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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"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> <metadata> <dcterms:identifier>ECHO:05TCTFNR.xml</dcterms:identifier> <dcterms:creator identifier="GND:118872621">Cavalieri, Bonaventura</dcterms:creator> <dcterms:title xml:lang="la">Geometria indivisibilibus continuorum</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1653</dcterms:date> <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> <dcterms:rights>CC-BY-SA</dcterms:rights> <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> <figure> <image file="0001-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0001-01"/> </figure> <pb file="0002" n="2"/> <handwritten/> <handwritten/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">TURNER COLLECTION</head> <figure> <image file="0002-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0002-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head2" xml:space="preserve">THE LIBRARY <lb/>UNIVERSITY OF KEELE</head> <p> <s xml:id="echoid-s1" xml:space="preserve">Presented by <lb/>C. </s> <s xml:id="echoid-s2" xml:space="preserve">W. </s> <s xml:id="echoid-s3" xml:space="preserve">TURNER <lb/>1968</s> </p> <pb file="0003" n="3"/> <handwritten/> <pb file="0004" n="4"/> <pb file="0005" n="5"/> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head3" xml:space="preserve">GEOMETRIA <lb/>INDIVISIBILIBVS <lb/>CONTIN VOR VM</head> <head xml:id="echoid-head4" xml:space="preserve">Noua quadam ratione promota.</head> <head xml:id="echoid-head5" xml:space="preserve">_AVTHORE_</head> <head xml:id="echoid-head6" xml:space="preserve">P. BONAVENTVRA CAVALERIO <lb/>MEDIOLANEN</head> <head xml:id="echoid-head7" xml:space="preserve">_Ordinis S.Hieron. Olim in Almo Bononien. Archigym._ <lb/>_Prim. Mathematicarum Profeſſ._</head> <head xml:id="echoid-head8" xml:space="preserve">In hac poftrema edictione ab erroribus expurgata.</head> <head xml:id="echoid-head9" xml:space="preserve">_Ad Illuſtriſs. D. D._</head> <head xml:id="echoid-head10" xml:space="preserve">MARTIVM VRSINVM <lb/>PENNÆ MARCHIONEM &c.</head> <figure> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0005-01"/> </figure> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head11" xml:space="preserve">BONONIÆ, M. DC. LIII.</head> <pb file="0006" n="6"/> <handwritten/> <handwritten/> <handwritten/> <pb file="0007" n="7"/> <figure> <image file="0007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0007-01"/> </figure> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head12" xml:space="preserve">_ILLVSTRISSIME_</head> <head xml:id="echoid-head13" xml:space="preserve"><emph style="bf">MARCHIO</emph></head> <figure> <image file="0007-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0007-02"/> </figure> <p> <s xml:id="echoid-s4" xml:space="preserve">LItteris vſq; </s> <s xml:id="echoid-s5" xml:space="preserve">naſcentibus, eos co-<lb/>luit ſemper honoribus Antiqui-<lb/>tas, eoſque non morituris obſe-<lb/>quijs Poſteritas venerabitur He-<lb/>roas, quos vel cadentes diſcipli-<lb/>nas tollere, vel iniquis fortunæ <lb/>caſibus oppreſſos erigere fæpius conſpexit; <lb/></s> <s xml:id="echoid-s6" xml:space="preserve">quarè cum ego cmni tempore, quantum in <lb/>me fuit, ſtudiofæ iuuentuti conſulere cupie-<lb/>rim, & </s> <s xml:id="echoid-s7" xml:space="preserve">preſertim generoſę Nobilium propa-<lb/>gini Mathematicas diſciplinas adamanti, & </s> <s xml:id="echoid-s8" xml:space="preserve"><lb/>iamdudum enixè poſtulanti, vt ſecundis typis <lb/>præclariſsima eruditiſsimi Caualerij Geome-<lb/>tria mandaretur, illorum votis reſpondi, quip-<lb/>pe dolentibus tam pretiofum opus, tranſlatis <pb file="0008" n="8"/> aliunde ab huius diſciplinæ cultoribus cunctis <lb/>ferè voluminibus, in Bononienſi ſolo lucem na-<lb/>ctum, tam fubito noctẽ nanciſci, nec poffe de-<lb/>functi Auctoris mandata litteris Geometrica <lb/>ſchemata legere, quem viua voce in Archigy-<lb/>mnafio Bononienſi dictantem primarium Pro-<lb/>feſſorẽ ſilentès audierant; </s> <s xml:id="echoid-s9" xml:space="preserve">hinc eſt, Vir Genero-<lb/>ſiſsime, quod ego præfatum opus iterum prælo <lb/>commiſi, vtfæcundiſsimus parens filios produ-<lb/>ceret, quos formofos, & </s> <s xml:id="echoid-s10" xml:space="preserve">fine mendis ſpectabi-<lb/>les, omni adhibita diligentia, licet aſpicere, ac <lb/>intueri; </s> <s xml:id="echoid-s11" xml:space="preserve">vnde ſolùm deerat, cuihos tenues meos <lb/>labores ſacrarem, fortemque huic Geometrico <lb/>Cælo, ſorſan caſuro, Atlantem inuenirem, cum <lb/>ſtatim tui nominis fidei, inclyte Marti, ac tu-<lb/>telæ committere decreui; </s> <s xml:id="echoid-s12" xml:space="preserve">non enim maiori no-<lb/>mini, nobiliori Nurnini poteram hoc Nobili-<lb/>tatis opuſculum dicare, cum natus ſis ex nobi-<lb/>liſsima Vrſinorum familia inter Romanas fa-<lb/>cilè prima, cuius ſumoſæ Imagines patentia <lb/>vndequaq; </s> <s xml:id="echoid-s13" xml:space="preserve">arctant Palatia, monſtrantq; </s> <s xml:id="echoid-s14" xml:space="preserve">ni-<lb/>gricanti colore ſplendidiſsimam antiquiſsimę <lb/>gentis originem; </s> <s xml:id="echoid-s15" xml:space="preserve">& </s> <s xml:id="echoid-s16" xml:space="preserve">qui honorum gradus, qui <lb/>tituli inueniuntur, quosilla generoſiſsimè non <lb/>fubierit; </s> <s xml:id="echoid-s17" xml:space="preserve">quæ facta clariſsima perleguntur, quę <lb/>ipfa non peregerit; </s> <s xml:id="echoid-s18" xml:space="preserve">ab illa, ob ſeruatos Ciues <pb file="0009" n="9"/> ciuicę funt Coronæ habitæ, ab illa obſeſſum <lb/>ab hoſtibus ſuam libertatem agnouit Capito-<lb/>lium; </s> <s xml:id="echoid-s19" xml:space="preserve">illa prudentiſsimos Patriæ Conſules, <lb/>fapientiſsimos Romæ Senatores, ſpectatiſsi-<lb/>mos Vrbi Præfectos, Vigilantiſsimos Eccle-<lb/>ſiæ Vexilliferos peperit; </s> <s xml:id="echoid-s20" xml:space="preserve">nullus in tota Euro-<lb/>pa iacet angulus, in quo miranda Vrſinæ do-<lb/>mus non conſpiciantur monumẽta; </s> <s xml:id="echoid-s21" xml:space="preserve">illam Italicę <lb/>orę, Hiſpaniarũ Regna, Galliarum Prouincię, <lb/>remotiſsima Britanniæ Littora agnofcunt; </s> <s xml:id="echoid-s22" xml:space="preserve">ab <lb/>illa domo initæ cum primoribus Regũ Coro-<lb/>nis ſanguinis affinitates, potentiſsimaq; </s> <s xml:id="echoid-s23" xml:space="preserve">Germa-<lb/>nia ab illius nobiliſsimę familiæ germine Im-<lb/>perij Romani Electores vidit, quam infignem <lb/>dignitatem ducentis quadraginta annis, & </s> <s xml:id="echoid-s24" xml:space="preserve">vl-<lb/>tra magna cum maieſtate continuauit; </s> <s xml:id="echoid-s25" xml:space="preserve">inſi-<lb/>gnes titulorum honores, nàm alios Comites, <lb/>alios Marchiones perlegetis; </s> <s xml:id="echoid-s26" xml:space="preserve">Barones alios, <lb/>Principes alios, Duces mirabimini; </s> <s xml:id="echoid-s27" xml:space="preserve">conſpicuę <lb/>dignitates, cum Baiuliui alij, S. </s> <s xml:id="echoid-s28" xml:space="preserve">Michaelis alij, <lb/>Rhodi alij eximij Equites fuerint; </s> <s xml:id="echoid-s29" xml:space="preserve">ſpectatiſsi-<lb/>ma curarum munia, cum ab illa familia ſuos <lb/>Cancellarios Sicilia, Priores Aquitania, Cam-<lb/>piductores Reſpublica Veneta, Patriarchas <lb/>Antiochia, Magnos Inſula Melitenſis habue-<lb/>rit Magiſtros; </s> <s xml:id="echoid-s30" xml:space="preserve">quid dicam de numeroſa ſegete <pb file="0010" n="10"/> Epiſcoporum, Archiepiſcoporum, qui non tã <lb/>ſubiectis ſibi Ciuibus dicuntur præfuiſſe, quàm <lb/>fumma animi liberalitate, optimis vitæ mori-<lb/>bus, præfuiſſe; </s> <s xml:id="echoid-s31" xml:space="preserve">quoties verò, & </s> <s xml:id="echoid-s32" xml:space="preserve">quot purpu-<lb/>ratos Principes admirata eſt Roma, cui nil mi-<lb/>rum ſolet accidere, ac vna totius Chriſtiani <lb/>orbis ſumma Capita eſt adorata; </s> <s xml:id="echoid-s33" xml:space="preserve">erubeſcebat <lb/>illa domus eximios Terris Proceres produxiſſe, <lb/>ni etiam Cœlis ſimiles genuiſſet, fulgentibus in-<lb/>ter illos Beatos ſpiritus ſummis martyrio Prin-<lb/>cipibus; </s> <s xml:id="echoid-s34" xml:space="preserve">nec in procreandis fœminis voluit ſte-<lb/>rilis eſſe in Viris ſæcundiſsima, cum enituerit <lb/>miraculis Margherita Virgo, quæ Reginam <lb/>Vrſinam, Vngariæ Regis Coniugem, ſuam <lb/>agnoſcens Parentem, ſui generis nobilitatẽ cum <lb/>animi ſanctitate coniunxit; </s> <s xml:id="echoid-s35" xml:space="preserve">longior eſſem in <lb/>recenſenda tantę ſtirpis nobilitate, ſi hiſtoriam, <lb/>non epiſtolam ſcriberem, & </s> <s xml:id="echoid-s36" xml:space="preserve">ſi illam, me ta-<lb/>cente, ſaxa ipſa, ac monumenta illius celſitu-<lb/>dinem non proclamarent; </s> <s xml:id="echoid-s37" xml:space="preserve">nam Venetijs eque-<lb/>ſtris ſtatua Nicolæ Vrſino ob ſeruatum ab ob-<lb/>ſidione Patauium videtur, teſtantur antiquiſſsi-<lb/>mi lapides à familia Vrſina adæquatum olim <lb/>Capitolium fuiſſe reconditum, Tabularum le-<lb/>ges ſuiſſe ſeruatas, liberatam à Faliſcis Rem-<lb/>publicam, reſectos Pontes, Plebem placatam;</s> <s xml:id="echoid-s38" xml:space="preserve"> <pb file="0011" n="11"/> hic in tuas laudes, Marchio inclyte, libentiſsi-<lb/>mè deſcenderem, ni mihi ſilentium tua impo-<lb/>neret modeſtia, quæ mauult egregia ſacere, <lb/>quàm benèfacta palam audire; </s> <s xml:id="echoid-s39" xml:space="preserve">hoc tamen ſo-<lb/>lum dicam, te his virtutibus, quas in clariſsimis <lb/>tuæ Domus diſperſas Viris intellexiſti, non de-<lb/>generaſſe, teq; </s> <s xml:id="echoid-s40" xml:space="preserve">vnum inſignibus tum animi tum <lb/>Corporis dotibus tot hominibus reſpondere; <lb/></s> <s xml:id="echoid-s41" xml:space="preserve">nam ſi animi celſitudinem ſpectas, quiste ma-<lb/>gnificentior, ſi liberalitatem, quis munificen-<lb/>tior, ſi in rebus peragendis dexteritatem, quis <lb/>te eruditior, tu enim, omitto cæteras artes, ac <lb/>ſcientias, quibus ab ineunte ætate cum maxi-<lb/>mo progreſſus cenſu operam impendiſti; </s> <s xml:id="echoid-s42" xml:space="preserve">ma-<lb/>thematica theoremata calles optimè, tu loco-<lb/>rum diſtantias, diſsitas littorum regiones, Tellu-<lb/>ris, ac Pelagi menſuram apprimè cognoſcis; </s> <s xml:id="echoid-s43" xml:space="preserve"><lb/>non tibi ignoti ſunt ortus, & </s> <s xml:id="echoid-s44" xml:space="preserve">interitus ſyderum, <lb/>agnoſcis qua parte Cœli ſerenitates, qua tem-<lb/>peſtates ſint ſuturę;</s> <s xml:id="echoid-s45" xml:space="preserve">, nec te in immenſo latentes <lb/>Oceano latent arenę;</s> <s xml:id="echoid-s46" xml:space="preserve">; accipe igitur qualecunq; </s> <s xml:id="echoid-s47" xml:space="preserve"><lb/>hoc tuum munus, quod tibi libens, ac lubens <lb/>offero, tuæq; </s> <s xml:id="echoid-s48" xml:space="preserve">auctoritatis clypeo contra mali-<lb/>gniliuoris dentem tutor, ac pater defende, nec <lb/>doni tenuitatem, ſed animum dantis plura da-<lb/>turi, ſi poſſet, conſpice; </s> <s xml:id="echoid-s49" xml:space="preserve">oblatum à Mife gra- <pb file="0012" n="12"/> tum punicum malum; </s> <s xml:id="echoid-s50" xml:space="preserve">ita Deus te diù inco-<lb/>lumem feruet, vt ex Patriæ bono, Domus, ami-<lb/>corum in terris diù poſsis viuere, dũ me tibi per-<lb/>petuum clientem polliceor, tuumque patroci-<lb/>nium hoc perenni in te animi mei monumen-<lb/>to fummoperè expoſco.</s> <s xml:id="echoid-s51" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s52" xml:space="preserve">D. </s> <s xml:id="echoid-s53" xml:space="preserve">T. </s> <s xml:id="echoid-s54" xml:space="preserve">Ill. </s> <s xml:id="echoid-s55" xml:space="preserve"><emph style="sub">me</emph></s> </p> <pb file="0013" n="13"/> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head14" xml:space="preserve">PRÆFATIO</head> <p style="it"> <s xml:id="echoid-s56" xml:space="preserve">_N_Eminem profectò mathamaticarum demon-<lb/>ſtrationum dulceainem, vel primoribus la-<lb/>bris vix attigiſſe puto, qui (non ſccus ac, <lb/>mellis in arbore latentis deguſtata paululũ <lb/>fuauitate, innumera licet ferientibus certa-<lb/>tim aculeis apium caterua deglutientẽ Vr-<lb/>ſum agrè arcere poſſunt) ſummarum, qua illas commitantur dif-<lb/>ficult atum copia crebris velut ictibus obliſtenterepulſus, ad ſatie-<lb/>catem vſq; </s> <s xml:id="echoid-s57" xml:space="preserve">eadem vbiq; </s> <s xml:id="echoid-s58" xml:space="preserve">perfundi totis viribus non contendat. <lb/></s> <s xml:id="echoid-s59" xml:space="preserve">Talia tibi amice Lector, qui melleos hoſce fructus depaſcere con-<lb/>ſueſti, cuiſdam in Geometriarei admiranda caſu in me orta ſpe-<lb/>culationis occaſione, parta, huiuſce dulcedinis amore flagranti, <lb/>libanda propone. </s> <s xml:id="echoid-s60" xml:space="preserve">Cum ergo ſolidorum, quæ ex reuolutione circa <lb/>axim oriuntur, genefim aliquando meditarer, rationemq; </s> <s xml:id="echoid-s61" xml:space="preserve">gignẽ. </s> <s xml:id="echoid-s62" xml:space="preserve"><lb/>tium planarum figurarum cum genitis ſolidis compararem, maxi-<lb/>mè ſanè admirabar quod à propriorum parẽtum conditione adeò <lb/>natæ figuræ degenerarent, vt aliam omninò ab eiſdem rationem <lb/>ſequi viderentur. </s> <s xml:id="echoid-s63" xml:space="preserve">Cylindrus enim exempli gratia, in eadem ba-<lb/>ſi, & </s> <s xml:id="echoid-s64" xml:space="preserve">circa eundem axim, cum cono conſtitutus, eſt eiuſdem <anchor type="note" xlink:href="" symbol="a"/> tri-<lb/> <anchor type="note" xlink:label="note-0013-01a" xlink:href="note-0013-01"/> vlus, cum tamen ex parallelogrammo trianguli dictum conum <lb/> <anchor type="note" xlink:label="note-0013-02a" xlink:href="note-0013-02"/> generantis <anchor type="note" xlink:href="" symbol="b"/> duplo per reuolutionem oriatur. </s> <s xml:id="echoid-s65" xml:space="preserve">Similiter ſi in eadẽ <lb/>baſi, & </s> <s xml:id="echoid-s66" xml:space="preserve">circa eundem axim, bæmiſphærium, vel hamuphæroides, <lb/>necnon conoides parabolicum, atq; </s> <s xml:id="echoid-s67" xml:space="preserve">cylindcus, extiterint, hic erit <lb/>hæmiſphery, vel hæmiſphæroidis ſexquialter, conoidis <anchor type="note" xlink:href="" symbol="d"/> verò du-<lb/> <anchor type="note" xlink:label="note-0013-03a" xlink:href="note-0013-03"/> plus, cum tamen gignens par allelogrammum dictum cylindrum <lb/> <anchor type="note" xlink:label="note-0013-04a" xlink:href="note-0013-04"/> ad inſcriptum gignentem circulum, ſeu ellipſim, proximèrationẽ <lb/> <anchor type="note" xlink:label="note-0013-05a" xlink:href="note-0013-05"/> habeat, quam quatuordecim, ad vndecim ad parabolã verò ſit in <lb/>ratione <anchor type="note" xlink:href="" symbol="f"/> ſexquialteræ. </s> <s xml:id="echoid-s68" xml:space="preserve">Quinimmò & </s> <s xml:id="echoid-s69" xml:space="preserve">in planis figuris per reuolu. <lb/></s> <s xml:id="echoid-s70" xml:space="preserve"> <anchor type="note" xlink:label="note-0013-06a" xlink:href="note-0013-06"/> tionẽ rectarum linearum circa punctum genitis, quales ſunt cir-<lb/>culi, eandem varietatem licet experiri. </s> <s xml:id="echoid-s71" xml:space="preserve">Sicnim plures circuli <lb/>concentrici intelligantur expoſiti radios habentes ex. </s> <s xml:id="echoid-s72" xml:space="preserve">g. </s> <s xml:id="echoid-s73" xml:space="preserve">in pro-<lb/>portione numcrorum ab vnitate deinceps expoſitorum, ipſi circuli <lb/>non eandem radiorum proportionem conſeruabunt, ſedeam, quã <pb file="0014" n="14"/> eorum g quadrata inuicem habebunt. </s> <s xml:id="echoid-s74" xml:space="preserve">His verò perſpectis cum <lb/> <anchor type="note" xlink:label="note-0014-01a" xlink:href="note-0014-01"/> ad planarum, ac ſolidarum figurarum quoq; </s> <s xml:id="echoid-s75" xml:space="preserve">grauitatum centra <lb/>reſpicerem, ſimilemque varietatem nactus eſſem, adhuc auge-<lb/>batur admiratio; </s> <s xml:id="echoid-s76" xml:space="preserve">in cono enim centrum grauitatis <anchor type="note" xlink:href="" symbol="h"/> eſt in axe per <lb/> <anchor type="note" xlink:label="note-0014-02a" xlink:href="note-0014-02"/> quartam partem diſtant à baſi, in triangulo verò ipſum gignen-<lb/> <anchor type="note" xlink:label="note-0014-03a" xlink:href="note-0014-03"/> te eſt in eodem axæ, dictans ab eadem per <anchor type="note" xlink:href="" symbol="i"/> tertiam partem eiuſ-<lb/>dem axis. </s> <s xml:id="echoid-s77" xml:space="preserve">Similiter in conoide parabolico illiud ect in axe per <anchor type="note" xlink:href="" symbol="k"/> <lb/> <anchor type="note" xlink:label="note-0014-04a" xlink:href="note-0014-04"/> tertiam partem diſtans à baſi, in parabola verò ipſum generante <lb/> <anchor type="note" xlink:label="note-0014-05a" xlink:href="note-0014-05"/> per duas tertias eiuſdem axis remouetur ab ipſabaſi. </s> <s xml:id="echoid-s78" xml:space="preserve">cum er-<lb/>go talem varietatem in plurimis alijs ſiguris ſæpius, ac ſæpè fuiſsẽ <lb/>meditatus, vbi prius ex. </s> <s xml:id="echoid-s79" xml:space="preserve">g. </s> <s xml:id="echoid-s80" xml:space="preserve">cylindrum ex indefinitis numero pa-<lb/>rallelogrammis, conum verò in eadem baſi, & </s> <s xml:id="echoid-s81" xml:space="preserve">circa eundẽ axim <lb/>cum cylindro conſtitutum, ex indefinitis numero triangulis per <lb/>axem tranſeuntibus veluti compactum effingens, habita dictorũ <lb/>planorum mutua ratione, illicò & </s> <s xml:id="echoid-s82" xml:space="preserve">ipſorum ſolidor um ab ipſis ge-<lb/>nitorum emergere rationem exiſtimabam, cum iam planè conſta-<lb/>ret planorum rationi genitorum ab ijſdem ſolidorum rationem <lb/>minimè concordare, figurarum menſuram tali ratione inquiren-<lb/>temoleum, & </s> <s xml:id="echoid-s83" xml:space="preserve">operam perdere, ac ex inanibus paleis trituram fa-<lb/>cturum eſſe, mihi iure cenſendum videbatur. </s> <s xml:id="echoid-s84" xml:space="preserve">Verum paulò pro-<lb/>fundius rem contemplatus in hanc tandem deueni ſententiam, <lb/>nempè ad rem noſtram lineas, & </s> <s xml:id="echoid-s85" xml:space="preserve">plana, non ad inuicem coinci-<lb/>dentia, ſed æquidiſtantia aſſumenda eſſe; </s> <s xml:id="echoid-s86" xml:space="preserve">ſic enim in plurimis ra-<lb/>tione inueſtigata reperij tum corporum proportioni ipſorum plano-<lb/>rum, tum planorum proportiom ipſarum linearum proportionem <lb/>(ſi eo modo ſumantur, quo <anchor type="note" xlink:href="" symbol="m"/> lib. </s> <s xml:id="echoid-s87" xml:space="preserve">2. </s> <s xml:id="echoid-s88" xml:space="preserve">explicatur) ad amuſſim in <lb/> <anchor type="note" xlink:label="note-0014-06a" xlink:href="note-0014-06"/> omnibus reſpondere. </s> <s xml:id="echoid-s89" xml:space="preserve">Cylindrumigitur, & </s> <s xml:id="echoid-s90" xml:space="preserve">conum, iam dictos <lb/>non amplius per axem ſed æquidiſt anter baſiceu ſectos cõtempla-<lb/>tus, eandem ſanè rationem habere illa comperij, quæ lib. </s> <s xml:id="echoid-s91" xml:space="preserve">2. </s> <s xml:id="echoid-s92" xml:space="preserve">voco <lb/>omnia <anchor type="note" xlink:href="" symbol="n"/> plana cylindri ad omnia plana coni, <anchor type="note" xlink:href="" symbol="o"/> regula communi <lb/> <anchor type="note" xlink:label="note-0014-07a" xlink:href="note-0014-07"/> baſi (nempè circulorum congeriem, quæ intra cylindrũ, & </s> <s xml:id="echoid-s93" xml:space="preserve">conum, <lb/>veluti vefligia plani à b aſi ad oppoſitam baſim continuò illi æqui-<lb/> <anchor type="note" xlink:label="note-0014-08a" xlink:href="note-0014-08"/> diſtanter fluentis quodammodo relinqui intelliguntur) ei, quam <lb/>babet cylindrus ad conum. </s> <s xml:id="echoid-s94" xml:space="preserve">Optimam ergo methodum figurarum <lb/>ſcrutanda menſuræ indicaui prius line arum pro planis, & </s> <s xml:id="echoid-s95" xml:space="preserve">plano-<lb/>rum pro ſolidis rationes indagare, vt illicò ipſarum figurarum <pb file="0015" n="15"/> menſuram mihi co mpar arem, res, puto, iuxta vota ſucceſſit, vt <lb/>perlegenti patebit. </s> <s xml:id="echoid-s96" xml:space="preserve">Artificio autem tali v ſus ſum, quale ad pro-<lb/>poſitas quæſtiones ab ſoluendas Algebratici adhibere ſolent; </s> <s xml:id="echoid-s97" xml:space="preserve">qui <lb/>quidem numerorum radices, quamuis ineffabiles, ſurdas, ac igno-<lb/>tas, nihilominus ſimul aggregantes, ſubtrahentes, multiplican-<lb/>tes, ac diuidentes, dummodo propoſitę rei exoptatam ſibi notitiã <lb/>enucleare valeant, ſua ſatis obyſſe munera ſibi perſuadent, Non <lb/>aliter ipſe ergo indiuiſibilium ſine linearum, liue planorum con-<lb/>gerie (ijſdem vt in lib. </s> <s xml:id="echoid-s98" xml:space="preserve">2. </s> <s xml:id="echoid-s99" xml:space="preserve">explicatur aſſumptis) licet quoad eo-<lb/>rundem numerum innominabili, ſurda, ac ignota, quoad ma-<lb/>gnitudinem tamen conſpicuis limitibus clauſa, ad continuorum <lb/>inueſtigandam menſuram vſus ſum, vt legenti in proceſſu ope-<lb/>ris apparebit. </s> <s xml:id="echoid-s100" xml:space="preserve">Propoſitum mihi eſt autem ò Geometra in his ſe-<lb/>ptem libris quamplurium tam planarum, quam ſolidarum figu-<lb/>rarum dimenſionem adinuenire, quarum aliquæ etiam ab alijs, <lb/>ac præcipuè ab Euclide, & </s> <s xml:id="echoid-s101" xml:space="preserve">Archimede pertractatę fuerunt, reli-<lb/>qua verò nemini, quod ſciam bucuſq; </s> <s xml:id="echoid-s102" xml:space="preserve">attentatæ; </s> <s xml:id="echoid-s103" xml:space="preserve">vno tamem ex-<lb/>cepto Keplero, quioccaſionę Dolij Auſtriaci per virgam menſo-<lb/>riam dimetiendi, poſtquam in ſua <anchor type="note" xlink:href="" symbol="p"/> Stereometria Archimedea <lb/> <anchor type="note" xlink:label="note-0015-01a" xlink:href="note-0015-01"/> ſummariè ipſius Archimedis adinuenta ſibi opportuna recenſuit, <lb/>nouis aliquando, qualeſcumq; </s> <s xml:id="echoid-s104" xml:space="preserve">ſint, adiectis rationibus, tandem <lb/>cam partem ſuperaddidit, quam Stereometria Archimedeæ ſup-<lb/>plementum nuncupauit, in qua multiplicem Sectionum conica-<lb/>rum, Circuli nempè, Parabolæ, Hyperbolæ, & </s> <s xml:id="echoid-s105" xml:space="preserve">Ellipſis, necnon ea-<lb/>rundem portionum circa diu erſos axes reuolutionem contempla-<lb/>tus, ſolida numero octuaginta ſeptem, præter quinque Archime-<lb/>dea, Sphæram ſcilicet, Conoides parabolicum, Conoides hyperbo-<lb/>licum, Sphæroides oblongum, & </s> <s xml:id="echoid-s106" xml:space="preserve">prolatum Geometris perquam <lb/>eleganti præconio promulgauit. </s> <s xml:id="echoid-s107" xml:space="preserve">Cum ergo iam expoſitam me-<lb/>tiendarum figurarum nouam, ac, ſi dicere fas ſit, valde compen-<lb/>dioſam methodum adinueniſſem, fæliciter mecum actum eſſe exi-<lb/>ſtimaui, vt hæc ſolida, præter illa Archimedea, mihi ſuppedita-<lb/>rentur, circa quæ illius vim ac energiam, experiri liceret. </s> <s xml:id="echoid-s108" xml:space="preserve">Ne <lb/>quis tamen putet me omnium dictorum ſolidorum dimenſionem <lb/>fuiſſe conſequutum, ſicuti neq; </s> <s xml:id="echoid-s109" xml:space="preserve">Keplero contingere potuit, niſi <lb/>paucorum, nec ſatis fęliciter, vt prædictam Stereometriam, ac <pb file="0016" n="16"/> ſupplementum perlegenti conſtare poterit: </s> <s xml:id="echoid-s110" xml:space="preserve">ſatis mihi fuit eorum <lb/>aliqua certiori tamem, ni fallor ratione, inueſtigare, quæ circi-<lb/>ter numero pluſquam viginti ennumerari poterunt, præcipuè ſi <lb/>Archimedea in numero computentur, quinq; </s> <s xml:id="echoid-s111" xml:space="preserve">ſcilicet pro ſingulis <lb/>quatuor Coni ſectionibus, & </s> <s xml:id="echoid-s112" xml:space="preserve">amplius alia quædam inferius recẽ-<lb/>fenda. </s> <s xml:id="echoid-s113" xml:space="preserve">Velenim reuolutio fit circa axem dictarum ſectionum, & </s> <s xml:id="echoid-s114" xml:space="preserve"><lb/>ſic fiunt ſolida Archimedea, ex circulo nempe Sphæra, ex parabo-<lb/>la Conoides parabolicum, ex hyperbola hyperbolicum, & </s> <s xml:id="echoid-s115" xml:space="preserve">ex ellip-<lb/>ſi ſphæroides oblongum, ſeu prloatum. </s> <s xml:id="echoid-s116" xml:space="preserve">Velreuolutio fit circa pa-<lb/>rallelam axi, extra figuram, ſed minimè eandem tangentem con-<lb/> <anchor type="note" xlink:label="note-0016-01a" xlink:href="note-0016-01"/> ctitutam, & </s> <s xml:id="echoid-s117" xml:space="preserve">ſic ex circulo fit <anchor type="note" xlink:href="" symbol="q"/> anulus latus circularis, ex para-<lb/> <anchor type="note" xlink:label="note-0016-02a" xlink:href="note-0016-02"/> bola ſemianulus latus parabolicus, ex hyperbola hyperbolicus <lb/> <anchor type="note" xlink:label="note-0016-03a" xlink:href="note-0016-03"/> (hos Keplerus tanquam montis Aetna cauitatis ſimiles Crate-<lb/>res vocat) & </s> <s xml:id="echoid-s118" xml:space="preserve">ex Ellipſi Anulus latus ellipticus, quemidem Ke-<lb/> <anchor type="note" xlink:label="note-0016-04a" xlink:href="note-0016-04"/> plerus, velut ſerto ruſticarum puellarum ſimilem, Anulum arduũ <lb/> <anchor type="note" xlink:label="note-0016-05a" xlink:href="note-0016-05"/> appellat. </s> <s xml:id="echoid-s119" xml:space="preserve">Velveuolutio fit circa parallelam axi, ac figuram tan-<lb/>gentem, & </s> <s xml:id="echoid-s120" xml:space="preserve">tunc ex circulo fit Anulus ſtrictus circularis, ex pa <lb/> <anchor type="note" xlink:label="note-0016-06a" xlink:href="note-0016-06"/> rabola ſemianulus ſtrictus parabolicus, ex hyperbola hyperbo-<lb/> <anchor type="note" xlink:label="note-0016-07a" xlink:href="note-0016-07"/> licus, & </s> <s xml:id="echoid-s121" xml:space="preserve">tandem ex ellipſi, qui pariter Anulus ſtrictus <anchor type="note" xlink:href="" symbol="z"/> ellipti-<lb/> <anchor type="note" xlink:label="note-0016-08a" xlink:href="note-0016-08"/> cus nuncupatur. </s> <s xml:id="echoid-s122" xml:space="preserve">Denique reuolutione facta circa parallelam <lb/>axi, ſec antemq; </s> <s xml:id="echoid-s123" xml:space="preserve">figuram in duas portiones inæquales, ex circuli <lb/> <anchor type="note" xlink:label="note-0016-09a" xlink:href="note-0016-09"/> portione maiori fit Malum roſeum, ex minori Malum citrium. </s> <s xml:id="echoid-s124" xml:space="preserve">In <lb/> <anchor type="note" xlink:label="note-0016-10a" xlink:href="note-0016-10"/> ellipſi verò ex maiori Malum cotoneum, & </s> <s xml:id="echoid-s125" xml:space="preserve">ex minori fit Oli-<lb/>ua. </s> <s xml:id="echoid-s126" xml:space="preserve">Exparabolæ portione maiori fit <anchor type="note" xlink:href="" symbol="c"/> Accruus maior paraboli-<lb/> <anchor type="note" xlink:label="note-0016-11a" xlink:href="note-0016-11"/> cus, ex minori Aceruus minor: </s> <s xml:id="echoid-s127" xml:space="preserve">Ex hyperbola portione maiori <lb/>fit Aceruus maior <anchor type="note" xlink:href="" symbol="e"/> hyperbolicus, ex minori Aceruus <anchor type="note" xlink:href="" symbol="f"/> minor. <lb/></s> <s xml:id="echoid-s128" xml:space="preserve"> <anchor type="note" xlink:label="note-0016-12a" xlink:href="note-0016-12"/> Hos autem Aceruos minores parabolicos, & </s> <s xml:id="echoid-s129" xml:space="preserve">hyperbolicos, idem <lb/> <anchor type="note" xlink:label="note-0016-13a" xlink:href="note-0016-13"/> Keplerus cornibus rectis ſimiles exiſtimat, quorum alia ſunt acu-<lb/>ta, & </s> <s xml:id="echoid-s130" xml:space="preserve">alia obtuſa, vt in pecudibus, quando primum, inquit, cor-<lb/> <anchor type="note" xlink:label="note-0016-14a" xlink:href="note-0016-14"/> mbus coniſcunt. </s> <s xml:id="echoid-s131" xml:space="preserve">Hæc verò ſunt ſolida numero viginti, quibus <lb/>etiam Anulus ſtrictus ellipticus altera parte latior, & </s> <s xml:id="echoid-s132" xml:space="preserve">Anulus <lb/> <anchor type="note" xlink:label="note-0016-15a" xlink:href="note-0016-15"/> latus ellipticus alteraparte flrictior, addipoſſunt, quem Keple-<lb/>rus Tiara, ſeu Globo Turcico ſimilem putat, necnon ea ſolida, quæ <lb/> <anchor type="note" xlink:label="note-0016-16a" xlink:href="note-0016-16"/> ex ſectionibus oppoſitis ariuntur, ſeu præfaia videntur conco-<lb/> <anchor type="note" xlink:label="note-0016-17a" xlink:href="note-0016-17"/> mitantia. </s> <s xml:id="echoid-s133" xml:space="preserve">Hæc inquam ſunt, quæ ex ennumeratis ab ipſo ex-<lb/>cerpſimus examinanda, à quo præter aliqua nomina nibil aliud à <pb file="0017" n="17"/> nobis deſumptum eſt, vt inſpicienti manifeſtum erit. </s> <s xml:id="echoid-s134" xml:space="preserve">Sciat verò <lb/>lector nos præter dicta ſolida alia pariter quamplurima, quæ non <lb/>ſunt exgrege ſuperius enumeratorum, etiam contemplari. </s> <s xml:id="echoid-s135" xml:space="preserve">Præ <lb/>cæteris autem maximam buiuſce demonſtrandi methodi vniuer-<lb/>ſalitatem non reticebo, quod enim alij de vna, vel ſaltem paucis <lb/>ſolidorum ſpeciebus, nos de infinitis continuò demonctramus, ne-<lb/> <anchor type="note" xlink:label="note-0017-01a" xlink:href="note-0017-01"/> dum enim hic ex. </s> <s xml:id="echoid-s136" xml:space="preserve">g. </s> <s xml:id="echoid-s137" xml:space="preserve">oſtenditur <anchor type="note" xlink:href="" symbol="k"/> cylindrum coni, vel <anchor type="note" xlink:href="" symbol="l"/> priſma <lb/> <anchor type="note" xlink:label="note-0017-02a" xlink:href="note-0017-02"/> pyramidis, in eadem baſi, & </s> <s xml:id="echoid-s138" xml:space="preserve">altitudine cum eo exiſtentis, triplũ <lb/> <anchor type="note" xlink:label="note-0017-03a" xlink:href="note-0017-03"/> eſſe, ſed quacumq, in baſi variatione facta, quæ nullo aſſignato <lb/>numero coarctatur, ſolidum eidem inſiſtens, quod <anchor type="note" xlink:href="" symbol="n"/> cylindricum <lb/> <anchor type="note" xlink:label="note-0017-04a" xlink:href="note-0017-04"/> vocumus, eſſe triplum eius, quod in eadem baſi, & </s> <s xml:id="echoid-s139" xml:space="preserve">altitudine <lb/>cum eo conſtitutum, conicum appellamus; </s> <s xml:id="echoid-s140" xml:space="preserve">quorum quidem ſo-<lb/> <anchor type="note" xlink:label="note-0017-05a" xlink:href="note-0017-05"/> lidorũ ſpecies numero indefinitas eſſe manifeſtò apparet: </s> <s xml:id="echoid-s141" xml:space="preserve">Ex hoc <lb/>autẽ vnico exemplo, tamquam ex vngue Leonem, dignoſcet ſtu-<lb/>dioſus, quanto Geometricus ager per hac fertilior, & </s> <s xml:id="echoid-s142" xml:space="preserve">amplior fiat, <lb/>hanc vniuer ſalitatem namq; </s> <s xml:id="echoid-s143" xml:space="preserve">circa omnia penè ſolida à nobis hic <lb/>conſiderata iugiter proſequemur. </s> <s xml:id="echoid-s144" xml:space="preserve">In primoigitur, & </s> <s xml:id="echoid-s145" xml:space="preserve">ſecundum <lb/>Libro, vt plurimum lemmata proponuntur, quæ ad ſequentium <lb/>librorum doctrinam capiendam neceſſaria videntur, licet in eiſ-<lb/>dem plurima quoque ſint ſuigratia ſimpliciter demonſtrata: </s> <s xml:id="echoid-s146" xml:space="preserve">In <lb/>3.</s> <s xml:id="echoid-s147" xml:space="preserve">4. </s> <s xml:id="echoid-s148" xml:space="preserve">& </s> <s xml:id="echoid-s149" xml:space="preserve">5. </s> <s xml:id="echoid-s150" xml:space="preserve">Libro ſolida e xaminantur, quæ ex conicis ſectionibus <lb/>ſuamgeneſim agnoſcunt. </s> <s xml:id="echoid-s151" xml:space="preserve">In 6. </s> <s xml:id="echoid-s152" xml:space="preserve">agitur de ſpatijs helicis, hac ſo-<lb/>lidis ab eiſdem genitis, problemataq; </s> <s xml:id="echoid-s153" xml:space="preserve">circa predemonſtrata con-<lb/>ſtruuntur. </s> <s xml:id="echoid-s154" xml:space="preserve">In ſeptimo deniq; </s> <s xml:id="echoid-s155" xml:space="preserve">Lib noſtram infinitatis indiuiſibi-<lb/>lium Oceanum emenſamratem, alia inſtituta methodo, in portũ <lb/>deducimus, vt in illius infinitatis ſcopulis periclitandi omnis tã-<lb/>dem tollatur ambiguitas. </s> <s xml:id="echoid-s156" xml:space="preserve">Scio tamen hæc prima fronte leuiter <lb/>perpendentibus, quippe quæ per iamdiù tritam Geometriæ ſemi-<lb/>tam haud fuerint inquiſita, minus eſſe probanda; </s> <s xml:id="echoid-s157" xml:space="preserve">at quinauſeã-<lb/>tis ſtomaci tumentes flatus initio ſupprimentes ad extremam hui-<lb/>us doctrinæ metam peruenire haud dedignabuntur, forte ſuper <lb/>hæc minimè amplius nauſeabunt; </s> <s xml:id="echoid-s158" xml:space="preserve">Ne quis igitur hanc rogo me-<lb/>thodum prius damnare velit, quam hæc omnia puro mentis ocu-<lb/>lo, ſinceroq; </s> <s xml:id="echoid-s159" xml:space="preserve">illius affectu fuerit perluſtratus, hic enim talir atio-<lb/>ne de monſtrata cum aliorum inuentis ad vnguem concordare iu-<lb/>giter animaduertet. </s> <s xml:id="echoid-s160" xml:space="preserve">Nemo autem hæc aggrediatur, qui ſex ſal- <pb file="0018" n="18"/> tem priores Libros, & </s> <s xml:id="echoid-s161" xml:space="preserve">vndecimum Elementorum non calluerit, <lb/>quod ſi in Apollony, & </s> <s xml:id="echoid-s162" xml:space="preserve">Archimedis Operibus Lector pariter ver-<lb/>ſatus fuerit, facilius hæc apprehendet, ſin minus, quædam pau-<lb/> <anchor type="note" xlink:label="note-0018-01a" xlink:href="note-0018-01"/> ca, quæ ab ipſis deſumpta fuere, poterit ſupponere. </s> <s xml:id="echoid-s163" xml:space="preserve">Qui verò <lb/>viderit Com. </s> <s xml:id="echoid-s164" xml:space="preserve">de Motu Martis præfati Kepleri per has noſtras <lb/>fpeculationes planè intelliget, quam facilè in dimenſione plani el-<lb/>lipſis potuerit ipſe hallucinari, dum omnium diſtantiarum Pla-<lb/>netę à Sole, per ellipticam lineam circumuoluti, menſuram pu-<lb/>tat æquipollere plani eilipſis menſuræ (quod ect quoddam ſimile <lb/>errori, in quem initio præſentis ſpeculationis & </s> <s xml:id="echoid-s165" xml:space="preserve">ipſæ lapſus eram, <lb/>putans coincidentia lineas, vel plana, proportionem planorum, <lb/>ſou ſolidorum, eandem conſeruare) licet poſt modum & </s> <s xml:id="echoid-s166" xml:space="preserve">ipſe erro-<lb/>rem proprium detegat, & </s> <s xml:id="echoid-s167" xml:space="preserve">quomodo poſſit illum emendare conten-<lb/>dat. </s> <s xml:id="echoid-s168" xml:space="preserve">His igitur ritè conſideratis, neminem fore exictimo, qui <lb/>hanc nouam methodum duxerit aſpernendam, quin potius eandẽ <lb/>veluit auream clauem, qua ſumma arcis Geometriæ nonnullas <lb/>hucuſq; </s> <s xml:id="echoid-s169" xml:space="preserve">occluſas fores reſerantes, ſummis pulcherrimarum ſpe-<lb/>culationum the ſauris ditiſſimi fieri valeamus, albo adiecto cal-<lb/>culo, poſtmodum fortè ſatius comprobabit</s> </p> <div xml:id="echoid-div7" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0013-01" xlink:href="note-0013-01a" xml:space="preserve">_10. Duod._ <lb/>_Elem._</note> <note symbol="b" position="right" xlink:label="note-0013-02" xlink:href="note-0013-02a" xml:space="preserve">_41. Pri._ <lb/>_Elem._</note> <note symbol="c" position="right" xlink:label="note-0013-03" xlink:href="note-0013-03a" xml:space="preserve">_Coro. I_ <lb/>_34.l 3._</note> <note symbol="d" position="right" xlink:label="note-0013-04" xlink:href="note-0013-04a" xml:space="preserve">_Cor, I._ <lb/>_51.l.4._</note> <note symbol="e" position="right" xlink:label="note-0013-05" xlink:href="note-0013-05a" xml:space="preserve">_A ch._ <lb/>_de Dim._ <lb/>_Cuc._</note> <note symbol="f" position="right" xlink:label="note-0013-06" xlink:href="note-0013-06a" xml:space="preserve">_Piop. I._ <lb/>_l. 4._</note> <note symbol="g" position="left" xlink:label="note-0014-01" xlink:href="note-0014-01a" xml:space="preserve">_Cor. 1._ <lb/>_11.l.3._</note> <note symbol="h" position="left" xlink:label="note-0014-02" xlink:href="note-0014-02a" xml:space="preserve">_Luc. Val_ <lb/>_39 l.1_</note> <note symbol="i" position="left" xlink:label="note-0014-03" xlink:href="note-0014-03a" xml:space="preserve">_Idem 19._ <lb/>_l.1._</note> <note symbol="k" position="left" xlink:label="note-0014-04" xlink:href="note-0014-04a" xml:space="preserve">_Idë 41._ <lb/>_l.2._</note> <note symbol="l" position="left" xlink:label="note-0014-05" xlink:href="note-0014-05a" xml:space="preserve">_Arch. 8._ <lb/>_Se. ęquep_</note> <note symbol="m" position="left" xlink:label="note-0014-06" xlink:href="note-0014-06a" xml:space="preserve">_Def. I._ <lb/>_& 2.l.2._</note> <note symbol="n" position="left" xlink:label="note-0014-07" xlink:href="note-0014-07a" xml:space="preserve">_Def. 1. &_ <lb/>_2.l.2._</note> <note symbol="o" position="left" xlink:label="note-0014-08" xlink:href="note-0014-08a" xml:space="preserve">_Def. 1. &_ <lb/>_2.l.2._</note> <note symbol="p" position="right" xlink:label="note-0015-01" xlink:href="note-0015-01a" xml:space="preserve">_Kepleri_ <lb/>_Storcome_ <lb/>_tria Dolio_ <lb/>_rum._</note> <note symbol="q." position="left" xlink:label="note-0016-01" xlink:href="note-0016-01a" xml:space="preserve">_Cor. 14._ <lb/>_34.1.3._</note> <note symbol="r" position="left" xlink:label="note-0016-02" xlink:href="note-0016-02a" xml:space="preserve">_Cor. 12._ <lb/>_51.l.4._</note> <note symbol="f" position="left" xlink:label="note-0016-03" xlink:href="note-0016-03a" xml:space="preserve">_Cor. 16._ <lb/>_50.l.5._</note> <note symbol="t" position="left" xlink:label="note-0016-04" xlink:href="note-0016-04a" xml:space="preserve">_Cor. 14._ <lb/>_34.l.3._</note> <note symbol="u" position="left" xlink:label="note-0016-05" xlink:href="note-0016-05a" xml:space="preserve">_Cor. 13._ <lb/>_34.l.3._</note> <note symbol="x" position="left" xlink:label="note-0016-06" xlink:href="note-0016-06a" xml:space="preserve">_Cor. 10._ <lb/>_51.l.4._</note> <note symbol="y" position="left" xlink:label="note-0016-07" xlink:href="note-0016-07a" xml:space="preserve">_Cor. 15._ <lb/>_30.l.5._</note> <note symbol="z" position="left" xlink:label="note-0016-08" xlink:href="note-0016-08a" xml:space="preserve">_Cor. 3_ <lb/>_34.l.3._</note> <note symbol="a" position="left" xlink:label="note-0016-09" xlink:href="note-0016-09a" xml:space="preserve">_Cor. 19._ <lb/>_34.l.3._</note> <note symbol="b" position="left" xlink:label="note-0016-10" xlink:href="note-0016-10a" xml:space="preserve">_Cor. 20._ <lb/>_& 22 34._ <lb/>_l.2._</note> <note symbol="c" position="left" xlink:label="note-0016-11" xlink:href="note-0016-11a" xml:space="preserve">_Cor, 21,_ <lb/>_51.l.4._</note> <note symbol="d" position="left" xlink:label="note-0016-12" xlink:href="note-0016-12a" xml:space="preserve">_Cor. 24._ <lb/>_51.l.4._</note> <note symbol="e" position="left" xlink:label="note-0016-13" xlink:href="note-0016-13a" xml:space="preserve">_Parabo-_ <lb/>_licis con-_ <lb/>_formiter._</note> <note symbol="f" position="left" xlink:label="note-0016-14" xlink:href="note-0016-14a" xml:space="preserve">_Paraboli_ <lb/>_cis cõfor-_ <lb/>_miter._</note> <note symbol="g" position="left" xlink:label="note-0016-15" xlink:href="note-0016-15a" xml:space="preserve">_Cor. 28._ <lb/>_34.l.3._</note> <note symbol="h" position="left" xlink:label="note-0016-16" xlink:href="note-0016-16a" xml:space="preserve">_Cor 29._ <lb/>_34.l.3._</note> <note symbol="i" position="left" xlink:label="note-0016-17" xlink:href="note-0016-17a" xml:space="preserve">_Cor. 21._ <lb/>_30.l.5._</note> <note symbol="K" position="right" xlink:label="note-0017-01" xlink:href="note-0017-01a" xml:space="preserve">_10. Duos_ <lb/>_Elem._</note> <note symbol="l" position="right" xlink:label="note-0017-02" xlink:href="note-0017-02a" xml:space="preserve">_Cord. 5._ <lb/>_Duod. Elẽ._</note> <note symbol="m" position="right" xlink:label="note-0017-03" xlink:href="note-0017-03a" xml:space="preserve">_Def. 3._ <lb/>_l.1._</note> <note symbol="n" position="right" xlink:label="note-0017-04" xlink:href="note-0017-04a" xml:space="preserve">_Sect. 9._ <lb/>_Cor. 4. 34._ <lb/>_l.2._</note> <note symbol="o" position="right" xlink:label="note-0017-05" xlink:href="note-0017-05a" xml:space="preserve">_Def. 4.l.1._</note> <note symbol="p" position="left" xlink:label="note-0018-01" xlink:href="note-0018-01a" xml:space="preserve">_Kepleti_ <lb/>_Co de mo_ <lb/>_tu Martis._</note> </div> <pb file="0019" n="19"/> </div> <div xml:id="echoid-div9" type="section" level="1" n="8"> <head xml:id="echoid-head15" xml:space="preserve">In huius Libri Autorem.</head> <p style="it"> <s xml:id="echoid-s170" xml:space="preserve">_E_Xerit cece nouos ſapies C AV ALERIVS auſus <lb/>Archimedæa deficiente manu; <lb/></s> <s xml:id="echoid-s171" xml:space="preserve">Nempè geometricas ex vmbris cruit artes, <lb/>Queis metiare ſolum, queis metiare ſolum. </s> <s xml:id="echoid-s172" xml:space="preserve"><lb/>Egregium mirata VIRI decus, Ars ſtupet, indè <lb/>Sit ait, ergo meas exuperabis opes?</s> <s xml:id="echoid-s173" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s174" xml:space="preserve">Anonymus.</s> <s xml:id="echoid-s175" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div10" type="section" level="1" n="9"> <head xml:id="echoid-head16" xml:space="preserve">In Librum Geometriæ.</head> <p style="it"> <s xml:id="echoid-s176" xml:space="preserve">_D_Vm nou@ peruoluis C AV ALERI ſchemata, deque <lb/>Arte Geometrica prima trophæa refers; <lb/></s> <s xml:id="echoid-s177" xml:space="preserve">Applaudit dignis tibi Felſina laudibus, & </s> <s xml:id="echoid-s178" xml:space="preserve">quam <lb/>Suſpicit ingenio, voce per aſtra vehit.</s> <s xml:id="echoid-s179" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s180" xml:space="preserve">Hinc Archimedis ſileant monumenta, reuixit <lb/>Eſſe Syracusij qui premit acta ſenis.</s> <s xml:id="echoid-s181" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s182" xml:space="preserve">Co. </s> <s xml:id="echoid-s183" xml:space="preserve">Franc. </s> <s xml:id="echoid-s184" xml:space="preserve">Carolus Caprara Coll. </s> <s xml:id="echoid-s185" xml:space="preserve">Nob Alum,</s> </p> </div> <div xml:id="echoid-div11" type="section" level="1" n="10"> <head xml:id="echoid-head17" xml:space="preserve">Ad Libri Auctorem.</head> <p style="it"> <s xml:id="echoid-s186" xml:space="preserve">_V_Era Geometriæ recte documenta recludis, <lb/>Quæ minus antiquis emicuere viris. <lb/></s> <s xml:id="echoid-s187" xml:space="preserve">Sufficis illius noua ſchemata ſcilicet artis, <lb/>Percipis vndè dccus tu quoquè in orbe nouum. </s> <s xml:id="echoid-s188" xml:space="preserve"><lb/>Emenſæ ſpatium terræ dumque exprimis; </s> <s xml:id="echoid-s189" xml:space="preserve">indè <lb/>Arripis immenſi limina ſumma Poli.</s> <s xml:id="echoid-s190" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s191" xml:space="preserve">Petrus Franc. </s> <s xml:id="echoid-s192" xml:space="preserve">Coruinus Coll Nob. </s> <s xml:id="echoid-s193" xml:space="preserve">Alum.</s> <s xml:id="echoid-s194" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div12" type="section" level="1" n="11"> <head xml:id="echoid-head18" xml:space="preserve">Ad Librum Geometriæ.</head> <p style="it"> <s xml:id="echoid-s195" xml:space="preserve">_O_Ptima ſi cupias cognoſcere ſchemata Lector, <lb/>Firma Geometrici percipeiura libri. <lb/></s> <s xml:id="echoid-s196" xml:space="preserve">Acquoris, atquè ſoli diſces ſpatia alma metiri, <lb/>Ingenij miros arripie ſquè modos.</s> <s xml:id="echoid-s197" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s198" xml:space="preserve">Felſina plaudit ouans, tantoquè ſuperba triumpho, <lb/>Gaudia non vnquam deperitura cict.</s> <s xml:id="echoid-s199" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s200" xml:space="preserve">Co. </s> <s xml:id="echoid-s201" xml:space="preserve">Franc. </s> <s xml:id="echoid-s202" xml:space="preserve">Carolus Caprara Coll. </s> <s xml:id="echoid-s203" xml:space="preserve">Nob. </s> <s xml:id="echoid-s204" xml:space="preserve">Alum.</s> <s xml:id="echoid-s205" xml:space="preserve"/> </p> <pb file="0020" n="20"/> </div> <div xml:id="echoid-div13" type="section" level="1" n="12"> <head xml:id="echoid-head19" xml:space="preserve">DeLibro Geometriæ.</head> <p style="it"> <s xml:id="echoid-s206" xml:space="preserve">_E_Xprimit egregiam nobis CAV ALERIVS, artem <lb/>ingenioquè refert abdita ſenſa nouo.</s> <s xml:id="echoid-s207" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s208" xml:space="preserve">Huic veterum penitus cedunt monumenta virorum, <lb/>Vt longè meritis inferiora ſuis.</s> <s xml:id="echoid-s209" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s210" xml:space="preserve">Co. </s> <s xml:id="echoid-s211" xml:space="preserve">Marcus Antonius Herculanus Coll. </s> <s xml:id="echoid-s212" xml:space="preserve">Nob. </s> <s xml:id="echoid-s213" xml:space="preserve">Alum.</s> <s xml:id="echoid-s214" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div14" type="section" level="1" n="13"> <head xml:id="echoid-head20" xml:space="preserve">De Libro Geometriæ.</head> <p style="it"> <s xml:id="echoid-s215" xml:space="preserve">_P_Lena Geometricis ſunt hęc monumenta figuris, <lb/>Quæ BON AVENTV RAE condidit alma manus, <lb/>Ingeny vires, & </s> <s xml:id="echoid-s216" xml:space="preserve">ſuſpice mentis acumen, <lb/>Quod meritò æternum concelebrare licet.</s> <s xml:id="echoid-s217" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s218" xml:space="preserve">Sola latere nequit VIRTVS: </s> <s xml:id="echoid-s219" xml:space="preserve">hæc ſidera tranat, <lb/>Imaquè deſpiciens limina, ſumma petit.</s> <s xml:id="echoid-s220" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s221" xml:space="preserve">Marcus à Cartis Coll. </s> <s xml:id="echoid-s222" xml:space="preserve">Nob. </s> <s xml:id="echoid-s223" xml:space="preserve">Alum.</s> <s xml:id="echoid-s224" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div15" type="section" level="1" n="14"> <head xml:id="echoid-head21" xml:space="preserve">Ad Autorem Libri Geometriæ.</head> <p style="it"> <s xml:id="echoid-s225" xml:space="preserve">_I_Am noua lux ſplendet, iam ſplendor prænitet omnis, <lb/>Arte Geometrica, dum noua iura refers.</s> <s xml:id="echoid-s226" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s227" xml:space="preserve">Lux fuit Architas, lux Archimedis opuſque, <lb/>Lux ea ſed tenebris conſociata fuit; <lb/></s> <s xml:id="echoid-s228" xml:space="preserve">Lux tua pellucet nulla caligine preſſa, <lb/>Inctar Apollinei ſideris inſtar adeſt.</s> <s xml:id="echoid-s229" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s230" xml:space="preserve">Petrus Franc. </s> <s xml:id="echoid-s231" xml:space="preserve">Coruinus Coll. </s> <s xml:id="echoid-s232" xml:space="preserve">Nob. </s> <s xml:id="echoid-s233" xml:space="preserve">Alum.</s> <s xml:id="echoid-s234" xml:space="preserve"/> </p> <pb o="1" file="0021" n="21" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div16" type="section" level="1" n="15"> <head xml:id="echoid-head22" xml:space="preserve">CAVALERII <lb/>LIBER PRIMVS.</head> <head xml:id="echoid-head23" style="it" xml:space="preserve">In quo præcipuè de ſectionibus Cylindricorum, & <lb/>Conicorum, nec non ſimilibus figuris quædam <lb/>element aria præmittuntur; ac aliquæ Pro-<lb/>poſitiones lemmaticæ pro ſequen-<lb/>tibus Libris oſtenduntur.</head> <head xml:id="echoid-head24" xml:space="preserve">DIFINITIONES.</head> <head xml:id="echoid-head25" xml:space="preserve">A. I.</head> <note position="right" xml:space="preserve">A</note> <p> <s xml:id="echoid-s235" xml:space="preserve">CVM duæ rectæ lineæ inuicem paralle-<lb/>læ aliquam tetigerint figuram pla-<lb/>nam cum illis in eodem plano con-<lb/>ſtitutam, vnumquodq;</s> <s xml:id="echoid-s236" xml:space="preserve">punctum con-<lb/>tactus illius vertex dicatur, & </s> <s xml:id="echoid-s237" xml:space="preserve">oppo-<lb/>ſiti vertices puncta contactuum <lb/>vtriuſque dictarum tangentium pa-<lb/>rallelarum ſimul comparata; </s> <s xml:id="echoid-s238" xml:space="preserve">quilibei <lb/>autem vertices ſemper intelligentur aſſumpti reſpectu cu-<lb/>iuſcunque rectæ lineæ dictis tangentibus æquidiſtantis, <lb/>quæ infra regula appellatur.</s> <s xml:id="echoid-s239" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div17" type="section" level="1" n="16"> <head xml:id="echoid-head26" xml:space="preserve">B.</head> <note position="right" xml:space="preserve">B</note> <p> <s xml:id="echoid-s240" xml:space="preserve">LIneæ tangentes dicantur, oppoſitæ tangentes eiuſdem <lb/>figuræ reſpectu cuiuſcumque rectæ lineę eiſdem tan-<lb/>gentibus æquidiſtanter ductæ.</s> <s xml:id="echoid-s241" xml:space="preserve"/> </p> <pb o="2" file="0022" n="22" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div18" type="section" level="1" n="17"> <head xml:id="echoid-head27" xml:space="preserve">C.</head> <note position="left" xml:space="preserve">C</note> <p> <s xml:id="echoid-s242" xml:space="preserve">CVm earum vnius contactus fuetit in linea, tunc linea <lb/>contactus vocabitur baſis eiuſdem figuræ, reſpectu <lb/>cuius poterunt dici vertices puncta contactuum alterius <lb/>tangentis: </s> <s xml:id="echoid-s243" xml:space="preserve">vel ſi iſtius contactus pariter ſit in linea, ambæ <lb/>lineæ contactus, oppoſitæ baſes, ſumptæ reſpectu <lb/>cuiuſcumq; </s> <s xml:id="echoid-s244" xml:space="preserve">lineæ, cuiſint æquidiſtantes.</s> <s xml:id="echoid-s245" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div19" type="section" level="1" n="18"> <head xml:id="echoid-head28" xml:space="preserve">A. II.</head> <note position="left" xml:space="preserve">A</note> <p> <s xml:id="echoid-s246" xml:space="preserve">CVm plana inuicem parallela tetigerint aliquod ſoli-<lb/>dum, vnumquodq; </s> <s xml:id="echoid-s247" xml:space="preserve">punctum contactus illius vertex <lb/>dicatur; </s> <s xml:id="echoid-s248" xml:space="preserve">& </s> <s xml:id="echoid-s249" xml:space="preserve">oppoſiti vertices puncta contactuum vtriuſque <lb/>dictorum tangentium planorum ſimul comparata: </s> <s xml:id="echoid-s250" xml:space="preserve">quilibet <lb/>autem vertices ſemper intelligantur aſſumpti reſpectu cu-<lb/>inſcumq. </s> <s xml:id="echoid-s251" xml:space="preserve">plani dictis tangentibus æquidiſtantis, quod in-<lb/>fra regula pariter appellatur.</s> <s xml:id="echoid-s252" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div20" type="section" level="1" n="19"> <head xml:id="echoid-head29" xml:space="preserve">B.</head> <note position="left" xml:space="preserve">B</note> <p> <s xml:id="echoid-s253" xml:space="preserve">IPſa tengentia plana dicantur, oppoſita tangentia plana <lb/>eiuſdem ſolidi, reſpectu dicti plani tangentibus æqui-<lb/>diſtantis aſſumpta.</s> <s xml:id="echoid-s254" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div21" type="section" level="1" n="20"> <head xml:id="echoid-head30" xml:space="preserve">C.</head> <note position="left" xml:space="preserve">C</note> <p> <s xml:id="echoid-s255" xml:space="preserve">CVm dictorum tangentium contactus fuerit in plano, <lb/>tunc vtriuſuis tangentium planorum plana conta-<lb/>ctus baſes dicantur, cuius reſpectu puncta contactus reli-<lb/>quitangentis plani poterunt vertices appellari, & </s> <s xml:id="echoid-s256" xml:space="preserve">vtriuſq; <lb/></s> <s xml:id="echoid-s257" xml:space="preserve">tangentium planorum contactus plana dicentur, oppoſitæ <lb/>baſes:</s> <s xml:id="echoid-s258" xml:space="preserve">cum verò vtriuſque contactus fuerit in linea, oppoſi-<lb/>tæ baſes lineares ipſæ lineæ contactus vocabuntur.</s> <s xml:id="echoid-s259" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div22" type="section" level="1" n="21"> <head xml:id="echoid-head31" xml:space="preserve">D.</head> <note position="left" xml:space="preserve">D</note> <p> <s xml:id="echoid-s260" xml:space="preserve">CVm figuræ planæ oppoſitis tangentibus vtcumq. </s> <s xml:id="echoid-s261" xml:space="preserve">du-<lb/>ctis, & </s> <s xml:id="echoid-s262" xml:space="preserve">ſolidę oppoſitis planis tangentibus, inciderit <lb/>perpendiculariter recta linea in eadem tangentia termina-<lb/>ta, dicetur hæc altitudo propoſitæ figuræ planæ, vel ſolidę, <lb/>reſpectu dictorum tangentium, vel cuiuſcumque eidem <lb/>æquidiſtantis, aſſumpta.</s> <s xml:id="echoid-s263" xml:space="preserve"/> </p> <pb o="3" file="0023" n="23" rhead="LIBERI."/> </div> <div xml:id="echoid-div23" type="section" level="1" n="22"> <head xml:id="echoid-head32" xml:space="preserve">E.</head> <note position="right" xml:space="preserve">E</note> <p> <s xml:id="echoid-s264" xml:space="preserve">REgula appellabitur in planis recta linea, cui quædam <lb/>lineæ ducuntur æquidiſtantes, & </s> <s xml:id="echoid-s265" xml:space="preserve">in ſolidis, planum, <lb/>cui quædam plana ducuntur æquidiſtantia, qualis in ſu-<lb/>perioribus eſt recta linea, vel planum, cuius reſpectu fu-<lb/>muntur vertices, vel oppoſita tangentia, cui vel vtraq; </s> <s xml:id="echoid-s266" xml:space="preserve">vel <lb/>alterum tangentium æquidiſtat.</s> <s xml:id="echoid-s267" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div24" type="section" level="1" n="23"> <head xml:id="echoid-head33" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s268" xml:space="preserve">_H_Aec minimè diſcrepant ab bis, quæ in Euclide, Archimede, <lb/>& </s> <s xml:id="echoid-s269" xml:space="preserve">Apollonio, circa vertices, baſes, altitudines, & </s> <s xml:id="echoid-s270" xml:space="preserve">tangen-<lb/>tia, ſiuelineas, ſine plana, aſſamuntur; </s> <s xml:id="echoid-s271" xml:space="preserve">cum, licet vniuerſalius, idem, <lb/>quod ipſi, declarent, vt ijs, qui in ſupra dictorum auctorum opert-<lb/>bus verſati ſunt innoteſcet facilè, vnde ſine ſcrupulo aſſumemus <lb/>aliquando ex dictis auctoribus, quæ ex conſimilibus difinitionibus <lb/>pendent, illis commiſcentes, prout opus fuerit, quæ ex bis dedu-<lb/>cuntur.</s> <s xml:id="echoid-s272" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div25" type="section" level="1" n="24"> <head xml:id="echoid-head34" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s273" xml:space="preserve">EXpoſita quacumque figura plana, & </s> <s xml:id="echoid-s274" xml:space="preserve">in eiuſdem ambitu <lb/>ſumpto vt cumque puncto, ab eoque ad alteram eiuf-<lb/>dem partium ducta quadam recta linea terminata, & </s> <s xml:id="echoid-s275" xml:space="preserve">ſuper <lb/>planum propoſitæ figuræ eleuata, ſihæc per ambitum talis <lb/>figuræ ſemper æquidiſtanter cuidam rectæ lineæ moueri <lb/>intelligatur, donec omnem percurrerit ambitum, alterum <lb/>eiuſdem extremum punctum, quod non fertur per ambi-<lb/>tum propoſitæ figuræ, deſcribet circuitum planæ figuræ <lb/>ipſi propoſitæ æquidiſtantis, vt probabitur. </s> <s xml:id="echoid-s276" xml:space="preserve">Solidum er-<lb/> <anchor type="note" xlink:label="note-0023-02a" xlink:href="note-0023-02"/> go, quod compræhenditur vtriſq. </s> <s xml:id="echoid-s277" xml:space="preserve">figuris iam dictis, & </s> <s xml:id="echoid-s278" xml:space="preserve">ſu-<lb/>perficie linea quæ reuoluitur, deſcripta, dicetur: </s> <s xml:id="echoid-s279" xml:space="preserve">Cylin-<lb/>dricus; </s> <s xml:id="echoid-s280" xml:space="preserve">ſuperficies in reuolutione deſcripta, nec non quod <lb/>libet illius fruſtum, ſuperficies cylindracea. </s> <s xml:id="echoid-s281" xml:space="preserve">Cylindrici <lb/>oppofitæ baſes dictæ figuræ planæ interſe æquidiſtantes; <lb/></s> <s xml:id="echoid-s282" xml:space="preserve">latus autem cylindrici, quæuis recta in ſuperficie cylindra-<lb/>cea oppoſitas baſes pertingens, cui congruit in reuolutio- <pb o="4" file="0024" n="24" rhead="GEOMETRIÆ"/> ne ipſa linea reuoluta; </s> <s xml:id="echoid-s283" xml:space="preserve">& </s> <s xml:id="echoid-s284" xml:space="preserve">tandem regula lateris cylindrici <lb/>dicetur illa, cui reuoluta ſemper manet æquidiſtans.</s> <s xml:id="echoid-s285" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="1"> <note position="right" xlink:label="note-0023-02" xlink:href="note-0023-02a" xml:space="preserve">6.huius.</note> </div> </div> <div xml:id="echoid-div27" type="section" level="1" n="25"> <head xml:id="echoid-head35" xml:space="preserve">A. IV.</head> <note position="left" xml:space="preserve">A</note> <p> <s xml:id="echoid-s286" xml:space="preserve">EXpoſita plana quacumq, figura; </s> <s xml:id="echoid-s287" xml:space="preserve">extra cuius planum ad <lb/>vtramuis eiuſdem partium quodcũque fit aſlumptum <lb/>punctum, ſi ab eo ad quoduis punctum illius ambitus recta <lb/>linea ducatur, quæ indefinitè quoq; </s> <s xml:id="echoid-s288" xml:space="preserve">ſit producta, & </s> <s xml:id="echoid-s289" xml:space="preserve">hęc per <lb/>eiuſdem ambitum moueatur donec ipſum totum percur-<lb/>rerit ambitum; </s> <s xml:id="echoid-s290" xml:space="preserve">ſumptum punctum erit vertex ſolidi, quod <lb/>compræhenditur ſuperficie deſcripta à linea, quæ reuolui-<lb/>tur inter ambitum propoſitæ figurę, & </s> <s xml:id="echoid-s291" xml:space="preserve">ſumptum punctum <lb/>clauſa, vertex, inquam ſumptus reſpectu propoſitę figuræ, vt <lb/>probabitur. </s> <s xml:id="echoid-s292" xml:space="preserve">Tale ſolidum autem dicatur; </s> <s xml:id="echoid-s293" xml:space="preserve">Conicus, cuius <lb/> <anchor type="note" xlink:label="note-0024-02a" xlink:href="note-0024-02"/> baſis propoſita figura, & </s> <s xml:id="echoid-s294" xml:space="preserve">ver tex dictum punctum; </s> <s xml:id="echoid-s295" xml:space="preserve">ſuperfi-<lb/>cies deſcripta linea, quę reuoluitur, & </s> <s xml:id="echoid-s296" xml:space="preserve">iacet inter ambitum <lb/>propoſitę figuræ, & </s> <s xml:id="echoid-s297" xml:space="preserve">dictum punctum, & </s> <s xml:id="echoid-s298" xml:space="preserve">quodlibet illius <lb/>fruſtum dicatur; </s> <s xml:id="echoid-s299" xml:space="preserve">ſuperficies. </s> <s xml:id="echoid-s300" xml:space="preserve">Conicularis; </s> <s xml:id="echoid-s301" xml:space="preserve">illæ verò rectæ <lb/>lineæ, quæ in eadem reperiuntur, quibus congruit reuolu-<lb/>tainter verticem, & </s> <s xml:id="echoid-s302" xml:space="preserve">ambitum baſis concluſa, vocentur, la-<lb/>tera eiuſdem Conici.</s> <s xml:id="echoid-s303" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="1"> <note position="left" xlink:label="note-0024-02" xlink:href="note-0024-02a" xml:space="preserve">IS.huius.</note> </div> </div> <div xml:id="echoid-div29" type="section" level="1" n="26"> <head xml:id="echoid-head36" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s304" xml:space="preserve">_E_X hac, & </s> <s xml:id="echoid-s305" xml:space="preserve">autecedentidefinitione, petet cylindrum eſſe cylindri-<lb/>cum, & </s> <s xml:id="echoid-s306" xml:space="preserve">conum eſſe conicum, eos ſcilicet, qui ab Apollonio, <lb/>& </s> <s xml:id="echoid-s307" xml:space="preserve">Sereno definiuntur.</s> <s xml:id="echoid-s308" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div30" type="section" level="1" n="27"> <head xml:id="echoid-head37" xml:space="preserve">B.</head> <note position="left" xml:space="preserve">B</note> <p> <s xml:id="echoid-s309" xml:space="preserve">CYlindricirecti dicentur, cum eorum latera fuerint ad <lb/>rectos angulos baſibus, ſcaleni verò, cum non fue-<lb/>rint ad rectos angulos eiſdem: </s> <s xml:id="echoid-s310" xml:space="preserve">Conicorum verò, & </s> <s xml:id="echoid-s311" xml:space="preserve">cylin-<lb/>dricorum fruſta vocabuntur, quę per plana baſibus pa-<lb/>rallela (pro conſcis verſus ipſas baſes) ab ijſdem abſcin-<lb/>duatur.</s> <s xml:id="echoid-s312" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div31" type="section" level="1" n="28"> <head xml:id="echoid-head38" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s313" xml:space="preserve">AXis, diameter, figuræ planę, vel ſolidę, ordinatim ap-<lb/>plicatę adeaſdem, lineæ, iuxta quas poſſunt, &</s> <s xml:id="echoid-s314" xml:space="preserve">c.</s> <s xml:id="echoid-s315" xml:space="preserve"> <pb o="5" file="0025" n="25" rhead="LIBERI."/> nomina fectionum conicorum latera recta, ſeu tranſuerſa, <lb/>ſumantur, prout ab Apollonio definiuntur, hoctantum ani-<lb/>maduerſo, me in ſequentibus aliquando abuti eiſdem no-<lb/>minibus ſectionum coni, Parabolæ .</s> <s xml:id="echoid-s316" xml:space="preserve">ſ. </s> <s xml:id="echoid-s317" xml:space="preserve">Hyperbolæ, Ellipſis, <lb/>& </s> <s xml:id="echoid-s318" xml:space="preserve">oppoſitarum ſectionum, ſpatia videlicet intelligens ſub <lb/>illis, & </s> <s xml:id="echoid-s319" xml:space="preserve">earum baſibus, compręhenſa, quod ex modo lo-<lb/>quendi tunc euidenter cognoſcitur. </s> <s xml:id="echoid-s320" xml:space="preserve">Cætera deniq Apol-<lb/>lonij, & </s> <s xml:id="echoid-s321" xml:space="preserve">quæ ab Archimede circa Sphęroides, & </s> <s xml:id="echoid-s322" xml:space="preserve">Conoides, <lb/>definiuntur, niſi alia afferatur à me definitio, ſumantur, <lb/>prout ab ipſis vſurpantur.</s> <s xml:id="echoid-s323" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div32" type="section" level="1" n="29"> <head xml:id="echoid-head39" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s324" xml:space="preserve">FIguram planam circa diametrum, vocat Apollonius, <lb/>Conicorum, cum in ea ductis quotuis lineis cuidam <lb/>æquidiſtantibus, omnes bifariam à quadam recta linea di-<lb/>uiduntur, quam vocat diametrum, ſieas oblique ſecet, & </s> <s xml:id="echoid-s325" xml:space="preserve"><lb/>axem, ſi eas rectè diuidat, & </s> <s xml:id="echoid-s326" xml:space="preserve">ipſam figuram circa diame-<lb/>trum, vel axem.</s> <s xml:id="echoid-s327" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s328" xml:space="preserve">Siergo figura circa axem, reuoluatur circa eundem do-<lb/>nec redeat, vnde diſceſſit, deſcripta in tali reuolutione ab <lb/>eadem ſolida figura dicatur: </s> <s xml:id="echoid-s329" xml:space="preserve">ſolidum rotundum, eiuſdem <lb/>verò axis, circa quem fit reuolutio.</s> <s xml:id="echoid-s330" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div33" type="section" level="1" n="30"> <head xml:id="echoid-head40" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s331" xml:space="preserve">SImiles Cylindrici, & </s> <s xml:id="echoid-s332" xml:space="preserve">Conicidicantur, quorum baſes <lb/>ſunt ſimiles (iuxta definitionem 10. </s> <s xml:id="echoid-s333" xml:space="preserve">ſimilium figura-<lb/>rum infra poſitam, ſubint ellige, veliuxta aliorum defini-<lb/>tiones, quas cum prędictam concordare infra oſtendemus) <lb/>in quibus ſumptis duabus homologis lineis, vel lateribus <lb/>vtcumque, & </s> <s xml:id="echoid-s334" xml:space="preserve">per ipſas, & </s> <s xml:id="echoid-s335" xml:space="preserve">latera extenſis planis ipſa ad ean-<lb/>dem partem ęquè ad baſes inclinantur, horumq. </s> <s xml:id="echoid-s336" xml:space="preserve">conceptę <lb/>in eiſdem figurę ſunt ſimiles, nempè ſimilia parallelogram-<lb/>ma in cylindricis, & </s> <s xml:id="echoid-s337" xml:space="preserve">ſimilia triangula in conicis, quorum ho-<lb/>mologa latera ſint ſumptę in baſibus homologę.</s> <s xml:id="echoid-s338" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div34" type="section" level="1" n="31"> <head xml:id="echoid-head41" xml:space="preserve">VIII.</head> <p> <s xml:id="echoid-s339" xml:space="preserve">SImiles ſphęroides dieentur, quę ex ſimilium ellipſium <lb/>reuolutione oriuntur.</s> <s xml:id="echoid-s340" xml:space="preserve"/> </p> <pb o="6" file="0026" n="26" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div35" type="section" level="1" n="32"> <head xml:id="echoid-head42" xml:space="preserve">IX.</head> <p> <s xml:id="echoid-s341" xml:space="preserve">SImiles portiones ſpherarum, vel ſpęroidum, & </s> <s xml:id="echoid-s342" xml:space="preserve">ſimiles <lb/>Conoides, ſiue Conoidum portiones appellabimus, <lb/>quando per axes ductis planis ad rectos angulos baſibus <lb/>conceptę in eiſdem ſolidis figurę ſimiles erunt (iuxta de-<lb/>finit. </s> <s xml:id="echoid-s343" xml:space="preserve">10. </s> <s xml:id="echoid-s344" xml:space="preserve">ſubſequentem, vel etiam iuxta aliorum definitio-<lb/>nes de ſimilibus figuris planis allatas, ſubintellige) qua-<lb/>rum, & </s> <s xml:id="echoid-s345" xml:space="preserve">baſium communes ſectiones ſint homologe baſium <lb/>diametri, quę vel circuliſint, vel ſimiles ellipſes.</s> <s xml:id="echoid-s346" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div36" type="section" level="1" n="33"> <head xml:id="echoid-head43" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s347" xml:space="preserve">_C_Aetræ d finitiones ab Euclide ſimilium planarum figurarum, <lb/>& </s> <s xml:id="echoid-s348" xml:space="preserve">ſolidarum, & </s> <s xml:id="echoid-s349" xml:space="preserve">ſimilium Cylindrorum, & </s> <s xml:id="echoid-s350" xml:space="preserve">Conorum, & </s> <s xml:id="echoid-s351" xml:space="preserve">quæ <lb/>ab Apollonio lib.</s> <s xml:id="echoid-s352" xml:space="preserve">6. </s> <s xml:id="echoid-s353" xml:space="preserve">Conicorum, referente Eutocio, fiunt ſimilium ſe-<lb/>ctionum Coniportionum, ſumantur, vt abipſis afferuntur, adtuncto <lb/>tamen definitioni ſimilium ſectionum Coni portionum ibidem ab Apol-<lb/>lonio allatæ, ſi pro ſpatijs vſurpetur quam infr a dicetur.</s> <s xml:id="echoid-s354" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div37" type="section" level="1" n="34"> <head xml:id="echoid-head44" xml:space="preserve">A. X.</head> <note position="left" xml:space="preserve">A</note> <p> <s xml:id="echoid-s355" xml:space="preserve">SImiles figurę planę in vniuerſum vocentur, in quarum <lb/>ſingulis oppoſitę tangentes ita duci poſlunt, & </s> <s xml:id="echoid-s356" xml:space="preserve">in eaſ-<lb/>dem tangentes ita incidere ad eundem angulum, ex eadem <lb/>parte rectę lineæ in illis terminatę, vt, ſi intra dictas op-<lb/>poſitas tangentes eiſdem æquidiſtantes vtcumq; </s> <s xml:id="echoid-s357" xml:space="preserve">ductę fue <lb/>rint rectę lineæ, eas, quę incidunt dictis tangentibus, ſimi-<lb/>liter ad eandem partem ſecantes; </s> <s xml:id="echoid-s358" xml:space="preserve">reperiamus harum paral-<lb/>lelarum, nec non & </s> <s xml:id="echoid-s359" xml:space="preserve">oppoſitarum tangentium eas portiones, <lb/>quę inter dictas incidentes, & </s> <s xml:id="echoid-s360" xml:space="preserve">circuitus figurarum ad ean-<lb/>dem partem ſitę funt, eodem ordine ſumptas, eandem inter <lb/>ſe rationem habere, quam rectæ lineæ, quę dictis tangenti-<lb/>bus inciderunt, & </s> <s xml:id="echoid-s361" xml:space="preserve">in eaſdem terminantur.</s> <s xml:id="echoid-s362" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div38" type="section" level="1" n="35"> <head xml:id="echoid-head45" xml:space="preserve">B.</head> <note position="left" xml:space="preserve">B</note> <p> <s xml:id="echoid-s363" xml:space="preserve">IPſę autem quę dictis tangentibus incidunt, & </s> <s xml:id="echoid-s364" xml:space="preserve">in easter-<lb/>minantur, dicentur; </s> <s xml:id="echoid-s365" xml:space="preserve">Incidentes dictarum tangentium <lb/>oppoſitarum, & </s> <s xml:id="echoid-s366" xml:space="preserve">figurarum.</s> <s xml:id="echoid-s367" xml:space="preserve"/> </p> <pb o="7" file="0027" n="27" rhead="LIBERI."/> </div> <div xml:id="echoid-div39" type="section" level="1" n="36"> <head xml:id="echoid-head46" xml:space="preserve">C.</head> <note position="right" xml:space="preserve">C</note> <p> <s xml:id="echoid-s368" xml:space="preserve">QVę verò dictis tangentibus oppoſitis ęquidiſtant, & </s> <s xml:id="echoid-s369" xml:space="preserve"><lb/>diuidunt productę, ſi opus ſit, ſimiliter ad eandem <lb/>partem ipſas incidentes, necnon oppoſitarum <lb/>tangentium portiones, quę in ſimilibus figuris iam dictis <lb/>reperiuntur, vocentur; </s> <s xml:id="echoid-s370" xml:space="preserve">homologæ earumdem, ſumptę re-<lb/>gula qualibet earum; </s> <s xml:id="echoid-s371" xml:space="preserve">dicantur autem lineæ homologę, quę <lb/>funt intra ambitum ſimilium figurarum, quę verò in ambi-<lb/>tu, latera homologa. </s> <s xml:id="echoid-s372" xml:space="preserve">Ipſę verò tangentes etiam, tangentes <lb/>earumdem homologarum.</s> <s xml:id="echoid-s373" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div40" type="section" level="1" n="37"> <head xml:id="echoid-head47" xml:space="preserve">D. IV.</head> <note position="right" xml:space="preserve">D</note> <p> <s xml:id="echoid-s374" xml:space="preserve">CVm verò duę ſimiles figuræ planæ in eodem plano, vel <lb/>in planis ęquidiſtantibus ita poſitę fuerint, vt earum, <lb/>& </s> <s xml:id="echoid-s375" xml:space="preserve">oppoſitarum tangentium, quę ſunt regulę homologarum <lb/>earumdem incidentes vel ſint ſuperpoſitę, vel ſibi inuicem <lb/>ęquidiſtent, homologis earumdem figurarum, & </s> <s xml:id="echoid-s376" xml:space="preserve">homolo-<lb/>gis partibus ipſarum incidentium, ad eandem partem con-<lb/>ſtitutis, ipſæ figurę ſimiles dicantur etiam, inter ſe ſimiliter <lb/>poſitę; </s> <s xml:id="echoid-s377" xml:space="preserve">ſiue à ſuis lineis, vel lateribus homologis ſimiliter <lb/>deſcriptæ.</s> <s xml:id="echoid-s378" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div41" type="section" level="1" n="38"> <head xml:id="echoid-head48" xml:space="preserve">E.</head> <note position="right" xml:space="preserve">E</note> <p> <s xml:id="echoid-s379" xml:space="preserve">SIverò fuerint quotcumq; </s> <s xml:id="echoid-s380" xml:space="preserve">& </s> <s xml:id="echoid-s381" xml:space="preserve">qualeſcumq; </s> <s xml:id="echoid-s382" xml:space="preserve">figurę planę in <lb/>eodem plano vtcumq; </s> <s xml:id="echoid-s383" xml:space="preserve">diſpoſitę; </s> <s xml:id="echoid-s384" xml:space="preserve">fuerint autem alię tot <lb/>numero figurę in quouis plano, cum prędictis ita ſe haben-<lb/>tes, vt binę ſint ſimiles, & </s> <s xml:id="echoid-s385" xml:space="preserve">earum omnium lineę homologę <lb/>duabus quibuſdam ſint æquidiſtantes: </s> <s xml:id="echoid-s386" xml:space="preserve">ductis verò oppoſi-<lb/>tis tangentibus ſingularum ſimilium figurarum, quę ſint <lb/>parallelę illis duabus, quibus homologę earumdem ęqui-<lb/>diſtant, & </s> <s xml:id="echoid-s387" xml:space="preserve">repertis incidentibus duarum ex dictis ſimilibus <lb/>figuris, & </s> <s xml:id="echoid-s388" xml:space="preserve">earum tangentium, illę productę fuerint vſq; </s> <s xml:id="echoid-s389" xml:space="preserve">ad <lb/>extremas tangentes, reperiamus autem eaſdem à tangenti-<lb/>bus ſimilium figurarum ſimiliter ad eandem partem diuidi, <lb/>quarum portiones inter oppoſitas tangentes ſimilium figu-<lb/>rarum iacentes ſint earundem oppoſitarum tangentium, &</s> <s xml:id="echoid-s390" xml:space="preserve"> <pb o="8" file="0028" n="28" rhead="GEOMETRIÆ"/> ſimilium figurarum incidentes. </s> <s xml:id="echoid-s391" xml:space="preserve">Tales figuræ dicentur bi-<lb/>næ ſimiles, & </s> <s xml:id="echoid-s392" xml:space="preserve">ſimiliter inter ſe poſitę primò dictæ, ac ſecun-<lb/>dò dictæ, & </s> <s xml:id="echoid-s393" xml:space="preserve">earum, ac exremarum tangentium etiam dicen-<lb/>tur incidentes, quæ in tangentium extremas terminan-<lb/>tur.</s> <s xml:id="echoid-s394" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div42" type="section" level="1" n="39"> <head xml:id="echoid-head49" xml:space="preserve">APPENDIX PRIOR <lb/>Pro explicatione Definit. 10. antecedentis.</head> <p style="it"> <s xml:id="echoid-s395" xml:space="preserve">_S_Int duæ figuræ planæ. </s> <s xml:id="echoid-s396" xml:space="preserve">ABCD, KLγP, in quibus ſupponantur <lb/>ductæ oppoſitæ tangentes, AE, CG, in figura, ABCD, & </s> <s xml:id="echoid-s397" xml:space="preserve">KQ, <lb/> <anchor type="note" xlink:label="note-0028-01a" xlink:href="note-0028-01"/> γ℟, in fig. </s> <s xml:id="echoid-s398" xml:space="preserve">KLγP, quibus incidant duæ rectæ lineæ, EG, Q℟, ad <lb/>eundem angulum ex eadem parte, ſiue ſecent figuras, ſiue non, du-<lb/>ctis autem vtcumq dictis tangentibus parallelis, BF, L&</s> <s xml:id="echoid-s399" xml:space="preserve">, quæ <lb/>in puctis F, &</s> <s xml:id="echoid-s400" xml:space="preserve">, diuidant ſimiliter ad eandem partem ipſas, EC, <lb/>Q℟, & </s> <s xml:id="echoid-s401" xml:space="preserve">circuitus figu-<lb/> <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> rarum in punctis, B, I, <lb/>S, D, L, T, X, P. </s> <s xml:id="echoid-s402" xml:space="preserve">repe-<lb/>riamus, DF, ad, P&</s> <s xml:id="echoid-s403" xml:space="preserve">, <lb/>eſſe vt, EC, ad, Q℟, <lb/>& </s> <s xml:id="echoid-s404" xml:space="preserve">ita eſſe, SF, ad, X&</s> <s xml:id="echoid-s405" xml:space="preserve">, <lb/>IF, ad, T&</s> <s xml:id="echoid-s406" xml:space="preserve">, BF, ad <lb/>L&</s> <s xml:id="echoid-s407" xml:space="preserve">, ita nempè, vt, quæ <lb/>ſunt ad eandem partem <lb/>ipſarum, EG, Q℟, eo-<lb/>dem ordine ſumptæ, ſint, <lb/>vt ipſæ, EG, Q℟, ſic <lb/>etiam tangentes, AE, KQ, CG, γ℟, ſint vt, FQG, ℟, & </s> <s xml:id="echoid-s408" xml:space="preserve">ſic cæ-<lb/>teræ conſimiliter ſumptæ, tunc voco figuras, ABCD, KLγP ſimi-<lb/> <anchor type="note" xlink:label="note-0028-02a" xlink:href="note-0028-02"/> les, & </s> <s xml:id="echoid-s409" xml:space="preserve">ipſas, EG, Q℟, incidentes ſimiles figurarum, ABCD, KLγP, <lb/> <anchor type="note" xlink:label="note-0028-03a" xlink:href="note-0028-03"/> & </s> <s xml:id="echoid-s410" xml:space="preserve">oppoſitarum tangentium, AE, CG, KQ, γ℟; </s> <s xml:id="echoid-s411" xml:space="preserve">ipſas, BI, SD, <lb/>LT, XP, quæ clanduntur perimetris figurarum, & </s> <s xml:id="echoid-s412" xml:space="preserve">diuidunt pro-<lb/>ductæ, ſiopus ſit, ipſas, EG, Q℟. </s> <s xml:id="echoid-s413" xml:space="preserve">ſimiliter ad eandem partem, <lb/>voco, homologas earumdem figurarum, quarum dictæ oppoſitæ <lb/> <anchor type="note" xlink:label="note-0028-04a" xlink:href="note-0028-04"/> tangentes dicuntur tangentes, ſiue regulæ.</s> <s xml:id="echoid-s414" xml:space="preserve"/> </p> <div xml:id="echoid-div42" type="float" level="2" n="1"> <note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">_B.Def.1._</note> <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a"> <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0028-01"/> </figure> <note position="left" xlink:label="note-0028-02" xlink:href="note-0028-02a" xml:space="preserve">_A Def.10._</note> <note position="left" xlink:label="note-0028-03" xlink:href="note-0028-03a" xml:space="preserve">_B.Def.10._</note> <note position="left" xlink:label="note-0028-04" xlink:href="note-0028-04a" xml:space="preserve">_C.Def.10_</note> </div> <p> <s xml:id="echoid-s415" xml:space="preserve">Cum verò figuræ, ABCD, KLγP, fuerint in eodem plano, vel <pb o="9" file="0029" n="29" rhead="LIBERI."/> in planis æquidiſt antibus, ita conſtitutæ, vtipſæincidentes, EG, <lb/>Q℟, ſint vel ſuperpoſitæ adinuicem, vel parallelæ, & </s> <s xml:id="echoid-s416" xml:space="preserve">homolo-<lb/>gæ, BI, SD, LT, XP, ad eandem partem ipſarum, EG, Q℟, <lb/>& </s> <s xml:id="echoid-s417" xml:space="preserve">partes homologæ incidentium (per dictas homologas, produ-<lb/>ctas, ſi opusſit, ſimiliter ad eandem partem dtuiſarum) fuerint <lb/>pariter adeandem partem conſtitutæ, tunc voco figuras, ABC <lb/> <anchor type="note" xlink:label="note-0029-01a" xlink:href="note-0029-01"/> D, KLγP, nedum ſimiles, ſedetiam ſimiliter poſitas.</s> <s xml:id="echoid-s418" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="2"> <note position="right" xlink:label="note-0029-01" xlink:href="note-0029-01a" xml:space="preserve">_D.Def.10._</note> </div> <p> <s xml:id="echoid-s419" xml:space="preserve">Sint nunc quetcumque figuræ planæ in eodem plano vtcumque <lb/>diſpoſitæ, ABCD, ΟRΩV, & </s> <s xml:id="echoid-s420" xml:space="preserve">aliæ tot numero in quouis plano, <lb/>KLγP, Ζ9βΣ, quæbinæ ſint ſimiles, ſcilicet, ABCD, ipſi, <lb/>KLγP, &</s> <s xml:id="echoid-s421" xml:space="preserve">, ΟRΩV, tpſi, Ζ9βΣ, quarum omnium homologæ <lb/> <anchor type="note" xlink:label="note-0029-02a" xlink:href="note-0029-02"/> duabus quibuſdam reperiantur æquidiſtantes, ſint autem reſpe-<lb/>ctu ipſarum, quibus dictæ homologæ æquidistant, ductæ in figu-<lb/>ris, ABCD, KLγP, oppoſitæ tangentes, AE, CG, KQ, γ <lb/> <anchor type="note" xlink:label="note-0029-03a" xlink:href="note-0029-03"/> ℟, & </s> <s xml:id="echoid-s422" xml:space="preserve">in figuris, ΟRΩV, Ζ9βΣ, oppoſitæ tangentes, OH, Ω <lb/>M, ΖΓ, βΛ, quæ tangentes eruntregulæ homologarum ſimilium <lb/>figurarum iam dictarum; </s> <s xml:id="echoid-s423" xml:space="preserve">Sint deinde incidentes duarum ex di-<lb/> <anchor type="note" xlink:label="note-0029-04a" xlink:href="note-0029-04"/> ctis ſimilibus figuris vtcumq; </s> <s xml:id="echoid-s424" xml:space="preserve">vt ipſarum, ABCD, KLγP, & </s> <s xml:id="echoid-s425" xml:space="preserve"><lb/>oppoſitarum tangentium, AE, CG, ipſæ, EG, Q℟, quæ pro-<lb/>ducantur vſque ad extremas tangentes, SM, βΛ, quibus inci-<lb/>dant in punctis, M, Λ, reperiamus autem integras, EM, QΛ, <lb/>ſimiliter ad eandem partem ſecaritum à tangentibus, CG, γ℟, <lb/>tum ab, OH, ΖΓ, & </s> <s xml:id="echoid-s426" xml:space="preserve">inſuper portiones, HM, ΓΛ, eſſe etiam <lb/>incidentes oppoſitarum tangentium, OH, ΩΜ, ΖΓ, βΛ, & </s> <s xml:id="echoid-s427" xml:space="preserve"><lb/>ſimilium figurarum, ΟRΩV, Ζ9β;</s> <s xml:id="echoid-s428" xml:space="preserve">Σ, velutiipſæ, EG, Q℟, <lb/>ſunt incidentes oppoſitarum tangentium, AE, CG, KQ, γR, <lb/>& </s> <s xml:id="echoid-s429" xml:space="preserve">ſimilium figurarum, ABCD, KLγP. </s> <s xml:id="echoid-s430" xml:space="preserve">Tunc igitur has fi-<lb/>guras voco binas ſimiles, & </s> <s xml:id="echoid-s431" xml:space="preserve">vnas, ſcilicet ipſas, ABCD, OR <lb/>ΩV, ſimiliter, ac alias inter ſe diſpoſitas, ideſt vtipſæ, KLγP, <lb/>Ζ9βΣ, & </s> <s xml:id="echoid-s432" xml:space="preserve">earum, ac extremarum tangentium, AE, ΩΜ, K <lb/>Q, βΛ, ipſas, EM, QΛ, voco etiam incidentes.</s> <s xml:id="echoid-s433" xml:space="preserve"/> </p> <div xml:id="echoid-div44" type="float" level="2" n="3"> <note position="right" xlink:label="note-0029-02" xlink:href="note-0029-02a" xml:space="preserve">_C.Def.10._</note> <note position="right" xlink:label="note-0029-03" xlink:href="note-0029-03a" xml:space="preserve">_B. Def. 10._</note> <note position="right" xlink:label="note-0029-04" xlink:href="note-0029-04a" xml:space="preserve">_B.Def.10._</note> </div> </div> <div xml:id="echoid-div46" type="section" level="1" n="40"> <head xml:id="echoid-head50" xml:space="preserve">A. XI.</head> <p> <s xml:id="echoid-s434" xml:space="preserve">SImiles figuræ ſolidæ, vel ſimilia ſolida, in vniuerſum <lb/>vocentur, in quorum ſingulis oppofita plana tangen-<lb/>tia ita duci poſſunt, & </s> <s xml:id="echoid-s435" xml:space="preserve">in eadem ita incidere ad eundem an-<lb/>gulum ex eadem parte duo plana in ijſdem terminata, vt ſi <pb o="10" file="0030" n="30" rhead="GEOMETRIÆ"/> deinde inter eadem plana tangentia eiſdem æquidiſtantia <lb/> <anchor type="note" xlink:label="note-0030-01a" xlink:href="note-0030-01"/> vtcumque plana ducta fuerint, altitudines ſolidorum, re-<lb/>ſpectu dictorum tangentium ſumptas, ſimiliter ad eandem <lb/>partem diuidentia, reperiamus figuras exhis planis in di-<lb/> <anchor type="note" xlink:label="note-0030-02a" xlink:href="note-0030-02"/> ctis ſolidis conceptas eſſe ſimiles, vel ſi plures producan-<lb/>tur, tot numero in vno, quot in alio ſolido produci, quæ <lb/> <anchor type="note" xlink:label="note-0030-03a" xlink:href="note-0030-03"/> fint binæ ſimiles, & </s> <s xml:id="echoid-s436" xml:space="preserve">quæ ſunt vnius ſolidi ſimiliter inter ſe <lb/>diſpoſitę, ac quę ſunt alterius, & </s> <s xml:id="echoid-s437" xml:space="preserve">omnium homologas dua-<lb/>bus quibuſdam rectis lineis communiter, tamquam earum-<lb/>dem regulis, æquidiſtare. </s> <s xml:id="echoid-s438" xml:space="preserve">(ſic.</s> <s xml:id="echoid-s439" xml:space="preserve">n. </s> <s xml:id="echoid-s440" xml:space="preserve">earum homologæ cum <lb/>quibuſuis alijs duabus regulis angulos æquales cum præ-<lb/>dictis facientibus, vt infra Prop. </s> <s xml:id="echoid-s441" xml:space="preserve">23. </s> <s xml:id="echoid-s442" xml:space="preserve">huius oſtendetur, e-<lb/>tiam haberi poterunt) Vnde ſiregulæ homologarum acci-<lb/>piantur cum incidentibus planis concurrentes, & </s> <s xml:id="echoid-s443" xml:space="preserve">conce-<lb/>ptarum in ſolidis ſimilium figurarum ductæ in ſingulis op-<lb/>poſitæ tangentes præfatis regulis Parallelę producantur, ſt <lb/>opus ſit, quouſq; </s> <s xml:id="echoid-s444" xml:space="preserve">prædictis incidentibus planis occurrant, <lb/>& </s> <s xml:id="echoid-s445" xml:space="preserve">binarum quarumcumque oppoſitarum tangentium pun-<lb/>cta occurſuum iungantur rectis lineis, etiam has iungentes <lb/>reperiamus ſingulas eſſe incidentes ſuarum ſimilium figu-<lb/>rarum, & </s> <s xml:id="echoid-s446" xml:space="preserve">oppoſitarum tangentium, ac omnes dictas inci-<lb/>dentes concipi in figuris ſimilibus, quarum, & </s> <s xml:id="echoid-s447" xml:space="preserve">ipſæ inci-<lb/>dentes ſint homologæ, & </s> <s xml:id="echoid-s448" xml:space="preserve">omnium regulæ communes ſe-<lb/>ctiones planorum incidentium, & </s> <s xml:id="echoid-s449" xml:space="preserve">oppoſitorum planorum <lb/>tangentium. </s> <s xml:id="echoid-s450" xml:space="preserve">Has omnes, inquam, conditiones ſimilia ſo-<lb/>lida in vniuerſum habere ſuppono.</s> <s xml:id="echoid-s451" xml:space="preserve"/> </p> <div xml:id="echoid-div46" type="float" level="2" n="1"> <note position="left" xlink:label="note-0030-01" xlink:href="note-0030-01a" xml:space="preserve">D. Def.2.</note> <note position="left" xlink:label="note-0030-02" xlink:href="note-0030-02a" xml:space="preserve">A.Def.10.</note> <note position="left" xlink:label="note-0030-03" xlink:href="note-0030-03a" xml:space="preserve">E.Def.10.</note> </div> </div> <div xml:id="echoid-div48" type="section" level="1" n="41"> <head xml:id="echoid-head51" xml:space="preserve">B.</head> <note position="left" xml:space="preserve">B</note> <p> <s xml:id="echoid-s452" xml:space="preserve">IPſæ autem figuræ planæ ſimiles, quæ capiunt omnes di-<lb/>ctas incidentes, vocentur. </s> <s xml:id="echoid-s453" xml:space="preserve">Figuræ incidentes dictorum <lb/>ſimilium ſolidorum, & </s> <s xml:id="echoid-s454" xml:space="preserve">oppoſitorum tangentium iam du-<lb/>ctorum.</s> <s xml:id="echoid-s455" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div49" type="section" level="1" n="42"> <head xml:id="echoid-head52" xml:space="preserve">C.</head> <note position="left" xml:space="preserve">C</note> <p> <s xml:id="echoid-s456" xml:space="preserve">FIguræ verò ex planis dictis tangentibus Parallelis in <lb/> <anchor type="note" xlink:label="note-0030-06a" xlink:href="note-0030-06"/> eiſdem ſolidis conceptæ, quotcumque ſint, altitudi-<lb/>nes eorumdem reſpectu dictorum tangentium ſumptas ſi-<lb/>militer ad eandem partem diuidentes, quæ ſimiles eſſe re- <pb o="11" file="0031" n="31" rhead="LIBER I."/> periuntur, ſiue binæ ſimiles, & </s> <s xml:id="echoid-s457" xml:space="preserve">vnæ, ac aliæ ſimiliter inter <lb/> <anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> ſe diſpoſitæ, vocentur: </s> <s xml:id="echoid-s458" xml:space="preserve">Figuræ homologæ dictorum ſimi-<lb/>lium ſolidorum, ſumptæ regula vna ipſarum, vel oppoſito-<lb/>rum tangentium, quæ homologarum figurarum plana tan-<lb/>gentia, ſi libeat, etiam vocentur.</s> <s xml:id="echoid-s459" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="1"> <note position="left" xlink:label="note-0030-06" xlink:href="note-0030-06a" xml:space="preserve">D.Def.2.</note> <note position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">A.E. Def. <lb/>10.</note> </div> </div> <div xml:id="echoid-div51" type="section" level="1" n="43"> <head xml:id="echoid-head53" xml:space="preserve">APPENDIX POSTERIOR <lb/>Pro declaratione Definit. II.</head> <p style="it"> <s xml:id="echoid-s460" xml:space="preserve">_S_Int ſolida, Γ β 3 Φ, AHBM, quorum ſint oppoſita tangen-<lb/>tia plana, Δ ℟ Ζ &</s> <s xml:id="echoid-s461" xml:space="preserve">, ſolidi, Γ β 3 Φ, &</s> <s xml:id="echoid-s462" xml:space="preserve">, QP, L Π, ſθ-<lb/> <anchor type="note" xlink:label="note-0031-02a" xlink:href="note-0031-02"/> lidi, AHBM, ſint autem alia duo plana, quæ iſtis incidant ad <lb/>eundem angulum ex eadem parte, Δ Υ QK, illa nempè quo-<lb/>rum, et dictorum tangentium ſint communes ſictiones, ΔΧ Ζ <lb/> <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/> Τ, QF, LK, ſecentur nunc dicta ſolida planis tangentibus pa-<lb/>rallelis, quæ diuidant eorum altitudinesreſpectu duitorum tan-<lb/> <anchor type="note" xlink:label="note-0031-03a" xlink:href="note-0031-03"/> gentium ſump@as ſimiliter ad eandem partem: </s> <s xml:id="echoid-s463" xml:space="preserve">Sint autem eo-<lb/>rum in dictis ſolidis conceptæ ſiguræ planæ ſimiles, ſivna in vno <lb/> <anchor type="note" xlink:label="note-0031-04a" xlink:href="note-0031-04"/> quoq; </s> <s xml:id="echoid-s464" xml:space="preserve">ſolido figura producantur, velſiplures, binæ ſimiles, et fi-<lb/>militer inter ſe diſpoſitæ, quæ fiunt in vno, ac quæ fiunt in alio ſo-<lb/>lido, ex. </s> <s xml:id="echoid-s465" xml:space="preserve">g. </s> <s xml:id="echoid-s466" xml:space="preserve">ipſæ, β Λ, Σ Φ, HE, CM, quæ ſint binæ ſimiles, ideſt, <lb/>β Λ, ipſi, HE, &</s> <s xml:id="echoid-s467" xml:space="preserve">, Σ Φ, ipſi, CM, &</s> <s xml:id="echoid-s468" xml:space="preserve">, β λ, Σ Φ, ſimiliter inter <lb/>ſe diſpoſitæ, ac ipſæ, HE, CM, quarum ſimilium figurarum bo-<lb/>mologæ duabus quibuſcumque regulis, vt ipſis, ℟ Ω, PR, æqui-<lb/>dicto<unsure/>nt; </s> <s xml:id="echoid-s469" xml:space="preserve">vel ſihæ non ſint cum planis, Δ Υ, QK, concurrentes, <pb o="12" file="0032" n="32" rhead="GEOMETRIÆ"/> alias, Ω Δ RQ, cum prædictis angulos æquales continentes, ℟ <lb/>Ω Δ, PRQ, proregulis homolog arum accipiemus, hoc .</s> <s xml:id="echoid-s470" xml:space="preserve">n. </s> <s xml:id="echoid-s471" xml:space="preserve">fieri <lb/>pθſſe demonſtrabitur in Prop. </s> <s xml:id="echoid-s472" xml:space="preserve">23. </s> <s xml:id="echoid-s473" xml:space="preserve">buius, quę erunt cum planis, Δ <lb/>Υ, QK, concurrentes. </s> <s xml:id="echoid-s474" xml:space="preserve">Siergo ducantur prædictarum ſimilium <lb/> <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/> figurarum, β λ, HE, Σ Φ, CM, oppoſitæ tangentes, parallelæ <lb/>regulis, Ω Δ, RQ, ex. </s> <s xml:id="echoid-s475" xml:space="preserve">g. </s> <s xml:id="echoid-s476" xml:space="preserve">ſiguræ, β Λ, oppoſitæ tangentes, β Τ, <lb/>Λ S, &</s> <s xml:id="echoid-s477" xml:space="preserve">, HE, ipſæ, HD, EI, &</s> <s xml:id="echoid-s478" xml:space="preserve">, Σ Φ, ipſæ, Σ Ο, Φ V, & </s> <s xml:id="echoid-s479" xml:space="preserve">tan-<lb/>demipſius, CM, ipſæ, CN, MG, quæ productæ ſi opus ſit, oc-<lb/>currant planis, Δ Υ, QK, inpunctis, T, S; </s> <s xml:id="echoid-s480" xml:space="preserve">OV; </s> <s xml:id="echoid-s481" xml:space="preserve">D, I; </s> <s xml:id="echoid-s482" xml:space="preserve">N, G; <lb/></s> <s xml:id="echoid-s483" xml:space="preserve">iungantur autem, TS, OV, DI, NG, et ipſęreperiantur eſſe <lb/>incidentes ſimilium figurarum, β λ, HE, Σ Φ, CM, et dicta-<lb/>rum oppoſitarum tangentium.</s> <s xml:id="echoid-s484" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="1"> <note position="right" xlink:label="note-0031-02" xlink:href="note-0031-02a" xml:space="preserve">_B. Def. 2._</note> <figure xlink:label="fig-0031-01" xlink:href="fig-0031-01a"> <image file="0031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0031-01"/> </figure> <note position="right" xlink:label="note-0031-03" xlink:href="note-0031-03a" xml:space="preserve">_D. Def. 2._</note> <note position="right" xlink:label="note-0031-04" xlink:href="note-0031-04a" xml:space="preserve">_A. E. Def._ <lb/>_10._</note> <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a"> <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0032-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s485" xml:space="preserve">Conſimiliter, ſectis eiſdem ſolidis alijs planis dictis planis tan-<lb/>gentibus paralle@is, altitudine ſque dictas ſimiliter ad candem, <lb/>partem ſecantibus, ſemper conceptæ in ſolidis figuræ ſint ſimiles, <lb/>velbinę ſimiles, &</s> <s xml:id="echoid-s486" xml:space="preserve">c. </s> <s xml:id="echoid-s487" xml:space="preserve">& </s> <s xml:id="echoid-s488" xml:space="preserve">earumdem homologarum oppoſitę tangen-<lb/>t<unsure/>es parallelę pręfatis regulis, Ω Δ, RQ, ſint productę vſque ad <lb/>plana, Δ Υ, QK; </s> <s xml:id="echoid-s489" xml:space="preserve">occurrantque illis in punctis, quę ſiiungan-<lb/>tur rectis lineis, ipſę iungentes ſint dictarum ſimilium ſigurarum, <lb/>& </s> <s xml:id="echoid-s490" xml:space="preserve">ductarum oppoſitarum tangētium ſemper incidentes, quæ om-<lb/>nes iaceantin planis, Δ Υ, QK, per quarum extrema tranſeant <lb/>lineę, ΧVΥΤ, FGKD, et interius lineæ, 4 N 6 I, 7085, <lb/>& </s> <s xml:id="echoid-s491" xml:space="preserve">conting at figuras, ΧVΥΤ, FGKD, eſſe ſimiles, earumque <lb/>homologas dictas incidentes, & </s> <s xml:id="echoid-s492" xml:space="preserve">ipſarum regulas eſſe communes <lb/>ſectiones planorum, in quibus iacent, & </s> <s xml:id="echoid-s493" xml:space="preserve">oppoſitorum planor@m, <pb o="13" file="0033" n="33" rhead="LIBER I."/> iangentium, ideſt ipſas, Χ Δ, Υ Ζ, FQ, KL. </s> <s xml:id="echoid-s494" xml:space="preserve">Hìsìgitur <lb/>poſitis, voco ſolida, Γ β 3 Φ, AHBM, ſimilia; </s> <s xml:id="echoid-s495" xml:space="preserve">ſiguras verò, F <lb/> <anchor type="note" xlink:label="note-0033-01a" xlink:href="note-0033-01"/> GKD, ΧVΥΤ; </s> <s xml:id="echoid-s496" xml:space="preserve">dicθ figuras incidentes ſimilium ſolidorum iam <lb/> <anchor type="note" xlink:label="note-0033-02a" xlink:href="note-0033-02"/> dictorum, et oppoſitorum tangentium planorum, ℟ Δ, & </s> <s xml:id="echoid-s497" xml:space="preserve">Z; </s> <s xml:id="echoid-s498" xml:space="preserve">PQ, <lb/>Π L; </s> <s xml:id="echoid-s499" xml:space="preserve">ipſas autem figuras, β λ, Σ Φ, HE, CM, et eas, quarum <lb/>extenſa plana ſimiliter ad eandem partem diuidunt altitudines <lb/>ſolidorum, Γ β 3 Φ, AHBM, reſpectu dictorum tangentium <lb/>planorum ſumptas, & </s> <s xml:id="echoid-s500" xml:space="preserve">ſunt ſimiles, velbinæ ſimiles, & </s> <s xml:id="echoid-s501" xml:space="preserve">ſimiliter <lb/>inter ſe diſpoſitæ, voco figuras homologas dictorum ſolidorum, <lb/> <anchor type="note" xlink:label="note-0033-03a" xlink:href="note-0033-03"/> ſumptas, regulis earum duabus, vel dictis tangentibus planis.</s> <s xml:id="echoid-s502" xml:space="preserve"/> </p> <div xml:id="echoid-div52" type="float" level="2" n="2"> <note position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">_A. Def. @@._</note> <note position="right" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">_B. Def. <gap/>_</note> <note position="right" xlink:label="note-0033-03" xlink:href="note-0033-03a" xml:space="preserve">_C. Def. 11._</note> </div> </div> <div xml:id="echoid-div54" type="section" level="1" n="44"> <head xml:id="echoid-head54" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s503" xml:space="preserve">_A_Duertendum eſt autem pro ſimilium figurarum nominatione, <lb/>dum voco eas ſimiles figuras ſiue planas, ſiue ſolidas, me intel-<lb/>ligere in eis d finitiones generales ſuperius allatas; </s> <s xml:id="echoid-s504" xml:space="preserve">dum verò eas <lb/>particulari nomine appello, intelligere definitiones particulares pro <lb/>ipſarum ſimilitudine ab alijs, vel à me allatas, vt cum dicam, ſimi-<lb/>les coni ſectionum portiones, intelligam particularem in eis definitio-<lb/>nem, & </s> <s xml:id="echoid-s505" xml:space="preserve">cum dicam (ſimilta parallelogramma) intelligam in eis par-<lb/>ticularem definitionem ſimilium rectilinearum figurarum, & </s> <s xml:id="echoid-s506" xml:space="preserve">fic in <lb/>cæteris, licet vtramq; </s> <s xml:id="echoid-s507" xml:space="preserve">definitionem tum particularem, tum genera-<lb/>lem, de eiſdem figuris verificari inferius oſtendemus.</s> <s xml:id="echoid-s508" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div55" type="section" level="1" n="45"> <head xml:id="echoid-head55" xml:space="preserve">XII.</head> <p> <s xml:id="echoid-s509" xml:space="preserve">CVm fuerint quotcumque magnitudines eiuſdem ge-<lb/>neris vtcumque diſpoſitæ, prima ad vltimam dicitur <lb/>habere rationem compoſitam exrationibus primæ ad ſe-<lb/>cundam, ſecundæ ad tertiam, tertiæ ad quartam, & </s> <s xml:id="echoid-s510" xml:space="preserve">ſic de-<lb/>inceps vſq; </s> <s xml:id="echoid-s511" xml:space="preserve">ad vltimam.</s> <s xml:id="echoid-s512" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div56" type="section" level="1" n="46"> <head xml:id="echoid-head56" xml:space="preserve">XIII.</head> <p> <s xml:id="echoid-s513" xml:space="preserve">CVm vnum, & </s> <s xml:id="echoid-s514" xml:space="preserve">idem antecedens ad plura conſequen-<lb/>tia comparatum fuerit, ſingillatim ad vnumquodq; <lb/></s> <s xml:id="echoid-s515" xml:space="preserve">comparare idem ad eadem conſequentia ſimul collecta, di-<lb/>cemus, colligere, vel, colligendo.</s> <s xml:id="echoid-s516" xml:space="preserve"/> </p> <pb o="14" file="0034" n="34" rhead="GEOMETRI Æ"/> </div> <div xml:id="echoid-div57" type="section" level="1" n="47"> <head xml:id="echoid-head57" xml:space="preserve">XIV.</head> <p> <s xml:id="echoid-s517" xml:space="preserve">PArallelogrammum dicetur expoſitę cuicumque planę <lb/>figuræ circumſcriptum, ſi eius ſingula latera tangant <lb/>dictam figuram, quæ illi pariter dicetur inſcripta.</s> <s xml:id="echoid-s518" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div58" type="section" level="1" n="48"> <head xml:id="echoid-head58" xml:space="preserve">XV.</head> <p> <s xml:id="echoid-s519" xml:space="preserve">PArallelepipedum dicetur expoſito ſolido circumſcri-<lb/>prum, ſi eius ſingula plana tangant dictum ſolidum, <lb/>quod illi pariter dicetur inſcriptum.</s> <s xml:id="echoid-s520" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div59" type="section" level="1" n="49"> <head xml:id="echoid-head59" xml:space="preserve">POSTVLATA</head> <head xml:id="echoid-head60" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s521" xml:space="preserve">QVamlibet rectam lineam indefinitè ita poſſe moueri, <lb/>vt ſemper vni cuidam fixæ ſit parallela, ſiue in eo-<lb/>dem, ſiue in plut<unsure/>ibus planis, in tali motu exiſtat.</s> <s xml:id="echoid-s522" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div60" type="section" level="1" n="50"> <head xml:id="echoid-head61" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s523" xml:space="preserve">QVodlibet planum indefinitè ita poſſe moueri, vt fem-<lb/>per vni cuidam fixo ſit æquidiſtans.</s> <s xml:id="echoid-s524" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div61" type="section" level="1" n="51"> <head xml:id="echoid-head62" xml:space="preserve">PROBLEMA I. PROPOS. 1.</head> <p> <s xml:id="echoid-s525" xml:space="preserve">CViuslibet propoſitæ figuræ planæ, vel ſolidæ, verti-<lb/>cem inuenire, reſpectu datæ, pro figura plana rectæ <lb/>lineæ pro ſolida verò, reſpectu dati plani.</s> <s xml:id="echoid-s526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s527" xml:space="preserve">Sit ſigura plana quæcumque, ABC, & </s> <s xml:id="echoid-s528" xml:space="preserve">in ea ducta recta linea, <lb/>BC, oportet reſpectu ipſius, BC, verticem figuræ, ABC, inue-<lb/> <anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/> <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> nire. </s> <s xml:id="echoid-s529" xml:space="preserve">Sumatur in plano figuræ, AB <lb/>C, indefinitè producto, vtcumq; </s> <s xml:id="echoid-s530" xml:space="preserve">pun-<lb/>ctum, N, & </s> <s xml:id="echoid-s531" xml:space="preserve">per, N, ipſi, BC, du-<lb/>catur parallela, KV, indefinitè hinc <lb/>inde producta, vel igitur, KV, tan-<lb/>git figuram, BAC, & </s> <s xml:id="echoid-s532" xml:space="preserve">ſic inuentum <lb/>eſſet, quod quæritur, vel non; </s> <s xml:id="echoid-s533" xml:space="preserve">igitur <lb/>erit, KV, vel intra, vel extra figu-<lb/>ram, vbicumq; </s> <s xml:id="echoid-s534" xml:space="preserve">ſit, moueatur, KV, <lb/>ſemper manens in eiuſdem figuræ <lb/> <anchor type="note" xlink:label="note-0034-02a" xlink:href="note-0034-02"/> plano, & </s> <s xml:id="echoid-s535" xml:space="preserve">æquidiſtans ipſi, BC, re-<lb/>cedendo ab eadem, BC, ſi intra figu-<lb/>ram reperiebatur, vel accedendo, ſi erat extra, tandem .</s> <s xml:id="echoid-s536" xml:space="preserve">n. </s> <s xml:id="echoid-s537" xml:space="preserve">con- <pb o="15" file="0035" n="35" rhead="LIBER I."/> tinget figuram, ABC, contingatin ſitu ipſius, FG, & </s> <s xml:id="echoid-s538" xml:space="preserve">in pun-<lb/>cto, A, igitur, A, erit vertex figuræ, ABC, reſpectu ipſius, B <lb/> <anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> C, à nobis inuentus, qui in huius Problematis priori parte inue-<lb/>niendus proponebatur.</s> <s xml:id="echoid-s539" xml:space="preserve"/> </p> <div xml:id="echoid-div61" type="float" level="2" n="1"> <note position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">A. Def, 1.</note> <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0034-01"/> </figure> <note position="left" xlink:label="note-0034-02" xlink:href="note-0034-02a" xml:space="preserve">Poſtul. 1.</note> <note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">A. Def. 1.</note> </div> <p> <s xml:id="echoid-s540" xml:space="preserve">Sit nunc figura ſolida, ſiue ſolidum, ADE, in quo reſpectu <lb/>plani, BECD, ſit vertex inueniendus, ſumpto igitur exrra pla-<lb/>num figuræ, vtcumque puncto, N, per ipſum agatur planum, K <lb/>HVX, ipſi, BECD, æquidiſtans, quod vel continget ſolidum, <lb/>BAC, vel non, ſi autem non contingat, moueatur accedendo, <lb/> <anchor type="note" xlink:label="note-0035-02a" xlink:href="note-0035-02"/> velrecedendo, à plano, BECD, tandem igitur contingetipſum, <lb/>tangatin, A, puncto, igitur punctum, A, erit vertex ſolidi, AD <lb/> <anchor type="note" xlink:label="note-0035-03a" xlink:href="note-0035-03"/> E, reſpectu plani, BECD, qui inueniendus proponebatur.</s> <s xml:id="echoid-s541" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="2"> <note position="right" xlink:label="note-0035-02" xlink:href="note-0035-02a" xml:space="preserve">Poſtul. 2.</note> <note position="right" xlink:label="note-0035-03" xlink:href="note-0035-03a" xml:space="preserve">A. Def. 2.</note> </div> </div> <div xml:id="echoid-div64" type="section" level="1" n="52"> <head xml:id="echoid-head63" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s542" xml:space="preserve">_H_Inc manifeſtum eſt, ſi recta, BC, tangat planam figuram, AB <lb/> <anchor type="note" xlink:label="note-0035-04a" xlink:href="note-0035-04"/> C, quod ductæ erunt oppoſitę tangentes ipſius figuræ, ABC, <lb/>reſpectu datæ rectæ lineæ, quæ fuit vna ex eiſdem tangentibus, nem-<lb/> <anchor type="note" xlink:label="note-0035-05a" xlink:href="note-0035-05"/> pè, BC; </s> <s xml:id="echoid-s543" xml:space="preserve">& </s> <s xml:id="echoid-s544" xml:space="preserve">ita ſi figura, BDCE, tangit ſolidum, ADE, ducta erunt <lb/>oppoſita tangentia plana ſolidi, ADE, reſpectu plani, BECD, in <lb/> <anchor type="note" xlink:label="note-0035-06a" xlink:href="note-0035-06"/> quibus puncta contactuum erunt oppoſiti vertices earumdem figura-<lb/>rum, boc pacto inuenti: </s> <s xml:id="echoid-s545" xml:space="preserve">Siverò recta linea, BC, ſecaret figuran, A <lb/>BC, vel planum, BECD, ſecaret ſolidum, ADE, eodem pacto ex <lb/>alia parte lineæ, BC, vel plani, BDCE, inueniemus verticem, vn-<lb/>de inuenti erun@ propoſitæ figuræ planæ oppoſiti vertices, & </s> <s xml:id="echoid-s546" xml:space="preserve">ductæ op-<lb/>poſitæ tangentes reſpectu datæ lineæ BC; </s> <s xml:id="echoid-s547" xml:space="preserve">& </s> <s xml:id="echoid-s548" xml:space="preserve">in ſolido iuuenti erunt <lb/>oppoſiti vertices, & </s> <s xml:id="echoid-s549" xml:space="preserve">ducta oppoſita tangentia plana reſpectu dati <lb/>plani, BDCE, quæ cum tangunt in figuris planis, figuræ contactuum <lb/> <anchor type="note" xlink:label="note-0035-07a" xlink:href="note-0035-07"/> vocantur etiam oppoſitæ baſes, & </s> <s xml:id="echoid-s550" xml:space="preserve">earum ſingulæ baſes, & </s> <s xml:id="echoid-s551" xml:space="preserve">baſes li-<lb/>neares, ſi contactus fieret in lineis: </s> <s xml:id="echoid-s552" xml:space="preserve">binc ergo diſcimus inuenire op-<lb/> <anchor type="note" xlink:label="note-0035-08a" xlink:href="note-0035-08"/> poſitos vertices figuræ planæ, vel ſolidæ cuiuſcumque, & </s> <s xml:id="echoid-s553" xml:space="preserve">eorum op-<lb/>poſita tangentia ducere reſpectu datæ in figura plana rectæ lineæ, & </s> <s xml:id="echoid-s554" xml:space="preserve"><lb/>dati plani par@ter in ſolida figura.</s> <s xml:id="echoid-s555" xml:space="preserve"/> </p> <div xml:id="echoid-div64" type="float" level="2" n="1"> <note position="right" xlink:label="note-0035-04" xlink:href="note-0035-04a" xml:space="preserve">_B. Def. 1._</note> <note position="right" xlink:label="note-0035-05" xlink:href="note-0035-05a" xml:space="preserve">_B. Def. 2._</note> <note position="right" xlink:label="note-0035-06" xlink:href="note-0035-06a" xml:space="preserve">_A. Def. 2._</note> <note position="right" xlink:label="note-0035-07" xlink:href="note-0035-07a" xml:space="preserve">_C. Def. 2._</note> <note position="right" xlink:label="note-0035-08" xlink:href="note-0035-08a" xml:space="preserve">_A. B. Def._ <lb/>_1. & 2._</note> </div> </div> <div xml:id="echoid-div66" type="section" level="1" n="53"> <head xml:id="echoid-head64" xml:space="preserve">PROBLEMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s556" xml:space="preserve">CVilibct figuræ planæ parallelogrammum circumſcri-<lb/>bere, cuius latera duabus datis rectis lineis, in pro-<lb/>poſitæ figuræ plano ſe ſecantibus, ſint parallela.</s> <s xml:id="echoid-s557" xml:space="preserve"/> </p> <pb o="16" file="0036" n="36" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s558" xml:space="preserve">Sit propoſita quæcumque figura plana, AOVE, & </s> <s xml:id="echoid-s559" xml:space="preserve">in ipſiu@ <lb/>plano duæ rectæ lineæ, BD, IT, vtcumq; </s> <s xml:id="echoid-s560" xml:space="preserve">ſe inuicem fecante@ <lb/> <anchor type="note" xlink:label="note-0036-01a" xlink:href="note-0036-01"/> <anchor type="figure" xlink:label="fig-0036-01a" xlink:href="fig-0036-01"/> in puncto; </s> <s xml:id="echoid-s561" xml:space="preserve">C, oportet illi parallelogrammum <lb/>circumſcribere, cuius latera rectis, SD, IT, <lb/>ſint parallela. </s> <s xml:id="echoid-s562" xml:space="preserve">Ducantur ergo oppoſitæ tan-<lb/>gentes figuræ, AOVE, reſpectu ipſius, IT, <lb/>quæ ſint, KM, HL, & </s> <s xml:id="echoid-s563" xml:space="preserve">aliæ duæ reſpectu ip-<lb/> <anchor type="note" xlink:label="note-0036-02a" xlink:href="note-0036-02"/> ſius, BD, quæ ſint, KH, ML, quæ cum <lb/>prædictis concurrent, nam ſunt parallelæ ip-<lb/>ſis, BD, IT, quæ inuicem concurrunr, ſit <lb/>ergo concurſus in punctis, K, M, L, H, igi-<lb/>tur, KL, erit parallelogrammum, cuius ſin-<lb/>gula latera tangent ambitum ſiguræ, vt in <lb/>punctis, A, O, V, E, & </s> <s xml:id="echoid-s564" xml:space="preserve">ideò erit figuræ, AOVE, circumfcri-<lb/> <anchor type="note" xlink:label="note-0036-03a" xlink:href="note-0036-03"/> ptum, habens latera duabus datis rectis lineis, BD, IT, in figu-<lb/>ræ, AOVE, plano ſe inuicem ſecantibus, parallela; </s> <s xml:id="echoid-s565" xml:space="preserve">quod effi-<lb/>cere, &</s> <s xml:id="echoid-s566" xml:space="preserve">c.</s> <s xml:id="echoid-s567" xml:space="preserve"/> </p> <div xml:id="echoid-div66" type="float" level="2" n="1"> <note position="left" xlink:label="note-0036-01" xlink:href="note-0036-01a" xml:space="preserve">Def. 14.</note> <figure xlink:label="fig-0036-01" xlink:href="fig-0036-01a"> <image file="0036-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0036-01"/> </figure> <note position="left" xlink:label="note-0036-02" xlink:href="note-0036-02a" xml:space="preserve">Cor. ant.</note> <note position="left" xlink:label="note-0036-03" xlink:href="note-0036-03a" xml:space="preserve">Def. 14.</note> </div> </div> <div xml:id="echoid-div68" type="section" level="1" n="54"> <head xml:id="echoid-head65" xml:space="preserve">PROBLEMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s568" xml:space="preserve">CVilibet ſolido parallelepipedum circumſcribere, cu-<lb/>ius plana oppoſita tribus datis planis, ſe inuicem ſe-<lb/>cantibus, ſint parallela.</s> <s xml:id="echoid-s569" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s570" xml:space="preserve">Sit ſolidum, <lb/> <anchor type="figure" xlink:label="fig-0036-02a" xlink:href="fig-0036-02"/> ACBD, quod-<lb/>cumq; </s> <s xml:id="echoid-s571" xml:space="preserve">in quo <lb/>tria plana, A <lb/>CBD, AB, C <lb/>D, ſe inuicem <lb/>fecent, quæli-<lb/>bet duo, opor-<lb/>tetſolido, AC <lb/> <anchor type="note" xlink:label="note-0036-04a" xlink:href="note-0036-04"/> BD, paralle-<lb/>lepipedum cir-<lb/>cūſcribere, cu-<lb/>ius oppoſita <lb/>plana prædi-<lb/>ctis planis ſint <lb/>parallela. </s> <s xml:id="echoid-s572" xml:space="preserve">Du-<lb/>cātur duo pla-<lb/>na oppoſita <lb/> <anchor type="note" xlink:label="note-0036-05a" xlink:href="note-0036-05"/> tangentia dictum ſolidum reſpectu cuiuſuis planorum ſe ſecan- <pb o="17" file="0037" n="37" rhead="LIBER I."/> tium, ACBD, AB, CD, & </s> <s xml:id="echoid-s573" xml:space="preserve">producantur donec ſibi occurrant, oc-<lb/>current autem, quia hæc planis ſe inuicem ſecantibus ſunt parallela, <lb/>& </s> <s xml:id="echoid-s574" xml:space="preserve">ſit ab illis comprehenſum ſolidum, ZF, erit igitur, ZF, paralle-<lb/>ſepipedum, cum eius oppoſita plana ſint inuicem parallela, quæ tan-<lb/>gunt ſolidum, ACBD, vt in punctis, A, C, B, D, E, X, & </s> <s xml:id="echoid-s575" xml:space="preserve">ideò <lb/>erit ſolido, ACBD, circumſcriptum, habens plana oppoſita pro-<lb/> <anchor type="note" xlink:label="note-0037-01a" xlink:href="note-0037-01"/> poſitis planis ſe ſecantibus parallela; </s> <s xml:id="echoid-s576" xml:space="preserve">quod efficere opus erat.</s> <s xml:id="echoid-s577" xml:space="preserve"/> </p> <div xml:id="echoid-div68" type="float" level="2" n="1"> <figure xlink:label="fig-0036-02" xlink:href="fig-0036-02a"> <image file="0036-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0036-02"/> </figure> <note position="left" xlink:label="note-0036-04" xlink:href="note-0036-04a" xml:space="preserve">Def. 1@.</note> <note position="left" xlink:label="note-0036-05" xlink:href="note-0036-05a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="right" xlink:label="note-0037-01" xlink:href="note-0037-01a" xml:space="preserve">Def. 15.</note> </div> </div> <div xml:id="echoid-div70" type="section" level="1" n="55"> <head xml:id="echoid-head66" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s578" xml:space="preserve">_P_Oteſt autem contingere in antecedentis Propoſ. </s> <s xml:id="echoid-s579" xml:space="preserve">figura ipſam eſſepa-<lb/>railelogrammum, & </s> <s xml:id="echoid-s580" xml:space="preserve">line as rectas ſe ſecantes, qutbus parallelo-<lb/>grammi circumſcriptibilis latera debent eſſe parallela, eſſe ipſa paralle-<lb/>logrammi latera, in quo caſu idem eſſet parallelogrammum circumſcri-<lb/>ptum, & </s> <s xml:id="echoid-s581" xml:space="preserve">cui circumſcriberetur: </s> <s xml:id="echoid-s582" xml:space="preserve">V eluti hic etiam ſi ſolidum, ACBD, <lb/>eſſet parallelepipedum, cuius oppoſitis planis, plana circum ſcrip tibilis <lb/>deberent eſſe pacallela, tunc enim idem eſſet parallelepipedum circum-<lb/>ſcriptum, & </s> <s xml:id="echoid-s583" xml:space="preserve">cui cir cumſcriberetur: </s> <s xml:id="echoid-s584" xml:space="preserve">Contactus autem in antecedenti po-<lb/>teſt etiam eſſe in linea, & </s> <s xml:id="echoid-s585" xml:space="preserve">in bac tum in linea, tum in planis, licet con-<lb/>tactus, qui fit in punctis tantum expoſitus fuerit.</s> <s xml:id="echoid-s586" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div71" type="section" level="1" n="56"> <head xml:id="echoid-head67" xml:space="preserve">THEOREMA I. PROPOS. IV.</head> <p> <s xml:id="echoid-s587" xml:space="preserve">DAta quacumq; </s> <s xml:id="echoid-s588" xml:space="preserve">figura plana, vel ſolida, & </s> <s xml:id="echoid-s589" xml:space="preserve">in plana da-<lb/>ta recta linea, in ſolida verò dato plano; </s> <s xml:id="echoid-s590" xml:space="preserve">qualibet li-<lb/>nea, vel planum, quod indefinitè productum non tangat fi-<lb/>guram dictam planam, vel ſolidam, in vertice ſumpto reſpe-<lb/>ctu dictæ lineæ, vel plani, vel totum extra, vel aliquid eius <lb/>intra figuram cadit, nempè figuram ſecat, ſi linea lineæ, vel <lb/>planum plano æquidiſtet.</s> <s xml:id="echoid-s591" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s592" xml:space="preserve">Sit data figura plana, CARB, <lb/> <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> & </s> <s xml:id="echoid-s593" xml:space="preserve">in ea recta, AB, ſit vertex vnus <lb/>reſpectuipſius, AB, punctus, C, <lb/>& </s> <s xml:id="echoid-s594" xml:space="preserve">ſit recta, HM, parallela ipſi, <lb/>AB, quæ etiam indefinitè produ-<lb/>cta non tangat figuram, ARBC, <lb/>in, C, vertice. </s> <s xml:id="echoid-s595" xml:space="preserve">Dico, HM, vel <lb/>totam extra figuram cadere, vel <lb/>eandem ſecare. </s> <s xml:id="echoid-s596" xml:space="preserve">Neutrum efficiat <lb/>ſi poſſibile eſt, igitur, HM, tan-<lb/>get figuram, CARB, & </s> <s xml:id="echoid-s597" xml:space="preserve">non in, <lb/>C, igitur in alio puncto, vt in, E, <lb/>igitur, E, erit vertex figuræ, CA <lb/>RB, reſpectu ipſius, AB, eſt e-<lb/> <anchor type="note" xlink:label="note-0037-02a" xlink:href="note-0037-02"/> tiam, C, vertēx eiuſdem reſpectu eiuſdem, AB, ergo ſi per, C, <pb o="18" file="0038" n="38" rhead="GEOMETRIÆ"/> ducamus rectam, VN, parallelam ipſi, AB, tranſibit hæc per <lb/> <anchor type="note" xlink:label="note-0038-01a" xlink:href="note-0038-01"/> punctum, E, qui eſt etiam vertex rcſpectu ipſius, AB, igitur ſeca-<lb/>bit, HM, quod eſt abſurdum, nam vtræque ſunt parallelæ eidem, <lb/>AB, & </s> <s xml:id="echoid-s598" xml:space="preserve">ideò inter ſe ſunt parallelæ, vel, VN, extendetur ſuper, H <lb/>M, & </s> <s xml:id="echoid-s599" xml:space="preserve">ſic, HM, tranſiret per, C, in ipſoq; </s> <s xml:id="echoid-s600" xml:space="preserve">tangeret figuram con-<lb/>tra ſuppoſitum, quod etiam eſt abſurdum, non igitur, HM, tanget <lb/> <anchor type="note" xlink:label="note-0038-02a" xlink:href="note-0038-02"/> figuram, CARB, ſed erit tota extra figuram, ſi nullibi concurrat <lb/>cum ambitu figuræ, vel, tranſiens per aliquem punctum, eandem <lb/>ſecabit, ſi is punctus non ſit ex illis, qui funt vertices ipſius figuræ ex <lb/>hac parte, vel ex oppofito reſpectu ipſius, AB; </s> <s xml:id="echoid-s601" xml:space="preserve">quod ſimiliter in ſo-<lb/>lidis oſtendemus pro rectis lineis, AB, HM, VN, plana intelligen-<lb/>tes, & </s> <s xml:id="echoid-s602" xml:space="preserve">ipſam, CARB, eſſe figuram ſolidam ſupponentes, quæ <lb/>oſtendere opus erat.</s> <s xml:id="echoid-s603" xml:space="preserve"/> </p> <div xml:id="echoid-div71" type="float" level="2" n="1"> <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0037-01"/> </figure> <note position="right" xlink:label="note-0037-02" xlink:href="note-0037-02a" xml:space="preserve">A. Def. @ <lb/>huius.</note> <note position="left" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">Vide di-<lb/>cta lib 7. <lb/>Annot. <lb/>Prop. 3.</note> <note position="left" xlink:label="note-0038-02" xlink:href="note-0038-02a" xml:space="preserve">Ex A. De-<lb/>fin. 1. hu-<lb/>ius.</note> </div> </div> <div xml:id="echoid-div73" type="section" level="1" n="57"> <head xml:id="echoid-head68" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s604" xml:space="preserve">_H_Inc patet à quolibet puncto ambitus datæ figuræ planæ, vel ſolidæ <lb/>ductam lineam, vel planum æquidiſtans illi, reſpectu cuius ſumi-<lb/>tur vertex (ſi ſumptus punctus non ſit vnus ex verticalibus dictis) ſeca-<lb/>rè figuram, cum, vt oſtenſum eſt, tangens eſſe non poſſit, & </s> <s xml:id="echoid-s605" xml:space="preserve">ideò ſem-<lb/>per inter duo oppoſita tangentia, reſpectu regulæ, penes quam ſumitur <lb/>vertex, aſſumpta linea cadet, licet indefinitè producatur.</s> <s xml:id="echoid-s606" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div74" type="section" level="1" n="58"> <head xml:id="echoid-head69" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s607" xml:space="preserve">_E_T quia ſi recta linea, vel planum, ſecat duas parallelas, vel duo <lb/>æquidiſtantia plana, ſecat etiam omnia intermedia illis æquidi-<lb/>ſtantia; </s> <s xml:id="echoid-s608" xml:space="preserve">ideò ſi recta linea, vel planum, tranſeat per verticem, & </s> <s xml:id="echoid-s609" xml:space="preserve">baſim, <lb/>ſiue per oppoſitos vertices datæ figuræ planæ, vel ſolidæ, ſecabit etiam om-<lb/>nes in figura oppoſitis tangentibus æquidiſtantes intra figuram, vel ea-<lb/>ſdem productas extra figuram.</s> <s xml:id="echoid-s610" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div75" type="section" level="1" n="59"> <head xml:id="echoid-head70" xml:space="preserve">THEOREMA II. PROPOS. V.</head> <p> <s xml:id="echoid-s611" xml:space="preserve">SI à quocumque puncto circuitus cylindrici, per quam fit <lb/>reuolutio verſus cylindricum ducta fuerit recta linea <lb/> <anchor type="note" xlink:label="note-0038-03a" xlink:href="note-0038-03"/> paralleìa regulæ lateris cylindrici, hæc eritlatus cylindrici <lb/>in talibaſi conſtituti.</s> <s xml:id="echoid-s612" xml:space="preserve"/> </p> <div xml:id="echoid-div75" type="float" level="2" n="1"> <note position="left" xlink:label="note-0038-03" xlink:href="note-0038-03a" xml:space="preserve">D. fin. 3.</note> </div> <p> <s xml:id="echoid-s613" xml:space="preserve">Sit cylindricus, CB, In baſi, AFB, in cuius circuitu ſumpto vt-<lb/>cumq; </s> <s xml:id="echoid-s614" xml:space="preserve">puncto, F, ab eo ducta ſit verſus cylindricum quædam paral- <pb o="19" file="0039" n="39" rhead="LIBER I."/> lela ipſi, HM, quæ ſit regula lateris cylindrici. </s> <s xml:id="echoid-s615" xml:space="preserve">Dico eam eſſe la-<lb/>tus huius cylindrici: </s> <s xml:id="echoid-s616" xml:space="preserve">Intelligatur per punctum, F, ductum latus cy-<lb/> <anchor type="figure" xlink:label="fig-0039-01a" xlink:href="fig-0039-01"/> lindrici, quod ſit, FE, veligitur du-<lb/>cta ab, F, parallela ipſi, HM, ca-<lb/>dit ſuper, FE, & </s> <s xml:id="echoid-s617" xml:space="preserve">ſic erit, & </s> <s xml:id="echoid-s618" xml:space="preserve">ipſa la-<lb/>tus cylindrici, vel non, nempè ſi ca-<lb/>deret, vt, FG, tunc quia, FE, eſt <lb/>parallela ipſi, HM, & </s> <s xml:id="echoid-s619" xml:space="preserve">etiam, FG, <lb/>eſt ipſi, HM, parallela ſequitur, F <lb/>E, ipſi, FG, eſſe parallelam, & </s> <s xml:id="echoid-s620" xml:space="preserve">ſunt <lb/>FE, FG, eductæ ab eodem pun-<lb/>cto, F, In quo ſunt concurrentes, <lb/>quod eſt abſurdum, igitur quæ du-<lb/>citur à puncto, F, parallela ipſi, H <lb/>M, cadet ſuper, FE, latus cylindri-<lb/>ci, igitur erit latus huius cylindrici, quod erat oſtendendum.</s> <s xml:id="echoid-s621" xml:space="preserve"/> </p> <div xml:id="echoid-div76" type="float" level="2" n="2"> <figure xlink:label="fig-0039-01" xlink:href="fig-0039-01a"> <image file="0039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0039-01"/> </figure> </div> </div> <div xml:id="echoid-div78" type="section" level="1" n="60"> <head xml:id="echoid-head71" xml:space="preserve">THEOREMA III. PROPOS. VI.</head> <p> <s xml:id="echoid-s622" xml:space="preserve">SVperficies, quæ clauditur ambitu deſcripto ab extremo <lb/>puncto lateris cylindrici, quod per circuitum eiuſdem <lb/>baſis non properat, eſt ſuperficies plana, & </s> <s xml:id="echoid-s623" xml:space="preserve">æquidiſtans bafi; <lb/></s> <s xml:id="echoid-s624" xml:space="preserve">ſi ea ſumatur, in qua iacent iungentes duo quæuis puncta de-<lb/>ſcripti ambitus.</s> <s xml:id="echoid-s625" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s626" xml:space="preserve">Sit quilibet cylindricus, AE, cuius baſis, CDEV, latus, MV, <lb/>cuius punctum extremum, M, quod non properat per ambitum ba-<lb/> <anchor type="figure" xlink:label="fig-0039-02a" xlink:href="fig-0039-02"/> ſis, in reuolutione deſeribat circuitum, MANH. <lb/></s> <s xml:id="echoid-s627" xml:space="preserve">Dico figuram hoc circuitu comprehenſam, in qua <lb/>iacent iungentes duo quęuis puncta deſcripti am-<lb/>bitus eſſe ſuperficiem planam, ęquidiſtantem baſi, <lb/>CDEV, & </s> <s xml:id="echoid-s628" xml:space="preserve">ideò ſingula puncta huius circuitusre-<lb/>periri in tali plano. </s> <s xml:id="echoid-s629" xml:space="preserve">Sumatur ergo in tali circuitu <lb/>vtcumq; </s> <s xml:id="echoid-s630" xml:space="preserve">punctum, M, & </s> <s xml:id="echoid-s631" xml:space="preserve">per, M, ducatur baſi, <lb/>CE, æquidiſtans planum, MBOF. </s> <s xml:id="echoid-s632" xml:space="preserve">Dico om-<lb/>nia puncta deſcripti circuitus eſſe in hoc, plano: </s> <s xml:id="echoid-s633" xml:space="preserve">ſi <lb/>enim non ſint, aliquod erit extra, ſit hoc pun-<lb/>ctum, N, & </s> <s xml:id="echoid-s634" xml:space="preserve">per, N, ſit ductum latus cylindrici, <lb/>quod ſit, ND, ſecans circuitum figuræ planæ, B <lb/>F, in, O, & </s> <s xml:id="echoid-s635" xml:space="preserve">circuitum baſis in, D, deinde per, N <lb/>D, MV, quæ ſunt æquidiſtantes, cum fint cylindrici latera, exten- <pb o="20" file="0040" n="40" rhead="GEOMETRIÆ"/> datur planum, quod baſimiſecet in recta, DV, figuram planam, M <lb/> <anchor type="note" xlink:label="note-0040-01a" xlink:href="note-0040-01"/> BOF, in recta, OM, & </s> <s xml:id="echoid-s636" xml:space="preserve">iungantur, MN, puncta, quia ergo plana <lb/>parallela, BF, CE, ſecantur plano quodam, communes eorum ſe-<lb/>ctiones, nempè, OM, DV, erunt inuicem parallelæ, ſed etiam, O <lb/>D, MV, ſunt parallelæ, ergo, OV, erit parallelogrammum, &</s> <s xml:id="echoid-s637" xml:space="preserve">, O <lb/>D, æqualis ipſi, MV, eſt autem, MV, æqualis ipſi, ND, quia am-<lb/>bo ſunt latera eiuſdem cylindrici, ergo, DO, æqualis erit ipſi, DN, <lb/>pars toti, quod eſt abſurdum, non igitur aliquod punctum circuitus <lb/>deſcripti a puncto, M, eſt extra planum æquidiſtans baſi, CE, igi-<lb/>tur omnia ſunt in tali plano, iuncta igitur, NM, ipſa erit in eodem <lb/>cum illis plano, in quo pariter iacebunt duo quæuis puncta iungen-<lb/>tes eiuſdem circuitus, & </s> <s xml:id="echoid-s638" xml:space="preserve">ideò figura tali ambitu contenta eſt ſuper-<lb/>ficies plana ipſi baſi, CE, æquidiſtans, quod erat oſtendendum: </s> <s xml:id="echoid-s639" xml:space="preserve">iſte <lb/>autem vocantur cylindrici oppoſitæ baſes.</s> <s xml:id="echoid-s640" xml:space="preserve"/> </p> <div xml:id="echoid-div78" type="float" level="2" n="1"> <figure xlink:label="fig-0039-02" xlink:href="fig-0039-02a"> <image file="0039-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0039-02"/> </figure> <note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">16. Vnde. <lb/>cimi Ele.</note> </div> </div> <div xml:id="echoid-div80" type="section" level="1" n="61"> <head xml:id="echoid-head72" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s641" xml:space="preserve">_Q_Voniam vero ſuppoſito ipſam, MBOF, eſſe ſuperficiem planam <lb/>baſi æquidictantem, & </s> <s xml:id="echoid-s642" xml:space="preserve">ducto per latera, OD, MV, plano oſten-<lb/>dimus, OV, eſſe parallelogrammum, ideò cum ſciamus, MANH, <lb/>eſſe ſuperficiem planam baſi, CE, æquidiſtantem, ducto per latera vtcum-<lb/>que plano cylindricum ſecante, oſtendemus eodem pacto, ducti plani ſe-<lb/>cantis in cylindrico conceptam figuram eſſe parallelogrammum, cum ſci-<lb/>licet planum ducitur tantum per duo latera, vel parallelogramma, cum <lb/> <anchor type="note" xlink:label="note-0040-02a" xlink:href="note-0040-02"/> per plura duobus, ipſum in eorum aliquo non tangens.</s> <s xml:id="echoid-s643" xml:space="preserve"/> </p> <div xml:id="echoid-div80" type="float" level="2" n="1"> <note position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">_Defin. 3._</note> </div> </div> <div xml:id="echoid-div82" type="section" level="1" n="62"> <head xml:id="echoid-head73" xml:space="preserve">THEOREMA IV. PROPOS. VII.</head> <p> <s xml:id="echoid-s644" xml:space="preserve">SI cylindricus ſecetur, vel tangatur à duobus planis per <lb/>eiuſdem latera ductis, quę non fint inter ſe parallela, ſint <lb/>autem illa producta donec ſibi occurrant, communis eorum <lb/>ſectio erit eiuſdem cylindrici lateribus parallela.</s> <s xml:id="echoid-s645" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s646" xml:space="preserve">Sit quilibet cylindricus, FG, per cuius latera ſint ducta duo pla-<lb/>na non parallela, quæ ita ſint producta, donec ſibi occurrant, ſint <lb/>autem illa plana, AM, DN, quorum, & </s> <s xml:id="echoid-s647" xml:space="preserve">oppoſitarum baſium cy-<lb/>lyndrici, FG, communes ſectiones, AC, HM, DE, SN, erunt <lb/> <anchor type="note" xlink:label="note-0040-03a" xlink:href="note-0040-03"/> igitur, AM, DN, parallelogramma, intelligantur oppoſitarum <lb/>baſium, FL, GK, indefinitè productarum plana ſecarià planis di-<lb/>ctorum parallelogrammorum pariter indefinitè productis, in rectis, <lb/>AR, DR, HO, SO, & </s> <s xml:id="echoid-s648" xml:space="preserve">eadem ſe inuicem ſecare in recta, RO.</s> <s xml:id="echoid-s649" xml:space="preserve"> <pb o="21" file="0041" n="41" rhead="LIBER I."/> Dico, RO, eſſe parallelam lateri cylindrici, FG. </s> <s xml:id="echoid-s650" xml:space="preserve">Iungantur, CE. <lb/></s> <s xml:id="echoid-s651" xml:space="preserve"> <anchor type="note" xlink:label="note-0041-01a" xlink:href="note-0041-01"/> MN, quoniam ergo, CE, MN, coniungunt extrema laterum cy <lb/> <anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/> lindrici, CM, EN, quæ ſunt æqualia, & </s> <s xml:id="echoid-s652" xml:space="preserve"><lb/>parallela, erunt & </s> <s xml:id="echoid-s653" xml:space="preserve">ipſæ æquales, & </s> <s xml:id="echoid-s654" xml:space="preserve">paralle-<lb/>læ, ſunt etiam parallelę ipſæ, CR, MO, er-<lb/>goangulus, ECR, erit æqualis angulo, N <lb/>MO, eodem pacto oſtendemus angulum, C <lb/>ER, eſſe æqualem angulo, MNO, vnde <lb/> <anchor type="note" xlink:label="note-0041-02a" xlink:href="note-0041-02"/> etiam, CR, MO, erunt æquales, & </s> <s xml:id="echoid-s655" xml:space="preserve">funt pa-<lb/>rallelæ, ergo eas iungentes, quæ ſunt, RO, <lb/>CM, erunt ęquales, & </s> <s xml:id="echoid-s656" xml:space="preserve">parallelę, eſt autem, <lb/>CM, latus cylindrici, FG, ergo, RO, com-<lb/>munis ſectio duorum planorum dictum cy-<lb/>lindricum ſecantium, erit eiuſdem lateribus <lb/>parallela. </s> <s xml:id="echoid-s657" xml:space="preserve">Idem oſtendemus, ſi ſectio contingat fieri intra cylindri <lb/>cum, ſiautem fiat in ſuperficie, patet non poſſe fieri, niſi in latere <lb/>cylindrici, nam plana ſecantia ducuntur per latera, quodfibet autem <lb/>latus eſt cęteris eiuſdem cylindrici lateribus æquidiſtans, & </s> <s xml:id="echoid-s658" xml:space="preserve">ideò vbi-<lb/>cumq; </s> <s xml:id="echoid-s659" xml:space="preserve">contingat ſectionem fieri ſemper communis ſectio planorum <lb/>perlatera cylindrici ductorum ſe inuicem ſecantium, eſt parallela la-<lb/>teribus cylindrici. </s> <s xml:id="echoid-s660" xml:space="preserve">Idem ſequetur de tangentibus planis, quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s661" xml:space="preserve"/> </p> <div xml:id="echoid-div82" type="float" level="2" n="1"> <note position="left" xlink:label="note-0040-03" xlink:href="note-0040-03a" xml:space="preserve">Corol. n <lb/>teced.</note> <note position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">33. p. Pri-<lb/>mi Elem. <lb/>p. 16. Vn-<lb/>dec. Elem. <lb/>10. Vnde-<lb/>cimi Ele.</note> <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a"> <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0041-01"/> </figure> <note position="right" xlink:label="note-0041-02" xlink:href="note-0041-02a" xml:space="preserve">26. Primi <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div84" type="section" level="1" n="63"> <head xml:id="echoid-head74" xml:space="preserve">THEOREMA V. PROPOS. VIII.</head> <p> <s xml:id="echoid-s662" xml:space="preserve">SI quilibet cylindricus ſecetur planis parallelis perlatera <lb/>ductis conceptæ in cylindrico figuræ erunt parallelo-<lb/>gramma æquiangula.</s> <s xml:id="echoid-s663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s664" xml:space="preserve">Sit quilibet cylindricus, BF, planis ſectus <lb/> <anchor type="figure" xlink:label="fig-0041-02a" xlink:href="fig-0041-02"/> parallelis per latera ductis, ſit autem vnius <lb/>in cylindrico, AF, concepta figuræ paral-<lb/>lelogrammum, BH, alterius autem paral-<lb/>lelogramma, AN, QF. </s> <s xml:id="echoid-s665" xml:space="preserve">Dico hæc eſſe <lb/>ęquiangula, quod enim ſint parallelogram-<lb/> <anchor type="note" xlink:label="note-0041-03a" xlink:href="note-0041-03"/> ma, patet, quia plana ſecantia ducuntur per <lb/>latera, quod verò fint æquiangula patet e-<lb/>tiam, nam in parallelogrammo, AN, la-<lb/>tus, AD, æquidiſtat lateri, BO, &</s> <s xml:id="echoid-s666" xml:space="preserve">, AP, <lb/>ipſi, BC, nam ſunt communes fectiones pla-<lb/>ni, ABCR, & </s> <s xml:id="echoid-s667" xml:space="preserve">æquidiftantium planorum, <lb/>AN, BH, & </s> <s xml:id="echoid-s668" xml:space="preserve">ideo angulus, PAD, æqua-<lb/> <anchor type="note" xlink:label="note-0041-04a" xlink:href="note-0041-04"/> tur angulo, CBO, ergo parallelogramma, AN, BH, erunt equi- <pb o="22" file="0042" n="42" rhead="GEOMETRIÆ"/> angula, eodem pacto oſtendemus parallelogramma, QF, BH, eſſe <lb/>æquiangula, vnde concludetur etiam parallelogramma, AN, QF, <lb/>eſſeinter ſe æquiangula, quod oſtendere opus erat.</s> <s xml:id="echoid-s669" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="1"> <figure xlink:label="fig-0041-02" xlink:href="fig-0041-02a"> <image file="0041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0041-02"/> </figure> <note position="right" xlink:label="note-0041-03" xlink:href="note-0041-03a" xml:space="preserve">ExCor. 6. <lb/>huius.</note> <note position="right" xlink:label="note-0041-04" xlink:href="note-0041-04a" xml:space="preserve">10. Vnde-<lb/>cimi Ele.</note> </div> </div> <div xml:id="echoid-div86" type="section" level="1" n="64"> <head xml:id="echoid-head75" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s670" xml:space="preserve">_S_I autem intelligamus oppoſitarum baſium cylindrici, AF, ita pra-<lb/>ducta plana, vt ſecentur à plano per latera, AD, PN, QM, RF, <lb/>ducto in rectis, AR, DF, quarum portiones extra cylindricum manen-<lb/>tes ſint, PQ, NM, manifeſtum eſt etiam parallelogrammum, PM, <lb/>quod extra cylindricum conſtituitur, & </s> <s xml:id="echoid-s671" xml:space="preserve">quod integratur ex parallelo-<lb/>grammis, AN, PM, QF, .</s> <s xml:id="echoid-s672" xml:space="preserve">i. </s> <s xml:id="echoid-s673" xml:space="preserve">AF, eſſe prædictis æquiangulum.</s> <s xml:id="echoid-s674" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div87" type="section" level="1" n="65"> <head xml:id="echoid-head76" xml:space="preserve">THEOREMA VI. PROPOS. IX.</head> <p> <s xml:id="echoid-s675" xml:space="preserve">SI planum æquidiſtans plano perlatera cylindrici ducto <lb/>tangat cylindricum, contactus fiet in recta linea, velre-<lb/>ctis lineis, quæ erunt latera eiuſdem cylindrici: </s> <s xml:id="echoid-s676" xml:space="preserve">Vel ſi tan-<lb/>gat in plano, aut planis, plana contactus erunt parallelo-<lb/>gramma, æquiangula perlatera ducto.</s> <s xml:id="echoid-s677" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s678" xml:space="preserve">Sit cylindricus, AC, per cuius latera ducatur planum in eo pro-<lb/>ducens parallelogrammum, AC, ſit autem ductum aliud plannm <lb/> <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> huic æquidiſtans, quod tangat cy-<lb/>lindricum, AC. </s> <s xml:id="echoid-s679" xml:space="preserve">Dico eiuſdem con-<lb/>tactum fieri in recta linea, vel rectis <lb/>lineis, quę erunt latera cylindrici, A <lb/>C, vel ſi tangat in plano, aut planis, <lb/>plana contactus eſſe parallelogram-<lb/>ma, æquiangula ipſi, AC. </s> <s xml:id="echoid-s680" xml:space="preserve">Primò <lb/>igitur non tangat ipſum in plano, <lb/>quia ergo tangit cylindricum, ali-<lb/>quid ſuperficiei cylindrici commune <lb/>eſt ipſi, & </s> <s xml:id="echoid-s681" xml:space="preserve">plano tangenti, ſit is pun-<lb/>ctus, O, exiſtens, & </s> <s xml:id="echoid-s682" xml:space="preserve">in plano tangen-<lb/>te, & </s> <s xml:id="echoid-s683" xml:space="preserve">in ſuperficie cylindracea, & </s> <s xml:id="echoid-s684" xml:space="preserve"><lb/>per, O, ſit ductum latus cylindrici, <lb/>quod ſit, EM. </s> <s xml:id="echoid-s685" xml:space="preserve">Dico totum, EM, <lb/>reperiri in plano tangente cylindri-<lb/>cumin, O, ęquidiſtante ipſi, AC. </s> <s xml:id="echoid-s686" xml:space="preserve">Ducatur per, M, ipſi, BC, pa-<lb/>rallela, XR, quia ergo, XR, ęquidiſtatipſi, BC, & </s> <s xml:id="echoid-s687" xml:space="preserve">EM, ipſi, AB, <pb o="23" file="0043" n="43" rhead="LIBER I."/> vel, DC, planum per, EM, XR, ductum æquidiſtabit plauo, AC, <lb/>eſt autem planum, quod tangit cylindricum in, O, æquidiſtans pla-<lb/>no, AC, & </s> <s xml:id="echoid-s688" xml:space="preserve">tranſit per idem punctum, O, per quod tranſit planum <lb/>per, EM, XR, ductum, ergo illa duo plana fiunt vnum planum, <lb/>iacet autem, EM, in plano per, EM, XR, ducto, ergo iacet etiam <lb/>in plano ęquidiſtante ipſi, AC, & </s> <s xml:id="echoid-s689" xml:space="preserve">cylindricum, AC, tangente, igi-<lb/>tur tangit cylindricum in recta, EM. </s> <s xml:id="echoid-s690" xml:space="preserve">Eodem pacto ſi in alio pun-<lb/>cto extra, EM, in ſuperficie cylindracea ſumpto tangeret cylindri-<lb/>cum, AC, oſtenderemus tangere ipſum in latere, quod per tale pun-<lb/>ctum tranſiret; </s> <s xml:id="echoid-s691" xml:space="preserve">in quo caſu tangeret cylindricum in lateribus vno <lb/>pluribus, vt contingere poteſt. </s> <s xml:id="echoid-s692" xml:space="preserve">Tangat autem ſecundò ipſum in <lb/>plano, igitur in eo plano ſnmpto vtcumque puncto, tanget cylindri-<lb/>cum in latere tranſeunte per tale punctum, igitur planum contactus <lb/>tale eſt, vt in eo omnes ductæ rectæ lineæ æquidiſtantes ipſi, EM, <lb/>ſint latera cylindrici, AC, & </s> <s xml:id="echoid-s693" xml:space="preserve">ſubinde eidem, EM, æqualia, vnde <lb/>ſuperſicies, in qua iacent erit parallelogrammum, igitur planum <lb/>contactus in hoc caſu erit parallelogrammum, & </s> <s xml:id="echoid-s694" xml:space="preserve">erit æquiangulum <lb/>parallelogrammo, AC, nam eius latera ſunt parallela lateribus pa-<lb/>rallelogrammi, AC, & </s> <s xml:id="echoid-s695" xml:space="preserve">ideò continent angulos æquales contentis <lb/>à lateribus parallelogrammi, AC, vnde talia parallelogramma ſunt <lb/>æquiangula, igitur contactus plani æquidiſtantis plano per latera <lb/>cylindrici ducto, vel fit in latere, aut lateribus contacti cylindrici, <lb/>vel in parallelogrammo, ſiue parallelogrammis, in eiuſdem ſuper-<lb/>ficie iacentibus, & </s> <s xml:id="echoid-s696" xml:space="preserve">æquiangulis ei, quod fit a plano per latera ducto, <lb/>quod oſtendendum erat.</s> <s xml:id="echoid-s697" xml:space="preserve"/> </p> <div xml:id="echoid-div87" type="float" level="2" n="1"> <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a"> <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0042-01"/> </figure> </div> </div> <div xml:id="echoid-div89" type="section" level="1" n="66"> <head xml:id="echoid-head77" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s698" xml:space="preserve">_H_Inc babetur communes ſectiones plani tangentis, & </s> <s xml:id="echoid-s699" xml:space="preserve">cylindrici op-<lb/>poſitarum baſium productorum planorum, quæ ſint, VF, XR, <lb/>eſſe inter ſe parallelas, & </s> <s xml:id="echoid-s700" xml:space="preserve">tangere eaſdem baſes; </s> <s xml:id="echoid-s701" xml:space="preserve">ſcilicet, VF, ipſam <lb/>baſim, EAD, &</s> <s xml:id="echoid-s702" xml:space="preserve">, XR, ipſam, MBC.</s> <s xml:id="echoid-s703" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div90" type="section" level="1" n="67"> <head xml:id="echoid-head78" xml:space="preserve">THEOREMA VII. PROPOS. X.</head> <p> <s xml:id="echoid-s704" xml:space="preserve">SI cylindricus quomodocumque ſecetur per latera, diui-<lb/>ditur in cylindricos à ſecantibus planis, ſi autem ſecetur <lb/>planis omnibus eiuſdem lateribus coincidentibus inter ſe <lb/>parallelis; </s> <s xml:id="echoid-s705" xml:space="preserve">ſolidum compræhenſum conceptis in cylindrico <lb/>figuris, & </s> <s xml:id="echoid-s706" xml:space="preserve">incluſa ſuperficie cylindracea, erit cylindricus.</s> <s xml:id="echoid-s707" xml:space="preserve"/> </p> <pb o="24" file="0044" n="44" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s708" xml:space="preserve">Sit cylindricus, AE, ſectus a planis quomodocumque per latera. <lb/></s> <s xml:id="echoid-s709" xml:space="preserve">Dico per eadem diuidi in cylindricos; </s> <s xml:id="echoid-s710" xml:space="preserve">ſint autem ſecantia plana, quę <lb/>in cylindrico, AE, producant parallelogramma, AE, ME. </s> <s xml:id="echoid-s711" xml:space="preserve">Quia <lb/>igitur, AE, eſt parallelogrammum, ſi in ipſo ducantur rectæ lineæ <lb/>ipſi, AD, HE, parallelæ, & </s> <s xml:id="echoid-s712" xml:space="preserve">in, AH, DE, terminatæ, erunt ei-<lb/>ſdem, AD, HE, æquales, & </s> <s xml:id="echoid-s713" xml:space="preserve">ſubinde erunt æquales, & </s> <s xml:id="echoid-s714" xml:space="preserve">parallelæ <lb/> <anchor type="note" xlink:label="note-0044-01a" xlink:href="note-0044-01"/> regulælateris cylindrici, AE, vnde erit, AE, ſuperficies cylindra-<lb/>cea deſcripta latere, AD, ſiue latere cylindrici, AE, ergo ſolidum, <lb/>ARXE, erit cylindricus. </s> <s xml:id="echoid-s715" xml:space="preserve">Eodem pacto oſtendemus ſolida, AM <lb/>HDVE, MZHVIE, eſſe cylindricos, talibus igitur planis cy-<lb/>lindricus, AE, ſemper diuiditur in cylindricos, quæ eſt prior pars <lb/>huius Theorematis.</s> <s xml:id="echoid-s716" xml:space="preserve"/> </p> <div xml:id="echoid-div90" type="float" level="2" n="1"> <note position="left" xlink:label="note-0044-01" xlink:href="note-0044-01a" xml:space="preserve">Ex def. 3.</note> </div> <p> <s xml:id="echoid-s717" xml:space="preserve">Secetur nunc duobus planis vtcumque inter ſe parallelis coinci-<lb/>dentibus cum omnibus ciuſdem lateribus, quæ in cylindrico, AE, <lb/>producant figuras, BNGK, COFL. </s> <s xml:id="echoid-s718" xml:space="preserve">Dico ſolidum compræhen-<lb/> <anchor type="figure" xlink:label="fig-0044-01a" xlink:href="fig-0044-01"/> ſum inter has figuras, & </s> <s xml:id="echoid-s719" xml:space="preserve">ijs incluſam ſuperficiem <lb/>cylindraceam, eſſe cylindricum. </s> <s xml:id="echoid-s720" xml:space="preserve">Sintadhuc pla-<lb/>na per latera cylindrici, AE, vtcumque ducta, A <lb/>E, ME, quæ ſecent figuras, BNGK, COFL, <lb/>in rectis, BG, CF, NG, OF, igitur eiuſdem pla <lb/>ni, & </s> <s xml:id="echoid-s721" xml:space="preserve">ipſarum, BNGK, COFL, communes ſe-<lb/>ctiones erunt parallelæ, quę ſint, BG, CF, ſicut <lb/>etiam ipſæ, NG, OF, ſunt autem parallelę etiam <lb/>ipſæ, BC, NO, GF, ergo, BF, NF, erunt pa-<lb/>rallelogramma, & </s> <s xml:id="echoid-s722" xml:space="preserve">latera eorumdem, BC, GF, <lb/>NO, inter ſe æqualia, & </s> <s xml:id="echoid-s723" xml:space="preserve">æquidiſtantia; </s> <s xml:id="echoid-s724" xml:space="preserve">ſi igitur <lb/>eorum quoduis, vt, GF, ſtatuatur pro regula lateris ylindrici, ſu-<lb/>perficies incluſa duabus figuris, BNGK, COFL, erit deſcripta <lb/>vno laterum, BC, vel, NO, properante per circuitum figuræ, C <lb/>OFL, ſemper ipſi, GF, æquidiſtante, donec redeat vnde diſceſſit, <lb/>igitur hæc erit ſuperſicies cylindracea, cuius oppoſitæ baſes ipſæ fi-<lb/>guræ, BNGK, COFL, & </s> <s xml:id="echoid-s725" xml:space="preserve">ſolidum eiſdem incluſum erit cylindri-<lb/>cus, quod erat poſterior pars huius Theorematis à nobis demon-<lb/> <anchor type="note" xlink:label="note-0044-02a" xlink:href="note-0044-02"/> ſtranda.</s> <s xml:id="echoid-s726" xml:space="preserve"/> </p> <div xml:id="echoid-div91" type="float" level="2" n="2"> <figure xlink:label="fig-0044-01" xlink:href="fig-0044-01a"> <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0044-01"/> </figure> <note position="left" xlink:label="note-0044-02" xlink:href="note-0044-02a" xml:space="preserve">Def. 3.</note> </div> </div> <div xml:id="echoid-div93" type="section" level="1" n="68"> <head xml:id="echoid-head79" xml:space="preserve">THEOREMA VIII. PROPOS. XI.</head> <p> <s xml:id="echoid-s727" xml:space="preserve">CViuſuis cylindrici oppoſitæ baſes ſunt fimiles, æquales, <lb/>& </s> <s xml:id="echoid-s728" xml:space="preserve">ſimiliter poſitæ.</s> <s xml:id="echoid-s729" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s730" xml:space="preserve">Sit cylindricus, PN, cuius oppoſitæ baſes, APK, OZN. </s> <s xml:id="echoid-s731" xml:space="preserve">Dico <lb/>eas eſſe ſimiles, æquales, & </s> <s xml:id="echoid-s732" xml:space="preserve">ſimiliter poſitas. </s> <s xml:id="echoid-s733" xml:space="preserve">Ducantur vtcumque <pb o="25" file="0045" n="45" rhead="LIBER I."/> duo plana oppoſita tangentia cylindricum, PN, parallela cuidam <lb/> <anchor type="note" xlink:label="note-0045-01a" xlink:href="note-0045-01"/> per latera tranſeunti, quorum, & </s> <s xml:id="echoid-s734" xml:space="preserve">oppoſitarum baſium productarum <lb/>communes ſectiones ſint ex vna parteipſæ, VF, XL, ex alia verò, <lb/>AB, ZG, quæ tangent vel in latere, ſiue lateribus, vt in, VX, AZ, <lb/> <anchor type="note" xlink:label="note-0045-02a" xlink:href="note-0045-02"/> vel in planis, quæ erunt parallelogramma, ſint autem dicta plana, & </s> <s xml:id="echoid-s735" xml:space="preserve"><lb/>communes ſectiones, indefinitè producta, & </s> <s xml:id="echoid-s736" xml:space="preserve">in qualibet dictarum <lb/> <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> communium ſectionum, vt <lb/>in, AB, ſumpto vtcumque <lb/>puncto, B, ducatur vſque <lb/>ad oppoſitam tangentem <lb/>vtcumque in earum plano <lb/>recta, BF, illi incidens in, <lb/>F, & </s> <s xml:id="echoid-s737" xml:space="preserve">per, B, ducatur in pla-<lb/>no tangente ipſa, BG, pa-<lb/>rallela vni laterum cylin-<lb/>drici, PN, per ipſas autem, <lb/>FB, BG, intelligatur ex-<lb/>tenſum planum, quod ſe-<lb/>cet aliud planum tangens <lb/>in recta, FL, & </s> <s xml:id="echoid-s738" xml:space="preserve">planum <lb/>per, ZG, XL, ductum in <lb/>recta, GL, erunt igituripſę, <lb/>BE, GL, parallelæ, vt & </s> <s xml:id="echoid-s739" xml:space="preserve"><lb/>ipſæ, BG, FL, & </s> <s xml:id="echoid-s740" xml:space="preserve">erit, FG, <lb/>parallelogrammum. </s> <s xml:id="echoid-s741" xml:space="preserve">Du-<lb/>catur nunc intra dicta op-<lb/>poſita tangentia plana eiſdem ęquidiſtans planum, quod erit ductum <lb/>perlatera, cylindricumque, PN, ſecabit, ſit ductum perlatera, P <lb/> <anchor type="note" xlink:label="note-0045-03a" xlink:href="note-0045-03"/> O, CI, EM, KN, & </s> <s xml:id="echoid-s742" xml:space="preserve">productum ſecet planum, FG, in recta, D <lb/>H, & </s> <s xml:id="echoid-s743" xml:space="preserve">planum, quod tranſit per, AB, VF, in recta, PD, & </s> <s xml:id="echoid-s744" xml:space="preserve">quod <lb/>tranſit per, ZG, XL, in recta, OH: </s> <s xml:id="echoid-s745" xml:space="preserve">erit ergo, DH, parallela <lb/>ipſi, BG, &</s> <s xml:id="echoid-s746" xml:space="preserve">, BG, eſt parallela vni laterum cylindrici, ergo &</s> <s xml:id="echoid-s747" xml:space="preserve">, D <lb/>H, erit parallela ipſi, KN, EM, CI, PO, & </s> <s xml:id="echoid-s748" xml:space="preserve">erunt ipſa, PI, CM, <lb/>EN, KH, FH, DG, parallelogramma, & </s> <s xml:id="echoid-s749" xml:space="preserve">eorum latera oppo-<lb/>ſita inter ſe ęqualia, nempè, FD, ipſi, LH, & </s> <s xml:id="echoid-s750" xml:space="preserve">DB, ipſi, HG, D <lb/>K, ipſi, HN, DE, ipſi, HM, DC, ipſi, HI, &</s> <s xml:id="echoid-s751" xml:space="preserve">, DP, ipſi, H <lb/>O, ſunt igitur ipſæ, BF, GL, ductæ inter oppoſitas tangentes fi-<lb/>gurarum, APK, ZON, ad eundem angulum ex eadem parte, <lb/>quia angulus, BFV, eſt æqualis angulo, GLX, nam, BF, eſt pa-<lb/>rallela ipſi, GL, &</s> <s xml:id="echoid-s752" xml:space="preserve">, FV, ipſi, LX, & </s> <s xml:id="echoid-s753" xml:space="preserve">ſunt ipſæ, BF, GL, ſimili-<lb/> <anchor type="note" xlink:label="note-0045-04a" xlink:href="note-0045-04"/> ter ad eandem partem diuiſæ in punctis, D, H, per rectas, PD, O <lb/>H, parallelas ipſis oppoſitis tangentibus, quæ cum ſint vtcumque <pb o="26" file="0046" n="46" rhead="GEOMETRIÆ"/> ductæ, reperitur tamen earumdem portiones, quæiacent inter ip-<lb/>ſas, GL, BF, ex eadem parte, eodem ordine ſumptas, eſſe, vtip-<lb/>ſas, BF, GL, nam quia, DK, eſt æqualis ipſi, HN, &</s> <s xml:id="echoid-s754" xml:space="preserve">, BF, ipſi, <lb/>GL, vt, BF, ad, GL, ita eſt, DK, ad, HN, & </s> <s xml:id="echoid-s755" xml:space="preserve">ita eſſe oſtende-<lb/>mus, DE, ad, HM, DC, ad, HI, &</s> <s xml:id="echoid-s756" xml:space="preserve">, DP, ad, HO, nam iſtæ <lb/>ſunt æquales. </s> <s xml:id="echoid-s757" xml:space="preserve">Idem demonſtrabitur in cæteris, quæ ſimiliter ad ean-<lb/> <anchor type="note" xlink:label="note-0046-01a" xlink:href="note-0046-01"/> dem partem diuidunt ipſas, BF, GL, igitur figuræ, APK, ZON, <lb/>ſunt ſimiles: </s> <s xml:id="echoid-s758" xml:space="preserve">Et quia earum homologæ, tum, PC, OI, tum, EK, <lb/>MN, ſunt ęquales, quod etiam de cæteris oſtendetur eodem pacto, <lb/>ſunt enim ſemper parallelogrammorum oppoſita latera, ideò figu-<lb/>ræ, APK, ZON, nedum erunt ſimiles, ſed etiam æquales, & </s> <s xml:id="echoid-s759" xml:space="preserve">re-<lb/> <anchor type="note" xlink:label="note-0046-02a" xlink:href="note-0046-02"/> gulæ homologarum erunt ipſæ oppoſitæ tangentes, & </s> <s xml:id="echoid-s760" xml:space="preserve">ipſę, BF, G <lb/>L, earum incidentes. </s> <s xml:id="echoid-s761" xml:space="preserve">Quia verò figuræ, APK, ZON, ſunt in pla-<lb/>nis æquidiſtantibus ita conſtitutæ, vt earum incidentes ſint paralle-<lb/>læ, & </s> <s xml:id="echoid-s762" xml:space="preserve">homologæ figurarum, ZON, APK, ſunt ad eandem par-<lb/>tem incidentium poſitę, & </s> <s xml:id="echoid-s763" xml:space="preserve">item homologæ partes incidentium, B <lb/>F, GL, vt ipſæ, BD, GH, ſunt ad eandem partem pariter conſti-<lb/>tutæ, ideò figuræ, APK, ZON, nedum erunt ſimiles, & </s> <s xml:id="echoid-s764" xml:space="preserve">æqua-<lb/>les, ſed etiam ſimiliter poſitæ, quod oſtendere opus erat.</s> <s xml:id="echoid-s765" xml:space="preserve"/> </p> <div xml:id="echoid-div93" type="float" level="2" n="1"> <note position="right" xlink:label="note-0045-01" xlink:href="note-0045-01a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="right" xlink:label="note-0045-02" xlink:href="note-0045-02a" xml:space="preserve">9. Huius.</note> <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a"> <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0045-01"/> </figure> <note position="right" xlink:label="note-0045-03" xlink:href="note-0045-03a" xml:space="preserve">Ex Lem. <lb/>ſeq.</note> <note position="right" xlink:label="note-0045-04" xlink:href="note-0045-04a" xml:space="preserve">10. Vnde-<lb/>cimi Ele.</note> <note position="left" xlink:label="note-0046-01" xlink:href="note-0046-01a" xml:space="preserve">Def.10.</note> <note position="left" xlink:label="note-0046-02" xlink:href="note-0046-02a" xml:space="preserve">Aequales <lb/>homolo-<lb/>gas argue <lb/>reęquales <lb/>ſimiles fi-<lb/>guras, &è <lb/>contra, <lb/>patebit <lb/>infra in <lb/>Cor.25. <lb/>huius, ab <lb/>hac inde <lb/>pendẽter. <lb/>D. Defin. <lb/>10.</note> </div> </div> <div xml:id="echoid-div95" type="section" level="1" n="69"> <head xml:id="echoid-head80" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s766" xml:space="preserve">_M_Anifeſtum eſt autem, quia plana oppoſita tangentia cylindrici, <lb/>PN, ducta ſunt vtcumque, & </s> <s xml:id="echoid-s767" xml:space="preserve">eorum, & </s> <s xml:id="echoid-s768" xml:space="preserve">oppoſitarum baſium <lb/>productarum communes ſectiones ſunt regulæ homologarum earumdem, <lb/>quod ſi duxerimus duo alia oppoſita tangentia plana, habebimus etiam <lb/>earumdem figurarum homologas, regulis adbuc communibus ſectionibus <lb/>horum tangentium planorum poſtremò ductorum, & </s> <s xml:id="echoid-s769" xml:space="preserve">earumdem baſium <lb/>productarum, quæ communes ſectiones cum primò dictis angulos æqua-<lb/>les continebunt, nam quæ exiſtent ex. </s> <s xml:id="echoid-s770" xml:space="preserve">gr. </s> <s xml:id="echoid-s771" xml:space="preserve">in plano figuræ, APK, erunt <lb/> <anchor type="note" xlink:label="note-0046-03a" xlink:href="note-0046-03"/> parallelæ exiſtentibus in plano figuræ, ZON, igitur in oppoſitis cylin-<lb/>dricorumbaſibus homologas babebimus etiam cum quibuſuis rectis lineis <lb/>æquales angulos cum duabus quibuſuis homologarum earumdem inuen-<lb/>tis regulis continentibus, quæ igitur cum regulis homologarum oppoſi-<lb/>tarum baſium cylindrici angulos ad eandem partem continent æquales, <lb/>ſunt & </s> <s xml:id="echoid-s772" xml:space="preserve">ipſæ homologarum earumdem regulæ, neonon earundem oppoſi-<lb/>tarum baſium, & </s> <s xml:id="echoid-s773" xml:space="preserve">oppoſitarum tangentium æquè ad prædictas inclinata-<lb/>rum, etiam incidentes licebit, vt ſupra, inuenire.</s> <s xml:id="echoid-s774" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="1"> <note position="left" xlink:label="note-0046-03" xlink:href="note-0046-03a" xml:space="preserve">_10. Vnde-_ <lb/>_cimi Ele._</note> </div> <pb o="27" file="0047" n="47" rhead="LIBER I."/> </div> <div xml:id="echoid-div97" type="section" level="1" n="70"> <head xml:id="echoid-head81" xml:space="preserve">LEMMA PRO ANTECED. PROP.</head> <p> <s xml:id="echoid-s775" xml:space="preserve">DEſiderari tantum videtur huius euidentia, quod ſcilicet planum <lb/>inter oppoſita tangentia plana eiſdem a quidiſtanter ductum <lb/>tranſeat perlatera cylindrici, quod aſſumpra eiuſdem figura nunc <lb/> <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/> fiet manifeſtum; </s> <s xml:id="echoid-s776" xml:space="preserve">intelliga-<lb/>tur ergo in ambitu vtriuſuis <lb/>oppoſitarum baſium cylin-<lb/>drici, PN, ſumptum pun-<lb/>ctum, vt, O, in ambitu fi-<lb/>guræ, ZON. </s> <s xml:id="echoid-s777" xml:space="preserve">Dico pla-<lb/>num, quod tranſit per, O, <lb/>æquidiſtans planis tangen-<lb/>tibus, AG, VL, tranſire <lb/>per latera cylindrici, PN. <lb/></s> <s xml:id="echoid-s778" xml:space="preserve">Ducatur ergo à puncto, O, <lb/>latus cylindrici, PO, & </s> <s xml:id="echoid-s779" xml:space="preserve">ab <lb/>eodem pundo, O, in baſi, <lb/>ZON, recta, ON, paral-<lb/>lela ipſi, XL, igitur pla-<lb/>num, quod tranſit per, P <lb/>O, ON, æquidiſtat plano, <lb/> <anchor type="note" xlink:label="note-0047-01a" xlink:href="note-0047-01"/> VL, nam, PO, ipſi, VX, <lb/>lateri cylindrici, &</s> <s xml:id="echoid-s780" xml:space="preserve">, ON, <lb/>ipſi, XL, ęquidiſtat, quod <lb/>ergo ducitur per, O, eidem plano tangenti æquidiſtans tranſit per <lb/>ipſas, PO, ON, ſi. </s> <s xml:id="echoid-s781" xml:space="preserve">n. </s> <s xml:id="echoid-s782" xml:space="preserve">non, erunt duo plana eidem plano, VL, <lb/>æquidiſtantia, & </s> <s xml:id="echoid-s783" xml:space="preserve">ideò inter ſe æquidiſtantia, quibus communis erit <lb/>punctus, O, igitur in eo concurrent, quod eſt abſurdum, non ergo <lb/>illa ſunt duo plana, ſed vnum tantum, illud nempè, quod ducitur <lb/>per punctum, O, ipſi plano, VL, æquidiſtans, tranſitque per, P <lb/>O, ON, neceſſariò: </s> <s xml:id="echoid-s784" xml:space="preserve">Siverò à punctis, I, M, N, erigantur latera <lb/>cylindrici, CI, ME, NK, erunt cuncta in plano per, PO, ON, <lb/>tranſeunte, ergo planum, quod ducitur per punctum, O, æquidi-<lb/>ſtans plano, VL, cylindricum tangenti tranſit per latera, PO, CI, <lb/>EM, KN, quod oſtendendum erat.</s> <s xml:id="echoid-s785" xml:space="preserve"/> </p> <div xml:id="echoid-div97" type="float" level="2" n="1"> <figure xlink:label="fig-0047-01" xlink:href="fig-0047-01a"> <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0047-01"/> </figure> <note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">15. Vnde-<lb/>cimi Ele.</note> </div> </div> <div xml:id="echoid-div99" type="section" level="1" n="71"> <head xml:id="echoid-head82" xml:space="preserve">THEOREMA IX. PROPOS. XII.</head> <p> <s xml:id="echoid-s786" xml:space="preserve">SI cylindricus planis ſecetur quomodocumque pet latera <lb/>ductis, eiuſdem oppoſitę baſes in figuras ſimiles, ęqua-<lb/>les, & </s> <s xml:id="echoid-s787" xml:space="preserve">ſimiliter poſitas diuiduntur, tales autem erunt, quæ <pb o="28" file="0048" n="48" rhead="GEOMETRIÆ"/> ad eandem partem ſecantium planorum exiſtent: </s> <s xml:id="echoid-s788" xml:space="preserve">Et ſi idem <lb/>ſecetur planis parallelis quomodocumq; </s> <s xml:id="echoid-s789" xml:space="preserve">omnibus eiuſdem <lb/>lateribus coincidentibus, conceptæ in cylindrico figuræ e-<lb/>runt ſimiles, æquales, & </s> <s xml:id="echoid-s790" xml:space="preserve">ſimiliter poſitæ.</s> <s xml:id="echoid-s791" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s792" xml:space="preserve">Conſpiciatur figura Propoſit. </s> <s xml:id="echoid-s793" xml:space="preserve">10. </s> <s xml:id="echoid-s794" xml:space="preserve">in qua iam propoſitas ſectiones <lb/>habemus, plana enim, AE, ME, tranſeuntia per cylindrici latera <lb/> <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/> ipſum ſecant, & </s> <s xml:id="echoid-s795" xml:space="preserve">plana, BNG, COF, omni-<lb/>bus eiuſdem lateribus coincidunt, & </s> <s xml:id="echoid-s796" xml:space="preserve">ſunt paral-<lb/>lela. </s> <s xml:id="echoid-s797" xml:space="preserve">Dico ergo figuras, MZH, EIV, eſſe ſi-<lb/>miles, & </s> <s xml:id="echoid-s798" xml:space="preserve">æquales, & </s> <s xml:id="echoid-s799" xml:space="preserve">ſimiliter poſitas, quod pa-<lb/>tet, nam illæ ſunt cylindrici, MHZI, oppoſi-<lb/>tæ baſes; </s> <s xml:id="echoid-s800" xml:space="preserve">idem eodem modo probabitur de figu-<lb/>ris, AMH, DVE, & </s> <s xml:id="echoid-s801" xml:space="preserve">de, ARH, DXE, & </s> <s xml:id="echoid-s802" xml:space="preserve"><lb/>tandem oſtendemus pariter figuras, BNGK, C <lb/>OFL, eſſe ſimiles, æquales, & </s> <s xml:id="echoid-s803" xml:space="preserve">ſimiliter poſitas, <lb/> <anchor type="note" xlink:label="note-0048-01a" xlink:href="note-0048-01"/> quia ſunt cylindrici, BF, oppoſitæ baſes, quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s804" xml:space="preserve"/> </p> <div xml:id="echoid-div99" type="float" level="2" n="1"> <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a"> <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0048-01"/> </figure> <note position="left" xlink:label="note-0048-01" xlink:href="note-0048-01a" xml:space="preserve">11. Huius.</note> </div> </div> <div xml:id="echoid-div101" type="section" level="1" n="72"> <head xml:id="echoid-head83" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s805" xml:space="preserve">_H_Inc apparet, quamuis figuram planam ex ſectione plani, oppofi-<lb/>tis baſibus cylindrici æquidiſtantis, in eo productam, eiſdem op-<lb/>poſitis baſibus eſſe ſimilem, æqualem, & </s> <s xml:id="echoid-s806" xml:space="preserve">ſimiliter poſitam.</s> <s xml:id="echoid-s807" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div102" type="section" level="1" n="73"> <head xml:id="echoid-head84" xml:space="preserve">THEOREMA X. PROPOS. XIII.</head> <p> <s xml:id="echoid-s808" xml:space="preserve">SI quis cylindricus ſecetur plano per latera, deinde ſece-<lb/>tur planis oppoſitis eiuſdem baſibus æquidiſtantibus: <lb/></s> <s xml:id="echoid-s809" xml:space="preserve">Communes ſectiones plani per latera ducti, & </s> <s xml:id="echoid-s810" xml:space="preserve">planorum ba-<lb/>ſibus æquidiſtantium, erunt lineæ, vellatera homologa fi-<lb/>gurarum ſimilium, quæ ex ſectione æquidiſtantium plano-<lb/>rum in cylindrico effecta in eodem producuntur.</s> <s xml:id="echoid-s811" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s812" xml:space="preserve">Sit cylindricus, ADM, cuius oppoſitæ baſes, ABC, TDF, ſe-<lb/>cetur autem plano vtcumque per latera ducto, quod in eo producat <lb/>parallelogrammum, BF, & </s> <s xml:id="echoid-s813" xml:space="preserve">alio vtcumque plano oppoſitis baſibus <lb/>æquidiſtante, quodin eo producat figuram, YNXO, & </s> <s xml:id="echoid-s814" xml:space="preserve">in paral-<lb/>lelogrammo, BF, rectam, NO. </s> <s xml:id="echoid-s815" xml:space="preserve">Dicorectas, DF, NO, BC, eſſe <lb/>lineas, vellatera homologa figurarum, TDF, YNO, ABC, ſi- <pb o="29" file="0049" n="49" rhead="LIBER I."/> milium. </s> <s xml:id="echoid-s816" xml:space="preserve">Ducantur plana oppoſita tangentia cylindrici, AM, re-<lb/> <anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/> ſpectu plani, BF, in eo ducti, vnius quorum, & </s> <s xml:id="echoid-s817" xml:space="preserve">planorum figura-<lb/>rum, YNO, TDF, productorum, communes ſectiones ſint, XS, <lb/>MG, alterius autem, & </s> <s xml:id="echoid-s818" xml:space="preserve">eorundem planorum ſint rectæ, YP, TQ, <lb/>indefinitè ambæ productæ, ſumpto autem in, YP, vtcumque pun-<lb/>cto, P, ducatur per, P, ipſi, CF, æquidiſtans, PQ, & </s> <s xml:id="echoid-s819" xml:space="preserve">ab eodem <lb/>in plano per, YP, XS, tranſeunte vſquead, XS, ducatur vtcum-<lb/>queipſa, PS, per ipſas autem, QP, PS, intelligatur extenſum pla-<lb/>num, quod ſecetaliud tangens planum in, SG, & </s> <s xml:id="echoid-s820" xml:space="preserve">planum per, T <lb/> <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/> Q, MG, ductum in, QG, producan-<lb/>tur autem ipſæ, NO, DF, verſus, P <lb/>S, QG, quibus occurrant in, V, R, <lb/>& </s> <s xml:id="echoid-s821" xml:space="preserve">iungatur, VR, erunt igitur, VR, <lb/>PQ, communes ſectiones æquidiſtan-<lb/>tium planorum, YQ, NR, & </s> <s xml:id="echoid-s822" xml:space="preserve">plani, <lb/>PR, & </s> <s xml:id="echoid-s823" xml:space="preserve">ideò erunt parallelæ, vt & </s> <s xml:id="echoid-s824" xml:space="preserve">ip-<lb/>ſæ, PV, QR, &</s> <s xml:id="echoid-s825" xml:space="preserve">, PR, erit paralle-<lb/>logrammum: </s> <s xml:id="echoid-s826" xml:space="preserve">Similiter, vt in Prop. </s> <s xml:id="echoid-s827" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s828" xml:space="preserve">oſtendemus eſſe parallelogramma ip-<lb/>ſa, VG, PG, NF, OR, NR, & </s> <s xml:id="echoid-s829" xml:space="preserve">an-<lb/>gulum, PSX, æqualem eſſe angulo, <lb/>QGM, & </s> <s xml:id="echoid-s830" xml:space="preserve">tandem, PS, QG, eſſe in-<lb/> <anchor type="note" xlink:label="note-0049-02a" xlink:href="note-0049-02"/> cidentes ſimilium figurarum, YNO, <lb/>TDF, & </s> <s xml:id="echoid-s831" xml:space="preserve">oppofitarum tangentium, YP, XS, TQ, MG, & </s> <s xml:id="echoid-s832" xml:space="preserve">tan-<lb/>gentes eſſe homologarum earundem regulas, & </s> <s xml:id="echoid-s833" xml:space="preserve">quia eiſdem æqui-<lb/>diſtant ipſæ, NO, DF, & </s> <s xml:id="echoid-s834" xml:space="preserve">productæ ſimiliter, & </s> <s xml:id="echoid-s835" xml:space="preserve">ad eandem par-<lb/>tem ipſas incidentes, PS, QG, diuidunt; </s> <s xml:id="echoid-s836" xml:space="preserve">nam, PV, æquatur ipſi, <lb/>QR, &</s> <s xml:id="echoid-s837" xml:space="preserve">, VS, ipſi, RG, ideò ipſæ, NO, DF, erunt lineæ homo-<lb/>logæ figurarum, YNO, TDF, ſimilium, quæ in plures homolo-<lb/>gas ſecari contingere poteſt, prout ſe habet ambitus ſuperficiei cy-<lb/>lindraceæ huius cylindrici, AM, ſunt lineæ homologæ inquam, ſi <lb/> <anchor type="note" xlink:label="note-0049-03a" xlink:href="note-0049-03"/> ſint intra ambitum figurarum, quarum ſunt homologæ, ſunt verò <lb/>latera homologa, ſi ſint in earundem ambitu, veluti contingeret ſi <lb/>planum per latera ductum eſſet planum contactus vnius oppoſito-<lb/>rum tangentium, veluti ſi cylindricus fuiſſet, cuius oppoſitæ baſes <lb/>ſunt, ABC, TDF, excluſis reſiduis figuris, quæ ab ipſis, BC, D <lb/>F, abſcinduntur, tunc enim eodem modo facta fuiſſet demonſtra-<lb/>tio, vt conſideranti facilè patebit; </s> <s xml:id="echoid-s838" xml:space="preserve">idem oſtendemus in recta, BC, <lb/>& </s> <s xml:id="echoid-s839" xml:space="preserve">in quibuſuis alijs, quæ ſunt communes ſectiones planorum baſi-<lb/>bus æquidiſtantium, & </s> <s xml:id="echoid-s840" xml:space="preserve">parallelogrammi, BF, probantes ſcilicet <lb/>eaſdem eſſe lineas, vellatera homologa figurarum in cylindrico per <lb/>baſibus æquidiſtantia plana productarum, quod oſtendere opus erat.