s xml:space="preserve">Quod <lb/>tamen fi posſibile eſſet, quid ergo fricatio ſuperficierum orbis ſui, cum iis, quæ funt <lb/>deferentium augis efficeret? </s> <s xml:space="preserve"><choice><ex>Quando</ex><am>Quãdo</am></choice> tamen hoc fieret, vt ſcilicet à fricatione fuper <lb/>ficierum procederet calor, nil planè diferiminis inter hyemen, & æftatem intercede <lb/>ret, nec inter calorem diei, & noctis, nec inter unam horam, aut alteram; </s> <s xml:space="preserve">fed fecun-<lb/>dum Ariftotelis rationes, Venus, <choice><ex>Mercuriusque</ex><am>Mercuriusq́;</am></choice>, magis calefacere quam fol de <choice><ex>berent</ex><am>berẽt</am></choice>, <lb/>cum ita ſint veloces vt ipſe Sol, & eodem magis propinqua terræ. </s> <s xml:space="preserve">Verum Ari-<lb/>ſtotelis <choice><ex>temporibus</ex><am>tẽporibus</am></choice>, <choice><ex>nullum</ex><am>nullũ</am></choice> <choice><ex>alium</ex><am>aliũ</am></choice> <choice><ex>planetam</ex><am>planetã</am></choice> quam folem <choice><ex>putabant</ex><am>putabãt</am></choice> philofophi ſupra <choice><ex>Lunam</ex><am>Lunã</am></choice> eſ-<lb/>ſe. </s> <s xml:space="preserve">Atque etiam <choice><ex>contigeret</ex><am>cõtigeret</am></choice> menfe Decembri, quam Iunio, magis inualeſceret calor, <lb/>cum huiuſmodi menſe ſolad nos propius accedat, quam menfe Iunii. per differen-<lb/>tiam maiorem diametro regionis elementaris, (nam folaris eccentricitas maior eft <lb/>ſemidiametro <choice><ex>elementaris</ex><am>elemẽtaris</am></choice> regionis) non confiderans Ariftoteles differentiam ca-<lb/>loris, quæ naſcitur ex eo, <choice><ex>qui</ex><am>ꝗ</am></choice> Sol aut altius ſupra orizontem excurrat, aut infra <choice><ex>eundem</ex><am>eundẽ</am></choice> <lb/>feratur; </s> <s xml:space="preserve">neque eam, quę à longitudine, aut breuitate diei proficiſcitur. </s> <s xml:space="preserve">Sed quia Ari <lb/>ſtoteles eodem cap tertio Metheororum intelligit de motu rapto, ideſt diurno, ſiue <lb/>dicamus vniuerfali, hinc ſequi deberet, <choice><ex>qui</ex><am>ꝗ</am></choice> Sol maiorem caloris uim menſe Martij & <lb/>Septembris, quàm aliis menfibus, profunderet, quia in iiſdem temporibus, ſol virtu <lb/>te huiuſmodi motus velocior exiftat, quàm alio quolibet tempore anni, cum tunc <lb/>per æquatorem circuũoluatur. </s> <s xml:space="preserve">Multa quoque alia incommoda ſequerentur ſi Ari <lb/>ſtorelis rationes admitteremus. </s> <s xml:space="preserve">Sed clarè uidemus, mediante refl exione aut refra-<lb/>ctione radiorum folarium, <choice><ex>qui</ex><am>ꝗ</am></choice> vniente ſeſe lumine, unitur quoque, & augetur calor, <lb/>atque omnis res ad comburendum apta accenditur, & inflammatur. </s> <s xml:space="preserve">In lumine igi- <pb facs="0200" n="188"/><fw type="head">IO. BAPT BENED.</fw> tur continetur calor, & non in motu ipſius ſolis, & ita in lumine ſedem habet, ut fi ſol <lb/>quieſceret, neque in orbe ſuo circumager etur, infęliciſſima eſſet ea regio, in cuiu; <lb/></s> <s xml:space="preserve">Zenith ipſe reperiretur.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Vnde caloris ſolis prode at incrementum & state, et byeme <lb/>decrementum.</head> <head xml:space="preserve">CAP. XXXI.</head> <p> <s xml:space="preserve">CVm capite ſuperiore oſtenderim calorem ſolis non aliunde, quàm à lumine <lb/>prouenire, oſtendam nunc ex ordine, ex quot, <choice><ex>quibusque</ex><am>quibusq́;</am></choice> cauſis oriatur magna <lb/>differentia eius caloris æſtatis ad hyemem, quarum nonnullæ ab antiquis obſerua-<lb/>tæ fuerunt, aliæ autem à nemine, quod ſciam. </s> <s xml:space="preserve">Sunt autem quinque ad minus eæ cau <lb/>ſæ, quarum vna eft diuturna folis mora, tempore æſtatis ſupra orizontem, quæ cau-<lb/>ſa ab antiquis pofita, & citata fuit. </s> <s xml:space="preserve">Aliam quoque huius rei cauſam iidem antiqui <lb/>dicebant eſſe propinquitatem ſolis noftro Zenith, ſed hæc cauſa immediata non <lb/>eſt, quia ab ea tres caufæ immediatæ naſcuntur; </s> <s xml:space="preserve">quarum vna eft maior unio radij re <lb/>flexi cum radio incidenti; </s> <s xml:space="preserve">ſecunda maior quantitas luminis in ſuperficie terrę; <lb/></s> <s xml:space="preserve">tertia, minor <choice><ex>reſiſtentia</ex><am>reſiſtẽtia</am></choice> vaporum ipſi radio luminoſo facta; </s> <s xml:space="preserve">quarta verò eft impresfio <lb/>caloris facta in terra, quæ cum aliis caufis coniuncta calorem adauget. quæ quidem <lb/>caufæ nemini adhuc, quod fciam, in <choice><ex>mentem</ex><am>mentẽ</am></choice> venerunt. </s> <s xml:space="preserve">Quòd autem attinet ad ma-<lb/>iorem coniunctionem radii reſlexi cum incidente, quiſque, uel ſaltem mediocriter <lb/>in cathoptricę cognitione verſatus hoc verum eſſe cognoſcet. </s> <s xml:space="preserve">Vt hoc tamen in-<lb/>noteſcat facilius. </s> <s xml:space="preserve">Imaginemur <seg type="var">.q.p.</seg> et <seg type="var">.b.d.</seg> eſſe duas particulas ęquales ſuperficiei <lb/>ipfius terræ, ſuper quas cadant duo radii luminofi ſolis <seg type="var">.e.q.</seg> et <seg type="var">.n.d.</seg> quorum <seg type="var">.e.q.</seg> fit ad <lb/>modum obliquus, et <seg type="var">.n.d.</seg> quaſi perpendicularis, vnde radii reſlexi <seg type="var">.p.a.</seg> et <seg type="var">.b.u.</seg> aſcen <lb/>dent cum angulis æqualibus eis, qui funt ſuorum cadentium, cum omnis angulus re-<lb/>flexionis femper æqualis ſit angulo ſuæ incidentię, vt cuilibet in cathoptrica, vel me <lb/>diocriter verfato pater. </s> <s xml:space="preserve">Mixtio autem primorum obliquorum erit <seg type="var">.q.o.p.</seg> & ea, quæ <lb/>eft minus obliquorum <seg type="var">.b.i.d.</seg> quorum duorum triangulorum nullus unquam erit, qui <lb/>dubitare posfit <seg type="var">.q.o.p.</seg> non eſſe minorem <seg type="var">.b.i.d.</seg> cum anguli <seg type="var">.q.</seg> et <seg type="var">.p.</seg> trianguli <seg type="var">.q.o.p.</seg> a-<lb/>cutiores ſint angulis <seg type="var">.b.</seg> et <seg type="var">.d.</seg> trianguli <seg type="var">.b.i.d.</seg> ex ſuppoſito. </s> <s xml:space="preserve">Quòd uero attinet ad ma <lb/>iorem quantitatem luminis ſuper terræ ſuperſiciem; </s> <s xml:space="preserve">Imaginemur radium <seg type="var">.a.q.</seg> cuius <lb/>reſpectu etiam imagine mur duos ſuperficiei terræ ſitus, quorum vnus fit <seg type="var">.q.o</seg>: cui di-<lb/>ctus radius fit perpendicularis, & alter <seg type="var">.q.p.</seg> cui radius <seg type="var">.a.q.</seg> ex obliquo incidat. </s> <s xml:space="preserve">Ima-<lb/>ginemur ergo triangulum <seg type="var">.q.o.p.</seg> cuius angulus <seg type="var">.o.</seg> rectus eſt ex ſuppoſito, unde <seg type="var">.q.o.</seg> <lb/>minor erit <seg type="var">.q.p.</seg> ex .18. primi Euclidis. </s> <s xml:space="preserve">hinc fit, vt ſuper <seg type="var">.q.o.</seg> cadat vniuerſum lumen, <lb/>quod ſuper <seg type="var">.q.p.</seg> diffunditur. </s> <s xml:space="preserve">Sit <seg type="var">.q.u.</seg> æqualis <seg type="var">.q.o.</seg> & fit imaginatione protracta <seg type="var">.u.n.</seg> <lb/>æquidiftans <seg type="var">.p.o.a.</seg> vnde <seg type="var">.q.u.</seg> illuminata erit à radio <seg type="var">.n.q.</seg> minore radio <seg type="var">.a.q.</seg> ergo mi-<lb/>nus calida erit ſuperficies <seg type="var">.q.u.</seg> ipſius terræ, quàm.q.o. quia maius lumen in ſe maio-<lb/>rem calorem includit: </s> <s xml:space="preserve">quod manifeſtè apparet in radiorum vnione mediante refle-<lb/>xione, aut refractione. </s> <s xml:space="preserve">Sed quod attinet ad minorem refiſtentiam vaporum ad ip-<lb/>ſum radium luminoſum, etfi primo capite meæ Gnomonicæ leuiter id attigerim, ni <lb/>hilominus tamen, & idem ipſum hoc loco proponam. </s> <s xml:space="preserve">Denotetur, exempli gratia, <lb/>ſuperficies terræ ab <seg type="var">.o.g.</seg> et ea, quæ eft vaporum ab <seg type="var">.n.a.</seg> ſupponatur etiam ſol in fitu.</s> <pb facs="0201" n="189"/> <fw type="head">DISPVTATIONES.</fw> <s xml:space="preserve">q. qui ſit Zenith <choice><ex>puncti</ex><am>pũcti</am></choice> <seg type="var">.o.</seg> & <choice><ex>etiam</ex><am>etiã</am></choice> in <seg type="var">.p.</seg> ipſi <choice><ex>orizonti</ex><am>orizõti</am></choice> propinquus, aut extra Zenith, cuius <lb/>duos radios <seg type="var">.q.o.</seg> et <seg type="var">.p.o</seg>. </s> <s xml:space="preserve">Imaginemur, quorum duæ partes <seg type="var">.a.o.</seg> et <seg type="var">.n.o.</seg> erunt aliquo <lb/>modo ab ipſis vaporibus offuſcatæ, ſed <seg type="var">.o.n.</seg> breuior eſt <seg type="var">.o.a.</seg> ex .7. lib. 3. Eucli. </s> <s xml:space="preserve">mino-<lb/>rem ergo reſiſtentiam habebit à vaporibus ſol in Zenith, quàm extra cundem com -<lb/>morans, & quantò longius erit idem ab ipſo Zenith, tanto maiorem reſiſtentiam à <lb/>dictis vaporibus inferri ex eadem .7. lib. 3. Eucli. dicemus.</s> </p> <figure place="here"> <graphic url="0201-01"/> </figure> <figure place="here"> <graphic url="0201-02"/> </figure> </div> <div type="section"> <head rend="italics" xml:space="preserve">Nullum corpus ſenſus expers à ſono offendi, præterquam <lb/>Aristoteles crediderit.</head> <head xml:space="preserve">CAP. XXXII.</head> <p> <s xml:space="preserve">POſſe ſonum corpus aliquod, quod ſenſu ſit deſtitutum, vt Ariſtoteles .9. cap. li-<lb/>br .2. de cælo putauit, offendere, eſt falſum.</s> </p> <p> <s xml:space="preserve">Corpus enim non niſi à corpore poteſt lædi, non ergo à ſono, cum ſonus corpus <lb/>non ſit. </s> <s xml:space="preserve">Sed aer, & ignis, cum è contra ſint corpora, hoc facilè præſtare poſſunt im-<lb/>plendo aliquem locum velociter ad excludendum vacuum; </s> <s xml:space="preserve">vnde generatur ſonus. <lb/></s> <s xml:space="preserve">Quod hucuſque à nemine animaduerſum fuiſſe comperio.</s> </p> <pb facs="0202" n="190"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head rend="italics" xml:space="preserve">Pytagoreorum opinionem de ſonitu corporum cælestium non <lb/>fuiſſe ab Aristotele ſublatam.</head> <head xml:space="preserve">CAP. XXXIII.</head> <p> <s xml:space="preserve">SEnſerunt Pythagorici orbes cæleſtes dum circunuoluuntur, non autem corpora <lb/>ſtellarum ſolum, æd ere ſonu. </s> <s xml:space="preserve">Quibus dum Ariſtoteles contradicere cogitat, <lb/>maximè fauet. </s> <s xml:space="preserve">Eatamen opinio è phyloſophorum ſcholis eſt explodenda, quia aut <lb/>orbes ſunt ſibi ipſis contigui, aut inuicem diſtantes: </s> <s xml:space="preserve">ſi ab inuicem diſtant (quod <lb/>nemo adhuc conceſſit, quia hac ratione vacuum introduceretur) clarum eſt, quod <lb/>cum ſe minime tangant, ſonum edere nequeunt: </s> <s xml:space="preserve">Si verò eorum vnus alteri ſit conti <lb/>guus, <choice><ex>neque</ex><am>neq;</am></choice> etiam ab ipſis ſonus reſultare poterit, quia pro certo putandum eſt, <choice><ex>ipſorum</ex><am>ipſorũ</am></choice> <lb/>ſuperficies tam politas eſſe, ac lenas, vt nihil omnino aſperitatis, aut inæqualitatis <lb/>contineant. </s> <s xml:space="preserve">Vt exempli gratia, ſi aliquis duo ſpecula plana inuicem confricaret, nul <lb/>lum planè ſonum audiret, ſed ſi hoc faceret cum duabus ſuperficiebus a ſperis, <choice><ex>ſonum</ex><am>ſonũ</am></choice> <lb/>perſentiret, & tanto maiorem; </s> <s xml:space="preserve">quantò aſperiores eſſent dictæ ſuperficies, & qui vult <lb/>vtarcus lirę, ex corda ſonum eliciat, colophonia dictum arcum illinet, vt aſperiorem <lb/>reddat. </s> <s xml:space="preserve">Neceſſarium quoque eſt vt tremat ſiue trepidet corpus, quod <choice><ex>ſonum</ex><am>ſonũ</am></choice> edere <lb/>debet; </s> <s xml:space="preserve">Neque etiam abſque aere ſonus efficipotelt, quia aer ſonat ingrediendo <lb/>velociter ad implendum locum, vt non remaneat vacuus. </s> <s xml:space="preserve">Sed ſupponendo in æche <lb/>rea regione neque aerem eſſe, neque corpus aliquod fluidum, clarè patebit orbes <lb/>cœleſtes ex ſeſe nullum emittere ſonum. </s> <s xml:space="preserve">Idem affirmo de fricatione ſuperficiei con <lb/>cauæ infimi orbis lunaris cum conuexa materiæ à dicto orbe contentæ, ſuperioribus <lb/>rationibus fultus, vt etiam experientia à corpore aliquo fluido, quod in alio velociſ <lb/>ſimè moueretur deſumpta fretus, cuius corporis ſuperficies tamen lenis eſſet, à quo <lb/>ſonus non gigneretur. </s> <s xml:space="preserve">Et non minus dicere poſſum, corpus fluidum moueri in con-<lb/>tinente loco immobili, quam dictum corpus continens illud eſſe, quod moueretur, <lb/>& non fluidum corpus. </s> <s xml:space="preserve">Cuius rei poſſumus etiam exemplum habere à quouis <lb/>corpore perfectè rotundo, quod circa ſuum axem velociſſimè moueatur, nullum ſo-<lb/>num efficiet, quia nullam aeris partem extra ſuum <choice><ex>locum</ex><am>locũ</am></choice> impellit dum mouetur non <lb/>ſecundum totum, ſed ſecundum ſuas partes, quarum quælibet abſque reſiſtentia im-<lb/>mediatè ſubintrat locum alterius, abſque temporis interpoſitione. </s> <s xml:space="preserve">nec huiuſmodi <lb/>locum aliquo modo eadem materia dicti corporis, quod circunuoluitur: </s> <s xml:space="preserve">deſtitutum <lb/>dimittat. </s> <s xml:space="preserve">Sed ſi Pythagorici de alia quadam harmoniæ ſpecie ab ea, quæ eſt ſono-<lb/>rum, vt à diuerſis velocitatibus motuum, aut à diuerſis magnitudinibus aut diſtantiis, <lb/>aut ſtellarum influxibus intellexiſſent, rectè ſenſiſſent exparte, non autem omnino, <lb/>quia ea harmoniam efficere nequeunt, quæ ad <choice><ex>inuicem</ex><am>inuicẽ</am></choice> ſecundum interualla harmoni-<lb/>ca proportionata non ſunt, vt ſunt dupla, ſeſquial tera, ſeſquitertia, ſeſquiquarta, ſeſ-<lb/>quiquinta, ſupertripartientia quintas, <choice><ex>ſuperbipartientia</ex><am>ſuperbipartiẽtia</am></choice> tertias, & quę ab ijs <choice><ex>dependent</ex><am>dependẽt</am></choice> <lb/>ideſt coniuncta ſunt cum duplis; </s> <s xml:space="preserve">de conſonantijs loquendo. de diſſonantiis idem di <lb/>co, quæ harmonicis inſeruiunt modulationibus, vt <choice><ex>ſeſquioctauum</ex><am>ſeſquioctauũ</am></choice>, <choice><ex>ſeſquinonum</ex><am>ſeſquinonũ</am></choice>, ſeſqui <lb/><choice><ex>quintundecimum</ex><am>quintũdecimũ</am></choice>, <choice><ex>ſequiuigeſimunquartum</ex><am>ſequiuigeſimũquartũ</am></choice>, <choice><ex>ſeſquioctuogeſimum</ex><am>ſeſquioctuogeſimũ</am></choice>, & ſuperbipartiens vigeſi <lb/>masquintas. </s> <s xml:space="preserve"><choice><ex>Verum</ex><am>Verũ</am></choice> quidem eſt nonnulla harmonica interualla in aſpectibus <choice><ex>comperta</ex><am>cõperta</am></choice> <lb/>fuiſſe, vt Prolomeus oſtendit, & alii quoque aſſerunt. </s> <s xml:space="preserve">ineſt tamen huic rei nonnihil <lb/>difficultatis. </s> <s xml:space="preserve">vt exempli gratia, ſi ſubtrahamus diateſſaron extra diapaſon, remanet <lb/>diapente, & ſi à diapente ſubtrahamus ſemiditonum, remanet ditonum (quæ duæ <pb facs="0203" n="191"/><fw type="head">DISPVTATIONES.</fw> conſonantiæ, eum habent reſpectum ad inuicem, quem habent diateſſaron, & dia <lb/>pente, quia quemadmodum ſemiditonum, & ditonum ſimul coniuncta, compo <lb/>nunt diapente, ſic diateſſaron, & diapente ſimul vnita componunt diapaſon; </s> <s xml:space="preserve">& <choice><ex>quem</ex><am>quẽ</am></choice> <lb/>admodum terminus, qui diuidit diapaſon in <choice><ex>diateſſaron</ex><am>diateſſarõ</am></choice>, & diapente, eſt mediator ha@ <lb/>monicus inter extrema diapaſon diuiſi, ſic etiam terminus, qui diuidit <choice><ex>diapente</ex><am>diapẽte</am></choice> in ſe-<lb/>miditonum, & ditonum, mediator eſt harmonicus inter extrema ipſius diapente diui <lb/>ſi) ſubtrahendo deinde à diapaſon ſemiditonum remanet exachordum maius, & ab <lb/>eodem diapaſon ſubtrahendo ditonum remanet exachordum minus, quę <choice><ex>quidem</ex><am>quidẽ</am></choice> <choice><ex>non</ex><am>nõ</am></choice> <lb/>accidunt aſpectuum circulo, quia ſubtrahendo aſpectum quadratum ab oppoſito, <lb/>remanet aliud quadratum, & ſubtrahendo ſextilem à trino remanet quoque alius <lb/>ſextilis. </s> <s xml:space="preserve">Quòd autem attinet ad motus, ad magnitudines, ad diſtantias, & ad influ-<lb/>xus, nihil eſt, quod hiſce proportionibus conueniat, ſed quia hæc omnia <choice><ex>dependent</ex><am>depẽdent</am></choice> <lb/>ab <choice><ex>infinita</ex><am>ĩfinita</am></choice>, & diuina <choice><ex>prouidentia</ex><am>ꝓuidẽtia</am></choice> Dei, neceſſariò fit vt iſtæ velocitates, eæ magnitudines, <lb/>diſtantiæ, & influxus, talem ordinem, & reſpectum inter ſeipſa, & vniuerſum <choice><ex>habeant</ex><am>habeãt</am></choice>, <lb/>qualis perfectiſſimus ſit.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Deraro et denſo nonnulla, minus diligenter à Peripateticis <lb/>perpenſa.</head> <head xml:space="preserve">CAP. XXXIIII.</head> <p> <s xml:space="preserve">ANtiqui Peripatetici de videndo in hyeme animalium halitu. </s> <s xml:space="preserve">Id, quod in æſta <lb/>te non euenit, malè diſputauerunt, quia hoc naſciturà condenſatione hali <lb/>tus, quę ab ambiente frigore fit. </s> <s xml:space="preserve">quia halitus is abore, aut naſo animalis <choice><ex>exiens</ex><am>exiẽs</am></choice> <lb/>non eſt purus aer attractus primò, ſed mixtus eſt cum quodam vapore excrementi-<lb/>tio, & ſubtili, quo ſemper ab ea parte <choice><ex>euacuatur</ex><am>euacuat̃</am></choice> corpus, qui ſtatim ab aere frigido cir-<lb/>cundatur, & denſatur, quam ob cauſam ab ipſo ea luminis pars reflectitur, quæ eum <lb/>penetrare non poteſt, quod in hypocauſtis, <choice><ex>huiuſmodique</ex><am>huiuſmodiq́;</am></choice> calidis locis non fit. </s> <s xml:space="preserve">Idem <lb/>exemplo ab aqua ſtatim à ciſternis, aut profundis puteis in hyeme extracta compro <lb/>bari poteſt, quia tunc temporis, huiuſmodi aqua, cum magis calida ſit, quàm fri-<lb/>gida, emittit vaporem, qui facillimè videtur, ob rationem iam dictam, quod <lb/>in æſtate non cernitur in aqua, etſi ea magis calida eſſet, quam ea, quæ in hyeme <lb/>hauritur.</s> </p> <p> <s xml:space="preserve">Ratio autem, quam ab antiperiſtaſi deſumptam citarunt iidem ad inquirendum, <lb/>cur aqua ſubterranea magis calida, aut minus frigida, hyberno tempore, quàm ea, <lb/>quæ eſt ſupra terram ſit, vana eſt, quia hoc non aliunde fit, quàm ab eo, <choice><ex>quod</ex><am>ꝙ</am></choice> terræ por-<lb/>ri à frigoris ſiccitate ſint clauſi, vnde vapores & exalationes non tam facilè exire poſ <lb/>ſunt. </s> <s xml:space="preserve">quamobrem calefiunt ſubterraneæ partes. </s> <s xml:space="preserve">Fimum, fœnum, frumentum hac in <lb/>re ſunt nobis exemplo, in quibus ſępiſſimè viſum eſt ignem accendi.</s> </p> <p> <s xml:space="preserve">Priore illa quoque ratione de antiperiſtaſi dicta, volunt philoſophi maiorem ca-<lb/>liditatem hyęme, quàm ęſtate in animalium ſtomacho contineri, non animaduerten <lb/>tes ſiccitatem, frigiditatis partes ſuperficiales corporis, <choice><ex>reſtringentem</ex><am>reſtringẽtem</am></choice>, ſanguinem ver <lb/>ſus originem ſuam impellere, qui in eo loco copioſior cum ſit, eas partes tunc tem-<lb/>poris calefacit magis.</s> </p> <p> <s xml:space="preserve">Neque etiam ijdem nouerunt cauſam, vnde fiat, ut in æſtate impleto vaſe vitreo, <lb/>aut argenteo, aut ex materia non poroſa conſtante, aqua frigida, vas ſudet, quod <pb facs="0204" n="192"/><fw type="head">IO. BAPT. BENED.</fw> tempore hyemis, non niſi in calidis locis euenit, quem ſudorem, dicebantipſi, eſſe <lb/>eandem aquam, quæ per porros vaſis exiret, quod falſiſſimum eſt, quia ſi per porros <lb/>aqua frigida exiret multò magis exiret calida, cum ſubtilior ſit, & ad penetrandum <lb/>aptior. </s> <s xml:space="preserve">Sed hoc non aliunde oritur, quàm à condenſatione aeris vas circundantis, <lb/>cauſata à frigiditate vaſis refrigerati ab aqua, quemadmodum tempore hyberno <lb/>clarè videmus mane ſuperficies interiores vitri feneſtrarum ſudare, quia <choice><ex>extrinſecum</ex><am>extrinſecũ</am></choice> <lb/>frigus refrigerando vitrum, intrinſecum aerem ſibi contiguum congelat.</s> </p> <p> <s xml:space="preserve">Neque ſilentio inuoluendum eſt, nec Ariſtotelem, neque alium ex ſuis fautoribus <lb/>animaduertiſſe denſum, & rarum eſſe cauſam ventorum. </s> <s xml:space="preserve"><choice><ex>Rarum</ex><am>Rarũ</am></choice> autem & denſum, me <lb/>diante calore & frigore fit, & ſi à partibus, in omogeneis, licet <choice><ex>argumentari</ex><am>argumẽtari</am></choice>, de toto <lb/>deducat conſequentiam qui velit, obſeruans in calidis æſtatis diebus, dum aliqua nu <lb/>becula ad Solem cooperiendum incedit, ibi ſtatim agitationem aeris ſentiri; </s> <s xml:space="preserve">ea verò <lb/>nubecula prætergreſſa cum fuerit, & in ea parte, aer ad priſtinam raritatem cauſa-<lb/>tam à calore Solis redierit, quieſcit; </s> <s xml:space="preserve">huiuſmodi autem aeris agitatio, à nulla certè ex <lb/>halatione proficiſcitur, ſed à motu ſolum locali, quem dum condenſatur, facit. </s> <s xml:space="preserve">Om <lb/>ne denſum natura ſua frigidum eſt; </s> <s xml:space="preserve">omne rarum calidum, & è conuerſo. </s> <s xml:space="preserve">Et frigida <lb/>aura, quæ à flabellis cauſatur, non ſolum à nouo aere qui nos tangit, ſed etiam à den-<lb/>ſo, quod in agitatione eiuſdem aeris fit, naſcitur.</s> </p> <p> <s xml:space="preserve">Cum autem de raritate & denſitate diſputationem ſuſceperim, non ſine ratione <lb/>mihi <choice><ex>videtur</ex><am>videt̃</am></choice> <choice><ex>illorum</ex><am>illorũ</am></choice> <choice><ex>opinionem</ex><am>opinionẽ</am></choice> <choice><ex>explodendam</ex><am>explodẽdã</am></choice> eſſe, qui Lunę maculas <choice><ex>non</ex><am>nõ</am></choice> aliud eſſe dixerunt, <lb/>quàm aliquas partes rariores aliis eiuſdem Lunæ partibus, non obſeruantes rarum, & <lb/>denſum, proportionabilia lumini, quod ab huiuſmodi corporibus reflectitur, non eſ<lb/>ſe. </s> <s xml:space="preserve">quia corpus aliquod rarum aliquando aptum erit ad reflectendum maius lumen, <lb/>quàm corpus minus rarum ut manifeſtè apparet à nubibus reflecti lumen: </s> <s xml:space="preserve">quod <lb/>ab aere non fit. </s> <s xml:space="preserve">Non defuerunt qui contrarium dixerunt, ideſt, eas Lunę partes, den <lb/>ſiores eſſe; </s> <s xml:space="preserve">neque unquam aliquis fuit qui de diaphano, aut opaco mentionem fece <lb/>rit, quia melius eſt credere, eas partes diaphanas, ſiue perſpicuas magis eſſe, quàm a-<lb/>lias, quę per aliquod ſpatium, ſolis radio ingreſſum permittant, & alię partes <choice><ex>cum</ex><am>cũ</am></choice> ſint <lb/>opacæ ipſum à ſuperficie reflectant. </s> <s xml:space="preserve">diuerſa tamen ratione à ſpeculo, cum in pleni-<lb/>lunio tota ferè Lunę pars illuminata cernatur, quamuis dictum lumen extenſiuè & in <lb/>tenſiuè ſit minus eo, quod ipſa in nouilunio recipit. </s> <s xml:space="preserve">Indignum autem mihi videtur <lb/>ijs reſpondere, qui dixerunt huiuſmodi maculas, terræ vmbras exiſtere, cum craſſiſſi-<lb/>mæ ignorantiæ tenebris ſint circunfuſi, vt <choice><ex>etiam</ex><am>etiã</am></choice> fuit Cornelius Agrippa, qui primo de <lb/>occulta philoſophia dicens ſe noſſe modum <choice><ex>quendam</ex><am>quendã</am></choice> naturalem à Pythagora inuen-<lb/>tum, quo in Luna id totum, quod ipſe ſuper ſpeculum ſcripſiſſet, videretur. </s> <s xml:space="preserve">oſtendit <lb/>manifeſtè ſe ignorare luminum <choice><ex>vmbrarumque</ex><am>vmbrarumq́;</am></choice> naturam. </s> <s xml:space="preserve">quia nulla vnquam vmbra ge <lb/>nerari poteſt à corpore, quod aut opacum non ſit, aut officio opaci non fungatur, <lb/>vt nunc dicemus de diaphaneitate aquæ. </s> <s xml:space="preserve">Neque corpus opacum illuminatum <choice><ex>adum- brare</ex><am>adũ-brare</am></choice> poteſt, niſi opacum illud in linea recta ſitum obtineat, quæ inter lucidum & il <lb/>luminatum extenditur. </s> <s xml:space="preserve">Neque etiam reſpondebimus ijs, qui ſentiunt quotieſcun-<lb/>que nulla eſſet terra, ſed totus hic globus eſſet aqua, toties non futuram eclipſim lu-<lb/>narem, ratione diaphaneitatis aquæ. </s> <s xml:space="preserve">Quod <choice><ex>falſiſſimum</ex><am>falſiſſimũ</am></choice> eſt, quia omne corpus <choice><ex>ſphae- ricum</ex><am>ſphę-ricum</am></choice> quantumuis diaphanum ſit, dummodo ſit denſius aere, luminoſos radios re-<lb/>frangit, & eos ad inuicem interſecare facit, qui deinde vltra interſectionis <choice><ex>punctum</ex><am>punctũ</am></choice> di-<lb/>ſgregantur, ita vt amittant illuminationis actum. </s> <s xml:space="preserve">Adde <choice><ex>quod</ex><am>ꝙ</am></choice> etſi huiuſmodi corpus <lb/>aqueum, ſphęricum non eſſet, ſed cubicum, illud ſuper <choice><ex>aliquam</ex><am>aliquã</am></choice> ex eius ſuperficiebus <lb/>ad angulos rectos radius ſolaris percuteret, non eum tamen penetraret, quia dictus <lb/>radius perpetuò debilitatur, & eò magis, quo maiorem profunditatem in diaphano <pb facs="0205" n="193"/><fw type="head">DISPVTATIONES.</fw> eius corporis, quod ſit denſius aere acquirit, nec totus radius vnquam dictum corpus <lb/>ingr editur, cum ab eius ſuperficie magna pars reflectatur. </s> <s xml:space="preserve">Reſiſtit ergo huiuſmo-<lb/>di corpus lumini, & quantò magis ſpiſſum aut profundum exiſtit, tantò, validius reſi <lb/>ſtit. </s> <s xml:space="preserve">Habemus huius rei teſtes, piſcatores vnionum, in ijs mundi partibus, quæ pau-<lb/>cis ab hinc annis Hiſpanorum opera nobis innotuerunt, qui aſfirmanu ad maris <choice><ex>vſque</ex><am>vſq;</am></choice> <lb/>fundum lumen Solis non peruenire.</s> </p> <p> <s xml:space="preserve">Immediata ratio, cur nebulę in ijs locis in quibus <choice><ex>conſpiciuntur</ex><am>cõſpiciuntur</am></choice> permaneant, & <choice><ex>uunc</ex><am>uũc</am></choice> <lb/>altiores, nunc vero depręſſiores cernantur, non ea eſt, quam Ariſtoteles cap .3. lib. 1. <lb/>metheororum proponit, ſed inde oritur, quòd ſint eædem denſiores ea parte aeris, <lb/>quæ ipſis ſupereminet, & rariores e a, quæ ijſdem ſubiacet. </s> <s xml:space="preserve">Quòd autem alicuius cor <lb/>poris denſitas maior ea, quæ eſt medij, in quo reperitur, cauſa ſit, vt ipſum corpus de <lb/>ſcendat, & maior raritas eiuſdem corporis, ea, quæ eſt quoque me dij, efficiat, vt di-<lb/>ctum corpus aſcendat, iam Archimedes in lib. de inſidentibus aquæ docuit.</s> </p> <p> <s xml:space="preserve"><choice><ex>Rectiſſimem</ex><am>Rectiſſimẽ</am></choice> inſtituit natura, vt corpora denſiora verſus loca anguſtiora, & minora <lb/>(intelligendo ea loca orbicularis figuræ) quæ ad centrum propius accedunt, & rario <lb/>ra ad ampliora loca, & maius ſpatium occupantia, ſeſe reciperent. </s> <s xml:space="preserve">tum quia eadem <lb/>quantitas materiæ condenſatæ, eget minori loco quam rarefacta, <choice><ex>cum</ex><am>cũ</am></choice> etiani, quia <choice><ex>cum</ex><am>cũ</am></choice> <lb/>corpus denſum non ita ad velocitatem motus localis, vt rarum, idoneum ſit, ad eas <lb/>partes accedat, quæ motibus tardioribus magis ſunt aptæ, corpora <choice><ex>autem</ex><am>autẽ</am></choice> rara ad eas, <lb/>quæ velocioribus motibus ſunt aptiores ſeſe transferant. </s> <s xml:space="preserve">præterquam <choice><ex>quod</ex><am>ꝙ</am></choice> reuera ap-<lb/>pareat pro maiori parte, corpus magis denſum, minus diaphanum; </s> <s xml:space="preserve">aut magis <choice><ex>opacum</ex><am>opacũ</am></choice> <lb/>futurum, quàm rarum, licet ſæpiſſimè videamus contrarium, vt ſuperius innuimus. <lb/></s> <s xml:space="preserve">eſt tamen naturale <choice><ex>proportionatumque</ex><am>proportionatumq́;</am></choice> magis opacum denſo, & diaphanum raro, <lb/>quàm è contra. </s> <s xml:space="preserve">Quamobrem ſumma ratione inducta natura voluit, vt corpora ma <lb/>gis opaca, aut minus diaphana, magis vicina centro colligantur, vt ſpatium, quòd re <lb/>manet, abſque vllo impedimento à radijs ſolaribus penetrari poſſit. </s> <s xml:space="preserve">Tres autem eæ <lb/>cauſæ, quas hoc loco poſui, propriæ ſunt, immediatæ, & per ſe, ex quibus fit, vt corpo <lb/>ra denſiora deſcendant, & rariora aſcendant in mediis minus denſis, aut minus raris <lb/>dictorum corporum, quæ à nemine, <choice><ex>quod</ex><am>ꝙ</am></choice> ſciam, hucuſque propoſitæ fuerunt.</s> </p> <p> <s xml:space="preserve">Qui autem aſſerunt cucurbitæ, quam apponunt chirurgi, effectum ex eo naſci, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>calidi ſit attrahere, valdè aberrant à vero quia hoc, non niſi à raro, & à denſo imme-<lb/>diatè, à calido & frigido cauſatis efficitur, quia aer in cucurbita rarefactus à calore <lb/>& per conſequens dilatatus, ſtatim vt à dicto calore deſeritur, iterum condenſatur & <lb/>tantò citius, quantò aer ambiens frigidior exiſtet, & quia eadem materia cum con-<lb/>denſata fuerit minorem ſemper occupat locum, reſtringens igitur ſeſe in cucurbi-<lb/>ta aer dum condenſatur, neceſſariò fit, ne ulla, ſcilicet pars vacua remaneat, <choice><ex>quod</ex><am>ꝙ</am></choice> cum <lb/>alius aer ingredi cucurbitam nequeat aliud corpus ingrediatur. </s> <s xml:space="preserve">Idem cum amphora <lb/>in qua nullum aliud, quam aèreum ſit corpus experiri poſſumus, ſi <choice><ex>eam</ex><am>eã</am></choice> ad ignem pri-<lb/>mò calefactam, deinde <choice><ex>cum</ex><am>cũ</am></choice> ore in amplo aliquo cyatho, aut alio vaſe. </s> <s xml:space="preserve">vino, aut aqua <lb/>pleno vbi videbimus huiuſmodi liquorem ſtatim ſurſum ferri, quia dum calefit am-<lb/>phora, rarefit quoque aer qui in ea continetur, & quia rateſcit dilaratur, & quia <lb/>dilatatur, eget maiore loco; </s> <s xml:space="preserve">& ideo magna pars eius foras exit; </s> <s xml:space="preserve">Cum verò ea aeris <lb/>portio, quæ intus remanſerit, iterum condenſatur ob defectum caloris, reſtringitur, <lb/><choice><ex>minorique</ex><am>minoriq́;</am></choice> indiget loco; </s> <s xml:space="preserve">Quod cum ita ſe habeat, neceſſarium eſt, ne aliquis locus va <lb/>cuus remaneat, vt aliud quoddam corpus ingrediatur, cum ad <choice><ex>ingrediendum</ex><am>ingrediẽdum</am></choice> aeri non <lb/>patuerit aditus. </s> <s xml:space="preserve">quod ſi corpus admodum non erit fluxile, aut humidum, ita vt ingre <lb/>di amphoram poſſit ita amphorę hærebit, vt non cito diuelli poſſit, & eo modo ſępe <pb facs="0206" n="194"/><fw type="head">IO. BAPT. BENED.</fw> <choice><ex>cum</ex><am>cũ</am></choice> admiratione <choice><ex>videmus</ex><am>videmꝰ</am></choice> fragile vas <choice><ex>vitreum</ex><am>vitreũ</am></choice> <choice><ex>magnum</ex><am>magnũ</am></choice>, & graue <choice><ex>lapideum</ex><am>lapideũ</am></choice> corpus eleuare. <lb/></s> <s xml:space="preserve">Sed vt ad denſum & ad rarum redeamus, mihi videtur frigidum eſſe conſequentem <lb/>qualitatem denſi, & calidum rari, quia quæuis res dum calefit, rarefit, & quælibet <lb/>materia dum refrigeratur, ſimul condenſatur. </s> <s xml:space="preserve">Qua ratione fit, vt terra frigidior <lb/>ſit aqua, & ignis calidior ſit aere.</s> </p> <p> <s xml:space="preserve">Nec propriè locutus eſt Ariſtoteles .9. & .10. capite primi lib. & ſecundo ſecundi <lb/>metheororum cum dixerit <choice><ex>calorem</ex><am>calorẽ</am></choice> Solis eum eſſe, qui ſurſum humores, <choice><ex>vaporesque</ex><am>vaporesq́;</am></choice> eue <lb/>hat, quia Sol nil aliud facit, quam calefacere, cuius caloris ratione, ea materia rarefit, <lb/>& ob rarefactionem leuior facta aſcendit, non quia ſurſum à Sole feratur.</s> </p> <p> <s xml:space="preserve">Quę ſubſequuntur, cum raro ac denſo ſimbolum habere videntur. </s> <s xml:space="preserve">cum raro, ſcili-<lb/>cet calidum, humidum, leue, ſublime, diaphanum, lumen, clarum, lux, <choice><ex>album</ex><am>albũ</am></choice>, dies, mo-<lb/>tus, velox, ſimplex, diſgregatum, molle, lene, acutum, ſubtile, coctum, <choice><ex>ſpaciosum</ex><am>ſpaciosũ</am></choice>, <lb/>dulce, voluptas, audacia, lætitia, liberalitas, veritas, induſtria, amor, miſericordia, hu-<lb/>manitas, ſanitas, vita, & iis ſimilia. </s> <s xml:space="preserve">Cum denſo verò frigidum, ſiccum, graue, imum, <lb/>opacum, vmbra, obſcurum, tenebræ, nigrum, nox, quies, tardum, mixtum, congrega <lb/>tum, durum, aſperum, ob tuſum, craſſum, crudum, anguſtum, amarum, dolor, cimor, <lb/><choice><ex>melancholia</ex><am>melãcholia</am></choice>, auaritia, mendacium, inertia, odium, crudelitas, feritas, infirmitas, mors, <lb/>& ijs ſimilia.</s> </p> <p> <s xml:space="preserve">Verum eſt quod ea ratio, qua Ariſtoteles ait aerem humidum eſſe, parui eſt mo-<lb/>menti, quia ſimiliter deigne inferri poſſet, qui facilius à termino alieno, <choice><ex>quam</ex><am>quã</am></choice> aer, aut <lb/>aqua terminari poteſt.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Motum rectum curuo poſſe comparari etiam diſentiente <lb/>Ariſtotele.</head> <head xml:space="preserve">CAP. XXXV.</head> <p> <s xml:space="preserve">SEd vt ad <choice><ex>Ariſtotelem</ex><am>Ariſtotelẽ</am></choice> redeamus, rectè dicere non poteſt motum rectum ad <choice><ex>curuum</ex><am>curuũ</am></choice> <lb/>comparabilem non eſſe .4. cap. lib. 7. phyſicorum, vbi errat quoque dicens repe <lb/>riri non poſſe lineam aliquam rectam alicuius circuli circunferentiæ æqualem. </s> <s xml:space="preserve">quia <lb/>Archimedes iam probauit in lib. de quadratura circuli, triangulum illum orthogo-<lb/>nium, cuius vnum ex lateribus circundantibus angulum rectum æquale eſſet ſemi-<lb/>piametro alicuius circuli, & aliud circunferentiæ, æqualem futurum dicto circulo. </s> <s xml:space="preserve">Il <lb/>lud igitur triangulum orthogonium, quod æquale erit alicui circulo, & habebit ali-<lb/>quod ex ſuis lateribus circundantibus angulum rectum æquale ſemidiametro dicti <lb/>circuli, aliud quoque latus ipſum angulum rectum circundans, ex neceſſitate, <choice><ex>circun- ferentiæ</ex><am>circũ-ferentiæ</am></choice> dicti circuli habebit æquale. </s> <s xml:space="preserve">Poteſt igitur dari vna quædam recta linea <choice><ex>ae- qualis</ex><am>ę-qualis</am></choice> circulari contra Ariſtotelis opinionem, qui non benè reuocauit in mentem, <lb/>quod ſcripſit de relatiuis, cum dixit quadraturam circuli poſſe quidem dari, etſi <choice><ex>tunc</ex><am>tũc</am></choice> <lb/><choice><ex>temporis</ex><am>tꝑis</am></choice> de ea <choice><ex>non</ex><am>nõ</am></choice> <choice><ex>haberetur</ex><am>haberet̃</am></choice> ſcientia. </s> <s xml:space="preserve">Si <choice><ex>igitur</ex><am>igit̃</am></choice> dicta quadratura dari <choice><ex>pont</ex><am>põt</am></choice>, poteſt <choice><ex>etiam</ex><am>etiã</am></choice> dari vna <lb/>recta linea ęqualis circunferentiæ <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> circuli, ob rationes <choice><ex>iam</ex><am>iã</am></choice> dictas. </s> <s xml:space="preserve">Sed ſi Ariſt. <lb/>dixiſſet, circularem corporum cęleſtium motum, comparabilem non eſſerecto cor-<lb/>porum elementarium, verum dixiſſet, non quia eorum alter circularis, alter ve-<lb/>rò ſit rectus, ſed quia cœleſtis regularis ſit, neque modò tardus, modò velox, <lb/>ſed vnam ſemper & eandem velocitatem retinens, <choice><ex>motus</ex><am>motꝰ</am></choice> <choice><ex>autem</ex><am>aũt</am></choice>, qui eſt <choice><ex>corporum</ex><am>corporũ</am></choice> elemen <pb facs="0207" n="195"/><fw type="head">DISPVTATIO NES.</fw> tarium è contrà ſe habeat, præter id, <choice><ex>quod</ex><am>ꝙ</am></choice> nunquam fuit neque ſit futurus aliquis <choice><ex>horum</ex><am>horũ</am></choice> <lb/>rectorum, qui naturales dicuntur, qui tam velociter moueatur, ut motus cœli, quia ſi <lb/>voluerimus conſiderare motum diurnum .24. horarum, ſecundum opinionem com-<lb/>munem, reperiemus calculando, Lunam in quadraturis cum Sole, dum inuenitur in <lb/>æquatore, ſingulis horarum minutis moueri per .500. milliaria Italica vel circa, & in <lb/>coniunctionibus, & oppoſitionibus ipſius Solis .1000. vel circa, & Solem tempore <choice><ex>ae- quinoctiorum</ex><am>ę-quinoctiorũ</am></choice> .18000. & <choice><ex>Saturnum</ex><am>Saturnũ</am></choice> circa æquatoris <choice><ex>ſitum</ex><am>ſitũ</am></choice> .260000. & <choice><ex>amplius</ex><am>ampliꝰ</am></choice> de ſtellis <choice><ex>autem</ex><am>aũt</am></choice> <lb/>fixis circa æquatorem poſitis quiuis cogitet; </s> <s xml:space="preserve">quod reuera diffi cillimum quibuſdam <lb/>videbitur, quod quidem non occurrit <choice><ex>ſecundum</ex><am>ſecũdum</am></choice> pulcherrimam Ariſtarchi ſamij opi-<lb/>nionem, diuinitus à Nicolao Copernico expreſſam, contra quam nil planè valent <lb/>rationes ab Ariſtotele; </s> <s xml:space="preserve">neque etiam à Ptolomeo propoſitę. </s> <s xml:space="preserve">Motu verò proprio, quo <lb/>libet horę minuto, Sol <choice><ex>mouetur</ex><am>mouet̃</am></choice> per milliaria circa .48. </s> <s xml:space="preserve">Luna <choice><ex>quando</ex><am>quãdo</am></choice> <choice><ex>coniuncta</ex><am>cõiuncta</am></choice> eſt, aut op <lb/>poſita reperitur Soli .36. milliaria, & in quadraturis .18. Saturnus .24. Iupiter .40: <lb/>Mars .100: Venus .26: Mercur .5. </s> <s xml:space="preserve">Sed Saturnus motu rapido, vno horæ minuto <choice><ex>mo- uetur</ex><am>mo-uet̃</am></choice> circa .260000. milliaria, vt diximus Iupiter circa .170000. Mars .75000. Venus. <lb/>10000, Mercurius .2000. corpus <choice><ex>autem</ex><am>autẽ</am></choice> elementare, & ſi <choice><ex>moueretur</ex><am>moueret̃</am></choice> motu recto hoc <choice><ex>mon</ex><am>mõ</am></choice>, <lb/>& velocius etiam corpore cęleſti, non obſeruans <choice><ex>tamen</ex><am>tamẽ</am></choice> uniformitatem, ut dictum cœ <lb/>leſte facit, cum eodem nullo modo comparari poſſet, quia rectus dictus naturalis, <lb/>ſuam ſemper velocitatem adauget, ob continuam impreſſionem, quam recipit à cau <lb/>ſa perpetuò coniuncta cum ipſo corpore, quę eſt propenſio illa naturalis eundi bre-<lb/>uiori quadam via ad locum ſuum, ita vt etiam ſi dictum corpus elementare à motu <lb/>tardiore ad velociorem, ſuperare poſſet <choice><ex>motum</ex><am>motũ</am></choice> alicuius corporis cęleſtis, ij duo motus <lb/>interſecarent ſeſe in vno ſolo <choice><ex>puncto</ex><am>pũcto</am></choice>, quod diuidi <choice><ex>diſtribuique</ex><am>diſtribuiq́;</am></choice> in partes nequiret, ideſt <lb/>non niſi in vno ſolo temporis inſtanti redderentur æquales, vt ita dicam. </s> <s xml:space="preserve"><choice><ex>Neque</ex><am>Neq;</am></choice> <choice><ex>ſolum</ex><am>ſolũ</am></choice> <lb/>loquor de circulari cœleſti cum recto elementari, ſed de qualibet alia motuum ſpe-<lb/>cie, ſiue ſint ambo recti, ſiue ambo curui, quando aliquis eorum irregularis erit.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Minus ſufficienter exploſam fuiſſe ab Ariſtotele opinionem cre-<lb/>dentium plures mundos exiſtere.</head> <head xml:space="preserve">CAP. XXXVI.</head> <p> <s xml:space="preserve">MAior ratio, qua Ariſtoteles eorum opinionem, qui plures eſſe mundos dixe <lb/>runt, refutare nititur, in eo conſiſtit, quod is credat partes terræ, quæ alijs <lb/>mundis aſſignarentur, ad huius mundi centrum inclinationem habere, & ſic ignem <lb/>illorum, propenſionem habiturum ad circunferentiam huius.</s> </p> <p> <s xml:space="preserve">Quæ certè ratio tam debilis eſt, vt per ſe cadat, non conſideransipſe, quòd ſi <lb/>eſſent dicti mundi, eorum quilibet ſuum proprium centrum, <choice><ex>ſuamque</ex><am>ſuamq́;</am></choice> propriam cir-<lb/>cunferentiam haberet, <choice><ex>terrasque</ex><am>terrasq́;</am></choice> & ignes haberent inclinationem ad centra circunfe-<lb/><choice><ex>rentiasque</ex><am>rentiasq́;</am></choice> ſuorum mundorum, abſque eo, <choice><ex>quod</ex><am>ꝙ</am></choice> vna terra, alterius centrum appeteret; </s> <s xml:space="preserve">vt <lb/>exempli gratia, ſi doctiſſimi Ariſtarchi opinio eſt vera, rationi <choice><ex>quoque</ex><am>quoq;</am></choice> conſentaneum <lb/>erit maximè, vt quod Lunæ contingit, cuilibet <choice><ex>etiam</ex><am>etiã</am></choice> ex aliis quinque planetis eue-<lb/>niat, ideſt, vt quemadmodum Luna ſuorum epicyclorum ope <choice><ex>circum</ex><am>circũ</am></choice> terram voluitur, <lb/>quaſi per circunferentiam alterius cuiuſdam epicycli, in quo terra ſit inſtar centri <lb/>naturalis (ideſt ſit in medio) delati ab orbe annuo circa Solem; </s> <s xml:space="preserve">Sic etiam Saturnus, <lb/>Iupiter, Mars, Venus, atque Mercurius, cir cum aliquod corpus in medio ſui epici- <pb facs="0208" n="196"/><fw type="head">IO. BABPT. BENED.</fw> cli maioris, ſitum habens, voluantur; </s> <s xml:space="preserve">quod quidem corpus, & aliquem quoque ha-<lb/>beat motum circa ſuum axem, ſit opacum, ijs conditionibus, quæ terræ ſunt ſimi-<lb/>les, præditum exiſtat, & in dicto epyciclo ſint res ſimiles iſtis lunaribus.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Anrectè loquutus ſit Phyloſopbus de extenſione luminis <lb/>per uacuum.</head> <head xml:space="preserve">CAP. XXXVII.</head> <p> <s xml:space="preserve">ARriſtoteles ſecundo lib. de anima ſentit <choice><ex>quod</ex><am>ꝙ</am></choice> per vacuum non extenderetur <choice><ex>lu- mem</ex><am>lu-mẽ</am></choice>, quod procederet à corpore lucido. </s> <s xml:space="preserve">Quod veriſimile <choice><ex>non</ex><am>nõ</am></choice> eſt; </s> <s xml:space="preserve"><choice><ex>quia</ex><am>ꝗa</am></choice> <choice><ex>quemadmo dum</ex><am>quẽadmodum</am></choice> quantò rarius eſt aliquod corpus, tanto aptius eſt vt diaphanum exiſtat; </s> <s xml:space="preserve">& <choice><ex>quan- tò</ex><am>quã-tò</am></choice> rarius eſt dictum corpus, tantò minorem quantitatem materiæ contineat; </s> <s xml:space="preserve">ſic <choice><ex>quam</ex><am>quã</am></choice> <lb/>tò magis diaphanum eſt, cum ex perexigua materia conſtet, tantò magis liber tran-<lb/>ſitus luminis patet; </s> <s xml:space="preserve">Vnde quantò minor quantitas materiæ erit in dicto ſpatio, tan <lb/>tò nitidius pertranſibit lumen. </s> <s xml:space="preserve">Sequitur ergo, quòd vbi nulla eſſet materia, totum <lb/>lumen libere tranſiret. </s> <s xml:space="preserve">Color cęruleus quem videmus in profunditate aquæ, & ae-<lb/>ris, color eſt a quæ & aeris, qui denotat reſiſtentiam factam ab aere & ab aqua ipſi lu <lb/>mini; </s> <s xml:space="preserve">Quod quidem lumen ubi corpus aliquod non eſſet, minime reflecteretur, ſed <lb/>abſque vllo impedimento rectà tranſiret.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">An rectè phyloſophiœ<unclear reason="illegible"/> penus Ariſtoteles ſenſerit de loco im-<lb/>pellendo à pyramide.</head> <head xml:space="preserve">CAP. XXXVIII.</head> <p> <s xml:space="preserve">ARiſtoteles .8. cap. lib. 3. de cœlo, diſputans contra antiquos de elementorum <lb/>figuris, ait pyramidem implere poſſe locum corporeum. </s> <s xml:space="preserve">quod verum non <lb/>eſt. </s> <s xml:space="preserve">Cubus quidem id facit ab .8. enim cubis perfectè impletur locus, ſed non <lb/>item .12. pyramides, ut Ariſtoteles ſenſit (ideſt ſex ſuper aliquam exagonam figu-<lb/>ram ſuperficialem & ſexſub eadem) id præſtant, cum potius maius vacuum rema-<lb/>neatad quamlibet partium ſupra, & infra, quam plenum. </s> <s xml:space="preserve">Rectius Ariſtoteles <lb/>egiſſet, ſi probaſſet ratione immobilitatis conuenire pyramidem terræ, quam cu-<lb/>bum. </s> <s xml:space="preserve">quamuis, de horum corporum altero, ſit ſtultum hoc credere. </s> <s xml:space="preserve">decepti tamen <lb/>fuerunt antiqui, credentes cubum ad motum minus idoneum eſſe, quam reliqua <lb/>quatuor corpora regularia (loquor autem habita volubilitatis ratione) quia pyra-<lb/>midale eſt illud, quod ita ſe habet, vt multis rationibus probari poteſt, quarum vna <lb/>hæc nobis ſufficiet. </s> <s xml:space="preserve">Scimus iam ex communi conceptu corpus ſphęricum eſſe ma-<lb/>gis volubile, <choice><ex>inſtabileque</ex><am>inſtabileq́;</am></choice>, quàm alia ſint. </s> <s xml:space="preserve">Illud ergo corpus, cuius figura ad ſphæri-<lb/>cam magis accedet, ad uoluendum, & ad mouendum facilius erit quouis alio, quod <lb/>æqualis ſit quantitatis, & ſibi omogeneum materia, vt exempli gratia corpus .20. ba <lb/>ſium ad voluendum, & ad mouendum promptius erit eo, quod ex .12. conſtat, & id, <lb/>quod eſt .12. eo, quod eſt .8. & id, quod eſt .8. eo, quod eſt .6. & id, quod eſt .6. vt <lb/>cubus eſt, eo, quod eſt .4. cuiuſmodi eſt pyramidale. </s> <s xml:space="preserve">Huc accedit, quòd pyrami-<lb/>dale corpus aliam conditionem habet, quàm cubicum, cum in quauis facie inalte- <pb facs="0209" n="197"/><fw type="head">DISPVTATIONES.</fw> rabile ſit, cubicum autem econtrà ſit alterabile vndequaque, <choice><ex>ſuaque</ex><am>ſuaq́;</am></choice> quadrata in <choice><ex>rhum- bos</ex><am>rhũ-bos</am></choice> mutare poſſit, iiſdem exiſtentibus lateribus.</s> </p> </div> <div type="section"> <head rend="italics" xml:space="preserve">Examinatur quam ualida ſit ratio Aristotelis de <lb/>inalterabilitate Cœli.</head> <head xml:space="preserve">CAP. XXXIX.</head> <p> <s xml:space="preserve">ARriſtoteles textu .22. primi lib. de Cœlo ita inquit.</s> </p> <p> <s xml:space="preserve">Accidit autem, & hoc per ſenſum ſufficienter, quo ad humanam dixiſſe fi-<lb/>dem, & omni pręterito tempore ſecundum traditam inuicem memoriam, nihil vi-<lb/>detur tranſmutatum neque ſecundum totum vltimum cęlum, neque ſecundum par-<lb/>tem ipſius propriam vllam.</s> </p> <p> <s xml:space="preserve">Hoc autem in loco Ariſto. non conſiderauit, <choice><ex>quod</ex><am>ꝙ</am></choice> ſimiliter de terra dici poſſet, quan <lb/>do ipſa ita eminus proſpiceretur, imo abſque dubio putandum eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> ſi terra luce So <lb/>lis prædita eſſet, & aliquis ipſam ab octauo orbe vellet videre, nullo pacto cerne-<lb/>ret, cum ſidera illa quæ primæ magnitudinis vocantur, & quæ pluſquam centies ma <lb/>iora ip<unclear reason="illegible"/>ſa terra putantur non niſi vt puncta videantur.</s> </p> <pb facs="0210" n="198"/> </div> </div> <div type="chapter"> <head xml:space="preserve">IN QVINTVM <lb/>EVCLIDIS LIBRVM</head> <p rend="italics"> <s xml:space="preserve">QVamuis omnia libri quinti Euclid. uerißima ſint. <lb/></s> <s xml:space="preserve">Animaduertimus tamen permultos ſumma <choice><ex>cum</ex><am>cũ</am></choice> <lb/>difficultate <choice><ex>eorum</ex><am>eorũ</am></choice> demonstr ationes percipere. </s> <s xml:space="preserve">Prœ<unclear reason="illegible"/>-<lb/>cipuè ubi quint a, aut ſeptima deffinitiones eiuſ-<lb/>dem libri neceſſariœ<unclear reason="illegible"/> ſunt. </s> <s xml:space="preserve">Illœ enim adeo obſcurœ <lb/>uidentur, ut longè facilius admißuri ſint hœc no-<lb/>ſtra poſtulat at anquam clarior a. </s> <s xml:space="preserve">At que etiam tanquam intellectui <lb/>commodiora, quam ſit illud quintum <choice><ex>idemque</ex><am>idemq́ꝫ</am></choice> ultimum postulatum <lb/>eiuſdem in primo libro poſitum, de line a duas alias ſecante. </s> <s xml:space="preserve">Quan-<lb/>doquidem <choice><ex>ijs</ex><am>ijs</am></choice> noſtris postulatis admißis, ſequentia Theoremata per <lb/>facillima reddentur.</s> </p> <div type="section"> <div type="section"> <head rend="italics" xml:space="preserve">Horum autem primum est.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvod</hi> tota compoſita ex æquali numero partium æqualium, ſunt inuicem <lb/>æqualia.</s> </p> <p> <s xml:space="preserve">Vtſi quis diceret omnes proportiones quæ <choice><ex>compoſitæ</ex><am>cõpoſitæ</am></choice> ſunt ex æquali numero alia-<lb/>rum proportionum inuicem æqualium, ſunt etiam inuicem æquales, quod Eucli-<lb/>des conatur demonſtrare in .22. et .23. quinti libri.</s> </p> </div> <div type="section"> <head xml:space="preserve">SECVNDVM.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvod</hi> ſi à totis æqualibus detractæ fuerint æquales partes, quæ remanent erunt <lb/>partes inuicem æquales.</s> </p> <p> <s xml:space="preserve">Et è conuerſo ſi æqualibus æqualia addas compoſita erunt inuicem æqualia.</s> </p> <p> <s xml:space="preserve">Quod in ipſis proportionibus hoc loco ſemper intelligendum eſt.</s> </p> </div> <div type="section"> <head xml:space="preserve">TERTIVM.</head> <head rend="italics" xml:space="preserve">Quę est εuclidis ſeptima propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvod</hi> ſi fuerint plures termini æquales inuicem, ratio ſeu proportio vnius ip-<lb/>ſorum ad alium tertium terminum maiorem, minoremúe, ſed eiuſdem generis, erit <lb/>cadem quæ cuiuſuis alterius termini ad eundem tertium. </s> <s xml:space="preserve">Et è conuerſo, quæ fuerit <lb/>proportio tertij termini ad vnum prædictorum æqualium, eadem erit, ſpecie, cum <lb/>alio eorundem terminorum.</s> </p> <pb facs="0211" n="199"/> <fw type="head">IN QVINT. LIB. EVCLI.</fw> </div> <div type="section"> <head xml:space="preserve">QVARTVM.</head> <head rend="italics" xml:space="preserve">εuclidis uerò nona propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvotiescvnqve</hi> proportio vnius plurium terminorum collatorum cum ali <lb/>quo tertio eiuſdem generis, eadem fuerit cum ea quæ eſt cuiuſuis alterius dictorum <lb/>terminorum cum eodem tertio, aut proportio dicti tertij, cum aliquo dictorum, ea-<lb/>dem fuerit cum ea quæ ipſius eſt ad aliquem alium eorundem <choice><ex>terminorum</ex><am>terminorũ</am></choice>, tunc eiuſ-<lb/>modi termini, æquales erunt inter ſe.</s> </p> </div> <div type="section"> <head xml:space="preserve">QVINTVM.</head> <head rend="italics" xml:space="preserve">Euclidis uerò octaua propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvoties</hi> plures erunt termini, quorum vnus fuerit maior altero, ſi compa-<lb/>rentur alicui tertio eiuſdem generis, proportio maioris adtertium illum, maior erit <lb/>ea, quæ eſt minoris ad prædictum tertium, & proportio illius tertij ad maiorem, mi-<lb/>nor erit ea quæ eiuſdem tertij ad minorem terminum comparati.</s> </p> </div> <div type="section"> <head xml:space="preserve">SEXTVM.</head> <head rend="italics" xml:space="preserve">εuclidis uerò decima propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvoties</hi> proportio vnius, ex pluribus terminis comparatis ad aliquem ter-<lb/>tium, maior fuerit proportione alicuius alterius dictorum cum eodem tertio, primus <lb/>ille terminus, altero maior erit. </s> <s xml:space="preserve">Et quoties proportio tertij termini ad vnum <choice><ex>quam</ex><am>quã</am></choice> <lb/>ad alterum terminum maior fuerit, eiuſmodi terminus altero minor erit.</s> </p> </div> <div type="section"> <head xml:space="preserve">SEPTIMVM.</head> <head rend="italics" xml:space="preserve">Euclidis uerò undecima propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Proportiones</hi>, quarum vnaquęque cum aliqua tertia æqualis eſt, ipſæ quo-<lb/>que inter ſe ſunt æquales. </s> <s xml:space="preserve">Vtillud, Quæ vni & eidem ſunt æqualia, ſibi inuicem <lb/>ſunt æqualia.</s> </p> </div> <div type="section"> <head xml:space="preserve">OCTAVVM.</head> <head rend="italics" xml:space="preserve">εuclidis uerò duodecima propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvotiescvnqve</hi> proportio vnius ex pluribus antecedentibus cum ſuo ex <lb/>pluribus conſequentibus, æqualis fuerit ei cuiuſuis alterius dictorum <choice><ex>antecedentium</ex><am>antecedentiũ</am></choice>, <lb/>cum ſuo plurium <choice><ex>conſequentium</ex><am>cõſequentium</am></choice>, proportio totius aggregati antecedentium cum to-<lb/>to aggregato conſequentium, dictæ primę proportioni ęqualis erit, nempe illius an <lb/>tecedentis ad ſuum conſequens.</s> </p> <pb facs="0212" n="200"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="section"> <head xml:space="preserve">NONVM.</head> <head rend="italics" xml:space="preserve">Euclidis uero tertiadecima propoſitio.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvotiescvnqve</hi> aliqua proportio plurium proportionum inuicem æqua-<lb/>lium, tertia aliqua proportione, maior aut minor fuerit, quælibet prædictarum æqua <lb/>lium inter ſe, tertia illa proportione maior aut minor pariter erit.</s> </p> </div> <div type="section"> <head xml:space="preserve">DECIMVM.</head> <p> <s xml:space="preserve"><hi rend="small caps">Qvotiescvnqve</hi> fuerint ex vna parte plurestermini (ſiue coniuncti ſiue di-<lb/>ſiuncti ſint) æquales ſinguli vni tertio termino; </s> <s xml:space="preserve">ex altera verò parte totidem fuerint <lb/>alteri tertio termino æquales, proportio aggregati priorum terminorum ad <choice><ex>ſuum</ex><am>ſuũ</am></choice> ter-<lb/>tium, æqualis erit proportioni aggregati reliquorum terminorum ad ſuum tertium, <lb/>& è conuerſo, ita ſe habebit primus tertius terminus ad ſuos multos terminos, ſicut <lb/>ſe habet ſecundus tertius terminus ad ſuos ſimul ſumptos.</s> </p> </div> <div type="section"> <head xml:space="preserve">VNDECIMVM.</head> <p> <s xml:space="preserve">Aggregatum ex partibus proportiona litatis continuæ, quod inter maximum, & <lb/>minimum terminum omnium terminorum proportionalium compræhenditur, ſem <lb/>per multiplex eſt ad ſingulas partiales proportiones, ex quibus ipſum componitur.</s> </p> </div> <div type="section"> <head xml:space="preserve">DVODECIMVM.</head> <p> <s xml:space="preserve">Quæuis proportio quocunque modo diuiſa fuerit, ex iis partibus componitur, in <lb/>quas diuiditur.</s> </p> <p rend="italics"> <s xml:space="preserve">Cum enim bæ præpoſitiones ſint ita conſpicuæ ipſi intellectui, ut <choice><ex>abſque</ex><am>abſq;</am></choice> dubio inter obie <lb/>ct a ipſius intellectus connumerari poſſint, nullus ſanæ mentis eas negabit.</s> </p> </div> </div> <div type="section"> <div type="section"> <head xml:space="preserve">THEOR.I. II. ET III.</head> <p> <s xml:space="preserve">PRimum, ſecundum, & tertium theorema quinti Euclidis ab ipſo ſatis exactè de <lb/>monſtratur, ſtudioſus itaque autorem conſulat.</s> </p> </div> <div type="section"> <head xml:space="preserve">THEOREM. IIII.</head> <p> <s xml:space="preserve">QVartum vero Theorema Eu-<lb/> <ptr xml:id="fig-0212-01a" corresp="fig-0212-01" type="figureAnchor"/> clidis ego ſic <choice><ex>demonſtrarem</ex><am>demonſtrarẽ</am></choice>. <lb/></s> <s xml:space="preserve">ſit, verbi gratia, proportio <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> <lb/>quæ eſt <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> ſumptis multiplici-<lb/>bus <seg type="var">.e.</seg> et <seg type="var">.f.</seg> ad <seg type="var">.a.</seg> et <seg type="var">.c.</seg> æqualiter, item <lb/>multiplicibus <seg type="var">.g.</seg> et <seg type="var">.h.</seg> ad <seg type="var">.b.</seg> et <seg type="var">.d.</seg> dico <lb/>proportionem <seg type="var">.e.</seg> ad <seg type="var">.g.</seg> eſſe eandem <lb/>quæ eſt <seg type="var">.f.</seg> ad <seg type="var">.h</seg>. </s> <s xml:space="preserve">Habemus enim ex .10 <lb/>poſtulato præmiſſo, eandem futuram <lb/>proportionem <seg type="var">.e.</seg> ad <seg type="var">.a.</seg> quæ eſt <seg type="var">.f.</seg> ad <seg type="var">.c.</seg> <lb/>& ita <seg type="var">.b.</seg> ad <seg type="var">.g.</seg> quæ eſt <seg type="var">.d.</seg> ad <seg type="var">.h.</seg> ex præ-<lb/>ſuppoſito verò <choice><ex>cum</ex><am>cũ</am></choice> ſic ſe habeat <seg type="var">.a.</seg> ad <lb/>b. ſicut <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> erit ex primo poſtula-<lb/>to <choice><ex>eadem</ex><am>eadẽ</am></choice> proportio <seg type="var">.e.</seg> ad <seg type="var">.g.</seg> quæ eſt <seg type="var">.f.</seg> <lb/>ad <seg type="var">.h</seg>. </s> <s xml:space="preserve">Nam proportio <seg type="var">.e.</seg> ad <seg type="var">.g.</seg> compo <lb/>nitur ex eis quæ ſunt <seg type="var">.e.</seg> ad .a: et <seg type="var">.a.</seg> ad <seg type="var">. <pb facs="0213" n="201"/><fw type="head">IN QVINT. LIB. EVCLI.</fw> b.</seg> et <seg type="var">.b.</seg> ad <seg type="var">.g.</seg> & ſimiliter proportio <seg type="var">.f.</seg> ad <seg type="var">.h.</seg> <choice><ex>componitur</ex><am>cõponitur</am></choice> ex eis quæſunt <seg type="var">.f.</seg> ad <seg type="var">.c.</seg> et <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> <lb/>et <seg type="var">.d.</seg> ad <seg type="var">.h</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0212-01" corresp="fig-0212-01a"> <graphic url="0212-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOR.V. ET VI.</head> <p> <s xml:space="preserve"><hi rend="small caps">Circa</hi> 5. et .6. theorema nihil notandum occurrit.</s> </p> </div> <div type="section"> <head xml:space="preserve">THEOR. VII. VIII. IX.X. XI. XII. XIII.</head> <p> <s xml:space="preserve">THeoremata à .6. in .13. cum ſint de obiectis intelligibilibus, ſine vllo medio, <lb/>ab intellectu cognitis, inter axiomata à me relata fuerunt .7. inquam quinti <lb/>Euclid. fecimus tertium Poſtulatum, .8. quintum, .9. quartum, .10. ſextum, .11. ſepti<lb/>mum, .12. octauum, .13. nonum.</s> </p> </div> <div type="section"> <head xml:space="preserve">THEOREM. XIIII.</head> <p> <s xml:space="preserve">QVartumdecimum Theorema ex Euclide demonſtrabitur, mutatis tantum <lb/>theorematibus ab interprete notatis, ita vt loco .7. 8. noni, & decimi citetur <lb/>tertium .5. 4. et .6. poſtulatum à me propoſitum.</s> </p> </div> <div type="section"> <head xml:space="preserve">THEOR. XV.</head> <p> <s xml:space="preserve">QVintumdecimum Theorema ſic demonſtrabo; </s> <s xml:space="preserve">Sit, exempli gratia, a. termi-<lb/>nus antecedens. et <seg type="var">.b.</seg> conſequens, qui-<lb/>bus duo multiplices ſumantur <seg type="var">.c.</seg> et <seg type="var">.d</seg>. </s> <s xml:space="preserve">Dico <lb/> <ptr xml:id="fig-0213-01a" corresp="fig-0213-01" type="figureAnchor"/> eandem proportionem habiturum <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> <lb/>quam <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> habet. </s> <s xml:space="preserve">In primis enim manife-<lb/>ſtè patet quamlibet partem ipſius <seg type="var">.c.</seg> habitu-<lb/>ram eandem proportionem cum qualibet par <lb/>te <seg type="var">.d.</seg> quam habet <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> quare ex .7. et .8. po <lb/>ſtulato propoſitum eluceſcet.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0213-01" corresp="fig-0213-01a"> <graphic url="0213-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOREM. XVI.</head> <p> <s xml:space="preserve">SExtumdecimum theorema ſic demonſtrabitur. </s> <s xml:space="preserve">Sit, exempli cauſa, eadem pro <lb/>portio <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> quæ eſt <seg type="var">.c.</seg> ad <seg type="var">.d</seg>. </s> <s xml:space="preserve">Dico <choice><ex>quod</ex><am>ꝙ</am></choice> ita ſe habebit <seg type="var">.a.</seg> ad <seg type="var">.c.</seg> ſicut <seg type="var">.b.</seg> ad <seg type="var">.d</seg>. </s> <s xml:space="preserve">Cogi-<lb/>temus itaque alterum iſtorum terminorum <seg type="var">.c.</seg> aut <seg type="var">.b.</seg> medium inter <seg type="var">.a.</seg> et <seg type="var">.d</seg>. </s> <s xml:space="preserve">quare <lb/>primum intelligamus <seg type="var">.b.</seg> inter <seg type="var">.a.</seg> et. d proportio ipſius <seg type="var">.a.</seg> ad <seg type="var">.d.</seg> componetur ex ea quę <lb/>eſt <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> & ea quæ eſt <seg type="var">.b.</seg> ad <seg type="var">.d.</seg> ex .12. poſtulato. </s> <s xml:space="preserve">Et ex eodem, illa ipſa proportio <seg type="var">.<lb/>a.</seg> ad <seg type="var">.d.</seg> pariter componetur ex ea quæ eſt <seg type="var">.a.</seg> ad <seg type="var">.c.</seg> & ea quæ eſt <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> ſumpto <seg type="var">.c.</seg> pro <lb/>medio termino. </s> <s xml:space="preserve">Ex quo ſequitur, aggregatum duarum proportionum, videlicet <seg type="var">.a.</seg> <lb/>ad <seg type="var">.b.</seg> et <seg type="var">.b.</seg> ad <seg type="var">.d.</seg> æquale eſſe aggregato <seg type="var">.a.</seg> ad <seg type="var">.c.</seg> et <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> ex quibus aggregatis æqua-<lb/>libus ſi duas proportiones æquales ſubtraxerimus, eam videlicet quæ eſt <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> & il <lb/>lam quæ eſt <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> ſupererunt duæ proportiones <lb/>inter ſe æquales. </s> <s xml:space="preserve">erit enim proportio <seg type="var">.a.</seg> ad <seg type="var">.c.</seg> æqua <lb/> <ptr xml:id="fig-0213-02a" corresp="fig-0213-02" type="figureAnchor"/> lis proportioni <seg type="var">.b.</seg> ad <seg type="var">.d.</seg> ex prima parte ſecundi po <lb/>ſtulati diuiſim.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0213-02" corresp="fig-0213-02a"> <graphic url="0213-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Alia etiam ratione idipſum <choice><ex>demonſtrari</ex><am>demõſtrari</am></choice> poteſt, <lb/>ſumpto <seg type="var">.b.</seg> pro medio termino inter <seg type="var">.a.</seg> et .c: et <seg type="var">.c.</seg> <lb/>pro termino medio inter <seg type="var">.b.</seg> et <seg type="var">.d</seg>. </s> <s xml:space="preserve">quare propor-<lb/>tio <seg type="var">.a.</seg> ad <seg type="var">.c.</seg> componetur ex <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> et <seg type="var">.b.</seg> ad <seg type="var">.c.</seg> illa <lb/>verò quæ eſt <seg type="var">.b.</seg> ad <seg type="var">.d.</seg> ex <seg type="var">.b.</seg> ad <seg type="var">.c.</seg> et <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> ex .12. <pb facs="0214" n="202"/><fw type="head">IO. BAPT. BENED.</fw> poſtulato. </s> <s xml:space="preserve">Sed cum proportio <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> ęqualis ſit <lb/> <ptr xml:id="fig-0214-01a" corresp="fig-0214-01" type="figureAnchor"/> proportioni <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> communis autem <seg type="var">.b.c</seg>: propor <lb/>tio. </s> <s xml:space="preserve">itaque <seg type="var">.a.</seg> ad <seg type="var">.c.</seg> æqualis erit <seg type="var">.b.</seg> ad <seg type="var">.d.</seg> ex ſecunda <lb/>parte .2. poſtulati compoſitè, & ſic habebimus pro <lb/>poſitum, ita quòd quotieſcunque <choice><ex>dabuntur</ex><am>dabũtur</am></choice> .4. <choice><ex>quam</ex><am>quã</am></choice> <lb/>titates ex una parte proportionales, illæ ipſæ ex <lb/>altera proportionales erunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0214-01" corresp="fig-0214-01a"> <graphic url="0214-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOR. XVII.</head> <p> <s xml:space="preserve">DEcimiſeptimi theorematis hæc eſt demonſtratio. </s> <s xml:space="preserve">Ita ſe ha beat <seg type="var">a.c.b.</seg> ad <seg type="var">.c.<lb/>b.</seg> ſicut ſe habet <seg type="var">.d.f.e.</seg> ad <seg type="var">.f.e</seg>. </s> <s xml:space="preserve">Probo ita ſe habere <seg type="var">.a.c.</seg> ad <seg type="var">.c.b.</seg> ſicut ſe habet <seg type="var">.d.<lb/>f.</seg> ad <seg type="var">.f.e</seg>. </s> <s xml:space="preserve">Cogitemus itaque alterum terminum ſcilicet <seg type="var">.n.f.</seg> qui ſic ſe habeat. ad <seg type="var">.f.e.</seg> <lb/>ſicut ſe habet <seg type="var">.a.c.</seg> ad <seg type="var">.c.b</seg>. </s> <s xml:space="preserve">Quare ex præcedenti theoremate ita ſe habebit <seg type="var">.a.c.</seg> ad <seg type="var">.n.<lb/>f.</seg> ſicut ſe habet <seg type="var">.c.b.</seg> ad <seg type="var">.f.e.</seg> & ex .8 poſtulato ita ſe habebit <seg type="var">.a.c.b.</seg> ad <seg type="var">.n.f.e.</seg> ſicut ſe ha-<lb/>bet <seg type="var">.c.b.</seg> ad <seg type="var">.f.e</seg>. </s> <s xml:space="preserve">Sed cum ex præſuppoſito ita ſe habeat <seg type="var">.a.c.b.</seg> ad <seg type="var">.c.b.</seg> ſicut ſe habet <seg type="var">.<lb/>d.f.e.</seg> ad <seg type="var">.f.e.</seg> ideo ex præcedenti theoremate ita ſe habebit <seg type="var">.a.c.b.</seg> ad <seg type="var">.d.f.e.</seg> ſicut ſe ha <lb/>bet <seg type="var">.c.b.</seg> ad <seg type="var">.f.e.</seg> demonſtratum autem eſt ita ſe habere <seg type="var">.c.b.</seg> ad <seg type="var">.f.e.</seg> ſicut ſe habet <seg type="var">.a.c.b.</seg> <lb/>ad <seg type="var">.n.f.e</seg>. </s> <s xml:space="preserve">Quare ex .7. poſtulato proportio <seg type="var">.a.c.b.</seg> ad <seg type="var">.d.f.</seg> e, æqualis erit proportioni <seg type="var">.a.<lb/>c.b.</seg> ad <seg type="var">.n.f.e.</seg> & ex .4. poſtulato <seg type="var">.d.f.e.</seg> æqualis erit <seg type="var">.n.f.e</seg>. </s> <s xml:space="preserve">Itaque ex 3. poſtulato primi <lb/>Euclidis <seg type="var">.f.d.</seg> æqualis erit <seg type="var">.n.f</seg>. </s> <s xml:space="preserve">Quamob <lb/>rem proportio <seg type="var">.a.c.</seg> ad <seg type="var">.d.f.</seg> ęqualis erit <lb/> <ptr xml:id="fig-0214-02a" corresp="fig-0214-02" type="figureAnchor"/> proportioni <seg type="var">.a.c.</seg> ad <seg type="var">.n.f.</seg> ex ſecunda par-<lb/>te tertij axiomatis præmiſſi. </s> <s xml:space="preserve">Igitur ita <lb/>ſe habebit <seg type="var">.a.c.</seg> ad <seg type="var">.d.f.</seg> ſicut <seg type="var">.c.b.</seg> ad <seg type="var">.f.e.</seg> ex <lb/>7. poſtulato. </s> <s xml:space="preserve">& ſic ex præcedenti theo-<lb/>remate ita ſe habebit <seg type="var">.a.c.</seg> ad <seg type="var">.c.b.</seg> ſicut <seg type="var">.d.f.</seg> ad <seg type="var">.f.e.</seg> quod erat propoſitum: </s> <s xml:space="preserve">Quotieſ-<lb/>cunque igitur dabuntur .4. quantitates coniunctim proportionales, diuiſim quoque <lb/>proportionales erunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0214-02" corresp="fig-0214-02a"> <graphic url="0214-02"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOREM. XVIII.</head> <p> <s xml:space="preserve">THeorema .18. hac ratione demonſtrari poteſt. </s> <s xml:space="preserve">Detur proportio <seg type="var">.a.c.</seg> ad <seg type="var">.c.b.</seg> ſi-<lb/>milis ei quæ eſt <seg type="var">.d.f.</seg> ad <seg type="var">.f.e.</seg> probo ita ſe habere <seg type="var">.a.c.b.</seg> ad <seg type="var">.c.b.</seg> ſicut ſe habet <seg type="var">.d.f.<lb/>e.</seg> ad <seg type="var">.f.e</seg>. </s> <s xml:space="preserve">In primis notum eſt ex .16. theoremate ita ſe habiturum, <seg type="var">a.c.</seg> ad <seg type="var">.d.f.</seg> ſi <lb/>cut <seg type="var">.c.b.</seg> ad <seg type="var">.f.e</seg>. </s> <s xml:space="preserve">Quare ex .8. poſtulato ita <lb/>ſe habebit <seg type="var">.a.c.b.</seg> ad <seg type="var">.d.f.e.</seg> ſicut <seg type="var">.c.b.</seg> ad <seg type="var">.f.e.</seg> <lb/> <ptr xml:id="fig-0214-03a" corresp="fig-0214-03" type="figureAnchor"/> </s> <s xml:space="preserve">Itaque ex .16. theoremate ita ſe habebit <seg type="var">.<lb/>a.c.b.</seg> ad <seg type="var">.c.b.</seg> ſicut <seg type="var">.d.f.e.</seg> ad <seg type="var">.f.e</seg>. </s> <s xml:space="preserve">Quod erat <lb/>propoſitum. </s> <s xml:space="preserve">Quotieſcunque igitur .4. <lb/>quantitates dabuntur vnius <choice><ex>eiuſdemque</ex><am>eiuſdemq́;</am></choice> generis diſiunctim proportionales, coniun-<lb/>ctim quoque proportionales erunt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0214-03" corresp="fig-0214-03a"> <graphic url="0214-03"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOREM. XIX.</head> <p> <s xml:space="preserve">THeorema .19. ſatis quidem apud Euclidem demonſtratur: </s> <s xml:space="preserve">eius tamentertia <lb/>pars commodius hac ratione demonſtrari poterit (nempe) quod cum ſit pro- <pb facs="0215" n="203"/><fw type="head">IN QVINT. LIB. EVCLI.</fw> portio <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> quæ eſt <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> probabo ita ſe habituram proportionem <seg type="var">.b.</seg> ad <seg type="var">.a.</seg> ſicut <lb/>ſe habet <seg type="var">.d.</seg> ad <seg type="var">.c.</seg> hoc argumento: </s> <s xml:space="preserve">ſi <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> ita ſe <lb/>habet ſicut <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> ex .16. theoremate ita ſe ha <lb/> <ptr xml:id="fig-0215-01a" corresp="fig-0215-01" type="figureAnchor"/> bebit <seg type="var">.a.</seg> ad <seg type="var">.c</seg>, ſicut <seg type="var">.b.</seg> ad <seg type="var">.d</seg>. </s> <s xml:space="preserve">Quare ſic ſe habebit <lb/>b. ad <seg type="var">.d.</seg> ſicut <seg type="var">.a.</seg> ad <seg type="var">.c</seg>. </s> <s xml:space="preserve">Itaque ex eodem .16. ita ſe <lb/>ſe habebit <seg type="var">.b.</seg> ad <seg type="var">.a.</seg> ſicut <seg type="var">.d.</seg> ad <seg type="var">.c</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0215-01" corresp="fig-0215-01a"> <graphic url="0215-01"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOREM. XX.</head> <p> <s xml:space="preserve">QVamuis .20. theorema apud Eucli. perfectè demonſtratum fuerit, poteſt ni-<lb/>hilominus & hac via demonſtrari. </s> <s xml:space="preserve">Sic ſe habeat proportio <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> ſicut ſe <lb/>habet <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> & proportio <seg type="var">.b.</seg> ad <seg type="var">.e.</seg> ſicut <seg type="var">.d.</seg> ad <seg type="var">.<lb/>f</seg>. </s> <s xml:space="preserve">Dico <choice><ex>quod</ex><am>ꝙ</am></choice> ſi <seg type="var">.a.</seg> maius fuerit <seg type="var">.e.</seg> pariter <seg type="var">.c.</seg> maius <lb/> <ptr xml:id="fig-0215-02a" corresp="fig-0215-02" type="figureAnchor"/> erit <seg type="var">.f.</seg> & ſi <seg type="var">.a.</seg> minus fuerit .e: c. <choice><ex>quoque</ex><am>quoq;</am></choice> minus erit <lb/>f. ſin verò ęquale, <choice><ex>ent</ex><am>ẽt</am></choice> æquale erit. </s> <s xml:space="preserve">Nam ex pri <lb/>mo poſtulato certi ſumus ita ſe habere pro <lb/><choice><ex>portionem</ex><am>portionẽ</am></choice> <seg type="var">.a.</seg> ad <seg type="var">.e.</seg> ſicut ſe habet proportio <seg type="var">.c.</seg> ad <lb/>p. </s> <s xml:space="preserve">Quare ex .12. theor <choice><ex>propoſitum</ex><am>ꝓpoſitũ</am></choice> <choice><ex>manifeſtum</ex><am>manifeſtũ</am></choice> erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0215-02" corresp="fig-0215-02a"> <graphic url="0215-02"/> </figure> </div> </body> </floatingText> </div> <div type="section"> <head xml:space="preserve">THEOREM. XXI.</head> <p> <s xml:space="preserve">VIgeſimum primum theorema, ſatis apud Eucli. probatum, nihilominus præ-<lb/>ſcripto nunc modo demonſtrari poterit.</s> </p> </div> <div type="section"> <head xml:space="preserve">THEOREM. XXII. XXIII.</head> <p> <s xml:space="preserve">DVO hæc theoremata in primum poſtulatum collegimus. <lb/></s> <s xml:space="preserve">Sequentia verò cum exactè apud Eucli. demonſtrentur non eſt cur nos in <lb/>ijs immoremur.</s> </p> <pb facs="0216" n="204"/> </div> </div> </div> <div type="chapter"> <head xml:space="preserve">PHYSICA, <lb/>ET MATHEMATICA <lb/>RESPONSA.</head> <head rend="italics" xml:space="preserve">FO. BAPTISTAE BεNεDICTI PATRITII <lb/>Veneti, Philoſophi Mathematici.</head> <div type="preface"> <head xml:space="preserve">Ad Lectorem.</head> <p> <s xml:space="preserve">VT Nilmagis virtutis eſt proprium, quàm <lb/>agitari, & inceßabili motu prodeße. </s> <s xml:space="preserve">Ac velu <lb/>ti fulgidum ſydus ante oculos <choice><ex>ſpectantium</ex><am>ſpectantiũ</am></choice> com <lb/>micare. </s> <s xml:space="preserve">Ita mihi mathematicis <choice><ex>ijsque</ex><am>ijsq;</am></choice> maxi <lb/>mè philoſophicis ſpeculationibus dedito, ſapiſ-<lb/>ſimè, ut in principium ſummorum aulis, & <lb/>amplißimis ciuitatibus degenti, ubi multa ſem <lb/>per Nobilium mir a curioſitate, ſciendi deſiderio, & conferendicu <lb/>piditate referta, uerſantur ingenia, contigit, modo ab his, modo ab <lb/>illis, aut uerbis tentari, aut literis prouocari ad diſſerendum, de <lb/>his, in quorum ſtudijs uerſamur. </s> <s xml:space="preserve">Quarum concertationum & re <lb/>ſponſionum, quoniam non omnino indigna exiſtimaui, quæmemoriæ<unclear reason="illegible"/> <lb/>comendarentur, partem aliquam apud me conſeruaui. </s> <s xml:space="preserve">Vbi uerò <lb/>per ocium licuit, collegi, relegi, ac tandem de manu mittere decreui. <lb/></s> <s xml:space="preserve">Tum ut ſcientia ipſa quo magis diffundetur, creſcat; </s> <s xml:space="preserve">& quicquid <lb/>ualeo, ſine inuidia in communem utilitatem conferam. </s> <s xml:space="preserve">Tum ut ui-<lb/>rorum præctantiβimorum, qui me ſuis interrogationibus excitaue <lb/>runt, quantum in me erit, gratitudinis ergo, nomina reddam im-<lb/>mortalia, & eorum exemplo alios, ocio ſordidiore abiecto, quod ſolet <lb/>ourialium præcipuè excelſa ingenia corrumpere, ad ſciſcit andum <lb/>conferendum, & diſſerendum, derebus <choice><ex>ſerijs</ex><am>ſerijs</am></choice>, & quæuſui aliquan-<lb/>do eße poßint, & <choice><ex>quandoque</ex><am>quandoq;</am></choice> euulgari mereantur, alliciam. </s> <s xml:space="preserve">Tuinte-<lb/>rim nostris laboribus fruere, & nostram diligentiam boni, & æqui <lb/>conſule, & Vale.</s> </p> <pb facs="0217" n="205"/> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DETEMPORVM <lb/>EMENDATIONE <lb/>IO. BAPTIST AE BENEDICTI <lb/>Patritij Veneti, Philoſophi <lb/>Mathematici.</head> <head rend="italics" xml:space="preserve">AD SERENISS. CMANVELEM PHILIB. <lb/>Allobrogum & ſubalpinarum gentium Ducem <lb/>Inuictiβimum.</head> <head xml:space="preserve">EPISTOLA.</head> <p> <s xml:space="preserve"><hi rend="small caps">MIrvm</hi>, Quàm lectione epiſtolæſeu (vt vocant) Breuis <seg type="var">.S.D.<lb/>N</seg>. </s> <s xml:space="preserve">Gregorij XIII. Pont. Max. quod ad me nuper tua Celſitu <lb/>do miſit ex Nicea, vt meam de ea re ſententiam proferrem, <lb/>delectatus ſim; </s> <s xml:space="preserve">ex quo, non tantum recta illius mens ac verè <lb/>ſancta cogitatio, ſed etiam aperta <choice><ex>maximaque</ex><am>maximaq́;</am></choice>, ſi ad exitum per <lb/><choice><ex>ducatur</ex><am>ducat̃</am></choice>, imo ſummè neceſſaria vniuerſo orbi vtilitas percipi <lb/>poteſt; </s> <s xml:space="preserve">qua de re memini cum Celſitudine tua aliquando ſer-<lb/>monem habuiſſe. </s> <s xml:space="preserve">Vidi præterea cum ipſo breui tranſ-<lb/>miſſum compendium Domini Aloiſij Lilij: </s> <s xml:space="preserve">cuius mihi ſententia perplacet, de corre <lb/>ctione eius diei, qui 134. quoque anno præter, neceſſitatem, gignitur. </s> <s xml:space="preserve">qui ſanè dies <lb/>perpetuæ retrogradationis ingreſſus Solis in Zodiaci ſigna, cauſa fuit. </s> <s xml:space="preserve">quod ita per-<lb/>ſpicuè patebit. </s> <s xml:space="preserve">Cum Numa Pompilius anni curſum correxit <choice><ex>emendauitque</ex><am>emendauitq́;</am></choice>, ea ſanè <lb/>mente id videtur præſtitiſſe, vt principium Ianuarij primi menſis anni, præcisè in ip <lb/>ſo hyemalis ſoltitij puncto collocaretur. </s> <s xml:space="preserve">quod hac tempeſtate, dictam ob cauſam <lb/>adeò retroceſſit, vt circa vndecimam diem Decembris eſſe reperiatur. </s> <s xml:space="preserve">quod ſi cen-<lb/>teſimo trigeſimo quarto quoque anno detractus dies vnus fuiſſet, nihil erroris pror-<lb/>ſus accidiſſet. </s> <s xml:space="preserve"><choice><ex>Atque</ex><am>Atq;</am></choice> dies hic (vt alias Celſit. tuæ ſignificaui) inde generatur, quod quar <lb/>to quoque anno addentes nos ad quarti anni dies .365. diem horarum .24. ob erro-<lb/>rem annuum horarum quinque minutorum .49. ſecundorum ferè .16. (anni æqualis <lb/>ſiue medij) fallimur quarto quoque anno in minutis .42. <choice><ex>ſecundis</ex><am>ſecũdis</am></choice> propè .56. amplius <lb/>quàm par ſit minutis ſcilicet .10. ſecundis ferè .44. ſingulis annis; </s> <s xml:space="preserve">qui numerus .134. <lb/>multiplicatus, diem penè horarum .24. conſtituit; </s> <s xml:space="preserve">penè inquam, quia minutum <choice><ex>vnum</ex><am>vnũ</am></choice> <lb/>deeſſet <choice><ex>tantummodo</ex><am>tm̃modo</am></choice>, & ſecunda .44. ſi <choice><ex>decem</ex><am>decẽ</am></choice> illa minuta, & .44. ſe cunda annua, exquiſita eſ <lb/>ſent atque perfecta; </s> <s xml:space="preserve">quæ tamen differentia nullius adeo eſſet momenti, aut certè pe-<lb/>rexigui, vt vix exactis .111086. annis, diem vnum afferret. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> planè neceſſaria <lb/>eiuſmodi eſſet emendatio, <choice><ex>aptaque</ex><am>aptaq́;</am></choice> eius ratio à D. </s> <s xml:space="preserve">Lilio oſtenditur, prout etiam Pe-<lb/>trus Pitatus Veronenſis tradidit, in eo, quem de vera anni quantitate tractatu con-<lb/>ſcripſit, nempe vt tribus primis centeſimis annis, centeſimus quiſque annus commu-<lb/>nis ſit, quartus ſubſequens centeſimus intercalaris: </s> <s xml:space="preserve">quod ſanè fierineceſſe eſt. </s> <s xml:space="preserve">Nam <pb facs="0218" n="206"/> cùm tribus centeſimis <choice><ex>communibus</ex><am>cõmunibus</am></choice>, tres quartas diei partes plus æquo detraxerimus, <lb/>non enim centeſimo quoque anno, ſed centeſimo trigeſimo quarto, dictus dies de-<lb/>trahi debet, poſtquam tres integros dies, qui quadringentis detrahendi erant, tre-<lb/>centorum annorum ſpacio detraxerimus; </s> <s xml:space="preserve">ſitq́ue 134. penè tertia pars .400. quarto <lb/>annorum centenario, tres quartæ diei partes recuperabuntur; </s> <s xml:space="preserve">atque ita in fine qua-<lb/>dringentorum annorum omnia exactè ſuo loco reſtituta erunt. </s> <s xml:space="preserve">Idcirco dictus iam <lb/>quadringenteſimus annus intercalaris & non communis conſtituendus erit, non alia <lb/>de cauſa, quam vt biſſexti ordinem ſequamur.</s> </p> <p> <s xml:space="preserve">Is verò modus, qui à D. </s> <s xml:space="preserve">Lilio traditus eſt, de ratione inueniendi ſingulis menſibus <lb/>Nouilunij diem, interdum fallere nos poſſet vno die; </s> <s xml:space="preserve">prout Ianuario proximè <lb/>lapſo accidit; </s> <s xml:space="preserve">quo ex præſcripto modo nouilunij, dies nonus illius menſis eſſe debuiſ <lb/>ſet, qui fuit tamen dies ſeptimus, ſexta decima hora cum dimidia poſt meridiem. </s> <s xml:space="preserve">Ne <lb/>que etiam tutum eſt, via integrorum dierum, nulla habita horarum aut minutorum <lb/>ratione, nec minus ea, quæ à Pitato tradita eſt, mediorum ſeu æqualium <choice><ex>motuum</ex><am>motuũ</am></choice> pro <lb/>gredi: </s> <s xml:space="preserve">At cenſerem potius veros motus ſequendos eſſe ex calculis exactarum tabu-<lb/>larum, quales Prutenicas eſſe iudico; </s> <s xml:space="preserve">Et cum ſolius Paſchæ cauſa laboremus hac in <lb/>re, pleniluniorum verorum, in multos annos tabulas formarem, quæ æquinoctia ver <lb/>nalia ſequuntur, cum aſſignatione diei Paſchatis præcisè, prout fecit Pitatus; </s> <s xml:space="preserve">non <lb/>via tamen æqualium pleniluniorum ſed verorum. </s> <s xml:space="preserve">Porrò quod ad Paſchatis cele-<lb/>brationem attinet, rationi conſentaneum eſt, concilij Niceni decretum ea de re ſer <lb/>uari, prima ſcilicet dominica die poſt primum plenilunium, quod æquinoctium ver-<lb/>nale ſequitur; </s> <s xml:space="preserve">hoc tamen animaduerſo, ſi dictum plenilunium primum poſt æquino-<lb/>ctium contingens, <choice><ex>diem</ex><am>diẽ</am></choice> dominicum ſortiretur; </s> <s xml:space="preserve">nulla ratione tali die Paſcha celebran-<lb/>dum eſſe; </s> <s xml:space="preserve">verum ſubſequenti, ne cum Hębreis conſentiat Eccleſia Chriſti: </s> <s xml:space="preserve">quæ fuit <lb/>cauſa, vt in decreto concilij Niceni ſtatutum ſit, à quartadecima, in vigeſimam pri-<lb/>mam celebrari debere: </s> <s xml:space="preserve">Quod mihi Petrus Pitatus non animaduertiſſe videtur, cum <lb/>ex <choice><ex>eius</ex><am>eiꝰ</am></choice> <choice><ex>ſententia</ex><am>ſentẽtia</am></choice> in ſuis tabulis die Paſchate declarata, huiuſce anni Paſca <choice><ex>celebrandum</ex><am>celebrandũ</am></choice> <lb/>fuerit .23. </s> <s xml:space="preserve">Martij, ipſomet de plenilunij non tantum æqualis, ſed veri.</s> </p> <p> <s xml:space="preserve">Dies autem Paſchatum elapſorum, quos hactenus examinaui, reperi omnes con <lb/>cordare cum ea regula, quam nonnulli de die carnis priuij tradiderunt. </s> <s xml:space="preserve">nempe pri-<lb/>mum diem martis poſt nouilunium Februarij, carnis priuij diem eſſe; </s> <s xml:space="preserve">non autem <choice><ex>cum</ex><am>cũ</am></choice> <lb/><choice><ex>sanctione</ex><am>sãctione</am></choice> Patrum concilij Niceni, qua ſtatuerunt à vigeſima prima Martii dirigen-<lb/>dum eſſe Paſchatis diem, vt potè qui ſibi perſuaſerunt, circa eum diem <choice><ex>æquinoctium</ex><am>æquinoctiũ</am></choice> <lb/>perpetuò eſſe debere; </s> <s xml:space="preserve">prout tunc temporis erat. </s> <s xml:space="preserve">Non <choice><ex>itaque</ex><am>itaq;</am></choice> error accidit, quod Pa <lb/>ſcha ex huiuſmodi ſuppoſitione concilij, poſt vigeſimam primam lunę celebretur, <lb/>cum ſeruata regula concilij non fuerit. </s> <s xml:space="preserve">Prout <choice><ex>manifeſtum</ex><am>manifeſtũ</am></choice> eſt de Paſchate anni .1566. <lb/>celebrato .14. </s> <s xml:space="preserve">Aprilis (quę fuit .24. lunę) quod .7. dicti menſis <choice><ex>celebrandum</ex><am>celebrãdum</am></choice> erat. </s> <s xml:space="preserve">Tum <lb/>anno .1569. 10. </s> <s xml:space="preserve">Aprilis ſolenne fuit Paſca, quod tertia eiuſdem eſſe debuerat. </s> <s xml:space="preserve">Anno <lb/>deinde 1572. 6. Aprilis, dies fuit Paſchatis, quæ .30. Martij futura erat, anno vero <lb/>1575. in tertiam Aprilis Paſcha incidit, caſurum in .27. Martii.</s> </p> <p> <s xml:space="preserve">Cum igirur (vt ex diplomate ad Celſit. tuam miſſo patet) <seg type="var">S.D.N.</seg> mens ſit <choice><ex>atque</ex><am>atq;</am></choice> vo <lb/>luntas, ut quiſque liberè in medium proferat quid hac dere ſentiat: </s> <s xml:space="preserve">quædam mihi <lb/>non omnino præmittenda occurrunt, quæ tantis cœptis non nihil adiumenti for-<lb/>taſſe adferre queant.</s> </p> <p> <s xml:space="preserve">Atque illud in primis non tantum ut corrigatur Calendarium ob Paſcha cætera-<lb/>q́ue feſta mobilia ab illo manantia, vt decreto concilij Niceni ſancitum eſt, ſcilicet <lb/>vt ipſum Paſcha celebretur prima dominica poſt primum plenilunium, quod æqui- <pb facs="0219" n="207"/> noctium vernale proximè ſequitur; </s> <s xml:space="preserve">verum etiam quò anni principium emendetur, <lb/>ſcilicet vt ad ſuum verum principium reuocetur annus. </s> <s xml:space="preserve">Nempè ad diem hyemalis <lb/>ſolſtitij, quæ prima Ianuarij dies eſſe debet.</s> </p> <p> <s xml:space="preserve">Deinde, tot dierum menſes conſtituantur, quot hac noſtra tempeſtate, ſol in ipſis <lb/>Zodiaci ſignis verſatur. </s> <s xml:space="preserve">Poſtremò, quædam feſta immobilia in alios dies <choice><ex>transferam</ex><am>transferã</am></choice> <lb/>tur, <choice><ex>celebrenturque</ex><am>celebrenturq́;</am></choice> aptis temporibus: </s> <s xml:space="preserve">quod à <seg type="var">.S.D.N.</seg> mente diſſentire minimè vide-<lb/>tur. </s> <s xml:space="preserve">cum non magis de feſtis mobilibus quam immobilibus agat, imo etiam planè <lb/>æquum ſit, vt habeatur vtrorunque ratio, quò ſtatutis temporibus celebrentur.</s> </p> <p> <s xml:space="preserve">Vt autem ad primam Ianuarij <choice><ex>diem</ex><am>diẽ</am></choice> verum principium anni reuocetur; </s> <s xml:space="preserve">cenſerem <lb/>ex eo anno, quem corrigere voluerimus, non modò dies .10. eſſe detrahendos, <choice><ex>verum</ex><am>verũ</am></choice> <lb/>etiam vnum & uiginti, illo ipſo anno; </s> <s xml:space="preserve"><choice><ex>idque</ex><am>idq́;</am></choice> duplici via; </s> <s xml:space="preserve">aut partiendo menſes, atque <lb/>ex illis demendo eos dies, qui minus ad rem hanc facere videbuntur, ac tum rema-<lb/>neat annus trecentorum quadraginta quatuor dierum ita vt decem menſes ſint die. <lb/></s> <s xml:space="preserve">rum duorum ſpatio ſolito breuiores, alter menſis vno deficiat: </s> <s xml:space="preserve">aut conſtituto <choice><ex>Decem- bri</ex><am>Decẽ-bri</am></choice> dicti anni dierum decem, dies autem ille, qui decimum <choice><ex>proximem</ex><am>proximẽ</am></choice> ſequitur, ſit & <lb/>primus Ianuarij, & dies ſolſtitij ob quam cauſam exiſtimarem conſultiſſimum eiuſ <lb/>modi annum eſſe mileſimum quingenteſimum ſeptuageſimum nonum. </s> <s xml:space="preserve">Quo quam <lb/>primum <seg type="var">.S.D.N</seg>. </s> <s xml:space="preserve">Pontifex max. ſuis temporibus huius correctionis manifeſtos effe-<lb/>ctus experiri & perpendere, atque diſpoſitionem anni non ſolum principio, ſed <choice><ex>cae teris</ex><am>cęteris</am></choice> partibus ſuis in vniuerſum tam concinnè <choice><ex>apteque</ex><am>apteq́;</am></choice> reſpondere, & aſtrorum moti-<lb/>bus, & Eccleſiæ ſacroſanctæ ſanctionibus, ſe authore lætari poſſit.</s> </p> <p> <s xml:space="preserve">Omnino <choice><ex>itaque</ex><am>itaq;</am></choice> iudico detrahendos eſſe vnum & viginti dies elapſi erroris: </s> <s xml:space="preserve">non de <lb/>cem tantum, quo hyemmalis conuerſio ad initium Ianuarij reuocetur; </s> <s xml:space="preserve">idq́ue ne à <lb/>communi opinione de ipſo anni principio veritas diſcrepet, quæ principium Ianua-<lb/>rij, anni principium arbitratur. </s> <s xml:space="preserve">etenim cum credant omnes <choice><ex>annum</ex><am>annũ</am></choice> à Ianuario inchoa-<lb/>ri, veritas autem ipſa ſic ſe habeat, vt nobis ſeptentrionalibus tunc inchoet annus, <lb/>cum ad nos Sol accedere incipit, aut dies augetur; </s> <s xml:space="preserve">non conuenit principia eiuſmo-<lb/>di ſeparata & diſcrepantia eſſe. </s> <s xml:space="preserve">Et hanc fuiſſe Numæ Pompilio mentem credibile <lb/>eſt, qui ad annum Romuli decem menſium, Ianuarium & Februarium addidit, vt <lb/>principium Ianuarij principium eſſet anni: </s> <s xml:space="preserve">cuius rei argumentum eſſe poteſt, quod <lb/>C. </s> <s xml:space="preserve">Iulij Cæſaris temporibus (qui multis annis poſt Numam fuit) <choice><ex>atque</ex><am>atq;</am></choice> vti Pont. </s> <s xml:space="preserve">Max. <lb/></s> <s xml:space="preserve">corrigendorum feſtorum curam ſuſcepit hyemale ſolſtitium per aliquot dies retro-<lb/>ceſſerat; </s> <s xml:space="preserve">nec mirum tamen eſſet, ſi Numæ temporibus, exactè prima Ianuarij die non <lb/>fuiſſet hyemale ſolſtitium, adhuc pubeſcente in Italia Aſtronomia.</s> </p> <p> <s xml:space="preserve">Huiuſmodi autem correctio dierum .21. poſt .2300. annos à Numa, quæ ſit per-<lb/>petuo ſeruitura, media emendatione ea, quæ de tribus centeſimis annis communi-<lb/>bus, & quarto intercalari, ſuperius propoſita fuit, non repudianda ei videatur, qui <lb/>ſciet, qua ratione Numæ Pompilij annus corrigeretur, octauo quoque anno, inter-<lb/>calando annum vltimum medijs diebus .90. quo prima dies Ianuarij ad verum prin <lb/>cipium anni, hoc eſt hyemale ſolſtitium, reduceretur.</s> </p> <p> <s xml:space="preserve">Alio item argumento cuique patere poteſt, priſcos Romanos ſtatuiſſe annum ab <lb/>hyemali ſolſtitio initium ſumere, vt inquit Ouidius primo Faſtorum.</s> </p> <p> <s xml:space="preserve">Bruma noui prima eſt, <choice><ex>veterisque</ex><am>veterisq́;</am></choice> nouiſſima Solis.</s> </p> <p> <s xml:space="preserve">Principium capiunt Phębus, & annus idem.</s> </p> <p> <s xml:space="preserve">co quod diem naturalem à medio noctis inchoarent, ab eo puncto ſcilicet, quo Sol <lb/>ad noſtrum hemiſpherium accedere incipit.</s> </p> <p> <s xml:space="preserve">Tribuebant igitur veteres diei, atque anno principium ab eo puncto, quo Sol <pb facs="0220" n="208"/> ad nos accedit: </s> <s xml:space="preserve">cum punctum Zodiaci, quod tropicum hyemalem Capricorni nobis <lb/>producit, reſpondeat puncto meridiani ſub terra, in quo Sol ſemel in die reperitur: <lb/></s> <s xml:space="preserve">Quòd apertè norunt hi, qui ſub polo boreali conſtituti ſunt. </s> <s xml:space="preserve"><choice><ex>Atque</ex><am>Atq;</am></choice> facilè diſcerne <lb/>re poſſumus, diem ſcilicet & annum, quaſi ſibi ad inuicem medio ſuarum partium <lb/>reſpondere; </s> <s xml:space="preserve">ſolſtitium inquam hyemale, mediæ nocti, æſtiuum meridiei, æquino-<lb/>ctium vernale ortui Solis, autumnale occaſui. </s> <s xml:space="preserve">Quam tamen ſimilitudinem, multò <lb/>quam nos manifeſtius deprehendunt, hi qui (ut diximus) ſub polo borcali verſantur.</s> </p> <p> <s xml:space="preserve">Quod ſi quis dubitet hac ratione correcto anno, quo nam pacto ad calculos coe-<lb/>leſtes motus medijs tabulis aſtronomicis hactenus in lucem æditis redigi poſſint, id <lb/>facilimum ſanè erit, exempli gratia; </s> <s xml:space="preserve">aliquis planetę ſitum, aut alicuius ſtellę fixæ, quo <lb/>cunque die menſis anni correcti inuenire cupit, detrahat ex huiuſmodi <choice><ex>tempore</ex><am>tẽpore</am></choice> dies <num value="21">.<lb/>21.</num> ab Aera Chriſti, cum reſiduo ſupputet ſtellam, cuius ſitum ſcire deſiderat; </s> <s xml:space="preserve">ſum-<lb/>pta quacunque tabula, ſupputatio erit exacta: </s> <s xml:space="preserve">Cuius ratio cuilibet manifeſta erit, <lb/>qui ſciet annum vt potè .1579. dierum .344. tantummodo conſtitutum fuiſſe. </s> <s xml:space="preserve">Nam <lb/>in ijſdem locis cœli prima die Ianuarij correcti, erunt ſtellæ quibus eſſe ſolebant .11 <lb/>Decembris præcedentis anni ex ſupputatione tabularum: </s> <s xml:space="preserve">atque ita deinceps. </s> <s xml:space="preserve">Alia <lb/>præterea via idem perfici poſſet inuentione omnium motuum cęleſtium ipſo princi <lb/>pio anni .1580. correcti: </s> <s xml:space="preserve">hoc ſtatuto, vt hi motus radices eſſent Aeræ <seg type="var">S.D.N.</seg> Grego <lb/>rij XIII. quod ſi alio <choice><ex>tempore</ex><am>tẽpore</am></choice> quiſpiam motus cęleſtes ad calculos redigere voluerit, <lb/>ſupputabit ab Aera huiuſmodi, quæ anno .1580. principium habuerit: </s> <s xml:space="preserve">Quæ vt nobi <lb/>lius nomen ſortiatur, <choice><ex>idque</ex><am>idq́;</am></choice> merito ex nomine Gregorij. XIII. Pont. Max. appelletur; <lb/></s> <s xml:space="preserve">exemplo antiquarum, quæ ex Principum nominibus ſunt appellate: </s> <s xml:space="preserve">vt tanto Pontifi <lb/>ci, cùm ex alijs multi, tum etiam ex hac non infima re, inter mortales immortale no <lb/>men comparetur. </s> <s xml:space="preserve">Ei verò ſummæ, quæ ex huiuſmodi Aera Gregoriana ex tabu-<lb/>lis colligetur, ipſiuſmet Aeræ radices addantur, vt exactus calculus habeatur. </s> <s xml:space="preserve">Et <lb/>hæc ſit primæ ſententiæ noſtræ explicatio.</s> </p> <p> <s xml:space="preserve">Altera erit numerum dierum menſium anni alia ratione quam nunc ſe habeat, or <lb/>dinandum eſſe: </s> <s xml:space="preserve">nempe vt Ianuarius, Nouember <choice><ex>atque</ex><am>atq;</am></choice> December dies .29. ſinguli con <lb/>tineant, Februarius, Martius, & October .30. </s> <s xml:space="preserve">Aprilis, Maius, Auguſtus, & September <lb/>dies .31. Iunius, ac Iulius .32. atque id hac potiſſimum de cauſa, vt Sol unum <choice><ex>quodque</ex><am>quodq́;</am></choice> <lb/>ſignum calendis menſrum ingredi poſſit. </s> <s xml:space="preserve">Nam detractis (ut dictum eſt) diebus .21. <lb/>& reuocato ingreſſu Solis in principium Capricorni ad principium Ianuarij, in quo <lb/>ſigno hac noſtra tempeſtate, Sol, dies propè .29. & quartam vnam verſatur: </s> <s xml:space="preserve">ſi Ianua <lb/>rius .29. dies continebit, exactis hiſce diebus, ingredietur Aquarium circa princi-<lb/>pium Februarii; </s> <s xml:space="preserve">hæret autem hoc noſtro ſæculo in Aquario Sol dies propè .29. cum <lb/>dimidio; </s> <s xml:space="preserve">quare ſi Februarius erit .30. dierum, elapſis ipſis diebus, Sol ingredietur pi-<lb/>ſces circa principium Martii: </s> <s xml:space="preserve">& ſic de cæteris.</s> </p> <p> <s xml:space="preserve">Quamobrem ſi generali correctione annus emendandus erit, pulcherrimè acci-<lb/>det, ſi menſes anni cum duodecim partibus cœleſtibus, itineris annui Solis, concor-<lb/>dauerint; </s> <s xml:space="preserve"><choice><ex>eiſque</ex><am>eiſq́;</am></choice> aptè reſponderint. </s> <s xml:space="preserve">Qua ex re, varię vtilitates promanabunt, pręſertim <lb/>Nautis, Agricolis, Medicis, & alijs qui vera principia, & interualla temporum per <lb/>ſpecta habebunt: </s> <s xml:space="preserve">terminos item & interualla incrementi & diminutionis dierum & <lb/>noctium, & eorundem æqualitatis. </s> <s xml:space="preserve">Exempli cauſa, ſcient omnes principium Ianua-<lb/>rij, eſſe non modo anni principium, verum etiam hyemis, eſſe minimam anni diem, <lb/>& eius noctem maximam; </s> <s xml:space="preserve">principium incrementi diei, & diminutionis noctis; </s> <s xml:space="preserve">atque <lb/>etiam omnia illa, quæ ex huiuſmodi conuerſione Solis ad nos dependent. </s> <s xml:space="preserve">Pariter <lb/>ſcient omnes primam diem Iulij, non tantum æqualiter annum diuidere, ſed prin <pb facs="0221" n="209"/><fw type="head">EPISTOLA.</fw> cipìum quoque eſſe ęſtatis, maximam diem, noctem minimam totius anni; </s> <s xml:space="preserve">princi-<lb/>pium diminutionis diei & incrementi noctis, vnà <choice><ex>etiam</ex><am>etiã</am></choice> ea, quę Solis conuerſionem <lb/>ad auſtrales ſequuntur.</s> </p> <p> <s xml:space="preserve">Neconon intelliget vnuſquiſque primam diem Aprilis, <choice><ex>primamque</ex><am>primamq́;</am></choice> Octobr. æqui-<lb/>noctiorum dies eſſe; </s> <s xml:space="preserve">primam autem diem Aprilis, initium veris; </s> <s xml:space="preserve">Octobris Autumni; <lb/></s> <s xml:space="preserve">Item Aprilis diem eſſe eum, quo dies noctis prolixitatem vincere incipit: </s> <s xml:space="preserve">Octobris, <lb/>quo nox diei longitudinem ſuperat, & alia huiuſmodi, quæ ab æquinoctijs <choice><ex>dependent</ex><am>depẽdẽt</am></choice>.</s> </p> <p> <s xml:space="preserve">Si vero quiſpiam obijciat, modum hunc noſtrum & ordinem perpetuum eſſe non <lb/>poſſe, ob motum augis Solis; </s> <s xml:space="preserve">quod punctum cum fuerit in principio Capricorni, <choice><ex>tunc</ex><am>tũc</am></choice> <lb/>Sol hærebit in ſigno Sagittarij .32. diebus, totidem in Capricorno, in Geminis vero <lb/>29. totidem in Cancro; </s> <s xml:space="preserve">ex quo ſequetur prioribus <choice><ex>contrarius</ex><am>cõtrarius</am></choice> effectus; </s> <s xml:space="preserve">huic ego reſpon <lb/>debo, tale <choice><ex>quidpiam</ex><am>quidpiã</am></choice> non euenturum, niſi exactis ab hoc anno annis .240@0. quod ſi <lb/>mundus poſthac totidem annis, quot fuit antehac, perdurauerit, punctus augis non <lb/>amplius à ſitu præſenti, quàm .45. gradibus diſtabit. </s> <s xml:space="preserve">Verum demus <choice><ex>modum</ex><am>modũ</am></choice> noſtrum <lb/>& regulam in annos ter, aut quater mille ſubſeruire poſſe, nec amplius, certè hoc <lb/>toto tempore nullius momenti penè erit, quæ accidere poterit mutatio, tametſi ela-<lb/>pſis quatuor millibus annorum Februarius eſſe debebit .29. dierum. Aprilis & No-<lb/>uember .30. Iunius & October .31. Auguſtus .32. in aliis verò menſibus nihil mutan-<lb/>dum erit. </s> <s xml:space="preserve">Ecce quam ſit nullius momenti mutatio.</s> </p> <p> <s xml:space="preserve">Quæ ſi Iulij Cæſaris temporibus fuiſſent animaduerſa nunquam omiſſa fuiſſent, <lb/>ſed ſcientiæ Aſtronomicæ nondum (vt ita dicam) confirmata ætas, cum alibi, <choice><ex>tum</ex><am>tũ</am></choice> maxi <lb/>mè in Italia, quo minus hæc aut ſcirentur aut ſtatuerentur impediebat.</s> </p> <p> <s xml:space="preserve">Tertia ratio eſt, vt non <choice><ex>ſolum</ex><am>ſolũ</am></choice> feſta mobilia, verum <choice><ex>ent</ex><am>ẽt</am></choice> immobilia ad meliorem regu <lb/>lam (ut dictum eſt) reuocentur, ſi ſuis temporibus celebranda erunt. </s> <s xml:space="preserve">Quorum <choice><ex>primum</ex><am>primũ</am></choice> <lb/>eſt Natiuitas Domini, & quæ ab ea pendent; </s> <s xml:space="preserve">nempe Circuncifio, Epiphania, Purifi-<lb/>catio, Annunciatio, & Natiuitas Io. Baptiſtæ. </s> <s xml:space="preserve">ita vt dies Natalis Domini celebretur <lb/>prima die anni, cum Dei filius naſci voluerit circa verum principium anni, quod à <lb/>ſolſtitio hyemali initium ducit, & in ipſo principio diei naturalis ex <choice><ex>Romanorum</ex><am>Romanorũ</am></choice> ſen <lb/>tentia, media ſcilicet nocte, tanquam qui ſummæ lætitiæ principium, poſt longos & <lb/>graues filiorum Adæ mærores, eſſet allaturus. </s> <s xml:space="preserve">Nec forſan Ianuarij nomini, à vete-<lb/>ribus Iano bifronti dicati hæc mutatio non conueniret, cum in ip ſo ſeruatore, duæ <lb/>veluti frontes & formæ vnitæ ſint, duæ ſcilicet naturæ diuina & humana. </s> <s xml:space="preserve">Hac ratio <lb/>ne abuſus tolletur, natus ex diuerſis moribus Tabulariorum, quorum alij monumen-<lb/>ta, ſeu quæ uocant Inſtrumenta, à die Natiuitatis Domini incohant, alij à Circunci <lb/>ſione, alij à Calendis Martij, nonnulli à Paſchate; </s> <s xml:space="preserve">quæ varietas innumerabiles lites <lb/>affert & abuſus propè infinitos, ob dubiam & ancipitem ſcripturam. </s> <s xml:space="preserve">Indictionum <lb/>præterea ordini, hic noſter modus nihil officiet; </s> <s xml:space="preserve">celebrato Natali celebrabitur Cir-<lb/>cunciſio octaua Ianuarij. </s> <s xml:space="preserve">Epiphania .13. eiuſdem. </s> <s xml:space="preserve">Purificatio .11. Februarij quæ erit <lb/>40: </s> <s xml:space="preserve">dies à Natiuitate ſeruatoris. </s> <s xml:space="preserve">Prima Aprilis Annunciatio Virginis ſolennis erit, <lb/>ipſo nempè die æquinoctij, natiuitas Diui Io: </s> <s xml:space="preserve">Baptiſtę celebrabitur Prima Iulij die <lb/>quæ erit ſolſtitij æſtiui, cum illa diminutionem capit. vtrectè Diuus Auguſtinus il-<lb/>la verba Io: </s> <s xml:space="preserve">Baptiſtę interpretatus fuerit. </s> <s xml:space="preserve">Illum opportet creſcere, me autem minui: <lb/></s> <s xml:space="preserve">in quibus ſic tantus Doctor philoſophatur, vt tempus etiam natiuitatis ſerui & do-<lb/>mini præclare notet dicens, natus eſt ſeruus cum decreſcunt dies, natus eſt Dominus <lb/>cùm creſcere incipiunt.</s> </p> <p> <s xml:space="preserve">Inſignes etiam Theologi admonuerunt habendam rationem eſſe nonnullorum <lb/>feſtorum, vt Diui Antonij, diuorum Fabiani & Sebaſtiani, & aliorum ſanctorum, <pb facs="0222" n="210"/><fw type="head">IO. BAPT. BENED.</fw> fi forte in octauam Epiphaniæ inciderint: </s> <s xml:space="preserve">Verum hęc <seg type="var">.S.D.N.</seg> curæ erunt, ut in aptiſ <lb/>ſima tempora transferantur.</s> </p> <p> <s xml:space="preserve">Admonuerunt præterea transferendos eſſe dies feſtos Beati Stephani, Ioannis, <lb/>& Innocentium, vt quemadmodum factum eſt hactenus, diem natalis proximè ſe <lb/>quantur, ob multorum Doctorum, non recentium modo, ſed etiam antiquorum ob <lb/>ſeruantiam; </s> <s xml:space="preserve">qui ſuis omelijs & concionibus multa piè, de myſteriis ſucceſſionis Feſto <lb/>rum huiuſimodi tradiderunt.</s> </p> <p> <s xml:space="preserve">Cuperent etiam præclari Theologi diem Aſſumptionis Beatæ virginis incidere <lb/>in primam Septembris, Natiuitatem autem in .25. vt quemadmodum toto illo men <lb/>ſe in ſigno Virginis ſol verſabitur, ita Eccleſia Der in cęlebrandi tantæ Virginis ma <lb/>tris Deilaudibus occupetur.</s> </p> <p> <s xml:space="preserve">Atque hęc ſunt Serenisſime Princeps, quę longa & attenta cogitatione à me exa <lb/>minata, atque perpenſa fuerunt; </s> <s xml:space="preserve">quæſitam diligenter & accuratè expendentur ab <lb/>his, quorum intereſt, quam mihi apta & rationi conſentanea, ac vera penitus, imo <lb/>(quod me magis afficit) etiam tibi viſa fuerunt; </s> <s xml:space="preserve">non dubito quin placitura ſint; </s> <s xml:space="preserve">& vo <lb/>tis ſummi Pont. aliqua ex parte ſatisfactura. </s> <s xml:space="preserve">eò magis quòd te iubente, & cogitata à <lb/>me, & ſcripta fuerint. </s> <s xml:space="preserve">Vale Princeps Sereniſſime, & qua ſoles hylaritate cętera no <lb/>ſtra, etiam has breues vigilias ſuſcipe & foue. </s> <s xml:space="preserve">Dat. Auguſtæ Taurinorum Kal. <lb/></s> <s xml:space="preserve">Aprilis. <num value="1578">MDLXXVIII</num>. </s> </p> <p> <s xml:space="preserve">T. Celſitudinis.</s> </p> <p> <s xml:space="preserve">Deditiſſimus Mathematicus.</s> </p> <p> <s xml:space="preserve">Io. Bap. Benedictus.</s> </p> <pb facs="0223" n="211"/> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE CIRCVLO <lb/>AMBIENTE QVADRILATERVM.</head> <head xml:space="preserve">AD SERENISS. CAROLVM EMANVELEM <lb/>Pedemontis Principem.</head> <p> <s xml:space="preserve">PRoblema quod à celſitudine tua nobis proponitur non ſolum poſſibile eſt, ſed <lb/>facile etiam ad ſoluendum, hoc eſt quod circulus talis inueniatur, qui poſſit cir <lb/>cunſcribere, ſeu capere quadrilaterum ex quatuor datis rectis lineis terminatum, vel <lb/>ſic, datis quatuor rectis lineis ex quibus quadrilaterum poſſit eftici, tale efficiatur vt <lb/>circa ipſum, circulus poſſit circunſcribi.</s> </p> <p> <s xml:space="preserve">Sint igitur .4. lineæ propoſitæ <seg type="var">.b.d</seg>: <seg type="var">q.b</seg>: <seg type="var">a.q</seg>: et <seg type="var">.a.d.</seg> ex <choice><ex>quibus</ex><am>quibꝰ</am></choice> poſſibile ſit quadrilate <lb/>rum conſtitui, tale vero conſtituatur, vt aliquis circulus poſſit ipſum circunſcribere. <lb/></s> <s xml:space="preserve">imaginemur autem hoc factum eſſe, quod quidem quadrilaterum ſit <seg type="var">.a.d.q.b.</seg> cuius <lb/> <ptr xml:id="fig-0223-01a" corresp="fig-0223-01" type="figureAnchor"/> <ptr xml:id="fig-0223-02a" corresp="fig-0223-02" type="figureAnchor"/> <ptr xml:id="fig-0223-03a" corresp="fig-0223-03" type="figureAnchor"/> <ptr xml:id="fig-0223-04a" corresp="fig-0223-04" type="figureAnchor"/> <ptr xml:id="fig-0223-05a" corresp="fig-0223-05" type="figureAnchor"/> <pb facs="0224" n="212"/><fw type="head">IO. BAPT. BENED.</fw> diametri ſint <seg type="var">.q.d.</seg> et <seg type="var">.a.b.</seg> quæ ſe inuicem interſecent in puncto <seg type="var">.o.</seg> vnde cum anguli <lb/>contra ſe poſiti circa <seg type="var">.o.</seg> æquales inuicem ſint ex .15. primi Eucli. </s> <s xml:space="preserve">& angulus <seg type="var">.a.q.d.</seg> æ-<lb/>qualis angulo <seg type="var">.a.b.d.</seg> & angulus <seg type="var">.q.b.a.</seg> æqualis angulo <seg type="var">.q.d.a.</seg> et <seg type="var">.b.q.d.</seg> angulo <seg type="var">.b.a.d.</seg> <lb/>ex .20. tertij tunc triangulus <seg type="var">.a.o.q.</seg> ſimilis erit triangulo <seg type="var">.d.o.b.</seg> et <seg type="var">.q.o.b.</seg> ſimilis trian-<lb/>gulo <seg type="var">.a.o.d.</seg> ex definitione. </s> <s xml:space="preserve">Vnde eadem proportio erit ipſius <seg type="var">.q.o.</seg> ad <seg type="var">.b.o.</seg> quæ ipſius <lb/><seg type="var">q.a.</seg> ad <seg type="var">.b.d.</seg> & ipſius <seg type="var">.b.o.</seg> ad <seg type="var">.o.d.</seg> eadem quæ <seg type="var">.q.b.</seg> ad <seg type="var">.a.d.</seg> & ipſius <seg type="var">.q.o.</seg> ad <seg type="var">.o.a.</seg> eadem <lb/>quæ <seg type="var">.q.b.</seg> ad <seg type="var">.a.d.</seg> proportio igitur <seg type="var">.q.o.</seg> ad <seg type="var">.o.d.</seg> cognita nobis erit, vt compoſita ex <lb/>ea quæ eſt <seg type="var">.q.o.</seg> ad <seg type="var">.o.b.</seg> ex <seg type="var">.o.b.</seg> ad <seg type="var">.o.d.</seg> quæ nobis cognitę ſunt, mediante <lb/>proportione ipſius <seg type="var">.q.a.</seg> ad <seg type="var">.b.d.</seg> & ipſius <seg type="var">.q.b.</seg> ad <seg type="var">.a.d.</seg> proportio ſimiliter ipſius <seg type="var">.b.o.</seg> <lb/>ad <seg type="var">.o.a.</seg> nobis cognita erit, vt compoſita ex proportione ipſius <seg type="var">.b.o.</seg> ad <seg type="var">.o.q.</seg> & <lb/>ipſius <seg type="var">.o.q.</seg> ad <seg type="var">.o.a.</seg> cognitis, mediante proportione ipſius <seg type="var">.b.d.</seg> ad <seg type="var">.q.a.</seg> & ipſius <seg type="var">.q.b.</seg> ad <lb/><seg type="var">a.d.</seg> cum <choice><ex>autem</ex><am>autẽ</am></choice> proportio ipſius <seg type="var">.q.o.</seg> ad <seg type="var">.o.b.</seg> nobis cognita ſit, </s> <s xml:space="preserve">tunc nobis cognita erit <lb/>proportio ipſius <seg type="var">.q.d.</seg> ad <seg type="var">.a.b</seg>. </s> <s xml:space="preserve">Nam ut <seg type="var">.q.o.</seg> ad <seg type="var">.o.b.</seg> eſt vt <seg type="var">.a.o.</seg> ad <seg type="var">.o.d.</seg> ex ſimilitudine, <lb/></s> <s xml:space="preserve">quare proportio compoſiti ex primo, & quarto terminorum ad compoſitum ex .2. & <lb/>tertio, cognita erit. </s> <s xml:space="preserve">ſed quod fit ex <seg type="var">.q.d.</seg> in <seg type="var">.a.b.</seg> cognitum nobis eſt, vt æquale duobus <lb/>productis, hoc eſt ex <seg type="var">.q.a.</seg> in <seg type="var">.d.b.</seg> & ex <seg type="var">.q.b.</seg> in <seg type="var">.d.a.</seg> ex ſecunda primi Almageſti. </s> <s xml:space="preserve">quæ <lb/>producta nobis cognita ſunt, cum nobis data ſint eorum latera. </s> <s xml:space="preserve">Quapropter facta <lb/>cum fuerit figura quadrilatera rectangula ſimilis alicui alterirectangulæ figuræ pro <lb/>ductæ à duobus lateribus inuicem ita proportionatis, vt ſe habet <seg type="var">.q.d.</seg> ad <seg type="var">.a.b.</seg> æqua-<lb/>lis tamen duobus productis, hoc eſt producto ex <seg type="var">.q.a.</seg> in <seg type="var">.d.b.</seg> & ex <seg type="var">.q.b.</seg> in <seg type="var">.d.a.</seg> ex <lb/>doctrina, 25. ſexti Eucli quæ quidem figura, exempli gratia, ſit <seg type="var">.u.t.</seg> eius verò latera <lb/>ſint <seg type="var">.u.n.</seg> et <seg type="var">.n.t.</seg> Hæc enim dico æqualia eſſe <seg type="var">.q.d.</seg> et <seg type="var">.b.a.</seg> hoc eſt <seg type="var">.n.t.</seg> maius maio-<lb/>ri <seg type="var">.b.a.</seg> et <seg type="var">.u.n.</seg> minus minori <seg type="var">.q.d</seg>. </s> <s xml:space="preserve">Quod ita probabo. </s> <s xml:space="preserve">cogitemus rectangulum <seg type="var">.s.r.</seg> <lb/>productum eſſe ex duobus lateribus <seg type="var">.q.d.</seg> et <seg type="var">.a.b.</seg> ſed, <seg type="var">s.n.</seg> æqualis ſit <seg type="var">.q.d.</seg> et <seg type="var">.n.r.</seg> æqua-<lb/>lis <seg type="var">.a.b.</seg> <choice><ex>ſintque</ex><am>ſintq́;</am></choice> duæ lineæ <seg type="var">.s.n.</seg> et <seg type="var">.n.t.</seg> inuicem directè coniunctæ, vnde <seg type="var">.u.n.</seg> directè <lb/>coniuncta etiam erit cum <seg type="var">.n.r.</seg> ex quo rectangulum <seg type="var">.u.t.</seg> æquale erit rectangulo <seg type="var">.s.r.</seg> ex <lb/>communi conceptu, <choice><ex>eademque</ex><am>eademq́</am></choice> proportio erit <seg type="var">.u.n.</seg> ad <seg type="var">.n.t.</seg> quę <seg type="var">.s.n.</seg> ad <seg type="var">.n.r.</seg> eo <choice><ex>quod</ex><am>ꝙ</am></choice> ita fa-<lb/>ctum fuit, cum autem ita ſit <seg type="var">.u.n.</seg> ad <seg type="var">.n.t.</seg> vt <seg type="var">.s.n.</seg> ad <seg type="var">.n.r.</seg> </s> <s xml:space="preserve">tunc permutando ita erit <seg type="var">.n.t.</seg> ad <lb/><seg type="var">n.r.</seg> vt <seg type="var">.u.n.</seg> ad <seg type="var">.n.s.</seg> ſed quia ita eſt <seg type="var">.u.n.</seg> ad <seg type="var">.n.r.</seg> vt <seg type="var">.s.n.</seg> ad <seg type="var">.n.t.</seg> ex 15. ſexti, </s> <s xml:space="preserve">tunc permutan <lb/>do ita erit <seg type="var">.n.r.</seg> ad <seg type="var">.n.t.</seg> vt <seg type="var">.n.u.</seg> ad <seg type="var">.n.s.</seg> </s> <s xml:space="preserve">quare ex 11. quinti ita erit <seg type="var">.n.t.</seg> ad <seg type="var">.n.r.</seg> vt <seg type="var">.n.r.</seg> ad <seg type="var">.n.<lb/>t.</seg> quapropter ex neceſſitate ſequitur <seg type="var">.n.t.</seg> et <seg type="var">.n.r.</seg> inuicem æquales eſſe, et <seg type="var">.u.n.</seg> ſimiliter <lb/>cum <seg type="var">.n.s</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0223-01" corresp="fig-0223-01a"> <graphic url="0223-01"/> </figure> <figure xml:id="fig-0223-02" corresp="fig-0223-02a"> <graphic url="0223-02"/> </figure> <figure xml:id="fig-0223-03" corresp="fig-0223-03a"> <graphic url="0223-03"/> </figure> <figure xml:id="fig-0223-04" corresp="fig-0223-04a"> <graphic url="0223-04"/> </figure> <figure xml:id="fig-0223-05" corresp="fig-0223-05a"> <graphic url="0223-05"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Inuentæ nunc cum fuerint duæ diametri <seg type="var">.q.d.</seg> et <seg type="var">.a.b.</seg> ipſius quadrilateri, difficile <lb/>non erit eius angulos inuenire, eo <choice><ex>quod</ex><am>ꝙ</am></choice> mediante <seg type="var">.a.b.</seg> cognita, ſimul cum <seg type="var">.b.d.</seg> et <seg type="var">.a.d.</seg> da <lb/>tis, faciemus triangulum <seg type="var">.a.b.d.</seg> vel <choice><ex>mediante</ex><am>mediãte</am></choice> <seg type="var">.q.d.</seg> et <seg type="var">.q.a.</seg> et <seg type="var">.a.d.</seg> cognitis faciemus <choice><ex>triam</ex><am>triã</am></choice> <lb/>gulum <seg type="var">.a.q.d.</seg> ex .22. primi. </s> <s xml:space="preserve">Vnde cum centrum circuli circunſcriptibilis cuiuſuis di-<lb/>ctorum triangulorum ex quinta quarti inuentum fuerit, triangulum reliquum, ab eo <lb/>dem circulo circunſcriptum erit, ex communi ſcientia.</s> </p> <p> <s xml:space="preserve">SEd vt ipſa operatio facilior fiat, Sint eędem lineæ <seg type="var">.b.d</seg>: <seg type="var">b.q</seg>: <seg type="var">a.q.</seg> et <seg type="var">.a.d.</seg> ex quibus <lb/>poſſit <choice><ex>quadrilaterum</ex><am>quadrilaterũ</am></choice> effici. </s> <s xml:space="preserve">Videatur deinde primò quas volumus oppoſitas ſibi <lb/>inuicem eſſe, ponatur ergò ut <seg type="var">.q.a.</seg> et <seg type="var">.b.d.</seg> velimus oppoſitas inuicem facere, et <seg type="var">.q.b.</seg> <lb/>cum <seg type="var">.a.d.</seg> ſimiliter, accipiemus nunc <seg type="var">.K.</seg> cuiuſuis magnitudinis, cui comparetur <seg type="var">.e.</seg> <lb/>ita proportionata, vt <seg type="var">.q.b.</seg> eſt ipſi <seg type="var">.a.d.</seg> ex doctrina .10. ſexti Eucli. vel accipiatur <seg type="var">.a.d.</seg> <lb/>vice <seg type="var">.K.</seg> et <seg type="var">.q.b.</seg> vice <seg type="var">.e.</seg> quod idem erit, & expeditius, inuenietur ſimiliter <seg type="var">.h.</seg> ita pro-<lb/>portionata ad <seg type="var">.e.</seg> et <seg type="var">.g.</seg> ad <seg type="var">.k.</seg> vt <seg type="var">.b.d.</seg> eſt ad <seg type="var">.q.a.</seg> vel <seg type="var">.g.</seg> ad <seg type="var">.h.</seg> vt <seg type="var">.a.d.</seg> ipſi <seg type="var">.q.b.</seg> quod <choice><ex>idem</ex><am>idẽ</am></choice> erit.</s> </p> <p> <s xml:space="preserve">Hoc facto coniungantur inuicem directè <seg type="var">.g.</seg> et <seg type="var">.e.</seg> quarum compoſitum ſit <seg type="var">.g.e.</seg> & <lb/>ita duæ <seg type="var">.K.</seg> et <seg type="var">.h.</seg> ex quibus ſit <seg type="var">.K.h</seg>. </s> <s xml:space="preserve">Nunc ex iſtis duabus lineis <seg type="var">.e.g.</seg> et <seg type="var">K.h.</seg> fiat paral- <pb facs="0225" n="213"/><fw type="head">EPISTOLAE</fw> lelogrammum <seg type="var">.Z.</seg> deinde fiant alia duo parallelogramma rectangula quorum vnum <lb/>ſit ex <seg type="var">.q.a.</seg> in <seg type="var">.b.d.</seg> reliquum verò ſit ex <seg type="var">.q.b.</seg> in <seg type="var">.a.d.</seg> quæ quidem ſint <seg type="var">.f.m</seg>.</s> </p> <p> <s xml:space="preserve">Quo facto deſignetur rectangulum <seg type="var">.u.t.</seg> ex .25. ſexti, quod æquale ſit duobus re-<lb/>ctangulis <seg type="var">.f.</seg> et <seg type="var">.m.</seg> ſimile tamen <seg type="var">.Z.</seg> cuius rectanguli vnum latus correſpondet <seg type="var">.e.g.</seg> reli-<lb/>quum verò <seg type="var">.K.h.</seg> in proportione, ſed in æqualitate, vnum correſpondet <seg type="var">.q.d.</seg> <choice><ex>reliquum</ex><am>reliquũ</am></choice> <lb/>vero <seg type="var">.a.b.</seg> diametris ipſius quadrilateri.</s> </p> <p> <s xml:space="preserve">Accipiatur nunc latus illud quod correſpondet <seg type="var">.K.h.</seg> hoc eſt ipſi <seg type="var">.a.b.</seg> maius ſcili-<lb/>cet, & ſimul cum <seg type="var">.b.d.</seg> et <seg type="var">.a.d.</seg> formetur <choice><ex>triangulum</ex><am>triãgulũ</am></choice> <seg type="var">.a.b.d.</seg> ex .22. primi Eucli. circa quod <lb/>circunſcribatur circulus ex .5. quarti. </s> <s xml:space="preserve">& inuentum erit quod quęrebamus.</s> </p> <figure place="here"> <graphic url="0225-01"/> </figure> <figure place="here"> <graphic url="0225-02"/> </figure> <figure place="here"> <graphic url="0225-03"/> </figure> <figure place="here"> <graphic url="0225-04"/> </figure> <figure place="here"> <graphic url="0225-05"/> </figure> <pb facs="0226" n="214"/> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">PER EVNDEM PARALLELVM <lb/>abſque correctione ſemper nauigari <lb/>non poſſe.</head> <head rend="italics" xml:space="preserve">Vbi not antur Petri <choice><ex>Nonij</ex><am>Nonij</am></choice> lapſus in correctione erroris nauis. <lb/>Et alij Petri Medinæ errores.</head> <head xml:space="preserve">ILLVSTRISSIMO ANDREAE PROVANAE <lb/>Leinici Domino, Fruzaſci comiti, Aequiti Torquato, inthimo Sere-<lb/>niſsimi Sabaudiæ Ducis Conſiliario, <choice><ex>eiuſque</ex><am>eiuſq́;</am></choice>, & ſacræ religio-<lb/>nis ſanctorum Mauritij, & Lazari Claſsi Præfecto.</head> <p> <s xml:space="preserve"><hi rend="small caps">INter</hi> Eximiastuas virtutes, reinauticæ peritia Illuſtris emicat <lb/>merito ad te ſcribendum duxi, quod ad eam facultatem perti-<lb/>nens excogitaui, ſimul cum quibuſdam alijs inſtrumentis, vt non-<lb/>nihil commodi attuliſſe videar maritimis negotijs, & aliqua ex <lb/>parte animi mei erga te propenſionem indicauiſſe.</s> </p> <p> <s xml:space="preserve">PEr vnum <choice><ex>eum</ex><am>eũ</am></choice><choice><ex>demque</ex><am>demq́;</am></choice> parallelum in primis <choice><ex>abſque</ex><am>abſq;</am></choice> aliqua corre <lb/>ctione ſemper nauigari poſſe, omninò nego. </s> <s xml:space="preserve"><choice><ex>Nam</ex><am>Nã</am></choice>, <choice><ex>verum</ex><am>verũ</am></choice> eſtid quod Petrus Nonius in <lb/>initio ſui operis oſtendit, ideſt nauim verſus æquatorem ſemper declinare: </s> <s xml:space="preserve">qui <choice><ex>cum</ex><am>cũ</am></choice> <lb/>corrigit errorem, fallitur, cum ipſe, eandem nauim, parallelam æquatori in vno ver <lb/>ticali ipſi æquatori propinquiori, & non in primo parallelo dirigit, itaque exiſtimat <lb/>in fine itineris, vbi deſcribit punctum <seg type="var">.o.</seg> eam in eodem parallelo priori repe-<lb/>riri debere, quod <choice><ex>verum</ex><am>verũ</am></choice> <choice><ex>non</ex><am>nõ</am></choice> eſt, quia ea correctio efficit, vt motus nauis <choice><ex>effectum</ex><am>effectũ</am></choice> <choice><ex>cuiuſdam</ex><am>cuiuſdã</am></choice> <lb/><choice><ex>deſcenſus</ex><am>deſcẽſus</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> ſcaligradum pręſtet, in quo à gradu in <choice><ex>gradum</ex><am>gradũ</am></choice> fiat deſcenſus, ſed ſi per gra <lb/>dus <choice><ex>tantum</ex><am>tm̃</am></choice> aſcenderet <choice><ex>quantum</ex><am>quãtũ</am></choice> <choice><ex>deſcendit</ex><am>deſcẽdit</am></choice>, <choice><ex>dubium</ex><am>dubiũ</am></choice> <choice><ex>non</ex><am>nõ</am></choice> eſt quin in fine ita eſſet ſe habitura <choice><ex>quem</ex><am>quẽ</am></choice> <lb/>ad modum in medio & in principio, cum verò ſemper deſcendat, abſque vlla aſcen-<lb/>ſione, neceſſariò ſic ſemper procedens, remota cum eſſent impedimenta terræ, ſub <lb/>æquatore reperiretur, ſub quo perpetuò circuiret globum.</s> </p> <p> <s xml:space="preserve">Idem ſub quolibet meridiano præſtare poteſt, ideſt vno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> vento circun-<lb/>uerti: </s> <s xml:space="preserve">ſed per alios circulos quam per hos duos (ſiue circulus magnus ſiue paruus) id <lb/>nunquam perfectè efficere poteſt, de parallelis iam manifeſtum eſt, cum impetus na <lb/>turalis corporum, quæ mota ſunt ſint ſemper in ſuperficiebus circulorum maiorum, <lb/>quorum circunferentię cum circunferentijs minorum, præter quam per vnum quod <lb/>dam punctum quando adinuicem contiguæ ſunt, aut per duo ideſt cum ſe ſe interſe-<lb/>cant non communicant, ita quod ad efficiendum, vt triremis aliqua, aut nauis, per <lb/>aliquem ex parallelis ad æquatorem moueatur, neceſſario ſit futurum, vt ratione <choice><ex>con- tiguitatis</ex><am>cõ-tiguitatis</am></choice> & non continuitatis eam moueri curemus. </s> <s xml:space="preserve">quia ratione continuitatis om-<lb/>ninò fieri non <choice><ex>pont</ex><am>põt</am></choice>, aut conſtet virtus mouens remis, aut velis. </s> <s xml:space="preserve">Sed per quemlibet <choice><ex>alium</ex><am>aliũ</am></choice> <lb/>circulum maiorem, qui non ſit aut æquator aut aliquis ex meridianis, eſt penitus im <lb/>poſſibile. </s> <s xml:space="preserve">ideſt vt vnius venti vi nauis impellatur. </s> <s xml:space="preserve">Quod vt clarè pateat, ſit orizon <seg type="var">.<lb/>a.c.b.d.</seg> & æquator <seg type="var">.c.q.t.d.</seg> & vnus meridianorum ſit <seg type="var">.a.r.n.t.b.</seg> in quo <seg type="var">.n.</seg> ſit Zenit ſub <lb/>quo primum nauis reperiatur et <seg type="var">.r.</seg> ſit polus ſeptentrionalis. </s> <s xml:space="preserve">Ponamus etiam quod <pb facs="0227" n="215"/><fw type="head">EPISTOLA.</fw> azimut <seg type="var">.f.q.u.n.p.</seg> conſtituat angulum <seg type="var">.a.n.p.</seg> ſeu <seg type="var">.f.n.b.</seg> cum meridiano graduum .45. <lb/>vnde tot graduum eruntarcus <seg type="var">.a.p.</seg> et <seg type="var">.b.f.</seg> orizontis, quapropter punctum <seg type="var">.f.</seg> commu-<lb/>ne ipſius orizontis cum azimut, erit medio in loco inter <seg type="var">.b.</seg> et <seg type="var">.c.</seg> & ideo quarta <seg type="var">.n.f.</seg> ip <lb/>ſius azimut ſecabit quartam <seg type="var">.c.t.</seg> ipſius æquatoris in puncto <seg type="var">.q.</seg> & habebimus triangu-<lb/>Ium <seg type="var">.q.t.n.</seg> cuius angulus <seg type="var">.t.</seg> rectus erit, & angulus <seg type="var">.n.</seg> cognitus ſimul cum latere <seg type="var">.n.t.</seg> la-<lb/>titudinis loci, quibus rebus mediantibus deueniemus in cognitionem lateris <seg type="var">.q.n.</seg> la-<lb/>teris <seg type="var">.q.t.</seg> & anguli <seg type="var">.q.</seg> ex .4. primi Copernici ſi voluerimus.</s> </p> <p> <s xml:space="preserve">Ponamus nunc nauem à puncto <seg type="var">.n.</seg> diſcedere ſeu iter facere verſus <seg type="var">.u.</seg> punctum, & <lb/> <ptr xml:id="fig-0227-01a" corresp="fig-0227-01" type="figureAnchor"/> in ipſo <seg type="var">.u.</seg> reperiri, iam in hoc ſitu ha-<lb/>bebimus angulum huius ſecundi me-<lb/>ridiani <seg type="var">.r.u.p.</seg> qui quidem in hoc caſu <lb/>minor eſſet angulo <seg type="var">.r.n.p.</seg> extrinſeco <lb/>trianguli <seg type="var">.r.u.n.</seg> ex conuerſo ſecundæ <lb/>partis .48. <choice><ex>propoſitionis</ex><am>ꝓpoſitionis</am></choice> tertij lib. de <choice><ex>trian gulis</ex><am>triãgulis</am></choice> Monteregij, ſeu ex .13. primi Me <lb/>nelai, cuius anguli <seg type="var">.u.</seg> arcus orizontalis <lb/>ſit <seg type="var">.x.e.</seg> qui quidem minor erit arcu <seg type="var">.a.<lb/>p.</seg> vt patet ratione anguli <seg type="var">.r.u.e.</seg> mino-<lb/>ris, ergo alius ventus nauem impellet à <lb/>puncto .11. verſus <seg type="var">.q.</seg> diuerſus ab illo qui <lb/>prius ab <seg type="var">.n.</seg> verſus <seg type="var">.u.</seg> eam impellebat.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0227-01" corresp="fig-0227-01a"> <graphic url="0227-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Vnde clarè patet verum eſſe quod <lb/>dico, hoc eſt quod aliquo modo fieri <lb/>non poteſt, vt nauis ab aliquo loco ad <lb/>alium, breuiſſimo interuallo ire poſſit <lb/>ideſt per gyrum circuli maioris ſphæræ vno tantummodo <choice><ex>vento</ex><am>vẽto</am></choice> eam impellente, præ <lb/>ter quam in ęquatore, ſeu in aliquo quouis meridianorum, nos autem ire per gyrum <lb/>alicuius paralleli dementia eſſet, niſi neceſſitas cogeret.</s> </p> <p> <s xml:space="preserve">Huiuſmodi demonſtrationis ope, quantum decipiatur Petrus Medina cap .6. lib. <lb/>3. cognoſcitur, vbi ſic ſcribit; </s> <s xml:space="preserve">Vbicunque locorum reperiatur homo, aliquem circu-<lb/>lum qui vniuerſum ambiat imaginatione ſibi confingens, per totum eum circulum <lb/>vno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> vento nauigatio ſuſcipitur. </s> <s xml:space="preserve">Ex hac etiam demonſtratione, quàm fal <lb/>ſa ſit charta maritima patet, cuius beneficio exiſtimant nautę ſe per breuiſſimum iter <lb/>a loco ad locum vehi etiamſi dicti loci non ſint ambo in æquatore, aut in aliquo me <lb/>ridiano, ſed extra dictos circulos vnico tantum vento impellente & ſi in paruis æquo <lb/>fibus hic error parum depræhenditur, forte tamen in magno Oceano clarè pateret. <lb/></s> <s xml:space="preserve">In ſuperius igitur dicta demonſtratione iam oſtendi, quod ſi velimus vehi ab vno lo <lb/>co ad alium beneficio alicuius circuli maioris, præter duos iam dictos, hoc fieri non <lb/>poteſt vno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> <choice><ex>vento</ex><am>vẽto</am></choice> impellente. </s> <s xml:space="preserve">Vnde ſequitur, omnia ea interualla quæ vno <lb/><choice><ex>eodemque</ex><am>eodemq́</am></choice> vento tranſibimus futura longiora, præterquam in duobus dictis circulis <lb/>æquinoctiali & meridiano.</s> </p> <p> <s xml:space="preserve">Cum verò Petrus Medina cap .7. volens probare chartam maritimam bonam eſ-<lb/>ſe, planiſphęrium Prolomei & Iordani citat, non animaduertit quam diuerſo mo-<lb/>do a charta maritima huiuſmodi inſtrumentum ſit fabricatum, cum exceptis orizon <lb/>te recto, & meridiano in dicto inſtrumento quilibet alius circulus ſit circulus, ſiue ſit <lb/>almicantarat, ſiue azimur, ſiue æquator, ſiue tropicus, ſiue zodiacus, ſiue alius quiuis <lb/>circulus, eum in charta maritima ne vna quidem ſit linea, quę non ſit recta, quolibet <lb/>nomine vocetur.</s> </p> <pb facs="0228" n="216"/> <fw type="head">IO. BABPT. BENED.</fw> <p> <s xml:space="preserve">Superius poſitæ meæ demonſtrationis ope, deuenimus in cognitionem magnitu <lb/>dinis arcus <seg type="var">.n.q.</seg> cognoſcimus etiam angulum <seg type="var">.n.q.t.</seg> vnde nobis <choice><ex>manifeſtum</ex><am>manifeſtũ</am></choice> eſſet quo <lb/>vento oporteret iter facere. cum à puncto <seg type="var">.q.</seg> nauis aliqua diſceſſura eſſet, in eodem <lb/>azimut propoſito. </s> <s xml:space="preserve">Idem etiam dico de puncto <seg type="var">.u.</seg> cum cogniti eſſent arcus <seg type="var">.n.u.</seg> et <seg type="var">.n.<lb/>r.</seg> vt ſupponitur, ſimul cum angulo <seg type="var">.r.n.u.</seg> vnde cognitus eſſet nobis angulus <seg type="var">.n.u.r.</seg> ex <lb/>11. primi lib. Copernici, ex quo ventus nobis cognitus foret.</s> </p> <p> <s xml:space="preserve">Modus autem quem idem Medina cap .9. lib. tertij ad cognoſcendam diſtantiam <lb/>vnius meridiani ab alio præſcribit, in genere eſt falſus, etiam ſi is ab antiquis eum de <lb/>ſumat, qui, hic non viderunt quam magna inter meridianos differentia ſit interuallo <lb/>rum eorum quæ ſunt vicina polis & eorum quæ ſunt circa æquatorem.</s> </p> <p> <s xml:space="preserve">Falſus eſt etiam modus ab eo traditus ad cognoſcendos gradus longitudinis per <lb/>medium itineris cogniti in quouis parallelo extra æquatorem facti, & hoc cap .14. li <lb/>bri tertij eiuſdem, & primo cap. lib. 4. <choice><ex>continetur</ex><am>cõtinetur</am></choice>, vbi .17. leucas cum dimidia cuilibet <lb/>gradui tam paralleli quàm meridiani aſſignat.</s> </p> <p> <s xml:space="preserve">Falſum eſt etiam quod ab eo aſſeritur, Solem, cum reperiretur in æquatore, circa <lb/>eos qui ſub ipſo æquatore habitant, vnius diei <choice><ex>noctisque</ex><am>noctisq́;</am></choice> ſpatio per omnes uentos cir-<lb/>cunuolui. </s> <s xml:space="preserve">quia illis æquator idem eſt cum verticali, qui duos tantum rhumbos pro-<lb/>ducit, ideſt orientis, & occidentis: </s> <s xml:space="preserve">hic verò error, in ſecundo cap. lib. 6. habetur.</s> </p> <p> <s xml:space="preserve">Falſum eſt etiam quod profert Solem ijs qui habitant ſphæram obliquam, qua-<lb/>libet hora tertia, regulariter ab vno rhumbo ad alium ex præcipuis ideſt ab vno azi <lb/>mut ad alium progredi, quemadmodum eadem cap .2. lib. 6. et .7. cap. ſeptimi libr. <lb/>ſcribit. </s> <s xml:space="preserve">Huius autem rei falſitas ita facile depræhendetur, ponamus hemiſphęrium <lb/>orientale, verbi gratia, cuius meridianus ſit <seg type="var">.p.z.b.</seg> æquator <choice><ex>autem</ex><am>aũt</am></choice> <seg type="var">.e.m.</seg> vnus verò paral <lb/>lelorum ſeptentrionalium ſit <seg type="var">.c.a.</seg> in quo Solem exiſtere ponamus, orizon autem ſit <seg type="var">.<lb/>b.m.</seg> zenit vero <seg type="var">.z.</seg> polus arcticus <seg type="var">.p.</seg> ſit poſtea azimut <seg type="var">.z.q.</seg> à meridiano diſtans per gra <lb/>dus .45. qui quidem azimut in hoc hemiſphęrio erit rhumbus illius venti, quem uul-<lb/>go Itali Sirocum dicunt, et <seg type="var">.z.m.</seg> ſit azimut verticalis qui in hoc hemiſphærio erit <choice><ex>ron</ex><am>rõ</am></choice> <lb/>bus venti orientalis, ita <choice><ex>quod</ex><am>ꝙ</am></choice> ſecundum Medinam à rhumbo <seg type="var">.z.m.</seg> ad <seg type="var">.z.q</seg>. </s> <s xml:space="preserve">Solabſoluet <lb/>ſpatium <choice><ex>temporis</ex><am>tẽporis</am></choice> trium horarum, & aliud <lb/>æquale temporis ſpatium abſoluet à <choice><ex>rhum</ex><am>rhũ</am></choice> <lb/> <ptr xml:id="fig-0228-01a" corresp="fig-0228-01" type="figureAnchor"/> bo <seg type="var">.z.q.</seg> ad <seg type="var">.z.b.</seg> ex ipſo Medina, vnde ar-<lb/>cus <seg type="var">.a.o.</seg> paralleli eſſet graduum .45. & <lb/>item arcus <seg type="var">.o.c</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0228-01" corresp="fig-0228-01a"> <graphic url="0228-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Ponamus <choice><ex>nunc</ex><am>nũc</am></choice> Solem reperiri in ęqua <lb/>tore, vbi per ipſum Medinam arcus <seg type="var">.u.m.</seg> <lb/>ſimiliter eſſet graduum .45. & ſic <seg type="var">.u.e</seg>: pro <lb/>tracto ergo arcu <seg type="var">.p.o.f.</seg> palam erit ar-<lb/>cum <seg type="var">.f.e.</seg> fore graduum .45. ſed cum <lb/>arcus <seg type="var">.e.u.</seg> ſit graduum .45. ex ſuppoſito <lb/>ipſius Medinæ, ſequeretur arcum <seg type="var">.e.f.</seg> <lb/>æqualem eſſe arcui <seg type="var">.e.u.</seg> pars igitur æqua <lb/>lis erit ſuo toto.</s> </p> <p> <s xml:space="preserve">Id etiam quod Petrus Nonius pagina <lb/>124. et .125. lib. de arte nauigandi con-<lb/>tra nautas de diſtantijs Solis à meridiano ſcribit, hanc opinionem Petrià Medina & <lb/>corum qui idem ei perſuaſerunt falſam eſſe demonſtrat.</s> </p> <pb facs="0229" n="217"/> <fw type="head">EPISTOLAE.</fw> <p> <s xml:space="preserve">Falſum eſt etiamid quod cap .3. lib. 6. pronuntiat, ita dicens.</s> </p> <p> <s xml:space="preserve">Quod cum verum eſſet à parte <choice><ex>orientali</ex><am>oriẽtali</am></choice> inſularum quæ azore dicuntur, pyxidem <lb/>verſus eum ventum qui vulgò Græcus dicitur, & ab occidentali verſus eum qui Ma <lb/>giſter dicitur, vergere, huius rei nulla eſt ratio.</s> </p> <p> <s xml:space="preserve">Ego enim huiuſmodi rationem reperiri poſſe contendo, quæ talis eſt, quia pars <lb/>roſæ (ut vocant) à magnete tacta, ad aliquod punctum, aut ſitum globi terrę, in eo-<lb/>dem meridiano inſularum, quæ Azore dicuntur, vltra ſitum poli arctici in terra diri-<lb/>geretur, ita vt ſitus dicti poli in terra eſſet in dicto meridiano, inter locum qui ab in <lb/>dice roſæ aut pyxidis reſpiceretur, & dictas inſulas, id quod ſuperius ſcripto meridia <lb/>no facile cognoſci poteſt, ſumendo pro inſulis ſitum <seg type="var">.e.</seg> in meridiano et <seg type="var">.z.</seg> pro polo, <lb/>et <seg type="var">.p.</seg> pro loco qui à pyxide ſit viſus, <choice><ex>imaginando</ex><am>imaginãdo</am></choice> deinde pyxidem in <seg type="var">.f.</seg> magis orienta <lb/>li quam eſt <seg type="var">.e.</seg> clarum eſt lineam quæ reſpicit (ponamus) <seg type="var">f.p.</seg> verſus Græcum & ab alia <lb/>parte verſus Magiſtrum declinare.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De Armilla Nautica.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">CVm ſæpe viderim quam in magnis æquoribus nos fallant, <choice><ex>atque</ex><am>atq;</am></choice> decipiant mari <lb/>timæ, ſeu nauigatoriæ chartę, quemadmodum aliquoties inter nos ſermonem <lb/>habuimus: </s> <s xml:space="preserve">in id totus incubui vt aliquam machinam excogitarem, quæ difficilis non <lb/>eſſet, <choice><ex>efficeretque</ex><am>efficeretq́;</am></choice> vt nauis ſuper aquę globum, beneficio circulorum maiorum, quam <lb/>optimè poſſet, ideſt breuiſſimo itinere ab uno loco <choice><ex>and</ex><am>ãd</am></choice> alium ferretur. </s> <s xml:space="preserve">Id <choice><ex>quod</ex><am>ꝙ</am></choice> mihi ex <lb/>animi voto ſucceſſurum putaui, beneficio <choice><ex>quinque</ex><am>quinq;</am></choice> circulorum circundantium <choice><ex>aliquem</ex><am>aliquẽ</am></choice> <lb/>globum <choice><ex>terreſtrem</ex><am>terreſtrẽ</am></choice> & maritimum, quales ij ſunt qui in inferiori Germania à Gerardo <lb/>Mercatore ſtruuntur, qui vno pede cum dimidio diametri conſtet, ideſtſeſquipede.</s> </p> <p> <s xml:space="preserve">Sit ergo, exempli gratia, huiuſmodi globus <seg type="var">.a.b.d.</seg> circa quem duo circuli, aut cir<lb/> <ptr xml:id="fig-0229-01a" corresp="fig-0229-01" type="figureAnchor"/> <pb facs="0230" n="218"/><fw type="head">IO. BAPT. BENED.</fw> culares lineæ ex aurichalco applicentur inuicem coniuncti per medium ad angulos <lb/>rectos, quorum prior <seg type="var">.f.e.g.</seg> in ſe globi polos mediantibus extremitatibus axis mun-<lb/>di contineat, qui quidem poli à punctis ſuarum interſectionum per <choice><ex>quartam</ex><am>quartã</am></choice> ex æquo <lb/>in punctis <seg type="var">.f.</seg> et <seg type="var">.g.</seg> ita diſtent, vt globus circa eoſdem, in ſitu longitudinis mundi vol-<lb/>ui poſſit. </s> <s xml:space="preserve">Huiuſmodi autem circulus, æquatoris deferens appelletur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0229-01" corresp="fig-0229-01a"> <graphic url="0229-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Secundus autem circulus ſit <seg type="var">.h.e.K.</seg> cum primo ad angulos rectos in puncto <seg type="var">.e.</seg> & <lb/>in ſuo oppoſito connexus, & is appellabitur æquator, & poli <seg type="var">.f.g.</seg> primi poli dicentur.</s> </p> <p> <s xml:space="preserve">Circa huiuſmodi duos circulos, alios <choice><ex>etiam</ex><am>etiã</am></choice> duos exiſtere <choice><ex>vellem</ex><am>vellẽ</am></choice> ſimul <choice><ex>coniunctos</ex><am>cõiũctos</am></choice> medio <lb/>ad angulos rectos. </s> <s xml:space="preserve">In quibus <choice><ex>quidem</ex><am>quidẽ</am></choice> interſectionis punctis ſint duo poli, qui hos duos <lb/>circulos cum ſecundo priorum ideſt cum æquatore in duobus punctis inuicem op-<lb/>poſitis connectant; </s> <s xml:space="preserve">quæ æquatoris puncta à punctis interſectionis eiuſdem cum ſuo <lb/>deferente, ratione vna quarta diſtent, quorum duorum circulorum primus ſit <seg type="var">.n.i.<lb/>m.</seg> quem deferentem azimut appellabimus; </s> <s xml:space="preserve">ſecundus <seg type="var">.r.n.s.m.</seg> azimut locorum no<lb/>minabimus. </s> <s xml:space="preserve">eorundem interſectionis rectæ, puncta ſint <seg type="var">.n.</seg> et <seg type="var">.m.</seg> à quibus duo poli ex <lb/>aurichalco confecti ſimiles primis <seg type="var">.n.h.</seg> et <seg type="var">.m.K.</seg> vſque ad puncta <seg type="var">.h.</seg> et <seg type="var">.K.</seg> æquatoris <lb/>perueniant, qui ſpisſitudinem æquatoris diſtantem à puncto <seg type="var">.e.</seg> vna quarta <choice><ex>penetrent</ex><am>penetrẽt</am></choice>, <lb/>ita vt æquator circum circa <seg type="var">.n.h.</seg> et <seg type="var">.m.K.</seg> in ſitu latitudinis mundi verti queat. </s> <s xml:space="preserve">Et <lb/>hos, ſecundos polos nominabimus.</s> </p> <p> <s xml:space="preserve">Alius deinde circulus <seg type="var">.q.i.p.</seg> duos poſteriores circulos ambiat, cum deferente ta-<lb/>men azimut mediantibus duobus polis in puncto <seg type="var">.i.</seg> & in ſuo oppoſito exęquo diſtan <lb/>tibus à ſecundis polis vnius quartæ ſpatio iungatur. </s> <s xml:space="preserve">Ita vt dictum deferens azimut <lb/>circa hos tertios polos volui poſſit, atque hunc circulum <seg type="var">.q.i.p.</seg> orizontem vniuerſa <lb/>lem vocabimus. </s> <s xml:space="preserve">Hic vero orizon ſuper quatuor quartas circuli, aut ſuper quatuor <lb/>paruis columnis, ut fieri ſolet innixis ſuæ baſi, ita ponatur, vt moueri non poſſit.</s> </p> <p> <s xml:space="preserve">Primus autem circulus <seg type="var">.f.e.g.</seg> deferens æquatoris in .4. partes æquales diuidatur, <lb/>quarum quælibet .90. gradibus conſtet, incipiendo ab interſectionibus <seg type="var">.e.</seg> & eius op <lb/>poſito æquatoris, & numeri in polis <seg type="var">.f.</seg> et <seg type="var">.g.</seg> globi finem ſortiantur. </s> <s xml:space="preserve">Diuidatur etiam <lb/>æquator <seg type="var">.h.e.K.</seg> in .360. partes incipientes à puncto <seg type="var">.e.</seg> verſus <seg type="var">.K.</seg> deferens autem azi <lb/>mut <seg type="var">.n.i.m.</seg> ab omni diuiſione liber maneat, ſed azimut <seg type="var">.n.s.m.r.</seg> in .360. gradus inci <lb/>piendo à puncto <seg type="var">.n.</seg> verſus <seg type="var">.r.</seg> diuidatur.</s> </p> <p> <s xml:space="preserve">Orizon autem <seg type="var">.q.i.p.</seg> diuidatur in quartas, quarum quælibet ſit nonaginta <choice><ex>graduum</ex><am>graduũ</am></choice> <lb/>incipiendo à puncto <seg type="var">.i.</seg> & eius oppoſito ideſt à polis poſtremis & terminando in pun-<lb/>ctis <seg type="var">.q.</seg> et <seg type="var">.p.</seg> in medio ipſorum polorum, & quarta <seg type="var">.i.p.</seg> orientalis ſeptentrionalis, et <seg type="var">.i.<lb/>q.</seg> orientalis meridiana appellentur. </s> <s xml:space="preserve">& ſic ordine ſeruato occidentales.</s> </p> <p> <s xml:space="preserve">Præterea pręparata ſit quædam quarta, ex aurichalco, circuli æqualis ipſi orizon-<lb/>ti, & in .90. gradus diſtincta quæ cum quauis ſuarum extremitatum ipſi zenit, in azi-<lb/>mut applicari poſſit, quemadmodum circa globos cęleſtes fieri ſolet; </s> <s xml:space="preserve">quę quidem ad <lb/>cognoſcendam altitudinem poli ipſius globi ab orizonte nobis inſeruiet.</s> </p> <p> <s xml:space="preserve">Atque hac ratione hanc noſtram machinam perfectè abſoluemus <choice><ex>quam</ex><am>quã</am></choice> appellan-<lb/>dam eſſe Armillam nauticam ſentio. </s> <s xml:space="preserve">Hic autem illud non omittam, concauum <lb/>duorum priorum circulorum à ſuperficie globi non nimis diſtare debere & con-<lb/>cauum aliorum à ſuperficie conuexa priorum longe poſitos eſſe <choice><ex>non</ex><am>nõ</am></choice> debere, & con <lb/>cauam orizontis à conuexa ſecundorum procul abeſſe non debere.</s> </p> <p> <s xml:space="preserve">Neque illud etiam prætermittendum eſt, opere pretium fore ſi in interſectione <lb/>e. priorum, erit foramen elicum, vt clauo elico ex aurichalco confecto, poſſimus <lb/>ſiſtere globum, quando oportuerit, ne amplius circa primos ſuos polos <seg type="var">.f.g.</seg> circun-<lb/>uoluatur, cum <choice><ex>ſitus</ex><am>ſitꝰ</am></choice> fuerit. </s> <s xml:space="preserve">Inde etiam laudo vt in azimut <seg type="var">.r.n.s.m.</seg> è regione deferen <lb/>tis æquatoris, ideſt <seg type="var">.f.e.g.</seg> aliud quoddam foramen huiuſmodi ſit poſitum, in quo <pb facs="0231" n="219"/><fw type="head">EPISTOL AE.</fw> clauus elicus vſque ad circulum <seg type="var">.f.e.g.</seg> perueniens, æquatorem ſiſtere poſſit, ne circa <lb/>ſecundos polos <seg type="var">.h.</seg> et <seg type="var">.K.</seg> amplius moueatur quum noluerimus eum mutare ſitum.</s> </p> <figure place="here"> <graphic url="0231-01"/> </figure> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Deuſu Armillæ nauticæ.</head> <p> <s xml:space="preserve">V Tautem noſtra Armilla nautica vti poſſimus pyxidem nos prius oportebit <lb/>habere, diuerſam tamen ab ijs, quibus nautæ hactenus vſi fuere: </s> <s xml:space="preserve">nolo <lb/>enim vt <choice><ex>tam</ex><am>tã</am></choice> craſſa minerua beneficio <choice><ex>ventorum</ex><am>vẽtorum</am></choice> communium circa hanc rem nos gera <lb/>mus, ſed ratione graduum orizontis in .360. partes diſtincti, atque ob hanc cauſam <lb/>ſentio, vt ima pars pyxidis penitus detecta videatur, & in .360. partes dinidatur, <choice><ex>nilque</ex><am>nilq́</am></choice> <lb/>aliud quam quandam lanceo lam ſupra eius acum eſſe volo, quæ dum mouebitur na <lb/>uis, per gradus quamlibet orizontis partem oſtendet; </s> <s xml:space="preserve">hos autem .360. gradus, ita ſe <lb/>habere volo, vt quęlibet quarta .90. contineat, <choice><ex>ſupputatioque</ex><am>ſupputatioq́;</am></choice> à linea meridiana inci-<lb/>piat, & in verticali deſinat, vt huiuſmodi diuiſio cum ea, quæ eſt orizontis Armillæ <lb/>eadem ſit.</s> </p> <p> <s xml:space="preserve"><choice><ex>Pręſupponantur</ex><am>Pręſupponãtur</am></choice> nunc in globo duo loci extra æquatorem, & in diuerſis meridia-<lb/>nis quomodolibet a dinuicem diſtantes, à quorum vno ad alium ſit nauigandum iti-<lb/>nere quo ad fieri poterit breuiori, ideſt per gyrum circuli maioris, dixi autem extra <lb/>æquatorem, ideſt vt ambo, nec in æquatore, nec in uno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> meridiano <choice><ex>exiſtant</ex><am>exiſtãt</am></choice>, <lb/>quia vt aliàs dixi in huiuſmodi locis, vnico tantum vento comite, iter conficere <lb/>poſſumus.</s> </p> <pb facs="0232" n="220"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Volo primum vt mediante <choice><ex>circunuolutione</ex><am>circũuolutione</am></choice> globi circa primos polos <seg type="var">.f.g.</seg> & æqua <lb/>toris circa ſecundos <seg type="var">.h.K.</seg> hoc eſt per longitudinem, & latitudinem, hi duo loci in <lb/>globo propoſiti ſub azimut <seg type="var">.r.n.s.m.</seg> ſecundorum circulorum <choice><ex>ſitum</ex><am>ſitũ</am></choice> ſortiantur, qui azi-<lb/>mut orizontem in punctis <seg type="var">.q.</seg> et <seg type="var">.p.</seg> ſemper ad angulos rectos diſpeſcit <choice><ex>ibique</ex><am>ibiq;</am></choice> globum <lb/>ita quieſcere vt circa polos <seg type="var">.f.g.</seg> non voluatur, & æquatorem etiam ſic ſirmare, vt cir-<lb/>ca ſecundos polos <seg type="var">.h.K.</seg> non vertatur faciamus.</s> </p> <p> <s xml:space="preserve">Quod cum factum fuerit, ſecundorum circulorum primus, qui eſt <seg type="var">.n.i.m.</seg> deferens <lb/>azimut, circa tertios polos <seg type="var">.i.</seg> & eius oppoſitum, eo uſque voluatur quouſque prior <lb/>globi locus, ideſt is a quo iter eſt incohandum per .90. gradus azimut diſtet ab ori-<lb/>zonte, ideſt ſub zenit orizontis <seg type="var">.q.i.p.</seg> ſit poſitus, quemadmodum, exempli gratia, ſi <lb/>punctum <seg type="var">.a.</seg> dicti primi loci globi rationem indueret, & borealius eſſet, mediante <lb/>circunuolutione circuli <seg type="var">.n.i.m.</seg> circa dictos tertios polos æqualiter diſtans ab <seg type="var">.q.</seg> et <seg type="var">.p.</seg> <lb/>ideſt per .90. gradus poneretur ſub <seg type="var">.r</seg>.</s> </p> <p> <s xml:space="preserve">Conſideretur deinde vbi æquator <seg type="var">.h.e.K.</seg> ſecundus circulus duorum primorum, <lb/>ab orizonte <seg type="var">.q.i.p.</seg> ſecabitur, exempli gratia, in puncto <seg type="var">.c.</seg> quartę orientalis ſepten-<lb/>trionalis eiuſdem orizontis. </s> <s xml:space="preserve">Videatur deinde quot nam gradibus conſtabit ar-<lb/>cus <seg type="var">.i.c.</seg> & per totidem gradus conſtituatur extremitas ſeptentrionalis lineæ meridia-<lb/>nę pyxidis nauticę, diſtantis à cuſpide ſeptentrionali ipſius lanceolæ orientem ver-<lb/>ſus, mediante nauis circunuolutione. </s> <s xml:space="preserve">vnde ipſamet nauis in huiuſmodi ſitu azimut, <lb/>qui per duos hos loc<unclear reason="illegible"/>os tranſit, dirigetur, eſſiciendo vt eius prora verſus <choice><ex>locum</ex><am>locũ</am></choice> ad <choice><ex>quem</ex><am>quẽ</am></choice> <lb/>voluerimus tendere dirigatur. </s> <s xml:space="preserve">Cum verò vela ventis dabimus, tot milliarium <lb/>ſeu leucarum iter conficiemus, quot quarta pars vnius gradus requirit. </s> <s xml:space="preserve">& dum <lb/>hociter abſoluitur, ille qui pręeſt naui, defferentem azimut <seg type="var">.n.i.m.