</s> <s xml:id="echoid-s841" xml:space="preserve"/> </p> <div xml:id="echoid-div102" type="float" level="2" n="1"> <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <figure xlink:label="fig-0049-01" xlink:href="fig-0049-01a"> <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0049-01"/> </figure> <note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">B. Def. 10. <lb/>huius.</note> <note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">C. Def. 10. <lb/>huius.</note> </div> <pb o="30" file="0050" n="50" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div104" type="section" level="1" n="74"> <head xml:id="echoid-head85" xml:space="preserve">THEOREMA XI. PROPOS. XIV.</head> <p> <s xml:id="echoid-s842" xml:space="preserve">SI duæ figuræ planæ non exiſtentes in eodem plano fue-<lb/>rint ſimiles, æquales, & </s> <s xml:id="echoid-s843" xml:space="preserve">ſim iliter poſitæ, illæ erunt cu-<lb/>iuſdam cylindrici oppoſitæ baſes.</s> <s xml:id="echoid-s844" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s845" xml:space="preserve">Sint duæ ſimiles figuræ planæ, & </s> <s xml:id="echoid-s846" xml:space="preserve">æquales, AQTO, FDNC, <lb/>non exiſtentes in eodem plano, & </s> <s xml:id="echoid-s847" xml:space="preserve">ſimiliter poſitæ. </s> <s xml:id="echoid-s848" xml:space="preserve">Dico eas eſſe <lb/>cuiuſdam cylindrici oppoſitas baſes. </s> <s xml:id="echoid-s849" xml:space="preserve">Quoniam enim ſunt ſimiliter <lb/>poſitæ erunt inter ſe æquidiſtantes, & </s> <s xml:id="echoid-s850" xml:space="preserve">earum incidentes pariter inter <lb/> <anchor type="note" xlink:label="note-0050-01a" xlink:href="note-0050-01"/> ſe æquidiſtantes, ducantur oppoſitæ tangentes figuræ, AQTO, <lb/>quæ ſint, TP, AB, & </s> <s xml:id="echoid-s851" xml:space="preserve">figuræ, FDNC, quæ ſint, FH, NL, <lb/> <anchor type="note" xlink:label="note-0050-02a" xlink:href="note-0050-02"/> quæque ſint regulæ homologarum earumdem ſimilium figurarum, <lb/>& </s> <s xml:id="echoid-s852" xml:space="preserve">ſint incidentes earum, & </s> <s xml:id="echoid-s853" xml:space="preserve">ſimilium figurarum ipſę, BP, HL, quę <lb/>erunt parallelæ, & </s> <s xml:id="echoid-s854" xml:space="preserve">quia ſunt incidentes ſimilium figurarum, AT, <lb/> <anchor type="note" xlink:label="note-0050-03a" xlink:href="note-0050-03"/> FN, & </s> <s xml:id="echoid-s855" xml:space="preserve">oppoſitarum tangentium iam ductarum, ideò ad eaſdem ex <lb/>eadem parte efficient angulos æquales, igitur angulus, BPT, erit <lb/> <anchor type="note" xlink:label="note-0050-04a" xlink:href="note-0050-04"/> æqualis angulo, HLN, & </s> <s xml:id="echoid-s856" xml:space="preserve">ideò etiam, PT, ęquidiſtabitipſi, LN, <lb/> <anchor type="note" xlink:label="note-0050-05a" xlink:href="note-0050-05"/> <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> &</s> <s xml:id="echoid-s857" xml:space="preserve">, BA, ipſi, FH, iungantur, BH, PL, <lb/>quoniam ergo, AT, FN, ſunt ſimiles, <lb/>& </s> <s xml:id="echoid-s858" xml:space="preserve">æquales, earum homologæ erunt pa-<lb/>riter æquales, ſunt autem incidentes, BP, <lb/>HL, vt ipſæ homologæ, vt colligitur in <lb/>Coroll. </s> <s xml:id="echoid-s859" xml:space="preserve">1. </s> <s xml:id="echoid-s860" xml:space="preserve">ſequentis Propoſit. </s> <s xml:id="echoid-s861" xml:space="preserve">22. </s> <s xml:id="echoid-s862" xml:space="preserve">indepen-<lb/>denter ab hac Propoſitione, ergo, BP, H <lb/>L, erunt æquales, & </s> <s xml:id="echoid-s863" xml:space="preserve">ſunt æquidiſtantes, <lb/>ergo eas iungentes, BH, PL, erunt ęqua-<lb/>les, & </s> <s xml:id="echoid-s864" xml:space="preserve">æquidiſtantes. </s> <s xml:id="echoid-s865" xml:space="preserve">Diuidantur ipſę in-<lb/>cidentes, BP, HL, ſimiliter ad eandem <lb/> <anchor type="note" xlink:label="note-0050-06a" xlink:href="note-0050-06"/> partem in punctis, E, M, G, K, & </s> <s xml:id="echoid-s866" xml:space="preserve">iun-<lb/>gantur, EG, MK, erit ergo, MP, ęqua-<lb/>lis ipſi, KL, &</s> <s xml:id="echoid-s867" xml:space="preserve">, EM, ipſi, GK, &</s> <s xml:id="echoid-s868" xml:space="preserve">, BE, <lb/>ipſi, HG, nam quia, BP, HL, ſimiliter <lb/>diuiduntur in his punctis, earum partes ſunt, vt ipſæ integræ, illæ <lb/>verò ſunt æquales, & </s> <s xml:id="echoid-s869" xml:space="preserve">ideò etiam homologæ partes ſunt æquales, & </s> <s xml:id="echoid-s870" xml:space="preserve"><lb/>eas iungentes, PL, MK, EG, BH, erunt æquales, & </s> <s xml:id="echoid-s871" xml:space="preserve">æquidiſtan-<lb/> <anchor type="note" xlink:label="note-0050-07a" xlink:href="note-0050-07"/> tes, ducatur à puncto, K, verſus figuram, FN, ipſa, KR, æquidi-<lb/>ſtans ipſi, NL, quia ergo, MK, æquidiſtat ipſi, PL, &</s> <s xml:id="echoid-s872" xml:space="preserve">, RK, <lb/>ipſi, NL, planum per, MK, KR, tranſiens æquidiſtat tranſeunti <lb/>per, PL, LN, ſecet hoc planum tranſiens per, MK, KR, pla-<lb/>num, AT, productum, in recta, SM, & </s> <s xml:id="echoid-s873" xml:space="preserve">iungantur, SR, VI, erit <pb o="31" file="0051" n="51" rhead="LIBERI."/> ergo, SM, æquidiſtans ipſi, TP, regulæ homologarum figurę, A <lb/>T, veluti, RK, æquidiſtat ipſi, NL, regulæ homologarum figu-<lb/>ræ, F N, & </s> <s xml:id="echoid-s874" xml:space="preserve">ſecant incidentes, BP, HL, ſimiliter ad eandem par-<lb/>tem in punctis, M, K, ergo ipſæ, SV, RI, erunt homologę di-<lb/>ctarum figurarum ſimilium, & </s> <s xml:id="echoid-s875" xml:space="preserve">ęqualium, quę ideò erunt æquales, <lb/>ſicut etiam ipſæ, VM, IK. </s> <s xml:id="echoid-s876" xml:space="preserve">& </s> <s xml:id="echoid-s877" xml:space="preserve">ſunt ęquidiſtantes, ergo eas iungen-<lb/>tes erunt ęquales, & </s> <s xml:id="echoid-s878" xml:space="preserve">ęquidiſtantes, ſcilicet, SR, VI, MK, eſtau-<lb/>tem, MK, parallela, & </s> <s xml:id="echoid-s879" xml:space="preserve">ęqualis ipſi, PL, ergo, SR, VI, erunt <lb/>ęquales, & </s> <s xml:id="echoid-s880" xml:space="preserve">parallelęipſi, PL: </s> <s xml:id="echoid-s881" xml:space="preserve">Eodem pacto per, EG, extendentes <lb/>planum ęquidiſtans plano, TL, quod ſecet figurarum, AT, FN, <lb/>productarum plana in rectis, QE, DG, oſtendemus ipſas, QO, D <lb/>C, eſſe homologas figurarum ſimilium, & </s> <s xml:id="echoid-s882" xml:space="preserve">ęqualium, AT, FN, & </s> <s xml:id="echoid-s883" xml:space="preserve"><lb/>ideò eas eſſe ęquales, vt & </s> <s xml:id="echoid-s884" xml:space="preserve">ipſas, OE, CG, ergo ſi iungantur, QD, <lb/>OC, iſtę erunt ęquales, & </s> <s xml:id="echoid-s885" xml:space="preserve">parallelę ipfi, EG, ideſt ipſi, PL; </s> <s xml:id="echoid-s886" xml:space="preserve">ſimi-<lb/>liter in cæteris planis procedemus, quæ inter plana, TL, AH, ip-<lb/>ſis æquidiſtantia ducuntur, oſtendentes, quæ iungunt extrema ho-<lb/>mologarum earundem figurarum, AT, FN, eſſe æquales, & </s> <s xml:id="echoid-s887" xml:space="preserve">ęqui-<lb/>diſtantesipſi, PL, ſi igitur, PL, regula ſtatuatur, erunt omnes di-<lb/>ctæ iungentes in ſuperficie quadam, per quam ipſi, PL, properan-<lb/>te quadam recta linea æquali ſemper ęquidiſtanter, eiuſdem extrema <lb/> <anchor type="note" xlink:label="note-0051-01a" xlink:href="note-0051-01"/> iugiter manent in ambitu ſigurarum, AT, FN, ergo hæc erit ſu-<lb/>perficies cylindrici, cuius oppoſitę baſes erunt ipſę, AT, FN, ſunt <lb/>igitur, AT, FN, cylindrici cuiuſdam (nempè cuius latus eſt quod-<lb/>uis ipſorum, QD, SR, VI, OC,) oppoſirę baſes, quod erat no-<lb/>bis oſtendendum.</s> <s xml:id="echoid-s888" xml:space="preserve"/> </p> <div xml:id="echoid-div104" type="float" level="2" n="1"> <note position="left" xlink:label="note-0050-01" xlink:href="note-0050-01a" xml:space="preserve">D. Def. 10</note> <note position="left" xlink:label="note-0050-02" xlink:href="note-0050-02a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="left" xlink:label="note-0050-03" xlink:href="note-0050-03a" xml:space="preserve">D. Def. 10.</note> <note position="left" xlink:label="note-0050-04" xlink:href="note-0050-04a" xml:space="preserve">B. Def. 10.</note> <note position="left" xlink:label="note-0050-05" xlink:href="note-0050-05a" xml:space="preserve">Excõuer. <lb/>ſa 10. Vn-<lb/>dec. Ele.</note> <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a"> <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0050-01"/> </figure> <note position="left" xlink:label="note-0050-06" xlink:href="note-0050-06a" xml:space="preserve">10. Sexti <lb/>Elem.</note> <note position="left" xlink:label="note-0050-07" xlink:href="note-0050-07a" xml:space="preserve">15. Vnde-<lb/>cimi El.</note> <note position="right" xlink:label="note-0051-01" xlink:href="note-0051-01a" xml:space="preserve">Def.3.</note> </div> </div> <div xml:id="echoid-div106" type="section" level="1" n="75"> <head xml:id="echoid-head86" xml:space="preserve">THEOREMA XII. PROPOS. XV.</head> <p> <s xml:id="echoid-s889" xml:space="preserve">PVnctus manens, cui in reuolutione innititur latus coni-<lb/>ci, eſt vnicus vertex conicireſpectu eiuſdem baſis.</s> <s xml:id="echoid-s890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s891" xml:space="preserve">Sit conicus, ABD, baſis, BD, punctus, eui innititur latus co-<lb/> <anchor type="note" xlink:label="note-0051-02a" xlink:href="note-0051-02"/> nici, ABD, in reuolutione, quę ab eo fit per circuitum baſis, BD, <lb/> <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> ſit, A. </s> <s xml:id="echoid-s892" xml:space="preserve">Dico, A, eſſe vnicum verticem conici, A <lb/>BD, reſpectu baſis, BD. </s> <s xml:id="echoid-s893" xml:space="preserve">Intelligatur per pun-<lb/>ctum, A, ductum planum ęquidiſtans baſi, dico <lb/>hoc planum tantummodo in hoc puncto tangere <lb/>conicum, ſi enim poſſibile eſt eundem tangat, ſeu <lb/>ſecet in duobus punctis, vt in, C, A, iuncta ergo, <lb/>AC, illa erit in ſuperficie coniculari, & </s> <s xml:id="echoid-s894" xml:space="preserve">cum de-<lb/>ſcendat à puncto, A, per ipſum tranſiet aliquando <lb/>latus conici, vt, AB, igitur, AB, erit in plano ducto per, A, baſi, <pb o="32" file="0052" n="52" rhead="GEOMETRIÆ"/> BD, ęquidiſtante, & </s> <s xml:id="echoid-s895" xml:space="preserve">quia latus, AB, indefinitè productum oc-<lb/>currit baſi, etiam dictum baſi ęquidiſtans planum occurret indefini-<lb/>tè productum ipſi baſi, quod eſt abſurdum, non igitur planum du-<lb/>ctum per, A, baſi, BD, ęquidiſtans conicum tangit vel ſecat in a-<lb/>lio, quam in puncto, A, ergo, A, erit illius vnicus vertex reſpectu <lb/>baſis, BD, quod erat oſtendendum.</s> <s xml:id="echoid-s896" xml:space="preserve"/> </p> <div xml:id="echoid-div106" type="float" level="2" n="1"> <note position="right" xlink:label="note-0051-02" xlink:href="note-0051-02a" xml:space="preserve">A. Def.4.</note> <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a"> <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0051-01"/> </figure> </div> </div> <div xml:id="echoid-div108" type="section" level="1" n="76"> <head xml:id="echoid-head87" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s897" xml:space="preserve">_C_Vm autem dicemus verticem alicuius conici, intelligemus ſemper <lb/>ipſum reſpectu baſis aſſumptum, ideſt punctum, curin reuolutie-<lb/>ne innititur latus cylindrict, niſi aliud explicetur.</s> <s xml:id="echoid-s898" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div109" type="section" level="1" n="77"> <head xml:id="echoid-head88" xml:space="preserve">THEOREMA XIII. PROPOS. XVI.</head> <p> <s xml:id="echoid-s899" xml:space="preserve">SI conicus ſecetur vtcumque per verticem ducto plano, <lb/>concepta in ipſo ſigura, vel figuræ, erit triangulus, vel <lb/>trianguli.</s> <s xml:id="echoid-s900" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s901" xml:space="preserve">Secetur quilibet conicus, ABF, plano vtcumque per verticem <lb/>ducto, quod in eo producat figuram, ſiue figuras, ABC, AEF. <lb/></s> <s xml:id="echoid-s902" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> Dico eas eſſe triangulos, Sit com-<lb/>munis ſectio illius, & </s> <s xml:id="echoid-s903" xml:space="preserve">baſis pro-<lb/>ducti plani, tota, BF, cuius, CE, <lb/>portio maneat extra baſim, eſt igi-<lb/>tur, BF, recta linea, dico etiam <lb/>eſſe rectas ipſas, AB, AC, AE, <lb/>AF, ſienim non eſt, AB, recta, <lb/>ducatur in plano figurę, ABC, re-<lb/>cta, AOB, igitur, AOB, quę <lb/>iungit punctum, B, & </s> <s xml:id="echoid-s904" xml:space="preserve">verticem coni eſt latus conici, ABF, ergo <lb/>eſt in ſuperficie coniculari, & </s> <s xml:id="echoid-s905" xml:space="preserve">eſt etiam in plano figurę, ABC, ergo <lb/>eſt in eorum communi ſectione, ideſt cadit ſuper, AB, igitur, AB, <lb/>erit recta linea, eodem modo oſtendemus ipſas, AC, AE, AF, eſſe <lb/>rectas, & </s> <s xml:id="echoid-s906" xml:space="preserve">ideò erit, ABC, triangulus, vt etiam, AEF, quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s907" xml:space="preserve"/> </p> <div xml:id="echoid-div109" type="float" level="2" n="1"> <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a"> <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0052-01"/> </figure> </div> </div> <div xml:id="echoid-div111" type="section" level="1" n="78"> <head xml:id="echoid-head89" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s908" xml:space="preserve">_E_Odem modo nobis innoteſcit figuras, quæ extra conicum fiunt eſſe <lb/>triangulos, ideſt, ACE, eſſe triangulum, & </s> <s xml:id="echoid-s909" xml:space="preserve">qui ex ipſis inte-<lb/>gratur, ſcilicet, ABF, pariter eſſe triangulum.</s> <s xml:id="echoid-s910" xml:space="preserve"/> </p> <pb o="33" file="0053" n="53" rhead="LIBERI."/> </div> <div xml:id="echoid-div112" type="section" level="1" n="79"> <head xml:id="echoid-head90" xml:space="preserve">THEOREMA XIV. PROPOS. XVII.</head> <p> <s xml:id="echoid-s911" xml:space="preserve">SI conicus ſecetur vtcumque planis per verticem, diuidi-<lb/>tur ab eiſdem in conicos: </s> <s xml:id="echoid-s912" xml:space="preserve">Etſiſecetur vtcumque planis <lb/>coincidentibus omnibus eiuſdem lateribus, ſolida ab ijſdem <lb/>abſciſſa verſus verticem erunt pariter conici, & </s> <s xml:id="echoid-s913" xml:space="preserve">eorum baſes <lb/>ipſæ ſiguræ abſcindentes.</s> <s xml:id="echoid-s914" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s915" xml:space="preserve">Sit quilibet conicus, AMV, ſectus plano vtcum que per verticem <lb/>ducto, quod in eo producat triangulum, ACD. </s> <s xml:id="echoid-s916" xml:space="preserve">Dico ab hoc pla-<lb/>no ſecante in conicos, ACVD, ACMD, fuiſſe diuiſum. </s> <s xml:id="echoid-s917" xml:space="preserve">Sin. </s> <s xml:id="echoid-s918" xml:space="preserve">in-<lb/>telligamus latus trianguli, ACD, quod ſit, AC, vel, AD, inni-<lb/>xum puncto, A, indefinitè productum ferri per rectam, CD, ipſa <lb/>deſcribet ſuperriciem trianguli, ACD, ad modum ſuperficiei coni-<lb/>cularis, eſt autem reliqua, quę inſiſtit ambitui, CVD, ſic deſcripta, <lb/> <anchor type="figure" xlink:label="fig-0053-01a" xlink:href="fig-0053-01"/> ergo tota ſuperficies, ACDV, eſt co-<lb/> <anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/> nicularis deſcripta latere, AC, vel, AD, <lb/>properante per circuitum figuræ planæ, <lb/>CVD, ergo erit, ACVD, conicus, <lb/>cuius baſis ipſa figura, CVD, & </s> <s xml:id="echoid-s919" xml:space="preserve">ver-<lb/>tex, A. </s> <s xml:id="echoid-s920" xml:space="preserve">Eodem modo oſtendemus, A <lb/>CMD, eſſe conicum, cuius baſis, CM <lb/>D, vertex, A. </s> <s xml:id="echoid-s921" xml:space="preserve">Secetur nunc plano vt-<lb/>cumque omnibus conici, AMV, late-<lb/>ribus concidente, quod in eo producat <lb/>figuram, BNEO. </s> <s xml:id="echoid-s922" xml:space="preserve">Dico, ANO, eſſe <lb/>conicum, cuius baſis figura, BNEO, <lb/>vertex, A, nam dum latus conici, AM <lb/>V, properat per circuitum baſis, CMD <lb/>V, vt deſcribat eius conicularem ſuperficiem, properat etiam per <lb/>circuitum figurę, BNEO, & </s> <s xml:id="echoid-s923" xml:space="preserve">deſcribit ſupra ipiam ſuperficiem co-<lb/>nicularem, igitur ſuperficies ab eadem figura, BE, ablciſia verſus, <lb/>A, eſt conicularis, & </s> <s xml:id="echoid-s924" xml:space="preserve">ſolidum comprehenſum ab ipſa, & </s> <s xml:id="echoid-s925" xml:space="preserve">figura pla-<lb/>na, BNEO, erit conicus, & </s> <s xml:id="echoid-s926" xml:space="preserve">eiuſdem baſis ipſa figura, BNEO, <lb/>vertex autem, A, quod oſtendere opus erat.</s> <s xml:id="echoid-s927" xml:space="preserve"/> </p> <div xml:id="echoid-div112" type="float" level="2" n="1"> <figure xlink:label="fig-0053-01" xlink:href="fig-0053-01a"> <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0053-01"/> </figure> <note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">AD, ef.4.</note> </div> </div> <div xml:id="echoid-div114" type="section" level="1" n="80"> <head xml:id="echoid-head91" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s928" xml:space="preserve">_H_Inc habetur, ſi planum tranſeat per verticem conici, & </s> <s xml:id="echoid-s929" xml:space="preserve">quamli-<lb/>bet rectam lineam intra baſim conici exiſtentem, qui quidem ſe-<lb/>cetur alio plano coincidente cum omnibus eiuſdem conici lateribus, <pb o="34" file="0054" n="54" rhead="GEOMETRIÆ"/> communem ſectionem borum duorum planorum fore intra figuram in, <lb/>conico productan à plano omnibus eiuſdem lateribus coincidente, vt <lb/>patet in conico, ACD qui ſecatur plano, ACD, & </s> <s xml:id="echoid-s930" xml:space="preserve">alio, BNEO, <lb/>quorum com nunis ſectio ſit, BE. </s> <s xml:id="echoid-s931" xml:space="preserve">Dico n. </s> <s xml:id="echoid-s932" xml:space="preserve">ſi, CD, ſit intra figuram, C <lb/>MDV, etiam, BE, fore intra figuram, BNEO, nam, ACVD, e§t <lb/>conicus, & </s> <s xml:id="echoid-s933" xml:space="preserve">quia latera non vniuntur, niſi in puncto, A, ideo, BOE, <lb/>eſt aliqua figura, vt etiam, BNE, & </s> <s xml:id="echoid-s934" xml:space="preserve">ideò, BE, cadit intra figuram, <lb/>BNEO.</s> <s xml:id="echoid-s935" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div115" type="section" level="1" n="81"> <head xml:id="echoid-head92" xml:space="preserve">THEOREMA XV. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s936" xml:space="preserve">SI per verticem conici, & </s> <s xml:id="echoid-s937" xml:space="preserve">rectam tangentem eius baſim <lb/>extendatur planum, hoc tanget ipſum conicum in vna, <lb/>vel pluribus rectis lineis, quę erunt latera conici, velin pla-<lb/>no tranſeunte per eiuſdem latera, quod erit triangulum, ſiue <lb/>in plurib us triangulis.</s> <s xml:id="echoid-s938" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s939" xml:space="preserve">Sit conicus, cuius vertex, A, baſis, BCE, quam tangat recta, <lb/>DF, in puncto, vel punctis, ſiue in linea. </s> <s xml:id="echoid-s940" xml:space="preserve">Dico planum, ADF, <lb/>tangere dictum conicum in recta linea, ſiue in pluribus rectis lineis, <lb/>ſiue in plano, quod erit triangulum per eiuſdem latera tranſiens. <lb/></s> <s xml:id="echoid-s941" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/> Tangat igitur, DF, figuram, BCE, <lb/>in puncto, B, & </s> <s xml:id="echoid-s942" xml:space="preserve">iungatur, AB, perq; <lb/></s> <s xml:id="echoid-s943" xml:space="preserve">AB, &</s> <s xml:id="echoid-s944" xml:space="preserve">, DF, dictum ſit extenſum pla-<lb/>num, ergo, AB, erit latus conici, A <lb/>CE, nam latus, quod reuoluitur tran-<lb/>ſiens per, B, congruit rectę, AB, alio-<lb/>quin duæ rectæ lineæ clauderent ſuper-<lb/>ficiem, eſt ergo, AB, in ſuperficie co-<lb/>niculari, eſt etiam in plano per, A, &</s> <s xml:id="echoid-s945" xml:space="preserve">, <lb/>DF, tranſeunte, ergo, AB, eſt com-<lb/>munis tum ſuperſiciei coniculari, tum <lb/>plano per, A, &</s> <s xml:id="echoid-s946" xml:space="preserve">, DF, ducto, nullus <lb/>autem punctus rectę, AB, eſt intra ſu-<lb/>perſiciem cylindraceam, ergo planum <lb/>per, AB, DF, ductum tangit conicum <lb/>in recta, AB: </s> <s xml:id="echoid-s947" xml:space="preserve">Eodem pacto oſtendemus idem tangere conicum in <lb/>quibuſuis alijs lateribus, quæ ducuntur à punctis contactus rectæ li-<lb/>neæ, DF, qui fi ſint plures, fit etiam contactus in omnibus lineis, <lb/>fi vero contactus rectæ, DF, fiat in recta linea tunc contactus plani <lb/>per, AB, DF, fit in ſingulis rectis lineis, quæ à recta talis contactus <pb o="35" file="0055" n="55" rhead="LIBERI."/> ad verticem, A, duci poſſunt, iacent autem omnes illæ in plano <lb/>trianguli, cuius baſis eſt linea contactus vertex reſpectu eius, pun-<lb/>ctus, A, igitur, contactus plani per, AB, DF, ductifit vel in vna, <lb/>vel pluribus rectis lineis, vel in plano, quod eſt triangulum, ſiue <lb/>plura triangula, non ſecabit autem alicubi tale planum ipſum coni-<lb/>cum, tunc enim aliquis punctus talis plani per, AB, DF, tranſeun-<lb/>tis eſſet intra ſuperficiem conicularem, ſit is punctus, I, iuncta igi-<lb/>tur, AI, & </s> <s xml:id="echoid-s948" xml:space="preserve">producta verſus baſim incidet intra baſim, vt facilè o-<lb/>ſtendi poteſt, & </s> <s xml:id="echoid-s949" xml:space="preserve">quia eſt, AX, in plano per, AB, DF, ducto, & </s> <s xml:id="echoid-s950" xml:space="preserve"><lb/>punctus, X, eſt etiam in plano baſis, erit in communi fectione, ideſt <lb/>in linea, DF, igitur aliquis punctus rectæ, DF, erit intra baſim, <lb/>igitur illam ſecabit, quod eſt abſurdum, ergo falſum eſt planum per, <lb/>A, DF, ductum ſecare alicubi ipſum conicum, igitur illum tanget <lb/>in his, quæ dicta ſunt, quod oſtendere oportebat.</s> <s xml:id="echoid-s951" xml:space="preserve"/> </p> <div xml:id="echoid-div115" type="float" level="2" n="1"> <figure xlink:label="fig-0054-01" xlink:href="fig-0054-01a"> <image file="0054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0054-01"/> </figure> </div> </div> <div xml:id="echoid-div117" type="section" level="1" n="82"> <head xml:id="echoid-head93" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s952" xml:space="preserve">_E_X hoc habetur, ſi conicus ſecetur plano baſi æquidiſtante, commu-<lb/>nem ſectionem huius, & </s> <s xml:id="echoid-s953" xml:space="preserve">plani per verticem, & </s> <s xml:id="echoid-s954" xml:space="preserve">tangentem baſim <lb/>ducti, tangere figuram à plano æquidiſtante baſi in conico productam, ſi <lb/>enim eam ſecaret, etiam tangens planum ſecaret conicum, quod eſt ab-<lb/>ſurdum.</s> <s xml:id="echoid-s955" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div118" type="section" level="1" n="83"> <head xml:id="echoid-head94" xml:space="preserve">THEOREMA XVI. PROPOS. XIX.</head> <p> <s xml:id="echoid-s956" xml:space="preserve">SI conicus planoſecetur baſi æquidiſtante, concepta in <lb/>eo figura erit ſimilis baſi, & </s> <s xml:id="echoid-s957" xml:space="preserve">eidem ſimiliter poſita.</s> <s xml:id="echoid-s958" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s959" xml:space="preserve">Sit conicus, cuius vertex, A, baſis, TDF, ſecetur autem plano <lb/>baſi æquidiſtante, quod in eo producat figura, VBO. </s> <s xml:id="echoid-s960" xml:space="preserve">Dico hanc <lb/>eſſe ſimilem baſi, & </s> <s xml:id="echoid-s961" xml:space="preserve">eidem ſimiliter poſitam. </s> <s xml:id="echoid-s962" xml:space="preserve">Ducantur ipſius ba-<lb/>ſis duæ vtcumque oppoſitæ tangentes, quæ ſint, TH, SP, indefi-<lb/> <anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/> nitè productæ, deinde per verticem, & </s> <s xml:id="echoid-s963" xml:space="preserve">quamlibet dictarum tangen-<lb/>tium extendatur planum, erunt ergo hęc plana tangentia conicum, <lb/>ADF, ſecent autem figuræ, VBO, productum planum in rectis, <lb/> <anchor type="note" xlink:label="note-0055-02a" xlink:href="note-0055-02"/> VK, XN, quæ erunt ipſius figuræ, VBO, oppoſitæ tangentes, <lb/> <anchor type="note" xlink:label="note-0055-03a" xlink:href="note-0055-03"/> ſumatur deinde in altera ipſarum, TH, SP, vtin, TH, vtcumq; <lb/></s> <s xml:id="echoid-s964" xml:space="preserve">punctum, vt, H, à quo verſus reliquam tangentem eiuſdem figurę, <lb/>TDF, in eiuſdem plano ducatur vtcumque, HP, in, SP, termi-<lb/>nata, deinde intelligatur extenſum planum per, A, &</s> <s xml:id="echoid-s965" xml:space="preserve">, HP, tran-<lb/>ſiens ita, vtſecet plana conicum tangentia in rectis, AH, AP, &</s> <s xml:id="echoid-s966" xml:space="preserve"> <pb o="36" file="0056" n="56" rhead="GEOMETRIÆ"/> planumper, VK, XN, ductum in recta, KN, rurſus diuidatur, H <lb/>P, vtcumq; </s> <s xml:id="echoid-s967" xml:space="preserve">in puncto, G, à quo ducatur ipſi, SP, parallela, GD, <lb/>ſecans baſis ambitum in punctis, F, E, C, D, deinde extendatur <lb/>planum per, A, verticem, & </s> <s xml:id="echoid-s968" xml:space="preserve">rectam, DG, quod per conici latera <lb/> <anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/> tranſibit, & </s> <s xml:id="echoid-s969" xml:space="preserve">producet triangula ſiueintus, ſiue extra conicum, quæ <lb/>ſint, ADC, ACE, AEF, AFG, ſecabitque figuram, VBO, ſe-<lb/>cet eius productum planum in recta, BM, quæ ambitum eiuſdem, <lb/>VBO, diuidat in punctis, B, R, I, O, habebimus etiam triangula, <lb/>ABR, ARI, AIO, AOM, quorum latera erunt portiones late-<lb/>rum inferiorum triangulorum, per planum autem, ADG, ſiue per <lb/>rectam, AG, ſit ſecta, KN, in puncto, M. </s> <s xml:id="echoid-s970" xml:space="preserve">Quia ergo plana, quę <lb/> <anchor type="figure" xlink:label="fig-0056-01a" xlink:href="fig-0056-01"/> per rectas, VK, XN, & </s> <s xml:id="echoid-s971" xml:space="preserve"><lb/>per, TH, SP, tranſeunt <lb/>ſunt parallela, & </s> <s xml:id="echoid-s972" xml:space="preserve">ſecan-<lb/>tur à plano, APH, com-<lb/> <anchor type="note" xlink:label="note-0056-02a" xlink:href="note-0056-02"/> munes eorum ſectionese-<lb/>runt parallelę.</s> <s xml:id="echoid-s973" xml:space="preserve">ſ.</s> <s xml:id="echoid-s974" xml:space="preserve">KN, ipſi, <lb/>HP, igitur triangulus, A <lb/>MN, æquiangulus erit <lb/>triangulo, AGP, & </s> <s xml:id="echoid-s975" xml:space="preserve">ideo <lb/>circa æquales angulos e-<lb/> <anchor type="note" xlink:label="note-0056-03a" xlink:href="note-0056-03"/> runt latera proportiona-<lb/>lia, ergo vt, PG, ad, G <lb/>A, ſic erit, NM, ad, M <lb/>A, eodem modo oſtende-<lb/>mus, vt, AG, ad, GH, ita eſſe, AM, ad, MK, ergo ex æquali <lb/>PG, ad, GH, erit vt, NM, ad, MK, ſunt igitur, PH, NK, ſi-<lb/>militer ad eandem partem diuiſæ in punctis, M, G: </s> <s xml:id="echoid-s976" xml:space="preserve">Eodem modo <lb/>oſtendemus triangulum, AMO, eſſe ęquiangulum ipſi, AGF, &</s> <s xml:id="echoid-s977" xml:space="preserve">, <lb/>AMI, ipſi, AGE, &</s> <s xml:id="echoid-s978" xml:space="preserve">, AMR, ipſi, AGC, & </s> <s xml:id="echoid-s979" xml:space="preserve">tandem, AMB, <lb/>ipſi, AGD, igitur, vt, GA, ad, AM, ſic erit, permutando, FG, <lb/>ad, OM, vt verò, GA, ad, AM, ſic permutando eſt, PG, ad, <lb/>NM, ideſt, PH, ad, NK, ergo, FG, ad, OM, eſt vt, PH, ad, <lb/>NK, ſimiliter oſtendemus, EG, ad, IM, &</s> <s xml:id="echoid-s980" xml:space="preserve">, CG, ad, RM, & </s> <s xml:id="echoid-s981" xml:space="preserve"><lb/>tandem, DG, ad, BM, eſſe vt, PH, ad, NK, & </s> <s xml:id="echoid-s982" xml:space="preserve">quia, KN, eſt <lb/>parallela ipſi, HP, &</s> <s xml:id="echoid-s983" xml:space="preserve">, NX, ipſi, PS, ideò angulus, KNX, eſt <lb/> <anchor type="note" xlink:label="note-0056-04a" xlink:href="note-0056-04"/> æqualis angulo, HPS; </s> <s xml:id="echoid-s984" xml:space="preserve">habemus igitur duas figuras planas, VBO, <lb/>TDF, quarum ductæ ſunt oppoſitæ tangentes, VK, XN, vnius, <lb/>&</s> <s xml:id="echoid-s985" xml:space="preserve">, TH, SP, alterius, inuenimus autem rectas, KN, HP, inter <lb/>eaſdem poſitas, cum eis ad eandem partem angulos æquales conti-<lb/>nentes, ita ſe habere, vt ductis duabus vtcumque ipſis tangentibus <lb/>parallelis, quæ diuidant ipſas ſimiliter ad eandem partem, repertum <pb o="37" file="0057" n="57" rhead="LIBERI."/> fit eas, quæ inter taliter incidentes, & </s> <s xml:id="echoid-s986" xml:space="preserve">perimetrum figurarum con-<lb/>tinentur, eodem ordine ſumptas, eſſe vt ipſas, HP, KN, inciden-<lb/> <anchor type="note" xlink:label="note-0057-01a" xlink:href="note-0057-01"/> tes, ſunt igitur figuræ planę, BVO, DTF, inter ſe ſimiles, & </s> <s xml:id="echoid-s987" xml:space="preserve">ho-<lb/>mologarum earundem regulæ ipſæ tangentes, dictæ figuræ ſunt in <lb/>planis æquidiſtantibus, quarum incidentes fibi inuicem ęquidiſtant, <lb/>& </s> <s xml:id="echoid-s988" xml:space="preserve">homologæ earundem figurarum ſunt ad eandem partem inciden-<lb/>tium, & </s> <s xml:id="echoid-s989" xml:space="preserve">ipſarum incidentium partes homologæ pariter ad eandem <lb/>partem conſtitutæ, igitur figuræ, VBO, TDF, nedum erunt ſimi-<lb/>les, ſed etiam ſimiliter poſitæ, quod oſtendendum erat.</s> <s xml:id="echoid-s990" xml:space="preserve"/> </p> <div xml:id="echoid-div118" type="float" level="2" n="1"> <note position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="right" xlink:label="note-0055-02" xlink:href="note-0055-02a" xml:space="preserve">Penãtes. 1</note> <note position="right" xlink:label="note-0055-03" xlink:href="note-0055-03a" xml:space="preserve">Corol. an-<lb/>teced.</note> <note position="left" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">16. Huius.</note> <figure xlink:label="fig-0056-01" xlink:href="fig-0056-01a"> <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0056-01"/> </figure> <note position="left" xlink:label="note-0056-02" xlink:href="note-0056-02a" xml:space="preserve">10. Vnde-<lb/>cimi El.</note> <note position="left" xlink:label="note-0056-03" xlink:href="note-0056-03a" xml:space="preserve">4. Sexti <lb/>Elem.</note> <note position="left" xlink:label="note-0056-04" xlink:href="note-0056-04a" xml:space="preserve">10. Vnde-<lb/>Fimi El.</note> <note position="right" xlink:label="note-0057-01" xlink:href="note-0057-01a" xml:space="preserve">A. Def. 10.</note> </div> </div> <div xml:id="echoid-div120" type="section" level="1" n="84"> <head xml:id="echoid-head95" xml:space="preserve">COROLLARIVMI.</head> <p style="it"> <s xml:id="echoid-s991" xml:space="preserve">_E_T quia oſtenſum eſt ipſas tangentes, SP, XN, eſſe bomologárum <lb/>earundem ſimilium figurarum regulas, & </s> <s xml:id="echoid-s992" xml:space="preserve">ductæ ſunt vtcumque, <lb/>patet ſi duxerimus alias duas eiuſdem baſis oppoſitas tangentes, quæ cum <lb/>primò ductis angulos efficient æquales, & </s> <s xml:id="echoid-s993" xml:space="preserve">per ipſas, & </s> <s xml:id="echoid-s994" xml:space="preserve">verticem, A, <lb/>extenderimus duo plana (quorum & </s> <s xml:id="echoid-s995" xml:space="preserve">plani figuræ, BVO, producti com-<lb/>munes ſectiones erunt aliæ duæ figuræ, BVO, oppoſitæ tangentes) quod <lb/>eodem modo oſtendemus has ſecundas tangentes eſſe homologarum earun-<lb/>dem ſimilium figurarum regulas, & </s> <s xml:id="echoid-s996" xml:space="preserve">intra ipſas contineri earundem <lb/>quoq; </s> <s xml:id="echoid-s997" xml:space="preserve">incidentes, ſacient autem ſecunda tangentes cum primis angu-<lb/>los æquales, prima. </s> <s xml:id="echoid-s998" xml:space="preserve">n. </s> <s xml:id="echoid-s999" xml:space="preserve">ex. </s> <s xml:id="echoid-s1000" xml:space="preserve">gr. </s> <s xml:id="echoid-s1001" xml:space="preserve">tangens figuræ, BVO, quæ eſt, XN, eſt <lb/>parallela ipſi, SP, primæ tangenti figuræ, DTF, & </s> <s xml:id="echoid-s1002" xml:space="preserve">ſecundatangens <lb/>figuræ, BVO, eſt pariter parallela ſecundæ tangenti figuræ, DTF, nam <lb/>tum primæ, tum ſecundæ tangentes ſunt communes ſectiones æquidiſtan-<lb/>tium planorum, ipſarum nempè figurarum, BVO, DTF, productorum <lb/>planorum, & </s> <s xml:id="echoid-s1003" xml:space="preserve">ideò ſunt parallelæ, & </s> <s xml:id="echoid-s1004" xml:space="preserve">angulos continent æquales, vnde <lb/> <anchor type="note" xlink:label="note-0057-02a" xlink:href="note-0057-02"/> in figuris, quæ à planis baſi conici parallelis producuntur, ſi babeamus <lb/>bomologas cum àuabus quibuſdam regulis, eaſdem etiam babebimus cum <lb/>duabus quibaſuis alijs angulos æquales cum prædictis ad eandem partem <lb/>continentibus.</s> <s xml:id="echoid-s1005" xml:space="preserve"/> </p> <div xml:id="echoid-div120" type="float" level="2" n="1"> <note position="right" xlink:label="note-0057-02" xlink:href="note-0057-02a" xml:space="preserve">_10. Vnde-_ <lb/>_cimi El._</note> </div> </div> <div xml:id="echoid-div122" type="section" level="1" n="85"> <head xml:id="echoid-head96" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s1006" xml:space="preserve">_P_Atet in ſuper ex bac, & </s> <s xml:id="echoid-s1007" xml:space="preserve">11. </s> <s xml:id="echoid-s1008" xml:space="preserve">ac 12. </s> <s xml:id="echoid-s1009" xml:space="preserve">huius ſimilium planarum figu-<lb/>rarum, quæex ſectione planorum baſi cylindrici, vel conici æqui-<lb/>diſtantium in illis producuntur, vel ſunt oppoſitæ baſes cylindrici, aut <lb/>fruſti conici, poſſibile eſſe inuenire incidentes, quæ ſint & </s> <s xml:id="echoid-s1010" xml:space="preserve">ductarum vt-<lb/>cumq; </s> <s xml:id="echoid-s1011" xml:space="preserve">oppoſitarum earundem tangentium incidentes, & </s> <s xml:id="echoid-s1012" xml:space="preserve">quia punctum, <lb/>H, ſumptum eſt vtcumque, & </s> <s xml:id="echoid-s1013" xml:space="preserve">ab ipſo ducta quælibet incidens, HP, pa-<lb/>tet, quod, ducta vtcumque in dictis figuris incidente earum tangenti- <pb o="38" file="0058" n="58" rhead="GEOMETRIÆ"/> bus, quæ ſunt regulæ homologarum earundem, poſſunt reperiri duæ in-<lb/>cidentes earundem, quarum altera ſit iam ducta; </s> <s xml:id="echoid-s1014" xml:space="preserve">veluti, acta, HP, vt-<lb/>cumque inuentæ ſunt duæ incidentes, KN, HP, quarum altera fuit, <lb/>HP. </s> <s xml:id="echoid-s1015" xml:space="preserve">Et quia bomologarum in eaſdem incidentes productarum, & </s> <s xml:id="echoid-s1016" xml:space="preserve">ad <lb/>eas terminatarum, portiones, eodem ordine ſumpcæ, ſunt proportiona-<lb/>les, ſunt enim, vt ipſæ incidentes, ideò per homologarum productarum, <lb/>talia extremaſemper tranſeunt aliquæ incidentes.</s> <s xml:id="echoid-s1017" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div123" type="section" level="1" n="86"> <head xml:id="echoid-head97" xml:space="preserve">THEOREMA XVII. PROPOS. XX.</head> <p> <s xml:id="echoid-s1018" xml:space="preserve">SI conicus ſecetur quomodocumq; </s> <s xml:id="echoid-s1019" xml:space="preserve">planis parallelis, cum <lb/>omnibus eiuſdem lateribus coincidentibus, conceptæ <lb/>in ipſo figuræ erunt inter ſe ſimiles, & </s> <s xml:id="echoid-s1020" xml:space="preserve">ſimiliter poſitæ.</s> <s xml:id="echoid-s1021" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1022" xml:space="preserve">Sit conicus, cuius baſis, FHG, ver-<lb/> <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> tex, A, ſecetur autein vtcumque planis <lb/>parallelis, quæ cum omnibus eiuſdem la-<lb/>teribus coincidant, & </s> <s xml:id="echoid-s1023" xml:space="preserve">ſint conceptæ in <lb/>ipſo figuræ, DME, BNC. </s> <s xml:id="echoid-s1024" xml:space="preserve">Dico has <lb/>eſſe ſimiles, & </s> <s xml:id="echoid-s1025" xml:space="preserve">ſimiliter poſitas: </s> <s xml:id="echoid-s1026" xml:space="preserve">Nam <lb/>quia planum figuræ, DME, coincidit <lb/>omnibus lateribus conici, AFHG, ideo <lb/> <anchor type="note" xlink:label="note-0058-01a" xlink:href="note-0058-01"/> eſt etiam conicus ipſe, ADME, ſecatur <lb/>autem plano eius baſi, DME, æquidi-<lb/> <anchor type="note" xlink:label="note-0058-02a" xlink:href="note-0058-02"/> ſtante, eo ſcilicet, quod producit figu-<lb/>ram, BNC, ergo figura, BNC, erit ſi-<lb/>milis baſi, DME, & </s> <s xml:id="echoid-s1027" xml:space="preserve">eidem ſimiliter po-<lb/>ſita, quod erat demonſtrandum.</s> <s xml:id="echoid-s1028" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="1"> <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a"> <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0058-01"/> </figure> <note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">17. Huius.</note> <note position="left" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div125" type="section" level="1" n="87"> <head xml:id="echoid-head98" xml:space="preserve">THE OREMA XVIII. PROPOS. XXI.</head> <p> <s xml:id="echoid-s1029" xml:space="preserve">SI quilibet conicus ſecetur plano per verticem, ſiue ab <lb/>eodem tangatur in plano, nempe in triangulo, veltrian-<lb/>gulis, ſecetur autem alijs planis vtcumq; </s> <s xml:id="echoid-s1030" xml:space="preserve">baſi parallelis, com-<lb/>munes ſectiones, quæ ab eodem plano ſecante ſiunt in dictis <lb/>planis baſi parallelis, erunt homologę lineę, vel latera figu-<lb/>rarum, quæ ab eiſdem æquidiſtantibus planis in eodem co-<lb/>nico producuntur.</s> <s xml:id="echoid-s1031" xml:space="preserve"/> </p> <pb o="39" file="0059" n="59" rhead="LIBERI."/> <p> <s xml:id="echoid-s1032" xml:space="preserve">Videatur figura Propoſ.</s> <s xml:id="echoid-s1033" xml:space="preserve">16. </s> <s xml:id="echoid-s1034" xml:space="preserve">huius, in qua conicus, ATDF, in-<lb/>telligatur ſectus plano vtcumque per verticem, A, ducto efficiente <lb/>triangulum, ſiue triangulos, ADC, AEF, intra, extra autem trian-<lb/>gulum, ACE, & </s> <s xml:id="echoid-s1035" xml:space="preserve">qui ex illis integratur, ADF, ſecetur autem alio <lb/>plano baſi parallelo, quod in conico producat figuram, VBO, & </s> <s xml:id="echoid-s1036" xml:space="preserve"><lb/>ſint earum, & </s> <s xml:id="echoid-s1037" xml:space="preserve">plani per verticem communes ſectiones, BR, DC, <lb/>IO, EF. </s> <s xml:id="echoid-s1038" xml:space="preserve">Dico eaſdem eſſe lineas homologas earundem figurarum, <lb/>VBO, TDF. </s> <s xml:id="echoid-s1039" xml:space="preserve">Intelligantur in baſi ductæ oppoſitæ tangentes, T <lb/>H, SP, per quas, & </s> <s xml:id="echoid-s1040" xml:space="preserve">verticem, A, extendantur plana, quæ pariter <lb/> <anchor type="note" xlink:label="note-0059-01a" xlink:href="note-0059-01"/> tangent conicum, ATDF, ſint autem eorum, & </s> <s xml:id="echoid-s1041" xml:space="preserve">plani figurę, VB <lb/>O, producti communes ſectiones, VK, XN, quas, vt ibi, oſten-<lb/> <anchor type="note" xlink:label="note-0059-02a" xlink:href="note-0059-02"/> demus eſſe oppoſitas tangentes ipſius, VBO, reſpectu, BO, ſum-<lb/>ptas, accipiatur deinde in, TH, vtcumq; </s> <s xml:id="echoid-s1042" xml:space="preserve">punctum, H, à quo vſq; <lb/></s> <s xml:id="echoid-s1043" xml:space="preserve">ad aliam oppoſitam tangentem, SP, ducatur vtcumque, HP, & </s> <s xml:id="echoid-s1044" xml:space="preserve"><lb/>peripſam, & </s> <s xml:id="echoid-s1045" xml:space="preserve">punctum, A, extendatur planum, quod ſecet tangen-<lb/>tia plana in rectis, AH, AP, & </s> <s xml:id="echoid-s1046" xml:space="preserve">planum parallelarum, VK, XN, <lb/>in recta, KN, erunt ergo ipſæ, KN, HP, parallelæ, extendatur <lb/>planum trianguli, ADF, ita vt ſecet triangulum, APH, in recta, <lb/>AG, & </s> <s xml:id="echoid-s1047" xml:space="preserve">planum figuræ, TDF, productum, ſi opus ſit, in recta, D <lb/>G, Eodem modo igitur, quo vti ſumus in Propoſ. </s> <s xml:id="echoid-s1048" xml:space="preserve">19. </s> <s xml:id="echoid-s1049" xml:space="preserve">quia, KN, <lb/>HP, ſunt parallelæ, oſtendemus ipſas, KN, HP, eſſe ab ipſis, B <lb/>M, DG, (quę ſunt communes ſectiones trianguli, ADF, & </s> <s xml:id="echoid-s1050" xml:space="preserve">ęqui-<lb/>diſtantium planorum, VBO, TDF, & </s> <s xml:id="echoid-s1051" xml:space="preserve">ideò ſunt parallelæ) ſimi-<lb/>liter diuiſas, & </s> <s xml:id="echoid-s1052" xml:space="preserve">ad eandem partem in punctis, M, G, vnde, vt ibi <lb/>oſtendemus figuras, VBO, TDF, eſſe ſimiles, & </s> <s xml:id="echoid-s1053" xml:space="preserve">earum, & </s> <s xml:id="echoid-s1054" xml:space="preserve">tan-<lb/>gentium oppoſitarum, XN, VK; </s> <s xml:id="echoid-s1055" xml:space="preserve">SP, TH, incidentes eſſe ipſas, <lb/>KN, HP, & </s> <s xml:id="echoid-s1056" xml:space="preserve">tangentes eſſe regulas homologarum earundem, qua-<lb/>rum duæ ſunt ipſæ, BRIO, DCEF, coniunctæ, ſiue ipſæ, BR, <lb/>DC; </s> <s xml:id="echoid-s1057" xml:space="preserve">IO, EF. </s> <s xml:id="echoid-s1058" xml:space="preserve">Eodem modo, ſi propoſitus conicus fuiſſet, cuius <lb/>vertex, A, baſis altera figurarum a bafi, TDF, per rectam, DF, <lb/>abſciſſarum, vt ipſa, DTF, oſtenſum eſſet ipſas, BR, DC; </s> <s xml:id="echoid-s1059" xml:space="preserve">IO, <lb/>EF, communes ſectiones plani conicum tangentis in triangulis, A <lb/> <anchor type="note" xlink:label="note-0059-03a" xlink:href="note-0059-03"/> DC, AEF, & </s> <s xml:id="echoid-s1060" xml:space="preserve">planorum æquidiſtantium, BVO, DTF, eſſe ea-<lb/>rundem homologas, erunt autem in hoc caſu latera homologa, ve-<lb/>lut cum ſunt intra figuras ſunt lineæ homologæ earumdem, quode-<lb/>rat oſtendendum.</s> <s xml:id="echoid-s1061" xml:space="preserve"/> </p> <div xml:id="echoid-div125" type="float" level="2" n="1"> <note position="right" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">Coroll.1. <lb/>huius.</note> <note position="right" xlink:label="note-0059-02" xlink:href="note-0059-02a" xml:space="preserve">18. Huius.</note> <note position="right" xlink:label="note-0059-03" xlink:href="note-0059-03a" xml:space="preserve">Sed hoc <lb/>etiam per <lb/>modũ Co-<lb/>rollar. ex <lb/>Prop. 19. <lb/>deducipo <lb/>tuiſſet.</note> </div> </div> <div xml:id="echoid-div127" type="section" level="1" n="88"> <head xml:id="echoid-head99" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1062" xml:space="preserve">_H_Inc habetur, ſi propoſitum fuiſſet fruſtum conici, BTF, quod eius <lb/>omnia latera producta coincidiſſent in vno puncto, A vnde, <lb/>oſtenſum pariter fuiſſet communes ſectiones plani per eius latera tran- <pb o="40" file="0060" n="60" rhead="GEOMETRIÆ"/> ſeuntis, vt ipſius, BDFO, quod ſemper eſt trapezium, & </s> <s xml:id="echoid-s1063" xml:space="preserve">ipſarum, V <lb/>BO, TDF, ſiue eiſdem æquidiſtantium inter eaſdem ductarum, eſſe ea-<lb/>rundem lineas, vel latera homologa, vnde patet communes ſectiones <lb/>planiper latera fruſti conici ducti, & </s> <s xml:id="echoid-s1064" xml:space="preserve">eiuſdem baſium oppoſitarum, ſiue <lb/>eiſdem æquidiſtantium inter eas productarum figurarum, eſſe earundem <lb/>lineas, vel latera homologa; </s> <s xml:id="echoid-s1065" xml:space="preserve">lineas, inquam, cum ſunt intra figuras, <lb/>nec ſumuntur in plano tangente: </s> <s xml:id="echoid-s1066" xml:space="preserve">latera, cum ſunt in earum circuitu, <lb/>cum nempè ſunt in eodem plano tangente, in eo præcisè, quod eſt pla-<lb/>num contactus fruſti conici (contactus ſcilicet cius plani, quod per ver-<lb/>ticem ducitur) quod ſemper erit trapezium, vel trapezia, vt patere po-<lb/>teſt in trapezijs, BDCR, IEFO, quæ eſſent planum contactus fruſti <lb/>conici, ſiidem fruſtum tangeretur à plano trianguli, ADF.</s> <s xml:id="echoid-s1067" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div128" type="section" level="1" n="89"> <head xml:id="echoid-head100" xml:space="preserve">THEOREMA XIX. PROPOS. XXII.</head> <p> <s xml:id="echoid-s1068" xml:space="preserve">SI duæ figuræ planę ſimiles, non exiſtentes in eodem pla-<lb/>no, fuerint inæquales, & </s> <s xml:id="echoid-s1069" xml:space="preserve">ſimiliter poſitæ; </s> <s xml:id="echoid-s1070" xml:space="preserve">erunt cuiu-<lb/>ſdam fruſticonici oppoſitæ baſes.</s> <s xml:id="echoid-s1071" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1072" xml:space="preserve">Vtamuradhuc figura Propoſ. </s> <s xml:id="echoid-s1073" xml:space="preserve">19. </s> <s xml:id="echoid-s1074" xml:space="preserve">& </s> <s xml:id="echoid-s1075" xml:space="preserve">ſint duæ figuræ planæ quæ-<lb/>cumque ſimiles, inæquales, & </s> <s xml:id="echoid-s1076" xml:space="preserve">ſimiliter poſitę, non tamen exiſten-<lb/>tesin eodem plano, ipſæ, VBO, TDF. </s> <s xml:id="echoid-s1077" xml:space="preserve">Dico, quod erunt am-<lb/>bæ cuiuſdam fruſti conici oppoſitę baſes. </s> <s xml:id="echoid-s1078" xml:space="preserve">Quoniam ergo figure, V <lb/>BO, TDF, ſunt ſimiliter poſitæ, & </s> <s xml:id="echoid-s1079" xml:space="preserve">non in eodem plano, erunt in <lb/> <anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/> planis ęquidiſtantibus, & </s> <s xml:id="echoid-s1080" xml:space="preserve">quia ſunt ſimiles ſint earum incldentes, & </s> <s xml:id="echoid-s1081" xml:space="preserve"><lb/>oppoſitarum tangentium, quæ ſunt earundem homologarum regu-<lb/>læ, ipſæ, KN, HP; </s> <s xml:id="echoid-s1082" xml:space="preserve">KN, ipſius, VBO, &</s> <s xml:id="echoid-s1083" xml:space="preserve">, HP, ipſius, TDF, <lb/>& </s> <s xml:id="echoid-s1084" xml:space="preserve">prædictæ tangentes figuræ, VBO, ſint ipſæ, VK, XN, & </s> <s xml:id="echoid-s1085" xml:space="preserve">fi-<lb/>guræ, TDF, ipſæ, TH, SP, erunt ergo ipſæ, KN, HP, æqui-<lb/>diſtantes, & </s> <s xml:id="echoid-s1086" xml:space="preserve">quia ad tangentes, quæ ſunt regulæ homologarum, illę <lb/> <anchor type="note" xlink:label="note-0060-02a" xlink:href="note-0060-02"/> efficiunt ad eandem partem angulos æquales, erit angulus, KNX, <lb/>æqualis angulo, HPS, & </s> <s xml:id="echoid-s1087" xml:space="preserve">quia, KN, eſt parallela ipſi, HP, erit <lb/>etiam, XN, parallela ipſi, SP. </s> <s xml:id="echoid-s1088" xml:space="preserve">Eodem pacto oſtendemus, VK, <lb/>eſſe parallelam ipſi, TH; </s> <s xml:id="echoid-s1089" xml:space="preserve">ducantur in figuris, VBO, TDF, duæ <lb/>earum homologæ regulis dictis tang entibus, quæ ſint ipſæ, BR, I <lb/>O, DC, EF, ſint autem totæ, BO, DF, productæ, ſi opus ſit, vt <lb/>ſecent ipſas, KN, HP, quas diuident ſimiliter ad eandem partem, <lb/>vt in punctis, M, G, & </s> <s xml:id="echoid-s1090" xml:space="preserve">quia figuræ propoſitæ ſunt inæquales, ſit <lb/>maior ipſa, TDF, igitur etiam maior erit, DC, ipſa, BR, vel, E <lb/>F, ipſa, IO, ſi, n. </s> <s xml:id="echoid-s1091" xml:space="preserve">eſſent eiſdem æquales, etiam reliquæ homologæ <lb/>his parallelæ eſſent ęquales, cum omnes ſint proportionales (ſunt.</s> <s xml:id="echoid-s1092" xml:space="preserve">n.</s> <s xml:id="echoid-s1093" xml:space="preserve"> <pb o="41" file="0061" n="61" rhead="LIBERI."/> vt incidentes) ynde etiam figuræ eſſent æquales, & </s> <s xml:id="echoid-s1094" xml:space="preserve">ſi minores, etiam <lb/>ipſa figura, TDF, eſſet minor figura, BVO, contra ſuppoſitum, <lb/>eſtigitur, DC, maior ipſa, BR, eſt autem, vt, DC, ad, BR, ita, <lb/>PH, ad, NK, nam vt, DG, ad, BM, itaeſt, PH, ad, NK, & </s> <s xml:id="echoid-s1095" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0061-01a" xlink:href="note-0061-01"/> etiamita, CG, ad, RM, ergo reliqua, DC, ad reliquam, BR, <lb/>erit vt, PH, ad, NK, ſic etiam eſſe oſtendemus, EF, ad, IO, vt, <lb/>PH, ad, NK, & </s> <s xml:id="echoid-s1096" xml:space="preserve">quia, DC, eſt maior ipſa, BR, vel, EF, ipſa, <lb/>IO, ideò, HP, erit maior, KN, ſi igitur iunxerimus puncta, PN, <lb/>HK, ipſæ, PN, HK, ſi producantur ad partes ipſius, NK, con-<lb/>current, vt in, A. </s> <s xml:id="echoid-s1097" xml:space="preserve">Dico, A, eſſe verticem conici, cuius eſt baſis ipſa, <lb/> <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> TDF, & </s> <s xml:id="echoid-s1098" xml:space="preserve">explano ipſi, <lb/>TDF, ęquidiſtanter du-<lb/>cto eſt in ipſo concepta <lb/>figura, VBO. </s> <s xml:id="echoid-s1099" xml:space="preserve">Quia er-<lb/> <anchor type="note" xlink:label="note-0061-02a" xlink:href="note-0061-02"/> go, NK, eſt parallela <lb/>ipſi, PH, erunt triangu-<lb/>la, ANK, APH, ęqui-<lb/>angula, & </s> <s xml:id="echoid-s1100" xml:space="preserve">circa æquales <lb/>angulos latera propor-<lb/>tionalia, igitur, HP, ad, <lb/>PA, erit vt, KN, ad, N <lb/>A, &</s> <s xml:id="echoid-s1101" xml:space="preserve">, permutando, H <lb/>P, ad, NK, erit vt, PA, <lb/>ad, AN, vt autem, PH, <lb/>ad, NK, ita eſt, PG, ad, NM, nam ipſa, HP, KN, ſimiliter <lb/>ſunt diuiſæ in punctis, G, M, ergo, PA, ad, AN, erit vt, PG, <lb/>ad, NM, & </s> <s xml:id="echoid-s1102" xml:space="preserve">ſunt parallelæ ipſæ, PG, NM, ergo puncta, G, M, <lb/>A, erunt in vna recta linea, ſit illa, AG, igitur, vt, PG, ad, NM, <lb/> <anchor type="note" xlink:label="note-0061-03a" xlink:href="note-0061-03"/> vel, PH, ad, NK, ita erit, GA, ad, AM, eſt autem, PH, ad, <lb/>NK, vt, FG, ad, OM, & </s> <s xml:id="echoid-s1103" xml:space="preserve">vt, EG, ad, IM, & </s> <s xml:id="echoid-s1104" xml:space="preserve">tandem, vt, D <lb/>G, ad, BM, ergo, vt, GA, ad, AM, ita erit, FG, ad, OM; </s> <s xml:id="echoid-s1105" xml:space="preserve">E <lb/>G, ad, IM; </s> <s xml:id="echoid-s1106" xml:space="preserve">CG, ad, RM; </s> <s xml:id="echoid-s1107" xml:space="preserve">&</s> <s xml:id="echoid-s1108" xml:space="preserve">, DG, ad, BM, ergo, cum ſint <lb/>parallelæ, erunt tum puncta, AOF, tum, AIE, ARC, tum etiam, <lb/>ABD, in vna recta linea, extendantur ergo dictæ rectæ lineæ, quę <lb/> <anchor type="note" xlink:label="note-0061-04a" xlink:href="note-0061-04"/> erunt, AF, AE, AD, AC. </s> <s xml:id="echoid-s1109" xml:space="preserve">Eodem modo, ſi per duas quaslibet <lb/>homologas figurarum, VBO, TDF, planum extendamus, fiet in <lb/>cæteris demonſtratio; </s> <s xml:id="echoid-s1110" xml:space="preserve">igitur ſi ſumantur in ambitu figurę, TDF, <lb/>quęcumq; </s> <s xml:id="echoid-s1111" xml:space="preserve">puncta, quę iungantur cum puncto, A, ſemper iungen-<lb/>tes tranſibunt per circuitum figuræ, VBO, ergo figurę, TDF, &</s> <s xml:id="echoid-s1112" xml:space="preserve">, <lb/>VBO, erunt fruſti conici oppoſitę baſes, quod à conico, ATDF, <lb/> <anchor type="note" xlink:label="note-0061-05a" xlink:href="note-0061-05"/> abſcinditur per figuram, VBO, quod erat demonſtrandum.</s> <s xml:id="echoid-s1113" xml:space="preserve"/> </p> <div xml:id="echoid-div128" type="float" level="2" n="1"> <note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">D.Def.0. <lb/>huius.</note> <note position="left" xlink:label="note-0060-02" xlink:href="note-0060-02a" xml:space="preserve">Conuerla <lb/>10. Vnde-<lb/>cimi El</note> <note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">A. Defin. <lb/>10.</note> <figure xlink:label="fig-0061-01" xlink:href="fig-0061-01a"> <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0061-01"/> </figure> <note position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">4. Sexti <lb/>Elem.</note> <note position="right" xlink:label="note-0061-03" xlink:href="note-0061-03a" xml:space="preserve">Ex Lem-<lb/>mate leq.</note> <note position="right" xlink:label="note-0061-04" xlink:href="note-0061-04a" xml:space="preserve">Ex Lem-<lb/>mate leq.</note> <note position="right" xlink:label="note-0061-05" xlink:href="note-0061-05a" xml:space="preserve">Defin. 4.</note> </div> <pb o="42" file="0062" n="62" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div130" type="section" level="1" n="90"> <head xml:id="echoid-head101" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s1114" xml:space="preserve">QVoniam oſtendimus, tum, DC, BR, tum etiam, EF, IO, eſſe vt <lb/>ipſas incidentes, PH, NK, habetur ſimilium figurar um homo-<lb/>logas pariter eſſe, vt incidentes earundem, & </s> <s xml:id="echoid-s1115" xml:space="preserve">oppoſitarum tangentium, <lb/>quæ ſunt earundem regulæ, quod in diffinitione aſſumitur contingere <lb/>tantum ijs, quæ inter circuicum figurarum, & </s> <s xml:id="echoid-s1116" xml:space="preserve">ipſas incidentes, eodem <lb/>ordine ſumptæ, continentur.</s> <s xml:id="echoid-s1117" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div131" type="section" level="1" n="91"> <head xml:id="echoid-head102" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s1118" xml:space="preserve">PAtet etiam ex hac, & </s> <s xml:id="echoid-s1119" xml:space="preserve">14. </s> <s xml:id="echoid-s1120" xml:space="preserve">huius, omnes ſimiles figuras planas poſſe <lb/>eſſe alicuius cylindrici, vel fruſti conici, oppoſitas baſes; </s> <s xml:id="echoid-s1121" xml:space="preserve">vnde qua <lb/>pro illis in Coroll. </s> <s xml:id="echoid-s1122" xml:space="preserve">2. </s> <s xml:id="echoid-s1123" xml:space="preserve">19. </s> <s xml:id="echoid-s1124" xml:space="preserve">huius colliguntur, pro omnibus ſimilibus figu-<lb/>ris planis etiam colligi poſſunt.</s> <s xml:id="echoid-s1125" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div132" type="section" level="1" n="92"> <head xml:id="echoid-head103" xml:space="preserve">LEMMA PRO ANTECED. PROP.</head> <p> <s xml:id="echoid-s1126" xml:space="preserve">SI in recta linea ſignenturtria puncta, primum, medium, & </s> <s xml:id="echoid-s1127" xml:space="preserve">po-<lb/>ſtremum, à primo autem, & </s> <s xml:id="echoid-s1128" xml:space="preserve">medio ducantur ad eandem par-<lb/>tem duę inuicem parallelę ita ſe habentes, vt educta à primo ad edu-<lb/>ctam à ſecundo, ſit veluti recta inter primum, & </s> <s xml:id="echoid-s1129" xml:space="preserve">poſtremum pun-<lb/>ctum poſita, ad eam, quę inter medium, & </s> <s xml:id="echoid-s1130" xml:space="preserve">idem poſtremum ſita eſt; <lb/></s> <s xml:id="echoid-s1131" xml:space="preserve">Extrema puncta parallelarum, quę non ſunt in propoſita linea, & </s> <s xml:id="echoid-s1132" xml:space="preserve"><lb/>illius poſtremum, eruntin recta linea.</s> <s xml:id="echoid-s1133" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1134" xml:space="preserve">Sit propoſita recta, AC, in qua ſignatis vt-<lb/> <anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/> cumque tribus punctis, C, primo, B, medio, <lb/>&</s> <s xml:id="echoid-s1135" xml:space="preserve">, A, poſtremo, à punctis, C, B, educantur <lb/>ad eandem partem duę inuicem parallelę, quę <lb/>ſint, CE, BD, ita ſe habentes, vt, CE, ad, <lb/>BD, ſit, vt, CA, ad, AB. </s> <s xml:id="echoid-s1136" xml:space="preserve">Dico puncta, <lb/>A, D, E, eſſe in recta linea, ſi enim (iuncta, <lb/>ED,) ipſa, ED, producta non tranſit per, <lb/>A, tranſibit ſupra, vel infra, A, ſecans, CA, <lb/>(nam, BD, eſt minor ipſa, CE, vt eſt, AB, <lb/>minor, AC,) tranſeat, vt per, M, quia igi-<lb/>tur, EDM, eſt recta erit, MCE, triangu-<lb/>lus, in quo lateri, CE, ducitur parallela, B <lb/>D, ergo trianguli, ECM, DBM, erunt æ-<lb/> <anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/> quianguli, & </s> <s xml:id="echoid-s1137" xml:space="preserve">circa æquales angulos latera proportionalia, ergo, per-<lb/>mutando, CE, ad, BD, erit vt, CM, ad, MB, eſt autem vt, C <pb o="43" file="0063" n="63" rhead="LIBERI."/> E, ad, BD, ita, CA, ad, AB, ergo vt, CM, ad, MB, ita erit, C <lb/>A, ad, AB, diuidendo, CB, ad, BM, erit vt, CB, ad, BA, er-<lb/>go, MB, erit æqualis ipſi, BA, totum parti, quod eſt abiurdum, <lb/>non igitur, ED, producta tranſit ſupra, A, eodem modo oſtende-<lb/>mus non tranfire infra, A, ergo tranſibit per, A, ergo tria puncta, <lb/>A, D, E, erunt in recta linea, AE, quod erat oſtendendum.</s> <s xml:id="echoid-s1138" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="1"> <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a"> <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0062-01"/> </figure> <note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">4. Sexti <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div134" type="section" level="1" n="93"> <head xml:id="echoid-head104" xml:space="preserve">THEOREMA XX. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s1139" xml:space="preserve">SI duarum quarumlibet ſimilium figurarum habeamus <lb/>homologas cum duabus quibuſdam regulis, habebi-<lb/>mus etiam homologas earundem cum duabus quibuſuis a-<lb/>lijs, cum prædictis angulos æquales ad eandem partem fa-<lb/>cientibus.</s> <s xml:id="echoid-s1140" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1141" xml:space="preserve">Patet hęc propoſitio, nam quæcunq; </s> <s xml:id="echoid-s1142" xml:space="preserve">figuræ planę ſimiles, ſi ſint <lb/>æquales, & </s> <s xml:id="echoid-s1143" xml:space="preserve">ſimiliter poſitæ, poſſunt eſſe cuiuſdam cylindrici oppo-<lb/> <anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/> ſitæ baſes, ſi ſint inæquales, oppoſitæ bales fruſti conici, in his au-<lb/> <anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/> tem contingit, ſi habeamus homologas cum duabus quibuſdam re-<lb/> <anchor type="note" xlink:label="note-0063-03a" xlink:href="note-0063-03"/> gulis, nos eaſdem habere cum alijs duabus quibuſcumque cum præ-<lb/>dictis angulos æquales ad eandem partem conſtituentibus, ergo hoc <lb/>in quibuſcumque planis ſimilibus figuris verificatur, quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s1144" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="1"> <note position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">14. Huius.</note> <note position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">22. Huius</note> <note position="right" xlink:label="note-0063-03" xlink:href="note-0063-03a" xml:space="preserve">Corol. 9. <lb/>& 11. hus <lb/>ius.</note> </div> </div> <div xml:id="echoid-div136" type="section" level="1" n="94"> <head xml:id="echoid-head105" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1145" xml:space="preserve">_E_T quia incidentes ad homologarum ſimilium figurarum regulas an-<lb/> <anchor type="note" xlink:label="note-0063-04a" xlink:href="note-0063-04"/> gulos ad eandem partem efficiunt æquales, ideò & </s> <s xml:id="echoid-s1146" xml:space="preserve">ipſæ incidentes <lb/>erunt homologarum earundem ſimilium figurarum regulæ, & </s> <s xml:id="echoid-s1147" xml:space="preserve">vice verſa <lb/>in quibuſdam regulis homologarum poterunt ſumi earum incidentes.</s> <s xml:id="echoid-s1148" xml:space="preserve"/> </p> <div xml:id="echoid-div136" type="float" level="2" n="1"> <note position="right" xlink:label="note-0063-04" xlink:href="note-0063-04a" xml:space="preserve">_B. Def. 10._</note> </div> </div> <div xml:id="echoid-div138" type="section" level="1" n="95"> <head xml:id="echoid-head106" xml:space="preserve">THEOREMA XXI. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s1149" xml:space="preserve">SI in duarum ſimilium figurarum oppoſitas tangentes, quę <lb/>earundem homologarum ſint regulæ, incidant duæ re-<lb/>ctæ lineæ ad eundem angulum ex eadem parte eaſdem ſe-<lb/>cantes, ductis verò quibuſdam duabus, prædictis tangenti-<lb/>bus parallelis, in dictis figuris, quæ ſecantes diuidant ſimi-<lb/>liter ad eandem partem, vel aſſumptis ipſis oppoſitis tangen-<lb/>tibus, reperiamus harum portiones inter incidentes, & </s> <s xml:id="echoid-s1150" xml:space="preserve">cir- <pb o="44" file="0064" n="64" rhead="GEOMETRIÆ"/> cuitum figurarum eodem ordine ſumptas, ita ſe habere, ve-<lb/>lut illæ, quæ dictis tangentibus inciderunt, iſtæ, quæ illis <lb/>inciderunt, erunt tum ſimilium propoſitarum figurarum, tum <lb/>ductarum tangentium, incidentes.</s> <s xml:id="echoid-s1151" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1152" xml:space="preserve">Sint duę quęcumq; </s> <s xml:id="echoid-s1153" xml:space="preserve">ſimiles planę figurę, ACEI, MTVS, qua-<lb/>rum ſint ductæ oppoſitæ tangentes homologarum earundem regu-<lb/>læ, AB, EF, figuræ, AE, &</s> <s xml:id="echoid-s1154" xml:space="preserve">, MN, VR, figurę, MV, incidant <lb/>autem eiſdem ad eundem angulum ex eadem parte duæ, BF, NR, <lb/>& </s> <s xml:id="echoid-s1155" xml:space="preserve">ductæ ſint quædam duæ ipſis tangentibus parallelæ, CD, TO, <lb/>ſecantes ipſas, BF, NR, (& </s> <s xml:id="echoid-s1156" xml:space="preserve">coniequenter incidentes, vt facilè <lb/>patet) ſimiliter ad eandem partem, reperiamus, CD, ad, TO, & </s> <s xml:id="echoid-s1157" xml:space="preserve"><lb/>pariter, ID, ad, SO, eſſe vt, BF, ad, NR. </s> <s xml:id="echoid-s1158" xml:space="preserve">Dico ipſas, BF, N <lb/>R, eſſe incidentes ſimilium figurarum, AE, MV, & </s> <s xml:id="echoid-s1159" xml:space="preserve">ductarum op-<lb/> <anchor type="figure" xlink:label="fig-0064-01a" xlink:href="fig-0064-01"/> poſitarum tangentium, VR, MN; <lb/></s> <s xml:id="echoid-s1160" xml:space="preserve">EF, AB. </s> <s xml:id="echoid-s1161" xml:space="preserve">Ex dictis igitur ipſę, CI, T <lb/> <anchor type="note" xlink:label="note-0064-01a" xlink:href="note-0064-01"/> S, erunt homologę earundem ſimilium <lb/>figurarum, AE, MV, & </s> <s xml:id="echoid-s1162" xml:space="preserve">quia, CD, <lb/> <anchor type="note" xlink:label="note-0064-02a" xlink:href="note-0064-02"/> ad, TO, eſt vt, BF, ad, NR, &</s> <s xml:id="echoid-s1163" xml:space="preserve">, B <lb/>F, ad, NR, vt, ID, ad, SO, erit, C <lb/>D, ad, TO, vt, ID, ad, SO, igitur <lb/>puncta, D, O, reperientur in duabus <lb/>dictarum fimilium figurarum, & </s> <s xml:id="echoid-s1164" xml:space="preserve">op-<lb/>poſitarum tangentium, incidentibus, <lb/>ſint illæ ipſæ, HG, PL, quę cum ip-<lb/>ſis, TO, CD, æquales angulos ad <lb/>eandem partem continebunt. </s> <s xml:id="echoid-s1165" xml:space="preserve">Dico ta-<lb/>men etiam ipſas, NR, BF, eſſe ea-<lb/>rundem figurarum, & </s> <s xml:id="echoid-s1166" xml:space="preserve">tangentium, in-<lb/>cidentes: </s> <s xml:id="echoid-s1167" xml:space="preserve">Sint puncta contactus tangentium, FE, RV, proxima ip-<lb/>ſis, NR, BF, ipſa, V, E. </s> <s xml:id="echoid-s1168" xml:space="preserve">Dico, EF, ad, VR, eſſe vt, FB, ad, <lb/> <anchor type="note" xlink:label="note-0064-03a" xlink:href="note-0064-03"/> RN, nam, EL, ad, VG, eſt vt, LP, ad, GH, quia verò angu-<lb/>lus, CDP, æquatur angulo, TOH, &</s> <s xml:id="echoid-s1169" xml:space="preserve">, CDB, ipſi, TON, re-<lb/>liquus, PDB, ęquabitur reliquo, HON, & </s> <s xml:id="echoid-s1170" xml:space="preserve">ſic etiam, FDL, ipſi, <lb/>ROG, eſt etiam angulus, PLE, ęqualis angulo, HGV, ideò re-<lb/> <anchor type="note" xlink:label="note-0064-04a" xlink:href="note-0064-04"/> liquus in triangulo, DFL, ideſt angulus, DFL, erit ęqualis angu-<lb/>lo, ORG, & </s> <s xml:id="echoid-s1171" xml:space="preserve">ſic triangula, FDL, ORG, erunt æquiangula, vt <lb/>etiam probabimus triangula, DPB, OHN, eſſe ęquiangula, ſicut <lb/>ſunt ęquiangula inter ſetriangula, FDL, PDB, &</s> <s xml:id="echoid-s1172" xml:space="preserve">, ROG, HO <lb/>N, vnde vt, LD, ad, DF, ſic erit, PD, ad, DB, permutando, <lb/>LD, ad, DP, erit vt, FD, ad, DB, componendo, LP, ad, PD, <lb/>erit vt, FB, ad, BD, permutando, LP, ad, FB, erit vt, PD, ad, <pb o="45" file="0065" n="65" rhead="LIBERI."/> DB, ideſt vt, HO, ad, ON, at, vt ſupra, oſtendemus, HO, ad, <lb/>ON, eſſe vt, HG, ad, NR, ergo, PL, ad, BF, erit vt, HG, <lb/>ad, NR, erat autem, EL, ad VG, vt, PL, ad, HG, ergo, EL, <lb/>ad, VG, erit vt, BF, ad, NR, quia verò, BF, ad, NR, eſt vt, <lb/>DF, ad, OR, (nam, BF, NR, ſunt ſimiliter diuiſæ in punctis, <lb/>D, O,) ideſt vt, FL, ad, RG, ergo, EL, ad, VG, erit vt, FL, <lb/>ad, RG, ergo reliqua, EF, ad, VR, erit vt tota, EL, ad, VG, <lb/>ideſt vt, BF, ad, NR. </s> <s xml:id="echoid-s1173" xml:space="preserve">Idem oſtendemus de quibuslibet ductis ip-<lb/>ſis, EF, VG, parallelis, quę diuidant, BF, NR, ſimiliter ad ean-<lb/>dem partem, nempè eas, quæ inter ipſas, BF, NR, & </s> <s xml:id="echoid-s1174" xml:space="preserve">circuitum <lb/>figurarum, AE, MV, eodem ordine ſumptæ continentur, eſſe vt <lb/>ipias, BF, NR, ergo, BF, NR, ſunt incidentes ſimilium figura-<lb/> <anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/> rum, MV, AE, & </s> <s xml:id="echoid-s1175" xml:space="preserve">ductarum tangentium, quod oſtendere opus erat.</s> <s xml:id="echoid-s1176" xml:space="preserve"/> </p> <div xml:id="echoid-div138" type="float" level="2" n="1"> <figure xlink:label="fig-0064-01" xlink:href="fig-0064-01a"> <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0064-01"/> </figure> <note position="left" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve">C. 10. defi. <lb/>mitionis.</note> <note position="left" xlink:label="note-0064-02" xlink:href="note-0064-02a" xml:space="preserve">Ex cor. 2. <lb/>19. & 22. <lb/>huius.</note> <note position="left" xlink:label="note-0064-03" xlink:href="note-0064-03a" xml:space="preserve">3. Defin. <lb/>10.</note> <note position="left" xlink:label="note-0064-04" xlink:href="note-0064-04a" xml:space="preserve">4. Sexti <lb/>Elem.</note> <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">B. Def. 10.</note> </div> </div> <div xml:id="echoid-div140" type="section" level="1" n="96"> <head xml:id="echoid-head107" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1177" xml:space="preserve">INnoteſcit exhoe conſequenter duarum ſimilium figurarum, & </s> <s xml:id="echoid-s1178" xml:space="preserve">ea-<lb/>rundem oppoſitarum tangentium, quæ ſuntregulæ homologarum, <lb/>tum incidentes ſimiliter diuidi ab homologis earundem figurarum, pro-<lb/>ductis, ſi opus ſit, tum quaſcumque alias, quæ cum homologis angulos <lb/>continent æquales, vt exempli gratia ipſæ, NR, BF. </s> <s xml:id="echoid-s1179" xml:space="preserve">Et vlterius ip-<lb/>ſas homologas eſſe tum vt quaſuis incidentes, tum vt eiſdem parallelas, <lb/>ideſt ex. </s> <s xml:id="echoid-s1180" xml:space="preserve">gr. </s> <s xml:id="echoid-s1181" xml:space="preserve">CI, ad, TS, mdum erit vt, PE, ad, HG, ſiue vt, BF, ad, <lb/>NR, ſed etiam vt, BF, ad quamcumque aliam parallolam ipſi, NR, <lb/>ductam inter parattelas, MN, VR, nam illa erit æqualis ipſi, NR. <lb/></s> <s xml:id="echoid-s1182" xml:space="preserve">Patet igitur duarum ſimilium figurarum homologas nedum eſſe vt ea-<lb/>rum, & </s> <s xml:id="echoid-s1183" xml:space="preserve">oppoſitarum earundem tangentium, quæ ſunt regulæ homolo-<lb/>garum, incidentes, ſed etiam vt quaſuis alias inter eaſdem tangentes <lb/>ductas ipſis incidentibus æquidiſtantes, ſiue ad homologas ſimilium figu-<lb/>rarum æqualiter inclinatas.</s> <s xml:id="echoid-s1184" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div141" type="section" level="1" n="97"> <head xml:id="echoid-head108" xml:space="preserve">THEOREMA XXII. PROPOS. XXV.</head> <p> <s xml:id="echoid-s1185" xml:space="preserve">SI quæcunque ſimiles figuræ planæ à rectis lineis deſcti-<lb/>bantur, quæ ſint earundem homologæ, & </s> <s xml:id="echoid-s1186" xml:space="preserve">inter ſe æqua-<lb/>les; </s> <s xml:id="echoid-s1187" xml:space="preserve">ſuperponantur autem ad inuicem ipſæ figuræ, ita vt ea-<lb/>ſdem deſcribentes rectæ lineæ ſibi congruant, figuræq; </s> <s xml:id="echoid-s1188" xml:space="preserve">ſint <lb/>fimiliter poſitæ, illæ quoque erunt ad inuicem congruentes.</s> <s xml:id="echoid-s1189" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1190" xml:space="preserve">Sint ſimiles figuræ planæ, ABXC, EFPG, quæcunq; </s> <s xml:id="echoid-s1191" xml:space="preserve">deſcri-<lb/>ptæ ab earundern homologis, & </s> <s xml:id="echoid-s1192" xml:space="preserve">æqualibus rectis lineis, BC, PG, <pb o="46" file="0066" n="66" rhead="GEOMETRIÆ"/> quæ ita inuicem ſuperponantur, vt, BC, FG, ſibi congruant, & </s> <s xml:id="echoid-s1193" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0066-01a" xlink:href="note-0066-01"/> ipſæ ſint ſimiliter poſitæ. </s> <s xml:id="echoid-s1194" xml:space="preserve">Dico etiam ipſas figuras ad inuicem fore <lb/>congruentes. </s> <s xml:id="echoid-s1195" xml:space="preserve">Sint oppoſitæ tangentes ductæ pro figura, ABXC, <lb/>ipſæ, AD, XQ, regula, BC, & </s> <s xml:id="echoid-s1196" xml:space="preserve">pro figura, EFPG, regula, F <lb/> <anchor type="note" xlink:label="note-0066-02a" xlink:href="note-0066-02"/> G, ipſæ, EM, PN, quarum figurarum, ac oppoſitarum tangen-<lb/>tium ſint quoque incidentes ipſę, DQ, MN, productis verò, BC, <lb/> <anchor type="note" xlink:label="note-0066-03a" xlink:href="note-0066-03"/> FG, verſus, DQ, MN, illis incidant in punctis, O, R, & </s> <s xml:id="echoid-s1197" xml:space="preserve">ſuper-<lb/>ponatur figura, ABXC, figuræ, EFPG, ita vt, BC, congruat <lb/>ipſi, FG, & </s> <s xml:id="echoid-s1198" xml:space="preserve">ſint ſimiliter poſitę: </s> <s xml:id="echoid-s1199" xml:space="preserve">Erunt ergo ipſæ incidentes, DQ, <lb/> <anchor type="note" xlink:label="note-0066-04a" xlink:href="note-0066-04"/> MN, ad eandem partem figurarum iam ſuperpoſitarum, & </s> <s xml:id="echoid-s1200" xml:space="preserve">inuicem <lb/>parallelæ, vel congruentes, ſed in noſtro caſu erunt congruentes, <lb/>cum enim vt, BC, ad, FG, ita ſit, DQ, ad, MN, ipſæ verò, B <lb/>C, FG, ſint ęquales, etiam, DQ, MN, ęquales erunt, ſicut etiam, <lb/> <anchor type="note" xlink:label="note-0066-05a" xlink:href="note-0066-05"/> CO, GR, (quæ ſunt inter ſe vt, DQ, MN,) ergo cum punctus, <lb/>B, poſitus ſit in, F, erit, O, in, R, & </s> <s xml:id="echoid-s1201" xml:space="preserve">DQ, extenſa ſuper, MN, <lb/> <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> & </s> <s xml:id="echoid-s1202" xml:space="preserve">cum etiam, DO, MR, ſint ęquales <lb/>punctus, D, erit in, M, ſic autem oſten-<lb/>demus quoque punctum, Q, cadere in, <lb/>N, & </s> <s xml:id="echoid-s1203" xml:space="preserve">conſequenter, XQ, cadere ſuper, <lb/>PN, &</s> <s xml:id="echoid-s1204" xml:space="preserve">, AD, ſuper, EM, ſi ergo figu-<lb/>ra, ABXC, cadens ſuper, EFPG, non <lb/>congruit illi, eſto quod ceciderit, ſi poſ-<lb/>ſibile eſt vt, FVIG, ita vt ambitus ex-<lb/>tra ambitum cadat, ſumpto autem quo-<lb/>cunque puncto, I, qui ſit in ambitu fi-<lb/>guræ, VFPGI, ſed cadens non in am-<lb/>bitu figuræ, EFPG, per ipſum duca-<lb/>tur, TZ, parallela, EM, ſecans, MN, <lb/>in, Z, ambitum figuræ, VPI, in, V, I, & </s> <s xml:id="echoid-s1205" xml:space="preserve">ambitum figuræ, EF <lb/>PG, in, T, S, erunt autem homologę, VI, TS, & </s> <s xml:id="echoid-s1206" xml:space="preserve">inter ſe ęqua-<lb/> <anchor type="note" xlink:label="note-0066-06a" xlink:href="note-0066-06"/> les cum ſint, vt incidentes, DQ, MN, quæ ſunt ęquales, necnon <lb/>æquales reliquæ vſq; </s> <s xml:id="echoid-s1207" xml:space="preserve">ad incidentes, nempè, SZ, IZ, quod eſt ab-<lb/> <anchor type="note" xlink:label="note-0066-07a" xlink:href="note-0066-07"/> ſurdum, punctus enim, I, non eſt in S, non ergo cadet ambitus fi-<lb/>guræ, ABXC, ſuperpoſitę, ipſi, EFPG, vt dictum eſt, extra am-<lb/>bitum eiuſdem figurę, EFPG, igitur cadet ſuper illius ambitum, & </s> <s xml:id="echoid-s1208" xml:space="preserve"><lb/>ipſę figurę erunt ſibi inuicem congruentes, quod oſtendendum erat.</s> <s xml:id="echoid-s1209" xml:space="preserve"/> </p> <div xml:id="echoid-div141" type="float" level="2" n="1"> <note position="left" xlink:label="note-0066-01" xlink:href="note-0066-01a" xml:space="preserve">D. Defin. <lb/>10.</note> <note position="left" xlink:label="note-0066-02" xlink:href="note-0066-02a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="left" xlink:label="note-0066-03" xlink:href="note-0066-03a" xml:space="preserve">B. Def. 10.</note> <note position="left" xlink:label="note-0066-04" xlink:href="note-0066-04a" xml:space="preserve">D. Defin. <lb/>10.</note> <note position="left" xlink:label="note-0066-05" xlink:href="note-0066-05a" xml:space="preserve">A. Def. 10.</note> <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a"> <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0066-01"/> </figure> <note position="left" xlink:label="note-0066-06" xlink:href="note-0066-06a" xml:space="preserve">C. Defin. <lb/>10.</note> <note position="left" xlink:label="note-0066-07" xlink:href="note-0066-07a" xml:space="preserve">A. Defin. <lb/>10.</note> </div> </div> <div xml:id="echoid-div143" type="section" level="1" n="98"> <head xml:id="echoid-head109" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1210" xml:space="preserve">EX hoc inſuper colligitur figuras quaſcumq; </s> <s xml:id="echoid-s1211" xml:space="preserve">planas ſimiles ab æqua-<lb/>libus rectis lineis, tanquam ab bomologis, deſcriptas interfé æqua-<lb/>les eſſe, cum ita ad inuicem ſuperponi poſſint, vt ſibi congruant, velut <pb o="47" file="0067" n="67" rhead="LIBERI."/> in Prop. </s> <s xml:id="echoid-s1212" xml:space="preserve">demonſtratum eſt. </s> <s xml:id="echoid-s1213" xml:space="preserve">Et vice verſaſi figuræ ſint ſimiles, & </s> <s xml:id="echoid-s1214" xml:space="preserve">æqua-<lb/>les, etiam homologas æquales eſſe, ſi enim inæquales eſſent, etiam ipſæ <lb/>figuræ inæquales eſſent, quod eſt abſurdum. </s> <s xml:id="echoid-s1215" xml:space="preserve">Vlterius autem patet, ſi <lb/>ſint inuicem ſuperpoſitæ, ita vt ſimiliter ſint conſtitutæ, ac duæ quæuis <lb/>homologæ inuicem fuerint congruentes, etiam ipſas figuras fore con-<lb/>gruentes, alioquin ſequerentur abſurda ſuperius demonſtrata, cum quę-<lb/>uis aliæ homologæ neceſſariò quoque ſint æquales, quæ enim congruerunt <lb/>ſunt æquales, & </s> <s xml:id="echoid-s1216" xml:space="preserve">ſubinde etiam incidentes, & </s> <s xml:id="echoid-s1217" xml:space="preserve">quæuis aliæ homologæ in-<lb/>ter ſe ſunt æquales.</s> <s xml:id="echoid-s1218" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div144" type="section" level="1" n="99"> <head xml:id="echoid-head110" xml:space="preserve">THEOREMA XXIII. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s1219" xml:space="preserve">SI duobus parallelis quibuſcumque planis inciderint duo <lb/>plana ſe ſe interſecantia, primum nempè, & </s> <s xml:id="echoid-s1220" xml:space="preserve">ſecundum; <lb/></s> <s xml:id="echoid-s1221" xml:space="preserve">fuerint autem alia duo parallela quæcumque plana, quibus <lb/>pariter incidant duo alia plana ſe ſe diuidentia, primum ſi-<lb/>militer, & </s> <s xml:id="echoid-s1222" xml:space="preserve">ſecundum: </s> <s xml:id="echoid-s1223" xml:space="preserve">Eorum autem cum paralielis planis <lb/>communes ſectiones angulos ęquales comprehenderint, nec-<lb/>non primorum, ac ſecundorum planorum mutuæ ſectiones <lb/>ad communes ſectiones primorum planorum cum planis pa-<lb/>rallelis effe ct as angulos æquales conſtituerint, ipſa verò pri-<lb/>ma plana ad plana parallela æquè fuerint ad eandem partem <lb/>inclinata: </s> <s xml:id="echoid-s1224" xml:space="preserve">Eędem communes ſectiones ad communes ſectio-<lb/>nes ſecundorum planorum cum planis parallelis effe ctas an-<lb/>gulos pariter conſtituent æquales, necnon ſecunda plana e-<lb/>runt ad eadem plana parallela æqualiter ad eandem partem <lb/>inclinata.</s> <s xml:id="echoid-s1225" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1226" xml:space="preserve">Sint duo parallela quæcunque plana, BD, HV, quibus incidat <lb/>duo plana, HA, primum, AV, ſecundum ſe ſe ſecantia in recta, <lb/>AG. </s> <s xml:id="echoid-s1227" xml:space="preserve">Sint nunc alia duo plana quęcunq; </s> <s xml:id="echoid-s1228" xml:space="preserve">parallela, LQ, & </s> <s xml:id="echoid-s1229" xml:space="preserve">Λ, qui-<lb/>bus pariter incidant alia duo plana, LY, primum, &</s> <s xml:id="echoid-s1230" xml:space="preserve">, Κ Λ, ſecun-<lb/>dum, ſe ſe pariter ſecantia in recta, KY, communes vero ſectiones, <lb/>BA, AD; </s> <s xml:id="echoid-s1231" xml:space="preserve">LK, KQ, incidentium planorum cum planis paralle-<lb/>lis contineant angulos æquales, ſit nempè, BAD, angulus æqua-<lb/>lis angulo, LKQ, (erit. </s> <s xml:id="echoid-s1232" xml:space="preserve">n. </s> <s xml:id="echoid-s1233" xml:space="preserve">&</s> <s xml:id="echoid-s1234" xml:space="preserve">, HGV, ęqualis ipſi, & </s> <s xml:id="echoid-s1235" xml:space="preserve">Υ Λ,) ſimili-<lb/> <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> ter ipſæ, AG, KY, cum ipſis, GH, Y &</s> <s xml:id="echoid-s1236" xml:space="preserve">, angulos conſtituant æ-<lb/>quales, & </s> <s xml:id="echoid-s1237" xml:space="preserve">prima plana, BG, LY, ad plana parallela, BD, HV; <lb/></s> <s xml:id="echoid-s1238" xml:space="preserve">LQ, & </s> <s xml:id="echoid-s1239" xml:space="preserve">Λ, ſint æquè ad eandem partem inclinata. </s> <s xml:id="echoid-s1240" xml:space="preserve">Dico angulos, <lb/>AGV, Κ Υ Λ, ęquales eſſe, necnonſecunda plana, AV, Κ Λ, ad <pb o="48" file="0068" n="68" rhead="GEOMETRIÆ"/> eadem parallela plana eſſe æqualiter ad eandem partem inclinata. <lb/></s> <s xml:id="echoid-s1241" xml:space="preserve"> <anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> Siigitur, AG, KY, eſſent dictis planis parallelis perpendiculares, <lb/> <anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/> manifeſtum eſt, quod anguli, AGV, Κ Υ Λ, eſſent æquales, ideſt <lb/>recti, & </s> <s xml:id="echoid-s1242" xml:space="preserve">plana, AV, Κ Λ, eiſdem planis parallelis erecta; </s> <s xml:id="echoid-s1243" xml:space="preserve">ſed non <lb/>ſint perpendiculares, & </s> <s xml:id="echoid-s1244" xml:space="preserve">à punctis, A, K, demittanturipſę, AE, K <lb/> <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> T, quæ eiſdem <lb/>ſint perpẽdicu-<lb/>lares, incidant <lb/>autem ſubiectis <lb/>planis in pun-<lb/>ctis, E, T; </s> <s xml:id="echoid-s1245" xml:space="preserve">de-<lb/>inde à puncto, <lb/>A, ad, HG, V <lb/>G, productas, <lb/>ducantur per-<lb/>pendiculares, A <lb/>F, quidem ipſi; <lb/></s> <s xml:id="echoid-s1246" xml:space="preserve">HG, & </s> <s xml:id="echoid-s1247" xml:space="preserve">AP, <lb/> <anchor type="note" xlink:label="note-0068-03a" xlink:href="note-0068-03"/> ipſi, VG, inci-<lb/>dentes in pun-<lb/> <anchor type="note" xlink:label="note-0068-04a" xlink:href="note-0068-04"/> ctis, V, P, niſi <lb/>fortè, AG, eſſet <lb/>alteri earũ per-<lb/>pendicularis, vt <lb/>contingere po-<lb/>teſt, & </s> <s xml:id="echoid-s1248" xml:space="preserve">iungan-<lb/>tur, EP, EG, <lb/>EF; </s> <s xml:id="echoid-s1249" xml:space="preserve">ſimiliter in <lb/>alia figura ca-<lb/>dant à puncto, <lb/>K, perpendicu-<lb/>lariter ſuper ip-<lb/>ſas, & </s> <s xml:id="echoid-s1250" xml:space="preserve">Y, Λ Υ, <lb/>productas, ſi o-<lb/>pus ſit, ipſæ, K <lb/>Z, KX, & </s> <s xml:id="echoid-s1251" xml:space="preserve">iun-<lb/> <anchor type="note" xlink:label="note-0068-05a" xlink:href="note-0068-05"/> gantur ſimiliter, TX, TY, &</s> <s xml:id="echoid-s1252" xml:space="preserve">, TZ. </s> <s xml:id="echoid-s1253" xml:space="preserve">Quoniam ergo anguli, AF <lb/>G, KZY, ſunt recti, ideò quadratum, AG, erit æquales duobus <lb/>quadratis, AF, FG, ſicut quadratum, KY, æquale duobus, KZ, <lb/>ZY, eſt autem etiam quadratum, AF, ęquale duobus quadratis, A <lb/>E, EF, quia angulus, AEF. </s> <s xml:id="echoid-s1254" xml:space="preserve">rectus eſt, & </s> <s xml:id="echoid-s1255" xml:space="preserve">quadratum, KZ, pari-<lb/>teræquale quadratis, KT, TZ, ergo quadratum, AG, ideſt duo <pb o="49" file="0069" n="69" rhead="LIBER I."/> quadrata, AE, EG, (quia etiam, AEG, rectus eſt) æquabuntur <lb/>tribus quadratis, AE, EF, FG, vnde, ablato communiquadrato, <lb/> <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> AE, quadratum, GE, ęquabitur duobus quadratis, GF, FE; </s> <s xml:id="echoid-s1256" xml:space="preserve">pari <lb/>ratione autem probabimus quadratum, YT, æquari quadratis, Y <lb/> <anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/> Z, ZT, vnde anguli, GFE, YZT, recti erunt; </s> <s xml:id="echoid-s1257" xml:space="preserve">& </s> <s xml:id="echoid-s1258" xml:space="preserve">eodem modo <lb/> <anchor type="note" xlink:label="note-0069-03a" xlink:href="note-0069-03"/> probabimus eſſe rectos, EPG, TXY, ergo anguli, AFE, KZT, <lb/> <anchor type="note" xlink:label="note-0069-04a" xlink:href="note-0069-04"/> erunt anguli inclinationis primorum planorum, BG, LY, cum iu-<lb/>biectis planis, HV, & </s> <s xml:id="echoid-s1259" xml:space="preserve">Λ, & </s> <s xml:id="echoid-s1260" xml:space="preserve">ideò inter ſe æquales: </s> <s xml:id="echoid-s1261" xml:space="preserve">Similiter anguli, <lb/>APE, KXT, erunt inclinationes ſecundorum planorum, AV, K <lb/>Λ, cum eiſdem ſubiectis planis. </s> <s xml:id="echoid-s1262" xml:space="preserve">Quia ergo anguli, AFE, KZT, <lb/>ſunt æquales, &</s> <s xml:id="echoid-s1263" xml:space="preserve">, AEF, KTZ, recti, erunt triangula, AFE, K <lb/>ZT, inter ſe ſimilia, vt etiam triangula, AFG, KZY, inter ſe, <lb/>nam anguli, AGF, KYZ, ſunt quoque æquales, &</s> <s xml:id="echoid-s1264" xml:space="preserve">, AFG, KZ <lb/>Y, recti; </s> <s xml:id="echoid-s1265" xml:space="preserve">erit ergo, vt, EF, ad, FA, ſic, TZ, ad, ZK, & </s> <s xml:id="echoid-s1266" xml:space="preserve">vt, A <lb/>F, ad, FG, ſic, KZ, ad, ZY, ergo ex æquali, vt, EF, ad, FG, <lb/>itaerit, TZ, ad, ZY, & </s> <s xml:id="echoid-s1267" xml:space="preserve">ſunt circa rectos, nempè æquales angu-<lb/> <anchor type="note" xlink:label="note-0069-05a" xlink:href="note-0069-05"/> los, GFE, YZT, ergo triangula, GFE, YZT, pariter ſimilia <lb/>erunt, anguli igitur, EGF, TYZ, adæquabuntur, totus autem, <lb/>PGF, toti, XYZ, æquatur, ergo reliquus, EGP, erit ęqualis re-<lb/>liquo, TYX, & </s> <s xml:id="echoid-s1268" xml:space="preserve">ſunt recti, EPG, TXY, vt probatum eſt, ergo <lb/>erunt, GPE, YXT, ſimilia triangula, igitur, vt, PG, ad, GE, <lb/>ſic erit, XY, ad, YT, vt verò, GE, ad, GF, ſic eſt, YT, ad, YZ, <lb/>& </s> <s xml:id="echoid-s1269" xml:space="preserve">vt, GF, ad, GA, ſic, YZ, ad, YK, ergo ex æquali, PG, ad, <lb/>GA, erit vt, XY, ad, YK, habemus ergo duo triangula, APG, <lb/>KXY, habentia duos angulos, APG, KXY, ęquales, ſunt. </s> <s xml:id="echoid-s1270" xml:space="preserve">n. </s> <s xml:id="echoid-s1271" xml:space="preserve">re-<lb/>cti, circa verò duos, PGA, XYK, latera proportionalia, & </s> <s xml:id="echoid-s1272" xml:space="preserve">reli-<lb/>quorum vtrumq; </s> <s xml:id="echoid-s1273" xml:space="preserve">ſimul, PAG, XKY, minorem recto, ergo erunt <lb/> <anchor type="note" xlink:label="note-0069-06a" xlink:href="note-0069-06"/> ſimilia, & </s> <s xml:id="echoid-s1274" xml:space="preserve">anguli, PGA, XYK, ęquales, vnde reliqui, AGV, Κ <lb/>Υ Λ, pariter æquales erunt, quod eſt vnum propoſitorum.</s> <s xml:id="echoid-s1275" xml:space="preserve"/> </p> <div xml:id="echoid-div144" type="float" level="2" n="1"> <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">10. Vnde. <lb/>cimi El.</note> <note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">Defin. 3. <lb/>Vndec. El.</note> <note position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">18. Vnde-<lb/>cimi El.</note> <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a"> <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0068-01"/> </figure> <note position="left" xlink:label="note-0068-03" xlink:href="note-0068-03a" xml:space="preserve">Vide di-<lb/>cta lib. 7. <lb/>annot.</note> <note position="left" xlink:label="note-0068-04" xlink:href="note-0068-04a" xml:space="preserve">Prop. 3.</note> <note position="left" xlink:label="note-0068-05" xlink:href="note-0068-05a" xml:space="preserve">47. Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">48. Primi <lb/>El. Def. 6.</note> <note position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">Vndec.</note> <note position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">Elem.</note> <note position="right" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">Defin. 6. <lb/>Vnd. El.</note> <note position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">6. Sexti <lb/>Elem.</note> <note position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">7. Sexti <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s1276" xml:space="preserve">Rurſus, quia, PE, ad, EF, eſt vt, XT, ad, TZ; </s> <s xml:id="echoid-s1277" xml:space="preserve">EF, autem <lb/>ad, EA, vt, TZ, ad, TK, ergo ex æquali, PE, ad, EA, erit vt, <lb/>XT, ad; </s> <s xml:id="echoid-s1278" xml:space="preserve">TK, & </s> <s xml:id="echoid-s1279" xml:space="preserve">ſunt circa ęquales angulos, PEA, XTK, latera <lb/> <anchor type="note" xlink:label="note-0069-07a" xlink:href="note-0069-07"/> proportionalia, ergo triangula, APE, KXT, ſimil a erunt, nec-<lb/>non anguli, APE, KXT, inclinationis ſecundorum planorum, A <lb/>V, Κ Λ, cum ſubiectis planis inter ſe æquales, & </s> <s xml:id="echoid-s1280" xml:space="preserve">ad eandem partem <lb/>quod etiam demonſtrare propoſitum fuit.</s> <s xml:id="echoid-s1281" xml:space="preserve"/> </p> <div xml:id="echoid-div145" type="float" level="2" n="2"> <note position="right" xlink:label="note-0069-07" xlink:href="note-0069-07a" xml:space="preserve">6. Sexti <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div147" type="section" level="1" n="100"> <head xml:id="echoid-head111" xml:space="preserve">THEOREMA XXIV. PROPOS XXVII.</head> <p> <s xml:id="echoid-s1282" xml:space="preserve">POſita definitione, quam affert Euclides lib 6. </s> <s xml:id="echoid-s1283" xml:space="preserve">El. </s> <s xml:id="echoid-s1284" xml:space="preserve">de ſimi-<lb/>libus figuris rectilineis, ſequitur pro ipſis etiam defini-<lb/>tio geneneralis, quam de omnibus ſimilibus figuris planis <lb/>ipſe attuli.</s> <s xml:id="echoid-s1285" xml:space="preserve"/> </p> <pb o="50" file="0070" n="70" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s1286" xml:space="preserve">Sint duæ vtcumque figuræ rectilineæ, ABDEH, MTRPN; <lb/></s> <s xml:id="echoid-s1287" xml:space="preserve"> <anchor type="note" xlink:label="note-0070-01a" xlink:href="note-0070-01"/> ſimiles iuxta definitionem E@clidis, ideſt ſingulos habentes angulos <lb/>æquales, A, M; </s> <s xml:id="echoid-s1288" xml:space="preserve">B, T; </s> <s xml:id="echoid-s1289" xml:space="preserve">D, R; </s> <s xml:id="echoid-s1290" xml:space="preserve">P, E; </s> <s xml:id="echoid-s1291" xml:space="preserve">HN, & </s> <s xml:id="echoid-s1292" xml:space="preserve">circa æquales angu-<lb/>los latera proportionalia. </s> <s xml:id="echoid-s1293" xml:space="preserve">Dico eaidem eſſe ſimiles iuxta meam de-<lb/>finitionem: </s> <s xml:id="echoid-s1294" xml:space="preserve">Ducantur duæ vtcum que oppoſitæ earum tangentes, <lb/>quæ cum duobus ex lateribus homologis earumdem angulos æqua-<lb/>les ab eadem parte contineant, ſint autem ex vna parte tangentes <lb/>ipſæ, AH, MN, quæ cum ipſis, HE, NP, lateribus homologis <lb/>angulos continent ęquales, AHG, MNO, & </s> <s xml:id="echoid-s1295" xml:space="preserve">ſint ex alia parte tan-<lb/>gentes ipſæ, DF, RQ, quæ cum ipſis, HE, NP, productis con-<lb/> <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> currant in punctis, F, Q, ducantur deinde à <lb/>punctis angulorum, qui ſunt, B, E; </s> <s xml:id="echoid-s1296" xml:space="preserve">TP, di-<lb/>ctis tangentibus parallelæ, BG, CE, TO, S <lb/>P, & </s> <s xml:id="echoid-s1297" xml:space="preserve">iungantur, BH, BE, TN, TP. </s> <s xml:id="echoid-s1298" xml:space="preserve">Quia <lb/>ergo anguli, MNQ, AHF, ſunt æquales, <lb/>etiam anguli, NQR, HFD, erunt ęquales, <lb/>& </s> <s xml:id="echoid-s1299" xml:space="preserve">quia anguli, NPR, HED, ſunt quoque <lb/>æquales, etiam anguli, RPQ, DEF, erunt <lb/>æquales, & </s> <s xml:id="echoid-s1300" xml:space="preserve">reliqu reliquis, vnde trianguli, R <lb/>PQ, DEF, erunt æquianguli, & </s> <s xml:id="echoid-s1301" xml:space="preserve">ideò, QP, <lb/> <anchor type="note" xlink:label="note-0070-02a" xlink:href="note-0070-02"/> ad, PR, erit vt, FE, ad, ED, eſt autem, R <lb/>P, ad, PN, vt, DE, ad, EH, ergo, ex ęquali, <lb/> <anchor type="note" xlink:label="note-0070-03a" xlink:href="note-0070-03"/> QP, ad, PN, erit vt, FE, ad, EH, igitur, <lb/>NQ, HF, ſunt ſimiliter ad eandem partem <lb/>diuiſæ in punctis, E, P, quia verò angulus, NPS, æquatur angu-<lb/>lo, NQR.</s> <s xml:id="echoid-s1302" xml:space="preserve">. HFD.</s> <s xml:id="echoid-s1303" xml:space="preserve">. HEC, &</s> <s xml:id="echoid-s1304" xml:space="preserve">, NPR, ipſi, HED, ideo reli-<lb/>quus, SPR, æquabitur reliquo, CED, eſt autem angulus, TR <lb/>P, ęqualis angulo, BDE, ergo trianguli, PSR, ECD, erunt æ-<lb/>quianguli, & </s> <s xml:id="echoid-s1305" xml:space="preserve">ideò, CE, ad, ED, erit vt, SP, ad, PR, &</s> <s xml:id="echoid-s1306" xml:space="preserve">, ED, <lb/>ad, EF, erit vt, RP, ad, PQ; </s> <s xml:id="echoid-s1307" xml:space="preserve">ergo ex æquali, & </s> <s xml:id="echoid-s1308" xml:space="preserve">permutando, C <lb/>E, ad, SP, erit vt, EF, ad, PQ .</s> <s xml:id="echoid-s1309" xml:space="preserve">i. </s> <s xml:id="echoid-s1310" xml:space="preserve">vt, HF, ad, NQ. </s> <s xml:id="echoid-s1311" xml:space="preserve">S militer <lb/> <anchor type="note" xlink:label="note-0070-04a" xlink:href="note-0070-04"/> quia anguli, BDE, TRP, ſunt æquales, & </s> <s xml:id="echoid-s1312" xml:space="preserve">circa eos latera ſunt <lb/>proportionalia, ideò trianguli, BDE, TRP, erunt æquianguli, <lb/>vnde anguli, DBE, RTP, &</s> <s xml:id="echoid-s1313" xml:space="preserve">, BED, TPR, erunt ęquales, ſunt <lb/>autem ęquales ipſi, CED, SPR, ergo reliqui, BEC, TPS, erunt <lb/>æquales, & </s> <s xml:id="echoid-s1314" xml:space="preserve">ideò trianguli, BCE, TSP, erunt ęquianguli, & </s> <s xml:id="echoid-s1315" xml:space="preserve">quia <lb/>angulus, BEF, eſt ęqualis ipſi, TPQ, reliquns, BEH, erit ęqua-<lb/>lis reliquo, TPN, eſt autem, BGE, ęqual sipſi, TOP, ergo trian-<lb/>guli, BGE, TOP, erunt ęquianguli, ergo, BG, ad, TO, erit vt, <lb/>BE, ad, TP, ideſt vt, CE, ad, SP, ideſt vt, HF, ad, NQ, per-<lb/>mucando, & </s> <s xml:id="echoid-s1316" xml:space="preserve">conuertendo, HF, ad, GB, erit vt, NQ, ad, OT; <lb/></s> <s xml:id="echoid-s1317" xml:space="preserve">quia verò anguli, HAB, NMT, ſunt ęquales, & </s> <s xml:id="echoid-s1318" xml:space="preserve">circa eoſdem la- <pb o="51" file="0071" n="71" rhead="LIBER I."/> tera proportionalia, ideòtriang. </s> <s xml:id="echoid-s1319" xml:space="preserve">HAB, NMT, ſuntæquiangun, <lb/> <anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/> & </s> <s xml:id="echoid-s1320" xml:space="preserve">anguli, AHB, MNT; </s> <s xml:id="echoid-s1321" xml:space="preserve">ABH, MTN, interſe ęquales, ergo <lb/>cum anguli, AHG, MNO, ſint æquales, reliqui, BHG, TN <lb/>O, erunt æquales, ſunt etiam æquales anguli, HGB, NOT, ergo <lb/>trianguli, HBG, NTO, ſunt æquianguli, ergo, BG, ad, GH, <lb/>erit vt, TO, ad, ON, erat autem, FH, ad, GB, vt, QN, ad, O <lb/>T, ergo ex ęquali, FH, ad, HG, erit vt, QN, ad, NO, ſunt@gi-<lb/>tur ipiæ, HF, NQ, ſimiliter diuiſæ, & </s> <s xml:id="echoid-s1322" xml:space="preserve">ad eandem partem in pun-<lb/>ctis, G, O, & </s> <s xml:id="echoid-s1323" xml:space="preserve">ipſæ diuidentes, BG, TO, ſunt vt ipſæ, HF, NQ.</s> <s xml:id="echoid-s1324" xml:space="preserve"/> </p> <div xml:id="echoid-div147" type="float" level="2" n="1"> <note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">Prima <lb/>Def. Sex-<lb/>ti Elem.</note> <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a"> <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0070-01"/> </figure> <note position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">4. Sexti <lb/>Elem.</note> <note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">Ex Defin. <lb/>Eucl.</note> <note position="left" xlink:label="note-0070-04" xlink:href="note-0070-04a" xml:space="preserve">6. Sexti <lb/>Elem.</note> <note position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">6. Sexei <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s1325" xml:space="preserve">Ducantur nunc inter dictas oppoſitas tangentes elſdem parallelæ <lb/>duæ v@cumque, VK, XY, inter circuitum figurarum iam propoſi-<lb/>tarum, & </s> <s xml:id="echoid-s1326" xml:space="preserve">rectas, HF, NQ, comprehenſę, ſimiliter ad eandem par-<lb/>tem diu@dentes ipſas, HF, NQ, in punctis, K, Y, ſecanteſque ip-<lb/>ſas, BE, TP, in punctis, 3, 4, eſt ergo, FK, ad, QY, permutan-<lb/>do, vt, HF, ad, QN, ideſt vt, FE, ad, QP, ergo, FK, ad, Q <lb/>Y, erit vt, FE, ad, QP, & </s> <s xml:id="echoid-s1327" xml:space="preserve">reliqua, EK, ad reliquam, PY, vt, F <lb/>K, ad, QY, ideſt vt, FH, ad, QN; </s> <s xml:id="echoid-s1328" xml:space="preserve">Similiter oſtendenius, vt, F <lb/>H, ad, QN, ſic eſſe, GK, ad, OY, ergo, GK, ad, OY, erit vt, <lb/>KE, ad, YP, &</s> <s xml:id="echoid-s1329" xml:space="preserve">, permutando, GK, ad, KE, erit vt, OY, ad, Y <lb/>P, componendoque, GE, ad, FK, erit vt, OP, ad, PY, eſt verò, <lb/>vt, GE, ad, EK, ita, BG, ad, 3K, & </s> <s xml:id="echoid-s1330" xml:space="preserve">vt, OP, ad, PY, ita, T <lb/>O, ad, Y4, ergo, BG, ad, 3K, erit vt, TO, ad, Y4, & </s> <s xml:id="echoid-s1331" xml:space="preserve">permu-<lb/>tando, BG, ad, TO, erit vt, 3K, ad, 4Y, eſt verò vt, BG, ad, T <lb/>O, ita, HF, ad, NQ, ergo, 3K, ad, 4Y, erit vt, HF, ad, NQ, <lb/>ſimiliter, quia ipſæ, VK, XY, diuidunt ſimiliter ad eandem partem <lb/>ipſas, BC, TS, in punctis, V, X, ac diuiduntur ipſę, GE, OP, in <lb/>punctis, K, Y, ideò eodem modo oſtendemus ipſas, V3, X4, eſſe <lb/>vt ipſas, CE, SP, ideſt vt ipſas, HF, NQ, erant autem, 3K, 4 <lb/>X, vt ipſæ, HF, NQ, ergo totæ, VK, XY, erunt vt ipiæ, HF, <lb/>NQ, habemus igitur figuras, ADE, MRP, in quibus ductę ſunt <lb/>oppoſitæ tangentes, AH, DF, MN, RQ, quibus inciderunt ip-<lb/>ſæ, HF, NQ, ad eundem angulum ex eadem parte, inuentum eſt <lb/>autem eas, quæ inter dictas, HF, NQ, & </s> <s xml:id="echoid-s1332" xml:space="preserve">circuitum figurarum ei-<lb/>ſdem tangentibus vtcumq; </s> <s xml:id="echoid-s1333" xml:space="preserve">ducuntur ęquidiſtantes, & </s> <s xml:id="echoid-s1334" xml:space="preserve">ſecant dictas, <lb/>HF, NQ, ſimiliter ad eandem partem, eodem ordine ſumptas, eſſe <lb/>vt ipſas, HF, NQ, ergo figuræ, ADE, MRP, quæ erant ſimi-<lb/>les iuxta definitionem Euclidis, erunt etiam ſimiles iuxta definitio-<lb/>nem meam, & </s> <s xml:id="echoid-s1335" xml:space="preserve">erunt dictæ tangentes regulæ homologarum earum-<lb/> <anchor type="note" xlink:label="note-0071-02a" xlink:href="note-0071-02"/> dem, & </s> <s xml:id="echoid-s1336" xml:space="preserve">ipſarum, ac dictarum ſimilium figurarum incidentes ipſę, H <lb/>F, NQ, quod erat oſtendendum.</s> <s xml:id="echoid-s1337" xml:space="preserve"/> </p> <div xml:id="echoid-div148" type="float" level="2" n="2"> <note position="right" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">Deſin. 10, <lb/>huius.</note> </div> <pb o="52" file="0072" n="72" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div150" type="section" level="1" n="101"> <head xml:id="echoid-head112" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1338" xml:space="preserve">_Q_Via verò oppoſitæ tangentes, AH, DF, MN, RQ, ductæ ſunt <lb/>vtcumque, angulos tamen æquales ad eandem partem cum homo-<lb/>logis lateribus continentes, ideò quaſcumq; </s> <s xml:id="echoid-s1339" xml:space="preserve">duxerimus oppoſitas <lb/>tangentes in figuris rectilineis ſimilibus iuxta Euclidem, dummodo fa-<lb/>ciant angulos æquales ad eandem partem cum lateribus homologis, ea-<lb/>ſdem eſſe regulas homologarum ſimilium figurarum poterit probari.</s> <s xml:id="echoid-s1340" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div151" type="section" level="1" n="102"> <head xml:id="echoid-head113" xml:space="preserve">THEOREMA XXV. PROPOS. XXVIII.</head> <p> <s xml:id="echoid-s1341" xml:space="preserve">POſita infraſcripta definitione ſimilium portionum ſectio-<lb/>num coni, illi adiuncti, quod infra dicetur, ſequitur <lb/>pro ipſis etiam mea definitio generalis ſimilium planarum fi-<lb/>gurarum. </s> <s xml:id="echoid-s1342" xml:space="preserve">Hoc autem dico pro ſpatijs ſub ipſis ſectionibus, <lb/>& </s> <s xml:id="echoid-s1343" xml:space="preserve">rectis lineis contentis, non autem pro ipſis tanquam lineis, <lb/>licet crediderim Apolloniũ ipſarum ſimilium ſectionum tan-<lb/>quam linearum, non autem figurarum, quę fiunt ab ipſis, ſi-<lb/>militudinem attendiſſe, ego verò ipſam recipio tanquam ip-<lb/>ſarum figurarum ſimilitudini congruam, dum illi adiungitur, <lb/>quod in ipſa Propoſ. </s> <s xml:id="echoid-s1344" xml:space="preserve">explicatur.</s> <s xml:id="echoid-s1345" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div152" type="section" level="1" n="103"> <head xml:id="echoid-head114" xml:space="preserve">DEFINITIO.</head> <p> <s xml:id="echoid-s1346" xml:space="preserve">SImiles portiones ſectionum coni ſunt, in quarum ſingulis ductis <lb/>lineis baſi parallelis numero æqualibus, ſunt ipſæ parallelæ, & </s> <s xml:id="echoid-s1347" xml:space="preserve"><lb/>baſes ad abſciſſas diametrorum partes ſumptas à verticibus, in ijſdem <lb/>rationibus, tumabſciſſæ ipſæ ad abſciſſas: </s> <s xml:id="echoid-s1348" xml:space="preserve">Apollonius lib.</s> <s xml:id="echoid-s1349" xml:space="preserve">6. </s> <s xml:id="echoid-s1350" xml:space="preserve">Coni-<lb/>corum, vt refert Eutocius.</s> <s xml:id="echoid-s1351" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1352" xml:space="preserve">Sint ſimiles portiones ſectionum coni, DAF, QRK, in baſibus, <lb/>DF, QK, quarum diametri ſint ipſæ, AE, RG, ſecentur autem <lb/>ſimiliter ipſæ diametri in punctis, N, O; </s> <s xml:id="echoid-s1353" xml:space="preserve">V, X; </s> <s xml:id="echoid-s1354" xml:space="preserve">& </s> <s xml:id="echoid-s1355" xml:space="preserve">ſit, D F, ad, E <lb/>A, vt, Q K, ad, G R, &</s> <s xml:id="echoid-s1356" xml:space="preserve">, C H, ad, O A, vt, T L, ad, X R, &</s> <s xml:id="echoid-s1357" xml:space="preserve">, P <lb/>M, ad, N A, vt, S P, ad, V R; </s> <s xml:id="echoid-s1358" xml:space="preserve">has igitur Apollonius in ſupradicta <lb/>definitione ſimiles vocat, mihi autem hoc opus eſt illi adiungere.</s> <s xml:id="echoid-s1359" xml:space="preserve">ſ. <lb/></s> <s xml:id="echoid-s1360" xml:space="preserve">quod anguli baſibus, & </s> <s xml:id="echoid-s1361" xml:space="preserve">diametris, ad eandem partem contenti ſint <lb/>æquales, vt angulus, A E D, ipſi, R G Q, ſi.</s> <s xml:id="echoid-s1362" xml:space="preserve">u. </s> <s xml:id="echoid-s1363" xml:space="preserve">hoc non ponatur <lb/>poſſet contingere eſſe baſes, D F, Q K, æquales, & </s> <s xml:id="echoid-s1364" xml:space="preserve">ipſas, A E, R <lb/>G, in quo caſu tot figuras ſimiles, & </s> <s xml:id="echoid-s1365" xml:space="preserve">æquales, ex. </s> <s xml:id="echoid-s1366" xml:space="preserve">gr. </s> <s xml:id="echoid-s1367" xml:space="preserve">ipſi, A D F, <pb o="53" file="0073" n="73" rhead="LIBER I."/> poſſemus habere, quot ſunt variationes inclinationum diametrorum <lb/>ad baſes, quam tamen variationem per definitionem ſupradictam <lb/>excludere neceſſarium eſſe exiſtimaui. </s> <s xml:id="echoid-s1368" xml:space="preserve">Suppoſito igitur, quod tali <lb/>definitioni hoc adiungatur, dico eam cum mea concordare, ſi pro <lb/> <anchor type="figure" xlink:label="fig-0073-01a" xlink:href="fig-0073-01"/> ipſis ſectionibus tanquam figuris in-<lb/>telligatur. </s> <s xml:id="echoid-s1369" xml:space="preserve">Ductis enim per vertices, <lb/>A, R, baſibus, DF, QK, paralle-<lb/>lis, illæ tangent dictas portiones, & </s> <s xml:id="echoid-s1370" xml:space="preserve"><lb/>inter eaſdem ductas habebimus ip-<lb/>ſas, AE, RG, illis ad eundem an-<lb/>gulum incidentes ex eadem parte, <lb/>quibus ſimiliter ad eandem partem <lb/>diuiſis, vt in punctis, N, O; </s> <s xml:id="echoid-s1371" xml:space="preserve">V, X; <lb/></s> <s xml:id="echoid-s1372" xml:space="preserve">& </s> <s xml:id="echoid-s1373" xml:space="preserve">per eadem ductis ipſis tangenti-<lb/>bus parallelis, BM, CH, SP, TL, <lb/>inuenimus eas, quęinter ipſas, AE, <lb/>RG, & </s> <s xml:id="echoid-s1374" xml:space="preserve">circuitum figurarum, ADF, RQK, ad eandem partem <lb/>continentur, & </s> <s xml:id="echoid-s1375" xml:space="preserve">diuidunt ipſas ſimiliter ad eandem partem, eodem <lb/>ordine ſumptas, eſſe in proportione ipſarum, AE, RG, nam quia, <lb/>DF, ad, EA, eſt vt, QK, ad, GR, permutando, DF, ad, QK, <lb/>erit vt, EA, ad, GR, & </s> <s xml:id="echoid-s1376" xml:space="preserve">quia ipſæ, AE, RG, ſunt diametri, ad <lb/>quas ordinatim applicantur dictæ parallelę, ideò ab eiſdem bifariam <lb/>diuidentur, ergo, &</s> <s xml:id="echoid-s1377" xml:space="preserve">, DE, ad, QG, &</s> <s xml:id="echoid-s1378" xml:space="preserve">, EF, ad, GK, erit vt, E <lb/>A, ad GR, eodem modo oſtendemus tum, CO, ad, TX, tum, O <lb/>H, ad, XL, eſſe vt, OA, ad, XR .</s> <s xml:id="echoid-s1379" xml:space="preserve">i. </s> <s xml:id="echoid-s1380" xml:space="preserve">vt, EA, ad, GR, & </s> <s xml:id="echoid-s1381" xml:space="preserve">ſic, B <lb/>N, ad, SV, &</s> <s xml:id="echoid-s1382" xml:space="preserve">, NM, ad, VP, eſſe vt, NA, ad, VR .</s> <s xml:id="echoid-s1383" xml:space="preserve">i. </s> <s xml:id="echoid-s1384" xml:space="preserve">vt, EA, <lb/>ad, GR, ſunt igitur figurę, ADF, RQK, ſimiles iuxta meam de-<lb/> <anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/> finitionem, earum verò, & </s> <s xml:id="echoid-s1385" xml:space="preserve">tangentium oppoſitarum (quarum duæ <lb/>ex vna parte ſuntipſæ, DF, QK,) incidentes ſunt ipſæ, AE, RG.</s> <s xml:id="echoid-s1386" xml:space="preserve"/> </p> <div xml:id="echoid-div152" type="float" level="2" n="1"> <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a"> <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0073-01"/> </figure> <note position="right" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">Defin.10.</note> </div> </div> <div xml:id="echoid-div154" type="section" level="1" n="104"> <head xml:id="echoid-head115" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s1387" xml:space="preserve">_A_Ffert Commandinus ali@m definitionem ſimilium hyperbolarum, <lb/>ſcilicet ſimiles eſſe, quarum coniuncta diametri inter ſe, vel qua-<lb/>rum figuræ latera eandem proportionem babent, quam Dauid Riualtus <lb/>in Com. </s> <s xml:id="echoid-s1388" xml:space="preserve">in Arch. </s> <s xml:id="echoid-s1389" xml:space="preserve">lib de Conoidib. </s> <s xml:id="echoid-s1390" xml:space="preserve">& </s> <s xml:id="echoid-s1391" xml:space="preserve">Sphæroidibus ad Defin.</s> <s xml:id="echoid-s1392" xml:space="preserve">18. </s> <s xml:id="echoid-s1393" xml:space="preserve">oſten-<lb/>dit concordare cum ſupradicta Apollonij, quam videat, qui voluerit: <lb/></s> <s xml:id="echoid-s1394" xml:space="preserve">Hæc igitur eodem modo, quo illa Apollonij, cummea pariter concorda-<lb/>bit (ſumpta tamen hyp@rbola tanquam figura) vnde hac quoque hypo-<lb/>teſi ſi opus fuerit, pariter vtemur ad paſſiones inde dependentes demon-<lb/>ſtranda<unsure/>s.</s> <s xml:id="echoid-s1395" xml:space="preserve"/> </p> <pb o="54" file="0074" n="74" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div155" type="section" level="1" n="105"> <head xml:id="echoid-head116" xml:space="preserve">LEMMA I.</head> <p> <s xml:id="echoid-s1396" xml:space="preserve">SI ſint duæ ſimiles ſolidæ figuræ iuxta definit.</s> <s xml:id="echoid-s1397" xml:space="preserve">9. </s> <s xml:id="echoid-s1398" xml:space="preserve">Vndec. </s> <s xml:id="echoid-s1399" xml:space="preserve">Elem. </s> <s xml:id="echoid-s1400" xml:space="preserve">& </s> <s xml:id="echoid-s1401" xml:space="preserve"><lb/>in earum altera duæ aſſumantur in ambitu quæcumque figuræ <lb/>coincidentes, illæ erunt ad inuicem æquè ad eandem partem incli-<lb/>natæ, ac aliæ duæ, quæ in reliqua ſolida figura eiſdem ſimiles eſſe <lb/>ſupponuntur.</s> <s xml:id="echoid-s1402" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1403" xml:space="preserve">Sint ſim les ſolidæ figuræ, AN, KR, in earum autem altera, A <lb/>N, ſumantur duæ quæcumq; </s> <s xml:id="echoid-s1404" xml:space="preserve">figuræ inuicem coincidentes, AV, V <lb/> <anchor type="figure" xlink:label="fig-0074-01a" xlink:href="fig-0074-01"/> H, quibus in <lb/>reliqua ſimiles <lb/>ſint, Κ Λ, qui-<lb/>dem, AV, &</s> <s xml:id="echoid-s1405" xml:space="preserve">, <lb/>Λ &</s> <s xml:id="echoid-s1406" xml:space="preserve">, ipſi, HV: <lb/></s> <s xml:id="echoid-s1407" xml:space="preserve">Dico vtraſque, <lb/>AV, VH, ęquè <lb/>ad inuicem, & </s> <s xml:id="echoid-s1408" xml:space="preserve"><lb/>ad eandempar-<lb/>tem eſſe incli-<lb/>natas, ac ſunt <lb/>ipſę, Κ Λ, Λ &</s> <s xml:id="echoid-s1409" xml:space="preserve">. <lb/>Velergo, AG, <lb/> <anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/> KY, ſunt ſubie-<lb/>ctis planis per-<lb/>pẽdiculares, & </s> <s xml:id="echoid-s1410" xml:space="preserve"><lb/>tunc, AV, Κ Λ, <lb/>erunt ipſis, H <lb/>V, & </s> <s xml:id="echoid-s1411" xml:space="preserve">Λ, erecta, <lb/>vel nõ, & </s> <s xml:id="echoid-s1412" xml:space="preserve">tunc <lb/>demittantur à <lb/>punctis, A, K, <lb/>ſubiectis planis <lb/>perpẽdiculares, <lb/>AE, KT, & </s> <s xml:id="echoid-s1413" xml:space="preserve">ſu-<lb/>per ipſas, HG, <lb/>VG, productas <lb/>(ſi opus ſit, & </s> <s xml:id="echoid-s1414" xml:space="preserve"><lb/>niſi, AG, KY, <lb/>ſint vel ipſis, H <lb/>G, & </s> <s xml:id="echoid-s1415" xml:space="preserve">Y, vel ip-<lb/>ſis, GV, Υ Λ, perpendiculares) ſimiliter ad angulos rectos cadant, <pb o="55" file="0075" n="75" rhead="LIBER I."/> AP, KX, quidem ipſis, VG, Λ Υ, &</s> <s xml:id="echoid-s1416" xml:space="preserve">, AF, KZ, ipſis, HG, <lb/>& </s> <s xml:id="echoid-s1417" xml:space="preserve">Y, perpendiculares, iunganturque, PE, XT, PF, XZ, &</s> <s xml:id="echoid-s1418" xml:space="preserve">, F <lb/>E, ZT. </s> <s xml:id="echoid-s1419" xml:space="preserve">Quoniam ergo, APG, eſt angulus rectus, erit quadra-<lb/> <anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/> tum, AG, æquale quadratis, GP, PA, quadratum verò, PA, <lb/>æquatur duobus quadratis, PE, EA, propter angulum rectum, A <lb/> <anchor type="note" xlink:label="note-0075-02a" xlink:href="note-0075-02"/> EP, ergo quadratum, AG, hoc eſt duo quadrata, GE, EA, ęqua-<lb/>buntur tribus quadratis, GP, PE, EA, & </s> <s xml:id="echoid-s1420" xml:space="preserve">ablato communi qua-<lb/>drato, EA, quadratum, GE, æquabitur quadratis, GP, PE, er-<lb/>go, EP, erit perpendicularis ipſi, PV, cui etiam eſt perpendicula-<lb/> <anchor type="note" xlink:label="note-0075-03a" xlink:href="note-0075-03"/> ris, AP, ergo, APE, erit inclinatio planorum, AV, VH. </s> <s xml:id="echoid-s1421" xml:space="preserve">Eo-<lb/>dem modo oſtendemus, KXT, eſſe inclinationem planorum, Κ Λ, <lb/> <anchor type="note" xlink:label="note-0075-04a" xlink:href="note-0075-04"/> Λ &</s> <s xml:id="echoid-s1422" xml:space="preserve">, & </s> <s xml:id="echoid-s1423" xml:space="preserve">angu os, EFG, TZY, eſſe rectos. </s> <s xml:id="echoid-s1424" xml:space="preserve">Quoniam verò angu-<lb/>lus, AGV, æquatur ipſi, Κ Υ Λ, (ſunt. </s> <s xml:id="echoid-s1425" xml:space="preserve">n. </s> <s xml:id="echoid-s1426" xml:space="preserve">figuræ, AV, Κ Λ, ſimi-<lb/>les ex hypoteſi) etiam, AGP, æquabitur, KYX, &</s> <s xml:id="echoid-s1427" xml:space="preserve">, APG KX <lb/>Y, recti ſunt, ergo triangula, APG, KXY, ſimil a erunt. </s> <s xml:id="echoid-s1428" xml:space="preserve">Eodem <lb/>modo probabimus etiam triangula, AGF, KYZ, eſſe ſimilia, er-<lb/>go, PG, ad, GA, erit vt, XY, ad, YK, &</s> <s xml:id="echoid-s1429" xml:space="preserve">, GA, ad, GF, vt, Y <lb/>K, ad, YZ, ergo ex æqual@, PG, ad, GF, erit vt, XY, ad, YZ, <lb/>& </s> <s xml:id="echoid-s1430" xml:space="preserve">ſunt latera proportionalia circa æquales augulos, PGF, XYZ, <lb/>(ſunt.</s> <s xml:id="echoid-s1431" xml:space="preserve">n. </s> <s xml:id="echoid-s1432" xml:space="preserve">æquales ijs, qui ſunt ad verticem, nempè, HGV, & </s> <s xml:id="echoid-s1433" xml:space="preserve">Υ Λ, <lb/>qui adęquantur, cum ſint ſimilium figurarum, HGV, & </s> <s xml:id="echoid-s1434" xml:space="preserve">Υ Λ,) er-<lb/> <anchor type="note" xlink:label="note-0075-05a" xlink:href="note-0075-05"/> go triangula, PGF, XYZ, erunt ſimilia, & </s> <s xml:id="echoid-s1435" xml:space="preserve">anguli, GPF, YXZ, <lb/>vt &</s> <s xml:id="echoid-s1436" xml:space="preserve">, GFP, YZX, inter ſe æquales, ergo ipſi, FPE, ZXT; </s> <s xml:id="echoid-s1437" xml:space="preserve">P <lb/>FE, ZXT, inter ſe quoque erunt æquales, cum ſint reſiduirectc-<lb/>rum, GPE, GFE, YXT, YZT; </s> <s xml:id="echoid-s1438" xml:space="preserve">ergo triangula, PEF, XTZ, <lb/>pariter ſimilia erunt. </s> <s xml:id="echoid-s1439" xml:space="preserve">Erit ergo, AP, ad, PG, vt, KX, ad, XY; <lb/></s> <s xml:id="echoid-s1440" xml:space="preserve"> <anchor type="note" xlink:label="note-0075-06a" xlink:href="note-0075-06"/> PG, ad, PF, vt, XY, ad, XZ; </s> <s xml:id="echoid-s1441" xml:space="preserve">&</s> <s xml:id="echoid-s1442" xml:space="preserve">, PF, ad, PE, vt, XZ, ad, X <lb/>T, ergo ex ęquali, AP, ad, PE, erit vt, KX, ad, XT, & </s> <s xml:id="echoid-s1443" xml:space="preserve">ſunt an-<lb/>guli, AEP, KTX, rect@, ergo triangula, APF, KXT, ſitnilia <lb/> <anchor type="note" xlink:label="note-0075-07a" xlink:href="note-0075-07"/> erunt, & </s> <s xml:id="echoid-s1444" xml:space="preserve">angu@i, APE, KXT, ęqual@s, qu@@unt inclinationes pla-<lb/>norum, AV, Κ Λ, ad plana, VH, Λ &</s> <s xml:id="echoid-s1445" xml:space="preserve">, ad eandem partein, quod <lb/>oſſendendum erat.</s> <s xml:id="echoid-s1446" xml:space="preserve"/> </p> <div xml:id="echoid-div155" type="float" level="2" n="1"> <figure xlink:label="fig-0074-01" xlink:href="fig-0074-01a"> <image file="0074-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0074-01"/> </figure> <note position="left" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">18. Vnde-<lb/>cimi El.</note> <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">47. Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">Defin. 3. <lb/>Vndec. <lb/>Elem.</note> <note position="right" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">48. Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0075-04" xlink:href="note-0075-04a" xml:space="preserve">Defin. 6. <lb/>Vndec. <lb/>Elem.</note> <note position="right" xlink:label="note-0075-05" xlink:href="note-0075-05a" xml:space="preserve">6. Sexti <lb/>Elem.</note> <note position="right" xlink:label="note-0075-06" xlink:href="note-0075-06a" xml:space="preserve">4. Sexti <lb/>Elem.</note> <note position="right" xlink:label="note-0075-07" xlink:href="note-0075-07a" xml:space="preserve">7. Sexti <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div157" type="section" level="1" n="106"> <head xml:id="echoid-head117" xml:space="preserve">LEMMA II.</head> <p> <s xml:id="echoid-s1447" xml:space="preserve">IN eadem antecedentis ſigura ſi @upponamus propoſitas eſſe duas <lb/>ſimiles quaſcumque rectihneas ſiguras, AV, Κ Λ, interſe, nec-<lb/> <anchor type="note" xlink:label="note-0075-08a" xlink:href="note-0075-08"/> non, HV, & </s> <s xml:id="echoid-s1448" xml:space="preserve">Λ, conuen@entes in homologis lateribus vtriſq; </s> <s xml:id="echoid-s1449" xml:space="preserve">com-<lb/>munibus, GV, Υ Λ, ſint autem homologæ inter ſe, AG, KY; </s> <s xml:id="echoid-s1450" xml:space="preserve">H <lb/>G, & </s> <s xml:id="echoid-s1451" xml:space="preserve">Y; </s> <s xml:id="echoid-s1452" xml:space="preserve">& </s> <s xml:id="echoid-s1453" xml:space="preserve">ipſæ figuræ æquè ad eandem partem inuicem inclinatæ. <lb/></s> <s xml:id="echoid-s1454" xml:space="preserve">D@co angulos, AGH, KY &</s> <s xml:id="echoid-s1455" xml:space="preserve">, ęquales eſſe, & </s> <s xml:id="echoid-s1456" xml:space="preserve">circa eo@dem latera <lb/>pr@portionalia, quod etiam de angulis, DVN, Q Λ ℟, pariter ve-<lb/>rum eſſe oſtendemus.</s> <s xml:id="echoid-s1457" xml:space="preserve"/> </p> <div xml:id="echoid-div157" type="float" level="2" n="1"> <note position="right" xlink:label="note-0075-08" xlink:href="note-0075-08a" xml:space="preserve">Iux.def.1. <lb/>S@xti El.</note> </div> <pb o="56" file="0076" n="76" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s1458" xml:space="preserve">Hocautem ex Propoſ.</s> <s xml:id="echoid-s1459" xml:space="preserve">26. </s> <s xml:id="echoid-s1460" xml:space="preserve">huius facilè comprehendemus, ſunt.</s> <s xml:id="echoid-s1461" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s1462" xml:space="preserve">(ijſdem vtibi conſtructis) duo oppoſita plana parallela tangentia fi-<lb/>guras, AV, Κ Λ, ipſa, BD, HV; </s> <s xml:id="echoid-s1463" xml:space="preserve">LQ, & </s> <s xml:id="echoid-s1464" xml:space="preserve">Λ, quibus incidunt pla-<lb/>na figurarum ſimilium, AV, Κ Λ, ęquè ad eandem partem inclina-<lb/>ta, quæ ſint nobis tanquam prima, ijſdem autem incidunt etiam ſe-<lb/>cunda plana prima diuidentia, nempè plana, AGH, KYT, anguli <lb/>autem, HGV, & </s> <s xml:id="echoid-s1465" xml:space="preserve">Υ Λ, ſunt æquales, qui nempè continentur com-<lb/>munibus ſectionibus primorum, & </s> <s xml:id="echoid-s1466" xml:space="preserve">ſecundorum planorum cum pla-<lb/>nis, HV, & </s> <s xml:id="echoid-s1467" xml:space="preserve">Λ, quæ ſunt duo parallelorum planorum, ſimiliter an-<lb/>guli, AGV, Κ Υ Λ, (contenti communibus ſectionibus primorum, <lb/>& </s> <s xml:id="echoid-s1468" xml:space="preserve">ſecundorum planorum, & </s> <s xml:id="echoid-s1469" xml:space="preserve">communibus ſectionib. </s> <s xml:id="echoid-s1470" xml:space="preserve">primorum pla-<lb/>norum, & </s> <s xml:id="echoid-s1471" xml:space="preserve">ipſorum, HV, & </s> <s xml:id="echoid-s1472" xml:space="preserve">Λ,) ſunt æquales, ſunt.</s> <s xml:id="echoid-s1473" xml:space="preserve">n. </s> <s xml:id="echoid-s1474" xml:space="preserve">fimilium fi-<lb/>gurarum, AV, Κ Λ, ergo etiam anguli, AGH, KY &</s> <s xml:id="echoid-s1475" xml:space="preserve">, æquales <lb/>erunt, vt in Propoſ.</s> <s xml:id="echoid-s1476" xml:space="preserve">26. </s> <s xml:id="echoid-s1477" xml:space="preserve">iam oſtenſum eſt. </s> <s xml:id="echoid-s1478" xml:space="preserve">Cum autem figurę, AV, <lb/>Κ Λ, ſint ſimiles, &</s> <s xml:id="echoid-s1479" xml:space="preserve">, AG, KY, latera homologa, erit, AG, ad, <lb/>GV, vt, KY, ad, Υ Λ, oſtendemus autem eadem ratione, VG, ad, <lb/>GH, eſſe vt, Λ Υ, ad, Y &</s> <s xml:id="echoid-s1480" xml:space="preserve">, ergo ex ęquali, AG, ad, GH, erit vt, <lb/>KY, ad, Y &</s> <s xml:id="echoid-s1481" xml:space="preserve">. Eodem modo probabimus angulos, DVN, Q Λ ℟, <lb/>ęquales eſſe (ſiue plana, AH, DN; </s> <s xml:id="echoid-s1482" xml:space="preserve">K &</s> <s xml:id="echoid-s1483" xml:space="preserve">, Q ℟, ſint parallela, ſiue <lb/>non, hoc.</s> <s xml:id="echoid-s1484" xml:space="preserve">n. </s> <s xml:id="echoid-s1485" xml:space="preserve">nihil refert) & </s> <s xml:id="echoid-s1486" xml:space="preserve">circa eos latera eſſe proportionalia, quod <lb/>oſtendere opus erat.</s> <s xml:id="echoid-s1487" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div159" type="section" level="1" n="107"> <head xml:id="echoid-head118" xml:space="preserve">LEMMA III.</head> <p> <s xml:id="echoid-s1488" xml:space="preserve">SI in ſimilibus rectilineis figuris, iuxta Euclidem, ducantur rectæ <lb/>lineæ quęcumque, earundem latera homologa ſimiliter ad ean-<lb/>dem partem diuidentes, ipſę diuident eaſdem in ſimiles figuras, ſimi-<lb/>les autem erunt, quę ad eandem partem diuidentium linearum con-<lb/>ſtituentur, & </s> <s xml:id="echoid-s1489" xml:space="preserve">ipſæ ſecantes earundem erunt homologa latera.</s> <s xml:id="echoid-s1490" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1491" xml:space="preserve">Sint ſimiles rectilineæ figuræ iuxta Euclidem, <lb/> <anchor type="figure" xlink:label="fig-0076-01a" xlink:href="fig-0076-01"/> ACED, GMNH, quibus incidant rectæ, B <lb/>F, IO, ſecantes latera homologa, AC, GM; <lb/></s> <s xml:id="echoid-s1492" xml:space="preserve">necnon, DE, HN, ſimiliter ad eandem pa-<lb/>tem, vt, AC, GM, in punctis, B, I, &</s> <s xml:id="echoid-s1493" xml:space="preserve">, DE, <lb/>HN, in punctis, F, O. </s> <s xml:id="echoid-s1494" xml:space="preserve">Dico figuras ab eiſdem <lb/>conſtitutas ad eandem partem, nempè, BAD <lb/>F, IGHO; </s> <s xml:id="echoid-s1495" xml:space="preserve">BCEF, IMNO, inter ſe ſimi-<lb/>les eſſe. </s> <s xml:id="echoid-s1496" xml:space="preserve">Ducantur à punctis, B, I, ad angulos <lb/>oppoſitos rectæ lineæ, BD, BE, IH, IN, vt <lb/>ſi figuræ ſint quadrilateræ, vel multilateræ, in <lb/>triangula diſceparentur. </s> <s xml:id="echoid-s1497" xml:space="preserve">Quoniam ergo, AC, <lb/>GM, ſimiliter diuiduntur in, B, I, erit, BA, ad, IG, vt, AC, ad, <pb o="57" file="0077" n="77" rhead="LIBER I."/> GM, ideſt, vt, AD, ad, GH, ergo permutando, BA, ad, AD, <lb/> <anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/> erit vt, IG, ad, GH, & </s> <s xml:id="echoid-s1498" xml:space="preserve">anguli, BAD, IGH, ſunt æquales, er-<lb/>go, BAD, IGH, erunt triangula ſimilia, ergo anguli, ADB, G <lb/>HI, æquales erunt, ſunt autem ęquales etiam, ADF, GHO, er-<lb/>go reliqui, BDF, IHO, erunt æquales, eſt verò, BD, ad, DA, <lb/>vt, IH, ad, HG, &</s> <s xml:id="echoid-s1499" xml:space="preserve">, AD, ad, DF, vt, GH, ad, HO, ergo ex <lb/>æquali, BD, ad, DF, eſt vt, IH, ad, HO, ergo triangula, BD <lb/>F, IHO, pariter ſimilia erunt, & </s> <s xml:id="echoid-s1500" xml:space="preserve">anguli, DFB, HOI, inter ſe, <lb/>necnon, DBF, HIO, inter ſe ęquales, ergo anguli, ABF, GIO, <lb/>ADF, GHO, erunt etiam ęquales, & </s> <s xml:id="echoid-s1501" xml:space="preserve">figurę, ABFD, GIOH, <lb/>æquiangulę, & </s> <s xml:id="echoid-s1502" xml:space="preserve">cum, BA, ad, DF, FB, binæ ſint in eadem ratio-<lb/>ne cum, IG, GH, HO, OI, patet, quod etiam circa ęquales an-<lb/>gulos ſunt latera proportionalia, ergo ipſæ figuræ, BADF, IG <lb/>HO, ſimiles erunt. </s> <s xml:id="echoid-s1503" xml:space="preserve">Eodem autem modo oſtendemus ſimiles eſſe, B <lb/>CEF, IMNO, patet autem ipſas, BF, IO, eſſe earum latera ho-<lb/>mologa, quod erat demonſtrandum.</s> <s xml:id="echoid-s1504" xml:space="preserve"/> </p> <div xml:id="echoid-div159" type="float" level="2" n="1"> <figure xlink:label="fig-0076-01" xlink:href="fig-0076-01a"> <image file="0076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0076-01"/> </figure> <note position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">6. Sextr <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div161" type="section" level="1" n="108"> <head xml:id="echoid-head119" xml:space="preserve">LEMMA IV.</head> <p> <s xml:id="echoid-s1505" xml:space="preserve">SI in ſimilibus ſolidis planis contentis iuxta def.</s> <s xml:id="echoid-s1506" xml:space="preserve">9. </s> <s xml:id="echoid-s1507" xml:space="preserve">vndec. </s> <s xml:id="echoid-s1508" xml:space="preserve">Elem. <lb/></s> <s xml:id="echoid-s1509" xml:space="preserve">quatuor quęlibet puncta ſumantur in vnoquoq; </s> <s xml:id="echoid-s1510" xml:space="preserve">eorundem (non <lb/>tamen in eodem plano conſtituta) ad quę anguli ſolidi æquales ter-<lb/>minantur, illaq; </s> <s xml:id="echoid-s1511" xml:space="preserve">iungatur rectis lineis, fient ſimiles pyramides trian-<lb/> <anchor type="note" xlink:label="note-0077-02a" xlink:href="note-0077-02"/> gulatæ comprehenſæ ſub triangulis, ijſdem rectis lineis iungentibus <lb/>contentis.</s> <s xml:id="echoid-s1512" xml:space="preserve"/> </p> <div xml:id="echoid-div161" type="float" level="2" n="1"> <note position="right" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">Defin, 1. <lb/>@ti El.</note> </div> <p> <s xml:id="echoid-s1513" xml:space="preserve">Sint ſimilia ſolida, AHCD, FO <lb/> <anchor type="figure" xlink:label="fig-0077-01a" xlink:href="fig-0077-01"/> GL, iuxta def.</s> <s xml:id="echoid-s1514" xml:space="preserve">9. </s> <s xml:id="echoid-s1515" xml:space="preserve">vndec. </s> <s xml:id="echoid-s1516" xml:space="preserve">Elem. </s> <s xml:id="echoid-s1517" xml:space="preserve">& </s> <s xml:id="echoid-s1518" xml:space="preserve">in <lb/>ijs accepta quatuor quęcumq; </s> <s xml:id="echoid-s1519" xml:space="preserve">pun-<lb/>cta, nempè, A, H, C, D, in vno, <lb/>&</s> <s xml:id="echoid-s1520" xml:space="preserve">, F, O, G, L, in alio ſolido, quæ <lb/>non ſint in eodem plano, ſed ad an-<lb/>gulos ęquales conſtituta, iungantur-<lb/>querectis lineis, AH, AC, CD, C <lb/> <anchor type="figure" xlink:label="fig-0077-02a" xlink:href="fig-0077-02"/> H, HD; </s> <s xml:id="echoid-s1521" xml:space="preserve">FO, FG, FL, OG, G <lb/>L, LO, ſiue hęc iungentia ſint ipſo-<lb/>rum ſimilium ſolidorum latera. </s> <s xml:id="echoid-s1522" xml:space="preserve">Di-<lb/>copyramides, AHCD, FOGL, <lb/>ſimiles eſſe. </s> <s xml:id="echoid-s1523" xml:space="preserve">Vel ergo plana has py-<lb/>ramides continentia ſunt in ambitu <lb/>ſolidorum, vt ex.</s> <s xml:id="echoid-s1524" xml:space="preserve">gr. </s> <s xml:id="echoid-s1525" xml:space="preserve">CHD, GOL, & </s> <s xml:id="echoid-s1526" xml:space="preserve">tunc erunt ſimilia, ex ipſa de-<lb/>finitione, vel non ſunt in ambitu, tunc autem probandum eſt nihi-<lb/>lominus eſſe ſimilia, vt non ſint in ambitu ipſa triangula, ACH, F <pb o="58" file="0078" n="78" rhead="GEOMETRIÆ"/> GO, ſint verò in ambitu triangula, ABC, FIG; </s> <s xml:id="echoid-s1527" xml:space="preserve">ABH, FIO, H <lb/>BC, OIG, ergo tria hęc tribus iam dictis ſimilia erunt, ergo & </s> <s xml:id="echoid-s1528" xml:space="preserve">ba-<lb/>ſes, ACH, FGO, ſimiles erunt, nam cum ſit, AC, ad, CB, vt, <lb/>FG, ad, GI; </s> <s xml:id="echoid-s1529" xml:space="preserve">BC, ad, CH, vt, IG, ad, GO, erit ex ęquali, AC, <lb/>ad, CH, vt, FG, ad, GO, eadem ratione oſtendemus, CH, ad, <lb/> <anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/> HA, eſſe vt, GO, ad, OF, ex quo habebitur ex ęquali, CA, ad, <lb/>AH, eſſe vt, GF, ad, FO, ergo triangula, ACH, FGO, ſimilia <lb/>erunt. </s> <s xml:id="echoid-s1530" xml:space="preserve">Eodem modo probabimus triangula, AHD, FLO, ACD, <lb/>FGL, eſſe ſimilia, ex quo concludemus ipſas pyramides ſimiles eſſe. <lb/></s> <s xml:id="echoid-s1531" xml:space="preserve">Quod ſi tria triangula ad, B, I, terminantia omnia non ſint in am-<lb/>bitu, oſtendemus tamen illa eſſe ſimilia, erunt.</s> <s xml:id="echoid-s1532" xml:space="preserve">n. </s> <s xml:id="echoid-s1533" xml:space="preserve">vel baſes pyrami-<lb/>dum, quarum tria triangula verticalia erunt in ambitu, vel ſaltem <lb/>aliarum pyramidum, quarum triangula ſimilia eſſe probabuntur, <lb/>quia erunt baſes pyramidum tria triangula verticalia in ambitu ha-<lb/>bentium, ad hæc.</s> <s xml:id="echoid-s1534" xml:space="preserve">n. </s> <s xml:id="echoid-s1535" xml:space="preserve">tandem deuenire neceſſe erit: </s> <s xml:id="echoid-s1536" xml:space="preserve">Igitur oſtenſum <lb/>eſt, quod proponebatur.</s> <s xml:id="echoid-s1537" xml:space="preserve"/> </p> <div xml:id="echoid-div162" type="float" level="2" n="2"> <figure xlink:label="fig-0077-01" xlink:href="fig-0077-01a"> <image file="0077-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0077-01"/> </figure> <figure xlink:label="fig-0077-02" xlink:href="fig-0077-02a"> <image file="0077-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0077-02"/> </figure> <note position="left" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">5. Sexti <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div164" type="section" level="1" n="109"> <head xml:id="echoid-head120" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1538" xml:space="preserve">_Q_Via verò in pyramidibus triangulatis, BAHC, IFGO, exiſten-<lb/>tibus ſimilibus illarum triangulis verticalibus, baſes, ACH, F <lb/>GO, neceſſiriò ſimiles eſſe oſtenſæ ſunt, ideò ex hoc colligimus ſi <lb/>in duabus pyramidibus triangulatis tria verticalia triangulatribus ver-<lb/>ticalibus triangulis ſimilia ſint, etiam baſes ſimiles eſſe.</s> <s xml:id="echoid-s1539" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div165" type="section" level="1" n="110"> <head xml:id="echoid-head121" xml:space="preserve">LEMMA V.</head> <p> <s xml:id="echoid-s1540" xml:space="preserve">SI duo ſimilia triangula fuerint ſubiectis planis æquè ad eandem <lb/>partem inclinata, ita vt communes cum illis ſectiones ſint ea-<lb/>rum latera homologa, quæ tanquam baſes aſſumantur; </s> <s xml:id="echoid-s1541" xml:space="preserve">ab eorum <lb/>autem verticibus rectæ lineæ in ſublimi fuerint conſtitutæ, angulos <lb/>æquales cum eorum lateribus homologis continentes, illæ erunt ſu-<lb/>biectis planis æqualiter inclinatæ, vel eiſdem ambo parallelæ; </s> <s xml:id="echoid-s1542" xml:space="preserve">ſi au-<lb/>tem fuerint inclinatæ, & </s> <s xml:id="echoid-s1543" xml:space="preserve">vſque ad ſubiecta plana producantur, iun-<lb/>ganturq; </s> <s xml:id="echoid-s1544" xml:space="preserve">pucta occurſuum cum extremis baſium dictorum triangu-<lb/>lorum, pariter hinc conſtitutæ pyramides ſimiles erunt.</s> <s xml:id="echoid-s1545" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1546" xml:space="preserve">Sint ſimilia triangula, ABD, HPO, ſubiectis planis ęquè incli-<lb/>nata, in baſibus, BD, PO, à quorum verticibus, A, H, rectæ li-<lb/>neæ, AC, HN, in ſublimi conſtitutæ contineant cum homologis <lb/>eorum lateribus angulos æquales, ſint nempè anguli, CAB, NH <lb/>P, necnon, CAD, VHO, inter ſe æquales. </s> <s xml:id="echoid-s1547" xml:space="preserve">Dico ipſas, AC, H <lb/>N, ſubiectis planis eſſe ęqualiter inclinatas, vel eiſdem ambo paral- <pb o="59" file="0079" n="79" rhead="LIBER I."/> lelas, ac (ſi ſint inclinatę, incidantque ipſis in punctis, C, N, iun-<lb/>ganturque, CB, CD, NP, NO,) pyramides, ACDB, HNO <lb/>P, ſimiles eſſe. </s> <s xml:id="echoid-s1548" xml:space="preserve">Sumatur ergo in, AD, etiam quantumuis protenſa <lb/>vbicumq; </s> <s xml:id="echoid-s1549" xml:space="preserve">punctum, F, & </s> <s xml:id="echoid-s1550" xml:space="preserve">accipiatur in, HO, producta, ſi opus ſit, <lb/>HL, ęqualis, AF, & </s> <s xml:id="echoid-s1551" xml:space="preserve">indefinitè extenſis lineis, AC, AB, HN, H <lb/> <anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/> P, ducantur in planis, FAC, FAG, LHN, LHP, à punctis, F, <lb/> <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/> L, ipſis, AF, HL, per-<lb/>pendiculares, FE, FG, <lb/>LI, LM, occurrentes <lb/>ipſis, AE, AG, HI, H <lb/>M, in punctis, E, G, I, <lb/>M, & </s> <s xml:id="echoid-s1552" xml:space="preserve">iungantur, EG, <lb/>IM. </s> <s xml:id="echoid-s1553" xml:space="preserve">Quoniam ergo duo <lb/>anguli, AFG, HLM, <lb/>recti, &</s> <s xml:id="echoid-s1554" xml:space="preserve">, FAG, LH <lb/>M, ſunt æquales, & </s> <s xml:id="echoid-s1555" xml:space="preserve">la-<lb/>tera, AF, HL, ęqua-<lb/>lia, erunt etiam, FG, <lb/> <anchor type="note" xlink:label="note-0079-02a" xlink:href="note-0079-02"/> LM, GA, MH, ęqua-<lb/>lia; </s> <s xml:id="echoid-s1556" xml:space="preserve">eodem modo oſten-<lb/>demus æqualia eſſe, F <lb/>E, LI, EA, IH, vnde cum ſint ęquales, EA, IH, AG, HM, & </s> <s xml:id="echoid-s1557" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0079-03a" xlink:href="note-0079-03"/> anguli, EAG, IHM, pariter ęquales, etiam baſes, EG, IM, æ-<lb/>quales erunt, & </s> <s xml:id="echoid-s1558" xml:space="preserve">pyramides, AEFG, HILM, ſimiles, & </s> <s xml:id="echoid-s1559" xml:space="preserve">ęquales <lb/>ad inuicem exiſtent. </s> <s xml:id="echoid-s1560" xml:space="preserve">Suſpendatur nunc pyramis, AEFG, & </s> <s xml:id="echoid-s1561" xml:space="preserve">pona-<lb/> <anchor type="note" xlink:label="note-0079-04a" xlink:href="note-0079-04"/> tur punctum, F, in, L, demittaturque, FG, ſuper, LM, cui con-<lb/>gruet, ſed & </s> <s xml:id="echoid-s1562" xml:space="preserve">triangulo, EFG, cadente ſuper, ILM, punctum, F, <lb/>erit in, I, ac latus, AF, in, HL, alioquin duę eidem plano, ILM, <lb/> <anchor type="note" xlink:label="note-0079-05a" xlink:href="note-0079-05"/> perpendiculares eſſent eductæ ab eodem puncto, L, quod eſt abſur-<lb/>dum (ſunt autem, AF, HL, perpendiculares planis, EFG, ILM, <lb/> <anchor type="note" xlink:label="note-0079-06a" xlink:href="note-0079-06"/> hoc eſt ſolo plano, ILM, cum ſuperponuntur, ex eo, quod duabus, <lb/>IL, IM, ſint perpendiculares in puncto, L,) ergo, FA, cadet ſu-<lb/>per, LH, & </s> <s xml:id="echoid-s1563" xml:space="preserve">punctum, A, in, H, vnde etiam, EA, cadet in, IH, <lb/> <anchor type="note" xlink:label="note-0079-07a" xlink:href="note-0079-07"/> &</s> <s xml:id="echoid-s1564" xml:space="preserve">, AG, in, HM, punctum, B, verò eſto, quod ſit in, T, D, in, <lb/>S, &</s> <s xml:id="echoid-s1565" xml:space="preserve">, C, in, V, erit et@am, DB, congruens ipſi, ST, CD, VS, <lb/>&</s> <s xml:id="echoid-s1566" xml:space="preserve">, CB, ipſi, VT, & </s> <s xml:id="echoid-s1567" xml:space="preserve">quia angulus, ABD, æquatur ipſi, HPO, <lb/>ABD, autem eſt etiam æqualis, HTS, ergo, HTS, HPO, ſunt <lb/>æquales, &</s> <s xml:id="echoid-s1568" xml:space="preserve">, ST, parallela, OP. </s> <s xml:id="echoid-s1569" xml:space="preserve">Dico etiam triangulum, VST, <lb/> <anchor type="note" xlink:label="note-0079-08a" xlink:href="note-0079-08"/> æquidiſtare ipſi, NOP, ſi .</s> <s xml:id="echoid-s1570" xml:space="preserve">n. </s> <s xml:id="echoid-s1571" xml:space="preserve">hoc non ſit, quia, ST, eſt parallela <lb/>ipſi, OP, poterit per, ST, duci planum ipſi, NOP, parallelum, <lb/>ducatur, & </s> <s xml:id="echoid-s1572" xml:space="preserve">producat in pyramide triangulum, KST, acta autem à <lb/>puncto, H, ipſi, OP, perpendiculari, quę ſit, HQ, ſecante, ST, <pb o="60" file="0080" n="80" rhead="GEOMETRIÆ"/> in, X, ducatur in plano, NOP, recta, QR, à puncto, Q, perpen-<lb/>dicularis ipſi, OP, & </s> <s xml:id="echoid-s1573" xml:space="preserve">iungatur, HR, triangulumque, HRQ, ſe-<lb/>cet duo triangula, VST, KST, in rectis, YX, ZX. </s> <s xml:id="echoid-s1574" xml:space="preserve">Quia ergo <lb/> <anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/> triangula, VST, NOP, ſunt parallela, erunt etiam ipſę, ZX, R <lb/>Q, parallelæ, ſed &</s> <s xml:id="echoid-s1575" xml:space="preserve">, ST, OP, ſunt parallelę, ergo anguli, ZXS, <lb/>RQO, erunt æquales, rectus ergo eſt etiam ipſe, ZXS, ſed etiam, <lb/> <anchor type="note" xlink:label="note-0080-02a" xlink:href="note-0080-02"/> SXH, rectus eſt, ergo, SX, eſt duabus, ZX, XH, perpendicula-<lb/>ris, & </s> <s xml:id="echoid-s1576" xml:space="preserve">ſubinde plano per ipſas tranſeunti, & </s> <s xml:id="echoid-s1577" xml:space="preserve">conſequenter, SXY, eſt <lb/> <anchor type="note" xlink:label="note-0080-03a" xlink:href="note-0080-03"/> rectus, vnde, HXZ, erit inclinatio planorum, HST, KST, &</s> <s xml:id="echoid-s1578" xml:space="preserve">, H <lb/>XY, inclinatio planorum, HST, SVT, hæc autem eſt æqualis in-<lb/>clinationi planorum, HOP, NOP, ex hypoteſi, ideſt angulo, H <lb/> <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> QR, ideſt angulo, H <lb/>XZ, ergo angulus, H <lb/>XY, qui eſt totum, eſt <lb/>ęqualis augulo, HXZ, <lb/>eiuſdem parti, quod eſt <lb/>abſurdũ, ergo abſurdum <lb/>etiam eſt dicere trian-<lb/>gulum, VST, non æ-<lb/>quidiſtare ipſi, NOP, <lb/>æquidiſtat ergo, & </s> <s xml:id="echoid-s1579" xml:space="preserve">ip-<lb/>ſæ, VS, VT, ſunte-<lb/> <anchor type="note" xlink:label="note-0080-04a" xlink:href="note-0080-04"/> tiam parallelæ ipſis, N <lb/>O, NP, & </s> <s xml:id="echoid-s1580" xml:space="preserve">triangula, <lb/>VHS, ipſi, NHO, V <lb/>HT, ipſi, NHP, nec-<lb/>non, VST, ipſi, NOP, ſunt ſimilia, ergo pyramides, HVST, <lb/>HNOP, ſunt ſimiles, eſt autem pyramis, HVST, ſimilis, immo <lb/>& </s> <s xml:id="echoid-s1581" xml:space="preserve">ęqualis, ipſi, ACDB, ergo pyramides, ACDB, HNOP, in-<lb/>ter ſe ſimiles erunt, & </s> <s xml:id="echoid-s1582" xml:space="preserve">anguli, ACB, HVT, ACD, HVS, inter <lb/>ſe æquales, ergo, AC, HV, rectę lineę ſtantes in ſublimi, & </s> <s xml:id="echoid-s1583" xml:space="preserve">cum <lb/>ipſis, CD, CB, VS, VT, angulos æquales continentes (à quibus <lb/>etiam contenti anguli, DCB, SVT, ſunt ęquales) erunt ad plana <lb/> <anchor type="note" xlink:label="note-0080-05a" xlink:href="note-0080-05"/> triangulorum, CDB, NOP, æqualiter inclinata, & </s> <s xml:id="echoid-s1584" xml:space="preserve">ſunt ipſæ py-<lb/>ramides, ACDB, HNOP, ſimiles, vt propoſitum fuit demon-<lb/>ſtrare.</s> <s xml:id="echoid-s1585" xml:space="preserve"/> </p> <div xml:id="echoid-div165" type="float" level="2" n="1"> <note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">Vidi dicta <lb/>lib.7. An-<lb/>not. Pro-<lb/>poſ.3.</note> <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a"> <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0079-01"/> </figure> <note position="right" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">26.Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0079-03" xlink:href="note-0079-03a" xml:space="preserve">4. Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0079-04" xlink:href="note-0079-04a" xml:space="preserve">Defin. 10. <lb/>vndec. El.</note> <note position="right" xlink:label="note-0079-05" xlink:href="note-0079-05a" xml:space="preserve">7. Pri. El.</note> <note position="right" xlink:label="note-0079-06" xlink:href="note-0079-06a" xml:space="preserve">13. Vnd. <lb/>Elem.</note> <note position="right" xlink:label="note-0079-07" xlink:href="note-0079-07a" xml:space="preserve">4. Vnd. El.</note> <note position="right" xlink:label="note-0079-08" xlink:href="note-0079-08a" xml:space="preserve">4. Primi <lb/>Elem.</note> <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">16. Vnd. <lb/>Elem.</note> <note position="left" xlink:label="note-0080-02" xlink:href="note-0080-02a" xml:space="preserve">10.Vnd. <lb/>Elem.</note> <note position="left" xlink:label="note-0080-03" xlink:href="note-0080-03a" xml:space="preserve">4.Vndec. <lb/>Elem.</note> <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a"> <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0080-01"/> </figure> <note position="left" xlink:label="note-0080-04" xlink:href="note-0080-04a" xml:space="preserve">16. Vnd. <lb/>Elem.</note> <note position="left" xlink:label="note-0080-05" xlink:href="note-0080-05a" xml:space="preserve">35. Vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s1586" xml:space="preserve">Si verò rectæ lineæ angulos æquales cum ipſis, DA, AB, OH, <lb/>HP, continentes eſſent ipſæ, AT, Η Λ, quarum, Λ Η, eſſet paral-<lb/>lela plano, VST, probaremus etiam, TA, eſſe parallelam plano, <lb/>CDB, alioquin ſi cum ipſo producta concurreret, etiam, Λ Η, ex <lb/>ſupra oſtenſis, producta concurreret cum plano trianguli, VST. </s> <s xml:id="echoid-s1587" xml:space="preserve">Vel <lb/>præintellectis duabus iam datis, AC, HN, & </s> <s xml:id="echoid-s1588" xml:space="preserve">ſuppoſita ſuperiori <pb o="61" file="0081" n="81" rhead="LIBER I."/> conſtructione, oſtenderemus, vt ſupra, tria latera, ΓΑ, Λ Η; </s> <s xml:id="echoid-s1589" xml:space="preserve">AD, <lb/>HO; </s> <s xml:id="echoid-s1590" xml:space="preserve">AB, HP; </s> <s xml:id="echoid-s1591" xml:space="preserve">eſſe ad inuicem ſuperpoſita, vnde ſi, Λ Η, æquidi-<lb/>ſtat plano, NOP, etiam neceſſe eſſe concluderetur, Λ Η, ſeu, ΓΑ, <lb/>in ea conſtitutam, æquidiſtare plano, NOP, vel ipſi, VST, ſeu, <lb/>ΓΑ, ipſi, CDB, quod erat oſtendendum.</s> <s xml:id="echoid-s1592" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div167" type="section" level="1" n="111"> <head xml:id="echoid-head122" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s1593" xml:space="preserve">_E_X boc Lemmate colligitur ſimilium ſolidorum, iuxta Euclidis de-<lb/>finitionem, latera bomologa quœcunque, vel (duabus in ambitu <lb/>quibuſcumque figuris ſimilibus aſſumptis) iacere in plano ſimilium di-<lb/>ctarum figurarum, aut illis œquidiſtare, vel œqualiter eiſdem inclinari; <lb/></s> <s xml:id="echoid-s1594" xml:space="preserve">Vt in figura Lemmatis 4. </s> <s xml:id="echoid-s1595" xml:space="preserve">ex. </s> <s xml:id="echoid-s1596" xml:space="preserve">gr. </s> <s xml:id="echoid-s1597" xml:space="preserve">CD, GL, (aſſumptis ſimilibus figuris, <lb/>HCD, OGL,) iacent in earum plano, BA, IF, autem vel ambo illi <lb/>œquidiſtant, vel eiſdem ſunt œqualiter inclinata, namiunctis, AC, A <lb/> <anchor type="note" xlink:label="note-0081-01a" xlink:href="note-0081-01"/> H, FG, FO, niſi bœc ſint lateradictorum ſolidorum, fiunt anguli, BA <lb/>H, IFO, BAC, IFG, œquales, & </s> <s xml:id="echoid-s1598" xml:space="preserve">triangula, ACH, FGO, ſimilia, <lb/>nam pyramides, ABCH, FIGO, ſunt inter ſe ſimiles, ipſa verò trian-<lb/> <anchor type="note" xlink:label="note-0081-02a" xlink:href="note-0081-02"/> gula, ACH, FGO, œquè ad eandem partem inclinantur ipſis, HCD, <lb/>OGL, cum etiam, ACHD, FGLO, pyramides ſint ſimiles ex eodem <lb/>Lemmate 4. </s> <s xml:id="echoid-s1599" xml:space="preserve">vnde vel, AB, FI, œquidiſtant baſibus, CHD, GOL, vel <lb/>ſunt eiſdem œqualiter inclinata, idem de cœteris bomologis quibuſcum-<lb/>que lateribus, quibuslibet ſimilibus figuris in ambitu aſſumptis compa-<lb/>ratis, pariter intelligendum erit.</s> <s xml:id="echoid-s1600" xml:space="preserve"/> </p> <div xml:id="echoid-div167" type="float" level="2" n="1"> <note position="right" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve">_Lemma 4._</note> <note position="right" xlink:label="note-0081-02" xlink:href="note-0081-02a" xml:space="preserve">_Lemma 1._</note> </div> </div> <div xml:id="echoid-div169" type="section" level="1" n="112"> <head xml:id="echoid-head123" xml:space="preserve">LEMMA VI.</head> <p> <s xml:id="echoid-s1601" xml:space="preserve">SI in ſimilibus ſolidis iuxta Euclidem ducantur plana duabus qui-<lb/>buſcumque ſimilibus figuris in eorum ambitu aſſumptis paralle-<lb/>la, quæ vt eorum baſes accipiantur; </s> <s xml:id="echoid-s1602" xml:space="preserve">diuidant autem ducta plana eo-<lb/>rum altitudines, reſpectu dictarum baſium captas, ſimiliter ad ean-<lb/>dem partem, quęcumque latera homologa ab eiſdem ſecabuntur, ſi-<lb/>militer ad eandem partem diuidentur.</s> <s xml:id="echoid-s1603" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1604" xml:space="preserve">Sint in ſimilibus ſolidis iuxta Euclidis definition. </s> <s xml:id="echoid-s1605" xml:space="preserve">9. </s> <s xml:id="echoid-s1606" xml:space="preserve">Vndec. </s> <s xml:id="echoid-s1607" xml:space="preserve">Elem. <lb/></s> <s xml:id="echoid-s1608" xml:space="preserve">aſſumptę in ambitu duæ ſimiles figurę tanquam baſes, ex. </s> <s xml:id="echoid-s1609" xml:space="preserve">gr. </s> <s xml:id="echoid-s1610" xml:space="preserve">trian-<lb/>gula ſimilia, ADB, MKN, ſint verò de ambitu etiam deſcripta <lb/>triangula ſimilia, AHI, MSQ; </s> <s xml:id="echoid-s1611" xml:space="preserve">AHD, MSK; </s> <s xml:id="echoid-s1612" xml:space="preserve">&</s> <s xml:id="echoid-s1613" xml:space="preserve">, IHD, QS <lb/>K; </s> <s xml:id="echoid-s1614" xml:space="preserve">quibus etiam adiungantur latera bomologa, IF, QP, ad verti-<lb/>ces, F, P, reſpectu dictarum baſium captos, pertingentia, reliquis <lb/>dimiſſis figuris eorum ambitum complentibus, ne nimia fieret Sche-<lb/>matis confuſio, ſint autem à verticibus, F, P, demiſſæ altitudines <lb/>reſpectu baſium, ADB, MKN, ipſę, FC, PO, planis baſium in <pb o="62" file="0082" n="82" rhead="GEOMETRIÆ"/> punctis, P, O, occurrentes, ductis autem duobus planis quomodo-<lb/>cumque baſibus parallelis, & </s> <s xml:id="echoid-s1615" xml:space="preserve">ſecantibus altitudines, FC, PO, ſimi-<lb/>liter ad eandem partem in punctis, Λ Γ, eadem ſecent latera homo-<lb/>loga ex. </s> <s xml:id="echoid-s1616" xml:space="preserve">gr. </s> <s xml:id="echoid-s1617" xml:space="preserve">IH, QS, in punctis, Π Z. </s> <s xml:id="echoid-s1618" xml:space="preserve">Dico in eiſdem ſecari ſimi-<lb/>liter ad eandem partem. </s> <s xml:id="echoid-s1619" xml:space="preserve">Producantur ergo, HI, SQ, hinc inde, <lb/>ita vt (niſi hocipſis contingat abſque eo, quod producantur) ad pla-<lb/>na baſium, DAB, KMN, & </s> <s xml:id="echoid-s1620" xml:space="preserve">eiſdem æquidiſtantia plana per ver-<lb/>tices, F, P, ducta, terminentur, vt in punctis, L, T, G, R, à pun-<lb/>ctis verò, G, R, demittantur ad plana dictarum baſium perpendicu-<lb/>lares, GE, RX, illis incidentes in, E, X, & </s> <s xml:id="echoid-s1621" xml:space="preserve">iungantur, L, E, TX. <lb/></s> <s xml:id="echoid-s1622" xml:space="preserve">Sìmiliter à verticibus, F, P, ad puncta baſium, B, N, ducantur, F <lb/>B, PN, & </s> <s xml:id="echoid-s1623" xml:space="preserve">iungantur, BC, NO. </s> <s xml:id="echoid-s1624" xml:space="preserve">Quoniam ergo latera homolo-<lb/>ga, HI, SQ, continent cumhomologis lateribus ſimilium trian-<lb/>gulorum, AID, MQK, ad eandem partem baſibus, DAB, KM <lb/>N, inclinatorum (quia, IADB, QMNK, eſſent ſimiles pyrami-<lb/>des) angulos ęquales, & </s> <s xml:id="echoid-s1625" xml:space="preserve">producta incidunt in plana dictarum baſium <lb/> <anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/> in, L, T, erunt eiſdem æqualiter inclinata, ergo anguli, GLE, R <lb/> <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> TX, erunt ęquales, <lb/>&</s> <s xml:id="echoid-s1626" xml:space="preserve">, GEL, RXT, <lb/>ſunt recti, ergo triã-<lb/>gula, GLE, RT <lb/>X, ſimilia erunt, er-<lb/>go, GL, ad, RT, <lb/>erit vt, GE, ad, R <lb/>X, ideſt vt, FC, ad <lb/>PO. </s> <s xml:id="echoid-s1627" xml:space="preserve">Vlterius ſi iun-<lb/>geremus, FA, FD, <lb/>PM, PK, fierent <lb/>ſimiles pyramides, <lb/>FDAB, PKMN, vnde pateret, FB, PN, eſſe ad plana baſium, <lb/> <anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/> DAB, KMN, ſimiliter inclinata, & </s> <s xml:id="echoid-s1628" xml:space="preserve">ſubinde angulos, FBC, P <lb/>NO, eſſe ęquales, & </s> <s xml:id="echoid-s1629" xml:space="preserve">cum ſint recti, FCB, PNO, triangula, FB <lb/>C, PNO, eſſe æquiangula, & </s> <s xml:id="echoid-s1630" xml:space="preserve">vt, FC, ad, PO, ita eſſe, FB, ad, <lb/>PN, etiam manifeſtum eſſet, ſed vt, FC, ad, PO, ita eſt, GL, ad, <lb/>RT, & </s> <s xml:id="echoid-s1631" xml:space="preserve">vt, FB, ad, PN, ita, BD, ad, NK, & </s> <s xml:id="echoid-s1632" xml:space="preserve">ita quodcunque <lb/>latus in ſolido, FHB, ad latus ſibi homologum in ſolido, PSN, <lb/>ideſt ita, IH, ad, QS, ergo vt, GL, ad, RT, ita, HI, ad, SQ, <lb/> <anchor type="note" xlink:label="note-0082-03a" xlink:href="note-0082-03"/> & </s> <s xml:id="echoid-s1633" xml:space="preserve">ita compoſitum ex reſiduis, LH, IG, ad compoſitum ex reſiduis, <lb/>TS, QR, ſunt autem, LH, TS, latera homologa ſimilium pyra-<lb/> <anchor type="note" xlink:label="note-0082-04a" xlink:href="note-0082-04"/> midum, HLAD, STMK, ergo vt, HA, ad, SM, velvt, HI, <lb/>ad, SQ, ita, HL, ad, ST, ergo etiam reliqua, IG, ad reliquam, <lb/>QR, eſt vt, HI, ad, SQ, vel vt altitudo, FC, ad, PO, vel vt, G <pb o="63" file="0083" n="83" rhead="LIBER I."/> L, ad, RT, vel vt, GΠ, ad, RZ, ſunt enim & </s> <s xml:id="echoid-s1634" xml:space="preserve">ipſæ, GL, RT, <lb/>ſimiliter ad eandem partem ſectę in punctis, Π, Z, nam ſimiliter ſe-<lb/>cantur ac, FC, PO, in punctis, Λ, Γ, ergo etiam reliqua, I Π, ad, <lb/>QZ, erit vt tota, GΠ, ad totam, RZ, ideſt vt, FC, ad, PO. </s> <s xml:id="echoid-s1635" xml:space="preserve">Eo-<lb/>dem modo oſtendemus, ΠH, ad, ZS, eſſe vt, FC, ad, PO, er-<lb/>go, IΠ, ad, QZ, erit vt, ΠH, ad, ZS, & </s> <s xml:id="echoid-s1636" xml:space="preserve">permutando, IΠ, ad, <lb/>ΠH, erit vt, QZ, ad, ZS, ſunt ergo latera homologa, IH, QS, <lb/>ſimiliter ad eandem partem ſecta à præfatis planis, quod eodem mo-<lb/>do de quibuſcumq; </s> <s xml:id="echoid-s1637" xml:space="preserve">homologis lateribus, quæ contingat dictis planis <lb/>ſecari, pariter oſtendemus, hoc verò demonſtrare propoſitum fuit.</s> <s xml:id="echoid-s1638" xml:space="preserve"/> </p> <div xml:id="echoid-div169" type="float" level="2" n="1"> <note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">Ex Lem. <lb/>5.</note> <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0082-01"/> </figure> <note position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">Ex Lem. <lb/>4.</note> <note position="left" xlink:label="note-0082-03" xlink:href="note-0082-03a" xml:space="preserve">19. Quin. <lb/>Elem.</note> <note position="left" xlink:label="note-0082-04" xlink:href="note-0082-04a" xml:space="preserve">Ex Lem. <lb/>5.</note> </div> </div> <div xml:id="echoid-div171" type="section" level="1" n="113"> <head xml:id="echoid-head124" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1639" xml:space="preserve">_E_X boc autem Lemmate inſuper habetur nedum latera bomologa ſi-<lb/>milium ſolidorum, ſed etiam, ſi illa producantur vſq; </s> <s xml:id="echoid-s1640" xml:space="preserve">ad oppoſita <lb/>tangentia plana, eorum reſidua, vel ipſa tota, eſſe vt eorum dictas al-<lb/>titudines.</s> <s xml:id="echoid-s1641" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div172" type="section" level="1" n="114"> <head xml:id="echoid-head125" xml:space="preserve">THEOREMA XXVI. PROPOS. XXIX.</head> <p> <s xml:id="echoid-s1642" xml:space="preserve">SI in duobus ſimilibus ſolidis iuxta defin. </s> <s xml:id="echoid-s1643" xml:space="preserve">9. </s> <s xml:id="echoid-s1644" xml:space="preserve">vndec. </s> <s xml:id="echoid-s1645" xml:space="preserve">Elem. <lb/></s> <s xml:id="echoid-s1646" xml:space="preserve">accipiantur, ac in eorumdem ambitu, duæ quæcumq; </s> <s xml:id="echoid-s1647" xml:space="preserve"><lb/>ſimiles figurę planę tanquam baſes, quibus parallela ducan-<lb/>tur quæcumq; </s> <s xml:id="echoid-s1648" xml:space="preserve">plana eadem ſecantia, necnon corum altitu-<lb/>dines, reſpectu dictarum baſium aſſumptas, ſimiliter ad ean-<lb/>dem partem diuidentia. </s> <s xml:id="echoid-s1649" xml:space="preserve">Productæ ijſdem in ſolidis figuræ <lb/>ſimiles erunt iuxta definitionem 10. </s> <s xml:id="echoid-s1650" xml:space="preserve">huius, & </s> <s xml:id="echoid-s1651" xml:space="preserve">omnium ho-<lb/>mologæ duabus quibuſdam regulis æquidiſtabunt.</s> <s xml:id="echoid-s1652" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1653" xml:space="preserve">Sint ſimilia ſolida iuxta defin. </s> <s xml:id="echoid-s1654" xml:space="preserve">9. </s> <s xml:id="echoid-s1655" xml:space="preserve">vndec. </s> <s xml:id="echoid-s1656" xml:space="preserve">Elem. </s> <s xml:id="echoid-s1657" xml:space="preserve">ipſa, AEFSOGo, <lb/>Tl & </s> <s xml:id="echoid-s1658" xml:space="preserve">p f8s, in eorum autem ambitu capiantur ſimiles quæcumque <lb/>figurę planę, OGFS, f8 & </s> <s xml:id="echoid-s1659" xml:space="preserve">p, quibus parallela ducantur duo quę-<lb/>cumque plana eadem ſecantia, necnon & </s> <s xml:id="echoid-s1660" xml:space="preserve">altitudines reſpectu dicta-<lb/>rum baſium aſſumptas ſimiliter ad eandem partem diuidentia, ac in <lb/>ipſis ſolidis figuras, LHMP, YVZd, producentia. </s> <s xml:id="echoid-s1661" xml:space="preserve">Dico has eſſe <lb/>ſimiles figuras planas icxta defin. </s> <s xml:id="echoid-s1662" xml:space="preserve">10. </s> <s xml:id="echoid-s1663" xml:space="preserve">huius, omniumque ſic produ-<lb/>ctarum in dictis ſolidis homologas duabus quibuſdam regulis, vtex. <lb/></s> <s xml:id="echoid-s1664" xml:space="preserve">gr. </s> <s xml:id="echoid-s1665" xml:space="preserve">ipſis, OS, fp, æquidiſtare. </s> <s xml:id="echoid-s1666" xml:space="preserve">Igitur figurarum ambientium dicta <lb/>ſolida duæ aliæ ſimiles quæcumq; </s> <s xml:id="echoid-s1667" xml:space="preserve">capiantur cum baſibus concurren-<lb/>tes, vt ex. </s> <s xml:id="echoid-s1668" xml:space="preserve">gr. </s> <s xml:id="echoid-s1669" xml:space="preserve">oOS, sfp, ſimilia triangula, ducantur autem pręfa-<lb/>tis baſibus oppoſita tangentia plana, AC, TR, ſecantia producta <pb o="64" file="0084" n="84" rhead="GEOMETRIÆ"/> plana figurarum, oOS, spf, in rectis, BC, QR, quibus occur-<lb/>rant, Oo, fs, productæ vt in punctis, B, Q, & </s> <s xml:id="echoid-s1670" xml:space="preserve">iungantur, SB, p <lb/>Q, eſto autem, quod plana figurarum, LHMP, YVZd, diuiſe-<lb/>rint plana figurarum, oOS, sfp, producta in rectis, KN, ug, quę <lb/>ab ipſis, BS, Qp, BO, Qf, ſecentur in, I, X, Ku, & </s> <s xml:id="echoid-s1671" xml:space="preserve">iungantur, <lb/>LK, PI, Yu, dX. </s> <s xml:id="echoid-s1672" xml:space="preserve">Quoniam ergo plana figurarum, HMPL, V <lb/> <anchor type="figure" xlink:label="fig-0084-01a" xlink:href="fig-0084-01"/> ZdY, prędictas altitudines ſi-<lb/>militer ad eandem partem di-<lb/>uidentia, ſecant latera homo-<lb/>loga, ao, Ts, ſimiliter ad <lb/>eandem partem in punctis, L, <lb/> <anchor type="note" xlink:label="note-0084-01a" xlink:href="note-0084-01"/> Y, vt etiam, AG, T8, in, H <lb/>V, erunt figurę, ALH, TY <lb/>V, ad eandem partem ſecan-<lb/> <anchor type="note" xlink:label="note-0084-02a" xlink:href="note-0084-02"/> tium, HL, VY, conſtitutæ <lb/>inter ſe ſimiles, & </s> <s xml:id="echoid-s1673" xml:space="preserve">earum late-<lb/>ra homologa ipſę, HL, VY; <lb/></s> <s xml:id="echoid-s1674" xml:space="preserve">eodem modo oſtendemus ſi-<lb/>miles eſſe ipſas, EALP, lT <lb/>Yd, & </s> <s xml:id="echoid-s1675" xml:space="preserve">earum latera homolo-<lb/>ga ipſas, LP, Yd, ſunt autem <lb/>figuræ, AEPL, ALH, in-<lb/>uicem ad eandem partem æ. </s> <s xml:id="echoid-s1676" xml:space="preserve"><lb/>què inclinatæ, acipſæ, Tld <lb/>Y, TYV, cum ſint in planis <lb/> <anchor type="note" xlink:label="note-0084-03a" xlink:href="note-0084-03"/> ſimilium figurarum, AESo, <lb/>Tlps, AGOo, T8fs, quę <lb/> <anchor type="note" xlink:label="note-0084-04a" xlink:href="note-0084-04"/> ſunt inuicem ad eandem par-<lb/>tem æquè inclinatę, ergo an-<lb/>guli, HLP, VYd, homolo-<lb/>gis lateribus contenti eruntę-<lb/>quales, & </s> <s xml:id="echoid-s1677" xml:space="preserve">circa eoſdem latera <lb/>erunt proportionalia. </s> <s xml:id="echoid-s1678" xml:space="preserve">Eodem <lb/>modo oſtedemus cæteros an-<lb/>gulos, LPM, YdZ, interſe, <lb/>necnon, PMH, dZV, ac, <lb/>MHL, ZVY, æquales eſ-<lb/>ſe, & </s> <s xml:id="echoid-s1679" xml:space="preserve">circa æquales angulos <lb/>latera exiſtere proportionalia, <lb/>ergo figuræ, LHMP, YVZd, ſimiles erunt iuxta Euclidem, er-<lb/> <anchor type="note" xlink:label="note-0084-05a" xlink:href="note-0084-05"/> go etiam ſimiles erunt iuxta definit. </s> <s xml:id="echoid-s1680" xml:space="preserve">10. </s> <s xml:id="echoid-s1681" xml:space="preserve">huius.</s> <s xml:id="echoid-s1682" xml:space="preserve"/> </p> <div xml:id="echoid-div172" type="float" level="2" n="1"> <figure xlink:label="fig-0084-01" xlink:href="fig-0084-01a"> <image file="0084-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0084-01"/> </figure> <note position="left" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">Ex Lem. <lb/>ant.</note> <note position="left" xlink:label="note-0084-02" xlink:href="note-0084-02a" xml:space="preserve">Ex Lem, <lb/>3.</note> <note position="left" xlink:label="note-0084-03" xlink:href="note-0084-03a" xml:space="preserve">Ex Lem. <lb/>1.</note> <note position="left" xlink:label="note-0084-04" xlink:href="note-0084-04a" xml:space="preserve">Ex Lem. <lb/>2.</note> <note position="left" xlink:label="note-0084-05" xlink:href="note-0084-05a" xml:space="preserve">Defin. 1. <lb/>Sex. El.</note> </div> <p> <s xml:id="echoid-s1683" xml:space="preserve">Reliquum eſt, vt demonſtremus earum homologas duabus aſſum- <pb o="65" file="0085" n="85" rhead="LIBER I."/> ptis regulis, OS, fp, omnes ęquidiſtare: </s> <s xml:id="echoid-s1684" xml:space="preserve">Et quidem ſi plana ſecent <lb/>figuras, oOS, sfp, hoc manifeſtum eſt, etenim productę lineę ip-<lb/>ſis baſibus, OS, sp, erunt parallelæ, & </s> <s xml:id="echoid-s1685" xml:space="preserve">latera homologa ſimilium <lb/>figurarum ex traiectis planis in ſolidis productarum. </s> <s xml:id="echoid-s1686" xml:space="preserve">Siverò plana <lb/>parallela ſecent duas figuras ipſis, oOS, sfp, continuatas, vti ſa-<lb/>ciunt plana figurarum, HMPL, VZdY, quæ etiam ſecant plana <lb/>figurarum, oOS, sfp, producta in rectis, KN, ug, oſtendemus <lb/>ipſas, KN, ug, eſſe regulas homologarum ſimilium figurarum, L <lb/>HMP, VZdY, iunctis, PK, du. </s> <s xml:id="echoid-s1687" xml:space="preserve">Quia enim, Oo, fs, ſunt ip-<lb/>ſorum ſimilium ſolidorum latera homologa, producta, ac terminata <lb/>ad baſium plana, & </s> <s xml:id="echoid-s1688" xml:space="preserve">oppoſitorum tangentium, in punctis, O, B; </s> <s xml:id="echoid-s1689" xml:space="preserve">f, <lb/> <anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/> Q, ideò, BO, Qf, ſunt ſimiliter ad eandem partem ſectæ in, o, s, <lb/>& </s> <s xml:id="echoid-s1690" xml:space="preserve">nedum, Oo, fs, ſed etiam, oB, sQ, ſunt vt eorum altitudines <lb/>ſumptæ reſpectu dictarum baſium, ſed ſic etiam ſunt ipſæ, oS, sp, <lb/>latera homologa, ergo, Bo, ad, oS, eſt vt, Qs, ad, sp, & </s> <s xml:id="echoid-s1691" xml:space="preserve">angu-<lb/>los æquales, BoS, Qsp, complectuntur latera proportionalia, er <lb/> <anchor type="note" xlink:label="note-0085-02a" xlink:href="note-0085-02"/> go triangula, BoS, Qsp, ſunt ſimilia, cum verò ſint in planis trian-<lb/>gulorum, oOS, sfp, ſunt etiam ſimilibus figuris, LPSo, Ydps, <lb/>ęquè ad eandem partem inclinata, quibus communia ſunt homolo-<lb/> <anchor type="note" xlink:label="note-0085-03a" xlink:href="note-0085-03"/> galatera, oS, sp, ergo anguli, KoL, usY, interſe, necnon, PS <lb/> <anchor type="note" xlink:label="note-0085-04a" xlink:href="note-0085-04"/> I, dpX, æquales erunt; </s> <s xml:id="echoid-s1692" xml:space="preserve">cum verò, BS, Qp, ſint vt dictæ altitudi-<lb/>nes, & </s> <s xml:id="echoid-s1693" xml:space="preserve">ſic etiam, IS, Xp, necnon, PS, dp, (etenim, BS, Qp, in, <lb/> <anchor type="note" xlink:label="note-0085-05a" xlink:href="note-0085-05"/> I, X, &</s> <s xml:id="echoid-s1694" xml:space="preserve">, ES, lp, ſimiliter ſecantur, & </s> <s xml:id="echoid-s1695" xml:space="preserve">ad eandem partem, in pun-<lb/>ctis, P, d,) erit, IS, ad, SP, vt, Xp, ad, pd, & </s> <s xml:id="echoid-s1696" xml:space="preserve">circumſtant an-<lb/> <anchor type="note" xlink:label="note-0085-06a" xlink:href="note-0085-06"/> gulos æquales, ISP, Xpd, ergo triangula, ISP, Xpd, ſunt ſimi-<lb/>lia. </s> <s xml:id="echoid-s1697" xml:space="preserve">Eodem modo oſtendemus ſimilia eſſe triangula, LoK, Y su. <lb/></s> <s xml:id="echoid-s1698" xml:space="preserve">Vlterius, quia eſt, Ko, ad, oS, vt, us, ad, sp, &</s> <s xml:id="echoid-s1699" xml:space="preserve">, oS, ad, SI, <lb/>vt, sp, ad, pX, & </s> <s xml:id="echoid-s1700" xml:space="preserve">anguli, KoS, usp, necnon, oSI, spX, ſunt <lb/>æquales, ideò trapezia, KoSI, uspX, erunt ſimilia, ſed etiam fi-<lb/>guræ, LPSo, Ydps, ſunt ſimiles, eſt autem, KL, ad, Lo, vt, <lb/>uY, ad, Ys, &</s> <s xml:id="echoid-s1701" xml:space="preserve">, oL, ad, LP, vt, sY, ad, Yd, ergo, KL, ad, L <lb/>P, erit vt, uY, ad, Yd, eodem modo autem oſtendemus, LP, PI, <lb/>IK, KL, binas eſte in eadem proportione cum ipſis, Yd, dX, Xu. </s> <s xml:id="echoid-s1702" xml:space="preserve"><lb/>uY. </s> <s xml:id="echoid-s1703" xml:space="preserve">Manifeſtum eſt autem ſi iungeremus, AO, Tf, AS, Tp, quod <lb/>fierent ſimiles pyramides triangulatæ ipſæ, AOoS, Tfsp, ſimili-<lb/>bus n. </s> <s xml:id="echoid-s1704" xml:space="preserve">triangulis comprehenderentur, vt meditanti compertum fiet, <lb/> <anchor type="note" xlink:label="note-0085-07a" xlink:href="note-0085-07"/> ideò plana, AoO, Tsf, ideſt triangula ſimilia, LKo, Yus, ſunt <lb/> <anchor type="note" xlink:label="note-0085-08a" xlink:href="note-0085-08"/> æquè ad eandem partem ipſis ſimilibus figuris, LPSo, Ydps, in-<lb/>clinata, cum quibus coincidunt in lateribus homologis, Lo, Ys, <lb/> <anchor type="note" xlink:label="note-0085-09a" xlink:href="note-0085-09"/> ergo anguli, KLP, uYd, erunt æquales, quibus circumſtant latera <lb/>proportionalia, vt probatum eſt, ergo triangula, KLP, uYd, ſi-<lb/>milia erunt, & </s> <s xml:id="echoid-s1705" xml:space="preserve">erit, KP, ad, PL, vt, ud, ad, dY, eſt verò, PL, <pb o="66" file="0086" n="86" rhead="GEOMETRIÆ"/> ad, PI, vt, dY, ad, dX, ergo ex æquali, KP, ad, PI, erit vt, u <lb/>d, ad, dX, eſt autem, PI, ad, IK, vt, dX, ad, Xu, ergo trian-<lb/>gula quoque, PKI, duX, pariter ſimilia erunt, vnde anguli, LP <lb/>I, YdX; </s> <s xml:id="echoid-s1706" xml:space="preserve">PIK, dXu, &</s> <s xml:id="echoid-s1707" xml:space="preserve">, IKL, XuY, æquales erunt. </s> <s xml:id="echoid-s1708" xml:space="preserve">Ducantur <lb/>nunc in planis figurarum, LHMP, YVZd, à punctis, L, Y, pa-<lb/> <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> rallelæ, KN, ug, ipſæ, L3, <lb/>Y4. </s> <s xml:id="echoid-s1709" xml:space="preserve">Cum igitur anguli, LK <lb/>I, YuX, ſint ęquales, etiam, <lb/>KL3, uY4, æquales erunt, <lb/> <anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/> ſed &</s> <s xml:id="echoid-s1710" xml:space="preserve">, KLP, uYd, ſuntæ-<lb/>quales, ergo reſidui quoque, 3 <lb/>LP, 4Yd, erunt ęquales, vn-<lb/>de cum ipſæ, L3, Y4, con-<lb/>tineant cum lateribus homo-<lb/>logis, LP, Yd, ad eandem <lb/>partem angulos æquales, e-<lb/>runt regulę homologarum ſi-<lb/>milium figurarum, LHMP, <lb/>YVZd, vnde etiam ipſæ, K <lb/>N, ug, velipſæ OS, fp, e-<lb/>runt regulæ homologarum <lb/>earundem, ſunt. </s> <s xml:id="echoid-s1711" xml:space="preserve">n. </s> <s xml:id="echoid-s1712" xml:space="preserve">OS, KN, <lb/>parallelæ, vt etiam, ug, fp, <lb/>vnde omnes homologæ ſimi-<lb/>lium figurarum, LHMP, Y <lb/>VZd, ipſis regulis, OS, fp, <lb/>æquidiſtabunt, quod & </s> <s xml:id="echoid-s1713" xml:space="preserve">de cę-<lb/>teris eodem modo oſtende-<lb/>tur, dumſectio fiet in figuris, <lb/>AESo, Tlps. </s> <s xml:id="echoid-s1714" xml:space="preserve">Quod ſi fi-<lb/>guris, AESO, Tlps, aliæ <lb/>figuræ planæ continuarentur <lb/>citra cõtactum planorum ba-<lb/>ſibus oppoſitorum, cum his <lb/>in lateribus homologis, AE, <lb/>Tl, conuenientes, quibus eſ-<lb/>ſent inclinatę, parum diſſimili <lb/>methodo, producentes, OB, <lb/>fQ, vſq; </s> <s xml:id="echoid-s1715" xml:space="preserve">ad tangentia plana, & </s> <s xml:id="echoid-s1716" xml:space="preserve">occurſuum puncta cum ipſis, S, p, <lb/>iungentes, necnon extrema laterum homologorum, qualia fuerunt, <lb/>LP, Yd, cum extremis rectarum in triangulis (qualia fuerunt, BO <lb/>S, Qfp,) productarum, panter iungentes, vt fecimus cum ipſis, KI, <pb o="67" file="0087" n="87" rhead="LIBER I."/> uX, oſtenderemus figuras his ductis comprehenſas, quales fuerunt, <lb/>LPIK, YdXu, eſſe ſimiles, ex quo propoſitum quoque noſtrum <lb/>haberemus. </s> <s xml:id="echoid-s1717" xml:space="preserve">Similiter ſi anguli ſolidi, Q, f, pluribus, quam tribus <lb/>angulis planis contineantur, currit tamen demonſtratio, cum trian-<lb/> <anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/> gula, GOo, 8fs, licet non ſint in ambitu ſolidorum ſint tamen ſi-<lb/>milia, & </s> <s xml:id="echoid-s1718" xml:space="preserve">ęquè ad eandem partem inclinata figuris, cum quibus con-<lb/>currunt, etenim ex. </s> <s xml:id="echoid-s1719" xml:space="preserve">gr. </s> <s xml:id="echoid-s1720" xml:space="preserve">pyramides, SGOo, p8fs, ſi eorum latera <lb/>iungerentur, ſimiles eſſent, quapropter ipſius demonſtrationis vis <lb/>non eneruatur. </s> <s xml:id="echoid-s1721" xml:space="preserve">Similiter ſi, oOS, sfp, non eſſent triangula, ſed <lb/>aliæ quæcumque figuræ ſimiles, pro, oS, sp, acceptis lateribus ip-<lb/>ſis, Oo, fs, adiuncta, os, conterminantibus, & </s> <s xml:id="echoid-s1722" xml:space="preserve">planis ad hæc la-<lb/>tera pariter terminantibus, eodem modo demonſtratio abſoluere-<lb/>tur: </s> <s xml:id="echoid-s1723" xml:space="preserve">hæc omnia autem ſingillatim proſequi nimis longum, ac ſche-<lb/>matibus rem aperire, res tricis plena eſſet, quapropter Lectoris in-<lb/>duſtriæ hoc relinquo, ſi enim ea rectè percepcrit, quę ſuperius expli-<lb/>cata ſunt, circa huius veritatem minimè hæſitabit, infinita autem ſi-<lb/>milium ſolidorum planis contentorum varietas efficit, vt ægrè ip-<lb/>ſius demonſtrationis vniuerſalitatem oculis ſubijcere poſſim, quod <lb/>Lector æqui, bonique faciat, hæc verò oſtendenda proponebantur.</s> <s xml:id="echoid-s1724" xml:space="preserve"/> </p> <div xml:id="echoid-div173" type="float" level="2" n="2"> <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">Elicitur <lb/>ex Corol. <lb/>Lem. 6.</note> <note position="right" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">6. Sex. El.</note> <note position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">Ex Lem. <lb/>1.</note> <note position="right" xlink:label="note-0085-04" xlink:href="note-0085-04a" xml:space="preserve">Corollar. <lb/>Lem. 6.</note> <note position="right" xlink:label="note-0085-05" xlink:href="note-0085-05a" xml:space="preserve">Corol. 26 <lb/>huius.</note> <note position="right" xlink:label="note-0085-06" xlink:href="note-0085-06a" xml:space="preserve">6. Sex. El.</note> <note position="right" xlink:label="note-0085-07" xlink:href="note-0085-07a" xml:space="preserve">Ex Lem. <lb/>4.</note> <note position="right" xlink:label="note-0085-08" xlink:href="note-0085-08a" xml:space="preserve">Ex Lem. <lb/>1.</note> <note position="right" xlink:label="note-0085-09" xlink:href="note-0085-09a" xml:space="preserve">Ex Lem. <lb/>2.</note> <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a"> <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0086-01"/> </figure> <note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">Corol. 27 <lb/>huius.</note> <note position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">Ex Lem. <lb/>4. & 1.</note> </div> </div> <div xml:id="echoid-div175" type="section" level="1" n="115"> <head xml:id="echoid-head126" xml:space="preserve">THEOREMA XXVII. PROPOS. XXX.</head> <p> <s xml:id="echoid-s1725" xml:space="preserve">POſita definit. </s> <s xml:id="echoid-s1726" xml:space="preserve">9. </s> <s xml:id="echoid-s1727" xml:space="preserve">vndec. </s> <s xml:id="echoid-s1728" xml:space="preserve">Elem. </s> <s xml:id="echoid-s1729" xml:space="preserve">ſimilium ſolidarum figura-<lb/>rum, ſequitur, & </s> <s xml:id="echoid-s1730" xml:space="preserve">mea definitio generalis ſimilium ſo-<lb/>lidorum.</s> <s xml:id="echoid-s1731" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1732" xml:space="preserve">Aſſumptis denuò antec. </s> <s xml:id="echoid-s1733" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s1734" xml:space="preserve">figuris, ſint adhuc ſimilia ſolida <lb/>iuxta Euclidem ipſa, AGS, T8p. </s> <s xml:id="echoid-s1735" xml:space="preserve">Dico eadem eſſe etiam ſimilia <lb/>iuxta definit. </s> <s xml:id="echoid-s1736" xml:space="preserve">11. </s> <s xml:id="echoid-s1737" xml:space="preserve">huius, quam de ſimilibus ſolidis generaliter attuli. <lb/></s> <s xml:id="echoid-s1738" xml:space="preserve">Sint autem ducta eadem oppoſita tangentia plana, vt ibi, ita vt duę <lb/>ſimiles figurę, GFSO, f8 & </s> <s xml:id="echoid-s1739" xml:space="preserve">p, ſint plana contactuum ex vna par-<lb/>te, ex alia verò ſint plana tangentia, AC, TR, captis autem alijs <lb/>duabus ſimilibus figuris cum baſibus concurrentibus, ipſis nempè, O <lb/>So, fps, illarum plana extendantur, ita vt ſecent oppoſita tangen-<lb/>tia plana, in rectis nempè, BC, OD; </s> <s xml:id="echoid-s1740" xml:space="preserve">QR, f℟; </s> <s xml:id="echoid-s1741" xml:space="preserve">extenſis autem vl-<lb/>terius planis, GOo, 8fs, quęnunc ſint in ambitibus ſolidorum ipſa <lb/>ſecent oppoſita tangentia plana in rectis, AB, TQ; </s> <s xml:id="echoid-s1742" xml:space="preserve">GO, 8f, & </s> <s xml:id="echoid-s1743" xml:space="preserve"><lb/>plana figurarum, OSO, fps, producta in rectis, BO, Qf, ſecen-<lb/>tur verò hęc ſolida duobus planis oppoſitis tangentibus parallelis vt-<lb/>cumque, & </s> <s xml:id="echoid-s1744" xml:space="preserve">ſint illa eadem, quę in ſolidis produxerunt figuras, LH <lb/>MP, YVZd, iſtæ ergo ſimiles erunt, & </s> <s xml:id="echoid-s1745" xml:space="preserve">earum homologæ, ſi pro <pb o="68" file="0088" n="88" rhead="GEOMETRIÆ"/> regulis aſſumamus iterum ipſas, OS, fp, eiſdem quoque æquidiſta-<lb/> <anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/> bunt, ergo in eiſdem figuris habebimus etiam homologas alias ęqui-<lb/>diſtantes regulis quibuſcumque cum ipſis, OS, fp, angulos æqua-<lb/>les ad eandem partem continentibus, cum ergo ipſæ, GO, 8f, an-<lb/> <anchor type="note" xlink:label="note-0088-02a" xlink:href="note-0088-02"/> gulos æquales cum ipſis, OS, fp, ad eandem partem contineant, <lb/> <anchor type="figure" xlink:label="fig-0088-01a" xlink:href="fig-0088-01"/> ideò omnium homologæ pa-<lb/>riter duabus, GO, 8f, tan-<lb/>quam nouis regulis ęquidiſta-<lb/>bunt, iſtis autem, quæ tan-<lb/>gunt ex vna parte figuras, G <lb/>FSO, 8 & </s> <s xml:id="echoid-s1746" xml:space="preserve">pf, ducantur ex <lb/>alia parte oppoſitæ tangen <lb/>tes, FD, & </s> <s xml:id="echoid-s1747" xml:space="preserve">℟, ita vt incidant <lb/>duę, GO, FD, plano, BD, <lb/>in punctis, O, D, & </s> <s xml:id="echoid-s1748" xml:space="preserve">duæ, 8 <lb/>f, & </s> <s xml:id="echoid-s1749" xml:space="preserve">℟, plano, Q℟, in pun-<lb/>ctis, f, ℟, ſint autem iunctæ, <lb/>OD, f℟. </s> <s xml:id="echoid-s1750" xml:space="preserve">Similiter figurarum, <lb/>HMPL, VZdY, ſint ductę <lb/>oppoſitæ tangentes præfatis <lb/>regulis, GO, 8f, parallelæ, <lb/>planis, BD, Q℟, occurren-<lb/>tes in punctis, K, N; </s> <s xml:id="echoid-s1751" xml:space="preserve">ug, iun-<lb/>gantur autem, KN, ug, & </s> <s xml:id="echoid-s1752" xml:space="preserve"><lb/>ita cæterarum ſic producibi-<lb/>lium figurarum intelligantur <lb/>ductæ oppoſitæ tangentes ip-<lb/>ſis, GO, 8f, parallelæ, & </s> <s xml:id="echoid-s1753" xml:space="preserve"><lb/>productæ vſque ad plana, B <lb/>D, Q℟, punctaq; </s> <s xml:id="echoid-s1754" xml:space="preserve">occurſuum <lb/>iuncta rectis lineis, per qua-<lb/>rum omnium extrema tran. <lb/></s> <s xml:id="echoid-s1755" xml:space="preserve">ſeant lineæ, BO, CD, Qf, <lb/>R℟. </s> <s xml:id="echoid-s1756" xml:space="preserve">Cum ergo, GO, 8f, <lb/>ſint homologarum regulæ, ac <lb/>oppoſitę tangentes figurarum <lb/>ſimilium, OGFS, f8 & </s> <s xml:id="echoid-s1757" xml:space="preserve">p, <lb/>incidant autem illis ad eun-<lb/>dem angulum ex eadem parte, OD, f℟, & </s> <s xml:id="echoid-s1758" xml:space="preserve">ſit, GO, ad, f8, vt, O <lb/>D, ad, f℟, ideo, OD, f℟, erunt incidentes ſimilium figurarum, <lb/> <anchor type="note" xlink:label="note-0088-03a" xlink:href="note-0088-03"/> OGFS, f8&</s> <s xml:id="echoid-s1759" xml:space="preserve">p, & </s> <s xml:id="echoid-s1760" xml:space="preserve">oppoſitarum tangentium, GO, FD, 8f, & </s> <s xml:id="echoid-s1761" xml:space="preserve"><lb/>℟. </s> <s xml:id="echoid-s1762" xml:space="preserve">Similiter in figuris, HMPL, VZdY, oſtendemus eſſe ipſarum <pb o="69" file="0089" n="89" rhead="LIBER I."/> incidentes, ac oppoſitarum tangentium, HK, MN, Vu, Zg, ip-<lb/>ſas, KN, ug, ſi. </s> <s xml:id="echoid-s1763" xml:space="preserve">n. </s> <s xml:id="echoid-s1764" xml:space="preserve">iungeremus, MK, Zu, probaretur, MH, ad, <lb/>HK, eſſe vt, ZV, ad, Vu, (ſunt. </s> <s xml:id="echoid-s1765" xml:space="preserve">n. </s> <s xml:id="echoid-s1766" xml:space="preserve">ſimiles figuræ, HMPL, V <lb/>ZdY, necnon, LPK, Ydu, circumſtant autem latera proportio-<lb/>nalia angulos æquales, MHK, ZVu, & </s> <s xml:id="echoid-s1767" xml:space="preserve">ideò oſtenderemus trian-<lb/>gula, MHK, ZVu, eſſe ſimilia, vnde pateret angulos, HKM, <lb/>VuZ, eſſe æquales, ſed etiam, HKN, Vug, ſunt ęquales, ergo <lb/> <anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/> pateret angulos, MKN, Zug, eſſe æquales, ſunt autem etiam æ-<lb/>quales, MNK, Zgu, ergo triangula, MKN, Zug, eſſent æ-<lb/>quiangula, vnde, MN, ad, Zg, eſſet vt, KN, ad, ug, incidunt <lb/>autem, KN, ug, oppoſitis tangentibus, HK, MN, Vu, Zg, ad <lb/>eundem angulum ex eadem parte, ergo ipſarum tangentium, ac fi-<lb/> <anchor type="note" xlink:label="note-0089-02a" xlink:href="note-0089-02"/> gurarum ſunt incidentes, KN, ug, cum verò, KN, ad, ug, ſit vt, <lb/>MK, ad, Zu, ideſt vt, MH, ad, ZV, vel vt quoduis ſolidorum <lb/>latus homologum ad quoduis latus homologum, ideſt vt, GO, ad, <lb/>8f, ideſt vt, OD, ad, f℟; </s> <s xml:id="echoid-s1768" xml:space="preserve">OD, autem ad, f℟, ſit vt, Bo, ad, Qf, <lb/>ideò, KN, ad, ug, erit vt, BO, ad, Qf, & </s> <s xml:id="echoid-s1769" xml:space="preserve">diuidunt ſimiliter ad <lb/>eandem partem ipſas, BO, Qf, in punctis, Ku, quæ incidunt ip-<lb/>ſis, BC, OD, Qf, R℟, ad eundem angulum ex eadem parte, ſunt <lb/>.</s> <s xml:id="echoid-s1770" xml:space="preserve">n. </s> <s xml:id="echoid-s1771" xml:space="preserve">anguli, BOD, Qf℟, æquales, quod & </s> <s xml:id="echoid-s1772" xml:space="preserve">de cæteris incidentibus <lb/>probabitur, ergo figurę, BODC, Qf℟R, quę capiunt omnes di-<lb/> <anchor type="note" xlink:label="note-0089-03a" xlink:href="note-0089-03"/> ctas incidentes, ſunt ſimiles, & </s> <s xml:id="echoid-s1773" xml:space="preserve">arum homologę ipſę incidentes, qua-<lb/>rum omnium regulæ ſunt, OD, f℟, & </s> <s xml:id="echoid-s1774" xml:space="preserve">ſunt ipſę figurę, BD, Q℟, <lb/>æquè ad eandem partem ipſis baſibus inclinatę, cum ſint in planis fi-<lb/> <anchor type="note" xlink:label="note-0089-04a" xlink:href="note-0089-04"/> gurarum, oOS, sfp, ergo dicta ſolida ſunt etiam ſimilia iuxta de-<lb/>fin. </s> <s xml:id="echoid-s1775" xml:space="preserve">11. </s> <s xml:id="echoid-s1776" xml:space="preserve">huius. </s> <s xml:id="echoid-s1777" xml:space="preserve">quod ſi plana, GOo, 8fs, non eſſent in ambitu ſi-<lb/>milium dictorum ſolidorum, facilè tamen oſtenderemus portiones <lb/>ſolidorum vltra eadem plana exiſtentes eſſe ſimiles, ac ipſarum, & </s> <s xml:id="echoid-s1778" xml:space="preserve"><lb/>oppoſitorum tangentium planorum iam dictorum incidentes repe-<lb/>riri in planis figurarum, BD, Q℟, cum eiſdem integrantes figuras <lb/>incidentes integrorum ſimilium ſolidorum, ac dictorum oppoſito-<lb/>rum tangentium, quod ſpeculanti facilè innoteſcet, hoc autem erat <lb/>oſtendendum.</s> <s xml:id="echoid-s1779" xml:space="preserve"/> </p> <div xml:id="echoid-div175" type="float" level="2" n="1"> <note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">Fuxta def. <lb/>10. huius.</note> <note position="left" xlink:label="note-0088-02" xlink:href="note-0088-02a" xml:space="preserve">Jmin s. 2</note> <figure xlink:label="fig-0088-01" xlink:href="fig-0088-01a"> <image file="0088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0088-01"/> </figure> <note position="left" xlink:label="note-0088-03" xlink:href="note-0088-03a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">6. Sex. El.</note> <note position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">24. huius.</note> <note position="right" xlink:label="note-0089-03" xlink:href="note-0089-03a" xml:space="preserve">Deſio. 10. <lb/>huius.</note> <note position="right" xlink:label="note-0089-04" xlink:href="note-0089-04a" xml:space="preserve">Lem. 1.</note> </div> </div> <div xml:id="echoid-div177" type="section" level="1" n="116"> <head xml:id="echoid-head127" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s1780" xml:space="preserve">CIrculi omnes, necnon femicirculi ſunt ſimiles iuxta meam deſi-<lb/>nitionem ſimilium planarum figurarum, & </s> <s xml:id="echoid-s1781" xml:space="preserve">eorum, & </s> <s xml:id="echoid-s1782" xml:space="preserve">tangen-<lb/>tium oppoſitarum, quæ ab extremitatibus diamertrorum ducuntur, <lb/>incidentes ſunt ipſi diametri.</s> <s xml:id="echoid-s1783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1784" xml:space="preserve">Sint circuli, ABCD, ONQ, quorum diametri, AC, OQ, per <lb/>quorum extrema ducantur tangentes, FA, GC, HO, LQ. </s> <s xml:id="echoid-s1785" xml:space="preserve">Dico <pb o="70" file="0090" n="90" rhead="GEOMETRIÆ"/> hos circulos eſſe ſimiles iuxta meam definitionem ſimilium plana-<lb/>rum figurarum, & </s> <s xml:id="echoid-s1786" xml:space="preserve">eorum, & </s> <s xml:id="echoid-s1787" xml:space="preserve">ductarum oppoſitarum tangentium in-<lb/>cidentes eſſe ipſas diametros, AC, OQ, quæ etiam de ſemicirculis <lb/>verificantur. </s> <s xml:id="echoid-s1788" xml:space="preserve">Diametri ergo, AC, OQ, diuidantur fimiliter ad ean-<lb/>dem partem in punctis, E, M, à quibus vſque ad circumferentiam <lb/>ducantur ipſæ, EB, MN, parallelæ dictis tangentibus, quæ cum <lb/>ad angulos rectos diametros diuidant, etiam ipſę, BE, NM, erunt <lb/> <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> illis perpendiculares, igitur quadratum, <lb/>BE, erit ęquale rectangulo, AEC, ſi-<lb/>cuti quadratum, NM, æquale rectan-<lb/>gulo, OMQ, rectangulum autem, A <lb/>EC, ad quadratum, EC, eſt vt, AE, <lb/>ad, EC, ideſt vt, OM, ad, MQ, ideſt <lb/> <anchor type="note" xlink:label="note-0090-01a" xlink:href="note-0090-01"/> vt rectangulum, OMQ, ad quadratum, <lb/>MQ, ideſt vt quadratum, NM, ad qua-<lb/>dratum, MQ, ergo quadratum, BE, <lb/>ad quadratum, EC, eſt vt quadratum, <lb/>NM, ad quadratum, MQ, (quæ au-<lb/>tem hic ſupponuntur, vel petantur ex <lb/>Eucl. </s> <s xml:id="echoid-s1789" xml:space="preserve">lib. </s> <s xml:id="echoid-s1790" xml:space="preserve">Elem. </s> <s xml:id="echoid-s1791" xml:space="preserve">vel ex ſequenti meo lib. <lb/></s> <s xml:id="echoid-s1792" xml:space="preserve"> <anchor type="note" xlink:label="note-0090-02a" xlink:href="note-0090-02"/> in quo, quæ hic aſſumuntur indepen-<lb/>denter ab hoc Lemmate demonſtratur) <lb/>ergo, BE, ad, EC, erit vt, NM, ad, <lb/>MQ, permutando, BE, ad, NM, e-<lb/>rit vt, EC, ad, MQ, vel vt, AC, ad, OQ, igitur, quæ æquidi-<lb/>ſtant ipſis tangentibus, FA, HO, & </s> <s xml:id="echoid-s1793" xml:space="preserve">ſimiliter ad eandem partem <lb/>vtcumque diuidunt ipſas, AC, OQ, & </s> <s xml:id="echoid-s1794" xml:space="preserve">iacent inter ipſas, & </s> <s xml:id="echoid-s1795" xml:space="preserve">circui-<lb/>tus ſemicirculorum, ABC, ONQ, ad eandem partem, eodem or-<lb/>dine ſumptæ, ſunt vt ipſæ, AC, OQ, quæ dictis tangentibus inci-<lb/> <anchor type="note" xlink:label="note-0090-03a" xlink:href="note-0090-03"/> dunt ad eundem angulum ex eadem parte, quęideò ſunt earum inci-<lb/>dentes, ergo ſemicirculi, ABC, ONQ, ſunt figuræ planæ fimiles <lb/>ſuxta meam definitionem, quarum & </s> <s xml:id="echoid-s1796" xml:space="preserve">oppoſitarum tangentium, quę <lb/>ab extremitate diametrorum ducuntur, incidentes ſunt ipſi diame-<lb/>tri; </s> <s xml:id="echoid-s1797" xml:space="preserve">ſic etiam patebit ſemicirculos, ADC, OZQ, necnon circu-<lb/>los, AC, OQ, eſſe ſimiles, iuxta eandem definitionem; </s> <s xml:id="echoid-s1798" xml:space="preserve">quod oſten-<lb/>dendum erat.</s> <s xml:id="echoid-s1799" xml:space="preserve"/> </p> <div xml:id="echoid-div177" type="float" level="2" n="1"> <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a"> <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0090-01"/> </figure> <note position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">1. Sex. El.</note> <note position="left" xlink:label="note-0090-02" xlink:href="note-0090-02a" xml:space="preserve">8. Lib. 2. <lb/>ſequen. <lb/>vel 20. <lb/>Sex. El.</note> <note position="left" xlink:label="note-0090-03" xlink:href="note-0090-03a" xml:space="preserve">Defin. 10.</note> </div> </div> <div xml:id="echoid-div179" type="section" level="1" n="117"> <head xml:id="echoid-head128" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXXI.</head> <p> <s xml:id="echoid-s1800" xml:space="preserve">POſitis infraſcriptis definitionibus ſimilium cylindro-<lb/>rum, & </s> <s xml:id="echoid-s1801" xml:space="preserve">conorum, ſequitur definitio generalis, quam <lb/>de ſimilibus ſolidis ipſe attuli.</s> <s xml:id="echoid-s1802" xml:space="preserve"/> </p> <pb o="71" file="0091" n="91" rhead="LIBERI."/> </div> <div xml:id="echoid-div180" type="section" level="1" n="118"> <head xml:id="echoid-head129" xml:space="preserve">DEFINITIO.</head> <p> <s xml:id="echoid-s1803" xml:space="preserve">SImiles coni, & </s> <s xml:id="echoid-s1804" xml:space="preserve">cylindriſunt, quorum & </s> <s xml:id="echoid-s1805" xml:space="preserve">axes, & </s> <s xml:id="echoid-s1806" xml:space="preserve">baſium diame-<lb/>tri eandem inter ſe proportionem habent. </s> <s xml:id="echoid-s1807" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s1808" xml:space="preserve">lib. </s> <s xml:id="echoid-s1809" xml:space="preserve">11. </s> <s xml:id="echoid-s1810" xml:space="preserve">Elem. <lb/></s> <s xml:id="echoid-s1811" xml:space="preserve">defin. </s> <s xml:id="echoid-s1812" xml:space="preserve">24.</s> <s xml:id="echoid-s1813" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1814" xml:space="preserve">Verum, quia ſupradicta definitonon <lb/> <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> eſt niſi cylindrorum, & </s> <s xml:id="echoid-s1815" xml:space="preserve">conorum re-<lb/>ctorum, ideo aliam, quę affertur à Com-<lb/>mandino tum de rectis, tum etiam de <lb/>ſcalenis illi ſubiungo, quam ſufficiet o-<lb/>ſtendere cum mea ſuptadicta concor-<lb/>dare, nam hęcCommandinieam, quam <lb/>Euclides attulit, inuoluit.</s> <s xml:id="echoid-s1816" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="1"> <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a"> <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0091-01"/> </figure> </div> </div> <div xml:id="echoid-div182" type="section" level="1" n="119"> <head xml:id="echoid-head130" xml:space="preserve">DEFINITIO.</head> <p> <s xml:id="echoid-s1817" xml:space="preserve">SImiles coni, & </s> <s xml:id="echoid-s1818" xml:space="preserve">cylindri, ſiue recti, ſiue ſcaleni ſunt, quando per <lb/>axes ductis planis ad rectos angulos baſibus, communes ſectio-<lb/>nes eorum, & </s> <s xml:id="echoid-s1819" xml:space="preserve">baſium cum axibus æquales angulos continentes, ean-<lb/>dem inter ſe, quam axes, proportionem habent: </s> <s xml:id="echoid-s1820" xml:space="preserve">Commandinus lo-<lb/>co definitionis ſupra citatæ.</s> <s xml:id="echoid-s1821" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1822" xml:space="preserve">Sint coni, BEC, GMN, & </s> <s xml:id="echoid-s1823" xml:space="preserve">cylindri, AC, FN, ſimiles iuxta <lb/> <anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/> proximam definitionem. </s> <s xml:id="echoid-s1824" xml:space="preserve">Dico eoſdem eſſe ſimiles iuxta meam ſu-<lb/>pradictam. </s> <s xml:id="echoid-s1825" xml:space="preserve">Vt autem in ſimul pro conis, & </s> <s xml:id="echoid-s1826" xml:space="preserve">cylindris fiat demon-<lb/>ſtratio, ſupponanturconi, & </s> <s xml:id="echoid-s1827" xml:space="preserve">cylindri iam dicti eſſe in eiſdem baſi-<lb/>bus, & </s> <s xml:id="echoid-s1828" xml:space="preserve">circa eoſdem axes; </s> <s xml:id="echoid-s1829" xml:space="preserve">ducantur ergo in ipſis plana per axes, qui <lb/>ſint, EO, MR, quoniam ergo latera cylindrorum ſunt ſuis axibus <lb/>parallela, ideò dicta plana tranſibunt per latera cylindrorum, ſiue <lb/> <anchor type="note" xlink:label="note-0091-02a" xlink:href="note-0091-02"/> cylindricorum, AC, FN, & </s> <s xml:id="echoid-s1830" xml:space="preserve">per latera conorum, ſiue conicorum, <lb/>EBC, MGN, quia per eorum vertices intra ipſos ducuntur, ſint <lb/>autem dicta plana ea, quę ſint ad rectos angulos baſibus, quorum & </s> <s xml:id="echoid-s1831" xml:space="preserve"><lb/>baſium communes ſectiones, quæ ſint, BC, GN, cum axibus æ-<lb/>quales angulos continentes eandem interſe, quam axes proportio-<lb/>nem habeant, vt fert definitio, fient igitur in cylindricis parallelo-<lb/>gramma, vt, AC, FN, & </s> <s xml:id="echoid-s1832" xml:space="preserve">in conicis triangula, vt, BEC, GMN, <lb/> <anchor type="note" xlink:label="note-0091-03a" xlink:href="note-0091-03"/> & </s> <s xml:id="echoid-s1833" xml:space="preserve">quia anguli, BOE, GRM, ſunt ęquales, ideò etiam ipſi, BCD, <lb/>GNH, ſunt æquales, & </s> <s xml:id="echoid-s1834" xml:space="preserve">eſt, BC, ad, CD, vt, GN, ad, NH, <lb/>ideò parallelogramma, AC, FN, & </s> <s xml:id="echoid-s1835" xml:space="preserve">triangula, BEC, GMN, e-<lb/>runt ſimilia iuxta definitionem Euclidis, & </s> <s xml:id="echoid-s1836" xml:space="preserve">ideò etiam iuxta meam, <lb/>& </s> <s xml:id="echoid-s1837" xml:space="preserve">quia ipſæ, AD, BC, FH, GN, tangunt figuras, AC, FN, <lb/> <anchor type="note" xlink:label="note-0091-04a" xlink:href="note-0091-04"/> <pb o="72" file="0092" n="92" rhead="GEOMETRIÆ"/> quibus incidunt ad eundem angulum ex eadem parte, EO, MR, & </s> <s xml:id="echoid-s1838" xml:space="preserve"><lb/>quę diuidunt ipſas, EO, MR, ſimiliter ad eandem partem exiſten-<lb/>tes parallelæ ipſis, BC, GN, ſunt vtipſæ, EO, MR, ad eandem <lb/>partem eodem ordine inter ipſas, & </s> <s xml:id="echoid-s1839" xml:space="preserve">circuitum dictarum figurarum <lb/>compræhenſæ, quia quæ ſunt ex vna parte ſunt æquales ipſis, BO, <lb/>GR, & </s> <s xml:id="echoid-s1840" xml:space="preserve">quæ ex alia ipſis, OC, RN, in triangulis autem ſunt, vt <lb/>ipſæ, BO, GR, vel, OC, RN, .</s> <s xml:id="echoid-s1841" xml:space="preserve">i. </s> <s xml:id="echoid-s1842" xml:space="preserve">vt, OE, RM, & </s> <s xml:id="echoid-s1843" xml:space="preserve">ideo, earum <lb/> <anchor type="note" xlink:label="note-0092-01a" xlink:href="note-0092-01"/> incidentes, & </s> <s xml:id="echoid-s1844" xml:space="preserve">oppoſitarum tangentium dictarum erunt ipſæ, EO, <lb/>MR, quę tangentes ſunt regulæ homologarum ſimilium figurarum, <lb/>AC, FN, vel, EBC, MGN. </s> <s xml:id="echoid-s1845" xml:space="preserve">Vlterius, quia, BXC, GYN, <lb/>ſunt ſemicirculi, erunt figurę planę ſimiles iuxta meam definitionem, <lb/>quarum & </s> <s xml:id="echoid-s1846" xml:space="preserve">tangentium, quæ per extrema, BC, GN, ducuntur e-<lb/> <anchor type="note" xlink:label="note-0092-02a" xlink:href="note-0092-02"/> runt incidentes ipſi diametri, BC, GN, vt probatum fuit, veluti <lb/>idem patet de ſemicirculis, B ℟ C, GZN, & </s> <s xml:id="echoid-s1847" xml:space="preserve">de quibuſcumq; </s> <s xml:id="echoid-s1848" xml:space="preserve">alijs, <lb/>quæ diuident ipſas, EO, MR, ſimiliter ad eandem partem, & </s> <s xml:id="echoid-s1849" xml:space="preserve">con-<lb/>ſequenter diuidunt etiam altitudines eorũdem reſpectu baſium ſum-<lb/> <anchor type="note" xlink:label="note-0092-03a" xlink:href="note-0092-03"/> ptas ſimiliter ad eandem partem, & </s> <s xml:id="echoid-s1850" xml:space="preserve">deijs, quæ per extrema, E, M, <lb/>ducuntur, habemus igitur cylindros, AC, FN, ſiue conos, BEC, <lb/>GMN, quorum ducta ſunt plana oppoſita tangentia dictorum ſo-<lb/>lidorum homologis figuris parallela, quæ ſunt plana, B ℟ CX, A <lb/>D; </s> <s xml:id="echoid-s1851" xml:space="preserve">GYNZ, FH, quibus inciderunt duo plana ad ęquales angulos <lb/>ex eadem parte, illa nempè, in quibus ſunt ipſa parallelogramma, <lb/>AC, FN, vel triangula, BEC, quia ſunt recta ad baſes .</s> <s xml:id="echoid-s1852" xml:space="preserve">i. </s> <s xml:id="echoid-s1853" xml:space="preserve">ad dicta <lb/>tangentia, ipſæ autem ſiguræ .</s> <s xml:id="echoid-s1854" xml:space="preserve">i. </s> <s xml:id="echoid-s1855" xml:space="preserve">parallelogramma, vel triangula in-<lb/>uenta ſunt eſſe ſimilia, quarum homologarum regulæ oppoſitę tan-<lb/>gentes, AD, BC; </s> <s xml:id="echoid-s1856" xml:space="preserve">FH, GN, quarum ſunt incidentes, EO, MR, <lb/>earum autem lineæ homologæ, ſumptæ regulis dictis tangentibus, <lb/>repertæ ſunt eſſe incidentes figurarum planarum ſimilium, quæ di-<lb/>uidunt altitudines dictorum ſolidorum iam dictas ſimiliter ad ean-<lb/>dem partem, & </s> <s xml:id="echoid-s1857" xml:space="preserve">oppoſitarum tangentium, quæ omnes ijs, quæ du-<lb/>cuntur per extrema, BC, GN, tangentes circulos, B ℟ CX, GY <lb/>NZ, ſunt ęquidiſtantes, vt facilè conſideranti patebit, ergo cylin-<lb/>dri, AC, FN, vel coni, BEC, GMN, ſunt ſimiles iuxta meam defi-<lb/> <anchor type="note" xlink:label="note-0092-04a" xlink:href="note-0092-04"/> nitionem generalem ſimilium ſolidorum, quod oſtendere opus erat.</s> <s xml:id="echoid-s1858" xml:space="preserve"/> </p> <div xml:id="echoid-div182" type="float" level="2" n="1"> <note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">Defin. 11.</note> <note position="right" xlink:label="note-0091-02" xlink:href="note-0091-02a" xml:space="preserve">Ex. def. 3. <lb/>& 4. Cor.</note> <note position="right" xlink:label="note-0091-03" xlink:href="note-0091-03a" xml:space="preserve">Ex Cor. 5. <lb/>& ex 16. <lb/>huius.</note> <note position="right" xlink:label="note-0091-04" xlink:href="note-0091-04a" xml:space="preserve">27. huius.</note> <note position="left" xlink:label="note-0092-01" xlink:href="note-0092-01a" xml:space="preserve">B. defin. <lb/>10.</note> <note position="left" xlink:label="note-0092-02" xlink:href="note-0092-02a" xml:space="preserve">Ex Lem. <lb/>ant.</note> <note position="left" xlink:label="note-0092-03" xlink:href="note-0092-03a" xml:space="preserve">17. Vnde-<lb/>cimi El.</note> <note position="left" xlink:label="note-0092-04" xlink:href="note-0092-04a" xml:space="preserve">Defin. 11.</note> </div> </div> <div xml:id="echoid-div184" type="section" level="1" n="120"> <head xml:id="echoid-head131" xml:space="preserve">THEOREMA XXIX. PROPOS. XXXII.</head> <p> <s xml:id="echoid-s1859" xml:space="preserve">DEfinitio mea ſimilium conicorum, & </s> <s xml:id="echoid-s1860" xml:space="preserve">cylindricorum <lb/>concordat cum definitione generali ſimilium ſolido-<lb/>rum.</s> <s xml:id="echoid-s1861" xml:space="preserve"/> </p> <pb o="73" file="0093" n="93" rhead="LIBERI."/> <p> <s xml:id="echoid-s1862" xml:space="preserve">Sint cylindrici quicunque, AH, KY; </s> <s xml:id="echoid-s1863" xml:space="preserve">ſeu conici in ijſdem baſibus, <lb/>& </s> <s xml:id="echoid-s1864" xml:space="preserve">altitudinibus (vt vna vice vtriuſq; </s> <s xml:id="echoid-s1865" xml:space="preserve">demonſtrationem abſoluamus) <lb/>NLH, VXY, ſimiles iuxta definit. </s> <s xml:id="echoid-s1866" xml:space="preserve">7. </s> <s xml:id="echoid-s1867" xml:space="preserve">huius. </s> <s xml:id="echoid-s1868" xml:space="preserve">Dico eoſdem etiam <lb/>eſſe ſimiles iuxta definit. </s> <s xml:id="echoid-s1869" xml:space="preserve">11. </s> <s xml:id="echoid-s1870" xml:space="preserve">Quoniam ergo vtraque prædicta ſolida <lb/> <anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/> ſunt ſimilia, erunt baſes, LH, XY, ſimiles, ducantur earum oppo-<lb/>ſitę tangentes, quę ſint homologarum regulę, ipſę, LD, HG, X f, <lb/> <anchor type="note" xlink:label="note-0093-02a" xlink:href="note-0093-02"/> <anchor type="figure" xlink:label="fig-0093-01a" xlink:href="fig-0093-01"/> Y l, quarum, & </s> <s xml:id="echoid-s1871" xml:space="preserve">prædictarum ſimi-<lb/> <anchor type="note" xlink:label="note-0093-03a" xlink:href="note-0093-03"/> lium figurarum incidentes ſint ipſæ, <lb/>DG, f l, quæ etiam pro regulis alia-<lb/> <anchor type="note" xlink:label="note-0093-04a" xlink:href="note-0093-04"/> rum homologarum ſumi poterunt, <lb/>ſint ergo duę quæcunque homologę <lb/>parallelę incidentibus, D G, fl, ip-<lb/>ſæ, LH, XY, ſi ergo per has, & </s> <s xml:id="echoid-s1872" xml:space="preserve">la-<lb/> <anchor type="note" xlink:label="note-0093-05a" xlink:href="note-0093-05"/> tera cylindricorum, vel conicorum <lb/>iam dictorum extendantur plana, ab <lb/>ijs producentur in cylindricis ſimilia <lb/>parallelogramma, & </s> <s xml:id="echoid-s1873" xml:space="preserve">in conicis ſimi-<lb/>lia triangula, quę etiam erunt ad ba-<lb/>ſes æquè ad eandem partem inclina-<lb/>ta. </s> <s xml:id="echoid-s1874" xml:space="preserve">Extendantur ergo per oppoſitas <lb/>tangentes, LD, HG; </s> <s xml:id="echoid-s1875" xml:space="preserve">Xf, Yl, pla-<lb/>na tangentia tam cylindricos, quam <lb/>conicos iam dictos, & </s> <s xml:id="echoid-s1876" xml:space="preserve">hęc ſimul cum <lb/>planis baſium indefinitè producan-<lb/>tur ad partes incidentium, DG, fl, <lb/>& </s> <s xml:id="echoid-s1877" xml:space="preserve">tandem per, DG, fl, cum ſint <lb/>parallelæ, extendantur plana ipſis, <lb/>AH, KY, parallela ſecantia iam pro-<lb/>ducta plana in rectis, DG, GE, E <lb/>B, BD, DE, fl, l &</s> <s xml:id="echoid-s1878" xml:space="preserve">, & </s> <s xml:id="echoid-s1879" xml:space="preserve">Z, Zf, f <lb/> <anchor type="note" xlink:label="note-0093-06a" xlink:href="note-0093-06"/> &</s> <s xml:id="echoid-s1880" xml:space="preserve">, erunt ergo parallelepipeda, AG, <lb/>Kl, & </s> <s xml:id="echoid-s1881" xml:space="preserve">priſmata, LNGD, XVlf, <lb/> <anchor type="note" xlink:label="note-0093-07a" xlink:href="note-0093-07"/> ergo erit parallelogrammum, BG, <lb/>ſimile ipſi, AH, &</s> <s xml:id="echoid-s1882" xml:space="preserve">, Zl, ſimile, K <lb/>Y, quæ cum ſint inter ſe ſimilia, e-<lb/>tiam, BG, Zl, erunt ſimilia, ſic e-<lb/>tiam oſtendemus triangula, EDG, <lb/>& </s> <s xml:id="echoid-s1883" xml:space="preserve">fl, eſſe ſimilia, ſub intellige iuxta <lb/>definitionem Euclidis, ergo erunt e-<lb/>tiam ſimilia iuxta defin. </s> <s xml:id="echoid-s1884" xml:space="preserve">10. </s> <s xml:id="echoid-s1885" xml:space="preserve">Ducantur duo plana oppoſitis tangenti-<lb/> <anchor type="note" xlink:label="note-0093-08a" xlink:href="note-0093-08"/> bus intermedia, ac parallela, altitudines dictorum ſolidorum reipectu <lb/>baſium, LH, XY, ſumptas, ſimiliter ad eandem partem diuidentia, <pb o="74" file="0094" n="94" rhead="GEOMETRIÆ"/> quæ in cylindricis producant figuras, IM, RT, in conicis verò, O <lb/>M, ST, ſecent verò plana tangentia in rectis, IC, MF, Od; </s> <s xml:id="echoid-s1886" xml:space="preserve">r ℟, <lb/>Tp, So, iſtæ ergo erunt ad inuicem parallelæ, & </s> <s xml:id="echoid-s1887" xml:space="preserve">tangent figuras, <lb/> <anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/> IM, RT, OM, ST, eadem verò planaſecent plana, BG, Zl, in <lb/> <anchor type="note" xlink:label="note-0094-02a" xlink:href="note-0094-02"/> <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> rectis, CF, ℟ p. </s> <s xml:id="echoid-s1888" xml:space="preserve">Quod ergo figuræ, <lb/> <anchor type="note" xlink:label="note-0094-03a" xlink:href="note-0094-03"/> IM, RT, vel, OM, ST, ſint ſimi-<lb/>les baſibus, & </s> <s xml:id="echoid-s1889" xml:space="preserve">ijſdem ſimiliter poſitę <lb/> <anchor type="note" xlink:label="note-0094-04a" xlink:href="note-0094-04"/> iam oſtenſum fuit, ex quo fit, vt & </s> <s xml:id="echoid-s1890" xml:space="preserve"><lb/>ipſarum, & </s> <s xml:id="echoid-s1891" xml:space="preserve">quarumcunq; </s> <s xml:id="echoid-s1892" xml:space="preserve">ſic in prę-<lb/>fatis ſolidis producibilium ſimilium <lb/>figurarum homologæ duabus qui-<lb/>buſdam regulis, vt ex. </s> <s xml:id="echoid-s1893" xml:space="preserve">gr. </s> <s xml:id="echoid-s1894" xml:space="preserve">ipſis, HG, <lb/>Yl, ſemper æquidiſtent. </s> <s xml:id="echoid-s1895" xml:space="preserve">Reliquum <lb/>eſt autem, vt probemus, CF, ℟ p, <lb/>vel, dF, op, eſſe prædictarum in-<lb/>cidentes. </s> <s xml:id="echoid-s1896" xml:space="preserve">Cumergo duę, IC, CF, <lb/>duabus, LD. </s> <s xml:id="echoid-s1897" xml:space="preserve">DG, ęquidiſtentan-<lb/> <anchor type="note" xlink:label="note-0094-05a" xlink:href="note-0094-05"/> guli, ICF, LDG, æquales erunt, <lb/>ſic etiam probabimus eſſe æquales, <lb/>R ℟ p, Xfl, cum verò, IC, ſit e-<lb/>tiam æqualis, LD, & </s> <s xml:id="echoid-s1898" xml:space="preserve">R ℟, ipſi, <lb/>Xf, necnon, CF, ipſi, DG, &</s> <s xml:id="echoid-s1899" xml:space="preserve">, <lb/>℟ p, ipſi, fl, erit, IC, ad, R ℟, vt, <lb/>CF, ad, ℟ p, & </s> <s xml:id="echoid-s1900" xml:space="preserve">incidunt ipſis, IC, <lb/>MF, R ℟, Tp, ad eundem angu-<lb/>lum ex eadem parte, ergo, CF, ℟ <lb/>p, erunt incidentes ſimilium figura-<lb/>rum, IM, RT, & </s> <s xml:id="echoid-s1901" xml:space="preserve">oppoſitarum tan-<lb/> <anchor type="note" xlink:label="note-0094-06a" xlink:href="note-0094-06"/> gentium, IC, MF; </s> <s xml:id="echoid-s1902" xml:space="preserve">R ℟, Tp, ea-<lb/>dem ratione demonſtrabimus, dF, <lb/>op, eſſe incidentes ſimilium figura-<lb/>rum, OM, ST, & </s> <s xml:id="echoid-s1903" xml:space="preserve">oppoſitarum tan-<lb/>gentium, Od, MF; </s> <s xml:id="echoid-s1904" xml:space="preserve">So, Tp, eſt <lb/>autem, dF, ad, op, vt, dE, ad, <lb/>o &</s> <s xml:id="echoid-s1905" xml:space="preserve">, ſcilicet, vt, DE, ad, f &</s> <s xml:id="echoid-s1906" xml:space="preserve">, nam, <lb/>DE, f &</s> <s xml:id="echoid-s1907" xml:space="preserve">, ſunt ſimiliter ad eandem <lb/>partem diuiſæ in punctis, do, (ete-<lb/> <anchor type="note" xlink:label="note-0094-07a" xlink:href="note-0094-07"/> nim altitudines dictorum ſolidorum per plana, IF, Rp, ſimiliter ad <lb/>eandem partem diuiduntur) ergo, dF, op, æquidiſtantes oppoſitis <lb/>tangentibus, BE, DG, Z &</s> <s xml:id="echoid-s1908" xml:space="preserve">, fl, ſunt homologæ figurarum ſimi-<lb/>lium, EDG, & </s> <s xml:id="echoid-s1909" xml:space="preserve">fl, quarum & </s> <s xml:id="echoid-s1910" xml:space="preserve">oppoſitarum tangentium incidentes <lb/>erunt ipſæ, ED, & </s> <s xml:id="echoid-s1911" xml:space="preserve">f. </s> <s xml:id="echoid-s1912" xml:space="preserve">Eodem modo oſtendemus, CF, ℟ p, eſſe <pb o="75" file="0095" n="95" rhead="LIBERI."/> homologas ſimilium figurarum, BG, Zl, quarum & </s> <s xml:id="echoid-s1913" xml:space="preserve">oppoſitarum <lb/>tangentium, BE, DG, Z &</s> <s xml:id="echoid-s1914" xml:space="preserve">, fl, incidentes ſunt ipſæ, BD, Zf, <lb/>hæc autem etiam in cęteris traiectis planis, vt dictum eſt contingere <lb/>oſtendemus, ergo, BG, Zl, EDG, & </s> <s xml:id="echoid-s1915" xml:space="preserve">fl, erunt figurę incidentes <lb/>ſimilium cylindricorum, ſeu conicorum iam dictorum, & </s> <s xml:id="echoid-s1916" xml:space="preserve">oppoſito-<lb/>rum tangentium planorum, AE, LG, K &</s> <s xml:id="echoid-s1917" xml:space="preserve">, XL, ergo in his ſoli-<lb/>dis adſunt omnes conditiones defin. </s> <s xml:id="echoid-s1918" xml:space="preserve">11. </s> <s xml:id="echoid-s1919" xml:space="preserve">vt recolenti eafdem patefiet, <lb/>igitur erunt iuxta eandem pariter ſimilia. </s> <s xml:id="echoid-s1920" xml:space="preserve">Aduerte autem, quod ſup-<lb/>poſui planum, NG, tangere tam cylindricum, quam conicum, vt <lb/>etiam, Vl, ne figura nimis confunderetur, & </s> <s xml:id="echoid-s1921" xml:space="preserve">vt fierent latera, E <lb/>G, & </s> <s xml:id="echoid-s1922" xml:space="preserve">l, communia parallelogrammis, BG, Zl, & </s> <s xml:id="echoid-s1923" xml:space="preserve">triangulis, D <lb/>EG, f & </s> <s xml:id="echoid-s1924" xml:space="preserve">l, valebit tamen eadem demonſtratio etiamſi plana ducta <lb/>per, HG, Yl, tangentia cylindricos, diuerſa ſint à planis per eaſdem, <lb/>HG, Yl, tranſeuntibus, ac tangentibus ipſos conicos, fient enim <lb/>ſemper ſimilia triangula, EDG, & </s> <s xml:id="echoid-s1925" xml:space="preserve">fl, etiamſi non adiaceant late-<lb/>ribus, EG, & </s> <s xml:id="echoid-s1926" xml:space="preserve">l, vt conſideranti facilè patebit, hæc autem nobis o-<lb/>ſtendenda erant.</s> <s xml:id="echoid-s1927" xml:space="preserve"/> </p> <div xml:id="echoid-div184" type="float" level="2" n="1"> <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">Defin. 7.</note> <note position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">Coroll. 1.</note> <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a"> <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0093-01"/> </figure> <note position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">B. Def. 10.</note> <note position="right" xlink:label="note-0093-04" xlink:href="note-0093-04a" xml:space="preserve">Coroll. <lb/>23.</note> <note position="right" xlink:label="note-0093-05" xlink:href="note-0093-05a" xml:space="preserve">Defin. 7.</note> <note position="right" xlink:label="note-0093-06" xlink:href="note-0093-06a" xml:space="preserve">Defin. 13. <lb/>vndec. El.</note> <note position="right" xlink:label="note-0093-07" xlink:href="note-0093-07a" xml:space="preserve">24. Vnd. <lb/>Elem.</note> <note position="right" xlink:label="note-0093-08" xlink:href="note-0093-08a" xml:space="preserve">27. huius.</note> <note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">16. Vnd. <lb/>Elem.</note> <note position="left" xlink:label="note-0094-02" xlink:href="note-0094-02a" xml:space="preserve">Corol. 9.</note> <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a"> <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0094-01"/> </figure> <note position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">Corol. 18.</note> <note position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">@2. Et 19. <lb/>huius.</note> <note position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">10. Vnd. <lb/>Elem.</note> <note position="left" xlink:label="note-0094-06" xlink:href="note-0094-06a" xml:space="preserve">@4. huius.</note> <note position="left" xlink:label="note-0094-07" xlink:href="note-0094-07a" xml:space="preserve">@7. Vnd. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div186" type="section" level="1" n="121"> <head xml:id="echoid-head132" xml:space="preserve">THEOREMA XXX. PROPOS. XXXIII.</head> <p> <s xml:id="echoid-s1928" xml:space="preserve">SI ſolidum rotundum ſecetur plano per axem, producta in <lb/>eo figura erit, quæ per reuolutionem ipſum genuit.</s> <s xml:id="echoid-s1929" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1930" xml:space="preserve">Sit ſolidum rotundum, cuius axis, AM, baſis circulus, HDEF, <lb/> <anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> hoc autem plano per axem, AM, ducto ſecetur, quod in eo produ-<lb/>cat figuram, ACDFG. </s> <s xml:id="echoid-s1931" xml:space="preserve">Dico hanc <lb/>eſſe eam, quæ per reuolutionem ip-<lb/> <anchor type="figure" xlink:label="fig-0095-01a" xlink:href="fig-0095-01"/> ſum ſolidum genuit. </s> <s xml:id="echoid-s1932" xml:space="preserve">Intelligatur re-<lb/>uolui circa, AM, figura, quę dictum <lb/>ſolidum genuit, donec reperiatur po-<lb/>ſita in plano figuræ, ACDFG, igi-<lb/>tur vel harum figurarum perimetri <lb/>congruunt, vel non, ſi ſic ex illis fa-<lb/>cta erit vna figura, ea nempè, quæ <lb/>per reuolutionem generat dictum ſo-<lb/>lidum, ſi verò non congruant, ali-<lb/>quis punctus alterius ambituum di-<lb/>ctarum figurarum non reperietur in <lb/>reliquæ ambitu, ſit is punctus, B, qui <lb/>reperiatur in ambitu figuræ, quæ per reuolutionem dictum ſolidum <lb/>deſcripſit, quæ ſit ipſa, ABDFG, & </s> <s xml:id="echoid-s1933" xml:space="preserve">non in ambitu figurę, ACD <lb/>FG, cuius ambitus eſt communis ſectio plani ducti per axem, & </s> <s xml:id="echoid-s1934" xml:space="preserve">ſu- <pb o="76" file="0096" n="96" rhead="GEOMETRIÆ"/> perſiciei dictum ſolidum ambientis, quia gitur, B, non eſt in commu-<lb/>ni ſectione iam dicta, & </s> <s xml:id="echoid-s1935" xml:space="preserve">eſt in plano figuræ, ACDFG, igitur erit <lb/>intra, vel extra ſuperficiem ambientem dictum ſolidum, eſt autem in <lb/>ambitu figuræ, quæ tali ambitu dictam ſuperficiem deſcribit, ergo <lb/>erit in ipſa ſuperficie ambiente, & </s> <s xml:id="echoid-s1936" xml:space="preserve">non erit, quod eſt abſurdum, non <lb/>igitur aliquis punctus ambitus figuræ, quæ dictam ſolidum per reuo-<lb/>lutionem generat eſt extra ambitum figuræ, ACDFG, igitur iſti <lb/>ambitus, & </s> <s xml:id="echoid-s1937" xml:space="preserve">conſequenter ipſæ figuræ ſibi inuicem congruunt, & </s> <s xml:id="echoid-s1938" xml:space="preserve">fit <lb/>vna figura, ea ſcilicet, quæ per reuolutionem dictum ſolidum rotun-<lb/>dum generat, quod erat demonſtrandum.</s> <s xml:id="echoid-s1939" xml:space="preserve"/> </p> <div xml:id="echoid-div186" type="float" level="2" n="1"> <note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">Defin. 6.</note> <figure xlink:label="fig-0095-01" xlink:href="fig-0095-01a"> <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0095-01"/> </figure> </div> </div> <div xml:id="echoid-div188" type="section" level="1" n="122"> <head xml:id="echoid-head133" xml:space="preserve">THEOREMA XXXI. PROPOS. XXXIV.</head> <p> <s xml:id="echoid-s1940" xml:space="preserve">SI ſolidum rotũdum ſecetur plano ad axem recto, fiet con-<lb/>cepta in ipſo figura circulus, cuius centrum erit in axe.</s> <s xml:id="echoid-s1941" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1942" xml:space="preserve">Sit ſolidum rotundum, cuius axis, AC, & </s> <s xml:id="echoid-s1943" xml:space="preserve">figura, quæ ipſum per <lb/>reuolut onem genuit ipſa, ABCD, ſecetur autem plano ad axem <lb/>recto, ex quo in ipſo producatur figura, MBND. </s> <s xml:id="echoid-s1944" xml:space="preserve">Dico hanc eſſe <lb/>circulum, cuius centrum erit in axe, vt, E, ſit autem communis ſe-<lb/>ctio plani recti ad axem, & </s> <s xml:id="echoid-s1945" xml:space="preserve">figuræ, ABCD, recta, BD, quia er-<lb/>go figura, ABCD, eſt circa axem, ipſa autem, BD, quæ rectè a-<lb/> <anchor type="note" xlink:label="note-0096-01a" xlink:href="note-0096-01"/> xim ſecat, vna eſt ex ordinatim ad ipſam <lb/>axim applicatis, ideò ab ea bifariam diui-<lb/> <anchor type="figure" xlink:label="fig-0096-01a" xlink:href="fig-0096-01"/> ditur in puncto, E, ducatur nunc aliud pla-<lb/>num per axem, quod in dicto ſolido pro-<lb/>ducat figuram, AMCN, quæ ſecet figu-<lb/> <anchor type="note" xlink:label="note-0096-02a" xlink:href="note-0096-02"/> ram, MBND, in recta, MN, erit ergo <lb/>hæc figura eadem ei, quæ per reuolutio-<lb/>nem dictum genuit ſolidum, & </s> <s xml:id="echoid-s1946" xml:space="preserve">ideò erit fi-<lb/>gura circa axem, ad quam ordinatim ap-<lb/>plicatur, MN, cum ipſa rectè axem, AC, <lb/>diu dat, ergo, MN, bifariam diuiditur in, <lb/>E, eodem pacto quaſcumq; </s> <s xml:id="echoid-s1947" xml:space="preserve">alias commu-<lb/>nes ſectiones figurarum per axem, AC, <lb/>tranſeuntium, & </s> <s xml:id="echoid-s1948" xml:space="preserve">figurę, BNDM, oſten-<lb/>demus bifariam diuidi in, E. </s> <s xml:id="echoid-s1949" xml:space="preserve">Vlterius, quia figuræ, ABCD, AM <lb/>CN, ſunt eædem illi, quæ per reuolutionem generat ſolidum, AB <lb/>CD, &</s> <s xml:id="echoid-s1950" xml:space="preserve">, BD, MN, tranſeunt per idem punctum axis, AC, rectè <lb/>eundem ſecantes, ideo ſi ipſa, AMCN, reuolueretur, donec eſſet <lb/>in plano figurę, ABCD, illi congrueret, &</s> <s xml:id="echoid-s1951" xml:space="preserve">, MN, ipſi, BD, vn-<lb/>de, MN, BD, ſunt æquales, & </s> <s xml:id="echoid-s1952" xml:space="preserve">ideò earum dimidię, NE, EB; </s> <s xml:id="echoid-s1953" xml:space="preserve">M <pb o="77" file="0097" n="97" rhead="LIBER I."/> E, ED, erunt æquales, eodem pacto oſtendemus quaſcumque du-<lb/>ctas à puncto, E, ad lineam ambientem, MBND, eſſe æquales <lb/>cuilibet ipſarum, BE, EN, ED, EM, ergo figura, MBND, erit <lb/>circulus, cuius centrum, E, in axe reperitur, quod erat oſtendendum.</s> <s xml:id="echoid-s1954" xml:space="preserve"/> </p> <div xml:id="echoid-div188" type="float" level="2" n="1"> <note position="left" xlink:label="note-0096-01" xlink:href="note-0096-01a" xml:space="preserve">Defin. 6.</note> <figure xlink:label="fig-0096-01" xlink:href="fig-0096-01a"> <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0096-01"/> </figure> <note position="left" xlink:label="note-0096-02" xlink:href="note-0096-02a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div190" type="section" level="1" n="123"> <head xml:id="echoid-head134" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1955" xml:space="preserve">_C_olligimus autem ipſas, BD, MN, communes ſectiones figurã-<lb/>rum per axem ductarum, & </s> <s xml:id="echoid-s1956" xml:space="preserve">circulorum, qui per ſectionem dicti <lb/>ſolidi per plana ad axem recta in eo produsuntur, eſſe eorum diametros, <lb/>cum per centrum tranſeant.</s> <s xml:id="echoid-s1957" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div191" type="section" level="1" n="124"> <head xml:id="echoid-head135" xml:space="preserve">THEOREMA XXXII. PROPOS. XXXV.</head> <p> <s xml:id="echoid-s1958" xml:space="preserve">SI quicunq; </s> <s xml:id="echoid-s1959" xml:space="preserve">conus ſecetur plano baſi æquidiſtante conce-<lb/>pta in cono figura erit circulus centrum in axe habens.</s> <s xml:id="echoid-s1960" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1961" xml:space="preserve">Si conus ſit rectus patet hoc ex antecedenti Propoſ. </s> <s xml:id="echoid-s1962" xml:space="preserve">cæterum ſi ſit <lb/>ſcalenus, qualis ſit conus, ACFD, qui ſecetur plano baſi, CFD, <lb/>æquidiſtante, quod in eo producat figuram, BRE. </s> <s xml:id="echoid-s1963" xml:space="preserve">Dico ipſam eſſe <lb/>circulum, centrum in axe habentem. </s> <s xml:id="echoid-s1964" xml:space="preserve">Secetur ergo plano per axem, <lb/> <anchor type="note" xlink:label="note-0097-01a" xlink:href="note-0097-01"/> quod in eo producat triangulum, ACD, cuius & </s> <s xml:id="echoid-s1965" xml:space="preserve">circuli, CFD, <lb/>communis ſectio ſit, CD, quę erit diameter dicti circuli; </s> <s xml:id="echoid-s1966" xml:space="preserve">eius autem <lb/>& </s> <s xml:id="echoid-s1967" xml:space="preserve">figuræ, BRE, communis ſectio, BE; </s> <s xml:id="echoid-s1968" xml:space="preserve">ſunt igitur trianguli, AB <lb/> <anchor type="note" xlink:label="note-0097-02a" xlink:href="note-0097-02"/> l, ACN, ſimiles, quia, BI, ęquidiſtat ipſi, C <lb/>N, ergo, CN, ad, NA, erit vt, BI, ad, IA, <lb/> <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/> eodem modo oſtendemus, AN, ad, ND, eſſe <lb/>vt, BI, ad, IE, ergo, ex æquo, CN, ad, N <lb/>D, erit vt, BI, ad, IE, ſed, CN, eſt ęqualis, <lb/>ND, ergo &</s> <s xml:id="echoid-s1969" xml:space="preserve">, BI, ipſi, IE. </s> <s xml:id="echoid-s1970" xml:space="preserve">Ducatur nunc <lb/>aliud planum per axem, quod producat trian-<lb/>gulum, ANF, quodq; </s> <s xml:id="echoid-s1971" xml:space="preserve">ſecet figuram, BRE, <lb/>in, IR, fient ergo trianguli, AIR, ANF, æ-<lb/>quianguli, ergo, FN, NA, NC, erunt lineæ <lb/>in eadem proportione cum ipſis, RI, IA, IB, <lb/>ergo, ex ęquo, FN, ad, NC, erit vt, RI, ad, <lb/>IB, ſed, FN, eſt æqualis ipſi, NC, ergo, R <lb/>I, erit æqualis ipſi, IB, eodem modo oſtende <lb/>mus quaſcunque ductas à puncto, I, ad lineam ambientem, BRE, <lb/>eſſe æquales ipſi, BI, ergo figura, BRE, erit circulus, cuius, cen-<lb/>trum, I, quod oſtendere oportebat.</s> <s xml:id="echoid-s1972" xml:space="preserve"/> </p> <div xml:id="echoid-div191" type="float" level="2" n="1"> <note position="right" xlink:label="note-0097-01" xlink:href="note-0097-01a" xml:space="preserve">16. huius.</note> <note position="right" xlink:label="note-0097-02" xlink:href="note-0097-02a" xml:space="preserve">4. Sex. El.</note> <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a"> <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0097-01"/> </figure> </div> <pb o="78" file="0098" n="98" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div193" type="section" level="1" n="125"> <head xml:id="echoid-head136" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s1973" xml:space="preserve">_H_Inc patet ipſam, BE, communem ſectionem trianguli por axem <lb/>ducti, & </s> <s xml:id="echoid-s1974" xml:space="preserve">circuli, BRE, eſſe eiuſdem diametrum, cum per eius <lb/>centrum tranſedt.</s> <s xml:id="echoid-s1975" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div194" type="section" level="1" n="126"> <head xml:id="echoid-head137" xml:space="preserve">THEOREMA XXXIII. PROPOS. XXXVI.</head> <p> <s xml:id="echoid-s1976" xml:space="preserve">SI ſolidum rotundum, vel conus ſcalenus ſecentur plano <lb/>per axem, deinde ſecetur ſolidum rotundum (niſi baſim <lb/>habeat, quę circulus erit) plano ad axem recto circulum pro-<lb/>ducente, in cuius plano, & </s> <s xml:id="echoid-s1977" xml:space="preserve">illius, qui eſt coni baſis perpen-<lb/>dicularis ducta ſit baſi figurę per axim ductę; </s> <s xml:id="echoid-s1978" xml:space="preserve">deinde ſumpto <lb/>puncto in ambitu figuræ per axem, per illum æquidiſtans di-<lb/>ctæ perpendiculari ducta fueritrecta linea, hæc tanget dicta <lb/>ſolida, at ſi ſumptus punctus ſit extra talem ambitum, ſed in <lb/>ſuperficie ambiente dicta ſolida, quæ per ipſum ducitur ei-<lb/>dem æquidiſtans intra dicta ſolida cadet, & </s> <s xml:id="echoid-s1979" xml:space="preserve">producta vſque <lb/>ad ſuperficiem ambientem à figura ducta per axem bifariam <lb/>diuidetur.</s> <s xml:id="echoid-s1980" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1981" xml:space="preserve">Sit ſolidum rotundum, ABTF, vel conus ſcalenus, APR, in <lb/>baſi circulo, PXRZ, quorum axis, AT, & </s> <s xml:id="echoid-s1982" xml:space="preserve">ſi ſolidum rotundum <lb/>non habeat baſim, ſe-<lb/> <anchor type="figure" xlink:label="fig-0098-01a" xlink:href="fig-0098-01"/> cetur plano recto ad <lb/>axem, quod in eo pro-<lb/>ducat circulum, PXR <lb/>Z, ſecentur autem am-<lb/>bo planis per axem-<lb/>quæ producant in ſo-<lb/>lido rotundo figuram, <lb/>APTF, & </s> <s xml:id="echoid-s1983" xml:space="preserve">in cono <lb/>triangulum, APR, <lb/>deinde in plano circu-<lb/>li, PZRX, ducatur <lb/>ipſi, PR, communi <lb/>ſectioni dicti circuli, & </s> <s xml:id="echoid-s1984" xml:space="preserve"><lb/>figuræ per axem, perpendicularis, ZX, & </s> <s xml:id="echoid-s1985" xml:space="preserve">ſumpto puncto in ambitu <lb/>figurę per axem, vt, 2, per ipſum ducatur recta linea parallela ipſi, <pb o="79" file="0099" n="99" rhead="LIBERI."/> ZX. </s> <s xml:id="echoid-s1986" xml:space="preserve">Dico hanc tangere dicta ſolida, ſi enim non tangit ſecet, ve-<lb/>luti, D2N, in puncto, N, igitur punctus. </s> <s xml:id="echoid-s1987" xml:space="preserve">n. </s> <s xml:id="echoid-s1988" xml:space="preserve">erit extra planum figu-<lb/>ræper axem, nam ipſa, D2N, eſt parallela ipſi, ZX, quæ eſt ad <lb/>rectos angulos figuræ per axem tranſeunti, & </s> <s xml:id="echoid-s1989" xml:space="preserve">ideò etiam, D2N, <lb/> <anchor type="note" xlink:label="note-0099-01a" xlink:href="note-0099-01"/> eſt illi ad rectos angulos, occurrit autem illi in puncto, 2, ergo non <lb/>occurret illi in alio puncto, ergo, N, eſt extra planum figuræ per a-<lb/>xem, ducatur per, N, planum æquidiſtans plano, PXRZ, circuli, <lb/> <anchor type="note" xlink:label="note-0099-02a" xlink:href="note-0099-02"/> quod producat circulum, BNFC, & </s> <s xml:id="echoid-s1990" xml:space="preserve">ſit, BF, communisſectio ip-<lb/>ſius circuli, & </s> <s xml:id="echoid-s1991" xml:space="preserve">figuræ per axem, quæ erit ipſius circuli diameter, &</s> <s xml:id="echoid-s1992" xml:space="preserve">, <lb/> <anchor type="note" xlink:label="note-0099-03a" xlink:href="note-0099-03"/> N, non erit aliquis punctorum, BF, ergo ſi ab, N, duxerimus ipſi, <lb/>ZX, parallelam, vt, NC, cum etiam, BF, ſit parallela ipſi, PR, <lb/>continebunt angulos æquales, ſed, ZX, ſecat perpendiculariter, P <lb/> <anchor type="note" xlink:label="note-0099-04a" xlink:href="note-0099-04"/> R, ergo, NC, ſecabit perpendiculariter, BF, ducta non ab extre-<lb/>mitate diametri, ergo intra circulum, BCFN, erit, & </s> <s xml:id="echoid-s1993" xml:space="preserve">bifariam ſe-<lb/>cabitur ab ipſa, BF, ergo non tranſibit per circuitum figuræ per a-<lb/>xem ductæ, & </s> <s xml:id="echoid-s1994" xml:space="preserve">per ipſum tranſit, D2N, ergo, NC, N2D, ſunt <lb/>duæ rectæ lineæ eidem, ZX, parallelę, ergo etiam inter ſe erunt pa-<lb/>rallelæ, quod eſt abſurdum, cum tranſeant per idem punctum, N, <lb/>ergo ducta per punctum ambitus figuræ per axem parallela ipſi, ZX, <lb/>tanget dicta ſolida: </s> <s xml:id="echoid-s1995" xml:space="preserve">Sit nobis nunc punctus, N, pro puncto vtcung; <lb/></s> <s xml:id="echoid-s1996" xml:space="preserve">in ſuperficie ambiente ſumpto extra circuitum figuræ per axem, à <lb/>quo ducta ipſi, ZX, parallela, occurrat producta ſuperficiei ambienti <lb/>in puncto, C, oſtendemus ergo eodem modo ſupra adhibito (poſt-<lb/>quam duxerimus per, N, planum circulo, PXRZ, æquidiſtans, <lb/>quod in ſolido producat circulum, BNFC,) ipſam, NC, intra cir-<lb/>culum, BNFC, cadere, & </s> <s xml:id="echoid-s1997" xml:space="preserve">bifariam diuidi à recta, BF, ſiue à figu@a <lb/>per axem ducta (nam eſt, NC, perpendicularis ipſi, BF,) quod o-<lb/>ſtendere opus erat.</s> <s xml:id="echoid-s1998" xml:space="preserve"/> </p> <div xml:id="echoid-div194" type="float" level="2" n="1"> <figure xlink:label="fig-0098-01" xlink:href="fig-0098-01a"> <image file="0098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0098-01"/> </figure> <note position="right" xlink:label="note-0099-01" xlink:href="note-0099-01a" xml:space="preserve">8. Vndec. <lb/>Elem.</note> <note position="right" xlink:label="note-0099-02" xlink:href="note-0099-02a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-0099-03" xlink:href="note-0099-03a" xml:space="preserve">Corol. 34 <lb/>huius.</note> <note position="right" xlink:label="note-0099-04" xlink:href="note-0099-04a" xml:space="preserve">10. Vnd. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div196" type="section" level="1" n="127"> <head xml:id="echoid-head138" xml:space="preserve">THEOREMA XXXIV. PROPOS. XXXVII.</head> <p> <s xml:id="echoid-s1999" xml:space="preserve">SI ſolidum rotundum, vel conus ſcalenus, ſecentur plano <lb/>per axem, & </s> <s xml:id="echoid-s2000" xml:space="preserve">deinde alio plano ſecentur, cuius, & </s> <s xml:id="echoid-s2001" xml:space="preserve">vnius <lb/>planorum rectè axem ſecantium communis ſectio ſit recta li-<lb/>nea perpendicularis communi ſectioni eiuſdem, & </s> <s xml:id="echoid-s2002" xml:space="preserve">plani per <lb/>axem; </s> <s xml:id="echoid-s2003" xml:space="preserve">figura à ſecundo ſecante plano in ſolido producta erit <lb/>circa axem, in cono ſcaleno autem erit circa axem, vel dia-<lb/>metrum, & </s> <s xml:id="echoid-s2004" xml:space="preserve">axis, vel diameter erit communis ſectio per dicta <lb/>ſecantia plana productarum figurarum.</s> <s xml:id="echoid-s2005" xml:space="preserve"/> </p> <pb o="80" file="0100" n="100" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s2006" xml:space="preserve">Sit ſolidum rotundum, APCQ, & </s> <s xml:id="echoid-s2007" xml:space="preserve">conusicalenus, APEQM, <lb/>vtraque autem ſecentur plano per axem, quod producat figuram, A <lb/>PCQ, in ſolido, & </s> <s xml:id="echoid-s2008" xml:space="preserve">triangulum, APQ, in cono, deinde ſecentur <lb/>altero plano, cuius, & </s> <s xml:id="echoid-s2009" xml:space="preserve">plani recti ad axem (quo productus ſit circu-<lb/>lus, PMQE,) communis ſectio ſit, EM, perpendicularis ipſi, PQ, <lb/>communi ſectioni eiuſdem, & </s> <s xml:id="echoid-s2010" xml:space="preserve">plani per axem ducti. </s> <s xml:id="echoid-s2011" xml:space="preserve">Dico figuram, <lb/>BEDM, in ſolido rotundo eſſe circa axem, & </s> <s xml:id="echoid-s2012" xml:space="preserve">in cono circa axem, <lb/> <anchor type="note" xlink:label="note-0100-01a" xlink:href="note-0100-01"/> vel diametrum, & </s> <s xml:id="echoid-s2013" xml:space="preserve">axem, vel diametrum eſſe, BD, communem ſe-<lb/>ctionem productarum figurarum. </s> <s xml:id="echoid-s2014" xml:space="preserve">Si ergo ſecundò producta figura <lb/>per axem pariter ducta eſſet, manifeſtum eſt in ſolido rotundo fore <lb/> <anchor type="note" xlink:label="note-0100-02a" xlink:href="note-0100-02"/> figuram talem circa axem, & </s> <s xml:id="echoid-s2015" xml:space="preserve">in cono fore triangulum, in quo axis, <lb/> <anchor type="note" xlink:label="note-0100-03a" xlink:href="note-0100-03"/> AC, ſi ſecaret æquidiſtantes baſi talis trianguli ad angulos rectos, <lb/>cum illas bifariam diuidat, eſſet talis triangulus figura circa axem, ſi <lb/>verò ad angulos non rectos, eſſet figura circa diametrum, nempè <lb/>circa, AC. </s> <s xml:id="echoid-s2016" xml:space="preserve">Sed non tranſeat hęc ſecunda figura per axem, ſint au-<lb/>tem puncta, B, D, extrema communis ſectionis primæ, & </s> <s xml:id="echoid-s2017" xml:space="preserve">ſecundę <lb/>figuræ, ideſt ip-<lb/> <anchor type="figure" xlink:label="fig-0100-01a" xlink:href="fig-0100-01"/> ſius, BD, ergo <lb/>in ſolido rotun-<lb/>do (& </s> <s xml:id="echoid-s2018" xml:space="preserve">in-cono, <lb/>dum triangulus, <lb/>APQ, per axem <lb/>ductus tranſit e-<lb/>tiam per ductam <lb/>à vertice, A, per-<lb/>pẽdicularem ipſi <lb/>baſi, PEQM, <lb/>ideſt cum trian-<lb/>gulus, APQ, eſt <lb/>erectus baſi, PE <lb/>QM,) ipſa, EM, communis ſectio ſecundi plani ſecantis, &</s> <s xml:id="echoid-s2019" xml:space="preserve">, PQ, <lb/> <anchor type="note" xlink:label="note-0100-04a" xlink:href="note-0100-04"/> plani rectè axim ſecantis, cum ſit perpendicularis, PQ, communi ſe-<lb/>ctioni planorum, PEQM, APQ, ad inuicem erectorum, erit etiam <lb/>perpendicularis plano per axem, & </s> <s xml:id="echoid-s2020" xml:space="preserve">ideò erit perpendicularis ad om-<lb/>nes per eam in tali plano tranſeuntes, ideò, BD, rectè ſecabit ipſam, <lb/>EM, & </s> <s xml:id="echoid-s2021" xml:space="preserve">quæ ducuntur per extrema, BD, æquidiſtantes ipſi, EM, <lb/>tangent ipſa ſolida, vnde, B, D, erunt oppoſiti vertices figurarum, <lb/>BEDM, reſpectu ipſius, EM, ſumptarum, quare, BD, ſecabit <lb/> <anchor type="note" xlink:label="note-0100-05a" xlink:href="note-0100-05"/> omnes illi æquidiſtantes in figura, BEDM, ductas, & </s> <s xml:id="echoid-s2022" xml:space="preserve">quia ſumpto <lb/> <anchor type="note" xlink:label="note-0100-06a" xlink:href="note-0100-06"/> in figura, BEDM, puncto, qui non ſit vertex reſpectu ipſius, EM, <lb/>& </s> <s xml:id="echoid-s2023" xml:space="preserve">ab eo ducta eidem, EM, parallela intra figuram cadit, ſit is pun-<lb/> <anchor type="note" xlink:label="note-0100-07a" xlink:href="note-0100-07"/> ctus, O, à quo ipſi, EM, ſit ducta parallela ipſa, OR, igitur, OR, <pb o="81" file="0101" n="101" rhead="LIBER I."/> terminans in ambientem ſuperficiem bifariam diuidetur ab ipſa, BD, <lb/> <anchor type="note" xlink:label="note-0101-01a" xlink:href="note-0101-01"/> vtin, N; </s> <s xml:id="echoid-s2024" xml:space="preserve">Sie oſtendemus, BD, diuidere cæteras omnes ipfi, EM, <lb/>æquidiſtantes in ſuperficiem ambientem hinc inde terminatas, & </s> <s xml:id="echoid-s2025" xml:space="preserve"><lb/>quia, BD, ſecat, EM, adangulos rectos, cæteras omnes iam di-<lb/>ctas bifariam, & </s> <s xml:id="echoid-s2026" xml:space="preserve">ad angulos rectos ſecabit, igitur tunc figura, BED <lb/>M, erit circa axem, BD, ſiue in ſolido rotundo, ſiue in cono: </s> <s xml:id="echoid-s2027" xml:space="preserve">Siau-<lb/>tem triangulus, APQ, non tranſeat per ductam ipſi plano perpen-<lb/>dicularem, tunc eodem modo, quoſupra oſtendemus, BD, ſecare <lb/>omnes ęquidiſtantes ipſi, EM, bifariam, & </s> <s xml:id="echoid-s2028" xml:space="preserve">quia triangulus, APQ, <lb/>non tranſit per perpendicularem baſi, neque erit erectus ipſi baſi, P <lb/>EQM, ergo angulus, EDB, non erit rectus, nam ſi eſſet rectus, <lb/>cum ſit etiam rectus, EDP, planum circuli, PEQM, eſſet erectum <lb/>triangulo, APQ, & </s> <s xml:id="echoid-s2029" xml:space="preserve">ille huic, quod eſt contra ſuppoſitum, igitur, <lb/> <anchor type="note" xlink:label="note-0101-02a" xlink:href="note-0101-02"/> BD, ſecabit, EM, & </s> <s xml:id="echoid-s2030" xml:space="preserve">conſequenter cæteras iam dictas illi æquidi-<lb/>ſtantes bifariam, & </s> <s xml:id="echoid-s2031" xml:space="preserve">ad angulos non rectos, igitur figura, EBM, tunc <lb/>erit circa diametrum, & </s> <s xml:id="echoid-s2032" xml:space="preserve">erit diameter ipſa, BD, ſiue axis, in ſupra-<lb/>dicto caſu tum in cono, tum etiam in ſolido rotundo, quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s2033" xml:space="preserve"/> </p> <div xml:id="echoid-div196" type="float" level="2" n="1"> <note position="left" xlink:label="note-0100-01" xlink:href="note-0100-01a" xml:space="preserve">6. Defin.</note> <note position="left" xlink:label="note-0100-02" xlink:href="note-0100-02a" xml:space="preserve">33. huius.</note> <note position="left" xlink:label="note-0100-03" xlink:href="note-0100-03a" xml:space="preserve">16. huius.</note> <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a"> <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0100-01"/> </figure> <note position="left" xlink:label="note-0100-04" xlink:href="note-0100-04a" xml:space="preserve">4. Defin. <lb/>vndec. El.</note> <note position="left" xlink:label="note-0100-05" xlink:href="note-0100-05a" xml:space="preserve">1. Defin.</note> <note position="left" xlink:label="note-0100-06" xlink:href="note-0100-06a" xml:space="preserve">Corol. 2. <lb/>4. Huius.</note> <note position="left" xlink:label="note-0100-07" xlink:href="note-0100-07a" xml:space="preserve">Coroll. 1. <lb/>4. Huius.</note> <note position="right" xlink:label="note-0101-01" xlink:href="note-0101-01a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0101-02" xlink:href="note-0101-02a" xml:space="preserve">4. Vndec. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div198" type="section" level="1" n="128"> <head xml:id="echoid-head139" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2034" xml:space="preserve">_H_Inc colligitur in cono, ſi triangulus per axem ductus ſit erectus <lb/>baſi, fieri dictam figuram circa axem; </s> <s xml:id="echoid-s2035" xml:space="preserve">ſi verò non ſit erectus, ſed <lb/>inclinatus eidem, fieri figuram circa diametrum; </s> <s xml:id="echoid-s2036" xml:space="preserve">in ſolido rotundo au-<lb/>tem fieri ſemper figuram circa axem.</s> <s xml:id="echoid-s2037" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div199" type="section" level="1" n="129"> <head xml:id="echoid-head140" xml:space="preserve">THEOREMA XXXV. PROPOS. XXXVIII.</head> <p> <s xml:id="echoid-s2038" xml:space="preserve">SI conus ſecetur plano per axem, ſecetur deinde altero pla-<lb/>no ſecante baſim coni ſecundum rectam lineam, quę ad <lb/>baſim trianguli per axem ſit perpendicularis, cuius & </s> <s xml:id="echoid-s2039" xml:space="preserve">trian-<lb/>guli per axem cõmunis ſectio ſit parallela vni laterum trian-<lb/>guli per axem; </s> <s xml:id="echoid-s2040" xml:space="preserve">quadrata ordinatim applicatarum ad axim, <lb/>vel diametrum figurę in cono ſecundo plano productę, æqui-<lb/>diſtantium eiuſdem, & </s> <s xml:id="echoid-s2041" xml:space="preserve">baſis communi ſe ctioni erunt inter fe, <lb/>vt abíciſſæ per eaidem ordinatim applicatas verſus verticem <lb/>ſumptæ ab eiſdem axibus, vel diametris iam dictis.</s> <s xml:id="echoid-s2042" xml:space="preserve"/> </p> <pb o="82" file="0102" n="102" rhead="GEOMETRI Æ"/> <p> <s xml:id="echoid-s2043" xml:space="preserve">Sit conus, cuius vertex, A, baſis circulus, CEFD, ſecetur autem <lb/>prius plano per axem, quod in eo producat triangulum, ACF, ſe-<lb/> <anchor type="note" xlink:label="note-0102-01a" xlink:href="note-0102-01"/> cetur deinde altero plano baſim ſecante ſecundum rectam, ED, per-<lb/>pendicularem ipfi, CF, cuius in cono concepta ſit figura, BED, <lb/> <anchor type="note" xlink:label="note-0102-02a" xlink:href="note-0102-02"/> erit ergo hæc figura circa axem, vel diametrum, BV, quę ſit paral-<lb/>lela ipſi, AF, cuius vertex reſpectu ipſius, ED, erit, B; </s> <s xml:id="echoid-s2044" xml:space="preserve">ducaturà <lb/>puncto, M, qui non ſit punctus, B, ſed vtcumque ſumptus in linea, <lb/>EBD, extra baſim, ED, ipſi, ED, recta ęquidiſtans, MO, pro-<lb/>ducta vſq; </s> <s xml:id="echoid-s2045" xml:space="preserve">ad ambientem ſuperficiem, cui occurrat in, O, igitur hęc <lb/>erit vna ex ordinatim applicatis ad axim, vel diametrum, BV, ęqui-<lb/>diſtans ipſi, ED, quę bifariam diuidetur ab ipſa, BV, in puncto, N, <lb/>ducatur per, N, ipſi, CF, parallela, HR, eſt verò etiam, MO, ipſi, <lb/>ED, parallela, ergo planum tranſiens per, HR, MO, æquidiſta-<lb/> <anchor type="note" xlink:label="note-0102-03a" xlink:href="note-0102-03"/> bit baſi, CEFD, & </s> <s xml:id="echoid-s2046" xml:space="preserve">quatuor puncta, H, M, R, O, erunt in circuli <lb/> <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/> periphæria, cuius diameter, HR, quem <lb/> <anchor type="note" xlink:label="note-0102-04a" xlink:href="note-0102-04"/> ſecat, MO, perpendiculariter, nam an-<lb/>gulus, HNM, æquatur angulo, CVE, <lb/> <anchor type="note" xlink:label="note-0102-05a" xlink:href="note-0102-05"/> quirectus eſt, ergo quadratum, MN, æ-<lb/>quatur rectangulo, HNR, & </s> <s xml:id="echoid-s2047" xml:space="preserve">quadra-<lb/>tum, EV, rectangulo, CVF, eſt autem <lb/>rectangulum, CVF, ad rectangulum, H <lb/>NR, (quia eorum altitudines, VF, NR, <lb/>ſunt æ quales, cum ſint parallelogrammi, <lb/>NF, oppoſita latera) vt baſis, CV, ad, <lb/>HN, ex prima Sexti Elem. </s> <s xml:id="echoid-s2048" xml:space="preserve">vel ex quinta <lb/>libro ſequentis independénter ab hac de-<lb/>monſtrata, & </s> <s xml:id="echoid-s2049" xml:space="preserve">quia, HN, eſt parallela <lb/>ipſi, CV, trianguli, BHN, BCV, ſunt æquianguli, ideò, vt, C <lb/> <anchor type="note" xlink:label="note-0102-06a" xlink:href="note-0102-06"/> V, ad, HN, ita, VB, ad, BN, ergo rectangulum, CVF, ad re-<lb/>ctangulum, HNR, ideſt quadratum, EV, ad quadratum, MN, <lb/>erit vt, VB, ad, BN, eſt autem quadratum, ED, quadruplum <lb/>quadrati, EV, nam eſt æquale quadratis, EV, VD, & </s> <s xml:id="echoid-s2050" xml:space="preserve">rectangulis <lb/> <anchor type="note" xlink:label="note-0102-07a" xlink:href="note-0102-07"/> fub, EVD, bis, ideſt duobus quadratis, EV, quæ cum prædictis <lb/>conficiunt quatuor quadrata, EV, & </s> <s xml:id="echoid-s2051" xml:space="preserve">eadem ratione quadratum, M <lb/>O, eſt quadruplum quadrati, MN, ergo quadratum, ED, ad qua-<lb/>dratum, MO, erit vt, BV, ad, BN, quæſunt abſciſſæ ab ipſa axi, <lb/>vel diametro, BV, verſus verticem, B, per ipſas, ED, MO, ordi-<lb/>natim adipſam, BV, applicatas, quod oſtendere opus erat; </s> <s xml:id="echoid-s2052" xml:space="preserve">hęc au-<lb/>tem vocatur ab Apolonio Parabola.</s> <s xml:id="echoid-s2053" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="1"> <note position="left" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve">16, huius.</note> <note position="left" xlink:label="note-0102-02" xlink:href="note-0102-02a" xml:space="preserve">Ex antec.</note> <note position="left" xlink:label="note-0102-03" xlink:href="note-0102-03a" xml:space="preserve">15. Vnde-<lb/>cim. El.</note> <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a"> <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0102-01"/> </figure> <note position="left" xlink:label="note-0102-04" xlink:href="note-0102-04a" xml:space="preserve">15. huius.</note> <note position="left" xlink:label="note-0102-05" xlink:href="note-0102-05a" xml:space="preserve">14. Secun. <lb/>Elem.</note> <note position="left" xlink:label="note-0102-06" xlink:href="note-0102-06a" xml:space="preserve">4. Sexti <lb/>Elem.</note> <note position="left" xlink:label="note-0102-07" xlink:href="note-0102-07a" xml:space="preserve">4. Secun. <lb/>Elem.</note> </div> <pb o="83" file="0103" n="103" rhead="LIBERI."/> </div> <div xml:id="echoid-div201" type="section" level="1" n="130"> <head xml:id="echoid-head141" xml:space="preserve">THEOREMA XXXVI. PROPOS. XXXIX.</head> <p> <s xml:id="echoid-s2054" xml:space="preserve">IIſdem poſitis, præterquamquod, BV, ſit parallela ipſi, A <lb/>F, ſed poſito, quod concurrat cum eodem latere, FA, ver-<lb/>ſus verticem producto, vt in, Z. </s> <s xml:id="echoid-s2055" xml:space="preserve">Dico quadratum, ED, ad <lb/>quadratum, MO, eſſe vt rectangulum, ZVB, ad rectangu-<lb/>lum, ZNB.</s> <s xml:id="echoid-s2056" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2057" xml:space="preserve">Quia enim quadratum, EV, eſt æquale rectangulo, CVF, & </s> <s xml:id="echoid-s2058" xml:space="preserve"><lb/>quadratum, MN, rectangulo, HNR, ideò quadratum, EV, ad <lb/> <anchor type="note" xlink:label="note-0103-01a" xlink:href="note-0103-01"/> <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/> quadratum, MN, erit vt rectangulum, CV <lb/>F, HNR, rectangulum verò, CVF, ad, <lb/>HNR, habet rationem compoſitam ex ea, <lb/>quam habet, CV, ad, HN, (vt infra inde-<lb/>pendenter ab hac Propoſit. </s> <s xml:id="echoid-s2059" xml:space="preserve">probatur) .</s> <s xml:id="echoid-s2060" xml:space="preserve">i. </s> <s xml:id="echoid-s2061" xml:space="preserve">VB, <lb/> <anchor type="note" xlink:label="note-0103-02a" xlink:href="note-0103-02"/> ad, BN, quia trianguli, CVB, HNB, ſunt <lb/>æquianguli, & </s> <s xml:id="echoid-s2062" xml:space="preserve">ex ea, quam habet, VF, ad, <lb/>NR, ideſt, VZ, ad, ZN, quia trianguli, V <lb/>FZ, NRZ, ſunt æquianguli, duę verò ra-<lb/>tiones, VB, ad, BN, &</s> <s xml:id="echoid-s2063" xml:space="preserve">, VZ, ad, ZN, <lb/> <anchor type="note" xlink:label="note-0103-03a" xlink:href="note-0103-03"/> componunt rationem rectanguli, ZVB, ad <lb/>rectangulum, ZNB, ergo rectangulum, C <lb/>VF, ad rectangulum, HNR, .</s> <s xml:id="echoid-s2064" xml:space="preserve">i. </s> <s xml:id="echoid-s2065" xml:space="preserve">quadratum, <lb/>EV, ad quadratum, MN, vel quadratum, ED, ad quadratum, M <lb/>O, erit vt rectangulum, ZVB, ad rectangulum, ZNB, quod oſten-<lb/>dere opus erat; </s> <s xml:id="echoid-s2066" xml:space="preserve">hæc autem ab Apollonio vocatur Hyperbola.</s> <s xml:id="echoid-s2067" xml:space="preserve"/> </p> <div xml:id="echoid-div201" type="float" level="2" n="1"> <note position="right" xlink:label="note-0103-01" xlink:href="note-0103-01a" xml:space="preserve">14. Secun. <lb/>Elem.</note> <figure xlink:label="fig-0103-01" xlink:href="fig-0103-01a"> <image file="0103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0103-01"/> </figure> <note position="right" xlink:label="note-0103-02" xlink:href="note-0103-02a" xml:space="preserve">Ex Sexta <lb/>lib. 2. ſeq. <lb/>vel ex 23. <lb/>Sexti El.</note> <note position="right" xlink:label="note-0103-03" xlink:href="note-0103-03a" xml:space="preserve">Ex Sexta <lb/>lib. 2. ſeq. <lb/>velex 23. <lb/>Sexti El.</note> </div> </div> <div xml:id="echoid-div203" type="section" level="1" n="131"> <head xml:id="echoid-head142" xml:space="preserve">THEOREMA XXXVII. PROPOS. XL.</head> <p> <s xml:id="echoid-s2068" xml:space="preserve">TAndem eiſdem poſitis, preterquam dicto concurſu, po-<lb/>ſito, inquam, BV, concurrere cum vtroq; </s> <s xml:id="echoid-s2069" xml:space="preserve">latere trian-<lb/>guli per axem, & </s> <s xml:id="echoid-s2070" xml:space="preserve">productum, etiam cum baſi trianguli per <lb/>axem conuenire, vt in, 2. </s> <s xml:id="echoid-s2071" xml:space="preserve">Dico quadratum, RD, ad qua-<lb/>dratum, MO, eſſe vt rectangulum, VSB, ad rectangulum, <lb/>VNB.</s> <s xml:id="echoid-s2072" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2073" xml:space="preserve">Sit ergo talis hic appoſitum ſchema, in quo planum figuræ B ℟ <lb/>VD, (cuius axis, vel diameter ſecat vtraque latera, AC, AF, & </s> <s xml:id="echoid-s2074" xml:space="preserve"><lb/>producta incidit in baſim, CF, productam in, 2,) extenſum indefi- <pb o="84" file="0104" n="104" rhead="GEOMETRI Æ"/> nitè ſecat baſis productum planum in recta, 2, Z, perpendiculari <lb/>triangulo per axem, ACF, & </s> <s xml:id="echoid-s2075" xml:space="preserve">ſint adhuc per puncta, N, S, ipſi, C <lb/>F, ductæ parallelæ, TL, HR, igitur quadratum, ℟ S, erit ęquale <lb/> <anchor type="note" xlink:label="note-0104-01a" xlink:href="note-0104-01"/> <anchor type="figure" xlink:label="fig-0104-01a" xlink:href="fig-0104-01"/> rectangulo, TSL, & </s> <s xml:id="echoid-s2076" xml:space="preserve">quadra-<lb/>tum, MN, æquale rectangulo, <lb/> <anchor type="note" xlink:label="note-0104-02a" xlink:href="note-0104-02"/> HNR, at rectangulum, TSL, <lb/>ad, HNR, habet rationem com-<lb/>poſitam ex ea, quam habet, T <lb/>S, ad, HN, .</s> <s xml:id="echoid-s2077" xml:space="preserve">i. </s> <s xml:id="echoid-s2078" xml:space="preserve">SB, ad, BN, <lb/>quia trianguli, BTS, BHN, <lb/>ſunt æquianguli, & </s> <s xml:id="echoid-s2079" xml:space="preserve">ex ea, quam <lb/>habet, SL, ad, NR, .</s> <s xml:id="echoid-s2080" xml:space="preserve">i. </s> <s xml:id="echoid-s2081" xml:space="preserve">SV, <lb/>ad, VN, quia pariter trianguli, <lb/>SVL, NVR, ſunt æquiangu-<lb/>li, duę autem rationes, SB, ad, <lb/>BN, &</s> <s xml:id="echoid-s2082" xml:space="preserve">, SV, ad, VN, componunt rationem rectanguli, BSV, <lb/> <anchor type="note" xlink:label="note-0104-03a" xlink:href="note-0104-03"/> ad rectangulum, BNV, ergo rectangulum, TSL, ad, HNR, .</s> <s xml:id="echoid-s2083" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s2084" xml:space="preserve">quadratum, ℟ S, ad quadratum, MN, vel quadratum, ℟ D, ad <lb/>quadratum, MO, erit vt rectangulum, VSB, ad rectangulum, V <lb/>NB, quod oſtendere opu erat; </s> <s xml:id="echoid-s2085" xml:space="preserve">hæc autem ab Apollonio vocatur <lb/>Ellipſis.</s> <s xml:id="echoid-s2086" xml:space="preserve"/> </p> <div xml:id="echoid-div203" type="float" level="2" n="1"> <note position="left" xlink:label="note-0104-01" xlink:href="note-0104-01a" xml:space="preserve">14. Secunn. <lb/>Elem.</note> <figure xlink:label="fig-0104-01" xlink:href="fig-0104-01a"> <image file="0104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0104-01"/> </figure> <note position="left" xlink:label="note-0104-02" xlink:href="note-0104-02a" xml:space="preserve">Ex Sexta <lb/>lib. 2. feq. <lb/>velex 23. <lb/>Sext. El.</note> <note position="left" xlink:label="note-0104-03" xlink:href="note-0104-03a" xml:space="preserve">Ex Sexta <lb/>lib. 2. feq. <lb/>vel ex 23. <lb/>Sexti El.</note> </div> </div> <div xml:id="echoid-div205" type="section" level="1" n="132"> <head xml:id="echoid-head143" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s2087" xml:space="preserve">_H_Aec circa ſectiones conicas appoſui, tum vt quod menti meæ ſuc-<lb/>currit in lucem proferrem, tum vt eluceſcat, quam facilè paſſio-<lb/>nes, quæ ab. </s> <s xml:id="echoid-s2088" xml:space="preserve">Apollonio in Elementis conicis circa earundem diametros, <lb/>vel axes quoſcumque demonſtrantur, circa eos, qui axes, vel diametri <lb/>princibales, ſiue ex generatione vocantur modo ſupradicto oſtendantur. <lb/></s> <s xml:id="echoid-s2089" xml:space="preserve">His tamen contenti ex Apollonio recipiemus pro dictarum ſectionum <lb/>axibus, vel diametris quibuſcumq; </s> <s xml:id="echoid-s2090" xml:space="preserve">quod ipſe colligit ad finem Trop. </s> <s xml:id="echoid-s2091" xml:space="preserve">51. </s> <s xml:id="echoid-s2092" xml:space="preserve"><lb/>primi Conicorum, ſcilicet.</s> <s xml:id="echoid-s2093" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2094" xml:space="preserve">In Parabola vnamquamque rectarum linearum, quę diametro ex <lb/>generatione ducuntur æquidiſtantes, diametrum eſſe: </s> <s xml:id="echoid-s2095" xml:space="preserve">In hyperbola <lb/>verò, & </s> <s xml:id="echoid-s2096" xml:space="preserve">ellipſi, & </s> <s xml:id="echoid-s2097" xml:space="preserve">oppoſitis ſectionibus vnamquamque earum, quę <lb/>per centrum ducuntur, & </s> <s xml:id="echoid-s2098" xml:space="preserve">in parabola quidem applicatas ad vnam-<lb/>quamq; </s> <s xml:id="echoid-s2099" xml:space="preserve">diametrum, ęquidiſtantes contingentibus, poſte rectangula <lb/>ipſi adiacentia: </s> <s xml:id="echoid-s2100" xml:space="preserve">In hyperbola, & </s> <s xml:id="echoid-s2101" xml:space="preserve">oppoſitis poſſe rectangula adiacen-<lb/>tia ipſi, quę excedunt eadem figura: </s> <s xml:id="echoid-s2102" xml:space="preserve">In ellipſi autem, quę eadem de-<lb/>ficiunt: </s> <s xml:id="echoid-s2103" xml:space="preserve">Poſt@@mò quęcumque circa ſectiones adhibitis principalibus <lb/>diametris demonſtrata ſunt, & </s> <s xml:id="echoid-s2104" xml:space="preserve">alijs diametris aſſumptis eadem con-<lb/>tingere.</s> <s xml:id="echoid-s2105" xml:space="preserve"/> </p> <pb o="85" file="0105" n="105" rhead="LIBER I."/> <p style="it"> <s xml:id="echoid-s2106" xml:space="preserve">Tres autem proximæ Propoſitiones etiam in meo Speculo Vſtorio de-<lb/>ſcriptæ fuerunt, cum & </s> <s xml:id="echoid-s2107" xml:space="preserve">ibi ijſdem indigerem, has verò hic repetere <lb/>volui, vt qui meum illud Speculum non viderunt, etiam ijſdem potiri <lb/>poſſint: </s> <s xml:id="echoid-s2108" xml:space="preserve">Aliqua tamen ex infraſcriptis nunc ex Archimede, & </s> <s xml:id="echoid-s2109" xml:space="preserve">eiuſdem <lb/>Commentatoribus ſumemus, vt iam oſtenſa, ne has demonſtrationes, quæ <lb/>apud præfatos Auctores videri poſſunt, fruſtra repetamus.</s> <s xml:id="echoid-s2110" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div206" type="section" level="1" n="133"> <head xml:id="echoid-head144" xml:space="preserve">THEOREMA XXXVIII. PROPOS. XLI.</head> <p> <s xml:id="echoid-s2111" xml:space="preserve">SI ſphęra, vel ſphęroides, conoides parabolicum, vel hy-<lb/>perbolicum planis ſecentur ad axem rectis, communes <lb/>ſectiones erunt circuli diametros in eadem figura ducta per <lb/>axem (quæ eſt illa, quę per reuolutionem creat dictum ſoli-<lb/>dum) ſitas habentes.</s> <s xml:id="echoid-s2112" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2113" xml:space="preserve">Patet hæc Propoſitio, nam ſupradicta ſunt ſolida rotunda, na-<lb/> <anchor type="note" xlink:label="note-0105-01a" xlink:href="note-0105-01"/> ſcuntur .</s> <s xml:id="echoid-s2114" xml:space="preserve">n. </s> <s xml:id="echoid-s2115" xml:space="preserve">ex reuolutione figurarum circa axem.</s> <s xml:id="echoid-s2116" xml:space="preserve"/> </p> <div xml:id="echoid-div206" type="float" level="2" n="1"> <note position="right" xlink:label="note-0105-01" xlink:href="note-0105-01a" xml:space="preserve">Defin. 6. <lb/>34. huius.</note> </div> </div> <div xml:id="echoid-div208" type="section" level="1" n="134"> <head xml:id="echoid-head145" xml:space="preserve">THEOREMA XXXIX PROPOS. XLII.</head> <p> <s xml:id="echoid-s2117" xml:space="preserve">SI conoides parabolicum plano ſecetur non quidem per a-<lb/>xem, neque æquidiſtanter axi, neque ad rectos angulos <lb/>cum axe, communis ſectio erit ellipſis, diameter verò ipſius <lb/>maior erit linea concepta in conoide ab interſectione facta <lb/>planorum, eius ſcilicet, quod ſecat figuram, & </s> <s xml:id="echoid-s2118" xml:space="preserve">eius, quod <lb/>ducitur recto per axem ad planum ſecans, minor verò diame-<lb/>ter æqualis erit diſtantiæ linearum ductarum æquidiſtanter <lb/>axi ab extremis diametri maioris.</s> <s xml:id="echoid-s2119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2120" xml:space="preserve">Hæc oſtenditur ab Archimede lib. </s> <s xml:id="echoid-s2121" xml:space="preserve">de Conoidibus, & </s> <s xml:id="echoid-s2122" xml:space="preserve">Sphæroidi-<lb/>bus p. </s> <s xml:id="echoid-s2123" xml:space="preserve">13.</s> <s xml:id="echoid-s2124" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div209" type="section" level="1" n="135"> <head xml:id="echoid-head146" xml:space="preserve">THEOREMA XL. PROPOS. XLIII.</head> <p> <s xml:id="echoid-s2125" xml:space="preserve">SI conoides hyperbolicum plano ſecetur coincidente in <lb/>omnia conilatera conoides compræhendentis non recto <lb/>ad axem; </s> <s xml:id="echoid-s2126" xml:space="preserve">ſectio erit ellipſis, diameter verò maior ipſius erit <lb/>concepta in conoide à ſectione facta planorum, alterius qui- <pb o="86" file="0106" n="106" rhead="GEOMETRIE"/> dem ſecantis figuram, & </s> <s xml:id="echoid-s2127" xml:space="preserve">alterius acti per axem recto ad pla-<lb/>num ſecans. </s> <s xml:id="echoid-s2128" xml:space="preserve">Archim. </s> <s xml:id="echoid-s2129" xml:space="preserve">ibid. </s> <s xml:id="echoid-s2130" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s2131" xml:space="preserve">14.</s> <s xml:id="echoid-s2132" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div210" type="section" level="1" n="136"> <head xml:id="echoid-head147" xml:space="preserve">THEOREMA XLI. PROPOS. XLIV.</head> <p> <s xml:id="echoid-s2133" xml:space="preserve">SI ſphæroides plano ſecetur non recto ad axem, ſectio erit <lb/>ellipſis, diameter verò ipſius maior erit concepta in ſphę-<lb/>roide ſectio duorum planorum, eius ſcilicet, quod ſecat figu-<lb/>ram, & </s> <s xml:id="echoid-s2134" xml:space="preserve">eius, quod ducitur per axem recto ad planum ſecans. <lb/></s> <s xml:id="echoid-s2135" xml:space="preserve">Arch. </s> <s xml:id="echoid-s2136" xml:space="preserve">ibid. </s> <s xml:id="echoid-s2137" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s2138" xml:space="preserve">15.</s> <s xml:id="echoid-s2139" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2140" xml:space="preserve">Minor verò diameter ſic habetur. </s> <s xml:id="echoid-s2141" xml:space="preserve">Sit Sphæroides, vel conoides <lb/>hyperbolicum, BDMF, axis, BM, centrum, A, ellipſis verò per <lb/> <anchor type="figure" xlink:label="fig-0106-01a" xlink:href="fig-0106-01"/> axem tranſiens in <lb/>ſphæroide, BDM <lb/>F, in conoide verò <lb/>hyperbola, NCO. <lb/></s> <s xml:id="echoid-s2142" xml:space="preserve">Secetur autem ſphę-<lb/>roides, vel conoides <lb/>plano non recto ad <lb/>axem, ſed erecto fi-<lb/>guræ, BDMF, ex <lb/>quo fiat in ipſis ſe-<lb/>ctio, DF, hæc erit <lb/>ellipſis, cuius maior <lb/>diameter, DF. </s> <s xml:id="echoid-s2143" xml:space="preserve">In-<lb/>ueniatur nunc ver-<lb/>tex ellipſis, ſeu hy-<lb/>perbolæ, BDMF, <lb/>reſpectu ipſius, DF, qui ſit, C, & </s> <s xml:id="echoid-s2144" xml:space="preserve">iungatur, CB, ac per, B, aga-<lb/>tur, BG, tangens in, B, ipſam ellipſim, ſeu hyperbolam, tandem à <lb/>puncto, D, parallela ipſi, BG, & </s> <s xml:id="echoid-s2145" xml:space="preserve">à puncto, F, parallela ipſi, CB, <lb/>produc antur, DE, FE, quæ inuicem concurrent vt in, E. </s> <s xml:id="echoid-s2146" xml:space="preserve">Dico <lb/>igitur, quod erit, ED, minor diameter eiuſdem ellipſis, DF.</s> <s xml:id="echoid-s2147" xml:space="preserve"/> </p> <div xml:id="echoid-div210" type="float" level="2" n="1"> <figure xlink:label="fig-0106-01" xlink:href="fig-0106-01a"> <image file="0106-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0106-01"/> </figure> </div> <p> <s xml:id="echoid-s2148" xml:space="preserve">Hoc autem demonſtrat ibid. </s> <s xml:id="echoid-s2149" xml:space="preserve">Dauid Riualtus in Commentarijs in <lb/>Archim. </s> <s xml:id="echoid-s2150" xml:space="preserve">ad Propoſ. </s> <s xml:id="echoid-s2151" xml:space="preserve">14. </s> <s xml:id="echoid-s2152" xml:space="preserve">& </s> <s xml:id="echoid-s2153" xml:space="preserve">15.</s> <s xml:id="echoid-s2154" xml:space="preserve"/> </p> <pb o="87" file="0107" n="107" rhead="LIBERI."/> </div> <div xml:id="echoid-div212" type="section" level="1" n="137"> <head xml:id="echoid-head148" xml:space="preserve">THEOREMA XLII. PROPOS. XLV.</head> <p> <s xml:id="echoid-s2155" xml:space="preserve">SI ſphæroides, vel conoides parabolicum, ſeu hyperboli-<lb/>cum ſecentur quomodocumq; </s> <s xml:id="echoid-s2156" xml:space="preserve">planis parallelis ad axem <lb/>rectis, ſiue inclinatis, communes ſectiones ſimiles erunt, & </s> <s xml:id="echoid-s2157" xml:space="preserve"><lb/>diametri eiuſdem rationis erunt omnes in eadem figura per <lb/>axem tranſeunte, rectè eaſdem ſecante.</s> <s xml:id="echoid-s2158" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2159" xml:space="preserve">Hæc colliguntur in Coroll. </s> <s xml:id="echoid-s2160" xml:space="preserve">2. </s> <s xml:id="echoid-s2161" xml:space="preserve">Prop. </s> <s xml:id="echoid-s2162" xml:space="preserve">15. </s> <s xml:id="echoid-s2163" xml:space="preserve">lib. </s> <s xml:id="echoid-s2164" xml:space="preserve">Arch. </s> <s xml:id="echoid-s2165" xml:space="preserve">de Conoidibus, <lb/>& </s> <s xml:id="echoid-s2166" xml:space="preserve">Sphæroidibus, & </s> <s xml:id="echoid-s2167" xml:space="preserve">ibidem etiam à Federico Commandino in ſuis in <lb/>Arch. </s> <s xml:id="echoid-s2168" xml:space="preserve">Comment. </s> <s xml:id="echoid-s2169" xml:space="preserve">demonſtrantur. </s> <s xml:id="echoid-s2170" xml:space="preserve">Hęc verò circa ipſas ſectionum fi-<lb/>guras verificari pariter manifeſtum eſt, hoc autem dico, vtor enim <lb/>ijſdem ſectionum nominibus tamquam figuras ſub ipſis comprehen-<lb/>fas ſignificantibus.</s> <s xml:id="echoid-s2171" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div213" type="section" level="1" n="138"> <head xml:id="echoid-head149" xml:space="preserve">THEOREMA XLIII. PROPOS. XLVI.</head> <p> <s xml:id="echoid-s2172" xml:space="preserve">EXpoſitis prædictis coni ſectionibus, circulo nempè, El-<lb/>lipſi, Parabola, & </s> <s xml:id="echoid-s2173" xml:space="preserve">Hyperbola, ſi, quę ad earundem axes <lb/>ordinatim applicantur, diametri eſſe intelligantur circulo-<lb/>rum ab ipſis deſcriptorum, qui ſint erecti pianis ipſarum figu-<lb/>rarum, periphærię deſcriptorum circulorum in ſectione, quę <lb/>eſt circulus, erunt omnes in ſuperficie ſphęrę, in Ellipſi verò <lb/>in ſuperficie ſphæroidis, in Parab. </s> <s xml:id="echoid-s2174" xml:space="preserve">in ſuperficie conoidis pa-<lb/>rabolici, & </s> <s xml:id="echoid-s2175" xml:space="preserve">in Hyperbola in ſuperficie conoidis Hyperbolici.</s> <s xml:id="echoid-s2176" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2177" xml:space="preserve">Sint prædictę ſectiones figurę <lb/> <anchor type="figure" xlink:label="fig-0107-01a" xlink:href="fig-0107-01"/> ſcilicet, ipſæ, ABCD, earum <lb/>axes, AC, vna ex ordinatim ad <lb/>axim applicatis, BD, quæ in-<lb/>telligatur eſſe diameter ab ea <lb/>deſcripti circuli, BNDE. </s> <s xml:id="echoid-s2178" xml:space="preserve">Di-<lb/>co circumferentiam, BNDE, <lb/>eſſe in dicta ſuperficie. </s> <s xml:id="echoid-s2179" xml:space="preserve">Intelli-<lb/>gantur dictę figuræ reuolui circa <lb/>ſuos axes, vt ex circulo fiat ſphę-<lb/>ra, ex ellipſi ſphæroides, ex pa-<lb/>rabola conoides parabolicum, <lb/>& </s> <s xml:id="echoid-s2180" xml:space="preserve">ex hyperbola hyperbolicum, <lb/>ſecentur autem planis ad axem <lb/>rectis, eundem axem ſecantibus in eodem puncto, in quo deſcriptus <pb o="88" file="0108" n="108" rhead="GEOMETRI Æ"/> circulus eum ſecat, producetur ergo ab hoc ſecante plano in ipſis ſo-<lb/>lidis circulus centrum in axehabens, cuius diameter erit, BD, ha-<lb/> <anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/> bemus igitur duos circulos in eodem plano, circa eandem diametrum, <lb/> <anchor type="note" xlink:label="note-0108-02a" xlink:href="note-0108-02"/> ergo illi erunt congruentes, periphæria autem circuli dicto ſecante <lb/>plano in dicto ſolido producti eſt in ſuperficie ambiente dictum ſoli-<lb/>dum, ergo, & </s> <s xml:id="echoid-s2181" xml:space="preserve">periphęria circuli, BNDE, deſcripti, vt dictum eſt, <lb/>erit in tali ſuperficie, ſcilicet in ſuperficie ſphæræ in figura circuli, <lb/>ſphæroidis in figura ellipſis, conoidis parabolici in figura parabolæ, <lb/>& </s> <s xml:id="echoid-s2182" xml:space="preserve">hyperbolici in figura hyperbolę, idem oſtendemus de alijs quibuſ-<lb/>cumque ſic deſcriptis circulis ab ordinatim applicatis ad dictos axes <lb/>tanquam à diametris, qui ſint erecti eiſdem ſectionibus, igitur quod <lb/>proponebatur demonſtratum fuit.</s> <s xml:id="echoid-s2183" xml:space="preserve"/> </p> <div xml:id="echoid-div213" type="float" level="2" n="1"> <figure xlink:label="fig-0107-01" xlink:href="fig-0107-01a"> <image file="0107-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0107-01"/> </figure> <note position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-0108-02" xlink:href="note-0108-02a" xml:space="preserve">Corol. 34 <lb/>huius.</note> </div> </div> <div xml:id="echoid-div215" type="section" level="1" n="139"> <head xml:id="echoid-head150" xml:space="preserve">THEOREMA XLIV. PROPOS. XLVII.</head> <p> <s xml:id="echoid-s2184" xml:space="preserve">INFRASCRIPTIS poſitis, eadem adhuc ſequi oſten-<lb/>demus.</s> <s xml:id="echoid-s2185" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2186" xml:space="preserve">Ijſdem enim expoſitis figuris, præter circulum, ſupponamus ip-<lb/>fam, AC, non eſſe axem, ſed diametrum, & </s> <s xml:id="echoid-s2187" xml:space="preserve">ad ipſam ordinatim ap-<lb/>plicari vtcumque, BD, intelligatur autem, BD, diameter cuiuſdam <lb/>ellipſis ab eadem deſcriptæ, quæ ſit erecta plano propoſitæ figuræ, <lb/>ſit autem, in figura ellipſis, deſcriptæ ellipſis ſecunda diameter per-<lb/>pendicularis ipſi, BD, & </s> <s xml:id="echoid-s2188" xml:space="preserve">æqualis ductæ à puncto, B, parallelę tan-<lb/> <anchor type="figure" xlink:label="fig-0108-01a" xlink:href="fig-0108-01"/> genti ellipſim, ABCD, in ex-<lb/>tremitate eiuſdem axis (quæ <lb/>tangat in, S,) interiectæ in-<lb/>ter, BD, & </s> <s xml:id="echoid-s2189" xml:space="preserve">eam, quę ducitur <lb/> <anchor type="note" xlink:label="note-0108-03a" xlink:href="note-0108-03"/> à puncto, D, parallela iun-<lb/>genti puncta, S, A. </s> <s xml:id="echoid-s2190" xml:space="preserve">In figura <lb/>verò hyperbolæ ſit ſecunda <lb/>diameter perpendicularis, BD, <lb/>& </s> <s xml:id="echoid-s2191" xml:space="preserve">æqualis ei, quæ ducitur à <lb/>puncto, D, parallela tangenti <lb/>hyperbolam in extremitate a-<lb/>xis (vt in, S,) interiectæ in-<lb/>ter, BD, & </s> <s xml:id="echoid-s2192" xml:space="preserve">eam, quę ducitur <lb/>à puncto, B, parallela iungenti <lb/>puncta, S, A, & </s> <s xml:id="echoid-s2193" xml:space="preserve">tandem in párabola ſit ſecunda diameter perpendi-<lb/>cularis quoque ipſi, BD, & </s> <s xml:id="echoid-s2194" xml:space="preserve">æqualis diſtantiæ parallelarum eiuſdem <lb/> <anchor type="note" xlink:label="note-0108-04a" xlink:href="note-0108-04"/> axi, quę ducuntur ab extremitatibus ip ſius, B, D. </s> <s xml:id="echoid-s2195" xml:space="preserve">Intelligantur dein- <pb o="89" file="0109" n="109" rhead="LIBER I."/> de conſtituta conoides, & </s> <s xml:id="echoid-s2196" xml:space="preserve">ſphæroides, in quibus planis per eorum <lb/>axes ductis, productæ ſint figuræ iam dictæ, ſecentur deinde planis <lb/>ad axem obliquis, ſed erectis ad dictas figuras, & </s> <s xml:id="echoid-s2197" xml:space="preserve">ſint eadem plana <lb/>deſcriptarum ellipſium dicta ſolida ſecantia, erunt ergo ex his ſecan-<lb/>tibus planis conceptæ in ipſis figuræ pariter ellipſes, quarum diame-<lb/>trierunt, BD, quidem prima, ſecunda autem in ſpha roide æqualis <lb/>ductæ à puncto, B, parallelæ tangenti ellipſim in, S, interiectæ in-<lb/>ter ipſam, BD, & </s> <s xml:id="echoid-s2198" xml:space="preserve">ductam à puncto, D, parallelam iungenti pun-<lb/>cta, S, A, (in cęteris autem ſolidis eadem ſuo modo verificabuntun) <lb/> <anchor type="note" xlink:label="note-0109-01a" xlink:href="note-0109-01"/> ergo in ſphæroide ipſa, BD, eſt prima diameter dictæ ellipſis, quæ <lb/>à dicto ſecante plano producitur, & </s> <s xml:id="echoid-s2199" xml:space="preserve">eſt etiam prima diameter ellipſis, <lb/>quę deſcribitur modo ſupradicto, ſunt autem ſecundę diametri vtriuſ-<lb/>que ellipſis ęquales, immo communes, quia ad rectos angulos ſecant <lb/>ipſam, BD, ergo habemus in eodem plano duas ellipſes circa ea-<lb/>ſdem diametros coniugatas, ergo neceſſario erunt congruentes, ſed <lb/>linea ellipſis, quę eſt communis ſectio dicti plani, & </s> <s xml:id="echoid-s2200" xml:space="preserve">ſuperſiciei ſphe-<lb/> <anchor type="note" xlink:label="note-0109-02a" xlink:href="note-0109-02"/> roidis eſt in ſuperficie ſphæroidis, ergo, & </s> <s xml:id="echoid-s2201" xml:space="preserve">linea ellipſis vt ſupra de-<lb/>ſcriptæ erit in ſuperficie dicti ſphæroidis. </s> <s xml:id="echoid-s2202" xml:space="preserve">Eodem modo idem de cæ-<lb/>teris ellipſibus ſimiliter deſcriptis demonſtrabimus tum in ſphæroide, <lb/>tum etiam in conoidibus parabolicis, & </s> <s xml:id="echoid-s2203" xml:space="preserve">hyperbolicis, quę oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s2204" xml:space="preserve"/> </p> <div xml:id="echoid-div215" type="float" level="2" n="1"> <figure xlink:label="fig-0108-01" xlink:href="fig-0108-01a"> <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0108-01"/> </figure> <note position="left" xlink:label="note-0108-03" xlink:href="note-0108-03a" xml:space="preserve">44. huius.</note> <note position="left" xlink:label="note-0108-04" xlink:href="note-0108-04a" xml:space="preserve">42. huius.</note> <note position="right" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">44. huius.</note> <note position="right" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">Elicietur <lb/>ex Corol. <lb/>25. huius.</note> </div> </div> <div xml:id="echoid-div217" type="section" level="1" n="140"> <head xml:id="echoid-head151" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2205" xml:space="preserve">_H_Inc patet propoſito aliquo ex ſupradictis ſolidis, eoq; </s> <s xml:id="echoid-s2206" xml:space="preserve">ſecto planis <lb/>vtcumque parallelis ad axem rectis, ſiue obliquis figuras, quæ ex <lb/>ſectione planorum in ipſis ſolidis producuntur, eaſdem eſſe illis, quæ de-<lb/>ſcribuntur lineis rectis, tamquam homologis diametris, & </s> <s xml:id="echoid-s2207" xml:space="preserve">primis, ijs, <lb/>inquam, quæ-ſunt communes ſectiones dictarum æquidiſtantium figura-<lb/>rum, & </s> <s xml:id="echoid-s2208" xml:space="preserve">figuræ, quæ produceretur ducto plano per axem rectè eas ſecan-<lb/>te, quæ deſcribentes eſſent, quæ ondinatim applicantur ad axes, vel dia-<lb/>metros dictarum figurarum, ſecundis autem diametris deſcriptarum fi-<lb/>gurarum exiſtentibus, ijs, quæ ſupradictæ ſunt, prout poſtul at varietas <lb/>ſolidorum, iuxta Prop.</s> <s xml:id="echoid-s2209" xml:space="preserve">42. </s> <s xml:id="echoid-s2210" xml:space="preserve">43. </s> <s xml:id="echoid-s2211" xml:space="preserve">& </s> <s xml:id="echoid-s2212" xml:space="preserve">44. </s> <s xml:id="echoid-s2213" xml:space="preserve">huius.</s> <s xml:id="echoid-s2214" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div218" type="section" level="1" n="141"> <head xml:id="echoid-head152" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s2215" xml:space="preserve">_A_Duerte tamen licet ſupra vocentur diametri, quæ dictas figuras de-<lb/>ſcribunt, deberetamen intelligi ſemper eſſe axes deſeriptarum fi-<lb/>gurarum, cum .</s> <s xml:id="echoid-s2216" xml:space="preserve">n. </s> <s xml:id="echoid-s2217" xml:space="preserve">nomen diametri ſit commune diametro, & </s> <s xml:id="echoid-s2218" xml:space="preserve">axi, @li-<lb/>quando vice axis vtimur nomine diametri, vt in circulo apparet, cuius <lb/>tamen omnes diametri ſunt axes: </s> <s xml:id="echoid-s2219" xml:space="preserve">Inſuper ſciendum eſt etiam, quæ circa <pb o="90" file="0110" n="110" rhead="GEOMETRI Æ"/> byperbolam hic babentur, circa ſectiones oppoſitas, quirum communes <lb/>ſunt dictæ paſſiones, quoq; </s> <s xml:id="echoid-s2220" xml:space="preserve">intelligi poſſe. </s> <s xml:id="echoid-s2221" xml:space="preserve">Eadem verò nedum in dictis <lb/>integris ſolidis, ſed etiam in eorum portionibus, ſiue in portionibus ſe-<lb/>ctionum coni abſciſſis per line as ad earum axim, vel diametrum ordina-<lb/>tim ductas, pariter verificari manifeſtum eſt.</s> <s xml:id="echoid-s2222" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div219" type="section" level="1" n="142"> <head xml:id="echoid-head153" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s2223" xml:space="preserve">PRopoſitis duabus quibuſcumq; </s> <s xml:id="echoid-s2224" xml:space="preserve">ſimilibus figuris, duæ quæuis re-<lb/>ctæ lineæ earum homologę poterunt eſſe incidentes, vel in ipſis <lb/>productis reperientur ſaltem earum incidentes, & </s> <s xml:id="echoid-s2225" xml:space="preserve">oppoſitarum tan-<lb/>gentium, quibus ipſæ incidunt ad eundem angulum ex eadem parte, <lb/>erunt autem dictæ homologæ ſemper inuentarum incidentium par-<lb/>tes proportionales.</s> <s xml:id="echoid-s2226" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2227" xml:space="preserve">Sint duæ quæcunq; </s> <s xml:id="echoid-s2228" xml:space="preserve">ſimiles figuræ planæ, IQsP, 487R, in eiſ-<lb/>que duæ quælibet homologę, Is, 47. </s> <s xml:id="echoid-s2229" xml:space="preserve">Dico has eſſe vel incidentes, <lb/>vel in eiſdem productis reperiri poſſe incidentes prædictarum figura-<lb/>rum, & </s> <s xml:id="echoid-s2230" xml:space="preserve">oppoſitarum tangentium, quibus occurrant ipſæ homolo-<lb/>gæ, productæ, ad eundem angulum ex eadem parte, quales ſint, D <lb/>L, dO; </s> <s xml:id="echoid-s2231" xml:space="preserve">pu, g Y. </s> <s xml:id="echoid-s2232" xml:space="preserve">Ducantur autem vlterius oppoſitę tangentes, quę <lb/> <anchor type="figure" xlink:label="fig-0110-01a" xlink:href="fig-0110-01"/> ſunt regulæ homologarum, <lb/>ls, 47, ipſæ, Ad, Co; </s> <s xml:id="echoid-s2233" xml:space="preserve">F <lb/>g, KZ, quarum, & </s> <s xml:id="echoid-s2234" xml:space="preserve">dicta-<lb/> <anchor type="note" xlink:label="note-0110-01a" xlink:href="note-0110-01"/> rum figurarum incidentes <lb/>ſint, AC, FK, parallelæ <lb/> <anchor type="note" xlink:label="note-0110-02a" xlink:href="note-0110-02"/> ipſis, DL, pu, hoc .</s> <s xml:id="echoid-s2235" xml:space="preserve">n. </s> <s xml:id="echoid-s2236" xml:space="preserve">fieri <lb/> <anchor type="note" xlink:label="note-0110-03a" xlink:href="note-0110-03"/> poteſt; </s> <s xml:id="echoid-s2237" xml:space="preserve">erunt autem etiam <lb/>ipſæ, d O, g Y, regulæ ho-<lb/>mologarum, cum faciant <lb/>angulos æquales cum regu-<lb/>lis, dA, gF, vt ſuppono, <lb/>& </s> <s xml:id="echoid-s2238" xml:space="preserve">inueniri poterunt earum, <lb/>& </s> <s xml:id="echoid-s2239" xml:space="preserve">dictarum figurarum inci-<lb/>dentes parallelæ eiſdem, A <lb/>d, Fg, ſint ipſę, LO, uY, <lb/>tales incidentes: </s> <s xml:id="echoid-s2240" xml:space="preserve">Vel ergo <lb/>homologæ datæ, ls, 47, <lb/>terminantur ad oppofitas <lb/>tangentes, DL, dO; </s> <s xml:id="echoid-s2241" xml:space="preserve">pu, <lb/>gY, velnon, & </s> <s xml:id="echoid-s2242" xml:space="preserve">tunc producantur, & </s> <s xml:id="echoid-s2243" xml:space="preserve">ipſis incidant in punctis, E, <lb/>f; </s> <s xml:id="echoid-s2244" xml:space="preserve">℟, &</s> <s xml:id="echoid-s2245" xml:space="preserve">, & </s> <s xml:id="echoid-s2246" xml:space="preserve">vlterius productæ vſque ad, AC, FK, ſecent ipſas in <lb/>punctis, B, G. </s> <s xml:id="echoid-s2247" xml:space="preserve">Vlterius vel, Ho, XZ, tangunt ſetotis, vel aliqua <pb o="91" file="0111" n="111" rhead="LIBER I."/> tantum ſui parte, velin vno puncto tantum, prędictas figuras, tan-<lb/>gant in punctis tantum, P, R, & </s> <s xml:id="echoid-s2248" xml:space="preserve">ab ipſis ducantur parallelę regulis, <lb/>dO, gY, ipſæ, PN, RT, occurrentes incidentibus, LO, uY, in <lb/>punctis, N, T, dico, LN, uY, ſimiliter ad eandem partem ſecari <lb/>in, N, T, ſi.</s> <s xml:id="echoid-s2249" xml:space="preserve">n. </s> <s xml:id="echoid-s2250" xml:space="preserve">hoc non ſit, diuidatur, LO, in, M, ſimiliter ad ean-<lb/>dem partem, acdiuiditur, uY, in, T, & </s> <s xml:id="echoid-s2251" xml:space="preserve">per, M, extendatur, MI, <lb/>parallela, dO, incidentes ambitui figuræ in, I, & </s> <s xml:id="echoid-s2252" xml:space="preserve">rurſus ſecetur, u <lb/>Y, in, V, ſimiliter ad eandem partem, vt ſecatur, LO, in, N, quia <lb/>ergo, N, eſt intra puncta, M, O, etiam, V, erit inter puncta, T, <lb/>Y; </s> <s xml:id="echoid-s2253" xml:space="preserve">ducatur tandem, VS, parallela, gY, incidens ambitui figurę in, <lb/>S. </s> <s xml:id="echoid-s2254" xml:space="preserve">Quia igitur, MI, non incidit in punctum contactus rectæ, H o, <lb/>cum figura, erit, MI, maior, NP, eadem ratione oſtendemus, SV, <lb/>fore maiorem ipſa, RT, eſt enim, RT, minima earum, quæ abin-<lb/>cidente, uY, ad ambitum figuræ duci poſſunt æquidiſtanter ipſi, g <lb/>Y. </s> <s xml:id="echoid-s2255" xml:space="preserve">Cum verò, IM, RT, ſimiliter diuidant, & </s> <s xml:id="echoid-s2256" xml:space="preserve">ad eandem partem <lb/>ipſas incidentes, LO, uY, erit, IM, ad, RT, vt, LO, ad, uY, <lb/> <anchor type="note" xlink:label="note-0111-01a" xlink:href="note-0111-01"/> ideſt vt, PN, ad, SV, ergo, permutando, IM, ad, PN, erit vt, <lb/>RT, ad, SV, eſt autem, IM, maior, PN, ergo etiam, RT, erit <lb/>maior, SV, ſed etiam minor, quod eſt abſurdum, ergo falſum eſt ip-<lb/>ſas, PN, RT, non ſecare ſimiliter ad eandem partem ipſas, LO, <lb/>uY, ſic igitur eaſdem diuidunt, eritque, PN, ad, RT, hoc eſt, H <lb/>L, ad, Xu, vt, LO, ad, uY, idem oſtendemus etiam ſi contactus <lb/>eſſet in parte linearum, Ho, XZ, ſeu in totis eiſdem lineis, vt conſi-<lb/>deranti facilè innoteſcet. </s> <s xml:id="echoid-s2257" xml:space="preserve">Eadem autem methodo probabimus etiam, <lb/>DL, pu, eſſe vt ipſas, LO, uY, ergo reſiduæ, DH, pX, hoc eſt, <lb/>AC, FK, erunt vt, LO, uY, ideſt vt, E4, ℟ &</s> <s xml:id="echoid-s2258" xml:space="preserve">, ſed, AC, FK, <lb/>ſimiliter ſunt diuiſę ab homologis, sl, 7 4, productis, in punctis, B, <lb/> <anchor type="note" xlink:label="note-0111-02a" xlink:href="note-0111-02"/> G, ergo, AB, ad, FG, ideſt, DE, ad, p℟, erit vt, AC, ad, F <lb/>K, ideſt vt, E 4, ad, ℟ &</s> <s xml:id="echoid-s2259" xml:space="preserve">. Extendantur, NP, TR, quę diuidunt, <lb/>LO, uY, ſimiliter ad eandem partem, ſecentque ipſos, E4, ℟ &</s> <s xml:id="echoid-s2260" xml:space="preserve">, <lb/>in punctis, 2, 3, incidat autem, NQ, in, Q, punctum contactus <lb/>lineæ, Ad, cum figura, oſtendemus, vt factum eſt circa ipſas, NP, <lb/>TR, etiam, T8, incidere in punctum contactus rectę, p g, cum fi-<lb/>gura, quod ſit ipſum, 8, quoniam ergo probatum eſt, DE, ad, p <lb/>℟, eſſe vt, E4, ad, ℟ &</s> <s xml:id="echoid-s2261" xml:space="preserve">, erit etiam, Q2, ad, 83, vt, E4, ad, <lb/>℟ &</s> <s xml:id="echoid-s2262" xml:space="preserve">. Similiter probabimus, 2P, ad, 3R, eſſe vt, E4, ad, ℟ &</s> <s xml:id="echoid-s2263" xml:space="preserve">, <lb/>& </s> <s xml:id="echoid-s2264" xml:space="preserve">diuidunt ipſas, E4, ℟ &</s> <s xml:id="echoid-s2265" xml:space="preserve">, ſimiliter ad eandem partem, à quibus <lb/>viciſſim ſecantur ad eundem angulum ex eadem parte, cum, E4, ℟ <lb/>&</s> <s xml:id="echoid-s2266" xml:space="preserve">, ſint parallelæ ipſis, LO, uY, ergo, E4, ℟ &</s> <s xml:id="echoid-s2267" xml:space="preserve">, erunt incidentes <lb/>ſimilium figurarum, PlQs, R487, & </s> <s xml:id="echoid-s2268" xml:space="preserve">oppoſitarum tangentium, <lb/> <anchor type="note" xlink:label="note-0111-03a" xlink:href="note-0111-03"/> DH, do, pX, gZ, quod etiam veriſicaretur de ipſis homologis, ls, <lb/>47, ſi fuiſſent ad oppoſitas tangentes terminatę in punctis, E, 4, ℟, <pb o="92" file="0112" n="112" rhead="GEOMETRIÆ"/> &</s> <s xml:id="echoid-s2269" xml:space="preserve">. Modòetiam ſi ad illa puncta non terminentur dico tamen, ls, <lb/>ad, E4, eſſe vt, 47, ad, ℟ &</s> <s xml:id="echoid-s2270" xml:space="preserve">, etenim, ls, ad, 47, eſt vt, AC, <lb/>ad, FK, ideſt vt, LO, ad, uY, vel vt, E4, ad, ℟ &</s> <s xml:id="echoid-s2271" xml:space="preserve">, vt probatum <lb/>eſt, ergo permutando, ls, ad, E4, erit vt, 47, ad, ℟ &</s> <s xml:id="echoid-s2272" xml:space="preserve">, quod o-<lb/>ſtendere oportebat.</s> <s xml:id="echoid-s2273" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="1"> <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a"> <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0110-01"/> </figure> <note position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">Corol. 19. <lb/>& p. 24.</note> <note position="left" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">23. huius.</note> <note position="left" xlink:label="note-0110-03" xlink:href="note-0110-03a" xml:space="preserve">Coroll. <lb/>2. 19. & p. <lb/>24.</note> <note position="right" xlink:label="note-0111-01" xlink:href="note-0111-01a" xml:space="preserve">Defin. 10. <lb/>huius.</note> <note position="right" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">Defin. 10. <lb/>huius.</note> <note position="right" xlink:label="note-0111-03" xlink:href="note-0111-03a" xml:space="preserve">24, huius.</note> </div> </div> <div xml:id="echoid-div221" type="section" level="1" n="143"> <head xml:id="echoid-head154" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2274" xml:space="preserve">_E_T quoniam probatum eſt, l s, ad, 4 7, eſſe vt, E 4, ad, ℟ &</s> <s xml:id="echoid-s2275" xml:space="preserve">, ſeu <lb/>vt, LO, ad, u γ, vt autem, LO, ad, u γ, ita duæ homologæ, QP, <lb/>ad, 83, ideò duæ homologæ, ls, 47, ſunt inter ſe, vt duæ homologæ, <lb/>QP, 8R, & </s> <s xml:id="echoid-s2276" xml:space="preserve">cum oppoſitæ tangentes, DL, dO, pu, g γ, ductæ ſint vt-<lb/>cumque, licet ad eundem angulum cx eadem parte cum ipſis, E4, ℟ &</s> <s xml:id="echoid-s2277" xml:space="preserve">, <lb/>ideò duæhomologæ, ls, 4 7, erunt vt quæcumq; </s> <s xml:id="echoid-s2278" xml:space="preserve">aliæ duæ homologæ qui-<lb/>buſuis regulis aſſimptæ, vel vt ecrum incidentes, immo & </s> <s xml:id="echoid-s2279" xml:space="preserve">ipſæ inciden-<lb/>tes, crunt inter ſe, vt quæuis aliæ duæ incidentes, oſtenſum. </s> <s xml:id="echoid-s2280" xml:space="preserve">n. </s> <s xml:id="echoid-s2281" xml:space="preserve">eſt, A <lb/>C, ad, FK, eſſe vt, LO, ad, u γ.</s> <s xml:id="echoid-s2282" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div222" type="section" level="1" n="144"> <head xml:id="echoid-head155" xml:space="preserve">THEOREMA XLV. PROPOS. XLVIII.</head> <p> <s xml:id="echoid-s2283" xml:space="preserve">SI ſint duæ ſimiles figuræ planæ, quarum ſint ductæ oppo-<lb/>ſitæ tangentes, quæ ſunt homologarum earundem regu-<lb/>læ, per quas extendantur duo plana vtcumque inuicem pa-<lb/>rallela ęquè ad eandem partem ijſdem inclinata, deinde ſum-<lb/>ptis duabus quibuslibet homologis illæ deſcribere intelli-<lb/>gantur figuras planas ſimiles, ductis primò planis æquidi-<lb/>ſtantes, ita vt ſint ſimiliter deſcriptæ, & </s> <s xml:id="echoid-s2284" xml:space="preserve">deſcribentes earum <lb/>lineæ, vel latera homologa, idem autem contingat cæteris <lb/>homologis, etiam ſi omnes figuræ deſcriptæ ſeorſim in vna-<lb/>quaque propoſitarum figurarum non eſſent ſimiles; </s> <s xml:id="echoid-s2285" xml:space="preserve">Solida, <lb/>quę ab ijſdem tanguntur oppoſitis planis, in quibus ex traie-<lb/>ctione planorum præfatis oppoſitis tangentibus æquidiſtan-<lb/>tium eædem figuræ produci poſſunt, erunt ſimilia, & </s> <s xml:id="echoid-s2286" xml:space="preserve">figuræ <lb/>deſcriptæ eorundem homologæ figuræ, & </s> <s xml:id="echoid-s2287" xml:space="preserve">earum regulę ipſa <lb/>oppoſita tangentia plana, quorum & </s> <s xml:id="echoid-s2288" xml:space="preserve">dictorum ſolidorum fi-<lb/>guræ incidentes erunt primò propoſitæ figuræ.</s> <s xml:id="echoid-s2289" xml:space="preserve"/> </p> <pb o="93" file="0113" n="113" rhead="LIBER I."/> <p> <s xml:id="echoid-s2290" xml:space="preserve">Hæc Propoſitio manifeſta eſt, inuoluit. </s> <s xml:id="echoid-s2291" xml:space="preserve">n. </s> <s xml:id="echoid-s2292" xml:space="preserve">requiſita omnia defini-<lb/> <anchor type="note" xlink:label="note-0113-01a" xlink:href="note-0113-01"/> tionis ſimilium ſolidorum; </s> <s xml:id="echoid-s2293" xml:space="preserve">nam hic habemus duo ſolida, ea nempè, <lb/>quæ ſecantur planis dictarum figurarum, quorum duo extrema ſiue <lb/>primo ducta æquidiſtantia plana talia ſunt, vt illis incidant duo pla-<lb/>na (in quibus nempè reperiuntur propoſitæ figuræ ſimiles, quarum <lb/>homologarum regulę ſunt communes ſectiones earum, & </s> <s xml:id="echoid-s2294" xml:space="preserve">dictorum <lb/>oppoſitorum planorum tangentium) ad eundem angulum ex eadem <lb/>parte, ſunt autem figuræ planę deſcriptæ lineis, vel lateribus homo-<lb/>logis propoſitarum figurarum inter ſe ſimiles, illæ. </s> <s xml:id="echoid-s2295" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2296" xml:space="preserve">quæ ſecant in-<lb/>cidentes propoſitarum figurarum, & </s> <s xml:id="echoid-s2297" xml:space="preserve">ſubinde altitudines dictorum ſc-<lb/> <anchor type="note" xlink:label="note-0113-02a" xlink:href="note-0113-02"/> lidorum ſimiliter ad eandem partem, & </s> <s xml:id="echoid-s2298" xml:space="preserve">æquidiſtant dictis tangenti-<lb/>bus planis, reſpectu quorum altitudines dictas aſſumptas intelligo; <lb/></s> <s xml:id="echoid-s2299" xml:space="preserve">& </s> <s xml:id="echoid-s2300" xml:space="preserve">quia ſupponimus omnium deſcriptarum ſimilium figurarum late-<lb/>ra homologa deſcribentia eſſe lineas, vel latera homologa ſimilium <lb/>figurarum, quæ omnia ſunt inter ſe æquidiſtantia, ideò omnes ea-<lb/> <anchor type="note" xlink:label="note-0113-03a" xlink:href="note-0113-03"/> rum lineæ homologæ duabus quibuſdam regulis æquidiſtabunt, & </s> <s xml:id="echoid-s2301" xml:space="preserve"><lb/>ipſa latera deſcribentia erunt etiam lineæ incidentes, vel in eiſdem <lb/>productis ſaltem reperiri poterunt incidentes deſcriptarum ſimilium <lb/>figurarum, & </s> <s xml:id="echoid-s2302" xml:space="preserve">oppoſitarum tangentium duabus quibuſdam ſemper <lb/>ęquidiſtantium, ſcilicet eis, quę cum dictis incidentibus angulos con-<lb/>tinent ęquales (erunt autem dicta latera homologa incidentes, ſi di-<lb/>ctæ tangentes tranſeant per extrema laterum deſcribentium, ſi au-<lb/>tem non, poterunt tamen in ipſis lateribus productis aſſumi earun-<lb/> <anchor type="note" xlink:label="note-0113-04a" xlink:href="note-0113-04"/> dem incidentes, quæ erunt, vtipſa latera homologa) & </s> <s xml:id="echoid-s2303" xml:space="preserve">cum ipſæ <lb/>propoſitæ figuræ ſint ſimiles, ſubinde etiam erunt ſimiles illæ, quæ <lb/>capient omnes dictas incidentes, ſi fortè accidat ipſa latera homolo-<lb/>ga deſcribentia non eſſe incidentes, vt dictum eſt, igitur adſunt hic <lb/>omnes conditiones definitionis meæ ſimilium ſolidorum, ergo ſoli-<lb/>da, in quibus dictæ ſimiles deſcriptæ figuræ ex traiectione dictorum <lb/>planorum producuntur, erunt ſimilia, & </s> <s xml:id="echoid-s2304" xml:space="preserve">regulæ figurarum homo-<lb/>logarum erunt dicta plana tangentia, & </s> <s xml:id="echoid-s2305" xml:space="preserve">eorum, ac dictorum ſolido-<lb/>rum figuræ incidentes, propoſitæ primò figuræ, vel aliæ in eiſdem <lb/>planis inuentæ, illæ ſcilicet, in quibus iacent omnium ſimilium de-<lb/>ſcriptarum figurarum lineæ incidentes, quod oſtendere opus erat.</s> <s xml:id="echoid-s2306" xml:space="preserve"/> </p> <div xml:id="echoid-div222" type="float" level="2" n="1"> <note position="right" xlink:label="note-0113-01" xlink:href="note-0113-01a" xml:space="preserve">Defin. 21. <lb/>huius.</note> <note position="right" xlink:label="note-0113-02" xlink:href="note-0113-02a" xml:space="preserve">17. Vnd. <lb/>Elem.</note> <note position="right" xlink:label="note-0113-03" xlink:href="note-0113-03a" xml:space="preserve">Corol. 23. <lb/>huius.</note> <note position="right" xlink:label="note-0113-04" xlink:href="note-0113-04a" xml:space="preserve">Ex Lem. <lb/>antec.</note> </div> </div> <div xml:id="echoid-div224" type="section" level="1" n="145"> <head xml:id="echoid-head156" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2307" xml:space="preserve">_H_Inc apparet ſi deſcriptæ figuræ omnes ſint inter ſe ſimiles, dicta, <lb/>ſolida pariter eſſe ſimilia. </s> <s xml:id="echoid-s2308" xml:space="preserve">Vnde ſi intelligamus ſimiles coni ſe-<lb/>ctionum portiones, ſiue eaſdem integras, circa axes, vel diametros, & </s> <s xml:id="echoid-s2309" xml:space="preserve"><lb/>ab ordinatim applicatis ad axim, vel diametrum, earundem deſcribi ſi-<lb/>mil. </s> <s xml:id="echoid-s2310" xml:space="preserve">s figuras planas eiſdem ſectionum portionibus erectas, tanquam à <pb o="94" file="0114" n="114" rhead="GEOMETRIÆ"/> lineis, vel lateribus homologis deſcriptarum figurarum; </s> <s xml:id="echoid-s2311" xml:space="preserve">ſolida, in qui-<lb/>bus deſcriptæ figuræ ex traiectis planis producentur (quæ in ſequenti li-<lb/>bro dicuntur, ſolida ad inuicem ſimilaria genita ex dictis ſectionum por-<lb/>tionibus) erunt ſimilia, & </s> <s xml:id="echoid-s2312" xml:space="preserve">figurarum homologarum eorundem regulæ <lb/> <anchor type="note" xlink:label="note-0114-01a" xlink:href="note-0114-01"/> oppoſita tangentia plana dictis iam deſcriptis figuris æquidiſtantia, quo-<lb/>rum & </s> <s xml:id="echoid-s2313" xml:space="preserve">dictorum ſolidorum figuræ incidentes erunt dictæ ſectionum por-<lb/>tiones, vel in earum planis iacebunt. </s> <s xml:id="echoid-s2314" xml:space="preserve">V nde colligimus omnes ſphæras <lb/>eſſe ſimiles, nam ſi ſecentur planis per axem, conceptæ figuræ fiunt ſimi-<lb/>les, ideſt circuli, quod ſi ſecentur adhuc planis ad horum circulorum pla-<lb/> <anchor type="note" xlink:label="note-0114-02a" xlink:href="note-0114-02"/> na erectis, productæ figuræ fiunt pariter circuli deſcripti tanquam dia-<lb/>metris eiſdem rectis lineis, in quibus coincidunt circulis per axem du-<lb/> <anchor type="note" xlink:label="note-0114-03a" xlink:href="note-0114-03"/> ctis, quæ diametri ſunt etiam incidentes eorundem deſcriptorum circu-<lb/> <anchor type="note" xlink:label="note-0114-04a" xlink:href="note-0114-04"/> lorum, & </s> <s xml:id="echoid-s2315" xml:space="preserve">oppoſitarum tangentium per eorum extrema ductarum, quæ <lb/> <anchor type="note" xlink:label="note-0114-05a" xlink:href="note-0114-05"/> tangentes omnes inter ſe æquidiſtant, vt facilè patet, & </s> <s xml:id="echoid-s2316" xml:space="preserve">ſunt iſtæ inci-<lb/>dentes, ſiue diametri deſcriptorum circulorum, quæ axem diuidunt fi-<lb/>militer ad eandem partem, vt ipſi axes, igitur ſpbæræ omnes ſunt ſimi-<lb/>les, & </s> <s xml:id="echoid-s2317" xml:space="preserve">ductis duobus planis oppoſitis tangentibus vtcumq; </s> <s xml:id="echoid-s2318" xml:space="preserve">& </s> <s xml:id="echoid-s2319" xml:space="preserve">per axem, <lb/> <anchor type="note" xlink:label="note-0114-06a" xlink:href="note-0114-06"/> qui iungit puncta contactuum ductis planis, hinc effecti circuli erunt <lb/>figuræ incidentes dictorum tangentium, & </s> <s xml:id="echoid-s2320" xml:space="preserve">ſphærarum, & </s> <s xml:id="echoid-s2321" xml:space="preserve">dicta plana <lb/>tangentia erunt regulæ homologarum figurarum earundem, vnde tan-<lb/>dem patet quoſuis circulos in ſphæris per centrum tranſeuntes poſſe eſſe <lb/>figuras incidentes earundem ſphærarum, & </s> <s xml:id="echoid-s2322" xml:space="preserve">planorum oppoſitorum tan-<lb/>gentium ſphæras in extremis punctis diametrorum quorumuis dictorum <lb/>circulorum per centrum tr anſeuntium.</s> <s xml:id="echoid-s2323" xml:space="preserve"/> </p> <div xml:id="echoid-div224" type="float" level="2" n="1"> <note position="left" xlink:label="note-0114-01" xlink:href="note-0114-01a" xml:space="preserve">_C. Def. 8._ <lb/>_lib. 2._</note> <note position="left" xlink:label="note-0114-02" xlink:href="note-0114-02a" xml:space="preserve">_Lẽma 31._ <lb/>_huius pr._</note> <note position="left" xlink:label="note-0114-03" xlink:href="note-0114-03a" xml:space="preserve">_33. huius._</note> <note position="left" xlink:label="note-0114-04" xlink:href="note-0114-04a" xml:space="preserve">_34. huius._</note> <note position="left" xlink:label="note-0114-05" xlink:href="note-0114-05a" xml:space="preserve">_Lẽma 31._ <lb/>_huius._</note> <note position="left" xlink:label="note-0114-06" xlink:href="note-0114-06a" xml:space="preserve">_Lẽma 31._ <lb/>_huius._</note> </div> </div> <div xml:id="echoid-div226" type="section" level="1" n="146"> <head xml:id="echoid-head157" xml:space="preserve">THEOREMA XLVI. PROPOS. XLIX.</head> <p> <s xml:id="echoid-s2324" xml:space="preserve">POſita definitione particulari ſimilium ſphæroidum, ſe-<lb/>quitur & </s> <s xml:id="echoid-s2325" xml:space="preserve">generalis ſimilium ſolidorum.</s> <s xml:id="echoid-s2326" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2327" xml:space="preserve">Sint ſimiles ſphæroides <lb/> <anchor type="figure" xlink:label="fig-0114-01a" xlink:href="fig-0114-01"/> iuxta definitionem particu-<lb/>larem de ipſis allatam, AB <lb/>CD, FEHG. </s> <s xml:id="echoid-s2328" xml:space="preserve">Dico has <lb/>eſſe ſimiles iuxta definitio. <lb/></s> <s xml:id="echoid-s2329" xml:space="preserve">nem generalem ſimilium <lb/>ſolidorum; </s> <s xml:id="echoid-s2330" xml:space="preserve">ductis enim pla-<lb/>nis per axes, AC, FH, <lb/>producantur in eiſdem el-<lb/>lipſes, ABCD, FEHG, <lb/> <anchor type="note" xlink:label="note-0114-07a" xlink:href="note-0114-07"/> quæ erunt eædem illis, ex quarum reuolutione circa axes, AC, FH, <lb/> <anchor type="note" xlink:label="note-0114-08a" xlink:href="note-0114-08"/> <pb o="95" file="0115" n="115" rhead="LIBER I."/> oriuntur dictæ ſphæroides, & </s> <s xml:id="echoid-s2331" xml:space="preserve">proinde erunt ſimiles tum iuxta defi-<lb/>nit. </s> <s xml:id="echoid-s2332" xml:space="preserve">Apollonij, tum iuxta definit. </s> <s xml:id="echoid-s2333" xml:space="preserve">10. </s> <s xml:id="echoid-s2334" xml:space="preserve">huius. </s> <s xml:id="echoid-s2335" xml:space="preserve">Et quoniam ſi ſecentur <lb/>planis ad axem rectis in dictis ſphæroidibus gignuntur circuli, vt ex. <lb/></s> <s xml:id="echoid-s2336" xml:space="preserve">gr. </s> <s xml:id="echoid-s2337" xml:space="preserve">BNDO, EXGV, (qui ſecent axes, AC, FH, ſimiliter ad ean-<lb/> <anchor type="note" xlink:label="note-0115-01a" xlink:href="note-0115-01"/> dem partem in punctis, M, I,) quorum diametri ſunt communes ſe-<lb/>ctiones cum figuris per axem tranſeuntibus, vt ipſę, BD, BG, ideò <lb/>iſtæ erunt incidentes ipſorum circulorum, BNDO, EXGV, & </s> <s xml:id="echoid-s2338" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0115-02a" xlink:href="note-0115-02"/> oppoſitarum tangentium in punctis, B, D; </s> <s xml:id="echoid-s2339" xml:space="preserve">E, G; </s> <s xml:id="echoid-s2340" xml:space="preserve">quod etiam de <lb/>cæteris intelligemus. </s> <s xml:id="echoid-s2341" xml:space="preserve">Ergo ſi per axium, AC, FH, extrema ducta <lb/>ſint duo oppoſita tangentia plana, quæ erunt circulis, BNDO, E <lb/>XGV, parallela, habebimus plana ellipſium, ABCD, FEHG, <lb/>illis incidentia ad eundem angulum ex eadem parte; </s> <s xml:id="echoid-s2342" xml:space="preserve">nam adilla ſunt <lb/>erecta, in quibus reperientur ſimiles figuræ, ellipſes nempè iam di-<lb/>ctæ, & </s> <s xml:id="echoid-s2343" xml:space="preserve">homologarum earundem regulæ erunt communes ſectiones <lb/>earundem productorum planorum cum oppoſitis tangentibus pla-<lb/>nis, quæ homologę erunt incidentes homologarum figurarum (qua-<lb/>rum regulæ ſunt dicta tangentia plana) & </s> <s xml:id="echoid-s2344" xml:space="preserve">oppoſitarum tangentium <lb/>per earundem extrema ductarum, quæ ſemper duabus quibuſdam re-<lb/>gulis æquidiſtabunt. </s> <s xml:id="echoid-s2345" xml:space="preserve">Ergo dictæ ſphæroides ſimiles erunt iuxta de-<lb/>fin. </s> <s xml:id="echoid-s2346" xml:space="preserve">10. </s> <s xml:id="echoid-s2347" xml:space="preserve">huius, & </s> <s xml:id="echoid-s2348" xml:space="preserve">earum, ac dictorum oppoſitorum tangentium pla-<lb/>norum figuræ incidentes erunt eædem ellipſes, ABCD, FEHG, <lb/>per axes tranſeuntes, quod &</s> <s xml:id="echoid-s2349" xml:space="preserve">c.</s> <s xml:id="echoid-s2350" xml:space="preserve"/> </p> <div xml:id="echoid-div226" type="float" level="2" n="1"> <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a"> <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0114-01"/> </figure> <note position="left" xlink:label="note-0114-07" xlink:href="note-0114-07a" xml:space="preserve">33 huius.</note> <note position="left" xlink:label="note-0114-08" xlink:href="note-0114-08a" xml:space="preserve">38. huius.</note> <note position="right" xlink:label="note-0115-01" xlink:href="note-0115-01a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-0115-02" xlink:href="note-0115-02a" xml:space="preserve">Lẽma 31. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div228" type="section" level="1" n="147"> <head xml:id="echoid-head158" xml:space="preserve">THEOREMA XLVII. PROPOS. L:</head> <p> <s xml:id="echoid-s2351" xml:space="preserve">P Oſita definitione ſimilium portionum ſphæràrum, vel <lb/>ſphæroidum, aut conoidum, ſiue earundem portionum, <lb/>ſequitur etiam definitio generalis ſimilium 4olidorum.</s> <s xml:id="echoid-s2352" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2353" xml:space="preserve">Sint ſolida, FMH, BAC, ſimiles <lb/> <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> portiones ſphęrarum, vel ſphæroidum, <lb/>vel ſimiles conoides, ſeu conoidum por-<lb/>tiones iuxta particularem definitionem <lb/> <anchor type="note" xlink:label="note-0115-03a" xlink:href="note-0115-03"/> de illis allatam. </s> <s xml:id="echoid-s2354" xml:space="preserve">Dico eadem eſſe ſimi-<lb/>lia iuxta definitionem generalem ſimi-<lb/>lium ſolidorum. </s> <s xml:id="echoid-s2355" xml:space="preserve">Baſes ergo erunt vel <lb/>circuli, vel ſimiles ellipſes, nempè, F <lb/>GHN, BDCE, ductis autem planis <lb/>per axes ad rectos angulos baſibus fiant <lb/>in ipſis figuræ, FMH, BAC, quæ e-<lb/>runt ſimiles ſectionum coni portiones, & </s> <s xml:id="echoid-s2356" xml:space="preserve">earum baſes, FH, BC, <pb o="96" file="0116" n="116" rhead="GEOMETRIÆ"/> erunt axes baſium eorundem ſolidorum, ipſarum nempè figurarum, <lb/> <anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/> FGHN, BDCE, ſunt. </s> <s xml:id="echoid-s2357" xml:space="preserve">n. </s> <s xml:id="echoid-s2358" xml:space="preserve">ſolida rotunda, & </s> <s xml:id="echoid-s2359" xml:space="preserve">plana, FMH, BA <lb/>C, per axes tranſeuntia ſunt baſibus erecta. </s> <s xml:id="echoid-s2360" xml:space="preserve">Sint autem ſolidorum <lb/>iam dictorum axes, necnon axes, ſeu diametri figurarum, FMH, <lb/>BAC, ipſæ, OM, XA. </s> <s xml:id="echoid-s2361" xml:space="preserve">Qura ergo ſiguræ, FMH, BAC, ſunt <lb/>fimiles portionum coni ſectiones, quarum baſes, ſiue ad earum axes, <lb/>vel diametros, MO, AX, ordinatim applicatæ ſunt, FH, BC, e-<lb/>runt homologarum earundem regulæ, ac tangentes ipſas figuras ex <lb/>vna parte, ex alia verò, quo per vertices, M, A, eiſdem ducentur æ-<lb/>quidiſtantes, earundem verò oppoſitarum tangentium, acipſarum <lb/>figurarum incidentes, MO, AX, eritque, FH, ad, BC, vt, MO, <lb/> <anchor type="note" xlink:label="note-0116-02a" xlink:href="note-0116-02"/> ad, AX. </s> <s xml:id="echoid-s2362" xml:space="preserve">Si ergo baſes, FGHN, BDCE, ſint circuli erunt figurę <lb/>ſimiles, quarum & </s> <s xml:id="echoid-s2363" xml:space="preserve">oppoſitarum tangentium per extrema, FH, du-<lb/> <anchor type="note" xlink:label="note-0116-03a" xlink:href="note-0116-03"/> ctarum incidentes fient diametri, FH, BC. </s> <s xml:id="echoid-s2364" xml:space="preserve">Si verò ſint ſimiles el-<lb/> <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> lipſes, quoniam, FH, BC, ſunt axes, <lb/>facilè probabimus, ſicut pro circulo fa-<lb/>ctum eſt ad Lemma Propoſ. </s> <s xml:id="echoid-s2365" xml:space="preserve">31. </s> <s xml:id="echoid-s2366" xml:space="preserve">huius, <lb/>auxilio Propoſ. </s> <s xml:id="echoid-s2367" xml:space="preserve">40. </s> <s xml:id="echoid-s2368" xml:space="preserve">huius, ipſas, FH, <lb/>BC, eſſe incidentes ſimilium figurarum, <lb/>FGHN, BDCE, & </s> <s xml:id="echoid-s2369" xml:space="preserve">oppoſitarum <lb/>tangentium, quę per puncta, F, H; </s> <s xml:id="echoid-s2370" xml:space="preserve">B, <lb/>C, ducuntur (quę ipſis, FH, BC, exi-<lb/>ſtent perpendiculares, cum ſint axes ea-<lb/>rundem figurarum.) </s> <s xml:id="echoid-s2371" xml:space="preserve">Et eodem modo <lb/>ſi dicta ſolida ſecentur alijs planis præ-<lb/>fatis baſibus parallelis (ita tamen vt illa <lb/>diuidant ſimiliter ad eandem partem ip-<lb/>ſas, MO, AX, & </s> <s xml:id="echoid-s2372" xml:space="preserve">ſubinde etiam altitudines ipſorum ſolidorum re-<lb/> <anchor type="note" xlink:label="note-0116-04a" xlink:href="note-0116-04"/> ſpectu dictarum baſium aſſumptas) oſtendemus & </s> <s xml:id="echoid-s2373" xml:space="preserve">productas in ſo-<lb/>lidis figuras eſſe ſimiles, & </s> <s xml:id="echoid-s2374" xml:space="preserve">earum, ac oppoſitarum tangentium (æ-<lb/>quidiſtantium tanquam regulis duabus oppoſitis tangentibus ba-<lb/>ſium, FH, BC, per extrema, F, H; </s> <s xml:id="echoid-s2375" xml:space="preserve">B, C, iam ductarum) inci-<lb/>dentes eſſe communes ipſarum ſectiones cum figuris, FMH, BAC, <lb/>quæ omnes erunt lineæ homologæ ſimilium figurarum, FMH, B <lb/>AC, quarum regulę, FH, BC. </s> <s xml:id="echoid-s2376" xml:space="preserve">Ergo, ductis per, M, A, duobus <lb/>planis baſibus parallelis, quæ ipſa ſolida contingent, incidunt hiſce <lb/>oppoſitis tangentibus planisad eundem angulum ex eadem parte <lb/>plana figurarum, FMH, BAC, ſectis autem ſolidis planis paralle-<lb/>lis, vt dictum eſt, fiunt in ipſis ſimiles figuræ planæ, & </s> <s xml:id="echoid-s2377" xml:space="preserve">earum inci-<lb/>dentes capiuntur omnes in ſimilibus figuris, FMH, BAC, quarum <lb/>ſunt homologæ, earumque regulæ ipſæ, FH, BC, & </s> <s xml:id="echoid-s2378" xml:space="preserve">lineæ homo-<lb/>logæ figurarum homologarum duabus quibuſdam regulis, vtpotè <pb o="97" file="0117" n="117" rhead="LIBER I."/> oppoſitis tangentibus baſium, FGHN, BDCE, iam dictis, om-<lb/>nes æquidiſtant, ergo ſolida, FMH, BAC, ſunt ſimilia iuxta de-<lb/>fin. </s> <s xml:id="echoid-s2379" xml:space="preserve">11. </s> <s xml:id="echoid-s2380" xml:space="preserve">huius, & </s> <s xml:id="echoid-s2381" xml:space="preserve">earum, ac oppoſitorum tangentium planorum iam <lb/>dictorum, figuræ incidentes ſunt ipſæ, FMH, BAC, quod erat o-<lb/>ſtendendum.</s> <s xml:id="echoid-s2382" xml:space="preserve"/> </p> <div xml:id="echoid-div228" type="float" level="2" n="1"> <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a"> <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0115-01"/> </figure> <note position="right" xlink:label="note-0115-03" xlink:href="note-0115-03a" xml:space="preserve">Def. 9.</note> <note position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">Elicitur <lb/>ex 37. hu-<lb/>ius.</note> <note position="left" xlink:label="note-0116-02" xlink:href="note-0116-02a" xml:space="preserve">28. huius.</note> <note position="left" xlink:label="note-0116-03" xlink:href="note-0116-03a" xml:space="preserve">Lẽma 31. <lb/>huius.</note> <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a"> <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0116-01"/> </figure> <note position="left" xlink:label="note-0116-04" xlink:href="note-0116-04a" xml:space="preserve">17. Vnd. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div230" type="section" level="1" n="148"> <head xml:id="echoid-head159" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s2383" xml:space="preserve">_H_Inc etiam non difficile intelligi poteſt, propoſitis duabus coniſi-<lb/>milibus ſectionibus, FMH, BAC, quarum axes, vel diametri <lb/>ſint, MO, AX, ac poſito ipſas, FH, BC, tanquam axes deſcribere cir-<lb/>culos, ſeu ſimiles ellipſes erectas planis figurarum, FMH, BAC, & </s> <s xml:id="echoid-s2384" xml:space="preserve"><lb/>cæteras omnes ordinatim applicatas ad ipſas, MO, AX, vel circulos, <lb/>vel ſemper ſimiles ellipſes deſcribere, vt dictum eſt, ſolida in cuius ſu-<lb/>perficie capiuntur omnes peripbæriæ circulorum, vel ſimilium ellipſi-<lb/>um, eſſe ſimiles portiones ſphærarum, vel ſimiles ſphæroides, vel conoi-<lb/>des, earumuè portiones, ſimiles inquam nedum iuxta defin. </s> <s xml:id="echoid-s2385" xml:space="preserve">11. </s> <s xml:id="echoid-s2386" xml:space="preserve">huius, <lb/>hoc. </s> <s xml:id="echoid-s2387" xml:space="preserve">n. </s> <s xml:id="echoid-s2388" xml:space="preserve">habetur ex 48. </s> <s xml:id="echoid-s2389" xml:space="preserve">huius, ſed etiam iuxta defin. </s> <s xml:id="echoid-s2390" xml:space="preserve">9. </s> <s xml:id="echoid-s2391" xml:space="preserve">habentur. </s> <s xml:id="echoid-s2392" xml:space="preserve">n. </s> <s xml:id="echoid-s2393" xml:space="preserve">hic <lb/>omnes iſtius conditiones, vt examinanti facilè apparebit, quod eſt con-<lb/>uerſum eius, quod in præſenti Theor. </s> <s xml:id="echoid-s2394" xml:space="preserve">propoſitum fuit. </s> <s xml:id="echoid-s2395" xml:space="preserve">Hoc autem con-<lb/>uerſum etiam in reliquis Theorematibus, in quibus definitiones particu-<lb/>lares ſimilium planarum, vel ſolidarum figurarum cum generalibus o-<lb/>ſtendimus cencordare, poterat demonſtrari, ſedcum in ſequentibus libris <lb/>vel nullam, vel ſaltem non neceſſariam occaſionem viderem me huius <lb/>habiturum eſſe, & </s> <s xml:id="echoid-s2396" xml:space="preserve">cum etiam facilè hoc ſtudioſus, quirectè priores pro-<lb/>poſitiones intellexit, deducere poſſit, proptereane longior fierem, con-<lb/>ſultò hoc prætermiſi, quod tamen verum eſſe minimè dubito, & </s> <s xml:id="echoid-s2397" xml:space="preserve">propte-<lb/>rea hoc etiam pro vero ſuppoſito infraſcriptum Coroll. </s> <s xml:id="echoid-s2398" xml:space="preserve">ſubiungere volui.</s> <s xml:id="echoid-s2399" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div231" type="section" level="1" n="149"> <head xml:id="echoid-head160" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s2400" xml:space="preserve">_V_Lterius ergo cum hucuſque ſatis manifeſtum ſit, definitiones par-<lb/>ticulares ſimilium planarum, vel ſolidarum figurarum, cum de-<lb/>finitionibus generalibus 10. </s> <s xml:id="echoid-s2401" xml:space="preserve">nempè, & </s> <s xml:id="echoid-s2402" xml:space="preserve">11. </s> <s xml:id="echoid-s2403" xml:space="preserve">huius concordare, ideò in ſe-<lb/>quentibus vtriuſq; </s> <s xml:id="echoid-s2404" xml:space="preserve">definitionis, tam particularis ſcilicet quam generalis, <lb/>prout libuerit, hypoteſi nos vti poſſe ex hoc colligemus.</s> <s xml:id="echoid-s2405" xml:space="preserve"/> </p> <pb o="98" file="0118" n="118" rhead="GEOMETRIÆ LIBER I."/> </div> <div xml:id="echoid-div232" type="section" level="1" n="150"> <head xml:id="echoid-head161" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s2406" xml:space="preserve">_N_E miretur autem Lector ſi in hoc quaſdam propoſitiones aſſum-<lb/>pſerim tamquam veras, quæ in ſequenti Libro demon§trantur, <lb/>quales præcipuè eſſe potuerunt Propoſ. </s> <s xml:id="echoid-s2407" xml:space="preserve">5. </s> <s xml:id="echoid-s2408" xml:space="preserve">6. </s> <s xml:id="echoid-s2409" xml:space="preserve">7. </s> <s xml:id="echoid-s2410" xml:space="preserve">& </s> <s xml:id="echoid-s2411" xml:space="preserve">8. </s> <s xml:id="echoid-s2412" xml:space="preserve">lib. </s> <s xml:id="echoid-s2413" xml:space="preserve">ſequentis, has <lb/>.</s> <s xml:id="echoid-s2414" xml:space="preserve">n accepitamquam in Elementis iam demonſtratas, licet potuiſſent etiam <lb/>deſumi ex ſeq. </s> <s xml:id="echoid-s2415" xml:space="preserve">Lib. </s> <s xml:id="echoid-s2416" xml:space="preserve">2. </s> <s xml:id="echoid-s2417" xml:space="preserve">cum ipſæ non penderent ex hic demonſtrandis, ne <lb/>fieret petitio principij, vt ſuis locis admonui in præſenti Libro; </s> <s xml:id="echoid-s2418" xml:space="preserve">placuit <lb/>tamen eaſdem Propoſ. </s> <s xml:id="echoid-s2419" xml:space="preserve">noua mea methodo indiuiſibilium etiam demon-<lb/>ſtrare, vt ex ea, tamquam ex herculeo cornu, quanta ſit manans demon-<lb/>ſtrationum affluentia paſſin digito demonſtrarem.</s> <s xml:id="echoid-s2420" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div233" type="section" level="1" n="151"> <head xml:id="echoid-head162" xml:space="preserve">Finis Primi Libri.</head> <figure> <image file="0118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0118-01"/> </figure> <pb o="99" file="0119" n="119" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div234" type="section" level="1" n="152"> <head xml:id="echoid-head163" xml:space="preserve">CAVALER II</head> <head xml:id="echoid-head164" xml:space="preserve">LIBER SECVNDVS.</head> <p style="it"> <s xml:id="echoid-s2421" xml:space="preserve">In quo de Triangulo præcipuè, & </s> <s xml:id="echoid-s2422" xml:space="preserve">Parallelogram-<lb/>mo, ac Solidis ab eiſdem genitis plura de-<lb/>monſtrantur, necnon aliæ quædam <lb/>Propoſitiones lemmaticæ pro ſe-<lb/>quentibus Libris oſten-<lb/>duntur.</s> <s xml:id="echoid-s2423" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div235" type="section" level="1" n="153"> <head xml:id="echoid-head165" xml:space="preserve">DIFINITIONES.</head> <head xml:id="echoid-head166" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s2424" xml:space="preserve">SI per oppoſitas tangentes cuiuſcunq; </s> <s xml:id="echoid-s2425" xml:space="preserve">da-<lb/>tæ planæ figuræ ducantur duo plana in-<lb/>uicem parallela, recta, ſiue inclinata ad <lb/>planum datæ figuræ, hinc inde indefini-<lb/>tè producta; </s> <s xml:id="echoid-s2426" xml:space="preserve">quorum alterum moueatur <lb/>verſus reliquum eidem ſemper æquidi-<lb/> <anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/> ſtans donec illi congruerit: </s> <s xml:id="echoid-s2427" xml:space="preserve">ſingulæ re-<lb/>ctæ lineæ, quæ in toto motu fiunt com-<lb/>munes ſectiones plani moti, & </s> <s xml:id="echoid-s2428" xml:space="preserve">datæ figuræ, ſimul collectæ <lb/> <anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/> vocentur: </s> <s xml:id="echoid-s2429" xml:space="preserve">Omnes lineæ talis figuræ, ſumptæ regula vna ea-<lb/>rundem; </s> <s xml:id="echoid-s2430" xml:space="preserve">& </s> <s xml:id="echoid-s2431" xml:space="preserve">hoc cum plana ſunt recta ad datam figuram: </s> <s xml:id="echoid-s2432" xml:space="preserve">Cum <lb/>verò ad illam ſunt inclinata vocentur. </s> <s xml:id="echoid-s2433" xml:space="preserve">Omnes lineę eiuſdem <lb/>obliqui tranſitus datæ figuræ, regula pariter earundem vna; <lb/></s> <s xml:id="echoid-s2434" xml:space="preserve">libeat tamen, cum expediet, etiam prædictas vocare, recti <lb/>tranſitus, ſicuti has, obliqui tanſitus, eius nempè, qui fit in <lb/>tali ęquidiſtãtium planorum ad datam figuram inclinatione.</s> <s xml:id="echoid-s2435" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="1"> <note position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">Poſt Se-<lb/>cund. lib. <lb/>1.</note> <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">E. Defin. <lb/>Sec. lib. 1.</note> </div> <pb o="100" file="0120" n="120" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div237" type="section" level="1" n="154"> <head xml:id="echoid-head167" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2436" xml:space="preserve">_H_ Inc patet, quoniam oppoſitæ tangentes regula quacunque in data <lb/> <anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/> figura duci poſſunt, etiam omnes lineas datæ figuræ regula qua-<lb/>cunq; </s> <s xml:id="echoid-s2437" xml:space="preserve">recta linea propoſita haberi poſſe, tum recti, tum etiam eiuſdem <lb/>obliquitranſitus.</s> <s xml:id="echoid-s2438" xml:space="preserve"/> </p> <div xml:id="echoid-div237" type="float" level="2" n="1"> <note position="left" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve">_Corol. 1._ <lb/>_lib. 1._</note> </div> </div> <div xml:id="echoid-div239" type="section" level="1" n="155"> <head xml:id="echoid-head168" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s2439" xml:space="preserve">SI, propoſito quocunq; </s> <s xml:id="echoid-s2440" xml:space="preserve">ſolido, eiuſdem oppoſita plana tan-<lb/> <anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/> gentia regula quacunque ducta fuerint, hinc inde inde-<lb/>finitè producta, quorum alterum verſus reliquum moueatur <lb/>ſemper eidem ęquidiſtans, donec illi congruerit; </s> <s xml:id="echoid-s2441" xml:space="preserve">ſingula pla-<lb/> <anchor type="note" xlink:label="note-0120-03a" xlink:href="note-0120-03"/> na, quę in toto motu concipiuntur in propoſito ſolido, ſimul <lb/>collecta, vocentur: </s> <s xml:id="echoid-s2442" xml:space="preserve">Omnia plana propoſiti ſolidi, ſumpta, re-<lb/> <anchor type="note" xlink:label="note-0120-04a" xlink:href="note-0120-04"/> gula eorundem vno.</s> <s xml:id="echoid-s2443" xml:space="preserve"/> </p> <div xml:id="echoid-div239" type="float" level="2" n="1"> <note position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">Corol. 1. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0120-03" xlink:href="note-0120-03a" xml:space="preserve">Poſt Se-<lb/>cund. l. 1.</note> <note position="left" xlink:label="note-0120-04" xlink:href="note-0120-04a" xml:space="preserve">E. Defin. <lb/>Sec. lib. 1.</note> </div> </div> <div xml:id="echoid-div241" type="section" level="1" n="156"> <head xml:id="echoid-head169" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2444" xml:space="preserve">_H_ Inc etiam diſcimus, veluti propoſiti ſolidi oppoſita tangentia pla-<lb/>na quacunque regula duci poſſunt, ita eiuſdem omnia plana re-<lb/>gula quocunq; </s> <s xml:id="echoid-s2445" xml:space="preserve">plano haberi poſſe.</s> <s xml:id="echoid-s2446" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div242" type="section" level="1" n="157"> <head xml:id="echoid-head170" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s2447" xml:space="preserve">SI oppoſitis tangentibus planis occurrant interius duæ re-<lb/> <anchor type="note" xlink:label="note-0120-05a" xlink:href="note-0120-05"/> ctę lineę, vna perpendiculariter, reliqua obliquè pun-<lb/>cta, quę ſunt comm unes ſectiones propoſitæ lineæ perpendi-<lb/>culariter incidentis, & </s> <s xml:id="echoid-s2448" xml:space="preserve">ſingulorum planorum, quæ collecta <lb/>dicuntur, omnia plana (ita tamen producta, vt eaſdem ſecare <lb/>poſſint) ſiue puncta, quę ſunt communes ſectiones eiuſdem, <lb/>& </s> <s xml:id="echoid-s2449" xml:space="preserve">moti plani, fiuntq; </s> <s xml:id="echoid-s2450" xml:space="preserve">in toto motu, ſimul collecta vocentur: <lb/></s> <s xml:id="echoid-s2451" xml:space="preserve">Omnia puncta recti tranſitus propoſitæ lineæ perpendicula-<lb/>riter incidentis; </s> <s xml:id="echoid-s2452" xml:space="preserve">quę in obliquè incidente vocentur, eiuſdem <lb/>obliqui tranſitus.</s> <s xml:id="echoid-s2453" xml:space="preserve"/> </p> <div xml:id="echoid-div242" type="float" level="2" n="1"> <note position="left" xlink:label="note-0120-05" xlink:href="note-0120-05a" xml:space="preserve">Corol. 1. <lb/>lib. 1.</note> </div> <pb o="101" file="0121" n="121" rhead="LIBER II."/> </div> <div xml:id="echoid-div244" type="section" level="1" n="158"> <head xml:id="echoid-head171" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2454" xml:space="preserve">_E_X hoc habetur ſingula puncta recti tranſitus, vel obliqui, inciden-<lb/>tis lineæ, nedum eſſe communes ſectiones illius, & </s> <s xml:id="echoid-s2455" xml:space="preserve">ſingulorum, <lb/>quæ collecta dicuntur, omnia plana propoſiti ſolidi, ſed etiam, ſi per <lb/>talem incidentem extendatur planum, eſſe communes ſectiones illius, <lb/>& </s> <s xml:id="echoid-s2456" xml:space="preserve">ſingularum, quæ collectæ dicuntur : </s> <s xml:id="echoid-s2457" xml:space="preserve">Omnes lineæ planæ figuræ, cuius <lb/>oppoſitætangentes ſunt communes ſectiones plani eiuſdem figuræ, & </s> <s xml:id="echoid-s2458" xml:space="preserve"><lb/>oppoſitarum tangentium dicti ſolidi : </s> <s xml:id="echoid-s2459" xml:space="preserve">nam motum planum deſignat in <lb/>plano ſecante rectam lineam, & </s> <s xml:id="echoid-s2460" xml:space="preserve">inſimul punctum in ineidente, quod <lb/>reperitur in illa recta linea, & </s> <s xml:id="echoid-s2461" xml:space="preserve">ideò idem punctum eſt communis ſectio <lb/>tum moti plani & </s> <s xml:id="echoid-s2462" xml:space="preserve">rectæ incidentis, tum vnius earum, quæ dicuntur om-<lb/>nes lineæ datæ figuræ planæ (ita tamen productæ, vt hanc incidentem ſe-<lb/>care poſſint) & </s> <s xml:id="echoid-s2463" xml:space="preserve">eiuſdem incidentis.</s> <s xml:id="echoid-s2464" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div245" type="section" level="1" n="159"> <head xml:id="echoid-head172" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s2465" xml:space="preserve">SI inter alterum extremorum punctorum propoſitæ rectæ <lb/>lineæ, & </s> <s xml:id="echoid-s2466" xml:space="preserve">ſingula puncta, quæ ſimul collecta dicuntur <lb/>omnia puncta recti, veleiuſdem obliqui tranſitus eiuſdem, <lb/>fumamus interiacentes lineas, dicantur iſtæ ſimul collectæ: <lb/></s> <s xml:id="echoid-s2467" xml:space="preserve">Omnes abſciſſæ propoſitæ lineæ, quas (etiam ſi non expri-<lb/>matur) vocari ſupponemus recti tranſitus, ſi puncta ſint recti <lb/>tranſitus, vel eiuſdem obliqui tranſitus, ſi puncta ſint eiu-<lb/>ſdem obliqui tranſitus.</s> <s xml:id="echoid-s2468" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div246" type="section" level="1" n="160"> <head xml:id="echoid-head173" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s2469" xml:space="preserve">REctæ lineæ verò in antecedentis deſinitionis propoſita <lb/>linea inter eadem puncta, & </s> <s xml:id="echoid-s2470" xml:space="preserve">reliquum extremorum in-<lb/>teria centes, dicentur: </s> <s xml:id="echoid-s2471" xml:space="preserve">Reſiduæ omnium abſciſſarum propo-<lb/>ſitæ lineæ recti tranſitus, ſi puncta ſint rectitranſitus, vel eiu-<lb/>idem obliquitranſitus, ſi ſumpta puncta ſint eiuſdem obliqui <lb/>tranſitus.</s> <s xml:id="echoid-s2472" xml:space="preserve"/> </p> <pb o="102" file="0122" n="122" rhead="GEOMETRIE"/> </div> <div xml:id="echoid-div247" type="section" level="1" n="161"> <head xml:id="echoid-head174" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2473" xml:space="preserve">_H_Inc liquet cuilibet abſciſſæ in proximis definitionibus propoſitæ li-<lb/>neæ reſpondere vnam ex reſiduis, ita vt tot ſint illæ, quæ dicun-<lb/>tur reſiduæ omnium abſciſſarum propoſitæ lineæ quot illæ, quæ dicun-<lb/>tur eiuſdem omnes abſciſſæ, ſiue recti, ſiue eiuſdem obliqui tranſitus, nam <lb/>reſiduæ omnium abſciſſarum propoſitælineæ interiacent inter reliquum <lb/>extremum eiuſdem punctum, & </s> <s xml:id="echoid-s2474" xml:space="preserve">eadem illa puncta, inter quæ, & </s> <s xml:id="echoid-s2475" xml:space="preserve">ex-<lb/>tremum primò dictum, interiacent omnes abſciſſæ.</s> <s xml:id="echoid-s2476" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div248" type="section" level="1" n="162"> <head xml:id="echoid-head175" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s2477" xml:space="preserve">SI pro qualibet earum, quæ dicuntur omnes abſciſſæ pro-<lb/>poſitæ rectæ lineæ, ipſa propoſita linea, ſiue eidem æ-<lb/>qualis, ſemel aſſumpta intelligatur, iſtæ ſimul collectæ di-<lb/>centur: </s> <s xml:id="echoid-s2478" xml:space="preserve">Maximæ omnium abſciſſarum propoſitæ lineæ, vel <lb/>ſubintelligentur ſemper eſſe omnium, etiam ſi dicerentur ſo-<lb/>lummodò: </s> <s xml:id="echoid-s2479" xml:space="preserve">Maximæ abſciſſarum.</s> <s xml:id="echoid-s2480" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div249" type="section" level="1" n="163"> <head xml:id="echoid-head176" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2481" xml:space="preserve">_E_T quia omnes abſciſſæ tot ſunt, quot omnes reſidue, maximè verò <lb/>omnium abſciſſtrum tot ſunt, quot omnes abſciſſæ, nam cuilibet <lb/>abſciſſæ reſpondet vna maximarum, ideò maximæ omnium abſciſſarum <lb/>propoſitæ lineæ tot erunt, quot etiam reſiduæ omnium abſciſſarum, quot-<lb/>cumque ſint omnes abſciſſæ, vel reſiduæ : </s> <s xml:id="echoid-s2482" xml:space="preserve">ideſt pro qualibet reſidua ha-<lb/>bebimus quoque vnam maximarum; </s> <s xml:id="echoid-s2483" xml:space="preserve">ijs ſemper recti, vel eiuſdem obli-<lb/>qui tranſitus aſſumptis.</s> <s xml:id="echoid-s2484" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div250" type="section" level="1" n="164"> <head xml:id="echoid-head177" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s2485" xml:space="preserve">SI cuilibet omnium abſciſſarum propoſitæ rectæ lineæ ad-<lb/>iuncta intelligatur alia recta linea cuidam equalis, com-<lb/>poſitæ ex omnibus abſciſſis, & </s> <s xml:id="echoid-s2486" xml:space="preserve">adiunctis, ſinul colle ctæ di-<lb/>centur : </s> <s xml:id="echoid-s2487" xml:space="preserve">Omnes abſciſſæ propoſitæ lineæ adiuncta tali, nem-<lb/>pè adiuncta illa, cui, quę adiunguntur, ſunt ęquales. </s> <s xml:id="echoid-s2488" xml:space="preserve">Sive-<lb/>rò fieret hęc adiunctio reſiduis, vel maximis omnium abſciſ-<lb/>ſarum, pariter dicerentur : </s> <s xml:id="echoid-s2489" xml:space="preserve">Reſiduæ, vel Maximæ omnium <lb/>abſciſſarum adiuncta eadem; </s> <s xml:id="echoid-s2490" xml:space="preserve">rectiſemper, veleiuſdem ob-<lb/>liqui tranſitus.</s> <s xml:id="echoid-s2491" xml:space="preserve"/> </p> <pb o="103" file="0123" n="123" rhead="LIBER II."/> </div> <div xml:id="echoid-div251" type="section" level="1" n="165"> <head xml:id="echoid-head178" xml:space="preserve">A. VIII.</head> <note position="right" xml:space="preserve">A.</note> <p> <s xml:id="echoid-s2492" xml:space="preserve">PRopoſita quacunque plana figura, & </s> <s xml:id="echoid-s2493" xml:space="preserve">in ea ducta vtcun-<lb/>que recta linea vſque ad ambitum hinc inde terminata, <lb/>ſi ipſa recta linea deſcribere quamcumque figuram planam <lb/>intelligatur, non exiſtentem in plano propoſitę figurę, ac de-<lb/>inde reliquæ earum, quæ dicuntur omnes lineæ propoſitæ fi-<lb/>guræ, ſumptę regula iam ducta linea (& </s> <s xml:id="echoid-s2494" xml:space="preserve">recti tranſitus ſi de-<lb/>ſcripta figura ſit erecta plano propoſitę, veleiuſdem obliqui <lb/>tranſitus, ſi illi ſit inclinata, eius nempè tranſitus, qui fit in <lb/>tali inclinatione) deſcribere intelligantur figuras planas ſi-<lb/>miles, ac ſimiliter poſitas, & </s> <s xml:id="echoid-s2495" xml:space="preserve">æquidiſtantes primò deſcriptę, <lb/>ita vt omnes deſcribentes ſint deſcriptarum ſigurarum lineę, <lb/>vel latera homologa; </s> <s xml:id="echoid-s2496" xml:space="preserve">omnes deſcriptæ figuræ ſimul ſumptæ <lb/>dicentur. </s> <s xml:id="echoid-s2497" xml:space="preserve">Omnes figuræ planæ ſimiles talis propoſitæ figurę, <lb/>ſumptæ regula earum vna, vel regula etiam ipſa linea, vel <lb/>latere deſcribente; </s> <s xml:id="echoid-s2498" xml:space="preserve">vt ſi deſcriptæ figuræ eſſent quadrata, hæ <lb/>dicerentur. </s> <s xml:id="echoid-s2499" xml:space="preserve">Omnia quadrata talis propoſitæ figuræ, vel ſi eſ-<lb/>ſent triangula ęquilatera dicerentur. </s> <s xml:id="echoid-s2500" xml:space="preserve">Omnia triangula ęqui-<lb/>latera eiuſdem.</s> <s xml:id="echoid-s2501" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div252" type="section" level="1" n="166"> <head xml:id="echoid-head179" xml:space="preserve">B.</head> <note position="right" xml:space="preserve">B.</note> <p> <s xml:id="echoid-s2502" xml:space="preserve">Solidum, cuius omnes deſcriptæ figuræ ſimiles ſunt om-<lb/>nia plana, dicetur: </s> <s xml:id="echoid-s2503" xml:space="preserve">Solidum ſimilare genitum ex propoſita fi-<lb/>gura iuxta eandem regulam, iuxta quam ſumptæ omnes di-<lb/>ctæ figuræ ſimiles fuerunt, quæ igitur ex figuris propoſitis, <lb/>vt ſic generantur, dicentur abſque alio addito: </s> <s xml:id="echoid-s2504" xml:space="preserve">Solida ſimi-<lb/>laria genita ex propoſitis figuris iuxta regulas omnium ſimi-<lb/>lium figurarum, quæ ipſorum euadunt omnia plana, propo-<lb/>ſitæ autem figuræ, eorundem genitrices figuræ vocabuntur.</s> <s xml:id="echoid-s2505" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div253" type="section" level="1" n="167"> <head xml:id="echoid-head180" xml:space="preserve">C.</head> <note position="right" xml:space="preserve">C.</note> <p> <s xml:id="echoid-s2506" xml:space="preserve">Cum verò duarum genitricium vtcunq: </s> <s xml:id="echoid-s2507" xml:space="preserve">figurarum omnes <lb/>deſcriptæ figuræ nedum ſimiles erunt: </s> <s xml:id="echoid-s2508" xml:space="preserve">quę reperientur in ea-<lb/>rum vnaquaque, ſed etiam quæ ſunt vnius, inuenientur ſimi-<lb/>les omnibus figuris ſimilibus alterius propoſitæ figuræ, fue-<lb/>rint autem in vtroque ſolido figuræ æquè eleuatæ ſuper pla-<lb/>na genitricium figurarum, tunc ſolida genita ex propoſitis fi- <pb o="104" file="0124" n="124" rhead="GEOMETRIÆ"/> guris iuxta regulas eas, quę ſunt regulę omnium ſimilium fi-<lb/>gurarum earundem propoſitarum genitricium figurarum, di-<lb/>centur ſolida inter ſe, vel ad inuicem ſimilaria, genita ex di-<lb/>ctis figuris iuxta dictas regulas, vel intelligentur ſemper eſſe <lb/>inter ſe, ſeu ad inuicem ſimilaria, licet hoc non exprimatur, <lb/>quotieſcunq; </s> <s xml:id="echoid-s2509" xml:space="preserve">contrarium aliquid non adijciatur.</s> <s xml:id="echoid-s2510" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div254" type="section" level="1" n="168"> <head xml:id="echoid-head181" xml:space="preserve">D.</head> <note position="left" xml:space="preserve">D.</note> <p> <s xml:id="echoid-s2511" xml:space="preserve">Cum autem duas figuras in eodem plano habuerimus in <lb/>eadem altitudine exiſtentes, rectangula ſub ſingulis earum, <lb/>quæ dicuntur omnes lineæ vnius propoſitarum figurarum, & </s> <s xml:id="echoid-s2512" xml:space="preserve"><lb/>illis in directum reſpondentibus in alia figura ſimul ſumpta <lb/>ſic vocabimus, nempè Rectangula ſub eiſdem figuris, regu-<lb/>la eadem, quæ eſt omnium ſumptarum linearum regula.</s> <s xml:id="echoid-s2513" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div255" type="section" level="1" n="169"> <head xml:id="echoid-head182" xml:space="preserve">E.</head> <note position="left" xml:space="preserve">E.</note> <p> <s xml:id="echoid-s2514" xml:space="preserve">Cum verò propoſitarum figurarum altera fuerit parallelo-<lb/>grammum, cuius baſis, iuxta quam altitudo ſumitur, ſit ſum-<lb/>pta pro regula, dicta rectangula vocabuntur etiam: </s> <s xml:id="echoid-s2515" xml:space="preserve">Omnia <lb/>rectangula reliquæ figuræ æquè alta ac eorum vnum.</s> <s xml:id="echoid-s2516" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div256" type="section" level="1" n="170"> <head xml:id="echoid-head183" xml:space="preserve">APPENDIX.</head> <head xml:id="echoid-head184" xml:space="preserve">Pro antecedentium Definitionum explicatione.