</seg> circa ſuos polos <seg type="var">.i.</seg> <lb/>& eius oppoſitum, ſic circunuoluat, vt interſe ctio azimut <seg type="var">.r.n.s.m.</seg> cum orizonte <seg type="var">.q.i.<lb/>p.</seg> diſtet à prima ratione dictæ quartę partis vnius gradus, conſtituendo ſecundum lo <lb/>cum, proximiorem zenit, ratione dictæ quartæ partis gradus azimut. </s> <s xml:space="preserve">Hiſce ita pera <lb/>ctis, obſeruetur deinde vbiæquator <seg type="var">.h.e.K.</seg> hac ſecunda vice interſecabit orizontem <lb/><seg type="var">q.i.p.</seg> quod quidem interſectionis punctum ſemper appelletur <seg type="var">.c.</seg> quod dico non am <lb/>plius in eadem diſtantia manſurum, ut prius à puncto <seg type="var">.i.</seg> ſed aut longius diſtabit, aut <lb/>propius accedet, vt in præſenti exemplo. </s> <s xml:space="preserve">quemadmodum ex ſe manifeſtum eſt, cú <lb/>poli globi, ideſt ęquatoris ſint extra azimut, vt præſupponitur, quia loci ſunt in diuer <lb/>ſis meridianis.</s> </p> <p> <s xml:space="preserve">Pro huiuſmodi autem diſtantiæ ratione denuo dirigatur nauis prout æquator <seg type="var">.h.<lb/>e.K.</seg> in orizonte <seg type="var">.q.i.p.</seg> nobis oſtendet, atque hoc modo omnium iter quaſi breuiſſi-<lb/>mum fiet. </s> <s xml:space="preserve">dico autem, quaſi, quia omnibus modis neceſſariò conficitur iter contor-<lb/>tum & in formam ſerpigineæ lineæ. </s> <s xml:space="preserve">Applicantes deinde per vices extremitatem <lb/>quartæ appoſitæ (de qua ſuperius mentionem fecimus) ipſi zenit <seg type="var">.r.</seg> efficientes ut per <lb/>ſitum poli globi pertranſeat, deueniemus in cognitionem altitudinis eiuſdem ab <lb/>orizonte, & per conſequens quantum itineris per latitudinem eiuſdem globi pere-<lb/>gerit. </s> <s xml:space="preserve">mediante deinde interſectione orizontis <seg type="var">.q.i.p.</seg> cum æquatore, cognoſcemus <lb/><choice><ex>quantum</ex><am>quãtum</am></choice> itineris per longitudinem eiuſdem globi, in ipſo ęquatore fuerit peractum,.</s> </p> <pb facs="0233" n="221"/> <fw type="head">EPISTOLAE.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Inctrumentum adortum, & occaſum Lunacognoſcendum <lb/>qualibet anni die.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">ECce tibi vir Illuſtriſs. <choice><ex>modum</ex><am>modũ</am></choice> <choice><ex>conficiendi</ex><am>conficiẽdi</am></choice> inſtrumenti nuper à me <choice><ex>inuenti</ex><am>inuẽti</am></choice>, vt tibi ſi <lb/>gnificaui, quo ſcire poſſis fermè in dies, qua hora (de aſtronomicis loquor) ad <lb/>determinatum parallelum & abſque multa ſupputatione, etiam abſque Aſtrolabio <lb/>Luna oriatur <choice><ex>occidatque</ex><am>occidatq́;</am></choice>. </s> <s xml:space="preserve">In quo inſtrumento poteris etiam videre quo in ſigno Sol, <lb/>& ſæpius itidem Luna permeat, & huiuſce aſpectus cum Sole, atque longitudinem <lb/>diei <choice><ex>noctisque</ex><am>noctisq́;</am></choice> toto anni tempore exactè diſcernere.</s> </p> <p> <s xml:space="preserve">Circularis lamina ex argento, aut ære, aliaúe materia paranda eſt, in cu-<lb/>ius ſuperficie ambarum facierum Zodiacus delineabitur, modo inferius depicto, <lb/>deinde pro anno quinque circuli ſibi inuicem <choice><ex>concentrici</ex><am>cõcentrici</am></choice>, at reſpectu Zodiaci excen <lb/>trici cęlabuntur in ea, adeo vt vtriuſque centri diſtantia ſit pro .32. parte ſemidia-<lb/>metri concauitatis Zodiaci è regione locis augis, temporis qui noſtra ætate circa ſi-<lb/>nem ſecundi gradus cancri inuenitur, eandem viam, in hoc, ſequuti, quam Stofle-<lb/>rus in dorſo Aftrolabij docet. </s> <s xml:space="preserve">At nomina menſium media ponantur inter duos <lb/>maiores circulos, poſtea inter ſecundum, & tertium ab vna facierum laminæ, ar-<lb/>cus ſemidiurni, ab altera vero arcus ſeminocturni, per quinos quoſque dies collo-<lb/>centur, ita exactè, vt hic ſubtus videbis. </s> <s xml:space="preserve">adeo vt numeri dierum & ipſorum dierum <lb/>ſigna ſint in interuallis vicinioribus centro communi dictorum quinque circulorum.</s> </p> <p> <s xml:space="preserve">Poſteaquam ab vna & altera <choice><ex>facierum</ex><am>facierũ</am></choice> laminæ hæc inſculpra fuerint, aliæ duæ circu <lb/>lares laminæ, magnitudinis ſemidiametri minimi quinque circulorum accipiantur: <lb/></s> <s xml:space="preserve">quarum vna pro ortu, & altera pro occaſu Lunæ deſeruiet. </s> <s xml:space="preserve">In qualibet ipſarum <lb/><choice><ex>conſtituentur</ex><am>conſtituẽtur</am></choice> circuli quatuor, eo modo qui paulo inferius cernitur, quos omnes diui-<lb/>demus in triginta ſpacia æqualia: </s> <s xml:space="preserve">& in interuallo <choice><ex>quod</ex><am>qđ</am></choice> inter duos primos circulos poſi <lb/>tum eſt, triginta dies <choice><ex>annotabimus</ex><am>annotabimꝰ</am></choice> qui ipſos Lunætriginta dies <choice><ex>pręſcribent</ex><am>pręſcribẽt</am></choice>, vt in figara.</s> </p> <p> <s xml:space="preserve">Poſtmodum in lamina quæ ortus Lunæ indicabit, ac duorum maiorum <choice><ex>circulo- rum</ex><am>circulo-rũ</am></choice> interuallo è regione numeri .1. videlicet primi diei, ponemus horas .12. & minuta <num value="48">.<lb/>48.</num> ex aduerſo diei ſecundi ho .13. et min .36. ex oppoſito tertij ho 14. min .24. & ſic <lb/>ſucceſſiuè augendo per min .48. & indicem è diuerſo diei .30. <choice><ex>ſtatuendo</ex><am>ſtatuẽdo</am></choice>, qui coitus <lb/>Lunæ cum Sole ſigniſicabit: </s> <s xml:space="preserve">atque lineas aſpectuum, vt inferius videre eſt facilè in <lb/>ueniemus.</s> </p> <p> <s xml:space="preserve">Altera in lamina quæ occaſum Lunæ indicabit, poſtquam diſtincta fuerit, vt alte-<lb/>ra .30. dies ac cęteræ lineæ, eo modo quo in ſuperiori collocabuntur, at numeri in-<lb/>terualli maioris, aliter diſponentur, vt potè ex aduerſo diei primi ſolum .48. minu-<lb/>ta deſcribi debent, è directo ſecundi diei ponenda erit hora vna cum minutis .36. & <lb/>è regione tertij inſcribentur .2. horæ, & min .24. & ſic ex ordine per .48. minuta au-<lb/>gendo.</s> </p> <p> <s xml:space="preserve">Nunc lamina ortus Lunę, cum anno arcuum ſeminocturnorum, & illa occaſus cum <lb/>anno arcuum ſemidiurnorum <choice><ex>concentrari</ex><am>concẽtrari</am></choice> debet, & ita noſtrum inſtrumentum perfe-<lb/>ctum erit & abſolutum.</s> </p> <p> <s xml:space="preserve">Quoties igitur voluerimus medio inſtrumento dignoſcere fermè in tali orizonte <lb/>qua hora Luna oriatur, ita neceſſe erit volubilem rotam ortus flectere, ut index ve <lb/>niat è regione diei menſis in quo talis operatio fit & talirota ſirma manente perſpi- <pb facs="0234" n="222"/><fw type="head">IO. BAPT. BENED.</fw> cere ex aduerſo diei Lunæ, numerum horarum & minutorum in maiori interuallo <lb/>ipſius rotæ notatorum, qui cum arcu ſeminocturno anni, quo cum in ipſa rectitudi-<lb/>ne centri conueniet colligetur, & ſumma quæ ex tali ſupputatione proueniet aper-<lb/>tas faciet horas aſtronomicas, quibus ferè etſi non exactè in die propoſito Luna <lb/>orietur. </s> <s xml:space="preserve">Idipſum fiet pro occaſu Lunæ.</s> </p> <p> <s xml:space="preserve"><hi rend="small caps">DIem</hi> ætatis Lunæ iam totus orbis ſcit inuenire, media ſupputatione nu-<lb/>meri Epactę currentis cum numero menſium, ſumpto principio à Martio, <lb/>adiunctis diebus menſis currentis, & detracto numero .30. à ſumma prędicta, ſi ab ip <lb/>ſa dictus numerns .30. ſuperatur.</s> </p> <p> <s xml:space="preserve">Sed ne aliquis putet ſufficere tantummodo additionem quatuor quintarum ho-<lb/>rę qualibet die. à nouilunio inchoando, ſciendum eſt huiuſmodi receſſum Lunæ <lb/>(quamuis non ita exactæ fiat) non <choice><ex>computandum</ex><am>computandũ</am></choice> eſſe ab orizonte aliquo, ſed à recto, <lb/>ſeu à meridiano quod idem eſt, quemadmodum <choice><ex>vnicuique</ex><am>vnicuiq;</am></choice> mediocriter erudito pa-<lb/>tere poteſt. </s> <s xml:space="preserve">At propoſitum nobis non eſt ſcire qua hora Luna in meridiano repe-<lb/>riatur, ſed in noſtro obliquo orizonte, in parte orientali ſeu occidentali, </s> <s xml:space="preserve">propterea <lb/>igitur addendus eſt, ei ſummæ temporis, qua Luna diſtat à meridiano, arcus ſemi-<lb/>diurnus, vel ſeminocturnus illius loci Zodiaci, in quo Luna reperitur illa die in pro <lb/>poſito parallelo, vt ſciatur proximę, qua hora (ex aſtronomicis) Luna erit in ori-<lb/>zonte <choice><ex>orientali</ex><am>oriẽtali</am></choice>, vel occidentali dicti paralleli. </s> <s xml:space="preserve">ſupra dicta enim additio quatuor quin <lb/>tarum horæ tantummodo, ſufficiens erit temporibus æquinoctij, ſed aliis anni tem-<lb/>poribus falli ratione iam dicta.</s> </p> <pb facs="0235" n="223"/> <figure place="here"> <graphic url="0235-01"/> <head rend="italics" xml:space="preserve">Pro Lunæ ortu. <lb/>Ad lati .45.</head> </figure> <figure place="here"> <graphic url="0235-02"/> </figure> <pb facs="0236" n="224"/> <figure place="here"> <graphic url="0236-01"/> <head rend="italics" xml:space="preserve">Pro Lunæ occaſu. <lb/>Ad lati .45.</head> </figure> <figure place="here"> <graphic url="0236-02"/> </figure> <pb facs="0237" n="225"/> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE LVCERNA SPIRITALI QVAM SERENISS. <lb/>Sabaudiæ Duce <seg type="var">.D.</seg> meo collendiſs. anno .1570. conſtruxi.</head> <head rend="italics" xml:space="preserve">CLARISS. FRANCISCO BARBARO VENETORVM <lb/>apud Sereniſſimum Sabaudiæ Ducem Oratori Illuſtriſſimo.</head> <p> <s xml:space="preserve"><hi rend="small caps">HEron</hi> varias ac diuerſas hydraulicas, & ſpiritales machinas propoſuit, in <lb/>ter quastamen nullam ſimilem, ei <choice><ex>quam</ex><am>quã</am></choice> ego Sereniſſimo Sabaudiæ Duci <lb/>conſtruxi, deſcribit, quæ quidem fuit Lucerna, & erat huiuſmodi, vt à <lb/>magno aliquo vaſe oleo pleno ſupra alicuius triclinij tabulatum poſito, <lb/>ſubtilis quidam tubus perpendi-<lb/> <ptr xml:id="fig-0237-01a" corresp="fig-0237-01" type="figureAnchor"/> culariter per tabulatum exiret, <lb/>& in dictum triclinium vſque ad <lb/>medium deſcenderet, ita tamcn <lb/>vt hic ſolus tubus, non item <lb/>vas oleo plenum cerneretur, cu-<lb/>ius <choice><ex>quidem</ex><am>quidẽ</am></choice> tubi inferiori extremi-<lb/>tati iunctum eſſet quoddam par-<lb/>uum receptaculum olei, ſimile co <lb/>operculo alicuius pyxidis, è cuius <lb/>ambitu prope baſim multi diuer-<lb/>ſi quæ tubi æquales & orizonta-<lb/>les, cuiuſuis longitudinis proſili-<lb/>rent, quorum quilibet in extremi <lb/>tate ſua, exiguam quandam pyra <lb/><choice><ex>midem</ex><am>midẽ</am></choice>, <choice><ex>appensam</ex><am>appẽsã</am></choice> haberet, in qua elli <lb/><choice><ex>chnum</ex><am>chnũ</am></choice> eſſet <choice><ex>cum</ex><am>cũ</am></choice> mixo. </s> <s xml:space="preserve"><choice><ex>oleum</ex><am>oleũ</am></choice> deinde <lb/>medio <choice><ex>perpendicularis</ex><am>ꝑpẽdicularis</am></choice> tubi ad rece <lb/><choice><ex>ptaculum</ex><am>ptaculũ</am></choice> extrinſece deſcendebat, <lb/>& per alios tubos ad nutriendas <lb/>flammas dum <choice><ex>arderent</ex><am>arderẽt</am></choice> ferebatur: <lb/></s> <s xml:space="preserve">at vero <choice><ex>cum</ex><am>cũ</am></choice> eædem erant extinctæ <lb/>ne minima quidem olei gutta de <lb/>ſcendebat: </s> <s xml:space="preserve">id quod eos qui aſta-<lb/>bant in admirationem trahebat.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0237-01" corresp="fig-0237-01a"> <graphic url="0237-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Hæcautem lucerna ſic erat <choice><ex>con</ex><am>cõ</am></choice> <lb/>ſtructa. </s> <s xml:space="preserve">Vas <choice><ex>olearium</ex><am>oleariũ</am></choice> cylindricum <lb/>vt in ſubſcripta figura patet, cuiuſ <lb/>uis magnitudinis, omni ex parte <lb/>clauſum faciendum curaui, ita ta <lb/>men vt eius coopereulum <choice><ex>aliquam</ex><am>aliquã</am></choice> <lb/><choice><ex>tulum</ex><am>tulũ</am></choice> concauum eſſet, in cuius me <lb/>dio erat foramen <seg type="var">.e.</seg> quod erat os <lb/>tubi <seg type="var">.e.g.</seg> qui ſub <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> vaſis fun-<lb/>do vſque ad <seg type="var">.g.</seg> tranſibat, ſed po-<lb/>ſtea ſurſum, quaſi <choice><ex>vſque</ex><am>vſq;</am></choice> ad cooper <lb/>culum in ſitu <seg type="var">.c.</seg> ab inferiori parte <lb/>reflectebatur, & ibi <choice><ex>terminabatur</ex><am>terminabat̃</am></choice>.</s> <pb facs="0238" n="226"/> <fw type="head">IO. BABPT. BENED.</fw> <s xml:space="preserve">Vnde oleum quod in vas infundebatur per foramen <seg type="var">.e.</seg> dictum vas poſtea ingredie-<lb/>batur per foramen <seg type="var">.c</seg>. </s> <s xml:space="preserve">Habebat deinde tubum <seg type="var">.n.u.</seg> rectum, qui à ſitu <seg type="var">.n.</seg> propinquo co <lb/>operculo ad libellam extremi <seg type="var">.c.</seg> incipiebat, & per fundum <choice><ex>contignationemque</ex><am>contignationemq́;</am></choice> vſque <lb/>ad centrum ſupradictireceptaculi (circa quod <choice><ex>tuborum</ex><am>tuborũ</am></choice> ope appenfæ erant ellychnio-<lb/>rum pyramides) tranſibat, atque huiuiuſmodi tubi <seg type="var">.n.u.</seg> extremitates tam ſuperius <lb/>quam inferius erant apertæ, & hic tubus aeris erat. </s> <s xml:space="preserve">Præterea aliud quoddam fora-<lb/> <ptr xml:id="fig-0238-01a" corresp="fig-0238-01" type="figureAnchor"/> men in vaſis fundo feceram, cui <lb/>paruum <choice><ex>tubum</ex><am>tubũ</am></choice> <seg type="var">.a.o.t.</seg> reflexum, ita <lb/>tamen, vt <seg type="var">.o.</seg> altius eſſet quam <seg type="var">.g.</seg> <lb/><choice><ex>aptaueram</ex><am>aptauerã</am></choice>, <choice><ex>atque</ex><am>atq;</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> hunc reflexum tu <lb/>bum <seg type="var">.a.o.t.</seg> oleum vaſis exibat, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>per oſculum <seg type="var">.t.</seg> in quendam cana-<lb/>lem tubo <seg type="var">.n.u.</seg> inſertum, ab extra <lb/>oleum effundebat, & ab exteriori <lb/>parte arundinis <seg type="var">.n.u.</seg> ingredieba-<lb/>tur receptaculum appenſum ex-<lb/>tremo dicti tubi <seg type="var">.n.u.</seg> quod extre-<lb/>mum <choice><ex>apertum</ex><am>apertũ</am></choice> erat, ut dixi, & à fun-<lb/>do receptaculi tantum <choice><ex>diſtans</ex><am>diſtãs</am></choice>, <choice><ex>quam</ex><am>quã</am></choice> <lb/>tum volebamus oleum ab ipſius <lb/>receptaculi fundo altum exiſtere <lb/>quod quidem oleum ſtatim vt ad <lb/>oſculum dicti tubi <seg type="var">.n.u.</seg> accedebat <lb/>id claudebat. </s> <s xml:space="preserve">Vnde aeri <choice><ex>ingrediem</ex><am>ingrediẽ</am></choice> <lb/>di vas <seg type="var">.q.b.</seg> non amplius patebat <lb/>aditus, & per conſequens, neque <lb/>amplius <choice><ex>oleum</ex><am>oleũ</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> <choice><ex>tubum</ex><am>tubũ</am></choice> <seg type="var">.a.o.t.</seg> eflue-<lb/>bat, nec etiam per <choice><ex>tubum</ex><am>tubũ</am></choice> <seg type="var">.e.g.c.</seg> aer <lb/>ingredi poterat, cum <seg type="var">.c.g.</seg> ſemper <lb/>oleo exiſteret <choice><ex>plenum</ex><am>plenũ</am></choice> <choice><ex>quum</ex><am>quũ</am></choice> <seg type="var">.g.</seg> magis <lb/><choice><ex>quam</ex><am>quã</am></choice> <seg type="var">.t.</seg> <choice><ex>adimum</ex><am>adimũ</am></choice> deuergeret.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0238-01" corresp="fig-0238-01a"> <graphic url="0238-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quoties deinde oleum in vas <lb/>infundere volebamus, oportebat <lb/><choice><ex>cum</ex><am>cũ</am></choice> fumitate digiti claudere oſcu-<lb/>lum <seg type="var">.t.</seg> exiguitubi <seg type="var">.a.o.t.</seg> vnde aer <lb/><choice><ex>impulsus</ex><am>impulsꝰ</am></choice> ab oleo tubi <seg type="var">.e.g.c.</seg> extra <lb/>per tubum <seg type="var">n.u.</seg> quouſque oleum <lb/>vaſis ad ęquilibrium ipſius <seg type="var">.n.</seg> per-<lb/>ueniebat, <choice><ex>pertubum</ex><am>pertubũ</am></choice> <seg type="var">.n.u.</seg> ingredie-<lb/>batu. </s> <s xml:space="preserve">& quando dictum oleum <choice><ex>qui</ex><am>ꝗ</am></choice> <lb/>dictum tubum <seg type="var">.n.u.</seg> extrinſecè in-<lb/>trabat in receptaculum <seg type="var">.d.p.</seg> nil <lb/>amplius olei in vas infundendum <lb/>erat, & oportebat alicuius digito <lb/>foramen <seg type="var">.u.</seg> inferius arundinis <seg type="var">.n.<lb/>u.</seg> claudi, & foramen <seg type="var">.t.</seg> aperiri, vt <lb/><choice><ex>per</ex><am>ꝑ</am></choice> ipſum <seg type="var">.t.</seg> aliqua portio olei exi-<lb/>ret, quia tunc quædam pars tubi <seg type="var">.e.g.</seg> vacua reddebatur, & cum per <seg type="var">.t.</seg> nil amplius <pb facs="0239" n="227"/><fw type="head">EPISTOLAE.</fw> olei egrediebatur, aperiebatur <seg type="var">.u.</seg> & peripſum <seg type="var">.t.</seg> denuo tantum olei exire permitte-<lb/>bamus, quantum in receptaculo ad claudendum foramen <seg type="var">.u.</seg> idoneum exiſteret. </s> <s xml:space="preserve">Ra-<lb/>tio vero, quę me mouit, ut punctum <seg type="var">.g.</seg> inferius ipſius <seg type="var">.t.</seg> conſtituam, eſt, quia <choice><ex>cum</ex><am>cũ</am></choice> clau-<lb/>ſum erit <seg type="var">.u.</seg> per dictum <seg type="var">.t.</seg> oleum non amplius egredietur, quia pondus olei in tubo <seg type="var">.c.<lb/>g.</seg> maius euadet oleo quod vſque ad <seg type="var">.t.</seg> progrederetur, tubum autem <seg type="var">.e.g.c.</seg> reflexum <lb/>facio, ne cogamur claudere foramen <seg type="var">.e.</seg> quia hoc difficile præſtaretur, tubum etiam <lb/><seg type="var">a.o.t.</seg> ſurſum verſusreflexum conſtitui, vt aerem ab ingreſſu per foramen <seg type="var">.t.</seg> <choice><ex>arcerem</ex><am>arcerẽ</am></choice>, <lb/>quia huiuſmodi aer nunquam deſcendit ſi corpus magis denſum non deſcendat.</s> </p> <p> <s xml:space="preserve">Verum eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> melius erit, vt maiores difficultates euitemus, ſtatuere dictum <choice><ex>tubum</ex><am>tubũ</am></choice> <lb/><seg type="var">a.o.t.</seg> ita curuum vt eſt .ω qui cum ſuo extremo inferiori ipſi <seg type="var">.n.u.</seg> ſit contiguus ita ta-<lb/>men ut dictum extremum inferius <choice><ex>non</ex><am>nõ</am></choice> ſit inferius quam <seg type="var">.o.</seg> quia totum oleum exiret.</s> </p> <p> <s xml:space="preserve">Volui etiam vt ſuperior extremitas <seg type="var">.n.</seg> tubi <seg type="var">.n.u.</seg> ſit in aere vaſis & non in oleo, ne <lb/>per eam oleum exeat, quia cum extremitas <seg type="var">.u.</seg> inferior ſit <seg type="var">.g.</seg> totum oleum quod ſu-<lb/>peraret oſculum <seg type="var">.n.</seg> per dictum tubum <seg type="var">.n.u.</seg> ratione maioris ponderis egrederetur, <lb/><choice><ex>quemadmodum</ex><am>quẽadmodum</am></choice> cuilibet, vel mediocriter in philoſophicis rebus verſato innoteſcet.</s> </p> <pb facs="0240" n="228"/> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DEFENSIO EPHEMERIDVM.</head> <head rend="italics" xml:space="preserve">AD JLLVST.D. BERNARDVM <lb/>Trottum.</head> <p> <s xml:space="preserve"><hi rend="small caps">EDita</hi> erant ſcripta quædam, quorum titulus <hi rend="small caps">animadver-<lb/>siones in ephemeridas</hi>. </s> <s xml:space="preserve">& breuis alia diſputatio de er <lb/>roribus calculorum Aſtronomicorum. ac <choice><ex>demum</ex><am>demũ</am></choice> Theſes quæ-<lb/>dam typis datæ .11. Auguſti .1581. quę omnia cum ad manus <lb/>meas perueniſſent, non potui, non eis animum admouere, <choice><ex>cum</ex><am>cũ</am></choice> <lb/>ibi de his ſtudijs <choice><ex>ageretur</ex><am>ageret̃</am></choice>, in quibus partem non exiguam an-<lb/>norum meorum conſumpſi. </s> <s xml:space="preserve">nec tamen ſcribere aliquid ſta-<lb/>tueram; </s> <s xml:space="preserve">tum quod exiſtimarem viros Aſtronomiæ peritos <lb/>facile, quanti facienda eſſent ea quæ edita erant iudicaturos, alijs verò haud gratam <lb/>futuram harum rerum tractationem. </s> <s xml:space="preserve">Tum quod ſi ingenue meam ſententiam profer <lb/>re voluiſſem<unclear reason="illegible"/>, non poteram, ſine maxima authoris moleſtia ferè omnia reprobare. <lb/></s> <s xml:space="preserve">Quandoquidem ſolet vnuſquiſque indignationem concipere ex his, quæ ſuæ opi-<lb/>nioni repugnant, id omne maleuolentiæ potius quam veritatis ſtudio tribuens. </s> <s xml:space="preserve">Qui <lb/>nimo cum nec deeſſent qui dicerent in meipſum directa ea tela fuiſſe, nullam fidem <lb/>eis adhibendam duxi, nec enim qui in ephemeri das inuehitur, me arguere poteſt, <lb/>qui nullas ephemeridas ſcripſi, nec tabulas compoſui. </s> <s xml:space="preserve">Nec ſi author quidpiam ex <lb/>his quæ à nobis edita fuiſſent impugnare voluiſſet, ægrè ferre debuiſſem, modo à ve <lb/>ritate nuſquam deuiaſſet. </s> <s xml:space="preserve">Liberum enim eſt cuique ſcribere quodlibet. </s> <s xml:space="preserve">nec Ari-<lb/>ſtotelem afficit iniuria, quicunque illi fidem ſuam non accommodat, & ſi valdè ini-<lb/>quus ſit, quiſquis maiorum opiniones veras, & ab omnibus merito comprobatas <choice><ex>non</ex><am>nõ</am></choice> <lb/>admittit. </s> <s xml:space="preserve">Hinc mihi ſatis <choice><ex>oimbus</ex><am>oĩbus</am></choice> feciſſe videbar, cum his qui de ſcriptis illis me <choice><ex>inter- rogauerant</ex><am>inter-rogauerãt</am></choice> reſpondiſſem, ea non ſatis firmis eſſe innixa <choice><ex>fundamentis</ex><am>fundamẽtis</am></choice>, & quod ad talia <lb/>tractanda opus fuiſſet exercitatiore iudicio. </s> <s xml:space="preserve">Verumtamen cum tu vnus maxime om <lb/>nium deſideres tibi clarius, quæ nam de his mea ſit ſententia explicari, non tam tuis <lb/>precibus deuictus quam mea ipſius cupiditate de te benemerendi impulſus, non ſu-<lb/>ſtineo diutius animum tuum hęſitantem relinquere. </s> <s xml:space="preserve">Atque vt tibi adeo honeſta cu <lb/>pienti <choice><ex>morem</ex><am>morẽ</am></choice> geram paucis hiſce ſcriptis incultis quidem, vt ab homine omni pror-<lb/>ſus facundia deſtituto exaratis, ſed ex quibus nihilominus facile, atque perſpicue, vt <lb/>ſpero, conceptum animi noſtri percipere poſſis, ſi tamen eam præſtantis ingenij tui <lb/>aciem adhibueris, qua ſoles intima quæque ſcientiarum penetrare, noſtræ opinio-<lb/>nis ſummam perſtrinxi, quę ad te mittere decreui. </s> <s xml:space="preserve">& quamuis ipſa res de qua agitur, <lb/>quæ exactiorem deſiderat expoſitionem, prolixiorem me eſſe cogerit quam voluiſ-<lb/>ſem. </s> <s xml:space="preserve">multa tamen me obmiſiſſe intelliges, non admodum neceſſaria his quibus <lb/>Aſtrologiæ noti ſunt termini, vt tuarum occupationum rationem me etiam habere <lb/>intelligeres, at que vt ſummam oblectationem <choice><ex>concipiam</ex><am>cõcipiam</am></choice> animo ſi me tibi aliqua ex <lb/>parte ſatisfeciſſe intellexero ita humanitati tuæ gratiam habebo, quę mihi occaſio <lb/><choice><ex>nem</ex><am>nẽ</am></choice> <choice><ex>praebuit</ex><am>p̃buit</am></choice>, imò verò me impulit, ad ea <choice><ex>proferenda</ex><am>proferẽda</am></choice>: </s> <s xml:space="preserve">quæ grata eſſe poſſint tui ſimilibus, <lb/>ideſt <choice><ex>praeclaro</ex><am>p̃claro</am></choice> & <choice><ex>candido</ex><am>cãdido</am></choice> ingenio præditis, <choice><ex>atque</ex><am>atq;</am></choice> ad <choice><ex>euellendam</ex><am>euellẽdã</am></choice> ex <choice><ex>eorum</ex><am>eorũ</am></choice> animis falſam opi- <pb facs="0241" n="229"/><fw type="head">EPISTOL AE.</fw> nionem, ſi quam fortaſſe ex <choice><ex>illorum</ex><am>illorũ</am></choice> ſcriptorum lectione <choice><ex>conceperunt</ex><am>cõceperunt</am></choice> circa ea, de qui-<lb/>bus nunc ſum acturus.</s> </p> <p> <s xml:space="preserve"><choice><ex>Quemadmodum</ex><am>Quemadmodũ</am></choice> igitur ab hoc authore ter ſcriptum fuit de <choice><ex>contradictionibus</ex><am>cõtradictionibus</am></choice>, ſiue <lb/>erroribus Ephemeridum, & earum calculos ſequentium, & de ratione qua cognoſci <lb/>poteſt ſitus & locus alicuius ſuperioris planetæ, diuerſus ab eo, qui ab ipſis Epheme <lb/>ridibus aſſignatus eſt, ita diſputationem hanc meam diuidam in tres partes, quo ſci <lb/>licet minus confusè, & magis diſtinctè à me ſcribatur, <choice><ex>praeſupponendo</ex><am>p̃ſupponendo</am></choice>, vt animaduer-<lb/>tere potes, huius ſcriptoris intentionem, aliam non fuiſſe, <choice><ex>quam</ex><am>quã</am></choice> <choice><ex>oſtendere</ex><am>oſtẽdere</am></choice>, quod ſcripto <lb/>res Ephemeridum diuerſimode eiuſdem temporis locum planetæ aſſignauere, & <lb/>quod eum faciant modo nimium velociter currere, modo nimium in vno ſigno mo-<lb/>rari, vt (exempli gratia) Martem interdum faciunt morari ſex, aut ſeptem menſibus <lb/>in vno ſigno. </s> <s xml:space="preserve"><choice><ex>Idque</ex><am>Idq;</am></choice> poſtea in cauſa eſſe ait, vt Aſtrologi indiciarij <choice><ex>fallantur</ex><am>fallãtur</am></choice>, & ſimul <lb/>careant certis fundamentis rationum quibus futura indicent, & prædicant. </s> <s xml:space="preserve">Primum <lb/>ergo videndum eſt, quam rectè hic vſus ſit arte, & ſcientia, vt aliorum opiniones, & <lb/>ſcripta redarguere poſſet. </s> <s xml:space="preserve">Deinde videbimus quomodo verum ſit, & poſſibile id <lb/>quod ab Aſtrologis hactenus creditum, atque traditum eſt, & qua ratione poſſint ſie <lb/>ri veri calculi à peritis regularum ſcientiæ.</s> </p> <p> <s xml:space="preserve">In primo igitur tractatu inſcripto Animadnerſiones, præſupponit Author pro-<lb/>feſſores huius ſcientiæ neſcire inuenire vera loca planetarum, quia vtuntur Ephe-<lb/>meridibus, in quibus eorum loca non rectè ſunt notata. </s> <s xml:space="preserve">Quod ſecundum ipſum ori <lb/>tur, ex errore calculatorum, ſeu computiſtarum, potius quam ex varietate tabula-<lb/>rum, à quibus Ephemerides ſumptæ ſunt, hoc tamen verum non eſt, Ephemeridas, <lb/>ſcilicet, ita inter ſe differre, ratione errorum computiſtarum tantummodo, ſed po-<lb/>tius ratione ipſarum tabularum, & ſi interdum contingere poſſit error aliquorum mi <lb/>nutorum, nec non graduum, non propterea Ephemerides ita ſpernendæ ſunt. </s> <s xml:space="preserve">In <lb/>multis enim calculis, tales errores excuſabiles ſunt, cum ab innumerabilibus propè <lb/>accidentibus oriri poſſint, præſertim in calculis prutenicis.</s> </p> <p> <s xml:space="preserve">Videatur deinde vbi is profert quinquageſimum enuntiatum centiloquij Ptole-<lb/>męi, ſatis mendoſe. </s> <s xml:space="preserve">Ptolemęus enim ibi ſic ait.</s> </p> <quote> <s xml:space="preserve">Non obliuiſcaris eſſe centum viginti coniunctiones, quæ ſunt in ſtellis erraticis, <lb/>in illis enim eſt maior ſcientia eorum quæfiunt in hunc <choice><ex>mundum</ex><am>mũdum</am></choice> ſuſcipiendi incre-<lb/>mentum, & decrementum.</s> </quote> <p> <s xml:space="preserve">Nam, neque eo in loco, neque alibi, Ptolemęus quidquam eius dicere voluit <lb/>quod ab hoc profertur.</s> </p> <p> <s xml:space="preserve">Pergatur poſtea in pag .2. & videbitur hunc exiſtimare abſurdum quod Saturni, <lb/>& Iouis coniunctio vera anni .1563. potuerit eſſe in Leone ſigno igneæ triplicitatis <lb/>cum eorum coniunctio vera anni .1544. fuerit in Scorpione, ſigno triplicitatis <lb/>aqueæ, & cum coitus eorum anni .1583. futurus ſit in Piſcibus, ſigno pariter tripli-<lb/>citatis aqueæ. </s> <s xml:space="preserve">Ita enim ait.</s> </p> <quote> <s xml:space="preserve">Nam poſtquam duæ ſtellæ coiuerint, non prius ſub alio alterius triplicitatis ſigno <lb/>inter ſe ſunt conuenturæ, quam per omnia ſigna quæ eiuſdem ternarij cum primo ex <lb/>titerint prius <choice><ex>coniungantur</ex><am>coniungãtur</am></choice>. </s> <s xml:space="preserve">Ita ſentit Ptolemęus, <choice><ex>cæterique</ex><am>cæteriq́;</am></choice> non aſpernendi nominis <lb/>Aftronomi.</s> </quote> <p> <s xml:space="preserve">Et <choice><ex>tamen</ex><am>tamẽ</am></choice> Ptolemęus nunquam <choice><ex>quidquam</ex><am>quidquã</am></choice> huius rei attigit, & quamuis Albumaſar <lb/>& Alchibitius de eo <choice><ex>loquantur</ex><am>loquãtur</am></choice>, is tamen eosnon intellexit, cum illi ibi <choice><ex>non</ex><am>nõ</am></choice> agant de <pb facs="0242" n="230"/><fw type="head">IO. BAPT. BENED.</fw> periodis apparentibus, aut veris, ſed de mediocribus aut æqualibus, & quidem re-<lb/>ctè dicunt, quia lineæ eorum mediorum motuum non coeunt in aliquo ſigno alte-<lb/>rius triplicitatis, prius quam pertranſiuerint omnia ſigna illius, in qua incęperunt. <lb/></s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> nullum inconueniens ſequitur, ſi in veris <choice><ex>coniunctionibus</ex><am>cõiunctionibus</am></choice> non reperitur hæc re-<lb/>gula. </s> <s xml:space="preserve">Fieri enim poteſt, vt lineæ mediorum motuum coniungantur in vno ſigno, cor <lb/>pora verò eorum planetarum coeant in alio, cum rarò eueniat, vt linea medij mo-<lb/>tus, eadem ſit cum linea veri.</s> </p> <p> <s xml:space="preserve">Nunc quidem <choice><ex>tamen</ex><am>tamẽ</am></choice> non affirmauerim, nec ne gauerim eorum coniunctionem an <lb/>ni .1563. fuiſſe potius in Cancro, quam in Leone. </s> <s xml:space="preserve">Sed tantum dicam <choice><ex>vanum</ex><am>vanũ</am></choice> eſſe cre <lb/>dere id eueniſſe propter ſimilem naturam, aut qualitatem ſignorum. </s> <s xml:space="preserve">Hunc enim <lb/>reſpectum non habent illi planetæ in verisſuis coniunctionibus. </s> <s xml:space="preserve">Exempli autem <lb/>cauſa ponamus, quod rectè ſupputatæ fuerint coniunctiones annorum .1484. 1504. <lb/>& 1524. quod attinet ad differentiam duodecatemorij, ſcilicet prima in .24. gradu <lb/>Scorpij, ſecunda in .20. Cancri, tertia in .10. Piſcium. </s> <s xml:space="preserve">Cum ſecunda anticipauerit <lb/>trigonum perfectum cum prima, gradibus .4. & tertia anticipauerit trigonum perfe-<lb/>ctum <choice><ex>cum</ex><am>cũ</am></choice> ſecunda gradibus .10. ſi forte prima vt facta fuit in .24. gradu Scorpij facta <lb/>fuiſſet in .2. gradu eiuſdem, <choice><ex>planum</ex><am>planũ</am></choice> eſt <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>ſecunda</ex><am>ſecũda</am></choice> facta fuiſſet in .28. gradu geminorum <lb/>& tertia in .18. Aquarij, quæ ſigna ſunt diuerſæ triplicitatis ab illa Cancri. </s> <s xml:space="preserve">Inſuper <lb/>ſi coniunctio anni .1544. quæ fuit in .28. gradu Scorpij fuerit recta <choice><ex>correſpondens</ex><am>correſpondẽs</am></choice> <choice><ex>prae cedenti</ex><am>pręcedenti</am></choice>, anni .1524. per gra .18. ſine dubio ſi coniunctio anni .1524. facta fuiſſet in <lb/>18. gradu Aquarij, illa anni .1544. fuiſſet in .6. Scorpij ſigni alterius triplicitatis <choice><ex>quam</ex><am>quã</am></choice> <lb/>ſint Gemini. </s> <s xml:space="preserve">Præterea, vt anno .1544. <choice><ex>coniunctio</ex><am>cõiunctio</am></choice> facta eſt in .28. gra. Scorpij, & 1563 <lb/>in .29. Cancri, ponendo eas eſſe rectas, quod attinet ad ſuperandum trigonum vno <lb/>gradu, ſi anno .1544. facta fuiſſet in .30. Scorpij, anno .1563. proculdubio facta fuiſ-<lb/>ſet in primo gradu Leonis. </s> <s xml:space="preserve">Et ſuppoſitis ijs interuallis, quæ ſuperſunt, aut deſunt per <lb/>fectis trigonis, ſi coniunctio anni .1524. fuiſſet in .20. gradu Piſcium, anno .1544. <lb/>fuiſſet in .8. Sagittarij. </s> <s xml:space="preserve">Quæ quidem omnia aduerſantur opinioni huius ſcriptoris.</s> </p> <p> <s xml:space="preserve">Quodautem opinatur coniunctionem anni .1583. fore in Ariete, ſic dicens pagi <lb/>na ſecunda.</s> </p> <quote> <s xml:space="preserve">Non erit ab re ſi & eandem Saturni, & Iouis coniunctionem in primo igneæ tri-<lb/>plicitatis ſigno, quod eſt Aries futuram afferamus anno .1583. ſi ab accidentibus no <lb/>bis licet, vt ab omnibus paſſim conceditur, planetarum loca diſcernere.</s> </quote> <p> <s xml:space="preserve">In eo fallitur, <choice><ex>nam</ex><am>nã</am></choice> <choice><ex>neque</ex><am>neq;</am></choice> Saturnus, <choice><ex>neque</ex><am>neq;</am></choice> Iupiter, <choice><ex>errant</ex><am>errãt</am></choice> à vero per .9: nec .8. gra. ac ne <choice><ex>per</ex><am>ꝑ</am></choice> <lb/>4. <choice><ex>quidem</ex><am>quidẽ</am></choice> in <choice><ex>qui</ex><am>ꝗ</am></choice> buſuis Ephemeridibus aut tabulis. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> videbit eiuſmodi <choice><ex>coniunctionem</ex><am>cõiunctionẽ</am></choice> <lb/>contra <choice><ex>ſententiam</ex><am>ſententiã</am></choice> <choice><ex>ſuam</ex><am>ſuã</am></choice> fieri in Piſcibus, <choice><ex>non</ex><am>nõ</am></choice> <choice><ex>autem</ex><am>aũt</am></choice> in Ariete. (vt poſtea res ipſa nos docuit <lb/>ſub <choice><ex>menſe</ex><am>mẽſe</am></choice> Aprili. poſt <choice><ex>quidem</ex><am>ꝗdẽ</am></choice> <choice><ex>ſcriptam</ex><am>ſcriptã</am></choice> <choice><ex>hanc</ex><am>hãc</am></choice> <choice><ex>epiſtolam</ex><am>epiſtolã</am></choice>, <choice><ex>vulgarique</ex><am>vulgariq́;</am></choice> ſermone <choice><ex>tranſmiſsam</ex><am>trãſmiſsã</am></choice>, ſed <choice><ex>an- tequam</ex><am>an-tequã</am></choice> in <choice><ex>latinum</ex><am>latinũ</am></choice> <choice><ex>tranſlata</ex><am>trãſlata</am></choice>, & huic volumini inſerta <choice><ex>cum</ex><am>cũ</am></choice> alijs Typographo <choice><ex>committeretur</ex><am>cõmitteretur</am></choice>.)</s> </p> <p> <s xml:space="preserve">Vbi poſtea meminit magnæ periodi annorum .960. non tantum ei cogitandum <lb/>erat <choice><ex>hanc</ex><am>hãc</am></choice> fuiſſe opinionem <choice><ex>antiquorum</ex><am>antiquorũ</am></choice>, vt videri <choice><ex>pont</ex><am>põt</am></choice> apud <choice><ex>Albumaſarem</ex><am>Albumaſarẽ</am></choice> & <choice><ex>Alchibicium</ex><am>Alchibiciũ</am></choice>, <lb/>ſed etiam perpendendum an eſſet vera, <choice><ex>priuſquam</ex><am>priuſquã</am></choice> ei adhæreret. </s> <s xml:space="preserve">Hic enim fuit vnus <lb/>ex erroribus illius ætatis, quæ nondum penetrauerat intima huius ſcientiæ. </s> <s xml:space="preserve">Sunt <lb/>tamen illi antiqui excuſatione aliqua digni. </s> <s xml:space="preserve">Ponebant enim vigeſimo quoque an-<lb/>no præcisè fieri mediam coniunctionem Saturni <choice><ex>cum</ex><am>cũ</am></choice> Ioue, & in quolibet ſigno eiuſ-<lb/>dem triplicitatis <choice><ex>coniungi</ex><am>cõiungi</am></choice> quater. </s> <s xml:space="preserve">Itaque in qualibet triplicitate dicebant eos coire <lb/>duodecies.</s> </p> <pb facs="0243" n="231"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Quod ſecundum primum ſuppoſitum finiebatur ſpacio annorum .240. qui nume <lb/>rus fit. ex .20. duodecies multiplicatis. </s> <s xml:space="preserve">Et quia triplicitates ſunt .4. ideo credebant <lb/>in ſpacio annorum .960. qui numerus fit ex .240. quater multiplicato, perfici .48. <choice><ex>con- iunctiones</ex><am>cõ-iunctiones</am></choice>, priuſquam redirent ad ſe coniungendos in eodem loco, ubi prius iun-<lb/>cti fuiſſent. </s> <s xml:space="preserve">Primum autem ſuppoſitum, quod vigeſimo quoque anno <choice><ex>iungerentur</ex><am>iũgerentur</am></choice>, <lb/>colligebant ſic ratiocinantes. </s> <s xml:space="preserve">Si Saturnus annis .30. peragit ſuum curſum per om-<lb/>nia ſigna Zodiaci, Iupiter autem peragit eum annis .12. Saturnus ambulauerit .4. ſi <lb/>gna, et .4. quintas partes ſigni, ſiue gra .24. dum Iupiter peragit integrum ambitum <lb/>ideſt annis .12. </s> <s xml:space="preserve">Itaque deſunt ei anni .8. ad perueniendum ad .20. quibus .8. annis Sa-<lb/>turnus <choice><ex>perambulat</ex><am>ꝑambulat</am></choice> ſigna tria & <choice><ex>quintam</ex><am>quintã</am></choice> <choice><ex>partem</ex><am>partẽ</am></choice> unius ſigni <seg type="var">.i.</seg> gradus .6. qui iuncti dictis ſi <lb/>gnis .4. & gra .24. faciunt ſigna .8. quæ Iupiter item percurrit in annis .8. atque ita in <lb/>annis .20. </s> <s xml:space="preserve">Iupiter percurrit .20. ſigna <choice><ex>antequam</ex><am>antequã</am></choice> perueniat ad <choice><ex>Saturnum</ex><am>Saturnũ</am></choice>, cum Saturnus eo <lb/>dem tempore perfecerit curſum ſignorum .8. </s> <s xml:space="preserve">Eandem concluſionem etiam fortaſ <lb/>ſe collegerant ex dictis ſuppoſitis, dicentes, ſi Saturnus annis .30. ambulat .12. ſigna <lb/>proculdubio annis .20. ambulat .8. ſigna, quo tempore Iupiter perambulat .20. ad ra <lb/>tionem .12. ſignorum in annis .12.</s> </p> <p> <s xml:space="preserve">Verum hoc ſuppoſitum non eſt bonum, quoniam, ſi ita eſſet, coniunctiones <choice><ex>horum</ex><am>horũ</am></choice> <lb/>duorum planetarum nunquam exirent ex vna triplicitate, & non modo .960. <choice><ex>quoque</ex><am>quoq;</am></choice> <lb/>anno, ſed etiam ſexageſimo rurſus coniungerentur in eodem puncto. </s> <s xml:space="preserve">nec coniunctio <lb/>nes eorum (ſemper autem intelligo de medijs) unquam egrederentur ex illis tribus <lb/>ſignis Zodiaci.</s> </p> <p> <s xml:space="preserve">Sed periodus æqualis Saturni, eſt dierum circiter .10740. atque ita minor an .30. <lb/>atque etiam .29. cum dimidio periodus autem æqualis Iouis, eſt circiter .4328. vt <lb/>ego eam comperio, quidquid alij <choice><ex>dicant</ex><am>dicãt</am></choice>, <choice><ex>vtque</ex><am>vtq́;</am></choice> planius alias oſtendam. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> hæc per <lb/>iodus Iouis, etiam minor eſt ann .12. prætermittendo in ſupputatione <choice><ex>tam</ex><am>tã</am></choice> Saturni <choice><ex>quam</ex><am>quã</am></choice> <lb/>Iouis quaſdam minutias horarum & earum partium, quæ hac in re pto nihilo habe-<lb/>ri poſſunt. </s> <s xml:space="preserve">Atque his duabus periodis eccentricorum duorum planetarum poſſu-<lb/>mus cognoſcere interuallum quod erit inter vtramque mediam coniunctionem, hoc <lb/>modo agendo, & ratiocinando.</s> </p> <p> <s xml:space="preserve">Si Saturnus diebus .10740. circuit gradus .360. diebus .4328. qui ſunt periodus <lb/>Iouis, conficiet gradus .145. & min .4. ideſt min .8704. & eadem regula inueniemus <lb/><choice><ex>quod</ex><am>ꝙ</am></choice> Saturnus .30. <choice><ex>quibuſque</ex><am>quibuſq́;</am></choice> diebus, conſiciet min .60. & ſecunda .20. </s> <s xml:space="preserve">Iupiter autem ſin-<lb/>gulis .30. diebus, conficiet min .149. & ſecunda .43. vnde ſubtrahendo minuta Satur-<lb/>ni à minutis Iouis, ſupererunt min .89. cum ſecun .23. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> Iupiter .30. <choice><ex>quibuſque</ex><am>quibuſq́;</am></choice> die-<lb/>bus velocitate curſus, ſuperabit Saturnum minutis .89. cum ſecundis .23. </s> <s xml:space="preserve"><choice><ex>Atque</ex><am>Atq;</am></choice> dicen-<lb/>do, ſi minuta .89. cum ſecundis .23 dant nobis dies .30. ſupradicta, minuta .8704. da-<lb/>bunt nobis dies .2921. quibus iunctis cum diebus .4328. periodi Iouis, efficientur dies <lb/>7249. ideſt anni Aegiptij .19. cum diebus .314. & hæc erit æqua periodus temporis <lb/>inter vtranque coniunctionem horum duorum Altiorum planetarum. </s> <s xml:space="preserve">Vt autem pla <lb/>nius oſtendatur hanc operationem rectam eſſe (nam demonſtrationem <choice><ex>ſpeculatiuam</ex><am>ſpeculatiuã</am></choice> <lb/>huius operationis in .113. Theoremate noſtræ Arithmeticę <choice><ex>cuique</ex><am>cuiq;</am></choice> videre licet) fieri <lb/>poteſt his alijs calculis.</s> </p> <p> <s xml:space="preserve">Si Saturnus diebus .10740. tranſit per gra .360. in ſpacio dierum .2921. tranſibit <lb/>per gradus .97. min .54. quibus iunctis cum gra .145. min .4. ſupra notatis, efticientur <lb/>gra .242. min .58. </s> <s xml:space="preserve">Deinde, ſi Iupiter ſpatio dierum .4328. tranſit per gra .360. igitur <lb/>ſpatio .2921. per eandem regulam inueniemus eum tranſire gradus .242. mi .58. qui <lb/>numerus par eſt illi Saturni. </s> <s xml:space="preserve">Cum ergo Iupiter confecerit vnum ambitum poſt con <pb facs="0244" n="232"/><fw type="head">IO. BAPT. BENED.</fw> iunctionem cum Saturno, vt rurſus aſſequatur Saturnum, tranſeundum ei erit <choice><ex>per</ex><am>ꝑ</am></choice> gra <num value="242">.<lb/>242.</num> min .58. iter confectum à Saturno toto tempore annorum .19. & dierum .314. <lb/>ad rationem graduum .360. diebus .10740. (poſſumus <choice><ex>etiam</ex><am>etiã</am></choice> dicere gra .243. quia præ <lb/>termiſimus quaſdam exiguas particulas periodorum perfectorum cuiuſque planetę <lb/>in ſuperioribus ſupputationibus.) </s> <s xml:space="preserve">Illos verò gradus .43. Iupiter conſiciet diebus <num value="2921">.<lb/>2921.</num> ad rationem graduum .360. diebus .4328. </s> <s xml:space="preserve"><choice><ex>Atque</ex><am>Atq;</am></choice> ita vt diximus, ab vna coniun-<lb/>ctione ad aliam intererunt anni .19. </s> <s xml:space="preserve">Aegiptij cum diebus .314. vel circa.</s> </p> <p> <s xml:space="preserve">Nunc autem vt videatur an tabulæ Alfonſi conueniant cum hoc <choice><ex>numero</ex><am>nr̃o</am></choice> calculo, <choice><ex>con- ſiderabimus</ex><am>cõ-ſiderabimus</am></choice>, <choice><ex>quod</ex><am>ꝙ</am></choice> Era (vt vocant) dicti temporis annorum .19. cum diebus .314. eſt dua-<lb/>rum tertiarum ſexagenarum, ſecundæ nullius, & 53. <choice><ex>primarum</ex><am>primarũ</am></choice> ſiue dierum. </s> <s xml:space="preserve">Et per <choice><ex>hanc</ex><am>hãc</am></choice> <lb/>Eram colligendo motum mediocrem, tum Saturni, tum Iouis, omiſſis radicibus, & in <lb/>cipiendo ab Ariete, comperiemus <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>vtriuſque</ex><am>vtriuſq;</am></choice> planetæ lineæ eiuſmodi motus tranſi <lb/>bunt per min .56. tertij gradus Sagittarij, ideſt coniunctæ erunt.</s> </p> <p> <s xml:space="preserve">In fine poſtea ſecundæ periodi, cuius era erit .4. tertiarum, ſecundæ .1. et .47. pri-<lb/>marum ſexagenarum, locus mediocris <choice><ex>vtriuſque</ex><am>vtriuſq;</am></choice> erit in min .56. gra. ſexti Leonis. </s> <s xml:space="preserve">In <lb/>fine verò tertiæ periodi, cuius era erit .6. tertiarum .2. ſecundarum, et .41. primæ, lo-<lb/>cus eorum mediocris inuenietur in .56. minuto gradus .9. Arietis. </s> <s xml:space="preserve"><choice><ex>Atque</ex><am>Atq;</am></choice> ita deinceps <lb/>in fine <choice><ex>cuiusque</ex><am>cuiusq́;</am></choice> periodi, locus <choice><ex>eorum</ex><am>eorũ</am></choice> mediocris coniunctim ſemper diſtabit à loco me <lb/>diocri præcedentis coniunctionis gradibus .117. ideſt in trigono antecedenti, minus <lb/>gra .3. </s> <s xml:space="preserve">Vnde apparet has coniunctiones procedere in contrariam partem reſpectu or <lb/>dinis <choice><ex>ſignorum</ex><am>ſignorũ</am></choice> Zodiaci, ſed reſpectu ordinis graduum <choice><ex>ſignorum</ex><am>ſignorũ</am></choice>, ſemper <choice><ex>progrediuntur</ex><am>progrediunt̃</am></choice> or <lb/>dine per ternos gradus nunquam retrogradientes. </s> <s xml:space="preserve">Hinc ſe quitur, vt non duodecies <lb/>in omni triplicitate coniungantur hi duo planetę, vt antiqui putauerunt, ſed decies <lb/>tantum. </s> <s xml:space="preserve">& ad ſummum ter in ſingulo ſigno, ſpatio annorum .198. & dierum .220. aut <lb/>circiter, non autem .240. nec .242. </s> <s xml:space="preserve">Atque decem vices comprehendunt gra .27. & <lb/>vltima vice inueniuntur in ſigno ſequenti alterius triplicitatis. </s> <s xml:space="preserve">Exempli gratia, po-<lb/>namus <choice><ex>quod</ex><am>ꝙ</am></choice> prima vice <choice><ex>coniungantur</ex><am>cõiungant̃</am></choice> in gra .2. Arietis, ſecunda coniunctio erit in .5. </s> <s xml:space="preserve">Sagit-<lb/>tarij, tertia. in .8. Leonis. quarta in .11. Arietis, quinta in .14. Sagit .6. in .17. Leonis. <lb/>ſeptima in .20. Arietis, octaua in .23. Sagittarij, nona in .26. Leonis, decima in .29. <lb/>Arietis, et vndecima erit in gra .2. Capricorni ſigni <choice><ex>ſequentis</ex><am>ſequẽtis</am></choice> triplicitatis. </s> <s xml:space="preserve">Decem igi<lb/>tur interualla ſingula annorum .19. & dierum .314. faciunt annos .198. & dies .220. <lb/>Immo pertabulas Alfonſi, eiuſmodi periodus non modo non reperitur <choice><ex>annorum</ex><am>annorũ</am></choice> .242 <lb/>nec .240. vt antiqui credidere, ſed tribus diebus minor annis .198. & diebus .220. id-<lb/>eſt per dictas tabulas inuenitur eſſe annorum .198. & dierum .217. tantum, qui nume <lb/>rus multiplicatus per .4. triplicitates, efficiet periodum maiorem, quæ erit annorum <lb/>794. & dierum .138. quo tempore dicti planetæ redeunt ad eundem locum vbi pri-<lb/>mum ſe coniunxere.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia, locus mediocris Saturni & Iouis in fine annorum .198. dierum <lb/>217. reperitur in gradu .30. Sagittarij. </s> <s xml:space="preserve">Si quæſiuerimus hunc locum per <choice><ex>aggregatum</ex><am>aggregatũ</am></choice> <lb/>annorum .794. & dierum .138. cum annis .198. & diebus .217. quorum ſumma eſt <num value="992">.<lb/>992.</num> & dies .355. inuenietur locus mediocris ipſorum <choice><ex>planetarum</ex><am>planetarũ</am></choice> in dicto vltimo gra <lb/>du Sagittarij. </s> <s xml:space="preserve">Sed ſi quęſiuerimus eorum locum mediocrem per aggregatum anno <lb/>rum .198. & dierum .217. cum annis .960. quod erit ſumma annorum .1158. & <choice><ex>dierum</ex><am>dierũ</am></choice> <lb/>217. reperiemus Iouem in gradu .18. </s> <s xml:space="preserve">Sagittarij & Saturnum in .16. Leonis diſtanti-<lb/>bus inter ſe duabus eorum lineis motuum mediocrium gra. circiter .122. </s> <s xml:space="preserve"><choice><ex>Atque</ex><am>Atq;</am></choice> Iupi-<lb/>ter præcedet, & oportebit <choice><ex>quod</ex><am>ꝙ</am></choice> coniunctio eorum mediocris fuerit multis annis ante <lb/>omittendo (vt dixi) radices, quia ſatis eſt inuenire interuallum inter lineas eorum me <lb/>diorum motuum.</s> </p> <pb facs="0245" n="233"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Debebat igitur author animaduerſionum non quaſi cæcus eæcos ſequi, ſed prius <lb/>laborare, vt certior fieret, an interuallum annorum .960. </s> <s xml:space="preserve">Verum eſſet.</s> </p> <p> <s xml:space="preserve">Sed peius eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> idem author paulo inferius citat coniunctiones horum duorum <lb/>planetarum anni .1493. et .1512. quas neſcio vnde ſumpſerit.</s> </p> <p> <s xml:space="preserve">Nam, etſi inter hos annos eſt interuallum annorum .19. tamen tantum abeſt, vt <lb/>coiuerint dictis annis, vt Saturnus anno .1493. ante finem Auguſti fuerit in .28. gra-<lb/>du Aquarij, Iupiter verò in .28. Leonis ex diametro oppoſiti. </s> <s xml:space="preserve">Et anno .1512. per to <lb/>tum menſem Iunium & Auguſtum, Saturnus fuerit in Libra, Iupiter verò in Ariete, <lb/>itaque inter ſe ſimiliter oppoſiti, & ſi perfecta oppoſitio non fuit poſtea niſi ad <choice><ex>finem</ex><am>finẽ</am></choice> <lb/>Iunij ann .1513. & locus Monteregij ab eo citatus, vbi ait eum ponere coniunctio-<lb/>nem anni .1484. in gra .23. min .4. </s> <s xml:space="preserve">Scorpij, eſt mendoſus. </s> <s xml:space="preserve">Nam ipſe Montere gius po <lb/>nit dictam coniunctionem in mi .42. gra .24. non autem in min .4. ipſius gradus. </s> <s xml:space="preserve">Sed <lb/>hic error nullius eſt momenti, fortaſſe qui impræſſorum incuria irrepſit.</s> </p> <p> <s xml:space="preserve">Pergatur poſtea obſecro ad paginam .3. ipſarum Animaduerſionum, vbi hic co-<lb/>natur oſtendere calculatores non obſeruaſſe verum modum, ſic dicens.</s> </p> <quote> <s xml:space="preserve">Anno .1484. Nouembris .25. Saturno locum conſtituit Monteregius in grad .23. <lb/>min .4. Scorpij. </s> <s xml:space="preserve">Anno poſtmodum ſubſequenti qui eſt .1485. eundem in min .7. Sa-<lb/>gitarij collocat .21. Februarij die. </s> <s xml:space="preserve"><choice><ex>Interque</ex><am>Interq́;</am></choice> tempora duo interſunt menſes dies .26. <lb/></s> <s xml:space="preserve">At cum ex motus ſui natura Saturnus hoctemporis ſpacio gradus .4. non debeat <choice><ex>tran- ſcendere</ex><am>trã-ſcendere</am></choice>, ſit tamen inter <choice><ex>vtrunque</ex><am>vtrunq;</am></choice> tempus differentia graduum .7. minutorum .3. quæ <lb/>ratione ſui motus requirunt menſes .6. vt eos perficiat, conſtat pluſquam tribus men <lb/>ſibus fallere nos Saturnum.</s> </quote> <p> <s xml:space="preserve">Hic videre licet quam veram viam hic ſecutus ſit ad aperiendos errores Epheme <lb/>ridum, & miſeri Monteregij, qui Saturnum claudum facit tantum itineris conficere <lb/>tribus <choice><ex>menſibus</ex><am>mẽſibus</am></choice>, <choice><ex>quantum</ex><am>quãtũ</am></choice> vix confeciſſet <choice><ex>menſib</ex><am>mẽſib</am></choice>. ſex. </s> <s xml:space="preserve">Sed fortaſſe <choice><ex>ratiocinatur</ex><am>ratiocinat̃</am></choice> hoc modo.</s> </p> <p> <s xml:space="preserve">Si motus naturalis Saturni facit vt circumeat totum cęlum annis .30. igitur menſi-<lb/>bus .30. conficiet duodecimam partem circuitus, cum menſes .30. ſint duodecima <lb/>pars annorum .30. & quia duodecima pars circuitus cęli intelligitur conſtare ex .30. <lb/>gradibus, igitur quilibet menſis poſtulabit gradum vnum. </s> <s xml:space="preserve">Ideo illi .6. aut .7. gradus <lb/>poſtulant tempus, amplius menſium ſex.</s> </p> <p> <s xml:space="preserve">Atque eiuſmodi mira ratiocinatio poteſt in .2. exemplo eius, inſcripto.</s> </p> <quote> <s xml:space="preserve">Deeodem ex eodem</s> </quote> <p> <s xml:space="preserve">Vbi miratur, <choice><ex>quod</ex><am>ꝙ</am></choice> Monteregius faciat Saturnum ambulare gra .9. min .10. in menſi-<lb/>bus .7. & diebus .6. </s> <s xml:space="preserve">Ad quod iter Saturnc ſeni opus eſſet ſaltem menſibus .9. eius <lb/>iudicio.</s> </p> <p> <s xml:space="preserve">Sed ſi hoc miratur, quid dicturus fuiſſet, ſi animaduertiſſet, quod idem calculator <lb/>Monteregius facit Saturnum ambulare immo volare gra .9. min .48. non in 7. ſed in <lb/>2. menſibus cum dimidio, videlicet à .10. die Iunij vſque ad .26. Auguſti <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> an-<lb/>ni .1504.</s> </p> <p> <s xml:space="preserve">Quid ſi etiam animaduertiſſet <choice><ex>quod</ex><am>ꝙ</am></choice> à .10. die Iunij ſupradicti vſque ad .16. Ianuarij <lb/>anni ſequentis, faciunt Saturnum, ſurſum, deorſum curſitare amplius gra .17. mi .54. <lb/>Imino ſi animaduertiſſet, quod anno .1524. </s> <s xml:space="preserve">Stoflerinus ab initio anni, <choice><ex>vſquem</ex><am>vſquẽ</am></choice> ad <lb/>medium Maium, ideſt menſib .4. cum dimidio, facit Saturnum ambulare gra .15. </s> <s xml:space="preserve">Pro <lb/>fectò ob has velocitates, eius iudicio, tam abſurdas, obſtupuiſſet.</s> </p> <p> <s xml:space="preserve">Vbi autem in tergo eiuſdem paginę ait, quod gradibus .13. min .42. reſpondent <lb/>menſes .19. errauit in calculo, nam ex eiuſmodi tempore ſecundum eius regulam ef- <pb facs="0246" n="234"/><fw type="head">IO. BAPT. BENED.</fw> ficerentnr ſinguli ambitus Saturni ad rationem annorum amplius .40.</s> </p> <p> <s xml:space="preserve">Videamus nunc vbi agit de Ioue, & reperiemus <choice><ex>quod</ex><am>ꝙ</am></choice> in primo exemplo circa <choice><ex>annum</ex><am>annũ</am></choice>. <lb/>1484. repręhendit Monteregium, quia facit Iouem ambulare gradus .14. cum min <num value="6">.<lb/>6.</num> in menſibus .2. diebus .4. ad quod iter, vt ipſe ait, opus eſſet ſaltem <choice><ex>menſibus</ex><am>mẽſibus</am></choice> .11. <choice><ex>atque</ex><am>atq;</am></choice> <lb/>ita ſecundum ipſum, Ioui opus eſſet anno vno pro ſingulo medio ſigno. </s> <s xml:space="preserve">Vbi bonus <lb/>hic vir pariter cæcutit.</s> </p> <p> <s xml:space="preserve">Idem in ſecundo exemplo ſumpto à Stoflerino ait, <choice><ex>quod</ex><am>ꝙ</am></choice> Ioui ad curſum vnius gra-<lb/>dus, & min .5. opus eſt diebus .30. non autem menſibus .7. & diebus .28. vbi oſtendit, <lb/>ſe paruum diſcrimen facere inter Iouem, & Saturnum.</s> </p> <p> <s xml:space="preserve">Miratur poſtea <choice><ex>quod</ex><am>ꝙ</am></choice> Stoflerinus faciat laborare generoſum Iouem ferè menſibus <lb/>ſex in vno gradu. </s> <s xml:space="preserve">Sed multo magis, vt puto, miratus eſſet, ſi vidiſſet, quod idem <lb/>Stoflerus in eodem anno facit, quod Iupiter die .4. </s> <s xml:space="preserve">Ianuarij ſit in eodem puncto, in <lb/>quo poſtea reperitur die vltima Auguſti. </s> <s xml:space="preserve">At fortaſſe dici poſſet, quod Iupiter pro-<lb/>pter prudentiam, & bonitatem ſuam factus eſt R ex omnium Deorum, vt ait Home-<lb/>rus, & ideo expulit è ſede Saturnum, & aſcendit in altiori cœlo. </s> <s xml:space="preserve">Vnde euenit vt fa-<lb/>ctus fuerit lentior in curſu, Saturnus autem velocior. </s> <s xml:space="preserve">Aut iam tot annos eſſe na-<lb/>tum Iouem, vt iure credi poſſit eum iam factum eſſe ſenem, & pariter tardio-<lb/>rem in ſe <choice><ex>mouendo</ex><am>mouẽdo</am></choice>. </s> <s xml:space="preserve">aut <choice><ex>tunc</ex><am>tũc</am></choice> temporis illum detentum fuiſſe in ſibi dilecta Arcadia <choice><ex>cum</ex><am>cũ</am></choice> <lb/>Caliſto. </s> <s xml:space="preserve">Aut fortaſſe erat in alta ſpecula intentus audiendo ingenti certami-<lb/>ni Timoclis & Damidis, vnde pendebat exitium aut gloria familiæ ſuæ, nam alio-<lb/>quin Stoflerus non depræhendiſſet eum tam otioſum & morantem. </s> <s xml:space="preserve">Sediam relin-<lb/>quamus Saturnum & Iouem, & ad Martem veniamus.</s> </p> <p> <s xml:space="preserve">Ferox & inquietus Mars, qui ſemper bella & ignes ſpirare ſolet, etiam, & ipſe ab <lb/>Aſtrologis factus eſt piger, & languidus, vt velint eum nonnunquam commorari <lb/>in vno ſigno ſex aut ſeptem menſibus; </s> <s xml:space="preserve">quod nullo pacto placet authori Animaduer-<lb/>ſionum, cum pag .4. ita ſcribat.</s> </p> <quote> <s xml:space="preserve">Quod citra notam, ab omnibus creditur poſſe obſeruari, quamuis à nobis non ac <lb/>cipiatur.</s> </quote> <p> <s xml:space="preserve">Itaque ei videtur impoſſibile. </s> <s xml:space="preserve">Quia Mars peragit ſuum eircuitum minus .2. an-<lb/>nis. </s> <s xml:space="preserve">Sed audacior fuiſſe videtur, qui voluerit arguere tot egregios viros antiquos, <lb/>& recentiores, qui vti diligentes rerum cœleſtium obſeruatores, ipſis oculis certi fa <lb/>cti ſunt tam de his effectibus Martis, quam aliorum, vnde coacti ſunt fingere tantam <lb/>magnitudinem eius epicycli, cum ipſe nunquam obſeruauerit motum, nec huius nec <lb/>alterius planetæ, ſed tantum viderit eius moram in Ephemeride ſcriptam. </s> <s xml:space="preserve">Si enim <lb/>ſaltem diceret, ſe aliquo tempore obſeruaſſe iter Martis, & comperuiſſe aliorum opi <lb/>nionem falſam, attuliſſet aliquem colorem ſententiæ ſuæ. </s> <s xml:space="preserve">Sed ſi obſeruaſſet, non <lb/>ſcripſiſſet poſtea contra, vt puto. </s> <s xml:space="preserve">Res enim ita ſe habet, quod Mars in omni circui <lb/>tu ſui epicycli tranſiens per inferiorem partem ipſius epicycli, ſemper commoratur <lb/>multis menſibus in vno duodecatemorio Zodiaci, ſcilicet .6. et .7. menſibus, atque <lb/>etiam amplius, quod quidem ego ſæpe obſeruaui, præſertim anno .1565. et .1566. <lb/>hoc ordine. </s> <s xml:space="preserve">Primum inſpiciens Ephemeridas ſtadij, reperi <choice><ex>quod</ex><am>ꝙ</am></choice> Mars ſecundum eum <lb/>egrediebatur retrogradationem circa diem .12. Ianuarij anni 1566. in .16. grad. <lb/>Geminorum. </s> <s xml:space="preserve">Et ſimiliter quod anno .1565. die vltima Auguſti Mars futurus erat in <lb/>eodem ſupradicto loco, priuſquam retrogradi inciperet. </s> <s xml:space="preserve">Poſtea inueni, quod poſt <lb/>retrogradationem die .11. Aprilis .1566. </s> <s xml:space="preserve">Idem Mars futurus erat in gra .16. Cancri, <lb/><choice><ex>itaque</ex><am>itaq;</am></choice> in his .30. gradibus à .16. Geminorum ad .16. Cancri conſumebatur ſpatium <lb/>menſium .7. & dierum .11.</s> </p> <pb facs="0247" n="235"/> <fw type="head">EPISTOLAE.</fw> <p> <s xml:space="preserve">Quo ſupputato, ſumpſi inſtrumenta, & ad experimentum me paraui, & vltima <lb/>nocte menſis Auguſti anni .1565. reperi Martem eſſe in dicto gradu geminorum vt <lb/>ſcribebat Stadius. </s> <s xml:space="preserve">Deinde ſingulis ebdomadibus obſeruans retrogradationem; </s> <s xml:space="preserve">vidi <lb/>circa finem Octobris quod retrogradi incipiebat, & ea retrogradatio perſeuerauit <lb/>vſque ad medium menſem Ianuarium, aut circiter, anni .1566. obſeruaui poſtea <lb/>etiam ſitum eiuſdem planetæ die .11. </s> <s xml:space="preserve">Aprilis ſequentis <choice><ex>eumque</ex><am>eumq́;</am></choice><unclear reason="illegible"/> inueni in gradu .16. <lb/>Cancri, vti eum poſuerat Stadius. </s> <s xml:space="preserve">Atque ita experimentum meum conuenit cum <lb/>calculo Stadij, <choice><ex>comperique</ex><am>comperiq́;</am></choice> eum non erraſſe: </s> <s xml:space="preserve">Et ſic quiſque binis <choice><ex>quibusque</ex><am>quibusq́;</am></choice> annis pote-<lb/>rit certior fieri de veritate. </s> <s xml:space="preserve">Si autem delectationis cauſa id experiri volueris, expe-<lb/>ctato primam retrogradationem Martis, cuius initium ſecundum Stadium futurum <lb/>eſt circa diem .20. </s> <s xml:space="preserve">Nouembris anni .1582. & finis circa diem .10. Februar .1583. <lb/>circa grad .9. Cancri, & animaduerte quando Mars erit circa dictum gra .9. Can-<lb/>cri prius quam retrogradi incipiat, quod erit circa diem .19. Septem .1582. </s> <s xml:space="preserve">Dein-<lb/>de aſpice quum erit in grad .9. Leonis, quod erit circa diem .7. Mai .1583. & vide-<lb/>bis <choice><ex>quod</ex><am>ꝙ</am></choice> ipſe Mars in his gra .30. <choice><ex>morabitur</ex><am>morabit̃</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> menſes .7. & dies .18. <choice><ex>atque</ex><am>atq;</am></choice> vt eius rei pericu-<lb/>lum facias, obſerua noctem præcedentem diei .19. </s> <s xml:space="preserve">Septem .1582. locum <choice><ex>longitudinis</ex><am>lõgitudinis</am></choice> <lb/>eius ſtellæ, & idem poſtea obſerua nocte præcedente diei .7. Mai, aut nocte ſequen-<lb/>ti .1583. & inter duos hoſce terminos obſerua aliqua alia nocte ſtatum eius. </s> <s xml:space="preserve">Mani-<lb/><choice><ex>feſtoque</ex><am>feſtoq́;</am></choice> videbis Martem conſumere totum dictum tempus in hoc duodecatemorio. <lb/></s> <s xml:space="preserve">Et quicunque aliquid intelligit in hac facultate quamuis non viderit Ptolomęi <lb/>Almageſtum, minori labore poſſet per calculos ſcientificos colligere verita-<lb/>tem, ſuppoſitis tamen terminis ſcriptis in theoricis planetarum. </s> <s xml:space="preserve">Qui enim vidit <lb/>Almageſtum vel reuolutiones orbium cœleſtium Nicolai Copernici, non poteſt de <lb/>hoc vllo pacto dubitare. </s> <s xml:space="preserve">Sed qui nondum tantopere progreſſus eſt, <choice><ex>ſaltem</ex><am>ſaltẽ</am></choice> capiat <choice><ex>huius</ex><am>huiꝰ</am></choice> <lb/>rei notitiam vniuerſalem, hoc modo. </s> <s xml:space="preserve">Supponat primum eccentricitatem deferen-<lb/>tis epicycli Martis, eſſe .6. partium <choice><ex>talium</ex><am>taliũ</am></choice>, quales ſunt ſexageſimæ ſemidiametri ipſius <lb/>deferentis, & ſemidiametrum epicycli eſſe, partium <choice><ex>ſupradictarum</ex><am>ſupradictarũ</am></choice> .39. cum dimidia, <lb/>& quod argumenta vera, in temporibus primarum ſtationum ( cum epicyclus eſt in <lb/>auge, aut in eius oppoſito, aut in <choice><ex>lungitudinib</ex><am>lũgitudinib</am></choice>. me dio cribus ) <choice><ex>iam</ex><am>iã</am></choice> ab antiquis rectè ſuppu <lb/>tata ſint, ſicuti ſunt. </s> <s xml:space="preserve">Et præſupponat motum diurnum centri epicycli. min .31. cum di <lb/>midio, quamuis reuera ſit min .31. & ſecundorum .27. aut circiter, nunc <choice><ex>quidem</ex><am>quidẽ</am></choice> præter <lb/>mittens, quod vnus habeat reſpectum ad augem mediam epicycli, & alter ad cen-<lb/>trum æquantis. </s> <s xml:space="preserve">Atque his præſuppoſitis fingat ( exempli gratia ) quod centrum <lb/>epicycli ſit in quauis longitudinum mediarum, & Mars in prima maxima æqua-<lb/>tione argumenti, ſcilicet in prima linea, quæ attingens epicyclum, à centro mundi <lb/>pergat ad circunferentiam Zodiaci, quæ erit illa linea <choice><ex>contingentiæ</ex><am>cõtingentiæ</am></choice> a qua <choice><ex>proficiſcens</ex><am>proficiſcẽs</am></choice> <lb/>Mars perget ad lineam primæ ſtationis, vt poſtea retrogradiatur, veluti ſi in infrapo <lb/>ſita figura maiori, <choice><ex>centrum</ex><am>cẽtrũ</am></choice> <choice><ex>mundi</ex><am>mũdi</am></choice> eſſet <seg type="var">.o.</seg> & vnus arcus <choice><ex>eccentrici</ex><am>eccẽtrici</am></choice> eſſet <seg type="var">.a.b.c.d.</seg> & vna ex li <lb/>neis medio cribus longitudinum eſſet <seg type="var">.o.c.f.</seg> & centrum epicycli <seg type="var">.c.</seg> qui notabitur per <lb/><seg type="var">a.f.e.g.</seg> & lineæ contingentes epicyclum in punctis <seg type="var">.i.</seg> et <seg type="var">.t.</seg> ſint notatæ <seg type="var">.o.i.</seg> et <seg type="var">.o.t.</seg> & li-<lb/>nea primæ ſtationis <seg type="var">.o.n.b.</seg> & linea ſecundæ <seg type="var">.o.u.d.</seg> ſi igitur Mars eſſet in puncto <seg type="var">.i.</seg> an-<lb/>gulus <seg type="var">.i.o.e.</seg> maximæ æquationis argumenti eſſet gra .40. minut .55. <choice><ex>quanuis</ex><am>quãuis</am></choice> talis maxi <lb/>ma æquatio argumenti in longitudinibus mediocribus Alfonſi ponatur eſſe gra .41. <lb/>minut .10. quod euenit quia calculatores ipſarum tabularum interuallum <seg type="var">.o.c.</seg> quod <lb/>in eo ſitu epicycli interponitur inter centrum mundi, & centrum dicti epicycli, ac-<lb/>ceperunt partium ſexaginta præcisè, nihili facientes minuta illa .18. aut circiter, quę <lb/>verè ſunt præter dictas partes .60. quandoquid em euenit vt dictum interuallum in <pb facs="0248" n="236"/><fw type="head">IO. BAPT. BENED.</fw> tali ſitu epicycli ſit baſis vnius trianguli orthogonij, cuius vnum ex illis duobus late-<lb/>ribus eſt ſemidiameter eccentrici partium .60. pręcisè, aliud eſt interuallum eccen-<lb/>tricitatis partium .6. eiuſmodi. </s> <s xml:space="preserve">Angulus ergo <seg type="var">.i.o.c.</seg> vt dixi, erit partium .40. minu <num value="55">.<lb/>55.</num> qui angulus continuò variatur ſecundum ſitum epicycli. </s> <s xml:space="preserve">& cum centrum <lb/>eius eſt in auge eccentrici. eſt minimus <choice><ex>quam</ex><am>quã</am></choice> eſſe poſſit. </s> <s xml:space="preserve"><choice><ex>eſtque</ex><am>eſtq́;</am></choice> tantum grad .36. min <num value="46">.<lb/>46.</num> & in oppoſito ipſius augis eſt grad .47. min .1. maximus quam alibi vnquam ſit, <lb/>& ſic continuò variatur, ſecundum ſitum, quem habet epicyclus in eccentrico. </s> <s xml:space="preserve">Qui <lb/>quidem angulus inuenitur per doctrinam .27. et .28. libri primi Monteregij de trian <lb/>gulis. </s> <s xml:space="preserve">Nam triangulus <seg type="var">.c.i.o.</seg> eſt ſemper rectangulus in puncto <seg type="var">.i.</seg> & latus <seg type="var">.c.i.</seg> reſpectu <lb/>ſemidiametri eſt datum. </s> <s xml:space="preserve">Quod <seg type="var">.c.i.</seg> erit veluti partium .39. cum dimidia, et dictum <lb/>interuallum <seg type="var">.o.c.</seg> veluti pat<unclear reason="illegible"/> cium .60. min .18. & quia datur nobis etiam eccentricitas <lb/>veluti partium .60. talium, & cum <seg type="var">.c.o.</seg> ſit linea veri motus epicycli, & latus ſimiliter <lb/>vnius trian guli, cuius duo latera ſunt ſupradicta, ſcilicet ſemidiameter eccentrici, & <lb/>eccentricitas, inter ſe compræhendentes angulum datum. </s> <s xml:space="preserve">Nam ſemper præſuppo <lb/>nitur datus locus centri ipſius epicycli, cum ipſe eſt extra augem aut oppoſitum eius <lb/>quia in auge linea <seg type="var">.o.c.</seg> conſtat ex ſemidiametro eccentrici & interualli eccentricita-<lb/>tis. </s> <s xml:space="preserve">& in eius oppoſito, ipſa linea <seg type="var">.o.c.</seg> eſt minor dicto ſemidiametro eccentrici per in <lb/>teruallum dictæ eccentricitatis. </s> <s xml:space="preserve">Vnde etiam poſſumus extra augem, vel oppoſitum <lb/>eius cognoſcere <seg type="var">.o.c.</seg> tanquam latus dicti trianguli duorum laterum <choice><ex>cum</ex><am>cũ</am></choice> angulo cogni <lb/>torum. </s> <s xml:space="preserve"><choice><ex>Idque</ex><am>Idq́;</am></choice> per .49. propoſitionem libri primi <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> Monteregij cum ſcilicet dictus <lb/>angulus <choice><ex>non</ex><am>nõ</am></choice> fuerit rectus. </s> <s xml:space="preserve">Nam ſi fuerit rectus videbitur per .27. et .28. ſupra citatas.</s> </p> <p> <s xml:space="preserve">Cum igitur hab eamus angulum <seg type="var">.c.o.i.</seg> gra .40. mi .55. angulus <seg type="var">.o.c.i.</seg> tanquam reli-<lb/>quus exrecto, erit grad .49. mi .5. cui reſpondet arcus <seg type="var">.i.g.</seg> epicycli confectus à Marte <lb/>in diebus circiter .105. ad rationem min .28. aut circiter in ſingulos dies, prætermiſ-<lb/>ſis nunc quidem minutijs cum exigui momenti ſit error .15. aut .20. dierum ad verifi <lb/>cationem longæ morę Martis in vno duodecatemorio, atque per hoc tempus cen-<lb/>trum epicycli conficit gradus .55. min .7. aut circiter, ad rationem minutorum .31. <choice><ex>cum</ex><am>cũ</am></choice> <lb/>dimidio in ſingulos dies. qui numerus graduum .55. min .7: differt à numero <choice><ex>graduum</ex><am>graduũ</am></choice>. <lb/>40. min .55. maximæ æquationis argumenti gradibus .14. mi .12. nec refert quod gra <num value="55">.<lb/>55.</num> min .7. habeant reſpectum ad centrum æquantis, magis quam ad centrum <choice><ex>mundi</ex><am>mũdi</am></choice>, <lb/>quia differentia non eſt tanta, vt poſſit inducere errorem menſium. </s> <s xml:space="preserve">Hinc ſequitur <lb/>quod in fine dictorum dierum .105. </s> <s xml:space="preserve">Mars erit in linea <seg type="var">.o.c.</seg> veri motus epicycli, ſed <lb/>gradibus .14. min .12. vlterius quam in primo loco, in quo erat in Zodiaco, & erit in <lb/>medio ſuæ retrogradationis. </s> <s xml:space="preserve">Sed quoniam Mars manifeſtè retrogradi non incipit <lb/>in puncto <seg type="var">.i.</seg> conting entiæ, imo ab illo puncto vſque ad terminum primæ ſtationis li <lb/>neæ <seg type="var">.o.n.</seg> interponitur arcus <seg type="var">.i.n.</seg> epicycli, qui eſt graduum .32. minu .14. </s> <s xml:space="preserve"><choice><ex>Idque</ex><am>Idq́;</am></choice> cogno-<lb/>ſcitur ſubtrahendo arcum <seg type="var">.f.i.n.</seg> graduum .163. mi .9. qui eſt inter augem, & primam <lb/>ſtationem, à gradibus .180. ( qui arcus <seg type="var">.f.i.n.</seg> erit verum argumentum, quod ſi-<lb/>militer variatur ſecundum ſitum epicycli, etſi eiuſmodi varietas, nobis <choice><ex>non</ex><am>nõ</am></choice> eſt magni <lb/>momenti, vnde poſſumus præſupponere, quod <seg type="var">.c.</seg> centrum epicycli non alteret <choice><ex>in- teruallum</ex><am>in-teruallũ</am></choice> <seg type="var">.c.o.</seg> à centro <choice><ex>mundi</ex><am>mũdi</am></choice>, <choice><ex>cum</ex><am>cũ</am></choice> non posſit intercedere, error <choice><ex>menſium</ex><am>mẽſiũ</am></choice> reliquum verò <seg type="var">.g.<lb/>n.</seg> graduum .16. min .51. ſubtrahendo ex arcu <seg type="var">.g.i.</seg> graduum .49. minuti .5. vnde reli-<lb/>quus nobis erit arcus <seg type="var">.n.i.</seg> graduum .32. min .14. in eiuſmodi tamen ſitu mediocrium <lb/>longitudinum. </s> <s xml:space="preserve">Nunc hic arcus epicycli graduum .32. mi .14. fit à ſtella Martis die-<lb/>bus .69. ad rationem ſupradictam, omittendo quod ipſa ſtella habeat reſpectum ad <lb/>augem mediocrem epicycli, & quod dicta aux mediocris mutet diſtantiam à vera <lb/>propter motum epicycli, quod nunc quidem parui refert, in quibus diebus .69. cen- <pb facs="0249" n="237"/><fw type="head">EPISTOLAE.</fw> trum epicycli conficit gra .36. min .13. ad rationem ſupradictam. </s> <s xml:space="preserve">Reſtat nunc no-<lb/>bis inuenire angulum <seg type="var">.b.o.c.</seg> in centro <choice><ex>mundi</ex><am>mũdi</am></choice> inter duas lineas, <seg type="var">b.o.</seg> et <seg type="var">.c.o.</seg> <choice><ex>quarum</ex><am>quarũ</am></choice> prior <lb/>eſt primæ ſtationis, altera eſt veri motus epicycli, quod facilè intelligemus per di-<lb/>ctam .49. lib. 1. Monteregij, cum duo latera <seg type="var">.n.c.</seg> et <seg type="var">.c.o.</seg> & angulus <seg type="var">.n.c.o.</seg> ſint nobis no <lb/>ta. </s> <s xml:space="preserve">Hoc autem fiet fingendo lineam <seg type="var">.n.h.</seg> perpendiculerem ad <seg type="var">.o.c.</seg> quæ tanquam ſi-<lb/>nus anguli <seg type="var">.n.c.h.</seg> erit partium .28986. talium qualium <seg type="var">.n.c.</seg> eſſet partium .100000. & <lb/><seg type="var">c.h.</seg> tanquam ſinus anguli <seg type="var">.c.n.h.</seg> reſtantis ex vno recto, erit partium .95706. dicendo <lb/>poſtea ſi <seg type="var">.n.e.</seg> tanquam ſinus totalis partium .100000. dat nobis <seg type="var">.n.h.</seg> partium .28986 <lb/>quid dabit nobis diameter <seg type="var">.n.c.</seg> tanquam partium .39. mi .30. inueniemus <seg type="var">.n.h.</seg> venire <lb/>nobis ex partibus 11. mi .27. & idem faciendo de <seg type="var">.c.h.</seg> inueniemus quod veniet no-<lb/>bis partium .37. mi .48. quibus ſubtractis extota <seg type="var">.c.o.</seg> quę eſt <choice><ex>partium</ex><am>partiũ</am></choice> .60. mi .18. reliqua <lb/>erit nobis <seg type="var">.h.o.</seg> partium .22. min .30. capiendo poſtea radicem <choice><ex>quadratam</ex><am>quadratã</am></choice> ſummæ qua <lb/>drati <seg type="var">.n.h.</seg> cum quadrato <seg type="var">.h.o.</seg> veniet nobis <seg type="var">.n.o.</seg> partium .25. min .12. talium qualis <seg type="var">.n.<lb/>h.</seg> eſt partium .11. min .27. ſi igitur ad <seg type="var">.o.n.</seg> tanquam partium .25. min .12. reſpondet <seg type="var">.<lb/>n.h.</seg> partium .11. minuti .27. linea <seg type="var">.n.h.</seg> ad <seg type="var">.o.n.</seg> tanquam partium 100000. reſpon <lb/>debit part .45436. tanquam ſinus anguli <seg type="var">.n.o.h.</seg> qui angulus erit gra .27. minut .1. ſub <lb/>tracto poſtea hoc angulo ab angulo <seg type="var">.c.o.i.</seg> graduum .40. minut .55. remanebit an <lb/>gulus <seg type="var">.n.o.i.</seg> graduum .13. minut .54. inter lineam contingentiæ, & lineam primæ <lb/>ſtationis in eiuſmodi ſitu. </s> <s xml:space="preserve">Et ideo Mars acceſſerit ad lineam <seg type="var">.o.c.</seg> veri motus epi-<lb/>cycli. </s> <s xml:space="preserve">Sed quia linea <seg type="var">.o.i.</seg> contingentiæ, propter motum centri epicycli, in dictis die-<lb/>bus .69. confecerit gradus .36. minut .13. ( præſuppoſita ſemper eadem di-<lb/>ſtantia <seg type="var">.o.c.</seg> quamuis nonnulla ſit differentia, quam nunc prætermittemus ) & Mars in <lb/>dicto tempore retrogreſſus fuerit per dictum angulum gra .13. mi .54. quibus dedu-<lb/>ctis, ex .36. & min .13. reſtabunt gra .22. min .19. itaque in diebus .69. </s> <s xml:space="preserve">Mars promo-<lb/>tus fuerit a primo ſitu gra .22. min .19. aut circiter, prius quam retrogradatio eius in-<lb/>cipiat eſſe appa-<lb/>rens.</s> </p> <figure place="here"> <graphic url="0249-01"/> </figure> <p> <s xml:space="preserve">Nunc à prima <lb/>ſtatione vſque ad <lb/><choice><ex>lineam</ex><am>lineã</am></choice> veri motus <lb/>epicycli ſunt gra <num value="16">.<lb/>16.</num> min .51. ipſius <lb/>epicycli, vt ſupra <lb/><choice><ex>vidimus</ex><am>vidimꝰ</am></choice> quos Mars <lb/>tranſit in diebus <lb/>36. aut circiter ad <lb/>rationem min .28. <lb/>in ſingulos dies, <lb/>quo tempore cen <lb/>trum epicycli, in <lb/>tali diſtantia à <choice><ex>cen- tro</ex><am>cẽ-tro</am></choice> mundi confice <lb/>ret gra .18. mi .54. <lb/>ad rationem min <num value="31">.<lb/>31.</num> cum dimidio <lb/>in ſingulos dies, <lb/><choice><ex>quibus</ex><am>quibꝰ</am></choice> deductis ex <lb/>gra .27. min .1. an- <pb facs="0250" n="238"/><fw type="head">IO. BAPT. BENED.</fw> guli <seg type="var">.c.o.n.</seg> remanebunt gra .8. min .7. pro numero dimidiæ retrogradationis quum <lb/>Mars erit in linea <seg type="var">.o.c.</seg> veri motus epicycli. </s> <s xml:space="preserve">Quibus gradibus .8 min .7. ſubtractis à <lb/>grad .22. minu .19. per quos Mars progreſſus erat, ſupererunt grad .14. minut .12. <lb/>quibus ipſe in media retrogradatione exiſtens in linea <seg type="var">.o.c.</seg> veri motus epicycli pro-<lb/>motus erit à principio primi ſitus. </s> <s xml:space="preserve">Quod cum eo concordat quod ſupra diximus cir <lb/>ca hos gradus .14. min .12. vltra primum ſitum in ſpatio dierum .105. vt ſupra, ad <choice><ex>tam</ex><am>tã</am></choice> <lb/>tum enim aſcendunt .69. et .36. nunc fingendo <choice><ex>quod</ex><am>ꝙ</am></choice> Mars pergat in ſuo motu compo-<lb/>ſito qui conſtat ex his duobus circulis, vi eccentrici, & epicycli ( quanquam <lb/>vt dixi omittimus illam ſummam ſubtilitatem ſeu ſcrupuloſi<unclear reason="illegible"/> atem <choice><ex>continuæ</ex><am>cõtinuæ</am></choice> inæqua-<lb/>litatis diſtantiæ centri epicycli à centro mundi, & præterimus etiam <choice><ex>irregularitatem</ex><am>irregularitatẽ</am></choice> <lb/>eius circa centrum mundi, propter regularitatem eius circa centrum æquantis, <lb/>atque etiam miſſum facimus motum epicycli recti à ſua media auge ) fingendo <lb/>inquam, quod Mars dicto motu ſuo pergat, vſque ad punctum ſecundæ ſta-<lb/>tionis, præteribunt alij dies .36. vt prius, quibus iunctis cum .105. fiet ſum-<lb/>ma dierum .141. & Mars retrogreſſus erit per alios grad .8. minut .7. quibus ſub-<lb/>tractis a gradibus .14. m in .12. per quos progreſſus erat ſupererunt. gra .6. min .5. qui <lb/>bus ipſe Mars in fine ſuæ retrogradationis promotus erit à primo loco vnde moueri <lb/>cœpit. </s> <s xml:space="preserve">Inter hanc igitur ſecundam ſtationem lineæ <seg type="var">.o.u.d.</seg> & lineam <seg type="var">.o.t.</seg> ſecundæ con <lb/>tingentiæ, Mars diebus .69. vt prius, confecerit gradus .32. min .14. ſui epicycli, & eo-<lb/>dem tempore linea contingens <seg type="var">.o.t.</seg> ambulauerit gra .36. min .13. vt prius, à quo itine <lb/>re ſubtracto angulo ſecundo <seg type="var">.d.o.t.</seg> graduum .13. min .54. ſupererunt gra .22. min .19 <lb/>vti prius, quos Mars ambulauerit directè & apparenter, quibus additis ad gra .6. mi <lb/>nut .5. quibus Mars progreſſus erat à principio motus, fient gra .28. min. circiter .24. <lb/>quibus proceſſerit à priori loco in diebus .177. ideſt in .141. et .36. qui ſunt ferè m. n <lb/>ſes .6. </s> <s xml:space="preserve">Itaque Mars partim ſurſum partim, deorſum ambulans detentus erit <choice><ex>menſibus</ex><am>mẽſibus</am></choice> <lb/>6. in grad .28. </s> <s xml:space="preserve">Zo-<lb/>diaci, <choice><ex>atque</ex><am>atq;</am></choice> ſi finxe <lb/> <ptr xml:id="fig-0250-01a" corresp="fig-0250-01" type="figureAnchor"/> rimas quod epicy <lb/>clus <choice><ex>moueatur</ex><am>moueat̃</am></choice> ver-<lb/>ſus oppoſitum au-<lb/>gis, <choice><ex>longior</ex><am>lõgior</am></choice> erit mo <lb/>ra planetæ in eiuſ <lb/>modi duodecate-<lb/>morio, propter au <lb/>gumentum æqua-<lb/>tionis argumenti. <lb/></s> <s xml:space="preserve">Itaque probata à <lb/>nobis eſt poſſibi-<lb/>litas huius moræ <lb/>Mattis. </s> <s xml:space="preserve">Quod qui <lb/>dem mihi ſuffice-<lb/>re videtur <choice><ex>non</ex><am>nõ</am></choice> mo-<lb/>do cibi, ſed etiam <lb/>cuiuis, qui harum <lb/>ſcientiarum prin-<lb/>cipia teneat. </s> <s xml:space="preserve">Ne-<lb/>que enim <choice><ex>nunc</ex><am>nũc</am></choice> do-<lb/>cere volo eos qui <pb facs="0251" n="239"/><fw type="head">EPISTOLAE.</fw> in ijs ſunt conſumati, nec curam mihi ſuſcipere erudiendi imperitos. </s> <s xml:space="preserve">Satis igitur ſit <lb/>oſtendiſſe, quod qui ſcripſit Martem commorari poſſe tam multos menſes in vno ſi <lb/>gno, non impoſſibilem rem tradidit. </s> <s xml:space="preserve">Immo per obſeruationes huius veritatis mil-<lb/>lies factas, Aſtrologi fecere ſupradictas ſuppoſitiones neceſſarias ad <choice><ex>reducendum</ex><am>reducẽdum</am></choice> in <lb/>ſuas cauſas, & ad regulam, eiuſmodi veriſſimos effectus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0250-01" corresp="fig-0250-01a"> <graphic url="0250-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Non oportebat autem ſcriptorem harum animaduerſionum tantopere eiuſmodi <lb/>mora commoueri, ſed cogitare <choice><ex>quod</ex><am>ꝙ</am></choice> fortaſſe calculi facti fuerunt eo tempore quo mi-<lb/>ſer Mars à Vulcano rete vinctus erat. </s> <s xml:space="preserve">Vnde cum nonita celeriter ſe expedire poſſet <lb/>iter eius ſegnius peractum fuit. </s> <s xml:space="preserve">Aut <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>quum</ex><am>quũ</am></choice> vulneratus fuit in bello Troiano, vis eius <lb/>& agilitas per aliquantulum temporis imminuta fuit. </s> <s xml:space="preserve">Atque ſi hic etiam intellexiſ <lb/>ſet eum <choice><ex>aliquando</ex><am>aliquãdo</am></choice> fuiſſe in poteſtate Othi, & Ephialtis <choice><ex>vinctum</ex><am>vinctũ</am></choice> & carceri incluſum <choice><ex>men- ſes</ex><am>mẽ-ſes</am></choice> tredecim, dum ab Eribea ſolutus fuit, vt tu, antiquos ſequens, eleganter ſcribis <lb/>in illis tuis pulcherimis dialogis. non exiſtimaſſet, credo, tam abſurdum quod alius <lb/><choice><ex>eum</ex><am>eũ</am></choice> detinuiſſet ſex aut ſeptem menſibus, ſed operam dediſſet vt a te intelligeret quid <lb/>ſibi vellet tam longa captiuitas.</s> </p> <p> <s xml:space="preserve">Sed vt ad rem redeamus. </s> <s xml:space="preserve">Idem pag .4. ait, quod verus motus Martis diſtat à me-<lb/>dio circiter dies .8. ſupponens medium motum eſſe dierum .683. <choice><ex>quod</ex><am>ꝙ</am></choice> etiam falſum eſt. <lb/></s> <s xml:space="preserve">Sed vtcunque ſit, fallitur. </s> <s xml:space="preserve">Solet enim periodus veri motus Martis eſſe die-<lb/>rum circiter .708. modo paulo plus, modo paulo minus, & interdum poteſt etiam <lb/>eſſe multo breuior, ſicuti erit à die 3. </s> <s xml:space="preserve">Decembris anni .1593. <choice><ex>vſque</ex><am>vſq;</am></choice> ad initium Iunij <num value="1595">.<lb/>1595.</num> </s> <s xml:space="preserve">Tunc enim erit tantum dierum .545. & non quidem ſine ratione, nam dicto <lb/>initio Decembris Mars paulo ante cæperiteſſe directus, cum centrum epicycli erit <lb/>circa medium Tauri, & eius ſtella in principio Arietis & initio Iunij .1595. </s> <s xml:space="preserve">Mars pa <lb/>rum diſtabit ab initio retrogradationis, regreſſus tamen ad initium ipſius Arietis, & <lb/>centrum epicycli erit circa medium Aquarij, in cuius ſigni medio, hac ætate repe-<lb/>ritur oppoſitum augis, & in quo ſitu, æquationes <choice><ex>argumenti</ex><am>argumẽti</am></choice> ſunt, quam maximę eſſe <lb/>poſſint, quum centrum epicycli circuiuerit ſolum circiter tres quartas totius ambi-<lb/>tus, & Mars circuiuerit per partem ſuperiorem epicycli circiter gradus .252. </s> <s xml:space="preserve">Hoc au <lb/>tem dico, vt oſtendam poſſibilitatem huius eius extraordinariæ velocitatis. </s> <s xml:space="preserve">Nam <lb/><choice><ex>quicunque</ex><am>quicunq;</am></choice> voluerit poterit certior fieri, per calculum partium motus Martis.</s> </p> <p> <s xml:space="preserve">Vbiautem poſtea idem author miratur interualla, quæ <choice><ex>ponuntur</ex><am>ponũtur</am></choice> inter coniunctio <lb/>nes Iouis, & Martis in eodem ſigno, <choice><ex>eaque</ex><am>eaq́;</am></choice> vocat errores maximos, oſtendit ſe non re <lb/>ctè conſideraſſe motus eorum. </s> <s xml:space="preserve">Et præcipuè primum miratur <choice><ex>quod</ex><am>ꝙ</am></choice> inter annum .1528. <lb/>et .1553. </s> <s xml:space="preserve">Iupiter & Mars nunquam coeant in Leone, cum hæ duæ coniunctiones in <lb/>ter ſe diſtent ann .25. afferens pro ratione, quod hæc duo ſydera, altero quoque anno <lb/>coniunguntur, ſic dicens.</s> </p> <quote> <s xml:space="preserve">Qui ſciet has duas ſtellas ſecundo quoque anno inter ſe coniungendas, mirabitur <lb/>quomodo non poterunt numeratores, huiuſmodi animaduertere errores.</s> </quote> <p> <s xml:space="preserve">Et præter hanc <choice><ex>rationem</ex><am>rationẽ</am></choice> fortaſſe <choice><ex>ent</ex><am>ẽt</am></choice> conſiderauit, <choice><ex>quod</ex><am>ꝙ</am></choice> in dicto temporis interuallo Iu <lb/>piter <choice><ex>semper</ex><am>sẽper</am></choice> fuit in Leone, vt ann .1540. et .1541. </s> <s xml:space="preserve">Mars <choice><ex>autem</ex><am>aũt</am></choice> in eo ſæpe fuit. </s> <s xml:space="preserve">Vnde im <lb/>poſſibile eilvidetur eos non conueniſſe in dicto ſigno. </s> <s xml:space="preserve"><choice><ex>Idemque</ex><am>Idemq́;</am></choice> dici poteſt de alijs <lb/>coniunctionibus eorundem planetarum, atque has differentias temporum inter di-<lb/>ctas coniunctiones ipſe tribuit erroribus calculorum Ephemeridum, non autem ta-<lb/>bularum, vt ſupra dixit. </s> <s xml:space="preserve">ſed neſcio quare vellet dictos planetas coire in Leone, ſi <lb/>quum Iupiter in eo erat anno .1540. et .1541. & in eo <choice><ex>deambulabatur</ex><am>deambulabat̃</am></choice>, Mars interea <lb/>erat <choice><ex>mon</ex><am>mõ</am></choice> in Libra, modo in Scorpione, Sagittario, Capricorno, & alijs ſignis <choice><ex>vſque</ex><am>vſq;</am></choice> ad <lb/>Cancrum, in quo cum repertus fuit anno .1541. cogitans congredi cum loue in Leo <pb facs="0252" n="240"/><fw type="head">IO. BAPT. BENED.</fw> ne, comperit eum inde aufugiſſe. </s> <s xml:space="preserve"><choice><ex>Idque</ex><am>Idq́;</am></choice> fortaſſe, Iupiter data opera fecit, vt huiuſ-<lb/>modi Aſtrologos in admirationem induceret.</s> </p> <p> <s xml:space="preserve">Idem dico de alijs coniunctionibus horum duorum.</s> </p> <p> <s xml:space="preserve">Quod poſtea ait, eos ſecundo quoque anno coniungi, animaduertendum eſt, <choice><ex>quia</ex><am>ꝗa</am></choice> <lb/>(vt <choice><ex>iam</ex><am>iã</am></choice> dixi) duæ ſunt ſpecies coniunctionum, quarum vna eſt linearum eorum me-<lb/>diorum motuum, altera corporum eorum, ſaltem in longitudine, cum ambo inue-<lb/>niuntur in eodem circulo, qui tranſit per polos ecclipticæ, nam eos inueniri in <choice><ex>eadem</ex><am>eadẽ</am></choice> <lb/>linea recta <choice><ex>tranſeunte</ex><am>trãſeũte</am></choice> per centrum mundi, rarisſimum eſt. </s> <s xml:space="preserve">Atque coniunctio ſupra di-<lb/>ctarum linearum vocatur media, & inter Iouem & Martem fieri ſolet ſpatio <choice><ex>dierum</ex><am>dierũ</am></choice>. <lb/>816. cum dimidio, aut circiter. </s> <s xml:space="preserve">Altera <choice><ex>dicitur</ex><am>dicit̃</am></choice> vera, ſiue apparens, & irregulatiſſima, <lb/>quæ quidem non ſeruat tempus determinatum. </s> <s xml:space="preserve">Quare quamuis altero <choice><ex>quoque</ex><am>quoq;</am></choice> an-<lb/>no coniungantur; </s> <s xml:space="preserve">& Iupiter duodenis annis tranſeat per totum Zodiacum, non ideo <lb/>neceſſe eſt, vt in ſpatio .24. annorum coniungantur in ſingulis ſignis, nunquam in eo <lb/>deſicientes, vtipſe credit loquens de veris coniunctionibus apparentibus, eo quod <lb/>ſint irregulatiſſimæ, vt dixi.</s> </p> <p> <s xml:space="preserve">Atque ſi quis velit inuenire periodum coniunctionum mediocrium horum duo-<lb/>rum planetarum, ita faciendum erit. </s> <s xml:space="preserve">Sumat periodum motus mediocris Iouis, quę <lb/>eſt dierum .4328. & Martis, quæ eſt dierum .687. in quo tempore Martis, Iupiter am <lb/>bulat gra .57. min .8. & diebus .30. conficit. grad .2. minut .29. & ſecun .23. ad ratio-<lb/>nem gra .360. in diebus .4328. </s> <s xml:space="preserve">Mars verò ad rationem <choice><ex>graduum</ex><am>graduũ</am></choice> .360. in diebus .687. <lb/>ſingulis .30. diebus conficit. gra .15. mi .43. <choice><ex>ſecum</ex><am>ſecũ</am></choice> .14. vnde differentia inter eos eſt gra <lb/>duum .13. mi .15. <choice><ex>ſecum</ex><am>ſecũ</am></choice> .51. per quam diuidendo productum graduum .57. min .8. in <lb/>dies .30. <choice><ex>obuenient</ex><am>obueniẽt</am></choice> dies .129. & duæ tertiæ. </s> <s xml:space="preserve">quibus addendo periodum Martis fient <num value="816">.<lb/>816.</num> cum dimidio, aut circiter. </s> <s xml:space="preserve">Atque hęc eſt periodus infallibilis mediarum con <lb/>iunctionum Iouis cum Marte.</s> </p> <p> <s xml:space="preserve">Nunc venientes ad tabulas Animaduerſionum, videbimus hæc mirabilia eius, in <lb/>quo conſiſtant & vbi ſint tam multi inſignes errores.</s> </p> <p> <s xml:space="preserve">Primum igitur neminem later quod calculus Saturni, à Leouitio editus, difert à <lb/>calculo Stadij circiter gra .2. aut .3. cum Leouitius faciat eum progredi per <choice><ex>tantum</ex><am>tãtum</am></choice> in <lb/>teruallum, modo plus, modo minus, & ſimiliter Iouem. </s> <s xml:space="preserve">ſed longe minori diffe-<lb/>rentia, & ſępe gra .1. minus, atque in alijs planetis differunt, modo plus, modo minus. <lb/></s> <s xml:space="preserve">Huic igitur <choice><ex>mirum</ex><am>mirũ</am></choice> videtur, quod vnus ex his calculatoribus detineat Saturnum plu-<lb/>ribus menſibus in vno ſigno, & alter in alio, non animaduertens dictam <choice><ex>differentiam</ex><am>differentiã</am></choice> <lb/>eſſe eius rei cauſam. </s> <s xml:space="preserve">Miratur item, quod vnus ex is faciat Saturnum morari paucis <lb/>menſibus in vno ſigno, alter vero eum ibi detineat integris annis. </s> <s xml:space="preserve">Vt exempli gra-<lb/>tia, verſus finem ſuæ tabulæ Saturni, dicit quod Leouitius eum carceri includit in <lb/>geminis annis .2. menſe vno, & diebus .9. </s> <s xml:space="preserve">Stadius vero clementior eum liberat intra <lb/>menſes .3. & dies .14. </s> <s xml:space="preserve">Sed hic non cogitat, quod Stadius facit eum ingredi in gemi-<lb/>nos anno .1559. die .10. Iunii, & ambulare directum vſque ad diem .6. Septembris, <lb/>eiuſdem anni gra .6. min .34. <choice><ex>eumque</ex><am>eumq́;</am></choice> poſtea retrogradum inde exire die .22. Decem. <lb/>eiuſdem anni, cum ingreditur in Taurum, vbi partim retrogradus, & partim dire-<lb/>ctus manet vſque ad diem .20. Februa .1560. rediens poſtea in geminos, in quibus <lb/>manet vſque ad diem Iunii .1561. & inde ingreditur in Cancrum, <choice><ex>ambulatque</ex><am>ambulatq́;</am></choice> dire-<lb/>ctus. gra .4. min .59. vſque ad diem .4. Octob. </s> <s xml:space="preserve">Vnde retrogradiens rurſum intratin Ge <lb/>minos die .28. Decemb. eiuſdem anni, at que ibi partim retrogradus partim directus <lb/>manet vſque ad diem .12. Apr .1562. itaque in pluribus vicibus facit eum morari in <lb/>Geminis dies circiter .816. ideſt circiter menſes .27. ſumpſit autem hic ſcriptor bre <pb facs="0253" n="241"/><fw type="head">EPISTOL AE.</fw> uiſſimam moram cauſa comparationis cum calculo Leouitij, vt faceret differentiam <lb/>apparere maiorem. </s> <s xml:space="preserve">Tamen in quouis dictorum temporum nunquam inuenietur <lb/>Leouitius differre à Stadio plus gradibus tribus integris. </s> <s xml:space="preserve">Idem fecit in multis alijs lo <lb/>cis dictorum virorum eos conferens tum in Saturno tum in Ioue, & Marte, <choice><ex>putans</ex><am>putãs</am></choice> ma <lb/>gnum eſſe errorem, <choice><ex>quod</ex><am>ꝙ</am></choice> planeta non perambulet totum ſignum, in quod eſt ingreſſus <lb/>vel directus vel totum retrogradus. </s> <s xml:space="preserve">Atque hæc opinio ſimilis eſt ſuperiori de con <lb/>iunctionibus veris Saturni, & Iouis, vbi dicit quod nunquam coniunguntur in vno ſi <lb/>gno alterius triplicitatis, niſi perfecerit coniunctionem in omnibus ſignis primæ tri <lb/>plicitatis. </s> <s xml:space="preserve">Verum vt ſuperſedeam vlterius diſputare, mihi videtur, quod hactenus <lb/>dixi, poſſe tibi ſatisfacere, quod attinet ad ſciendam ſententiam mean ſuper dictis <lb/>Animaduerſionibus latinè ſcriptis. </s> <s xml:space="preserve">Hoc tamen non prætermittam, <choice><ex>quod</ex><am>ꝙ</am></choice> hic non ani-<lb/>maduertit, <choice><ex>nempe</ex><am>nẽpe</am></choice> <choice><ex>quod</ex><am>ꝙ</am></choice> <choice><ex>differentiæ</ex><am>differẽtiæ</am></choice> locorum <choice><ex>planetarum</ex><am>planetarũ</am></choice> quæ <choice><ex>suntinter</ex><am>sũtinter</am></choice> ephemeridas Leouitij <lb/>& Stadij, euenere, quia vnus ſupputat <choice><ex>cum</ex><am>cũ</am></choice> radicibus, & <choice><ex>fundamentis</ex><am>fundamẽtis</am></choice> Alfonſi alter verò <lb/>Reinoldi ex Copernico recentius obſeruatis, ita idem euenire poterit futuris tem-<lb/>poribus, ſi ſupputati fuerint dicti motus, & loci cum recentioribus obſeruationibus <lb/>cum impoſſibile ſit tam ſubtiliter, tanq́ue perfectè ſupputare loca & <choice><ex>motus</ex><am>motꝰ</am></choice> eorum, <lb/>vtlungo interuallo temporis non comperiantur in eis aliquæ differentię, cuius rei re <lb/>medium eſt ſemper ſequi recentiores obſeruationes & tabulas.</s> </p> <p> <s xml:space="preserve">Atque vt tibi ſatisfaciam etiam circa alia ſcripta vulgari lingua edita menſibus .4 <lb/>poſt latina, etſi intelligere potes, qualia poſſint eſſe alia eius ſcripta, ex ijs quæ ſupra <lb/>dicta ſunt, atque etiam ex eo, quod dicit ſemiſiſſe multa exempla ſuarum Animad <lb/>uerſionum in varias terras, illis qui profitentur has ſcientias, aut earum ſtudioſi ſunt, <lb/>nec <choice><ex>quenquam</ex><am>quenquã</am></choice> inueniſſe qui ad <choice><ex>tam</ex><am>tã</am></choice> laudabilem prouinciam motus ſit, nec vidiſſe, <choice><ex>quod</ex><am>ꝙ</am></choice> ali <lb/>quis reſponderit eius rationibus; </s> <s xml:space="preserve">laudabilem prouinciam, autem puto, <choice><ex>quod</ex><am>ꝙ</am></choice> intelligat <lb/>correctionem ephemeridum, verens, ne culpa calculatorum, qui eas ſumpſere e ta-<lb/>bulis, tam differentes ſint, vt <choice><ex>quibuſdam</ex><am>quibuſdã</am></choice> locis cap .1. </s> <s xml:space="preserve">Videtur, & præcipuè vbi ſic ait.</s> </p> <quote xml:lang="it"> <s xml:space="preserve">Perche eſſendo impoſſibile alli ſtudioſi di dette ſcientiæ di non ſeruirſi delle <lb/>ephemeridi, maggiormente a quelli che non ſanno ſeruirſi delle tauole, e cono-<lb/>ſcendo d'incorrere in errori ſenza hauerui altro rimedio, ſarebbono forzati di ab <lb/>bandonare i ſtudij loro.</s> </quote> <p> <s xml:space="preserve">Quanquam circa finem dicti capitis redeat in meliorem viam & aduerſetur ſi-<lb/>bijpſi vbi ſic ait.</s> </p> <quote xml:lang="it"> <s xml:space="preserve">Che poi eſſi poſſeſſori della ſcienza, & c.</s> </quote> <p> <s xml:space="preserve">Etiam aperiam tibi, quæ mea ſit de ijs ſententia.</s> </p> <p> <s xml:space="preserve">Hicigitur in ſcriptis Italicis, vt morderet aliquem ex ijs, qui eius ſuperiora ſcripta <lb/>non laudauerant, occaſionem capit aperiendi aliquos illius errores, per editionem <lb/>collationis quorundam calculorum a ſe collectorum illius, atque etiam aliorum, cu <lb/>ius calculi ſunt in ſecunda, & ſeptima figura. </s> <s xml:space="preserve">Sed prius quam veniamus ad defenſio <lb/>nem harum duarum figurarum vide obſecro quam alienum ei videatur, quod alij <lb/>dixerint differentiam ephemeridum non eſſe magni momenti, non afferens reſpe-<lb/>ctum vllum, qui enim dixerunt eiuſmodi differentiam non eſſe magni momen-<lb/>ti id dixerunt habito reſpectu ad ſignum in quo eſt planeta, vt (exempli gra-<lb/>tia) quamuis in ponendo loco Saturni Leouitius interdum differat à Stadio gra <lb/>dibus .3. quum vterque eum ponat in eodem ſigno, tuncid nullius momenti eſt, <lb/>& ſic in coniunctionibus aut alijs aſpectibus duo, aut .3. gradus non faciunt alteratio <lb/>nem ſenſibilem, cum virtus coniunctionum, & aſpectuum inſit, & duret per mul-<lb/>tos gradus ante aut poſt ipſum punctum. </s> <s xml:space="preserve">Nec quicquam <choice><ex>tamen</ex><am>tamẽ</am></choice> eſt qui dubitet, quin <lb/>præſtaret ſcire ſubtiliter ipſum punctum. </s> <s xml:space="preserve">Nec vnquam fuit aliquis qui negauerit <choice><sic>re ferre</sic><corr resp="paul">re ferre</corr></choice> vt anni directionum correſpondeant gradibus æquatoris. </s> <s xml:space="preserve">Et præterea in ephe <lb/>meridibus videntur certè motus & aſpectus luminarium, quamuis inſit <choice><ex>differentia</ex><am>differẽtia</am></choice> mi <lb/>nutorum. </s> <s xml:space="preserve">Nam non differunt gradibus, præter ſitum parum diſtantem à vero om-<lb/>nium planetarum, quorum cognitio in cœlo, quamuis circa eorum locum error eſ-<lb/>ſet gra .10. tamen in hoc prodeſſet, & tempus aſpectus eorum, etſinon diei præcisè, <lb/>quia influentia eiuſmodi a ſpectuum, præterquam Lunæ durat multis diebus, & non <lb/>vno tantum. </s> <s xml:space="preserve">præterquam quod ipſæ ephemerides <choice><ex>oſtendunt</ex><am>oſtendũt</am></choice> nobis tempus <choice><ex>ecclipſium</ex><am>ecclipſiũ</am></choice>, <lb/>in quo certènon differunt nec diebus nec multis horis, & itidem multa alia.</s> </p> <p> <s xml:space="preserve">Non ſunt igitur contemnendæ ephemerides, nec habendæ pro re nullius pretij, <lb/>vt hic ait.</s> </p> <p> <s xml:space="preserve">Quod attinet ad illa alia, quæ hic vocat errores ephemeridum, tam de apparenti <lb/>coniunctione Saturni cum Ioue in ſignis alterius triplicitatis prius quam peręgerit <lb/>præcedentem, quam de faciendo currere <choice><ex>Saturnum</ex><am>Saturnũ</am></choice>, & de retinendo Ioue, de <choice><ex>detinem</ex><am>detinẽ</am></choice> <lb/>do Marte .6. aut .7. menſibus in vno ſigno, de Marte, & Ioue non coeuntibus ſingu-<lb/>lis .24. annis in quolibet ſigno, & cius generis alia, minime verum eſt quod ſint er-<lb/>rores, quamuis huic præbuerint occaſionem toties errandi.</s> </p> <p> <s xml:space="preserve">Comparatio poſtea inter eius calculos ſumptos partim ex tabulis Iunctini, & par <lb/>tim ex ephemeridibus Stadij tan quam calculis Copernici, & calculos figurarum ſu-<lb/>per eis poſitarum ſupputatarum à diuerſis per ephemeridas Alfonſinas, etiam pro-<lb/>poſita ab eo eſt ad oſtendendum magnam & monſtruoſam differentiam, vt ait cap <num value="2">.<lb/>2.</num> vbi miratur, quod cum ex communi ſententia calculi Copernici meliores ſint, cal <lb/>culatores dictarum figurarum potius eos ſumpſerint à tabulis Alfonſi, quam Coper <lb/>nici. </s> <s xml:space="preserve">Quæ admiratio quam aliena ſit, conſiderandum permittam cuiuis intelligen-<lb/>ti harum ſacultatum, cum ſæpe accidere poſſit. </s> <s xml:space="preserve">vt cum aliquis velit ſcire ſolum vni <lb/>uerſalia alicuius geneſis, ſiue natiuitatis, cum non inueniantur ephemerides Coper-<lb/>nici, ſed tantum Alfonſi, calculator vtatur tantum ephemeridibus, quas inuenit, <choice><ex>tum</ex><am>tũ</am></choice> <lb/>cauſa vitandi tædij calculi tabularum, qui magni laboris eſt, pręcipuè in tabulis Pru <lb/>tenicis Reinoldi. </s> <s xml:space="preserve">tum quia ſuperflua ei eſt ſumma ſubtilitas, cum non curet laborare <lb/>circa directiones vt <choice><ex>factum</ex><am>factũ</am></choice> eſt pro <choice><ex>ſecunda</ex><am>ſecũda</am></choice> figura ab hoc propoſita, quæ erat anni .1551 <lb/>quo non <choice><ex>inueniebantur</ex><am>inueniebãtur</am></choice> ephemerides Copernicæ, quæ non editæ ſunt ante annum <lb/>1554. præter quam quod ille nobilis vir pro quo ſupputata fuit dicta ſecunda natiui <lb/>tas dubitabat de anno, vt hic ſimiliter ſcit. </s> <s xml:space="preserve">quare potuiſſet perdi tempus, & labor, ſi <lb/>ſupputata fuiſſet per tabulas Reinoldi, nam Iunctini tabulæ nondum editæ fuerant. <lb/></s> <s xml:space="preserve">Calculus poſtea ſeptimæ figuræ, qui erat reuolutio dictæ ſecundæ natiuitatis, <choice><ex>duabus</ex><am>duabꝰ</am></choice> <lb/>de cauſis non factus eſt per tabulas prutenicas, primum, quia eius anni .1580. non <lb/>inueniebantur amplius ephemerides Copernicæ. </s> <s xml:space="preserve">Ephemerides enim Stadij <choice><ex>in- cipientes</ex><am>in-cipiẽtes</am></choice> ab ann .1554. <choice><ex>deſinunt</ex><am>deſinũt</am></choice> ann .1576. & <choice><ex>continuatæ</ex><am>cõtinuatæ</am></choice> poſtea quæ perueniunt vſque <lb/>ad annum .1600. non peruenere ad manus calculatoris ante hunc annum .1581. </s> <s xml:space="preserve">Al-<lb/>tera ratio eſt, quia in reuolutionibus, quoniam in eis non fiunt directiones, non po-<lb/>nuntur à doctis, ne minuta quidem. </s> <s xml:space="preserve">quare non ſolum non curant eas ſupputare per <lb/>tabulas, ſed nec exquiſitè quidem per ephemeridas. </s> <s xml:space="preserve">Calculi poſtea ab hoc ſumpti <lb/>ex tabulis Iunctini, & poſiti ſub dicta ſecunda figura, adeò rectè facti ſunt, vt cum ſe <lb/>cundum ip ſas tabulas oporteat Saturnum eſſe circa .32. minutum gradus .23. Aqua-<lb/>rij, ipſe eum ſcribat in gra .11. mi .3. dicti ſigni. Iupiter ſimiliter qui ſecundum dictas <lb/>tabulas inuenitur circa finem gradus .5. Cancri, ab eo ponitur in min .28. gra .19. eiuſ <lb/>dem. ex quibus planetis Saturnus in figura poſitus eſt in min .27. grad .23. Arietis, Iu <lb/>piter autem in min .3. gra .6. Cancri, </s> <s xml:space="preserve">Vnde ſecun dum verum, inter calculum Alfon <pb facs="0255" n="243"/><fw type="head">EPISTOLAE.</fw> ſi & Iunctini in Saturno non erat differentia plus quam minu .5. & in Ioue min .4. ſed <lb/>ſecundum calculum huius in Saturno fuiſſet differentia gra .11. minu .54. & in Ioue <lb/>grad .13. min .35. </s> <s xml:space="preserve">Atque hæ ſunt quidem differentiæ magnæ, & monſtruoſæ vtipſe <lb/>eas vocat, vt etiam eſtilla Veneris, & Mercurij inter tertiam figuram, & eius calcu-<lb/>lum ſumptum, non quidem à tabulis laborioſis, ſed à ſimplicibus ephemeridibus Sta <lb/>dij, quæ differentia <choice><ex>non</ex><am>nõ</am></choice> eſt quidem paucorum graduum, cum ſit tertiæ partis cœli in <lb/>quolibet dictotum planetarum. </s> <s xml:space="preserve"><choice><ex>Huiuſmodique</ex><am>Huiuſmodiq́;</am></choice> monſtra certè non ſunt orta à ta-<lb/>bulis ſiue ephemeridibus diuerſis, ſed ſunt partus huius authoris.</s> </p> <p> <s xml:space="preserve">Pergens poſtea aſſiduè bonus hic vir hominibus dare ſpecimen doctrinæ ſuæ ape <lb/>riendo (vt conatur) aliorum errores, proponit duas differentias inter primam fi-<lb/>guram, & ſuum calculum ſuppoſitum Saturni, & Iouis. </s> <s xml:space="preserve">Primum de Saturno ait, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>cum differentia ſit gra .1. min .30. oſtendit in directione, <choice><ex>quod</ex><am>ꝙ</am></choice> accidens ſit euenturum <lb/>anno vno, & menſibus ſex ante, aut poſt, quaſi eiuſmodi differentia eſſet partium æ-<lb/>quatoris, ſicuti eſt partium Zodiaci. </s> <s xml:space="preserve">Idem dico de differentia Iouis. </s> <s xml:space="preserve">Quod quidem, <lb/>manifeſtum eſt inditium ſcientiæ ſuæ, & quantum ea intelligat de quibus loquitur.</s> </p> <p> <s xml:space="preserve">Quod poſtea attinet ad differentiam inter Copernicum & Alfonſum, circa <choice><ex>Solem</ex><am>Solẽ</am></choice>, <lb/>nullus eſt harum ſcientiarum peritus, qui id neſciat, & ſimiliter de differentia ſitus <choice><ex>cae li</ex><am>cęli</am></choice> in reuolutionibus annuis.</s> </p> <p> <s xml:space="preserve">Quod vero ait ſeptimam ſiguram malè <choice><ex>ſupputatam</ex><am>ſupputatã</am></choice> fuiſſe, ſi non eſt maximus cer-<lb/>tè non eſt minimus monſtruoſorum eius errorum. </s> <s xml:space="preserve">Vbi itidem videri poteſt, quam <lb/>alienus hic ſit ab hac ſcientia. </s> <s xml:space="preserve">Nam ſi ſaltem curaſſet ſibi ab aliquo ſupputandum <lb/>locum Solis per tabulas Alfonſi in inſtanti minutorum .36. pomeridianorum, certior <lb/>factus eſſet quod in illo puncto Sol inueniebatur in minu 54. grad .11. </s> <s xml:space="preserve">Geminorum, <lb/>ideſt præterierat gra .10. cum min .54. vel ſi curaſſet ſibi inueniendum tempus, per <lb/>dictas tabulas cum grad .10. min .54. </s> <s xml:space="preserve">Geminorum vt faciendum eſt, ſequendo <choice><ex>tamen</ex><am>tamẽ</am></choice> <lb/>Alfonſum, & non per calculum Solis poſitum in ephemeridibus, vt parum periti fa <lb/>cere ſolent, vidiſſet <choice><ex>quod</ex><am>ꝙ</am></choice> inuenta eſſent min .36. pomeridiana. </s> <s xml:space="preserve">Leuis tamen occaſio hu <lb/>ic fuit ſuſpicandi eiuſmodi tempus eſſe falſum, quod viderit in illa figura <choice><ex>Solem</ex><am>Solẽ</am></choice> po <lb/>ſitum eſſe cum gra .11. & non cum gra .10. min .54. non animaduertens ita notatum <lb/>fuiſſe Solem vt omnes alios planetas, ſcilicet ſine minutis, quum, vt dixi, in reuolu <lb/>tionibus non adhibeatur tanta ſcruploſitas.</s> </p> <p> <s xml:space="preserve">Quod deinde ait, in illa figura Solem poſitum eſſe in decima domo, & non in .9. <lb/>id relinquam iudicio eorum qui ſciunt numerare domos, ſaltem poſuiſſet authori-<lb/>tate ſua Solem in dicta decima diuersè ab exemplo ei dato ab amico, vt oſtenderet <lb/>ſe dicere verum, vt in ſecunda figura diſcrepat ab ipſo exemplo in collocando Leo-<lb/>ne, Virgine, & Libra, & Scorpio, quos malè locauit, & ſi alii bene ſe habent.</s> </p> <p> <s xml:space="preserve">Atque quod hactenus à me dictum eſt, ſatis ſit ad <choice><ex>intelligendum</ex><am>intelligẽdum</am></choice> quale ſit <choice><ex>reliquum</ex><am>reliquũ</am></choice> <lb/>dictæ eius diſputationis. </s> <s xml:space="preserve">Sienim velim pergere notare omnia eius errorum loca, eſ <lb/>ſet mihi inanis labor, & tibi nimia moleſtia. </s> <s xml:space="preserve">Et quamuis non defuerint præſtantiſſi-<lb/>mi viri, qui viſis eius ſcriptis familiariter eum monuere, & tu ipſe, vt audiui, cum <choice><ex>in- ſtrumento</ex><am>in-ſtrumẽto</am></choice> theoricę in <choice><ex>manibus</ex><am>manibꝰ</am></choice> ei <choice><ex>oſtenderis</ex><am>oſtẽderis</am></choice> quo <choice><ex>mon</ex><am>mõ</am></choice> Mars poſſit morari amplius ſex <choice><ex>mem</ex><am>mẽ</am></choice> <lb/>ſibus in vno ſigno. </s> <s xml:space="preserve">& præterea cum iam ab initio Taurinum aduenit, mecum com-<lb/>municauerit illa ſua prima ſcripta, <choice><ex>egoque</ex><am>egoq́;</am></choice> eum monuerim, quod in varijs <choice><ex>rebus</ex><am>rebꝰ</am></choice> falleba <lb/>tur, <choice><ex>diſſuaſerimque</ex><am>diſſuaſerimq́;</am></choice> ne ea imprimenda curaret, quia nullum honorem inde referret, <lb/>eum hortans, vt potius alijs rebus operam daret, atque ei dixerim quod ad animad-<lb/>uerſiones differentiarum ephemeridum attinet, quod id iam <choice><ex>oens</ex><am>oẽs</am></choice> animaduerterant. <lb/></s> <s xml:space="preserve">Mihi reſpondit ſe decreuiſſe illa edere, vt poſtea fecit, & tot admonitionibus non <pb facs="0256" n="244"/><fw type="head">IO. BABPT. BENED.</fw> acqu<unclear reason="illegible"/>ieſcens, die .11. </s> <s xml:space="preserve">Auguſti edidit chartam illam impreſſam inuitans ad diſputa-<lb/>tionem quotquot adhęrerent contrariæ ſententiæ, volens ſuſtinere <choice><ex>Martem</ex><am>Martẽ</am></choice> non poſ <lb/>ſe commorari in vno ſigno amplius duobus menſibus, ſupponens partem princi-<lb/>piorum ab omnibus admiſſorum, & in fine paginæ exponens modum, quo vtitur ad <lb/>probationem ſuæ intentionis. </s> <s xml:space="preserve">Puto autem quodſecum ratiocinabatur de Marte, <lb/>vt fecit de Saturno in ſcripto latino, hoc modo. </s> <s xml:space="preserve">Si Mars in duobus annis ambulat <lb/>per omnia 12. ſigna, neceſſe eſt igitur, vtin menſibus duobus ambulet per vnum ſi-<lb/>gum, cum menſes .2. ſint duo decima pars annorum duorum. </s> <s xml:space="preserve">Sedibi ſtatim in ipſo <lb/>initio commitcit errorem graduum ferè .7. dicens, quod medius motus Martis inue-<lb/>niebatur ſignorum .4. & gra .17. cum eo tempore dictus medius motus non eſſet reue <lb/>ra plus quam ſign .4. grad .10. mi .36. verum hoc ad ea, quæ ſequuntur exigui eſt mo <lb/>menti. </s> <s xml:space="preserve">Is poſtea particulatim colligit medium motum Martis ad diem .29. </s> <s xml:space="preserve">Mai an-<lb/>ni .15 14. quem ait eſſe ſignorum .9. gra .27. min .53. & tamen reuera erat <choice><ex>tantum</ex><am>tantũ</am></choice> ſigno <lb/>rum .9. gra .21. mi .29. ſed miſſum faciamus etiam hunc errorem <choice><ex>tanquam</ex><am>tanquã</am></choice> à primo pen <lb/>dentem. </s> <s xml:space="preserve">Cum deinde ibidem ponit centrum epicycli, ſimiliter errat, nam <lb/>centrum epicycli nunquam poni debet vbi eſt linea medij motus, niſi ſit in auge, aut <lb/>in oppoſito aug is eccentrici, quia debebat collocare ipſum centrum <choice><ex>tanto</ex><am>tãto</am></choice> poſt linec<unclear reason="illegible"/><choice><ex>am</ex><am>ã</am></choice> <lb/>medij motus, quanta erat æquatio centri, quia medium centrum Martistunc erat mi <lb/>nus ſignis ſex, & aux eccentrici eius erat in ſexto minuto grad .16. Leonis. </s> <s xml:space="preserve">Tamen <lb/>hoc etiam leue eſt. </s> <s xml:space="preserve">Præſupponamus igitur quod centrum epicycli cſſet in grad .28 <lb/>Capricorni, vt ipſe credidit, ideſt gradibus .7. vlterius quam erat reuera. </s> <s xml:space="preserve">Ait poſtea <lb/>ſe comperiſſe Martem ambulaſſe ſigna .4. & grad .22. eius epicycli, ſed non explicat <lb/>an intelligat de argumento medio, an de vero, quod vocatur æquatum, nam ſi intel <lb/>ligatur de medio, hoc eſſe non poteſt, cum <choice><ex>medium</ex><am>mediũ</am></choice> eſſet ſignorum .4. gra .24. mi .35. <lb/>ſed ſi intelligatur de vero, vt iure credendum eſt (alioquin etiam erraſſet) cer-<lb/>tè falſum eſt. </s> <s xml:space="preserve"><choice><ex>Nam</ex><am>Nã</am></choice>, verum, erat ſignorum .4. grad .29. minu .39. </s> <s xml:space="preserve">Itaque Mars non di-<lb/>ſtabat à linea veri motus epicycli amplius gradibus .30. & minu .21. ipſius epicycli, <lb/>& æquatio argumentiſecundo correcta erat gra .44. minu .2. à quo ſubtracta æquatio <lb/>ne centri, quæ erat gr .5. minu .4. (cum centrum epicycli deberet tanto ſpacio eſſe <lb/>poſt lineam medij motus quantum ſupra dixi) ſupererant gra .38. minu .58. adden-<lb/>di gradibus, & minu. medij motus, qui cum reuera eſſent grad .21. & minu .29. Ca-<lb/>pricorni, perueniebant ad minu .27. grad .1. Piſcium. </s> <s xml:space="preserve">Sed præſuppoſito <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> <lb/>ipſum, quod medius motus eſſet grad .28. </s> <s xml:space="preserve">Capricorni, & quod Mars eſſet non <choice><ex>ſolum</ex><am>ſolũ</am></choice> <lb/>vbi hic ait, ſed etiam in prima linea contingentiæ epicycli, ideſt in prima linea ma <lb/>ximæ æquationis argumenti, & præſuppoſito etiam quod dicta æquatio eſſet æqua-<lb/>lis illi, quam haberet ad medium Aquarij ſcilicet grad .47. quum centrum epicycli <lb/>eſt in oppoſito augis, manifeſtum eſt, quod eiuſmodi linea contingentiæ non tranſi <lb/>ret vltra grad .15. Piſcium, & tamen hic ait, quodlinea veri motus Martis vadit ad <lb/>grad .16. Arietis. </s> <s xml:space="preserve">vnde oporteret, quod æquatio argumenti eſſet plus quam grad <num value="78">.<lb/>78.</num> </s> <s xml:space="preserve">Quod ſi verum eſſet, & <seg type="var">.o.c.</seg> etiam eſſet partium .54. ſecundum diſtantiam pro-<lb/>ximiorem centro mundi, ſemidiameter epicycli eſſet eiuſmodi partium .52. minut <num value="49">.<lb/>49.</num> & quum Mars eſſet in <seg type="var">.g.</seg> ideſt in oppoſito veræ augis epicycli, dum centrum epi <lb/>cycli eſſet in eiuſmodi diſtantia à terra, diſtantia <seg type="var">.o.g.</seg> ideſt à terra ad Martem non <lb/>eſſet plus, quam vna ſola pars ex dictis, cum minut .11. cum partes .52. minu .49. <lb/>ad .54. ſint vt ſinus anguli gra .78. qui eſt partium .97814. ad ſinum totalem partium <lb/>100000. </s> <s xml:space="preserve">Nam iam ſupra dixi, quod triangulus <seg type="var">.o.c.i.</seg> eſt rectangulus. </s> <s xml:space="preserve">Hinc ſeque- <pb facs="0257" n="245"/><fw type="head">EPISTOLAE.</fw> retur, quod in in-<lb/>teruallo <seg type="var">.o.g.</seg> <choice><ex>vnius</ex><am>vniꝰ</am></choice> <lb/> <ptr xml:id="fig-0257-01a" corresp="fig-0257-01" type="figureAnchor"/> partis, & min .11. <lb/>reſpectu <seg type="var">.o.c.</seg> par-<lb/>tium .54. </s> <s xml:space="preserve">Colloca <lb/>retur ſemidiame-<lb/>ter terrę cum ſpiſ-<lb/>ſitudine aeris, i-<lb/>gnis, cęlorum Lu-<lb/>næ, Mercurij, Ve-<lb/>neris, & Solis, <choice><ex>prae terquam</ex><am>pręterquam</am></choice> quod vt <lb/>inter Solem, & <lb/>terram ſunt circa <lb/>605. diametri ip <lb/>ſius terræ, inter <lb/>terram, & Mar-<lb/>tem cum eſſet in <lb/>auge ſui epicycli, <lb/>& epicyclus in au <lb/>ge eccentrici, in-<lb/>uenirentur cir--<lb/>ca .60000. dia--<lb/>metri eiuſdem ter <lb/>ræ, & tamen ea diſtantia ſiue interuallum non poteſt continere .5000. diametri ter-<lb/>ræ. </s> <s xml:space="preserve">Et quod plus eſt, hic tam vaſtum facit hunc ſuum epicyclum, vt ambiente Mar-<lb/>te per inferiorem eius partem, neceſſe ei eſſet manere in vno duodecatemorio mul-<lb/>to plus quam .7. aut .8. menſ. </s> <s xml:space="preserve">vnde hic multo magis miraretur quam prius. </s> <s xml:space="preserve">Hinc cer-<lb/>nere licet quam rectè facti ſint hi eius calculi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0257-01" corresp="fig-0257-01a"> <graphic url="0257-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Vt autem etiam hinc aliqua vtilitas capiatur (prætermiſſis inconuenientibus vna <lb/>cum falſis ſuppoſitis huius) Videamus ordine ſcientifico vbi poterat eſſe verus lo-<lb/>cus Martis, aut vero proximus, die .29. Mai anni .1514. quem hic exempli cauſa ſu-<lb/>mit. </s> <s xml:space="preserve"><choice><ex>Idque</ex><am>Idq́</am></choice> tam ad defenſionem tabularum Alfonſi, quam ephemeridum ex eis col <lb/>lectarum. </s> <s xml:space="preserve">quæ quidem exactæ ſunt, vt quiſque peritus ſacile videre poterit, non au-<lb/>tem calculatæ à tam ſtupidis hominibus, vt à vero aberrent etiam gradibus .46. vt <lb/>hic ait ſe depræhendiſſe.</s> </p> <p> <s xml:space="preserve">Primum igitur ſupponemus eoſdem illos terminos, quos ipſe nec d@bet, nec po <lb/>teſt negare, præter ea quæ ſupra ſuppoſita ſunt, nempe quod ſemidiameter epicycli <lb/>ſit partium .39. minu .30. & eccentricitas partium .6. talium qualium eſt ſemidiame-<lb/>ter deferentis diuiſus in .60. & quod dicto tempore aux eccentrici Martis eſſet cir-<lb/>ca minutum .5. grad .16. Leonis, ſcilicet graduum .135. min .5. & quod linea motus <lb/>me<unclear reason="illegible"/>diocris eſſer circa minu .30. gradus .22. </s> <s xml:space="preserve">Capricorni, & quod verum <choice><ex>centrum</ex><am>centrũ</am></choice> Mar <lb/>tis eſſet grad .151 minut .20. & quod argumentum verum eſſet grad .149. minu .39. <lb/><choice><ex>atque</ex><am>atq;</am></choice> ita oſtendam, neque tabulas, neque ephemerides errare, ne quidem vno gra-<lb/>du, ac ne quidem multis minutis, non modò tam monſtruoſa differentia, vt ipſe <lb/>ait.</s> </p> <p> <s xml:space="preserve">Quare primum nobis ſcientificè inueniendum eſt, quanta eſſet diſtantia <seg type="var">.o.c.</seg> <pb facs="0258" n="246"/><fw type="head">IO. BAPT. BENED.</fw> præciſe ideſt interuallum inter centrum mundi, & centrum epicycli Martis in huiuſ-<lb/>modi ſitu.</s> </p> <p> <s xml:space="preserve">Fingemus igitur eccenticum Martis ſignificatum per <seg type="var">.p.c.m.</seg> cuius centrum ſit <seg type="var">.r.</seg> <lb/>& lineam augis <seg type="var">.p.r.o.m.</seg> in qua <choice><ex>centrum</ex><am>centrũ</am></choice> mundi ſit <seg type="var">.o.</seg> centrum autem verum epicycli, <lb/>comprehendatur ab angulo <seg type="var">.p.o.c.</seg> qui ſit graduum .151. min .30. ſecundum ſuppoſi-<lb/>tum. </s> <s xml:space="preserve">Quare in puncto <seg type="var">.c.</seg> erit centrum epicycli. </s> <s xml:space="preserve">Imaginemur ergo <seg type="var">.c.o.</seg> productam à <lb/>parte <seg type="var">.o.</seg> quouſque ab <seg type="var">.r.</seg> centro deferentis veniat linea <seg type="var">.r.k.</seg> perpendiculariter, faciens <lb/>angulum rectum in puncto. k & quoniam angulus <seg type="var">.r.o.c.</seg> datur nobis graduum .151. <lb/>min .30. ideo cognoſcemus angulum <seg type="var">.r.o.k.</seg> tanquam reliquum ex duobus rectis, qui <lb/>erit gra .28. min .30. & ſimiliter angu-<lb/> <ptr xml:id="fig-0258-01a" corresp="fig-0258-01" type="figureAnchor"/> lum <seg type="var">.o.r.k.</seg> tanquam reſiduum vnius <lb/>recti, qui erit gra .61. min .30. cuius ſi-<lb/>nus ideſt <seg type="var">.o.k.</seg> erit partium .8788 1. et <seg type="var">.k.<lb/>r.</seg> vt ſinus anguli <seg type="var">.r.o.k.</seg> partium .47715 <lb/>talium qualium <seg type="var">.o.r.</seg> eſſet 100000. ſed <lb/>vt <seg type="var">.o.r.</seg> eſt .6. latus <seg type="var">.o.k.</seg> erit .5. & min .16 <lb/>et <seg type="var">.r.k.</seg> partium .2. min .52. & quia <seg type="var">.r.c.</seg> <lb/>cſt <choice><ex>partium</ex><am>partiũ</am></choice> 60. eiuſmodi, ſi ab eius qua-<lb/>drato ſubtractum fuerit quadratum ip <lb/>ſius <seg type="var">.r.k.</seg> reliquum erit nobis <choice><ex>quadratum</ex><am>quadratũ</am></choice> <lb/>ipſius <seg type="var">.k.c.</seg> cuius radix, ideſt <seg type="var">.k.</seg> erit par-<lb/>tium .59. min .56. à qua <seg type="var">.c.k.</seg> ſubtrahen-<lb/>do poſtea <seg type="var">.k.o.</seg> partium .5. minu .16. re-<lb/>manebit <seg type="var">.o.c.</seg> partium .54. min .40. pro <lb/>diſtantia quæſita.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0258-01" corresp="fig-0258-01a"> <graphic url="0258-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Fingamus poſtea epicyclum <seg type="var">.f.n.g.</seg> <lb/>in quo argumentum verum graduum <num value="149">.<lb/>149.</num> minu .39. ſit arcus <seg type="var">.f.n.</seg> vbi Mars inueniatur in <seg type="var">.n.</seg> per quem punctum tranſeat li-<lb/>nea <seg type="var">.o.n.</seg> veri motus Martis. </s> <s xml:space="preserve">Deinde inueniamus angulum <seg type="var">.c.o.n.</seg> æquationis <choice><ex>argumem</ex><am>argumẽ</am></choice> <lb/>ti, modo iam dicto, ideſt ducendo ſinum <seg type="var">.n.h.</seg> arcus <seg type="var">.n.g.</seg> qui arcus tanquam reliquus <lb/>argumenti veri, iam præſuppoſiti, ex dimidio circulo, erit graduum 30. minu .21. & <lb/><seg type="var">n.h.</seg> eius ſinus partium .50528. ſinus ſimiliter anguli <seg type="var">.n.c.h.</seg> et <seg type="var">.c.h.</seg> tanquam ſinus an-<lb/>guli <seg type="var">.c.n.h.</seg> reſtantis ex uno recto grad .59. minu .39. erit partium .86295. <choice><ex>talium</ex><am>taliũ</am></choice> qua-<lb/>lium <seg type="var">.c.n.</seg> ſinus totus eſſet partium .100000. ſed vt partium .39. & min .30. ſinus <seg type="var">.c.h.</seg> <lb/>erit partium .34. min .5. et <seg type="var">.n.h.</seg> partium .19. mi .57. reliquum poſtea <seg type="var">.h.o.</seg> ex <seg type="var">.o.c.</seg> par-<lb/>tium .20. min .35. quia iam ſupra inuenimus <seg type="var">.o.c.</seg> eſſe partium eiuſmodi .54. minu .40. <lb/>vnde <seg type="var">.o.n.</seg> vt radix quadrata ſummæ duorum <seg type="var">.n.h.</seg> et <seg type="var">.h.o.</seg> erit partium .28. minu .41. <lb/>talium qualium <seg type="var">.n.h.</seg> inuenta fuit partium .19. min .57. quæ <seg type="var">.n.h.</seg> erit poſtea partium, <lb/>69552. talium qualium <seg type="var">.n.o.</seg> partium .100000. & ſumpta dicta <seg type="var">.n.h.</seg> vt ſinus dictarum <lb/>partium, dabit nobis angulum <seg type="var">.n.o.h.</seg> quæſitum gra .44. min .4. qui per tabulas Alfon <lb/>ſi inuentus eſt gra .44. min .2. par huic, vt dici poteſt. </s> <s xml:space="preserve">Quiangulus gra .44. minu .4. <lb/>collectus cum angulo veri centri iam ſuppoſito graduum .151. minu .20. & cum an-<lb/>gulo augis eccentrici Martis, ſimiliter ſuppoſitæ grad .135. min .5. dabit nobis ſum-<lb/>mam veræ diſtantiæ Martis à principio Arietis grad .330. min .29. quod aliud non <lb/>ſignificat, niſi quod Mars inuenietur in minu .29. primi gradus Piſcium. </s> <s xml:space="preserve">Et Stofle-<lb/>rus in ſuis ephemeridibus ponit eum in .22. minuto dicti primi gradus, cuius diffe- <pb facs="0259" n="247"/><fw type="head">EPISTOL AE.</fw> rentia à tabulis <lb/>eſt minut .5. tan <lb/> <ptr xml:id="fig-0259-01a" corresp="fig-0259-01" type="figureAnchor"/> <choice><ex>tum</ex><am>tũ</am></choice>, & à me<unclear reason="illegible"/>o cal-<lb/>culo min .7. vide <lb/>licet minima.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0259-01" corresp="fig-0259-01a"> <graphic url="0259-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nunc autem <lb/>nolui ſumere ip-<lb/>ſum angulum æ-<lb/>quationis à tabu <lb/>lis propter duas <lb/>rationes, <choice><ex>primum</ex><am>primũ</am></choice> <lb/>quia ne hic qui-<lb/>dem repræhen-<lb/>for in hoc voluit <lb/>credere dictis ta <lb/>bulis. </s> <s xml:space="preserve">Sed id vo-<lb/>luit videre pro-<lb/>prijs oculis <choice><ex>in</ex><am>ĩ</am></choice> ſua <lb/>theorica Martis. <lb/></s> <s xml:space="preserve">Vbi <choice><ex>inuenit</ex><am>ĩuenit</am></choice> quod <lb/>linea <seg type="var">.o.n.</seg> tranſit <lb/>per gra .16. </s> <s xml:space="preserve">Arie <lb/>tis. </s> <s xml:space="preserve">Secunda ra-<lb/>tio eſt, vt videa-<lb/>tur quod dictæ tabulæ rectè ſupputatæ ſunt, ſuper dictis ſuppoſitis.</s> </p> <p> <s xml:space="preserve">Sed vt videat quantus ſit medius motus Martis die .29. </s> <s xml:space="preserve">Mai colligit fruſtatim, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>eleganter colligere poterat, vna opera in columellis ipſius medij motus eiuſmodi <lb/>ſtellæ per eram eiuſdem temporis, quæ erat .2. primarum ſexagenarum .33. ſecunda <lb/>rum .32. tertiarum, et .52. quartarum.</s> </p> <p> <s xml:space="preserve">Primum deinde ſuppoſitum quod ſcribit, ſcilicet, quod diameter epicycli ſum-<lb/>ptus in longitudine media ſit ſignorum .2. & grad .19. vti ſuperfluum eſt, ita etiam fal <lb/>ſum, nam eiuſmodi diameter in dicto loco, non occupat ad centrum mundi plus <choice><ex>quam</ex><am>quã</am></choice> <lb/>gra .66. min .28. ideſt ſigna .2. gra .6. min .28. quia proportio <seg type="var">.o.c.</seg> ad ſemidiametrum <lb/>epicycli in eiuſmodi loco eſt ut partium .60. minu .18. ad partes .39. min .30. quę duę <lb/>lineæ intellectę, vt latera vnius trianguli rectanguli, habebunt pro baſi aliam lineam <lb/>partium ſimilium .72. mi .5. </s> <s xml:space="preserve">Quæ intellecta vt ſinus totus dabit ſemidiametrum epi-<lb/>cycli partium .54798. tanquam ſinum ſubiectum angulo gra .33. min .14. pro medie <lb/>tate illius, quod quæritur.</s> </p> <p> <s xml:space="preserve">Nec prætermittenda mihi videtur ratio, qua credere poſſumus, hunc cogi-<lb/>taſſe, quod diameter epicycli compleat ſpatium duorum ſignorum cum gradibus. <lb/>19 quæ quidem ratio alia eſſe non poteſt, niſi quod cum iſte inuenerit, in <choice><ex>comentarijs</ex><am>comẽtarijs</am></choice> <lb/>Theoricarum, ſemidiametr um huiuſmodi epicycli eſſe partium .39 min .30. talium <lb/>qualium ſunt .60. illæ quæ ſunt ſemidiametri huius eccentrici, dictas igitur partes .39 <lb/>mi .30. hic putauit eſſe gradus Zodiaci, & propterea dixit <choice><ex>diametrum</ex><am>diametrũ</am></choice> huiuſmodi epicy <lb/>cli eſſe <choice><ex>ſignorum</ex><am>ſignorũ</am></choice> <choice><ex>duorum</ex><am>duorũ</am></choice>, & gra .19. qui numerus .79. duplus eſt numero .39. <choice><ex>cum</ex><am>cũ</am></choice> dimidio <lb/>hoc autem dixit accidere in longitudinibus medijs, quia ſi hic intellexiſſet de pro-<lb/>portione horum duorum diametrorum, quæ eſt ut .120. ad .79. non ſpecificaſſet lo- <pb facs="0260" n="248"/><fw type="head">IO. BAPT. BENED.</fw> cum epicycli, cum ipſa proportio nullo modo alteratur exiſtente epicyclo vbi volue <lb/>ris ipſius circunferentiæ eccentrici, ſed angulus in centro mundi, cui ſubiacet dictus <lb/>diameter epicycli, bene alteratur, propter inæqualem diſtantiam centri epicycli <lb/>ab ipſo centro mundi. </s> <s xml:space="preserve">At ſi de tali angulo inferre voluiſſet, iam probaui ipſum <choice><ex>con- tinere</ex><am>cõ-tinere</am></choice> ſolum gra .66. minu .28. exiſtente centro epicycli in longitudinibus medijs & <lb/>non gra .79. vt ipſe dicit.</s> </p> <p> <s xml:space="preserve">Omitto poſtea, quod vbi mentionem facit coniunctionum Solis cum Marte au-<lb/>gium & earum oppoſitorum, non explicat an intelligat de veris an de medijs. </s> <s xml:space="preserve">Nam <lb/>ſi ex eius modo loquendi accipiatur eum loqui de veris multum erraret.</s> </p> <p> <s xml:space="preserve">Sed quia iam tibi moleſtum eſſe inciperet ſi diutius te detinerem in his conten-<lb/>tionibus aſtronomicis, vlterius non diſputabo. </s> <s xml:space="preserve">Satis enim hactenus explicaui <choice><ex>ſenten- tiam</ex><am>ſentẽ-tiam</am></choice> meam, vt oſtendiſſe videor quam mihi iucundum ſit tibi <choice><ex>morem</ex><am>morẽ</am></choice> gerere. </s> <s xml:space="preserve">In quo <lb/>etiam hnmanitati tuæ gratiam habebo, <choice><ex>quum</ex><am>quũ</am></choice> petitione tua occaſionem mihi de deris <lb/>efficiendi, vt tum amici tui (amant enim te omnia ſublimia ingenia) tum alij, ſi <choice><ex>quam</ex><am>quã</am></choice> <lb/>falſam opinionem ex huius Benedicti Altæuillæ ſcriptis ſumpſiſſent, <choice><ex>eam</ex><am>eã</am></choice> relinquant, <lb/>& per te hoc beneficium à me conſequantur, & huiuſmodi occaſionem, & iuuandi <lb/>hominum ſtudia & tibi gratum faciendi, honorificum, & per gratum mihi fuiſſe in-<lb/>telligant. </s> <s xml:space="preserve">Vale & me vt ſoles ama.</s> </p> <p> <s xml:space="preserve">Taurini pridie Kal. </s> <s xml:space="preserve">Octobris .1581.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De probatione diuiſionis numerorum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">INter alia quæ à me ſcire cupis, vir doctiſſime, hoc vnum eſt, vt ex literis tuis ac-<lb/>cepi, vnde ſit vt priſci noſtri probatione numeri nouenarij potius quam ſepte-<lb/>narij vſi fuerint, & qua ratione non idem proueniat ex probatione numerorum <lb/>octonarij, ſenarij, vel quinarij, aut cuiuslibet alterius: </s> <s xml:space="preserve">Vnde pariter oriatur quod in <lb/>partitionis probatione neceſſum ſit probationum euentus multiplicare cum proba-<lb/>tione diuiſoris, ac eam quæ eſt producti poſtea cum probatione fractionis in ſum-<lb/>mam colligere, & c. </s> <s xml:space="preserve">Ad hæc in primis reſpondeo, cum aliquoties accidere poſſit ta <lb/>les probationes nos fallere poſſe, <choice><ex>idque</ex><am>idq́;</am></choice> fi in tali ſumma ſimilis numerus, ut puta ſe-<lb/>ptem, aut nouem, plus vel minus æquo iuſtouè poſitus fuerit, attamen per raro eueni <lb/>re poteſt, vt quis per nouenarium potius quam per ſeptenarium decipiatur. </s> <s xml:space="preserve">Exem-<lb/>pli gratia, ponamus ſummam eſſe .100. quam numerus nouenarius vndecies ſolum <lb/>ingreditur, at ſeptenarius quatuordecies, vnde quis <choice><ex>ſępius</ex><am>ſępiꝰ</am></choice> ex ſeptenario, hacratione, <lb/>quam ex nouenario numero ſe poſſe errare facile depræhendet, efſi ex probatione <lb/>nouenarij magis quam ſeptenarij, vt practici ſcribunt, duabus de cauſis errare poſſi-<lb/>mus. </s> <s xml:space="preserve">Alia tamen ratio mihi ſuppetit, ob quam credibile eſt ipſos potius nouena-<lb/>rio adiutos fuiſſe, quam ſeptenario, quæ eſt ob ſui cum velocitatem tum <choice><ex>facilitatem</ex><am>facilitatẽ</am></choice>, <lb/><choice><ex>neque</ex><am>neq;</am></choice> enim in ſeptenario eſt adeo facilis. </s> <s xml:space="preserve"><choice><ex>Nam</ex><am>Nã</am></choice> quamuis, tam vna quam altera aliud <lb/>non ſit, quam numerorum ordines diuidere (ſi de ſummis primo loquamur) aut è <lb/>ſumma ſuperſluum ordinum colligere, & videre an idemmet ſuperfluum ex eadem <lb/>ſumma emanet, attamen cum modus, qui in hoc adhiberi poteſt in nouenario <choice><ex>quam</ex><am>quã</am></choice> <lb/>in ſeptenario velocior ſit, & ob id probationem nouenarij ſeligunt potius quam <lb/>ſeptenarij.</s> </p> <pb facs="0261" n="249"/> <fw type="head">EPISTOLAE.</fw> <p> <s xml:space="preserve">Verum nolo te in ea, quæfalſa eſt, opinione conſiſtere, nonidem, & cum octona-<lb/>rio, ſenario, vel quinario, aut quouis alio numero poſſe efficere, cum eademmet ra <lb/>tio, quæ in ſeptenario, aut nouenario, <choice><ex>ent</ex><am>ẽt</am></choice> in cæteris perhibeatur. </s> <s xml:space="preserve">Ponamus <choice><ex>exemplum</ex><am>exemplũ</am></choice> <lb/>hos tres or dinum numeros velle ſupputare, quorum primus ſit .679. ſecundus .846. <lb/>& tertius .935. & illorum <choice><ex>ſummam</ex><am>ſummã</am></choice> .2460. nunc maiorem numerum primi ordinis ab <lb/>octonario menſi, proijciendo, remanebit .7. deinde maiorem numerum demendo à <lb/>ſecundo or dine, reſiduum erit .6. ac ſi idem in tertio ordine fecerimus, erit nobis re-<lb/>liquum .7. </s> <s xml:space="preserve">Demum tria hæc reſidua in vnum collecta .20. efficient, à quibus ſi nume <lb/>rum maiorem ab octonario menſum dempſeris, ſupererunt .4. & totidem à ſumma <num value="2460">.<lb/>2460.</num> remanebunt, reiecto maiori numero ab octonario menſo. </s> <s xml:space="preserve">Atque idem me-<lb/>dio quouis alio numero, euenire poteſt.</s> </p> <p> <s xml:space="preserve">Cuius ratio tam perſe clara atque euidens eſt, quod ſi ſummam trium <choice><ex>reliquorum</ex><am>reliquorũ</am></choice>, <lb/>quæ eſt .20. à ſumma .2460. ſubduxeris, remanebunt .2440. pro ſumma trium nume <lb/>rorum dictorum trium ordinum ab octonario menſorum, cui numero addito .16. pro <lb/>maiori numero ſummę <choice><ex>reliquorum</ex><am>reliquorũ</am></choice>, qui ab octonario menſus ſit, ſupererunt .4. </s> <s xml:space="preserve">At ſi per <lb/><choice><ex>ſenarium</ex><am>ſenariũ</am></choice> <choice><ex>experimentum</ex><am>experimẽtũ</am></choice> feceris, remanebit <seg type="var">.o.</seg> & ſic de reliquis per ordinem <choice><ex>procedendo</ex><am>procedẽdo</am></choice>.</s> </p> <p> <s xml:space="preserve">Verum poſſes ſciſcitari, quare velocius, exceſſus ordinum, potius per <choice><ex>noue- narium</ex><am>noue-nariũ</am></choice>, quam per cæteros numeros, prout <choice><ex>docent</ex><am>docẽt</am></choice> practici, inueniri queat, videlicet ag <lb/>gregando prius duas figuras numerorum primæ ſummæ, deinde alias duas. </s> <s xml:space="preserve">Exem-<lb/>plum ſit primus ordo .679. colligendo .6. et .7. faciunt 13. & cum hæc ſumma ſit dua <lb/>rum figurarum, ſupputantur & ipſæ, è quibus prodeunt .4. & conſimilis erit proba-<lb/>tio numeri .67. facta per .9. quod idem eſt, ac ſi quis diuidat .67. per .9. ex quo reli-<lb/>qui erunt ſemper .4.</s> </p> <p> <s xml:space="preserve">At quo ratio huiuſce perſpicuè dignoſci poſſit, in primis ſciendum eſt, cuique <lb/>ex ſe cognitum, atque exploratum eſſe, denarium numerum vnitate nouenarium ſu <lb/>perare, & ex hoc ſequitur, ſex denarios continere in ſe ſex nouenarios, & ſex vni-<lb/>tates.</s> </p> <p> <s xml:space="preserve">At ſex vnitates, vna cum .7. faciunt .13. & quia in .13. eſt denarius, igitur in illo erit <lb/>vnitas ſupra .9. </s> <s xml:space="preserve">Quæ vnitas addita ternario, præbet nobis ſuperfluum, per quod .67. <lb/>ſuperat .54. iunctum cum .9. ſcilicet ſummam .63.</s> </p> <p> <s xml:space="preserve">Idem dicinon poteſt de octonario, ſeptenario, vel ſenario, & de reliquis, quo-<lb/>niam numerus denariorum, in cæteris minoribus nouenario non præbet illico nu-<lb/>merum exceſſus maioris numeri, qui à numero probationis menſus eſt. </s> <s xml:space="preserve">Et quod di <lb/>co de probatione aggregationis, idem intelligo de alijs operationibus, vt puta ſub-<lb/>tractionis, multiplicationis, & partitionis ſeu diuiſionis.</s> </p> <p> <s xml:space="preserve">Vnde autem oriatur, vt in partitionis probatione opus ſit probationem euentus <lb/>cum diuiſionis probatione multiplicare, & productum cum fractionis probatione <lb/>ſupputare, ſeu aggregare, tibi non erit ignotum, quoties animaduerteris, quod <lb/>productum ipſius euentus cum diuiſore, adiunctum fractioni, perpetuo ſe æquat nu <lb/>mero diuiſibili. </s> <s xml:space="preserve">Et quoniam numeri probationum ſunt partes, quæ remanent ex <lb/>ipſis totis, detractis maioribus numeris ab eo dimenſis, quo pro communi men-<lb/>ſura vtimur (prout .7. vel .9. aut alium numerum, quem voluerimus) par eſt vt ex ip-<lb/>ſarum remanentibus partibus, velut ex ipſis totis idem fiat.</s> </p> <pb facs="0262" n="250"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De falacia operationis triangulorum ſphericorum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod diebus præteris tibi ſignificaui, idem nunc confirmo, ſcilicet ſphærico-<lb/>rum triangulorum operationem ſæpe nos fallere, vt exempli gratia, ſi pro <lb/>poſitus nobis fuiſſet triangulus <seg type="var">.A.B.C.</seg> cuius angulus <seg type="var">.A.</seg> nobis datus eſſet graduum <num value="114">.<lb/>114.</num> mi <seg type="var">.o.</seg> & eius latus <seg type="var">.A.</seg> B, graduum .67. min .5. & latus <seg type="var">.A.C.</seg> graduum .45. mi .10. <lb/>ſi reliquos angulos cum tertio latere etiam cognoſcere voluerimus, ex methodo .11 <lb/>primi Copernici propoſitum obtinebimus. </s> <s xml:space="preserve">vnde latus <seg type="var">.B.C.</seg> eſſet graduum .89. min <num value="30">.<lb/>30.</num> angulus vero <seg type="var">.C.</seg> graduum .57. min .14. angulus autem <seg type="var">.B.</seg> grad .48. min .38. </s> <s xml:space="preserve">Qua-<lb/>re vltimus hic angulus <seg type="var">.B.</seg> falſus eſſet, eo quod operatio paruorum triangulorum in <lb/>cauſa eſt, quotieſcunque eorum latera tam breuia ſint, ut non eccedant vnum gra-<lb/>dum, quare ipſorum angulorum veram quantitatem non tribuunt. </s> <s xml:space="preserve">propterea igitur <lb/>cum voluerimus veram <choice><ex>quantitatem</ex><am>quãtitatem</am></choice> ipſius anguli <seg type="var">.B.</seg> oportet poſt quam inuenerimus <lb/>angulum <seg type="var">.C.</seg> mediante arcu <seg type="var">.D.E.</seg> ſupponere alium polum in <seg type="var">.B.</seg> deinde producere. </s> <s xml:space="preserve">B <lb/>A. vſque ad <seg type="var">.d.</seg> et <seg type="var">.B.C.</seg> vſque ad <seg type="var">.e.</seg> imaginando <seg type="var">.B.d.</seg> et <seg type="var">.B.e.</seg> duas quartas eſſe magno-<lb/>rum circulorum, extendendo poſtea <seg type="var">.d.e.</seg> vſque ad interſectionem cum <seg type="var">.A.C.</seg> & <choice><ex>eum</ex><am>eũ</am></choice> <lb/>dem ordinem proſequendo, </s> <s xml:space="preserve">tunc <seg type="var">.e.d.</seg> nobis oſtendet angulum <seg type="var">.B.</seg> eſſe gra .40. mi .22 <lb/>quæ erit eius vera quantitas. </s> <s xml:space="preserve">Cuius quidem rei experientiam poſſumus etiam fa-<lb/>cere, hoc modo, eſto, exempli gratia, quod nobis datus ſit angulus <seg type="var">.C.</seg> graduum .57. <lb/>min .14. cum latere <seg type="var">.A.C.</seg> gra .45. min .10. & latus <seg type="var">.B.C.</seg> gra .89. min .30. </s> <s xml:space="preserve">Tunc ſi ordi-<lb/>nem .11. dicti lib. ſe quemur, obtinebimus intentum, hoc modo ſcilicet ſupponendo <lb/>in <seg type="var">.A.</seg> polum, & non in <seg type="var">.B.</seg> ducendo etiam <seg type="var">.A.B.</seg> et <seg type="var">.A.C.</seg> ſed <seg type="var">.A.B.</seg> <choice><ex>vſque</ex><am>vſq;</am></choice> ad gra .90. du-<lb/>cendo poſtea <seg type="var">.D.E.</seg> ita quod ab omni parte concurrat cum latere <seg type="var">.B.C.</seg> producto, vn <lb/>de tam <seg type="var">.f.C.B.F.</seg> quam <seg type="var">.f.D.E.F.</seg> erunt ſemicirculi magnorum circulorum. </s> <s xml:space="preserve">quare <seg type="var">.C.<lb/>D.</seg> nobis cognitus erit gra .44. min .50. & ſic etiam angulus <seg type="var">.D.C.f.</seg> gra .57. min .14. ex <lb/>4. dicti lib. poſtea habebimus <seg type="var">.F.l.</seg> gra .60. min .54. & angulum <seg type="var">.f.</seg> gra .53. mi .24. aggre <lb/>gatum poſtea <seg type="var">.f.C.</seg> cum <seg type="var">.C.B.</seg> habebimus <seg type="var">.f.B.</seg> gra .150. min .24. qui ſi a ſemicirculo <choice><ex>dem</ex><am>dẽ</am></choice> <lb/>ptus fuerit, nobis remanebit <seg type="var">.B.F.</seg> gra .29. mi .36. cum angulo <seg type="var">.F.</seg> cognito <choice><ex>cum</ex><am>cũ</am></choice> ſit æqua-<lb/> <ptr xml:id="fig-0262-01a" corresp="fig-0262-01" type="figureAnchor"/> <ptr xml:id="fig-0262-02a" corresp="fig-0262-02" type="figureAnchor"/> lis <seg type="var">.f.</seg> eius oppoſito. </s> <s xml:space="preserve">Vnde ex dicta .4. co-<lb/>gnoſcemus angulum <seg type="var">.B.</seg> gra .40. min .31. <lb/>qui ferè æ qualis eſt ſuperiori iam inuen-<lb/>to, nec ab ipſo differt niſi per min .9. quæ <lb/>quidem differentia parua eſt reſpectu al <pb facs="0263" n="251"/><fw type="head">EPISTOLAE.</fw> terius differentiæ quam ſupra inuenerimus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0262-01" corresp="fig-0262-01a"> <graphic url="0262-01"/> </figure> <figure xml:id="fig-0262-02" corresp="fig-0262-02a"> <graphic url="0262-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Superius enim dixinon eſſe ponendum polum in <seg type="var">.B.</seg> eo quod <seg type="var">.B.C.</seg> ſit gra .89. mi <num value="30">.<lb/>30.</num> vnde nobis prodijſſet triangulus <seg type="var">.f.C.D.</seg> trium valde paruorum laterum, quorum <lb/>latus <seg type="var">.C.D.</seg> eſſet gra <seg type="var">.o.</seg> mi .30. & latus <seg type="var">.f.l.</seg> gra <seg type="var">.o.</seg> mi .55. & latus <seg type="var">.F.D.</seg> gra <seg type="var">.o.</seg> mi .47. vn-<lb/>de angulus <seg type="var">.f.</seg> gra .32. min .40. falſus eſſet, qui <choice><ex>quidem</ex><am>quidẽ</am></choice> poſtea nobis daret <seg type="var">.D.E.</seg> gra .45 <lb/>minu .16. falſum ſimiliter.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De paßione circuli bactenus incognita.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">DVbitandum quidem <choice><ex>non</ex><am>nõ</am></choice> eſt quin paſſiones circuli innumerabiles penè ſint, quę <lb/>quidem omnes ferè caſu inueniuntur, vt mihi nunc accidit, quam tibi mitto, <lb/>hæc autem eſt, quòd quadratum lineæ <seg type="var">.a.g.</seg> in figura hic ſubſcripta ſemper æquale <lb/>eſt ei producto, quod fit ex <seg type="var">.a.e.</seg> in diametro circuli <seg type="var">.g.c.b.</seg> ſimul ſumpto cum quadra <lb/>to inſcriptibili in dicto circulo, & ſimul cum quadrato lineæ <seg type="var">.a.b.</seg> <choice><ex>contingentis</ex><am>contingẽtis</am></choice> ipſum <lb/>circulum, ſupponendo <seg type="var">.a.g.</seg> per centrum ipſius circuli tranſire.</s> </p> <p> <s xml:space="preserve">Pro cuius demonſtratione à centro <seg type="var">.e.</seg> duco ſemidiametrum <seg type="var">.e.c.</seg> <choice><ex>perpendicularem</ex><am>perpendicularẽ</am></choice> <lb/>ipſi <seg type="var">.g.a.</seg> & à puncto <seg type="var">.c.</seg> ad <seg type="var">.a.</seg> duco <seg type="var">.c.a.</seg> quæ ſecabit circunferentiam ipſius circuli in <choice><ex>pum</ex><am>pũ</am></choice> <lb/>cto <seg type="var">.d.</seg> eo, quod angulus <seg type="var">.c.</seg> acutus eſt. </s> <s xml:space="preserve">Nunc ex .35. tertij, productum <seg type="var">.c.a.</seg> in <seg type="var">.a.d.</seg> æqua <lb/>le eſt quadrato <seg type="var">.a.b.</seg> productum autem <seg type="var">.a.c.</seg> in <seg type="var">.d.c.</seg> æquale eſt quadrato inſcriptibili in <lb/>circulo <seg type="var">.g.c.b.</seg> ex .130. primi Vitellionis, <choice><ex>in</ex><am>ĩ</am></choice> qua propoſitione ipſe Vitellio ſupplet pro <lb/>eo, quod in quinta propoſitione libri de lineis ſpirabilibus Archimedis deſideratur, <lb/>ſed quadratum <seg type="var">.a.c.</seg> æquale eſt ijs duobus productis. per .2. ſecundi Eucli. ergo qua-<lb/>dratum <seg type="var">.a.c.</seg> æquale erit quadrato inſcriptibili in circulo <seg type="var">.d.c.g.</seg> & quadrato <seg type="var">.a.b.</seg> ſed <lb/>quadratum lineæ <seg type="var">.a.c.</seg> æquale eſt duobus quadratis, hoc eſt lineæ <seg type="var">.a.e.</seg> & lineæ <seg type="var">.e.c.</seg> ex <lb/>pitagorica, </s> <s xml:space="preserve">quare ex communi conceptu duo quadrata lineæ <seg type="var">.a.e.</seg> & lineę <seg type="var">.e.c.</seg> hoc eſt <lb/>lineæ <seg type="var">.e.g.</seg> quod idem eſt, æqualia erunt duobus iam dictis, hoc eſt inſcriptibili, <lb/>& ei, quod fit ex <seg type="var">.a.b.</seg> ſed quadratum lineæ <seg type="var">.a.g.</seg> æquale eſt quadrato lineæ <seg type="var">.a.e.</seg> & qua <lb/>drato quod fit ex <seg type="var">.e.g.</seg> & duplo illius quod fit ex <seg type="var">.a.e.</seg> in <seg type="var">.e.g.</seg> hoc eſt producto <seg type="var">.a.e.</seg> in <lb/>diametrum. </s> <s xml:space="preserve">Quare quadratum lineæ <seg type="var">.a.g.</seg> æquale eſt quadrato circunſcriptibili, & <lb/>quadrato lineæ <seg type="var">.a.b.</seg> & producto lineæ <seg type="var">.a.e.</seg> in diametrum circuli <seg type="var">.d.c.g</seg>.</s> </p> <p> <s xml:space="preserve">Breuiori etiam methodo demonſtrare poſſu <lb/> <ptr xml:id="fig-0263-01a" corresp="fig-0263-01" type="figureAnchor"/> mus quadrata lineæ <seg type="var">.a.e.</seg> et <seg type="var">.e.g.</seg> æqualia eſ-<lb/>ſe quadrato circunſcriptibili, & quadrato lineæ <seg type="var">.<lb/>a.b.</seg> ducendo lineam <seg type="var">.e.b.</seg> quæ æqualis eſt lineæ <seg type="var">.<lb/>e.g.</seg> tali methodo, hoc eſt, conſiderando, quod <lb/>quadratum inſcriptibile ſemper duplum eſt qua <lb/>drato ſemidiametri, vel medietati circumſcri-<lb/>ptibili, quod quidem nihil aliud eſt, niſi æquale <lb/>eſſe ijs duobus quadratis, hoc eſt lineæ <seg type="var">.e.b.</seg> & li-<lb/>neæ <seg type="var">.e.g.</seg> ſed quadratum lineæ <seg type="var">.a.e.</seg> æquale eſt iis <lb/>duobus quadratis, hoc eſt lineæ <seg type="var">.a.b.</seg> & lineæ <seg type="var">.b.e.</seg> vnde quadrat um lineæ <seg type="var">.a.e.</seg> cum <lb/>quadrato lineæ <seg type="var">.e.g.</seg> æquale eſt quadrato circunſcriptibili, ſimul collecto cum qua-<lb/>drato lineæ <seg type="var">.a.b</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0263-01" corresp="fig-0263-01a"> <graphic url="0263-01"/> </figure> </div> </body> </floatingText> <pb facs="0264" n="252"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Demonstrationes quarundam propoſitionum de quibus agit <lb/>Cardanus capite primo libro .16. de <lb/>ſubtilitate.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">EA quæ Cardanus in primo cap. lib. 16. de ſubtilitate ita ſcribit, quod ſi diame-<lb/>tros producatur extra quantumlibet, alia verò diametro in centro ſecetur ad <lb/>rectos, ex huius fine <choice><ex>&c.</ex><am>&c.</am></choice> quæ quidem ſecundum illum eſt vndecima proprietas cir <lb/>culi, quoniam te id non intelligere ſcribis, <choice><ex>idemque</ex><am>idemq́;</am></choice> dicis etiam de duodecima, & ſi-<lb/>militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, ſeu <lb/>ellipſi, & hyperboli, tibi breuiter reſpondebo.</s> </p> <p> <s xml:space="preserve">Circa vndecimam proprietatem circuli verum dicit. </s> <s xml:space="preserve">Imaginemur circulum <seg type="var">.p.<lb/>d.q.</seg> à duabus diametris, inuicem ad angulos rectos coniunctis, diuiſum <seg type="var">.p.d.</seg> et <seg type="var">.d.g.</seg> di <lb/>uidatur enim quarta <seg type="var">.q.d.</seg> per quot partes æquales volueris, mediantibus punctis <seg type="var">.b.a.<lb/>o.</seg> <choice><ex>ducanturque</ex><am>ducanturq́;</am></choice> ab ijſdem punctis tot perpendiculares diametro <seg type="var">.d.g.</seg> quæ ſint <seg type="var">.b.m.a.n.</seg> <lb/>et <seg type="var">.o.s.</seg> quæ quidem erunt parallelæ diametro <seg type="var">.q.p.</seg> coniungatur deinde extremitas <seg type="var">.d.</seg> <lb/>diametri <seg type="var">.d.g.</seg> cum primo puncto <seg type="var">.b.</seg> & protrahatur <seg type="var">.d.b.</seg> vſque ad concurſum cum diz <lb/>metro <seg type="var">.p.q.</seg> protracto in puncto, h. </s> <s xml:space="preserve">Nunc dico <seg type="var">.q.h.</seg> quæ adiacet diametro <seg type="var">.q.p.</seg> æqua-<lb/>lem eſſe omnibus dictis perpendicularibus, quapropter coniungantur puncta <seg type="var">.m.a</seg>: <lb/><seg type="var">n.o.</seg> et <seg type="var">.s.q.</seg> & producantur vſque ad adiacentem diametro <seg type="var">.q.p.</seg> in punctis <seg type="var">.c.</seg> et <seg type="var">.e.</seg> vn <lb/>de habebimus angulos <seg type="var">.b.a.o.q.</seg> inuicem æquales ex .26. tertij, cum verò <seg type="var">.o.s.a.n.</seg> et <lb/><seg type="var">b.m.</seg> parallelæ ſint ipſi <seg type="var">.p.h</seg>. </s> <s xml:space="preserve">tunc anguli <seg type="var">.b.h.c</seg>: <seg type="var">a.c.e</seg>: et <seg type="var">.o.e.q.</seg> æquales erunt angulis <seg type="var">.d.