</head> <p style="it"> <s xml:id="echoid-s2517" xml:space="preserve">_S_It ſigura plana quæcunque, ABC, duæ eiuſdem oppoſitæ tan-<lb/>gentes vtcunque ductæ, EO, BC, intelligantur autem per, E <lb/> <anchor type="note" xlink:label="note-0124-03a" xlink:href="note-0124-03"/> O, BC, indefinitè extenſa duæ plana inuicem parallela, quorum <lb/>quod tranſit per, EO, ex. </s> <s xml:id="echoid-s2518" xml:space="preserve">gr. </s> <s xml:id="echoid-s2519" xml:space="preserve">moueatur verſus planum per, BC, <lb/>ſemper illi æquidiſtans, donec illi congruat, igitur communes ſe-<lb/>ctiones talis moti, ſiue fluentis plani, & </s> <s xml:id="echoid-s2520" xml:space="preserve">figuræ, ABC, quæ in toto <lb/>motu ſiunt, ſimul collectæ à me vocantur: </s> <s xml:id="echoid-s2521" xml:space="preserve">Omnes lineæ figuræ, AB <lb/> <anchor type="note" xlink:label="note-0124-04a" xlink:href="note-0124-04"/> C, quarum aliquæ ſint ipſæ, LH, PF, BC, ſumptæ regula earum <lb/>vna, vt, BC, recti tranſitus, cum plana parallela rectè ſecant fi-<lb/>guram, ABC, eiuſdem obliqui tranſitus, cum illam obliquè ſecant, <lb/>eius ſcilicet tranſitus, qui in tali inclinatione ſit.</s> <s xml:id="echoid-s2522" xml:space="preserve"/> </p> <div xml:id="echoid-div256" type="float" level="2" n="1"> <note position="left" xlink:label="note-0124-03" xlink:href="note-0124-03a" xml:space="preserve">_Coroll.i._ <lb/>_lib.I._</note> <note position="left" xlink:label="note-0124-04" xlink:href="note-0124-04a" xml:space="preserve">_Defin.1._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2523" xml:space="preserve">Intelligamus nunc, ABC, eſſe ſolidum, cuius duo oppoſita pla-<lb/>na tangentia ſint, quæ tranſeunt per, EO, BC, moueatur autem <lb/>adhuc planum, per, EO, extenſum, verſus planum per, BC, ſem- <pb o="105" file="0125" n="125" rhead="LIBER II."/> per illi æquidiſtans, igitur huius plani moti, ſiue fluèntis conceptæ <lb/>in ſolido, ABC, figuræ, quæ in toto motu fieri intelliguntur, voco: <lb/></s> <s xml:id="echoid-s2524" xml:space="preserve">Omnia plana ſolidi, ABC, ſumpta regula corum vno, quarum ali-<lb/> <anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/> qua repræſentare poſſunt plana, LH, PF, BC.</s> <s xml:id="echoid-s2525" xml:space="preserve"/> </p> <div xml:id="echoid-div257" type="float" level="2" n="2"> <note position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve">_Defin.2._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2526" xml:space="preserve">Vlterius duæ rectæ lineæ, ON, EM, occurrant planis per, EO, <lb/>BC, tranſeuntibus iam dictis in punctis, O, N; </s> <s xml:id="echoid-s2527" xml:space="preserve">EM, quarum, O <lb/>N, perpendiculariter, EM, verò obliquè illis incidat, puncta igi-<lb/>tur, quæ ſunt communes ſectiones omnium planorum ſ lidi, ABC, <lb/>productorum, ſiopus ſit, & </s> <s xml:id="echoid-s2528" xml:space="preserve">rectæ, ON, vocantur ipſius omnia pun-<lb/>cta recti tranſirus, quarum aliqua ſunt puncta, H, I, N, quæ in-<lb/>teripſa, & </s> <s xml:id="echoid-s2529" xml:space="preserve">extremum punctum, O, continentur, vt ipſæ, OH, OI, <lb/> <anchor type="note" xlink:label="note-0125-02a" xlink:href="note-0125-02"/> ON, dicuntur abſciſſæ, quæ inter eadem puncta, & </s> <s xml:id="echoid-s2530" xml:space="preserve">aliud extre-<lb/> <anchor type="note" xlink:label="note-0125-03a" xlink:href="note-0125-03"/> mum, quod eſt, N, continentur, vt ipſæ, NI, NH, NO, reſiduæ <lb/>omnium abſciſſarum; </s> <s xml:id="echoid-s2531" xml:space="preserve">tot æquales ipſi, ON, quot ſunt omnes ab-<lb/> <anchor type="note" xlink:label="note-0125-04a" xlink:href="note-0125-04"/> ſciſſæ, ſiue reſiduæ omnium abſciſſarum, ON, dicuntur maximæ <lb/>abſciſſarum, ſiue omnium abſciſſarum, ON, quibus ſi adiung atur <lb/> <anchor type="note" xlink:label="note-0125-05a" xlink:href="note-0125-05"/> aliqua recta linea, dicuntur abſciſſæ, reſiduæ, ſiue maximæ adiun-<lb/>cta tali linea, omnes quidem recti tranſitus in recta, ON, in, EM, <lb/> <anchor type="note" xlink:label="note-0125-06a" xlink:href="note-0125-06"/> verò dicuntur eiuſdem obliqui tranſitus, eius nempè, qui in tali in-<lb/>clinatione fit.</s> <s xml:id="echoid-s2532" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="3"> <note position="right" xlink:label="note-0125-02" xlink:href="note-0125-02a" xml:space="preserve">_Defin. 3._ <lb/>_huius._</note> <note position="right" xlink:label="note-0125-03" xlink:href="note-0125-03a" xml:space="preserve">_Def. 4._ <lb/>_huius._</note> <note position="right" xlink:label="note-0125-04" xlink:href="note-0125-04a" xml:space="preserve">_Def. 5._ <lb/>_huius._</note> <note position="right" xlink:label="note-0125-05" xlink:href="note-0125-05a" xml:space="preserve">_Defin. 6._ <lb/>_huius._</note> <note position="right" xlink:label="note-0125-06" xlink:href="note-0125-06a" xml:space="preserve">_Defin. 7._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2533" xml:space="preserve">Dicitur autem in Coroll. </s> <s xml:id="echoid-s2534" xml:space="preserve">Defin. </s> <s xml:id="echoid-s2535" xml:space="preserve">3. </s> <s xml:id="echoid-s2536" xml:space="preserve">eadem puncta recti tranſitus, <lb/>ſiue obliqui, fieri tum ab omnibus planis propoſiti ſolidi, vt, ABC, <lb/> <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> tum ab omnibus lineis <lb/>planiper eaſdem inciden-<lb/>tes extenſi, vt ex. </s> <s xml:id="echoid-s2537" xml:space="preserve">gr. </s> <s xml:id="echoid-s2538" xml:space="preserve">pla-<lb/>ni, quod tranſit per, EO, <lb/>BC, quod quidem ctiam <lb/>tranſeat per ipſas, ON, <lb/>EM, idem enim planum, <lb/>quod in ſolidum, ABC, <lb/>producit figuram, LH, in <lb/>figura plana, ABC, producit rectam, LH, & </s> <s xml:id="echoid-s2539" xml:space="preserve">in recta, ON, pun-<lb/>ctum, H, in, EM, verò punctum, γ, quod tranſit, HL, produ-<lb/>cta, & </s> <s xml:id="echoid-s2540" xml:space="preserve">ideò dico puncta, H, γ, poſſe dici etiam effecta àresta, γ, <lb/>H, & </s> <s xml:id="echoid-s2541" xml:space="preserve">ſic omnia puncta recti tranſitus quę nempè ſunt in, ON, ne-<lb/>dum fieri à dictis planis parallelis ſed etiam à lineis parallelis fi- <pb o="106" file="0126" n="126" rhead="GEOMETRIÆ"/> guræ, ABC, productis ſi opus ſit, idem intellige in recta, EM, cuius <lb/>omnia puncta dicuntur eiuſdem obliqui tranſitus, eius nempè, qui <lb/>in tali inclinatione fit.</s> <s xml:id="echoid-s2542" xml:space="preserve"/> </p> <div xml:id="echoid-div259" type="float" level="2" n="4"> <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a"> <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0125-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s2543" xml:space="preserve">Pro intelligentia Defin. </s> <s xml:id="echoid-s2544" xml:space="preserve">_8._ </s> <s xml:id="echoid-s2545" xml:space="preserve">ſupponatur in figura plana propoſita, <lb/>ABC, vtcunque recta, BC, quæ deſcribat figuram planam, BC, <lb/> <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> eleuatam ſuper, ABC, ſin-<lb/>gulæ autem lineæ, quæ di-<lb/>cuntur omnes lineæ figuræ, <lb/>ABC, ſumptæ regula, B <lb/>C, recti tranſitus, ſi figu-<lb/>ra, BC, ſit erecta figuræ, <lb/>ABC, veleiuſdem obliqui <lb/>tranſitus (qui nempè in in-<lb/>clinatione deſcriptæ figuræ <lb/>ad planum, ABC, fit, ſi figura, BC, ſit inclinata ad figuram, AB <lb/>C,) deſcribere intelligantur figuras planas ſimiles ſimiliter poſitas, <lb/>& </s> <s xml:id="echoid-s2546" xml:space="preserve">æquidiſtantes ipſi figuræ, BC, ita vt deſcribentes ſint deſcripta-<lb/> <anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/> rum figurarum lineæ, vel later a bomologa, quarum figurarum ali-<lb/>quæ ſint ipſæ, BC, PF, LH, iſtæ igitur omnes ſimul ſumptæ vocan-<lb/> <anchor type="note" xlink:label="note-0126-02a" xlink:href="note-0126-02"/> tur, omnes figuræ ſimiles ipſius figuræ, ABC, ſumptæ regula figura, <lb/>BC, vellinea, aut latere, BC.</s> <s xml:id="echoid-s2547" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="2" n="5"> <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0126-01"/> </figure> <note position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">_D. Defin._ <lb/>_10. lib.1._</note> <note position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve">_A. Def. 8._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2548" xml:space="preserve">Solidum, cuius omnes dictæ figuræ ſimiles ipſius, ABC, ſunt <lb/>omnia plana, dicitur, ſolidum ſimilare genitum ex figura plana, A <lb/>BC, iuxta regulam ipſam figuram, vel lineam, BC, & </s> <s xml:id="echoid-s2549" xml:space="preserve">ipſa figu-<lb/> <anchor type="note" xlink:label="note-0126-03a" xlink:href="note-0126-03"/> ra, ABC, appellatur genitrix eiuſdem ſolidi, quod eſſe intelliga-<lb/>tur ipſum, ABC.</s> <s xml:id="echoid-s2550" xml:space="preserve"/> </p> <div xml:id="echoid-div261" type="float" level="2" n="6"> <note position="left" xlink:label="note-0126-03" xlink:href="note-0126-03a" xml:space="preserve">_B. Def. 8._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2551" xml:space="preserve">Si verò adſit alia figura plana, cuius omnes lineæ, quædam re-<lb/>gula ſumptæ, deſcribant ſimiles figuras planas, & </s> <s xml:id="echoid-s2552" xml:space="preserve">ſimiliter poſitas, <lb/>omnes vni cuidam æquidiſtantes, & </s> <s xml:id="echoid-s2553" xml:space="preserve">ſimiles figuræ, BC, & </s> <s xml:id="echoid-s2554" xml:space="preserve">æquè <lb/> <anchor type="note" xlink:label="note-0126-04a" xlink:href="note-0126-04"/> eleuatas ſuper plana genitricium figur arum, ſolida ſimilaria genita <lb/>ex iſtis figuris, iuxta dictas regulas vocabuntur vlterius inter ſe, <lb/>vel ad inuicem ſimilaria, licet cum dicemus, ſolida ſimilaria geni-<lb/>ta ex talibus, & </s> <s xml:id="echoid-s2555" xml:space="preserve">talibus figuris, & </s> <s xml:id="echoid-s2556" xml:space="preserve">hoc etiam ſine alio addito, in-<lb/>telligemus ſemper ea eſſe inter ſe, vel ad inuicem ſimilaria, etiam <lb/>ſinon exprimatur, hoc autem niſi aliter explicetur.</s> <s xml:id="echoid-s2557" xml:space="preserve"/> </p> <div xml:id="echoid-div262" type="float" level="2" n="7"> <note position="left" xlink:label="note-0126-04" xlink:href="note-0126-04a" xml:space="preserve">_C. Def. 8._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2558" xml:space="preserve">Pro declarandis D. </s> <s xml:id="echoid-s2559" xml:space="preserve">& </s> <s xml:id="echoid-s2560" xml:space="preserve">E. </s> <s xml:id="echoid-s2561" xml:space="preserve">Defin. </s> <s xml:id="echoid-s2562" xml:space="preserve">_8._ </s> <s xml:id="echoid-s2563" xml:space="preserve">exponantur duæ figuræ in eo- <pb o="107" file="0127" n="127" rhead="LIBER II."/> dem plano, & </s> <s xml:id="echoid-s2564" xml:space="preserve">in eadem altitudine, quæ ſint, BCDA, ADE, ſit <lb/>autem altitudo figuræ, ABCD, ſumpta reſpectu ipſius rectæ, CD, <lb/>& </s> <s xml:id="echoid-s2565" xml:space="preserve">altitudo figuræ, ADE, reſpectu ipſius, DE, quæ intelligantur <lb/>abſcindere ex eadem parte à communi altitudine partes æquales, <lb/>quæ ſibi in directum erunt, ſint verò ambæ communis regula om-<lb/>nium linearum dictarum figurarum, & </s> <s xml:id="echoid-s2566" xml:space="preserve">ſit ducta alia vtcumq; </s> <s xml:id="echoid-s2567" xml:space="preserve">ei-<lb/> <anchor type="figure" xlink:label="fig-0127-01a" xlink:href="fig-0127-01"/> dem, CE, parallela, MN, <lb/>cuius portio manens in figu-<lb/>ra, BD, ſit, MO, & </s> <s xml:id="echoid-s2568" xml:space="preserve">ma-<lb/>nens in figura, ADC, ſit, <lb/>ON; </s> <s xml:id="echoid-s2569" xml:space="preserve">rectangula igitur, C <lb/>DE, MON, & </s> <s xml:id="echoid-s2570" xml:space="preserve">reliqua re-<lb/> <anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/> ctangula, quæ ſub qualibet <lb/>earum, quæ dicuntur omnes lineæ figuræ, BD, (regula, CE, vel, <lb/>CD,) & </s> <s xml:id="echoid-s2571" xml:space="preserve">illi in directum poſita in figura, ADE, continentur (erit <lb/>autem ſemper aliqua eidem in directum, præterquam fortè illi, qua <lb/>tangit figuram, vt, BA, potest .</s> <s xml:id="echoid-s2572" xml:space="preserve">n. </s> <s xml:id="echoid-s2573" xml:space="preserve">in figura, ADE, illi vice lineæ <lb/>vnum punctum tantum reſpondere, vt, A, hoc tamen rect augulum <lb/>non computatur, quia nihil illis adiungit, erit inquam bæc linea-<lb/>rum reſpondentia in figura, ADE, eis, quæ ſumuntur in, BD, nam <lb/>ſunt in eadem altitudine ſumpta aſpectu earundem linearum, ſub <lb/>quibus rectangula continentur) ſimul ſumpta vocamus: </s> <s xml:id="echoid-s2574" xml:space="preserve">Rectangu-<lb/> <anchor type="note" xlink:label="note-0127-02a" xlink:href="note-0127-02"/> la ſub figuris, BCDA, ADE.</s> <s xml:id="echoid-s2575" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="8"> <figure xlink:label="fig-0127-01" xlink:href="fig-0127-01a"> <image file="0127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0127-01"/> </figure> <note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">_Definit. i._ <lb/>_huius._</note> <note position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">_D. Def. 8._ <lb/>_huius._</note> </div> <p style="it"> <s xml:id="echoid-s2576" xml:space="preserve">Siverò contingeret alteram earundem ſigurarum eſſe par allelo-<lb/>grammum, & </s> <s xml:id="echoid-s2577" xml:space="preserve">regulam omnium eiuſdem linearum eſſe vnum eiu-<lb/>ſdem laterum, vt, CD, reſpectu cuius ſumitur altitudo, tune quia <lb/>illæ, quæ æquidiſtantipſi, CD, in parallelogrammo, BD, ſunt ei-<lb/>dem, CD, æquales, & </s> <s xml:id="echoid-s2578" xml:space="preserve">ſunt latera dictorum rectangulorum, ideò <lb/>dico, nos ea vocare poſſe nedum rectangula ſub his figuris, ſed etiam <lb/>ſic appellare, nempè. </s> <s xml:id="echoid-s2579" xml:space="preserve">Omnia rectangula figuræ ADE, (quæ non, <lb/>eſt neceſſariò parallelogrammum) æquè alta, ac vnum eorum .</s> <s xml:id="echoid-s2580" xml:space="preserve">i. </s> <s xml:id="echoid-s2581" xml:space="preserve">ac <lb/>rectangulum, CDE, altitudinis ſcilicet æqualis ipſi, CD, prout <lb/>libuerit autem nominentur.</s> <s xml:id="echoid-s2582" xml:space="preserve"/> </p> <pb o="108" file="0128" n="128" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div265" type="section" level="1" n="171"> <head xml:id="echoid-head185" xml:space="preserve">POSTVLATA</head> <head xml:id="echoid-head186" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s2583" xml:space="preserve">COngruentium planarum figurarum omnes lineæ, ſum-<lb/> <anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/> ptæ vna earundem vt regula communi, ſunt congruen-<lb/>tes; </s> <s xml:id="echoid-s2584" xml:space="preserve">Et congruentium ſolidorum omnia plana, ſumpta eorum <lb/>vno, vt regula communi, ſunt pariter congruentia.</s> <s xml:id="echoid-s2585" xml:space="preserve"/> </p> <div xml:id="echoid-div265" type="float" level="2" n="1"> <note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">Def. 1. & <lb/>2. huius.</note> </div> </div> <div xml:id="echoid-div267" type="section" level="1" n="172"> <head xml:id="echoid-head187" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s2586" xml:space="preserve">Omnes figuræ ſimiles alicuius figuræ planæ ſunt omnia <lb/>plana ſolidi, quod terminatur ſuperficie, in qua iacent peri-<lb/> <anchor type="note" xlink:label="note-0128-02a" xlink:href="note-0128-02"/> metri omnium dictarum ſimilium figurarum.</s> <s xml:id="echoid-s2587" xml:space="preserve"/> </p> <div xml:id="echoid-div267" type="float" level="2" n="1"> <note position="left" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">A. Def. 8. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div269" type="section" level="1" n="173"> <head xml:id="echoid-head188" xml:space="preserve">THEOREMA I. PROPOS. I.</head> <p> <s xml:id="echoid-s2588" xml:space="preserve">QVarumlibet planarum figurarum omnes lineæ recti <lb/>tranſitus; </s> <s xml:id="echoid-s2589" xml:space="preserve">& </s> <s xml:id="echoid-s2590" xml:space="preserve">quarumlibet ſolidarum omnia plana, ſunt <lb/>magnitudines inter ſe rationem habentes.</s> <s xml:id="echoid-s2591" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2592" xml:space="preserve">Sint duæ planæ vtcumque figuræ, EAG, GOQ, quarum re-<lb/>gulæ, EG, GQ, vtcumq; </s> <s xml:id="echoid-s2593" xml:space="preserve">ſit autem figuræ, EAG, altitudo ſum-<lb/>pta reſpectu, EG, ipſa, A ℟, & </s> <s xml:id="echoid-s2594" xml:space="preserve">figuræ, GOQ, altitudo ſumpta <lb/>reſpectu, GQ, ipſa, OP. </s> <s xml:id="echoid-s2595" xml:space="preserve">Dico ergo omnes lineas recti tranſitus fi-<lb/>guræ, EAG, ſumptas cum regula, EG, ad omnes lineas rectitran-<lb/> <anchor type="figure" xlink:label="fig-0128-01a" xlink:href="fig-0128-01"/> ſitus figuræ, GOQ, ſum-<lb/>ptas cum regula, GQ, ra-<lb/>tionem habere. </s> <s xml:id="echoid-s2596" xml:space="preserve">Conſtitu-<lb/>antur regulæ, EG, GQ, <lb/>ſibi in directum, & </s> <s xml:id="echoid-s2597" xml:space="preserve">ſint to-<lb/>tæ figuræ ſupra ipſas regu-<lb/>las in eodem plano, vel igi-<lb/>tur altitudines, A ℟, OP, <lb/>ſunt æquales, vel non, ſupponamus primò ipſas eſſe ęquales, abſcin-<lb/>dantur nuncab altitudinibus, A ℟, OP, ex hypoteſi ęqualibus, por-<lb/>tiones, I ℟, RP, æquales verſus regulas, EG, GQ, ſi ergo per <lb/>punctum, I, duxerimus regulæ, EG, parallelam, LM, hæc pro-<lb/>ducta tranſibit per punctum, R, fiet ergo, LM, quę clauditur peri- <pb o="109" file="0129" n="129" rhead="LIBER II."/> metro figuræ, EAG, vna ex ijs, quæ dicuntur omnes lineæ figurę, <lb/>EAG, &</s> <s xml:id="echoid-s2598" xml:space="preserve">, NS, clauſa perimetro figuræ, GOS, vna ex omnibus <lb/>lineis figurę, GOQ, ſumptis omnibus lineis iam dictis, regula com-<lb/>muni, EQ, & </s> <s xml:id="echoid-s2599" xml:space="preserve">recti tranſitus, vti ſemper intelligemus, niſi aliter ex-<lb/>plicetur, etiamſi id non exprimatur. </s> <s xml:id="echoid-s2600" xml:space="preserve">Quoniam igitur ſi recta, NS, <lb/>ſit minor recta, LM, poteſt indefinitè producta aliquando fieri ma-<lb/>ior, ſi hoc intelligamus fieri de cæteris lineis, quæ ab altitudinibus <lb/>portiones abſcindunt ęquales verſus regulas, EG, GQ, patet, quod <lb/>ſingulę, quę erunt in figura, GOQ, productę fient maiores ijs, quę <lb/>erunt in figura, EAG, ſit autem ita facta productio cuiuſuis om-<lb/>nium linearum figuræ, GOQ, regula, EQ, vt quæ illi in directum <lb/>conſtituitur in figura, EAG, ſit portio eiuſdem productæ, vt ex. </s> <s xml:id="echoid-s2601" xml:space="preserve">gr. <lb/></s> <s xml:id="echoid-s2602" xml:space="preserve">ita ſit producta, SN, verſus, ML, vt ipſam pertranſeat perueniens <lb/>verbi gratia vſque ad, T, ita vt, LM, ſit portio ipſius, TS, patet <lb/>ergo, quod omnes lineæ figuræ, EAG, erunt pars omnium linea-<lb/>rum figuræ, GOQ, ſic productarum, & </s> <s xml:id="echoid-s2603" xml:space="preserve">iſtę erunt totum, nam illę <lb/>iſtis claudentur, ſiue in his totæ reperientur, & </s> <s xml:id="echoid-s2604" xml:space="preserve">aliquid amplius .</s> <s xml:id="echoid-s2605" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2606" xml:space="preserve"><lb/>quod de omnibus lineis figuræ, GOQ, ſic productis manet extra fi-<lb/>guram, EAG, totum autem eſt maius ſua parte, ergo omnes lineę <lb/>figuræ, GOQ, ſic productę fuerunt, vt maiores effectę fuerint om-<lb/>nibus lineis figuræ, EAG; </s> <s xml:id="echoid-s2607" xml:space="preserve">eadem methodo omnes lineas figurę, E <lb/>AG, ſic producemus, vt complectantur omnes lineas figuræ, GO <lb/>Q, iam productas, vt dictum eſt, & </s> <s xml:id="echoid-s2608" xml:space="preserve">ideò maiores eiſdem fiant, ma-<lb/>gnitudines autem rationem habere inter ſe dicuntur, quæ multipli-<lb/> <anchor type="note" xlink:label="note-0129-01a" xlink:href="note-0129-01"/> catæ ſe inuicem ſuperare poſiunt, ergo patet omnes lineas figura-<lb/>rum, EAG, GOQ, cum altitudines, A ℟, OP, fuerint æquales, <lb/>inter ſe rationem habere.</s> <s xml:id="echoid-s2609" xml:space="preserve"/> </p> <div xml:id="echoid-div269" type="float" level="2" n="1"> <figure xlink:label="fig-0128-01" xlink:href="fig-0128-01a"> <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0128-01"/> </figure> <note position="right" xlink:label="note-0129-01" xlink:href="note-0129-01a" xml:space="preserve">Diffin. 4. <lb/>1. 5. Elem.</note> </div> <p> <s xml:id="echoid-s2610" xml:space="preserve">Non ſint autem æquales, ſed altitudo, A ℟, ſit maior altitudine, <lb/>OP, & </s> <s xml:id="echoid-s2611" xml:space="preserve">ab, A ℟, ſit abſciſſa verſus, EG, ipſa, C ℟, ęqualis ipſi, O <lb/>P, & </s> <s xml:id="echoid-s2612" xml:space="preserve">per, C, ducta, BD, parallela, EG, intelligatur per, BD, à <lb/>figura, EAG, abſciſſa figura, BAD, & </s> <s xml:id="echoid-s2613" xml:space="preserve">ea conſtituta, vt, HFE, <lb/>ita vt ſit in eodem plano ad eandem partem cum figuris, EBDG, <lb/>(quæ remanſit) &</s> <s xml:id="echoid-s2614" xml:space="preserve">, GOQ, exiſtente, HE, in directum ipſi, EQ, <lb/>quod ſi adhuc altitudo, FX, ſit maior altitudine, OP, abſcindatur <lb/>illi æqualis, & </s> <s xml:id="echoid-s2615" xml:space="preserve">ſic ſemper fiat, & </s> <s xml:id="echoid-s2616" xml:space="preserve">diſponantur figuræ reſiduæ, vt ea-<lb/>rum baſes ſint in directum ipſi, EQ, & </s> <s xml:id="echoid-s2617" xml:space="preserve">figuræ conſtitutæ in eodem <lb/>plano, & </s> <s xml:id="echoid-s2618" xml:space="preserve">ad eandem partem cum figuris, EAG, GOQ, in altitu-<lb/>dinibus vel ęqualibus, vel non maioribus altitudine, OP, Intelliga-<lb/>tur nunc ducta vtcumque in figura, GOQ, recta, NS, parallela, <lb/>GQ, quæ erit vna ex omnibus lineis figura, GOQ, regula, GQ, <lb/>producaturq; </s> <s xml:id="echoid-s2619" xml:space="preserve">ita, vt pertranſeat omnes ſic diſpoſitas figuras, vt vſq; <lb/></s> <s xml:id="echoid-s2620" xml:space="preserve">in, Z, complectetur ergo, SZ, ipſas, LM, YT, & </s> <s xml:id="echoid-s2621" xml:space="preserve">ſic quæuis om- <pb o="110" file="0130" n="130" rhead="GEOMETRIÆ"/> nium linearum figurę, GOQ, hac lege producta, complectetur eas, <lb/>quæ de ip. </s> <s xml:id="echoid-s2622" xml:space="preserve">a manent in figuris iam diſpoſitis, ergo omnes lineæ figu-<lb/>ræ, GOQ, ſic productæ complectentur omnes lineas figurarum ſic <lb/>diſpoſitarum, ergo erunt ad illas ſimul ſumptas, vt totum ad partem, <lb/>nam illæ in his reperientur, & </s> <s xml:id="echoid-s2623" xml:space="preserve">aliquid amplius, ergo erunt illis ma-<lb/>iores, omnes lineę autem figurarum ſic diſpoſitarum ſunt non mino-<lb/>res omnibus lineis figuræ, EAG, ex qua deſumptæ ſunt, ergo om-<lb/>nes lineę figurę, GOQ, ſic productæ ſunt, vt effectæ fuerint maio-<lb/>res omnibus lineis figurę, EAG; </s> <s xml:id="echoid-s2624" xml:space="preserve">eodem pacto oſtendemus nos poſ-<lb/>ſe vice verſa iſtas illis efficere maiores, ergo omnes lineæ figurarum, <lb/>EAG, GOQ, ſumptæ cum regulis vtcumque ſuppoſitis, cuiuſuis <lb/> <anchor type="note" xlink:label="note-0130-01a" xlink:href="note-0130-01"/> <anchor type="figure" xlink:label="fig-0130-01a" xlink:href="fig-0130-01"/> ſint altitudinis ſumptę iux-<lb/>ta eaſdem regulas, ſunt <lb/>magnitudines inter ſe ra-<lb/>tionem habentes, quod ſi <lb/>ſubter rectam, HQ, ad-<lb/>huc eſſent portiones con-<lb/>ſideratarum à nobis figu-<lb/>rarum, EAG, GOQ, eo-<lb/>dem modo oſtenderemus omnes lineas earundem ſumptas, cum ijſ-<lb/>dem regulis eſſe magnitudines rationem inter ſe habentes, vnde inte-<lb/>grarum figurarum omnes lineę eſſent magnitudines inter ſe rationem <lb/>habentes, quod in fig. </s> <s xml:id="echoid-s2625" xml:space="preserve">planis oſtendere opus erat.</s> <s xml:id="echoid-s2626" xml:space="preserve"/> </p> <div xml:id="echoid-div270" type="float" level="2" n="2"> <note position="left" xlink:label="note-0130-01" xlink:href="note-0130-01a" xml:space="preserve">Diffin. 4. <lb/>1.5. Elem.</note> <figure xlink:label="fig-0130-01" xlink:href="fig-0130-01a"> <image file="0130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0130-01"/> </figure> </div> <p> <s xml:id="echoid-s2627" xml:space="preserve">In ſiguris autem ſolidis conſimiliter procedemus; </s> <s xml:id="echoid-s2628" xml:space="preserve">nam ſi in ſupe-<lb/>riori figura intellexerimus, EAG, GOQ, eſſe figuras ſolidas, & </s> <s xml:id="echoid-s2629" xml:space="preserve"><lb/>pro rectis lineis æquidiſtantibus intellexerimus plana æquidiſtantia, <lb/>vt pro rectis, EG, GQ, plana, EG, GQ, quibus plana, LM, N <lb/>S, ſint æquidiſtanter ducta, ſumptis pro regulis planis, EG, GQ, <lb/>ijſque in directum ſibi conſtitutis.</s> <s xml:id="echoid-s2630" xml:space="preserve">i. </s> <s xml:id="echoid-s2631" xml:space="preserve">ita vt iaceant regulæ in eodem <lb/>plano, oſtendemus nos poſſe ita producere omnia plana ſolidæ figu-<lb/>ræ, GOQ, vt eadem complectantur omnia plana figuræ, EAG, <lb/>(ſi ſint eiuſdem altitudinis dictæ figuræ) integræ exiſtentis, vel (ſi <lb/>non ſint) diuiſæ in figuras ſolidas, ex. </s> <s xml:id="echoid-s2632" xml:space="preserve">gr. </s> <s xml:id="echoid-s2633" xml:space="preserve">EBDG, BAD, ſic di-<lb/>ſpoſitas, vt baſes, ſiue regulę iaceant in eodem plano, & </s> <s xml:id="echoid-s2634" xml:space="preserve">ita, vt om-<lb/>nia plana dictarum ſigurarum ſolidarum, vel ſint intra oppoſita pla-<lb/>na dictas figuras tangentia, vel nihil eorum extra, vnde omnia pla-<lb/>na figuræ ſolidæ, GOQ, ſic producta fient totum, & </s> <s xml:id="echoid-s2635" xml:space="preserve">portiones ab <lb/>eiſdem captæ in figura ſolida, EAG, integra, vel diuiſa, vt dictum <lb/>eſt.</s> <s xml:id="echoid-s2636" xml:space="preserve">i. </s> <s xml:id="echoid-s2637" xml:space="preserve">omnia plana figuræ, EAG, fient pars omnium planorum fi-<lb/>guræ, GOQ, ſic productorum, nam hæc in illis tota reperientur, <lb/>& </s> <s xml:id="echoid-s2638" xml:space="preserve">aliquid amplius, vnde omnia plana figuræ, GOQ, ſic producta <lb/>erunt, vt effecta ſint maiora omnibus planis figuræ, EAG; </s> <s xml:id="echoid-s2639" xml:space="preserve">eodem <pb o="111" file="0131" n="131" rhead="LIBER II."/> modo oſtendemus nos poſſe ſic producere omnia plana figuræ, EA <lb/>G, vt fiant maiora omnibus planis figuræ, GOQ, ita productis, & </s> <s xml:id="echoid-s2640" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0131-01a" xlink:href="note-0131-01"/> ſic deinceps; </s> <s xml:id="echoid-s2641" xml:space="preserve">ergo omnia plana ſolidarum figurarum, EAG, GO <lb/>Q, ſunt magnitudines inter ſe rationem habentes, quod oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s2642" xml:space="preserve"/> </p> <div xml:id="echoid-div271" type="float" level="2" n="3"> <note position="right" xlink:label="note-0131-01" xlink:href="note-0131-01a" xml:space="preserve">Diffin. 4. <lb/>1. 5. Elem.</note> </div> </div> <div xml:id="echoid-div273" type="section" level="1" n="174"> <head xml:id="echoid-head189" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s2643" xml:space="preserve">_P_Oſſet fortè quis circa hanc demonſtrationem dubitare, nonrectè per-<lb/>cipiens quomodo ind finitæ numero lineæ, vel plana, quales eſſe <lb/>exiſtimari poſſunt, quæà me vocantur, omnes linea, vel omnia plana <lb/>talium, vel talium figurarum poſſint ad inuicem comparari: </s> <s xml:id="echoid-s2644" xml:space="preserve">Propter <lb/>quod innuendum mihi videtur, dum conſidero omnes lineas, vel omnia <lb/>plana alicuius figuræ, me non numerum ipſarum comparare, quem igno-<lb/>ramus, ſed tantum magnitudinem, quæ adæquatur ſpatio ab eiſdem li-<lb/>neis occupato, cum illi congruat, & </s> <s xml:id="echoid-s2645" xml:space="preserve">quoniam illud ſpatium terminis <lb/>comprehenditur, ideò & </s> <s xml:id="echoid-s2646" xml:space="preserve">earum magnitudo eſt terminis eiſdem compre-<lb/>henſa, quapropter illi poteſt fieri additio, vel ſubtractio, licet numerum <lb/>earundem ignoremus; </s> <s xml:id="echoid-s2647" xml:space="preserve">quod ſufficere dico, vt illa ſint ad inuicem compa-<lb/>rabilia, alioquin neque ipſa ſpatia figurarum eſſent ad inuicem compa-<lb/>rabilia: </s> <s xml:id="echoid-s2648" xml:space="preserve">Vel enim continuum nihil ali ud eſt pręter ipſa indiuiſibilia, vel <lb/>aliquid aliud, ſi nihil eſt præter indiuiſibilia, profectò ſi eorum conge-<lb/>ries nequit comparari, neque ſpatium, ſiue continuum, erit comparabi-<lb/>le, cum illud nihil aliud eſſe ponatur, quam ipſa indiuiſibilia: </s> <s xml:id="echoid-s2649" xml:space="preserve">Si Verò <lb/>continuum eſt aliquid aliud præter ipſa indiuiſibilia, fateri æquum eſt hoc <lb/>aliquid aliud interiacere ipſa indiuiſibilia, habemus ergo continuum, <lb/>diſſeparabile in quædam, quæ continuum componunt, numero adbuc in-<lb/>definita, inter quælibet enim duo indiuiſibilia æquum eſt interiacere ali-<lb/>quid illius, quod dictum eſt eſſe aliquid aliud in ipſo continuo præter in-<lb/>diuiſibilia, qua enim ratione tolleretur à medio duarum, à medijs quo-<lb/>que oæterarum tolleretur; </s> <s xml:id="echoid-s2650" xml:space="preserve">hoc cum ita ſit comparare nequibimus ipſa, <lb/>continua, ſiue ſpatia adinuicem, cum ea, quæ colliguntur, & </s> <s xml:id="echoid-s2651" xml:space="preserve">ſimul col-<lb/>lecta comparantur, ſcilicet, quæ continuum componunt, ſint numero in-<lb/>definita, abſurdum, autem eſt dicere coutinua terminis comprehenſa non <lb/>eſſe ad muicem comparabilia, ergo abſurdum eſt dicere congeriem om-<lb/>nium linearum ſiue planorum, duarum quarumlibet figurarum non eſſe <lb/>ad inuicem comparabilem, non obſtante, quod quæ colliguntur, & </s> <s xml:id="echoid-s2652" xml:space="preserve">il-<lb/>lam congeriem componunt ſint numero indefinita, veluti hoc non obſtat <lb/>in continuo, ſiue ergo continuum ex indiuiſibilibus componatur, ſiue, <lb/>non, indiuiſibilium congeries ſunt adinuicem comparabiles, & </s> <s xml:id="echoid-s2653" xml:space="preserve">propor-<lb/>tionem habent.</s> <s xml:id="echoid-s2654" xml:space="preserve"/> </p> <pb o="112" file="0132" n="132" rhead="GEOMETRIÆ"/> <p style="it"> <s xml:id="echoid-s2655" xml:space="preserve">Non inutile autem mibi videtur eſſe animaduerter e pro huius confir-<lb/>matione, hoc pro vero ſuppoſito, quam plurima, quæ ab Euclide, Ar-<lb/>chimede, & </s> <s xml:id="echoid-s2656" xml:space="preserve">alijs oſtenſa ſunt, à me pariter fuiſſe demonſtrata, meaſq; <lb/></s> <s xml:id="echoid-s2657" xml:space="preserve">concluſiones ad vnguem cum illorum concluſionibus concordare, quod <lb/>euidens ſignum eſſe poteſt, me in principijs vera aſſumpſiſſe, licet ſciam, <lb/>& </s> <s xml:id="echoid-s2658" xml:space="preserve">ex falſis principijs ſophiſticè vera aliquando deduci poſſe, quod ta-<lb/>men in tot, & </s> <s xml:id="echoid-s2659" xml:space="preserve">tot concluſionibus, methodo geometrica demonſtratis mihi <lb/>accidiſſe abſurdum putarem: </s> <s xml:id="echoid-s2660" xml:space="preserve">Hoc tamen addo, non tanquam præfatæ ve-<lb/>ritatis legitimum fundamentum, ſed vt non negligendum, immò ſummè <lb/>expendendum illius argumentum, quod ſequentia percurrenti continuò <lb/>magis, ac magis eluceſcet.</s> <s xml:id="echoid-s2661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div274" type="section" level="1" n="175"> <head xml:id="echoid-head190" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s2662" xml:space="preserve">AEqualium planarum figurarum omnes lineæ ſunt ęqua-<lb/>les, & </s> <s xml:id="echoid-s2663" xml:space="preserve">æqualium ſolidarum omnia plana ſunt æqua-<lb/>lia, regula quauis affumpta.</s> <s xml:id="echoid-s2664" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2665" xml:space="preserve">Sint duę æquales planę figuræ, ADC, AEB, in figura, ADC, <lb/>ſit regula, AC, vtcunque, & </s> <s xml:id="echoid-s2666" xml:space="preserve">in figura, AEB, regula vtcunque ſit, <lb/>AB. </s> <s xml:id="echoid-s2667" xml:space="preserve">Dico omnes lineas figuræ, ADC, regula, AC, ęquales eſſe <lb/>omnibus lineis figurę, AEB, regula, AB; </s> <s xml:id="echoid-s2668" xml:space="preserve">intelligatur ſiguram, A <lb/>EB, ita ſuperponi figuræ, ADC, vt regulæ ſint ad inuicem ſuper-<lb/>poſitę, velut eſt, AB, in, AC, vel ſaltem ęquidiſtent, vel ergo tota <lb/>figura congruit toti, vel pars parti, congruat pars parti, ergo con-<lb/> <anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/> <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> gruentium harum partium omnes lineæ erunt <lb/>pariter congruentes, ſcilicet omnes lineę, AD <lb/>B, partis figurę, AEB, erunt congruentes om-<lb/>nibus lineis, ADB, partis figuræ, ADC, ſu <lb/>perponantur adhuc reſiduæ harum figurarum <lb/>partes, hac lege tamen, vt omnes earundem li-<lb/>neæ regulis, AB, AC, fiue regulę communi, <lb/>AB, vel, AC, ſemper ſituentur æquidiſtantes, <lb/>& </s> <s xml:id="echoid-s2669" xml:space="preserve">hoc ſemper fiat, donec omnes refiduę partes ad inuicem ſuperpo-<lb/>ſitæ fuerint, quia ergo integræ figuræ ſunt æquales erunt dictæ par-<lb/>tes ſuperpoſitæ inuicem congruentes, ergo & </s> <s xml:id="echoid-s2670" xml:space="preserve">earum omnes lineæ <lb/>erunt pariter congruentes, magnitudines autem congruentes ſunt <lb/>ad inuicem æquales, ergo omnes lineæ partium figuræ, AEB, ſi-<lb/>mul ſumptarum.</s> <s xml:id="echoid-s2671" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2672" xml:space="preserve">omnes lineæ figuræ, AEB, ſumptæ regula, A <lb/>B, erunt ęquales omnibus lineis partium figurę, ADC, quibus prę-<lb/>dictæ partes congruerunt, ſimul ſumptarum.</s> <s xml:id="echoid-s2673" xml:space="preserve">. omnibus lineis figu- <pb o="113" file="0133" n="133" rhead="LIBER II."/> ræ, ADC, ſumptis, regula, AC, quod in figuris planis oſtenden <lb/>dum erat.</s> <s xml:id="echoid-s2674" xml:space="preserve"/> </p> <div xml:id="echoid-div274" type="float" level="2" n="1"> <note position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">Poſtul. 1. <lb/>huius.</note> <figure xlink:label="fig-0132-01" xlink:href="fig-0132-01a"> <image file="0132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0132-01"/> </figure> </div> <p> <s xml:id="echoid-s2675" xml:space="preserve">Ita ſuperpoſitis æqualibus figuris ſolidis, ita vt duæ in ipſis aſſum-<lb/>ptæ vtcunq; </s> <s xml:id="echoid-s2676" xml:space="preserve">regulæ ſint ad inuicem ſuperpoſitæ, vel æquidiſtantes, <lb/>& </s> <s xml:id="echoid-s2677" xml:space="preserve">reſiduorum facta ſemper ſuperpoſitione ita, vt omnia eorum pla-<lb/>na regulis iam iuperpoſitis ęquidiſtent, tandem, quia figurę ſunt æ-<lb/>quales, dictæ partes erunt ad inuicem congruentes, & </s> <s xml:id="echoid-s2678" xml:space="preserve">conſequen-<lb/>ter integrę quoq; </s> <s xml:id="echoid-s2679" xml:space="preserve">figuræ erunt congruentes, ergo earum omnia pla-<lb/>na ſumpta cum dictis regulis erunt ad inuicem congruentia, ergo & </s> <s xml:id="echoid-s2680" xml:space="preserve"><lb/>æqualia, quod in figuris ſolidis oſtendere quoque opus erat.</s> <s xml:id="echoid-s2681" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div276" type="section" level="1" n="176"> <head xml:id="echoid-head191" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s2682" xml:space="preserve">_H_Inc patet in eadem figura plana, omnes lineas ſumptas cum qua-<lb/>damregula æquart omnibus lineis ſumptis cum alia quauis regu-<lb/>la; </s> <s xml:id="echoid-s2683" xml:space="preserve">& </s> <s xml:id="echoid-s2684" xml:space="preserve">in figuris ſolidis omnia plana vnius ſumpta cum quadam regula <lb/>æquari omnibus planis eiuſdem, regula quauis aſſumpta; </s> <s xml:id="echoid-s2685" xml:space="preserve">vnde ex. </s> <s xml:id="echoid-s2686" xml:space="preserve">gr. <lb/></s> <s xml:id="echoid-s2687" xml:space="preserve">ſecto planis cylindro æquidiſtanter axi, qua ſectione in ipſo creantur pa-<lb/> <anchor type="note" xlink:label="note-0133-01a" xlink:href="note-0133-01"/> rallelogramma, & </s> <s xml:id="echoid-s2688" xml:space="preserve">ſecto eodem planis æquidistanter baſi ductis, qua ſe-<lb/>ctione creantur in eodem circuli, patet ex hoc, omnia parallelogramma <lb/> <anchor type="note" xlink:label="note-0133-02a" xlink:href="note-0133-02"/> dicti cylindri, regula eorundem vno, eſſe æqualia omnibus circulis eiu-<lb/>ſdem, regula baſi.</s> <s xml:id="echoid-s2689" xml:space="preserve"/> </p> <div xml:id="echoid-div276" type="float" level="2" n="1"> <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">_Coroll. 6._ <lb/>_lib. 1._</note> <note position="right" xlink:label="note-0133-02" xlink:href="note-0133-02a" xml:space="preserve">_Corol. 12._ <lb/>_lib. 1._</note> </div> </div> <div xml:id="echoid-div278" type="section" level="1" n="177"> <head xml:id="echoid-head192" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s2690" xml:space="preserve">FIguræ planæ habent inter ſe eandem rationem, quam <lb/>eorum omnes lineæ iuxta quaniuis regulam aſſumptæ; <lb/></s> <s xml:id="echoid-s2691" xml:space="preserve">Et figuræ ſolidæ, quam eorum omnia plana iuxta quamuis <lb/>regulam aſſumpta.</s> <s xml:id="echoid-s2692" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2693" xml:space="preserve">Sint figuræ planæ vtcunque, A, D. </s> <s xml:id="echoid-s2694" xml:space="preserve">Dico, <lb/> <anchor type="figure" xlink:label="fig-0133-01a" xlink:href="fig-0133-01"/> A, figuram ad nguram, D, eſſe vt omnes lineę <lb/>figuræ, A, iuxta quamuis regulam aſſumptæ <lb/>ad omnes lineas figuræ, D, iuxta quamuis re-<lb/>gulam aſſumptas. </s> <s xml:id="echoid-s2695" xml:space="preserve">Intelligantur ergo omnes <lb/>lineæ figuræ, A, &</s> <s xml:id="echoid-s2696" xml:space="preserve">, D, aſlumptæ iuxta qua-<lb/>ſdam regulas, deinde capiantur quotcunque fi-<lb/>guræ, BC, ſingulæ æquales figuræ, A, & </s> <s xml:id="echoid-s2697" xml:space="preserve">fi <lb/>guræ, D, quotcunque ęquales figurę, vt, E; </s> <s xml:id="echoid-s2698" xml:space="preserve">nunc, ſi continuum <lb/>componitur ex indiuiſibilibus, patet abſque alia demonſtratione fi-<lb/>guram, A, ad figuram, D, eſſe vt omnes lineæ figurę, A, ad om- <pb o="114" file="0134" n="134" rhead="GEOMETRIÆ"/> nes lineas figurę, D, tunc enim comparare continuum ad continuum <lb/>non eſſet niſi ipſa indiuiſibilia comparare; </s> <s xml:id="echoid-s2699" xml:space="preserve">ſed eſto, quod hoc ſit fal-<lb/>ſum, vel quod, etiamſi verum ſit, tamen legitima ratione ad hoc pro-<lb/>bandum nondum peruenerimus; </s> <s xml:id="echoid-s2700" xml:space="preserve">nihilominus adhuc dico ipſa indi-<lb/>uiſibilia. </s> <s xml:id="echoid-s2701" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2702" xml:space="preserve">omnes lineas figurę, A, ad omnes lineas figuræ, D, eſſe <lb/>vt figuram, A, ad figuram, D. </s> <s xml:id="echoid-s2703" xml:space="preserve">Quoniam ergo aſſumpſimus figu-<lb/>ras, B, C, ſingulas æquales figuræ, A, &</s> <s xml:id="echoid-s2704" xml:space="preserve">, E, æqualem figuræ, D, <lb/>omnes lineæ ſingularum figurarum, A, B, C, erunt æquales omni-<lb/> <anchor type="note" xlink:label="note-0134-01a" xlink:href="note-0134-01"/> bus lineis figuræ, A, ſumptis iuxta dictam regulam (quacunque re-<lb/>gula dictæ omnes lineæ ſint aſſumptæ) & </s> <s xml:id="echoid-s2705" xml:space="preserve">ideò quotuplex erit com-<lb/>poſitum ex figuris, ABC, figuræ, A, totuplex erit compoſitum ex <lb/>omnibus lineis figurarum, ABC, omnium linearum figuræ, A, & </s> <s xml:id="echoid-s2706" xml:space="preserve"><lb/>ideò habebimus æquè multiplicia primæ, & </s> <s xml:id="echoid-s2707" xml:space="preserve">tertiæ vtcunq; </s> <s xml:id="echoid-s2708" xml:space="preserve">ſumpta; <lb/></s> <s xml:id="echoid-s2709" xml:space="preserve">ſimiliter oſtendemus compoſitum ex figuris, E, D, æquè multiplex <lb/> <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> eſſe figuræ, D, ac compoſitum ex omnibus li-<lb/>neis figurarum, E, D, multiplex eſt omnium <lb/>linearum figuræ, D, quæ ſunt æquè multipli-<lb/>cia ſecundæ, & </s> <s xml:id="echoid-s2710" xml:space="preserve">quartę vtcunque ſumpta, quia <lb/>ergo ſi multiplex primę. </s> <s xml:id="echoid-s2711" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2712" xml:space="preserve">compoſitum ex figu <lb/>ris, ABC, ſuperauerit multiplex ſecundę, ſci-<lb/>licet compoſitum ex figuris, DE, etiam mul-<lb/>tiplex tertiæ. </s> <s xml:id="echoid-s2713" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2714" xml:space="preserve">compoſitum ex omnibus lineis <lb/>figurarum, ABC, ſuperabit multiplex quartæ <lb/>.</s> <s xml:id="echoid-s2715" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2716" xml:space="preserve">compoſitum ex omnibus lineis figurarum, DE, & </s> <s xml:id="echoid-s2717" xml:space="preserve">ſi multiplex pri-<lb/> <anchor type="note" xlink:label="note-0134-02a" xlink:href="note-0134-02"/> mæ fuerit æquale multiplici ſecundæ, etiam multiplex tertię erit æ-<lb/>quale multiplici quarte, ſcilicet ſi compoſitum ex figuris, ABC, <lb/>fuerit æquale compoſito ex figuris, DE, etiam eorundem compo-<lb/>ſitorum omnes lineæ erunt æquales, & </s> <s xml:id="echoid-s2718" xml:space="preserve">ſi minus, minus, ideò prima <lb/> <anchor type="note" xlink:label="note-0134-03a" xlink:href="note-0134-03"/> ad ſecundam erit, vt tertia ad quartam, ſcilicet figura, A, ad figu-<lb/>ram, D, erit vt omnes lineæ figuræ, A, ad omnes lineas figuræ, D, <lb/>ſumptas iuxta datas regulas. </s> <s xml:id="echoid-s2719" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2720" xml:space="preserve">iuxta quaſcunq; </s> <s xml:id="echoid-s2721" xml:space="preserve">regulas, quod in fig. <lb/></s> <s xml:id="echoid-s2722" xml:space="preserve"> <anchor type="note" xlink:label="note-0134-04a" xlink:href="note-0134-04"/> planis erat oſtendendum.</s> <s xml:id="echoid-s2723" xml:space="preserve"/> </p> <div xml:id="echoid-div278" type="float" level="2" n="1"> <figure xlink:label="fig-0133-01" xlink:href="fig-0133-01a"> <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0133-01"/> </figure> <note position="left" xlink:label="note-0134-01" xlink:href="note-0134-01a" xml:space="preserve">Perante-<lb/>ctd.</note> <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a"> <image file="0134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0134-01"/> </figure> <note position="left" xlink:label="note-0134-02" xlink:href="note-0134-02a" xml:space="preserve">Elicitur <lb/>ex antec.</note> <note position="left" xlink:label="note-0134-03" xlink:href="note-0134-03a" xml:space="preserve">Defin. 5. <lb/>Qui. El.</note> <note position="left" xlink:label="note-0134-04" xlink:href="note-0134-04a" xml:space="preserve">Coroll. I. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s2724" xml:space="preserve">Verum ſi intellexerimus, A, D, eſſe figuras ſolidas, aſſumentes, <lb/>C, B, ſingulas æquales ipſi, A, &</s> <s xml:id="echoid-s2725" xml:space="preserve">, E, ipſi, D, oſtendemus com-<lb/>poſitum ex figuris, ABC, tam multiplex eſſe figurę, A, ac compo-<lb/>ſitum ex omnibus planis figurarum, A, B, C, multiplex eſt omnium <lb/>planorum figurę, A, & </s> <s xml:id="echoid-s2726" xml:space="preserve">ſic compoſitum ex figuris, D, E, tam mul-<lb/>tiplex eſſe figuræ, D, ac compoſitum ex omnibus planis figurarum, <lb/>DE, multiplex eſt omnium planorum figuræ, D, & </s> <s xml:id="echoid-s2727" xml:space="preserve">tandem per <lb/>antecedentem Propoſitionem oſtendemus, ſi multiplex primæ ſupe-<lb/>rauerit multiplex ſecundę, etiam multiplex tertiæ ſuperaturum mul-<lb/>tiplex quartæ, & </s> <s xml:id="echoid-s2728" xml:space="preserve">ſi minus, minus, vel ſi æquale, & </s> <s xml:id="echoid-s2729" xml:space="preserve">ęquale fore, er- <pb o="115" file="0135" n="135" rhead="LIBER II."/> go prima ad ſecundam erit, vt tertia ad quartam, ſcilicet figura ſo-<lb/> <anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/> lida, A, ad figuram ſolidam, D, erit vt omnia plana, A, ad om-<lb/>nia plana, D, cum quibuſuis regulis aſſumpta, quod & </s> <s xml:id="echoid-s2730" xml:space="preserve">in figuris ſo-<lb/>lidis oſtendere opus erat.</s> <s xml:id="echoid-s2731" xml:space="preserve"/> </p> <div xml:id="echoid-div279" type="float" level="2" n="2"> <note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">Def. Qui. <lb/>5. Elem.</note> </div> </div> <div xml:id="echoid-div281" type="section" level="1" n="178"> <head xml:id="echoid-head193" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2732" xml:space="preserve">_L_Iquet ex hoc, quod, vt inueniamus, quam rationem habeant inter <lb/>ſe duæ figuræ planæ, vel ſolidæ, ſuſſiciet nobis reperire, quam, in <lb/>figuris planis, inter ſe rationem habeant earundem omnes lineæ, &</s> <s xml:id="echoid-s2733" xml:space="preserve">, in <lb/>figuris ſolidis, earundem omnia plana iuxta quamuis regulam aſſumpta, <lb/>quod nouæ huius meæ Geometriæ veluti maximum iacio fundamentum,.</s> <s xml:id="echoid-s2734" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div282" type="section" level="1" n="179"> <head xml:id="echoid-head194" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s2735" xml:space="preserve">SI duæ figuræ planæ, vel ſolidæ, in eadem altitudine fue-<lb/>rint conſtitutæ, ductis autem in planis rectis lineis, & </s> <s xml:id="echoid-s2736" xml:space="preserve"><lb/>in figuris ſolidis ductis planis vtcumque inter ſe parallelis, <lb/>quorum reſpectu prædicta ſumpta ſit altitudo, repertum fue-<lb/>rit ductarum linearum portiones figuris planis interceptas, <lb/>ſeu ductorum planorum portiones figuris ſolidis interceptas, <lb/>eſſe magnitudines proportionales, homologis in eadem figu-<lb/>ra ſemper exiſtentibus, dictæ figuræ erunt inter ſe, vt vnum <lb/>quodlibet eorum antecedentium, ad ſuum conſequens in a-<lb/>lia figura eidem correſpondens.</s> <s xml:id="echoid-s2737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2738" xml:space="preserve">Sint primò duæ figurę planæ in eadem altitudine conſtitutæ, CA <lb/>M, CME, in quibus duæ vtcunque rectæ lineæ inuicem parallelæ <lb/>ductæ intelligantur, AE, BD, reſpectu quarum communis altitu-<lb/> <anchor type="figure" xlink:label="fig-0135-01a" xlink:href="fig-0135-01"/> do aſſumpta intelligatur, ſint au-<lb/>tem portiones figuris interceptæ <lb/>ipſæ, AM, BR, in fig. </s> <s xml:id="echoid-s2739" xml:space="preserve">CAM, <lb/>&</s> <s xml:id="echoid-s2740" xml:space="preserve">, ME, RD, in fig. </s> <s xml:id="echoid-s2741" xml:space="preserve">CME, <lb/>reperiatur autem, vt, AM, ad, <lb/>ME, ita eſſe, BR, ad, RD. <lb/></s> <s xml:id="echoid-s2742" xml:space="preserve">Dico figu am, CAM, ad figu-<lb/>ram, CME, eſſe vt, AM, ad, <lb/>ME, vel, BR, ad, RD, quoniam enim, BD, AE, vtcumq; </s> <s xml:id="echoid-s2743" xml:space="preserve">du-<lb/>ctæ ſunt inter ſe æquidiſtantes, patet, quod quęlibet earum, quę di-<lb/>cuntur omnes lineæ figuræ, CAM, ſumptæ regula altera ipſarum, <pb o="116" file="0136" n="136" rhead="GEOMETRIÆ"/> AM, BR, ad eam, quæ illi indirectum iacet in figura, CME, erit <lb/>vt, BR, ad, RD, vel vt, AM, ad, ME, vt igitur, AM, ad, M <lb/>E, vnum .</s> <s xml:id="echoid-s2744" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2745" xml:space="preserve">antecedentium ad vnum conſequentium, ita erunt om-<lb/>nia antecedentia, nempè omnes lineę figurę, CAM, regula, AM <lb/> <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> ad omnia conſequentia, ſcilicet <lb/>ad omnes lineas figuræ, CME, <lb/>regula, ME; </s> <s xml:id="echoid-s2746" xml:space="preserve">indefinitus .</s> <s xml:id="echoid-s2747" xml:space="preserve">n. </s> <s xml:id="echoid-s2748" xml:space="preserve">nu-<lb/>merus omnium antecedentium, <lb/>& </s> <s xml:id="echoid-s2749" xml:space="preserve">conſequentium, qui pro vtriſ-<lb/>que hic idem eſt, quicunque ſit <lb/>(& </s> <s xml:id="echoid-s2750" xml:space="preserve">hoc nam figuræ ſunt in ea-<lb/>dem altitudine, & </s> <s xml:id="echoid-s2751" xml:space="preserve">cuilibet ante-<lb/>cedenti in figura, CAM, aſſumpto reſpondet ſuum conſequens illi <lb/>in directum in alia figura conſtitutum) non obſtat quin omnes lineę <lb/>figurę, CAM, ſint comparabiles omnibus lineis figurę, CME, cum <lb/> <anchor type="note" xlink:label="note-0136-01a" xlink:href="note-0136-01"/> ad illas rationem habeant, vt probatum eſt, & </s> <s xml:id="echoid-s2752" xml:space="preserve">ideò omnes lineæ fi-<lb/>guræ, CAM, regula, AM, ad omnes lineas figurę, CME, regu-<lb/>la, ME, erunt vt, AM, ad, ME, verum, vt omnes lineæ figuræ, <lb/>CAM, ad omnes lineas figurę, CME, ita fig. </s> <s xml:id="echoid-s2753" xml:space="preserve">CAM, eſt ad figu-<lb/> <anchor type="note" xlink:label="note-0136-02a" xlink:href="note-0136-02"/> ram, CME, ergo figura, CAM, ad figuram, CME, erit vt, B <lb/>R, ad, RD, vel, AM, ad, ME, quod in figuris planis oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s2754" xml:space="preserve"/> </p> <div xml:id="echoid-div282" type="float" level="2" n="1"> <figure xlink:label="fig-0135-01" xlink:href="fig-0135-01a"> <image file="0135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0135-01"/> </figure> <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a"> <image file="0136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0136-01"/> </figure> <note position="left" xlink:label="note-0136-01" xlink:href="note-0136-01a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-0136-02" xlink:href="note-0136-02a" xml:space="preserve">3. huius.</note> </div> <p> <s xml:id="echoid-s2755" xml:space="preserve">Si verò ſupponamus, CAM, CME, eſſe figuras ſolidas, & </s> <s xml:id="echoid-s2756" xml:space="preserve">vice <lb/>rectarum, AM, BR, ME, RD, plana intelligamus figuris, CA <lb/>M, CME, intercepta inuicem parallela, & </s> <s xml:id="echoid-s2757" xml:space="preserve">ita conſtituta, vt plana, <lb/>AM, ME, iaceant in eodem plano, veluti ſe habeant etiam plana, <lb/>BR, RD, reſpectu quorum præfata altitudo aſſumpta quoq; </s> <s xml:id="echoid-s2758" xml:space="preserve">intel-<lb/>ligatur, eadem methodo procedentes oſtendemus omnia plana figu-<lb/>ræ, CAM, ad omnia plana figuræ, CME, ideſt figuram ſolidam, <lb/>CAM, ad figuram ſolidam, CME, eſſe vt planum, BR, ad pla-<lb/> <anchor type="note" xlink:label="note-0136-03a" xlink:href="note-0136-03"/> num, RD, vel vt planum, AM, ad planum, ME, quod & </s> <s xml:id="echoid-s2759" xml:space="preserve">in ſoli-<lb/>dis oſtendere opus erat.</s> <s xml:id="echoid-s2760" xml:space="preserve"/> </p> <div xml:id="echoid-div283" type="float" level="2" n="2"> <note position="left" xlink:label="note-0136-03" xlink:href="note-0136-03a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div285" type="section" level="1" n="180"> <head xml:id="echoid-head195" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2761" xml:space="preserve">_C_Olligitur ex hoc in figuris planis, vel ſolidis, ſi magnitudines com-<lb/>paratæ ſint lineæ rectæ, vel plana, ſint autem illæ, quæ dicuntur <lb/>omnes lineæ, vel omnia plana dictarum figurarum, de illis quoq; </s> <s xml:id="echoid-s2762" xml:space="preserve">verifi-<lb/>cari, vt vnum antecedentium ad vnum conſequentium, ita eſſe omnia, <lb/>antecedentia ad omnia conſequentia; </s> <s xml:id="echoid-s2763" xml:space="preserve">& </s> <s xml:id="echoid-s2764" xml:space="preserve">in ſupradictis figuris planis <lb/>omnes lineas vnius ad omnes lineas alterius, vel in ſolidis omnia plana <lb/>vnius ad omnia plana alterius, eſſe vt vnum antecedentium ad vnum, <pb o="117" file="0137" n="137" rhead="LIBER II."/> conſequentium, iuxta quæ, tanquam regulas, dictæ omnes lineæ, vel <lb/>omnia plana intelliguntur aſſumpta.</s> <s xml:id="echoid-s2765" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div286" type="section" level="1" n="181"> <head xml:id="echoid-head196" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s2766" xml:space="preserve">PArallelogramma in eadem altitudine exiſtentia inter ſe <lb/>ſunt, vt baſes; </s> <s xml:id="echoid-s2767" xml:space="preserve">& </s> <s xml:id="echoid-s2768" xml:space="preserve">quę in eadem baſi, vt altitudines, vel, <lb/>vt latera æqualiter baſibus inclinata.</s> <s xml:id="echoid-s2769" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2770" xml:space="preserve">Sint parallelogramma quæcunque, AM, MC, in eadem altitu-<lb/>dine conftituta, ſumpta altitudine iuxta baſes, GM, MH. </s> <s xml:id="echoid-s2771" xml:space="preserve">Dico <lb/>parallelogrammum, AM, ad parallelogrammum, MC, eſſe vt, G <lb/>M, ad, MH. </s> <s xml:id="echoid-s2772" xml:space="preserve">Ducatur quęcunq; </s> <s xml:id="echoid-s2773" xml:space="preserve">intra parallelogramma, AM, M <lb/> <anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/> C, parallela ipſis, GM, MH, cu-<lb/>ius portiones parallelogrammis, <lb/>AM, MC, interceptę ſint, DE, <lb/>EI. </s> <s xml:id="echoid-s2774" xml:space="preserve">Quoniam ergo, DM, eſt <lb/>parallelogrammum, ſicut &</s> <s xml:id="echoid-s2775" xml:space="preserve">, E <lb/>H, erit, DE, ęqualis ipſi, GM, <lb/>&</s> <s xml:id="echoid-s2776" xml:space="preserve">, EI, ipſi, MH, erit igitur, G <lb/>M, ad, MH, vt, DE, ad, EI, & </s> <s xml:id="echoid-s2777" xml:space="preserve">DE, EI, ductæ ſunt vtcunq; <lb/></s> <s xml:id="echoid-s2778" xml:space="preserve">parallelæ ipſis, GM, MH, ergo parallelogramma, AM, MC, e-<lb/>runt ex genere figurarum Theorematis anteced. </s> <s xml:id="echoid-s2779" xml:space="preserve">ergo, AM, ad, M <lb/>C, erit vt, DE, ad, EI, vel vt, GM, ad, MH, quæ ſunt eorun-<lb/>dem baſes. </s> <s xml:id="echoid-s2780" xml:space="preserve">Hæc autem verificabuntur etiam ſi altitudines æquales <lb/>fuerint, vt facilè patet.</s> <s xml:id="echoid-s2781" xml:space="preserve"/> </p> <div xml:id="echoid-div286" type="float" level="2" n="1"> <figure xlink:label="fig-0137-01" xlink:href="fig-0137-01a"> <image file="0137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0137-01"/> </figure> </div> <p> <s xml:id="echoid-s2782" xml:space="preserve">Sint nunc parallelogramma, QP, LP, in eadem baſi, NP, con-<lb/>ſtituta. </s> <s xml:id="echoid-s2783" xml:space="preserve">Dico eadem eſſe, vt altitudines ſumptæ iuxta baſim, NP, <lb/>demittantur ergo, OR, TS, altitudines in, NP, productam, in <lb/>punctis, RS, illi occurrentes (niſi fortè, TP, OP, eſſent ipſæ alti-<lb/>tudines, vel intra parallelogramma inciderent baſi, NP,) & </s> <s xml:id="echoid-s2784" xml:space="preserve">à pun-<lb/>ctis, Q, L, illis parallelæ, QX, LV, in punctis, V, X, baſi, NP, <lb/>incidentes, ſuntigitur parallelogramma, QS, LR, in ęqualibus al-<lb/>titudinibus, QT, LO, ſumptis iuxta baſes, TS, OR, ergo paral-<lb/> <anchor type="note" xlink:label="note-0137-01a" xlink:href="note-0137-01"/> lelogramma, QS, LR, erunt inter ſe, vt baſes, TS, OR, eſt au-<lb/>tem parallelogrammum, QS, æquale parallelogrammo, QP, &</s> <s xml:id="echoid-s2785" xml:space="preserve">, <lb/>LR, ipſi, LP, ergo parallelogramma, QP, LP, erunt inter ſe, vt, <lb/>TS, OR, quæ pro ipſis ſunt altitudines ſumptæ iuxta baſim, NP. <lb/></s> <s xml:id="echoid-s2786" xml:space="preserve">Si autem latus, OP, extenderetur ſuper latus, PT, ideſt latera, O <lb/>P, PT, eſſent ęqualiter inclinata communi baſi, NP, tunc ſumptis <lb/>pro baſibus ipſis, TP, OP, haberemus parallelogramma, QP, L <pb o="118" file="0138" n="138" rhead="GEOMETRIÆ"/> P, in eadem altitudine ſumpta iuxta baſes, TP, OP, & </s> <s xml:id="echoid-s2787" xml:space="preserve">ideò eſſent, <lb/> <anchor type="note" xlink:label="note-0138-01a" xlink:href="note-0138-01"/> vt ipſæ baſes, TP, OP, ideſt vt latera, TP, OP, æqualiter baſi, <lb/>NP, inclinata, hæc autem pariter verificabuntur etiamſi baſis, NP, <lb/>non ſit communis, ſint tamen duæ baſes æquales, quæ oſtendere o-<lb/>pus erat.</s> <s xml:id="echoid-s2788" xml:space="preserve"/> </p> <div xml:id="echoid-div287" type="float" level="2" n="2"> <note position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">Ex prima <lb/>parte hu-<lb/>ius Prop.</note> <note position="left" xlink:label="note-0138-01" xlink:href="note-0138-01a" xml:space="preserve">Ex prima <lb/>par. huius <lb/>Propoſ.</note> </div> </div> <div xml:id="echoid-div289" type="section" level="1" n="182"> <head xml:id="echoid-head197" xml:space="preserve">THEOREMA VI. PROPOS. VI.</head> <p> <s xml:id="echoid-s2789" xml:space="preserve">PArallelogramma habent rationem compoſitam ex ra-<lb/>tione baſium, & </s> <s xml:id="echoid-s2790" xml:space="preserve">altitudinum iuxta eaſdem baſes ſum-<lb/>ptarum, ſiue laterum æqualiter baſibus inclinatorum, cum <lb/>ſcilicetilla ſunt æquiangula.</s> <s xml:id="echoid-s2791" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2792" xml:space="preserve">Sint parallelogramma vtcunque, AD, FM. </s> <s xml:id="echoid-s2793" xml:space="preserve">Dico eadem habe. <lb/></s> <s xml:id="echoid-s2794" xml:space="preserve">re inter ſe rationem compoſitam ex rationibus baſium, quæ ſint, C <lb/> <anchor type="note" xlink:label="note-0138-02a" xlink:href="note-0138-02"/> D, GM, & </s> <s xml:id="echoid-s2795" xml:space="preserve">altitudinum, quę ſint, BV, ON, ſumptæ iuxta baſes, <lb/>CD, GM, illiſque productis, ſi opus ſit, in punctis, V, N, occur-<lb/>rentes, ſiue ex ratione laterum, BD, OM, ſi ſint æquiangula: </s> <s xml:id="echoid-s2796" xml:space="preserve">Ab-<lb/> <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> ſcindatur à, BV, verſus, V, ipſa, <lb/>XV, æqualis ipſi, ON, & </s> <s xml:id="echoid-s2797" xml:space="preserve">per, X, <lb/>ducatur, XP, parallela, CD, ſe-<lb/>cans, BD, in, R, vt fiat paralle-<lb/>logrammum, PD, in eadem altitu-<lb/>dine cum parallelogrammo, FM, <lb/>& </s> <s xml:id="echoid-s2798" xml:space="preserve">in eadẽ baſi cum parallelogram-<lb/>mo, AD. </s> <s xml:id="echoid-s2799" xml:space="preserve">Parallelogrammum er-<lb/>go, AD, ad parallelogrammum, <lb/>FM, ſumpto medio de foris parallelogrammo, PD, habet ratio-<lb/> <anchor type="note" xlink:label="note-0138-03a" xlink:href="note-0138-03"/> nem compoſitam ex ratione parallelogrammi, AD, ad parallelo-<lb/>grammum, PD, ideſt ex ratione, quam habet, BV, ad, VX, vel, <lb/>ON, ſiue, BD, ad, DR, quoniam, AD, PD, ſunt æquiangu-<lb/>la, ideſt ex ratione, BD, ad, OM, & </s> <s xml:id="echoid-s2800" xml:space="preserve">hoc quotieſcunque, PD, F <lb/> <anchor type="note" xlink:label="note-0138-04a" xlink:href="note-0138-04"/> M, ſint pariter æquiangula, & </s> <s xml:id="echoid-s2801" xml:space="preserve">inſuper eſt compoſita ex ea, quam <lb/>habet parallelogrammum, PD, ad parallelogrammum, FM, ideſt <lb/>ex ea, quam habet, CD, ad, GM, ergo parallelogrammum, AD, <lb/> <anchor type="note" xlink:label="note-0138-05a" xlink:href="note-0138-05"/> ad parallelogrammum, FM, habet rationem compoſitam ex ea, <lb/>quam habet, BV, ad, ON, quæ ſunt altitudines, vel etiam ex ea, <lb/>quam habet, BD, ad, OM, ſi, AD, FM, ſint æquiangula; </s> <s xml:id="echoid-s2802" xml:space="preserve">& </s> <s xml:id="echoid-s2803" xml:space="preserve">ex <lb/>ea, quam habet, CD, ad, GM, quod oſtendere opus erat.</s> <s xml:id="echoid-s2804" xml:space="preserve"/> </p> <div xml:id="echoid-div289" type="float" level="2" n="1"> <note position="left" xlink:label="note-0138-02" xlink:href="note-0138-02a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a"> <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0138-01"/> </figure> <note position="left" xlink:label="note-0138-03" xlink:href="note-0138-03a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0138-04" xlink:href="note-0138-04a" xml:space="preserve">Exſecun. <lb/>par. ant.</note> <note position="left" xlink:label="note-0138-05" xlink:href="note-0138-05a" xml:space="preserve">Ex prima <lb/>parte an-<lb/>teced.</note> </div> <pb o="119" file="0139" n="139" rhead="LIBER II."/> </div> <div xml:id="echoid-div291" type="section" level="1" n="183"> <head xml:id="echoid-head198" xml:space="preserve">THEOREMA VII. PROPOS. VII.</head> <p> <s xml:id="echoid-s2805" xml:space="preserve">PArallelogramma, quorum baſes altitudinibus, vellate-<lb/>ribus æqualiter baſibus inclinatis, reciprocantur, ſunt <lb/>æqualia, & </s> <s xml:id="echoid-s2806" xml:space="preserve">quæ ſunt æqualia, baſes habent altitudinibus, <lb/>vel lateribus æqualiter baſibus inclinatis, reciprocas.</s> <s xml:id="echoid-s2807" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2808" xml:space="preserve">Sint parallelogramma, HX, AD, quorum baſes, VX, BD, re-<lb/>ciprocentur eorum altitudinibus, CO, RZ, vel lateribus, CD, R <lb/>X, quotieſcunq; </s> <s xml:id="echoid-s2809" xml:space="preserve">ſint æqualiter baſibus inclinata. </s> <s xml:id="echoid-s2810" xml:space="preserve">Dico hæc paral-<lb/>lelogramma eſſe æqualia; </s> <s xml:id="echoid-s2811" xml:space="preserve">etenim parallelogrammum, HX, ad pa-<lb/>rallelogrammum, AD, habet rationem compoſitam ex ea, quam <lb/>habet, VX, ad, BD, &</s> <s xml:id="echoid-s2812" xml:space="preserve">, RZ, ad, CO, ſiue, RX, ad, CD, cum <lb/> <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> illa ſunt æquiangula, eſt autem, vt, <lb/>VX, ad, BD, ita, CO, ad, RZ, <lb/>vel, CD, ad, RX, cum illa ſunt <lb/> <anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> æquiangula, ergo parallelogram-<lb/>mum, HX, ad parallelogrammum, <lb/>AD, habet rationem compoſitam <lb/>ex ea, quam habet, CO, ad, RZ, <lb/>&</s> <s xml:id="echoid-s2813" xml:space="preserve">, RZ, ad, CO, ſiue ex ea, quam <lb/>habet, CD, ad, RX, &</s> <s xml:id="echoid-s2814" xml:space="preserve">, RX, ad, CD, quæ eſt eadem ei, quam <lb/> <anchor type="note" xlink:label="note-0139-02a" xlink:href="note-0139-02"/> habet, CD, ad, CD, vt illa eſt eadem ei, quam habet, CO, ad, <lb/>CO, ſuntque proportiones æqualitatis, ergo parallelogrammum, <lb/>HX, erit æquale parallelogrammo, AD.</s> <s xml:id="echoid-s2815" xml:space="preserve"/> </p> <div xml:id="echoid-div291" type="float" level="2" n="1"> <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a"> <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0139-01"/> </figure> <note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">Ex ante-<lb/>ced.</note> <note position="right" xlink:label="note-0139-02" xlink:href="note-0139-02a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> </div> <p> <s xml:id="echoid-s2816" xml:space="preserve">Sint nunc parallelogrammum, HX, æquale parallelogrammo, A <lb/>D. </s> <s xml:id="echoid-s2817" xml:space="preserve">Dico, vt, VX, ad, BD, ita eſſe, CO, ad, RZ, vel, CD, ad, <lb/>RX, cum ſunt æquiangula. </s> <s xml:id="echoid-s2818" xml:space="preserve">Quoniam ergo parallelogrammum, H <lb/>X, eſt æquale parallelogrammo, AD, erit ad illud, vt, CO, ad, <lb/>CO, vel vt, CD, ad, CD, ideſt (de foris ſumpto, RZ, vel pro <lb/>ſecunda ratione, RX,) inratione compoſita ex ea, quam habet, C <lb/> <anchor type="note" xlink:label="note-0139-03a" xlink:href="note-0139-03"/> O, ad, RZ, & </s> <s xml:id="echoid-s2819" xml:space="preserve">ex ea, quam habet, RZ, ad, CO, vel ex ea, quam <lb/>habet, CD, ad, RX, &</s> <s xml:id="echoid-s2820" xml:space="preserve">, RX, ad, CD, verum, HX, ad, AD, <lb/> <anchor type="note" xlink:label="note-0139-04a" xlink:href="note-0139-04"/> habet etiam rationem compoſitam ex ea, quam habet, VX, ad, B <lb/>D, &</s> <s xml:id="echoid-s2821" xml:space="preserve">, RZ, ad, CO, vel, RX, ad, CD, cum ſunt æquiangula, <lb/>ergo duæ rationes, CO, ad, RZ, &</s> <s xml:id="echoid-s2822" xml:space="preserve">, RZ, ad, CO, vel, CD, ad, <lb/>RX, &</s> <s xml:id="echoid-s2823" xml:space="preserve">, RX, ad, CD, componunt eandem rationem, quam iſtę <lb/>duæ.</s> <s xml:id="echoid-s2824" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2825" xml:space="preserve">VX, ad, BD, &</s> <s xml:id="echoid-s2826" xml:space="preserve">, RZ, ad, CO, vel, RX, ad, CD, eſt <lb/>autem communis ratio, quam habet, RZ, ad, CO, vel, RX, ad, <lb/>CD, ergo reliqua ratio .</s> <s xml:id="echoid-s2827" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2828" xml:space="preserve">quam habet, VX, ad, BD, erit eadem <pb o="120" file="0140" n="140" rhead="GEOMETRIÆ"/> ei, quam habet, CO, ad, RZ, vel, CD, ad, RX, cum ſunt æqui-<lb/>angula, ergo æqualia parallelogramma baſes habent altitudinibus, <lb/>vel lateribus æqualiter baſibus inclinatis reciprocas, quod oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s2829" xml:space="preserve"/> </p> <div xml:id="echoid-div292" type="float" level="2" n="2"> <note position="right" xlink:label="note-0139-03" xlink:href="note-0139-03a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0139-04" xlink:href="note-0139-04a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div294" type="section" level="1" n="184"> <head xml:id="echoid-head199" xml:space="preserve">THEOREMA VIII. PROPOS. VIII.</head> <p> <s xml:id="echoid-s2830" xml:space="preserve">SImilia parallelogramma ſunt in dupla ratione laterum <lb/>homologorum.</s> <s xml:id="echoid-s2831" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2832" xml:space="preserve">Sint ſimilia parallelogramma, AC, EG. </s> <s xml:id="echoid-s2833" xml:space="preserve">Dico eadem eſſe in du-<lb/> <anchor type="note" xlink:label="note-0140-01a" xlink:href="note-0140-01"/> plarat one laterum homologorum: </s> <s xml:id="echoid-s2834" xml:space="preserve">Quoniam enim ſunt ſimilia illa <lb/>ſunt æquiangula, ſint anguli, BCD, FGH, æquales, & </s> <s xml:id="echoid-s2835" xml:space="preserve">latera ho-<lb/> <anchor type="figure" xlink:label="fig-0140-01a" xlink:href="fig-0140-01"/> mologa, BC, FG; </s> <s xml:id="echoid-s2836" xml:space="preserve">CD, GH, ſi ergo pro <lb/>baſibus ſumpierimus ipſas, BC, FG, erit, <lb/> <anchor type="note" xlink:label="note-0140-02a" xlink:href="note-0140-02"/> AC, ad, EG, in ratione compoſita ex ea, <lb/>quam habet, BC, ad, FG, & </s> <s xml:id="echoid-s2837" xml:space="preserve">ex ea, quam <lb/>habet, DC, ad, HG, quæ eſt eadem ei, <lb/>quam habet, BC, ad, FG, vel, FG, ad <lb/>tertiam proportionalem duaru, primę nem-<lb/>pè, BC, & </s> <s xml:id="echoid-s2838" xml:space="preserve">ſecundæ, FG, ergo, AC, ad, EG, erit vt, BC, ad ter-<lb/>tiam proportionalem duarum primę, nempè, BC, & </s> <s xml:id="echoid-s2839" xml:space="preserve">@ecundę, FG, <lb/>.</s> <s xml:id="echoid-s2840" xml:space="preserve">i. </s> <s xml:id="echoid-s2841" xml:space="preserve">erit in dupla ratione eius, quam habet, BC, ad, FG, vel, CD, <lb/> <anchor type="note" xlink:label="note-0140-03a" xlink:href="note-0140-03"/> ad, GH, quod oſtendere opus erat.</s> <s xml:id="echoid-s2842" xml:space="preserve"/> </p> <div xml:id="echoid-div294" type="float" level="2" n="1"> <note position="left" xlink:label="note-0140-01" xlink:href="note-0140-01a" xml:space="preserve">lux diff. <lb/>Sex. El.</note> <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a"> <image file="0140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0140-01"/> </figure> <note position="left" xlink:label="note-0140-02" xlink:href="note-0140-02a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-0140-03" xlink:href="note-0140-03a" xml:space="preserve">Defin. 10. <lb/>5. Elem.</note> </div> </div> <div xml:id="echoid-div296" type="section" level="1" n="185"> <head xml:id="echoid-head200" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2843" xml:space="preserve">_H_Inc patet, quæ de parallelogrammis in ſuperioribus Propoſitioni-<lb/>bus oſtenſa ſunt, eadem de eorundem omnibus lineis cum quibuſ. <lb/></s> <s xml:id="echoid-s2844" xml:space="preserve">uis regulis aſſumptis pariter verificari, nam illa ſunt, vt ipſa paralle-<lb/> <anchor type="note" xlink:label="note-0140-04a" xlink:href="note-0140-04"/> logramma.</s> <s xml:id="echoid-s2845" xml:space="preserve"/> </p> <div xml:id="echoid-div296" type="float" level="2" n="1"> <note position="left" xlink:label="note-0140-04" xlink:href="note-0140-04a" xml:space="preserve">_3. huius._</note> </div> </div> <div xml:id="echoid-div298" type="section" level="1" n="186"> <head xml:id="echoid-head201" xml:space="preserve">THEOREMA IX. PROPOS. IX.</head> <p> <s xml:id="echoid-s2846" xml:space="preserve">PArallelogrammorum in eadem altitudine exiſtentium <lb/>omnia quadrata, regula baſi, iuxta quam altitudo ſum-<lb/> <anchor type="note" xlink:label="note-0140-05a" xlink:href="note-0140-05"/> pta eſt, ſunt inter ſe, vt quadrata baſium.</s> <s xml:id="echoid-s2847" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="2" n="1"> <note position="left" xlink:label="note-0140-05" xlink:href="note-0140-05a" xml:space="preserve">A. Def. 8. <lb/>huius.</note> </div> <pb o="121" file="0141" n="141" rhead="LIBER II."/> <p> <s xml:id="echoid-s2848" xml:space="preserve">Sint igitur parallelogramma, AM, MC, in eadem altitudine. </s> <s xml:id="echoid-s2849" xml:space="preserve">Di-<lb/> <anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/> co omnia quadrata parallelogrammi, AM, ad omnia quadrata pa-<lb/>rallelogrammi, MC, regula, GH, eſſe vt quadratum, GM, ad <lb/>quadratum, MH. </s> <s xml:id="echoid-s2850" xml:space="preserve">Sit intra parallelogramma, AM, MC, ducta <lb/>vtcunque, DI, parallela ipſi, GH, cuius portio, DE, maneat in, <lb/> <anchor type="figure" xlink:label="fig-0141-01a" xlink:href="fig-0141-01"/> AM, &</s> <s xml:id="echoid-s2851" xml:space="preserve">, EI, in, BH, quoniam ergo, D <lb/>E, eſt æqualis ipſi, GM, figurę autem pla-<lb/>næ ſimiles deicriptæ à lateribus, vel lineis <lb/> <anchor type="note" xlink:label="note-0141-02a" xlink:href="note-0141-02"/> homologis æqualibus ſunt æquales, & </s> <s xml:id="echoid-s2852" xml:space="preserve">ideò <lb/>quadratum, DE, erit æquale quadrato, G <lb/>M, & </s> <s xml:id="echoid-s2853" xml:space="preserve">quadratum, EI, quadrato, MH, <lb/>ergo, vt quadratum, GM, ad quadratum, <lb/>MH, ita erit quadratum, DE, ad quadratum, EI, & </s> <s xml:id="echoid-s2854" xml:space="preserve">quia, DI, vt-<lb/>cunq; </s> <s xml:id="echoid-s2855" xml:space="preserve">ducta eſt parallela ipſi, GH, ideò, vt vnum ad vnum, ita om-<lb/> <anchor type="note" xlink:label="note-0141-03a" xlink:href="note-0141-03"/> nia ad omnia idelt vt quadratum, GM, ad quadratum, MH, ita <lb/>erunt omnia quadrata parallelogrammi, AM, ad omnia quadrata <lb/>parallelogrammi, MC, regula, GH, quod erat oſtendendum.</s> <s xml:id="echoid-s2856" xml:space="preserve"/> </p> <div xml:id="echoid-div299" type="float" level="2" n="2"> <note position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">A. Deſ. 8. <lb/>huius.</note> <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a"> <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0141-01"/> </figure> <note position="right" xlink:label="note-0141-02" xlink:href="note-0141-02a" xml:space="preserve">25. lib. 1.</note> <note position="right" xlink:label="note-0141-03" xlink:href="note-0141-03a" xml:space="preserve">Coroll. 4. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div301" type="section" level="1" n="187"> <head xml:id="echoid-head202" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2857" xml:space="preserve">_H_Inc patet, ſi vice quadratorum ſumamus alias quaſcunque figuras <lb/>ſimiles, quod eodem pacto oſtendemus omnes figuras ſimiles pa-<lb/> <anchor type="note" xlink:label="note-0141-04a" xlink:href="note-0141-04"/> rallelogrammi, AM, ad omnes ſimiles figuras parallelogrammi, MC, <lb/>vt ex. </s> <s xml:id="echoid-s2858" xml:space="preserve">gr. </s> <s xml:id="echoid-s2859" xml:space="preserve">omnes circutos parallelogrammi, AM, ad omnes circulos pa-<lb/>rallelogrammi, MC, eſſe vt ſimiles ſiguras ab ipſis b ſibus, GM, MH, <lb/>d@ſcriptas, nam fi uræ planæ ſimiles quæcunq; </s> <s xml:id="echoid-s2860" xml:space="preserve">vt dictum eſt, deſcriptæ à <lb/>lateribus, vel lineis homologis æqualibus ſunt æquales; </s> <s xml:id="echoid-s2861" xml:space="preserve">omnibus pari-<lb/> <anchor type="note" xlink:label="note-0141-05a" xlink:href="note-0141-05"/> ter aſſumptis figuris ſimilibus, regula eadem, GH.</s> <s xml:id="echoid-s2862" xml:space="preserve"/> </p> <div xml:id="echoid-div301" type="float" level="2" n="1"> <note position="right" xlink:label="note-0141-04" xlink:href="note-0141-04a" xml:space="preserve">_A. Def. 8._ <lb/>_huius._</note> <note position="right" xlink:label="note-0141-05" xlink:href="note-0141-05a" xml:space="preserve">_25. lib. 1._</note> </div> </div> <div xml:id="echoid-div303" type="section" level="1" n="188"> <head xml:id="echoid-head203" xml:space="preserve">THEOREMA X. PROPOS. X.</head> <p> <s xml:id="echoid-s2863" xml:space="preserve">PArellelogrammorum in eadem baſiexiſtentium omnia <lb/>quadrata, regula ipſa baſi, ſunt vt altitudines, vel vt la-<lb/>tera, quę æqualiter baſi ſunt inclinata, ſi illa ſint ęquiangula.</s> <s xml:id="echoid-s2864" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2865" xml:space="preserve">Sint parallelogramma, AD, BD, in eadem b ſi, CD, exiſten-<lb/>tia, quorum ſint altitudines iuxta baſim, CD, ſumptæ, AO, CN. <lb/></s> <s xml:id="echoid-s2866" xml:space="preserve">Dico omnia quadrata parallelogrammi, AD, adomnia quadrata <lb/>parallelogrammi, BD, regula, CD, eſſe vt, AO, ad, CN, vel <lb/>etiam vt, AC, ad, CB, ſi parallelogramma, BD, DA, fuerint æ-<lb/>quiangula, producantur autem, CA, CB, indefinitè ad partes op- <pb o="122" file="0142" n="142" rhead="GEOMETRIÆ"/> poſitas, ex quibus ſumantur quotcunque partes æquales, AI, IH, <lb/>nempè æquales ipſi, CA, &</s> <s xml:id="echoid-s2867" xml:space="preserve">, BP, æqualis ipſi, BC, & </s> <s xml:id="echoid-s2868" xml:space="preserve">complean-<lb/>tur parallelogramma, AM, IK, BQ; </s> <s xml:id="echoid-s2869" xml:space="preserve">ſunt igitur parallelogramma, <lb/>CF, AM, IK, in æqualibus altitudinibus, ac baſibus, & </s> <s xml:id="echoid-s2870" xml:space="preserve">ideò ſin-<lb/>gulorum omnia quadrata regulis eiſdem baſibus, erunt æqualia, & </s> <s xml:id="echoid-s2871" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0142-01a" xlink:href="note-0142-01"/> pari ratione omnia quadrata parallelogrammorum, BQ, CQ, e-<lb/>runt ęqualia, regula, CD, altitudines autem parallelogrammorum, <lb/>CF, AM, IK, ſunt æquales ipſi, AO, & </s> <s xml:id="echoid-s2872" xml:space="preserve">altitudines parallelogram-<lb/>morum, CE, BQ, ſunt æquales, nempè ipſi, CN, habemus ergo <lb/>æquèmultiplices primę, & </s> <s xml:id="echoid-s2873" xml:space="preserve">tertiæ .</s> <s xml:id="echoid-s2874" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2875" xml:space="preserve">compoſitum ex altitudinibus pa-<lb/>rallelogrammorum, CF, AM, IK, quod tam multiplex eſt altitu-<lb/> <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> dinis, AO, quam compoſitum ex omnibus qua-<lb/>dratis, CF, AM, IK, multiplex eſt omnium <lb/>quadratorum parallelogrammi, CF, & </s> <s xml:id="echoid-s2876" xml:space="preserve">ſic com-<lb/>poſitum ex altitudinibus parallelogrammorum, <lb/>CE, BQ, tam multiplex eſt altitudinis, CN, <lb/>ac compoſitum ex omnibus quadratis parallelo. <lb/></s> <s xml:id="echoid-s2877" xml:space="preserve">grammorum, BQ, CE, multiplex eſt omnium <lb/>quadratorum, CE; </s> <s xml:id="echoid-s2878" xml:space="preserve">ideſt quam multiplicia ſunt <lb/>omnia quadrata parallelogrammi, HD, omnium quadratorum pa-<lb/>rallelogrammi, AD, tam altitudo parallelogrammi, HD, multi-<lb/>plex eſt altitudinis parallelogrammi, AD, ſiue tam ipſa, CH, mul-<lb/>tiplex eſt ipfius, CA, dum ſunt æquiangula, & </s> <s xml:id="echoid-s2879" xml:space="preserve">quam omnia qua-<lb/>drata parallelogrammi, PD, multiplicia ſunt omnium quadratorum <lb/>parallelogrammi, BD, tam altitudo parallelogrammi, PD, mul-<lb/>tiplex eſt altitudinis, CN, vel tam, PC, multiplex eſt ipſius, CB: </s> <s xml:id="echoid-s2880" xml:space="preserve"><lb/>Si autem multiplex primæ fuerit æquale multiplici ſecundæ, etiam <lb/>multiplex tertiæ erit æquale multiplici quartæ, ſi maius maius, & </s> <s xml:id="echoid-s2881" xml:space="preserve">ſi <lb/>minus minus, nam ſi altitudo parallelogrammi, HD, fuerit æqua-<lb/>lis altitudini parallelogrammi, DP, omnia quadrata, HD, erunt <lb/>æqualia omnibus quadratis, DP, nam parallelogramma, HD, D <lb/>P, ſunt in eadem baſi, CD, ſi illa maior, & </s> <s xml:id="echoid-s2882" xml:space="preserve">hæc maiora, & </s> <s xml:id="echoid-s2883" xml:space="preserve">ſi mi-<lb/> <anchor type="note" xlink:label="note-0142-02a" xlink:href="note-0142-02"/> nor minora, ergo prima ad ſecundam erit, vt tertia ad quartam, <lb/> <anchor type="note" xlink:label="note-0142-03a" xlink:href="note-0142-03"/> nempè vt altitudo parallelogrammi, AD, ad altitudinem paralle-<lb/>logrammi, DB, .</s> <s xml:id="echoid-s2884" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2885" xml:space="preserve">AO, ad, CN, vel, AC, ad, CB, dum ſunt <lb/>æquiangula, ita erunt omnia quadrata, AD, ad omnia quadrata, <lb/>DB, ſunt ergo, vt altitudines ipſorum parallelogrammorum, vel <lb/>vt latera ęqualiter baſi inclinata, cum nempè parallelogramma ſunt <lb/>æquiangula: </s> <s xml:id="echoid-s2886" xml:space="preserve">hæc autem etiam verificarentur ſi parallelogramma <lb/>eſſent in æqualibus baſibus, quod oſtendere opus erat.</s> <s xml:id="echoid-s2887" xml:space="preserve"/> </p> <div xml:id="echoid-div303" type="float" level="2" n="1"> <note position="left" xlink:label="note-0142-01" xlink:href="note-0142-01a" xml:space="preserve">9. Huius</note> <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a"> <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0142-01"/> </figure> <note position="left" xlink:label="note-0142-02" xlink:href="note-0142-02a" xml:space="preserve">Exautec.</note> <note position="left" xlink:label="note-0142-03" xlink:href="note-0142-03a" xml:space="preserve">5. Quinti <lb/>Elem.</note> </div> <pb o="123" file="0143" n="143" rhead="LIBER II."/> </div> <div xml:id="echoid-div305" type="section" level="1" n="189"> <head xml:id="echoid-head204" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2888" xml:space="preserve">_E_Ademratione, ſi vice quadratorum ſumamus alias figuras ſimiles, <lb/>oſtendemus omnus figuras ſimiles parallelogram morum in eadem, <lb/> <anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/> baſi exiſtentium eſſe, vt altitudines, vel vt latera baſi æqualiter incli-<lb/>nata, dum illa ſunt æquiangula.</s> <s xml:id="echoid-s2889" xml:space="preserve"/> </p> <div xml:id="echoid-div305" type="float" level="2" n="1"> <note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">A. Def 8. <lb/>Huius.</note> </div> </div> <div xml:id="echoid-div307" type="section" level="1" n="190"> <head xml:id="echoid-head205" xml:space="preserve">THEOREMA XI. PROPOS. XI.</head> <p> <s xml:id="echoid-s2890" xml:space="preserve">QVorumlibet parallelogrammorum omnia quadrata te-<lb/>gulis duobus quibuſuis in eiſdem aſſumptis lateribus, <lb/>habent inter ſe rationem compoſitam exratione <lb/>quadratorum dictorum laterum, & </s> <s xml:id="echoid-s2891" xml:space="preserve">altitudinum, vel laterum, <lb/>quę cum prędictis ęqualiter inclinãtur, ſi illa ſint ęquiangula.</s> <s xml:id="echoid-s2892" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2893" xml:space="preserve">Sint parallelogramma vtcunq; </s> <s xml:id="echoid-s2894" xml:space="preserve">AD, FM, in quibus regulæ ex-<lb/>tent latera vtcunque, CD, GM, altitudines autem iuxta dictas re-<lb/>gulas ſumptæ, BV, ON. </s> <s xml:id="echoid-s2895" xml:space="preserve">Dico omnia quadrata, AD, ad omnia <lb/>quadrata, FM, habere rationem compoſitam ex ea, quam habet <lb/>quadratum, CD, ad quadratum, GM, & </s> <s xml:id="echoid-s2896" xml:space="preserve">ex ea, quam habet, B <lb/>V, altitudo ad altitudinem, ON, vel etiam, BD, ad. </s> <s xml:id="echoid-s2897" xml:space="preserve">OM, ſi illa <lb/>ſint æquiangula, lateraq; </s> <s xml:id="echoid-s2898" xml:space="preserve">BD, OM, æqualiter ſint inclinata cum <lb/> <anchor type="figure" xlink:label="fig-0143-01a" xlink:href="fig-0143-01"/> lateribus, CD, GM; </s> <s xml:id="echoid-s2899" xml:space="preserve">abicindatur <lb/>à, BV, verſus, V, ipſa, XV, æ-<lb/>qualis, ON, & </s> <s xml:id="echoid-s2900" xml:space="preserve">per, X, ducatur, X <lb/>P, parallela ipſi, CD, ſecans, BD, <lb/>in, R, erit autem, DR, æqualis <lb/>ipſi, OM, ſi ſint æquiangula, quod <lb/>facilè probari poteſt, erit etiam pa-<lb/>rallelogrammum, PD, in eadem <lb/>baſi cum parallelogrammo, AD, <lb/>ſed in eadem altitudine cum parallelogrammo, FM, omnia ergo <lb/> <anchor type="note" xlink:label="note-0143-02a" xlink:href="note-0143-02"/> quadrata parallelogrammi, AD, ad omnia quadrata, FM, habent <lb/>rationem compoſitam ex ea, quam habent omnia quadrata, AD, <lb/>ad omnia quadrata, DP, .</s> <s xml:id="echoid-s2901" xml:space="preserve">i. </s> <s xml:id="echoid-s2902" xml:space="preserve">ex ea, quam habet, BV, ad, VX, ſiue, <lb/> <anchor type="note" xlink:label="note-0143-03a" xlink:href="note-0143-03"/> ON, vel ex ea, quam habet, BD, ad, DR, ſiue, OM, ſi ſint æ-<lb/>quiangula parallelogramma, AD, DP; </s> <s xml:id="echoid-s2903" xml:space="preserve">& </s> <s xml:id="echoid-s2904" xml:space="preserve">componitur ex ea, quam <lb/>habent omnia quadrata, PD, ad omnia quadrata, FM, .</s> <s xml:id="echoid-s2905" xml:space="preserve">. ex ea, <lb/> <anchor type="note" xlink:label="note-0143-04a" xlink:href="note-0143-04"/> quam habet quadratum, CD, ad quadratum, GM, ergo omnia <lb/>quadrata, AD, ad omnia quadrata, FM, habent rationem com- <pb o="124" file="0144" n="144" rhead="GEOMETRIÆ"/> poſitam ex ea, quam habet, BV, ad, ON, vel, BD, ad, OM, cum <lb/>ſunt æquiangula, & </s> <s xml:id="echoid-s2906" xml:space="preserve">ex ea, quem habet quadratum, CD, ad qua-<lb/>dratum, GM, quod oſtendendum erat.</s> <s xml:id="echoid-s2907" xml:space="preserve"/> </p> <div xml:id="echoid-div307" type="float" level="2" n="1"> <figure xlink:label="fig-0143-01" xlink:href="fig-0143-01a"> <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0143-01"/> </figure> <note position="right" xlink:label="note-0143-02" xlink:href="note-0143-02a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0143-03" xlink:href="note-0143-03a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0143-04" xlink:href="note-0143-04a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div309" type="section" level="1" n="191"> <head xml:id="echoid-head206" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2908" xml:space="preserve">_H_Inc patet, ſi vice quadratorum ſumamus alias figuras planas ſimi-<lb/>les, quod eodem pacto oſtendemus omnes figuras ſimiles, AD, F <lb/>M, habere inter ſerationem compoſitam ex ratione quadratorum, CD, <lb/>GM, & </s> <s xml:id="echoid-s2909" xml:space="preserve">altitudinum, BV, ON, vel laterum, BD, OM, æqualiter <lb/>baſibus inclinatorum, cum parallelogramma ſunt æquiangula.</s> <s xml:id="echoid-s2910" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div310" type="section" level="1" n="192"> <head xml:id="echoid-head207" xml:space="preserve">THEOREMA XII. PROPOS. XII.</head> <p> <s xml:id="echoid-s2911" xml:space="preserve">P Arallelogrammorum, quorum baſium quadrata altitu-<lb/>dinibus iuxta eaſdem baſes ſumptis reciprocantur, vel <lb/>lateribus æqualiter dictis baſibus inclinatis; </s> <s xml:id="echoid-s2912" xml:space="preserve">omnia quadra-<lb/>ta, regulis eiſdem baſibus, ſunt æqualia: </s> <s xml:id="echoid-s2913" xml:space="preserve">Et quorum paral-<lb/>lelogrammorum, regulis baſibus, omnia quadrata ſunt æ-<lb/>qualia, baſium quadrata altitudinibus, vellateribus æqua-<lb/>liter dictis baſibus inclinatis, reciprocantur.</s> <s xml:id="echoid-s2914" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2915" xml:space="preserve">Sint parallelogramma, HX, AD, quorum baſium, VX, BD, <lb/>quadrata altitudinibus iuxta ipſas baſes ſumptis, vel lateribus, RX, <lb/>CD, ſi hæc baſibus, VX, BD, æqualiter ſint inclinata, recipro-<lb/> <anchor type="figure" xlink:label="fig-0144-01a" xlink:href="fig-0144-01"/> centur. </s> <s xml:id="echoid-s2916" xml:space="preserve">Dico omnia quadrata pa-<lb/>rallelogrammorum, HX, AD, eſſe <lb/>inter ſe æqualia. </s> <s xml:id="echoid-s2917" xml:space="preserve">Nam omnia qua-<lb/> <anchor type="note" xlink:label="note-0144-01a" xlink:href="note-0144-01"/> drara, HX, ad omnia quadrata, A <lb/>D, habent rationem compoſitam ex <lb/>ea, quam habet quadratum, VX, ad <lb/>quadratum, BD, .</s> <s xml:id="echoid-s2918" xml:space="preserve">i. </s> <s xml:id="echoid-s2919" xml:space="preserve">ex ea, quam <lb/>habet, CO, ad, RZ, vel, CD, ad, <lb/>RX, cum ſunt æquiangula, & </s> <s xml:id="echoid-s2920" xml:space="preserve">ex ea, quam habet, RZ, ad, CO, <lb/>vel, RX, ad, CD, quæ duæ rationes componunt rationem, CO, <lb/>ad, CO, vel, CD, ad, CD, quæ eſt ratio æqualitatis, & </s> <s xml:id="echoid-s2921" xml:space="preserve">ideò om-<lb/>nia quadrata, HX, erunt æqualia omnibus quadratis, AD.</s> <s xml:id="echoid-s2922" xml:space="preserve"/> </p> <div xml:id="echoid-div310" type="float" level="2" n="1"> <figure xlink:label="fig-0144-01" xlink:href="fig-0144-01a"> <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0144-01"/> </figure> <note position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">Ex auce-<lb/>ced.</note> </div> <p> <s xml:id="echoid-s2923" xml:space="preserve">Sint nunc omnia quadrata, HX, æqualia omnibus quadratis, A <lb/>D, regulis eiſdem, VX, BD. </s> <s xml:id="echoid-s2924" xml:space="preserve">Dico quadratum, VX, ad quadra-<lb/>tum, BD, eſſe vt, CO, ad, RZ, vel, CD, ad, RX, cum ſunt æ- <pb o="125" file="0145" n="145" rhead="LIBER II."/> quiangula, etenim, CO, ad, CO, habet rationem compoſitam ex <lb/> <anchor type="note" xlink:label="note-0145-01a" xlink:href="note-0145-01"/> ea, quam habet, CO, ad, RZ, & </s> <s xml:id="echoid-s2925" xml:space="preserve">RZ, ad, CO, & </s> <s xml:id="echoid-s2926" xml:space="preserve">ſic, CD, ad, <lb/>CD, ex ea, quam habet, CD, ad, RX, &</s> <s xml:id="echoid-s2927" xml:space="preserve">, RX, ad, CD, quia <lb/>verò omnia quadrata, HX, ſunt æqualia omnibus quadratis, AD, <lb/>ideò ſunt ad illa, vt, CO, ad, CO, vel vt, CD, ad, CD, .</s> <s xml:id="echoid-s2928" xml:space="preserve">i. </s> <s xml:id="echoid-s2929" xml:space="preserve">in ra-<lb/>tione compoſita ex ratione, CO, ad, RZ, &</s> <s xml:id="echoid-s2930" xml:space="preserve">, RZ, ad, CO, vel, <lb/>CD, ad, RX, &</s> <s xml:id="echoid-s2931" xml:space="preserve">, RX, ad, CD, ſunt autem omnia quadrata, H <lb/>X, ad omnia quadrata, AD, in ratione compoſita ex ea, quam ha-<lb/> <anchor type="note" xlink:label="note-0145-02a" xlink:href="note-0145-02"/> bet quadratum, VX, ad quadratum, BD, &</s> <s xml:id="echoid-s2932" xml:space="preserve">, RZ, ad, CO, ſiue, <lb/>RX, ad, CD, cum ſunt æquiangula, ideò duæ rationes, CO, ad, <lb/>RZ, &</s> <s xml:id="echoid-s2933" xml:space="preserve">, RZ, ad, CO, ſiue aliæ duę rationes, CD, ad, RX, &</s> <s xml:id="echoid-s2934" xml:space="preserve">, <lb/>RX, ad, CD, componunt eandem rationem, quam iſtę duę .</s> <s xml:id="echoid-s2935" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2936" xml:space="preserve">ra-<lb/>tio quadrati, VX, ad quadratum, BD, &</s> <s xml:id="echoid-s2937" xml:space="preserve">, RZ, ad, CO, vel, R <lb/>X, ad, CD, eſt autem communis ratio, RZ, ad, CO, vel, RX, <lb/>ad, CD, ergo reliqua ratio, quam habet quadratum, VX, ad qua-<lb/>dratum, BD, erit eadem reliquę, quam nempè habet, CO, ad, <lb/>RZ, vel, CD, ad, RX, cum ſunt æquiangula, quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s2938" xml:space="preserve"/> </p> <div xml:id="echoid-div311" type="float" level="2" n="2"> <note position="right" xlink:label="note-0145-01" xlink:href="note-0145-01a" xml:space="preserve">Defin. 13. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0145-02" xlink:href="note-0145-02a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div313" type="section" level="1" n="193"> <head xml:id="echoid-head208" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2939" xml:space="preserve">_I_Dem eodem modo de omnibus figuris ſimilibus quibuſuis parallelo-<lb/>grammorum, HX, AD, regulis ijſdem, VX, BD, oſtendi poſſe ex <lb/>ſuperiori methodo colligitur.</s> <s xml:id="echoid-s2940" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div314" type="section" level="1" n="194"> <head xml:id="echoid-head209" xml:space="preserve">THEOREMA XIII. PROPOS. XIII.</head> <p> <s xml:id="echoid-s2941" xml:space="preserve">SImilium parallelogrammorum omnia quadrata, regulis <lb/>homologis lateribus, ſunt in tripla ratione laterum ho-<lb/>mologorum.</s> <s xml:id="echoid-s2942" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2943" xml:space="preserve">Sint ſimilia parallelogramma, AC, EG, quorum latera homo-<lb/> <anchor type="note" xlink:label="note-0145-03a" xlink:href="note-0145-03"/> loga, BC, FG, ſint ſumpta pro regula. </s> <s xml:id="echoid-s2944" xml:space="preserve">Dico omnia quadrata, A <lb/> <anchor type="figure" xlink:label="fig-0145-01a" xlink:href="fig-0145-01"/> C, ad omnia quadr. </s> <s xml:id="echoid-s2945" xml:space="preserve">EG, eſſe in tripla ra-<lb/>tione eius, quam habet, BC, ad, FG. </s> <s xml:id="echoid-s2946" xml:space="preserve">Quo-<lb/> <anchor type="note" xlink:label="note-0145-04a" xlink:href="note-0145-04"/> niam enim parallelogramma, AC, EG, <lb/>ſunt ſimilia, ideò ſunt æquiangula, & </s> <s xml:id="echoid-s2947" xml:space="preserve">circa <lb/>æquales angulos latera habent proportio-<lb/>nalia, &</s> <s xml:id="echoid-s2948" xml:space="preserve">, BC, CD; </s> <s xml:id="echoid-s2949" xml:space="preserve">FG, GH, ſunt late-<lb/>ra ad inuicem æqualiter inclinata, quorum, <lb/>BC, FG, ſunt regulę, ideò omnia quadrata, AC, regula, BC, ad <pb o="126" file="0146" n="146" rhead="GEOMETRIÆ"/> omnia quadrata, EG, regula, FG, ſunt in ratione compoſita ex <lb/>ratione quadrati, BC, ad quadratum, FG, & </s> <s xml:id="echoid-s2950" xml:space="preserve">ex ratione, DC, ad, <lb/> <anchor type="note" xlink:label="note-0146-01a" xlink:href="note-0146-01"/> HG, ſiue, BC, ad, FG, .</s> <s xml:id="echoid-s2951" xml:space="preserve">i. </s> <s xml:id="echoid-s2952" xml:space="preserve">in ratione compoſita ex tribus rationi-<lb/>bus, BC, ad, FG, ideſt habent eandem rationem, quam, BC, ad <lb/> <anchor type="note" xlink:label="note-0146-02a" xlink:href="note-0146-02"/> quartam propo tionalem duarum, quarum prima, BC, ſecunda eſt, <lb/>FG, .</s> <s xml:id="echoid-s2953" xml:space="preserve">. ſunt in tripla ratione eius, quam habet, BC, ad, FG, quod <lb/> <anchor type="note" xlink:label="note-0146-03a" xlink:href="note-0146-03"/> erat oſtendendum.</s> <s xml:id="echoid-s2954" xml:space="preserve"/> </p> <div xml:id="echoid-div314" type="float" level="2" n="1"> <note position="right" xlink:label="note-0145-03" xlink:href="note-0145-03a" xml:space="preserve">Tux. diff. 1. <lb/>Sex. El.</note> <figure xlink:label="fig-0145-01" xlink:href="fig-0145-01a"> <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0145-01"/> </figure> <note position="right" xlink:label="note-0145-04" xlink:href="note-0145-04a" xml:space="preserve">Ex def. 1. <lb/>Sex. El.</note> <note position="left" xlink:label="note-0146-01" xlink:href="note-0146-01a" xml:space="preserve">11. huius.</note> <note position="left" xlink:label="note-0146-02" xlink:href="note-0146-02a" xml:space="preserve">Defin. 11. <lb/>Quin. El.</note> <note position="left" xlink:label="note-0146-03" xlink:href="note-0146-03a" xml:space="preserve">Defin. 11. <lb/>Quin. El.</note> </div> </div> <div xml:id="echoid-div316" type="section" level="1" n="195"> <head xml:id="echoid-head210" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2955" xml:space="preserve">_H_Inc patet, quod eodem modo idem oſtendemus de omnibus quibuſ-<lb/>uis alijs figuris ſimilibus parallelogrammorum, AC, EG vice <lb/>quadratorum ſumptis, regulis eiſdem, ex ſuperioribus Corollarijs id de-<lb/>ducentes.</s> <s xml:id="echoid-s2956" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div317" type="section" level="1" n="196"> <head xml:id="echoid-head211" xml:space="preserve">THEOREMA XIV. PROPOS. XIV.</head> <p> <s xml:id="echoid-s2957" xml:space="preserve">SI duo parallelogramma fuerint in eadem altitudine con-<lb/>ſtituta, omnes figuræ ſimiles vnius ad omnes figuras ſi-<lb/> <anchor type="note" xlink:label="note-0146-04a" xlink:href="note-0146-04"/> miles alterius, etiamſi ſint diffimiles primò dictis, regulis ba-<lb/>ſibus, iuxta quas altitudo ſumitur, erunt, vt figura deſcripta <lb/>à baſi parallelogrammi primò dicti ad figuram deſcriptam à <lb/>baſi parallelogrammi ſecundò dicti.</s> <s xml:id="echoid-s2958" xml:space="preserve"/> </p> <div xml:id="echoid-div317" type="float" level="2" n="1"> <note position="left" xlink:label="note-0146-04" xlink:href="note-0146-04a" xml:space="preserve">A. Def. 8. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s2959" xml:space="preserve">Sint parallelogramma in eadem altitudine conſtituta, AE, EC. <lb/></s> <s xml:id="echoid-s2960" xml:space="preserve">Dico omnes figuras ſimiles parallelogrammi, AE, ad omnes figu-<lb/> <anchor type="note" xlink:label="note-0146-05a" xlink:href="note-0146-05"/> ras ſimiles parallelogrammi, EC, etiamſi ſint diſſimiles prædictis, <lb/>eſſe vt figura deſcripta à, DE, ad figuram deſcriptam ab, EF, quæ <lb/> <anchor type="figure" xlink:label="fig-0146-01a" xlink:href="fig-0146-01"/> ſunt baſes, iuxta quas ſumitur dictorum paral-<lb/>lelogrammorum altitudo .</s> <s xml:id="echoid-s2961" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2962" xml:space="preserve">ex. </s> <s xml:id="echoid-s2963" xml:space="preserve">g. </s> <s xml:id="echoid-s2964" xml:space="preserve">omnia qua-<lb/>drata, AE, ad omnes circulos, EC, eſſe vt <lb/>quadratum, DE, ad circulum deſcriptum ab, <lb/>EF. </s> <s xml:id="echoid-s2965" xml:space="preserve">Ducta enim ipſa, HN, vtcunque paral-<lb/>lela, DF, reperiemus, vt figura, DE, ad fi-<lb/>guram, EF, ita eſſe figuram, HM, ad figu-<lb/>ram, MN, quia quæ deſcribuntur lateribus, <lb/>HM, DE, equalibus ſunt ęquales, veluti de <lb/> <anchor type="note" xlink:label="note-0146-06a" xlink:href="note-0146-06"/> ſcriptę à lateribus, MN, EF, pariter ſunt æquales, & </s> <s xml:id="echoid-s2966" xml:space="preserve">ideò, vt vnum <lb/>ad vnum, ſic omnia ad omnia .</s> <s xml:id="echoid-s2967" xml:space="preserve">ſ. </s> <s xml:id="echoid-s2968" xml:space="preserve">vt figura deſcripta à, DE, ad figu-<lb/> <anchor type="note" xlink:label="note-0146-07a" xlink:href="note-0146-07"/> rain deſcriptam ab, EF, ſic erunt omnes figuræ ſimiles parallelo- <pb o="127" file="0147" n="147" rhead="LIBER II."/> grammi, AE, ſimiles, inquam, figuræ deſcriptæ à, DE, ad omnes <lb/>figuras ſimiles parallelogrammi, EC, ſimiles, inquam, figuræ de-<lb/>ſcriptæ ab, EF, quod oſtendere opus erat.</s> <s xml:id="echoid-s2969" xml:space="preserve"/> </p> <div xml:id="echoid-div318" type="float" level="2" n="2"> <note position="left" xlink:label="note-0146-05" xlink:href="note-0146-05a" xml:space="preserve">A. Def. 8. <lb/>huius.</note> <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a"> <image file="0146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0146-01"/> </figure> <note position="left" xlink:label="note-0146-06" xlink:href="note-0146-06a" xml:space="preserve">25. lib. 1.</note> <note position="left" xlink:label="note-0146-07" xlink:href="note-0146-07a" xml:space="preserve">Coroll. 4 <lb/>huius.</note> </div> </div> <div xml:id="echoid-div320" type="section" level="1" n="197"> <head xml:id="echoid-head212" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s2970" xml:space="preserve">_H_Inc in figura Propoſ. </s> <s xml:id="echoid-s2971" xml:space="preserve">II. </s> <s xml:id="echoid-s2972" xml:space="preserve">colligemus omnes figuras ſimiles pàral-<lb/>lelogrammi, AD, ad omnes figuras ſimiles parallelogrammi, F <lb/>M, etiam tamen diſſimiles prædictis, habere rationem compoſitam ex ra-<lb/>tione figurarum, quæ à baſibus, CD, GM, deſcribuntur, & </s> <s xml:id="echoid-s2973" xml:space="preserve">altitudi-<lb/>num, vel laterum æqualiter baſibus inclinatorum; </s> <s xml:id="echoid-s2974" xml:space="preserve">quia omnes figuræ ſi-<lb/>miles, AD, ad omnes figuras ſimiles, FM, diſſimiles prædictis, ha-<lb/>bent rationem compoſitam ex ea, quam habent omnes figuræ ſimiles, A <lb/> <anchor type="note" xlink:label="note-0147-01a" xlink:href="note-0147-01"/> D, ad omnes figuras ſimiles, FM, ideſt compoſitam ex ratione figuræ de-<lb/>ſcriptæ à, CD, ad ſibi ſimilem figuram deſcriptam à, GM, & </s> <s xml:id="echoid-s2975" xml:space="preserve">ex ra-<lb/> <anchor type="note" xlink:label="note-0147-02a" xlink:href="note-0147-02"/> tione, BV, ad, ON, vel, BD, ad, OM, cum ſunt parallelogramma <lb/>æquiangula, & </s> <s xml:id="echoid-s2976" xml:space="preserve">eſt compoſita ex ratione omnium figurarum ſimilium, F <lb/>M, ad omnes figuras ſimiles ipſius, FM, diſſimiles tamen proximè di-<lb/>ctis, quæ eſt eadem ei, quam habet figura, GM, ſimiles figuræ, CD ad <lb/> <anchor type="note" xlink:label="note-0147-03a" xlink:href="note-0147-03"/> figuram, GM, vltimò deſcriptam, duæ verò rationes figuræ CD, ad fi-<lb/>guram, GM, ſibi ſimilem, & </s> <s xml:id="echoid-s2977" xml:space="preserve">huius ad figuram, GM, ſibi diſſimilem, <lb/> <anchor type="note" xlink:label="note-0147-04a" xlink:href="note-0147-04"/> componunt rationem figuræ, CD, ad figuram, GM, ſibi diſſimilem, & </s> <s xml:id="echoid-s2978" xml:space="preserve"><lb/>ideò habebimus omnes figuras ſimiles, AD, ad omnes figuras ſimiles ip. <lb/></s> <s xml:id="echoid-s2979" xml:space="preserve">ſius, FM, diſſimiles tamen prædictis habere rationem compoſitam ex <lb/>ea, quam habet figura ipſius, CD, ad figuram, GM, ſibi diſſimilem, & </s> <s xml:id="echoid-s2980" xml:space="preserve"><lb/>ex ea, quam habet, BV, ad, ON, vel, BD, ad, OM, cum parallelo-<lb/>gramma ſunt æquiangula. </s> <s xml:id="echoid-s2981" xml:space="preserve">Conſimili methado in figura Propoſ. </s> <s xml:id="echoid-s2982" xml:space="preserve">12. </s> <s xml:id="echoid-s2983" xml:space="preserve">col-<lb/>ligemus omnes parallelogrammi, HX, figuras ſimiles, omnibus figuris <lb/>ſimilibus parallelogrammi, AD, etiamſi prædictis ſint diſſimiles, eſſe <lb/>tamen æquales; </s> <s xml:id="echoid-s2984" xml:space="preserve">Et ſi ſint æquales, figuras deſcriptas ab, VX, BD, li-<lb/>cet diſſimiles, altitudinibus, CO, RZ, vel lateribus, CD, RX, baſi-<lb/>bus æqualiter inclinatis, reciprocè reſpondere.</s> <s xml:id="echoid-s2985" xml:space="preserve"/> </p> <div xml:id="echoid-div320" type="float" level="2" n="1"> <note position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">_Defin. 12._ <lb/>_lib. 1._</note> <note position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">_Corol. 11._ <lb/>_huius._</note> <note position="right" xlink:label="note-0147-03" xlink:href="note-0147-03a" xml:space="preserve">_Ex ſuper,_ <lb/>_Prop._</note> <note position="right" xlink:label="note-0147-04" xlink:href="note-0147-04a" xml:space="preserve">_Defin. 12._ <lb/>_lib. 1._</note> </div> </div> <div xml:id="echoid-div322" type="section" level="1" n="198"> <head xml:id="echoid-head213" xml:space="preserve">THEOREMA XV. PROPOS. XV.</head> <p> <s xml:id="echoid-s2986" xml:space="preserve">OMNES figuræ planæ ſimiles ſunt inter ſe in dupla <lb/>ratione linearum, ſiue laterum homologorum, ea-<lb/>rundem.</s> <s xml:id="echoid-s2987" xml:space="preserve"/> </p> <pb o="128" file="0148" n="148" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div323" type="section" level="1" n="199"> <head xml:id="echoid-head214" xml:space="preserve">A. DEMONSTRATIONIS SECTIO I.</head> <p> <s xml:id="echoid-s2988" xml:space="preserve">SInt duæ quæcunque figuræ planæ ſimiles, ABD, ΦΣΛ. </s> <s xml:id="echoid-s2989" xml:space="preserve">Dico <lb/>eaſdem eſſe in dupla ratione linearum, vel laterum homologo-<lb/> <anchor type="note" xlink:label="note-0148-01a" xlink:href="note-0148-01"/> rum, earundem. </s> <s xml:id="echoid-s2990" xml:space="preserve">Ducantur ipſarum oppoſitę tangentes, AK, FM, <lb/>figuræ, ABD, &</s> <s xml:id="echoid-s2991" xml:space="preserve">, Φ Π, Δ Ω, ſiguræ, Φ Σ Λ, quæ homologis ea-<lb/>rum lineis æquidiſtent, deinde ſint intradictas oppoſitas tangentes <lb/>ductæ, KM, Π Ω, taliter, vt illæ ſint incidentes dictarum ſimilium <lb/> <anchor type="note" xlink:label="note-0148-02a" xlink:href="note-0148-02"/> figurarum, & </s> <s xml:id="echoid-s2992" xml:space="preserve">tangentium, hoc facto, diuidantur ipſæ incident@s, K <lb/> <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> M, Π Ω, ſimil@ter, & </s> <s xml:id="echoid-s2993" xml:space="preserve">ad eandem <lb/>partem vtcunq; </s> <s xml:id="echoid-s2994" xml:space="preserve">in punctis, L, Γ, <lb/>per quæ puncta ſint ductæ ipſæ, B <lb/>L, Σ Γ, quarum portiones figuris <lb/>interceptę ſint, BE, ID, in figu-<lb/>ra, ABD, &</s> <s xml:id="echoid-s2995" xml:space="preserve">, Σ 2, 3 Λ, in figu-<lb/>ra, Φ Σ Λ, ſumatur autem ex, BL, <lb/>recta æqualis vtriſque ſimul, BE, <lb/>ID, ter ninans in, KM, quæ ſit, <lb/>QL, & </s> <s xml:id="echoid-s2996" xml:space="preserve">pariter ipſius, Σ 2, 3 Λ, ſit <lb/>ſumpta æqualis, Τ Γ, term nans <lb/>in, Π Ω, & </s> <s xml:id="echoid-s2997" xml:space="preserve">in puncto, Γ, ſicq; </s> <s xml:id="echoid-s2998" xml:space="preserve">fiat <lb/>de cæteris, quæ ipſis tangentibus <lb/>æquidiſtant, & </s> <s xml:id="echoid-s2999" xml:space="preserve">manent intra figu-<lb/>rarum ambitum, quibusn m è in <lb/>eadem rectitudine ſun antur rectæ <lb/>æquales in ipſis, KM, ΠΩ, termi-<lb/>natæ, erunt igitur omnium in@en-<lb/>tarum linearum reliqui termini in <lb/>alia quadam inea, quæ inc pi@t in <lb/>puncto, K, & </s> <s xml:id="echoid-s3000" xml:space="preserve">deſinet in, M, pro <lb/>figura, ABD, & </s> <s xml:id="echoid-s3001" xml:space="preserve">quæ incip e in Π, & </s> <s xml:id="echoid-s3002" xml:space="preserve">deſinet in, Ω, pro figura, <lb/>Φ Σ Λ, ſint iſtę lineę, KQM, Π Ω; </s> <s xml:id="echoid-s3003" xml:space="preserve">patet igitur figuram, KQM, <lb/> <anchor type="note" xlink:label="note-0148-03a" xlink:href="note-0148-03"/> eſſe æqualem ipſi, ABD, & </s> <s xml:id="echoid-s3004" xml:space="preserve">Π Ω, ipſi, Φ Σ Λ, nam omnes earum <lb/>lineæ ſumptæ regulis, FM, Δ Ω, ſunt æquales, quod ex ip a con-<lb/>ſtructione patet; </s> <s xml:id="echoid-s3005" xml:space="preserve">dicantur autem iſtæ conſtruct ones, translationes <lb/>omn um linearum ſ gurarum, ABD, ΦΣΛ, in figuras, KQA, Π <lb/>Τ Ω, ipſi, KM, ΠΩ adiacentes, effectę regulis dictis tangent bus. <lb/></s> <s xml:id="echoid-s3006" xml:space="preserve">Patet vlterius figuras, KQM, ΠΤΩ, eſſe ſim les, nam homologę <lb/>figurarum, ABD, ΦΣΛ, (quia illę ſunt ſimiles) ſunt vt incidentes, <lb/> <anchor type="note" xlink:label="note-0148-04a" xlink:href="note-0148-04"/> KM, ΠΩ, eæden autem in figuras, KQM,ΠΤΩ, modo dicto, <lb/>translatæ ſunt (ſimui in vnam rectam coniunctis, quæ diuiſæ erant, <pb o="129" file="0149" n="149" rhead="LIBER II."/> veluti, BE, ID, iunctæ ſunt in linea, QL, &</s> <s xml:id="echoid-s3007" xml:space="preserve">, Σ 2, 3 Λ, in linea, <lb/>Τ Γ,) ergo quę tangentibus dictis æquidiſtant in ſigur s, KQM, Π <lb/>Τ Ω, & </s> <s xml:id="echoid-s3008" xml:space="preserve">diuidunt incidentes, KM, ΠΩ, ſimiliter ad eandem par-<lb/>tem, & </s> <s xml:id="echoid-s3009" xml:space="preserve">iacent inter ipſas incidentes, & </s> <s xml:id="echoid-s3010" xml:space="preserve">circuitum figurarum ad ean-<lb/>dem partem eodem ordine ſumptæ, ſunt vt ipſæ incidentes, ergo fi-<lb/>gurę, KQM, ΠΤΩ, ſunt ſimiles, & </s> <s xml:id="echoid-s3011" xml:space="preserve">earundem homologarum re-<lb/> <anchor type="note" xlink:label="note-0149-01a" xlink:href="note-0149-01"/> gulæ eædem tangentes, & </s> <s xml:id="echoid-s3012" xml:space="preserve">earum incidentes ipſæ, KM, Π Ω.</s> <s xml:id="echoid-s3013" xml:space="preserve"/> </p> <div xml:id="echoid-div323" type="float" level="2" n="1"> <note position="left" xlink:label="note-0148-01" xlink:href="note-0148-01a" xml:space="preserve">Coroll. 1. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0148-02" xlink:href="note-0148-02a" xml:space="preserve">Coroll. 2. <lb/>9. lib. 1.</note> <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a"> <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0148-01"/> </figure> <note position="left" xlink:label="note-0148-03" xlink:href="note-0148-03a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0148-04" xlink:href="note-0148-04a" xml:space="preserve">Coroll. 1. <lb/>22. lib. 1.</note> <note position="right" xlink:label="note-0149-01" xlink:href="note-0149-01a" xml:space="preserve">Defin. 10. <lb/>lib. 1.</note> </div> </div> <div xml:id="echoid-div325" type="section" level="1" n="200"> <head xml:id="echoid-head215" xml:space="preserve">B. SECTIO SECVNDA.</head> <p> <s xml:id="echoid-s3014" xml:space="preserve">PRoducantur nunc ipſę, KM, ΠΩ, indefinitè verſus puncta, M, <lb/>Ω, & </s> <s xml:id="echoid-s3015" xml:space="preserve">ab ipſis productis ſumantur partes æquales, MP, ipſi, K <lb/>M, &</s> <s xml:id="echoid-s3016" xml:space="preserve">, ω &</s> <s xml:id="echoid-s3017" xml:space="preserve">, ipſi, ΠΩ, & </s> <s xml:id="echoid-s3018" xml:space="preserve">per puncta, P, &</s> <s xml:id="echoid-s3019" xml:space="preserve">, ducantur dictis tangen-<lb/>tibus parallelę, ZP, ℟ &</s> <s xml:id="echoid-s3020" xml:space="preserve">, quoniam ergo, KM, ΠΩ, ſunt inciden-<lb/> <anchor type="note" xlink:label="note-0149-02a" xlink:href="note-0149-02"/> tes ſimilium figurarum, KQM, ΠΤΩ, ideò habebimus etiam ho-<lb/>mòlogas earundem regulis ipſis incidentibus, KM, ΠΩ, ductis er-<lb/>go ex oppoſito tangentibus eaſdem figuras, KQM, ΠΤΩ, paral-<lb/>lelis ipſis, KP, Π &</s> <s xml:id="echoid-s3021" xml:space="preserve">, quæ ſint, XZ, β ℟, poterimus transferre om-<lb/>nes lineas figura@um, KQM, ΠΤΩ, in figuras ipſis, ZP, ℟ &</s> <s xml:id="echoid-s3022" xml:space="preserve">, ad-<lb/>iacentes, translatione facta regulis, KP, Π &</s> <s xml:id="echoid-s3023" xml:space="preserve">, fiant ergo dictę tran-<lb/> <anchor type="note" xlink:label="note-0149-03a" xlink:href="note-0149-03"/> ſlationes, vnde reſultent figu@æ, MZP, Ω℟ &</s> <s xml:id="echoid-s3024" xml:space="preserve">, quæ erunt æqua-<lb/>les ipſis, KQM, ΠΤΩ, & </s> <s xml:id="echoid-s3025" xml:space="preserve">ſubinde ipſis, ABD, ΦΣΛ, probab-<lb/>mus autem etiam eaſdem eſſe ſimiles (veluti in figuris, KQM, Π <lb/> <anchor type="note" xlink:label="note-0149-04a" xlink:href="note-0149-04"/> Τ Ω, factum eſt) &</s> <s xml:id="echoid-s3026" xml:space="preserve">, ZP, ℟ &</s> <s xml:id="echoid-s3027" xml:space="preserve">, eſſe dictarum figurarum incidentes, <lb/>& </s> <s xml:id="echoid-s3028" xml:space="preserve">homologarum regulas ipſas, MP, Ω &</s> <s xml:id="echoid-s3029" xml:space="preserve">, patet autem ex conſtru-<lb/>ctione integras eſſe in figuris, MZP Π℟ &</s> <s xml:id="echoid-s3030" xml:space="preserve">, tum quæ æquidiſtant <lb/>ipſis, ZP, ℟ &</s> <s xml:id="echoid-s3031" xml:space="preserve">, tum ipſis, MP, Π ℟, nam ex prima translatione <lb/>integras habuimus, quę in figuris, KQM, ΠΤΩ, ipſis, FM, ΔΩ, <lb/>erant æquidiſtantes, & </s> <s xml:id="echoid-s3032" xml:space="preserve">ſubinde etiam integras, quæ in figuris, MZ <lb/>P, Ω ℟ &</s> <s xml:id="echoid-s3033" xml:space="preserve">, ipſis, ZP, ℟ &</s> <s xml:id="echoid-s3034" xml:space="preserve">, æquidiſtant, ex ſecunda translatione ve-<lb/>rò integras habuimus eas, quę ipſis, MP, Ω &</s> <s xml:id="echoid-s3035" xml:space="preserve">, æquidiſtant, & </s> <s xml:id="echoid-s3036" xml:space="preserve">hęc <lb/>per conſtructionem, quæ omnia ſeruare opus eſt.</s> <s xml:id="echoid-s3037" xml:space="preserve"/> </p> <div xml:id="echoid-div325" type="float" level="2" n="1"> <note position="right" xlink:label="note-0149-02" xlink:href="note-0149-02a" xml:space="preserve">Corollar. <lb/>23. lib. 1.</note> <note position="right" xlink:label="note-0149-03" xlink:href="note-0149-03a" xml:space="preserve">Iux. Sect. <lb/>A. huius <lb/>Propoſ.</note> <note position="right" xlink:label="note-0149-04" xlink:href="note-0149-04a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div327" type="section" level="1" n="201"> <head xml:id="echoid-head216" xml:space="preserve">C. SECTIO III.</head> <p> <s xml:id="echoid-s3038" xml:space="preserve">NVncin figuris, MZP, Ω ℟ &</s> <s xml:id="echoid-s3039" xml:space="preserve">, à maiori homologarum, MP, <lb/>Ω &</s> <s xml:id="echoid-s3040" xml:space="preserve">, quæ ſit, MP, aicindatur, OP, æqualis ipſi, Ω &</s> <s xml:id="echoid-s3041" xml:space="preserve">, & </s> <s xml:id="echoid-s3042" xml:space="preserve"><lb/>vt, MP, ad, PO, ita ſit quælibet in figura, MZP, parallela ipſi, <lb/>MP, adeius portionem, & </s> <s xml:id="echoid-s3043" xml:space="preserve">portionum termini ſint ex vna parte in <lb/>recta, ZP, ex alia verò inlinea, ZO, erit ergo, vt vna ad vnam .</s> <s xml:id="echoid-s3044" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s3045" xml:space="preserve">vt, MP, ad, PO, ita omnia ad omnia, .</s> <s xml:id="echoid-s3046" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3047" xml:space="preserve">ita omnes lineæ figuræ, <pb o="130" file="0150" n="150" rhead="GEOMETRIÆ"/> MZP, adomnes lineas figuræ, OZP, regula, MP, & </s> <s xml:id="echoid-s3048" xml:space="preserve">ideò vt, M <lb/> <anchor type="note" xlink:label="note-0150-01a" xlink:href="note-0150-01"/> P, ad, PO, vel ad, Ω &</s> <s xml:id="echoid-s3049" xml:space="preserve">, ita figura, MZP, ad figuram, OZP, <lb/> <anchor type="note" xlink:label="note-0150-02a" xlink:href="note-0150-02"/> quod etiam ſerua.</s> <s xml:id="echoid-s3050" xml:space="preserve"/> </p> <div xml:id="echoid-div327" type="float" level="2" n="1"> <note position="left" xlink:label="note-0150-01" xlink:href="note-0150-01a" xml:space="preserve">Goroll. 4. <lb/>huius.</note> <note position="left" xlink:label="note-0150-02" xlink:href="note-0150-02a" xml:space="preserve">4. huius.</note> </div> </div> <div xml:id="echoid-div329" type="section" level="1" n="202"> <head xml:id="echoid-head217" xml:space="preserve">D. SECTIO IV.</head> <p> <s xml:id="echoid-s3051" xml:space="preserve">VLterius ab ipſis, OP, Ω &</s> <s xml:id="echoid-s3052" xml:space="preserve">, abſcindantur partes æquales, O <lb/>R, ΩΥ, & </s> <s xml:id="echoid-s3053" xml:space="preserve">per puncta, R, Y, ducantur ipſis, ZP, ℟ &</s> <s xml:id="echoid-s3054" xml:space="preserve">, æ-<lb/>quidiſtantes, SR, VY, & </s> <s xml:id="echoid-s3055" xml:space="preserve">per, S, vbi, RS, ſecat lineam, ZO, du-<lb/> <anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/> catur, HC, æquidiſtans ipſi, M <lb/>P, & </s> <s xml:id="echoid-s3056" xml:space="preserve">per, C, vbi, HC, ſecat li-<lb/>neain, ZM, ducatur, CN, pa-<lb/>rallelaipſi, ZP, ſecans, MP, in, <lb/>N; </s> <s xml:id="echoid-s3057" xml:space="preserve">eſt igitur vt, MP, ad, PO, <lb/>ita, CH, ad, HS, per conſtru-<lb/>ctionem .</s> <s xml:id="echoid-s3058" xml:space="preserve">i. </s> <s xml:id="echoid-s3059" xml:space="preserve">ita, NP, ad, PR, & </s> <s xml:id="echoid-s3060" xml:space="preserve"><lb/>permutando, vt, MP, ad, PN, <lb/>ita, OP, ad, PR, diuidendo, vt, <lb/>MN, ad, NP, ita, OR, ad, R <lb/>P, .</s> <s xml:id="echoid-s3061" xml:space="preserve">i. </s> <s xml:id="echoid-s3062" xml:space="preserve">ita, ΩΥ, ad, Y &</s> <s xml:id="echoid-s3063" xml:space="preserve">, igitur ip-<lb/>ſæ, CN, VY, æquidiſtant regu-<lb/>lis homologarum, quę ſunt, ZP, <lb/>℟ &</s> <s xml:id="echoid-s3064" xml:space="preserve">, & </s> <s xml:id="echoid-s3065" xml:space="preserve">diuidunt ad eandem par-<lb/>tem ſimiliter ipſas incidentes, M <lb/>P, Ω &</s> <s xml:id="echoid-s3066" xml:space="preserve">, (ſi .</s> <s xml:id="echoid-s3067" xml:space="preserve">n. </s> <s xml:id="echoid-s3068" xml:space="preserve">ZP, ℟ &</s> <s xml:id="echoid-s3069" xml:space="preserve">, ſtatue-<lb/> <anchor type="note" xlink:label="note-0150-03a" xlink:href="note-0150-03"/> ris regulas homologarum ipſę, M <lb/>P, Ω &</s> <s xml:id="echoid-s3070" xml:space="preserve">, ſunt incidentes, ſi verò <lb/>has ſtatueris regulas, illæ erunt in-<lb/>cidentes, ambę .</s> <s xml:id="echoid-s3071" xml:space="preserve">n. </s> <s xml:id="echoid-s3072" xml:space="preserve">terminant in <lb/>oppoſitas tangentes, quæ ſunt re-<lb/>gulæ homologarum earundem) ergo, CN, ad, VY, erit vt, MP, <lb/>ad, Ω &</s> <s xml:id="echoid-s3073" xml:space="preserve">, .</s> <s xml:id="echoid-s3074" xml:space="preserve">i. </s> <s xml:id="echoid-s3075" xml:space="preserve">vt, ZP, ad, ℟ &</s> <s xml:id="echoid-s3076" xml:space="preserve">, & </s> <s xml:id="echoid-s3077" xml:space="preserve">ſunt, CN, SR, æquales, &</s> <s xml:id="echoid-s3078" xml:space="preserve">, S <lb/>R, VY, vtcunque ductæ ipſis, ZP, ℟ &</s> <s xml:id="echoid-s3079" xml:space="preserve">, æquidiſtantes, ergo ve, <lb/>ZP, ad, ℟ &</s> <s xml:id="echoid-s3080" xml:space="preserve">, ita, SR, ad, VY, ergo vt, ZP, ad, ℟ &</s> <s xml:id="echoid-s3081" xml:space="preserve">, ita erit <lb/> <anchor type="note" xlink:label="note-0150-04a" xlink:href="note-0150-04"/> figura, OZP, ad figuram, Ω ℟ &</s> <s xml:id="echoid-s3082" xml:space="preserve">, quod pariter ſerua.</s> <s xml:id="echoid-s3083" xml:space="preserve"/> </p> <div xml:id="echoid-div329" type="float" level="2" n="1"> <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a"> <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0150-01"/> </figure> <note position="left" xlink:label="note-0150-03" xlink:href="note-0150-03a" xml:space="preserve">Corollar. <lb/>123. lib. 1.</note> <note position="left" xlink:label="note-0150-04" xlink:href="note-0150-04a" xml:space="preserve">4. huius.</note> </div> </div> <div xml:id="echoid-div331" type="section" level="1" n="203"> <head xml:id="echoid-head218" xml:space="preserve">E. SECTIO V. ET VLTIMA.</head> <p> <s xml:id="echoid-s3084" xml:space="preserve">QVoniam verò figura, MZP, ad figuram, Ω ℟ &</s> <s xml:id="echoid-s3085" xml:space="preserve">, habet ratio-<lb/> <anchor type="note" xlink:label="note-0150-05a" xlink:href="note-0150-05"/> nem compoſitam ex ratione figuræ, MZP, ad figuram, O <lb/>ZP, .</s> <s xml:id="echoid-s3086" xml:space="preserve">i. </s> <s xml:id="echoid-s3087" xml:space="preserve">ex ratione, MP, ad, Ω &</s> <s xml:id="echoid-s3088" xml:space="preserve">, & </s> <s xml:id="echoid-s3089" xml:space="preserve">ex ratione figuræ, OZ <lb/>P, ad figuram, Ω ℟ &</s> <s xml:id="echoid-s3090" xml:space="preserve">, .</s> <s xml:id="echoid-s3091" xml:space="preserve">i. </s> <s xml:id="echoid-s3092" xml:space="preserve">ex ratione ipſius, ZP, ad, ℟ & </s> <s xml:id="echoid-s3093" xml:space="preserve">a .</s> <s xml:id="echoid-s3094" xml:space="preserve">i. </s> <s xml:id="echoid-s3095" xml:space="preserve">ex ra- <pb o="131" file="0151" n="151" rhead="LIBER II."/> tione ipſius, MP, ad, Ω &</s> <s xml:id="echoid-s3096" xml:space="preserve">, ideò figura, MZP, ad figuram, Ω ℟ <lb/> <anchor type="note" xlink:label="note-0151-01a" xlink:href="note-0151-01"/> & </s> <s xml:id="echoid-s3097" xml:space="preserve">habebit rationem compofitam ex duabus rationibus ipſius, MP, <lb/>ad, Ω &</s> <s xml:id="echoid-s3098" xml:space="preserve">, . </s> <s xml:id="echoid-s3099" xml:space="preserve">i. </s> <s xml:id="echoid-s3100" xml:space="preserve">duplam eius, quam habet, MP, ad, Ω &</s> <s xml:id="echoid-s3101" xml:space="preserve">, ſiue, KM, <lb/>ad, Π Ω, quæ illis ſunt æquales, ſed & </s> <s xml:id="echoid-s3102" xml:space="preserve">figuræ, ABD, φ Σ Λ, ſunt <lb/>æquales figuris, MZP, Ω ℟ &</s> <s xml:id="echoid-s3103" xml:space="preserve">, ergo ſigura, ABD, ad figuram, Φ <lb/>Σ Λ, duplam rationem habebit eius, quam habet, KM, ad, Π Ω, <lb/>quia vero, KM, &</s> <s xml:id="echoid-s3104" xml:space="preserve">, Π Ω, ſunt incidentes ſimilium figurarum, AB <lb/> <anchor type="note" xlink:label="note-0151-02a" xlink:href="note-0151-02"/> D, Φ Σ Λ, ideò, vt, KM, ad, Π Ω, ita erit, BEID, ſimul ad, Σ 2, <lb/>3 Λ, ſimul, vel ita, BE, ad, Σ 2, ſiue, ID, ad, 3 Λ, ergo figura, <lb/> <anchor type="note" xlink:label="note-0151-03a" xlink:href="note-0151-03"/> ABD, ad figuram, Φ Σ Λ, duplam rationem habebit eius, quam <lb/>habet, BE, ad, Σ 2, vel, ID, ad, 3 Λ, .</s> <s xml:id="echoid-s3105" xml:space="preserve">i. </s> <s xml:id="echoid-s3106" xml:space="preserve">erunt iſtæ ſimiles figuræ <lb/>in dupla ratione linearum, vel laterum homologorum, BE, Σ 2, vel, <lb/>ID, 3 Λ, vel aliarum quarumcumque homologarum præfatis regu-<lb/>lis æquidiſtantium, quod oſtendere opus erat.</s> <s xml:id="echoid-s3107" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="2" n="1"> <note position="left" xlink:label="note-0150-05" xlink:href="note-0150-05a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0151-01" xlink:href="note-0151-01a" xml:space="preserve">Defnilo, <lb/>Quin. El.</note> <note position="right" xlink:label="note-0151-02" xlink:href="note-0151-02a" xml:space="preserve">B. Def. 10. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0151-03" xlink:href="note-0151-03a" xml:space="preserve">Coroll. 1. <lb/>22. lib. @.</note> </div> </div> <div xml:id="echoid-div333" type="section" level="1" n="204"> <head xml:id="echoid-head219" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s3108" xml:space="preserve">_E_T quia dictæ figuræ planæ ſimiles oſtenſæ ſunt eſſe in dupla ratione <lb/>linearum, vel laterum homologorum, quæ æquidiſtant regulis vt-<lb/>cunque ſumptis, patet eaſdem eſſe in dupla ratione quarumuis homolo-<lb/>garum, & </s> <s xml:id="echoid-s3109" xml:space="preserve">duas quaſdam homologas ſumptas cum quibuſdam regulis, eſſe <lb/>inter ſe, vt alias quaslibet duas homologas, cum alijs quibuſuis regulis, eſſe <lb/>aſſumptas, quod etiam in Corollario Lemmatis 48. </s> <s xml:id="echoid-s3110" xml:space="preserve">Lib. </s> <s xml:id="echoid-s3111" xml:space="preserve">1. </s> <s xml:id="echoid-s3112" xml:space="preserve">aliunde dedu-<lb/>ctum eſt.</s> <s xml:id="echoid-s3113" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div334" type="section" level="1" n="205"> <head xml:id="echoid-head220" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s3114" xml:space="preserve">_V_Niuersè inſuper manifeſtum eſt, ſitres rectæ lineæ deinceps pro-<lb/>portionales fuerint, vt prima ad tertiam, ita eſſe figuram planam <lb/>deſcriptam à prima ad eam, quæ à ſecunda de ſcribitur; </s> <s xml:id="echoid-s3115" xml:space="preserve">& </s> <s xml:id="echoid-s3116" xml:space="preserve">huius conuer-<lb/>ſum, dummodò deſcribentes ſint ſimilium deſcriptarum figurarum lineæ, <lb/>ſiue latera homologa.</s> <s xml:id="echoid-s3117" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div335" type="section" level="1" n="206"> <head xml:id="echoid-head221" xml:space="preserve">THEOREMA XVI. PROPOS. XVI.</head> <p> <s xml:id="echoid-s3118" xml:space="preserve">SI quatuor rectæ lineæ proportionales fuerint, prima au-<lb/>tem, & </s> <s xml:id="echoid-s3119" xml:space="preserve">ſecunda ſimiles figuras planas deſcripſerint, & </s> <s xml:id="echoid-s3120" xml:space="preserve"><lb/>tertia, & </s> <s xml:id="echoid-s3121" xml:space="preserve">quarta alias figuras planas ſimiles, licet etiam præ-<lb/>dictis diſſimiles eſſent, ita vt deſcribentes ſint earum lineæ, <lb/>vel latera homologa, figura primæ ad figuram lecundæ erit, <pb o="132" file="0152" n="152" rhead="GEOMETRIÆ"/> vt figura tertiæ ad figuram quartæ. </s> <s xml:id="echoid-s3122" xml:space="preserve">Et ſi fuerint quatuor fi-<lb/>guræ planæ proportionales, ita vt quæ ſunt termini eiuſdem <lb/>proportionis ſint figuræ ſimiles, deſcriptæ ab eorundem li-<lb/>neis, vel lateribus homologis; </s> <s xml:id="echoid-s3123" xml:space="preserve">lineæ, vel latera homologa <lb/>deſcribentia erunt proportionalia.</s> <s xml:id="echoid-s3124" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3125" xml:space="preserve">Sint quatuor rectæ lineæ proportionales, AB, CD, FG, HM, <lb/>prima verò, & </s> <s xml:id="echoid-s3126" xml:space="preserve">ſecunda deſcribant fig. </s> <s xml:id="echoid-s3127" xml:space="preserve">planas ſimiles, AXB, CVD, <lb/> <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> &</s> <s xml:id="echoid-s3128" xml:space="preserve">, FG, HM, ſimiles figuras pla-<lb/>nas, FOG, HNM, licet prędictis <lb/>diſſimiles eſſent, & </s> <s xml:id="echoid-s3129" xml:space="preserve">ſint deſcribentes <lb/>figurarum deſcriptarum lineæ, vel <lb/>latera homologa. </s> <s xml:id="echoid-s3130" xml:space="preserve">Dico, AXB, ad, <lb/>CVD, eſſe vt, FOG, ad, HNM. <lb/></s> <s xml:id="echoid-s3131" xml:space="preserve">Sit, R, tertia proportionalis ipſa-<lb/>rum, AB, CD, &</s> <s xml:id="echoid-s3132" xml:space="preserve">, I, tertia pro-<lb/>portionalis ipſarum, FG, HM; </s> <s xml:id="echoid-s3133" xml:space="preserve">eſt <lb/>igitur, AXB, ad, CVD, vt, AB, <lb/>ad, R, .</s> <s xml:id="echoid-s3134" xml:space="preserve">i. </s> <s xml:id="echoid-s3135" xml:space="preserve">in ratione dupla eius, quam habet, AB, ad, CD, .</s> <s xml:id="echoid-s3136" xml:space="preserve">i. </s> <s xml:id="echoid-s3137" xml:space="preserve">eius, <lb/> <anchor type="note" xlink:label="note-0152-01a" xlink:href="note-0152-01"/> quam habet, FG, ad, HM, .</s> <s xml:id="echoid-s3138" xml:space="preserve">i. </s> <s xml:id="echoid-s3139" xml:space="preserve">vt, FG, ad, I, quę eſt, vt, FOG, <lb/>ad, HNM, ergo vt, AXB, ad, CVD, ita erit, FOG, ad, H <lb/>NM.</s> <s xml:id="echoid-s3140" xml:space="preserve"/> </p> <div xml:id="echoid-div335" type="float" level="2" n="1"> <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0152-01"/> </figure> <note position="left" xlink:label="note-0152-01" xlink:href="note-0152-01a" xml:space="preserve">Coroll. 2. <lb/>antec.</note> </div> <p> <s xml:id="echoid-s3141" xml:space="preserve">Sit nunc figura, AXB, ad, CVD, ſibi ſimilem, vt, FOG, ad, <lb/>HNM, ſibi ſimilem, licet iſtæ eſſent prædictis diſſimiles, & </s> <s xml:id="echoid-s3142" xml:space="preserve">eas de-<lb/>ſcribentes ſint earum lineæ, vel latera homologa. </s> <s xml:id="echoid-s3143" xml:space="preserve">Dico, AB, ad, <lb/>CD, eſſe vt, FG, ad, HM, ſit adhuc, R, tertia proportionalis ip-<lb/>ſarum, AB, CD, &</s> <s xml:id="echoid-s3144" xml:space="preserve">, I, tertia proportionalis ipſarum, FG, HM, <lb/> <anchor type="note" xlink:label="note-0152-02a" xlink:href="note-0152-02"/> eſt ergo, vt figura, AXB, ad, CVD, ita, AB, ad, R, vt verò fi-<lb/>gura, FOG, ad, HNM, ita, FG, ad, I, eſt verò, vt, AXB, ad, <lb/>CVD, ita, FOG, ad, HNM, ergo vt, AB, ad, R, ſic, FG, ad, <lb/>I, eſtautem, AB, ad, R, dupla rationis ipſius, AB, ad, CD, &</s> <s xml:id="echoid-s3145" xml:space="preserve">, <lb/>FG, ad, I, dupla rationis ipſius, FG, ad, HM, ergo vt, AB, ad, <lb/>CD, ita, FG, ad, HM, quæ oſtendere opus erat.</s> <s xml:id="echoid-s3146" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="2"> <note position="left" xlink:label="note-0152-02" xlink:href="note-0152-02a" xml:space="preserve">Coroll. 2. <lb/>antec.</note> </div> </div> <div xml:id="echoid-div338" type="section" level="1" n="207"> <head xml:id="echoid-head222" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s3147" xml:space="preserve">_P_Ropoſitionis proximè ſubſequentis nimia fortaſſe prolixitas faſti-<lb/>dium potius Lectori, quam delectationem pariet, veruntamen, qui <lb/>hoc veretur, ac tantum otij, aut tolerantiæ habere nequit, vt illius ſa-<lb/>tis tongam texturam percurrere Valeat, ipſam ſupponat, ac prætereat, <lb/>ijs enim præcipuè à me dirigitur, quibus nec otium deeſt, nec ingenium, <pb o="133" file="0153" n="153" rhead="LIBER II."/> ac voluntas, pulchras demonſtrationes etſi difficiles, ac longas infracto <lb/>quodam animi vigore ſuperandi, potius quam ab ipſis ſuperari velint. <lb/></s> <s xml:id="echoid-s3148" xml:space="preserve">Poterat quidem in plures Propoſitiones commodius diſtribui, ſed cum-<lb/>illæ omnes in hanc ſimpliciſſimam eſſent conſpiraturæ, eas omnes ſub <lb/>hac vna Propoſit. </s> <s xml:id="echoid-s3149" xml:space="preserve">colligaui, quamtamen in Sectiones ceu in tot mem-<lb/>bra distinguere placuit, ne Lectoris mens nimium defatigaretur. </s> <s xml:id="echoid-s3150" xml:space="preserve">Porrò <lb/>quanti hæc Propoſitio ſit momenti, ſicut & </s> <s xml:id="echoid-s3151" xml:space="preserve">præcedens Propoſ. </s> <s xml:id="echoid-s3152" xml:space="preserve">15. </s> <s xml:id="echoid-s3153" xml:space="preserve">atten-<lb/>ta præcipuè earum vniuerſalitate, neminem, qui eaſdem intellex erit, <lb/>fore puto, qui itidem non agnoſcat; </s> <s xml:id="echoid-s3154" xml:space="preserve">quid enim fuit, quo ad figuras pla-<lb/>nas, Euclidem lib. </s> <s xml:id="echoid-s3155" xml:space="preserve">6. </s> <s xml:id="echoid-s3156" xml:space="preserve">Elementorum in Propoſ 19. </s> <s xml:id="echoid-s3157" xml:space="preserve">demonſtraſſe ſimilia-<lb/>triangula, & </s> <s xml:id="echoid-s3158" xml:space="preserve">in Propoſ. </s> <s xml:id="echoid-s3159" xml:space="preserve">20. </s> <s xml:id="echoid-s3160" xml:space="preserve">ſimilia Polygona eſſe in dupla ratione la-<lb/>te um homologorum, necnon lib. </s> <s xml:id="echoid-s3161" xml:space="preserve">12. </s> <s xml:id="echoid-s3162" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s3163" xml:space="preserve">2. </s> <s xml:id="echoid-s3164" xml:space="preserve">Circulos eſſe, vt diame-<lb/>trorum quadrata, hoc eſt in dupla ratione diametrorum? </s> <s xml:id="echoid-s3165" xml:space="preserve">Similiter in eo, <lb/>quod ſpectat ad ſolida, quid fuit ipſum nobis in lib. </s> <s xml:id="echoid-s3166" xml:space="preserve">12. </s> <s xml:id="echoid-s3167" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s3168" xml:space="preserve">8. </s> <s xml:id="echoid-s3169" xml:space="preserve">oſten-<lb/>diſſe ſimiles Pyramides eſſe in tripla ratione laterum homologorum, & </s> <s xml:id="echoid-s3170" xml:space="preserve"><lb/>in Prop. </s> <s xml:id="echoid-s3171" xml:space="preserve">12. </s> <s xml:id="echoid-s3172" xml:space="preserve">ſimiles conos, & </s> <s xml:id="echoid-s3173" xml:space="preserve">cylindros eſſe in triplaratione diametro-<lb/>rum quæ ſunt in baſibus, & </s> <s xml:id="echoid-s3174" xml:space="preserve">in Propoſ. </s> <s xml:id="echoid-s3175" xml:space="preserve">18. </s> <s xml:id="echoid-s3176" xml:space="preserve">Sphæras itidem eſſe in tri-<lb/>pla proportione diametrorum? </s> <s xml:id="echoid-s3177" xml:space="preserve">Quid tandem fuit alios quoq; </s> <s xml:id="echoid-s3178" xml:space="preserve">demonſtraſ-<lb/>ſe, quædam alia ſimilia ſolida, vt portiones Sphærarum, necnon Sphæ-<lb/>roidearum, & </s> <s xml:id="echoid-s3179" xml:space="preserve">Conoide arum figurarum, eſſe in tripla ratione linearum, <lb/>vel laterum homologorum? </s> <s xml:id="echoid-s3180" xml:space="preserve">Præ huius comparatione, quod in his duabus <lb/>tantum Propoſitionibus edocemur; </s> <s xml:id="echoid-s3181" xml:space="preserve">omnes .</s> <s xml:id="echoid-s3182" xml:space="preserve">n. </s> <s xml:id="echoid-s3183" xml:space="preserve">ſimiles figuras planas in <lb/>Prop. </s> <s xml:id="echoid-s3184" xml:space="preserve">15. </s> <s xml:id="echoid-s3185" xml:space="preserve">& </s> <s xml:id="echoid-s3186" xml:space="preserve">omnes ſolidas in ſubſequenti Propoſ. </s> <s xml:id="echoid-s3187" xml:space="preserve">17. </s> <s xml:id="echoid-s3188" xml:space="preserve">comprebendimus, <lb/>quod mebercle conſideratione dignum videtur.</s> <s xml:id="echoid-s3189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div339" type="section" level="1" n="208"> <head xml:id="echoid-head223" xml:space="preserve">THEOREMA XVII. PROPOS. XVII.</head> <p> <s xml:id="echoid-s3190" xml:space="preserve">OMnia ſimilia ſolida ſunt in tripla ratione linearum, vel <lb/>laterum homologorum, quę ſunt in eorundem homo-<lb/>logis figuris.</s> <s xml:id="echoid-s3191" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div340" type="section" level="1" n="209"> <head xml:id="echoid-head224" xml:space="preserve">A. DEMONSTRATIONIS SECTIO I.</head> <p> <s xml:id="echoid-s3192" xml:space="preserve">SInt duo vtcunq; </s> <s xml:id="echoid-s3193" xml:space="preserve">ſimilia ſolida, V &</s> <s xml:id="echoid-s3194" xml:space="preserve">, AP. </s> <s xml:id="echoid-s3195" xml:space="preserve">Dico hæc eſſe in tri-<lb/>pla ratione linearum, ſiue laterum homologorum, quæ ſunt in <lb/>eorundem homologis figuris. </s> <s xml:id="echoid-s3196" xml:space="preserve">Quia ergo dicta ſolida ſunt ſimilia, po-<lb/>terunt duci duo plana oppoſita tangentia in vnoquoque propoſito-<lb/>rum ſolidorum (quæ in ſolido, AP, repræſententur peripſas, AH, <lb/> <anchor type="note" xlink:label="note-0153-01a" xlink:href="note-0153-01"/> P {00/ }, & </s> <s xml:id="echoid-s3197" xml:space="preserve">in ſolido, V &</s> <s xml:id="echoid-s3198" xml:space="preserve">, peripſas, V Σ, & </s> <s xml:id="echoid-s3199" xml:space="preserve">2,) homologis eorundem <lb/>figuris æquidiſtantia, inter quæ etiam ducibilia erunt alia duo plana <lb/> <anchor type="note" xlink:label="note-0153-02a" xlink:href="note-0153-02"/> æqualiter ad ipſa, & </s> <s xml:id="echoid-s3200" xml:space="preserve">ad eandem partem inclinata, in quibus iacebunt <pb o="134" file="0154" n="154" rhead="GEOMETRIÆ"/> figuræ, quæ erunt dictorum ſimilium ſolidorum, & </s> <s xml:id="echoid-s3201" xml:space="preserve">tangentium op <lb/>poſitorum, figuræ incidentes, ſint igitur talia duo plana, quorum, <lb/>& </s> <s xml:id="echoid-s3202" xml:space="preserve">oppoſitorum planorum tangentium in ſolido, AP, communes ſe-<lb/>ctiones, HL, OO, G, & </s> <s xml:id="echoid-s3203" xml:space="preserve">ſolidi, V &</s> <s xml:id="echoid-s3204" xml:space="preserve">, Σ 3, 28, in his autem pla-<lb/> <anchor type="note" xlink:label="note-0154-01a" xlink:href="note-0154-01"/> nis ſint eorum incidentes figuræ, H {00/ }, Σ 2, iſtæ igitur erunt figuræ <lb/>fimiles, & </s> <s xml:id="echoid-s3205" xml:space="preserve">tangentur à dictis communibus ſectionibus, quæ erunt li-<lb/> <anchor type="note" xlink:label="note-0154-02a" xlink:href="note-0154-02"/> nearum homologarum earundem etiam regulæ, ſint earum inciden, <lb/> <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> tes vtcunque inter eaſdem ductæ, LG, 38, & </s> <s xml:id="echoid-s3206" xml:space="preserve">extendantur inter di-<lb/>cta oppoſita tangentia vtcunque plana eiſdem æquidiſtantia, altitu-<lb/>dines propoſitorum ſolidorum reſpectu dictorum tangentium ſum-<lb/>ptas ſimiliter ad eandem partem diuidentia, ſit igitur vnius ductorum <lb/>planorum concepta in ſolido, AP, figura, BC, eiuſdem autem, & </s> <s xml:id="echoid-s3207" xml:space="preserve"><lb/>figuræ, H {00/ }, communis ſectio, OX, quod etiam ſecet incidentem <lb/>figuræ, H {00/ }, quæ eſt, LG, in, E; </s> <s xml:id="echoid-s3208" xml:space="preserve">pariter alterius planiconcepta <lb/>in ſolido, V &</s> <s xml:id="echoid-s3209" xml:space="preserve">, figura ſit, Π Ω, idem verò planum ſecet figuram, Σ 2, <pb o="135" file="0155" n="155" rhead="LIBER II."/> inrecta, Φ Λ, & </s> <s xml:id="echoid-s3210" xml:space="preserve">incidentem eiuſdem figuræ, nempè ipſam, 38, in <lb/>puncto, 4, igitur figuræ, BC, Π Ω, erunt duæ figurarum homolo-<lb/> <anchor type="note" xlink:label="note-0155-01a" xlink:href="note-0155-01"/> garum ſolidorum, AP, V &</s> <s xml:id="echoid-s3211" xml:space="preserve">, &</s> <s xml:id="echoid-s3212" xml:space="preserve">, OX, Φ Λ, earum incidentes, &</s> <s xml:id="echoid-s3213" xml:space="preserve">, <lb/>LG, 38, erunt ſimiliter diuiſæ in punctis, E, 4, nam etiam altitu-<lb/> <anchor type="note" xlink:label="note-0155-02a" xlink:href="note-0155-02"/> dines propoſitorum ſimilium ſolidorum ſunt ſimiliter diuiſæ (ad ean-<lb/>dem partem ſub intellige) ſi igitur à punctis, O, Φ, duxerimus tan-<lb/>gentes figuras, BC, Π Ω, erunt iſtæ regulis homologarum earun-<lb/>dem figurarum parallelæ, vel pro regulis aliarum etiam aſſumi pote-<lb/>runt, & </s> <s xml:id="echoid-s3214" xml:space="preserve">quę à punctis, X, Λ, ducentur prędictis parallelę occurrent <lb/>eiſdem figuris, & </s> <s xml:id="echoid-s3215" xml:space="preserve">illas ex oppoſito prędictarum contingent, ita vt ha-<lb/>beamus (ſi & </s> <s xml:id="echoid-s3216" xml:space="preserve">iſtæ ductæ intelligantur, quæ ſint, XC, Λ Ω,) oppc-<lb/>fitas tangentes figuræ, BC, quæ erunt, BO, CX, & </s> <s xml:id="echoid-s3217" xml:space="preserve">figuræ, Π Ω, <lb/>quæ erunt, Π Φ, Ω Λ, necnon pro regulis homologarum earundem <lb/>haberi poterunt; </s> <s xml:id="echoid-s3218" xml:space="preserve">vel igitur figuræ, BC, Π Ω, adiacent ſuis inciden-<lb/>tibus, OX, Φ Λ, totę ad eandem partem, & </s> <s xml:id="echoid-s3219" xml:space="preserve">interius integrę exiſten-<lb/>tes, vel non, ſi ſic factum erit, quod volumus, ſi non transferantur <lb/>omnes lineæ figurarum, BC, Π Ω, regulis eiſdem tangentibus, in <lb/> <anchor type="note" xlink:label="note-0155-03a" xlink:href="note-0155-03"/> figuras ipſis, OX, Φ Λ, adiacentes, pro vt in Prop. </s> <s xml:id="echoid-s3220" xml:space="preserve">15. </s> <s xml:id="echoid-s3221" xml:space="preserve">effectum eſt, <lb/>hinc autem reſultantes figurę ſint, OZX, Φ Γ Λ, quę per talem con-<lb/>ſtructionem ad eandem partem incidentium, & </s> <s xml:id="echoid-s3222" xml:space="preserve">interius integrę nc-<lb/>bis proueniunt. </s> <s xml:id="echoid-s3223" xml:space="preserve">Similiter ſi intelligamus ducta alia duo plana prędi-<lb/>ctis ęquidiſtantia, quę ſolida propoſita ita ſecent, vt fiant in ipſis non <lb/>vnica in ſingulis figura, ſed plures, ex. </s> <s xml:id="echoid-s3224" xml:space="preserve">gr. </s> <s xml:id="echoid-s3225" xml:space="preserve">in ſolido, AP, figurę, R, <lb/>I, & </s> <s xml:id="echoid-s3226" xml:space="preserve">in, V &</s> <s xml:id="echoid-s3227" xml:space="preserve">, figuræ, ℟, N, eadem autem ſecent figuras inciden-<lb/>tes in rectis, SY, Β Δ, & </s> <s xml:id="echoid-s3228" xml:space="preserve">rectas, LG, 38, in punctis, K, {10/ }, dum-<lb/>modo hæc plana pariter ſecent altitudines dictas propoſitorum ſoli-<lb/>dorum ſimiliter ad eandem partem, erunt figurę, R, I, binę ſimiles, & </s> <s xml:id="echoid-s3229" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0155-04a" xlink:href="note-0155-04"/> ſimiliter poſitę, ac figurę, ℟, N, .</s> <s xml:id="echoid-s3230" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3231" xml:space="preserve">I, ſimilis ipſi, N, &</s> <s xml:id="echoid-s3232" xml:space="preserve">, R, ipſi, ℟, & </s> <s xml:id="echoid-s3233" xml:space="preserve"><lb/>linearum homologarum earundem regulæ ipſis, CX, Ω Λ, æquidi-<lb/>ſtabunt, ipſę autem rectę, S, Y; </s> <s xml:id="echoid-s3234" xml:space="preserve">β, Δ, erunt earundem incidentes, <lb/>vt, S, β, ipſarum, R, ℟, &</s> <s xml:id="echoid-s3235" xml:space="preserve">, Y, Δ, ipſarum, I, N, ſi igitur figurę, <lb/>R, I, ℟, N, non adiacent ſuis incidentibus, transferantur ſingula-<lb/>rum omnes lineę, regula ſemper, pro figuris, RI, ipſa, CX, & </s> <s xml:id="echoid-s3236" xml:space="preserve">pro <lb/> <anchor type="note" xlink:label="note-0155-05a" xlink:href="note-0155-05"/> figuris, ℟, N, ipſa, Ω Λ, in figuras adiacentes lineis homologis figu-<lb/> <anchor type="note" xlink:label="note-0155-06a" xlink:href="note-0155-06"/> rarum, H {00/ }, Σ 2, vt ſint nobis inuentæ figuræ, S, Y, β Δ, quæ <lb/>adiaceant homologis lineis figurarum incidentium, H, {00/ }, Σ 2: </s> <s xml:id="echoid-s3237" xml:space="preserve">Si <lb/>igitur eandem methodum ſeruemus in cæteris figuris, quæ ex lectio-<lb/>ne planorum tangentibus æquidiſtantium in dictis ſolidis producun-<lb/>tur, transferentes nempè omnes earum lineas homologas, regulis <lb/>ſemper ipſis, CX, Ω Λ, in figuras adiacentes lineis homologis figu-<lb/>rarum incidentium, H, {00/ }, Σ 2, quę reperientur totę ad eandem par-<lb/>tem, & </s> <s xml:id="echoid-s3238" xml:space="preserve">interius integrę, tandem nobis erunt comparata duo ſolida, <pb o="136" file="0156" n="156" rhead="GEOMETRIÆ"/> quæ prædictis ſimilibus ſolidis æquabuntur ea nempè, quorum om-<lb/>nes prædictæ adiacentes figuræ erunt omnia plana, nam hæ omnes <lb/>adiacentes erunt æquales omnibus homologis figuris dictorum ſimi-<lb/>lium ſolidorum, quarum omnes lineę in ipſas figuras adiacentes mo-<lb/> <anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/> dò dicto translatę funt, ſint hęc ſolida, HZ, {00/ }, Σ Γ 2, igitur, AP, <lb/>erit æquale ipſi, HZ {00/ }, &</s> <s xml:id="echoid-s3239" xml:space="preserve">, V &</s> <s xml:id="echoid-s3240" xml:space="preserve">, ipſi, Σ 2. </s> <s xml:id="echoid-s3241" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3242" xml:space="preserve">hæc ſolida, H <lb/> <anchor type="note" xlink:label="note-0156-02a" xlink:href="note-0156-02"/> Z {00/ }, Σ Γ 2, eruntinter ſe ſimilia, nam figurę planę in eiſdem captę, <lb/> <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> æquidiſtantes dictis tangentibus planis, & </s> <s xml:id="echoid-s3243" xml:space="preserve">altitudines reſpectu dicto-<lb/>rum tangentium ſumptas ſimiliter, & </s> <s xml:id="echoid-s3244" xml:space="preserve">ad eandem partem di uidentes, <lb/>ſunt inter ſe ſimiles, & </s> <s xml:id="echoid-s3245" xml:space="preserve">in ipſis linearum homologarum regulæ om-<lb/>nes vni cuidam æquidiſtant, illi nempè, qua regula translationes fa-<lb/>ctæ ſunt, & </s> <s xml:id="echoid-s3246" xml:space="preserve">earundem figurarum ſi milium, incidentes ſunt lineę ho-<lb/>mologæ duarum planarum ſimilium figurarum, nempè, H {00/ }, Σ 2, <lb/>æqualiter ad figuras adiacentes, & </s> <s xml:id="echoid-s3247" xml:space="preserve">ad eandem partem inclinatarum, <lb/>quarum regulæ ſunt communes ſectiones oppoſitorum tangentium <pb o="137" file="0157" n="157" rhead="LIBER II."/> planorum, necnon planorum earundem figurarum incidentium, <lb/>nempè, HL, 3 Σ, quod ſerua.</s> <s xml:id="echoid-s3248" xml:space="preserve"/> </p> <div xml:id="echoid-div340" type="float" level="2" n="1"> <note position="right" xlink:label="note-0153-01" xlink:href="note-0153-01a" xml:space="preserve">Coroll. 1. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0153-02" xlink:href="note-0153-02a" xml:space="preserve">Defin. 11. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve">Defin. 11. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve">B. Def. 10. <lb/>Lib. 1.</note> <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0154-01"/> </figure> <note position="right" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve">Defin. 11. <lb/>huius.</note> <note position="right" xlink:label="note-0155-02" xlink:href="note-0155-02a" xml:space="preserve">17. Vndec. <lb/>Elem.</note> <note position="right" xlink:label="note-0155-03" xlink:href="note-0155-03a" xml:space="preserve">Vide A. <lb/>15. huius <lb/>propèfin.</note> <note position="right" xlink:label="note-0155-04" xlink:href="note-0155-04a" xml:space="preserve">E. Def. 10. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0155-05" xlink:href="note-0155-05a" xml:space="preserve">Videad <lb/>fig. A.</note> <note position="right" xlink:label="note-0155-06" xlink:href="note-0155-06a" xml:space="preserve">Piop. 15. <lb/>huius.</note> <note position="left" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0156-02" xlink:href="note-0156-02a" xml:space="preserve">Defin. 11. <lb/>ib. 1.</note> <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a"> <image file="0156-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0156-01"/> </figure> </div> </div> <div xml:id="echoid-div342" type="section" level="1" n="210"> <head xml:id="echoid-head225" xml:space="preserve">B. SECTIO II.</head> <p> <s xml:id="echoid-s3249" xml:space="preserve">NVnc quia figuræ iam dictæ adiacentes homologis lineis figura-<lb/>rum, H {00/ }, Σ 2, plurificari poſſunt, quę ſunt in eodem plano, <lb/>vti apparet in figuris, S, Y, β, Γ, quę cum ſint in eodem plano ſunt <lb/>tamen duæ figuræ, ideò vt ex duobus fiat vna tantum, adhuc om-<lb/>nium linearum harum adiacentium figurarum aliam translationem <lb/>regulis, HL, Σ 3, faciemus; </s> <s xml:id="echoid-s3250" xml:space="preserve">ducantur ergo per ipſas, LG, 38, duo <lb/>plana, quorum & </s> <s xml:id="echoid-s3251" xml:space="preserve">oppoſitorum planorum tangentium communes <lb/>ſectiones ſint ipſæ, 3 {12/ }, 8 {13/ }, LQ, GT, cum ipſis, 3 Σ, 82, LH, <lb/>G {00/ }, angulos æquales continentes, & </s> <s xml:id="echoid-s3252" xml:space="preserve">agantur duę ex oppoſito tan-<lb/>gentes figuras, OZX, Φ Γ Λ, parallelę ipſis, OX, Φ Λ, quę ſint ip-<lb/>iæ, ZF, Γ 7, productæ cum reliquis tangentibus oppoſitis, OX, Φ <lb/>Λ, donec occurrant planis, LT, 3 {13/ }, vt in punctis, E, F; </s> <s xml:id="echoid-s3253" xml:space="preserve">4, 7, iun-<lb/>ctis rectis lineis, EF, 47. </s> <s xml:id="echoid-s3254" xml:space="preserve">Quia ergo, DE, ęquidiſtat ipſi, {00/ }G, &</s> <s xml:id="echoid-s3255" xml:space="preserve">, <lb/> <anchor type="note" xlink:label="note-0157-01a" xlink:href="note-0157-01"/> EF, ipſi, GT, angulus, DEF, æquatur angulo, {00/ } GT, & </s> <s xml:id="echoid-s3256" xml:space="preserve">eadem <lb/>ratione angulus, 647, probabitur æqualis ipſi, 28 {13/ }, vnde, quia, <lb/>{00/ }GT, æquatur ipſi, 28 {13/ }, angulus, FED, erit æqualis angulo, <lb/>746, & </s> <s xml:id="echoid-s3257" xml:space="preserve">cum ſit, vt, OX, ad, Φ Λ, vel vt, OE, ad, Φ 4, quia, L <lb/>G, 38, ſunt lineæ incidentes ſimilium planarum figurarum, H {00/ }, <lb/> <anchor type="note" xlink:label="note-0157-02a" xlink:href="note-0157-02"/> Σ 2, vel vt, XE, ad, Λ 4, ita, EF, ad, 47, ſint autem, XE, Λ 4, <lb/>comprehenſæ inter eaidem extremitates rectarum, EF, 47, & </s> <s xml:id="echoid-s3258" xml:space="preserve">peri-<lb/>metrum figurarum, OZX, Φ Γ Λ, eaſdem tangentes, ergo, EF, 4 <lb/>7, erunt incidentes ſimilium figurarum, OZX, Φ Γ Λ, & </s> <s xml:id="echoid-s3259" xml:space="preserve">oppoſita-<lb/>rum tangentium, OE, ZF; </s> <s xml:id="echoid-s3260" xml:space="preserve">Φ 4, Γ 7. </s> <s xml:id="echoid-s3261" xml:space="preserve">Similiter ſi ſic producantur <lb/> <anchor type="note" xlink:label="note-0157-03a" xlink:href="note-0157-03"/> oppoſitæ tangentes figurarum, S, Y; </s> <s xml:id="echoid-s3262" xml:space="preserve">β Δ, quarum duæ incidant ip-<lb/>ſis, LG, 38, vt in, K, {10/ }, reliquæ vero in punctis, {11/ } {14/ }, planis, L <lb/>T, 3 {13/ }, occurrant, iunctis, K {14/ }, {10/ } {11/ }, oſtendemus pariter ipſas, K <lb/>{14/ }, {10/ } {11/ }, eſſe incidentes ſimilium figurarum, Y, Δ, vel ſimilium, S, <lb/>β, & </s> <s xml:id="echoid-s3263" xml:space="preserve">oppoſitarum tangentium extremarum, quæ ad puncta, K 14, <lb/>&</s> <s xml:id="echoid-s3264" xml:space="preserve">, {10/ }, {11/ }, terminantur. </s> <s xml:id="echoid-s3265" xml:space="preserve">Si igitur transferamus omnes lineas tum fi-<lb/> <anchor type="note" xlink:label="note-0157-04a" xlink:href="note-0157-04"/> gurarum, S, Y, tum, β, Δ, regulis eiſdem tangentibus, vel ſemper <lb/>regulis ipſis, OE, Φ 4, prius compoſitis illis, quę ſibi in directum e-<lb/>runt, tum in figuris, S, Y, tum, β Δ, vt ex illis fiat vnica compoſita <lb/>recta linea, prędictis in directum poſita in figura adiacente, qualis ſit, <lb/>9 {10/ }, æqualis .</s> <s xml:id="echoid-s3266" xml:space="preserve">i. </s> <s xml:id="echoid-s3267" xml:space="preserve">compoſitæ ex his, quibus adiacent figuræ, β Δ, &</s> <s xml:id="echoid-s3268" xml:space="preserve">, <lb/>MK, æqualis compoſitæ ex his, quibus adiacent figuræ, S, Y; </s> <s xml:id="echoid-s3269" xml:space="preserve">tan-<lb/>dem habebimus figuras adiacentes ipſis incidentibus .</s> <s xml:id="echoid-s3270" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3271" xml:space="preserve">MK {14/ }, 9 {10/ } <lb/>{11/ }, in quibus plures figuræ, S, Y, in vnam, MK {14/ }, & </s> <s xml:id="echoid-s3272" xml:space="preserve">β, Δ, in v- <pb o="138" file="0158" n="158" rhead="GEOMETRIÆ"/> nam, 9 {10/ } {11/ }, collectę erunt. </s> <s xml:id="echoid-s3273" xml:space="preserve">Si igitur hoc fiat in cęteris figuris, quę <lb/>in ſolidis, HZ {00/ }, Σ Γ 2, ipſis tangentibus planis æquidiſtant, tandem <lb/>habebimus duo ſolida, quæ ſint, LDFG, 3687, æqualia duobus <lb/>ſolidis, HZ {00/ }, Σ Γ 2, ſeu duobus, AP, V &</s> <s xml:id="echoid-s3274" xml:space="preserve">, LDGF, nempè ipſi, <lb/>AP, &</s> <s xml:id="echoid-s3275" xml:space="preserve">, 3687, ipſi, V &</s> <s xml:id="echoid-s3276" xml:space="preserve">, nam omnia eorum plana, regulis oppo-<lb/> <anchor type="note" xlink:label="note-0158-01a" xlink:href="note-0158-01"/> ſitis tangentibus planis, ſunt inter ſe æqualia ex conſtructione. </s> <s xml:id="echoid-s3277" xml:space="preserve">Sed <lb/>& </s> <s xml:id="echoid-s3278" xml:space="preserve">hæc ſolida, LDGF, 3687, dico eſſe inter ſe ſimilia: </s> <s xml:id="echoid-s3279" xml:space="preserve">Cum .</s> <s xml:id="echoid-s3280" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s3281" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> præfatis oppoſitis tangentibus planis (quæ ſunt etiam oppoſita tan-<lb/>gentia plana ſolidorum, LDGF, 3687,) incidant quoq; </s> <s xml:id="echoid-s3282" xml:space="preserve">duo pla-<lb/>na, LT, 3, {13/ }, ad eundem angulum ex eadem parte (ſunt .</s> <s xml:id="echoid-s3283" xml:space="preserve">n. </s> <s xml:id="echoid-s3284" xml:space="preserve">prima <lb/> <anchor type="note" xlink:label="note-0158-02a" xlink:href="note-0158-02"/> plana, HG, Σ 8, oppoſitis tangentibus planis æquè, & </s> <s xml:id="echoid-s3285" xml:space="preserve">ad eandem <lb/>partem, inclinata, & </s> <s xml:id="echoid-s3286" xml:space="preserve">anguli, TG {00/ }, {13/ } 82, æquales inter ſe, nec-<lb/>non anguli, LG {00/ }, 382, vnde etiam ſecunda plana ad eadem tan-<lb/>gentia plana ſunt ad eundem angulum ex eadem parte. </s> <s xml:id="echoid-s3287" xml:space="preserve">Sint verò <lb/> <anchor type="note" xlink:label="note-0158-03a" xlink:href="note-0158-03"/> figuræ ex planis inclinata oppoſitis tangentibus parallelis, altitudi- <pb o="139" file="0159" n="159" rhead="LIBER II."/> neſque ipſorum ſolidorum, LDGF, 3687, ſimiliter ad eandem <lb/>partem diuidentibus, conceptę (vt probatum eſt) inter ſe ſimiles, vt <lb/>ipſæ, DEF, 647, necnon, MK {14/ }, 9 {10/ } {11/ }, & </s> <s xml:id="echoid-s3288" xml:space="preserve">omnium earundem <lb/>linearum homologarum regulæ duabus quibuſdam, nempe ipſis, O <lb/>E, Φ 4, æquid ſtantes, & </s> <s xml:id="echoid-s3289" xml:space="preserve">earum incidentes ipſæ, EF, 47, necnon, <lb/>K {14/ }, {10/ } {11/ }, quę omnes incidentes iacent in plano ſiminum figurarum, <lb/>LFG, 378, & </s> <s xml:id="echoid-s3290" xml:space="preserve">ſunt earum homologæ, æquidiſtantes ipſis, LQ, <lb/>3 {12/ }, communibus ſectionibus planorum incidentium figurarum, L <lb/>FG, 378, & </s> <s xml:id="echoid-s3291" xml:space="preserve">oppoſitorum tangentium, quarum quidem figurarum <lb/> <anchor type="note" xlink:label="note-0159-01a" xlink:href="note-0159-01"/> plana ſunt ad plana tangentia, vt dictum eſt, æquè ad eandem par-<lb/>tem inclinata, & </s> <s xml:id="echoid-s3292" xml:space="preserve">cum ipſæ, inquam, figuræ, LFG, 378, ſint ſi-<lb/>miles interſe, nam ex. </s> <s xml:id="echoid-s3293" xml:space="preserve">g. </s> <s xml:id="echoid-s3294" xml:space="preserve">eſt, EF, ad, 47, vt, OE, ad, Φ 4, ideſt <lb/> <anchor type="note" xlink:label="note-0159-02a" xlink:href="note-0159-02"/> vt, LG, ad, 38, quæ diuidunt ſimiliter ad eandem partem ipſas, L <lb/>G, 38, (quod etiam de cæteris probabitur) & </s> <s xml:id="echoid-s3295" xml:space="preserve">cum anguli, LGT, <lb/> <anchor type="note" xlink:label="note-0159-03a" xlink:href="note-0159-03"/> 38 {13/ }, ſint etiam æquales ſuperius dictis conſequenter, & </s> <s xml:id="echoid-s3296" xml:space="preserve">ijs, quæ <lb/>lib. </s> <s xml:id="echoid-s3297" xml:space="preserve">1. </s> <s xml:id="echoid-s3298" xml:space="preserve">Prop. </s> <s xml:id="echoid-s3299" xml:space="preserve">26. </s> <s xml:id="echoid-s3300" xml:space="preserve">oſtenia ſunt, ideò, inquam, & </s> <s xml:id="echoid-s3301" xml:space="preserve">ipſæ figuræ, LFG, <lb/> <anchor type="note" xlink:label="note-0159-04a" xlink:href="note-0159-04"/> 378, & </s> <s xml:id="echoid-s3302" xml:space="preserve">ipſa iolida, LDGF, 3687, pariter ſimilia erunt.</s> <s xml:id="echoid-s3303" xml:space="preserve"/> </p> <div xml:id="echoid-div342" type="float" level="2" n="1"> <note position="right" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve">16. Vnd@ <lb/>Elem.</note> <note position="right" xlink:label="note-0157-02" xlink:href="note-0157-02a" xml:space="preserve">Corollat. <lb/>24. lib. 1.</note> <note position="right" xlink:label="note-0157-03" xlink:href="note-0157-03a" xml:space="preserve">24. lib. 1.</note> <note position="right" xlink:label="note-0157-04" xlink:href="note-0157-04a" xml:space="preserve">Vide ad <lb/>finem A. <lb/>p. 15. hu-<lb/>ius.</note> <note position="left" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve">3. huius.</note> <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a"> <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0158-01"/> </figure> <note position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">26. lib. 1.</note> <note position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">26. lib. 1.</note> <note position="right" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve">26. lib. 1.</note> <note position="right" xlink:label="note-0159-02" xlink:href="note-0159-02a" xml:space="preserve">A. Def. 10. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0159-03" xlink:href="note-0159-03a" xml:space="preserve">26. lib. 1.</note> <note position="right" xlink:label="note-0159-04" xlink:href="note-0159-04a" xml:space="preserve">Defin. 11. <lb/>lib. 1.</note> </div> <p> <s xml:id="echoid-s3304" xml:space="preserve">Nunc ſolidum, 3678, planis, oppoſitis tangentibus parallelis, <lb/>in talia fruſta diuiſum intelligatur, vt quæ in ipſis ducuntur rectæ li-<lb/>neę ipſi, 38, æquidiſtantes, in ciſdem fruſtis ſingulę integrę habean-<lb/>tur .</s> <s xml:id="echoid-s3305" xml:space="preserve">i. </s> <s xml:id="echoid-s3306" xml:space="preserve">ita vt ductarum ſic linearum, quæ ad fruſtorum ambientem ſu-<lb/>perficiem terminantur, pars quidein non ſit intra fruſta, pars verò <lb/>extra, ſed totæ intra, vel ſaltem nihil earum extra reperiatur, hanc <lb/>etenim ſectionem ſupponere ſieri poſſe nullam inuoluit repugnan-<lb/>tiam, cum hoc totum ſolidum ex duabus linearum translationibus re-<lb/>ſultans ſit interius integrum, enim vero ſi præfatum ſolidum in fru-<lb/>ſta quæcunque per dicta plana parallela icinderetur, nec in ipſis con-<lb/>tingeret, quod attentamus, denuò facta fruſta planis prædictis pa-<lb/>rallelis continuò reſecaremus, vt tandem omnis lmearum, ipſi, 38, <lb/>æquidiſtanter in d ctis fruſtis ducibilium, fractura tolleretur: </s> <s xml:id="echoid-s3307" xml:space="preserve">Eſto igi-<lb/>tur, quod hoc obtinuerimus per duo plana, 9 {10/ } {11/ }, 647, oppoſitis <lb/>plan@s tangentibus parallela, qu@bus ſolidum, 3678, intria fruſta, <lb/>3647, 6 {11/ }, &</s> <s xml:id="echoid-s3308" xml:space="preserve">, 9 {10/ } {11/ } 8, ſectum habeatur eius rationis, qualem di-<lb/>ximus, in his ergo ſingul s fruſtis ductæ quæcunque ipſi, 38, æqui-<lb/>diſtantes, & </s> <s xml:id="echoid-s3309" xml:space="preserve">ad eorum ſuperficiem terminatæ, integræ habebuntur. <lb/></s> <s xml:id="echoid-s3310" xml:space="preserve">Sit vlterius in alio ſolido, LDFG, diuifa, LG, ſimiliter ac, 38, in <lb/>punctis, E, K, per quæ tranſeant plana, DEF, MK {14/ }, oppoſitis <lb/>planis tangentibus parallela, quibus ſolidum, LDFG, in tria fru-<lb/>ſta ſcindatur, LDEF, D {14/ }, MK {14/ } G, erunt ergo etiam hęc fruſta <lb/>eius rationis, qualem cupimus .</s> <s xml:id="echoid-s3311" xml:space="preserve">i. </s> <s xml:id="echoid-s3312" xml:space="preserve">omnes ductę ipſi, LG, æquidiſtan-<lb/>tes, in ipſis fruſtis conceptæ, integræ erunt; </s> <s xml:id="echoid-s3313" xml:space="preserve">quodex eorum ſimili-<lb/>tudine facilè oſtendi poteſt, ſi enim aliqua ex. </s> <s xml:id="echoid-s3314" xml:space="preserve">gr. </s> <s xml:id="echoid-s3315" xml:space="preserve">in fruſto, LDEF, <pb o="140" file="0160" n="160" rhead="GEOMETRIÆ"/> ductarum ſic linearum fracta per ſuperficiem ambientem inueniri <lb/>poſtet, etiam illi homologa in fruſto, 3647, fracta eſſe deberet, quod <lb/>eſt abſurdum, nullam .</s> <s xml:id="echoid-s3316" xml:space="preserve">n. </s> <s xml:id="echoid-s3317" xml:space="preserve">ducibilium ipſi, 38, in ſolido, 3678, ęqui-<lb/>diſtanter linearum fractam eſſe iam ex conſtructione manifeſtum eſt, <lb/>fruſta autem, 3647, LDEF, eſſe inter ſe ſimilia, ſicut etiam, 6 <lb/>{11/ }, D {14/ }, necnon, 9 {10/ } {11/ }8, MK {14/ } G 2 ex diffinitione ſimilium ſoli-<lb/>dorum liquidò apparet.</s> <s xml:id="echoid-s3318" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div344" type="section" level="1" n="211"> <head xml:id="echoid-head226" xml:space="preserve">D. SECTIO IV.</head> <p> <s xml:id="echoid-s3319" xml:space="preserve">EX his fruſtis autem duo accipiamus, quę ſimul cum homologis <lb/>partibus ipſarum, LG, 38, detruncantur, vt ipſa, LDEF, <lb/>3647, & </s> <s xml:id="echoid-s3320" xml:space="preserve">ponamus eadem ſeorſim, deinde ex maiori ipſarum, LE, <lb/>34, vt ex, LE, abſcindatur æquali minori .</s> <s xml:id="echoid-s3321" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3322" xml:space="preserve">OE, æqualis ipſi, 34, <lb/>hoc facto intelligamus ſingulas, quæ tum in figura, LDE, tum in <lb/>figura, LFE, ipſi, LE, æquidiſtant, & </s> <s xml:id="echoid-s3323" xml:space="preserve">ſunt exiam dictis totæ in-<lb/> <anchor type="figure" xlink:label="fig-0160-01a" xlink:href="fig-0160-01"/> terius integræ ſi-<lb/>militer, & </s> <s xml:id="echoid-s3324" xml:space="preserve">ad ean-<lb/>dem partem diui-<lb/>di, ac ſecatur, LE, <lb/>in, O, & </s> <s xml:id="echoid-s3325" xml:space="preserve">per di-<lb/>ctas ſectiones ex-<lb/>tenſas lineas, OD, <lb/>OF, vlterius ſecto <lb/>ſolido, LDEF, <lb/>plano vtcunq; </s> <s xml:id="echoid-s3326" xml:space="preserve">ip-<lb/>ſi, LFE, æquidi-<lb/>ſtante, quod in eo <lb/>producat figuram, <lb/>QMY, & </s> <s xml:id="echoid-s3327" xml:space="preserve">in ſigu-<lb/>ra, LDE, rectam, QY, in figura verò, DEF, rectam, YM, & </s> <s xml:id="echoid-s3328" xml:space="preserve"><lb/>in ſuperficie, LDF, lineam, QAM, intelligantur ſingulæ in figu-<lb/>ra, QYM, parallelæ ipſi, QY, ſimiliter, & </s> <s xml:id="echoid-s3329" xml:space="preserve">ad eandem partem di-<lb/>uidi, ac ſecatur, QY, in, T, & </s> <s xml:id="echoid-s3330" xml:space="preserve">per ipſas ſectiones concipiatur ex-<lb/>tenſa linea, TIM; </s> <s xml:id="echoid-s3331" xml:space="preserve">ſie autem fiat in cæteris figuris, quę in ſolido, L <lb/>DEF, ipſi, LEF, æquidiſtant, inuentis lineis, qualis eſt ipſa, TI <lb/>M, quorum termini erunt in lineis, DTO, DMF, per eaſdem au-<lb/>tem lineas ſic ſe habentes intelligamus extenſam ſuperficiem, cuius <lb/>termini erunt lineæ, DO, OF, FD, vt habeamus ſolidum, ODE <lb/>F, figuris, ODE, OEF, DEF, & </s> <s xml:id="echoid-s3332" xml:space="preserve">ſuperficie, DOF, comprehen-<lb/>ſum. </s> <s xml:id="echoid-s3333" xml:space="preserve">Quoniam ergo linea, OF, diuidit omnes ipfi, LE, in figura, <lb/>LEF, æquidiſtantes ſimiliter ad eandem partem, ac diuiditur, LE, <pb o="141" file="0161" n="161" rhead="LIBER II."/> in, O, ideò, vt vna ad vnam, ſic omnes ad omnes. </s> <s xml:id="echoid-s3334" xml:space="preserve">i. </s> <s xml:id="echoid-s3335" xml:space="preserve">vt, LE, ad, E <lb/> <anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/> O, ſic omnes lineæ figuræ, LEF, erunt ad omnes lineas figuræ, O <lb/>EF, regula, LE, .