<lb/>b.m</seg>: <seg type="var">m.a.n.</seg> et <seg type="var">.n.o.s.</seg> ex .29. primi: </s> <s xml:space="preserve">quare anguli <seg type="var">.h.c.e.q.</seg> erunt inuicem æquales, vnde <lb/>ex .28. eiuſdem <seg type="var">.b.h</seg>: <seg type="var">m.c</seg>: <seg type="var">n.e.</seg> et <seg type="var">.s.q.</seg> erunt <choice><ex>inuicem</ex><am>inuicẽ</am></choice> parallelę, & ex .34. <seg type="var">e.q.</seg> æqualis erit <seg type="var">.<lb/>o.s.</seg> et <seg type="var">.e.c.</seg> æqualis <seg type="var">.n.a.</seg> et <seg type="var">.m.b.</seg> æqualis <seg type="var">.c.h.</seg> verum eſt igitur propoſitum.</s> </p> <p> <s xml:space="preserve">Duodecima vero <choice><ex>proprietas</ex><am>ꝓprietas</am></choice> eſt, ut ſi fuerit circulus <seg type="var">.a.b.e.q.</seg> cuius duo diametriad <lb/>rectos coniuncti ſint <seg type="var">.a.e.</seg> et <seg type="var">.q.b.</seg> & diameter <seg type="var">.a.e.</seg> protractus indeterminatè ad partem <lb/>e. </s> <s xml:space="preserve">tunc ſi ab extremo <seg type="var">.b.</seg> diametri <seg type="var">.q.b.</seg> ducta fuerit <seg type="var">.b.n.u.</seg> extra circulum, ſeu <seg type="var">.b.u.n.</seg> in <lb/>tra circulum, vt in ſubiecta figura patet, ita vt ſecta ſit à circunferentia circuli in <choice><ex>pum</ex><am>pũ</am></choice> <lb/>cto <seg type="var">.n.</seg> vel à diametro in puncto <seg type="var">.u.</seg> ſemper id quod fit ex <seg type="var">.u.b.</seg> in <seg type="var">.b.n.</seg> æquale erit qua-<lb/>drato inſcriptibili in dicto circulo, hoc autem diuerſimodè cognoſci poteſt, tribus <lb/>enim modis ego inueni, quorum primus ita ſe habet. </s> <s xml:space="preserve">Nam ſi punctus <seg type="var">.u.</seg> fuerit ex-<lb/>tra circulum, ducantur <seg type="var">.b.e.</seg> et <seg type="var">.e.n.</seg> & habebimus duos triangulos <seg type="var">.b.n.e.</seg> et <seg type="var">.b.e.u.</seg> ſimi <lb/>les inuicem, eo, quod angulus <seg type="var">.b.</seg> communis ambobus exiſtit, & angulus <seg type="var">.b.n.e.</seg> æqua <lb/>lis eſt angulo <seg type="var">.b.e.u.</seg> quod ita probatur, nam angulus <seg type="var">.b.n.e.</seg> cum angulo <seg type="var">.b.a.e.</seg> (ducta <lb/>cum fuerit <seg type="var">.b.a.</seg>) æquatur duobus rectis ex .21. tertij, ſed ex quinta primi angulus <seg type="var">.b.<lb/>e.a.</seg> ęqualis eſt angulo <seg type="var">.b.a.e</seg>: </s> <s xml:space="preserve">quare angulus <seg type="var">.b.n.e.</seg> cum angulo <seg type="var">.b.e.a.</seg> ęquatur duobus <lb/>rectis, ſed ex .13. eiuſdem angulus <seg type="var">.b.n.e.</seg> cum angulo etiam <seg type="var">.e.n.u.</seg> æquatur duobus re <lb/>ctis, ergo angulus <seg type="var">.e.n.u.</seg> æquatur angulo <seg type="var">.b.e.a</seg>. </s> <s xml:space="preserve">quare angulus <seg type="var">.b.n.e.</seg> æquatur <choice><ex>etiam</ex><am>etiã</am></choice> an-<lb/>gulo <seg type="var">.b.e.u.</seg> vnde ex .32. eiuſdem reliquus angulus <seg type="var">.b.u.e.</seg> æqualis erit reliquo angulo <lb/><seg type="var">b.e.n.</seg> latera igitur erunt proportionalia ex .4. ſexti, vnde ita ſe habebit <seg type="var">.u.b.</seg> ad <seg type="var">.b.<lb/>e.</seg> vt <seg type="var">.b.e.</seg> ad <seg type="var">.b.n.</seg> ex .16. ſexti igitur <choice><ex>verum</ex><am>verũ</am></choice> erit propoſitum.</s> </p> <p> <s xml:space="preserve">Sed ſi punctus <seg type="var">.u.</seg> intra circulum fuerit, triangulus <seg type="var">.b.e.n.</seg> ſimilis erit triangulo <seg type="var">.b.u.<lb/>e.</seg> nam angulus <seg type="var">.b.</seg> ambobus communis erit. </s> <s xml:space="preserve">Angulus vero <seg type="var">.b.n.e.</seg> ęqualis eſt angulo <seg type="var">.<lb/>b.e.u.</seg> ex .26. tertij, </s> <s xml:space="preserve">quare ex .32. primi reliquus angulus <seg type="var">.b.e.n.</seg> æqualis erit reliquo <pb facs="0265" n="253"/><fw type="head">EPISTOLAE.</fw> angulo <seg type="var">.b.u.e.</seg> vnde ex .4. ſexti eadem proportio erit ipſius <seg type="var">.b.n.</seg> ad <seg type="var">.b.e.</seg> quæ <seg type="var">.b.e.</seg> ad <lb/><seg type="var">b.u</seg>. </s> <s xml:space="preserve">quare ex .16. eiuſdem patebit propoſitum.</s> </p> <p> <s xml:space="preserve">Secundus autem modus ita ſe habet, ducta <seg type="var">.q.n.</seg> habebimus duo triangula ortho-<lb/>gonia ſimilia inuicem <seg type="var">.b.q.n.</seg> et <seg type="var">.b.u.o.</seg> eo quod angulus <seg type="var">.b.</seg> communis ambobus exi-<lb/>ſtit, </s> <s xml:space="preserve">quare ex .4. ſexti ita ſe habebit <seg type="var">.u.b.</seg> ad <seg type="var">.b.o.</seg> vt <seg type="var">.q.b.</seg> ad <seg type="var">.b.n.</seg> vnde ex .15. eiuſdem <lb/>quod fit ex <seg type="var">.u.b.</seg> in <seg type="var">.b.n.</seg> æquale erit ei, quod fit ex <seg type="var">.q.b.</seg> in <seg type="var">.b.o</seg>. </s> <s xml:space="preserve">Sed ex .16. eiuſdem, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>fit ex <seg type="var">.q.b.</seg> in <seg type="var">.b.o.</seg> ęquatur quadrato <seg type="var">.b.e.</seg> quia <seg type="var">.b.e.</seg> media proportionalis eſt inter dia <lb/>metrum & ſemidiametrum eiuſdem circuli. ex .4. eiuſdem, </s> <s xml:space="preserve">quare quod fit ex <seg type="var">.u.b.</seg> in <lb/><seg type="var">b.n.</seg> æquale erit quadrato ipſius <seg type="var">.b.e</seg>.</s> </p> <p> <s xml:space="preserve">Tertius modus adiungitur, & eſt quod cum quadratum <seg type="var">.u.b.</seg> exiſtente <seg type="var">.u.</seg> extra cir-<lb/>culum æquale ſit ei, quod ſit ex <seg type="var">.u.b.</seg> in <seg type="var">.b.n.</seg> ſimul ſumpto cum eo, <choice><ex>quod</ex><am>ꝙ</am></choice> fit ex <seg type="var">.u.b.</seg> in <seg type="var">.u.n.</seg> <lb/>ex ſecunda ſecundi, & idem quadratum <seg type="var">.u.b.</seg> æquale duobus quadratis <seg type="var">.u.o.</seg> et <seg type="var">.o.b.</seg> ex <lb/>penultima primi, ideo duo dicta producta æqualia erunt dictis duobus quadratis <seg type="var">.o.</seg> <lb/> <ptr xml:id="fig-0265-01a" corresp="fig-0265-01" type="figureAnchor"/> u. ſcilicet et <seg type="var">.o.b.</seg> ſed quadratum <lb/>o u. æquatur ei, quod fit ex <seg type="var">.a.u.</seg> <lb/>in <seg type="var">.e.u.</seg> & ei quod fit. ex <seg type="var">.o.e.</seg> in ſe <lb/>ipſam ex .6. ſecundi, </s> <s xml:space="preserve">quare duo <lb/><choice><ex>iam</ex><am>iã</am></choice> dicta producta æqualia erunt <lb/>duobus dictis quadratis, <seg type="var">o.b.</seg> ſci <lb/>licet. et <seg type="var">.o.e.</seg> & ei quod fit ex <seg type="var">.a.<lb/>u.</seg> in <seg type="var">.u.e.</seg> ſed quod fit ex <seg type="var">b.u.</seg> in <seg type="var">.u <lb/>n.</seg> æquale eſt ei quod fit ex <seg type="var">.a.u.</seg> <lb/>in <seg type="var">.u.e.</seg> ex .35. 3. <choice><ex>relinquitur</ex><am>relinquit̃</am></choice> ergo <lb/>vt id <choice><ex>quod</ex><am>qđ</am></choice> fit ex <seg type="var">.u.b.</seg> in <seg type="var">.b.n.</seg> æqua-<lb/>le ſit <choice><ex>duobus</ex><am>duobꝰ</am></choice> quadratis <seg type="var">.o.b.</seg> et <seg type="var">.o.<lb/>e</seg>. </s> <s xml:space="preserve">quare & quadrato ipſius <seg type="var">.b.e.</seg> <lb/>ex Pitagorica.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0265-01" corresp="fig-0265-01a"> <graphic url="0265-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Siautem <choice><ex>punctum</ex><am>pũctũ</am></choice> <seg type="var">.u.</seg> fuiſſet intra <lb/>circulum idem eueniret. </s> <s xml:space="preserve">Nam <lb/>quadrato <seg type="var">.b.e.</seg> <choice><ex>æquantur</ex><am>æquãtur</am></choice> duo qua <lb/>drata <seg type="var">.o.b.</seg> et <seg type="var">.o.e.</seg> ſed vice qua-<lb/>drati <seg type="var">.o.e.</seg> dicemus <choice><ex>quadratum</ex><am>quadratũ</am></choice> <seg type="var">.o.<lb/>u.</seg> cum eo quod fit ex <seg type="var">.a.u.</seg> in <seg type="var">.u.e.</seg> <lb/>ex .5. ſecundi, id eſt quadratum <seg type="var">.<lb/>o.u.</seg> <choice><ex>cum</ex><am>cũ</am></choice> eo quod fit ex <seg type="var">.b.u.</seg> in <seg type="var">.u.<lb/>n.</seg> ex .34. tertij, vnde quadratum <lb/><seg type="var">b.e.</seg> æquale erit quadrato <seg type="var">.o.b.</seg> <lb/>& quadrato <seg type="var">.o.u.</seg> ideſt quadrato <lb/><seg type="var">b.u.</seg> ex Pitagorica ſimul <choice><ex>cum</ex><am>cũ</am></choice> pro-<lb/>ducto <seg type="var">.b.u.</seg> in <seg type="var">.u.n.</seg> ideſt producto <lb/><seg type="var">n.b.</seg> in <seg type="var">.b.u.</seg> quod æquale eſt qua <lb/>drat <seg type="var">o.b.u.</seg> cum producto <seg type="var">.b.u.</seg> in <lb/><seg type="var">u.n.</seg> ex .3. ſecundi.</s> </p> <p> <s xml:space="preserve">Circa tres paſſiones commu-<lb/>nes poſtea circulo hyperboli, & <lb/>defectioni notandum eſt <choice><ex>primam</ex><am>primã</am></choice> <lb/>patere ex .36: primi Pergei, ſe- <pb facs="0266" n="254"/><fw type="head">IO. BABPT. BENED.</fw> cundam verò ex .37. et .38. eiuſdem, </s> <s xml:space="preserve">propterea quod in .37. probat mediante maiori <lb/>diametro ipſius hyperbolis & defectionis, In .38. autem mediante minori diametro <lb/>ordinatè ad maiorem.</s> </p> <p> <s xml:space="preserve">Tertia autem paſſio, non niſi circulo conuenit; </s> <s xml:space="preserve">pace ipſius Cardani dictum ſit.</s> </p> <p> <s xml:space="preserve">Quapropter ſit circulus <seg type="var">.q.o.b.</seg> cuius diameter ſit <seg type="var">.q.b.</seg> contingentes vero ab extre <lb/>mitate diametri ſint <seg type="var">.d.b.</seg> et <seg type="var">.q.g.</seg> per punctum autem <seg type="var">.o.</seg> quoduis, ipſius <choice><ex>circunferentiæ</ex><am>circũferentiæ</am></choice>, <lb/>tranſeant <seg type="var">.b.o.g.</seg> et <seg type="var">.q.o.d</seg>. </s> <s xml:space="preserve">tunc dico productum <seg type="var">.q.o.</seg> in <seg type="var">.q.d.</seg> vel <seg type="var">.b.o.</seg> in <seg type="var">.b.g.</seg> ęquale eſ-<lb/>ſe quadrato <seg type="var">.q.b.</seg> quod ita probo.</s> </p> <p> <s xml:space="preserve">Nam angulus <seg type="var">.q.b.d.</seg> ſeu <seg type="var">.b.q.g.</seg> rectus eſt ex .17. tertij Eucli. et <seg type="var">.b.o.q.</seg> ſimiliter re-<lb/>ctus ex .30. ipſius lib. angulus verò <seg type="var">.b.q.d.</seg> ſeu <seg type="var">.q.b.g.</seg> communis eſt. </s> <s xml:space="preserve">quare <seg type="var">.b.q.</seg> media <lb/>proportionalis erit inter dictas lineas <seg type="var">.q.d.</seg> et <seg type="var">.q.o.</seg> & inter <seg type="var">.b.g.</seg> et <seg type="var">.b.o</seg>. </s> <s xml:space="preserve">Vnde ſequetur <lb/>propoſitum ex .16.6. Eucli.</s> </p> <p> <s xml:space="preserve">Sed ſi circa diametrum <seg type="var">.q.b.</seg> mente fingamus aliquam elipſim, quætangat ipſum <lb/> <ptr xml:id="fig-0266-01a" corresp="fig-0266-01" type="figureAnchor"/> circulum duobus punctis me-<lb/>diantibus <seg type="var">.q.</seg> et <seg type="var">.b.</seg> (nam pluribus <lb/>eſſet impoſſibile, ex .27. quarti <lb/>Pergei) clarè patebit, quod <choice><ex>pum</ex><am>pũ</am></choice> <lb/>ctus <seg type="var">.o.</seg> erit extra <choice><ex>circunferentiam</ex><am>circunferentiã</am></choice> <lb/>ipſius defectionis, </s> <s xml:space="preserve">quare ipſa cir <lb/>cunferentia ſecabit <seg type="var">.b.g.</seg> vel <seg type="var">.q.<lb/>d.</seg> in alio puncto, vnde ipſi non <lb/>occurret id quod probauimus <lb/>de circulo.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0266-01" corresp="fig-0266-01a"> <graphic url="0266-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Admiratus etiam ſum, ipſum <lb/>Cardanum dicere hyperbolem <lb/>ita vocari, eo quod angulus con <lb/>tentus ab axe ipſius figuræ, & à <lb/>latere trigoni in hyperbole ma-<lb/>ior ſit quam in parabole, quod <lb/>eriam confirmat paulo inferius, <lb/>nam hoc verum non eſt, imo fal <lb/>ſiſſimum. </s> <s xml:space="preserve">Talis enim ſectio ita <lb/>nominata fuit, hoc eſt hyperbo <lb/>les, ſimili ratione, qua elipſis ſeu <lb/>defectio etiam vocata fuit, nam <lb/>ſicut in ipſa defectione quadra-<lb/>tum ordinatę <seg type="var">.l.m.</seg> minor eſt pro <lb/>ducto lineæ <seg type="var">.e.m.</seg> in <seg type="var">.e.t.</seg> per figu <lb/>ram ſimilcm producto <seg type="var">.d.e.</seg> in <seg type="var">.e.<lb/>t.</seg> quæ eandem obtineat <choice><ex>altitu- dinem</ex><am>altitu-dinẽ</am></choice> ipſius <seg type="var">.e.m.</seg> vt ipſe Pergeus <lb/>monſtrat in .13. primi lib. ita in <lb/>hyperbole <choice><ex>dictum</ex><am>dictũ</am></choice> quadratum ex <lb/>cedit quantitatem illius figuræ, <lb/>per ſimilem dictæ vt in .12. <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <lb/>Pergei facilè videre eſt. </s> <s xml:space="preserve">ſed <choice><ex>prae ter</ex><am>pręter</am></choice> illas paſſiones, quas notat <pb facs="0267" n="255"/><fw type="head">EPISTOL AE.</fw> Cardanus in ſupradicto capite, multæ aliæ ſunt, cum corollarium primæ tertij Eu-<lb/>cli. ſit paſſio propria ipſius circuli, & idem dico de propoſitione .3. 4. 7. 8. 9. 11. 12. <lb/>13. 14. 15. 17. 18. 19. 20. 30. 31. ipſius tertij lib. nec non de .8. 9. ct .10. tertijdecimi, & <lb/>de prima .3. 4. 5. 6. et .7. quartidecimi eiuſdem. </s> <s xml:space="preserve">Idem infero de ea quod ſcrip ſi Ma <lb/>rio Nizzolio, Franciſco Vimercato, Franciſco Contareno, Angelo Agrimenſori, & <lb/>de alijs nonnullis à me excogitatis.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE FINE CORPORVM COELESTIVM, <lb/>& eorum motu.</head> <head rend="italics" xml:space="preserve">Illuſtri vire, Philiberto Pingonio Sabaudo Cuſiacenſium <lb/>Baroni.</head> <p> <s xml:space="preserve">CVm antea meo nomine Sebaſtianus noſter omnia ferè tibi retuliſſet, inter alia, <lb/>quæ relin quebantur tibi <choice><ex>dicenda</ex><am>dicẽda</am></choice>, hoc vnum erat, quod ſi abſque lumine ſupe-<lb/>riori, in quem finem facta fuerint corpora cœleſtia ſcire deſideras, & humanam ra-<lb/>tionem ſequi volueris, putandum tibi non erit ea ſolum effecta eſſe, vt tam vile cor <lb/>pus, vt eſt terra aquis irrigata, animalia, & plantas regant, cum ea corpora ſint diuina, <lb/>in numero incompręhenſibilia, maximis ma gnitudinibus, & motibus velocisfimis, <lb/>prædita, id etiam minus putabunt hij, qui opinionem Ariſtarchi Samij, & Nicolai <lb/>Copernici ſequuntur, quorum ratione fieri non poteſt, vt credant, eius, quod ex vni <lb/>uerſo reliquum eſt, alium finem non habere, quam regimen huius centri epicycli Lu <lb/>naris, vt illorum more loquar. </s> <s xml:space="preserve"><choice><ex>Quam</ex><am>Quã</am></choice> enim turpe eſſet ſi centra aliorum epicyclorum <lb/>planetarum tali regimine priuarentur, id quod nullo modo cum ratione conſentit, <lb/>ſi tam vera eſt ea opinio, quemadmodum <choice><ex>rationabiliorem</ex><am>rationabiliorẽ</am></choice> eam <choice><ex>exiſtimant</ex><am>exiſtimãt</am></choice>. </s> <s xml:space="preserve"><choice><ex>Neque</ex><am>Neq;</am></choice> quid <lb/>quam valet opinio Ariſtotelis, qui corpora cœleſtia, ab ortu, & interitu libera eſſe <lb/>fentit. </s> <s xml:space="preserve">dicens ſuperioribus fęculis, à noſtris antiquis nullam vnquam animaduerſam <lb/>fuiſſe alterationem in cœlo, cum non videat ſi quis eſſet in cœlo, <choice><ex>neque</ex><am>neq;</am></choice> etiam obſerua <lb/>re poſſet alterationes quæ in terra, & circa terram fiunt, quæ in partibus, & non in to <lb/>to ſpectantur: </s> <s xml:space="preserve">vnde etiam fieri poteſt, vt in cœlo ſint particulares alterationes, <lb/>quæ à nobis tamen, qui ab illis longè diſtamus, non compręhendantur, terra, <choice><ex>mareque</ex><am>mareq́;</am></choice> <lb/>(quamuis minimum reſpectu ipſius terræ) ratione totius ita ſe ſemper <choice><ex>habuerunt</ex><am>habuerũt</am></choice> <choice><ex>quem</ex><am>quẽ</am></choice> <lb/>admodum ſeſe habere corpora cœleſtia videmus, ſed alteratio, ratione tantum ali-<lb/>quarum minimarum partium quaſi inſenſibilium, ſi cum toto comparentur fit. </s> <s xml:space="preserve">Quis <lb/>enim ſcit, vt iam tibi dixi, quin, quemadmodum Luna circa terram voluitur, <choice><ex>ipſaque</ex><am>ipſaq́;</am></choice> <lb/>terra ſit veluti centrum epicycli maioris eiuſdem, vt Ariſtarchus Samius, & Nico-<lb/>laus Copernicus cenſuerunt; </s> <s xml:space="preserve">ſic etiam Saturnus, Iupiter, Mars, Venus, <choice><ex>atque</ex><am>atq;</am></choice> Mercu <lb/>rius circa alia huiuſmodi corpora, huic terræ ſimilia, in orbem agantur, quaſi ſpecu-<lb/>la, lumen Solis ſuo centro ex reflexione, deferentia (fuppoſita dico vera illorum opi <lb/>nione) <choice><ex>Nollem</ex><am>Nollẽ</am></choice> tamen tibi è mente excidere, vt aliàs te monui, <choice><ex>quod</ex><am>ꝙ</am></choice> ſi communis opinio <lb/>vera eſt, neceſſario <choice><ex>fatendum</ex><am>fatendũ</am></choice> ſit corpus ſolare, <choice><ex>dum</ex><am>dũ</am></choice> in æquatore reperitur moru diurno <lb/>quolibet horę minuto, magis <choice><ex>quam</ex><am>quã</am></choice> <choice><ex>decem</ex><am>decẽ</am></choice> & <choice><ex>ſeptem</ex><am>ſeptẽ</am></choice> mille milliaria <choice><ex>peragere</ex><am>ꝑagere</am></choice>, ideſt paulo mi <lb/>nus quam .18000. milliaria, Saturnum verò cum ſimiliter eſt in æquatore, eodem <choice><ex>tem- poris</ex><am>tẽ-poris</am></choice> ſpatio, quaſi tercentamille milliaria Italica conficere, & ſic per gradus alia cor <lb/>pora velociora alijs moueri; </s> <s xml:space="preserve">quæ quidem omnia, <choice><ex>cum</ex><am>cũ</am></choice> ſimplici gyro terræ circa ſuum <pb facs="0268" n="256"/><fw type="head">IO. BABPT. BENED.</fw> axem (vt dicunt) tolluntur, quod ſufficit ad recipiendum lumen, & influentias illo-<lb/>rum corporum. </s> <s xml:space="preserve">Et ita, veluti princeps corporum vniuerſi, intra vnum an-<lb/>num circa eam vertitur. </s> <s xml:space="preserve">Ita etiam ſuſſiceret, vt ipſa terra circa dictum diuinum <lb/>corpus ſolare, interſecando axem diurnum cum axe annuali (cum ab eo lumen, ca-<lb/>lorem, & influentiam ſuſcipere debeat) <choice><ex>circunuolueretur</ex><am>circunuolueret̃</am></choice>. </s> <s xml:space="preserve">Rationes autem a Ptolo-<lb/>meo in contrarium adductæ apud ipſos, nullę ſunt, quia quęlibet pars (vt inquiunt) <lb/>retinet naturam totius, præterquam <choice><ex>quod</ex><am>ꝙ</am></choice> aer, & aqua, quæ ipſam terram <choice><ex>circundant</ex><am>circundãt</am></choice>, pla <lb/>nè eundem naturalem impetum motus obtineant, quitanto lentior eſt, quanto lon <lb/>gius diſtat aer ab ipſa terra, ſecundum etiam talem opinionem, nulla neceſſitas fo-<lb/>ret, vt locus fixarum terminaretur aliquibus ſuperficiebus, conuexa ſcilicet, & de-<lb/>uexa.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De Luce, Lumine, & Colore, De obiectuoculi, De lumine <lb/>Luna, & Rubedine nubium.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod proximè quærebas, an ſit lux aliqua, quæ à corpore lucido non proue-<lb/>niat, mihi facilè ad conſiderandum videtur. </s> <s xml:space="preserve">hic enim oportet, vt nos ad id <lb/>quod perpetuò videmus referamus, exiſtimo autem te velle dicere lumen, non lu-<lb/>cem, quia propriè lux, qualitas ea viſibilis appellatur, quæ eſt in corpore lucido, à <lb/>quo quidem corpore lumen effunditur; </s> <s xml:space="preserve">lumen verò, ea qualitas eſſe dicitur, quæ ex <lb/>tra ipſum corpus reperitur, à luce, quæ in dicto corpore manet emanans. </s> <s xml:space="preserve">vnde pa-<lb/>tet, nullam lucem abſque corpore ſubiecto eſſe poſſe, id quod cum fieri quîret, <choice><ex>idem</ex><am>idẽ</am></choice> <lb/>de quolibet alio accidente dici poſſet, id eſt quod ex ſe, & abſque aliquo ſubiecto <lb/>ſubſiſteret.</s> </p> <p> <s xml:space="preserve">Lumen deinde à luce proficiſci patet, <choice><ex>quod</ex><am>ꝙ</am></choice> penetrat diaphanum, neque aliquo mo-<lb/>do ſuum actum oſtendit, niſi, aut per incidentiam, aut ratione opaci, ex reflexione, <lb/>cuius ſuperficiei colorem induit. </s> <s xml:space="preserve">Atque hæc eſt cauſa, vt inter crepuſculum matu-<lb/>tinum, aut veſpertinum, nox etiam ſi ſit ſerena, adeo obſcura nobis appareat, quam-<lb/>uis totum vniuerſum diaphanum, extra conum vmbræ, quæ ex terra prouenit ſit vn <lb/><choice><ex>dique</ex><am>diq;</am></choice> radijs luminoſis Solis colluſtratum; </s> <s xml:space="preserve">qui quidem radij, non niſi à ſuamet reflexio <lb/>ne à Luna, & ab alijs ſtellis (vt corporibus opacis, quæ reſiſtunt lumini, ne vlterius <lb/>penetrare poſſit, vnde retrò redit) comprehenduntur.</s> </p> <p> <s xml:space="preserve">Ais etiam propria viſus obiecta plura eſſe, nominans pro vno, colorem, & lucem <lb/>pro alio. </s> <s xml:space="preserve">Ego autem reſpondeo, obiectum oculi eſſe vnicum tantum, ideſt lumen. <lb/></s> <s xml:space="preserve">Quod ad lucem ſpectat, iam tibi dixi, eam eſſe quandam qualitatem in corpore luci <lb/>do, & non extra ipſum poſitam, à quo quidem corpore, cum non exeat, oculi obie-<lb/>ctum eſſe nequit, ſed lumen quidem ab ipſa luce productum. </s> <s xml:space="preserve">Color etiam, qui eſt in <lb/>corpore colorato, obiectum oculi eſſe non poteſt, cum dictum corpus non deſerat, <lb/>ſed lumen quidem ab eodem corpore reflexum, & huiuſmodi corporis colore tin-<lb/>ctum: </s> <s xml:space="preserve">vnde tam lumen incidens, quam reflexum colore eſt ſemper imbutum.</s> </p> <p> <s xml:space="preserve">Illud quidem coloratum eſt qualitate lucis corporis lucidi, a ut medij, per quod <lb/>tranſit, ſed hoc colore corporis, à quo reflectitur.</s> </p> <p> <s xml:space="preserve">Neque etiam te ignorare volo, lumina reflexa colorata, non reflecti à puris pro-<lb/><choice><ex>priisque</ex><am>priisq́;</am></choice> ſuperficiebus corporum coloratorum, eo <choice><ex>quod</ex><am>qđ</am></choice> pauca corpora tam opaca repe-<lb/>riuntur, ut immediatè lumen à ſuperficie propriè <choice><ex>reflectant</ex><am>reflectãt</am></choice>, ſed lumen penetrat alim <pb facs="0271" n="259"/><fw type="head">EPISTOL AE.</fw> quantulum dicta corpora, & ita illorum colore afficitur, vbi verò non penetrat, non <lb/>coloratur colore corporis illius.</s> </p> <p> <s xml:space="preserve">Sed vt ad propoſitum redeamus, dico lum en tantum eſſe viſus obiectum, quod ſi <lb/>colore eſt imbutum, aut tale eſt ratione color ris lucis, quæ eum mittit, aut ratione me <lb/>dij per quem tranſit, aut ratione corporis, vnde reflectitur, etſi ſuperficies corporis <lb/>vnde lumen reflectitur eſſet omnino priuata colore, ſub aſpectum non caderet, vt <lb/>etiam cum huiuſmodi ſuperficies læuigata, & polita eſt ſecundum <choice><ex>continuitatem</ex><am>continuitatẽ</am></choice> ſua-<lb/>rum partium, videlicet, vt ſpeculi radio tamen non profundante, & ideo perfectiffi <lb/>morum quorundam ſpeculorum ſuperficies non cernuntur, ſed lumen tantum re-<lb/>flexum, colore alicuius alterius ſuperficiei, aut à luce, corporis lucidi, aut à me-<lb/>dio per quem tranſit, conſpicitur. </s> <s xml:space="preserve">Ego verò non aſſero colorem non eſſe quid di-<lb/>uerſum à lumine, ſed imagineris lumen eſſe veluti animam, aut ſubſtantiam & colo <lb/>rem corporis formam accidentalem, cum nullum lumen à ſenſu viſus percipi poſ-<lb/>ſit, quod aliquo modo colore non ſit imbutum: </s> <s xml:space="preserve">& eundem reſpectum quem ſonus <lb/>ad auditum, lumen ad oculum habet, quia vt ſonus ſecundum eam velocitatem, quæ <lb/>à motione aeris, aut aquæ, ex colliſione <choice><ex>aliorum</ex><am>aliorũ</am></choice> corporum producitur ad euitan dum <lb/>vacuum, a cutus, vel grauis ſentitur, ita lumen originem ducens à corpore lucido per <lb/>medium diaphanum aeris, aut aquæ, aut alterius huiuſmodi corporis ad oculum tran <lb/>ſit colorem lucis, aut medij per quod tranſit, aut vnde reflectitur induit.</s> </p> <p> <s xml:space="preserve">Quod verò Luna nullum ex ſe habeat lumen, ſufficiens inditium eſt nos ipſam <lb/>tantò magis obſcuram videre, quantò magis in cono vmbræ terræ immergitur, & <lb/>ſi eo tempore ipſam videmus rubeo colore affectam, hoc enim accidit, quia radij ſo <lb/>lares vndequaque refranguntur à vaporibus ipſam terram circundantibus, quæ qui-<lb/>dem refractio fit verſus axem coni vmbræ terræ, </s> <s xml:space="preserve">& propterea vmbra dicti coni non <lb/>eſt æqualiter obſcura, ſeu tenebroſa, circa vero <choice><ex>axem</ex><am>axẽ</am></choice> ipſius coni, magis quam circa eius <lb/><choice><ex>circunferentiam</ex><am>circũferentiã</am></choice>, obſcura <choice><ex>verum</ex><am>vr̃</am></choice>, & quia corpus lunare tale eſt, vt facillimè recipiat <choice><ex>qualecum</ex><am>qualecũ</am></choice> <lb/>que lumen, quod etiam manifeſtè videtur dum ipſa Luna reperitur ſecundum lon-<lb/>gitudinem inter Solem, & Venerem, quod pars Lunæ lumine Solis deſtituta, à lumi <lb/>ne Veneris aliquantulum illuſtratur, quod ego ſæpè vidi, & multis oſtendi. </s> <s xml:space="preserve">Propte-<lb/>rea dum ipſa Luna in cono vmbræ terræ reperitur adhuc videtur. </s> <s xml:space="preserve">Rubedo etiam il-<lb/>la nubium poſt Solis occaſum, vel ante ortum, aliunde non prouenit, niſi à qualitate <lb/><choice><ex>vaporum</ex><am>vaporũ</am></choice>, per quos ſolares radij tranſeunt, à quibus vaporibus, tali colore ipſi radij <lb/>afficiuntur, eomet modo quo radius, cuiuſuis corporis lucidit, <choice><ex>tranſiens</ex><am>trãſiens</am></choice> per vitrum, ſeu <lb/>aliud diaphanum coloratum.</s> </p> <pb facs="0270" n="258"/> <fw type="head">IO. BABPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE ICTV BOMBARDAE SECVNDVM <lb/>diuerſas eleuationes. Et de quibuſdam erroribus Nico-<lb/>lai Tartaleæ, circa idem.</head> <head rend="italics" xml:space="preserve">Fllustri D. Ioſepho Cambiano ex Ruffia Dominis, aquiti <lb/>ſtrenuo, & tormentis bellicis Serenißimi Ducis <lb/>Sabaudia Prafecto.</head> <p> <s xml:space="preserve">EXcogitaui quędam dum ocio frui licuit per abſentiam Ducis Sereniſſimi, <lb/>quæ ad te ſcribere placuit, vt ſi probaueris in lucem quandoque profer-<lb/>re non dubitem, ſi deſpexeris, ocius ſupprimam, ſunt autem huiuſmodi.</s> </p> <p> <s xml:space="preserve">Vnde fiat vt tormentum bellicum vehementi<unclear reason="illegible"/> feriat ictu ſuperius delato <lb/>quam orizontali, vt Tartalea ſcribit, quæſito ſecundo libr. primi quæſitorum, à ne-<lb/>mine adhuc (quod ſciam) traditum eſt.</s> </p> <p> <s xml:space="preserve">Rationes verò Tartaleæ nullius ſunt momenti, quia ſi validæ eſſent, ſequeretur <lb/>vt inclinata bombarda, adeo vt angulus ſub orizonte factus æqualis eſſet ei, qui ſu <lb/>pra orizontem eſt, ictum bombardę in vtroque huiuſmodi ſitu eundem eſſe <choice><ex>futurum</ex><am>futurũ</am></choice>. <lb/></s> <s xml:space="preserve">& ſi aliqua differentia oriretur ratione granitatis pilæ ab ipſa bombarda emiſſæ, hoc <lb/>fieret, vt ſcilicet velocior eſſet in motu inclinato quam in eleuato cum pondus, mo-<lb/>tui adeo non opponatur. </s> <s xml:space="preserve">Id quod non ita se habet, vera enim cauſa vnde fiat, vt bom <lb/>barda eleuata vehementius feriat, quàm ea quæ eſt minus alta, eadem eſt ferè, in ge-<lb/>nere, cum ea, qua aliquod corpus materia magis denſa, ſed ſimile & ęquale alteri cor <lb/>pori materiæ minus denſæ velocius mouetur ab vna eademq́ue, aut æquali vi <lb/>compulſum. </s> <s xml:space="preserve">Eſt eadem etiam in ſpecie ei, qua maiorem effectum producit <lb/>puluis, qui in locis ſubterraneis ponitur quum vaſis optimè colligatis ferro in-<lb/>cluditur. </s> <s xml:space="preserve">Eſt etiam ſimilis ei, qua longius impellitur pila, qua ludimus, ab ali-<lb/>quo inſtrumento ligneo, quando percutitur contra, quam cum ſecundum ſuum mo-<lb/>tum proijcitur. </s> <s xml:space="preserve">Id quod inde fit, quia virtus mouens maiori vi, & intenſiori huiuſ-<lb/>modi corpus percutit, quia corpus quod moueri debet, quanto magis reſiſtit virtu-<lb/>ti mouenti (certum tamen terminum præſcribendo) in exiguo eo temporis ſpatio, <lb/>tanto maiorem virtutem colligit, quæ ipſum deinde tanto cum impetu mo-<lb/>uet, & tanto magis impellens concomitatur, vt maiorem effectum efficiat, quam ſi <lb/>ad mouendum ſeſe facilè reddidiſſet. </s> <s xml:space="preserve">Atque hoc ſupradictis ictibus eleuatis acci-<lb/>dit, quia grauitas pilæ, ea eſt quæ reſiſtens virtuti mouenti, dat ei commoditatem <lb/>colligendi dictam virtutem, multo magis quam eſſet ea, quæ ad depreſſiorem eleua <lb/>uationem eam <choice><ex>immpelleret</ex><am>ĩmpelleret</am></choice>. </s> <s xml:space="preserve">Et quia huiuſmodi multiplicatio virtutis, nullam propor <lb/>tionem cum pondere pilæ gerit, volo inferre quod dum colligitur tanta virtus, col-<lb/>ligitur multo plus eo, quod ad impellendam dictam pilam ſufficeret, ratione magnæ <lb/>velocitatis augumenti, quia quanto plus temporis ei conceditur ad commutandam <lb/>puluerem in ignem, tanto maior quantitasignis progignitur, vnde fit, vt tanto ma-<lb/>iori loco indigeat, quamobrem tanto magis impellit, ſed vt dixi, tanta cum veloci-<lb/>tate <choice><ex>adaugetur</ex><am>adauget̃</am></choice>, vt huiuſmodi virtus longè ſuperet <choice><ex>reſiſtentiam</ex><am>reſiſtẽtiã</am></choice> <choice><ex>ponderis</ex><am>põderis</am></choice> pilæ, & ſic eſt cau <lb/>fa, ut effectus, quod <choice><ex>experientia</ex><am>experiẽtia</am></choice> innot eſcit <choice><ex>producatur</ex><am>producat̃</am></choice>. </s> <s xml:space="preserve">Sed ea ratio, qua ſeſe <choice><ex>idem</ex><am>idẽ</am></choice> author <lb/>in tertio quæſito ad aliquod impoſſibile, circa iter ipſius pilæ Legatum Hiſpanum <pb facs="0271" n="259"/><fw type="head">EPISTOL AE.</fw> reducere putat, nullo fundamento nititur, quia non eſt ſemper dicendum, quod <choice><ex>quam</ex><am>quã</am></choice> <lb/>to velocior ſit quædam pila, tanto rectius moueatur, quia ei dici poſſet, vſque ad cer <lb/>tum quendam terminum velocitatis, per tantum ſpatij eam aptam eſſe, vt recta per <lb/>fectè moueatur, ſed ſi velocius iret, non tamen futurum, vt per idem ſpatium re-<lb/>ctius moueretur, ſed quod per longius ſpatium recta motum perageret, & ſic nihil <lb/>haberet quod replicaret, præter quam quod ipſe ſupponit id quod in 18. quæſito <lb/>negat, in quo ait pilam uicinam orificio, non adeo uelocem eſſe, quam cum aliquan-<lb/>tulum ab eodem eſt rem ota, ratione reſiſtentiæ ſui cyllindriaerei. </s> <s xml:space="preserve">Sed quod pila, <lb/>recta eat quanto altior, aut depreſſior <choice><ex>bombarda</ex><am>bõbarda</am></choice> erit, <choice><ex>ion</ex><am>iõ</am></choice> fit, quia linea inclinationis na <lb/>turalis cum linea inclinationis uiolentæ angulum rectum non facit, </s> <s xml:space="preserve">unde quanto lon <lb/>gius diſtat à recto huiuſmodi <choice><ex>angulus</ex><am>angulꝰ</am></choice>, ſiue ſit acutus ſiue obtuſus, tanto minorem uim <lb/>habet, eodem planè ferè modo quem tertio capite mei tractatus de rebus mechani <lb/>cis deſcripſi. </s> <s xml:space="preserve">Quia in ictibus eleuatis, iter inclinationis violentæ ipſius pilæ verſus <lb/>terminum ad quem, incipiendo à loco ipſius pilæ cum itinere inclinationis natura-<lb/>lis, angulum obtuſum, & in ictibus inclinatis acutum conſtituit. </s> <s xml:space="preserve">Neque etiam hic <lb/>prætermittam notatu dignum errorem, quem Tartalea eodem loco committit, <choice><ex>cum</ex><am>cũ</am></choice> <lb/>putet indifferenter aliquod corpus impellere, aut percutere maiori <choice><ex>cum</ex><am>cũ</am></choice> impetu cum <lb/>eſt in itinere recto. </s> <s xml:space="preserve">Quia ſequeretur quod aliquod corpus graue perpendiculari-<lb/>ter ſurſum verſus proiectum, in qualibet parte ſui itineris, ſemper fortius percute-<lb/>ret, quam in qualibet parte itineris alterius cuiuſuis eleuationis obliquæ, quod <choice><ex>quam</ex><am>quã</am></choice> <lb/>ſit falſum, tibi conſiderandum relinquo.</s> </p> <p> <s xml:space="preserve">Eſt etiam falſa ea ratio, quam in quarto quæſito idem adducit, quia aer in motu <lb/>non tantum durat, quantum ipſe putat, imò huiuſmodi violenta agitatio, citò ceſſat <lb/>& citius etiam, quam ſi extra aliquam bombardam cum tanta violentia impulliſſet <lb/>ſaceum plumis plenum.</s> </p> <p> <s xml:space="preserve">Ratio etiam quam in .18. quæſito de eo, quod pila pertran ſeat illud corpus cyllin <lb/>dricum aereum adducit, eſt planè vana, quia ſtatim aer, qui prius in <choice><ex>bombarda</ex><am>bõbarda</am></choice> erat <lb/>incluſus, extra ipſam <choice><ex>erumpit</ex><am>erũpit</am></choice>, cedit, à <choice><ex>pilaque</ex><am>pilaq́;</am></choice> diuiditur, vt ſi nunquam eam figuram in-<lb/>duiſſet, neque aer ambiens ei reſiſtit. </s> <s xml:space="preserve">Sed quod velocior ſit in certa quadam diſtan <lb/>tia, quam in principio erat, ſi hoc <choice><ex>verum</ex><am>verũ</am></choice> eſſet, ab alia cauſa dependeret, quæ partim ſi <lb/>milis eſſet ei, quæ efficit, vt corpora in motibus naturalibus, cum longius diſtant à ter <lb/>mino vnde naturaliter ſeſe mouerunt, ſint velociora, quia per aliquod ſpatium hu-<lb/>iuſmodi corpus moueretur quemadmodum motu naturali cietur.</s> </p> <p> <s xml:space="preserve">Ratio autem eius quare pila, aut globus bombardæ ſibiletab eodem in ſeptimo <lb/>quæſito nil valet, quia hoc fit cum pila aliquam paruam concauitatem habet.</s> </p> <p> <s xml:space="preserve">In .27. autem quæſito ait, quod retrotrahendo ſignum, ictus altius tenderet, quod <lb/>poteſt etiam eſſe falſum, cum hocnon ſit neceſſarium, quia pila dum deſcendit, for-<lb/>taſſe tangeret ſcopum.</s> </p> <pb facs="0272" n="260"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Deerroribus Ioannis Stadij.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">FIguram quam ponit Ioannes Stadius pag .147. in lib. ſuarum tabularum Prute-<lb/>nicarum, à Nicolao Copernico ſumpſit pag .64. à tergo in libr. reuolutionum <lb/>cœleſtium, ſed ipſe Stadius eam non intellexit, omitto, quod mutauerit characte-<lb/>res ipſius figurę, vt illa ſua videatur, quod nihil refert, alterat etiam <choice><ex>demonſtrationem</ex><am>demonſtrationẽ</am></choice>, <lb/>ſed ipſum putare <seg type="var">.i.K.</seg> perpendicularem à centro circuli ſemper dependere, eſt intol <lb/>lerabilis error; </s> <s xml:space="preserve">nec vnquam verificatur hoc, niſi quando punctum <seg type="var">.K.</seg> interſectionis <lb/>diametrorum parallelorum, forte reperitur in axe mundi. </s> <s xml:space="preserve">Reliqua verò ſuæ demon <lb/>ſtrationis, ſi non intelligis, minimè miror, eo quod ipſemet Stadius ſeipſum confun <lb/>dit. </s> <s xml:space="preserve">Veram autem demonſtratio nem huiuſmodi figuræ in dicto libr. Copernici cla-<lb/>rè videbis. </s> <s xml:space="preserve">Quod verò diuersè cogitaui nunc acciptito.</s> </p> <p> <s xml:space="preserve">Cum nobis cognita ſit maxima ecclipticæ declinatio, vt puta <seg type="var">.a.c.</seg> ſi latitudo <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>ſtellæ nobis data fuerit, vt puta <seg type="var">.c.e.</seg> cognitus nobis erit totalis arcus <seg type="var">.a.e.</seg> & eius ſinus <seg type="var">.<lb/>e.m.</seg> & quia notus etiam nobis eſt ſinus arcus <seg type="var">.a.c.</seg> hoc eſt <seg type="var">.c.n.</seg> & corda <seg type="var">.e.f.</seg> medio eius <lb/>arcus <seg type="var">.e.p.f.</seg> minoris media circunferentia, per duplum latitudinis datæ, vnde <seg type="var">.e.l.</seg> eius <lb/>dimidium nobis cognitum erit, vel vt ſinus arcus <seg type="var">.e.p.</seg> cognitus etiam nobis eſt ſinus <seg type="var">.<lb/>q.g.</seg> declinationis <seg type="var">.a.g.</seg> datæ, cui æqualis eſt <seg type="var">.m.t.</seg> ex .34. primi Euclid. </s> <s xml:space="preserve">vnde <seg type="var">.e.t.</seg> nobis <lb/>cognita remanet, cum verò duo trianguli <seg type="var">.i.c.n.</seg> et <seg type="var">.t.e.K.</seg> <choice><ex>æquianguli</ex><am>æquiãguli</am></choice> ſint, propter duas <lb/>parallelas <seg type="var">.e.m.</seg> et <seg type="var">.n.c.</seg> ex .28. primi, & propter duas <seg type="var">.a.b.</seg> et <seg type="var">.g.h.</seg> & propter duas <seg type="var">.c.d.</seg> <lb/>et <seg type="var">.e.f.</seg> eo quod ex communi ſcientia anguli <seg type="var">.c.</seg> et <seg type="var">.e.</seg> ſunt æquales, cum ex .29. dicti lib. <lb/><choice><ex>vnuſquiſque</ex><am>vnuſquiſq;</am></choice> æqualis ſit angulo <seg type="var">.m.ω.i.</seg> ita etiam infero de angulis <seg type="var">.e.K.t.</seg> et <seg type="var">.c.i.n.</seg> <choice><ex>quorum</ex><am>quorũ</am></choice> <lb/>vnuſquiſque æqualis eſt angulo <seg type="var">.ω.x.t.</seg> & ſic de alijs dico, co quod vnuſquiſque <choice><ex>eorum</ex><am>eorũ</am></choice> <lb/>æqualis eſt angulo <seg type="var">.m.</seg> vnde cum cognitum nobis ſit latus <seg type="var">.n.c.</seg> et <seg type="var">.c.i.</seg> et <seg type="var">.t.e.</seg> <choice><ex>notum</ex><am>notũ</am></choice> etiam <lb/>nobis erit <seg type="var">.e.K.</seg> ex .19. ſeptimi, eo <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>ex .4. ſexti ſunt inuicem proportio-<lb/> <ptr xml:id="fig-0272-01a" corresp="fig-0272-01" type="figureAnchor"/> nalia, detrahendo poſtea <seg type="var">.e.K.</seg> ab <seg type="var">.<lb/>e.l.</seg> cognito, vel ècontra, hoc ab il-<lb/>lo, nobis innoteſcet <seg type="var">.K.l.</seg> ſinus longi <lb/>tudinis ſtellæ.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0272-01" corresp="fig-0272-01a"> <graphic url="0272-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Valde etiam miror id, quod di-<lb/>ctus Stadius pag .9. illius libr. ſcri-<lb/>bit, hoc eſt, Solem maiorem <lb/>eſſe Luna, ſolum .1644. vici-<lb/>bus, </s> <s xml:space="preserve">propterea <choice><ex>quod</ex><am>ꝙ</am></choice> cum affirmet So-<lb/>lem maiorem eſſe terra (vt etiam <lb/>in Almageſto videre eſt) 166. vici <lb/>bus cum tribus quartis, terram ve-<lb/>ro maiorem Luna .39. vicibus cum <lb/>quarta parte, </s> <s xml:space="preserve">tunc Solem oporte-<lb/>ret maiorem eſſe Luna .6545. vici-<lb/>bus, & non .1644.</s> </p> <pb facs="0273" n="261"/> <fw type="head">EPISTOL AE.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Decognitione latitudinum stellarum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">AD cognoſcendam latitudinem ſtellæ, <choice><ex>eiusque</ex><am>eiusq́</am></choice> declinationem, Monteregius in <lb/>10. propoſitione .8. li. </s> <s xml:space="preserve">Almageſti <choice><ex>methodum</ex><am>methodũ</am></choice> ſatis docuit, ſed ſi alia aliqua metho <lb/>do hoc idem cognoſcere voluerimus, oportebit nos prius altitudinem poli cogno-<lb/>ſcere, deinde altitudinem meridianam ipſius ſtellæ, nec non horam, <choice><ex>quando</ex><am>quãdo</am></choice> ipſa ſtel <lb/>la in meridiano ſupra terram reperitur, qua hora mediante, illicò cognoſcemus pun <lb/>ctum ecclipticæà meridiano interſecto, eo tempore, quo ſtella cœlum mediat ſu-<lb/>pra terram. </s> <s xml:space="preserve">Et quia ex cognita altitudine poli, illico cognoſcitur altitudo æqua-<lb/>toris, cuius altitudinis differentia ab altitudine ſtellæ eſt declinatio ipſius ſtellæ, ha-<lb/>bebimus ideo eius declinationem cognitam; </s> <s xml:space="preserve">qua mediante ad <choice><ex>cognoſcendum</ex><am>cognoſcendũ</am></choice> etiam <lb/>latitudinem ita faciemus.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia <seg type="var">.p.o.u.</seg> meridianus <seg type="var">.u.a.</seg> verò æquator <seg type="var">.e.a.</seg> autem eccliptica, & <lb/>o. centrum aſtri <seg type="var">.u.o.</seg> verò eius declinatio ab æquatore, et <seg type="var">.e.a.</seg> arcus ęcclipticæ inter <lb/>æquatorem, & meridianum, hoc eſt minor quarta, et <seg type="var">.a.u.</seg> aſcenſio recta ipſius arcus, <lb/>et <seg type="var">.u.e.</seg> ſit declinatio puncti <seg type="var">.e.</seg> ęcclipticę ab æquatore, <choice><ex>reſiduum</ex><am>reſiduũ</am></choice> vero declinationis ſtel-<lb/>lę ſit <seg type="var">.o.e.</seg> quæ <choice><ex>omnia</ex><am>oĩa</am></choice> nobis cognita erunt, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.t.</seg> polus ęcclipticus, à quo per <seg type="var">.o.</seg> vſque ad <lb/>ęcclipticam tranſeat quarta <seg type="var">.t.i.</seg> in qua quęrendus erit arcus <seg type="var">.o.i.</seg> hoc modo.</s> </p> <p> <s xml:space="preserve">Primum arcus <seg type="var">.o.u</seg>: <seg type="var">e.u</seg>: <seg type="var">e.o</seg>: <seg type="var">a.e</seg>: et <seg type="var">.a.u.</seg> nobis cogniti ſunt, cum angulo <seg type="var">.a.</seg> declinatio <lb/>nis ęclipticę, & cum angulo <seg type="var">.u.</seg> recto, vnde ex .4. primi Copernici, cognoſcemus angu <lb/>lum <seg type="var">.a.e.u.</seg> collateralem, & eius <seg type="var">.o.e.i</seg>. </s> <s xml:space="preserve">quare in triangulo <seg type="var">.o.e.i.</seg> cognoſcemus <choice><ex>angulum</ex><am>angulũ</am></choice> <lb/>e. et deinde <seg type="var">.i.</seg> vt <choice><ex>rectum</ex><am>rectũ</am></choice>, & latus <seg type="var">.o.e.</seg> ergò ex <choice><ex>eadem</ex><am>eadẽ</am></choice> .4. cognoſcemus <choice><ex>arcum</ex><am>arcũ</am></choice> <seg type="var">.o.i.</seg> quæſitum, <lb/>& ſimiliter arcum <seg type="var">.e.i.</seg> qui coniunctus vel <choice><ex>demptus</ex><am>dẽptus</am></choice> ab <seg type="var">.a.e.</seg> tribuet nobis longitudinem <lb/>ſtellę, ſed quia huiuſmodi operatio in paruis triangulis valde fallit. </s> <s xml:space="preserve">Ideo tibi ſua-<lb/>deo alia methodo, hoc facere, hoc eſt inuenire angulum <seg type="var">.o.</seg> trianguli <seg type="var">.t.p.o.</seg> cuius duo <lb/>latera <seg type="var">.t.p.</seg> et <seg type="var">.p.o.</seg> cognita nobis ſunt, cum angulo <seg type="var">.p</seg>. </s> <s xml:space="preserve">Nam <seg type="var">.o.p.</seg> eſt complementum de <lb/>clinationis ſtellæ, et <seg type="var">.p.t.</seg> eſt arcus coluri ſolſtitiorum inter duos polos, & angulus <seg type="var">.p.</seg> <lb/>reſiduum ex recto <seg type="var">.t.p.a.</seg> duorum colurum dempto angulo. a, <seg type="var">p.u.</seg> cognito aſcenſionis <lb/>recte, vnde angulus <seg type="var">.u.o.s.</seg> vt contrapoſitus cognitus remanet. </s> <s xml:space="preserve">angulus verò <seg type="var">.u.</seg> rectus <lb/>eſt, & arcus <seg type="var">.o.u.</seg> cognitus, </s> <s xml:space="preserve">quare cognitus <lb/>nobis erit arcus <seg type="var">.u.s.</seg> & angulus <seg type="var">.u.s.o.</seg> vnde <lb/> <ptr xml:id="fig-0273-01a" corresp="fig-0273-01" type="figureAnchor"/> arcus <seg type="var">.a.s.</seg> nobis cognitus remanebit <choice><ex>cum</ex><am>cũ</am></choice> an-<lb/>gulo <seg type="var">.a.s.i.</seg> reſiduo ex duobus rectis. </s> <s xml:space="preserve">Et quia <lb/>etiam angulus <seg type="var">.s.a.i.</seg> cognitus eſt, cum ſit an <lb/>gulus maximę declinationis Zodiaci ab <lb/>æquatore. </s> <s xml:space="preserve">Ideo in triangulo <seg type="var">.a.s.i.</seg> cuius <lb/>duo anguli <seg type="var">.a.</seg> et <seg type="var">.s.</seg> cum latere <seg type="var">.a.s.</seg> dantur, fa <lb/>cilè inueniemus arcum <seg type="var">.s.i.</seg> <choice><ex>cum</ex><am>cũ</am></choice> arcu <seg type="var">.a.i.</seg> ſed <lb/><seg type="var">a.i.</seg> erit longitudinis ſtellæ dempto poſtea <seg type="var">.<lb/>s.i.</seg> ex <seg type="var">.s.o.</seg> iam inuento habebimus arcum <seg type="var">.i.<lb/>o.</seg> latitudinis ipſius ſtellæ.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0273-01" corresp="fig-0273-01a"> <graphic url="0273-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Hæc autem tibi ſcribo non vt ipſis vta-<lb/>ris, ſed potius vt tibi morem <choice><ex>geram</ex><am>gerã</am></choice>, cum bre <lb/>uiſſima methodus ſit illa, <choice><ex>quam</ex><am>quã</am></choice> Monteregius <lb/>ſcripſit <choice><ex>in</ex><am>ĩ</am></choice> .10. <choice><ex>propoſitione</ex><am>ꝓpoſitione</am></choice> .8. li. in Almageſt.</s> </p> <pb facs="0274" n="222"/> <fw type="head">IO. BAPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head rend="italics" xml:space="preserve">Qualiter circulus deſignari poßit alios duos circulos <lb/>propoſitos includens.</head> <head xml:space="preserve">CLARISS. PETRO PIZZAMANO.</head> <p> <s xml:space="preserve">SVperioribus diebus per tuas literas à me quæſiuiſti, vt modum tibi ſcribere vel-<lb/>lem, quo circulus deſignari poſſit circunſcribens alios duos propoſitos circulos. <lb/></s> <s xml:space="preserve">Qua in re vt tibi ſatisfaciam quod maximè cupio ita rem accipe.</s> </p> <p> <s xml:space="preserve">Propoſiti circuli ſint, aut inter ſe contigui, aut interſecantes vel ſeparati. </s> <s xml:space="preserve">Eſto <choice><ex>pri- mum</ex><am>pri-mũ</am></choice> contiguos eſſe, qui ſint <seg type="var">.d.b.</seg> et <seg type="var">.f.q.</seg> <choice><ex>quorum</ex><am>quorũ</am></choice> <seg type="var">.d.b.</seg> maior ſit et <seg type="var">.f.q.</seg> minor, <choice><ex>eorum</ex><am>eorũ</am></choice> vero <lb/>centra ſint .a et <seg type="var">.o.</seg> <choice><ex>punctum</ex><am>punctũ</am></choice> autem <choice><ex>contingentię</ex><am>cõtingentię</am></choice> ſit <seg type="var">.i</seg>. </s> <s xml:space="preserve"><choice><ex>Nunc</ex><am>Nũc</am></choice> <choice><ex>protrahatur</ex><am>ꝓtrahat̃</am></choice>. <seg type="var">b.a.o.q.</seg> per <choice><ex>centra</ex><am>cẽtra</am></choice> eo <lb/>rum ab vna circunferentia ad aliam, quę quidem linea tranſibit per punctum <seg type="var">.i.</seg> ex <lb/>11, tertij Eucli. </s> <s xml:space="preserve">deinde à diametro maiori abſcindatur <seg type="var">.i.e.</seg> ad æqualitatem minoris <lb/>ſemidiametri, quo facto ſumatur diſtantia inter <seg type="var">.e.</seg> et <seg type="var">.b.</seg> circino mediante <choice><ex>factoque</ex><am>factoq́</am></choice> cen <lb/>tro <seg type="var">.o.</seg> ſcindatur, alio circini pede, circunferentia maioris circuli in puncto <seg type="var">.u.</seg> à quo ſi <lb/>mente concipiemus duas lineas <seg type="var">.u.a.d.</seg> et <seg type="var">.u.o.f.</seg> tranſeuntes per eorum centra <seg type="var">.a.</seg> et <seg type="var">.o.</seg> <lb/>vſque ad circunferentias in punctis <seg type="var">.d.</seg> et <seg type="var">.f.</seg> ipſę <choice><ex>erunt</ex><am>erũt</am></choice> inuicem ęquales, eo quod <seg type="var">.e.i.</seg> <choice><ex>sum- pta</ex><am>sũ-pta</am></choice> fuit æqualis <seg type="var">.o.f.</seg> et <seg type="var">.o.u.</seg> æqualis <seg type="var">.e.b.</seg> </s> <s xml:space="preserve">quare <seg type="var">.u.f.</seg> æqualis erit <seg type="var">.b.i.</seg> ſed <seg type="var">u.d.</seg> <choice><ex>etiam</ex><am>etiã</am></choice> æqua <lb/>lis <seg type="var">.b.i.</seg> ergo <seg type="var">.u.d.</seg> æqualis erit <seg type="var">u.f.</seg> & circulus, cuius <seg type="var">u.d.</seg> vel <seg type="var">.u.f.</seg> erit ſemidiameter, con-<lb/>tiguus erit ipſis propoſitis circulis ex conuerſo .11. iam dictæ. </s> <s xml:space="preserve">Idem dico pro circu-<lb/>lis ſe inuicem ſecantibus.</s> </p> <figure place="here"> <graphic url="0274-01"/> </figure> <figure place="here"> <graphic url="0274-02"/> </figure> <pb facs="0275" n="263"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Sed ſi circuli propoſiti ſeiuncti fuerint, ſumatur <seg type="var">.b.i.</seg> diameter maioris, qui fiat ſe-<lb/>midiameter vnius circuli circa centrum <seg type="var">.o.</seg> & hic circulus vocetur <seg type="var">.h.x.</seg> coniunga-<lb/>tur deinde ſemidiameter <seg type="var">.o.i.</seg> minoris circuli cum ſemidiametro <seg type="var">.a.i.</seg> circuli maio-<lb/>ris, & ex huiuſmodi compoſita linea, fiat vnus ſemidiameter <seg type="var">.a.x.</seg> circuli <seg type="var">.x.n.</seg> concen <lb/>trici cum maiori, & à puncto <seg type="var">.x.</seg> interſectionis horum circulorum (poſito quod ſe in-<lb/>uicem interſecent) ducantur per eorum centra <seg type="var">.x.a.</seg> et <seg type="var">.x.o.</seg> vſque ad ipſorum circun-<lb/>ferentias in punctis <seg type="var">.d.</seg> et <seg type="var">.f.</seg> duę <lb/>lineæ, vnde habebimus <seg type="var">.x.d.</seg> <lb/>æqualem <seg type="var">.x.f.</seg> eo quod tam in <lb/> <ptr xml:id="fig-0275-01a" corresp="fig-0275-01" type="figureAnchor"/> <seg type="var">x.d.</seg> quam in <seg type="var">.x.f.</seg> reperiuntur <lb/>diametri, & ſemidiametri am-<lb/>borum circulorum, facto deni <lb/>que centro <seg type="var">.x.</seg> vnius circuli, cu <lb/>ius ſemidiameter ęqualis ſit <lb/>vni earum <seg type="var">.x.d.</seg> vel <seg type="var">.x.f.</seg> folu-<lb/>tum erit problema, dicta ra-<lb/>tione.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0275-01" corresp="fig-0275-01a"> <graphic url="0275-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Si verò diſtantia duorum <lb/>propoſitorum circulorum tanta fuerit, quod ſecundi circuli nequeant ſe inuicem <lb/>tangere, vel ſecare, tunc alia via incedendum erit, quę talis eſt & generalis. </s> <s xml:space="preserve">Diuida-<lb/>tur tota <seg type="var">.q.b.</seg> per æqualia in puncto <seg type="var">.z.</seg> circa quod <choice><ex>ſignentur</ex><am>ſignẽtur</am></choice> duo puncta ab ipſo ęquidi <lb/>ſtantia <seg type="var">.K.</seg> et <seg type="var">.p.</seg> diſtantia vero <seg type="var">.a.K.</seg> facta ſit ſemidiameter eſſe vnius circuli <seg type="var">.K.x.</seg> circa <lb/>centrum <seg type="var">.a.</seg> diſtantia autem <seg type="var">.o.p.</seg> ſemidiameter alterius circuli <seg type="var">.p.x.</seg> circa cen-<lb/>trum <seg type="var">.o.</seg> qui quidem circuli ſe inuicem ſecent in puncto <seg type="var">.x.</seg> à quo cum ductę fue-<lb/>rinc <seg type="var">.x.a.d.</seg> et <seg type="var">.x.o.f.</seg> per centra dictorum circulorum, ipſe erunt <choice><ex>inuicem</ex><am>inuicẽ</am></choice> ęquales, eo <choice><ex>quod</ex><am>qđ</am></choice> <lb/>cum <seg type="var">.b.K.</seg> æqualis ſit <seg type="var">.q.p.</seg> igitur <seg type="var">.x.d.</seg> et <seg type="var">.q.p.</seg> erunt inuicem ęquales, ſed <seg type="var">.f.x.</seg> æqualis eſt <lb/><seg type="var">q.p</seg>. </s> <s xml:space="preserve">quare <seg type="var">.x.f.</seg> æqualis erit <seg type="var">.x.d.</seg> tunc ſi <seg type="var">.x.</seg> centrum fuerit vnius circuli, cuius ſemidia-<lb/>mer ſit vna dictarum, problema ſolutum erit.</s> </p> <p> <s xml:space="preserve">Talis etiam ſoiutio commo-<lb/>da erit ad inueniendum dictum <lb/> <ptr xml:id="fig-0275-02a" corresp="fig-0275-02" type="figureAnchor"/> circulum cuiuſuis magnitudinis, <lb/>dato tamen <choice><ex>quod</ex><am>ꝙ</am></choice> eius diameter, ma <lb/>ior ſit <seg type="var">.b.z.</seg> cum in noſtra poteſta <lb/>te ſit accipere puncta <seg type="var">.K.</seg> et <seg type="var">.p.</seg> pro <lb/>xima vel remota ab ipſo <seg type="var">.z.</seg> ad li-<lb/>bitum. </s> <s xml:space="preserve">Vnde abſque vlla diuiſio <lb/>neipſius <seg type="var">.q.b.</seg> per medium, ſatis <lb/>erit ſignare puncta <seg type="var">.K.</seg> et <seg type="var">.p.</seg> dua-<lb/>bus diſtantijs mediantibus <seg type="var">.b.K.</seg> <lb/>et <seg type="var">.q.p.</seg> inuicem æqualibus, & <lb/>etiam propoſitis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0275-02" corresp="fig-0275-02a"> <graphic url="0275-02"/> </figure> </div> </body> </floatingText> <note/> <pb facs="0276" n="264"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Figuram ſuperficialem ellipſi ſimilem, ex datis axibus cir-<lb/>cino mediante delineari poſſe.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">FIguram ſuperficialem ellipſi ſimilem, ex datis axibus, circino mediante delinea <lb/>re cum volueris, ita facito.</s> </p> <p> <s xml:space="preserve">Sit <seg type="var">.e.c.</seg> ſemiaxis maior <seg type="var">.a.e.</seg> verò minor, ad angulum rectum inuicem coniuncti, <lb/>tunc <seg type="var">.a.e.</seg> producatur vſque ad <seg type="var">.o</seg>. </s> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> <seg type="var">.a.o.</seg> maior ſit quam diſtantia inter <seg type="var">.o.</seg> et <seg type="var">.c.</seg> quę <lb/>quidem <seg type="var">.a.o.</seg> poſſet etiam dari, deſcribatur poſtea circulus <seg type="var">.a.d.b.</seg> circa centrum <seg type="var">.o.</seg> à <lb/>quo puncto protrahatur ſemidiameter <seg type="var">.o.b.</seg> quæ cum <seg type="var">.a.o.</seg> angulum rectum conſti-<lb/>tuat, quę <seg type="var">.o.b.</seg> erit æquidiſtans <seg type="var">.e.c.</seg> ex .28. primi, ducatur poſtea <seg type="var">.b.c.d.</seg> et <seg type="var">.o.t.d.</seg> vnde <lb/>angulus <seg type="var">.t.c.d.</seg> ęqualis erit angulo <seg type="var">.o.b.d.</seg> ex .29. eiuſdem. </s> <s xml:space="preserve">ex quinta autem anguli <seg type="var">.b.</seg> <lb/>et <seg type="var">.d.</seg> ſunt inuicem æquales, </s> <s xml:space="preserve">quare etiam <lb/>& anguli <seg type="var">.d.</seg> et <seg type="var">.c.</seg> inuicem ęquales erunt, <lb/> <ptr xml:id="fig-0276-01a" corresp="fig-0276-01" type="figureAnchor"/> & ex .6. eiuſdem <seg type="var">.t.c.</seg> ęqualis erit <seg type="var">.t.d.</seg> duca <lb/>tur poſtea <seg type="var">.d.x.h.</seg> perpendicularis lineæ <seg type="var">.c.<lb/>e.</seg> ita diſtans ſub ipſa <seg type="var">.c.e.</seg> vt arcus circula-<lb/>ris circa <seg type="var">.t.</seg> delineatus ex ſemidiametro <seg type="var">.t.<lb/>d.</seg> aptus ſit eam ſecare, ſumpto poſtea <seg type="var">.r.</seg> <lb/>tam diſtante ab <seg type="var">.e.</seg> vt <seg type="var">.t.</seg> reperitur ab ipſo <lb/>e. et <seg type="var">.z.</seg> ab <seg type="var">.e.</seg> vt <seg type="var">.o.</seg> ab eodem, ducendo po-<lb/>ſtea duos alios arcus magnitudinis <choice><ex>priorum</ex><am>priorũ</am></choice> <lb/>circa centra <seg type="var">.r.</seg> et <seg type="var">.z.</seg> habebimus propoſi-<lb/>tum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0276-01" corresp="fig-0276-01a"> <graphic url="0276-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed cum quis voluerit prius arcus mi-<lb/>norum circulorum delineare circa maio-<lb/>rem axem, fiant cuiuſuis magnitudinis, vt <lb/>in ſecunda figura videre eſt, poſito tamen quod eorum diameter, minor ſit minore <lb/>axe ipſius figurę, quorum circulorum vnus ſit <seg type="var">.c.d.</seg> circa <seg type="var">.t.</seg> eius centrum, deinde in axe <lb/>minori ſumatur <seg type="var">.a.x.</seg> æqualis <seg type="var">.c.t.</seg> & protrahatur <seg type="var">.t.x.</seg> quę per ęqualia diuidatur in pun-<lb/>cto <seg type="var">.n.</seg> à quo poſtea ducatur <seg type="var">.n.o.</seg> ad angulos rectos <lb/> <ptr xml:id="fig-0276-02a" corresp="fig-0276-02" type="figureAnchor"/> cum <seg type="var">.t.x.</seg> vſque ad interſectionem cum <seg type="var">.a.e.</seg> in pun-<lb/>cto <seg type="var">.o.</seg> minori axi producta cum oportuerit, quod <lb/>quidem punctum <seg type="var">.o.</seg> centrum erit arcus <seg type="var">.d.a.</seg> maio-<lb/>ris, eo quod <seg type="var">.o.t.</seg> æqualis eſſet <seg type="var">.o.x.</seg> ex .4. primi Eu-<lb/>cli. </s> <s xml:space="preserve">vnde <seg type="var">.o.d.</seg> æqualis eſſet <seg type="var">.o.a.</seg> & circuli etiam in-<lb/>uicem contingentes in puncto <seg type="var">.d.</seg> ex .11. tertij tam <lb/>in prima, quam in ſecunda figura, ſumpto <choice><ex>denique</ex><am>deniq;</am></choice> <lb/>puncto <seg type="var">.s.</seg> tam remoto ab <seg type="var">.e.</seg> quam <seg type="var">.o.</seg> reperitur ab <lb/>eodem, ipſum, centrum erit alterius arcus oppoſi-<lb/>ti, poſſemus etiam <choice><ex>abſque</ex><am>abſq;</am></choice> diuiſione ipſius, <seg type="var">t.x.</seg> conſti <lb/>tuere angulum <seg type="var">.x.t.o.</seg> <choice><ex>æqualem</ex><am>æqualẽ</am></choice> angulo <seg type="var">.t.x.o.</seg> vnde ex <lb/>6. primi haberemus <seg type="var">.o.t.</seg> æqualem <seg type="var">.o.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0276-02" corresp="fig-0276-02a"> <graphic url="0276-02"/> </figure> </div> </body> </floatingText> <pb facs="0277" n="265"/> <fw type="head">EPISTOLAE.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De inuentione axis propoſite portionis datæ ſphæræ.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">VTaxem propoſitæ alicuius datæ ſphæræ inuenire poſſis ita tibi operandum eſt <lb/>vt gratia exempli. </s> <s xml:space="preserve">Propoſita nobis eſt ſphæra <seg type="var">.c.i.e.t.</seg> diametri cognitæ. </s> <s xml:space="preserve">pro <lb/>poſita etiam eſt nobis eius portio <seg type="var">.n.e.u.</seg> axis <seg type="var">.e.a.</seg> cognitæ minoris ſemidiametro, da-<lb/>ta etiam nobis eſt proportio alterius portionis minoris hemiſphærio <seg type="var">.i.e.t.</seg> ad por-<lb/>tionem <seg type="var">.n.e.u.</seg> quæritur nunc quantus ſit axis <seg type="var">.e.x.</seg> ſecundæ portionis hoc eſt deſidera-<lb/>mus cognoſcere proportionem <seg type="var">.e.x.</seg> ad <seg type="var">.e.a.</seg> vel ad diametrum ipſius ſpheræ.</s> </p> <p> <s xml:space="preserve">Cuius gratia reperiatur primò proportio <choice><ex>circunferentiæ</ex><am>circũferentiæ</am></choice> maioris circuli ipſius <choice><ex>ſphae ræ</ex><am>ſphęræ</am></choice> adeius diametrum, quæ ferè eſt vt .22. ad .7. ex Archimede.</s> </p> <p> <s xml:space="preserve">Quo facto, inueniatur quantitas ſuperficialis huiuſmodi maioris circuli, quæ ſem-<lb/>per æqualis eſt producto quod fit ex ſemidiametro in dimidium circunferentiæ ip-<lb/>fius circuli, ex eodem Archimede. </s> <s xml:space="preserve">Et ſic cognoſcemus quartam partem ſuperficiei <lb/>ſphæricæ ſphærę propoſite ex .31. primi lib. de ſphæra, & cyllindro Archimedis.</s> </p> <p> <s xml:space="preserve">Deinde ſumatur tertia pars producti, quod fit ex ſemidiametro in ſuperficiem <lb/>maioris circuli, & habebimus conum, cuius baſis erit circulus maior, altitudo verò <lb/>ſemidiameter propoſitæ ſphæræ ex .9. duodecimi Eucli.</s> </p> <p> <s xml:space="preserve">Quadruplum poſtea huiuſmodi coni, erit quantitas ſoliditatis, ſeu corporeitas to <lb/>tius ſphærę ex .32. dicti lib. Archimedis.</s> </p> <p> <s xml:space="preserve">Imaginemur poſtea <choice><ex>in</ex><am>ĩ</am></choice> ſphærica portione <seg type="var">.n.e.u.</seg> <choice><ex>lineam</ex><am>lineã</am></choice> <seg type="var">.e.u.</seg> à <choice><ex>summitate</ex><am>sũmitate</am></choice> ad <choice><ex>extremitatem</ex><am>extremitatẽ</am></choice> <lb/>baſis, cuius <seg type="var">.e.u.</seg> quantitatem cognoſcemus, hoc modo ſcilicet, fumendo <choice><ex>radicem</ex><am>radicẽ</am></choice> qua-<lb/>dratam producti <seg type="var">.c.e.</seg> in <seg type="var">.e.a.</seg> eo quod <lb/>quadratum <seg type="var">.e.u.</seg> æquale eſt quadrato <lb/> <ptr xml:id="fig-0277-01a" corresp="fig-0277-01" type="figureAnchor"/> <seg type="var">a.u.</seg> & quadrato <seg type="var">.a.e.</seg> ex penultima <lb/>primi Eucli. </s> <s xml:space="preserve">hoc eſt producto quod <lb/>fit ex <seg type="var">.c.a.</seg> in <seg type="var">.a.e.</seg> ex .34. tertij <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice>, <lb/>& quadrato <seg type="var">.a.e.</seg> hoc eſt producto, <lb/>quod fit ex <seg type="var">.c.e.</seg> in <seg type="var">.e.a.</seg> ex .3. ſecundi <lb/>eiuſdem.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0277-01" corresp="fig-0277-01a"> <graphic url="0277-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Inuenta poſtea <seg type="var">.e.u.</seg> ponamus eam <lb/>vnius circuli ſemidiametrum eſſe, cu <lb/>ius ſuperficialis quantitas etiam inue <lb/>niatur, vt ſupra dictum eſt, quæ qui <lb/><choice><ex>dem</ex><am>dẽ</am></choice> æqualis erit ſuperficiei portionis <lb/><seg type="var">n.e.u.</seg> ex .40. primi li. </s> <s xml:space="preserve">Archimedis de <lb/>ſphæra, & cyllindro.</s> </p> <p> <s xml:space="preserve">Hæc autem quantitas vltimo <choice><ex>inuem</ex><am>inuẽ</am></choice> <lb/>ta multiplicetur cum tertia parte ſe-<lb/>midiametri datæ ſphæræ, & habebi-<lb/>mus ſoliditatem vnius coni æqualis <lb/>aggregato ſoliditatis portionis <seg type="var">.n.e.<lb/>u.</seg> ſimul ſumptę<unclear reason="illegible"/>, <choice><ex>cum</ex><am>cũ</am></choice> ſoliditate vnius co <lb/>ni, cuius axis ſit <seg type="var">.a.o.</seg> <choice><ex>reſiduum</ex><am>reſiduũ</am></choice> ſemidia-<lb/>metri noſtræ ſphæræ dempta <seg type="var">.a.e.</seg> ba <pb facs="0278" n="266"/><fw type="head">IO. BABPT. BENED.</fw> ſis verò eadem quæ eſt portionis, cuius diameter eſt <seg type="var">.n.u.</seg> ex .9. 12. Eucli. & ex .42. id-<lb/>eſt vltima primi Archimedis de ſphæra, & cyllindro.</s> </p> <p> <s xml:space="preserve">Nunc autem ex hoc aggregato iam vltimo dicto detrahatur conus, cuius <seg type="var">.o.a.</seg> eſt <lb/>axis et <seg type="var">.n.u.</seg> diameter baſis, qui quidem conus nobis cognitus eſt, cum <seg type="var">.a.n.</seg> ſemidia-<lb/>meter eius baſis, nobis cognita ſit ex .34. 3. Eucli. </s> <s xml:space="preserve">& ſic quantitas eius baſis, & ita ter-<lb/>tia pars <seg type="var">.a.o.</seg> eius axis, quę multiplicata cum dicta baſi, cuius <seg type="var">.n.u.</seg> eſt diameter, produ <lb/>cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-<lb/>gre gato, relinquet nobis ſoliditatem portionis <seg type="var">.n.e.u.</seg> vnde cognoſcemus proportio <lb/>nem iſtius portionis ad totam ſphæram propoſitam.</s> </p> <p> <s xml:space="preserve">Sed cum nobis propoſita ſit proportio portionis <seg type="var">.n.e.u.</seg> ad portionem <seg type="var">.i.e.t.</seg> cogno <lb/>ſcemus etiam ſoliditatem huius ſecundę portionis <seg type="var">.i.e.t.</seg> & ſimiliter <choice><ex>proportionem</ex><am>proportionẽ</am></choice> hu-<lb/>ius ad totam ſphęram, & ad <choice><ex>reſiduum</ex><am>reſiduũ</am></choice> <choice><ex>etiam</ex><am>etiã</am></choice> ipſius ſphęrę hoc eſt portioni <seg type="var">.i.c.t</seg>.</s> </p> <p> <s xml:space="preserve">Protrahatur nunc diameter <seg type="var">.c.e.</seg> à parte <seg type="var">.e.</seg> <choice><ex>vſque</ex><am>vſq;</am></choice> quo <seg type="var">.e.f.</seg> æqualis ſit <seg type="var">.e.o.</seg> ſemidiame <lb/>tro ſphęrę, quæ quidem <seg type="var">.f.e.</seg> diuidatur in puncto <seg type="var">.h.</seg> ita vt proportio <seg type="var">.f.h.</seg> ad <seg type="var">.h.e.</seg> æqua-<lb/>lis ſit proportioni portionis <seg type="var">.i.c.t.</seg> ad portionem <seg type="var">.i.e.t.</seg> quod quidem hoc modo efficie<lb/>tur. </s> <s xml:space="preserve">applicabimus lineam <seg type="var">.f.q.</seg> (indeterminatam) cum <seg type="var">.f.e.</seg> ad quemuis angulum in <choice><ex>pun- cto</ex><am>pũ-cto</am></choice> <seg type="var">.f.</seg> in qua accipiemus duas lineas <seg type="var">.f.p.</seg> et <seg type="var">p.q.</seg> inuicem ita relatas, vt ſe habent in pro <lb/>portione duæ iam dictæ portiones, hoc eſt, vt <seg type="var">.i.c.t.</seg> portio ad portionem <seg type="var">.i.e.t.</seg> ducen <lb/>do poſtea <seg type="var">.q.e.</seg> et <seg type="var">.p.h.</seg> parallelam ad ipſam <seg type="var">.q.e.</seg> diuiſam habebimus <seg type="var">.f.e.</seg> in eadem pro <lb/>portione vt dictum eſt ex .2. ſexti, & .11 quinti Euclidis, vnde <seg type="var">.c.e</seg>: <seg type="var">e.f.</seg> et <seg type="var">.f.h.</seg> nobis co <lb/>gnitę erunt.</s> </p> <p> <s xml:space="preserve">Oportebit nos nunc cognoſcere quantitatem <seg type="var">.c.x.</seg> hoc modo, videlicet, quęramus <lb/>quadratum, cuius <seg type="var">.c.x.</seg> eius ſit radix, cui quadratum lineę <seg type="var">.c.e.</seg> cognitum, ita ſit propor-<lb/>tionatum, vt eſt linea <seg type="var">.x.f.</seg> ad lineam <seg type="var">.f.h.</seg> quę nobis cognita eſt, quod rectè factum erit <lb/>ex eo, quod ſcripſit Archimedes in .4. ſecundi de ſphęra, & cyllindro.</s> </p> <p> <s xml:space="preserve">Sed quia Archimedes eo in loco ſupponit id, quod necipſe, nec alius adhuc inue <lb/>nit, niſi via naturali, hoc eſt tres partes ęquales ex proportione data effici, non erit in <lb/>conueniens etiam nobis hac via, circa hoc aliquid dicere.</s> </p> <p> <s xml:space="preserve">Accipiemus igitur diametrum <seg type="var">.c.e.</seg> cum addita <seg type="var">.e.f.</seg> eius ſemidiametro, diuidemus<lb/>q́ue <seg type="var">.f.e.</seg> in puncto <seg type="var">.h.</seg> vt ſupra factum fuit, applicabimus poſtea <seg type="var">.c.m.</seg> indeterminatam <lb/>angulariter ad <seg type="var">.c.e.</seg> à qua <seg type="var">.c.m.</seg> accipiemus <seg type="var">.c.g.</seg> æqualem <seg type="var">.f.h.</seg> quęremus deinde natu-<lb/>rali via punctum <seg type="var">.b.</seg> ita ut protrahendo à puncto <seg type="var">.e.</seg> (altero extremo diametri) <seg type="var">e.m.</seg> pa <lb/>rallelam ad <seg type="var">.b.g.</seg> ductam, erigendo <seg type="var">.b.d.</seg> perpendicularem ad <seg type="var">.c.e.</seg> in puncto <seg type="var">.b.</seg> protra <lb/><choice><ex>ctaque</ex><am>ctaq́;</am></choice> <seg type="var">.d.c.</seg> quæ à diametro <seg type="var">.e.c.</seg> deducta ab <seg type="var">.c.</seg> incohando vſque ad <seg type="var">.x.</seg> relinquat nobis <seg type="var">.<lb/>x.f.</seg> ęqualem <seg type="var">.c.m</seg>.</s> </p> <p> <s xml:space="preserve">Cuius rei ratio eſt, quia quadratum <seg type="var">.c.e.</seg> ſe habet ad quadratum <seg type="var">.c.d.</seg> vt <seg type="var">.c.e.</seg> ad <seg type="var">.c.<lb/>b.</seg> ex .4. et .18. ſexti Eucl. </s> <s xml:space="preserve">ſed ex .4. ita ſe habet <seg type="var">.m.c.</seg> ad <seg type="var">.c.g.</seg> vt <seg type="var">.e.c.</seg> ad <seg type="var">.b.c.</seg> & cum ſit <seg type="var">.c.<lb/>g.</seg> ęq alis <seg type="var">.f.h.</seg> ſi <seg type="var">.c.m.</seg> ęqualis fuerit <seg type="var">.f.x.</seg> habebimus propoſitum. </s> <s xml:space="preserve">Quod ſi quis per di-<lb/>ſcretum vel et hoc facere, ita ei agendum erit.</s> </p> <p> <s xml:space="preserve">Ponamus exempli gratia totum diametrum <seg type="var">.c.e.</seg> propoſitæ ſphæræ eſſe ut decem, <lb/><choice><ex>proportionemque</ex><am>proportionemq́;</am></choice> reſiduę portionis <seg type="var">.i.c.t.</seg> ad ſecundam <seg type="var">.i.e.t.</seg> hoc eſt <seg type="var">.f.h.</seg> ad <seg type="var">.h.e.</seg> ſeſqui-<lb/>alteram eſſe, vnde <seg type="var">.e.h.</seg> bis tertia erit ìpſius <seg type="var">.f.h.</seg> <choice><ex>totaque</ex><am>totaq́;</am></choice> linea <seg type="var">.c.f.</seg> erit .15. et <seg type="var">.f.h.</seg> erit .3. <lb/>& quadratum lineæ <seg type="var">.c.e.</seg> erit .100.</s> </p> <p> <s xml:space="preserve">Quærendo poſtea quadratum lineæ <seg type="var">.c.x.</seg> cui quadratum <seg type="var">.c.e.</seg> hoc eſt .100. ita pro-<lb/>portionatum ſit vt <seg type="var">.f.x.</seg> ad <seg type="var">.f.h.</seg> hoc eſt ad .3. ſi autem cogitauerimus <seg type="var">.c.x.</seg> eſſe nouem <lb/>partium talium qualium <seg type="var">.c.e.</seg> eſt decem, eius quadratum erit .81. et <seg type="var">.x.f.</seg> erit .6. par-<lb/>tium talium qualium <seg type="var">.c.f.</seg> eſt .15. dicendo poſtea ſi .100. dat .81. (ex regula de tribus) <pb facs="0279" n="267"/><fw type="head">EPISTOL AE.</fw> <seg type="var">x.f.</seg> hoc eſt .6. dabit .4. integra cum <num value="86">.<lb/> <ptr xml:id="fig-0279-01a" corresp="fig-0279-01" type="figureAnchor"/> 86.</num> centeſimis, ſed nos vellemus no-<lb/>bis prouenire tria, eo <choice><ex>quod</ex><am>ꝙ</am></choice> ita eſt <seg type="var">.f.h.</seg> <lb/>qua propter deſcendere nos oporte-<lb/>bit à nouem ad .8. & ab .8. ad .7. & à. <lb/>7. ad .6. </s> <s xml:space="preserve">tunc inueniemus <seg type="var">.c.x.</seg> oporte-<lb/>re eſſe circiter quinque cum duabus <lb/>tertijs, <choice><ex>operando</ex><am>operãdo</am></choice> poſtea ex regula de <lb/>tribus, ſi dixerimus quando .100. no-<lb/>bis dat .32. cum nona parte integri, <lb/></s> <s xml:space="preserve">tunc nouem cum tertia parte integri <lb/>dabit .2. <choice><ex>cum</ex><am>cũ</am></choice> 296. de .300. hoc eſt .2. <lb/>cum circa .49. quinquageſimis, quæ <lb/>quidem quantitas, cum propinquiſſi <lb/>ma ſit lineæ <seg type="var">.f.h.</seg> trium integrorum di <lb/>cemus <seg type="var">.c.x.</seg> eſſe quinque integrorum <lb/>cum duabus tertijs partibus vnius in <lb/>tegri, et <seg type="var">.e.x.</seg> reſiduum, hoc eſt axem <lb/>quæſitum portionis <seg type="var">.i.e.t.</seg> eſſe circa .4 <lb/>integra cum tertia parte vnius inte-<lb/>gri.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0279-01" corresp="fig-0279-01a"> <graphic url="0279-01"/> </figure> </div> </body> </floatingText> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE ERRORIBVS THOMAE PORCACHII <lb/>& Benedicti Bordonij in eorum inſularijs.</head> <head rend="italics" xml:space="preserve">Excellentißimo D. lo. Baptiſtæ Fæmello Ciui <choice><ex>Decurionique</ex><am>Decurioniq́ꝫ</am></choice> Tau-<lb/>rinenſi Philoſopho, Medico, & in Accademia eius <lb/>Ciuitatis Medicinæ Practicæ Ordinario, <choice><ex>Pri- marioque</ex><am>Pri-marioq́ꝫ</am></choice> profeßori celeberrimo.</head> <p> <s xml:space="preserve">DIj perdant tuas adeo moleſtas, & aſſiduas curas, quæ te nimis à ſuauiori-<lb/>bus ſtudijs diſtrahunt, & à nobis longius abducunt. </s> <s xml:space="preserve">Nam, ut tibi <choice><ex>quietem</ex><am>quietẽ</am></choice>, <lb/>ita mihi ingentem adimunt voluptatem. </s> <s xml:space="preserve">Sed ne in aliquo erga te defi-<lb/>cere videar, quæ tibi olim promiſi, nunc mitto.</s> </p> <p> <s xml:space="preserve">Negari quidem non poteſt, quin fuerit laborioſum opus Porcachij, & Benedicti <lb/>Bordonij, hoc eſt inſularium, qui rectè etiam feciſſent, cum loqui eos oportebat de <lb/>terminis ſphæræ ratione ſitus locorum, ſi ſeipſos alicuius excellentis Coſmographi <lb/>conſilio ſubmiſiſſent. </s> <s xml:space="preserve">Conſidera quæſo, quomodo admitti poſſit, id quod ait Por-<lb/>cachius initio ſui operis, ideſt Iſlandiam ſub Polo arctico iacere, inter auſtrum, & <lb/>boream: </s> <s xml:space="preserve">omittamus etiam quod idem in Proęmio lib. ſecundi, vbi ait Biarmiam, <lb/>(& non Iſlandiam) eſſe ſub dicto polo arctico: </s> <s xml:space="preserve">in <choice><ex>eodemque</ex><am>eodemq́;</am></choice> principio repetit ipſam <lb/>Iſlandiam inter auſtrum, & boream per centum leucas Germanicas extendi, dein-<lb/>de verſus occidentem, ea duo ſtupenda miracula conſpici. </s> <s xml:space="preserve">Vide quæſo, quomodo <lb/>incolę ſub aliquo ex polis, habere poſſint occidentem, orientem, magiſtrum, <choice><ex>auſtrum</ex><am>auſtrũ</am></choice>, <pb facs="0280" n="268"/><fw type="head">IO. BAPT. BENDE.</fw> & boream, & vt melius dicam aliquem rhombum. </s> <s xml:space="preserve">Sed quomodo fieri poteſt, vt in-<lb/>ſula Iſlandiæ ſit ſub polo, eius tamen dies, & nox maior non ſit longior ſpatio <choice><ex>trium</ex><am>triũ</am></choice> <lb/>menſium? </s> <s xml:space="preserve">vt ipſe pagina .62. in proęmio ſecundi lib. affirmat, quamuis hoc à Bordo <lb/>ne deſumat. </s> <s xml:space="preserve">In quo <choice><ex>vterque</ex><am>vterq;</am></choice> fallitur, <choice><ex>ſentientes</ex><am>ſentiẽtes</am></choice> huiuſmodi diem ab ingreſſu Solis, in <lb/>principium geminorum incipere, & in egreſſu à Leone terminari, ideſt à .12. Maij ad <lb/>14. Auguſti, quaſi ſi ab æquatore finis Leonis ita declinaret, vt principium gemino-<lb/>rum, & finis Aquarij, vt initium Sagittarij, nam ratio poſtulat, tantum de-<lb/>clinari ab æquatore finem quantum initium diei, vbi maximus dies .24. horas ex <lb/>cedit, & ſic dico de noctibus: </s> <s xml:space="preserve">vnde in huiuſmodi regione, vbi per tres menſes conti <lb/>nuos Sol radios emittit, huiuſmodi dies à medietate Tauri incipit, & in medietate <lb/>Leonis terminatur, quæ quidem loca æqualem declinationem habent, & ſic nox <lb/>trium menſium incipit à medietate Scorpionis, & in medietate Aquarij, eadem ra-<lb/>tione finitur.</s> </p> <p> <s xml:space="preserve">Septima verò pag. <choice><ex>idem</ex><am>idẽ</am></choice> ait, dies ſolſtitiales eſſe circa .24. Iunij. </s> <s xml:space="preserve"><choice><ex>quod</ex><am>Qđ</am></choice>, an <choice><ex>tunc</ex><am>tũc</am></choice> eſſet <choice><ex>verum</ex><am>verũ</am></choice>, <lb/>tu ipſe videto. </s> <s xml:space="preserve">Is præterca modus <choice><ex>quem</ex><am>quẽ</am></choice> ad <choice><ex>inueniendum</ex><am>inueniẽdũ</am></choice> <choice><ex>orientem</ex><am>orientẽ</am></choice>, & <choice><ex>occidentem</ex><am>occidẽtem</am></choice> præſcribit <lb/>in eodem proęmio pag .63. eſt tædioſus, cum ſemper expectare nos cogat æquino-<lb/>ctij tempus, cum alij modi reperiantur breuiores, qui in qualibet reuolutione primi <lb/>mobilis obſeruari poſſunt, quorum vnus erit mediante inuentione lineæ meridiane <lb/>orizontalis, eo modo, quo ſcriptum eſt ab antiquis mediante Sole, aut Luna, quæ <lb/>luminaria in quolibet alio loco, <choice><ex>præterquam</ex><am>præterquã</am></choice> ſub polo efficiunt, vt extremitas vmbræ <lb/>rectæ <choice><ex>gnomonum</ex><am>gnomonũ</am></choice> <choice><ex>gyrum</ex><am>gyrũ</am></choice> <choice><ex>oxigonium</ex><am>oxigoniũ</am></choice>, ſeu <choice><ex>eclipticum</ex><am>eclipticũ</am></choice> ducat, ideſt in ijs locis, <choice><ex>quorum</ex><am>quorũ</am></choice> zenit eſt <lb/>inter polum, & circulum arcticum, quemadmodum facit, vt alijs, exiſtentibus ipſis <lb/>luminaribus extra æquatorem, & circulos arcticos gyrum hyperbolicum reddant. <lb/></s> <s xml:space="preserve">Sed id quod eidem Porcachio impoſſibile eſſe apud eos, qui habitant ſub polo vi-<lb/>detur, ideſt vt multis rationibus, vt ipſe dicit, fieri non poſſit, ut fiat immediata quę <lb/>dam, & ſubita mutatio à continuo die ad continuam noctem abſque eo quod ijs, <lb/>ſaltem ſemel conceſſa ſint dies, & nox terminata duodecim horarum, eſt magis ad <lb/>mirandum impoſſibile, quod imaginari poſſimus, nam neceſſarium eſſet, ut orizon-<lb/>habitatorum ſub polo ſecaret æquatorem contra id, quod ſuperius admiſerat, id-<lb/>eſt <choice><ex>orizontem</ex><am>orizõtẽ</am></choice> Biarmiæ, eſſe <choice><ex>eundem</ex><am>eũdẽ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> circulo æquinoctiali. </s> <s xml:space="preserve">Vide etiam quid is ab anti-<lb/>quis colligat, loquens de iis, quæ in inſula Taprobana ad finem pag .186. admirabi <lb/>lia ſunt, ſcribens eiuſdem inſulę habitatoribus, Lunam ſuper terram non apparere <lb/>ab octauo uſque ad decimumſextum diem: </s> <s xml:space="preserve">pręter quam, quod etiam ſcribit, in <lb/>eadem inſula, tramuntanam non uideri, quod falſum eſt, quia hæc à polo arctico <lb/>circiter quatuor gradibus diſtat noſtris temporibus. </s> <s xml:space="preserve">unde ab ijs qui ſunt ſub æqua-<lb/>tore, cum ea ſupra orizontem eſt, conſpici poteſt, cum ijſdem ſingulis diebus oria-<lb/>tur, & occidat. </s> <s xml:space="preserve">Idem etiam pro re admirabili ſcribit, uideri Canopum, qui à po-<lb/>lo antarctico plus quam quadraginta gradibus diſtat.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De erroribus Lucilli Philalthæi.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod Lucillus Philalthęus tam eximius Mathematicus ſit, ut ipſum Anto-<lb/>nius Berga facit, ego quidem non uideo. </s> <s xml:space="preserve">In ſuis enim commentariis de <lb/>Cœlo, dicit primum, Pyramidem, quę inter corpora regularia primum locum tenet <pb facs="0281" n="269"/><fw type="head">EPISTOL AE.</fw> ſex baſibus conſtare, pag.15. 583. 632. et .647. </s> <s xml:space="preserve">Omitto errorem ab eodem com-<lb/>miſſum in fine pag .39. ubi oleum grauius eſſe quam aquam fatetur, cum id ad res <lb/>mathematicas non ſpectet: </s> <s xml:space="preserve">Omitto etiam quod idem neget aſtrologiam pag .74. <lb/>79. & quod etiam dicat pag .89. </s> <s xml:space="preserve">Deum eſſe ad orientem, non conſiderans aliqui-<lb/>bus populis noſtrum orientem eſſe occidentem.</s> </p> <p> <s xml:space="preserve">Quod idem ait pag .241. </s> <s xml:space="preserve">Aſtrologiam eſſe antiquiorem Aſtronomia eſt falſiſſi-<lb/>mum, quia iudiciaria ſemper præſupponit cognitionem ſitus ſtellarum, quæ ab A-<lb/>ſtronomia petitur. </s> <s xml:space="preserve">Mouebit tibi riſum quod ait pag .307. his verbis.</s> </p> <quote> <s xml:space="preserve">Verum propriè media dicitur illa, quæ rectam ſphæram omninò habet, quæ eun <lb/>dem polum orizontis & mundi obtinet, quæ orizontem habet diuidentem <choice><ex>ſphæram</ex><am>ſphærã</am></choice> <lb/>æquè ſecundum angulos rectè.</s> </quote> <p> <s xml:space="preserve">Paulo inferius continuans ſermonem de ſphæra recta, ait.</s> </p> <quote> <s xml:space="preserve">Et niſi tumor terræ, & gibum eſſet, ijs perpetuus eſſet dies ſine nocte.</s> </quote> <p> <s xml:space="preserve">Linea verò .56. ait habitatores ſphęrę rectè habere .4. ſolſtitia, ſeſe ipſum huius <lb/>rei planè ignarum prodens .310. autem pag. ſic ſcribit.</s> </p> <quote> <s xml:space="preserve">Quoniam repercutiuntur radij, & peridem centrum tranſeunt, ob id ſtupam ap <lb/>poſitam centro radius accendit.</s> </quote> <p> <s xml:space="preserve">Quem quidem errorem ab Euclide deſumit, et .15. linea pag .636. repetit.</s> </p> <p> <s xml:space="preserve">Si vis ridere, legito .16. primas lineas .357. pag. </s> <s xml:space="preserve">Quod idem deinde dicat circa fi-<lb/>nem 396. pag. lucem eſſe ſubſtantiam corporis lucidi & corpoream, ſubijciam tuo <lb/>iudicio, vt etiam quod ait .397. pag. his vetbis vtens.</s> </p> <quote> <s xml:space="preserve">Idcirco animalia illa, quæ nocte vagantur perpolita, dum volant, aerem terunt <lb/>nocturnum, & fulgent.</s> </quote> <p> <s xml:space="preserve">Et pag .398.</s> </p> <quote> <s xml:space="preserve">Multitudo radiorum non admodum facit ad excitandum calorem ſi ſolum inci-<lb/>dat ſine repercuſſu, neorecta incidere iuuerit.</s> </quote> <p> <s xml:space="preserve">Quod falſum eſt cum radius incidens longè magis quam reflexus calefaciat. </s> <s xml:space="preserve">In fi <lb/>ne autem .405. ſic ſcribit.</s> </p> <quote> <s xml:space="preserve">Sol in ortu & in occaſu longius apparet, iccircò reuolui creditur. </s> <s xml:space="preserve">Hinc etiam in <lb/>abſide ſtare putatur, & in oppoſito abſidis, vnde ſolſtitia vocant, ſed nobis in Can-<lb/>cro, antipodibus verò in Capricorno tum Sol abeſſe longius apparet vtriſque.</s> </quote> <p> <s xml:space="preserve">An hoc quid peius dici poteſt? </s> <s xml:space="preserve">Circa vero .40. lineam pag .459. ſic ſcribit.</s> </p> <quote> <s xml:space="preserve">Si enim alij planetæ, & ſtellæ fixę reciperent à Sole lumen, dum accederent ad So <lb/>lem, vel recederent, aut contra, Sol ad eas appropinquaret, & abſcederet, eaſdem-<lb/>lucis viciſſitudinis ſubiret, quas Luna.</s> </quote> <quote> <s xml:space="preserve">Hoc autem nondum depręhenſum eſt, quin etiam Mercurius, Venus, ſuo interpo <lb/>ſitu, Solem occultarent nobis, vt Luna.</s> </quote> <quote> <s xml:space="preserve">Paulo inferius ſic ait. </s> <s xml:space="preserve">Rurſus æquè Saturnus, Jupiter, Mars, ſubire deliquium, <lb/>more Lunæ, aut ſaltem obiectu terræ inter Solem & ipſos, quia tum ob interpoſitam <lb/>terram non poſſent haurire lumen à Sole.</s> </quote> <p> <s xml:space="preserve">Hæc verò omnia, talia ſunt, qualia ab ijs qui incipiunt intelligere ſphæram non <lb/>proferrentur. </s> <s xml:space="preserve">Omittamus, quod ait deinde.</s> </p> <quote> <s xml:space="preserve">Accedit quod ſi aſtra lumen à Sole acciperent eiuſdem caloris eſſent. </s> <s xml:space="preserve">Itaque om <lb/>nia ſiccarent, & nulla eſſent frigidæ conſtitutionis contra Aſtrologos.</s> </quote> <p> <s xml:space="preserve">Quia hac ratione, Luna, quæ negari non poteſt, quin ab ipſo Sole <choice><ex>lumem</ex><am>lumẽ</am></choice> accipiat, <lb/>eiuſdem caloris eſſet cum eodem Sole. </s> <s xml:space="preserve">Sunt ea etiam ridenda, quæ idem ait pag.<lb/>460. lineis .18. 19. 23. 26. 27. 29. quaſi ea lux infinita (vt ita dicam) magni Solis, non <pb facs="0282" n="270"/><fw type="head">IO. BAPT. BENED.</fw> in alium finem ſit effecta quam ad illuminandam ſuperficiem huius excrementi ip-<lb/>ſius vniuerſi ad vtilitatem hominum, imò, vt rectius dicam, <choice><ex>animalium</ex><am>animaliũ</am></choice>. </s> <s xml:space="preserve">vide etiam <lb/>pag .632. et .633. vbi Ariſtotelem de implendo loco non intellexit, cum citet ſphæ-<lb/>ram, loco pyramidis, & inter .46. et .47. lineas dicat <choice><ex>quadratum</ex><am>quadratũ</am></choice> eſſe quid multiplex, <lb/>cum ſit vnicum tantum in ſpecie, quia ſpecies eſt quadrilateri, & quadranguli, ſed <lb/>vbi in .6. linea pag .633. ait.</s> </p> <p> <s xml:space="preserve">Item hexagonus.</s> </p> <p> <s xml:space="preserve">Magnum errorem committit, vt etiam cum .12. linea .636. pag. ſcribens.</s> </p> <p> <s xml:space="preserve">Pyramis, ſiue planum, ſiue ſolidum, habet acutiſſimum, & in .2. libr. de anima <lb/>pag .215. dicat de die poſſe videri ſtellas in ſpeculo poſito in vaſe aqua pleno, quod <lb/>reuera eſt valde abſurdum. </s> <s xml:space="preserve">Alios eiuſdem errores tibi non patefacio, quia iam ni-<lb/>hil amplius otij mihi eſt, ſed eos tu ipſe perſpicere, & cognoſcere facilè poteris, & <lb/>multò plures quidem, quam putas.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head rend="italics" xml:space="preserve">Cur maius lumen extenuet minus.</head> <head xml:space="preserve">PIRRO DE ARZONIS.</head> <p> <s xml:space="preserve">EX tuis literis intellexi id, quod etiam ſine ijs exploratum mihi erat. </s> <s xml:space="preserve">Sed conce <lb/>do tantum eſſe dicere vbi eſt maius lumen, minus non diſcerni, quantum inter <lb/>diu ſtellas non videri: </s> <s xml:space="preserve">immo eſt etiam magis vniuerſale, quia idem multis aliis lumi-<lb/>nibus, præter ea quæ ſunt ſtellarum, ea ratione contingit, quia ingrediente per pupil-<lb/>lam, tam lumine maiori, quam minori, reflexum ipſius maioris in oculo, in ſitu mino <lb/>ris, efficit, vt ipſum minus confundatur, & diſtingui nequeat, quemadmodum aper-<lb/>te cognoſci poteſt in aliquo cubiculo, cuius parietes dealbati ſint, in quo, vnicum <lb/>tantum ſit exiguum foramen, per quod aliqua lumina reflexa ab obiectis extrinſecis <lb/>intra ipſum cubiculum ingredi poſſint, vnde imagines <choice><ex>obiectorum</ex><am>obiectorũ</am></choice> in parietibus con-<lb/>ſpiciuntur, ſed ſi per idem foramen ingrederetur etiam primarius radius Solis, re-<lb/>flexus huiuſmodi radij efficeret, vt dictæ imagines, magis aut minus euaneſcerent, <lb/>prout dictus reflexus radij ſolaris, maiori, minoríue vi polleret.</s> </p> <p> <s xml:space="preserve">Ad hoc tamen propoſitum, nolo tibi ſilentio inuolui mirabilem quendam effe-<lb/>ctum eiuſmodi rei. </s> <s xml:space="preserve">Hoc eſt vt fiat foramen illud rotundum, magnitudinis tamen <lb/>vnius ſpecilli, quod foramen obturetur mediante vno illorum ſpecillorum, quæ pro <lb/>ſenibus (non breuis viſionis) conficiuntur, hoc eſt quorum ambæ ſuperficies con <lb/>uexæ ſunt, non autem concauæ. </s> <s xml:space="preserve">Deinde opponatur folium album papiri, adeo di <lb/>ſtans à foramine, vt extrinſeca obiecta in eo appareant. </s> <s xml:space="preserve">Quæ quidem obiecta ſi à <lb/>Sole illuſtrata fuerint, tam clara, & diſtincta videbuntur, vt nihil pulchrius dele-<lb/><choice><ex>ctabiliusque</ex><am>ctabiliusq́;</am></choice> videri poterit, inuerſa tamen. </s> <s xml:space="preserve">Sed ſi ea directa videre voluerimus. </s> <s xml:space="preserve">hoc <lb/>optimè faciemus, mediante reflexione alicuius ſpeculi plani.</s> </p> <pb facs="0283" n="271"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head rend="italics" xml:space="preserve">Cur byems valde frigida ſequatur ac<unclear reason="illegible"/>tatem in qua <lb/>calor viguerit.</head> <head xml:space="preserve">NOBILISSIMO, NECNON INGENIOSISSIMO <lb/>Gabrieli Buſchæ, Mediolanenſi.</head> <p> <s xml:space="preserve">QVod dixi hyemem valde frigidam ſequi <choice><ex>æſtatem</ex><am>æſtatẽ</am></choice>, in qua calor viguerit, inde na <lb/>ſcitur, quia calor terrę, aquæ, & aeris, non eſt naturalis horum corporum, vt <lb/>eſt frigus, cum calor à Sole procedat, qui ea calefacit ſuo lumine, vnde quod æſtate <lb/>Sol præter modum calefaciat <choice><ex>terram</ex><am>terrã</am></choice>, ideo <choice><ex>contingit</ex><am>cõtingit</am></choice>, quod minora <choice><ex>impedimenta</ex><am>impedimẽta</am></choice> contra <lb/>ria ſortiatur, & cum eandem poſtea deſerit, ad aliam partem æquatoris <choice><ex>tranſmigrans</ex><am>tranſmigrãs</am></choice> <lb/>terra ad ſuam qualitatem reddit, maiori cum impetu, eo modo, quo res in mo-<lb/>tibus localibus naturalibus, qui etiam terminos ſibi pręfixos, & conſtitutos exce-<lb/>dunt, hinc etiam hyeme fit glacies, ex calefacta prius aqua, quæ durior poſtea eſt <lb/>atque frigidior alia. </s> <s xml:space="preserve">Aeſtas etiàm quæ ſequitur hyemem valde frigidam, non <lb/>erit admodum calida, quia Sol inueniens contrarium naturale valde potens, non <lb/>tam facile illud pellere poteſt, vnde etiam ſi in Geminis, Cancro, & Leone, moram <lb/>trahat, non ſufficit tamen ut magnum calorem imprimere poſſit. </s> <s xml:space="preserve">Vnde ſequitur duas <lb/>æſtates quarum una ſequatur aliam, in eodem loco, uehementi calore præditas eſ-<lb/>ſe non poſſe, quemadmodum nec duas hyemes exceſſiuo frigore, remotis tamen <lb/>accidentibus uentorum, pluuiarum, & niuium.</s> </p> </div> <div type="letter"> <head xml:space="preserve">QVOD MALE SENSERIT NICOLAVS TARTA-<lb/>lea circa attractionem machinæ tormentalis.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">EFfectus, quem ſcribit Tattalea quęſito quinto primi lib. necnon quæſito <lb/>21. et .24. maxima cum ratione eſſe uidetur, non tamen ea quam ipſe in <lb/>quinto profert, quia uerum non eſt, vt quanto aliquid fit calidius, <choice><ex>tanto</ex><am>tãto</am></choice> ue-<lb/>hementius attrahat, eo quod ſi etiam huiuſmodi res, in eodem calore, in <lb/>quo ſemel reperitur, firma maneret; </s> <s xml:space="preserve">neque attraheret, neque aliquid impelleret. <lb/></s> <s xml:space="preserve">Nam dum aliquod corpus calefit, dilatatur, & per conſequens circumcirca <choice><ex>undique</ex><am>undiq;</am></choice> <lb/>trudit, & partes uaſis debiliores cedunt. </s> <s xml:space="preserve">dum uerò dictum corpus re frigeratur, re-<lb/>ſtringitur, & dum in unum cogitur, ſi reperiatur in uaſe, quod aer, aqua, aut aliud <lb/>aliquod corpus ingredi nequeat, dictum uas à quo circundatur frangit, ne aliqua <lb/>pars loci uacua remaneat, ſed ſi aliquod corpus ingredi poteſt, illud ipſum ad ſe at-<lb/>trahit, quemadmodum uidere licet in cucurbitulis. </s> <s xml:space="preserve">Vnde ſequitur eam propoſi-<lb/>tionem, qua dicitur, calidi eſt attrahere, ueram non eſſe, quia ſi hoc fieret, quanto <lb/>aliquid calidus efficeretur, tanto magis attraheret, & ècontra, cum tamen planè <lb/>contrarium appareat, cum quanto magis aliquid calefit, tanto uehementius impel-<lb/>lat, & quanto magis frigefit, tanto plus attrahat. </s> <s xml:space="preserve">Quapropter uerius dicemus, fri-<lb/>gidi eſſe attrahere, calidi uerò expellere, quamuis per accidens. </s> <s xml:space="preserve">Ex quo ſequitur, ut <lb/>quanto calidior facta fuerit materia aliqua, aliquo loco determinata, redeundo po-<lb/>ſtea ad ſuam priorem frigiditatem, tanto minori loco indigeat, ſimiliter etiam <lb/>è conuerſo accidit, ut quanto frigidior <choice><ex>reperitur</ex><am>reꝑitur</am></choice> talis materia, tanto maioriloco, po- <pb facs="0284" n="272"/><fw type="head">IO. BAPT. BENED.</fw> ſtea egeat ipſa ualde calefacta. </s> <s xml:space="preserve">Quod Tartalea in quinto quęſito non animaduer-<lb/>terat.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Solutiones aliqua, circa altimetriam.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">TVas literas accepi, <choice><ex>tuasque</ex><am>tuasq́;</am></choice> dubitationes conſideraui, quas quidem non inutiles <lb/>inueni, quo uerò ad primam, dico te oportere illud Theorema ſpeculari or <lb/>dine huiuſmodi methodi, uidelicet quod <choice><ex>quotieſcunque</ex><am>quotieſcunq;</am></choice> habuerimus <choice><ex>angulum</ex><am>angulũ</am></choice> <choice><ex>aliquem</ex><am>aliquẽ</am></choice> <lb/>cuiufuis amplitudinis, puta <seg type="var">.A.R.V.</seg> cuius duo latera <seg type="var">.R.A.</seg> et <seg type="var">.R.V.</seg> indeterminata <lb/>intelligantur, ſi ab aliquo puncto inter ipſas poſito, puta <seg type="var">.u.</seg> quod etiam uocetur <seg type="var">.i.</seg> du <lb/>ctę fuerint .4. lineę ipſis dictis lateribus, hac ſcilicet <choice><ex>conditione</ex><am>cõditione</am></choice>, <choice><ex>quod</ex><am>qđ</am></choice> duę ex dictis .4. ſint <lb/>parallelę ipfis <choice><ex>lateribus</ex><am>lateribꝰ</am></choice>, puta <lb/> <ptr xml:id="fig-0284-01a" corresp="fig-0284-01" type="figureAnchor"/> <seg type="var">u.e.</seg> et <seg type="var">.u.E.</seg> reliquę uero duę <lb/>ſeccent ipſa latera, ut <seg type="var">V.u.<lb/>a.</seg> et <seg type="var">.I.u.A</seg>. </s> <s xml:space="preserve">Dico nunc pro-<lb/>portionem <seg type="var">.e.A.</seg> ad <seg type="var">.e.a.</seg> ean <lb/>dem eſſe, quę <seg type="var">.E.V.</seg> ad <seg type="var">.E.I.</seg> <lb/>Nam ſcimus proportionem <lb/><seg type="var">E.i.</seg> ad <seg type="var">.E.i.</seg> eandem eſſe quę <lb/><seg type="var">e.i.</seg> ad <seg type="var">.e.A.</seg> ex fimilitudine <lb/><choice><ex>triangulorum</ex><am>triangulorũ</am></choice>, ſimiliter <choice><ex>propor</ex><am>ꝓpor</am></choice> <lb/><choice><ex>tionem</ex><am>tionẽ</am></choice> <seg type="var">.E.u.</seg> ad <seg type="var">.E.V.</seg> <choice><ex>eandem</ex><am>eãdẽ</am></choice> quę <lb/><seg type="var">e.a.</seg> ad <seg type="var">.e.u.</seg> </s> <s xml:space="preserve">quare aggregata <lb/>ex iſtis erunt inuicem <choice><ex>aequa- lia</ex><am>ęqua-lia</am></choice>, uel ſi mauis ex ęqua pro <lb/>portionalitate, quod idem <lb/>eſt, ita ſe habebit <seg type="var">.E.I.</seg> ad <seg type="var">.<lb/>E.V.</seg> ut <seg type="var">.e.a.</seg> ad <seg type="var">.e.A</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0284-01" corresp="fig-0284-01a"> <graphic url="0284-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Suppoſito nunc plano orizontali <seg type="var">.V.E</seg>. </s> <s xml:space="preserve"><choice><ex>Altitudineque</ex><am>Altitudineq́;</am></choice> inacceſſibili <seg type="var">.A.E</seg>. </s> <s xml:space="preserve">Duę ue-<lb/>rò ſtationes oculorum ſint <seg type="var">.V.</seg> et <seg type="var">.I.</seg> lineę autem uiſuales ſint <seg type="var">.V.A.</seg> et <seg type="var">.I.A</seg>. </s> <s xml:space="preserve">Et quadra-<lb/>tum geometricum ſit <seg type="var">.b.e</seg>. </s> <s xml:space="preserve">Supponatur nunc pro prima dubitatione, quod in am-<lb/>babus ſtationibus filum perpendiculare ſeccet latus <seg type="var">.e.c.</seg> non autem <seg type="var">.b.c.</seg> (nam quan-<lb/>do in ambabus ſtationibus filum ſecat latus <seg type="var">.b.c.</seg> nullum tibi dubium oritur, imo ma <lb/>nifeſtè patent partes lateris <seg type="var">.b.c.</seg> terminatas à <seg type="var">.b.</seg> & à filo proportionales eſſe <seg type="var">.V.E.</seg> & <lb/><seg type="var">I.E.</seg> ſumpto <seg type="var">.E.</seg> pro <seg type="var">.b.</seg> et <seg type="var">.I.V.</seg> pro punctis ſecatis à filo, ex <choice><ex>euidenti</ex><am>euidẽti</am></choice> ſimilitudine trian-<lb/>gulorum quadrati cum triangulis <seg type="var">.A.E.V.</seg> et <seg type="var">.A.E.I.</seg>) Sed cum in pręſenti caſu repe-<lb/>riatur triangulum <seg type="var">.u.e.a.</seg> minus, in ſtatione remotiori, ſimile triangulo maiori <seg type="var">.V.E.<lb/>A.</seg> & triangulum maius <seg type="var">.i.e.a.</seg> proximioris ſtationis, ſimile triangulo minori <seg type="var">.I.E.A.</seg> <lb/>(quod in alio iam dicto, caſu non accidit, ut unum triangulorum, minus ſcilicet, ſi-<lb/>mile ſit uno triangulorum, maiori ſcilicet & è conuerſo) Non omnino abſque ratio <lb/>ne dubitas quo pacto fieri poſſit ut <seg type="var">.a.e.</seg> remotioris ſtationis ad <seg type="var">.a.e.</seg> propinquioris ita <lb/>ſe habeat quema dmodum <seg type="var">.I.E.</seg> ad <seg type="var">.E.V</seg>. </s> <s xml:space="preserve">Quapropter ſi pręcedentem figuram dili- <pb facs="0285" n="273"/><fw type="head">EPISTOL AE.</fw> genter inſpexeris, omnis tua dubitatio euaneſcet. in qua figura apertè vi debis cor-<lb/>reſpondentiam talium triangulorum inter ſe, nec magis, nec minus quam in infra-<lb/>ſcripta hic figura cernere licet, quamuis in hac, triangula quadrati, ſeparata ſint ab <lb/>imaginarijs <seg type="var">.A.E.V.</seg> et <seg type="var">.A.E.I.</seg> in ſupradicta vero coniuncta, & inuicem communican <lb/>tia in puncto <seg type="var">.u.i.</seg> quod quidem nihil refert. </s> <s xml:space="preserve">Dempta igitur <seg type="var">.a.e.</seg> minori ex <seg type="var">.a.e.</seg> ma-<lb/>iori, reliquum ita ſe habebit ad <seg type="var">.a.e.</seg> minorem, vt, <seg type="var">V.I.</seg> ad <seg type="var">.I.E.</seg> quod nunc tibi <lb/>clarè patebit. </s> <s xml:space="preserve">Vnde ex te poteris ordinem operationis proſequi, vt in cognitionem <lb/>peruenias ipſius <seg type="var">.I.E.</seg> ipſius <seg type="var">.A.E.</seg> & ipſius <seg type="var">.I.A.</seg> vel <seg type="var">.V.A</seg>.</s> </p> <figure place="here"> <graphic url="0285-01"/> </figure> <p> <s xml:space="preserve">Sed <choice><ex>quando</ex><am>quãdo</am></choice> in proximiori ſtatione latus <seg type="var">.b.c.</seg> in remotiori vero latus <seg type="var">.c.e.</seg> ſecatur à fi <lb/>lo (pro ſecunda dubitatione) </s> <s xml:space="preserve">Tunc oportet imaginatione conſiderare latus <seg type="var">.b.c.</seg> in <lb/>re motiori ſtatione diſtentum eſſe vſque ad filum in puncto <seg type="var">.n.</seg> vbi videbis <choice><ex>triangulum</ex><am>triangulũ</am></choice> <seg type="var">.<lb/>u.b.n.</seg> ſimile triangulo <seg type="var">.A.E.V.</seg> ita vt <seg type="var">.i.b.a.</seg> ſimile ſuo <seg type="var">.A.E.I.</seg> reperitur, vbi tam in vno <lb/> <ptr xml:id="fig-0285-02a" corresp="fig-0285-02" type="figureAnchor"/> <pb facs="0286" n="274"/><fw type="head">IO. BAPT. BENED.</fw> quam in altero <seg type="var">.i.b.</seg> et <seg type="var">u.b.</seg> correſpondebit ipſi <seg type="var">.A.E.</seg> et <seg type="var">.b.n.</seg> ipſi <seg type="var">.E.V.</seg> et <seg type="var">.b.a.</seg> ipſi <seg type="var">.E.I.</seg> <lb/>quapropter iubeo, vt quæras quantum ſit latus <seg type="var">.b.n.</seg> ex regula de tribus, dicens ſi <seg type="var">.a.e.</seg> <lb/>tribuit mihi <seg type="var">.e.u.</seg> quid mihi dabit <seg type="var">.u.b</seg>? </s> <s xml:space="preserve">eo quod <seg type="var">.a.e.u.</seg> ſimile eſt <seg type="var">.u.b.n.</seg> reperto autem <lb/>latere <seg type="var">.b.n.</seg> ex quo dempto <seg type="var">.b.a.</seg> breuioris diſtantię, reſiduum reſpondebit ipſi <seg type="var">.I.V.</seg> vt <lb/>ſcis, vnde proſequendo operationem tibi cognitam, obtinebis intentum, hoc eſt co <lb/>gnoſces reliqua interualla. </s> <s xml:space="preserve">Nihil enim miror demonſtrationem Tartaleæ circa hu <lb/>iuſmodi operationem te minime ſatisfeciſſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0285-02" corresp="fig-0285-02a"> <graphic url="0285-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quod autem quarta propoſitio illius ſcriptoris, de quo nuper <choice><ex>mecum</ex><am>mecũ</am></choice> locutus es, vt <lb/>mihi dixiſti, tua ſit, hoc enimego, nec affirmare, nec negare audeo, quamuis in mul <lb/>tis cum tua manuſcripta concordet. </s> <s xml:space="preserve">Nam ſępæ cogitationes hominum in idem co-<lb/>incidunt, vt pluries cenſuit Ariſto.</s> </p> <figure place="here"> <graphic url="0286-01"/> </figure> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Demonstrationes quorundam problematum Nicolai Tartalea <lb/>cum <choice><ex>alijs</ex><am>alijs</am></choice> operationibus circa eadem ſubiecta.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">AMor erga te meus ſanè ſingularis, nullo modo <choice><ex>permittit</ex><am>ꝑmittit</am></choice>, vt ea quæ Tartaleę ſcri <lb/>pta <choice><ex>examinando</ex><am>examinãdo</am></choice> inuenerim, non tibi <choice><ex>communicem</ex><am>cõmunicem</am></choice>. </s> <s xml:space="preserve">Hæc autem ſunt circa quæ <lb/>dam illius Authoris problemata, quorum primum ab ipſo Tartalea <lb/>ſcriptum in .3. quæſito libr .4. tale eſt, is vult locare .3500. homines, eodem modo, <lb/>quo præſupponit locatos eſſe .1000. ita vt quilibet hominum ordo ſiue vt vulgo di-<lb/>citur filtia ſit .49. quapropter multiplicat quadratum ipſius .49. quod eſt .2401. <choice><ex>cum</ex><am>cũ</am></choice> <lb/>numero .3500. propoſito, productum verò .8403500. diuidit per .1000. vt proue-<lb/>niat .8403. cuius radix quadrata eſt .91. pro numero hominum vniuſcuiuſque ordinis <lb/>propoſiti numeri .3500.</s> </p> <p> <s xml:space="preserve">Pro cuius operationis ratione, cogitemus rectangulum <seg type="var">.a.b.</seg> 1000. hominum, et <seg type="var">.d.<lb/>b.</seg> ſit vna filtia ſiue ordo .49. <choice><ex>hominum</ex><am>hoĩum</am></choice>, cuius <choice><ex>quadratum</ex><am>quadratũ</am></choice> ſit <seg type="var">.b.c.</seg> 2401. imaginemur <choice><ex>etiam</ex><am>etiã</am></choice> re <lb/>ctangulum <seg type="var">.A.B.</seg> 3500. hominum, quod ſupponemus ſimile rectangulo <seg type="var">.a.b.</seg> et <seg type="var">.B.C.</seg> <pb facs="0287" n="275"/><fw type="head">EPISTOL AE.</fw> ſit quadratum ipſius <seg type="var">.D.B</seg>. </s> <s xml:space="preserve">Nunc ſupponendo <seg type="var">.A.B.</seg> ſimile <seg type="var">.a.b.</seg> clarum erit ex diffini-<lb/>tione ſimilium figurarum, quod eadem proportio erit <seg type="var">.A.D.</seg> ad <seg type="var">.D.B.</seg> quę <seg type="var">.a.d.</seg> ad <seg type="var">.d.<lb/>b.</seg> hoc eſt <seg type="var">.A.D.</seg> ad <seg type="var">.D.C.</seg> vt <seg type="var">.a.d.</seg> ad <seg type="var">.d.c.</seg> hoc eſt <seg type="var">.A.B.</seg> ad <seg type="var">.B.c.</seg> vt <seg type="var">.a.b.</seg> ad <seg type="var">.b.c.</seg> ex prima <lb/>ſexti, vel .18. ſeu .19. ſeptimi, </s> <s xml:space="preserve">tunc cum dixerimus ſi <seg type="var">.a.b.</seg> ita reſpondet ad <seg type="var">.b.c.</seg> ergo <seg type="var">.A.<lb/>B.</seg> correſpondet etiam ita ad <seg type="var">.B.C.</seg> </s> <s xml:space="preserve">quare ex regula de tribus rectè fit multiplicando <seg type="var">.<lb/>A.B.</seg> per <seg type="var">.b.c.</seg> productum verò diuidendo per <seg type="var">.a.b.</seg> ex .15. ſexti vel .20. ſeptimi, cuius <lb/>prouentus radix quadrata erit quod quærebatur.</s> </p> <p> <s xml:space="preserve">Sed aliter idem poſſe fieri ſpeculatus ſum, hoc eſt multiplicando numerum .49. <lb/>ordinis .1000. hominum <choice><ex>cum</ex><am>cũ</am></choice> radice quadrata numeri .3500. propoſiti, productum ve-<lb/>rò diuidere per radicem quadratam ipſius .1000. vnde prouentus .91. erit numerus <lb/>vnius ordinis .3500. numeri <choice><ex>propoſiti</ex><am>ꝓpoſiti</am></choice>.</s> </p> <p> <s xml:space="preserve">Cuius <choice><ex>operationis</ex><am>oꝑationis</am></choice> ſpeculatio eſt iſta. <lb/> <ptr xml:id="fig-0287-01a" corresp="fig-0287-01" type="figureAnchor"/> </s> <s xml:space="preserve">Sit <seg type="var">.a.b.</seg> quadratum .1000. et <seg type="var">.a.c.</seg> ſua <lb/>radix et <seg type="var">.a.d.</seg> rectangulum propoſi-<lb/>tum ipſius .1000. et <seg type="var">.a.e.</seg> vnus ordo. <lb/></s> <s xml:space="preserve">Sit etiam <seg type="var">.A.B.</seg> quadratum .3500. & <lb/><seg type="var">A.C.</seg> eius radix et <seg type="var">.A.D.</seg> <choice><ex>rectangulum</ex><am>rectangulũ</am></choice> <lb/>ipſius numeri .3500. propoſiti, ſimile <lb/>tamen rectangulo <seg type="var">.a.d.</seg> et <seg type="var">.A.E.</seg> eius <lb/>vnus ordo. </s> <s xml:space="preserve"><choice><ex>Cum</ex><am>Cũ</am></choice> enim <seg type="var">.a.b.</seg> æquale ſit <lb/><seg type="var">a.d.</seg> et <seg type="var">.A.B</seg>: <seg type="var">A.D.</seg> <choice><ex>tunc</ex><am>tũc</am></choice> <seg type="var">.a.c.</seg> erit media <lb/>proportionalis inter <seg type="var">.a.e.</seg> et <seg type="var">.e.d.</seg> & ſic <lb/><seg type="var">A.C.</seg> erit etiam media proportiona <lb/>lis inter <seg type="var">.A.E.</seg> et <seg type="var">.E.D.</seg> per .16. ſexti, <lb/>ſeu .20. ſeptimi, & quia proporrio. A <lb/>E. ad <seg type="var">.E.D.</seg> æqualis eſt proportioni <seg type="var">.<lb/>a.e.</seg> ad <seg type="var">.e.d.</seg> cum <seg type="var">.A.D.</seg> ſupponatur ſi-<lb/>mile <seg type="var">.a.d.</seg> ergo proportio <seg type="var">.A.E.</seg> ad <seg type="var">.A<lb/>C.</seg> ęqualis erit proportioni <seg type="var">.a.e.</seg> ad <seg type="var">.a.<lb/>c.</seg> quę medietates ſunt <choice><ex>totorum</ex><am>totorũ</am></choice> æqua-<lb/>lium, rectè igitur fiet ſi procedamus <lb/>ex regula de tribus, dicendo ſi <seg type="var">.a.c.</seg> <lb/><choice><ex>correſpondet</ex><am>correſpõdet</am></choice> <seg type="var">.a.e.</seg> tùc <seg type="var">.A.C.</seg> <choice><ex>correſpon</ex><am>correſpõ</am></choice> <lb/>det <seg type="var">.A.E.</seg> ex ſupradictis .15. ſexti. vel <lb/>20. ſeptimi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0287-01" corresp="fig-0287-01a"> <graphic url="0287-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Ratio verò quarti quæſiti per ſe <lb/>patet, quod eſt inuenire <choice><ex>pauimentum</ex><am>pauimentũ</am></choice> <lb/>ſeu aream quadratam, in qua poſſint <lb/>locari quot homines volueris, ita in <lb/>ter ſe ſiti, ut vnuſquiſque occupet <num value="7">.<lb/>7.</num> pedes ipſius areę in longitudinem <lb/>et .3. per latitudinem à lateribus.</s> </p> <p> <s xml:space="preserve">Seu ex propoſito hominum nume <lb/>ro inuenire numerum ipſorum loca-<lb/>bilem in aliqua area quadrata, ita, <lb/>vt vnuſquiſque occupet .21. pedes <lb/>quadratos ipſius areæ.</s> </p> <pb facs="0288" n="276"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Sed aliter idem fieri poſſe inueni, hoc eſt <choice><ex>multiplicando</ex><am>multiplicãdo</am></choice> radicem quadratam pro-<lb/>poſiti numeri hominum per .21. & productum item multiplicando per eandem radi <lb/>cem, & huiuſmodi producti radicem diuiden do per .3. vnde prouentus eſſet nume-<lb/>rus hominum vnius ordinis. </s> <s xml:space="preserve"><choice><ex>Exempli</ex><am>Exẽpli</am></choice> gratia proponu<unclear reason="illegible"/>ntur .3600. homines, multiplica <lb/>bimus huiuſmodi numeri radicem <lb/>quadratam hoc eſt .60. per .21. hoc <lb/> <ptr xml:id="fig-0288-01a" corresp="fig-0288-01" type="figureAnchor"/> eſt per productum quod fit ex .7. <choice><ex>cum</ex><am>cũ</am></choice> <lb/>3. & reſultabit nobis .1260. quod ſi <lb/>multiplicabitur, per .60. hoc eſt per <lb/>eandem radicem, reſultabit nobis <num value="75600">.<lb/>75600.</num> cuius producti radix qua-<lb/>drata eſt ferè .275. qua diuiſa per .3 <lb/>proueniet nobis .91. pro hominum <lb/>numero vnius ordinis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0288-01" corresp="fig-0288-01a"> <graphic url="0288-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Cuiusratio eſt iſta, cogitemus nu <lb/>merum .3600. propoſitum eſſe qua <lb/>dratum <seg type="var">.a.b.</seg> (ſed non areæ) cuius ra <lb/>dix .60. ſit <seg type="var">.a.c.</seg> & quia hic numerus <num value="60">.<lb/>60.</num> intelligitur eſſe hominum, quo-<lb/>rum <choice><ex>vnuſquiſque</ex><am>vnuſquiſq;</am></choice> occupat .21. pedes <lb/>quadratos ſuperficiales ex ſuppoſi-<lb/>to, </s> <s xml:space="preserve">& propterea multiplicatur, 60. <lb/>cum .21. vnde nobis veniat .1260. <lb/>quadrati ſuperficiales pro vnoquo-<lb/>que ordine, & <choice><ex>quia</ex><am>ꝗa</am></choice>.b.c. vt. latus qua-<lb/>drati <seg type="var">.a.b.</seg> habet tot ordines homi-<lb/>num ſimiliter, hoc eſt .60. igi-<lb/>tur multiplicando .60. cum .1260. <lb/>habebimus totalem ſuperficiem <seg type="var">.a.<lb/>b.</seg> ex .75600. quadratis ſuperficiali-<lb/>bus, quæ quadrata imaginemur lo-<lb/>cata eſſe in quodam totali quadra-<lb/>to, quod ſit <seg type="var">.e.f.</seg> cuius radix ſit <seg type="var">.e.</seg> g <num value="275">.<lb/>275.</num> pedum qui diuidantur per .3. <lb/>hoc eſt per numerum pedum latitu-<lb/>dinis & prouenient nobis .91. pro <lb/>numero <choice><ex>hominum</ex><am>hominũ</am></choice> <choice><ex>vniuſcuiuſque</ex><am>vniuſcuiuſq;</am></choice> ordi-<lb/>nis, diuidendo poſtea latus <seg type="var">.f.g.</seg> per <lb/>numerum ſpatij inter vnum, & <choice><ex>alium</ex><am>aliũ</am></choice> <lb/>ordinem, quod eſt .7. proueniet <lb/>nobis .39. pro numero ordinum.</s> </p> <p> <s xml:space="preserve">Aliter, & breuius etiam poſſumus idem inuenire, hoc eſt multiplicando <choice><ex>nume- rum</ex><am>nume-rũ</am></choice> <choice><ex>propoſitum</ex><am>propoſitũ</am></choice> <choice><ex>hominum</ex><am>hominũ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> rectangulo .21. vnde venietnobis <choice><ex>productum</ex><am>ꝓductũ</am></choice> .75600 quod pro <lb/><choice><ex>ductum</ex><am>ductũ</am></choice> ſi accipiemus vt <choice><ex>quadratum</ex><am>quadratũ</am></choice>, cuius radix erit .275. quæ diuidatur <choice><ex>per</ex><am>ꝑ</am></choice> .3. habebi-<lb/>mus <choice><ex>propofitum</ex><am>ꝓpofitũ</am></choice>. </s> <s xml:space="preserve">Cuius ratio <choice><ex>pendet</ex><am>pẽdet</am></choice> à ſupradicta, eo <choice><ex>quod</ex><am>qđ</am></choice> loco <choice><ex>multiplicandi</ex><am>multiplicãdi</am></choice> <seg type="var">.a.c.</seg> (hoc eſt <num value="60">. <pb facs="0289" n="277"/><fw type="head">EPISTOL AE.</fw> 60.</num>) per .21. deinde <choice><ex>productum</ex><am>productũ</am></choice> etiam multiplicare per <seg type="var">.b.c.</seg> (hoc eſt .60.) breuius erit <lb/>multiplicare totum numerum .3600. per .21. cętera verò facere, vt diximus.</s> </p> <p> <s xml:space="preserve">Sed <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> iſtarum operationum, aliquid imperfectionis patitur, eo quod <choice><ex>cum</ex><am>cũ</am></choice> <lb/>aliquis cuperet quadratum perfectum ſuperficiale habere, <choice><ex>abſque</ex><am>abſq;</am></choice> aliquo defectu, vel <lb/>exceſfu, aliquid aliud adhuc facere oporteret, hoc eſt, inuentum cum fuerit quadra <lb/>tum <seg type="var">.e.f.</seg> cum ſuis radicibus <seg type="var">.e.g.</seg> et <seg type="var">.g.f.</seg> pedum .275. vnaquaque, vt in dicto exemplo <lb/>factum eſt, oportebit <choice><ex>numerum</ex><am>numerũ</am></choice> quærere minorem ipſo .275. ſed proximiorem men-<lb/>furabilem ab .3. & ab .7. quod facilè fiet ſi diuiſerimus .275. per .21. detrahendo fra-<lb/>cta diuiſionis ab ipſo .275. quæ quidem fracta in hoc exemplo ſunt .2. vnde remane-<lb/>bit .273. pro numero laterum quadrati ſuperficialis, in quo poſſent locari .3549. ho-<lb/>mines, eo ordine quo ſupra dictum eſt, quorum ſcilicet vnuſquiſque obtineat .21. <lb/>pedes ſuperſiciales.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE INTERVALLIS MVSICIS.</head> <head rend="italics" xml:space="preserve">Cypriano Rorè Muſico celeberrimo.</head> <p> <s xml:space="preserve"><hi rend="small caps">OPinio</hi> Hectoris Euſonij Cypriane mi dilectiſſime, vera non eſt, quod ali <lb/>quisrectè poſſit intelligere rationes conſonantiarum muſicæ, ablque co <lb/>gnitione illarum mediante ipſo ſenſu, imo nemo <choice><ex>pont</ex><am>põt</am></choice> calere <choice><ex>theoriam</ex><am>theoriã</am></choice> mu <lb/>ſices, niſi aliquo <choice><ex>mon</ex><am>mõ</am></choice> verſatus ſit in praxi. </s> <s xml:space="preserve"><choice><ex>Quando</ex><am>Qũo</am></choice> enim cognoſci <choice><ex>poterunt</ex><am>poterũt</am></choice> <lb/>quid nam ſint diapaſon, diapente, diateſſeron, ditonus, ſemiditonus, hexacordum <lb/>maius, aut minus, & conſonantiæ ex ijs cum diapaſon compoſitæ, abſque earum <lb/>praxi? </s> <s xml:space="preserve">vnde ſequetur <choice><ex>neque</ex><am>neq;</am></choice> etiam cognoſci poſſe interualla diſſonantia. </s> <s xml:space="preserve">Et purus <lb/>practicus non intelliget quid ſit octaua, quinta, quarta, tertia maior, tertia minor, <lb/>ſexta maior, ſexta minor, decima maior, decima minor, vndecima, duode-<lb/>cima, decimatertia maior, aut minor, aut decimaquinta, & aliæ, ita vt ad <lb/>comparandam perfectionem muſicæ neceſſarium ſit, & thęoriam & praxim ad-<lb/>diſcere. </s> <s xml:space="preserve">Cum pręterea Ludouicus Folianus apertè monſtrarit (etiam ſi id à diato-<lb/>nico ſintono Ptolomei deſumpſerit) reperiri duos tonos, maiorem, & minorem, id-<lb/>eſt ſeſquioctauum, & ſeſquinonum, & tria ſemitonia, maius, minus, & mini-<lb/>mum, ideſt ſeſquiquintumdecimum, qui eſt maius, ſeſquiuigeſimum quartum id-<lb/>eſt minimum, & mediocre, vt .27. ad .25. quæ proportio ſuperbipartiens vigeſi-<lb/>maſquintas appellatur, & cum cognouerit ſemiditonum conſonantem eſſe ſeſqui-<lb/>quintum, ditonum ſeſquiquartum, & hexachordum minus, vt .8. ad .5. quæ propor-<lb/>tio dicitur ſupertripartiens quintas, & hexachordum maius, vt .5. ad .3. hęc autem vo <lb/>catur ſuperbipartienstertias; </s> <s xml:space="preserve">omnium ſimplicium conſonantiarum cognitioni, ex-<lb/>tremam impoſuit manum. </s> <s xml:space="preserve">Et quia tibi etiam oſtendere promiſi in modulationibus <pb facs="0290" n="278"/><fw type="head">IO. BAPT. BENED.</fw> hæc omnia interualla ſeruari, ideo ad te mitto ſeptem hic ſubſcripta exempla, in <lb/>quorum primo, & ſecundo, inter dieſim, et <seg type="var">.b.</seg> in ſuperiori, agnoſces interuallum mi <lb/>nimi ſemitonij, & ſi ibi ſit dieſis, tanquam terminus ad quem, et <seg type="var">.b.</seg> tanquam termi-<lb/>nus à quo: </s> <s xml:space="preserve">quod autem inter dieſim et <seg type="var">.b.</seg> ſit ſemitonium minimum, facilè agnoſces <lb/>ſi ſubtraxeris <choice><ex>decimam</ex><am>decimã</am></choice> <choice><ex>minorem</ex><am>minorẽ</am></choice> à maiori, <choice><ex>quam</ex><am>quã</am></choice> facit ſuperius <choice><ex>cum</ex><am>cũ</am></choice> inferiori, ideſt <choice><ex>cum</ex><am>cũ</am></choice> baſſu.</s> </p> <p> <s xml:space="preserve">Qua quidem modulatione tu etiam vſus es in cantilena illa, quæ Galica lingua <lb/>incipit. </s> <s xml:space="preserve">Hellas comment. </s> <s xml:space="preserve">Eadem, ego quoque in meis cantilenis latino ſermo-<lb/>ne compoſitis, quæ Moreta vocantur aliquando vſus ſum.</s> </p> <p> <s xml:space="preserve">Sed in tertio exemplo inuenies ſemitonium maius, neceſſariò genitum in ſupe-<lb/>riori, ſi ſextam maiorem cum baſſu eſſicere volueris, quia tenor, à ditono cum <lb/>ſuperiori ad diapentem, & ad vniſonum cum baſſu procedit, vbi quieſcit, progre-<lb/>diendo poſtea baſſus ad ſemiditonum cum tenore, </s> <s xml:space="preserve">tunc ſi à proportione huius ſep-<lb/>timæ, quæ eſt vt .9. ad .5. hoc eſt ſuperquadripartiensquintas demptum fuerit hexa-<lb/>chordum maius, ſeu ſexta maior, quæ eſt vt .5. ad .3. remanebit proportio .27. ad .25. <lb/>quæ maior eſt quam .32. ad .30.</s> </p> <p> <s xml:space="preserve">In quarto <choice><ex>exemplo</ex><am>exẽplo</am></choice> habebis ſemitonium minus in ſuperiori, quod quidem remanet <lb/>ex ſubtractione ditoni <choice><ex>conſonantis</ex><am>cõſonãtis</am></choice> ab diateſſaron <choice><ex>compręhenſa</ex><am>cõpręhenſa</am></choice> à ſuperiori cum tenore.</s> </p> <p> <s xml:space="preserve">In quinto exemplo videbis tonum minorem, & tonum maiorem ſucceſſiuè vnum <lb/>poſt alium in tenore, detrahendo primo <choice><ex>ſemiditonum</ex><am>ſemiditonũ</am></choice> à diateſſaron, quod ſuperius fa-<lb/>cit cum tenore, vel detrahendo diapente ab hexachordo maiori, quod facit tenor <lb/>cum baſſu, vnde remanet tonus minor ſeſquinonus, detrahendo poſtea diateſſaron <lb/>à diapente, quod ſuperius facit cum tenore, remanebit tonus maior ſeſquioctauus.</s> </p> <p> <s xml:space="preserve">In ſexto exemplo deinde videbis tenorem aſcendere per duos tonos minores ſuc <lb/>ceſſiuè vnum poſt alium in tenore, ſi <choice><ex>demp</ex><am>dẽp</am></choice> ſeris <choice><ex>ſemiditonum</ex><am>ſemiditonũ</am></choice> à diateſſaron <choice><ex>cum</ex><am>cũ</am></choice> ſuperiori.</s> </p> <p> <s xml:space="preserve">In .7. <choice><ex>exemplo</ex><am>exẽplo</am></choice> demum videbis <choice><ex>ſuperiorem</ex><am>ſuperiorẽ</am></choice> aſcendere per duos tonos maiores ſucceſ-<lb/>ſiuè <choice><ex>vnum</ex><am>vnũ</am></choice> poſt <choice><ex>alium</ex><am>aliũ</am></choice>, ſi dempſeris diateſſaron à diapente, quod facittenor <choice><ex>cum</ex><am>cũ</am></choice> ſuperiori.</s> </p> <figure place="here"> <graphic url="0290-01"/> </figure> <pb facs="0291" n="279"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head rend="italics" xml:space="preserve">De eodem ſubiecto.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod aliàs tibi dixi, verum eſt, quod neceſſarium nullo modo ſit, vt modulan-<lb/>do, deſinat cantilena in eodem tono (quod Græci phthongum appel-<lb/>lant) à quo incępit. </s> <s xml:space="preserve">immo neceſſariò ſemper ferè, altius, aut depræſ-<lb/>ſius terminatur, per differentiam alicuius interualli æqualis, vel multiplicis ipſi com <lb/>mati ſeſquioctuageſimæ, quod quidem comma, quamuis cantabile non ſit, inſenſi-<lb/>biliter tamen generatur, & toties ab aliqua parte ipſius cantilenæ poſſet <choice><ex>dictum</ex><am>dictũ</am></choice> com-<lb/>ma gcnerari, verſus acutum, vel graue, quod in fine ipſius cantilenę, vocis phtongus <lb/>reperiatur diſtans à primo per interuallum alicuius toni ſeſquinoni, ſeu ſeſquioctaui <lb/>plus, minúsue, vt in ſubſcripto exemplo clarè videre potes in prima figura, vbi ſu-<lb/>perius à <seg type="var">.g.</seg> primę cellulæ ad <seg type="var">.g.</seg> ſecundæ, intereſt vnum <choice><ex>comma</ex><am>cõma</am></choice>, eo quod progrediens <lb/>ſuperius in prima cellula ipſius cantilenæ à quarta ad quintam cum tenore, aſcendit <lb/>per tonum ſeſquioctauum, à prima cellula deinde ad ſecundam, tenor aſcendit ſimi-<lb/>liter per tonum ſeſquioctauum cum tranſeat à quinta ad quartam, quod facit cum <lb/>ſuperiori, in ſecunda cellula poſtea, cum ſuperius deſcendat à maiori ſexta ad quin <lb/>tam, quod facit cum baſſu, ſeu à quarta ad tertiam minorem, quod facit cum teno-<lb/>re, </s> <s xml:space="preserve">tunc deſcendit per tonum ſeſquinonum, ita quod non reuertitur ad <choice><ex>eundem</ex><am>eundẽ</am></choice> <choice><ex>phthon</ex><am>phthõ</am></choice> <lb/>gum, vbi prius erat in prima cellula, ſed reperitur per <choice><ex>vnum</ex><am>vnũ</am></choice> <choice><ex>coomma</ex><am>coõma</am></choice> altius, <choice><ex>quod</ex><am>qđ</am></choice> <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/><choice><ex>comma</ex><am>cõma</am></choice> eſt differentia inter <choice><ex>tonum</ex><am>tonũ</am></choice> <choice><ex>ſeſquioctauum</ex><am>ſeſquioctauũ</am></choice> & <choice><ex>ſeſquinonum</ex><am>ſeſquinonũ</am></choice>, vt alias tibi <choice><ex>demonſtraui</ex><am>demõſtraui</am></choice>.</s> </p> <p> <s xml:space="preserve">Progrediendo igitur hoc modo, videbis quod cum tenor à ſecunda cellula ad ter <lb/>tiam tranſeat à tertia minori ad quartam, quod facit cum ſuperiori, deſcendit per <lb/>tonum ſeſquinonum, vnde in tertia cellula altius remanet quam in prima per <choice><ex>vnum</ex><am>vnũ</am></choice> <lb/>comma, in qua tertia cellula, cum iterum tranſeat ſuperius à quarta ad quintam, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>facit cum tenore, eleuatur per tonum ſeſquioctauum, proſequendo deinde tali ordi <lb/>ne, vidc<unclear reason="illegible"/>bis in quarta cellula cantilenam auctam per duo commata, in ſexta, <choice><ex>autem</ex><am>aũt</am></choice> cel-<lb/>lula per tria commata, in octaua verò per .4. commata, vnde hac merhodo, ſi can-<lb/>tilena prolixior debito eſſet, vel ſi talia interualla frequentiora reperirentur, poſſet <lb/>cantilena à principio ad finem differre per .9. commata, & plus etiam, quæ quidem <lb/> <ptr xml:id="fig-0291-01a" corresp="fig-0291-01" type="figureAnchor"/> <pb facs="0292" n="280"/><fw type="head">IO. BABPT. BENED.</fw> interualla ſuperant tonum ſeſquinonum, & ſi eſſent .10. commata<unclear reason="illegible"/> ſuperarent tonum <lb/>ſeſquioctauum, eo quod aggregatum ex .9. commatibus continetur ſub iſtis duobus <lb/>terminis hoc eſt .150094635296999121. et .134217728000000000. quæ qui-<lb/>dem proportio maior eſt proportione ſeſquinona, ſumma verò .10. commatum con <lb/>tinetur ſub .12157665459056928801. et .10737418240000000000. quæ pro <lb/>portio maior eſt tono ſeſquioctauo, quod autem dico de aſcenſu cantilenæ, idem aſ-<lb/>ſero de eiuſdem deſcenſu, & hoc non tantum per interuallum illius commatis, quod <lb/>eſt differentia toni maioris à minori, ſed etiam per illud quod eſt differentia ſemito <lb/>nij maioris à minori, vt in ſecundo exemplo hic ſubſcripto videre eſt in deſcenſu <lb/>cantilenæ per comma & comma, vt differentia inter ſemitonia maiora<unclear reason="illegible"/> & minora, <lb/>vbi in prima cellula diſcedens baſſus à quinta cum ſuperiori, & ab vniſono cum te-<lb/>nore deſcendens ad tertiam minorem cum ipſo tenore, facit cum ſuperiori <choice><ex>ſeptimam</ex><am>ſeptimã</am></choice> <lb/>maiorem, quæ eſt vt .9. ad .5. ſuperquadripartiensquintas ſcilicet, à qua diſcedens <lb/>poſtea ſuperius, vt faciat cum baſſu ſextam maiorem, deſcendit per ſemitonium ma <lb/>ius, à qua ſexta maiori deſcendens baſſus, & aſcendens per quartam, efficit cum di-<lb/>cto ſuperiori <choice><ex>tertiam</ex><am>tertiã</am></choice> maiorem, à qua diſcedens ſuperius, vt efficiat quartam cum ipſo <lb/>baſſu (qui quidem baſſus tranſit in tenorem) aſcendit per ſemitonium minus, diffe-<lb/>rens à ſemitonio maiori per vnum comma, vnde cantilena remanet depreſſa per <lb/>vnum comma. </s> <s xml:space="preserve">cum deinde idem faciat inter tertiam, & quartam cellulam, per a-<lb/>liud comma deſcendit, & ſic toties facere poſſet, vt poſtremo valde deprimatur <lb/>cantilena à primo phthongo.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0291-01" corresp="fig-0291-01a"> <graphic url="0291-01"/> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0292-01"/> </figure> <p> <s xml:space="preserve">Quod autem hic ſupradictum eſt, clrca inſtrumenta artificialia non accidit, qua <lb/>propter organa, & clauicimbula concordantur certo quodam ordine, ita vt omnes <lb/>conſonantiæ, excepta diapaſon, ſeu octaua, ſint imperfectæ, hoc eſt, aut diminutę, <lb/>aut ſuperantes à inſto, vt exempli gratia, omnes quintæ ſunt diminutæ, quartæ verò <lb/>ſunt exceſſiuę, quod quidem fit, vt tertiæ, & ſextæ, non multum auribus diſſonent, <lb/>eo quod ſi quintæ omnes, & quartæ, perfectæ eſſent, </s> <s xml:space="preserve">tunc omnes ſextę, & tertiæ in-<lb/>tollerabiles eſſent, & à perfectis differrent per vnum comma, quod manifeſtum no-<lb/>bis erit hoc modo, accipiamus tres diapentes, ſeu quintas, conſequenter ſucceſſiuas <lb/>vnam poſt aliam, hoc eſt tres proportiones ſeſquialteras, quarum aggregatum erit <lb/>vt .27. ad .8. quæ proportio, dicitur tripla ſupertripartiensoctauas, & quæ à practicis <pb facs="0293" n="281"/><fw type="head">EPISTOL AE.</fw> appellaretur tertia decima maior, vt exempli gratia, eſſet Gamaut cum ſecundo ela <lb/>mi, </s> <s xml:space="preserve">tunc talis tertiadecima valde odioſa eſſet ſenſui auditus, à qua, ſi dempta ſuerit <lb/>diapaſon, ſeu octaua, remaneret quoddam hexachordum maius, ſeu ſexta maior, au-<lb/>ribus valde inimica, ſub proportione .13. ad 8. ſed hæc proportio differret à propor <lb/>tione ſuperbipartientetertias perfecti hexachordi maioris, hoc eſt ſextæ maioris <lb/>conſonantis, per proportionem ſeſquioctuageſimam, hoc eſt per vnum comma, <lb/>quod quidem eſt etiam differentia aggregati trium ſeſquialterarum, à tertiadeci-<lb/>ma maiori conſonanti, hoc eſt exceſſus proportionis triplæ ſupertripartientis octa-<lb/>uas, ſupra triplam ſeſquitertiam, quæ eſt ſumma ipſius duplæ cum ſuperbi partien-<lb/>tetertias.</s> </p> <p> <s xml:space="preserve">A tali ſumma igitur trium ſeſquialterarum efficitur tertiadecima maior diſſonans <lb/>excedens conſonantem per vnum comma (cuius proportio eſt .81. ad .80.) quæ con-<lb/>fonans continetur in proportione .10. ad .3. vt ſupra dixi.</s> </p> <p> <s xml:space="preserve">Hæcigitur eſt vera ratio, propter quam debemus comma diſtribuere in organis <lb/>& clauicymbalis, cum ab aggregato trium quintarum producatur talis exceſſus ſu-<lb/>pra perfectam, ſeu conſonantem tertiamdecimam maiorem, quod quidem aggre-<lb/>gatum, cum demptum fuerit à quintadecima, relinquet nobis tertiam minorem <lb/>diſſonantem, & mancam, per eundem exceſſum à conſonanti. </s> <s xml:space="preserve">quæ quidem tertia <lb/>minor diſſonans ſubtracta à diapente ſeu quinta perfecta, relinquet nobis tertiam <lb/>maiorem diſſonantem, qu@ conſonantem excedit per eundem exceſſum comma-<lb/>tis, & hæc demum tertia maior diſſonans, dempta ex diapaſon, ſeu octaua, relin-<lb/>quet nobis hexachordum minus, hoc eſt ſextam minorem diſſonantem, & muti-<lb/>lam à conſonanti per eundem exceſſum commatis. </s> <s xml:space="preserve">De huiuſmodi verò commatis <lb/>diſtributione doctiſſimè ſcripſit Excellentiſſimus Zarlinus in ſecunda parte Inſtitu-<lb/>tionum Harmonicarum.</s> </p> <p> <s xml:space="preserve">Sed quia ſenſus auditus non poteſt exactè cognoſcere debitam quantitatem ex-<lb/>ceſſus, vel defectus, intendendo vel remittendo chordas inſtrumentorum, ideo hanc <lb/>viam ſequutus ſum.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia, hic ſubſcriptus ordo lignorum tangentium ſeu pinarum inci-<lb/>piens ab <seg type="var">.G.</seg> deſinens ad <seg type="var">.g.</seg> ita quod inter ipſos terminos ſit ea conſonantia quæ vo-<lb/>catur vigeſimaſecunda, quæro primum <seg type="var">.b.</seg> inter <seg type="var">.D.E.</seg> quod eſt nigrum ipſius Ela-<lb/>mi grauiſſimum, quod groſſo modo facio conſonans cum <seg type="var">.G.</seg> grauiſſimo per ſex-<lb/>tam <choice><ex>minorem</ex><am>minorẽ</am></choice>, deinde <choice><ex>cum</ex><am>cũ</am></choice> ipſo pt<unclear reason="illegible"/>imo <seg type="var">.b.</seg> ipſius elami concordo ſuum octauum & quin-<lb/>tumdecimum, quo perfectius poſſum, deinde accipio <seg type="var">.b.</seg> molle ſecundum ipſius. b <lb/>fabmi quod concordo cum <seg type="var">.b.</seg> primo ipſius Elami per quintam imperfectam, dein-<lb/>de cum hoc <seg type="var">.b.</seg> ſecundo ipſius bſabmi concordo ſecundum <seg type="var">.f.</seg> per quintam ſimiliter <lb/>imperfectam, cum quo <seg type="var">.f.</seg> poſtea concordo tertium <seg type="var">.c.</seg> per ſimilem quintam, quem <lb/>tertium <seg type="var">.c.</seg> poſtea confero <choice><ex>cum</ex><am>cũ</am></choice> ſecundo <seg type="var">.b.</seg> ipſius elami, ita quod inter ſe conſonent per <lb/>ſextam maiorem tolerabilem, & ſi ſic inuenio, tunc nihil muto has treschordas hoc. <lb/> <ptr xml:id="table-0293-01a" corresp="table-0293-01" type="tableAnchor"/> <pb facs="0294" n="288"/><fw type="head">IO. BAPT. BENED.</fw> eſt <seg type="var">.b.</seg> ſecundum ipſius bfabmi, f. ſecundum, et <seg type="var">.c.</seg> tertium, ſed ſi dictum tertium <seg type="var">.c.</seg> <lb/>valde diſſonans eſſet cum <seg type="var">.b.</seg> ſecundo ipſius elami, </s> <s xml:space="preserve">tunc ipſum <seg type="var">.c.</seg> intendo, aut re-<lb/>mitto, quouſque aliquo modo ſit conſonans per ſextam maiorem aliquantulum ex <lb/>ceſſiuam cum <seg type="var">.b.</seg> ſecundo ipſius elami, cum quo poſtea <seg type="var">.c.</seg> conſonare aliquantulum fa <lb/>cio <seg type="var">.f.</seg> ſecundum per quintam defectiuam, & cum hoc<unclear reason="illegible"/> demum <seg type="var">.b.</seg> ſecundum ipſius <lb/>bfabmi, quo facto concordo ſecundum <seg type="var">.c.</seg> cum tertio per octauam, cum quo ſecun-<lb/>do <seg type="var">.c.</seg> poſtea concordo tertium <seg type="var">.g.</seg> per talem quintam, quod ipſum tertium <seg type="var">.g.</seg> cum ſe-<lb/>cundo <seg type="var">.b.</seg> ipſius bfabmi conſonet tolerabiliter per ſextam maiorem <choice><ex>aliquantulum</ex><am>aliquãtulum</am></choice> ex-<lb/>ceſſiuam, </s> <s xml:space="preserve">deinde cum iſto tertio <seg type="var">.g.</seg> concordo tertium <seg type="var">.d.</seg> per talem quintam, ita quod <lb/>ipſum <seg type="var">.3.d.</seg> concordet tolerabiliter cum <seg type="var">.2.f.</seg> per ſextam maiorem exceſſiuam, poſtea <lb/>cum hoc <seg type="var">.3.d.</seg> concordo <seg type="var">.2.d.</seg> per octauam perfecte, cum quo <seg type="var">.2.d.</seg> poſtea concordo <seg type="var">.<lb/>3.a.</seg> per quintam, vt in alijs <choice><ex>factum</ex><am>factũ</am></choice> eſt, ita vt <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.2.c.</seg> conſonet talis ſexta maior, vt ſupra <lb/>dictum eſt, cum quo <seg type="var">.3.a.</seg> poſtea concordo <seg type="var">.3.e.</seg> per quintam, vt dictum eſt, ita quod <lb/>cum <seg type="var">.3.g.</seg> faciat ſextam maiorem vt ſupra, poſtea cum hoc <seg type="var">.e.</seg> concordo <seg type="var">.2.e.</seg> per octa <lb/>uam, cum quo concordo <seg type="var">.b.</seg> quadrum tertium per quintam, vt dictum eſt, ita quod <choice><ex>cum</ex><am>cũ</am></choice> <lb/><seg type="var">2.d.</seg> faciat ſextam maiorem ſimilem alijs ſuperius dictis, cum quo <seg type="var">.b.</seg> quadrato tertio <lb/>concordo tertium nigrum ipſius <seg type="var">.f.</seg> per quintam, ita quod cum <seg type="var">.3 a.</seg> faciat ſextam ma-<lb/>iorem, vt ſupra, deinde cum hoc concordo <seg type="var">.2.f.</seg> nigrum per octauam, cum quo, per <lb/>quintam concordo <seg type="var">3.c.</seg> nigrum ita quod cum <seg type="var">.2.e.</seg> faciat ſextam dictam, demum <choice><ex>cum</ex><am>cũ</am></choice> <lb/>hoc concordo <seg type="var">.4.g.</seg> nigrum per quintam, ita quod faciat cum <seg type="var">.3.b.</seg> quadrato ſextam <lb/>dictam, & ſic ad vltimam quintam peruenio, ſupra quod <seg type="var">.g.</seg> nigrum nulla quinta am-<lb/>plius reperitur, poſtea cum iſtis chordis concordo per octauas omnes alias ab acutis <lb/>ad graues.</s> </p> <floatingText> <body> <div type="float"> <table xml:id="table-0293-01" corresp="table-0293-01a" rows="2" cols="1"> <row> <cell xml:space="preserve">G A B b C * D b E F * g * a b b c * d b e f * g * a b b c * d b e f * g *</cell> </row> <row> <cell xml:space="preserve">1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4.</cell> </row> </table> </div> </body> </floatingText> <table rows="2" cols="1"> <row> <cell xml:space="preserve">G A B b C * D b E F * g * a b b c * d b e f * g * a b b c * d b e f * g *</cell> </row> <row> <cell xml:space="preserve">1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4.</cell> </row> </table> <p> <s xml:space="preserve">Valde etiam admiratione dignum eſt, quod perſectiores quæque conſonan <lb/>tiæ, ita in harmonica diuiſione ſibi inuicem conueniant, vt dia paſon cum diapente, <lb/>cum diapaſondiapente, cum ditono, cum hexachordo maiori cum bisdiapaſon, <choice><ex>cum</ex><am>cũ</am></choice> <lb/>decimaleptima maiori. </s> <s xml:space="preserve">Nam in ipſa diapaſon, harmonicè <choice><ex>locantur</ex><am>locãtur</am></choice> diapente in par <lb/>te grauiori, & diateſlaron in acutiori. </s> <s xml:space="preserve">In diapente verò harmonicè locantur ditonus <lb/>in parte grauiori, & ſemiditonus in acutiori. </s> <s xml:space="preserve">In ditono harmonicè locantur tonus <lb/>maior in parte grauiori, & tonus minor in acutiori. </s> <s xml:space="preserve">In hexachordo maiori, harmo-<lb/>nicè locantur diateſſaron in parte grauiori, & ditonus in acutiori. </s> <s xml:space="preserve">In diapaſondia-<lb/>pente, harmonic<unclear reason="illegible"/>è locantur diapaſon in parte grauiori, & diapente in acutiori. <lb/></s> <s xml:space="preserve">In bisdiapaſon, ha@monicè locantur decima maior in parte grauiori & hexachor-<lb/>dum minus in acutiori. </s> <s xml:space="preserve">In decimaſeptima maiori, harmonicè locantur diapaſondia-<lb/>pente in parte grauiori, & hexachordum maius in parte acutiori. </s> <s xml:space="preserve">Ita quod ronus <lb/>ſeſquioctauus in ditono, proportionalis eſt ipſi ditono in diapente. </s> <s xml:space="preserve">Tonus verò ſeſ-<lb/>quinonus in ipſo ditono, proportionalis eſt triemitonio, vel ſeſquitonio ſeu ſemidi-<lb/>tono (quod idem eſt) in diapente. </s> <s xml:space="preserve">Ditonus autem in diapente, proportionalis eſt <lb/>ipſi diapente in diapaſon. </s> <s xml:space="preserve">Seſquitonus verò in diapente, proportionalis eſt diateſ-<lb/>ſaron in diapaſon. </s> <s xml:space="preserve">Et ſic de ſingulis. </s> <s xml:space="preserve">Ita quod tonus ſeſquioctauus in ditono, dito-<lb/>mus in diapente, diateſſaron in hexachordo maiori, diapente in diapaſon, diapaſon <lb/>in diapaſondiapente, decimamaior in bisdiapaſon, diapaſondiapente in decima- <pb facs="0295" n="283"/><fw type="head">EPISTOL AE.</fw> ſeptima maiori, omnia ſibi inuicem ſunt proportionalia, idem etiam dico de reli-<lb/>quis partibus, cum relatæ fuerint ad ſua tota.</s> </p> <p> <s xml:space="preserve">Nec alienum mihi videtur à propoſito inſtituto, ſpeculari modum generationis <lb/>ipſarum ſimplicium <choice><ex>conſonantiarum</ex><am>conſonantiarũ</am></choice>; </s> <s xml:space="preserve">qui quidem modus fit ex quadam æquatione per <lb/>@uſſionum, ſeu æquali concurſu vndarum aeris, vel conterminatione earum.</s> </p> <p> <s xml:space="preserve">Nam, nulli dubium eſt, quin vniſonus ſit prima principalis <choice><ex>audituque</ex><am>audituq́</am></choice> amiciſſima, <lb/>nec non magis propria conſonantia; </s> <s xml:space="preserve">& ſi intelligatur, vt punctus in linea, vel vnitas <lb/>in numero, quam immediate ſequitur diapaſon, ei ſimillima, poſt hanc verò diapen <lb/>te, <choice><ex>cæteræque</ex><am>cæteræq́;</am></choice>. </s> <s xml:space="preserve">Videamus igitur ordinem concurſus percuſſionum terminorum, ſeu <lb/>vndarum aeris, vnde ſonus generatur.</s> </p> <p> <s xml:space="preserve">Concipiatur igitur mente monochordus, hoc eſt chorda diſtenta, quæ cum diuiſa <lb/>fuerit in duas æquales partes à ponticulo, </s> <s xml:space="preserve">tunc <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> pars eundem ſonum pro-<lb/>feret, & ambæ formabunt vniſonum, quia eodem tempore, tot percuſſiones in aere <lb/>faciet vna partium illius chordæ, quot & altera: </s> <s xml:space="preserve">ita vt vndæ aeris ſimul eant, & æqua <lb/>liter concurrant, abſque ulla interſectione, vel fractione illarum inuicem.</s> </p> <p> <s xml:space="preserve">Sed cum ponticulus ita diuiſerit chordam, vt relicta ſit eius tertia pars ab vno la-<lb/>tere, ab alio vero, duę tertię, </s> <s xml:space="preserve">tunc maior pars, dupla erit minori, & <choice><ex>ſonabunt</ex><am>ſonabũt</am></choice> ipſam dia <lb/>paſon conſonantiam, percuſſiones vero terminorum ipſius, tali proportione ſe inui-<lb/>cem habebunt, ut in qualibet ſecunda percuſſione minoris portionis ipſius chordæ, <lb/>maior percutiet, ſeu concurret cum minori, eodem temporis inſtanti, cum ne-<lb/>mo ſit qui neſciat, quod quo longior eſt chorda, etiam tardius moueatur, </s> <s xml:space="preserve">quare <lb/>cum longior dupla ſit breuiori, & eiuſdem intenſionis tam vna quam altera, tunc eo <lb/>tempore, quo longior vnum interuallum tremoris perfecerit, breuior duo interual-<lb/>la conficiet.</s> </p> <p> <s xml:space="preserve">Cum autem ponticulus ita diuiſerit chordam, ut ab uno latere relinquantur duæ <lb/>quintæ partes, ab alio verò tres quintæ, ex quibus partibus generatur conſonantia <lb/>diapente; </s> <s xml:space="preserve">tunc clarè patet, quod eadem proportione tardius erit vnum interuallu<unclear reason="illegible"/>m <lb/>tremoris maioris portionis, vno interuallo tremoris minoris portionis, quam ma-<lb/>i<unclear reason="illegible"/>or portio habet ad minorem; </s> <s xml:space="preserve">hoc eſt tempus maioris interualli ad tempus minoris <lb/>erit <choice><ex>ſeſquialterum</ex><am>ſeſquialterũ</am></choice> quare non <choice><ex>conuenient</ex><am>cõuenient</am></choice> ſimul, niſi perfectis tribus interuallis mino-<lb/>ris portionis, & duobus maioris; </s> <s xml:space="preserve">ita quod eadem proportio erit numeri interuallo-<lb/>rum minoris portionis ad interualla maioris, quæ longitudinis maioris portionis ad <lb/>longitudinem minoris; </s> <s xml:space="preserve">vnde productum numeri portionis minoris ipſius chordæ <lb/>in numerum interuallorum motus ipſius portionis, æquale erit producto numeri <lb/>portionis maioris in numerum interuallorum ipſius maioris portionis; </s> <s xml:space="preserve">quæ quidem <lb/>producta ita ſe habebunt, vt in diapaſon, ſit binarius numerus; </s> <s xml:space="preserve">in diapente verò <lb/>ſenarius; </s> <s xml:space="preserve">in diateſſaron duodenarius, in hexachordo maiori quindenarius; </s> <s xml:space="preserve">in di-<lb/>tono vicenarius, in ſemiditono tricenarius, demum in hexachordo minori quadra <lb/>genarius: </s> <s xml:space="preserve">qui quidem numeri non abſque mirabili analogia conueniunt inuicem.</s> </p> <p> <s xml:space="preserve">Voluptas autem, quam auditui afferunt conſonantiæ fit, quia leniuntur ſenſus, <lb/>quemadmodum <choice><ex>contra</ex><am>cõtra</am></choice>, dolor qui à diſſonantijs oritur, ab aſperitate naſcitur, id quod <lb/>facilè vide<unclear reason="illegible"/>re poteris cum conchordantur organorum fiſtulæ.</s> </p> <pb facs="0296" n="284"/> <fw type="head">IO. BAPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE IVSTITIA COMMVTATIVA.</head> <head rend="italics" xml:space="preserve">Franciſco Ferrario Anciſa Iuriſconſulto <choice><ex>ſenatorique</ex><am>ſenatoriq́ꝫ</am></choice> apud <lb/>ſubalpinos grauißimo.</head> <p> <s xml:space="preserve"><hi rend="small caps">SAepivs</hi> inter nos dum oportunitas vicinarum ædium, & amoris mutui <lb/>vis, ad familiaria trahunt colloquia ego de meis mathematicis, tu de tuis <lb/>legibus, in quibus tractandis magnum tibi nomen comparaſti loquuti ſu <lb/>mus. </s> <s xml:space="preserve">Cum vero nonnunquam de mirabili iuſtitiæ commutatíuæ inſtitu <lb/>to non ingratus incidiſſet ſermo, dixi modum, quo formam ſuam à proportionali-<lb/>tate arithmetica diſiuncta, & non a coniuncta deſumat, à nemine literis proditum <lb/>eſſe, libet autem nunc per otium latius explicare. </s> <s xml:space="preserve">dixi enim à diſiuncta, & non con-<lb/>iuncta proportionalitate, quia in coniuncta, ſeu continua nullo pacto fieri poteſt <lb/>talis commutatio, cum ſemper quatuor terminos ad minus tranſeat, vt nunc vide-<lb/>bimus.</s> </p> <p> <s xml:space="preserve">Exempli gratia, Petrus ex ſuis bonis tribuat Ioanni aliquid valoris quinquagin <lb/>ta aureorum.</s> </p> <p> <s xml:space="preserve">Vnde priuſquam Ioannes aliquid ex ſuis bonis retribuat Petro, bona ipſius Pe-<lb/>tri diminuta erunt per quinquaginta aureos, bona verò ipſius Ioannis, aucta toti-<lb/>dem aureis.</s> </p> <p> <s xml:space="preserve">Ecce nunc quo pacto conftituti ſunt .4. termini in proportionalitate aritmetica, <lb/>per quos ſit talis permutatio, ſed nondum æquata, niſi fiat æqualis retributio à Ioan-<lb/>ne ad Petrum, vt videbimus.</s> </p> <p> <s xml:space="preserve">Cogitentur itaque .4. termini aritmeticè proportionales <seg type="var">.C.A.B.D</seg>. </s> <s xml:space="preserve">Ita quod <seg type="var">.A.</seg> <lb/>mediante ſignificentur bona Ioannis <seg type="var">.B.</seg> vero Petri, prius quam Petrus aliquid ex bo <lb/>nis ſuis tribuat Ioanni. </s> <s xml:space="preserve">Tunc Petrus ſecat partem vnam ex <seg type="var">.B.</seg> <choice><ex>eamque</ex><am>eamq́;</am></choice> dat ipſi Ioan-<lb/>ni, vnde ipſi Petro remanet <seg type="var">.D</seg>. </s> <s xml:space="preserve">Ioanni autem <seg type="var">.C.</seg> quatuor igitur termini conſtituti <lb/>ſunt <seg type="var">.B.D.C.A.</seg> quorum <seg type="var">.B.</seg> primus <seg type="var">.A.</seg> quartus <seg type="var">.C.</seg> uero tertius <seg type="var">.D.</seg> <choice><ex>autem</ex><am>aũt</am></choice> ſecundus, ſed <lb/>B. et <seg type="var">.A.</seg> ſunt in ſua naturali mediocritate abſque defectu vel exceſſu ſui ipſius. </s> <s xml:space="preserve">Non <lb/>ita tamen ſe habet <seg type="var">.C.</seg> et <seg type="var">.D.</seg> quia <seg type="var">.D.</seg> deficit <seg type="var">.C.</seg> autem excedit à ſua priori quantitate. <lb/></s> <s xml:space="preserve">Nihilominus iſti .4. termini conſtituti ſunt in ipſa aritmetica proportionalitate, nam <lb/>eadem quantitate qua <seg type="var">.D.</seg> diminuta eſt à <seg type="var">.B.</seg> eadem <seg type="var">.C.</seg> aucta eſt ſupra <seg type="var">.A</seg>.</s> </p> <p> <s xml:space="preserve">Sed quia <seg type="var">.B.</seg> et <seg type="var">.A.</seg> tantummodo iuſti ſunt termini <seg type="var">.C.</seg> uerò et <seg type="var">.D.</seg> iniuſti, vt ad ſuam <lb/>priorem æqualitatem reuertantur, oportebit ex <seg type="var">.C.</seg> ſecare aliquam partem æqualis <lb/>valoris ei, qua <seg type="var">.C.</seg> ſuperat <seg type="var">.A.</seg> vel qua <seg type="var">.D.</seg> minor eſt <seg type="var">.B.</seg> & ipſam partem addere ipſi <seg type="var">.D.</seg> <lb/>vt bona Petri reuertantur ad priorem ſuam quantitatem ipſius <seg type="var">.B.</seg> & bona Ioannis <lb/>remaneant æqualia <seg type="var">.A.</seg> vt prius.</s> </p> <p> <s xml:space="preserve">Quare neceſſarium non eſt, vt talis proportionalitas ſit coniuncta (vt inquit Eu <lb/>f<gap reason="illegible"/>atius ſeu Michael Epheſius, <lb/>ſuper quinto capite libr. quin-<lb/> <ptr xml:id="fig-0296-01a" corresp="fig-0296-01" type="figureAnchor"/> ti Ethicorum) tribus terminis <lb/>contenta, imò oportet ut ipſa <lb/>diſiuncta ſit, ut diximus, vbi <lb/>non eſt neceſſe quod <seg type="var">.A.</seg> æqualis ſit <seg type="var">.B.</seg> aliquo modo.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0296-01" corresp="fig-0296-01a"> <graphic url="0296-01"/> </figure> </div> </body> </floatingText> <pb facs="0297" n="285"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE MOTV MOLAE, ET TROCHI, DE AMPVL-<lb/>lis aquæ, de claritate aeris, & Lunæ noctu fulgentis, de æter-<lb/>nitate temporis, & infinito ſpacio extra <lb/>Cœlum, Cœ<choice><ex>lique</ex><am>liq́;</am></choice> figura.</head> <head rend="italics" xml:space="preserve">Illust. Ioanni Paulo Capra Nouarienſi Sabaudia Ducis boſpicij <lb/>Magistro, viro ingeny praſtantia, & morum cando-<lb/>re, non minus quam familia nobili-<lb/>tate conſpicuo.</head> <p> <s xml:space="preserve">SI vera eſſet animorum illa tranſmigratio quam ſibi Italicæ ſapientiæ Pa-<lb/>ter Pythagoras effinxerat, tuam, <choice><ex>meamque</ex><am>meamq́;</am></choice> exiſtimarem animam canis, <lb/>quandoque venatici fuiſſe.</s> </p> <p> <s xml:space="preserve">Quæris à me literis tuis, an motus circularis alicuius molæ molendina <lb/>rie, ſi ſuper aliquod punctum, quaſi <choice><ex>mathematicum</ex><am>mathematicũ</am></choice>, quieſceret, poſſet eſſe perpetuus, <lb/>cum aliquando eſſet mota, ſupponendo etiam eandem eſſe perfectè rotundam, & <lb/>lęuigatam. </s> <s xml:space="preserve">Reſpondeo huiuſmodi motum nullo modo futurum perpetuum, nec <lb/>etiam multum duraturum, quia præterquam quod ab aere qui ei circumcirca <choice><ex>aliquam</ex><am>aliquã</am></choice> <lb/>reſiſtentiam facit ſtringitur, eſt etiam reſiſtentia partium illius corporis moti, quæ <lb/>cum motæ ſunt, natura, impetum habent efficiendi iter directum, vnde cum ſimul <lb/>iunctæ ſint, & earum vna continuata cum alia. </s> <s xml:space="preserve">dum circulariter mouentur patiuntur <lb/>violentiam, & in huiuſmodi motu per vim vnitæ manent, quia quanto magis mo-<lb/>uentur, tanto magis in ijs creſcit naturalis inclinatio recta eundi, vnde tanto magis <lb/>contra ſuammet naturam voluuntur, ita vt ſecundum naturam quieſcant, quia cum <lb/>eis proprium ſit, quando ſunt motæ, eundi recta, quanto violentius voluuntur, tan-<lb/>to magis vna reſiſtit alteri, & quaſi retrò reuocat eam, quam antea reperitur habere.