</s> <s xml:id="echoid-s3336" xml:space="preserve">i. </s> <s xml:id="echoid-s3337" xml:space="preserve">vt, LE, ad, EO, ita figura, LEF, ad figu-<lb/>ram, OEF; </s> <s xml:id="echoid-s3338" xml:space="preserve">eodem modo oſtendemus, vt, QY, ad, YT, ſic eſſe <lb/> <anchor type="note" xlink:label="note-0161-02a" xlink:href="note-0161-02"/> figuram, QYM, ad figuram, TYM, eſt autem vt, QY, ad, YT, <lb/>ita, LE, ad, EO, ergo figura, LEF, ad, OEF, erit vt, QYM, <lb/>ad, TYM, & </s> <s xml:id="echoid-s3339" xml:space="preserve">ſic erit quælibet alia figura in ſolido, LEDF, ipſi, <lb/> <anchor type="note" xlink:label="note-0161-03a" xlink:href="note-0161-03"/> LEF, æquidiſtans, ad eius portionem in ſolido, OEDF, manen-<lb/>tem, ergo vt vna ad vnam, ſic omnes ad omnes .</s> <s xml:id="echoid-s3340" xml:space="preserve">i. </s> <s xml:id="echoid-s3341" xml:space="preserve">vt figura, LEF, <lb/>ad figuram, OEF, ſic omnia plana ſolidi, LEDF, ad omnia pla-<lb/> <anchor type="note" xlink:label="note-0161-04a" xlink:href="note-0161-04"/> na ſolidi, OEDF, regula plano, LEF, & </s> <s xml:id="echoid-s3342" xml:space="preserve">ita ſolidum, LEDF, <lb/>ad ſolidum, OEDF, eſt autem figura, LEF, ad figuram, OEF, <lb/>vt, LE, ad, EO, vel ad, 34, ergo ſolidum, LEDF, ad ſolidum, <lb/>OEDF, erit vt, LE, ad, 34, quod pariterſerua.</s> <s xml:id="echoid-s3343" xml:space="preserve"/> </p> <div xml:id="echoid-div344" type="float" level="2" n="1"> <figure xlink:label="fig-0160-01" xlink:href="fig-0160-01a"> <image file="0160-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0160-01"/> </figure> <note position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve">Coroll. 4. <lb/>huius.</note> <note position="right" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-0161-03" xlink:href="note-0161-03a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-0161-04" xlink:href="note-0161-04a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div346" type="section" level="1" n="212"> <head xml:id="echoid-head227" xml:space="preserve">E. SECTIO V.</head> <p> <s xml:id="echoid-s3344" xml:space="preserve">DVcatur nunc intra ſolidum, OEDF, planum ipſi, DEF, æ-<lb/>quidiſtans, quod in eo producat figuram, CNX, quæ ſecet <lb/>figuram, ODE, in recta, CN, &</s> <s xml:id="echoid-s3345" xml:space="preserve">, OFE. </s> <s xml:id="echoid-s3346" xml:space="preserve">in recta, NX, & </s> <s xml:id="echoid-s3347" xml:space="preserve">ſuper-<lb/>ficiem, ODF, in linea, CX, ſecet autem & </s> <s xml:id="echoid-s3348" xml:space="preserve">lineas, DO, in, C, O <lb/>E, in, N, &</s> <s xml:id="echoid-s3349" xml:space="preserve">, OF, in, X, ſimiliter in ſolido, 3467, ducatur pla-<lb/>num ipſi, 647, æquidiſtans, quod abipſa, 34, abſcindat, 35, æqua-<lb/>lem ipſi, ON, & </s> <s xml:id="echoid-s3350" xml:space="preserve">producat in eo figuram, RSP; </s> <s xml:id="echoid-s3351" xml:space="preserve">vlterius per pun-<lb/>cta, C, X, ducantur, BH, G Ω, parallele ipſi, LE, & </s> <s xml:id="echoid-s3352" xml:space="preserve">occurrentes <lb/>lineis, DL, LF, in, B, G, & </s> <s xml:id="echoid-s3353" xml:space="preserve">rectis, DE, EF, in, H, Ω, deinde <lb/>à puncto, B, ducatur, BV, parallela ipſi, DE, ſiue, CN, (nam, <lb/>DE, CN, ſunt communes ſectiones planorum æquidiſtantium, C <lb/>NX, DEF, & </s> <s xml:id="echoid-s3354" xml:space="preserve">plani, ODE, eadem ſecantis, vnde, CN, DE, ſunt <lb/>parallelæ, veluti patebit etiam, NX, æquidiſtare ipſi, EF,) & </s> <s xml:id="echoid-s3355" xml:space="preserve">iun-<lb/>gatur, VG, quia ergo, NX, eſt parallela ipſi, Ε Ω, &</s> <s xml:id="echoid-s3356" xml:space="preserve">, Χ Ω, ipſi, <lb/>NE, erit, Χ Ω, æqualis ipſi, NE, & </s> <s xml:id="echoid-s3357" xml:space="preserve">quia, LE, ad, EO, eſt vt, B <lb/>H, ad, HC, .</s> <s xml:id="echoid-s3358" xml:space="preserve">i. </s> <s xml:id="echoid-s3359" xml:space="preserve">vt, VE, ad, EN, eſt autem, G Ω, ad, Ω Χ, vt, <lb/>LE, ad, EO, quia eſt illi parallela, & </s> <s xml:id="echoid-s3360" xml:space="preserve">ſecatur à linea, OF, in, X, <lb/>ergo, G Ω, ad, Ω Χ, erit vt, VE, ad, EN, ſunt autem, Ω Χ, EN, <lb/>inter ſe æquales, ergo &</s> <s xml:id="echoid-s3361" xml:space="preserve">, G Ω, VE, erunt æquales, & </s> <s xml:id="echoid-s3362" xml:space="preserve">ſunt paralle-<lb/>læ, ergo etiam eas iungentes, VG, Ε Ω, erunt æquales, & </s> <s xml:id="echoid-s3363" xml:space="preserve">paralle-<lb/>læ. </s> <s xml:id="echoid-s3364" xml:space="preserve">Sumatur nunc intra lineam, CX, vtcunq; </s> <s xml:id="echoid-s3365" xml:space="preserve">punctum, I, per quod <lb/>ipſi, LE, parallela ducatur, AK, quæ ſuperficiei, LDF, occurrat <lb/>in, A, & </s> <s xml:id="echoid-s3366" xml:space="preserve">plano, DEF, in, K, quia ergo, AK, æquidiſtat ipſi, L <lb/> <anchor type="note" xlink:label="note-0161-05a" xlink:href="note-0161-05"/> E, poterit per, AK, planum duci æquidiſtans plano, LEF, ſit du-<lb/>ctum idem, quod prius, quod adhuc ſecet figura, LDE, in recta, <pb o="142" file="0162" n="162" rhead="GEOMETRIÆ"/> QY, DEF, in recta, YM, ſuperſiciem, LDF, in linea, QM, ſu-<lb/>perficiem, ODF, in linea, TM, & </s> <s xml:id="echoid-s3367" xml:space="preserve">figuram, CNX, in recta, ZI, <lb/>ſecet autem, QY, ipſam, BV, in puncto, ℟, & </s> <s xml:id="echoid-s3368" xml:space="preserve">iungatur, A ℟, erit <lb/>ergo, ZI, ipſi, YK, æquidiſtans, eſt autem etiam, AK, æquidiſtans <lb/>ipſi, QY, ergo, YI, erit parallelogrammum, & </s> <s xml:id="echoid-s3369" xml:space="preserve">ideò, IK, erit æ. <lb/></s> <s xml:id="echoid-s3370" xml:space="preserve"> <anchor type="note" xlink:label="note-0162-01a" xlink:href="note-0162-01"/> qualis ipſi, ZY, & </s> <s xml:id="echoid-s3371" xml:space="preserve">quia, AK, ad, KI, eſt vt, QY, ad, YT, .</s> <s xml:id="echoid-s3372" xml:space="preserve">i. </s> <s xml:id="echoid-s3373" xml:space="preserve">vt, <lb/>BH, ad, HC, .</s> <s xml:id="echoid-s3374" xml:space="preserve">i vt, ℟ Y, ad, YZ, erit, AK, ad, KI, vt, ℟ Y, ad, <lb/>YZ, ſunt verò, IK, ZY, æquales, ergo &</s> <s xml:id="echoid-s3375" xml:space="preserve">, AK, ℟ Y, erunt æqua-<lb/>les, & </s> <s xml:id="echoid-s3376" xml:space="preserve">ſunt parallelæ, quia ambo ſunt parallelæ eidem, LE, ergo <lb/>eas iungentes, quæ ſunt, ℟ A, YK, erunt æquales, & </s> <s xml:id="echoid-s3377" xml:space="preserve">parallelę, eſt <lb/> <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> autem, YK, pa-<lb/>rallela ipſi, Ε Ω, <lb/>&</s> <s xml:id="echoid-s3378" xml:space="preserve">, Ε Ω, ipſi, VG, <lb/>ergo, ℟ A, erit pa-<lb/>rallela ipſi, VG. <lb/></s> <s xml:id="echoid-s3379" xml:space="preserve">Similiterautẽ pro-<lb/>cedemus in reli-<lb/>quis, quę per pun-<lb/>cta lineę, CX, ipſi, <lb/>LE, ducuntur æ-<lb/>quidiſtantes, do-<lb/>nec occurrant ſu-<lb/>perſiciei, LDF, <lb/>& </s> <s xml:id="echoid-s3380" xml:space="preserve">plano, DEF, <lb/>harum autem patet nihil extra ſuperficiem, LDF, manere, ex iam <lb/>dictis, ſint ergo omnium earum termini ex vna parte in linea, BAG, <lb/>ex alia in linea, Η Κ Ω, veluti ergo oſtenſum eſt, A ℟, eſſe paralle-<lb/>lam ipſi, GV, ſic oſtendemus reliquas, quę iungunt puncta, quibus <lb/>iam ductæ occurrunt lineæ, BG, cum punctis, in quibus plana per <lb/>dictas lineas ducta, ipſi, LEF, æquidiſtantia, ſecant ipſam, BV, eſſe <lb/>ipſi, VG, paralſelas ergo omnes erunt in eodem plano, in eo ſcili-<lb/>cet quod tranſit per, BV, VG, omnes .</s> <s xml:id="echoid-s3381" xml:space="preserve">n. </s> <s xml:id="echoid-s3382" xml:space="preserve">dictæ parallelæ tranſeunt <lb/>per puncta rectæ lineæ, BV, ſunt igiturd cta occurſuum puncta, & </s> <s xml:id="echoid-s3383" xml:space="preserve"><lb/>in ſuperficie, LDF, & </s> <s xml:id="echoid-s3384" xml:space="preserve">in plano, BVG, erunt ergo in eorum com-<lb/>muni ſectione, linea ergo, BAG, eſt communis ſectio plani per, B <lb/>V, VG, tranſeuntis, & </s> <s xml:id="echoid-s3385" xml:space="preserve">ſuperficiei, LDF; </s> <s xml:id="echoid-s3386" xml:space="preserve">habemus ergo ſolidum, <lb/>Β Ω, in cuius ambiente ſuperficie ſunt duæ figuræ planæ inuicem pa-<lb/>rallelæ, BVG, Η Ε Ω, in quarum circuitu ſumptis vtcunque duo-<lb/>bus punctis, V, E, & </s> <s xml:id="echoid-s3387" xml:space="preserve">iuncta, VE, cæteræ iungentes quælibet aliæ <lb/> <anchor type="note" xlink:label="note-0162-02a" xlink:href="note-0162-02"/> duo puncta earundem circuitus eidem ſemper, VE, parallelę reper-<lb/>tæ ſunt æquales, ergo, Β Ω erit cylindricus, cu@us oppoſitæ baſes <lb/>ipſæ, BVG, Η Ε Ω, hoc autem ſecatur plano eildem oppoſit@s ba- <pb o="143" file="0163" n="163" rhead="LIBER II."/> ſibus æquidiſtante, eo nempè, quod producit figuram, CNX, er-<lb/> <anchor type="note" xlink:label="note-0163-01a" xlink:href="note-0163-01"/> go, CNX, erit æqualis ipſi, BVG, quod cum alijs adhuc ſerua.</s> <s xml:id="echoid-s3388" xml:space="preserve"/> </p> <div xml:id="echoid-div346" type="float" level="2" n="1"> <note position="right" xlink:label="note-0161-05" xlink:href="note-0161-05a" xml:space="preserve">Exis: @@ <lb/>Elem.</note> <note position="left" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve">Percõſtru <lb/>ctionem.</note> <figure xlink:label="fig-0162-01" xlink:href="fig-0162-01a"> <image file="0162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0162-01"/> </figure> <note position="left" xlink:label="note-0162-02" xlink:href="note-0162-02a" xml:space="preserve">Def. Cy-<lb/>lindrici <lb/>confor-<lb/>miter.</note> <note position="right" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">Corol. 12. <lb/>hb. 1.</note> </div> </div> <div xml:id="echoid-div348" type="section" level="1" n="213"> <head xml:id="echoid-head228" xml:space="preserve">F. SECTIO VI.</head> <p> <s xml:id="echoid-s3389" xml:space="preserve">QVia verò, LE, ad, EO, eſt vt, BH, ad, HC, .</s> <s xml:id="echoid-s3390" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3391" xml:space="preserve">vt, VE, <lb/>ad, EN, permutando, & </s> <s xml:id="echoid-s3392" xml:space="preserve">diuidendo, LV, ad, VE, erit vt, <lb/>ON, ad, NE, .</s> <s xml:id="echoid-s3393" xml:space="preserve">i. </s> <s xml:id="echoid-s3394" xml:space="preserve">vt, 35, ad, 54, ergo, LE, 34, ſunt ſimi-<lb/>liter ad eandem partem diuiſæ a figuris, BVG, RSP, ergo ſunt ip-<lb/>ſæ figuræ inter ſe ſimiles, quarum latera homologa ipſæ, VG, SP, <lb/> <anchor type="note" xlink:label="note-0163-02a" xlink:href="note-0163-02"/> lineæ homologę figurarum ſimilium, LFE, 374, quarum inciden-<lb/>tes ſuntipſę, LE, 34, vnde eſt, EF, ad, 47, vt, LE, ad, 34, .</s> <s xml:id="echoid-s3395" xml:space="preserve">ſ. <lb/></s> <s xml:id="echoid-s3396" xml:space="preserve">vt, VG, ad, SP, ſunt verò figuræ, DEF, 647, quia ſimiles, in <lb/>dupla ratione ipſarum, EF, 47, & </s> <s xml:id="echoid-s3397" xml:space="preserve">ipſæ, BVG, RSP, in dupla <lb/> <anchor type="note" xlink:label="note-0163-03a" xlink:href="note-0163-03"/> ratione ipſarum, VG, SP, ergo vt figura, DEF, ad figuram, 64 <lb/>7, ita erit figura, BVG, vel, CNX, eidem æqualis ad figuram, R <lb/>SP, Quoniam verò ſolida, LEDF, 3647, ſunt ſimilia, vt facilè <lb/>oſtendi poreſt, & </s> <s xml:id="echoid-s3398" xml:space="preserve">eorum figuræ incidentes, & </s> <s xml:id="echoid-s3399" xml:space="preserve">oppoſitorum plano-<lb/>rum tangentium (quorum ex vna parte duo ſunt ipſa, 647, DEF,) <lb/>ſunt figuræ, LEF, 347 quarum lineæ incidentes, LE, 34, ideò <lb/>plana ipſis, DEF, 647, æquidiſtantia, quæ ſimiliter ad eandem <lb/>partem diu dunt incidentes, LE, 34, diuidunt etiam altitudines di-<lb/>ctorum ſolidorum reſpectu dictorum tangentium ſumptas ſimiliter <lb/> <anchor type="note" xlink:label="note-0163-04a" xlink:href="note-0163-04"/> ad eandem partem (hocdico quotieſcunque, non contingat, LE, <lb/>34, eſſe perbendiculares ipſis, DEF, 647, tunc enim ſiunt eædem <lb/>incidentes altitudines dictorum ſolidorum) cum igitur, vt, LE, ad, <lb/>34, .</s> <s xml:id="echoid-s3400" xml:space="preserve">i. </s> <s xml:id="echoid-s3401" xml:space="preserve">ad, EO, ita ſit altitudo ſolidi, LEDF, tum adabſciſſam al-<lb/>titudinem per planum tangens in, O, ipſi, DEF, æquidiſtans .</s> <s xml:id="echoid-s3402" xml:space="preserve">i. </s> <s xml:id="echoid-s3403" xml:space="preserve">ad <lb/>altitudinem ſolidi, OEDF, tum ad altitudinem ſolidi, 3467, ideò <lb/>ſolida, OEDF, 347, erunt in eadem altitudine ſumpta reſpectu <lb/>baſium, DEF, 647, & </s> <s xml:id="echoid-s3404" xml:space="preserve">plana ipſis baſibus æquidiſtantia partes æ-<lb/>quales ab ipſis, OE, 34, abſcindentia, et am ab eorum altitudini-<lb/>bus abſcindent partes æquales, oſtendimus autem figuras, quę ab ip-<lb/>ſis, OE, 34, abſcindunt partes æquales, eſſe proportionales, ergo <lb/>in ſolidis, OEDF, 3467, in eadem altitudine exiſtentibus ſumpta <lb/>reſpectu baſium, DEF, 647, figuræ, quę ab eiſdem altitudinibus <lb/>vtcunque abſcindunt partes æquales, ſunt ſemper, vt ipſæ baſes, er-<lb/>go vt vna ad vnam, ſic omnes ad omnes, & </s> <s xml:id="echoid-s3405" xml:space="preserve">ſic ſolida ad ſolida .</s> <s xml:id="echoid-s3406" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3407" xml:space="preserve">vt <lb/> <anchor type="note" xlink:label="note-0163-05a" xlink:href="note-0163-05"/> baſis, DEF, ad baſun, 647, ita erit ſolidum, OEDF, ad ſolidum, <lb/> <anchor type="note" xlink:label="note-0163-06a" xlink:href="note-0163-06"/> 3467, eſt autem, DEF, ad, 647, in ratione dupla eius, quam <lb/>habet, EF, ad, 47, .</s> <s xml:id="echoid-s3408" xml:space="preserve">i. </s> <s xml:id="echoid-s3409" xml:space="preserve">in ratione compoſita ex duabus rationibus <lb/> <anchor type="note" xlink:label="note-0163-07a" xlink:href="note-0163-07"/> ipſius, EF, ad, 47, velipſius, LE, ad, 34, ergo ſolidum, ODE <pb o="144" file="0164" n="164" rhead="GEOMETRIÆ"/> F, ad ſolidum, 3467, habebit rationem compoſitam ex duabus ra-<lb/>tionibus ipſius, LE, ad, 34, quod etiam ſerua.</s> <s xml:id="echoid-s3410" xml:space="preserve"/> </p> <div xml:id="echoid-div348" type="float" level="2" n="1"> <note position="right" xlink:label="note-0163-02" xlink:href="note-0163-02a" xml:space="preserve">Ex diffin. <lb/>Emilium <lb/>ſolid.</note> <note position="right" xlink:label="note-0163-03" xlink:href="note-0163-03a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0163-04" xlink:href="note-0163-04a" xml:space="preserve">17. Vnd. <lb/>Elem.</note> <note position="right" xlink:label="note-0163-05" xlink:href="note-0163-05a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-0163-06" xlink:href="note-0163-06a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0163-07" xlink:href="note-0163-07a" xml:space="preserve">Deſin. 12. <lb/>lib. 1.</note> </div> </div> <div xml:id="echoid-div350" type="section" level="1" n="214"> <head xml:id="echoid-head229" xml:space="preserve">G. SECTIO VII.</head> <p> <s xml:id="echoid-s3411" xml:space="preserve">SI igitur inter ſolida, LEDF, 3467, medium ſumamus ſoll-<lb/> <anchor type="note" xlink:label="note-0164-01a" xlink:href="note-0164-01"/> dum, OEDF, habebit ſolidum, LEDF, ad ſolidum, 3467, <lb/>rationem compoſitam ex ratione ſolidi, LEDF, ad ſolidum, OE <lb/>DF, .</s> <s xml:id="echoid-s3412" xml:space="preserve">i. </s> <s xml:id="echoid-s3413" xml:space="preserve">ex ratione ipſius, LE, ad, 34, & </s> <s xml:id="echoid-s3414" xml:space="preserve">ex ratione ſolidi, OED <lb/>F, ad ſolidum, 3467, .</s> <s xml:id="echoid-s3415" xml:space="preserve">i. </s> <s xml:id="echoid-s3416" xml:space="preserve">compoſitam ex duabus rationibus ipſius, <lb/>LE, ad, 34, igitur ſolidum, LEDF, ad ſolidum, 3467, habe-<lb/>bit rationem compoſitam ex tribus rationibus ipſius, LE, ad, 34, <lb/>.</s> <s xml:id="echoid-s3417" xml:space="preserve">i. </s> <s xml:id="echoid-s3418" xml:space="preserve">triplam rationem habebit eius, quam habet, LE, ad, 34, quia <lb/> <anchor type="note" xlink:label="note-0164-02a" xlink:href="note-0164-02"/> verò, LE, 34, ſunt homologæ partes integrarum incidentium, L <lb/>G, 38, quæ ſunt in prima huius Propoſ. </s> <s xml:id="echoid-s3419" xml:space="preserve">figura, ideò his fruſtis ibi-<lb/>dem conſpectis iam oſtenſum erit fruſtum, LEDF, ad fruſtum, 34 <lb/>67, triplam rationem habere eius, quam habet, LE, ad, 34, ideſt, <lb/>LG, ad, 38.</s> <s xml:id="echoid-s3420" xml:space="preserve"/> </p> <div xml:id="echoid-div350" type="float" level="2" n="1"> <note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve">Deſin. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0164-02" xlink:href="note-0164-02a" xml:space="preserve">Def. Vnd. <lb/>6. Elem.</note> </div> </div> <div xml:id="echoid-div352" type="section" level="1" n="215"> <head xml:id="echoid-head230" xml:space="preserve">H. SECTIO VIII. ET VLTIMA.</head> <p> <s xml:id="echoid-s3421" xml:space="preserve">EOdem modo ſumptis alijs duobus fruſtis, D {14/ }, 6 {11/ }, oſtendemus <lb/>eadem habere triplam rationem duarum, LG, 38, & </s> <s xml:id="echoid-s3422" xml:space="preserve">ſimiliter <lb/>reliqua fruſta pariter triplam rationem habere duarum, LG, 38, & </s> <s xml:id="echoid-s3423" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0164-03a" xlink:href="note-0164-03"/> vt vnum ad vnum, ſic omnia ad omnia .</s> <s xml:id="echoid-s3424" xml:space="preserve">i. </s> <s xml:id="echoid-s3425" xml:space="preserve">vt fruſtum, LEDF, ad <lb/>fruſtum, 3467, ita eſſe omnia fruſta ſolidi, LG, ad omnia fruſta <lb/>ſolidi, 38, ſed fruſtum, LEDF, ad fruſtum, 3467, triplam ratio-<lb/>nem habere oſtenſum eſt eius, quam habet, LG, ad, 38, ergo ſo-<lb/>lidum, LG, ad ſolidum, 38, triplam rationem habebit eius, quam <lb/> <anchor type="note" xlink:label="note-0164-04a" xlink:href="note-0164-04"/> habet, LG, ad, 38, eſt autem ſolidum, LG, æquale ſolido, AP, <lb/>&</s> <s xml:id="echoid-s3426" xml:space="preserve">, 38, ipſi, V &</s> <s xml:id="echoid-s3427" xml:space="preserve">, ergo ſolidum, AP, ad, V &</s> <s xml:id="echoid-s3428" xml:space="preserve">, triplam rationem <lb/>habebit eius, quam, LG, ad, 38, quia verò, LG, 38, ſunt inci-<lb/>dentes ſimilium planarum figurarum, H {00/ }, Σ 2, & </s> <s xml:id="echoid-s3429" xml:space="preserve">oppoſitarum <lb/>tangentium, HL, {00/ } G, Σ 3, 28, ideò, vt, LG, ad, 38, ita erunt <lb/>lineæ homologæ figurarum, H {00/ }, Σ 2, ſumptæ regulas, HL, Σ 3, <lb/>ex. </s> <s xml:id="echoid-s3430" xml:space="preserve">gr. </s> <s xml:id="echoid-s3431" xml:space="preserve">ita, OX, ad, ΦΛ, iſtæ verò ſunt incidentes ſimilium figura-<lb/> <anchor type="note" xlink:label="note-0164-05a" xlink:href="note-0164-05"/> rum, BC, ΠΩ, & </s> <s xml:id="echoid-s3432" xml:space="preserve">oppoſitarum tangentium, BO, CX, ΠΦ, ΩLamp;</s> <s xml:id="echoid-s3433" xml:space="preserve">, <lb/>ideò, vt ipſæ, OX, ΦΛ, ita erunt quælibet homologæ figurarum, <lb/>BC, ΠΩ, ſumptę regulis ipſis, CX, ΩΛ, at ſolidum, AP, ad, V &</s> <s xml:id="echoid-s3434" xml:space="preserve">, <lb/> <anchor type="note" xlink:label="note-0164-06a" xlink:href="note-0164-06"/> triplam rationem habet eius, quam, LG, ad, 38, ergo etiam tri-<lb/>plam rationem habebit eius, quam, OX, ad, ΦΛ, & </s> <s xml:id="echoid-s3435" xml:space="preserve">conſequenter <lb/>etiam triplam rationem eius, quam habebit quælibet in figura, BC, <pb o="145" file="0165" n="165" rhead="LIBER II."/> ipſi, CX, æquidiſtans ad ſibi homologam in figura, ΠΩ, ipſi, ΩΛ, <lb/>æquidiſtantem, vel quælibet in quacunque figurarum ipſi, BC, in <lb/>ſolido, AP, æquidiſtantium, ad ſibi homologam in ſolido, V & </s> <s xml:id="echoid-s3436" xml:space="preserve"><lb/>lgitur ſimilia ſolida ſunt in tripla ratione linearum, vel laterum ho-<lb/>mologorum, quæ ſunt in eorundem homologis figuris, quod nobis <lb/>oſtendendum erat.</s> <s xml:id="echoid-s3437" xml:space="preserve"/> </p> <div xml:id="echoid-div352" type="float" level="2" n="1"> <note position="left" xlink:label="note-0164-03" xlink:href="note-0164-03a" xml:space="preserve">12. Quin. <lb/>Elem.</note> <note position="left" xlink:label="note-0164-04" xlink:href="note-0164-04a" xml:space="preserve">B. Huius <lb/>Propoſ.</note> <note position="left" xlink:label="note-0164-05" xlink:href="note-0164-05a" xml:space="preserve">Ex diffin. <lb/>linearum <lb/>incident.</note> <note position="left" xlink:label="note-0164-06" xlink:href="note-0164-06a" xml:space="preserve">Vt patet <lb/>in A. hu-<lb/>ius.</note> </div> </div> <div xml:id="echoid-div354" type="section" level="1" n="216"> <head xml:id="echoid-head231" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s3438" xml:space="preserve">_E_T quia iam dicta ſimilia ſolida oſtenſa ſunt eſſe in tripla ratione li-<lb/>nearum bomologarum, quæ ſunt in homologis figuris, æquidiſtan-<lb/>tibus oppoſitis planis tangentibus vtcunque ſumptis, ideò clarum eſt ea-<lb/>dem ſimilia ſolida eſſe in tripla ratione quarumuis homologarum in ipſis <lb/>ſolidis deſoriptibilium, & </s> <s xml:id="echoid-s3439" xml:space="preserve">duas quaſuis homologas ſumptas iuxta quæ-<lb/>dam oppoſita tangentia plana, eſſe vt duas quaſuis homologas ſumptas <lb/>iuxta alia oppoſita tangentia plana.</s> <s xml:id="echoid-s3440" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div355" type="section" level="1" n="217"> <head xml:id="echoid-head232" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s3441" xml:space="preserve">_V_Niuersè inſuper habetur, ſi fuerint quatuor rectæ lineæ deinceps <lb/>proportionales, vt prima ad quartam, ita eſſe ſolidum deſcriptum <lb/>à prima ad ſolidum illi ſimile deſ criptum à ſecunda, & </s> <s xml:id="echoid-s3442" xml:space="preserve">huius conuer-<lb/>ſim; </s> <s xml:id="echoid-s3443" xml:space="preserve">dummodò deſcribentes ſint lineæ, vel latera homologa ſimilium fi-<lb/>gurarum, quæ in ipſis homologæ vocantur.</s> <s xml:id="echoid-s3444" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div356" type="section" level="1" n="218"> <head xml:id="echoid-head233" xml:space="preserve">THE OREMA XVIII. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s3445" xml:space="preserve">SI quaturor rectę lineę proportionales fuerint, ſolidum de-<lb/>ſeriptum à prima ad ſolidum ſibi ſimile deſcriptum à ſe-<lb/>cunda, erit, vt ſolidum deſcriptum à tertia ad ſibi ſimile de-<lb/>ſcriptum à quarta. </s> <s xml:id="echoid-s3446" xml:space="preserve">Et ſi fuerint quatuor ſolida proportiona-<lb/>lia, quorum quæ ſunt eiuſdem proportionis termini ſint ſimi-<lb/>lia, eadem deſcribentia erunt proportionalia; </s> <s xml:id="echoid-s3447" xml:space="preserve">dummodò ta-<lb/>men ſemper deſcribentia ſint vel lineæ, vel latera homologa <lb/>figurarum, quæ in ipſis homologæ vocantur.</s> <s xml:id="echoid-s3448" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3449" xml:space="preserve">Sint ergo quatuor rectę lineę proportionales, AB, CD, FG, H <lb/>M, & </s> <s xml:id="echoid-s3450" xml:space="preserve">ſint ab ipſis, AB, CD, deſcripta ſimilia ſolida, AXB, CV <lb/>D, & </s> <s xml:id="echoid-s3451" xml:space="preserve">ab, FG, HM, ſimilia ſolida, OFPG, NHQM, ita vt <lb/>duæ, AB, CD, ſint homologę figurarum, AEBY, DKC ℟, &</s> <s xml:id="echoid-s3452" xml:space="preserve">, <pb o="146" file="0166" n="166" rhead="GEOMETRIÆ"/> FG, HM, homologæ figurarum, FGP, HMQ, quę figurę vo-<lb/>cantur in ipſis ſolidis homologæ. </s> <s xml:id="echoid-s3453" xml:space="preserve">Dico hæc ſolida eſſe proportiona-<lb/>lia; </s> <s xml:id="echoid-s3454" xml:space="preserve">ſit duarum, AB, CD, tertia proportionalis, R, quarta, S, & </s> <s xml:id="echoid-s3455" xml:space="preserve"><lb/>duarum, FG, HM, tertia, I, quarta, T, eſt igitur ſolidum, AXB, <lb/>ad, CVD, vt, AB, ad, S, .</s> <s xml:id="echoid-s3456" xml:space="preserve">i. </s> <s xml:id="echoid-s3457" xml:space="preserve">vt, FG, ad, T, (quia vt, AB, ad, <lb/> <anchor type="note" xlink:label="note-0166-01a" xlink:href="note-0166-01"/> CD, ita eſt, FG, ad, HM,) .</s> <s xml:id="echoid-s3458" xml:space="preserve">i. </s> <s xml:id="echoid-s3459" xml:space="preserve">vt ſolidum, FOGP, ad, HNM <lb/>Q, quod eſt propoſitum.</s> <s xml:id="echoid-s3460" xml:space="preserve"/> </p> <div xml:id="echoid-div356" type="float" level="2" n="1"> <note position="left" xlink:label="note-0166-01" xlink:href="note-0166-01a" xml:space="preserve">Ex Corol. <lb/>2. antec.</note> </div> <p> <s xml:id="echoid-s3461" xml:space="preserve">Sit nunc ſolidum, AXB, ad ſibi ſimile, CVD, vt, FOGP, ad <lb/> <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> ſibi ſimile, HNMQ, & </s> <s xml:id="echoid-s3462" xml:space="preserve">ſint ea-<lb/>dem deſcribentes, AB, CD, li-<lb/>neæ, vel latera homologa figura-<lb/>rum homologarum, AEBY, C <lb/>KD ℟, &</s> <s xml:id="echoid-s3463" xml:space="preserve">, FG, HM, duo po <lb/>ſtrema deſcribentes ſint lineæ, vel <lb/>latera homologa figurarum ho-<lb/>mologarum, FGP, HMQ. </s> <s xml:id="echoid-s3464" xml:space="preserve">Di-<lb/>co has eſſe proportionales; </s> <s xml:id="echoid-s3465" xml:space="preserve">ſint <lb/>adhuc duarum, AB, CD, tertia <lb/>proportionalis, R, quarta, S, & </s> <s xml:id="echoid-s3466" xml:space="preserve"><lb/>duarum, FG, HM, tertia, I, <lb/>quarta, T, quia ergo ſolida, AX <lb/>B, CVD, ſunt ſimilia erit, AXB, ad, CVD, vt, AB, ad, S, &</s> <s xml:id="echoid-s3467" xml:space="preserve">, <lb/> <anchor type="note" xlink:label="note-0166-02a" xlink:href="note-0166-02"/> FOGP, ad, HNMP, vt, FG, ad, T, ſunt autem hæc quatuor <lb/>ſolida proportionalia, ergo &</s> <s xml:id="echoid-s3468" xml:space="preserve">, AB, ad, S, erit vt, FG, ad, T, er-<lb/>go, AB, ad, CD, erit vt, FG, ad, HM, quod oſtendendum erat.</s> <s xml:id="echoid-s3469" xml:space="preserve"/> </p> <div xml:id="echoid-div357" type="float" level="2" n="2"> <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a"> <image file="0166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0166-01"/> </figure> <note position="left" xlink:label="note-0166-02" xlink:href="note-0166-02a" xml:space="preserve">Ex Corol. <lb/>2. antec.</note> </div> </div> <div xml:id="echoid-div359" type="section" level="1" n="219"> <head xml:id="echoid-head234" xml:space="preserve">THE OREMA XIX. PROPOS. XIX.</head> <p> <s xml:id="echoid-s3470" xml:space="preserve">SI in parallelogrammo diameter ducta fuerit, parallelo-<lb/>grammum duplum eſt cuiuſuis triangulorum per ipſam <lb/>diametrum conſtitutorum.</s> <s xml:id="echoid-s3471" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3472" xml:space="preserve">Sit parallelogrammum vtcunque, AD, in <lb/> <anchor type="figure" xlink:label="fig-0166-02a" xlink:href="fig-0166-02"/> quo ducta diameter, FC, ipſum diuidat in tri-<lb/>angula, FAC, CDF. </s> <s xml:id="echoid-s3473" xml:space="preserve">Dico parallelogram-<lb/>mum, AD, duplum eſſe cuiuſuis triangulo-<lb/>rum, FAC, CDF; </s> <s xml:id="echoid-s3474" xml:space="preserve">abſcindanturab, FD, C <lb/>A, verſus puncta, F, C, partes æquales, FE, <lb/>CB, & </s> <s xml:id="echoid-s3475" xml:space="preserve">per puncta, B, E, parallelæ ipſi baſi, <lb/>CD, ducantur, EH, BM, incidentes diame-<lb/>tro, FC, in punctis, H, M; </s> <s xml:id="echoid-s3476" xml:space="preserve">quoniam ergo in triangulis, FHE, C <pb o="147" file="0167" n="167" rhead="LIBER II."/> BM, angulus, HFE, æqualis eſt angulo illi coalterno, BCM, &</s> <s xml:id="echoid-s3477" xml:space="preserve">, <lb/>HEF, ipſi, FDC, qui eſt æqualis angulo illi oppoſito, FAC, qui <lb/>tandem æquatur angulo, MBC; </s> <s xml:id="echoid-s3478" xml:space="preserve">interior exteriori, ideò anguius, <lb/>FEH, æquatur angulo, MBC, ſunt igitur in triangulis, FEH, M <lb/>BC, duo anguli duobus angulis æquales, & </s> <s xml:id="echoid-s3479" xml:space="preserve">latera illis adiacentia <lb/>ſunt æqualia, nempè, FE, ipſi, BC, ergo reliqua latera erunt æ-<lb/> <anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> qualia, .</s> <s xml:id="echoid-s3480" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3481" xml:space="preserve">HE, ipſi, BM, eodem modo oſtendemus de cæteris pa-<lb/>rallelis ipſi, CD, eas nempè, quæ verſus puncta, F, C, abſcindunt <lb/>à lateribus, FD, CA, partes æquales, eſſe pariter inter ſe æquales, <lb/>veiuti ſunt extremæ, AF, CD, æquales, ergo omnes lineæ trian-<lb/>guli, CAF, æquabuntur omnibus lineis trianguli, FDC, ſumptis <lb/> <anchor type="note" xlink:label="note-0167-02a" xlink:href="note-0167-02"/> in vtriſq; </s> <s xml:id="echoid-s3482" xml:space="preserve">omnibus lineis regula, CD, ergo triangulus, ACF, erit <lb/>æqualis triangulo, FDC, ergo duo trianguli, ACF, FDC, ſcili-<lb/>cet parallelogrammum, AD, erit duplum cuiuſuis triangulorum, A <lb/>CF, FCD, quod oſtendere opus erat.</s> <s xml:id="echoid-s3483" xml:space="preserve"/> </p> <div xml:id="echoid-div359" type="float" level="2" n="1"> <figure xlink:label="fig-0166-02" xlink:href="fig-0166-02a"> <image file="0166-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0166-02"/> </figure> <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">26. Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div361" type="section" level="1" n="220"> <head xml:id="echoid-head235" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s3484" xml:space="preserve">_H_Inc patet, quæcunq; </s> <s xml:id="echoid-s3485" xml:space="preserve">de parallelogrammis in Prop. </s> <s xml:id="echoid-s3486" xml:space="preserve">5.</s> <s xml:id="echoid-s3487" xml:space="preserve">6.</s> <s xml:id="echoid-s3488" xml:space="preserve">7. </s> <s xml:id="echoid-s3489" xml:space="preserve">& </s> <s xml:id="echoid-s3490" xml:space="preserve">8. <lb/></s> <s xml:id="echoid-s3491" xml:space="preserve">huius Librioſtenſaſunt, eadem de triangulis vt verarecipi poſſe, <lb/>ſi in triangulis conditiones ibi oppoſitæ repertæ fuerint, nam in vnoquo-<lb/>que expoſitorum triangulorum ſumptis duobus quibuſuis lateribus ſieri <lb/>poteſt ſub illis in eodem angulo parallelogr ammum, cuius triangulum <lb/>erit dimidium. </s> <s xml:id="echoid-s3492" xml:space="preserve">Triangula ergo, quæ in eadem ſunt altitudine inter ſe <lb/>ſunt, vt baſes: </s> <s xml:id="echoid-s3493" xml:space="preserve">Et quæ in eadem baſi mierſe ſunt, vt altitudines, vel vt <lb/>latera æqualiter baſibus inclinata; </s> <s xml:id="echoid-s3494" xml:space="preserve">Item babent rationem compoſitam ex <lb/>ratione baſium, & </s> <s xml:id="echoid-s3495" xml:space="preserve">altitudinum, ſine laterum æqualiter baſibus inclina-<lb/>torum, cum ſunt æquiangulæ: </s> <s xml:id="echoid-s3496" xml:space="preserve">Item triangula, quorum baſes altitudi-<lb/>nibus, vel lateribus æqualiter baſibus inclinatis, reciprocantur ſunt æ-<lb/>qualia; </s> <s xml:id="echoid-s3497" xml:space="preserve">& </s> <s xml:id="echoid-s3498" xml:space="preserve">quæ ſunt æqualia baſes habent altitudinibus, vel lateribus <lb/>æqualiter baſibus inclmatis, reciprocas: </s> <s xml:id="echoid-s3499" xml:space="preserve">Et tandem habetur ſimilia trian-<lb/> <anchor type="note" xlink:label="note-0167-03a" xlink:href="note-0167-03"/> gula eſſe in dupla ratione laterum homologorum, quæ omnia ex præſenti <lb/>Propoſ. </s> <s xml:id="echoid-s3500" xml:space="preserve">pendent.</s> <s xml:id="echoid-s3501" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="1"> <note position="right" xlink:label="note-0167-03" xlink:href="note-0167-03a" xml:space="preserve">_Iux. diff._ <lb/>_1. Sexti_ <lb/>_Elem._</note> </div> </div> <div xml:id="echoid-div363" type="section" level="1" n="221"> <head xml:id="echoid-head236" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s3502" xml:space="preserve">_C_Olligitur in ſuper, ſi ſupponamur, CD, eſſe æqualem ipſi, DF, <lb/>quamlibet ductam in triangulo, FCD, parallelam ipſi, CD, æqua-<lb/>lem eſſe ei, quam ipſa abſcindit ab, FD, verſus, F, nempè ipſi abſciſſæ, <lb/>FE, & </s> <s xml:id="echoid-s3503" xml:space="preserve">producta, EH, verjus, AC, cui incidat in, N, ipſam, HN, <lb/>æquari reſiduæ abſciſſæ, FE, .</s> <s xml:id="echoid-s3504" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3505" xml:space="preserve">ipſi, ED, &</s> <s xml:id="echoid-s3506" xml:space="preserve">, NE, integram æquari <pb o="148" file="0168" n="168" rhead="GEOMETRIÆ"/> ipſi, FD, quæ eſi vna maximarum abſciſſarum ipſius, FD, vnde hac via <lb/>colligemus omnes lineas trianguli, FCD, regula, CD, dum latus, FD, <lb/>æquatur ipſi, DC, eſſe æquales omnibus abſciſſis ipſius, FD; </s> <s xml:id="echoid-s3507" xml:space="preserve">& </s> <s xml:id="echoid-s3508" xml:space="preserve">omnes <lb/> <anchor type="note" xlink:label="note-0168-01a" xlink:href="note-0168-01"/> lineas trianguli, AFC, eſſe æquales reſiduis omnium abſciſſarum, FD, <lb/> <anchor type="note" xlink:label="note-0168-02a" xlink:href="note-0168-02"/> & </s> <s xml:id="echoid-s3509" xml:space="preserve">omnes lineas par allelogrammi. </s> <s xml:id="echoid-s3510" xml:space="preserve">AD, æquari maximis abſciſſarum, <lb/>FD, quæ dicuntur eiuſdem obliqui tranſitus, ſi angulus, CDF, non ſit <lb/> <anchor type="note" xlink:label="note-0168-03a" xlink:href="note-0168-03"/> rectus, & </s> <s xml:id="echoid-s3511" xml:space="preserve">rectitranſitus, ſi ſit rectus; </s> <s xml:id="echoid-s3512" xml:space="preserve">vnde ſicuti oſtendimus, paralle, <lb/>logrammum, AD, duplum eſſe trianguli, FCD, vel, ACF, & </s> <s xml:id="echoid-s3513" xml:space="preserve">ſubin-<lb/>de etiam omnes lineas, AD, regula CD duplas eſſe omnium linearum <lb/>trianguli, FCD, vel, ACF, ſic etiam vt demonſtratum recipi potest <lb/> <anchor type="note" xlink:label="note-0168-04a" xlink:href="note-0168-04"/> propoſitæ linearectæ, vt ipſius, FD, vtcunque, maximas abſciſſarum, <lb/>duplas eſſe omnium abſciſſarum eiuſdem, vel reſiduarum omnium abſciſ-<lb/>ſarum, vnde & </s> <s xml:id="echoid-s3514" xml:space="preserve">omnes abſctſſas patebit æquari reſiduis omnium abſciſ-<lb/>ſarum eiuſdem lineæ, ijs vel recti, vel eiuſdem obliqui tranſitus ſumptis, <lb/>quæ ad ſequentium intelligentiam diligenter ſunt adnotanda.</s> <s xml:id="echoid-s3515" xml:space="preserve"/> </p> <div xml:id="echoid-div363" type="float" level="2" n="1"> <note position="left" xlink:label="note-0168-01" xlink:href="note-0168-01a" xml:space="preserve">_Defin. 4._ <lb/>_huius._</note> <note position="left" xlink:label="note-0168-02" xlink:href="note-0168-02a" xml:space="preserve">_Defin. 5._ <lb/>_huius._</note> <note position="left" xlink:label="note-0168-03" xlink:href="note-0168-03a" xml:space="preserve">_Defin. 6._ <lb/>_huius._</note> <note position="left" xlink:label="note-0168-04" xlink:href="note-0168-04a" xml:space="preserve">_3. huius._</note> </div> </div> <div xml:id="echoid-div365" type="section" level="1" n="222"> <head xml:id="echoid-head237" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s3516" xml:space="preserve">SIt magnitudo, A, ad quotcunque magnitudines, E, O, ſingil-<lb/>latim ad vnamquamque, vt magnitudo, V, ad tot alias, P, S, <lb/> <anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/> ſingillatim ad vnamquamq; </s> <s xml:id="echoid-s3517" xml:space="preserve">nempè ſit, A, ad, E, <lb/>vt, V, ad, P; </s> <s xml:id="echoid-s3518" xml:space="preserve">A, ad, O, vt, V, ad, S. </s> <s xml:id="echoid-s3519" xml:space="preserve">Dico, A, <lb/>ad, E, O, ſimul eſſe, vt, V, ad, P, S, ſimul iun-<lb/>ctas. </s> <s xml:id="echoid-s3520" xml:space="preserve">Etenim conuertendo erit prima, E, ad ſecun-<lb/>dam, A, vttertia, P, ad quartam, V, ſed etiam <lb/>conuertendo quinta, O, eſt ad ſecundam, A, vt <lb/> <anchor type="note" xlink:label="note-0168-05a" xlink:href="note-0168-05"/> ſexta, S, ad quartam, V, ergo compoſita ex prima, <lb/>E, & </s> <s xml:id="echoid-s3521" xml:space="preserve">quinta, O, erit ad ſecundam, A, vt compo-<lb/>ſita ex tertia, P, & </s> <s xml:id="echoid-s3522" xml:space="preserve">ſexta, S, ad quartam, V, ergo <lb/>conuertendo, A, ad, EO, ſimul erit, vt, V, ad, P, <lb/> <anchor type="note" xlink:label="note-0168-06a" xlink:href="note-0168-06"/> S, ſimul iunctas, qui arguendi modus dicitur à me, <lb/>colligere, ſeu colligendo.</s> <s xml:id="echoid-s3523" xml:space="preserve"/> </p> <div xml:id="echoid-div365" type="float" level="2" n="1"> <figure xlink:label="fig-0168-01" xlink:href="fig-0168-01a"> <image file="0168-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0168-01"/> </figure> <note position="left" xlink:label="note-0168-05" xlink:href="note-0168-05a" xml:space="preserve">24. Quin. <lb/>Elem.</note> <note position="left" xlink:label="note-0168-06" xlink:href="note-0168-06a" xml:space="preserve">Defin. 13. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div367" type="section" level="1" n="223"> <head xml:id="echoid-head238" xml:space="preserve">THE OREMA XX. PROPOS. XX.</head> <p> <s xml:id="echoid-s3524" xml:space="preserve">ASſumpta Propoſ. </s> <s xml:id="echoid-s3525" xml:space="preserve">antecedentis figura, dimiſſa, BM, re-<lb/>tineatur, NE, pro vna ex ductis vtcunque parallela <lb/>ipſi, CD, producta autem, CD, vtcunque in, M, comple-<lb/>toque parallelogrammo, OD. </s> <s xml:id="echoid-s3526" xml:space="preserve">Dico parallelogrammum, A <lb/>M, ad trapezium, FCMO, eſſe vt, CM, ad, MD, ſimul <lb/>cum {1/2}, CD.</s> <s xml:id="echoid-s3527" xml:space="preserve"/> </p> <pb o="149" file="0169" n="169" rhead="LIBER II."/> <p> <s xml:id="echoid-s3528" xml:space="preserve">Erit enim, AM, parallelogrammum, vnde, MA, ad, AD, erit <lb/> <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> vt, CM, ad, CD, AD, verò ad trian <lb/> <anchor type="note" xlink:label="note-0169-01a" xlink:href="note-0169-01"/> gulum, FCD; </s> <s xml:id="echoid-s3529" xml:space="preserve">eſt vt, CD, ad, {1/2}, C <lb/> <anchor type="note" xlink:label="note-0169-02a" xlink:href="note-0169-02"/> D, ergo, AM, ad triangulum, FCD, <lb/>erit vt, MC, ad, {1/2}, CD, eſt autem, <lb/>AM, ad, FM, vt, CM, ad, MD, <lb/>ergo, colligendo, AM, ad, FM, cum <lb/> <anchor type="note" xlink:label="note-0169-03a" xlink:href="note-0169-03"/> triangulo, FCD, ideſt ad trapezium, <lb/>OFCM, erit vt, CM, ad, MD, cum, <lb/>{1/2}, DC, quod oſtendendum erat.</s> <s xml:id="echoid-s3530" xml:space="preserve"/> </p> <div xml:id="echoid-div367" type="float" level="2" n="1"> <figure xlink:label="fig-0169-01" xlink:href="fig-0169-01a"> <image file="0169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0169-01"/> </figure> <note position="right" xlink:label="note-0169-01" xlink:href="note-0169-01a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-0169-02" xlink:href="note-0169-02a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0169-03" xlink:href="note-0169-03a" xml:space="preserve">5. huius</note> </div> </div> <div xml:id="echoid-div369" type="section" level="1" n="224"> <head xml:id="echoid-head239" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s3531" xml:space="preserve">_M_Anifeſtnm eſt autem, ſi, CD, ſit æqualis ipſi, DF, omnes lineas <lb/> <anchor type="note" xlink:label="note-0169-04a" xlink:href="note-0169-04"/> parallelogrammi, AD, regula, CD, eſſe æquales maximis ab-<lb/>ſciſſarum, FD, & </s> <s xml:id="echoid-s3532" xml:space="preserve">omnes lineas trianguli, FCD, regula eadem æquari <lb/>omnibus abſciſſis, FD. </s> <s xml:id="echoid-s3533" xml:space="preserve">Nunc ſi intelligamus cuilibet earum, quæ dicun-<lb/>tur maximæ abſciſſarum, vel abſciſſæ, adiungirectam, DM, vocantur <lb/>tunc maximæ abſciſſarum, vel abſciſſæ adiuncta, DM, hæc autem ſunt <lb/> <anchor type="note" xlink:label="note-0169-05a" xlink:href="note-0169-05"/> eædem illis, quæ habentur in parallelogrammo, AM, & </s> <s xml:id="echoid-s3534" xml:space="preserve">trapezio, FC <lb/>MO, nam ſi produxeris, NE, vſq; </s> <s xml:id="echoid-s3535" xml:space="preserve">ad, OM, in, X, ſiet, EX, adiun-<lb/>cta tum ipſi, NE, vni ex maximis abſciſſarum, FD, tum ipſi, HE, <lb/>vni ex omnibus abſciſſis, FD, &</s> <s xml:id="echoid-s3536" xml:space="preserve">, EX, adiuncta eſt æqualis ipſi, DM, <lb/>vnde omnes linea, AD, adiuncta, DM, ſunt omnes lineæ parallelo-<lb/>grammi, AM, & </s> <s xml:id="echoid-s3537" xml:space="preserve">ſunt æquales maximis abſciſſarum ipſius, FD, ad-<lb/>iuncta, DM, & </s> <s xml:id="echoid-s3538" xml:space="preserve">omnes lineæ trianguli, FCD, adiuncta, DM, ſunt om-<lb/>nes lineæ trapezij, FCMO, & </s> <s xml:id="echoid-s3539" xml:space="preserve">ſunt æquales omnibus abſciſſis ipſius, F <lb/>D, adiuncta, DM. </s> <s xml:id="echoid-s3540" xml:space="preserve">Quiaergo, AM, ad trapezium, FCMO, eſt vt, C <lb/>M, ad, MD, cum, {1/2}, DC, ideò omnes lineæ, AM, ad omnes lineas <lb/> <anchor type="note" xlink:label="note-0169-06a" xlink:href="note-0169-06"/> trapezij, FCMO, (regulam hic ſemperintelligeipſam, CM,) .</s> <s xml:id="echoid-s3541" xml:space="preserve">i. </s> <s xml:id="echoid-s3542" xml:space="preserve">ma-<lb/>ximæ abſciſſarum, FD, adiuncta, DM, ad omnes abſciſſas, FD, adiun-<lb/>cta, DM, erunt vt, CM, compoſita nempè ex propoſita linea, CD, ſiue <lb/>ex propoſita, FD, illi æquali, & </s> <s xml:id="echoid-s3543" xml:space="preserve">adiuncta, DM, ad compoſitam ex ad-<lb/>iuncta, MD, &</s> <s xml:id="echoid-s3544" xml:space="preserve">, {1/2}, propoſitæ lineæ, CD, vel, DF.</s> <s xml:id="echoid-s3545" xml:space="preserve"/> </p> <div xml:id="echoid-div369" type="float" level="2" n="1"> <note position="right" xlink:label="note-0169-04" xlink:href="note-0169-04a" xml:space="preserve">_ExCor. 2._ <lb/>_antec._</note> <note position="right" xlink:label="note-0169-05" xlink:href="note-0169-05a" xml:space="preserve">_Defin. 7._ <lb/>_huius._</note> <note position="right" xlink:label="note-0169-06" xlink:href="note-0169-06a" xml:space="preserve">_3. huius._</note> </div> </div> <div xml:id="echoid-div371" type="section" level="1" n="225"> <head xml:id="echoid-head240" xml:space="preserve">THE OREMA XXI. PROPOS. XXI.</head> <p> <s xml:id="echoid-s3546" xml:space="preserve">IN expoſita ſuperioris Propoſ. </s> <s xml:id="echoid-s3547" xml:space="preserve">figura, ſiproducatur, CD, <lb/>ad partes, C, vtcunque, vt in, R, & </s> <s xml:id="echoid-s3548" xml:space="preserve">compleatur parallc-<lb/>logrammum, GC, oſtendemus trapezium, FGRC, ad tra- <pb o="150" file="0170" n="170" rhead="GEOMETRI Æ"/> pezium, FCMO, eſſe vt compoſita ex, RC, &</s> <s xml:id="echoid-s3549" xml:space="preserve">, {1/2}, CD, ad <lb/>compoſitam ex, MD, &</s> <s xml:id="echoid-s3550" xml:space="preserve">, {1/2}, CD.</s> <s xml:id="echoid-s3551" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3552" xml:space="preserve">Nam trapezium, CRGF, ad, GD, eſt vt compoſita ex, RC, <lb/>&</s> <s xml:id="echoid-s3553" xml:space="preserve">, {1/2}, CD, ad, RD, inſuper, GD, ad, AM, eſt vt, RD, ad, C <lb/>M, & </s> <s xml:id="echoid-s3554" xml:space="preserve">tandem, AM, ad trapezium, FCMO, eſt vt, CM, ad, M <lb/>D, cum, {1/2}, CD, ergo ex æquali trapezium, FGRC, ad trape-<lb/>zium, FCMO, erit vt, RC, cum, {1/2}, CD, ad, MD, cum, {1/2}, D <lb/>C, quod erat demonſtrandum.</s> <s xml:id="echoid-s3555" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div372" type="section" level="1" n="226"> <head xml:id="echoid-head241" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s3556" xml:space="preserve">_H_Inc patet omnes lineas trapezij, FGRC, regula, RM, ad omnes <lb/> <anchor type="note" xlink:label="note-0170-01a" xlink:href="note-0170-01"/> lineas trapezij, FCMO, regula eadem eſſe, vt, RC, cum, {1/2}, <lb/>CD, ad, MD, cum, {1/2}, DC, veluti autem in antecedenti oſtendimus, <lb/>ſi, CD, ſit æqualis ipſi, DF, omnes lineas trapezij, FGMO, regula, <lb/>CM, æquari omnibus abſciſſis ipſius, FD, adiuncta, DM, ita in præ-<lb/>ſenti oſtendemus omnes lineas trapezij, FGRC, regula, RD, æquari re-<lb/>ſiduis omnium abſciſſarum ipſius, AC, vel, FD, adiuucta, RC; </s> <s xml:id="echoid-s3557" xml:space="preserve">vnde <lb/>patebit reſiduas abſciſſarum propoſitæ lineæ, vt, FD, adiuncta, RC, ad <lb/>omnes abſciſſas eiuſdem, adiuncta alia linea, vt, DM, eſſe vt compoſi-<lb/>tum ex prima adiuncta, &</s> <s xml:id="echoid-s3558" xml:space="preserve">, {1/2}, propoſitæ, CD, ſiue, FD, illi æqualis, <lb/>ad compoſitum ex ſecunda adiuncta, &</s> <s xml:id="echoid-s3559" xml:space="preserve">, {1/2}, propoſitæ lineæ, ideſt vt, R <lb/>C, cum, {1/2}, CD, vel, DF, ad, MD, cum, {1/2}, CD, vel, DF.</s> <s xml:id="echoid-s3560" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="2" n="1"> <note position="left" xlink:label="note-0170-01" xlink:href="note-0170-01a" xml:space="preserve">_3. huius._</note> </div> </div> <div xml:id="echoid-div374" type="section" level="1" n="227"> <head xml:id="echoid-head242" xml:space="preserve">THE OREMA XXII. PROPOS. XXII.</head> <p> <s xml:id="echoid-s3561" xml:space="preserve">EXpoſitis duobus vtcunq; </s> <s xml:id="echoid-s3562" xml:space="preserve">parallelogrammis, in eiſdem-<lb/>que ductis diametris, & </s> <s xml:id="echoid-s3563" xml:space="preserve">duobus vtcunq; </s> <s xml:id="echoid-s3564" xml:space="preserve">lateribus pro <lb/>regula ſumptis, nempè in vnoquoq; </s> <s xml:id="echoid-s3565" xml:space="preserve">eorum vno: </s> <s xml:id="echoid-s3566" xml:space="preserve">Omnia qua-<lb/>drata cuiuſuis dictorum parallelogrãmorum ad omnia qua-<lb/>drata cuiuſuis triangulorum per diametrum in ipſo conſtitu-<lb/>torum, erunt vt omnia quadrata reliqui parallelogrammi ad <lb/>omnia quadrata cuiuſuis triangulorum per diametrum in <lb/>iſto ductam pariter conſtitutorum.</s> <s xml:id="echoid-s3567" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3568" xml:space="preserve">Sint expoſita vtcunque parallelogramma, AS, Τ β, in ijſque du-<lb/>ctæ diametri, EO, Z &</s> <s xml:id="echoid-s3569" xml:space="preserve">, regulis ſumptis, ES, Ζβ. </s> <s xml:id="echoid-s3570" xml:space="preserve">Dico omnia <lb/>quadrata, AS, ad omnia quadrata trianguli, OES, eſſe vt omnia <lb/>quadrata, Τ β, ad omnia quadrata, & </s> <s xml:id="echoid-s3571" xml:space="preserve">Ζ β. </s> <s xml:id="echoid-s3572" xml:space="preserve">Sienim, vtomnia qua- <pb o="151" file="0171" n="171" rhead="LIBER II."/> drata, Τ β, ad omnia quadrata trianguli, & </s> <s xml:id="echoid-s3573" xml:space="preserve">Ζ β, ita non ſunt om-<lb/>nia quadrata, AS, ad omnia quadrata trianguli, OES, erunt igi-<lb/>tur ita omnia quadrata, AS, ad maius, vel ad minus omnibus qua-<lb/>dratis trianguli, OES, ſint exceſſus, vel defectus, omnia quadrata <lb/>figuræ planæ, Ω, diuidatur autem latus, OS, bifariam, in, Q, &</s> <s xml:id="echoid-s3574" xml:space="preserve">, <lb/>OQ, QS, bifariam in, P, R, & </s> <s xml:id="echoid-s3575" xml:space="preserve">ſic deinceps fiat, ita vt ductis per <lb/>puncta diuiſionum parallelis ipſi, ES, DR, CQ, BP, tandem de-<lb/>uentum ſit ad parallelogrammum, DS, cuius omnia quadrata, re-<lb/> <anchor type="note" xlink:label="note-0171-01a" xlink:href="note-0171-01"/> gula, ES, ſint minora omnibus quadratis figurę, Ω, per puncta au-<lb/>tem, in quibus dictæ parallelę ipſam, OE, ſecant, ducantur vſque <lb/>ad proximas parallelas æquidiſtantes lateribus, AE, OS, ipſę, LN, <lb/>GK, EM, erit igitur triangulo, OES, circumſcripta figura quæ-<lb/> <anchor type="figure" xlink:label="fig-0171-01a" xlink:href="fig-0171-01"/> dam cõpoſita ex <lb/>parallelogrãmo, <lb/>LP, GQ, FR, <lb/>DS, & </s> <s xml:id="echoid-s3576" xml:space="preserve">alia in-<lb/>ſcripta compoſi-<lb/>ta ex parallelo-<lb/>grammis, 9 Q, I <lb/>R, HS, ita vt <lb/>omnia quadrata <lb/>figuræ circũſcri-<lb/>ptę, regula, ES, <lb/>excedant omnia <lb/>quadrata inſcri-<lb/>ptæ, regula ea-<lb/>dẽ, minori quan-<lb/>titate, quam ſint <lb/>omnia quadrata figuræ, Ω; </s> <s xml:id="echoid-s3577" xml:space="preserve">nam in parallelogrammo, DS, recta, <lb/>HM, diuidit omnia quadrata, DS, in omnia quadrata, DM, in <lb/>omnia quadrata, HS, & </s> <s xml:id="echoid-s3578" xml:space="preserve">in rectangula bis ſub, DM, MR, veluti <lb/>punctum, H, diuidit quadratum, DR, in quadrat. </s> <s xml:id="echoid-s3579" xml:space="preserve">DH, quad at-<lb/>HR, & </s> <s xml:id="echoid-s3580" xml:space="preserve">rectangulum bis ſub, DHR, ſiue ex 23. </s> <s xml:id="echoid-s3581" xml:space="preserve">ſeq. </s> <s xml:id="echoid-s3582" xml:space="preserve">ab hac inde-<lb/>pendente, & </s> <s xml:id="echoid-s3583" xml:space="preserve">ideò omnia quadrat. </s> <s xml:id="echoid-s3584" xml:space="preserve">DS, excedunt omnia quadrata, <lb/>HS, omnibus quadratis, DM, & </s> <s xml:id="echoid-s3585" xml:space="preserve">rectangulis bis ſub, DM, MR, <lb/>eodem pacto oſtendemus omnia quadrata, FR, excedere omnia <lb/>quadrata, IR, omnibus quadratis, FK, & </s> <s xml:id="echoid-s3586" xml:space="preserve">rectangulis bis ſub, FK, <lb/>KQ, & </s> <s xml:id="echoid-s3587" xml:space="preserve">ſic omnia quadrata, GQ, excedere omnia quadrata, 9 Q, <lb/>omnibus quadratis, GN, cum rectangulis bis ſub, GN, NP, & </s> <s xml:id="echoid-s3588" xml:space="preserve">in <lb/>figura circumſcripta ſuperſunt adhuc omnia quadrata, LP, porro ſi <lb/>hos exceſſus ſimul colligamus fient omnia quadrata, DS, nam ſi <lb/>omnia quadrata, LP, vel, 9 Q, iunxeris omnibus quadratis, GN, <pb o="152" file="0172" n="172" rhead="GEOMETRIÆ"/> & </s> <s xml:id="echoid-s3589" xml:space="preserve">rectangulis bis ſub, GN, NP, fient omnia quadrata, GQ, hęc <lb/>ſi iunxeris omnibus quadratis, FK, cum rectangulis bis ſub, FK, K <lb/>Q, fient omnia quadrata, FR, quę tandem ſi iunxeris omnibus qua-<lb/>dratis, DM, cum rectangulis bis ſub, DM, MR, fient omnia qua-<lb/>drata, DS, quę cum ſint minora omnibus quadratis figurę, Ω, hinc <lb/>figuræ circumſcriptæ omnia quadrata excedunt omnia quadrata in-<lb/>ſcriptę minori quantitate, quam ſint omnia quadrata, Ω, & </s> <s xml:id="echoid-s3590" xml:space="preserve">ideò ex-<lb/>cedunt omnia quadrata trianguli, OES, multò minon quantitate: <lb/></s> <s xml:id="echoid-s3591" xml:space="preserve">Quia ergo omnia quadrata, AS, ad omnia quadrata trianguli, OE <lb/>S, cum omnibus quadratis, Ω, erant vt omnia quadrata, Τ β, ad om-<lb/>nia quadrata trianguli, & </s> <s xml:id="echoid-s3592" xml:space="preserve">Ζ β, hinc omnia quadrata, AS, ad om-<lb/>nia quadrata figurę circumſcriptę triangulo, OES, habebunt maio-<lb/>rem rationem, quam omnia quadrata, Τ β, ad omnia quadrata tri-<lb/>anguli, & </s> <s xml:id="echoid-s3593" xml:space="preserve">Ζ β.</s> <s xml:id="echoid-s3594" xml:space="preserve"/> </p> <div xml:id="echoid-div374" type="float" level="2" n="1"> <note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">Iux. prim. <lb/>10. Elem.</note> <figure xlink:label="fig-0171-01" xlink:href="fig-0171-01a"> <image file="0171-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0171-01"/> </figure> </div> <p> <s xml:id="echoid-s3595" xml:space="preserve">Nunc diuidatur ſimiliter, & </s> <s xml:id="echoid-s3596" xml:space="preserve">β in punctis, ℟, Δ Σ, ac, OS, in <lb/>punctis, P, Q, R, & </s> <s xml:id="echoid-s3597" xml:space="preserve">per puncta, ℟ Δ Σ, parallelæ ipſi, Ζ β, du-<lb/>cantur, ℟ V, Δ Χ, Σ Υ, ſecantes, & </s> <s xml:id="echoid-s3598" xml:space="preserve">Ζ, in punctis, r, 3, 6, per quę <lb/>vſque ad proximas parallelas ipſis, & </s> <s xml:id="echoid-s3599" xml:space="preserve">β, ΤΖ, æquidiſtantes ducan-<lb/>tur, Φ Γ, Λ 3, 46, vt triangulo, & </s> <s xml:id="echoid-s3600" xml:space="preserve">Ζ β, ſit circumſcripta figura ex <lb/> <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> parallelogrãmis, <lb/>Φ ℟, Δ Δ, 4 Σ, Υ <lb/>β, cõpoſita, quia <lb/>ergo, vt, OS, ad, <lb/>SR, ita eſt, & </s> <s xml:id="echoid-s3601" xml:space="preserve">β, <lb/>ad, β Σ, vt au-<lb/>tem, OS, ad, S <lb/>R, ita ſunt om-<lb/> <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/> nia quadrata, A <lb/>S, ad omnia qua-<lb/>drata, DS, & </s> <s xml:id="echoid-s3602" xml:space="preserve"><lb/>vt, & </s> <s xml:id="echoid-s3603" xml:space="preserve">β, ad, β <lb/>Σ, rta ſunt omnia <lb/>quadrata, Τ β, ad <lb/>omnia quadrata, <lb/>Υ β, ergo omnia <lb/>quadrata, AS, ad omnia quadrata, DS, ſunt vt omnia quadrata, <lb/>Τ β, ad omnia quadrata, Υ β, quia verò omnia quadrata, Υ β, ad <lb/>omnia quadrata, 6 β, .</s> <s xml:id="echoid-s3604" xml:space="preserve">@. </s> <s xml:id="echoid-s3605" xml:space="preserve">ad omnia quadrata, 4 Σ, ſunt vt quadra-<lb/> <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/> tum, Ζ β, ad quadratum, 7 β, .</s> <s xml:id="echoid-s3606" xml:space="preserve">@. </s> <s xml:id="echoid-s3607" xml:space="preserve">ad quadratum, 6 Σ, .</s> <s xml:id="echoid-s3608" xml:space="preserve">@. </s> <s xml:id="echoid-s3609" xml:space="preserve">vt quadra-<lb/>tum, β &</s> <s xml:id="echoid-s3610" xml:space="preserve">, ad quadratum, & </s> <s xml:id="echoid-s3611" xml:space="preserve">Σ, .</s> <s xml:id="echoid-s3612" xml:space="preserve">@. </s> <s xml:id="echoid-s3613" xml:space="preserve">vt quadrarum, SO, ad quadra-<lb/>tum, OR, ideſt vt quadratum, ES, ad quadratum, HR, ideſt, vt <lb/>omnia quadrata, DS, ad omnia quadrata, FR, ergo ex æquali om- <pb o="153" file="0173" n="173" rhead="LIBER II."/> nia quadrata, AS, ad omnia quadrata, FR, erunt vt omnia qua-<lb/> <anchor type="note" xlink:label="note-0173-01a" xlink:href="note-0173-01"/> drata, Τ β, ad omnia quadrata, 4 Σ: </s> <s xml:id="echoid-s3614" xml:space="preserve">Eodem pacto oſtendemus om-<lb/>nia quadrata, AS, ad omnia quadrata, GQ, elle vt omnia quadra-<lb/>ta, Τ β, ad omnia quadrata, Λ Δ, & </s> <s xml:id="echoid-s3615" xml:space="preserve">tandem omnia quadrata, AS, <lb/>ad omnia quadrata, LP, eſſe vt omnia quadrata, Τ β, ad omnia <lb/>quadrata, Φ ℟, vnde, colligendo, omnia quadrata, AS, ad omnia <lb/>quadrata parallelogrammorum, DS, FR, GQ, LP, ideſt figurę <lb/>circumſcriptæ, erunt vt omnia quadrata, Τ β, ad omnia quadrata <lb/> <anchor type="note" xlink:label="note-0173-02a" xlink:href="note-0173-02"/> parallelogrammorum, Φ ℟, Λ Δ, 4 Σ, Υ β, ideſt ad omnia quadrata <lb/>figuræ circumicriptæ triangulo, & </s> <s xml:id="echoid-s3616" xml:space="preserve">Ζ β, ſed omnia quadrata, AS, <lb/>ad omnia quadrata figuræ circumſcriptæ triangulo, OES, oſtenſa <lb/>ſunt habere maiorem rationem, quam omnia quadrata, Τ β, ad om-<lb/>nia quadrata trianguli, & </s> <s xml:id="echoid-s3617" xml:space="preserve">Ζ β, ergo omnia quadrata, Τ β, ad om-<lb/>nia quadrata figuræ circumſcriptæ triangulo, & </s> <s xml:id="echoid-s3618" xml:space="preserve">Ζ β, habebunt ma-<lb/>iorem rationem, quam ad omnia quadrata trianguli, & </s> <s xml:id="echoid-s3619" xml:space="preserve">Ζ β, ergo <lb/>omnia quadrata figuræ circumſcriptæ triangulo, & </s> <s xml:id="echoid-s3620" xml:space="preserve">Ζ β, minora c-<lb/>runt omnibus quadratis trianguli, & </s> <s xml:id="echoid-s3621" xml:space="preserve">Ζ β, quod eſt abſurdum, non <lb/>ergo omnia quadrata, AS, ad maius, quam ſint omnia quadrata <lb/>trianguli, OES, habenteandem rationem, quam omnia quadrata, <lb/>Τ β, ad omnia quadrata trianguli, & </s> <s xml:id="echoid-s3622" xml:space="preserve">Ζ β.</s> <s xml:id="echoid-s3623" xml:space="preserve"/> </p> <div xml:id="echoid-div375" type="float" level="2" n="2"> <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a"> <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0172-01"/> </figure> <note position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-0173-01" xlink:href="note-0173-01a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-0173-02" xlink:href="note-0173-02a" xml:space="preserve">Defin. @@ <lb/>lib. 10</note> </div> <p> <s xml:id="echoid-s3624" xml:space="preserve">Dico autem neque ad minus eiuſdem habere eandem rationem, <lb/>ſint enim defectus adhuc omnia quadra a figurę, Ω, & </s> <s xml:id="echoid-s3625" xml:space="preserve">ſit circumſcri-<lb/>pta triangulo, OES, figura ex parallelogrammis, LP, GQ, FR, <lb/>DS, & </s> <s xml:id="echoid-s3626" xml:space="preserve">al@a inſcripta ex parallelogrammis, MQ, IR, HS, com-<lb/>poſita, ita vt omnia quadrata circumſcriptæ ſuperent omnia qua-<lb/>drata inſcriptę minori quantitate, quam ſint omnia quadrata, Ω, er-<lb/>go omnia quadrata trianguli, OES, ſuperabunt omnia quadrata in-<lb/>icriptæ figuræ multo minoriquan@tate, ſunt autem omnia quadra-<lb/>ta, AS, ad omnia quadrata trianguli, OES, detractis omnibus qua-<lb/>drat@s, Ω, vt omnia quadrata, Τ β, ad omnia quadrata trianguli, & </s> <s xml:id="echoid-s3627" xml:space="preserve"><lb/>Ζ β, ergo omnia quadrata, AS, ad omnia quadrata inſcriptæ figu-<lb/>ræ habebunt minorem rationem, quam omnia quadrata, Τ β, ad <lb/>omnia quadrata trianguli, & </s> <s xml:id="echoid-s3628" xml:space="preserve">Ζ β. </s> <s xml:id="echoid-s3629" xml:space="preserve">Diuidatur nunc pariter latus, & </s> <s xml:id="echoid-s3630" xml:space="preserve"><lb/>β, in punctis, ℟, Δ, Σ, ſimiliter ac, OS, diuiditur in, P, Q, R, & </s> <s xml:id="echoid-s3631" xml:space="preserve"><lb/>cæ@era, vt ſupra, fiant, vt habeamus figuram inſcriptam ex paralle-<lb/>logrammis, Τ Δ, 3 Σ, 6 β, compoſitam, oſtendemus igitur, vt ſu-<lb/>pra, omnia quadrata, AS, ad omnia quadrata figurę inſcriptę trian-<lb/>gulo, OES, eſſe vt omnia quadrata, Τ β, ad omnia quadrata figu-<lb/>ræ inſcriptæ triangulo, & </s> <s xml:id="echoid-s3632" xml:space="preserve">Ζ β, ſunt autem omnia quadrata, AS, ad <lb/>omnia quadrata figuræ inſcriptæ triangulo, OES, in minori ratio-<lb/>ne, quam ſint omnia quadrata, Τ β, ad omnia quadrata trianguli, <lb/>& </s> <s xml:id="echoid-s3633" xml:space="preserve">Ζ β, ergo omnia quadrata, Τ β, ad omnia quadrata figurę inſcri- <pb o="154" file="0174" n="174" rhead="GEOMETRIÆ"/> ptæ triangulo, & </s> <s xml:id="echoid-s3634" xml:space="preserve">Ζ β, erunt in minori ratione, quam omnia qua-<lb/>drata, Τ β, ad omnia quadrata trianguli, & </s> <s xml:id="echoid-s3635" xml:space="preserve">Ζ β, ergo figurę inſcri-<lb/>ptæ triangulo, & </s> <s xml:id="echoid-s3636" xml:space="preserve">Ζ β, omnia quadrata maiora erunt omnibus qua-<lb/>dratis trianguli, & </s> <s xml:id="echoid-s3637" xml:space="preserve">Ζ β, quod eſt abſurdum, igitur omnia quadrata, <lb/>AS, non ad minus, quam ſint omnia quadrata trianguli, OES, erunt <lb/>vt omnia quadrata, Τ β, ad omnia quadrata trianguli, & </s> <s xml:id="echoid-s3638" xml:space="preserve">Ζ β, ſed <lb/>neque ad maius, vt oſtenſum eſt ergo ad ipſa erunt, vt omnia qua-<lb/>drata, Τ β, ad omnia quadrata, & </s> <s xml:id="echoid-s3639" xml:space="preserve">Ζ β. </s> <s xml:id="echoid-s3640" xml:space="preserve">Si autem comparentur om-<lb/>nia quadrata, AS, Τ β, ad omnia quadrata triangulorum, AEO, <lb/>TZ &</s> <s xml:id="echoid-s3641" xml:space="preserve">, eodem modo fiet demonſtratio, igitur oſtenſum eſt, quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s3642" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div377" type="section" level="1" n="228"> <head xml:id="echoid-head243" xml:space="preserve">A. COROLLARII SECTIO I.</head> <p style="it"> <s xml:id="echoid-s3643" xml:space="preserve">_H_Inc patet quæcunque de omnibus quadratis parallelogrammorum <lb/>tales, vel tales conditiones habentium in Propoſ. </s> <s xml:id="echoid-s3644" xml:space="preserve">9. </s> <s xml:id="echoid-s3645" xml:space="preserve">10. </s> <s xml:id="echoid-s3646" xml:space="preserve">11. </s> <s xml:id="echoid-s3647" xml:space="preserve">12. <lb/></s> <s xml:id="echoid-s3648" xml:space="preserve">13. </s> <s xml:id="echoid-s3649" xml:space="preserve">14. </s> <s xml:id="echoid-s3650" xml:space="preserve">buius Libri oſtenſa ſunt, eadem de omnibus quadratis triangulo-<lb/>rum, tanquam de eorundem partibus proportionalibus verificari, regu-<lb/>la vno latere ſumpta, dum triangula circa altitudines, & </s> <s xml:id="echoid-s3651" xml:space="preserve">baſes, ſiue à <lb/>baſibus de ſcriptas figuras, & </s> <s xml:id="echoid-s3652" xml:space="preserve">latera æqualiter baſibus inclinata, eaſdem <lb/>obtinuerint conditiones ibi notatas.</s> <s xml:id="echoid-s3653" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div378" type="section" level="1" n="229"> <head xml:id="echoid-head244" xml:space="preserve">B. SECTIO II.</head> <p style="it"> <s xml:id="echoid-s3654" xml:space="preserve">_I_Gitur triangulorum in eadem altitudine exiſtentium omnia quadra-<lb/>ta, vel omnes figuræ ſimiles (ſiue ſint ſimiles ad inuicem, quæ ſunt <lb/> <anchor type="note" xlink:label="note-0174-01a" xlink:href="note-0174-01"/> vtriuſque trianguli, ſiue diſſimiles) er unt vt figuræ à baſibus deſcriptæ.</s> <s xml:id="echoid-s3655" xml:space="preserve"/> </p> <div xml:id="echoid-div378" type="float" level="2" n="1"> <note position="left" xlink:label="note-0174-01" xlink:href="note-0174-01a" xml:space="preserve">_9. huius._</note> </div> </div> <div xml:id="echoid-div380" type="section" level="1" n="230"> <head xml:id="echoid-head245" xml:space="preserve">C. SECTIO III.</head> <p style="it"> <s xml:id="echoid-s3656" xml:space="preserve">_E_T ſi triangula fuerint in eadem, vel æqualibus baſibus, omnes figu-<lb/> <anchor type="note" xlink:label="note-0174-02a" xlink:href="note-0174-02"/> ræ ſimiles, vtriuſque ad inuicem, erunt vt altitudines, vel vt la-<lb/>tera baſibus æqualiter in clinata.</s> <s xml:id="echoid-s3657" xml:space="preserve"/> </p> <div xml:id="echoid-div380" type="float" level="2" n="1"> <note position="left" xlink:label="note-0174-02" xlink:href="note-0174-02a" xml:space="preserve">_10. huius._</note> </div> </div> <div xml:id="echoid-div382" type="section" level="1" n="231"> <head xml:id="echoid-head246" xml:space="preserve">D. SECTIO IV.</head> <p style="it"> <s xml:id="echoid-s3658" xml:space="preserve">_I_Tem triangulorum omnia quadrata, ſiue omnes figuræ ſimiles, etiamſi <lb/> <anchor type="note" xlink:label="note-0174-03a" xlink:href="note-0174-03"/> ſint diſſimiles, quæ ſunt vtriuſq; </s> <s xml:id="echoid-s3659" xml:space="preserve">trianguli, habebunt rationem com-<lb/>poſitam ex ratione figurarum à baſibus deſcriptarum, & </s> <s xml:id="echoid-s3660" xml:space="preserve">altitudinum, <lb/>ſiue laterum baſibus æqualiter inclinatorum.</s> <s xml:id="echoid-s3661" xml:space="preserve"/> </p> <div xml:id="echoid-div382" type="float" level="2" n="1"> <note position="left" xlink:label="note-0174-03" xlink:href="note-0174-03a" xml:space="preserve">_11. huius._</note> </div> <pb o="155" file="0175" n="175" rhead="LIBER II."/> </div> <div xml:id="echoid-div384" type="section" level="1" n="232"> <head xml:id="echoid-head247" xml:space="preserve">E. SECTIO V.</head> <p style="it"> <s xml:id="echoid-s3662" xml:space="preserve">_E_T triangulorum, quorum baſium figuræ altitudinibus, vel lateri-<lb/>bus æqualiter bafibus inclinatis reciprocantur, omnes figuræ, ſi-<lb/> <anchor type="note" xlink:label="note-0175-01a" xlink:href="note-0175-01"/> miles baſium figuris, ſunt æquales: </s> <s xml:id="echoid-s3663" xml:space="preserve">Et ſi omnes figuræ, ſimiles baſium fi-<lb/>guris, ſint æquales, figuras baſium altitudinibus, vel latoribus æquali-<lb/>ter baſibus inclinatis reciprocè reſpondentes habebunt.</s> <s xml:id="echoid-s3664" xml:space="preserve"/> </p> <div xml:id="echoid-div384" type="float" level="2" n="1"> <note position="right" xlink:label="note-0175-01" xlink:href="note-0175-01a" xml:space="preserve">_12. huius._</note> </div> </div> <div xml:id="echoid-div386" type="section" level="1" n="233"> <head xml:id="echoid-head248" xml:space="preserve">F. SECTIO VI.</head> <p style="it"> <s xml:id="echoid-s3665" xml:space="preserve">_E_T tandem ſimilium triangulorum omnia quadrata erunt in tripla <lb/> <anchor type="note" xlink:label="note-0175-02a" xlink:href="note-0175-02"/> ratione laterum bomotogorum, ſiue vt eorum cubi; </s> <s xml:id="echoid-s3666" xml:space="preserve">regulas verò <lb/>in ſupradictis ſuppono ſemper duo illorum triangulorum latera, quæ ba-<lb/>ſes voco; </s> <s xml:id="echoid-s3667" xml:space="preserve">hic verò intellige illorum triangulorum latera bomologa. </s> <s xml:id="echoid-s3668" xml:space="preserve">His <lb/>autem ſequentem Tropoſitionem ſubiungam, tum buius gratia, tum eo-<lb/> <anchor type="note" xlink:label="note-0175-03a" xlink:href="note-0175-03"/> rum, quæ ſequentur.</s> <s xml:id="echoid-s3669" xml:space="preserve"/> </p> <div xml:id="echoid-div386" type="float" level="2" n="1"> <note position="right" xlink:label="note-0175-02" xlink:href="note-0175-02a" xml:space="preserve">_Iuxt. dif-_ <lb/>_fin. 1. Sex-_ <lb/>_ti Elem._</note> <note position="right" xlink:label="note-0175-03" xlink:href="note-0175-03a" xml:space="preserve">_12. huius._</note> </div> </div> <div xml:id="echoid-div388" type="section" level="1" n="234"> <head xml:id="echoid-head249" xml:space="preserve">THEOR EMA XXIII. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s3670" xml:space="preserve">SI, expoſita quacunque figura plana, in ea ducatur vtcun-<lb/>que recta linea, quæ ſit ſumpta pro regula, eadem verò <lb/>in puncto, vel punctis diuiſa, prout lib. </s> <s xml:id="echoid-s3671" xml:space="preserve">2. </s> <s xml:id="echoid-s3672" xml:space="preserve">Elem. </s> <s xml:id="echoid-s3673" xml:space="preserve">ſupponitur <lb/>ſecari, per puncta diuiſionum lineas duxerimus rectas, ſiue <lb/>curuas, figuram diuidentes, & </s> <s xml:id="echoid-s3674" xml:space="preserve">ſemeltantum ſecantes quam-<lb/>uis aliam regulæ parallelam, ſiregula in vno puncto tantum <lb/>diuiſa ſit, vel toties, quot ſunt puncta diuiſionum regulę (ex-<lb/>ceptis tamen extremis, in quibus linearum ſectæ partes in <lb/>puncta aliquando degenerare poſſunt.) </s> <s xml:id="echoid-s3675" xml:space="preserve">Quæcunq; </s> <s xml:id="echoid-s3676" xml:space="preserve">in dict, 2. <lb/></s> <s xml:id="echoid-s3677" xml:space="preserve">lib. </s> <s xml:id="echoid-s3678" xml:space="preserve">demonſtrantur hac diuiſione ſuppoſita circa vel quadra-<lb/>ta, vel rectangula eidem rectæ lineæ applicata, eadem de <lb/>omnibus quadratis dictæ figuræ, vel eiuſdem partium, vel <lb/> <anchor type="note" xlink:label="note-0175-04a" xlink:href="note-0175-04"/> de rectangulis ſub ipſis pariter verificabuntur.</s> <s xml:id="echoid-s3679" xml:space="preserve"/> </p> <div xml:id="echoid-div388" type="float" level="2" n="1"> <note position="right" xlink:label="note-0175-04" xlink:href="note-0175-04a" xml:space="preserve">D. Diff. 8. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s3680" xml:space="preserve">Sit expoſita vtcunq; </s> <s xml:id="echoid-s3681" xml:space="preserve">figura plana, ABCD, in qua ducta, BD, <lb/>recta linea vtcunq; </s> <s xml:id="echoid-s3682" xml:space="preserve">ſit illa ſumpta pro regula, & </s> <s xml:id="echoid-s3683" xml:space="preserve">ea diuiſa in vno, vel <lb/>pluribus punctis, prout poſtulant Propoſ. </s> <s xml:id="echoid-s3684" xml:space="preserve">2. </s> <s xml:id="echoid-s3685" xml:space="preserve">lib. </s> <s xml:id="echoid-s3686" xml:space="preserve">Elem. </s> <s xml:id="echoid-s3687" xml:space="preserve">per puncta di-<lb/>uifionum ducantur lineæ fiue rectę, ſiue curuę, AEC, AFI, toties <lb/>quamuis aliam ipſi, BD, parallelam in figura, BADC, ſecantes <pb o="156" file="0176" n="176" rhead="GEOMETRIÆ"/> quoties, BD, ſecta eſſe ſupponitur, exceptis tamen extremis, vt ex. <lb/></s> <s xml:id="echoid-s3688" xml:space="preserve">gr. </s> <s xml:id="echoid-s3689" xml:space="preserve">ipſa, CI, in qua parte, CI, quæin recta, CI, feparari debuiſ-<lb/>ſent per lineas, AEC, AFI, in puncta, C, I, partibus, BE, FD, <lb/>reſpondentia degenerauerunt. </s> <s xml:id="echoid-s3690" xml:space="preserve">Dico quæcunque demonſtrantur in <lb/>linea, BD, circa quadrata, vel rectangula, illi, vel illius partibus ap-<lb/>plicata, verificari de omnibus quadratis totius figurę, BADC, ſiue <lb/>partium eiuſdem figuræ per dictas lineas conſtitutarum, ſiue de re-<lb/>ctangulis ſub eiuſdem partibus. </s> <s xml:id="echoid-s3691" xml:space="preserve">Vtex gr. </s> <s xml:id="echoid-s3692" xml:space="preserve">quia in 3. </s> <s xml:id="echoid-s3693" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s3694" xml:space="preserve">lib. </s> <s xml:id="echoid-s3695" xml:space="preserve">2. </s> <s xml:id="echoid-s3696" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0176-01a" xlink:href="note-0176-01"/> Elem. </s> <s xml:id="echoid-s3697" xml:space="preserve">oſtenditur rectangulum ſub, BD, DF, æquari rectangulo <lb/>ſub, BFD, cum quadrato, FD, ſic dico verum eſſe rectangula ſub <lb/> <anchor type="figure" xlink:label="fig-0176-01a" xlink:href="fig-0176-01"/> figura, ABCD, & </s> <s xml:id="echoid-s3698" xml:space="preserve">figura, ADI, æquari re-<lb/>ctangulis ſub figuris, ABIF, ADIF, cum om-<lb/>nibus quadratis figuræ, ADIF, ſi enim aliam <lb/>vtcunque duxerimus regulæ, BD, parallelam, <lb/>vt, HO, ſecantem lineas, AC, in, M, &</s> <s xml:id="echoid-s3699" xml:space="preserve">, A <lb/>I, in, N, verum eſſe comperiemus rectangulum, <lb/>HON, æquari rectangulo, HNO, cum qua-<lb/>drato, NO, & </s> <s xml:id="echoid-s3700" xml:space="preserve">idem in cęteris regulę, BD, pa-<lb/>rallelis in figura, ABCD, ductis reperiemus, <lb/>ergo verum erit rectangula illa ſimul collecta, ideſt rectangula ſub fi-<lb/> <anchor type="note" xlink:label="note-0176-02a" xlink:href="note-0176-02"/> gura, ABID, & </s> <s xml:id="echoid-s3701" xml:space="preserve">figura, ADI, æquari rectangulis ſub figuris, AB <lb/>I, ADI, cum omnibus quadratis, ADI, quod 3. </s> <s xml:id="echoid-s3702" xml:space="preserve">Propoſit. </s> <s xml:id="echoid-s3703" xml:space="preserve">2. </s> <s xml:id="echoid-s3704" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s3705" xml:space="preserve">Elem. </s> <s xml:id="echoid-s3706" xml:space="preserve">reſpondet.</s> <s xml:id="echoid-s3707" xml:space="preserve"/> </p> <div xml:id="echoid-div389" type="float" level="2" n="2"> <note position="left" xlink:label="note-0176-01" xlink:href="note-0176-01a" xml:space="preserve">Vide D. <lb/>Defin. 8. <lb/>huius.</note> <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a"> <image file="0176-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0176-01"/> </figure> <note position="left" xlink:label="note-0176-02" xlink:href="note-0176-02a" xml:space="preserve">Coroll. 4. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s3708" xml:space="preserve">Similiter ſi ſupponamus, BF, bifariam ſecari in, E, cui adiunga-<lb/>tur, FD, ſuppoſuerimus etiam lineam, AC, bifariam ſecare quam-<lb/>libet omnium linearum figuræ, ABI, regula, BD, ſupradictarum, <lb/>quarum ſingulis aditur, quę in directum manetin figura, ADI, ve-<lb/>luti Propoſ. </s> <s xml:id="echoid-s3709" xml:space="preserve">6. </s> <s xml:id="echoid-s3710" xml:space="preserve">oſtenditur rectangulum, BDF, cum quadrato, FE, <lb/>æquari quadrato, ED, ita hic ad modum ſuperioris oſtendemus re-<lb/>ctangula ſub figura, ABID, &</s> <s xml:id="echoid-s3711" xml:space="preserve">, ADI, cum omnibus quadratis fi-<lb/>guræ, ACI, æquari omnibus quadratis figuræ, ACD, quod re-<lb/>ſpondet Prop. </s> <s xml:id="echoid-s3712" xml:space="preserve">6. </s> <s xml:id="echoid-s3713" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s3714" xml:space="preserve">Conſimiliter reliqua demonſtrabimus, <lb/>vnde iuxta I. </s> <s xml:id="echoid-s3715" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s3716" xml:space="preserve">Secundi Elem. </s> <s xml:id="echoid-s3717" xml:space="preserve">colligemus.</s> <s xml:id="echoid-s3718" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div391" type="section" level="1" n="235"> <head xml:id="echoid-head250" xml:space="preserve">A. COROLLARII SECTIO I.</head> <p style="it"> <s xml:id="echoid-s3719" xml:space="preserve">_R_Ectangula ſub figura indiuiſa, ABID, & </s> <s xml:id="echoid-s3720" xml:space="preserve">ſub diuiſa, ACD, per <lb/>lineam, AI, æquari rectangulis ſub indiuiſa, ABID, & </s> <s xml:id="echoid-s3721" xml:space="preserve">ſub <lb/>partibus diuiſæ, quæ ſunt, ACI, AID.</s> <s xml:id="echoid-s3722" xml:space="preserve"/> </p> <pb o="157" file="0177" n="177" rhead="LIBER II."/> </div> <div xml:id="echoid-div392" type="section" level="1" n="236"> <head xml:id="echoid-head251" xml:space="preserve">B. SECTIO II.</head> <p style="it"> <s xml:id="echoid-s3723" xml:space="preserve">_I_V xta ſecundam habebimus omnia quadrata figuræ, ABID, æquar@ <lb/>rectang. </s> <s xml:id="echoid-s3724" xml:space="preserve">ſub, ABID, & </s> <s xml:id="echoid-s3725" xml:space="preserve">ſingulis partibus, ABI, AID.</s> <s xml:id="echoid-s3726" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div393" type="section" level="1" n="237"> <head xml:id="echoid-head252" xml:space="preserve">C. SECTIO III.</head> <p style="it"> <s xml:id="echoid-s3727" xml:space="preserve">_I_Vxta tertiam iam dictum eſt in Propoſitione quid colligamus.</s> <s xml:id="echoid-s3728" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div394" type="section" level="1" n="238"> <head xml:id="echoid-head253" xml:space="preserve">D. SECTIO IV.</head> <p style="it"> <s xml:id="echoid-s3729" xml:space="preserve">_I_V xta quartam habemus omnia quadrata figuræ, ABID, pe@ vnicãm <lb/>lineam, AFI, diuiſæ, æquari omnibus quadratis fi u@arum, AB <lb/>I, AID, & </s> <s xml:id="echoid-s3730" xml:space="preserve">rectangulis bis ſub dictis fig. </s> <s xml:id="echoid-s3731" xml:space="preserve">ABI, AID.</s> <s xml:id="echoid-s3732" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div395" type="section" level="1" n="239"> <head xml:id="echoid-head254" xml:space="preserve">E. SECTIO V.</head> <p style="it"> <s xml:id="echoid-s3733" xml:space="preserve">_I_V xta quintam, ſi ſupponamus lineam, AI, bifariam diuidere omnes <lb/>lineas figuræ, ABID, regula, BD, ſumptas, & </s> <s xml:id="echoid-s3734" xml:space="preserve">eaſdem lineam, <lb/>AC. </s> <s xml:id="echoid-s3735" xml:space="preserve">non bifariam diuidere, colligemus rectangula ſub inæqualibus par-<lb/>tibus, ABC, ACD, cum omnibus quadratis figuræ, ACI, æquari om-<lb/>nibus quadratis figuræ, ABI.</s> <s xml:id="echoid-s3736" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div396" type="section" level="1" n="240"> <head xml:id="echoid-head255" xml:space="preserve">F. SECTIO VI.</head> <p style="it"> <s xml:id="echoid-s3737" xml:space="preserve">_I_V xta ſextam quid colligatur iam dictum eſt in Propoſitione.</s> <s xml:id="echoid-s3738" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div397" type="section" level="1" n="241"> <head xml:id="echoid-head256" xml:space="preserve">G. SECTIO VII.</head> <p style="it"> <s xml:id="echoid-s3739" xml:space="preserve">_I_V xta ſeptimam colligemus, ſuppoſito, quodfigura, ABID, ſeoetur <lb/>à ſola linea, AI, vtcunque, dummodo eadem ſecet omnes æquidi-<lb/>ſtantes ipſi regulæ, BD, in figura, ABID, ductas, & </s> <s xml:id="echoid-s3740" xml:space="preserve">in vno tantum <lb/>puncto, colligemus inquam omnia quadrata figuræ, ABID, & </s> <s xml:id="echoid-s3741" xml:space="preserve">omnia <lb/>quadrata figuræ, ADI, æquari rectangulis bis ſub figuris, ABID, A <lb/>DI, vna cum omnibus quadratis, ABI.</s> <s xml:id="echoid-s3742" xml:space="preserve"/> </p> <pb o="158" file="0178" n="178" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div398" type="section" level="1" n="242"> <head xml:id="echoid-head257" xml:space="preserve">H. SECTIO VIII.</head> <p style="it"> <s xml:id="echoid-s3743" xml:space="preserve">_I_V xta octauam, ſi ſupponamus figuram, ABCD, vtcunque ſectam <lb/>per lineam, AC, (quæ tamen ſecet omnes ipſi, BD, æquidiſtantes <lb/>in figura, ABCD, ductas, & </s> <s xml:id="echoid-s3744" xml:space="preserve">in vno tantum puncto vti dictum eſt) col-<lb/>ligemus rectangula quater ſub figuris, ABCD, ABC, cum omnibus <lb/>quadratis, ACD, aquari omnibus quadratis figuræ compoſitæ ex figu-<lb/>ra, ABCD, &</s> <s xml:id="echoid-s3745" xml:space="preserve">, ABC, ita vt omnium linearum figuræ, ABCD ſin-<lb/>gulis intelligatur adiecta, quænunc in figura, ABC, eſicum illa in ea-<lb/>dem rectitudine.</s> <s xml:id="echoid-s3746" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div399" type="section" level="1" n="243"> <head xml:id="echoid-head258" xml:space="preserve">I. SECTIO IX.</head> <p style="it"> <s xml:id="echoid-s3747" xml:space="preserve">_I_V xta nonam, ſi ſupponamus lineam, AI, ſecare omnes æquidiſtan-<lb/>tes ipſi, BD, in figura, ABID, ductas bifariam, & </s> <s xml:id="echoid-s3748" xml:space="preserve">lineam, AC, <lb/>eaſdem bifariam non ſecare, colligemus omnia quadr ata figuræ, ACD, <lb/>cum omnibus quadratis figuræ, ABC, dupla eſſe omnium quadratorum <lb/>figuræ, AID, cum omnibus quadratis figuræ, ACI, intermediæ.</s> <s xml:id="echoid-s3749" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div400" type="section" level="1" n="244"> <head xml:id="echoid-head259" xml:space="preserve">K. SECTIO X.</head> <p style="it"> <s xml:id="echoid-s3750" xml:space="preserve">_I_V xta decimam, ſi ſupponamus, AC, lineam bifariam ſecare omnes <lb/>æquidiſtantes ipſi, BD, in figura, ABI, ductas, & </s> <s xml:id="echoid-s3751" xml:space="preserve">illis addi, quæ <lb/>in directum illas iacent in figura, AID; </s> <s xml:id="echoid-s3752" xml:space="preserve">colligemus omnia quadrata fi-<lb/>guræ, ABCD, cum omnibus quadratis figuræ, ADI, dupla eſſe om-<lb/>nium quadratorum figuræ, ABC, cum omnibus quadratis figuræ, ACD.</s> <s xml:id="echoid-s3753" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div401" type="section" level="1" n="245"> <head xml:id="echoid-head260" xml:space="preserve">L. SECTIO XI.</head> <p style="it"> <s xml:id="echoid-s3754" xml:space="preserve">_I_V xta vndecimam, ſi ſupponamus, BD, in, E, itaſectam eſſe, vtre-<lb/>ctangulum, DBE, ſit æquate quadrato, ED, quælibet autem æqui-<lb/>diſtantium ipſi, BD, in figura, ABCD, tali modo, & </s> <s xml:id="echoid-s3755" xml:space="preserve">ad eandem par-<lb/>tem diuid itur per lineam, AEC, patet, quod etiam rectangula ſub fi-<lb/>guris, ABCD, ABC, æquabuntur omnibus quadratis figuræ, ACD, <lb/>regula, BD, igitur linea, AC, diuidet ſuperſiciem planam, ABCD, <lb/>(ſic dicere liceat) ſecundum extremam, ac mediam rationem, hæc au-<lb/>tempro ſequentibus accuratè memoriæ commendetur.</s> <s xml:id="echoid-s3756" xml:space="preserve"/> </p> <pb o="159" file="0179" n="179" rhead="LIBER II."/> </div> <div xml:id="echoid-div402" type="section" level="1" n="246"> <head xml:id="echoid-head261" xml:space="preserve">THEOREMA XXIV. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s3757" xml:space="preserve">EXpoſito parallelogrammo quocunq; </s> <s xml:id="echoid-s3758" xml:space="preserve">in eoque ducta dia-<lb/>metro; </s> <s xml:id="echoid-s3759" xml:space="preserve">omnia quadrata parallelogrammiad omnia qua-<lb/>drata cuiuſuis triangulorum per dictam diametrum conſtitu-<lb/>torum erunt in ratione tripla, vno laterum parallelogrammi <lb/>communiregula exiſtente.</s> <s xml:id="echoid-s3760" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3761" xml:space="preserve">Sit parallelogrammum, AG, in eo ducta diameter, CE, regula <lb/>vtcunque latus, EG. </s> <s xml:id="echoid-s3762" xml:space="preserve">Dico omnia quadrata, AG, eſſe tripla om-<lb/>nium quadratorum trianguli cuiuſuis, AEC, ſiue, CEG. </s> <s xml:id="echoid-s3763" xml:space="preserve">Diui-<lb/>dantur bifariam latera, AC, CG, in punctis, B, H, & </s> <s xml:id="echoid-s3764" xml:space="preserve">per, B, ip-<lb/>ſi, CG, perque, H, ipſi, CA, parallelę ducantur, BF, DH, quę <lb/>ſe cum recta, CE, communiter bifariam ſecabuntin puncto, M. <lb/></s> <s xml:id="echoid-s3765" xml:space="preserve">Quia igitur in figura, ſiue parallelogrammo, AG, ducitur linea, B <lb/>F, quę omnes æquidiſtantes ipſi, EG, bifariam ſecat, &</s> <s xml:id="echoid-s3766" xml:space="preserve">, CE, quæ <lb/> <anchor type="figure" xlink:label="fig-0179-01a" xlink:href="fig-0179-01"/> eaſdem in partes inæquales diuidit, pręter-<lb/>quam, DH, omnia quadrata trianguli, A <lb/> <anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/> EC, cum omnibus quadratis trianguli, C <lb/>EG, & </s> <s xml:id="echoid-s3767" xml:space="preserve">cum omnibus quadratis duorum <lb/> <anchor type="note" xlink:label="note-0179-02a" xlink:href="note-0179-02"/> triangulorum, CBM, EMF, dupla erunt <lb/>omnium quadratorum, AF, licet enim, D <lb/>H, perlineam, CE, fit non bifariam diui-<lb/>ſa, nihil tamen hoc obſtat noſtro propoſi-<lb/>to, nam & </s> <s xml:id="echoid-s3768" xml:space="preserve">ipſi, DH, contingit, veluti ijs, <lb/>quæ inæqualiter ſecantur, quadratum ſe-<lb/>ctarum partium, ſcilicet quadrata, DM, <lb/>MH, dupla eſſe quadratorum dimidiæ, nempè quadrati, DM, & </s> <s xml:id="echoid-s3769" xml:space="preserve"><lb/>eius, quæ inter ſectiones interijcitur, quæ hic nulla eſt, cum duę ſe-<lb/>cantes, BF, CE, vniantur in puncto, M: </s> <s xml:id="echoid-s3770" xml:space="preserve">Sunt autem omnia qua-<lb/>drata trianguli, AEC, æqualia omnibus quadratis trianguli, CE <lb/>G, quia ſunt triangula in æqualibus baſibus, EG, AC, & </s> <s xml:id="echoid-s3771" xml:space="preserve">eadem al-<lb/> <anchor type="note" xlink:label="note-0179-03a" xlink:href="note-0179-03"/> titudine licet euersè poſita, & </s> <s xml:id="echoid-s3772" xml:space="preserve">ideò omnia quadrata trianguli, CE <lb/>G, ſunt æqualia omnibus quadratis, AF, cum omnibus quadratis <lb/>triangulorum, CBM, MEF. </s> <s xml:id="echoid-s3773" xml:space="preserve">Quoniam verò omnia quadrata tri-<lb/>anguli, BMC, funt æqualia omnibus quadratis trianguli, CMH, <lb/>omnia verò quadrata trianguli, CEG, ad omnia quadrata triangu-<lb/>li, CMH, ſunt in tripla ratione eius, quam habet, GC, ad, CH, <lb/>quæ eſt dupla .</s> <s xml:id="echoid-s3774" xml:space="preserve">i. </s> <s xml:id="echoid-s3775" xml:space="preserve">in ratione octupla, & </s> <s xml:id="echoid-s3776" xml:space="preserve">hoc, quia triangula, CEG, <lb/>CMH, ſunt ſimilia, ideò omnia quadrata, CEG, erunt octupla <pb o="160" file="0180" n="180" rhead="GEOMETRIÆ"/> omnium quadratorum, CMH, & </s> <s xml:id="echoid-s3777" xml:space="preserve">quadrupla omnium quadrato-<lb/>rum, CMH, vel, CBM, &</s> <s xml:id="echoid-s3778" xml:space="preserve">, MEF, ſunt autem omnia quadrata <lb/>trianguli, CEG, æqualia omnibus quadratis, AF, cum omnibus <lb/>quadratis triangulorum, CBM, MEF, ergo hæc erunt quadrupla <lb/>omnium quadratorum triangulorum, CBM, MEF, & </s> <s xml:id="echoid-s3779" xml:space="preserve">diuidendo <lb/> <anchor type="figure" xlink:label="fig-0180-01a" xlink:href="fig-0180-01"/> omnia quadrata, AF, eruntillorum tripla, <lb/> <anchor type="note" xlink:label="note-0180-01a" xlink:href="note-0180-01"/> ſunt autem omnia quadrata, AG, ad om-<lb/>nia quadrata, AF, vt quadratum, GE, ad <lb/>quadratum, EF, ideſt quadrupla .</s> <s xml:id="echoid-s3780" xml:space="preserve">i. </s> <s xml:id="echoid-s3781" xml:space="preserve">vt 12. <lb/></s> <s xml:id="echoid-s3782" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s3783" xml:space="preserve">& </s> <s xml:id="echoid-s3784" xml:space="preserve">omnia quadrata, AF, ſunt omnium <lb/>quadratorum triangulorum, BMC, ME <lb/>F, tripla, ergo omnia quadrata, AG, e-<lb/>runt duodecupla omnium quadratorum <lb/>triangulorum, BMC, MEF, & </s> <s xml:id="echoid-s3785" xml:space="preserve">ſunt ad <lb/>omnia quadrata, AF, vt 12. </s> <s xml:id="echoid-s3786" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s3787" xml:space="preserve">ergo om-<lb/>nia quadrata, AG, ad omnia quadrata, A <lb/>F, cum omnibus quadratis triangulorum, CBM, MEF, erunt vt <lb/>12. </s> <s xml:id="echoid-s3788" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s3789" xml:space="preserve">ſunt autem omnia quadrata, AF. </s> <s xml:id="echoid-s3790" xml:space="preserve">cum omnibus quadra-<lb/>tis triangulorum, CBM, MEF, æqualia omnibus quadratis trian-<lb/>guli, CEG, vel, AEC, vt oſtenſum eſt, ergo omnia quadrata, A <lb/>G, ad omnia quadrata trianguli, CEG, vel, AEC, ſunt vt 12. </s> <s xml:id="echoid-s3791" xml:space="preserve"><lb/>ad 4. </s> <s xml:id="echoid-s3792" xml:space="preserve">.</s> <s xml:id="echoid-s3793" xml:space="preserve">i. </s> <s xml:id="echoid-s3794" xml:space="preserve">ſunt eorum tripla. </s> <s xml:id="echoid-s3795" xml:space="preserve">quod oſtendendum erat.</s> <s xml:id="echoid-s3796" xml:space="preserve"/> </p> <div xml:id="echoid-div402" type="float" level="2" n="1"> <figure xlink:label="fig-0179-01" xlink:href="fig-0179-01a"> <image file="0179-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0179-01"/> </figure> <note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">Per I. Co-<lb/>rol. antec.</note> <note position="right" xlink:label="note-0179-02" xlink:href="note-0179-02a" xml:space="preserve">Vide D. <lb/>lib. 7. An-<lb/>not. Pro-<lb/>pofit. 8.</note> <note position="right" xlink:label="note-0179-03" xlink:href="note-0179-03a" xml:space="preserve">Ex B. vel <lb/>C. Corol. <lb/>Prop. 22. <lb/>huius.</note> <figure xlink:label="fig-0180-01" xlink:href="fig-0180-01a"> <image file="0180-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0180-01"/> </figure> <note position="left" xlink:label="note-0180-01" xlink:href="note-0180-01a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div404" type="section" level="1" n="247"> <head xml:id="echoid-head262" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s3797" xml:space="preserve">_H_Inc patet, ſi ducamus intra parallelogrammum, AG, æquidiftan-<lb/>tem ipſi, EG, vtcunque, RV, ſec antem, CE, in, T, &</s> <s xml:id="echoid-s3798" xml:space="preserve">, BF, <lb/>in, S, quod veluti oſtendimus, RV, æquari vni maximarum abſciſſarum. <lb/></s> <s xml:id="echoid-s3799" xml:space="preserve">CG, dum, EG, eſt æqualis ipſi, GC, ita namc oſtendemus quadratum, <lb/>RV, æquari quadrato vnius maxim trum abſciſſarum, CG, & </s> <s xml:id="echoid-s3800" xml:space="preserve">quadra-<lb/>tum, TV, æquari quadrato vnius omnium abſciſſarum, CG, ideſt qua-<lb/>drato, VC; </s> <s xml:id="echoid-s3801" xml:space="preserve">quadratum verò, RT, æquari quadrato @nius reſiduarum <lb/>omnium abſciſſirum, CG, ideſt quadrato, VG, vnde concludemus om-<lb/>nia quadrata, AG, regula, EG, æquari quadratis maximarum abſciſ-<lb/>ſarum, CG, & </s> <s xml:id="echoid-s3802" xml:space="preserve">omnia quadrata triangult, CEG, æquari quadratis om-<lb/>nium abſciſſarum, CG, & </s> <s xml:id="echoid-s3803" xml:space="preserve">omnia quadrata trianguli, AEC, æquari <lb/>quadratis reſiduarum omnium abſciſſarum, CG, & </s> <s xml:id="echoid-s3804" xml:space="preserve">rectangula ſub tri-<lb/>angulis, AEC, CEG, æquari rectangu is ſub omnibus abſctſſis, & </s> <s xml:id="echoid-s3805" xml:space="preserve">re-<lb/>ſiduis omnium abſciſſarum, CG, ita ſumptis, vt quoduts rectangulum <lb/>intelligatur ſub vna abſciſſirum, & </s> <s xml:id="echoid-s3806" xml:space="preserve">eius reſidua: </s> <s xml:id="echoid-s3807" xml:space="preserve">Vnde veluti oſtendi-<lb/>mus omnia quadrata, AG, tripla eſſe omnium quadratorum trianguli, <pb o="161" file="0181" n="181" rhead="LIBER II."/> CEG, veltrianguli, CAE, ex quo patet tripla etiam eſſe rectangulo-<lb/>rum bis ſub triangulis, AEC, CEG, (ſunt enim omnia quadrata, AG, <lb/>æqualia omnibus quadratis triangulorum, AEC, CEG, & </s> <s xml:id="echoid-s3808" xml:space="preserve">rectangulis <lb/> <anchor type="note" xlink:label="note-0181-01a" xlink:href="note-0181-01"/> bis ſub eiſdem triangulis) ita apparebit quadrata maximarum abſciſſa-<lb/>rum, C G, tripla eſſe quadratorum omnium abſciſſarum, bel quadrato-<lb/>rum reſiduarum omnium abſciſſarum, CG, & </s> <s xml:id="echoid-s3809" xml:space="preserve">tripla etiam eſſe rectan-<lb/>gulorum fub dictis omnibus abſciſſis, reſiduiſque bis ſumptis, ſexcupla <lb/>berò eorundem rectangulorum ſemel ſumptorum, ſunt autem maximæ <lb/>abſciſſarum, abſciſſæ, & </s> <s xml:id="echoid-s3810" xml:space="preserve">reſiduærecti tranſitus ſi angulus, EGC, ſitre-<lb/> <anchor type="note" xlink:label="note-0181-02a" xlink:href="note-0181-02"/> ctus, vel eiuſdem obliquitranſitus, ſi ille non ſit angulus rectus.</s> <s xml:id="echoid-s3811" xml:space="preserve"/> </p> <div xml:id="echoid-div404" type="float" level="2" n="1"> <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve">_D. Corol._ <lb/>_23. huius._</note> <note position="right" xlink:label="note-0181-02" xlink:href="note-0181-02a" xml:space="preserve">_Ex diff._ <lb/>_huius._</note> </div> </div> <div xml:id="echoid-div406" type="section" level="1" n="248"> <head xml:id="echoid-head263" xml:space="preserve">THEOREMA XXV. PROPOS. XXV.</head> <p> <s xml:id="echoid-s3812" xml:space="preserve">SI in duobus parallelogrammis ſumptis duobus lateribus <lb/>pro baſibus, & </s> <s xml:id="echoid-s3813" xml:space="preserve">regulis, ipſa parallelogramma fuerint in <lb/>eadem altitudine ſumpta reſpectu dictarum baſium; </s> <s xml:id="echoid-s3814" xml:space="preserve">in ei-<lb/>ſdem autem baſibus, & </s> <s xml:id="echoid-s3815" xml:space="preserve">altitudine fuerint aliæ duæ planæ fi-<lb/>guræ ita ſe habentes, vt ſi ducatur vtcunque parallela dictis <lb/>baſibus (quæ in directum ſint conſtitutæ) recta linea, eiu-<lb/>ſdem portiones dictis parallelogrammis, & </s> <s xml:id="echoid-s3816" xml:space="preserve">figuris interce-<lb/>ptæ, vel abeiſdem deſcriptę planæ figuræ ſint proportiona-<lb/>les, homologis exiſtentibus, quæ ſunt in parallelogrammis, <lb/>& </s> <s xml:id="echoid-s3817" xml:space="preserve">pariter quę ſunt in figuris, in ijſdem baſibus, & </s> <s xml:id="echoid-s3818" xml:space="preserve">altitudine <lb/>cum illis conſtitutis, dictorum parallelogrammorum, ac fi-<lb/>gurarum omnes lineæ, ſi lineæ, vel omnes figurę planę ſimi-<lb/>les, ſi iſtæ comparentur (fimiles in quam exiſtentibus, quæ <lb/>ſunt in vnaquaque figura) erunt proportionales.</s> <s xml:id="echoid-s3819" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3820" xml:space="preserve">Sint parallelogramma, AE, <lb/> <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> ED, in baſibus, CE, EF, in <lb/>directum iacentibus, & </s> <s xml:id="echoid-s3821" xml:space="preserve">in eadem <lb/>altitudine reſpectu dictarum ba-<lb/>ſium conſtituta, AE, ED, ſit <lb/>autem regula, CE, vel, EF, & </s> <s xml:id="echoid-s3822" xml:space="preserve"><lb/>in eiuſdem tanquam in baſibus, <lb/>& </s> <s xml:id="echoid-s3823" xml:space="preserve">eadem altitudine cum paral. <lb/></s> <s xml:id="echoid-s3824" xml:space="preserve">lelogrammis, AE, ED, ſint fi-<lb/>guræ, BCE, BEF, eiuſmodi, vt ſi duxerimus vtcunqueipſi, CF, <lb/>parallelam, vt, MQ, cuius portiones interceptę parallelogrammis, <pb o="162" file="0182" n="182" rhead="GEOMETRIÆ"/> AE, ED, ſint, MO, OQ, & </s> <s xml:id="echoid-s3825" xml:space="preserve">interceptę figuris ſint, IO, OP, le-<lb/>periamus, MO, ad, OI, eſſe vt, QO, ad, OP. </s> <s xml:id="echoid-s3826" xml:space="preserve">Dico omnes li-<lb/>neas, AE, ad omnes lineas figurę, BCE, eſſe vt omnes lineę, BF, <lb/>ad omnes lineas figuræ, B E F, ſi verò vice linearum comparentur <lb/> <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> ab eiſdem deſcriptę figurę, ſimi-<lb/>libus exiſtentibus, quę ab omni-<lb/>bus lineis vniuſcuiuſque propo-<lb/>ſitarum figurarum deſcribuntur, <lb/>cuius deſcribentes ſint earum li-<lb/>neę, vel latera homologa. </s> <s xml:id="echoid-s3827" xml:space="preserve">Di-<lb/>co omnes figuras ſimiles ipſius, <lb/>A E, ad omnes figuras ſimiles <lb/>ſigurę, BCE, eſſe vt omnes fi-<lb/>guras ſimiles ipſius, BF, ad omnes figuras ſimiles figuræ, BEF, <lb/>quia enim, MQ, vtcunque ducta eſt parallela ipſi, CF, & </s> <s xml:id="echoid-s3828" xml:space="preserve">eſt, M <lb/>O, ad, OI, vt, QO, ad, OP, permutando erit, vt, MO, ad, O <lb/>Q, ſic, IO, ad, OP, i. </s> <s xml:id="echoid-s3829" xml:space="preserve">vt, CE, ad, EF, ſic, IO, ad, OP, & </s> <s xml:id="echoid-s3830" xml:space="preserve">ſic <lb/>oſtendemus, vt, CE, ad, EF, ita eſſe quaſlibet alias duas in figu-<lb/>ris, BCE, BEF, exiſtentes ipſi, CF, parallelas, & </s> <s xml:id="echoid-s3831" xml:space="preserve">vt vna ad vnam <lb/>ſic omnia ad omnia .</s> <s xml:id="echoid-s3832" xml:space="preserve">i. </s> <s xml:id="echoid-s3833" xml:space="preserve">vt, CE, ad, EF, ita omnes lineæ figurę, B <lb/> <anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/> CE, ad omnes lineas figurę, BEF, vt autem, CE, ad, EF, ita ſunt <lb/>omnes lineæ, AE, ad omnes lineas, ED, ergo omnes lineę, AE, <lb/>ad omnes lineas, ED, erunt vt omnes lineę figurę, BCE, ad om-<lb/>nes lineas figuræ, BEF.</s> <s xml:id="echoid-s3834" xml:space="preserve"/> </p> <div xml:id="echoid-div406" type="float" level="2" n="1"> <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a"> <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0181-01"/> </figure> <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a"> <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0182-01"/> </figure> <note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">Coroll. 4. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s3835" xml:space="preserve">Si verò vice linearum ſumamus deſcriptas, vt dictum eſt, ab eiſdem <lb/>figuras, ex. </s> <s xml:id="echoid-s3836" xml:space="preserve">gr. </s> <s xml:id="echoid-s3837" xml:space="preserve">ſi, vt quadratum, MO, ad triangulum &</s> <s xml:id="echoid-s3838" xml:space="preserve">ae;</s> <s xml:id="echoid-s3839" xml:space="preserve">quilaterum, <lb/>cuius latus, IO, ita reperiamus eſſe circulum, cuius diameter, OQ, <lb/>ad polygonum, cuius latus, OQ, omnium autem linearum, AE, <lb/>fingulæ deſcribant quadrata, & </s> <s xml:id="echoid-s3840" xml:space="preserve">omnium linearum figuræ, BCE, <lb/>ſingulę deſcribant, triangula &</s> <s xml:id="echoid-s3841" xml:space="preserve">ae;</s> <s xml:id="echoid-s3842" xml:space="preserve">quilatera, & </s> <s xml:id="echoid-s3843" xml:space="preserve">omnium linearum, BF, <lb/>ſingulæ deſcribant circulos, & </s> <s xml:id="echoid-s3844" xml:space="preserve">figuræ, B E F, ſingulę deſcribant po-<lb/>lygona prædicto ſimilia, ita vt quæ in eadem figura ſunt lineæ, vel <lb/>latera deſcribentia ſint homologa, erit vt quadratum, MO, permu-<lb/>tando, ad circulum, OQ, ita triangulum &</s> <s xml:id="echoid-s3845" xml:space="preserve">ae;</s> <s xml:id="echoid-s3846" xml:space="preserve">quilaterum, IO, ad po-<lb/>lygonum, OP, quia verò, MO, &</s> <s xml:id="echoid-s3847" xml:space="preserve">ae;</s> <s xml:id="echoid-s3848" xml:space="preserve">quaturipſi, CE, &</s> <s xml:id="echoid-s3849" xml:space="preserve">, OQ, ipſi, <lb/>EF, ideò quadratum, MO, &</s> <s xml:id="echoid-s3850" xml:space="preserve">ae;</s> <s xml:id="echoid-s3851" xml:space="preserve">quatur quadrato, CE, & </s> <s xml:id="echoid-s3852" xml:space="preserve">circulus, <lb/>OQ, circulo, EF, & </s> <s xml:id="echoid-s3853" xml:space="preserve">ideò, vt quadratum, CE, ad circulum, EF, <lb/> <anchor type="note" xlink:label="note-0182-02a" xlink:href="note-0182-02"/> ita erit triangulum &</s> <s xml:id="echoid-s3854" xml:space="preserve">ae;</s> <s xml:id="echoid-s3855" xml:space="preserve">quilaterum, IO, ad polygonum, OP, vnde, <lb/>quia, MQ, vtcunq; </s> <s xml:id="echoid-s3856" xml:space="preserve">ducta eſt parallelaipſi, CF, concludemus om-<lb/>nia quadrata, AE, ad omnes circulos, BF, eſſe, vt omnia triangu-<lb/>la &</s> <s xml:id="echoid-s3857" xml:space="preserve">ae;</s> <s xml:id="echoid-s3858" xml:space="preserve">quilatera figuræ, BCE, ad omnia polygona vni ſimilia figuræ, <lb/> <anchor type="note" xlink:label="note-0182-03a" xlink:href="note-0182-03"/> BEF, & </s> <s xml:id="echoid-s3859" xml:space="preserve">permutando omnia quadrata, AE, ad omnia triangula a- <pb o="163" file="0183" n="183" rhead="LIBER II."/> quilatera figuræ, BCE, eſſe, vt omnes circuli, BF, ad omnia po-<lb/>lygona vniſimilia figuræ, BEF.</s> <s xml:id="echoid-s3860" xml:space="preserve"/> </p> <div xml:id="echoid-div407" type="float" level="2" n="2"> <note position="left" xlink:label="note-0182-02" xlink:href="note-0182-02a" xml:space="preserve">@5. lib. 1.</note> <note position="left" xlink:label="note-0182-03" xlink:href="note-0182-03a" xml:space="preserve">Ex 4. hu-<lb/>ius.</note> </div> <p> <s xml:id="echoid-s3861" xml:space="preserve">Eodem modo fiet demonſtratio, ſi vice iſtarum aliæ aſſumantur <lb/>figuræ planæ, quarum poſſunt etiam, quæ ſunt duarum figurarum <lb/>eſſe ſimiles, vt ſi comparentur omnia quadrata parallelogrammo-<lb/>rum, AE, ED, & </s> <s xml:id="echoid-s3862" xml:space="preserve">omnia triangula &</s> <s xml:id="echoid-s3863" xml:space="preserve">ae;</s> <s xml:id="echoid-s3864" xml:space="preserve">quilatera figurarum, BCE, <lb/>BEF, vel ſi comparentur omnia quadrata, AE, & </s> <s xml:id="echoid-s3865" xml:space="preserve">figuræ, BCE, <lb/>& </s> <s xml:id="echoid-s3866" xml:space="preserve">omnia triangula &</s> <s xml:id="echoid-s3867" xml:space="preserve">ae;</s> <s xml:id="echoid-s3868" xml:space="preserve">quilatera, BF, & </s> <s xml:id="echoid-s3869" xml:space="preserve">figuræ, B E F; </s> <s xml:id="echoid-s3870" xml:space="preserve">poteſt etiam <lb/>eſſe omnium quatuor figurarum omnes figuras eſſe ſimiles, vt ſi com-<lb/>parentur omnia quadrata eorundem, vel omnes circuli, &</s> <s xml:id="echoid-s3871" xml:space="preserve">c. </s> <s xml:id="echoid-s3872" xml:space="preserve">patet <lb/>autem hic demonſtrationem currere quotieſconque ea, quæ compa-<lb/>rantur ſunt eiuſdem generis .</s> <s xml:id="echoid-s3873" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3874" xml:space="preserve">vel lineæ, vel ſuperficies, ſi verò con-<lb/>tingat magnitudines diuerſi generis comparari, vt ſi compararentur <lb/>omnes lineæ, AE, & </s> <s xml:id="echoid-s3875" xml:space="preserve">figuræ, BCE, & </s> <s xml:id="echoid-s3876" xml:space="preserve">omnia quadrata, BF, & </s> <s xml:id="echoid-s3877" xml:space="preserve">fi-<lb/>gurę, BEF, tunc quia &</s> <s xml:id="echoid-s3878" xml:space="preserve">a4; </s> <s xml:id="echoid-s3879" xml:space="preserve">permutata ratione non poſſumus argumen-<lb/>tari, cum lineam ſuperficiei comparare ſit abſurdum, ideò demon-<lb/>ſtratio pro his non currit, quapropter aliud Theorema pro hoc ſut-<lb/>iungemus.</s> <s xml:id="echoid-s3880" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div409" type="section" level="1" n="249"> <head xml:id="echoid-head264" xml:space="preserve">THE OREMA XXVI. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s3881" xml:space="preserve">IN eadem antecedentis Propoſ. </s> <s xml:id="echoid-s3882" xml:space="preserve">figura ſi comparentur ma-<lb/>gnitudines diuerſi generis, adhuc comparatæ magnitu-<lb/>dines erunt proportionales.</s> <s xml:id="echoid-s3883" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3884" xml:space="preserve">Comparentur ex. </s> <s xml:id="echoid-s3885" xml:space="preserve">gr. <lb/></s> <s xml:id="echoid-s3886" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> omnes lineæ, AE, re-<lb/>gula, CE, ad omnes li-<lb/>neas fi uræ, BCE, & </s> <s xml:id="echoid-s3887" xml:space="preserve"><lb/>omnia quadrata, BF, <lb/>regula, EF, ad omnia <lb/>quadrata figurę, BEF, <lb/>ita vt ducta vtcunq. </s> <s xml:id="echoid-s3888" xml:space="preserve">ipſi, <lb/>CF, paraliela, MQ, <lb/>reperiamus, MO, ad, <lb/>OI, eſſe vt quadratum, <lb/>QO, ad quadratum, O <lb/>P. </s> <s xml:id="echoid-s3889" xml:space="preserve">Dieo adhuc omnes <lb/>lineas, AE, ad omnes <lb/>lineas figurę, BCE, eſ-<lb/>ſe vt omnia quadrata, B <lb/>F, ad omnia quadrata figurę, BEF; </s> <s xml:id="echoid-s3890" xml:space="preserve">ponatur ſeorſim parallelogram- <pb o="164" file="0184" n="184" rhead="GEOMETRIÆ"/> mum, AE, ſimul cum figura, BCE, ſed, ne fiat confuſio, ſint ſub <lb/> <anchor type="note" xlink:label="note-0184-01a" xlink:href="note-0184-01"/> ampliori forma, & </s> <s xml:id="echoid-s3891" xml:space="preserve">inipſis tanquam in baſibus conſtituti intelligan-<lb/>tur duo cylindrici recti, FE, nempè in baſi, AE, &</s> <s xml:id="echoid-s3892" xml:space="preserve">, DGE, in baſi <lb/>figura, BCE, & </s> <s xml:id="echoid-s3893" xml:space="preserve">in eadem altitudine, quorum quod inſiſtit ipſi, A <lb/>E, eſt parallelepipedum, vt facilè oſtendetur, intelligatur nunc pa-<lb/>rallelepipedum, FE, ſecari vtcunque plano ipſi, GE, &</s> <s xml:id="echoid-s3894" xml:space="preserve">ae;</s> <s xml:id="echoid-s3895" xml:space="preserve">quidiſtante, <lb/>producetut ergo ex hac ſectione in ipſo parallelogrammum rectan-<lb/> <anchor type="note" xlink:label="note-0184-02a" xlink:href="note-0184-02"/> gulum, quod ſit, KO, eodem autem plano fiat in cylindrico, DG <lb/>E, rectangulum, LO, fiet autem & </s> <s xml:id="echoid-s3896" xml:space="preserve">in hoc cylindrico rectangulum, <lb/>quia dictum planum ducitur per latera baſi, BCE, rectè inſiſten-<lb/> <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> tia, cum ducatur &</s> <s xml:id="echoid-s3897" xml:space="preserve">ae;</s> <s xml:id="echoid-s3898" xml:space="preserve">qui-<lb/>diſtanter ipſi, GE, quod <lb/>ducitur perlatera, GC, <lb/>SE, erit ergo rectange-<lb/>lum, KO, vnum ex ijs, <lb/>quę dicũtur omnia pla-<lb/>na parallelepipedi, FE, <lb/>regula, GE, & </s> <s xml:id="echoid-s3899" xml:space="preserve">rectan-<lb/>gulum, LO, erit vnum <lb/>ex ijs, quę dicuntur om-<lb/>nia plana cylindrici, G <lb/>DE, regula, GE, quę <lb/>rectangula erunt &</s> <s xml:id="echoid-s3900" xml:space="preserve">ae;</s> <s xml:id="echoid-s3901" xml:space="preserve">què <lb/>alta, ac rectangulum, <lb/>GE, omnia igitur pla-<lb/>na parallelepipedi, FE, <lb/>(regula, GE,) ſunt omnia rectangula &</s> <s xml:id="echoid-s3902" xml:space="preserve">ae;</s> <s xml:id="echoid-s3903" xml:space="preserve">què alta, ac, GE, ipſius pa-<lb/>rallelogrammi, AE, (regula, CE,) & </s> <s xml:id="echoid-s3904" xml:space="preserve">omnia plana cylindrici, G <lb/> <anchor type="note" xlink:label="note-0184-03a" xlink:href="note-0184-03"/> DE, ſunt omnia rectangula figuræ, BCE, &</s> <s xml:id="echoid-s3905" xml:space="preserve">ae;</s> <s xml:id="echoid-s3906" xml:space="preserve">quiangula, & </s> <s xml:id="echoid-s3907" xml:space="preserve">&</s> <s xml:id="echoid-s3908" xml:space="preserve">ae;</s> <s xml:id="echoid-s3909" xml:space="preserve">què <lb/>alta, ac ipſum, GE, regula eadem, CE: </s> <s xml:id="echoid-s3910" xml:space="preserve">Secentur nunc dicti cylin-<lb/>drici planis baſibus &</s> <s xml:id="echoid-s3911" xml:space="preserve">ae;</s> <s xml:id="echoid-s3912" xml:space="preserve">quidiſtantibus, fient ergo communes corum ſe-<lb/>ctiones ſimiles, & </s> <s xml:id="echoid-s3913" xml:space="preserve">&</s> <s xml:id="echoid-s3914" xml:space="preserve">ae;</s> <s xml:id="echoid-s3915" xml:space="preserve">quales baſibus, ſit in parallelepipedo, FE, pro-<lb/> <anchor type="note" xlink:label="note-0184-04a" xlink:href="note-0184-04"/> ducta, NP, & </s> <s xml:id="echoid-s3916" xml:space="preserve">in cylindrico, GDE, producta figura, HQP, erit <lb/>ergo vt, AE, ad figuram, BCE, ita, NP, ad figuram, HQP, & </s> <s xml:id="echoid-s3917" xml:space="preserve"><lb/>ita etiam quælibet alię figurę in ipſis per plana &</s> <s xml:id="echoid-s3918" xml:space="preserve">ae;</s> <s xml:id="echoid-s3919" xml:space="preserve">quidiſtanter baſibus <lb/>eoſdem ſecantia productæ, & </s> <s xml:id="echoid-s3920" xml:space="preserve">vt vna ad vnam, ſic omnes ad omnes <lb/> <anchor type="note" xlink:label="note-0184-05a" xlink:href="note-0184-05"/> .</s> <s xml:id="echoid-s3921" xml:space="preserve">i. </s> <s xml:id="echoid-s3922" xml:space="preserve">vt, AE, ad figuram, CBE, ita omnia plana parallelepipedi, F <lb/>E, regula, AE, ad omnia plana cylindrici, GDE, regula eadem <lb/> <anchor type="note" xlink:label="note-0184-06a" xlink:href="note-0184-06"/> baſi, ſunt autem omnia plana parallelepipedi, FE, regula, AE, &</s> <s xml:id="echoid-s3923" xml:space="preserve">ae;</s> <s xml:id="echoid-s3924" xml:space="preserve">-<lb/>qualia omnibus eiuſdem planis, regula, GE, quæ ſunt omnia re-<lb/> <anchor type="note" xlink:label="note-0184-07a" xlink:href="note-0184-07"/> ctangula ipſius, AE, regula, CE, &</s> <s xml:id="echoid-s3925" xml:space="preserve">ae;</s> <s xml:id="echoid-s3926" xml:space="preserve">què alta, acipſum, GF, & </s> <s xml:id="echoid-s3927" xml:space="preserve"><lb/>omnia plana cylindrici, GDE, regula baſi, CBE, ſunt &</s> <s xml:id="echoid-s3928" xml:space="preserve">ae;</s> <s xml:id="echoid-s3929" xml:space="preserve">qualia om- <pb o="165" file="0185" n="185" rhead="LIBER II."/> nibus eiuſdem planis, regula, GE, quæ & </s> <s xml:id="echoid-s3930" xml:space="preserve">ipſa ſunt omnia rectan-<lb/> <anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/> gula figuræ, CBE, regula, CE, &</s> <s xml:id="echoid-s3931" xml:space="preserve">ae;</s> <s xml:id="echoid-s3932" xml:space="preserve">què alta, acipſum, GE, ergo <lb/>omnia rectangula ipſius, AE, regula, CE, &</s> <s xml:id="echoid-s3933" xml:space="preserve">ae;</s> <s xml:id="echoid-s3934" xml:space="preserve">què alta, acipſum, G <lb/>E, ad omnia rectangula figuræ, CBE, regula, CE, &</s> <s xml:id="echoid-s3935" xml:space="preserve">ae;</s> <s xml:id="echoid-s3936" xml:space="preserve">què alta, ac <lb/> <anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/> ipſum, GE, erunt vt, AE, ad figuram, BCE, .</s> <s xml:id="echoid-s3937" xml:space="preserve">ſ. </s> <s xml:id="echoid-s3938" xml:space="preserve">vt omnes lineę, <lb/>AE, ad omnes lineas, BCE, regula, CE, quod ſerua.</s> <s xml:id="echoid-s3939" xml:space="preserve"/> </p> <div xml:id="echoid-div409" type="float" level="2" n="1"> <figure xlink:label="fig-0183-01" xlink:href="fig-0183-01a"> <image file="0183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0183-01"/> </figure> <note position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve">B. Def. 4. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0184-02" xlink:href="note-0184-02a" xml:space="preserve">Coroll. 6. <lb/>lib. 1.</note> <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a"> <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0184-01"/> </figure> <note position="left" xlink:label="note-0184-03" xlink:href="note-0184-03a" xml:space="preserve">E. Def. 8. <lb/>huius.</note> <note position="left" xlink:label="note-0184-04" xlink:href="note-0184-04a" xml:space="preserve">Corol. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0184-05" xlink:href="note-0184-05a" xml:space="preserve">ExCorol. <lb/>A. huius.</note> <note position="left" xlink:label="note-0184-06" xlink:href="note-0184-06a" xml:space="preserve">ExCorol. <lb/>x. huius.</note> <note position="left" xlink:label="note-0184-07" xlink:href="note-0184-07a" xml:space="preserve">E. Def. 8. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">ExCor. 2. <lb/>huius.</note> <note position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">3. huius.</note> </div> <p> <s xml:id="echoid-s3940" xml:space="preserve">Conſpiciatur nunc figura Theorematis anteced. </s> <s xml:id="echoid-s3941" xml:space="preserve">in qua diximus, <lb/>MO, ad, OI, eſſe vt quadratum, QO, ad quadratum, OP. </s> <s xml:id="echoid-s3942" xml:space="preserve">Di-<lb/>co omnes lineas, AE, ad omnes lineas figurę, BCE, regula, CE, <lb/>eſſe vt omnia quadrata, BF, ad omnia quadrata figurę, B E F, quia <lb/>enim, vt, MO, ad, OI, ita (ſumpta quauis communi altitudine, <lb/>nempè ex. </s> <s xml:id="echoid-s3943" xml:space="preserve">gr. </s> <s xml:id="echoid-s3944" xml:space="preserve">altitudine conſtitutorum parallelepipedorum, quę eſt, <lb/>SE,) rectangulum ſub, MO, &</s> <s xml:id="echoid-s3945" xml:space="preserve">, SE, ad rectangulum ſub, IO, S <lb/>E, ideò, vt rectangulum ſub, MO, SE, ad rectangulum ſub, IO, <lb/>SE, ita erit quadratum, OQ, ad quadratum, OP, ſunt autem hæ <lb/>magnitudines eiuſdem generis, nempè omnes ſuperficies, ergo om. <lb/></s> <s xml:id="echoid-s3946" xml:space="preserve">nia rectangulaipſius, AE, regula, CE, &</s> <s xml:id="echoid-s3947" xml:space="preserve">ae;</s> <s xml:id="echoid-s3948" xml:space="preserve">què alta, ac vnum eorum, <lb/> <anchor type="note" xlink:label="note-0185-03a" xlink:href="note-0185-03"/> nempè, vt rectangulum ſub, CE, ES, ad omnia rectangula figurę, <lb/>BCE, regula eadem, CE, &</s> <s xml:id="echoid-s3949" xml:space="preserve">ae;</s> <s xml:id="echoid-s3950" xml:space="preserve">què alta, ac vnum eorum, vt, GE, <lb/>erunt vt omnia quadrata, BF, ad omnia quadrata figuræ, BEF, <lb/>omnia verò rectangulaipſius, AE, &</s> <s xml:id="echoid-s3951" xml:space="preserve">ae;</s> <s xml:id="echoid-s3952" xml:space="preserve">què alta, ac vnum eorum, vt, <lb/>GE, ad omnia rectangula figuræ, BCE, &</s> <s xml:id="echoid-s3953" xml:space="preserve">ae;</s> <s xml:id="echoid-s3954" xml:space="preserve">què alta, acipſum, G <lb/>E, ſunt vt omnes lineæ ipſius, AE, ad omnes lineas figuræ, BCE, <lb/> <anchor type="note" xlink:label="note-0185-04a" xlink:href="note-0185-04"/> regula, CE, ergo omnes lineæ, AE, ad omnes lineas figuræ, BC <lb/>E, regula, CE, erunt vt omnia quadrata, BF, ad omnia quadrata <lb/>figuræ, BEF, ſunt ergo proportionales, licet ſint magnitudines di-<lb/>uerſi generis, nempè lineę, & </s> <s xml:id="echoid-s3955" xml:space="preserve">ſuperficies, quod oſtendere opus erat.</s> <s xml:id="echoid-s3956" xml:space="preserve"/> </p> <div xml:id="echoid-div410" type="float" level="2" n="2"> <note position="right" xlink:label="note-0185-03" xlink:href="note-0185-03a" xml:space="preserve">Exantee.</note> <note position="right" xlink:label="note-0185-04" xlink:href="note-0185-04a" xml:space="preserve">Ex proxi-<lb/>mè dictis.</note> </div> </div> <div xml:id="echoid-div412" type="section" level="1" n="250"> <head xml:id="echoid-head265" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s3957" xml:space="preserve">_H_Inc igitur primò habetur, ſi fuerint parallel ogrammum, & </s> <s xml:id="echoid-s3958" xml:space="preserve">figurá <lb/>plana in eadem baſi, & </s> <s xml:id="echoid-s3959" xml:space="preserve">altitudine, regula ſumpta baſi, omnia, <lb/>rectangula parallelogrammi &</s> <s xml:id="echoid-s3960" xml:space="preserve">ae;</s> <s xml:id="echoid-s3961" xml:space="preserve">què alta ad omnia rectangula illius figu-<lb/>ræ &</s> <s xml:id="echoid-s3962" xml:space="preserve">ae;</s> <s xml:id="echoid-s3963" xml:space="preserve">què alta ac prædicta, eſſe vt dictum parallelogrammum ad dictam, <lb/>figuram, quod patuit, dum oſtenſum eſt omnia rectangulaipſius, AE, <lb/>altitudinis, SE, ad omnia rectangula figuræ, BCE, altitudinis eiuſdem, <lb/>SE, eſſe vt, AE, ad figuram, BCE.</s> <s xml:id="echoid-s3964" xml:space="preserve"/> </p> <pb o="166" file="0186" n="186" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div413" type="section" level="1" n="251"> <head xml:id="echoid-head266" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s3965" xml:space="preserve">_H_Abetur ſecundò cylindricos in eadem altitudine exiſtentes eſſe in-<lb/>terſe, vt baſes, quod de cæteris, veluti de ſupradictis, FE, GD <lb/>E, oſiendetur, quamuis aliter etiam id aliundè infra colligetur.</s> <s xml:id="echoid-s3966" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div414" type="section" level="1" n="252"> <head xml:id="echoid-head267" xml:space="preserve">COROLLARIVM III.</head> <p style="it"> <s xml:id="echoid-s3967" xml:space="preserve">_H_Abetur tertiò, ſi non ſint in ſupradictis duobus Theorematibus ex-<lb/>poſita duo parallelogramma, & </s> <s xml:id="echoid-s3968" xml:space="preserve">duæ figuræ, ſed vnum tantum, <lb/>& </s> <s xml:id="echoid-s3969" xml:space="preserve">vna figurain eadem baſi, & </s> <s xml:id="echoid-s3970" xml:space="preserve">altitudine cumipſo, cuius baſi poſita pro <lb/>regula, & </s> <s xml:id="echoid-s3971" xml:space="preserve">ſumpto vteunque puncto in vno laterum baſi inſi§tentium, <lb/>perque ipſum baſi ducta parallela, reperiatur eam, quæ intercipitur pa-<lb/>rallelogrammo ad eam, quæ intercibitur figura, vel figuras ſimiles ab <lb/>ipſis deſcriptas, tanquam homologis lineis, vel lateribus, eſſe vt vnam <lb/>ex maximis abſciſſarum lateris, in quo ſumptum eſt punctum, ad abſciſ-<lb/>ſam per ductam baſi &</s> <s xml:id="echoid-s3972" xml:space="preserve">ae;</s> <s xml:id="echoid-s3973" xml:space="preserve">quidiſtantem, vel vt iſtas adiuncta quadam recta, <lb/>linea, vel vt iſtarum figuras ſimiles ab ipſis tanquam lineis, vel lateri-<lb/>bus homologis deſcriptas, ita vt figuræ deſcriptæ &</s> <s xml:id="echoid-s3974" xml:space="preserve">a4; </s> <s xml:id="echoid-s3975" xml:space="preserve">ſingulis earum, quæ <lb/>dicuntur omnes lineæ parallelogrammi, & </s> <s xml:id="echoid-s3976" xml:space="preserve">dictæ figuræ, ſint ſimiles, vt <lb/>pariter, quæ deſcribuntur &</s> <s xml:id="echoid-s3977" xml:space="preserve">a4; </s> <s xml:id="echoid-s3978" xml:space="preserve">ſingulis earum, quæ dicuntur maximæ ab-<lb/>ſciſſarum, vel abſciſſæ dicti lateris, quod adbuc dictæ magnitudines col-<lb/>lectæ erunt proportionales: </s> <s xml:id="echoid-s3979" xml:space="preserve">Vt ex. </s> <s xml:id="echoid-s3980" xml:space="preserve">gr. </s> <s xml:id="echoid-s3981" xml:space="preserve">ſi in Theorematis antecedentis fi-<lb/>gura habeamus tantum parallelogrammum, BF, & </s> <s xml:id="echoid-s3982" xml:space="preserve">in eiuſdem baſi, E <lb/>F, & </s> <s xml:id="echoid-s3983" xml:space="preserve">eadem altitudine, figuram, BEF, & </s> <s xml:id="echoid-s3984" xml:space="preserve">ſumpto in vno laterum, B <lb/>E, DF, vtcunque puncto, O, & </s> <s xml:id="echoid-s3985" xml:space="preserve">per, O, ducta, OQ, parallela ipſi, E <lb/>F, reperiamus, QO, ad, OP, eſſe vt, EB, ad, BO, vel figuras ſimiles <lb/>deſcriptas ab, OQ, OP, tanquam lineis, vel lateribus homologis, vt <lb/>ex. </s> <s xml:id="echoid-s3986" xml:space="preserve">gr. </s> <s xml:id="echoid-s3987" xml:space="preserve">quadratum, QO, ad quadratum, OP, eſſe vt, EB, ad, BO, vel <lb/>vt, EB, adiuncta quadam linea ad, BO, adiuncta eadem, vel vt abiſtis <lb/>deſcriptas ſimiles figuras, dico collectas magnitudines, quæ comparan-<lb/>tur eſſe proportionales: </s> <s xml:id="echoid-s3988" xml:space="preserve">Nam ſi ipſi, BE, intelligatur applicatum pa-<lb/>rallelogrammum, AE, cuius baſis ſit, CE, in directum ipſi, EF, con-<lb/>ſtituta, &</s> <s xml:id="echoid-s3989" xml:space="preserve">, CE, &</s> <s xml:id="echoid-s3990" xml:space="preserve">ae;</s> <s xml:id="echoid-s3991" xml:space="preserve">qualis ipſi, EB, tunc omnes lineæ, AE, regula, C <lb/> <anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/> E, ſunt &</s> <s xml:id="echoid-s3992" xml:space="preserve">ae;</s> <s xml:id="echoid-s3993" xml:space="preserve">quales maximis abſciſſarum, BE, vt probatum eſt, & </s> <s xml:id="echoid-s3994" xml:space="preserve">omnes <lb/>abſciſſæ &</s> <s xml:id="echoid-s3995" xml:space="preserve">ae;</s> <s xml:id="echoid-s3996" xml:space="preserve">quales omnibus lineis trianguli BCE, ſi ſit iuncta, BC, (quæ <lb/>ſecet, MO, in, X,) vnde vice earum, quæ dicuntur maximæ abſciſſa-<lb/>rum, vel abſciſſæ ipſius, BE, rectè ſumemus omnes lineas, AE, & </s> <s xml:id="echoid-s3997" xml:space="preserve">tri-<lb/>anguli, BCE, & </s> <s xml:id="echoid-s3998" xml:space="preserve">itareperiemus quadratum, QO, ad quadratum, OP, <lb/>ex. </s> <s xml:id="echoid-s3999" xml:space="preserve">gr. </s> <s xml:id="echoid-s4000" xml:space="preserve">eſſe vt, MO, ad, OX, vel vt quadratum, M, O, ad quadratum, <pb o="167" file="0187" n="187" rhead="LIBER II."/> OX, vel vt aliæ figuræ ſimiles ab ipſis deſcriptæ, ſiue abipſis ſimplici-<lb/>bus, ſiue ab ipſis adiuncta quadam linea, vnde caſus iſte ad caſum Theo-<lb/>rematis præſentis, vel antecedentis deductus erit, & </s> <s xml:id="echoid-s4001" xml:space="preserve">ideò patebit, om-<lb/>nes lineas, AE, ad omnes lineas trianguli, BCE, vel omnes figuras ſi-<lb/>miles, AE, ad omnes figuras ſimiles trianguli, BCE, ideſt vel maxi-<lb/>mas abſciſſarum, BE, ad abſciſſas omnes ipſius, BE, vel earum figuras <lb/>ſimiles eſſe, vt omnia quadrata, B F, ad omnia quadrata figuræ, BEF. <lb/></s> <s xml:id="echoid-s4002" xml:space="preserve">Vocabuntur autem iſtæ; </s> <s xml:id="echoid-s4003" xml:space="preserve">Quatuor ordinum magnitudines collectæ iuxta <lb/>quatuor magnitudines proportionales vtcunque inuentas, quæ fuerunt <lb/>ex. </s> <s xml:id="echoid-s4004" xml:space="preserve">gr. </s> <s xml:id="echoid-s4005" xml:space="preserve">prima quadratum, OQ, ſecunda quadratum, OP, tertia, EB, <lb/>quarta, BO, magnitudines autem collecta iuxta primam. </s> <s xml:id="echoid-s4006" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4007" xml:space="preserve">ex. </s> <s xml:id="echoid-s4008" xml:space="preserve">gr. </s> <s xml:id="echoid-s4009" xml:space="preserve">om <lb/>nia quadrata, BF, dicentur primi ordinis, collectæ verò iuxta ſecun-<lb/>dam. </s> <s xml:id="echoid-s4010" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4011" xml:space="preserve">omnia quadrata figuræ, BEF, magnitudines ſecundi ordinis, col-<lb/>lectæ verò iuxta tertiam magnitudines tertij ordinis, & </s> <s xml:id="echoid-s4012" xml:space="preserve">tandem collectæ <lb/>iuxta quartam magnitudines quarti ordinis, ſic igitur appellabimus hos <lb/>quatuor magnitudinum ordines. </s> <s xml:id="echoid-s4013" xml:space="preserve">In ſupradictis autem, quod dicimus de <lb/>abſciſſis, idem intellige de reſiduis abſciſſarum, & </s> <s xml:id="echoid-s4014" xml:space="preserve">quod de ipſis ſimpli-<lb/>cibus, idem de eiſdem adiunctis alijs, ſiue ſint recti, ſiue eiuſdem obli-<lb/>qui tranſitus: </s> <s xml:id="echoid-s4015" xml:space="preserve">Hoc autem Corollarium præ cæteris ſummè animaduerten-<lb/>dumeſt, ac memoriæ diligentiſſimè commendandum, ex hoc enim potiſ-<lb/>ſimas demonſtrationes tanquam ex fonte dermari ſtudioſus in ſequen-<lb/>tium Librorum lectione ſacilè comprehendet.</s> <s xml:id="echoid-s4016" xml:space="preserve"/> </p> <div xml:id="echoid-div414" type="float" level="2" n="1"> <note position="left" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve">_Corol. 2._ <lb/>_19. huius._</note> </div> </div> <div xml:id="echoid-div416" type="section" level="1" n="253"> <head xml:id="echoid-head268" xml:space="preserve">THEOREMA XXVII. PROPOS. XXVII.</head> <p> <s xml:id="echoid-s4017" xml:space="preserve">SI duo trapezia fuerint in eadem baſi, ſumpto vnolate-<lb/>rum &</s> <s xml:id="echoid-s4018" xml:space="preserve">ae;</s> <s xml:id="echoid-s4019" xml:space="preserve">quidiſtantium pro baſi, & </s> <s xml:id="echoid-s4020" xml:space="preserve">regula, & </s> <s xml:id="echoid-s4021" xml:space="preserve">fuerint etiam <lb/>in eadem altitudine reſpectu illius baſis, & </s> <s xml:id="echoid-s4022" xml:space="preserve">latera baſi paral-<lb/>lela fuerint &</s> <s xml:id="echoid-s4023" xml:space="preserve">ae;</s> <s xml:id="echoid-s4024" xml:space="preserve">qualia, trapezia erunt &</s> <s xml:id="echoid-s4025" xml:space="preserve">ae;</s> <s xml:id="echoid-s4026" xml:space="preserve">qualia, & </s> <s xml:id="echoid-s4027" xml:space="preserve">omnia eo-<lb/>rundem quadrata erunt &</s> <s xml:id="echoid-s4028" xml:space="preserve">ae;</s> <s xml:id="echoid-s4029" xml:space="preserve">qualia.</s> <s xml:id="echoid-s4030" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4031" xml:space="preserve">Sint duo trapezia, AERB, IABD, in eadem baſi, AB, quæ <lb/>ſit ſumpta pro regula, cuiq; </s> <s xml:id="echoid-s4032" xml:space="preserve">latera, ER, ID, ſint parallela, & </s> <s xml:id="echoid-s4033" xml:space="preserve">in-<lb/>ter ſe &</s> <s xml:id="echoid-s4034" xml:space="preserve">ae;</s> <s xml:id="echoid-s4035" xml:space="preserve">qualia, Dico trapezia eſſe &</s> <s xml:id="echoid-s4036" xml:space="preserve">ae;</s> <s xml:id="echoid-s4037" xml:space="preserve">qualia, & </s> <s xml:id="echoid-s4038" xml:space="preserve">omnia eorundem qua-<lb/>drata eſſe &</s> <s xml:id="echoid-s4039" xml:space="preserve">ae;</s> <s xml:id="echoid-s4040" xml:space="preserve">qualia. </s> <s xml:id="echoid-s4041" xml:space="preserve">Producantur, AE, BR, donec ſibi occurrant, <lb/>vt in, O, &</s> <s xml:id="echoid-s4042" xml:space="preserve">, AI, BD, donec ſimul incidant, vt in, X, & </s> <s xml:id="echoid-s4043" xml:space="preserve">iunga-<lb/>tur, OX, quia ergo, ER, parallela eſt ipſi, AB, erunt triangula, <lb/> <anchor type="note" xlink:label="note-0187-01a" xlink:href="note-0187-01"/> AOB, EOR, ſimilia, & </s> <s xml:id="echoid-s4044" xml:space="preserve">eadem ratione ſimilia erunt triangula, A <lb/>XB, IXD, ergo vt, AB, ad, ER, velad, ID, illiæqualem, ita <lb/>erit, BO, ad, OR, vt autem, AB, ad, ID, ita eſt, BX, ad, XD, <lb/> <anchor type="note" xlink:label="note-0187-02a" xlink:href="note-0187-02"/> ergo vt, BO, ad, OR, ita eſt, BX, ad, XD, ergo, OX, paral- <pb o="168" file="0188" n="188" rhead="GEOMETRIÆ"/> lela eſt ipſi, ED. </s> <s xml:id="echoid-s4045" xml:space="preserve">Ducaturintra trapezia parallela ipſi, AB, vtcun-<lb/> <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> que, VC, ſecans, XA, in, S, &</s> <s xml:id="echoid-s4046" xml:space="preserve">, O <lb/>B, in, T, ſunt igitur triangula, AO <lb/>B, VOT, ſimilia, & </s> <s xml:id="echoid-s4047" xml:space="preserve">pariter ſunt ſi-<lb/>milia triangula, AXB, SXC, ergo, <lb/>AB, ad, VT, erit vt, BO, ad, OT, <lb/>.</s> <s xml:id="echoid-s4048" xml:space="preserve">i. </s> <s xml:id="echoid-s4049" xml:space="preserve">vt, BX, ad, XC, (quia, VC, pa-<lb/>rallela eſtipſi, AB, & </s> <s xml:id="echoid-s4050" xml:space="preserve">conſequenter <lb/>ipſi, OX,) .</s> <s xml:id="echoid-s4051" xml:space="preserve">i. </s> <s xml:id="echoid-s4052" xml:space="preserve">vt, AB, ad, SC, er-<lb/>go, VT, SC, erunt &</s> <s xml:id="echoid-s4053" xml:space="preserve">ae;</s> <s xml:id="echoid-s4054" xml:space="preserve">quales. </s> <s xml:id="echoid-s4055" xml:space="preserve">& </s> <s xml:id="echoid-s4056" xml:space="preserve">eo-<lb/>rum quadrata pariter &</s> <s xml:id="echoid-s4057" xml:space="preserve">ae;</s> <s xml:id="echoid-s4058" xml:space="preserve">qualia, ſic au-<lb/>tem de cæteris ipſi, AB, parallelis <lb/>idem oſtendetur, ergo omnes lineæ <lb/>trapezij, AERB, erunt &</s> <s xml:id="echoid-s4059" xml:space="preserve">ae;</s> <s xml:id="echoid-s4060" xml:space="preserve">quales omnibus lineis trapeZij, AIDB, <lb/>regula, AB, & </s> <s xml:id="echoid-s4061" xml:space="preserve">conſequenter ipſa trapezia erunt &</s> <s xml:id="echoid-s4062" xml:space="preserve">ae;</s> <s xml:id="echoid-s4063" xml:space="preserve">qualia, & </s> <s xml:id="echoid-s4064" xml:space="preserve">omnia <lb/>eorundem quadrata pariter &</s> <s xml:id="echoid-s4065" xml:space="preserve">ae;</s> <s xml:id="echoid-s4066" xml:space="preserve">qualia, quod oſtendere opus erat.</s> <s xml:id="echoid-s4067" xml:space="preserve"/> </p> <div xml:id="echoid-div416" type="float" level="2" n="1"> <note position="right" xlink:label="note-0187-01" xlink:href="note-0187-01a" xml:space="preserve">Iux. diff. 1. <lb/>Sexti Ele-<lb/>ment.</note> <note position="right" xlink:label="note-0187-02" xlink:href="note-0187-02a" xml:space="preserve">4. Sex. El.</note> <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a"> <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0188-01"/> </figure> </div> </div> <div xml:id="echoid-div418" type="section" level="1" n="254"> <head xml:id="echoid-head269" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXVIII:</head> <p> <s xml:id="echoid-s4068" xml:space="preserve">SI parallelogrammum, & </s> <s xml:id="echoid-s4069" xml:space="preserve">trapezium habuerint commu-<lb/>nem baſim vnum ęquidiſtantium laterum trapezij, quod <lb/>ſit ſumptum pro regula; </s> <s xml:id="echoid-s4070" xml:space="preserve">Omnia quadrata parallelogrammi <lb/>ad omnia quadrata trapezij erunt, vt quadratum dictæ baſis <lb/>ad rectangulum ſub parallelis lateribus trapezij, cum, {1/3}, <lb/>quadrati differentiæ dictorum laterum &</s> <s xml:id="echoid-s4071" xml:space="preserve">ae;</s> <s xml:id="echoid-s4072" xml:space="preserve">quidiſtantium.</s> <s xml:id="echoid-s4073" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4074" xml:space="preserve">Sit parallelogrammum, AC, & </s> <s xml:id="echoid-s4075" xml:space="preserve">trapezium, IBCO, cuius late-<lb/>rum &</s> <s xml:id="echoid-s4076" xml:space="preserve">ae;</s> <s xml:id="echoid-s4077" xml:space="preserve">quidiſtantium alterum, vt, BC, ſit communis baſis ipſi, & </s> <s xml:id="echoid-s4078" xml:space="preserve"><lb/>trapezio, & </s> <s xml:id="echoid-s4079" xml:space="preserve">regula. </s> <s xml:id="echoid-s4080" xml:space="preserve">Dico ergo omnia quadrata, AC, ad omnia qua-<lb/> <anchor type="figure" xlink:label="fig-0188-02a" xlink:href="fig-0188-02"/> drata trapezij, IBCO, eſſe vt quadratum, <lb/>BC, ad rectangulum ſub, BC, IO, vna <lb/>cum, {1/3}, quadrati differentiæ ipſarum, B <lb/>CIO. </s> <s xml:id="echoid-s4081" xml:space="preserve">Sumatur in, DA, ipſa, ED, &</s> <s xml:id="echoid-s4082" xml:space="preserve">ae;</s> <s xml:id="echoid-s4083" xml:space="preserve">-<lb/>qualis ipſi, IO, & </s> <s xml:id="echoid-s4084" xml:space="preserve">iungatur, BE, & </s> <s xml:id="echoid-s4085" xml:space="preserve">per, <lb/>E, ipſis, AB, DC, parallela ducatur, E <lb/> <anchor type="note" xlink:label="note-0188-01a" xlink:href="note-0188-01"/> M: </s> <s xml:id="echoid-s4086" xml:space="preserve">Omnia ergo quadrata trapezij, EBC <lb/>D, perlineam, EM, diuiduntur in omnia <lb/>quadrata trianguli, EBM, & </s> <s xml:id="echoid-s4087" xml:space="preserve">in omnia <lb/>quadrata, MD, & </s> <s xml:id="echoid-s4088" xml:space="preserve">in rectangula ſub tri-<lb/>angulo, EBM, &</s> <s xml:id="echoid-s4089" xml:space="preserve">, EC, bis ſumpta; </s> <s xml:id="echoid-s4090" xml:space="preserve">ad horum ergo ſingula com-<lb/>paremus omnia quadrata, AC. </s> <s xml:id="echoid-s4091" xml:space="preserve">Igitur omnia quadrata, AC, ad <lb/> <anchor type="note" xlink:label="note-0188-02a" xlink:href="note-0188-02"/> <pb o="169" file="0189" n="189" rhead="LIBER II."/> omnia quadrata, CE, ſunt vt quadratum, BC, ad quadratum, C <lb/>M, quod ſerua. </s> <s xml:id="echoid-s4092" xml:space="preserve">Inſuper omnia quadrata, AC, ad omnia quadra-<lb/> <anchor type="note" xlink:label="note-0189-01a" xlink:href="note-0189-01"/> ta, AM, ſunt vt quadratum, CB, ad quadratum, BM, item om-<lb/>nia quadrata, AM, ſunt tripla omnium quadratorum trianguli, EB <lb/> <anchor type="note" xlink:label="note-0189-02a" xlink:href="note-0189-02"/> M, .</s> <s xml:id="echoid-s4093" xml:space="preserve">l. </s> <s xml:id="echoid-s4094" xml:space="preserve">ſunt ad illa, vt quadratum, BM, ad, {1/3}, quadrati, BM, er-<lb/>go, ex &</s> <s xml:id="echoid-s4095" xml:space="preserve">ae;</s> <s xml:id="echoid-s4096" xml:space="preserve">quali, omnia quadrata, AC, ad omnia quadrata trianguli, <lb/> <anchor type="note" xlink:label="note-0189-03a" xlink:href="note-0189-03"/> EBM, erunt vt, BC, ad, {1/3}, quadrati, BM, quod pariter ſerua. </s> <s xml:id="echoid-s4097" xml:space="preserve">Tan-<lb/>dem omnia quadrata, AC, ad rectangula ſub, AM, MD, erunt vt <lb/>quadratum, BC, ad rectangulum, BMC, rectangula verò ſub, A <lb/> <anchor type="note" xlink:label="note-0189-04a" xlink:href="note-0189-04"/> M, MD, ad rectangula ſub triangulo, EBM, & </s> <s xml:id="echoid-s4098" xml:space="preserve">ſub, MD, ſunt vt, <lb/>AM, ad triangulum, EBM, (quia illa ſunt omnia rectangula pa-<lb/>rallelogrammi, AM, & </s> <s xml:id="echoid-s4099" xml:space="preserve">trianguli, EBM, &</s> <s xml:id="echoid-s4100" xml:space="preserve">ae;</s> <s xml:id="echoid-s4101" xml:space="preserve">què alta, altitudinis <lb/>nempè &</s> <s xml:id="echoid-s4102" xml:space="preserve">ae;</s> <s xml:id="echoid-s4103" xml:space="preserve">qualis ipſi, MC, ſumpta regula, BM,) .</s> <s xml:id="echoid-s4104" xml:space="preserve">i. </s> <s xml:id="echoid-s4105" xml:space="preserve">dupla .</s> <s xml:id="echoid-s4106" xml:space="preserve">i. </s> <s xml:id="echoid-s4107" xml:space="preserve">vtre-<lb/>ctangulum, BMC, ad eiuſdem dimidium, ergo, ex &</s> <s xml:id="echoid-s4108" xml:space="preserve">ae;</s> <s xml:id="echoid-s4109" xml:space="preserve">quali, omnia <lb/>quadrata, AC, ad rectangula ſub triangulo, EBM, & </s> <s xml:id="echoid-s4110" xml:space="preserve">ſub, MD, <lb/>erunt vt quadratum, BC, ad dimidium rectanguli, BMC, ad ea-<lb/>dem verò bis ſumpta erunt, vt quadratum, BC, ad rectangulum, B <lb/>MC, ergo, colligendo, omnia quadrata, AC, ad omnia quadra-<lb/>ta, EC, ad omnia quadrata trianguli, EBM, & </s> <s xml:id="echoid-s4111" xml:space="preserve">ad rectangula bis <lb/>ſub triangulo, EBM, & </s> <s xml:id="echoid-s4112" xml:space="preserve">ſub, EC, erunt vt quadratum, BC, ad qua-<lb/>dratum, CM, cum rectangulo, CMB, &</s> <s xml:id="echoid-s4113" xml:space="preserve">, {1/3}, quadrati, BM, ſed <lb/>rectangulum, BMC, cum quadrato, MC, conficit rectangulum <lb/>ſub, BC, CM, ergo omnia quadrata, AC, ad omnia quadrata tri-<lb/>anguli, EBM, parallelogrammi, MD, & </s> <s xml:id="echoid-s4114" xml:space="preserve">rectangula bis ſub eiſdem, <lb/>.</s> <s xml:id="echoid-s4115" xml:space="preserve">i. </s> <s xml:id="echoid-s4116" xml:space="preserve">ad omnia quadrata trapezij, EDCB, .</s> <s xml:id="echoid-s4117" xml:space="preserve">i. </s> <s xml:id="echoid-s4118" xml:space="preserve">ad omnia quadrata tra-<lb/> <anchor type="note" xlink:label="note-0189-05a" xlink:href="note-0189-05"/> pezij, IBCO, (quia, O, ED, ſunt &</s> <s xml:id="echoid-s4119" xml:space="preserve">ae;</s> <s xml:id="echoid-s4120" xml:space="preserve">qu@les) erunt, vt quadratum, <lb/>BC, ad rectangulum ſub, BC, CM, .</s> <s xml:id="echoid-s4121" xml:space="preserve">i. </s> <s xml:id="echoid-s4122" xml:space="preserve">ſub, BC, ED, vel, IO, <lb/>vna cum, {1/3}, quadrati, BM, quę eſt differentia parallelarum, BC, E <lb/>D, ſiue, BC, IO, ipſius trapezij, IBCO, quod oſtendere opus erat.</s> <s xml:id="echoid-s4123" xml:space="preserve"/> </p> <div xml:id="echoid-div418" type="float" level="2" n="1"> <figure xlink:label="fig-0188-02" xlink:href="fig-0188-02a"> <image file="0188-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0188-02"/> </figure> <note position="left" xlink:label="note-0188-01" xlink:href="note-0188-01a" xml:space="preserve">PerD. Co <lb/>toll. 23. <lb/>huius.</note> <note position="left" xlink:label="note-0188-02" xlink:href="note-0188-02a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-0189-02" xlink:href="note-0189-02a" xml:space="preserve">24. huius.</note> <note position="right" xlink:label="note-0189-03" xlink:href="note-0189-03a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-0189-04" xlink:href="note-0189-04a" xml:space="preserve">Coroll. 1. <lb/>26. huius.</note> <note position="right" xlink:label="note-0189-05" xlink:href="note-0189-05a" xml:space="preserve">PerD. Co <lb/>roll. 23. <lb/>huius. <lb/>Exantec.</note> </div> </div> <div xml:id="echoid-div420" type="section" level="1" n="255"> <head xml:id="echoid-head270" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4124" xml:space="preserve">_P_Atet autem ſi ipſi, ME adiungamus in directum, EF, &</s> <s xml:id="echoid-s4125" xml:space="preserve">ae;</s> <s xml:id="echoid-s4126" xml:space="preserve">qualem, <lb/>ipſi, MC, & </s> <s xml:id="echoid-s4127" xml:space="preserve">ſi ſupponamus, BM, &</s> <s xml:id="echoid-s4128" xml:space="preserve">ae;</s> <s xml:id="echoid-s4129" xml:space="preserve">quari ipſi, ME, facillimè pro-<lb/>bari poſſe omnia quadrata, AC, &</s> <s xml:id="echoid-s4130" xml:space="preserve">ae;</s> <s xml:id="echoid-s4131" xml:space="preserve">quari quadratis maximarum abſciſ-<lb/>ſarum ipſius, ME, adiuncta, EF, & </s> <s xml:id="echoid-s4132" xml:space="preserve">omnia quadrata trapezij, EBC <lb/>D, &</s> <s xml:id="echoid-s4133" xml:space="preserve">ae;</s> <s xml:id="echoid-s4134" xml:space="preserve">quari quadratis omnium abſciſſarum, ME, adiuncta, EF, nam ex. <lb/></s> <s xml:id="echoid-s4135" xml:space="preserve">gr. </s> <s xml:id="echoid-s4136" xml:space="preserve">ducta ipſi, BC, parallela vtcunque, VR, quæ ſecet, EB, in, S, &</s> <s xml:id="echoid-s4137" xml:space="preserve">, <lb/>EM, in, T, patet, quod, VT, eſt &</s> <s xml:id="echoid-s4138" xml:space="preserve">ae;</s> <s xml:id="echoid-s4139" xml:space="preserve">qualis ipſi, ME, &</s> <s xml:id="echoid-s4140" xml:space="preserve">, TR, adiun-<lb/>ctæ, EF, & </s> <s xml:id="echoid-s4141" xml:space="preserve">ideò tota, VR, &</s> <s xml:id="echoid-s4142" xml:space="preserve">ae;</s> <s xml:id="echoid-s4143" xml:space="preserve">qualis toti, MF; </s> <s xml:id="echoid-s4144" xml:space="preserve">ſimiliter, ST, eſt &</s> <s xml:id="echoid-s4145" xml:space="preserve">ae;</s> <s xml:id="echoid-s4146" xml:space="preserve">qua-<lb/>lis ipſi, TE, &</s> <s xml:id="echoid-s4147" xml:space="preserve">, TR, adiunctæ, EF, vnde patet, SR, &</s> <s xml:id="echoid-s4148" xml:space="preserve">ae;</s> <s xml:id="echoid-s4149" xml:space="preserve">quari compo-<lb/>ſitæ ex, TE, vnaabſciſſarum, & </s> <s xml:id="echoid-s4150" xml:space="preserve">adiuncta: </s> <s xml:id="echoid-s4151" xml:space="preserve">Conſimiliter in cæteris fa- <pb o="170" file="0190" n="190" rhead="GEOMETRIÆ"/> cta demon tratione propoſitum oſtendemus; </s> <s xml:id="echoid-s4152" xml:space="preserve">vnde patebit pariter quadra-<lb/>ta maxim trum abſciſſarum propoſitæ rectæ lineæ, vt ipſius, EM, adiun-<lb/>cta quædam, vt, EF, ad quadrata omnium abſciſſarum eiuſdem adiuncta <lb/>eadem, eſſe vt quadratum vnius maximarum abſciſſarum adiuncta iam <lb/>dicta .</s> <s xml:id="echoid-s4153" xml:space="preserve">i. </s> <s xml:id="echoid-s4154" xml:space="preserve">vt quadratum compoſitæ ex propoſita, & </s> <s xml:id="echoid-s4155" xml:space="preserve">adiuncta, adrectangu-<lb/>lum ſub hac compoſita, & </s> <s xml:id="echoid-s4156" xml:space="preserve">ſub adiuncta, vnacum, {1/3}, quadrati differen-<lb/>tiæhuius compoſitæ, & </s> <s xml:id="echoid-s4157" xml:space="preserve">adiunctæ .</s> <s xml:id="echoid-s4158" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4159" xml:space="preserve">vt quadratum, MF, ad rectangu-<lb/>lum ſub, MF, FE, vnacum, {1/3}, quadrati, EM, quæ eſt differentia ea-<lb/>rundem, & </s> <s xml:id="echoid-s4160" xml:space="preserve">eſt etiam propoſita linea.</s> <s xml:id="echoid-s4161" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div421" type="section" level="1" n="256"> <head xml:id="echoid-head271" xml:space="preserve">THEOREMA XXIX. PROPOS. XXIX.</head> <p> <s xml:id="echoid-s4162" xml:space="preserve">CViuſcunque parallelogrammi omnia quadrata regula <lb/>vno laterum ad omnia quadrata eiuſdem regula altero <lb/>laterum cum prædicto angulum continentium, erunt vt pri-<lb/>ma regula ad ſecundam.</s> <s xml:id="echoid-s4163" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4164" xml:space="preserve">Sit quodcunq; </s> <s xml:id="echoid-s4165" xml:space="preserve">parallelogrammum, AD. </s> <s xml:id="echoid-s4166" xml:space="preserve">Dico omnia quadrata <lb/>eiuſdem, regula, DB, eſſe vt, CD, ad, DB: </s> <s xml:id="echoid-s4167" xml:space="preserve">Omnia enim quadra-<lb/>ta, AD, regula, CD, ad omnia quadrata, AD, regula, DB, ha-<lb/>bent rationem compoſitam ex ea, quam habet quadratum, CD, ad <lb/> <anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/> <anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/> quadratum, DB, & </s> <s xml:id="echoid-s4168" xml:space="preserve">ex ea, <lb/>quam habet, BD, ad, DC, <lb/>(quia, BD, &</s> <s xml:id="echoid-s4169" xml:space="preserve">ae;</s> <s xml:id="echoid-s4170" xml:space="preserve">qualiter inclina-<lb/>tur baſi, CD, ac, CD, ipſi <lb/>baſi, DB, nam ſunt circa eun-<lb/>dem angulum) .</s> <s xml:id="echoid-s4171" xml:space="preserve">i. </s> <s xml:id="echoid-s4172" xml:space="preserve">ex ea, quam <lb/> <anchor type="note" xlink:label="note-0190-02a" xlink:href="note-0190-02"/> habet quadratum, BD, ad re-<lb/>ctangulum ſub, BD, DC, duæ autem rationes, nempè quadrati, C <lb/>D, ad quadratum, BD, & </s> <s xml:id="echoid-s4173" xml:space="preserve">quadrati, BD, ad rectangulum ſub, B <lb/>D, DC, componunt rationem quadrati, CD, ad rectangulum ſub, <lb/> <anchor type="note" xlink:label="note-0190-03a" xlink:href="note-0190-03"/> BD, DC, quę eſt eadem ei, quam habet, CD, ad, DB, ergo om-<lb/> <anchor type="note" xlink:label="note-0190-04a" xlink:href="note-0190-04"/> nia quadrata, AD, regula, CD, ad omnia quadrata eiuſdem, AD, <lb/>regula, DB, erunt vt, CD, prima regula ad, DB, ſecundam, quod <lb/>oſtendere opus erat.</s> <s xml:id="echoid-s4174" xml:space="preserve"/> </p> <div xml:id="echoid-div421" type="float" level="2" n="1"> <note position="left" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve">11. huius.</note> <figure xlink:label="fig-0190-01" xlink:href="fig-0190-01a"> <image file="0190-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0190-01"/> </figure> <note position="left" xlink:label="note-0190-02" xlink:href="note-0190-02a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-0190-03" xlink:href="note-0190-03a" xml:space="preserve">Diffin. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0190-04" xlink:href="note-0190-04a" xml:space="preserve">5. huius.</note> </div> </div> <div xml:id="echoid-div423" type="section" level="1" n="257"> <head xml:id="echoid-head272" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4175" xml:space="preserve">_H_Inc patet, ſi iungamus, CB, omnia quadrata trianguli, CBD, <lb/>regula, CD, ad omnia quadratratrianguli eiuſdem, regula, DB, <lb/>eſſe vt, CD, primam regulam ad, D B, ſecundam, nam omnia quadrata <pb o="171" file="0191" n="191" rhead="LIBER II."/> trìangulorum in eadem baſi, & </s> <s xml:id="echoid-s4176" xml:space="preserve">altitudine cum parallelogrammis conſti-<lb/>tutorum ſunt omnium quadratorum dictorum parallelogrammorum ſub-<lb/> <anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/> tripla, ſumpto communi latere pro regula, vt probatum eſt.</s> <s xml:id="echoid-s4177" xml:space="preserve"/> </p> <div xml:id="echoid-div423" type="float" level="2" n="1"> <note position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">24. huius.</note> </div> </div> <div xml:id="echoid-div425" type="section" level="1" n="258"> <head xml:id="echoid-head273" xml:space="preserve">THEOREMA XXX. PROPOS. XXX.</head> <p> <s xml:id="echoid-s4178" xml:space="preserve">SI intra parallelogrammum agatur à puncto baſis lateri-<lb/>bus oppoſitis parallela, & </s> <s xml:id="echoid-s4179" xml:space="preserve">conſtitutorum hinc parallelo-<lb/>grammorum vnius ducatur diameter: </s> <s xml:id="echoid-s4180" xml:space="preserve">Rectangula ſub factis <lb/>parallelogrammis ad rectangula ſub trapezio, & </s> <s xml:id="echoid-s4181" xml:space="preserve">triangulo in <lb/>toto parallelogrammo per dictam diametrum conſtitutis, re-<lb/>gula baſi, habebunt eandem rationem, quam baſis paralle-<lb/>logrammi, in quo non ducitur diameter ad compoſitam ex, <lb/>{1/2}, eiuſdem, &</s> <s xml:id="echoid-s4182" xml:space="preserve">, {1/6}, baſis alterius: </s> <s xml:id="echoid-s4183" xml:space="preserve">Rectangula verò ſub toto <lb/>parallelogrammo, & </s> <s xml:id="echoid-s4184" xml:space="preserve">ſub eo, in quo ducitur diameter, ad re-<lb/>ctangula ſub dicto trapezio, & </s> <s xml:id="echoid-s4185" xml:space="preserve">ſub triangulo, qui eſt trape-<lb/>zijportio, erunt vt baſis totius parallelogrammi ad compoſi-<lb/>tam ex, {1/2}, baſis parallelogrammi, in quo non ducitur diame-<lb/>ter, & </s> <s xml:id="echoid-s4186" xml:space="preserve">ex, {1/3}, baſis alterius.</s> <s xml:id="echoid-s4187" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4188" xml:space="preserve">Sit ergo parallelogrammum, AF, in baſi, DF, quæ ſit regula, <lb/>intra quam ſumptum ſit punctum, E, & </s> <s xml:id="echoid-s4189" xml:space="preserve">per, E, ipſis, AD, CF, <lb/>acta parallela, BE, ducatur autem in alterutro parallelogrammo-<lb/>rum, AE, EC, vtin, EC, diameter, EC. </s> <s xml:id="echoid-s4190" xml:space="preserve">Dico ergo rectangula <lb/> <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> ſub, AE, EC, ad rectangula ſub tra-<lb/>pezio, ADEC, & </s> <s xml:id="echoid-s4191" xml:space="preserve">triangulo, CEF, <lb/>eſſe vt, DE, ad compoſitam ex, {1/2}, D <lb/>E, &</s> <s xml:id="echoid-s4192" xml:space="preserve">, {1/6}, EF. </s> <s xml:id="echoid-s4193" xml:space="preserve">Rectangula enim ſub <lb/> <anchor type="note" xlink:label="note-0191-02a" xlink:href="note-0191-02"/> trapezio, ADEC, diuiſo per lineam, <lb/>BE, & </s> <s xml:id="echoid-s4194" xml:space="preserve">ſub triangulo, CEF, indiui-<lb/>ſo, æquantur rectangulis ſub, AE, & </s> <s xml:id="echoid-s4195" xml:space="preserve"><lb/>triangulo, CEF, vel triangulo, BE <lb/>C, & </s> <s xml:id="echoid-s4196" xml:space="preserve">rectangulis ſub triangulo, BE <lb/>C, & </s> <s xml:id="echoid-s4197" xml:space="preserve">triangulo, CEF, nunc patet <lb/>rectangula ſub, AE, EC, ad rectan-<lb/> <anchor type="note" xlink:label="note-0191-03a" xlink:href="note-0191-03"/> gula ſub, AE, & </s> <s xml:id="echoid-s4198" xml:space="preserve">triangulo, BCE, eſſe vt, BF, ad triangulum, ſt <lb/>EC, .</s> <s xml:id="echoid-s4199" xml:space="preserve">i. </s> <s xml:id="echoid-s4200" xml:space="preserve">dupla .</s> <s xml:id="echoid-s4201" xml:space="preserve">i. </s> <s xml:id="echoid-s4202" xml:space="preserve">vt, DE, ad, {1/2}, DE, quod ſerua.</s> <s xml:id="echoid-s4203" xml:space="preserve"/> </p> <div xml:id="echoid-div425" type="float" level="2" n="1"> <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a"> <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0191-01"/> </figure> <note position="right" xlink:label="note-0191-02" xlink:href="note-0191-02a" xml:space="preserve">Per A. <lb/>Coroll. <lb/>@3. huius.</note> <note position="right" xlink:label="note-0191-03" xlink:href="note-0191-03a" xml:space="preserve">Corol. 1. <lb/>26. huius.</note> </div> <p> <s xml:id="echoid-s4204" xml:space="preserve">Item rectangula ſub, AE, EC, ad omnia quadrata, BF, ſunt vt <lb/>rectangulum, DEF, ad quadratum, EF, .</s> <s xml:id="echoid-s4205" xml:space="preserve">i. </s> <s xml:id="echoid-s4206" xml:space="preserve">vt, DE, ad, EF, om-<lb/> <anchor type="note" xlink:label="note-0191-04a" xlink:href="note-0191-04"/> nia verò quadrata, BF, ſunt ſexcupla rectangulorum ſub triangulis, <pb o="172" file="0192" n="192" rhead="GEOMETRIÆ"/> BEC, CEF, .</s> <s xml:id="echoid-s4207" xml:space="preserve">i. </s> <s xml:id="echoid-s4208" xml:space="preserve">ſunt ad illa, vt, EF, ad, {1/6}, eiuſdem, EF, ergo ex <lb/> <anchor type="note" xlink:label="note-0192-01a" xlink:href="note-0192-01"/> æquali, rectingula ſub, AE, EC, ad rectangula ſub triangulis, BE <lb/> <anchor type="note" xlink:label="note-0192-02a" xlink:href="note-0192-02"/> C, CEF, erunt vt, DE, ad, {1/6}, EF, eadem verò ad rectangula ſub, <lb/>AE, & </s> <s xml:id="echoid-s4209" xml:space="preserve">triangulo, BEC, ſiue, CEF, oſtenſa ſunt eſſe, vt, DE, <lb/>ad, {1/2}, DE, ergo, colligendo, rectangula ſub, AE, EC, ad rectan-<lb/>gula ſub, AE, & </s> <s xml:id="echoid-s4210" xml:space="preserve">triangulo, CEF, & </s> <s xml:id="echoid-s4211" xml:space="preserve">ſub triangulo, BEC, & </s> <s xml:id="echoid-s4212" xml:space="preserve">eo-<lb/>dem, CEF, .</s> <s xml:id="echoid-s4213" xml:space="preserve">i. </s> <s xml:id="echoid-s4214" xml:space="preserve">ad rectangula ſub trapezio, ADEC, & </s> <s xml:id="echoid-s4215" xml:space="preserve">triangulo, <lb/> <anchor type="note" xlink:label="note-0192-03a" xlink:href="note-0192-03"/> CEF, erunt vt, DE, ad compoſitam ex, {1/2}, DE, &</s> <s xml:id="echoid-s4216" xml:space="preserve">, {1/6}, EF, quę <lb/>eſt Theorematis prima pars.</s> <s xml:id="echoid-s4217" xml:space="preserve"/> </p> <div xml:id="echoid-div426" type="float" level="2" n="2"> <note position="right" xlink:label="note-0191-04" xlink:href="note-0191-04a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-0192-01" xlink:href="note-0192-01a" xml:space="preserve">Elicitur <lb/>ex.</note> <note position="left" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">24. huius.</note> <note position="left" xlink:label="note-0192-03" xlink:href="note-0192-03a" xml:space="preserve">Per A. Co <lb/>roll. 23. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s4218" xml:space="preserve">Dico vlterius rectangula ſub, AF, FB, ad rectangula ſub trape-<lb/>zio, ADEC, & </s> <s xml:id="echoid-s4219" xml:space="preserve">triangulo, BEC, eſſe vt, DF, ad compoſitam ex, <lb/>{1/6}, DE, &</s> <s xml:id="echoid-s4220" xml:space="preserve">, {1/3}, EF; </s> <s xml:id="echoid-s4221" xml:space="preserve">rectangula .</s> <s xml:id="echoid-s4222" xml:space="preserve">n. </s> <s xml:id="echoid-s4223" xml:space="preserve">ſub, AF, FB, ad rectangula ſub, <lb/>AE, EC, ſunt vt rectangulum, DFE, ad rectangulum, DEF, .</s> <s xml:id="echoid-s4224" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s4225" xml:space="preserve"> <anchor type="note" xlink:label="note-0192-04a" xlink:href="note-0192-04"/> vt, FD, ad, DE, rectangula vero ſub, AE, EC, ad rectangula ſub, <lb/> <anchor type="note" xlink:label="note-0192-05a" xlink:href="note-0192-05"/> <anchor type="figure" xlink:label="fig-0192-01a" xlink:href="fig-0192-01"/> AE, & </s> <s xml:id="echoid-s4226" xml:space="preserve">triangulo, BEC, ſunt vt, B <lb/> <anchor type="note" xlink:label="note-0192-06a" xlink:href="note-0192-06"/> F, ad triangulum, BEC, .</s> <s xml:id="echoid-s4227" xml:space="preserve">i. </s> <s xml:id="echoid-s4228" xml:space="preserve">dupla .</s> <s xml:id="echoid-s4229" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s4230" xml:space="preserve">vt, DE, ad, {1/2}, ipſius, DE, ergo, ex <lb/>æquali rectangula ſub, AF, FB, ad <lb/>rectangula ſub, AE, & </s> <s xml:id="echoid-s4231" xml:space="preserve">triangulo, B <lb/>EC, erunt vt, FD, ad, {1/2}, DE, quod <lb/>ſerua. </s> <s xml:id="echoid-s4232" xml:space="preserve">Item rectangula ſub, AF, FB, <lb/> <anchor type="note" xlink:label="note-0192-07a" xlink:href="note-0192-07"/> ad omnia quadrata, BF, ſunt vt re-<lb/>ctangulum, DFE, ad quadratum, F <lb/>E, .</s> <s xml:id="echoid-s4233" xml:space="preserve">i. </s> <s xml:id="echoid-s4234" xml:space="preserve">vt, DF, ad, FE: </s> <s xml:id="echoid-s4235" xml:space="preserve">Omnia verò <lb/>quadrata, BF, ſunt tripla omnium <lb/>quadratorum trianguli, BEC, .</s> <s xml:id="echoid-s4236" xml:space="preserve">i. </s> <s xml:id="echoid-s4237" xml:space="preserve">ſunt vt, FE, ad, {1/3}, FE, ergo ex <lb/>æquali rectangula ſub, AF, FB, ad omnia quadrata trianguli, BE <lb/>C, ſunt vt, DF, ad, {1/3}, FE, erant autem eadem ad rectangula ſub, <lb/>AE, & </s> <s xml:id="echoid-s4238" xml:space="preserve">triangulo, BEC, vt, DF, ad, {1/2}, DE, ergo, colligendo, <lb/>rectangula ſub, AF, FB, ad rectangula ſub, AE, & </s> <s xml:id="echoid-s4239" xml:space="preserve">triangulo, BE <lb/>C, vna cum omnibus quadratis trianguli, BEC, .</s> <s xml:id="echoid-s4240" xml:space="preserve">i. </s> <s xml:id="echoid-s4241" xml:space="preserve">ad rectangula <lb/>ſub trapezio, ADEC, & </s> <s xml:id="echoid-s4242" xml:space="preserve">triangulo, BEC, erunt vt, DF, ad com-<lb/> <anchor type="note" xlink:label="note-0192-08a" xlink:href="note-0192-08"/> poſitam ex, {1/2}, DE, &</s> <s xml:id="echoid-s4243" xml:space="preserve">, {1/3}, EF, quę eſt Theorematis ſecunda pars; <lb/></s> <s xml:id="echoid-s4244" xml:space="preserve">hæc autem erant demonſtranda.</s> <s xml:id="echoid-s4245" xml:space="preserve"/> </p> <div xml:id="echoid-div427" type="float" level="2" n="3"> <note position="left" xlink:label="note-0192-04" xlink:href="note-0192-04a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-0192-05" xlink:href="note-0192-05a" xml:space="preserve">3. huius.</note> <figure xlink:label="fig-0192-01" xlink:href="fig-0192-01a"> <image file="0192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0192-01"/> </figure> <note position="left" xlink:label="note-0192-06" xlink:href="note-0192-06a" xml:space="preserve">Coroll. 1. <lb/>26. huius.</note> <note position="left" xlink:label="note-0192-07" xlink:href="note-0192-07a" xml:space="preserve">14. huius. <lb/>3. huius. <lb/>24. huius.</note> <note position="left" xlink:label="note-0192-08" xlink:href="note-0192-08a" xml:space="preserve">Per C. <lb/>Coroll. <lb/>23. huius.</note> </div> </div> <div xml:id="echoid-div429" type="section" level="1" n="259"> <head xml:id="echoid-head274" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4246" xml:space="preserve">_C_Olligimus autem ex hoc Theoremate rectangula ſub maximis ab-<lb/>ſciſſarum propoſitæ lineæ, adiunctis eiſdem tot vni cuidam æquali-<lb/>bus, ad rectangula ſub omnibus abſciſſis eiuſdem adiunctaiam dicta li-<lb/>nea, & </s> <s xml:id="echoid-s4247" xml:space="preserve">ſub reſiduis abſciſſarum eiuſdem, eſſe vt adiuncta ad compoſitam <lb/>ex, {1/2}, adiunctæ, & </s> <s xml:id="echoid-s4248" xml:space="preserve">{1/2}, propoſitæ lineæ, & </s> <s xml:id="echoid-s4249" xml:space="preserve">hoc ex prima parte huius <pb o="173" file="0193" n="193" rhead="LIBER II."/> Theorematis, nam, vt alibi oſtendimus, ſi ſupponamus ipſi, BE, adiun-<lb/>girectam, EM, æqualem ipſi, DE, & </s> <s xml:id="echoid-s4250" xml:space="preserve">BE, eſſe æqualem ipſi, EF, om-<lb/>nes lineæ trapezij, ADEC, erunt æquales omnibus abſciſſis ipſius, BE, <lb/>(quæ ſit propoſita linea) adiuncta tamen, EM, & </s> <s xml:id="echoid-s4251" xml:space="preserve">omnes lineæ triangu-<lb/>li, CEF, (intellige ſemper regulam, DF,) erunt æquales reſiduis om-<lb/>nium abſciſſarum prop@ſitæ lineæ, BE, item omnes lineæ, AE, erunt <lb/>æquales ijs, quæ adiunguntur maximis abſciſſarum, BE, nam earum ſin-<lb/>gulæ ſunt æquales ipſi, DE, vel, EM, & </s> <s xml:id="echoid-s4252" xml:space="preserve">omnes lineæ, EC, maximis <lb/>abſciſſarum, BE, pariter æquales erunt, vnde patet propoſitum. </s> <s xml:id="echoid-s4253" xml:space="preserve">Exſe-<lb/>cunda verò parte conſimili ratione colligemus rectangula ſub maximis <lb/>abſciſſ rum propoſitæ lineæ, vt, BE, adiuncta quadam, vt, EM, & </s> <s xml:id="echoid-s4254" xml:space="preserve"><lb/>ſub maximis abſciſſarum eiuſdem propoſitæ, BE, ad rectangula ſub om-<lb/>nibus abſciſſis, ſumptis verſus, E, eiuſdem propoſitæ, BE, adiuncta, <lb/>EM, & </s> <s xml:id="echoid-s4255" xml:space="preserve">ſub eiuſdem omnibus abſciſſis propoſitæ, BE, eſſe vt compoſita <lb/>ex propoſita, & </s> <s xml:id="echoid-s4256" xml:space="preserve">adiecta .</s> <s xml:id="echoid-s4257" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4258" xml:space="preserve">vt, BM, ad compoſitam ex, {1/2}, adiectæ, quæ <lb/>eſt, ME, & </s> <s xml:id="echoid-s4259" xml:space="preserve">{1/3}, propoſitæ, quæ eſt, BE.</s> <s xml:id="echoid-s4260" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div430" type="section" level="1" n="260"> <head xml:id="echoid-head275" xml:space="preserve">THEOREMA XXXI. PROPOS. XXXI.</head> <p> <s xml:id="echoid-s4261" xml:space="preserve">EXpoſita Propoſit. </s> <s xml:id="echoid-s4262" xml:space="preserve">antecedentis figura, & </s> <s xml:id="echoid-s4263" xml:space="preserve">intra parallelas, <lb/>AC, DF, eiſdem æquidiſtanter ducta recta linea, H <lb/>O, quæ ſecet, BE, in, M, &</s> <s xml:id="echoid-s4264" xml:space="preserve">, CE, in, N, oſtendemus, re-<lb/>gula eadem, DF, rectangula ſub parallelogrammis, AO, O <lb/>B, ad rectangula ſub trapezijs, HACN, MBCN, eſſe vt <lb/>rectangulum, HOM, ad rectangulum ſub, HO, MN, cum <lb/>rectangulo ſub compoſita ex, {1/2}, HM, &</s> <s xml:id="echoid-s4265" xml:space="preserve">, {1/5}, NO, & </s> <s xml:id="echoid-s4266" xml:space="preserve">ſub, NO.</s> <s xml:id="echoid-s4267" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4268" xml:space="preserve">Rectangula enim ſub parallelogram-<lb/> <anchor type="figure" xlink:label="fig-0193-01a" xlink:href="fig-0193-01"/> mis, AO, OB, ad rectangula ſub paral-<lb/> <anchor type="note" xlink:label="note-0193-01a" xlink:href="note-0193-01"/> lelogrammis, AM, MC, ſunt vt re-<lb/>ctangulum, HOM, ad rectangulum, <lb/> <anchor type="note" xlink:label="note-0193-02a" xlink:href="note-0193-02"/> HMO, rectangula verò ſub, AM, M <lb/>C, ad rectangula ſub parallelogrammo, <lb/>AM, & </s> <s xml:id="echoid-s4269" xml:space="preserve">trapezio, BMNC, ſunt vt, B <lb/> <anchor type="note" xlink:label="note-0193-03a" xlink:href="note-0193-03"/> O, ad trapezium, BMNC, .</s> <s xml:id="echoid-s4270" xml:space="preserve">i. </s> <s xml:id="echoid-s4271" xml:space="preserve">vt, MO, <lb/>ad, MN, cum, {1/2}, NO, vel vt rectan-<lb/> <anchor type="note" xlink:label="note-0193-04a" xlink:href="note-0193-04"/> gulum, HMO, ad rectangulum ſub, H <lb/>M, & </s> <s xml:id="echoid-s4272" xml:space="preserve">ſub compoſita ex, MN, &</s> <s xml:id="echoid-s4273" xml:space="preserve">, {1/2}, N <lb/>O, ergo, ex æquali, rectangula ſub, AO, OB, ad rectangula ſub, <lb/>AM, & </s> <s xml:id="echoid-s4274" xml:space="preserve">trapezio, BMNC, ſunt vt rectangulum, HOM, ad rectan-<lb/>gulum ſub, HM, & </s> <s xml:id="echoid-s4275" xml:space="preserve">compoſita ex, MN, &</s> <s xml:id="echoid-s4276" xml:space="preserve">, {1/2}, NO, quod ſerua.</s> <s xml:id="echoid-s4277" xml:space="preserve"/> </p> <div xml:id="echoid-div430" type="float" level="2" n="1"> <figure xlink:label="fig-0193-01" xlink:href="fig-0193-01a"> <image file="0193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0193-01"/> </figure> <note position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-0193-02" xlink:href="note-0193-02a" xml:space="preserve">Coroll. 1. <lb/>26. huius.</note> <note position="right" xlink:label="note-0193-03" xlink:href="note-0193-03a" xml:space="preserve">20. huius.</note> <note position="right" xlink:label="note-0193-04" xlink:href="note-0193-04a" xml:space="preserve">5. huius.</note> </div> <pb o="174" file="0194" n="194" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s4278" xml:space="preserve">Inſuper rectangula ſub, AO, OB, ad omnia quadrata, OB, ſunt <lb/> <anchor type="note" xlink:label="note-0194-01a" xlink:href="note-0194-01"/> vt rectangulum, HOM, ad quadratum, OM, & </s> <s xml:id="echoid-s4279" xml:space="preserve">omnia quadrata, <lb/>OB, ad omnia quadrata trapezij, BMNC, ſunt vt quadratum, O <lb/>M, ad rectangulum, OMN, cum, {1/3}, quadrati, NO, ergo, ex æ-<lb/> <anchor type="note" xlink:label="note-0194-02a" xlink:href="note-0194-02"/> quali rectangula ſub, AO, OB, ad omnia quadrata trapezij, BM <lb/>NC, ſunt vt rectangulum, HOM, ad rectangulum, OMN, cum, <lb/>{1/3}, quadrati, NO, oſtenſa ſunt autem rectangula ſub, AO, OB, ad <lb/>rectangula ſub, AM, & </s> <s xml:id="echoid-s4280" xml:space="preserve">trapezio, BMNC, eſſe vt rectangulum, <lb/>HOM, ad rectangulum ſub, HM, & </s> <s xml:id="echoid-s4281" xml:space="preserve">compoſita ex, MN, &</s> <s xml:id="echoid-s4282" xml:space="preserve">, {1/2}, <lb/> <anchor type="figure" xlink:label="fig-0194-01a" xlink:href="fig-0194-01"/> NO, ergo, colligendo, rectangula ſub, <lb/>AO, OB, ad rectangula ſub, AM, & </s> <s xml:id="echoid-s4283" xml:space="preserve"><lb/>trapezio, BMNC, cum omnibus <lb/> <anchor type="note" xlink:label="note-0194-03a" xlink:href="note-0194-03"/> quadratis eiuſdem trapezij, ideſt ad re-<lb/>ctangula ſub trapezijs, AHNC, BM <lb/>NC, erunt vt rectangulum, HOM, <lb/>ad rectangulum ſub, HM, & </s> <s xml:id="echoid-s4284" xml:space="preserve">compo-<lb/>ſita ex, MN, &</s> <s xml:id="echoid-s4285" xml:space="preserve">, {1/2}, NO, vna cum re-<lb/>ctangulo ſub, OM, &</s> <s xml:id="echoid-s4286" xml:space="preserve">, MN, &</s> <s xml:id="echoid-s4287" xml:space="preserve">, {1/3}, <lb/>quadrati, NO, rectangulum autem <lb/>ſub, HM, & </s> <s xml:id="echoid-s4288" xml:space="preserve">compoſita ex, MN, &</s> <s xml:id="echoid-s4289" xml:space="preserve">, <lb/>{1/2}, NO, diuiditur in rectangula ſub, H <lb/>M, &</s> <s xml:id="echoid-s4290" xml:space="preserve">, MN, & </s> <s xml:id="echoid-s4291" xml:space="preserve">ſub, HM, &</s> <s xml:id="echoid-s4292" xml:space="preserve">, {1/2}, NO, ſi ergo iunxeris rectangulum <lb/>ſub, HM, MN, cum rectangulo ſub, OM, MN, ſiet rectangulum <lb/> <anchor type="note" xlink:label="note-0194-04a" xlink:href="note-0194-04"/> ſub tota, HO, & </s> <s xml:id="echoid-s4293" xml:space="preserve">ſub, MN, & </s> <s xml:id="echoid-s4294" xml:space="preserve">remanebit rectangulum ſub, HM, <lb/>& </s> <s xml:id="echoid-s4295" xml:space="preserve">ſub, {1/2}, NO, cum, {1/3}, quadrati, NO, ideſt cum rectangulo ſub, <lb/>NO, &</s> <s xml:id="echoid-s4296" xml:space="preserve">, {1/3}, NO, eſt autem rectangulum ſub, HM, &</s> <s xml:id="echoid-s4297" xml:space="preserve">, {1/2}, NO, <lb/> <anchor type="note" xlink:label="note-0194-05a" xlink:href="note-0194-05"/> æquale rectangulo ſub, {1/2}, HM, & </s> <s xml:id="echoid-s4298" xml:space="preserve">ſub, NO, hoc ergo ſi iunxeris <lb/>rectangulo ſub, NO, &</s> <s xml:id="echoid-s4299" xml:space="preserve">, {1/3}, NO, conficiemus rectangulum ſub com-<lb/>poſita ex, {1/2}, HM, &</s> <s xml:id="echoid-s4300" xml:space="preserve">, {1/3}, NO, & </s> <s xml:id="echoid-s4301" xml:space="preserve">ſub, NO, totum igitur conſe-<lb/>quens iam dictum diuiſum eſt in hæc duo rectangula, nempè vnum <lb/>ſub, HO, MN, aliud ſub compoſita ex, {1/2}, HM, &</s> <s xml:id="echoid-s4302" xml:space="preserve">, {1/3}, NO, & </s> <s xml:id="echoid-s4303" xml:space="preserve"><lb/>ſub, NO; </s> <s xml:id="echoid-s4304" xml:space="preserve">ad hæc ergo ſimul ſumpta rectangulum, HOM, erit vt <lb/>rectangula ſub, AO, OB, ad rectangula ſub trapezijs, AHNC, B <lb/>MNC, quod oſtendendum erat.