</s> </p> <p> <s xml:space="preserve">Ab eiuſmodi inclinatione rectitudinis motus partium alicuius corporis rotundi <lb/>fit, vt per aliquod temporis ſpacium, trochus cum magna violentia ſeipſum circun-<lb/>agens, omninò rectus quieſcat ſuper illam cuſpidem ferri quam habet, non incli-<lb/>nans ſe verſus mundi centrum, magis ad vnam <choice><ex>partem</ex><am>partẽ</am></choice>, quam ad aliam, cum quælibet <lb/>ſuarum partium in huiuſmodi motu non inclinet omnino verſus <choice><ex>mundi</ex><am>mũdi</am></choice> centrum, ſed <lb/>multo magis per tranſuerſum ad angulos rectos cum linea directionis, aut verticali, <lb/>aut orizontis axe, ita vt neceſſariò huiuſmodi corpus rectum ſtare debeat. </s> <s xml:space="preserve">Et quod <lb/>dico ipſas partes non omninò inclinare verſus mundi centrum, id ea ratione dico, <lb/>quia non abſolutè ſunt unquam priuatæ huiuſmodi inclinatione, quę efficit vt ipſum <lb/>corpus eo puncto nitatur. </s> <s xml:space="preserve">Verum tamen eſt, quod quanto magis eſt velox, tan-<lb/>to minus premit ipſum punctum, imò ipſum corpus <choice><ex>tanto</ex><am>tãto</am></choice> magis leue remanet. </s> <s xml:space="preserve">Id <choice><ex>quod</ex><am>qđ</am></choice> <lb/>apertè patet <choice><ex>ſumendo</ex><am>ſumẽdo</am></choice> <choice><ex>exemplum</ex><am>exẽplũ</am></choice> pilę alicuius arcus, aut <choice><ex>alicuius</ex><am>alicuiꝰ</am></choice> alterius <choice><ex>inſtrumenti</ex><am>inſtrumẽti</am></choice>, ſeu ma <lb/>chinæ miſſilis, quæ pila quanto eſt velocior, in motu violento, tanto maiorem pro-<lb/>penſionem habet rectius eundi, vnde verſus mundi centrum tanto minus inclinat, <lb/>& hanc ob cauſam leuior redditur. </s> <s xml:space="preserve">Sed ſi clarius, hanc veritatem videre cupis, <lb/>cogita illud corpus, Trochum ſcilicet, dum velociſſime circunducitur ſecari, ſeu <lb/>diuidi in multas partes, vnde uidebis illas omnes, non illico uerſus mundi centrum <pb facs="0298" n="286"/><fw type="head">IO. BAPT. BENED.</fw> deſcendere, ſed recta orizontaliter, vt ita dicam, moueri. </s> <s xml:space="preserve">Id quod à nemine ad-<lb/>huc (quod ſciam) in trocho eſt obſeruatum. </s> <s xml:space="preserve">Ab huiuſmodi motu trochi, aut hu-<lb/>ius generis corporis, clarè perſpicitur, quàm errent peripatetici circa motum uio-<lb/>lentum alicuius corporis, qui exiſtimant aerem qui ſubintrat ad occupandum locum <lb/>à corpore relictum, ipſum corpus impellere, cum ab hoc, magis effectus contrarius <lb/>naſcatur.</s> </p> <p> <s xml:space="preserve">Quod deinde ampullæ iungantur in aqua, non fit ratione ſimpathiæ, de qua lo-<lb/>quitur Fracaſtorus, nam per accidens iunguntur, quia cum alia ad aliam accedit, quę <lb/>libet earum tentat aſcendere ab ea parte, à qua inuicem hærent, quemadmodum <lb/>efficiunt iuxta labrum vaſis, ea enim ſuperficies a quæ vicina circunferentiæ vaſis ali <lb/>quantulum aſcendit in vaſe, qui non eſt omnino plenus.</s> </p> <p> <s xml:space="preserve">Ad id deinde quod de claritate noctium ſcribis, miror cur non videas, quod <choice><ex>quam</ex><am>quã</am></choice> <lb/>to magis obſcura nox apparet, non dico ratione nubium, ſed diſtantiæ Solis ſub <lb/>orizonte ab eodem orizonte, tanto magis claram, & luminoſam ſeſe nobis oſten-<lb/>dit Luna in quintadecima, quia cum Sol eſt in Sagittario, & Capricorno, Luna eſt <lb/>in Geminis, & in Cancro, vnde in media nocte, eius radius per valde exiguam <choice><ex>quam</ex><am>quã</am></choice> <lb/>titatem vaporum tranſit, quia tunc ipſa eſt valde propinqua axi orizontis, & præ-<lb/>terea in huiuſmodi tempore anni & noctis, aer eſt magis purgatus, quàm in qualibet <lb/>alia temporis parte, quia hieme Sol non poteſt excitare multos vapores, & ij, qui <lb/>at tolluntur, nocte à frigore ſtatim congellati ratione grauitatis <choice><ex>decidunt</ex><am>decidũt</am></choice>, </s> <s xml:space="preserve">unde rema-<lb/>net aer multo clarior, qua ratione apparent ſtellæ minutæ, & Cœlum ijſdem ma-<lb/>gis ornatum, quàm in quolibet alio anni tempore.</s> </p> <p> <s xml:space="preserve">Dicere deinde, quemadmodum hic mundus eſt ætatis ſeptem, aut octomillium <lb/>annorum, ita nunc potuiſſet eſſe (ſi Deus voluiſſet) ætatis quinquagintamillium; </s> <s xml:space="preserve">er <lb/>go erat tempus; </s> <s xml:space="preserve">ita ſe habet, ac ſi diceremus, quemadmodum hic mundus eſt tan-<lb/>tæ magnitudinis, ita etiam quinquagies maior eſſe potuiſſet, ergo eſt ſpatium, aut <lb/>interuallum corporeum, quod eum capere potuiſſet.</s> </p> <p> <s xml:space="preserve">Illud, nihil, Ariſtotelis extra Cęlum, nullo modo nobis inſeruit pro eiuſdem Cœ <lb/>li ſphęrica rotunditate, cum cuiuſque alterius ex infinitis figuris Cęlum ipſum eſſe <lb/>poſſit, ſecundum ſuam ſuperficiem conuexam. </s> <s xml:space="preserve">Nam Cœlum ea ratione ſphęricum <lb/>non eſt, quod magis ſit capax, quia ei innumerahiles alias figuras adeo magnas po <lb/>terat concedere cauſa diuina: </s> <s xml:space="preserve">ſed ſphæricum eſt effectum, ne partem aliquam habe <lb/>ret ſui termini ſuperfluam, quia nullum corpus à breuiori termino quam à ſphærico <lb/>terminari poteſt.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Derèuolutione rota putealis & <choice><ex>alijs</ex><am>alijs</am></choice> <lb/>problematibus.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">FVnis cui appenſa eſt ſitula, longè facilius axi inuoluitur, ſi ipſi axi affixa ſit rota. <lb/></s> <s xml:space="preserve">atque item commodius eò fiet, quo amplior rota erit, & axis exilior. <lb/></s> <s xml:space="preserve">Commodiſſimè autem, ſi ipſa rotæ extrema circunferentia, ex materia minori, <lb/>& denſiori, ac proinde grauiori conſtabit. </s> <s xml:space="preserve">Cuius rei ratio multiplex eſt. </s> <s xml:space="preserve">Nem-<lb/>pe quia omne corpus graue, aut ſui natura, aut vi motum, in ſe recipit impreſſio- <pb facs="0299" n="287"/><fw type="head">EPISTOL AE.</fw> nem & impetum motus, ita vt ſeparatum à virtute mouente per aliquod temporis <lb/>ſpatium ex ſeipſo moueatur. </s> <s xml:space="preserve">nam ſi ſecundum naturam motu cieatur, ſuam veloci-<lb/>tatem ſemper augebit, cum in eo, impetus & impreſſio ſemper augeantur, quia <lb/>coniunctam habet per petuò virtutem mouentem. </s> <s xml:space="preserve">Vnde manu mouendo rotam, ab <lb/><choice><ex>eaque</ex><am>eaq́;</am></choice> eam remouendo rota ſtatim non quieſcet, ſed per aliquod temporis ſpatium <lb/>circunuertetur.</s> </p> <p> <s xml:space="preserve">Secunda cauſa eſt, quia quoduis gr aue corpus, aut per naturam, aut per vim mo-<lb/>tum, rectitudinem itineris naturaliter appetat, quod clarè cognoſcere poſſumus, <lb/>proijciendo lapides funda, & circunducentes brachium, nam funes tanto maius <lb/>pondus acquirunt, & manum tanto magis onerant, quanto velocius voluitur funda, <lb/>& incitatur motus, quod ab appetitu naturali inſito ei corpori per <choice><ex>lineam</ex><am>lineã</am></choice> rectam pro-<lb/>grediendi procedit. </s> <s xml:space="preserve">Vnde fit, vt pondus circunferentiæ ipſius rotæ, tanto facilius cir-<lb/>cunuoluatur, & ex ſeipſo tanto longiori tempore moueatur, quanto longius diſtat à <lb/>centro, cum eius iter tanto minus ſit curuum. </s> <s xml:space="preserve">Hanc igitur ob cauſam, rota, quanto <lb/>maior erit, <choice><ex>eiuſque</ex><am>eiuſq́;</am></choice> pondus tanto magis vicinum circunferentiæ, tanto magis durabit <lb/>impetus motus aſſumptus.</s> </p> <p> <s xml:space="preserve">Tertia cauſa eſt, quod funis dum circunuoluitur, vicinius axi mathematico reuo-<lb/>lutionis, quam corpus graue circunferentiæ rotæ, ratione vectis, cum rota eſt in mo <lb/>tu, eius impetus non obtinet reſiſtentiam æqualem à contrario pondere aquæ in ſitu <lb/>la poſitæ.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De machina, qua aquam impellit & ſubleuat.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">VNde ſit vt in fonte mandauerim, <lb/> <ptr xml:id="fig-0299-01a" corresp="fig-0299-01" type="figureAnchor"/> vas ſeu mortarium in quod in-<lb/>greditur inſtrumentum, quod aquam <lb/>impellit, diametrum ſuæ concauitatis, <lb/>habere non oportere maiorem dia-<lb/>metro fiſtulæ, per quam debet aſcende <lb/>re aqua, ratio eſt, quia ſi maius eſſet, <lb/>neceſſarium eſſet aliquod inſtrumen-<lb/>tum quo aqua impelleretur multo gra <lb/>uius toto corpore aqueo, quod aptum <lb/>eſſet implere aliquam fiſtulam adeo <lb/>altam, vt eſt fons, quæ tamen eſſet <lb/>adeo lata vt eſt mortarium.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0299-01" corresp="fig-0299-01a"> <graphic url="0299-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sit exempli gratia, tota fiſtula, ſeu <lb/>hirundo, per quam aſcendit aqua <seg type="var">.f.</seg> <lb/>mortarium verò ſit <seg type="var">.a.u.</seg> quod tam <choice><ex>altum</ex><am>altũ</am></choice> <lb/>ſit vt <seg type="var">.f.</seg> ſed <seg type="var">.f.</seg> anguſtior ipſo <seg type="var">.a.u</seg>. </s> <s xml:space="preserve">Nunc <lb/> <ptr xml:id="fig-0299-02a" corresp="fig-0299-02" type="figureAnchor"/> cum repleta fuerint hæc duo vaſa, ma-<lb/>nifeſtum erit, quod aqua ipſius <seg type="var">.f.</seg> ſuffi-<lb/>ciens erit ad <choice><ex>reſiſtendum</ex><am>reſiſtẽdum</am></choice> toti aquæ <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <lb/><seg type="var">a.u.</seg> & aqua <seg type="var">.a.u.</seg> reſiſtet aquæ <seg type="var">.f.</seg> quam-<lb/>uis aqua <seg type="var">.a.u.</seg> maioris quantitatis ſit, & <lb/>ponderis ipſa <seg type="var">.f.</seg> hoc autem euenit ex <lb/>eo quod aqua <seg type="var">.a.u.</seg> <choice><ex>non</ex><am>nõ</am></choice> impellit aquam <pb facs="0300" n="288"/><fw type="head">IO. BAPT. BENED.</fw> f. toto ſuo pondere, </s> <s xml:space="preserve">propterea quod pondus diuiditur proportionaliter ſupra ba-<lb/>ſim vaſis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0299-02" corresp="fig-0299-02a"> <graphic url="0299-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sit exempli <choice><ex>gratia</ex><am>gr̃a</am></choice> vas aliquod <seg type="var">.b.d.n.m.</seg> conicæ figuræ, ſeu <choice><ex>truncus</ex><am>trũcus</am></choice> coni concaui aqua <lb/>plenus, cuius orificij diameter ſit <seg type="var">.b.d.</seg> & multiplex diametro <seg type="var">.m.n.</seg> infimæ baſis. </s> <s xml:space="preserve">co-<lb/>gitemus etiam <seg type="var">.b.d.</seg> diuiſum in tot partes, quarum <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> æqualis ſit <seg type="var">.m.n.</seg> imagi-<lb/><choice><ex>nemurque</ex><am>nemurq́;</am></choice> tot lineas perpendiculares deſcendere verſus mundi centrum ad puncta <seg type="var">r.<lb/>c.m.</seg> et <seg type="var">.t.x.m.</seg> vt in ſubſcripta hic figura videre eſt, per quas cogitemus tot ſuperfi-<lb/>cies curuas <choice><ex>conicasque</ex><am>conicasq́;</am></choice>, inter quas, mente concipienda eſt aqua, quę pondere ſuo quie <lb/>ſcet ſupra maiorem ſuperficiem illa, quæ æque diſtans eſſet mundi centro, ſeu quam <lb/>ſupra baſim <seg type="var">.m.n.</seg> vt exempli gratia conſideretur aqua inter <seg type="var">.g.m.</seg> et <seg type="var">.s.r.</seg> cuius pondus <lb/>diſtribuitur fecundum latitudinem <seg type="var">.m.r.</seg> quæ maior eſt <seg type="var">.g.s.</seg> cogitemus igitur <seg type="var">.m.c.</seg> æ-<lb/>qualem eſſe <seg type="var">.g.s.</seg> manifeſtum erit, quod <seg type="var">.m.c.</seg> non ſuſtinebit totum pondus a quæ, quæ <lb/>inter <seg type="var">.g.m.</seg> et <seg type="var">.s.r.</seg> reperitur, eo quod omnis pars aquæ ad perpendiculum inclinat ver-<lb/>ſus mundi centrum, quapropter fundus ſeu baſis <seg type="var">.m.n.</seg> non ſuſtinet aliud pondus <choice><ex>quam</ex><am>quã</am></choice> <lb/>aquæ <seg type="var">.f.m.</seg> ſed ſi quis hoc in dubium reuocaret dicens, quod aqua circunſcribens ſi-<lb/>tum corporis aquei <seg type="var">.f.m.</seg> impellit lateraliter dictum corpus aqueum, reſpondendum <lb/>eſt, quod ex æquo huius corporis <seg type="var">.f.m.</seg> aqua impellit etiam aquam circunſtantem, <lb/>eo, quod ſunt corpora homogenea, cum in corporibus homogeneis æquales partes <lb/>habeant æquales vires.</s> </p> <p> <s xml:space="preserve">Sed redeundo ad vaſa <seg type="var">.a.u.</seg> et <seg type="var">.f.</seg> dico quod ſicut aqua <seg type="var">.f.</seg> ſufficit ad <choice><ex>reſiſtendum</ex><am>reſiſtendũ</am></choice> aquæ <lb/><seg type="var">a.u.</seg> ita quodlibet aliud pondus ęquale <seg type="var">.f.</seg> cuiuſuis materiæ, in fiſtula <seg type="var">.f.</seg> poſitum, ſuffi-<lb/>ciens erit, dummodo illud corpus ita ſit adæquatum concauitati fiſtulæ <seg type="var">.f.</seg> quod non <lb/>permittat tranſitum aliquem aquæ vel <lb/> <ptr xml:id="fig-0300-01a" corresp="fig-0300-01" type="figureAnchor"/> aeris inter conuexum ipſius corporis, <lb/>& deuexum fiſtulæ <seg type="var">.f.</seg> & hoc ex ſe ſatis <lb/>patet, ſed in vaſe <seg type="var">.a.u.</seg> cum ex hypothe <lb/>ſi latius ſit ipſo <seg type="var">.f.</seg> nullum aliud corpus <lb/>ſufficiens erit ad reſiſtendum aquæ ip-<lb/>ſius <seg type="var">.f.</seg> quin tam graue ſit, quam tota <lb/>aqua <seg type="var">.a.u.</seg> exiſtente <seg type="var">.a.u.</seg> tam alto quam <lb/>f. </s> <s xml:space="preserve">Vnde ſi aqua ipſius <seg type="var">.f.</seg> nil plus eſſet <lb/>quam vna tantummodo libra, & vas <seg type="var">.a.<lb/>u.</seg> exiſteret latius ipſo <seg type="var">.f.</seg> in decupla pro <lb/>portione, </s> <s xml:space="preserve">tunc in ipſo <seg type="var">.a.u.</seg> oporteret <lb/>corpus adæquatum ipſi concauitati po <lb/>nere, cuius pondus eſſet decem libra-<lb/>rum, vt ſufficeret ad ſuſtinendum <choice><ex>aquam</ex><am>aquã</am></choice> <lb/>ipſius <seg type="var">.f.</seg> & ad im<unclear reason="illegible"/><choice><ex>pellendum</ex><am>pellendũ</am></choice> ipſam <choice><ex>aquam</ex><am>aquã</am></choice> <seg type="var">.<lb/>f.</seg> deberet eſſe plus quam decem libra-<lb/>rum. </s> <s xml:space="preserve">Ponamus nunc illud corpus, ita <lb/> <ptr xml:id="fig-0300-02a" corresp="fig-0300-02" type="figureAnchor"/> denſius eſſe aqua, vt maius <choice><ex>interuallum</ex><am>interuallũ</am></choice> <lb/>non occupet, quam <seg type="var">.o.e.</seg> corpus igitur <lb/><seg type="var">o.e.</seg> ſufficiens erit ad impellendum <lb/>aquam <seg type="var">.f.</seg> & non eo minus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0300-01" corresp="fig-0300-01a"> <graphic url="0300-01"/> </figure> <figure xml:id="fig-0300-02" corresp="fig-0300-02a"> <graphic url="0300-02"/> </figure> </div> </body> </floatingText> <pb facs="0301" n="289"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">NOVA SOLVTIO PROBLEMATIS DE VASE <lb/>pleno liquoris.</head> <head rend="italics" xml:space="preserve">Nicolao Caluxio Serenißimi Ducis Sabaudia à ſecretis.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> à me poſtulas eſt problema ab alijs iam ſcriptum, ſed illud tibialio <lb/>medio ſoluam.</s> </p> <p> <s xml:space="preserve">Proponitur vas <choice><ex>plenum</ex><am>plenũ</am></choice> liquore aliquo, puta aqua, <choice><ex>quod</ex><am>ꝙ</am></choice> tres habeat fiſtulas <lb/>ad baſim, quarum vnaquæque poſſit euacuare ipſum vas, inæquales ta-<lb/>men, ita quod prima tam lata ſit, vt ſpatio vnius horæ poſſit ipſum euacuare to-<lb/>tum; </s> <s xml:space="preserve">ſecunda vero ſpatio duarum horarum, tertia autem ſpatio trium hora-<unclear reason="illegible"/>-<lb/>rum. </s> <s xml:space="preserve">Tunc quæritur quanto tempore omnes tres fiſtulæ ſimul apertæ euacua-<lb/>bunt ipſum vas. </s> <s xml:space="preserve">Ad hoc volo vt quæratur primo quanta pars aquæ vnaquęquę fi-<lb/>ſtula euacuabit in aliquo dato tempore, quod facilè eſt, vt puta, prima fiſtu-<lb/>la, ſpatio dimidiæ horæ euacuabit dimidium vas, eo quod ſpatio integræ horæ po-<lb/>teſt totum euacuare, ſecunda fiſtula, eodem temporis ſpatio, euacuabit quartam <lb/>partem ipſius vaſis, tertia verò fiſtula, eodemmet ſpatio temporis dimidiæ horæ, <lb/>euacuabit ſextam partem ipſius vaſis, quæ omnia fracta ſimul collecta faciunt vnde-<lb/>cim duodecimas partes totius vaſis, vnde manifeſtum erit, quod omnes fiſtulæ pari-<lb/>ter apertæ, ſpatio dimidię horæ euacuabunt vndecim duodecimas partes totius a-<lb/>quæ, ſed nos cupimus ſcire, quanto tempore, totum vas euacuabitur, apertis omni <lb/>bus fiſtulis, quapropter dicemus ita; </s> <s xml:space="preserve">Si vndecim duodecimæ partes conſumunt mi-<lb/>nuta .30. temporis, quantum conſument omnes partes aquæ? </s> <s xml:space="preserve">quæ ſunt .12. quare ex <lb/>regula de tribus prouenient nobis minuta .32. cum .8. vndecimis vnius minuti, hoc <lb/>eſt cum .43. ſecundis horæ ferè, vel ſiaccipiemus tres quartas vnius horæ, </s> <s xml:space="preserve">tunc pri-<lb/>ma fiſtula emittet tres quartas partes totius aquæ, ſecunda, tres octauas <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> aquę, <lb/>tertia verò, quarta pars, tunc omnia, hæc collecta, faciunt vnum integrum cum tri <lb/>bus octauis. </s> <s xml:space="preserve">Si dixerimus igitur quando vnum integrum cum tribus octauis abſu-<lb/>mit .45. minuta temporis, ergo illud ſolum integrum abſumet idem vt ſupra hoc eſt <lb/>min .32. cum .8. vndecimis vnius minuti vel .43. ſecundis. </s> <s xml:space="preserve">Cuius rei ſpeculatio <choice><ex>tam</ex><am>tã</am></choice> con <lb/>iuncta eſt operationi, quòd vna cognita, reliqua ſtatim cognoſcitur.</s> </p> <p> <s xml:space="preserve">Idem eueniet de implendo vaſe tribus ſimilibus fiſtulis mediantibus.</s> </p> <p> <s xml:space="preserve">Secundum quæſitum ab alijs traditum, tuum etiam, aliter quoque poteſt ſolui, <lb/>propterea non prętermittam tibi ſatisfacere.</s> </p> <p> <s xml:space="preserve">Problema itaque tale eſt, vt ſit vas aliquod in <choice><ex>quod</ex><am>qđ</am></choice> infunditur aqua per tres fiſtu-<lb/>las, ſed dum infunditur aqua, eadem egreditur per duas alias fiſtulas in fundo <lb/>vaſis poſitas, ſed tres ſuperiores ſint inuicem proportionatæ, vt ſupradictum <lb/>eſt, primaq́ue inferiorum talis ſit, vt ſpatio .4. horarum poſſit totum vas euacua-<lb/>re, ſecunda autem poſſit ſpatio .6. horarum idem facere, vnde ex ſupradictis, vas im <lb/>plebitur à tribus fiſtulis ſuperioribus, clauſis exiſtentibus inferioribus, ſpatio tempo <lb/>ris <choice><ex>minutorum</ex><am>minutorũ</am></choice> .32. <choice><ex>cum</ex><am>cũ</am></choice> .8. vndecimis hoc eſt min .32. cum .43. ſecundis, deinde per duas <lb/>fiſtulas inferiores poſſet euacuari ſpatio <choice><ex>temporis</ex><am>tẽporis</am></choice> horarum .2. et. mi .24. exſupradictis.</s> </p> <p> <s xml:space="preserve">Supponamus igitur omnes fiſtulas operari ſpatio temporis minutorum .32. cum <lb/>ſecundis .43. </s> <s xml:space="preserve">tunc manifeſtum eſt quod vas non implebitur, eo ſpatio min .32. cum <lb/>ſecundis .43. ſed tanta aqua deficiet, quanta ab inferioribus fiſtulis eo ſpatio tempo <lb/>ris min .32. ſecun .43. poteſt euacuari, </s> <s xml:space="preserve">quare proportio partis vaſis vacuæ, ad totum <lb/>vas, erit vt min .33. ferè ad horas .2. min .24. quod per ſe patet, </s> <s xml:space="preserve">tunc ſi demptum fue- <pb facs="0302" n="290"/><fw type="head">IO. BAPT. BENED.</fw> rit tempus .33. minutorum ex h oris .2. min .24. reliquum erit hora .1. min .51. vnde <lb/>proportio aquæ, quæ in vaſe reperitur, ad eam, quæ totum vas implet, erit vt .111. <lb/>ad .144. </s> <s xml:space="preserve">Quare nunc poſſumus rectè dicere ex regula de tribus ſi .111. indigent mi-<lb/>nuta .33. temporis, ergo .144. indigent min .43. horæ, in quo tempore implebitur to-<lb/>tum vas omnibus fiſtulis operantibus.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Aliæ circuli noua paßiones.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">VTad aſcendendum ignis, & ad <choice><ex>deſcendendum</ex><am>deſcendendũ</am></choice> quicquid graue natum eſt, ita ad <lb/>ſpeculandum humanus intellectus. </s> <s xml:space="preserve">nec quieſcit, dum poteſt, eſt enim ver-<lb/>ſatile, <choice><ex>agitandoque</ex><am>agitandoq́;</am></choice> ſeſe cauſis rerum immiſcere, & abditum aliquid rimari, <lb/>conatur, & eſt in nobis, quaſi Diogenes quidam in Dolio.</s> </p> <p> <s xml:space="preserve">Tibi igitur mitto quod vltimò inueni, alias ſcilicet nouas circuli paſſiones, <lb/>quæ ita ſe <choice><ex>habent</ex><am>habẽt</am></choice>. </s> <s xml:space="preserve">Sit circulus <seg type="var">.a.b.c.</seg> in quo ſit <seg type="var">.a.d.</seg> latus quadrati inſcriptibilis in ipſo <lb/>circulo, ct <seg type="var">.b.c.</seg> ſit diameter ad rectos cum <seg type="var">.a.d.</seg> in puncto <seg type="var">.e.</seg> quod medium erit inter <lb/>a. et <seg type="var">.d.</seg> ex .3. tertij Eucli. ſit ſimiliter <seg type="var">.a.f.</seg> contingens ipſum circulum in puncto <seg type="var">.a.</seg> quæ <lb/>protracta ſit vſque ad punctum <seg type="var">.f.</seg> interſectionis cum diametro protracto, quod ita <lb/>eueniet cum anguli <seg type="var">.a.e.f.</seg> et <seg type="var">.f.a.e.</seg> minores ſint duobus rectis, eo quod angulus <seg type="var">.f.a.e.</seg> <lb/>acutus ſit, cum <seg type="var">.a.d.</seg> tranſeat inter centrum et <seg type="var">.f</seg>.</s> </p> <p> <s xml:space="preserve">Dico nunc quod productum diametri <seg type="var">.b.c.</seg> in parte <seg type="var">.c.e.</seg> ipſius, æqualis erit produ-<lb/>cto ipſius <seg type="var">.c.f.</seg> in <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Protrahatur imaginatione <seg type="var">.b.a.</seg> et <seg type="var">.a.c.</seg> </s> <s xml:space="preserve">vnde ex .26. tertij Euclid. <lb/>habebimus angulum <seg type="var">.d.a.c.</seg> æqualem angulo <seg type="var">.a.b.c</seg>. </s> <s xml:space="preserve">ſed ex .31. eiuſdem angulus <seg type="var">.f.a.<lb/>c.</seg> æqualis eſt angulo <seg type="var">.b</seg>. </s> <s xml:space="preserve">quare æqualis erit angulo <seg type="var">.d.a.c.</seg> & ita habebimus per .3. ſexti <lb/>eandem proportionem <seg type="var">.f.c.</seg> ad <seg type="var">.c.e.</seg> quæ <seg type="var">.f.a.</seg> ad <seg type="var">.a.e.</seg> ſed <seg type="var">.a.f.</seg> eſt æqualis ſemidiametro <lb/>circuli propoſiti, </s> <s xml:space="preserve">propterea quod ſi producta fuerit à puncto <seg type="var">.a.</seg> ad centrum <seg type="var">.o.</seg> ſemi <lb/>diameter <seg type="var">.a.o.</seg> hæc cum <seg type="var">.o.e.</seg> faciet dimidium angulirecti, cum ex ſuppoſito <seg type="var">.a.d.</seg> la-<lb/>tus ſit quadrati inſcriptibilis in ipſo circulo. </s> <s xml:space="preserve">& cum <seg type="var">.a.f.</seg> rectum ex .17. tertij, vnde an <lb/>gulus <seg type="var">.f.</seg> erit ſimiliter medietas recti ex .32. primi, </s> <s xml:space="preserve">quare ex .6. eiuſdem <seg type="var">.a.f.</seg> æqualis <lb/>erit <seg type="var">.a.o</seg>. </s> <s xml:space="preserve">Ergo cum proportio <seg type="var">.f.c.</seg> ad <seg type="var">.c.e.</seg> ſit. vt <seg type="var">.f.a.</seg> ad <seg type="var">.a.e.</seg> erit ſimiliter vt <seg type="var">.b.c.</seg> ad <seg type="var">.a.d.</seg> <lb/>hoc eſt ut dupli ad duplum, vnde ex .15. ſexti <lb/>manifeſtum erit propoſitum, ex quo alia paſ-<lb/> <ptr xml:id="fig-0302-01a" corresp="fig-0302-01" type="figureAnchor"/> ſio oritur, hoc eſt, quod productum <seg type="var">.f.c.</seg> in <seg type="var">.a.<lb/>d.</seg> æ quale ſit qua drato ipſius <seg type="var">.a.c.</seg> ratio eſt, quia <lb/>quadratum <seg type="var">.a.c.</seg> æ quale eſt producto <seg type="var">.b.c.</seg> in <seg type="var">.c.<lb/>e.</seg> eo quod <seg type="var">.a.c.</seg> media proportionalis eſt inter <seg type="var">.<lb/>b.c.</seg> et <seg type="var">.c.e.</seg> ex ſimilitudine triangulorum <seg type="var">.a.b.c.</seg> <lb/>et <seg type="var">.e.a.c.</seg> nam anguli <seg type="var">.b.a.c.</seg> et <seg type="var">.a.e.c.</seg> recti ſunt <lb/>et <seg type="var">.c.</seg> <choice><ex>communis</ex><am>cõmunis</am></choice>, vnde <seg type="var">.b.</seg> erit æqualis <seg type="var">.e.a.c.</seg> ex .32 <lb/>primi, ſequitur etiam, quod <seg type="var">.a.c.</seg> ſit media pro <lb/>portionalis inter <seg type="var">.a.d.</seg> et <seg type="var">.f.c.</seg> & hæc etiam erit <lb/>alia circuli paſſio, & quia <seg type="var">.a.c.</seg> eſt latus octago-<lb/>ni igitur tale latus <choice><ex>medium</ex><am>mediũ</am></choice> proportionale erit <lb/>inter latus quadrati. et <seg type="var">.f.c.</seg> <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> circuli, quę <lb/>quidem <seg type="var">.f.c.</seg> eſt una portio diametri quadrati circunſcriptibilis ipſum circulum inter <lb/>circulum & angulum ipſius quadrati.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0302-01" corresp="fig-0302-01a"> <graphic url="0302-01"/> </figure> </div> </body> </floatingText> <pb facs="0303" n="291"/> <fw type="head">EPISTOLAE.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Quod incendium, ex reflexione radiorum ſolarium, non fiat in cen <lb/>tro ſpeculi ſpharici, & aliquid contra Cardanum, & <lb/>de moturadiorum ſolarium.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">ITerum tibi dico, quod radij illi ſolares, quià diuerſis punctis ipſius ſolaris corpo-<lb/>ris veniunt, tranſeuntes per centrum ſpeculi ſphærici concaui, quamuis à ſuper-<lb/>ficie ſpeculi ad centrum ipſum reflectantur, vt alíâs tibi dixi, nihilominus nullo mo <lb/>do poſſunt aliquod obiectum incendere duabus ex cauſis, quarum vna eſt, quia cum <lb/>Sol valde remotus ſit à nobis, valde etiam acutus generatur angulus coni radiorum <lb/>in centro ſpeculi, vnde à parua ſuperficie ipſius ſpeculi reflectuntur, </s> <s xml:space="preserve">quare pauciſſi-<lb/>mi radij ſunt qui reflectantur in ipſo centro, & propterea non ſufficiunt ad combu <lb/>ſtionem alicuius obiecti. </s> <s xml:space="preserve">Alia verò cauſa eſt, quod quamuis multi, & ſufficientes <lb/>radij fuiſſentad <choice><ex>comburendum</ex><am>cõburendũ</am></choice> velociter quoduis obiectum. </s> <s xml:space="preserve">impoſſibile tamen omnino <lb/>eſset, vt aliquod obiectum comburerent, propterea quod cum radij incidentes de-<lb/>beant per centrum tranſire, obiectum combuſtibile, vt opacum, obſtaret ipſis radijs, <lb/>ne vlterius tranſirent, vnde nulla fieret reflexio, ſed etiam ſi dicti radij in centro re <lb/>flexi, ſufficerent ad combuſtionem, incidentes hoc magis eſſicerent. </s> <s xml:space="preserve">& ita abſque <lb/>vllo ſpeculo, omnia & in quolibet loco comburerentur, quod manifeſtè falſum eſt. <lb/></s> <s xml:space="preserve">Deſine igitur mihi citate Lucillum Philalteum, qui in philoſophia mathematica <lb/>fuit omnium imperitiſſimus. </s> <s xml:space="preserve">Verum ſpeculum vſtorium illud eſt quod ab Alhazem <lb/></s> <s xml:space="preserve">Deinde à Vitellìone deſcribitur.</s> </p> <p> <s xml:space="preserve">Quod deinde verum ſit, <choice><ex>vmbram</ex><am>vmbrã</am></choice> vniuſcuiuſque corporis opaci à Sole productam <lb/>ſemper eſſe centum <choice><ex>nouemque</ex><am>nouemq́;</am></choice> vicibus maiorem diametro eiuſdem corporis, nego.</s> </p> <p> <s xml:space="preserve">Imaginemur <seg type="var">.s.</seg> 1. diametrum eſſe illius circuli, quo vltimi radij ſolares veniunt tan <lb/>gentes corpus cuius diameter ſit <seg type="var">.c.e.</seg> et <seg type="var">.a.i.</seg> ſit diameter alterius circuli eiuſdem cor-<lb/>poris ſolaris à quo vltimi radij veniunt tangentes corpus, cuius diameter ſit <seg type="var">.f.g.</seg> in <lb/>eadem diſtantia, & eodem ſitu prioris corporis. </s> <s xml:space="preserve">Tunc conus vmbræ ipſius <seg type="var">.f.g.</seg> ſit <seg type="var">.f.<lb/>g.q.</seg> & ipſius <seg type="var">.c.e.</seg> ſit <seg type="var">.c.n.e.</seg> centrum autem ſolare ſit <seg type="var">.o.</seg> conorum verò axes ſint <seg type="var">.t.n.q</seg>. <lb/></s> <s xml:space="preserve">tunc ex ſuppoſito <seg type="var">.q.f.a</seg>: <seg type="var">n.c.s</seg>: <seg type="var">n.e.l</seg>: et <seg type="var">.q.g.i.</seg> erunt omnes contigui corpori ſolari, vn-<lb/>de ex .17. tertij Eucli. anguli <seg type="var">.o.a.q.</seg> et <seg type="var">.o.s.n.</seg> erunt recti. </s> <s xml:space="preserve">protracta deinde cum fu erit <lb/><seg type="var">a.s.</seg> habebimus angulos <seg type="var">.u.a.s.</seg> et <seg type="var">.u.s.a.</seg> minores duobus rectis. </s> <s xml:space="preserve">Quare <seg type="var">.n.s.</seg> concurret <lb/> <ptr xml:id="fig-0303-01a" corresp="fig-0303-01" type="figureAnchor"/> <pb facs="0304" n="292"/><fw type="head">IO. BAPT. BENED.</fw> cum <seg type="var">.a.q.</seg> in puncto <seg type="var">.u</seg>. </s> <s xml:space="preserve">Nunc verò ſi vmbra <seg type="var">.t.q.</seg> tanto maior eſt <seg type="var">.f.g.</seg> quanto .109. eſt <lb/>vno et <seg type="var">.t.n.</seg> etiam <choice><ex>tanto</ex><am>tãto</am></choice> maior <seg type="var">.c.e.</seg> ergò eadem proportio erit <seg type="var">.q.t.</seg> ad <seg type="var">.t.f.</seg> quę <seg type="var">.n.t.</seg> ad <seg type="var">.t.<lb/>c.</seg> ſed cum angulus <seg type="var">.t.</seg> communis ſit ambobus triangulis <seg type="var">.q.t.f.</seg> et <seg type="var">.n.t.c.</seg> ſequitur ex .6. <lb/>ſexti dictos triangulos æ quiangulos eſſe. </s> <s xml:space="preserve">Vnde ſi anguli <seg type="var">.t.n.c.</seg> et <seg type="var">.t.q.f.</seg> æ quales inui <lb/>cem ſunt, ergo <seg type="var">.q.f.</seg> æquidiſtans erit <seg type="var">.n.c.</seg> quod eſt impoſſibile, quia nunc demonſtra-<lb/>uimus ipſas concurrere in puncto <seg type="var">.u</seg>. </s> <s xml:space="preserve">Quare non eſt eadem proportio <seg type="var">.q.t.</seg> ad <seg type="var">.t.f.</seg> quæ <lb/><seg type="var">n.t.</seg> ad <seg type="var">.t.c.</seg> decipitur ergo Cardanus in .4. lib. de ſubtilitate.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0303-01" corresp="fig-0303-01a"> <graphic url="0303-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Circa illud deinde quod à me quæris, hoc eſt, quæ ſit cauſa, quod nos videmus <lb/>radium ſolarem tardiſſimè moueri, cum alias tibi dixerim ipſum qualibet hora cir-<lb/>ca terram quindecim gradus perficere, reſpondeo, quod radius ille quem videmus, <lb/>exempli gratia, in aliquo cubiculo, nunquam eſt idem numero, ſed quia ipſi radij <lb/>nullo modo differunt inter ſe, niſi in numero, proptera putamus eundem ſemper eſſe, <lb/>cum ſemper alius, atque alius ſit, quorum vnuſquiſque (de illis loquor, qui ad hunc <lb/>terræ globum perueniunt) circa terram reuoluitur ſpatio .24. horarum, & cum quili <lb/>bet circulus diuidatur in .360. gradus, quorum vigeſimaquarta pars eſt .15. verum <lb/>eſt igitur, quod tibi iam dixeram.</s> </p> <figure place="here"> <graphic url="0304-01"/> </figure> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">OPERATIONES DIVERSAE AB ALIIS <lb/>Michaelis Stifelij.</head> <head rend="italics" xml:space="preserve">Conrado Terl.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> in .2. exemplo. II. cap. Stifelius ſcribit in .3. lib. pag .282. non nego <lb/>quin pulchrum ſit, ſed alijs pulchrioribus modis poſſumus illud idem de-<lb/>monſtrare; </s> <s xml:space="preserve">cogita igitur ſuperficiem rectangulam, cuius medietas ſit <choice><ex>triam</ex><am>triã</am></choice> <lb/>gulus rectangulus <seg type="var">.a.b.g.</seg> vnde ex ſuppoſito nobis cognita erit ſuperficies <lb/>ipſius trianguli, tanquam dimidium totius parallelogrammi rectanguli cogniti. <lb/></s> <s xml:space="preserve">Quare ex .25. ſecundi triangulorum <choice><ex>Monteregij</ex><am>Mõteregij</am></choice>, cognita nobis <choice><ex>erunt</ex><am>erũt</am></choice> latera <seg type="var">.a.b.</seg> et <seg type="var">.b.g</seg>.</s> </p> <p> <s xml:space="preserve">Alia etiam breuiori methodo idem poſſumus eſſicere, mediante angulo <seg type="var">.b.</seg> recto, <lb/>eo quod cum nobis cognita ſit ſuperficies trianguli ſimul <choice><ex>cum</ex><am>cũ</am></choice> baſi <seg type="var">.a.g.</seg> cognita etiam <lb/>nobis fit perpendicularis <seg type="var">.b.d.</seg> à puncto <seg type="var">.b.</seg> ad baſim, & conſequenter cognitum no-<lb/>bis erit productum ipſius <seg type="var">.a.d.</seg> in <seg type="var">.d.g.</seg> & quia nobis cognita eſt <seg type="var">.a.g.</seg> & eius medietas, <pb facs="0305" n="293"/><fw type="head">EPISTOL AE.</fw> ideo vnaquæque eius pars <seg type="var">.a.d.</seg> et <seg type="var">.d.g.</seg> ſimiliter nobis cognita erit ex quinta ſecundi <lb/>Eucl. </s> <s xml:space="preserve">vnde ex penultima primi habebimus propoſitum.</s> </p> <p> <s xml:space="preserve">Poſſumus item circulum mente concipere cuius <seg type="var">.a.g.</seg> ſit diameter, & ab eius cen-<lb/>tro <seg type="var">.e.</seg> protracta cum fuerit <seg type="var">.e.b.</seg> quæ nobis cognita erit, vt medietas ipſius <seg type="var">.a.g.</seg> de cu <lb/>ius potentia, dempta <choice><ex>cum</ex><am>cũ</am></choice> fuerit potentia <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">b.o.</seg> remanebit nobis potentia ipſius <seg type="var">.d.<lb/>e.</seg> & ita eius longitudo, quæ addita medietati <seg type="var">.e.g.</seg> & detracta à dimidio <seg type="var">.e.d.</seg> erunt <lb/>nobis cognitæ <seg type="var">.a.d.</seg> et <seg type="var">.d.g.</seg> vnde <seg type="var">.b.g.</seg> et <seg type="var">.b.d.</seg> remanebunt nobis cognitæ ex dicta pe-<lb/>nultima primi Eucli. </s> <s xml:space="preserve">huiuſmodi figuram videbis in dicto .25. problemate .2. li. Mon-<lb/>tisregij.</s> </p> <p> <s xml:space="preserve">Aliter etiam poſſumus hoc idem efficere.</s> </p> <p> <s xml:space="preserve">Sit rectangulus hic ſubſcriptus <seg type="var">.a.b.c.u.</seg> ſuperficiei cognitę ſimul cum diametro <seg type="var">.a.<lb/>c.</seg> extendatur imaginatione <seg type="var">.b.c.</seg> vſque ad, f. ita quod <seg type="var">.c.f.</seg> æqualis ſit <seg type="var">.c.u.</seg> intelligan-<lb/><choice><ex>turque</ex><am>turq́;</am></choice> quadrata <seg type="var">.g.f</seg>: <seg type="var">g.u.</seg> ct <seg type="var">.u.f.</seg> vnde <choice><ex>summa</ex><am>sũma</am></choice> <choice><ex>quadratorum</ex><am>quadratorũ</am></choice> <seg type="var">.g.u</seg>:u.f. cognita nobis erit ex <lb/>penultima primi. </s> <s xml:space="preserve">nam <seg type="var">.a.c.</seg> data nobis fuit, quare <choice><ex>ſummam</ex><am>ſummã</am></choice> <seg type="var">.g.u</seg>:u.b: et <seg type="var">.u.f.</seg> cognoſce-<lb/>mus, cui <choice><ex>summæ</ex><am>sũmæ</am></choice> addito ſuplemento <seg type="var">.d.e.</seg> æ quali <seg type="var">.u.<lb/>b.</seg> dabit nobis <choice><ex>cognitum</ex><am>cognitũ</am></choice> quadrarum <seg type="var">.g.f.</seg> totale, qua <lb/> <ptr xml:id="fig-0305-01a" corresp="fig-0305-01" type="figureAnchor"/> re cognoſcetur eius radix <seg type="var">.b.f.</seg> cognita igitur <seg type="var">.b.f.</seg> <lb/>cum pro ducto <seg type="var">.b.u.</seg> illico ex .5. ſecundi cognoſce-<lb/>tur <seg type="var">.b.c.</seg> et <seg type="var">.c.f.</seg> forte cognita <seg type="var">.b.f.</seg> diuiſa <choice><ex>per</ex><am>ꝑ</am></choice> æqualia <lb/>in puncto <seg type="var">.t.</seg> & per inæqualiz in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.c</seg>. </s> <s xml:space="preserve">Nam qua <lb/><choice><ex>dratum</ex><am>dratũ</am></choice> ipſius <seg type="var">.t.f.</seg> cognitum, ęquatur <choice><ex>rectangulo</ex><am>rectãgulo</am></choice> <seg type="var">.b.u.</seg> <lb/><choice><ex>cum</ex><am>cũ</am></choice> quadrato ipſius <seg type="var">.t.c.</seg> <choice><ex>dempto</ex><am>dẽpto</am></choice> igitur rectangulo, <seg type="var">b.<lb/>u.</seg> ex quadrato ipſius <seg type="var">.t.f.</seg> relinquetur quadratum <lb/><choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.t.c.</seg> cognitum & eius radix <seg type="var">.t.c.</seg> qua addita ipſi <lb/>medietati <seg type="var">.b.t.</seg> & <choice><ex>dempta</ex><am>dẽpta</am></choice> ex medietate <seg type="var">.f.t.</seg> relinque-<lb/>tur propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0305-01" corresp="fig-0305-01a"> <graphic url="0305-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Similiter de tertio exemplo eiuſdem Stifelij <lb/>infero.</s> </p> <p> <s xml:space="preserve">Sit rectangulus <seg type="var">.a.b.c.u.</seg> cuius diametri <seg type="var">.a.c.</seg> quantitas, ſimul cum proportione late <lb/>rum <seg type="var">.b.c.</seg> et <seg type="var">.b.a.</seg> nobis data ſit. </s> <s xml:space="preserve">cum autem ſcire voluerimus eius ſuperficiem <seg type="var">.b.u.</seg> cla-<lb/>rum eſt, quod cum nobis data ſit proportio <seg type="var">.b.c.</seg> ad <seg type="var">.b.a.</seg> illico cognoſcemus <choice><ex>etiam</ex><am>etiã</am></choice> pro-<lb/>portionem quadrati ipſius <seg type="var">.b.c.</seg> ad quadratum ip-<lb/> <ptr xml:id="fig-0305-02a" corresp="fig-0305-02" type="figureAnchor"/> ſius <seg type="var">.b.a.</seg> cum dupla ſit ei quæ <seg type="var">.b.c.</seg> ad <seg type="var">.b.a.</seg> ita etiam <lb/>& aggregati dictorum quadratorum ad quadra-<lb/>tum ipſius <seg type="var">.b.a.</seg> hoc eſt nota erit nobis proportio <lb/>quadrati ipſius <seg type="var">.a.c.</seg> diagonalis ad quadratum ip-<lb/>ſius <seg type="var">.a.b.</seg> idem dico de quadrato <seg type="var">.b.c.</seg> ideſt quod <lb/>proportio quadrati ipſius <seg type="var">.a.c.</seg> ad quadratum <seg type="var">.b.c.</seg> <lb/>cognita nobis erit, ſed <seg type="var">.a.c.</seg> data nobis fuit, qua-<lb/>re cognoſcemus etiam omnia dicta quadrata eo-<lb/><choice><ex>rumque</ex><am>rumq́;</am></choice> radices <seg type="var">.a.b.</seg> et <seg type="var">.b.c.</seg> </s> <s xml:space="preserve">quare & ſuperficiem re-<lb/>ctanguli quæſitam.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0305-02" corresp="fig-0305-02a"> <graphic url="0305-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quartum exemplum etiam faciliori via poteſt <lb/>ſolui, propterea, quod cum nobis cognita ſit ba-<lb/>ſis trianguli cum ſumma reliquorum laterum, & <lb/><choice><ex>cum</ex><am>cũ</am></choice> angulo oppoſito baſi ipſius reliqua cognita no <lb/>bis emergunt ex .15. problemate ſecundi lib. de Triangulis ipſius Monteregii.</s> </p> <pb facs="0306" n="294"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Vel ſi tibi placet, accipe hanc aliam methodum à me excogitatum.</s> </p> <p> <s xml:space="preserve">Duplicetur <choice><ex>triangulum</ex><am>triangulũ</am></choice> <seg type="var">.a.b.c.</seg> <choice><ex>orthogonium</ex><am>orthogoniũ</am></choice>, & fiat <choice><ex>rectangulum</ex><am>rectangulũ</am></choice> <seg type="var">.b.u.</seg> vt in mea figura <lb/>ſecundi exempli hic vides. </s> <s xml:space="preserve"><choice><ex>producaturque</ex><am>producaturq́;</am></choice> <seg type="var">.b.c.</seg> quouſque <seg type="var">.c.f.</seg> æqualis ſit <seg type="var">.c.u.</seg> vnde <seg type="var">.b.f.</seg> <lb/>cognita nobis erit ex hypotheſi, </s> <s xml:space="preserve">quare cognoſcemus etiam quadratum <seg type="var">.g.f.</seg> à quo <lb/><choice><ex>demptum</ex><am>demptũ</am></choice> cum fuerit <choice><ex>aggregatum</ex><am>aggregatũ</am></choice> quadratorum <seg type="var">.g.u.</seg> et <seg type="var">.u.f.</seg> nobis <choice><ex>cognitum</ex><am>cognitũ</am></choice> (nam quadra <lb/>ta <seg type="var">.g.u.</seg> et <seg type="var">.u.f.</seg> æqualia ſunt quadrato ipſius <seg type="var">.a.c.</seg> diagonalis datę) remanebit aggrega-<lb/>tum <choice><ex>ſupplementorum</ex><am>ſupplemẽtorũ</am></choice> cognitum, </s> <s xml:space="preserve">quare eius medietas cognoſcetur ideſt <seg type="var">.b.u.</seg> vndæ ex <num value="5">.<lb/>5.</num> ſecundi Eucli. vt ſuperius diximus cognoſcetur etiam <seg type="var">.b.c.</seg> et <seg type="var">.c.f.</seg> diſtinctæ.</s> </p> <p> <s xml:space="preserve">Idem aſſero de <choice><ex>exemplo</ex><am>exẽplo</am></choice> Gemmæ Friſij à Stifelio citato in Appendice regulæ falſi.</s> </p> <p> <s xml:space="preserve">Sit gratia exempli rectangulum hicſubſcriptum <seg type="var">.a.b.</seg> datæ ſuperficiei data etiam <lb/>nobis ſit proportio <seg type="var">.a.e.</seg> ad <seg type="var">.e.b.</seg> laterum producentium, <choice><ex>cogitemusque</ex><am>cogitemusq́;</am></choice> <seg type="var">.a.e.</seg> producta <lb/>vſque ad <seg type="var">.o.</seg> ita vt <seg type="var">.e.o.</seg> æqualis ſit ipſi <seg type="var">.e.b.</seg> imagine <lb/> <ptr xml:id="fig-0306-01a" corresp="fig-0306-01" type="figureAnchor"/> mus <choice><ex>etiam</ex><am>etiã</am></choice> perfectum eſſe quadratum <seg type="var">.b.o.</seg> vnde ex <lb/>prima ſexti ſeu .18. vel .19. ſeptimi vel .15. quinti <lb/>eadem proportio erit ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.b.o.</seg> vt <seg type="var">.a.e.</seg> ad <lb/><seg type="var">e.o.</seg> vel ad <seg type="var">.e.b.</seg> </s> <s xml:space="preserve">quare ex regula de tribus, cogno-<lb/>ſcemus quadratum <seg type="var">.b.o.</seg> & eius <choice><ex>radicem</ex><am>radicẽ</am></choice> <seg type="var">.e.o.</seg> & ex ea <lb/>demregula cognoſcemus <seg type="var">.a.e.</seg> cum cognita nobis ſit <seg type="var">.e.o.</seg> ſimul cum proportione <seg type="var">.e.o.</seg> <lb/>ad <seg type="var">.e.a</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0306-01" corresp="fig-0306-01a"> <graphic url="0306-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Quod circulus ſit figura infinitorum angulorum hoc eſt <lb/>ultima poligoniarum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">SEd quod idem Stifelius in Appendice ſecundi libri dicat circulum eſſe figuram <lb/>poligoniam, non eſt ita mirandum, nam & alij multi doctiſſimi viri hanc <lb/>veritatem cognouerunt, de Leone Baptiſta Alberto nihil dicam, cum ipſe fateatur <lb/>hoc accepiſſe à philoſophis, vt etiam refert Ariſt. de ſphæratertio de cœlo. </s> <s xml:space="preserve">conſi-<lb/>dera quæſo in circulo, quod cum angulus contingentiæ ſit angulus, quamuis <choice><ex>omnium</ex><am>omniũ</am></choice> <lb/>acutorum rectilineorum anguſtiſſimus, vnde ex communi ratione ſequitur reliquum <lb/>ex duobus rectis rectilineis eſſe angulum, & ſi omnium obtuſorum rectilineorum ſit <lb/>ampliſſimum, tanto magis igitur erit angulus, id quod remanet ex duobus rectis re <lb/>ctilineis, detractis <choice><ex>cum</ex><am>cũ</am></choice> fuerint duobus angulis contingentiæ, qui quidem angulus erit <lb/>in quouis puncto circunferentiæ ipſius circuli, idem intelligendum eſt de ſphæra, <lb/>cuius angulus eſt reſiduum ex quatuor rectis ſolidis, detractis cum fuerint quatuor <lb/>angulis contingentiæ <choice><ex>ſolidisque</ex><am>ſolidisq́;</am></choice>.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Explanatio .25. Problematis lib. 2. Monteregij.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod in .25. problemate .2. lib. de triangulis Monteregium non intelligas, mi-<lb/>rum non eſt, eo quod quandoque bonus dormitat Homerus. </s> <s xml:space="preserve">Puto enim il-<lb/>lud problema ab ipſo Monteregio non fuiſſe viſitatum. </s> <s xml:space="preserve">Sed ne me aliquo modo <lb/>culpes, accipe hanc <choice><ex>aliam</ex><am>aliã</am></choice> <choice><ex>methodum</ex><am>methodũ</am></choice> a me aliter <choice><ex>etiam</ex><am>etiã</am></choice> <choice><ex>excogitatam</ex><am>excogitatã</am></choice> in eadem ipſius figura.</s> </p> <pb facs="0307" n="295"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve"><choice><ex>Propoſitum</ex><am>Propoſitũ</am></choice> ſit nobis triangulum <seg type="var">.a.b.g.</seg> cuius baſis data ſit cum area, ſeu perpendi-<lb/>culari <seg type="var">.a.d.</seg> cum angulo etiam <seg type="var">.a.</seg> ad cognoſcendum autem <seg type="var">.a.b.</seg> et <seg type="var">.b.g.</seg> cogitemus circu <lb/>lum <seg type="var">.a.b.q.g.</seg> circunſcribere ipſum triangulum cuius diameter <seg type="var">.p.q.</seg> ad rectos ſe-<lb/>cet baſim <seg type="var">.b.g.</seg> in puncto <seg type="var">.m.</seg> cogitemus etiam <seg type="var">.b.p.</seg> et <seg type="var">.p.g.</seg> vnde ex .20. ter-<lb/>tij Euclid. angulus <seg type="var">.b.p.g.</seg> æqualis erit <lb/> <ptr xml:id="fig-0307-01a" corresp="fig-0307-01" type="figureAnchor"/> angulo <seg type="var">.a.</seg> & angulus <seg type="var">.m.p.b.</seg> erit eius di <lb/>midium, quod ex te ipſo cognoſces, & <lb/><choice><ex>angulus</ex><am>angulꝰ</am></choice> <seg type="var">.p.b.m.</seg> ſimiliter cognoſcetur, <lb/></s> <s xml:space="preserve">quare ex .29. primi eiuſdem Montere <lb/>gij cognoſcemus <seg type="var">.p.m.</seg> et <seg type="var">.p.b.</seg> (nam <seg type="var">.b.<lb/>m.</seg> datum fuit, vt dimidium totius ba-<lb/>ſis <seg type="var">.b.g.</seg>) ducta poſtea <seg type="var">.b.q.</seg> ex <choice><ex>eadem</ex><am>eadẽ</am></choice> .29. <lb/>cognoſcemus <seg type="var">.p.q.</seg> cum <seg type="var">.p.b.</seg> iam cogni <lb/>ta fuerit, à qua <seg type="var">.p.q.</seg> (diametro) <choice><ex>dempta</ex><am>dẽpta</am></choice> <lb/><seg type="var">p.m.</seg> remanebit <seg type="var">.q.m.</seg> cognita, <choice><ex>cum</ex><am>cũ</am></choice> qua <lb/>iuncta cum fuerit <seg type="var">.m.t.</seg> æquali <seg type="var">.a.d.</seg> per <lb/>pendiculari, dabitur <seg type="var">.q.t.</seg> et <seg type="var">.t.p.</seg> inter <lb/>quas <seg type="var">.a.t.</seg> media proportionalis loca-<lb/>tur, </s> <s xml:space="preserve">quare cognoſcemus <seg type="var">.a.t.</seg> quæ ſinus <lb/>eſt arcus <seg type="var">.a.p.</seg> vnde cognitus erit arcus <lb/><seg type="var">a.p.</seg> ſed arcus <seg type="var">.p.g.</seg> cognitus eſt median <lb/>te angulo <seg type="var">.p.b.g.</seg> cognito, qui quidem <lb/>arcus <seg type="var">.p.g.</seg> ſi coniunctus fuerit cum arcu <seg type="var">.p.a.</seg> cognoſcemus compoſitum <seg type="var">.a.g.</seg> & eius <lb/>chorda ſimiliter (hoc eſt <choice><ex>ſecundum</ex><am>ſecundũ</am></choice> latus) qua cognita, illico cognoſcemus chordam <lb/><seg type="var">a.b.</seg> hoc eſt tertium latus trianguli propoſiti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0307-01" corresp="fig-0307-01a"> <graphic url="0307-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Quæ<unclear reason="illegible"/>dam not and a in Federicum Comandinum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">PVtabas enim me ioco dixiſſe Federicum Comandinum non omnino irrepræ-<lb/>henſibilem eſſe, vide igitur, quod ſcribit in quinto lemmate in decimam <lb/>propoſitionem libr .2. de inſidentibus aquæ Archimedis, volens demonſtra-<lb/>re eandem eſſe proportionem <seg type="var">.l.b.</seg> ad <seg type="var">.b.m.</seg> quæ <seg type="var">.c.e.</seg> ad <seg type="var">.e.a.</seg> vbi eſt aliquo modo pro-<lb/>lixum, mediante linea <seg type="var">.c.p.</seg> cum ſuis partibus, citans etiam antecedens lemma extra <lb/>propoſitum, eo quod nec in antecedente lemmate, nec in alio, ipſe vnquam proba <lb/>uerit proportionem <seg type="var">.c.d.</seg> ad <seg type="var">.d.q.</seg> eſſe, vt <seg type="var">.l.b.</seg> @d.b.m. ſed ne putes me falli, tibi demon <lb/>ſtrabo non eſſe neceſſarium ducere lineam <seg type="var">.c.m.p.</seg> vel <seg type="var">.q.p.</seg> eo quod <choice><ex>cum</ex><am>cũ</am></choice> per quintam <lb/>lib. de quadratura parabolę Archimedis, ita ſit <seg type="var">.c.d.</seg> ad <seg type="var">.d.e.</seg> vt <seg type="var">.l.b.</seg> ad <seg type="var">.b.m.</seg> exiſtente <lb/><seg type="var">a.c.</seg> dupla ipſi <seg type="var">.d.c.</seg> et <seg type="var">.e.c.</seg> dupla ipſi <seg type="var">.g.c.</seg> et <seg type="var">.l.d.</seg> dupla ipſi <seg type="var">.l.b</seg>: erit, primo componen-<lb/>do <seg type="var">.c.e.</seg> ad <seg type="var">.e.d.</seg> vt <seg type="var">.l.d.</seg> ad <seg type="var">.d.m.</seg> & per æqualitatem proportionum, ita erit <seg type="var">.e.g.</seg> ad <seg type="var">.e.d.</seg> <lb/>vt <seg type="var">.b.d.</seg> 2d.d.m. & per .19. quinti Eucli. ita erit <seg type="var">.e.g.</seg> ideſt <seg type="var">.g.c.</seg> ad <seg type="var">.g.d.</seg> vt <seg type="var">.b.d.</seg> ideſt <seg type="var">.l.b.</seg> <lb/>ad <seg type="var">.b.m.</seg> ſed <seg type="var">.c.g.</seg> ad <seg type="var">.g.d.</seg> eft vt <seg type="var">.c.e.</seg> ad <seg type="var">.e.a.</seg> ratio eſt, quia componendo ita eſt <seg type="var">.c.d.</seg> ad <seg type="var">.d.<lb/>g.</seg> vt <seg type="var">.c.a.</seg> ad <seg type="var">.a.e.</seg> & hoc eſt, quia permutando, ita eſt <seg type="var">.a.c.</seg> ad <seg type="var">.d.c.</seg> vt <seg type="var">.a.e.</seg> ad <seg type="var">.d.g.</seg> & hoc <lb/>verum eſt ex .19. quinti eo quod totius <seg type="var">.a.c.</seg> ad totum <seg type="var">.d.c.</seg> eft vt abſciſſi <seg type="var">.e.c.</seg> ad abſciſ <lb/>ſum <seg type="var">.g.c.</seg> vt ſupradixi.</s> </p> <pb facs="0308" n="296"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Sed etiam alio vniuerſaliori modo potes probare, quod ita ſit <seg type="var">.u.x.</seg> ad <seg type="var">.x.y.</seg> vt <seg type="var">.c.e.</seg> <lb/>ad <seg type="var">.e.a.</seg> cogitando in linea <seg type="var">.c.a.</seg> punctum quoddam quod vocabimus ſimiliter <seg type="var">.y.</seg> in <lb/>tali ſitu locatum, quod diuidat <seg type="var">.c.a.</seg> eadem proportione qua <seg type="var">.y.</seg> diuidit <seg type="var">.u.s.</seg> vnde cum <lb/><seg type="var">e.s.</seg> diuiſa eodem modo etiam ſit à puncto <seg type="var">.s.</seg> ex ſupradicta quinta lib. de quadratura <lb/>parabolæ, erit igitur proportio <seg type="var">.a.y.</seg> ad <seg type="var">.y.c.</seg> vt <seg type="var">.e.s.</seg> ad <seg type="var">.s.c.</seg> per .11. quinti Eucli. </s> <s xml:space="preserve">& com <lb/>ponendo ita erit <choice><ex>totius</ex><am>totiꝰ</am></choice> <seg type="var">.a.c.</seg> ad totum <seg type="var">.y.c.</seg> vt abſcisſi <seg type="var">.s.c.</seg> ad abſciſsum <seg type="var">.s.c.</seg> </s> <s xml:space="preserve">quare reſidui <lb/><seg type="var">a.e.</seg> ad reſiduum <seg type="var">.y.s.</seg> erit vt totius <seg type="var">.a.c.</seg> ad totum <seg type="var">.y.c.</seg> & permutando, ita erit <seg type="var">.a.c.</seg> ad <seg type="var">.a.<lb/>e.</seg> vt <seg type="var">.y.c.</seg> ad <seg type="var">.y.s.</seg> & diuidendo, ita erit <seg type="var">.<lb/>c.e.</seg> ad <seg type="var">.e.a.</seg> ut <seg type="var">.c.s.</seg> ad <seg type="var">.s.y.</seg> & quia pun-<lb/> <ptr xml:id="fig-0308-01a" corresp="fig-0308-01" type="figureAnchor"/> ctum <seg type="var">.s.</seg> diuidit <seg type="var">.c.a.</seg> eodem modo, quo <lb/>x. diuidit <seg type="var">.u.s.</seg> per ſupradictam <choice><ex>quintam</ex><am>quintã</am></choice>, <lb/>ergo ita erit <seg type="var">.c.s.</seg> ad <seg type="var">.s.y.</seg> in linea <seg type="var">.c.a.</seg> vt <lb/><seg type="var">u.x.</seg> ad <seg type="var">.x.y.</seg> </s> <s xml:space="preserve">vnde ex .11. quinti <seg type="var">.c.e.</seg> ad <lb/><seg type="var">e.a.</seg> erit, vt <seg type="var">.u.x.</seg> ad <seg type="var">.x,y</seg>. </s> <s xml:space="preserve">quare ſequitur, <lb/>primum, ſecundum, tertium, & quartum lemma ſuperflua eſſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0308-01" corresp="fig-0308-01a"> <graphic url="0308-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quod deinde ponit pro corellario in fine .6. lemmatis, aliter quam per .6. lemma <lb/>poteſt demonſtrari, hoc mode. </s> <s xml:space="preserve">Nam ſuperius demonſtrauimus eandem propor-<lb/>tionem eſſe <seg type="var">.l.b.</seg> ad <seg type="var">.b.m.</seg> quæ <seg type="var">.c.e.</seg> ad <seg type="var">.e.a.</seg> <choice><ex>idem</ex><am>idẽ</am></choice> dico de proportione <seg type="var">.u.x.</seg> ad <seg type="var">.x.y.</seg> & om-<lb/>nium æquidiſtantium ad <seg type="var">.h.e.</seg> quibus rationibus mediantibus codem modo ſcies, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/><seg type="var">u.y.</seg> ad <seg type="var">.y.r.</seg> erit, vt <seg type="var">.c.d.</seg> ad <seg type="var">.d.c.</seg> & ita dico de omnibus <choice><ex>æquidiſtantibus</ex><am>æquidiſtãtibus</am></choice>. ad <seg type="var">.h.e.</seg> </s> <s xml:space="preserve">vnde <seg type="var">.l.b.</seg> <lb/>ad <seg type="var">.b.m.</seg> erit vt <seg type="var">.u.x.</seg> ad <seg type="var">.x.y.</seg> et <seg type="var">.l.m.</seg> ad <seg type="var">.m.d.</seg> vt <seg type="var">.u.y.</seg> ad <seg type="var">.y.r.</seg> per .11. quinti, ſed cum ſit <seg type="var">.l.<lb/>b.</seg> ad <seg type="var">.b.m.</seg> vt <seg type="var">.u.x.</seg> ad <seg type="var">.x.y.</seg> componendo erit <seg type="var">.l.m.</seg> ad <seg type="var">.b.m.</seg> vt <seg type="var">.u.x.</seg> ad <seg type="var">.x.y.</seg> & euerſim <seg type="var">.b.<lb/>m.</seg> ad <seg type="var">.m.b.</seg> erit, vt <seg type="var">.x.y.</seg> ad <seg type="var">.y.u.</seg> & per æquam proportionalitatem erit <seg type="var">.b.m.</seg> ad <seg type="var">.m.d.</seg> vt<unclear reason="illegible"/> <lb/><seg type="var">x.y.</seg> ad <seg type="var">.y.r.</seg> quod eſt propoſitum.</s> </p> <p> <s xml:space="preserve">Non video etiam, quare ipſe ducat lineam <seg type="var">.s.r.</seg> cum in ipſo contextu nihil ſaciac <lb/>de dicta <seg type="var">.s.r</seg>.</s> </p> <p> <s xml:space="preserve">Comentum poſtea contextus <seg type="var">.P.</seg> pulchrius eſſet, ſi diceret, quod cum ita ſit totius, <lb/><seg type="var">l.a.</seg> ad totum <seg type="var">.a.d.</seg> ſic ſe habebit abſciſſum <seg type="var">.a.i.</seg> ad abſciſſum <seg type="var">.a.z.</seg> eo quod ita eſt, vt ſcis, <lb/>hoc eſt in proportione dupla, ergo reſidui <seg type="var">.i.l.</seg> ad reſiduum <seg type="var">.d.z.</seg> erit vt totius <seg type="var">.a.l.</seg> ad <lb/>totum <seg type="var">.a.d.</seg> hoc eſt in proportione dupla.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De Viſu.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">RAtio vnde ſiat, vt videamus diſtinctè omnes eolores, cum in qualibet aeris par <lb/>te, quo lumina reſlexa poſſunt peruenire mixta ſint, & non diſtincta, oritur à <lb/>paruitate ipſius pupillæ oculorum, & à magna expanſione virtutis viſiuæ in ſuperſi-<lb/>cie concaua orbis continentis humores diaphanos oculorum per ramuſculos nerui <lb/>optici remotè ab ipſa pupilla. </s> <s xml:space="preserve">& quamuis radii luminoſi frangantur ab vnoquoque <lb/>humore diuerſimodè, hoc nihilominus maximè iuuat ad diſtinctionem radiorum, <lb/>ſed & ſi directè procederent, idem ferè eueniret, non tamen ſuis locis, cogita exem-<lb/>pli gratia lineam <seg type="var">.a.u.e.</seg> vt communis ſectio cuiuſdam plani ſecantis ſphæram oculi, <lb/>per centrum ipſius, & pupillæ, et <seg type="var">.o.</seg> punctum ſit proximum centro ipſius pupillæ, <lb/>ſed interius aliquantulum, extra <choice><ex>autem</ex><am>autẽ</am></choice> <choice><ex>oculum</ex><am>oculũ</am></choice>, ſint varij colores, vt <seg type="var">.c.n.t.</seg> in dicto plano.</s> </p> <p> <s xml:space="preserve">Iam nulli dubium eſt quod lumina quæ producuntur ab <seg type="var">.c.n.t.</seg> ad <seg type="var">.o.</seg> in ipſo <seg type="var">.o.</seg> mi- <pb facs="0309" n="297"/><fw type="head">EPISTOL AE.</fw> xta, & non diſtincta, procedendo igitur vlteriusipſi radij citra <seg type="var">.o</seg>. </s> <s xml:space="preserve">tunc <choice><ex>diſgregantur</ex><am>diſgregãtur</am></choice>, <lb/>& ſeparantur abinuicem, & <choice><ex>cum</ex><am>cũ</am></choice> perueniunt ad lineam <seg type="var">.a.u.e.</seg> ſentiuntur diſtincti alij <lb/>ab alijs. </s> <s xml:space="preserve">Cuius quidem rei, exemplum manifeſtum accipere poſſumus à quouis <lb/>cubiculo ex omni parte clauſo, quod tranſitum <choice><ex>nullum</ex><am>nullũ</am></choice> permittat radijs luminoſis, ni <lb/>ſi per aliquod paruum foramen, in quo foramine, & extra ipſum cubiculum, omnes <lb/>radij mixti erunt, ſed in obiecto pariete ipſius cu-<lb/> <ptr xml:id="fig-0309-01a" corresp="fig-0309-01" type="figureAnchor"/> biculi videb untur diſtincti, vnde ſequitur, quòd <lb/>quo remotius erit obiectum <seg type="var">.c.n.t.</seg> ab <seg type="var">.o.</seg> tanto acu-<lb/>tior erit angulus <seg type="var">.c.o.t.</seg> & ſuus contrapoſitus ſimili-<lb/>ter, & per conſequens linea <seg type="var">.e.u.a.</seg> breuior erit, & <lb/><choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.o.</seg> propinquius etiam erit ipſi lineæ <seg type="var">.a.u.e.</seg> <lb/>quæ omnia efficiunt, vt nobis obiectum <seg type="var">.c.t.</seg> <choice><ex>paruum</ex><am>paruũ</am></choice>, <lb/>& minus diſtinctum, ſeu magis confuſum appareat.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0309-01" corresp="fig-0309-01a"> <graphic url="0309-01"/> </figure> </div> </body> </floatingText> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE APPARENTI DISTANTIA PARTIV M <lb/>hæmiſphærij.</head> <head rend="italics" xml:space="preserve">Anſelmo Fucaro.</head> <p> <s xml:space="preserve">GRatæ mihi tuæ literæ <choice><ex>fuerunt</ex><am>fuerũt</am></choice>, quibus <choice><ex>oſtendis</ex><am>oſtẽdis</am></choice> non <choice><ex>paruum</ex><am>paruũ</am></choice> <choice><ex>deſiderium</ex><am>deſideriũ</am></choice> <choice><ex>ſciendi</ex><am>ſciẽdi</am></choice> vnde <lb/>fiat, quod cum dies illuceſcit, & eſt ſerena pars Cœli, circa axem orizontis <lb/>demiſſior appareat, quam aliæ partes, <choice><ex>cum</ex><am>cũ</am></choice> ab alijs (quod <choice><ex>ſciam</ex><am>ſciã</am></choice>) ſatis expreſſum <choice><ex>non</ex><am>nõ</am></choice> fue <lb/>rit, ſed quia de eo à me aliquid ſcire deſideras dicam quod mihi <choice><ex>verum</ex><am>vr̃</am></choice>. </s> <s xml:space="preserve">Scias non <choice><ex>ſolum</ex><am>ſolũ</am></choice> <lb/><choice><ex>multitudinem</ex><am>multitudinẽ</am></choice> <choice><ex>obiectorum</ex><am>obiectorũ</am></choice> <choice><ex>oppoſitorum</ex><am>oppoſitorũ</am></choice> efficere, vt aliqua res alia longius diſtare videat, <lb/>vt alij <choice><ex>putarunt</ex><am>putarũt</am></choice>, ſed etiam diuerſitates colorum, quamobrem cum decipiamur, cre-<lb/>dentes Cœlum eſſe præditum colore cęruleo, cum is color, aeri, non Cœlo <lb/>conueniat, & videntes huiuſmodi colorem circa axem <choice><ex>orizontis</ex><am>orizõtis</am></choice> magis denſum, <choice><ex>quam</ex><am>quã</am></choice> <lb/>verſus ipſum orizontem, ratione exiguæ reſlexionis, à pauca quantitate vaporum <lb/>inter noſtrum ſitum, & reſlexionis locum, iudicamus Cœlum proximiorem eſſe cir-<lb/>ca dictum axem, quam ſint aliæ partes; </s> <s xml:space="preserve">præterquam, quod is color, qui videtur <lb/>terminare, aut impedire radium viſualem (aduertas tamen me hac in re platonicum <lb/>non eſſe) eo ſemper propinquior eſſe videtur, qui ei locum dat, & hanc ob cauſam <lb/>videntes nos <choice><ex>denſitatem</ex><am>dẽſitatẽ</am></choice> cęrulei circa axem orizontis, & cernentes amplitudinem gy <lb/>ri aliarum partium, adducimur, vt putemus <choice><ex>eam</ex><am>eã</am></choice> <choice><ex>partem</ex><am>partẽ</am></choice> <lb/> <ptr xml:id="fig-0309-02a" corresp="fig-0309-02" type="figureAnchor"/> viciniorem eſſe. </s> <s xml:space="preserve"><choice><ex>Neque</ex><am>Neq;</am></choice> illud <choice><ex>etiam</ex><am>etiã</am></choice> <choice><ex>omittam</ex><am>omittã</am></choice> hoc <choice><ex>etiam</ex><am>etiã</am></choice> fie <lb/>ri ratione imaginationis, vnde <choice><ex>etiam</ex><am>etiã</am></choice> multis <choice><ex>contrarium</ex><am>contrariũ</am></choice> <lb/>euenire poteſt, ideſt vt eis magis profundum videa <lb/>tur <choice><ex>Cœlum</ex><am>Cœlũ</am></choice>, circa axem orizontis, quam vicinum gy <lb/>ro <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> <choice><ex>orizontis</ex><am>orizõtis</am></choice>, iudicantibus <choice><ex>eam</ex><am>eã</am></choice> <choice><ex>partem</ex><am>partẽ</am></choice> <choice><ex>longinquio</ex><am>lõginquio</am></choice> <lb/><choice><ex>rem</ex><am>rẽ</am></choice> eſſe, quæ ſeſe magis <choice><ex>obſcuram</ex><am>obſcurã</am></choice> oculo <choice><ex>demonſtrat</ex><am>demõſtrat</am></choice>, & <lb/><choice><ex>eam</ex><am>eã</am></choice> <choice><ex>propinquiorem</ex><am>propinquiorẽ</am></choice> quę ſeſe <choice><ex>clariorem</ex><am>clariorẽ</am></choice> oſtendit, vt ei <choice><ex>ent</ex><am>ẽt</am></choice> <lb/>cont<unclear reason="illegible"/>ingere poteſt, qui <choice><ex>ſubſcriptam</ex><am>ſubſcriptã</am></choice> <choice><ex>ſiguram</ex><am>ſigurã</am></choice> <choice><ex>cubicam</ex><am>cubicã</am></choice> non <lb/><choice><ex>quidem</ex><am>quidẽ</am></choice> <choice><ex>ductam</ex><am>ductã</am></choice> ſecundú <choice><ex>ordinem</ex><am>ordinẽ</am></choice> opticè, ſed ita, vt om-<lb/>nia latera oppoſita <choice><ex>inuicem</ex><am>inuicẽ</am></choice> ſint parallela, proſpiciet, <lb/>ideſt <seg type="var">.a.i.</seg> ad <seg type="var">.e.t.</seg> et <seg type="var">.c.u.</seg> ad <seg type="var">.o.n.</seg> et <seg type="var">.a.i.</seg> ad <seg type="var">.c.u.</seg> et <seg type="var">.e.t.</seg> ad <lb/><seg type="var">o.n.</seg> vnde ſequitur, vt aliquando quadratum <seg type="var">.a.o.</seg> <lb/>videbitur citra, et <seg type="var">.i.n.</seg> vltra <choice><ex>dictum</ex><am>dictũ</am></choice> cubum aliquando <lb/>verò èconuerſo.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0309-02" corresp="fig-0309-02a"> <graphic url="0309-02"/> </figure> </div> </body> </floatingText> <pb facs="0310" n="298"/> <fw type="head">IO. BAPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE PHILOSOPHIA MATHEMA TICA.</head> <head rend="italics" xml:space="preserve">Dominico Piſano.</head> <p> <s xml:space="preserve">SI omnia vno colore conſtarent, & corporum vmbræ à luminibus non di-<lb/>ſtinguerentur, neque diuerſitas ſitus, lumina, quæ veniunt ad oculum non al-<lb/>teraret; </s> <s xml:space="preserve">perinde eſſet, ac ſi eſſemus cœci. </s> <s xml:space="preserve">Miror quod cum in Ariſtotele ſis <lb/>verſatus, in tuis tamen ſcriptis philoſophum à Mathematico ſepares, quaſi mathe-<lb/>maticus non ſit adeò philoſophus, vt eſt naturalis, & metaphyſicus, cum multo ma <lb/>gis quam ij philoſophus ſit appellandus, ſi ad veritatem ſuarum concluſionum reſpi <lb/>ciamus. </s> <s xml:space="preserve">Verum <choice><ex>quidem</ex><am>quidẽ</am></choice> eſt, te in huiuſmodi errore <choice><ex>ſolum</ex><am>ſolũ</am></choice> non verſari; </s> <s xml:space="preserve">ſed grauius eſt, <lb/>quod cum vos videatis etiam res morales ſub philoſophię <choice><ex>appellationem</ex><am>appellationẽ</am></choice> cadere, non <lb/>animaduertatis diuinas ſcientias mathematicas etiam philoſophiæ nomine ornan-<lb/>das eſſe. </s> <s xml:space="preserve">Quod ſi eiuſdem nomen penitius conſiderare velimus, inueniemus aper-<lb/>tè, mathematico magisillud ipſum quàm cuilibet alio conuenire, cum nullus ex alijs <lb/>tam certo ſciat id quod affirmat quam mathematicus, neque aliquis ſit, qui in co-<lb/>gnitionis, & ſcientiæ cupiditatem magis ducatur, vt apertè patet, cum nec etiam ipſi <lb/>ſenſui det locum, neque aliquid præſupponat, quod non ſit ita verum & intellectui <lb/>notum, vt nulla quæuis porentia, illud eſſe falſum oſtendere queat. </s> <s xml:space="preserve">Sed quia <lb/>Græci, qui ad placitum nomina rebus impoſuerunt, voluerunt etiam, non ſolum <lb/>mathematica, ſed etiam naturalia, metaphyſica, & moralia, ſub communi philoſo-<lb/>phiæ nomine contineri. </s> <s xml:space="preserve"><choice><ex>Vtaunt</ex><am>Vtaũt</am></choice> tibi ſatisfaciam authoritate Ariſtotelis, quem tanto-<lb/>pere colis, primum conſidera, nunquam eum de philoſopho <choice><ex>mentionem</ex><am>mẽtionem</am></choice> facere quin <lb/>prius aperiat de quo philoſopho loquatur, atque hoc ſemper præſtat, exceptis qui-<lb/>buſdam locis, vt cap .2. lib. 4. <choice><ex>Metaphyſicorum</ex><am>Metaphyſicorũ</am></choice>, vbi de philoſopho in genere <choice><ex>loquens</ex><am>loquẽs</am></choice>, <lb/>ait, proprium eſſe philoſophi. vt res omnes ſpeculetur atque hoc in principio quin <lb/>ti textus aſſerit, cum in quarto iam oſtenderit mathematicum eſſe philoſophum: <lb/></s> <s xml:space="preserve">omitto quod in .2. textu ſecundi phyſicorum idem affirmet, æquum eſſe appellare <lb/>philoſophiam ſcientiam veritatis, & finem ſpeculatiuæ exiſtere veritatem. </s> <s xml:space="preserve">An non <lb/>idem in primo cap .6. metaphiſicæ philoſophiam ſpeculatiuam, mathematicis phy <lb/>ſicis & ſupernaturalibus rebus contineri? </s> <s xml:space="preserve">An non idem paulo inferius ſcribit phyſi-<lb/>cam primam futuram, ſi aliæ ſubſtantiæ quam naturales non reperirentur? </s> <s xml:space="preserve">conſidera <lb/>deinde quid dicat in fine tertij cap. lib. 11. quo loco nil clarius eſſe poteſt, lege etiam <lb/>quæ .6. cap. eiuſdem libri ab eodem adducuntur, & quæ in .8. cap .12. libri textu .44. <lb/>apertè ponuntur. </s> <s xml:space="preserve">Quod ſi hæc tibi non ſufficiunt, vereor ne tuus morbus deſpe-<lb/>ratus euadat.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De imaginatione ſpecierum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod dixi domino Tadeo eſt, quod aliquas particularium ſpecies, perfectè & <lb/>integrè imaginari poſſumus, alias non item, id tibi melius exemplo innote-<lb/>ſc<unclear reason="illegible"/>et. </s> <s xml:space="preserve">Proponatur tibi triangulus æquilaterus datæ magnitudinis, <choice><ex>datique</ex><am>datiq́;</am></choice> coloris, hu <lb/>iuſmodi enim particularis, potes imaginatione tibi fingere integram ſpeciem, tota <lb/><choice><ex>lemque</ex><am>lemq́;</am></choice> ei adæquatam, ſed ſi aliquam ſpeciem aliquando vniuerſaliorem imaginatio <pb facs="0311" n="299"/><fw type="head">EPISTOL AE.</fw> ne concipere velles, quemadmodum vnius trianguli ęquilateri, tali magnitudine, <lb/>ſed non præfinito colore conſtantis, hoc minime præſtare poſſes. </s> <s xml:space="preserve">quia nullam rem <lb/>viſibilem priuatam colore imaginari poſſumus. </s> <s xml:space="preserve">nec etiam potes imaginari <choice><ex>ſpeciem</ex><am>ſpeciẽ</am></choice> ali <lb/>cuius trianguli æquilateri, indeterminatæ magnitudinis, & indefiniti coloris, quæ <lb/>cuilibet particulari cuiuſuis magnitudinis, & coloris poſtea applicari queat. </s> <s xml:space="preserve">Species <lb/>deinde alicuius trianguli ęquicruri, aut vnius trianguli laterum inęqualium, aut <choice><ex>triam</ex><am>triã</am></choice> <lb/>guli in genere, aut tandem figuræ, conſiderato tu ipſe, an poſſit ſub imaginationem <lb/>cadere. </s> <s xml:space="preserve">Poſſumus quidem huiuſmodi ſpeciem (ratione mediante) intelligere, vn <lb/>de quamlibet ſpeciem rei particularis viſibilis, compoſitæ, ex figura, magnitudine, <lb/>& colore, perfectè imaginari poſſimus, & huiuſmodi conceptus erit ſpecialiſſima <lb/>ſpecies, quia in infinito ſuorum indiuiduorum, nunquam fiet, vt aliquod eorum, ali <lb/>quo modo ab alijs differre poſſit; </s> <s xml:space="preserve">admonens te, nil reperiri, quod differat, aut in ſe <lb/>partem aliquam habeat, quod aliquid aliud non obtineat, quin dicta differentia ſit <lb/>ſpecifica, eius tamen ſolum partis quæ differt ab alia duorum indiuiduorum, vnius, <lb/><choice><ex>eiuſdemque</ex><am>eiuſdemq́;</am></choice> ſpeciei. </s> <s xml:space="preserve">quia ſi eſt in magnitudine nulla planè magnitudo reperitur, quę <lb/>ſua ſpecie non ſit dotata, quod ſi non eſſet, inter res omnes nulla æqualitas <lb/>eluceret: </s> <s xml:space="preserve">& ſi in figura, & colore, idem affirmo, aliter nulla res fimilis eſſet <lb/>alteri, neque aliqua ſimilitu lo reperiretur. </s> <s xml:space="preserve">Idem de quolibet alio obiecto <lb/>ſenſibili dico. </s> <s xml:space="preserve">Ratio autem eorum omnium quæ dixi eſt, quia imaginatiua nihil <lb/>aliud intellectui oſtendere poteſt, quam id quod recipit à ſenſu, & cum ſenſus, <lb/>alio modo moueri non poſſit quam ſupradicto, hanc ob cau ſam verum eſt, <choice><ex>quicquid</ex><am>ꝗcquid</am></choice> <lb/>ſcripſi. </s> <s xml:space="preserve">Vnde triangulum ęquilaterum datę magnitudinis, erit genus triangulorum <lb/><choice><ex>ęquilaterum</ex><am>ęquilaterũ</am></choice> <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> datæ magnitudinis, ſed <choice><ex>diuerſorum</ex><am>diuerſorũ</am></choice> <choice><ex>colorum</ex><am>colorũ</am></choice>, erit <choice><ex>etiam</ex><am>etiã</am></choice> ſpecies trianguli <lb/>æquilateri indeterminatæ magnitudinis, & hic deinde erit ſpecies trianguli, & hic <lb/>poſtea ſpecies figuræ. </s> <s xml:space="preserve">Idem de alijs omnibus rebus per gradus dico, quę ſicut à ſen <lb/>ſu, ita etiam ab imaginatione longè recedunt, adeo vt has ſpecies ſpecialiſſimas tan-<lb/>tum, ideſt eas ſolùm, quas hic ſuperius deſcripſi, integrè capere poſſit: </s> <s xml:space="preserve">at verò gene <lb/>ra, quanto vniuerſaliora ſunt, ab eadem imaginatione, tanto longius diſtant.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De maculis Lunæ, & eius lumine.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MAculæ Lunæ, nihil aliud ſunt, quàm partes ipſius Lunę magis perſpicuæ, à qui <lb/>bus, lumen non refleſſum, ſed penetrans, nobis occultatur; </s> <s xml:space="preserve"><choice><ex>quemadmodum</ex><am>quemadmodũ</am></choice> <lb/>via lactea, nihil aliud eſt, quam pars octaui orbis magis opaca, à qua lumen Solis re-<lb/>fleſſum, ſeſe nobis oſtendit. </s> <s xml:space="preserve">Quod autem Maurolicus ſcribit folio .64. cap. de aſtro <lb/>rum fulſionibus, circa <choice><ex>Lunam</ex><am>Lunã</am></choice>, eſt falſum; </s> <s xml:space="preserve">primo, quia non conſiderat <choice><ex>differentiam</ex><am>differentiã</am></choice> inten-<lb/>ſionis luminum inter Venerem, & Lunam, cum <choice><ex>lumem</ex><am>lumẽ</am></choice> illius ſit magis intenſum, quam <lb/>Lunæ, quia quilibet qui ſano ſit <choice><ex>onculo</ex><am>õculo</am></choice>, facile poteſt compræhendere, ſi Lu-<lb/>na eſſet, vbi eſt Venus, aut Venus vbireperitur Luna (quibus in locis eiuſdem ma <lb/>gnitudinis nobis apparerent) ipſa Luna à Venere longè ſuperaretur, & excedere-<lb/>tur ſplendore, & lumine, ita vt ſi etiam verum eſſet, quod per tres gradus inter-<lb/>ualli ſeſe nobis proderet ſexageſima pars luminis (quod in quadraturis nec in vllo <lb/>alio ſitu verum e<unclear reason="illegible"/>uadit, reſpectu ad Solem, ideſt vt tres gradus differentiæ ſitus, con <lb/>ſtituant ſexag eſimam partem differentiæ luminis reſpectu noſtri) non ideo tamen <pb facs="0312" n="300"/><fw type="head">IO. BAPT. BENED.</fw> dictum lumen conſpiceretur, quia non ſufficit extenſio luminis, cum eiuſdem inten <lb/>ſio ſit etiam neceſſaria. </s> <s xml:space="preserve">Sed id quoque tibi dico, quod etiam ſi dicta ſexageſima <lb/>pars totius luminis lunaris, eadem intenſione ſplendoris, & luminis Veneris, in tali <lb/>diſtantia trium graduum à Sole prædita eſſet, non eam <choice><ex>tamen</ex><am>tamẽ</am></choice> videremus, ratione ob <lb/>liquitatis curuę, & ſphæricę ſuperficiei Lunæ, reſpectu noſtri, in huiuſmodi ſitu: </s> <s xml:space="preserve">id <choice><ex>qui</ex><am>ꝗ</am></choice> <lb/>tibi ita demonſtratum volo.</s> </p> <p> <s xml:space="preserve">Pars ſuperficialis lunaris globi, quæ nos reſpicit ſit <seg type="var">.a.p.u.</seg> quam accipere poſſu-<lb/>mus pro medietate ipſius ſuperficiei totalis, eo quod reſpectu noſtri viſus, inſenſibi <lb/>liter, ab ipſa medietate differat, pars autem à Sole viſa ſit <seg type="var">.u.q.a.</seg> cogitemus etiam cir <lb/>culum <seg type="var">.a.p.u.q.</seg> vnum eſſe ex maioribus ipſius globi, cuius ſuperficies <choice><ex>tranſeat</ex><am>trãſeat</am></choice> per ocu <lb/>lum vidontis, vnde pars eius <seg type="var">.a.p.u.</seg> diuidet vmbram per æqualia, reliqua verò pars <seg type="var">.<lb/>a.q.u.</seg> diuidet per æqualia lumen ipſius Lunæ à Sole receptum, ita quod pars illumi <lb/>nata, erit medietas <seg type="var">.u.q.a.</seg> exceſſus verò, cum noſtro viſui incompræhenſibilis ſit, pro <lb/>nihilo reputetur, cuius cauſa eſt, maxima illa diſtantia, quæ inter Solem, & Lunam <lb/>reperitur, quamuis Sol maior ſit Luna multis millibus vicium, eo quod tunc inter So <lb/>lem, & Lunam reperiantur plus quam .570. diametri terræ.</s> </p> <p> <s xml:space="preserve">Supponamus nunc Lunam remotam eſſe à loco ipſius <choice><ex>coniunctionis</ex><am>cõiunctionis</am></choice> cum Sole per <lb/>3. gradus. </s> <s xml:space="preserve">vnde <choice><ex>quemadmodum</ex><am>quẽadmodum</am></choice> prius <lb/> <ptr xml:id="fig-0312-01a" corresp="fig-0312-01" type="figureAnchor"/> lumen erat in gyro <seg type="var">.a.q.u.</seg> nunc re-<lb/>periatur in gyro <seg type="var">.x.q.t.</seg> ita quod <seg type="var">.t.u.</seg> <lb/>erit ſexageſima pars ipſius <seg type="var">.a.p.u.</seg> <choice><ex>quod</ex><am>qđ</am></choice> <lb/>à vero ſenſibiliter non diſcedit. <lb/></s> <s xml:space="preserve">Imaginentur nunc duæ rectæ lineæ <lb/>ductæ ab oculo <seg type="var">.d.</seg> ad puncta <seg type="var">.t.</seg> et <seg type="var">.u.</seg> <lb/>verum tamen eſt quod linea <seg type="var">.d.u.</seg> ſe-<lb/>cabit <choice><ex>arcum</ex><am>arcũ</am></choice> <seg type="var">.t.u.</seg> ſed ita propinqua <choice><ex>pum</ex><am>pũ</am></choice> <lb/>cto <seg type="var">.u.</seg> quod erit ei ferè contingens, <lb/>vnde abſque ſenſibili errore poſſu-<lb/>mus arcum <seg type="var">.t.u.</seg> intelligere inter duas <lb/>lineas <seg type="var">.d.t.</seg> et <seg type="var">.d.u.</seg> quapropter tale lu-<lb/>men compræhendetur, ferè, ſub an-<lb/>gulo <seg type="var">.t.d.u.</seg> quem quidem angulum <lb/>oportet nos videre, cuius magnitu-<lb/>dinis exiſtat, reſpectu totalis anguli <lb/><seg type="var">a.d.u.</seg> protracta cum fuerit <seg type="var">.d.a</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0312-01" corresp="fig-0312-01a"> <graphic url="0312-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Producatur primo <seg type="var">.d.t.</seg> vſque ad <lb/>diametrum in puncto <seg type="var">.i.</seg> deinde per <lb/>puncta <seg type="var">.a.</seg> et <seg type="var">.u.</seg> ducatur arcus <seg type="var">.a.e.u.</seg> cir <lb/>ca <seg type="var">.d.</seg> <choice><ex>centrum</ex><am>cẽtrum</am></choice>, ad quem ducatur linea <seg type="var">.<lb/>d.t.i.</seg> in puncto <seg type="var">.e.</seg> ſed quia, cum dia-<lb/>meter <seg type="var">.a.u.</seg> tam breuis ſit reſpectu di <lb/>ſtantiæ à terra, tempore interlunij, <lb/>vnde minor <choice><ex>centeſima</ex><am>cẽteſima</am></choice> parte ipſius di-<lb/>ſtantiæ exiſtit, <choice><ex>ſequitur</ex><am>ſequit̃</am></choice> nos poſſe <choice><ex>abſque</ex><am>abſq;</am></choice> <lb/>ſenſibili errore cogitare, à puncto <seg type="var">.d.</seg> <lb/>ad quoduis punctum ipſius diametri <lb/>omnes lineas ad angulos rectos cum <lb/>ipſo diametro, & inſenſibilis inæqua <pb facs="0313" n="301"/><fw type="head">EPISTOL AE</fw> litatis à linea <seg type="var">.d.o</seg>. </s> <s xml:space="preserve">Accipiemus igitur <seg type="var">.t.i.</seg> pro ſinu arcus <seg type="var">.t.u.</seg> qui eſt graduum .3. hoc <lb/>eſt ſexageſima pars ſemicirculi graduum .180. quapropter <seg type="var">.t.i.</seg> erit partium .5233. ta-<lb/>lium qualium <seg type="var">.o.u.</seg> eſt .100000. cuius <seg type="var">.t.i.</seg> quadratum demptum cum fuerit à quadra-<lb/>to ſemidiametri <seg type="var">.o.t.</seg> relinquet nobis quadratum ipſius <seg type="var">.o.i.</seg> quæ quidem <seg type="var">.o.i.</seg> vt radix <lb/>quadrata, erit partium .99862. talium qualium ſemidiameter eſt .100000. vnde <seg type="var">.i.u.</seg> <lb/>reſiduum diametri, remanebit partium .138. </s> <s xml:space="preserve">Vel ſic, cum cognitus ſit nobis arcus <seg type="var">.<lb/>t.u.</seg> illicò cognoſcemus ſinum arcus <seg type="var">.p.t.</seg> <choice><ex>complementum</ex><am>complementũ</am></choice> vnius quartæ, qui ſinus æ qua-<lb/>lis erit ferè arcui <seg type="var">.o.i.</seg> partium .99862. vnde <seg type="var">.i.u.</seg> erit, vt dictum eſt, partium .138. quę <lb/>quidem <seg type="var">.i.u.</seg> æqualis eſt ferè ſinui arcus <seg type="var">.u.e.</seg> & ita etiam <seg type="var">.u.e.</seg> </s> <s xml:space="preserve">quare ſi diuiſa fuerit to-<lb/>ta <seg type="var">.a.u.</seg> <choice><ex>partium</ex><am>partiũ</am></choice> .200000. per .138. proueniet nobis .1449. & ſic angulus <seg type="var">.t.d.u.</seg> erit vna <lb/>partium .1449. anguli <seg type="var">.a.d.u</seg>. </s> <s xml:space="preserve">Confideremus igitur quomodo fieri poteſt, vt oculo <lb/>compræhendatur hæc tam parua particula luminis lunaris.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">SOLVTIONES ALIQVAE.</head> <head rend="italics" xml:space="preserve">Paulo Aemilio Raifestaim.</head> <p> <s xml:space="preserve">Poſt eas literas quas proximè ad te dedi, Franciſcus Monardus mihi retulit tuas <lb/>nonnullas dubitationes circa noſtrum Theorema Arithmeticum .116. quarum <lb/>prima eſt, quod ſi numerus <seg type="var">.a.</seg> <choice><ex>cogitatus</ex><am>cogitatꝰ</am></choice>, eſſet æqualis .4. </s> <s xml:space="preserve">tunc ipſe non eſſet multiplex <lb/>ipſi .4. de quo tamen nullam mentionem feci. </s> <s xml:space="preserve">Idem etiam inquis, ſi <seg type="var">.a.</seg> fuiſſet .5. 6. 7. <lb/>nec non .1. 2. et .3. </s> <s xml:space="preserve">Cui reſpondi, quod <choice><ex>quanuis</ex><am>quãuis</am></choice> nullam fecerim mentionem de æqua <lb/>litate ipſius <seg type="var">.a.</seg> cum .4. nihiltamen refert, </s> <s xml:space="preserve">propterea quod quando ita fuiſſet, nihi-<lb/>lominus eaſdem conditiones ſubiret, <choice><ex>quemadodum</ex><am>quemadodũ</am></choice> ſi fuiſſet duplus, triplus, aut qua <lb/>druplus. eo quod à genere multiplici, æqualitas, formam diuerſam non induat. </s> <s xml:space="preserve">Qua <lb/>re idem eueniet ſi <seg type="var">.a.</seg> fuerit .4. 5. 6. 7. vt ſi eſſet .8. 9. 10. et .11. & ſic de cæteris, excepto <lb/>quod in proprijs multiplicibus, vel in ſuperantibus ipſis multiplicibus <seg type="var">.a.</seg> menſurare <lb/>tur ab ipſo .4. plus quam ſemel. </s> <s xml:space="preserve">Quod autem dicis. de .1. 2. et .3. nihil eſt, quia, <lb/>vt in ſecunda ſumma, hoc eſt in tertio termino maximo, reliquus tertius terminus, <lb/>ideſt .9. non compræhendetur, ita nobis indicabit primum numerum ſumptum mi <lb/>norem eſſe quaternario. </s> <s xml:space="preserve">Quæ omnia, exipſa noſtra thęoria ibidem expreſſa ma-<lb/>nifeſtantur. </s> <s xml:space="preserve">Quid autem circa hoc Frater Lucas dicat, neſcio, quia ipſius opus <lb/>ad manus meas nunquam peruenit, ſatis enim mihi fuit, in Tartalea hanc praxim <lb/>vidiſſe, ratio vero nullibi à me reperta fuit. </s> <s xml:space="preserve">Tartalea enim multos citat authores, <lb/>quorum ſcripta ego nunquam vidi, vt Leonardi Piſani, Proſdocimi, Petri Borghi, <lb/>Fratris Lucæ, Ioannis Sfortunati, <choice><ex>cæterorumque</ex><am>cæterorumq́;</am></choice> ſimilium.</s> </p> <pb facs="0314" n="302"/> <fw type="head">IO. BABPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">ELIPSIM PROPOSITAM QVALITER</head> <head xml:space="preserve">quadrare valeamus.</head> <head rend="italics" xml:space="preserve">Illuſtri Uiro Franciſco Mendo Zzæ</head> <p> <s xml:space="preserve">QVod antea tuo nomine fecerat Marcus Antonius amicus noſter ſufficie-<lb/>bat. </s> <s xml:space="preserve">Sed quia, quæ nunc à me petis, talia ſunt, vt ſine tripartita <choice><ex>aequa- liter</ex><am>ęqua-liter</am></choice> aliqua data proportione non poſſit aliquis exactè intentum perfice-<lb/>re, nihilominus, ſuppoſita di <lb/> <ptr xml:id="fig-0314-01a" corresp="fig-0314-01" type="figureAnchor"/> cta diuiſione, reliqua facilia <choice><ex>erunt</ex><am>erũt</am></choice>. </s> <s xml:space="preserve"><choice><ex>Primum</ex><am>Primũ</am></choice> <lb/>enim eſt. </s> <s xml:space="preserve">Propoſitam Ellipſim qua-<lb/>drare.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0314-01" corresp="fig-0314-01a"> <graphic url="0314-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sit <choice><ex>igitur</ex><am>igit̃</am></choice> Ellipſis propoſita <seg type="var">.a.b.d.c.</seg> cu-<lb/>ius axes ſint <seg type="var">.a.b.</seg> et <seg type="var">.d.c.</seg> dati, ſeu <choice><ex>reperti</ex><am>reꝑti</am></choice> ex <lb/>47. <choice><ex>ſecundi</ex><am>ſecũdi</am></choice> Pergei, <choice><ex>ſintque</ex><am>ſintq́;</am></choice> duo circuli <seg type="var">.a.e.<lb/>b.f.</seg> et <seg type="var">.g.d.h.c.</seg> circa eaſdem diametros, <lb/><choice><ex>tunc</ex><am>tũc</am></choice> proportio <seg type="var">.a.b.</seg> ad <seg type="var">.d.c.</seg> <choice><ex>dimidium</ex><am>dimidiũ</am></choice> erit <lb/>proportionis circulorum ex .2. 12. Eu-<lb/>clid. </s> <s xml:space="preserve">ſed proportio <seg type="var">.a.b.</seg> ad <seg type="var">.d.c.</seg> æqualis <lb/>eſt proportioni maioris circuli ad Elli <lb/>pſim .ex .5. Archimedis in lib. de cono<lb/>idalibus, quapropter proportio Elli-<lb/>pſis ad minorem circulum altera me-<lb/>dietas erit totius proportionis circulo-<lb/>rum, hoc eſt maioris ad minorem, qua <lb/>re Ellipſis media proportionalis erit <lb/>inter eos circulos. </s> <s xml:space="preserve">Nunc verò cum <lb/>ex Archimede repertę fuerint duæ fi-<lb/>guræ rectilineæ æquales duobus circu <lb/>lis iam dictis, & inter has, reperta fue <lb/>rit alia media proportionalis propoſi-<lb/>tum obtinebimus.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Spheroidem propoſitam cubare.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">PRopoſita ſphæroides erit, aut prolata, aut oblonga, ſit prius prolata, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.a.b.</seg> <lb/>diameter circuli, qui eam per æqualia ſecat, circa quam <seg type="var">.a.b.</seg> vt circa axem in-<lb/>telligatur ſphæroides oblonga, cuius ſpiſſitudo ſit <seg type="var">.d.c.</seg> axis prolatæ, cogitemus <choice><ex>nunc</ex><am>nũc</am></choice> <lb/>duas ſphæras <seg type="var">.a.e.b.f.</seg> et <seg type="var">.g.d.h.c.</seg> circa dictos axes. </s> <s xml:space="preserve">Vnde quatuor corpora habebi-<lb/>mus, hoc eſt duas ſphæras, & duas ſphæroides, quas probabo continuas proportio-<lb/>nales inuicem eſſe.</s> </p> <p> <s xml:space="preserve">Conſideremus igitur duos conos rectos, quorum <seg type="var">.a.b.</seg> diameter ſit eorum baſium, <lb/>altitudo autem maioris, æqualis ſit ſemidiametro majori, hoc eſt medietati <seg type="var">.a.b.</seg> al- <pb facs="0315" n="303"/><fw type="head">EPISTOLAE.</fw> titudo verò minoris, æqualis ſit ſemidiametro minori, hoc eſt medietati <seg type="var">.d.c.</seg> vnde <lb/>habebimus proportionem coni maioris ad conum minorem, <choice><ex>eandem</ex><am>eãdem</am></choice> quæ eſt diame <lb/>tri maioris ad diametrum minorem, quod ex .2. parte .11. duodecimi Eucli. </s> <s xml:space="preserve">nec non <lb/>ex .9. eiuſdem manifeſtum eſt, ſed conus minor, eſt quarta pars ſphæroidis prolatæ <lb/>ex .29. </s> <s xml:space="preserve">Archimedis in lib. de conoidalibus, & conus maior, eſt etiam quarta pars <lb/>ſphæræ, ex .32. primi lib. de ſphæra, & cyllindro, </s> <s xml:space="preserve">quare ex communi ſcientia, <choice><ex>eadem</ex><am>eadẽ</am></choice> <lb/>proportio erit ſphæræ maioris ad ſphæroidem prolatam, quæ <seg type="var">.a.b.</seg> ad <seg type="var">.d.c.</seg> ſed pro-<lb/>portio <seg type="var">.a.b.</seg> ad <seg type="var">.d.c.</seg> eſt tertia pars proportionis maioris ſphæræ ad <choice><ex>minorem</ex><am>minorẽ</am></choice>. </s> <s xml:space="preserve">Conſidere <lb/>mus <choice><ex>nunc</ex><am>nũc</am></choice> alios duos conos rectos, vnius & <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> baſis, <choice><ex>cuius</ex><am>cuiꝰ</am></choice> diameter ſit <seg type="var">.d.c.</seg> ſed altitu <lb/>do maioris, æqualis ſit ſemidiametroſphęrę maioris, altitudo verò minoris, ſit æqua <lb/>lis ſemidiametro minoris ſphæræ, vnde ex dictis rationibus habebimus <choice><ex>proportio- nem</ex><am>proportio-nẽ</am></choice> maioris coni ad <choice><ex>minorem</ex><am>minorẽ</am></choice>, vt quæ eſt <seg type="var">.o.b.</seg> ad <seg type="var">.o.d.</seg> hoc eſt vt <seg type="var">.a.b.</seg> ad <seg type="var">.d.c.</seg> & ex dictis <choice><ex>pro poſitionibus</ex><am>ꝓpoſitionibus</am></choice> ita ſe habebit ſphæroides oblonga ad ſphęram minorem vt <seg type="var">.a.b.</seg> ad <seg type="var">.d.<lb/>c.</seg> hoc eſt tertia pars proportionis ſphæræ maioris ad minorem. </s> <s xml:space="preserve">Quare proportio <lb/>ſphæroidis prolatæ ad oblongam, erit reliqua tertia pars proportionis maioris <choice><ex>ſphae ræ</ex><am>ſphęræ</am></choice> ad minorem. </s> <s xml:space="preserve">Quapropter hæc quatuor corpora continua proportionalia inui-<lb/>cem erunt.</s> </p> <p> <s xml:space="preserve">Nunc verò quærenda eſt inter <seg type="var">.a.b.</seg> & ſuas duas tertias partes vna media pro por-<lb/>tionalis, quæ ſit <seg type="var">.K.</seg> & ex Archimede, inuentum ſit quadratum ęquale circulo, cuius <lb/>ſit <seg type="var">.K.</seg> diameter. </s> <s xml:space="preserve">Vnde proportio circuli (cuius <seg type="var">.a.b.</seg> eſt diameter) ad circulum cu-<lb/>ius <seg type="var">.K.</seg> eſt diameter, ſeſquialtera erit ex .2. 12. Eucli.</s> </p> <p> <s xml:space="preserve">Ducatur deinde quadratum lineæ <seg type="var">.K.</seg> in lineam <seg type="var">.a.b.</seg> & proueniet nobis cor-<lb/>pus quoddam, quod æquale erit ſphærę maiori, ex corellario .32. primi de ſphęra & <lb/>cyllindro, cuius corporis, latus cubus ſit <seg type="var">.m</seg>.</s> </p> <p> <s xml:space="preserve">Idem facere oportebit mediante <seg type="var">.d.c.</seg> minoris ſphærę, cuius corporis cubica ra-<lb/>dix ſit <seg type="var">.n</seg>.</s> </p> <p> <s xml:space="preserve">Nunc verò inter <seg type="var">.m.</seg> et <seg type="var">.n.</seg> inueniantur duę medię proportionales <seg type="var">.s.t.</seg> & ex <seg type="var">.s.</seg> pro-<lb/>ducatur cubus, qui ęqualis erit ſphęroidi prolatæ propoſiti, cubus vero <seg type="var">.t.</seg> æqualis <lb/>erit ſphęroidi oblongę, cuius axis eſſet <seg type="var">.a.b</seg>.</s> </p> <p> <s xml:space="preserve">Si autem ſphęroides oblonga nobis propoſita fuiſſet, eodem methodo ſoluere-<lb/>tur problema.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Quadratum circulis mediantibus deſignare.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MOdus autem conficiendi quadratum ex circulis ſupra datam lineam, vt Do-<lb/>minum Gaſparem docui, facillimus eſt.</s> </p> <p> <s xml:space="preserve">Sit enim linea <seg type="var">.b.a.</seg> 46. propoſitionis primi Euclidis, <choice><ex>poſitoque</ex><am>poſitoq́;</am></choice> pede immobli circi-<lb/>ni in puncto <seg type="var">.a.</seg> ſecundum quantitatem lineæ <seg type="var">.a.b.</seg> propoſitę fiat circulus, ſimiliter cir-<lb/>ca punctum <seg type="var">.b.</seg> alius circulus eiuſdem magnitudinis, </s> <s xml:space="preserve">erecta deinde ſola <seg type="var">.a.c.</seg> perpendi <lb/>culari ipſi <seg type="var">.a.b.</seg> ex puncto <seg type="var">.a.</seg> ipſa ſecabitur à circunferentia circuli. cuius centrum eſt <seg type="var">.<lb/>a.</seg> in puncto <seg type="var">.c.</seg> vnde <seg type="var">.a.c.</seg> æqualis erit <seg type="var">.a.b.</seg> poſito demum pede immobili ipſius circi <lb/>ni in puncto <seg type="var">.c.</seg> ſecundum longitudinem ipſius <seg type="var">.c.a.</seg> fiat alius circulus, qui æqualis erit <lb/>reliquis duobus circulis cum eorum ſemidiametri æquales ſint, & hic vltimo factus <lb/>ſecabit circulum, cuius <choice><ex>centrum</ex><am>centrũ</am></choice> eſt <seg type="var">.b.</seg> in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.d.</seg> à quo cum ductæ fuerint <seg type="var">.d.c.</seg> et <seg type="var">.d.b.</seg> <pb facs="0316" n="304"/><fw type="head">IO. BAPT. BENED.</fw> rectè habebimus quod volumus. </s> <s xml:space="preserve">nam omnia latera ſunt inuicem ęqualia ex condi-<lb/>ti onibus circuli, angulus autem <seg type="var">.a.</seg> rectus effectus fuit, </s> <s xml:space="preserve">tunc ſi imaginatione cogita-<lb/>ta fuerit diameter <seg type="var">.b.c.</seg> ex .8. primi, concludemus angulum <seg type="var">.d.</seg> eſſe rectum </s> <s xml:space="preserve">deinde ex .5 <lb/>et .32. eiuſdem concludemus etiam reliquos angulos rectos eſſe.</s> </p> <p> <s xml:space="preserve">Circa verò id quod mihi ſcripſiſti de igne perpetuo putans nugas eſſe, quod Ro-<lb/>mæ inuentæ fuerint lucernę ardentes in ſepulchris antiquorum. </s> <s xml:space="preserve">Ego quid em mi-<lb/>nimè puto eas nugas eſſe, propterea quod tales lucernas non vnus tantum aut duo <lb/>viderint, ſed multi homines fide digniſſimi. </s> <s xml:space="preserve">Prętera cum aisid nulla ratione poſſe <lb/>fieri. </s> <s xml:space="preserve">Reſpondeo quod maxima ratione poſſibile eſſe puto, quam quidem ra-<lb/>tionem ita eſſe oportet, quod primum lucerna ſit perfectè circuncluſa, vt <lb/>materia in ea conſtituta nullo modo exire poſſit, </s> <s xml:space="preserve">deinde quod materia inflamabilis <lb/>talis ſit, vt excrementum fuliginoſum ex flamma tranſmiſſum, tangendo ſuperfi-<lb/>ciem deuexam ipſius lucernæ, aptum ſit in <choice><ex>priſtinum</ex><am>priſtinũ</am></choice> <choice><ex>humorem</ex><am>humorẽ</am></choice> conge<unclear reason="illegible"/>lari, ſiue transfor-<lb/>mari, vnde materia prima per tres formas perpetuò tranſibit, hoc eſt per humorem, <lb/>ſiue oleum tale, vt diximus, per ignem, ſeu flammam, & per vaporem, ſeu exhala-<lb/>tionem fuliginoſam aptam condenſari, atque in priorem humorem illicò reuerti.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE DIVISIONE TRIANGVLI SECVNDVM <lb/>propoſitam proportionem.</head> <head rend="italics" xml:space="preserve">Michaeli Angelo Muciaſco.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> mihi proponis, tale eſt, vt ſcilicet tibi modum ſcribam diuidendi <lb/>triangulum propoſitum ſecundum datam proportionem à linea tranſeun <lb/>te per punctum notatum extra triangulum.</s> </p> <p> <s xml:space="preserve">Tr<unclear reason="illegible"/><choice><ex>iangulum</ex><am>iangulũ</am></choice> <choice><ex>igitur</ex><am>igit̃</am></choice> à te mihi propoſitum ſit <seg type="var">.n.o.u.</seg> conſidero <choice><ex>primum</ex><am>primũ</am></choice> quod ſi <lb/>quis ipſum diuiſerit in duas partes mediante <seg type="var">.e.s.</seg> parallela ad <seg type="var">.n.u.</seg> ea proportione, <lb/>quam mihi proponis. </s> <s xml:space="preserve">deinde inuenerit in dicta <seg type="var">.e.s.</seg> punctum <seg type="var">.r.</seg> per quod tranſiens <lb/>alia linea à puncto <seg type="var">.p.</seg> propoſito, ita quod efficiat duo triangula <seg type="var">.m.r.e.</seg> et <seg type="var">.r.s.x.</seg> inui-<lb/>cem æqualia, problema ſolutum erit. <lb/> <ptr xml:id="fig-0316-01a" corresp="fig-0316-01" type="figureAnchor"/> eo quod triangulum <seg type="var">.m.o.x.</seg> æquale <lb/>eſſet triangulo <seg type="var">.e.o.s.</seg> & quadrilate-<lb/>rum reſiduum <seg type="var">.m.n.u.x.</seg> etiam ęquale <lb/>eſſet quadrilatero <seg type="var">.e.n.u.s</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0316-01" corresp="fig-0316-01a"> <graphic url="0316-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed dum punctum <seg type="var">.r.</seg> uenarer, alia <lb/> <ptr xml:id="fig-0316-02a" corresp="fig-0316-02" type="figureAnchor"/> via mihi in mentem venit, cognoui <lb/>igitur quod quum propoſitum expe-<lb/>ditum fuiſſer, hoc eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> ſi à puncto p. <lb/>protracta eſſet linea <seg type="var">.p.m.</seg> quę trian-<lb/>gulum <seg type="var">.n.o.u.</seg> in duas partes inuicem <lb/>ita proportionatas diuiſiſſet, vt ſe ha <lb/>bet <seg type="var">.A.</seg> et <seg type="var">.B.</seg> ita ſe haberet <choice><ex>productum</ex><am>productũ</am></choice> <lb/> <ptr xml:id="fig-0316-03a" corresp="fig-0316-03" type="figureAnchor"/> <seg type="var">n.o.</seg> in <seg type="var">.o.u.</seg> ad productum <seg type="var">.m.o.</seg> in <seg type="var">.o.<lb/>x.</seg> vt trianguli <seg type="var">.n.o.u.</seg> ad triangulum <lb/><seg type="var">m.o.x.</seg> quod quidem non eſt diffi-<lb/>cile ſpeculari, ex methodo .24. ſexti, <pb facs="0317" n="305"/><fw type="head">EPISTOLAE.</fw> eo quod tam proportio producti <seg type="var">.n.o.</seg> in <seg type="var">.o.u.</seg> ad productum <seg type="var">.m.o.</seg> in <seg type="var">.o.x.</seg> quam pro-<lb/>portio trianguli <seg type="var">.n.o.u.</seg> ad triangulum <seg type="var">.m.o.x.</seg> componitur ex proportione <seg type="var">.u.o.</seg> ad <seg type="var">.o.<lb/>x.</seg> & ex proportion <seg type="var">e.n.o.</seg> ad <seg type="var">.m.o.</seg> vnde proportio dictorum productorum nobis co-<lb/>gnita erit, eo quod cum nobis cognita ſit proportio <seg type="var">.A.</seg> ad <seg type="var">.B.</seg> vt data, cognita etiam <lb/>nobis erit coniuncta, hoceſt <seg type="var">.A.B.</seg> ad <seg type="var">.B</seg>. </s> <s xml:space="preserve">& propterea ea quæ trianguli <seg type="var">.n.o.u.</seg> ad <choice><ex>trian- gulum</ex><am>triã-gulum</am></choice> <seg type="var">.m.o.x.</seg> & ſimiliter productorum. </s> <s xml:space="preserve">Quæſiui poſtea modum inueniendi duas <lb/>dictas lineas <seg type="var">.m.o.</seg> et <seg type="var">.o.x.</seg> & cognoui quod ſi producta fuerit <seg type="var">.p.i.</seg> æquidiſtans li-<lb/>neæ <seg type="var">.o.x.</seg> <choice><ex>producendoque</ex><am>producendoq́</am></choice> <seg type="var">.o.n.</seg> quouſque cum <seg type="var">.p.i.</seg> ſe interſecarent in puncto <seg type="var">.i.</seg> inuenien <lb/>do poſtea lineam quandam, quæ ducta cum <seg type="var">.p.i.</seg> efficeret rectangulum æquale rectan <lb/>gulo cognito quod ex <seg type="var">.m.o.</seg> in <seg type="var">.o.x.</seg> poteſt fieri, quod cognitum dico, eo quod nobis <lb/>cognita eſt proportio data, & rectangulum etiam <seg type="var">.n.o.</seg> in <seg type="var">.o.u.</seg> deinde ſecando ab <seg type="var">.o.<lb/>n.</seg> partem æqualem lineæ iam inuentæ, quæ ſit <seg type="var">.o.t</seg>. </s> <s xml:space="preserve">Inueniendo poſtea, ex .28. ſexti <lb/>lineam <seg type="var">.o.m.</seg> cuius productum in <seg type="var">.m.t.</seg> æquale ſit producto <seg type="var">.t.o.</seg> in <seg type="var">.o.i.</seg> vnde ex .15. eiuſ <lb/>dem proportio <seg type="var">.o.i.</seg> ad <seg type="var">.m.o.</seg> eadem eſſet, quæ <seg type="var">.m.t.</seg> ad <seg type="var">.o.t.</seg> & componendo, ita ſe ha-<lb/>beret <seg type="var">.m.i.</seg> ad <seg type="var">.m.o.</seg> vt <seg type="var">.m.o.</seg> ad <seg type="var">.o.t.</seg> ſed ex .4. ſexti, ita eſſet <seg type="var">.p.i.</seg> ad <seg type="var">.o.x.</seg> vt <seg type="var">.m.i.</seg> ad <seg type="var">.m.o</seg>. <lb/></s> <s xml:space="preserve">quare ex .11. quinti, ita eſſet <seg type="var">.p.i.</seg> ad <seg type="var">.o.x.</seg> vt <seg type="var">.m.o.</seg> ad <seg type="var">.o.t.</seg> vnde ex .15. ſexti productum <seg type="var">.<lb/>o.x.</seg> in <seg type="var">.m.o.</seg> æquale eſſet producto. p, i. in <seg type="var">.o.t.</seg> & ſic haberemus intentum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0316-02" corresp="fig-0316-02a"> <graphic url="0316-02"/> </figure> <figure xml:id="fig-0316-03" corresp="fig-0316-03a"> <graphic url="0316-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi punctum <seg type="var">.m.</seg> caderet in punctum <seg type="var">.n.</seg> idem eſſet, ſi vorò punctum <seg type="var">.m.</seg> tranſiret <lb/>n. oporteret nos facere hoc in latere <seg type="var">.n.u.</seg> ipſum quærendo in linea <seg type="var">.n.u.</seg> ducendo pri <lb/>mum lineam <seg type="var">.p.i.</seg> <choice><ex>æquidiſtantem</ex><am>æquidiſtantẽ</am></choice> <seg type="var">.u.x.</seg> & producendo <seg type="var">.u.n.</seg> ad partem <seg type="var">.u.</seg> proſequendo, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>ſuperius iam dictum eſt.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Idem facere de parallelogr ammo.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">DAtum parallelogrammum in duas partes diuidere, ſecundum aliquam datam <lb/>proportionem à linea tranſeunte per punctum propoſitum.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia, datum parallelogrammum <seg type="var">.b.u.</seg> datum verò punctum <seg type="var">.o.</seg> extra <lb/>figuram, proportio autem ea ſit, quæ <seg type="var">.A.</seg> ad <seg type="var">.B.</seg> vt ſupra. </s> <s xml:space="preserve">Nunc diuidatur primò re-<lb/>ctangulum datum per æqualia, mediante linea <seg type="var">.r.c.</seg> parallela ambobus lateribus <seg type="var">.b.x.</seg> <lb/>et <seg type="var">.s.u.</seg> quæ quidem linea diuidatur in puncto <seg type="var">.i.</seg> ita quod eadem proportio ſit <seg type="var">.r.i.</seg> ad <seg type="var">.<lb/>i.c.</seg> vt <seg type="var">.A.</seg> ad <seg type="var">.B.</seg> protrahatur deinde à puncto <seg type="var">.o.</seg> linea <seg type="var">.o.i.q.</seg> quæ ſecabit ambo duo la-<lb/>tera <seg type="var">.b.x.</seg> vel <seg type="var">.s.u.</seg> intra terminos eorum, vel tantum <seg type="var">.b.x.</seg> reliquum verò extra termi-<lb/>nos <seg type="var">.s.u</seg>.</s> </p> <p> <s xml:space="preserve">Nunc autem ſi intra dictos terminos tranſibit, vt in prima figura videre potes, <lb/>problema ſolutum erit, eo quod <lb/> <ptr xml:id="fig-0317-01a" corresp="fig-0317-01" type="figureAnchor"/> ſi à puncto <seg type="var">.i.</seg> protracta fuerit <seg type="var">.p.<lb/>d.</seg> pa rallela ad <seg type="var">.u.x.</seg> habebimus <lb/>ex prima ſexti eandem propor-<lb/>tionem <seg type="var">.s.d.</seg> ad <seg type="var">.p.x.</seg> ut <seg type="var">.r.i.</seg> ad <seg type="var">.i.c.</seg> <lb/>hoc eſt vt <seg type="var">.A.</seg> ad <seg type="var">.B.</seg> ſed <choice><ex>triangulus</ex><am>triãgulus</am></choice> <lb/><seg type="var">i.e.d.</seg> æqualis eſt triangulo <seg type="var">.i.q.p.</seg> <lb/>vt tibi facilè patebit, vnde qua-<lb/>drilaterum <seg type="var">.e.q.u.x.</seg> æquale erit <lb/>quadrilatero <seg type="var">.d.u.</seg> ex communi <pb facs="0318" n="306"/><fw type="head">IO. BAPT. BENED.</fw> ſcientia. </s> <s xml:space="preserve">Quare ex .9. quinti, ita erit <seg type="var">.s.d.</seg> ad dictum <seg type="var">.d.u.</seg> vt ad quadrilaterum <seg type="var">.e.q.u.<lb/>x.</seg> hoc eſt vt <seg type="var">.A.</seg> ad <seg type="var">.B.</seg> ex .11. eiuſdem.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0317-01" corresp="fig-0317-01a"> <graphic url="0317-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi punctum <seg type="var">.q.</seg> fuerit extra ut in .2. figura videre eſt. </s> <s xml:space="preserve">tunc manifeſtum erit, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>triangulus <seg type="var">.e.x.t.</seg> maior erit pa-<lb/>rallelogrammo <seg type="var">.d.u.</seg> per triangu <lb/> <ptr xml:id="fig-0318-01a" corresp="fig-0318-01" type="figureAnchor"/> lum <seg type="var">.q.t.u.</seg> cum triangulus <seg type="var">.q.i.p.</seg> <lb/>æqualis triangulo <seg type="var">.d.i.e.</seg> excedat <lb/>quadrilaterum <seg type="var">.i.t.u.p.</seg> per trian <lb/>gulum <choice><ex>dictum</ex><am>dictũ</am></choice> <seg type="var">.q.t.u.</seg> quapropter <lb/>cum diuiſus fuerit triangulus <seg type="var">.e.<lb/>x.t.</seg> mediante linea <seg type="var">.o.n.K.</seg> ita <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/><choice><ex>quadrilaterum</ex><am>quadrilaterũ</am></choice> <seg type="var">.e.n.K.t.</seg> ſit æquale <lb/>triangulo <seg type="var">.q.t.u.</seg> ex doctrina præ <lb/>cedenti, habebimus propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0318-01" corresp="fig-0318-01a"> <graphic url="0318-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Idem de frusto trianguli.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">SEd ſi quadrilaterum dictum eſſet fruſtum alicuius <choice><ex>trianguli</ex><am>triãguli</am></choice> ut in figura <seg type="var">.A.</seg> hic ſub <lb/>ſcripta videre eſt, ſuppoſita, <seg type="var">b.d.</seg> parallela ad <seg type="var">.u.p.</seg> ita faciendum eſſet, ducendo <lb/>ſcilicet parallelam <seg type="var">.u.x.</seg> ad <seg type="var">.b.p.</seg> quæ producatur vſque ad concurſum cum <seg type="var">.b.d.</seg> <lb/>in puncto <seg type="var">.x.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> proportio data inter <seg type="var">.t.a.</seg> et <seg type="var">.a.e.</seg> quas duas lineas cogitemus inuicem <lb/>directè coniunctas, </s> <s xml:space="preserve">tunc diuidatur tota <seg type="var">.t.e.</seg> <lb/> <ptr xml:id="fig-0318-02a" corresp="fig-0318-02" type="figureAnchor"/> in puncto <seg type="var">.i.</seg> ita vt <seg type="var">.t.i.</seg> ad <seg type="var">.i.e.</seg> ſit vt quadrilate <lb/>ri <seg type="var">.p.d.</seg> ad trigonum <seg type="var">.u.d.x</seg>. </s> <s xml:space="preserve">deinde diuidatur <lb/><seg type="var">t.i.</seg> in puncto r. tali modo vt <seg type="var">.t.r.</seg> ad <seg type="var">.r.i.</seg> ſe ha-<lb/>beat vt <seg type="var">.t.a.</seg> ad <seg type="var">.a.e.</seg> quo facto ex doctrina <choice><ex>prae cedenti</ex><am>pręcedenti</am></choice> diuidatur totum parallelogram--<lb/>mum <seg type="var">.p.x.</seg> mediante linea <seg type="var">.o.q.</seg> ſecundum <lb/>quod ſe habet <seg type="var">.t.r.</seg> ad <seg type="var">.r.e</seg>. </s> <s xml:space="preserve">Atque ita ſolu-<lb/>tum erit problema, vt exte ipſo ratiotina-<lb/>ri facile potes.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0318-02" corresp="fig-0318-02a"> <graphic url="0318-02"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Fdem de quadrilatero in genere.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">SEd ſi nullum latus parallelum reliquo erit, ita faciendum erit. </s> <s xml:space="preserve">ſi ſit tale quadrila <lb/>terum <seg type="var">.b.d.u.p.</seg> oportet vt ipſum conuertamus in triangulum, producendo duo <lb/>quęuis eius latera oppoſita uſque ad interſectionem ut pote <seg type="var">.u.p.</seg> et <seg type="var">.d.b.</seg> in puncto <seg type="var">.x.</seg> <lb/>quo facto, ſupponemus <seg type="var">.o.</seg> eſſe punctum datum, proportio verò data ſit <seg type="var">.t.r.</seg> ad <seg type="var">.r.i.</seg> ad <lb/>iungatur deinde <seg type="var">.i.e.</seg> ad <seg type="var">.t.i.</seg> ad quam <seg type="var">.e.i.</seg> ipſa <seg type="var">.t.i.</seg> ſe habeat vt quadrilaterum <seg type="var">.b.d.u.p.</seg> <pb facs="0319" n="307"/><fw type="head">EPISTOLAE.</fw> ſe habet ad triangulum <seg type="var">b.p.x.</seg> ducatur poſtea <seg type="var">.o.q.</seg> quæ diuidat totale triangulum <seg type="var">.d.<lb/>u.x.</seg> in duas partes inuicem ita proportionatas, ut ſe habent <seg type="var">t.r.</seg> et <seg type="var">.r.e.</seg> quæ quidem <lb/>partes ſint <seg type="var">.c.d.u.q.</seg> et <seg type="var">.c.q.x.</seg> ut in primo problemate tibi monſtraui, & habebis pro-<lb/>poſitum, dato quod punctum <seg type="var">.c.</seg> ſit inter <lb/>b. et <seg type="var">.d</seg>.</s> </p> <p> <s xml:space="preserve">Sed ſi forte linea <seg type="var">.o.q.</seg> ſecabit <seg type="var">.b.x.</seg> hoc <lb/> <ptr xml:id="fig-0319-01a" corresp="fig-0319-01" type="figureAnchor"/> eſt ſi punctum <seg type="var">.c.</seg> eſſet inter <seg type="var">.b.</seg> et <seg type="var">.x.</seg> mani-<lb/>feſtum eſt, quod <seg type="var">.c.q.</seg> ſecaret <seg type="var">.b.p.</seg> in pun-<lb/>cto <seg type="var">.y.</seg> vnde in tali caſu, alio modo ope-<lb/>randum eſſet, hoc eſt ducendo <seg type="var">.b.u.</seg> quæ <lb/>diuideret quadrilaterum in duo triangu-<lb/>la, & ut ſe haberet triangulum <seg type="var">.b.d.u.</seg> ad <lb/>triangulum <seg type="var">.b.p.u.</seg> vellem vt ita ſecaretur <lb/><seg type="var">t.i.</seg> in puncto <seg type="var">.n.</seg> vt ita ſe haberet <seg type="var">.t.n.</seg> ad <seg type="var">.n.<lb/>i.</seg> ut dictum eſt de iſtis duobus triangulis, <lb/></s> <s xml:space="preserve">deinde prout ſe habet <seg type="var">.n.r.</seg> ad <seg type="var">.r.i.</seg> ita ſeca-<lb/>res triangulum <seg type="var">.b.p.u.</seg> mediante linea <seg type="var">.o.<lb/>K.</seg> ex doctrina primi problematis, & ita haberes propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0319-01" corresp="fig-0319-01a"> <graphic url="0319-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Idem de Pentagono, Exagono, & de reliquis.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">PEntagonum, ſeu hexagonum, vel alias quaſuis multilateras figuras propoſitas its <lb/>diuidere, vt dictum eſt de trilateris, & quadrilateris.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia pentagonus <seg type="var">.a.d.u.p.b.</seg> quem ſecare volumus <choice><ex>mediante</ex><am>mediãte</am></choice> linea <seg type="var">.o.<lb/>q.</seg> in duas partes inuicem ſe habentes, vt ſe habent <seg type="var">.t.r.</seg> et <seg type="var">.r.i.</seg> oportet igitur ut ipſum <lb/>pentagonum reducas ad quadrilaterum <seg type="var">.x.a.d.u.</seg> quod diuidatur ſecundum præce-<lb/>dentem doctrinam, vt ſe habet <seg type="var">.t.r.</seg> ad <seg type="var">.r.e.</seg> <lb/>vnde ſi punctum <seg type="var">.q.</seg> incidit inter <seg type="var">.p.</seg> et <seg type="var">.u</seg>. </s> <s xml:space="preserve">tunc <lb/>habebis propoſitum, ſi verò incidet inter <seg type="var">.<lb/> <ptr xml:id="fig-0319-02a" corresp="fig-0319-02" type="figureAnchor"/> p.</seg> et <seg type="var">.x.</seg> clarum erit quod linea <seg type="var">.o.q.</seg> ſecabit <lb/>latus <seg type="var">.p.b.</seg> trianguli <seg type="var">.b.x.p.</seg> in puncto <seg type="var">.y.</seg> qua-<lb/>propter duces lineam <seg type="var">.a.p.</seg> vt claudat trian-<lb/>gulum <seg type="var">.a.b.p.</seg> <choice><ex>diuidaturque</ex><am>diuidaturq́;</am></choice> <seg type="var">.t.i.</seg> in puncto <seg type="var">.n.</seg> ita <lb/>vt <seg type="var">.t.n.</seg> ad <seg type="var">.n.i.</seg> ſe habeat, vt quadrilaterum. a<unclear reason="illegible"/> <seg type="var">.<lb/>d.u.p.</seg> ad <choice><ex>triangulum</ex><am>triãgulum</am></choice> <seg type="var">.a.b.p</seg>. </s> <s xml:space="preserve">deinde <choice><ex>hunc</ex><am>hũc</am></choice> trian <lb/>gulum <seg type="var">.a.b.p.</seg> diuidas mediante linea <seg type="var">.o.K.</seg> <lb/>vt <seg type="var">.n.r.</seg> ad <seg type="var">.r.i.</seg> ex doctrina primi problematis <lb/>& habebis propoſitum. </s> <s xml:space="preserve">Idem dico de hexa <lb/>gono, reducendo ipſum ad pentagonum, & <lb/>item de eptagono, ipſum reducendo ad exa <lb/>gonum, & idem infero de infinito ipſarum <lb/>ſuperficialium figurarum rectilinearum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0319-02" corresp="fig-0319-02a"> <graphic url="0319-02"/> </figure> </div> </body> </floatingText> <pb facs="0320" n="308"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De duobus triangulis equalibus inter lineas <lb/>inuicem inclinatas.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">TV mihi vltimò proponis duas lineas rectas <seg type="var">.b.f.</seg> et <seg type="var">.q.s.</seg> in eadem ſuperficie pla-<lb/>na, non tamen inuicem æqu idiſtantes, proponis etiam <seg type="var">.n.t.</seg> in eadem ſuperfi-<lb/>cie, quæ vnamquamque priorum ſecat, proponis etiam lineam <seg type="var">.h.</seg> tali conditione, <lb/>quod nulli dictarum ſit parallela, </s> <s xml:space="preserve">deinde ſcire cupis qua arte aliquis poſſet ducere <seg type="var">.<lb/>c.u.</seg> parallelam ad <seg type="var">.h.</seg> ita quod ſecando <seg type="var">.n.t.</seg> conſtituat duos triangulos <seg type="var">.n.o.u.</seg> et <seg type="var">.t.o.e.</seg> <lb/>inuicem æquales.</s> </p> <p> <s xml:space="preserve">Facita, producas primò duas primas lineas à parte, in qua inuicem inclinantur, <lb/>vſque ad concurſum in puncto <seg type="var">.i</seg>. </s> <s xml:space="preserve">deinde à puncto <seg type="var">.n.</seg> duces <seg type="var">.n.c.</seg> <choice><ex>parallelam</ex><am>parallelã</am></choice> ad <seg type="var">.h.</seg> poſtea <lb/>ex .25. ſexti Eucli. conſtitues <choice><ex>triangulum</ex><am>triãgulum</am></choice> <seg type="var">.i.u.e.</seg> ſimile triangulo <seg type="var">.i.c.n.</seg> æquale tamen <lb/>triangulo <seg type="var">.i.t.n.</seg> & ſolutum erit problema. <lb/></s> <s xml:space="preserve">Velſic, inuenies <seg type="var">.i.e.</seg> mediam <lb/> <ptr xml:id="fig-0320-01a" corresp="fig-0320-01" type="figureAnchor"/> proportionalem inter <seg type="var">.i.c.</seg> & <lb/><seg type="var">i.t.</seg> duces poſtea <seg type="var">.e.u.</seg> paralle-<lb/>lam lineę <seg type="var">.h.</seg> vel <seg type="var">.c.n.</seg> quod <choice><ex>idem</ex><am>idẽ</am></choice> <lb/>erit ex .30. primi Eucli. </s> <s xml:space="preserve">& ſo-<lb/>lutum erit problema.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0320-01" corresp="fig-0320-01a"> <graphic url="0320-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nam ex .17. ſexti eadem <lb/>proportio erittrianguli <seg type="var">.i.c.<lb/>n.</seg> ad triangulum <seg type="var">.i.e.u.</seg> ut <seg type="var">.i.c.</seg> <lb/>ad <seg type="var">.i.t</seg>. </s> <s xml:space="preserve">Quare ut trianguli <seg type="var">.i.<lb/>c.n.</seg> ad <choice><ex>triangulum</ex><am>triangulũ</am></choice> <seg type="var">.i.t.n.</seg> ex pri-<lb/>ma ſexti, et .11. quinti. </s> <s xml:space="preserve">Vnde <lb/>ex .9. eiuſdem <seg type="var">.i.e.u.</seg> æqualis <lb/>erit <seg type="var">.i.t.n</seg>. </s> <s xml:space="preserve">Quapropter <seg type="var">.o.n.u.</seg> <lb/>æqualis etiam erit <seg type="var">.o.e.t</seg>.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">SOLVTIONES NONNVLLAE QVOR VNDAM <lb/>problematum.</head> <head rend="italics" xml:space="preserve">Thaodoſio à Raifestaim.</head> <p> <s xml:space="preserve"><hi rend="small caps">DVritandvm</hi> profecto non eſt, quin quotidie hominibus ſtudioſis ali-<lb/>quid noui deſit, quemadmodum, quod tibi nunc occurrit, mihi non-<lb/>nunquam accidit, hoc eſt inuenire orizontem, cui aliqua propoſita ſtel <lb/>la oriatur cum gradu ipſius longitudinis. </s> <s xml:space="preserve">pro <choice><ex>cuius</ex><am>cuiꝰ</am></choice> rei operatione te prius <lb/>ſcire oportebit vtrum ſtella in ſignis aſcendentibus, vel deſcendentibus reperiatur, <lb/>hoc eſt in ſignis, quę à Capricorno ad Cancrum procedunt, vel in illis, quę à Can-<lb/>cro ad Capricornum numerantur, </s> <s xml:space="preserve">propterea quod ſi in ſignis aſcendentibus inue-<lb/>nitur, ſciendum eſt, quod ſupra talem orizontem polus mundi auſtralis attollitur, <lb/>ſed ſi in ſignis deſcendentibus reperitur, </s> <s xml:space="preserve">tunc polus borealis eleuatur ſupra dictum <pb facs="0321" n="309"/><fw type="head">EPISTOL AE.</fw> orizontem, vt exempli gratia, canicula quæ à Græcis <choice><ex>Prochyon</ex><am>Prochyõ</am></choice> vocatur, reperitur in <lb/>24. minuto vigeſimi gradus Cancri, quapropter polus borealis eleuatur ſupra ori-<lb/>zontem, cui ipſa oritur cum eodem gradu, & minuto eclipticę illius ſigni. </s> <s xml:space="preserve">ſed <lb/>quia volumus etiam ſcire veram quantitatem arcus eleuationis huiuſmodi poli, pro <lb/>pterea accipiemus in tabula generali Monteregij numerum qui vocatur radix aſcen <lb/>ſionum, èregione numeri longitudinis ipſius ſtellæ, qui quidem numerus in præſen <lb/>ti exemplo erit gra .107. cum minutis .53. qui eſt <choice><ex>cuiuſdam</ex><am>cuiuſdã</am></choice> arcus æquatoris, qui inci-<lb/>pit in principio Arietis, & in circulo latitudinis deſinit, hoc eſt ab orizonte quæſi-<lb/>to, ita quod talis numerus erit aſcenſio obliqua huiuſmodi puncti eclipticæ illi ori-<lb/>zonti, qua aſcenſione mediante, ſimul cum gradu, & minuto longitudinis in tabulis <lb/>aſcenſionum obliquarum, inueniemus gradum, & minutum altitudinis pollaris, <choice><ex>quod</ex><am>qđ</am></choice> <lb/>quærebatur, eodem ordine ac methodo, quo vtimur ad inueniendum in tabulis po-<lb/>ſitionum, polum circuli poſitionis alicuius aſtri, mediante declinatione & diſtantia <lb/>à meridiano ciuſdem aſtri, vt ſcis. </s> <s xml:space="preserve">Vnde in præſenti exemplo eleuatio poli borea <lb/>lis ſupra talem orizontem erit gra .7. cum minutis .45.</s> </p> <p> <s xml:space="preserve">Sed ſi ſtella fuerit in medietate aſcendente, tunc certi erimus polum auſtralem ſu <lb/>per dictum orizontem attolli, nam idem eſt quærere altitudinem vnius <choice><ex>polorum</ex><am>polorũ</am></choice> mun <lb/>di à tali orizonte, quod diſtantiam dicti poli à circulo ſecundum quem longitudo <lb/>terminatur, qui etiam latitudinis dicitur, eo quod tunc temporis talis circulus vnus <lb/>& idem eſt cum orizonte. </s> <s xml:space="preserve">Sumatur ergo exempli gratia ſtella, quæ in ore piſcis au <lb/>ſtralis eſt, quę, pro nunc, ſit in gradu .20. cum minutis .14. </s> <s xml:space="preserve">Aquarij longitudinis, & <lb/>in gradu .23. cum nullo minuto meridianæ latitudinis. </s> <s xml:space="preserve">Tunc certi erimus orizon-<lb/>tem, cui dicta ſtella oritur cum eiuſmodi puncto eclipticæ, depreſſum eſſe à parte <lb/>auſtrali ſub <choice><ex>illoque</ex><am>illoq́</am></choice> polo, ſed quia propoſitum eſt ſcire etiam quantitatem huiuſmo-<lb/>di depræſſionis, reperiemus in tabula generali gradum, & minutum æquatoris, cor-<lb/>reſpondentem tali puncto longitudinis à circulo latitudinis terminato, qui quidem <lb/>numerus in præſenti exemplo erit gra .317. cum minutis .46. & hic numerus, vt dixi <lb/>mus eſt aſcen. obli. ad dictum orizontem, vbi polus auſtralis attollitur, & deſcenſio <lb/>obliqua, vbi polus borealis eleuatur. </s> <s xml:space="preserve">Quapropter ſi à .317. gradibus cum minutis <lb/>46. demptus fuerit dimidius circulus gra .180. remanebunt gra .137. cum minutis .46 <lb/>& punctus oppoſitus gradibus .20. cum .14. minutis Aquarij eſt in eodem numero <lb/>Leonis, & mediantibus iſtis gradibus .137. min .46. aſcenſionis, cum grad .20. min <num value="14">.<lb/>14.</num> Leonis inueniemus eleuationem poli borealis ab orizonte in tabulis aſcenſio-<lb/>num obliquarum Monteregij, hoc eſt gra .17. min .53. & eadem altitudo erit poli <lb/>auſtralis ſupra orizontem à quo Fomahant cum dicto puncto eclipticæ oritur, in qua <lb/>longitudine dicta ſtella reperitur.</s> </p> <p> <s xml:space="preserve">Sed ſi propoſitus nobis fuerit punctus eclipticæ, cum quo aliqua ſtella oritura ſit, <lb/>& oporteat inuenire vbi, hoc eſt orizontem huiuſmodi ortus, eleuatione poli arti <lb/>ci, ſeu antarctici ſupra talem orizontem, ita operandum eſſet.</s> </p> <pb facs="0322" n="310"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Sit exempli gratia ſtella <seg type="var">.o.</seg> ecli <lb/> <ptr xml:id="fig-0322-01a" corresp="fig-0322-01" type="figureAnchor"/> ptica verò <seg type="var">.d.q.</seg> æquator autem <seg type="var">.<lb/>p.q.</seg> punctus verò eclipticæ, cum <lb/>quo ſtella oritura ſit <seg type="var">.e.</seg> orizon de <lb/><choice><ex>mum</ex><am>mũ</am></choice> <seg type="var">.o.e.</seg> vbi ſtella oriri poſſit <choice><ex>cum</ex><am>cũ</am></choice> <lb/>puncto <seg type="var">.e</seg>. </s> <s xml:space="preserve">Nam cum ſtella pro-<lb/>ponitur, datur etiam eius longi-<lb/>tudo, nec non latitudo, </s> <s xml:space="preserve">quare ar-<lb/>cus <seg type="var">.a.q.</seg> & arcus <seg type="var">.a.o.</seg> nobis cogni <lb/>tus erit, cum ſupponatur arcus <seg type="var">.a.<lb/>o.</seg> eſſe circuli latitudinis, et <seg type="var">.a.o.</seg> <lb/>Iatitudo ipſius ſtellæ, & angulus <lb/>a. rectus erit, & quia punctum <seg type="var">.e.</seg> <lb/>datur, ergo arcus <seg type="var">.a.o.</seg> & arcus <seg type="var">.a.<lb/>e.</seg> ſimul <choice><ex>cum</ex><am>cũ</am></choice> angulo <seg type="var">.a.</seg> recto cogni-<lb/>ti ſunt, vnde ex .11. primi lib. co<lb/>pernici, angulus <seg type="var">.a.e.o.</seg> cognoſce-<lb/>tur, & angulus <seg type="var">.q.e.o.</seg> ſimiliter, vt <lb/>reſiduum ex duobus rectis quo. e <lb/>mediante cum angulo <seg type="var">.q.</seg> declina <lb/>tionis ab æquatore, <choice><ex>medianteque</ex><am>medianteq́;</am></choice> <lb/>latere <seg type="var">.q.e.</seg> cognito, cognitus quo <lb/>que nobis erit angnlus <seg type="var">.e.t.q.</seg> ex <lb/>12. eiuſdem. </s> <s xml:space="preserve">qui quidem angulus <lb/>erit altitudinis æquatoris ab ori-<lb/>zonte quæſito, qui demptus à <lb/>90. gradibus, dabit altitudinem <lb/>poli ab orizonte quæſito.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0322-01" corresp="fig-0322-01a"> <graphic url="0322-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Inuenire poſtea gradum eclipticę, cum quo ſtella data oriatur ad orizontem pro <lb/>poſitum, nullius eſt difficul@atis.</s> </p> <p> <s xml:space="preserve">Ponamus exempli gratia, aliquem ſcire velle gradum eclipticæ, cum quo canicu-<lb/>la oritur ad orizontem, cui polus boreus eleuatur per gradus .44. quæ canicula ſup-<lb/>ponatur habere gradus .19. cum min .24. </s> <s xml:space="preserve">Cancri longitudinis, & gra .16. min .10. lati-<lb/>tudinis meridianæ, quærere primum oportet eius declinationem ex doctrina .2. pro <lb/>blematis tabularum directionum Monteregij, quæ erit graduum .6. cum minutis .5. <lb/>ſeptentrionalis, </s> <s xml:space="preserve">deinde inuenire eius aſcenſionem rectam ex doctrina .4. problema <lb/>tis eiuſdem Monteregij, quæ erit gra .108. mi .42. </s> <s xml:space="preserve">deinde <choice><ex>mediante</ex><am>mediãte</am></choice> declinatione iam <lb/>inuenta in tabulis differentiarum aſcenſionalium ſub polo .44. accipiemus differen-<lb/>tiam aſcenſionum, qua differt recta ab obliqua, quæ in præſenti exemplo erit gra <num value="5">.<lb/>5.</num> min .55. quæ dempta ab aſcenſione recta ſtellæ, vt præſens exemplum exigit, relin <lb/>quet nobis aſcenſionem obliquam ſtellæ propoſitæ ad polum. gra .44. quæ erit gra <num value="102">.<lb/>102.</num> minu .47. qua mediante, in tabulis aſcenſionum obliquarum poli .44. habebi-<lb/>mus gradum & minutum eclipticę cum quo ſtella oritur. </s> <s xml:space="preserve">quod in caſu noſtro erit <lb/>gra .1. min .8. Leonis, ſed ſi tecum non fuerint tabulæ dictæ, potes eleganter omnia <lb/>hæc perficere via triangulorum ſphæricorum.</s> </p> <pb facs="0323" n="311"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Via triangul@rum<unclear reason="illegible"/> idem facere.</s> </p> <p> <s xml:space="preserve">Sit <choice><ex>exempli</ex><am>exẽpli</am></choice> gr@tia <seg type="var">.q.b.</seg> æquator, ecliptica verò <seg type="var">.q.a.</seg> propoſitus <choice><ex>autem</ex><am>aũt</am></choice> orizon ſit <seg type="var">.o.c.d.</seg> <lb/>& ſtella data ſit <seg type="var">.o.</seg> in orientali parte orizontis, circulus verò <seg type="var">.a.o.</seg> ille ſit, qui <choice><ex>tranſiens</ex><am>tranſiẽs</am></choice> <lb/>per polos eclipti<unclear reason="illegible"/>cæ & per centrum ſtellæ terminat longitudinem ipſius ſtellæ, & in <lb/>ipſo ſit eius latitudo. </s> <s xml:space="preserve">Nunc propoſitum ſit inuenire arcum <seg type="var">.d.q.</seg> eo quod illicò ſcie <lb/>mus punctum <seg type="var">.d.</seg> qua propter oportet nos prius cognoſcere arcum <seg type="var">.d.a.</seg> qui demptus, <lb/>vel additus arcui <seg type="var">.a.q.</seg> prius cognito ex ſuppoſito (nam data nobis eſt longitudo, & <lb/>latitudo ſtellæ) dabit nobis <seg type="var">.d.q</seg>.</s> </p> <p> <s xml:space="preserve">Cum igitur voluerimus arcum <seg type="var">.d.a.</seg> cognoſcere, ita faciemus. </s> <s xml:space="preserve">nam <seg type="var">.q.a.</seg> cognitus <lb/>nobis eſt ex ſuppoſito vt dictum eſt. </s> <s xml:space="preserve">angulus quoque <seg type="var">.a.q.b.</seg> qui declin tionis eclipti <lb/>cæ ab æquatore eſt, angulus deinde <seg type="var">.a.</seg> (trianguli <seg type="var">.a.b.q.</seg>) rectus eſt, ergo ex .4. primi <lb/>copernici cogn@tus nobis erit arcus <seg type="var">.<lb/>a.b.</seg> nec non angulus <seg type="var">.a.b.q.</seg> vnde an-<lb/> <ptr xml:id="fig-0323-01a" corresp="fig-0323-01" type="figureAnchor"/> gulus <seg type="var">.o.b.e.</seg> reſiduus ex duobus re-<lb/>ctis in duobus primis hic ſubſcriptis <lb/>figuris nobis itidem cognitus erit, <lb/>etiam & arcus <seg type="var">.b.o.</seg> reſiduus ſiue com <lb/>poſitus ex ar cu <seg type="var">.a.o.</seg> cognito ex ſup-<lb/>poſito <choice><ex>cum</ex><am>cũ</am></choice> ſit arcus latitud nis ab ecli-<lb/>ptica. </s> <s xml:space="preserve">Tunc<unclear reason="illegible"/> in triangulo <seg type="var">.o.b.e.</seg> co-<lb/>gnoſcimus latus <seg type="var">.o.b.</seg> & angulum <seg type="var">.o.<lb/>b.e.</seg> nec non angulum <seg type="var">.b.e.o.</seg> qui eſt <lb/>altitudinis æquatoris ab orizonte , <lb/></s> <s xml:space="preserve">quare ex .12. dicti lib. cognitus nobis <lb/>erit angulus <seg type="var">.b.o.e</seg>. </s> <s xml:space="preserve">Conſideremus <lb/>deinde triangulum <seg type="var">.a.o.d.</seg> cuius angu <lb/>lus <seg type="var">.a.</seg> rectus eſt, & angulus <seg type="var">.a.o.d.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <lb/>latere <seg type="var">.a.o.</seg> etiam cognitus, vnde ex <lb/>ſupradicta .4. nobis cognitus erit ar-<lb/>cus <seg type="var">.a.d.</seg> & conſequenter cognoſce-<lb/>mus at cum <seg type="var">.d.q.</seg> eius reſiduum, ſeu <lb/>compoſitum, quem quærebamus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0323-01" corresp="fig-0323-01a"> <graphic url="0323-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi hac via inuenire deſideras, <lb/>cui orizonti propoſita ſtella oriatur <lb/>cum eodem eclipticę puncto <seg type="var">.a.</seg> lon-<lb/>gitudinis, hoc aliud nihil eſſet, quam <lb/>cognoſcere amplitudinem anguli <seg type="var">.a.<lb/>b.q.</seg> eo quod talis orizon, idem cir-<lb/>culus eſſet <seg type="var">.a.b.o.</seg> vnde cum quis ſci-<lb/>ret vnum illorum angulorum quem <lb/>æquator efficit cum orizonte, reli-<lb/>qua illicò ei innoteſcent, ſed dictus <lb/>angulus <seg type="var">.b.</seg> iam diximus quomodo <lb/>cognoſcatur.</s> </p> <pb facs="0324" n="312"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Ponamus nos ſcire velle <choice><ex>punctum</ex><am>pũctum</am></choice> <lb/> <ptr xml:id="fig-0324-01a" corresp="fig-0324-01" type="figureAnchor"/> eclipticę, cum quo Procyon oritur <lb/>polo .44. o. dato, quod ſtella in gra <num value="19">.<lb/>19.</num> cum min .24. </s> <s xml:space="preserve">Cancri, reperiatur <lb/>diſtans ab ecliptica per gra .16. min <num value="10">.<lb/>10.</num> meridiem verſus. </s> <s xml:space="preserve">vnde <choice><ex>arcus</ex><am>arcꝰ</am></choice> <seg type="var">.a.q.</seg> <lb/>erit gra .70. min .36. <choice><ex>eiusque</ex><am>eiusq́;</am></choice> ſinus par-<lb/>tium 94321. talium qualium totalis <lb/>eſt .100000. arcus verò <seg type="var">.a.o.</seg> gra .16. <lb/>minut .10. ſinus erit 27845. angu-<lb/>lus autem <seg type="var">.a.q.e.</seg> declinationis zodia <lb/>ci ab ęquatore grad .23. min .30. cu-<lb/>ius ſinus eſt .39875. </s> <s xml:space="preserve">Quare ex ſupra-<lb/>dictis rationibus angulus <seg type="var">.a.b.q.</seg> erit <lb/>gra .82. mi .24. cuius <choice><ex>ſinus</ex><am>ſinꝰ</am></choice> erit .99122. <lb/>arcus vero <seg type="var">.a.b.</seg> gra .22. minu .17. cu-<lb/>ius ſinus erit .37945. </s> <s xml:space="preserve">angulus deinde <lb/><seg type="var">o.e.b.</seg> trianguli <seg type="var">.o.e.b.</seg> eſt gra .46. mi <seg type="var">.<lb/>o.</seg> altitudinis æquatoris ab orizonte, <lb/>cuius ſinus eſt .71934. angulus ſimili <lb/>ter <seg type="var">.o.b.e.</seg> medio coniuncti, quibus <lb/>rectus perſicitur, arcus etiam <seg type="var">.o.b.</seg> no <lb/>tus eſt grad .6. min .7. cuius ſinus eſt <num value="10655">.<lb/>10655.</num> cum ſit differentia inter ar-<lb/>cus <seg type="var">.a.b.</seg> et <seg type="var">.a.o.</seg> cognitos.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0324-01" corresp="fig-0324-01a"> <graphic url="0324-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quare ex .12. iam ſupradicta an-<lb/>gulus <seg type="var">.e.o.b.</seg> hoc eſt <seg type="var">.a.o.d.</seg> erit. grad <num value="36">.<lb/>36.</num> min .39. cuius ſinus erit .59693. <lb/></s> <s xml:space="preserve">deinde per .4. cognitus erit nobis an <lb/>gulus <seg type="var">.a.d.o.</seg> gra .55. min .5. cuius <choice><ex>ſinus</ex><am>ſinꝰ</am></choice> <lb/>erit .81998. arcus verò <seg type="var">.d.o.</seg> gra .19. <lb/>min .51. cuius ſinus erit .33957 ar-<lb/>cus autem gra .11. min .42. cuius <choice><ex>ſinus</ex><am>ſinꝰ</am></choice> <lb/>erit .20270. vnde <choice><ex>arcus</ex><am>arcꝰ</am></choice> <seg type="var">.d.q.</seg> reſiduus <lb/>ex <seg type="var">.a.q.</seg> erit gra .58. min .54. complementum <choice><ex>autem</ex><am>aũt</am></choice> quartæ erit gra .31. mi .6. hoc eſt gra <num value="1">.<lb/>1.</num> ſigni Leonis. cum min .6.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De ſphæroide duplæ ſpbær æ propoſit æ.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MOdus autem inueniendi ſphæroidem ex dato axe, quod duplum ſit ſphæ-<lb/>ra propoſita, talis eſt.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia <seg type="var">.a.b.c.</seg> ſphæra propoſita. cuius ſemidia meter ſit <seg type="var">.o.c.</seg> ſemiaxis <lb/>vero ſphæroidis ſit <seg type="var">.d.x.</seg> cuius dimidium ſit <seg type="var">.u.x.</seg> </s> <s xml:space="preserve">tunc ex doctrina .9. ſexti Euclid. inue <lb/>niatur <seg type="var">.g.h.</seg> media proportionalis inter <seg type="var">.u.x.</seg> et <seg type="var">.c.o.</seg> </s> <s xml:space="preserve">deinde ſicut ſe habet <seg type="var">.u.x.</seg> ad <seg type="var">.g.h.</seg> <pb facs="0325" n="313"/><fw type="head">EPISTOL AE.</fw> faciemus, quod diameter <seg type="var">.a.b.</seg> dictæ ſphæræ ita ſe habcat ad <seg type="var">.e.f.</seg> ex .10. ſexti, quæ <lb/><seg type="var">e.f.</seg> erit reliqua axis quæſita. </s> <s xml:space="preserve">Vnde conſtituta cum fuerit ellipſis <seg type="var">.d.f.t.e.</seg> ex dictis axi-<lb/>bus, </s> <s xml:space="preserve">deinde circumuertendo ellipſim circa maiorem axem, conſtituemus ſphæroi-<lb/>dem oblongam, ſi autem circumuertemus ipſam circa minorem axim conſtituemus <lb/>ſphæroidem prolatam.</s> </p> <p> <s xml:space="preserve">Quod autem talis operatio rationalis ſit, nulli dubium erit, que<unclear reason="illegible"/>tieſcunque co-<lb/>gnoſcet conum rectum <seg type="var">.e.u.f.</seg> æqualem eſſe cono recto <seg type="var">.a.c.b.</seg> ex .2. parte .12. duodeci <lb/>mi Euclid. </s> <s xml:space="preserve">& quod cum conus <seg type="var">.e.d.f.</seg> duplus ſit cono <seg type="var">.e.u.f.</seg> ex lemmate collecto ab <lb/>11. duodecimi, conus <seg type="var">.e.d.f.</seg> duplus exiſtit etiam cono <seg type="var">.a.c.b.</seg> ex .7. quinti. </s> <s xml:space="preserve">Cum de-<lb/>inde ex .32. primi lib. de ſphæra, & cyllindro ſphæra <seg type="var">.a.c.b.q.</seg> quadrupla ſit cono <seg type="var">.a.<lb/>c.b.</seg> ipſa conſequenter dupla erit cono <seg type="var">.e.d.f.</seg> ſed ex .29. primi de conoidalibus, dimi <lb/>dium ſphæroidis <seg type="var">.e.d.f.t.</seg> hoc eſt <seg type="var">.e.d.f.</seg> dupla eſt cono <seg type="var">.e.d.f</seg>. </s> <s xml:space="preserve">Quare talis medietas <lb/>æqualis eſt ſphæræ propoſitæ, totaq́ue ſphæroides dupla erit ſphærę datæ. </s> <s xml:space="preserve">Quod <lb/>autem dico de proportione dupla, idem infero de qualibet alia, ſumendo <seg type="var">.u.x.</seg> ita pro <lb/>portionatam ad <seg type="var">.d.x.</seg> vt proponitur.</s> </p> <p> <s xml:space="preserve">Sphęram autem inuenire quæ dimidia ſit ſphæroidis propoſitæ nullius erit nego-<lb/>tij, quotieſcunque inuentus fuerit modus diuidendi vnam datam proportionem in <lb/>tres æquales partes.</s> </p> <p> <s xml:space="preserve">Sit propoſita ſphæroides <seg type="var">.e.f.d.t.</seg> cuius axes ex conſequentia dantur <seg type="var">.e.f.</seg> et <seg type="var">.d.t.</seg> quę <lb/>quidem ſphæroides ſit primo oblonga, et <seg type="var">.u.x.</seg> ſit dimidium axis maioris. </s> <s xml:space="preserve">imagine-<lb/>tur etiam conus <seg type="var">.e.u.f.</seg> vt ſupra. </s> <s xml:space="preserve">Imaginetur etiam factum eſſe, quod proponitur, hoc <lb/>eſt, vt ſphæra <seg type="var">.a.b.c.q.</seg> ſit dimidium ipſius ſphæroidis, vnde conus <seg type="var">.a.c.b.</seg> æqualis erit <lb/>cono <seg type="var">.e.u.x.</seg> vt ſupra demonſtratum eſt, & ſit <seg type="var">.g.h.</seg> media proportionalis inter <seg type="var">.u.x.</seg> et <lb/><seg type="var">o.c</seg>. </s> <s xml:space="preserve">Iam viſum ſuperius fuit, quod eadem proportio erat ipſius <seg type="var">.u.x.</seg> ad <seg type="var">.g.h.</seg> quæ <seg type="var">.a.b.</seg> <lb/>ad <seg type="var">.e.f.</seg> </s> <s xml:space="preserve">quare eadem quæ <seg type="var">.o.b.</seg> ad <seg type="var">.e.x.</seg> ſed <seg type="var">.u.x.</seg> et <seg type="var">.e.x.</seg> dantur. </s> <s xml:space="preserve">inter quas <seg type="var">.g.h.</seg> et <seg type="var">.o.b.</seg> vel <lb/><seg type="var">o.c.</seg> (nam <seg type="var">.o.c.</seg> æqualis eſt <seg type="var">.o.b.</seg>) medię proportionales ſunt, eo quod cum <seg type="var">.g.h.</seg> media <lb/>proportionalis ſit inter <seg type="var">.u.x.</seg> et <seg type="var">.o.c.</seg> & proportio <seg type="var">.o.b.</seg> ad <seg type="var">.e.x.</seg> æqualis ſit ei, quæ <seg type="var">.u.x.</seg> <lb/>ad <seg type="var">.g.h.</seg> hoc eſt ei quæ <seg type="var">.g.h.</seg> ad <seg type="var">.o.c.</seg> vel. ad <seg type="var">.o.b.</seg> </s> <s xml:space="preserve">quare quotieſcunque inuentæ fuerint <seg type="var">.<lb/>g.h.</seg> et <seg type="var">.o.c.</seg> vel <seg type="var">.o.b.</seg> mediæ proportionales inter <seg type="var">.d.x.</seg> et <seg type="var">.x.e.</seg> ipſa <seg type="var">.o.c.</seg> vel <seg type="var">.o.b.</seg> erit ſemi <lb/>diameter ſphæræ quæſitę. </s> <s xml:space="preserve">eodem modo faciendum erit ſi ſphęroides fuerit prolata.</s> </p> <figure place="here"> <graphic url="0325-01"/> </figure> <pb facs="0326" n="314"/> <fw type="head">IO. BABPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Modus inueniendi duo triangula <choice><ex>varijs</ex><am>varijs</am></choice> conditionibus <lb/>affecta.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod etiam quæris ita ſe habet, duo ſcilicet triangula inuenire, æqualia dua-<lb/>bus ſuperficiebus rectilineis propoſitis, quę quidem triangula ſint eiuſdem <lb/>alritudinis, & quod <choice><ex>vnunquodque</ex><am>vnũquodque</am></choice> habeat angulum æqualem angulo pro<unclear reason="illegible"/>poſito, & <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>alius angulus vnius, cum alio alterius, æquetur duobus rectis.</s> </p> <p> <s xml:space="preserve">Sint exempli gratia duæ propoſitæ ſuperflcies <seg type="var">.c.y.</seg> duo verò anguli dati ſint <seg type="var">.r.s.</seg> <lb/>cum voluerimus inuenire duo triangula (quæ ſint <seg type="var">.a.i.u.</seg> et <seg type="var">.n.t.x.</seg> ) tali conditio-<lb/>ne prædita, quod angulus, a. æqualis ſit angulo <seg type="var">.s.</seg> & angulus <seg type="var">.t.</seg> angulo <seg type="var">.r.</seg> & quod <lb/>angulus <seg type="var">.x.</seg> ſimul cum angulo <seg type="var">.u.</seg> <choice><ex>æ- quentur</ex><am>æ-quẽtur</am></choice> duobus rectis, & quod <choice><ex>triam</ex><am>triã</am></choice> <lb/> <ptr xml:id="fig-0326-01a" corresp="fig-0326-01" type="figureAnchor"/> <choice><ex>gulum</ex><am>gulũ</am></choice> <seg type="var">.a.i.u.</seg> æquale ſit ſuperficiei <seg type="var">.<lb/>c.</seg> reliquum verò ſuperficiei <seg type="var">.y.</seg> <lb/>Ex duabus ſuperficiebus <seg type="var">.c.</seg> et <seg type="var">.y.</seg> <lb/>conſtituemus duo quadrata, per vl <lb/>timam ſecundi Eucli. accipiemus, <lb/></s> <s xml:space="preserve">deinde duo latera tetragonica ip-<lb/>ſorum quadratorum, & inuenie-<lb/>mus tertiam lineam in continua <lb/>proportionalitate cum illis lateri-<lb/>bus ex .10. ſexti, ſeruabimus po-<lb/>ſtea extremas illarum, quæ ſint <seg type="var">.z.</seg> <lb/>et <seg type="var">.l.</seg> quarum proportio, <choice><ex>eadem</ex><am>eadẽ</am></choice> erit, <lb/>quæ inter duas propoſitas ſuperfi-<lb/>cies reperitur ex .18. ſexti, accipie <lb/>mus, deinde lineam aliquam cu-<lb/>inſuis longitudinis, quæ ſit <seg type="var">.q.g.</seg> ſu-<lb/>pra quam conſtituemus in puncto <lb/>q. angulum <seg type="var">.m.q.g.</seg> ęqualem angu-<lb/>lo <seg type="var">.s.</seg> & angulum <seg type="var">.m.q.K.</seg> æqualem <lb/>angulo <seg type="var">.r.</seg> ex .23. primi, poſtea ve-<lb/>rò à quouis puncto ipſius lineæ <seg type="var">.q.<lb/>m.</seg> puta <seg type="var">.o.</seg> ducetur <seg type="var">.o.f.</seg> vſque ad <seg type="var">.q.<lb/>g.</seg> quorſum volueris, producendo <lb/>ipſam <choice><ex>vſque</ex><am>vſq;</am></choice>. ad <seg type="var">.d.</seg> ita quod propor-<lb/>tio <seg type="var">f.o.</seg> ad <seg type="var">.o.d.</seg> ſit vt <seg type="var">.z.</seg> ad <seg type="var">.l.</seg> ex .10. <lb/>ſexti, ducendo poſtea à puncto <seg type="var">.d.</seg> <lb/>lineam <seg type="var">.d.h.E.</seg> parallelam lineæ <seg type="var">.q.<lb/>g.</seg> & quia ex .2. primi Vitellionis <seg type="var">.<lb/>h.E.</seg> ſecatur ab <seg type="var">.q.K.</seg> in puncto <seg type="var">.b.</seg> <lb/>protrahemus <seg type="var">.b.o.p.</seg> vnde ex ſimi-<lb/>litudine triangulorum habebimus <lb/>proportionem <seg type="var">.p.o.</seg> ad <seg type="var">.o.b.</seg> vt <seg type="var">.f.o.</seg> <pb facs="0327" n="315"/><fw type="head">EPISTOL AE.</fw> ad <seg type="var">.o.d.</seg> hoc eſt vt <seg type="var">.z.</seg> ad <seg type="var">.l.</seg> hoc eſt vt <seg type="var">.c.</seg> ad <seg type="var">.y</seg>. </s> <s xml:space="preserve">quare <choice><ex>triangulum</ex><am>triangulũ</am></choice> <seg type="var">.p.q.o.</seg> ita erit proportio <lb/><choice><ex>natum</ex><am>natũ</am></choice> triangulo <seg type="var">.o.q.b.</seg> vt <seg type="var">.c.</seg> ad <seg type="var">.y.</seg> conſtituo deinde ex .25. fexti duo triangula ſimi-<lb/>lia duobus <seg type="var">.p.q.o.</seg> et <seg type="var">.o.q.b.</seg> <choice><ex>æqualiaque</ex><am>æqualiaq́</am></choice> <seg type="var">.c.</seg> et <seg type="var">.y.</seg> quę ſint <seg type="var">.a.i.u.</seg> et <seg type="var">.n.t.x.</seg> ſecetur poſtea <seg type="var">.q.<lb/>g.</seg> in puncto <seg type="var">.æ.</seg> ita, quod <seg type="var">.q.æ.</seg> æqualis ſit <seg type="var">.i.a.</seg> duco poſtea <seg type="var">.æ.e.</seg> æquidiſtantem. ad <seg type="var">.p.b.</seg> <lb/>& ſic habebimus duo triangula <seg type="var">.q.x.æ.</seg> ct <seg type="var">.q.x.</seg> e, vt quærebantur, quamuis duo trian <lb/>gula <seg type="var">.a.i.u.</seg> et <seg type="var">.t.n.x.</seg> eaſdem habeant conditiones.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0326-01" corresp="fig-0326-01a"> <graphic url="0326-01"/> </figure> </div> </body> </floatingText> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE IMPERFECTA SOLVTIONE PROBLE-<lb/>matis Nicolai Tartaleæ ad Cardanum. De animad-<lb/>uerſione in Ptolomeum. Deincendio carbo-<lb/>num à vento.</head> <head rend="italics" xml:space="preserve">Clariβimo Dominico Moreſino.</head> <p> <s xml:space="preserve"><hi rend="small caps">SCio</hi> propoſitam tibi quæſtionem te diu agitauiſſe, nectamen ſolutio-<lb/>nem aſſequi potuiſſe, aduerte igitur ipſam falſam, ideſt impoſſibilem <lb/>eſſe, quemadmodum etiam decimumoctauum quæſitum propoſitum à <lb/>Cardano Tartaleæ, ab ipſo Tarralea ſolutum minimè fuit. </s> <s xml:space="preserve">Quiquidem <lb/>Tartalea vult circulum deſcribi circa triangulum per quintam libri quarti Euclidis, <lb/>vt in fine ferè quintæ partis ſuarum menſurarum affirmat, neque videt in quinta <lb/>quarti Euclidem vti vndecima primi, & in vndecima primi, quarta aut octaua eiuſ-<lb/>dem, quas, ipſe Euclides oſtenſiuè non demonſtrauit. </s> <s xml:space="preserve">Quapropter oportebat Tar-<lb/>taleam demonſtraſſe omnes propoſitiones ad hoc neceſſarias oſtenſiuè <choice><ex>vſque</ex><am>vſq;</am></choice> ad pri-<lb/>mas indemonſtrabiles, quia ad demonſtrandam ſcientificè <choice><ex>aliquam</ex><am>aliquã</am></choice> propoſitionem, <lb/>aut à propoſitione in propoſitionem vſque ad prima principia vniuerſalia ( vt ali-<lb/>quando ego feci) eſt retrogradandum, aut ab ipſis principijs incipiendum ſucceſſi-<lb/>uè eouſque progrediendo donec ad propoſitionem quam demonſtrare volumus <lb/>perueniamus.</s> </p> <p> <s xml:space="preserve">Quod ad Ptolomeum in geographia attinet, dico eum mihi non ſatisfacere, cum <lb/>ſumit portionem arcus circuli maioris inter vnam ciuitatem, & aliam, ea ratione <lb/>quam deſcribit. </s> <s xml:space="preserve">Quod ſi vſus fuiſſet modo Menelai, ab ipſomet deinde in <choice><ex>ſuum</ex><am>ſuũ</am></choice> Al-<lb/>mageſtum vſurpato, aut Monteregij triangulorum ſphęricorum, quem Copernicus <lb/>adhibuit (qui tamen modus, tempore Ptolomei, nondum fortaſſe in lucem vene-<lb/>rat) bene egiſſet.</s> </p> <p> <s xml:space="preserve">Quod deinde ad ſuum illud inſtrumentum geometricum attinet, eſt <choice><ex>imperfectum</ex><am>imperfectũ</am></choice>, <lb/>vt oſtendi domino Pandulfo.</s> </p> <p> <s xml:space="preserve">Motum autem aeris, aut mauis ventum, accendere ignem, non ſolum ratione an <lb/>tiperiſtaſis, quam affers euenit, ſed etiam quia à carbonibus accenſis totam excre <lb/>mentitiam materiam, quæ eos circundat, auferat.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Alia dilucidatio propoſitionis .25. lib. 2. Monteregij.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">SCribiste non intelligere .25. propoſitionem lib. 2. Monteregij. cum necſcias <lb/>reperire diametrum circuli circunſcriptibilis circa propoſitum triangu- <pb facs="0328" n="316"/><fw type="head">IO. BAPT. BENED.</fw> lum, cuius data ſit b aſis tantummodo ſimul cum angulo, qui ipſi baſi opponitur.</s> </p> <p> <s xml:space="preserve">Imagineris igitur triangulum datum eſſe obtuſiangulum <seg type="var">.a.b.g.</seg> cuius baſi <seg type="var">.b.<lb/>g.</seg> ſit nobis data ſimul cum angulo <seg type="var">.a.</seg> ei oppoſito, obtuſoq́ue; </s> <s xml:space="preserve">Conſidera etiam cir-<lb/>culum <seg type="var">.a.b.g.q.</seg> ipſum trian gulum circunſcribentem, cuius diameter <seg type="var">.q.e.p.</seg> tranſeat <lb/>per <seg type="var">.m.</seg> punctum medium ipſius <seg type="var">.b.g.</seg> <choice><ex>tunc</ex><am>tũc</am></choice> protractis imaginatione <seg type="var">.e.g.</seg> et <seg type="var">.g.p.</seg> certi eri-<lb/>mus angulos. circa <seg type="var">.m.</seg> rectos eſſe ex .3 tertij Eucli. <choice><ex>angulumque</ex><am>angulumq́</am></choice> <seg type="var">.q.e.g.</seg> duplum eſſe an <lb/>gulo <seg type="var">.q.p.g.</seg> ex .19. eiuſdem, vnde æqualem angulo <seg type="var">.a.</seg> qui etiam duplus eſt angulo <seg type="var">.q.<lb/>p.g.</seg> quapropter proportio arcus <seg type="var">.q.g.</seg> ad arcum <lb/> <ptr xml:id="fig-0328-01a" corresp="fig-0328-01" type="figureAnchor"/> <seg type="var">g.p.</seg> tibi cognita erit, & proportio etiam chor-<lb/>de <seg type="var">.p.g.</seg> ad ſinum <seg type="var">.m.g.</seg> arcus <seg type="var">.g.p.</seg> & quia <seg type="var">.m.g.</seg> vt <lb/>dimidium ipſius <seg type="var">.b.g.</seg> tibi data eſt, cognoſces <lb/>etiam <seg type="var">.p.g.</seg> vt <seg type="var">.m.g.</seg> & ſic tertium latus <seg type="var">.m.p.</seg> trian-<lb/>guli orthogonij <seg type="var">.p.m.g.</seg> & q<unclear reason="illegible"/>a ex .34. tertij quod <lb/>fit ex <seg type="var">.p.m.</seg> in <seg type="var">.m.q.</seg> eſt æquale ei quod fit ex <seg type="var">.b.m.</seg> <lb/>in <seg type="var">.m.g.</seg> ideo cum diuiſum fuerit productum <seg type="var">.b.</seg> <lb/>m in <seg type="var">.m.g.</seg> per <seg type="var">.p.m.</seg> proueniet <seg type="var">.m.q.</seg> quapropter <lb/>habebis totum <seg type="var">.q.p</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0328-01" corresp="fig-0328-01a"> <graphic url="0328-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Idem efficies, ſi <choice><ex>cum</ex><am>cũ</am></choice> angulus <seg type="var">.a.</seg> acutus fuiſſet.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Modus inueniendi puncta elliptica via Pergei.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MOdus inueniendi puncta elliptica, via .21. primi lib. Pergei ex datis axibus, <lb/>vt vbi alias ſignificati, talis eſt. <lb/> <ptr xml:id="fig-0328-02a" corresp="fig-0328-02" type="figureAnchor"/> </s> <s xml:space="preserve">Sit exempli gratia maior axis propo-<lb/>ſitus <seg type="var">.a.c.</seg> minor autem <seg type="var">.b.d.</seg> cum ergo <lb/>volueris inuenire punctum circunfe-<lb/>rentiæ correſpondentem puncto <seg type="var">.e.</seg> <lb/>maioris axis, inueniemus primò la-<lb/>tus tetragonicum producti <seg type="var">.a.g.</seg> in <seg type="var">.g.<lb/>c.</seg> quod ſit <seg type="var">.h.</seg> <choice><ex>latusque</ex><am>latusq́</am></choice> <choice><ex>tetragonicum</ex><am>tetragonicũ</am></choice> pro-<lb/>ducti <seg type="var">.a.e.</seg> in <seg type="var">.e.c.</seg> quod ſit <seg type="var">.i</seg>. </s> <s xml:space="preserve">deinde in-<lb/>ueniemus lineam <seg type="var">.K.</seg> tertiam in con-<lb/>tinua proportionalitate cum <seg type="var">.h.</seg> et <seg type="var">.i.</seg> <lb/>vnde <seg type="var">.i.</seg> erit media proportionalis in-<lb/>ter <seg type="var">.h.</seg> et <seg type="var">.K.</seg> & vt <seg type="var">.h.</seg> proportionalis erit <lb/>ad <seg type="var">.K.</seg> inueniemus <seg type="var">.e.f.</seg> cui <seg type="var">.g.d.</seg> medie-<lb/>tas ſecundi axis ita ſe habeat, quæ po <lb/>ſtea iuncta axi maiori, ad angulosrectos in puncto <seg type="var">.e.</seg> dabit ſitum puncti <seg type="var">.f.</seg> quæſiti ex <lb/>dicta .21. primi lib. Pergei, ſed talis modus prolixus eſt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0328-02" corresp="fig-0328-02a"> <graphic url="0328-02"/> </figure> </div> </body> </floatingText> <pb facs="0329" n="317"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Accipeigitur huncalium.</s> </p> <p> <s xml:space="preserve">Sit propoſitus maior axis <seg type="var">.q.p.</seg> minor verò <seg type="var">.e.c.</seg> ad angulos rectos ſe inuicem <lb/>ſecantes in puncto <seg type="var">.o.</seg> deſcribatur circulus <seg type="var">.q.n.p.a.</seg> cuius diameter ſit axis maior, in <lb/>quo accipiatur punctum, quod volueris, vt puta <seg type="var">.u.</seg> à quo protrahatur <seg type="var">.u.b.</seg> paralle-<lb/>la ad <seg type="var">.o.c.n.</seg> deſignetur poſtea ſeparatim circulus <seg type="var">.u.b.n.</seg> cuius diameter æqualis ſit ſe <lb/>midiametro prioris circuli, ita etiam fiat circulus <seg type="var">.u.i.c.</seg> contingens circulum <seg type="var">.u.b.n.</seg> <lb/>in puncto <seg type="var">.u.</seg> cuius diameter ſit <seg type="var">.u.c.</seg> æqualis dimidio axi minori. </s> <s xml:space="preserve">accipiatur deinde in <lb/>circulo maximo longitudo <seg type="var">.u.b.</seg> quæ collocetur in circulo mediocri à puncto <seg type="var">.u.</seg> quæ <lb/>ſecabitur à minimo circulo in puncto <seg type="var">.i.</seg> cum itaque longitudo <seg type="var">.u.i.</seg> menſurata fue-<lb/>rit in <seg type="var">.u.b.</seg> maximi circuli à puncto <seg type="var">.u.</seg> habebimus propoſitum.</s> </p> <p> <s xml:space="preserve">Cuius reiratio eſt, quia <seg type="var">.u.b.</seg> mediocris circuli diuiditur à gyro minimi in puncto <lb/>i. eadem proportione, qua diuiſa eſt <seg type="var">.u.n.</seg> in puncto <seg type="var">.c.</seg> quod manifeſtum eſt exſimi-<lb/>litudine triangulorum <seg type="var">.u.b.n.</seg> et <seg type="var">.u.i.c.</seg> imaginatæ cum fuerint duæ <seg type="var">.b.n.</seg> et <seg type="var">.i.c.</seg> ſed ita <lb/>eſſe oportet parallelas maximi circuli, quotieſcunque circunferentia ipſius ellipſis <lb/>tranſitura ſit per <seg type="var">.c.</seg> vt in .51. cap. meæ gnomonicæ oſtenſum fuit.</s> </p> <figure place="here"> <graphic url="0329-01"/> </figure> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Modus deſignandi angulum, certo modo conditionatum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">NVllius reuera difficultatis mihi videtur eſſe, quotieſcunque nobis propoſita <lb/>fuerint duo puncta <seg type="var">.a.</seg> et <seg type="var">.b.</seg> ſimul cum <lb/>angulo <seg type="var">.d.</seg> <choice><ex>necnon</ex><am>necnõ</am></choice> linea <seg type="var">.g.</seg> ducere duas lineas <lb/> <ptr xml:id="fig-0329-02a" corresp="fig-0329-02" type="figureAnchor"/> à dictis punctis terminatas, quæ <choice><ex>conſtituant</ex><am>conſtituãt</am></choice> <lb/>angulum æqualem dato, & ipſæ directè <choice><ex>con</ex><am>cõ</am></choice> <lb/>iunctæ conſtituant lineam æqualem da-<lb/>tæ. </s> <s xml:space="preserve">Nam ducatur linea indefinita per <lb/>puncta propoſita, cuius lineæ, pars illa, quę <lb/>intercepta fuerit inter dicta puncta, diui-<lb/>datur per æqualia in puncto <seg type="var">.o.</seg> etiam & li-<lb/>nea data, quarum medietates accipio in <lb/>linea indefinitè protracta à puncto <seg type="var">.o.</seg> me- <pb facs="0330" n="318"/><fw type="head">IO. BAPT. BENED.</fw> dio, vt vna earum ſit <seg type="var">.o.c.</seg> reliqua verò ſit <seg type="var">.o.e.</seg> </s> <s xml:space="preserve">deinde aperiatur circinus quantum <seg type="var">.<lb/>o.c.</seg> <choice><ex>poſitoque</ex><am>poſitoq́;</am></choice> vno pede in <seg type="var">.b.</seg> <choice><ex>deſignentur</ex><am>deſignẽtur</am></choice> cum altero duo arcus <seg type="var">.n.K.</seg> poſito iterum vno <lb/>pede in <seg type="var">.a.</seg> deſignentur alij duo arcus inter-<lb/>ſecantes primos in punctis <seg type="var">.n.K</seg>. </s> <s xml:space="preserve">Deinde à <lb/> <ptr xml:id="fig-0330-01a" corresp="fig-0330-01" type="figureAnchor"/> puncto <seg type="var">.n.</seg> ad <seg type="var">.K.</seg> ducetur linea <seg type="var">.n.K.</seg> quæ per <lb/>punctum <seg type="var">.o.</seg> tranſibit, quam <seg type="var">.n.K.</seg> mente <choice><ex>con- cipio</ex><am>cõ-cipio</am></choice>, vt axis minor <choice><ex>vnius</ex><am>vniꝰ</am></choice> ellipſis, cuius <seg type="var">.e.c.</seg> <lb/>ſit axis maior, quibus axibus mediantibus <lb/>deſignetur ellipſis <seg type="var">.n.c.K.e.</seg> conſidero dein <lb/>de <seg type="var">.a.b.</seg> vt <choice><ex>chordanvnius</ex><am>chordãvnius</am></choice> circuli, ſeu portio-<lb/>nis circularis, quæ capax ſit vnius anguli <lb/>æqualis angulo <seg type="var">.d.</seg> propoſito, ex .32. tertij <lb/>Euclid. cuius circunferentia, circunferen-<lb/>tiam ipſius ellipſis ſecabit in <choice><ex>duobus</ex><am>duobꝰ</am></choice> punctis <lb/>quorum <choice><ex>vnum</ex><am>vnũ</am></choice> ſit <seg type="var">.i.</seg> à quo protractæ cum fue <lb/>rint duæ lineæ <seg type="var">.a.i.</seg> et <seg type="var">.i.b.</seg> habebis propoſitum, cum <seg type="var">.a.i.</seg> iuncta cum <seg type="var">.i.b.</seg> æquetur <seg type="var">.e.</seg> c, <lb/>ex .52. tertij Pergei.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0329-02" corresp="fig-0329-02a"> <graphic url="0329-02"/> </figure> <figure xml:id="fig-0330-01" corresp="fig-0330-01a"> <graphic url="0330-01"/> </figure> </div> </body> </floatingText> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">EMENDATIO CVIVSDAM FALSI MODI <lb/>delineandi horologia Italica orizontalia.</head> <head rend="italics" xml:space="preserve">Foanni Paulo Dardano.</head> <p> <s xml:space="preserve"><hi rend="small caps">MOdvs</hi> delineandi horologia Italica orizontalia, quem tibi monſtrauit <lb/>neſcio quis, ni fallor, talis eſt. </s> <s xml:space="preserve">Deſignato meridiano <seg type="var">.l.b.m.q.</seg> <choice><ex>ductisque</ex><am>ductisq́;</am></choice> duo <lb/>bus diametris <seg type="var">.l.m.</seg> et <seg type="var">.b.</seg> q, inuicem ad rectos in centro <seg type="var">.g.</seg> quorum <seg type="var">.l.m.</seg> ſit <lb/>verticalis <seg type="var">.b.q.</seg> vero <choice><ex>orizontalis</ex><am>orizõtalis</am></choice>, <choice><ex>ductoque</ex><am>ductoq́;</am></choice> diametro <seg type="var">.f.h.</seg> tropici Cancri ſe-<lb/>cundum altitudinem poli datam, <choice><ex>deſcriptoque</ex><am>deſcriptoq́;</am></choice> dimidio circulo <seg type="var">.f.z.h.</seg> ipſius paralleli, <lb/><choice><ex>inuentoque</ex><am>inuentoq́;</am></choice> puncto <seg type="var">.z.</seg> horæ propoſitę, & ab eo ducta per pendiculari <seg type="var">.z.r.</seg> ad <seg type="var">.f.h.</seg> & à <lb/>puncto <seg type="var">.r.</seg> ducta <seg type="var">.r.o.y.</seg> parallela ad diametrum <seg type="var">.q.b.</seg> orizontalem, ducis poſtea <seg type="var">.f.ω.</seg> et <seg type="var">.<lb/>r.t.</seg> vſque ad orizontalem <seg type="var">.q.b.</seg> parallelas ad diametrum <seg type="var">.l.m.</seg> verticalem. </s> <s xml:space="preserve">Determi-<lb/>nato poſtea gnomone <seg type="var">.g.s.</seg> in orizontis axe, <choice><ex>ductaque</ex><am>ductaq́;</am></choice> vmbrarum linea <seg type="var">.s.K.</seg> parallela <lb/>orizontali, <choice><ex>ductaque</ex><am>ductaq́;</am></choice> <seg type="var">.y.g.K.</seg> ad terminandam <seg type="var">.s.K.</seg> delineas deinde ſeparatim circulum <lb/><seg type="var">q.x.b.n.</seg> magnitudinis prioris, qui quidem circulus ſignificet orizontem ipſum, in <lb/>quo ductis diametris <seg type="var">.q.g.b.</seg> et <seg type="var">.l.g.m.</seg> accipis in diametro <seg type="var">.q.g.b.</seg> puncta <seg type="var">.a.</seg> et <seg type="var">.ω.</seg> ita à <choice><ex>cen tro</ex><am>cẽtro</am></choice> <seg type="var">.g.</seg> diſtantia, vt ſunt in diametro orizontali prioris circuli, ducis poſtea per pun-<lb/>ctum <seg type="var">.a.</seg> lineam <seg type="var">.x.a.n.</seg> ad rectos cum dicto diametro, </s> <s xml:space="preserve">deinde per tria puncta <seg type="var">.n.ω.x.</seg> <lb/>tranſire facis circunferentiam circuli per <choice><ex>quintam</ex><am>quintã</am></choice> quarti Euclidis, poſtea in dicto dia-<lb/>metro accipis punctum <seg type="var">.t.</seg> ita diſtans à centro, & ex eadem parte, vt in priori circulo, <lb/>à quo puncto ducis <seg type="var">.t.u.</seg> parallelam <seg type="var">.x.n.</seg> vſque ad circunferentiam <seg type="var">.x.ω.n.</seg> in puncto <seg type="var">.u.</seg> <lb/>quo facto, ducis à centro <seg type="var">.g.</seg> per punctum <seg type="var">.u.</seg> ipſius circularis circunferentiæ <seg type="var">.g.u.</seg> inde-<lb/>terminatam, quam poſtea terminas in puncto <seg type="var">.K.</seg> ita quod <seg type="var">.g.K.</seg> æqualis ſit <seg type="var">.s.K</seg>. </s> <s xml:space="preserve">Dicis <lb/>poſtea punctum <seg type="var">.K.</seg> in eodem ſitu reperiri, reſpectu duorum diametrorum <seg type="var">.q.b.</seg> me-<lb/>ridiani. et <seg type="var">.l.m.</seg> verticalis, vt decet, & oportet punctum horæ propoſitę exiſtere.</s> </p> <p> <s xml:space="preserve">Quod quidem dico eſſe falſum, propterea quod perpendiculares quas cogita-<lb/>mus cadere à punctis circunferentiæ cuiuſuis paralleli ſupra quemuis orizontem ob <pb facs="0331" n="319"/><fw type="head">EPISTOL AE.</fw> liquum ſecantem æquatorem, omnes caduntin gyro elliptico, oxygonio, ſeu de-<lb/>fectionali, & non circulari. </s> <s xml:space="preserve">Vnde per ſupradicta tria puncta <seg type="var">.n.ω.x.</seg> oporteret tranſi <lb/>re talem circunferentiam, & non <choice><ex>circularem</ex><am>circularẽ</am></choice>, quæ circunferentia eſſet vnius ellipſis, <lb/>cuius minor axis in diametro <seg type="var">.b.q.</seg> eſſet .ab <seg type="var">.ω.</seg> vſque ad <seg type="var">.i.</seg> terminum ſini <seg type="var">h.i.</seg> arcus <seg type="var">.h.b.</seg> <lb/>in analemate, maior verò axis eſſet magnitudinis <seg type="var">.f.h.</seg> diametri paralleli, quæ <choice><ex>tranſiſ- ſet</ex><am>trãſiſ-ſet</am></choice> per punctum <seg type="var">.c.</seg> medium inter <seg type="var">.ω.</seg> et <seg type="var">.i.</seg> quę quidem circunfere ntia tota eſſet intra cir <lb/>culum <seg type="var">.q.n.b.x.</seg> <choice><ex>con</ex><am>cõ</am></choice><choice><ex>tiguaque</ex><am>tiguaq́;</am></choice> gyro <seg type="var">.q.n.b.x.</seg> in punctis <seg type="var">.n.x</seg>.</s> </p> <p> <s xml:space="preserve">Si ergò circunferentia <seg type="var">.n.ω.x.</seg> eſſet elliptica tunc punctum <seg type="var">.u.</seg> in orizonte illud eſſet <lb/>vbi caderet ſinus altitudinis horę, et <seg type="var">.t.u.</seg> æqualis eſſet <seg type="var">.r.z.</seg> communi ſectioni paralle <lb/>li cum almicantarat ex .34. primi Euclid. et <seg type="var">.u.g.</seg> æqualis eſſet <seg type="var">.o.y.</seg> communi ſectioni <lb/>almicantarat cum meridiano, vel cum azimut illius horæ ex .4. primi, cum <seg type="var">.g.t.</seg> æqua <lb/>lis ſit ipſi <seg type="var">.o.r.</seg> et <seg type="var">.t.u.</seg> ipſi <seg type="var">.r.z.</seg> & angu <lb/> <ptr xml:id="fig-0331-01a" corresp="fig-0331-01" type="figureAnchor"/> lus. t trianguli <seg type="var">.g.t.u.</seg> rectus, quem-<lb/>admodum <seg type="var">.r.</seg> qui compræhenditur <lb/>ab <seg type="var">.z.r.</seg> et <seg type="var">.r.o.</seg> vnde anguli <seg type="var">.K.g.m.</seg> <lb/>et <seg type="var">.K.g.b.</seg> rectè ſe haberent, diſtan-<lb/>tia verò inter <seg type="var">.K.</seg> et <seg type="var">.g.</seg> <choice><ex>iam</ex><am>iã</am></choice> rectè ſum-<lb/>pta fuit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0331-01" corresp="fig-0331-01a"> <graphic url="0331-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed quia punctum <seg type="var">.u.</seg> vt <choice><ex>plurimum</ex><am>plurimũ</am></choice> <lb/>(in gyro circulari ſumptum) extra <lb/>puncta interſectionum ipſius circu <lb/>laris gyri cum elliptico reperìtur, <lb/></s> <s xml:space="preserve">propterea efficit angulos <seg type="var">.K.g.m.</seg> <lb/>et <seg type="var">.K.g.b.</seg> falſos, & non æquales il-<lb/>lis, qui fiunt ab azimut horæ cum <lb/>verticali, & cum meridiano, quæ <lb/>omnia ex cap .52. meæ gnomonicę <lb/>facilè videre potes.</s> </p> <p> <s xml:space="preserve">Nectacere volo quod punctum <lb/>u. verum, hoc eſt ellipticum, inue-<lb/>niri poſſet ea via quam ſcripſi in <lb/>eodem .52. cap. qua mediante do-<lb/>cui demum inuenire punctum <seg type="var">.π.</seg> <lb/>orizontis, quamuis in præſenti ca<lb/>ſu <seg type="var">.ω.λ.</seg> perpendicularis eſſet ſupra <lb/>minorem axem ipſius ellipſis, <choice><ex>quan- uis</ex><am>quã-uis</am></choice> ſupra maiorem axem, quod ta-<lb/>men minimè mutat ordinem, imò <lb/>rationes eędem ſunt, tam in vna, <lb/>quam in alia operatione, ſed vt il-<lb/>licò <choice><ex>idipsum</ex><am>idipsũ</am></choice> habeas, fac vt <seg type="var">.t.u.</seg> æqua <lb/>lis. ſit <seg type="var">.r.z</seg>. </s> <s xml:space="preserve">& tunc punctum <seg type="var">.K.</seg> erit <lb/><choice><ex>quæſitum</ex><am>quæſitũ</am></choice>, quod ego in .52. cap. meę <lb/>gnomonicę, ijs verbis ſignificaui.</s> </p> <quote> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> mediis binis triangulis ijs, <lb/><choice><ex>medioque</ex><am>medioq́;</am></choice> azimut Solis pariter ho-<lb/>rologia fabricari poterunt.</s> </quote> <pb facs="0332" n="320"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De Horologio perpendiculari ad oriz ontem rectum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MOdus quem tibi ſcribere promiſi delineandi lineas horarias communes in <lb/>pariete perpendiculariter ad orizontem rectum, declinantem à meridiano, <lb/>ſumendus eſt ex .46. cap. meæ gnomonicæ, hocſcilicet ordine.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia, orizon hic ſubſcriptus. or. oc <seg type="var">.M.S.</seg> diuiſus à meridiana <seg type="var">.M.<lb/>S.</seg> et verticali ſeu æquinoctiali. or. oc. </s> <s xml:space="preserve">Sitq́ue <seg type="var">.e.t.</seg> communis ſectio muri cum ori-<lb/>zonte, et <seg type="var">.g.n.</seg> ſit gnomon perpendicularis ipſi muro, vnde ex dictis in mea gnomo-<lb/>nica, cognoſcemus in ipſa murali orizontali totam <seg type="var">.e.t.</seg> inter meridianam orizonta-<lb/>lem, & æquinoctialem orizontalem, cognoſcemus etiam partem <seg type="var">.g.t.</seg> ipſius æ-<lb/>quinoctialis orizontalis, quam quidem accipiamus in rectitudine ipſius muralis ori <lb/>zontalis, quæ quidem ſit <seg type="var">.t.G.</seg> quo facto erigatur <seg type="var">.G.A.</seg> ad rectos cum <seg type="var">.G.t.e.</seg> & cir-<lb/>cum <seg type="var">.G.e.</seg> deſignetur vna medietas circuli verſus <seg type="var">.e.</seg> cuiuſuis magnitudinis, quæ di-<lb/>uiſa in .12. partes ęquales, ſignificabit <choice><ex>medietatem</ex><am>medietatẽ</am></choice> æquatoris, <choice><ex>protrahanturque</ex><am>protrahanturq́;</am></choice> lineæ oc <lb/>cultæ à centro <seg type="var">.G.</seg> per ſectiones circunferentię dimidij circuli, quæ fignificabunt <choice><ex>con</ex><am>cõ</am></choice> <lb/>munes ſectiones æquatoris cum circulis horarijs communibus, quo facto oportet, vt <lb/>à puncto <seg type="var">.t.</seg> protrahatur <seg type="var">.t.s.</seg> ad rectos cum murali orizontali, quæ quidem <seg type="var">.t.s.</seg> ſignifi-<lb/>cabit communem ſectionem æquatoris cum muro propoſito, & erit ęquędiſtans me <lb/>ridianæ murali ex .6. vndecimi Eucli. eo quod ex .19. eiuſdem <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> illarum, per <lb/>pendicularis eſt tali orizonti. </s> <s xml:space="preserve">Videantur nunc puncta communia iſti <seg type="var">.t.s.</seg> & occultis <lb/>protractis à centro <seg type="var">.G.</seg> medietatis circularis, per quæ puncta protrahantur à puncto <seg type="var">.<lb/>e.</seg> tot lineæ, punctum enim <seg type="var">.e.</seg> ſignificat punctum axis mundi, & meridianæ in mu-<lb/>ro propoſito, eo quod in tali ſitu ſphæræ rectæ, dictum punctum reperitur in orizon <lb/>re, cum <seg type="var">.M.s.</seg> non ſolum ſit meridiana orizontalis, ſed etiam axis mundi, </s> <s xml:space="preserve">deinde nul <lb/>li dubium eſt quin meridiana muralis ſit perpendicularis orizontali murali <seg type="var">.e.t.</seg> à <lb/>puncto <seg type="var">.e</seg>. </s> <s xml:space="preserve">Sed quia dimidium <lb/> <ptr xml:id="fig-0332-01a" corresp="fig-0332-01" type="figureAnchor"/> harum <choice><ex>linearum</ex><am>linearũ</am></choice> horariarum erit <lb/>ſub orizontali <seg type="var">.e.t.G.</seg> alterum <lb/>vero <choice><ex>dimidium</ex><am>dimidiũ</am></choice> ſupra ipſam, opor <lb/>tet quod quę ſupra ſunt produ-<lb/>cantur à parte .oe. ſub orizonta-<lb/>li, ab alia parte meridianæ, & <lb/>talis erit effigies horologij mura <lb/>lis in hoc ſphęrę ſitu, hoc eſt ver <lb/>ſus quartam orientalem auſtra-<lb/><choice><ex>lemque</ex><am>lemq́;</am></choice>, vnde orizontalis <seg type="var">.e.t.</seg> <lb/>erit ſemper horæ .6. matutinæ, <lb/>ſecunda verò ab ipſa erit horæ. <lb/>7. tertia <choice><ex>autem</ex><am>autẽ</am></choice> horæ .8. & ſic dein-<lb/>ceps.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0332-01" corresp="fig-0332-01a"> <graphic url="0332-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quotieſcunque verò angu-<lb/>lus <seg type="var">.n.g.e.</seg> minor erit maxima <lb/>Solis declinatione, & Sol fuerit <lb/>in parte auſtrali ab æquatore <choice><ex>cum</ex><am>cũ</am></choice> <pb facs="0333" n="321"/><fw type="head">EPISTOL AE.</fw> maiori numero declinationis quam fuerit angulus <seg type="var">.n.g.e</seg>. </s> <s xml:space="preserve">tunc talis paries illumina-<lb/>bitur ab ipſo Sole à mane vſque ad veſperam.</s> </p> <p> <s xml:space="preserve">Huius quidem rei ſpeculatio, vnicuique manifeſta erit, qui rationes .46. cap. no-<lb/>ſtræ gnomonicæ prius intellexerit, vbi manifeſtè apparet proportionem ſemidiame <lb/>tri horologij (ſi ita eam appellare licet) ad ſemidiametrum æquatoris horarij ſem-<lb/>per eſſe, vt <seg type="var">.e.t.</seg> ad <seg type="var">.t.g.</seg> hoc eſt proportio maioris inæqualitatis. </s> <s xml:space="preserve">nolo etiam prætermic <lb/>tere. </s> <s xml:space="preserve">quin te admoneam, vt nullo pacto confidas in longioribus vmbris, eo quod val <lb/>de nos decipiant, cum ſemper iuſto breuiores ſint.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Declar atio quorundam verborum noſtræ<unclear reason="illegible"/> Gnomonicæ. <lb/><choice><ex>Defenſioque</ex><am>Defenſioq́ꝫ</am></choice> nostra contra Christophorum Clauium.</head> <head xml:space="preserve">AD EVNDEM DARDANVM.</head> <p> <s xml:space="preserve">TVas demum accepi literas, qui <lb/> <ptr xml:id="fig-0333-01a" corresp="fig-0333-01" type="figureAnchor"/> bus mihi ſignificas te totum <num value="52">.<lb/>52.</num> caput meæ gnomonicæ intelle-<lb/>xiſſe, præter illa verba, quæ etiam <lb/>ſuperioribus diebus ad te ſcripſi, <lb/>hoc eſt.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0333-01" corresp="fig-0333-01a"> <graphic url="0333-01"/> </figure> </div> </body> </floatingText> <quote> <s xml:space="preserve"><choice><ex>Itaque</ex><am>Itaq;</am></choice> medijs binis triangulis ijs, <lb/><choice><ex>medioque</ex><am>medioq́;</am></choice> azimut Solis, pariter ho-<lb/>rologia fabricari poterunt.</s> </quote> <p> <s xml:space="preserve">Quapropter nealiquid tibi de-<lb/>ſit, ſcire debes, menihil aliud, eo <lb/>in loco inferre voluiſſe, quàm <choice><ex>quod</ex><am>qđ</am></choice> <lb/>punctum horæ propoſitæ in plano <lb/>horologij orizontali reperiri po-<lb/>teſt, ope longitudinis vmbræ gno-<lb/>monis, & eius declinationis à ver-<lb/>ticali linea, ſeu à meridiana orizon <lb/>tali, iam in ipſo horologij plano <lb/>ductis.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſit analemma <seg type="var">.<lb/>l.q.m.b.</seg> in quo <seg type="var">.l.m.</seg> ſit verticalis <seg type="var">.q.<lb/>b.</seg> verò orizontalis <seg type="var">.f.n.h.</seg> autem ſit <lb/>ſemicirculus, cuiuſuis paralleli æqui <lb/>noctiali, cuius diameter fit <seg type="var">.f.h.</seg> et <seg type="var">.<lb/>n.</seg> ſit Solis locus in ipſo parallelo: </s> <s xml:space="preserve">n.<lb/>r. autem ſit rectus ſinus arcus <seg type="var">.f.n</seg>: et <seg type="var">.<lb/>r.o.z.</seg> ſectio communis ipſius almi-<lb/>cantarat cum meridiano, et <seg type="var">.s.a.</seg> <choice><ex>con</ex><am>cõ</am></choice> <lb/>munis ſectio azimut Solis cum pla-<lb/>no horologij, et <seg type="var">.s.g.</seg> gnomon, et <seg type="var">.x.<lb/>g.a.</seg> radius Solis <seg type="var">.z.u.</seg> verò ſinus alti-<lb/>tudinis ipſius Solis, vbi videre po-<lb/>tes duo triangula dicta eſſe <seg type="var">.z.u.g.</seg> <pb facs="0334" n="322"/><fw type="head">IO. BAPT. BENED.</fw> et <seg type="var">.g.s.a.</seg> quibus mediantibus cognoſcitur longitudo vmbræ gnomonis hoc eſt <seg type="var">.s.a</seg>.</s> </p> <p> <s xml:space="preserve">Cum autem dico, <choice><ex>medioque</ex><am>medioq́;</am></choice> azimut Solis, nihil aliud ſigniſicare volo, niſi angu-<lb/>lum, quem terminat linea azimutalis horologij, hoc eſt vmbra gnomonis cum li-<lb/>nea meridiana, ſeu cum verticali in ipſo plano horologij. </s> <s xml:space="preserve">qui quidem anguli, æqua <lb/>les ſunt ijs, qui in triangulo conſtituto ex <seg type="var">.n.r.</seg> ex <seg type="var">.r.o.</seg> & ex <seg type="var">.o.z.</seg> reperiuntur, cuius qui <lb/>dem trianguli, angulus puncti <seg type="var">.r.</seg> rectus eſt, angulus verò terminatus ab <seg type="var">.n.r.</seg> et <seg type="var">.o.z.</seg> il <lb/>le eſt quem conſtituit azimut cum verticali, vel ipſi æqualis, vt coalternus, reliquus <lb/>verò in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.o.</seg> ille eſt <choice><ex>quem</ex><am>quẽ</am></choice> azimut facit <choice><ex>cum</ex><am>cũ</am></choice> meridiano, vel ipſi ęqualis vt coalternus.</s> </p> <p> <s xml:space="preserve">Vnde quotieſcunque volueris in aliquo plano, orizonti parallelo, lineas hora-<lb/>rias ducere, iudico optimum fore ſi ſeparatim deſignatæ fuerint hæ tres figuræ, hoc <lb/>eſt analemma meridianum, vel azimutale, vt ita dicam, </s> <s xml:space="preserve">deinde parallelus <choice><ex>inſeruiens</ex><am>inſeruiẽs</am></choice> <lb/>pro tropicis, vt ego feci cap .51. meæ gnomonicæ, quæ duæ figuræ, <choice><ex>ſufficientes</ex><am>ſufficiẽtes</am></choice> <choice><ex>erunt</ex><am>erũt</am></choice> <lb/>pro omnibus horologijs, tam ori-<lb/> <ptr xml:id="fig-0334-01a" corresp="fig-0334-01" type="figureAnchor"/> zontalibus quam muralibus, non <lb/>tamen omninò, ideo pro orizon-<lb/>talibus, tertiam figuram ſeparatam <lb/>deſignaui, quę erit circulus <seg type="var">.H.I.<lb/>K.</seg> eiuſdem magnitudinis cum ana-<lb/>lemmate, in quo, ductis duobus <lb/>dia metris inuicom ad rectos, quo-<lb/>rum vnus <seg type="var">.H.I.</seg> ſignificet orizon-<lb/>talem lineam, reliqua verò <choice><ex>ver- ticalem</ex><am>ver-ticalẽ</am></choice>, ducatur poſtea <seg type="var">.s.a.</seg> tam di-<lb/>ſtans ab <seg type="var">.H.I.</seg> quanta eſt longitudo <lb/>gnomonis horologij orizontalis, <lb/>cogitemus, </s> <s xml:space="preserve">deinde hunc circulum <lb/>communem eſſe omnibus azimut <lb/>necnon plano horologij, in cuius <lb/>circunferentia à puncto <seg type="var">.k.</seg> nadir ip-<lb/>ſius zenit, accipiantur arcus æqua-<lb/>les ijs ipſorum azimut, quos termi-<lb/>nat zenit, & ipſi almicantarat, vt <lb/>exempli gratia, accipiemus arcum <lb/><seg type="var">k.L.</seg> æqualem arcui <seg type="var">.L.z.</seg> ipſius ana-<lb/>lemmatis, ducta poſtea linea occul <lb/>ta <seg type="var">.o.L.</seg> ſignabimus azimutalem <seg type="var">.s.a.</seg> <lb/>in puncto <seg type="var">.a.</seg> vbi hæ duæ lineæ ſein <lb/>uicem ſecant, & ſic habebimus iu-<lb/>ſtam <choice><ex>quantitatem</ex><am>quãtitatem</am></choice> ipſius vmbræ gno <lb/>monis <seg type="var">.s.o.</seg> tali hora, </s> <s xml:space="preserve">deinde in ori-<lb/>zontali <seg type="var">.H.I.</seg> ſumatur <seg type="var">.o.r.</seg> à centro <seg type="var">.<lb/>o.</seg> ęqualis ei quę in analemate repe <lb/>ritur, quæ vna portio eſt commu-<lb/>nis ſectionis meridiani cum almi-<lb/>cantarat, terminata ab axe orizon <lb/>tis, & à diametro paralleli. </s> <s xml:space="preserve">Deinde <pb facs="0335" n="323"/><fw type="head">EPISTOL AE.</fw> <choice><ex>ducatur</ex><am>ducat̃</am></choice>. <seg type="var">r.V.</seg> ad rectos cum <seg type="var">.H.I.</seg> vſque ad circunferentiam, in qua accipiatur <seg type="var">.r.n.</seg> æqua <lb/>lis ei quæ eſt in parallelo, ducatur poſtea <seg type="var">.o.n.M.</seg> & habe bimus triangulum <seg type="var">.o.r.n.</seg> ſi-<lb/>milem <choice><ex>æqualemque</ex><am>æqualemq́;</am></choice> triangulo iam ſupradicto. </s> <s xml:space="preserve">Vnde angulus <seg type="var">.H.o.M.</seg> ei æqualis erit, <lb/>quem azimut facit cum meridiano, & angulus <seg type="var">.M.o.k.</seg> ei ęqualis, quem azimut con-<lb/>ſtituit cum verticali, ita quod ſi talis circulus <seg type="var">.H.k.I.</seg> eſſet planum horologij orizon-<lb/>talis, ſuppoſito <seg type="var">.o.</seg> pro pede gnomonis, ſecando poſtea <seg type="var">.o.M.</seg> in puncto <seg type="var">.i.</seg> ita vt <seg type="var">.o.i.</seg> <lb/>æqualis eſſet <seg type="var">.s.a.</seg> dato quod <seg type="var">.o.M.</seg> ducta ſit ad partem ſibi conuenientem, reſpectu <seg type="var">.o.<lb/>k.</seg> ipſa pro verticali ſuppoſita, quod tibirelinquo, cum hoc facillimum ſit, tunc pun <lb/>ctum <seg type="var">.i.</seg> eſſet quod quærebamus. </s> <s xml:space="preserve">Quod verò de vno puncto dico, idem de omni-<lb/>bus infero.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0334-01" corresp="fig-0334-01a"> <graphic url="0334-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Vbi verò mihi ſignificas Chriſtophorum Clauium, me duobus in locis meæ gno <lb/>monicæ redarguere, iam vidi. </s> <s xml:space="preserve">Circa primum locum igitur, qui eſt in pagin .161. <lb/>ita inquit.</s> </p> <quote> <s xml:space="preserve">Non enim deſunt, qui vel omninò negent, inter quos eſt Ioannes Baptiſta Be-<lb/>nedictus in ſua gnomonica cap .70. et .71. vbialia, & multo longiore ratione cona-<lb/>tur arcus ſignorum deſcribere, vel certe dubitent, hoc modo rectè poſſe deſcribi ar-<lb/>cus ſignorum, cum rationem non videant, qua hæc noſtra deſcriptio quam quidem <lb/>omnes ſcriptores ſine vlla demonſtratione tradunt nitatur.</s> </quote> <p> <s xml:space="preserve">Abſque dubio raptim tranſcurrit illa capita .70. 71. </s> <s xml:space="preserve">Reuerendus Clauius alio quin <lb/>non ſcripſiſſet, quòd ego alia & multo longiore ratione conatus ſim arcus ſignorum <lb/>deſcribere & c. præſertim cum eadem prorſus ratio, quæ ibi à me tradita eſt, illa ſit, <lb/>quam ip ſe ſuis ſcriptis inſeruit.</s> </p> <p> <s xml:space="preserve">Meus igitur modus in dictis capitibus traditus, minime diſcrepat ab eo, ſed ab il-<lb/>lorum modo, quorum opinio eſt interualla <seg type="var">.e.h</seg>: <seg type="var">h.u</seg>: u. n; </s> <s xml:space="preserve">n.m. et <seg type="var">.m.d.</seg> meæ figuræ in <lb/>pagi .75. poſitæ, æqualia eſſe interuallis <seg type="var">.e.h</seg>:h.u:u.n:n.m. et <seg type="var">.m.d.</seg> præcedentis figuræ, <lb/>qui etiam ſupponunt <seg type="var">.t.e.</seg> meæ figuræ .75. eſſe directè coniuncta cum linea <seg type="var">.e.h.u.n.<lb/>m.d.</seg> & propterea verſus finem .73. pag. dixi.</s> </p> <quote> <s xml:space="preserve">Aduertat autem quam diligentiſſime quiſque ne ſe decipi patiatur à ſubſcripta fi <lb/>gura ſemicirculi <seg type="var">.Q.æ.m.</seg> cum reliquis lineis ductis, ex antiquorum more, & c.</s> </quote> <p> <s xml:space="preserve">Eo quod non defuerunt aliqui, ex vetuſtioribus (quorum ſcripta ad meas manus <lb/>peruenerunt) qui ſumentes interualla <seg type="var">e.h</seg>:h.u. & c. figuræ. pag .75. æqualia illis figu-<lb/>ræ pag .74. <choice><ex>putauerunt</ex><am>putauerũt</am></choice> lineam <seg type="var">.t.e.</seg> directè coniunctam eſſe cum <seg type="var">.e.h.</seg> & c. quod quidem <lb/>maximi erroris cauſa erat, </s> <s xml:space="preserve">& propterea cap .71. verum modum oſtendi, ſeruando il <lb/>lam eandem ſuppoſitionem, hoc eſt quod interſtitia <seg type="var">.e.h</seg>: <seg type="var">h.u</seg>: & c. figurę pag .75. æqua <lb/>lia ſint interſtitijs <seg type="var">.e.h</seg>: <seg type="var">h.u.</seg> & c. præcedentis figuræ, & ideò in dicto cap .71. dixi.</s> </p> <quote> <s xml:space="preserve">Suppoſito deinde <seg type="var">.f.e.b.</seg> lineam meridianam eſſe in plano orizontali, cęterę lineę <lb/>horarię erunt prędictę.</s> </quote> <p> <s xml:space="preserve">Stantibus igitur his ſuppoſitis, vt habeantur omnia ſcientificè, volui, vt intellige-<lb/>retur pyra mis qua drilatera, eo modo quo dixi, cap .71. vbi clarè patet eandem pyra <lb/>midem eſſe, quam Pater Clauius (tacitè) poſuit in figura horologij, vt ipſe docuit <lb/>propoſitione ſecunda, lib. ſecundi, cuius baſis eſt triangulum <seg type="var">.H.I.F.</seg> ſuæ figurę (exem <lb/>pli gratia pro quinta hora poſt meridiana) Alterum verò triangulum à me cogita-<lb/>tum, terminatum ab <seg type="var">.t.e</seg>: <seg type="var">e.d</seg>: et. ab <seg type="var">.t.d.</seg> eleuata in mea figura, eſt in ſua <choice><ex>triangulum</ex><am>triãgulum</am></choice> <seg type="var">.D.<lb/>I.F.</seg> & propterea dixi.</s> </p> <quote> <s xml:space="preserve">Nam <seg type="var">.t.e.</seg> et <seg type="var">.e.d.</seg> <choice><ex>vtræque</ex><am>vtræq;</am></choice> in plano horologii non ſunt, quamuis in plano æquatoris <lb/>tres ſint, & c.</s> </quote> <p> <s xml:space="preserve">Angulus verò <seg type="var">.e.</seg> quem dico rectum eſſe, in ſua figura eſt angulus <seg type="var">.D.I.F.</seg> & mea <pb facs="0336" n="324"/><fw type="head">IO. BAPT. BENED.</fw> <seg type="var">t.d.</seg> imaginata, eſt ſua <seg type="var">.D.F</seg>. </s> <s xml:space="preserve">Tertium deinde triangulum, quod in mea figura ter-<lb/>minatur ab <seg type="var">.t.d.</seg> ab <seg type="var">.f.d.</seg> & ab <seg type="var">f.t.</seg> in ſua eſt triangulum <seg type="var">.D.F.H.</seg> vnde mea <seg type="var">.f.t.</seg> reſpon-<lb/>det ſuę <seg type="var">.H.D.</seg> & mea <seg type="var">.f.d.</seg> ſuæ <seg type="var">.H.F.</seg> & mea <seg type="var">.t.d.</seg> ſuę <seg type="var">.D.F</seg>. </s> <s xml:space="preserve">Quartum autem triangulum <lb/><seg type="var">f.t.e.</seg> in mea figura, reſpondet ſuo <seg type="var">.H.D.I.</seg> & meum punctum <seg type="var">.t.</seg> ſuo. </s> <s xml:space="preserve">D, Nunc triangu <lb/>lum rectangulum, quod dico ſeparatim conſtituere, eſt illud tertium dictum corre-<lb/>ſpondens ſuo <seg type="var">.D.F.H.</seg> vt ipſe facit in <choice><ex>ſequenti</ex><am>ſequẽti</am></choice> figura, quod ipſe vocat <seg type="var">.D.C.H.</seg> & <choice><ex>meus</ex><am>meꝰ</am></choice><unclear reason="illegible"/> <lb/>radius <seg type="var">.t.x.</seg> in ſua figura, ille eſt qui terminatur ab <seg type="var">.D.</seg> & ab initio Tauri, & Virginis.</s> </p> <p> <s xml:space="preserve">Et quamuis ego non ſcripſerim talem ſiguram, vt ipſe fecit, nihilominus ipſam <lb/>verbis deſcripſi eomet modo, & propterea dixi.</s> </p> <quote> <s xml:space="preserve">Quam <choice><ex>diuiſionem</ex><am>diuiſionẽ</am></choice>, ſi in triangulo ſeorſum deſcripto inuenire voluerimus, res erit <lb/>inuentu facillima, cum rectum angulum <seg type="var">.f.t.d.</seg> (reſpondentem ſuo <seg type="var">.H.D.C.</seg>) prędicti <lb/>trianguli tertij ea ratione diuiſerimus, & c.</s> </quote> <p> <s xml:space="preserve">Quapropter Reuerendus Clauius non animaduertit meam rationem aliam non <lb/>eſſe, nec puncto longiorem ſua, cum eademmet ipſa ſit.</s> </p> <p> <s xml:space="preserve">Citaui etiam Munſterum cap .30. eo quod in ea impreſſione, quam tunc prę mani <lb/>bus habui, vidi in ea figura, quam ipſe vocat fundamentum horologiorum, literam <lb/>c. poſitam eſſe loco <seg type="var">.f.</seg> et <seg type="var">.f.</seg> loco <seg type="var">.c.</seg> quod cauſæ fuit, vt omnia mendoſa viderentur, re <lb/>centiores autem impreſſiones correctæ ſunt.</s> </p> <p> <s xml:space="preserve">Rurſus alio in loco mihi accidit vt repręhenderim Alexandrum Piccolomineum <lb/>in libris de ſphęra, qui quidem dicebat eas figuras ſuperficiales, quæ paucioribus an <lb/>gulis circunſcriberentur, capaciores eſſe alijs, dummodo earum periphæriæ eſſent <lb/>æquales.</s> </p> <p> <s xml:space="preserve">Nunc autem correctę ſunt eo in loco impreſſiones, & qui non viderit primas, pu-<lb/>tabit me immeritò ipſum repræhendere.</s> </p> <p> <s xml:space="preserve">Idem etiam dico de eo capite ipſius Piccolominei, in ijſdem libris, vbi tractat de <lb/>modo, quo vſi ſunt antiqui ad diuidendum zodiacum in .12. ſigna, quod erat circa <lb/>finem quarti libri.</s> </p> <p> <s xml:space="preserve">Nunc verò, in recentioribus impreſſionibus, illud caput poſitum non eſt. </s> <s xml:space="preserve">Impreſ <lb/>ſiones autem illæ, vbi talia dixit, duæ fuerunt, quarum prima erat anni .1540. ſecun <lb/>da verò .1552 Venetijs apud Andream Puteum.</s> </p> <p> <s xml:space="preserve">Alius verò locus ipſius Reuerendi Clauij, contra meas repręhenſiones, eſt circa <lb/>finem pag .298. & circa .299. vbi ita ſcribit.</s> </p> <quote> <s xml:space="preserve">Ex his liquido conſtat, non rectè à Ioan. Baptiſta Benedicto in ſua gnomonica ca <lb/>pit .49. repręhendi hancrationem deſcribendi horologij declinantis, qua omnes fe-<lb/>rè alij ſcriptores vtuntur, quoniam, vt ex demonſtratione à nobis allata conſtat, re-<lb/>ctè per eam lineæ ho rarię in plano, quod à verticali declinat ducuntur. </s> <s xml:space="preserve">Modus au <lb/>tem quem eo loco pręſcribit differentem ab eo, quem nos tradidimus certus etiam <lb/>eſt, ſed nulla ratione noſtro contrarius, quia nos conſtituimus <seg type="var">.D.E.F.</seg> angulum de-<lb/>clinationis plani à verticali circulo propriè dicto, ipſe autem loco huius anguli aſſu-<lb/>mit angulum declinationis eiuſdem plani à Meridiano circulo, vnde mirum non eſt <lb/>modum ipſius à noſtro diſcrepare. </s> <s xml:space="preserve">Quod ſi <choice><ex>conſtitueremus</ex><am>cõſtitueremus</am></choice> <seg type="var">.D.E.F.</seg> angulum decli-<lb/>natio nis plani à Meridiano, ut ipſe (quemadmodum forſitan ab alijs putauit fieri) <lb/>& in reliqua deſcriptione progrederemur, vt tradidimus, proculdubio <choice><ex>horologium</ex><am>horologiũ</am></choice> <lb/>declinans perperam deſcriberetur, vt rectè docet.</s> </quote> <p> <s xml:space="preserve">Optimè ſcripſiſſet Reuerendus Clauius, ſi verum fuiſſet, quod antiqui ſumerent <lb/>declinationem ſuperius <choice><ex>dictam</ex><am>dictã</am></choice> à verticali propriè dicto, & non à meridiano. </s> <s xml:space="preserve">Sed ego <lb/>dico, authores à me citatos. capit .49. meę gnomonicę ſumere dictam declinatio- <pb facs="0337" n="325"/><fw type="head">EPISTOLAE.</fw> nem planià meridiano, & non à dicto verticali.</s> </p> <p> <s xml:space="preserve">Con ſidera primum in Munſtero cap .16. ſuæ horologiographiæ, vbi clarè docet <lb/>accipere angulum compræhenſum inter meridianum, & planum propoſitum, vbi <lb/>etiam ponit quandam figuram ædificij cum pariete ſuper quo deſignatum eſt quod <lb/>dam horologium, & vbi ſe manifeſtè declarar, ita dicens.</s> </p> <quote> <s xml:space="preserve">Nam ipſarum partium complementum. propoſitum indicabit angulum, quan-<lb/>tus videlicet fuerit arcus eiuſdem circuli <seg type="var">.d.e.f.g.</seg> à puncto <seg type="var">.g.</seg> vſque ad productam li-<lb/>neam meridianam interceptus, qui vnà cum ipſo <seg type="var">.f.g.</seg> quadrantem integrare videtur, <lb/>vt in ſequenti figura: </s> <s xml:space="preserve">quoniam arcus <seg type="var">.f.g.</seg> eſt ſexaginta partium, qualium <seg type="var">.e.</seg> f<unclear reason="illegible"/>. <choice><ex>quadrans</ex><am>quadrãs</am></choice> <lb/>nonaginta, vnde concluditur reliquam partem hoc eſt, datum inclinationis <choice><ex>angulum</ex><am>angulũ</am></choice>, <lb/>fore partium triginta ſimilium.</s> </quote> <p> <s xml:space="preserve">Orontius verò cap .13. <choice><ex>ijſdem</ex><am>ijſdẽ</am></choice> vtitur verbis, cum figura ſimili ad reliqua autem ipſius <lb/>R. Clauij, videnda nondum mihi otium fuit. </s> <s xml:space="preserve">quod ſi dabitur, tibi libenter<unclear reason="illegible"/> dicam <lb/>quid ſentiam.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE MODO DVCENDI LINEAS HORARIAS <lb/>ſuper cyllindro immobili.</head> <head rend="italics" xml:space="preserve">Hieronymo Ferrerio artium & Medicina Doctori peritißimo.</head> <p> <s xml:space="preserve"><hi rend="small caps">DEsignare</hi> horarias lineas ſuper cyllindro immobili, ad orizontemq́ue <lb/>perpendiculariter erecto difficile tibi non erit, (quod à me poſtulaſti) ſi <lb/>modum .53. cap. meæ gnomonicæ obſeruaueris, accipiendo tamen pro <lb/>linea orizontali in tabula non aliquam rectam lineam, ſed circularem, <lb/>ſimilemque circunferentiæ ipſius cyllindri, dico autem ſimilem, eo quod ſi gn o-<lb/>mon <seg type="var">.o.x.</seg> ſupra tabulam ſignatus, & perpendicularis ipſi orizontali circulari <seg type="var">.b.i.x.</seg> <lb/>eſſet dimidia, vel tertia vel quarta pars gnomonis cyllindro infixi, oporteret, vt <lb/>ſemidiameter circuli <seg type="var">.b.i.x.</seg> etiam <lb/>eſſet medietas, vel tertia, aut quar <lb/> <ptr xml:id="fig-0337-01a" corresp="fig-0337-01" type="figureAnchor"/> ta pars ſemidiametri cyllindri, vt <lb/>omnes arcus huiuſmodi circuli in <lb/>ter ipſos azimut intercepti ſimiles <lb/>ſint arcubus cyllindri, quod à <lb/>te ipſo facilè videre ſcientificè po <lb/>teris. </s> <s xml:space="preserve">reliqua nihil mutanda erunt <lb/>ab eo, quod ſcripſi circa figuram <num value="53">.<lb/>53.</num> cap. vt dixi. </s> <s xml:space="preserve">Vnde inuenta <lb/>cum fuerit diſtantia orizontalis <lb/>puncti <seg type="var">.b.</seg> à pede gnomonis <seg type="var">.x.</seg> nec <lb/>non quantitas azi mutalis muralis <lb/><seg type="var">b.t.</seg> quæ ſemper ab orizontali per <lb/>pendiculariter deſcendit, illicò <lb/>punctum <seg type="var">.t.</seg> horæ propoſitæ in cy-<lb/>lindro inuenietur.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0337-01" corresp="fig-0337-01a"> <graphic url="0337-01"/> </figure> </div> </body> </floatingText> <pb facs="0338" n="326"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Nunc verò cum duo puncta alicuius horarię lineæ inuenta fuerint, quæ à Solis ſi-<lb/>tu in diuerſis parallelis efficiuntur, ſi voluerimus ipſam lineam <choice><ex>horariam</ex><am>horariã</am></choice> ducere, ſcien <lb/>dum primò eſt ipſam lineam horariam eſſe communem ſectionem circuli horarij, <lb/>illius horæ cum ſuperficie cyllindrica, </s> <s xml:space="preserve">& propterea ellipticam, vt oſtendit Serenus <lb/>in .19. primi lib. quod etiam ellicere poſſumus ab eo, quod Archimedes in .10. pro-<lb/>poſitione libr. de conoidalibus, ſcribit. </s> <s xml:space="preserve">Quapropter oporter nos inſtrumen-<lb/>tum prius componere, modo circini, ſed trium crurum, quæ omnia in eadem <lb/>plana ſuperficie ſint, ea tamen arte factum, vt quodlibet illorum poſſimus pro-<lb/>longare, necnon contrahere, ut cum duo extrema firmata fuerint, media poſ-<lb/>ſit circunduci circa centrum, ſeu punctum commune illarum interſectionum <choice><ex>ſimulque</ex><am>ſimulq́;</am></choice> <lb/>poſſit produci, necnon abbreuiari vel augeri, & diminui, vt mediante ſua extremi-<lb/>tate inſeriori poſſimus delineare gyrum ellipticum horarium, dum <choice><ex>centrum</ex><am>cẽtrum</am></choice> ipſorum <lb/>crurum adhæreat extremitati gnomonis, reliquæ vero extremitates ipſorum <choice><ex>crurum</ex><am>crurũ</am></choice> <lb/>ſint ſupra puncta inuenta ipſius horæ. </s> <s xml:space="preserve">oportet etiam vt hoc inſtrumentum à tergo <lb/>ipſorum crurum habeat in ſuperiori parte ſuperficiem quandam <choice><ex>ſemicircularem</ex><am>ſemicircularẽ</am></choice>, quę <lb/>ſit vice vnius partis illius ſuperficiei, in qua ſupponuntur omnia crura inſtrumenti, <lb/>& hoc quantum fieri poteſt, quod quidem fieri debet, ne crus medium, hoc eſt mo <lb/>bile, exeat à tali ſuperficie, ſeu declinet ab ea, quæ ſemper ſupponitur in ſitu circuli <lb/>horarij talis horæ. </s> <s xml:space="preserve">oportet etiam, vt iuxta circunferentiam dimidij circuli ſint duo <lb/>gyri eiuſdem materiæ inter ſe parum diſtantes, ita ut crura poſſint moueri, intra hos <lb/>gyros, & dimidium circulum, & quod inter hos gyros locatæ ſint duæ cochleæ, ſeu <lb/>duo helices, vt quando voluerimus, poſſimus fir-<lb/>mare ipſa crura extrema, dum eorum extremitates <lb/> <ptr xml:id="fig-0338-01a" corresp="fig-0338-01" type="figureAnchor"/> fuerint ſupra puncta inuenta illius horæ, </s> <s xml:space="preserve">deinde in <lb/>dorſo iſtius inſtrumenti, circa centrum coniunctio <lb/>nis, rectè factum erit ſi aliqua concauitas fuerit, in <lb/>qua, extremitas gnomonis poſſit locari, dum duce-<lb/>re voluerimus aliquam horariam lineam.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0338-01" corresp="fig-0338-01a"> <graphic url="0338-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Tale inſtrumentum excogitaui ad fugiendum <lb/>tædium inueniendi dictam ellipticam ex punctis.</s> </p> <p> <s xml:space="preserve">Nunc autem ſciendum eſt, quod vnus tantum-<lb/>modo gnomon ſufficiens non erit pro tota die æſti-<lb/>ua, neque duo, niſi valde breues fuerint reſpectu <lb/>ſemidiametri cyllindri, & in ſitu medio quartarum <lb/>meridionalium noſtro orizonti, quorum autem <lb/>longitudo ita inuenienda eſſet.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia circulus <seg type="var">.a.b.e.u.</seg> cyllindri ori <lb/>zontis vice, <choice><ex>diuiſusque</ex><am>diuiſusq́;</am></choice> à duobus diametris <seg type="var">.d.e.</seg> et <seg type="var">.c.<lb/>f.</seg> quarum <seg type="var">.c.f.</seg> ſit pro meridiana: </s> <s xml:space="preserve">d.e. autem pro verticali, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> e. punctus orientalis: </s> <s xml:space="preserve">d. <lb/>verò <choice><ex>occidentalis</ex><am>occidẽtalis</am></choice> <seg type="var">.f.</seg> autem meridionalis. et <seg type="var">.c.</seg> ſeptentrionalis, <choice><ex>computeturque</ex><am>computeturq́;</am></choice> maxima. <lb/></s> <s xml:space="preserve">Solis amplitudo ab <seg type="var">.f.</seg> verſus <seg type="var">.e.</seg> quæ terminetur ab <seg type="var">.q.</seg> ita <choice><ex>quodarcus</ex><am>quodarcꝰ</am></choice> <seg type="var">.f.q.</seg> minor ſit <choice><ex>quam</ex><am>quã</am></choice> <lb/>graduum .45. aliter impoſſibile eſſet duobus <choice><ex>tantummodo</ex><am>tantũmodo</am></choice> gnomonibus mediantibus <lb/>tota die æſtiua horas videre.</s> </p> <p> <s xml:space="preserve">Quo facto ducatur ab .q: <seg type="var">q.p.</seg> contingens circulum & à centro circuli <seg type="var">.o.</seg> per pun-<lb/>ctum <seg type="var">.u.</seg> medium quartæ ducatur <seg type="var">.o.u.i.</seg> vſque ad contingentem <seg type="var">.q.p.</seg> vnde <seg type="var">.u.i.</seg> longitu <lb/>do erit vniuſcuiuſque gnomonis, qui gnomones infixi erunt in medio dictarum <lb/>quartarum.</s> </p> <pb facs="0339" n="327"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Huiuſmodi rei ratio per ſe nota erit quotieſcunque cogitauerimus verum arcum <lb/><seg type="var">e.b.</seg> amplitudinis æſtiuæ, <choice><ex>protractaque</ex><am>protractaq́;</am></choice> <seg type="var">.o.b.</seg> quę parallela erit <seg type="var">.q.p.</seg> vnde cum Sol tem-<lb/>pore æſtiuo orietur, tunc radios ſuos emittet via iſtarum æquidiſtantium linearum.</s> </p> <p> <s xml:space="preserve">Sed ſi longiores gnomones cuperes, oportebit eos tres eſſe, quorum vnus erit <lb/>orientalis in puncto <seg type="var">.e.</seg> alter occi-<lb/>dentalis in puncto <seg type="var">.d.</seg> reliquus ve-<lb/> <ptr xml:id="fig-0339-01a" corresp="fig-0339-01" type="figureAnchor"/> rò meridionalis in puncto <seg type="var">.f.</seg> quo-<lb/>rum <choice><ex>vnuſquiſque</ex><am>vnuſquiſq;</am></choice> poteſt eſſe maior <lb/>tertia parte ſemidiametri cyllin-<lb/>dri, ſed ſi voluerimus ſcire <choice><ex>quan- tum</ex><am>quan-tũ</am></choice> ad plus poſſit eſſe longus vnuſ-<lb/>quiſque illorum, ita faciendum <lb/>erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0339-01" corresp="fig-0339-01a"> <graphic url="0339-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Faciemus <choice><ex>quadratum</ex><am>quadratũ</am></choice> <seg type="var">.o.a.h.u.</seg> ex <lb/>ſemidiametro dicti circuli, a dia-<lb/>metro poſtea <seg type="var">.o.h.</seg> huiuſmodi qua <lb/>drati ſubtrahatur ſemidiameter <seg type="var">.<lb/>o.e.</seg> circuli, reſiduum verò <seg type="var">.e.h.</seg> ip-<lb/>ſius diametri <seg type="var">.o.h.</seg> quadrati, erit <choice><ex>lon</ex><am>lõ</am></choice> <lb/>gitudo gnomonis, vbi ſimul appa <lb/>ret huiuſmodi rei ratio, eo quod <lb/>cum gnomon <seg type="var">.e.h.</seg> orientalis deſi-<lb/>net operari, illico meridionalis <seg type="var">.<lb/>f.g.</seg> ſubintrabit, poſt hunc verò occidentalis <seg type="var">.d.K.</seg> monſtrabit reliquum diei.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Earundem line arum deſcriptio ſaper conum rectum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">CVm ſuper datum conum re-<lb/> <ptr xml:id="fig-0339-02a" corresp="fig-0339-02" type="figureAnchor"/> ctum idem facere volueris <lb/>eſto conus <seg type="var">.A.</seg> & R. qui diuiſus ima <lb/>ginatione ſit à quodam plano per <lb/>axem, & communis ſectio ſit trian <lb/>gulus <seg type="var">.A.</seg> & R. in quo plano cogite <lb/>mus gnomonem infixum ad rectos <lb/>vbi volueris, qui ſit <seg type="var">.p.t.o.</seg> cogitem <lb/>etiam <seg type="var">.l.t.m.</seg> aliud eſſe planum (in <lb/>quo ſit gnomon) quod conum ſe <lb/>cet, quæ quidem ſectio, circularis <lb/>erit, ex .4. primi Pergei. </s> <s xml:space="preserve">imagine-<lb/>mur etiam ſuperficiem <seg type="var">.p.s.</seg> eſſe azi <lb/>mut in quo gnomonreperirur, ſu-<lb/><choice><ex>perficiemque</ex><am>perficiemq́;</am></choice> <seg type="var">.e.s.</seg> azimut propoſitæ <lb/>horę, <choice><ex>angulumque</ex><am>angulumq́;</am></choice> <seg type="var">.e.o.a.</seg> contrapoſi <lb/><choice><ex>tum</ex><am>tũ</am></choice> angulo altitudinis Solis ab ori- <pb facs="0340" n="328"/><fw type="head">IO. BAPT. BENED.</fw> zonte; </s> <s xml:space="preserve">cogitemus etiam lineam <seg type="var">.A.t.i.x.</seg> illud coni latus eſſe, qu od à ſummitate ver<lb/>ſus baſim tranſit per medium latitudinis ipſius gnomonis, concipiamus etiam mente <lb/><seg type="var">e.a.</seg> communem ſectionem eſſe trianguli ſupra dicti cum azimut horæ, necnon pun-<lb/>ctum <seg type="var">.K.</seg> eſſe commune radio Solis <seg type="var">.o.a.</seg> & ſuperficiei conicæ, quod quidem eſt illud <lb/>quod quæritur, hoc ſcilicet modo. </s> <s xml:space="preserve">Primum cognoſcimus angulum <seg type="var">.p.A.t.</seg> vt medie <lb/>tas anguli totius coni, & angulum <seg type="var">.p.</seg> rectum, vnde <seg type="var">.t.</seg> tam intrinſecus, quam extrinſe-<lb/>custrianguli <seg type="var">.A.p.t.</seg> nobis cognitus erit. </s> <s xml:space="preserve">Nunc cum angulus <seg type="var">.A.t.o.</seg> cognoſcatur, ſi <lb/>gnomon <seg type="var">t.o.</seg> fixus fuerit in ſuperficie conica, ita qd cum latere <seg type="var">.A.t.</seg> eſſiciat <choice><ex>angulum</ex><am>angulũ</am></choice> <lb/><seg type="var">A.t.o.</seg> & lateraliter faciat angulosrectos cum ſuperficie conica, ad quod efficiendum <lb/>nulla eſt difficultas, cognoſcendo deinde <seg type="var">.A.t.</seg> ſimul cum angulis <seg type="var">.A.</seg> et <seg type="var">.t.</seg> intrinſecis <lb/>trianguli ortogonij <seg type="var">.A.p.t.</seg> cognoſcemus <seg type="var">.p.t.</seg> et <seg type="var">.A.p.</seg> vnde etiam tota <seg type="var">.o.p.</seg> ſed cogno <lb/>ſcendo <seg type="var">.o.p.</seg> cum angulo <seg type="var">.p.o.e.</seg> (angulus enim <seg type="var">.p.o.e.</seg> cognoſcitur ex hypotheſi cum <lb/>ſit inter azimut Solis & azimut gnomonis) cum angulo <seg type="var">.o.p.e.</seg> recto cognoſcemus <seg type="var">.p.<lb/>e.</seg> et <seg type="var">.o.e.</seg> </s> <s xml:space="preserve">deinde cum nobis nota ſit <seg type="var">.o.e.</seg> cum angulo altitudinis Solis <seg type="var">.e.o.a.</seg> & angu-<lb/>lo <seg type="var">.o.e.a.</seg> recto cognoſc emus longitudinem azimutalis <seg type="var">.e.a.</seg> necnon quantitatem <seg type="var">.a.o.</seg> <lb/>Imaginata poſtea <seg type="var">.a.q.</seg> æquidiſtante <seg type="var">.e.p.</seg> habebimus <seg type="var">.p.q.</seg> æqualem <seg type="var">.a.e.</seg> ex .34. primi <lb/>Eucli. </s> <s xml:space="preserve">Vnde duabus <seg type="var">.o.p.</seg> et <seg type="var">.p.q.</seg> mediantibus, <choice><ex>cognitiſque</ex><am>cognitiſq́;</am></choice> cum angulo recto <seg type="var">.p.</seg> cogno <lb/>ſcemus <seg type="var">.o.q.</seg> nec non angulum <seg type="var">.o.q.<lb/>p.</seg> quo mediante, necnon median-<lb/>te angulo <seg type="var">.q.A.t.</seg> et <seg type="var">.A.q.</seg> cognita, co <lb/> <ptr xml:id="fig-0340-01a" corresp="fig-0340-01" type="figureAnchor"/> gnoſcemus <seg type="var">.A.i.</seg> et <seg type="var">.q.i.</seg> quę <seg type="var">.q.i.</seg> dem <lb/>pta à <seg type="var">.q.o.</seg> relinquet nobis <choice><ex>cognitam</ex><am>cognitã</am></choice> <lb/><seg type="var">i.o</seg>. </s> <s xml:space="preserve">Et quia <seg type="var">.o.i.q.</seg> et <seg type="var">.o.K.a.</seg> ſemper <lb/>ſunt in eadem ſuperficie ſecante co <lb/>num, quæ etiam ſecat ſuperficiem <lb/>trianguli <seg type="var">.A.q.x.</seg> ad rectos ex .18. vn <lb/>decimi, cum linea <seg type="var">.u.n.</seg> perpendicu <lb/>laris ſit ſuperficiei trianguli <seg type="var">.A.q.i.</seg> <lb/>ex .8. dicti, quia parallela eſt <seg type="var">.l.p.</seg> quę <lb/>perpendicularis eſt ſuperficiei <choice><ex>trian- guli</ex><am>triã-guli</am></choice> <seg type="var">.o.p.q.</seg> ex .4. eiuſdem, ſequitur, <lb/>quod talis ſectio ( quæ intelligatur <lb/>per <seg type="var">.u.K.i.n.</seg>) ſemper erit elliptica, <lb/>vel parabole, ſeu hyperbole, <choice><ex>prout</ex><am>ꝓut</am></choice> <lb/>linea <seg type="var">.o.i.q.</seg> ſecabit latus coni, oppo <lb/>ſitum lateri <seg type="var">.A.i.</seg> diſtento in ipſa ſuperficie conica, ſeu ad ſuperiorem partem produ <lb/>ctum, velipſi parallelum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0339-02" corresp="fig-0339-02a"> <graphic url="0339-02"/> </figure> <figure xml:id="fig-0340-01" corresp="fig-0340-01a"> <graphic url="0340-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Supponamus nunc dictam lineam <seg type="var">.o.q.</seg> ſecare dictum oppoſitum latus lateri <seg type="var">.A.i.</seg> <lb/>verſus baſim, vnde ſectio <seg type="var">.u.K.i.n.</seg> erit elliptica. </s> <s xml:space="preserve">quod facile cognitu eſt <choice><ex>mediante</ex><am>mediãte</am></choice> com <lb/>paratione angulorum <seg type="var">.A.q.i.</seg> et <seg type="var">.q.A.i.</seg> interſe, eo quod ſi eſſent ęquales, dicta ſect o <lb/>barabola eſſet ex .27. primi Eucli. et .11. primi Pergei, ſed ſi angulus <seg type="var">.A.q.i.</seg> maior eſ-<lb/>ſet angulo <seg type="var">.q.A.i.</seg> ſectio eſſet ellipſis, ex ultimo poſtulato primi Euclid. </s> <s xml:space="preserve">& ex .13. pri-<lb/>mi Pergei, ſed ſi dictus angulus <seg type="var">.A.q.i.</seg> minor eſſet angulo <seg type="var">.A.</seg> tunc ſectio eſſet hyper-<lb/>bole ex dicto poſtulato & ex .12. primi Pergei. </s> <s xml:space="preserve">Sit ergo primum vt <choice><ex>dictum</ex><am>dictũ</am></choice> eſt, hoc <lb/>eſt, quod ſectio eſſet oxygonia, ideſt elliptica, ſeu defectio (quod idem eſt,) ſepa-<lb/>ratim oportebit nos ellipſim deſignare <choice><ex>ſimilem</ex><am>ſimilẽ</am></choice> <choice><ex>ęqualemque</ex><am>ęqualẽq́;</am></choice> ei, quæ eſt <seg type="var">.u.K.i.n.</seg> <choice><ex>quod</ex><am>qđ</am></choice> <choice><ex>quidem</ex><am>quidẽ</am></choice> <lb/>difficile non erit, quotieſcunque ſuos axes inuenerimus, maiorem ſcilicet, & mino- <pb facs="0341" n="329"/><fw type="head">EPISTOL AE.</fw> rem, quæ ita reperientur, efficiemus primo anguium coni, qui ſit <seg type="var">.i.A.b.</seg> quem diui-<lb/>demus per æqualia mediante <seg type="var">.A.q.</seg> conſtituendo <seg type="var">.A.i.</seg> huius anguli æqualem <seg type="var">.A.i.</seg> ſu-<lb/>perficiei conicæ et <seg type="var">.A.q.</seg> diuidentem, æqualem parti <seg type="var">.A.q.</seg> axis coni, ducendo poſtea <lb/>ab <seg type="var">.i.</seg> per <seg type="var">.q.</seg> lineam vnam quouſque concurrat <seg type="var">.A.b.</seg> in puncto <seg type="var">.b.</seg> habebimus <seg type="var">.i.b.</seg> pro <lb/>maiori axi ipſi ellipſis, quod per ſe clarum eſt, cuius medietas ſit <seg type="var">.i.c.</seg> ſed <seg type="var">.i.q.</seg> ipſius <seg type="var">.i.<lb/>b.</seg> æqualis eſt ipſi <seg type="var">.q.i.</seg> ipſius coni, ex quarta primi Eucli. et <seg type="var">.q.b.</seg> ipſius <seg type="var">.i.b.</seg> æqualis alte <lb/>ri parti inuiſibili. </s> <s xml:space="preserve">Reliquum eſt, vt reperiamus minorem axem, quem vocabimus <seg type="var">.<lb/>f.r.</seg> ducatur ergo primum <seg type="var">.q.a.u.n.</seg> ad rectos cum <seg type="var">.i.b.</seg> <choice><ex>æqualisque</ex><am>æqualisq́;</am></choice> ei quæ eſt coni, & diui <lb/>ſa ſimiliter in <seg type="var">.a.</seg> quæ <seg type="var">.u.n.</seg> ipſius coni nobis cognita eſt ex lateribus <seg type="var">.A.u.</seg> et <seg type="var">.A.n.</seg> & ex <lb/>angulo coni, et <seg type="var">.a.q.</seg> æqualis eſt <seg type="var">.e.p.</seg> ex .34. primi. </s> <s xml:space="preserve">Nunc certi erimus ex .21. primi <lb/>Pergei, quod eadem proportio erit quadrati <seg type="var">.u.q.</seg> ad quadratum ipſius <seg type="var">.f.c.</seg> quæ pro-<lb/>ducti ipſius <seg type="var">.i.q.</seg> in <seg type="var">.q.b.</seg> ad productum ipſius <seg type="var">.i.c.</seg> in <seg type="var">.c.b.</seg> & cum cognita nobis ſint <lb/>hæc tria producta hoc eſt <seg type="var">.i.q.</seg> in <seg type="var">.q.b.</seg> et <seg type="var">.i.c.</seg> in <seg type="var">.c.b.</seg> et <seg type="var">.u.q.</seg> in ſeipſa, cognoſcemus <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>quartum ipſius <seg type="var">.f.c.</seg> & fic <seg type="var">.f.c.</seg> <choice><ex>eiuſque</ex><am>eiuſq́;</am></choice> duplum <seg type="var">.f.r.</seg> cogniti nobis itaque cum ſint hi duo <lb/>axes <seg type="var">.i.b.</seg> et <seg type="var">.f.r.</seg> formabimus ellipſim. </s> <s xml:space="preserve">Deinde producemus axim <seg type="var">.b.i.</seg> à part <seg type="var">e.i.</seg> quo-<lb/>uſque <seg type="var">.i.o.</seg> æqualis ſit ei quæ extra conum eſt, dein-<lb/> <ptr xml:id="fig-0341-01a" corresp="fig-0341-01" type="figureAnchor"/> de ducemus <seg type="var">.o.a.</seg> quæ circunferentiam ellipticam <lb/>ſecabit in puncto <seg type="var">.K.</seg> vnde habebimus quantita-<lb/>tem ipſius <seg type="var">.o.K.</seg> et <seg type="var">.K.i.</seg> rectam. </s> <s xml:space="preserve">inde mediante cir-<lb/>cino ſi acceperimus rectam diſtantiam ab <seg type="var">.i.</seg> ad <seg type="var">.K.</seg> <lb/>in ellipſi, </s> <s xml:space="preserve">deinde firmando pedem circini in pun-<lb/>cto <seg type="var">.i.</seg> in ſuperficie conica, & cum alio ſignando <lb/>lineam vnam curuam ad partem <seg type="var">.K.</seg> in ſuperficie <lb/>conica, ſumendo poſtea interuallum <seg type="var">.o.K.</seg> extra el <lb/>lipſim, </s> <s xml:space="preserve">deinde firmando vnum pedem circini in <lb/>extre mitate gnomonis, cum alio poſtea ſignan-<lb/>do aliam lineam curuam in ſuperficie ipſius coni, <lb/>quæ primam ſe cet in puncto <seg type="var">.K.</seg> hoc erit punctum <lb/>quæſitum horę propoſitæ in ſuperficie conica <lb/>propoſita.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0341-01" corresp="fig-0341-01a"> <graphic url="0341-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi talis ſectio fuerit parabole, vel hyperbo <lb/>le, tunc mediante ſuo diametro <seg type="var">.i.q.</seg> cum baſi <seg type="var">.u.<lb/>q.n.</seg> cognita, deſignabimus ipſam ſectionem <seg type="var">.u.i.</seg> n <lb/>ope mei <choice><ex>inſtrumenti</ex><am>inſtrumẽti</am></choice> in calce meę gnomonicæ de <lb/>ſcripti, </s> <s xml:space="preserve">deinde diuiſa <seg type="var">.u.q.</seg> in <seg type="var">.a.</seg> <choice><ex>pro</ex><am>ꝓ</am></choice><choice><ex>ductaque</ex><am>ductaq́;</am></choice> <seg type="var">q.i.</seg> <choice><ex>vſque</ex><am>vſq;</am></choice> <lb/> <ptr xml:id="fig-0341-02a" corresp="fig-0341-02" type="figureAnchor"/> ad <seg type="var">.o.</seg> <choice><ex>ductaque</ex><am>ductaq́;</am></choice> <seg type="var">.o.a.</seg> habebimus punctum <seg type="var">.K</seg>. </s> <s xml:space="preserve">Reli-<lb/>qua facienda ſunt, vt dictum eſt de ellipſi.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0341-02" corresp="fig-0341-02a"> <graphic url="0341-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Inuenta modo cum fuerint duo puncta eiuſ-<lb/>dem horæ propoſitę, ducemus ab vno ad a-<lb/>liud, lineam horariam mediante circino trium <lb/>crurum, quem tibi ſcripſi nudius tertius pro cyl <lb/>lindro, quæ <choice><ex>quidem</ex><am>quidẽ</am></choice> linea crit portio gyri ellipſis, <lb/>ſeu hyperbolę, vel parabolę, vt à te ipſo cogi-<lb/>tare potes.</s> </p> <pb facs="0342" n="330"/> <fw type="head">IO. BAPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">QVAEDAM NOTATV DIGNA IN <lb/>Ptolomeum.</head> <head rend="italics" xml:space="preserve">Bartolomeo Christino Serenißimi Sabaudiœ Ducis apparitore.</head> <p> <s xml:space="preserve">EX tuis literis cognoui quo erga me animo eſſes, <choice><ex>qualique</ex><am>qualiq́;</am></choice> voluntate, ſed ne <lb/>tua pulcherrima ſtudia aliquo modo imperfecta <choice><ex>relinquantur</ex><am>relinquant̃</am></choice>, vel ego tibi <lb/>deeſſe videar, dum Problemata geographica Magni Ptolomei conſi-<lb/>deras, aduerte, quod ſi putares in figura .6. cap. libr .7. geographię eiuſ-<lb/>dem (vt multi credunt) lineam <seg type="var">.V.*.</seg> ſecare circunferentiam <seg type="var">.A.D.</seg> in puncto <seg type="var">.G.</seg> ita <lb/>vt punctus <seg type="var">.G.</seg> ſit tropici æſtiui, ideſt arcum <seg type="var">.D.G.</seg> eſſe graduum .24. cum illis inci-<lb/>deres in maximum errorem. </s> <s xml:space="preserve">Quapropter conſidera quæ nunc tibi ſcribo.</s> </p> <p> <s xml:space="preserve">Sit circulus <seg type="var">.A.B.C.D.</seg> huius centrum <seg type="var">.E.</seg> <choice><ex>ſupponaturque</ex><am>ſupponaturq́;</am></choice> ſemidiameter <seg type="var">.E.D.</seg> eſſe <lb/>partium 120. quarum <seg type="var">.E.Z.</seg> in alio ſemidiametro <seg type="var">.C.E.</seg> ei orthogonaliter coniuncto, <lb/>talium ſit .17. in ſemidiametro vero <seg type="var">.E.A.</seg> accipiatur <seg type="var">.E.S.</seg> talium .24. et <seg type="var">.E.V.</seg> 64. vn <lb/>de <seg type="var">.S.V.</seg> erit partium 40. ſimilium.</s> </p> <p> <s xml:space="preserve">Erigatur deinde <seg type="var">.S.*.</seg> ad rectos cum <seg type="var">.E.A.</seg> in puncto <seg type="var">.S.</seg> quæ terminetur ab inter-<lb/>ſectione lineę ductæ per puncta <seg type="var">.Z.D.</seg> in puncto <seg type="var">.*.</seg> ducatur demum <seg type="var">.V.*.</seg> quæ ſeca-<lb/>bit circunf rentiam <seg type="var">.A.D.</seg> in puncto <seg type="var">.G</seg>. </s> <s xml:space="preserve">Quæratur nunc quantitas ipſius <seg type="var">.G.D.</seg> <lb/>Ad quod efficiendum quærenda primum eſt quantitas ipſius <seg type="var">.S.*.</seg> quam illico co <lb/>gnoſcemus ex regula de tribus, cum dixerimus, ſi@ 17. dat nobis .120. quid dabit .41. <lb/>(nam duo triangula <seg type="var">.Z.E.D.</seg> et <seg type="var">.Z.S.*.</seg> ſunt inuicem ſimilia, cum <seg type="var">.S.*.</seg> parallela ſit <lb/>ipſi <seg type="var">.E.D.</seg>) vnde <seg type="var">.S.*.</seg> proueniet nobis ex ſimilibus partibus .289. cum fracto, quod <lb/>r<unclear reason="illegible"/>eijciamus ob minorem laborem.</s> </p> <p> <s xml:space="preserve">Producantur poſtea <seg type="var">.V.*.</seg> et <seg type="var">.E.D.</seg> vſque ad eorum concurſum in puncto <seg type="var">.ω.</seg> quæ-<lb/><choice><ex>remusque</ex><am>remusq́;</am></choice> quanta ſit <seg type="var">.E.ω.</seg> ex eadem regula, cum dixerimus, ſi .40. dat nobis .289. quid <lb/> <ptr xml:id="fig-0342-01a" corresp="fig-0342-01" type="figureAnchor"/> <pb facs="0343" n="331"/><fw type="head">EPISTOL AE.</fw> dabit .64. (nam duo triangula <seg type="var">.V.S.*.</seg> et <seg type="var">.V.E.ω.</seg> ſunt inuicem ſimilia eadem ratio-<lb/>ne) vnde <seg type="var">.E.ω.</seg> veniet nobis extalibus partibus .462.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0342-01" corresp="fig-0342-01a"> <graphic url="0342-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Coniungatur nunc quadratum ipſius <seg type="var">.E.V.</seg> quod eſt .4096. cum quadrato ipſius <seg type="var">.<lb/>E.ω.</seg> quod eſt .213444. & habebimus quadratum ipfius <seg type="var">.V.ω.</seg> talium <choice><ex>partium</ex><am>partiũ</am></choice> .217540.</s> </p> <p> <s xml:space="preserve">Dicemus poſtea ſi .217540. dat nobis .4096. quid dabit quadratum ipſius <seg type="var">.V.ω.</seg> vt <lb/>ſinus totus quod eſt .10000000000. vnde veniet pro quadrato ipſius <seg type="var">.V.E.</seg> talium <lb/>partium, ſuperficialium ſcilicet .18827211. cuius radix erit .13721. & erit ſinus an-<lb/>guli <seg type="var">.V.ω.E.</seg> qui erit grad .7. min .53. vnde angulus <seg type="var">.ω.V.E.</seg> erit grad .82. min .7. eius <lb/>vero ſinus erit partium .99054.</s> </p> <p> <s xml:space="preserve">Nunc autem quia angulus <seg type="var">.E.V.ω.</seg> eſt acutus, imaginemur <seg type="var">.E.Ŕ.</seg> ductam eſſe ad re<lb/>ctos ipſi <seg type="var">.V.ω.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> etiam ducta ipſa <seg type="var">.E.G</seg>. </s> <s xml:space="preserve">Vnde habebimus angulum <seg type="var">.Ŕ.E.V.</seg> gra-<lb/>duum .7. min .53. eius vero ſinus <seg type="var">.Ŕ.V.</seg> partium .13721. (propter ſimilitudinem trian<lb/>gulorum <seg type="var">.E.Ŕ.V.</seg> et <seg type="var">.ω.E.V.</seg>) talium ſcilicet, qualium <seg type="var">.E.V.</seg> fuerit .100000. </s> <s xml:space="preserve">Sed qua-<lb/>lium <seg type="var">.E.V.</seg> eſt .64. talium erit .8. cum tribus quartis, cuius <seg type="var">.Ŕ.V.</seg> quadratum erit par<lb/>tium .76. cum dimidio ſimilium ſed ſuperſicialium, quo quidem quadrato dempto <lb/>ex quadrato ipſius .64. quod eſt .4096. remanebit quadratum ipſius <seg type="var">.E.Ŕ.</seg> partium <num value="2871">.<lb/>2871.</num> quo etiam quadrato <seg type="var">.E.Ŕ.</seg> dempto ex quadrato <seg type="var">.E.G.</seg> partium .14400. remane<lb/>bit quadratum ipſius <seg type="var">.Ŕ.G.</seg> partium .11529. cuius radix <seg type="var">.Ŕ.G.</seg> erit partium .107. <choice><ex>talium</ex><am>taliũ</am></choice> <lb/>qualium <seg type="var">.E.G.</seg> eſt .120. ſed qualium <seg type="var">.E.G.</seg> erit .100000. talium <seg type="var">.Ŕ.G.</seg> erit partium <num value="89166">.<lb/>89166.</num> quæ vt ſinus anguli <seg type="var">.Ŕ.E.G.</seg> habebit pro ipſo angulo, gra .63. min .5. qui colle<lb/>cti cum gra .7. min .53. anguli <seg type="var">.V.E.Ŕ.</seg> dabunt totum angulum <seg type="var">.A.E.G.</seg> grad .70. <lb/>min .58. cuius complementum ex grad .90. erit <seg type="var">.G.D.</seg> graduum .19. min .2. & non .24. <lb/>vt omnes ferè putant.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE REFLEXIONIBVS RADIORVM.</head> <head rend="italics" xml:space="preserve">Excellentißimo Philoſopbo Franciſco Vimercato.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVoniam</hi> non videbatur quieſcere animus tuus, cum paucis ab hinc die-<lb/>bus tibi ſiſcitanti reſpondiſſem, nec tamen rationem omnium, quæ dixe-<lb/>ram exactè explicare per tem poris anguſtiam potuiſſem, cogitaui ad te <lb/>per hanc occaſionem ſcribens, & iam dicta repetere, & omnium tibi ra-<lb/>tionem ſubiungere, & vt mihi plenius ſatisfaciam, & tibi commodè perlegenti faci <lb/>lius ſit veritatem intueri. </s> <s xml:space="preserve">Scripſiſti enim in tuis diſputationibus, vir doctiſſime, quod <lb/>omnis res viſa per <choice><ex>ſpeculum</ex><am>ſpeculũ</am></choice> <choice><ex>quodcunque</ex><am>quodcũque</am></choice>, ſub breuiſſimis lineis <choice><ex>compræhendatur</ex><am>cõpræhendatur</am></choice> à vifu.</s> </p> <p> <s xml:space="preserve">Propoſitio hæc non eſt vniuerſaliter vera (quamuis etiam ab alijs omnibus pro ta <lb/>li poſita ſit) cum in ſpeculis concauis non ſemper verificetur, vt nunc tibi demon-<lb/>ſtrabo.</s> </p> <p> <s xml:space="preserve">Eſto quod linea recta <seg type="var">.b.d.</seg> tangat circulum <lb/> <ptr xml:id="fig-0343-01a" corresp="fig-0343-01" type="figureAnchor"/> <seg type="var">b.o.q.n.</seg> qui ſit communis ſectionis ſup crficiei re <lb/>flexionis, & ſphæricę alicuius ſpeculi ſphærici <lb/>concaui, & punctum contingentiæ ſit <seg type="var">.b.</seg> à quo <lb/>exeant duæ lineæ <seg type="var">.b.q.</seg> et <seg type="var">.b.n.</seg> efficientes duos an <lb/>gulos inuicem æquales circa perpendicularem <seg type="var">.<lb/>b.c.</seg> res autem viſa primò ſit in ipſa circunferen-<lb/>tia huiuſmodi circuli in puncto <seg type="var">.n.</seg> oculus vero in <lb/>puncto <seg type="var">.q.</seg> ipſius circunferentię. </s> <s xml:space="preserve">Dico nunc duas <pb facs="0344" n="332"/><fw type="head">IO. BAPT. BENED.</fw> lineas <seg type="var">.b.q.</seg> et <seg type="var">.b.n.</seg> ſimul ſumptas longiores eſſe omnibus alijs lineis exeuntibus ab ip <lb/>ſis punctis <seg type="var">.q.n.</seg> quæ in aliquo puncto dictæ circunferentiæ ſimul concurrant.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0343-01" corresp="fig-0343-01a"> <graphic url="0343-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sint igitur aliæ duæ <seg type="var">.q.o.</seg> et <seg type="var">.n.o.</seg> quas probare volo ſimul ſumptas, eſſe minores dua <lb/>bus ſimul ſumptis <seg type="var">.q.b.</seg> et <seg type="var">.n.b</seg>. </s> <s xml:space="preserve">Nam ex .20. tertij Eucli. cognoſcimus angulos <seg type="var">.q.b.n.</seg> <lb/>et <seg type="var">.q.o.n.</seg> inuicem æquales eſſe, & ſimiliter angulos <seg type="var">.b.n.o.</seg> et <seg type="var">.b.q.o</seg>. </s> <s xml:space="preserve">deinde ex .15. pri <lb/>mi eiuſdem habemus angulos contra ſe poſitos, <lb/>circa <seg type="var">.a.</seg> eſſe etiam inuicem ęquales. </s> <s xml:space="preserve">Vnde ex .4 <lb/> <ptr xml:id="fig-0344-01a" corresp="fig-0344-01" type="figureAnchor"/> ſexti, habebimus proportionem <seg type="var">.a.b.</seg> ad .a<lb/>o. eandem eſſe, quæ <seg type="var">.a.n.</seg> ad <seg type="var">.a.q.</seg> & ſic .b<lb/>n. ad <seg type="var">.o.q</seg>. </s> <s xml:space="preserve">Quare ita erit <seg type="var">.a.b.n.</seg> ad <seg type="var">.a.o.q.</seg> vt <seg type="var">.a.n</seg> <lb/>ad <seg type="var">.a.q.</seg> ſed cum <seg type="var">.a.n.</seg> maior ſit <seg type="var">.q.a.</seg> ex .18. primi, <lb/>eo quod angulus <seg type="var">.b.q.n.</seg> (qui æqualis eſt angulo <seg type="var">.<lb/>b.n.q.</seg> ex .5. eiuſdem) maior eſt angulo <seg type="var">.a.n.q.</seg> <lb/>qui pars eſt ipſius <seg type="var">.b.n.q.</seg> ergo latera ſimul ſum-<lb/>pta <seg type="var">.a.b.n.</seg> maiora erunt lateribus <seg type="var">.a.o.q.</seg> ſed ex <num value="20">.<lb/>20.</num> primi <seg type="var">.a.b.n.</seg> <choice><ex>etiam</ex><am>etiã</am></choice> maior erit <seg type="var">.a.n.</seg> vnde ex .25. <lb/>quinti <seg type="var">.q.a.b.n.</seg> maior erit <seg type="var">.n.a.o.q</seg>. </s> <s xml:space="preserve">quare ſequi-<lb/>tur verum eſſe propofitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0344-01" corresp="fig-0344-01a"> <graphic url="0344-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi oculus eſſet in <seg type="var">.u.</seg> quemadmodum in ſubſcripta hic <choice><ex>ſecunda</ex><am>ſecũda</am></choice> figura videre eſt, <lb/>res autem viſibilis in <seg type="var">.n.</seg> ambo extra dictum circulum, eſto etiam primum <seg type="var">.b.u.</seg> æqua-<lb/>lis <seg type="var">.b.n.</seg> probabo ſimiliter <seg type="var">.u.b.n.</seg> maiores eſſe <seg type="var">.u.o.n</seg>. </s> <s xml:space="preserve">Nam angulus <seg type="var">.o.</seg> maior eſt angu-<lb/>lo <seg type="var">.b.</seg> eo quod ſi circulum <seg type="var">.u.b.n.</seg> cogitemus circunſcribere triangulum <seg type="var">.u.b.n.</seg> ducen-<lb/>do vſque ad ſuam circunferentiam <seg type="var">.o.n.</seg> in puncto <seg type="var">.s.</seg> deinde ducendo <seg type="var">.u.s.</seg> habebimus <lb/>ex .20. tertij angulum <seg type="var">.u.s.n.</seg> <choice><ex>æqualem</ex><am>æqualẽ</am></choice> angulo <seg type="var">.u.b.n.</seg> ſed <choice><ex>cum</ex><am>cũ</am></choice> angulus <seg type="var">.u.o.n.</seg> exterior trian <lb/>guli <seg type="var">.u.o.s.</seg> exiſtat, ipſe maior erit angulo <seg type="var">.s.</seg> ex .16. primi. </s> <s xml:space="preserve">duco poſtea <seg type="var">.o.q.</seg> parallelam <lb/>ad <seg type="var">.u.s.</seg> quæ ſecabit <seg type="var">.a.u.</seg> in puncto <seg type="var">.q.</seg> & habebimus angulum <seg type="var">.a.o.q.</seg> ęqualem angulo <seg type="var">.<lb/>n.s.u.</seg> ex .29. eiuſdem, hoc eſt angulo <seg type="var">.n.b.u.</seg> fed ex ſu-<lb/> <ptr xml:id="fig-0344-02a" corresp="fig-0344-02" type="figureAnchor"/> pradictis rationibus, lineæ <seg type="var">.q.b.n.</seg> ſimul ſumptæ maio-<lb/>rem efficient longitudinem, quam <seg type="var">.q.o.n</seg>. </s> <s xml:space="preserve">Nunc cum <lb/>ipſi <seg type="var">.q.b.</seg> addita fuerit <seg type="var">.u.q.</seg> & vice <seg type="var">.q.o.</seg> ſumpta fuerit ali-<lb/>qua linea minor ipſa <seg type="var">.u.q.o.</seg> eo amplius <seg type="var">.u.q.b.n.</seg> maior <lb/>erit, quod quidem hoc modo faciendum. </s> <s xml:space="preserve">Acci-<lb/>piatur <seg type="var">.o.u.</seg> vt comes <seg type="var">.o.n.</seg> quæ minor eſt ambabus <seg type="var">.o.<lb/>q.</seg> et <seg type="var">.q.u.</seg> ex .20. primi, ita enim habebimus <choice><ex>propoſitum</ex><am>propoſitũ</am></choice>. <lb/></s> <s xml:space="preserve">ſed breuiori modo hoc ipſum videbis ex pręcedenti, <lb/>& ex .21. primi Euclid. </s> <s xml:space="preserve">Nam ex præcedenti <seg type="var">.u.b.n.</seg> lon-<lb/>gior eſt ipſa <seg type="var">.u.s.n.</seg> ex .21. autem primi <seg type="var">.u.s.n.</seg> longior eſt <lb/>ipſa <seg type="var">.u.o.n.</seg> ergo verum eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0344-02" corresp="fig-0344-02a"> <graphic url="0344-02"/> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0344-03"/> </figure> <p> <s xml:space="preserve">Si verò radius incidentiæ <choice><ex>non</ex><am>nõ</am></choice> fuerit æqualis radio <lb/>reflexionis, ſit vt in hac ſubſcripta tertia figura vide <lb/>re eſt <seg type="var">.u.b.p</seg>.</s> </p> <p> <s xml:space="preserve">Cum autem probauerim longitudinem <seg type="var">.u.b.n.</seg> ma <lb/>iorem eſſe longitudine <seg type="var">.u.o.n.</seg> coniungatur <seg type="var">.n.p.</seg> cum <lb/><seg type="var">u.b.n</seg>. </s> <s xml:space="preserve">deinde. ab <seg type="var">.o.</seg> ad <seg type="var">.p.</seg> ducatur <seg type="var">.o.p.</seg> quæ minor <lb/>erit longitudine <seg type="var">.o.n.p.</seg> ex .20. primi, & illicò <lb/>manifeſtabitur verum eſſe propoſitum, etiam hoc <lb/>tertio modo.</s> </p> <pb facs="0345" n="333"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Si <choice><ex>autem</ex><am>autẽ</am></choice> res viſibilis <choice><ex>oculusque</ex><am>oculusq́;</am></choice> ambo fuerint intra circulum, <choice><ex>tunc</ex><am>tũc</am></choice> poſſibile eſſet quod <lb/><choice><ex>longitudo</ex><am>lõgitudo</am></choice> <seg type="var">.u.b.n.</seg> modo maior, modo minor, modo verò æqualis eſſet ipſa <seg type="var">.u.o.n.</seg> <choice><ex>nunc</ex><am>nũc</am></choice>. <lb/></s> <s xml:space="preserve">Quod etiam affirmo de <seg type="var">.u.b.p.</seg> ſimiliter etiam eueniet ſi vnus terminorum <seg type="var">.u.</seg> vel <seg type="var">.n.</seg> <lb/>fuerit intra circunferentiam, reliquus verò extra ipſam.</s> </p> <p> <s xml:space="preserve">Conſideremus nunc hic inſraſcriptam .4. figuram vbi <seg type="var">.d.b.p.</seg> ſit circunferentia oxy <lb/>gonia ſeu elliptica (quod idem eſt) cuius maior axis ſit <seg type="var">.d.p.</seg> in quo, duo termini <seg type="var">.u.n.</seg> <lb/>ſint centra eius generationis: </s> <s xml:space="preserve">b.x. verò ſit minor axis. </s> <s xml:space="preserve">Imaginemur etiam circulum <seg type="var">.<lb/>b.o.x.</seg> cuius ſemidiameter ſit <seg type="var">.c.b.</seg> non maior medietate minoris axis, ne circunferen-<lb/>tia huiuſmodi circuli ſecet circunferentiam oxygoniam. </s> <s xml:space="preserve">Cogitemus etiam circu-<lb/>lum <seg type="var">.b.e.</seg> cuius ſemidiameter, minor non ſit minori axe <seg type="var">.b.x.</seg> ipſius oxygoniæ, ne ſe <lb/>inuicem ſecent huiuſmodi circunferentiæ, ſint etiam ambo eorum centra in linea <seg type="var">.b.<lb/>x.</seg> minoris axis, & punctum <seg type="var">.b.</seg> ſit commune vnicuique earum periphæriarum, vnde <lb/>minor circulus, totus intra, maior autem, totus extra ipſam <choice><ex>figuram</ex><am>figurã</am></choice> oxygoniam erit. <lb/></s> <s xml:space="preserve">Nunc ad partem <seg type="var">.o.r.e.</seg> vbi non communicant inuicem ipſæ circunferentiæ ducan-<lb/>tur <seg type="var">.n.o.r.e</seg>: <seg type="var">u.o</seg>: <seg type="var">u.r</seg>: et <seg type="var">.u.e.</seg> & per <seg type="var">.b.</seg> et <seg type="var">.r.</seg> cogitetur tranſire alium circulum, cuius cen-<lb/>trum in axe <seg type="var">.b.x.</seg> ſit <seg type="var">.t.</seg> <choice><ex>omnesque</ex><am>omnesq́;</am></choice> iſti circuli imaginentur trium diuerſorum ſphærico-<lb/>rum ſpeculorum, vnde pro genera <lb/>tione <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> oxygonię, ſeu ex .52. ter <lb/>tij Pergei, habebis longitudinem <seg type="var">.<lb/> <ptr xml:id="fig-0345-01a" corresp="fig-0345-01" type="figureAnchor"/> u.r.n.</seg> ęqualem eſſe longitudini <seg type="var">.u.b.<lb/>n.</seg> & ei, quæ eſt <seg type="var">.u.o.n.</seg> (vt minor ip <lb/>ſa <seg type="var">.u.r.n.</seg> ex .21. primi Euclidis) mi-<lb/>nor ipſa <seg type="var">.u.b.n.</seg> & longitudinem <seg type="var">.u.<lb/>e.n.</seg> (vt maior ipſa <seg type="var">.u.r.n.</seg> ex eadem <num value="21">.<lb/>21.</num> primi Eucli.) maior ipſa <seg type="var">.u.b.n</seg>. <lb/></s> <s xml:space="preserve">Sed ſi quis vellet hoc demonſtrare <lb/>ope circuli, <choice><ex>vnius</ex><am>vniꝰ</am></choice> <choice><ex>tantummodo</ex><am>tãtũmodo</am></choice> ſpeculi, <lb/><choice><ex>multiplicando</ex><am>multiplicãdo</am></choice> ipſas oxygonias <choice><ex>quem- admodum</ex><am>quẽ-admodum</am></choice> de ipſis circulis fecimus, obtineret ſimiliter propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0345-01" corresp="fig-0345-01a"> <graphic url="0345-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Solutio dubitationis.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">RAtionalis eſt dubitatio tua, <lb/> <ptr xml:id="fig-0345-02a" corresp="fig-0345-02" type="figureAnchor"/> vtrum (<choice><ex>cum</ex><am>cũ</am></choice> circulus minor hoc <lb/>eſt <seg type="var">.b.o.</seg> habeat ſuum centrum in mi <lb/>nori axe inter centrum oxygoniæ, <lb/>et .b: exiſtente <seg type="var">.b.</seg> extremo axis mi-<lb/>noris, <choice><ex>communeque</ex><am>communeq́;</am></choice> ambobus circun-<lb/>ferentijs circuli ſcilicet & oxigonię) <lb/>dictus circulus minor, plura puncta <lb/>communia habeat cum ipſis circun-<lb/>ferentijs.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0345-02" corresp="fig-0345-02a"> <graphic url="0345-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Cui dubitationi <choice><ex>reſpondeo</ex><am>reſpõdeo</am></choice> quod <lb/>quotieſcunque centrum alicuius cir <lb/>culi fuerit idem cum <seg type="var">.c.</seg> centro oxy-<lb/>goniæ, vel inter <seg type="var">.c.</seg> et <seg type="var">.b.</seg> in interual-<lb/>lo ſcilicet minoris axis, exiſtente <seg type="var">.b.</seg> <lb/>ſua extremitate communi ambabus <pb facs="0346" n="334"/><fw type="head">IO. BABPT. BENED.</fw> circunferentijs, ipſas circunferentias inuicem contiguas eſſe oportebit in puncto <seg type="var">.b.</seg> <lb/>tantummodo.</s> </p> <p> <s xml:space="preserve">Eſto primum quod centrum <seg type="var">.c.</seg> commune exiſtat, vt dictum eſt. </s> <s xml:space="preserve">ſit etiam centrum <lb/>vnius circuli, cuius diameter ſit <choice><ex>idem</ex><am>idẽ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> maiori axe <seg type="var">.d.p.</seg> & in gyro oxygoniæ accipia-<lb/>tur punctum <seg type="var">.f.</seg> proximum <seg type="var">.b.</seg> quantum fieri poterit, </s> <s xml:space="preserve">tunc protrahatur <seg type="var">.f.a.e.</seg> parallela <lb/>ipſi <seg type="var">.g.c.</seg> vſque ad gyrum maioris circuli in puncto <seg type="var">.e.</seg> quæ cum <seg type="var">.d.p.</seg> rectos efficiec <lb/>angulos. ex .29. primi Eucli. </s> <s xml:space="preserve"><choice><ex>ſecabitque</ex><am>ſecabitq́;</am></choice> gyrum circuli <seg type="var">.b.o.</seg> minoris in puncto <seg type="var">.t.</seg> quod di <lb/>co eſſe intra oxygoniam, <choice><ex>ſeparatumque</ex><am>ſeparatumq́;</am></choice> ab <seg type="var">.f</seg>. </s> <s xml:space="preserve">Quapropter duco <seg type="var">.c.e.</seg> quæ ſecabit cir-<lb/>cunferentiam circuli minoris in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.o.</seg> à quo puncto duco etiam <seg type="var">.o.i.</seg> parallelam ad <lb/><seg type="var">e.a</seg>. </s> <s xml:space="preserve">Deinde conſidero, quod ex ra-<lb/>tionibus ab Archimede adductis in <lb/> <ptr xml:id="fig-0346-01a" corresp="fig-0346-01" type="figureAnchor"/> quinta propoſitione libri de conoi-<lb/>dalibus, & ſphæroidibus, eadem <lb/>proportio erit <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.g.c.</seg> ad <seg type="var">.b.c.</seg> quę <lb/>ipſius <seg type="var">.e.a.</seg> ad <seg type="var">.f.a.</seg> vnde permutando <lb/>ita erit ipſius <seg type="var">.g.c.</seg> ad <seg type="var">.e.a.</seg> vel <seg type="var">.b.c.</seg> ad <lb/><seg type="var">f.a.</seg> hoc eſt ipſius <seg type="var">.e.c.</seg> ad <seg type="var">.e.a.</seg> vt <seg type="var">.o.c.</seg> <lb/>ad <seg type="var">.f.a.</seg> ſed ex ſimilitudine triangu-<lb/>lorum, & ex .11. quinti, ita <choice><ex>etiam</ex><am>etiã</am></choice> erit <lb/>ipſius <seg type="var">.o.c.</seg> ad <seg type="var">.o.i.</seg> vt <seg type="var">.o.c.</seg> ad <seg type="var">.f.a</seg>. </s> <s xml:space="preserve">Vn-<lb/>de ſequitur <seg type="var">.o.i.</seg> æqualem eſſe <seg type="var">.f.a.</seg> <lb/>ſed ex .14. tertij Eucli <seg type="var">.t.a.</seg> minor eſt <seg type="var">.<lb/>o.i</seg>. </s> <s xml:space="preserve">Quare minor etiam erit ipſa <seg type="var">.f.<lb/>a</seg>. </s> <s xml:space="preserve">Vnde punctum <seg type="var">.t.</seg> intra oxygo-<lb/>niam erit, & conſequenter ſepara-<lb/>tum .ab <seg type="var">.f</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0346-01" corresp="fig-0346-01a"> <graphic url="0346-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed ſi centrum circuli minoris <lb/>fuerit inter <seg type="var">.c.</seg> et <seg type="var">.b.</seg> hoc eſt eccentri-<lb/>cum ipſius oxygoniæ, ipſe tanget concentricum in puncto <seg type="var">.b.</seg> tantummodò, vt in .3. <lb/>Euclidis libro probatur. </s> <s xml:space="preserve">Vnde tanto magis diſtans erit punctum <seg type="var">.t.</seg> à puncto <seg type="var">.f.</seg> quod <lb/>erit propoſitum.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Alterius dubitationis ſolutio.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">VNde autem fiat, quod à ſpeculis planis, obiectorum imagines, ita diſtantes <lb/>vltra ſuperficiem ipſius ſpeculi videantur, vt obiecta citra ipſam ſuperficiem <lb/>reperiuntur.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ſcientia, tres cognitiones nos primum habere oportet, quarum pri-<lb/>ma eſt. </s> <s xml:space="preserve">Vnde fiat, quod obiecti imago in catheto incidentiæ videatur. </s> <s xml:space="preserve"><choice><ex>Secunda</ex><am>Secũda</am></choice>. </s> <s xml:space="preserve">vn-<lb/>de efficiatur, quod angulus reflexionis, ſemper æqualis ſit angulo incidentiæ.</s> </p> <p> <s xml:space="preserve">Terria demum. </s> <s xml:space="preserve">Vnde naſcatur quod radius incidentiæ ſimul cum radio reflexio-<lb/>nis ſit in quodam plano ſecante ſuperficiem ſpeculi ſemper ad rectos, quod qui-<lb/>dem planum vocatur ſuperficies reflexionis. </s> <s xml:space="preserve">Huiuſmodi tres paſſiones, ab omnibus <lb/>ſpecularijs conſideratæ ſunt, ſed rationes ab illis traditæ, mihi non ſatisfaciunt.</s> </p> <pb facs="0347" n="335"/> <fw type="head">EPISTOLAE.</fw> <p> <s xml:space="preserve">Nam circa æqualitatem angulorum reflexionis & incidentiæ, iam tibi probaui <lb/>illud non vniuerſaliter euenire à breuitate aggregati radiorum incidentiæ reflexio-<lb/><choice><ex>nisque</ex><am>nisq́;</am></choice>. </s> <s xml:space="preserve">Sed hoc naſcitur potius ab eo, quod cum radius incidentiæ non poſſit ſuper <lb/>ficiem corporis opaci penetrare, reflectit, vt citra ipſam <choice><ex>cum</ex><am>cũ</am></choice> angulo æquali ei, quem <lb/>faceret cum eadem ſuperficie vltra ipſam ſi tranſiuiſſet.</s> </p> <p> <s xml:space="preserve">Exempli gratia ſit <seg type="var">.a.</seg> obiectum <seg type="var">.b.</seg> <choice><ex>autem</ex><am>autẽ</am></choice> oculus in figura <seg type="var">.A.</seg> et <seg type="var">.c.e.</seg> ſuperficies ipſius <lb/>ſpeculi <seg type="var">.d.</seg> verò ſit punctum ipſius ſuperficiei, à quo ad oculum reflectitur imago ip-<lb/> <ptr xml:id="fig-0347-01a" corresp="fig-0347-01" type="figureAnchor"/> ſius <seg type="var">.a</seg>. </s> <s xml:space="preserve">Nunc ſi radius <seg type="var">.a.d.</seg> incidentiæ, recta <lb/>incederet ſub <seg type="var">.c.e.</seg> efficeret angulum <seg type="var">.e.d.h.</seg> <lb/>æqualem angulo <seg type="var">.c.d.a.</seg> eius contrapoſito, <lb/>ſed quia impeditur ipſæ radius ab opacitate <lb/>ipſius ſpeculi <seg type="var">.c.e.</seg> ne vlterius incedat, propte <lb/>rea reflectitur ab ipſa ſuperficie ſpeculi, con-<lb/>ſtituens cum ipſa angulum <seg type="var">.e.d.b.</seg> æqualem <lb/>angulo <seg type="var">.e.d.h.</seg> ſed quia angulus <seg type="var">.c.d.a.</seg> eſt <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>ęqualis ipſi angulo <seg type="var">.e.d.h.</seg> propterea angulus <seg type="var">.e.d.b.</seg> ęqualis exiſtit angulo <seg type="var">.c.d.</seg> a; </s> <s xml:space="preserve">per <lb/>accidens igitur ſequitur <seg type="var">.a.d.</seg> et <seg type="var">.d.b.</seg> ſimul ſumptas, breuiorem facere longiludinem <lb/>omni alia, quæ ab ipſa ſuperficie <seg type="var">.c.e.</seg> ad eadem puncta <seg type="var">.a.b.</seg> ducta eſſet, </s> <s xml:space="preserve">quare natu-<lb/>ræintentio eſt efficere angulum <seg type="var">.e.d.b.</seg> æqualem angulo <seg type="var">.e.d.h.</seg> vnde ex accidenti po <lb/>ſtea ſequitur, ipſum æqualem eſſe angulo <seg type="var">.c.d.a.</seg> & deinde <choice><ex>quod</ex><am>qđ</am></choice> lineæ <seg type="var">.a.d.</seg> et <seg type="var">.d.b.</seg> con-<lb/>ſtituant longitudinem breuiorem. </s> <s xml:space="preserve">Quare illud quod omnes putabant eſſe primum <lb/>& perſe, vltimum eſt, & exaccidenti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0347-01" corresp="fig-0347-01a"> <graphic url="0347-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quare vero ſuperficies, quæ vocatur reflexionis, in qua ſunt duæ lineę, hoc eſt <lb/>incidentię, <choice><ex>reflexionisque</ex><am>reflexionisq́;</am></choice>, ſemper ſit perpendicularis ſuperficiei ipſius ſpeculi: </s> <s xml:space="preserve">Hæc <lb/>eſt ratio, quia cum quilibet radius incidentiæ, perpendicularis ipſi ſuperficiei ſpe-<lb/>culi, in ſeipſo reflectit, ex ijſdem dictis rationibus, hoc eſt, quia cum tali angulo vult <lb/>reflecti, cum quali tranſiret, ita etiam purandum eſt, quodradius incidens obliquus, <lb/>cum in ſeipſum non poſſit redire, quia non eſt perpendicularis ſuperficiei ſpeculi, <lb/>reflectitur tamen per planum erectum ipſi ſuperficiei ſpeculi, vt in eo, cui magis re-<lb/>ſiſtit ſuperficies corporis opaci, quàm alicui alij plano ipſius infiniti inclinatorum <lb/>planorum, ab vtraque parte ipſius plani perpendicularis, quod vnum etiam tan-<lb/>tummodo eſt, & in quo, radius maiorem vim obtinet reflectendi, ſeu in eo, in quo <lb/>radius ipſe cum maiori reſiſtentia repercutitur à ſuperficie corporis opaci.</s> </p> <p> <s xml:space="preserve">Poſtremo <choice><ex>ſciendum</ex><am>ſciẽdũ</am></choice> vnde oriatur, <choice><ex>quod</ex><am>ꝙ</am></choice> rei viſibilis imago, à ſpeculo plano reflexa, ſem <lb/>per in catheto incidentiæ videatur.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ratione cognoſcendum primò eſt, quo modo fit perfecta <choice><ex>ſimplexque</ex><am>ſimplexq́;</am></choice> <lb/>viſio, & non reflexa, deinde proſequemur ad reliqua huius tertiæ propoſitionis.</s> </p> <p> <s xml:space="preserve">Animaduertendum igitur eſt, quod <choice><ex>quotieſcunque</ex><am>quotieſcunq;</am></choice> obiectum aliquod viſibile aſpi <lb/>cimus, nos nunquam perfectè illud comprehendere poſſumus, niſi in puncto con-<lb/>curſus, ſeu interſectionis axium viſualium, ſeu radialium ( vt ita loquar ) <choice><ex>quam</ex><am>quã</am></choice> <choice><ex>inter- ſectionem</ex><am>inter-ſectionẽ</am></choice>, nos efficimus ope reuolutionis oculorum <choice><ex>adinuicem</ex><am>adinuicẽ</am></choice>, hoc eſt voluendo vnum <lb/>verſus alium, ita vt in ſitu ipſius obiecti, ſeinuicem ſecent axes iam dicti, </s> <s xml:space="preserve">tunc enim <lb/>vtroque oculo mediante, exacte rem perſpicimus, cęteris .8. circunſtantijs non ob-<lb/>ſtantibus.</s> </p> <p> <s xml:space="preserve">Vnde ſtantibus oculis in tali ſitu, altero reſpectu alterius, ſi eorum alter tectus; <lb/></s> <s xml:space="preserve">ſeu velatus fuerit, tune alio tantummodo oculo mediante, videbimus obiectum, <lb/>in ea diſtantia, exactius, quam in quauis alia propinquiori, & remotiori.</s> </p> <pb facs="0348" n="336"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Animal igitur, ſecundum diſtantiam obiecti, oculum accommodat ad recipien-<lb/>dum quam exactiſſimè ſpeciem ipſius obiecti, & hoc voluendo ambos oculos, vnum <lb/>verſus alium, ita quod interſectio axium ſit in ſitu ſeu loco dicti obiecti, nam tunc vi <lb/>dent ambo vel aliquis eorum ſolus, in tali diſtantia exactè obiectum videbit.</s> </p> <p> <s xml:space="preserve">Vnde ſequitur obiectum viſibile, compræhenſibile non eſſe ab vno tantummodo <lb/>oculo in quolibet ſitu axis ipſius oculi, ſed in eo, vbi alius axis interſecatur à dicto. <lb/></s> <s xml:space="preserve">Quæ quidem interſectio poteſt fieri propinqua, vel remota à viſu, ad certos tamen <lb/>terminos vſque.</s> </p> <p> <s xml:space="preserve">De huiuſmodi axium viſualium interſectione ſcribit Alhazem in .2. et .15. propo <lb/>ſitione tertij lib. Vitellio verò in .32. et .45. eiuſdem.</s> </p> <p> <s xml:space="preserve">Quod igitur dico, verum eſt, ideſt, quod ſi vno tantummodo oculo aſpiciemus <lb/>obiectum aliquod, ipſum nunquam perfectè proſpicietur, niſi cum oculus ita ſitus <lb/>fuerit, vt eius axis cum axe alterius in loco obiecti ſe inuicem ſecent, quamuis alter <lb/>oculus nihil videat, <choice><ex>cum</ex><am>cũ</am></choice> <choice><ex>autem</ex><am>aũt</am></choice> duobus oculis in tali ſitu <choice><ex>conſtitutis</ex><am>cõſtitutis</am></choice> <choice><ex>obiectum</ex><am>obiectũ</am></choice> videmus, vnum <lb/>tantummodo nobis cernere videbimur, & ſi extra talem punctum interſectionis ip-<lb/>ſum obiectum poſitum fuerit, tunc duo talia, obiecta nobis apparebunt, ſed huiuſ <lb/>modi rei cauſam alias tibi manifeſtabo.</s> </p> <p> <s xml:space="preserve">His igitur cognitis, ponamus aliquam <lb/> <ptr xml:id="fig-0348-01a" corresp="fig-0348-01" type="figureAnchor"/> ſpeculi ſuperficiem eſſe <seg type="var">.g.h.</seg> in figura <seg type="var">.B.</seg> <lb/>obiectum autem viſibile <seg type="var">.b.</seg> oculos vero <seg type="var">.a.</seg> <lb/>et <seg type="var">.u.</seg> punctum autem <seg type="var">.n.</seg> in ſuperficie ſpecu <lb/>li, à quo imago ipſius <seg type="var">.b.</seg> reflectit ad <seg type="var">.a.</seg> & <lb/>punctum <seg type="var">.t.</seg> à quo reflectitur ad <seg type="var">.u.</seg> et <seg type="var">.c.e.</seg> <lb/>ſit <choice><ex>communis</ex><am>cõmunis</am></choice> ſectio ſuperficiei reflexionis <lb/>radiorum <seg type="var">.b.n.a.</seg> et <seg type="var">.c.f.</seg> ſit communis ſectio <lb/>ſuperficiei reflexionis radiorum <seg type="var">.b.t.u.</seg> qua <lb/>rum <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> ſuperficies reflexionis, ere-<lb/>cta eſt ad ſuperficiem ſpeculi <seg type="var">.g.h.</seg> vt ſupra <lb/>diximus. </s> <s xml:space="preserve">Nunc ex .19. vndecimi Eucl. ſequitur communem ſectionem harum dua-<lb/>rum ſuperficierum. (b.c.d. ſcilicet) ad rectos etiam eſſe ſupra ſuperficiem ſpeculi <seg type="var">.g.<lb/>h.</seg> cum qua <seg type="var">.b.c.</seg> quælibet linearum <seg type="var">.a.n.</seg> vel <seg type="var">.u.t.</seg> reflexarum ( productę cum fuerint ) <lb/>ſeinuicem interſecabunt eo quod duo anguli <seg type="var">.d.c.n.</seg> et <seg type="var">.d.n.c.</seg> ſimul collecti minores <lb/>ſunt duobus rectis, & ita <seg type="var">.d.c.t.</seg> cum <seg type="var">.d.t.c.</seg> cum anguli <seg type="var">.a.n.e.</seg> et <seg type="var">.u.t.f.</seg> reflexi, ipſis con-<lb/>trapoſiti, æquales ſint angulis <seg type="var">.b.n.c.</seg> et <seg type="var">.b.t.c.</seg> incidentiæ, quorum <choice><ex>vnuſquiſque</ex><am>vnuſquiſq;</am></choice> ex .32. <lb/>primi, minor eſt recto.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0348-01" corresp="fig-0348-01a"> <graphic url="0348-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Dico etiam quod in eodem puncto huiuſmodi catheti <seg type="var">.b.c.d.</seg> in quo interſecabi-<lb/>tur à linea <seg type="var">.a.n.</seg> in eodem ſecabitur à linea <seg type="var">.u.t.</seg> & quod punctum dicti concurſus, tan-<lb/>tum depreſſum erit ſub ſuperficie ſpeculi <seg type="var">.g.h.</seg> quantum <seg type="var">.b.</seg> ſupra ipſam reperietur. <lb/></s> <s xml:space="preserve">Nam anguli <seg type="var">.b.n.c.</seg> et <seg type="var">.d.n.c.</seg> ſunt inuicem æquales, <choice><ex>angulique</ex><am>anguliq́;</am></choice> <seg type="var">.b.c.n.</seg> et <seg type="var">.d.c.n.</seg> recti <seg type="var">.c.n.</seg> <lb/>verò communis ambobus triangulis <seg type="var">.b.c.n.</seg> et <seg type="var">.d.c.n.</seg> vnde ex .26. primi Eucli. latus <seg type="var">.d.<lb/>c.</seg> commune, vt trianguli <seg type="var">.d.c.n.</seg>æquale erit lateri communi <seg type="var">.b.c.</seg> vt trianguli <seg type="var">.b.c.n.</seg> <lb/>Idem etiam dico de latere <seg type="var">.d.c.</seg> vt ipſius trianguli <seg type="var">.d.c.t.</seg> quod æquatur lateri <seg type="var">.b.c.</seg> vt <lb/>trianguli <seg type="var">.b.c.t</seg>. </s> <s xml:space="preserve">Vnde cum <seg type="var">.b.c.</seg> vnum, & idem ſit: </s> <s xml:space="preserve">d.c. igitur etiam erit, & ipſum <choice><ex>vnum</ex><am>vnũ</am></choice> <lb/>& idem, quod erit propoſitum.</s> </p> <p> <s xml:space="preserve">Nunc autem cum hi duo radij ſeinuicem ſecent in puncto <seg type="var">.d.</seg> ergo in ipſo puncto <seg type="var">.<lb/>d.</seg> videbimur nobis videre <choice><ex>imaginem</ex><am>imaginẽ</am></choice> obiecti .b: <choice><ex>cum</ex><am>cũ</am></choice> ope <choice><ex>duorum</ex><am>duorũ</am></choice> <choice><ex>iſtorum</ex><am>iſtorũ</am></choice> <choice><ex>radiorum</ex><am>radiorũ</am></choice> <seg type="var">.n.a.</seg> et <seg type="var">.t.<lb/>u.</seg> ita inuicem <choice><ex>ſitorum</ex><am>ſitorũ</am></choice>, videamur nobis <choice><ex>imaginem</ex><am>imaginẽ</am></choice> proſpicere. </s> <s xml:space="preserve">Vnde ſi in tali caſu, vnus <pb facs="0349" n="337"/><fw type="head">EPISTOLAE.</fw> oculorum clauderetur, nihilominus cum reliquo obiectum vidiſſemus in <choice><ex>eodem</ex><am>eodẽ</am></choice> ipſo <lb/>loco <seg type="var">.d.</seg> & non in alio ex ſuperius dictis rationibus.</s> </p> <p> <s xml:space="preserve">Et ſi ſtantibus ijs terminis volueremus pupillam oculi <seg type="var">.u.</seg> verſus aliam <seg type="var">.a.</seg> ad aſpi-<lb/>ciendum punctum <seg type="var">.n.</seg> in ſuperficie <seg type="var">.g.h.</seg> ipſius ſpeculi, hoc eſt ſi fecerimus quod axes <lb/>viſuales ſeinuicem ſecarent in ipſo puncto <seg type="var">.n</seg>. </s> <s xml:space="preserve">tunc videremur nobis videre duas <lb/>imagines ipſius obiecti <seg type="var">.b.</seg> intra ſpeculum, eo quod obiectum, propter hoc non <lb/>ceſſaret reflectere ad oculos ab ipſis punctis <seg type="var">.n.</seg> et <seg type="var">.t.</seg> quapropter recipiendo ra-<lb/>dium <seg type="var">.t.u.</seg> in ſitu axis oculi <seg type="var">.u.</seg> & radium <seg type="var">.n.a.</seg> in ſitu axis oculi <seg type="var">.a.</seg> hi axes ex neceſſitate <lb/>(vt probauimus ) ſeinuicem ſecant in puncto <seg type="var">.d.</seg> vnde vnam tantummodo imaginem <lb/>ipſius obiecti nobis apparebit.</s> </p> <p> <s xml:space="preserve">Ex his igitur omnibus potes facilè videre omnem imaginem, cuiuſuis obiecti, re-<lb/>flexam à ſpeculo, reperiri in ipſo catheto incidentiæ, cum ipſe ſemper ſit communis <lb/>ſectio duarum ſuperficierum reflexionis, in quo catheto concurrunt ipſæ axes vi-<lb/>ſuales.</s> </p> <p> <s xml:space="preserve">Exijſdem etiam dictis rationibus facile compræhendere poteris, vnde fiat, vt vi-<lb/>deamus imaginem reflexam à ſpeculis ſphęricis concauis citra <choice><ex>ipſorum</ex><am>ipſorũ</am></choice> ſuperficiem, & <lb/>non vltra. </s> <s xml:space="preserve">Quod <choice><ex>nunquam</ex><am>nunquã</am></choice> euenit, niſi quando <choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.d.</seg> interſectionis <choice><ex>ipſorum</ex><am>ipſorũ</am></choice> <choice><ex>radiorum</ex><am>radiorũ</am></choice> <lb/>viſualium (quod alio in loco non fit, niſi in catheto incidentiæ hoc eſt in communi <lb/>ſectione duarum ſuperficierum reflexionis. </s> <s xml:space="preserve">Dato quod obiectum non ſit in vna ea-<lb/>demq́ue ſuperficie, in qua reperti fuerint axes viſuales, hoc eſt dato, <choice><ex>quod</ex><am>ꝙ</am></choice> ambo axes <lb/>viſuales non ſint in vna <choice><ex>eademque</ex><am>eademq́;</am></choice> ſuperficie reflexionis) reperitur citra & non vltra ſu <lb/>perficiem ipſius ſpeculi.</s> </p> <p> <s xml:space="preserve">Ad cuius rei euidentiam non <choice><ex>prætermittam</ex><am>prætermittã</am></choice> dicere, quod cum debeant ſemper ſu-<lb/>perficies reflexionum perpendiculares eſſe, velad rectos ſecare ſuperficiem ipſius <lb/>ſpeculi, ipſarum communes ſectiones cum ſuperficie ſpeculi ſphęrici, ſemper erunt <lb/>circunferentiæ magnorum circulorum illius ſphæræ, cuius portio eſt ſpeculum <lb/>propoſitum, vt etiam Vitellio affirmat in prima ſexti libri. </s> <s xml:space="preserve">Vnde vnuſquiſque ca-<lb/>thetus incidentiæ tranſibit per centrum ſpeculi, cum ipſe ſit communis ſectio dua-<lb/>rum ſuperficierum reflexionis, </s> <s xml:space="preserve">quare in ipſo catheto erit punctum interſectionis ip <lb/>ſorum axium viſualium ex neceſſitate, vt videbimus, ſi vnam tantummodo <choice><ex>imaginem</ex><am>imaginẽ</am></choice> <lb/>obiecti nobis videremur videre.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ſint duæ ſuperficies reflexionis ſpeculi ſphærici concaui <seg type="var">.b.n.c.a.</seg> <lb/>et <seg type="var">.b.t.c.u.</seg> <choice><ex>obiectumque</ex><am>obiectumq́;</am></choice> ſit <seg type="var">.b.</seg> oculi autem ſint <seg type="var">.a.u.</seg> punctum verò ſuperficiei ſpeculi, à <lb/>quo obiectum emittit reflexionem ſuę <lb/>imaginis ad oculum <seg type="var">.a.</seg> ſit <seg type="var">.n.</seg> <choice><ex>punctum</ex><am>pũctum</am></choice> au-<lb/> <ptr xml:id="fig-0349-01a" corresp="fig-0349-01" type="figureAnchor"/> <ptr xml:id="fig-0349-02a" corresp="fig-0349-02" type="figureAnchor"/> tem à quo eandem reflectit oculo <seg type="var">.u.</seg> ſit <lb/>t. communis autem ſectio harum dua-<lb/>rum ſuperficierum ſit <seg type="var">.b.c.</seg> ſed <seg type="var">.x.</seg> <choice><ex>centrum</ex><am>centrũ</am></choice> <lb/>ſit ſpeculi, radius verò incidentię ſuper <lb/>ficiei <seg type="var">.b.n.c.</seg> erit <seg type="var">.b.n.</seg> cuius reflexus ſit <seg type="var">.n.<lb/>a.</seg> radij autem alterius ſuperficiei erunt <lb/><seg type="var">b.t.</seg> et <seg type="var">.t.u</seg>. </s> <s xml:space="preserve">Imaginemur nunc duos ſemi <lb/>diametros <seg type="var">.x.n.</seg> et <seg type="var">.x.t.</seg> quæ angulos <seg type="var">.b.n.<lb/>a.</seg> et <seg type="var">.b.t.u.</seg> per æqualia diuidant ex ſup-<lb/>poſito.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0349-01" corresp="fig-0349-01a"> <graphic url="0349-01"/> </figure> <figure xml:id="fig-0349-02" corresp="fig-0349-02a"> <graphic url="0349-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nunc ijs ſuppoſitis, ſi vnam tantum-<lb/>modo obiecti imaginem videbimus, <pb facs="0350" n="338"/><fw type="head">IO. BAPT. BENED.</fw> clarum erit ex rationibus ſupradictis nos ipſam videre in <choice><ex>communi</ex><am>cõmuni</am></choice> concurſu ipſorum <lb/>axium viſualium, qui axes cum reperiantur vnà cum ipſis radijs reflexis <seg type="var">.n.a.</seg> et <seg type="var">.t.u.</seg> <lb/>ex neceſſitate ſeinuicem <choice><ex>ſecabunt</ex><am>ſecabũt</am></choice> in catheto <seg type="var">.b.c.</seg> cum extendantur in ipſis ſuperficie-<lb/>bus reflexionum, quæ ſuperficies nihil aliud commune inuicem habent, quam cathe <lb/>tum dictum <seg type="var">.b.c.</seg> ſit igitur in puncto <seg type="var">.d</seg>.</s> </p> <p> <s xml:space="preserve">Ex his dictis alia oritur neceſſitas, hoc eſt, quod quotieſcunque vnam tantummo <lb/>do imaginem obiecti <seg type="var">.b.</seg> videmus, dato quod duæ ſuperficies reflexionis ſint, & non <lb/>vna tantum, tunc angulos <seg type="var">.n.</seg> et <seg type="var">.t.</seg> ſemper inuicem æquales eſſe oportebit. </s> <s xml:space="preserve">Vnde ar-<lb/>cus <seg type="var">.n.c.</seg> et <seg type="var">.t.c.</seg> ex neceſſitate inuicem æquales erunt.</s> </p> <p> <s xml:space="preserve">Scimas enim ex .3. ſexti Euclid. quod eadem proportio erit ipſius <seg type="var">.b.n.</seg> ad <seg type="var">.n.<lb/>d.</seg> quę ipſius <seg type="var">.b.x.</seg> ad <seg type="var">.x.d.</seg> & ipſius <seg type="var">.b.t.</seg> ad <seg type="var">.t.d.</seg> ſimiliter, </s> <s xml:space="preserve">quare ipſiusb <seg type="var">.n.</seg> ad <seg type="var">.n.<lb/>d.</seg> erit vt ipſius <seg type="var">.b.t.</seg> ad <seg type="var">.t.d</seg>. </s> <s xml:space="preserve">Vnde ſequitur <seg type="var">.b.n.</seg> æqualem eſſe ipſi <seg type="var">.b.t.</seg> et <seg type="var">.n.d.</seg> <lb/>ipſi <seg type="var">.t.d.</seg> vt à medio circulo <seg type="var">.E.</seg> potes videre, quamuis etiam <seg type="var">.b.</seg> non eſſet extremum <lb/>diametri, ſed vbicunque volueris in ipſo diametro, vel <choice><ex>etiam</ex><am>etiã</am></choice> protracta, eo quod pun-<lb/>ctum <seg type="var">.n.</seg> & punctum <seg type="var">.t.</seg> in eodem ſemicirculo, vel in æqualibus ſemicirculis, non <choice><ex>poſsent</ex><am>poſsẽt</am></choice> <lb/>aliter in ipſa circunferentia locari, <choice><ex>eandem</ex><am>eãdem</am></choice> ſeruando proportionem <seg type="var">.b.n.</seg> ad <seg type="var">.n.d.</seg> vt <seg type="var">.b.<lb/>t.</seg> ad <seg type="var">.t.d.</seg> </s> <s xml:space="preserve">propterea quod in omni alio ſitu exiſtente puncto <seg type="var">.t.</seg> ipſa <seg type="var">.b.t.</seg> eſſet aut maior <lb/>aut minor ipſa <seg type="var">.b.n.</seg> et <seg type="var">.t.d.</seg> aut minor, aut maior ipſa <seg type="var">.t.d.</seg> ex .7. & 14. tertij Eucli. </s> <s xml:space="preserve">vnde <lb/>aut maior, aut minor proportio eſſet ipſius <seg type="var">.b.t.</seg> ad <seg type="var">.t.d.</seg> quam ipſius <seg type="var">.b.n.</seg> ad <seg type="var">.n.d.</seg> & non <lb/>eadem.</s> </p> <p> <s xml:space="preserve">Nunc è conuerſo ſi <seg type="var">.b.n.</seg> et <seg type="var">.b.t.</seg> ſunt ſibi inuicem æquales, & ſic <seg type="var">.n.d.</seg> cum <seg type="var">.t.d.</seg> ſequi-<lb/>tur ex .8. primi Eucli. angulos <seg type="var">.n.</seg> et <seg type="var">.t.</seg> inuicem æquales eſſe.</s> </p> <p> <s xml:space="preserve">Ab ijſdem ſpeculationibus potes etiam videre vnde accidat quod partes ſuperio <lb/>res alicuius obiecti reflexæ à tali ſpeculo concauo videntur nobis inferiores eſſe, & <lb/>inferiores appareant ſuperiores, & dextræ ſiniſtræ, & ſiniſtræ dextræ. </s> <s xml:space="preserve">quod autem <lb/>hucuſque demonſtraui de ſpeculis planis, & ſphæricis concauis, ratiocinare tu ijſdem <lb/>medijs circa ſphærica conuexa, vbi clarè videbis puncta huiuſmodi ſpeculi conuexi, <lb/>à quibus reflectitur imago obiecti ad ambos oculos, ſemper oportere æquidiſtantia <lb/>eſſe à <choice><ex>puncto</ex><am>pũcto</am></choice> communi ipſius ſuperficiei ſpeculi, & catheto incidentiæ, dum unam tan <lb/>tummodo imaginem ipſius obiecti videmus, & à diuerſis ſuperficiebus reflexionum.</s> </p> <p> <s xml:space="preserve">Nolo etiam prætermittere, quod nunc mihi ſuccurrit, hoc eſt quod poſſet ali-<lb/>quis duos ſitus inuenire, vnum pro oculo, alterum verò pro obiecto, reſpectu alicu-<lb/>ius ſpeculi concaui, ſphęroidis prolatæ, vt reflexio ipſius obiecti videretur, vt linea <lb/>diuidens per æqualia ipſum ſpeculum. </s> <s xml:space="preserve">Reſpectu verò alicuius ſpeculi concaui ſphæ-<lb/>roidis oblongæ, vt reflexio obiecti ad oculum veniret à tota ſuperficie ipſius ſpecu-<lb/>li, vnde tota ſuperficies ipſius ſpeculi videretur colorata illo colore cuius eſſet <lb/>obiectum, quæ quidem paſſiones <choice><ex>pendent</ex><am>pendẽt</am></choice> à .48. tertij lib. ipſius Pergei, vt ex te ipſo fa <lb/>cile videre potes, propter æqualitatem angulorum reflexionis, & incidentiæ.</s> </p> <p> <s xml:space="preserve">Opinio autem mea, quam ſcire cupis de imagine obiecti reflexa, quam putas eſ-<lb/>ſe in ſuperficie ſpeculi, hæc eſt, quod nec in ſuperficie, nec ultra, nec citra eam eſt ip <lb/>ſa imago, quod autem vltra non ſit, hoc puto nulli dubium eſſe. </s> <s xml:space="preserve">eadem etiam ra-<lb/>tione non erit citra ſuperficiem ſpeculi concaui, quamuis ipſam nos compræhenda-<lb/>mus in concurſu radiorum viſualium, tam ab vno ſpeculo quam ab alio reflexione <lb/>facta. </s> <s xml:space="preserve">Sed quòd ipſa neque ſit in ipſa ſpeculi ſuperficie, manifeſtum erit ex hoc, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>duo ſpectantes in eodem ſpeculo, duas diuerſas imagines vident, tres, <choice><ex>autem</ex><am>aũt</am></choice> tres, qua-<lb/>tuor, quatuor, & ſic deinceps, vnde tot eſſent imagines ſupra ſuperficiem ſpeculi, <lb/>quot obiecta, <choice><ex>quod</ex><am>ꝙ</am></choice> tamen ita non eſt, nec plus eſt in vno loco ipſa imago, quam in alio, <pb facs="0351" n="339"/><fw type="head">EPISTOLAE.</fw> niſi in obiecto ipſo, lumen enim abipſo obiecto reflexum, ſeipſum diffundit vndi-<lb/>que, & radijipſius lu<unclear reason="illegible"/>minis reflexi, vt plurimum ſeinuicem ſecant. </s> <s xml:space="preserve">Vnde in ipſo ae-<lb/>re funt omnes miſti. </s> <s xml:space="preserve">Quapropter natura ſagacifſima pupillam oculi animalibus tam <lb/>paruam conſtruxit ad ſuperficiem tam amplæ ſphæræ ipſius oculi, vt diſtinctæ vide-<lb/>rentur omnia obiecta.</s> </p> <p> <s xml:space="preserve">Nolo etiam tibi tacere, quod <choice><ex>quotieſcunque</ex><am>quotieſcunq;</am></choice> oculorum pupillæ poſitæ fuerint inter <lb/>cathetum incidentiæ, & ſuperficiem <lb/> <ptr xml:id="fig-0351-01a" corresp="fig-0351-01" type="figureAnchor"/> <ptr xml:id="fig-0351-02a" corresp="fig-0351-02" type="figureAnchor"/> ſpeculi ſphærici concaui, vt puta in li-<lb/>neis <seg type="var">.d.t.</seg> et <seg type="var">.t.n.</seg> in figura <seg type="var">.D</seg>. </s> <s xml:space="preserve">tunc nullo <lb/>pacto poſſemus videre vnam imagi-<lb/>nem obiecti, ſed duas nec non confu-<lb/>sè, </s> <s xml:space="preserve">propterea <choice><ex>quod</ex><am>ꝙ</am></choice> nullo pacto radij <seg type="var">.t.d.</seg> <lb/>et <seg type="var">.n.t.</seg> reflexi poterint. </s> <s xml:space="preserve">ambo vniri <choice><ex>cum</ex><am>cũ</am></choice> <lb/>ambobus axibus viſualibus, eo quod <lb/>axes vifuales nunquam poſſunt inui-<lb/>cem interſecari poſt viſum, ſed ſem-<lb/>per ante ipſum, vnde nec inuicem pa-<lb/>ralleli poſſunt eſſe.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0351-01" corresp="fig-0351-01a"> <graphic url="0351-01"/> </figure> <figure xml:id="fig-0351-02" corresp="fig-0351-02a"> <graphic url="0351-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Dico etiam, quod ſi obiectum inci <lb/>derit in eadem ſuperficie, in qua duo <lb/>axes viſuales, vel radij reflexi <choice><ex>reperiuntur</ex><am>reperiũtur</am></choice>, hoc eſt in vna <choice><ex>eademque</ex><am>eademq́;</am></choice> ſuperficie reflexio-<lb/>nis, </s> <s xml:space="preserve">tunc locus imaginis non erit in catheto incidentiæ, eo quod interfectio axium <lb/>uifualium non erit in ipſo catheto ſed extra, in qua interſectione fit viſio vnius tan-<lb/>tummodo imaginis, quod antiqui non animaduerterunt. </s> <s xml:space="preserve">Hoc autem dico deſpe-<lb/>culo ſphærico concauo.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Speculatio cuiuſdam propoſitionis aritbmetica.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">SPeculatio vltimæ propoſitionis quam numerorum via inueni, hæc eſt. </s> <s xml:space="preserve">Imagi-<lb/>nemur triangulum <seg type="var">.r.e.o.</seg> abſciſum à circulo, in cuius circunferentia ſit punctum <lb/>r. ſuperioris anguli ipſius trianguli, vel etiam non ſit abſciſum dummodo protrahan <lb/>tur lineæ <choice><ex>vſque</ex><am>vſq;</am></choice> ad circunferentiam, à quo ad oppoſitum latus <choice><ex>defcendant</ex><am>defcẽdant</am></choice> duæ <seg type="var">.r.K.</seg> et <seg type="var">.r.<lb/>f.</seg> ita <choice><ex>quod</ex><am>ꝙ</am></choice>. <seg type="var">K.o.</seg> æqualis ſit <seg type="var">.f.e.</seg> vnde hæc .4. lineæ ſecabuntur à circulo dicto in punctis <seg type="var">.n.<lb/>c.b.u</seg>. </s> <s xml:space="preserve">Dico nunc <choice><ex>quod</ex><am>ꝙ</am></choice> producta <seg type="var">.o.r.n.</seg> et <seg type="var">.e.r.u.</seg> æqualia erunt productis <seg type="var">.K.r.c.</seg> et <seg type="var">.f.r.b.</seg> <lb/> <ptr xml:id="fig-0351-03a" corresp="fig-0351-03" type="figureAnchor"/> <ptr xml:id="fig-0351-04a" corresp="fig-0351-04" type="figureAnchor"/> <pb facs="0352" n="340"/><fw type="head">IO. BAPT. BENED.</fw> quapropter cogitemus <seg type="var">.r.a.</seg> indeterminatam tranſire per centrum <seg type="var">.s.</seg> ipſius circuli, ſi-<lb/>militer etiam <seg type="var">.r.i.</seg> ad punctum medium lateris <seg type="var">.e.o.</seg> deinde à tribus punctis, <seg type="var">e.i.o.</seg> ima-<lb/>ginemur tres perpendiculares ad <seg type="var">.r.a.</seg> hoc eſt <seg type="var">.e.a</seg>: <seg type="var">i.d.</seg> et <seg type="var">.o.q.</seg> & vbi circulus ſecat <seg type="var">.r.a.</seg> <lb/>fit punctum <seg type="var">.g.</seg> protractis deinde <seg type="var">.g.n</seg>: <seg type="var">g.x</seg>: et <seg type="var">.g.u.</seg> habebimus triangulum <seg type="var">.a.e.r.</seg> ſimi-<lb/>lem triangulo <seg type="var">.g.u.r.</seg> vnde clarum erit productum <seg type="var">.g.r.a.</seg> æquale eſſe producto <seg type="var">.e.r.u.</seg> <lb/><choice><ex>productumque</ex><am>productumq́</am></choice> <seg type="var">.g.r.q.</seg> æquale eſſe producto <seg type="var">.o.r.n.</seg> nam trianguli <seg type="var">.g.r.n.</seg> et <seg type="var">.o.r.q.</seg> ſunt in-<lb/>u<unclear reason="illegible"/>icem ſimiles, ſed productum <seg type="var">.g.r.a.</seg> ſimul cum producto <seg type="var">.g.r.q.</seg> duplum eſt producto <seg type="var">.<lb/>g.r.d.</seg> ex prima ſexti, eo quod <seg type="var">.a.r.q.</seg> dupla eſt <seg type="var">.d.r.</seg> & ideo productum <seg type="var">.e.r.u.</seg> ſimul <choice><ex>cum</ex><am>cũ</am></choice> <lb/>producto <seg type="var">.o.r.n.</seg> duplum erit producto <seg type="var">.i.r.x.</seg> quod quidem æquale eſt producto <seg type="var">.g.r.<lb/>d.</seg> ex ſimilibus rationibus iam ſupradictis. </s> <s xml:space="preserve">Nunc ex ſimilibus rationibus producta <seg type="var">.f.<lb/>r.b.</seg> et <seg type="var">.K.r.c.</seg> dupla erunt producto <seg type="var">.i.r.x</seg>. </s> <s xml:space="preserve">quare prima producta æqualia erunt ſecun-<lb/>dis. </s> <s xml:space="preserve">Quod eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0351-03" corresp="fig-0351-03a"> <graphic url="0351-03"/> </figure> <figure xml:id="fig-0351-04" corresp="fig-0351-04a"> <graphic url="0351-04"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Ab huiuſmodi demonſtrat ione facilè videre poteris non eſſe generaliter verum, <lb/>id quod Nicolaus Tartalea inquit .43. quæſito vltimæ partis ſuorum tractatuum, hoc <lb/>eſt centrum circuli <seg type="var">.r.n.g.</seg> ſemper eſſe in perpendiculari, quæ à puncto <seg type="var">.r.</seg> ad lineam <seg type="var">.e.<lb/>o.</seg> tranſit, protracta ipſa <seg type="var">.e.o.</seg> quantum volueris, imò in quacunque alia linea ipſum eſ <lb/>ſe poteſt, nec non in aliqua parallela ipſi <seg type="var">.e.o.</seg> quemadmodum ex te ipſo, medianti-<lb/>bus, hic ſupradictis rationibus videre poteris, vnde ex neceſſitate ſequitur illud pro <lb/>blema ſemper ferè falſum eſſe.</s> </p> <figure place="here"> <graphic url="0352-01"/> </figure> <figure place="here"> <graphic url="0352-02"/> </figure> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Alia ſpeculatio circa breuitatem radiorum incidentium <lb/>& reflexorum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">ALius modus quem exercitationis gratia vltimò cogitaui, ad demonſtrandum <lb/>breuitatem radiorum incidentium, & reflexorum in ſpeculo plano, nunc ad <lb/>te ſcribo, quamuis prolixior ali quantulum ſit eo, quod ab antiquis traditus eſt.</s> </p> <p> <s xml:space="preserve">Imaginemur itaque <choice><ex>lineam</ex><am>lineã</am></choice> <seg type="var">.p.h.</seg> pro <choice><ex>communi</ex><am>cõmuni</am></choice> ſectione ſuperficiei reflexionis <choice><ex>cum</ex><am>cũ</am></choice> ſpe-<lb/>culo <seg type="var">.r.a.</seg> verò et <seg type="var">.a.b.</seg> pro radijs dictis, qui ſemper <choice><ex>faciunt</ex><am>faciũt</am></choice> angulos <seg type="var">.b.a.h.</seg> et <seg type="var">.r.a.p.</seg> <choice><ex>inuicem</ex><am>inuicẽ</am></choice> <pb facs="0353" n="341"/><fw type="head">EPISTOL AE.</fw> æquales. </s> <s xml:space="preserve">Nunc protrahantur duæ <seg type="var">.r.o.</seg> et <seg type="var">.b.o.</seg> ab iiſdem punctis <seg type="var">.b.r.</seg> ad aliud punctum, <lb/>quod volueris ipſius lineæ <seg type="var">.p.h.</seg> quas probabo <choice><sic>longiores'</sic><corr>longiores</corr></choice> (ſimul ſumptas) eſſe priori-<lb/>bus. </s> <s xml:space="preserve">Imaginemur igitur duas perpendiculares, ſeu cathetos <seg type="var">.b.i.</seg> et <seg type="var">.q.r.a.</seg> punctis <seg type="var">.b.<lb/>r.</seg> ad <seg type="var">.p.h.</seg> <choice><ex>abſciſſaque</ex><am>abſciſſaq́</am></choice> ſit linea <seg type="var">.o.b.</seg> in puncto <seg type="var">.x.</seg> ita quod <seg type="var">.b.x.</seg> æqualis ſit ipſi <seg type="var">.b.a.</seg> quod <lb/>nulli dubium erit poſſe effici, cum <seg type="var">.o.b.</seg> <choice><ex>longiot</ex><am>lõgiot</am></choice> ſit <seg type="var">.b.a.</seg> co quod opponatur angulo ob-<lb/>tuſo ipſius trianguli <seg type="var">.b.a.o.</seg> quę <seg type="var">.o.b.</seg> ſimiliter protrahatur vſque ad <seg type="var">.d.</seg> ita quod <seg type="var">.b.d.</seg> <lb/>æqualis ſit <seg type="var">.x.b.</seg> </s> <s xml:space="preserve">deinde protrahatur <seg type="var">.o.i.</seg> quouſque <seg type="var">.i.h.</seg> æqualis ſit <seg type="var">.a.i</seg>. </s> <s xml:space="preserve">In alia parte po-<lb/>ſtea idem faciendum eſt ſecando <seg type="var">.a.r.</seg> in puncto <seg type="var">.u.</seg> ita quod <seg type="var">.u.r.</seg> æqualis ſit <seg type="var">.r.o.</seg> efficien <lb/>do <seg type="var">.r.s.</seg> æqualem <seg type="var">.r.u.</seg> et <seg type="var">.q.p.</seg> æquale <seg type="var">.q.o.</seg> vnde habebimus <choice><ex>productum</ex><am>productũ</am></choice> <seg type="var">.o.d.</seg> in <seg type="var">.o.x.</seg> æqua <lb/>le producto <seg type="var">.o.h.</seg> in <seg type="var">.o.a.</seg> & productum <seg type="var">.a.s.</seg> in <seg type="var">.a.u.</seg> æquale producto <seg type="var">.a.p.</seg> in <seg type="var">.a.o.</seg> exiſtis <lb/>rationibus. </s> <s xml:space="preserve">Nam cum quadratum ipſius <seg type="var">.o.b.</seg> æquale ſit duobus quadratis <seg type="var">.o.i.</seg> et <seg type="var">.i.<lb/>b.</seg> ex penultima primi Eucli. ipſa quadrata <seg type="var">.o.i.</seg> et <seg type="var">.i.b.</seg> æqualia erunt producto <seg type="var">.o.d.</seg> in <lb/><seg type="var">o.x.</seg> ſimul ſumpto cum quadrato <seg type="var">.b.x.</seg> ex .6. ſecundi, hoc eſt ipſi producto ſimul ſum-<lb/>pto cum quadrato <seg type="var">.b.a.</seg> hoc eſt ipſi producto ſimul ſumpto cum duobus quadratis <seg type="var">.a.<lb/>i.</seg> et <seg type="var">.i.b.</seg> ſed quia productum <seg type="var">.o.h.</seg> in <seg type="var">.o.a.</seg> ſimul ſumpto cum quadrato <seg type="var">.a.i.</seg> ęquatur qua <lb/>drato <seg type="var">.o.i.</seg> ideo productum <seg type="var">.o.h.</seg> in <seg type="var">.o.a.</seg> ſimul ſumptum cum quadrato <seg type="var">.a.i.</seg> & cum qua-<lb/>drato <seg type="var">.i.b.</seg> æquale erit producto <seg type="var">.o.d.</seg> in <seg type="var">.o.x.</seg> ſimul ſumpto <choice><ex>cum</ex><am>cũ</am></choice> duobus quadratis dictis <lb/>hoc eſt ipſius <seg type="var">.a.i.</seg> et <seg type="var">.i.b.</seg> quę quadrata dempta cum fuerint ab vtraque parte, tunc cer <lb/>ti erimus producta eſſe inuicem æqualia. </s> <s xml:space="preserve">Idem dico de alijs ex altera parte. </s> <s xml:space="preserve">Nunc <lb/>imaginemur protractam eſſc <seg type="var">.a.e.</seg> parallelam ipſi <seg type="var">.o.b.</seg> & habebimus proportionem <lb/>ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.a.i.</seg> maiorem eſſe ea quæ eſt ipſius <seg type="var">.a.e.</seg> ad eandem <seg type="var">.a.i.</seg> cum <seg type="var">.a.b.</seg> maior <lb/>ſit ipſa <seg type="var">.a.e.</seg> vt oppoſita angulo obtuſo, quapropter proportio <seg type="var">.x.b.</seg> ad <seg type="var">.a.i.</seg> maior erit <lb/>ea quæ eſt <seg type="var">.o.b.</seg> ad <seg type="var">.o.i</seg>. </s> <s xml:space="preserve">Iam enim ſcis proportionem <seg type="var">.o.b.</seg> ad <seg type="var">.o.i.</seg> eſſe, vt <seg type="var">.a.e.</seg> ad <seg type="var">.a.i.</seg> ex <lb/>ſimilitudine triangulorum. </s> <s xml:space="preserve">quare proportio <seg type="var">.b.d.</seg> ad <seg type="var">.i.h.</seg> maior erit proportione <seg type="var">.o.b.</seg> <lb/>ad <seg type="var">.o.i.</seg> <choice><ex>tunc</ex><am>tũc</am></choice> ex .27. quinti <choice><ex>permutando</ex><am>ꝑmutãdo</am></choice> <choice><ex>proportio</ex><am>ꝓportio</am></choice> <seg type="var">.b.d.</seg> ad <seg type="var">.b.o.</seg> maior erit proportione <seg type="var">.i.h.</seg> <lb/>ad <seg type="var">.i.o.</seg> & ex .26. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> <choice><ex>componendo</ex><am>cõponẽdo</am></choice> maior <choice><ex>proportio</ex><am>ꝓportio</am></choice> erit <seg type="var">.o.d.</seg> ad <seg type="var">.o.b.</seg> ea quę eſt <seg type="var">.o.h.</seg> ad. o <lb/>i. & <choice><ex>permutando</ex><am>permutãdo</am></choice> maior ipſius <seg type="var">.o.d.</seg> ad <seg type="var">.o.h.</seg> ea quæ <seg type="var">.o.b.</seg> ad <seg type="var">.o.i.</seg> & ex .33. maior ipſius <seg type="var">.b.<lb/>d.</seg> ad <seg type="var">.i.h.</seg> ea quæ <seg type="var">.o.d.</seg> ad <seg type="var">.o.h</seg>. </s> <s xml:space="preserve">Sed vt <seg type="var">.b.a.</seg> ad <seg type="var">.a.i.</seg> ita eſt <seg type="var">.a.r.</seg> ad <seg type="var">.a.q.</seg> ex ſimilitudine <choice><ex>triam</ex><am>triã</am></choice> <lb/>gulorum. </s> <s xml:space="preserve">Erit igitur <seg type="var">.a.r.</seg> ad <seg type="var">.a.q.</seg> maior proportio, ea quæ eſt <seg type="var">.o.b.</seg> ad <seg type="var">.o.i.</seg> & exijſdem <lb/>ſupradictis rationibus maior erit proportio ipſius <seg type="var">.s.a.</seg> ad <seg type="var">.p.a.</seg> ea quæ eſt <seg type="var">.a.r.</seg> ad <seg type="var">.a.q.</seg> <lb/>ſed cum iam probatum fuit proportio <lb/> <ptr xml:id="fig-0353-01a" corresp="fig-0353-01" type="figureAnchor"/> nem <seg type="var">.b.d.</seg> ad <seg type="var">.i.h.</seg> hoc eſt <seg type="var">.a.b.</seg> ad <seg type="var">.a.i.</seg> ma <lb/>iorem eſſe <seg type="var">.o.d.</seg> ad <seg type="var">.o.h.</seg> ergo eo ma-<lb/>gis maior erit proportio ipſius <seg type="var">.a.s.</seg> ad <lb/><seg type="var">a.p.</seg> ca quæ <seg type="var">.o.d.</seg> ad <seg type="var">.o.h.</seg> ſed cum ex .15 <lb/>ſexti, eadem ſit proportio <seg type="var">.o.d.</seg> ad <seg type="var">.o.a.</seg> <lb/>quæ <seg type="var">.o.h.</seg> ad <seg type="var">.o.x.</seg> et <seg type="var">.s.a.</seg> ad <seg type="var">.o.a.</seg> quę <seg type="var">a.p.</seg> <lb/>ad <seg type="var">.a.u.</seg> </s> <s xml:space="preserve">tunc erit <choice><ex>permutando</ex><am>permutãdo</am></choice> eadem <lb/>proportio ipſius <seg type="var">.o.d.</seg> ad <seg type="var">.o.h.</seg> quæ <seg type="var">.o.a.</seg> <lb/>ad <seg type="var">.o.x.</seg> & ipſius <seg type="var">.a.o.</seg> ad <seg type="var">.a.u.</seg> quemad-<lb/>modum ipſius <seg type="var">.a.s.</seg> ad <seg type="var">.a.p</seg>. </s> <s xml:space="preserve">Quare maior proportio erit ipſius <seg type="var">.a.o.</seg> ad <seg type="var">.a.u.</seg> quam <seg type="var">.a.</seg> o<unclear reason="illegible"/>. <lb/>ad <seg type="var">.o.x</seg>. </s> <s xml:space="preserve">Vnde ſequitur <seg type="var">.o.x.</seg> maiorem eſſe <seg type="var">.a.u.</seg> ex .8. quinti, ergo <seg type="var">.b.x.o.r.</seg> longior erit <lb/>ipſa <seg type="var">.b.a.u.r</seg>. </s> <s xml:space="preserve">Quod eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0353-01" corresp="fig-0353-01a"> <graphic url="0353-01"/> </figure> </div> </body> </floatingText> <pb facs="0354" n="342"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Alia etiam via poſſumus idem concludere. </s> <s xml:space="preserve">Imaginemur maiorem axem alicu-<lb/>ius ellipſis tranſire per duo puncta <seg type="var">.r.</seg> et <seg type="var">.b.</seg> ſupponendo ipſa puncta, ea eſle, quæ ita <lb/>axem diuidunt, vt ſingula produ-<lb/> <ptr xml:id="fig-0354-01a" corresp="fig-0354-01" type="figureAnchor"/> cta fectionum ſint, vt inquit Per-<lb/>geus. </s> <s xml:space="preserve">imaginemur, etiam <seg type="var">.p.h.</seg> con <lb/>tiguam eſſe ipſi ellipſi in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.a.</seg> <lb/>vnde ſi protractæ fuerint duæ <seg type="var">.r.a.</seg> <lb/>et <seg type="var">.b.a.</seg> habebimus ex .48. tertijip-<lb/>ſius Pergei angulos <seg type="var">.b.a.h.</seg> et <seg type="var">.r.a.<lb/>p.</seg> inuicem æquales. </s> <s xml:space="preserve">Ducendo <lb/>poſtea ad quoduis punctum ipſius <lb/><seg type="var">p.h.</seg> duas <seg type="var">.b.o.</seg> et <seg type="var">.r.o.</seg> certi erimus, <lb/>quod ſecabuntur à gyro oxygo-<lb/>nio, quarum vna ſecta ſit in pun-<lb/>cto <seg type="var">.i.</seg> ducta poſtea <seg type="var">.i.r.</seg> clarum erit ex .52. dicti, quod longitudo <seg type="var">.b.i.r.</seg> æqualis erit lon <lb/>gitudini <seg type="var">.b.a.r.</seg> & minor ipſa <seg type="var">.b.o.r.</seg> ex .21. primi Euclid.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0354-01" corresp="fig-0354-01a"> <graphic url="0354-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Deerrore Euclidis circa ſpeculum vstorium.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">VErum ſpeculum vſtorium, illud non eſt, quod ab Euclide traditum fuit, & <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>tu etiam putas, Nam Euclides errat, cum credat radios reflexos à ſuperficie <lb/>ſphærica concaua ſeinuicem in centro ſpeculi interſecare. </s> <s xml:space="preserve">Nam cum omnes lineę <lb/>recte à centro, & cir cunferentia alicuius ſphæræ terminatæ, ſint eidem circunferen-<lb/>tiæ perpendiculares, ſequeretur ex neceſſitate radios incidentiæ etiam perpendicu <lb/>lares eidem ſuperficiei eſſe, cum anguli incidentiæ ſemper æquales ſint angulis re-<lb/>flexionis, vnde etiam ex neceſſitate ſequeretur punctum corporis lucidi, à quo radij <lb/>luminoſi excunt, in centro ſpeculi reperiri. </s> <s xml:space="preserve">quod quidem falſiſſimum eſt.</s> </p> <p> <s xml:space="preserve">Alia etiam via poſſum hanc oſtendere impoſſibilitatem, & tibi probabo, quod <lb/>in nullo aliquo puncto poſſunt inuicem conuenire ipſi radijrefle xi omnes.</s> </p> <p> <s xml:space="preserve">Sit igitur <seg type="var">.l.a.c.</seg> <choice><ex>conis</ex><am>cõis</am></choice> ſectio ſuperficiei reflexionis cum ſpeculo, cuius centrum ſit <seg type="var">.o.</seg> <lb/>punctum verò lucidum ſit <seg type="var">.g.</seg> <choice><ex>protrahaturque</ex><am>protrahaturq́</am></choice> <seg type="var">.g.o.a</seg>. </s> <s xml:space="preserve">Nunc autem primum dico, quod <lb/>radij reflexi à punctis diuerſarum <choice><ex>diſtantiarum</ex><am>diſtantiarũ</am></choice> ab <seg type="var">.a.</seg> non <choice><ex>coincident</ex><am>coincidẽt</am></choice> inuicem in aliquo <lb/>puncto lineę <seg type="var">.g.o.a</seg>: ſint ergo duo puncta <seg type="var">.u.</seg> et <seg type="var">.r.</seg> diuerſarum <choice><ex>diſtantiarum</ex><am>diſtantiarũ</am></choice> ab <seg type="var">.a.</seg> à quibus <lb/>veniant duo radij incidentiæ <seg type="var">.g.r.</seg> et <seg type="var">.g.u.</seg> radius verò reflexus ab <seg type="var">.r.</seg> ſit <seg type="var">.r.e.</seg> protrahatur <lb/><seg type="var">u.e.</seg> quam dico effe non poſſe radium reflexum ab <seg type="var">.u.</seg> quotieſcunque eius incidens <lb/>deſcendat ab <seg type="var">.g</seg>. </s> <s xml:space="preserve">Protrahantur ergo duæ lineæ <seg type="var">.o.r.</seg> et <seg type="var">.o.u.</seg> vnde cum dixerit aliquis <lb/><seg type="var">u.e.</seg> <choice><ex>reflexum</ex><am>reflexũ</am></choice> eſſe ipſius <seg type="var">.g.u.</seg> igitur anguli <seg type="var">.g.u.o.</seg> et <seg type="var">.o.u.e.</seg> erunt inuicem æquales, & ſic <lb/>etiam erunt duo <seg type="var">.g.r.o.</seg> et <seg type="var">.o.r.e.</seg> vnde ex tertia ſexti & .11. quinti Eucli. proportio <seg type="var">.g.<lb/>u.</seg> ad <seg type="var">.u.e.</seg> æqualis eſſet ei, quæ <seg type="var">.g.r.</seg> ad <seg type="var">.r.e.</seg> quod quidem impoſſibile eſſe demonſtra-<lb/>bo, eo quod cum <seg type="var">.g.u.</seg> maior ſit <seg type="var">.g.r.</seg> ex .8. tertij, erit ex .8. quinti proportio ipſius <seg type="var">.g.u.</seg> <lb/>ad <seg type="var">.r.e.</seg> maior proportione ipſius <seg type="var">.g.r.</seg> ad <seg type="var">.r.e.</seg> ſed ex .7. tertij <seg type="var">.u.e.</seg> minor eſt <seg type="var">.r.e.</seg> erit igi-<lb/>tur ex dicta .8. quinti maior proportio <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.g.u.</seg> ad <seg type="var">.u.e.</seg> quam <seg type="var">.g.u.</seg> ad <seg type="var">.r.e.</seg> vnde eo ma <pb facs="0355" n="343"/><fw type="head">EPISTOL AE.</fw> gis erit maior proportio ipſius <seg type="var">.g.u.</seg> ad <seg type="var">.u.e.</seg> <lb/> <ptr xml:id="fig-0355-01a" corresp="fig-0355-01" type="figureAnchor"/> quam ipſius <seg type="var">.g.r.</seg> ad <seg type="var">.r.e.</seg> ergo non æqualis, <lb/>quapropter impoſſibile eſt <seg type="var">.u.e.</seg> eſſe radium <lb/>reflexum incidentis radij <seg type="var">.g.u</seg>. </s> <s xml:space="preserve">Vnde ſequi <lb/>tur concurſum radiorum reflexorum à ſpe-<lb/>culo ſphærico concauo non eſſe in vno, & <lb/>e<unclear reason="illegible"/>odem puncto ipſius catheti incidentiæ, <lb/>quando à ſitu non æquidiſtanti ab ipſo ca-<lb/>theto <choice><ex>reflectuntur</ex><am>reflectũtur</am></choice>, ex hac ſpeculatione <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>videre licet, verum eſſe id quod in .3. Epiſto <lb/>la tibi ſcripſi nempe, quod quotieſcunque <lb/>axes viſuales, vel radij reflexi, in vna ea-<lb/><choice><ex>demque</ex><am>demq́</am></choice> ſuperficie reflexionis fuerint, </s> <s xml:space="preserve">tunc <lb/>imago obiecti nullo modo videbitur in ca-<lb/>theto incidentiæ, in ſpeculo ſphærico con-<lb/>cauo.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0355-01" corresp="fig-0355-01a"> <graphic url="0355-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Alterius dubit ationis ſolutio.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">NOn abſque ratione dubitas, vtrum etiam in ſphæricis ſpeculis conuexis idem <lb/>accidat, hoc eſt, an radij reflexi à punctis inęqualis diſtantiæ à catheto inciden <lb/>tiæ conueniant inuicem in eodem catheto.</s> </p> <p> <s xml:space="preserve">Ad quod reſpondeo, non concurrere in dicto catheto, ſed extra ipſum, & ſimi-<lb/>liter extra ipſum vide bitur imago.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ratione, imaginemur ſuperficiem reflexionis alicuius ſpeculi ſphæ-<lb/>rici conuexi <seg type="var">.b.d.h.g.</seg> cuius communis ſectio cum ſuperficie ſphærica ſit linea <lb/>circularis <seg type="var">.d.e.h.</seg> et <seg type="var">.o.</seg> eius <choice><ex>centrum</ex><am>cẽtrum</am></choice>, à quo protrahatur <seg type="var">.g.b.</seg> indeterminata, et <seg type="var">.o.g.</seg> ſit ſe <lb/>midiameter circuli <seg type="var">.d.g.h.</seg> et <seg type="var">.o.c.</seg> ſit plus medietate ipſius <seg type="var">.o.g.</seg> <choice><ex>accipiaturque</ex><am>accipiaturq́;</am></choice> linea <seg type="var">.e.c.</seg> <lb/>minor ipſa <seg type="var">.o.c.</seg> ſed maior ipſa <seg type="var">.c.g.</seg> <lb/> <ptr xml:id="fig-0355-02a" corresp="fig-0355-02" type="figureAnchor"/> quod difficile non erit, locando im <lb/>mobilem pedem circini in puncto <seg type="var">.<lb/>c.</seg> aperiendo ipſum aliquantulum <lb/>plus quam <seg type="var">.c.g.</seg> ſed minus quam <seg type="var">.c.<lb/>o.</seg> ſignando circunferentiam <seg type="var">.d.e.h.</seg> <lb/>in puncto <seg type="var">.e.</seg> quod ex .7. tertij poſſi-<lb/>bile eſt, protrahatur poſtea <seg type="var">.o.e.f.</seg> <lb/>indeterminatè. </s> <s xml:space="preserve">Facicmus deinde <lb/>angulum <seg type="var">.f.e.b.</seg> æqualem angulo <seg type="var">.o.<lb/>e.c.</seg> protracta poſtea cum fuerit <seg type="var">.c.<lb/>e.K.</seg> indeterminatè, <choice><ex>habebimus</ex><am>habebimꝰ</am></choice> duos <lb/>angulos <seg type="var">.b.e.f.</seg> et <seg type="var">.f.e.K.</seg> æquales in-<lb/>uicem mediante .15. primi, ita <choice><ex>quod</ex><am>qđ</am></choice> ſi <lb/>radius incidens veniet à puncto <seg type="var">.b.</seg> <lb/>ad <seg type="var">.e.</seg> reflexus erit <seg type="var">.e.K.</seg> qui quidem <pb facs="0356" n="344"/><fw type="head">IO. BAPT. BENED.</fw> refleyus ſecabit cathetum <seg type="var">.b.o.</seg> in puncto <seg type="var">.c.</seg> intra ſpeculum, nec dubitandum eſt quin <lb/>linea <seg type="var">.e.b.</seg> ſectura ſit <seg type="var">.b.o.</seg> eo quod cum angulus <seg type="var">.o.e.c.</seg> ſit maior angulo <seg type="var">.e.o.c.</seg> ex .19. <lb/>primi, & ſimiliter angulus <seg type="var">.b.e.f.</seg> ſequitur ex .13. dicti, angulos <seg type="var">.b.e.o.</seg> et <seg type="var">.e.o.b.</seg> eſſe mi <lb/>nores duobus rectis, vnde ex penultima petitione primi, duæ lineæ <seg type="var">.b.e.</seg> et <seg type="var">.o.b.</seg> <choice><ex>inuicem</ex><am>inuicẽ</am></choice> <lb/>concurrent. </s> <s xml:space="preserve">Quare poſſumus ex hoc, quoddam corollarium extrahere, hoc eſt <lb/><choice><ex>neceſſarium</ex><am>neceſſariũ</am></choice> <choice><ex>semper</ex><am>sẽper</am></choice> exiſtat, vt linea <seg type="var">.c.e.</seg> minor eſſe linea <seg type="var">.c.o</seg>. </s> <s xml:space="preserve">Sed vnde eueniat quod ip <lb/>ſa neceſſariò debeat ſemper maior eſſe ipſa <seg type="var">.c.g.</seg> clarum eſt ex .7. tertij Eucli. </s> <s xml:space="preserve">Nunc <lb/>imaginemur ductas eſſe duas <choice><ex>tangentes</ex><am>tãgentes</am></choice> <seg type="var">.b.d.</seg> et <seg type="var">.b.h.</seg> & ab <seg type="var">.e.</seg> <choice><ex>ipsam</ex><am>ipsã</am></choice> <seg type="var">.e.i.</seg> vnde certi erimus, <lb/>quod ab interuallo inter <seg type="var">.h.</seg> et <seg type="var">.d.</seg> punctum <seg type="var">.b.</seg> <choice><ex>ponſſibile</ex><am>põſſibile</am></choice> ſit vt reflectatur. </s> <s xml:space="preserve">Accipiamus <lb/>nunc <seg type="var">.p.c.</seg> minorem medietate ipſius <seg type="var">.b.c.</seg> & à puncto <seg type="var">.p.</seg> imaginemur tangentem <seg type="var">.p.q.</seg> <lb/>in puncto <seg type="var">.q.</seg> prorractaq́ue ſit <seg type="var">.b.q.</seg> vt radius incidentiæ, </s> <s xml:space="preserve">tunc dico, radium reflexum <lb/>ipſius <seg type="var">.b.q.</seg> <choice><ex>non</ex><am>nõ</am></choice> concurrere in eodem puncto <seg type="var">.c.</seg> ipſius catheti, ſi vero dixeris <choice><ex>quod</ex><am>ꝙ</am></choice> ſic. </s> <s xml:space="preserve">Eſto <lb/><choice><ex>igitur</ex><am>igit̃</am></choice> radius dictus <seg type="var">.c.q.s</seg>. </s> <s xml:space="preserve">Imaginemur <choice><ex>tangentem</ex><am>tãgentẽ</am></choice> <seg type="var">.e.i.</seg> in puncto <seg type="var">.e.</seg> vnde ex .18. quinti Alha <lb/>zem, vel .12. ſexti Vitellionis proportio <seg type="var">.b.i.</seg> ad <seg type="var">.i.c.</seg> erit, vt <seg type="var">.b.o.</seg> ad <seg type="var">.o.c.</seg> & ſimiliter erit <lb/>ipſius <seg type="var">.b.p.</seg> ad <seg type="var">.p.c.</seg> vt <seg type="var">.b.o.</seg> ad <seg type="var">.o.c.</seg> ex eadem. </s> <s xml:space="preserve">Quare ex .11. quinti Eucli. proportio ip <lb/>ſius <seg type="var">.b.p.</seg> ad <seg type="var">.p.c.</seg> erit vt ipſius <seg type="var">.b.i.</seg> ad <seg type="var">.i.c.</seg> ſed quia <seg type="var">.p.b.</seg> vt pars ipſius <seg type="var">.b.i.</seg> minor eſt ip-<lb/>ſa, ergo ex .14. dicti <seg type="var">.p.c.</seg> minor erit ipſa <seg type="var">.c.i.</seg> hoc eſt totum minus ſua parte, quod eſt <lb/>impoſſibile, </s> <s xml:space="preserve">quare non in ipſo catheto videbitur imago ipſius obiecti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0355-02" corresp="fig-0355-02a"> <graphic url="0355-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Aliud notandum etiam cernere potes ex ipſis ſpeculis ſphæricis conuexis, hoc eſt <lb/>quod poſſibile ſit aliquoties, radium reflexum concurrere cum catheto incidentiæ <lb/>extra ſpeculum inter puncta <seg type="var">.g.</seg> et <seg type="var">.p.</seg> vt exempli gratia .ſi punctus <seg type="var">.p.</seg> eſſet exactè <lb/>in medio inter <seg type="var">.b.</seg> et g. </s> <s xml:space="preserve">tunc punctum <seg type="var">.c.</seg> ipſius concurſus cum catheto incidentiæ eſſet <lb/>inter <seg type="var">.g.</seg> et <seg type="var">.p.</seg> eo quod <choice><ex>cum</ex><am>cũ</am></choice> linea <seg type="var">.p.q.</seg> debeat @iui lere <choice><ex>angulum</ex><am>angulũ</am></choice> <seg type="var">.b.</seg> q, c. <choice><ex>per</ex><am>ꝑ</am></choice> ęqualia, oportebit <lb/>c. poſitum eſſe inter <seg type="var">.g.</seg> et <seg type="var">.p.</seg> quia angulus <seg type="var">.g.q.p.</seg> maior eſt angulo <seg type="var">.p.q.b.</seg> vt per te faci <lb/>le potes ratiotinari, imaginando cir <lb/> <ptr xml:id="fig-0356-01a" corresp="fig-0356-01" type="figureAnchor"/> culum circa <choice><ex>triangulum</ex><am>triãgulum</am></choice> <seg type="var">.g.q.b.</seg> & dia <lb/>merrum perpendicularem .ad <seg type="var">.g.b.</seg> <lb/>in puncto <seg type="var">.p.</seg> producendo poſtea <seg type="var">.q.<lb/>p.</seg> <choice><ex>vſque</ex><am>vſq;</am></choice> ad <choice><ex>alteram</ex><am>alterã</am></choice> <choice><ex>partem</ex><am>partẽ</am></choice> circunferen-<lb/>tiæ ipſius circuli. </s> <s xml:space="preserve"><choice><ex>argumentando</ex><am>argumẽtãdo</am></choice> dein-<lb/>de mediante vltima ſexti, illud <choice><ex>idem</ex><am>idẽ</am></choice> <lb/>po@es etiam ſcire ex .22. quinti Alha <lb/>zeni. & ex .26. ſexti Vitellionis. </s> <s xml:space="preserve">vn-<lb/>de ſi ad ambas pupillas venerint ra <lb/>dij reflexi ipſius obiecti <seg type="var">.b.a.</seg> duobus <lb/>punctis huiuſmodi ſpeculi, ita di-<lb/>ſtantibus à puncto <seg type="var">.g.</seg> vt <seg type="var">.q</seg>. </s> <s xml:space="preserve">tunc com <lb/>mune punctum concurſus axium vi <lb/>ſualium erit in catheto inter <seg type="var">.g.p.</seg> <lb/>vbi apparebit imago ex ſuperius di <lb/>ctis rationibus, ita vt <choice><ex>non</ex><am>nõ</am></choice> ſolum con <lb/>cauis, ſed etiam conuexis hoc accidere poſſit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0356-01" corresp="fig-0356-01a"> <graphic url="0356-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">In planis autem <choice><ex>nunquam</ex><am>nunquã</am></choice> hoc poteſt euenire, vt tibi alias dixi, eo quod ſi <choice><sic>accéperi-mus</sic><corr>acceperi- mus</corr></choice> <choice><ex>rectam</ex><am>rectã</am></choice> <seg type="var">.m.r.</seg> pro <choice><ex>coni</ex><am>cõi</am></choice> ſectione <choice><ex>ſuperficiei</ex><am>ſuꝑficiei</am></choice> <seg type="var">.l.t.x.</seg> reflexionis & <choice><ex>ſuperficiei</ex><am>ſuꝑficiei</am></choice> ſpeculi, <choice><ex>punctumque</ex><am>pũctũq́;</am></choice> <lb/>lucidum <seg type="var">.l.</seg> <choice><ex>protractoque</ex><am>protractoq́;</am></choice> catheto <seg type="var">.l.r.t.</seg> <choice><ex>lineisque</ex><am>lineisq́;</am></choice> incidentiæ <seg type="var">.l.x.</seg> et <seg type="var">.l.m.</seg> reflexionis etiam <lb/><seg type="var">x.y.</seg> et <seg type="var">.m.z.</seg> cum anguli <seg type="var">.l.x.r.</seg> et <seg type="var">.y.x.h.</seg> et <seg type="var">.r.x.t.</seg> æquales inuicem ſint, & ſic anguli <seg type="var">.l.m.<lb/>r.</seg> et <seg type="var">.z.m.h.</seg> et <seg type="var">.r.m.t.</seg> erit <seg type="var">.r.t.</seg> tam pro triangulo <seg type="var">.r.x.t.</seg> quam pro triangulo <seg type="var">.r.m.t.</seg> æqua <lb/>lis <seg type="var">.r.l.</seg> ex .26. primi, ita quod ſemper in puncto <seg type="var">.t.</seg> <choice><ex>conuenient</ex><am>conueniẽt</am></choice> omnes radij reflexi ipſius <pb facs="0357" n="345"/><fw type="head">EPISTOL AE.</fw> puncti <seg type="var">.l.</seg> clarum igitur nunc habes, quod in ſphærico concauo, ſeu conuexo, non <lb/>omnes radij reflexi conueniunt in vno, <choice><ex>eodemque</ex><am>eodemq́;</am></choice> puncto catheti incidentiæ, quemad <lb/>modum in planis accidit, in quibus ſemper vnum, & idem punctum eſt ipſis commu <lb/>ne in ipſo incidentiæ catheto.</s> </p> <p> <s xml:space="preserve">Non prætermittam etiam hunc alium breuiorem modum ſpeculandi <choice><ex>æqualita- tem</ex><am>æqualita-tẽ</am></choice> depreſſionis imaginis ſub ſpeculo plano, ei quæ ſupra reperitur ipſius obiecti, in ca <lb/>theto incidentiæ, quemadmodum nu nc <lb/>vltimò diximus, hoc eſt quod cum <lb/> <ptr xml:id="fig-0357-01a" corresp="fig-0357-01" type="figureAnchor"/> imago obiecti <seg type="var">.l.</seg> reflexa à puncto <seg type="var">.<lb/>x.</seg> reperiatur in linea <seg type="var">.y.x.t.</seg> & ima-<lb/>go eiuſdem obiecti reflexa à pun-<lb/>cto <seg type="var">.m.</seg> reperiatur in linea <seg type="var">.z.m.t.</seg> & <lb/>iſtæ duæ lineę ſeinuicem ſecent in <lb/>puncto <seg type="var">.t.</seg> ipſius catheti, exiſtente <seg type="var">.<lb/>r.t.</seg> æquali <seg type="var">.r.l.</seg> vt nunc vidimus, er-<lb/>go ſemper imago reflexa à ſpecu-<lb/>lo plano, nobis apparebit <choice><ex>in</ex><am>ĩ</am></choice> ipſo ca <lb/>theto, tam vltra ſpeculum, quam ci <lb/>tra ipſum, <choice><ex>repertum</ex><am>reꝑtũ</am></choice> fuerit <choice><ex>ipsum</ex><am>ipsũ</am></choice> <choice><ex>obiectum</ex><am>obiectũ</am></choice> <lb/>quod nec Alhazem, nec Vitellio, <lb/>nec alius aliquis (quod ſciam) ad huc ſcientificè demonſtrauit. </s> <s xml:space="preserve">exempla enim vel ex <lb/>perientia non faciunt ſcire. </s> <s xml:space="preserve">Credo etiam te non dubitare quin duæ lineæ <seg type="var">.y.x.</seg> et <seg type="var">.z.<lb/>m.</seg> inuicem concurrant, cum anguli <seg type="var">.t.x.m.</seg> et <seg type="var">.t.m.x.</seg> minores ſint duobus rectis cum <lb/>æquales ſint angulis <seg type="var">.l.x.m.