</s> <s xml:id="echoid-s4305" xml:space="preserve"/> </p> <div xml:id="echoid-div431" type="float" level="2" n="2"> <note position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-0194-02" xlink:href="note-0194-02a" xml:space="preserve">18. huius.</note> <figure xlink:label="fig-0194-01" xlink:href="fig-0194-01a"> <image file="0194-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0194-01"/> </figure> <note position="left" xlink:label="note-0194-03" xlink:href="note-0194-03a" xml:space="preserve">PerC. Co <lb/>rol. 23. hu <lb/>ius.</note> <note position="left" xlink:label="note-0194-04" xlink:href="note-0194-04a" xml:space="preserve">i. Secundi <lb/>Elem.</note> <note position="left" xlink:label="note-0194-05" xlink:href="note-0194-05a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div433" type="section" level="1" n="261"> <head xml:id="echoid-head276" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4306" xml:space="preserve">_H_Inc etiam patet, ſi ſupponamus, FE, eſſe æqualem ipſi, EB, & </s> <s xml:id="echoid-s4307" xml:space="preserve">ipſi, <lb/>EB, in directum adiunctam ipſam, EZ, ſumamus tamen cum, <lb/>EZ, ipſam, EM, ex quibus conſiciamus, MZ, adiunctam maximis ab-<lb/>ſciſſarum, vel abſciſſis ipſius, BM, propoſitæ vtcunque lineæ, quod fa-<lb/> <anchor type="note" xlink:label="note-0194-06a" xlink:href="note-0194-06"/> cilè oſtendemus omnes lineas parallelogrammi, AO, æquari maximis <pb o="175" file="0195" n="195" rhead="LIBER II."/> abſciſſarum, BM, adiuncta, MZ, & </s> <s xml:id="echoid-s4308" xml:space="preserve">omnes lineas, BO, æquari ma-<lb/>ximis abſciſſarum, BM, adiuncta, ME, & </s> <s xml:id="echoid-s4309" xml:space="preserve">omnes lineas trapezij, A <lb/>HNC, æquari omnibus abſciſſis, BM, adiuncta, MZ, & </s> <s xml:id="echoid-s4310" xml:space="preserve">omnes lineas <lb/>trapezij, BMNC, æquari omnibus abſciſſis ipſius, BM, adiuncta, M <lb/>E, quorum exemplum patere poteſt in recta, HO, in qua, HO, æquatur <lb/>ipſi, BZ, &</s> <s xml:id="echoid-s4311" xml:space="preserve">, HN, ipſi, MZ, &</s> <s xml:id="echoid-s4312" xml:space="preserve">, MN, ipſi, ME, æquari autem ſu-<lb/>pradicta ſic intellige, vt ſemper cuilibet aſſumptæ in parallelogrammo, <lb/>AO, reperiatur ſibi æqualis reſpondens in recta, BZ, & </s> <s xml:id="echoid-s4313" xml:space="preserve">ſic cuilibet aſ-<lb/>ſumptæ in trapezijs iam dictis, reperiatur illi æqualis correſpondens in <lb/>recta, BZ, quæ erit vna abſciſſarum, BM, adiuncta, MZ, vel, ME, <lb/>ea nempè, que terminatur ad idem punctum, per quod tranſit ea, quæ <lb/>æquidiſtat ipſi, DF, & </s> <s xml:id="echoid-s4314" xml:space="preserve">cum eadem comparata illi reperitur æqualis (ſic <lb/>autem intellige in cæteris, cum dicimus omnes lineas alicuius figuræ, <lb/>quæ eſt parallelogrammum, vel trapezium, vel triangulum æquari om-<lb/>nibus abſciſſis, vel maximis, vel reſiduis omnium abſciſſarum alicuius <lb/>lineæ, adiuncta, vel non adiuncta aliqua linea.) </s> <s xml:id="echoid-s4315" xml:space="preserve">Rectangula ergo ſub <lb/>maximis abſciſſarum, BM, adiuncta, MZ, & </s> <s xml:id="echoid-s4316" xml:space="preserve">ſub maximis abſciſſarum, <lb/>BM, adiuncta, ME, ad rectangula ſub omnibus abſciſſis, BM, adiun-<lb/>cta, MZ, & </s> <s xml:id="echoid-s4317" xml:space="preserve">ſub omnibus abſciſſis, BM, adiuncta, ME, erunt vt re-<lb/>ctangulum ſub, HO, OM, ideſt ſub, ZB, BE, ad rectangulum ſub, H <lb/>O, MN, vnà cum rectangulo ſub compoſita ex, {1/2}, HM, &</s> <s xml:id="echoid-s4318" xml:space="preserve">, {1/3}, NO, <lb/>& </s> <s xml:id="echoid-s4319" xml:space="preserve">ſub, NO, ideſt ad rectangulum ſub, ZB, ME, vna cum rectangulo <lb/>ſub compoſita ex, {1/2}, ZE, &</s> <s xml:id="echoid-s4320" xml:space="preserve">, {1/3}, MB, & </s> <s xml:id="echoid-s4321" xml:space="preserve">ſub, MB.</s> <s xml:id="echoid-s4322" xml:space="preserve"/> </p> <div xml:id="echoid-div433" type="float" level="2" n="1"> <note position="left" xlink:label="note-0194-06" xlink:href="note-0194-06a" xml:space="preserve">_Vt in Cor._ <lb/>_21. huius._</note> </div> </div> <div xml:id="echoid-div435" type="section" level="1" n="262"> <head xml:id="echoid-head277" xml:space="preserve">THEOREMA XXXII. PROPOS. XXXII.</head> <p> <s xml:id="echoid-s4323" xml:space="preserve">EXpoſita adhuc antecedentis Theorematis figura, ſi ipſi, <lb/>EF, ad punctum, F, iungatur in directum quædam re-<lb/>cta linea, vt, FS, & </s> <s xml:id="echoid-s4324" xml:space="preserve">compleatur parallelogrammum, FR, re-<lb/>gula ſumpta, DS, oſtendemus rectangula ſub, AE, ER, ad <lb/>rectangula ſub trapezijs, ADEC, CESR, eſſe vt rectan-<lb/>gulum, DES, ad rectangulum ſub, DE, & </s> <s xml:id="echoid-s4325" xml:space="preserve">compoſita ex, S <lb/>F, &</s> <s xml:id="echoid-s4326" xml:space="preserve">, {1/2}, FE, vna cum rectangulo ſub, EF, & </s> <s xml:id="echoid-s4327" xml:space="preserve">compoſita ex, <lb/>{1/6}, EF, &</s> <s xml:id="echoid-s4328" xml:space="preserve">, {1/2}, FS.</s> <s xml:id="echoid-s4329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4330" xml:space="preserve">Rectangula enim ſub, AE, ER, ad rectangula ſub, AE, & </s> <s xml:id="echoid-s4331" xml:space="preserve">tra-<lb/> <anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/> pezio, CESR, ſunt vt, ER, ad trapezium, CESR, .</s> <s xml:id="echoid-s4332" xml:space="preserve">i. </s> <s xml:id="echoid-s4333" xml:space="preserve">vt, ES, <lb/>ad, SF, cum, {1/2}, FE, .</s> <s xml:id="echoid-s4334" xml:space="preserve">i. </s> <s xml:id="echoid-s4335" xml:space="preserve">tumpta, DE, communi altitudine, vt re-<lb/> <anchor type="note" xlink:label="note-0195-02a" xlink:href="note-0195-02"/> ctangulum, DES, ad rectangulum ſub, DE, & </s> <s xml:id="echoid-s4336" xml:space="preserve">ſub compoſita ex, <lb/>SF, &</s> <s xml:id="echoid-s4337" xml:space="preserve">, {1/2}, FE, quod ſerua.</s> <s xml:id="echoid-s4338" xml:space="preserve"/> </p> <div xml:id="echoid-div435" type="float" level="2" n="1"> <note position="right" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">Coroll .1. <lb/>26. huius.</note> <note position="right" xlink:label="note-0195-02" xlink:href="note-0195-02a" xml:space="preserve">20. huius. <lb/>5. huius.</note> </div> <pb o="176" file="0196" n="196" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s4339" xml:space="preserve">Inſuper rectangula ſub, AE, ER, ad rectangula ſub, BF, FR, <lb/> <anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/> ſunt vt rectangulum, DES, ad rectangulum, EFS; </s> <s xml:id="echoid-s4340" xml:space="preserve">item rectan-<lb/>gula ſub, BF, FR, ad rectangula ſub triangulo, BEC, & </s> <s xml:id="echoid-s4341" xml:space="preserve">trapezio, <lb/>CESR, ſunt vt, FS, ad compoſitam ex, {1/2}, SF, &</s> <s xml:id="echoid-s4342" xml:space="preserve">, {1/6}, FE, ideſt <lb/> <anchor type="note" xlink:label="note-0196-02a" xlink:href="note-0196-02"/> ſumpta, EF, communi altitudine, vt rectangulum, EFS, ad rectan-<lb/> <anchor type="figure" xlink:label="fig-0196-01a" xlink:href="fig-0196-01"/> gulum ſub, EF, & </s> <s xml:id="echoid-s4343" xml:space="preserve">compoſita ex, <lb/>{1/6}, EF, &</s> <s xml:id="echoid-s4344" xml:space="preserve">, {1/2}, FS, ergo ex æquali <lb/> <anchor type="note" xlink:label="note-0196-03a" xlink:href="note-0196-03"/> rectangula ſub, AE, ER, ad re-<lb/>ctangula ſub triangulo, BEC, & </s> <s xml:id="echoid-s4345" xml:space="preserve"><lb/>trapezio, CESR, erunt vt rectan-<lb/>gulum, DES, ad rectangulum ſub, <lb/>EF, & </s> <s xml:id="echoid-s4346" xml:space="preserve">compoſita ex, {1/6}, EF, &</s> <s xml:id="echoid-s4347" xml:space="preserve">, <lb/>{1/2}, FS; </s> <s xml:id="echoid-s4348" xml:space="preserve">erant autem eadem rectan-<lb/>gula ſub, AE, ER, ad rectangula <lb/>ſub, AE, & </s> <s xml:id="echoid-s4349" xml:space="preserve">trapezio, CESR, vt <lb/>idem rectangulum, DES, ad re-<lb/>ctangulum ſub, DE, & </s> <s xml:id="echoid-s4350" xml:space="preserve">compoſita <lb/>ex, SF, &</s> <s xml:id="echoid-s4351" xml:space="preserve">, {1/2}, FE, ergo, colligen-<lb/>do, rectangula ſub, AE, ER, ad rectangula ſub, AF, & </s> <s xml:id="echoid-s4352" xml:space="preserve">trapezio, <lb/>CESR, & </s> <s xml:id="echoid-s4353" xml:space="preserve">ſub triangulo, BEC, & </s> <s xml:id="echoid-s4354" xml:space="preserve">eodem trapezio, CESR, .</s> <s xml:id="echoid-s4355" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s4356" xml:space="preserve">ad rectangula ſub trapezio, ADEC, & </s> <s xml:id="echoid-s4357" xml:space="preserve">trapezio, CESR, erunt <lb/> <anchor type="note" xlink:label="note-0196-04a" xlink:href="note-0196-04"/> vt rectangulum, DES, ad rectangulum ſub, DE, & </s> <s xml:id="echoid-s4358" xml:space="preserve">compoſita ex, <lb/>SF, &</s> <s xml:id="echoid-s4359" xml:space="preserve">, {1/2}, FE, vna cum rectangulo ſub, EF, & </s> <s xml:id="echoid-s4360" xml:space="preserve">compoſita ex, {1/6}, <lb/>EF, &</s> <s xml:id="echoid-s4361" xml:space="preserve">, {1/2}, FS, quod oſtendere opus erat.</s> <s xml:id="echoid-s4362" xml:space="preserve"/> </p> <div xml:id="echoid-div436" type="float" level="2" n="2"> <note position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve">@4. huius.</note> <note position="left" xlink:label="note-0196-02" xlink:href="note-0196-02a" xml:space="preserve">@@. huius.</note> <figure xlink:label="fig-0196-01" xlink:href="fig-0196-01a"> <image file="0196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0196-01"/> </figure> <note position="left" xlink:label="note-0196-03" xlink:href="note-0196-03a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-0196-04" xlink:href="note-0196-04a" xml:space="preserve">Per A. <lb/>Corol. <lb/>23. huius.</note> </div> </div> <div xml:id="echoid-div438" type="section" level="1" n="263"> <head xml:id="echoid-head278" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4363" xml:space="preserve">_Q_Voniam verò ſi ſupponamus, FE, eſſe æqualem ipſi, EB, faci’e, <lb/> <anchor type="note" xlink:label="note-0196-05a" xlink:href="note-0196-05"/> modo vſitato oſtendemus omnes lineas trapezij. </s> <s xml:id="echoid-s4364" xml:space="preserve">ADEC, æquari <lb/>reſiduis amnium abſciſſarum, BE, ſumptis verſus, E, adiuncti, <lb/>EZ, & </s> <s xml:id="echoid-s4365" xml:space="preserve">omnes lineas trapezij, CESR, æquari omnibus abſciſſis, EB, <lb/>adiuncta recta linea æquali ipſi, FS, ad punctum, B, quæ ſit, BV, & </s> <s xml:id="echoid-s4366" xml:space="preserve"><lb/>omnes lineas, AE, æquari tot æqualibus adiunctæ, ZE, quot ſunt <lb/>omnes abſciſſæ, BE, & </s> <s xml:id="echoid-s4367" xml:space="preserve">omnes lineas, ER, æquari maximis abſciſſa-<lb/>rum, EB, adiuncta, BV, ideò rectangula ſub iſtis erunt etiam æqualia <lb/>rectangulis ſub dictis trapezijs, & </s> <s xml:id="echoid-s4368" xml:space="preserve">parallelogrammis, vnde propoſita <lb/>vtcunq; </s> <s xml:id="echoid-s4369" xml:space="preserve">linea, VZ, eaq; </s> <s xml:id="echoid-s4370" xml:space="preserve">vtcunq; </s> <s xml:id="echoid-s4371" xml:space="preserve">ſecta in duobus punctis, BE, pate-<lb/>bit rectangula ſub tot æqualibus, ZE, quot ſunt omnes abſciſſæ, ſiue, <lb/>maximæ abſciſſarum, EB, & </s> <s xml:id="echoid-s4372" xml:space="preserve">ſub maximis abſciſſirum, EB, adiuncta, <lb/>BV, ad rectangula ſub reſiduis omnium abſciſſarum, BE, adiuncta, <lb/>EZ, & </s> <s xml:id="echoid-s4373" xml:space="preserve">ſub omnibus abſciſſis, EB, adiuncta, BV, eſſe vt rectangu- <pb o="177" file="0197" n="197" rhead="LIBER II."/> lum, DES, ad rectangulum ſub, DE, & </s> <s xml:id="echoid-s4374" xml:space="preserve">compoſita ex, SF, &</s> <s xml:id="echoid-s4375" xml:space="preserve">, {1/2}, FE, <lb/>vna cum rectangulo ſub, EF, & </s> <s xml:id="echoid-s4376" xml:space="preserve">compoſita ex, {1/6}, EF, &</s> <s xml:id="echoid-s4377" xml:space="preserve">, {1/2}, FS .</s> <s xml:id="echoid-s4378" xml:space="preserve">i. </s> <s xml:id="echoid-s4379" xml:space="preserve">vt <lb/>rectangulum, ZEV, quod eſt vnum rectangulorum maximis æqualium, <lb/>ad rectangulum ſub, ZE, & </s> <s xml:id="echoid-s4380" xml:space="preserve">ſub compoſita ex, VB, &</s> <s xml:id="echoid-s4381" xml:space="preserve">, {1/2}, BE, vna <lb/>cum rectangulo ſub, EB, & </s> <s xml:id="echoid-s4382" xml:space="preserve">compoſita ex, {1/6}, EB, &</s> <s xml:id="echoid-s4383" xml:space="preserve">, {1/2}, BV, regulam <lb/>autem bic pariter ſuppone ipſam, DS, & </s> <s xml:id="echoid-s4384" xml:space="preserve">abſciſſas, reſiduas & </s> <s xml:id="echoid-s4385" xml:space="preserve">maxi-<lb/>mas abſciſſarum tum bic, tum in ſupradictis, & </s> <s xml:id="echoid-s4386" xml:space="preserve">ſequentibus, niſi aliud <lb/>dicatur, ſemper intellige, vel recti, vel ei uſdem obliqui tranſitus, recti <lb/> <anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/> nempè, cum parallelogramma ſunt rectangula, obliqui autem, cum <lb/>non ſuerint rectangula, cum diffinitiones de his allatas.</s> <s xml:id="echoid-s4387" xml:space="preserve"/> </p> <div xml:id="echoid-div438" type="float" level="2" n="1"> <note position="left" xlink:label="note-0196-05" xlink:href="note-0196-05a" xml:space="preserve">_Corol._ <lb/>_20. huius._</note> <note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">_Hux. diff .i._ <lb/>_huius._</note> </div> </div> <div xml:id="echoid-div440" type="section" level="1" n="264"> <head xml:id="echoid-head279" xml:space="preserve">THEOREMA XXXIII. PROPOS. XXXIII.</head> <p> <s xml:id="echoid-s4388" xml:space="preserve">EXpoſitis duabus vtcunq; </s> <s xml:id="echoid-s4389" xml:space="preserve">figuris planis, & </s> <s xml:id="echoid-s4390" xml:space="preserve">in earum vna-<lb/>quaque ſumpta vtcumque regula, vt omnia quadrata <lb/>earumdem figurarum ſumpta iuxta dictas regulas, ita erunt <lb/>ſolida quæcumq; </s> <s xml:id="echoid-s4391" xml:space="preserve">ad inuicem ſimilaria ex eiſ dem figuris ge-<lb/>nita iuxta eaſdem regulas.</s> <s xml:id="echoid-s4392" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4393" xml:space="preserve">Sint duæ vtcunque ſiguræ planæ, ABC, DEF, in quibus duæ <lb/>vtcunque ſint iumptæ regulæ, BC, EF, rectæ lineæ. </s> <s xml:id="echoid-s4394" xml:space="preserve">Dico igitur, <lb/>vt omnia quadrata figuræ, ABC, regula, BC, ad omnia quadrata <lb/>figuræ, DEF, regula, EF, ita eſſe ſolidum fimilare quodcunque <lb/> <anchor type="note" xlink:label="note-0197-02a" xlink:href="note-0197-02"/> genitum ex figura, ABC, iuxta regulam, BC, ad ſibi ſimilare ge-<lb/>nitum ex figura, DEF, iuxta regulam, EF. </s> <s xml:id="echoid-s4395" xml:space="preserve">Ducatur in altera di-<lb/>ctarum figurarum, vtin, DEF, vtcumque regulæ, EF, parallela, <lb/> <anchor type="figure" xlink:label="fig-0197-01a" xlink:href="fig-0197-01"/> HM, quia ergo quadrata habent inter ſe du-<lb/>plam rationem laterum, à quibus deſcribun-<lb/> <anchor type="note" xlink:label="note-0197-03a" xlink:href="note-0197-03"/> tur, ideò quadratum, EF, ad quadratum, <lb/>HM, habebit duplam rationem eius, quam <lb/>habet, EF, ad, HM, ſed etiam aliæ duæ <lb/>quæcumque figuræ planę ſimiles ab eiſdem <lb/>tanquam lineis, vel lateribus homologis ea <lb/> <anchor type="note" xlink:label="note-0197-04a" xlink:href="note-0197-04"/> rumdem deſcriptę habent duplam rationem <lb/>earumdem, ergo, vt quadratum, EF, ad <lb/>quadratum, HM, ita erit alia quælibet figura plana deſcripta ab, E <lb/>F, ad ſimilem ſibi deicriptam ab, HM, ua vt, EF, HM, ſint ea-<lb/>rum homologæ, &</s> <s xml:id="echoid-s4396" xml:space="preserve">, permutando, quadratum, EF, ad aliam ngu-<lb/>ram deſcriptam ab, EF, erit vt quadratum, HM, ad figuram præ-<lb/>dictę ſimilem ab, HM, deſcriptam. </s> <s xml:id="echoid-s4397" xml:space="preserve">Sic etiam eſſe oſtendemus qua-<lb/>dratum cuiuſcumque in figura, DEF, ductæ ipſi, EF, æquidiſtan- <pb o="178" file="0198" n="198" rhead="GEOMETRIÆ"/> tis, ergo, vt vnum ad vnum, ſic omnia ad omnia .</s> <s xml:id="echoid-s4398" xml:space="preserve">i. </s> <s xml:id="echoid-s4399" xml:space="preserve">vt quadratum, <lb/> <anchor type="note" xlink:label="note-0198-01a" xlink:href="note-0198-01"/> EF, ad figuram aliam quamcumq; </s> <s xml:id="echoid-s4400" xml:space="preserve">deſcriptam ab, EF, ſic erunt om-<lb/>nia quadrata figuræ, DEF, regula, EF, ad omnes figuras ſimiles <lb/>eiuſdem figuræ, DEF, regula eadem, EF, ſimiles inquam deſcri-<lb/>ptæ ab, EF, vt autem quadratum, EF, ad figuram deſcriptam ab, <lb/>EF, ita quadratum, BC, ad figuram, quę deſcribitur à, BC, ſimi-<lb/>lis ei, quę deſcripta eſt ab, EF, ita vt deſcribentes ſint earumdem ho-<lb/>mologę, vt autem quadratum, BC, ad figuram deſcriptam à, BC, <lb/>ſic eſſe oſtendemus omnia quadrata figuræ, ABC, regula, BC, ad <lb/>omnes figuras ſimiles eiuſdem figurę, ABC, ſimiles inquam deſcri-<lb/>ptæ à, BC, vel ab, EF, eodem modo, quo id oſtendimus in figura, <lb/>DEF, ergo omnia quadrata figurę, ABC, ad omnes figuras ſimi-<lb/>les alias quaſcunque eiuſdem figuræ, ABC, erunt, vt omnia qua-<lb/> <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> drata figuræ, DEF, ad omnes figuras ſimi-<lb/>les prædictis eiuſdem figuræ, DEF, regulis <lb/>prædictis, BC, EF, ergo permutando, vt <lb/>omnia quadrata figuræ, ABC, ad omnia <lb/>quadrata figurę, DEF, ita erunt omnes fi-<lb/>guræ ſimiles quæcumque figurę, ABC, ad <lb/>omnes figuras ſimiles prædictis figuræ, DE <lb/>F, quia verò omnes figuræ ſimiles alicuius <lb/>figuræ planæ regula quadam ſumptæ, ſunt <lb/>omnia plana ſolidi, quod dicitur ſimilare, & </s> <s xml:id="echoid-s4401" xml:space="preserve"><lb/>genitum ex tali figura iuxta eandem regulam, ideò, vt omnes figurę <lb/> <anchor type="note" xlink:label="note-0198-02a" xlink:href="note-0198-02"/> ſimiles quæcumque ipſius figuræ, ABC, regula, BC, ad omnes fi-<lb/>guras ſimiles ipſius figuræ, DEF, regula, EF, ſimiles inquam præ-<lb/> <anchor type="note" xlink:label="note-0198-03a" xlink:href="note-0198-03"/> dictis .</s> <s xml:id="echoid-s4402" xml:space="preserve">i. </s> <s xml:id="echoid-s4403" xml:space="preserve">vt omnia quadrata figuræ, ABC, regula, BC, ad omnia <lb/>quadrata figuræ, DEF, regula, EF, ita erunt omnia plana ſolidi <lb/>ſimilaris cuiuſcumque geniti ex figura, ABC, iuxta regulam, BC, <lb/>ad omnia plana ſolidi ſimilaris geniti ex figura, DEF, iuxta regu-<lb/>lam, EF, vt au@em omnia plana duorum ſolidorum ſic & </s> <s xml:id="echoid-s4404" xml:space="preserve">ipſa ſoli-<lb/> <anchor type="note" xlink:label="note-0198-04a" xlink:href="note-0198-04"/> da, ergo etiam ſolida ſimilaria genita ex figuris, ABC, DEF, (quę <lb/>ſunt ſimilaria ad inuicem, quia omnes figuræ ſimiles figuræ, ABC, <lb/>ſunt etiam ſimiles omnibus figuris ſimilibus figuræ, DEF,) iuxta <lb/>regulas, BC, EF, erunt ad inuicem, vt omnia quadrata earumdem <lb/>figurarum eiſdem regulis ſumpta, quod erat oſtendendum.</s> <s xml:id="echoid-s4405" xml:space="preserve"/> </p> <div xml:id="echoid-div440" type="float" level="2" n="1"> <note position="right" xlink:label="note-0197-02" xlink:href="note-0197-02a" xml:space="preserve">Vide B. <lb/>Definit. 8. <lb/>huius.</note> <figure xlink:label="fig-0197-01" xlink:href="fig-0197-01a"> <image file="0197-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0197-01"/> </figure> <note position="right" xlink:label="note-0197-03" xlink:href="note-0197-03a" xml:space="preserve">8. & 15. <lb/>huius.</note> <note position="right" xlink:label="note-0197-04" xlink:href="note-0197-04a" xml:space="preserve">15. huius.</note> <note position="left" xlink:label="note-0198-01" xlink:href="note-0198-01a" xml:space="preserve">Coroll. 4. <lb/>huius.</note> <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a"> <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0198-01"/> </figure> <note position="left" xlink:label="note-0198-02" xlink:href="note-0198-02a" xml:space="preserve">B. Diff. 8. <lb/>huius.</note> <note position="left" xlink:label="note-0198-03" xlink:href="note-0198-03a" xml:space="preserve">Poſtulatũ <lb/>2. huius.</note> <note position="left" xlink:label="note-0198-04" xlink:href="note-0198-04a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div442" type="section" level="1" n="265"> <head xml:id="echoid-head280" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s4406" xml:space="preserve">_H_Inc patet, ſi in figura, ABC, vtcumq; </s> <s xml:id="echoid-s4407" xml:space="preserve">regula, BC, deſcripſerit <lb/>duas quaſcumque figuras, quod vt vna ad aliam, ita erunt om-<lb/>nes figuræ ipſius, ABC, ſimiles vni deſcriptarum ad omnes figuras eiu- <pb o="179" file="0199" n="199" rhead="LIBER II."/> ſdem Fimiles alteri deſcriptarum .</s> <s xml:id="echoid-s4408" xml:space="preserve">i. </s> <s xml:id="echoid-s4409" xml:space="preserve">ita omnia plana ſolidi ſimilaris ge-<lb/>niti ex, ABC, iuxta regulam, BC, (ſimilibus exiſtentibus eiuſdem fi-<lb/>guris vni deſcriptarum) ad omnia plana ſolidi ſimilaris geniti ex eadem <lb/>figura iuxta eandem regulam (huius ſimilibus exiſtentibus figuris alteri <lb/>deſcriptarum) ideſt ita erunt ſolida ſimilaria genita ex eadem figura, A <lb/>BC, iuxta communem regulam, BC, non tamen ſimilaria ad inuicem. <lb/></s> <s xml:id="echoid-s4410" xml:space="preserve">ſed, quarum omnia plana ſunt omnes figuræ ſimiles eiuſdem, ABC, ſi-<lb/>miles inquam, quæ ſunt vnius dictorum ſolidorum vni deſcriptar um à, <lb/>BC, figurarum, & </s> <s xml:id="echoid-s4411" xml:space="preserve">quæ ſunt alterius, ſimiles alteri deſcriptarum fi-<lb/>gurarum.</s> <s xml:id="echoid-s4412" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div443" type="section" level="1" n="266"> <head xml:id="echoid-head281" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s4413" xml:space="preserve">_V_Nde ſolida ſimilaria, ſed non ad inuicem, genita ex. </s> <s xml:id="echoid-s4414" xml:space="preserve">gr. </s> <s xml:id="echoid-s4415" xml:space="preserve">à figuris, <lb/>ABC, DEF, iuxta regulas, BC, EF, quæ duas figuras planas <lb/>diſſimiles deſcripſerint, quibus ſint ſimiles figuræ, quæ dicuntur omnia <lb/>plana dictorum ſimilarium ſolidorum, erunt ad inuicem, vt ipſæ figuræ <lb/>diſſimiles deſcriptæ ab ipſis, BC, EF, nam ſolidum ſimilare genitum ex, <lb/>DEF, iuxta regulam, EF, ad ſibi ſimilare genitum ex figura, ABC, <lb/>iuxta regulam, BC, erit vt figura deſcripta ab, EF, ad ſibi ſimilem fi-<lb/>guram deſcriptam à, BC, item ſolidum boc ſimilare genitum ex figu-<lb/>ra, ABC, iuxta regulam, BC, ad ſolidum ſimilare, ſed non fibi, <lb/>genitum ex eadem figura iuxta eandem regulam, BC, erit vt figura de-<lb/>ſcripta à, BC, ſimilis deſcriptæ ab, EF, ad figuram ſibi diſſimilem de-<lb/>ſcriptam ab eadem, BC, (quibus figuris diſſimilibus ſint ſimiles figuræ, <lb/>quæ dicuntur omnia plana ſolidorum ſimilarium genitorum ex eadem fi-<lb/>gura, ABC, iuxta communem regulam, BC,) ergo, ex æquali ſolidum <lb/>ſimilare genitum ex figura, DEF, ad ſolidum ſimilare, ſed non ſibi, ge-<lb/>nitum ex figura, ABC, (genita intellige iuxta regulas, EF, BC,) erit <lb/>vt figura deſcripta ab, EF, cui ſunt ſimiles figuræ huius ſolidi, ad figu-<lb/>ram deſcriptam à, BC, prædictæ diſſimilem, cui ſunt ſimiles figuræ ſoli-<lb/>di ex, BAC, geniti iuxta regulam, BC.</s> <s xml:id="echoid-s4416" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div444" type="section" level="1" n="267"> <head xml:id="echoid-head282" xml:space="preserve">THEOREMA XXXIV. PROPOS. XXXIV.</head> <p> <s xml:id="echoid-s4417" xml:space="preserve">SOlida ſimilaria genita ex parallelogrammis iuxta regu-<lb/>lam vnum eorundem laterum, ſunt cylindrici; </s> <s xml:id="echoid-s4418" xml:space="preserve">& </s> <s xml:id="echoid-s4419" xml:space="preserve">ſoli-<lb/>da ſimilaria genita ex triangulis iuxta regulam vnum eorun-<lb/>dem laterum, ſunt conici, quorum baſes erunt figuræ à re-<lb/>gulis deſcriptæ, & </s> <s xml:id="echoid-s4420" xml:space="preserve">latera, eorundem parallelogrammorum, <lb/>vel triangulorum latera regulis inſiſtentia.</s> <s xml:id="echoid-s4421" xml:space="preserve"/> </p> <pb o="180" file="0200" n="200" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s4422" xml:space="preserve">Sit ergo expoſitum quodcumq; </s> <s xml:id="echoid-s4423" xml:space="preserve">parallelogrammum, AC, & </s> <s xml:id="echoid-s4424" xml:space="preserve">tri-<lb/>angulum, FGH, in quibus pro regulis ſumantur latera, BC, GH. <lb/></s> <s xml:id="echoid-s4425" xml:space="preserve">Dico ſolidum quodcumq; </s> <s xml:id="echoid-s4426" xml:space="preserve">ſimilare genitum ex parallelogrammo, A <lb/>C, (iuxta regulam, BC,) eſſe cylindricum, cuius baſis erit a, BC, <lb/>deſcripta figura, & </s> <s xml:id="echoid-s4427" xml:space="preserve">latus, vtrumuis ipſorum, AB, LC, laterum re-<lb/>gulæ, BC, inſiſtentium; </s> <s xml:id="echoid-s4428" xml:space="preserve">Et genitum ex triangulo, FGH, iuxta <lb/>regulam, GH, eſſe conicum, cuius baſis erit à, GH, deicripta fi-<lb/>gura, & </s> <s xml:id="echoid-s4429" xml:space="preserve">latus vtrumuis duorum, FG, FH, regulæ, GH, inſiſten-<lb/>tium. </s> <s xml:id="echoid-s4430" xml:space="preserve">Deſcribantur à regulis, BC, GH, figuræ vtcunque planæ, <lb/>BCE, GHP, æqualiter inclinatę planis, AC, FGH, deinde per <lb/>circuitum figuræ, BCE, feratur latus, AB, cui ſit æqualelatus, E <lb/>X, quodque puncto, A, deſcribat circuitum figuræ, AXL, & </s> <s xml:id="echoid-s4431" xml:space="preserve">per <lb/>circuitum figuræ, GHP, feratur vtrumuis laterum, FG, FH, in-<lb/> <anchor type="figure" xlink:label="fig-0200-01a" xlink:href="fig-0200-01"/> definitè productum verſus fi-<lb/>guram, GHP, cuius portio <lb/>inter, F, & </s> <s xml:id="echoid-s4432" xml:space="preserve">punctum, P, fit, <lb/>FP, erit ergo ſolidum quod <lb/>clauditur ſuperficie cylindri-<lb/> <anchor type="note" xlink:label="note-0200-01a" xlink:href="note-0200-01"/> ca, deſcripta latere, AB, & </s> <s xml:id="echoid-s4433" xml:space="preserve"><lb/>duabus figuris, ALX, BC <lb/>E, cylindricus; </s> <s xml:id="echoid-s4434" xml:space="preserve">& </s> <s xml:id="echoid-s4435" xml:space="preserve">quod c au-<lb/>ditur ſuperficie conica de-<lb/>ſcripta altero laterum, FG, <lb/>FH, indefinitè producto, & </s> <s xml:id="echoid-s4436" xml:space="preserve"><lb/>figura, GHP, erit conicus. <lb/></s> <s xml:id="echoid-s4437" xml:space="preserve">Dico autem ſolidum ſimilare genitum ex, AC, iuxta regulam, BC, <lb/>cuius omnia plana ſint omnes figuræ ipſius, AC, ſimiles figuræ, B <lb/>CE, eſſe hunc cylindricum, ACE, nam quælibet earum, quæ di-<lb/>cuntur omnes figuræ ſimiles parallelogrammi, AC, regula, BC, <lb/> <anchor type="note" xlink:label="note-0200-02a" xlink:href="note-0200-02"/> ſimiles inquam figuræ, BCE, eſt etiam ſimiliter poſita, ac, BCE, <lb/>deſcripta latere homologo ipſi, BC, igitur eius perimeter erit in ſu-<lb/>perficie cylindrica deſcripta latere, AB, ſi enim aliquid eius eſſet in-<lb/>tra, vel extra illam ſuperficiem, aliquid eius eſſet intra, vel extra com-<lb/>munem ſectionem talis aſſumptę figurę, & </s> <s xml:id="echoid-s4438" xml:space="preserve">ſuperficiei cylindricę, ta-<lb/>lis autem communis ſectio eſt perimeter figuræ ſimilis, & </s> <s xml:id="echoid-s4439" xml:space="preserve">ſimiliter <lb/>poſitę, ac, BCE, (quia ſi cylindricus plano ſecetur baſi æquidiſtan-<lb/> <anchor type="note" xlink:label="note-0200-03a" xlink:href="note-0200-03"/> te concepta figura erit ſimilis, & </s> <s xml:id="echoid-s4440" xml:space="preserve">ſimiliter poſita, ac baſis) ergo ha-<lb/>beremus duas figuras ab eodem latere homologo deſcriptas, ſimiles <lb/>æquales, & </s> <s xml:id="echoid-s4441" xml:space="preserve">ſimiliter poſitas, & </s> <s xml:id="echoid-s4442" xml:space="preserve">non congruentes, quod eſt abſur-<lb/> <anchor type="note" xlink:label="note-0200-04a" xlink:href="note-0200-04"/> dum, congruent igitur, erit ergo aſſumpta figura, quæ eſt vna ea-<lb/>rum, quę dicuntur omnes figurę parallelogrammi, AC, ſimiles ipſi, <lb/>BCE, regula, BC, & </s> <s xml:id="echoid-s4443" xml:space="preserve">eſt vnum eorum, quæ dicuntur omnia pla- <pb o="181" file="0201" n="201" rhead="LIBER II."/> na ſolidi ſimilaris geniti ex, AC, iuxta regulam, BC, erit, inquam, <lb/>aſſumpta figura etiam vnum eorum, quæ dicuntur omnia plana cy-<lb/>lindrici, ACE, regula, BCE, quod etiam de cæteris ſimili modo <lb/>oſtendetur, ergo ſolidum ſimilare genitum ex, AC, iuxta regulam, <lb/>BC, & </s> <s xml:id="echoid-s4444" xml:space="preserve">cylindricus, ACE, habebunt omnia plana (regula, BCE, <lb/>aſiumpta) communia, ergo ſolidum ſimilare genitum ex, AC, iux-<lb/>ta regulam, BC, erit idem cylindrico, ACE, cuius baſis eſt figura, <lb/>BCE, & </s> <s xml:id="echoid-s4445" xml:space="preserve">latus alterum laterum, AB, LC. </s> <s xml:id="echoid-s4446" xml:space="preserve">Similiter oſtendemus ſo-<lb/>lidum ſimilare genitum ex triangulo, FGH, iuxta regulam, GH, <lb/>eſſe dem conico, FGHP, cuius latus alterum laterum, FG, FH, <lb/>& </s> <s xml:id="echoid-s4447" xml:space="preserve">baſis eſt figura, GHP, conſimili via ſupradictæ procedentes, quę <lb/>erant demonſtranda.</s> <s xml:id="echoid-s4448" xml:space="preserve"/> </p> <div xml:id="echoid-div444" type="float" level="2" n="1"> <figure xlink:label="fig-0200-01" xlink:href="fig-0200-01a"> <image file="0200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0200-01"/> </figure> <note position="left" xlink:label="note-0200-01" xlink:href="note-0200-01a" xml:space="preserve">Ex def. 3. <lb/>324. lib. 1.</note> <note position="left" xlink:label="note-0200-02" xlink:href="note-0200-02a" xml:space="preserve">A. def. 8. <lb/>huius.</note> <note position="left" xlink:label="note-0200-03" xlink:href="note-0200-03a" xml:space="preserve">Corol. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0200-04" xlink:href="note-0200-04a" xml:space="preserve">Corollar. <lb/>25. l. 1.</note> </div> </div> <div xml:id="echoid-div446" type="section" level="1" n="268"> <head xml:id="echoid-head283" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s4449" xml:space="preserve">_H_Inc manifeſtum eſt, ſi figuræ deſcriptæ à, BC, GH, ſint circuli, <lb/>quod ſolidum ſimilare genitum ex, AC, erit cylindrus, & </s> <s xml:id="echoid-s4450" xml:space="preserve">geni-<lb/>tum ex triangulo, FGH, conus ſiue recti, ſiue ſcaleni, ſi verò deſ criptæ <lb/>figuræ ſint rectilineæ, genitum ex, AC, erit priſma, ex, FGH, autem <lb/>pyramis, ſiue recta, ſiue ſcalena cętera autem nomine communi vo-<lb/>cantur ſolida ſimilaria genita ex eiſdem fig. </s> <s xml:id="echoid-s4451" xml:space="preserve">iuxta regulas, intellige, <lb/>BC, GH.</s> <s xml:id="echoid-s4452" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div447" type="section" level="1" n="269"> <head xml:id="echoid-head284" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s4453" xml:space="preserve">_S_I intra triangulum, FGH, ducamus ipſi, GH, parallelam vtcunq; <lb/></s> <s xml:id="echoid-s4454" xml:space="preserve">quæ ſit, DI, abſcindens à triangulo, FGH, trapezium, DH, oſten-<lb/>demus eodem modo, quo ſupra, ſolidum ſimilare genitum ex trape-<lb/>zio, DH, iuxta regulam, GH, eſſe fruſtum ſolidi ſimilaris geniti ex <lb/>triangulo, FGH, iuxta eandem regulam, ideſt fruſtum conici, FGHP, <lb/> <anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/> .</s> <s xml:id="echoid-s4455" xml:space="preserve">i. </s> <s xml:id="echoid-s4456" xml:space="preserve">cont, cum figura deſcripta à GH, eſt circulus, vel fruſtum pyra-<lb/>midis rectæ, ſiue ſcalenæ, cum illa eſi figura rectilinea, quæ facilè <lb/>oſtendentur.</s> <s xml:id="echoid-s4457" xml:space="preserve"/> </p> <div xml:id="echoid-div447" type="float" level="2" n="1"> <note position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">_B. def. 4.<unsure/>_ <lb/>_l. I._</note> </div> </div> <div xml:id="echoid-div449" type="section" level="1" n="270"> <head xml:id="echoid-head285" xml:space="preserve">COROLLARIVM III.</head> <p style="it"> <s xml:id="echoid-s4458" xml:space="preserve">_T_Andem patet vice verſa, ſi quiuis cylindricus, vel conicus, vel <lb/>eius fruſium, intelligatur ſecari per latera, de illo plano ſecante <lb/> <anchor type="note" xlink:label="note-0201-02a" xlink:href="note-0201-02"/> conceptam in ſecto ſolido figuram eſſe genitricem earumdem per deſcri-<lb/>ptionem ſimitium figurarum, & </s> <s xml:id="echoid-s4459" xml:space="preserve">ipſa eſſe ſolida ſimilaria genita ex ei-<lb/>ſdem figuris geni@ricibus iuxta regulas communes ſectiones planorum, <pb o="182" file="0202" n="202" rhead="GEOMETRI Æ"/> ſecantium, & </s> <s xml:id="echoid-s4460" xml:space="preserve">baſium, quæ figuræ genitrices in cylindricis erunt paral-<lb/> <anchor type="note" xlink:label="note-0202-01a" xlink:href="note-0202-01"/> lelogramma in conicis triangula, & </s> <s xml:id="echoid-s4461" xml:space="preserve">in fruſtis conicis trapezia; </s> <s xml:id="echoid-s4462" xml:space="preserve">igitur <lb/>verum eſt quodlibet ſolidum ſimilare genitum ex parallelogrammo iuxta <lb/>regulam vnum laterum eſſe cylindricum, & </s> <s xml:id="echoid-s4463" xml:space="preserve">genitum ex triangulo iux-<lb/>taregulam vnum laterum eſſe conicum, & </s> <s xml:id="echoid-s4464" xml:space="preserve">ex trapezio eſſe fruſtum co-<lb/>nicum; </s> <s xml:id="echoid-s4465" xml:space="preserve">& </s> <s xml:id="echoid-s4466" xml:space="preserve">vice verſa, quemlibet cylindrum eſſe ſolidum ſimilare geni-<lb/>tum ex parallelogrammo in ipſo producto per planum per latera ductum, <lb/>genitum inquam iuxta communem ſectionem eius, & </s> <s xml:id="echoid-s4467" xml:space="preserve">baſis cylindrici; <lb/></s> <s xml:id="echoid-s4468" xml:space="preserve">& </s> <s xml:id="echoid-s4469" xml:space="preserve">quemlibet conicum eſſe ſolidum ſimilare genitum ex triangulo in eo-<lb/> <anchor type="figure" xlink:label="fig-0202-01a" xlink:href="fig-0202-01"/> dem producto per traiectio-<lb/>nem plani per latera, geni-<lb/>tum, inquam iuxta commu-<lb/>nem ſectionem eius, & </s> <s xml:id="echoid-s4470" xml:space="preserve">baſis <lb/>dicticonici; </s> <s xml:id="echoid-s4471" xml:space="preserve">& </s> <s xml:id="echoid-s4472" xml:space="preserve">quodlibet fru-<lb/>ſtum conicum eſſe ſolidum ſi. <lb/></s> <s xml:id="echoid-s4473" xml:space="preserve">milare genitum ex trapezio in <lb/>ipſo producto per traiectionem <lb/>plani per latera eiuſdem fru-<lb/>ſti, genitum inquam iuxtare-<lb/>gulam communem ſectionem-<lb/>eius, & </s> <s xml:id="echoid-s4474" xml:space="preserve">vnius baſium eiu-<lb/>ſdem: </s> <s xml:id="echoid-s4475" xml:space="preserve">Siue ergo, expoſito pa-<lb/>rallelogrammo, & </s> <s xml:id="echoid-s4476" xml:space="preserve">triangulo intellexeris iuxta diffin 8. </s> <s xml:id="echoid-s4477" xml:space="preserve">huius, deſcribi <lb/>omnes figuras ſimiles eis quæ deſcribuntur à baſibus dicti parallelogram-<lb/>mi, & </s> <s xml:id="echoid-s4478" xml:space="preserve">trianguli, & </s> <s xml:id="echoid-s4479" xml:space="preserve">ſic conceperis effici ſolidum, cuius illæ ſunt omnia <lb/>plana; </s> <s xml:id="echoid-s4480" xml:space="preserve">ſiue intellexeris latus dicti parallelogrammi, vel trianguli in-<lb/> <anchor type="note" xlink:label="note-0202-02a" xlink:href="note-0202-02"/> definitè productum reuolui per circuitum figurarum à baſibus deſcripta-<lb/>rum, vt babeas ſolidum dicta ſuperficie deſcripta, & </s> <s xml:id="echoid-s4481" xml:space="preserve">baſi, vel baſibus <lb/>comprehenſum, idem vtroque modo tibi obuenit ſolidum, poteſt autem <lb/>prior vocarigeneratio ſolidorum per deſcriptionem figurarum; </s> <s xml:id="echoid-s4482" xml:space="preserve">poſteriov <lb/>autem, generatio ſolidorum per reuolutionem facta, quæ maioris diluci-<lb/>dationis gratia bic appoſui, vt ex hac declaratione aliqualiter pateat, <lb/>in plurimis etiam alijs vtramq; </s> <s xml:id="echoid-s4483" xml:space="preserve">gener ationemritè nos imaginari poſſe, <lb/>vt in ſpbęra, ſphęroide, & </s> <s xml:id="echoid-s4484" xml:space="preserve">conoidibus, & </s> <s xml:id="echoid-s4485" xml:space="preserve">eiuſdem fruſtis, & </s> <s xml:id="echoid-s4486" xml:space="preserve">alijs quam-<lb/>plurimis, vt ſuo loco animaduertetur.</s> <s xml:id="echoid-s4487" xml:space="preserve"/> </p> <div xml:id="echoid-div449" type="float" level="2" n="1"> <note position="right" xlink:label="note-0201-02" xlink:href="note-0201-02a" xml:space="preserve">_13. l. I._ <lb/>_2. & Cor._ <lb/>_l. I._</note> <note position="left" xlink:label="note-0202-01" xlink:href="note-0202-01a" xml:space="preserve">_Cor. 6. l. I._ <lb/>_16. lib. l._</note> <figure xlink:label="fig-0202-01" xlink:href="fig-0202-01a"> <image file="0202-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0202-01"/> </figure> <note position="left" xlink:label="note-0202-02" xlink:href="note-0202-02a" xml:space="preserve">_Diff. 3. &_ <lb/>_4. lib. I._</note> </div> <figure> <image file="0202-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0202-02"/> </figure> <pb o="183" file="0203" n="203" rhead="LIBER II."/> </div> <div xml:id="echoid-div451" type="section" level="1" n="271"> <head xml:id="echoid-head286" xml:space="preserve">A. COROLLARII IV. GENERALIS.</head> <head xml:id="echoid-head287" xml:space="preserve">SECTIO I.</head> <p style="it"> <s xml:id="echoid-s4488" xml:space="preserve">_E_T quoniam, vt oſtenſum eſt Prop. </s> <s xml:id="echoid-s4489" xml:space="preserve">33. </s> <s xml:id="echoid-s4490" xml:space="preserve">huius Libri, vt omnia qua-<lb/>drata duarum figurarum inter ſe ſumpta cum datis regulis, ita <lb/>ſunt ſolida ſimilaria genita ex ijſdem figuris iuxta eaſdem regulas, ideò <lb/>cumin Propoſitionibus huius Libri inuenta eſt ratio omnium quadrato-<lb/>rum par allelogrammorum, vel triangulorum, vel trapeziorum, regu-<lb/>lis eorum lateribus, eandem rationem comperiemus habere ſolida ſimi-<lb/>laria genita ex parallelogrammis, ideſt cylindricos, vel ex triangulis, <lb/>ideſt conicos, vel ex trapezijs, ideſt fruſta conica, genitainquam iuxta <lb/>eaſdem regulas, quæ amplius dilucidabimus ſingula, quæ opportuna <lb/>fuerint, Theoremata denuò aſſumentes.</s> <s xml:id="echoid-s4491" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div452" type="section" level="1" n="272"> <head xml:id="echoid-head288" xml:space="preserve">B. SECTIO II.</head> <p style="it"> <s xml:id="echoid-s4492" xml:space="preserve">_I_N Propoſ. </s> <s xml:id="echoid-s4493" xml:space="preserve">9. </s> <s xml:id="echoid-s4494" xml:space="preserve">igitur expoſita denuò eius figura, intelligantur baſes, <lb/>GM, MH, deſcribere ſimiles figuras planas, quæ ſint, GIMR, M <lb/>THS, vt eorum lineæ, vel latera homologa, æquè erectas planis, AM, <lb/> <anchor type="figure" xlink:label="fig-0203-01a" xlink:href="fig-0203-01"/> MC, & </s> <s xml:id="echoid-s4495" xml:space="preserve">in ijs, tanquam in baſibus conſiſte-<lb/>re cylindricos, AM, BH, quorum latera <lb/>ſint, AG, CH, erunt igitur hi cylindrici <lb/>ſolida ſimilaria genita ex parallelogram-<lb/> <anchor type="note" xlink:label="note-0203-01a" xlink:href="note-0203-01"/> mis, AM, MC, iuxta regulas, GM, M <lb/>H, igitur erunt, vt omnia quadrata eorun-<lb/>dem regulis eiſdem, GM, MH, ſunt au-<lb/> <anchor type="note" xlink:label="note-0203-02a" xlink:href="note-0203-02"/> tem omnia eorum quadrata, vt quadrata baſium, GM, MH, .</s> <s xml:id="echoid-s4496" xml:space="preserve">i. </s> <s xml:id="echoid-s4497" xml:space="preserve">ergo cy-<lb/>lindrici, AM, MC, erunt vt quadrata baſium, GM, MH, .</s> <s xml:id="echoid-s4498" xml:space="preserve">i. </s> <s xml:id="echoid-s4499" xml:space="preserve">vt figu-<lb/>ræ ſimiles, GIMR, MSHT, igitur cylindrici in eadem altitudine, & </s> <s xml:id="echoid-s4500" xml:space="preserve"><lb/>ſimilibus baſibus inſiſtentes ſunt, vt ipſæ baſes.</s> <s xml:id="echoid-s4501" xml:space="preserve"/> </p> <div xml:id="echoid-div452" type="float" level="2" n="1"> <figure xlink:label="fig-0203-01" xlink:href="fig-0203-01a"> <image file="0203-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0203-01"/> </figure> <note position="right" xlink:label="note-0203-01" xlink:href="note-0203-01a" xml:space="preserve">_Coroll. 3._ <lb/>_ant._</note> <note position="right" xlink:label="note-0203-02" xlink:href="note-0203-02a" xml:space="preserve">_9. huius._</note> </div> </div> <div xml:id="echoid-div454" type="section" level="1" n="273"> <head xml:id="echoid-head289" xml:space="preserve">C. SECTIO III.</head> <p style="it"> <s xml:id="echoid-s4502" xml:space="preserve">_I_N Propoſ. </s> <s xml:id="echoid-s4503" xml:space="preserve">10. </s> <s xml:id="echoid-s4504" xml:space="preserve">conſimiliter procedentes collige@@us, cylindricos in <lb/>eadem, vel æqualibus, ac ſimilibus baſibus conſiſtentes eſſe, vt alti-<lb/>tudines, vel vt latera ęqualiter eorundem baſibus inclinata.</s> <s xml:id="echoid-s4505" xml:space="preserve"/> </p> <pb o="184" file="0204" n="204" rhead="GEOMETRI Æ"/> </div> <div xml:id="echoid-div455" type="section" level="1" n="274"> <head xml:id="echoid-head290" xml:space="preserve">D. SECTIO IV.</head> <p style="it"> <s xml:id="echoid-s4506" xml:space="preserve">_I_N Propoſ. </s> <s xml:id="echoid-s4507" xml:space="preserve">II. </s> <s xml:id="echoid-s4508" xml:space="preserve">deducemus cylindricos ſimilibus baſibus inſiſtentes <lb/>habere inter ſe rationem compoſitam ex ratione baſium, & </s> <s xml:id="echoid-s4509" xml:space="preserve">altitu-<lb/>dinum, vel laterum æqualiter dictis baſibus inclinatorum.</s> <s xml:id="echoid-s4510" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div456" type="section" level="1" n="275"> <head xml:id="echoid-head291" xml:space="preserve">E. SECTIO V.</head> <p style="it"> <s xml:id="echoid-s4511" xml:space="preserve">_I_N Propoſ. </s> <s xml:id="echoid-s4512" xml:space="preserve">12. </s> <s xml:id="echoid-s4513" xml:space="preserve">colligemus cylindricos, quorum ſimiles baſes altitù-<lb/>dinibus, vel lateribus æqualiter baſibus inclinatis reciprocè re-<lb/>ſpondent, eſſe æquales; </s> <s xml:id="echoid-s4514" xml:space="preserve">& </s> <s xml:id="echoid-s4515" xml:space="preserve">cylindricos æquales, ſimilibus baſibus inſi-<lb/>ſtentes, baſes habere altitudinibus, vel lateribus æqualiter baſibus in-<lb/>clinatis, reciprocas.</s> <s xml:id="echoid-s4516" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div457" type="section" level="1" n="276"> <head xml:id="echoid-head292" xml:space="preserve">F. SECTIO VI.</head> <p style="it"> <s xml:id="echoid-s4517" xml:space="preserve">_I_N Prop. </s> <s xml:id="echoid-s4518" xml:space="preserve">13. </s> <s xml:id="echoid-s4519" xml:space="preserve">habebimus ſimiles cylindricos eſſe in tripla ratione la-<lb/>terum homologorum.</s> <s xml:id="echoid-s4520" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div458" type="section" level="1" n="277"> <head xml:id="echoid-head293" xml:space="preserve">G. SECTIO VII.</head> <p style="it"> <s xml:id="echoid-s4521" xml:space="preserve">_E_X Prop. </s> <s xml:id="echoid-s4522" xml:space="preserve">14. </s> <s xml:id="echoid-s4523" xml:space="preserve">colligimus ſi prædicti cylindrici inſiſtant baſibus diſ-<lb/>ſimilibus, adh uc prædictas paſſiones de ipſis ver@ficari; </s> <s xml:id="echoid-s4524" xml:space="preserve">in quibus <lb/>tamen non numerant ur ſimiles cylindrici, cum oporteat eoſdem ſimiles <lb/>baſes habere.</s> <s xml:id="echoid-s4525" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div459" type="section" level="1" n="278"> <head xml:id="echoid-head294" xml:space="preserve">H. SECTIO VIII.</head> <p style="it"> <s xml:id="echoid-s4526" xml:space="preserve">_I_N Prop. </s> <s xml:id="echoid-s4527" xml:space="preserve">22. </s> <s xml:id="echoid-s4528" xml:space="preserve">habemus in eius figura, ſolida ſimilaria genita ex pa-<lb/>rallelogrammis, AS, Τβ, iuxta regulas, ES, Ζβ, ea@dem ratio-<lb/>nem babere ad ſolidi ſimilaria genita ex triangulis, OES, & </s> <s xml:id="echoid-s4529" xml:space="preserve">Ζβ, ideſt <lb/> <anchor type="note" xlink:label="note-0204-01a" xlink:href="note-0204-01"/> cyliadricos @enitos ex, AS, Τβ ad conicos genitos ex triangulis, <lb/>OES, & </s> <s xml:id="echoid-s4530" xml:space="preserve">Ζβ, eaadem rationem babere, vnde, cum conica<unsure/> ſint partes <lb/>proportionales cylindricorum in eadem altitudine cum ipſis exiſten-<lb/>tium, quæeunq; </s> <s xml:id="echoid-s4531" xml:space="preserve">de cylindricis in huius Coroll. </s> <s xml:id="echoid-s4532" xml:space="preserve">Sectionibus 2. </s> <s xml:id="echoid-s4533" xml:space="preserve">3. </s> <s xml:id="echoid-s4534" xml:space="preserve">4. </s> <s xml:id="echoid-s4535" xml:space="preserve">5. </s> <s xml:id="echoid-s4536" xml:space="preserve">6. <lb/></s> <s xml:id="echoid-s4537" xml:space="preserve">& </s> <s xml:id="echoid-s4538" xml:space="preserve">7. </s> <s xml:id="echoid-s4539" xml:space="preserve">collecta ſunt, eadem & </s> <s xml:id="echoid-s4540" xml:space="preserve">pro conicis tamquam collecta recipiemus.</s> <s xml:id="echoid-s4541" xml:space="preserve"/> </p> <div xml:id="echoid-div459" type="float" level="2" n="1"> <note position="left" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">_Ex hae_ <lb/>_Propoſ._</note> </div> <pb o="185" file="0205" n="205" rhead="LIBER II."/> </div> <div xml:id="echoid-div461" type="section" level="1" n="279"> <head xml:id="echoid-head295" xml:space="preserve">I. SECTIO IX.</head> <p style="it"> <s xml:id="echoid-s4542" xml:space="preserve">_I_N Propoſ. </s> <s xml:id="echoid-s4543" xml:space="preserve">24. </s> <s xml:id="echoid-s4544" xml:space="preserve">habemus quemcumque cylindricum eſſe triplum coni-<lb/>ci in eadem baſi, & </s> <s xml:id="echoid-s4545" xml:space="preserve">altitudine cum ipſo. </s> <s xml:id="echoid-s4546" xml:space="preserve">Sit cxpoſitus quicunq; </s> <s xml:id="echoid-s4547" xml:space="preserve">cy-<lb/>lindricus, AE, in baſi, DHEF, in eadem autem baſi, & </s> <s xml:id="echoid-s4548" xml:space="preserve">altitudine ſit <lb/>conicus, DBE, ſic tamen baſi inſiſtens, vt ducto plano per latera conici, <lb/>idem tranſeat per latera cylindrici, AE, ſit autem ductum tale planum, <lb/> <anchor type="figure" xlink:label="fig-0205-01a" xlink:href="fig-0205-01"/> quod faciat in conico, DBE, triangulum, <lb/>DBE, & </s> <s xml:id="echoid-s4549" xml:space="preserve">in cylindrico, AE, parallelo-<lb/> <anchor type="note" xlink:label="note-0205-01a" xlink:href="note-0205-01"/> grammum, AE, erunt igitur, AE, & </s> <s xml:id="echoid-s4550" xml:space="preserve">tri-<lb/>angulum, DBE, genitrices figuræ eorum-<lb/> <anchor type="note" xlink:label="note-0205-02a" xlink:href="note-0205-02"/> dem ſolidorum, quæ ſimilaria ad inuicem, <lb/>vocantur, genita iuxta communem regulam, <lb/>DE, quod ergo gignitur ex, AE, ad geni-<lb/>tum ex triangulo, DBE, erit vt omnia qua-<lb/>drata, AE, ad omnia quadrata trianguli, <lb/>DBE, regula, DE, ideſt triplum, ſolidum <lb/>verò ſimilare genitum ex, AE, iuxta re-<lb/> <anchor type="note" xlink:label="note-0205-03a" xlink:href="note-0205-03"/> gulam, DE, cuius figuræ ſint ſimiles figuræ, DFEH, eſt cylindricus, <lb/>AE, & </s> <s xml:id="echoid-s4551" xml:space="preserve">ſolidum ſimilare genitum ex triangulo, DBE, iuxta regulam, <lb/>DE, cuius figuræ ſint ſimiles pariter figuræ, DFEH, eſt conicus, DBE, <lb/>ergo cylindricus, AE, triplus erit conici, DBE, & </s> <s xml:id="echoid-s4552" xml:space="preserve">conſequenter tri-<lb/>plus erit cuiuſuis alij in eadem baſi, DFEH, & </s> <s xml:id="echoid-s4553" xml:space="preserve">altitudine, cum coni-<lb/> <anchor type="note" xlink:label="note-0205-04a" xlink:href="note-0205-04"/> co, DBE, exiſtentis, quoniam, vt oſtenſum eſt, conici in eadem alti-<lb/>tudme ſiantes ſunt, vt baſes, vnde cum baſes ſunt æquales, & </s> <s xml:id="echoid-s4554" xml:space="preserve">conici <lb/>ſunt æquales, verum ergo eſt, quod proponebatur.</s> <s xml:id="echoid-s4555" xml:space="preserve"/> </p> <div xml:id="echoid-div461" type="float" level="2" n="1"> <figure xlink:label="fig-0205-01" xlink:href="fig-0205-01a"> <image file="0205-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0205-01"/> </figure> <note position="right" xlink:label="note-0205-01" xlink:href="note-0205-01a" xml:space="preserve">_Cor. 6. &_ <lb/>_16. lib. I._</note> <note position="right" xlink:label="note-0205-02" xlink:href="note-0205-02a" xml:space="preserve">_Corol. 3._ <lb/>_34. huius._</note> <note position="right" xlink:label="note-0205-03" xlink:href="note-0205-03a" xml:space="preserve">_24. huius._</note> <note position="right" xlink:label="note-0205-04" xlink:href="note-0205-04a" xml:space="preserve">_34. huius._ <lb/>_Per B. Co-_ <lb/>_rollar. 27._ <lb/>_huius._</note> </div> </div> <div xml:id="echoid-div463" type="section" level="1" n="280"> <head xml:id="echoid-head296" xml:space="preserve">K. SECTIO X.</head> <p style="it"> <s xml:id="echoid-s4556" xml:space="preserve">_I_N Prop. </s> <s xml:id="echoid-s4557" xml:space="preserve">27. </s> <s xml:id="echoid-s4558" xml:space="preserve">habemus ſolida ad inuicem ſimilaria genita ex trape-<lb/>zijs in eadem baſi (quæ ſit vnum laterum æquidiſtantium) & </s> <s xml:id="echoid-s4559" xml:space="preserve">altitu-<lb/>dine conſtitutis, quorum oppoſitæ baſes ſint æquales, genita, inquam, <lb/>iuxta communem regulam ipſam baſim, ideſt fruſta conicorum quorum <lb/>oppoſitæ baſes ſunt figuræ deſcriptæ à lateribus dictorum trapeziorum <lb/>æquidiſtantibus, eſſe æqualia.</s> <s xml:id="echoid-s4560" xml:space="preserve"/> </p> <pb o="186" file="0206" n="206" rhead="GEOMETRI Æ"/> </div> <div xml:id="echoid-div464" type="section" level="1" n="281"> <head xml:id="echoid-head297" xml:space="preserve">L. SECTIO XI.</head> <p style="it"> <s xml:id="echoid-s4561" xml:space="preserve">_I_N Prop. </s> <s xml:id="echoid-s4562" xml:space="preserve">28. </s> <s xml:id="echoid-s4563" xml:space="preserve">habetur cylindricum in ea dem baſi, & </s> <s xml:id="echoid-s4564" xml:space="preserve">altitudine cum <lb/>fruſto conici conſtitutum, ad idem, eſſe (ſumptis duabus homologis <lb/>in oppoſitis fruſti conici baſibus) vt quadratum maioris dictarum homo-<lb/>logarum ad rectangulum ſub dictis homologis vna cum, {1/3}, quadrati dif-<lb/>ferentiæ earumdem homologarum. </s> <s xml:id="echoid-s4565" xml:space="preserve">Sit eylindricus, AC, in baſi figura <lb/>quacumque plana, BC, in eadem autem baſi, & </s> <s xml:id="echoid-s4566" xml:space="preserve">altitudine ſit fruſtum <lb/>conici, EBCI, ſic tamen ſe habens, vt ducto plano per latera cylindri-<lb/> <anchor type="figure" xlink:label="fig-0206-01a" xlink:href="fig-0206-01"/> ci, AC, idemtranſeat per latera fruſti conici <lb/>BEIC, ſit autem ductum tale planum, quod <lb/>faciat in cylindrico, AC, parallelogram-<lb/>mum, AC, & </s> <s xml:id="echoid-s4567" xml:space="preserve">in fruſto, BEIC, trapezium, <lb/>BEIC, erunt igitur rectæ, BC, EI, lineæ <lb/>oppoſitarum baſium fructi inter ſe bomologæ, <lb/> <anchor type="note" xlink:label="note-0206-01a" xlink:href="note-0206-01"/> & </s> <s xml:id="echoid-s4568" xml:space="preserve">quia cylindricus, AC, eſt ſolidum ſimi-<lb/>lare genitum ex, AC, iuxta regulam, BC, <lb/> <anchor type="note" xlink:label="note-0206-02a" xlink:href="note-0206-02"/> & </s> <s xml:id="echoid-s4569" xml:space="preserve">fruſtum, EBCI, eſt ſolidum prædicto ſimilare genitum ex trapezio, <lb/>EBCI, ſunt autem h æc ſolida ſimilaria, vt omnia eorumdem quadrata, <lb/>& </s> <s xml:id="echoid-s4570" xml:space="preserve">omnia quadrata, AC, regula, BC, ad omnia quadrata trapezij, E <lb/>BCI, regula eadem ſunt, vt quadratum, BC, ad rectangulum ſub, BC, <lb/>EI, vna cum, _{1/3},_ quadrati differentiæ earumdem, ergo cylindricus, A <lb/>C, ad fruſtum conicum, EBCI, & </s> <s xml:id="echoid-s4571" xml:space="preserve">ad quoduis aliud in eadem baſi, & </s> <s xml:id="echoid-s4572" xml:space="preserve">al-<lb/>titudine cum hoc conſtitutum (quo niam exiſtet huic æquale) erit vt qua-<lb/> <anchor type="note" xlink:label="note-0206-03a" xlink:href="note-0206-03"/> dratum, BC, ad rectangulum ſub, BC, EI, vna cum, _{1/3}_, quadrati dif-<lb/>ferentiæ earu mdem, BC, EI, quæ ſunt duarum oppoſitarum baſium, E <lb/>I, BC, bomologæ vtcumque ſumptæ, nam planum eadem ſolida ſecans <lb/> <anchor type="note" xlink:label="note-0206-04a" xlink:href="note-0206-04"/> ductum eſt vtcumque, dummodo per eorumdem latera tranſeat.</s> <s xml:id="echoid-s4573" xml:space="preserve"/> </p> <div xml:id="echoid-div464" type="float" level="2" n="1"> <figure xlink:label="fig-0206-01" xlink:href="fig-0206-01a"> <image file="0206-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0206-01"/> </figure> <note position="left" xlink:label="note-0206-01" xlink:href="note-0206-01a" xml:space="preserve">_Corol. 21._ <lb/>_lib. 1._</note> <note position="left" xlink:label="note-0206-02" xlink:href="note-0206-02a" xml:space="preserve">_Coroll. 3._ <lb/>_34. huius._ <lb/>_33. huius._ <lb/>_27. huius._</note> <note position="left" xlink:label="note-0206-03" xlink:href="note-0206-03a" xml:space="preserve">_K. Huius._ <lb/>_Coroll._ <lb/>_Gener._</note> <note position="left" xlink:label="note-0206-04" xlink:href="note-0206-04a" xml:space="preserve">_Corol. 21._ <lb/>_lib. I._</note> </div> </div> <div xml:id="echoid-div466" type="section" level="1" n="282"> <head xml:id="echoid-head298" xml:space="preserve">M. SECTIO XII.</head> <p style="it"> <s xml:id="echoid-s4574" xml:space="preserve">_H_Inc pátet ſi in eadem baſi, BC, figura, fuerit conicus, & </s> <s xml:id="echoid-s4575" xml:space="preserve">eadem <lb/>altitudine cum fruſto, ideſt cum cylindrico, AC, qui ſit conicus, <lb/> <anchor type="note" xlink:label="note-0206-05a" xlink:href="note-0206-05"/> BOC, quod hic erit, _{1/3}_, cylindrici, AC, & </s> <s xml:id="echoid-s4576" xml:space="preserve">ideò ad fruſtum, EBCI, erit <lb/>vt, _{1/3}_, quadrati, BC, ad rect angulum ſub, BC, EI, vna cum, _{1/3}_, qua-<lb/>drati differentiæ, BC, EI, ideſt vtt otum quadr atum, BC, ad rectangu-<lb/>lum ſub, BC, & </s> <s xml:id="echoid-s4577" xml:space="preserve">tripla, EI, vna cumtoto quadrato differentiæ earum-<lb/>dem, BC, EI. </s> <s xml:id="echoid-s4578" xml:space="preserve">Vide igitur quam ſit amplior hæc demonſtratio ea, qua <lb/>alij oſtenderunt cylindrum eſſe triplum coni, & </s> <s xml:id="echoid-s4579" xml:space="preserve">priſma piramidis in ea-<lb/>dem baſi, & </s> <s xml:id="echoid-s4580" xml:space="preserve">al<unsure/>titudine cum ipſo conſtitute, nam ad tot varia ſolida hęc <pb o="187" file="0207" n="207" rhead="LIBER II."/> ſe extendit, quot ſunt figurarum planarum variationes, quæ nulto aſſi-<lb/>gnato coarctantur numero, cuius modi demonſtrationis vniuerſalitatem <lb/>in alijs figuris quoque in poſterum conſiderandis proſequemur, vt am-<lb/>plius infra patebit.</s> <s xml:id="echoid-s4581" xml:space="preserve"/> </p> <div xml:id="echoid-div466" type="float" level="2" n="1"> <note position="left" xlink:label="note-0206-05" xlink:href="note-0206-05a" xml:space="preserve">_I. Huius._ <lb/>_Corollar._ <lb/>_Gener._</note> </div> </div> <div xml:id="echoid-div468" type="section" level="1" n="283"> <head xml:id="echoid-head299" xml:space="preserve">N. SECTIO XIII.</head> <p style="it"> <s xml:id="echoid-s4582" xml:space="preserve">_I_N Prop. </s> <s xml:id="echoid-s4583" xml:space="preserve">29. </s> <s xml:id="echoid-s4584" xml:space="preserve">& </s> <s xml:id="echoid-s4585" xml:space="preserve">eius Corollario tandem edocemur ſolida ſimilaria <lb/>genita ex parallelogrammo, vel triangulo eodem, iuxta duas regu-<lb/>las latera angulum continentia, ideſt cylindricos ab eodem parallelo-<lb/>grammo, & </s> <s xml:id="echoid-s4586" xml:space="preserve">conicos ab eodem triangulo genitos, iuxta dictas regulas, <lb/>eſſe inter ſe, vt eaſdem regulas.</s> <s xml:id="echoid-s4587" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div469" type="section" level="1" n="284"> <head xml:id="echoid-head300" xml:space="preserve">THEOREMA XXXV. PROPOS. XXXV.</head> <p> <s xml:id="echoid-s4588" xml:space="preserve">PArallelepipedum ſub baſi rectangulo quodam, altitu-<lb/>dine autem quadam recta linea æquatur parallelepipe-<lb/>dis ſub baſi eodem rectangulo, & </s> <s xml:id="echoid-s4589" xml:space="preserve">ſub quotcumq; </s> <s xml:id="echoid-s4590" xml:space="preserve">paitibus, <lb/>in quas altitudo vtcumq; </s> <s xml:id="echoid-s4591" xml:space="preserve">diuidui poteſt. </s> <s xml:id="echoid-s4592" xml:space="preserve">Et ſi rectangulum, <lb/>quod eſt baſis, intelligatur vtcumq; </s> <s xml:id="echoid-s4593" xml:space="preserve">diuiſum in quotcumq; <lb/></s> <s xml:id="echoid-s4594" xml:space="preserve">rectangula, dictum parallelepipedum æquatur parallelepi-<lb/>pedis ſub ſingulis partibus altitudinis, & </s> <s xml:id="echoid-s4595" xml:space="preserve">ſingulis partibus <lb/>bafis.</s> <s xml:id="echoid-s4596" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4597" xml:space="preserve">Sit parallelepipedum rectangulum, AP, cuius baſis rectangulum, <lb/>TH, ſupponatur pro nunc indiuiſa, & </s> <s xml:id="echoid-s4598" xml:space="preserve">altitudo, DT, diuiſa vtcum-<lb/>quein quotcumq; </s> <s xml:id="echoid-s4599" xml:space="preserve">partes, DS, ST. </s> <s xml:id="echoid-s4600" xml:space="preserve">Dico parallelepipedum, AP, <lb/>æquari parallelepipedis ſub, DS, TH, & </s> <s xml:id="echoid-s4601" xml:space="preserve">ſub, ST, TH. </s> <s xml:id="echoid-s4602" xml:space="preserve">Duca-<lb/>tur per, S, planum æquidiſtans baſi, TH, quodin eo producet re-<lb/> <anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/> ctangulum, vt, SG, ſuntigitur, AM, NP, parallelepipeda, &</s> <s xml:id="echoid-s4603" xml:space="preserve">, A <lb/>M, eſt ſub, DS, SG, vel, IH, (quia, SG, TH, ſunt figuræ ſi-<lb/> <anchor type="note" xlink:label="note-0207-02a" xlink:href="note-0207-02"/> mdes, & </s> <s xml:id="echoid-s4604" xml:space="preserve">æquales) NP, vero ſub, ST, TH, continetur, eit autem <lb/>parallelepipedum, AP, contentum ſub, DT, TH, æquale paral-<lb/>lelepipedis, AM, NP, ſuis partibus ſimul ſumptis, ergo parallele-<lb/>pipedum ſub, DT, TH, æquatur parallelepipedis ſub, DS, TH, <lb/>& </s> <s xml:id="echoid-s4605" xml:space="preserve">ſub, ST, TH.</s> <s xml:id="echoid-s4606" xml:space="preserve"/> </p> <div xml:id="echoid-div469" type="float" level="2" n="1"> <note position="right" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve">Corol. 12. <lb/>lib. I.</note> <note position="right" xlink:label="note-0207-02" xlink:href="note-0207-02a" xml:space="preserve">Corol. 12. <lb/>lib. I.</note> </div> <p> <s xml:id="echoid-s4607" xml:space="preserve">Sit nunc bafis, TH, diuiſa vtcumque in quotcumque rectangula, <lb/>TV, VP. </s> <s xml:id="echoid-s4608" xml:space="preserve">Dico parallelepipedum ſub, DT, TH, æquari paral-<lb/>lelepipedis ſub, DS, TV, ſub, DS, VP, ſub, ST, TV, & </s> <s xml:id="echoid-s4609" xml:space="preserve">ſub, <lb/>ST, VP. </s> <s xml:id="echoid-s4610" xml:space="preserve">Ducatur per rectam, QV, planum æquidiſtans planis, <pb o="188" file="0208" n="208" rhead="GEOMETRI Æ"/> DX, FH, quod producat in parallelepipedo, AP, rectangulum, E <lb/> <anchor type="figure" xlink:label="fig-0208-01a" xlink:href="fig-0208-01"/> V, in parallelepipedo, AM, rectan-<lb/> <anchor type="note" xlink:label="note-0208-01a" xlink:href="note-0208-01"/> gulum, EO, & </s> <s xml:id="echoid-s4611" xml:space="preserve">in parallelepipedo, <lb/>NP, rectangulum, RV, per pla-<lb/> <anchor type="note" xlink:label="note-0208-02a" xlink:href="note-0208-02"/> num igitur, EV, diuiduntur paral-<lb/>lelepipeda, AM, NP, in paralle-<lb/>pipeda, AR, BM, NQ, OP, eſt <lb/>autem totum parallelepipedum, A <lb/>P, æquale parallelepipedis, AR, B <lb/>M, NQ, OP, & </s> <s xml:id="echoid-s4612" xml:space="preserve">eſt parallelepipe-<lb/>dum, AR, ſub, DS, SO, ideſt ſub, <lb/>DS, TV, & </s> <s xml:id="echoid-s4613" xml:space="preserve">parallelepipedum, B <lb/>M, ſub, ER, RG, hoc eſt ſub, D <lb/>S, QH, & </s> <s xml:id="echoid-s4614" xml:space="preserve">parallelepipedum, NQ, <lb/>eſt ſub, ST, TV, &</s> <s xml:id="echoid-s4615" xml:space="preserve">, OP, eſt ſub, <lb/>RQ, QH, hoc eſt ſub, ST, QH, <lb/>ergo parallelepipedum, AP, ideſt <lb/>ſub, DT, TH, eſt æquale paralle-<lb/>lepipedis ſub, DS, &</s> <s xml:id="echoid-s4616" xml:space="preserve">, TV, & </s> <s xml:id="echoid-s4617" xml:space="preserve">ſub, <lb/>DS, VP, & </s> <s xml:id="echoid-s4618" xml:space="preserve">ſub, ST, TV, & </s> <s xml:id="echoid-s4619" xml:space="preserve">ſub, <lb/>ST, QH, ideſt parallelepipedis ſub <lb/>fingulis partibus altitudinis, & </s> <s xml:id="echoid-s4620" xml:space="preserve">ſingulis partibus baſis content@s.</s> <s xml:id="echoid-s4621" xml:space="preserve"/> </p> <div xml:id="echoid-div470" type="float" level="2" n="2"> <figure xlink:label="fig-0208-01" xlink:href="fig-0208-01a"> <image file="0208-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0208-01"/> </figure> <note position="left" xlink:label="note-0208-01" xlink:href="note-0208-01a" xml:space="preserve">Coroll. 6. <lb/>lib. I.</note> <note position="left" xlink:label="note-0208-02" xlink:href="note-0208-02a" xml:space="preserve">10. Lib. 1.</note> </div> </div> <div xml:id="echoid-div472" type="section" level="1" n="285"> <head xml:id="echoid-head301" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s4622" xml:space="preserve">_C_Ontineri autem parallelepipedum voco ſub tribus rectis eiuſdem <lb/>angulum ſolidum continentibus, quarum dua qualibet rectum <lb/>angulum conſtituunt, ſiue ſub earum quauis, & </s> <s xml:id="echoid-s4623" xml:space="preserve">parallelogrammo re-<lb/>ctangulo ſub reliquis duabus; </s> <s xml:id="echoid-s4624" xml:space="preserve">ita vt, cum dico parallelepipedum ſub <lb/>tali recta linea, & </s> <s xml:id="echoid-s4625" xml:space="preserve">tali rectangulo, ſiue ſub talibus tribus rectis lineis, <lb/>intelligam illud parallelepipedum habere angulum ſolidum rectis an-<lb/>gulis conſtitutum, veluti in iſtis Theorematibus ipſum aſſumo, igitur <lb/>patet nos ex tribus rectis parallelepipedum continentibus quamlibet <lb/>poſſe pro altitudine ſumere, & </s> <s xml:id="echoid-s4626" xml:space="preserve">rectangulum ſub reliquis duabus pro <lb/>baſi.</s> <s xml:id="echoid-s4627" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div473" type="section" level="1" n="286"> <head xml:id="echoid-head302" xml:space="preserve">THEOREMA XXXVI. PROPOS. XXXVI.</head> <p> <s xml:id="echoid-s4628" xml:space="preserve">SI recta linea in vno puncto ſecta ſit vtcumq; </s> <s xml:id="echoid-s4629" xml:space="preserve">parallelepi-<lb/>pedum ſub tota linea, & </s> <s xml:id="echoid-s4630" xml:space="preserve">quadrato vnius factarum par-<lb/>tium erit æquale parallelepipedo ſub tali parte, & </s> <s xml:id="echoid-s4631" xml:space="preserve">rectan- <pb o="189" file="0209" n="209" rhead="LIBER II."/> gulo totius in talem partem ductæ. </s> <s xml:id="echoid-s4632" xml:space="preserve">Idem autem parallelepi-<lb/>pedum ſub tota, & </s> <s xml:id="echoid-s4633" xml:space="preserve">talis partis quadrato, erit æquale paral-<lb/>lelepipedo ſub reliqua, & </s> <s xml:id="echoid-s4634" xml:space="preserve">quadrato talis partis, vna cum <lb/>cubo eiuſdem partis.</s> <s xml:id="echoid-s4635" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4636" xml:space="preserve">Sit ergo recta linea, AC, vtcumque ſectain, B, dico parallelepi-<lb/>pedum ſub, AC, & </s> <s xml:id="echoid-s4637" xml:space="preserve">quadrato, CB, & </s> <s xml:id="echoid-s4638" xml:space="preserve">quari parallelepipedo ſub, B <lb/> <anchor type="figure" xlink:label="fig-0209-01a" xlink:href="fig-0209-01"/> C, & </s> <s xml:id="echoid-s4639" xml:space="preserve">rectangulo, BCA, hoc autem patet <lb/>ex ſuperiori Scholio, nam parallelepipedum <lb/>ſub, AC, & </s> <s xml:id="echoid-s4640" xml:space="preserve">quadrato, CB, continetur ſub <lb/>tribus his rectis lineis, nempè, AC, & </s> <s xml:id="echoid-s4641" xml:space="preserve">dua-<lb/>bus, CB, & </s> <s xml:id="echoid-s4642" xml:space="preserve">ideòidem contìnetur ſub, CB, & </s> <s xml:id="echoid-s4643" xml:space="preserve">rectangulo, ACB, <lb/>ſiue eſt æquale contento ſub, BC, & </s> <s xml:id="echoid-s4644" xml:space="preserve">rectangulo, ACB.</s> <s xml:id="echoid-s4645" xml:space="preserve"/> </p> <div xml:id="echoid-div473" type="float" level="2" n="1"> <figure xlink:label="fig-0209-01" xlink:href="fig-0209-01a"> <image file="0209-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0209-01"/> </figure> </div> <p> <s xml:id="echoid-s4646" xml:space="preserve">Dico inſuper parallelepipedum ſub, AC, & </s> <s xml:id="echoid-s4647" xml:space="preserve">quadrato, CB, æ-<lb/>quari parallelepipedo ſub, AB, & </s> <s xml:id="echoid-s4648" xml:space="preserve">quadrato, CB, vna cum cubo, C <lb/>B, quod patet nam parallelepipedum ſub diuiſa altitudine, AC, & </s> <s xml:id="echoid-s4649" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0209-01a" xlink:href="note-0209-01"/> indiuiſa baſi, nempè quadrato, CB, æquatur parallelepipedis ſub <lb/>partibus ſingulis, & </s> <s xml:id="echoid-s4650" xml:space="preserve">baſi, ſcilicet ſub, AB, & </s> <s xml:id="echoid-s4651" xml:space="preserve">quadrato, BC, & </s> <s xml:id="echoid-s4652" xml:space="preserve">ſub, <lb/>BC, & </s> <s xml:id="echoid-s4653" xml:space="preserve">quadrato, BC, ideſt cubo, BC, quod erat oſtendendum.</s> <s xml:id="echoid-s4654" xml:space="preserve"/> </p> <div xml:id="echoid-div474" type="float" level="2" n="2"> <note position="right" xlink:label="note-0209-01" xlink:href="note-0209-01a" xml:space="preserve">Ex antec.</note> </div> </div> <div xml:id="echoid-div476" type="section" level="1" n="287"> <head xml:id="echoid-head303" xml:space="preserve">THEOREMA XXXVII. PROPOS. XXXVII.</head> <p> <s xml:id="echoid-s4655" xml:space="preserve">SI recta linea in vno puncto ſecta ſit vtcumq; </s> <s xml:id="echoid-s4656" xml:space="preserve">cubus totius <lb/>æquabitur parall elepipedis ſub partibus, & </s> <s xml:id="echoid-s4657" xml:space="preserve">quadrato <lb/>eiuſdem. </s> <s xml:id="echoid-s4658" xml:space="preserve">Idem etiam erit æquale parallelepipedis ſub tota, <lb/>& </s> <s xml:id="echoid-s4659" xml:space="preserve">partibus quadrati totius per talem diuiſtonem factis, ideſt <lb/>parallelepipedis ſub tota, & </s> <s xml:id="echoid-s4660" xml:space="preserve">quadratis partium, & </s> <s xml:id="echoid-s4661" xml:space="preserve">rectan-<lb/>gulo ſub partibus bis contento.</s> <s xml:id="echoid-s4662" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4663" xml:space="preserve">Sit recta linea, AC, vtcumq; </s> <s xml:id="echoid-s4664" xml:space="preserve">ſecta in, B, dico cubum, AC, æquari <lb/>parallelepipedis ſub partibus, AB, BC, & </s> <s xml:id="echoid-s4665" xml:space="preserve">quadrato totius, quod <lb/>patet nam cubus, AC, ideſt parallelepipedum ſub diuiſa, AC, & </s> <s xml:id="echoid-s4666" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0209-02a" xlink:href="note-0209-02"/> indiuiſa baſi quadrato, AC, eſt æquale parallelepipedis ſub partibus, <lb/>AB, BC, eiuſdem, AC, diuiſæ, & </s> <s xml:id="echoid-s4667" xml:space="preserve">ſub eadem baſi quadrato, AC.</s> <s xml:id="echoid-s4668" xml:space="preserve"/> </p> <div xml:id="echoid-div476" type="float" level="2" n="1"> <note position="right" xlink:label="note-0209-02" xlink:href="note-0209-02a" xml:space="preserve">35. huius.</note> </div> <p> <s xml:id="echoid-s4669" xml:space="preserve">Dico etiam cubum, AC, æquari parallelepipedis ſub, AC, & </s> <s xml:id="echoid-s4670" xml:space="preserve"><lb/>quadrato, AB, quadrato, BC, & </s> <s xml:id="echoid-s4671" xml:space="preserve">rectangulo bis ſub, ABC, nam <lb/>cubus, AC, ideſt parallelepipedum ſub indiuiſa altitudine, AC, & </s> <s xml:id="echoid-s4672" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0209-03a" xlink:href="note-0209-03"/> diuiſa baſi in dicta quattuor ſpatia, æquatur parallelepipedis ſub ea-<lb/>dem indiuiſa altitudine, AC, & </s> <s xml:id="echoid-s4673" xml:space="preserve">ſub dictis baſis partibus, nempè ſub <lb/>quadrato, AB, quadrato, BC, & </s> <s xml:id="echoid-s4674" xml:space="preserve">rectangulo bis ſub, ABC, quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s4675" xml:space="preserve"/> </p> <div xml:id="echoid-div477" type="float" level="2" n="2"> <note position="right" xlink:label="note-0209-03" xlink:href="note-0209-03a" xml:space="preserve">35. huius.</note> </div> <pb o="190" file="0210" n="210" rhead="GEOMETRI Æ"/> </div> <div xml:id="echoid-div479" type="section" level="1" n="288"> <head xml:id="echoid-head304" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4676" xml:space="preserve">_H_Inc patet cubùm totius, AC, æquari parallelepipedis ſub ſingulis <lb/>partibus i pſius, AC, & </s> <s xml:id="echoid-s4677" xml:space="preserve">ſingulis partibus quadrati, AC, quod <lb/>etiam patet ex Theoremate 35.</s> <s xml:id="echoid-s4678" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div480" type="section" level="1" n="289"> <head xml:id="echoid-head305" xml:space="preserve">THEOREMA XXXVIII. PROPOS. XXXVIII.</head> <p> <s xml:id="echoid-s4679" xml:space="preserve">SI recta linea in vno puncto ſecta ſit vtcumq; </s> <s xml:id="echoid-s4680" xml:space="preserve">cubus totius <lb/>æquatur cubis partium, vna cum parallel@ pipedis rer <lb/>ſub qualibet partium, & </s> <s xml:id="echoid-s4681" xml:space="preserve">quadrato reliquæ. </s> <s xml:id="echoid-s4682" xml:space="preserve">Vel æquatur <lb/>cubis partium vna cum tribus parallelepipedis, ſub tota, & </s> <s xml:id="echoid-s4683" xml:space="preserve"><lb/>eiuſdem partibus contentis.</s> <s xml:id="echoid-s4684" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4685" xml:space="preserve">Sit recta linea, AC, vtcumque ſecta in puncto, B. </s> <s xml:id="echoid-s4686" xml:space="preserve">Dico cubum, <lb/>AC, æquari cubis, AB, BC, & </s> <s xml:id="echoid-s4687" xml:space="preserve">parallelepipedis ter ſub, AB, & </s> <s xml:id="echoid-s4688" xml:space="preserve"><lb/>quadrato, BC, & </s> <s xml:id="echoid-s4689" xml:space="preserve">ter ſub, BC, & </s> <s xml:id="echoid-s4690" xml:space="preserve">quadrato, AB. </s> <s xml:id="echoid-s4691" xml:space="preserve">Nam parallele-<lb/>pipedum ſub, AC, & </s> <s xml:id="echoid-s4692" xml:space="preserve">quadrato, AC, (qui eſt cubus, AC,) æqua-<lb/> <anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/> tur parallelepipedis ſub ſingulis partibus ipſius, AC, & </s> <s xml:id="echoid-s4693" xml:space="preserve">ſub ſingulis <lb/>partibus quadrati, AC, ab hac diuiſione prouenientibus, ideſt pa-<lb/>rallelepipedo ſub, AB, & </s> <s xml:id="echoid-s4694" xml:space="preserve">quadrato, AB, qui eſt cubus, AB, item <lb/>ſub, AB, & </s> <s xml:id="echoid-s4695" xml:space="preserve">quadrato, BC, item ſub, AB, & </s> <s xml:id="echoid-s4696" xml:space="preserve">rectangulo, ABC, bis, <lb/> <anchor type="note" xlink:label="note-0210-02a" xlink:href="note-0210-02"/> ideſt ſub, CB, & </s> <s xml:id="echoid-s4697" xml:space="preserve">quadrato, BA, bis ſumpto, habemus ergo vnum <lb/> <anchor type="figure" xlink:label="fig-0210-01a" xlink:href="fig-0210-01"/> cubum, AB, vnum parallelepipedum ſub, <lb/>AB, & </s> <s xml:id="echoid-s4698" xml:space="preserve">quadrato, BC, & </s> <s xml:id="echoid-s4699" xml:space="preserve">duo ſub, BC, & </s> <s xml:id="echoid-s4700" xml:space="preserve"><lb/>quadrato, BA; </s> <s xml:id="echoid-s4701" xml:space="preserve">tranſeamus nunc ad aliam <lb/>partem, BC, remanent ergo parallelepipe. <lb/></s> <s xml:id="echoid-s4702" xml:space="preserve">da ſub, BC, & </s> <s xml:id="echoid-s4703" xml:space="preserve">quadrato, BC, ideſt vnus cubus, BC, item ſub, C <lb/>B, & </s> <s xml:id="echoid-s4704" xml:space="preserve">quadrato, AB, & </s> <s xml:id="echoid-s4705" xml:space="preserve">tandem ſub, CB, & </s> <s xml:id="echoid-s4706" xml:space="preserve">rectangulo, CBA, bis, <lb/>ideſt ſub, AB, & </s> <s xml:id="echoid-s4707" xml:space="preserve">quadrato, BC, bis, ſi igitur hæc poſteriora pa-<lb/> <anchor type="note" xlink:label="note-0210-03a" xlink:href="note-0210-03"/> rallelepipeda prioribus iunxeris habebis cubum, AB, cubum, BC, <lb/>parallelepipedum ter ſub, AB, & </s> <s xml:id="echoid-s4708" xml:space="preserve">quadrato, BC, & </s> <s xml:id="echoid-s4709" xml:space="preserve">ter ſub, BC, & </s> <s xml:id="echoid-s4710" xml:space="preserve"><lb/>quadrato, BA, quibus æquale erit parallelepipedum ſub, CA, & </s> <s xml:id="echoid-s4711" xml:space="preserve"><lb/>quadrato, CA, ideſt cubus, CA. </s> <s xml:id="echoid-s4712" xml:space="preserve">Quia verò parallelepipedum ſub, <lb/>CB, & </s> <s xml:id="echoid-s4713" xml:space="preserve">quadrato, BA, ideſt ſub, AB, & </s> <s xml:id="echoid-s4714" xml:space="preserve">rectangulo, ABC, cum <lb/>parallelepipedo ſub, AB, & </s> <s xml:id="echoid-s4715" xml:space="preserve">quadrato, BC, ideſt ſub, CB, & </s> <s xml:id="echoid-s4716" xml:space="preserve">re-<lb/>ctangulo, ABC, æquatur, ex 35. </s> <s xml:id="echoid-s4717" xml:space="preserve">huius, parallelepipedo ſub tota, <lb/>AC, & </s> <s xml:id="echoid-s4718" xml:space="preserve">rectangulo ſub partibus, AB, BC, ideò dicta ſex parallelepi-<lb/>peda tribus ſub tota, AC, & </s> <s xml:id="echoid-s4719" xml:space="preserve">partibus eiuidem, AB, BC, æqualia <lb/>c<unsure/>runt, quod demonſtrare propoſitum fuit.</s> <s xml:id="echoid-s4720" xml:space="preserve"/> </p> <div xml:id="echoid-div480" type="float" level="2" n="1"> <note position="left" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve">ExCorol. <lb/>ant. vel ex <lb/>35. huius.</note> <note position="left" xlink:label="note-0210-02" xlink:href="note-0210-02a" xml:space="preserve">36. huius. <lb/>Pars I.</note> <figure xlink:label="fig-0210-01" xlink:href="fig-0210-01a"> <image file="0210-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0210-01"/> </figure> <note position="left" xlink:label="note-0210-03" xlink:href="note-0210-03a" xml:space="preserve">30. huius. <lb/>Pars I.</note> </div> <pb o="191" file="0211" n="211" rhead="LIBER II."/> </div> <div xml:id="echoid-div482" type="section" level="1" n="290"> <head xml:id="echoid-head306" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s4721" xml:space="preserve">_Q_Voniam poſterior pars Propoſ. </s> <s xml:id="echoid-s4722" xml:space="preserve">antec. </s> <s xml:id="echoid-s4723" xml:space="preserve">addita fuit poſt impreſionem <lb/>Lib. </s> <s xml:id="echoid-s4724" xml:space="preserve">3 4. </s> <s xml:id="echoid-s4725" xml:space="preserve">& </s> <s xml:id="echoid-s4726" xml:space="preserve">5. </s> <s xml:id="echoid-s4727" xml:space="preserve">ideò ne mireris, benigne Lector, ſi in eiſdem ali-<lb/>quando Propoſitiones offenderis nonnibil prolixiores, quam ſi per banc <lb/>poſteriorem partem fuiſſent demonſtrate, cum illaex priori parte tunc <lb/>deductæ fuerint, quod ſolerti Geometræ haud difficile erit in illis propo-<lb/>ſitionibus animaducrtere, in quibus banc viderit adhiberi.</s> <s xml:id="echoid-s4728" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div483" type="section" level="1" n="291"> <head xml:id="echoid-head307" xml:space="preserve">THEOREMA XXXIX. PROPOS. XXXIX:</head> <p> <s xml:id="echoid-s4729" xml:space="preserve">SI recta linea bifariam, & </s> <s xml:id="echoid-s4730" xml:space="preserve">non bifariam ſecta fuerit, pa-<lb/>rallelepipedum ſub medietate propoſitæ lineæ, & </s> <s xml:id="echoid-s4731" xml:space="preserve">ſub <lb/>rectangulo inæqualibus partibus contento, cum parallele-<lb/>pipedo ſub eadem medietate, & </s> <s xml:id="echoid-s4732" xml:space="preserve">ſub quadrato ſectionibus <lb/>intermediæ, æquabitur cubo eiuſdem medietatis propoſi-<lb/>cæ lineæ.</s> <s xml:id="echoid-s4733" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4734" xml:space="preserve">Sit recta linea, AE, bifariam diuiſa in, B, non bifariam in C. </s> <s xml:id="echoid-s4735" xml:space="preserve">Di-<lb/>co parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s4736" xml:space="preserve">rectangulo, ACE, vna cum pa-<lb/>rallelepipedo ſub, BE, & </s> <s xml:id="echoid-s4737" xml:space="preserve">ſub quadrato, BC, cubo eiuſdem, BE, <lb/>æquale eſſe; </s> <s xml:id="echoid-s4738" xml:space="preserve">Nam rectangulum, ACE, cum quadrato, BC, qua-<lb/> <anchor type="note" xlink:label="note-0211-01a" xlink:href="note-0211-01"/> drato, BE, eſt æquale, vt autem rectangulum, ACE, cum qua <lb/>drato, BC, ad quadratum, BE, ita (ſumpta communi altitudine, <lb/> <anchor type="note" xlink:label="note-0211-02a" xlink:href="note-0211-02"/> BE,) parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s4739" xml:space="preserve">rectangulo, ACE, & </s> <s xml:id="echoid-s4740" xml:space="preserve">ſub, B <lb/>E, & </s> <s xml:id="echoid-s4741" xml:space="preserve">quadrato, BC, ad parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s4742" xml:space="preserve">quadrato, <lb/>BE, ideſt ad cubum, BE, ergo parallelepipedum ſub, EB, & </s> <s xml:id="echoid-s4743" xml:space="preserve">ſub <lb/>rectangulo, ACE, vna cum parallelepipedo ſub eadem, EB, & </s> <s xml:id="echoid-s4744" xml:space="preserve">ſub <lb/>quadrato, BC, erit æquale cubo, EB, quod oſtendendum erat.</s> <s xml:id="echoid-s4745" xml:space="preserve"/> </p> <div xml:id="echoid-div483" type="float" level="2" n="1"> <note position="right" xlink:label="note-0211-01" xlink:href="note-0211-01a" xml:space="preserve">5. Secũdi <lb/>Elem.</note> <note position="right" xlink:label="note-0211-02" xlink:href="note-0211-02a" xml:space="preserve">5. huius,</note> </div> <figure> <image file="0211-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0211-01"/> </figure> <pb o="192" file="0212" n="212" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div485" type="section" level="1" n="292"> <head xml:id="echoid-head308" xml:space="preserve">THEOREMA XL. PROPOS. XL.</head> <p> <s xml:id="echoid-s4746" xml:space="preserve">SI recta linea bifariam ſecta fuerit, & </s> <s xml:id="echoid-s4747" xml:space="preserve">illi in directum ad-<lb/>iuncta quæuis recta linea; </s> <s xml:id="echoid-s4748" xml:space="preserve">parallelepipedum ſub com-<lb/>poſita ex dimidia propoſitæ, & </s> <s xml:id="echoid-s4749" xml:space="preserve">ex adiuncta linea, & </s> <s xml:id="echoid-s4750" xml:space="preserve">ſub re-<lb/>ctangulo ſub compoſita ex tota, & </s> <s xml:id="echoid-s4751" xml:space="preserve">adiuncta, & </s> <s xml:id="echoid-s4752" xml:space="preserve">ſub adiun-<lb/>cta, vna cum parallelepipedo ſub compoſito ex eadem pro-<lb/>poſitæ medietate, & </s> <s xml:id="echoid-s4753" xml:space="preserve">ex adiuncta, & </s> <s xml:id="echoid-s4754" xml:space="preserve">ſub quadrato eiuſdem <lb/>medietatis, æquabitur cubo compoſitæ ex dicta medietate, <lb/>& </s> <s xml:id="echoid-s4755" xml:space="preserve">adiuncta.</s> <s xml:id="echoid-s4756" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4757" xml:space="preserve">Sit recta linea propoſita, AC, bifariam in, B, diuiſa, cui in dire-<lb/>ctum ſit adiuncta vtcumq; </s> <s xml:id="echoid-s4758" xml:space="preserve">CE. </s> <s xml:id="echoid-s4759" xml:space="preserve">Dico parallelepipedum ſub, BE, <lb/>& </s> <s xml:id="echoid-s4760" xml:space="preserve">rectangulo, AEC, vna cum parallelepipedo ſub, BE, & </s> <s xml:id="echoid-s4761" xml:space="preserve">qua-<lb/>drato, BC, æquari cubo ipſius, BE. </s> <s xml:id="echoid-s4762" xml:space="preserve">Nam rectangulum, AEC, cum <lb/>quadrato, CB, æquatur quadrato, BE, igitur (ſumpta communi <lb/> <anchor type="note" xlink:label="note-0212-01a" xlink:href="note-0212-01"/> altitudine, BE,) parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s4763" xml:space="preserve">rectangulo, AEC, <lb/>vna cum parallelepipedo ſub, BE, & </s> <s xml:id="echoid-s4764" xml:space="preserve">quadrato, BC, æquabitur pa-<lb/>rallelepipedo ſub, BE, & </s> <s xml:id="echoid-s4765" xml:space="preserve">quadrato, BE, ideſt cubo, BE, quod eran<unsure/> <lb/> <anchor type="note" xlink:label="note-0212-02a" xlink:href="note-0212-02"/> oſtendendum.</s> <s xml:id="echoid-s4766" xml:space="preserve"/> </p> <div xml:id="echoid-div485" type="float" level="2" n="1"> <note position="left" xlink:label="note-0212-01" xlink:href="note-0212-01a" xml:space="preserve">6. Secũdi <lb/>Elem.</note> <note position="left" xlink:label="note-0212-02" xlink:href="note-0212-02a" xml:space="preserve">5. huius.</note> </div> </div> <div xml:id="echoid-div487" type="section" level="1" n="293"> <head xml:id="echoid-head309" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4767" xml:space="preserve">_E_X methodo in ſuperioribus demonſtrationibus adhibita manifeſtum <lb/>eſt nos ſimiliter cęteras Propoſitiones ſecundi Elementorum de-<lb/>monſtrare poſſe, in quibus linea ſecta in vno, vel pluribus punctis con-<lb/>ſideratur, ad parallelepipeda eadem traducentes, nam ſi ſuper ſpatia <lb/>in illis conſiderata intelligantur conſtitui æquè alta parallelepipeda, <lb/>erunt illa, vt ipſę baſes, propterea quę ibi de baſibus demonſtrantur, <lb/>de parallelepipedis æquè altis eiſdem baſibus inſiſtentibus rectè colligi <lb/>poſſunt, quæ ob claritatem, & </s> <s xml:id="echoid-s4768" xml:space="preserve">facilitatem à me relinquuntur.</s> <s xml:id="echoid-s4769" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div488" type="section" level="1" n="294"> <head xml:id="echoid-head310" xml:space="preserve">THEOREMA XLI. PROPOS. XLI.</head> <p> <s xml:id="echoid-s4770" xml:space="preserve">PArallelepipedum, quod ſub tribus rectis lineis propor-<lb/>tionalibus continetur, æquale eſt cubo mediæ.</s> <s xml:id="echoid-s4771" xml:space="preserve"/> </p> <pb o="193" file="0213" n="213" rhead="LIBER II."/> <p> <s xml:id="echoid-s4772" xml:space="preserve">Hæc manifeſta eſt, nam habebunt baſes ipſis altitudinibus recipro-<lb/>cas, quod etiam vniuerſalius oſtenditur Vndecimo Elementorum <lb/>Propoſ. </s> <s xml:id="echoid-s4773" xml:space="preserve">36.</s> <s xml:id="echoid-s4774" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div489" type="section" level="1" n="295"> <head xml:id="echoid-head311" xml:space="preserve">THEOREMA XLII. PROPOS. XLII.</head> <p> <s xml:id="echoid-s4775" xml:space="preserve">DAta recta linea terminata, vtcumque in puncto diuiſa, <lb/>poſſibile eſt ipſam ad alteram eiuſdem partium ita <lb/>producere, vt cubus compoſitæ ex propoſita linea, & </s> <s xml:id="echoid-s4776" xml:space="preserve">adiun-<lb/>cta, ſit æqualis cubo propoſitæ lineæ, ſimul cum cubo com-<lb/>poſitæ ex adiecta, & </s> <s xml:id="echoid-s4777" xml:space="preserve">illi conterminante portione ſectæ lineę.</s> <s xml:id="echoid-s4778" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4779" xml:space="preserve">Sit data recta linea, AC, terminata, diuiſaq; </s> <s xml:id="echoid-s4780" xml:space="preserve">vtcumque in pun-<lb/>cto, B, oftendendum eſt poſſibile eſſe ipſam ita producere ad alteram <lb/>illius partium, vt ad, C, vt cubus compoſitę ex, AC, & </s> <s xml:id="echoid-s4781" xml:space="preserve">adiecta, ſit <lb/>æqualis cubo, AC, cum cubo compoſitæ ex eadem adiecta, & </s> <s xml:id="echoid-s4782" xml:space="preserve">ex, <lb/>BC, portione, AC, adiectæ conterminante. </s> <s xml:id="echoid-s4783" xml:space="preserve">Producatur ergo, C <lb/>A, ad partes, A, vt in, N, ita quod, NB, ſit tripla, BA, ſiat dein-<lb/>de, vt, NB, ad, BC, ita quadratum, BC, ad quadratum rectæ li-<lb/>neę, M, ſeorſim poſitæ: </s> <s xml:id="echoid-s4784" xml:space="preserve">Vlterius exponatur recta, EF, æqualis com-<lb/>poſitæ ex, AC, CB, cui applicetur rectangulum æquale quadrato, <lb/> <anchor type="figure" xlink:label="fig-0213-01a" xlink:href="fig-0213-01"/> M, excedens figura quadrata, <lb/> <anchor type="note" xlink:label="note-0213-01a" xlink:href="note-0213-01"/> cuius latus ſit, FH, producatur <lb/>autem, AC, verſus, C, vt in, <lb/>D, ita nempè, vt, CD, ſit <lb/>æqualis, FH. </s> <s xml:id="echoid-s4785" xml:space="preserve">Dico cubum to-<lb/>tius, AD, æquari duobus cubis, <lb/>AC, BD. </s> <s xml:id="echoid-s4786" xml:space="preserve">Cum.</s> <s xml:id="echoid-s4787" xml:space="preserve">n. </s> <s xml:id="echoid-s4788" xml:space="preserve">ſit, vt, N <lb/>B, ad, BC, ita quadratum, BC, <lb/>ad quadratum, M, ideò parallelepipedum ſub altitudine, AB, (qu <lb/>eſt, {1/3}, prædictę altitudinis, NB,) & </s> <s xml:id="echoid-s4789" xml:space="preserve">quadrato, M, æquabitur ter-<lb/>tiæ parti cubi, BC. </s> <s xml:id="echoid-s4790" xml:space="preserve">Quoniam verò quadratum, M, æquatur rectan-<lb/> <anchor type="note" xlink:label="note-0213-02a" xlink:href="note-0213-02"/> gulo, EHF, ideſt rectangulo ſub compoſita ex, AD, BC, & </s> <s xml:id="echoid-s4791" xml:space="preserve">ſub, <lb/>CD, ideò parallelepipedum ſub altitudine, AB, & </s> <s xml:id="echoid-s4792" xml:space="preserve">baſi rectangulo <lb/>ſub compoſita ex, AD, BC, & </s> <s xml:id="echoid-s4793" xml:space="preserve">ſub, DC, æquabitur tertiæ parti <lb/>cubi, BC, addatur commune parallelepipedum ſub, BC, & </s> <s xml:id="echoid-s4794" xml:space="preserve">baſi re-<lb/>ctangulo, BDC, erit ex vna parte hoc parallelepipedum cum, {1/3}, cu-<lb/>bi, BC, ex alia verò hæcſumma; </s> <s xml:id="echoid-s4795" xml:space="preserve">ſcilicet parallelepipedum ſub, AB, <lb/>& </s> <s xml:id="echoid-s4796" xml:space="preserve">ſub rectangulo ſub compoſita ex, AD, BC, & </s> <s xml:id="echoid-s4797" xml:space="preserve">ſub, DC, vna <lb/>cum parallelepipedo ſub, BC, & </s> <s xml:id="echoid-s4798" xml:space="preserve">rectangulo, BDC, quæ quidem <lb/>ſumma efficit parallelepipedum ſub, AC, & </s> <s xml:id="echoid-s4799" xml:space="preserve">rectangulo, ADC, iam <pb o="194" file="0214" n="214" rhead="GEOMETRIÆ"/> .</s> <s xml:id="echoid-s4800" xml:space="preserve">n. </s> <s xml:id="echoid-s4801" xml:space="preserve">nabemus parallelepipedum ſub, AB, & </s> <s xml:id="echoid-s4802" xml:space="preserve">rectangulo, ADC, & </s> <s xml:id="echoid-s4803" xml:space="preserve"><lb/>ſub, AB, & </s> <s xml:id="echoid-s4804" xml:space="preserve">rectingulo, BCD,.</s> <s xml:id="echoid-s4805" xml:space="preserve">. ſub, BC, & </s> <s xml:id="echoid-s4806" xml:space="preserve">rectangulo ſub, A <lb/>B, CD, cui ſi iunxeris parallelepipedum ſub, BC, & </s> <s xml:id="echoid-s4807" xml:space="preserve">rectangulo ſub, <lb/>BD, DC, componeour parallelepipedum ſub, BC, & </s> <s xml:id="echoid-s4808" xml:space="preserve">rectangulo, <lb/> <anchor type="figure" xlink:label="fig-0214-01a" xlink:href="fig-0214-01"/> ADC, quod additum parallele-<lb/>pipedo ſub, AB, & </s> <s xml:id="echoid-s4809" xml:space="preserve">eodem re-<lb/>ctangulo, ADC, componet pa-<lb/> <anchor type="note" xlink:label="note-0214-01a" xlink:href="note-0214-01"/> rallelepipedum ſub, AC, & </s> <s xml:id="echoid-s4810" xml:space="preserve">re-<lb/>ctangulo, ADC, quod quidem <lb/>æquale erit alteri ſummæ prædi-<lb/>ctæ, nempè parallelepipedo ſub, <lb/>BC, & </s> <s xml:id="echoid-s4811" xml:space="preserve">rectangulo ſub, BD, D <lb/>C, vna cum, {1/3}, cubi, BC, ergo & </s> <s xml:id="echoid-s4812" xml:space="preserve">eorum tripla æqualia erunt ſci-<lb/>licet parallelepipedum ter ſub, AC, & </s> <s xml:id="echoid-s4813" xml:space="preserve">rectangulo, ADC, ſeu ter <lb/> <anchor type="note" xlink:label="note-0214-02a" xlink:href="note-0214-02"/> ſub, AD, & </s> <s xml:id="echoid-s4814" xml:space="preserve">rectangulo, ACD, æquabitur parallelepipedo ter ſub, <lb/>BC, & </s> <s xml:id="echoid-s4815" xml:space="preserve">rectangulo, BDC, ſeu ter ſub, BD, & </s> <s xml:id="echoid-s4816" xml:space="preserve">rectangulo, BCD, <lb/>cum cubo, BC, additis verò communibus cubis, AC, CD, fiet pa-<lb/> <anchor type="note" xlink:label="note-0214-03a" xlink:href="note-0214-03"/> rallelepipedum ter ſub, AD, & </s> <s xml:id="echoid-s4817" xml:space="preserve">rectangulo, ACD, cum cubis, A <lb/>C, CD, ideſt totus cubus, AD, æqualis parallelepipedo ter ſub, B <lb/> <anchor type="note" xlink:label="note-0214-04a" xlink:href="note-0214-04"/> D, & </s> <s xml:id="echoid-s4818" xml:space="preserve">rectangulo, BCD, cum cubis, BC, CD, (quæ integrant <lb/>cubum, BD,) & </s> <s xml:id="echoid-s4819" xml:space="preserve">cum cubo, AC, eſt igitur cubus, AD, æqualis <lb/>duobus cubis, AC, BD. </s> <s xml:id="echoid-s4820" xml:space="preserve">Poſſibile eſt ergo facere, quod propoſi-<lb/>tum fuit.</s> <s xml:id="echoid-s4821" xml:space="preserve"/> </p> <div xml:id="echoid-div489" type="float" level="2" n="1"> <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a"> <image file="0213-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0213-01"/> </figure> <note position="right" xlink:label="note-0213-01" xlink:href="note-0213-01a" xml:space="preserve">29. Sex. <lb/>lem,</note> <note position="right" xlink:label="note-0213-02" xlink:href="note-0213-02a" xml:space="preserve">E. Cor. 4 <lb/>Gen. 34. <lb/>huius.</note> <figure xlink:label="fig-0214-01" xlink:href="fig-0214-01a"> <image file="0214-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0214-01"/> </figure> <note position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve">35. huius.</note> <note position="left" xlink:label="note-0214-02" xlink:href="note-0214-02a" xml:space="preserve">Schol. 35. <lb/>huius.</note> <note position="left" xlink:label="note-0214-03" xlink:href="note-0214-03a" xml:space="preserve">38. huius.</note> <note position="left" xlink:label="note-0214-04" xlink:href="note-0214-04a" xml:space="preserve">38. huius.</note> </div> </div> <div xml:id="echoid-div491" type="section" level="1" n="296"> <head xml:id="echoid-head312" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4822" xml:space="preserve">EX hoc manifeſtum eſt, ſi, AC, ſit latus dati cubi, & </s> <s xml:id="echoid-s4823" xml:space="preserve">ſit etiam da-<lb/>tarecta linea, vt, AB, minor, AC, poſſibile eſſe inuenire duos <lb/>eubos, vt, AD, DB, ita vt eorum differentia ſit æqualis cubo dato, <lb/>AC, & </s> <s xml:id="echoid-s4824" xml:space="preserve">laterun cubicorum, AD, DB, ſcilicet, AB, pariter diffe-<lb/>rentia ſit data, eſt. </s> <s xml:id="echoid-s4825" xml:space="preserve">n. </s> <s xml:id="echoid-s4826" xml:space="preserve">cubus, AC, æqualis dictæ cuborum, AD, DB, <lb/>differentiæ, vt eſtenſum eſt. </s> <s xml:id="echoid-s4827" xml:space="preserve">Cum verò ſimilia ſolida quæunq; </s> <s xml:id="echoid-s4828" xml:space="preserve">ſint in <lb/>tripla ratione linearum, ſeu later um bomologorum eorumdem, ideò <lb/>erunt, vt cubi ipſarum linearum, ſeu laterum bomologoroum, & </s> <s xml:id="echoid-s4829" xml:space="preserve">ideò <lb/>eandem rationem, quam babet cubus, AD, ad cubum, DB, babebit <lb/>ex. </s> <s xml:id="echoid-s4830" xml:space="preserve">gr. </s> <s xml:id="echoid-s4831" xml:space="preserve">Icoſaedrum deſcriptum latere, AD, ad Icoſaedrum deſoriptum <lb/>latere, BD, prædicto bomologo, & </s> <s xml:id="echoid-s4832" xml:space="preserve">vt cubus, AD, ad cubum, AC, <lb/>ita erit Icoſaedrum, AD, ad Icoſaedrum, AC, nec non colligendo, vt <lb/>cubus, AD, ad cubos, AC, BD, ita erit Icoſaedrum, AD, ad Ico-<lb/>ſaedra, AC, BD, ergo Icoſaedrum, AD, æquabitur Icoſaedris, AC, <lb/>BD, & </s> <s xml:id="echoid-s4833" xml:space="preserve">ſuperabit Icoſaedrum, BD, Icoſaedro, AC, ergo ſi datum fuiſ- <pb o="195" file="0215" n="215" rhead="LIBER II."/> ſct Icoſaedrum, AC, &</s> <s xml:id="echoid-s4834" xml:space="preserve">, AB, recta linea ipſius latere minor, non diſ-<lb/>ſimiliter, ac in cubis inuenta eſſent Icoſaedra, AD, DB, quorum diffe-<lb/>rentia eſſet æqualis dato Icoſaedro, AC, nec non eorumdem laterum bo-<lb/>mologorum differentia æqualis datæ rectæ lineæ, AB. </s> <s xml:id="echoid-s4835" xml:space="preserve">Sic etiam datæ <lb/>Sphæræ Orbem datæ craſſitici, minoris tamen illius ſemidiametro, æqua-<lb/>lem poſſibile erit inuenire. </s> <s xml:id="echoid-s4836" xml:space="preserve">Vniuerſaliſſimè autem dato quocumq. </s> <s xml:id="echoid-s4837" xml:space="preserve">ſolido, <lb/>duorum ipſi dato ſimilium differentiam æqualem poſſibile erit inuenire, <lb/>quorum pariter linearum, ſeu laterum bomologorum differentia ſit da-<lb/>ta, dummodo ea ſit minor linea, ſeu latere propoſiti ſolidi prædictis <lb/>bomologo, quodex ſuperius dictis facilè conſiare poteſt.</s> <s xml:id="echoid-s4838" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div492" type="section" level="1" n="297"> <head xml:id="echoid-head313" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s4839" xml:space="preserve">NOnnulla autem ex præfatis proximis Propoſitionibus etiam ab <lb/>alijs oſtenſæ fuerunt, ſed ne Lectori ad alios Libros pro barum <lb/>captu eſſet recurrendum, bic eas adiungere placuit, pręcipuè cum ea-<lb/>rum adductę demonſtrationes abaliorum Auctorum rationibus, ni fal-<lb/>lor, non parum ſint differentes, cum ferè omnes ex vnica Propoſ. </s> <s xml:id="echoid-s4840" xml:space="preserve">35. <lb/></s> <s xml:id="echoid-s4841" xml:space="preserve">via ſatis compendioſa deductæ ſint; </s> <s xml:id="echoid-s4842" xml:space="preserve">qued olim me circa Propoſitiones <lb/>Secundi Elem. </s> <s xml:id="echoid-s4843" xml:space="preserve">à prima nempè vſq; </s> <s xml:id="echoid-s4844" xml:space="preserve">ad 10. </s> <s xml:id="echoid-s4845" xml:space="preserve">præſtitiſſe memini, eas omnes <lb/>ex prima compendioſiſſimè demonſtrando, vt etiam poſtmodum, & </s> <s xml:id="echoid-s4846" xml:space="preserve">Pa-<lb/>trem Clauium feciſſe animaduerti.</s> <s xml:id="echoid-s4847" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div493" type="section" level="1" n="298"> <head xml:id="echoid-head314" xml:space="preserve">Finis Secundi Libri.</head> <figure> <image file="0215-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0215-01"/> </figure> <pb file="0216" n="216"/> <pb o="197" file="0217" n="217" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div494" type="section" level="1" n="299"> <head xml:id="echoid-head315" xml:space="preserve">CAVALERII <lb/>LIBER TERTIVS.</head> <head xml:id="echoid-head316" xml:space="preserve">In quo de circulo, & Ellipſi, ac ſolidis ab <lb/>eiſdem genitis, traditur doctrina.</head> <figure> <image file="0217-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0217-01"/> </figure> </div> <div xml:id="echoid-div495" type="section" level="1" n="300"> <head xml:id="echoid-head317" xml:space="preserve">THEOREMA I. PROPOS. I.</head> <p> <s xml:id="echoid-s4848" xml:space="preserve">OMnia quadrata portionis circuli, vel El-<lb/>lipſis, ad omnia quadrata parallelo-<lb/>grammi in eadem baſi, & </s> <s xml:id="echoid-s4849" xml:space="preserve">altitudine <lb/>cum portione conſtituti, regula baſi, <lb/>erunt, vt compoſita ex ſexta parte axis, <lb/>vel diametri eiuſdem, & </s> <s xml:id="echoid-s4850" xml:space="preserve">dimidia reli-<lb/>quæ portionis, ad axim, vel diame-<lb/>trum reliquæ portionis: </s> <s xml:id="echoid-s4851" xml:space="preserve">Eadem verò <lb/>ad omnia quadrata trianguli in ijſdem <lb/>exiſtentis erunt, vt compoſita ex dimidia totius, & </s> <s xml:id="echoid-s4852" xml:space="preserve">reliquæ <lb/>portionis axi, vel diametro, ad axim, vel diametrum reliquæ <lb/>portionis.</s> <s xml:id="echoid-s4853" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4854" xml:space="preserve">Sit circulus, vel ellipſis, EDRP, cuius axis, vel diameter, ER, <lb/>ad quem ordinatim applicetur, DP, abſcindens vtcumque portio-<lb/>nem, DEP, quæ ſumatur quoq; </s> <s xml:id="echoid-s4855" xml:space="preserve">pro regula, & </s> <s xml:id="echoid-s4856" xml:space="preserve">centrum ſit, A, ac <lb/>parallelogrammum, FP, in eadem baſi, DP, cum portione, & </s> <s xml:id="echoid-s4857" xml:space="preserve">ea-<lb/>dem altitudine; </s> <s xml:id="echoid-s4858" xml:space="preserve">ſint autem primò, DF, PH, latera parallelogram-<lb/>mi, FP, parallela ipſi, ER. </s> <s xml:id="echoid-s4859" xml:space="preserve">Dico ergo omnia quadrata portionis, <lb/>DEP, ad omnia quadrata parallelogrammi, FP, eſſe, vt compo-<lb/>ſita ex ſexta parte, EB, & </s> <s xml:id="echoid-s4860" xml:space="preserve">dimidia, BR, adipſam, BR. </s> <s xml:id="echoid-s4861" xml:space="preserve">Sumatur <pb o="198" file="0218" n="218" rhead="GEOMETRIÆ"/> ergo intra, EB, vtcumque punctum, C, & </s> <s xml:id="echoid-s4862" xml:space="preserve">per, C, ducaturipſi, D <lb/>P, parallela, CM, ſecans curuam circuli, vel ellipſis, EDRP, in, <lb/>N; </s> <s xml:id="echoid-s4863" xml:space="preserve">Eſt igitur quadratum, BP, vel, MC, ad quadratum, CN, vt <lb/>rectangulum, RBE, ad rectangulum, RCE; </s> <s xml:id="echoid-s4864" xml:space="preserve">eſt autem, EP, pa-<lb/> <anchor type="figure" xlink:label="fig-0218-01a" xlink:href="fig-0218-01"/> rallelogrammum in eadem baſi, & </s> <s xml:id="echoid-s4865" xml:space="preserve">alti-<lb/>tudine, cum ſemiportione, EBP, regu-<lb/>la eſt ipſa baſis, &</s> <s xml:id="echoid-s4866" xml:space="preserve">, CM, ducta vtcum-<lb/>que parallela ipſi baſi, repertumque eſt <lb/>quadratum, CM, ad quadratum, CN, <lb/>eſſe vt rectangulum, RBE, ad rectan <lb/>gulum, RCE, ergo magnitudines ho-<lb/>rum quatuor ordinum erunt proportio-<lb/>nales.</s> <s xml:id="echoid-s4867" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4868" xml:space="preserve">omnia quadrata parallelogram-<lb/> <anchor type="note" xlink:label="note-0218-01a" xlink:href="note-0218-01"/> mi, EP, magnitudines primi ordinis col-<lb/>lectæ, iuxta primam, nempè iuxta qua-<lb/>dratum, CM, ad omnia quadrata ſemi-<lb/>portionis, EBP, magnitudines ſecundi <lb/>ordinis collectas, iuxta ſecundam. </s> <s xml:id="echoid-s4869" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4870" xml:space="preserve">iux-<lb/>ta quadratum, CN, erunt vt rectangu-<lb/>la ſub maximis abſciſſarum, EB, & </s> <s xml:id="echoid-s4871" xml:space="preserve">ſub <lb/>adiunctis, BR, magnitudines tertij or-<lb/>dinis collectæ, iuxta tertiam .</s> <s xml:id="echoid-s4872" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4873" xml:space="preserve">iuxta re-<lb/>ctangulum, RBE, ad rectangula ſub <lb/>omnibus abſciſſis, EB, & </s> <s xml:id="echoid-s4874" xml:space="preserve">reſiduis earun-<lb/>dem, adiuncta, BR, (recti, vel obliqui <lb/>tranſitus ſupradictis exiſtentibus) quæ <lb/>ſunt magnitudines quarti ordinis colle-<lb/>ctæ, iuxta quartam.</s> <s xml:id="echoid-s4875" xml:space="preserve">ſ.</s> <s xml:id="echoid-s4876" xml:space="preserve">iuxta rectangulum, <lb/>R CE; </s> <s xml:id="echoid-s4877" xml:space="preserve">quoniam verò rectangula ſub <lb/>maximis abſciſſarum, EB, & </s> <s xml:id="echoid-s4878" xml:space="preserve">ſub adiun-<lb/>ctis, BR, ad rectangula ſub omnibus ab-<lb/>ſciſſis, EB, adiuncta, BR, & </s> <s xml:id="echoid-s4879" xml:space="preserve">ſub earum <lb/>reſiduis, ſunt vt, BR, ad compoſitam ex <lb/> <anchor type="note" xlink:label="note-0218-02a" xlink:href="note-0218-02"/> dimidia, BR, & </s> <s xml:id="echoid-s4880" xml:space="preserve">ſexta parte, EB, ergo conuertendo omnia quadrata <lb/>ſemiportionis, BEP, ad omnia quadrata parallelogrammi, EP, vel <lb/> <anchor type="note" xlink:label="note-0218-03a" xlink:href="note-0218-03"/> iſtorum quadrupla .</s> <s xml:id="echoid-s4881" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4882" xml:space="preserve">omnia quadrata portionis, DEP, ad omnia <lb/>quadrata parallelogrammi, FP, erunt vt compoſita ex, {1/6}, BE, &</s> <s xml:id="echoid-s4883" xml:space="preserve">, <lb/>{1/2}, BR, ad eandem, BR; </s> <s xml:id="echoid-s4884" xml:space="preserve">Iungantur nunc, DE, EP.</s> <s xml:id="echoid-s4885" xml:space="preserve"/> </p> <div xml:id="echoid-div495" type="float" level="2" n="1"> <figure xlink:label="fig-0218-01" xlink:href="fig-0218-01a"> <image file="0218-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0218-01"/> </figure> <note position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve">Coroll.3. <lb/>26.lib. 2.</note> <note position="left" xlink:label="note-0218-02" xlink:href="note-0218-02a" xml:space="preserve">Cor. 30. <lb/>lib.2.</note> <note position="left" xlink:label="note-0218-03" xlink:href="note-0218-03a" xml:space="preserve">8. lib. 2.</note> </div> <p> <s xml:id="echoid-s4886" xml:space="preserve">Dico vlterius omnia quadrata portionis, EDP, ad omnia qua-<lb/>drata trianguli, DEP, eſſe vt compoſita ex dimidia totius, ER, & </s> <s xml:id="echoid-s4887" xml:space="preserve"><lb/>ipſa, BR, ad eandem, BR. </s> <s xml:id="echoid-s4888" xml:space="preserve">Cum enim oſtenderimus omnia qua-<lb/>drata parallelogrammi, FP, ad omnia quadrata portionis, DEP, <pb o="199" file="0219" n="219" rhead="LIBER III."/> eſſe vt, BR, ad compoſitam ex, {1/2}, BR, &</s> <s xml:id="echoid-s4889" xml:space="preserve">, {1/6}, BE, ideò omnia <lb/>quadrata trianguli, DEP, cum ſint, {1/3}, omnium quadratorum pa-<lb/> <anchor type="note" xlink:label="note-0219-01a" xlink:href="note-0219-01"/> rallelogrammi, FP, erunt ad omnia quadrata portionis, DEP, vt, <lb/>{1/3}, RB, ad compoſitam ex, {1/2}, RB, &</s> <s xml:id="echoid-s4890" xml:space="preserve">, {1/6}, BE, ideſt vt tota, RB, <lb/>ad compoſitam ex, {3/2}, RB, &</s> <s xml:id="echoid-s4891" xml:space="preserve">, {3/6}, BE, ſed, {1/2}, RB, .</s> <s xml:id="echoid-s4892" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4893" xml:space="preserve">{3/6}, RB, cum, <lb/>{3/6}, BE, conſtituunt, {3/6}, integrę, ER, ſcilicet, {1/2}, eiuſdem, ER, quę <lb/>ideò cum, {2/2}, ipſius, BR, .</s> <s xml:id="echoid-s4894" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4895" xml:space="preserve">cum, BR, ad ipſam, BR, erit, vt om-<lb/>nia quadrata (conuertendo) portionis, DEP, ad omnia quadrata <lb/>trianguli, DEP.</s> <s xml:id="echoid-s4896" xml:space="preserve"/> </p> <div xml:id="echoid-div496" type="float" level="2" n="2"> <note position="right" xlink:label="note-0219-01" xlink:href="note-0219-01a" xml:space="preserve">24.Lib.2</note> </div> <p> <s xml:id="echoid-s4897" xml:space="preserve">Quoniam verò, ſi in parallelogrammi, vel trianguli dicti, baſi, D <lb/> <anchor type="note" xlink:label="note-0219-02a" xlink:href="note-0219-02"/> P, ſit parallelogrammum, vel triangulum, & </s> <s xml:id="echoid-s4898" xml:space="preserve">in eadem altitudine, <lb/> <anchor type="note" xlink:label="note-0219-03a" xlink:href="note-0219-03"/> omnia quadrata dictorum parallelogrammorum inter ſe æquantur, <lb/>ficut etiam omnia quadrata triangulorum, regula eorundem baſi, <lb/>ideò oſtenſum eſt omnia quadrata portionis, DEP, ad omnia qua-<lb/>drata parallelogrammi in eadem baſi, & </s> <s xml:id="echoid-s4899" xml:space="preserve">altitudine cum ipſa conſti-<lb/>tuti eſſe, vt compoſita ex, {1/6}, BE, &</s> <s xml:id="echoid-s4900" xml:space="preserve">, {1/2}, BR, ad eandem, BR, ad <lb/>omnia verò quadrata trianguli in ijſdem poſiti, vt compoſita ex, B <lb/>R, & </s> <s xml:id="echoid-s4901" xml:space="preserve">dimidia, RE, ad ipſam, BR, quod oſtendere opus erat.</s> <s xml:id="echoid-s4902" xml:space="preserve"/> </p> <div xml:id="echoid-div497" type="float" level="2" n="3"> <note position="right" xlink:label="note-0219-02" xlink:href="note-0219-02a" xml:space="preserve">9. Lib. 2.</note> <note position="right" xlink:label="note-0219-03" xlink:href="note-0219-03a" xml:space="preserve">Per B. Co <lb/>roll. 22. <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div499" type="section" level="1" n="301"> <head xml:id="echoid-head318" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s4903" xml:space="preserve">_H_INC patet in figura, in qua baſis portionis conſtitutæ per cen-<lb/>trum circuli, vel ellipſis tranſeat, quoniam omnia quadrata pa-<lb/>rallelogrammi, FP, ad omnia quadrata portionis, DEP, ſunt vt, <lb/>A R, ad compoſitam ex, {1/2}, AR, &</s> <s xml:id="echoid-s4904" xml:space="preserve">, {1/6}, AE, ſcilicet, {1/6}, AR, quia, <lb/>E A, eſt æqualis ipſi, AR, {1/2}, AR, autem, &</s> <s xml:id="echoid-s4905" xml:space="preserve">, {1/6}, AR, conſtituunt, <lb/>{4/6}, vel, {2/3}, ipſius, AR, ideò omnia quadrata parallelogrammi, FP, eſ-<lb/>ſe ad omnia quadrata portionis, DEP, vt, AR, ad, {2/3}, AR, ideſt eſſe <lb/>eorundem ſexquialtera; </s> <s xml:id="echoid-s4906" xml:space="preserve">quia verò omnia quadrata trianguli, DEP, <lb/> <anchor type="note" xlink:label="note-0219-04a" xlink:href="note-0219-04"/> ſunt, {1/3}, omnium quadratorum parallelogrammi, FP, ideò omnia qua-<lb/>drata trianguli, DEP, ad omnia quadrata portionis, DEP, ſunt vt 1. <lb/></s> <s xml:id="echoid-s4907" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s4908" xml:space="preserve">& </s> <s xml:id="echoid-s4909" xml:space="preserve">conuertendo omnia quadrata portionis, DEP, ſunt dupla om-<lb/>uium quadratorum trianguli, DEP, & </s> <s xml:id="echoid-s4910" xml:space="preserve">ſub ſexquialtera omnium qua-<lb/>dratorum parallelogrammi, FP, dummodo in eadem baſi, & </s> <s xml:id="echoid-s4911" xml:space="preserve">altitudine <lb/>cum portione ſint conſtituti parallelogrammum, & </s> <s xml:id="echoid-s4912" xml:space="preserve">triangulum, vt pau-<lb/>lò ſupr a in fine demonſtrationis ſubiunximus.</s> <s xml:id="echoid-s4913" xml:space="preserve"/> </p> <div xml:id="echoid-div499" type="float" level="2" n="1"> <note position="right" xlink:label="note-0219-04" xlink:href="note-0219-04a" xml:space="preserve">_24.Lib.2._</note> </div> <pb o="200" file="0220" n="220" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div501" type="section" level="1" n="302"> <head xml:id="echoid-head319" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s4914" xml:space="preserve">SI à circulo, vel ellipſi per lineam ad eorum axim, vel dia-<lb/>metrum ordinatim applicatam vtcunque portio abſcin-<lb/>datur, ſit autem parallelogrammum in eadem altitudine cum <lb/>dicta portione, ſed in baſi æquali ſecundę diametro, & </s> <s xml:id="echoid-s4915" xml:space="preserve">regula <lb/>baſis ipſius portionis: </s> <s xml:id="echoid-s4916" xml:space="preserve">Omnia quadrata dicti parallelogram-<lb/>miad omnia quadrata dictę pottionis erunt, vt rectangulum <lb/>ſub dimidia eiuſdem axis, vel diametri, & </s> <s xml:id="echoid-s4917" xml:space="preserve">ſub eiuſdem dimi-<lb/>diæ tripla, ad rectangulum ſub axi, vel diametro abſciſſæ <lb/>portionis, & </s> <s xml:id="echoid-s4918" xml:space="preserve">ſub compoſita ex axe, vel diametro reliquę por-<lb/>tionis, & </s> <s xml:id="echoid-s4919" xml:space="preserve">dimidia totius axis, vel diametri.</s> <s xml:id="echoid-s4920" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4921" xml:space="preserve">Sit igitur circulus, vel ellipſis, BVOR, eius axis, vel diameter, <lb/>BO, ordinatim ad ipſum applicata, VR, vtcumq; </s> <s xml:id="echoid-s4922" xml:space="preserve">abſcindens por-<lb/>tionem, VBR, ſit verò ſecunda diameter, CF, & </s> <s xml:id="echoid-s4923" xml:space="preserve">producta, VR, <lb/>ita vt, PN, ſit æqualis ipſi, CF, &</s> <s xml:id="echoid-s4924" xml:space="preserve">, PM, ipſi, CA, in baſi, PN, <lb/>& </s> <s xml:id="echoid-s4925" xml:space="preserve">altitudine portionis, VBR, ſit parallelogrammum, DN, & </s> <s xml:id="echoid-s4926" xml:space="preserve">cir-<lb/>ca axim, vel diametrum, BM. </s> <s xml:id="echoid-s4927" xml:space="preserve">Dico ergo omnia quadrata paralle-<lb/>logrammi, DN, regula, VR, ad omnia quadrata portionis, VBR, <lb/>eſſe vt rectangulum ſub, BA, & </s> <s xml:id="echoid-s4928" xml:space="preserve">tripla, AO, ad rectangulum ſub, B <lb/> <anchor type="figure" xlink:label="fig-0220-01a" xlink:href="fig-0220-01"/> M, & </s> <s xml:id="echoid-s4929" xml:space="preserve">ſub compoſita ex, MO, OA; </s> <s xml:id="echoid-s4930" xml:space="preserve">iun <lb/>gantur, VB, PB; </s> <s xml:id="echoid-s4931" xml:space="preserve">Omńia ergo quadrata <lb/>ſemiportionis, BCVM, ad omnia qua-<lb/>drata trianguli, BVM, ſunt vt, AO, O <lb/>M, ad, OM, .</s> <s xml:id="echoid-s4932" xml:space="preserve">i. </s> <s xml:id="echoid-s4933" xml:space="preserve">ſumpta, BM, commu-<lb/> <anchor type="note" xlink:label="note-0220-01a" xlink:href="note-0220-01"/> ni altitudine, vt rectangulum ſub, BM, <lb/>MOA, ad rectangulum, BMO, omnia <lb/> <anchor type="note" xlink:label="note-0220-02a" xlink:href="note-0220-02"/> autem quadrata trianguli, BVM, ad <lb/> <anchor type="note" xlink:label="note-0220-03a" xlink:href="note-0220-03"/> omnia quadrata trianguli, BPM, ſunt <lb/>vt quadratum, VM, ad quadratum, P <lb/>M, velad quadratum, CA, .</s> <s xml:id="echoid-s4934" xml:space="preserve">i. </s> <s xml:id="echoid-s4935" xml:space="preserve">vt rectan-<lb/> <anchor type="note" xlink:label="note-0220-04a" xlink:href="note-0220-04"/> gulum, OMB, ad rectangulum, OAB, ergo ex æquali, & </s> <s xml:id="echoid-s4936" xml:space="preserve">conuer-<lb/>tendo omnia quadrata trianguli, BPM, ad omnia quadrata ſemi-<lb/>portionis, BVM, erunt vt rectangulum, BAO, ad rectangulum <lb/> <anchor type="note" xlink:label="note-0220-05a" xlink:href="note-0220-05"/> ſub, BM, &</s> <s xml:id="echoid-s4937" xml:space="preserve">, MOA, & </s> <s xml:id="echoid-s4938" xml:space="preserve">antecedentium tripla.</s> <s xml:id="echoid-s4939" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4940" xml:space="preserve">omnia quadrata <lb/>parallelogrammi, DM, ad omnia quadrata ſemiportionis, BVM, <lb/> <anchor type="note" xlink:label="note-0220-06a" xlink:href="note-0220-06"/> vel omnia quadrata parallelogrammi, DN, ad omnia quadrata <lb/>portionis, VBR, erunt vt rectangulum ſub, BA, & </s> <s xml:id="echoid-s4941" xml:space="preserve">tripla, AO, <pb o="201" file="0221" n="221" rhead="LIBER III."/> ad rectangulum ſub, BM, &</s> <s xml:id="echoid-s4942" xml:space="preserve">, MOA, quod verum eſſe oſtendetur, <lb/>vt in antecedente, etiam ſi parallelogrammum, DN, non ſit circa <lb/>axim, vel diametrum, BM, vnde patet, &</s> <s xml:id="echoid-s4943" xml:space="preserve">c.</s> <s xml:id="echoid-s4944" xml:space="preserve"/> </p> <div xml:id="echoid-div501" type="float" level="2" n="1"> <figure xlink:label="fig-0220-01" xlink:href="fig-0220-01a"> <image file="0220-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0220-01"/> </figure> <note position="left" xlink:label="note-0220-01" xlink:href="note-0220-01a" xml:space="preserve">Exant.</note> <note position="left" xlink:label="note-0220-02" xlink:href="note-0220-02a" xml:space="preserve">5. Lib.2.</note> <note position="left" xlink:label="note-0220-03" xlink:href="note-0220-03a" xml:space="preserve">PerB.Co. <lb/>rollar.22. <lb/>lib.2.</note> <note position="left" xlink:label="note-0220-04" xlink:href="note-0220-04a" xml:space="preserve">Ex 40. l.1. <lb/>& eiuſdẽ <lb/>Scholio.</note> <note position="left" xlink:label="note-0220-05" xlink:href="note-0220-05a" xml:space="preserve">24. Lib. 2.</note> <note position="left" xlink:label="note-0220-06" xlink:href="note-0220-06a" xml:space="preserve">8. Lib.2.</note> </div> </div> <div xml:id="echoid-div503" type="section" level="1" n="303"> <head xml:id="echoid-head320" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s4945" xml:space="preserve">SI intra circulum, velellipſim, duæ ad axim, vel diame-<lb/>trum ordinatim applicentur rectæ lineę, ſit autem paral-<lb/>lelogrammum, & </s> <s xml:id="echoid-s4946" xml:space="preserve">triangulum in eadem altitudine cum por-<lb/>tione inter applicatas concluſa, ſed in baſi altera applicata-<lb/>rum: </s> <s xml:id="echoid-s4947" xml:space="preserve">Omnia quadrata dicti parallelogrammia ad omnia qua-<lb/>drata concluſę portionis (regula baſi) erunt, vt rectangulum <lb/>ſub partibus axis, vel diametri per baſim conſtitutis ad re-<lb/>ctangulum ſub abſciſſa per baſim ab extremitate axis, vel <lb/>diametri, & </s> <s xml:id="echoid-s4948" xml:space="preserve">ſub compoſita ex medietate portionis axis, vel <lb/>diametri eiſdem applicatis intermedię, & </s> <s xml:id="echoid-s4949" xml:space="preserve">abſciſſa per aliam <lb/>applicatam ab eiuſdem extremitate, vna cum rectangulo ſub <lb/>eadem intermedia, & </s> <s xml:id="echoid-s4950" xml:space="preserve">ſub compoſita ex, @, eiuſdem, &</s> <s xml:id="echoid-s4951" xml:space="preserve">, {1/2}, ab-<lb/>ſciſſæ per eandem applicatam ab eiuſdem extremitate: </s> <s xml:id="echoid-s4952" xml:space="preserve">Om-<lb/>nia verò quadrata incluſę portionis ad omnia quadrata dicti <lb/>trianguli erunt, vt rectangulum ſub compoſita ex abſciſſis ab <lb/>axi, vel diametro per ordinatim applicatas verſus terminum, <lb/>cui baſis propinquior eſt, & </s> <s xml:id="echoid-s4953" xml:space="preserve">ſub ſexquialtera abſciſſæ ab alio <lb/>extremo per applicatam, quę non eſt baſis, vna cum rectan-<lb/>gulo ſub huius reliqua, & </s> <s xml:id="echoid-s4954" xml:space="preserve">ſub dupla abſciſſę per baſim ab ex-<lb/>tremo, cui ipſa baſis propinquior eſt, ad rectangulum ſub <lb/>partibus axis, vel diametri per baſim conſtitutis.</s> <s xml:id="echoid-s4955" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4956" xml:space="preserve">Sit ergo circulus, velellipſis, ACDF, <lb/> <anchor type="figure" xlink:label="fig-0221-01a" xlink:href="fig-0221-01"/> centrum, O, axis, vel diameter, AD, duæ <lb/>ad ipſam ordinatim applicatæ ſint, IS, C <lb/>F, intercipientes portionem, ICFS, ſit <lb/>autem parallelogrammum, BF, in baſi <lb/>vtrauis applicatarum, vt in, CF, & </s> <s xml:id="echoid-s4957" xml:space="preserve">eadem <lb/>altitudine cum fruſto, CISF, ſit etiam <lb/>nunc circa axim, vel diametrum, MR, re-<lb/>gula verò, CF; </s> <s xml:id="echoid-s4958" xml:space="preserve">Dico ergo omnia quadra-<lb/>ta parallelogrammi, BF, ad omnia qua-<lb/>drata portionis, ICFS, eſſe vt rectangulum, DRA, ad rectangu- <pb o="202" file="0222" n="222" rhead="GEOMETRIÆ"/> lum ſub, DR, & </s> <s xml:id="echoid-s4959" xml:space="preserve">ſub compoſita ex, {1/2}, RM, & </s> <s xml:id="echoid-s4960" xml:space="preserve">ex, MA, vna cum <lb/>rectangulo ſub, RM, & </s> <s xml:id="echoid-s4961" xml:space="preserve">ſub compoſita ex, {1/6}, RM, &</s> <s xml:id="echoid-s4962" xml:space="preserve">, {1/2}, MA. </s> <s xml:id="echoid-s4963" xml:space="preserve">Su-<lb/>matur in, MR, vtcunque punctum, T, per quod agatur ipſi, CF, <lb/>parallela, TX, fecans curuam, SF, in, V, erit ergo quadratum, R <lb/>F, vel quadratum, TV, ad quadratum, TX, vt rectangulum, D <lb/>RA, ad rectangulum, DTA; </s> <s xml:id="echoid-s4964" xml:space="preserve">& </s> <s xml:id="echoid-s4965" xml:space="preserve">quoniam, MF, eſt parallelogram-<lb/>mum in eadem baſi, RF, & </s> <s xml:id="echoid-s4966" xml:space="preserve">altitudine cum ſemiportione, MRFX <lb/> <anchor type="figure" xlink:label="fig-0222-01a" xlink:href="fig-0222-01"/> S, &</s> <s xml:id="echoid-s4967" xml:space="preserve">, TX, ducta fuit vtcunque parallela <lb/>ipſi, RF, repertumque eſt quadratum, T <lb/>V, ad quadratum, TX, eſſe vt rectangu-<lb/>lum, DRA, ad rectangulum, DTA, con-<lb/> <anchor type="note" xlink:label="note-0222-01a" xlink:href="note-0222-01"/> ſtructis quatuor magnitudinum ordinibus, <lb/>vt in antecedente, cocludemus omnia qua-<lb/>drata parallelogrammi, MF, ad omnia <lb/>quadrata ſemiportionis, MRFS, eſſe vt <lb/>rectangula, DRA, tot, quotſunt omnes <lb/>abſciſſæ ipſius, MR, ad rectangula ſub re-<lb/>ſiduis omnium abſciſſarum, MR, ad@un-<lb/>cta, RD, & </s> <s xml:id="echoid-s4968" xml:space="preserve">ſub omnibus abſciſſis, MR, adiuncta, MA; </s> <s xml:id="echoid-s4969" xml:space="preserve">quia ve-<lb/> <anchor type="note" xlink:label="note-0222-02a" xlink:href="note-0222-02"/> rò, DA, diuiſa eſt vtcumque in duobus punctis, R, M, rectangula <lb/>ſub, DRA, tot, quot ſunt omnes abſciſſę, RM, ad rectangula ſub <lb/>refiduis omnium abſciſſarum, MR, adiuncta, RD, & </s> <s xml:id="echoid-s4970" xml:space="preserve">ſub omnibus <lb/> <anchor type="note" xlink:label="note-0222-03a" xlink:href="note-0222-03"/> abſciſſis, MR, adiuncta, MA, ſunt vt rectangulum, DRA, ad re-<lb/>ctangulum ſub, DR, & </s> <s xml:id="echoid-s4971" xml:space="preserve">ſub compoſita ex, {1/2}, RM, &</s> <s xml:id="echoid-s4972" xml:space="preserve">, MA, vna <lb/>cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s4973" xml:space="preserve">ſub compoſita ex, {1/6}, RM, &</s> <s xml:id="echoid-s4974" xml:space="preserve">, {1/2}, M <lb/>A, ergo omnia quadrata parallelogramini, MF, ad omnia quadrata <lb/>ſemiportionis, MRFS, vel omnia quadrata parallelogrammi, BF, <lb/>ad omnia quadrata portionis, ICFS, erunt vt rectangulum, DR <lb/>A, ad rectangulum ſub, DR, & </s> <s xml:id="echoid-s4975" xml:space="preserve">ſub compoſita ex, {1/2}, MR, & </s> <s xml:id="echoid-s4976" xml:space="preserve">ex, <lb/>MA, vna cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s4977" xml:space="preserve">ſub compoſita ex, {1/6}, RM, <lb/>&</s> <s xml:id="echoid-s4978" xml:space="preserve">, {1/2}, MA.</s> <s xml:id="echoid-s4979" xml:space="preserve"/> </p> <div xml:id="echoid-div503" type="float" level="2" n="1"> <figure xlink:label="fig-0221-01" xlink:href="fig-0221-01a"> <image file="0221-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0221-01"/> </figure> <figure xlink:label="fig-0222-01" xlink:href="fig-0222-01a"> <image file="0222-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0222-01"/> </figure> <note position="left" xlink:label="note-0222-01" xlink:href="note-0222-01a" xml:space="preserve">Coroll. 3. <lb/>26. Lib.2.</note> <note position="left" xlink:label="note-0222-02" xlink:href="note-0222-02a" xml:space="preserve">Cor. 32. <lb/>lib. 2.</note> <note position="left" xlink:label="note-0222-03" xlink:href="note-0222-03a" xml:space="preserve">@ Lib. 2.</note> </div> <p> <s xml:id="echoid-s4980" xml:space="preserve">Iungantur nunc, CM, MF. </s> <s xml:id="echoid-s4981" xml:space="preserve">Dico inſuperomnia quadrata por-<lb/>tionis, ICFS, ad omnia quadrata trianguli, MCF, eſſe vt rectan-<lb/>gulum ſub compoſita ex, MD, DR, & </s> <s xml:id="echoid-s4982" xml:space="preserve">ſub ſexquialtera, MA, vna <lb/>cum rectangulo ſub compoſita ex, MD, & </s> <s xml:id="echoid-s4983" xml:space="preserve">dupla, DR, & </s> <s xml:id="echoid-s4984" xml:space="preserve">ſub, {1/2}, <lb/>MR, ad rectangulum, DRA; </s> <s xml:id="echoid-s4985" xml:space="preserve">omnia .</s> <s xml:id="echoid-s4986" xml:space="preserve">n. </s> <s xml:id="echoid-s4987" xml:space="preserve">quadrata parallelogram-<lb/>mi, BF, ad omnia quadrata portionis, ICFS, oſtenſa ſunt eſſe vt <lb/>rectangulum, DRA, ad rectangulum ſub, DR, & </s> <s xml:id="echoid-s4988" xml:space="preserve">ſub compoſita <lb/>ex, {1/2}, RM, & </s> <s xml:id="echoid-s4989" xml:space="preserve">ex, MA, vna cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s4990" xml:space="preserve">ſub com-<lb/>poſita ex, {1/6}, RM, &</s> <s xml:id="echoid-s4991" xml:space="preserve">, {1/2}, MA, ergo eorum tertia pars ad eadem con-<lb/>ſequentia erunt vt tertia pars rectanguli, DRA, ad eadem conſe-<lb/>quentia rectangula .</s> <s xml:id="echoid-s4992" xml:space="preserve">ſ. </s> <s xml:id="echoid-s4993" xml:space="preserve">vt integrum rectangulum, DRA, ad illa re- <pb o="203" file="0223" n="223" rhead="LIBER III."/> ctangula triplicata, rectangulum autem ſub, DR, & </s> <s xml:id="echoid-s4994" xml:space="preserve">ſub compoſi-<lb/> <anchor type="note" xlink:label="note-0223-01a" xlink:href="note-0223-01"/> ta ex, {1/2}, RM, &</s> <s xml:id="echoid-s4995" xml:space="preserve">, MA, diuiditur in rectangula ſub, DR, &</s> <s xml:id="echoid-s4996" xml:space="preserve">, {1/2}, R <lb/>M, & </s> <s xml:id="echoid-s4997" xml:space="preserve">ſub, DR, &</s> <s xml:id="echoid-s4998" xml:space="preserve">, MA, triplicetur rectangulum ſub, DR, &</s> <s xml:id="echoid-s4999" xml:space="preserve">, <lb/> <anchor type="note" xlink:label="note-0223-02a" xlink:href="note-0223-02"/> {1/2}, RM, fit rectangulum ſub tripla, DR, & </s> <s xml:id="echoid-s5000" xml:space="preserve">ſub, {1/2}, RM, cui ſi ad-<lb/>datur rectangulum ſub, MR, &</s> <s xml:id="echoid-s5001" xml:space="preserve">, {1/2}, RM, fit rectangulum ſub com-<lb/>poſita ex tripla, RD, & </s> <s xml:id="echoid-s5002" xml:space="preserve">ex, RM, .</s> <s xml:id="echoid-s5003" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5004" xml:space="preserve">ſub compoſita ex, MD, & </s> <s xml:id="echoid-s5005" xml:space="preserve"><lb/>dupla, RD, & </s> <s xml:id="echoid-s5006" xml:space="preserve">ſub, {1/2}, RM, quod ſerua: </s> <s xml:id="echoid-s5007" xml:space="preserve">Remanent rectangula ad-<lb/> <anchor type="note" xlink:label="note-0223-03a" xlink:href="note-0223-03"/> huc ſub, DR, MA, & </s> <s xml:id="echoid-s5008" xml:space="preserve">ſub, MR, &</s> <s xml:id="echoid-s5009" xml:space="preserve">, {1/2}, MA, triplicanda, quod <lb/>ſic fiet; </s> <s xml:id="echoid-s5010" xml:space="preserve">rectangulum ſub, DR, MA, æquatur rectangulo ſub dupla, <lb/> <anchor type="note" xlink:label="note-0223-04a" xlink:href="note-0223-04"/> DR, &</s> <s xml:id="echoid-s5011" xml:space="preserve">, {1/2}, MA, cui ſi addatur rectangulum ſub, {1/2}, MA, & </s> <s xml:id="echoid-s5012" xml:space="preserve">ſub, <lb/>MR, fiet rectangulum ſub, {1/2}, MA, & </s> <s xml:id="echoid-s5013" xml:space="preserve">ſub compoſita ex, MR, & </s> <s xml:id="echoid-s5014" xml:space="preserve"><lb/>dupla, RD, .</s> <s xml:id="echoid-s5015" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5016" xml:space="preserve">ſub compoſita ex, MD, DR, quod triplicatum fit <lb/>rectangulum ſub compoſita ex, MD, DR, & </s> <s xml:id="echoid-s5017" xml:space="preserve">ſub ſexquialtera, M <lb/>A, quod ſimul cum rectangulo ſub compoſita ex, MD, & </s> <s xml:id="echoid-s5018" xml:space="preserve">dupla, D <lb/>R, & </s> <s xml:id="echoid-s5019" xml:space="preserve">ſub, {1/2}, MR, ad rectangulum, DRA, conuertendo, habe-<lb/>bit eandem rationem, quam omnia quadrata portionis, ICFS, ad <lb/>omnia quadrata trianguli, CMF; </s> <s xml:id="echoid-s5020" xml:space="preserve">quod etiam verificabitur, ſi di-<lb/> <anchor type="note" xlink:label="note-0223-05a" xlink:href="note-0223-05"/> ctum parallelogrammum, & </s> <s xml:id="echoid-s5021" xml:space="preserve">triangulum, ſint quidem in eadem baſi <lb/>cum portione, ſed non circa eundem axim, vel diametrum cum ea-<lb/>dem portione, vt ſupra patere poteſt in antecedentibus, quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s5022" xml:space="preserve"/> </p> <div xml:id="echoid-div504" type="float" level="2" n="2"> <note position="right" xlink:label="note-0223-01" xlink:href="note-0223-01a" xml:space="preserve">1. 2. elem.</note> <note position="right" xlink:label="note-0223-02" xlink:href="note-0223-02a" xml:space="preserve">1. 2. elem.</note> <note position="right" xlink:label="note-0223-03" xlink:href="note-0223-03a" xml:space="preserve">7. Lib. 2.</note> <note position="right" xlink:label="note-0223-04" xlink:href="note-0223-04a" xml:space="preserve">1. 2. ele@.</note> <note position="right" xlink:label="note-0223-05" xlink:href="note-0223-05a" xml:space="preserve">Ex 9. & @. <lb/>Coroll. <lb/>22. lib. 2@</note> </div> </div> <div xml:id="echoid-div506" type="section" level="1" n="304"> <head xml:id="echoid-head321" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s5023" xml:space="preserve">IN eadem antecedentis figura ſi parallelogrammum ſit <lb/>quidem in eadem altitudine cum portione, ſed in baſi æ-<lb/>quali ſecundæ diametro; </s> <s xml:id="echoid-s5024" xml:space="preserve">omnia quadrata dicti parallelo-<lb/>grammiad omnia quadrata dictę portionis erunt, vt quadra-<lb/>tum dimidijaxis, vel diametri eorumdem ad eadem conſe-<lb/>quentia rectangula, retenta eadem regula.</s> <s xml:id="echoid-s5025" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5026" xml:space="preserve">Exponatur denuò antece@entis figura, <lb/> <anchor type="figure" xlink:label="fig-0223-01a" xlink:href="fig-0223-01"/> & </s> <s xml:id="echoid-s5027" xml:space="preserve">producatur, CF, ita vt, V @, ſit æqua-<lb/>lis ſecundæ diametro, quæ ſit, EH, &</s> <s xml:id="echoid-s5028" xml:space="preserve">, <lb/>VR, æqualis, RX, & </s> <s xml:id="echoid-s5029" xml:space="preserve">in, VX, baſi ſit <lb/>conſtructum parallelogrammum, GX, <lb/>in altitudine eadem cum portione, ICF <lb/>S, ſit etiam circa eandem axim, vel dia-<lb/>metrum, MR, cum portione, IECFH <lb/>S: </s> <s xml:id="echoid-s5030" xml:space="preserve">Omnia ergo quadrata parallelogram-<lb/>mi, GR, ad omnia quadrata parallelogrammi, BR, (regula, CF,) <lb/> <anchor type="note" xlink:label="note-0223-06a" xlink:href="note-0223-06"/> <pb o="204" file="0224" n="224" rhead="GEOMETRIÆ"/> ſunt vt quadratum, VR, ad quadratum, CR, .</s> <s xml:id="echoid-s5031" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5032" xml:space="preserve">vt rectangulum, <lb/> <anchor type="note" xlink:label="note-0224-01a" xlink:href="note-0224-01"/> AOD, vel quadratum, AO, ad rectangulum, DRA, omnia au-<lb/>tem quadrata parallelogrammi, BR, ad omnia quadrata ſemipor-<lb/> <anchor type="figure" xlink:label="fig-0224-01a" xlink:href="fig-0224-01"/> tionis, ICRM, ſunt vt rectangulum, D <lb/>RA, ad rectangulum ſub, DR, & </s> <s xml:id="echoid-s5033" xml:space="preserve">ſub <lb/>compoſita ex, {1/2}, RM, & </s> <s xml:id="echoid-s5034" xml:space="preserve">ex, MA, vna <lb/>cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s5035" xml:space="preserve">ſub com-<lb/>poſita ex, {1/6}, RM, &</s> <s xml:id="echoid-s5036" xml:space="preserve">, {1/2}, MA, ergo ex <lb/> <anchor type="note" xlink:label="note-0224-02a" xlink:href="note-0224-02"/> æquali omnia quadrata parallelogram-<lb/>mi, GR, ad omnia quadrata ſemiportio-<lb/>nis, ICRM, vel omnia quadrata paral-<lb/>lelogrammi, GX, ad omnia quadrata <lb/>portionis, ICFS, erunt vt quadratum, <lb/>AO, ad rectangulum ſub, DR, & </s> <s xml:id="echoid-s5037" xml:space="preserve">ſub compoſita ex, {1/2}, RM, & </s> <s xml:id="echoid-s5038" xml:space="preserve"><lb/>ex, MA, vna cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s5039" xml:space="preserve">ſub compoſita ex, {1/6}, R <lb/> <anchor type="note" xlink:label="note-0224-03a" xlink:href="note-0224-03"/> M, &</s> <s xml:id="echoid-s5040" xml:space="preserve">, {1/2}, MA; </s> <s xml:id="echoid-s5041" xml:space="preserve">quod etiam pater, ſi parallelogrammum, GX, non <lb/>ſit circa axim, vel diametrum, MR, quod erat oſtendendum.</s> <s xml:id="echoid-s5042" xml:space="preserve"/> </p> <div xml:id="echoid-div506" type="float" level="2" n="1"> <figure xlink:label="fig-0223-01" xlink:href="fig-0223-01a"> <image file="0223-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0223-01"/> </figure> <note position="right" xlink:label="note-0223-06" xlink:href="note-0223-06a" xml:space="preserve">9. Lib. 2.</note> <note position="left" xlink:label="note-0224-01" xlink:href="note-0224-01a" xml:space="preserve">Ex 40. l. 1. <lb/>& eiuidẽ <lb/>Scholio.</note> <figure xlink:label="fig-0224-01" xlink:href="fig-0224-01a"> <image file="0224-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0224-01"/> </figure> <note position="left" xlink:label="note-0224-02" xlink:href="note-0224-02a" xml:space="preserve">Ex anter.</note> <note position="left" xlink:label="note-0224-03" xlink:href="note-0224-03a" xml:space="preserve">Ex 9. & B. <lb/>Cor. 22. <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div508" type="section" level="1" n="305"> <head xml:id="echoid-head322" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s5043" xml:space="preserve">SI in circulo, vel ellipſi ducantur coniugati axes, vel dia-<lb/>metri, in altera autem eorundem ſit tamquam in baſi pa-<lb/>rallelogrammum circa eundem axim, vel diametrum cum cir-<lb/>culo, vel ellipſi, circa quæm ſit etiam triangulus, ſed in baſi <lb/>oppoſita baſi parallelogrammi, ſumatur autem in dicta axi, <lb/>vel diametro vtcunq; </s> <s xml:id="echoid-s5044" xml:space="preserve">punctum, per quod baſibus dictis aga-<lb/>tur parallela; </s> <s xml:id="echoid-s5045" xml:space="preserve">quadratum eiuſdem parallelæ trianguli lateri-<lb/>bus interceptæ æquabitur reliquo quadrati eius, quæ inter-<lb/>cipitur lateribus parallelogrammi, dempto quadrato eius, <lb/>quæ intra circulum, vel ellipſim concludetur.</s> <s xml:id="echoid-s5046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5047" xml:space="preserve">Sit circulus, vel ellipſis, BDHF, eius coniugati axes, vel diame-<lb/>tri, BH, DF, in altera autem earum, vtin, DF, tanquam in baſi, <lb/>& </s> <s xml:id="echoid-s5048" xml:space="preserve">circa axim, vel diametrum, BE, ſit parallelogrammum, AF, cir-<lb/>ca eundem verſo4;</s> <s xml:id="echoid-s5049" xml:space="preserve">, ſed in baſi, AC, ſit triangulum, AEC, ſumatur <lb/>autem in, BE, vtcunque punctum, M, per quodipſi, DF, agatur <lb/>parallela, VR, ſecans curuam, DBF, in, T, I, & </s> <s xml:id="echoid-s5050" xml:space="preserve">latera trianguli, <lb/>AEC, in, S, N. </s> <s xml:id="echoid-s5051" xml:space="preserve">Dico ergo quadratum, SN, æquari reliquo qua-<lb/>drati, VR, dempto quadrato, TI. </s> <s xml:id="echoid-s5052" xml:space="preserve">Nam rectangulum, HEB, ad <lb/>rectangulum, HMB, eſt vt quadratum, FE, vel quadratum, RM, <pb o="205" file="0225" n="225" rhead="LIBER III."/> ad quadratum, IM, ergo per conuerſionem rationis rectangulum, <lb/> <anchor type="note" xlink:label="note-0225-01a" xlink:href="note-0225-01"/> <anchor type="figure" xlink:label="fig-0225-01a" xlink:href="fig-0225-01"/> HEB, .</s> <s xml:id="echoid-s5053" xml:space="preserve">i. </s> <s xml:id="echoid-s5054" xml:space="preserve">quadratum, BE, ad qua-<lb/>dratum, ME, (quod eſt exceſſus re <lb/>ctanguli, HEB, ſub rectangulum, H <lb/> <anchor type="note" xlink:label="note-0225-02a" xlink:href="note-0225-02"/> MB,) erit vt quadratum, RM, ad ſui <lb/>reliquum, dempto quadrato, MI, ſed <lb/>vt quadratum, BE, ad quadratum, E <lb/>M, ita quadratum, BC, ideſt quadra-<lb/>tum, MR, ad quadratum, MN, quia <lb/> <anchor type="note" xlink:label="note-0225-03a" xlink:href="note-0225-03"/> triangula, BEC, MEN, ſunt æquian-<lb/>gula; </s> <s xml:id="echoid-s5055" xml:space="preserve">ergo quadratum, BC, ideſt qua-<lb/>dratum, MR, ad quadratum, MN, <lb/>erit vt idem quadratum, MR, ad ſui <lb/>reliquum, dempto quadrato, MI, & </s> <s xml:id="echoid-s5056" xml:space="preserve"><lb/>eorum quadrupla .</s> <s xml:id="echoid-s5057" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5058" xml:space="preserve">quadratum, SN, æquabitur reliquo quadrati, <lb/>VR, dempto quadrato, TI, quod erat oſtendendum.</s> <s xml:id="echoid-s5059" xml:space="preserve"/> </p> <div xml:id="echoid-div508" type="float" level="2" n="1"> <note position="right" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve">Ex 40. l. 1. <lb/>& ex eius <lb/>Scholio.</note> <figure xlink:label="fig-0225-01" xlink:href="fig-0225-01a"> <image file="0225-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0225-01"/> </figure> <note position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">5. 2. elem.</note> <note position="right" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">4. 6. elem.</note> </div> </div> <div xml:id="echoid-div510" type="section" level="1" n="306"> <head xml:id="echoid-head323" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s5060" xml:space="preserve">_Q_VONIAM autem punctum, M, ſumptum eſt vtcumque hinc <lb/>patet, quod omnia quadrata trianguli, AEC, (regula, DF,) <lb/>æquantur reliquo omnium quadratorum parallelogrammi, AF, dem-<lb/>ptis omnibus quadratis ſemicirculi, vel ſemiellipſis, DBF, & </s> <s xml:id="echoid-s5061" xml:space="preserve">duabus <lb/>vtcunq; </s> <s xml:id="echoid-s5062" xml:space="preserve">ductis ipſis, DF, parallelis, vt, XG, VR, patet, quod om-<lb/>nia quadra@a trapezij, γSN℟, æquabuntur reſiduo omnium quadra-<lb/>torumpar allelogrammi, XR, demptis omnibus quadratis portionis ſe-<lb/>micirculi, vel ſemiellipſis inter, ZL, TI, concluſæ: </s> <s xml:id="echoid-s5063" xml:space="preserve">Quia verò oſten-<lb/>ſa eſtratio omnium quadratorum cuiuſuis parallelogrammorum in alti-<lb/>tudine eadem cum portionibus, baſi autem æquali ſecundæ diametre, <lb/> <anchor type="note" xlink:label="note-0225-04a" xlink:href="note-0225-04"/> ad omnia quadrata trapeziorum, vel triangulorum in ijſdem exiſten-<lb/>tium, hine manifeſta eſt ratio eorundem ad dictareſidua, & </s> <s xml:id="echoid-s5064" xml:space="preserve">conſequen-<lb/>ter ad omnia quadrata portionum ſemicirculi, vel ſemiellipſis, DBF, <lb/>dictis parallelis interpoſitarum, vt ex. </s> <s xml:id="echoid-s5065" xml:space="preserve">gr. </s> <s xml:id="echoid-s5066" xml:space="preserve">nota erit ratio, quam babent <lb/>omnia quadrata parallelogrammi, XR, ad omnia quadrata portionis, <lb/>ZTIL, & </s> <s xml:id="echoid-s5067" xml:space="preserve">ſic in reliquis. </s> <s xml:id="echoid-s5068" xml:space="preserve">Quia verò omnia quadrata trianguli, AE <lb/> <anchor type="note" xlink:label="note-0225-05a" xlink:href="note-0225-05"/> C, ad omnia quadrata trianguli, SEN, ſunt in tripla ratione ipſius, <lb/>BE, ad, EM, ideò etiam patebit, quod omnia quadrata parallelogram-<lb/>mi, AF, demptis omnibus quadratis ſemicirculi, vel ſemiellipſis, D <lb/>BF, ad omnia quadrata parallelogrammi, VF, demptis omnibus qua-<lb/>dratis fruſti, TDFR, ſint in tripla ratione ipſius, BE, ad, EM, ideſt <lb/>vt cubus, BE, ad cubum, EM.</s> <s xml:id="echoid-s5069" xml:space="preserve"/> </p> <div xml:id="echoid-div510" type="float" level="2" n="1"> <note position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">_24, & 28._ <lb/>_lib. 2._</note> <note position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">_F. Cor. 22_ <lb/>_lib. 2._</note> </div> <pb o="206" file="0226" n="226" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div512" type="section" level="1" n="307"> <head xml:id="echoid-head324" xml:space="preserve">THEOREMA VI. PROPOS. VI.</head> <p> <s xml:id="echoid-s5070" xml:space="preserve">SI in circulo, vel ellipſiad axim, vel diametrum eiuſdem <lb/>ordinatim applicetur vtcumque recta linea, quæ ſuma-<lb/>tur pro regula; </s> <s xml:id="echoid-s5071" xml:space="preserve">Omnia quadrata eiuſdem ad omnia quadra-<lb/>ta alterutrius portionis peream conſtitutæ, erunt vt paralle-<lb/>lepipedum ſub quadrato totius axis, vel diametri, altitudi-<lb/>ne eiuſdem dimidia, ad parallele pipedum ſub quadrato aſ-<lb/>ſumptæ portionis, altitudine autem linea compoſita ex reli-<lb/>quæ portionis axi, vel diametro, & </s> <s xml:id="echoid-s5072" xml:space="preserve">dimidia totius: </s> <s xml:id="echoid-s5073" xml:space="preserve">Vel e-<lb/>runt, vt cubus totius axis, vel diametri ad parallelepipe-<lb/>dum ſub quadrato aſſumptæ portionis axis, vel diametri, & </s> <s xml:id="echoid-s5074" xml:space="preserve"><lb/>ſub altitudine linea compoſita ex tripla axis, vel diametri <lb/>reliquæ portionis, cum cubo axis, vel diametri reliquæ por-<lb/>tionis.</s> <s xml:id="echoid-s5075" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5076" xml:space="preserve">Sit circulus, vel ellipſis, ABCD, cuius axis, vel diameter, AC, <lb/>centrum, O, & </s> <s xml:id="echoid-s5077" xml:space="preserve">ordinatim vtcunq; </s> <s xml:id="echoid-s5078" xml:space="preserve">ad ipſam applicata, BD, con-<lb/>ſtituens duas portiones, BAD, BCD, quæ quoque ſit regula. <lb/></s> <s xml:id="echoid-s5079" xml:space="preserve">Dico ergo omnia quadrata circuli, vel ellipſis, ABCD, ad omnia <lb/>quadrata portionis, BAD, ex duabus portionibus, BAD, BC <lb/>D, ad libitum ſumptæ, eſſe, vt parallelepipedum ſub baſi quadra-<lb/> <anchor type="figure" xlink:label="fig-0226-01a" xlink:href="fig-0226-01"/> to, AC, altitudine, CO, vel, CX, quæ ſit <lb/>æqualis, CO, & </s> <s xml:id="echoid-s5080" xml:space="preserve">illi in directum conſtituta, ad <lb/>parallelepipedum ſub baſi quadrato, AE, al-<lb/>titudine, EX, vel vt cubus, AC, ad parallele-<lb/>pipedum ſub baſi quadrato, AE, altitudine tri-<lb/>pla. </s> <s xml:id="echoid-s5081" xml:space="preserve">EC, cum cubo, AE; </s> <s xml:id="echoid-s5082" xml:space="preserve">iungantur, BA, A <lb/>D, BC, CD: </s> <s xml:id="echoid-s5083" xml:space="preserve">Omnia ergo quadrata portio-<lb/>nis, BCD, ad omnia quadrata portion@s, BA <lb/> <anchor type="note" xlink:label="note-0226-01a" xlink:href="note-0226-01"/> D, habent rationem compoſitam ex ea, quam <lb/>habent omnia quadrata portionis, BCD, ad <lb/>omnia quadrata trianguli, BCD, & </s> <s xml:id="echoid-s5084" xml:space="preserve">ex ea, <lb/>quam habent hæc ad omnia quadrata trianguli, BAD, & </s> <s xml:id="echoid-s5085" xml:space="preserve">ex ratio-<lb/>ne iſtorum ad omnia quadrata portionis, BAD: </s> <s xml:id="echoid-s5086" xml:space="preserve">Omnia verò qua-<lb/> <anchor type="note" xlink:label="note-0226-02a" xlink:href="note-0226-02"/> drata portionis, BCD, ad omnia quadrata trianguli, BCD, ſunt <lb/> <anchor type="note" xlink:label="note-0226-03a" xlink:href="note-0226-03"/> vt compoſita ex, OA, AE, ad, AE: </s> <s xml:id="echoid-s5087" xml:space="preserve">Omnia item quadrata trian-<lb/>guli, BCD, ad omnia quadrata trianguli, BAD, (quia triangula <lb/>ſunt in eadem baſi, BD,) ſunt vt, CE, ad, EA: </s> <s xml:id="echoid-s5088" xml:space="preserve">Omnia denique <pb o="207" file="0227" n="227" rhead="LIBER III."/> quadrata trianguli, BAD, ad omnia quadrata portionis, BAD, <lb/> <anchor type="note" xlink:label="note-0227-01a" xlink:href="note-0227-01"/> ſunt vt, EC, ad compoſitam ex, EC, CO; </s> <s xml:id="echoid-s5089" xml:space="preserve">harum autem trium ra-<lb/>tionum componentium rationem ſupradictam illa, quam habet, C <lb/> <anchor type="note" xlink:label="note-0227-02a" xlink:href="note-0227-02"/> E, ad, EA, &</s> <s xml:id="echoid-s5090" xml:space="preserve">, CE, ad, ECO, componit rationem quadrati, C <lb/>E, ad rectangulum ſub, AE, & </s> <s xml:id="echoid-s5091" xml:space="preserve">fub, ECO, habemus ergo illas tres <lb/>rationes in has duas reſolutas .</s> <s xml:id="echoid-s5092" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5093" xml:space="preserve">in eam, quam habet quadratum, E <lb/>C, ad rectangulum ſub, AE, &</s> <s xml:id="echoid-s5094" xml:space="preserve">, ECO, & </s> <s xml:id="echoid-s5095" xml:space="preserve">in eam, quam habet com-<lb/>poſita ex, OA, AE, ad, AE; </s> <s xml:id="echoid-s5096" xml:space="preserve">ratio autem quadrati, EC, ad rectan-<lb/>gulum ſub, AE, &</s> <s xml:id="echoid-s5097" xml:space="preserve">, ECO, & </s> <s xml:id="echoid-s5098" xml:space="preserve">ratio ipſius, OAE, ſumptę pro al-<lb/>titudine ad, AE, pariter pro altitudine ſumptam, componunt ratio-<lb/>nem parallelepipedi ſub baſi quadrato, CE, altitudine autem, EA <lb/> <anchor type="note" xlink:label="note-0227-03a" xlink:href="note-0227-03"/> O, ad parallepipedum ſub baſi quadrato, AE, altitudine autem, E <lb/>CO, quod ſerua.</s> <s xml:id="echoid-s5099" xml:space="preserve"/> </p> <div xml:id="echoid-div512" type="float" level="2" n="1"> <figure xlink:label="fig-0226-01" xlink:href="fig-0226-01a"> <image file="0226-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0226-01"/> </figure> <note position="left" xlink:label="note-0226-01" xlink:href="note-0226-01a" xml:space="preserve">Diff. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0226-02" xlink:href="note-0226-02a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-0226-03" xlink:href="note-0226-03a" xml:space="preserve">PerC. Co <lb/>rollar. 22. <lb/>lib. 2.</note> <note position="right" xlink:label="note-0227-01" xlink:href="note-0227-01a" xml:space="preserve">1. Huius.</note> <note position="right" xlink:label="note-0227-02" xlink:href="note-0227-02a" xml:space="preserve">6. Lib. 2.</note> <note position="right" xlink:label="note-0227-03" xlink:href="note-0227-03a" xml:space="preserve">Per D. Co <lb/>rollar. 4. <lb/>Gen. 34. <lb/>lib. 2.</note> </div> <p> <s xml:id="echoid-s5100" xml:space="preserve">Duplicentur nunchorum parallelepipedorum altitudines, omnia <lb/>ergo quadrata portionis, BCD, ad omnia quadrata portionis, BA <lb/>D, erunt vt parallelepipedum ſub quadrato, EC, altitudine verò <lb/>dupla, EA, & </s> <s xml:id="echoid-s5101" xml:space="preserve">dupla, AO, quæ eſt, AC, ad parallelepipedum ſub <lb/>baſi quadrato, AE, altitudine dupla, EC, & </s> <s xml:id="echoid-s5102" xml:space="preserve">dupla, CO, quę eſt, <lb/>AC; </s> <s xml:id="echoid-s5103" xml:space="preserve">parallelepipedum autem ſub quadrato, CE, & </s> <s xml:id="echoid-s5104" xml:space="preserve">ſub compoſita <lb/> <anchor type="note" xlink:label="note-0227-04a" xlink:href="note-0227-04"/> ex dupla, AE, &</s> <s xml:id="echoid-s5105" xml:space="preserve">, AC, æquatur parallelepipedis ſub quadrato, C <lb/>E, & </s> <s xml:id="echoid-s5106" xml:space="preserve">ſub, AE, bis, vna cum parallelepipedo ſub, AC, & </s> <s xml:id="echoid-s5107" xml:space="preserve">ſub qua-<lb/>drato, CE, ideſt vna cum parallelepipedo ſub, AE, adhuc ſemel, <lb/> <anchor type="note" xlink:label="note-0227-05a" xlink:href="note-0227-05"/> & </s> <s xml:id="echoid-s5108" xml:space="preserve">ſub quadrato, EC, cum cubo, EC, quę ſimul cum prædictis con-<lb/>ficiunt parallelepipedum ter ſub, AE, & </s> <s xml:id="echoid-s5109" xml:space="preserve">ſub quadrato, EC, cum <lb/>cubo ipſius, EC. </s> <s xml:id="echoid-s5110" xml:space="preserve">Similiter oſtendemus parallelepipedum ſub qua-<lb/>drato, AE, & </s> <s xml:id="echoid-s5111" xml:space="preserve">ſub compoſita ex, CA, & </s> <s xml:id="echoid-s5112" xml:space="preserve">dupla, CE, æquari paral-<lb/>lelepipedis ter ſub, CE, & </s> <s xml:id="echoid-s5113" xml:space="preserve">ſub quadrato, EA, cumcubo, EA, er-<lb/>go omnia quadrata portionis, BCD, ad omnia quadrata portionis, <lb/>BAD, erunt vt parallelepipedum ter ſub quadra@o, CE, altitudi-<lb/>ne, EA, cum cubo, CE, ad parallelepipedum ter ſub quadrato, A <lb/>E, altitudine, EC, cum cubo, AE, ergo, componendo, omnia qua-<lb/>drata circuli, vel ellipſis, ABCD, ad omnia quadrata portionis, B <lb/>AD, erunt vt parallelepipedum ter ſub altitudine, AE, & </s> <s xml:id="echoid-s5114" xml:space="preserve">quadra-<lb/>to, EC, cum cubo, EC, ſimul cum parallelepipedo ter ſub altitu-<lb/>dine, CE, & </s> <s xml:id="echoid-s5115" xml:space="preserve">ſub quadrato, EA, cum cubo, EA, ad parallelepipe-<lb/>dum ter ſub quadrato, AE, altitudine, EC, cum cubo, AE, illa <lb/> <anchor type="note" xlink:label="note-0227-06a" xlink:href="note-0227-06"/> autem ſimul ſumpta conficiunt cubum, AC, ergo omnia quadrata <lb/>circuli, vel ellipſis, ABCD, ad omnia quadrata portionis, BAD, <lb/>erunt vt cubus, AC, ad parallelepipedum ſub baſi quadrato, AE, <lb/>altitudine linea compoſita ex dupla, EC, & </s> <s xml:id="echoid-s5116" xml:space="preserve">ex, AC, ergo (dimi-<lb/>diatis huius rationis terminis) omnia quadrata circuli, vel ellipſis, A <lb/>BCD, ad omnia quadrata portionis, BAD, erunt vt parallelepi- <pb o="208" file="0228" n="228" rhead="GEOMETRIÆ"/> pedum ſub baſi quadrato, AC, altitudine, CO, vel, CX, (quod <lb/> <anchor type="note" xlink:label="note-0228-01a" xlink:href="note-0228-01"/> eſt dimidium cubi, AC,) ad parallelepipedum ſub baſi quadrato, A <lb/>E, altitudine, EX, (quæ eſt dimidia altitudinis parallelepipedi ſub <lb/>baſi quadrato, AE, altitudine dupla, EC, & </s> <s xml:id="echoid-s5117" xml:space="preserve">ipſa, CA, ſimul) pa-<lb/>tet ergo, quod omnia quadrata circuli, vel ellipſis, ABCD, ad om-<lb/>nia quadrata portionis, BAD, erunt vt parallelepipedum ſub baſi <lb/>quadrato, AC, altitudine, CX, ad parallelepipedum ſub baſi qua-<lb/>drato, AE, altitudine, EX, vel (vt probauimus) vt cubus, AC, ad <lb/>parallelepipedum ſub baſi quadrato, AE, altitudine linea compo-<lb/>ſita ex dupla, EC, & </s> <s xml:id="echoid-s5118" xml:space="preserve">ex, AC, .</s> <s xml:id="echoid-s5119" xml:space="preserve">i. </s> <s xml:id="echoid-s5120" xml:space="preserve">ad parallelepipedum ſub baſi qua-<lb/>drato, AE, altitudine tripla, EC, cum cubo, AE, quæ erant de-<lb/>monſtranda.</s> <s xml:id="echoid-s5121" xml:space="preserve"/> </p> <div xml:id="echoid-div513" type="float" level="2" n="2"> <note position="right" xlink:label="note-0227-04" xlink:href="note-0227-04a" xml:space="preserve">35. Lib. 2.</note> <note position="right" xlink:label="note-0227-05" xlink:href="note-0227-05a" xml:space="preserve">36. Lib. 2.</note> <note position="right" xlink:label="note-0227-06" xlink:href="note-0227-06a" xml:space="preserve">38. Lib. 2.</note> <note position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve">Per C. Co <lb/>rollar. 4 <lb/>G@n. 34. <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div515" type="section" level="1" n="308"> <head xml:id="echoid-head325" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5122" xml:space="preserve">HInc etiam patet portionis, BCD, omnia quadrata ad omnia qua-<lb/>drata portionis, BAD, eſſe vt parallelepipedum ſub baſi qua-<lb/>drato, CE, altitudine autem, EAO, ad parallelepipedum ſub baſi qua-<lb/>drato, AE, altitudine autem, ECO, patet ergo ſi circulus, vel ellipſis <lb/>per applicatam ad eorum axim, vel di@metrum in duas portiones vt-<lb/>cumq; </s> <s xml:id="echoid-s5123" xml:space="preserve">diuidantur, quæq; </s> <s xml:id="echoid-s5124" xml:space="preserve">ſumatur pro regula, quod nota erit ratio om-<lb/>nium quadratorum vtriuſque portionis inter ſe.</s> <s xml:id="echoid-s5125" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div516" type="section" level="1" n="309"> <head xml:id="echoid-head326" xml:space="preserve">THEOREMA VII. PROPOS. VII.</head> <p> <s xml:id="echoid-s5126" xml:space="preserve">SI in circulo, vel ellipſi duæ ad eundem axim, vel diame-<lb/>trum ordinatim applicentur rectæ lineæ; </s> <s xml:id="echoid-s5127" xml:space="preserve">Omnia qua-<lb/>drata vnius portionis (regula baſi) ad omnia quadrata alte-<lb/>rius portionis erunt, vt parallelepipedum ſub baſi quadrato <lb/>axis, vel diametri illius, & </s> <s xml:id="echoid-s5128" xml:space="preserve">ſub compoſita ex axi, vel dia-<lb/>metro reliquæ portionis, & </s> <s xml:id="echoid-s5129" xml:space="preserve">dimidia totius, ad parallelepi-<lb/>pedum ſub baſi quadrato axis, vel diametri alterius portio-<lb/>nis, & </s> <s xml:id="echoid-s5130" xml:space="preserve">ſub compoſita ex axi, vel diametro reliquæ portio-<lb/>nis, & </s> <s xml:id="echoid-s5131" xml:space="preserve">dimidia totius.</s> <s xml:id="echoid-s5132" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5133" xml:space="preserve">Sit circulus, vel ellipſis, ACND, cuius axis, vel diameter, AN, <lb/>centrum, O, duæ ad ipſum vtcunq; </s> <s xml:id="echoid-s5134" xml:space="preserve">ordinatim applicatæ ſint, BF, <lb/>CD, ſit autem producta, AN, in, X, ita vt, XN, ſit æqualis, N <lb/>O; </s> <s xml:id="echoid-s5135" xml:space="preserve">regula vero alterutra applicatarum, vt, CD. </s> <s xml:id="echoid-s5136" xml:space="preserve">Dico ergo omnia <lb/>quadrata portionis, BAF, ad omnia quadrata portionis, CAD, <pb o="209" file="0229" n="229" rhead="LIBER III."/> eſſe, vt parallelepipedum ſub baſi quadrato, AE, altitudine autem, <lb/> <anchor type="figure" xlink:label="fig-0229-01a" xlink:href="fig-0229-01"/> EX, ad parallelepipedum ſub baſi quadrato, AM, <lb/>altitudine, MX. </s> <s xml:id="echoid-s5137" xml:space="preserve">Nam omnia quadrata portionis, <lb/> <anchor type="note" xlink:label="note-0229-01a" xlink:href="note-0229-01"/> BAF, ad omnia quadrata circuli, vel ellipſis, A <lb/>CND, ſunt vt parallelepipedum ſub baſi quadra-<lb/>to, AE, altitudine, EX, ad parallelepipedum ſub <lb/>baſi quadrato, AN, altitudine, NX, item om-<lb/>nia quadrata circuli, vel ellipſis, ACND, ad om-<lb/> <anchor type="note" xlink:label="note-0229-02a" xlink:href="note-0229-02"/> nia quadrata portionis, CAD, ſunt vt parallele-<lb/>pipedum ſub baſi quadrato, AN, altitudine, N <lb/>X, ad parallelepipedum ſub baſi quadrato, AM, <lb/>altitudine, MX, ergo ex æquali omnia quadrato portionis, BAF, <lb/>ad omnia quadrata portionis, CAD, erunt vt parallelepipedum ſub <lb/>baſi quadrato, AE, altitudine, EX, ad parallelepipedum ſub baſi <lb/>quadrato, AM, altitudine, MX, quod erat oſtendendum.</s> <s xml:id="echoid-s5138" xml:space="preserve"/> </p> <div xml:id="echoid-div516" type="float" level="2" n="1"> <figure xlink:label="fig-0229-01" xlink:href="fig-0229-01a"> <image file="0229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0229-01"/> </figure> <note position="right" xlink:label="note-0229-01" xlink:href="note-0229-01a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0229-02" xlink:href="note-0229-02a" xml:space="preserve">Ex antec.</note> </div> </div> <div xml:id="echoid-div518" type="section" level="1" n="310"> <head xml:id="echoid-head327" xml:space="preserve">PROBLEMA I PROPOS. VIII.</head> <p> <s xml:id="echoid-s5139" xml:space="preserve">ADato circulo, vel ellipſi portionem abſcindere per li-<lb/>neam ad eiuſdem axim, vel diametrum ordinatim ap-<lb/>plicatam, cuiusomnia quadrata ad omnia quadrata trian-<lb/>guli in eadem baſi, & </s> <s xml:id="echoid-s5140" xml:space="preserve">altitudine cum ipſa portione, habeant <lb/>rationem datam; </s> <s xml:id="echoid-s5141" xml:space="preserve">oportet autem hanc eſſe maiorem ſexqui-<lb/>altera, exiſtente regula ipſa ordinatim applicata.</s> <s xml:id="echoid-s5142" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5143" xml:space="preserve">Sit circulus, vel ellipſis, ADME, axis, vel diameter, AM, cen-<lb/>trum, F, oportet igitur ad ipſum axim, vel diametrum, lineam or-<lb/> <anchor type="figure" xlink:label="fig-0229-02a" xlink:href="fig-0229-02"/> dinatim applicare, quæ ab ipſo circulo, <lb/>vel ellipſi abſcindat, portionem, cuius <lb/>omnia quadrata (regula ipſa applicata) <lb/>ad omnia quadrata trianguli in eadem <lb/>baſi, & </s> <s xml:id="echoid-s5144" xml:space="preserve">altitudine cum ipſa habeant ra-<lb/>tionem datam; </s> <s xml:id="echoid-s5145" xml:space="preserve">hanc dico prius oporte-<lb/>re eſſe maiorem ſexquialtera, nam cu-<lb/>iuslibet abſciſſæ portionis (vt oſtenſum <lb/>eſt) omnia quadrata ad omnia quadrata <lb/>trianguli in eadem baſi, & </s> <s xml:id="echoid-s5146" xml:space="preserve">altitudine <lb/> <anchor type="note" xlink:label="note-0229-03a" xlink:href="note-0229-03"/> cum ipſa ſunt, vt compoſita ex dimidia <lb/>totius axis, vel diametri, & </s> <s xml:id="echoid-s5147" xml:space="preserve">ex diametro <lb/>reliquæ portionis, ad axim, vel diame-<lb/>trum reliquæ portionis, & </s> <s xml:id="echoid-s5148" xml:space="preserve">diuidendo exceſſus omnium quadratorum <pb o="210" file="0230" n="230" rhead="GEOMETRIÆ"/> dictæ portionis ſuper omnia quadrata dicti trianguli, ad omnia qua-<lb/>drata dicti trianguli, ſunt vt d@midia totius axis, vel diametri ad axim, <lb/> <anchor type="figure" xlink:label="fig-0230-01a" xlink:href="fig-0230-01"/> vel diametrum reliquæ portionis, opor-<lb/>tet ergo, quod dicta ratio diuiſa ſit ma-<lb/>ior ea, quam habet, FM, ad, MA, quæ <lb/>componendo euadit ſexquialtera: </s> <s xml:id="echoid-s5149" xml:space="preserve">ſit er-<lb/>go data ratio, quam habet, BH, ad, N <lb/>R, maior ſexquialtera, & </s> <s xml:id="echoid-s5150" xml:space="preserve">abſcindatur, <lb/>HS, æqualis ipſi, NR, & </s> <s xml:id="echoid-s5151" xml:space="preserve">fiat, vt, BS, <lb/>ad, SH, ita, FM, ad, MO, & </s> <s xml:id="echoid-s5152" xml:space="preserve">ducatur <lb/>per, O, ipſa, DE, ad axim, vel diame-<lb/>trum, AM, ordinatim applicata, & </s> <s xml:id="echoid-s5153" xml:space="preserve">iun-<lb/>gantur, DA, AE; </s> <s xml:id="echoid-s5154" xml:space="preserve">quoniam ergo, vt, <lb/>BS, ad, SH, ita eſt, FM, ad, MO, <lb/>componendo, BH, ad, HS, vel, NR, <lb/>crit, vt, FM, MO, ad, MO, ſunt autem omnia quadrata portio-<lb/>nis, DAE, (regula, DE,) ad omnia quadrata trianguli, DAE, <lb/> <anchor type="note" xlink:label="note-0230-01a" xlink:href="note-0230-01"/> vt, FM, MO, ad, MO, & </s> <s xml:id="echoid-s5155" xml:space="preserve">ideò ſunt ad ea in ratione data, in ea .</s> <s xml:id="echoid-s5156" xml:space="preserve">ſ. <lb/></s> <s xml:id="echoid-s5157" xml:space="preserve">quam habet, BH, ad, NR, quod efficere opus erat.</s> <s xml:id="echoid-s5158" xml:space="preserve"/> </p> <div xml:id="echoid-div518" type="float" level="2" n="1"> <figure xlink:label="fig-0229-02" xlink:href="fig-0229-02a"> <image file="0229-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0229-02"/> </figure> <note position="right" xlink:label="note-0229-03" xlink:href="note-0229-03a" xml:space="preserve">1. Huius.</note> <figure xlink:label="fig-0230-01" xlink:href="fig-0230-01a"> <image file="0230-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0230-01"/> </figure> <note position="left" xlink:label="note-0230-01" xlink:href="note-0230-01a" xml:space="preserve">@. Huius.</note> </div> </div> <div xml:id="echoid-div520" type="section" level="1" n="311"> <head xml:id="echoid-head328" xml:space="preserve">THEOREMA VIII. PROPOS. IX.</head> <p> <s xml:id="echoid-s5159" xml:space="preserve">OMnia quadrata circuli, vel ellipſis, regula altero axium, <lb/>vel diametrorum, ad omnia quadrata eiuſdem, re-<lb/>gula reliquo axium, vel diametrorum, erunt, vt dictus pri-<lb/>mus axis, vel diameter, ad dictum ſecundum axim, vel dia-<lb/>metrum.</s> <s xml:id="echoid-s5160" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5161" xml:space="preserve">Sit circulus, vel ellipſis, MPCF, cuius axes, vel diametri coniu-<lb/> <anchor type="figure" xlink:label="fig-0230-02a" xlink:href="fig-0230-02"/> gatæ, MC, PF. </s> <s xml:id="echoid-s5162" xml:space="preserve">Dico ergo omnia qua-<lb/>drata circuli, vel ellipſis, MP, CF, regu-<lb/>la, MC, ad omnia quadrata eiuſdem, re-<lb/>gula, PF, eſſe, vt, MC, ad, PF; </s> <s xml:id="echoid-s5163" xml:space="preserve">ducan-<lb/>tur per puncta, M, P, C, F, tangentes cir-<lb/>culum, vel ellipſim, MPCF, quę ſint, A <lb/>N, ND, DH, HA, conſtituentes paral-<lb/>lelogrammum, AD, circulo, vel ellipſi, <lb/>MPCF, circumſcriptum, cuius latera <lb/>parallela ſint ipſis, PF, MC, axibus, vel <lb/>diametris coniugatis: </s> <s xml:id="echoid-s5164" xml:space="preserve">Omnia ergo qua-<lb/>drata circuli, vel ellipſis, MPCF, regula, MC, ſunt ſubſexquial- <pb o="211" file="0231" n="231" rhead="LIBER III."/> tera omnium quadratorum parallelogrammi, AD, regula eadem, <lb/> <anchor type="note" xlink:label="note-0231-01a" xlink:href="note-0231-01"/> MC, omnia verò quadrata eiuſdem circuli, vel ellipſis, regula, PF, <lb/>ſunt ſubſexquialtera omnium quadratorum parallelogrammi, AD, <lb/> <anchor type="note" xlink:label="note-0231-02a" xlink:href="note-0231-02"/> regula eadem, PF, ergo omnia quadrata circuli, vel ellipſis, MP <lb/>CF, regula, MC, ad omnia quadrata eiuſdem regula, PF, erunt, <lb/>vt omnia quadrata parallelogrammi, AD, regula, MC, ad omnia <lb/> <anchor type="note" xlink:label="note-0231-03a" xlink:href="note-0231-03"/> quadrata eiuſdem, regula, PF, ſed omnia quadrata parallelogram-<lb/>mi, AD, regula, MC, ad omnia quadrata eiuſdem, regula, PF, <lb/>ſunt, vt, MC, ad, PF, ergo omnia quadrata circuli, vel ellipſis, M <lb/>PCF, regula, MC, ad omnia quadrata eiuſdem, regula, PF, erunt, <lb/>vt, MC, ad, PF, quod oſten ler oportebat.