</seg> et <seg type="var">.l.m.x</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0357-01" corresp="fig-0357-01a"> <graphic url="0357-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De rotunditate vmbræterræ in ecclipſibus Lunaribus.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">ROtunditas vmbræ in ecclipſi-<lb/> <ptr xml:id="fig-0357-02a" corresp="fig-0357-02" type="figureAnchor"/> bus lunaribus oritur <choice><ex>tam</ex><am>tã</am></choice> à rotun <lb/>ditate maris, <choice><ex>quam</ex><am>quã</am></choice> terræ, & ſi terra eſ-<lb/>ſet <choice><ex>etiam</ex><am>etiã</am></choice> cuiuſuis alterius figurę, <choice><ex>quam</ex><am>quã</am></choice> <lb/>ſphæricę, dummodo aqua impleret <lb/><choice><ex>locum</ex><am>locũ</am></choice> ſphęriceitatis à terra <choice><ex>derelictum</ex><am>derelictũ</am></choice>, <lb/>nihilominus vmbra eſſet rotunda, <lb/>quę quidem ab aqua produceretur, <lb/><choice><ex>quanuis</ex><am>quãuis</am></choice> Alexander Piccolhomineus <lb/> <ptr xml:id="fig-0357-03a" corresp="fig-0357-03" type="figureAnchor"/> aliter ſentiat in libro de magnitudi-<lb/>ne terrę, & aquæ.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0357-02" corresp="fig-0357-02a"> <graphic url="0357-02"/> </figure> <figure xml:id="fig-0357-03" corresp="fig-0357-03a"> <graphic url="0357-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve"><choice><ex>Sciendum</ex><am>Sciẽdũ</am></choice> enim eſt, quod omne cor <lb/>pus in ſe habens <choice><ex>aliquantulum</ex><am>aliquantulũ</am></choice> opaci-<lb/>tatis, ſemper debilitat <choice><ex>radium</ex><am>radiũ</am></choice> lumino <lb/>ſum, & <choice><ex>tanto</ex><am>tãto</am></choice> magis, <choice><ex>quanto</ex><am>quãto</am></choice> magis in ip <lb/>ſo corpore radius penetrat, <choice><ex>etiam</ex><am>etiã</am></choice> & ſi <lb/>ad rectos incideret ipſe radius ſupra <lb/><choice><ex>ſuperficiem</ex><am>ſuperficiẽ</am></choice> <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> corporis. </s> <s xml:space="preserve"><choice><ex>Exempli</ex><am>Exẽpli</am></choice> gra <lb/>tia, eſto <seg type="var">.q.p.</seg> corpus a <choice><ex>queum</ex><am>queũ</am></choice>, cuius pro <lb/>funditas diuidatur in partibus <seg type="var">.d.K</seg>: <lb/><seg type="var">K.s</seg>: et <seg type="var">.s.f.</seg> à puncto verò lucido <seg type="var">.b.</seg> <pb facs="0358" n="346"/><fw type="head">IO. BABPT. BENED.</fw> deſcendat radius <seg type="var">.b.d.K.s.f.</seg> ad <choice><ex>libitum</ex><am>libitũ</am></choice> hoc eſt rectè vel obliquè, cuius pars <seg type="var">.b.d.</seg> in ipſo <lb/>aere exiſtat. </s> <s xml:space="preserve">Nunc manifeſtum erit partem <seg type="var">.b.d.</seg> ipſius radij clariorem ſeu minus im <lb/><choice><ex>peditam</ex><am>peditã</am></choice> eſſe quam <seg type="var">.d.K.</seg> quod ex eo etiam cognoſcere poſſumus quia <seg type="var">.b.d.</seg> reflectitur à <lb/>puncto <seg type="var">.d.</seg> ſuperficiei corporis a quei, quapropter minus luminoſa remanebit pars <seg type="var">.d.<lb/>K.</seg> cum non tota claritas <seg type="var">.b.d.</seg> deſcendat in corpore aqueo, ſed vna eius pars reflecta-<lb/>tur, reliqua verò tantummodò deſcendat, </s> <s xml:space="preserve">deinde pars <seg type="var">.K.s.</seg> ex neceſſitate debilior <lb/>erit ipſa <seg type="var">.d.K.</seg> eo quod ſuccedit poſt ipſam <seg type="var">.d.K.</seg> propter hoc etiam, quia cum corpus <lb/>aqueum habeat aliquantulum opacitatis, radius <seg type="var">.d.K.</seg> ab omni puncto ipſius ſpiſſitu-<lb/>dinis a quæ continuo reflectitur, quę quidem reflexio eſt illud lumen cęruleum, quod <lb/>in profunditate ipſius aquę nobis apparet. </s> <s xml:space="preserve">Cum igitur reflexio ipſa ſemper detra-<lb/>hat ab ipſo radio luminoſo, reſiduum verò ſit id quod penetrat, ideo <seg type="var">.K.s.</seg> erit vna <lb/>pars tantummodò luminis ipſius <seg type="var">.d.K</seg>: in <seg type="var">.s.f.</seg> verò aliqua pars luminis ipſius <seg type="var">.K.s.</seg> & ſic <lb/>continuò debilitatur radius, ita quod ad nihilum vſque deuenit, & vltra tale cor-<lb/>pus remanebit vmbra, quaſi ſi ipſum corpus eſſet perfectè opacum, cuius rei cauſa, <lb/>eſt illa continua reflexio, vt diximus, quæ continuò adimit aliquid ex ipſo radio, <lb/>nec permittit eum totum tranſire.</s> </p> <p> <s xml:space="preserve">Quapropter mirandum non eſt eos, qui margaritas quærunt in fundo maris nul-<lb/>lum ibi videre lumen. </s> <s xml:space="preserve">Nihilominus vmbra maris, quam dico nos poſſe videre in <lb/>ſuperficie corporis lunaris, ab alia etiam ratione prouenire poſſet. </s> <s xml:space="preserve">Imaginemur enim <lb/>aggregatum terrę, <choice><ex>marisque</ex><am>marisq́;</am></choice> eſſe tantummodò aqueum, quod quidem eſſet perfectè <lb/>ſphæricum ratione centri grauitatis, <choice><ex>ſupponamusque</ex><am>ſupponamusq́;</am></choice> <choice><ex>ipsum</ex><am>ipsũ</am></choice> eſſe valde diaphanum, ita <lb/>quod radij ſolares ipſum penetraſſent. </s> <s xml:space="preserve">Tunc dico quod in ſuperficie corporis luna-<lb/>ris produceret vmbram. </s> <s xml:space="preserve">Pro cuius intelligentia cogitemus ſubſcriptam hic figuram <lb/><seg type="var">b.h.q.a.e.</seg> eſſe ſphęram aliquam cryſtallinam, & ad partem <seg type="var">.b.h.q.</seg> ſit radius lumino-<lb/>ſus ſolaris qui ipſam illuminet, cuius radij extremitates ſint <seg type="var">.d.b.l.</seg> et <seg type="var">.p.q.r.</seg> ſupponen-<lb/>do <seg type="var">.d.l.</seg> et <seg type="var">.p.r.</seg> terminos eſſe vnius plani ſecantis ipſum radium per axem, </s> <s xml:space="preserve">tunc vide-<lb/>bis ipſum radium <seg type="var">.b.p.q.d.</seg> <choice><ex>tranſeun- tem</ex><am>tranſeũ- tem</am></choice> ipſam ſphæram, congregari ſeu <lb/><choice><ex>condenſari</ex><am>condẽſari</am></choice>, ob vniformem refractio-<lb/>nem, vſque ad punctum <seg type="var">.o.</seg> deinde; <lb/>propter rectitudinem ipſius diffu-<lb/>ſionis, vltra punctum <seg type="var">.o.</seg> ipſum dila-<lb/>tari, diſgregari, ſeu rarefieri, <choice><ex>quouſque</ex><am>quouſq;</am></choice> <lb/>nullius illuminationis actum habeat .<lb/>vt: exempli gratia <seg type="var">.o.t.</seg> et <seg type="var">.o.s.</seg> eius par <lb/>tes, ita quod interualla <seg type="var">.c.o.b.</seg> et <seg type="var">.u.</seg> <lb/> <ptr xml:id="fig-0358-02a" corresp="fig-0358-02" type="figureAnchor"/> <seg type="var">o.q.</seg> relinquerentur priuata lumini-<lb/>bus, vnde vmbroſa remanerent. di-<lb/><choice><ex>ſtantiaque</ex><am>ſtantiaq́;</am></choice> ab <seg type="var">.o.</seg> ad ſuperficiem ſphęri <lb/>cam corporis <seg type="var">.b.e.d.q.</seg> non ſolum <choice><ex>non</ex><am>nõ</am></choice> <lb/>maior eſt diametro ipſius ſphæræ; <lb/></s> <s xml:space="preserve">imo minor, vt à te ipſo experiri po-<lb/>tes. </s> <s xml:space="preserve">Poſito igitur aliquo obiecto <lb/>opaco in loco <seg type="var">.K.o.g.</seg> eius ſuperficies <lb/>intercepta inter <seg type="var">.K.</seg> et <seg type="var">.g.</seg> adumbrata <lb/>erit, excepto puncto <seg type="var">.o</seg>. </s> <s xml:space="preserve">Poſito dein <lb/>de ipſo obiecto in loco <seg type="var">.n.y.x.m.</seg> <choice><ex>eius</ex><am>eiꝰ</am></choice> <lb/>partes <seg type="var">.y.n.</seg> et <seg type="var">.x.m.</seg> remanebunt lu- <pb facs="0359" n="347"/><fw type="head">EPISTOL AE.</fw> mine deſtitutæ <choice><ex>interuallumque</ex><am>interuallumq́;</am></choice> tantummodò inter <seg type="var">.y.x.</seg> illuminatum erit, ſed ſi in <lb/>loco <seg type="var">.c.u.</seg> poſitum fuerit, </s> <s xml:space="preserve">tunc totum <seg type="var">.c.u.</seg> illuminatum erit, ſed debili modo propter <lb/>detractionem factam à reflexione in ſuperficie corporis ſphærici, vt ſupra diximus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0358-01" corresp="fig-0358-01a"> <graphic url="0358-01"/> </figure> <figure xml:id="fig-0358-02" corresp="fig-0358-02a"> <graphic url="0358-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Poſito deinde obiecto in loco <seg type="var">.i.z.H.f.</seg> tunc partes <seg type="var">.z.i.</seg> et <seg type="var">.H.f.</seg> rectos Solis radios <lb/>habebunt cum aliquibus refractis, ſed <seg type="var">.z.H.</seg> pauciſſimum habebit lumen, pro-<lb/>pter diſgregationem radiorum. </s> <s xml:space="preserve">Poſito poſtea ipſo obiecto in loco <seg type="var">.t.l.r.s.</seg> tanto <lb/>minus lumen habebit pars <seg type="var">.l.r.</seg> propter dictam <choice><ex>diſgregationem</ex><am>diſgregationẽ</am></choice>, ſeu <choice><ex>diſſipationem</ex><am>diſſipationẽ</am></choice> radio <lb/>rum, & ſic ſucceſſiuè quanto remotius poſitum fuerit ipſum obiectum, tanto minus <lb/>illuminabitur. </s> <s xml:space="preserve">vnde ita remotum poterit locari, ut nullus actus luminis in eo <lb/>videatur, de radijs ſcilicet, qui per ſphæram chryſtallinam tranſibunt, ſed videbi-<lb/>tur vmbra ipſius ſphęrę in obiecto propoſito, cum nullum actum illuminationis in <lb/>eo loco obiecti habeant radij tranſeuntes per dictam ſphęram. </s> <s xml:space="preserve">quapropter partes <seg type="var">.<lb/>t.l.</seg> et <seg type="var">.r.s.</seg> illuminatæ erunt à Sole, et <seg type="var">.l.r.</seg> omnino lumine deſtituta.</s> </p> <p> <s xml:space="preserve">Quòd vero tolerabilior ſit oculis radius reflexus Solis à ſuperſicie aquæ, quàm <lb/>à ſuperficie alicuius ſpeculi, oritur ab eo, quod ſupra diximus, hoc eſt, quod ma-<lb/>gna parsipſius luminis penetrat in aquam, & non totum reflectit, quod quidem non <lb/>accidit ſpeculis opacis.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE LONGITVDINE DVORVM LATERVM <lb/>cuiuſuis trianguli ſupra tertium.</head> <head rend="italics" xml:space="preserve">Hieronymo Fenarolo.</head> <p> <s xml:space="preserve"><hi rend="small caps"><choice><sic>QVo'd</sic><corr>QVod</corr></choice></hi> quælibet duo latera continentia rectum angulum cuiuſuis triangu-<lb/>li orthogonij, longiora ſint tertio latere, per diametrum circuli in eo in-<lb/>ſcripti, ab alijs iam demonſtratum fuit. </s> <s xml:space="preserve">Sed quòd quælibet duo latera <lb/>cuiuſuis trianguli longiora ſint tertio per latus tetragonicum, quadrupli <lb/>producti cuiuſuis lineæ deſcendentis ab angulo contento à dictis duobus lateribus <lb/>ad oppoſitam partem circuli inſcripti, in partem extrinſecam ipſius lineæ, nullus <lb/>(quod ſciam) vnquam ſcripſit, vel animaduertit.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia triangulus <seg type="var">.a.b.c.</seg> quem volueris, in quo deſcribatur circulus <seg type="var">.<lb/>u.s.n.</seg> & puncta contingentiæ ſint eadem <seg type="var">.u.s.n.</seg> à puncto vero <seg type="var">.a.</seg> deſcendat linea <seg type="var">.a.<lb/>i.e.</seg> quæ terminetur à circunferentia in puncto <seg type="var">.e.</seg> ipſius circunferentiæ, vbi volue-<lb/>ris. </s> <s xml:space="preserve">Dico nunc latera <seg type="var">.a.b.</seg> et <seg type="var">.a.c.</seg> longiora eſſe latere <seg type="var">.b.c.</seg> per latus <choice><ex>tetragonicum</ex><am>tetragonicũ</am></choice> qua-<lb/>drupli producti ipſius <seg type="var">.a.e.</seg> in <seg type="var">.a.i</seg>. </s> <s xml:space="preserve">Nam certi ſamus ex vltima parte penultimæ ter-<lb/>tij Eucli <seg type="var">.n.c.</seg> et <seg type="var">.s.c.</seg> æquales inuicem eſſe, & ſimiliter <seg type="var">.b.s.</seg> et <seg type="var">.b.u.</seg> vnde ex communi <lb/>conceptu dicta latera maiora erunt <lb/> <ptr xml:id="fig-0359-01a" corresp="fig-0359-01" type="figureAnchor"/> ipſo <seg type="var">.b.c.</seg> per <seg type="var">.a.u.</seg> et <seg type="var">.a.n.</seg> quæ duæ <lb/>partes ſunt inuicem æquales di-<lb/>cta ratione, & quadratum lineæ <lb/>æqualis aggregato earum, eſſet qua <lb/>druplum quadrato cuiuſuis earum <lb/>ex .4. ſecundi, ſed ex penultima ter <lb/>tij, productum <seg type="var">.a.e.</seg> in <seg type="var">.a.i.</seg> æquale eſt <lb/>quadrato ipſius <seg type="var">.a.u.</seg> vel ipſius <seg type="var">.a.n</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0359-01" corresp="fig-0359-01a"> <graphic url="0359-01"/> </figure> </div> </body> </floatingText> <pb facs="0360" n="348"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Verum eſt igitur quod <seg type="var">.a.b.</seg> cum <seg type="var">.a.c.</seg> longiores ſint ipſa <seg type="var">.b.c.</seg> per latus terrago. <lb/>nicum quadrupli eius quod fit. ex <seg type="var">.a.e.</seg> in <seg type="var">.a.i.</seg> quod fuit propoſitum.</s> </p> <p> <s xml:space="preserve">Illud etiam non eſt ſpernendum, quod quotieſcunque data fuerint omnia latera <lb/>alicuius trianguli, illicò poſſumus cognoſcere puncta <seg type="var">.u.n.s.</seg> contingentiæ circuli in <lb/>ſcripti, ope vltimæ partis penultimæ tertij, eo quod ex illa iam ſcimus, quod de-<lb/>trahendo <seg type="var">.b.c.</seg> ex aggregato aliorum duorum laterum, remanebit <seg type="var">.u.a.</seg> et <seg type="var">.a.n.</seg> qua-<lb/>rum vnaquęque nota erit, cum illarum quælibet, medietas ſit reſidui cogniti, detra <lb/>hendo poſtea vnam <choice><ex>illarum</ex><am>illarũ</am></choice> ab altero <lb/> <ptr xml:id="fig-0360-01a" corresp="fig-0360-01" type="figureAnchor"/> duorum laterum <seg type="var">.a.b.</seg> vel <seg type="var">.a.c.</seg> rema <lb/>nebit <seg type="var">.u.b.</seg> vel <seg type="var">.c.n.</seg> ęqualis <seg type="var">.b.s.</seg> vel <seg type="var">.c.<lb/>s.</seg> vnde ſimiliter nobis innoteſcet <lb/>punctum <seg type="var">.s.</seg> cum duobus punctis <seg type="var">.u.</seg> <lb/>ct <seg type="var">.n.</seg> à quibus duobus punctis, ſi <lb/>duę perpendiculares ad talia latera <lb/>ductæ fuerint, vbi hæe perpendicu <lb/>lares ſeinuicem ſecabunt, ibi <choice><ex>cen- trum</ex><am>cen-trũ</am></choice> circuli inſcriptibilis erit in trian <lb/>gulo propoſito.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0360-01" corresp="fig-0360-01a"> <graphic url="0360-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Inter alia, quæ tibi dixi de Iride, quod memoria non tenes, nihil aliud eſt niſi <lb/>quod cum Iris videtur, non eodem loco ab omnibus videtur, quia reflexio eſt, & <lb/>vt reflexio luminis à ſpeculo non omnibus ab eodem puncto fit, ita etiam tibi dixi <lb/>de Iride.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De Inſtrumento oxygonio, ſeu elliptico.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod aliquando à me audiuiſti falſum non eſt, ſcilicet poſſibile eſſe (vt <lb/>ſpeculatus ſum) particulare inſtrumentum fabricari ad deſignandum oxy-<lb/>goniam, ſeu ellipticam ſectionem, quæ à Pergeo defectio appellatur, quod quidem <lb/>inſtrumentum valde diuerſum eſt ab alijs, quę aliàs inueni, pro ipſis conicis ſectio <lb/>nibus delineandis. </s> <s xml:space="preserve">Occaſionem <choice><ex>autem</ex><am>aũt</am></choice> huiuſimodi inſtrumenti inueniendi mihi præ <lb/>buit <choice><ex>ſecunda</ex><am>ſecũda</am></choice> dubij ſolutio <choice><ex>quam</ex><am>quã</am></choice> feci ann .1568. grauiſſ. philoſopho Franciſco Vimer <lb/>cato, <choice><ex>nam</ex><am>nã</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> viderim in ea figura <seg type="var">.f.a.</seg> ſemper <choice><ex>æqualem</ex><am>æqualẽ</am></choice> eſſe <seg type="var">.o.i.</seg> ſuæ parallelæ ſcilicet, <lb/>vnde cum recta linea fuerit protracta per <seg type="var">.o.</seg> et <seg type="var">.f.</seg> ipſa foret ſemper <choice><ex>ęquidiſtans</ex><am>ęquidiſtãs</am></choice> <seg type="var">.d.p.</seg> ex <lb/>33. primi Eucli. </s> <s xml:space="preserve">Venit mihi in mentem modus conſtruendi hoc ſubſcriptum inſtru-<lb/>mentum, tali ordine, videlicet, <choice><ex>coniungendo</ex><am>coniungẽdo</am></choice> ſeptem hic ſubnotatas lineas materia-<lb/>les <seg type="var">.z.r</seg>: <seg type="var">u.n</seg>: <seg type="var">e.h</seg>: <seg type="var">e.c</seg>: <seg type="var">c.l</seg>: <seg type="var">l.s.</seg> et <seg type="var">.s.e.</seg> ſimul, hoc modo, ſcilicet <choice><ex>ſabricando</ex><am>ſabricãdo</am></choice> quadrila-<lb/>terum æquilaterum <seg type="var">.c.e.s.l.</seg> hac conditione, quod immobili exiſtente puncto <seg type="var">.c.</seg> in li <lb/>nea <seg type="var">.z.r.</seg> reliqua omnia mobilia exiſtant, hoc eſt quod <choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.s.</seg> moueatur per di-<lb/>ctam lineam <seg type="var">.z.r.</seg> & immobili exiſtente puncto <seg type="var">.e.</seg> vt extremum lineæ <seg type="var">.e.h.</seg> hoc eſt <lb/>coniuncto extremo <seg type="var">.e.</seg> lineæ <seg type="var">.e.h.</seg> cum angulo <seg type="var">.c.e.s.</seg> reliqua puncta lineæ ipſius <seg type="var">.e.h.</seg> <lb/>moueantur per <seg type="var">.l.</seg> & per duas parallelas <seg type="var">.u.n.</seg> et <seg type="var">.z.r.</seg> longitudo vero <seg type="var">.e.h.</seg> ſit compo-<lb/>ſita ex duplo vnius lateris ipſius quadrilateris. </s> <s xml:space="preserve">Oportet deinde quod punctum <seg type="var">.f.</seg> <lb/>ſemper vnum, & idem ſit ipſius parallelæ <seg type="var">.u.n.</seg> moueatur tamen per <seg type="var">.e.h.</seg> quod qui-<lb/>dem punctum illud erit, quod vnam <choice><ex>portionem</ex><am>portionẽ</am></choice> circunferentiæ oxygoniæ ſectonis <pb facs="0361" n="349"/><fw type="head">EPISTOL AE.</fw> deſignabit, puncta verò <seg type="var">.o.</seg> et <seg type="var">.K.</seg> vt puncta laterum <seg type="var">.c.e.</seg> et <seg type="var">.s.e.</seg> æquædiſtantia à <lb/>punctis <seg type="var">.c.</seg> et <seg type="var">.s.</seg> <choice><ex>eadem</ex><am>eadẽ</am></choice> ſemper ſint, ita tamen vt puncta lineæ <seg type="var">.u.n.</seg> ſemper diuerſa exi <lb/><choice><ex>ſtant</ex><am>ſtãt</am></choice>, & quodlibet ipſius quadrilateri latus, æquale ſit medietati maioris axis ipſius <lb/>oxygoniæ ſectionis delineandæ, et <seg type="var">.c.o.</seg> ſeu <seg type="var">.s.K.</seg> (quod idem eſt) ſit æqualis medie <lb/>tati axis minoris dictæ ſectionis, et <seg type="var">.z.r.</seg> æqualis duplo <seg type="var">.e.h.</seg> vnde, quando puncta <seg type="var">.e.</seg> et <seg type="var">.<lb/>l.</seg> coniuncta ſimul erunt, ſimiliter coniunctæ ſimul erunt <seg type="var">.c.e.</seg> et <seg type="var">.e.s.</seg> cum <seg type="var">.c.l.</seg> et <seg type="var">.l.s.</seg> <ptr xml:id="fig-0361-01a" corresp="fig-0361-01" type="figureAnchor"/> <ptr xml:id="fig-0361-02a" corresp="fig-0361-02" type="figureAnchor"/> </s> <pb facs="0362" n="350"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve">Quapropter puncta <seg type="var">.e.l.f.</seg> et <seg type="var">.p.</seg> extremum axis maioris, in eodem met loco erunt, <lb/>hoc eſt in aliquo extremorum maioris axis, & cum punct is <seg type="var">.s.</seg> coniunct is fuerit cum <lb/>centro <seg type="var">.c.</seg> punctus <seg type="var">.f.</seg> parallelę <seg type="var">.u.p.</seg> in extremo axis minoris erit, & in eodem loco erit <lb/>cum <seg type="var">.o.</seg> & cum <seg type="var">.K</seg>. </s> <s xml:space="preserve">In extremitatibus verò lineæ <seg type="var">.z.r.</seg> neceſſe eſt, vt ſint duo <choice><ex>puncta</ex><am>pũcta</am></choice> fer <lb/>rea, ad firmandum ipſam <seg type="var">.z.r.</seg> ſuper ſubiectam lineam ſignificantem maiorem axem <lb/>propoſitę ſectionis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0361-01" corresp="fig-0361-01a"> <graphic url="0361-01"/> </figure> <figure xml:id="fig-0361-02" corresp="fig-0361-02a"> <graphic url="0361-02"/> <head rend="italics" xml:space="preserve">Instrumentum <lb/>oxigonium</head> </figure> </div> </body> </floatingText> <figure place="here"> <graphic url="0362-01"/> </figure> <figure place="here"> <graphic url="0362-02"/> </figure> <pb facs="0363" n="351"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Volo etiam quod ad partem <seg type="var">.c.l.s.</seg> quadrilateri conſtituta ſit alia parallela ad <seg type="var">.z.<lb/>r.</seg> & in æquali diſtantia ab ipſa quemadmodum <seg type="var">.u.n.</seg> diſtat ad eademmet <seg type="var">.z.r.</seg> ad ean <lb/>dem operationem faciendam. </s> <s xml:space="preserve">Vnde in vno tantummodo itinere puncti <seg type="var">.s.</seg> ab <seg type="var">.r.</seg> <choice><ex>vſque</ex><am>vſq;</am></choice> <lb/>ad <seg type="var">.c.</seg> deſignabimus quartam partem ſectionis, conuerſo poſtea inſtrumento, hoc eſt <lb/>poſito puncto <seg type="var">.r.</seg> vbi prius erat <seg type="var">.z.</seg> et <seg type="var">.z.</seg> vbi erat <seg type="var">.r.</seg> aliam delineabimus quartam, & <lb/>ſic ad oppoſitam partem ipſius <seg type="var">.z.r.</seg> faciendum erit. </s> <s xml:space="preserve">Hoc inſtrumentum poſſumus <lb/>etiam ita conſtruere, vt puncta <seg type="var">.o.</seg> et <seg type="var">.K.</seg> poſſint collocari in laterihus <seg type="var">.c.e.</seg> et <seg type="var">.e.s.</seg> vbi no <lb/>bis magis libuerit, ita vt licebit in qualibet proportione <choice><ex>axium</ex><am>axiũ</am></choice> propoſita, oxygoniam <lb/>deſignare. </s> <s xml:space="preserve">Nam <seg type="var">.c.o.</seg> erit longitudo dimidij axis minoris, et <seg type="var">.c.e.</seg> dimidij maioris.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE CONSTITVTIONE TRIANGVLI <lb/>orthogonij conditionati.</head> <head rend="italics" xml:space="preserve">Domino Ludouico de Rocchaforte.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> à me poſtulas, non eſt admodum difficile, cupis enim triangulum <lb/>orthogonium, exempli gratia <seg type="var">.o.i.e.</seg> in figura <seg type="var">.A.</seg> ita conſtituere, vt di-<lb/>uiſum ſit à perpendiculari <seg type="var">.a.i.</seg> & quod proportio <seg type="var">.o.e.</seg> ad <seg type="var">.o.i.</seg> ſit vt <seg type="var">.o.i.</seg> ad <lb/><seg type="var">i.e.</seg> & quod quadrati <seg type="var">.o.i.</seg> ad quadratum <seg type="var">.o.a.</seg> ſit vt <seg type="var">.e.i.</seg> ad <seg type="var">.e.a.</seg> & quadra <lb/>tum <seg type="var">.o.i.</seg> ad quadratum <seg type="var">.e.i.</seg> ſit .ut <seg type="var">.o.a.</seg> ad <seg type="var">.e.a</seg>. </s> <s xml:space="preserve">Quæ omnia in promptu veniunt, quo <lb/>tieſcunque <seg type="var">.o.e.</seg> fuerit diameter alicuius circuli, <choice><ex>diuiſaque</ex><am>diuiſaq́;</am></choice> in puncto <seg type="var">.a.</seg> ſecundum pro <lb/>portionem habentem medium <choice><ex>duoque</ex><am>duoq́;</am></choice> extrema, protracta deinde perpendiculari <seg type="var">.a.<lb/>i.</seg> ad <seg type="var">o.e.</seg> uſque ad circunferentiam, <choice><ex>coniunctæque</ex><am>coniunctæq́;</am></choice> <seg type="var">.o.i.</seg> et <seg type="var">.i.e</seg>: tale triangulum, omnia <lb/>ſupradicta in ſe continebit.</s> </p> <p> <s xml:space="preserve">Nam ex .30. tertij angulus <seg type="var">.i.</seg> rectus erit, & ex .8. ſexti <seg type="var">.o.i.</seg> erit media proportio-<lb/>nalis inter <seg type="var">.o.e.</seg> et <seg type="var">.o.a.</seg> et <seg type="var">.e.i.</seg> inter <seg type="var">.o.e.</seg> <lb/> <ptr xml:id="fig-0363-01a" corresp="fig-0363-01" type="figureAnchor"/> et <seg type="var">.a.e.</seg> ſed quia ex diuiſione facta in <choice><ex>pum</ex><am>pũ</am></choice> <lb/>cto <seg type="var">.a.</seg> etiam <seg type="var">.o.a.</seg> erit media proportio-<lb/>nalis inter totum & reſiduum, ideo ex <num value="11">.<lb/>11.</num> quinti ita erit <seg type="var">.o.e.</seg> ad <seg type="var">.e.i.</seg> vt <seg type="var">.o.e.</seg> ad <seg type="var">.<lb/>o.a.</seg> vnde ex .9. eiuſdem <seg type="var">.a.o.</seg> erit æqua-<lb/>lis <seg type="var">.e.i.</seg> & ideo <seg type="var">.o.i.</seg> erit media proportio <lb/>nalis inter <seg type="var">.o.e.</seg> et <seg type="var">.e.i</seg>. </s> <s xml:space="preserve">Sed quia propor-<lb/>tio <seg type="var">.e.i.</seg> ad <seg type="var">.a.e.</seg> <choice><ex>eadem</ex><am>eadẽ</am></choice> eſt, quę ipſius <seg type="var">.o.e.</seg> ad <lb/><seg type="var">o.a</seg>. </s> <s xml:space="preserve">tunc videbis ex .18. ſexti, quod pro <lb/>portio quadrati <seg type="var">.o.i.</seg> ad quadratum <seg type="var">.o.a.</seg> <lb/>erit vt <seg type="var">.e.i.</seg> ad <seg type="var">.e.a.</seg> cum vero duo trian-<lb/>guli <seg type="var">.o.i.a.</seg> et <seg type="var">.a.i.e.</seg> ſint inuicem ſimiles <lb/>ex ſupradicta .8. ſexti, </s> <s xml:space="preserve">tunc videbis ex <lb/>18. et .17. eiuſdem dictos <choice><ex>triangulos</ex><am>triãgulos</am></choice> ean <lb/>dem habere inter ſe proportionem, quę <lb/>eſt inrer quadrata ipſius <seg type="var">.o.i.</seg> et <seg type="var">.i.e.</seg> vnde <lb/>ex prima ſexti ita ſe inuicem habebunt <seg type="var">.<lb/>a.o.</seg> et <seg type="var">.a.e</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0363-01" corresp="fig-0363-01a"> <graphic url="0363-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Circa eam verò difficultatem quam <pb facs="0364" n="352"/><fw type="head">I O. BAPT. BENED.</fw> habes in circulo .ω vbi fateris te non videre qua ratione eadem proportio ſit qua-<lb/>drati <seg type="var">.u.o.</seg> ad quadratum <seg type="var">.o.n.</seg> vt lineæ <seg type="var">.o.a.</seg> ad lineam <seg type="var">.o.e.</seg> partes diametri <seg type="var">.o.i.</seg> ipſius <lb/>circuli, terminatæ à perpendicularibus <seg type="var">.u.a.</seg> et <seg type="var">.n.e</seg>.</s> </p> <p> <s xml:space="preserve">Hoc neceſſario contingit, propterea quod cum fuerint protractæ <seg type="var">.u.i.</seg> et <seg type="var">.n.i.</seg> <choice><ex>tunc</ex><am>tũc</am></choice> <lb/>habebimus ad partem <seg type="var">.o.u.i.</seg> triangulum <seg type="var">.o.u.i.</seg> diuiſum in duo triangula ſimilia ipſi <lb/>totali triangulo. </s> <s xml:space="preserve">Idem etiam dico ad partem <seg type="var">.o.n.i.</seg> vnde ex tali ſimilitudine habe-<lb/>bimus <seg type="var">.o.u.</seg> mediam proportionalem inter <seg type="var">.o.i.</seg> et <seg type="var">.o.a.</seg> et ſic <seg type="var">.o.n.</seg> erit media proportio <lb/>nalis inter <seg type="var">.o.i.</seg> et <seg type="var">.o.e.</seg> </s> <s xml:space="preserve">quare ex .16. ſexti, quadratum <seg type="var">.o.u.</seg> æquale erit producto ipſius <lb/><seg type="var">o.i.</seg> in <seg type="var">.o.a.</seg> & quadratum <seg type="var">.o.n.</seg> æquale producto <seg type="var">.o.i.</seg> in <seg type="var">.o.e.</seg> ſed ex prima eiuſdem, ea <lb/>dem proportio eſt ipſius <seg type="var">.o.a.</seg> ad <seg type="var">.o.e.</seg> quæ producti ipſius <seg type="var">.o.i.</seg> in <seg type="var">o.a.</seg> ad productum <seg type="var">.o.<lb/>i.</seg> in <seg type="var">.o.e.</seg> </s> <s xml:space="preserve">quare, ex <choice><ex>communi</ex><am>cõmuni</am></choice> conceptu, ita erit quadrati <seg type="var">.o.u.</seg> ad quadratúm.o.n. </s> <s xml:space="preserve">Et hęc <lb/>eſt alia circuli paſſio.</s> </p> <p> <s xml:space="preserve">Reliqua verò difficultas quam te habere ſcribis, eſt, quare cum duæ lineę <lb/><seg type="var">a.u.</seg> et <seg type="var">.b.s.i.</seg> ſint inuicem ęquales, diuiſæ verò non æquali modo, ſed tali, quod <seg type="var">.a.</seg> <lb/>maior ſit quam <seg type="var">.u.</seg> et <seg type="var">.b.s.</seg> maior quam <seg type="var">.i.</seg> quomodo poteſt fieri, quod ſi <seg type="var">.u.</seg> maior fue-<lb/>rit <seg type="var">.i.</seg> proportio <seg type="var">.a.</seg> ad <seg type="var">.i.</seg> maior ſit quam ipſius <seg type="var">.b.s.</seg> ad <seg type="var">.u</seg>.</s> </p> <p> <s xml:space="preserve">Hoc etiam ex neceſſitate cuenit, eo <lb/>quod ſi accepta fuerit <seg type="var">.t.n.</seg> æqualis <seg type="var">.u.</seg> ab <lb/> <ptr xml:id="fig-0364-01a" corresp="fig-0364-01" type="figureAnchor"/> <choice><ex>ipſaque</ex><am>ipſaq́;</am></choice> abſciſa fuerit <seg type="var">.t.</seg> æqualis <seg type="var">.i.</seg> & ab <seg type="var">.<lb/>b.s.</seg> abſciſa <seg type="var">.s.</seg> æqualis <seg type="var">.n.</seg> habebimus <seg type="var">.a.</seg> et <lb/>b. inuicem æ quales, vnde habebis ma-<lb/>iorem propor tionem ipſius <seg type="var">.b.</seg> ad <seg type="var">.t.</seg> <choice><ex>quam</ex><am>quã</am></choice> <lb/>s. ad <seg type="var">.n.</seg> quod cum clarum per ſe ſit, tibi <lb/>relinquo. </s> <s xml:space="preserve">ſed ex .27. quinti, proportio <lb/>b. ad. s, maior erit quam <seg type="var">.t.</seg> ad <seg type="var">.n.</seg> & ex <lb/>28. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> <choice><ex>proportio</ex><am>ꝓportio</am></choice> <seg type="var">.b.s.</seg> ad <seg type="var">.s.</seg> maior erit, <lb/>quam <seg type="var">.t.n.</seg> ad <seg type="var">.n.</seg> & ex .27. maior propor <lb/>tio erit ipſius <seg type="var">.b.s.</seg> ad <seg type="var">.n.t.</seg> quam <seg type="var">.s.</seg> ad <seg type="var">.n.</seg> <lb/>ergo ex .33. maior erit ipſius <seg type="var">.b.</seg> ad <seg type="var">.t.</seg> <choice><ex>quam</ex><am>quã</am></choice> <lb/><seg type="var">b.s.</seg> ad <seg type="var">.n.t.</seg> hoc eſt maior ipſius <seg type="var">.a.</seg> ad <lb/>i. quam <seg type="var">.b.s.</seg> ad <seg type="var">.u.</seg> quod eſt propo-<lb/>ſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0364-01" corresp="fig-0364-01a"> <graphic url="0364-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Id verò de quo me interrogas <choice><ex>nempe</ex><am>nẽpe</am></choice> de <lb/>diſtinctione orbium cęleſtium, ortum <lb/>habet à communi opinione motuum <lb/>fixarum. </s> <s xml:space="preserve">Nam cum putauerint philo-<lb/>ſophi ipſas moueri, ſemper eandem <choice><ex>ſeruando</ex><am>ſeruãdo</am></choice> inuicem diſtantiam, non ſine ratione <lb/>crediderunt eas fixas eſſe eodem in orbe, idem etiam poſtea de planetis opinaue-<lb/>runt. </s> <s xml:space="preserve">Hoc eſt, vnumquemque, aliquo in orbe, fixo exiſtere.</s> </p> <pb facs="0365" n="353"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE MODO DIVIDENDI PARABOLAM <lb/>propoſitam ſecundum datam proportionem.</head> <head rend="italics" xml:space="preserve">Pamphilo Gothfrid.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> à me quæris, eſt quidem poſſibile, non tamen adhuc inuentum, quo <lb/>niam nemo ad <choice><ex>hunc</ex><am>hũc</am></choice> vſque diem diuiſit vnam datam proportionem in tres <lb/>æquales partes, ſed ſi hoc pro facto conceſſeris, nunc tibi morem geram. <lb/></s> <s xml:space="preserve">Nam proponis n. ihi parabolem <seg type="var">.x.b.e.</seg> cum proportione <seg type="var">.p.</seg> ad <seg type="var">.q.</seg> <choice><ex>cupiſque</ex><am>cupiſq́;</am></choice> <lb/>ſcire modum diuidendi ipſam parabolem vna mediante linea parallela ipſi baſi, ita <lb/>vt eandem habeat proportionem tota parabola ad partem abſciſſam, quæ eſt inter <seg type="var">.<lb/>p.</seg> et <seg type="var">.q</seg>. </s> <s xml:space="preserve">Ad quod faciendum, ſupponendum primò datam proportionem inter <seg type="var">.<lb/>p.</seg> et <seg type="var">.q.</seg> diuiſam eſſe in tres partes æquales, duabus lineis mediantibus <seg type="var">.n.</seg> et <seg type="var">.u.</seg> quæ me <lb/>diæ proportionales vocabuntur inter <seg type="var">.p.</seg> et <seg type="var">.q</seg>. </s> <s xml:space="preserve">deinde à quouis puncto circunferentię <lb/>ipſius figuræ ducatur parallela baſi <seg type="var">.x.e.</seg> poſtea verò per puncta media harum dua-<lb/>rum <choice><ex>æquidiſtantium</ex><am>æquidiſtantiũ</am></choice> protrahatur <seg type="var">.g.b.</seg> quæ diameter erit ſectionis, ex 28. ſecundi Per-<lb/>gei, </s> <s xml:space="preserve">diuidatur deinde hæc diameter in puncto <seg type="var">.a.</seg> ita quod eadem proportio ſit ipſius <lb/><seg type="var">b.g.</seg> ad <seg type="var">.b.a.</seg> quæ ipſius <seg type="var">.p.</seg> ad <seg type="var">.u.</seg> quod tibi facile erit, ſecando à linea <seg type="var">.p.</seg> partem <seg type="var">.i.</seg> æqua <lb/>lem ipſi <seg type="var">.u.</seg> tali modo poſtea diuidendo <seg type="var">.b.g.</seg> ex .12. ſexti, ducatur a puncto <seg type="var">.a.</seg> ipſa <seg type="var">.d.<lb/>h.</seg> parallclam ipſi <seg type="var">.x.e.</seg> & habebitur propoſitum.</s> </p> <p> <s xml:space="preserve">Pro cuius reiratione, ſcies primum quod <seg type="var">.h.d.</seg> diuiſa erit à diametro <seg type="var">.b.g.</seg> per æqua <lb/>lia ex .7. primi Pergei, vel ſi cogitabimus aliquam lineam tangentem ipſam parabo <lb/>lam in puncto <seg type="var">.b</seg>. </s> <s xml:space="preserve">tunc ex quinta ſecundi ipſius Pergei habebimus ipſam eſſe paralle-<lb/>lam <seg type="var">.e.x.</seg> & ex .30. primi Eucli. erit ſimiliter æquidiſtans <seg type="var">.d.h.</seg> vnde ex .46. primi eiuſ-<lb/>dem Pergei <seg type="var">.h.a.</seg> æqualis erit <seg type="var">.d.a</seg>. </s> <s xml:space="preserve">Protrahatur deinde <seg type="var">.e.b</seg>: d b: <seg type="var">x.b.</seg> et <seg type="var">.h.b.</seg> vnde ex .17 <lb/>lib. de quadratura parabolæ Archimedis, habebimus eandem proportionem ſuper <lb/>ficiei totalis parabolæ <seg type="var">.x.b.e.</seg> ad trigonum <seg type="var">.x.b.e.</seg> quæ portionis <seg type="var">.h.b.d.</seg> ad ſuum <choice><ex>tri- gonum</ex><am>tri-gonũ</am></choice>, eo quod <choice><ex>tam</ex><am>tã</am></choice> vna quàm alia erit ſeſquitertia, <choice><ex>eius</ex><am>eiꝰ</am></choice> <choice><ex>etiam</ex><am>etiã</am></choice> medietates ſic ſe <choice><ex>habebunt</ex><am>habebũt</am></choice>.</s> </p> <p> <s xml:space="preserve">Vnde permutando, proportio medietatis totalis parabolę ad medietatem partia <lb/>lem ipſius, æqualis erit proportioni trianguli <lb/><seg type="var">g.b.e.</seg> ad triangulum <seg type="var">.a.b.d.</seg> ſed ex .20. primi <lb/>Pergei, eadem eſt proportio quadrati ipſius <seg type="var">.<lb/> <ptr xml:id="fig-0365-01a" corresp="fig-0365-01" type="figureAnchor"/> g.e.</seg> ad quadratum ipſius <seg type="var">.a.d.</seg> quæ <seg type="var">.b.g.</seg> ad <seg type="var">.b.a.</seg> <lb/>hoc eſt, vt <seg type="var">.g.e.</seg> ad <seg type="var">.a.o.</seg> ex ſimilitudine triangu-<lb/>lorum, & quia <seg type="var">.b.g.</seg> ad <seg type="var">.b.a.</seg> eſt ſicut <seg type="var">.p.</seg> ad <seg type="var">.u.</seg> ita <lb/>igitur erit quadrati ipſius <seg type="var">.g.e.</seg> ad quadratum <lb/>ipſeus <seg type="var">.a.d.</seg> </s> <s xml:space="preserve">quare <seg type="var">.g.e.</seg> ad <seg type="var">.a.d.</seg> erit ut p. ad <seg type="var">.n.</seg> <lb/>ex .18. ſexti Euclid. </s> <s xml:space="preserve">ſed cum ex .24. eiuſdem <lb/>proportio trianguli <seg type="var">.b.g.e.</seg> ad triangulum <seg type="var">.b.<lb/>a.d.</seg> compoſita ſit ex proportione <seg type="var">.g.e.</seg> ad <seg type="var">.a.<lb/>d.</seg> er. ex <seg type="var">.g.b.</seg> ad <seg type="var">.b.a.</seg> hoc eſt <seg type="var">.g.e.</seg> ad <seg type="var">.a.o.</seg> & <lb/>quia <choice><ex>proportio</ex><am>ꝓportio</am></choice> <seg type="var">.g.e.</seg> ad <seg type="var">.a.o.</seg> æqualis eſt ei quæ <seg type="var">.p.</seg> <lb/>ad. u ex .11. quinti Euclid. </s> <s xml:space="preserve">& proportio <seg type="var">.g.e.</seg> <lb/>ad <seg type="var">.a.d.</seg> æqualis eſt ei quæ <seg type="var">.p.</seg> ad <seg type="var">.n.</seg> hoc eſt vt <seg type="var">.u.</seg> <lb/>ad <seg type="var">.q.</seg> ergo proportio trianguli <seg type="var">.b.g.e.</seg> ad trian-<lb/>gulum <seg type="var">.b.a.d.</seg> compoſita erit ex ca quę <seg type="var">.p.</seg> ad <seg type="var">.u.</seg> <lb/>& ex ea quæ <seg type="var">.u.</seg> ad <seg type="var">.q.</seg> æqualis ergo erit ei, quæ <lb/>p. ad <seg type="var">.q.</seg> & ita medietates parabolarum, & eorum dupla.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0365-01" corresp="fig-0365-01a"> <graphic url="0365-01"/> </figure> </div> </body> </floatingText> <pb facs="0366" n="354"/> <fw type="head">IO. BABPT. BENED.</fw> </div> <div type="unknown"> <head rend="small caps" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:space="preserve">Proportio maioris portionis ad minorem ſemper erit ſeſquialtera proportioni <lb/>ipſius <seg type="var">.b.g.</seg> ad <seg type="var">.a.b.</seg> eo quod cum ſit proportio totalis portionis ad partialem vt trian-<lb/>guli <seg type="var">.b.g.e.</seg> ad <seg type="var">.b.a.d.</seg> & hæc ſeſquialtera proportioni ipſius <seg type="var">.g.e.</seg> ad <seg type="var">.a.o.</seg> hoc eſt vt ip-<lb/>ſius <seg type="var">.b.g.</seg> ad <seg type="var">.b.a.</seg> ideo proportio ipſarum portionum erit ſimiliter ſeſquialtera pro-<lb/>portioni diametrorum.</s> </p> <p> <s xml:space="preserve">Deinde ſi protractæ fuerint <seg type="var">.b.d.</seg> et <seg type="var">.g.e.</seg> quouſque conueniant in puncto <seg type="var">.z.</seg> habe <lb/>bis inter <seg type="var">.g.z.</seg> et <seg type="var">.a.o.</seg> duas <seg type="var">.g.e.</seg> et <seg type="var">.a.d.</seg> medias proportionales in proportionalitate con <lb/>tinua, eo quod cum (ex ijs quæ ſupra diximus.). <seg type="var">a.d.</seg> media proportionalis ſit inter <seg type="var">.<lb/>g.e.</seg> et <seg type="var">.a.o.</seg> & proportio <seg type="var">.g.z.</seg> ad <seg type="var">.g.e.</seg> vt ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.a.o.</seg> eo quodipſius <seg type="var">.g.z.</seg> ad <seg type="var">.a.d.</seg> <lb/>& ipſius <seg type="var">.g.e.</seg> ad <seg type="var">.a.o.</seg> eſt vt ipſius <seg type="var">.b.g.</seg> ad <seg type="var">.b.a.</seg> ex ſimilitudine triangulorum, ideo di-<lb/>ctæ <choice><ex>proportiones</ex><am>ꝓportiones</am></choice> erunt <choice><ex>inuicem</ex><am>inuicẽ</am></choice> æquales. </s> <s xml:space="preserve">Vnde permutatim ita erit ipſius <seg type="var">.g.z.</seg> ad <seg type="var">.g.e.</seg> <lb/>vt ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.a.o.</seg> & ut ipſius <seg type="var">.g.e.</seg> ad <seg type="var">.a.d</seg>.</s> </p> <p> <s xml:space="preserve">Amplius etiam dico, quod proportio pa <lb/> <ptr xml:id="fig-0366-01a" corresp="fig-0366-01" type="figureAnchor"/> rabolæ totalis ad partialem, eadem eſt, quę <lb/>cubi ipſius <seg type="var">.g.e.</seg> ad cubum ipſius <seg type="var">.a.d.</seg> & ex <choice><ex>con</ex><am>cõ</am></choice> <lb/>ſequenti, vt cuborum earundem baſium, eo <lb/>quod cum ſit, ex .36. vndecimi Euclid. pro-<lb/>portio cubi ipſius <seg type="var">.g.e.</seg> ad cubum ipſius <seg type="var">.a.d.</seg> <lb/>tripla ei quæ ipſius <seg type="var">.g.e.</seg> ad <seg type="var">.a.d.</seg> ideo æqualis <lb/>erit ei quę trianguli <seg type="var">.b.g.e.</seg> ad triangulum <seg type="var">.b.<lb/>a.d.</seg> cum proportio horum duorum triangu <lb/>lorum compoſita ſit (vt ſupra vidimus) ex <lb/>ea quæ <seg type="var">.g.e.</seg> ad <seg type="var">.a.o.</seg> & ex ea quæ <seg type="var">.g.e.</seg> ad <seg type="var">.a.d.</seg> <lb/>& hæc medietas illius, ſed trianguli ita ſe in <lb/>uicem habenr, vt parabolę, </s> <s xml:space="preserve">quare ipſæ para-<lb/>bolæ ſeinuicem habebunt, vt cubi ipſarum <lb/>baſium.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0366-01" corresp="fig-0366-01a"> <graphic url="0366-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Cubum fabricare æqualem pyramidi propoſitæ.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">CVbum fabricare æqualem propoſitæ pyramidi quadrilateræ, nullius erit diffi-<lb/>cultatis, ſuppoſita tamen pro reperta diuiſione cuiuſuis datæ proportionis in <lb/>tres partes æquales. </s> <s xml:space="preserve">Nam ex .6. duodecimi Eucli. patet omne corpus ſerratile d-ui <lb/>ſibile eſſe in tres pyramides quadrilateras æquales, ſcimus etiam quod cuilibet py-<lb/>ramidi quadrilateræ poteſt reperiri ſuum ſerratile. </s> <s xml:space="preserve">Sit igitur propoſita pyramis qua <lb/>drilatera <seg type="var">.m.g.f.h.</seg> cuius ſerratile ita inueniemus, ducendo primum <seg type="var">.h.i.</seg> parallelam <lb/>ipſi <seg type="var">.g.f.</seg> et <seg type="var">.f.i.</seg> ipſi <seg type="var">.g.h.</seg> in ſuperficie trianguli <seg type="var">.f.g.h.</seg> et <seg type="var">.m.K.</seg> ipſi <seg type="var">.g.h.</seg> in ſuperficie <lb/>trianguli <seg type="var">.m.g.h.</seg> & æqualem dictæ <seg type="var">.g.h.</seg> ducetur poſtea <seg type="var">.K.h.</seg> et <seg type="var">.K.i.</seg> & habebimus cor <lb/>pus <seg type="var">.f.K.g.</seg> ſerratile, & triplum pyramidi propoſitæ. </s> <s xml:space="preserve">Nunc duplicemus ipſum, du-<lb/>cendo <seg type="var">.K.x.</seg> in ſuperficie trianguli <seg type="var">.i.k.h.</seg> parallelam, <choice><ex>æqualemque</ex><am>æqualemq́;</am></choice> ipſi <seg type="var">.i.h.</seg> et <seg type="var">.m.y.</seg> <lb/>in ſuperficie trianguli <seg type="var">.f.m.g.</seg> parallelam, ę<choice><ex>qualemque</ex><am>qualemq́;</am></choice> ipſi <seg type="var">.f.g.</seg> ducatur poſtea <seg type="var">.g.y.</seg> et <seg type="var">.h.<lb/>x.</seg> quarum <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> æqualis erit ipſi <seg type="var">.f.m.</seg> vnde habebimus corpus <seg type="var">.f.x.</seg> parallelepe-<lb/>pidum, & ſexcuplum ipſi pyramidi propoſitæ.</s> </p> <pb facs="0367" n="355"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Inueniatur nunc quadratum <seg type="var">.u.n.</seg> æquale ſextæ parti ſuperficiei <seg type="var">.f.i.g.h.</seg> quod per <lb/>ſe facile erit, </s> <s xml:space="preserve">deinde accipiatur altitudo corporis <seg type="var">.f.x.</seg> ducendo vnam perpendicula <lb/>rem à puncto <seg type="var">.m.</seg> ad baſim <seg type="var">.f.g.h.</seg> quę ſit <seg type="var">.n.e.</seg> qua mediante, cum quadrato <seg type="var">.u.n.</seg> fabri <lb/>cetur ſolidum parallelepepidum <seg type="var">.u.e.</seg> quod erit æquale dictæ pyramidi ex .33. vnde-<lb/>cimi Euclid.</s> </p> <p> <s xml:space="preserve">Repertæ nunc ſint duæ mediæ proportionales <seg type="var">.r.s.</seg> inter <seg type="var">.n.e.</seg> et <seg type="var">.n.p.</seg> quarum <seg type="var">.s.</seg> ſit <lb/>proximior ipſi <seg type="var">.u.p.</seg> ex qua <seg type="var">.s.</seg> ſi conſtitutus fuerit cubus, habebimus propoſitum.</s> </p> <p> <s xml:space="preserve">Pro cuius rei ratione, cogitemus corpus <seg type="var">.u.e.</seg> productum eſſe vſque ad <seg type="var">.a.o.</seg> per lon-<lb/>gitudem <seg type="var">.s.</seg> latus dicti cubi, qui quidem cubus ſit <seg type="var">.d.b.</seg> vnde proportio corporis <seg type="var">.u.e.</seg> <lb/>ad corpus <seg type="var">.e.o.</seg> erit, vt ſuperficiei <seg type="var">.p.e.</seg> ad ſuperficiem <seg type="var">.t.e.</seg> ex .33. undecimi, ipſæ verò <lb/>ſuperficies ſibi inuicem erunt vt <seg type="var">.n.e.</seg> ad <seg type="var">.e.a.</seg> ex prima ſexti, </s> <s xml:space="preserve">quare proportio corpo <lb/>ris <seg type="var">.u.e.</seg> ad corpus <seg type="var">.e.o.</seg> dupla erit proportioni ipſius <seg type="var">.s.</seg> ad <seg type="var">.n.p.</seg> ſed cum ex .33 vndeci-<lb/>mi, proportio cubi <seg type="var">.d.b.</seg> ad corpus <seg type="var">.e.o.</seg> ſit vt <choice><ex>quadratum</ex><am>quadratũ</am></choice> <seg type="var">.q.b.</seg> ad quadratum <seg type="var">.o.a.</seg> & cum <lb/>proportio <seg type="var">.q.b.</seg> ad <seg type="var">.o.a.</seg> dupla ſit ei quæ <seg type="var">.q.o.</seg> ad <seg type="var">.o.t.</seg> ex .18. ſexti, erit igitur proportio <lb/>cubi <seg type="var">.d.b.</seg> ad corpus <seg type="var">.e.o.</seg> dupla ei quæ <seg type="var">.q.o.</seg> ad <seg type="var">.o.t.</seg> hoc eſt ei quæ <seg type="var">.s.</seg> ad <seg type="var">.n.p.</seg> ſed ita erat <lb/>corporis <seg type="var">.u.e.</seg> ad corpus <seg type="var">.e.o.</seg> </s> <s xml:space="preserve">quare ex .9. quinti, cubus <seg type="var">.d.b.</seg> æqualis erit corpor<unclear reason="illegible"/>i.u.e. <lb/>hoc eſt pyramidi propoſitæ.</s> </p> <p> <s xml:space="preserve">Sed ſi oportebit cubum maiorem vel minorem ipſa pyramide reperire, in qua <lb/>proportione tibi placuerit, </s> <s xml:space="preserve">tunc opus erit aliud quadratum inuenire, quod in ea <lb/>proportione ſe habeat ad quadratum <seg type="var">.u.n.</seg> quam volueris, quo mediante ſimul cum <lb/>altitudine pyramidis conſequemur propoſitum.</s> </p> <p> <s xml:space="preserve">Aduertendum tamen quod fabri-<lb/>care ipſum corpus ſerratile <seg type="var">.k.f.h.</seg> & ſo <lb/> <ptr xml:id="fig-0367-01a" corresp="fig-0367-01" type="figureAnchor"/> lidum <seg type="var">.f.x.</seg> neceſſarium non eſt, niſi pro <lb/>demonſtratione. </s> <s xml:space="preserve"><choice><ex>idemque</ex><am>idemq́;</am></choice> dico de alijs <lb/>ſolidis, nam pro ſimplici operatione <lb/>huiuſmodi problematis, abſque ali-<lb/>qua re neceſſaria ad ſpeculandum, ita <lb/>faciendum erit.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0367-01" corresp="fig-0367-01a"> <graphic url="0367-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Data pyramide <seg type="var">.m.f.g.h.</seg> accipe <choice><ex>eius</ex><am>eiꝰ</am></choice> <lb/>alitudinem à <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.m.</seg> vſque ad ſuper <lb/>ficiem baſis <seg type="var">.f.g.h.</seg> quæ ſit <seg type="var">.n.e.</seg> accipe <lb/>deinde latus letragonicum quadrati <seg type="var">.<lb/> <ptr xml:id="fig-0367-02a" corresp="fig-0367-02" type="figureAnchor"/> u.n.</seg> æqualis tertiæ partis ipſius baſis <seg type="var">.f.<lb/>g.h.</seg> quod latus ſit <seg type="var">.n.p.</seg> inter quod, et <seg type="var">.<lb/>n.e.</seg> inuentæ cum fuerint duæ lineæ <lb/>mediæ proportiona es <seg type="var">.s.</seg> et <seg type="var">.r.</seg> <choice><ex>quarum</ex><am>quarũ</am></choice> <seg type="var">.<lb/>s.</seg> proximior ſit <seg type="var">.n.p.</seg> quæ <choice><ex>quidem</ex><am>quidẽ</am></choice> <seg type="var">.s.</seg> erit <lb/>latus cubi quæſiti.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0367-02" corresp="fig-0367-02a"> <graphic url="0367-02"/> </figure> </div> </body> </floatingText> <pb facs="0368" n="356"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Duplex modus par allelam orizontalem alicui muro propoſito <lb/>una tantummodo statione ducendi.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">DVcere parallelam orizontalem alicui muro recto propoſito vna tantummodò <lb/>ſtatione, non ſolum poſſibile eſt ſed etiam facile.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia murus rectus <seg type="var">.a.d.</seg> ſitus verò <seg type="var">.o.n</seg>. </s> <s xml:space="preserve">Si cupimus ducere <seg type="var">.n.u.</seg> <lb/>parallelam dicto muro, accipiatur quadratum geometricum, ſeu ſcala altimetra <lb/>vel aliquod ſimile inſtrumentum, quo mediante à ſitu <seg type="var">.o.</seg> videbimus punctum <seg type="var">.q.</seg> <lb/>quod volueris ipſius muri, <choice><ex>dexteram</ex><am>dexterã</am></choice> <lb/>verſus, inferius tamen. ipſo <seg type="var">.o.</seg> vnde <lb/> <ptr xml:id="fig-0368-01a" corresp="fig-0368-01" type="figureAnchor"/> formatum habebimus triangulum <seg type="var">.<lb/>n.o.q</seg>. </s> <s xml:space="preserve">Quo facto ad partem <choice><ex>ſiniſtram</ex><am>ſiniſtrã</am></choice> <lb/>cum eodem angulo <seg type="var">.n.o.q.</seg> oporte-<lb/>bit nos inuenire punctum aliquod <seg type="var">.<lb/>p.</seg> in dicta ſuperficie muri, </s> <s xml:space="preserve">& tunc <lb/>habebimus angulum <seg type="var">.n.o.p.</seg> æqua-<lb/>lem angulo <seg type="var">.n.o.q.</seg> vnde angulus <seg type="var">.q.<lb/>n.p.</seg> nobis cognitus erit, <choice><ex>duoque</ex><am>duoq́;</am></choice> late <lb/>ra <seg type="var">.n.q.</seg> et <seg type="var">.n.p.</seg> erunt inuicem æqua-<lb/>lia, ex .26. primi Euclid. cum angu-<lb/>li <seg type="var">.q.o.n.</seg> et <seg type="var">.q.n.o.</seg> ſint æquales angu <lb/>lis <seg type="var">.p.o.n.</seg> et <seg type="var">.p.n.o.</seg> & latus <seg type="var">.o.n.</seg> com <lb/>mune, vnde angulus <seg type="var">.q.n.g.</seg> extrinſe <lb/>cus trianguli <seg type="var">.p.q.n.</seg> <choice><ex>reſiduusque</ex><am>reſiduusq́;</am></choice> ex <lb/>duobus rectis nobis cognitus erit, <lb/>etiam & eius medictas <seg type="var">.q.n.u.</seg> æqua <lb/>lis angulo <seg type="var">.p.q.n.</seg> eo quod ex .5. pri-<lb/>mi, anguli <seg type="var">.q.p.</seg> ſunt inuicem æquales, & ex .32. eiuſdem, æquales ſunt extrinſeco <seg type="var">.q.n.<lb/>g.</seg> & ex 27. <seg type="var">n.u.</seg> erit parallela ipſi <seg type="var">.q.p</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0368-01" corresp="fig-0368-01a"> <graphic url="0368-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Aliter etiam poſſumus idem efficere, ſumendo duo illa puncta in ſuprem a linea <lb/>orizontali ipſius muri ad ſuperiorem partem aſpiciendo, quemadmodum ad infe-<lb/>riorem, quod vnum & idem erit, dummodò non aſpiciamus orizontaliter, eo quod <lb/>nos oportet ſuperficiem conicam producere, linea viſuali mediante. </s> <s xml:space="preserve">cognoſcere au<lb/>tem angulum <seg type="var">.q.n.p.</seg> facile erit, conſtituendo primò inſtrumentum in ſitu trianguli <seg type="var">.<lb/>o.n.q.</seg> <choice><ex>aſpiciendoque</ex><am>aſpiciendoq́;</am></choice> punctum <seg type="var">.c.</seg> in ſuperficie <seg type="var">.n.q.o.</seg> & ſic in alia parte, exiſtente in-<lb/>ſtrumento in ſitu trianguli <seg type="var">.o.p.n.</seg> aſpicere oportet punctum <seg type="var">.e.</seg> proximum puncto <seg type="var">.n.</seg> <lb/>vbi poſſit metiri angulum <seg type="var">.c.n.e</seg>.</s> </p> <p> <s xml:space="preserve">Sed ſi ſitus puncti <seg type="var">.n.</seg> talis eſſet, vt ab eo non poſſet aliquis murum videre ad re-<lb/>ctos angulos, aſpiceremus punctum <seg type="var">.q.</seg> ſub orizontali ab oculis noſtris, in orizontali <lb/>tamen puncti <seg type="var">.n.</seg> ita quod angulus <seg type="var">.o.n.q.</seg> rectus exiſtat, quo facto obſeruando angu-<lb/>lum <seg type="var">.n.o.q.</seg> eo mediante, medianteq́ue <seg type="var">.n.o.</seg> cum angulo <seg type="var">.o.n.q.</seg> cognoſcemus <lb/>quantitatem diſtantiæ <seg type="var">.n.q.</seg> idem etiam faciendum eſt cum alio puncto <seg type="var">.p.</seg> quod <lb/>volueris, & mediantibus duobus punctis inuicem proximis <seg type="var">.c.e.</seg> cognoſcatur an- <pb facs="0369" n="357"/><fw type="head">EPISTOL AE.</fw> gulus <seg type="var">.p.n.q.</seg> vnde ex methodo .56. <lb/> <ptr xml:id="fig-0369-01a" corresp="fig-0369-01" type="figureAnchor"/> primi triangulorum Monteregij, <lb/>cognoſcemus reliqua trianguli <seg type="var">.<lb/>q.p.n</seg>. </s> <s xml:space="preserve">Conſtituendo poſtea angu-<lb/>lum <seg type="var">.q.n.u.</seg> æqualem angulo <seg type="var">.n.q.p.</seg> <lb/>propoſitum habebimus.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0369-01" corresp="fig-0369-01a"> <graphic url="0369-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Si etiam puncta <seg type="var">.q.p.</seg> lineæ <seg type="var">.q.p.</seg> <lb/>orizontali in eodem plano non exi <lb/>ſterent cum puncto <seg type="var">.n.</seg> nihil refer-<lb/>ret, dummodo in pauimento <choice><ex>notem</ex><am>notẽ</am></choice> <lb/>tur <choice><ex>puncta</ex><am>pũcta</am></choice> <seg type="var">.c.e.</seg> proxima <seg type="var">.n.</seg> in ijſdem <lb/>ſuperficiebus triangulorum <seg type="var">.n.o.p.</seg> <lb/>et <seg type="var">.n.o.q.</seg> vnde <seg type="var">.n.c.</seg> et <seg type="var">.n.e.</seg> erunt <choice><ex>com- munes</ex><am>cõ-munes</am></choice> ſectiones dictarum ſuperficierum cum ſuperficie pauimenti ſupra quam fit <lb/>ſtatio.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">CONI RECTI DIVISIO A PLANO <lb/>parallelo baſi ſecundum datam proportionem.</head> <head rend="italics" xml:space="preserve">Rapbaeli de Auria.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVotiescvnqve</hi> volueris conum rectum diuidere à plano parallelo ba-<lb/>ſi ſecundum vnam datam proportionem, nullius tibi erit difficultatis, con <lb/>ceſſa <choice><ex>tamen</ex><am>tamẽ</am></choice> pro inuenta diuiſione cuiuſuis propoſitę proportionis per tres <lb/>æquales partes.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia conus rectus <seg type="var">.a.b.c.</seg> ſecandus vt dictum eſt, accipiatur latus <lb/>ipſius, quod ſit <seg type="var">.a.c.</seg> <choice><ex>ipſumque</ex><am>ipſumq́;</am></choice> diuidatur in puncto <seg type="var">.d.</seg> ſecundum illam proportionem <lb/>quam deſideras, hoc eſt ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.d.</seg> quo facto, inter totum <seg type="var">.a.c.</seg> et <seg type="var">.a.d.</seg> inuenian <lb/>tur duæ lineæ proportionales, quarum maior ſit <seg type="var">.a.i.</seg> </s> <s xml:space="preserve">tunc ſi conus <seg type="var">.a.b.c.</seg> ſectus fue-<lb/>rit à plano per punctum <seg type="var">.i.</seg> parallelo baſi, habebimus quod quærebamus.</s> </p> <p> <s xml:space="preserve">Cuius rei ratio, primò eſt, quia quotieſcunque conus aliquis ſectus fuerit ab ali-<lb/>quo plano parallelo baſi ipſius, pars ſuperior ſimilis ſemper erit totali cono, quod <lb/>ita probo, cogitemus conum ſectum eſſe <lb/>à plano per axem <seg type="var">.a.l.</seg> vnde ex .3. primi <lb/> <ptr xml:id="fig-0369-02a" corresp="fig-0369-02" type="figureAnchor"/> Pergei, talis ſectio triangularis erit, quæ <lb/>ſit <seg type="var">.a.b.c.</seg> et <seg type="var">.b.c.</seg> diameter erit baſis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0369-02" corresp="fig-0369-02a"> <graphic url="0369-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Imaginemur deinde <seg type="var">.K.i.</seg> communem <lb/>eſſe ſectionem huiuſmodi trianguli cum <lb/>plano parallelo ipſi baſi, </s> <s xml:space="preserve">tunc tale <choice><ex>planum</ex><am>planũ</am></choice>, <lb/>circulare erit ex .4. primi ipſius Pergei <seg type="var">.K.<lb/>i.</seg> verò, eius diameter erit, et <seg type="var">.a.m.</seg> <choice><ex>ſuus</ex><am>ſuꝰ</am></choice> axis.</s> </p> <p> <s xml:space="preserve">Cum verò <seg type="var">.a.l.</seg> ſit perpendicularis ipſi <lb/>baſi conitotalis, eo quod rectus ſupponi-<lb/>tur, ideo eadem <seg type="var">.a.m.l.</seg> erit perpendicula <lb/>ris eriam ipſi ſecundo plano circulari, ex <lb/>conuerſa .14. vndecimi Euclid. </s> <s xml:space="preserve">vnde ex <pb facs="0370" n="358"/><fw type="head">IO. BAPT. BENED.</fw> ſecunda definitione eiuſdem libr <seg type="var">.a.m.l.</seg> efficiet angulos rectos cum duabus <seg type="var">.b.c.</seg> et <seg type="var">.K.<lb/>i.</seg> in punctis <seg type="var">.m.</seg> et <seg type="var">.l.</seg> et <seg type="var">.k.i.</seg> parallela erit ipſi <seg type="var">.b.c.</seg> ex .28. primi, quod etiam poteſt con <lb/>cludi mediante .16. vndecimi, cum <seg type="var">.k.i.</seg> et <seg type="var">.b.c.</seg> ſint communes ſectiones duorum pla <lb/>norum cum triangulari. </s> <s xml:space="preserve">Deinde ex .29. primi anguli <seg type="var">.a.i.m.</seg> et <seg type="var">.a.c.l.</seg> erunt inuicem <lb/>æquales, idem etiam dico de angulis <seg type="var">.a.k.i.</seg> et <seg type="var">.a.b.c.</seg> anguli poſtea ad <seg type="var">.a.</seg> communes <lb/>ſunt triangulis <seg type="var">.l.a.c.</seg> et <seg type="var">.m.a.i.</seg> vt triangulis <seg type="var">.l.a.b.</seg> et <seg type="var">.m.a.k</seg>. </s> <s xml:space="preserve">Vnde ex .4. ſexti, eadem <lb/>proportio erit ipſius <seg type="var">.m.i.</seg> ad <seg type="var">.l.c.</seg> & ipſius <seg type="var">.m.k.</seg> ad <seg type="var">.l.b.</seg> vt ipſius <seg type="var">.a.m.</seg> ad <seg type="var">.a.l</seg>. </s> <s xml:space="preserve">Quare ex <lb/>vndecima quinti, ita erit ipſius <seg type="var">.m.k.</seg> ad <seg type="var">.l.b.</seg> vt ipſius <seg type="var">.m.i.</seg> ad <seg type="var">.l.c.</seg> & ex .13. eiuſdem, ita <lb/>erit ipſius <seg type="var">.k.i.</seg> ad <seg type="var">.b.c.</seg> vt <seg type="var">.m.i.</seg> ad <seg type="var">.l.c.</seg> ſed ipſius <seg type="var">.m.i.</seg> ad <seg type="var">.l.c.</seg> eſt vt ipſius <seg type="var">.a.m.</seg> ad <seg type="var">.a.l.</seg> quod <lb/>iam dictum eſt, vnde ex .11. dicta, ita erit ipſius <seg type="var">.k.i.</seg> ad <seg type="var">.b.c.</seg> vt ipſius <seg type="var">.a.m.</seg> ad <seg type="var">.a.l.</seg> & ex <lb/>16. dicti ita erit ipſius <seg type="var">.a.m.</seg> ad <seg type="var">.k.i.</seg> vt ipſius <seg type="var">.a.l.</seg> ad <seg type="var">.b.c</seg>. </s> <s xml:space="preserve">Quare ex definitione ab Eu-<lb/>cli. poſita in .11, lib. pars coni ſuperior ſimilis erit cono totali.</s> </p> <p> <s xml:space="preserve">Deinde ſciendum eſt illud quod Euclid. ſcribit in .10. duodecimi lib. hoc eſt, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>proportio duarum pyramidum inuicem <lb/>ſimilium, triplicata eſt ei diametrorum <lb/> <ptr xml:id="fig-0370-01a" corresp="fig-0370-01" type="figureAnchor"/> ſuarum baſium, hoc eſt, quod proportio <seg type="var">.<lb/>b.c.</seg> ad <seg type="var">.k.i.</seg> tertia pars erit proportionis to <lb/>tius pyramidis <seg type="var">.a.b.c.</seg> partiali pyramidi <seg type="var">.a.<lb/>k.i.</seg> ſed ita eſt ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.i.</seg> vt ipſius <seg type="var">.b.<lb/>c.</seg> ad <seg type="var">.k.i.</seg> ex .4. ſexti cum trianguli <seg type="var">.a.b.c.</seg> <lb/>et <seg type="var">.a.k.i.</seg> ſint æquianguli, quod ex ijs, quę <lb/>ſuperius diximus facile compręhenditur. <lb/></s> <s xml:space="preserve">Quare <choice><ex>proportio</ex><am>ꝓportio</am></choice> <seg type="var">.a.c.</seg> ad <seg type="var">.a.i.</seg> tertia pars erit <lb/>proportionis totius coni <seg type="var">.a.b.c.</seg> ad eius par <lb/>tem abſciſſam <seg type="var">.a.k.i.</seg> ſed eadem proportio <lb/>ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.i.</seg> erat etiam tertia pars pro <lb/>portionis ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Quare ex com <lb/>muni conceptu, proportio totius pyramidis, ad partem abſciſſam, æqualis erit pro-<lb/>portioni ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.a.d</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0370-01" corresp="fig-0370-01a"> <graphic url="0370-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De differentia caloris Solis propter vaporum<unclear reason="illegible"/> <lb/>altitudinem.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">NOlo, mihi credas, ſed ex rationibus, quas tibi ſcribo conſidera, quod quo <lb/><choice><ex>tieſcunque</ex><am>tieſcunq;</am></choice> craſſities vel <choice><ex>denſitas</ex><am>dẽſitas</am></choice> <choice><ex>vaporum</ex><am>vaporũ</am></choice>, ſeu altitudo, maior eſſet ea, quę nunc re-<lb/>peritur, </s> <s xml:space="preserve">tunc minor differentia eſſet inter maiorem <choice><ex>minoremque</ex><am>minoremq́;</am></choice> calorem Solis, quam <lb/>nunc ſentiamus. </s> <s xml:space="preserve">Pro cuius rei euidentia, imaginemur in hac ſubſcripta figura, li-<lb/>neam <seg type="var">.o.a.</seg> pro ſemidiametro terræ, et <seg type="var">.a.c.</seg> pro craſſitie vaporum, vt nunc ſe <lb/>habet, et <seg type="var">.a.d.</seg> pro maiori craſſitie, imaginemurq́ue lineam <seg type="var">.a.b.</seg> quaſi perpen-<lb/>dicularem ad <seg type="var">.o.a.</seg> quæ abſciſſa ſit in puncto u. à circunferentia <seg type="var">.c.u.</seg> inferiori prio-<lb/>rum vaporum.</s> </p> <p> <s xml:space="preserve">Tunc dico minorem eſſe proportionem ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.a.d.</seg> quam ipſius <seg type="var">.a.u.</seg> ad <seg type="var">.a.<lb/>c.</seg> cogitemus ergo protractas eſſe lineas <seg type="var">.o.b</seg>: <seg type="var">d.b</seg>: <seg type="var">c.u.</seg> et <seg type="var">.c.n.</seg> quæ <seg type="var">.c.n.</seg> ſecabit <seg type="var">.a.u.</seg> in <pb facs="0371" n="359"/><fw type="head">EPISTOL AE.</fw> puncto <seg type="var">.i.</seg> ex communi conceptu, & <lb/> <ptr xml:id="fig-0371-01a" corresp="fig-0371-01" type="figureAnchor"/> parallcla erit ipſi <seg type="var">.d.b.</seg> ex. ſecunda par-<lb/>te ſecundæ ſexti, vnde ex prima parte <lb/>ciuſdem, ita eritipſius <seg type="var">.b.i.</seg> ad <seg type="var">.i.a.</seg> vt <seg type="var">.d.<lb/>c.</seg> ad <seg type="var">.c.a.</seg> & coniunctim ita erit ipſius <seg type="var">.b.<lb/>a.</seg> ad <seg type="var">.a.i.</seg> vt ipſius <seg type="var">.d.a.</seg> ad <seg type="var">.a.c.</seg> & permu <lb/>tatim ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.a.d.</seg> erit, vt <seg type="var">.a.i.</seg> <lb/>ad <seg type="var">.a.c.</seg> ſed cum <seg type="var">.a.u.</seg> maior ſit ipſa <seg type="var">.a.i.</seg> <lb/>vt omne totum maius eſt ſua parte. <lb/></s> <s xml:space="preserve">maior proportio erit ipſius <seg type="var">.a.u.</seg> ad <seg type="var">.a.<lb/>c.</seg> quam ipſius <seg type="var">.a.i.</seg> ad <seg type="var">.a.c.</seg> hoc eſt quam <lb/>ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Verum igitur eſt <lb/>propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0371-01" corresp="fig-0371-01a"> <graphic url="0371-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De differentia caloris Solis reſpectu altitudinis ipſius.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVodà me poſtulas deinde, ita ſe habet. </s> <s xml:space="preserve">Inquis enim, quod cum differentia <lb/>inter maiorem, <choice><ex>minoremque</ex><am>minoremq́;</am></choice> calorem, oriatur etiam ex differentia maioris <lb/>quantitatis vaporum ad minorem, per quam quantitatem vaporum rranſit lumen <lb/>Solis (vt alias etiam tibi dixi) velles nunc ſcire quantitatem ipſius differentię, quæ <lb/>inter duas Solis datas altitudines ſupra orizontem reperitur.</s> </p> <p> <s xml:space="preserve">Quapropter imaginemur circulum <seg type="var">.a.e.</seg> pro magno terræ, et <seg type="var">.z.b.d.</seg> pro magno <lb/>vaporum, ſupponatur etiam quod angulus <seg type="var">.z.o.d.</seg> vel <seg type="var">.z.a.b.</seg> qui ſunt inuicem fe-<lb/>rè æquales, ſit angulus diſtantiæ Solis à zenit, <seg type="var">z.a.</seg> verò ſit ſpiſſitudo vaporum, et <seg type="var">.a.<lb/>b.</seg> radius tranſiens per vapores dictos. </s> <s xml:space="preserve">nunc <lb/> <ptr xml:id="fig-0371-02a" corresp="fig-0371-02" type="figureAnchor"/> quæratur proportio, quæ eſt inter <seg type="var">.a.b.</seg> et <seg type="var">.a.<lb/>z.</seg> qua inuenta, angulo <seg type="var">.z.a.b.</seg> mediante, <lb/>quæremus eandem mediante angulo <seg type="var">.z.a.b.</seg> <lb/>maiore priori, velipſo minore, vnde cogno <lb/>ſcemus differentiam duarum <seg type="var">.a.b.</seg> quæ qui-<lb/>dem inæquales inuicem erunt, eo quod ſup <lb/>ponatur <seg type="var">.a.z.</seg> immutabilis, & hoc ita facie-<lb/>mus. </s> <s xml:space="preserve">Imaginabimur <seg type="var">.o.b.</seg> quæ claudat trian <lb/>gulum <seg type="var">.a.b.o.</seg> & quia <seg type="var">.a.z.</seg> cognita eſt quam <lb/>Alhazem docetinuenire, cognoſcimus <choice><ex>etiam</ex><am>etiã</am></choice> <lb/><seg type="var">o.a.</seg> vt ſemidiametrum terræ, vnde <seg type="var">.o.b.</seg> et <seg type="var">.<lb/>o.a.</seg> duo latera trianguli <seg type="var">.a.o.b.</seg> cognita <choice><ex>erunt</ex><am>erũt</am></choice> <lb/>ſimul cum angulo <seg type="var">.o.a.b.</seg> reſiduo duorum re <lb/>ctorum, eo quod reliquus <seg type="var">.z.a.b.</seg> datus eſt. <lb/></s> <s xml:space="preserve">Quare <seg type="var">.a.b.</seg> cognita erit reſpectu <seg type="var">.o.a.</seg> et <seg type="var">.o.<lb/>b.</seg> et <seg type="var">.a.z.</seg> quæ eſt eorum differentia. </s> <s xml:space="preserve">Nunc <lb/>ſi idem faciemus cum alia <seg type="var">.a.b.</seg> ſub diuerſo <lb/>angulo, habebimus propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0371-02" corresp="fig-0371-02a"> <graphic url="0371-02"/> </figure> </div> </body> </floatingText> <pb facs="0372" n="360"/> <fw type="head">IO. BAPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">NOTABILES ERRORES ORONTII <lb/>& Tartaleæ.</head> <head rend="italics" xml:space="preserve">Cornelio Bitonto.</head> <p> <s xml:space="preserve"><hi rend="small caps">PArvvs</hi> error non fuit, vt putabat Orontius, quodanguli triangulorum <lb/>æquicrurium inuicem æqualium, baſibus oppoſiti, ijſdem baſibus propor <lb/>tionales eſſent, cuius opinionis cauſa fuit quod nunquam viderit vel me <lb/>minerit eius quod Ptolomeus ſcripſit lib. primo Almageſti, vbi de diſpro <lb/>portionalitate chordarum <choice><ex>arcuumque</ex><am>arcuumq́;</am></choice> tractat, vel quod ſcribit Vitellio lib. primo pro <lb/>poſitione .35. ſeu lib. quarto, propoſitione .21. quod idem eſt. </s> <s xml:space="preserve">Sed nec ego tibi pro <lb/>ponam id quod ſcribit Nicolaus Tartalea diuiſioni .28. quinti capitis quartæ partis <lb/>ſuorum tractatuum, eo quod non exactè ſcientificè ſcripſerit, nec vniuerſaliter, <choice><ex>quan- uis</ex><am>quã-uis</am></choice> talis propoſitio poſſit ſcientificè ſcribi, accipiendo <seg type="var">.b.c.</seg> in eius figura, pro latere <lb/>octagoni, vnde angulus <seg type="var">.a.e.b.</seg> duplum foret angulo <seg type="var">.b.e.c.</seg> collocato poſtea <seg type="var">.b.c.</seg> in <lb/>arcu <seg type="var">.a.b.</seg> punctum <seg type="var">.c.</seg> medium fuiſſet dicti arcus, et <seg type="var">.e.c.</seg> diuideret <seg type="var">.a.b.</seg> per æqualia, <lb/>ex quinta primi, nec non ad rectos ex .3. tertij, vnde ex .18. primi, clare vidiſſemus <lb/>non eſſe proportionem <seg type="var">.a.b.</seg> ad <seg type="var">.b.c.</seg> vt anguli ad angulum. </s> <s xml:space="preserve">Sed vniuerſaliori modo <lb/>poſſumus hoc ſpeculari. </s> <s xml:space="preserve">Nam manifeſtè ſcimus, eandem eſſe proportionem circun <lb/>ferentiæ ad diametrum in omnibus circulis tam maioribus, quam minoribus. <lb/></s> <s xml:space="preserve">Sint igitur duo anguli <seg type="var">.a.e.b.</seg> et <seg type="var">.c.e.b.</seg> cuiuſuis amplitudinis, quorum latera <seg type="var">.e.a</seg>: <seg type="var">e.b</seg>: <lb/>et <seg type="var">.e.c.</seg> ſint inuicem æqualia, protrahatur <seg type="var">.b.a.</seg> et <seg type="var">.b.c</seg>. </s> <s xml:space="preserve">Tunc dico maiorem proportio <lb/>nem eſſe anguli <seg type="var">.a.e.b.</seg> ad angulum <seg type="var">.b.e.c.</seg> quam <seg type="var">.a.b.</seg> ad <seg type="var">.c.b.</seg> ducatur enim <seg type="var">.b.g.</seg> ita <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>faciat angulum <seg type="var">.g.b.c.</seg> æqualem angulo <seg type="var">.e.b.a.</seg> protracta poſtea <seg type="var">.c.g.</seg> quæ idem faciat <lb/>in puncto <seg type="var">.c.</seg> vnde <seg type="var">.g.b.</seg> et <seg type="var">.g.c.</seg> æquales inuicem erunt ex .6. primi, & quia angulus <seg type="var">.a.</seg> <lb/>æqualis eſt angulo <seg type="var">e.b.a.</seg> ex quinta eiuſdem, ideo ex .32. dicti, et .4. ſexti, horum <lb/>duorum triangulorum latera, erunt inuicem proportionalia. </s> <s xml:space="preserve">Conſtituto deinde <seg type="var">.g.</seg> <lb/>centro, & ſecundum ſemidiametrum <seg type="var">.g.b.</seg> vel <seg type="var">.g.c.</seg> quod idem eſt, deſcripto circu-<lb/>lo <seg type="var">.b.i.c.</seg> necnon circulo <seg type="var">.b.c.a.</seg> circa centrum <seg type="var">.e.</seg> ope ſemidiametri <seg type="var">.e.b.</seg> et <seg type="var">.e.a.</seg> vn <lb/>de iſte circulus eritillo maior, cum <seg type="var">.e.b.</seg> maior ſit <seg type="var">.g.b.</seg> ex .14. quinti. cum ex .14. tertij <lb/><seg type="var">a.b.</seg> longior ſit <seg type="var">.c.b.</seg> ſed ex vltima definitione tertij, arcus <seg type="var">.b.i.c.</seg> et <seg type="var">.b.c.a.</seg> erunt in-<lb/>uicem ſimiles, hoc eſt proportio totius cir-<lb/>cunferentiæ circuli <seg type="var">.b.i.c.</seg> ad arcus <seg type="var">.b.i.c.</seg> ea-<lb/> <ptr xml:id="fig-0372-01a" corresp="fig-0372-01" type="figureAnchor"/> dem erit, quæ totius circunferentiæ circuli <lb/><seg type="var">b.c.a.</seg> ad arcus <seg type="var">.b.c.a.</seg> ſed proportio diame-<lb/>tri ad circunferentiam eſt vt diametri ad cir <lb/>cunferentiam, vt ſupra diximus; </s> <s xml:space="preserve">Quare ex <lb/>proportionum æqualitate, vt ſemidiametri <lb/>ad circunferentiam erit, vt ſemidiametri <lb/>ad circunferentiam, & per eandem propor <lb/>tionum ęqualitatem, proportio <seg type="var">.e.b.</seg> ad <choice><ex>arcum</ex><am>arcũ</am></choice> <lb/><seg type="var">b.c.a.</seg> erit, vt <seg type="var">.g.b.</seg> ad arcum <seg type="var">.b.i.c.</seg> & per ean <lb/>dem æqualitatem, ita erit <seg type="var">.a.b.</seg> chordæ ad ar <lb/>cum <seg type="var">.b.c.a.</seg> vt <seg type="var">.c.b.</seg> chordæ ad arcum <seg type="var">.b.i.c.</seg> <lb/>& permutando, ita erit chordæ <seg type="var">.a.b.</seg> ad chor <lb/>dam <seg type="var">.c.b.</seg> vt arcus <seg type="var">.b.c.a.</seg> ad arcum <seg type="var">.b.i.c.</seg> ſed <lb/>arcus <seg type="var">.b.i.c.</seg> maior eſt arcu <seg type="var">.b.d.c.</seg> ex commu <pb facs="0373" n="361"/><fw type="head">EPISTOL AE.</fw> ni ſcicntia. </s> <s xml:space="preserve">Quare maior proportio erit acus <seg type="var">.b.c.a.</seg> ad arcum <seg type="var">.b.d.c.</seg> quam ad arcum <lb/><seg type="var">b.i.c.</seg> ex .8. quinti. </s> <s xml:space="preserve">Vnde ex vltima ſexti et .12. quinti, proportio anguli <seg type="var">.a.e.b.</seg> ad an-<lb/>gulum <seg type="var">.c.e.b.</seg> maior erit quam chordæ, ſiue baſis <seg type="var">.a.b.</seg> ad chordam ſiue baſim <seg type="var">.c.b</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0372-01" corresp="fig-0372-01a"> <graphic url="0372-01"/> </figure> </div> </body> </floatingText> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE CAVSA SVSPENSIONIS NVBIVM <lb/>in aere contra Antonium Bergam.</head> <head rend="italics" xml:space="preserve">Clarißimo Franciſco Venerio.</head> <p> <s xml:space="preserve">EGo enim non tantum miror ea quæ mihi ſcripſiſti de opinione Ortenſij <lb/>quantum quod Antonius Berga putat nubes à Sole ſupenſas teneri, id pla <lb/>nè falſum eſt, vera cauſa huiuſmodi effectus, alia nulla eſt, niſi earundem <lb/>raritas hoc eſt, cum rariores ſint ipſo aere ſubiecto, </s> <s xml:space="preserve">propterea ſupra <choice><ex>ipsum</ex><am>ipsũ</am></choice> <lb/>natant & ſtant ſub eo qui rarior ipſis eſt, eo quod corpora rariora poſita in medio <lb/>non tam raro, aſcendunt, & denſiora in medio minus denſo deſcendunt. </s> <s xml:space="preserve">Nam ſi <lb/>Sol ipſas nubes ſuſpenſas in aere teneret, hoc interdiu tantummodo fieret, ſed no <lb/>ctu, cur non deſcendunt vſque ad terram, & in eodem loco ſemper manent? </s> <s xml:space="preserve">Scien-<lb/>dum igitur eſt nubes aſcendere in altum quouſque inueniant aerem eiuſdem ra-<lb/>ritatis cuius ipſæ ſunt. </s> <s xml:space="preserve">Raritas enim & denſitas non ſunt res viſibiles niſi per acci-<lb/>dens, quemadmodum etiam leuitas, & grauitas, opacitas verò & diaphaneitas ma <lb/>gis <choice><ex>compræhenduntur</ex><am>compræhendũtur</am></choice>, opacitas enim ex reflexione radiorum luminoſorum, diapha <lb/>neitas verò compræhenditur ex penetratione ipſorum radiorum, opacitas autem nu <lb/>bis non eſt denſitas, cum valde diuerſa ſit denſitas ab opacitate, ſicut raritas ab dia-<lb/>phaneitate, vt aliàs dixi. </s> <s xml:space="preserve">Et quando dicit, quod Sol calefaciendo aerem ipſam nu <lb/>bem ambientem, rarefaciat eum magis quam ipſam nubem reſpondeo, hoc verum <lb/>non eſſe, proptere<unclear reason="illegible"/>a quodradius Solis non multum calefacit ea corpora, quæ ip ſi per <lb/>mittunt liberum tranſitum. </s> <s xml:space="preserve">vnde corpora quanto magis diaphana ſunt tanto minus <lb/>ab ipſo radio luminoſo calefiunt, ſed ea quæ magis opaca ſunt, magis etiam calefiunt <lb/>& per conſequens magis rarefiunt, cum calidi ſit per ſe rarefacere, & non attrahere, <lb/>vt ipſe & ferè omnes alij putant.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE RATIONE EXTENSIONIS FVNIS <lb/>cuiuſdam libramenti, & de quadam ſimboleita-<lb/>te circuli cum ellipſi.</head> <head rend="italics" xml:space="preserve">Angelo Ferrario Serenißimi Ducis Sabaudia <lb/>Agrimenſori expertißimo.</head> <p> <s xml:space="preserve"><hi rend="small caps">TIbi</hi> in mentem veniet, quod cum ſuperioribus diebus in villa lucenti, in <lb/>qua degebat Sereniſſimus Dux noſter, dum viridarium ad æquilibrium <lb/>reducebas, eſſemus, à te quæſiui an ſcires vnde fieret, vt ſtante libramen-<lb/>to ad angulos rectos ſupra ſuum pedem, funis quæ extrema eiuſdem li-<lb/>bramenti cum pede in formam trianguli æquicruris coniungit, magis diſtentus exi-<lb/>ſteret, quam cum dictum libramentum cum pede obliquum remanet, ita vt huiuſ- <pb facs="0374" n="362"/><fw type="head">IO. BAPT. BENED.</fw> modifunis cum libramento triangulum ſcalenum conſtitueret.</s> </p> <p> <s xml:space="preserve">Exempli gratia, ponamus lineam <seg type="var">.d.b.c.</seg> eſſe libramentum .et <seg type="var">.b.e.u.</seg> eius pedem, <lb/>funem autem, qui aliquando cum libramento facit triangulum iſocellum, & aliquan <lb/>do ſcalenum, eſſe <seg type="var">.d.e.c.</seg> eſto etiam quod in figura <seg type="var">.A.</seg> dictus triangulus <seg type="var">.d.e.c.</seg> ſit iſo-<lb/>cellus, & in figura <seg type="var">.B.</seg> ſcalenus. </s> <s xml:space="preserve">Tunc quæſiui à te an ſcires rationem, quare <lb/>funis <seg type="var">.d.e.c.</seg> in figura <seg type="var">.A.</seg> eſſet diſtenſus, & in figura <seg type="var">.B.</seg> laxus quemadmodum vide-<lb/>bamus. </s> <s xml:space="preserve">cum mihireſponderis, neſcio quid, quod nunc memoria <choice><ex>non</ex><am>nõ</am></choice> teneo, ſed quia <lb/>pollicitus ſum metibi eam afferre, propterea nunc ad te mitto. </s> <s xml:space="preserve">Scias ergo huiuſ-<lb/>modirationem nihil aliud eſſe niſi quod in figura <seg type="var">.A.</seg> duæ lineæ <seg type="var">.c.e.</seg> et <seg type="var">.d.e.</seg> ſimul è <lb/>directo iunctæ longiores ſint illis, quę reperiuntur in figura <seg type="var">.B.</seg> ſed quia funis tam in <lb/>figura <seg type="var">.B.</seg> quam in figura <seg type="var">.A.</seg> vnus, & idem eſt, ideo in figura <seg type="var">.B.</seg> laxatus eſt, & non in <lb/>tenſus, ut in figura <seg type="var">.A</seg>. </s> <s xml:space="preserve">Sed vt huiuſmodi veritatis certam notitiam habeas, infraſcri <lb/>ptum circulum mente concipe <seg type="var">.f.e.i.</seg> cuius ſemidiameter, æqualis ſit <seg type="var">.b.e.</seg> & diame-<lb/>ter ſit <seg type="var">.f.i.</seg> in quo imaginare eſſe tuum <lb/>libramentum <seg type="var">.d.b.c.</seg> & figuras <seg type="var">.A.</seg> et <seg type="var">.B.</seg> <lb/> <ptr xml:id="fig-0374-01a" corresp="fig-0374-01" type="figureAnchor"/> <ptr xml:id="fig-0374-02a" corresp="fig-0374-02" type="figureAnchor"/> & pr obabo lineas <seg type="var">.d.e.c.</seg> figurę <seg type="var">.A.</seg> lon <lb/>giores eſſe lineis <seg type="var">.d.e.c.</seg> figuræ <seg type="var">.B</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0374-01" corresp="fig-0374-01a"> <graphic url="0374-01"/> </figure> <figure xml:id="fig-0374-02" corresp="fig-0374-02a"> <graphic url="0374-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Imaginemur igitur lineam <seg type="var">.b.e.</seg> eſſe <lb/>dimidium minoris axis <choice><ex>alicuius</ex><am>alicuiꝰ</am></choice> ellipſis <lb/>cuius quidem figuræ ponamus <seg type="var">.d.</seg> et <seg type="var">.c.</seg> <lb/>centra ipſius circunſcriptionis eſſe, cu <lb/>ius <choice><ex>circunferentia</ex><am>circunferẽtia</am></choice>, nullidubium eſt, quin <lb/>extra propoſitum circulum tranſitura, <lb/>& in vno tantummodo puncto ipſum <lb/>circulum tactura ſit, qui exiſtat <seg type="var">.e.</seg> <lb/> figuræ <seg type="var">.A.</seg> ſeparatum tamen à puncto <lb/>e. figuræ <seg type="var">.B</seg>. </s> <s xml:space="preserve">Tunc ſi protracta fue-<lb/>rit linea <seg type="var">.d.e.</seg> figuræ <seg type="var">.B.</seg> vſque ad gi <lb/> <ptr xml:id="fig-0374-03a" corresp="fig-0374-03" type="figureAnchor"/> rum ellipticum in puncto <seg type="var">.g.</seg> à quo <lb/>ad punctum <seg type="var">.c.</seg> ducta etiam ſit linea <lb/><seg type="var">g.c</seg>. </s> <s xml:space="preserve">tunc <choice><ex>manifeſtum</ex><am>manifeſtũ</am></choice> erit duas lineas <lb/><seg type="var">d.e.</seg> et <seg type="var">.e.c.</seg> figuræ <seg type="var">.A.</seg> ſimul iunctas, <lb/>æquales eſſe duabus <seg type="var">.d.g.</seg> et <seg type="var">.g.c.</seg> ſi-<lb/>mul poſitis, vt etiam ex .52. tertij <lb/>Pergei facilè videre eſt, ſed ex .21. <lb/>primi Euclid. iam certò ſcimus <seg type="var">.d.g.c.</seg> longiores eſſe <seg type="var">.d.e.c.</seg> ſiguræ <seg type="var">.B.</seg> ergo <seg type="var">.d.e.c.</seg> figu-<lb/>ræ <seg type="var">.A.</seg> longiores ſunt <seg type="var">.d.e.c.</seg> figuræ <seg type="var">.B.</seg> quod eſt propoſitum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0374-03" corresp="fig-0374-03a"> <graphic url="0374-03"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quod etiam mihinunc circa hoc ſuccurrit, tibi libenter ſignifico, hoc eſt, quod <lb/>ſicut in ellipſi duæ lineæ <seg type="var">.d.e.e.c.</seg> figuræ <seg type="var">.A.</seg> ſimul iunctæ, ſunt ſemper æquales duabus <lb/>lineis <seg type="var">.d.g.g.c.</seg> in longitudine, ita in circulo duæ <seg type="var">.d.e.e.c.</seg> figuræ <seg type="var">.A.</seg> æquales ſunt in <lb/>potentia duabus <seg type="var">.d.e.e.c.</seg> figurę <seg type="var">.B</seg>.</s> </p> <p> <s xml:space="preserve">Manifeſtum enim primum eſt ex penultima primi in figura <seg type="var">.A.</seg> quadratum <seg type="var">.e.c.</seg> <lb/>æquale eſſe duobus quadratis ſcilicet <seg type="var">.e.b.</seg> et <seg type="var">.b.c.</seg> & quadratum <seg type="var">.e.d.</seg> æquale duobus <seg type="var">.<lb/>e.b.</seg> et <seg type="var">.b.d</seg>. </s> <s xml:space="preserve">Quare quadrata <seg type="var">.e.c.</seg> et <seg type="var">.e.d.</seg> æqualia ſunt quadratis <seg type="var">.e.b.</seg> figuræ <seg type="var">.A.</seg> et <seg type="var">.e.<lb/>b.</seg> figurę. B et <seg type="var">.b.c.</seg> et <seg type="var">.b.d.</seg> hoc eſt duplo quadrati <seg type="var">.e.a.</seg> (ducta cum fuerit <seg type="var">.e.a.</seg> perpen-<lb/>dicularis ad <seg type="var">.c.b.d.a.</seg>) duplo quadrati <seg type="var">.a.b.</seg> ex penultima primi, & duplo quadrati <seg type="var">.b.<lb/>c</seg>. </s> <s xml:space="preserve">Sed quadrata <seg type="var">.d.e.</seg> et <seg type="var">.e.c.</seg> figurę <seg type="var">.B.</seg> æqualia ſunt duplo quadrati <seg type="var">.a.e.</seg> & quadrato <seg type="var">a.d.</seg> <pb facs="0375" n="363"/><fw type="head">EPISTOL AE.</fw> & qua drato <seg type="var">.a.c.</seg> ex <choice><ex>eadem</ex><am>eadẽ</am></choice>. </s> <s xml:space="preserve">Nunc videndum eſt <choice><ex>vtrum</ex><am>vtrũ</am></choice> <choice><ex>duplum</ex><am>duplũ</am></choice> quadrati <seg type="var">.a.e.</seg> <choice><ex>cum</ex><am>cũ</am></choice> duplo qua <lb/>drati <seg type="var">.b.a.</seg> <choice><ex>cum</ex><am>cũ</am></choice> duplo quadrati <seg type="var">.b.c.</seg> ſit æquale duplo quadrati <seg type="var">.a.e.</seg> <choice><ex>cum</ex><am>cũ</am></choice> quadrato <seg type="var">.a.d.</seg> & <lb/>cum quadrato <seg type="var">.a.c</seg>. </s> <s xml:space="preserve">Sed quia tam ex vna parte quàm ex alia habemus duplum qua-<lb/>drati <seg type="var">.a.e</seg>. </s> <s xml:space="preserve">Videndum igitur erit vtrum duplum quadrati <seg type="var">.a.b.</seg> ſimul cum duplo qua-<lb/>drati <seg type="var">.b.c.</seg> ęquale ſit quadrato <seg type="var">.a.c.</seg> cum quadrato <seg type="var">.a.d.</seg> ſed hoc manifeſtum eſt .ex .10. <lb/>ſecundi Euclidis, dato quod <choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.a.</seg> ſit inter <seg type="var">.f.</seg> et <seg type="var">.d.</seg> ſed ſi fuerit inter <seg type="var">.d.</seg> et <seg type="var">.b.</seg> hoc <lb/>manifeſtum erit ex .9. ſecundi dicti, nihilominus accipe hunc alium modum.</s> </p> <p> <s xml:space="preserve">Sit hic ſubſcriptum quadratum <seg type="var">.D.</seg> ex <seg type="var">.a.c.</seg> in ſeipſa producta, cuius diameter ſit <lb/><seg type="var">a.n.</seg> <choice><ex>protrahanturque</ex><am>protrahanturq́</am></choice> parallelę <seg type="var">.d.h</seg>: <seg type="var">b.K</seg>: <seg type="var">l.m.o.</seg> et <seg type="var">.r.q.s.</seg> <choice><ex>eique</ex><am>eiq́;</am></choice> addatur <seg type="var">.c.p.</seg> ad <seg type="var">.a.c.</seg> æqua-<lb/>lis tamen <seg type="var">.d.a.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> protracta <seg type="var">.p.u.</seg> vſque ad <seg type="var">.m.o.u.</seg> vnde habebimus <seg type="var">.a.n.</seg> pro totali <lb/>quadrato, et <seg type="var">.p.s.</seg> pro partiali, & æquali quadrato lineæ <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Videndum nunc eſt, <choice><ex>vtrum</ex><am>vtrũ</am></choice> <lb/>hęc duo quadrata æqualia ſint duobus quadratis lineæ <seg type="var">.a.b.</seg> & duobus lineæ <seg type="var">.b.c.</seg> <choice><ex>Nam</ex><am>Nã</am></choice> <lb/>duo quadrata lineæ <seg type="var">.b.c.</seg> ſint <seg type="var">.K.o.</seg> et <seg type="var">.h.l.</seg> videndum nunc eſt utrum reſiduum ęquale <lb/>ſit duobus quadratis lineę <seg type="var">.a.b.</seg> quorum vnum ſit <seg type="var">.m.b.</seg> alterum verò <seg type="var">.l.p.</seg> quod ſupe-<lb/>rat <seg type="var">.l.c.</seg> et <seg type="var">.s.p.</seg> figuræ <seg type="var">.D.</seg> per ſupplementum <seg type="var">.o.t.</seg> cui æquale eſt parallelogrammum <seg type="var">.h.<lb/>m.</seg> figuræ <seg type="var">.D.</seg> ſed ſi punctus <seg type="var">.a.</seg> poſitus fuerit inter <seg type="var">.d.</seg> et <seg type="var">.b.</seg> conſtituto quadrato <seg type="var">.d.u.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <lb/>omnibus parallelis, vtin figura <seg type="var">.C.</seg> viderelicet, in qua figura videbimus quadrata <seg type="var">.r.<lb/>n.</seg> et <seg type="var">.d.r.</seg> ęquari duplo quadratorum <seg type="var">.l.n.</seg> et <seg type="var">.r.l.</seg> nam in quadrato <seg type="var">.r.n.</seg> ipſa duo quadra-<lb/>ta <seg type="var">.l.n.</seg> et <seg type="var">.r.l.</seg> capiuntur, reliquum eſt igitur vt videamus an duo ſupplementa <seg type="var">.l.t.</seg> et <seg type="var">.l.<lb/>s.</seg> cum quadrato <seg type="var">.d.r.</seg> ſint æqualia dictis <choice><sic>q́uadratis</sic><corr>quadratis</corr></choice> <seg type="var">.l.n.</seg> et <seg type="var">.r.l.</seg> ſed quadratum <seg type="var">.d.l.</seg> æ qua-<lb/>tur quadrato <seg type="var">.l.n.</seg> videndum igitur eſt, <lb/>an duo ſupplementa <seg type="var">.l.t.</seg> et <seg type="var">.l.s.</seg> cum qua <lb/> <ptr xml:id="fig-0375-01a" corresp="fig-0375-01" type="figureAnchor"/> drato <seg type="var">.d.r.</seg> ſint æqualia duobus quadra <lb/>tis <seg type="var">.d.l.</seg> et <seg type="var">.r.l.</seg> ſed quadratum <seg type="var">.d.l.</seg> æqua-<lb/>tur quadrato <seg type="var">.d.r.</seg> & ſupplemento <seg type="var">.l.t.</seg> <lb/>mediante <seg type="var">.q.l.</seg> & ſupplemento <seg type="var">.r.b.</seg> ſup-<lb/>plementum verò <seg type="var">.l.s.</seg> ſuperat <choice><ex>ſupplemem</ex><am>ſupplemẽ</am></choice> <lb/>tum <seg type="var">.r.b.</seg> per quantitatem <choice><ex>æqualem</ex><am>æqualẽ</am></choice> qua-<lb/>drato <seg type="var">.r.l.</seg> </s> <s xml:space="preserve">quare duo ſupplementa <seg type="var">.l.t.</seg> <lb/>et <seg type="var">.l.s.</seg> cum quadrato <seg type="var">.d.r.</seg> æquantur qua <lb/>drato <seg type="var">.d.l.</seg> <choice><ex>cum</ex><am>cũ</am></choice> quadrato <seg type="var">.l.r.</seg> verum igitur eſt duas <seg type="var">.d.e.e.c.</seg> figuræ <seg type="var">.A.</seg> æquales eſſe in <lb/>potentia duabus <seg type="var">d.e.e.c.</seg> figurę <seg type="var">.D.</seg> quæ quidem affectio circuli, à nemine fuit adhuc <lb/>(quod ſciam) detecta.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0375-01" corresp="fig-0375-01a"> <graphic url="0375-01"/> </figure> </div> </body> </floatingText> <pb facs="0376" n="364"/> <fw type="head">IO. BAPT. BENED.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE AVGMENTO PONDERIS CORPORIS <lb/>ad ſtateram appenſi, & quadam alia demonſtratione, <lb/>& quibuſdam erroribus Tartaleæ.</head> <head rend="italics" xml:space="preserve">Mutio Groto.</head> <p> <s xml:space="preserve">SI ea quæ à me audiuiſti non credis, conſidera quæſo libram ſeu ſtateram <lb/><seg type="var">o.a.</seg> cuius centrum non longitudinis ſed ponderum ſit <seg type="var">.i.</seg> quę ſtatera, vt ori <lb/>zontaliter conſiſtat, oportebit pondus extremitatis <seg type="var">.o.</seg> ita ſe habere <lb/>ad pondus extremitatis <seg type="var">.a.</seg> ut <seg type="var">.a.i.</seg> ſe habet ad <seg type="var">.o.i.</seg> quod te ſcire puto, ima <lb/>ginemur nunc d uas lineas <seg type="var">.a.e.</seg> et <seg type="var">.o.n.</seg> paralle las <choice><ex>infinitasque</ex><am>infinitasq́;</am></choice> & à puncto <seg type="var">.n.</seg> immobili, <lb/>& fixo extra ſtateram, tranſeat per <seg type="var">.i.</seg> linea <seg type="var">.n.i.e</seg>. </s> <s xml:space="preserve">Cogitemus etiam punctum <seg type="var">.e.</seg> inter <lb/>ſectionis ipſius <seg type="var">.n.i.e.</seg> cum <seg type="var">.a.e.</seg> progredi vniformiter <choice><ex>continuòque</ex><am>continuòq́;</am></choice> ab <seg type="var">.a.</seg> per lineam <seg type="var">.a.e.</seg> <lb/>vnde punctum <seg type="var">.i.</seg> interſectionis ipſius <seg type="var">.n.i.e.</seg> cum <seg type="var">.a.i.o.</seg> ſemper vicinius fiet puncto <seg type="var">.o.</seg> <lb/>nec unquam cum illo vnum erit, quamuis moueatur tempore infinito. </s> <s xml:space="preserve">Nunc autem <lb/>dico, quod cum ſtateram <seg type="var">.o.i.a.</seg> oporteat ſemper orizontalem eſſe virtute ponderis, <lb/>o. oportebit pundus <seg type="var">.o.</seg> in infinitum etiam augeri, <choice><ex>quotieſcunque</ex><am>quotieſcunq;</am></choice> pondus <seg type="var">.a.</seg> nunquam <lb/>diminui voluerimus vel econtra hoc in infinitum diminui, ſi illud nunquam augeri <lb/>voluerimus.</s> </p> <p> <s xml:space="preserve">Sedre vera non putabam te indigere aliqua demonſtratione, quod linea <seg type="var">.b.h.</seg> di-<lb/>uiſa ſit per æqualia à<unclear reason="illegible"/> linea <seg type="var">.c.a.</seg> cum hæc perpendicularis ſit ab <seg type="var">.a.</seg> ad baſim <seg type="var">.g.d.</seg> in <choice><ex>triam</ex><am>triã</am></choice> <lb/>gulo orthogonio <seg type="var">.g.a.d.</seg> & cum ſit <seg type="var">.b.h.</seg> perpendicularis ad <seg type="var">.a.o.</seg> ex ſuppoſito quæ <seg type="var">.a.<lb/>o.</seg> in ſe habet punctum medium baſis <seg type="var">.g.d.</seg> nec <choice><ex>non</ex><am>nõ</am></choice> illud anguli recti <seg type="var">.a.</seg> quod per ſe cla <lb/>riſſimum eſt, cum iam ſcis <seg type="var">.o.</seg> eſſe centrum circuli circundantis triangulum <seg type="var">.g.a.d.</seg> or-<lb/>thogonium, et <seg type="var">.g.d.</seg> eius diameter, vnde <seg type="var">.o.a.</seg> æquabitur ipſi <seg type="var">.o.g.</seg> quapropter angulus <lb/>o. <choice><ex>am</ex><am>ã</am></choice>. g. æquabitur angulo <seg type="var">.g.</seg> ex quinta primi, </s> <s xml:space="preserve">deinde ex .32. eiuſdem, angulus <seg type="var">.h.</seg> æqua <lb/>bitur angulo <seg type="var">.d.</seg> eo quod an gulus <seg type="var">.e.</seg> rectus eſt, quemadmodum et <seg type="var">.a.</seg> ſed angulus <seg type="var">.d.</seg> <lb/>æqualis eſt angulo <seg type="var">.g.a.c</seg>. </s> <s xml:space="preserve">& propterea angulus <seg type="var">.h.</seg> erit etiam æqualis angulo <seg type="var">.h.a.u.</seg> <lb/>vnde <seg type="var">.h.u.</seg> æqualis erit ipſi <seg type="var">.u.<lb/>a.</seg> ex .6. primi, cum poſtea angulus <seg type="var">.<lb/> <ptr xml:id="fig-0376-01a" corresp="fig-0376-01" type="figureAnchor"/> o.a.d.</seg> æqualis ſitangulo <seg type="var">.d.</seg> ex quin<lb/>ta primi erit angulus <seg type="var">.a.b.e.</seg> æqua-<lb/>lis angulo <seg type="var">.g.</seg> ex .32. dicta, eo quod <lb/>e. rectus eſt, & ex eadem æqualis <lb/>erit angulo <seg type="var">.d.a.c.</seg> vnde <seg type="var">.u.b.</seg> erit <lb/>æqualis ipſi <seg type="var">.u.a.</seg> ex .6. dicti, & ideo <lb/>æqualis eric ipſi <seg type="var">.u.h</seg>. </s> <s xml:space="preserve">Reliqua ve-<lb/>rò illius propoſitionis credo ex te <lb/>omnia poſſe <choice><ex>intelligere</ex><am>ĩtelligere</am></choice>, excepto, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>vt tibi ſignificaui ſi à <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.i.</seg> com-<lb/>muni ipſi <seg type="var">.a.c.u.</seg> & circunferentiæ, <lb/>ducta fuerit <seg type="var">.i.x.</seg> ad <choice><ex>punctum</ex><am>pũctum</am></choice> <seg type="var">.x.</seg> com <lb/>mune vni parallelæ à <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.g.</seg> ipſi <lb/><seg type="var">h.b.</seg> & circunferentiæ, quod di-<lb/>cta <seg type="var">.i.x.</seg> ad rectos erit ipſi <seg type="var">.a.b.d.</seg> eo <lb/>quod cum angulus <seg type="var">.a.g.x.</seg> æqualis <pb facs="0377" n="365"/><fw type="head">EPISTOL AE.</fw> ſit angulo <seg type="var">.a.h.b.</seg> propter æquidiſtantiam dictam, æqualis etiam erit angulo <seg type="var">.d.</seg> & ar-<lb/>cus <seg type="var">.a.x.</seg> æqualis arcui <seg type="var">.a.g.</seg> vnde angulus <seg type="var">.a.i.x.</seg> æqualis erit <seg type="var">.d.</seg> ſed angulus <seg type="var">.i.a.d.</seg> com-<lb/>munis eſt triangulis <seg type="var">.c.a.d.</seg> et <seg type="var">.i.a.t</seg>. </s> <s xml:space="preserve">quare angulus <seg type="var">.a.t.i.</seg> rectus erit, vt <seg type="var">.c.</seg> hoc eſt <seg type="var">.i.x.</seg> per <lb/>pendicularis erit ipſi <seg type="var">.a.d</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0376-01" corresp="fig-0376-01a"> <graphic url="0376-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed vbitibi ſcripſi circa finem illius epiſtolæ, Tartaleam erraſſe in quinta propo-<lb/>ſitione primi lib. ſuæ nouæ ſcientiæ, non ſine ratione illud ſcripſi. </s> <s xml:space="preserve">Nam, inquit ipſe, <lb/>nullum corpus æquè graue poteſt in aliquo temporis ſpatio moueri motu naturali, <lb/><choice><ex>violentoque</ex><am>violentoq́;</am></choice> ſimul miſtis. </s> <s xml:space="preserve">Vbi decipitur, eo quod non animaduertit incrementum ve <lb/>locitatis vnius motus, ſimul eſſe cum decremento velocitatis alterius, <choice><ex>eodemque</ex><am>eodemq́;</am></choice> tem <lb/>pore, vt manifeſtè patet in itinere corporis, ab ipſo pro exemplo aſſumpto, hoc eſt <lb/>quod velocitas motus in ſpatio <seg type="var">.c.d.</seg> creſcit vt naturalis, & decreſcit vt violenta. </s> <s xml:space="preserve"><choice><ex>nam</ex><am>nã</am></choice> <lb/>creſcit orizontem verſus & decreſcit in remotione à linea <seg type="var">.a.b.</seg> ſed ſi à puncto <seg type="var">.c.</seg> ad <lb/>punctum <seg type="var">.d.</seg> motus eſſet purè violentus, vt putat Tartalea, corpus illud minimè de-<lb/>ſcenderet, eo quod uirtus mouens, in <seg type="var">.a.</seg> poſita, nullo pacto poteſt talem effectum ef-<lb/>ficere, vnde ab ipſa natura prouenit deſcenſio illius corporis propter <choice><ex>grauitatem</ex><am>grauitatẽ</am></choice>, <choice><ex>quam</ex><am>quã</am></choice> <lb/>dictum corpus habet in tali medio, aeris ſcilicet, & non ex violentia aliqua. </s> <s xml:space="preserve">Sed ſi <lb/>dixiſſet ipſe, illum motum eſſe purum naturalem, hoc eſſet falſum, eo quod purus <lb/>naturalis motus alicuius corporis non impediti, extra locum ſuum, ſit per lineam re <lb/>ctam, & non per curuam, vt videre eſt inter <seg type="var">.c.</seg> et <seg type="var">.d</seg>.</s> </p> <p> <s xml:space="preserve">In vltima propoſitione deinde eiuſdem lib. quæ .6. eſt decipitur ſimiliter, & hæc <lb/>deceptio oritur ab ignoratione quintæ, & à putando motum naturalem non eſſe cau <lb/>ſam ipſius deſcenſus per ſpatium <seg type="var">.c.d</seg>. </s> <s xml:space="preserve">Sed quia tibi ſignificaui expeditiorem viam <lb/>repeririad <choice><ex>cognoſcendam</ex><am>cognoſcendã</am></choice> proportionem inter <seg type="var">.a.h.</seg> et <seg type="var">.a.e.</seg> in vltima propoſitione ſe-<lb/>cundi lib. ipſius Tartaleæ, ipſam nunc tibi ſcribo. </s> <s xml:space="preserve"><choice><ex>Nam</ex><am>Nã</am></choice> iam ſcis angulum <seg type="var">.h.l.i.</seg> diui-<lb/>ſum eſſe per æqualia ab <seg type="var">.P.l.</seg> & quod <seg type="var">.a.h.</seg> et <seg type="var">.h.p.</seg> ęquales inuicem ſunt ex .6. primi Eu-<lb/>cli. </s> <s xml:space="preserve">vnde <seg type="var">.p.i.</seg> et <seg type="var">.a.h.</seg> æquales erunt inuicem ſimiliter, ſed ex .3. ſexti ita eſt ipſius <seg type="var">.a.l.</seg> <lb/>ad <seg type="var">.l.i.</seg> vt ipſius <seg type="var">.a.p.</seg> ad <seg type="var">.p.i.</seg> & coniunctim ita erit <seg type="var">.a.l.i.</seg> ad <seg type="var">.l.i.</seg> vt <seg type="var">.a.i.</seg> ad <seg type="var">.p.i.</seg> ſed <seg type="var">.a.l.</seg> cogni <lb/>ta eſt ex eius quadrato, et <seg type="var">.l.i.</seg> etiam, cum æqualis ſit ipſi <seg type="var">.a.i.</seg> vnde ex regula de tribus <lb/>notam habebimus <seg type="var">.p.i.</seg> reſpectu <seg type="var">.a.i.</seg> & ita reſpectu <seg type="var">.a.e.</seg> ſi hypotheſes ipſius Tartaleæ <lb/>veræ ſunt.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Alia demonstratio impoßibilitatis diuidendi per æqualia <lb/>proportionem ſuperparticularem in <lb/>diſcretis.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod à me poſtulas, hoc eſt ſcientiam impoſſibilitatis diuidendi per æqualia <lb/>proportionem ſuperparticularem in numeris ſatis à Campano in .8. octaui <lb/>potes habere, Iacobus Faber Stapulenſis etiam idem tractat<unclear reason="illegible"/> in libello ſuę muſicæ <lb/>demonſtratæ. </s> <s xml:space="preserve">Sed ſi etiam alia via idem deſideras, quamuis longiori, nih<unclear reason="illegible"/>ilomi-<lb/>nus vniuerſaliori, conſidera duos numeros <seg type="var">.g.</seg> et <seg type="var">.h.</seg> inuicem relatos ſecundum pro-<lb/>portionem ſuperparcicularem, quam volueris. </s> <s xml:space="preserve">Tunc dico impoſſibile eſle, vt per <lb/>æqualia diuidatur, quod ſi dixeris poſſibile eſſe, ſit per te <seg type="var">.K.</seg> medius numerus <pb facs="0378" n="366"/><fw type="head">IO. BAPT. BENED.</fw> proportionalis inter <seg type="var">.g.</seg> et <seg type="var">.h</seg>. </s> <s xml:space="preserve">quare <seg type="var">.g.</seg> et <seg type="var">.h.</seg> non erunt minimi in ea proportione, quia <lb/>vnitas diuiſibilis eſſet ſi <seg type="var">.g.h.</seg> minimi fuiſſent, quod non conceditur, ſint igitur mini <lb/>mi in dicta proportione <seg type="var">.a.</seg> et <seg type="var">.b.</seg> quorum differentia erit vnitas, vt ſcis, <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.c.</seg> quadra <lb/>tum ipſius <seg type="var">.g.</seg> et <seg type="var">.d.</seg> quadratum ipſius <seg type="var">.K</seg>. </s> <s xml:space="preserve">tunc clarum erit ex .11. octaui, quod propor-<lb/>tio ipſius c. ad <seg type="var">.d.</seg> eadem erit quæ <seg type="var">.g.</seg> ad <seg type="var">.h.</seg> hoc eſt vt ipſius <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> vnde ſi vnus termi. <lb/>norum <seg type="var">.a.</seg> vel <seg type="var">.b.</seg> eſſet quadratus, reliquus etiam quadratus eſſet ex .22. octaui, & ex <lb/>16. eiuſdem, inter <seg type="var">.a.</seg> et <seg type="var">.b.</seg> reperiretur aliquis medius numerus proportionalis, quod <lb/>fieri non poteſt ex hypotheſi, cum inter <seg type="var">.a.</seg> et <seg type="var">.b.</seg> nullus ſit numerus, quia differunt in <lb/>ter ſe per vnitatem tantummodo. </s> <s xml:space="preserve">Nunc autem cum nullus numerorum <seg type="var">.a.</seg> vel <seg type="var">.b.</seg> qua <lb/>dratus ſit, ponatur quod <seg type="var">.f.</seg> quadratus ſit ipſius <seg type="var">.b.</seg> et <seg type="var">.e.</seg> ſit productum ipſius <seg type="var">.a.</seg> in <seg type="var">.b.</seg> vn <lb/>de ex .18. ſeptimi, proportio ipſius <seg type="var">.e.</seg> ad <seg type="var">.f.</seg> erit vt. ipſius <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> hoc eſt vt ipſius <seg type="var">.c.</seg> ad <lb/>d. quapropter <seg type="var">.e.</seg> erit quadratus ex .22. octaui, cuius latus tetragonicum eſſet <choice><ex>medium</ex><am>mediũ</am></choice> <lb/>proportionale inter <seg type="var">.a.</seg> et <seg type="var">.b.</seg> ex .20. ſeptimi, quod eſt impoſſibile, vt iam dixi, cum <seg type="var">.a.</seg> <lb/>et <seg type="var">.b.</seg> ſint inui cem conſequentes, vnus poſt alium immediatè.</s> </p> <p> <s xml:space="preserve">Superius enim dixi hunc modum eſſe vniuerſalem, <lb/>hoc eſt quod hac methodo poſſumus in cognitionem <lb/>vcnire, quod non ſolum in duas æquales partes diui-<lb/> <ptr xml:id="fig-0378-01a" corresp="fig-0378-01" type="figureAnchor"/> di non poſſit, ſed nec in tres, nec quatuor nec quot vo <lb/>lueris. </s> <s xml:space="preserve">Primum enim quod non in tres diuidatur à te <lb/>ipſo cognoſces ope <choice><ex>cuborum</ex><am>cuborũ</am></choice> vice <choice><ex>quadratorum</ex><am>quadratorũ</am></choice>, opevero <lb/><choice><ex>cenſuum</ex><am>cenſuũ</am></choice> <choice><ex>cenſuum</ex><am>cẽſuũ</am></choice>, ve<unclear reason="illegible"/>l qui cognouerit eam <choice><ex>proportionem</ex><am>proportionẽ</am></choice> <lb/>eſſe indiuiſibilem per æqualia, illicò etiam cognoſcet <lb/>indiuiſibilem eſſe per quatuor partes, ope verò pri-<lb/>morum relatorum, cognoſcet non eſſe diuiſibilem per <lb/><choice><ex>quinque</ex><am>quinq;</am></choice> partes, & ſic de cęteris, ſed mediantibus ijs <lb/>quas ſcripſi de iſtis dignitatibus in libro <choice><ex>Thęorematum</ex><am>Thęorematũ</am></choice> <lb/>arithmeticorum.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0378-01" corresp="fig-0378-01a"> <graphic url="0378-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Id autem quod Illuſtriſſimus Daniel Barbarus ſcri <lb/>bit in quinta parte ſuæ perſpectiuæ, ſi ſupra aliquo im <lb/>mobili, atque magno pariete facere volueris, te opor <lb/>tebit hoc ex reflexione radij ſolaris à ſpeculo plano <lb/>perficere.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE INVENTIONE DIAMETRI <lb/>circuli circunſcribentis triangulum.</head> <head rend="italics" xml:space="preserve">Francbino Triuultio.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> mihi nunc proponis eſt triangulum, cuius baſis cum angulo ſibi op <lb/>poſito dantur. </s> <s xml:space="preserve"><choice><ex>Vellesque</ex><am>Vellesq́;</am></choice> diametrum circuli apti eum triangulum circnn-<lb/>ſcribere inuenire in diſcreto.</s> </p> <p> <s xml:space="preserve">Sit igitur triangulum <seg type="var">.a.b.g.</seg> cuius baſis <seg type="var">.b.g.</seg> ſimul cum angulo <seg type="var">.a.</seg> ei op-<lb/>poſito data ſit in numeris. </s> <s xml:space="preserve">Imaginetur ergo circulas circunſeribens ipſum triangu-<lb/>lum <seg type="var">.b.p.g.q.</seg> cuius diameter ſit <seg type="var">.q.p.</seg> perpendicularis eius baſi <seg type="var">.b.g.</seg> vnde <seg type="var">.b.g.</seg> diuiſa <lb/>erit per æqualia ab ipſo diametro in puncto <seg type="var">.m.</seg> per tertiam tertij, protrahatur etiam <pb facs="0379" n="367"/><fw type="head">EPISTOL AE.</fw> e<unclear reason="illegible"/> <seg type="var">.g.</seg> vnde angulus <seg type="var">.g.e.q.</seg> æqualis erit angulo <seg type="var">.b.a.g.</seg> portionis, cum duplus ſit angulo <lb/><seg type="var">q.p.g.</seg> medietati anguli ipſius portionis ex .19. tertij, ita quod angulus <seg type="var">.q.e.g.</seg> nobis <lb/>cognitus erit, & ſimiliter arcus <seg type="var">.g.q.</seg> & conſequenter ar-<lb/>cus <seg type="var">.p.g.</seg> reſiduum medij circuli, & ſic <seg type="var">.m.g.</seg> eius ſinus re <lb/> <ptr xml:id="fig-0379-01a" corresp="fig-0379-01" type="figureAnchor"/> ctus, & etiam chorda <seg type="var">.p.g.</seg> vt dupla ſinus dimidij arcus <seg type="var">.<lb/>p.g.</seg> & ſic <seg type="var">.p.m.</seg> eius ſinus verſus, vel vt tertium latus trian <lb/>guli orthogonij <seg type="var">.p.g.m.</seg> vnde nobis cognita erit propor <lb/>tio ipſius <seg type="var">.b.g.</seg> (quæ dupla eſt ipſi <seg type="var">.m.g.</seg>) ad <seg type="var">.m.p.</seg> & quia <lb/>productum <seg type="var">.p.m.</seg> in <seg type="var">.m.q.</seg> æquale eſt ei, quod fit ex <seg type="var">.b.m.</seg> <lb/>in <seg type="var">m.g.</seg> ex .34. tertij, quapropter nobis cognita erit pars <lb/><seg type="var">q.m.</seg> quæ cum <seg type="var">.p.m.</seg> complet totum diametrum <seg type="var">.q.p.</seg> vn <lb/>de nobis cognita erit proportio ipſius <seg type="var">.b.g.</seg> ad <seg type="var">.q.p.</seg> qua <lb/>mediante cognoſcemus diametrum ſecundum partes il <lb/>las quibus propoſita ſuerit <seg type="var">.b.g</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0379-01" corresp="fig-0379-01a"> <graphic url="0379-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Hoc autem problema non in numeris ſed in continuo ab Euclid. ponitur in .32<unclear reason="illegible"/>. <lb/>tertij.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De inuentione alterius trianguli conditionati.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVotieſcunque etiam inuenire voluerimus triangulum aliquem, puta <seg type="var">.n.q.o.</seg> <lb/>æqualem triangulo <seg type="var">.t.</seg> (exempli gratia) propoſito, qui habeat angulum <seg type="var">.n.</seg> æ-<lb/>qualem angalo <seg type="var">.a.</seg> dato, latera vero continentia ipſum angulum <seg type="var">.n.</seg> ſint inuicem pro-<lb/>portionata vt <seg type="var">.x.</seg> et <seg type="var">.y.</seg> ita faciemus, accipiemus lineam <seg type="var">.n.m.</seg> cuius volueris magnitu-<lb/>dinis, ſupra quam conſtituemus triangulum <seg type="var">.m.n.p.</seg> æqualem triangulo <seg type="var">.t.</seg> hac metho-<lb/>do, hoc eſt prolungando latus <seg type="var">.r.z.</seg> trianguli <seg type="var">.t.</seg> quod ſit <seg type="var">.r.e.</seg> ita vt duplum ſit ipſi <seg type="var">.r.z.</seg> <lb/>ducendo poſtea <seg type="var">.c.e.</seg> habebimus ex .38. primi triangulum <seg type="var">.t.</seg> eſſe dimidium totius <lb/>trianguli <seg type="var">.r.c.e.</seg> deſignabimus deinde ex .44. dicti ſuperficiem <seg type="var">.p.n.m.b.</seg> parallelo <lb/>grammam <choice><ex>æqualemque</ex><am>æqualemq́;</am></choice> triangu <lb/>lo <seg type="var">.r.c.e.</seg> habentem angulum <seg type="var">.<lb/> <ptr xml:id="fig-0379-02a" corresp="fig-0379-02" type="figureAnchor"/> n.</seg> æqualem angulo <seg type="var">.a.</seg> ducatur <lb/>poſtea <seg type="var">.p.m.</seg> & habebimus <choice><ex>triam</ex><am>triã</am></choice> <lb/>gulum <seg type="var">.m.n.p.</seg> æqualem <seg type="var">.t.</seg> cum <lb/>angulo <seg type="var">.n.</seg> æquali angulo <seg type="var">.a.</seg> pro <lb/>ducatur poſtea <seg type="var">.n.p.</seg> ita vt <seg type="var">.n.K.</seg> <lb/>ſe habeat .ad <seg type="var">.n.m.</seg> quemadmo <lb/>dum <seg type="var">.x.</seg> ad <seg type="var">.y.</seg> quod erit facilli-<lb/>mum producendo <seg type="var">.n.m.</seg> et <seg type="var">.n.<lb/>K.</seg> indeterminatè ſi oportuerit, <lb/></s> <s xml:space="preserve">deinde eas ad æqualitatem ſe-<lb/>can<unclear reason="illegible"/>do ipſis <seg type="var">.x.</seg> et <seg type="var">.y.</seg> efficiendo <lb/>exempli gratia quod <seg type="var">.n.i.</seg> ſit <lb/>æqualis ipſi <seg type="var">.x.</seg> et <seg type="var">.n.u.</seg> ipſi <seg type="var">.y.</seg> du <lb/>cendo poſtea <seg type="var">.u.i.</seg> deinde à puncto <seg type="var">.m.</seg> ducendo <seg type="var">.m.K.</seg> æquidiſtanter <seg type="var">.u.i.</seg> ex .31. <lb/>primi. </s> <s xml:space="preserve">& ſic habebimus ex .4. ſexti proportionem <seg type="var">.x.</seg> ad <seg type="var">.y.</seg> eſſe inter <seg type="var">.n.K.</seg> et <seg type="var">.n.</seg> <pb facs="0380" n="368"/><fw type="head">IO. BAPT. BENED.</fw> m. inuenies poſtea ex .9. eiuſ-<lb/> <ptr xml:id="fig-0380-01a" corresp="fig-0380-01" type="figureAnchor"/> dem lineam aliquam mediam <lb/>proportionalem inter <seg type="var">.n.K.</seg> et <seg type="var">.<lb/>n.p.</seg> quæ ſit <seg type="var">.n.o.</seg> duces poſtea <lb/><seg type="var">o.q.</seg> parallelam ipſi <seg type="var">.m.K.</seg> & ha <lb/>bebis propoſitum, eo quod <choice><ex>cum</ex><am>cũ</am></choice> <lb/>ſit proportio trianguli <seg type="var">.n.m.K.</seg> <lb/>ad triangulum <seg type="var">.n.m.p.</seg> vt <seg type="var">.n.K.</seg> <lb/>ad <seg type="var">.n.p.</seg> ex prima ſexti, duo <choice><ex>triam</ex><am>triã</am></choice> <lb/>guli <seg type="var">.m.p.n.</seg> et <seg type="var">.n.q.o.</seg> æquales <lb/>erunt inuicem, ex .17. eiuſdem <lb/>& ex .9. quinti, & proportio <seg type="var">.<lb/>o.n.</seg> ad <seg type="var">.n.q.</seg> erit, vt <seg type="var">.x.</seg><unclear reason="illegible"/> ad <seg type="var">.y.</seg> ex <num value="11">.<lb/>11.</num> dicti, cum ex .4. ſexti ſit vt <seg type="var">.<lb/>n.k.</seg> ad <seg type="var">.n.m</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0379-02" corresp="fig-0379-02a"> <graphic url="0379-02"/> </figure> <figure xml:id="fig-0380-01" corresp="fig-0380-01a"> <graphic url="0380-01"/> </figure> </div> </body> </floatingText> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De producto conditionato.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">PRoponis deinde mihi duas rectas lineas, vni quarum, vis vt aliam quandam di-<lb/>rectè coniungam, ita quod productum huius aggregati in lineam adiunctam <lb/>æquale ſit quadrato alterius.</s> </p> <p> <s xml:space="preserve">Vt exempli gratia ſi fuerint duæ lineæ <seg type="var">.e.d.</seg> et <seg type="var">.e.f.</seg> opor-<lb/><choice><ex>teretque</ex><am>teretq́;</am></choice> nos ad lineam <seg type="var">.e.f.</seg> aliam lineam puta <seg type="var">.f.c.</seg> vel <seg type="var">.e.b.</seg> <choice><ex>iun gere</ex><am>iũ gere</am></choice>, ita longam, vt productum totius compoſiti <seg type="var">.e.c.</seg> vel <seg type="var">.<lb/>f.b.</seg> in <seg type="var">.f.c.</seg> vel <seg type="var">.e.b.</seg> eſſet æquale quadrato ipſius <seg type="var">.e.d</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0380-02" corresp="fig-0380-02a"> <graphic url="0380-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Hoc enim nu llius eſſet difficultatis, eo quod <choice><ex>quotieſcun- que</ex><am>quotieſcũ-que</am></choice> <seg type="var">.e.d.</seg> coniuncta erit cum <seg type="var">.e.f.</seg> ad rectos, <choice><ex>diuiſaque</ex><am>diuiſaq́;</am></choice> per me <lb/>dium à puncto <seg type="var">.a.</seg> à quo ducta <seg type="var">.a.d.</seg> deinde ſecundum ſemi-<lb/>diametrum <seg type="var">.a.d.</seg> deſignato circulo <seg type="var">.b.d.c.</seg> & protracta <seg type="var">.e.f.</seg> <lb/>à qua volueris parte vſque ad circunferentiam in <choice><ex>puncto</ex><am>pũcto</am></choice> <seg type="var">.c.</seg> <lb/>ſeu in puncto <seg type="var">.b.</seg> habebimus intentum, eò quod ſi produ-<lb/>cta fuerit <seg type="var">.e.f.</seg> etiam ab alia parte, vſque ad circunferentiam, habebimus <seg type="var">.b.e.</seg> æqua-<lb/>lem ipſi <seg type="var">.f.c.</seg> ex communi conceptu, & productum <seg type="var">.e.c.</seg> in <seg type="var">.e.b.</seg> æqualem quadra-<lb/>to ipſius <seg type="var">.e.d.</seg> ex .34. tertij, cum ex .3. eiuſdem <seg type="var">.e.d.</seg> medietas ſit chordæ arcus dupli <lb/><seg type="var">b.d</seg>.</s> </p> <p> <s xml:space="preserve">De lapſu verò lapidis verſus mundi centrum, dum ipſum attingere, ac præterire <lb/>poſſet, de quo me interrogas. </s> <s xml:space="preserve">Dico Nicolaum Tartaleam, nec non Franciſcum <lb/>Maurolicum rectè ſenſiſſe, malè verò Alexandrum Piccolhomineum, & exemplum <lb/>Maurolici optimum eſſe, quod tamen ſi capere non potes, crede ſaltem authoritati <lb/>bus talium virorum, qui tantum in ijs ſcientijs ſuperant ipſum Alexandrum Piccol-<lb/>homineum, quantum à Sole cætera ſuperantur aſtra.</s> </p> <p> <s xml:space="preserve">Lapis igitur ille tranſiret centrum, <choice><ex>reddiretque</ex><am>reddiretq́;</am></choice>, cum diminutione tamen motus im <lb/>preſſi, eo fermè modo vt ſcribunt iudicioſiſſimi illi viri, donec poſt multas reddi-<lb/>tiones ſurſum, <choice><ex>deorſumque</ex><am>deorſumq́;</am></choice> quieſceret circa centrum mundi. </s> <s xml:space="preserve">Lucidioris tamen intelli <pb facs="0381" n="369"/><fw type="head">EPISTOL AE.</fw> gentiæ gratia cogita ſilum illum (exempli adducti ab illis doctiſſimis viris) cui pon <lb/>dus appenſum eſt, æqualem eſſe axi orizontis, hoc eſt eius extremitatem immobi-<lb/>lem eſſe in primo mobili, & in ipſo zenit tui orizontis, </s> <s xml:space="preserve">tunc arcus motionis ipſius la <lb/>pidis per tantum interuallum, quantum eſt diameter terræ, inſenſibiliter differret à <lb/>linea recta, & cum lapis diſtans à centro mundi per ſemidiametrum terræ, iret re-<lb/><choice><ex>diretque</ex><am>diretq́;</am></choice>, vt ſcis, ergo idem faceret ſi ſilum longius eſſet per dictum terræ ſemidia-<lb/>metrum, ita vt poſſetipſum centrum attingere, nam differentia illa ſemidiametri <lb/>terræ, ferè nulla eſt reſpectu ſemidiametri ipſius primi mobilis.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">AN PENTAGONVS AB ALBERTO DVRERO <lb/>deſcriptus æquiangulus ſit.</head> <head rend="italics" xml:space="preserve">Conrado Neubart.</head> <p> <s xml:space="preserve">SI non credis Pentagonum ab Alberto Durero ſuper datam lineam deſi-<lb/>gnatum, æquiangulum non eſſe. </s> <s xml:space="preserve">Fingamus hic ſubiectam ſiguram ſimi-<lb/>lem ei quæ à Durero ponitur, in qua primò, ducta ſit linea <seg type="var">.o.a.</seg> & habe <lb/>bimus angulum <seg type="var">.a.o.b.</seg> graduum .60. talium qualium duo recti fuerint gra <num value="360">.<lb/>360.</num> vel .30. talium qualium duo recti fuerint .180. nam ex ſuppoſito, arcus <seg type="var">.a.b.</seg> eſt <lb/>ſexta pars totius circunferentiæ, angulus vero <seg type="var">.b.o.d.</seg> rectus eſt, eo quod <seg type="var">.b.o.q.</seg> rectus <lb/>etiam ſit, </s> <s xml:space="preserve">quare angulus <seg type="var">.d.o.a.</seg> reſiduus ex recto erit graduum .60. talium, ut rectus <lb/>eſt .90. angulus verò <seg type="var">.o.a.c.</seg> erit gra .15. eorundem.</s> </p> <p> <s xml:space="preserve">Ducatur deinde perpendicularis <seg type="var">.a.e.</seg> ad <seg type="var">.o.d.</seg> quæ vt ſinus anguli <seg type="var">.a.o.e.</seg> erit par-<lb/>tium .86602. talium qualium <seg type="var">.a.o.</seg> erit .100000. quæ quidem <seg type="var">.o.a.</seg> vt chorda arcus <seg type="var">.a.<lb/>o.</seg> eſt partium .51762. talium qualium <seg type="var">.a.d.</seg> vel <seg type="var">.a.c.</seg> ſemidiameter eſt .100000.</s> </p> <p> <s xml:space="preserve">Nam ſinus dimidij arcus <seg type="var">.a.o.</seg> (exi <lb/>ſtente <seg type="var">.a.o.</seg> graduum .30.) eſt <choice><ex>partium</ex><am>partiũ</am></choice> <num value="25881">.<lb/> <ptr xml:id="fig-0381-01a" corresp="fig-0381-01" type="figureAnchor"/> 25881.</num> ex quo <seg type="var">.a.e.</seg> erit partium <num value="44827">.<lb/>44827.</num> talium qualium <seg type="var">.a.d.</seg> erit <lb/>100000. vnde angulus <seg type="var">.a.d.o.</seg> cuius ſi <lb/>nus eſt <seg type="var">.a.e.</seg> erit graduum .26. min .38 <lb/>qui quidem angulus, ſumptus cum an <lb/>gulo <seg type="var">.a.o.d.</seg> erit gra .86. min .38. </s> <s xml:space="preserve">Dem <lb/>pta denique hac ſumma ex duobus <lb/>rectis gra .180. reliquum erit gra .93. <lb/>min .22. ideſt angulus <seg type="var">.o.a.d.</seg> cui addi <lb/>tus cum fuerit angulus <seg type="var">.o.a.c.</seg> gra .15. <lb/>talium, habebimus angulum <seg type="var">.c.a.d.</seg> <lb/>graduum .108. min .22. exuperantem <lb/>verum angulum pentagoni per min <num value="22">.<lb/>22.</num> vel ſic, cum inuentus fuerit angu-<lb/>lus <seg type="var">.a.d.o.</seg> gra .26. min .38. ſi ex vno re <lb/>cto demptus fuerit, relinquetur an-<lb/>gulus <seg type="var">.d.a.e.</seg> gra .63. min .22. qui qui-<lb/>dem collectus cum fuer it cum angu-<lb/>lo <seg type="var">.e.a.o.</seg> reſiduo ex re cto dempto angulo <seg type="var">.a.o.e.</seg> grad .60. qui <seg type="var">.e.a.o.</seg> eſt grad .30. & <pb facs="0382" n="370"/><fw type="head">IO. BAPT. BENED.</fw> e<unclear reason="illegible"/>tiam collectus cum angulo <seg type="var">.o.a.c.</seg> grad .15. hi tres anguli efficient angulum <seg type="var">.d.a.c.<lb/>d.</seg> ctum grad .108. min .22.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0381-01" corresp="fig-0381-01a"> <graphic url="0381-01"/> </figure> </div> </body> </floatingText> </div> <div type="unknown"> <head rend="italics" xml:space="preserve">Examinatio anguli <seg type="var">.u.</seg></head> <p> <s xml:space="preserve">Ducatur <seg type="var">.d.n.</seg> quam quidem <seg type="var">.d.n.</seg> cognoſcemus vt ſinus anguli <seg type="var">.d.o.n.</seg> gra .45. nam <lb/>angulus ei contrapoſitus <seg type="var">.q.o.p.</seg> eſt dimidium recti, </s> <s xml:space="preserve">quare <seg type="var">.d.n.</seg> erit partium .70710. <lb/>talium qualium <seg type="var">.d.o.</seg> fuerit .100000. ſed <seg type="var">.d.o.</seg> eſt partium .115270. qualium <seg type="var">.a.d.</seg> eſt <num value="100000">.<lb/>100000.</num> nam <seg type="var">.e.d.</seg> vt ſinus anguli <seg type="var">.e.a.d.</seg> gra .63. min .22. eſt partium .89389. <seg type="var">o.e.</seg> ve-<lb/>ro eſt partium .50000. talium qualium <seg type="var">.a.o.</seg> eſt .100000. vt ſinus anguli <seg type="var">.e.a.o.</seg> gra .30. <lb/>ſed vt <seg type="var">.a.o.</seg> eſt <choice><ex>partium</ex><am>partiũ</am></choice> .51762. </s> <s xml:space="preserve">hoc eſt vt <seg type="var">.a.d.</seg> eſt .100000. ipſa <seg type="var">.o.e.</seg> erit <choice><ex>partium</ex><am>partiũ</am></choice> .25881 <lb/>quæ iuncta cum fuerit cum <seg type="var">.e.d.</seg> efficiet <seg type="var">.d.o.</seg> partium .115270. vt dictum eſt, quapro-<lb/>pter cum <seg type="var">.d.n.</seg> ſit partium .70710. talium qualium <seg type="var">.d.o.</seg> fuerit .100000. ipſa <seg type="var">.d.n.</seg> erit <lb/>partium .81507. talium qualium <seg type="var">.d.o.</seg> erit .115270. ideſt qualium <seg type="var">.d.a.</seg> vel <seg type="var">.d.u.</seg> erit <lb/>100000. quæ quidem <seg type="var">.d.n.</seg> eſt ſinus anguli <seg type="var">.d.u.n.</seg> graduum ſcilicet .54. 36. cuius du-<lb/>plum erit gra .109. mi .12. debebattamen eſſe .108. <seg type="var">m.o</seg>.</s> </p> </div> <div type="unknown"> <head rend="italics" xml:space="preserve">Examinatio anguli <seg type="var">.d.</seg></head> <p> <s xml:space="preserve">Accipe angulum <seg type="var">.a.d.o.</seg> gra .26. <lb/> <ptr xml:id="fig-0382-01a" corresp="fig-0382-01" type="figureAnchor"/> min .38. vt ſupra, cui applica angu-<lb/>lum <seg type="var">.o.d.n.</seg> gra .45. min <seg type="var">.o.</seg> ſimul cum <lb/>angulo <seg type="var">.u.d.n.</seg> reſiduo ex recto gra <lb/>duum .35. minu .24. & conficies an-<lb/>gulum <seg type="var">.a.d.u.</seg> grad .107. minu .2. & <lb/>habebis propoſitum, quem tamen <lb/>oportebat eſſe gra .108. min <seg type="var">.o</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0382-01" corresp="fig-0382-01a"> <graphic url="0382-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Quod autem omnia rectè ſuppu-<lb/>tata ſint, ex ſumma omnium angulo <lb/>rum patere poteſt. </s> <s xml:space="preserve">nam collectis om <lb/>nibus quinque angulis <seg type="var">.a.c.d.f.u.</seg> ſi-<lb/>mul, hoc eſt grad .108. minu .22. cum <lb/>gra .107. min .2. cum grad .12. effi-<lb/>cient grad .540. min <seg type="var">.o.</seg> ſumma æqua <lb/>lis ſex angulis rectis.</s> </p> <pb facs="0383" n="371"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">ALIA DEMONSTRATIO NONÆ, ET DECIMÆ <lb/>ſecundi Euclidis.</head> <head rend="italics" xml:space="preserve">Petro Catenæ.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVamvis</hi> nona ac decima ſecundi Euclid. aliter à Comandino & Mau-<lb/>rolico demonſtratæ fuerint, nihilominus mihi etiam viſum eſt non nihil <lb/>meo moræ in eas tibi ſcribere, vt ſenſibiliter quoque cognoſcas il-<lb/>las veras eſſe.</s> </p> <p> <s xml:space="preserve">Eſto linc<unclear reason="illegible"/>a.a.b. pro nona propoſitione, diuiſa per æqualia in <seg type="var">.c.</seg> per inæqualia verò <lb/>in <seg type="var">.d.</seg> quadratum autem <seg type="var">.a.d.</seg> ſit <seg type="var">.d.e.</seg> quadratum verò <seg type="var">.d.b.</seg> ſit <seg type="var">.d.i.</seg> quadratum <seg type="var">.a.c.</seg> <lb/>ſit <seg type="var">.c.f.</seg> & quadratum <seg type="var">.c.d.</seg> ſit <seg type="var">.c.K.</seg> clarum enim erit <seg type="var">.K.h.</seg> æqualem exiſtere ipſi <seg type="var">.a.c.</seg> <lb/>ſeccetur igitur <seg type="var">.e.h.</seg> in <seg type="var">.g.</seg> ita vt <seg type="var">.h.g.</seg> ęqualis exiſtat ipſi <seg type="var">.K.h.</seg> vnde <seg type="var">.g.e.</seg> æqualls erit <lb/><seg type="var">c.d.</seg> perficiatur etiam quadratum <seg type="var">.h.n.</seg> vnde in totali quadrato <seg type="var">.a.h.</seg> habe bis <choice><ex>duplum</ex><am>duplũ</am></choice> <lb/>quadrati partis <seg type="var">.c.d.</seg> nempe <seg type="var">.c.K.</seg> et <seg type="var">.f.g.</seg> & quadratum <seg type="var">.a.u.</seg> cum gnomone <seg type="var">.u.g.h.k.</seg> <lb/>cui deficit quadratum æquale <seg type="var">.d.i.</seg> quadrato, vt ſint etiam duo quadrata partis <seg type="var">.a.c</seg>.</s> </p> <p> <s xml:space="preserve">In decima <choice><ex>autem</ex><am>aũt</am></choice> propoſitione, <choice><ex>quadratum</ex><am>quadratũ</am></choice> totalis lineæ <seg type="var">.a.d.</seg> ſit <seg type="var">.d.e.</seg> & lineæ <seg type="var">.b.d.</seg> ſit <seg type="var">.b.i.</seg> et <lb/><seg type="var">c.d.</seg> ſit <seg type="var">.d.n.</seg> et <seg type="var">.a.c.</seg> ſit <seg type="var">.c.f.</seg> et <seg type="var">.f.e.</seg> ſit <seg type="var">.e.u.</seg> vnde <seg type="var">.n.u.</seg> æquale erit quadrato <seg type="var">.b.i.</seg> vnde in qua <lb/>drato totali <seg type="var">.a.h.</seg> videbis duo quadrata æqualia <seg type="var">.f.c.</seg> et <seg type="var">.g.k.</seg> partis <seg type="var">.a.c.</seg> & quadratum <seg type="var">.c.<lb/>K.</seg> cum gnomone <seg type="var">.n.f.e.g.</seg> cui addito quadrato <seg type="var">.b.i.</seg> habebis duplum quadrati partis <seg type="var">.<lb/>c.d</seg>.</s> </p> <figure place="here"> <graphic url="0383-01"/> </figure> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE STELLA CASSIOPEIÆ.</head> <head rend="italics" xml:space="preserve">Annibali Raymundo Aſtrologo Peritißimo.</head> <p> <s xml:space="preserve"><hi rend="small caps">POstqvam</hi> tua doctiſſima ſcripta perlegi, conſideraui, quod ſi à mul-<lb/>titudine exhalationum in regione elementari acciderit anno .1572. & <lb/>1573. vt totos ſex menſes ab omnibus per vniuerſum terrarum orbem <lb/>viſa fuerit ſtella illa, quæ eſt in angulo ſeptentrionali quadrilateri Caſſio <pb facs="0384" n="372"/><fw type="head">IO. BAPT. BENED.</fw> peiæ tam lucida, vt ipſo lucifero videretur rutilantior <choice><ex>atque</ex><am>atq;</am></choice> cæterarum (abſque vlla <lb/>aſpectus diuerſitate) magis ſcintillans. </s> <s xml:space="preserve">Quî fieri poterat, vt ſtellæ quæ ab illa pa-<lb/>rum diſtant, alioqui multo maiores, non etiam illa clariores apparuerint? </s> <s xml:space="preserve">ſed ſi ali-<lb/>quis diceret eam exhalationem non ita fortaſſe dilatari, vt inter nos, & aliam ali-<lb/>quam ſtellam interponeretur. </s> <s xml:space="preserve">Tunc ego reſponderem neceſſariò ſequi debere ta-<lb/>lem exhalationem, tantam latitudinem occupare, quod aliquibus populis aliam <lb/><choice><ex>aliquam</ex><am>aliquã</am></choice> <choice><ex>ſtellam</ex><am>ſtellã</am></choice> circunuicinãhac ipſa de qua mentionem ſecimus redderet lucidiorem. <lb/></s> <s xml:space="preserve">Sed cum hoc perſpectum fuerit nulli, ſequebatur lucem illam ab ipſis exhalatio-<lb/>nibus elementaribus haud poſſe oriri: </s> <s xml:space="preserve">quod nobis ſcintillatio illa maxima perma-<lb/>gno fuit inditio, ſi phas eſt credere, <choice><ex>nam</ex><am>nã</am></choice> quo magis aliquod coeleſte corpus ſcintillat, <lb/>eo longius à nobis diſtare.</s> </p> <p> <s xml:space="preserve">Verum quoniam efflagitaſti à me vt aliquid circa huiuſce rei ſpeculationem tibi <lb/>ſcribam, idcirco tibi morem gerere volens paucis ſubiungam.</s> </p> <p> <s xml:space="preserve">Conſidera primo hanc ſubſcriptam primam figuram, in qua <seg type="var">.c.a.e.</seg> ſignatur pro <lb/>Globo terreſtri cuius <seg type="var">.i.</seg> centrum ſit et <seg type="var">.u.o.n.</seg> pro conuexo ignis, ſed <seg type="var">.K.x.s.</seg> pro orbe <lb/>octauo <seg type="var">.x.</seg> autem pro ſtella iam ſuperius dicta, quæ ſemper fuit, eſt, & erit, quamuis <lb/>cæteris tribus nunc obſcurior ſit. </s> <s xml:space="preserve">Accipiantur deinde duo loca in ſuperſicie terræ, <lb/>quę ſint <seg type="var">.c.</seg> et <seg type="var">.e.</seg> diametraliter inuicem oppoſita, ita quod circa eorum orizontes poſ <lb/>ſibile ſit ſtellam <seg type="var">.x.</seg> videre, radijs ipſius ſtellæ mediantibus <seg type="var">.x.n.e.</seg> et <seg type="var">.x.u.c.</seg> <choice><ex>quorum</ex><am>quorũ</am></choice> par-<lb/>tes <seg type="var">.n.c.</seg> et <seg type="var">.u.e.</seg> ita breues ſint, reſpectu eorum <choice><ex>totorum</ex><am>totorũ</am></choice>, vt vix ſexcenteſima pars ſit vna <lb/><choice><ex>quæque</ex><am>quæq;</am></choice> illarum, nec non <seg type="var">.c.e.</seg> ita breuis reſpectu ſemidiametri octauæ ſphæræ, quod <lb/>vix ſit vna ex partibus decemmillibus, vt ſcis, ſequitur quod recta terminata ab <seg type="var">.u.</seg> <lb/>et <seg type="var">.n.</seg> minor ſenſibiliter non ſit ipſo terræ diametro <seg type="var">.c.e.</seg> cum duo hæc interualla ex <lb/>triangulorum ſimilitudine ſe habeant vt <seg type="var">.x.i.</seg> ad <seg type="var">.x.o.</seg> hoc eſt ferè vt .602. ad .601. vn-<lb/>de anguli <seg type="var">.n.e.c.</seg> et <seg type="var">.u.c.e.</seg> à rectis minime differre videbuntur, cum eorum differen-<lb/>tia certo modo minima ſit. </s> <s xml:space="preserve">ductę poſtea <choice><ex>cum</ex><am>cũ</am></choice> fuerint duæ diagonales <seg type="var">.e.u.</seg> et <seg type="var">.n.c.</seg> <choice><ex>termi- nabunt</ex><am>termi-nabũt</am></choice> angulos <seg type="var">.n.e.u.</seg> et <seg type="var">.e.n.c.</seg> <choice><ex>inuicem</ex><am>inuicẽ</am></choice> ferè ęquales, <choice><ex>idem</ex><am>idẽ</am></choice> aſſero de angulis <seg type="var">.u.c.n.</seg> et <seg type="var">.e.u.c</seg>.</s> </p> <p> <s xml:space="preserve">Supponatur nunc primò tuam exhalationem ſublimatam eſſe ad ſupremas par-<lb/>tes elementaris regionis circum circa lineam <seg type="var">.o.i</seg>: </s> <s xml:space="preserve">tunc clarum eſſet quod ſi ratione hu <lb/>iuſmodi exhalationis ſtella <seg type="var">.x.</seg> ita lucida viſa fuerit tam aſpicientibus ab <seg type="var">.e.</seg> quam ab <lb/>c. exhalatio minoris latitudinis quam <seg type="var">.u.n.</seg> eſſe non poterat, hoc eſt, quam terræ dia-<lb/>meter, cum idem in longitudine ferè ſit, ſed punctum <seg type="var">.u.</seg> ſatis videri poteſt ab oculo <lb/>in <seg type="var">.e.</seg> & punctum <seg type="var">.n.</seg> ab oculo in <seg type="var">.c.</seg> vt alias tibi probaui, ratione refractionis radiorum <lb/>per diuerſa diafana tranſeuntium. </s> <s xml:space="preserve">Nunc producti cum fuerint ij duo radij <seg type="var">.e.u.</seg> et <seg type="var">.c.<lb/>n.</seg> vſque ad octauum orbem ad puncta <seg type="var">.s.</seg> et <seg type="var">.K.</seg> reliquum erit nos videre quantitates <lb/>graduum arcus <seg type="var">.s.x.</seg> et <seg type="var">.k.x.</seg> ſed <seg type="var">.s.x.</seg> ſubiacet a ngulo <seg type="var">.s.e.x.</seg> et <seg type="var">.k.x.</seg> angulo <seg type="var">.k.c.x.</seg> qui qui <lb/>quidem anguli nihil differunt ſenſibiliter ac ſi eſſent in centro <seg type="var">.i</seg>. </s> <s xml:space="preserve">Et cum ſuperius di-<lb/>xerimus angulos <seg type="var">.s.e.x.</seg> et <seg type="var">.k.c.x.</seg> ſenſibiliter mi nime differre ab angulis <seg type="var">.c.n.e.</seg> et <seg type="var">.e.<lb/>u.c.</seg> ſi cognouerimus quantitatem iſtorum, cognita etiam nobis erit quantitas <lb/>ill orum.</s> </p> <p> <s xml:space="preserve">Cum igitur ſemidiameter elementaris regionis maior ſit ſemidiametro terræ, vt <lb/>33. ad vnum, & cogitata <seg type="var">.c.n.</seg> vt dicta ſemidiameter, quia ſenſibiliter ab ea minime <lb/>diffe<unclear reason="illegible"/>rt, nunc ſi ſupponatur dicta <seg type="var">.n.c.</seg> vt <choice><ex>baſistrianguli</ex><am>baſistriãguli</am></choice> orthogonij eſſe <choice><ex>partium</ex><am>partiũ</am></choice> .100000 <lb/>& dixerimus ſi <seg type="var">.c.n.</seg> vt partium .33. præbet nobis <seg type="var">.c.e.</seg> duarum partium, quid nobis <lb/>pręſtabit eadem <seg type="var">.c.n.</seg> vt partium .100000. vnde proueniet nobis <seg type="var">.c.e.</seg> vt <choice><ex>partium</ex><am>partiũ</am></choice> .6060. <lb/>cuius angulus <seg type="var">.c.n.e.</seg> erit graduum .3. & min .29. ita etiam erit angulus <seg type="var">.k.</seg> e<unclear reason="illegible"/> <seg type="var">.x.</seg> cuius ar- <pb facs="0385" n="373"/><fw type="head">EPISTOL AE.</fw> c<unclear reason="illegible"/>us <seg type="var">.s.x.</seg> eorundem graduum erit, <lb/> <ptr xml:id="fig-0385-01a" corresp="fig-0385-01" type="figureAnchor"/> & minutorum. </s> <s xml:space="preserve">idem dico de arcu <seg type="var">.<lb/>x.k</seg>. </s> <s xml:space="preserve">Sed circa dictam ſtellam om-<lb/>nes aliæ non diſtant huiuſmodi in <lb/>teruallo.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0385-01" corresp="fig-0385-01a"> <graphic url="0385-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nihilominus nec tu nec alij pe <lb/>ripatetici qui hanc ſequuti ſunt <lb/>opinionem exhalationum, ad ſer <lb/>uandam nullitatem diucrſitatis <lb/>aſpectus, affirmant poſſe tam lon <lb/>ge à terra <choice><ex>aſcendere</ex><am>aſcẽdere</am></choice> exhalationes, <lb/>imo nec attingere ſupremas tertię <lb/>regionis a eris partes, ita ut non <choice><ex>in- grediantur</ex><am>in-grediãtur</am></choice> ſuum <choice><ex>igneum</ex><am>igneũ</am></choice> orbem, qui <lb/>quidem orbis ſecundum illorum <lb/>opinionem incipit non valde lon-<lb/>gè à ſuperſicie terræ, vt <choice><ex>iam</ex><am>iã</am></choice> in mea <lb/>conſideratione contra Antonium <lb/>Bergam probaui, ſed demus, <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>dictæ exhalationes aſcenderint <lb/>per decem ſemidiametros terræ, <lb/>diſcurrendo poſtea ſic, cum <seg type="var">.c.n.</seg> ut <lb/>decem, nobis dat <seg type="var">.c.e.</seg> vt duo, <choice><ex>quid</ex><am>ꝗd</am></choice> <lb/>dabit nobisipſa <seg type="var">.c.n.</seg> vt .100000. <lb/>& proueniet nobis <seg type="var">.c.e.</seg> vt .200000. <lb/>cuius ſinus angulus erit gra .11. mi <num value="32">.<lb/>32.</num> & ita erunt anguli <seg type="var">.s.e.x.</seg> et <seg type="var">.k.<lb/>c.x.</seg> & ſic eorum arcus <seg type="var">.s.x.</seg> et <seg type="var">.k.x.</seg> <lb/>ſed quis vnquam dubitabit <choice><ex>quod</ex><am>ꝙ</am></choice> in <lb/>tanto interuallo à dicta ſtella non <lb/>fint aliæ multæ ipſa maiores? </s> <s xml:space="preserve">li-<lb/>neas vero <seg type="var">.e.o.r.</seg> et <seg type="var">.c.o.t.</seg> duxi, vt <lb/>videres effectum maioris aſpectus <lb/>diuerſitatis ab oculis <seg type="var">.e.</seg> et <seg type="var">.c.</seg> in cir-<lb/>culo altitudinis quando <seg type="var">.o.</seg> fuiſſet <lb/>punctum illud lucidiſſimum, & <lb/>non <seg type="var">.x</seg>.</s> </p> <pb facs="0386" n="374"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">At poterit aliquis mihi obijcere quod cum <seg type="var">.i.o.</seg> fuiſſet longior <seg type="var">.i.e.</seg> per decem vi-<lb/>ces tantummodo, exiſtente oculo in <seg type="var">.e.</seg> uel <seg type="var">.c.</seg> per gradus .90. ab <seg type="var">.a</seg>. </s> <s xml:space="preserve">tunc punctus <seg type="var">.u.</seg> vel <lb/>n. ab ipſo oculo non videretur ob terræ globoſitatem. </s> <s xml:space="preserve">Imaginemur igitur à puncto <lb/>u. recta <seg type="var">.u.b.</seg> tangens quartam <seg type="var">.a.e.</seg> in puncto <seg type="var">.b.</seg> vt in ſecunda figura videre eſt, in qua <lb/>ducantur <seg type="var">.c.b</seg>: <seg type="var">i.b</seg>: et <seg type="var">.i.u.</seg> quæ <seg type="var">.i.u.</seg> ſecabit arcum <seg type="var">.c.</seg> <lb/> <ptr xml:id="fig-0386-01a" corresp="fig-0386-01" type="figureAnchor"/> b. in puncto <seg type="var">.p.</seg> per æqualia et <seg type="var">.c.b.</seg> ſimiliter in pun-<lb/>cto <seg type="var">.y.</seg> quod nulli dubium eſt, cum <seg type="var">.c.u.</seg> æqualis ſit <seg type="var">.<lb/>u.b.</seg> ex .35. tertij Euclidis, </s> <s xml:space="preserve">unde ex octaua primi an-<lb/>gulus <seg type="var">.c.i.u.</seg> æqualis erit angulo <seg type="var">.u.i.b.</seg> & ideo arcus <seg type="var">.<lb/>c.p.</seg> æquabitur arcui <seg type="var">.p.b.</seg> ſed ex .4. primi <seg type="var">.c.y.</seg> æqua <lb/>lis erit ipſi <seg type="var">.y.b</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0386-01" corresp="fig-0386-01a"> <graphic url="0386-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Nunc ſuppoſita <seg type="var">.c.i.</seg> decima parte ipſius <seg type="var">.c.u.</seg> nemi <lb/>ni dubium erit quod cum <seg type="var">.u.i.</seg> ſubtendatur angulo <lb/>recto <seg type="var">.u.c.i.</seg> (iam ſupra diximus angulum <seg type="var">.c.</seg> ſenſibi-<lb/>liter minime differre à recto) ipſa vt ſinus totus erit <lb/>partium .100000. cuius quadratum cum diuiſum <lb/>fuerit in partes æquales centum & <choice><ex>vnam</ex><am>vnã</am></choice>, illarum vna <lb/>æqualis erit quadrto <seg type="var">.c.i.</seg> reliquę vero quadrato ip-<lb/>ſius <seg type="var">.u.c.</seg> ex proportione duplicata quadratorum ad <lb/>eam quam continent eorum latera. </s> <s xml:space="preserve">Sed quadra-<lb/>tum ipſius <seg type="var">.u.i.</seg> eſt partium .10000000000. </s> <s xml:space="preserve">quare <lb/>quadratum <seg type="var">.c.i.</seg> erit .99009900. cuius radix <seg type="var">.c.i.</seg> <lb/>erit partium .9950. vnde quadratum ipſius <seg type="var">.c.u.</seg> erit <lb/>partium .9900990100. cuius radix <seg type="var">.u.c.</seg> erit <choice><ex>partium</ex><am>partiũ</am></choice> <lb/>99500. vnde angulus <seg type="var">.c.i.u.</seg> erit graduum .84. & mi <lb/>nu .17. & angulus <seg type="var">.c.u.i.</seg> qui reſpondet ſinui <seg type="var">.c.i.</seg> erit <lb/>gra .5. & min .43. cuius duplum, hoc eſt angulus <seg type="var">.c.<lb/>u.b.</seg> erit grad .11. min .26. æqualis ferè angulo iam <lb/>ſupradicto. </s> <s xml:space="preserve">ſed <seg type="var">.c.y.</seg> ſinus anguli <seg type="var">.c.i.y.</seg> erit ſimiliter <lb/>partium .99500. talium vt <seg type="var">.c.i.</seg> ſunt .100000. ſed vt <lb/><seg type="var">c.i.</seg> eſt partium .9950. </s> <s xml:space="preserve">tunc <seg type="var">.c.y.</seg> erit partium .9900 <lb/>hoc eſt quaſi decima pars ipſius <seg type="var">.c.u.</seg> </s> <s xml:space="preserve">quare ſi ocu-<lb/>lus in <seg type="var">.e.</seg> non videbit punctum <seg type="var">.u.</seg> hoc punctum be-<lb/>ne videbitur ab oculo in <seg type="var">.b.</seg> abſque ſenſibili dimi-<lb/>nutione anguli in puncto <seg type="var">.u.</seg> vt probauimus.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE MAGNITVDINIBVS FIGVRARVM <lb/>iſoperimetrarum.</head> <head rend="italics" xml:space="preserve">Domino Ioanni Mariæ Agatio.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVamvis</hi> à Theone ſupra Ptolomei Almageſtum ſufficienter traditum <lb/>ſit de magnitudinibus figurarum iſopetimetrarum, nihilominus vt tibi <lb/>morem geram, ea nunc ſcribo, quæ mihi in mentem venerunt contra <lb/><choice><ex>Alexandrum</ex><am>Alexãdrum</am></choice> Piccolhomineum, <choice><ex>antequam</ex><am>antequã</am></choice> aliquid ipſius Theonis vidiſſem <pb facs="0387" n="375"/><fw type="head">EPISTOLAE.</fw> Alexander Piccolhomineus in libro primo de mundi ſphæra vbi tractat de <choice><ex>caeliro- tunditate</ex><am>cęliro-tunditate</am></choice>, ita inquit.</s> </p> <quote xml:lang="it"> <s xml:space="preserve">Oltre di queſto, douendo il decimo cielo contenere & in ſe chiudere tutte le co-<lb/>ſe, è conueneuol coſa il penſare, che foſſe fatto di quella più capace figura che eſ-<lb/>ſer poſſa, la qual è la figura rotunda, però che ſi può trar da molti luoghi d'Euclide <lb/>che ſi come ſe noi ciimmagineremo più figure ſuperficiali talmente che tutte le li-<lb/>nee de l'vna congionte inſieme, ſieno vguali à tutte le linee pur inſiememente com <lb/>poſte di qual ſi voglia de l'altre figure, ne ſeguirà, che quella figura ſarà più capa-<lb/>ce la qual haurà manco angoli, & quella capaciſſima che ſarà ſenza alcuno come è <lb/>la figura circolare, & c.</s> </quote> <p> <s xml:space="preserve">Cogitemus igitur primò de triangulo æquilate-<lb/>ro & quadrato iſoperimetris, ſit enim triangulus æ-<lb/>quilaterus <seg type="var">.o.b.g.</seg> quadratum verò <seg type="var">.b.l.</seg> quorum pe-<lb/>riferiæ inuicem æquales ſint. </s> <s xml:space="preserve">Dico quadratum ma-<lb/> <ptr xml:id="fig-0387-01a" corresp="fig-0387-01" type="figureAnchor"/> ioris ſuperficiei eſſe ipſo triangulo. </s> <s xml:space="preserve">Accipio pri-<lb/>mum lineam <seg type="var">.f.h.</seg> eiuſdem longitudinis quæ vnius <lb/>periferiæ dictarum figurarum, quam punctis <seg type="var">.r.K.</seg> <lb/>mediantibus diuido in tres ęquas partes, in quatuor <lb/>verò mediantibus punctis <seg type="var">.l.x.i.</seg> vnde proportio to-<lb/>tius <seg type="var">.f.h.</seg> ad <seg type="var">.K.h.</seg> erit vt <seg type="var">.l.h.</seg> ad <seg type="var">.i.h.</seg> ideſt tripla, & per <lb/>16. quinti erit <seg type="var">.f.h.</seg> ad <seg type="var">.l.h.</seg> vt <seg type="var">.k.h.</seg> ad <seg type="var">.i.h.</seg> per .19. verò <lb/><seg type="var">f.h.</seg> ad <seg type="var">.f.l.</seg> vt <seg type="var">.K.h.</seg> ad <seg type="var">.K.i.</seg> ſed <seg type="var">.f.l.</seg> eſt quarta pars ip-<lb/>ſius <seg type="var">.f.h.</seg> ergo <seg type="var">.k.i.</seg> erit quarta pars ipſius <seg type="var">.k.h</seg>. </s> <s xml:space="preserve"><choice><ex>Conium</ex><am>Coniũ</am></choice> <lb/>gantur enim ambo iſtæ figuræ vt hic inferius vides, <lb/>vnde <seg type="var">.a.g.</seg> erit quarta pars ipſius <seg type="var">.b.g.</seg> diuiſa poſtea <seg type="var">.<lb/>b.g.</seg> per æqualia in <seg type="var">.c.</seg> erit <seg type="var">.a.c.</seg> æqualis <seg type="var">.a.g</seg>. </s> <s xml:space="preserve">Ducatur <lb/>deinde <seg type="var">.o.c.</seg> quę per .8. primi, nec <choice><ex>non</ex><am>nõ</am></choice> ex definitione, <lb/>perpendicularis erit ipſi <seg type="var">.b.g.</seg> ergo etiam <choice><ex>quadratum</ex><am>quadratũ</am></choice> <lb/>b q. ſupra <seg type="var">.b.g.</seg> <choice><ex>producoque</ex><am>producoq́;</am></choice> <seg type="var">.o.c.</seg> vſque ad <seg type="var">.m.</seg> nam nul <lb/>li dubium eſt quin <seg type="var">.o.c.</seg> breuior ſit <seg type="var">.o.g.</seg> ex .18. vel .48 <lb/>primi cui æquatur <seg type="var">.q.g.</seg> diuido etiam <seg type="var">.c.m.</seg> per æqua <lb/>lia in puncto <seg type="var">.e.</seg> <choice><ex>ducoque</ex><am>ducoq́;</am></choice> <seg type="var">t.e.p.</seg> æquidiſtantem <seg type="var">.b.g.</seg> <lb/>vnde habebimus duo quadrata <seg type="var">.e.g.</seg> et <seg type="var">.e.b.</seg> ſed <lb/>quadratum <seg type="var">.b.l.</seg> æquatur quadrato ipſius <seg type="var">.c.a.</seg> <lb/>cum duplo illius quod fit ex <seg type="var">.b.c.</seg> in <seg type="var">.c.g.</seg> vt patet <lb/>ex .9. ſecundi, hoc eſt æquatur quadrato <seg type="var">.c.a.</seg> & re-<lb/>ctangulo <seg type="var">.t.g</seg>. </s> <s xml:space="preserve">Deinde vt ſe habet <seg type="var">.p.g.</seg> ad <seg type="var">.o.e.</seg> ita ſe habet <seg type="var">.u.p.</seg> ad <seg type="var">.u.e.</seg> ex ſimilitudine <lb/>triangulorum. </s> <s xml:space="preserve">Sed <seg type="var">.p.g.</seg> maior eſt ipſa <seg type="var">.o.e.</seg> cum <seg type="var">.p.g.</seg> æqualis ſit <seg type="var">.e.m.</seg> </s> <s xml:space="preserve">quare triangu-<lb/>lus <seg type="var">.u.g.p.</seg> maior erit triangulo <seg type="var">.o.e.u.</seg> ex .17. ſexti. </s> <s xml:space="preserve">Similiter dico maiorem eſſe trian <lb/>gulum <seg type="var">.b.d.t.</seg> triangulo <seg type="var">.e.o.d.</seg> vnde ſequitur rectangulum <seg type="var">.t.g.</seg> maiorem eſſe triangu-<lb/>lo <seg type="var">.b.o.g.</seg> ſed quadratum <seg type="var">.b.l.</seg> eſt etiam maior ipſo rectangulo <seg type="var">.t.g.</seg> ex quadrato ipſius <lb/><seg type="var">c.a.</seg> vt diximus, tanto igitur maior erit triangulo <seg type="var">.b.o.g</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0387-01" corresp="fig-0387-01a"> <graphic url="0387-01"/> </figure> </div> </body> </floatingText> <pb facs="0388" n="376"/> <fw type="head">IO. BABPT. BENED.</fw> <p> <s xml:space="preserve">Poſſumus etiam probare quod periferia quadrati æqualis triangulo æquilatero <lb/>minor ſit periferia ipſius trianguli æquilateri. </s> <s xml:space="preserve">Cogita triangulum æquilaterum hic <lb/>ſubſcriptum <seg type="var">.d.l.q.</seg> cuius baſis <seg type="var">.l.q.</seg> diuiſa ſit per æqualia à perpendiculari <seg type="var">.d.o.</seg> <choice><ex>deſcri- ptumque</ex><am>deſcriptũq́;</am></choice> ſit rectangulum <seg type="var">.o.g.</seg> quod æquale erit triangulo <seg type="var">.d.l.q.</seg> ſed periferia trianguli <lb/>maior eſt periferia rectanguli, nam <seg type="var">.l.q.</seg> æqualis eſt <seg type="var">.o.q.</seg> cum <seg type="var">.d.g.</seg> ſed <seg type="var">.q.d.</seg> maior eſt <seg type="var">.o.<lb/>d.</seg> ex .18. primi, vnde <seg type="var">.l.d.</seg> maior etiam <seg type="var">.q.g.</seg> cum ex .34. dicti latera oppoſita ipſius re <lb/>ctanguli ſint inuicem æqualia, accipiamus poſtea <seg type="var">.e.c.</seg> æqualem <seg type="var">.o.d.</seg> et <seg type="var">.c.h.</seg> indire-<lb/>ctum æqualem <seg type="var">.o.q.</seg> circa quem diametrum <seg type="var">.e.h.</seg> intelligatur circulus <seg type="var">.e.i.h.k.</seg> et. à pun<lb/>cto <seg type="var">.c.</seg> dirigatur perpendicularis <seg type="var">.k.i.</seg> ad <seg type="var">.e.h.</seg> vnde ex .3. tertij <seg type="var">.c.i.</seg> æqualis erit <seg type="var">.c.k.</seg> & ex <lb/>34. quod fit ex <seg type="var">.c.i.</seg> in <seg type="var">.c.k.</seg> hoc eſt quadratum ipſius <seg type="var">.c.i.</seg> æquale erit ei quod fit .ex <seg type="var">.e.c.</seg> <lb/>in <seg type="var">.c.h.</seg> hoc eſt rectangulo <seg type="var">.g.o.</seg> hoc eſt triangulo <seg type="var">.d.l.q.</seg> ſed <seg type="var">.e.h.</seg> eſt dimidium perife-<lb/>rię ipſius rectanguli <seg type="var">.g.o.</seg> quæ minor eſt di midio periferiæ trianguli <seg type="var">.d.l.q.</seg> vt vidimus <lb/>et <seg type="var">.i.k.</seg> eſt dimidium periferię quadrati ipſius <seg type="var">.i.c.</seg> & minor etiam ipſa <seg type="var">.e.h.</seg> ex .14. tertij <lb/></s> <s xml:space="preserve">quare verum eſt propoſitum.</s> </p> <figure place="here"> <graphic url="0388-01"/> </figure> <p> <s xml:space="preserve">Sed quando periferiæ ſunt inuicem æquales, poſſumus etiam breuiter videre id <lb/>quod ſupradiximus, hoc eſt, quod quadratum, maius ſit triangulo æquilatero. </s> <s xml:space="preserve">Nam <lb/>cum <seg type="var">.b.g.</seg> ſeſquitertia ſit ad <seg type="var">.b.a.</seg> ergo <seg type="var">.b.g.</seg> erit vt .4. et <seg type="var">.b.a.</seg> ut .3. vnde <seg type="var">.b.q.</seg> erit vt .16 <lb/>et <seg type="var">.b.l.</seg> vt .9. et <seg type="var">.c.q.</seg> vt .8. </s> <s xml:space="preserve">quare <seg type="var">.b.l.</seg> maius erit ipſo <choice><ex>rectangulo</ex><am>rectãgulo</am></choice> <seg type="var">.c.q.</seg> ſed <seg type="var">.c.q.</seg> maius eſt <choice><ex>triam</ex><am>triã</am></choice> <lb/>gulo <seg type="var">.b.o.g.</seg> cum <seg type="var">.q.g.</seg> quæ æqualis eſt <seg type="var">.o.g.</seg> maior ſit <seg type="var">.o.c.</seg> ex .18. vel penultima primi, <lb/>nam ſi <seg type="var">.q.g.</seg> æqualis eſſet <seg type="var">.o.c.</seg> </s> <s xml:space="preserve">tunc <seg type="var">.c.q.</seg> æqualis eſſet triangulo <seg type="var">.b.o.g.</seg> ex .41. primi.</s> </p> <p> <s xml:space="preserve">Alia etiam via maiores noſtri vſi ſunt quæ generalis eſt vt in Theone ſupra Al-<lb/>mageſtum videre eſt, medijs perpendicularibus à centris ad latera figurarum, ſed <lb/>quia <choice><ex>differentia</ex><am>differẽtia</am></choice> longitudinum ipſarum perpendicularium alio medio inueniri poteſt, <lb/>eo quo ipſi vſi ſunt, prætermittere nolo quin tibi ſcribam.</s> </p> <p> <s xml:space="preserve">Ego enim ita diſcurro.</s> </p> <p> <s xml:space="preserve">Sint duæ figuræ iſoperimetrę æquilaterę & æquiangulæ, puta primò trian-<lb/>gulum & quadratum quorum centra ſint <seg type="var">.e.</seg> et <seg type="var">.o.</seg> à quibus centris ad latera ſint per-<lb/>pendiculares <seg type="var">.e.n.</seg> et <seg type="var">.o.u.</seg> vnde <seg type="var">.n.</seg> et <seg type="var">.u.</seg> diuident latera per æqualia vt ſcis, ducantur <lb/>poſtea <seg type="var">.e.t.</seg> et <seg type="var">.o.a.</seg> ad angulos dictorum laterum, vnde habebimus angulum <seg type="var">.o.a.u.</seg> <choice><ex>di- midium</ex><am>di-midiũ</am></choice> recti, et <seg type="var">.e.t.n.</seg> tertia pars vnius recti, vt ex te ipſo videre potes, </s> <s xml:space="preserve">quare angulus <pb facs="0389" n="377"/><fw type="head">EPISTOLAE.</fw> a. ſeſquialter erit angulo <seg type="var">.t.</seg> quod vt clarius videas cogita lineam <seg type="var">.b.d.</seg> cuius medietas <lb/>ſit <seg type="var">.c.d.</seg> tertia verò pars illius ſit <seg type="var">.g.d.</seg> </s> <s xml:space="preserve">tunc dico <seg type="var">.c.d.</seg> ſeſquialteram eſſe ipſi <seg type="var">.g.d.</seg> ſit enim <lb/><seg type="var">f.d.</seg> duplum ipſius <seg type="var">.g.d.</seg> </s> <s xml:space="preserve">quare <seg type="var">.f.d.</seg> erunt duæ tertiæ totius lineę <seg type="var">.b.d.</seg> & quia eadem pro <lb/>portio eſt totius <seg type="var">.b.d.</seg> ad <seg type="var">.c.d.</seg> quæ <seg type="var">.f.d.</seg> ad <seg type="var">.g.d.</seg> ergo permutando eadem erit totius <seg type="var">.b.<lb/>d.</seg> ad <seg type="var">.f.d.</seg> quæ <seg type="var">.c.d.</seg> ad <seg type="var">.g.d</seg>. </s> <s xml:space="preserve">Sed <seg type="var">.b.d.</seg> ad <seg type="var">.f.d.</seg> ſeſquialtera eſt, verum igitur erit quod an-<lb/>gulus <seg type="var">.a.</seg> ſeſquialter ſit ipſi <seg type="var">.t.</seg> deinde <seg type="var">.t.n.</seg> eſt ſeſquitertia ipſi <seg type="var">.a.u.</seg> vt ſuperius vidimus .<lb/>in eorum duplis. </s> <s xml:space="preserve">ſcimus etiam <seg type="var">.n.e.</seg> eſſe dimidium ipſius <seg type="var">.t.e.</seg> co quod cum <seg type="var">.e.t.n.</seg> ſit <lb/>tertia pars vnius recti, angulus, <seg type="var">t.e.n.</seg> erit duo tertia vnius recti, vnde <seg type="var">.e.n.</seg> erit latus. <lb/>exagoni æquilateris inſcriptibilis circulo cuius diameter ſit <seg type="var">.e.t.</seg> </s> <s xml:space="preserve">quare <seg type="var">.e.t.</seg> dupla erit <lb/>ipſi <seg type="var">.e.n.</seg> in longitudine, ſed quadrupla in potentia: </s> <s xml:space="preserve">t.n. vero tripla in potentia ipſi <seg type="var">.n.<lb/>e.</seg> ex penultima primi, quæ omnia etiam ex .8. tertijdecimi. Eucli. elicere potes, ſed <lb/><seg type="var">c.n.</seg> erat ſexquitertia ipſi <seg type="var">.a.u.</seg> in longitudine, hoc eſt ipſi <seg type="var">.o.u.</seg> nam <seg type="var">.o.u.</seg> æqualis eſt ipſi <lb/><seg type="var">a.u</seg>. </s> <s xml:space="preserve">quare <seg type="var">.n.t.</seg> erit minus quam dupla in potentia ipſi <seg type="var">.o.u.</seg> hoc eſt, vt .16. ad .9. ergo <lb/>maior proportio erit ipſius <seg type="var">.t.n.</seg> in potentia ad <seg type="var">.n.e.</seg> quam ad <seg type="var">.o.u.</seg> </s> <s xml:space="preserve">quare etiam in lon <lb/>gitudine, maior proportio erit ipſius <seg type="var">.t.n.</seg> ad <seg type="var">.n.e.</seg> quam ad <seg type="var">.o.u.</seg> vnde <seg type="var">.o.u.</seg> longior erit <lb/>ipſa <seg type="var">.n.e.</seg> quod eſt propoſitum.</s> </p> <p> <s xml:space="preserve">Sed ſi <seg type="var">.o.a.u.</seg> eſſet pentagonus æquilaterus & æquiangulus, ſimiliter probabo per-<lb/>pendicularem <seg type="var">.o.u.</seg> longiorem eſſe <seg type="var">.n.e.</seg> ipſius trianguli æquilateri, dummodo ſint iſo-<lb/>perimetrę. </s> <s xml:space="preserve">Sit enim <seg type="var">.a.u.</seg> dimidium lateris pentagoni ex ſuppoſito, cuius centrum ſit <lb/>o. </s> <s xml:space="preserve">tunc proportio <seg type="var">.t.n.</seg> ad <seg type="var">.a.u.</seg> erit ſuperbipartienstertias, vt ex ordine iam hic ſupradi <lb/>cto à te facillimè elicere potes, hoc eſt, vt .5. ad .3. et <seg type="var">.a.u.</seg> minor erit <seg type="var">.o.u.</seg> eo quod <lb/>angulus <seg type="var">.o.</seg> minor erit angulo <seg type="var">.a.</seg> nam angulus <seg type="var">.o.</seg> erit quinta pars <choice><ex>duorum</ex><am>duorũ</am></choice> rectorum, hoc <lb/>eſt duæ quintæ vnius recti, vnde angulus <seg type="var">.a.</seg> reſiduum vnius recti erit tres quin-<lb/>tæ vnius recti, </s> <s xml:space="preserve">quare angulus <seg type="var">.a.</seg> maior ericangulo <seg type="var">.o.</seg> & conſequenter latus <seg type="var">.o.u.</seg> ma-<lb/>ius latere <seg type="var">.a.u.</seg> ſed <seg type="var">.t.n.</seg> minor eſt quam tripla in potentia ad <seg type="var">.a.u.</seg> eo quod erit vt .25. <lb/>ad .9. cum in longitudine ſit vt .5. ad .3. ſed dicta <seg type="var">.t.n.</seg> tripla eſt in potentia ad <seg type="var">.e.n.</seg> qua-<lb/>re <seg type="var">.a.u.</seg> maior erit ipſa <seg type="var">.e.n.</seg> ſed <seg type="var">.o.u.</seg> maior eſt ipſa <seg type="var">.a.u.</seg> vt diximus, igitur multo magis <seg type="var">.<lb/>o.u.</seg> maior eſt ipſa <seg type="var">.a.u.</seg> vt <choice><ex>diximus</ex><am>diximꝰ</am></choice> & <choice><ex>conſequenter</ex><am>cõſequẽter</am></choice> multo magis <seg type="var">.o.u.</seg> maior erit ipſa <seg type="var">.n.e</seg>.</s> </p> <p> <s xml:space="preserve">Quotieſcunque enim cognoſcimus proportionem anguli <seg type="var">.o.</seg> ad angulum <seg type="var">.a.</seg> quod <lb/>quidem facillimum eſt, nec non proportionem <seg type="var">.t.n.</seg> ad <seg type="var">.a.u.</seg> quod, etiam illico cogno-<lb/>ſcitur, </s> <s xml:space="preserve">tunc exſcientia cordarum & arcuum omnia etiam facillimè innueniuntur. <lb/></s> <s xml:space="preserve">Verum circa <choice><ex>triangulum</ex><am>triangulũ</am></choice> æquilaterum, & pentagonum, alium <choice><ex>modum</ex><am>modũ</am></choice> inueni, ſed aliquan <lb/>tulum prolixiorem.</s> </p> <figure place="here"> <graphic url="0389-01"/> </figure> <pb facs="0390" n="378"/> <fw type="head">IO. BAPT. BENED.</fw> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De incommenſur abilitate, in longitudine perpendicu-<lb/>laris trianguli æquilateri cum eiuſdem latere.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">ID quod à me poſtulas eſt omnino impoſſibile, velles enim duos numeros inueni <lb/>re inter ſe ita ſe habentes, vt ſe habent perpendicularis in triangulo æquilatero <lb/>cum vno eius laterum, quod vero hoc fieri non poſſit, conſidera in figura præcedenti <lb/>triangulum æquilaterum <seg type="var">.d.l.q.</seg> cuius perpendicularis ſit <seg type="var">.d.o.</seg> quæ diuidit <seg type="var">.l.q.</seg> per <lb/>æqualia in <seg type="var">.o.</seg> vnde ex .4. ſecundi Euclidis, quadratum <seg type="var">.l.q.</seg> (ideſt <seg type="var">.d.q.</seg>) quadruplum <lb/>erit quadrato <seg type="var">.o.q.</seg> & ex penultima primi ęquale quadratis <seg type="var">.d.o.</seg> et <seg type="var">.o.q.</seg> </s> <s xml:space="preserve">quare erit ſeſ-<lb/>quitertium quadrato ipſius <seg type="var">.d.o.</seg> & ita quadratum <seg type="var">.d.o.</seg> erit triplum quadrato ipſius <seg type="var">.<lb/>o.q.</seg> hæe autem proportiones non ſunt vt numeri quadrati ad numerum quadratum <lb/>quod ſi ita fuiſſent, ſequeretur ternarium numerum eſſe quadratum ex .22. octaui. <lb/></s> <s xml:space="preserve">Cum igitur non ſint vt numeri quadrati ad numerum quadratum, ſequitur ex ſepti-<lb/>ma decimi <seg type="var">.d.o.</seg> eſſe incommenſurabilem ipſi <seg type="var">.l.q.</seg> ſeu <seg type="var">.d.q.</seg> in longitudine.</s> </p> <p> <s xml:space="preserve">Vel dicamus ita, proportio quadrati ipſius <seg type="var">.l.q.</seg> ad quadratum ipſius <seg type="var">.o.d.</seg> eſt in ge <lb/>nere ſuperparticulari, cum ſit ſeſquitertia, vnde quadratum ipſius <seg type="var">.d.o.</seg> numeris da-<lb/>ri non poteſt, eo quod ſi dabilis fuiſſet, ſequeretur, quod inter quadratum ipſius. l<unclear reason="illegible"/> <seg type="var">.<lb/>q.</seg> & ipſius <seg type="var">.d.o.</seg> eſſet aliquis numerus medius proportionalis ex .16. octaui, vnde ex <lb/>octaua eiuſdem vnitas diuiſibilis eſſet, quod fieri non poteſt.</s> </p> <figure place="here"> <graphic url="0390-01"/> </figure> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De triangulo & Pentagono æquilatero</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MOdum quem conſideraui circa triangulum æquilaterum & pentagonum, vt <lb/>tibi ſignificaui ita ſe habet.</s> </p> <p> <s xml:space="preserve">Probandum primò eſt pentagonum altiorem eſſe triangulo ſibi iſoperimetro. <lb/></s> <s xml:space="preserve">Iam tibi notam puto proportionem lateris trianguli ad latus pentagoni eſſe vt .5. <lb/>ad .3.</s> </p> <p> <s xml:space="preserve">Sit igitur pentagonus <seg type="var">.b.d.m.g.v.</seg> cuius periferia diſtenta ſit <seg type="var">.K.z.</seg> baſis autem <seg type="var">.m.<lb/>g.</seg> bifariam diuiſa ſit in puncto <seg type="var">.a.</seg> <choice><ex>ductaque</ex><am>ductaq́;</am></choice> <seg type="var">.a.b</seg>: <seg type="var">b.g.</seg> et <seg type="var">.b.m.</seg> clarum erit <seg type="var">.a.b.</seg> perdicu-<lb/>larem eſſe ad <seg type="var">.m.g.</seg> ex .8. primi Eucli. cum <seg type="var">.b.m.</seg> et <seg type="var">.b.g.</seg> (baſes triangulorum <seg type="var">.b.d.m.</seg> <pb facs="0391" n="379"/><fw type="head">EPISTOLAE.</fw> et <seg type="var">.b.u.g.</seg>) ſint inuicem æquales ex .4. eiuſdem.</s> </p> <p> <s xml:space="preserve">Accipiatur deinde vel intelligatur <seg type="var">.g.p.</seg> æqualis duabus te<unclear reason="illegible"/>rtijs ipſius <seg type="var">.a.g.</seg> ducatur<lb/>q́ue <seg type="var">.b.p.</seg> quam probabo maiorem eſſe duplo ipſius <seg type="var">.a.p.</seg> vnde maior erit latere ipſius <lb/>trigoni æquilateris, cuius dimidium eſt <seg type="var">.a.p.</seg> ſcimus enim ipſum latus ſe habere ad <seg type="var">.m.<lb/>g.</seg> vt quinque ad .3. ita etiam <seg type="var">.a.p.</seg> ad <seg type="var">.a.g.</seg> vt diximus.</s> </p> <p> <s xml:space="preserve">Cum <choice><ex>autem</ex><am>autẽ</am></choice> angulus <seg type="var">.a.b.g.</seg> ſit quarta pars anguli <seg type="var">.b.g.a.</seg> ex .10. quarti & quinta pars <lb/>vnius recti ex .32. primi, dictus angulus erit graduum .18. et <seg type="var">.a.g.</seg> erit partium .30902. <lb/>et <seg type="var">.a.b.</seg> partium .95015 et <seg type="var">.a.p.</seg> 51503. vnde ex penultima primi latus <seg type="var">.b.p.</seg> erit par-<lb/>tium .108075. duplum vero ipſius <seg type="var">.a.p.</seg> erit .103006. latus igitur dicti trigoni, quod <lb/>ab <seg type="var">.p.</seg> erigitur, ſecabit perpendicularem <seg type="var">.a.b.</seg> ſub <seg type="var">.b.</seg> hoc eſt inter <seg type="var">.b.</seg> et <seg type="var">.a.</seg> ex penultima <lb/>primi. </s> <s xml:space="preserve">Finiatur enim triangulus æquicrurus <seg type="var">.b.q.p.</seg> quem probaui maiorem eſſe æ-<lb/>quilatero iſoperimetro pentagono propoſito, <choice><ex>ducaturque</ex><am>ducaturq́;</am></choice> <seg type="var">.u.p.</seg> ducatur etiam <seg type="var">.u.n.</seg> pa-<lb/>rallela ipſi <seg type="var">.b.g.</seg> quæ concludet triangulum <seg type="var">.g.u.n.</seg> ſimilem triangulo <seg type="var">.m.b.g.</seg> eo quod <lb/>cum angulus <seg type="var">.m.b.g.</seg> æqualis ſit angulo <seg type="var">.b.g.u.</seg> ex .16. tertij, per .27. primi <seg type="var">.m.b.</seg> et <seg type="var">.g.u.</seg> <lb/>erunt inuicem <choice><ex>æquidiſtantes</ex><am>æquidiſtãtes</am></choice>, vnde angulus <seg type="var">.b.m.g.</seg> æqualis erit angulo <seg type="var">.u.g.n.</seg> et. ex .29. <lb/>angulus <seg type="var">.g.u.n.</seg> æqualis erit angulo <seg type="var">.u.g.b</seg>. </s> <s xml:space="preserve">quare etiam angulo <seg type="var">.g.b.m.</seg> & angulus <seg type="var">.u.n.<lb/>g.</seg> angulo <seg type="var">.b.g.m.</seg> ex .32. eiuſdem, </s> <s xml:space="preserve">vnde ex .4. ſexti proportio <seg type="var">.g.n.</seg> ad <seg type="var">.g.m.</seg> erit .vt <seg type="var">.g.u.</seg> <lb/>ad <seg type="var">.m.b.</seg> ſed cum <seg type="var">.g.u.</seg> maior ſit dimidio ipſius <seg type="var">.b.g.</seg> ex .20. primi, hoc eſt maior dimi-<lb/>dio ipſius <seg type="var">.b.m.</seg> ergo <seg type="var">.g.n.</seg> etiam maior erit ipſa <seg type="var">.g.a.</seg> quapropter maior erit ipſa <seg type="var">.g.p.</seg> <lb/>cum <seg type="var">.g.p.</seg> minor ſit ipſa <seg type="var">.g.a.</seg> ex hypotheſi, ducta deinde cum fuerit <seg type="var">.b.n.</seg> habebimus <lb/>triangulum <seg type="var">.b.n.g.</seg> <choice><ex>æqualem</ex><am>æqualẽ</am></choice> triangulo <seg type="var">.b.u.g.</seg> & <choice><ex>maiorem</ex><am>maiorẽ</am></choice> <choice><ex>triangulo</ex><am>triãgulo</am></choice> <seg type="var">.b.p.g.</seg> ex prima ſexti <lb/>vel quia totum maius eſt ſua parte. </s> <s xml:space="preserve">Triangulus igitur <seg type="var">.b.u.g.</seg> maior eſt triangu-<lb/>lo <seg type="var">.b.p.g</seg>. </s> <s xml:space="preserve">quare triangulus <seg type="var">.b.u.o.</seg> maior erit triangulo <seg type="var">.g.o.p.</seg> ex communi conceptu, <lb/>idem infero ab alia parte dictarum figurarum. </s> <s xml:space="preserve">Quare pentagonus <seg type="var">.b.d.m.g.u.</seg> maior <lb/>erit triangulo <seg type="var">.b.q.p.</seg> quem probauimus maiorem eſſe triangulo æquilatero ſibi iſo-<lb/>perimetro.</s> </p> <figure place="here"> <graphic url="0391-01"/> </figure> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Comparatio periferiarum quadrati & trianguli aquilateri circunſcriptorum ab eodem circulo.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVod autem periferia quadrati in eodem circulo inſcripti, in quo ſit triangu-<lb/>lus æquilaterus, longior ſit periferia ipſius trianguli æquilateri, abſque vllo <pb facs="0392" n="380"/><fw type="head">IO. BAPT. BENED.</fw> negotio cordarum & arcuum poſſumus geometricè demonſtrare quod valde de-<lb/>ſideras.</s> </p> <p> <s xml:space="preserve">Quapropter ſit circulus <seg type="var">.b.a.e.q.</seg> in quo ſit <choice><ex>triangulum</ex><am>triangulũ</am></choice> æquilaterum <seg type="var">.b.e.n.</seg> & quadra <lb/>tum <seg type="var">.b.a.q.u.</seg> cuius periferiam probabo longiorem eſſe periferia trianguli. </s> <s xml:space="preserve">Sit enim <lb/>diameter circuli <seg type="var">.b.q.</seg> qui etiam erit diameter quadrati, vt à te ſcire potes. </s> <s xml:space="preserve">Sit etiam <lb/><choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.b.</seg> commune tam anguli quadrati quam trianguli. </s> <s xml:space="preserve">vnde ſequitur quod dictus <lb/>diameter ſecabit latus <seg type="var">.n.e.</seg> trianguli ad rectos & per æqualia in <seg type="var">.t</seg>. </s> <s xml:space="preserve">Nam cum arcus <seg type="var">.b.<lb/>e.</seg> æqualis ſit arcui <seg type="var">.b.n.</seg> ex .27. tertij, remanet vt arcus <seg type="var">.q.e.</seg> equalis ſit arcui <seg type="var">.q.n.</seg> vnde <lb/>angulus <seg type="var">.q.b.e.</seg> æqualis erit angulo <seg type="var">.q.b.n.</seg> ex .26. eiuſdem. </s> <s xml:space="preserve">quare ex .4. primi anguli <lb/>ad <seg type="var">.t.</seg> erunt recti, et <seg type="var">.n.t.</seg> æqualis erit ipſi <seg type="var">.t.e.</seg> vt diximus.</s> </p> <p> <s xml:space="preserve">Deinde <seg type="var">.b.e.</seg> et <seg type="var">.q.a.</seg> ſeinuicem <choice><ex>ſecant</ex><am>ſecãt</am></choice> in puncto <seg type="var">.o.</seg> vt ex ſe clarum patet, ducatur po <lb/>ſtea <seg type="var">.q.e.</seg> vnde habebimus angulum <seg type="var">.b.e.q.</seg> rectum ex .30. tertij, </s> <s xml:space="preserve">quare ex .18. primi <seg type="var">.q.<lb/>o.</seg> longior erit ipſa <seg type="var">.q.e.</seg> et <seg type="var">.q.e.</seg> longior erit ipſa <seg type="var">.e.t</seg>. </s> <s xml:space="preserve">quare <seg type="var">.q.o.</seg> longior erit ipſa <seg type="var">.t.e</seg>.</s> </p> <p> <s xml:space="preserve">Vt probemus poſtea <seg type="var">.b.a.o.</seg> longiorem eſſe ipſa <seg type="var">.b.e.</seg> producatur <seg type="var">.b.a.</seg> ita quod <seg type="var">.a.<lb/>p.</seg> æqualis ſit ipſi <seg type="var">.a.o.</seg> <choice><ex>ducaturque</ex><am>ducaturq́;</am></choice> <seg type="var">o.p.</seg> et <seg type="var">.a.e.</seg> cum autem ex iam dicta .30. tertij angulus <lb/><seg type="var">b.a.o.</seg> <choice><ex>rectus</ex><am>rectꝰ</am></choice> ſit, erit angulus <seg type="var">.o.a.p.</seg> ſimiliter <choice><ex>rectus</ex><am>rectꝰ</am></choice> ex .13. primi, vnde ex .5. et .32. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> <lb/>angulus <seg type="var">.a.p.o.</seg> erit dimidium recti, & ſimiliter, exijſdem, angulus <seg type="var">.b.q.a.</seg> eſt dimidium <lb/>recti </s> <s xml:space="preserve">quare angulus <seg type="var">.a.p.o.</seg> æqualis erit angulo <seg type="var">.a.q.b.</seg> ſed angulus <seg type="var">.a.e.b.</seg> æqualis eſt an <lb/>gulo <seg type="var">.a.q.b.</seg> ex .20. tertij, ergo angulus <seg type="var">.b.p.o.</seg> æqualis erit angulo .b, <seg type="var">e.a.</seg> angulus vero <lb/><seg type="var">a.b.e.</seg> communis eſt ambobus triangulis <seg type="var">.a.b.e.</seg> et <seg type="var">.o.b.p</seg>. </s> <s xml:space="preserve">quare ex .32. primi anguli <seg type="var">.<lb/>b.a.e.</seg> et <seg type="var">.b.o.p.</seg> reliqui ex duobus rectis æqua <lb/> <ptr xml:id="fig-0392-01a" corresp="fig-0392-01" type="figureAnchor"/> les inuicem erunt. </s> <s xml:space="preserve">Quare ex quarta ſexti, <lb/>et .18. quinti proportio <seg type="var">.b.o.</seg> ad <seg type="var">.b.p.</seg> erit, vt <lb/><seg type="var">b.a.</seg> ad <seg type="var">.b.e.</seg> ſed ex .18. primi <seg type="var">.b.o.</seg> maior eſt <lb/>ipſa <seg type="var">.b.a</seg>. </s> <s xml:space="preserve">quare ex .14. quinti <seg type="var">.b.p.</seg> maior erit <lb/>ipſa <seg type="var">.b.e.</seg> ſed <seg type="var">.b.p.</seg> æquatur ipſis <seg type="var">.b.a.</seg> cum <seg type="var">.a.</seg> o <lb/>ex hypoteſi, ergo <seg type="var">.b.a.</seg> cum <seg type="var">.a.o.</seg> maior erit <lb/>ipſa <seg type="var">.b.e.</seg> ſed <seg type="var">.q.o.</seg> maior erat ipſa <seg type="var">.t.e.</seg> vt ſupe <lb/>rius vidimus, </s> <s xml:space="preserve">quare <seg type="var">.b.a.</seg> cum <seg type="var">.a.o.</seg> et <seg type="var">.o.q.</seg> ma <lb/>ior eſt ipſa <seg type="var">.b.e.</seg> cum <seg type="var">.e.t.</seg> hoc eſt dimidium <lb/>periferię ipſius quadrati, <choice><ex>maius</ex><am>maiꝰ</am></choice> erit dimidio <lb/>periferię <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <choice><ex>trianguli</ex><am>triãguli</am></choice> propoſiti, </s> <s xml:space="preserve">quare ex 14. <lb/>dicta tota periferia dicti trianguli, ſimiliter <lb/>probarem de omnibus alijs figuris regulari <lb/>bus eodem circulo inſcriptis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0392-01" corresp="fig-0392-01a"> <graphic url="0392-01"/> </figure> </div> </body> </floatingText> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">CONSIDERATIONES NONNVLLÆ IN <lb/>Archimedem.</head> <head rend="italics" xml:space="preserve">Doct ßimo atque Reuerendo Domino Vincentio <lb/>Mercato.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> tibi aliàs dixi verum eſt, intellectum ſcilicet non omninò quieſcere cir <lb/>ca illas duas Archimedis propoſitiones, quæ in translatione Tartaleæ ſunt <lb/>ſub numeris .4. et .5. & in impreſſione Baſileæ ſub numeris .6. et .7. vbi <lb/>tractat <pb facs="0393" n="381"/><fw type="head">EPISTOLAE.</fw> tractat de centris libræ, ſeu ſtateræ: </s> <s xml:space="preserve">A ſpice igitur in .4. ſupradicta, quod cum appen-<lb/>ſæ fuerint omnes illæ partes ponderum, partibus longitudinis ipſius <seg type="var">.l.K.</seg> in qua volo <lb/>vt à punctis <seg type="var">.e.</seg> et <seg type="var">.d.</seg> imagineris duas lineas <seg type="var">.e.o.</seg> et <seg type="var">.d.u.</seg> inuicem æquales, & ferè per-<lb/>pendiculares ipſi <seg type="var">.l.K.</seg> hoc eſt reſpicientes mundi centrum; </s> <s xml:space="preserve">imagineris etiam <seg type="var">.o.u.</seg> <lb/> <ptr xml:id="hd-0393-01a" corresp="hd-0393-01" type="handwrittenAnchor"/> quæ ſit paralle la ipſi <seg type="var">.l.k.</seg> quæ diuiſa ſit in puncto <seg type="var">.i.</seg> ſupra <seg type="var">.g</seg>. </s> <s xml:space="preserve">Hinc nulli dubium erit, <lb/>cum <seg type="var">.g.</seg> fuerit centrum totius ponderis appenſi ipſi <seg type="var">.l.K.</seg> quod <seg type="var">.i.</seg> ſimiliter erit centrum <lb/>cum directe locatum ſit ſupra <seg type="var">.g.</seg> hoc eſt in eadem directionis linea, quod quidem <lb/>non indiget aliqua demonſtratione, cum per ſe ſatis pateat. </s> <s xml:space="preserve">Vnde ex communi <lb/>conceptu <seg type="var">.o.</seg> erit centrum ponderis appenſi ipſi <seg type="var">.l.h.</seg> et <seg type="var">.u.</seg> erit centrum ponderis ap-<lb/>penſi. ipſi <seg type="var">h.K</seg>. </s> <s xml:space="preserve">Scimus <choice><ex>igitur</ex><am>igit̃</am></choice> <seg type="var">.i.</seg> eſſe <choice><ex>centrum</ex><am>cẽtrum</am></choice> duorum, hoc eſt ipſius <seg type="var">.l.h.</seg> & ipſius <seg type="var">.h.k.</seg> con <lb/>tinuatorum per totam <seg type="var">.l.k</seg>. </s> <s xml:space="preserve">Nunc ergo ſi conſideremus <seg type="var">.l.k.</seg> diuiſam eſſe, hoc eſt di-<lb/>ſiunctam in puncto <seg type="var">.h.</seg> inueniemus nihilominus <seg type="var">.i.</seg> centrum eſſe dictorum ponderum, <lb/>& quod tantum eſt, ipſam eſſe <choice><ex>continuam</ex><am>continuã</am></choice>, quantum diuiſam in dicto puncto <seg type="var">.h.</seg> neque <lb/>ex hoc, punctum <seg type="var">.i.</seg> erit magis vel minus centrum duorum ponderum <seg type="var">.l.h.</seg> et <seg type="var">.h.k.</seg> quo <lb/>rum vnum pendet totum ab <seg type="var">.o.</seg> aliud verò totum ab <seg type="var">.u.</seg> & hoc modo in longitudine <seg type="var">.<lb/>o.u.</seg> diuiſa vt dictum eſt, habebimus propoſitum.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0393-01" corresp="hd-0393-01a"/> </div> </body> </floatingText> <p> <s xml:space="preserve">Reliquam propoſitionem tibi relinquo.</s> </p> <p> <s xml:space="preserve">Illa verò propoſitio, quam tibi dixi Archimedem tacuiſſe in huiuſmodi materia <lb/>eſt, quod ſi duo pondera æquilibrant ab extremis alicuius ſtateræ, in certis præfixis <lb/>diſtantijs à centro. </s> <s xml:space="preserve">Tunc dico ſi eorum vno manente alterum moueatur remotius <lb/>ab ipſo centro quod illud deſcendet, & ſi vicinius ipſi centro appenſum fuerit aſcen-<lb/>det. </s> <s xml:space="preserve">Hæc enim propoſitio quotidie omnibus in locis videtur, ipſam verſo4; </s> <s xml:space="preserve">puto Ar <lb/>chimedem prætermiſiſſe ob facilitatem, cum ab antedicta ferè dependeat.</s> </p> <p> <s xml:space="preserve">Sit exempli gratia ſtatera <seg type="var">.a.u.</seg> cuius centr um ſit <seg type="var">.i.</seg> & pondera <seg type="var">.u.a.</seg> appenſa, ſein-<lb/> <ptr xml:id="hd-0393-02a" corresp="hd-0393-02" type="handwrittenAnchor"/> uicem habeant vt <seg type="var">.i.u.</seg> et <seg type="var">.i.a.</seg> ſe inuicem habent. </s> <s xml:space="preserve">Nunc dico quod ſi pondus ipſius <seg type="var">.u.</seg> <lb/>poſitum fuerit vicinius centro vt puta in <seg type="var">.o.</seg> inmoto exiſtente pondere, a. quod bra-<lb/>chium <seg type="var">.i.o.u.</seg> aſcendet, & è conuerſo, ſi remotius poſitum fuerit, deſcendet.</s> </p> <floatingText> <body> <div type="float"> <note xml:id="hd-0393-02" corresp="hd-0393-02a"/> </div> </body> </floatingText> <p> <s xml:space="preserve"><choice><ex>Ponatur</ex><am>Ponat̃</am></choice> ergo vt <choice><ex>dictum</ex><am>dictũ</am></choice> eſt in <seg type="var">.o.</seg> vicinius <choice><ex>centro</ex><am>cẽtro</am></choice>, quapropter brachium <seg type="var">.i.o.</seg> <choice><ex>breuius</ex><am>breuiꝰ</am></choice> erit <lb/>brachio <seg type="var">.i.u.</seg> vnde minor proportio erit ipſius <seg type="var">.i.o.</seg> ad <seg type="var">.i.a.</seg> quàm.i.u. ad eundem <seg type="var">.a.i.</seg> & <lb/>conſequenter quam ponderis ipſius <seg type="var">.a.</seg> (quod ſit <seg type="var">.n.e.</seg>) ad pondus ipſius <seg type="var">.u</seg>. </s> <s xml:space="preserve">Quare ſi cx <lb/>pondere <seg type="var">.n.e.</seg> dempta fuerit <seg type="var">.e.</seg> pars eius, ita quod reliqua pars <seg type="var">.n.</seg> ſe habeat ad pondus <lb/>o. vt ſe habet. i<unclear reason="illegible"/> <seg type="var">.o.</seg> ad <seg type="var">.i.a.</seg> tunc ſtatera non mouebitur; </s> <s xml:space="preserve">addita verò parte <seg type="var">.e.</seg> ex com-<lb/>muni conceptu, a. deſcendet vnde <seg type="var">.o.</seg> aſcenderet conuerſum verò ex ſimilibus ratio-<lb/>nibus per te concludes.</s> </p> <figure place="here"> <graphic url="0393-01"/> </figure> <pb facs="0394" n="382"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">In eo quod à me petis, mittendo te ad Eutotium, tibi non ſatisfacerem, cum Eu-<lb/>totius citet ſextum librum Pergei, quem nunquam vidimus, <choice><ex>ſupponatque</ex><am>ſupponatq́;</am></choice> ea, quæ nec <lb/>ipſe nec alius vnquam quod ſcimus probauit.</s> </p> <p> <s xml:space="preserve">Deſideras enim demonſtrationem illius quod Archimedes dicit inter primam, <lb/>& ſecundam propoſitionem ſecundi libri, vbi tractat de centris grauium, propte-<lb/>rea quod illud ſupponit pro manifeſto.</s> </p> <p> <s xml:space="preserve">Sit enim figura hic ſubſcripta, ferè ſimilis parabolæ poſitæ in .2. propoſitione di <lb/>cti libri, vt in impreſſione Baſileenſi habetur, <choice><ex>ſintque</ex><am>ſintq́;</am></choice> diuiſæ duæ <seg type="var">.a.b.</seg> et <seg type="var">.b.c.</seg> per æqua <lb/>lia à punctis <seg type="var">.x.</seg> et <seg type="var">.u.</seg> <choice><ex>protractisque</ex><am>protractisq́;</am></choice> <seg type="var">.f.x.</seg> et <seg type="var">.u.i.</seg> ad <seg type="var">.b.d.</seg> quæ inuicem etiam erunt parallelę <lb/>ex .30. primi Eucli. </s> <s xml:space="preserve">vnde ipſæ etiam, diametri erunt ipſarum portionum: </s> <s xml:space="preserve">vt ex eo col <lb/>ligere eſt, quod in .49. primi lib. Pergei probatur. </s> <s xml:space="preserve">Imaginando poſtea ad puncta <seg type="var">.b.<lb/>f.</seg> er<unclear reason="illegible"/> <seg type="var">.i.</seg> tres contingentes, manifeſtum erit punctum <seg type="var">.b.</seg> illud eſſe quod terminat alti-<lb/>tudinem huiuſmodi portionis, et <seg type="var">.f.</seg> et <seg type="var">.i.</seg> terminantia altitudines partialium, ex .5. ſe<lb/>cundi ipſius Pergei, eo quod dictæ contingentes paralellæ erunt ipſis baſibus, vnde <lb/>trianguli inſcripti, eaſdem habebunt altitudines, quas portiones ipſæ, quod erit ex <lb/>mente Archimedis. </s> <s xml:space="preserve">Et ſic deinceps poteris multiplicare angulos ſiguræ rectilineæ <lb/>in parabola, quæ deſignata erit vt deſiderat Archimedes, qui quidem dicit, quod <lb/>protractæ cum fuerint aliæ deinceps poſt <seg type="var">.f.i.</seg> ipſæ inuicem ęquidiſtantes <choice><ex>erunt</ex><am>erũt</am></choice>, diuiſę-<lb/>q́ue peræqualia ab <seg type="var">.d.b.</seg> quod <choice><ex>quanuis</ex><am>quãuis</am></choice> <choice><ex>verum</ex><am>verũ</am></choice> ſit, <choice><ex>tantum</ex><am>tñ</am></choice> ab Eutotio non ſatis <choice><ex>demonſtratum</ex><am>demõſtratũ</am></choice> <lb/>eſt, cum ſupponat <seg type="var">.a.f.b.</seg> æqualem eſſe ipſi <seg type="var">.b.i.c.</seg> probare volens eius diametros æqua <lb/>les eſſe abſque aliqua citata ratione, quæ quidem ratio eſſet conuerſum .4. propoſi-<lb/>tionis libri de conoidalibus. </s> <s xml:space="preserve">Sed oporteret nos <choice><ex>etiam</ex><am>etiã</am></choice> videre .6. librum ipſius Pergei, <lb/>& propterea tibi non ſatisfacerem.</s> </p> <p> <s xml:space="preserve">Eſto igitur, ut inuenta ſit linea <seg type="var">.K.</seg> cuius productum in <seg type="var">.u.i.</seg> æquale ſit qua drato ip <lb/>ſius <seg type="var">.u.c.</seg> inuenta etiam ſit linea <seg type="var">.h.</seg> cuius productum cum <seg type="var">.f.x.</seg> æquale ſit quadrato ip-<lb/>ſius <seg type="var">.a.x.</seg> vnde ex conuerſo .49. primi ipſius Pergei, proportio ipſius <seg type="var">.K.</seg> ad <seg type="var">.b.c.</seg> erit ut <lb/>ipſius <seg type="var">.b.c.</seg> ad <seg type="var">.b.d.</seg> & ipſius <seg type="var">.h.</seg> ad <seg type="var">.a.b.</seg> vt ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.b.d</seg>. </s> <s xml:space="preserve">Erit igitur ex .16. ſexti Eucl. <lb/>quadratum <seg type="var">.b.c.</seg> æquale producto ipſius <seg type="var">.K.</seg> in <seg type="var">.b.d.</seg> & quadratum <seg type="var">.a.b.</seg> æquale produ-<lb/>cto ipſius <seg type="var">.h.</seg> in <seg type="var">.b.d.</seg> & ex prima ſexti, ita erit ipſius <seg type="var">.K.</seg> ad <seg type="var">.h.</seg> vt producti quod fit ex <seg type="var">.K.</seg> <lb/>in <seg type="var">.b.d.</seg> ad productum ipſius <seg type="var">.h.</seg> in <seg type="var">.b.d.</seg> hoc eſt vt quadrati ipſius <seg type="var">.b.c.</seg> ad quadratum ip <lb/>ſius <seg type="var">.b.a.</seg> ex .16. et .11. quinti, hoc eſt vt quadrati ipſius <seg type="var">.u.c.</seg> ad quadratum ipſius <seg type="var">.a.x.</seg> <lb/>hoc eſt ut productum ipſius <seg type="var">.k.</seg> in <seg type="var">.u.i.</seg> ad productnm ipſius <seg type="var">.h.</seg> in <seg type="var">.x.f</seg>. </s> <s xml:space="preserve">Nunc ſi ipſius <seg type="var">.k.</seg> <lb/>ad <seg type="var">.h.</seg> c<unclear reason="illegible"/>ſt vt producti ipſius <seg type="var">.K.</seg> in <seg type="var">.u.i.</seg> ad productum ipſius <seg type="var">.h.</seg> in <seg type="var">.f.x.</seg> ergo ex .24. ſexti, <lb/>& communi conceptu, proportio ipſius <seg type="var">.k.</seg> ad <seg type="var">.h.</seg> compoſita erit ex ea quæ ipſius <seg type="var">.u.i.</seg> <lb/>ad <seg type="var">.f.x.</seg> & ex ea quæ ipſius <seg type="var">.k.</seg> ad <seg type="var">.h</seg>. </s> <s xml:space="preserve">Cum ergo dempta fuerit proportio ipſius <seg type="var">.k.</seg> ad <seg type="var">.h.</seg> <lb/>(vt ſimplex) à proportione ipſius <seg type="var">.k.</seg> ad <seg type="var">.h.</seg> (vt compoſita) reliquum nihil erit. </s> <s xml:space="preserve">Qua-<lb/>re <seg type="var">.f.x.</seg> æqualis erit ipſi <seg type="var">.u.i</seg>.</s> </p> <p> <s xml:space="preserve">Sed quod <seg type="var">.f.m.</seg> æqualis ſit ipſi <seg type="var">.m.i</seg>. </s> <s xml:space="preserve">Videto in Eutotio, quia hoc ſatis ſui natura <lb/>facile eſt.</s> </p> <p> <s xml:space="preserve">Sed accipe alium modum breuiorem ad probandum <seg type="var">.f.x.</seg> eſſe æqualem ipſi <seg type="var">.u.i</seg>.</s> </p> <p> <s xml:space="preserve">Finge lineam <seg type="var">.e.b.g.</seg> conting entem in puncto <seg type="var">.b.</seg> prolungatisq́ue diametris <seg type="var">f.<lb/>x.</seg> et <seg type="var">.u.i.</seg> vſque ad contingentem ipſam, habebis <seg type="var">.f.e.</seg> æqualem ipſi <seg type="var">.f.x.</seg> et <seg type="var">.g.i.</seg> ipſi <seg type="var">.u.i.</seg> <lb/>Ex .35. primi Pergei, producta poſtea <seg type="var">.x.u.</seg> habeb is ex .2. ſexti Eucli <seg type="var">.x.u.</seg> parallelam <lb/>ipſi <seg type="var">.a.c.</seg> ſed <seg type="var">.e.g.</seg> parallela eſt ipſimet <seg type="var">.a.c.</seg> ex quinta ſecundi ipſius Pergei, </s> <s xml:space="preserve">quare ex .30 <lb/>primi Euclid <seg type="var">.e.g.</seg> parallela erit ipſi <seg type="var">.u.x.</seg> & ex .34. eiuſdem æqualis erit <seg type="var">.e.x.</seg> ipſi <seg type="var">.u.g.</seg> <lb/>vnde <seg type="var">.f.x.</seg> etiam æqualis erit <seg type="var">.u.i.</seg> ex communi conceptu.</s> </p> <p> <s xml:space="preserve">Sed ne quid deſideres probabo <seg type="var">.f.m.</seg> æqualem eſſe ipſi <seg type="var">.m.i</seg>. </s> <s xml:space="preserve">Iam igitur ſcis quod <pb facs="0395" n="383"/><fw type="head">EPISTOLAE.</fw> cum ſit <seg type="var">.f.x.</seg> æqualis ipſi <seg type="var">.u.i.</seg> vt tibi probaui, & inuicem parallelæ ideo <seg type="var">.f.i.</seg> parallela <lb/>erit ipſi <seg type="var">.x.u.</seg> ex .33. primi Euclidis. </s> <s xml:space="preserve">Vnde ex .30. eiuſdem, parallela erit etiam ipſi <seg type="var">.a.<lb/>c.</seg> ſed cum <seg type="var">.x.u.</seg> diuiſa ſit ab <seg type="var">.d.b.</seg> per æqualia, eo quod diuidit <seg type="var">.a.c.</seg> eodem modo, quę <lb/>ipſi parallela eſt ex .2. ſexti. </s> <s xml:space="preserve">Reliqua tibi conſideranda relinquo. </s> <s xml:space="preserve">cum verò ambæ <seg type="var">.f.<lb/>x.</seg> et <seg type="var">.u.i.</seg> parallelæ ſint ipſi <seg type="var">.b.d.</seg> ſequitur quod cum ex .34. primi <choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> <seg type="var">.f.m.</seg> et <seg type="var">.m.<lb/>i.</seg> æqualis ſit medietati ipſius <seg type="var">.x.u.</seg> erunt inuicem æquales.</s> </p> <figure place="here"> <graphic url="0395-01"/> </figure> <p> <s xml:space="preserve">Minime dubitabam tibi non ſatisfacere Eutocium in .3. propoſitione ſecundi <lb/>lib. de centris Grauium Archimedis, cum citet .6. librum de elementis conicis, ad-<lb/>de quod ſi aliud in ipſo .6. libro ab eo citato non eſſet magis ad propoſitum, quàm <lb/>ca quæ ab ipſo citata ſunt, nihilominus adhuc irreſolutus maneres.</s> </p> <p> <s xml:space="preserve">Conſidera igitur eandem ipſam figuram præcedentem; </s> <s xml:space="preserve">pro alia verò parabola ſi <lb/>mili dictæ, accipe ſecundam figuram ipſius tertiæ dictæ propoſitionis. </s> <s xml:space="preserve">Deinde ima <lb/>ginabis duo latera <seg type="var">.o.x.</seg> et <seg type="var">.o.p.</seg> diuiſa eſſe per æqualia in punct is <seg type="var">.g.</seg> et <seg type="var">.K.</seg> <choice><ex>protractisque</ex><am>protractisq́;</am></choice> <lb/>diametris <seg type="var">.g.y.</seg> et <seg type="var">.K.u.</seg> quæ, vt in præcedenti probaui, ſunt inuicem æquales, ſcire <lb/>debes quod ſimiles parabolæ inuicem aliæ non poſſunt eſſe, niſi eæ quæ diametros <lb/>proportionales ſuis baſibus habeant, <choice><ex>ſimiliterque</ex><am>ſimiliterq́;</am></choice> poſitæ, hoc eſt, ut proportio ipſius <lb/><seg type="var">b.d.</seg> ad <seg type="var">.a.c.</seg> ſit eadem quæ ipſius <seg type="var">.o.r.</seg> ad <seg type="var">.x.p.</seg> & quod anguli ad <seg type="var">.r.</seg> ſint æquales angulis <lb/>circa <seg type="var">.d</seg>. </s> <s xml:space="preserve">Notentur ergo primum puncta communia ip ſius <seg type="var">.o.g.</seg> cum <seg type="var">.y.t.</seg> & ipſius <seg type="var">.b.</seg> x <lb/>cum <seg type="var">.f.m.</seg> characteribus. ω<unclear reason="illegible"/>. et <seg type="var">.n</seg>. </s> <s xml:space="preserve">Nunc igitur ſcimus <seg type="var">.f.m.</seg> æqualem eſſe <seg type="var">.m.i.</seg> tota <choice><ex>mque</ex><am>mq́;</am></choice> <seg type="var">.f.<lb/>i.</seg> parallelam eſſe ipſi <seg type="var">.a.c</seg>. </s> <s xml:space="preserve">Idem dico de <seg type="var">.y.t.u.</seg> <choice><ex>triangulique</ex><am>trianguliq́;</am></choice> <seg type="var">.x.f.n.</seg> et <seg type="var">.g.y.</seg> ω<unclear reason="illegible"/>. eſſe ſimiles <lb/>triangulis <seg type="var">.n.m.b.</seg> et. ω<unclear reason="illegible"/> <seg type="var">.t.o.</seg> quod ita probatur, nam ex .15. primi Euclid. anguli ad <seg type="var">.n.</seg> <lb/>ſunt inuicem æquales, ex .29. verò eiuſdem anguli <seg type="var">.f.x.n.</seg> et <seg type="var">.n.b.m.</seg> ſimiliter æquales <lb/>ita etiam <seg type="var">.n.f.x.</seg> et <seg type="var">.n.m.b</seg>.</s> </p> <p> <s xml:space="preserve">Idem dico in ſecunda figura, vnde ex .4. ſexti Eucli. proportio <seg type="var">.n.f.</seg> ad <seg type="var">.m.n.</seg> erit ea <lb/>dem quę <seg type="var">.f.x.</seg> ad <seg type="var">.b.m.</seg> & ipſius <seg type="var">.n.f.</seg> ad <seg type="var">.x.f.</seg> vt <seg type="var">.n.m.</seg> ad <seg type="var">.m.b.</seg> ex .16. quinti. </s> <s xml:space="preserve">Quare ex .11. <lb/> <ptr xml:id="fig-0395-02a" corresp="fig-0395-02" type="figureAnchor"/> <pb facs="0396" n="384"/><fw type="head">IO. BAPT. BENED.</fw> eiuſdem erit vt <seg type="var">.a.d.</seg> ad <seg type="var">.d.b</seg>. </s> <s xml:space="preserve">Idem etiam dico in ſecunda parabola, ſed ipſius <seg type="var">.x.o.</seg> ad <lb/><seg type="var">o.r.</seg> eſt vt <seg type="var">.a.b.</seg> ad <seg type="var">.b.d.</seg> ex .6. ſexti Eucli. </s> <s xml:space="preserve">vnde ex .11. quinti <seg type="var">.n.f.</seg> ad <seg type="var">.f.x.</seg> erit vt <seg type="var">.ω.y.</seg> <lb/>ad <seg type="var">.y.g</seg>. </s> <s xml:space="preserve">Sed in precedenti iam tibi dixi <seg type="var">.a.b.</seg> mediam proportionalem eſſe inter <seg type="var">.h.</seg> <lb/>et <seg type="var">.b.d</seg>. </s> <s xml:space="preserve">Sit nunc <seg type="var">.z.</seg> pro ſecunda parabola, ita ut <seg type="var">.h.</seg> eſt pro prima, vnde <seg type="var">.o.x.</seg> crit media <lb/>proportionalis inter <seg type="var">.z.</seg> et <seg type="var">.o.r.</seg> & ex .11. quinti ita erit <seg type="var">.h.</seg> ad <seg type="var">.a.b.</seg> vt <seg type="var">.z.</seg> ad <seg type="var">.x.o.</seg> & ex .22. <lb/>h. ad <seg type="var">.a.x.</seg> ut z. ad <seg type="var">.x.g.</seg> & quia ex .16. ſexti <seg type="var">.a.x.</seg> media proportionalis eſt inter <seg type="var">.h.</seg> et <seg type="var">.f.<lb/>x.</seg> cum ſupponatur productum <seg type="var">.h.</seg> in <seg type="var">.f.x.</seg> æquale eſſe quadrato <seg type="var">.a.x</seg>. </s> <s xml:space="preserve">Idem dico <seg type="var">.x.g.</seg> <lb/>mediam eſſe proportionalem inter <seg type="var">.z.</seg> et <seg type="var">.g.y.</seg> </s> <s xml:space="preserve">quare ex .11. iam dicta, ita erit <seg type="var">.a.x.</seg> ad <seg type="var">.f.<lb/>x.</seg> vt <seg type="var">.y.g.</seg> ad <seg type="var">.x.o.</seg> & ex eadem, ita erit ipſius <seg type="var">.f.n.</seg> ad <seg type="var">.a.b.</seg> ut <seg type="var">.y.ω.</seg> ad <seg type="var">.x.o.</seg> & ſic <seg type="var">.f.n.</seg> ad <seg type="var">.d.a.</seg> <lb/>vt <seg type="var">.y.ω.</seg> ad <seg type="var">.x.r.</seg> ſed <seg type="var">.f.m.</seg> ad <seg type="var">f.n.</seg> eſt vt <seg type="var">.y.t.</seg> ad <seg type="var">.y.ω.</seg> ex .18. quinti vnde <seg type="var">.f.m.</seg> ad <seg type="var">.a.d.</seg> erit vt <lb/><seg type="var">y.t.</seg> ad <seg type="var">.x.r</seg>. </s> <s xml:space="preserve">Idem dico de eorum duplis.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0395-02" corresp="fig-0395-02a"> <graphic url="0395-02"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Ex ijſdem rationibus dico ita eſſe <seg type="var">.b.d.</seg> ad <seg type="var">.b.m.</seg> vt <seg type="var">.o.r.</seg> ad <seg type="var">.o.t.</seg> & ex .17. quinti <seg type="var">.d.m.</seg> <lb/>ad <seg type="var">.b.m.</seg> vt <seg type="var">.r.t.</seg> ad <seg type="var">.t.o</seg>. </s> <s xml:space="preserve">Reliqua tibi conſideranda relinquo.</s> </p> <figure place="here"> <graphic url="0396-01"/> </figure> <p> <s xml:space="preserve">In reliquis verò propoſitionibus illius lib. nullo pacto poteris dubitare: </s> <s xml:space="preserve">Verum ne <lb/>in .4. aliquid tibi noui exurgat, te ſcire volo <ref>corollarium .20. in libr. de quadratu<lb/>ra parabolę</ref> docere poſſibile eſſe inſcriptionem rectilineæ, ea tamen conditione <choice><ex>quam</ex><am>quã</am></choice> <lb/>dicit Archimedes.</s> </p> <p> <s xml:space="preserve">In quinta poſtea animaduertendum eſt, quod prima pars, probat tantummodo de <lb/>centro trianguli, et .2. pars probat de centro pentagoni, à te ipſo deinde potes pro-<lb/>bare de centro nonanguli: </s> <s xml:space="preserve">& ſic de cæteris: </s> <s xml:space="preserve">eo quod cum probatum fuerit de centro <lb/>figuræ in medio locatæ ſi conſtitutæ poſtea fuerint ſimiles figuræ in portionibus la-<lb/>teralibus habebitur propoſitum in infinitum.</s> </p> <p> <s xml:space="preserve">Idem intelligendum eſt in .3. propoſitione quamuis exemplum vlterius non ex-<lb/>tendatur quam ad pentagonos.</s> </p> <p> <s xml:space="preserve">Sexta verò <choice><ex>propoſitio</ex><am>ꝓpoſitio</am></choice> tibi ſacilis erit, quæ nihilominus <choice><ex>pont</ex><am>põt</am></choice> <choice><ex>demonſtrari</ex><am>demõſtrari</am></choice> hoc <choice><ex>mon</ex><am>mõ</am></choice> ſcili<lb/>cet. </s> <s xml:space="preserve">Sint .4. <choice><ex>quantitates</ex><am>quãtitates</am></choice> <seg type="var">.a.b.c.d.</seg> ipſius Archimedis <choice><ex>ſupponendo</ex><am>ſupponẽdo</am></choice> <seg type="var">.a.</seg> pro figura rectilinea <lb/>inſcripta in parabola, et <seg type="var">.b.</seg> pro reſiduo ipſius parabolę et <seg type="var">.c.</seg> pro triangulo <seg type="var">.a.b.c.</seg> in me <lb/>dio ipſius parabolę et <seg type="var">.d.</seg> pro triangulo <seg type="var">.r</seg>. </s> <s xml:space="preserve">Nunc cum <seg type="var">.a.</seg> maior ſit <seg type="var">.c.</seg> prout totum ma-<lb/>ius eſt ſua parte, ideo ex .8. quinti maior proportio habebit <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> quam <seg type="var">.c.</seg> ad <seg type="var">.b.</seg> <lb/>Cum autem <seg type="var">.b.</seg> minor ſit <seg type="var">.d.</seg> ex ſuppoſito, ideo ex eadem dicta, maior proportio habe <lb/>bit <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> quam <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> cum verò centrum cuiuſuis figuræ plenæ neceſſariò ſit intra <lb/>ipſam figuram, idcirco centrum reſidui ipſius parabolę intra ipſam reperietur. </s> <s xml:space="preserve">quod <lb/>ita <choice><ex>clarum</ex><am>clarũ</am></choice> <choice><ex>per</ex><am>ꝑ</am></choice> ſe eſt, <choice><ex>quemadmodum</ex><am>quẽadmodũ</am></choice> quoduis aliud axioma, & quia <choice><ex>dictum</ex><am>dictũ</am></choice> <choice><ex>centrum</ex><am>centrũ</am></choice> ex .8. primi <lb/>de centris, neceſſariò eſt in linea <seg type="var">.b.h.</seg> inter <seg type="var">.b.</seg> et <seg type="var">.h</seg>. </s> <s xml:space="preserve">Sit igitur <seg type="var">.g.</seg> vnde ex eadem .8. ita <lb/>erit <seg type="var">.g.h.</seg> ad <seg type="var">.h.e.</seg> vt <seg type="var">.a.</seg> ad <seg type="var">.b.</seg> ergo <seg type="var">.g.h.</seg> ad <seg type="var">.h.e.</seg> maior proportio erit <choice><ex>quam</ex><am>quã</am></choice> <seg type="var">.c.</seg> ad <seg type="var">.d.</seg> hoc eſt <lb/>quam <seg type="var">.b.h.</seg> ad <seg type="var">.f.</seg> ex .12. quinti. </s> <s xml:space="preserve">Sed <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.h.b.</seg> maior ſit ipſa <seg type="var">.h.g.</seg> prout omne totum ma-<lb/>ius eſt ſua parte, ideo maior proportio habebit <seg type="var">.h.b.</seg> ad <seg type="var">.h.e.</seg> quam <seg type="var">.h.g.</seg> ad <seg type="var">.h.e.</seg> vnde <lb/>multo <choice><ex>maiorem</ex><am>maiorẽ</am></choice> <choice><ex>quam</ex><am>quã</am></choice> <seg type="var">.h.b.</seg> ad <seg type="var">.f.</seg> ex <choice><ex>coni</ex><am>cõi</am></choice> <choice><ex>conceptu</ex><am>cõceptu</am></choice>, </s> <s xml:space="preserve">quare <seg type="var">.h.e.</seg> erit minor ipſa <seg type="var">.f.</seg> ex .10. <choice><ex>quinti</ex><am>quĩti</am></choice>.</s> </p> <p> <s xml:space="preserve">Septima verò et .8. propoſitio nullius tibi erit difficultatis.</s> </p> <pb facs="0397" n="385"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Quamuis Eutotius ſcribat ſuper duas vltimas lib. ſecundi de centris <choice><ex>grauium</ex><am>grauiũ</am></choice>, nihil <lb/>miror ipſum tibi non ſatisfacere. </s> <s xml:space="preserve">Accipe igitur quod ego nunc tibi mitto.</s> </p> <p> <s xml:space="preserve">Archimedes eo in loco <choice><ex>primum</ex><am>primũ</am></choice> ſupponit in penultima dicti libri quatuor lineas <lb/>proportionales <seg type="var">.a.b</seg>: <seg type="var">c.b</seg>: <seg type="var">d.b</seg>: et <seg type="var">.e.b</seg>: ſupponit etiam quod proportio quæ eſt ipſius <seg type="var">.<lb/>e.b.</seg> ad <seg type="var">.e.a.</seg> <choice><ex>eadem</ex><am>eadẽ</am></choice> ſit quæ ipſius <seg type="var">.f.g.</seg> ad tres quintas ipſius <seg type="var">.a.d.</seg> & quod proportio com <lb/>poſiti dupli ipſius <seg type="var">.a.b.</seg> cum quadruplo ipſius <seg type="var">.b.c.</seg> cum ſexcuplo ipſius <seg type="var">.b.d.</seg> cum triplo <lb/> <ptr xml:id="fig-0397-01a" corresp="fig-0397-01" type="figureAnchor"/> <pb facs="0398" n="386"/><fw type="head">IO. BAPT. BENED.</fw> ipſius <seg type="var">.b.e.</seg> ad compoſitum quintupli ipſius <seg type="var">.a.b.</seg> cum decuplo ipſius <seg type="var">.c.b.</seg> cum decuplo <lb/>ipſius <seg type="var">.b.d.</seg> cum quintuplo ipſius <seg type="var">.b.e.</seg> eadem ſit quæ ipſius <seg type="var">.g.h.</seg> ad <seg type="var">.a.d.</seg> & vult proba-<lb/>re <seg type="var">.f.h.</seg> eſſe duas quintas ipſius <seg type="var">.a.b</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0397-01" corresp="fig-0397-01a"> <graphic url="0397-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Cum autem dicit proportionem ipſius <seg type="var">.a.c.</seg> ad <seg type="var">.c.d.</seg> & ipſius <seg type="var">.c.d.</seg> ad <seg type="var">.d.e.</seg> eſſe vt ipſius <lb/><seg type="var">a.b.</seg> a<unclear reason="illegible"/>d.b.c. & cętera, verum dicit ex .19. quinti Eucli. eo quod cum ex hypotheſi ſit <lb/>ipſius, <seg type="var">a.b.</seg> totalis ad <seg type="var">.c.b.</seg> totalom vt ipſius <seg type="var">.c.b.</seg> partialis (ſumptæ vt pars abſciſa ab <seg type="var">.a.<lb/>b.</seg> pro nunc) ad <seg type="var">.d.b.</seg> partialem (abſciſam ab <seg type="var">.c.b.</seg>) erit ex .19. dicta ipſius <seg type="var">.a.c.</seg> (reſidui <lb/>ex <seg type="var">.a.b.</seg>) ad <seg type="var">.c.d.</seg> (reſiduum ex <seg type="var">.c.b.</seg>) vt ipſius <seg type="var">.a.b.</seg> ad <seg type="var">.c.b.</seg> & ita probabitur de pro-<lb/>portione ipſiuas <seg type="var">.c.d.</seg> ad <seg type="var">.d.e.</seg> eadem ratione.</s> </p> <p> <s xml:space="preserve">Cum verò ex .18. quinti ſit ipſius <seg type="var">.a.b.</seg> cum <seg type="var">.c.b.</seg> ad <seg type="var">.c.b.</seg> vt ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.d.e.</seg> ergo ex <lb/>22. eiuſdem, ita erit ipſius <seg type="var">.a.b.</seg> cum <seg type="var">.c.b.</seg> ad <seg type="var">.d.b.</seg> vt <seg type="var">.a.d.</seg> ad <seg type="var">.d.e.</seg> & exijſdem rationibus <lb/>eadem proportio erit ipſius <seg type="var">.c.b.</seg> cum <seg type="var">.d.b.</seg> ad <seg type="var">.b.e.</seg> vt <seg type="var">.a.d.</seg> ad <seg type="var">.d.e.</seg> quod inquit Archi. <lb/></s> <s xml:space="preserve">Verum etiam erit (ex .13. quinti) cum dicit eandem proportionem eſſe ipſius <seg type="var">.a.d.</seg> <lb/>ad <seg type="var">.d.e.</seg> quę dupli primi antecedentis cum ſimplo ſecundi antecedentis ad duplum <lb/>primi conſequentis cum ſimplo ſecundi conſequentis, hoc eſt dupli ipſius <seg type="var">.a.b.c.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <lb/>ſimplo <seg type="var">.c.b.d.</seg> ad duplum ipſius <seg type="var">.d.b.</seg> cum ſimplo <seg type="var">.e.b.</seg> hoc eft dupli <seg type="var">.a.b.</seg> cum triplo ip-<lb/>ſius <seg type="var">.b.c.</seg> cum ſimplo <seg type="var">.d.b.</seg> ad duplum ipſius <seg type="var">.d.b.</seg> cum ſimplo <seg type="var">.e.b</seg>. </s> <s xml:space="preserve">Nunc duplum <seg type="var">.a.b.</seg> <lb/>cum triplo <seg type="var">.b.c.</seg> cum ſimplo <seg type="var">.b.d.</seg> ſignatum ſit charactere <seg type="var">.D.</seg> ſuum verò conſequens, <lb/>hoc eſt duplum <seg type="var">.d.b.</seg> <choice><ex>cum</ex><am>cũ</am></choice> ſimplo <seg type="var">.e.b.</seg> ſignificetur à charactere <seg type="var">.B.</seg> hinc proportio ipſius <lb/> <ptr xml:id="note-0398-01a" corresp="note-0398-01" type="noteAnchor"/> <seg type="var">a.d.</seg> ad <seg type="var">.d.e.</seg> erit vt <seg type="var">.D.</seg> ad <seg type="var">.B</seg>.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0398-01" corresp="note-0398-01a" place="margin">M</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Inquit nunc Archimedes, ſi quis ſumeret aliquod maius antecedens æquale ſci-<lb/>licet duplo ipſius <seg type="var">.a.b.</seg> cum quadruplo ipſius <seg type="var">.b.e.</seg> cum quadruplo ipſius <seg type="var">.b.d.</seg> cum du-<lb/>plo ipſius <seg type="var">.b.e.</seg> <choice><ex>compararetque</ex><am>compararetq́;</am></choice> illud cum <choice><ex>conſequente</ex><am>cõſequente</am></choice> <seg type="var">.B.</seg> clarum eſſet ex .8. quinti quod <lb/>tale antecedens maiorem proportionem haberet ad <seg type="var">.B.</seg> quam ad <seg type="var">.D.</seg> hoc eſt maiorem <lb/>quàm ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.d.e.</seg> ex .12. quinti.</s> </p> <p> <s xml:space="preserve">Nunc ſi ſumpta fuerit aliqua linea, puta <seg type="var">.d.o.</seg> cui <seg type="var">.a.d.</seg> <choice><ex>dictam</ex><am>dictã</am></choice> habeat proportionem <lb/>maiorem, larum erit ex ſecunda parte decimę quinti quod <seg type="var">.d.o.</seg> minor erit ipſa <seg type="var">.d.e.</seg> <lb/>Corrige igitur impreſſionem Baſileę locando characterem <seg type="var">.o.</seg> inter <seg type="var">.d.</seg> et <seg type="var">.e.</seg> eo quod <lb/>ibi poſitum non fuit.</s> </p> <p> <s xml:space="preserve">Volo nunc quod dictum maius antecedens æquale ſcilicet duplo ipſius <seg type="var">.a.b.</seg> cum <lb/>quadruplo ipſius <seg type="var">.b.c.</seg> cum quadruplo ipſius <seg type="var">.b.d.</seg> cum duplo ipſius <seg type="var">.b.c.</seg> ſignificetur à <lb/>charactere <seg type="var">.A</seg>. </s> <s xml:space="preserve">Hinc habebimus proportionem ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.d.o.</seg> ut <seg type="var">.A.</seg> ad <seg type="var">.B</seg>.</s> </p> <note xml:space="preserve" place="margin">β</note> <p> <s xml:space="preserve">Ex .18. quinti poſtea habebimus <seg type="var">.A.B.</seg> ad <seg type="var">.B.</seg> vt <seg type="var">.a.o.</seg> ad <seg type="var">.d.o.</seg> & proportionalitate <lb/>euerſa in .19. dicti ita erit <seg type="var">.A.B.</seg> ad <seg type="var">.A.</seg> vt <seg type="var">.a.o.</seg> ad <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Sed hoc vltimum antecedens in <lb/> <ptr xml:id="note-0398-03a" corresp="note-0398-03" type="noteAnchor"/> ſe continet id quod Archimedes ſcribit, hoc eſt duplum ipſius <seg type="var">.a.b.</seg> <choice><ex>quadruplum</ex><am>quadruplũ</am></choice> ipſius <lb/><seg type="var">b.c.</seg> ſexcuplum ipſius <seg type="var">.b.d.</seg> & triplum ipſius <seg type="var">.b.e</seg>. </s> <s xml:space="preserve">Conſequens verò <seg type="var">.A.</seg> continet du <lb/>plum ipſius <seg type="var">.a.b.</seg> quadruplum ipſius <seg type="var">.b.c.</seg> quadruplum ipſius <seg type="var">.b.d.</seg> & duplum ipſius <seg type="var">.b.e</seg>.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0398-03" corresp="note-0398-03a" place="margin">T</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Ex ſuppoſito deinde ipſius Archimedis & ex conuerſa proportionalitate in .19. <lb/>dicta, verum eſt id quod dicit Archimedes, videlicet quod eadem proportio eſt <lb/>ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.g.h.</seg> quod quintupli ipſius <seg type="var">.a.b.</seg> cum quintuplo ipſius <seg type="var">.b.e.</seg> cum decuplo <lb/>ipſius <seg type="var">.b.c.</seg> cum decuplo ipſius <seg type="var">.b.d.</seg> (quod quidem antecedens ſignificetur per <seg type="var">.V.</seg>) <lb/>ad duplum ipſius <seg type="var">.a.b.</seg> cum quadruplo ipſius <seg type="var">.b.c.</seg> cum ſexcuplo ipſius <seg type="var">.b.d.</seg> cum triplo <lb/>ipſius <seg type="var">.b.e.</seg> hoc eſt ad <seg type="var">.A.B</seg>.</s> </p> <p> <s xml:space="preserve">Erit igitur <seg type="var">.V.</seg> ad <seg type="var">.A.B.</seg> vt ipſius <seg type="var">.a.d.</seg> ad <seg type="var">.g.h.</seg> ſed ſuperius vbi ſignatum eſt <seg type="var">.T.</seg> iam <lb/>probatum fuit ita eſſe <seg type="var">.A.B.</seg> ad <seg type="var">.A.</seg> vt ipſius <seg type="var">.a.o.</seg> ad <seg type="var">.a.d</seg>. </s> <s xml:space="preserve">Ergò ex .23. quinti Archime <lb/>des verum ſcribit, hoc eſt quod ita erit ipſius <seg type="var">.V.</seg> ad <seg type="var">.A.</seg> vt ipſius <seg type="var">.a.o.</seg> ad <seg type="var">.g.h</seg>.</s> </p> <p> <s xml:space="preserve">Clarum per ſe etiam eſt, id quod Archimed. dicit hoc eſt quod <seg type="var">.V.</seg> ad <seg type="var">.A.</seg> eſt vt <lb/> <ptr xml:id="note-0398-04a" corresp="note-0398-04" type="noteAnchor"/> <pb facs="0399" n="387"/><fw type="head">EPISTOL AE.</fw> quinque ad duo, cum quodlibet ingredientium in compoſito <seg type="var">.V.</seg> ad quodlibet in-<lb/>gredientium in compoſito <seg type="var">.A.</seg> ſit vt quinque ad duo. </s> <s xml:space="preserve">Quare ex .13. quinti verum <lb/>dicit. </s> <s xml:space="preserve">Vnde <seg type="var">.a.o.</seg> ad <seg type="var">.g.h.</seg> erit vt <choice><ex>quinque</ex><am>quinq;</am></choice> ad duo ex .11. <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> vt inquit Archimedes.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0398-04" corresp="note-0398-04a" place="margin">ω</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Corrige impreſſionem vbi ſcriptum eſt, rurſus quoniam <seg type="var">.o.a.</seg> quia oportet dicere <lb/>Rurſus quoniam <seg type="var">.o.d</seg>.</s> </p> <p> <s xml:space="preserve">Archimedes igitur verum dicit, quod ipſius <seg type="var">.o.d.</seg> ad <seg type="var">.d.a.</seg> eſt vt ipſius <seg type="var">.B.</seg> ad <seg type="var">.A.</seg> ex <lb/> <ptr xml:id="fig-0399-01a" corresp="fig-0399-01" type="figureAnchor"/> <pb facs="0400" n="388"/><fw type="head">IO. BABPT. BENED.</fw> conuerſa proportionalitate in .19. quinti, cum <seg type="var">.a.d.</seg> ad <seg type="var">.d.o.</seg> iam probatum fuit (vbi <lb/>B.) ita eſſe ut <seg type="var">.A.</seg> ad <seg type="var">.B</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0399-01" corresp="fig-0399-01a"> <graphic url="0399-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed in principio huius ſpeculationis probatum iam fuit ita eſſe ipſius <seg type="var">.d.a.</seg> ad <seg type="var">.d.e.</seg> <lb/>vt ipſius <seg type="var">.D.</seg> ad <seg type="var">.B.</seg> vbi notatum eſt <seg type="var">.M</seg>. </s> <s xml:space="preserve">quare ex .23. quinti, Archimedes verum dicit, <lb/>qu od <seg type="var">.d.o.</seg> ad <seg type="var">.d.e.</seg> erit vt <seg type="var">.D.</seg> ad <seg type="var">.A</seg>.</s> </p> <p> <s xml:space="preserve">Sed cum <seg type="var">.d.o.</seg> ad <seg type="var">.d.e.</seg> ſe habeat ut <seg type="var">.D.</seg> ad <seg type="var">.A.</seg> erit ex conuerſa proportionalitate iam <lb/>dicta <seg type="var">.d.e.</seg> ad <seg type="var">.d.o.</seg> vt <seg type="var">.A.</seg> ad <seg type="var">.D.</seg> per euerſam vero erit <seg type="var">.d.e.</seg> ad <seg type="var">.a.o.</seg> vt <seg type="var">.A.</seg> ad ſuum reſi-<lb/> <ptr xml:id="note-0400-01a" corresp="note-0400-01" type="noteAnchor"/> duum. </s> <s xml:space="preserve">quod reſiduum componitur ex ſimplo <seg type="var">.b.c.</seg> cum triplo <seg type="var">.b.</seg> cum duplo <seg type="var">.b.o.</seg> quod <lb/>à te ipſo videre poteris detrahendo numeros ipſarum quantitatum quæ in <seg type="var">.D.</seg> <lb/>reperiuntur, ex numeris earundem, quæ in <seg type="var">.A.</seg> quod quidem reſiduum ſigniſicetur <lb/>à charactere <seg type="var">.E</seg>. </s> <s xml:space="preserve">Vnde ex conuerſa proportionalitate verum dicit Archime. hoc eſt <lb/>quod ita ſe hab ebit <seg type="var">.o.e.</seg> ad <seg type="var">.d.e.</seg> vt <seg type="var">.E.</seg> ad <seg type="var">.A</seg>.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0400-01" corresp="note-0400-01a" place="margin">λ</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Cum autem ſit <seg type="var">.a.b.</seg> ad <seg type="var">.c.b.</seg> vt <seg type="var">.c.b.</seg> ad <seg type="var">.d.b.</seg> & ita <seg type="var">.d.b.</seg> ad <seg type="var">.e.b.</seg> ex ſuppoſito, ideo ex <lb/>17. quinti verum dicit Archim. </s> <s xml:space="preserve">hoc eſt quod ita erit ipſius <seg type="var">.d.e.</seg> ad <seg type="var">.e.b.</seg> vt <seg type="var">.a.c.</seg> ad <seg type="var">.c.b.</seg> <lb/>& vt <seg type="var">.c.d.</seg> ad <seg type="var">.d.b.</seg> & ex .13. eiuſdem eadem proportio erit tripli ipſius <seg type="var">.c.d.</seg> ad triplum <lb/>ipſius <seg type="var">.d.b.</seg> quæ dupli ipſius <seg type="var">.d.e.</seg> ad. duplum ipſius <seg type="var">.e.b.</seg> vt inquit Archi.</s> </p> <p> <s xml:space="preserve">Ex qua .13. compoſitum ex <seg type="var">.a.c.</seg> cum triplo ipſius <seg type="var">.c.d.</seg> cum duplo ipſius <seg type="var">.d.e.</seg> ean-<lb/>dem proportionem habebit ad <choice><ex>compoſitum</ex><am>compoſitũ</am></choice> ipſius <seg type="var">.c.b.</seg> cum triplo <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.d.b.</seg> cum duplo <lb/>ipſius <seg type="var">.e.b.</seg> quam ipſius <seg type="var">.d.e.</seg> ad <seg type="var">.e.b</seg>. </s> <s xml:space="preserve">Sed horum compoſitorum primum ſignificetur <lb/>per <seg type="var">.H.</seg> ſecundum verò ſignificatum fuit per <seg type="var">.E.</seg> vnde <seg type="var">.H.</seg> ad <seg type="var">.E.</seg> ſe habebit vt <seg type="var">.d.e.</seg> ad <lb/><seg type="var">e.b.</seg> ſed <seg type="var">.E.</seg> ad <seg type="var">.A.</seg> iam dictum eſt eſſe vt <seg type="var">.o.e.</seg> ad <seg type="var">.d.e.</seg> vbi ſignatum eſt <seg type="var">.λ</seg>. </s> <s xml:space="preserve">quare ex .23. <lb/>quinti eadem proportio erit ipſius <seg type="var">.o.e.</seg> ad <seg type="var">.e.b.</seg> quæ <seg type="var">.H.</seg> ad <seg type="var">.A.</seg> vt ipſe inquit.</s> </p> <note xml:space="preserve" place="margin">X</note> <p> <s xml:space="preserve">Ex .18. poſtea eiuſdem ita erit <seg type="var">.o.b.</seg> ad <seg type="var">.e.b.</seg> vt <seg type="var">.H.A.</seg> ad <seg type="var">.A</seg>.</s> </p> <p> <s xml:space="preserve">Notandum etiam eſt quod ſi collectæ fuerint omnes partes compoſiti <seg type="var">.H.A.</seg> hoc <lb/>eſt duplum <seg type="var">.a.b.</seg> cum duplo <seg type="var">.b.e.</seg> cum quadruplo <seg type="var">.b.c.</seg> cum quadruplo <seg type="var">.b.d.</seg> cum ſimplo <lb/><seg type="var">a.c.</seg> cum triplo <seg type="var">.c.d.</seg> cum duplo <seg type="var">.d.e.</seg> habebitur triplum <seg type="var">.a.b.</seg> triplum <seg type="var">.b.d.</seg> & ſexcuplum <lb/><seg type="var">b.c.</seg> vt ipſe dixit. </s> <s xml:space="preserve">Quod autem hoc verum ſit, cum diſtinctæ fuerint omnes partes, <lb/>vt in ſubſcriptis his lineis videre eſt, videbis quod ſi ex <seg type="var">.H.</seg> detracta fuerit ſimplex <seg type="var">.a.<lb/>c.</seg> quæ quidem poſtea iuncta vni ex partibus quadrupli <seg type="var">.b.c.</seg> ipſius <seg type="var">.A.</seg> reſultabit nobis <lb/>vna inte gra <seg type="var">.a.b</seg>. </s> <s xml:space="preserve">Vnde habebimus triplum ipſius <seg type="var">.a.b.</seg> & in <seg type="var">.A.</seg> remanebit triplum ip <lb/>ſius <seg type="var">.c.b</seg>. </s> <s xml:space="preserve">Deinde ſi ex <seg type="var">.H.</seg> auferatur triplum ipſius <seg type="var">.c.d.</seg> & ipſum addatur tribus parti-<lb/>bus quadrupli <seg type="var">.b.d.</seg> ipſius <seg type="var">.A.</seg> habebimus tres vices <seg type="var">.b.c.</seg> quæ ſi iungantur tribus, quæ <lb/>remanebant in <seg type="var">.A.</seg> vt dixi, habebimus ſexcuplum ipſius <seg type="var">.b.c.</seg> & in <seg type="var">.A.</seg> remanebit ſim <lb/>plum <seg type="var">.b.d.</seg> cum duplo ipſius <seg type="var">.b.e</seg>. </s> <s xml:space="preserve">Vnde ſi ex <seg type="var">.H.</seg> demptum fuerit duplum ipſius <seg type="var">.d.<lb/>e.</seg> quod quidem iungatur cum duplo ipſius <seg type="var">.b.e.</seg> habebimus duplum ipſius <seg type="var">.b.d.</seg> quod <lb/>coniunctum cum ſimplo <seg type="var">.b.d.</seg> quod in <seg type="var">.A.</seg> relictum fuerat, habebimus triplum ipſius <lb/><seg type="var">d.b</seg>. </s> <s xml:space="preserve">Verum igitur eſt quod inquit Archimedes, hoc eſt, quod <seg type="var">.H.A.</seg> eſt triplum ip-<lb/>ſius <seg type="var">.a.b.</seg> ſexcuplum ipſius <seg type="var">.b.c.</seg> & triplum ipſius <seg type="var">.b.d</seg>.</s> </p> <p> <s xml:space="preserve">Verum etiam dicit ex eo (vt ſupra probatum eſt) quod <seg type="var">.a.c</seg>: <seg type="var">c.d</seg>: et <seg type="var">.d.e.</seg> ſe <choice><ex>habebant</ex><am>habebãt</am></choice> <lb/>in continua proportionalitate, </s> <s xml:space="preserve">quare ex conuerſa proportionalitate erunt ſibi inui-<lb/>cem continuæ proportionales.</s> </p> <p> <s xml:space="preserve">Nunc autem cum <seg type="var">.a.c</seg>: <seg type="var">c.d.</seg> et <seg type="var">.d.e.</seg> ſint continuæ proportionales in ea proportione <lb/>in qua ſunt <seg type="var">.a.b</seg>: <seg type="var">c.b</seg>: <seg type="var">d.b</seg>: et <seg type="var">.e.b.</seg> vt in principio diximus, erit ex .22. quinti <seg type="var">.a.c.</seg> ad <seg type="var">.d.<lb/>e.</seg> vt <seg type="var">.a.b.</seg> ad <seg type="var">.d.b.</seg> & ſic etiam <seg type="var">.c.b.</seg> ad <seg type="var">.e.b</seg>. </s> <s xml:space="preserve">Vnde ex .24. eiuſdem <seg type="var">.a.d.</seg> ad <seg type="var">.d.e.</seg> erit vt <seg type="var">.a.<lb/>b.</seg> cum <seg type="var">.b.e.</seg> ad <seg type="var">.d.b.</seg> & vt <seg type="var">.c.b.</seg> cum <seg type="var">.b.d.</seg> ad <seg type="var">.e.b.</seg> & ex .13. dicti vt <seg type="var">.a.b.</seg> cum <seg type="var">.b.e.</seg> bis <lb/>ſumpto, & cum <seg type="var">.b.d.</seg> ad <seg type="var">.e.b</seg>. </s> <s xml:space="preserve">Quare ex conuerſa proportionalitate, vt ſe habet <seg type="var">.e.d.</seg> <lb/>ad <seg type="var">.d.a.</seg> ita ſe habebit <seg type="var">.e.b.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.d.b.</seg> ad <seg type="var">d.b.</seg> <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.b.c.</seg> duplicato & <choice><ex>cum</ex><am>cũ</am></choice> <seg type="var">.b.a.</seg> vt inquit Archi <lb/>medes. </s> <s xml:space="preserve">Nunc antecedens vocetur <seg type="var">.M.</seg> hoc eſt <seg type="var">.b.e.</seg> cum <seg type="var">.d.b.</seg> conſequens verò, hoc <pb facs="0401" n="389"/><fw type="head">EPISTOLAE.</fw> eſt <seg type="var">.d.b.</seg> cum duplo <seg type="var">.b.c.</seg> cum ſimplo <seg type="var">.b.a.</seg> vocetur <seg type="var">.N</seg>.</s> </p> <p> <s xml:space="preserve">Animaduertendum tamen eſt quod impreſſio mendoſa eſt ubi dicit. <lb/></s> <s xml:space="preserve"><hi rend="q">vnaquæque <seg type="var">.c.b</seg>: <seg type="var">b.d.</seg> & cætera,</hi> <lb/>propterea quod dicendum eſt ita <lb/><choice><ex>vnaquæque</ex><am>vnaquæq;</am></choice> <seg type="var">e.b</seg>: <seg type="var">b.d</seg>.</s> </p> <p> <s xml:space="preserve">Nunc ex .18. quinti, quemadmodum ſe habet <seg type="var">.a.e.</seg> ad <seg type="var">.d.a.</seg> ita ſe habebit <seg type="var">.M.N.</seg> ad <seg type="var">.N.</seg> <lb/> <ptr xml:id="fig-0401-01a" corresp="fig-0401-01" type="figureAnchor"/> </s> <pb facs="0402" n="390"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve">Vbiautem ſcriptum eſt <lb/><hi rend="q">ad vtrunque ſimul <seg type="var">.b.d</seg>: <seg type="var">d.a.</seg> cum dupla <seg type="var">.b.c.</seg></hi> <lb/>dicendum eſt ita, <lb/>ad vtranque ſimul <seg type="var">.b.d.b.a.</seg> cum dupla <seg type="var">.b.c</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0401-01" corresp="fig-0401-01a"> <graphic url="0401-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Inquit deinde Archi. quod ſicut ſe haber <seg type="var">.e.a.</seg> ad <seg type="var">.d.a.</seg> ita ſe habebit duplum <seg type="var">.M.N.</seg> <lb/>ad duplum <seg type="var">.N</seg>. </s> <s xml:space="preserve">Quod quidem verum eſt ex .13. quinti, huiuſmodi verò antecedons <lb/>& conſequens, Archi. manifeſtat ex ſuis partibus, ſumendo duplum <seg type="var">.e.b.</seg> cum duplo <lb/><seg type="var">b.d.</seg> pro duplo <seg type="var">.M.</seg> & duplum <seg type="var">.b.d.</seg> cum duplo <seg type="var">.a.b.</seg> cum quadruplo <seg type="var">.b.c.</seg> pro duplo <seg type="var">.N.</seg> <lb/>quę ſimul iuncta æquantur duplo <seg type="var">.e.b.</seg> cum duplo <seg type="var">.a.b.</seg> cum quadruplo <seg type="var">.b.d.</seg> cum qua-<lb/>druplo <seg type="var">.b.c.</seg> ex quo æquabuntur <seg type="var">.A.</seg> vocentur igitur hæc omnia <seg type="var">.A.</seg> potius quàm du-<lb/>plum ipſius <seg type="var">.M.N</seg>.</s> </p> <p> <s xml:space="preserve">Verum etiam ſcribit, vbi dicit, quod proportio <seg type="var">.e.a.</seg> ad tres quintas ipſius <seg type="var">.a.d.</seg> erit <lb/>vt <seg type="var">.A.</seg> ad tres quintas dupli <seg type="var">.N.</seg> ex .22. quinti. </s> <s xml:space="preserve">Sed cum ex ſuppoſito ita ſe habeat <seg type="var">.f.<lb/>g.</seg> ad tres quintas ipſius <seg type="var">.a.d.</seg> quemadmodum <seg type="var">.b.e.</seg> ad <seg type="var">.e.a.</seg> erit ex .16. quinti verum <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>dicit Archimed. </s> <s xml:space="preserve">hoc eſt, ita ſe habere <seg type="var">.b.e.</seg> ad <seg type="var">.f.g.</seg> vt <seg type="var">.e.a.</seg> ad tres quintas ipſius <seg type="var">.a.d</seg>.</s> </p> <p> <s xml:space="preserve">Et per .11. eiuſdem verum etiam erit quod ſicut ſe habet <seg type="var">.e.b.</seg> ad <seg type="var">.f.g.</seg> ita ſe habe-<lb/>bit <seg type="var">.A.</seg> ad tres quintas dupli <seg type="var">.N.</seg> quod quidem duplum <seg type="var">.N.</seg> ſignificetur per <seg type="var">.Q</seg>.</s> </p> <p> <s xml:space="preserve">Sed ſuperius iam demonſtratum fuit (vbi <seg type="var">.X.</seg>) quod <seg type="var">.o.b.</seg> ad <seg type="var">.b.e.</seg> ita ſe habebat vt <lb/><seg type="var">H.A.</seg> ad <seg type="var">.A.</seg> & <choice><ex>nunc</ex><am>nũc</am></choice> demum probatum fuit ita eſſe <seg type="var">.A.</seg> ad tres quintas ipſius <seg type="var">.Q.</seg> vt <seg type="var">.e.b.</seg> <lb/>ad <seg type="var">.f.g</seg>. </s> <s xml:space="preserve">Quare ex .22. quinti ita erit <seg type="var">.H.A.</seg> ad tres quintas ipſius <seg type="var">.Q.</seg> vt <seg type="var">.o.b.</seg> ad <seg type="var">.f.g.</seg> vt <lb/> <ptr xml:id="note-0402-01a" corresp="note-0402-01" type="noteAnchor"/> idem inquit.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0402-01" corresp="note-0402-01a" place="margin">Y</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Sed <seg type="var">.H.A.</seg> ad <seg type="var">.Q.</seg> (vt ex ſuis partibus videre eſt) ita ſe habet vt tres ad duo ex .13. <lb/>quinti, vt inquit Archimedes.</s> </p> <p> <s xml:space="preserve">Ipſe etiam dicit proportionem <seg type="var">.H.A.</seg> ad tres quintas ipſius <seg type="var">.Q.</seg> eſſe vt quinque <lb/>ad duo. </s> <s xml:space="preserve">Pro cuius rei euidentia imaginemur tam <seg type="var">.H.A.</seg> quam <seg type="var">.Q.</seg> diuiſa per <choice><ex>quinque</ex><am>quinq;</am></choice> <lb/>partes æquales, vnde ex .16. quinti habebimus quamlibet quintam <choice><ex>partem</ex><am>partẽ</am></choice> ipſius <seg type="var">.Q.</seg> <lb/><choice><ex>æqualem</ex><am>æqualẽ</am></choice> eſſe duabus tertijs vniuſcuiuſque quintæ partis <seg type="var">.H.A.</seg> vnde tres quintæ ipſius <lb/>Q. erunt, ex communi conceptu, ſex tertiæ vnius quintæ ipſius <seg type="var">.H.A.</seg> hoc eſt duæ <lb/>quintæ. ipſius <seg type="var">.H.A</seg>. </s> <s xml:space="preserve">Quare <seg type="var">.o.b.</seg> ita ſe habebit ad <seg type="var">.f.g.</seg> vt quinque ad duo ex commu <lb/>ni <choice><ex>conceptu</ex><am>cõceptu</am></choice>, cum <seg type="var">.o.b.</seg> ad <seg type="var">.f.g.</seg> probatum fuerit ſe habere vt <seg type="var">.H.A.</seg> ad tres quintas ipſius <lb/>Q. (vbi <seg type="var">.Y.</seg>) ſed iam probatum fuit (vbi. ω) quod <seg type="var">.o.a.</seg> ad <seg type="var">.h.g.</seg> erat etiam vt <lb/>quinque ad duo, hoc eſt quod <seg type="var">.f.h.</seg> erit duæ quintę ipſius <seg type="var">.a.b</seg>. </s> <s xml:space="preserve">Quod eſt propoſitum.</s> </p> <pb facs="0403" n="391"/> <fw type="head">EPISTOL AE.</fw> <figure place="here"> <graphic url="0403-01"/> </figure> <pb facs="0404" n="392"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">In vltima verò propoſitione ſecundi lib. de ponderibus Archi. hoc modo intelli<lb/>gendus eſt, vt ſi diceret, <lb/>Sit paraboles <seg type="var">.a.</seg> cuius baſis ſit <seg type="var">.a.c.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.d.e.</seg> recta parallela dictæ baſi <seg type="var">.a.c.</seg> <choice><ex>diameterque</ex><am>diameterq́;</am></choice> <lb/><seg type="var">b.f</seg>. <lb/></s> <s xml:space="preserve">Inquit deinde quod linea contingens in <seg type="var">.b.</seg> parallela erit ipſi <seg type="var">.a.c.</seg> et <seg type="var">.e.d.</seg> quod proba <lb/>bimus hoc modo. <lb/></s> <s xml:space="preserve">Cum <seg type="var">.b.f.</seg> diameter ſit et <seg type="var">.a.c.</seg> baſis, clarum erit ex definitione quod <seg type="var">.b.f.</seg> diuidet <seg type="var">.a.c.</seg> <lb/>per æqualia in <seg type="var">.g</seg>. </s> <s xml:space="preserve">Vnde ex .7. vel etiam ex .46. primi Pergei <seg type="var">.d.e.</seg> diuiſa erit per æqua <lb/>lia à diametro <seg type="var">.b.f</seg>. </s> <s xml:space="preserve">Quare verum dicit ex quinta ſecundi ipſius Pergei hoc eſt quod <lb/>dicta contingens in puncto. b parallela erit ambobus <seg type="var">.a.c.</seg> et <seg type="var">.e.d</seg>.</s> </p> <p> <s xml:space="preserve">Inquit poſtea quod diuiſa cum fuerit pars diametri quę inter <seg type="var">.d.e.</seg> et <seg type="var">.a.c.</seg> poſita eſt <lb/>(hoc eſt <seg type="var">.g.f.</seg>) per quinque partes æquales, <choice><ex>quarum</ex><am>quarũ</am></choice> partium media ſit <seg type="var">.h.k.</seg> diuiſa etiam <lb/>imaginatione ſit in puncto <seg type="var">.i.</seg> ita quod proportio ipſius <seg type="var">.h.i.</seg> ad <seg type="var">.i.K.</seg> eadem ſit quæ in-<lb/>ter duo ſolida quorum vnum (illud ſcilicet à quo relatio incipit, hoc eſt antecedens) <lb/>pro ſua baſi teneat quadratum ipſius <seg type="var">.a.f.</seg> cuius etiam ſolidi altitudo compoſita ſit ex <lb/> <ptr xml:id="note-0404-01a" corresp="note-0404-01" type="noteAnchor"/> duplo ipſius <seg type="var">.d.g.</seg> cum ſimplo <seg type="var">.a.f</seg>. </s> <s xml:space="preserve">Aliud verò ſolidum habeat pro ſua baſi quadra-<lb/>tum ipſius <seg type="var">.d.g.</seg> eius verò altitudo compoſita ſit ex duplo ipſius <seg type="var">.a.f.</seg> cum ſimplo <seg type="var">.d.g</seg>.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0404-01" corresp="note-0404-01a" place="margin">R</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Inquit nunc Archi. quod cum ita factum fuerit, oſtendet punctum <seg type="var">.i.</seg> centrum eſſe <lb/>portionis abſciſſę à tota ſectione, quod <choice><ex>fruſtum</ex><am>fruſtũ</am></choice> <choice><ex>nominatur</ex><am>nominat̃</am></choice> <choice><ex>ſignatum</ex><am>ſignatũ</am></choice> characteribus <seg type="var">.a.d.e.c</seg>.</s> </p> <p> <s xml:space="preserve">Sit igitur num@. <seg type="var">m.n.</seg> inquit, æqualis diametro <seg type="var">.b.f.</seg> et <seg type="var">.n.o.</seg> æqualis <seg type="var">.b.g.</seg> <choice><ex>ſitque</ex><am>ſitq́;</am></choice> <seg type="var">.x.n.</seg> me <lb/>dia proportionalis inter <seg type="var">.n.m.</seg> et <seg type="var">.n.o.</seg> et <seg type="var">.t.n.</seg> in continua proportionalitate poſt <seg type="var">.o.n.</seg> <lb/>hoc eſt quod ea proportio quæ eſt ipſius <seg type="var">.o.n.</seg> ad <seg type="var">.n.t.</seg> eadem ſit ipſius <seg type="var">.x.n.</seg> ad <seg type="var">.n.o</seg>. </s> <s xml:space="preserve">Hinc <lb/>habebimus .4. lineas in continua proportionalitate ſibi inuicem coniunctas <seg type="var">.m.n</seg>: <seg type="var">x.<lb/>n</seg>: <seg type="var">o.n.</seg> et <seg type="var">.t.n</seg>.</s> </p> <p> <s xml:space="preserve">Vult etiam quod à linea <seg type="var">.i.b.</seg> incipiens ab <seg type="var">.i.</seg> verſus <seg type="var">.g.</seg> alia linea abſciſſa ſit, cui li-<lb/> <ptr xml:id="note-0404-02a" corresp="note-0404-02" type="noteAnchor"/> neæ, ita proportionata ſit <seg type="var">.f.h.</seg> vt <seg type="var">.t.m.</seg> eſt ad <seg type="var">.t.n.</seg> quæ quidem linea ſignata ſit <seg type="var">.i.r</seg>.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0404-02" corresp="note-0404-02a" place="margin">A</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Dicit poſtea quod diameter <seg type="var">.b.f.</seg> erit fortaſſe a xis vel aliqua reliquarum diame-<lb/>trorum, quod quidem in .46. primi Pergei videre eſt, cum omnes diametri ſint in-<lb/>uicem paralleli ipſi axi.</s> </p> <p> <s xml:space="preserve">Cum poſtea dicit, quod <seg type="var">.a.f.</seg> et <seg type="var">.d.g.</seg> ſunt intentæ ductæq́ue, ibi vult id em infer-<lb/>re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipſius Pergei videre li-<lb/>cet, vnde ex .20. eiuſdem proportio <seg type="var">.b.f.</seg> ad <seg type="var">.b.g.</seg> erit vt quadrati <seg type="var">.a.f.</seg> ad quadratum <lb/>ipſius <seg type="var">.d.g.</seg> vt ipſe dicit.</s> </p> <p> <s xml:space="preserve">Sed ita erit quadrati <seg type="var">.m.n.</seg> ad qua <choice><ex>dratum</ex><am>dratũ</am></choice> <seg type="var">.x.n.</seg> ex .18. ſexti Eucli. </s> <s xml:space="preserve">Quare ex .11. quin-<lb/> <ptr xml:id="note-0404-03a" corresp="note-0404-03" type="noteAnchor"/> ti quadratum ipſius <seg type="var">.m.n.</seg> ad quadratum ipſius <seg type="var">.n.x.</seg> eandem habebit proportionem, <lb/>quam quadratum ipſius <seg type="var">.a.f.</seg> ad quadratum ipſius <seg type="var">.d.g</seg>. </s> <s xml:space="preserve">Vnde ex .18. & ex communi <lb/><choice><ex>ſcientia</ex><am>ſciẽtia</am></choice>, eadem proportio erit ipſius <seg type="var">.m.n.</seg> ad <seg type="var">.n.x.</seg> quę ipſius <seg type="var">.a.f.</seg> ad <seg type="var">.d.g.</seg> vt inquit Arch.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0404-03" corresp="note-0404-03a" place="margin">α</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Quaptopter proportio cubi ipſius <seg type="var">.m.n.</seg> ad cubum ipſius <seg type="var">.n.x.</seg> erit vt cubi ipſius <seg type="var">.a.<lb/>f.</seg> ad cubum ipſius <seg type="var">.d.g.</seg> vt etiam dicit ex communi ſcientia, nec non ex .36. vndecimi.</s> </p> <p> <s xml:space="preserve">Inquit poſtea quod proportio totius ſectionis <seg type="var">.a.b.c.</seg> ad portionem <seg type="var">.d.b.e.</seg> eadem <lb/>eſt quæ cubi ipſius <seg type="var">.a.f.</seg> ad cubum ipſius <seg type="var">.d.g.</seg> quod verum eſt, vt aliàs tibi monſtraui in <lb/>diuiſione parabolæ ſecundum aliquam propoſitam proportionem.</s> </p> <p> <s xml:space="preserve">Quando autem dicit quod proportio cubi ipſius <seg type="var">.m.n.</seg> ad cubum ipſius <seg type="var">.n.x.</seg> eadem <lb/> <ptr xml:id="note-0404-04a" corresp="note-0404-04" type="noteAnchor"/> eſt quæ ipſius <seg type="var">.m.n.</seg> ad <seg type="var">.n.t.</seg> verum dicit ex .36. vndecimi. </s> <s xml:space="preserve">Vnde ex .11. quinti ita ſe <lb/>habebit totalis ſectio <seg type="var">.a.b.c.</seg> ad portionem <seg type="var">.d.b.c.</seg> vt <seg type="var">.m.n.</seg> ad <seg type="var">.n.t.</seg> & ex .17. eiuſdem ita <lb/>erit ipſius <seg type="var">.m.t.</seg> ad <seg type="var">.t.n.</seg> vt fruſti <seg type="var">.a.d.e.c.</seg> ad ſectionem <seg type="var">.d.b.e.</seg> quemadmodum ipſe di-<lb/>cit. </s> <s xml:space="preserve">Sed quia ſuperius, vbi <seg type="var">.A.</seg> ipſa <seg type="var">.f.h.</seg> (quæ eſt tres quintæ ipſius <seg type="var">.f.g.</seg>) ad <seg type="var">.i.r.</seg> ita rela- <pb facs="0405" n="393"/><fw type="head">EPISTOL AE.</fw> ta fuit vt <seg type="var">.m.t.</seg> ad <seg type="var">.t.n.</seg> idcirco ex .11. quinti ita erit ipſius fruſti <seg type="var">.a.e.</seg> ad ſectionem <seg type="var">.d.b.<lb/>e.</seg> vt tres quintę ipſius <seg type="var">.f.g.</seg> ad <seg type="var">.i.r</seg>.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0404-04" corresp="note-0404-04a" place="margin">β</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Inquit deinde quod proportio corporis iam ſupradicti, quod pro ſua baſi habeat <lb/>quadratum ipſius <seg type="var">.a.f.</seg> altitudinem verò compoſitam ex duplo ipſius <seg type="var">.d.g.</seg> cum ſimplo <lb/><seg type="var">a.f.</seg> ad cubum ipſius <seg type="var">.a.f.</seg> eadem erit quæ dupli ipſius <seg type="var">.d.g.</seg> cum ſimplo <seg type="var">.a.f.</seg> ad <seg type="var">.a.f</seg>. </s> <s xml:space="preserve">Quod <lb/>quidem verum eſt ex .33. vndecimi & ex prima ſexti.</s> </p> <p> <s xml:space="preserve">Sed ſuperius (vbi. α.) iam probauimus eandem proportio nem eſſe inter <seg type="var">.m.n.</seg> & <lb/><seg type="var">n.x.</seg> quæ inter <seg type="var">.a.f.</seg> et <seg type="var">.d.g.</seg> ideo ex conuerſa pro portion alitate ita erit ipſius <seg type="var">.x.n.</seg> ad <seg type="var">.n.<lb/>m.</seg> vt ipſius <seg type="var">.d.g.</seg> ad <seg type="var">.a.f.</seg> ſed dupli <seg type="var">.x.n.</seg> ad ſimplum <seg type="var">.x.n.</seg> eſt vt dupli <seg type="var">.d.g.</seg> ad <seg type="var">.d.g</seg>. </s> <s xml:space="preserve">Qua <lb/>re ex .22. quinti dupli <seg type="var">.x.n.</seg> ad <seg type="var">.m.n.</seg> erit vt dupli <seg type="var">.d.g.</seg> ad <seg type="var">.a.f.</seg> & ex .18. eiuſdem ita erit <lb/>dupli <seg type="var">.x.n.</seg> cum ſimplo <seg type="var">.m.n.</seg> ad <seg type="var">.m.n.</seg> vt dupli <seg type="var">.d.g.</seg> cum ſimplo <seg type="var">.a.f.</seg> ad <seg type="var">.a.f</seg>. </s> <s xml:space="preserve">Quare ſolidi <lb/> <ptr xml:id="fig-0405-01a" corresp="fig-0405-01" type="figureAnchor"/> <pb facs="0406" n="394"/><fw type="head">IO. BAPT. BENED.</fw> iam dicti ad cubum inſius <seg type="var">.a.f.</seg> ex .11. quinti erit vt dupli <seg type="var">.x.n.</seg> <choice><ex>cum</ex><am>cũ</am></choice> ſimplo <seg type="var">.m.n.</seg> ad <seg type="var">.m.n</seg>.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0405-01" corresp="fig-0405-01a"> <graphic url="0405-01"/> </figure> </div> </body> </floatingText> <note xml:space="preserve" place="margin">δ</note> <p> <s xml:space="preserve">Superius autem vbi. β. demonſtratum fuit ita eſſe ipſius <seg type="var">.m.n.</seg> ad <seg type="var">.n.t.</seg> vt cubi <seg type="var">.m.n.</seg> <lb/>ad cubum <seg type="var">.x.n.</seg> & inter. α et. β probatum fuit ita eſſe cubi <seg type="var">.a.f.</seg> ad cubum <seg type="var">.d.g.</seg> vt <lb/>cubi <seg type="var">.m.n.</seg> ad cubum <seg type="var">.x.n</seg>. </s> <s xml:space="preserve">Vnde ex .11. quinti <seg type="var">.m.n.</seg> ad <seg type="var">.n.t.</seg> erit vt cubi <seg type="var">.a.f.</seg> ad cubum <lb/><seg type="var">d.g</seg>.</s> </p> <p> <s xml:space="preserve">Dicit poſtea quod eadem proportio erit inter cubum <seg type="var">.d.g.</seg> & corpus illud quod <lb/>pro baſi habeat quadratum inſius <seg type="var">.d.g.</seg> altitudinem verò vt dictum eſt, quæ eſt inter <lb/><seg type="var">d.g.</seg> & compoſitum ex duplo <seg type="var">.a.f.</seg> cum ſimplo <seg type="var">.d.g.</seg> quod compoſitum eſt altitudo di <lb/>cta, & <choice><ex>verum</ex><am>verũ</am></choice> dicit ex ratione ſuperius allegata pro reliquo corpore & cubo ipſius <seg type="var">.a.f</seg>. <lb/></s> <s xml:space="preserve">Quare etiam quemadmodum <seg type="var">.t.n.</seg> ſe habet ad duplum ipſius <seg type="var">.o.n.</seg> cum ſimplo <seg type="var">.t.n.</seg> <lb/>ex ijſdem rationibus ſupradictis, vbiloquuti ſumus de <seg type="var">.x.n.</seg> cum <seg type="var">.m.n</seg>.</s> </p> <p> <s xml:space="preserve">Diſponantur <choice><ex>nunc</ex><am>nũc</am></choice> omnia tali ordine, ita vt <seg type="var">.u.</seg> primum ſit corpus quod pro ſua ba <lb/>ſi habeat quadratum ipſius <seg type="var">.a.f.</seg> & c.</s> </p> <p> <s xml:space="preserve">Et <seg type="var">.y.</seg> ſit cubus ipſius <seg type="var">.a.f.</seg> et <seg type="var">.s.</seg> ſit cubus ipſius <seg type="var">.d.g.</seg> et <seg type="var">.z.</seg> ſit corpus quod baſim ha-<lb/>bet quadratum ipſius <seg type="var">.d.g.</seg> altitudinem verò vt ſupradictum eſt, et <seg type="var">.p.</seg> ſit compoſitum <lb/>dupli <seg type="var">.n.x.</seg> cum ſimplo <seg type="var">.m.n.</seg> et <seg type="var">.l.</seg> ſit compoſitum dupli ipſius <seg type="var">.n.o.</seg> cum ſimplo <seg type="var">.t.n.</seg> <lb/>Sed <seg type="var">.u.</seg> locata ſit è regione <seg type="var">.p.</seg> et <seg type="var">.y.</seg> è regione <seg type="var">.m.n.</seg> et <seg type="var">.s.</seg> è regione <seg type="var">.n.t.</seg> et <seg type="var">.z.</seg> è regione <seg type="var">.l.</seg> <lb/>& habebimus proportionem ipſius <seg type="var">.u.</seg> ad <seg type="var">.y.</seg> vt <seg type="var">.y.</seg> ad <seg type="var">.m.n.</seg> & ipſius <seg type="var">.y.</seg> ad <seg type="var">.s.</seg> vt <seg type="var">.m.n.</seg> ad <seg type="var">.<lb/>n.t.</seg> quod ſuperius iam demonſtratum fuit, vbi, δ. et <seg type="var">.s.</seg> ad <seg type="var">.z.</seg> ita ſe habebit vt <seg type="var">.n.t.</seg> ad <seg type="var">.<lb/>l.</seg> vt vltimò probatum fuit. </s> <s xml:space="preserve">Quare ex .22. quinti ita ſe habebit <seg type="var">.u.</seg> ad <seg type="var">.z.</seg> vt <seg type="var">.p.</seg> ad <seg type="var">.l.</seg> <lb/>quemadmodum dicit Archi.</s> </p> <p> <s xml:space="preserve">Et quia vt ſe habet <seg type="var">.u.</seg> ad <seg type="var">.z.</seg> ita facta fuit <seg type="var">.h.i.</seg> ad <seg type="var">.i.K.</seg> vbi <seg type="var">.R.</seg> ideo ex .11. quinti vt ſe <lb/>habet <seg type="var">.h.i.</seg> ad <seg type="var">.i.K.</seg> ita ſe habebit <seg type="var">.p.</seg> ad <seg type="var">.l.</seg> vt ipſe dicit: </s> <s xml:space="preserve">Et ex .18. quinti ita erit <seg type="var">.h.K.</seg> <lb/>ad <seg type="var">.K.i.</seg> vt <seg type="var">.p.l.</seg> ad <seg type="var">.l.</seg> & ex communi conceptu <seg type="var">.g.f.</seg> ſe habebit ad <seg type="var">.h.K.</seg> vt quintuplum <lb/>ipſius <seg type="var">.p.l.</seg> ad <seg type="var">.p.l.</seg> & ex .22. eiuſdem ita ſe habebit <seg type="var">.f.g.</seg> ad <seg type="var">.i.k.</seg> vt quintuplum ipſius <seg type="var">.p.<lb/>l.</seg> ad <seg type="var">.l.</seg> quintuplum autem ipſius <seg type="var">.p.l.</seg> compoſitum eſt ex quintuplo ipſius <seg type="var">.n.m.</seg> cum <lb/>decuplo ipſius <seg type="var">.n.x.</seg> cum quintuplo ipſius <seg type="var">.n.t.</seg> cum decuplo ipſius <seg type="var">.n.o.</seg> vt à te facilè <lb/>computare potes.</s> </p> <p> <s xml:space="preserve">Verum etiam erit ex communi ſcientia quod <seg type="var">.g.f.</seg> ad <seg type="var">.f.k.</seg> eſt ut quintuplum ipſius <lb/><seg type="var">p.l.</seg> ad duplum ipſius <seg type="var">.p.l.</seg> eo quod ſuperius ſuppoſitum fuit <seg type="var">.h.K.</seg> eſſe <choice><ex>quintam</ex><am>quintã</am></choice> mediam, <lb/>vnde <seg type="var">.k.f.</seg> relinquebatur pro duabus quintis inferioribus, duplum autem <seg type="var">.p.l.</seg> com-<lb/>poſitum eſt ex duplo ipſius <seg type="var">.m.n.</seg> cum duplo ipſius <seg type="var">.n.t.</seg> cum quadruplo ipſius <seg type="var">.n.x.</seg> & <lb/>cum quadruplo ipſius <seg type="var">.x.o</seg>.</s> </p> <p> <s xml:space="preserve">Ex conuerſa proportionalitate deinde ita ſe habet, <seg type="var">i.K.</seg> ad <seg type="var">.i.k.</seg> ad <seg type="var">.f.g.</seg> vt <seg type="var">.l.</seg> ad quin-<lb/>tuplum ipſius <seg type="var">.p.l.</seg> et <seg type="var">.k.f.</seg> ad <seg type="var">.f.g.</seg> vt duplum ipſius <seg type="var">.p.l.</seg> ad quintuplum ipſius <seg type="var">.p.l</seg>. </s> <s xml:space="preserve">Vnde <lb/>ex .24. quinti <seg type="var">.i.f.</seg> ſe habebit ad <seg type="var">.f.g.</seg> vt <choice><ex>duplum</ex><am>duplũ</am></choice> ipſius <seg type="var">.p.l.</seg> cum ſimplo <seg type="var">.l.</seg> ad quintuplum <lb/>ipſius <seg type="var">.p.l</seg>. </s> <s xml:space="preserve">Deinde ex conuerſa proportionalitate quintuplum ipſius <seg type="var">.p.l.</seg> ſe habebit <lb/> <ptr xml:id="note-0406-02a" corresp="note-0406-02" type="noteAnchor"/> ad duplum ipſius <seg type="var">.p.l.</seg> cum ſimplo <seg type="var">.l.</seg> vt <seg type="var">.f.g.</seg> ad <seg type="var">.f.i</seg>. </s> <s xml:space="preserve">Sed compoſitum dupli ipſius <seg type="var">.p.l.</seg> <lb/>cum ſimplo <seg type="var">.l.</seg> æquale eſt duplo ipſius <seg type="var">.m.n.</seg> cum quadruplo ipſius <seg type="var">.x.n.</seg> cum ſexcuplo <lb/>ipſius <seg type="var">.o.n.</seg> cum triplo ipſius <seg type="var">.n.t.</seg> vt per te computare potes.</s> </p> <floatingText> <body> <div type="float"> <note xml:space="preserve" xml:id="note-0406-02" corresp="note-0406-02a" place="margin">θ</note> </div> </body> </floatingText> <p> <s xml:space="preserve">Superius enim ſumpta fuit <seg type="var">.i.r.</seg> ad quam ita ſe haberet <seg type="var">.f.h.</seg> hoc eſt tres quintæ ip-<lb/>ſius <seg type="var">.f.g.</seg> vt <seg type="var">.m.t.</seg> ad <seg type="var">.t.n</seg>. </s> <s xml:space="preserve">Quare ex conuerſa proportionalitate ita ſe habebit <seg type="var">.i.r.</seg> ad tres <lb/>quintas ipſius <seg type="var">.f.g.</seg> vt <seg type="var">.t.n.</seg> ad <seg type="var">.t.m</seg>. </s> <s xml:space="preserve">Et quia <seg type="var">.o.n.</seg> ſumpta fuit æqualis ipſi <seg type="var">.b.g.</seg> et <seg type="var">.m.n.</seg> ipſi <lb/><seg type="var">b.f.</seg> ideo <seg type="var">.m.o.</seg> ex communi ſcientia æ qualis erit ipſi <seg type="var">.g.f</seg>. </s> <s xml:space="preserve">Vnde proportio <seg type="var">.r.i.</seg> ad tres <lb/>quintas ipſius <seg type="var">.m.o.</seg> erit vt <seg type="var">.n.t.</seg> ad <seg type="var">.t.m.</seg> vt inquit Archi.</s> </p> <p> <s xml:space="preserve">Sed vbi. θ. iam probauimus ita ſe habere <seg type="var">.i.f.</seg> ad <seg type="var">.f.g.</seg> vt duplum <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> <seg type="var">.p.l.</seg> cum ſim-<lb/>plo <seg type="var">.l.</seg> ſe habet ad quintuplum ipſius <seg type="var">.p.l.</seg> hoc eſt <seg type="var">.i.f.</seg> ad <seg type="var">.m.o.</seg> vt duplum ipſius <seg type="var">.p.l.</seg> cum <lb/>ſimplo <seg type="var">.l.</seg> ad quintuplum ipſius <seg type="var">.p.l</seg>.</s> </p> <pb facs="0407" n="395"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Habemus igitur <choice><ex>nuncomnem</ex><am>nuncomnẽ</am></choice><unclear reason="illegible"/>s illas conditiones quas Archimedes in præcedenti <lb/>propoſitione ſupponit. </s> <s xml:space="preserve">Vnde ex rationibus ibi allegatis ſequitur <seg type="var">.f.r.</seg> eſſe duas quin-<lb/>tas ipſius <seg type="var">.m.n.</seg> hoc eſt ipſius <seg type="var">.f.b</seg>. </s> <s xml:space="preserve">Quapropter punctum <seg type="var">.r.</seg> centrum erit ponderis to-<lb/>tius ſectionis parabolæ ex .8. ſecundi lib. de ponderibus eiuſdem Archimedis.</s> </p> <p> <s xml:space="preserve">Inquit nunc Archimedes, quod exiſtente <seg type="var">.q.</seg> centro ponderis ipſius parabolæ <seg type="var">.d.<lb/>b.e.</seg> partialis, centrum fruſti erit in linea recta <seg type="var">.q.r.f.</seg> ita remotum à centro <seg type="var">.r.</seg> quod <lb/>proportio <seg type="var">.q.r.</seg> ad partem illam ipſius <seg type="var">.r.f.</seg> quæ reperitur inter centrum <seg type="var">.r.</seg> & centrum <lb/>huius fruſti æqualis eſt proportioni totius parabolæ ad partialem. </s> <s xml:space="preserve">Quod quidem ve <lb/>rum eſt ex .8. primi libri eiuſdem.</s> </p> <p> <s xml:space="preserve">Inquit etiam punctum <seg type="var">.i.</seg> illud eſſe, eo quod cum probatum ſit <seg type="var">.f.r.</seg> duas quintas eſ-<lb/>ſe ipſius <seg type="var">.f.b.</seg> ideo <seg type="var">.b.r.</seg> tres quintas erit ipſius <seg type="var">.b.f.</seg> vt ipſe dicit.</s> </p> <figure place="here"> <graphic url="0407-01"/> </figure> <pb facs="0408" n="396"/> <fw type="head">IO, BAPT. BENED.</fw> <p> <s xml:space="preserve">Sed <seg type="var">.q.b.</seg> ſimiliter tres quintæ eſt ipſius <seg type="var">.d.b.</seg> ex .8. prædicta. </s> <s xml:space="preserve">Quare <seg type="var">.q.r.</seg> tres quintæ <lb/>erit ipſius <seg type="var">.f.g.</seg> ex .19. quinti.</s> </p> <p> <s xml:space="preserve">Dicamus igitur hoc modo cum <seg type="var">.f.b.</seg> totum ad totum <seg type="var">.b.r.</seg> ita ſe habeat vt abſciſ-<lb/>ſum <seg type="var">.b.g.</seg> ad abſciſſum <seg type="var">.q.b.</seg> ex .7. et .8. dicti primi libri eiuſdem ideo reſiduum <seg type="var">.f.g.</seg> ex <lb/><seg type="var">f.b.</seg> ad reſiduum <seg type="var">.r.q.</seg> ex <seg type="var">.r.b.</seg> erit vt totum <seg type="var">.f.b.</seg> ad. totum <seg type="var">.r.b.</seg> ex .19. quinti Eucli.</s> </p> <p> <s xml:space="preserve">Sed iam ſub. β. probauimus ita ſe habere fruſtum <seg type="var">.a.d.e.c.</seg> ad parabolam <seg type="var">.d.b.e.</seg> vt <lb/><seg type="var">m.t.</seg> ad <seg type="var">.t.n.</seg> ſed vt <seg type="var">.m.t.</seg> ad <seg type="var">.t.n.</seg> ita aſſ umpta fuit (vbi <seg type="var">.A.</seg>). <seg type="var">i.r.</seg> ad quam ſic ſe haberet <seg type="var">.f.<lb/>h.</seg> hoc eſt tres quintæ ipſius <seg type="var">.f.g.</seg> hoc eſt <seg type="var">.q.r</seg>. </s> <s xml:space="preserve">quare ex .11. quinti prop ortio fruſti <seg type="var">.a.<lb/>d.e.c.</seg> ad parabolam partialem erit vt <seg type="var">.q.r.</seg> ad <seg type="var">.r.i</seg>. </s> <s xml:space="preserve">Exiſtente igitur <seg type="var">.r.</seg> centro totius pa <lb/>rabolæ et <seg type="var">.q.</seg> centro partialis, ergo <seg type="var">.i.</seg> centrum erit fruſti propoſiti.</s> </p> <p> <s xml:space="preserve">Sed ſi nullo ſolido intercedente, voluerimus centrum <seg type="var">.i.</seg> fruſti <seg type="var">.a.e.</seg> citius inuenire, <lb/>inueniemus primò centrum <seg type="var">.r.</seg> totius figuræ ex .8. ſecundi eiuſdem conſtituendo <seg type="var">.b.r.</seg> <lb/>tres quintas totius axis <seg type="var">.b.f.</seg> & centrum <seg type="var">.q.</seg> parabolæ <seg type="var">.d.b.e.</seg> partialis ſimiliter.</s> </p> <p> <s xml:space="preserve">Nunc igitur manifeſtum eſt nobis, eandem proportionem fore ipſius <seg type="var">.q.r.</seg> <lb/>ad <seg type="var">.r.i.</seg> quæ fruſti <seg type="var">.a.e.</seg> ad portionem <seg type="var">.d.b.e.</seg> ex .8. dicta. </s> <s xml:space="preserve">Vnde ex coniuncta pro-<lb/>portionalitate ita ſe habebit <seg type="var">.q.i.</seg> ad <seg type="var">.i.r.</seg> vt <seg type="var">.a.b.c.</seg> ad <seg type="var">.d.b.e.</seg> ſed vt <seg type="var">.a.b.c.</seg> ad <seg type="var">.d.b.e.</seg> ita ſe <lb/>habet <seg type="var">.m.n.</seg> ad <seg type="var">.n.t.</seg> eo quod vnaquæque harum duarum proportionum ſeſquialtera <lb/>eſt proportioni <seg type="var">.f.b.</seg> ad <seg type="var">.b.g.</seg> eo. quod <seg type="var">.f.b.</seg> ad <seg type="var">.b.g.</seg> ita ſe habet. vt <seg type="var">.m.n.</seg> ad <seg type="var">.o.n.</seg> </s> <s xml:space="preserve">quare <lb/><seg type="var">m.n.</seg> ad <seg type="var">.t.n.</seg> ita ſe habebit vt <seg type="var">.g.i.</seg> ad <seg type="var">.r.i.</seg> vnde diſiunctim <seg type="var">.m.t.</seg> ad <seg type="var">.t.n.</seg> ita ſe habebit vt <lb/><seg type="var">q.r.</seg> ad <seg type="var">.r.i</seg>. </s> <s xml:space="preserve">Iungatur igitur <seg type="var">.r.i.</seg> quæ quidem <seg type="var">.r.i.</seg> ita ſe habeat ad <seg type="var">.r.q.</seg> vt <seg type="var">.t.n.</seg> ad <seg type="var">.t.m.</seg> vt <lb/>habeatur centrum fruſti.</s> </p> <pb facs="0409" n="397"/> <fw type="head">EPISTOL AE.</fw> <figure place="here"> <graphic url="0409-01"/> </figure> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DEFENSIO NOSTRA CONTRA ANTONIVM <lb/>Bergam, & Alexandrum Piccolhomineum.</head> <head rend="italics" xml:space="preserve">Illuſtri Domino Horatio Muto.</head> <p> <s xml:space="preserve"><hi rend="small caps">INter</hi> ea quæ olim contra Antonium Bergam, ſermone Italico ſcripſi, <lb/>hoc vnum erat, quod ip ſe Berga non viderat quendam notatu dignum <lb/>errorem ipſius Pi ccolhominei, vbi ipſe Alexander arguit quendam au-<lb/>thorem in tractatu de magnitudine terræ & aquæ pag .37. linea .26. ita di <lb/>cens, & erit maior aqua.</s> </p> <pb facs="0410" n="398"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Quo in loco clare videtur ipſum putare eandem proportionem inter diametros, <lb/>quæ inter ſphæras ipſas eſſe, nec amplius recordari eius quod ſcripſerat pag .24.</s> </p> <p> <s xml:space="preserve">Piccolhom. igitur ibi <choice><ex>ſupponens</ex><am>ſupponẽs</am></choice> centrum <seg type="var">.D.</seg> eſſe magnitudinis aquæ, & intra ſphæ <lb/>ram terreſtrem, putat omnino cauſam eſſe vt terra ſuperet aquam magnitudine, qua-<lb/>ſi quod ſi punctum <seg type="var">.D.</seg> vt centrum ſphæræ aquæ, vnum <choice><ex>idemque</ex><am>idemq́;</am></choice> eſſet cum puncto <seg type="var">.E.</seg> <lb/>extremo diametri ipſius terræ, ſphæra <seg type="var">.A.G.H.</seg> ſphæræ <seg type="var">.A.B.E.</seg> dupla eſſe deberet, <lb/>quod quidem nullo pacto fieri poteſt, quamuis etiam proportio <seg type="var">.A.H.</seg> ad <choice><ex>diametrum</ex><am>diametrũ</am></choice> <lb/><seg type="var">A.E.</seg> ſuperbipartiensſeptimas exiſteret, quæ minor eſſet quam ſeſquitertia, ita quod <lb/>quando etiam <seg type="var">.D.E.</seg> maior medietate ipſius <seg type="var">.D.H.</seg> fuiſſet, nihilominus tamen <lb/>terra minor eſſet aqua, eo quod proportio dupla minor eſt, quam tripla ad <choice><ex>propor- tionem</ex><am>propor-tionẽ</am></choice> ſuperbipartientenſeptimas, & maior <choice><ex>quam</ex><am>quã</am></choice> tripla ad proportionem ſeſquiquar-<lb/>tam. </s> <s xml:space="preserve">Vnde ſi Piccolhom. ſuppòſuiſſet proportionem ipſius <seg type="var">.D.H.</seg> ad <seg type="var">.C.E.</seg> eſſe <lb/>ſeſquiquartam, rectè profectò dixiſſet, ſed dicere quod ubicunque exiſtat <choice><ex>punctum</ex><am>punctũ</am></choice> <seg type="var">.<lb/>D.</seg> intra ſphæram terreſtrem, ſequitur ipſam eſſe maiorem aquea, verum non eſt.</s> </p> <p> <s xml:space="preserve">Scripſi etiam quod Piccoloho. decipiebatur vbi loquitur de diaphaneitate aquæ <lb/>pag .40. ita dicens.</s> </p> <quote> <s xml:space="preserve">Et cum rationabiliter aliquis exiſtimare non poteſt, quod vmbra quæ facit ori-<lb/>ri e cclipſes Lunæ, producta ſit à terra, & ab aqua ſimul, vt ab vno corpore aggre-<lb/>gato exijs duobus elementis, & ad vnam communem ſphæreceitatem reductis, pro <lb/>pterea quod cum vmbra produci debeat à corporibus opacis, quorum opacitas effi-<lb/>cit illa corpora vmbroſa, aqua autem, ſit corpus diaphanum, & tranſparens, nullam <lb/>vmbram poterit à ſe eminus producere.</s> </quote> <p> <s xml:space="preserve">Hic enim decipitur Piccolhom. duabus rationibus, quarum prima eſt, quod ra-<lb/>dius luminoſus non poteſt multum in profundum mergi, vt probaui in .8. epiſtola ad <lb/>Vimercatum, altera verò eſt, quod cum ſphærica ſit aqua maris, ſupponatur etiam <lb/>quod ſub ea nulla terræ portio eſſet, & quod radij ſolares ipſam, non ſecus ac pilam <lb/>ex criſtallo fabręfactam penetrarent, cum autem ipſi radij, tam ab una, quam ab alia <lb/>parte ſup erficiei huiuſmodi globi <choice><ex>frangantur</ex><am>frãgantur</am></choice>, ob diſſimilem diaphaneitatem inter ae <lb/>rem & aquam, ipſi ſeinuicem interſecarent, vt poſt pilam criſtallinam videre eſt, de-<lb/>inde procedentes, diſgregarentur, <choice><ex>diſciparenturque</ex><am>diſciparenturq́;</am></choice> quouſque nullam vim illuminatio <lb/>nis haberent, quod quilibet experiri poterit mediante aliquo vaſe uitreo ſphærico, <lb/>aqua pleno, cuiuſuis magnitudinis, ſoli expoſito.</s> </p> <p> <s xml:space="preserve">Rationes etiam quas eodem loco Piccolho. adducit ad probandum quod ſi quis <lb/>in fundo maris exiſteret, nullum uideret lumen, nihil ualent. </s> <s xml:space="preserve">Quarum prima eſt, <lb/>ubi ita dicit.</s> </p> <quote> <s xml:space="preserve">Ille qui ſe in aquam mergit, cum maiorem lucem, quæ ſupra aquam eſt, relin-<lb/>quat, iudicat pro magno temporis ſpatio locum illum obſcurum, quemadmodum <lb/>accidit quando per multum temporis ſpatium fixis oculis in corpore Solis intuiti ſu <lb/>mus, ab eodem poſtea eoſdem amouentes, omnia obſcura nobis videntur.</s> </quote> <p> <s xml:space="preserve">Ipſe autem non conſiderat quod talis obſcuritas quæ ſequitur viſionem maioris <lb/>luminis, parum durat, immo cito euaneſcit, ſed in aqua nunquam reuertimur ad vi-<lb/>dendum, ne que veſtigium aliquod luminis ibi videtur, in fundo maris dico, quem-<lb/>admodum nobis nuntiauerunt hi qui margaritas expiſcantur in imis partibus ingen <lb/>tium æquorum indicorum.</s> </p> <p> <s xml:space="preserve">Secunda uerò ratio ipſius Piccolhom. eſt ubi ita dicit.</s> </p> <quote> <s xml:space="preserve">Altera cauſa quod nobis obſcurus appareat locus ſub aqua, eſſe poteſt obſtacu-<lb/>lum quod aquæ habent ab opacitate terræ ſub eorum fundo, etenim ſicut <choice><ex>chriſtallum</ex><am>chriſtallũ</am></choice> <pb facs="0411" n="399"/><fw type="head">EPISTOLAE.</fw> quamuis <choice><ex>perſpicuum</ex><am>perſpicuũ</am></choice> ſiue <choice><ex>tranſparens</ex><am>tranſparẽs</am></choice> ſit, nihilominus propter obſtaculum plumbi ſub <lb/>ipſo poſiti, efficit vt radij viſuales repercuſſi reuertantur. ita etiam quamuis aqua ſit <lb/>corpus tranſparens, nihilominus propter obſtaculum terræ opacæ, quæ ſubſidet in <lb/>fundo maris efficere poteſt obſcuras partes illas ſub aqua, illis hominibus qui in <lb/>ipſa aqua mèrguntur.</s> </quote> <p> <s xml:space="preserve">In hac ſecunda ratione decipitur Piccolhom. </s> <s xml:space="preserve">Primum quia ſi vſque ad imam par <lb/>tem maris, Solis radius ferri poſſet, ille qui ibi eſſet, attollens oculos ſurſum Solem <lb/>cerneret, </s> <s xml:space="preserve">deinde aſpiciendo ipſum fundum Maris, videret illum, ratione reflexio-<lb/>nis luminis ab ipſo fundo, & ex eadem ratione ſpeculi ab ipſo adducta, quæ contra <lb/>ipſum eſt.</s> </p> <p> <s xml:space="preserve">Decipitur etiam cum dicatradios viſuales à ſpeculo ſeu plumbo repercuti, eo <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>non radij viſuales ſunt hi qui reflectuntur, ſed ſunt radij luminoſi primarij, ſeu ſecun <lb/>darij qui non ab oculis exeunt ſed à corpore lucido.</s> </p> <p> <s xml:space="preserve">Scripſi etiam quod ſi verum eſſet proportionalitatem continuam <choice><ex>quantitatum</ex><am>quãtitatum</am></choice> ele-<lb/>mentorum ex proportione decupla conſtare, ignem pro maximo, terram verò pro <lb/>minimo terminorum ſumentes, totum aggregatum ex terra, aqua, aere, & igne, ita <lb/>eſſet maius terra, quemadmodum mille centum & vndecim ad vnum, vnde ſemidia <lb/>meter regionis elementaris eſſet quaſi aut paulo maior decuplo ſolum ſemidiame-<lb/>tro terræ, vnde inter conuexum ignis, & concauum minimi, ſeu inferioris orbis lu-<lb/>naris, relinqueretur quidam orbis vacuus ſpiſſitudinis vnius interualli plus quam vi-<lb/>ginti terræ ſemidiametrorum, quod ſpatium vacuum orbiculariter, maius exi-<lb/>ſteret ipſa totali regione elementari plus quam trigeſies millies, immo ſi ſemidia <lb/>meter dicti primi orbislunaris maior eſſet terreſtri vt trigintanouem ad unum, <choice><ex>dictus</ex><am>dictꝰ</am></choice> <lb/>orbis vacuus maior eſſet elementari regione plus quam .58208. ad vnum, proportio <lb/>nalitatem igitur continuam quæ ex decupla proportionalitate reſultat in elementis <lb/>eſſe putare eſt maximus error.</s> </p> <p> <s xml:space="preserve">Subdit deinde Berga, hoc voluiſſe Platonem neceſſario requiri, vt extrema ele-<lb/>menta, nempeignis & terra cum duobus medijs aere, & aqua coniungerentur, cum in <lb/>corporibus ſolidis (quaſi Bergę ſint quædam corpora quæ ſolida non extent) poſſit <lb/>dari medium æquale in geometrica proportione.</s> </p> <p> <s xml:space="preserve">Sed vbi Plato ad ſermonem de numero elementorum ſe confert, poſtquam ra-<lb/>tione creationis ignis, & terrę ſe propoſuiſſe putat, vt <choice><ex>idem</ex><am>idẽ</am></choice> de alijs duobus corporibus <lb/>medijs probet, comparatione proportionalitatis continuæ geometricæ in tribus ter-<lb/>minis, ratione rerum ſuperficialium primò, deinde in quatuor, ratione corporearum <lb/>vtitur, ita dicens.</s> </p> <quote> <s xml:space="preserve">Vinculorum verò ideſt aptiſſimum atque pulcherrimum quod exſe, & ex ijs quę <lb/>aſtringunt, quam maximè vnum efficit, <choice><ex>&c.</ex><am>&c.</am></choice></s> </quote> <p> <s xml:space="preserve">Quo in loco Plato inſerre vult de proportionalitate geometrica trium termino-<lb/>rum, in qua ijdem ita ſe habent, vt medius, primi, <choice><ex>vltimique</ex><am>vltimiq́;</am></choice> vice fungatur, ita vt vtriuſ-<lb/>que ipſorum extremorum particeps fiat, cum productum quod à medio termino in <lb/>ſeipſo progignitur idem ſit ei quod ab extremis fuit, vnde medius, potentia idem eſt <lb/>quod productum ab extremis.</s> </p> <p> <s xml:space="preserve">Subdit deinde Plato dicens.</s> </p> <quote> <s xml:space="preserve">Quando enim in tribus numeris, aut molibus, aut viribus, medium ita ſe habet <lb/>ad poſtremum vt primum ad medium, <choice><ex>viciſſimque</ex><am>viciſſimq́;</am></choice> vt poſtremum cum medio, ita me-<lb/>dium cum primo congruit, </s> <s xml:space="preserve">tunc quod medium eſt, & primum fit & <choice><ex>poſtremum</ex><am>poſtremũ</am></choice>, po-<lb/>ſtremum quoque, & primum & media fiunt.</s> </quote> <pb facs="0412" n="400"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Hic <choice><ex>animaduertendum</ex><am>animaduertendũ</am></choice> eſt omnes interpretes falli, qui hoc loco Platonem de omni-<lb/>bus proportionalitatibus continuis quæ ternario numero (alia enim Arithmetica, <lb/>alia geometrica, alia harmonica dicitur) continentur, intelligendum eſſe cenſent, <lb/>quia de numeris, magnitudinibus, <choice><ex>viribusque</ex><am>viribusq́;</am></choice>, aut ut dici ſolet, virtutibus mentionem <lb/>fecerit. </s> <s xml:space="preserve">Plato enim nihil aliud inferre voluit, quam eandem paſſionem (ut ipſe reci-<lb/>tat) inter medium <choice><ex>extremaque</ex><am>extremaq́;</am></choice> vnius proportionalitatis continuæ geometricæ, tam <lb/>in quantitate, quam in qualitate <choice><ex>reſultaturam</ex><am>reſultaturã</am></choice>, cum tres termini eiuſdem eſſent ſpe-<lb/>ciei, & quia quantitas in duas principes <choice><ex>primariasque</ex><am>primariasq́;</am></choice> partes, ideſt in continuam, & <lb/>diſcretam diuiditur, hanc ob cauſam Plato hoc præcipuè ſignificat numerorum ma-<lb/><choice><ex>gnitudinisque</ex><am>gnitudinisq́;</am></choice> vocabulis vtens, quibus vniuerſum quantitatis genus complectitur.</s> </p> <p> <s xml:space="preserve">Cum verò ait vires, uniuerſum qualitatis genus inferre uult. </s> <s xml:space="preserve">Quia proportio & <lb/>proportionalitas tam continua quam diſcreta, non ſolum interterminos quanti, ſed <lb/>inter eos etiam qui quali attribuuntur elucet.</s> </p> <p> <s xml:space="preserve">Sed quod eo loco de harmonica proportionalitate quæ <choice><ex>cum</ex><am>cũ</am></choice> geometrica magis ſim <lb/>bola eſt quam cum Arithmetica Plato minime intelligat, ex eiuſdem uerbis cum ita <lb/>ſcribit manifeſtè patet.</s> </p> <quote> <s xml:space="preserve">Quando enim medium ita ſe habet ad poſtremum ut primum ad medium, <choice><ex>uiciſ- ſimque</ex><am>uiciſ-ſimq́;</am></choice> ut poſtremum cum medio ita medium cum primo congruit.</s> </quote> <p> <s xml:space="preserve">Id enim in harmonica proportionalitate non cernitur in qua primus terminus ad <lb/>poſtremum, & non ad medium, ita ſe habet geometricè ut differentia inter primum <lb/>& medium ad differentiam inter medium & ultimum.</s> </p> <p> <s xml:space="preserve">Quod ſi clarum eſt ipſum de harmonica proportionalitate nullo modo intellige-<lb/>re, quanto minus de Arithmetica, quæ cum geometrica nihil habet commune.</s> </p> <p> <s xml:space="preserve">Cum uerò Plato ait.</s> </p> <quote> <s xml:space="preserve">Tunc quod medium eſt & primum fit & poſtremum, poſtremum quoque, & pri-<lb/>mum media fiunt, <choice><ex>&c.</ex><am>&c.</am></choice></s> </quote> <p> <s xml:space="preserve">Nihil aliud oſtendere uult, quam ſimilitudinem quæ inter huiuſmodi medium & <lb/>extrema intercedit, cum ipſum medium ad poſtremum, quem primus ad ſeipſum, <lb/>eundem reſpectum habeat, in quo eſt ſimilis primo, & contra ad primum <choice><ex>terminum</ex><am>terminũ</am></choice>, <lb/>eundem reſpectum, quem poſtremum ad ſeipſum habet, </s> <s xml:space="preserve">unde hac ratione ultimum <lb/>repręſentat, uolens Plato inferre de conuenientia quę inter media elementa, & ex-<lb/>trema intercedit, ut aquæ inter aerem, & terram, cum aqua, ratione ſuæ frigiditatis, <lb/>terrę, ratione uero ſuæ humiditatis aeri ſimilis euadat. </s> <s xml:space="preserve">Aer uero qui inter ignem, <lb/><choice><ex>aquamque</ex><am>aquamq́;</am></choice> ponitur quod ad caliditatem attinet cum igne, quod uero ad humidita-<lb/>tem ſpectat cum aqua communicet.</s> </p> <p> <s xml:space="preserve">Sed quia Plato multis in rebus doctrinam Pythagoricam ſequutus eſt, Pythago-<lb/>rici aut em omnia numeris metiebantur, & de omnire ſecundum numerorum ratio <lb/>nem diſſerebant, <choice><ex>uidensque</ex><am>uidensq́;</am></choice> Plato quod inter duos numeros ſuperficiales, <choice><ex>inuicemque</ex><am>inuicemq́;</am></choice> <lb/>ſimiles exiſtentes, unum tantum numerum medium in proportionalitate continua <lb/>geometrica cadere poteſt, ideo ſubiungit.</s> </p> <quote> <s xml:space="preserve">Quod ſi uniuerſi corpus latitudinem habere debuiſſet, nullam uerò profundita-<lb/>tem, unum ſanè, tum ad ſeipſum, tum ad extrema uincienda interiectum medium <lb/>ſuffeciſſet.</s> </quote> <p> <s xml:space="preserve">Sequitur poſtea ſic.</s> </p> <quote> <s xml:space="preserve">Sed cum ſoliditatem mundus requireret, ſolida uerò non uno, ſed duobus ſem-<lb/>per modis copulentur, inter ignem, & terram, Deus, Aerem, Aquamq́ue loca-<lb/>uit, <choice><ex>&c.</ex><am>&c.</am></choice></s> </quote> <pb facs="0413" n="401"/> <fw type="head">EPISTOLAE.</fw> <p> <s xml:space="preserve">Volens inferre, quod quemadmodum inter duos numeros ſolidos, & inuicem <lb/>ſimiles, <choice><ex>vnus</ex><am>vnꝰ</am></choice> <choice><ex>tantum</ex><am>tãtũ</am></choice> medius proportionalis intercedere <choice><ex>non</ex><am>nõ</am></choice> poteſt, ſed duo neceſſariò re <lb/>quiruntur (vt exijs quæ Euclid .8. lib. 16. 17. 18. et .19. propoſitione proponit viden<lb/>tur) ita <choice><ex>dictante</ex><am>dictãte</am></choice> ratione inter igneum, <choice><ex>terreumque</ex><am>terreumq́;</am></choice> corpus duo corpora interiecta <choice><ex>eſsent</ex><am>eſsẽt</am></choice>, <lb/>non ratione proportionalitatis continuæ in quantitate <choice><ex>eorundem</ex><am>eorũdem</am></choice> corporum, ſed pro <lb/>pter ſimilitudinem connexionis, cum productum ex duobus medijs proportionali-<lb/>bus æquale ſit producto ab extremis, & idem reſpectus, quem primum ipſorum qua <lb/>tuor ad ſecundum habet, ſecundi ad tertium extet, vnde ſecundum primo ſimile <lb/>euadit, & contra, reſpectus qui eſt quarti ad tertium, ſit etiam tertij ad ſecundum, vn-<lb/>de ipſum tertium, ratione vltimi ſubit, & eius imaginem induit, & hanc ob cauſam <lb/>ſic ſcribit Plato.</s> </p> <quote> <s xml:space="preserve">Propterea ex huiuſmodi rebus numero quaternario concluſis, mundi corpus con <lb/>flatum eſt, ea connexum comparatione qua dixi. </s> <s xml:space="preserve">Ex quo ſeipſum amicitia concor-<lb/>di complectitur, <choice><ex>&c.</ex><am>&c.</am></choice></s> </quote> <p> <s xml:space="preserve">Vbi Platonem, elementa maiora, <choice><ex>minoranue</ex><am>minorãue</am></choice> in proportionalitate continua, nec <lb/>geometrica, nec alterius cuiuſuis generis eſſe noluiſſe, clarè perſpicitur, ſed huiuſmo <lb/>di ſimilitudine, in eo quod media elementa cum extremis conueniunt eſt vſus, quæ <lb/>quidem conuenientia, nullibi maior, quam in proportionalitate continua geome-<lb/>trica reperitur. </s> <s xml:space="preserve">Sed etiam ſi Plato de huiuſmodi corporea elementorum magnitu-<lb/>dine ſeipſum intelligi voluiſſet, ſi ſemidiameter regionis elementaris ex ęquo vt .39 <lb/>ad vnum, reſpectu ſemidiametri terræ fuiſſet, aqua, ipſam terram, magis quam tri-<lb/>geſies, & octies, non ſolum decies, & aer quoque eandem magis quam .1500. & <lb/>ignis magis quam .55000. partibus magnitudine ſuperaret.</s> </p> <p> <s xml:space="preserve">Subſtantia vero rerum quas ſcripſeram circa finem illius conſiderationis talis fuit.</s> </p> <p> <s xml:space="preserve">Nunc autem tempus eſſe videtur, vt ego etiam, ne tantum deſtruxiſſe, ſed etiam <lb/>conſtruxiſſe videar aliquid pro veritate diſſeram.</s> </p> <p> <s xml:space="preserve">Non eſt igitur dubium, ſolidæ doctrinæ viris, quin præſtantiſſimus Piccolo. ſe-<lb/>cutus ſit tutam viam ad explorandum, quod terra maior ſit quam aqua, metiendo <lb/>vtriuſque horum corporum ſuperficiem detectam. </s> <s xml:space="preserve">Omittamus autem compenſa-<lb/>tionem illam curuitatis, & concauitatis vallium, & montium, <choice><ex>&c.</ex><am>&c.</am></choice> quam ipſe Piccolo. <lb/>propè finem ſexti cap. vellet dare fluminibus, ſtagnis, fontibus, & eiuſmodi aquis. <lb/></s> <s xml:space="preserve">eo enim in loco labitur Piccolo. vbi non conſiderat, quod eiuſmodi obliquis ſuper-<lb/>ficiebus non reſpondent anguli ſolidi centri ſphæræ, qui reſpiciunt eorum baſim ad <lb/>rectos angulos. </s> <s xml:space="preserve">Sed poſtquam Piccolo. comperit ſuperficiem terrę detectam, eſſe <lb/>maiorem apparente ſuperficie ſphærica aquæ, proculdubio poterat concludere ter-<lb/>ram eſſe maiorem aqua, ſicuti fecit, <choice><ex>etiam</ex><am>etiã</am></choice> ſi aqua profunda eſſet pyramidaliter <choice><ex>vſque</ex><am>vſq;</am></choice> <lb/>ad mundi centrum, ideſt .3500. milliaria, ſupponendo tantum eſſe huius globi ſemi <lb/>diametrum.</s> </p> <p> <s xml:space="preserve">Verum quia poſſet aliquis dubitare circa diligentiam Piccolo. in hiſcæ duabus ſu <lb/>perficiebus dimetiendis, viſum eſt mihi non alienum ſequi aliam viam pro hac veri <lb/>tate probanda, ſupponendo verum eſſe, quod non vnus ſolus metitus fuerit, ſed mul <lb/>ti, ideſt ſupponendo <choice><ex>verum</ex><am>verũ</am></choice> eſſe quod maris profunditas menſurari poſſit, & præterea, <lb/>quod non modo ipſius maris maxima profunditas non perueniat ad quingentos paſ <lb/>ſus, ſicuti refert Piccolo. in fine ſui tractatus, & mihi aſſeruerunt Hiſpani multi, & <lb/>Luſitani præſtantiſſimi nautæ, tum Venetijs, tum Parmæ, in Aula Sereniſſimæ quon <lb/>dam Principis, inter quos, Venetijs fuit Illuſtris Rodericus Guzmanus, Dominus <lb/>Franciſcus Lopes, Dominus Garzias de Seuilia, <choice><ex>multique</ex><am>multiq́;</am></choice> alij. </s> <s xml:space="preserve">Parmæ autem varij <pb facs="0414" n="402"/><fw type="head">IO. BAPT. BENED.</fw> quos omnes <choice><ex>recenſerem</ex><am>recenſerẽ</am></choice> moleſtum eſſet. </s> <s xml:space="preserve">Sed etiam ſupponendo quod maxima <lb/>pelagi profunditas ſit, non modo .500. paſſuum, ſed etiam .500. millium paſſuum, vt <lb/>dixi, & quod mare ſit huius profunditatis, non vno in loco tantum, aut multis, ſed <lb/>quod ſupra totam etiam faciem terræ, mare tantę profunditatis ipſam terram vn-<lb/>dique operiret, ideſt, quod vbicunque nunc terra detecta eſt, eſſet aqua, ſpiſſitudi-<lb/>nis .500. millium paſſuum. </s> <s xml:space="preserve">Atque vt planius intelligar ſupponendo quod ſicuti to-<lb/>tus huius globi ſemidiameter eſt <choice><ex>milliarium</ex><am>milliariũ</am></choice> .3500. </s> <s xml:space="preserve">Terreſtris partis ſemidiameter <lb/>eſſet <choice><ex>tantum</ex><am>tm̃</am></choice> .3000. & reliquum ſemidiametri, id eſt quingenta milliaria eſſet craſſitudo <lb/>ſiue profunditas orbis aquei, in quo nihil neceſſe eſſet laborare in dimetiendis fon-<lb/>tibus, fluminibus, lacubus, ſtagnis, paludibus, & huiuſmodi particulis nullius momen <lb/>ti apud peritos, nec curare ſubterraneas aquas cauernarum, aut aliorum terræ cauo-<lb/>rum, ſeu terræ porroſitatum, quæ omnia ſunt circa ipſius terræ ſuperficiem. </s> <s xml:space="preserve">Quia ve <lb/>riſimile non eſt naturam eiuſmodi caua ſiue ſpong oſitates produxiſſe demiſſius li-<lb/>bramenti maris. </s> <s xml:space="preserve">Supponendo igitur ea quæ nunc dicta ſunt, terra tamen eſſet ferè <lb/>duplo maior aqua, hoc eſt, vt .12. ad .7. </s> <s xml:space="preserve">Quod quidem, cuiuis mathematicæ philoſo-<lb/>phiæ mediocriter perito, ſupputatu facillnnum eſt. </s> <s xml:space="preserve">Cum proportio diametrorum, <lb/>ſeu ſemidiametrorum, tertia pars exiſtat proportionis eorundem ſphærarum. </s> <s xml:space="preserve">Sed <lb/>vt parum periti minore labore ſupputare poſſint.</s> </p> <p> <s xml:space="preserve">Primum ſciendum eſt, quod ſupponendo diametrum globi, ex terra, & aqua com <lb/>poſiti, eſſe .3500. milliarium, & ſemidiametrum puræ terreſtris partis eſſe .3000. tan <lb/>tum, eiuſmodi proportio erit ut .7. ad .6. quia communis maior numerator horum <lb/>duum ſemidiametrorum erit .500. qui in maiorem ingredietur ſepties, in minorem <lb/>a utem ſexies. </s> <s xml:space="preserve">Et eiuſmodi proportio ſuperparticularis, vocatur ſeſquiſexta, cuius <lb/>triplum erit vt .57. cum ſexta parte ad .36. & idem erit inter dictum globum compo-<lb/>ſitum, & partem terreſtrem ſimplicem. </s> <s xml:space="preserve">Quare ſubtrahendo puram, ſeu ſimplicem <lb/>partem terreſtrem, ex compoſito, reliqua pars erit, vt .21. cum ſexta, pro quantitate <lb/>aquei orbis, ad quam, terreſtris quantitas .36. erit ferè in <choice><ex>eadem</ex><am>eadẽ</am></choice> proportione, quæ .12. <lb/>ad .7.</s> </p> <p> <s xml:space="preserve">Nunc fortaſſe alienum non erit videre quanto ferè maior eſſet terra, quam tota <lb/>aqua, non dico <choice><ex>autem</ex><am>autẽ</am></choice> ſolum de parte illa maximæ e ius profunditatis, quæ nuſquam <lb/>ad quingentos paſſus peruenit, ſed de ficto illo orbe aqueo, profunditatis .500. paſ-<lb/>ſuum, qui totum terreſtrem orbem circundaret, & tegeret, ſupponendo quod per <lb/>quingentos paſſus profunditatis, quidquid eſt terra, eſſet aqua, ideſt ſuppoſito quod <lb/>ex totius orbis compoſiti ſemidiametro exiſtente .3500. milliarium, purę terræ ſemi <lb/>diameter eſſet milliarium .3499. cum dimidio. </s> <s xml:space="preserve">Supponendo igitur, vt ſupradixi. <lb/></s> <s xml:space="preserve">Comperietur quod terra eſſet maior aqua amplius quam .2333. vicibus. </s> <s xml:space="preserve">Sed quia <lb/>partes terræ detectæ rumpunt eiuſmodi fictum orbem aqueum, quæ quidem partes, <lb/>ſunt ampliores ſuperficię aquæ, vt obſeruauit Piccolo. atque alij <choice><ex>præſtantes</ex><am>præſtãtes</am></choice> viri, ideo <lb/>ſequetur, vt terra ſit maior aqua amplius .4666. vicibus imo amplius quinquies mil-<lb/>lecuplo. </s> <s xml:space="preserve">Si autem quis diceret, in quantitate aquæ computari etiam illam, quæ gi-<lb/>gnatur ex vaporibus, qui globum hunc compoſitum circundant: </s> <s xml:space="preserve">reſpondeo quod <lb/>non modò ei concedo computari eiuſmodi aquam, ſed ſupponendo etiam quodto <lb/>tus locus à vaporibus occupatus, qui attolluntur .52. milliaria ſupra ſuperficiem <lb/>huius globi, vt iam ſupradictum eſt, totus eſſet aqueus, & amplius, ſupponendo quod <lb/>orbis hic aqueus eſſet ſpiſſitudinis, ſiue altitudinis quingentorum milliarium ſupra <lb/>totum ipſum globum compoſitum. </s> <s xml:space="preserve">Tamen terra eſſet maior ipſa aqua ferè duplo; <lb/></s> <s xml:space="preserve">qua dere, quiſque eiuſmodi ſupputationum peritus certior fieri poterit. </s> <s xml:space="preserve">Vnde iti- <pb facs="0415" n="403"/><fw type="head">EPISTOLAE.</fw> dem affirmare poſſemus, terram non ſolum maiorem eſſe aqua, ſed aqua & præte-<lb/>rea aere, ſi aer non tam altè pertingit, quam multi alij præter Piccolo. ſentiunt, qui <lb/>dicuntinde euenire quod aerea humiditas non tam altè aſcendere poteſt, quoniam <lb/>humiditas ipſa grauitatem ſecum affert, præterquam quod nubium ſitus oſtendit ſu <lb/>pra eas materiam eſſerariorem quam ſint ipſę nubes, infra vero denſiorem. </s> <s xml:space="preserve">Corpo-<lb/>ra enim eouſque aſcendunt donec inueniunt conſtitutionem mediam formæ æqua-<lb/>lis (vt ita dicam) ſuis. </s> <s xml:space="preserve">Quare materia illa quæ impropriè ignis vocatur (non enim <lb/>eſt ignis) incipit carere humiditate (qua mediante aer definitur) circa quinquage-<lb/>ſimum ſecundum milliarium ſupra ſuperficiem terræ, vt iam ſupradixi à Vitellione <lb/>demonſtratum fuiſſe. </s> <s xml:space="preserve">Ariſto. autem affert <choice><ex>rationem</ex><am>rationẽ</am></choice> quare nubes altius <choice><ex>non</ex><am>nõ</am></choice> <choice><ex>tranſcendant</ex><am>tranſcẽdãt</am></choice>. <lb/></s> <s xml:space="preserve">Vnde apparet tertiam aeris regionem impropriè aerem appellari, ſi humiditate ca-<lb/>ret, vt ait Ariſt. qua mediante aer definitur, immo potius retinet ignis naturam, vt <lb/>etiam aſſerunt interpretes Ariſtotelis in primum Meteororum. </s> <s xml:space="preserve">Qui Ariſto. in locis <lb/>ſupra citatis itidem oſtendit ſe etiam huius modi eſſe opinionis.</s> </p> <p> <s xml:space="preserve">Quod autem attinet ad probandum quod ſuperficies terrę detecta ſit altior quam <lb/>ſuperficies detecta aquæ, id tam clarum eſt ſua ſponte philoſophis, qui ſciunt quid ſit <lb/>altum, quidue demiſſum, quod ſuperfluum eſſet quidquid ſuper hoc dicerem præ <lb/>terquam, quòd conſtat ex demonſtratione ab Ariſto. ſacta textu 31. li .2. de cœlo, in <lb/>quo agit de corporibus in aqua poſitis, vnde eiuſmodi veritas planiſſimè aperitur. <lb/></s> <s xml:space="preserve">Omittimus etiam quod præſtantes Moderni omnes, eam pro manifeſtiſſima <choice><ex>ponunt</ex><am>ponũt</am></choice>, <lb/>ſicutiapud omnes ſani iudicij homines reuera exiſtimatur.</s> </p> <p> <s xml:space="preserve">Hæc enim ſunt quæ in fine illius conſiderationis ſcripſeram.</s> </p> <p> <s xml:space="preserve">Anno autem præterito editus in lucem fuit tractatus quidam Pulcherrimus, ab Ex <lb/>cellentiſſimo, nec non Doctiſſimo viro Auguſtino Michele, Patritio Veneto, ad cor <lb/>roborandam opinionem antiquorum, vbi tot authoritates, <choice><ex>totque</ex><am>totq́;</am></choice> rationes adducit, <lb/>vt nil amplius dici poſſit. </s> <s xml:space="preserve">Atego ſenſum, <choice><ex>rationemque</ex><am>rationemq́;</am></choice>, & non authoritatem <choice><ex>aliquam</ex><am>aliquã</am></choice> <lb/>ſequutus ſum: </s> <s xml:space="preserve">cum verò dico ſenſum, de ſenſu illorum intelligo, qui profunditatem <lb/>maris metiti ſunt, vt non mihi ſolum, ſed, & Piccolo. & alijs permultis retulerunt, <lb/>de ratione vero à me adducta, aliorum ſit iudicium.</s> </p> <quote> <s xml:space="preserve">Sediſte mirabilis & Excellentiſſimus vir, verba mea non accepit in eo ſenſu, vt <lb/>ego ſcripſi, ita vt omnino alienas conſequentias ſibi confingat, <choice><ex>quemadmodum</ex><am>quemadmodũ</am></choice> pag. <lb/>3. ſui tractatus inquit, me non concedere naturam produxiſſe in magna quantitate, <lb/>atque immenſa, id totum, quod bonum, & neceſſarium eſt. </s> <s xml:space="preserve">Hanc enim conſequen <lb/>tiam ipſe colligit ex eo, quod ego pag .19. meæ conſiderationis contra Antonium <lb/>Bergam ſcripſeram, quod videntur multa corpora alijs nobiliora, nihilominus mi-<lb/>nora, eo quod quantitas non ſequitur nobilitatem, neque ab ea pender, ita vt res <lb/>illa quæ nobilior eſt, neceſſarium ſit vt etiam maior exiſtat. </s> <s xml:space="preserve">Sed Excellentiſſimus <lb/>iſtæ vir ſcribit ita me dixiſſe.</s> </quote> <p> <s xml:space="preserve">Multa immo infinita corpora ſunt nobilia, & neceſſaria, nihilominus ſunt paruę <lb/>molis.</s> </p> <p> <s xml:space="preserve">Vide igitur quantum hoc diſtat ab illo.</s> </p> <p> <s xml:space="preserve">Præterea cap .12. aliam conſequentiam facit, quam ego non tam amplam facio. <lb/></s> <s xml:space="preserve">„ Ipſe enim me inferre vult in alijs terrę partibus cauernas non reperiri, eo quod Mon <lb/>„ tes ſint cauernoſi. </s> <s xml:space="preserve">Aſpice quæſo. pag .29. meæ conſiderationis, & clarè videbis me <lb/>nullo modo negare illas concauitates ſeu porroſitates terræ extra montana loca, <lb/>circa ſuperficiem terræ, vſque ad æquilibrium, orbiculariter, infimæ profunditatis <lb/>maris.</s> </p> <p> <s xml:space="preserve">Sed putare inferius has porroſitates reperiri, cum nulla ratio nobis perſuaſibilis <pb facs="0416" n="404"/><fw type="head">IO. BAPT. BENED.</fw> adhuc ab aliquo prodita ſit, idoneum nullo pacto eſſet. </s> <s xml:space="preserve">Rationes autem ab ipſo Ex <lb/>cellentiſſimo Auguſtino adductas circa huiuſmodi rem, alij dijudicent, de authori-<lb/>tatibus verò, nihil dicam, quia ab illis petendæ ſunt, qui profitentur tales facultates, <lb/>quorum vnius tantummodo authoritas præualere deberet, contra omnes alias <choice><ex>eorum</ex><am>eorũ</am></choice> <lb/>qui nunquam attigerunt ſummis labris orificia harum <choice><ex>ſcientiarum</ex><am>ſcientiarũ</am></choice>. </s> <s xml:space="preserve">Vt ſi exempli gra-<lb/>tia non ſolum authoritas illorum virorum, quos ipſe recenſuit, ſufficiens eſſet vt pu <lb/>ta Pioccolo. </s> <s xml:space="preserve">Naibodæ, Bordini, Clauij, <choice><ex>reliquorumque</ex><am>reliquorumq́;</am></choice> fautorum verę opinionis, <lb/>ſed Franciſci Maurolici tantummodo, qui in primo Dialogo ſuæ coſmographiæ ita <lb/>inquit.</s> </p> <quote> <s xml:space="preserve">Exiſtimo autem totum terræ corpus rigidum eſſe ſaxum, nam ſi arena eſſet, aue <lb/>gleba fragilis, ita humorem imbiberet, vt cum eo quaſi confunderetur; </s> <s xml:space="preserve">huc ac-<lb/>cedit, quod ſi mineræ, ac rupes, quæ ſunt grauiſſimæ partes in ipſa plerunque ſuper-<lb/>ficie comperiuntur, multo magis apud centrum eſſe debent. </s> <s xml:space="preserve">Videtur ita ratio exi-<lb/>gere, vt grauiora centro quoque ſint propinquiora.</s> </quote> <p> <s xml:space="preserve">Hæcigitur ſola authoritas, inſtar reliquarum omnium ſufficere poſſet. </s> <s xml:space="preserve">Verum <lb/>de authoritatibus minime curandum eſt, vbi ſenſus, <choice><ex>ratioque</ex><am>ratioq́;</am></choice> vera illis opponuntur.</s> </p> <p> <s xml:space="preserve">Quod autem numerus animalium aquatilium maior exiſtat numero terreſtrium, <lb/>ſatis reſpondimus pag .41. noſtræ conſiderationis.</s> </p> <p> <s xml:space="preserve">Sed in cap .14. Excellentiſſimus Auguſtinus ita inquit (vt etiam ſuperius dixerat) <lb/></s> <quote> <s xml:space="preserve">quod certiorem cognitionem homo non habet illa, quæ à ſenſu prouenit. </s> <s xml:space="preserve">Et quod <lb/>nemo eſt qui aſpiciat terram, & aquam, quod hanc maiorem illa non iudicet, & <choice><ex>non</ex><am>nõ</am></choice> <lb/>exiſtimet.</s> </quote> </p> <p> <s xml:space="preserve">Quod autem certiorem cognitionem homo non habeat illa, quæ à ſenſu proue-<lb/>nit, concedendum non cenſeo. </s> <s xml:space="preserve">Nam omnis cognitio mathematica (cum primum <lb/>gradum certitudinis obtineat) ab ipſo ſenſu fieret, quod omnino alienum <lb/>eſt à veritate. </s> <s xml:space="preserve">Senſus enim nunquam vidit incommenſurabilitates magnitudinum, <lb/>vel incoincidentias linearum non tangentium cum curuitate hyperbolica, aut angu <lb/>lum contingentiæ aliquem, nec (vt vno verbo dicam) aliquam concluſionem ma-<lb/>thematicam, quam volueris. </s> <s xml:space="preserve">Neque per ſenſum eſt ſcire, inquit Ariſtoteles. </s> <s xml:space="preserve">Co-<lb/>gnitio igitur ſenſitiua, certior non eſt illa, quæ per habitum ſcientiſicum acquiritur.</s> </p> <p> <s xml:space="preserve">Ad reliqua verò, ſupponamus nos tunc fuiſſe in Arca Noe, <choice><ex>cum</ex><am>cũ</am></choice> aquæ cooperiebant <lb/>omnia cacumina montium, vbi nullum terræ veſtigium videbatur, </s> <s xml:space="preserve">quare proculdu <lb/>bio aquam iudicaremus, atque exiſtimaremus maiorem terra, <choice><ex>dum</ex><am>dũ</am></choice> nulla aliare vtere-<lb/>mur niſi ſenſu abſque alio diſcurſu intellectuali, ut reliqua illa animalia irrationalia, <lb/>quæ nobiſcum erant in dicta arca. </s> <s xml:space="preserve"><choice><ex>Non</ex><am>Nõ</am></choice> ſufficit igitur ſuperficiem aquæ tantummodo <lb/>aſpicere, quia neque tunc temporis, aqua erat maior terra, etiam ſi non ſolum tot <lb/>cubitis attolleretur ſupra cacumina montium, ſed quingenta milliaria, vt ſupradi-<lb/>ximus.</s> </p> <p> <s xml:space="preserve">Ratio autem illa, ex infinitis, ab ipſo, eo in loco adducta, talis eſt.</s> </p> <quote> <s xml:space="preserve">Aqua eſt eccentrica ad terram, & pro <choice><ex>centro</ex><am>cẽtro</am></choice> habet centrum grauitatis terræ, aqua <lb/>igitur maioris eſt amplitudinis ipſa terra.</s> </quote> <quote> <s xml:space="preserve">Hanc etiam conſequentiam alijs relinquo Philoſophis dijudicandam.</s> </quote> <p> <s xml:space="preserve">Subſequitur poſtea dicens.</s> </p> <quote> <s xml:space="preserve">Præterea proprius locus terræ, eſt ſuperſicies aquæ, igitur terram oportet ab <lb/>aqua tegi.</s> </quote> <p> <s xml:space="preserve">Ad hoc etiam aliquis poſſet quærere, quis nam erit locus illius partis terræ de-<lb/>tectæ ab aqua? </s> <s xml:space="preserve">nulli dubium erit quin ſuperficies aeris, & non aquæ exiſtet.</s> </p> <pb facs="0417" n="405"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Nune autem ſi locus terræ eſt ſub aqua, ergo locus aquæ proprius eſt ſub aere, & <lb/>non ſub terra, vnde non erit rationabile putare maiorem copiam aquarum exiſtere <lb/>in cauernis ſubterraneis, quam ſupra ſuperficiem terræ. </s> <s xml:space="preserve">Adde quod locus illarum <lb/>aquarum non eſſet ſuperficies aeris, ſed terræ, vnde non minus locus aquę eſſet ter-<lb/>ra, quam locus terræ, aqua. </s> <s xml:space="preserve">Sed miſſa faciamus hæc.</s> </p> <p> <s xml:space="preserve">Cap. verò .20. ita inquit.</s> </p> <p> <s xml:space="preserve">Materia elementorum æqualis eſt. </s> <s xml:space="preserve">Ergo aqua maior eſt terra.</s> </p> <quote> <s xml:space="preserve">Hæc enim conſequentia veriſſima eſſet. </s> <s xml:space="preserve">Sed nullus vnquam Philoſophus (vt Phi-<lb/>loſophus dico) concedet totam materiam elementarem, in quatuor æquales partes <lb/>eſſe diuiſam.</s> </quote> <p> <s xml:space="preserve">Cap. verò .21. inquit me dixiſſe non ſuffecturam paucam ſpiſſitudinem. </s> <s xml:space="preserve">Eo enim <lb/>in loco pag .26. mei tractatus contradicens ipſi Bergæ, dixi, quod ſecundum ipſum <lb/>Bergam non ſufficeret pauca ſpiſſitudo.</s> </p> <p> <s xml:space="preserve">Similiter etiam dixi, quod ſecundum ipſum, quanto remotius diffunditur lumen <lb/>fortaſſe tantò magis illuminat. </s> <s xml:space="preserve">Putans ipſe Berga quod in propinquo debilius exi-<lb/>ſteret dictum lumen. </s> <s xml:space="preserve">Et propter ea dixi, quod apud ipſum fortaſſe nihil valet illa <lb/>propoſicio, quæ dicit. </s> <s xml:space="preserve">Agens in propinquo, fortius agit quam in remoto.</s> </p> <p> <s xml:space="preserve">Cap. autem .22. vbi Excellentiſſimus Auguſtinus inquit, vnum tantummodo ele <lb/>mentum non ſufficere ad generationem miſtorum. </s> <s xml:space="preserve">Hoc enim concedo, ſed hoc ni-<lb/>hil ad me ſpectat, eo quod meum reſponſum ad Bergam, erat circa tranſitum lumi-<lb/>nis, & non circa generationem elementorum.</s> </p> <p> <s xml:space="preserve">Cap. demum .23. pag .20. linea .10. vbi ſcribit me dixiſſe, iudicare, oportebat <lb/>ſcribere, dubitare.</s> </p> <p> <s xml:space="preserve">Puto tamen hoc vocabulum eſſe errorem Thypographi, quamuis in correctione <lb/>illud non inuenerim, quia vt ego multoties expertusſum, difficillimum omnes Thy <lb/>pographi errores corrigere, neque (vt fertur) Argi oculi ſufficerent.</s> </p> <p> <s xml:space="preserve">Hactenus enim in mei defenſionem hæc ſubiungere volui.</s> </p> <p> <s xml:space="preserve">Ad defenſionem autem Piccolo. <choice><ex>aliorumque</ex><am>aliorumq́;</am></choice> virorum meæ opinionis, nec non de <lb/>proportione duplicata profunditatis maris ad ſuam amplitudinem, ex conſequentia <lb/>pyramidali: </s> <s xml:space="preserve"><choice><ex>alijsque</ex><am>alijsq́;</am></choice> ſimilibus rationibus, prodeant alij. </s> <s xml:space="preserve">Huiuſmodi tamen Doctiſſi-<lb/>mi viriingenium, memoriam, nec non doctrinam valde admiror, atque obſeruo.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE METHODO PRODVCTIONIS FRACTORVM <lb/>qua vtuntur Pedemontani Agrimenſores.</head> <head rend="italics" xml:space="preserve">Anſelmo Roſemburg Agrimenſori Caſareo.</head> <p> <s xml:space="preserve"><hi rend="small caps">MEthodvs</hi> quàm mihi ſcribis in Prouincia tua maximè in vſu eſſe, nimis <lb/>longa atque prolixa eſt, Pedemontani verò Agrimenſores in produ-<lb/>ctione fractorum, valde breui methodo vti ſolent, quam libenter tibi <lb/>ſcribo, eo maxime, vt videas quam rationabiliter operentur.</s> </p> <p> <s xml:space="preserve">Scire igitur primum te oportet illos, maximam eorum communem menſuram <lb/>vocare Trabucum, cuius ſextam partem vocant Pedem, duodecimam verò pe-<lb/>dis, Vnciam, <choice><ex>duodecimam</ex><am>duodecimã</am></choice> <choice><ex>autem</ex><am>autẽ</am></choice> vnciæ <choice><ex>punctum</ex><am>punctũ</am></choice>, <choice><ex>duodecimam</ex><am>duodecimã</am></choice> demum puncti; </s> <s xml:space="preserve">Attomum.</s> </p> <p> <s xml:space="preserve">Quotieſcunque igitur multiplicant trabucum, per trabucum nulli dubium eſt <lb/>quin producant trabucum ſuperficialem ſcilicet.</s> </p> <pb facs="0418" n="406"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Similiter multiplicando pedes, vncias, puncta, & attoma per trabucum, produ-<lb/>cunt pedes, vncias, puncta, & attoma ſuperficialia rectangula oblonga, quorum lon <lb/>gitudo eſt ipſius trabuchi, latitudo vero lineæ dictarum ſpecierum.</s> </p> <p> <s xml:space="preserve">Dum vero multiplicant pedem per pedem, nulli dubium eſt quin producant pe-<lb/>dem quadratum, ſed apud ipſos non vocatur quadratum, quamuis reuera ita ſit, ſed <lb/>illud vocant duas vncias, quæ quidem ſunt rectangula oblonga iam hic ſupradicta, <lb/>quarum vniuſcuiuſque longitudo ſit vnius trabuchi, latitudo vero vnius duodecimæ <lb/>partis ipſius pedis linearis.</s> </p> <p> <s xml:space="preserve">Productum autem pedis per vnciam, vocant duo puncta, quæ etiam ſunt duo re-<lb/>ctangula oblonga, vt ſupra.</s> </p> <p> <s xml:space="preserve">Productum deinde vnciæ per vnciam, vocant duos attomos, qui <choice><ex>etiam</ex><am>etiã</am></choice> ſunt duo re-<lb/>ctangula oblonga, vt dictum eſt, quæ omnia ſcientificè videbimus.</s> </p> <p> <s xml:space="preserve">Pro cuius rei cognitione, ſit, exempli gratia <seg type="var">.a.e.</seg> vnus Trabuchus linearis <seg type="var">.e.i.</seg> ve-<lb/>ro vnus pes <seg type="var">.i.o.</seg> autem vna vncia, <seg type="var">o.u.</seg> poſtea vnum punctum, et <seg type="var">.u.t.</seg> vnus At-<lb/>tomus.</s> </p> <p> <s xml:space="preserve">Vnde <seg type="var">.e.i.</seg> erit ſexta pars ipſius <seg type="var">.a.e.</seg> et <seg type="var">.i.o.</seg> duodecima ipſius <seg type="var">.e.i.</seg> et <seg type="var">.o.u.</seg> duodecima <lb/>ipſius <seg type="var">.i.o.</seg> et <seg type="var">.u.t.</seg> duodecima ipſius <seg type="var">.o.u</seg>. </s> <s xml:space="preserve">Sit etiam <seg type="var">.a.b.</seg> æqualis <seg type="var">.a.e.</seg> lineæ & ſic <seg type="var">.e.d</seg>: <seg type="var">i.<lb/>f</seg>: <seg type="var">o.g</seg>: <seg type="var">o.n.</seg> & c. <choice><ex>terminenturque</ex><am>terminenturq́;</am></choice> parallelogramma <seg type="var">.b.e</seg>: <seg type="var">d.i</seg>: <seg type="var">f.o</seg>: <seg type="var">g.u.</seg> et <seg type="var">.c.t.</seg> vnde <seg type="var">.b.e.</seg> erit <lb/>trabuchum quadratum, et <seg type="var">.d.i.</seg> pes rectangulus oblongus vt ſupra, et <seg type="var">.f.o.</seg> vncia rectan <lb/>gula oblonga, et <seg type="var">.g.u.</seg> punctum rectangulum oblongum, et <seg type="var">.c.t.</seg> attomus rectangu-<lb/>lus oblongus.</s> </p> <p> <s xml:space="preserve">De producto igitur trabuchi per <choice><ex>trabuchum</ex><am>trabuchũ</am></choice>, nulli dubium eſt quin ſit quadratum <seg type="var">.<lb/>a.d.</seg> vt ſuperius diximus.</s> </p> <p> <s xml:space="preserve">Productum autem trabuchi cum pede erit <seg type="var">.d.i.</seg> ſexta pars ipſius <seg type="var">.a.d.</seg> cum <seg type="var">.e.i.</seg> ſit ſex <lb/>ta ipſius <seg type="var">.a.e.</seg> ex prima ſexti vel .18. aut .19. ſeptimi, ſiue etiam ex .15. quinti Eucli.</s> </p> <p> <s xml:space="preserve">Productum autem pedis cum pede erit <seg type="var">.e.K.</seg> quadratum, quod probandum eſt <lb/> <ptr xml:id="fig-0418-01a" corresp="fig-0418-01" type="figureAnchor"/> <pb facs="0419" n="407"/><fw type="head">EPISTOL AE.</fw> duplum eſſe rectangulo <seg type="var">.f.o.</seg> <choice><ex>Nam</ex><am>Nã</am></choice> <seg type="var">.K.i.</seg> ſexta pars eſt ipſius <seg type="var">.f.i.</seg> ex ſuppoſito, et <seg type="var">.i.o.</seg> duo-<lb/>decima ipſius <seg type="var">.e.i.</seg> proportio igitur <seg type="var">.e.i.</seg> ad <seg type="var">.o.i.</seg> dupla eſt proportioni ipſius <seg type="var">.f.i.</seg> ad <seg type="var">.K.i</seg>. <lb/></s> <s xml:space="preserve">quare <seg type="var">.K.e.</seg> duplo maius eſt ipſius <seg type="var">.f.o.</seg> eo quod ſi <seg type="var">.i.o.</seg> vel <seg type="var">.f.g.</seg> (quod idem eſt) duplo <lb/>maius eſſet ipſo latere pręſenti <seg type="var">.o.i.</seg> vel <seg type="var">.f.g.</seg> </s> <s xml:space="preserve">tunc <seg type="var">.f.o.</seg> æquale eſſet ipſi <seg type="var">.K.e.</seg> ex .15. ſexti <lb/>vel .20. ſeptimi quod quidem <seg type="var">.f.o.</seg> duplo maius eſſet ipſo præſenti <seg type="var">.f.o</seg>. </s> <s xml:space="preserve">Rectè igitur <lb/>inquiunt dicentes productum pedis cum pede eſſe duas vncias, vel ſi mauis, ita dicas <lb/><seg type="var">e.K.</seg> ſexta pars eſt ipſius <seg type="var">.d.i.</seg> ex iam dictis propoſitionibus <seg type="var">.f.o.</seg> autem eſt duodecima <lb/>ipſius <seg type="var">.d.i.</seg> ex ijſdem, cum exſuppoſito <seg type="var">.i.o.</seg> duodecima ſit ipſius <seg type="var">.e.i.</seg> </s> <s xml:space="preserve">quare <seg type="var">.e.K.</seg> <choice><ex>duplum</ex><am>duplũ</am></choice> <lb/>erit ipſius <seg type="var">.f.o.</seg> ex commu ni notione.</s> </p> <floatingText> <body> <div type="float"> <figure xml:id="fig-0418-01" corresp="fig-0418-01a"> <graphic url="0418-01"/> </figure> </div> </body> </floatingText> <p> <s xml:space="preserve">Productum verò pedis cum vncia. ſit <seg type="var">.K.o.</seg> quod probabimus ex ijſdem rationibus <lb/>duplum eſſe ipſius <seg type="var">.g.u.</seg> puncti rectanguli oblongi. </s> <s xml:space="preserve">Nam <seg type="var">.l.o.</seg> ſexta pars ſimiliter eſt <lb/>ipſius <seg type="var">.g.o.</seg> et <seg type="var">.o.u.</seg> duodecima ipſius <seg type="var">.o.i.</seg> </s> <s xml:space="preserve">quare proportio <seg type="var">.i.o.</seg> ad <seg type="var">.o.n.</seg> dupla eſt propor <lb/>tioni <seg type="var">.g.o.</seg> ad <seg type="var">.o.l.</seg> ſequitur ergo ex prædictis rationibus <seg type="var">.k.o.</seg> duplum eſſe ipſius <seg type="var">.g.u.</seg> <lb/>vel ſic, vtlin præcedenti, cum <seg type="var">.K.o.</seg> ſit ſexta pars ipſius <seg type="var">.f.o.</seg> ex dictis propoſitionibus <seg type="var">.<lb/>g.u.</seg> verò duodecima eiuſdem <seg type="var">.f.o.</seg> ex ijſdem, nam <seg type="var">.o.u.</seg> duodecima eſt ipſius <seg type="var">.o.i.</seg> ergo <lb/><seg type="var">K.o.</seg> duplo maius eſt ipſo <seg type="var">.g.u</seg>.</s> </p> <p> <s xml:space="preserve">Ex ijſdemmet rationibus productum <seg type="var">.l.u.</seg> pedis cum puncto duplum eſt ipſius <seg type="var">.c.t.</seg> <lb/>attomi rectanguli oblongi.</s> </p> <p> <s xml:space="preserve">Probandum nunc relinquitur productum <seg type="var">.o.n.</seg> vnciæ cum vncia, quod eſt quadra-<lb/>tum, duplum eſſe ipſius <seg type="var">.c.t.</seg> attomi rectanguli oblongi. </s> <s xml:space="preserve">Nam <seg type="var">.i.n.</seg> eſt pars vna ex .72. <lb/>ipſius <seg type="var">.c.u.</seg> et <seg type="var">.u.t.</seg> pars vna ex .144. ipſius <seg type="var">.o.i.</seg> ex ſuppoſito, </s> <s xml:space="preserve">quare proportio <seg type="var">.i.o.</seg> ad <seg type="var">.u.t.</seg> <lb/>dupla eſt proportioni ipſius <seg type="var">.c.u.</seg> ad <seg type="var">.n.i.</seg> ex dictis igitur rationibus <seg type="var">.o.n.</seg> duplo maius <lb/>eſt ipſo <seg type="var">.c.t</seg>. </s> <s xml:space="preserve">Vel ſi placet dicas <seg type="var">.n.o.