</s> <s xml:id="echoid-s5165" xml:space="preserve"/> </p> <div xml:id="echoid-div520" type="float" level="2" n="1"> <figure xlink:label="fig-0230-02" xlink:href="fig-0230-02a"> <image file="0230-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0230-02"/> </figure> <note position="right" xlink:label="note-0231-01" xlink:href="note-0231-01a" xml:space="preserve">Iuxt. 1. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0231-02" xlink:href="note-0231-02a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="right" xlink:label="note-0231-03" xlink:href="note-0231-03a" xml:space="preserve">29.Lib.2.</note> </div> </div> <div xml:id="echoid-div522" type="section" level="1" n="312"> <head xml:id="echoid-head329" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5166" xml:space="preserve">_H_INC patet, ſi ad, MC, PF, ordinatim applicentur rectæ lineæ <lb/>portiones abſcindentes à dicto circulo, vel ellipſi, quoniam oſten-<lb/>ſa eſtratio omnium quadratorum abſciſſæ portionis, regulabaſi, ad omnia <lb/> <anchor type="note" xlink:label="note-0231-04a" xlink:href="note-0231-04"/> quadrata circuli, vel ellipſis, MPCF, & </s> <s xml:id="echoid-s5167" xml:space="preserve">item oſtenſa eſt ratio om-<lb/> <anchor type="note" xlink:label="note-0231-05a" xlink:href="note-0231-05"/> nium quadratorum circuli, vel ellipſis, MPCF, regula altero axium, <lb/>vel diametrorum, ad omnia quadrata eiuſdem, regula reliquo axi, vel <lb/>diametro, & </s> <s xml:id="echoid-s5168" xml:space="preserve">deniq; </s> <s xml:id="echoid-s5169" xml:space="preserve">oſtenſa eſt ratio omnium quadratorum eiuſdem cir-<lb/>culi, vel ellipſis, ad omnia quadrata portionis per aliam ordinatim ap-<lb/>plicatam abſciſſæ, regula baſi dictæ portionis, quod ideo nota erit ratio <lb/> <anchor type="note" xlink:label="note-0231-06a" xlink:href="note-0231-06"/> omnium quadratorum duarum portionum per dictas applicatas abſciſſa-<lb/>rum, regulis dictarum portionum baſibus, quod, &</s> <s xml:id="echoid-s5170" xml:space="preserve">c.</s> <s xml:id="echoid-s5171" xml:space="preserve"/> </p> <div xml:id="echoid-div522" type="float" level="2" n="1"> <note position="right" xlink:label="note-0231-04" xlink:href="note-0231-04a" xml:space="preserve">_6. Huius._</note> <note position="right" xlink:label="note-0231-05" xlink:href="note-0231-05a" xml:space="preserve">_Exantec._</note> <note position="right" xlink:label="note-0231-06" xlink:href="note-0231-06a" xml:space="preserve">_6. Huius._</note> </div> </div> <div xml:id="echoid-div524" type="section" level="1" n="313"> <head xml:id="echoid-head330" xml:space="preserve">THEOREMA IX. PROPOS. X.</head> <p> <s xml:id="echoid-s5172" xml:space="preserve">SI circulus, & </s> <s xml:id="echoid-s5173" xml:space="preserve">ellipſis, vel duæ ellipſes ſuerint circa eun-<lb/>dem axim, vel diametrum, illi erunt interſe, vt eorum <lb/>ſecundiaxes, vel diametri.</s> <s xml:id="echoid-s5174" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5175" xml:space="preserve">Sint circulus, & </s> <s xml:id="echoid-s5176" xml:space="preserve">ellipſis, vel duæ ellipſes, AFVT, AGVS, cir-<lb/> <anchor type="note" xlink:label="note-0231-07a" xlink:href="note-0231-07"/> ca eundem axim, vel diametrum, AV, ſint verò ſecundi axes, vel <lb/>diametri, FT, GS. </s> <s xml:id="echoid-s5177" xml:space="preserve">Dico circulum, vel ellipſim, AFVT, ad cir-<lb/>culum, vel ellipſim, AGVS, eſſe, vt, FT, ad, GS; </s> <s xml:id="echoid-s5178" xml:space="preserve">duæ igitur, <lb/>DA, DF, tangentes eaſdem in terminis coniugatarum axium, vel <lb/>diametrorum, inter ſe conueniant in, D, erit ergo, DH, paralle-<lb/>logrammum, ducatur etiam per, G, ipſa, GC, parallela ipſi, AV, <lb/>quæ tanget ellipſim, AGVS, in, G, erit ergo etiam, CH, paral-<lb/> <anchor type="note" xlink:label="note-0231-08a" xlink:href="note-0231-08"/> lelogrammum in eadem baſi, & </s> <s xml:id="echoid-s5179" xml:space="preserve">altitudine cum ſemiportione, AG <pb o="212" file="0232" n="232" rhead="GEOMETRIÆ"/> H, vt etiam parallelogrammum, DH, eſtin eadem baſi, & </s> <s xml:id="echoid-s5180" xml:space="preserve">altitu-<lb/>dinecum ſemiportione, AFH; </s> <s xml:id="echoid-s5181" xml:space="preserve">ſumatur vtcunque in, AH, pun-<lb/>ctum, O, & </s> <s xml:id="echoid-s5182" xml:space="preserve">per ipſum ducatur ipſi, FT, parallela, OE, ſecans cur-<lb/>uam, AG, in, N, CG, in, I, curuam, AF, in, M, &</s> <s xml:id="echoid-s5183" xml:space="preserve">, DF, in, <lb/> <anchor type="figure" xlink:label="fig-0232-01a" xlink:href="fig-0232-01"/> E. </s> <s xml:id="echoid-s5184" xml:space="preserve">Igitur quadratum, FH, ad quadratum, <lb/> <anchor type="note" xlink:label="note-0232-01a" xlink:href="note-0232-01"/> MO, erit vt rectangulum, VHA, adre-<lb/>ctangulum, VOA, .</s> <s xml:id="echoid-s5185" xml:space="preserve">i. </s> <s xml:id="echoid-s5186" xml:space="preserve">vt quadratum, GH, <lb/>ad quadratum, NO, ergo quadratum, F <lb/> <anchor type="note" xlink:label="note-0232-02a" xlink:href="note-0232-02"/> H, vel quadratum, EO, ad quadratum, <lb/>MO, erit vt quadratum, IO, ad quadra-<lb/>tum, ON, ergo, EO, ad, OM, erit vt, <lb/>IO, ad, ON, eſt autem, EO, ducta vt-<lb/>cunque parallela, FT, & </s> <s xml:id="echoid-s5187" xml:space="preserve">ſunt parallelo-<lb/>gramma, DH, CH, in ijſdem baſibus, & </s> <s xml:id="echoid-s5188" xml:space="preserve"><lb/>altitudinibus cum ſemiportionibus, AFH, <lb/>AGH, ergo omnes lineæ parallelogram-<lb/> <anchor type="note" xlink:label="note-0232-03a" xlink:href="note-0232-03"/> mi, DH, ad omnes lineas ſemiportionis, FAH, erunt vt omnes li-<lb/>neæ parallelogrammi, CH, ad omnes lineas ſemiportionis, AG <lb/> <anchor type="note" xlink:label="note-0232-04a" xlink:href="note-0232-04"/> H, ergo parallelogrammum, DH, ad ſemiportionem, AFH, erit <lb/>vt parallelogrammum, CH, ad ſemiportionem, AGH, ergo, per-<lb/>mutando, DH, ad, CH, parallelogrammum erit, vt ſemiportio, <lb/>AFH, ad ſemiportionem, AGH, ergo vt, DH, ad, CH, .</s> <s xml:id="echoid-s5189" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5190" xml:space="preserve">vt <lb/> <anchor type="note" xlink:label="note-0232-05a" xlink:href="note-0232-05"/> baſis, FH, ad baſim, HG, vel vt, FT, ad, GS, ita erit ſemipor-<lb/>tio, AFH, adſemiportionem, AGH, vel ſic eorum quadrupla .</s> <s xml:id="echoid-s5191" xml:space="preserve">ſ. <lb/></s> <s xml:id="echoid-s5192" xml:space="preserve">ita erit circulus, vel ellipſis, AFVT, ad circulum, vel ellipſim, A <lb/>GVS, quod, &</s> <s xml:id="echoid-s5193" xml:space="preserve">c.</s> <s xml:id="echoid-s5194" xml:space="preserve"/> </p> <div xml:id="echoid-div524" type="float" level="2" n="1"> <note position="right" xlink:label="note-0231-07" xlink:href="note-0231-07a" xml:space="preserve">Vide d-<lb/>cta lib. 7. <lb/>Annot. <lb/>Prop. 21.</note> <note position="right" xlink:label="note-0231-08" xlink:href="note-0231-08a" xml:space="preserve">17. 1. Co-<lb/>nicorum.</note> <figure xlink:label="fig-0232-01" xlink:href="fig-0232-01a"> <image file="0232-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0232-01"/> </figure> <note position="left" xlink:label="note-0232-01" xlink:href="note-0232-01a" xml:space="preserve">Ex 40.1.1. <lb/>& eius <lb/>Scholio.</note> <note position="left" xlink:label="note-0232-02" xlink:href="note-0232-02a" xml:space="preserve">16. Lib.2.</note> <note position="left" xlink:label="note-0232-03" xlink:href="note-0232-03a" xml:space="preserve">Coroll.3. <lb/>26.lib.2.</note> <note position="left" xlink:label="note-0232-04" xlink:href="note-0232-04a" xml:space="preserve">3.Lib.2.</note> <note position="left" xlink:label="note-0232-05" xlink:href="note-0232-05a" xml:space="preserve">5.Lib.2.</note> </div> </div> <div xml:id="echoid-div526" type="section" level="1" n="314"> <head xml:id="echoid-head331" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5195" xml:space="preserve">_H_INC etiam habetur, quoniam quadratum, EO, ad quadratum, <lb/>OM, eſt vt quadratum, IO, ad quadratum, ON, idcircò, quòd <lb/>eodem pacto, iuxta Th. </s> <s xml:id="echoid-s5196" xml:space="preserve">antecedens, concludere poſſumus omnia qua-<lb/>drata, DH, ad omnia quadrata, CH, eſſe, vt omnia quadrata ſemi-<lb/>portionis, AFH, ad omnia quadrata ſemiportionis, AGH, vel vt <lb/>omnia quadrata circuli, vel ellipſis, AFVT, ad omnia quadrata cir-<lb/>culi, vel ellipſis, AGVS, ſunt autem omnia quadrata parallelogram-<lb/>mi, DH, ad omnia quadrata parallelogrammi, CH, vt quadratum, <lb/> <anchor type="note" xlink:label="note-0232-06a" xlink:href="note-0232-06"/> FH, ad quadratum, GH, habetur ergo inquam, quod omnia quadrata <lb/>circuli, vel ellipſis, AFVT, ad omnia quadrata circuli, vel elli-<lb/>pſis, AGVS, ſunt vt quadratum, FH, ad quadratum, HG, vel vt qua-<lb/>dratum, FT, ad quadratum, GS, ſcilicet ſunt vt quadrata ſecundorum <lb/>axium, vel diametrorum.</s> <s xml:id="echoid-s5197" xml:space="preserve"/> </p> <div xml:id="echoid-div526" type="float" level="2" n="1"> <note position="left" xlink:label="note-0232-06" xlink:href="note-0232-06a" xml:space="preserve">_9.Lib.2._</note> </div> <pb o="213" file="0233" n="233" rhead="LIBER III."/> </div> <div xml:id="echoid-div528" type="section" level="1" n="315"> <head xml:id="echoid-head332" xml:space="preserve">THEOREMAX. PROPOS. XI.</head> <p> <s xml:id="echoid-s5198" xml:space="preserve">C Irculus, vel ellipſis ad quemlibet circulum, vel ellipſim <lb/>habet eandem rationem, quam rectangulum ſub ipſius <lb/>coniugatis axibus, vel diametris, ad rectangulum ſub iſtius <lb/>coniugatis axibus, vel diametris, æquè tamen diametris ad <lb/>inuicem inclinatis.</s> <s xml:id="echoid-s5199" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5200" xml:space="preserve">Sit circulus, ABCD, cuius axes coniugati ſint, AC, BD, cen-<lb/>trum, O, ductis verò per puncta, A, C, parallelis ipſi, BD, FL, Q <lb/>G, & </s> <s xml:id="echoid-s5201" xml:space="preserve">per puncta, B, D, parallelis ipſi, AC, LG, FQ, vt ſit, FG, <lb/>rectangulum circulo, ABCD, circumſcriptum, ſit, STVI, qui-<lb/> <anchor type="figure" xlink:label="fig-0233-01a" xlink:href="fig-0233-01"/> libet circulus, vel ellipſis, cuirectangu-<lb/>lum, ER, ſit circumſcriptum, habens <lb/>latera parallela coniugatis axibus, SV, <lb/>TI. </s> <s xml:id="echoid-s5202" xml:space="preserve">Dico circulum, ABCD, ad elli-<lb/>pſem, STVI, eſſe vt rectangulum, FG, <lb/>ad rectangulum, ER; </s> <s xml:id="echoid-s5203" xml:space="preserve">producatur, SV, <lb/>hinc inde, ita vt, NK, ſit æqualis, OA, <lb/>&</s> <s xml:id="echoid-s5204" xml:space="preserve">, NM, ipſi, OC, & </s> <s xml:id="echoid-s5205" xml:space="preserve">circa, KM, TI, <lb/>axes intelligatur, KT, MI, ellipſis, vel <lb/>circulus, & </s> <s xml:id="echoid-s5206" xml:space="preserve">productis tangentibus, TE, <lb/>IR, vt occurrantipſis, HK, MP, ſit <lb/>rectangulum, HP, circumſcriptum ipſi, <lb/>KTMI, ellipſi, vel circulo, habens la-<lb/>tera coniugatis axibus, KM, TI, paral-<lb/>lela: </s> <s xml:id="echoid-s5207" xml:space="preserve">Eſt ergo vt rectangulum, FG, ad <lb/> <anchor type="note" xlink:label="note-0233-01a" xlink:href="note-0233-01"/> rectangulum, HP, ita circulus, ABC <lb/>D, ad circulum, vel ellipſim, KTMI, <lb/>quia ſunt ambo circa, AC, KM, axes <lb/> <anchor type="note" xlink:label="note-0233-02a" xlink:href="note-0233-02"/> æquales; </s> <s xml:id="echoid-s5208" xml:space="preserve">item parallelogrammum, H <lb/>P, ad parallelogrammum, ER, eſt vt <lb/>circulus, vel ellipſis, KTMI, ad circu-<lb/>lum, vel ellipſim, STVI, ergo ex ęqua-<lb/>lirectangulum, FG, ad rectangulum, ER, erit vt circulus, ABC <lb/>D, ad circulum, vel ell pſim, STVI.</s> <s xml:id="echoid-s5209" xml:space="preserve"/> </p> <div xml:id="echoid-div528" type="float" level="2" n="1"> <figure xlink:label="fig-0233-01" xlink:href="fig-0233-01a"> <image file="0233-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0233-01"/> </figure> <note position="right" xlink:label="note-0233-01" xlink:href="note-0233-01a" xml:space="preserve">Exantec.</note> <note position="right" xlink:label="note-0233-02" xlink:href="note-0233-02a" xml:space="preserve">Exantec.</note> </div> <p> <s xml:id="echoid-s5210" xml:space="preserve">Sit nunc, ABCD, ellipſis, vt etiam, STVI, poterit eſſe, quod, <lb/>AC, BD, ſint non axes, ſed coniugatæ diametri, &</s> <s xml:id="echoid-s5211" xml:space="preserve">, FG, pa-<lb/>rallelogrammum, oportet autem ſumere in ellipſi, ST, VI, <lb/>coniugatas diametros, SV, TI, itavt æqualiter ſint inclinatæ <lb/>ac ipiæ, AC, BD, tunc enim circumſcripta parallelogramma, <pb o="214" file="0234" n="234" rhead="GEOMETRIÆ"/> licet non ſint rectangula, tamen erunt æquiangula, vndeæquiangu-<lb/>lum erit parallelogrammi, HP, ipſi, FG, & </s> <s xml:id="echoid-s5212" xml:space="preserve">ellipſes, ABCD, K <lb/> <anchor type="figure" xlink:label="fig-0234-01a" xlink:href="fig-0234-01"/> TMI, erunt circa, AC, KM, <lb/>æquales diametros, ita vt ſi ſuper-<lb/>ponerentur ad inuicem iſti ellipſes, <lb/>vt, KM, eſſet in, AC, ipſa, TI, <lb/>eſſet in, BD, & </s> <s xml:id="echoid-s5213" xml:space="preserve">ideò eodem mo-<lb/>do oſtendemus, vt ſupra ellipſes, <lb/>ABCD, STVI, eſſe inter ſe, vt <lb/>parallelogramma illis eircumſcri-<lb/>pta, FG, ER, & </s> <s xml:id="echoid-s5214" xml:space="preserve">quia illa ſunt <lb/>ęquiangula habebunt rationem ex <lb/>ratione laterum compoſitam, ſed <lb/>etiam parallelogramma rectangu-<lb/> <anchor type="note" xlink:label="note-0234-01a" xlink:href="note-0234-01"/> la ſub eiſdem lateribus habent ra-<lb/>tionem cõpoſitam ex ratione eo-<lb/>rundem laterum, ergo ellipſis, A <lb/>BCD, ad ellipſim, STVI, erit <lb/>vt parallelogrammum, FG, ad <lb/>parallelogrammum, ER, ſibiæ-<lb/>quiangulum. </s> <s xml:id="echoid-s5215" xml:space="preserve">.</s> <s xml:id="echoid-s5216" xml:space="preserve">vt rectangulum ſub, <lb/>FL, LG, vel ſub, BD, AC, dia-<lb/>metris, ad rectangulum ſub, TI, <lb/>SV, diametris, patetigitur circu-<lb/>lum, vel ellipſim, ABCD, ad cir-<lb/>eulum, vel cllipſim, STVI, eſſe vt rectangulum ſub axibus, vel dia-<lb/>metris, AC, BD, ad rectangulum ſub axibus, vel diametris, SV, <lb/>TI, quæ diametri æquè ad inuicem inclinantur, quod oſtendere <lb/>opuserat.</s> <s xml:id="echoid-s5217" xml:space="preserve"/> </p> <div xml:id="echoid-div529" type="float" level="2" n="2"> <figure xlink:label="fig-0234-01" xlink:href="fig-0234-01a"> <image file="0234-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0234-01"/> </figure> <note position="left" xlink:label="note-0234-01" xlink:href="note-0234-01a" xml:space="preserve">6.Lib.2.</note> </div> </div> <div xml:id="echoid-div531" type="section" level="1" n="316"> <head xml:id="echoid-head333" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s5218" xml:space="preserve">_H_INC ergo colligitur, quod quando circulos comparatur ad cir-<lb/>culum, illi ſunt interſe, vt rectangula ſub eorum axibus. </s> <s xml:id="echoid-s5219" xml:space="preserve">i. </s> <s xml:id="echoid-s5220" xml:space="preserve">vt <lb/>quadrata axium, & </s> <s xml:id="echoid-s5221" xml:space="preserve">ideò ſunt in dupla ratione axium, ſiue diametro-<lb/>rum, quando verò circulus comparatur ad ellipſim, erit ad illum, vt <lb/>ſui axis quadratum adrectangulum ſub axibus ellipſis. </s> <s xml:id="echoid-s5222" xml:space="preserve">Denique, ſiel-<lb/>lipſis comparetur ad ellipſim, erit ad illum, vt rectangulum ſub axibus <lb/>illius ad rectangulum ſub axibus alterius, vel vt rectangulum ſub dia-<lb/>metris (coniugatis ſemper intellige, niſi aliud addatur) illius ad rectan-<lb/>gulum ſub diametris alterins, quæ vt prædicti æqualiter ad inuicem <lb/>ſunt inclinatæ; </s> <s xml:id="echoid-s5223" xml:space="preserve">vel tandem, vt parallelogramma illis circumſcripta, <pb o="215" file="0235" n="235" rhead="LIBER III."/> quorum latera ſint prædictis diametris parallela, quæ ideò ſunt æquian-<lb/>gula, vniuerſaliter igitur prædicta ſunt iter ſe, vt parallelogramna re-<lb/>ctangula, vel æquiangula illis circumſcripta; </s> <s xml:id="echoid-s5224" xml:space="preserve">Vnde etiam habetur pa-<lb/>rallelogramma rectangula illis circumſcripta eſſe, vt parallelogramma <lb/>æquiangula pariter illis circumſcripta.</s> <s xml:id="echoid-s5225" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div532" type="section" level="1" n="317"> <head xml:id="echoid-head334" xml:space="preserve">COROLL II. A. SECTIO I.</head> <note position="right" xml:space="preserve">A.</note> <p style="it"> <s xml:id="echoid-s5226" xml:space="preserve">_H_INC vlterius colligitur, quod quæcunque de binis parallelo-<lb/>grammis oſtenſa ſunt in Theorem. </s> <s xml:id="echoid-s5227" xml:space="preserve">5. </s> <s xml:id="echoid-s5228" xml:space="preserve">6. </s> <s xml:id="echoid-s5229" xml:space="preserve">7. </s> <s xml:id="echoid-s5230" xml:space="preserve">8. </s> <s xml:id="echoid-s5231" xml:space="preserve">lib 2. </s> <s xml:id="echoid-s5232" xml:space="preserve">præſuppoſitis <lb/>conditionibus illic conſideratis circa eorum baſes, & </s> <s xml:id="echoid-s5233" xml:space="preserve">altitudines, vel <lb/>circa eorum latera, eadem & </s> <s xml:id="echoid-s5234" xml:space="preserve">de ellipſibus verificabuntur eaſdem con-<lb/>ditiones in proprijs axibus, vel diametris habentibus; </s> <s xml:id="echoid-s5235" xml:space="preserve">nam his poſitis <lb/>parallelogrammaillis circumſcripta, & </s> <s xml:id="echoid-s5236" xml:space="preserve">æquiangula habent in ſuis la-<lb/>teribus, vel in baſi, & </s> <s xml:id="echoid-s5237" xml:space="preserve">altitudine eaſdem conditiones, vnde ſicuti di-<lb/>ctæ concluſiones ſequuntur pro parallelogrammis circumſcriptis, ita <lb/>etiam verificantur pro inſcriptis ellipſibus, ad quas dicta parallelo-<lb/>gramma habent eaſdem rationes, vt probatum eſt, quæ igitur hic non <lb/> <anchor type="note" xlink:label="note-0235-02a" xlink:href="note-0235-02"/> ſunt pro ellipſibus ad inuicem comparatis oſtenſa, per ſupracitata <lb/>Theoremata ſupplentur, pro circulis autem hoc tantum habemus, quod <lb/>ſint, vt eorum axium, vel (ſimanis dicere) diametrorum quadrata, <lb/>non aliaque circa eoſdem variatio contingit.</s> <s xml:id="echoid-s5238" xml:space="preserve"/> </p> <div xml:id="echoid-div532" type="float" level="2" n="1"> <note position="right" xlink:label="note-0235-02" xlink:href="note-0235-02a" xml:space="preserve">_11. Huius._</note> </div> </div> <div xml:id="echoid-div534" type="section" level="1" n="318"> <head xml:id="echoid-head335" xml:space="preserve">B. SECTIO II.</head> <note position="right" xml:space="preserve">B.</note> <p style="it"> <s xml:id="echoid-s5239" xml:space="preserve">_C_olliguntur ergo hæc de binis ellipſibus .</s> <s xml:id="echoid-s5240" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5241" xml:space="preserve">quod quæ ſunt circa ean. <lb/></s> <s xml:id="echoid-s5242" xml:space="preserve">dem diametrum, ſunt vt reliquæ ſecundæ diametri.</s> <s xml:id="echoid-s5243" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div535" type="section" level="1" n="319"> <head xml:id="echoid-head336" xml:space="preserve">C. SECTIO III.</head> <note position="right" xml:space="preserve">C.</note> <p style="it"> <s xml:id="echoid-s5244" xml:space="preserve">_Q_V æcunq; </s> <s xml:id="echoid-s5245" xml:space="preserve">ellipſes habent rationem ex axibus, vel diametris con-<lb/>iugatis, æqualiter ad inuicem inclinatis compoſitam.</s> <s xml:id="echoid-s5246" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div536" type="section" level="1" n="320"> <head xml:id="echoid-head337" xml:space="preserve">D. SECTIO IV.</head> <note position="right" xml:space="preserve">D.</note> <p style="it"> <s xml:id="echoid-s5247" xml:space="preserve">_E_Llipſes habentes axes, vel diametros coniugatas, quæ æqualiter <lb/>ſunt inclinatæ, reciprocè reſpondentes, ſunt æquales; </s> <s xml:id="echoid-s5248" xml:space="preserve">& </s> <s xml:id="echoid-s5249" xml:space="preserve">quæ <lb/>ſunt æquales, & </s> <s xml:id="echoid-s5250" xml:space="preserve">habent axes, vel diametros ad inuicem æqualiter in-<lb/>clinatas, eaſdem habent reciprocè reſpondentes.</s> <s xml:id="echoid-s5251" xml:space="preserve"/> </p> <pb o="216" file="0236" n="236" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div537" type="section" level="1" n="321"> <head xml:id="echoid-head338" xml:space="preserve">E. SECTIO V.</head> <note position="left" xml:space="preserve">E.</note> <p style="it"> <s xml:id="echoid-s5252" xml:space="preserve">_S_Imiles ellipſes ſunt in dupla ratione ſuorum axium, vel diametrc-<lb/>rum homologarum, vel vt corundem quadrata.</s> <s xml:id="echoid-s5253" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div538" type="section" level="1" n="322"> <head xml:id="echoid-head339" xml:space="preserve">F. SECTIO VI.</head> <note position="left" xml:space="preserve">F.</note> <p style="it"> <s xml:id="echoid-s5254" xml:space="preserve">_P_Ro circulis autem (vt ſupra dictum eſt) hoc tantum habetur, quod <lb/>ſint vt diametrorum quadrata, vel in dupla ratione diametrorum; <lb/></s> <s xml:id="echoid-s5255" xml:space="preserve">neque illis alia variatio contingit, ſicuti ellipſibus competere ex ſupe-<lb/>rioribus compertum eſt.</s> <s xml:id="echoid-s5256" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div539" type="section" level="1" n="323"> <head xml:id="echoid-head340" xml:space="preserve">THEOREMA XI. PROPOS. XII.</head> <p> <s xml:id="echoid-s5257" xml:space="preserve">QVęcunq; </s> <s xml:id="echoid-s5258" xml:space="preserve">de omnibus quadratis parallelogrammorum, <lb/>appoſitas ibi conditiones habentium, oſtenſa ſunt in <lb/>Theor. </s> <s xml:id="echoid-s5259" xml:space="preserve">9.</s> <s xml:id="echoid-s5260" xml:space="preserve">10.</s> <s xml:id="echoid-s5261" xml:space="preserve">11.</s> <s xml:id="echoid-s5262" xml:space="preserve">12.</s> <s xml:id="echoid-s5263" xml:space="preserve">13. </s> <s xml:id="echoid-s5264" xml:space="preserve">lib. </s> <s xml:id="echoid-s5265" xml:space="preserve">2 eadem de omnibus quadratis <lb/>circulorum, vel ellipſium illis inſcriptorum (regula in <lb/>vtriſque altero axium, vel diametrorum coniugatarum) ve-<lb/>rificabuntur.</s> <s xml:id="echoid-s5266" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5267" xml:space="preserve">Patet hæc propoſitio, nam omnia quadrata circulorum, vel el-<lb/>lipſium (regula altero axium, vel diametrorum) ſunt ſubſexquial-<lb/> <anchor type="note" xlink:label="note-0236-03a" xlink:href="note-0236-03"/> tera omnium quadratorum parallelogrammorum, quibus inſcri-<lb/>buntur, latera habentium dictis axibus, vel diametris parallela; </s> <s xml:id="echoid-s5268" xml:space="preserve">ha-<lb/>bentibus autem illis appoſitas ibi conditiones in ſuis lateribus, eędem <lb/>adſunt in axibus, vel diametris circulorum, vel ellipſium, quibus <lb/>circumſcribuntur, & </s> <s xml:id="echoid-s5269" xml:space="preserve">è contra; </s> <s xml:id="echoid-s5270" xml:space="preserve">& </s> <s xml:id="echoid-s5271" xml:space="preserve">ideò concluſiones, quæ collectæ <lb/>ſunt pro illis in dictis Theor. </s> <s xml:id="echoid-s5272" xml:space="preserve">etiam pro omnibus quadratis circulo-<lb/>rum, vel ellipſium illis inſcriptorum, vt demonſtratę recipi poſſunt, <lb/>cum fint eorum partes proportionales, ijſdem regulis pro omnibus <lb/>quadratis circulorum, vel ellipſium, & </s> <s xml:id="echoid-s5273" xml:space="preserve">pro omnibus quadra-<lb/>tis parallelogrammorum illis circumſeriptorum, aſſumptis, <lb/>quod, &</s> <s xml:id="echoid-s5274" xml:space="preserve">c.</s> <s xml:id="echoid-s5275" xml:space="preserve"/> </p> <div xml:id="echoid-div539" type="float" level="2" n="1"> <note position="left" xlink:label="note-0236-03" xlink:href="note-0236-03a" xml:space="preserve">Coroll.1. <lb/>buius.</note> </div> <figure> <image file="0236-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0236-01"/> </figure> <pb o="217" file="0237" n="237" rhead="LIBER III."/> </div> <div xml:id="echoid-div541" type="section" level="1" n="324"> <head xml:id="echoid-head341" xml:space="preserve">THEOREMA XII. PROPOS. XIII.</head> <p> <s xml:id="echoid-s5276" xml:space="preserve">SI circulum, vel ellipſim duæ rectæ lineæ in terminis <lb/>coniugatarum diametrorum tetigetint inter ſe conue-<lb/>nientes, eiſdem diametris ductis. </s> <s xml:id="echoid-s5277" xml:space="preserve">Omnia quadrata conſti-<lb/>tuti parallelogrammiad omnia quadrata trilinei à dictis tan-<lb/>gentibus, & </s> <s xml:id="echoid-s5278" xml:space="preserve">ab incluſa curua comprehenſi, regula altera <lb/>diametrorum, erunt vt dictum parallelogrammum ad ſui <lb/>reliquum, dempto quadrante circuli, vel ellipſis iam dictæ, <lb/>quod inſcribitur prædicto parallelogrammo, ſimul cum ex-<lb/>ceſſu dicti quadrantis ſuper duas tertias iam dicti parallelo-<lb/>grammi, quæ ratio erit proximè, vt 21. </s> <s xml:id="echoid-s5279" xml:space="preserve">ad 2.</s> <s xml:id="echoid-s5280" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5281" xml:space="preserve">Sit circulus, vel ellipſis, ABCD, cuius diametri coniugatæ, A <lb/>C, BD, in quorum terminis, C, D, duæ rectæ lineę ipſum tangen-<lb/>tesinter ſe conueniant in, V. </s> <s xml:id="echoid-s5282" xml:space="preserve">Dico ergo (ſumpta regula qualibet <lb/>diametrorum, vt, BD,) quod omnia quadrata parallelogrammi, O <lb/>V, ad omnia quadrata trilinei, DCV, duabus tangentibus, DV, <lb/>VC, & </s> <s xml:id="echoid-s5283" xml:space="preserve">ab ijs incluſa curua, DC, comprehenſi ſunt, vt idem paral-<lb/> <anchor type="figure" xlink:label="fig-0237-01a" xlink:href="fig-0237-01"/> lelogrammum, OV, ad ſui reliquum <lb/>dempto quadrante, OCD, circuli, <lb/>vel ellipſis, ABCD, ſimul cum eo <lb/>ſpatio, quo idem quadrans excedit <lb/>duas tertias parallelogrammi, OV. <lb/></s> <s xml:id="echoid-s5284" xml:space="preserve">Sumatur intra, OC, vtcunque pun <lb/>ctum, E, & </s> <s xml:id="echoid-s5285" xml:space="preserve">per, E, ducatur ipſi, B <lb/>D, parallela, EF, ſecans curuam, D <lb/>C, in, I. </s> <s xml:id="echoid-s5286" xml:space="preserve">Omnia ergo quadrata pa-<lb/>rallelogrammi, OV, ad rectangula <lb/>ſub parallelogrammo, OV, & </s> <s xml:id="echoid-s5287" xml:space="preserve">ſemi-<lb/>portione, OCD, ſunt vt parallelo <lb/> <anchor type="note" xlink:label="note-0237-01a" xlink:href="note-0237-01"/> grammum, OV, ad eandem ſemiportionem, OCD; </s> <s xml:id="echoid-s5288" xml:space="preserve">ſed eadem ad <lb/> <anchor type="note" xlink:label="note-0237-02a" xlink:href="note-0237-02"/> omnia quadrata ſemiportionis, OCD, ſunt ſexquialtera, ergo ad <lb/>reſiduum erunt vt parallelogrammum, OV, ad reſiduum ſemipor-<lb/>tionis, OCD, demptis ab ea, {2/3}, parallelogrammi, OV, quarum <lb/>idem parallelogrammum, OV, eſt ſexquialterum; </s> <s xml:id="echoid-s5289" xml:space="preserve">reſiduum autem <lb/>rectangulorum ſub parallelogrammo, OV, & </s> <s xml:id="echoid-s5290" xml:space="preserve">ſemiportione, OC <lb/>D, demptis omnibus quadratis ſemiportionis, OCD, ſunt rectan-<lb/> <anchor type="note" xlink:label="note-0237-03a" xlink:href="note-0237-03"/> gula ſub ſemiportione, OCD, & </s> <s xml:id="echoid-s5291" xml:space="preserve">trilineo, CDV, nam veluti in, <pb o="218" file="0238" n="238" rhead="GEOMETRIÆ"/> EF, ducta, vtcunque quadratum, EI, detractum à rectangulo ſub, <lb/>IE, EF, relinquit rectangulum ſub, EI, IF, ita in cæteris ſequitur; <lb/></s> <s xml:id="echoid-s5292" xml:space="preserve">& </s> <s xml:id="echoid-s5293" xml:space="preserve">illis ſimul collectis ſequitur etiam, quod detractis omnibus qua-<lb/> <anchor type="note" xlink:label="note-0238-01a" xlink:href="note-0238-01"/> dratis ſemiportionis, OCD, à rectangulis ſub parallelogrammo, O <lb/>V, & </s> <s xml:id="echoid-s5294" xml:space="preserve">ſemiportione, OCD, relinquantur rectangula ſub ſemipor-<lb/>tione, OCD, & </s> <s xml:id="echoid-s5295" xml:space="preserve">trilineo, DCV, ad hæc igitur, quæ ſunt dictum <lb/> <anchor type="figure" xlink:label="fig-0238-01a" xlink:href="fig-0238-01"/> reſiduum, omnia quadrata parallelo-<lb/>grammi, OV erunt vt parallelogram-<lb/>mum, OV, adreſiduum ſemiportio-<lb/>nis, OCD, ab ea demptis, {2/3}, paral-<lb/>lelogrammi, OV; </s> <s xml:id="echoid-s5296" xml:space="preserve">eadem autem om-<lb/>nia quadrata parallelogrammi, OV, <lb/>ad rectangula ſub parallelogrammo. <lb/></s> <s xml:id="echoid-s5297" xml:space="preserve">OV, & </s> <s xml:id="echoid-s5298" xml:space="preserve">ſemiportione, OCD, .</s> <s xml:id="echoid-s5299" xml:space="preserve">i. </s> <s xml:id="echoid-s5300" xml:space="preserve">ad <lb/> <anchor type="note" xlink:label="note-0238-02a" xlink:href="note-0238-02"/> omnia quadrata ſemiportionis, OC <lb/>D, vna cum rectangulis ſub ſemipor-<lb/>tione, OCD, & </s> <s xml:id="echoid-s5301" xml:space="preserve">trilineo, CVD, <lb/>ſunt vt parallelogrammum, OV, ad <lb/>ſemiportionem, OCD, vt paulò ſupra in hac demonſtratione oſten-<lb/>dimus, ergo, colligendo, omnia quadrata parallelogrammi, OV, <lb/>ad omnia quadrata ſemiportionis, OCD, vna cum rectangu is bis <lb/>ſub ſemiportione, OCD, & </s> <s xml:id="echoid-s5302" xml:space="preserve">trilineo, CVD, ſumptis, erunt vt pa-<lb/>rallelogrammum, OV, ad ſemiportionem, OCD, vna cum exceſ-<lb/>ſu, quo dicta ſemiportio, OCD, excedit, {2/3}, parallelogrammi, O <lb/>V, ergo, perconuerſionem rationis, omnia quadrata parallelogram-<lb/>mi, OV, ad omnia quadrata trilinei, DCV, quæ remanent detra-<lb/> <anchor type="note" xlink:label="note-0238-03a" xlink:href="note-0238-03"/> ctis omnibus quadratis ſemiportionis, OCD, vna cum rectangulis <lb/>ſub illa, & </s> <s xml:id="echoid-s5303" xml:space="preserve">ſub trilineo, DCV, bis ſumptis, ab omnibus quadratis <lb/>parallelogrammi, OV; </s> <s xml:id="echoid-s5304" xml:space="preserve">(veluti detracto quadrato, EI, vna cum re-<lb/>ctangulo bis ſub, EI, IF, remanet quadratum, IF,) ad omnia qua-<lb/>dratatrilinei, DCV, erunt vt parallelogrammum, OV, adreſiduum, <lb/>detracta ſemiportione, OCD, vna cum exceſſu, quoipſa ſuperat <lb/>duas tertias parallelogrammi, OV, à dicto parallelogrammo, <lb/>OV.</s> <s xml:id="echoid-s5305" xml:space="preserve"/> </p> <div xml:id="echoid-div541" type="float" level="2" n="1"> <figure xlink:label="fig-0237-01" xlink:href="fig-0237-01a"> <image file="0237-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0237-01"/> </figure> <note position="right" xlink:label="note-0237-01" xlink:href="note-0237-01a" xml:space="preserve">Coroll.1. <lb/>26.lib.2.</note> <note position="right" xlink:label="note-0237-02" xlink:href="note-0237-02a" xml:space="preserve">Coroll.1. <lb/>huius.</note> <note position="right" xlink:label="note-0237-03" xlink:href="note-0237-03a" xml:space="preserve">Vide ibid. <lb/>dicta.</note> <note position="left" xlink:label="note-0238-01" xlink:href="note-0238-01a" xml:space="preserve">Iux. dicta <lb/>pro C.23. <lb/>lib.2.</note> <figure xlink:label="fig-0238-01" xlink:href="fig-0238-01a"> <image file="0238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0238-01"/> </figure> <note position="left" xlink:label="note-0238-02" xlink:href="note-0238-02a" xml:space="preserve">Per C.23. <lb/>lib.2.</note> <note position="left" xlink:label="note-0238-03" xlink:href="note-0238-03a" xml:space="preserve">Per D.23. <lb/>lib. 2.</note> </div> <p> <s xml:id="echoid-s5306" xml:space="preserve">Eſt verò parallelogrammum, OV, ad dictum ſpatium reſiduum <lb/>proximè, vt 21. </s> <s xml:id="echoid-s5307" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s5308" xml:space="preserve">nam ſi ſupponamus parallelogrammum, OV, <lb/>eſſe 21. </s> <s xml:id="echoid-s5309" xml:space="preserve">erit ſemiportio, OCD, earumdem partium proximè 16. </s> <s xml:id="echoid-s5310" xml:space="preserve">{1/2}, <lb/>eſt .</s> <s xml:id="echoid-s5311" xml:space="preserve">n. </s> <s xml:id="echoid-s5312" xml:space="preserve">adeam, ſicut rectangulum, quod eſſet circulo, vel ellipſi, A <lb/> <anchor type="note" xlink:label="note-0238-04a" xlink:href="note-0238-04"/> BCD, circumſcriptum, habens latera ipſis, AC, BD, axibus pa-<lb/>rallela ad eundem circulum, vel ellipſim .</s> <s xml:id="echoid-s5313" xml:space="preserve">i. </s> <s xml:id="echoid-s5314" xml:space="preserve">vt 14. </s> <s xml:id="echoid-s5315" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s5316" xml:space="preserve">proximè, vt <lb/>oſtendit Archimedes lib. </s> <s xml:id="echoid-s5317" xml:space="preserve">de Dimenſione Circuli, eſt .</s> <s xml:id="echoid-s5318" xml:space="preserve">n. </s> <s xml:id="echoid-s5319" xml:space="preserve">vt 14. </s> <s xml:id="echoid-s5320" xml:space="preserve">ad 11. <lb/></s> <s xml:id="echoid-s5321" xml:space="preserve">ita 21. </s> <s xml:id="echoid-s5322" xml:space="preserve">ad 16. </s> <s xml:id="echoid-s5323" xml:space="preserve">{1/2}, rurſus duæ tertiæ parallelogrammi, OV, ſunt 14.</s> <s xml:id="echoid-s5324" xml:space="preserve"> <pb o="219" file="0239" n="239" rhead="LIBER III."/> ſemiportio verò. </s> <s xml:id="echoid-s5325" xml:space="preserve">OCD, quæ eſt pioximè 16. </s> <s xml:id="echoid-s5326" xml:space="preserve">{1/2}, excedit, {2/3}, paral-<lb/>lelogrammi, OV, ſcilicet 14. </s> <s xml:id="echoid-s5327" xml:space="preserve">per 2 {1/2}, ſi ergo ſemiportioni, OCD, <lb/>quæ eſt proximè 16 {1/2}, iunxerimus exceſſum eiuſdem ſemiportionis <lb/>ſuper, {2/3}, parallelogrammi, OV, .</s> <s xml:id="echoid-s5328" xml:space="preserve">i.</s> <s xml:id="echoid-s5329" xml:space="preserve">2 {1/2}, fiet totum conſequens pro-<lb/>ximè 19. </s> <s xml:id="echoid-s5330" xml:space="preserve">hoc ſi detrahatur a toto parallelogrammo, OV, quod eſt <lb/>21. </s> <s xml:id="echoid-s5331" xml:space="preserve">relinquentur 2. </s> <s xml:id="echoid-s5332" xml:space="preserve">erit ergo parallelogrammum, OV, ad hoc reſi-<lb/>duum proximè, vt 21. </s> <s xml:id="echoid-s5333" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s5334" xml:space="preserve">vnde & </s> <s xml:id="echoid-s5335" xml:space="preserve">omnia quadrata parallelogram-<lb/>mi, OV, ad omnia quadrata trilinei, DCV, erunt proximè vt 21. <lb/></s> <s xml:id="echoid-s5336" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s5337" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s5338" xml:space="preserve"/> </p> <div xml:id="echoid-div542" type="float" level="2" n="2"> <note position="left" xlink:label="note-0238-04" xlink:href="note-0238-04a" xml:space="preserve">11.huius.</note> </div> </div> <div xml:id="echoid-div544" type="section" level="1" n="325"> <head xml:id="echoid-head342" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5339" xml:space="preserve">_H_INC patet, ſi nos præcisè ſciamus, quam rationem habeant om-<lb/>nia quadrata parallelogrammi, OV, ad omnia quadrata trilinei, <lb/>DCV, quia etiam ſcimus, quam rationem habeant omnia quadrata, C <lb/>D, ad omnia quadrata ſemiportionis, OCD, ſciemus etiam, quam ra-<lb/>tionem habeant eadem ad rectangula ſub ſemiportione, OCD, & </s> <s xml:id="echoid-s5340" xml:space="preserve">trili-<lb/>neo, DCV, bis ſumpta, & </s> <s xml:id="echoid-s5341" xml:space="preserve">item nota erit ratio ad eadem ſemel ſum-<lb/>pta, quæ ſi iungantur omnibus quadratis ſemiportionis, OCD, compo-<lb/> <anchor type="note" xlink:label="note-0239-01a" xlink:href="note-0239-01"/> nentur rectangula ſub para lelogrammo, OV, & </s> <s xml:id="echoid-s5342" xml:space="preserve">ſemiportione, <lb/>OCD, & </s> <s xml:id="echoid-s5343" xml:space="preserve">fiet nota ratio omnium quadratorum, OV, ad rectangula <lb/>ſub parallelogrammo, OV, & </s> <s xml:id="echoid-s5344" xml:space="preserve">ſemiportione, OCD, quæ eſt eadem <lb/> <anchor type="note" xlink:label="note-0239-02a" xlink:href="note-0239-02"/> ei, quam habet parallelogrammum OV, ad ſemiportionem, OCD, <lb/>& </s> <s xml:id="echoid-s5345" xml:space="preserve">ideò hęc erit nota, ſicut etiam erit nota ratio parallelogrammi cir-<lb/>culo, vel ellipſi, ABCD, circumſcripti, habentis latera parallela <lb/>ipſis, AC, BD, ad eundem circulum, vel ellipſim, ABCD, & </s> <s xml:id="echoid-s5346" xml:space="preserve">hinc habere-<lb/>tur circuli quadratura; </s> <s xml:id="echoid-s5347" xml:space="preserve">ideò quærendum eſt, quam rationem habeant præ-<lb/>cise omnia quadrata, OV, ad omnia quadrata trilinei, CDV; </s> <s xml:id="echoid-s5348" xml:space="preserve">quod hucuſ-<lb/>que nec alijs, nec mibi compertum eſſe potuit.</s> <s xml:id="echoid-s5349" xml:space="preserve"/> </p> <div xml:id="echoid-div544" type="float" level="2" n="1"> <note position="right" xlink:label="note-0239-01" xlink:href="note-0239-01a" xml:space="preserve">_PerC.23._ <lb/>_lib.2._</note> <note position="right" xlink:label="note-0239-02" xlink:href="note-0239-02a" xml:space="preserve">_Coroll.1._ <lb/>_26.lib.2._</note> </div> </div> <div xml:id="echoid-div546" type="section" level="1" n="326"> <head xml:id="echoid-head343" xml:space="preserve">THEOREMA XIII. PROPOS. XIV.</head> <p> <s xml:id="echoid-s5350" xml:space="preserve">SI circa parallelogrammi rectanguli quodlibet laterum, <lb/>tamquam circa diametrum integrorum, ſemicirculus, <lb/>vel ſemiellipſis, etiam ipſo non exiſtente rectangulo, deſ-<lb/>cripti fuerint, circumferentia autem circuli, vel curua elli-<lb/>pſis non pertingant, neque ſecet oppoſitum prædicto latus, <lb/>ſit autem regula parallelogrammi baſis: </s> <s xml:id="echoid-s5351" xml:space="preserve">Omnia quadrata <lb/>dicti parallelogrammi ad omnia quadrata figuræ, quæ reli-<lb/>quis tribus parallelogrammi lateribus (dempto eo, quod <pb o="220" file="0240" n="240" rhead="GEOMETRIÆ"/> pro axiſumptum eſt) & </s> <s xml:id="echoid-s5352" xml:space="preserve">curua circuli, vel ellipſis contine-<lb/>tur, erunt proximè, vt baſis eiuſdem parallelogrammi ad ſui <lb/>reliquum, demptis ab ea, {11/14}, rectæ lineæ, quæ ſit æqualis <lb/>dimidiæ ſecundæ diametri Prædicti circuli, vel ellipſis, ſi-<lb/>mul cum exceſſu, quo dicti, {11/14}, excedunt, {2/3}, tertiæ propor-<lb/>tionalis duarum, quarum prima eſt dicta baſis, ſecunda au-<lb/>tem dicta ſecundæ diametri dimidia.</s> <s xml:id="echoid-s5353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5354" xml:space="preserve">Sit parallelogrammum, FD, & </s> <s xml:id="echoid-s5355" xml:space="preserve">circa latus, FB, vtcunque tam-<lb/>quam circa diametrum (intellige autem ſemper diametrum hic, & </s> <s xml:id="echoid-s5356" xml:space="preserve"><lb/>in ſequentibus, vt eſt nomen commune diametro, & </s> <s xml:id="echoid-s5357" xml:space="preserve">axi) integri ſic <lb/>deſcriptus ſemicirculus, vel ſemiellipſis, FQB, cuius curua, FQB, <lb/>neque tangat, neque ſecet latus, ZD, oppoſitum lateri, FB, bifa-<lb/>riam autem diuiſa, FB, in, A, & </s> <s xml:id="echoid-s5358" xml:space="preserve">per, A, ipſi, BD, baſi ducta pa-<lb/>rallela, AP, ſeceturà curua, FQB, vtcunq; </s> <s xml:id="echoid-s5359" xml:space="preserve">in, Q; </s> <s xml:id="echoid-s5360" xml:space="preserve">erit autem, A <lb/>Q, dimidia ſecundæ axis circuli, vel ellipſis, cuius centrum, A; </s> <s xml:id="echoid-s5361" xml:space="preserve">du-<lb/>catur inſuperper, Q, ipſi, FB, parallela, HC, quæ tanget circu-<lb/> <anchor type="note" xlink:label="note-0240-01a" xlink:href="note-0240-01"/> lum dictum, vel ellipſim, & </s> <s xml:id="echoid-s5362" xml:space="preserve">erit, BC, æqualis ipſi, AQ; </s> <s xml:id="echoid-s5363" xml:space="preserve">fiat dein-<lb/>de, vt, DB, ad, BC, ita, BC, ad, BI, & </s> <s xml:id="echoid-s5364" xml:space="preserve">ſumatur, BR, quæ ſit, <lb/>{2/3}, BI, &</s> <s xml:id="echoid-s5365" xml:space="preserve">, BE, quæ fit, {11/14}, ipſius, CB, &</s> <s xml:id="echoid-s5366" xml:space="preserve">, EV, quæ ſit æqualis <lb/> <anchor type="figure" xlink:label="fig-0240-01a" xlink:href="fig-0240-01"/> ipſi, ER, regula verò ſit, BD. </s> <s xml:id="echoid-s5367" xml:space="preserve">Dico er-<lb/>go omnia quadrata parallelogrammi, FD, <lb/>ad omnia quadrata figuræ, quæ compre-<lb/>henditur tribus lateribus, FZ, ZD, DB, <lb/>& </s> <s xml:id="echoid-s5368" xml:space="preserve">curua, FQB, eſſe, vt, BD, ad, DV, <lb/>proximè. </s> <s xml:id="echoid-s5369" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s5370" xml:space="preserve">n. </s> <s xml:id="echoid-s5371" xml:space="preserve">quadrata parallelo-<lb/>grammi, FD, ad rectangula ſub paralle-<lb/>logrammo, FD, & </s> <s xml:id="echoid-s5372" xml:space="preserve">ſemicirculo, vel ſemi-<lb/> <anchor type="note" xlink:label="note-0240-02a" xlink:href="note-0240-02"/> ellipſis, FQB, ſunt vt parallelogram-<lb/>mum, FD, ad eundem ſemicirculum, vel <lb/>ſemiellipſim, FQB; </s> <s xml:id="echoid-s5373" xml:space="preserve">quia verò parallelo-<lb/>grammum, FD, ad parallelogrammum, FC, eſt vt, DB, ad, BC, <lb/>& </s> <s xml:id="echoid-s5374" xml:space="preserve">item parallelogrammum, FC, ad ſemicirculum, vel ſemiellipſim, <lb/> <anchor type="note" xlink:label="note-0240-03a" xlink:href="note-0240-03"/> FQB, eſt proximè vt 14. </s> <s xml:id="echoid-s5375" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s5376" xml:space="preserve">ideſt vt, CB, ad, BE, ergo ex <lb/>æquali parallelogrammum, FD, ad ſemicirculum, vel ſemiellipſim, <lb/>FQB, erit vt, DB, ad, BE, & </s> <s xml:id="echoid-s5377" xml:space="preserve">ideò omnia quadrata parallelogram-<lb/>mi, FD, ad rectangula ſub parallelogrammo, FD, & </s> <s xml:id="echoid-s5378" xml:space="preserve">ſemicirculo, <lb/> <anchor type="note" xlink:label="note-0240-04a" xlink:href="note-0240-04"/> vel ſemiellipſi, FQB, erunt vt, DB, ad, BE,.</s> <s xml:id="echoid-s5379" xml:space="preserve">i. </s> <s xml:id="echoid-s5380" xml:space="preserve">ſumpta, DB, com-<lb/>muni altitudine erunt, vt quadratum, DB, ad rectangulum ſub, D <lb/>B, BE, quodſerua.</s> <s xml:id="echoid-s5381" xml:space="preserve"/> </p> <div xml:id="echoid-div546" type="float" level="2" n="1"> <note position="left" xlink:label="note-0240-01" xlink:href="note-0240-01a" xml:space="preserve">17.1.Con.</note> <figure xlink:label="fig-0240-01" xlink:href="fig-0240-01a"> <image file="0240-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0240-01"/> </figure> <note position="left" xlink:label="note-0240-02" xlink:href="note-0240-02a" xml:space="preserve">Coroll.1. <lb/>26.lib.2.</note> <note position="left" xlink:label="note-0240-03" xlink:href="note-0240-03a" xml:space="preserve">5. Lib.2. <lb/>Arch. de <lb/>Dim.Cir.</note> <note position="left" xlink:label="note-0240-04" xlink:href="note-0240-04a" xml:space="preserve">5.Lib.2.</note> </div> <p> <s xml:id="echoid-s5382" xml:space="preserve">Aduerte nunc, quodrectangula ſub parallelogrammo, FD, & </s> <s xml:id="echoid-s5383" xml:space="preserve">ſe- <pb o="221" file="0241" n="241" rhead="LIBER III."/> micirculo, vel ſemiellipſi, FQB, diuiduntur per curuam, FQB, in <lb/> <anchor type="note" xlink:label="note-0241-01a" xlink:href="note-0241-01"/> rectangula fub quadrilineo, FQBDZ, & </s> <s xml:id="echoid-s5384" xml:space="preserve">ſemicirculo, vel ſemiel-<lb/>lipſi, FQB, & </s> <s xml:id="echoid-s5385" xml:space="preserve">in omnia quadrata ſemicirculi, vel ſemiellipſis, FQ <lb/>B, videndum ergo nunceſt, quamrationem habeant omnia quadra-<lb/>ta, FD, ad omnia quadrata ſemicirculi, vel ſemiellipſis, FQB, <lb/> <anchor type="note" xlink:label="note-0241-02a" xlink:href="note-0241-02"/> quod fic patet; </s> <s xml:id="echoid-s5386" xml:space="preserve">omnia quadrata, FD, ad omnia quadrata, FC, <lb/> <anchor type="note" xlink:label="note-0241-03a" xlink:href="note-0241-03"/> ſunt vt quadratum, DB, ad quadratum, BC, .</s> <s xml:id="echoid-s5387" xml:space="preserve">i. </s> <s xml:id="echoid-s5388" xml:space="preserve">ad rectangulum ſub, <lb/>DB, BI, nam tres, DB, BC, BI, ſunt continuè proportionales, <lb/>omnia item quadrata, FC, omnium quadratorum ſemicirculi, vel <lb/> <anchor type="note" xlink:label="note-0241-04a" xlink:href="note-0241-04"/> ſemiellipſis, FQB, ſunt ſexquialtera .</s> <s xml:id="echoid-s5389" xml:space="preserve">i. </s> <s xml:id="echoid-s5390" xml:space="preserve">ſunt vt rectangulum, DBI, <lb/>ad rectangulum, DBR, quia, BR, eſt, {2/3}, BI, ergo ex æquali om <lb/>nia quadrata, FD, ad omnia quadrata ſemicirculi, vel ſemiellipfis, <lb/>FQB, ſunt vt quadratum, DB, ad rectangulum ſub, DB, BR, <lb/>omnia autem quadrata, FD, ad rectangula ſub, FD, & </s> <s xml:id="echoid-s5391" xml:space="preserve">ſemicircu-<lb/>lo, vel ſemiellipſi, FQB, erant vt idem quadratum, DB, ad rectan-<lb/>gulum ſub, DB, BE, ergo omnia quadrata, FD, ad rectangula ſub <lb/>ſemicirculo, vel ſemiell pſi, FQB, & </s> <s xml:id="echoid-s5392" xml:space="preserve">ſub quadrilineo, FQBDZ, <lb/>erunt vt idem quadratum, DB, ad rectangulum ſub, DB, &</s> <s xml:id="echoid-s5393" xml:space="preserve">, RE, <lb/>ad eadem verò bis ſumpta, vt idem quadratum, DB, ad rectangu-<lb/>lum ſub, DB, &</s> <s xml:id="echoid-s5394" xml:space="preserve">, RV, quia verò omnia quadrata, FD, ad omnia <lb/>quadrata ſemicirculi, vel ſemiellipſis, FQB, ſunt vt quadratum, D <lb/>B, ad rectangulum ſub, DB, BR, ergo colligendo omnia quadra-<lb/>ta, FD, ad omnia quadrata ſemicirculi, vel ſemiellipſis, FQB, vna <lb/>cum rectangulis ſub ſemicirculo, vel ſemiellipſi, FQB, & </s> <s xml:id="echoid-s5395" xml:space="preserve">quadri-<lb/>lineo, FQBDZ, bis ſumptis, erunt vt quadratum, DB, ad rectan-<lb/>gula ſub, DB, BR, DB, RV, .</s> <s xml:id="echoid-s5396" xml:space="preserve">i. </s> <s xml:id="echoid-s5397" xml:space="preserve">ad rectangulum ſub, DB, BV; <lb/></s> <s xml:id="echoid-s5398" xml:space="preserve">quia verò ſiab omnibus quadratis, FD, ſubtraxeris omnia quadrata <lb/>ſemicirculi, vel ſemiellipſis, FQB, vna cum rectangulis bis ſub eo-<lb/>dem ſemicirculo, vel ſemiellipſi, FQB, & </s> <s xml:id="echoid-s5399" xml:space="preserve">ſub quadrilineo, FQB <lb/> <anchor type="note" xlink:label="note-0241-05a" xlink:href="note-0241-05"/> DZ, remanent omnia quadrata quadrilinei, FQBDZ, ideò, per <lb/>conuerſionem rationis, omnia quadrata parallelogrammi, FD, ad <lb/> <anchor type="note" xlink:label="note-0241-06a" xlink:href="note-0241-06"/> omnia quadrata quadrilinei, FQBDZ, erunt vt quadratum, BD, <lb/>ad rectangulum ſub, BD, DV, .</s> <s xml:id="echoid-s5400" xml:space="preserve">i. </s> <s xml:id="echoid-s5401" xml:space="preserve">vt, BD, ad, DV, quod tantum <lb/>proximè verificatur, non .</s> <s xml:id="echoid-s5402" xml:space="preserve">n. </s> <s xml:id="echoid-s5403" xml:space="preserve">parallelogrammum, FC, ad ſemicir-<lb/>culum, vel ſemiellipſim, FQB, eſt pręcisè, vt 14. </s> <s xml:id="echoid-s5404" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s5405" xml:space="preserve">ſed tantum <lb/>proximè, ideò, &</s> <s xml:id="echoid-s5406" xml:space="preserve">c.</s> <s xml:id="echoid-s5407" xml:space="preserve"/> </p> <div xml:id="echoid-div547" type="float" level="2" n="2"> <note position="right" xlink:label="note-0241-01" xlink:href="note-0241-01a" xml:space="preserve">Per C.23. <lb/>lib. 2.</note> <note position="right" xlink:label="note-0241-02" xlink:href="note-0241-02a" xml:space="preserve">9. Lib. 2.</note> <note position="right" xlink:label="note-0241-03" xlink:href="note-0241-03a" xml:space="preserve">Elici etiã <lb/>poteſt ex <lb/>12. lib. 2.</note> <note position="right" xlink:label="note-0241-04" xlink:href="note-0241-04a" xml:space="preserve">Coroll. 1. <lb/>huius. <lb/>5. Lib. 2.</note> <note position="right" xlink:label="note-0241-05" xlink:href="note-0241-05a" xml:space="preserve">PerD.23. <lb/>lib, 2.</note> <note position="right" xlink:label="note-0241-06" xlink:href="note-0241-06a" xml:space="preserve">5. Lib. 2.</note> </div> <p> <s xml:id="echoid-s5408" xml:space="preserve">Defiderari nunctantum videtur in hac demonſtratione, quod pro. <lb/></s> <s xml:id="echoid-s5409" xml:space="preserve">betur punctum, R, non identificari puncto, E, ſed cadere inter, B <lb/>E, quod ſic facilè patet, cum .</s> <s xml:id="echoid-s5410" xml:space="preserve">n. </s> <s xml:id="echoid-s5411" xml:space="preserve">oſtenſum ſit omnia quadrata, FD, <lb/>ad rectangula ſub parallelogrammo, FD, & </s> <s xml:id="echoid-s5412" xml:space="preserve">ſemicirculo, vel ſemiel-<lb/>lipſi, FQB, eſſe vt quadratum, DB, ad rectangulum ſub, DB, B <lb/>E, inſuper oſtenſum ſit omnia quadrata, FD, ad omnia quadrata <pb o="222" file="0242" n="242" rhead="GEOMETRIÆ"/> ſemicirculi, vel ſemiellipſis, FQB, eſſe vt quadratum, DB, ad re-<lb/>ctangulum ſub, DB, BR, quoniam rectanguſa ſub, FD, & </s> <s xml:id="echoid-s5413" xml:space="preserve">ſemi-<lb/>circulo, vel ſemiellipſi, FQB, ſunt maiora omnibus quadratis ſemi-<lb/>circuli, vel ſemiellipſis, FQB, ideò etiam rectangulum ſub, DB, B <lb/>E, ſemper maius eſt rectangulo ſub, DB, BR, & </s> <s xml:id="echoid-s5414" xml:space="preserve">ideo punctum, R, <lb/>ſemper cadet inter punctum, B, & </s> <s xml:id="echoid-s5415" xml:space="preserve">punctum, E, quocunque deinde <lb/>cadat punctum, I, vnde patet, &</s> <s xml:id="echoid-s5416" xml:space="preserve">c.</s> <s xml:id="echoid-s5417" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5418" xml:space="preserve">Similiter, quia omnia quadrata ſemicirculi, vel ſemiellipſis, FQ <lb/>B, vna cum rectangulis ſub eodem, & </s> <s xml:id="echoid-s5419" xml:space="preserve">ſub quadrilineo, FQBDZ, <lb/>bis ſumptis, minora ſunt omnibus quadratis, FD, ideò, BV, com-<lb/>pofita ex tribus, BR, RE, EV, minor eſt ipſa, BD, nam, DB, ad, <lb/>BV, eſt, vt omnia quadrata, FD, ad compoſitum ex omnibus quadra-<lb/>tis ſemicirculi, vel ſemiellipſis, FQB, & </s> <s xml:id="echoid-s5420" xml:space="preserve">ex rectangulis ſub eodem, & </s> <s xml:id="echoid-s5421" xml:space="preserve"><lb/>ſub quadrilineo, FQBDZ, bis ſumptis, vnde omnia clarè patent.</s> <s xml:id="echoid-s5422" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div549" type="section" level="1" n="327"> <head xml:id="echoid-head344" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5423" xml:space="preserve">_H_INC habetur omnia quadrata, FD, ad reliquum ſui, demptis <lb/>omnibus quadratis quadrilinei, FQBDZ, eſſe, vt, DB, ad, BV.</s> <s xml:id="echoid-s5424" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div550" type="section" level="1" n="328"> <head xml:id="echoid-head345" xml:space="preserve">THEOREMA XIV. PROPOS. XV.</head> <p> <s xml:id="echoid-s5425" xml:space="preserve">SI circulo, vel ellipſi circumſcribatur parallelogrammum, <lb/>habebit latera eorundem diametris parallela; </s> <s xml:id="echoid-s5426" xml:space="preserve">ſumpto <lb/>autem quolibet laterum pro regula; </s> <s xml:id="echoid-s5427" xml:space="preserve">omnia quadrata dicti <lb/>parallelogrammi rectanguli, ad omnia quadrata circuli, <lb/>vel ellipſis inſcripti, vna cum rectangulis bis ſub codem cir-<lb/>culo, & </s> <s xml:id="echoid-s5428" xml:space="preserve">duobus trilineis cuiliber laterum adiacentibus, quæ <lb/>non fuerunt ſumpta pro regula, erunt, vt dictum parallelo-<lb/>grammum ad dictum circulum, vel ellipſim.</s> <s xml:id="echoid-s5429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5430" xml:space="preserve">Sit circulus, vel ellipſis, MBEG, cuius centrum, A, per quod <lb/>tranſeant diametri, ME, BG, ductis autem tangentibus circulum, <lb/>vel ellipſim in punctis, M, B, E, G, donec concurrant, ſit eidem cir-<lb/>cumſcriptum parallelogrammum, HF, quod habebit latera paral-<lb/> <anchor type="note" xlink:label="note-0242-01a" xlink:href="note-0242-01"/> lela ipſis axibus, ME, BG, ſit autem regula vtcunque, DF. </s> <s xml:id="echoid-s5431" xml:space="preserve">Di-<lb/>co ergo omnia quadrata parallelogrammi, HF, ad omnia quadra-<lb/>ta circuli, vel ellipſis, MBEG, vna cum rectangulis bis ſub eodem <lb/>circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5432" xml:space="preserve">ſub trilineis, MGN, GFE, adia-<lb/>centibus lateri, NF, ſumpto vtcunque ex duobus, HD, NF, quę <pb o="223" file="0243" n="243" rhead="LIBER III."/> non ſunt regula, eſſe vt parallelogrammum, HF, ad circulum, vel <lb/>ellipſim, MBEG. </s> <s xml:id="echoid-s5433" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s5434" xml:space="preserve">n. </s> <s xml:id="echoid-s5435" xml:space="preserve">quadrata parallelogrammi, HF, ſunt <lb/> <anchor type="note" xlink:label="note-0243-01a" xlink:href="note-0243-01"/> ſexquialtera omnium quadratorum circuli, vel ellipſis, MBEG, & </s> <s xml:id="echoid-s5436" xml:space="preserve"><lb/>ideò ſunt ad illa, vt parallelogrammum, HF, ad ſui ipſius duas ter-<lb/>tias, quod ſerua.</s> <s xml:id="echoid-s5437" xml:space="preserve"/> </p> <div xml:id="echoid-div550" type="float" level="2" n="1"> <note position="left" xlink:label="note-0242-01" xlink:href="note-0242-01a" xml:space="preserve">@.l. Con.</note> <note position="right" xlink:label="note-0243-01" xlink:href="note-0243-01a" xml:space="preserve">Coroll. 1. <lb/>huit S.</note> </div> <p> <s xml:id="echoid-s5438" xml:space="preserve">Quoniam verò omnia quadrata parallelogrammi, AF, ad rectan-<lb/>gula ſub eodem, & </s> <s xml:id="echoid-s5439" xml:space="preserve">ſub ſemiportione, AEG, ſunt vt parallelogram-<lb/> <anchor type="note" xlink:label="note-0243-02a" xlink:href="note-0243-02"/> mum, AF, ad ſemiportionem, AEG, eadem verò ad omnia qua-<lb/>drata ſemiportionis, AEG, ſunt ſexquialtera .</s> <s xml:id="echoid-s5440" xml:space="preserve">i. </s> <s xml:id="echoid-s5441" xml:space="preserve">ſunt vt parallelo-<lb/>grammum, AF, ad ſui ipſius, {2/3}, igitur eadem ad reliqua. </s> <s xml:id="echoid-s5442" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5443" xml:space="preserve">ad rectan-<lb/>gula ſub ſemiportione, AEG, & </s> <s xml:id="echoid-s5444" xml:space="preserve">trilineo, GEF, erunt vt parallelo-<lb/>grammum, AF, ad exceſium, quo ſemiportio, AEG, excedit, {2/3}, pa-<lb/>rallelogrammi, AF, omnia autem quadrata, BF, ſunt quadrupla om-<lb/>nium quadratorum, AF, ergo omnia quadrata, BF, ad rectangula <lb/> <anchor type="note" xlink:label="note-0243-03a" xlink:href="note-0243-03"/> ſub ſemiportione, AEG, & </s> <s xml:id="echoid-s5445" xml:space="preserve">trilineo, GEF, erunt vt quater paral-<lb/>lelogra nmum, AF, ad dictum exceſſum.</s> <s xml:id="echoid-s5446" xml:space="preserve">i. </s> <s xml:id="echoid-s5447" xml:space="preserve">vt parallelogrammum, H <lb/> <anchor type="figure" xlink:label="fig-0243-01a" xlink:href="fig-0243-01"/> F, ad dictum exceſſum, & </s> <s xml:id="echoid-s5448" xml:space="preserve">conſequentibus <lb/>quadruplicatis, omnia quadrata paralle <lb/>logrammi, BF, adrectangula quater ſub <lb/>ſemiportione, AEG, & </s> <s xml:id="echoid-s5449" xml:space="preserve">trilineo, GEF, .</s> <s xml:id="echoid-s5450" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s5451" xml:space="preserve">ad rectangula bis ſub portione, BEG, & </s> <s xml:id="echoid-s5452" xml:space="preserve"><lb/>trilineo, GEF, erunt vt, HF, ad dictum ex <lb/>ceſſum quater ſumptum, quia enim, AE, <lb/>eſt diameter bifariam diuidit in portione, <lb/>BEG, omnes ipſi, DF, æquidiſtantes, & </s> <s xml:id="echoid-s5453" xml:space="preserve"><lb/>ideò rectangula quater ſub ſemiportione, <lb/>AEG, & </s> <s xml:id="echoid-s5454" xml:space="preserve">trilineo, GEF, fiunt rectangula <lb/>bis ſub portione, BEG, & </s> <s xml:id="echoid-s5455" xml:space="preserve">trilineo, GEF, omnia ergo quadrata paral-<lb/>lelogrammi, BF, ad rectangula bis ſub portione, BEG, & </s> <s xml:id="echoid-s5456" xml:space="preserve">trilineo, G <lb/>EF, vel eorum dupla. </s> <s xml:id="echoid-s5457" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5458" xml:space="preserve">omnia quadrata parallelogrammi, HF, ad re-<lb/> <anchor type="note" xlink:label="note-0243-04a" xlink:href="note-0243-04"/> ctangula bis ſub circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5459" xml:space="preserve">ſub trilineis, MGN, GE <lb/>F, erunt vt parallelogrammum, HF, ad quatuor exceſſus ſemiportio-<lb/>nis, AEG, ſuper duas tertias parallelogrammi, AF, .</s> <s xml:id="echoid-s5460" xml:space="preserve">i. </s> <s xml:id="echoid-s5461" xml:space="preserve">ad exceſſum cir-<lb/>culi, vel ellipſis, MBEG, ſuper, {2/3}, parallelogrammi, HF, erant autem <lb/>omnia quadrata parallelogrammi, HF, ad omnia quadrata circuli, <lb/>vel ellipſis, MBEG, vtidem parallelogrammum, HF, ad, {2/3}, ſui ipſius, <lb/>ergo omnia quadrata parallelogrammi, HF, ad omnia quadrata cir-<lb/>culi, vel ellipſis, MBEG, ſimul cum rectangulis bis ſub eodem circulo, <lb/>vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5462" xml:space="preserve">ſub trilineis, MNG, GFE, erunt vt parallelo-<lb/>grammum, HF, ad ſui ipſius, {2/3}, vna cum exceſſu circuli, vel ellipſis, M <lb/>BEG, ſuper eaſdem duas tertias .</s> <s xml:id="echoid-s5463" xml:space="preserve">i. </s> <s xml:id="echoid-s5464" xml:space="preserve">erunt vt parallelogrammum, H <lb/>F, ad circulum, vel ellipſim, MBEG, quod erat oſtendendum.</s> <s xml:id="echoid-s5465" xml:space="preserve"/> </p> <div xml:id="echoid-div551" type="float" level="2" n="2"> <note position="right" xlink:label="note-0243-02" xlink:href="note-0243-02a" xml:space="preserve">Coroll. 1. <lb/>26.l.2.</note> <note position="right" xlink:label="note-0243-03" xlink:href="note-0243-03a" xml:space="preserve">7.l.2.</note> <figure xlink:label="fig-0243-01" xlink:href="fig-0243-01a"> <image file="0243-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0243-01"/> </figure> <note position="right" xlink:label="note-0243-04" xlink:href="note-0243-04a" xml:space="preserve">10.l.2.</note> </div> <pb o="224" file="0244" n="244" rhead="GEOMETRIE"/> </div> <div xml:id="echoid-div553" type="section" level="1" n="329"> <head xml:id="echoid-head346" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s5466" xml:space="preserve">OMnia quadrata, BF, ad rectangula ſub, BF, & </s> <s xml:id="echoid-s5467" xml:space="preserve">ſub portione, <lb/>BEG, ſunt vt, BF, ad portionem, BEG, rectangula verò <lb/> <anchor type="note" xlink:label="note-0244-01a" xlink:href="note-0244-01"/> ſub portione, BEG, & </s> <s xml:id="echoid-s5468" xml:space="preserve">parallelogrammo, BF, diuiduntur in re-<lb/>ctangula ſub, BEG, &</s> <s xml:id="echoid-s5469" xml:space="preserve">, BDE, trilineo .</s> <s xml:id="echoid-s5470" xml:space="preserve">i. </s> <s xml:id="echoid-s5471" xml:space="preserve">ſub trilineo, GEF, & </s> <s xml:id="echoid-s5472" xml:space="preserve"><lb/>ſub, BEG, & </s> <s xml:id="echoid-s5473" xml:space="preserve">trilineo, GEF, & </s> <s xml:id="echoid-s5474" xml:space="preserve">ſub, BEG, & </s> <s xml:id="echoid-s5475" xml:space="preserve">eadem portione, <lb/>BEG, .</s> <s xml:id="echoid-s5476" xml:space="preserve">i. </s> <s xml:id="echoid-s5477" xml:space="preserve">in omnia quadrata portionis, BEG, ergo omnia quadra-<lb/>ta, BF, ad omnia quadrata portionis, BEG, ſimul cum rectangu-<lb/>lis ſub portione, BEG, & </s> <s xml:id="echoid-s5478" xml:space="preserve">trilineo, GEF, bis ſumptis, vel omnia <lb/>quadrata, HF, ad omnia quadrata circuli, vel ellipſis, MBEG, <lb/>ſimul cum rectangulis ſub circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5479" xml:space="preserve">trilineis, <lb/>MNG, GFE, bis ſumptis, erunt vt, BF, ad portionem, BEG, <lb/>vel vt, HF, ad circulum, vel ellipſim, MBEG, quod erat oſten, <lb/>dendum.</s> <s xml:id="echoid-s5480" xml:space="preserve"/> </p> <div xml:id="echoid-div553" type="float" level="2" n="1"> <note position="left" xlink:label="note-0244-01" xlink:href="note-0244-01a" xml:space="preserve">Coroll. 1. <lb/>26.lib.2. <lb/>per A.23. <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div555" type="section" level="1" n="330"> <head xml:id="echoid-head347" xml:space="preserve">THEOREMA XV. PROPOS. XVI.</head> <p> <s xml:id="echoid-s5481" xml:space="preserve">SI à parallelogrammo per lineam lateribus parallelam <lb/>parallelogrammum abſcindatur, quod intelligatur cir-<lb/>culo, vel ellipſi circumſcriptum, regula autem ſit parallelo-<lb/>grammi baſis : </s> <s xml:id="echoid-s5482" xml:space="preserve">Omnia quadrata circumſcripti parallelo-<lb/>grammi, ſimul cum rectangulis bis ſub eodem, & </s> <s xml:id="echoid-s5483" xml:space="preserve">ſub reli-<lb/>quo parallelogrammo per dictam parallelam conſtituto, ad <lb/>omnia quadrata dicti circuli, vel ellipſis, ſimul cum rectan-<lb/>gulis bis ſub eodem circulo, vel ellipſi, & </s> <s xml:id="echoid-s5484" xml:space="preserve">ſub quadrilineo <lb/>duabus parallelis circulum, vel ellipſim tangentibus, inclu-<lb/>ſaque ab ijſdem curua, & </s> <s xml:id="echoid-s5485" xml:space="preserve">latere totius parallelogrammi, <lb/>quod circulum, vel ellipſim non tangit, comprehenſo, erunt, <lb/>vt dictum circumſcriptum parallelogrammum ad eundem <lb/>circulum, velellipſim.</s> <s xml:id="echoid-s5486" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5487" xml:space="preserve">Sit ergo parallelogrammum, HO, cuius baſis, & </s> <s xml:id="echoid-s5488" xml:space="preserve">regula, DO, <lb/>ductaque, NF, intra ipſum lateribus, HD, CO, parallela, ſit ab-<lb/>ſciſſum à toto parallelogrammo, HO, parallelogrammum, HF, in-<lb/>telligatur autem circumſcriptum circulo, vel ellipſi, MBEG, cuics <lb/>centrum, A, per quod tranſeant diametri, ME, &</s> <s xml:id="echoid-s5489" xml:space="preserve">, BG, quæ ſit <lb/>producta vſque in, P, erunt autem dictæ diametri parallelæ paralle-<lb/>logrammi, HO, lateribus, tranſibuntque per puncta contactuum, <pb o="225" file="0245" n="245" rhead="LIBER III."/> M,B, E, G. </s> <s xml:id="echoid-s5490" xml:space="preserve">Dico igitur omnia quadrata parallelogrammi, HF, <lb/>ſimul cum rectangulis bis ſub, HF, & </s> <s xml:id="echoid-s5491" xml:space="preserve">parallelogrammo, FC, ad <lb/>omnia quadrata circuli, vel ellipſis, MBEG, ſimul cum rectangu-<lb/>lis bis ſub eodem circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5492" xml:space="preserve">ſub quadrilineo, M <lb/>GEOC, eſſe vt parallelogrammum, HF, ad circulum, vel ellipſim, <lb/>MBEG : </s> <s xml:id="echoid-s5493" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s5494" xml:space="preserve">n. </s> <s xml:id="echoid-s5495" xml:space="preserve">quadrata parallelogrammi, HO, ad omnia <lb/> <anchor type="note" xlink:label="note-0245-01a" xlink:href="note-0245-01"/> quadrata parallelogrammi, MO, ſunt vt quadratum, DO, ad qua-<lb/>dratum, OE, omnia item quadrata parallelogrammi, MO, ad re-<lb/> <anchor type="note" xlink:label="note-0245-02a" xlink:href="note-0245-02"/> ctangula ſub parallelogrammo, MO, & </s> <s xml:id="echoid-s5496" xml:space="preserve">portione, MGE, ſunt vt, <lb/>MO, ad portionem, MGE, fiat vt, MF, ad portionem, MGE, <lb/>ita, FE, ad, EI, erit ergo vt, MO, ad portionem, MGE, ita, O <lb/>E, ad, EI, ergo omnia quadrata, MO, ad rectangula ſub, MO, & </s> <s xml:id="echoid-s5497" xml:space="preserve"><lb/>portione, MGE, erunt vt, OE, ad, EI, .</s> <s xml:id="echoid-s5498" xml:space="preserve">i. </s> <s xml:id="echoid-s5499" xml:space="preserve">vt quadratum, OE, <lb/> <anchor type="note" xlink:label="note-0245-03a" xlink:href="note-0245-03"/> ad rectangulum ſub, OE, EI, erant autem omnia quadrata, HO, <lb/>ad omnia quadrata, OM, vt quadratum, DO, ad quadratum, O <lb/>E, ergo ex æquali omnia quadrata, HO, ad rectangula ſub, MO, <lb/>& </s> <s xml:id="echoid-s5500" xml:space="preserve">ſub portione, MGE, erunt vt quadratum, DO, ad rectangu-<lb/>lum ſub, OE, EI, ad eadem verò quater ſumpta, vt quadratum, D <lb/>O, ad rectangulum ſub, OE, & </s> <s xml:id="echoid-s5501" xml:space="preserve">quadrupla, EI; </s> <s xml:id="echoid-s5502" xml:space="preserve">rectangula autem <lb/> <anchor type="figure" xlink:label="fig-0245-01a" xlink:href="fig-0245-01"/> ſub, MO, & </s> <s xml:id="echoid-s5503" xml:space="preserve">por-<lb/> <anchor type="note" xlink:label="note-0245-04a" xlink:href="note-0245-04"/> tione, MGE, æ-<lb/>quantur rectangulis <lb/>ſub quadrilineo, M <lb/>GEOC, & </s> <s xml:id="echoid-s5504" xml:space="preserve">por <lb/>tione, MGE, vna <lb/>cum omnibus qua <lb/>dratis portionis, M <lb/>GE, ilia ig tur qua-<lb/>ter ſumpta reddunt <lb/>quater rectangula ſub portione, MGE, & </s> <s xml:id="echoid-s5505" xml:space="preserve">quadrilineo, MGEO <lb/>C, vna cum omnibus quadratis portionis, MGE, quater ſumptis, <lb/>quia verò omnia quadrata circuli, vel ellipſis, MBEG, æquantur <lb/> <anchor type="note" xlink:label="note-0245-05a" xlink:href="note-0245-05"/> omnibus quadratis portionis, MBE, & </s> <s xml:id="echoid-s5506" xml:space="preserve">portionis, MGE, vna cum <lb/>rectangulis bis ſub vtriſq; </s> <s xml:id="echoid-s5507" xml:space="preserve">portionibus .</s> <s xml:id="echoid-s5508" xml:space="preserve">i. </s> <s xml:id="echoid-s5509" xml:space="preserve">vna cum omnibus quadratis <lb/>portionis, MGE, bis ſumptis, & </s> <s xml:id="echoid-s5510" xml:space="preserve">omnia quadrata portionis, MBE, <lb/>æquantur omnibus quadratis portionis, MGE, ideò omnia quadra-<lb/>ta portionis, MGE, quater ſumpta æquantur omnib. </s> <s xml:id="echoid-s5511" xml:space="preserve">quadratis cir-<lb/>culi, vel ellipſis, MBEG, item rectangula ſub portione, MGE, & </s> <s xml:id="echoid-s5512" xml:space="preserve"><lb/>quadrilineo, MGEOC, quater æquantur rectangulis ſub toto circu-<lb/>lo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5513" xml:space="preserve">ſub quadrilineo, MGEOC, bis ſumptis, itaut <lb/>hucuſq; </s> <s xml:id="echoid-s5514" xml:space="preserve">probauerimus rectangula ſub portione, MGE, & </s> <s xml:id="echoid-s5515" xml:space="preserve">paralle-<lb/>logrammo, MO, quater ſumpta æquari omnibus quadratis circuli, <pb o="226" file="0246" n="246" rhead="GEOMETRIÆ"/> vel ellipſis, MBEG, vna cum rectangulis bis ſub eodem circulo, vel <lb/>ellipſi, & </s> <s xml:id="echoid-s5516" xml:space="preserve">ſub quadrilineo, MGEOC, quoniam verò oſtenſum eſt <lb/>omnia quadrata, HO, ad rectangula ſub portione, MGE, & </s> <s xml:id="echoid-s5517" xml:space="preserve">pa-<lb/>rallelogrammo, MO, quater ſumpta eſſe, vt quadratum, DO, ad <lb/>rectangulum ſub, OE, & </s> <s xml:id="echoid-s5518" xml:space="preserve">quadrupla, EI, ideò ex æquali omnia <lb/>quadrata, HO, ad omnia quadrata circuli, vel ellipſis, MBEG, <lb/>vna cum rectangulis bis ſub eodem circulo, vel ellipſi, & </s> <s xml:id="echoid-s5519" xml:space="preserve">ſub quadri-<lb/>lineo, MGEOC, erunt vt quadratum, DO, ad rectangulum ſub, <lb/>OE, & </s> <s xml:id="echoid-s5520" xml:space="preserve">ſub quadrupla, EI, quod ſerua.</s> <s xml:id="echoid-s5521" xml:space="preserve"/> </p> <div xml:id="echoid-div555" type="float" level="2" n="1"> <note position="right" xlink:label="note-0245-01" xlink:href="note-0245-01a" xml:space="preserve">9. Lib. 2.</note> <note position="right" xlink:label="note-0245-02" xlink:href="note-0245-02a" xml:space="preserve">Coroll. 1. <lb/>26.lib.2.</note> <note position="right" xlink:label="note-0245-03" xlink:href="note-0245-03a" xml:space="preserve">5.Lib.2.</note> <figure xlink:label="fig-0245-01" xlink:href="fig-0245-01a"> <image file="0245-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0245-01"/> </figure> <note position="right" xlink:label="note-0245-04" xlink:href="note-0245-04a" xml:space="preserve">Per C, 23. <lb/>lib.2.</note> <note position="right" xlink:label="note-0245-05" xlink:href="note-0245-05a" xml:space="preserve">Per D.23. <lb/>lib,2.</note> </div> <p> <s xml:id="echoid-s5522" xml:space="preserve">Quoniam verò omnia quadrata, HF, vna cum rectangulis bis ſub <lb/> <anchor type="note" xlink:label="note-0246-01a" xlink:href="note-0246-01"/> parallelogrammis, HF, FC, ad omnia quadrata, HO, ſunt, vt <lb/>vnum, ad vnum .</s> <s xml:id="echoid-s5523" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5524" xml:space="preserve">vt quadratum, DF, vna cum rectangulo bis ſub, <lb/>DF, FO, ad quadratum, DO, omnia quadrata verò parallelogram-<lb/>mi, HO, ad omnia quadrata circuli, vel ellipſis, MBEG, vna cum <lb/>rectangulis bis ſub eodem circulo, vel ellipſi, & </s> <s xml:id="echoid-s5525" xml:space="preserve">ſub quadrilineo, M <lb/>GEOC, eſſe oſtenſa ſunt, vt idem quadratum, DO, ad rectangu-<lb/>lum ſub, OE, & </s> <s xml:id="echoid-s5526" xml:space="preserve">quadrupla, EI, ergo ex æquali omnia quadrata pa-<lb/>rallelogrammi, HF, vna cum rectangulis bis ſub parallelogrammis, <lb/>HF, FC, ad omnia quadrata circuli, vel ellipſis, MBEG, vna <lb/> <anchor type="figure" xlink:label="fig-0246-01a" xlink:href="fig-0246-01"/> cum rectangulis bis <lb/>ſub eodem circulo, <lb/>vel ellipſi, MBE <lb/>G, erunt vt quadra-<lb/>tum, DF, vna cum <lb/>rectangulo ſub, D <lb/>F, FO, bis, ad re-<lb/>ctangulum ſub, OE, <lb/>& </s> <s xml:id="echoid-s5527" xml:space="preserve">quadrupla, EI, <lb/>vel erunt, vt eorum <lb/>dimidia .</s> <s xml:id="echoid-s5528" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5529" xml:space="preserve">vt dimi-<lb/>dium quadrati, DF, <lb/>quod erit rectangulum, DFE, vna cum rectangulo ſub, DFO, ſe-<lb/>mel (ex quibus componetur rectangulum ſub, OE, FD,) ad rectan-<lb/>gulum ſub, OE, & </s> <s xml:id="echoid-s5530" xml:space="preserve">dupla, EI, vel, vt adhuc horum dimidia .</s> <s xml:id="echoid-s5531" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5532" xml:space="preserve">vt <lb/>rectangulum ſub, OE, &</s> <s xml:id="echoid-s5533" xml:space="preserve">, ED, ad rectangulum ſub, OE, &</s> <s xml:id="echoid-s5534" xml:space="preserve">, EI, <lb/> <anchor type="note" xlink:label="note-0246-02a" xlink:href="note-0246-02"/> .</s> <s xml:id="echoid-s5535" xml:space="preserve">i. </s> <s xml:id="echoid-s5536" xml:space="preserve">vt, DE, ad, EI, quia, OE, altitudo eſt communis, oſtenſum <lb/>ergo eſt omnia quadrata, HF, vna cum rectangulis bis ſub paralle-<lb/>logrammis, HF, FC, ad omnia quadrata circuli, vel ellipſis, MB <lb/>EG, vna cum rectangulis bis ſub eodem, & </s> <s xml:id="echoid-s5537" xml:space="preserve">ſub quadrilineo, MGE <lb/>OC, eſſe, vt, DE, vel, FE, ad, EI, .</s> <s xml:id="echoid-s5538" xml:space="preserve">i. </s> <s xml:id="echoid-s5539" xml:space="preserve">vt parallelogrammum, M <lb/>F, ad portionem, MGE, vel vt parallelogrammum, HF, ad cir-<lb/>culum, vel ellipſim, MBEG, quod erat propoſitum.</s> <s xml:id="echoid-s5540" xml:space="preserve"/> </p> <div xml:id="echoid-div556" type="float" level="2" n="2"> <note position="left" xlink:label="note-0246-01" xlink:href="note-0246-01a" xml:space="preserve">14.Lib.2.</note> <figure xlink:label="fig-0246-01" xlink:href="fig-0246-01a"> <image file="0246-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0246-01"/> </figure> <note position="left" xlink:label="note-0246-02" xlink:href="note-0246-02a" xml:space="preserve">5. Lib.2.</note> </div> <pb o="227" file="0247" n="247" rhead="LIBER III."/> </div> <div xml:id="echoid-div558" type="section" level="1" n="331"> <head xml:id="echoid-head348" xml:space="preserve">THEOREMA XVI. PROPOS. XVII.</head> <p> <s xml:id="echoid-s5541" xml:space="preserve">OMnia quadrata parallelogrammi circulo, vel ellipſi <lb/>circumſcripti (regula baſi) ad omnia quadrata figuræ <lb/>compoſitæ ex circulo, vel ellipſi, & </s> <s xml:id="echoid-s5542" xml:space="preserve">ex duobus trilineis ad-<lb/>iacentibus lateri, quod non eſt regula, nec ipſi parallelum, <lb/>veluti dicitur in Th. </s> <s xml:id="echoid-s5543" xml:space="preserve">14. </s> <s xml:id="echoid-s5544" xml:space="preserve">erunt, vt idem parallelogrammum <lb/>ad circulum, vel ellipſim, cui circumſcribitur, vna cum eo <lb/>ſpatio, quod relinquitur, dempto à quarta parte dicti paral-<lb/>lelogrammi circuli, vel ellipſis quadrante, ſimul cum exceſ-<lb/>ſu, quo idem quadrans ſuperat duas tertias dicti parallelo-<lb/>grammi ideſt erit, proximè, vt 21. </s> <s xml:id="echoid-s5545" xml:space="preserve">ad 17.</s> <s xml:id="echoid-s5546" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5547" xml:space="preserve">Exponatur denuò figura Theor.</s> <s xml:id="echoid-s5548" xml:space="preserve">14. </s> <s xml:id="echoid-s5549" xml:space="preserve">Dico omnia quadrata paral-<lb/>lelogrammi, HF, ad omnia quadrata figuræ compoſitæ ex circulo, <lb/>vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5550" xml:space="preserve">trilineis, MGN, EGF, eſſe vt, HF, ad <lb/>circulum, vel ellipſim, MBEG, vna cum reſiduo, dempto à paral-<lb/>lelogrammo, MG, circuli, vel ellipſis, quadrante, MGA, ſimul <lb/>cum eo exceſſu, quo idem quadrans ſuperat duas tertias parallelo-<lb/>grammi, MG. </s> <s xml:id="echoid-s5551" xml:space="preserve">Etenim oſtenſum eſt omnia quadrata, HF, ad om-<lb/> <anchor type="note" xlink:label="note-0247-01a" xlink:href="note-0247-01"/> nia quadrata circuli, vel ellipſis, MBEG, vna cum rectangulis bis <lb/>ſub eodem, & </s> <s xml:id="echoid-s5552" xml:space="preserve">ſub trilineis, MNG, GFE, eſſe vt, HF, ad circu-<lb/>lum, vel ellipſim, MBEG, quod lerua.</s> <s xml:id="echoid-s5553" xml:space="preserve"/> </p> <div xml:id="echoid-div558" type="float" level="2" n="1"> <note position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve">15. huius.</note> </div> <p> <s xml:id="echoid-s5554" xml:space="preserve">Vlterius, quia omnia quadrata, HG, ad omnia quadrata, MG, <lb/> <anchor type="note" xlink:label="note-0247-02a" xlink:href="note-0247-02"/> ſunt vt quadratum, BG, ad quadratum, GA, .</s> <s xml:id="echoid-s5555" xml:space="preserve">@. </s> <s xml:id="echoid-s5556" xml:space="preserve">vt parallelogram-<lb/>mum, HF, ad parallelogrammum, MG; </s> <s xml:id="echoid-s5557" xml:space="preserve">inſuper omnia quadrata, <lb/> <anchor type="note" xlink:label="note-0247-03a" xlink:href="note-0247-03"/> MG, ad omnia quadrata trilinei, MGN, ſunt vt, MG, ad reſi-<lb/> <anchor type="figure" xlink:label="fig-0247-01a" xlink:href="fig-0247-01"/> duum dempto quadrante, MAG, ſimul <lb/>cum eo ſpatio, quo idem ſuperat duas <lb/>tertias rectanguli, MG, ab eodem re-<lb/>ctangulo, MG, ergo ex æquali omnia <lb/>quadrata, HG, ad omnia quadrata tri-<lb/>linei, MGN, erunt vt, HF, ad reſi-<lb/>duum, dempto quadrante, MAG, ſimul <lb/>cum eo ſpatio, quo idem ſuperat, {2/3}, re-<lb/>ctanguli, MG, ab eodem rectangulo, M <lb/> <anchor type="note" xlink:label="note-0247-04a" xlink:href="note-0247-04"/> G, &</s> <s xml:id="echoid-s5558" xml:space="preserve">, duplicatis proportionis terminis, <lb/>omnia quadrata, HF, ad omnia quadra-<lb/>ta trilineorum, MNG, GFE, erunt vt duplum, HF, ad duplum <pb o="228" file="0248" n="248" rhead="GEOMETRIÆ"/> illius reſidui .</s> <s xml:id="echoid-s5559" xml:space="preserve">i. </s> <s xml:id="echoid-s5560" xml:space="preserve">vt, HF, ad vnum illud reſiduum ; </s> <s xml:id="echoid-s5561" xml:space="preserve">omnia autem qua-<lb/>drata eiuſdem, HF, ad omnia quadrata circuli, vel ellipſis, MBE <lb/>G, ſimul cum rectangulis bis ſub eodem, & </s> <s xml:id="echoid-s5562" xml:space="preserve">ſub trilineis, MNG, G <lb/>FE, ſunt vt, HF, ad circulum, vel ellipſim, MBEG, ergo, col-<lb/>ligendo, omnia quadrata, HF, ad omnia quadrata circuli, vel elli-<lb/>pſis, MBEG, & </s> <s xml:id="echoid-s5563" xml:space="preserve">ad omnia quadrata trilineorum, MNG, GFE, <lb/>ſimul cum rectangulis bis ſub circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5564" xml:space="preserve">trili-<lb/>neis, MNG, GFE, ideſt ad omnia quadrata figuræ, NMBEF, <lb/>erunt vt, HF, ad circulum, vel ellipſim, MBEG, ſimul cum reſi-<lb/> <anchor type="note" xlink:label="note-0248-01a" xlink:href="note-0248-01"/> duo, dempto à parallelogrammo, MG, quadrante, MAG, & </s> <s xml:id="echoid-s5565" xml:space="preserve">eo <lb/>ſpatio, quo idem excedit duas tertias parallelogrammi, MG.</s> <s xml:id="echoid-s5566" xml:space="preserve"/> </p> <div xml:id="echoid-div559" type="float" level="2" n="2"> <note position="right" xlink:label="note-0247-02" xlink:href="note-0247-02a" xml:space="preserve">9. Lib. 2.</note> <note position="right" xlink:label="note-0247-03" xlink:href="note-0247-03a" xml:space="preserve">13. huius.</note> <figure xlink:label="fig-0247-01" xlink:href="fig-0247-01a"> <image file="0247-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0247-01"/> </figure> <note position="right" xlink:label="note-0247-04" xlink:href="note-0247-04a" xml:space="preserve">10.Lib.2.</note> <note position="left" xlink:label="note-0248-01" xlink:href="note-0248-01a" xml:space="preserve">Per D.23. <lb/>lib.2.</note> </div> <p> <s xml:id="echoid-s5567" xml:space="preserve">Dico nunc hanc rationem eſſe, vt 21. </s> <s xml:id="echoid-s5568" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s5569" xml:space="preserve">proximè, parallelo-<lb/>grammum enim, MG, ad dictum reſiduum eſt, vt 21. </s> <s xml:id="echoid-s5570" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s5571" xml:space="preserve">proxi-<lb/>mè, vt oſtendimus Theor. </s> <s xml:id="echoid-s5572" xml:space="preserve">12. </s> <s xml:id="echoid-s5573" xml:space="preserve">parallelogrammum vero, HF, qua-<lb/>druplum eſt ipſius, MG, ergo, HF, ad illud reſiduum eſt, vt 84. </s> <s xml:id="echoid-s5574" xml:space="preserve">ad 2. <lb/></s> <s xml:id="echoid-s5575" xml:space="preserve">proximè, eſt autem idem, HF, ad circulum, vel ellipſim, MBEG, <lb/>vt 14. </s> <s xml:id="echoid-s5576" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s5577" xml:space="preserve">proximè .</s> <s xml:id="echoid-s5578" xml:space="preserve">i. </s> <s xml:id="echoid-s5579" xml:space="preserve">vt 84. </s> <s xml:id="echoid-s5580" xml:space="preserve">ad 66. </s> <s xml:id="echoid-s5581" xml:space="preserve">ergo parallelogrammum, H <lb/>F, ad compoſitum ex circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5582" xml:space="preserve">dicto reſiduo <lb/>eſt, vt 84. </s> <s xml:id="echoid-s5583" xml:space="preserve">ad 68. </s> <s xml:id="echoid-s5584" xml:space="preserve">proximè .</s> <s xml:id="echoid-s5585" xml:space="preserve">i. </s> <s xml:id="echoid-s5586" xml:space="preserve">vt 21. </s> <s xml:id="echoid-s5587" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s5588" xml:space="preserve">proximè, ideò omnia qua-<lb/>drata, HF, ad omnia quadrata figuræ, NMBEF, ſunt proximè, <lb/>vt 21. </s> <s xml:id="echoid-s5589" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s5590" xml:space="preserve">patet ergo propoſitum.</s> <s xml:id="echoid-s5591" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div561" type="section" level="1" n="332"> <head xml:id="echoid-head349" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s5592" xml:space="preserve">_H_INC patet, quoniam omnia quadrata, HF, omnium quadrato-<lb/> <anchor type="note" xlink:label="note-0248-02a" xlink:href="note-0248-02"/> rum, MF, ſunt quadrupla, quod ſunt ad illa, vt, HF, ad, MG, <lb/>& </s> <s xml:id="echoid-s5593" xml:space="preserve">ideò, ſi dempſeris omnia quadrata, MF, ab omnibus quadratis ſigu-<lb/>ræ, NMBEF, omnia quadrata, HF, ad reſiduum erunt, vt, HF, ad <lb/>illud, quod relinquitur, dempto, MG, à circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5594" xml:space="preserve"><lb/>reſiduo ſæpius dicto .</s> <s xml:id="echoid-s5595" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5596" xml:space="preserve">quod remanet ablato ab, MG, quadrante, MAG, <lb/>& </s> <s xml:id="echoid-s5597" xml:space="preserve">eo exceſſu, quo idem ſuperat, {2/3}, MG, eſt autem, HF, ad bac rema-<lb/>nentia ſpatia proximè, vt 84.</s> <s xml:id="echoid-s5598" xml:space="preserve">ad 47.</s> <s xml:id="echoid-s5599" xml:space="preserve"/> </p> <div xml:id="echoid-div561" type="float" level="2" n="1"> <note position="left" xlink:label="note-0248-02" xlink:href="note-0248-02a" xml:space="preserve">_@. Lib. 8._</note> </div> <p style="it"> <s xml:id="echoid-s5600" xml:space="preserve">Conſtitue .</s> <s xml:id="echoid-s5601" xml:space="preserve">n. </s> <s xml:id="echoid-s5602" xml:space="preserve">HF, 84. </s> <s xml:id="echoid-s5603" xml:space="preserve">erit circulus, vel ellipſis, MBEG, 66. </s> <s xml:id="echoid-s5604" xml:space="preserve">& </s> <s xml:id="echoid-s5605" xml:space="preserve">di-<lb/>ctum reſiduum 2. </s> <s xml:id="echoid-s5606" xml:space="preserve">vt ſupra oſtendimus (proximè ſemper intellige) eſt au-<lb/>tem, MG, 21. </s> <s xml:id="echoid-s5607" xml:space="preserve">demas ergo 21. </s> <s xml:id="echoid-s5608" xml:space="preserve">à compoſito ex 66. </s> <s xml:id="echoid-s5609" xml:space="preserve">& </s> <s xml:id="echoid-s5610" xml:space="preserve">2. </s> <s xml:id="echoid-s5611" xml:space="preserve">ideſt à 68. </s> <s xml:id="echoid-s5612" xml:space="preserve">re-<lb/>manent 47. </s> <s xml:id="echoid-s5613" xml:space="preserve">eſt ergo, HF, ad remanentia ſpatia, vt 84. </s> <s xml:id="echoid-s5614" xml:space="preserve">ad 47. </s> <s xml:id="echoid-s5615" xml:space="preserve">vnde om-<lb/>nia quadrata, HF, ad reſiduum, dempt is omnibus quadratis, MF, ab <lb/>omnibus quadratis ſiguræ, NMBEF, erunt, proximè, vt 84. </s> <s xml:id="echoid-s5616" xml:space="preserve">ad 47. <lb/></s> <s xml:id="echoid-s5617" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s5618" xml:space="preserve"/> </p> <pb o="229" file="0249" n="249" rhead="LIBER III."/> </div> <div xml:id="echoid-div563" type="section" level="1" n="333"> <head xml:id="echoid-head350" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s5619" xml:space="preserve">_H_INC etiam patet, quoniam omnia quadrata, MF, ad omnià <lb/>quadrata trilineorum, MNG, GFE, ſunt, vt 21. </s> <s xml:id="echoid-s5620" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s5621" xml:space="preserve">pro. <lb/></s> <s xml:id="echoid-s5622" xml:space="preserve">ximè, quod ad ſuireliquum erunt, vt 21. </s> <s xml:id="echoid-s5623" xml:space="preserve">ad 19. </s> <s xml:id="echoid-s5624" xml:space="preserve">proximè, ſunt autem <lb/>omnia quadrata, HF, quadrupla omnium quadratorum, MF, & </s> <s xml:id="echoid-s5625" xml:space="preserve">ideò <lb/>omnia quadrata, HF, ad reſiduum, demptis omnibus quadratis trili-<lb/>neorum, MNG, GFE, ab omnibus quadratis, MF, erunt proximè, <lb/>vt 84. </s> <s xml:id="echoid-s5626" xml:space="preserve">ad 19. </s> <s xml:id="echoid-s5627" xml:space="preserve">ſunt autem omnia quadrata, HF, ad reſiduum, demptis <lb/>omnibus quadratis, MF, ab omnibus quadratis figuræ, NMBEF, vt <lb/>84. </s> <s xml:id="echoid-s5628" xml:space="preserve">ad 47. </s> <s xml:id="echoid-s5629" xml:space="preserve">proximè, ergo reſiduum primum .</s> <s xml:id="echoid-s5630" xml:space="preserve">i. </s> <s xml:id="echoid-s5631" xml:space="preserve">quod relinquitur, demptis <lb/>omnibus quadratis trilineorum, MNG, GFE, ab omnibus quadratis, <lb/>MF, ad reſiduum ſecundum .</s> <s xml:id="echoid-s5632" xml:space="preserve">i. </s> <s xml:id="echoid-s5633" xml:space="preserve">ad id, quod relinquitur, demptis omni-<lb/>bus quadratis, MF, ab omnibus quadratis figuræ, NMBEF, erit pro-<lb/>ximè, vt 19.</s> <s xml:id="echoid-s5634" xml:space="preserve">ad 47. </s> <s xml:id="echoid-s5635" xml:space="preserve">vnde patet, &</s> <s xml:id="echoid-s5636" xml:space="preserve">c.</s> <s xml:id="echoid-s5637" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div564" type="section" level="1" n="334"> <head xml:id="echoid-head351" xml:space="preserve">THEOREMA XVII. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s5638" xml:space="preserve">EXponatur denuo figura Prop. </s> <s xml:id="echoid-s5639" xml:space="preserve">16. </s> <s xml:id="echoid-s5640" xml:space="preserve">Dico omnia quadra-<lb/>ta, HO, (regula eadem ibi ſumpta) ad omnia quadra-<lb/>ta figuræ compoſitæ ex parallelogrammo, MO, & </s> <s xml:id="echoid-s5641" xml:space="preserve">ſemicir-<lb/>culo, vel ſemiellipſi, MBE, eſſe, vt quadratum, DO, ad <lb/>rectangulum ſub, DO, OE, vna cum rectangulo ſub, OE, <lb/>& </s> <s xml:id="echoid-s5642" xml:space="preserve">ſub exceſlu, quo dupla, EI, ſuperat, EF, cum, {2/3}, quadra-<lb/>ti, DE.</s> <s xml:id="echoid-s5643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5644" xml:space="preserve">QVoniam ergo omnia quadrata figuræ, CMBEO, diuiduntur <lb/>per lineam, ME, in omnia quadrata parallelogrammi, MO, <lb/>in omnia quadrata ſemicirculi, vel ſemiellipſis, MBE, & </s> <s xml:id="echoid-s5645" xml:space="preserve">in re-<lb/> <anchor type="note" xlink:label="note-0249-01a" xlink:href="note-0249-01"/> ctangula bis ſub, MO, & </s> <s xml:id="echoid-s5646" xml:space="preserve">ſub ſemicirculo, vel ſemiellipſi, MBE, <lb/>patet primò, quod omnia quadrata, HO, ad omnia quadrata, MO, <lb/>ſunt, vt quadratum, DO, ad quadratum, OE. </s> <s xml:id="echoid-s5647" xml:space="preserve">Inſuper omnia <lb/>quadrata, HO, ad omnia quadrata, HE, ſunt vt quadratum, OD, <lb/>ad quadratum, DE, omnia verò quadrata, HE, ad omnia qua-<lb/> <anchor type="note" xlink:label="note-0249-02a" xlink:href="note-0249-02"/> drata ſemicirculi, vel ſemiellipſis, MBE, ſunt vt quadratum, DE, <lb/>ad ſui ipſius, {2/3}, ergo ex æquali omnia quadrata, HO, ad omnia <lb/>quadrata ſemicirculi, vel ſemiellipſis, MBE, ſunt vt quadratum, <lb/>OD, ad, {2/3}, quadrati, DE. </s> <s xml:id="echoid-s5648" xml:space="preserve">Vlterius omnia quadrata, HO, ad om- <pb o="230" file="0250" n="250" rhead="GEOMETRIÆ"/> nia quadrata, MO, ſunt vt quadratum, DO, ad quadratum, OE, <lb/>omnia item quadrata, MO, ad rectangula ſub, MO, & </s> <s xml:id="echoid-s5649" xml:space="preserve">ſemicir-<lb/>culo, vel ſemiellipſi, MBE, ſunt vt, OM, ad ſemicirculum, vel ſe-<lb/>miellipſim, MBE, .</s> <s xml:id="echoid-s5650" xml:space="preserve">i. </s> <s xml:id="echoid-s5651" xml:space="preserve">vt, OE, ad, EI, nam facta eſt, FE, ad, EI, <lb/>vt, MF, ad ſemicirculum, vel ſemiellipſim, MGE; </s> <s xml:id="echoid-s5652" xml:space="preserve">ad eadem verò <lb/> <anchor type="note" xlink:label="note-0250-01a" xlink:href="note-0250-01"/> rectangula bis ſumpta, erunt vt, OE, ad duplam, EI; </s> <s xml:id="echoid-s5653" xml:space="preserve">igitur, colli-<lb/>gendo, omnia quadrata, HO, ad omnia quadrata ſemicirculi, vel <lb/> <anchor type="figure" xlink:label="fig-0250-01a" xlink:href="fig-0250-01"/> ſemiellipſis, MBE, <lb/>ad omnia quadrata, <lb/>MO, & </s> <s xml:id="echoid-s5654" xml:space="preserve">ad rectan-<lb/>gula bis ſub ſemicir-<lb/>culo, vel ſemiellipſi, <lb/>MBE, & </s> <s xml:id="echoid-s5655" xml:space="preserve">ſub, MO, <lb/>ſimul ſumpta .</s> <s xml:id="echoid-s5656" xml:space="preserve">i. </s> <s xml:id="echoid-s5657" xml:space="preserve">ad <lb/>omnia quadrata fi-<lb/>guræ, CMBEO, <lb/>erunt vt quadratum, <lb/>OD, ad quadratum, OE, ad, {2/3}, quadrati, DE, cum rectangu-<lb/>lo ſub, OE, & </s> <s xml:id="echoid-s5658" xml:space="preserve">dupla, EI, ſimul ſumpta; </s> <s xml:id="echoid-s5659" xml:space="preserve">quia verò ſemicircu-<lb/>lus, vel ſemiellipſis, MGE, eſt pluſquam dimidium parallelogram-<lb/>mi, MF, etiam, EI, erit pluſquam dimidia, EF; </s> <s xml:id="echoid-s5660" xml:space="preserve">& </s> <s xml:id="echoid-s5661" xml:space="preserve">ideò dupla, <lb/>EI, excedet ipſam, EF, vel ipſam, DE, rectangulum ergo ſub, <lb/>OE, & </s> <s xml:id="echoid-s5662" xml:space="preserve">dupla, EI, poterimus diuidere in rectangulum ſub, OE, <lb/>&</s> <s xml:id="echoid-s5663" xml:space="preserve">, ED, & </s> <s xml:id="echoid-s5664" xml:space="preserve">in rectangulum ſub, OE, & </s> <s xml:id="echoid-s5665" xml:space="preserve">exceſſu, quo dupla, EI, <lb/>ſuperat, ED, iungamus rectangulum ſub, DE, EO, cum qua-<lb/>drato, EO, fiet rectangulum ſub, DO, OE; </s> <s xml:id="echoid-s5666" xml:space="preserve">quadratum ergo, <lb/>OE, {2/3}, quadrati, ED, & </s> <s xml:id="echoid-s5667" xml:space="preserve">rectangulum ſub, OE, & </s> <s xml:id="echoid-s5668" xml:space="preserve">dupla, EI, <lb/>commutata ſunt vt in, {2/3}, quadrati, ED, in rectangulum ſub, D <lb/>O, OE, cum rectangulo ſub, OE, & </s> <s xml:id="echoid-s5669" xml:space="preserve">ſub exceſſu duplæ, EI, <lb/>ſuper, ED. </s> <s xml:id="echoid-s5670" xml:space="preserve">Omnia ergo quadrata, HO, ad omnia quadrata fi-<lb/>guræ, CMBEO, erunt vt quadratum, DO, ad rectangulum <lb/>ſub, DO, OE, cum rectangulo ſub, OE, & </s> <s xml:id="echoid-s5671" xml:space="preserve">ſub exceſſu duplæ, <lb/>EI, ſuper, ED, vel, EF, cum, {2/3}, quadrati, DE, quod erat oſten, <lb/>dendum.</s> <s xml:id="echoid-s5672" xml:space="preserve"/> </p> <div xml:id="echoid-div564" type="float" level="2" n="1"> <note position="right" xlink:label="note-0249-01" xlink:href="note-0249-01a" xml:space="preserve">Per D.23. <lb/>lib.2.</note> <note position="right" xlink:label="note-0249-02" xlink:href="note-0249-02a" xml:space="preserve">Coroll.1. <lb/>huius.</note> <note position="left" xlink:label="note-0250-01" xlink:href="note-0250-01a" xml:space="preserve">Coroll.1. <lb/>26.lib.2.</note> <figure xlink:label="fig-0250-01" xlink:href="fig-0250-01a"> <image file="0250-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0250-01"/> </figure> </div> <figure> <image file="0250-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0250-02"/> </figure> <pb o="231" file="0251" n="251" rhead="LIBER III."/> </div> <div xml:id="echoid-div566" type="section" level="1" n="335"> <head xml:id="echoid-head352" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5673" xml:space="preserve">_P_ATET autem omnia quadrata, HT, ad omnia quadrata figure, <lb/>BMCP, eſſe pariter, vt quadratum, BP, ad rectangulum ſub, <lb/>BP, PA, vna cum rectangulo ſub, PA, & </s> <s xml:id="echoid-s5674" xml:space="preserve">ſub exceſſu, quo dupla, <lb/>EI, ſuperat, EF, cum, {2/3}, quadrati, BA. </s> <s xml:id="echoid-s5675" xml:space="preserve">Et quoniam, iuncta, BM, <lb/>oſtenſum eſt omnia quadrata, HP, ad omnia quadrata trapezij, MB, <lb/> <anchor type="note" xlink:label="note-0251-01a" xlink:href="note-0251-01"/> PC, eſſe vt quadratum, BP, adrectangulum, BPA, vna cum {1/3}, qua-<lb/>drati, AB, ideò eadem omnia quadrata, HP, ad reſiduum omnium <lb/>quadratorum figuræ, quæ eiſdem, MCPB, & </s> <s xml:id="echoid-s5676" xml:space="preserve">curua, MB, contine-<lb/>tur, demptis ab ijſdem omnibus quadratis trapezij, BMCP, erunt vt <lb/>idem quadratum, BP, ad rectangulum ſub, AP, & </s> <s xml:id="echoid-s5677" xml:space="preserve">ſub exceſſu du-<lb/>pla, EI, ſuper, EF, vnacum, {2/3}, quadrati, BA.</s> <s xml:id="echoid-s5678" xml:space="preserve"/> </p> <div xml:id="echoid-div566" type="float" level="2" n="1"> <note position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve">_28. Lib. 2._</note> </div> </div> <div xml:id="echoid-div568" type="section" level="1" n="336"> <head xml:id="echoid-head353" xml:space="preserve">THEOREMA XVIII. PROPOS. XIX.</head> <p> <s xml:id="echoid-s5679" xml:space="preserve">EXpoſita adhuc figura Propof. </s> <s xml:id="echoid-s5680" xml:space="preserve">15. </s> <s xml:id="echoid-s5681" xml:space="preserve">& </s> <s xml:id="echoid-s5682" xml:space="preserve">intra circulum, vel <lb/>ellipſim, MBEG, ducta, RV, vtcunq; </s> <s xml:id="echoid-s5683" xml:space="preserve">regulę, DF, <lb/>parallela, diuidente ipſum circulum, vel ellipſim in duas vt-<lb/>cunque portiones, SMT, SET. </s> <s xml:id="echoid-s5684" xml:space="preserve">Dico omnia quadrata <lb/>portionis, SMT, cumrectangulis bis ſub eadem portione, <lb/>& </s> <s xml:id="echoid-s5685" xml:space="preserve">ſub quadrilineo, MTVN, ad omnia quadrata portionis, <lb/>SET, cum rectangulis bis ſub eadem portione, & </s> <s xml:id="echoid-s5686" xml:space="preserve">ſub tri-<lb/>lineis, TGV, GEF, eſſe vt portio, SMT, ad portionem, <lb/>SET.</s> <s xml:id="echoid-s5687" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5688" xml:space="preserve">Quoniam .</s> <s xml:id="echoid-s5689" xml:space="preserve">n. </s> <s xml:id="echoid-s5690" xml:space="preserve">rectangula ſub portione, SMT, & </s> <s xml:id="echoid-s5691" xml:space="preserve">parallelogram-<lb/> <anchor type="note" xlink:label="note-0251-02a" xlink:href="note-0251-02"/> mo, HV, ad omnia quadrata, HV, ſunt vt portio, SMT, ad <lb/>parallelogrammum, HV, rectangula verò ſub, SMT, & </s> <s xml:id="echoid-s5692" xml:space="preserve">paralle-<lb/>logrammo, HV, diuiduntur in rectangula ſub, SMT, & </s> <s xml:id="echoid-s5693" xml:space="preserve">ſub, SM <lb/>T, ideſt in omnia quadrata, SMT, & </s> <s xml:id="echoid-s5694" xml:space="preserve">in rectangula ſub, SMT, & </s> <s xml:id="echoid-s5695" xml:space="preserve"><lb/>ſub quadrilineis, HRSM, MTVN, ideſt bis ſub, SMT, & </s> <s xml:id="echoid-s5696" xml:space="preserve">ſub <lb/>quadrilineo, MTVN; </s> <s xml:id="echoid-s5697" xml:space="preserve">namcum, ME, ſit diameter, bifariam di-<lb/>uidit tum ordinatim applicatas in parallelogrammo, HF, tum in <lb/>circulo, vel ellipſi, MBEG, & </s> <s xml:id="echoid-s5698" xml:space="preserve">ideò exceſſus earundem linearum <lb/>hinc inde relinquuntur æquales, vndein quadrilineis, HRSM, M <lb/>TVN, lineæ in eadem rectitudine ſumptæ ſunt æquales, ideò om-<lb/>nia quadrata portionis, SMT, & </s> <s xml:id="echoid-s5699" xml:space="preserve">rectangula ſub eadem, & </s> <s xml:id="echoid-s5700" xml:space="preserve">ſub qua- <pb o="232" file="0252" n="252" rhead="GEOMETRI Æ"/> drilineo, MTVN, bis ſumpta, ſunt ad omnia quadrata, HV, vt <lb/>portio, SMT, ad parallelogrammum, HV. </s> <s xml:id="echoid-s5701" xml:space="preserve">Omnia inſuper qua-<lb/> <anchor type="note" xlink:label="note-0252-01a" xlink:href="note-0252-01"/> <anchor type="figure" xlink:label="fig-0252-01a" xlink:href="fig-0252-01"/> drata, HV, ad omnia quadrata, VD, <lb/>ſunt vt, HR. </s> <s xml:id="echoid-s5702" xml:space="preserve">ad, RD, .</s> <s xml:id="echoid-s5703" xml:space="preserve">i. </s> <s xml:id="echoid-s5704" xml:space="preserve">vt, HV, <lb/>ad, VD; </s> <s xml:id="echoid-s5705" xml:space="preserve">eodem deniq; </s> <s xml:id="echoid-s5706" xml:space="preserve">modo, quo ſu-<lb/>pra, oſtendemus omnia quadrata, RF, <lb/>ad omnia quadrata portionis, SET, cum <lb/>rectangulis bis ſub eidem, & </s> <s xml:id="echoid-s5707" xml:space="preserve">ſub trilineis, <lb/>TGV, GEF, eſſe vt, RF, ad portio-<lb/>nem, SET, ergo ex æquali, omnia <lb/>quadrata portionis, SMT, cum rectan-<lb/>gulis bis ſub eadem, & </s> <s xml:id="echoid-s5708" xml:space="preserve">ſub quadrilineo, <lb/>MTVN, ad omnia quadrata portio-<lb/>nis, SET, cum rectangulis bis ſub eadem. </s> <s xml:id="echoid-s5709" xml:space="preserve">& </s> <s xml:id="echoid-s5710" xml:space="preserve">ſub trilineis, TGV, <lb/>GFE, erunt vt portio, SMT, ad portionem, SET. </s> <s xml:id="echoid-s5711" xml:space="preserve">quod oſten. <lb/></s> <s xml:id="echoid-s5712" xml:space="preserve">dere opus erat.</s> <s xml:id="echoid-s5713" xml:space="preserve"/> </p> <div xml:id="echoid-div568" type="float" level="2" n="1"> <note position="right" xlink:label="note-0251-02" xlink:href="note-0251-02a" xml:space="preserve">Coroll. 1. <lb/>26. lib. 2. <lb/>per A. 23. <lb/>lib. 2.</note> <note position="left" xlink:label="note-0252-01" xlink:href="note-0252-01a" xml:space="preserve">@o. l. 2.</note> <figure xlink:label="fig-0252-01" xlink:href="fig-0252-01a"> <image file="0252-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0252-01"/> </figure> </div> </div> <div xml:id="echoid-div570" type="section" level="1" n="337"> <head xml:id="echoid-head354" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5714" xml:space="preserve">_H_INC patet omniä quadrata parallelogrammorum in eadem al-<lb/>titudine cum portionibus, vel fruſtibus portionum exiſtentium, <lb/>ad omnia quadrata earundem ſimul cumrectangulis bis ſub ijſdem, & </s> <s xml:id="echoid-s5715" xml:space="preserve"><lb/>ſub quadrilineis, vel trilineis, quæ illis èregione reſpondent lateri, <lb/>NF, adiacentia, veluti ſupra fuerunt quadri ineum, MTVN, & </s> <s xml:id="echoid-s5716" xml:space="preserve"><lb/>trilineum, TGV, GEF, eſſe, vt eadem parallelogramma ad eaſdem <lb/>portiones, vel portionum fruſta, quod ex ſupra dictis clarè patet.</s> <s xml:id="echoid-s5717" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div571" type="section" level="1" n="338"> <head xml:id="echoid-head355" xml:space="preserve">THEOREMA XIX. PROPOS. XX.</head> <p> <s xml:id="echoid-s5718" xml:space="preserve">EXpoſita adhuc figura Propoſ. </s> <s xml:id="echoid-s5719" xml:space="preserve">16. </s> <s xml:id="echoid-s5720" xml:space="preserve">& </s> <s xml:id="echoid-s5721" xml:space="preserve">intra circulum, vel <lb/>ellipſim ducta quacunq; </s> <s xml:id="echoid-s5722" xml:space="preserve">regulæ parallela, RX, diui-<lb/>dente ipſum vtcunq; </s> <s xml:id="echoid-s5723" xml:space="preserve">in duas portiones, SMT, SET. </s> <s xml:id="echoid-s5724" xml:space="preserve">Di-<lb/>co omnia quadrata portionis, SMT, cum rectangulis bis <lb/>ſub eadem, & </s> <s xml:id="echoid-s5725" xml:space="preserve">ſub quadrilineo, MTXC, ad omnia quadra-<lb/>ta portionis, SET, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s5726" xml:space="preserve">ſubqua-<lb/>drilineo, TGEOX, eſſe vt portio, SMT, ad portionem, SET.</s> <s xml:id="echoid-s5727" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Coroll. 1. <lb/>26. l. 2.</note> <p> <s xml:id="echoid-s5728" xml:space="preserve">Fiat prius, vt, MV, ad ſemiportionem, MYT, ſic, VY, ad, <lb/>YZ. </s> <s xml:id="echoid-s5729" xml:space="preserve">Omnia ergo quadrata, MX, ad rectangula ſub, MX, & </s> <s xml:id="echoid-s5730" xml:space="preserve">ſe-<lb/>miportione, MYT, ſunt vt, MX, ad, MYT, diuide rectangula <lb/> <anchor type="note" xlink:label="note-0252-03a" xlink:href="note-0252-03"/> <pb o="233" file="0253" n="253" rhead="LIBER III."/> ſub, MX, &</s> <s xml:id="echoid-s5731" xml:space="preserve">, MYT, in omnia quadrata, MYT, & </s> <s xml:id="echoid-s5732" xml:space="preserve">in rectangula <lb/>ſub, MYT, & </s> <s xml:id="echoid-s5733" xml:space="preserve">ſub, MTXC, omnia ergo quadrata, MX, ad om-<lb/>nia quadrata, MYT, cum rectangulis ſub, MYT, & </s> <s xml:id="echoid-s5734" xml:space="preserve">ſub quadri-<lb/>lineo, MTXC, erunt vt, MX, ad, MYT, .</s> <s xml:id="echoid-s5735" xml:space="preserve">i. </s> <s xml:id="echoid-s5736" xml:space="preserve">vt, XY, ad, YZ, <lb/>.</s> <s xml:id="echoid-s5737" xml:space="preserve">i. </s> <s xml:id="echoid-s5738" xml:space="preserve">vt quadratum, XY, ad rectangulum ſub, XY, &</s> <s xml:id="echoid-s5739" xml:space="preserve">, YZ, eadem <lb/>verò ad hæc quater ſumpta erunt, vt quadratum, XY, adrectangu-<lb/>lum ſub, XY, & </s> <s xml:id="echoid-s5740" xml:space="preserve">quadrupla, YZ, ſunt autem omnia quadrata ſe-<lb/>miportionis, MYT, quater ſumpta æqualia omnibus quadratis por-<lb/> <anchor type="note" xlink:label="note-0253-01a" xlink:href="note-0253-01"/> tionis, MST, & </s> <s xml:id="echoid-s5741" xml:space="preserve">rectangula ſub, MYT, & </s> <s xml:id="echoid-s5742" xml:space="preserve">quadrilineo, MTXC, <lb/>quater ſumpta æqualia rectangulis ſub eodem quadrilineo, & </s> <s xml:id="echoid-s5743" xml:space="preserve">ſub <lb/>portione, SMT, bis ſumptis, nam portio, SMT, bis continet ſe-<lb/>miportionem, MYT, ergo conuertendo, omnia quadrata portio. <lb/></s> <s xml:id="echoid-s5744" xml:space="preserve">nis, SMT, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s5745" xml:space="preserve">quadrilineo, MTX <lb/>C, ad omnia quadrata, MX, erunt vt rectangulum ſub quadrupla, <lb/>YZ, & </s> <s xml:id="echoid-s5746" xml:space="preserve">ſub, YX, ad quadratum, YX, omnia autem quadrata, M <lb/>X, ad omnia quadrata, HV, cum rectangulis bis ſub parallelogram-<lb/> <anchor type="figure" xlink:label="fig-0253-01a" xlink:href="fig-0253-01"/> mis, HV, VC, ſunt <lb/>vt vnum ad vnum. <lb/></s> <s xml:id="echoid-s5747" xml:space="preserve">vt quadratum, YX, <lb/>ad quadratum, RV, <lb/>cum rectangulis bis <lb/>ſub, RV, VX, ergo <lb/>exęquali omnia qua-<lb/>drata portionis, SM <lb/>T, cum rectangulis <lb/>bis ſub eadem, & </s> <s xml:id="echoid-s5748" xml:space="preserve">ſub <lb/>quadrilineo, MTX, adomnia quadrata, HV, cum rectangulis bis <lb/>ſub parallelogrammis, HV, VC, erunt vt rectangulum ſub, XY, & </s> <s xml:id="echoid-s5749" xml:space="preserve"><lb/>quadrupla, YZ, ad quadratum, RV, cum rectangulis bis ſub, RV <lb/>X, vel vt eorum dim dia .</s> <s xml:id="echoid-s5750" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5751" xml:space="preserve">vt rectangulum ſub, XY, & </s> <s xml:id="echoid-s5752" xml:space="preserve">dupla, YZ, <lb/>ad dimidium quadrati, RV, ſeil cet ad rectangulum ſub, RV, VY, <lb/>cum rectangulo ſub, RV, VX, vel adhuc, vt horum dimidia (com-<lb/>pone autem rectangulum ſub, RV, VY, cumrectangulo ſub, RV, <lb/>VX, ex quibus fit rectangulum ſub, RV, YX,). </s> <s xml:id="echoid-s5753" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5754" xml:space="preserve">vtrectangulum <lb/>ſub, ZY, YX, ad rectangulum ſub, RY, YX, .</s> <s xml:id="echoid-s5755" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5756" xml:space="preserve">vt, ZY, ad, YR, <lb/>.</s> <s xml:id="echoid-s5757" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5758" xml:space="preserve">vt ſemiportio, MYT, ad, MV, vel vt portio, SMT, ad, HV.</s> <s xml:id="echoid-s5759" xml:space="preserve"/> </p> <div xml:id="echoid-div571" type="float" level="2" n="1"> <note position="left" xlink:label="note-0252-03" xlink:href="note-0252-03a" xml:space="preserve">C. 23. l. 2.</note> <note position="right" xlink:label="note-0253-01" xlink:href="note-0253-01a" xml:space="preserve">D.23. hu-<lb/>ius.</note> <figure xlink:label="fig-0253-01" xlink:href="fig-0253-01a"> <image file="0253-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0253-01"/> </figure> </div> <p> <s xml:id="echoid-s5760" xml:space="preserve">Inſuper omnia quadrata, HV, cum rectangulis bis ſub parallelo-<lb/>grammis, HV, VC, ad omnia quadrata, RF, cum rectangulis bis <lb/>ſub parallelogrammis, RF, FX, funt vt, HR, ad, RD, & </s> <s xml:id="echoid-s5761" xml:space="preserve">tan-<lb/>dem modo ſuperiori oſtendemus omnia quadrata, RF, cum rectan-<lb/>gulis bis ſub parallelogrammis, RF, FX, ad omnia quadrata <lb/>portionis, SET, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s5762" xml:space="preserve">ſub quadrili- <pb o="234" file="0254" n="254" rhead="GEOMETRI Æ"/> neo, TGEOX, eſſe ve, RF, adportion<unsure/>em, SET, ergo ex æquali <lb/>o nnia quadrata portionis, SMT, cum rectangulis bis ſub eadem, <lb/>& </s> <s xml:id="echoid-s5763" xml:space="preserve">ſub quadrilineo, MTXC, ad omnia quadrata portionis, SET, <lb/>cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s5764" xml:space="preserve">ſub quadrilineo, TGEOX, <lb/>erunt vt portio, SMT, ad portionem, SET, quod oſtendere o-<lb/>portebat.</s> <s xml:id="echoid-s5765" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div573" type="section" level="1" n="339"> <head xml:id="echoid-head356" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5766" xml:space="preserve">_H_INC patet omnia quadrata parallelogrammorum in eadem al-<lb/>titudine cum portionibus, vel portionum fruſtibus exiſtentium, <lb/>vna cum rectangulis bis ſub ijſdem parallelogrammis, & </s> <s xml:id="echoid-s5767" xml:space="preserve">reliquis pa-<lb/>rallelogram nis illis in directum exiſtentibus, ad omnia quadrata por. <lb/></s> <s xml:id="echoid-s5768" xml:space="preserve">tionum, vel fruſtorum eorundem, ſimul cumrectangulis bis ſub ijſdem, <lb/>& </s> <s xml:id="echoid-s5769" xml:space="preserve">ſub quadrilineis illis in directum iacentibus, veluti fuerunt quadri-<lb/>lineun, MTXC, TGEOX, eſſe, vt dicta parallelogramma ad di-<lb/>ctas portiones, vel portionum fruſta; </s> <s xml:id="echoid-s5770" xml:space="preserve">quodex prædictis clarè patet; </s> <s xml:id="echoid-s5771" xml:space="preserve"><lb/>Vnde ex. </s> <s xml:id="echoid-s5772" xml:space="preserve">g. </s> <s xml:id="echoid-s5773" xml:space="preserve">omnia quadrata, RG, ſimul cum rectangulis bis ſub paral-<lb/>lelogrammis, RG, GX, ad omnia quadrata fruſti, SBGT, cum re-<lb/>ctangulis bis ſub, SGBT, & </s> <s xml:id="echoid-s5774" xml:space="preserve">quadrilineo, TG, PX, erunt vt paral-<lb/>lelogrammum, RG, ad fruſtum, SBGT, hoc .</s> <s xml:id="echoid-s5775" xml:space="preserve">n. </s> <s xml:id="echoid-s5776" xml:space="preserve">pariter oſtendetur, <lb/>veluti probatum eſt omnia quadrata, HV, ſimul cum rectangulis bis <lb/>ſub, HV, VC, ad omnia quadrata portionis, SMT, ſimul cum re-<lb/>ctingulis bis ſub eadem, & </s> <s xml:id="echoid-s5777" xml:space="preserve">ſub quadrilineo, MTXC, eſſe vt, HV, <lb/>ad portionem, SMT, vnde manifeſtum eſt, quod in hoc Corollaris <lb/>colligitur.</s> <s xml:id="echoid-s5778" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div574" type="section" level="1" n="340"> <head xml:id="echoid-head357" xml:space="preserve">THEOREMA XX. PROPOS. XXI.</head> <p> <s xml:id="echoid-s5779" xml:space="preserve">SIin circulo, vel ellipſi apteturrecta linea, per cuius ex-<lb/>trema puncta ducantur duæ rectæ lineæ, quæ ſint (exi-<lb/>ſtente apta parallela vniaxium, vel diametrorum) paralle-<lb/>læ ſecundo axi, vel diametro, quæ ſumatur pro regula: </s> <s xml:id="echoid-s5780" xml:space="preserve">Re-<lb/>ctangula ſub portione minori abſciſſa per aptatam, & </s> <s xml:id="echoid-s5781" xml:space="preserve">ſub <lb/>quadrilineo, quodaptata, & </s> <s xml:id="echoid-s5782" xml:space="preserve">duabus dictis parallelis vſque <lb/>ad curuam circuli, vel ellipſis productis, & </s> <s xml:id="echoid-s5783" xml:space="preserve">ab ijſdem inclu-<lb/>ſa curua comprehenditur, in circulo, erunt æqualia rectan-<lb/>gulis ſub duobus triangulis per diametrum quadrati, vel <lb/>rhombi (& </s> <s xml:id="echoid-s5784" xml:space="preserve">hoc in ellipſicum diametri coniugatę ſe obliquę <lb/>ſecabunt, quibus latera dictirhombi ſint æquidiſtantia) ab <pb o="235" file="0255" n="255" rhead="LIBER III"/> eadem aptata deſcriptiin ijſdem conſtitutis: </s> <s xml:id="echoid-s5785" xml:space="preserve">In ellipſi verò <lb/>ad eadem rectangula, erunt vt quadratum ſecundiaxis, vel <lb/>diametri, ad quadratum primæ.</s> <s xml:id="echoid-s5786" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5787" xml:space="preserve">Sit primò circulus, ABFH, & </s> <s xml:id="echoid-s5788" xml:space="preserve">in eo vtcunque aptata recta, AB, <lb/> <anchor type="note" xlink:label="note-0255-01a" xlink:href="note-0255-01"/> parallela diametro, ER, & </s> <s xml:id="echoid-s5789" xml:space="preserve">per puncta, AB, producantur viq; </s> <s xml:id="echoid-s5790" xml:space="preserve">ad <lb/>circumferentiam, HF, duæ, AH, BF, parallelæ ſecundæ diame-<lb/>tro, ST, quæ ſumatur pro regula, quia autem circulus eſt, ESRT, <lb/>ideò coniugatæ diametri, ER, ST, ſeſecant ad angulos rectos, & </s> <s xml:id="echoid-s5791" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0255-01a" xlink:href="fig-0255-01"/> ſunt coniugati axes, & </s> <s xml:id="echoid-s5792" xml:space="preserve">ideò, AH, BF, <lb/>ſunt perpendiculares ipſi, AB; </s> <s xml:id="echoid-s5793" xml:space="preserve">ſuper, A <lb/>B, ergo ſit deſcriptum quadratum, AD, <lb/>& </s> <s xml:id="echoid-s5794" xml:space="preserve">in eo ducta diameter, AD. </s> <s xml:id="echoid-s5795" xml:space="preserve">Dico er-<lb/>go rectangula ſub portione, ASB, & </s> <s xml:id="echoid-s5796" xml:space="preserve"><lb/>quadrilineo, AB, FH, eſſe æqualia re-<lb/>ctangulis ſub duobus triangulis, ABD, <lb/>AVD, ſumatur enim in, AB, vtcunq; <lb/></s> <s xml:id="echoid-s5797" xml:space="preserve">punctum, m, & </s> <s xml:id="echoid-s5798" xml:space="preserve">per, M, ducatur ipſi, B <lb/>F, parallela, CG, ſecans, AD, in, N; </s> <s xml:id="echoid-s5799" xml:space="preserve"><lb/>VD, in, O, & </s> <s xml:id="echoid-s5800" xml:space="preserve">curuam circuli in, CG, <lb/>quia ergo duæ, AB, CG, in circulo ſe <lb/> <anchor type="note" xlink:label="note-0255-02a" xlink:href="note-0255-02"/> ſecant in puncto, M, rectangulum, G <lb/>MC, eſt æquale rectangulo, BMA, & </s> <s xml:id="echoid-s5801" xml:space="preserve"><lb/>quia, AM, eſt æqualis ipſi, MN, &</s> <s xml:id="echoid-s5802" xml:space="preserve">, M <lb/>B, ipſi, NO, rectangulum, AMB, eſt <lb/>æquale rectangulo, MNO; </s> <s xml:id="echoid-s5803" xml:space="preserve">ergo rectan-<lb/>gulum, CMG, erit æquale rectangulo, <lb/>MNO, idem de cæteris probabitur, er-<lb/>go rectangula ſub portione, ACB, & </s> <s xml:id="echoid-s5804" xml:space="preserve"><lb/>quadrilineo, ABFGH, erunt æqualia <lb/>rectangulis ſub triangulis, ABD, AVD, quod eſt propoſitum in <lb/>circulo.</s> <s xml:id="echoid-s5805" xml:space="preserve"/> </p> <div xml:id="echoid-div574" type="float" level="2" n="1"> <note position="right" xlink:label="note-0255-01" xlink:href="note-0255-01a" xml:space="preserve">Defin. <lb/>4. Elem.</note> <figure xlink:label="fig-0255-01" xlink:href="fig-0255-01a"> <image file="0255-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0255-01"/> </figure> <note position="right" xlink:label="note-0255-02" xlink:href="note-0255-02a" xml:space="preserve">35. 3. ele</note> </div> <p> <s xml:id="echoid-s5806" xml:space="preserve">Sit nunc in inferiori figura ellipſis, ESRT, centrum, X, axes, <lb/> <anchor type="note" xlink:label="note-0255-03a" xlink:href="note-0255-03"/> vel diametri coniugatę, ER, prima, ST, ſecunda, ſit autem in ipſo <lb/>aptata, AB, parallela ipſi, ER, per cuius extrema puncta, AB, <lb/>productæ ſint vſque ad curuam ellipſis duæ, AH, BF, parallelę ſe-<lb/>cundæ axi, vel diametro, ST; </s> <s xml:id="echoid-s5807" xml:space="preserve">ſit inſuper deſcriptum quadratum, <lb/>velrhombus, AD, cuius latera diametris, ER, ST, ſint paralle-<lb/>la, & </s> <s xml:id="echoid-s5808" xml:space="preserve">in eo ducta diameter, AD, & </s> <s xml:id="echoid-s5809" xml:space="preserve">per puncta, E, S, ſint etiam du-<lb/>ctæ tangentes, EY, SY, coincidentes in, Y, quæ erunt parallelæ <lb/>diametris, ER, ST, .</s> <s xml:id="echoid-s5810" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5811" xml:space="preserve">YE, ipſi, ST, &</s> <s xml:id="echoid-s5812" xml:space="preserve">, YS, ipſi, ER; </s> <s xml:id="echoid-s5813" xml:space="preserve">erit er- <pb o="236" file="0256" n="256" rhead="GEOMETRIÆ"/> go, vt quadratum, EY, ad quadratum, YS, ita rectangulum, TZ <lb/> <anchor type="note" xlink:label="note-0256-01a" xlink:href="note-0256-01"/> S, ad rectangulum, BZA, eodem modo (ſumptoin, AB, vtcun-<lb/>quepuncto, M, & </s> <s xml:id="echoid-s5814" xml:space="preserve">per, M, ducta, CMG, parallela ipſi, BF,) ſe-<lb/>quetur rectangulum, GMC, ad rectangulum, BMA, eſſe vt qua-<lb/>dratum, EY, ad quadratum, YS, ergo rectangulum, TZS, ad re-<lb/>ctangulum, BZA, erit vt rectangulum, GMC, ad rectangulum, <lb/>BMA, & </s> <s xml:id="echoid-s5815" xml:space="preserve">ſic dereliquis oſtendemus .</s> <s xml:id="echoid-s5816" xml:space="preserve">i. </s> <s xml:id="echoid-s5817" xml:space="preserve">rectangula ſub portione, AS <lb/>B, & </s> <s xml:id="echoid-s5818" xml:space="preserve">quadrilineo, AHTFB, adrectangula ſub omnibus abſciſſis, <lb/>AB, & </s> <s xml:id="echoid-s5819" xml:space="preserve">reſiduis abſciſſarum eiuſdem .</s> <s xml:id="echoid-s5820" xml:space="preserve">i. </s> <s xml:id="echoid-s5821" xml:space="preserve">adrectangula ſub triangulis, <lb/>ABD, AVD, (ſunt .</s> <s xml:id="echoid-s5822" xml:space="preserve">n. </s> <s xml:id="echoid-s5823" xml:space="preserve">rectangula ſub omnibus abſciſſis, AB, & </s> <s xml:id="echoid-s5824" xml:space="preserve"><lb/>reſiduis abſciſſarum eiuſdem, æqualia rectangulis ſub duobus triangu-<lb/> <anchor type="note" xlink:label="note-0256-02a" xlink:href="note-0256-02"/> lis, ABD, AVD,) erunt vt rectangulum, TZS, adrectangulum, <lb/> <anchor type="note" xlink:label="note-0256-03a" xlink:href="note-0256-03"/> AZB, ideſt vt quadratum, EY, ad quadratum, YS, vel vt quadra-<lb/>tum, SX, ad quadratum, XE, vel vt quadratum, ST, ad quadra-<lb/>tum, ER; </s> <s xml:id="echoid-s5825" xml:space="preserve">ergo rectangula ſub portione, ASB, & </s> <s xml:id="echoid-s5826" xml:space="preserve">quadrilineo, A <lb/>HTFB, ad rectangula ſub triangulis, ABD, AVD, erunt vt qua-<lb/>dratum, ST, ad quadratum, ER; </s> <s xml:id="echoid-s5827" xml:space="preserve">quod oſtendere oportebat.</s> <s xml:id="echoid-s5828" xml:space="preserve"/> </p> <div xml:id="echoid-div575" type="float" level="2" n="2"> <note position="right" xlink:label="note-0255-03" xlink:href="note-0255-03a" xml:space="preserve">Velut in <lb/>circulo.</note> <note position="left" xlink:label="note-0256-01" xlink:href="note-0256-01a" xml:space="preserve">Ex 3. Co-<lb/>nic. p. 17.</note> <note position="right" xlink:label="note-0256-02" xlink:href="note-0256-02a" xml:space="preserve">Coroll. 2.</note> <note position="right" xlink:label="note-0256-03" xlink:href="note-0256-03a" xml:space="preserve">Prop. 19. <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div577" type="section" level="1" n="341"> <head xml:id="echoid-head358" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s5829" xml:space="preserve">HINC patet, quoniam probauimus, omnia quadrata, AD, ſex-<lb/>cupla eſſe rectangulorum ſub triangulis, ABD, AVD, quod <lb/>in circulo eadem quadrat a ſint ſexcupla rectangulorum ſub portione, <lb/>ASB, & </s> <s xml:id="echoid-s5830" xml:space="preserve">quadrilineo, AHTFB. </s> <s xml:id="echoid-s5831" xml:space="preserve">In ellipſi verò, quia pariter om-<lb/>nia quadrata, AD, rectangulorum ſub triangulis, ABD, AVD, <lb/>ſunt ſexcupla .</s> <s xml:id="echoid-s5832" xml:space="preserve">i. </s> <s xml:id="echoid-s5833" xml:space="preserve">ſunt ad illa, vt cubus, AB, ad ſui ipſius ſextam par-<lb/>tem, inſuper rectangula ſub triangulis, ABD, AVD, adrectangula <lb/> <anchor type="note" xlink:label="note-0256-04a" xlink:href="note-0256-04"/> ſub portione, ASB, & </s> <s xml:id="echoid-s5834" xml:space="preserve">quadrilineo, AHTFB, ſunt vt quadra-<lb/>tum, ER, conuertendo ad quadratum, ST, .</s> <s xml:id="echoid-s5835" xml:space="preserve">i. </s> <s xml:id="echoid-s5836" xml:space="preserve">vt ſexta pars cubi, AB, <lb/>ad eiuſdein talem partem, ad quam ipſa ſextapars ſit, vt quadratum, E <lb/>R, ad quadratum, ST, binc ex æquali omnia quadrata, AD, in elli-<lb/>pſi, ad rectangula ſub portione, ASB, & </s> <s xml:id="echoid-s5837" xml:space="preserve">quadrilineo, AHTFB, <lb/>erunt vt cubus, AB, ad ſui ipſius eam partem, ad quam eiuſdem cubi, <lb/>AB, ſextapars ſit veluti quadratum, ER, ad quadratum, ST. </s> <s xml:id="echoid-s5838" xml:space="preserve">Ve-<lb/>rum ſi in ellipſi diametri non ſint axes, vice cubi, AB, concludemus <lb/>omnia quadrata, AD, adrectangula ſub portione, ASB, & </s> <s xml:id="echoid-s5839" xml:space="preserve">quadri-<lb/>lineo, AHTFB, eſſe vt parallelepipedum ſub altitudine, AB, baſi <lb/>rhombo quod ab ipſa, AB, deſeribitur, ad ſui ipſius eam partem, ad <lb/>quam eiuſdem parallelepipedi pars ſexta ſit veluti quadratum, ER, ad <lb/>quadratum, ST,</s> </p> <div xml:id="echoid-div577" type="float" level="2" n="1"> <note position="left" xlink:label="note-0256-04" xlink:href="note-0256-04a" xml:space="preserve">Cor. 24. <lb/>lib. 2.</note> </div> <pb o="237" file="0257" n="257" rhead="LIBER III."/> </div> <div xml:id="echoid-div579" type="section" level="1" n="342"> <head xml:id="echoid-head359" xml:space="preserve">THEOREMA XXI. PROPOS. XXII.</head> <p> <s xml:id="echoid-s5840" xml:space="preserve">SI intra parallelogrammum, quod circulo, vel ellipſi ſit <lb/>circumſcriptum, ducatur lateribus eiuſdem parallela <lb/>quædam rectalinea, per circuli, vel ellipſis centrum non <lb/>tranſiens, altero reliquorum laterum regula exiſtente. </s> <s xml:id="echoid-s5841" xml:space="preserve">Om-<lb/>nia quadrata parallelogrammi, quod maiori portioni circu-<lb/>li, vel ellipſis iam dicti, remanent circumſcriptum, ad om-<lb/>nia quadrata figuræ compoſitæ ex maiori portione, & </s> <s xml:id="echoid-s5842" xml:space="preserve">duo-<lb/>bus trilineis, quiad baſim eiuſdem hinc inde extra conſtitu-<lb/>untur, demptis eorundem trilineorum omnibus quadratis, <lb/>erunt in circulo, vt parallelepipedum ſub baſi parallelo-<lb/>grammo dictæ portioni maiori circumſcripto, altitudine <lb/>eiuſdem portionis diametro ad cylindricum ſub baſi eadem <lb/>maiori portione, altitudine differentia diametrorum maio-<lb/>ris, acminoris factarum portionum, vna cum ſexta parte cu-<lb/>bi baſis eiuſdem portionis In ellipſi verò erunt, vt paralle-<lb/>lepipedum ſub baſi parallelogrammo maiori portioni ſimili-<lb/>ter circumſcripto, altitudine eiuſdem portionis diametro, <lb/>ad cylindricum ſub baſi eadem maiori portione, altitudine <lb/>differentia diametrorum maioris, ac minoris factarum por-<lb/>tionum, vna cum ea porte cubi baſis eiuſdem portionis, ad <lb/>quam ſexta pars eiuſdem cubi ſit, vt quadratum primæ dia-<lb/>metriad quadratum ſecundæ, vel, ſi diametrinon ſint axes, <lb/>vna cum ea parte parallelepipedi ſub altitudine baſi eiu-<lb/>ſdem portionis, ac ſub baſi rhombo ab eadem deſcripto, ad <lb/>quam eiuſdem parallelepipedi pars ſexta ſit, vt quadratum <lb/>primę diametri ad quadratum ſecundæ.</s> <s xml:id="echoid-s5843" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5844" xml:space="preserve">Sit ergo circulus, vel ellipſis, CFEH, cui ſit circumſcriptum pa-<lb/>rallelogrammum, AQ, & </s> <s xml:id="echoid-s5845" xml:space="preserve">centrum ſit, N, diametriautem tranſeun-<lb/>tesper puncta contactuum laterum circumſcripti parallelogrammi, <lb/>& </s> <s xml:id="echoid-s5846" xml:space="preserve">per centrum, N. </s> <s xml:id="echoid-s5847" xml:space="preserve">ſint, CE, FH; </s> <s xml:id="echoid-s5848" xml:space="preserve">ſit autem, FH, regula, cui in-<lb/>ſiſtens, & </s> <s xml:id="echoid-s5849" xml:space="preserve">lateribus, AP, DQ, parallela intra ipſum ducta ſit, LG. <lb/></s> <s xml:id="echoid-s5850" xml:space="preserve">Dico ergo omnia quadrata parallelogrammi, AG, ad omnia qua-<lb/>drata figuræ, LCFEG, demptis omnibus quadratis trilineorum, <lb/>CLI, YGE, eſſe, in circulo, vt parallelepipedum ſub baſi <pb o="238" file="0258" n="258" rhead="GEOMETRIÆ"/> parallelogrammo, AG, altitudine, FI, ad cylindricum ſub baſi por-<lb/>tione, TCFEY, altitudine, IM, vna cum, {1/6}, cubi, TY. </s> <s xml:id="echoid-s5851" xml:space="preserve">In elli-<lb/>pſi verò, vt parallelepipedum ſub baſi parallelogrammo, AG, alti-<lb/>tudine, FI, ad cylindricum ſub baſi portione, TCFEY, altitudi-<lb/>ne, MI, vna cum ea parte cubi, TY, ad quam eiuſdem cubi ſexta <lb/>pars ſit, vt quadratum, CE, primę diametri, ad quadratum ſecun-<lb/>dæ .</s> <s xml:id="echoid-s5852" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5853" xml:space="preserve">ad quadratum, FH, vel, ſi diametri non ſint axes, vna cum <lb/>ea parte parallelepipedi ſub, TY, & </s> <s xml:id="echoid-s5854" xml:space="preserve">rhombo, RZ, ad quam illius <lb/>pars ſexta ſit, vt quadratum, CE, primæ diametri ad quadratum <lb/>ſecundæ. </s> <s xml:id="echoid-s5855" xml:space="preserve">Ducantur per, T, Y, ipſi, PQ, parallelæ, Τ Δ, Υ Φ, ſe-<lb/>cantes curuam, CFE, in punctis, R, V, quæ iungantur recta, R <lb/>V, producta in, B, K, quoniam ergo, EC, eſt diameter, ad quam <lb/> <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> ordinatim applicantur, RT, VY, eas <lb/>quoq; </s> <s xml:id="echoid-s5856" xml:space="preserve">bifariam ſecabit, eſt autem, ST, <lb/>æqualis, XY, ob parallelogrammum, <lb/>SY, ergo, VX, erit etiam æqualis ipſi, <lb/>RS, & </s> <s xml:id="echoid-s5857" xml:space="preserve">tota, VY, toti, RT, cui etiam <lb/>eſt parallela, ergo, RV, TY, ſunt etiam <lb/>æquales, & </s> <s xml:id="echoid-s5858" xml:space="preserve">parallelæ, eſtque, RV, in, <lb/>M, bifariam ſecta.</s> <s xml:id="echoid-s5859" xml:space="preserve"/> </p> <div xml:id="echoid-div579" type="float" level="2" n="1"> <figure xlink:label="fig-0258-01" xlink:href="fig-0258-01a"> <image file="0258-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0258-01"/> </figure> </div> <p> <s xml:id="echoid-s5860" xml:space="preserve">Diuidamus igitur omnia quadrata fi-<lb/>guræ, LCFEG, demptis omnibus <lb/>quadratis trilineorum, CLT, EGY, in <lb/>omnia quadrata figuræ, LCRT, dem-<lb/>ptis omnibus quadratis trilinei, LCT, <lb/>in omnia quadrata figuræ, GEVY, <lb/>demptis omnibus quadratis trilinei, E <lb/>GY, & </s> <s xml:id="echoid-s5861" xml:space="preserve">in omnia quadrata figuræ, TR <lb/>FVY. </s> <s xml:id="echoid-s5862" xml:space="preserve">Rurſus per rectam, RV, diui-<lb/> <anchor type="note" xlink:label="note-0258-01a" xlink:href="note-0258-01"/> duntur omnia quadrata figuræ, TRF <lb/>VY, in omnia quadrata, YR, in om-<lb/>nia quadrata portionis, RFV, & </s> <s xml:id="echoid-s5863" xml:space="preserve">in re-<lb/>ctangula bis ſub, YR, & </s> <s xml:id="echoid-s5864" xml:space="preserve">portione, R <lb/>FV, his ſeparatis, ad eorum ſingula comparemus nunc omnia qua-<lb/>drata parallelogrammi, KG.</s> <s xml:id="echoid-s5865" xml:space="preserve"/> </p> <div xml:id="echoid-div580" type="float" level="2" n="2"> <note position="left" xlink:label="note-0258-01" xlink:href="note-0258-01a" xml:space="preserve">Per D. 23. <lb/>lib. 2.</note> </div> <p> <s xml:id="echoid-s5866" xml:space="preserve">Igitur omnia quadrata, KG, ad omnia quadrata, RY, ſunt vt, <lb/> <anchor type="note" xlink:label="note-0258-02a" xlink:href="note-0258-02"/> KB, ad, RV, vel vt parallelogrammum, KG, ad parallelogram-<lb/>mum, RY; </s> <s xml:id="echoid-s5867" xml:space="preserve">omnia inſuper quadrata, KG, ad omnia quadrata, K <lb/>T, ſunt vt, BK, ad, KR, .</s> <s xml:id="echoid-s5868" xml:space="preserve">i. </s> <s xml:id="echoid-s5869" xml:space="preserve">vt, KG, ad, KT; </s> <s xml:id="echoid-s5870" xml:space="preserve">item omnia qua-<lb/>drata, KT, ad omnia quadrata figuræ, LCRT, demptis omnibus <lb/>quadratis trilinei, LCT, .</s> <s xml:id="echoid-s5871" xml:space="preserve">i. </s> <s xml:id="echoid-s5872" xml:space="preserve">ad omnia quadrata portionis, RCT, <lb/>cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s5873" xml:space="preserve">ſub trilineo, CLT, ſunt vt, KT, <pb o="239" file="0259" n="259" rhead="LIBER III."/> ad portionem, RCT, ergo ex æquali omnia quadrata, KG, adom-<lb/> <anchor type="note" xlink:label="note-0259-01a" xlink:href="note-0259-01"/> nia quadrata figuræ, LCRT, demptis omnibus quadratis trilinei, <lb/>CLT, erunt vt, KG, ad portionem, RCT. </s> <s xml:id="echoid-s5874" xml:space="preserve">Eodem modo oſten-<lb/>demus omnia quadrata, KG, ad omnia quadrata figuræ, VEGY, <lb/>demptis omnibus quadratis trilinei, EGY, eſſe vt, KG, ad portio-<lb/>nem, VEY, quæ conſerua.</s> <s xml:id="echoid-s5875" xml:space="preserve"/> </p> <div xml:id="echoid-div581" type="float" level="2" n="3"> <note position="left" xlink:label="note-0258-02" xlink:href="note-0258-02a" xml:space="preserve">@o. Lib. 2.</note> <note position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve">Cor. 19. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s5876" xml:space="preserve">Omnia inſuper quadrata, KG, ad omnia quadrata, RY, vt <lb/> <anchor type="note" xlink:label="note-0259-02a" xlink:href="note-0259-02"/> probauimus, ſunt vt, KG, ad, RY, item omnia quadrata, RY, <lb/>ad rectangula ſub, RY, R Φ, ſunt vt, RY, ad R Φ, & </s> <s xml:id="echoid-s5877" xml:space="preserve">tandem re-<lb/> <anchor type="note" xlink:label="note-0259-03a" xlink:href="note-0259-03"/> ctangula ſub, R Φ, RY, adrectangula ſub portione, RFV, & </s> <s xml:id="echoid-s5878" xml:space="preserve"><lb/>ſub, RY, ſunt vt, R Φ, ad portionem, RFV, ergo ex æquali <lb/>omnia quadrata, KG, adrectangula ſub portione, RFV, & </s> <s xml:id="echoid-s5879" xml:space="preserve">ſub, <lb/>RY, erunt vt, KG, ad portionem, RFV, ergo, colligendo, om-<lb/>nia quadrata, KG, ad omnia quadrata figurarum, LCRT, VE <lb/>GY, demptis omnibus quadratis trilineorum, CLT, EGY, & </s> <s xml:id="echoid-s5880" xml:space="preserve">ad <lb/>omnia quadrata, RY, & </s> <s xml:id="echoid-s5881" xml:space="preserve">ad rectangula ſemel ſub portione, RFV, <lb/>& </s> <s xml:id="echoid-s5882" xml:space="preserve">ſub, RY, erunt vt, KG, ad portiones, RCT, VEY, RFV, <lb/>& </s> <s xml:id="echoid-s5883" xml:space="preserve">ad rectangulum, RY, .</s> <s xml:id="echoid-s5884" xml:space="preserve">i. </s> <s xml:id="echoid-s5885" xml:space="preserve">vt, KG, ad portionem, TCFEY.</s> <s xml:id="echoid-s5886" xml:space="preserve"/> </p> <div xml:id="echoid-div582" type="float" level="2" n="4"> <note position="right" xlink:label="note-0259-02" xlink:href="note-0259-02a" xml:space="preserve">Coroll. <lb/>26. l. 2.</note> <note position="right" xlink:label="note-0259-03" xlink:href="note-0259-03a" xml:space="preserve">Caroll. 1. <lb/>26. l. 2.</note> </div> <p> <s xml:id="echoid-s5887" xml:space="preserve">Reliquum eſt, vt comparemus omnia quadrata, KG, ad omnia <lb/>quadrata portionis, RFV, & </s> <s xml:id="echoid-s5888" xml:space="preserve">ad rectangula ſub eadem, & </s> <s xml:id="echoid-s5889" xml:space="preserve">ſub, RY, <lb/>quia autem, RV, æquatur ipſi, TY, portio, RFV, æquatur por-<lb/>tioni, THY, etiam in ellipſi, quia, RV, TY, ſunt parallelæ, <lb/>ideò omnia quadrata portionis, RFV, ſunt rectangula ſub portio-<lb/>ne, RFV, & </s> <s xml:id="echoid-s5890" xml:space="preserve">ſub portione, THY, quibus ſi iunxeris rectangula <lb/> <anchor type="note" xlink:label="note-0259-04a" xlink:href="note-0259-04"/> ſub eadem portione, RFV, & </s> <s xml:id="echoid-s5891" xml:space="preserve">ſub, RY, componentur rectangu-<lb/>la ſub eadem portione, RFV, & </s> <s xml:id="echoid-s5892" xml:space="preserve">ſub quadrilineo, RTHYV. <lb/></s> <s xml:id="echoid-s5893" xml:space="preserve">Nuncvel, RV, eſt æqualis ipſi, VY, & </s> <s xml:id="echoid-s5894" xml:space="preserve">ſic, RY, erit quadratum, <lb/>ſiue rhombus, vel, RV, non eſt æqualis ipſi, VY, & </s> <s xml:id="echoid-s5895" xml:space="preserve">tunc in ipſa, <lb/>VY, producta, ſi opus ſit ſumatur, VZ, æqualis, ipſi, VR, & </s> <s xml:id="echoid-s5896" xml:space="preserve">du-<lb/>cta per, Z, Z Π, ipſi, RV, parallela, ſit conſtitutum, RZ, qua-<lb/>dratum, vel rhombus ipſius, RV: </s> <s xml:id="echoid-s5897" xml:space="preserve">Omnia ergo quadrata, KG, ad <lb/>omnia quadrata, RZ, habent rationem compoſitam ex ratione <lb/> <anchor type="note" xlink:label="note-0259-05a" xlink:href="note-0259-05"/> quadrati, KL, ad quadratum, R Π, vel ad quadratum, RV, & </s> <s xml:id="echoid-s5898" xml:space="preserve"><lb/>ex ratione ipſius, KB, ad, RV, quæ duæ rationes componunt ra-<lb/> <anchor type="note" xlink:label="note-0259-06a" xlink:href="note-0259-06"/> tionem parallelepipedi rectanguli ſub altitudine, BK, baſi autem <lb/> <anchor type="note" xlink:label="note-0259-07a" xlink:href="note-0259-07"/> quadrato, KL, ad cubum, RV. </s> <s xml:id="echoid-s5899" xml:space="preserve">Siautem, CE, FH, ſint tantum <lb/> <anchor type="note" xlink:label="note-0259-08a" xlink:href="note-0259-08"/> diametri, ſic dicemus, nempè, Omnia quadrata, KG, ad omnia <lb/>quadrata, RZ, rhombi habent rationem compoſitam ex ratione, <lb/>KL, ad, R Π, bis ſumpta, & </s> <s xml:id="echoid-s5900" xml:space="preserve">ex ratione, KB, ad, RV, quæ tres <lb/> <anchor type="note" xlink:label="note-0259-09a" xlink:href="note-0259-09"/> rationes componunt rationem parallelepiped ſub altitudine, KL, <lb/>baſi parallelogrammo, KG, ad parallelepipedum ſub altitudine, <lb/>RV, baſi autem rhombo, RZ: </s> <s xml:id="echoid-s5901" xml:space="preserve">Omnia verò quadrata, RZ, in <pb o="240" file="0260" n="260" rhead="GEOMETRIÆ"/> circulo ſunt ſexcupla rectangulorum ſub portione, RFV, & </s> <s xml:id="echoid-s5902" xml:space="preserve">qua-<lb/>drilineo, RTHYV, .</s> <s xml:id="echoid-s5903" xml:space="preserve">i. </s> <s xml:id="echoid-s5904" xml:space="preserve">ſunt adilla, vt cubus, RV, ad ſui ipſius <lb/>ſextam partem. </s> <s xml:id="echoid-s5905" xml:space="preserve">In ellipſi verò omnia quadrata, RZ, ad rectangu-<lb/>la ſub portione, RFV, & </s> <s xml:id="echoid-s5906" xml:space="preserve">quadrilineo, RTHYV, ſunt vt cubus, <lb/>RV, vel parallelepipedum ſub altitudine, RV, baſi rhombo, RZ, <lb/>ad ſui ipſius eam partem, ad quam ſexta pars eiuſdem cubi, vel pa-<lb/>rallelepipedi ſit, vt quadratum, CE, primæ diametri, ad quadra-<lb/>tum, FH, ſecundæ; </s> <s xml:id="echoid-s5907" xml:space="preserve">ergo ex æqualiin circulo omnia quadrata, K <lb/>G, adrectangula ſub portione, RFV, & </s> <s xml:id="echoid-s5908" xml:space="preserve">ſub quadrilineo, RTH <lb/>YV, erunt vt parallelepipedum ſub altitudine, BK, baſi quadra-<lb/>to, KL, vel (quodidem eſt) vt parallelepipedum ſub, LK, & </s> <s xml:id="echoid-s5909" xml:space="preserve">re-<lb/>ctangulo, KG, ad, {1/6}, cubi, RV. </s> <s xml:id="echoid-s5910" xml:space="preserve">In ellipſi verò eadem erunt, vt <lb/>parallelepipedum ſub altitudine, LK, baſi parallelogrammo, KG, <lb/>ad eam partem cubi, RV, vel dicti parallelepipedi ſub, RV, & </s> <s xml:id="echoid-s5911" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0260-01a" xlink:href="fig-0260-01"/> rhombo, RZ, ad quam eiuſdem cubi, <lb/>vel parallelepipedi ſexta pars ſit, vt <lb/>quad. </s> <s xml:id="echoid-s5912" xml:space="preserve">CE, ad quadratum, FH.</s> <s xml:id="echoid-s5913" xml:space="preserve"/> </p> <div xml:id="echoid-div583" type="float" level="2" n="5"> <note position="right" xlink:label="note-0259-04" xlink:href="note-0259-04a" xml:space="preserve">A. 23. l. 2.</note> <note position="right" xlink:label="note-0259-05" xlink:href="note-0259-05a" xml:space="preserve">Diffin. 12. <lb/>l. 1.</note> <note position="right" xlink:label="note-0259-06" xlink:href="note-0259-06a" xml:space="preserve">11. l. 2,</note> <note position="right" xlink:label="note-0259-07" xlink:href="note-0259-07a" xml:space="preserve">D. Cor. 4.</note> <note position="right" xlink:label="note-0259-08" xlink:href="note-0259-08a" xml:space="preserve">Gen. 34. <lb/>l. 2.</note> <note position="right" xlink:label="note-0259-09" xlink:href="note-0259-09a" xml:space="preserve">23. huius.</note> <figure xlink:label="fig-0260-01" xlink:href="fig-0260-01a"> <image file="0260-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0260-01"/> </figure> </div> <p> <s xml:id="echoid-s5914" xml:space="preserve">Sunt autem omnia quadrata, KG, ad <lb/>omnia quadrata figurarum, RCLT, <lb/>VEGY, demptis, omnibus quadratis <lb/>trilineorum, CLT, EGY, vna cum <lb/>omnibus quadratis, RY, & </s> <s xml:id="echoid-s5915" xml:space="preserve">cum re <lb/>δtangulis ſub portione, RFV, & </s> <s xml:id="echoid-s5916" xml:space="preserve">ſub, <lb/>RY, ſemel, vt, KG, ad portionem, <lb/>TCFEY, vt oſtendimus .</s> <s xml:id="echoid-s5917" xml:space="preserve">i. </s> <s xml:id="echoid-s5918" xml:space="preserve">ſumpta, K <lb/>L, communi altitudine, vt parallelepi-<lb/>pedum ſub altitudine, KL, baſi paral <lb/> <anchor type="note" xlink:label="note-0260-01a" xlink:href="note-0260-01"/> lelogrammo, KG, ad cylindricum ſub <lb/>eadem altitudine, KL, & </s> <s xml:id="echoid-s5919" xml:space="preserve">ſub baſi por <lb/>tione, TCFEY, ergo, colligendo, <lb/>omnia quadrata, KG, ad omnia qua <lb/>drata portionum, RCT, VEY, cum <lb/>rectangulis bis ſub ijſdem, & </s> <s xml:id="echoid-s5920" xml:space="preserve">ſub trili <lb/>neis, CLT, EGY, inſuper ad omnia <lb/>quadrata, RY, cum rectangulis ſub, R <lb/>Y, & </s> <s xml:id="echoid-s5921" xml:space="preserve">portione, RFV, ſemel, & </s> <s xml:id="echoid-s5922" xml:space="preserve">ad rectangula ſub eadem portio-<lb/>ne, RFV, & </s> <s xml:id="echoid-s5923" xml:space="preserve">quadrilineo, RTHYV, ideſt ad omnia quadrata <lb/>portionis, RFV, cum rectangulis iterum ſub eadem, & </s> <s xml:id="echoid-s5924" xml:space="preserve">ſub, RY, <lb/>quia rectangula ſub portione, RFV, & </s> <s xml:id="echoid-s5925" xml:space="preserve">quadrilineo, RTHYV, <lb/>ſeparantur per lineam, TY, inrectangula ſub, RY, & </s> <s xml:id="echoid-s5926" xml:space="preserve">portione, <lb/>RFV, & </s> <s xml:id="echoid-s5927" xml:space="preserve">ſub portione, THY, & </s> <s xml:id="echoid-s5928" xml:space="preserve">portione, RFV, quæ ſunt om-<lb/> <anchor type="note" xlink:label="note-0260-02a" xlink:href="note-0260-02"/> nia quadrata portionis, RFV, .</s> <s xml:id="echoid-s5929" xml:space="preserve">i. </s> <s xml:id="echoid-s5930" xml:space="preserve">(his omnibus in vnam ſummam <pb o="241" file="0261" n="261" rhead="LIBER III."/> collectis) ad omnia quadrata figuræ, LCFEG, demptis omnibus <lb/>quadratis trilineorum, CLT, EGY, erunt vt parallelepipedum ſub <lb/>altitudine, KL, baſi parallelogrammo, KG, ad cylindricum ſub <lb/>altitudine, KL, baſi portione, TCFEY, vna cum, {1/6}, cubi, RV, <lb/> <anchor type="note" xlink:label="note-0261-01a" xlink:href="note-0261-01"/> in circulo. </s> <s xml:id="echoid-s5931" xml:space="preserve">In ellipſi autem, vt idem parallepipedum ad eundem cy-<lb/>lindricum, vna cum ea parte cubi, RV, vel parallelepipedi ſub, R <lb/>V, & </s> <s xml:id="echoid-s5932" xml:space="preserve">rhombo, RZ, ad quam eiuſdem cubi, vel parallelepipedis ſex-<lb/>ta pars ſit, vt quadratum, CE, ad quadratum, FH: </s> <s xml:id="echoid-s5933" xml:space="preserve">Omnia autem <lb/>quadrata, AG, ad omnia quadrata, KG, ſunt vt parallelepipedum <lb/>ſub altitudine, AL, baſi parallelogrammo, AG, ad parallelepipe-<lb/>dum ſub altitudine, LK, baſi parallelogrammo, KG, ergo ex ęqua-<lb/>li pariter omnia quadrata, AG, ad omnia quadrata figurę, LCFE <lb/>G, demptis omnibus quadratis trilineorum, CLT, EGY, erunt in <lb/>circulo, vt parallelepipedum ſub altitudine, AL, vel, FI, baſi au-<lb/>tem parallelogrammo, AG, ad cylindricum ſub altitudine, LK, vel, <lb/>MI, baſi autem portione, TCFEY, vna cum, {1/6}, cubi, RV, vel, <lb/>TY. </s> <s xml:id="echoid-s5934" xml:space="preserve">In ellipſi verò erunt, vt parallelepipedum ſub altitudine, FI, <lb/>baſi autem parallelogrammo, AG, ad cylindricum ſub altitudine, <lb/>MI, baſi autem ipſa portione, TCFEY, vna cum ea parte cubi, <lb/>RV, vel, TY, ſiue parallelepipedi ſub altitudine, TY, & </s> <s xml:id="echoid-s5935" xml:space="preserve">baſi rhom-<lb/>bo, RZ, ad quam eiuſdem cubi, vel parallelepipedi ſexta pars ſit, vt <lb/>quadra@um, CE, ad quadratum, FH; </s> <s xml:id="echoid-s5936" xml:space="preserve">quod oſtendere oportebat.</s> <s xml:id="echoid-s5937" xml:space="preserve"/> </p> <div xml:id="echoid-div584" type="float" level="2" n="6"> <note position="left" xlink:label="note-0260-01" xlink:href="note-0260-01a" xml:space="preserve">G. B. Cor. <lb/>4. Gen. <lb/>34. l. 2.</note> <note position="left" xlink:label="note-0260-02" xlink:href="note-0260-02a" xml:space="preserve">A. 23. l. 2.</note> <note position="right" xlink:label="note-0261-01" xlink:href="note-0261-01a" xml:space="preserve">36. Lib. 2.</note> </div> </div> <div xml:id="echoid-div586" type="section" level="1" n="343"> <head xml:id="echoid-head360" xml:space="preserve">THEOREMA XXII. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s5938" xml:space="preserve">EXpoſita figura circuli Theorematis ſuperioris, & </s> <s xml:id="echoid-s5939" xml:space="preserve">in eo <lb/>ſumpta vtcunq;</s> <s xml:id="echoid-s5940" xml:space="preserve">portione minori, RFV, cæteris, prout <lb/>ſtant, ſuppoſitis. </s> <s xml:id="echoid-s5941" xml:space="preserve">Dico omnia quadrata, Δ V, ad omnia qua-<lb/>drata portionis, RFV, eſſe, vt ſexquialtera, FM, ad reli-<lb/>quum diametri, MH, maioris portionis, ab eodem dempta <lb/>recta linea, ad quam tripla, MN, ſit, vt parallelogrammum, <lb/>Δ V, ad portionem, RFV.</s> <s xml:id="echoid-s5942" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5943" xml:space="preserve">Rectangula enim ſub, Δ V, VT, ad omnia quadrata, RZ, ſunt vt <lb/> <anchor type="note" xlink:label="note-0261-02a" xlink:href="note-0261-02"/> vnum ad vnum. </s> <s xml:id="echoid-s5944" xml:space="preserve">i. </s> <s xml:id="echoid-s5945" xml:space="preserve">vt rectangulum, FMI, ad quadratum, VZ, vel <lb/>ad quadratum, RV, omnia item quadrata, RZ, ſunt ſexcupla re-<lb/> <anchor type="note" xlink:label="note-0261-03a" xlink:href="note-0261-03"/> ctangulorum ſub portione, RFV, & </s> <s xml:id="echoid-s5946" xml:space="preserve">quadrilineo, RTHYV, ideſt <lb/>ſunt ad illa, vt quadratum, RV, ad ſui, {1/6}, ergo ex æquali rectan-<lb/>gula ſub, Δ V, VT, ad rectangula ſub portione, RFV, & </s> <s xml:id="echoid-s5947" xml:space="preserve">quadri-<lb/>lineo, RTHYV, erunt vt rectang. </s> <s xml:id="echoid-s5948" xml:space="preserve">FMI, ad, {1/6}, quadrati, RV, <pb o="242" file="0262" n="262" rhead="GEOMETRIÆ"/> vel vtrectangulum, FMN, ad, {1/6}, quadratorum, RM, MV,.</s> <s xml:id="echoid-s5949" xml:space="preserve">. <lb/>ad, {1/3}, quadrati, RM, .</s> <s xml:id="echoid-s5950" xml:space="preserve">i. </s> <s xml:id="echoid-s5951" xml:space="preserve">ad rectangulum ſub; </s> <s xml:id="echoid-s5952" xml:space="preserve">FM, &</s> <s xml:id="echoid-s5953" xml:space="preserve">, {1/3}, MH, .</s> <s xml:id="echoid-s5954" xml:space="preserve">i. </s> <s xml:id="echoid-s5955" xml:space="preserve">vt, <lb/>MN, ad, {1/3}, MH, vel vt tripla, MN, ad, MH. </s> <s xml:id="echoid-s5956" xml:space="preserve">Inſuper eadem <lb/>rectangula ſub, Δ V, VT, ad rectangula ſub portione, RFV, & </s> <s xml:id="echoid-s5957" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0262-01a" xlink:href="fig-0262-01"/> ſub, RY, ſunt vt parallelogrammum, <lb/>Δ V, ad portionem, RFV, ergo ſi <lb/>fiat, vt, Δ V, ad portionem, RFV, <lb/> <anchor type="note" xlink:label="note-0262-01a" xlink:href="note-0262-01"/> ita tripla, MN, ad, H ω; </s> <s xml:id="echoid-s5958" xml:space="preserve">rectangula <lb/>ſub, Δ V, VT, ad reliquum, demptis <lb/>rectangulis ſub portione, RFV, & </s> <s xml:id="echoid-s5959" xml:space="preserve"><lb/>ſub, RY; </s> <s xml:id="echoid-s5960" xml:space="preserve">à rectangulis ſub eadem por-<lb/>tione, & </s> <s xml:id="echoid-s5961" xml:space="preserve">ſub quadrilineo, RTHYV, <lb/>.</s> <s xml:id="echoid-s5962" xml:space="preserve">i. </s> <s xml:id="echoid-s5963" xml:space="preserve">ad rectangula ſub portione, RFV, <lb/>& </s> <s xml:id="echoid-s5964" xml:space="preserve">portione, THY, .</s> <s xml:id="echoid-s5965" xml:space="preserve">i. </s> <s xml:id="echoid-s5966" xml:space="preserve">ad omnia qua-<lb/>drata portionis, RFV, erunt vetripla, <lb/>MN, ad, M ω, omnia autem quadra-<lb/>ta, Δ V, ad rectangula ſub, Δ V, VT, <lb/>ſunt vt quadratum, FM, ad rectangu-<lb/>lum, FMI, .</s> <s xml:id="echoid-s5967" xml:space="preserve">i. </s> <s xml:id="echoid-s5968" xml:space="preserve">vt, FM, ad, MI, vel <lb/>vt ſexquialtera, FM, ad ſexquialte-<lb/>ram, MI, .</s> <s xml:id="echoid-s5969" xml:space="preserve">i. </s> <s xml:id="echoid-s5970" xml:space="preserve">ad triplam, MN, rectan-<lb/>gula autem ſub, Δ V, VT, ad omnia <lb/> <anchor type="note" xlink:label="note-0262-02a" xlink:href="note-0262-02"/> quadrata portionis, RFV, ſunt vt <lb/>tripla, MN, ad, M ω, ergo ex æqua-<lb/> <anchor type="note" xlink:label="note-0262-03a" xlink:href="note-0262-03"/> li omnia quadrata, Δ V, ad omnia <lb/>quadrata portionis, RFV, erunt vt ſexquialtera ipſius, FM, ad, <lb/>M ω, quæ eſt reſiduumipſius, MH, dempta, H ω, ad quam tri-<lb/>pla, MN, eſt vt, Δ V, ad portionem, RFV, quod oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s5971" xml:space="preserve"/> </p> <div xml:id="echoid-div586" type="float" level="2" n="1"> <note position="right" xlink:label="note-0261-02" xlink:href="note-0261-02a" xml:space="preserve">5. Lib. 2.</note> <note position="right" xlink:label="note-0261-03" xlink:href="note-0261-03a" xml:space="preserve">Corol. 21. <lb/>huius.</note> <figure xlink:label="fig-0262-01" xlink:href="fig-0262-01a"> <image file="0262-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0262-01"/> </figure> <note position="left" xlink:label="note-0262-01" xlink:href="note-0262-01a" xml:space="preserve">Coroll. 1. <lb/>26. l. 2.</note> <note position="left" xlink:label="note-0262-02" xlink:href="note-0262-02a" xml:space="preserve">@4. l. 2.</note> <note position="left" xlink:label="note-0262-03" xlink:href="note-0262-03a" xml:space="preserve">@. l. 2.</note> </div> </div> <div xml:id="echoid-div588" type="section" level="1" n="344"> <head xml:id="echoid-head361" xml:space="preserve">THEOREMA XXIII. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s5972" xml:space="preserve">EXpoſita denuò figura circuli Th. </s> <s xml:id="echoid-s5973" xml:space="preserve">21. </s> <s xml:id="echoid-s5974" xml:space="preserve">oſtendendum eſt <lb/>omnia quadrata portionis minoris, RFV, vtcunque <lb/>ſumptæ regula diametro. </s> <s xml:id="echoid-s5975" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5976" xml:space="preserve">FM, ad omnia quadrata eiuſ-<lb/>dem regula baſi. </s> <s xml:id="echoid-s5977" xml:space="preserve">ſ. </s> <s xml:id="echoid-s5978" xml:space="preserve">RV, eſſe vt rectangulum ſub, M, & </s> <s xml:id="echoid-s5979" xml:space="preserve">ſub <lb/>baſi, RV, ad tria quadrata lineæ, RM, cum quad. </s> <s xml:id="echoid-s5980" xml:space="preserve">MF.</s> <s xml:id="echoid-s5981" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5982" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s5983" xml:space="preserve">n. </s> <s xml:id="echoid-s5984" xml:space="preserve">quadrata portionis, RFV, regula, FM, ad omnia <lb/> <anchor type="note" xlink:label="note-0262-04a" xlink:href="note-0262-04"/> quadrata, Δ V, regula eadem, ſunt vt, ω M, ad ſexquialteram, F <lb/>M, .</s> <s xml:id="echoid-s5985" xml:space="preserve">i. </s> <s xml:id="echoid-s5986" xml:space="preserve">vt, {2/3}, M ω, ad, FM; </s> <s xml:id="echoid-s5987" xml:space="preserve">omnia item quadrata, Δ V, regula, <pb o="243" file="0263" n="263" rhead="LIBER III."/> F M, ad omnia quadrata eiuſdem parallelogrammi; </s> <s xml:id="echoid-s5988" xml:space="preserve">Δ V, regula, <lb/> <anchor type="note" xlink:label="note-0263-01a" xlink:href="note-0263-01"/> R V, ſunt vt, FM, ad, RV, ergo ex æquali omnia quadrata por-<lb/>tionis, RFV, regula, FM, ad omnia quadrata, Δ V, regula, R <lb/>V, erunt vt, {2/3}, ω M, ad, RV, vel vt, {1/3}, ω M, ad, RM, .</s> <s xml:id="echoid-s5989" xml:space="preserve">i. </s> <s xml:id="echoid-s5990" xml:space="preserve">ſum-<lb/>pta, RM, communi altitudine, vt rectangulum ſub, {1/3}, ω M, & </s> <s xml:id="echoid-s5991" xml:space="preserve"><lb/>ſub, RM, ad quadratum, RM, vel ad rectangulum, FMH; <lb/></s> <s xml:id="echoid-s5992" xml:space="preserve">omnia vero quadrata, Δ V, regula, RV, ad omnia quadrata por-<lb/> <anchor type="note" xlink:label="note-0263-02a" xlink:href="note-0263-02"/> tionis, RFV, regula eadem, runt vt, HM, ad compoſitam ex, <lb/> <anchor type="note" xlink:label="note-0263-03a" xlink:href="note-0263-03"/> {1/2}, HM, &</s> <s xml:id="echoid-s5993" xml:space="preserve">, {1/6}, MF, .</s> <s xml:id="echoid-s5994" xml:space="preserve">i. </s> <s xml:id="echoid-s5995" xml:space="preserve">ſumpta, MF, communi altitudine, vt re-<lb/> <anchor type="note" xlink:label="note-0263-04a" xlink:href="note-0263-04"/> ctangulum, FMH, ad rectangulum ſub, FM, & </s> <s xml:id="echoid-s5996" xml:space="preserve">ſub compoſita ex, <lb/>{1/2}, HM, &</s> <s xml:id="echoid-s5997" xml:space="preserve">, {1/6}, MF, erant autem omnia quadrata portionis, RFV, <lb/>regula, FM, ad omnia quadrata, Δ V, regula, RV, vt rectangulum <lb/>ſub, {1/3}, M ω, & </s> <s xml:id="echoid-s5998" xml:space="preserve">ſub, RM, ad rectangulum, FMH, ergo ex æquali <lb/>omnia quadrata portionis, RFV, regula, FM, ad omnia quadrata <lb/>eiuſdem, regula, RV, erunt vt rectangulum ſub, {1/3}, M ω, & </s> <s xml:id="echoid-s5999" xml:space="preserve">ſub, R <lb/>M, ad rectangulum ſub, FM, & </s> <s xml:id="echoid-s6000" xml:space="preserve">ſub compoſita ex, {1/2}, HM, &</s> <s xml:id="echoid-s6001" xml:space="preserve">, {1/6}, <lb/>M F, .</s> <s xml:id="echoid-s6002" xml:space="preserve">i. </s> <s xml:id="echoid-s6003" xml:space="preserve">vt rectangulum ſub tota, M ω, & </s> <s xml:id="echoid-s6004" xml:space="preserve">ſub, RM, ad rectangu-<lb/>lum ſub, FM, & </s> <s xml:id="echoid-s6005" xml:space="preserve">ſub compoſita ex, {1/2}, FM, & </s> <s xml:id="echoid-s6006" xml:space="preserve">ſexquialtera, MH, <lb/>.</s> <s xml:id="echoid-s6007" xml:space="preserve">i. </s> <s xml:id="echoid-s6008" xml:space="preserve">& </s> <s xml:id="echoid-s6009" xml:space="preserve">ſub compoſita ex, {1/2}, FM, & </s> <s xml:id="echoid-s6010" xml:space="preserve">ſexquialtera, MI, & </s> <s xml:id="echoid-s6011" xml:space="preserve">ſexquial-<lb/>tera, IH, porrò ſexquialtera, IH, cum, {1/2}, FM, efficit duas, FM, <lb/>I H, quibus ſi iunxeris, MI, detractam de ſexquialtera ipſius, MI, <lb/>fiet tota, FH, cum, MN, æqualis dimidio, FM, & </s> <s xml:id="echoid-s6012" xml:space="preserve">ſexquialteræ, <lb/>M H: </s> <s xml:id="echoid-s6013" xml:space="preserve">Omnia ergo quadrata portionis, RFV, regula, FM, ad om-<lb/>nia quadrata eiuſdem portionis, regula, RV, erunt vt rectangulum <lb/>ſub, M ω, & </s> <s xml:id="echoid-s6014" xml:space="preserve">ſub, RM, ad rectangulum ſub, FM, & </s> <s xml:id="echoid-s6015" xml:space="preserve">ſub compoſi-<lb/>ta ex, FH, MN, .</s> <s xml:id="echoid-s6016" xml:space="preserve">i. </s> <s xml:id="echoid-s6017" xml:space="preserve">ad rectangulum ſub, FM, & </s> <s xml:id="echoid-s6018" xml:space="preserve">ſub, MN, ſub, <lb/> <anchor type="note" xlink:label="note-0263-05a" xlink:href="note-0263-05"/> F M, & </s> <s xml:id="echoid-s6019" xml:space="preserve">ſub, MH, & </s> <s xml:id="echoid-s6020" xml:space="preserve">ad quadratum, FM: </s> <s xml:id="echoid-s6021" xml:space="preserve">quia verò rectangulum, <lb/>F MH, æquatur quadrato, RM, erunt omnia illa quadrata, vt re-<lb/>ctangulum ſub, ω M, & </s> <s xml:id="echoid-s6022" xml:space="preserve">ſub, RM, ad quadratum, RM, quadra-<lb/>tum, MF, & </s> <s xml:id="echoid-s6023" xml:space="preserve">rectangulum ſub, FM, MN, vel vt iſtorum dupla. </s> <s xml:id="echoid-s6024" xml:space="preserve">ſ. <lb/></s> <s xml:id="echoid-s6025" xml:space="preserve">vt rectangulum ſub, ω M, & </s> <s xml:id="echoid-s6026" xml:space="preserve">ſub, RV, ad quadratum, RM, qua-<lb/>dratum, MV, duo quadrata, FM, & </s> <s xml:id="echoid-s6027" xml:space="preserve">duo rectangula ſub, FM, M <lb/>N, .</s> <s xml:id="echoid-s6028" xml:space="preserve">i. </s> <s xml:id="echoid-s6029" xml:space="preserve">vnum ſub, FM, MI, cui ſi iunxeris vnum de duobus quadra-<lb/> <anchor type="note" xlink:label="note-0263-06a" xlink:href="note-0263-06"/> tis ipſius, FM, componetur rectangulum, FMH, quod eſt æqua-<lb/>le quadrato, RM. </s> <s xml:id="echoid-s6030" xml:space="preserve">Sunt ergo omnia quadrata portionis, RFV, re-<lb/> <anchor type="note" xlink:label="note-0263-07a" xlink:href="note-0263-07"/> gula, FM, ad omnia quadrata eiuſdem portionis, regula, RV, vt <lb/>rectangulum ſub, ω M, & </s> <s xml:id="echoid-s6031" xml:space="preserve">ſub, RV, ad tria quadrata, R, M cum <lb/>vno quadrato, FM, quod oſtendere oportebat.</s> <s xml:id="echoid-s6032" xml:space="preserve"/> </p> <div xml:id="echoid-div588" type="float" level="2" n="1"> <note position="left" xlink:label="note-0262-04" xlink:href="note-0262-04a" xml:space="preserve">ex antec.</note> <note position="right" xlink:label="note-0263-01" xlink:href="note-0263-01a" xml:space="preserve">29. Lib. 2.</note> <note position="right" xlink:label="note-0263-02" xlink:href="note-0263-02a" xml:space="preserve">Vlt. 2. El.</note> <note position="right" xlink:label="note-0263-03" xlink:href="note-0263-03a" xml:space="preserve">1. Huius.</note> <note position="right" xlink:label="note-0263-04" xlink:href="note-0263-04a" xml:space="preserve">5. Lib. @</note> <note position="right" xlink:label="note-0263-05" xlink:href="note-0263-05a" xml:space="preserve">Ex vlt. 2. <lb/>Elem.</note> <note position="right" xlink:label="note-0263-06" xlink:href="note-0263-06a" xml:space="preserve">12. Elem.</note> <note position="right" xlink:label="note-0263-07" xlink:href="note-0263-07a" xml:space="preserve">Vlt. 2. El.</note> </div> <pb o="244" file="0264" n="264" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div590" type="section" level="1" n="345"> <head xml:id="echoid-head362" xml:space="preserve">THEOREMA XXIV. PROPOS. XXV.</head> <p> <s xml:id="echoid-s6033" xml:space="preserve">IN figura circuli, & </s> <s xml:id="echoid-s6034" xml:space="preserve">ellipſis eiuſdem Theor. </s> <s xml:id="echoid-s6035" xml:space="preserve">21. </s> <s xml:id="echoid-s6036" xml:space="preserve">oſtenden-<lb/>dum eſt, ibi appoſitis retentis, ſumpta tamen vtcunque <lb/>portione minori, RFV, & </s> <s xml:id="echoid-s6037" xml:space="preserve">regula diametro eiuſdem portio-<lb/>nis. </s> <s xml:id="echoid-s6038" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6039" xml:space="preserve">FM; </s> <s xml:id="echoid-s6040" xml:space="preserve">omnia quadrata parallelogrammi, Δ V, ad omnia <lb/>quadrata portionis, RFV, eſſe vt quadratum, FM, ad ſpa-<lb/>tium, quod remanet, dempto rectangulo ſub, IM, & </s> <s xml:id="echoid-s6041" xml:space="preserve">ſub, F <lb/>M, (ad quam, FM, ſit, vt, Δ V, ad portionem, RFV,) à re-<lb/>ctangulo ſub, FM, & </s> <s xml:id="echoid-s6042" xml:space="preserve">ſub, {2/3}, ipſius, MH.</s> <s xml:id="echoid-s6043" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6044" xml:space="preserve">Sit igitur vt, Δ V, ad, RFV, ita, FM, ad, MI; </s> <s xml:id="echoid-s6045" xml:space="preserve">omnia ergo qua-<lb/>drata, Δ V, adrectangula ſub, Δ V, VT, ſunt vt quadratum, FM, <lb/>ad rectangulum, FMI, rectangula inſuper ſub, Δ V, VT, ad re-<lb/> <anchor type="note" xlink:label="note-0264-01a" xlink:href="note-0264-01"/> ctangula ſub portione, RFV, & </s> <s xml:id="echoid-s6046" xml:space="preserve">ſub, VT, ſunt vt, Δ V, ad portionem, <lb/> <anchor type="figure" xlink:label="fig-0264-01a" xlink:href="fig-0264-01"/> RFV, .</s> <s xml:id="echoid-s6047" xml:space="preserve">i. </s> <s xml:id="echoid-s6048" xml:space="preserve">vt, FM, ad, MT, .</s> <s xml:id="echoid-s6049" xml:space="preserve">i. </s> <s xml:id="echoid-s6050" xml:space="preserve">ſumpta, MI, <lb/> <anchor type="note" xlink:label="note-0264-02a" xlink:href="note-0264-02"/> communi altitudine, vt rectangulum, F <lb/>MI, ad rectangulum, r MI, ergo ex æ-<lb/> <anchor type="note" xlink:label="note-0264-03a" xlink:href="note-0264-03"/> quali omnia quadrata, Δ V, ad rectangu-<lb/>la ſub portione, RFV, & </s> <s xml:id="echoid-s6051" xml:space="preserve">ſub, VT, erunt <lb/>vt quadratum, FM, ad rectangulum, <lb/>Γ Μ Ι, quod ſerua.</s> <s xml:id="echoid-s6052" xml:space="preserve"/> </p> <div xml:id="echoid-div590" type="float" level="2" n="1"> <note position="left" xlink:label="note-0264-01" xlink:href="note-0264-01a" xml:space="preserve">14. Lib. 2.</note> <figure xlink:label="fig-0264-01" xlink:href="fig-0264-01a"> <image file="0264-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0264-01"/> </figure> <note position="left" xlink:label="note-0264-02" xlink:href="note-0264-02a" xml:space="preserve">Coroll. 1. <lb/>26. lib. 2.</note> <note position="left" xlink:label="note-0264-03" xlink:href="note-0264-03a" xml:space="preserve">@. Lib. 2.</note> </div> <p> <s xml:id="echoid-s6053" xml:space="preserve">Vlterius omnia quadrata, ΔV, ad om-<lb/> <anchor type="note" xlink:label="note-0264-04a" xlink:href="note-0264-04"/> nia quadrata, VII, ſunt vt quadratum, F <lb/>M, ad quadratum, MO, vel ad quadra-<lb/>tum, RV, .</s> <s xml:id="echoid-s6054" xml:space="preserve">i. </s> <s xml:id="echoid-s6055" xml:space="preserve">ad quatuor rectangula ſub, <lb/>R M, MV: </s> <s xml:id="echoid-s6056" xml:space="preserve">Omnia inſuper quadrata, <lb/>V II, ad rectangula ſub portione, RFV, <lb/>& </s> <s xml:id="echoid-s6057" xml:space="preserve">quadrilineo, RTHY V, ſunt vt ſex <lb/>quadrata, CE, ad quadratũ, FH, nam in <lb/>circulo omnia quadrata, RZ, ſunt ſex-<lb/> <anchor type="note" xlink:label="note-0264-05a" xlink:href="note-0264-05"/> cupla rectangulorum ſub portione, RF <lb/>V, & </s> <s xml:id="echoid-s6058" xml:space="preserve">quadrilineo, RTHYV, & </s> <s xml:id="echoid-s6059" xml:space="preserve">ideo ſunt <lb/>ad illa, vt ſex quadrata, CE, ad quadra-<lb/>tum, CE, vel ad quadratum, FH, in ellipſi <lb/>verò omnia quadrata, RZ, ſunt ad re-<lb/>ctangula ſub portione, RFV, & </s> <s xml:id="echoid-s6060" xml:space="preserve">quadrilineo, RTHYV, vt ſex quadra-<lb/>ta, CE, ad quadratum, FH, quod elicitur ex Prop. </s> <s xml:id="echoid-s6061" xml:space="preserve">21. </s> <s xml:id="echoid-s6062" xml:space="preserve">huius. </s> <s xml:id="echoid-s6063" xml:space="preserve">Quia <lb/>vero rectangulum, RMV, ad rectangulum, FMH, (tum in cir-<lb/> <anchor type="note" xlink:label="note-0264-06a" xlink:href="note-0264-06"/> <pb o="245" file="0265" n="265" rhead="LIBER III."/> culo, tum in ellipſi) eſt vt quadratum, CN, ad quadratum, NF, vel <lb/>vt quadratum, CE, ad quadratum, FH, ideò ſex rectangula, RMV, <lb/>ad rectangulum, FMH, erunt vt ſex quadrata, CE, ad vnum qua-<lb/>dra um, FH, .</s> <s xml:id="echoid-s6064" xml:space="preserve">i. </s> <s xml:id="echoid-s6065" xml:space="preserve">erunt vt omnia quadrata, RZ, ad rectangula ſub <lb/>portione, RFV, & </s> <s xml:id="echoid-s6066" xml:space="preserve">quadrilineo, RTHY V, vt autem ſunt ſex re-<lb/>ctangula, RMV, ad rectangulum, FMH, ita quatuor rectangu-<lb/>la, RMV, ad, {2/3}, rectanguli, FMH, .</s> <s xml:id="echoid-s6067" xml:space="preserve">i. </s> <s xml:id="echoid-s6068" xml:space="preserve">ad rectangulum ſub, FM, <lb/>&</s> <s xml:id="echoid-s6069" xml:space="preserve">, {2/3}, MH, ergo omnia quadrata, RZ, ad rectangula ſub portio-<lb/>ne, RFV, & </s> <s xml:id="echoid-s6070" xml:space="preserve">quadrilineo, RTHY V, erunt vt quatuor rectangu-<lb/>la, RMV, ad rectangulum ſub, FM, &</s> <s xml:id="echoid-s6071" xml:space="preserve">, {2/3}, MH, erant autem om-<lb/>nia quadrata, Δ V, ad omnia quadrata, RZ, vt quadratum, FM, <lb/>ad quatuor rectangula ſub, RMV, ergo ex æquali omnia quadra-<lb/>ta, Δ V, ad rectangula ſub portione, RFV, & </s> <s xml:id="echoid-s6072" xml:space="preserve">quadrilineo, RTH <lb/>YV, erunt vt quadratum, FM, ad rectangulum ſub, FM, & </s> <s xml:id="echoid-s6073" xml:space="preserve">ſub, {2/3}, MH, <lb/>eadem verò omnia quadrata, Δ V, ad rectangula ſub portione, R <lb/>F V, & </s> <s xml:id="echoid-s6074" xml:space="preserve">ſub, VT, oſtenſa ſunt eſſe, vt quadratum, FM, ad rectan-<lb/>gulum, ΓΜΙ, (ex quibus habemus rectangulum ſub, ΓΜΙ, mi-<lb/>nus eſſe rectangulo ſub, FM, & </s> <s xml:id="echoid-s6075" xml:space="preserve">ſub, {2/3}, MH, nam rectangula ſub <lb/>portione, RFV, & </s> <s xml:id="echoid-s6076" xml:space="preserve">ſub, VT, minora ſunt rectangulis ſub eadem <lb/>portione, RFV, & </s> <s xml:id="echoid-s6077" xml:space="preserve">quadrilineo, RTHY V,) ergo omnia quadra-<lb/>ta, Δ V, ad reſiduum omnium rectangulorum ſub portione, RFV, <lb/>& </s> <s xml:id="echoid-s6078" xml:space="preserve">quadrilineo, RTHY V, demptis rectangulis ſub portione, RF <lb/>V, & </s> <s xml:id="echoid-s6079" xml:space="preserve">ſub, VT, .</s> <s xml:id="echoid-s6080" xml:space="preserve">i. </s> <s xml:id="echoid-s6081" xml:space="preserve">ad rectangula ſub vtriſq; </s> <s xml:id="echoid-s6082" xml:space="preserve">portionibus, RFV, THY, <lb/>.</s> <s xml:id="echoid-s6083" xml:space="preserve">i. </s> <s xml:id="echoid-s6084" xml:space="preserve">ad omnia quadrata portionis, RFV, erunt vt quadratum, FM, <lb/>ad reſiduum ſpatium, dempto rectangulo, ΓΜΙ, a rectangulo ſub, <lb/>F M, & </s> <s xml:id="echoid-s6085" xml:space="preserve">ſub, {2/3}, MH, (hoc autem vocetur reſiduum rectangulum <lb/>huius Theor.) </s> <s xml:id="echoid-s6086" xml:space="preserve">quod oſtendere opus erat.