</seg> eſt vna pars ex .72. ipſius <seg type="var">.f.o.</seg> exſupradictis, eo <lb/>quod <seg type="var">.n.i.</seg> ita ſe habet ad <seg type="var">.f.i.</seg> vt vnitas ad .72. ſed ex ijſdem rationibus <seg type="var">.c.t.</seg> pars vna ex <lb/>144. eſt ipſius <seg type="var">.f.o.</seg> eo quod ita ſe habet <seg type="var">.u.t.</seg> ad <seg type="var">.o.i.</seg> </s> <s xml:space="preserve">quare <seg type="var">.o.n.</seg> duplo maius erit ipſo <seg type="var">.<lb/>c.t</seg>.</s> </p> <p> <s xml:space="preserve">Propoſitum ſit nobis nunc, exercitij gratia, quærere ſuperficiem alicuius rectan <lb/>guli, cuius vnum latus ſit <choice><ex>trabuchorum</ex><am>trabuchorũ</am></choice> .3. pedum .2. & vnciarum .3. aliud vero latus ſit <lb/>trabuchorum .2. pedum .3. vnciarum vero .2.</s> </p> <p> <s xml:space="preserve">Huiuſmodi autem methodo mediante, multiplicando primum latus <choice><ex>dictum</ex><am>dictũ</am></choice> .3. 2. 3. <lb/>per numerum trabucorum ſecundi lateris .2. ſcilicet producentur nobis primò trabu <lb/>cha ſuperficialia .6. pedes .4. & vnciæ .6. omnia rectagula, vt dictum eſt. </s> <s xml:space="preserve">Multiplican-<lb/>do deinde idem primum latus .3. 2. 3. per pedes .3. ſecundi lateris. </s> <s xml:space="preserve">Ex trabuchis .3. <lb/>primi lateris cum .3. pedibus ſecundi, producentur .9. pedes rectanguli, hoc eſt <lb/>vnus trabuchus cum tribus pedibus rectangulis. </s> <s xml:space="preserve">Ex pedibus autem huius .2. cum <lb/>ijſdem alterius lateris .3. producentur .12. vnciæ rectangulæ ideſt vnus pes rectangu-<lb/>lus. </s> <s xml:space="preserve">Exijſdem pedibus .3. ſecundi lateris, cum .3. vncijs primi lateris producentur <num value="18">.<lb/> <ptr xml:id="table-0419-01a" corresp="table-0419-01" type="tableAnchor"/> <pb facs="0420" n="408"/><fw type="head">IO. BAPT. BENED.</fw> 18.</num> puncta rectangula, hoc eſt vna vncia cum .6. punctis rectangulis. </s> <s xml:space="preserve">Deinde ex <lb/>multiplicatione vnciarum .2. ſecundi lateris, cum .3. trabuchis primi lateris, produ-<lb/>centur .6. vnciæ. </s> <s xml:space="preserve">Ex multiplicatione poſtea dictarum .2. vnciarum ſecundi lateris <lb/>cum .2. pedibus primi, producentur .8. puncta.</s> </p> <floatingText> <body> <div type="float"> <table xml:id="table-0419-01" corresp="table-0419-01a" rows="8" cols="4"> <row> <cell role="label" xml:space="preserve">Trabucha.</cell> <cell role="label" xml:space="preserve">pedes.</cell> <cell role="label" xml:space="preserve">vnciæ.</cell> </row> <row> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">2.</cell> <cell xml:space="preserve">3.</cell> </row> <row> <cell xml:space="preserve">2.</cell> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">2.</cell> </row> <row> <cell xml:space="preserve">6.</cell> <cell xml:space="preserve">4.</cell> <cell xml:space="preserve">6.</cell> </row> <row> <cell xml:space="preserve">1.</cell> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">1.</cell> <cell xml:space="preserve">6.</cell> </row> <row> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">1.</cell> <cell xml:space="preserve">6.</cell> <cell xml:space="preserve">8.</cell> </row> <row> <cell xml:space="preserve"> </cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">1.</cell> </row> <row> <cell xml:space="preserve">8.</cell> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">2.</cell> <cell xml:space="preserve">3.</cell> </row> </table> </div> </body> </floatingText> <p> <s xml:space="preserve">Demum ex ijſdem .2. vncijs ſecundi lateris cum .3. primi, producentur .12. atto-<lb/>mi, ideſt vnum punctum. </s> <s xml:space="preserve">Quæ omnia collecta facient trabucha .8. pedes .3. uncias <lb/>2. & attomi .3. omnes rectanguli oblongi. </s> <s xml:space="preserve">Pulcherrima profecto operatio.</s> </p> <table rows="8" cols="4"> <row> <cell role="label" xml:space="preserve">Trabucha.</cell> <cell role="label" xml:space="preserve">pedes.</cell> <cell role="label" xml:space="preserve">vnciæ.</cell> </row> <row> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">2.</cell> <cell xml:space="preserve">3.</cell> </row> <row> <cell xml:space="preserve">2.</cell> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">2.</cell> </row> <row> <cell xml:space="preserve">6.</cell> <cell xml:space="preserve">4.</cell> <cell xml:space="preserve">6.</cell> </row> <row> <cell xml:space="preserve">1.</cell> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">1.</cell> <cell xml:space="preserve">6.</cell> </row> <row> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">1.</cell> <cell xml:space="preserve">6.</cell> <cell xml:space="preserve">8.</cell> </row> <row> <cell xml:space="preserve"> </cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">1.</cell> </row> <row> <cell xml:space="preserve">8.</cell> <cell xml:space="preserve">3.</cell> <cell xml:space="preserve">2.</cell> <cell xml:space="preserve">3.</cell> </row> </table> <p> <s xml:space="preserve">Videamus nunc exercitij cauſa, vt dixi, quomodo conueniat calculus iſte cum <lb/>calculo ordinario communi?</s> </p> <p> <s xml:space="preserve">Nam quotieſcunque dicta latera, fracta fuerint in vncias, primum latus erit <lb/>vnciarum .243. ſecundum autem .182. productum vero vnius in alterum erit vn-<lb/>ciarum quadratarum .44 226. quod quidem productum cum diuiſum fuerit per <num value="5184">.<lb/>5184.</num> vncias quadratas vnius trabuchi quadrati, prouentus erit .8. trabucho-<lb/>rum, reliquus verò numerus, ſiue fractus, erit vnciarum quadratarum .2754. qui <lb/>cum diuiſus fuerit per numerum .144. vnciarum vnius pedis quadrati, prouenient <lb/>pedes .19. quadrati cum vncijs .18. ſuperabundantibus, dicti autem pedes .19. ſignifi-<lb/>cant tres pedes rectangulos oblongos cum vno pede quadrato, hoc eſt cum duabus <lb/>vncijs rectangulis oblongis, vt ſupra.</s> </p> <p> <s xml:space="preserve">Videndum nunc eſt, vtrum illæ .18. vnciæ æquipolleant tribus punctis rectangu-<lb/>lis oblongis: </s> <s xml:space="preserve">ſed hoc manifeſtè videre eſt, ex hoc, quia quęlibet vncia rectangula <lb/>oblonga componitur ex .72. quadratis, punctum autem rectangulum oblongum, <choice><ex>cum</ex><am>cũ</am></choice> <lb/>ſit duodecima pars ipſius vnciæ rectangulæ oblongæ, ipſum componetur ex .6. vn-<lb/>cijs quadratis .18. igitur vncijs quadratis, triplum erit ipſius puncti rectanguli dicti. <lb/></s> <s xml:space="preserve">Vnde clarè patet, quod, quotieſcunque voluerimus ſcire proportionem ipſarum vn <lb/>ciarum quadratarum ſuperabundantium, ad punctum rectangulum oblongum, ſi <lb/>dixerimus ex regula de tribus, ſi .72. (vncia rectangula oblonga) dat .18. quid <choice><ex>dabunt</ex><am>dabũt</am></choice> <lb/>12? </s> <s xml:space="preserve">puncta rectangula oblonga, quarum vnaquæque eſt duodecima pars ipſius vn-<lb/>ciæ rectangulæ oblongæ, in præſenti autem caſu prouenient .3. pro quarto termino <lb/>quæſito, & habebimus propſitum.</s> </p> <pb facs="0421" n="409"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">SOLVTIO CVIVSDAM QVÆSITI.</head> <head rend="italics" xml:space="preserve">Magnifico Ludouico Fauzzoni amico cariβimo.</head> <p> <s xml:space="preserve">TVI quæſiti ſolutio quam neſcio quis te docuit, valde diuerſa eſt à vera. <lb/></s> <s xml:space="preserve">quæſitum enim tale fuit.</s> </p> <p> <s xml:space="preserve">Reperiuntur quatuor ſocij, Ludouicus, Hieronymus, Franciſcus, & Lau <lb/>rentius quorum primus, Ludouicus ſcilicet, poſuit aureos .6000. Hierony <lb/>mus verò aureos .5000. Franciſcus autem .2000. & Laurentius .1000. quorum ſum-<lb/>ma faciebat aureos .14000. </s> <s xml:space="preserve">interim tamen de tali ſumma Ludouicus recepit aureos <lb/>2000. Hieronymus verò .1000. Franciſcus autem .900. & Laurentius .800. quapro-<lb/>pter in ſumma reſidua Ludouicus non habebat niſi aureos .4000. Hieronymus <choice><ex>etiam</ex><am>etiã</am></choice> <lb/>4000. Franciſcus .1100. & Laurentius .200. quorum ſumma erat .9300. </s> <s xml:space="preserve">Nunc au-<lb/>tem iſti ſocij cupiunt augere hanc ſummam per aureos .20000. tali tamen conditio-<lb/>ne quod quilibet tantum tribuat vt in totali ſumma, tantam partem unus habeat, <lb/>quantam alter.</s> </p> <p> <s xml:space="preserve">Hoc autem problema tam ſacile eſt, & cum ſuo theoremate ita coniunctum, quod <lb/>miror amicum noſtrum illud illico non vidiſſe.</s> </p> <p> <s xml:space="preserve">Accipe igitur illos aureos .20000. & eos collige cum ſumma .9300. vnde habebis <lb/>aureos .29300. pro <choice><ex>summa</ex><am>sũma</am></choice> totali, cuius quarta pars erit .7325. <choice><ex>quam</ex><am>quã</am></choice> <choice><ex>vnuſquisque</ex><am>vnuſquisq;</am></choice> poſtea <lb/>habebit in dicta ſumma. </s> <s xml:space="preserve">Sed ut reperias quantitatem aureorum quam quilibet <lb/>prius debet contribuere, vt poſtea habeat aureos .7325. in dicta ſocietate. </s> <s xml:space="preserve">Iubeo, <lb/>vt Ludouicus demat illos aureos .4000. quos demum habebat, ex .7325. reliquum <lb/>autem erit .3325. qui quidem numerus erit aureorum nunc contribuendorum ipſius <lb/>Ludouici. </s> <s xml:space="preserve">Demptis ſimiliter aureis .4000. ex dictis .7325. <choice><ex>remanebunt</ex><am>remanebũt</am></choice> .3325. pro con <lb/>tributione ipſius Hieronymi. </s> <s xml:space="preserve">Deinde ſi ex .7325. extracti fuerint aurei .1100. relin-<lb/>quent .6225. pro contributione Franciſci. </s> <s xml:space="preserve">Demptis demum .200. ex .7325. reſidui <lb/>erunt .7125. pro contributione Laurentij, & ſic quilibet habebit æqualem portio-<lb/>nem in totaliſumma.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Speculatio cuiuſdam Methodire ductionis numiſmatum <lb/>unius ſpeciei in aliam.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">MIrum tibi videtur quo pacto verum ſit, quod ſumma <choice><ex>mendietatis</ex><am>mẽdietatis</am></choice> cuiuſuis <lb/>numeri illorum numiſmatum, quæ hic vocantur Blanci, cum ſexta parte eiuſ <lb/>dem medietatis, ſemper ſit numerus florenorum huius prouinciæ. </s> <s xml:space="preserve">Vt exempli gra <lb/>tia, quotieſcunque reducere voluerimus .48. Blancos in Florenos, ſi medietati ip-<lb/>ſius .48. hoc eſt .24. adiecta fuerit ſexta pars ipſius medietatis, quæ eſt .4. </s> <s xml:space="preserve">tunc habebi <lb/>mus .28. & ita dicemus quod .48. Blanci conſtituunt Florenos .28. quod quidem <lb/>verum eſt.</s> </p> <p> <s xml:space="preserve">Huiuſmodi autem rei ſpeculatio ita ſe habet. </s> <s xml:space="preserve">Nam vnuſquiſque Blancus diuidi-<lb/>tur in .7. æquales partes, quarum .12. conſtituunt vnum Florenum, horum verò nu-<lb/>miſmatum communis menſura, vocatur Groſſus, vt ſcis, ex quo ſequitur, quod ſi <pb facs="0422" n="410"/><fw type="head">IO. BAPT. BENED.</fw> 28. Floreni æquantur Blancis .48. tot Groſſi erunt in .28. Florenis quot in .48. Blan-<lb/>cis. </s> <s xml:space="preserve">Fingamus igitur, mente, noſtram figuram .79. Theorematis Arithmetici <seg type="var">.x.u.o.<lb/>e.n.</seg> ſupponendo ambo producta <seg type="var">.u.x.</seg> et <seg type="var">.n.e.</seg> inuicem ęqualia exiſtere, & vnumquod-<lb/>que eſſe groſſorum .336. ſit etiam <seg type="var">.o.x.</seg> vnus Florenus .12. groſſorum <seg type="var">.o.n.</seg> verò Blan-<lb/>cus .7. eorundem groſſorum <seg type="var">.o.e.</seg> autem Blancorum .48. </s> <s xml:space="preserve">Nunc certi erimus ex .15. <lb/>ſexti vel .20. ſeptimi Euclidis eandem fore proportionem <seg type="var">.o.u.</seg> ad <seg type="var">.o.e.</seg> quæ <seg type="var">.o.n.</seg> ad <seg type="var">.o.<lb/>x.</seg> ſed <seg type="var">.o.n.</seg> eſt ſumma medietatis ipſius <seg type="var">.o.x.</seg> cum ſexta parte dictæ medietatis, ita igi-<lb/>tur erit <seg type="var">.o.u.</seg> ipſius <seg type="var">.o.e.</seg> hoc eſt ſumma medietatis <seg type="var">.o.e.</seg> <choice><ex>cum</ex><am>cũ</am></choice> ſexta parte medietatis eiuſ-<lb/>dem, quæ ſumma in præſenti exemplo erit .28.</s> </p> <p> <s xml:space="preserve">Hac enim ſpeculatione mediante, poteris methodum inuenire conuertendi Flo-<lb/>renos in Blancos. </s> <s xml:space="preserve">Vt ſi nobis propoſiti fuerint Floreni .28. </s> <s xml:space="preserve">Voluerimusq́ue inuenire <lb/>quot Blancos faciant, ſuppoſita menſura communi, iam ſupradicta. </s> <s xml:space="preserve">Nam duplica-<lb/>bimus numerum Florenorum, à quo duplo detrahemus ſeptimam partem, <choice><ex>reliquum</ex><am>reliquũ</am></choice> <lb/>verò erit numerus quæſitus.</s> </p> <p> <s xml:space="preserve">Huiuſmodi autem rei ratio eſt, quia, cum in ſupradicta figura, proportio <seg type="var">.o.e.</seg> ad <lb/><seg type="var">o.u.</seg> ęqualis exiſtat ei, quæ <seg type="var">.o.x.</seg> ad <seg type="var">.o.n.</seg> atque etiam <seg type="var">.o.x.</seg> ſit minor duplo ipſius <seg type="var">.o.n.</seg> <lb/>per ſeptimam partem ipſius dupli <seg type="var">.o.n.</seg> minor erit <seg type="var">.o.e.</seg> duplo ipſius <seg type="var">.o.u.</seg> per <choice><ex>ſeptimam</ex><am>ſeptimã</am></choice> <lb/>partem eiuſdem dupli ipſius <seg type="var">.o.u</seg>.</s> </p> <p> <s xml:space="preserve">Idem affirmo de quauis conuerſione aliorum numiſmatum, quorum ſemper <seg type="var">.o.x.</seg> <lb/>maior ſit <seg type="var">.o.n.</seg> verò minor. </s> <s xml:space="preserve">Vt ſi <seg type="var">.o.x.</seg> æquiualeret .7: et <seg type="var">.o.n.</seg> valeret .4. et <seg type="var">.o.e.</seg> valeret <lb/>42. quæ quidem <seg type="var">.o.e.</seg> menſuraretur ab <seg type="var">.o.n</seg>.</s> </p> <p> <s xml:space="preserve">Si cuperemus ſcire quot <seg type="var">.o.x.</seg> ſint in <seg type="var">.o.n</seg>. </s> <s xml:space="preserve">Primo dicemus in <seg type="var">.o.n.</seg> reperiri ſummam <lb/>medietatis ſex ſeptimorum ipſius <seg type="var">.o.x.</seg> collectæ cum vna ſeptima parte ipſius <seg type="var">.o.x.</seg> ſeu <lb/>(vt ita dicam) cum tertia ipſius medietatis. </s> <s xml:space="preserve">Vnde dempta ſeptima parte ipſius .42. <lb/>quæ eſt .6. <choice><ex>collectaque</ex><am>collectaq́</am></choice> cum medietate reſidui, quæ eſt .18. habebimus .24. res, quarum <lb/>vnaquæque æqualis erit ipſi <seg type="var">.o.x</seg>.</s> </p> <p> <s xml:space="preserve">Sed ſi quis cupiat reperire <seg type="var">.o.e.</seg> dato <seg type="var">.o.u.</seg> duplicet <seg type="var">.o.u.</seg> à quo demat quartam <choice><ex>par- tem</ex><am>par-tẽ</am></choice> ipſius <seg type="var">.o.u.</seg> & habebit propoſitum. </s> <s xml:space="preserve">Nam ita ſe habere oportet <seg type="var">.o.e.</seg> ad <seg type="var">.o.u.</seg> quemad <lb/>modum <seg type="var">.o.x.</seg> ad <seg type="var">.o.n</seg>.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De lucro mercantili.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> demum ſcire à me deſideras, eſt, quod cum vendideris libram vnam <lb/>mercis pro .4. ſolidis, & lucratus fueris .2. cum quarta parte vnius pro ſingu-<lb/>lis decem libris, ſcire velles quantum lucri facturus eſſes in libris <choice><ex>decem</ex><am>decẽ</am></choice> dan-<lb/>do ſingulam libram pro .6. ſolidis.</s> </p> <p> <s xml:space="preserve">Nulli dubium eſt quin decima pars de .2. cum quarta vnius ſit lucrum libræ vnius. <lb/></s> <s xml:space="preserve">Quæ decima pars ſunt <choice><ex>nouem</ex><am>nouẽ</am></choice> quadrageſimæ partes, & hæc ſubducta à ſolidis .4. reli-<lb/>qui erunt ſolidi .3. cum .31. quadrageſimis partibus pro ſorte vnius libræ. </s> <s xml:space="preserve">Quę ſors <lb/>ſubtracta à ſolidis .6. remanebunt ſol .2. cum .9. quadrageſimis lucri pro libra, quod <lb/>multiplicatum per .10. proueniunt ſol .22. cum quarta parte vnius, & tantum aſcen-<lb/>deret lucrum, quod fieri poſſet in libris decem ſi quamlibet, ſol .3. cum .31. quadra <lb/>geſimis nobis conſtaret.</s> </p> <p> <s xml:space="preserve">Vel ſic multiplicemus ſortem vnius libræ per .10. productum erit .37. cum tribus <pb facs="0423" n="411"/><fw type="head">EPISTOL AE.</fw> quartis, iterum multiplicemus per .10. ſortem cum lucro vnius librę quod eſt .4. pro-<lb/>ductum erit .40. differens à primo ſol .2. cum quarta parte, multiplicemus pariter <lb/>per .10. precium .6. ſolidorum proueniens erit .60. à quo deducendo productum ſor-<lb/>tis librarum .10. quod erat ſol .37. cum tribus quartis ſupererunt ſol .22. cum quar-<lb/>ta parte, vt ſupra.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE DIGNITATIBVS PLANETARVM.</head> <head rend="italics" xml:space="preserve">Adriano Panetio.</head> <p> <s xml:space="preserve"><hi rend="small caps">QVod</hi> eam diſtinctionem orbium, quæiam inualuit, nonteneas, ſed putes <lb/>totum eſſe quoddam continuum excipiens corpora ſtellarum, nouum <choice><ex>non</ex><am>nõ</am></choice> <lb/>eſt, nam nonnulli ſolidæ doctrinæ Philoſophi idem cenſuerunt. </s> <s xml:space="preserve">Sed <lb/>quod attinet ad dignitates planetarum in ſignis zodiaci, ſcias huiuſmo-<lb/>di ordinem me compręhendere eſſe deſumptum ab ordine antiquo orbium <choice><ex>ipſorum</ex><am>ipſorũ</am></choice> <lb/>planetarum, quiquidem ordo erat, vt ſtatim poſt Lunam ſuccederet Sol, poſt So-<lb/>lem Mercurius, tum Venus deinde Mars, poſtea Iupiter, & tandem Saturnus per <lb/><choice><ex>eoſdemque</ex><am>eoſdemq́;</am></choice> orbes, retro redibant, atque hoc cognoſcitur conſtituendo Cancrum do <lb/>micilium Lunæ, Leonem, Solis, Virginem, Mercurij, Libram, Veneris, Scorpio-<lb/>nem, Martis, Sagittarium, Iouis, Capricornum, Saturni, Incipientes deinde ab <lb/>Aquario, quiad nos propius accedit <choice><ex>eundemque</ex><am>eundemq́;</am></choice> tribuentes Saturno, Piſces, Ioui, <lb/>Arietem, Marti, Taurum, Veneri, & Gemellos, Mercurio, ſeptem Planetas cum <lb/>duodecim ſignis zodiaci concordes reddebant.</s> </p> <p> <s xml:space="preserve">Quod deinde Ariſtoteles in libris de ſenſu & ijs quæ ſenſibus percipiuntur, dicit <lb/>pupillam oculi eſſe nigram, non ita ſe habet, nam idem eſt, ac ſi quis diceret <choice><ex>nigrum</ex><am>nigrũ</am></choice> <lb/>eſſe illud medium, quod permitteret tranſitum lumini per ſuam diaphaneitatem, nul <lb/>lum lumen à ſeipſo reflectens, & etiam ac ſi quis diceret nigrum eſſe aerem alicuius <lb/>cubiculi vndequaque clauſi tenebroſi.</s> </p> <p> <s xml:space="preserve">Quod etiam idem Ariſtoteles volens adducere cauſam, cur oculus magis mate-<lb/>riam aquæ, quam aeris participet, dicensidea ratione fieri, quod aqua magis quam <lb/>aer ſeruari poſſit, eodem libro ſcribit, eſt reuera admirandum. </s> <s xml:space="preserve">Ibi enim clarè de-<lb/>monſtrat ſe planè ignorare, & conſtructionem oculi, & cauſam diuerſitatis eorum <lb/>humorum tam in ſubſtantia, quam in figura, quæ non aliunde dependet quam quod <lb/>diuerſam refractionem radiorum luminoſorum producat, qui per pupillam ingre-<lb/>diuntur, vt ad proprios <choice><ex>ſibique</ex><am>ſibiq́;</am></choice> deſtinatos locos dirigantur radij, vt à virtute viſiua per <lb/>fectius ſen tiantur.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De ratione Frigiditatis locorum umbroſorum.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">VEra ratio vnde fiat, vt quanto magis ſentitur calor in locis expoſitis Soli, tan-<lb/>to minus ſentiatur in vmbra, vbi Solis radius non reflectitur, eſt quia cum ra <lb/>refactus eſt aer à vehementi calore radij ſolaris, ſeipſum colligit, & condenſatur in <lb/>locis, à quibus à calore, ratione rarefactionis, non expellitur, & quia naturaliter ca-<lb/>lor ſequitur rarum, rarum calorem, & frigidum <choice><ex>densum</ex><am>densũ</am></choice>, & <choice><ex>densum</ex><am>densũ</am></choice> frigidum, vt vnicui <lb/>que ſanę mentis patet, hanc ob cauſam ſequitur rem ita ſe habere vt diximus. </s> <s xml:space="preserve">Poſſu <lb/>mus etiam abſque dubio credere huiuſmodi ratione fieri, vt frigus matutini tempo <lb/>ris, in crepuſculo maius eſſe eo, quod noctu viguit. </s> <s xml:space="preserve">Nam materia conſiſtens in co-<lb/>no vmbræ terræ, ſemper denſior eſt ea, quæ extra reperitur, imo noua materia con <lb/>tinuo condenſatur, propter motum vmbrę, quæ ſemper corpori ſolari opponitur. </s> <s xml:space="preserve">hęc <pb facs="0424" n="412"/><fw type="head">IO. BAPT. BENED.</fw> autem noua condenſatio dico ſemper fit in crepuſculo matutino, hoc eſt in parte co <lb/>ni à Sole pulſa, in parte vero contrari a ipſius coni hoc eſt in parte crepuſculi ve-<lb/>ſpertini, contrarium accidit, quia potius aliquantulum in hac parte materia coni ra <lb/>rificatur, quia extrinſeca condenſatur, in parte vero matutina extrinſeca rarificatur; <lb/></s> <s xml:space="preserve">& propterea intrinſeca conde nſatur.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">QVOD RECTE ARIST. SENSERIT COELVM <lb/>caſu non eſſe productum.</head> <head rend="italics" xml:space="preserve">Hieronymo Condrumerio.</head> <p> <s xml:space="preserve">FErunt <choice><ex>Ariſtippum</ex><am>Ariſtippũ</am></choice> tempeſtate maris ad incognita littora delatum, cum in are-<lb/>na vidiſſet <choice><ex>quaſdam</ex><am>quaſdã</am></choice> figuras geometricas delineatas <choice><ex>exultantem</ex><am>exultantẽ</am></choice> lętitia dixiſſe: </s> <s xml:space="preserve">Hæc <lb/>ſunt hominum veſtigia. </s> <s xml:space="preserve">Nam conſonum rationi non erat, vt huiuſmodi figuræ ca-<lb/>ſu eſſent impreſſæ: </s> <s xml:space="preserve">neque etiam credendum eſt ingentem hanc ma chinam tanto or <lb/>dine conſtantem fortuitò eſſe productam, cum nulla quantumuis minima eiuſdem <lb/>particula, dummodo nitatur ordine, aliquo modo caſu effecta fuerit; </s> <s xml:space="preserve">cum caſus ni-<lb/>hil producat, quod regulam & ordinem ſeruet. </s> <s xml:space="preserve">Non eſt igitur producta caſu admi <lb/>randa correſpondentia, quæ eſt obiectorum cum potentijs, luminis cum oculo, ſo-<lb/>ni cum auditu, ſaporis cum guſtatu, odoris cum odoratu, qualitatum tangibilium <choice><ex>cum</ex><am>cũ</am></choice> <lb/>tactu. </s> <s xml:space="preserve">Si diligenter deinde cuiuſlibet rei naturalis operationem conſiderabimus, <lb/>eas tanta arte conſtructas videbimus, vt cogamur fateri aliquam prudentiſſimam, <lb/>& ſagaciſſimam mentem eas formaſſe, ſi ergo quælibet <choice><ex>mundi</ex><am>mũdi</am></choice> pars tanta cum ratione <lb/>& ordine eſt conſtructa: </s> <s xml:space="preserve">quomodo fieri poterit, vt de toto ipſo mundo id in dubium <lb/>vocemus, <choice><ex>certiſſimeque</ex><am>certiſſimeq́;</am></choice> non credamus diuiniſſimam aliquam <choice><ex>mentem</ex><am>mẽtem</am></choice> eſſe à qua exqui-<lb/>ſitiſſima huius vniuerſi harmonia, quæ ex tot <choice><ex>tantisque</ex><am>tantisq́;</am></choice> partibus, maximo ordine ni-<lb/>tentibus conficitur, non dependeat?</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">VARIA RESPONSA.</head> <head rend="italics" xml:space="preserve">Nicolao Petreio.</head> <p> <s xml:space="preserve">AD ea quæ mihi ſcribis dico, quod excrementa quæ ex corpore ſano prodeunt <lb/>in ſua <choice><ex>ipſorum</ex><am>ipſorũ</am></choice> qualitate ſenſibili ita ſe <choice><ex>habent</ex><am>habẽt</am></choice> ad <choice><ex>facultatem</ex><am>facultatẽ</am></choice> illius partis eiuſdem <lb/>corporis, ut <choice><ex>eam</ex><am>eã</am></choice> non lędant, <choice><ex>quemadmodum</ex><am>quẽadmodũ</am></choice> efficeret <choice><ex>ſputum</ex><am>ſputũ</am></choice>, ſi eſſet <choice><ex>amarum</ex><am>amarũ</am></choice>, aut quod ex <lb/>cernitur naſo <choice><ex>fętidum</ex><am>fętidũ</am></choice> eſſet. </s> <s xml:space="preserve">Imagineris igitur <choice><ex>quemadmodum</ex><am>quẽadmodũ</am></choice> poſſit eſſe <choice><ex>verum</ex><am>verũ</am></choice> id quod <choice><ex>idem</ex><am>idẽ</am></choice> <lb/><choice><ex>amicus</ex><am>amicꝰ</am></choice> noſter ait. </s> <s xml:space="preserve">Pręterea ſi aliquid tibi in <choice><ex>oculum</ex><am>oculũ</am></choice> inciderit, an neſcis quomodo ſtatim <lb/>affatim affluat humor, vt id foras <choice><ex>propellat</ex><am>ꝓpellat</am></choice>, vel abducat? </s> <s xml:space="preserve">(mirabile opus naturæ.) </s> <s xml:space="preserve">Dic <lb/><choice><ex>etiam</ex><am>etiã</am></choice> eidem non abſque myſterio naturam in tot miſerijs ſenectutem poſuiſſe, cum <lb/>ſæpiſſimè ſenex mori deſideret, ut huius vitæ calamitatibus liberetur, vnde fit, vt <lb/>cum eius aduentum ſentiat, minus affligatur. </s> <s xml:space="preserve">Dicito etiam eidem, naturam non <lb/>fuiſſe tam ſolicitam de quibuſdam partibus quemadmodum eſt de toto, vnde ma-<lb/>gis rotunda, & polita poterat eſſe ſuperficies terræ, quam nunc eſt, quia natura ma <lb/>gis reſpicit totum, quam partes, & magis maiores, quam minores.</s> </p> <p> <s xml:space="preserve">Dum tuas legerem, me continere non potui quin riſerim, id quod ſcribis te inter-<lb/>rogaſſe eum Philoſophum naturalem, vnde fit, vt ventus ſit frigidus, <choice><ex>eumque</ex><am>eumq́;</am></choice> tibi re <lb/>ſpondiſſe, quod à remotiſſimis partibus veniat, <choice><ex>genereturque</ex><am>genereturq́;</am></choice> à vaporibus terræ frigi-<lb/>dis. </s> <s xml:space="preserve">( cum ipſa ſit frigida.) </s> <s xml:space="preserve">Cæterum miror quod ab eo non quæſieris, vnde oriatur <lb/>frigiditas, quæ percipitur ab agitatione aeris, qui quidem à vaporibus terræ non <lb/>proſilit, nec à remotiſſimis partibus ad nos accedit. </s> <s xml:space="preserve">Sed quia de eadem re me in- <pb facs="0425" n="413"/><fw type="head">EPISTOL AE.</fw> rerrogas, ſcito <choice><ex>naturam</ex><am>naturã</am></choice> coniunxiſſe <choice><ex>frigiditatem</ex><am>frigiditatẽ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> denſitate, & <choice><ex>caliditatem</ex><am>caliditatẽ</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> raritate, <lb/>vt ſup. diximus, ita vt cum aliquod corpus <choice><ex>denſatur</ex><am>denſat̃</am></choice>, <choice><ex>frigidum</ex><am>frigidũ</am></choice> <choice><ex>reddatur</ex><am>reddat̃</am></choice>, & dum rarefit ma-<lb/>iorem caliditatem acquirat, & ſic econtra fit, vt quanto magis aliquod corpus refri <lb/>geratur, tanto denſius reddatur, & quanto calidius fit tanto rarius efficiatur. </s> <s xml:space="preserve">Quo-<lb/>ties igitur agitabitur aer, aut aliud corpus, quod ratione ſuæ ſubtilitatis, velociter <lb/>condenſari, & rarefieri poſſit, eius partes denſiores ſemper erunt frigidæ, & hanc <lb/>obrem quilibet ventus, qui per calida loca non tranſeat, natura ſua frigidus, calidus <lb/>autem per accidens erit. </s> <s xml:space="preserve">Hinc fit vt vaſa vitrea, & terrea tam in vehementi frigore, <lb/>quam in magno æſtu frangantur, quia horum vnum fit, ne aliquis locus vacuus rema <lb/>neat, & aliud ob loci neceſſitatem, ſed hoc non ſequeretur, ſi in materia, qua huiuſ <lb/>modi vas conſtat, aliqua aeris portio non contineretur.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE LVMINE LVNÆ, DE FINE LVMINIS, <lb/>de fine motus corporum cęleſtium, de albedine, <lb/>de ſphæra.</head> <head rend="italics" xml:space="preserve">Clariβimo Antonio Nauaiero.</head> <p> <s xml:space="preserve"><hi rend="small caps">LVmen</hi> Lunæ etiam ſi ſit lumen reflexum Solis ab ipſa Luna, ab ea tamen <lb/>non ita reflectitur, vt à ſuperficie polita ſpeculi, <choice><ex>cum</ex><am>cũ</am></choice> eius luminis <choice><ex>tantam</ex><am>tantã</am></choice> <choice><ex>quan titatem</ex><am>quãtitatem</am></choice> ſuper ipſum corpus lunare videamus, & eo modo <choice><ex>terminatam</ex><am>terminatã</am></choice> quo <lb/>conſpicimus. per ſe lumen, cauſa oculi eſt effectum, per accidens autem <lb/>puta quod vis. </s> <s xml:space="preserve">Terra deinde nunquam lunari lumine (<choice><ex>quanuis</ex><am>quãuis</am></choice> ſolaris reflexio exiſtat) <lb/>omnino deſtituta eſt, dico etiam, neque in ipſis ecclipſibus ſolaribus vel lunaribus, <lb/>in ſolaribus enim cum Soltot millia vices maior ſit Luna, Luna verò minor terra, ſe <lb/>quitur, vt terra non omnino priuata remancat lumine Lunæ, in ecclipſibus ve-<lb/>rò lunaribus Luna ſemper videtur, gratia luminis ſolaris, quamuis refracti. </s> <s xml:space="preserve">Mo-<lb/>tus corporum cœleſtium fit ratione ſitus, & varietatis virtutis ſtellæ in diuerſis locis, <lb/>hæc autem varietas abſque diuerſo ſitu eiuſdem ſtellæ, nec diuerſus hic ſitus abſque <lb/>motu fieri poſſet, ita vt motus ſtellarum ſit ratione diuerſitatis ſituum ipſarum, er-<lb/>go motus, & diuerſitas ſituum, fit, ob diuerſam influentiam. </s> <s xml:space="preserve">Quæ autem de albe-<lb/>dine fratri tuo dixeram, erant, quod inter <choice><ex>oens</ex><am>oẽs</am></choice> colores albedo, certo quodam modò, <lb/><choice><ex>maiorem</ex><am>maiorẽ</am></choice> ſimilitudinem habet cum lumine. </s> <s xml:space="preserve">Primò quia magis coniungitur cum lumi-<lb/>ne. </s> <s xml:space="preserve">Secundo quia magis afficit ſenſum. </s> <s xml:space="preserve">Tertiò quia abſque reſiſtentia magis reci-<lb/>pit qualitatem aliorum colorum, quam alij colores. </s> <s xml:space="preserve">Quartò quia maximus <lb/>eſt omnium colorum. </s> <s xml:space="preserve">Quintò quia ſimplicior eſt reliquis. </s> <s xml:space="preserve">Sextò quia diſgregat vi-<lb/>ſum. </s> <s xml:space="preserve">Septimò quia qualitas quæ in niue alba eſſe videtur, nihil aliud eſt quam mul-<lb/>titudo quædam luminum reflexorum, & non albedo, ſimilis ei, quæ eſt lactis, aut <lb/>panni, quæ quidem ſeptima cauſa effecit, vt ipſam albedinem, magis quam alium <lb/>quemuis colorem cum ipſo lumine compararem, cum nihil ſit, quod eſſe ſuum <choice><ex>tranſ</ex><am>trãſ</am></choice> <lb/>mutans, aut apparenter, aut eſſentialiter, illud ipſum prius non tranſmutet in for-<lb/>mam ſibi propin quiorem, vt manifeſtè patet. </s> <s xml:space="preserve">Eſt etiam huius rei octaua ratio, <lb/>magni ponderis, quia ſcilicet nullus ſit color, qui magis reſiſtat lumini, aut in quem <lb/>lumen minorem impreſſionem faciat, quam albedo. </s> <s xml:space="preserve">Vnde ſequitur, obiecta alba, <lb/>minus eſſe combuſtibilia quam alia, cum quælibetres in ſuum contrarium quam in <pb facs="0426" n="414"/><fw type="head">IO. BAPT. BENED.</fw> ſibi ſimile valentius agat, vtrectè vidit Ariſtoteles cum dixit, omne contrarium @ <lb/>ſuo contrario patinatum eſt.</s> </p> <p> <s xml:space="preserve">Inter corpora, multum ſimplicitatisretinet ſphæra.</s> </p> <p> <s xml:space="preserve">Circa quod, præter rationes adductas ab Ariſtotele in libris de Cœlo, poſſumus <lb/>etiam ratiocinarià facilitate motus <choice><ex>vndique</ex><am>vndiq́;</am></choice> ab eo quod violentiæ non reſiſtar, ab eo, <lb/>quod apta <choice><ex>nataque</ex><am>nataq́;</am></choice> ſit quieſcere ſupra quoduis punctum ſuę ſuperficiei, ab eo quod ab <lb/>aliqua ſuperficie alterius corporis ſeſe tangi non permittat, quæ curuitate concaua <lb/>non adæquetur, niſi medio vnius puncti. </s> <s xml:space="preserve">Verum eſt, quod licet hæc vltima ratio <choice><ex>non</ex><am>nõ</am></choice> <lb/>ſit propria ſphæræ, eſt tamen cauſa ſimplicitatis in eo, in quo reperitur, ſed proprię <lb/>paſſiones ſphæræ ſunt ſupradictæ, præter quam quod alia eiuſdem ſphæræ eſt pro-<lb/>prijſſima, quæ eſt diſtantia eiustermini ab vno tantummodo puncto ideſt ab <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> <lb/>centro, & etiam poſſe diuidere corpus aliquod medium, cum æquali reſiſtentia circa <lb/>punctum, quod prius in motu reperitur.</s> </p> <p> <s xml:space="preserve">Aequalitas autem rerum, eſt etiam valde ſimilis ſimplicitati, & vnitati.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Comparatio uiſus, & auditus.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">QVodad viſum & auditum attinet, magis neceſſarium eſſe viſum, & nobilio-<lb/>rem quam auditum exiſtimo, primò quia ſi quis viſu orbatus eſſet, contra <lb/>frigus, & calorem, contra famen, & ſitim nil prouidere poſſet, neque aliud quic-<lb/>quam hoc vocabulum prouidere ſignificat, neque abſque periculo vitæ ab vno loco <lb/>ad alium ferri poſſet, neque aliquid arte facere.</s> </p> <p> <s xml:space="preserve">Sed ſi quis deſtitutus eſſet facultate audiendi, ſupradictas tamen operationes <choice><ex>prae- ſtare</ex><am>prę-ſtare</am></choice> poſſet, neque modo careret, quo animi ſui ſenſa abſque beneficio ſoni, ſed <lb/>ope figurarum & characterum alteri aperiret: </s> <s xml:space="preserve">neque etiam munere ſpeculandi ſcien <lb/>tias (excepta muſica) deſtitueretur. </s> <s xml:space="preserve">Ad ſcientiam comparandam, longè magis ne <lb/>ceſſarius eſt viſus, quam auditus præterquam, quod viſus maiorem numerum obie-<lb/>ctorum, & differentiarum rerum percipit, & inter reliquos ſenſus velociſſimè imò <lb/>in inſtanti operatur, magis remotè quam alij, & exactius ſentit, <choice><ex>minusque</ex><am>minusq́;</am></choice> quam reli-<lb/>qui afficitur, præterquam quod ſemperagit, dummodò non dormiat animal. </s> <s xml:space="preserve">Præ-<lb/>terea ſeſe magis patefacit, & prodit anima per oculos, quam per aliud, cuiuslibet <lb/>ſenſus, inſtrumentum. </s> <s xml:space="preserve">Oculo magis quam alia corporis parte, hominis natura co-<lb/>gnoſcitur: </s> <s xml:space="preserve">& ſi aliquid ſpeculari volumus, quod ſine imaginatiua fieri non poteſt, <lb/>ſtatim imaginamur nos videre huiuſmodirem, ac ſi oculo fuiſſet compræhenſa, & <lb/>ab imagine quæ eſt vnum ex obiectis oculi, imaginatiua nuncupatur. </s> <s xml:space="preserve">Beneficio <lb/>oculorum omnes ferè ſcientiæ ſunt adinuentæ. </s> <s xml:space="preserve">Auditus nil aliud quam ſonum ca-<lb/>pit, auditus nunquam detulit intellectui figuram, corpus ſuperficiem, aut lineam, <lb/>materiam, formam, locum, dimenſionem, plenum inane, nec innumera alia acci-<lb/>dentia, quæ ab oculo compræhenduntur. </s> <s xml:space="preserve">Quæ verò viſui, & auditui ſunt commu-<lb/>nia, ſunt etiam tactui communia, vt numerus, motus, maius, & minus, ſunt tamen ali <lb/>qua oculo & tactui communia, quæ auditus non poteſt capere, vt durum, molle, acu <lb/>tum, obtuſum, aſperum, lene, planum, curuum, concauum, conuexum, magnum, <lb/>paruum, & ſupradicta, ideſt figura corpus & cętera, vt ctiam rectum, obliquum, & <lb/>ſimilia.</s> </p> <pb facs="0427" n="415"/> <fw type="head">EPISTOL AE.</fw> <p> <s xml:space="preserve">Ariſtoteles circa finem primi capitis libri de ſenſu ait mediante viſu, magis <choice><ex>quam</ex><am>quã</am></choice> <lb/>quolibet alio ſenſu, nos percipere ſenſibilia communia. </s> <s xml:space="preserve">Vbi eundem per ſe, & <lb/>non per accidens magis neceſſarium eſſe quam auditum, tam in ijs quæ ad victum, <lb/>quam in ijs quæ ad ſcientiam pertinent eſſe aſſerit, quia auditus intellectui confert <lb/>per accidens. </s> <s xml:space="preserve">Vide etiam quod idem ſcribit primo metaphyſicorum. </s> <s xml:space="preserve">Et ſi ad ali-<lb/>quid perfectè cognoſcendum, oculo ſeſe nobis offerrent ea omnia obiecta, quorum <lb/>ſpecies in imaginatiua formamus, ipſa imaginatiua non egeremus. </s> <s xml:space="preserve">Sed quia hoc <lb/>fieri non poteſt, hunc <choice><ex>theſaurum</ex><am>theſaurũ</am></choice> imaginatiuè, ſeu memoriæ ad conſeruandam imagi <lb/>nem omnium obiectorum ſenſibilium nobis dedit natura, vt ope diſcurſus <choice><ex>intellectus</ex><am>intellectꝰ</am></choice> <lb/>circa dictas imagines, rerum veritatem venari poſſimus. </s> <s xml:space="preserve">Sed vt ad propoſitum re-<lb/>deamus, beneficio oculi animal liberum eſt, cum ſine ipſo locum mutare nequeat, <lb/>vt ſit tutum. tenebræ, <choice><ex>priuatioque</ex><am>priuatioq́;</am></choice> viſus ſunt ferè vnum, & idem. </s> <s xml:space="preserve">Neque vllus eſt@ſen-<lb/>ſus, qui ſit magis ſimilis intellectui quam viſus: </s> <s xml:space="preserve">neque alij ſenſus habent obiecta vi-<lb/>ciſſim communia, quæ non ſint etiam oculo communia, ſed inter oculum, & quem <lb/>libet alium ex ſenſibus, inuenientur quidem obiecta communia, quæ cum alijs non <lb/>communicabunt, vt inter oculum & tactum, figura, acutum, obtuſum, & ſimilia, <lb/>quæ alijs ſenſibus non percipiuntur. </s> <s xml:space="preserve">Mediante viſu, & auditu etiam, <choice><ex>compræhendum</ex><am>compræhendũ</am></choice> <lb/>tur variæ diſtantiæ, <choice><ex>ſitusque</ex><am>ſitusq́;</am></choice> obiectorum, nec non proportiones, & alia quę ab alijs ſen-<lb/>ſibus non compræhenduntur. </s> <s xml:space="preserve">Multa obiecta deinde ſunt ſubiecta guſtatui, quę alijs <lb/>accidentibus prędita ſunt, vnde cum fuerint ſemel deguſtata, talia, qualia ſunt ab o-<lb/>culo percipiuntur, quod nullus ex alijs ſenſibus præſtabit. </s> <s xml:space="preserve">Idem de obiectis odora-<lb/>tus dico. </s> <s xml:space="preserve">Senſuum nullus eſt qui maiorem ſimilitudinem gerat cum vigilia & cum <lb/>vita, quam viſus, neque aliquid eſt, quod magis repræſentet imaginem ſomni, & <lb/>mortis, quàm cęcitas.</s> </p> <p> <s xml:space="preserve">Qui ſibi oculos eruit vt melius ſpecularetur maxima ſtultitia prius obcęcatus fuit <lb/>quia ſoni magis impediunt ſpeculationem quàm lumina, imò qui commodè vult <lb/>contemplari, quantum plus poteſt nititur longius eſſe ab omni ſtrepitu, magis quàm <lb/>à locis luminoſis, & animal magis lætatur lumine quam ſono: </s> <s xml:space="preserve">& ad ſpeculationem <lb/>nos magis inuitat harmonia luminum variorum colorum & figurarum, quàm har-<lb/>monia ſonorum, præterquam quod inſtrumentum viſus totius corporis partium eſt <lb/>pulcherrima, & in eminentiori loco locata, ſi de inſtrumentis ſenſuum loquamur, & <lb/>veluti fineſtræ animæ. </s> <s xml:space="preserve">Et ſi Ariſtoteles dicat oculos & aures in vno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> orbe <lb/>exiſtere, volens inferre quod in eodem æquilibrio ſint æqualiter alta non ita ſe ha-<lb/>bet, quia (ſi de homine loquamur) oculus eſt altior aure. </s> <s xml:space="preserve">Beneficio huius ſenſus, eo <lb/>rum quæ abſunt, & longo iam tempore ſunt mortui, animi ſenſa, & conceptus intel-<lb/>ligimus, neque alia ratione rerum omnium memoria ſeruari poteſt. </s> <s xml:space="preserve">Si cabala un-<lb/>quam vera fuit, nulla alia ratione eſt deleta, quam quia alicuius ſigni viſibilis medio <lb/>conſeruata non fuerit, & quæcunque non ſcribuntur, ideſt oculo non <choice><ex>commendantur</ex><am>cõmendantur</am></choice> <lb/>parum durant cito obliuioni <choice><ex>traduntur</ex><am>tradunt̃</am></choice>. </s> <s xml:space="preserve">In maiori ſemper pretio fuit pictura <choice><ex>quam</ex><am>quã</am></choice> muſi-<lb/>ca: </s> <s xml:space="preserve">obiectis viſibilibus magis quam ijs quæ ſub auditu cadunt, affectus animi, <choice><ex>atque</ex><am>atq;</am></choice> <lb/>alia quælibet res naturalis exprimi poſſunt. </s> <s xml:space="preserve">Aegyptij volentes ſignificare Deum, <lb/>oculi medio id præſtabant.</s> </p> <p> <s xml:space="preserve">Oculus, reſpectu aliorum inſtrumentorum ſenſuum, eſt quaſi epicyclus animæ, <lb/>neque defuit qui crederet oculum eſſe principem animi partem.</s> </p> <p> <s xml:space="preserve">Oculus à Sole, & à Luna ita dependet, vt qui tempore defectus cuiuslibet lumi-<lb/>naris naſcitur, ſtatim cæcus euadat, neque aliqua eſt corporis pars in qua magis ap- <pb facs="0428" n="416"/><fw type="head">IO. BABPT. BENED.</fw> pareat differentia vitæ à morte; </s> <s xml:space="preserve">quam in oculo. </s> <s xml:space="preserve">Ariſtoteles ad finem cap .15. lib. pri <lb/>mi poſteriorum ait, clarum eſſe quod ſi aliquis ſenſus deficiat, futurum vt aliqua <lb/>quoque ſcientia deſit. </s> <s xml:space="preserve">Conſidera, quot ſcientijs careret homo, ſi viſu orbaretur.</s> </p> <p> <s xml:space="preserve">Et in tertio de anima ait, eum qui non ſentit, nihil intelligere poſſe; </s> <s xml:space="preserve">id quod in-<lb/>de confirmat, quia nihil ſit in intellectu, quod prius non fuerit in ſenſu. </s> <s xml:space="preserve">Plato in ti <lb/>meo ait, oculos nobis attuliſſe rerum optimarum notitiam, & ſi oculus non fuiſſet ni <lb/>hil eorum, quæ ad cœlum ſpectant inueniri potuiſſe, & <choice><ex>cognitionem</ex><am>cognitionẽ</am></choice> diei ac noctis ab <lb/>oculis ortum duxiſſe, vt reuolutiones menſium, & annorum metiri, & tempus co-<lb/>gnoſcere, & inueſtigare ordinem naturæ vniuerſalis poſſemus; </s> <s xml:space="preserve">quibus <choice><ex>philoſophiam</ex><am>philoſophiã</am></choice> <lb/>nobis comparauimus, ut alia multa omittam, quæ ibi à Platone dicuntur. </s> <s xml:space="preserve">Addam <lb/>hic & aliam ſpecialem differentiam inter auditum & viſum, quæ eſt, vt obiectum vi <lb/>ſus ſit permanens, & obiectum auditus tranſitorium ſiue ſucceſſiuum aut, vt alio mo <lb/>do idem dicamus, obiectum viſus particpes ſit æternitatis, illud autem quod eſt au-<lb/>ditus non item, nam auditus tempori ſubiectus eſt, viſus autem minimè. </s> <s xml:space="preserve">Vel ſi di-<lb/>camus operationem auditus abſque tempore fieri non poſſe cum ſit motio, operatio <lb/>verò viſus, nullo indiget tempore, cum ip ſa ſit momentanea, & propterea inſtan-<lb/>tanea. </s> <s xml:space="preserve">Nam momentum non eſt motus, nec inſtans tempus.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">QVARE HYEME VIDEATVR HALITVS <lb/>animalium non autem æſtate, & de vento.</head> <head rend="italics" xml:space="preserve">Pancratio Mellano.</head> <p> <s xml:space="preserve"><hi rend="small caps">VNde</hi> fiat vt hyeme halitum noſtrum videamus, & non æſtate, ratio eſt ab <lb/>eiuſdem halitus congelatione, quæ ab extrinſeco frigore fit. </s> <s xml:space="preserve">Prius enim <lb/>ſcire debes aerem <choice><ex>attractum</ex><am>attractũ</am></choice> in pulmone, foras deinde erumpere cum alio <lb/>vapore aliquantulum craſſiore humido, & excrementitio expulſo à natu-<lb/>ra, quæ continuò noſtrum corpus euaporare facit, vnde ſequitur dum aer foras à pul <lb/>mone pellitur, maiorem ſemper materiæ portionem, ea quæ intus attracta eſt exire: <lb/></s> <s xml:space="preserve">vnde ſtatim vt dicta materia foras expulſa, frigidum aerem offendit, cum conſtet ex <lb/>partibus craſſis, & obnoxiis congelationi, condenſatur in formam vaporis, ad dif-<lb/>ferentiam aeris ambientis qui in ſe eas partes craſſas non habet, à quibus <choice><ex>quidem</ex><am>quidẽ</am></choice> par-<lb/>tibus condenſatis, & redditis opacis reflectitur lumen, atque hanc ob cauſam æſtate <lb/>hoc non fit, quia calor vim condenſandi non habet.</s> </p> <p> <s xml:space="preserve">Ventus nihil aliud eſt quam quidam aeris motus, cum condenſatur, ob defectum <lb/>caloris, neque (pace Ariſtotelis dicam) eſt exhalatio ſicca. </s> <s xml:space="preserve">Exemplum à Vitruuio <lb/>allatum nil planè valet, quantum ſpectatad venti naturam, cuius rationem à mere-<lb/>quiris. </s> <s xml:space="preserve">Exemplum etiam ventilabri quo tempore æſtate vtimur negligendum pe-<lb/>nitus non eſt, quia eius beneficio, non ſolum arcemus à nobis aerem ambientem <lb/>calidum, ſed alium etiam aerem circa nos condenſamus: </s> <s xml:space="preserve">& quia ordo naturæ eſt hu <lb/>iuſmodi quod quemadmodum calor ſequitur raritatem <choice><ex>corporum</ex><am>corporũ</am></choice>, ſic etiam frigus <lb/>eorundem denſitatem ſequatur. </s> <s xml:space="preserve">Quod ſi vis vt exemplo illuſtrem, diligenter ob-<lb/>ſeruato tempore æſtatis cum aliqua nubes nobis Solem adimit, vbiaer qui in eius <pb facs="0429" n="417"/><fw type="head">EPISTOL AE.</fw> vmbra reperitur, tantum quantum defectus caloris radij ſolaris fert, qui per vim, <lb/>dictum aerem rarefactum conſeruabat, ſtatim dictum aerem condenſari cognoſces: <lb/></s> <s xml:space="preserve">& quia ea condenſatio homogenea non eſt, ob diuerſas rationes, hanc ob cauſam <lb/>percipimus eam aeris impulſionem, & inæqualiter, dum verò eadem vmbra diſce-<lb/>dit, ventus, ferè, ſtatim ceſſat, & ſæpe ante quam dicta vmbra diſcedat; </s> <s xml:space="preserve">cuius rei cau <lb/>ſa eſt longa mora quam trahi vmbra, ita vt prius abſoluatur reditus aeris ad <choice><ex>formam</ex><am>formã</am></choice>, <lb/>quæ ei conuenit in huiuſmodi vmbra, quam faciet nubes dum Sol deregitur.</s> </p> <p> <s xml:space="preserve">Vera non ſunt ea, quæ tibi Arnoldus dixit, vt mihi tuis literis ſignificaſti. </s> <s xml:space="preserve">Nam ego <lb/>ita dixi, videlicet, quod quoti eſcunque aliquis aſpexerit aliquod punctum in ſuper-<lb/>ficie ſpeculi, </s> <s xml:space="preserve">tunc imaginem ipſius poſt dictam ſuperficie<unclear reason="illegible"/>m videbit duplicatam, ſi <lb/>verò aſpexerit imaginem intra ſpeculum, </s> <s xml:space="preserve">tunc illud punctum videbit duplicatum, <lb/>huiuſmodi autem rei ratio pendet ab hijs quę ad Franciſcum Vimercatum ſcri-<lb/>pſi, quæ ſi memoria tenes, nullum tibi dubium remanebit. </s> <s xml:space="preserve">Nam ea tibi omnia <lb/>oſtendi.</s> </p> <p> <s xml:space="preserve">Dum verò dicis omnem proportionem rationalem diuidi poſſe duobus numeris <lb/>mediantibus in tres æquas partes, mihi ad memoriam reuocas id quod quidam Vitru <lb/>uij commentator aſſerit ſuper primum cap. noni lib. eiuſdem Authoris, ita dicens.</s> </p> <quote> <s xml:space="preserve">Benè eſſe poteſt vt diagonalis (quadrati ſcilicet) numerorum via reperiatur, ſed <lb/>fortaſſe intercedent fracta.</s> </quote> <p> <s xml:space="preserve">Miror te non memoria tenere quid ſint numeri rationales quidúe ſurdi, <choice><ex>neque</ex><am>neq;</am></choice> con <lb/>ſideras, non ſolum non eſſe diuiſibilem in tres æquas partes omnem proportionem <lb/>rationabilem, ſed neque in duas, vt ſunt ſuperparticulares proportiones, necnon <lb/>aliæ innumeræ, ſed cum talia ſcribis te nimis parum verſatum in iſtis rebus oſtendis.</s> </p> <p> <s xml:space="preserve">Id verò quod tibi dicere volebam nudiustertius de Mercurio erat, quod nullo pa <lb/>cto confidendum eſt calculis qui fiunt de curſu Mercurij, eo quod eius ſitus nullo mo <lb/>do obſeruabilis eſt, nam ipſe nunquam nec vbiuis locorum orbis terrarum viſibilis <lb/>eſt altior .18. gradibus ſupra orizontem, ſed neque confidendum eſſet ſi <choice><ex>etiam</ex><am>etiã</am></choice> ipſum <lb/>videremus altum .20. gradibus, </s> <s xml:space="preserve">propterea quod magna refractio <choice><ex>radiorum</ex><am>radiorũ</am></choice> infra hos <lb/>gradus nos valde fallit, quæ quidem refractio, nec <choice><ex>vbique</ex><am>vbiq;</am></choice>, nec omni tempore vnifor-<lb/>mis eſt, propter diformem ſeu inæqualem craſſiciem vaporum quæ continuò muta <lb/>tur. </s> <s xml:space="preserve">Imo multoties eum. videre putabimus ſupra orizontem, exiſtente ipſo ſub <lb/>orizonte.</s> </p> </div> <div type="letter"> <head rend="italics" xml:space="preserve">Quod Ouidius tr anſcurrit à motu diurno, ad motum annuum <lb/>prater rem.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">TVus etiam Ouidius ceſpitauit, cum pro itinere vnius diei efficiat, vt Phaeton à <lb/>patre edoctus ſit etiam de itinere annuali.</s> </p> <p> <s xml:space="preserve">Nam, quod Phaeton petat pro curſu vnius diei, clarè patet <choice><sic>ẽx</sic><corr>ex</corr></choice> diuerſis locis, & pri-<lb/>mò vbi ita ſcribit Ouidius.</s> </p> <quote> <s xml:space="preserve">Currus petit ille paternos. <lb/></s> <s xml:space="preserve"><choice><ex>Inque</ex><am>Inq;</am></choice> diem alipedum ius & moderamen equorum.</s> </quote> <p> <s xml:space="preserve">Deinde vbi Pater ita loquitur.</s> </p> <quote> <s xml:space="preserve">Ardua prima via eſt, & qua vix <choice><ex>manem</ex><am>manẽ</am></choice><unclear reason="illegible"/> recentes. </s> <pb facs="0430" n="418"/> <fw type="head">IO. BAPT. BENED.</fw> <s xml:space="preserve">Enituntur equi medio eſt altiſſima cęlo. <lb/></s> <s xml:space="preserve">Vnde mare, & terras ipſi mihi ſæpe videre. <lb/></s> <s xml:space="preserve">Fit timor & pauida trepidat formidine pectus. <lb/></s> <s xml:space="preserve">Vltima prona via eſt & eget moderamine certo.</s> </quote> <p> <s xml:space="preserve">Etiam vbi dicit.</s> </p> <quote> <s xml:space="preserve"><choice><ex>Dumque</ex><am>Dumq́;</am></choice> ea magnanimus <choice><ex>Phaenton</ex><am>Phaẽton</am></choice> miratur, <choice><ex>opusque</ex><am>opusq́;</am></choice> <lb/>Perſpicit, ecce vigil nitido patefecit ab ortu. <lb/></s> <s xml:space="preserve">Purpureas aurora fores, & plena roſarum. <lb/></s> <s xml:space="preserve">Atria, diffugiunt ſtellæ, quarum agmina cogit. <lb/></s> <s xml:space="preserve">Lucifer, & coeli ſtatione nouiſſimus exit.</s> </quote> <p> <s xml:space="preserve">Necnon vbi ita inquit.</s> </p> <quote> <s xml:space="preserve">Et ſi (modo credimus) vnum <lb/>Iſſe diem ſine Sole ferunt, incendia lumen Præbebant.</s> </quote> <p> <s xml:space="preserve">Quod autem à Patre inſtruatur etiam de curſu annuali, <lb/>videbitur vbi ita dicit.</s> </p> <quote> <s xml:space="preserve">Nitor in aduerſum, nec me, qui cætera vincit. <lb/></s> <s xml:space="preserve">Impetus, & rapido contrarius euehor orbi.</s> </quote> <p> <s xml:space="preserve">Et vbi ita loquitur.</s> </p> <quote> <s xml:space="preserve">Forſitan & lucos illic, <choice><ex>vrbesque</ex><am>vrbesq́;</am></choice> Deorum. <lb/></s> <s xml:space="preserve">Concipias animo <choice><ex>delubraque</ex><am>delubraq́;</am></choice> ditia donis <lb/>Eſſe per inſidias iter eſt, <choice><ex>formasque</ex><am>formasq́;</am></choice> ferarum. <lb/></s> <s xml:space="preserve"><choice><ex>Vtque</ex><am>Vtq́;</am></choice> viam teneas, <choice><ex>nulloque</ex><am>nulloq́;</am></choice> errore traharis. <lb/></s> <s xml:space="preserve">Per tamen aduerſi, gradieris cornua Tauri. <lb/><choice><ex>Aemoniosque</ex><am>Aemoniosq́;</am></choice> arcus, <choice><ex>violentique</ex><am>violentiq́;</am></choice> ora Leonis. <lb/><choice><ex>Sæuaque</ex><am>Sæuaq́;</am></choice> circuitu curuantem brachia longo. <lb/>Scorpion atque aliter curuantem brachia cancrum. <lb/></s> <s xml:space="preserve">Nec tibi quadrupedes animoſos ignibus illis. <lb/></s> <s xml:space="preserve">Quos in pectore habent quos ore & naribus efflant, & c.</s> </quote> <p> <s xml:space="preserve">Sed lucidius etiam hoc videre eſt inferius vbi ita loquitur.</s> </p> <quote> <s xml:space="preserve">Nec tibi directos placeat via quinque per arcus. <lb/></s> <s xml:space="preserve">Sectus in obliquum eſt lato curuamine limes. <lb/></s> <s xml:space="preserve"><choice><ex>Zonarumque</ex><am>Zonarumq́;</am></choice> trium contentus fine, <choice><ex>polumque</ex><am>polumq́;</am></choice> <lb/>Effugit auſtralem <choice><ex>iunctamque</ex><am>iunctamq́;</am></choice> aquilonibus arcton. <lb/></s> <s xml:space="preserve">Hac ſit iter, manifeſta rotæ veſtigia cernes.</s> </quote> <p> <s xml:space="preserve">Et vbi etiam dicit.</s> </p> <quote> <s xml:space="preserve">Neute dexterior tortum declinet ad anguem. <lb/></s> <s xml:space="preserve">Ne ve ſiniſterior preſſam rota ducat ad aram.</s> </quote> </div> <div type="letter"> <head rend="italics" xml:space="preserve">De ſupputatione quinque corporum regularium. <lb/>De aliquibus etiam eorum ſympathijs.</head> <head xml:space="preserve">AD EVNDEM.</head> <p> <s xml:space="preserve">ID quod à me deſideras, ab alijs etiam factum eſt, ſed ne me putes laborem euita <lb/>re, non præter mittam aliquid tibi ſcribere, earum rerum quæ ab Euclide colle <pb facs="0431" n="419"/><fw type="head">EPISTOLAE.</fw> gi, methodo etiam qua vtebar dum in iſtisrebus me aliquo modo exercebam.</s> </p> <p> <s xml:space="preserve">Quotieſcunque igitur ſcire volueris quantitatem corpulentiæ <choice><ex>cuiuſque</ex><am>cuiuſq;</am></choice> <choice><ex>quinque</ex><am>quinq;</am></choice> cor-<lb/>porum regularium ab vna <choice><ex>eademque</ex><am>eademq́;</am></choice> ſphæra terminatorum ſeu <choice><ex>circunſcriptibilium</ex><am>circunſcriptibiliũ</am></choice> cu-<lb/>rabis primum, cognoſcere quantitatem lateris <choice><ex>cuiusque</ex><am>cuiusq́;</am></choice> eorum, talium partium, qua-<lb/>lium ſemidiameter dictæ ſphæræ ſit .100000. extabulis ſinuum Nicolai Copernici. <lb/></s> <s xml:space="preserve">Propone igitur tibiante oculos figuram ſemicircularem vltimæ propoſitionis .13. <lb/>lib. Eucli. & inuenies <seg type="var">.c.d.</seg> tertiam partem ſemidiametri <seg type="var">.d.b.</seg> eſſe partium .33333. æ-<lb/>qualem ſinui arcus <seg type="var">.f.e.</seg> graduum .19. mi .28. qui quidem arcus <choice><ex>demptus</ex><am>dẽptus</am></choice> <choice><ex>cum</ex><am>cũ</am></choice> fuerit à tota <lb/>quarta <seg type="var">.b.f.</seg> remanebitarcus <seg type="var">.e.b.</seg> gra .70. mi .32. cuius corda erit latus exaedri, quod<unclear reason="illegible"/> <lb/>latus ita cognoſces, ſumendo ſcilicet ſinum medietatis <seg type="var">.b.e.</seg> hoc eſt ſinum gra .35. mi <num value="16">.<lb/>16.</num> qui erit partium .57738. cuius duplum erit partium .115476. pro latere cubi.</s> </p> <p> <s xml:space="preserve">Dempto poſtea quadrato lateris exaedri, & quadrato totius diametri <seg type="var">.a.b.</seg> reſi-<lb/>dui radix quadrata, erit <seg type="var">.a.e.</seg> latus Tetraedri. </s> <s xml:space="preserve">Vel ſi duplicaueris ſinum dimidij ar-<lb/>cus <seg type="var">.a.e.</seg> qui quidem arcus, componitur ex quarta <seg type="var">.a.f.</seg> & ex arcu <seg type="var">.f.e.</seg> iam inuento, ſiue, <lb/>vt reſiduus totius dimidij circuli, dempto <seg type="var">.b.e.</seg> iam ſupra inuento, habebimus idem <lb/>latus <seg type="var">.a.e.</seg> partium .163294.</s> </p> <p> <s xml:space="preserve">Pro latere verò Octaedri accipere potes radicem quadratam dupli quadrati ip-<lb/>ſius <seg type="var">.d.b.</seg> & habebis <seg type="var">.f.b.</seg> latus quæſitum. </s> <s xml:space="preserve">Vel ſi malis accipe duplum ſinus medietatis <lb/>arcus <seg type="var">.b.f.</seg> quod duplum erit <seg type="var">.f.b.</seg> partium .14142.</s> </p> <p> <s xml:space="preserve">Pro latere verò Duodecaedri, diuide latus Exaedri ex methodo .11. ſecundi <lb/>Eucli. cuius maior pars erit latus quæſitum, partium .71368.</s> </p> <p> <s xml:space="preserve">Sed pro latere Icoſaedri, te primum oportebit inuenire quantitatem anguli <seg type="var">g.d.<lb/>a.</seg> hoc eſt ipſius arcus <seg type="var">.b.n.</seg> qui tali angulo ſubiacet, quod cum pluribus modis inue-<lb/>niri poſſit, nihilominus, hunc ſeruabis, inuenies primò quantitatem <seg type="var">.d.g.</seg> quæ eſt ra <lb/>dix quadrata ſummæ duorum quadratorum hoc eſt <seg type="var">.d.a.</seg> et <seg type="var">.a.g.</seg> quæ <seg type="var">.a.g.</seg> æqualis eſt <lb/>diametro <seg type="var">.a.b.</seg> vt ſcis, dices poſtea, ſi <seg type="var">.d.g.</seg> correſpondet ipſi <seg type="var">.g.a.</seg> cui correſpondet <seg type="var">.d.<lb/>h.</seg> ſemidiametro ſphæræ? </s> <s xml:space="preserve">tibi veniet <seg type="var">.h.k.</seg> ſinus arcus <seg type="var">.a.h.</seg> hoc eſt <seg type="var">.b.n.</seg> graduum .63 -<lb/>min .26. cuius medietas gra .31. mi .43. pro ſinu ſuo habet partes .52571. cuius ſinus du <lb/>plum eſt partium .105142. pro latere Icoſaedri.</s> </p> <p> <s xml:space="preserve">Incipiendo nunc à Tetraedro, ſcire debes, quod pars <seg type="var">.a.c.</seg> totius diametri <seg type="var">.a.b.</seg> æ-<lb/>qualis eſt axi ipſius Tetraedri, quæ quidem <seg type="var">.a.c.</seg> vt ſubſeſquialtera ipſius <seg type="var">.a.b.</seg> erit par <lb/>tium .13333.</s> </p> <p> <s xml:space="preserve">Quæres poſtea quantitatem ſuperficialem vnius faciei ipſius Tetraedri, hac me-<lb/>thodo, inueniendo primum radicem quadratam trium quartarum quadrati <lb/>ipſius <seg type="var">.a.e.</seg> lateris Tetraedri, eo quod latus hoc, ſeſquitertium in potentia eſt ipſi per<lb/>pendiculari terminatę ab vno angulorum trianguli æquilateris & à latere ei oppoſi-<lb/>to ex .11. tertijdecimi ipſius Eucli. quę quidem perpendicularis, erit <choice><ex>partium</ex><am>partiũ</am></choice> .141416. <lb/>& hæc multiplicata cum medietate lateris trianguli, hoc eſt cum .81647. tibi dabit <lb/>ſuperficiem quæſitam, hoc eſt baſim Tetraedri <choice><ex>partium</ex><am>partiũ</am></choice> <choice><ex>ſuperficialium</ex><am>ſuperficialiũ</am></choice> .11546192152. <lb/><choice><ex>Hanc</ex><am>Hãc</am></choice> demum baſim multiplicando cum tertia parte axis Tetraedri habebis corpu-<lb/>lentiam totius Tetraedri, quæ erit .513158964003488.</s> </p> <p> <s xml:space="preserve">Neque tibi hoc loco occultare volo quandam meam animaduerſionem, quæ eſt, <lb/>quod diameter ſeu perpendicularis (ſupradicta) faciei ipſius Tetraedri, ſemper æ-<lb/>qualis eſt lateri ipſius Octaedri circunſcriptibilis ab eadem ſphæra, hoc eſt ipſi <seg type="var">.b.f.</seg> <lb/>quapropter quotieſcunque ipſam perpendicularem habere voluerimus accipiendo <lb/><seg type="var">b.f.</seg> habebimus intentum. </s> <s xml:space="preserve">Et quod hoc verum ſit poſſumus ita demonſtrare.</s> </p> <p> <s xml:space="preserve">Primum, notum nobis eſt, ipſam perpendicularem, triplam eſſe eius parti, quæ <pb facs="0432" n="420"/><fw type="head">IO. BAPT. BENED.</fw> à centro circuli, ipſum triangulum circunſcribentis, terminatur, & à baſi, vt in tertio <lb/>propoſito decimæſeptimæ quartidecimi Eucli. probatur, ex quo ſequitur proportio<lb/>nem huiuſmodi perpendicularis ad axem Tetraedri, hoc eſt ad <seg type="var">.a.c.</seg> ſeſquioctauam <lb/>eſſe in potentia, ex penultima primi Eucli. </s> <s xml:space="preserve">Sed cum <seg type="var">.d.c.</seg> tertia pars ſit ipſius <seg type="var">.d.a.</seg> vt <lb/>etiam ex .2. propoſito, ſeu corollario decimæſeptimæ .14. lib. diſcurrere licet, cum ex <lb/>dicto corollario <seg type="var">.d.c.</seg> ſit ſexta pars ipſius <seg type="var">.a.b</seg>. </s> <s xml:space="preserve">Quare <seg type="var">.d.c.</seg> quarta pars erit ipſius <seg type="var">.a.c.</seg> vn <lb/>de <seg type="var">.a.c.</seg> ſeſquitertia erit ipſi <seg type="var">.a.d.</seg> in longitudine, <choice><ex>ideoque</ex><am>ideoq́;</am></choice> quadratum ipſius <seg type="var">.a.d.</seg> ad qua-<lb/>dratum ipſius <seg type="var">.a.c.</seg> erit vt .9. ad .16: </s> <s xml:space="preserve">& ita duplum quadrati ipſius <seg type="var">.a.d.</seg> hoc eſt quadra-<lb/>tum ipſius <seg type="var">.b.f.</seg> ad quadratum ipſius <seg type="var">.a.c.</seg> erit, vt .18. ad .16. hoc eſt ſeſquioctauum, er-<lb/>go <seg type="var">.b.f.</seg> æqualis erit dictæ perpendiculari, ex .9. quinti.</s> </p> <p> <s xml:space="preserve">Cubus poſtea ipſius <seg type="var">.b.e.</seg> erit partium .1539838576570176.</s> </p> <p> <s xml:space="preserve">Pro Octaedro deinde, accipies productum diametri in ſemidiametrum, quod <lb/>productum, æquale erit quadrato diuidenti per æqualia Octaedron, hocigitur pro-<lb/>ductum, multiplicando per .100000. ſemidiametrum ſphæræ, tibi dabit columnam <lb/>quadrilateram cuius tertia pars, erit partium .666666666666666. cuius duplum <lb/>erit ipſum Octaedron partium .1333333333333.</s> </p> <p> <s xml:space="preserve">Pro Icoſaedro autem, oportet prius quantitatem perpendicularis inuenire, quæ <lb/>perpendicularis, per æqualia diuidit baſim ipſius Icoſaedri, quæ vt radix quadrata <lb/>trium quartarum quadrati lateris ipſius baſis, erit partium .91055. talium, qualium <lb/>dictum latus erit partium .105142. cuius medietas eſt .52571. quæ medietas ſi mul-<lb/>tiplicata fuerit cum dicta perpendiculari, dabit totam baſim ſuperficialem, hoc eſt <lb/>ſuperficiem vnius trianguli æquilateris partium ſuperficialium .4786852405. quo <lb/>facto, accipe quadratum duarum tertiarum ipſius, hic ſupra dictæ perpendicularis, <lb/><choice><ex>ipſumque</ex><am>ipſumq́;</am></choice> deme ex quadrato ſemidiametri ſphæræ, hoc eſt, ex quadrato <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> .100000 <lb/>radix poſtea quadrata reſidui, erit partium .79468. & hæc erit perpendicularis à cen <lb/>tro ſphærę ad vnam baſim ipſius Icoſaedri, quam volueris, quam perpendicularem <lb/>ſi multiplicaueris cum quantitate ſuperficiali, hic ſuperius reperta, vnius baſis, con-<lb/>ſequeris columnam trilateram partium .380401586920540. cuius tertia pars, erit <lb/>partium .126800528973513. pro vna ex .20. </s> <s xml:space="preserve">Pyramidibus ipſum corpus compo-<lb/>nentibus. </s> <s xml:space="preserve">Breuius tamen hoc efficiens, ſi multiplicaueris baſim dictam, cum tertia <lb/>parte ipſius perpendicularis, hanc poſtea pyramidem multiplicando per .20. habebis <lb/>totam corpulentiam ipſius Icoſaedri partium .2536010579470260.</s> </p> <p> <s xml:space="preserve">Pro Duodecaedro demum, accipe ſinum gra .36. qui <choice><ex>gradus</ex><am>gradꝰ</am></choice> ſunt pro dimidio quin <lb/>tæ partis totius gyri circularis, <choice><ex>qui</ex><am>ꝗ</am></choice> <choice><ex>quidem</ex><am>quidẽ</am></choice> ſinus, erit partium .58778. cuius quadratum <lb/>ſi <choice><ex>dem</ex><am>dẽ</am></choice> pſeris ex quadrato <choice><ex>ipſius</ex><am>ipſiꝰ</am></choice> .100000. ſemidiametri circuli <choice><ex>circunſcribentis</ex><am>circũſcribentis</am></choice> <choice><ex>aliquem</ex><am>aliquẽ</am></choice> <choice><ex>pen- tago</ex><am>pẽ-tago</am></choice> num æquilaterum, & æquiangulum, </s> <s xml:space="preserve">tunc radix reſidui, erit perpendicularis du-<lb/>cta à centro dicti circuli ad medium vnius lateris ipſius pentagoni, quæ perp endicu <lb/>laris, erit partium .80902. talium qualium medietas lateris dicti fuerit .58778. <lb/>Nunc verò dicendo ſi .58778. dat .80902. quid nobis dabit .35684? </s> <s xml:space="preserve">medietas lateris <lb/>ipſius Duodecaedri, vnde da bit .49116. pro perpendiculari, à centro ipſius penta-<lb/>goni, ad latus ipſius Duodecaedri, quæ multiplicata cum me dietate ſupradicta ip-<lb/>ſius lat eris, hoc eſt cum .35684. producet vnum ex quinque triangulis componenti-<lb/>bus vn um pentagonum, ſeu vnam baſim ipſius Duodecaedri, quod quidem triangu <lb/>lum, erit partium .1752655344. ſu perficialium, quas ſi per quinque multiplicaueris <lb/>habeb is vnam baſim pentagonam dicti corporis partium .8763276720. </s> <s xml:space="preserve">Dicendum <lb/>poſtea eſt, ſi ad .80901. conuenit ſemidiameter circularis partium .100000. quid <choice><ex>con</ex><am>cõ</am></choice> <lb/>ueniet partibus .49116. dabit .60711. pro tali ſemidiametro circulari, cuius quadra- <pb facs="0433" n="421"/><fw type="head">EPISTOL AE.</fw> tum, ſi dempſeris ex quadrato ipſius .100000. ſemidiametro ſphęræ, </s> <s xml:space="preserve">tuncradix qua-<lb/>drata reſidui, erit perpendicularis à centro ſphæræ ad centrum pentagoni partium, <lb/>79461. cuius tertia pars, ſi multiplicata fuerit cum pentagono ſupra reperto dicti cor <lb/>poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron, <lb/>quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipſius Duo <lb/>decaedri partium .2785354925791680.</s> </p> <p> <s xml:space="preserve">Nunc verò ſi experiri voluerimus vtrum iſti calculi duorum corporum maiorum <lb/>ſint rectè ſupputati, <choice><ex>dicemus</ex><am>dicemꝰ</am></choice> ſi ad <choice><ex>corpus</ex><am>corpꝰ</am></choice> .12. <choice><ex>baſium</ex><am>baſiũ</am></choice>, <choice><ex>quod</ex><am>qđ</am></choice> eſt <choice><ex>partium</ex><am>partiũ</am></choice> .2785354925791680 <lb/>conuenit numerus partium .2536010579470260. ipſius Icoſaedri, quid conueniet <lb/>lateri cubi partium .115476. & inueniemus conuenire latus ipſius Icoſaedri partium <lb/>105138. eo quod probatum ſit in <ref>.10. propoſitione .14. li. Eucl.</ref> eandem <choice><ex>proportionem</ex><am>proportionẽ</am></choice> <lb/>eſſe corpulentiæ ipſius Duodecaedri ad corpulentiam ipſius Icoſaedri, quæ lateris <lb/>cubi ad latus Icoſaedri.</s> </p> <p> <s xml:space="preserve">Hæc autem corpora, ita ſibi inuicem, & cum eorum ſphæra harmonicè <choice><ex>conueniunt</ex><am>conueniũt</am></choice> <lb/>quemadmodum antiqui philoſophi inuenerunt, vt <choice><ex>mirandum</ex><am>mirandũ</am></choice> non ſit, ipſos credidiſ-<lb/>ſe omnia quæ natura conſtant, aliquo pacto exiſtis corporibus fieri. </s> <s xml:space="preserve">Conſidera quæ-<lb/>ſo quomodo conueniant inuicem Tetraedron, Octaedron, & Icoſaedron, cum uniuſ-<lb/>cuiuſque baſes ſint triangulares æquilateræ intelli gendo ſemper hæc corpora ab ea-<lb/>dem ſphæra circunſcriptibilia.</s> </p> <p> <s xml:space="preserve">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale <lb/>ſit ei perpendiculari, quæ diuidit baſim Tetraedri per æqualia, vtſupra demonſtra-<lb/>uimus.</s> </p> <p> <s xml:space="preserve">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baſis <lb/>Tetraedri ad baſim Octaedri ſeruet proportionem ſeſquitertiam, conſonantiæ dia-<lb/>teſſaron. </s> <s xml:space="preserve">Et proportio omnium ſuperficierum ſiue baſium Octaedri ſimul ſumpta-<lb/>rum, ad omnes baſes ipſius Tetraedri ſimul ſumptas ſit ſeſquialtera, conſonantiæ dia <lb/>pentis. </s> <s xml:space="preserve">Neque omittendum eſt, quod proportio Octaedriad triplum Tetraedri ſit, <lb/>vt latus Octaedri ad latus Tetraedri.</s> </p> <p> <s xml:space="preserve">Proportio verò lateris Octaedri, ad axem Tetraedri, potentia eſt ſeſquioctaua, <lb/>vt ſupra vidimus interuallum ſcilicet harmonicum toni maioris.</s> </p> <p> <s xml:space="preserve">Harmonia verò Tetraedri, & Exaedri <choice><ex>cum</ex><am>cũ</am></choice> eorum ſphæra, talis eſt, vt proportio dia <lb/>metriſphæræ, potentia, tripla ſit lateri Exaedri, & ſeſquialtera lateri Tetraedri, ex <lb/>quo ſequitur latus Tetraedri potentia duplum exiſtere lateri Exaedri. </s> <s xml:space="preserve">Interuallum <lb/>enim triplum in harmonicis, componitur ex diapaſon, & diapente, & ſonat ſpeciem <lb/>diapentis. </s> <s xml:space="preserve">Duplum verò eſt diapaſon, ſeſquialterum autem eſt di apente, quę con-<lb/>ſonantiæ perfectiſſimæ ſunt.</s> </p> <p> <s xml:space="preserve">Proportio verò diametri ſphæræ, potentia dupla eſt lat eri Octaedri, conſonantię <lb/>diapaſon. </s> <s xml:space="preserve">Ex quo ſequitur proportionem lateris Tetraedri ad latus Octaedri, po-<lb/>tentia, ſeſquitertiam eſſe, hoc eſt conſonantiæ diateſſaron, & proportionem lateris <lb/>Octaedri ad latus Exaedri, potentia, ſeſquialteram eſſe, ita quod quatuor iſtæ poten <lb/>tiæ, ideſt diametri ſphæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-<lb/>ſtituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenſam .6. 4. 3. 2. (dixi <lb/>ferè, quia ditonus ſupra terminum .3. vel ſemiditonus ſub termino .2. hoc loco non <lb/>reperitur, cuius quidem terminus eſſet .2. cum duabus quintis.)</s> </p> <p> <s xml:space="preserve">Adde quod diameter ſphæræ triplus eſt longitudine ad <choice><ex>perpendicularem</ex><am>perpendicularẽ</am></choice> ductam <lb/>à centro ſphæræ ad baſim Octaedri, quæ proportio, vt ſupra dictum eſt, dicitur dia-<lb/>paſondiapente, practici verò eam vocant duodecimam.</s> </p> <pb facs="0434" n="422"/> <fw type="head">IO. BAPT. BENED.</fw> <p> <s xml:space="preserve">Diameter verò ſphæræ ſeſquialter eſt longitudine axi Tetraedri, conſonantiæ <lb/>diapentis. </s> <s xml:space="preserve">Axis autem Tetraedri ſeſquitertius eſt longitudinis ſemidiametro ſphæ-<lb/>ræ conſonantiæ diateſſaron. </s> <s xml:space="preserve">Ita quod iſti tres termini, qui ſunt, diameter ſphæræ, <lb/>axis Tetraedri, & ſemidiameter ſphæræ conſtituunt etiam valde perfectam harmo-<lb/>niam huiuſmodi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-<lb/>lentiam Tetraedri tripla eſt, conſonantiæ iam ſupradictæ diapaſondiapente. </s> <s xml:space="preserve">Si ve-<lb/>rò de vniſono aliquid videre deſideras, conſidera æqualitatem dupli quadrati dia-<lb/>metri ipſius ſphæræ, cum omnibus baſibus Exaedri, vel potentia diametri ſphæræ <lb/>cum duabus potentijs ſimul ſumptis, quarum vna eſt lateris Tetraedri, reliqua verò <lb/>lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baſibus Exae <lb/>dri. </s> <s xml:space="preserve">Nec mihi videtur ſilentio inuoluendum eſſe, antequam vlterius progrediar no<lb/>tabilem ſympatiam inter triangulum æquilaterum, & Tetraedron (<choice><ex>quanuis</ex><am>quãuis</am></choice> <choice><ex>triangulum</ex><am>triangulũ</am></choice> <lb/>corpus non ſit) non ſolum ob <choice><ex>inalterabilitatem</ex><am>inalterabilitatẽ</am></choice> harum duarum figurarum. </s> <s xml:space="preserve">(nam omnes <lb/>aliæ alterabiles eſſe poſſunt, ijſdem lateribns exiſtentibus, cum ex quadrato rom-<lb/>bus, vel ex pentagono ęquiangulo, pentagonum non æquiangulum & c. efficiatur) <lb/></s> <s xml:space="preserve">ſed quod quemadmodum latus trianguli æquilateri ſeſquitertium potentia eſt per-<lb/>pendiculari ipſum per æqualia diuidenti, ita latus Tetraedri, ſeſquialterum eſt po-<lb/>tentia axi ipſius Tetraedri, vnde cum dempta fuerit illa proportio ſeſquitertia, ex <lb/>hac ſeſquialtera relinquetur nobis proportio ſeſquioctaua, inter perpendicularem <lb/>trianguli, & axem Tetraedri (quod etiam ſupra demonſtrauimus.) </s> <s xml:space="preserve">Tranſeamus nunc <lb/>hęc, nec omittamus tamen ſympatias quaſdam inter Exaedron, Octaedron, & Tetra <lb/>edron, hoc eſt quod eadem proportio ſit inter corpulentias Exaedri, & Octaedri, <lb/>quæinter eorum ſuperficies, nec non, vt latus Exaedri ad ſemidiametrum ſphæræ. <lb/></s> <s xml:space="preserve">Proportio verò baſis Exaedri ad baſim Tetraedri, vtlatus Tetraedri ad perpendicu <lb/>larem diuidentem per æqualia eius baſim.</s> </p> <p> <s xml:space="preserve">Hactenus ſatis dictum ſit de Tetraedro, Exaedro, & Octaedro cum ſphæra. </s> <s xml:space="preserve"><choice><ex>Dicem</ex><am>Dicẽ</am></choice> <lb/>dum nunc cenſeo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè <lb/>omnia hæc ab antiquis philoſophis inuenta ſint, quorum primum eſt, quod tam ba-<lb/>ſis Duodecaedri, quam Icoſaedri, ab vno <choice><ex>eodemque</ex><am>eodemq́;</am></choice> circulo circunſcriptibiles ſunt, ve <lb/>rùm, talis paſſio accidit etiam baſibus Exaedri & Octaedri. </s> <s xml:space="preserve">Præterea quemadmo-<lb/>dum in Duodecaedro, quilibet angulus ſolidus terminatur tribus angulis pentago-<lb/>norum æquiangulorum ita in Icoſaedro, quilibet angulus ſolidus viceuerſa termi-<lb/>natur quinque angulis triangulorum æquiangulorum. </s> <s xml:space="preserve">Et tam vnum, quam alte-<lb/>rum horum corporum, triginta lateribus continetur. </s> <s xml:space="preserve">Et tot ſolidos angulos trian-<lb/>gulares, habet Duodecaedron, quot baſes triangulares continet Icoſaedron.</s> </p> <p> <s xml:space="preserve">Et Icoſaedron, tot ſolidos angulos <choice><ex>pentagonos</ex><am>pẽtagonos</am></choice>, quot baſes <choice><ex>pentagonas</ex><am>pẽtagonas</am></choice> habet Duo <lb/>decaedron. </s> <s xml:space="preserve">Et tam vnum quam alterum habet .60. angulos ſuperficiales. </s> <s xml:space="preserve"><choice><ex>Eademque</ex><am>Eadẽq́;</am></choice> <lb/>proportio eſt omnium baſium ſimul <choice><ex>ſumptarum</ex><am>ſumptarũ</am></choice> Duodecaedri ad omnes baſes ſimul <lb/>ſumptas ipſius Icoſaedri, quæ corpulentiæ ipſius Duodecaedri ad corpulentiam <lb/>Icoſaedri (quamuis hęc paſſio accidat Exaedro cum Octaedro, vt ſpra diximus) quę <lb/>quidem proportio, eadem etiam eſt, quę lateris Exaedri ad latus Icoſaedri, vt ſu-<lb/>pra iam dictum fuit.</s> </p> <pb facs="0435" n="423"/> <fw type="head">EPISTOL AE.</fw> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">NOVA INVENTIO COMPONENDI ASTROLABIA <lb/>cum Horologijs artificialibus.</head> <head rend="italics" xml:space="preserve">Facobo Mayeto Ingenioſißimo Horologiorum Serenißimi <lb/>Sabaudiæ Ducis Artifici.</head> <p> <s xml:space="preserve"><hi rend="small caps">NOnnvnqvam</hi> conſideraui mirabilem pulchritudinem, ſimul cum vtili-<lb/>tate coniunctam, illorum horologiorum, quæin Germania conſtruuntur <lb/><choice><ex>cum</ex><am>cũ</am></choice> mobili Rete, ſeu Aranea Aſtrolabij <choice><ex>ſuper</ex><am>ſuꝑ</am></choice> <choice><ex>Tabulam</ex><am>Tabulã</am></choice> regionis, in <choice><ex>quibus</ex><am>ꝗbus</am></choice> <choice><ex>conti</ex><am>cõti</am></choice> <lb/>nuo <choice><ex>videntur</ex><am>vident̃</am></choice> oriri, <choice><ex>occidereque</ex><am>occidereq́;</am></choice> cæleſtia ſigna, cælum mediare ſupra orizon <lb/><choice><ex>tem</ex><am>tẽ</am></choice><unclear reason="illegible"/>, necnon ſub eo, & vt vno verbo dicam, continuo erecta videtur tota coelifigura. <lb/></s> <s xml:space="preserve">Sed quia talia horologia omnia eorum limbum diſtinctum habent in .24. horas, qua <lb/>propter diametrum limbi, minorem duobus palmis, ſeu ſemipede eſſe non oportet <lb/>neinterſtitia horarum iuſtò breuiora ſeu anguſtiora efficiantur, etiam ne interualla <lb/>dentium rotæ indicis nimis anguſta ſint. </s> <s xml:space="preserve">Sed quia talis magnitudo vt plurimum in-<lb/>commoda exiſtit. </s> <s xml:space="preserve">Ideo non inutile fore cogitaui, ſi modus aliquis inuentus fuerit, <lb/>vt ea omnia efficiantur in limbo diuiſo tantummodo in .12. horas æquales, <choice><ex>ipſumque</ex><am>ipſumq́;</am></choice> <lb/>inueni, qui quidem erit, efficiendo vt Tabula (in qua deſignantur cęleſtes domus, <lb/>cum almicantarat, atque azimut) Reti ſubiectæ, mobilis ſit, tardior tamen ipſo Re-<lb/>te cum indice, pro duplo temporis, hoc eſt, quod eo tempore, quo Aranea cum in <lb/>dice circunuoluetur ſpacio .12. horarum vno gyro perfecto, ipſa Tabula efficiat tan <lb/>tummodo ſexinterſtitia horarum. </s> <s xml:space="preserve">Ideſt dum Tabula dicta eſſicit vnam integram re <lb/>uolutionem, Aranea, ſeu Zodiacus cum indice, duas efficiat reuolutiones. </s> <s xml:space="preserve">Ita quod <lb/>Aranea cum indice perficiet vnam reuolutionem ſpaci o temporis .12. horarum, Ta-<lb/>bula verò perficiet eam ſpacio temporis .24. horarum. </s> <s xml:space="preserve">Vnde ſequetur quod Ara-<lb/>nea ſeu Zodiacus cum indice, ſpacio .24. horarum perfectè circunuoluetur ſupra Ta-<lb/>bulam, & ita huiuſmodi horologia, in hoc nihil differrent ab illis ſupradictis. </s> <s xml:space="preserve">Vt au <lb/>tem facias dictam tabulam tardiorem duplo temporis Araneæ cum indice, quamuis <lb/>diuerſis modis hoc fieri poſſit, pręſtantiorem tamen iudico, ſi cum Rota indicis, <choice><ex>aliam</ex><am>aliã</am></choice> <lb/>Rotam <choice><ex>concentricam</ex><am>concentricã</am></choice> coniunxeris, ita tamen, vt <choice><ex>vnaquęque</ex><am>vnaquęq;</am></choice> liberè poſſit volui, ſimiliter <lb/>ſi cum ea horologii particula (quę <choice><ex>circumagit</ex><am>circũagit</am></choice> Rotam indicis, quæ Italicè <hi rend="small caps">rochetto</hi> <lb/>Germanicè verò <hi rend="small caps">trib</hi> vocatur, Latinè <choice><ex>autem</ex><am>aũt</am></choice> ipſum vocabo, <hi rend="small caps">colinvm</hi>, qui ſubro-<lb/>ta fuſi reperitur) coniunxeris alium colinum quem, ſecundum vocabo, <choice><ex>concentricum</ex><am>concentricũ</am></choice> <lb/>verò cum primo, cum <choice><ex>eoque</ex><am>eoq́;</am></choice> conſolidato, numerum verò dentium, tam Rotę adiunctę <lb/>quam ſecundi colini, varijs modis poteris inuenire, quorum primus erit, vt numerus <lb/>dentium ſecundæ Rotę duplus exiſtat numero dentium primę, efficiendo ſecundum <lb/>colinum eiuſdem numeri dentium quo primum, ſed quia interualla dentium huiuſ-<lb/>modi Rotę, nimis anguſta fortaſſe reſultabunt, </s> <s xml:space="preserve">propterea alios etiam modos inue-<lb/>ni, quorum vnus erit (dum numerus dentium primi colini par fuerit) <choice><ex>efficiendo</ex><am>efficiẽdo</am></choice> ſecun <lb/>dam <choice><ex>Rotam</ex><am>Rotã</am></choice> <choice><ex>eiuſdem</ex><am>eiuſdẽ</am></choice> numeri <choice><ex>dentium</ex><am>dentiũ</am></choice> cuius eſt prima. <choice><ex>ſecundum</ex><am>ſecũdũ</am></choice> vero colinum, medietatis <lb/>numeri dentium cuius erit primus. </s> <s xml:space="preserve">Attamen ſi primus colinus eſſet .4. dentium, ſecun <lb/>dum oporteret eſſe duorum dentium, vnde motus ſecundę Rotę non eſſet ita conti-<lb/>nuus. </s> <s xml:space="preserve">Quapropter alium etiam mòdum excogitaui, hoc eſt, cupiendo vt ſecundus <lb/>colinus, extribus dentibus exiſtat, ſi primus ex .4. repertus fuerit, oportebit prius ex <lb/>regula de tribus, numerum quendam inuenire quo inuento ipſum duplicare, & hunc <lb/>duplicatum numerum conueniet ſecundam Rotam habere, vt ipſa poſſit ab illo co-<lb/>lino <choice><ex>trium</ex><am>triũ</am></choice> <choice><ex>dentium</ex><am>dentiũ</am></choice> circunuolui in duplo temporis, quo prima à ſuo colino quatuor den <pb facs="0436" n="424"/><fw type="head">IO. BAPT. BENED.</fw> tium. </s> <s xml:space="preserve">Exempli gratia, ſi prima Rota conſtaret ex .36. dentibus, dicendum eſſet, ſi <lb/>4. conuenit cum .36. cum quibus conuenient .3. & inueniemus .27. cum quo numero <lb/>dicta ſecunda Rota circunuolueretur eodem tempore à ſuo colino trium dentium, <lb/>quo prima à ſuo quatuor dentium, </s> <s xml:space="preserve">quare duplicando .27. haberemus .54. pro nume-<lb/>ro dentium dictę ſecundæ Rotæ, vt duplo temporis circunuoluatur quo prima. </s> <s xml:space="preserve">Sed <lb/>ſi primus colinus conſtaret ex .6. dentibus, exiſtente ſua Rota ex .36. <choice><ex>vellemusque</ex><am>vellemusq́;</am></choice> <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>ſecundus exiſteret ex .4. </s> <s xml:space="preserve">tunc ſuam Rotam oporteret habere dentes .48. ex dicta re-<lb/>gula. </s> <s xml:space="preserve">Si autem primus colinus conſtaret ex numero impari, nihil referret, dummo-<lb/>do huiuſmodi numerus impar, ſeu par, exiſteret pars propria numeri dentium, vel <lb/>ipſius dupli primæ Rotę, hoc eſt, eſſet pars aliquota numeri dentium ipſius primæ <lb/>Rotæ vel ipſius dupli. </s> <s xml:space="preserve">In ijs verò horologiis in quibus duplum numeri dentium di-<lb/>ctę primę Rotę non erit multiplex numero dentium primi colini, hoc fieri non pote <lb/>rit. </s> <s xml:space="preserve">Ratio enim tam clarè, tibi conſideranti, patebit, vt nullis verbis indigeat cum <lb/>ſemper numerus dentium ſecundę Rotę multiplex eſſe debeat numero dentium ſe-<lb/>cundi colini. </s> <s xml:space="preserve">Idem autem non dico de prima Rota cum ſuo colino, hoc eſt, vt nu-<lb/>merus primę multiplex ſit numero ſui colini, nam hoc neceſſarium non eſt. </s> <s xml:space="preserve">Pona-<lb/>mus exempli gratia primum colinum conſtare ſex dentibus, ſuam vero Rotam den-<lb/>tibus .21. cuius quidem numeri, 6. non eſt pars aliquota, ſed dupli ipſius .21. ipſe .6. <lb/>eſt pars aliquota. </s> <s xml:space="preserve">Nunc verò ſi voluerimus numerum dentium ſecundæ Rotę inue-<lb/>nire, cuius colinus ex quinque dentibus exiſtat (ſuppoſito primo ex .6. conſtare) </s> <s xml:space="preserve">tunc <lb/>ex regula de tribus, diuiſo producto, quod fit ex .21. in .5. per .6. exibit .17. cum di-<lb/>midio, cuius duplum eſſet .35. qui multiplex eſt ipſi quinque. </s> <s xml:space="preserve">Reperto igitur nume <lb/>ro ſecundę Rotę, cum numero ipſius colini, oportet nunc ſcire modum compoſitio-<lb/>nis, ſeu coniunctionis harum rerum, hoc eſt duorum colinorum <choice><ex>concentricorum</ex><am>concentricorũ</am></choice> (ſed <lb/>de ijs ſatis iam ſuperius dictum fuit) duarum Rotarum concentricarum cum Tabula, <lb/>cum Zodiaco, & cum indice, ſeu Oſtenſore, cuius quidem Oſtenſoris medietas tan <lb/>tummodo nobis ſufficiet. </s> <s xml:space="preserve">Sciendum igitur nunc eſt quod cum primus colinus re-<lb/>uoluat totam primam Rotam, ſpacio temporis .12. horarum, oportet vt eius axis, ſeu <lb/>arbor voluat oſtenſorem, <choice><ex>Zodiacumque</ex><am>Zodiacumq́;</am></choice>, eodem temporis ſpacio, & quia Rota hęc <lb/>inalterabis eſt, propter eius coniunctionem cum ſuo colino, & nos oporteat indicem <lb/><choice><ex>Zodiacumque</ex><am>Zodiacumq́;</am></choice>, quotidie ferè, dirigere, <choice><ex>ſuisque</ex><am>ſuisq́;</am></choice> locis collocare, ideo nos oportet, indi <lb/>cem, Zodiacum, & primam Rotam, ita cum axe, ſeu arbore coniungere, vt poſſimus <lb/>dicta omnia efficere. </s> <s xml:space="preserve">Pars igitur Arboris, ſeu axis dicti, quæ ingredi debet in prima <lb/>Rota, ſit rotunda, & contigua ipſi Rotæ, non autem continua, vel cum Rota conſoli-<lb/>data. </s> <s xml:space="preserve">Pars verò quę per foramen Zodiaci, ſeu Araneę tranſibit, ſit quadrata vſque <lb/>ad <choice><ex>ſummitatem</ex><am>ſummitatẽ</am></choice> ipſius axis (tali ſpiſſitudine, vt in claui ipſius horologij ingredi poſ-<lb/>ſit) & ita foramen ipſius Araneę, quadratum ſit, Oſtenſor autem circa axem, com <lb/>poſitus ſit tali ordine, vt circa paruum circulum volui poſſit, qui paruus circulus ha-<lb/>beat quadratum foramen, per quod tranſeat axis, qui axis aliquantulum emineat <lb/>ſupra <choice><ex>oſtenſorem</ex><am>oſtenſorẽ</am></choice>. </s> <s xml:space="preserve">Sub Aranea vero vel Zodiaco, locata erit Tabula, vt <choice><ex>nunc</ex><am>nũc</am></choice> dicemus, <lb/>ſed ſciendum eſt prius, quod inter Tabulam, & ſuam ſecundam Rotam, aliam lami-<lb/>nam immobilem interpoſitam eſſe oportet, quę circulare foramen habeat, per <choice><ex>quod</ex><am>qđ</am></choice> <lb/>quędam breuis fiſtula tranſeat circundans axem & coniungens <choice><ex>Tabulam</ex><am>Tabulã</am></choice> cum ſua Ro-<lb/>ta, cuius quidem fiſtulæ ſuperficies concaua, rotunda ſit, ſuperficies verò extrinſe-<lb/>ca, nontota, niſi ea pars, quę ſecundam Rotam ingreditur, vt in rotundo foramine <lb/>ipſius Rotę, dicta fiſtula volui poſſit, pars vero extrinſeca quę Tabulam ingredi de-<lb/>bet, ſit quadrata. </s> <s xml:space="preserve">Tabula vero quatuor paruiſſima foramina habeat in extremitati- <pb facs="0437" n="425"/><fw type="head">EPISTOL AE.</fw> bus linearum, meridianę, & verticalis, vt acu mediante volui poſſit, prout oportebit.</s> </p> <p> <s xml:space="preserve">Perfectum igitur cum fuerit op us hoc, te oportet ſcire modum ipſo vtendi. </s> <s xml:space="preserve">Qua-<lb/>propter quotieſcunque volueris, aſpice Solis locum in Zodiaco, Ephemeridibus me <lb/>diantibus, idem dico de vnoquoque reliquorum planetarum. </s> <s xml:space="preserve">Inuento poſtea So-<lb/>lis loco in noſtro Zodiaco horologij, manu mediante, volue oſtenſorem, ita, vt li-<lb/>nea fiduciæ tranſeat per gradum Solis, deinde, claui ipſius horologij mediante, vol-<lb/>ue indicem, ita cum Zodiaco coniunctum, vt linea fiducię, punctum, ſeu partem ho-<lb/>rę oſtendat in limbo horologii, quę quidem hora notanda eſt ſi fuerit ex ijs quę in-<lb/>cipiunt à meridie vſque ad mediam noctem, vel à media nocte vſque ad meri-<lb/>diem, </s> <s xml:space="preserve">tunc acu ſupradicta mediante, poſita in aliquo illorum quatuor foraminum, <lb/>circunuoluenda eſt Tabula, ita, vt extremitas lineę meridianę ſupra orizontem, ex <lb/>ęquo incidat inter duodecimam horam, & lineam fiducię, computum incipiendo à <lb/>duodecima hora, ſi vero dicta indicis hora fuerit ex ijs quę <choice><ex>incipiunt</ex><am>incipiũt</am></choice> à media nocte & <lb/>deſinunt poſtea in meridie, oportebit, acu mediante, circunuoluere Tabulam, quo-<lb/>uſque punctum extremum meridianæ ſub terra, medio loco exiſtat inter <choice><ex>duodecimam</ex><am>duodecimã</am></choice> <lb/>horam, & horam oſtenſam à linea fiducię. </s> <s xml:space="preserve">Quo facto continuo videbis erectam <choice><ex>cae- li</ex><am>cę-li</am></choice> figuram. </s> <s xml:space="preserve">& quia vidiſti loca planetarum in Ephemeridibus, videbis etiam <lb/>eorum loca accidentalia in domibus ſcilicetaccidentalibus, ſi aliquas fixarum in <lb/>Aranea deſiderabis, accipere poteris Ocu. ♉, cor. ♌, ſpi. ♍, Liram, Aquilam, & <lb/>Arcturum, dum locus fuerit capax. </s> <s xml:space="preserve">Nec te moueat, quod oportebit lineam fiducię <lb/>ſupra gra. Solis quotidie collocare, quod nihil refert. </s> <s xml:space="preserve">Nam oportet etiam quoti-<lb/>die cordam fuſo circunuoluere.</s> </p> </div> </div> <div type="section"> <div type="letter"> <head xml:space="preserve">DE DEMONSTRATIONIBVS PROPOSITIONVM <lb/>Mathematicarum, nec non de Aſtrologia Iudiciaria.</head> <head rend="italics" xml:space="preserve">Fu<unclear reason="illegible"/>llustriſſ <seg type="var">.D.</seg> Volfardo Aiſeſtain.</head> <p> <s xml:space="preserve"><hi rend="small caps">NIhil</hi> mihi gratius & iucundius afferri potuit tuis literis, quibus te cupi-<lb/>dum oſtendis ſciendi rationem, quare ego non vna methodo ad omnes <lb/>propoſitiones demonſtrandas vſus ſim, hoc eſt, </s> <s xml:space="preserve">quare non omnia ea Eucl. <lb/>Theoremata citem in vnaquaque propoſitione, quę ad <choice><ex>eam</ex><am>eã</am></choice> demonſtrandam <choice><ex>faciunt</ex><am>faciũt</am></choice>, <lb/>quemadmodum in mea Gnomonica vidiſti me aliquando omiſiſſe. </s> <s xml:space="preserve">Reſpondeo <choice><ex>quod</ex><am>ꝙ</am></choice> <lb/>mathematicę demonſtrationes, hominibus Euclidis Elementa poſſidentibus, non in <lb/>digent aliqua citatione numerorum Theorematum ipſius Euclidis, & ſi aliquando <lb/>vſus ſum aliqua citatione eorundem, hoc feci propter conſuetudinem noſtri tempo <lb/>ris, vel etiam ad faciliorem intelligentiam illorum, quibus ſcribebam. </s> <s xml:space="preserve">Sed omnia <lb/>quamuis minima citare, vt <choice><ex>faciunt</ex><am>faciũt</am></choice> nonnulli, mihi, nimis laborioſum, <choice><ex>ſuperfluumque</ex><am>ſuperfluumq́;</am></choice> <lb/>videtur, preſertim ijs (vt dixi) qui memoria tenent prima Elementa. </s> <s xml:space="preserve">Hęc igitur <lb/>eſt vna ratio. </s> <s xml:space="preserve">Alia, quia multoties, ita coniuncta eſt ſpeculatio cum ipſa concluſio <lb/>ne, vt mihi ſępius viſum ſit ſuperfluum, aliquid de ipſa theoria ſcribere. </s> <s xml:space="preserve">In iis <lb/>enim, quę dum puer eramſcripſi, videbis ſcrupuloſam illam methodum, ſed po-<lb/>ſtea, non niſi in arduis propoſitionibus me nihil eſſentiale prętermittere.</s> </p> <p> <s xml:space="preserve">Circa vero id de quo me interrogas, ſcilicet, vtrum putem omnia vera eſſe, ea <lb/>quę ſcripta reperiuntur in libris Aſtrologæ iudiciarię. </s> <s xml:space="preserve">Reſpondeo quod non, imo <pb facs="0438" n="426"/><fw type="head">IO. PAPT. BENED.</fw> puto plurima falſa eſſe. </s> <s xml:space="preserve">Nam illa multitudo partium, vt pars vitę, pars Hylech, pars <lb/>futurorum, & reliquę omnium domorum cœleſtium, ſalua parte fortunę, ſunt merę <lb/>nugę. </s> <s xml:space="preserve">Idem dico de faciebus, ſiue decanis, de terminis, & de gradibus ipſis, vt pu-<lb/>ta azemenis, puteis, vacuis, fumoſis, & de reliquis. </s> <s xml:space="preserve">De Domibus vero, Exaltationi <lb/>bus, nec non triplicitatibus, experientia <choice><ex>confirmat</ex><am>cõfirmat</am></choice> ea vera eſſe. </s> <s xml:space="preserve"><choice><ex>Idem</ex><am>Idẽ</am></choice> affirmo de Domi <lb/>bus accidentalibus, rationalibus tamen, non <choice><ex>autem</ex><am>autẽ</am></choice> de Domibus Campani, & Gazuli. <lb/></s> <s xml:space="preserve">Obſeruationes etiam complexionum ſeu inſluentiarum ipſorum Planetarum rectè <lb/>factæ ſunt, quę etiam à coloribus ipſorum Planetarum ferè iudicari poſſunt. </s> <s xml:space="preserve">Con-<lb/>iunctiones <choice><ex>aſpectusque</ex><am>aſpectusq́;</am></choice> ipſorum inuicem, ſimiliter mirabilia faciunt, & ex maiori par <lb/>te, ea, quę de iſtis ſcribuntur vera ſunt. </s> <s xml:space="preserve">Reuolutiones annuę ſimiliter, cum Domino <lb/>anni. </s> <s xml:space="preserve">Dominum verò orbis <choice><ex>Diuiſoremque</ex><am>Diuiſoremq́;</am></choice> non approbo, nam hic pendet à termino, <lb/>ille verò ab hora. </s> <s xml:space="preserve">Nouenarias autem Dodecathemoria, Alfridarias, & multa iis ſi <lb/>milia omnia nego. </s> <s xml:space="preserve">Antiſcia, vera ſunt, ideſt influunt, malos tamen effectus, alia <lb/>plus alia verò minus, prout aliqua eorum ſunt tetragona, alia verò trigona, alia ma-<lb/>gna, alia parua, magna ſunt, vt Arietis cum Virgine, & Librę cum Piſcibus, parua ve <lb/>rò, <choice><ex>debiliaque</ex><am>debiliaq́;</am></choice> Geminorum cum Cancro, & Sagittarij <choice><ex>cum</ex><am>cũ</am></choice> Capricorno. </s> <s xml:space="preserve">Sed difuſius <lb/>hęc <choice><ex>omnia</ex><am>oĩa</am></choice> videbis in meo illo particulari tractatu, de quo tibi aliàs dixi, in quo multa <lb/>videbis, quę omnia ab experientia, ex multis à me obſeruatis, comprobata ſunt, <lb/>quem quidem tractatum cum quibuſdam alijs meis ſpeculationibus in lucem prode <lb/>re cupio, ſi fieri poterit, antequam ad directionem mei Horoſcopi cum corpore <lb/>Martis Anęretę perueniam, quę quidem directio circa annum milleſimum quin-<lb/>genteſimum nonageſimum ſecundum eueniet.</s> </p> <fw type="footer">FINIS.</fw> </div> </div> </div> </body> <back> <div type="errata"> <pb facs="0439"/> <head xml:space="preserve">ERRATA CORRIGITO IVXTA INFRASCRIPTAM TABVLAM,</head> <head xml:space="preserve">Reliquos verò errores, qui orthographiam reſpiciunt, benignus lector corrigat, quiſciat multos errores in <lb/>editione irrepſiſſe, quod non paucos dies morbo fuerim detentus dum præſens opus excuderetur.</head> <table rend="italics" rows="72" cols="4"> <row> <cell role="label" xml:space="preserve">Pag.</cell> <cell role="label" xml:space="preserve">Lin.</cell> <cell role="label" xml:space="preserve">Errata</cell> <cell role="label" xml:space="preserve">Correcta</cell> </row> <row> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">29</cell> <cell xml:space="preserve">æqualis</cell> <cell xml:space="preserve">æquali</cell> </row> <row> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">35</cell> <cell xml:space="preserve">maius</cell> <cell xml:space="preserve">maior</cell> </row> <row> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">in vnitate ſuperficialis, erit ac</cell> <cell xml:space="preserve">in vnitate, ſupreficialis erit, ac</cell> </row> <row> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">proueuiens</cell> <cell xml:space="preserve">prouenientem</cell> </row> <row> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">futurum</cell> <cell xml:space="preserve">futurus</cell> </row> <row> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">31</cell> <cell xml:space="preserve">illæ nihil aliud ſunt</cell> <cell xml:space="preserve">illud nihil aliud est</cell> </row> <row> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">38</cell> <cell xml:space="preserve">diuidemus</cell> <cell xml:space="preserve">diuidamus</cell> </row> <row> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">tertiæ, ſint</cell> <cell xml:space="preserve">tertiæ ſint</cell> </row> <row> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">2</cell> <cell xml:space="preserve">producturn</cell> <cell xml:space="preserve">productus</cell> </row> <row> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">30</cell> <cell xml:space="preserve">proueniens</cell> <cell xml:space="preserve">prouenientem</cell> </row> <row> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">30</cell> <cell xml:space="preserve">productum æquale</cell> <cell xml:space="preserve">productus æqualis</cell> </row> <row> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">27</cell> <cell xml:space="preserve">eadem</cell> <cell xml:space="preserve">eædcm</cell> </row> <row> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">eſt</cell> <cell xml:space="preserve">eſſet</cell> </row> <row> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">39</cell> <cell xml:space="preserve">est</cell> <cell xml:space="preserve">eſſet</cell> </row> <row> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">quæ</cell> <cell xml:space="preserve">quæ</cell> </row> <row> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">31</cell> <cell xml:space="preserve">distinguendæ</cell> <cell xml:space="preserve">diſtinguendo</cell> </row> <row> <cell xml:space="preserve">59</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve">ſubſequens</cell> <cell xml:space="preserve">ſubſequentem</cell> </row> <row> <cell xml:space="preserve">61</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">hæc via tenendæ</cell> <cell xml:space="preserve">hanc viamtenere</cell> </row> <row> <cell xml:space="preserve">61</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">fuit</cell> <cell xml:space="preserve">fuerit</cell> </row> <row> <cell xml:space="preserve">61</cell> <cell xml:space="preserve">45</cell> <cell xml:space="preserve">numerum quæſitum</cell> <cell xml:space="preserve">numerus quæſitus</cell> </row> <row> <cell xml:space="preserve">62</cell> <cell xml:space="preserve">30</cell> <cell xml:space="preserve">quantum est <choice><ex>dimidiam</ex><am>dimidiã</am></choice> occupatam</cell> <cell xml:space="preserve">quanta est dimidia occupata</cell> </row> <row> <cell xml:space="preserve">64</cell> <cell xml:space="preserve">4</cell> <cell xml:space="preserve">ſpeculari</cell> <cell xml:space="preserve">conſiderari</cell> </row> <row> <cell xml:space="preserve">64</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve"><choice><ex>totque</ex><am>totq;</am></choice> ſunt termini</cell> <cell xml:space="preserve"><choice><ex>totque</ex><am>totq;</am></choice> eße terminos</cell> </row> <row> <cell xml:space="preserve">64</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">hoc est numerum</cell> <cell xml:space="preserve">hoc est per numerum</cell> </row> <row> <cell xml:space="preserve">64</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">primo quod vnus est</cell> <cell xml:space="preserve">primo qui vnus est</cell> </row> <row> <cell xml:space="preserve">66</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">minimum</cell> <cell xml:space="preserve">minimns</cell> </row> <row> <cell xml:space="preserve">67</cell> <cell xml:space="preserve">13</cell> <cell xml:space="preserve">maximum terminum <choice><ex>addendum</ex><am>addendũ</am></choice></cell> <cell xml:space="preserve">maximus terminus addendus</cell> </row> <row> <cell xml:space="preserve">72</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">numerum</cell> <cell xml:space="preserve">numerus</cell> </row> <row> <cell xml:space="preserve">72</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">ſingulos itinere</cell> <cell xml:space="preserve">ſingulos in itinere</cell> </row> <row> <cell xml:space="preserve">74</cell> <cell xml:space="preserve">7</cell> <cell xml:space="preserve">itinerarium</cell> <cell xml:space="preserve">itinerant ium</cell> </row> <row> <cell xml:space="preserve">78</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">morum</cell> <cell xml:space="preserve">modum</cell> </row> <row> <cell xml:space="preserve">78</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">noueratius</cell> <cell xml:space="preserve">nouenarius</cell> </row> <row> <cell xml:space="preserve">81</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">iuncta</cell> <cell xml:space="preserve">iunctæ</cell> </row> <row> <cell xml:space="preserve">88</cell> <cell xml:space="preserve">35</cell> <cell xml:space="preserve">armonicæ</cell> <cell xml:space="preserve">harmonicæ</cell> </row> <row> <cell xml:space="preserve">91</cell> <cell xml:space="preserve">6</cell> <cell xml:space="preserve">Quare <choice><ex>argumentande</ex><am>argumẽtãde</am></choice> permut ando</cell> <cell xml:space="preserve">Quare permutando</cell> </row> <row> <cell xml:space="preserve">95</cell> <cell xml:space="preserve">27</cell> <cell xml:space="preserve">vitium</cell> <cell xml:space="preserve">vicium</cell> </row> <row> <cell xml:space="preserve">97</cell> <cell xml:space="preserve">14</cell> <cell xml:space="preserve">ſumma</cell> <cell xml:space="preserve">ſummam</cell> </row> <row> <cell xml:space="preserve">97</cell> <cell xml:space="preserve">30</cell> <cell xml:space="preserve">illud verò quod con</cell> <cell xml:space="preserve">illud verò con</cell> </row> <row> <cell xml:space="preserve">98</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">deſideraremus</cell> <cell xml:space="preserve">deſiderauerimus</cell> </row> <row> <cell xml:space="preserve">103</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">diſpoſitis facit</cell> <cell xml:space="preserve">diſpoſitis, tantum facit</cell> </row> <row> <cell xml:space="preserve">104</cell> <cell xml:space="preserve">39</cell> <cell xml:space="preserve">diſpoſitis, facit</cell> <cell xml:space="preserve">diſpoſitis, tantum facit</cell> </row> <row> <cell xml:space="preserve">107</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">eccedit</cell> <cell xml:space="preserve">excedit</cell> </row> <row> <cell xml:space="preserve">110</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">habuerit eius cerebrum</cell> <cell xml:space="preserve">habuerit cerebrum</cell> </row> <row> <cell xml:space="preserve">113</cell> <cell xml:space="preserve">46</cell> <cell xml:space="preserve">ſufficeret</cell> <cell xml:space="preserve">ſuffecißet</cell> </row> <row> <cell xml:space="preserve">116</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">quarum ſecundam</cell> <cell xml:space="preserve">quorum ſecundum</cell> </row> <row> <cell xml:space="preserve">116</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">primæ, tertiam</cell> <cell xml:space="preserve">primi, tertium</cell> </row> <row> <cell xml:space="preserve">116</cell> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">ſecundæ</cell> <cell xml:space="preserve">ſecundi</cell> </row> <row> <cell xml:space="preserve">130</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">ducenda</cell> <cell xml:space="preserve">ducendus</cell> </row> <row> <cell xml:space="preserve">133</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">lineam qua</cell> <cell xml:space="preserve">lineam, qua</cell> </row> <row> <cell xml:space="preserve">133</cell> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">dratam</cell> <cell xml:space="preserve">dratum</cell> </row> <row> <cell xml:space="preserve">134</cell> <cell xml:space="preserve">16</cell> <cell xml:space="preserve">inueniet</cell> <cell xml:space="preserve">inuenimus</cell> </row> <row> <cell xml:space="preserve">137</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">distans</cell> <cell xml:space="preserve">diſtantius</cell> </row> <row> <cell xml:space="preserve">137</cell> <cell xml:space="preserve">45</cell> <cell xml:space="preserve">Notißimum igitur primum</cell> <cell xml:space="preserve">Notißimum primum</cell> </row> <row> <cell xml:space="preserve">139</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">est</cell> <cell xml:space="preserve">ſit</cell> </row> <row> <cell xml:space="preserve">139</cell> <cell xml:space="preserve">28</cell> <cell xml:space="preserve">duas</cell> <cell xml:space="preserve">duos</cell> </row> <row> <cell xml:space="preserve">141</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve">comperuiſſe</cell> <cell xml:space="preserve">comperiſſe</cell> </row> <row> <cell xml:space="preserve">142</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve"><choice><ex>quam</ex><am>quã</am></choice></cell> <cell xml:space="preserve">ac</cell> </row> <row> <cell xml:space="preserve">143</cell> <cell xml:space="preserve">17</cell> <cell xml:space="preserve">linea</cell> <cell xml:space="preserve">lineam</cell> </row> <row> <cell xml:space="preserve">144</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">patebit, ſi quis</cell> <cell xml:space="preserve">patebit, quòd ſi quis</cell> </row> <row> <cell xml:space="preserve">145</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">constante</cell> <cell xml:space="preserve">constantem</cell> </row> <row> <cell xml:space="preserve">146</cell> <cell xml:space="preserve">20</cell> <cell xml:space="preserve">paſtam</cell> <cell xml:space="preserve">maſſam</cell> </row> <row> <cell xml:space="preserve">146</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">pastam</cell> <cell xml:space="preserve">maſſam</cell> </row> <row> <cell xml:space="preserve">148</cell> <cell xml:space="preserve">13</cell> <cell xml:space="preserve"><choice><ex>proponit</ex><am>ꝓponit</am></choice> <choice><ex>non</ex><am>nõ</am></choice> <choice><ex>concludit</ex><am>cõcludit</am></choice> melius <choice><ex>autem</ex><am>aũt</am></choice></cell> <cell xml:space="preserve">proponit, <choice><ex>non</ex><am>nõ</am></choice> concludit, melius <choice><ex>autem</ex><am>aũt</am></choice></cell> </row> <row> <cell xml:space="preserve">14</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">prætergradiatur</cell> <cell xml:space="preserve">prætergredietur</cell> </row> <row> <cell xml:space="preserve">152</cell> <cell xml:space="preserve">6</cell> <cell xml:space="preserve">videtur</cell> <cell xml:space="preserve">videri</cell> </row> <row> <cell xml:space="preserve">153</cell> <cell xml:space="preserve">22 <lb/>23</cell> <cell xml:space="preserve">quia libra</cell> <cell xml:space="preserve">quia cum libræ</cell> </row> <row> <cell xml:space="preserve">153</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">materiales, cum ſuſtineantur</cell> <cell xml:space="preserve">materiales ſuſtineantur</cell> </row> <row> <cell xml:space="preserve">153</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">existente. vnde aliqua</cell> <cell xml:space="preserve">exiſtente, aliqua</cell> </row> <row> <cell xml:space="preserve">154</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">futurum</cell> <cell xml:space="preserve">effecturam</cell> </row> <row> <cell xml:space="preserve">155</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">carta</cell> <cell xml:space="preserve">charta</cell> </row> <row> <cell xml:space="preserve">156</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">ſufficere</cell> <cell xml:space="preserve">ſufficeret</cell> </row> </table> <table rend="italics" rows="73" cols="4"> <row> <cell role="label" xml:space="preserve">Pag.</cell> <cell role="label" xml:space="preserve">Lin.</cell> <cell role="label" xml:space="preserve">Errata</cell> <cell role="label" xml:space="preserve">Correcta</cell> </row> <row> <cell xml:space="preserve">158</cell> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">verſa</cell> <cell xml:space="preserve">verſam</cell> </row> <row> <cell xml:space="preserve">158</cell> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">ſit</cell> <cell xml:space="preserve">ſint</cell> </row> <row> <cell xml:space="preserve">162</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">cindenda</cell> <cell xml:space="preserve">ſcindenda</cell> </row> <row> <cell xml:space="preserve">163</cell> <cell xml:space="preserve">7</cell> <cell xml:space="preserve">oppeſitus</cell> <cell xml:space="preserve">oppoſitum</cell> </row> <row> <cell xml:space="preserve">164</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">tanta</cell> <cell xml:space="preserve">tantæ</cell> </row> <row> <cell xml:space="preserve">164</cell> <cell xml:space="preserve">37</cell> <cell xml:space="preserve">adiunctæ nobis eſſent duæ aliæ</cell> <cell xml:space="preserve">adiuncti nobis eßent duo alii</cell> </row> <row> <cell xml:space="preserve">164</cell> <cell xml:space="preserve">39</cell> <cell xml:space="preserve">ſestineret</cell> <cell xml:space="preserve">ſustineret</cell> </row> <row> <cell xml:space="preserve">164</cell> <cell xml:space="preserve">40</cell> <cell xml:space="preserve">dictæ</cell> <cell xml:space="preserve">dicti</cell> </row> <row> <cell xml:space="preserve">164</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">aliquam</cell> <cell xml:space="preserve">aliquem</cell> </row> <row> <cell xml:space="preserve">165</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">ſuffieiant</cell> <cell xml:space="preserve">ſufficiat</cell> </row> <row> <cell xml:space="preserve">165</cell> <cell xml:space="preserve">4</cell> <cell xml:space="preserve"> ſubſeſquialter<unclear reason="illegible"/> </cell> <cell xml:space="preserve">ſubſeſquialterum</cell> </row> <row> <cell xml:space="preserve">186</cell> <cell xml:space="preserve">32</cell> <cell xml:space="preserve">finit</cell> <cell xml:space="preserve">finitam</cell> </row> <row> <cell xml:space="preserve">187</cell> <cell xml:space="preserve">29</cell> <cell xml:space="preserve">propinqua</cell> <cell xml:space="preserve">propinqui</cell> </row> <row> <cell xml:space="preserve">187<unclear reason="illegible"/> </cell> <cell xml:space="preserve">30</cell> <cell xml:space="preserve">philoſophi ſupra</cell> <cell xml:space="preserve">philoſophi, proximè ſupra</cell> </row> <row> <cell xml:space="preserve">187<unclear reason="illegible"/> </cell> <cell xml:space="preserve">31</cell> <cell xml:space="preserve">contingeret menſe</cell> <cell xml:space="preserve">contingeret vt menſe</cell> </row> <row> <cell xml:space="preserve">187</cell> <cell xml:space="preserve">34</cell> <cell xml:space="preserve">regionis) non conſiderans</cell> <cell xml:space="preserve">regionis) in quo non conſiderauit</cell> </row> <row> <cell xml:space="preserve">190</cell> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">lenas</cell> <cell xml:space="preserve">lenæs</cell> </row> <row> <cell xml:space="preserve">190</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">lenis</cell> <cell xml:space="preserve">leuis</cell> </row> <row> <cell xml:space="preserve">191</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">aliud quadratum</cell> <cell xml:space="preserve">alius quadratus</cell> </row> <row> <cell xml:space="preserve">191</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">qualis</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">193</cell> <cell xml:space="preserve">17</cell> <cell xml:space="preserve">cum</cell> <cell xml:space="preserve">tum</cell> </row> <row> <cell xml:space="preserve">193</cell> <cell xml:space="preserve">36</cell> <cell xml:space="preserve">fit</cell> <cell xml:space="preserve">facit</cell> </row> <row> <cell xml:space="preserve">193</cell> <cell xml:space="preserve">40</cell> <cell xml:space="preserve">pleno vbi</cell> <cell xml:space="preserve">pleno poſuerimus, vbi</cell> </row> <row> <cell xml:space="preserve">195</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">valent</cell> <cell xml:space="preserve">valent apud ipſos</cell> </row> <row> <cell xml:space="preserve">196</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">phyloſophiæ</cell> <cell xml:space="preserve">philoſophiæ</cell> </row> <row> <cell xml:space="preserve">196</cell> <cell xml:space="preserve">20</cell> <cell xml:space="preserve">pellendo</cell> <cell xml:space="preserve">plendo</cell> </row> <row> <cell xml:space="preserve">198</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">ſunt etiam</cell> <cell xml:space="preserve">eße etiam</cell> </row> <row> <cell xml:space="preserve">204</cell> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">priñcipium</cell> <cell xml:space="preserve">principum</cell> </row> <row> <cell xml:space="preserve">204</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">indigna</cell> <cell xml:space="preserve">indignas</cell> </row> <row> <cell xml:space="preserve">204</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">diffundetur, creſcat</cell> <cell xml:space="preserve">diffundatur, & creſcat</cell> </row> <row> <cell xml:space="preserve">205</cell> <cell xml:space="preserve">16</cell> <cell xml:space="preserve">aliquando ſer</cell> <cell xml:space="preserve">aliquando me ſer</cell> </row> <row> <cell xml:space="preserve">205</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">anno præter, <choice><ex>neceßitatem</ex><am>neceßitatẽ</am></choice>, gignitur</cell> <cell xml:space="preserve">anno, præter <choice><ex>neceßitatem</ex><am>neceßitatẽ</am></choice> gignitur</cell> </row> <row> <cell xml:space="preserve">206</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">plenilunium, quod</cell> <cell xml:space="preserve">plenilunium fieri, quod</cell> </row> <row> <cell xml:space="preserve">207</cell> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">inchoet annus</cell> <cell xml:space="preserve">inchoetur annus</cell> </row> <row> <cell xml:space="preserve">209</cell> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">ſoli, quod punctum</cell> <cell xml:space="preserve">ſolis, cuius punctum</cell> </row> <row> <cell xml:space="preserve">210</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">cælebrandi</cell> <cell xml:space="preserve">celebrandis</cell> </row> <row> <cell xml:space="preserve">212</cell> <cell xml:space="preserve">33</cell> <cell xml:space="preserve">Inuentæ</cell> <cell xml:space="preserve">Inuenti</cell> </row> <row> <cell xml:space="preserve">212</cell> <cell xml:space="preserve">33</cell> <cell xml:space="preserve">duæ</cell> <cell xml:space="preserve">duo</cell> </row> <row> <cell xml:space="preserve">214</cell> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">claßi</cell> <cell xml:space="preserve">claßis</cell> </row> <row> <cell xml:space="preserve">214</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">Inter Eximias</cell> <cell xml:space="preserve">Quia inter Eximias</cell> </row> <row> <cell xml:space="preserve">214</cell> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">reperiretur</cell> <cell xml:space="preserve">reperietur</cell> </row> <row> <cell xml:space="preserve">214</cell> <cell xml:space="preserve">34</cell> <cell xml:space="preserve">neceſſario ſit futurum, vt</cell> <cell xml:space="preserve">neceßarium ſit, vt</cell> </row> <row> <cell xml:space="preserve">215</cell> <cell xml:space="preserve">37</cell> <cell xml:space="preserve">quod ſi velimus</cell> <cell xml:space="preserve">ſi velimus</cell> </row> <row> <cell xml:space="preserve">215</cell> <cell xml:space="preserve">39</cell> <cell xml:space="preserve"> poteſt vno <choice><ex>eodemque</ex><am>eodemq;</am></choice> </cell> <cell xml:space="preserve">vno <choice><ex>eodemque</ex><am>eodemq;</am></choice></cell> </row> <row> <cell xml:space="preserve">215</cell> <cell xml:space="preserve">45</cell> <cell xml:space="preserve">ſit circulus</cell> <cell xml:space="preserve">ſit verè circulus</cell> </row> <row> <cell xml:space="preserve">217</cell> <cell xml:space="preserve">2</cell> <cell xml:space="preserve">Quod cum verum</cell> <cell xml:space="preserve">Quod ſi verum</cell> </row> <row> <cell xml:space="preserve">217</cell> <cell xml:space="preserve">4</cell> <cell xml:space="preserve">nulla estratio</cell> <cell xml:space="preserve">nulla eſſet ratio</cell> </row> <row> <cell xml:space="preserve">221</cell> <cell xml:space="preserve">31</cell> <cell xml:space="preserve">inueniemus</cell> <cell xml:space="preserve">deſignabimus</cell> </row> <row> <cell xml:space="preserve">222</cell> <cell xml:space="preserve">20</cell> <cell xml:space="preserve">falli</cell> <cell xml:space="preserve">fallat</cell> </row> <row> <cell xml:space="preserve">225</cell> <cell xml:space="preserve">20</cell> <cell xml:space="preserve">ſi quæ</cell> <cell xml:space="preserve"><choice><ex>ſique</ex><am>ſiq;</am></choice> </cell> </row> <row> <cell xml:space="preserve">225</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">chnum</cell> <cell xml:space="preserve">chnium</cell> </row> <row> <cell xml:space="preserve">226</cell> <cell xml:space="preserve">14</cell> <cell xml:space="preserve">oleum effundebat</cell> <cell xml:space="preserve">effundebatur</cell> </row> <row> <cell xml:space="preserve">227</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve">euadit</cell> <cell xml:space="preserve">euadet</cell> </row> <row> <cell xml:space="preserve">231</cell> <cell xml:space="preserve">36</cell> <cell xml:space="preserve">ſupradicta, minuta</cell> <cell xml:space="preserve">ſupradicta minuta</cell> </row> <row> <cell xml:space="preserve">232</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">progrediuntur</cell> <cell xml:space="preserve">progredi</cell> </row> <row> <cell xml:space="preserve">234</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">aliiori cœlo</cell> <cell xml:space="preserve">altius cælum</cell> </row> <row> <cell xml:space="preserve">239</cell> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">dum</cell> <cell xml:space="preserve">cum</cell> </row> <row> <cell xml:space="preserve">241</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">afferens</cell> <cell xml:space="preserve">cogitans</cell> </row> <row> <cell xml:space="preserve">245</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">diametri</cell> <cell xml:space="preserve">diametros</cell> </row> <row> <cell xml:space="preserve">248</cell> <cell xml:space="preserve">42</cell> <cell xml:space="preserve">ſit & ob id</cell> <cell xml:space="preserve">ſit, ob id</cell> </row> <row> <cell xml:space="preserve">249</cell> <cell xml:space="preserve">42</cell> <cell xml:space="preserve">alium numerum</cell> <cell xml:space="preserve">alio numero</cell> </row> <row> <cell xml:space="preserve">250</cell> <cell xml:space="preserve">13</cell> <cell xml:space="preserve">um voluerimus</cell> <cell xml:space="preserve">cum inuestigare voluerimus</cell> </row> <row> <cell xml:space="preserve">251</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">inuenerimus</cell> <cell xml:space="preserve">inuenimus</cell> </row> <row> <cell xml:space="preserve">252</cell> <cell xml:space="preserve">6</cell> <cell xml:space="preserve">tros</cell> <cell xml:space="preserve">ter</cell> </row> <row> <cell xml:space="preserve">252</cell> <cell xml:space="preserve">6</cell> <cell xml:space="preserve">alia verò diametro</cell> <cell xml:space="preserve">alius verò diameter</cell> </row> <row> <cell xml:space="preserve">252</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">te id non</cell> <cell xml:space="preserve">te non</cell> </row> <row> <cell xml:space="preserve">252</cell> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">duabus</cell> <cell xml:space="preserve">duobus</cell> </row> <row> <cell xml:space="preserve">255</cell> <cell xml:space="preserve">16</cell> <cell xml:space="preserve">in numero</cell> <cell xml:space="preserve">numero</cell> </row> <row> <cell xml:space="preserve">255</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">cum non videat ſi</cell> <cell xml:space="preserve">cum non videat quod ſi</cell> </row> <row> <cell xml:space="preserve">255</cell> <cell xml:space="preserve">36</cell> <cell xml:space="preserve">ſuo centro</cell> <cell xml:space="preserve">ſuis centris</cell> </row> <row> <cell xml:space="preserve">255</cell> <cell xml:space="preserve">39</cell> <cell xml:space="preserve">mille milliaria</cell> <cell xml:space="preserve">mille milliariorum</cell> </row> <row> <cell xml:space="preserve">255</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">t<unclear reason="illegible"/>recent a mille</cell> <cell xml:space="preserve">tercenties mille</cell> </row> </table> <pb facs="0440"/> <table rend="italics" rows="72" cols="4"> <row> <cell role="label" xml:space="preserve">Pag.</cell> <cell role="label" xml:space="preserve">Lin.</cell> <cell role="label" xml:space="preserve">Errata</cell> <cell role="label" xml:space="preserve">Correcta</cell> </row> <row> <cell xml:space="preserve">257</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve">quem</cell> <cell xml:space="preserve">quod</cell> </row> <row> <cell xml:space="preserve">257</cell> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">quem</cell> <cell xml:space="preserve">quod</cell> </row> <row> <cell xml:space="preserve">257</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">quod</cell> <cell xml:space="preserve">nam</cell> </row> <row> <cell xml:space="preserve">258</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">ictum</cell> <cell xml:space="preserve">ictus</cell> </row> <row> <cell xml:space="preserve">258</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">eundem eſſe futurum</cell> <cell xml:space="preserve">idem eßet</cell> </row> <row> <cell xml:space="preserve">258</cell> <cell xml:space="preserve">27</cell> <cell xml:space="preserve">debet, quanto</cell> <cell xml:space="preserve">debet, in huiuſmodi ſitu, quanto</cell> </row> <row> <cell xml:space="preserve">258</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">quod</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">262</cell> <cell xml:space="preserve">17</cell> <cell xml:space="preserve">erunt</cell> <cell xml:space="preserve">ſupponuntur</cell> </row> <row> <cell xml:space="preserve">263</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">a.K. facta ſit</cell> <cell xml:space="preserve">a.K. ſit</cell> </row> <row> <cell xml:space="preserve">263</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">ſemidiameter eße vnius</cell> <cell xml:space="preserve">ſemidiameter vnius</cell> </row> <row> <cell xml:space="preserve">265</cell> <cell xml:space="preserve">4</cell> <cell xml:space="preserve">cognitæ</cell> <cell xml:space="preserve">cogniti</cell> </row> <row> <cell xml:space="preserve">272</cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">infigura vbieſt <seg type="var">.P.</seg></cell> <cell xml:space="preserve">ponatur. R</cell> </row> <row> <cell xml:space="preserve">272</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">E.i. ad <seg type="var">.E.i.</seg></cell> <cell xml:space="preserve">E.I. ad <seg type="var">.E.i.</seg></cell> </row> <row> <cell xml:space="preserve">273</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">licet</cell> <cell xml:space="preserve">liceret</cell> </row> <row> <cell xml:space="preserve">274</cell> <cell xml:space="preserve">7</cell> <cell xml:space="preserve">te</cell> <cell xml:space="preserve">ibi</cell> </row> <row> <cell xml:space="preserve">275</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">dividere</cell> <cell xml:space="preserve">diuidendo</cell> </row> <row> <cell xml:space="preserve">276</cell> <cell xml:space="preserve">47</cell> <cell xml:space="preserve">accipiemus</cell> <cell xml:space="preserve">accipiamus</cell> </row> <row> <cell xml:space="preserve">278</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">Moreta</cell> <cell xml:space="preserve">Moreta</cell> </row> <row> <cell xml:space="preserve">280</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">diminutæ, quartæ</cell> <cell xml:space="preserve">diminutæ ſeu defectiuæ, quarta</cell> </row> <row> <cell xml:space="preserve">281</cell> <cell xml:space="preserve">36</cell> <cell xml:space="preserve">grauißimum</cell> <cell xml:space="preserve">grauißimo</cell> </row> <row> <cell xml:space="preserve">281</cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve" cols="2">in fine pag. vnamquanque Zifram accommoda ſub vnoquoque al-<lb/>phabeti characterem ad priores Zifras prius addendo .1. Idem dico <lb/>in pag .282. & inter, D. et E. pone <seg type="var">.b.</seg></cell> </row> <row> <cell xml:space="preserve">285</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">antea</cell> <cell xml:space="preserve">ante</cell> </row> <row> <cell xml:space="preserve">289</cell> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">quanta pars</cell> <cell xml:space="preserve">quantam partem</cell> </row> <row> <cell xml:space="preserve">289</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">quanta pars</cell> <cell xml:space="preserve">quantam partem</cell> </row> <row> <cell xml:space="preserve">289</cell> <cell xml:space="preserve">42</cell> <cell xml:space="preserve">eſt</cell> <cell xml:space="preserve">erit</cell> </row> <row> <cell xml:space="preserve">290</cell> <cell xml:space="preserve">10</cell> <cell xml:space="preserve">ſatile</cell> <cell xml:space="preserve">ſatilis</cell> </row> <row> <cell xml:space="preserve">290</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">æqualis</cell> <cell xml:space="preserve">æquale</cell> </row> <row> <cell xml:space="preserve">290</cell> <cell xml:space="preserve">43</cell> <cell xml:space="preserve">circunſcriptibilis</cell> <cell xml:space="preserve">circunſcribentis</cell> </row> <row> <cell xml:space="preserve">291</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">od</cell> <cell xml:space="preserve">Quod</cell> </row> <row> <cell xml:space="preserve">294</cell> <cell xml:space="preserve">33</cell> <cell xml:space="preserve">detractis</cell> <cell xml:space="preserve">detracti</cell> </row> <row> <cell xml:space="preserve">294</cell> <cell xml:space="preserve">34</cell> <cell xml:space="preserve"> angulis contingentiæ <choice><ex>ſolidiſque</ex><am>ſolidiſq;</am></choice> </cell> <cell xml:space="preserve">anguli contingentiæ <choice><ex>ſolidique</ex><am>ſolidiq;</am></choice></cell> </row> <row> <cell xml:space="preserve">295</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">chorda</cell> <cell xml:space="preserve">chordam</cell> </row> <row> <cell xml:space="preserve">295</cell> <cell xml:space="preserve">32</cell> <cell xml:space="preserve">lixum</cell> <cell xml:space="preserve">lixus</cell> </row> <row> <cell xml:space="preserve">297</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">diſtincta, procedendo</cell> <cell xml:space="preserve">diſtincta ſint, procedendo</cell> </row> <row> <cell xml:space="preserve">299</cell> <cell xml:space="preserve">32</cell> <cell xml:space="preserve">refleßum</cell> <cell xml:space="preserve">reflexum</cell> </row> <row> <cell xml:space="preserve">299</cell> <cell xml:space="preserve">34</cell> <cell xml:space="preserve">fleſſum</cell> <cell xml:space="preserve">flexum</cell> </row> <row> <cell xml:space="preserve">302</cell> <cell xml:space="preserve">14</cell> <cell xml:space="preserve">eaſdem</cell> <cell xml:space="preserve">eoſdem</cell> </row> <row> <cell xml:space="preserve">304</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">ta</cell> <cell xml:space="preserve">tus</cell> </row> <row> <cell xml:space="preserve">305</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">Idem facere</cell> <cell xml:space="preserve">Idem poßumus facere</cell> </row> <row> <cell xml:space="preserve">312</cell> <cell xml:space="preserve">37</cell> <cell xml:space="preserve">duplæ</cell> <cell xml:space="preserve">duplo</cell> </row> <row> <cell xml:space="preserve">313</cell> <cell xml:space="preserve">6</cell> <cell xml:space="preserve">quotieſcunque</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">315</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">retrogradandum</cell> <cell xml:space="preserve">rotrogradiendum</cell> </row> <row> <cell xml:space="preserve">316</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">vbi</cell> <cell xml:space="preserve">tibi</cell> </row> <row> <cell xml:space="preserve">324</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">ſi conſtitueremus</cell> <cell xml:space="preserve">ſi nos conſtitueremus</cell> </row> <row> <cell xml:space="preserve">325</cell> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">ſimili ad</cell> <cell xml:space="preserve">ſimili. Ad</cell> </row> <row> <cell xml:space="preserve">326</cell> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">longare</cell> <cell xml:space="preserve">ducere</cell> </row> <row> <cell xml:space="preserve">329</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve">ipſi ellipſis</cell> <cell xml:space="preserve">ipſius ellipſis</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">cogitetur</cell> <cell xml:space="preserve">cogitemus</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">17</cell> <cell xml:space="preserve">vnde pro genera</cell> <cell xml:space="preserve">vnde ex genera</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">20</cell> <cell xml:space="preserve">æqualem eſſe longitudini</cell> <cell xml:space="preserve">æqualem longitudini</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">ei</cell> <cell xml:space="preserve">eam</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">21</cell> <cell xml:space="preserve">minor</cell> <cell xml:space="preserve">minorem</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">nor</cell> <cell xml:space="preserve">norem</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">maior</cell> <cell xml:space="preserve">maiorem</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">maior</cell> <cell xml:space="preserve">maiorem</cell> </row> <row> <cell xml:space="preserve">333</cell> <cell xml:space="preserve">37</cell> <cell xml:space="preserve"><choice><ex>communeque</ex><am>communeq;</am></choice> </cell> <cell xml:space="preserve"><choice><ex>communique</ex><am>communiq;</am></choice> </cell> </row> <row> <cell xml:space="preserve">335</cell> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">reflectit</cell> <cell xml:space="preserve">reflectitnr</cell> </row> <row> <cell xml:space="preserve">335</cell> <cell xml:space="preserve">47</cell> <cell xml:space="preserve"> & remotiori </cell> <cell xml:space="preserve">vel remotiori</cell> </row> <row> <cell xml:space="preserve">336</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">ſpeculi ſuperficiem</cell> <cell xml:space="preserve">ſpeculi planam ſuperficiem</cell> </row> <row> <cell xml:space="preserve">336</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">reflectit</cell> <cell xml:space="preserve">reflectitur</cell> </row> <row> <cell xml:space="preserve">337</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">vnam tantummodo imaginem</cell> <cell xml:space="preserve">vna tantummodo imago</cell> </row> <row> <cell xml:space="preserve">337</cell> <cell xml:space="preserve">12</cell> <cell xml:space="preserve">ipſæ</cell> <cell xml:space="preserve">ipſi</cell> </row> <row> <cell xml:space="preserve">337</cell> <cell xml:space="preserve">42</cell> <cell xml:space="preserve">quæ</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">338</cell> <cell xml:space="preserve">14</cell> <cell xml:space="preserve">protracta</cell> <cell xml:space="preserve">protracto</cell> </row> <row> <cell xml:space="preserve">338</cell> <cell xml:space="preserve">26</cell> <cell xml:space="preserve">ratiotinare</cell> <cell xml:space="preserve">apta</cell> </row> <row> <cell xml:space="preserve">340</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">Alius modus</cell> <cell xml:space="preserve">Alium modum</cell> </row> <row> <cell xml:space="preserve">341</cell> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">æquale</cell> <cell xml:space="preserve">æqualem</cell> </row> <row> <cell xml:space="preserve">343</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">videbitur</cell> <cell xml:space="preserve">videri</cell> </row> <row> <cell xml:space="preserve">344</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve">hoc eſt</cell> <cell xml:space="preserve">hoc est quod</cell> </row> <row> <cell xml:space="preserve">344</cell> <cell xml:space="preserve">@1</cell> <cell xml:space="preserve">punctus</cell> <cell xml:space="preserve">punctum</cell> </row> <row> <cell xml:space="preserve">345</cell> <cell xml:space="preserve">6</cell> <cell xml:space="preserve">ei</cell> <cell xml:space="preserve">cum altitudine</cell> </row> </table> <table rend="italics" rows="66" cols="4"> <row> <cell role="label" xml:space="preserve">Pag.</cell> <cell role="label" xml:space="preserve">Lin.</cell> <cell role="label" xml:space="preserve">Errata</cell> <cell role="label" xml:space="preserve">Correcta</cell> </row> <row> <cell xml:space="preserve">345</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">ipſum repertum</cell> <cell xml:space="preserve">ipſum, vt repertum</cell> </row> <row> <cell xml:space="preserve">347</cell> <cell xml:space="preserve">16</cell> <cell xml:space="preserve">reflectit</cell> <cell xml:space="preserve">reflectitur</cell> </row> <row> <cell xml:space="preserve">352</cell> <cell xml:space="preserve">38</cell> <cell xml:space="preserve">opinauerunt</cell> <cell xml:space="preserve">opinati ſune</cell> </row> <row> <cell xml:space="preserve">353</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">parabolem</cell> <cell xml:space="preserve">parabolam</cell> </row> <row> <cell xml:space="preserve">353</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">hæc</cell> <cell xml:space="preserve">hic</cell> </row> <row> <cell xml:space="preserve">353</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">parallelam</cell> <cell xml:space="preserve">parallela</cell> </row> <row> <cell xml:space="preserve">356</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">muri, &</cell> <cell xml:space="preserve">muri, in ori Zontali <seg type="var">.q.p.</seg> ev<unclear reason="illegible"/></cell> </row> <row> <cell xml:space="preserve">356</cell> <cell xml:space="preserve">33</cell> <cell xml:space="preserve">mediante. cognoſeere</cell> <cell xml:space="preserve"><choice><ex>mediante</ex><am>mediãte</am></choice> cuius axis ad ori Zontem <lb/>erectus ſit <seg type="var">.o.n</seg>: cognoſcere<unclear reason="illegible"/></cell> </row> <row> <cell xml:space="preserve">360</cell> <cell xml:space="preserve">13</cell> <cell xml:space="preserve">duplum</cell> <cell xml:space="preserve">duplus</cell> </row> <row> <cell xml:space="preserve">360</cell> <cell xml:space="preserve">28</cell> <cell xml:space="preserve">vnde iſte circulus</cell> <cell xml:space="preserve">iſte circulus</cell> </row> <row> <cell xml:space="preserve">361</cell> <cell xml:space="preserve">36</cell> <cell xml:space="preserve">quæ</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">364</cell> <cell xml:space="preserve">23</cell> <cell xml:space="preserve">diameter</cell> <cell xml:space="preserve">diametrum</cell> </row> <row> <cell xml:space="preserve">367</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">prolung ando</cell> <cell xml:space="preserve">produεendo</cell> </row> <row> <cell xml:space="preserve">369</cell> <cell xml:space="preserve">2</cell> <cell xml:space="preserve">æqualem</cell> <cell xml:space="preserve">æquale</cell> </row> <row> <cell xml:space="preserve">370</cell> <cell xml:space="preserve">4</cell> <cell xml:space="preserve">ſinus</cell> <cell xml:space="preserve">ſinum</cell> </row> <row> <cell xml:space="preserve">370</cell> <cell xml:space="preserve">30</cell> <cell xml:space="preserve">ſumma</cell> <cell xml:space="preserve">ſummam</cell> </row> <row> <cell xml:space="preserve">372</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve">cæterarum</cell> <cell xml:space="preserve">cæteris</cell> </row> <row> <cell xml:space="preserve">372</cell> <cell xml:space="preserve">36</cell> <cell xml:space="preserve">qui qui</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">378</cell> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">hæc</cell> <cell xml:space="preserve">hæ</cell> </row> <row> <cell xml:space="preserve">382</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">prolung atis</cell> <cell xml:space="preserve">productis</cell> </row> <row> <cell xml:space="preserve">383</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">quæ</cell> <cell xml:space="preserve">qui</cell> </row> <row> <cell xml:space="preserve">383</cell> <cell xml:space="preserve">13</cell> <cell xml:space="preserve">ginabis</cell> <cell xml:space="preserve">ginaberis</cell> </row> <row> <cell xml:space="preserve">383</cell> <cell xml:space="preserve">20</cell> <cell xml:space="preserve"><choice><ex>triangulique</ex><am>trianguliq;</am></choice> </cell> <cell xml:space="preserve"><choice><ex>triangulaque</ex><am>triangulaq́;</am></choice></cell> </row> <row> <cell xml:space="preserve">383 <lb/>384</cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">in prima parabola obliqua ex-<lb/>tremitatem inferiorem diametri <lb/>ſignabis charactere <seg type="var">.d.</seg> in ſecunda <lb/>vero charactere <seg type="var">.r.</seg></cell> <cell xml:space="preserve"> </cell> </row> <row> <cell xml:space="preserve">384</cell> <cell xml:space="preserve">29</cell> <cell xml:space="preserve">maior proportio</cell> <cell xml:space="preserve">maiorem proportionem</cell> </row> <row> <cell xml:space="preserve">384</cell> <cell xml:space="preserve">37</cell> <cell xml:space="preserve">proportio halebit</cell> <cell xml:space="preserve">proportionem habebis</cell> </row> <row> <cell xml:space="preserve">385</cell> <cell xml:space="preserve">5</cell> <cell xml:space="preserve"> <seg type="var">.c.b.</seg></cell> <cell xml:space="preserve"> <seg type="var">.e.b.</seg></cell> </row> <row> <cell xml:space="preserve">386</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">b.e.</cell> <cell xml:space="preserve"> <seg type="var">.b.c.</seg></cell> </row> <row> <cell xml:space="preserve">386</cell> <cell xml:space="preserve">29</cell> <cell xml:space="preserve">poſitum</cell> <cell xml:space="preserve">poſitus</cell> </row> <row> <cell xml:space="preserve">386</cell> <cell xml:space="preserve">31</cell> <cell xml:space="preserve"> <seg type="var">.b.c.</seg></cell> <cell xml:space="preserve"> <seg type="var">.b.e.</seg></cell> </row> <row> <cell xml:space="preserve">386</cell> <cell xml:space="preserve">33</cell> <cell xml:space="preserve">& proportion alitate</cell> <cell xml:space="preserve">& ex proportionalitate</cell> </row> <row> <cell xml:space="preserve">386</cell> <cell xml:space="preserve">34</cell> <cell xml:space="preserve">antecedens in</cell> <cell xml:space="preserve">antecedens <seg type="var">.A.B.</seg> in</cell> </row> <row> <cell xml:space="preserve">386</cell> <cell xml:space="preserve">40</cell> <cell xml:space="preserve">quod</cell> <cell xml:space="preserve">quæ</cell> </row> <row> <cell xml:space="preserve">388</cell> <cell xml:space="preserve">2</cell> <cell xml:space="preserve"> <seg type="var">.B.</seg>)</cell> <cell xml:space="preserve">ß)</cell> </row> <row> <cell xml:space="preserve">388</cell> <cell xml:space="preserve">8</cell> <cell xml:space="preserve">b. cum duplo <seg type="var">.b.e.</seg></cell> <cell xml:space="preserve"> <seg type="var">.b.d.</seg> cum duplo <seg type="var">.b.e.</seg></cell> </row> <row> <cell xml:space="preserve">388</cell> <cell xml:space="preserve">45</cell> <cell xml:space="preserve"> <seg type="var">.b.e.</seg></cell> <cell xml:space="preserve">b.c.</cell> </row> <row> <cell xml:space="preserve">388</cell> <cell xml:space="preserve">45</cell> <cell xml:space="preserve"> <seg type="var">.b.e.</seg></cell> <cell xml:space="preserve"> <seg type="var">.b.c.</seg></cell> </row> <row> <cell xml:space="preserve">388</cell> <cell xml:space="preserve">46</cell> <cell xml:space="preserve">ad <seg type="var">.e.b.</seg></cell> <cell xml:space="preserve">ad <seg type="var">.d.b.</seg> cum <seg type="var">.e.b.</seg></cell> </row> <row> <cell xml:space="preserve">390</cell> <cell xml:space="preserve">25</cell> <cell xml:space="preserve">diuiſa</cell> <cell xml:space="preserve">diuiſas</cell> </row> <row> <cell xml:space="preserve">392</cell> <cell xml:space="preserve">3</cell> <cell xml:space="preserve">ſit paraboles <seg type="var">.a.</seg></cell> <cell xml:space="preserve">ſit paraboles <seg type="var">.a.b.c.</seg></cell> </row> <row> <cell xml:space="preserve">392</cell> <cell xml:space="preserve">27</cell> <cell xml:space="preserve">aliqua reliquarum</cell> <cell xml:space="preserve">aliquis reliquorum</cell> </row> <row> <cell xml:space="preserve">393 <lb/>395 <lb/>397</cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">in parabola vbi ſub <seg type="var">.g.</seg><unclear reason="illegible"/> eſt <seg type="var">.i.</seg> pone <seg type="var">.r.</seg> <lb/>& ſupra ſolidum minus dicatur <lb/>cubus minor (leantur <lb/>duo vero lineæ ſine characterib. de</cell> <cell xml:space="preserve"> </cell> </row> <row> <cell xml:space="preserve">394</cell> <cell xml:space="preserve">18</cell> <cell xml:space="preserve">vt <seg type="var">.y.</seg> ad <seg type="var">.m.n.</seg></cell> <cell xml:space="preserve">vt p. ad <seg type="var">.m.n.</seg></cell> </row> <row> <cell xml:space="preserve">394</cell> <cell xml:space="preserve">33</cell> <cell xml:space="preserve">x.o.</cell> <cell xml:space="preserve">n.o.</cell> </row> <row> <cell xml:space="preserve">394</cell> <cell xml:space="preserve">34</cell> <cell xml:space="preserve"> <seg type="var">.i.k.</seg> ad <seg type="var">.i.k.</seg> ad <seg type="var">f.g.</seg></cell> <cell xml:space="preserve">i.k. ad <seg type="var">.f.g.</seg></cell> </row> <row> <cell xml:space="preserve">395</cell> <cell xml:space="preserve">11</cell> <cell xml:space="preserve">quintas</cell> <cell xml:space="preserve">quintæ</cell> </row> <row> <cell xml:space="preserve">396</cell> <cell xml:space="preserve">1</cell> <cell xml:space="preserve"> <seg type="var">.d.b.</seg></cell> <cell xml:space="preserve">g.b.</cell> </row> <row> <cell xml:space="preserve">397</cell> <cell xml:space="preserve"> </cell> <cell xml:space="preserve">Vbirubrica dicit. Defenſio nostra <lb/>contra Antonium Bergam, & <lb/>Alexandrum Piccol.</cell> <cell xml:space="preserve">dicatur. contra Anto. Bergam & <lb/>Alex Piccol atq, defenſio nostra <lb/>contra Excell. August. Michelem</cell> </row> <row> <cell xml:space="preserve">399</cell> <cell xml:space="preserve">32</cell> <cell xml:space="preserve">tione</cell> <cell xml:space="preserve">tionem</cell> </row> <row> <cell xml:space="preserve">400</cell> <cell xml:space="preserve">39</cell> <cell xml:space="preserve">vnum tantum numerum medium</cell> <cell xml:space="preserve">unus tantum numerus medius</cell> </row> <row> <cell xml:space="preserve">402</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">diametrum</cell> <cell xml:space="preserve">ſemidiametrum</cell> </row> <row> <cell xml:space="preserve">403</cell> <cell xml:space="preserve">7</cell> <cell xml:space="preserve">ſuis</cell> <cell xml:space="preserve">ſuæ</cell> </row> <row> <cell xml:space="preserve">403</cell> <cell xml:space="preserve">9</cell> <cell xml:space="preserve">iam ſupræ</cell> <cell xml:space="preserve">eo tractatu</cell> </row> <row> <cell xml:space="preserve">403</cell> <cell xml:space="preserve">17</cell> <cell xml:space="preserve">quidquid</cell> <cell xml:space="preserve">quidquam</cell> </row> <row> <cell xml:space="preserve">404</cell> <cell xml:space="preserve">37</cell> <cell xml:space="preserve">milliaria</cell> <cell xml:space="preserve">milliaribus</cell> </row> <row> <cell xml:space="preserve">406</cell> <cell xml:space="preserve">19</cell> <cell xml:space="preserve">trabuchum quadratum</cell> <cell xml:space="preserve">trabuchus quadratus</cell> </row> <row> <cell xml:space="preserve">410</cell> <cell xml:space="preserve">38</cell> <cell xml:space="preserve">quæ ſors</cell> <cell xml:space="preserve">qua ſorte</cell> </row> <row> <cell xml:space="preserve">410</cell> <cell xml:space="preserve">41</cell> <cell xml:space="preserve">quamlibet</cell> <cell xml:space="preserve">quælibet</cell> </row> <row> <cell xml:space="preserve">411</cell> <cell xml:space="preserve">27</cell> <cell xml:space="preserve">dicens</cell> <cell xml:space="preserve">dicat</cell> </row> <row> <cell xml:space="preserve">411</cell> <cell xml:space="preserve">28</cell> <cell xml:space="preserve">eodem libro</cell> <cell xml:space="preserve">vt eodem libre</cell> </row> <row> <cell xml:space="preserve">412</cell> <cell xml:space="preserve">24</cell> <cell xml:space="preserve">non dependeat?</cell> <cell xml:space="preserve">dependeat?</cell> </row> <row> <cell xml:space="preserve">413</cell> <cell xml:space="preserve">2</cell> <cell xml:space="preserve">vt ſupra</cell> <cell xml:space="preserve">vt alias</cell> </row> <row> <cell xml:space="preserve">413</cell> <cell xml:space="preserve">22</cell> <cell xml:space="preserve">millia vices</cell> <cell xml:space="preserve">millibus viciu<unclear reason="illegible"/>m</cell> </row> <row> <cell xml:space="preserve">415</cell> <cell xml:space="preserve">32</cell> <cell xml:space="preserve">fineſtræ</cell> <cell xml:space="preserve">fenestræ</cell> </row> <row> <cell xml:space="preserve">416</cell> <cell xml:space="preserve">15</cell> <cell xml:space="preserve">operatio</cell> <cell xml:space="preserve">operationem</cell> </row> </table> <fw type="footer">FINIS.</fw> <pb facs="0441"/> </div> <div type="cover"> <note/> <pb facs="0442"/> <pb facs="0443"/> <pb facs="0444"/> <p> <s xml:space="preserve">Test: Dieser Satz enthält ein Zeichen mit Unicode-Codepoint über FFFF, nämlich 𐆑 (U+10191; D800+DD91). </s> <s xml:space="preserve">Das gleiche Zeichen innerhalb eines Wortes: vorher𐆑nachher. </s> <s xml:space="preserve">Das Zeichen wird testweise zu X normalisiert. </s> <s xml:space="preserve">Das Zeichen 𐆒 (U+10192; D800+DD92) wird dagegen nicht normalisiert: vorher𐆒nachher. </s> </p> </div> </back> </text> </TEI>