</s> <s xml:id="echoid-s6087" xml:space="preserve"/> </p> <div xml:id="echoid-div591" type="float" level="2" n="2"> <note position="right" xlink:label="note-0264-04" xlink:href="note-0264-04a" xml:space="preserve">@. Lib. 2.</note> <note position="left" xlink:label="note-0264-05" xlink:href="note-0264-05a" xml:space="preserve">Elicitur <lb/>èx 21.hu-<lb/>ius.</note> <note position="left" xlink:label="note-0264-06" xlink:href="note-0264-06a" xml:space="preserve">17. 3. Con.</note> </div> </div> <div xml:id="echoid-div593" type="section" level="1" n="346"> <head xml:id="echoid-head363" xml:space="preserve">THEOREMA XXV. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s6088" xml:space="preserve">EXpoſita adhuc figura Theor. </s> <s xml:id="echoid-s6089" xml:space="preserve">antecedentis, oſtendemus <lb/>omnia quadrata portionis, RFV, regula, FM, ad om-<lb/>nia quadrata eiuſdem portionis, regula baſi, eſſe vt paralle-<lb/>lepipedum ſub b ſireſiduo rectangulo antecedentis Theor-<lb/>altitudine tripla, MH, ad parallelepipedum ſub baſi rectan-<lb/>gulo ipſius, FM, ductæ in, RV, altitudine linea compoſita <lb/>ex, MH, HN.</s> <s xml:id="echoid-s6090" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6091" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s6092" xml:space="preserve">n. </s> <s xml:id="echoid-s6093" xml:space="preserve">quadrata portionis, RFV, regula, FM, ad omnia <lb/>quacrata eluidem, regula, RV, habent rationem compoſitam ex <lb/> <anchor type="note" xlink:label="note-0265-01a" xlink:href="note-0265-01"/> ea, quam habent omnia quadrata, RFV, ad omnia quadrata, Δ <pb o="246" file="0266" n="266" rhead="GEOMETRIÆ"/> V. </s> <s xml:id="echoid-s6094" xml:space="preserve">regula, FM, .</s> <s xml:id="echoid-s6095" xml:space="preserve">i. </s> <s xml:id="echoid-s6096" xml:space="preserve">ex ea, quam habet reſiduum rectangulum Theor. <lb/></s> <s xml:id="echoid-s6097" xml:space="preserve">antecedentis ad quadratum, FM, & </s> <s xml:id="echoid-s6098" xml:space="preserve">ex ratione omnium quadrato-<lb/> <anchor type="note" xlink:label="note-0266-01a" xlink:href="note-0266-01"/> rum, Δ V, regula, FM, ad omnia quadrata eiuſdem, Δ V, regula, <lb/>R V, .</s> <s xml:id="echoid-s6099" xml:space="preserve">i. </s> <s xml:id="echoid-s6100" xml:space="preserve">ex ea, quam habet, Δ R, ad, RV, vel, ſumpta, Δ R, com-<lb/> <anchor type="note" xlink:label="note-0266-02a" xlink:href="note-0266-02"/> muni altitudine ex ea, quam habet quadratum, Δ R, vel quadra-<lb/> <anchor type="figure" xlink:label="fig-0266-01a" xlink:href="fig-0266-01"/> tum, FM, ad rectangulum ſub, FM, <lb/>R V; </s> <s xml:id="echoid-s6101" xml:space="preserve">& </s> <s xml:id="echoid-s6102" xml:space="preserve">tandem ex ea, quam habent <lb/> <anchor type="note" xlink:label="note-0266-03a" xlink:href="note-0266-03"/> omnia quadrata, Δ V, ad omnia qua-<lb/>drata portionis, RFV, .</s> <s xml:id="echoid-s6103" xml:space="preserve">i. </s> <s xml:id="echoid-s6104" xml:space="preserve">ex ea, quam <lb/>habet, MH, ad compoſitam ex, {1/2}, M <lb/>H, &</s> <s xml:id="echoid-s6105" xml:space="preserve">, {1/6}, FM. </s> <s xml:id="echoid-s6106" xml:space="preserve">Rationes autem re-<lb/>ctanguli reſidui Theor.</s> <s xml:id="echoid-s6107" xml:space="preserve">antecedentis ad <lb/> <anchor type="note" xlink:label="note-0266-04a" xlink:href="note-0266-04"/> quadratum, FM, & </s> <s xml:id="echoid-s6108" xml:space="preserve">quadrati, FM, <lb/>ad rectangulum ſub, FM, RV, re-<lb/>ſoluuntur in rationem rectanguli reſi-<lb/>dui Theor. </s> <s xml:id="echoid-s6109" xml:space="preserve">antecedentis ad rectangu-<lb/>lum ſub, FM, RV, quę iuncta rationi <lb/>ipſius, MH, ad compoſitam ex, {1/2}, M <lb/> <anchor type="note" xlink:label="note-0266-05a" xlink:href="note-0266-05"/> H, &</s> <s xml:id="echoid-s6110" xml:space="preserve">, {1/6}, FM, cõponit rationem paral-<lb/>lelepipedi ſub baſi reſiduo rectangulo <lb/>Theor. </s> <s xml:id="echoid-s6111" xml:space="preserve">antecedentis, altitudine, MH, <lb/>ad parallelepipedum ſub baſi rectan-<lb/>gulo ſub, FM, RV, & </s> <s xml:id="echoid-s6112" xml:space="preserve">ſub compoſita <lb/>ex, {1/2}, MH, &</s> <s xml:id="echoid-s6113" xml:space="preserve">, {1/6}, FM: </s> <s xml:id="echoid-s6114" xml:space="preserve">Triplicentur <lb/>horum parallelepipedoru altitudines, <lb/>ſiet pro an ecedentis altitudine tripla, <lb/>M H, & </s> <s xml:id="echoid-s6115" xml:space="preserve">pro altitudine parallelepipedi conſequentis tripla dimidiæ, <lb/>M H,.</s> <s xml:id="echoid-s6116" xml:space="preserve">. ſexquialtera ipſius, MH, . </s> <s xml:id="echoid-s6117" xml:space="preserve">. </s> <s xml:id="echoid-s6118" xml:space="preserve">ſexquialtera, MI, & </s> <s xml:id="echoid-s6119" xml:space="preserve">ſexquial <lb/>tera, IH, cum, {1/2}, FM, porro ſi ſexquialterę, MI, iunxeris ſexqui-<lb/>alteram, IH, cum dimidia, FM, .</s> <s xml:id="echoid-s6120" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6121" xml:space="preserve">duplam, IH, quoniam ſex-<lb/>quialtera, IH, eſt, MI, IN, ſi inquam illi iunxeris bis, IH, com-<lb/>ponetur altitudo conſequentis parailelepipedi, quę erit, MH, HN; <lb/></s> <s xml:id="echoid-s6122" xml:space="preserve">omnia ergo quadrata portionis, RFV, regula, FM, ad omnia qua-<lb/>drata eiuſdem, regula, RV, erunt vt parallelepipedum ſub bafi re-<lb/>ſiduo rectangulo Theor. </s> <s xml:id="echoid-s6123" xml:space="preserve">antecedentis, altitudine tripla, MH, ad <lb/>parallelepipedum ſub baſi rectangulo, ſub, FM, RV, altitudine li-<lb/>nea compoſita ex, MH, HN, tum in circuli, tum ellipſis figura, <lb/>quod oſtendere oportebat.</s> <s xml:id="echoid-s6124" xml:space="preserve"/> </p> <div xml:id="echoid-div593" type="float" level="2" n="1"> <note position="right" xlink:label="note-0265-01" xlink:href="note-0265-01a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">Ex antec.</note> <note position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">29. l. 2.</note> <figure xlink:label="fig-0266-01" xlink:href="fig-0266-01a"> <image file="0266-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0266-01"/> </figure> <note position="left" xlink:label="note-0266-03" xlink:href="note-0266-03a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-0266-04" xlink:href="note-0266-04a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> <note position="left" xlink:label="note-0266-05" xlink:href="note-0266-05a" xml:space="preserve">G. Cor. 4. <lb/>gen. 34. <lb/>l. 2.</note> </div> <pb o="247" file="0267" n="267" rhead="LIBER III."/> </div> <div xml:id="echoid-div595" type="section" level="1" n="347"> <head xml:id="echoid-head364" xml:space="preserve">THEOREMA XXVI. PROPOS. XXVII.</head> <p> <s xml:id="echoid-s6125" xml:space="preserve">ADhucetiam exponatur figura circuli, & </s> <s xml:id="echoid-s6126" xml:space="preserve">ellipſis Theor. <lb/></s> <s xml:id="echoid-s6127" xml:space="preserve">21. </s> <s xml:id="echoid-s6128" xml:space="preserve">oſtendemus, .</s> <s xml:id="echoid-s6129" xml:space="preserve">n. </s> <s xml:id="echoid-s6130" xml:space="preserve">omnia quadrata figuræ, LCFE <lb/>G, demptis omnibus quadratis trilineorum, CLT, YGE, <lb/>regula, FI, ad omnia quadrata portionis, TCFEY, regu-<lb/>la baſi, TY, eſſe, in circulo, vt cylindricus ſub, IM, & </s> <s xml:id="echoid-s6131" xml:space="preserve">por-<lb/>tione, TCFE Y, vna cum, {1/@}, cubi, TY, ad parallelepipe-<lb/>dum ſub altitudine, FI, baſi verò rectangulo ſub, FI, & </s> <s xml:id="echoid-s6132" xml:space="preserve"><lb/>ſexquitertia duarum, IH, HN. </s> <s xml:id="echoid-s6133" xml:space="preserve">In ellipſi verò habere ra-<lb/>tionem compoſitam ex ea, quam habet cylindricus ſub, IM, <lb/>& </s> <s xml:id="echoid-s6134" xml:space="preserve">portione, TCFE Y, vna cum ea parte cubi, TY, vel pa-<lb/>rallelepipedi ſub, RV, & </s> <s xml:id="echoid-s6135" xml:space="preserve">rhombo, RZ, ad quam eiuſdem <lb/>cubi, vel parallelepipedi ſexta pars fit, vt quadratum, CE, <lb/>ad quadratum, FH, ad parallclepipedum ſub altitudine, L <lb/>G, baſi parallelogrammo, AG; </s> <s xml:id="echoid-s6136" xml:space="preserve">& </s> <s xml:id="echoid-s6137" xml:space="preserve">ex ea, quam habet qua-<lb/>dratum, FH, ad rectangulum ſub, FI, & </s> <s xml:id="echoid-s6138" xml:space="preserve">ſub ſexquitertia <lb/>duarum, IH, HN.</s> <s xml:id="echoid-s6139" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6140" xml:space="preserve">Omnia quadrata namq; </s> <s xml:id="echoid-s6141" xml:space="preserve">figuræ, LCFG, demptis omnibus qua-<lb/> <anchor type="note" xlink:label="note-0267-01a" xlink:href="note-0267-01"/> dratis trilineorum, CLT, YGE, oſtenſa ſunt eſſe ad omnia qua-<lb/>drata, AG, regula, FI, vt cylindricum ſub, MI, & </s> <s xml:id="echoid-s6142" xml:space="preserve">ſub baſi por-<lb/>tione, TCFE Y, vna cum, {1/6}, cubi, TY, in circulo (in ellipſi ve-<lb/>rò vna cum ea parte cubi, TY, vel parallelepipedi ſub, RV, & </s> <s xml:id="echoid-s6143" xml:space="preserve"><lb/>rhombo, RZ, ad quam eiuſdem, {1/6}, ſit vt quadratum, CE, ad <lb/>quadratum, FH,) ad parallelepipedum ſub, LA, & </s> <s xml:id="echoid-s6144" xml:space="preserve">parallelo-<lb/>grammo, AG. </s> <s xml:id="echoid-s6145" xml:space="preserve">Vlterius omnia quadrata, AG, regula, FI, ad <lb/> <anchor type="note" xlink:label="note-0267-02a" xlink:href="note-0267-02"/> omnia quadrata eiuſdem, AG, regula, LG, ſunt vt, AL, ad, L <lb/>G, .</s> <s xml:id="echoid-s6146" xml:space="preserve">i. </s> <s xml:id="echoid-s6147" xml:space="preserve">vt parallelepipedum ſub, AL, & </s> <s xml:id="echoid-s6148" xml:space="preserve">parallelogrammo, ALG, <lb/>ad parallelepipedum ſub, LG, & </s> <s xml:id="echoid-s6149" xml:space="preserve">parallelogrammo eodem; </s> <s xml:id="echoid-s6150" xml:space="preserve">AL <lb/>G, ergo ex æquali omnia quadrata figuræ, LCFEG, demptis <lb/>omnibus quadratis trilineorum, CLT, YGE, regula, FI, ad om-<lb/>nia quadrata, AG, regula, TY, erunt vt cylindricus ſub, MI, & </s> <s xml:id="echoid-s6151" xml:space="preserve"><lb/>portione, TCFEY, vna cum, {1/6}, cubi, TY, in circulo, in ellipſi <lb/>verò vna cum dicta parte cubi, TY, vel parallelepipedi ſub, RV, <lb/>& </s> <s xml:id="echoid-s6152" xml:space="preserve">rhombo, RZ, ad parallelepipedum ſub, LG, & </s> <s xml:id="echoid-s6153" xml:space="preserve">parallelogram-<lb/>mo, ALG.</s> <s xml:id="echoid-s6154" xml:space="preserve"/> </p> <div xml:id="echoid-div595" type="float" level="2" n="1"> <note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">22. huius.</note> <note position="right" xlink:label="note-0267-02" xlink:href="note-0267-02a" xml:space="preserve">29. l. 2.</note> </div> <p> <s xml:id="echoid-s6155" xml:space="preserve">Tandem omnia quadrata, AG, ad omnia quadrata portionis, <lb/> <anchor type="note" xlink:label="note-0267-03a" xlink:href="note-0267-03"/> <pb o="248" file="0268" n="268" rhead="GEOMETRIÆ"/> T CFEY, regula, TY, ſunt vt rectangulum ſub, FN, & </s> <s xml:id="echoid-s6156" xml:space="preserve">tripla, <lb/>N H, .</s> <s xml:id="echoid-s6157" xml:space="preserve">i. </s> <s xml:id="echoid-s6158" xml:space="preserve">vt, {3/4}, quadrati, FH, ad rectangulum ſub, FI, & </s> <s xml:id="echoid-s6159" xml:space="preserve">ſub, I <lb/>H N, .</s> <s xml:id="echoid-s6160" xml:space="preserve">i. </s> <s xml:id="echoid-s6161" xml:space="preserve">vt totum quadratum, FH, ad rectangulum ſub, FI, & </s> <s xml:id="echoid-s6162" xml:space="preserve"><lb/>ſub ſexquitertia ipſarum, IHN, .</s> <s xml:id="echoid-s6163" xml:space="preserve">i. </s> <s xml:id="echoid-s6164" xml:space="preserve">in circulo, vt quadratum, AP, <lb/> <anchor type="figure" xlink:label="fig-0268-01a" xlink:href="fig-0268-01"/> (quod æquatur quadrato, FH, ) ad <lb/>idem rectangulum ideſt ſumpta, FI, <lb/>communi altitudine, vt parallelepipe-<lb/>dum ſub, FI, & </s> <s xml:id="echoid-s6165" xml:space="preserve">quadrato, AP, .</s> <s xml:id="echoid-s6166" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s6167" xml:space="preserve">vt parallelepipedum ſub, AP. </s> <s xml:id="echoid-s6168" xml:space="preserve">vel, L <lb/>G, & </s> <s xml:id="echoid-s6169" xml:space="preserve">parallelogrammo rectangulo <lb/>ſub, FI, ſiue, AL, &</s> <s xml:id="echoid-s6170" xml:space="preserve">, LG, ad pa-<lb/>rallelepipedum ſub, FI, & </s> <s xml:id="echoid-s6171" xml:space="preserve">ſub baſi <lb/>rectangulo ſub, FI, & </s> <s xml:id="echoid-s6172" xml:space="preserve">ſub ſexquiter-<lb/>tia, IHN, ergo ex æquali omnia <lb/>quadrata figuræ, LCFE G, dem-<lb/>ptis omnibus quadratis trilineorum, <lb/>CLT, YGE, regula, FI, ad omnia <lb/>quadrata portionis, TCFE Y, regu-<lb/>la, TY, erunt vt cylindricus ſub, M <lb/>I, & </s> <s xml:id="echoid-s6173" xml:space="preserve">ſub portione, TCFE Y, vna <lb/>cum, {1/6}, cubi, TY, ad parallelepipe-<lb/>dum ſub, FI, & </s> <s xml:id="echoid-s6174" xml:space="preserve">ſub rectangulo ſub, <lb/>F I, & </s> <s xml:id="echoid-s6175" xml:space="preserve">ſexquitertia, IHN; </s> <s xml:id="echoid-s6176" xml:space="preserve">& </s> <s xml:id="echoid-s6177" xml:space="preserve">hoc in <lb/>circulo.</s> <s xml:id="echoid-s6178" xml:space="preserve"/> </p> <div xml:id="echoid-div596" type="float" level="2" n="2"> <note position="right" xlink:label="note-0267-03" xlink:href="note-0267-03a" xml:space="preserve">2. huius.</note> <figure xlink:label="fig-0268-01" xlink:href="fig-0268-01a"> <image file="0268-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0268-01"/> </figure> </div> <p> <s xml:id="echoid-s6179" xml:space="preserve">In ellipſi autem eadem habebunt <lb/>rationem compoſitam exiam dicta ratione .</s> <s xml:id="echoid-s6180" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6181" xml:space="preserve">ex ratione cylindrici <lb/>ſub, MI, & </s> <s xml:id="echoid-s6182" xml:space="preserve">ſub portione, TGFE Y, vna cum ea parte cubi, vel <lb/>parallelepipedi ſub, RV, & </s> <s xml:id="echoid-s6183" xml:space="preserve">rhombo, RZ, ad quam eiuſdem, {1/6}, ſit <lb/>vt quadratum, CE, ad quadratum, FH, ad parallelepipedum, <lb/>ſub, LG, & </s> <s xml:id="echoid-s6184" xml:space="preserve">parallelogrammo, AG, & </s> <s xml:id="echoid-s6185" xml:space="preserve">ex ratione quadrati, FH, <lb/>ad rectangulum ſub, FI, & </s> <s xml:id="echoid-s6186" xml:space="preserve">ſub ſexquitertia ipſarum, IHN; </s> <s xml:id="echoid-s6187" xml:space="preserve">quas <lb/>duas rationes in circulo in vna reſoluimus, quia in eo quadratum, <lb/>F H, æquatur quadrato, AP, quod cum in ellipſi non verificetur, <lb/>ideò has duas rationes componentes pro ipſa ellipſi retinuimus; <lb/></s> <s xml:id="echoid-s6188" xml:space="preserve">quod oſtendere oportebat.</s> <s xml:id="echoid-s6189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div598" type="section" level="1" n="348"> <head xml:id="echoid-head365" xml:space="preserve">THEOREMA XXVII. PROPOS. XXVIII.</head> <p> <s xml:id="echoid-s6190" xml:space="preserve">IN eadem ſuperioris figura oſtendemus, tum in circulo, <lb/>tum in ellipſi, omnia quadrata figuræ, LCFEG, dem-<lb/>ptis omnibus quadratis trilineorum, CLT, YGE, regula, <pb o="249" file="0269" n="269" rhead="LIBER III."/> FI, ad omnia quadrata circuli, vel ellipſis, CFEH, eſſe <lb/>vt cylindricum ſub, MI, & </s> <s xml:id="echoid-s6191" xml:space="preserve">portione, TCFE Y, vna cum, <lb/>{1/6}, cubi, TY, pro circulo, pro ellipſi verò, vna cum ſæpius <lb/>dicta parte cubi, TY, vel parallelepipedi ſub, RV, & </s> <s xml:id="echoid-s6192" xml:space="preserve">rhom-<lb/>bo, RZ, ad, {2/3}, parallelepipedi ſub, AD, & </s> <s xml:id="echoid-s6193" xml:space="preserve">parallelogram-<lb/>mo, AQ, ideſt, in circulo ad, {1/6}, cubi, FH.</s> <s xml:id="echoid-s6194" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6195" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s6196" xml:space="preserve">n. </s> <s xml:id="echoid-s6197" xml:space="preserve">quadrata figurę, LCFE G, demptis omnibus quadra-<lb/>tis trilineorum, CLT, YGE, ad omnia quadrata, AG, ſunt vt cy-<lb/>lindricus ſub, MI, & </s> <s xml:id="echoid-s6198" xml:space="preserve">portione, TCFE Y, vna cum, {1/6}, cubi, TY, <lb/> <anchor type="note" xlink:label="note-0269-01a" xlink:href="note-0269-01"/> pro circulo, pro ellipſi verò, vna cum ſæpius dicta parte cubi, TY. <lb/></s> <s xml:id="echoid-s6199" xml:space="preserve">vel dicti parallelepipedi, ad parallelepipedum ſub, LA, & </s> <s xml:id="echoid-s6200" xml:space="preserve">paralle-<lb/>logrammo, AG; </s> <s xml:id="echoid-s6201" xml:space="preserve">omnia verò quadraca, AG, ad omnia quadrata, <lb/>AQ, ſunt vt quadratum, AL, ad quadratum, AD, .</s> <s xml:id="echoid-s6202" xml:space="preserve">i. </s> <s xml:id="echoid-s6203" xml:space="preserve">ſumpta, A <lb/> <anchor type="note" xlink:label="note-0269-02a" xlink:href="note-0269-02"/> P, communi altitudine, vt parallelepipedum ſub, PA, & </s> <s xml:id="echoid-s6204" xml:space="preserve">quadra-<lb/>to, AL, ad parallelepipedum ſub, PA, & </s> <s xml:id="echoid-s6205" xml:space="preserve">quadrato, AD, hoc eſt, <lb/>vt parallelepipedum ſub, LA, & </s> <s xml:id="echoid-s6206" xml:space="preserve">parallelogrammo, AG, ad paral-<lb/>lelepipedum ſub, DA, & </s> <s xml:id="echoid-s6207" xml:space="preserve">parallelogrammo, AQ, omnia autem qua-<lb/>drata, AQ, omnium quadratorum circuli, vel ellipſis, CFEH, ſunt <lb/> <anchor type="note" xlink:label="note-0269-03a" xlink:href="note-0269-03"/> ſexquialtera .</s> <s xml:id="echoid-s6208" xml:space="preserve">i. </s> <s xml:id="echoid-s6209" xml:space="preserve">ſunt ad ea, vt parallelepipedum ſub, AD, & </s> <s xml:id="echoid-s6210" xml:space="preserve">paral-<lb/>lelogrammo, AQ, ad eiuſdem, {2/3}, ergo ex æquali omnia quadrata <lb/>figuræ, LCFE G, demptis ommbus quadratis trilineorum, CLT, <lb/>YGE, ad omnia quadrata circuli, vel ellipſis, CFEH, erunt vt <lb/>cylindricus ſub, MI, & </s> <s xml:id="echoid-s6211" xml:space="preserve">portione, TCFEY, vna cum, {1/6}, cubi, T <lb/>Y, pro circulo, pro ellipſi verò, vna cum ſæpius dicta parte cubi, T <lb/>Y, vel parallelepipedi ſub, RV, & </s> <s xml:id="echoid-s6212" xml:space="preserve">rhombo, RZ, ad, {2/3}, parallele-<lb/>pipedi ſub, AD, & </s> <s xml:id="echoid-s6213" xml:space="preserve">parallelogrammo, AQ, .</s> <s xml:id="echoid-s6214" xml:space="preserve">i. </s> <s xml:id="echoid-s6215" xml:space="preserve">pro circulo ad, {2/3}, <lb/>cubi, AD, vel cubi, FH, quod oſtendere opus erat.</s> <s xml:id="echoid-s6216" xml:space="preserve"/> </p> <div xml:id="echoid-div598" type="float" level="2" n="1"> <note position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">22. huius.</note> <note position="right" xlink:label="note-0269-02" xlink:href="note-0269-02a" xml:space="preserve">9. Lib. 2.</note> <note position="right" xlink:label="note-0269-03" xlink:href="note-0269-03a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div600" type="section" level="1" n="349"> <head xml:id="echoid-head366" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXIX.</head> <p> <s xml:id="echoid-s6217" xml:space="preserve">SIparallelogrammo ſit inſcripta figura quæcunque, ita ta-<lb/>men, vt, ſumpto vno laterum parallelogrammi pro re-<lb/>gula, &</s> <s xml:id="echoid-s6218" xml:space="preserve">, ductis vtcunque ipſiregulæ parallelis intra paralle-<lb/>logrammum, earum quælibet, vel tota ſit intra figuram in-<lb/>ſcriptam, vel eiuſdem aliqua parte extra figuram exiſtente, <lb/>ac ad vnum laterum parallelogrammi terminante, ad latus <lb/>eiuſdem parallelogrammi prædicto oppoſitum terminet alia <lb/>portio eiuſdem, regulæ æquidiſtantis, ſint autem duæ quæ- <pb o="250" file="0270" n="270" rhead="GEOMETRIÆ"/> libet portiones extra figuram ad oppoſita latera terminan-<lb/>tes, & </s> <s xml:id="echoid-s6219" xml:space="preserve">in eadem recta linea conſtitutæ integræ, & </s> <s xml:id="echoid-s6220" xml:space="preserve">inter ſe <lb/>æquales: </s> <s xml:id="echoid-s6221" xml:space="preserve">Omnia quadrata dicti parallelogrammi ad omnia <lb/>quadrata inſcriptæ figuræ, cum rectangulis bis ſub eadem <lb/>figura, & </s> <s xml:id="echoid-s6222" xml:space="preserve">ſub dictarum portionum ijs omnibus, quę extra fi-<lb/>guram ad vnum dictorum laterum oppoſitorum eiuſdem pa-<lb/>rallelogrammi terminantur, erunt vt prædictum parallelo-<lb/>grammum ad inſcriptam figuram.</s> <s xml:id="echoid-s6223" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6224" xml:space="preserve">Sitigitur parallelogrammum, AN, & </s> <s xml:id="echoid-s6225" xml:space="preserve">illi inſcripta vtcunq; </s> <s xml:id="echoid-s6226" xml:space="preserve">figu-<lb/>ra, BDMO, & </s> <s xml:id="echoid-s6227" xml:space="preserve">ſumpta pro regula, EN, ſit ducta vtcunque intra <lb/>parallelogrammum, AN, ipſa, DO, quę cadat etiam tota intra fi-<lb/>guram, BDMO, ſit etiam ducta alia vtcunque parallela ipſi, EN, <lb/>nempè, VR, portiones autem eiuſdem, VR, ſint extra figuram, <lb/>ad latera oppoſita, AE, CN, terminantes .</s> <s xml:id="echoid-s6228" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6229" xml:space="preserve">VI, SR, quæ ſint in-<lb/>tegræ, & </s> <s xml:id="echoid-s6230" xml:space="preserve">inter ſe æquales. </s> <s xml:id="echoid-s6231" xml:space="preserve">Dico omnia quadrata, AN, ad omnia <lb/>quadrata figuræ, BDMO, cum rectangulis bis ſub figuræ, BDM <lb/>O, & </s> <s xml:id="echoid-s6232" xml:space="preserve">ſub trilineis, BCO, ONM, .</s> <s xml:id="echoid-s6233" xml:space="preserve">i. </s> <s xml:id="echoid-s6234" xml:space="preserve">ſub omnibus portionibus, quę <lb/>terminant ad latus, CN, extra figuram, BDMO, conſtitutis, elie <lb/> <anchor type="figure" xlink:label="fig-0270-01a" xlink:href="fig-0270-01"/> vt, AN, ad figuram, BDMO: </s> <s xml:id="echoid-s6235" xml:space="preserve">Omnia <lb/>enim quadrata, AN, ad rectangula ſub, <lb/>A N, & </s> <s xml:id="echoid-s6236" xml:space="preserve">ſub figura, BDMO, ſunt vt, A <lb/>N, ad figuram, BDMO, ſed rectangula <lb/>ſub, AN, & </s> <s xml:id="echoid-s6237" xml:space="preserve">ſub figura, BDMO, diui-<lb/> <anchor type="note" xlink:label="note-0270-01a" xlink:href="note-0270-01"/> duntur in rectangula ſub eadem figura, B <lb/>D MO, & </s> <s xml:id="echoid-s6238" xml:space="preserve">ſub trilineis, BAD, DEM, <lb/>ſub eadem, & </s> <s xml:id="echoid-s6239" xml:space="preserve">ſub trilineis, BCO, ON <lb/>M, & </s> <s xml:id="echoid-s6240" xml:space="preserve">in rectangula ſub eadem in eandem <lb/> <anchor type="note" xlink:label="note-0270-02a" xlink:href="note-0270-02"/> figuram .</s> <s xml:id="echoid-s6241" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6242" xml:space="preserve">in omnia quadrata eiuſdem fi-<lb/>guræ, BDMO, quia verò linearum æqui-<lb/>diſtantium, regulæ, EN, portiones, quæ <lb/>ſunt in eadem recta linea extra figuram adiacentes lateribus oppoſi-<lb/>tis, AE, CN, ſunt & </s> <s xml:id="echoid-s6243" xml:space="preserve">integræ, & </s> <s xml:id="echoid-s6244" xml:space="preserve">æquales, ideò ſicuti rectangu-<lb/>lum, VIS, eſt æquale rectangulo, ISR, ita rectangula ſub figura, <lb/>B DMO, & </s> <s xml:id="echoid-s6245" xml:space="preserve">trilineis, BAD, DEM, erunt æqualia rectangulis <lb/>ſub eadem figura, BDMO, & </s> <s xml:id="echoid-s6246" xml:space="preserve">ſub trilineis, BCO, ONM, ſunt <lb/>ergo rectangula ſub, AN, & </s> <s xml:id="echoid-s6247" xml:space="preserve">ſub figura, BDMO, æqualia om-<lb/>nibus quadratis figuræ, BDMO, cum rectangulis bis ſub eadem, <lb/>& </s> <s xml:id="echoid-s6248" xml:space="preserve">ſub trilineis, BCO, ONM; </s> <s xml:id="echoid-s6249" xml:space="preserve">omnia autem quadrata, AN, ad <lb/>rectangula ſub, AN, & </s> <s xml:id="echoid-s6250" xml:space="preserve">ſub figura, BDMO, ſunt vt, AN, ad fi-<lb/>guram, BDMO; </s> <s xml:id="echoid-s6251" xml:space="preserve">ergo omnia quadrata, AN, ad omnia quadra- <pb o="251" file="0271" n="271" rhead="LIBER III."/> ta figuræ, BDMO, cum rectangulis bis ſub eadem ſigura, & </s> <s xml:id="echoid-s6252" xml:space="preserve">ſub <lb/>trilineis, BCO, ONM, erunt vt, AN, ad figuram, BDMO, <lb/>quod oſtendere opus erat.</s> <s xml:id="echoid-s6253" xml:space="preserve"/> </p> <div xml:id="echoid-div600" type="float" level="2" n="1"> <figure xlink:label="fig-0270-01" xlink:href="fig-0270-01a"> <image file="0270-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0270-01"/> </figure> <note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">Coroll. 1. <lb/>26. lib. 2.</note> <note position="left" xlink:label="note-0270-02" xlink:href="note-0270-02a" xml:space="preserve">A. 23. l. 2.</note> </div> </div> <div xml:id="echoid-div602" type="section" level="1" n="350"> <head xml:id="echoid-head367" xml:space="preserve">THEOREMA XXIX. PROPOS. XXX.</head> <p> <s xml:id="echoid-s6254" xml:space="preserve">EXponatur figura Theor. </s> <s xml:id="echoid-s6255" xml:space="preserve">antecedentis, dimiſſis tamen re-<lb/>ctis lineis, DO, VR, & </s> <s xml:id="echoid-s6256" xml:space="preserve">ſit adhuc regula, EN, produ-<lb/>cantur autem ad eaſdem partes, AC, EN, in, HF, ita vt, C <lb/>H, ſit æqualis, NF, iuncta igitur, HF, erit, HF, parallela <lb/>ipſi, CN, quoniam, CH, NF, ſunt æquales, & </s> <s xml:id="echoid-s6257" xml:space="preserve">parallelæ, <lb/>& </s> <s xml:id="echoid-s6258" xml:space="preserve">erit parallelogrammum, AF, &</s> <s xml:id="echoid-s6259" xml:space="preserve">, CF. </s> <s xml:id="echoid-s6260" xml:space="preserve">Dico ergo omnia <lb/>quadrata, AN, cumrectangulis bis ſub, AN, NH, ad om-<lb/>nia quadrata figuræ, BDMO, cum rectangulis bis ſub ea-<lb/>dem, & </s> <s xml:id="echoid-s6261" xml:space="preserve">ſub quadrilineo, BOMFH, eſſe vt, AN, adfigu-<lb/>ram, BDMO.</s> <s xml:id="echoid-s6262" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6263" xml:space="preserve">Omnia quadrata. </s> <s xml:id="echoid-s6264" xml:space="preserve">n. </s> <s xml:id="echoid-s6265" xml:space="preserve">parallelogrammi, AN, ad omnia quadrata fi-<lb/>gurę, BDMO, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6266" xml:space="preserve">ſub trilineis, BC <lb/> <anchor type="note" xlink:label="note-0271-01a" xlink:href="note-0271-01"/> O, ONM, ſunt vt, AN, ad figuram, BDMO. </s> <s xml:id="echoid-s6267" xml:space="preserve">Item rectangula <lb/> <anchor type="note" xlink:label="note-0271-02a" xlink:href="note-0271-02"/> ſub, AN, NH, ad rectangula ſub figura, BDMO, & </s> <s xml:id="echoid-s6268" xml:space="preserve">ſub, NH, <lb/>ſunt vt, AN, ad f<unsure/>iguram, BDMO, & </s> <s xml:id="echoid-s6269" xml:space="preserve">eadem rectangula ſub, AN, <lb/>NH, bis ſumpta ad rectangula ſub figura, BDMO, & </s> <s xml:id="echoid-s6270" xml:space="preserve">ſub, NH, <lb/>bis ſumpta erunt pariter, vt, AN, ad figuram, BDMO, . </s> <s xml:id="echoid-s6271" xml:space="preserve">i. </s> <s xml:id="echoid-s6272" xml:space="preserve">vt omnia <lb/>quadrata, AN, ad rectangula bis ſub figura, BDMO, & </s> <s xml:id="echoid-s6273" xml:space="preserve">ſub trili-<lb/>neis, BCO, ONM, cum omnibus quadratis eiuſdem figurę, BDM <lb/>O, ergo vt vnum ad vnum, ita omnia ad omnia. </s> <s xml:id="echoid-s6274" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6275" xml:space="preserve">vt omnia quadra-<lb/> <anchor type="figure" xlink:label="fig-0271-01a" xlink:href="fig-0271-01"/> ta, AN, ad omnia quadrata figuræ, BD <lb/> <anchor type="figure" xlink:label="fig-0271-02a" xlink:href="fig-0271-02"/> MO, cum rectangulis bis ſub eadem figu-<lb/>ra, & </s> <s xml:id="echoid-s6276" xml:space="preserve">ſub trilineis, BCO, ONM, ita om-<lb/>nia quadrata, AN, cum rectangulis bis ſub, <lb/>AN, NH, ad omnia quadrata figuræ, B <lb/>DMO, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6277" xml:space="preserve"><lb/>ſub trilineis, BCO, ONM, & </s> <s xml:id="echoid-s6278" xml:space="preserve">bis ſub ea-<lb/>dem, & </s> <s xml:id="echoid-s6279" xml:space="preserve">ſub, NH, .</s> <s xml:id="echoid-s6280" xml:space="preserve">i. </s> <s xml:id="echoid-s6281" xml:space="preserve">ad rectangula bis ſub <lb/>eadem, & </s> <s xml:id="echoid-s6282" xml:space="preserve">ſub quadrilineo, BOMFH; <lb/></s> <s xml:id="echoid-s6283" xml:space="preserve">ſunt autem omnia quadrata, AN, ad om-<lb/>nia quadrata figurę, BDMO, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6284" xml:space="preserve">ſub <lb/>trilineis, BCO, ONM, vt, AN, ad, BDMO; </s> <s xml:id="echoid-s6285" xml:space="preserve">ergo omnia qua-<lb/>drata, AN, cum rectangulis bis ſub, AN, NH, ad omnia quadrata <lb/>figurę, BDMO, cum rectangulis bis ſub, eadem, & </s> <s xml:id="echoid-s6286" xml:space="preserve">ſub quadrilineo, <pb o="252" file="0272" n="272" rhead="GEOMETRIÆ"/> BOMFH, erunt vt, AN, ad figuram, BDMO, quod oſtendere <lb/>oportebat. </s> <s xml:id="echoid-s6287" xml:space="preserve">Per hanc autem, & </s> <s xml:id="echoid-s6288" xml:space="preserve">antecedentem Propoſit. </s> <s xml:id="echoid-s6289" xml:space="preserve">vniuerſa-<lb/>lius oſtenduntur Propoſ. </s> <s xml:id="echoid-s6290" xml:space="preserve">15. </s> <s xml:id="echoid-s6291" xml:space="preserve">16. </s> <s xml:id="echoid-s6292" xml:space="preserve">necnon Corollaria Prop. </s> <s xml:id="echoid-s6293" xml:space="preserve">19. </s> <s xml:id="echoid-s6294" xml:space="preserve">& </s> <s xml:id="echoid-s6295" xml:space="preserve">20.</s> <s xml:id="echoid-s6296" xml:space="preserve"/> </p> <div xml:id="echoid-div602" type="float" level="2" n="1"> <note position="right" xlink:label="note-0271-01" xlink:href="note-0271-01a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0271-02" xlink:href="note-0271-02a" xml:space="preserve">Coroll. 1. <lb/>26. lib. 2<unsure/>.</note> <figure xlink:label="fig-0271-01" xlink:href="fig-0271-01a"> <image file="0271-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0271-01"/> </figure> <figure xlink:label="fig-0271-02" xlink:href="fig-0271-02a"> <image file="0271-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0271-02"/> </figure> </div> </div> <div xml:id="echoid-div604" type="section" level="1" n="351"> <head xml:id="echoid-head368" xml:space="preserve">THEOREMA XXX. PROPOS. XXXI.</head> <p> <s xml:id="echoid-s6297" xml:space="preserve">SI parallelogrammum fuerit ellipſi circumſcriptum, ita ta-<lb/>men, vt eiuſdem latera non tangant ellipſim in extremis <lb/>punctis axium eiuſdem; </s> <s xml:id="echoid-s6298" xml:space="preserve">portiones coalternè tangentes erunt <lb/>æquales; </s> <s xml:id="echoid-s6299" xml:space="preserve">& </s> <s xml:id="echoid-s6300" xml:space="preserve">ſi duabus oppoſitis tangentibus ducantur paral-<lb/>lelę abſcindentes à reliquis coalternis tangentibus rectas li-<lb/>neas æquales, ſumptas verſus puncta contactuum; </s> <s xml:id="echoid-s6301" xml:space="preserve">rectangu-<lb/>lum, quod continetur ſub vnius parallelarum ea parte, quæ <lb/>manet intra curuam ellipſis, & </s> <s xml:id="echoid-s6302" xml:space="preserve">tangentem ex ea parte, & </s> <s xml:id="echoid-s6303" xml:space="preserve">ſub <lb/>reliqua illi in directum manente intra ellipſim, erit æquale <lb/>rectangulo ad coalternam tangentem ſimiliter ſumpto.</s> <s xml:id="echoid-s6304" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6305" xml:space="preserve">Sit ergo ellipſis, BDMG, cui ſit circumſcriptum parallelogram-<lb/>mum, AR, ita tamen, vt puncta contactuum non ſint puncta ex-<lb/>trema axium eiuſdem, tangant autem in punctis, BDMG, & </s> <s xml:id="echoid-s6306" xml:space="preserve">iun-<lb/>gantur, BM, DG, & </s> <s xml:id="echoid-s6307" xml:space="preserve">quoniam, AC, FR, ſunt tangentes paralle-<lb/>læ, vt etiam, AF, CR, ideò, BM, GD, per centrum ellipſis tran-<lb/> <anchor type="note" xlink:label="note-0272-01a" xlink:href="note-0272-01"/> ſibunt, ſit earum communis ſectio punctum, S, ergo, S, erit centrum <lb/> <anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/> ellipſis, cum, BM, GD, ſint diametri. <lb/></s> <s xml:id="echoid-s6308" xml:space="preserve">Dico ergo portiones laterum parallelo-<lb/>grammi, AR, coalternè tangentes eſſe <lb/>æquales. </s> <s xml:id="echoid-s6309" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6310" xml:space="preserve">AD, ipſi, GR, AB, ipſi, M <lb/>R, BC, ipſi, FM, &</s> <s xml:id="echoid-s6311" xml:space="preserve">, CG, ipſi, DF; </s> <s xml:id="echoid-s6312" xml:space="preserve"><lb/>iungantur, BG, DM; </s> <s xml:id="echoid-s6313" xml:space="preserve">in triangulis ergo, <lb/>BSG, DSM, latus, BS, æquatur late-<lb/>ri, SM, & </s> <s xml:id="echoid-s6314" xml:space="preserve">latus, GS, lateri, SD, item <lb/>angulus; </s> <s xml:id="echoid-s6315" xml:space="preserve">BSG, angulo, DSM, ergo ba-<lb/> <anchor type="note" xlink:label="note-0272-02a" xlink:href="note-0272-02"/> ſis, BG, æquatur baſi, DM, & </s> <s xml:id="echoid-s6316" xml:space="preserve">angulus, <lb/>SBG, angulo, SMD, &</s> <s xml:id="echoid-s6317" xml:space="preserve">, SGB, ipſi, S <lb/>DM, totus autem angulus, CBS, æquatur toti, FMS, ſibi coal-<lb/>terno, ergo reliquus angulus, CBG, æquatur reliquo angulo, DM <lb/>F, & </s> <s xml:id="echoid-s6318" xml:space="preserve">ſimiliter probabimus angulum, BGC, æquari angulo, MD <lb/>F, ergo reliquus, BCG, æquabitur reliquo, DFM, (qui etiam ſunt <lb/>æquales, quia ſunt anguli oppoſiti parallelogrammi, AR,) & </s> <s xml:id="echoid-s6319" xml:space="preserve">ideò <lb/>trianguli, BCG, DFM, erunt æquianguli, &</s> <s xml:id="echoid-s6320" xml:space="preserve">, BG, DM, latera <pb o="253" file="0273" n="273" rhead="LIBER III."/> homologa ſunt æqualia, ergo etiam, BC, æquabitur ipſi, FM, &</s> <s xml:id="echoid-s6321" xml:space="preserve">, <lb/>CG, ipſi, DF, eſt autem, AC, æqualis ipſi, FR, &</s> <s xml:id="echoid-s6322" xml:space="preserve">, AF, ipſi, C <lb/>R, ergo reliqua, AB, æquabitur reliquæ, MR, & </s> <s xml:id="echoid-s6323" xml:space="preserve">reliqua, AD, re-<lb/>liquæ, GR; </s> <s xml:id="echoid-s6324" xml:space="preserve">ſuntigitur portiones laterum parallelogrammi, AR, <lb/>coalterne tangentes inter ſe æquales.</s> <s xml:id="echoid-s6325" xml:space="preserve"/> </p> <div xml:id="echoid-div604" type="float" level="2" n="1"> <note position="left" xlink:label="note-0272-01" xlink:href="note-0272-01a" xml:space="preserve">Elicitur <lb/>ex 27. 2. <lb/>Con.</note> <figure xlink:label="fig-0272-01" xlink:href="fig-0272-01a"> <image file="0272-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0272-01"/> </figure> <note position="left" xlink:label="note-0272-02" xlink:href="note-0272-02a" xml:space="preserve">4. 1. Elem.</note> </div> <p> <s xml:id="echoid-s6326" xml:space="preserve">Sumantur nunc vtcunque duæ coalternè tangentes, AD, RG, <lb/>& </s> <s xml:id="echoid-s6327" xml:space="preserve">ab ipſis verſus puncta contactuum, DG, abſcindantur vtcunque <lb/>duæ rectæ æquales, PD, VG, & </s> <s xml:id="echoid-s6328" xml:space="preserve">per puncta, PV, ducantur baſi, <lb/>FR, parallelæ, PQ, EV, ſecantes curuam ellipſis in punctis, HI, <lb/>ipſa, PQ, & </s> <s xml:id="echoid-s6329" xml:space="preserve">in punctis, NO, ipſa, EV. </s> <s xml:id="echoid-s6330" xml:space="preserve">Quoniam ergo, AB, A <lb/>D, tangunt ellipſim, BDMG, coincidentes in puncto, A, eſt au-<lb/>tem, QP, parallela vni tangentium. </s> <s xml:id="echoid-s6331" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6332" xml:space="preserve">ipſi, AB, ſecans curuam el-<lb/>lipſis in, H, & </s> <s xml:id="echoid-s6333" xml:space="preserve">aliam tangentem in, P, rectangulum ergo, IPH, ad <lb/> <anchor type="note" xlink:label="note-0273-01a" xlink:href="note-0273-01"/> quadratum, PD, erit vt quadratum, BA, ad quadratum, AD,. </s> <s xml:id="echoid-s6334" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s6335" xml:space="preserve">vt quadratum, MR, ad quadratum, RG: </s> <s xml:id="echoid-s6336" xml:space="preserve">conſimili modo oſtende-<lb/>mus rectangulum, NVO, ad quadratum, VG, eſſe vt quadratum, <lb/>MR, ad quadratum, RG, . </s> <s xml:id="echoid-s6337" xml:space="preserve">i. </s> <s xml:id="echoid-s6338" xml:space="preserve">vt rectangulum, IPH, ad quadratum, <lb/>PD, vel ad quadratum, VG, ergo rectangulum, IPH, eſt æquale <lb/>rectangulo, NVO. </s> <s xml:id="echoid-s6339" xml:space="preserve">Nunc oſtendemus, PH, eſſe æqualem ipſi, O <lb/>V, conſideremus duo quadrilatera, APXB, MTVR, quæ ſunt ſi-<lb/>milia polygona, nam angulus, PAB, eſt æqualis angulo, MRV, <lb/>&</s> <s xml:id="echoid-s6340" xml:space="preserve">, ABX, ipſi, RMT, tum, BXP, ipſi, MTV, & </s> <s xml:id="echoid-s6341" xml:space="preserve">tandem, XP <lb/>A, ipſi, TVR, quę facilè apparent, & </s> <s xml:id="echoid-s6342" xml:space="preserve">duo latera, AB, MR, ſunt <lb/>æqualia, vt etiam, AP, RV, ergo reliqua latera erunt æqualia, quę <lb/>æqualibus adiacent angulis, vnde, TV, erit æqualis ipſi, PX, &</s> <s xml:id="echoid-s6343" xml:space="preserve">, <lb/>MT, ipſi, BX, vnde rectangulum, MTB, æquabitur rectangulo, <lb/>MXB, & </s> <s xml:id="echoid-s6344" xml:space="preserve">quoniam, vt rectangulum, MTB, ad rectangulum, M <lb/> <anchor type="note" xlink:label="note-0273-02a" xlink:href="note-0273-02"/> XB, ita quadratum, TO, ad quadratum, XI, quoniam, NO, H <lb/>I, ſunt parallelæ tangenti, AC, & </s> <s xml:id="echoid-s6345" xml:space="preserve">ideò ordinatim applicatę ad dia <lb/>metrum, BM, erit ergo quadratum, TO, æquale quadrato, XI, <lb/>&</s> <s xml:id="echoid-s6346" xml:space="preserve">, TO, ipſi, XI, vel, HX, ergo reliqua, OV, erit æqualis reli-<lb/>quæ, PH, & </s> <s xml:id="echoid-s6347" xml:space="preserve">quia rectangulum, IPH, eſt æquale rectangulo, N <lb/>VO, erit, IP, æqualis ipſi, NV, & </s> <s xml:id="echoid-s6348" xml:space="preserve">quia, PH, eſt æ qualis, OV, <lb/>erit, HI, æqualis, NO, & </s> <s xml:id="echoid-s6349" xml:space="preserve">ideò rectangulum, IHP, erit æquale re-<lb/>ctangulo, NOV.</s> <s xml:id="echoid-s6350" xml:space="preserve"/> </p> <div xml:id="echoid-div605" type="float" level="2" n="2"> <note position="right" xlink:label="note-0273-01" xlink:href="note-0273-01a" xml:space="preserve">16. 3. Con.</note> <note position="right" xlink:label="note-0273-02" xlink:href="note-0273-02a" xml:space="preserve">Ex 40, 1. <lb/>lib. & eiu. <lb/>ldẽ Scho-<lb/>lio.</note> </div> <p> <s xml:id="echoid-s6351" xml:space="preserve">Vel breuius ſic proceſſiſſet demonſtratio, dimiſſo Apollonij theo-<lb/>remate, oſtenſo. </s> <s xml:id="echoid-s6352" xml:space="preserve">n. </s> <s xml:id="echoid-s6353" xml:space="preserve">OV, eſſe æqualem ipſi, PH, &</s> <s xml:id="echoid-s6354" xml:space="preserve">, TO, ipſi, XI, <lb/>manet oſtenſum, NO, æquari ipſi, HI, quoniam, NO, HI, bi-<lb/>faria diuiduntur a diametro, BM, & </s> <s xml:id="echoid-s6355" xml:space="preserve">ideo illicò manifeſtum euadit <lb/>rectangulum, NOV, æquari rectangulo, IHP, quod oſtendere <lb/>oportebat.</s> <s xml:id="echoid-s6356" xml:space="preserve"/> </p> <pb o="254" file="0274" n="274" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div607" type="section" level="1" n="352"> <head xml:id="echoid-head369" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s6357" xml:space="preserve">_H_Inc patet nedum rectangulum, NOV, æquari rectangulo, IHP, <lb/>ſed etiam portiones interceptas tangentibus, & </s> <s xml:id="echoid-s6358" xml:space="preserve">curuaellipſis eſſe <lb/>inter ſe æquales, belut, OV, ipſi, PH.</s> <s xml:id="echoid-s6359" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div608" type="section" level="1" n="353"> <head xml:id="echoid-head370" xml:space="preserve">THEOREMA XXXI. PROPOS. XXXII.</head> <p> <s xml:id="echoid-s6360" xml:space="preserve">EXpoſita ellipſi, cum parallelogrammo illi circumſcripto <lb/>Theor. </s> <s xml:id="echoid-s6361" xml:space="preserve">antecedentis, cæteris omiſſis, oſtendemus, re-<lb/>gula, FR, omnia quadrata parallelogrammi, AR, ad om-<lb/>nia quadrata ellipſis, BDMG, cum rectangulis bis ſub ea-<lb/>dem ellipſi, & </s> <s xml:id="echoid-s6362" xml:space="preserve">ſub trilineis, BCG, GRM, eſſe, vt paralle-<lb/>logrammum, AR, ad ellipſim, BDMG.</s> <s xml:id="echoid-s6363" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6364" xml:space="preserve">Ducantur à punctis contactuum regulæ, FR, parallelę, GE, D <lb/> <anchor type="note" xlink:label="note-0274-01a" xlink:href="note-0274-01"/> V; </s> <s xml:id="echoid-s6365" xml:space="preserve">omnia ergo quadrata, AR, ad rectangula ſub ellipſi, BDMG, <lb/>& </s> <s xml:id="echoid-s6366" xml:space="preserve">ſub, AR, ſunt vt, AR, ad ellipſim, BDMG; </s> <s xml:id="echoid-s6367" xml:space="preserve">verum rectangula <lb/>ſub ellipſi, BDMG, & </s> <s xml:id="echoid-s6368" xml:space="preserve">ſub, AR, ſunt æqualia rectangulis ſub el-<lb/> <anchor type="note" xlink:label="note-0274-02a" xlink:href="note-0274-02"/> lipſi, BDMG, & </s> <s xml:id="echoid-s6369" xml:space="preserve">ſub duobus trilineis, BAD, DFM, item ſub el-<lb/>lipſi, BDMG, & </s> <s xml:id="echoid-s6370" xml:space="preserve">ſub eadem. </s> <s xml:id="echoid-s6371" xml:space="preserve">i. </s> <s xml:id="echoid-s6372" xml:space="preserve">omnibus quadratis ellipſis, BDM <lb/>G, & </s> <s xml:id="echoid-s6373" xml:space="preserve">ſub eadem ellipſi, BDMG, & </s> <s xml:id="echoid-s6374" xml:space="preserve">ſub duobus trilineis, BCG, <lb/>GRM, verum rectangula ſub ellipſi, BDMG, & </s> <s xml:id="echoid-s6375" xml:space="preserve">ſub trilineis, B <lb/> <anchor type="figure" xlink:label="fig-0274-01a" xlink:href="fig-0274-01"/> AD, DFM, æquantur rectangulis ſub ea-<lb/>dem ellipſi, & </s> <s xml:id="echoid-s6376" xml:space="preserve">ſub trilineis, BCG, GRM, <lb/>quod ſic patet, quoniam enim, AD, RG, <lb/> <anchor type="note" xlink:label="note-0274-03a" xlink:href="note-0274-03"/> coalternè tangentes ſunt æquales, & </s> <s xml:id="echoid-s6377" xml:space="preserve">ductis <lb/>ipſi, FR, parallelis intra ellipſim, ex ipſis <lb/>coalternè tangentibus, AD, RG, abſcin <lb/>dentibus portiones æquales verſus puncta <lb/>contactuum, rectangula ſumpta, vt dictum <lb/>eſt in antecedenti Theor. </s> <s xml:id="echoid-s6378" xml:space="preserve">ſunt æqualia, ideò <lb/>& </s> <s xml:id="echoid-s6379" xml:space="preserve">omnia omnibus erunt æqualia. </s> <s xml:id="echoid-s6380" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6381" xml:space="preserve">rectan-<lb/>gula ſub portione, OGBD, & </s> <s xml:id="echoid-s6382" xml:space="preserve">trilineo, BAD, erunt æqualia re-<lb/>ctangulis ſub portione, SMG, & </s> <s xml:id="echoid-s6383" xml:space="preserve">ſub trilineo, GMR, eadem ra-<lb/>tione rectangula ſub portione, OMD, & </s> <s xml:id="echoid-s6384" xml:space="preserve">trilineo, DFM, æquan-<lb/>tur rectangulis ſub portione, SBG, & </s> <s xml:id="echoid-s6385" xml:space="preserve">trilineo, BCG, ergo rectan-<lb/>gula ſub ellipſi, BDMG, & </s> <s xml:id="echoid-s6386" xml:space="preserve">duobus trilineis, BAD, DFM, ęquan-<lb/>tur rectangulis ſub ellipſi, BDMG, & </s> <s xml:id="echoid-s6387" xml:space="preserve">ſub trilineis, BCG, GRM;</s> <s xml:id="echoid-s6388" xml:space="preserve"> <pb o="255" file="0275" n="275" rhead="LIBER III."/> ergo omnia quadrata, AR, ad omnia quadrata ellipſis, BDMG, <lb/>cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6389" xml:space="preserve">ſub trilineis, BCG, GRM, erunt <lb/>vt, AR, ad ellipſim, BDMG. </s> <s xml:id="echoid-s6390" xml:space="preserve">Eodem modo, ſumpta pro regula, <lb/>CR, oſtendemus omnia quadrata, AR, ad omnia quadrata ellipſis, <lb/>BDMG, vna cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6391" xml:space="preserve">ſub trilineis, DAB, <lb/>CG, eſſe vt, AR, ad ellipſim, BDMG, quod oſtendere oportebat.</s> <s xml:id="echoid-s6392" xml:space="preserve"/> </p> <div xml:id="echoid-div608" type="float" level="2" n="1"> <note position="left" xlink:label="note-0274-01" xlink:href="note-0274-01a" xml:space="preserve">Coroll. 1. <lb/>26. lib. 2.</note> <note position="left" xlink:label="note-0274-02" xlink:href="note-0274-02a" xml:space="preserve">A. 23. 1. 2.</note> <figure xlink:label="fig-0274-01" xlink:href="fig-0274-01a"> <image file="0274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0274-01"/> </figure> <note position="left" xlink:label="note-0274-03" xlink:href="note-0274-03a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div610" type="section" level="1" n="354"> <head xml:id="echoid-head371" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s6393" xml:space="preserve">_H_Inc babetur omnia quadrata, AR, regula, FR, ad omnia quadra@ <lb/>ta ellipſis, BDGM, vna cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6394" xml:space="preserve">ſub <lb/>trilineis, BCG, GRM, eſſe vt omnia quadrata, AR, regula, CR, ad <lb/>omnia quadrata ellipſis, BDMG, vna cum rectangulis bis ſub eadem, <lb/>& </s> <s xml:id="echoid-s6395" xml:space="preserve">ſub trilineis, DAB, BCG; </s> <s xml:id="echoid-s6396" xml:space="preserve">vtraq; </s> <s xml:id="echoid-s6397" xml:space="preserve">enim oſtenſa ſunt eſſe, vt, AR, <lb/>ad ellipſim, BDMG.</s> <s xml:id="echoid-s6398" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div611" type="section" level="1" n="355"> <head xml:id="echoid-head372" xml:space="preserve">THEOREMA XXXII. PROPOS. XXXIII.</head> <p> <s xml:id="echoid-s6399" xml:space="preserve">ASſumpta antecedentis figura, dimiſſis lineis, EG, DV, <lb/>producantur ad eandem partem, AC, FR, in, TX, <lb/>ita vt, AT, ſit æqualis, FX, & </s> <s xml:id="echoid-s6400" xml:space="preserve">iungatur, TX, ergo ipſa, A <lb/>X, CX, erunt parallelogramma. </s> <s xml:id="echoid-s6401" xml:space="preserve">Dico igitur, omnia qua-<lb/>drata, AR, cum<unsure/> rectanguli<unsure/>s bis ſub, AR, RT, ad omnia <lb/>quadrata ellipſis, BDMG, cum rectangulis bis ſub eadem, <lb/>& </s> <s xml:id="echoid-s6402" xml:space="preserve">quadrilineo, BGMXT, eſſe vt, AR, ad ellipſim, BDM <lb/>G, regula, FX.</s> <s xml:id="echoid-s6403" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6404" xml:space="preserve">Omnia. </s> <s xml:id="echoid-s6405" xml:space="preserve">n. </s> <s xml:id="echoid-s6406" xml:space="preserve">quadrata, AR, ad rectangula bis ſub ellipſi, BDMG, <lb/> <anchor type="figure" xlink:label="fig-0275-01a" xlink:href="fig-0275-01"/> & </s> <s xml:id="echoid-s6407" xml:space="preserve">ſub trilineis, BCG, GRM, vna cum <lb/> <anchor type="note" xlink:label="note-0275-01a" xlink:href="note-0275-01"/> omnibus quadrati<unsure/>s eiuldem ellipſis, BD <lb/>MG, ſunt vt, AR, ad ellipſim, BDM <lb/> <anchor type="note" xlink:label="note-0275-02a" xlink:href="note-0275-02"/> G, item rectangula ſub, AR, RT, ad re-<lb/>ctangula ſub ellipſi, BDMG, & </s> <s xml:id="echoid-s6408" xml:space="preserve">ſub, R <lb/>T, ſunt vt, AR, ad ellipſim, BDMG, <lb/>& </s> <s xml:id="echoid-s6409" xml:space="preserve">eadem bis ſumpta ſunt pariter, vt, AR, <lb/>ad ellipſim, BDMG, ergo omnia qua-<lb/>drata, AR, cum rectangulis bis ſub, A <lb/>R, RT, ad omnia quadrata ellipſis, B <lb/>DMG, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6410" xml:space="preserve">ſub trilineis, BCG, <lb/>GRM, & </s> <s xml:id="echoid-s6411" xml:space="preserve">cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6412" xml:space="preserve">ſub, RT, ideſt cum <pb o="256" file="0276" n="276" rhead="GEOMETRIÆ"/> rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6413" xml:space="preserve">ſub quadrilineo, BGMXT, erunt vt, <lb/>AR, ad ellipſim, BDMG. </s> <s xml:id="echoid-s6414" xml:space="preserve">Sic etiam fiet demonſtratio, ſi produ-<lb/>cantur, FA, RC, ſimiliter ac productę ſunt, AC, FR, quarum al-<lb/>tera pro regula ſumatur.</s> <s xml:id="echoid-s6415" xml:space="preserve"/> </p> <div xml:id="echoid-div611" type="float" level="2" n="1"> <figure xlink:label="fig-0275-01" xlink:href="fig-0275-01a"> <image file="0275-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0275-01"/> </figure> <note position="right" xlink:label="note-0275-01" xlink:href="note-0275-01a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">Coroll. 1. <lb/>26. lib. @.</note> </div> </div> <div xml:id="echoid-div613" type="section" level="1" n="356"> <head xml:id="echoid-head373" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s6416" xml:space="preserve">_H_Inc patet, ſi, BDMG, non eſſet ellipſis, ſed alia vtcunque figu-<lb/>ra plana parallelogrammo, AR, inſcripta, dummodo portiones <lb/>laterum coalternè tangentes eſſent æquales, & </s> <s xml:id="echoid-s6417" xml:space="preserve">rectangula ſumpta ad <lb/>coalternè tangentes, eo modo, quo dictum eſt in heor. </s> <s xml:id="echoid-s6418" xml:space="preserve">antecedenti, eſ-<lb/>ſent quoque æqualia, quod omnia quadrata, AR, ad omnia quadrata, <lb/>talis figuræ, cum rectangulis bis ſub eadem, & </s> <s xml:id="echoid-s6419" xml:space="preserve">ſub trilineis adiacenti-<lb/>bus lateri, quod non ſumitur pro regula, erunt vt, AR, ad talem figu-<lb/>ram; </s> <s xml:id="echoid-s6420" xml:space="preserve">V eluti erunt etiam omnia quadrata, AR, cumrectangu<unsure/>lis bis ſub, <lb/>AR, RT, ad omnia quadrata talis figuræ, cum rectangulis bis ſub ea-<lb/>dem, & </s> <s xml:id="echoid-s6421" xml:space="preserve">ſub quadrilineo ſimili ipſi, BGMXT, bæc. </s> <s xml:id="echoid-s6422" xml:space="preserve">n. </s> <s xml:id="echoid-s6423" xml:space="preserve">eodem modo col-<lb/>ligentur, quo pro ellipſi, BDMG, per demonſtrationem collecta ſunt, <lb/>aderunt enim eadem principia, ex quibus demonſtratio pro ellipſi pende-<lb/>bat: </s> <s xml:id="echoid-s6424" xml:space="preserve">Exemplum facile baberi poteſt in figura ex duabus æqualibus cir-<lb/>culi, vel ellipſis portionibus minoribus compoſita tali pacto, vt baſis v-<lb/>nius portionis alterius baſi congruat, quæ quidem figura ſit inſoripta di-<lb/>cto rectangulo, cuiuſque latera eam tangant non in punctis extremis <lb/>axium, ſed in quatuor alijs vtcunq, vnde, &</s> <s xml:id="echoid-s6425" xml:space="preserve">c.</s> <s xml:id="echoid-s6426" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div614" type="section" level="1" n="357"> <head xml:id="echoid-head374" xml:space="preserve">THEOREMA XXXIII. PROPOS. XXXIV.</head> <p> <s xml:id="echoid-s6427" xml:space="preserve">QVæcunque ſolida ad inuicem ſimilaria, genita ex figu-<lb/>ris ſuperius in hoc Libro Tertio conſideratis, iuxta re-<lb/>gulas ibidem aſſumptas, quarum patefacta eſt ra-<lb/>tio omnium quadratorum, habent inter ſe rationem notam.</s> <s xml:id="echoid-s6428" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6429" xml:space="preserve">Quoniam enim alibi oſtenſum eſt, vt omnia quadrata duarum fi-<lb/>gurarum inter ſe ſumpta cum da<unsure/>tis regulis, ita eſſe ſoli<unsure/>da ad inuicem <lb/> <anchor type="note" xlink:label="note-0276-01a" xlink:href="note-0276-01"/> ſimilaria genita ex ijldem ſiguris, iuxta eaſdem regulas, ideò cum in <lb/>Theorematibus huius Libri inuenta eſt ratio omnium quadratorum <lb/>duarum quarundam figurarum cum talibus regulis, colligimus etiam <lb/>nunc eandem eſſe rationem duorum ad inuicem ſimilarium ſolido-<lb/>rum, quæ ex illis figuris iuxta eaſdem regulas genita dicuntur; </s> <s xml:id="echoid-s6430" xml:space="preserve">Vt <lb/>exempli gratia in Propoſ. </s> <s xml:id="echoid-s6431" xml:space="preserve">I. </s> <s xml:id="echoid-s6432" xml:space="preserve">conſpectis, iterum eiuſdem figuris, cum <pb o="257" file="0277" n="277" rhead="LIBER III."/> ibi oſtenſum eſt, ſumpta regula, DP, omnia quadrata portionis, D <lb/>EP, ad omnia quadrata parallelogrammi, FP, eſſe vt compoſita ex <lb/>ſexta parte, EB, & </s> <s xml:id="echoid-s6433" xml:space="preserve">dimidia, BR, ad ipſam, BR, oſtenſum etiam <lb/>eſt ſolidum ſimilare genitum ex portione, DEP, ad ſolidum ſibi ſi-<lb/>milare genitum ex parallelogrammo, FP, eſſe vt compoſita ex ſex-<lb/>ta parte, EB, & </s> <s xml:id="echoid-s6434" xml:space="preserve">dimidia, BR, adipſam, BR. </s> <s xml:id="echoid-s6435" xml:space="preserve">Cum verò oſten-<lb/>ſum eſt omnia quadrata portionis, EDP, ad omnia quadrata trian-<lb/>guli, DEP, eſſe vt compoſita ex dimidia totius, ER, & </s> <s xml:id="echoid-s6436" xml:space="preserve">ipſa, BR, <lb/>ad eandem, BR; </s> <s xml:id="echoid-s6437" xml:space="preserve">pariter oſtenſum eſt ſolidum ſimilare genitum ex <lb/>portione, EDP, ad ſibi ſimilare genitum ex triangulo, DEP, iux-<lb/>ta eaſdem regulas eſſe, vt compoſita ex dimidia totius, ER, & </s> <s xml:id="echoid-s6438" xml:space="preserve">ipſa, <lb/>BR, ad eandem, BR.</s> <s xml:id="echoid-s6439" xml:space="preserve"/> </p> <div xml:id="echoid-div614" type="float" level="2" n="1"> <note position="left" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve">33. Lib. 2.</note> </div> <p> <s xml:id="echoid-s6440" xml:space="preserve">Cum verò in Corollario eiuſdem Theorem. </s> <s xml:id="echoid-s6441" xml:space="preserve">collectum eſt, omnia <lb/>quadrata parallelogrammi, FP, eſſe ſexquialtera omnium quadra-<lb/>torum portionis, DEP, (ſi, DP, per centrum, A, tranſeat) hæc <lb/>verò eſſe dupla omnium quadratorum trianguli, DEP; </s> <s xml:id="echoid-s6442" xml:space="preserve">patet, quod <lb/>etiam ſolidum ſimilare genitum ex parallelogrammo, FP, ſexquial-<lb/>terum erit ſolidi ſibi ſimilaris geniti ex portione, DEP, iuxta ean-<lb/>dem regulam, DP, hoc verò erit duplum ſolidi ſibi ſimilaris geniti <lb/>ex triangulo, DEP, iuxta eandem regulam, DP.</s> <s xml:id="echoid-s6443" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div616" type="section" level="1" n="358"> <head xml:id="echoid-head375" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s6444" xml:space="preserve">_Q_Voniam verò ſolida ad inuicem ſimilaria genita ex duabus figuris <lb/> <anchor type="note" xlink:label="note-0277-01a" xlink:href="note-0277-01"/> planis, iuxta datas regulas, totuplicia ſunt, quotuplices ſunt fi-<lb/>guræ ſimiles, quæ dicuntur, omnes figuræ ſimiles duarum genitri-<lb/>cium figurarum, cum eiſdem regulis aſſu<unsure/>mpta, iuxta quas dicta ſolida <lb/>ſimilaria genita dicuntur, figurarum autem variationes nullo dato nu-<lb/>mero clauduntur, ideò nec horum ſimilarium ſolidorum variationes vl-<lb/>lo dato coarctantur numero, vnde euidentiſſimè apparet banc demon. <lb/></s> <s xml:id="echoid-s6445" xml:space="preserve">ſtrandi methodum, ipſamque demonſtrationem, infinitè (vt ita dicam) <lb/>locupletem eſſe; </s> <s xml:id="echoid-s6446" xml:space="preserve">vt igitur ad parttcularia ſolida ſimi aria deſcendamus, <lb/>expendendæ ſunt ipſæ figuræ, quæ dicuntur (omnes figuræ ſimiles, &</s> <s xml:id="echoid-s6447" xml:space="preserve">c.) </s> <s xml:id="echoid-s6448" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0277-02a" xlink:href="note-0277-02"/> patet igitur ex alibi à me o<unsure/>ſtenſis, ſi figuræ aſſumptæ ſint omnes figuræ ſi-<lb/>miles parall@logrammi, quod tunc i<unsure/>ſtæ erunt omnia plana cylindrici; </s> <s xml:id="echoid-s6449" xml:space="preserve">ſi <lb/>verò illæ ſint omnes figuræ ſimiles trianguli (intellige in parallelogram-<lb/>mo, & </s> <s xml:id="echoid-s6450" xml:space="preserve">triangulo vnum laterum pro regula) illæ erunt omnia plana Co-<lb/> <anchor type="note" xlink:label="note-0277-03a" xlink:href="note-0277-03"/> nici; </s> <s xml:id="echoid-s6451" xml:space="preserve">& </s> <s xml:id="echoid-s6452" xml:space="preserve">ſi parallelogrammum ſit rectangulum, & </s> <s xml:id="echoid-s6453" xml:space="preserve">figuræ eidem erectæ <lb/>erit cylindricus rectus, ſcalenus autem niſi ſit rectangulum, vel figuræ <lb/>non eidem erecta; </s> <s xml:id="echoid-s6454" xml:space="preserve">ex quo babes, qualeſcunque figuras intellexeris eſſe <lb/>eas, quæ dicuntur omnes figuræ ſimiles parallelogrammi, FP, regula, <lb/>DP, veltrianguli, EDP, regula eadem, ſolidum genitum ex paralle- <pb o="258" file="0278" n="278" rhead="GEOMETRIÆ"/> logrammo, FP, quod dicimus ſimilare, ſemper eſſe cylindricum, geni-<lb/>tum verò ex triangulo, DEP, ſemper eſſe conicum, vt etiam accidit in <lb/>omni parallelogrammo, & </s> <s xml:id="echoid-s6455" xml:space="preserve">triangulo, dummodò regula ſit vnum laterum <lb/>eorundem, ſolida igitur ſimilariagenita ex parallelogrammis ſunt cy-<lb/>lindrici, genita verò ex triangulis ſunt conici, genita inquam, regula, <lb/>vno laterum eorundem exiſtente; </s> <s xml:id="echoid-s6456" xml:space="preserve">quod ſi figuræ quæ dicuntur omnes fi-<lb/>guræ ſimiles par allelogrammidati, regula vno laterum, ſint eirculi, ille <lb/>cylindricus erit cylindrus; </s> <s xml:id="echoid-s6457" xml:space="preserve">& </s> <s xml:id="echoid-s6458" xml:space="preserve">ſi, quæ dicuntur omnes figuræ ſimiles dati <lb/>trianguli ſint pariter cirouli, regula vno laterum, conicus erit conus; <lb/></s> <s xml:id="echoid-s6459" xml:space="preserve">nomine ergo communi bic cylindrus, & </s> <s xml:id="echoid-s6460" xml:space="preserve">conus dicti ſunt ſolida ſimila-<lb/>ria nomine particulari dicti ſunt cylindricus, & </s> <s xml:id="echoid-s6461" xml:space="preserve">conicus, ſed nomine, <lb/>magis particulari, & </s> <s xml:id="echoid-s6462" xml:space="preserve">proprio dicuntur cylindrus, & </s> <s xml:id="echoid-s6463" xml:space="preserve">conus, quotieſcun-<lb/> <anchor type="note" xlink:label="note-0278-01a" xlink:href="note-0278-01"/> que dictæ figuræ ſint circuli, iuxta alibi à me oſtenſa.</s> <s xml:id="echoid-s6464" xml:space="preserve"/> </p> <div xml:id="echoid-div616" type="float" level="2" n="1"> <note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve">_A. Def. @_ <lb/>_lib. 1._</note> <note position="right" xlink:label="note-0277-02" xlink:href="note-0277-02a" xml:space="preserve">_34. Lib. 2._</note> <note position="right" xlink:label="note-0277-03" xlink:href="note-0277-03a" xml:space="preserve">_34. Lib. 2._</note> <note position="left" xlink:label="note-0278-01" xlink:href="note-0278-01a" xml:space="preserve">_34. Lib. 2._</note> </div> <p style="it"> <s xml:id="echoid-s6465" xml:space="preserve">Pariter ſi figuræ genitrices ſolidorum ſint circuli, vel ellipſes, illæ <lb/>autem, quæ dicuntur @omnes figuræ ſimiles earundem ſumptæ cum datis <lb/>regulis) ſint pariter circuli, quorum deſcribentes rectæ lineæ in figuris <lb/>genitricibus ſint eorundem diametri, ſolida ſimilaria genita ex eiſdem, <lb/>iuxta eaſdem regulas, erunt, alterum ſpbæra, quod ſcilicet gignitur ex <lb/> <anchor type="note" xlink:label="note-0278-02a" xlink:href="note-0278-02"/> circulo, alterum ſphæroides quod ſcilicet gignitur ex ellipſi, ſi figuræ <lb/>ſimiles rectè ſecent axem ellipſis, & </s> <s xml:id="echoid-s6466" xml:space="preserve">ſint erectæ tum circulo, tum ellipſi; <lb/></s> <s xml:id="echoid-s6467" xml:space="preserve"> <anchor type="note" xlink:label="note-0278-03a" xlink:href="note-0278-03"/> poterit etiam eſſe ſpbæroides, etiamſi figuræ ſimiles non ſint circuli, ſed <lb/>ellipſes iuxta alibi oſtenſa; </s> <s xml:id="echoid-s6468" xml:space="preserve">quæ igitur in boc caſu nomine communi di-<lb/>cuntur, ſolida ſimilaria genita ex circulo, vel ellipſi, iuxta datas regu-<lb/>las, nomine particulari, & </s> <s xml:id="echoid-s6469" xml:space="preserve">proprio, dicuntur ſphæra, vel ſphæroides: <lb/></s> <s xml:id="echoid-s6470" xml:space="preserve">Et quæ pariter dicerentur nomine communi ſolida ſimilaria genita ex <lb/>portione tali, vel tali, iuxta talem regulam, portione inquam circuli, <lb/>vel ellipſis, quotieſcunque figuræ, quæ dicuntur, omnes figuræ ſimiles <lb/>talis portionis iuxta eandem regulam, ſint circuli erecti genitricibus, <lb/>& </s> <s xml:id="echoid-s6471" xml:space="preserve">figura genitrix pontio circuli, erit, & </s> <s xml:id="echoid-s6472" xml:space="preserve">dicetur nomine particulari, & </s> <s xml:id="echoid-s6473" xml:space="preserve"><lb/>proprio, portio ſphæræ; </s> <s xml:id="echoid-s6474" xml:space="preserve">ſi verò ſigura genitrix ſit ellipſis portio, & </s> <s xml:id="echoid-s6475" xml:space="preserve">ſi-<lb/> <anchor type="note" xlink:label="note-0278-04a" xlink:href="note-0278-04"/> guræ ſimiles ſint circuli erecti genitricibus, rectè axem portionis ſecan-<lb/> <anchor type="note" xlink:label="note-0278-05a" xlink:href="note-0278-05"/> tes, ſiet portio ſphæroidis, quod ſi ſint ellipſes erectæ genitricibus, dia-<lb/>metros habentes, vt poſtulat Propoſ. </s> <s xml:id="echoid-s6476" xml:space="preserve">47. </s> <s xml:id="echoid-s6477" xml:space="preserve">Lib. </s> <s xml:id="echoid-s6478" xml:space="preserve">1. </s> <s xml:id="echoid-s6479" xml:space="preserve">fiet etiam portio ſphæ-<lb/>roidis: </s> <s xml:id="echoid-s6480" xml:space="preserve">Sic igitur nommibus particularibus hæc ſolida vocari conſuene-<lb/>runt. </s> <s xml:id="echoid-s6481" xml:space="preserve">Cum verò figuræ ſimiles non ſunt neq; </s> <s xml:id="echoid-s6482" xml:space="preserve">circuli, neq; </s> <s xml:id="echoid-s6483" xml:space="preserve">ellipſes ſum-<lb/>ptæ, vt dictum eſt, ſufficiet eadem vocare nomine communi ſolidi ſimi-<lb/>Laris, &</s> <s xml:id="echoid-s6484" xml:space="preserve">c. </s> <s xml:id="echoid-s6485" xml:space="preserve">licet ad variationem, & </s> <s xml:id="echoid-s6486" xml:space="preserve">nominationem ſimilium figurarum, <lb/>conſequenter & </s> <s xml:id="echoid-s6487" xml:space="preserve">eadem varia ſolida, variè nominari poſſent; </s> <s xml:id="echoid-s6488" xml:space="preserve">fortè autem <lb/>in ſequentibus ex genitricium figurarum variatione varia nomina com-<lb/>ponemus, interim hæc teneantur, hoc animaduerſo, quod in ſuperiori-<lb/>bus, dum ſit ſphæra, vel ſphæroides, vel eorundem portio, ſuppono li-<lb/>neas, quæ ſunt in genitricibus ſiguris, & </s> <s xml:id="echoid-s6489" xml:space="preserve">circulos, vel ellipſes deſcri- <pb o="259" file="0279" n="279" rhead="LIBER III."/> bunt, eſſe eorundem axes. </s> <s xml:id="echoid-s6490" xml:space="preserve">His autem prædemonſtratis ſequentia Corol-<lb/>laria colliguntur, quæ quidem cum Typographo deeſſent conſueti buiu-<lb/>ſcemodi caracteres, diuerſis imprimineceſſe fuit.</s> <s xml:id="echoid-s6491" xml:space="preserve"/> </p> <div xml:id="echoid-div617" type="float" level="2" n="2"> <note position="left" xlink:label="note-0278-02" xlink:href="note-0278-02a" xml:space="preserve">_@6. Lib. 1._</note> <note position="left" xlink:label="note-0278-03" xlink:href="note-0278-03a" xml:space="preserve">_47. Eiuſd._ <lb/>_lib. 1._</note> <note position="left" xlink:label="note-0278-04" xlink:href="note-0278-04a" xml:space="preserve">_46. Lib. 1._</note> <note position="left" xlink:label="note-0278-05" xlink:href="note-0278-05a" xml:space="preserve">_47. Lib. 1._</note> </div> </div> <div xml:id="echoid-div619" type="section" level="1" n="359"> <head xml:id="echoid-head376" xml:space="preserve">COROLLARIVMI.</head> <p> <s xml:id="echoid-s6492" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6493" xml:space="preserve">prima igitur, ſi ſuppoſuerimus, PRDE, eſſe circu-<lb/>lum, vel ellipſim, & </s> <s xml:id="echoid-s6494" xml:space="preserve">axem, ER, & </s> <s xml:id="echoid-s6495" xml:space="preserve">circa eandem in baſi, DP, <lb/>parallelogrammum, FP, quę quidem ſit baſis portionis, DEP, re-<lb/>ctè ſecans axim, ER, deinde intellexerimus omnes figuras ſimiles <lb/>po@@onis, DEP, eſſe circulos diametros in figura genitrice, DEP, <lb/>(cui ſut erecti) ſitos habentes, tunc, FP, erit figura genitrix ſolidi <lb/>ſimilaris, quod erit cylindrus rectus, & </s> <s xml:id="echoid-s6496" xml:space="preserve">DEP, erit figura genitrix <lb/>ſolidi præ@@ ſimilatis, quod erit portio ſphæræ, vel ſphæroidis, <lb/>cuius axis, ER, & </s> <s xml:id="echoid-s6497" xml:space="preserve">quia patet ex ſupradictis, quam rationem habeat <lb/>ſolidum ſimilare genitum ex, FP, ad ſibi ſimilare genitum ex, DE <lb/>P@uxta regulam, DP, ide@ patet ex ſupradictis, quam rationem ha-<lb/>beat cylindrus, FP, ad portionem ſphæræ, vel ſphæroidis, DEP, <lb/>ſiue, DP, tranſeat per centrum, A, ſiue non. </s> <s xml:id="echoid-s6498" xml:space="preserve">Similiter ſi ſuppoſue-<lb/>rimus, EDRP, eſſe ſphæroidem, &</s> <s xml:id="echoid-s6499" xml:space="preserve">, ER, non axim, ſed diame-<lb/>trum, & </s> <s xml:id="echoid-s6500" xml:space="preserve">ſecari per ellipſim, DSPR, obliquè ad diametrum, ER, <lb/>& </s> <s xml:id="echoid-s6501" xml:space="preserve">circa eandem diametrum, EB, in eadem baſi ellipſi, DSPR, <lb/>eſſe cylindricum, FP, ſecari autem cylindricum, & </s> <s xml:id="echoid-s6502" xml:space="preserve">portionem ſphæ-<lb/>roidis planis parallelis ipſi ellipſi, DP, quę intelligatur erecta plano, <lb/>DEPR, conceptæ in ipſo cylindrico figuræ erunt omnes figuræ ſi-<lb/> <anchor type="note" xlink:label="note-0279-01a" xlink:href="note-0279-01"/> miles parallelogrammi, FP, & </s> <s xml:id="echoid-s6503" xml:space="preserve">quæ ſiunt in portione ſphæroidis, D <lb/>EP, erunt omnes figuræ ſimiles portionis, DEP, omnes inquam <lb/>ellipſes ſimiles ellipſi, DP, (eſt enim idem, ſiue intelligas has figu-<lb/>ras ſimiles deſcribi omnibus lineis figurarum genitricium, FP, DE <lb/>P, ſiue percipias ealdem produci per ſectionem corporum per plana <lb/>factam ipſi, DP, parallela) & </s> <s xml:id="echoid-s6504" xml:space="preserve">quia patet ratio harum omnium ſimi-<lb/>lium ſigurarum, ſiue ellipſium inter ſe ex ſupradictis, & </s> <s xml:id="echoid-s6505" xml:space="preserve">ſubinde ſoli-<lb/>dorum ſimilarium genitorum ex, FP, & </s> <s xml:id="echoid-s6506" xml:space="preserve">portione, DEP, qucrum <lb/>vnum eſt cylmdricus, alterum eſt portio ſphæroidis ſecta plano, D <lb/>P, ideo patet, quam rationem habeat cylindricus, FP, ad portio-<lb/>nem, DEP, .</s> <s xml:id="echoid-s6507" xml:space="preserve">@. </s> <s xml:id="echoid-s6508" xml:space="preserve">eſſeeandem, quam habet compoſita ex ſexta parte, <lb/>EB, & </s> <s xml:id="echoid-s6509" xml:space="preserve">dimidia, BR, ad ipſam, BR, & </s> <s xml:id="echoid-s6510" xml:space="preserve">hoc, ſiue, ER, ſit axis, <lb/>ſiuenon; </s> <s xml:id="echoid-s6511" xml:space="preserve">Quod ſi, DP, tranſeat per centrum, A, cylindricum, FP, <lb/>eſſe ſexquialterum portionis ſphærę, vel ſphęroidis, DEP. </s> <s xml:id="echoid-s6512" xml:space="preserve">Ijſdem <lb/>vijs patebit conum, ſiue conicum, EDP, ad portionem ſphærę, vel <lb/>ſphæroidis, DEP, eſſe vt, BR, ad compoſitam ex, BR, & </s> <s xml:id="echoid-s6513" xml:space="preserve">dimi-<lb/>dia totius, ER, quod ſi, DP, per centrum tran@eat, conum, vel cc- <pb o="260" file="0280" n="280" rhead="GEOMETRIÆ"/> nicum, EDP, eſſe ſubduplum portionis ſphæræ, vel ſphæroidis, D <lb/>EP; </s> <s xml:id="echoid-s6514" xml:space="preserve">hæc autem etiam ab alijs oſteniſa ſunt. </s> <s xml:id="echoid-s6515" xml:space="preserve">Verum ſi ſiguræ ſimiles <lb/> <anchor type="figure" xlink:label="fig-0280-01a" xlink:href="fig-0280-01"/> iam dictæ non ſint circuli, vel elli-<lb/>pſes, ſed aliæ vtcunque figuræ, vt <lb/>ex. </s> <s xml:id="echoid-s6516" xml:space="preserve">g. </s> <s xml:id="echoid-s6517" xml:space="preserve">quadrata, veluti in figuris in <lb/>tra ellipſes exemplificare volu, dia <lb/>metros homologas in figuris gen -<lb/>tricibus habentia, adhuc eædem <lb/>rationes ſupradictis erunt inter hęc <lb/>ſolida ad inuicem ſimilaria genita <lb/>ex, FP, & </s> <s xml:id="echoid-s6518" xml:space="preserve">portione, DEP, ſiue <lb/>ex triangulo, EDP, & </s> <s xml:id="echoid-s6519" xml:space="preserve">portione, <lb/>DEP, baſes habentia quadratas; <lb/></s> <s xml:id="echoid-s6520" xml:space="preserve">patet autem hic, quod ſolidum ſi-<lb/>milare genitum ex, FP, baſem ha-<lb/>bens rectilineam, ſicuti eſt priſma, <lb/>ita & </s> <s xml:id="echoid-s6521" xml:space="preserve">hoc nomine vocari poteſt <lb/>magis particulari, veluti & </s> <s xml:id="echoid-s6522" xml:space="preserve">ſoli-<lb/>dum ſimilare genitum ex triangu-<lb/>lo, EDP, nomine piramidis vo-<lb/>cari poteſt, dum baſim habet recti-<lb/>lineam.</s> <s xml:id="echoid-s6523" xml:space="preserve"/> </p> <div xml:id="echoid-div619" type="float" level="2" n="1"> <note position="right" xlink:label="note-0279-01" xlink:href="note-0279-01a" xml:space="preserve">Corollar. <lb/>47. lib. @.</note> <figure xlink:label="fig-0280-01" xlink:href="fig-0280-01a"> <image file="0280-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0280-01"/> </figure> </div> <p> <s xml:id="echoid-s6524" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s6525" xml:space="preserve">vniuerſaliſſimè habetur <lb/>ratio quorumcumque duorum ſo-<lb/>lidorum genitorum ex, FP, & </s> <s xml:id="echoid-s6526" xml:space="preserve">por-<lb/>tione, DEP, ſiue ex triangulo, D <lb/>EP, & </s> <s xml:id="echoid-s6527" xml:space="preserve">portione, DEP, iuxta re-<lb/>gulam, DP, quacunque in ſimili-<lb/>bus figuris variatione facta. </s> <s xml:id="echoid-s6528" xml:space="preserve">Quæ <lb/>autem in huius Theorematis decla-<lb/>ratione animaduerſa ſunt, memo-<lb/>ria teneantur, nam & </s> <s xml:id="echoid-s6529" xml:space="preserve">ſequentia <lb/>conſimili methodo, ſed breuiori <lb/>declarabimus; </s> <s xml:id="echoid-s6530" xml:space="preserve">ſuſſiciat autem tot <lb/>figurarũ variationes in duabus tan-<lb/>tum exemplificaſſe, quas ſolido-<lb/>rum indicant baſes, nempè circu-<lb/>lus, & </s> <s xml:id="echoid-s6531" xml:space="preserve">quadratum, inſcriptum ei-<lb/>dem circulo, habens vtrunq; </s> <s xml:id="echoid-s6532" xml:space="preserve">dia-<lb/>metrum in figura genitrice, impo-<lb/>ſterum enim cuin ſine figurarum <lb/>confuſione id ægrè ſieri poſſit vna tantum poſitione contenti erimus, <pb o="261" file="0281" n="281" rhead="LIBER III."/> ea nempè, qua omnes figuræ ſimiles circuli eſſe ſupponuntur, cæte-<lb/>ras ergo variationes ex his facillimè auidus veritatis indagator pro-<lb/>prio marte comprehendere poterit, quæ pro huius Theor. </s> <s xml:id="echoid-s6533" xml:space="preserve">declarat. <lb/></s> <s xml:id="echoid-s6534" xml:space="preserve">adieq. </s> <s xml:id="echoid-s6535" xml:space="preserve">quoq; </s> <s xml:id="echoid-s6536" xml:space="preserve">dilucidat. </s> <s xml:id="echoid-s6537" xml:space="preserve">fatis effe reor.</s> <s xml:id="echoid-s6538" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div621" type="section" level="1" n="360"> <head xml:id="echoid-head377" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s6539" xml:space="preserve">IN Propofitione fecunda, expoſita figura Coroll. </s> <s xml:id="echoid-s6540" xml:space="preserve">antec. </s> <s xml:id="echoid-s6541" xml:space="preserve">confor-<lb/>miter, patet, quam rationem habeat folidum fimilare genitum <lb/>ex, DN, ideft cylindricus in bafi figura deſcripta à baſi, PN, cuius <lb/>latus eſt, HN, ad ſolidum ſibi ſimilare genitum ex portione, VCB <lb/>FR,. </s> <s xml:id="echoid-s6542" xml:space="preserve">i. </s> <s xml:id="echoid-s6543" xml:space="preserve">(ſi omnes figuræ ſimiles ipſius, DN, ſint circuli diametros <lb/>in, DN, habentes, & </s> <s xml:id="echoid-s6544" xml:space="preserve">omnes figuræ fimiles portionis, VCBFR, <lb/>ſint pariter circuli rectè axem, BO, ſecantes, & </s> <s xml:id="echoid-s6545" xml:space="preserve">diametros in eadem <lb/>portione ſitos habentes, quicirculi ſint genitricibus erecti) cylindrus, <lb/>DN, ad portionem ſphæræ, velſphæroidis, VCBFR, velſi figu-<lb/> <anchor type="figure" xlink:label="fig-0281-01a" xlink:href="fig-0281-01"/> ræ ſint recti lineę, patetratio, quam <lb/>habet prifma, DN, ad ſolidum ſibi <lb/>ſimilare genitum ex portione, VCB <lb/>FR, circuli, vel ellipſis, BCOF. <lb/></s> <s xml:id="echoid-s6546" xml:space="preserve">Ductis autem rectis, BP, PN, pa-<lb/>tet ſimiliter ratio, quam habet co-<lb/>nus, BPN, ad portionem ſphæræ, <lb/>vel ſphæroidis, VCBFR, fiue py <lb/>ramis, BPN, ad ſolldum ſimilare <lb/>genitum ex triangulo, BPN, (in-<lb/>telligeiemper hęc ſolida inuicem ge-<lb/>nita iuxta regulas in Theorematibus <lb/>aſſumptas, ne toties id repetatur) <lb/>ſiue tandem, quam habet vniuerſa-<lb/>liter ſolidum ſimilare genitum ex, DN, vel triangulo, BPN, ad ſo-<lb/>lidum ſibi ſimilare genitum ex portione, VCBFR, & </s> <s xml:id="echoid-s6547" xml:space="preserve">hocſi, BO, <lb/>ſit axis quod ſi tantum fit diameter eædem rationes colligentur ad <lb/>modum ſuperioris Theorematis. </s> <s xml:id="echoid-s6548" xml:space="preserve">Eſtergo in figura cylindricus, D <lb/>N, adportionem ſphæræ, vel ſphæroidis, VCBFR, vel priſma, <lb/>DN, ad ſolidum ſimilare genitum ex portione, VCBFR, veltan-<lb/>dem quodlibet ſolidum ſimilare genitum ex, DN, ſiue quilibet cy-<lb/>lindricus genitus ex, DN, ad ſolidum ſibi ſimilare genitum ex por-<lb/>tione, VCBFR, vt rectangulum ſub, BA, & </s> <s xml:id="echoid-s6549" xml:space="preserve">tripla, AO, adre-<lb/>ctangulum ſub, BM, & </s> <s xml:id="echoid-s6550" xml:space="preserve">ſub compofita ex, MO, OA. </s> <s xml:id="echoid-s6551" xml:space="preserve">Solidum ve-<lb/>rò ſimilare genitum ex triangulo, BPN, ſiue ſit conus, ſiue pyramis, <lb/>ſiue tantum conicus, ad ſibi ſimilare genitum ex portione, VCBFR, <pb o="262" file="0282" n="282" rhead="GEOMETRI E"/> ſiue hoc ſit portio ſphæræ, vel ſphæroidis, ſiue tantum ſolidum ſi-<lb/>milare genitum ex portione, VCBFR, vtrectangulum, BAO, ſiue <lb/>quadratum, BA, ad rectangulum ſub, BM, & </s> <s xml:id="echoid-s6552" xml:space="preserve">ſub compoſita ex, M <lb/>O, OA, nam ſicut rectangulum, BAO, eſt tertia pars rectanguli <lb/> <anchor type="note" xlink:label="note-0282-01a" xlink:href="note-0282-01"/> ſub, BA, & </s> <s xml:id="echoid-s6553" xml:space="preserve">tripla, AO, ita ſolidum ſimilare genitum ex triangulo, <lb/>BPN, eſt tertia pars ſolidi ſimilaris geniti ex, DN, vnde patet, &</s> <s xml:id="echoid-s6554" xml:space="preserve">c.</s> <s xml:id="echoid-s6555" xml:space="preserve"/> </p> <div xml:id="echoid-div621" type="float" level="2" n="1"> <figure xlink:label="fig-0281-01" xlink:href="fig-0281-01a"> <image file="0281-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0281-01"/> </figure> <note position="left" xlink:label="note-0282-01" xlink:href="note-0282-01a" xml:space="preserve">I. Propol. <lb/>24, lib. 2.</note> </div> </div> <div xml:id="echoid-div623" type="section" level="1" n="361"> <head xml:id="echoid-head378" xml:space="preserve">COROLLARIVM III.</head> <p> <s xml:id="echoid-s6556" xml:space="preserve">IN Propoſit. </s> <s xml:id="echoid-s6557" xml:space="preserve">tertia colligitur, quam rationem habeat ſolidum ſi-<lb/>milare genitum ex, BF, ſiue ſit cylindrus, ſiue priſma, ſiue tan-<lb/>tum conicus, ad ſibi ſimilare genitum ex portione circuli, vel ellipſis, <lb/>ICFS, ſiue hoc ſit fruſtum ſphærę, vel ſphæroidis, ſiue tantum ſo-<lb/>lidum ſimilare genitum ex portione, ICFS, ſiue, AD, ſit a@is ſiue <lb/> <anchor type="figure" xlink:label="fig-0282-01a" xlink:href="fig-0282-01"/> non, quę eadem eſt ei, quam habet re-<lb/>ctangulum, DRA, ad rectangulum <lb/>ſub, DR, & </s> <s xml:id="echoid-s6558" xml:space="preserve">ſub compoſita ex, {1/2}, R <lb/>M, & </s> <s xml:id="echoid-s6559" xml:space="preserve">ex, MA, vna cam rectangulo <lb/>ſub, RM, & </s> <s xml:id="echoid-s6560" xml:space="preserve">ſub compoſita ex, {1/6}, R <lb/>M, &</s> <s xml:id="echoid-s6561" xml:space="preserve">, {1/2}, MA. </s> <s xml:id="echoid-s6562" xml:space="preserve">Solidum autem ſimi-<lb/>lare genitum ex triangulo, MCF, ſiue <lb/>ſit conus, ſiue priſma, ſiue tantum co-<lb/>nicus, ad ſibi ſimilare genitum ex por-<lb/>tione, ICFS, ſiue hoc ſit fruſtum <lb/>ſphæræ, vel ſphæroidis, ſiue tantum <lb/>ſolidum ſimilare genitum ex tali por-<lb/>tione, ICFS, erit vt rectangulum, <lb/>DRA, ad rectangulum ſub compoſita ex, MD, DR, & </s> <s xml:id="echoid-s6563" xml:space="preserve">ſub ſex-<lb/>quialtera, MA, vna cum rectangulo ſub compoſita ex, MD, &</s> <s xml:id="echoid-s6564" xml:space="preserve">dupla, DR, & </s> <s xml:id="echoid-s6565" xml:space="preserve">ſub, {1/2}, MR, ſiue, AD, ſit axis, ſiue tantum dia-<lb/>meter, quę iuxta antecedẽtium declarationem facilè percipi poſſunt.</s> <s xml:id="echoid-s6566" xml:space="preserve"/> </p> <div xml:id="echoid-div623" type="float" level="2" n="1"> <figure xlink:label="fig-0282-01" xlink:href="fig-0282-01a"> <image file="0282-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0282-01"/> </figure> </div> </div> <div xml:id="echoid-div625" type="section" level="1" n="362"> <head xml:id="echoid-head379" xml:space="preserve">COROLLARIVM IV.</head> <p> <s xml:id="echoid-s6567" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6568" xml:space="preserve">quarta patet ratio, quam habet ſolidum ſimilare ge-<lb/>nitum ex, GX, quod apparet in ſuperioris figura, ad ſibi ſimilare <lb/>genitum ex portione, ICFS, quæratio ibi conſpiciatur.</s> <s xml:id="echoid-s6569" xml:space="preserve"/> </p> <pb o="263" file="0283" n="283" rhead="LIBER III."/> </div> <div xml:id="echoid-div626" type="section" level="1" n="363"> <head xml:id="echoid-head380" xml:space="preserve">COROLLARIVM V.</head> <p> <s xml:id="echoid-s6570" xml:space="preserve">IN Corollario Propof. </s> <s xml:id="echoid-s6571" xml:space="preserve">quintæ, ſi ſupponamus notam rationem, <lb/>quam habent omnia quadrata, AF, ad omnia quadrata trian-<lb/>guli, AEC, vel quam habent omnia quadrata, XR, ad omnia qua-<lb/>drata trapetij, YSN℟, veluti iam eam notam reddidimus, colligi-<lb/>mus, quam rationem habeant omnia quadrata, AF, ad reliquum, <lb/>demptis omnibus quadratis ſemicirculi, vel ſemiellipſis, DBF, & </s> <s xml:id="echoid-s6572" xml:space="preserve"><lb/>quam rationem habeant omnia quadrata, XR, adreliquum, dem-<lb/> <anchor type="figure" xlink:label="fig-0283-01a" xlink:href="fig-0283-01"/> ptis omnibus quadratis portionis, YT <lb/>I℟, & </s> <s xml:id="echoid-s6573" xml:space="preserve">ideò patet, quam rationem ha-<lb/>beant omnia quadrata, AF, ad om-<lb/>nia quadrata ſemicircult, vel ſemielli-<lb/>pſis, DBF, & </s> <s xml:id="echoid-s6574" xml:space="preserve">quam rationem habe-<lb/>ant omnia quadrata, XR, ad omnia <lb/>quadrata portionis, YTI℟, vnde ap-<lb/>paret, quam rationem habeat ſolidum <lb/>ſimilare genitum ex, AF, ſiue ſit cy-<lb/>lindrus, ſiue priſma, ſiue tantum cy-<lb/>lindricus, ad ſolidum ſibi ſimilare ge-<lb/>nitum ex ſemicirculo, vel ſemiellipſi, <lb/>DBF, ſiue hoc ſit hæmiſphærium, <lb/>ſiue hæmiſphæroides, ſiue tantum ſolidum ſimilare illi, genitum ex, <lb/>DBF. </s> <s xml:id="echoid-s6575" xml:space="preserve">Item patet, quam rationem habeat ſolidum ſimilare geni-<lb/>tum ex, XR, quodcunque illud ſit, ad ſibi ſimilare genitum ex por-<lb/>tione, YTI℟. </s> <s xml:id="echoid-s6576" xml:space="preserve">Eodem pacto manifeſta fieret ratio ſolidi ſimilaris <lb/>geniti ex, AG, ad ſibi ſimilare genitum ex portione, YB℟, & </s> <s xml:id="echoid-s6577" xml:space="preserve">ita <lb/>in reliquis. </s> <s xml:id="echoid-s6578" xml:space="preserve">Inuentæ igitur ſunt alio modoa prædictis, rationes fo-<lb/>lidorum inuicem ſimilarium genitorum ex parallelogrammis in bafi <lb/>æquali ſecundæ diametro conſtitutis. </s> <s xml:id="echoid-s6579" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6580" xml:space="preserve">in baſi æquali ipſi, DF, & </s> <s xml:id="echoid-s6581" xml:space="preserve"><lb/>circa eoidem axes, ſiue diametros vtcunque portionum, YB℟, T <lb/>Y℟I, DTIF, &</s> <s xml:id="echoid-s6582" xml:space="preserve">, DBF, quod explicare opus erat, & </s> <s xml:id="echoid-s6583" xml:space="preserve">in ſuprapo-<lb/>ſita ngura modo ſolito declaratum eſt, ſed tantum vnico exemplo ne <lb/>ipſa confonderetur.</s> <s xml:id="echoid-s6584" xml:space="preserve"/> </p> <div xml:id="echoid-div626" type="float" level="2" n="1"> <figure xlink:label="fig-0283-01" xlink:href="fig-0283-01a"> <image file="0283-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0283-01"/> </figure> </div> <figure> <image file="0283-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0283-02"/> </figure> <pb o="264" file="0284" n="284" rhead="GEOMETRI Æ"/> </div> <div xml:id="echoid-div628" type="section" level="1" n="364"> <head xml:id="echoid-head381" xml:space="preserve">COROLL. VI. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s6585" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6586" xml:space="preserve">ſexta, & </s> <s xml:id="echoid-s6587" xml:space="preserve">eiuſdem figura apparet ſolidum ſimilare ge-<lb/> <anchor type="figure" xlink:label="fig-0284-01a" xlink:href="fig-0284-01"/> nitum ex circulo, vel ellipſi, ABC <lb/>D, ſiue ſit ſphæra, vel ſphæroides, vel <lb/>tantum ſolidum ſimilare, ad ſolidum ſibi <lb/>ſimilare genitum ex altera portionum, B <lb/>AD, BCD, vtramuis, vtex, BAD, <lb/>ſiue hoc ſit portio ſphæræ, vel ſphæroi-<lb/>dis, vel tantum ſolidum ſimilare genitum <lb/>ex, BAD, eſſe vt parallelepipedum ſub <lb/>altitudine, XC, baſi quadrato, CA, ad <lb/>parallelepipedum ſub altitudine, XE, baſi <lb/>quadrato, EA, vel vt cubus, AC, ad pa-<lb/>rallelepipedum ſub altitudine tripla, EC, <lb/>baſi quadrato, AE, cum cubo, AE.</s> <s xml:id="echoid-s6588" xml:space="preserve"/> </p> <div xml:id="echoid-div628" type="float" level="2" n="1"> <figure xlink:label="fig-0284-01" xlink:href="fig-0284-01a"> <image file="0284-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0284-01"/> </figure> </div> </div> <div xml:id="echoid-div630" type="section" level="1" n="365"> <head xml:id="echoid-head382" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s6589" xml:space="preserve">VNde colligitur in Corollario ſolidum ſimilare genitum ex por-<lb/>tione, BAD, ad ſibi ſimilare genitum ex portione, BCD, <lb/>eſſe vt parallelepipedum ſub altitudine, XE, baſi quadrato, EA, ad <lb/>parallelepipedum ſub altitudine, OAE, baſi quadrato, EC, qua-<lb/>liacunque ſint illa ſolida ſimilaria, ſiue ſit, AC, axis, ſiue tantum <lb/>diameter.</s> <s xml:id="echoid-s6590" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div631" type="section" level="1" n="366"> <head xml:id="echoid-head383" xml:space="preserve">COROLLARIVM VII.</head> <p> <s xml:id="echoid-s6591" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6592" xml:space="preserve">ſeptima colligitur ſoli-<lb/> <anchor type="figure" xlink:label="fig-0284-02a" xlink:href="fig-0284-02"/> dum ſimilare genitum ex portio-<lb/>ne, BAF, ad ſibi ſim lare genitum ex <lb/>portione, CAD, ſiue hæciolida ſint <lb/>portiones ſphærę, vel ſphæroidis, ſiue <lb/>tantum ſolida ſimilaria, ſiue, AN, <lb/>ſit axis, ſiue tantum diameter, eſſe vt <lb/>parallelepipedum ſub altitudine, XE, <lb/>baſi quadrato, EA, ad parallelepipe-<lb/>dum ſub altitudine, XM, baſi qua-<lb/>drato, MA, vt exemplificaturin præ-<lb/>ſenti ſigura more ſolito.</s> <s xml:id="echoid-s6593" xml:space="preserve"/> </p> <div xml:id="echoid-div631" type="float" level="2" n="1"> <figure xlink:label="fig-0284-02" xlink:href="fig-0284-02a"> <image file="0284-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0284-02"/> </figure> </div> <pb o="265" file="0285" n="285" rhead="LIBER III."/> </div> <div xml:id="echoid-div633" type="section" level="1" n="367"> <head xml:id="echoid-head384" xml:space="preserve">COROLLARIVM VIII.</head> <p> <s xml:id="echoid-s6594" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6595" xml:space="preserve">octaua diſcimus à data ſphæra, vel ſphæroide, vel ſo-<lb/>lido quocunque genito ex circulo, vel ellipſi, iuxta regulam, quę <lb/>ſit vna ex ordinatim applicatis, abſcindere portionem, quæ ad ſoli-<lb/>dum ſimilare ſibi genitum ex triangulo in eadem baſi, & </s> <s xml:id="echoid-s6596" xml:space="preserve">circa eun-<lb/>dem axim, vel diametrum cum portione conſtituto, habeat datam <lb/>rationem, quam oportet eſſe maiorem ſexquialtera; </s> <s xml:id="echoid-s6597" xml:space="preserve">quæ omniaibi <lb/>clarè patent, & </s> <s xml:id="echoid-s6598" xml:space="preserve">ideo figuram non appono.</s> <s xml:id="echoid-s6599" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div634" type="section" level="1" n="368"> <head xml:id="echoid-head385" xml:space="preserve">COROLL. IX. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s6600" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6601" xml:space="preserve">nona patet ratio, quam <lb/> <anchor type="figure" xlink:label="fig-0285-01a" xlink:href="fig-0285-01"/> habet ſolidum ſimilare genitum ex <lb/>circulo, vel ellipſi, iuxta regulam pri-<lb/>mum axim, vel diametrum, ad ſolidum <lb/>ſimilare genitum ex eodem, iuxta ſe-<lb/>cundum axim, vel diametrum tamquam <lb/>regulam, ſiue hæc ſolida ſint ſphæra, vel <lb/>ſphæroides, vel tantum ſolida ſimilaria, <lb/>quæ in his appoſitis figuris clarè patent, <lb/>in quarum vna conſpici poteſt ſphęroi-<lb/>des prolatum, in altera oblongum, præ-<lb/>dicta autem ratio eſt ea, quam habet <lb/>prima axis, vel diameter ad ſecundam <lb/>axim, vel diametrum: </s> <s xml:id="echoid-s6602" xml:space="preserve">quę etiam pro re-<lb/>liquis ſolidis ad inuicem ſimilaribus ma-<lb/>nifeſta ſunt.</s> <s xml:id="echoid-s6603" xml:space="preserve"/> </p> <div xml:id="echoid-div634" type="float" level="2" n="1"> <figure xlink:label="fig-0285-01" xlink:href="fig-0285-01a"> <image file="0285-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0285-01"/> </figure> </div> </div> <div xml:id="echoid-div636" type="section" level="1" n="369"> <head xml:id="echoid-head386" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s6604" xml:space="preserve">IN Corollario autem eiuſdem Theorematis colligimus eſſe notam <lb/>rationem omnium quadratorum duarum portionum circuli, vel <lb/>ellipſis abſeiſlarum per lineas, quarum vna ſit parallela primo, alte-<lb/>ra ſecundo axi, vel diametro, quales ſint in appoſitis figuris portio-<lb/>nes, BMS, VPX, vnde etiam nota eritratio ſolidorum ſimilarium, <lb/>BMS, VPX, exipſis genitorum, vnumiuxta regulam, BS, alte-<lb/>rum iuxta regulam, VX, ſiue ſint hæc portiones ſphæræ, velſphæ-<lb/>roidis, ſiue ſolida ſimilaria genita ex portionibus, BMS, VFX.</s> <s xml:id="echoid-s6605" xml:space="preserve"/> </p> <pb o="266" file="0286" n="286" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div637" type="section" level="1" n="370"> <head xml:id="echoid-head387" xml:space="preserve">COROLL. X. SECTIO PRIMA.</head> <p> <s xml:id="echoid-s6606" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6607" xml:space="preserve">1 2. </s> <s xml:id="echoid-s6608" xml:space="preserve">dicimus, quod ſi circuli, vel ellipſes habuerint in <lb/>ſuis coniugatis axibus, vel diametris eas conditiones, quas ſup-<lb/>poſuimus in elſe lateribus parallelogrammorum in Theor.</s> <s xml:id="echoid-s6609" xml:space="preserve">9. </s> <s xml:id="echoid-s6610" xml:space="preserve">10. </s> <s xml:id="echoid-s6611" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s6612" xml:space="preserve">12. </s> <s xml:id="echoid-s6613" xml:space="preserve">13. </s> <s xml:id="echoid-s6614" xml:space="preserve">Lib. </s> <s xml:id="echoid-s6615" xml:space="preserve">2. </s> <s xml:id="echoid-s6616" xml:space="preserve">quod pro eorum circulorum, vel ellipſium omnibus <lb/>quadratis regula baſi ſequentur eædem concluſiones ibi collectæ, ſi <lb/>enim nis circumſcribantur parallelogramma latera habentia axibus, <lb/>vel diametris coniugatis circulorum, vel ellipſium parallela, habe-<lb/>bunt hæc parallelogramma requiſitas conditiones in ſuis lateribus, <lb/>& </s> <s xml:id="echoid-s6617" xml:space="preserve">ideò ſequentur iam dictæ concluſiones pro parallelogrammis, & </s> <s xml:id="echoid-s6618" xml:space="preserve"><lb/>conſequenter pro omnibus quadratis ellipſium illis inſcriptorum, <lb/>cum hæc ſint ſubſexq uialtera omnium quadratorum parallelogram-<lb/>morum illis circum ſcriptorum ideſt vt clarius loquar, ſi circulus, & </s> <s xml:id="echoid-s6619" xml:space="preserve"><lb/>ellipſis, vel duæ ellipſes fuerint circa eandem diametrum, vel circa <lb/>æquales diametros, velaxes, erunt omnia quadrata eorundem regu-<lb/>lis ſecundis axibus, vel diametris, vt omnia quadrata parallelogram-<lb/>morum illis circumſcriptibilium, latera habentium dictis axibus, vel <lb/>diametris parallela, regulis eildem retentis, & </s> <s xml:id="echoid-s6620" xml:space="preserve">quia omnia quadrata <lb/>parallelogrammorum, latera baſibus æquè inclinata & </s> <s xml:id="echoid-s6621" xml:space="preserve">qualia haben-<lb/>tium regulis baſibus, ſunt vt quadrata baſium, ideò omnia quadrata <lb/>circulorum, vel ellipſium circa eundem axim, vel diametrum, vel <lb/>æquales conſtitutorum, erunt vt quadrata ſecundorum axium, vel <lb/>diametrorum, & </s> <s xml:id="echoid-s6622" xml:space="preserve">ideò ſolida ſimilaria genita ex ipſis iuxta eaſdem <lb/>regulas, erunt vt quadrata ſecundorum axium, vel diametrorum, <lb/>quæ ſolida, veleruntſphæra, & </s> <s xml:id="echoid-s6623" xml:space="preserve">ſphæroides, vel ambo ſphæroides <lb/>circa eundem axim, vel diametrum, vel ſolida ſimilaria genita ex <lb/>dictis circulo, & </s> <s xml:id="echoid-s6624" xml:space="preserve">ellipſi, vel duabus ellipſibus iuxta dictas regulas, <lb/>quæ quoque erunt interſe, vt quadrata ſecundorum axium, vel dia-<lb/>metrorum.</s> <s xml:id="echoid-s6625" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div638" type="section" level="1" n="371"> <head xml:id="echoid-head388" xml:space="preserve">SECTIO II.</head> <p> <s xml:id="echoid-s6626" xml:space="preserve">QVod ſi in dictis figuris circulo, & </s> <s xml:id="echoid-s6627" xml:space="preserve">ellipſi, vel ellipſibus ſumatur <lb/>pro regula communis axis, vel diameter, erunt omnia qua-<lb/>drata eorundem inter ſe, vt ſecundi axes, vel diametri inter <lb/>ſe, & </s> <s xml:id="echoid-s6628" xml:space="preserve">ſic etiam erunt ſolida ſimilaria ex eiſdem genita iuxta dictam <lb/>regulam, in quibus includitur ſphæra, & </s> <s xml:id="echoid-s6629" xml:space="preserve">iphæroides.</s> <s xml:id="echoid-s6630" xml:space="preserve"/> </p> <pb o="267" file="0287" n="287" rhead="LIBER III."/> </div> <div xml:id="echoid-div639" type="section" level="1" n="372"> <head xml:id="echoid-head389" xml:space="preserve">SECTIO III.</head> <p> <s xml:id="echoid-s6631" xml:space="preserve">ITem colligimus ſolida ſimilaria genita excirculo, & </s> <s xml:id="echoid-s6632" xml:space="preserve">ellipſi, vel <lb/>ellipſibus, vtcunque iuxta datas regulas .</s> <s xml:id="echoid-s6633" xml:space="preserve">i. </s> <s xml:id="echoid-s6634" xml:space="preserve">ſphæram, & </s> <s xml:id="echoid-s6635" xml:space="preserve">ſphæroi-<lb/>des, & </s> <s xml:id="echoid-s6636" xml:space="preserve">alia quæcunque ſolida ſimilaria genita ex dictis figuris, ha-<lb/>bere inter ſe rationem ex eorum axibus, vel diametris coniugatis <lb/>compoſitam.</s> <s xml:id="echoid-s6637" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div640" type="section" level="1" n="373"> <head xml:id="echoid-head390" xml:space="preserve">SECTIO IV.</head> <p> <s xml:id="echoid-s6638" xml:space="preserve">ITem colligimus ſolida ſimilaria genita ex circulo, vel ellipſi, vel <lb/>ellipſibus, quæ habeant axes, vel diametros reciprocè quadratis <lb/>axium illis coniugatorum relpondentes iuxta quæ genita, intelligan-<lb/>tur, effe æqualia, dummodo vel vna in vtriſque ſumantur axes, vel <lb/>vna diametri æqualiter ad inuicem inclinatæ: </s> <s xml:id="echoid-s6639" xml:space="preserve">& </s> <s xml:id="echoid-s6640" xml:space="preserve">ſi hæc ſint æqualia, <lb/>illa eſſe reciprocè reſpondentia.</s> <s xml:id="echoid-s6641" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div641" type="section" level="1" n="374"> <head xml:id="echoid-head391" xml:space="preserve">SECTIO V.</head> <p> <s xml:id="echoid-s6642" xml:space="preserve">ITem habemus, quod ſphæræ, & </s> <s xml:id="echoid-s6643" xml:space="preserve">ſimilia ſphæroideia, & </s> <s xml:id="echoid-s6644" xml:space="preserve">in vni-<lb/>uerſum, quod ſolida ſimilaria genita ex circulis, vel ellipſibus ha-<lb/>bentibus axes, vel diametros in ratione ſecundorum axium, vel dia-<lb/>metrorum, cum quibus æqualiter ſintinclinati, quod, inquam, ſint <lb/>in tripla ratione axium, vel diametrorum. </s> <s xml:id="echoid-s6645" xml:space="preserve">i. </s> <s xml:id="echoid-s6646" xml:space="preserve">vt cubi eorundem. </s> <s xml:id="echoid-s6647" xml:space="preserve">Hæc <lb/>enim demonſtrata de omnibus quadratis parallelogrammorum pro <lb/>omnibus quadratis circulorum, vel ellipſium, tamquam eorundem <lb/>partibus proportionalibus (dum illis inſcripta intelliguntur) recipi <lb/>poſiunt.</s> <s xml:id="echoid-s6648" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div642" type="section" level="1" n="375"> <head xml:id="echoid-head392" xml:space="preserve">COROLLARIVM XI.</head> <p> <s xml:id="echoid-s6649" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s6650" xml:space="preserve">13. </s> <s xml:id="echoid-s6651" xml:space="preserve">colligemus ſolidum ſimilare genitum ex, OV, quod <lb/>poteſt eſſe vel cylindrus, vel prima, ad ſibi ſimilare genitum ex <lb/>trilineo, DCV, eſſe vt, OV, ad reliquum ſpatium, dempta quarta <lb/>circuli, vel ellipſis, OCD, cum exceſſu dicti quadrantis ſuper duas <lb/>tertias, rectanguli, OV, ideſt proximè, vt 21. </s> <s xml:id="echoid-s6652" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s6653" xml:space="preserve">Exponatur de <lb/>huius Theor. </s> <s xml:id="echoid-s6654" xml:space="preserve">figura tantum rectangulum, OV, cum quarta, OCD, <lb/>dimiffa, EF, ſi igitur intelligemus, OV, circa, DV, manentem re-<lb/>uolui, quoad redeat, vnde diſceſſit, defcribetur, ab, OV, cylindrus, <lb/>OA, ideſt ſolidum ſimilare genitum ex, OV, cuius omnes figuræ <pb o="268" file="0288" n="288" rhead="GEOMETRIÆ"/> ſimiles ſunt circuli, ſemidiametros in figura genitrice, OV, haben-<lb/>tes, à trilineo autem, DCV, deſcribetur quoddam ſolidum, quod <lb/> <anchor type="figure" xlink:label="fig-0288-01a" xlink:href="fig-0288-01"/> vocetur, Apex ſphæralis, ſi, OCD, ſit <lb/>quarta circuli; </s> <s xml:id="echoid-s6655" xml:space="preserve">vel ſphæroidalis, ſi, OC <lb/>D, ſit quarta ellipſis, ideſt ſolidum ſimila-<lb/>re, quod poteſt dici genitum ex trilineo, <lb/>DCV, habens omnes ſuas ſimiles figuras <lb/>circulos ſemidiametros in figura genitri-<lb/>ce, DCV, ſitos habentes, eſt igitur inter <lb/>hæc duo ſimilaria ſolida, quęin particulari <lb/>hoc exemplo ſunt cylindrus, & </s> <s xml:id="echoid-s6656" xml:space="preserve">apex ſphę-<lb/>ralis, vel ſphæroidalis, ratio eadem ſupradictę, quam breuitatis caufa <lb/>aliter exemplificare dimiſi. </s> <s xml:id="echoid-s6657" xml:space="preserve">Conſimili autem vnico exemplo.</s> <s xml:id="echoid-s6658" xml:space="preserve">f. </s> <s xml:id="echoid-s6659" xml:space="preserve">affu-<lb/>mendo pro ſiguris ſimilibus ipſos circulos, ſemidiametros in figuris <lb/>genitricibus habentibus, breuitatis cauſa, & </s> <s xml:id="echoid-s6660" xml:space="preserve">ob ſeruandam in figuris <lb/>claritatem impoſterum contenti erimus.</s> <s xml:id="echoid-s6661" xml:space="preserve"/> </p> <div xml:id="echoid-div642" type="float" level="2" n="1"> <figure xlink:label="fig-0288-01" xlink:href="fig-0288-01a"> <image file="0288-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0288-01"/> </figure> </div> </div> <div xml:id="echoid-div644" type="section" level="1" n="376"> <head xml:id="echoid-head393" xml:space="preserve">COROLLARIVM XII.</head> <p> <s xml:id="echoid-s6662" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6663" xml:space="preserve">14. </s> <s xml:id="echoid-s6664" xml:space="preserve">patet ratio ſolidi ſimilaris geniti ex, FD, ad ſoli-<lb/>dum ſimilare genitum ex figura, FQBDZ. </s> <s xml:id="echoid-s6665" xml:space="preserve">Appoſita. </s> <s xml:id="echoid-s6666" xml:space="preserve">n. </s> <s xml:id="echoid-s6667" xml:space="preserve">hic <lb/>illius figura, dimiffa, HC, &</s> <s xml:id="echoid-s6668" xml:space="preserve">, AP, & </s> <s xml:id="echoid-s6669" xml:space="preserve">retenta, BD, tantum in, V, <lb/>diuifa, reuoluatur, FD, circa manentem axim, ZD, modò ſupra-<lb/> <anchor type="figure" xlink:label="fig-0288-02a" xlink:href="fig-0288-02"/> dicto, ex, FD, igitur fiet cylin-<lb/>drus, FA, & </s> <s xml:id="echoid-s6670" xml:space="preserve">ex figura, FQBD <lb/>Z, fiet quoddam ſolidum rotun-<lb/>dum, quod vocetur, Tympanum <lb/>fphærale, ſi, FQB, ſit ſemicircu-<lb/>lus, vel ſphæroidale, ſi, FQB, ſit <lb/>ellipſis, erunt autem hæc duo ſo-<lb/>lida ſimilaria genita ex figuris, F <lb/>D, FQBDZ, figuras ſimiles cir-<lb/>culos habentia, quorum ſemidia-<lb/>metri iacent in ſuis genitricibus fi-<lb/>guris, & </s> <s xml:id="echoid-s6671" xml:space="preserve">patet, quod ratio cylin-<lb/>dri, FA, ad tympanum ſphærale, vel ſphæroidale, FQRA, eſt ea-<lb/>dem ei, quam habet, BD, ad, DV, eandem autem habet quodli-<lb/>bet ſolidum ſimilare genitum ex, FD, ad ſimilare ſibi genitum ex fi-<lb/>gura, FQBDZ, qualecunque ſit.</s> <s xml:id="echoid-s6672" xml:space="preserve"/> </p> <div xml:id="echoid-div644" type="float" level="2" n="1"> <figure xlink:label="fig-0288-02" xlink:href="fig-0288-02a"> <image file="0288-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0288-02"/> </figure> </div> <pb o="269" file="0289" n="289" rhead="LIBEN III."/> </div> <div xml:id="echoid-div646" type="section" level="1" n="377"> <head xml:id="echoid-head394" xml:space="preserve">COROLLARIVM XIII.</head> <p> <s xml:id="echoid-s6673" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6674" xml:space="preserve">13. </s> <s xml:id="echoid-s6675" xml:space="preserve">colligimus ſolidum ſimilare genitum ex, HF, ad <lb/>ſolidum ſimilare genitum ex figura, NMBEF, demptis ſolidis <lb/>ſimilaribus genitis à trilineis, MNG, GFE, eſſe vt, HF, ad circu-<lb/>lum, velellipſim, MBEG. </s> <s xml:id="echoid-s6676" xml:space="preserve">Reuoluatur, HF, circa, NF, manen-<lb/>tem, vt ſupra, ex, HF, igitur fiet cylindrus, H ℟, & </s> <s xml:id="echoid-s6677" xml:space="preserve">ex figura, N <lb/> <anchor type="figure" xlink:label="fig-0289-01a" xlink:href="fig-0289-01"/> MBEF, fiet quædam figu-<lb/>ra, à qua ſi auferantur ſoli-<lb/>da, quæ fiunt à duobus tri-<lb/>lineis, MNG, GFE, re-<lb/>manebit quædam figura ſo-<lb/>lida, quam vocabimus, Anu-<lb/>lum ſtrictum circularem, ſi, <lb/>MBEG, ſit circulus; </s> <s xml:id="echoid-s6678" xml:space="preserve">Elli-<lb/>pticum verò, ſi ſit ellipſis, & </s> <s xml:id="echoid-s6679" xml:space="preserve"><lb/>patebit, quam rationem ha-<lb/>beat cylindrus, H℟, ad hunc <lb/>anulum ſtrictum, AI, ſicuti vniuerſaliter patet ex ſupradictis, quam <lb/>rationem habeat ſolidum ſimilare genitum ex, HF, ad ſolidum ſibi <lb/>ſimilare genitum ex figura, NMBEF, demptis ſolidis ſimilaribus <lb/>genitis ex trilineis, MNG, GFE.</s> <s xml:id="echoid-s6680" xml:space="preserve"/> </p> <div xml:id="echoid-div646" type="float" level="2" n="1"> <figure xlink:label="fig-0289-01" xlink:href="fig-0289-01a"> <image file="0289-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0289-01"/> </figure> </div> </div> <div xml:id="echoid-div648" type="section" level="1" n="378"> <head xml:id="echoid-head395" xml:space="preserve">COROLLARIVM XIV.</head> <p> <s xml:id="echoid-s6681" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6682" xml:space="preserve">16. </s> <s xml:id="echoid-s6683" xml:space="preserve">patet, quam rationem habet ſolidum ſimilare ge-<lb/>nitum ex, HO, dempto ſolido ſimilari genito ex, NO, ad ſoli-<lb/> <anchor type="figure" xlink:label="fig-0289-02a" xlink:href="fig-0289-02"/> dum ſibi ſimilare genitum ex figura, MBEOC, dempto ſolido ſi-<lb/>milari genito ex figura, MGEOC, .</s> <s xml:id="echoid-s6684" xml:space="preserve">i. </s> <s xml:id="echoid-s6685" xml:space="preserve">effe eandem ei, quam habet, <pb o="270" file="0290" n="290" rhead="GEOMETRIÆ"/> HF, ad circulum, vel ellipſim, MB, EG. </s> <s xml:id="echoid-s6686" xml:space="preserve">Reuoluatur, HO, cir-<lb/>ca manentem axem, CO, modo ſuprad cto, ex, HO, igitur fiet cy-<lb/>lindrus, H ℟, & </s> <s xml:id="echoid-s6687" xml:space="preserve">ex figura, CMBEO, fiet quoddam ſolidum ſimi-<lb/>lare prædicto cylindro, auferatur à cylindro, H ℟, cylindrus, NL, <lb/>defcriptus ab, NO, & </s> <s xml:id="echoid-s6688" xml:space="preserve">à prædicto ſolido ſimilari auferatur ſolidum <lb/>ſimilare genitum ex figura, MGEOC. </s> <s xml:id="echoid-s6689" xml:space="preserve">Dico nos iam compertum <lb/>habere reſiduum primum. </s> <s xml:id="echoid-s6690" xml:space="preserve">i. </s> <s xml:id="echoid-s6691" xml:space="preserve">faſciam ſolidam cylindricam (vtita di-<lb/> <anchor type="figure" xlink:label="fig-0290-01a" xlink:href="fig-0290-01"/> cam) HFLK, ad reſiduum ſecundum, ad ſolidum, inquam, quod <lb/>gignitur ex reuolutione circuli, vel ellipſis, MBEG, effe vt, HF, <lb/>ad ipſum circulum, vel ellipſim, MBEG; </s> <s xml:id="echoid-s6692" xml:space="preserve">quod etiam patet de re-<lb/>fiduis quorumlibet ſimilarium ſolidorum ex, HO, & </s> <s xml:id="echoid-s6693" xml:space="preserve">figura, MBE <lb/>OC, genitorum, demptis ſolidis ſimilaribus genitis ex, NO, & </s> <s xml:id="echoid-s6694" xml:space="preserve">fi-<lb/>gura, MGEOC, vt ſupra diximus. </s> <s xml:id="echoid-s6695" xml:space="preserve">Vocetur autem ſolidum, quod <lb/>in ſupradicto exemplo, & </s> <s xml:id="echoid-s6696" xml:space="preserve">figura gignitur ex reuolutione circuli, vel <lb/>ellipſis, MBEG, Anulus latus circularis, ſi, MBEG, ſit circulus, <lb/>vel, Anulus latus ellipticus, ſi, MBEG, ſit ellipſis.</s> <s xml:id="echoid-s6697" xml:space="preserve"/> </p> <div xml:id="echoid-div648" type="float" level="2" n="1"> <figure xlink:label="fig-0289-02" xlink:href="fig-0289-02a"> <image file="0289-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0289-02"/> </figure> <figure xlink:label="fig-0290-01" xlink:href="fig-0290-01a"> <image file="0290-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0290-01"/> </figure> </div> </div> <div xml:id="echoid-div650" type="section" level="1" n="379"> <head xml:id="echoid-head396" xml:space="preserve">COROLL. XV. SECTIO PRIMA.</head> <p> <s xml:id="echoid-s6698" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s6699" xml:space="preserve">17. </s> <s xml:id="echoid-s6700" xml:space="preserve">colligitur ſolidum ſimilare genitum ex, HF, ad fo-<lb/>lidum ſimilare genitum ex figura, NMBEF, eſſe vt, HF, ad <lb/>circulum, vel ellipſim, MBEG, vna cum reſiduo, quod remanet, <lb/>ſi à rectangulo, MG, dematur quarta circuli, vel ellipſis, MAG, <lb/>ablato inſimul exceſſu, quo eadem quarta ſuperat duas tertias rectan-<lb/>guli, MG. </s> <s xml:id="echoid-s6701" xml:space="preserve">Conſpiciatur ergo exemplum in figura Coroll. </s> <s xml:id="echoid-s6702" xml:space="preserve">13. </s> <s xml:id="echoid-s6703" xml:space="preserve">huius, <lb/>patebit ergo cylindrum, H ℟, ad ſolidum genitum ex figura, NM <lb/>BEF, habere ſupradictam rationem, que eſt proximè, vt 21. </s> <s xml:id="echoid-s6704" xml:space="preserve">ad 17. <lb/></s> <s xml:id="echoid-s6705" xml:space="preserve">vt in Propof. </s> <s xml:id="echoid-s6706" xml:space="preserve">17. </s> <s xml:id="echoid-s6707" xml:space="preserve">huius oſtenditur. </s> <s xml:id="echoid-s6708" xml:space="preserve">Vocetur autem ſolidum ſimilare <pb o="271" file="0291" n="291" rhead="LIBER III."/> genitum ex figura, NMBEF, habens omnes ſuas figuras ſimiles, <lb/>quæ ſint circuli, ſiue quod fiet per reuolutionem dictæ figuræ, NM <lb/>BEF, vocetur, inquam. </s> <s xml:id="echoid-s6709" xml:space="preserve">Baſis columnaris ſtricta, & </s> <s xml:id="echoid-s6710" xml:space="preserve">circularis, ſi, <lb/>MBEG, ſit circulus, elliptica autem, ſi is fit ellipſis.</s> <s xml:id="echoid-s6711" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div651" type="section" level="1" n="380"> <head xml:id="echoid-head397" xml:space="preserve">SECTIO II.</head> <p> <s xml:id="echoid-s6712" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s6713" xml:space="preserve">i. </s> <s xml:id="echoid-s6714" xml:space="preserve">colligitur ſolidum ſimilare genitum ex, HF, ad ſo-<lb/>lidum ſimilare genitum ex figura, NMBEF, dempto ſolido ſi-<lb/>milari genito ex, MF, eſſe proximè, vt 84. </s> <s xml:id="echoid-s6715" xml:space="preserve">ad 47. </s> <s xml:id="echoid-s6716" xml:space="preserve">ideſt in noſtro <lb/>exemplo cylindrum, H ℟, ad baſem columnarem ſtrictam genitam <lb/>ex figura, NMBEF, dempto cylindro, MX, eſſe proximè, vt 84. <lb/></s> <s xml:id="echoid-s6717" xml:space="preserve">ad 47.</s> <s xml:id="echoid-s6718" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div652" type="section" level="1" n="381"> <head xml:id="echoid-head398" xml:space="preserve">SECTIO III.</head> <p> <s xml:id="echoid-s6719" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s6720" xml:space="preserve">2. </s> <s xml:id="echoid-s6721" xml:space="preserve">habetur ſolidum ſimilare genitum ex, MF, demptis <lb/>ſolidis ſimilaribus genitis ex trilineis, MNG, GFE, ad ſolidum <lb/>ſibi ſimilare genitum ex figura, NMBEF, dempto ſolido ſimilari <lb/>genito ex, MF, eſſe proximè, vt 19. </s> <s xml:id="echoid-s6722" xml:space="preserve">ad 47. </s> <s xml:id="echoid-s6723" xml:space="preserve">In exemplo autem no-<lb/>ſtro, dum reuoluitur, HF, apprehende ſuperficiem cylindricam de-<lb/>ſcriptam linea, ME, quę in duas partes diſſeparat anulum ſtrictum, <lb/>AI, ſcilicet in vnam, quam poſſumus vocare interiorem, & </s> <s xml:id="echoid-s6724" xml:space="preserve">in aliam <lb/>exteriorem; </s> <s xml:id="echoid-s6725" xml:space="preserve">interior eſt, quę gigniture ex reuolutione ſemicirculi, vel <lb/>ſemiellipſis, MGE; </s> <s xml:id="echoid-s6726" xml:space="preserve">exterior autem, quæ generatur ex ſemicirculo, <lb/>vel ſemiellipſi, MBE, eſtigitur hæc pars interior anuli ſtrictiad par-<lb/>tem exteriorem proximè, vt 19. </s> <s xml:id="echoid-s6727" xml:space="preserve">ad 47. </s> <s xml:id="echoid-s6728" xml:space="preserve">vtin cæteris ſolidis ſimilari-<lb/>bus ſupradictis contingere diximus.</s> <s xml:id="echoid-s6729" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div653" type="section" level="1" n="382"> <head xml:id="echoid-head399" xml:space="preserve">COROLL. XVI. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s6730" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6731" xml:space="preserve">18. </s> <s xml:id="echoid-s6732" xml:space="preserve">habemus ſolidum ſimilare genitum ex, HO, ad <lb/>ſolidum ſimilare genitum ex figura, CMBEO, eſſe vt quadra-<lb/>tum, DO, ad rectangulum ſub, DO, OE, vna cum rectangulo <lb/>ſub, OE, & </s> <s xml:id="echoid-s6733" xml:space="preserve">ſub exceſiu, quo dupla, EI, ſuperat, EF, &</s> <s xml:id="echoid-s6734" xml:space="preserve">, {2/3}, qua-<lb/>drati, DE. </s> <s xml:id="echoid-s6735" xml:space="preserve">Exemplum conſpici poteſt in figura Coroll. </s> <s xml:id="echoid-s6736" xml:space="preserve">14. </s> <s xml:id="echoid-s6737" xml:space="preserve">huius, <lb/>in qua ſolidum ſimilare genitum ex, HO, eſt cylindrus, H ℟, ſoli-<lb/>dum verò ſimilare genitum ex figura, CMBEO, eſt, quod naſcitur <lb/>ex reuolutione eiuſdem figurę circa, CO, quod ſolidum vocabimus. <lb/></s> <s xml:id="echoid-s6738" xml:space="preserve">Baſem columnarem latam, circularem, ſi, MBEG, ſit circulus, el-<lb/>lipticam verò, ſi ſit ellipſis.</s> <s xml:id="echoid-s6739" xml:space="preserve"/> </p> <pb o="272" file="0292" n="292" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div654" type="section" level="1" n="383"> <head xml:id="echoid-head400" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s6740" xml:space="preserve">IN huius Corollario colligitur ſolidum ſimilare genitum ex, HP, <lb/>ad ſolidum ſimilare genitum ex figura, CMSBP, dempto ſoli-<lb/>do ſimilari genito extrapezio, MBPC, ideſtin exemplo cylindrum <lb/>genitum ex reuolutione, HP, ad mediam baſem columnarem latam <lb/>genitam ex figura, MSBPC, dempto fruſto conico genito ex tra-<lb/>pezio, CMBP, eſſe vt quadratum, BP, adrectangulum ſub, AP, <lb/>& </s> <s xml:id="echoid-s6741" xml:space="preserve">ſub exceſſu duplæ, EI, ſuper, EF, vna cum, {1/3}, quadrati, BA. <lb/></s> <s xml:id="echoid-s6742" xml:space="preserve">Ex quibus etiam facilè inueniri poteſt, quam rationem habeat ſoli-<lb/>dum ſimilare genitum ex figura, MSBPC, ad ſolidum ſimilare ge-<lb/>nitum ex trapezio, MBPC, .</s> <s xml:id="echoid-s6743" xml:space="preserve">i. </s> <s xml:id="echoid-s6744" xml:space="preserve">quam rationem habeat, in exemplo, <lb/>media baſis columnaris lata genita ex reuolutione, MXBPC, ad <lb/>fruſtum conicum genitum ex reuolutione trapezij, MBPC.</s> <s xml:id="echoid-s6745" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div655" type="section" level="1" n="384"> <head xml:id="echoid-head401" xml:space="preserve">COROLL. XVII. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s6746" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6747" xml:space="preserve">19. </s> <s xml:id="echoid-s6748" xml:space="preserve">colligimus ſolidum ſimilare genitum ex figura, S <lb/>MNV, dempto ſolido ſimilari genito ex quadrilineo, MNV <lb/> <anchor type="figure" xlink:label="fig-0292-01a" xlink:href="fig-0292-01"/> T, ad ſolidum ſibi ſimilare <lb/>genitum ex figura, SBEG <lb/>T, demptis ſolidis ſimilari-<lb/>bus genitis ex trilineis, TV <lb/>G, GFE, eſſe vt portio, S <lb/>MT, ad portionem, SBE <lb/>GT, circuli, vel ellipſis, M <lb/>BEG; </s> <s xml:id="echoid-s6749" xml:space="preserve">ideſt in propoſito <lb/>exemplo, ſolidum, quod <lb/>generatur ex portione, SM <lb/>T, dum intelligimus, HF, <lb/>reuolui circa, NF, manentem axim, ad ſolidum, quod generatur ex <lb/>portione, SBEGT, erit vt portio, SMT, ad portionem, SBEGT.</s> <s xml:id="echoid-s6750" xml:space="preserve"/> </p> <div xml:id="echoid-div655" type="float" level="2" n="1"> <figure xlink:label="fig-0292-01" xlink:href="fig-0292-01a"> <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0292-01"/> </figure> </div> </div> <div xml:id="echoid-div657" type="section" level="1" n="385"> <head xml:id="echoid-head402" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s6751" xml:space="preserve">IN Corollario eiuſdem colligimus ſolida ſimilaria genita ex paral-<lb/>lelogrammis circa eoſdem axes, cum portionibus conſtitutos ad <lb/>ſolida ſibi ſimilaria genita ex eiſdem portionibus, eſſe vt dicta paral-<lb/>lelogramma ad dictas portiones. </s> <s xml:id="echoid-s6752" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6753" xml:space="preserve">in exemplo cylindrum, HO, 2 <lb/>fruſtum anuli ſtricti reſectum ſuperficie deſeripta linea, ST, erit v <lb/>HV, ad portionem, SMT, & </s> <s xml:id="echoid-s6754" xml:space="preserve">item cylindrus, R ℟, deſcriptus a<unsure/> <pb o="273" file="0293" n="293" rhead="LIBER III."/> RF, ad portionem anuliſtricti deſcriptam portionem, SBEGT. <lb/></s> <s xml:id="echoid-s6755" xml:space="preserve">erit vt, RF, adeandem portionem, quod patet etiam dereliquis ec-<lb/>rundem ſolidis ſimilaribus.</s> <s xml:id="echoid-s6756" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div658" type="section" level="1" n="386"> <head xml:id="echoid-head403" xml:space="preserve">COROLLARIVM XVIII.</head> <p> <s xml:id="echoid-s6757" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6758" xml:space="preserve">20. </s> <s xml:id="echoid-s6759" xml:space="preserve">expoſita figura, & </s> <s xml:id="echoid-s6760" xml:space="preserve">exemplo conſtructo, oſtendi-<lb/>mus pariter ſolidum deſcriptum à portione, SMT, ad ſolidum <lb/>deſcriptum à portione, SBEGT, dum, HO, reuoluitur circa ma-<lb/>nentem axim, CO, eſſe vt portio, SMT, ad portionem, SBGET. <lb/></s> <s xml:id="echoid-s6761" xml:space="preserve">quod etiam de reliquis ſolidis ſimilaribus ab eiſdem portionibus ge-<lb/>nitis patet.</s> <s xml:id="echoid-s6762" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6763" xml:space="preserve">In huius autem Corollario colligitur ſolida ſimilaria genita ex pa-<lb/>rallelogrammis, cum portionibus in eadem altitudine exiſtentibus, <lb/>& </s> <s xml:id="echoid-s6764" xml:space="preserve">ad rectas, HD, CO, terminantibus, demptis ſolidis ſimilaribus <lb/>genitis ex parallelogrammis in eadem altitudine cum dictis portioni-<lb/>bus exiſtentibus, ſed ad rectas, NF, CO, terminantibus, adſolida <lb/> <anchor type="figure" xlink:label="fig-0293-01a" xlink:href="fig-0293-01"/> ſibi ſimilaria genita ex figuris compoſitis ex dictis portionibus, & </s> <s xml:id="echoid-s6765" xml:space="preserve">re-<lb/>liquo ſpatio, viq; </s> <s xml:id="echoid-s6766" xml:space="preserve">ad, CO, dempto ſolido ſimilari genito ex hocre-<lb/>liquo ſpatio, eſſe vt dictorum parallelogrammorum reſiduum paral-<lb/>lelogrammum ad dictam portionem. </s> <s xml:id="echoid-s6767" xml:space="preserve">Vtin exemplo cylindrum, H <lb/>Y, dempto cylindro, NQ, ad portionem anulilati reſectam per ſu-<lb/>perficiem deſcriptam in reuolutione a linea, ST, eſſe vt, HV, ad <lb/>portionem, SMT.</s> <s xml:id="echoid-s6768" xml:space="preserve"/> </p> <div xml:id="echoid-div658" type="float" level="2" n="1"> <figure xlink:label="fig-0293-01" xlink:href="fig-0293-01a"> <image file="0293-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0293-01"/> </figure> </div> <pb o="274" file="0294" n="294" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div660" type="section" level="1" n="387"> <head xml:id="echoid-head404" xml:space="preserve">COROLLARIVM XIX.</head> <p> <s xml:id="echoid-s6769" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6770" xml:space="preserve">22. </s> <s xml:id="echoid-s6771" xml:space="preserve">expoſita figura, & </s> <s xml:id="echoid-s6772" xml:space="preserve">exemplo conſtituto, colligi-<lb/>mus ſolidum ſimilare genicum ex, AG, ad ſolidum ſimilare ge-<lb/>nitum ex figura, LCFEG, deinptis ſolidis ſimilaribus genitis ex tri-<lb/>lineis, CLT, YGE, eſſe (ſi, CFEH, ſit circulus) vt parallelepi-<lb/>pedum ſub baſi parallelogrammo, AG, altitudine, FI, ad cylindri-<lb/> <anchor type="figure" xlink:label="fig-0294-01a" xlink:href="fig-0294-01"/> cum ſub baſi maiori portione, TC <lb/>FEY, altitudine, IM, vna cum, <lb/>{3/6}, cubi, TY. </s> <s xml:id="echoid-s6773" xml:space="preserve">In ellipſi verò, vt <lb/>parallelepipedum ſub baſiparalle-<lb/>logrammo, AG, altitudine, FI, <lb/>ad cylindricum ſub baſi portione, <lb/>TCFEY, altitudine, MI, vna <lb/>cum ea parte cubi, TY, vel (rhom <lb/>bo ab eadem, TY, deſoripto, vt in <lb/>Theor. </s> <s xml:id="echoid-s6774" xml:space="preserve">21.) </s> <s xml:id="echoid-s6775" xml:space="preserve">parallelepipedi ſub, T <lb/>Y, & </s> <s xml:id="echoid-s6776" xml:space="preserve">dicto rhombo, ad quam eiu-<lb/>ſdem cubi, vel parallelepipedi ſex-<lb/>ta pars ſit, vt quadratum, CE, primæ axis, ad quadratum ſecundæ <lb/>.</s> <s xml:id="echoid-s6777" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6778" xml:space="preserve">ad quadratum, FH. </s> <s xml:id="echoid-s6779" xml:space="preserve">Sit ergo conſtitutum exemplum per reuolu-<lb/>tionem, AG, circa manentem axim, LG, ſiue ergo, CFEH, ſit <lb/>circulus, ſiue ellipſis, habebit genitus cylindrus ab, AG, ad genitum <lb/>ſolidum a portione, TCFEY, ſupradictam rationem. </s> <s xml:id="echoid-s6780" xml:space="preserve">Vocetur au-<lb/>tem ſolidum deſcriptum à portione, TCFEY, (ſi ſit portio circu-<lb/>li) malum roſeum; </s> <s xml:id="echoid-s6781" xml:space="preserve">ſi verò ſit portio ellipſis: </s> <s xml:id="echoid-s6782" xml:space="preserve">Malum cotoneum.</s> <s xml:id="echoid-s6783" xml:space="preserve"/> </p> <div xml:id="echoid-div660" type="float" level="2" n="1"> <figure xlink:label="fig-0294-01" xlink:href="fig-0294-01a"> <image file="0294-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0294-01"/> </figure> </div> </div> <div xml:id="echoid-div662" type="section" level="1" n="388"> <head xml:id="echoid-head405" xml:space="preserve">COROLLARIVM XX.</head> <p> <s xml:id="echoid-s6784" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s6785" xml:space="preserve">23. </s> <s xml:id="echoid-s6786" xml:space="preserve">ium-<lb/> <anchor type="figure" xlink:label="fig-0294-02a" xlink:href="fig-0294-02"/> pta ex fig. </s> <s xml:id="echoid-s6787" xml:space="preserve">Theor. <lb/></s> <s xml:id="echoid-s6788" xml:space="preserve">21. </s> <s xml:id="echoid-s6789" xml:space="preserve">portione minori <lb/>vtcunque, RFV, <lb/>quæ ſit portio circu-<lb/>li, cum illi circum-<lb/>ſcripto rectangulo, Δ <lb/>V, aſſumpto etiam <lb/>integro axi, FH, & </s> <s xml:id="echoid-s6790" xml:space="preserve"><lb/>puncto, @, in ea, vt <lb/>ibi ſumptum eſt, pa-<lb/>tet ſolidum ſimilare <lb/>genitum ex, Δ V, ad <lb/>ſolidum ſibi ſimilare <pb o="275" file="0295" n="295" rhead="LIBER III."/> genitum ex portione minori, RFV, eſſe vt ſexquialtera, FM, ad, M <lb/>w. </s> <s xml:id="echoid-s6791" xml:space="preserve">Reuoluatur igitur, vt ſiat noſtrum exemplum, Δ V, circa, RV, <lb/>manentem, cylindrus igitur deſcriptus à, Δ V, ad ſolidum deſcriptum <lb/>à portione, RFV, erit vt ſexquialtera, FM, ad, M w, & </s> <s xml:id="echoid-s6792" xml:space="preserve">ſic dere-<lb/>liquis ſolidis ſimilaribus ab ipſis genitis, &</s> <s xml:id="echoid-s6793" xml:space="preserve">c. </s> <s xml:id="echoid-s6794" xml:space="preserve">Vocetur autem ſolidum <lb/>deſcriptum per reuolutionem à portione circuli, RFV, minori; </s> <s xml:id="echoid-s6795" xml:space="preserve">Ma-<lb/>lum citrium.</s> <s xml:id="echoid-s6796" xml:space="preserve"/> </p> <div xml:id="echoid-div662" type="float" level="2" n="1"> <figure xlink:label="fig-0294-02" xlink:href="fig-0294-02a"> <image file="0294-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0294-02"/> </figure> </div> </div> <div xml:id="echoid-div664" type="section" level="1" n="389"> <head xml:id="echoid-head406" xml:space="preserve">COROLLARIVM XXI.</head> <p> <s xml:id="echoid-s6797" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6798" xml:space="preserve">24. </s> <s xml:id="echoid-s6799" xml:space="preserve">aſſumpta adhuc figura ſuperioris, quæ reuoluitur, <lb/>colligitur ſolidum ſimilare genitum ex portione, RFV, iuxta <lb/>axim, FM, regulam, ad ſolidum ſimilare genitum ex eadem, iuxta <lb/>baſim, RV, regulam eſſe, vt rectangulum ſub, w M, & </s> <s xml:id="echoid-s6800" xml:space="preserve">ſub baſi, R <lb/> <anchor type="figure" xlink:label="fig-0295-01a" xlink:href="fig-0295-01"/> V, ad tria quadrata <lb/>lineę, RM, cum qua-<lb/>drato, MF. </s> <s xml:id="echoid-s6801" xml:space="preserve">Fiat no-<lb/>ſtrum exemplum per <lb/>reuolutionẽ portio-<lb/>nis, RFV, ſemel cir-<lb/>ca, RV, & </s> <s xml:id="echoid-s6802" xml:space="preserve">iterum cir-<lb/>ca, FM, manentes <lb/>axes, primò igitur fit: <lb/></s> <s xml:id="echoid-s6803" xml:space="preserve">Malum citrium per <lb/>reuolutionem circa, <lb/>RV, ſecundò fit ſeg-<lb/>mentum ſphæræ per <lb/>reuolutionem circa, FM, patet ergo, quam rationem habeat Ma-<lb/>lum citrium, ad ſegmentum ſphæræ ab eadem circuli portione per <lb/>reuolutionem genitum, quod etiam de reliquis ſimilaribus ſolidis ab <lb/>eadem portione, iuxta dictas regulas genitis concluſum eſt.</s> <s xml:id="echoid-s6804" xml:space="preserve"/> </p> <div xml:id="echoid-div664" type="float" level="2" n="1"> <figure xlink:label="fig-0295-01" xlink:href="fig-0295-01a"> <image file="0295-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0295-01"/> </figure> </div> </div> <div xml:id="echoid-div666" type="section" level="1" n="390"> <head xml:id="echoid-head407" xml:space="preserve">COROLLARIVM XXII.</head> <p> <s xml:id="echoid-s6805" xml:space="preserve">IN Propof. </s> <s xml:id="echoid-s6806" xml:space="preserve">25. </s> <s xml:id="echoid-s6807" xml:space="preserve">ſi ſumamus ex figura Theorem. </s> <s xml:id="echoid-s6808" xml:space="preserve">21. </s> <s xml:id="echoid-s6809" xml:space="preserve">portionem cir-<lb/>culi, vel ellipſis, RFV, vtcunque, cum integra axi, FH, à quá <lb/>ſit abſciſſa, IH, æqualis, FM, ſumatur inſuper de, MH, ipſa, M <lb/>T, ad quam, FM, ſit vt, Δ V, ad portionem, RFV, coll gemus, ex-<lb/>poſita hic figura, ſolidum ſimilare genitum ex, Δ V, ad ſibi ſimilare <lb/>genitum ex portione, BFV, eſſe vt quadratum, FM, ad rectangu-<lb/>lûm, quod remanet, dempto rectangulo ſub, IM, & </s> <s xml:id="echoid-s6810" xml:space="preserve">ſub, TM, à <lb/>rectangulo ſub, FM, &</s> <s xml:id="echoid-s6811" xml:space="preserve">, {2/3}, ipſius, MH. </s> <s xml:id="echoid-s6812" xml:space="preserve">Fiat noſtrum exemplum;</s> <s xml:id="echoid-s6813" xml:space="preserve"> <pb o="276" file="0296" n="296" rhead="GEOMETRIÆ"/> reuoluatur, Δ V, circa manentem axim, RV, cylindrus igitur geni-<lb/> <anchor type="figure" xlink:label="fig-0296-01a" xlink:href="fig-0296-01"/> tus ex reuolutione, <lb/>Δ V, adſolidum ge-<lb/>nitum ex reuolutio-<lb/>ne portionis, RF <lb/>V, habebit ſupradi-<lb/>ctam rationem; </s> <s xml:id="echoid-s6814" xml:space="preserve">hoc <lb/>autem ſolidumiam <lb/>vocauimus: </s> <s xml:id="echoid-s6815" xml:space="preserve">Malum <lb/>citrium, ſi, RFV, <lb/>ſit portio circuli, ce-<lb/>terum, ſi ſit portio <lb/>ellipſis, vocetur; </s> <s xml:id="echoid-s6816" xml:space="preserve">O-<lb/>liua genita ex tali <lb/>portione;</s> <s xml:id="echoid-s6817" xml:space="preserve">eadem au-<lb/>tem rationem habe-<lb/>re ſolida ſimilaria genita ex, Δ V, & </s> <s xml:id="echoid-s6818" xml:space="preserve">portione, RFV, (intellige ſem-<lb/>pergenita iuxta regulam ibi aſſumptam, ſcilicet iuxta regulam, FM,) <lb/>iam ſuperius diximus.</s> <s xml:id="echoid-s6819" xml:space="preserve"/> </p> <div xml:id="echoid-div666" type="float" level="2" n="1"> <figure xlink:label="fig-0296-01" xlink:href="fig-0296-01a"> <image file="0296-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0296-01"/> </figure> </div> </div> <div xml:id="echoid-div668" type="section" level="1" n="391"> <head xml:id="echoid-head408" xml:space="preserve">COROLLARIVM XXIII.</head> <p> <s xml:id="echoid-s6820" xml:space="preserve">IN Propof. <lb/></s> <s xml:id="echoid-s6821" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0296-02a" xlink:href="fig-0296-02"/> 26. </s> <s xml:id="echoid-s6822" xml:space="preserve">peiſdẽ <lb/>antecedẽtis fi-<lb/>guris colligi-<lb/>tur ſolidu ſi-<lb/>milare genitũ <lb/>iuxta regulã, <lb/>FM, ad ſibi <lb/>ſimilare geni-<lb/>tum iuxta re-<lb/>gulam, RV, <lb/>eſſe vt paral-<lb/>lelepiped. </s> <s xml:id="echoid-s6823" xml:space="preserve">ſub <lb/>baſi rectãgu-<lb/>lo, qđ dicitur <lb/>reſiduum an-<lb/>reced. </s> <s xml:id="echoid-s6824" xml:space="preserve">Theor. <lb/></s> <s xml:id="echoid-s6825" xml:space="preserve">altitudine tri-<lb/>pla, MH, ad <lb/>parallelepipedum ſub baſi rectangulo ipſius, FM, ductæ in, RV, al-<lb/>titudine linea compoſita ex, MH, HN. </s> <s xml:id="echoid-s6826" xml:space="preserve">Pronoſtro exemplo ap- <pb o="277" file="0297" n="297" rhead="LIBER III."/> ponatur hic vtraq; </s> <s xml:id="echoid-s6827" xml:space="preserve">portio, quæ reuoluantur ſemel circa, FM, & </s> <s xml:id="echoid-s6828" xml:space="preserve">ſe-<lb/>mel circa, RV, pateb tergo, quam rationem habeat, Malum citrium <lb/>ad ſegmentum ſphæræ genitum ab eadem portione circuli, & </s> <s xml:id="echoid-s6829" xml:space="preserve">quam <lb/>habeat Oliua ad ſegmentum ſphęroidis genitum ex eadem portione.</s> <s xml:id="echoid-s6830" xml:space="preserve"/> </p> <div xml:id="echoid-div668" type="float" level="2" n="1"> <figure xlink:label="fig-0296-02" xlink:href="fig-0296-02a"> <image file="0296-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0296-02"/> </figure> </div> </div> <div xml:id="echoid-div670" type="section" level="1" n="392"> <head xml:id="echoid-head409" xml:space="preserve">COROLLARIVM XXIV.</head> <p> <s xml:id="echoid-s6831" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s6832" xml:space="preserve">27. </s> <s xml:id="echoid-s6833" xml:space="preserve">aſſumitur iterum fig. </s> <s xml:id="echoid-s6834" xml:space="preserve">Theor. </s> <s xml:id="echoid-s6835" xml:space="preserve">21. </s> <s xml:id="echoid-s6836" xml:space="preserve">tum circuli, tum el-<lb/>lipſis, & </s> <s xml:id="echoid-s6837" xml:space="preserve">nunc, jſdem figuris hicappoſitis, colligimus ſolidum ſi-<lb/>milare genitum ex portione, TCFEY, tuxta regulam, FI, ad ſolidum <lb/>fibi ſimilare genitum ex eadem portione, iuxta regulam, TY, eſſe, in <lb/>fig. </s> <s xml:id="echoid-s6838" xml:space="preserve">circul, vt cylindr cum ſub, IM, & </s> <s xml:id="echoid-s6839" xml:space="preserve">portione, TCFEY, vna cum, {1/6}, <lb/>cubi, TY, ad parallelepipedum ſub altitudine, FI, baſiverò rectangulo <lb/>ſub, FI, & </s> <s xml:id="echoid-s6840" xml:space="preserve">ſub ſexquitertia duarum, IH, HN. </s> <s xml:id="echoid-s6841" xml:space="preserve">In ellipſis verò fig. </s> <s xml:id="echoid-s6842" xml:space="preserve">habe-<lb/>re rationem cõpoſitam ex ea, quam habet cylindricus ſub, IM, & </s> <s xml:id="echoid-s6843" xml:space="preserve">por-<lb/> <anchor type="figure" xlink:label="fig-0297-01a" xlink:href="fig-0297-01"/> tione, TCFEY, vna cum ea <lb/>parte cubi, TY, vel parallelep-<lb/>pedi, ſub, RV, & </s> <s xml:id="echoid-s6844" xml:space="preserve">rhombo, RZ, <lb/>ad quam eiuſdem cubi, vel pa-<lb/>rallelepipedi ſexta pars ſit, vt <lb/>quadratũ, CE, ad quadratum, <lb/>FH, ad parallelepipedum ſub <lb/>altitudine, CE, baſi parallelo-<lb/>grammo, AG, in fig. </s> <s xml:id="echoid-s6845" xml:space="preserve">Th. </s> <s xml:id="echoid-s6846" xml:space="preserve">6. </s> <s xml:id="echoid-s6847" xml:space="preserve">& </s> <s xml:id="echoid-s6848" xml:space="preserve"><lb/>ex ea, quã habet quadratum, F <lb/>H, ad rectangulum ſub, FI, & </s> <s xml:id="echoid-s6849" xml:space="preserve"><lb/>ſub ſexquitertia duarum, IH, <lb/>HN. </s> <s xml:id="echoid-s6850" xml:space="preserve">Pro noſtro igitur exem-<lb/>plo reuoluantur portiones, T <lb/>CFEY, ſemel circa axes ma-<lb/>nentes, TY, & </s> <s xml:id="echoid-s6851" xml:space="preserve">ſemel circa axes <lb/>manentes, FI; </s> <s xml:id="echoid-s6852" xml:space="preserve">ex reuolutione <lb/>igitur facta à portione circuli <lb/>circa, TY, fit, Malum Roſeum, <lb/>ex reuolutione verò eiuſdẽ cir-<lb/>ca, FI, fit maius ſegmentum <lb/>ſphæræ: </s> <s xml:id="echoid-s6853" xml:space="preserve">Item ex reuolutione <lb/>facta à portione ellipſi, TCFE <lb/>Y, fit, malum cotoneum, circa <lb/>axim, TY, ex reuolutione verò eiuſdem circa, FI, fit maius ſegmentum <lb/>ſphæroidis: </s> <s xml:id="echoid-s6854" xml:space="preserve">Igicur malum roſeum ad ſegmentum maius ſphæræ, & </s> <s xml:id="echoid-s6855" xml:space="preserve"><lb/>malum cotoneum ad ſegmentum maius ſphæroidis iam dictum, ha-<lb/>bent ſupradictam rationem, vt & </s> <s xml:id="echoid-s6856" xml:space="preserve">ſolida ſimilaria, &</s> <s xml:id="echoid-s6857" xml:space="preserve">c.</s> <s xml:id="echoid-s6858" xml:space="preserve"/> </p> <div xml:id="echoid-div670" type="float" level="2" n="1"> <figure xlink:label="fig-0297-01" xlink:href="fig-0297-01a"> <image file="0297-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0297-01"/> </figure> </div> <pb o="278" file="0298" n="298" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div672" type="section" level="1" n="393"> <head xml:id="echoid-head410" xml:space="preserve">COROLLARIVM XXV.</head> <p> <s xml:id="echoid-s6859" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s6860" xml:space="preserve">28. </s> <s xml:id="echoid-s6861" xml:space="preserve">aſſumitur adhuc antecedentis figura, hic autem <lb/>colligimus nos, ſuperiores aſpicientes figuras pro noſtro exem-<lb/>plo, ſolidum ſimilare genitum ex portione circuli, vel ellipſis, TC <lb/>FEY, ad ſolidum ſimilare genitum ex circulo, velellipſi, iuxta com-<lb/>munem regulam, FH, (comparatis tamen genitis vel ambo ex ijs, <lb/>quæ ſunt circuli, vel ex ijs, quæ ſunt ipſius ellipſis) eſſe vt cylindri-<lb/>cum ſub altitudine, MI, baſi portione, TCFEY, vna cum, {1/6}, cu-<lb/>bi, TY, (quod tamen ſolum in circuli figu a contingit) in figura au-<lb/>tem ellipſis illud commutamus in hoc. </s> <s xml:id="echoid-s6862" xml:space="preserve">ſ. </s> <s xml:id="echoid-s6863" xml:space="preserve">vna cum ea parte cubi, TY, <lb/>vel parallelepipedi ſub, RV, & </s> <s xml:id="echoid-s6864" xml:space="preserve">rhombo, RZ, ad quam eiuſdem cu-<lb/>bi, vel parallelepipedi ſexta pars ſit, vt quadratum primi axis ad qua-<lb/>dratum ſecundi ad, {2/3}, parallelepipedi ſub, AD, & </s> <s xml:id="echoid-s6865" xml:space="preserve">parallelogram-<lb/>mo, AQ,. </s> <s xml:id="echoid-s6866" xml:space="preserve">i. </s> <s xml:id="echoid-s6867" xml:space="preserve">in figura circuli, ad, {2/3}, cubi, FH. </s> <s xml:id="echoid-s6868" xml:space="preserve">Dictam igitur ra-<lb/>tionem in ſuprapoſitis exemplis habet Malum Roſeum, ad ſphęram <lb/>genitam ex circulo, ex cuius portione maiori Malum Roſeum dici-<lb/>tur genitum iuxta regulam, FH; </s> <s xml:id="echoid-s6869" xml:space="preserve">& </s> <s xml:id="echoid-s6870" xml:space="preserve">eandem habet Malum Coto-<lb/>neum ad ſphæroides genitum ex ellipſi reuoluta circa axem, CE, pa-<lb/>rallelam axi, TY, circa quem reuoluitur portio, TCFEY, ad ge-<lb/>nerandum Malum Cotoneum, quam rationem pariter diximus ha-<lb/>bere ſupradicta ſimilaria ſolida, &</s> <s xml:id="echoid-s6871" xml:space="preserve">c.</s> <s xml:id="echoid-s6872" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div673" type="section" level="1" n="394"> <head xml:id="echoid-head411" xml:space="preserve">COROLLARIVM XXVI.</head> <p> <s xml:id="echoid-s6873" xml:space="preserve">IN Propoſit. </s> <s xml:id="echoid-s6874" xml:space="preserve">29. </s> <s xml:id="echoid-s6875" xml:space="preserve">habetur ſolidum ſimilare genitum ex, AN, ad <lb/> <anchor type="figure" xlink:label="fig-0298-01a" xlink:href="fig-0298-01"/> ſolidum ſimilare genitum ex figu-<lb/>ra, CBDMN, demptis ſolidis ſimila-<lb/>ribus genitis ex trilineis, ſiue figuris, B <lb/>CO, ONM, eſſe vt, AN, ad figuram, <lb/>BDMO. </s> <s xml:id="echoid-s6876" xml:space="preserve">Apponatur hie illa figura, <lb/>&</s> <s xml:id="echoid-s6877" xml:space="preserve">, vt fiat noſtrum exemplum, reuol-<lb/>uatur, AN, quod ſupponamus eſſe pa-<lb/>rallelogrammum rectangulum conue-<lb/>nienter ipſi reuolutioni, circa axim, C <lb/>N, manentem, fiet igitur ex, AN, cy-<lb/>lindrus, ex reuolutione verò figuræ, B <lb/>DMO, fiet ſolidum totupliciter varia-<lb/>bile, quotupliciter figura, BDMO, <lb/>variari poteſt, vocabimus autem ſolida <lb/>genita à figuris inſcriptis rectangulo, AN, genita inquam per reuo- <pb o="279" file="0299" n="299" rhead="LIBER III."/> lutionem circa, CN. </s> <s xml:id="echoid-s6878" xml:space="preserve">Solida anularia ſtricta, patet ergo cylindrum <lb/>genitum ab, AN, ad ſolidum anulare ſtrictum genitum ex figura, B <lb/>DMO, quæcunque ſit, eſſe vt, AN, ad eandem figuram, BDM <lb/>O; </s> <s xml:id="echoid-s6879" xml:space="preserve">ſicq; </s> <s xml:id="echoid-s6880" xml:space="preserve">eſſe cætera ſolida ſimilaria genita ex his, iuxta ſumptamre. <lb/></s> <s xml:id="echoid-s6881" xml:space="preserve">gulam ſiue, CN, ſiue, NE, vtrifq; </s> <s xml:id="echoid-s6882" xml:space="preserve">ſolidis communem.</s> <s xml:id="echoid-s6883" xml:space="preserve"/> </p> <div xml:id="echoid-div673" type="float" level="2" n="1"> <figure xlink:label="fig-0298-01" xlink:href="fig-0298-01a"> <image file="0298-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0298-01"/> </figure> </div> </div> <div xml:id="echoid-div675" type="section" level="1" n="395"> <head xml:id="echoid-head412" xml:space="preserve">COROLLARIVM XXVII.</head> <p> <s xml:id="echoid-s6884" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s6885" xml:space="preserve">30. </s> <s xml:id="echoid-s6886" xml:space="preserve">colligimus ſolidum ſimilare genitum ex, AF, dem-<lb/>pto ſolido ſimilari genito ex, CF, ad ſolidum ſimilare genitum <lb/>ex figura, HBDMF, dempto ſolido ſimilari genito ex figura, HB <lb/>OMF, eſſe vt, AN, ad figuram, BDMO. </s> <s xml:id="echoid-s6887" xml:space="preserve">Aſſumatur hic illius fi-<lb/> <anchor type="figure" xlink:label="fig-0299-01a" xlink:href="fig-0299-01"/> gura, & </s> <s xml:id="echoid-s6888" xml:space="preserve">pro noſtro exemplo <lb/>ſupponatur, AF, eſſe rectan-<lb/>gulum, reuoluaturq; </s> <s xml:id="echoid-s6889" xml:space="preserve">circa ma-<lb/>nentem axim, HF, cylindrus <lb/>ergo genitus ex, AF, dempto <lb/>cylindro genito ex, CF, ad ſo-<lb/>lidum genitum in reuolutione <lb/>ex figura, BDMO, erit vt, A <lb/>N, ad, BDMO; </s> <s xml:id="echoid-s6890" xml:space="preserve">ſolida autem <lb/>genita ex figuris inſcriptis re-<lb/>ctangulo, BDMO, cum con-<lb/>ditionibus ibirequiſitisvocabi-<lb/>mus communiter: </s> <s xml:id="echoid-s6891" xml:space="preserve">Solida anu-<lb/>laria lata; </s> <s xml:id="echoid-s6892" xml:space="preserve">eadem patent de cę-<lb/>teris ſolidis ſimilaribus genitis <lb/>ex, AN, & </s> <s xml:id="echoid-s6893" xml:space="preserve">figura, BDMO, etiamſi, AF, non ſit rectangulum, <lb/>quia tunc intelligo fieri generationem ſolidorum per deſcriptionem <lb/>ſimilium figumrum, & </s> <s xml:id="echoid-s6894" xml:space="preserve">non per reuolutionem, vt in exemplo ſolito <lb/>aſſumpſi, vnde patet, &</s> <s xml:id="echoid-s6895" xml:space="preserve">c.</s> <s xml:id="echoid-s6896" xml:space="preserve"/> </p> <div xml:id="echoid-div675" type="float" level="2" n="1"> <figure xlink:label="fig-0299-01" xlink:href="fig-0299-01a"> <image file="0299-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0299-01"/> </figure> </div> </div> <div xml:id="echoid-div677" type="section" level="1" n="396"> <head xml:id="echoid-head413" xml:space="preserve">COROLLARIVM XXVIII.</head> <head xml:id="echoid-head414" xml:space="preserve">SECTIO PRIOR.</head> <p> <s xml:id="echoid-s6897" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s6898" xml:space="preserve">32. </s> <s xml:id="echoid-s6899" xml:space="preserve">docemur ſolidum ſimilare genitum ex, AR, ad ſo-<lb/>lidum ſibi ſimilare genitum ex figura, BCGRMD, demptis <lb/>ſolidis ſimilaribus genitis ex trilineis, BCG, GRM, eſſe vt, AR, <pb o="280" file="0300" n="300" rhead="GEOMETRIÆ"/> ad ellipſim, BDMG; </s> <s xml:id="echoid-s6900" xml:space="preserve">ponatur hic illa figura, &</s> <s xml:id="echoid-s6901" xml:space="preserve">, vt fiatnoſtrum <lb/> <anchor type="figure" xlink:label="fig-0300-01a" xlink:href="fig-0300-01"/> exemplum, reuoluatur, AR, circa <lb/>manentem axim, CR; </s> <s xml:id="echoid-s6902" xml:space="preserve">cylindrus ergo <lb/>genitus ex, AR, ad ſolidum genitum <lb/>in reuolutione ex ellipſi, BDMG, erit <lb/>vt, AR, ad ellipſim, BDMG, ſic <lb/>etiam, vt diximus, cætera ſolida ſimi-<lb/>laria ex ijſdem per deſcriptionem ſimi-<lb/>lium figurarum genita: </s> <s xml:id="echoid-s6903" xml:space="preserve">Vocetur au <lb/>tem ſolidum in reuolutione genitum <lb/>ex ellipſi, BDMG; </s> <s xml:id="echoid-s6904" xml:space="preserve">Anulus ſtrictus <lb/>ellipticus altera parte latior.</s> <s xml:id="echoid-s6905" xml:space="preserve"/> </p> <div xml:id="echoid-div677" type="float" level="2" n="1"> <figure xlink:label="fig-0300-01" xlink:href="fig-0300-01a"> <image file="0300-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0300-01"/> </figure> </div> </div> <div xml:id="echoid-div679" type="section" level="1" n="397"> <head xml:id="echoid-head415" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s6906" xml:space="preserve">IN Corollario colligitur ſolidum ſimilare genitum ex, AR, ad ſo-<lb/>lidum ſibi ſimilare genitum ex ellipſi, BDMG, ambo iuxta <lb/>communem regulam, FR, eſſe vt ſolidum ſimilare genitum ex eo-<lb/>dem, AR, ad ſolidum ſimilare ſibi genitum ex eadem ellipſi, BDM <lb/>G, ſed ambo genita iuxta communem regulam, CR. </s> <s xml:id="echoid-s6907" xml:space="preserve">Exemplum <lb/>patebit, ſi concipies, AR, reuolui circa manentem axim, FR, cy-<lb/>lindrus enim tunc genitus, ab, AR, ad anulum ſtrictum ellipticum <lb/>altera parte latiorem, genitum ab ellipſi, BDMG, habebit ean-<lb/>dem rationem, quam ſupradictus cylindrus ad ſupradictum anulum, <lb/>Crideò (amplius colligemus) quoniam, permutando, cylindrus ge-<lb/>nitus in reuolutione circa, CR, facta, ad cylindrum genitum in re-<lb/>uolutione circa, FR, eſt vt anulus factus in illa reuolutione ad anu-<lb/>lum factum in hac, propterea ſicuti primus cylindrus ad ſecundum <lb/> <anchor type="note" xlink:label="note-0300-01a" xlink:href="note-0300-01"/> eſt, vt, FR, ad, RC, ita primus anulus ad ſecundum erit, vt, FR, <lb/>ad, RC, ſic etiam erunt ſolida ſimilaria genita ex eiſdem, iuxtare-<lb/>gulas, FR, RC.</s> <s xml:id="echoid-s6908" xml:space="preserve"/> </p> <div xml:id="echoid-div679" type="float" level="2" n="1"> <note position="left" xlink:label="note-0300-01" xlink:href="note-0300-01a" xml:space="preserve">N. Cor.4. <lb/>Gen. 34. <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div681" type="section" level="1" n="398"> <head xml:id="echoid-head416" xml:space="preserve">COROLL. XXIX. SECTIO PRIMA.</head> <p> <s xml:id="echoid-s6909" xml:space="preserve">IN Propoſit. </s> <s xml:id="echoid-s6910" xml:space="preserve">33. </s> <s xml:id="echoid-s6911" xml:space="preserve">colligimus ſolidum ſimilare genitum ex, AX, <lb/>dempto ſolido ſimilari genito ex, CX, ad ſolidum ſibi ſimilare <lb/>genitum ex figura, BDMXT, dempto ſolido ſimilari genito ex fi-<lb/>gura, BGMXT, eſſe vt, AR, ad ellipſim, BDMG; </s> <s xml:id="echoid-s6912" xml:space="preserve">quod ſi ſu-<lb/>mantur ſolida ſimilaria genita ex eiſdem iuxta communem regulam, <lb/>TX, vel, CR, eandem rationem inter ſe habere comperientur di-<lb/>cta reſidua ſcilicet quam habet, AR, ad ellipſim, BDMG. </s> <s xml:id="echoid-s6913" xml:space="preserve">Expo-<lb/>natur figura, &</s> <s xml:id="echoid-s6914" xml:space="preserve">, vt fiat exemplum, reuoluatur, AX, circa manen- <pb o="281" file="0301" n="301" rhead="LIBER III."/> tem axim, TX, igitur cyhndrus genitus in reuolutione ex, AX, <lb/> <anchor type="figure" xlink:label="fig-0301-01a" xlink:href="fig-0301-01"/> dempto cylindro genito <lb/>ex, CX, ad ſolidum ge-<lb/>nitum in reuolutione ex <lb/>ellipſi, BDMG, erit vt, <lb/>AR, ad ellipſim, BDM <lb/>G; </s> <s xml:id="echoid-s6915" xml:space="preserve">idem accidet, ſireuo-<lb/>lutio ſiat circa axem pa-<lb/>rallelam ipſi, AC, inclu-<lb/>ſam duabus, FA, RC, <lb/>verſus, A, C, puncta pro-<lb/>ductis: </s> <s xml:id="echoid-s6916" xml:space="preserve">vocetur autem ſo-<lb/>lidum genitum inreuolu-<lb/>tione ex ellipſi, BDMG, anulus latus ellipticus altera parte ſtrictior.</s> <s xml:id="echoid-s6917" xml:space="preserve"/> </p> <div xml:id="echoid-div681" type="float" level="2" n="1"> <figure xlink:label="fig-0301-01" xlink:href="fig-0301-01a"> <image file="0301-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0301-01"/> </figure> </div> </div> <div xml:id="echoid-div683" type="section" level="1" n="399"> <head xml:id="echoid-head417" xml:space="preserve">SECTIO II.</head> <p> <s xml:id="echoid-s6918" xml:space="preserve">HInc inſimul patet, quod faſcia ſolida cylindrica (vt ita dicam) <lb/>in reuolutione circa, TX, genita ex, AR, ad anulum geni-<lb/>tum ex ellipſi, BDMG, in eadem reuolutione, eſt vt cylindrus ge-<lb/>nitus ex, AR, dum reuolutio ſit circa, CR, ad anulum ſtrictum el-<lb/>lipticum altera parte latiorem in eadem reuolutione circa, CR, at-<lb/>ellipſi, BDMG, genitum; </s> <s xml:id="echoid-s6919" xml:space="preserve">nam ambo ſunt, vt, AR, ad ellipſim, <lb/>BDMG, dem patet pro ſolidis ſimilaribus, &</s> <s xml:id="echoid-s6920" xml:space="preserve">c. </s> <s xml:id="echoid-s6921" xml:space="preserve">Quia verò dicta fa-<lb/>ſcia ſolida genita ab, AR, ad cylindrum ab eodem, AR, genitum <lb/>eſt, vt reſiduum quadrati, FX, dempto quadrato, RX, ad quadra-<lb/>tum, FR, eſt. </s> <s xml:id="echoid-s6922" xml:space="preserve">n. </s> <s xml:id="echoid-s6923" xml:space="preserve">cylindrus genitus ab, AX, ad cylindrum genitum <lb/>ab, AR, vt quadratum, FX, ad quadratum, FR, cylindrus item <lb/>genitusà, CX, ad eundem cy ndrum genitum ab, AR, eſt vt qua-<lb/>dratum, RX, ad quadratum, RF, ergo hoc cylindro dempto à cy-<lb/>lindro genito ab, AX, reliqua faſcia ſolida genita ex, AR, ad cy-<lb/>lindrum genitum ex eodem, AR, erit vt reſiduum quadrati, FX, ab <lb/>eo dempto quadi<unsure/>ato, RX, ad quadratum, FR, hanc ergo ratio-<lb/>nem habebit etiam anulus latus ellipticus altera parte ſtrictior ad <lb/>anulum ſtrictum ellipticum altera parte latiorem ex eadem ellipſi, B <lb/>DMG, genitum; </s> <s xml:id="echoid-s6924" xml:space="preserve">quia vero reſiduum quadrati, FX, dempto qua-<lb/>drato, RX, eſtrectangulum ſub, XR, RF, bis cum quadrato, FR, <lb/>ideſt rectangulum ſub, XF, FR, cum rectangulo ſub, XR, RF, .</s> <s xml:id="echoid-s6925" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s6926" xml:space="preserve">rectangulum ſub compoſita ex, RX, XF, & </s> <s xml:id="echoid-s6927" xml:space="preserve">iub, FR, ideò dictus <lb/>anulus latus ad dictum anulum ſtrictum, erit vt rectangulum ſub <lb/>compoſita ex, RX, XF, & </s> <s xml:id="echoid-s6928" xml:space="preserve">ſub, FR, ad quadratum, FR, .</s> <s xml:id="echoid-s6929" xml:space="preserve">i. </s> <s xml:id="echoid-s6930" xml:space="preserve">erit <lb/>vt compoſita ex, FX, XR, ad, RF, nempè vt, VR, ad, RF.</s> <s xml:id="echoid-s6931" xml:space="preserve"/> </p> <pb o="282" file="0302" n="302" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div684" type="section" level="1" n="400"> <head xml:id="echoid-head418" xml:space="preserve">SECTIO III.</head> <p> <s xml:id="echoid-s6932" xml:space="preserve">VLterius habemus faſcias ſolidas cylindricas genitas exempligr. <lb/></s> <s xml:id="echoid-s6933" xml:space="preserve">ab eodem rectangulo, AR, dum ſit reuolutio ſemel circa, T <lb/>X, & </s> <s xml:id="echoid-s6934" xml:space="preserve">ſemel circa parallelam, AC, ad anulos latos ellipticos altera <lb/>parte ſtrictiores genitos in reuolutionibus ab ellipſi, BDMG, ha-<lb/>bere eandem rationem ſcilicet quam habet, AR, ad ellipſim, BDM <lb/>G, & </s> <s xml:id="echoid-s6935" xml:space="preserve">ideò inter ſe dictos anulos eſſe, vt dictas faſcias, dictæ autem <lb/>faſciæ ſolidæ cylindricæ ſunt, vt reſidua, demptis à quadratis ſemi-<lb/>diametrorum baſium integrorum cylindrorum quadratis ſemidiame-<lb/>trorum baſium cylindrorum, quas dictæ faſciæ complectuntur, & </s> <s xml:id="echoid-s6936" xml:space="preserve"><lb/>ideò dicti anuli inter ſe eandem rationem habebunt, quam dicta qua-<lb/>dratorum reſidua.</s> <s xml:id="echoid-s6937" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div685" type="section" level="1" n="401"> <head xml:id="echoid-head419" xml:space="preserve">SECTIO IV.</head> <p> <s xml:id="echoid-s6938" xml:space="preserve">IN Corollario huius tandem dicitur, quòd ſi, BDMG, non <lb/>eſſet ellipſis, tum in Schemate huius, tum Theorematis antece-<lb/>dentis, ſed alia vtcunque ſigura habens tamen prædictas conditiones <lb/>ibi appoſitas, quod de eadem dicta quoque de ellipſi, BDMG, ve-<lb/>riſicarentur, noſque hic colligimus, quod omnia ſupradicta æquè, <lb/>ac deſolidis genitis ab ellipſi, BDMG, de genitis abipſa figura pa-<lb/>riter veriſicarentur. </s> <s xml:id="echoid-s6939" xml:space="preserve">Poſſumus autem vocare ſolida deſcripta per <lb/>reuolutionem factam circa, CR, à ſigura, BDMG. </s> <s xml:id="echoid-s6940" xml:space="preserve">Solida anu-<lb/>laria ſtricta altera parte latiora: </s> <s xml:id="echoid-s6941" xml:space="preserve">quæ verò ſiunt ab eadem per reuolu-<lb/>tionem circa, TX. </s> <s xml:id="echoid-s6942" xml:space="preserve">Solida anularia lata altera parte ſtrictiora.</s> <s xml:id="echoid-s6943" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div686" type="section" level="1" n="402"> <head xml:id="echoid-head420" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s6944" xml:space="preserve">POſſent quidem plura alia circa bæc ſolida conſiderari; </s> <s xml:id="echoid-s6945" xml:space="preserve">bt ſi ſecentur <lb/>planis parallelis, ad axem, circa quem ſit reuolutio, exiſtentibus <lb/>rectis, quam inter ſerationem babeant reſecta ſegmenta. </s> <s xml:id="echoid-s6946" xml:space="preserve">Item reſtat <lb/>contemplandum ſolidum, quod naſceretur ex reuolutione dimidiæ elli-<lb/>pſis circa non axem, ſed diametrum, vel diametro parallelam; </s> <s xml:id="echoid-s6947" xml:space="preserve">quæ vo-<lb/>luta circa diametrum ſolidum deſcribit referens ſiguram Pyri; </s> <s xml:id="echoid-s6948" xml:space="preserve">circa, <lb/>berò parallelam diametro portionem maiorem ab ellipſireſecantem, de-<lb/>ſcribit quoddam ſolidum latius ex vna parte, quam ex alia, referens ſi-<lb/>guram Mali paradiſi, vt vulgò dicitur, circa berò parallelam diametro <pb o="283" file="0303" n="303" rhead="LIBER III."/> reuolutà, quæ ab ellipſi minorem abſcindat portionem, deſcribit quod-<lb/>dam ſolidum referens ſiguram Fici, pluraque bis ſimilia contemplandæ <lb/>remanerent, ſed bt ſtudioſo Lectori in agro hoc fertiliſſimo laborandi, il-<lb/>lumq; </s> <s xml:id="echoid-s6949" xml:space="preserve">excolendi non omnis bideatur ſublatus eſſe locus, illius hæc inqui-<lb/>ſitioni reſeruare bolui bis. </s> <s xml:id="echoid-s6950" xml:space="preserve">Aduerte autem in ſuperioribus licet ſigura-<lb/>rum aſſumpti ſuerint axes, bt circa eoſdem ſieret reuolutio, tamen ea-<lb/>dem beriſicart aſſumptis, quæ ſunt tantum diametri, nam paſſiones Se-<lb/>ctionum Conicarum eiſdem inſunt, ſiue ſint circa axes, ſiue circa tan-<lb/>tum diametros, bt babetur Libro Primo Scbolio Propoſitionis 40.</s> <s xml:id="echoid-s6951" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div687" type="section" level="1" n="403"> <head xml:id="echoid-head421" xml:space="preserve">Finis Tertij Libri.</head> <figure> <image file="0303-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0303-01"/> </figure> <pb file="0304" n="304"/> <pb o="285" file="0305" n="305" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div688" type="section" level="1" n="404"> <head xml:id="echoid-head422" xml:space="preserve">CAVALER II</head> <head xml:id="echoid-head423" xml:space="preserve">LIBER QVARTVS.</head> <head xml:id="echoid-head424" xml:space="preserve">In quo de Parabola, & ſolidis ab eadem <lb/>genitis enucleatur doctrina.</head> <figure> <image file="0305-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0305-01"/> </figure> </div> <div xml:id="echoid-div689" type="section" level="1" n="405"> <head xml:id="echoid-head425" xml:space="preserve">THEOREMAI. PROPOS. I.</head> <p> <s xml:id="echoid-s6952" xml:space="preserve">SI PARALLELOGRAMMVM, & </s> <s xml:id="echoid-s6953" xml:space="preserve">trian-<lb/>gulum fuerint in eadem baſi, & </s> <s xml:id="echoid-s6954" xml:space="preserve">circa <lb/>eundem axim, vel diametrum cum pa-<lb/>rabola; </s> <s xml:id="echoid-s6955" xml:space="preserve">parallelogrammum erit para-<lb/>bolæ ſexquialterum, triangulum au-<lb/>tem erit eiuſdem parabolæ ſubſexqui-<lb/>tertium.</s> <s xml:id="echoid-s6956" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6957" xml:space="preserve">Sit ergo parabola, FCH, in baſi, FH, circa axim, vel diame-<lb/>trum, CG, ſit autem in eadem baſi, FH, & </s> <s xml:id="echoid-s6958" xml:space="preserve">circa eundem axim, vel <lb/>diametrum parallelogrammum quoq; </s> <s xml:id="echoid-s6959" xml:space="preserve">AH, & </s> <s xml:id="echoid-s6960" xml:space="preserve">triangulum, CFH. <lb/></s> <s xml:id="echoid-s6961" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0305-02a" xlink:href="fig-0305-02"/> Dico ergo parallelogrammum, AH, <lb/>eſſe ſexquialterum parabolæ, FCH; <lb/></s> <s xml:id="echoid-s6962" xml:space="preserve">triangulum autem, CFH, eſſe eiuſdem <lb/>parabolæ, FCH, ſubſexquitertium. </s> <s xml:id="echoid-s6963" xml:space="preserve"><lb/>Sumatur ergoin, CE, quæ tangit pa-<lb/>rabolam in puncto, C, vtcunque pun-<lb/>ctum, N, & </s> <s xml:id="echoid-s6964" xml:space="preserve">per, N, ducatur ipſi, CG, <lb/>parallela, NO, producta vſque ad ba-<lb/>ſim, FH, cui occurrantin, O; </s> <s xml:id="echoid-s6965" xml:space="preserve">quæ pa-<lb/>riter ſecet curuam parabolæin, M, & </s> <s xml:id="echoid-s6966" xml:space="preserve"><lb/>per, M, ducatur ipſi baſi, FH, parallela, IL. </s> <s xml:id="echoid-s6967" xml:space="preserve">Eſtergo quadratum, <pb o="286" file="0306" n="306" rhead="GEOMETRIÆ"/> GH, vel quadratum, EC, ad quadratum, IM, vel ad quadratum, <lb/> <anchor type="note" xlink:label="note-0306-01a" xlink:href="note-0306-01"/> CN, vt, GC, ad, CI,.</s> <s xml:id="echoid-s6968" xml:space="preserve">. vt, ON, ad, NM, eſt autem, CH, pa-<lb/>rallelogrammum in eadem baſi, & </s> <s xml:id="echoid-s6969" xml:space="preserve">altitudine cum trilineo, CMH <lb/>E, & </s> <s xml:id="echoid-s6970" xml:space="preserve">punctum, N, vtcunq; </s> <s xml:id="echoid-s6971" xml:space="preserve">ſumptum, per quod acta eſt ipſi, CG, <lb/>parallela, NO, repertumque eſt, vt quadratum, EC, ad quadra. <lb/></s> <s xml:id="echoid-s6972" xml:space="preserve">tum, CN, itaeſſe, ON, ad, NM; </s> <s xml:id="echoid-s6973" xml:space="preserve">ergo horum quatuor ordinum <lb/> <anchor type="note" xlink:label="note-0306-02a" xlink:href="note-0306-02"/> magnitudines erunt proportionales ſcilicet omnia quadrata maxi-<lb/>marum abſciſſarum, EC, magnitudines primi ordinis collectæ iuxta <lb/>quadratum, CE, ad quadrata omnium abſciſſarum ipſius, CE, ſiue <lb/>ambo ſint recti, vel eiuſdem obliqui tranſitus, quæ ſunt magnitudi-<lb/>nes ſecundi ordinis collectæ, iuxta quadratum, CN, erunt vt om-<lb/> <anchor type="figure" xlink:label="fig-0306-01a" xlink:href="fig-0306-01"/> nes lineæ parallelogrammi, CH, ma-<lb/>gnitudines tertij ordinis collectæ, iux-<lb/>ta, NO, ad omnes lineas trilinei, CM <lb/>HE, magnitudines quarti ordinis col. <lb/></s> <s xml:id="echoid-s6974" xml:space="preserve">lectas, iuxta, NM, regula pro his om-<lb/>nibus lineis exiſtenteipſa, EH; </s> <s xml:id="echoid-s6975" xml:space="preserve">vt au-<lb/> <anchor type="note" xlink:label="note-0306-03a" xlink:href="note-0306-03"/> tem ſunt omnes lineæ parallelogram-<lb/>mi, CH, ad omnes lineas trilinei, CM <lb/>HE, ita eſt parallelogrammum, CH, <lb/>ad trilineum, CMHE, ergo paralle-<lb/>logrammum, CH, ad trilineum, CMHE, eſt vt quadrata maxi-<lb/>marum abſciſſarum ipſius, CE, ad quadrata omnium abſciſſarum ip-<lb/>ſius, CE, verum illa quadrata ſuntiſtorum tripla, ergo erit paralle-<lb/> <anchor type="note" xlink:label="note-0306-04a" xlink:href="note-0306-04"/> logrammum, CH, triplum ipſius trilinei, CMHE, ergo idem pa-<lb/>rallelogrammum, CH, erit ſexquialterum ſemiparabolæ, GCM <lb/>H, ideò etiam parallelogrammum, AH, erit parabole, FCH, ſex-<lb/>quialterum. </s> <s xml:id="echoid-s6976" xml:space="preserve">Quoniam verò triangulum, CFH, eſt dimidium pa-<lb/>rallelogrammi, AH, ideò quarum partium parallelogrammum, A <lb/>H, erit ſex, & </s> <s xml:id="echoid-s6977" xml:space="preserve">parabola, FCH, conſequenter ea undem quatuor, <lb/>triangulum, CFH, erit tria, & </s> <s xml:id="echoid-s6978" xml:space="preserve">ideò erit ad parabolam, FCH, vt <lb/>tria ad quatuor, & </s> <s xml:id="echoid-s6979" xml:space="preserve">idcircò erit eiuſdem ſubſexquitertium, quæ oſten-<lb/>dere oportebat.</s> <s xml:id="echoid-s6980" xml:space="preserve"/> </p> <div xml:id="echoid-div689" type="float" level="2" n="1"> <figure xlink:label="fig-0305-02" xlink:href="fig-0305-02a"> <image file="0305-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0305-02"/> </figure> <note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">Ex 38. & <lb/>Schol. 40. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0306-02" xlink:href="note-0306-02a" xml:space="preserve">Coroll. 3. <lb/>26. lib. 2.</note> <figure xlink:label="fig-0306-01" xlink:href="fig-0306-01a"> <image file="0306-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0306-01"/> </figure> <note position="left" xlink:label="note-0306-03" xlink:href="note-0306-03a" xml:space="preserve">3. Lib. 2.</note> <note position="left" xlink:label="note-0306-04" xlink:href="note-0306-04a" xml:space="preserve">Color. 25 <lb/>lib. 2.</note> </div> </div> <div xml:id="echoid-div691" type="section" level="1" n="406"> <head xml:id="echoid-head426" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s6981" xml:space="preserve">HInc patet ductas in trilineo, CMHE, æquidiſtantes axi, bel dia-<lb/>metro, CG, eſſe inter ſe, bt quadrata abſciſſarumper eaſdem d <lb/>tangente, CE, berſus berticem parabolæ, quieſt punctum, C; </s> <s xml:id="echoid-s6982" xml:space="preserve">nam oſten-<lb/>ſum eſt, ON, ſiue, HE, ad, NM, eſſe bt quadratum, EC, ad quadra-<lb/>tum, CN, & </s> <s xml:id="echoid-s6983" xml:space="preserve">punctum, N, ſumptum eſt btcunque, ideo, &</s> <s xml:id="echoid-s6984" xml:space="preserve">c.</s> <s xml:id="echoid-s6985" xml:space="preserve"/> </p> <pb o="287" file="0307" n="307" rhead="LIBER IV."/> </div> <div xml:id="echoid-div692" type="section" level="1" n="407"> <head xml:id="echoid-head427" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s6986" xml:space="preserve">SI intra parabolam ducantur vtcunque duæ ad axim, vel <lb/>diametrum eiuſdem ordinatim applicatę lineę, abſciſſæ <lb/>abijſdem parabolæ, erunt inter ſe, vt cubi dictarum linea-<lb/>rum ordinatim applicatarum.</s> <s xml:id="echoid-s6987" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6988" xml:space="preserve">Sint ergo intra parabolam circa axim, veldiametrum, CG, con-<lb/>ſtitutam, duæ adipſum ordinatim applicatæ rectæ lineæ, FH, OM, <lb/>parabolas, OCM, FCH, abſc ndentes. </s> <s xml:id="echoid-s6989" xml:space="preserve">Dico ergo parabolam, F <lb/>CH, ad parabolam, OCM, eſſe vt cubum, FH, ad cubum, OM; <lb/></s> <s xml:id="echoid-s6990" xml:space="preserve">conſtituantur circa axes, vel diametros, CI, CG, & </s> <s xml:id="echoid-s6991" xml:space="preserve">in eiſdem ba-<lb/>ſibus, OM, FH, cum dictis parabolis parallelogramma, AH, RM. </s> <s xml:id="echoid-s6992" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/> Quoniam ergo ęquiangula paral <lb/> <anchor type="note" xlink:label="note-0307-01a" xlink:href="note-0307-01"/> lelogramma habent rationem ex <lb/>lateribus compoſitam, ſunt au-<lb/>tem parallelogramma, AH, R <lb/>M, æquiangula, nam, OM, eſt <lb/>parallela ipſi, FH, ideò paralle-<lb/>logrammum, AH, ad parallelo-<lb/>grammum, RM, habebit ratio-<lb/>nem compoſitam ex ea, quam ha <lb/>bet, FA, ad, RO, .</s> <s xml:id="echoid-s6993" xml:space="preserve">i. </s> <s xml:id="echoid-s6994" xml:space="preserve">GC, ad, <lb/>CI, .</s> <s xml:id="echoid-s6995" xml:space="preserve">i. </s> <s xml:id="echoid-s6996" xml:space="preserve">quadratum, FH, ad qua-<lb/> <anchor type="note" xlink:label="note-0307-02a" xlink:href="note-0307-02"/> dratum, OM, & </s> <s xml:id="echoid-s6997" xml:space="preserve">ex ea, quam habet, FH, ad, OM, ſed etiam cu-<lb/>bus, FH, ad cubum, OM, habet rationem compoſitam ex ea, quam <lb/>habet quadratum, FH, ad quadratum, OM, & </s> <s xml:id="echoid-s6998" xml:space="preserve">ex ea, quam ha-<lb/> <anchor type="note" xlink:label="note-0307-03a" xlink:href="note-0307-03"/> bet, FH, ad, OM, ergo parallelogrammum, AH, ad parallelo-<lb/>grammum, RM, & </s> <s xml:id="echoid-s6999" xml:space="preserve">conſequenter parabola, FCH, ad parabolam, <lb/>OCM, (quia ſunt dictorum parallelogrammorum ſubſexquialterę) <lb/> <anchor type="note" xlink:label="note-0307-04a" xlink:href="note-0307-04"/> erit vt cubus, FH, ad cubum, OM, quodoſtendere opus erat.</s> <s xml:id="echoid-s7000" xml:space="preserve"/> </p> <div xml:id="echoid-div692" type="float" level="2" n="1"> <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a"> <image file="0307-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0307-01"/> </figure> <note position="right" xlink:label="note-0307-01" xlink:href="note-0307-01a" xml:space="preserve">11. Lib. 2.</note> <note position="right" xlink:label="note-0307-02" xlink:href="note-0307-02a" xml:space="preserve">38. Ec <lb/>Schol. 40. <lb/>lib. 1.</note> <note position="right" xlink:label="note-0307-03" xlink:href="note-0307-03a" xml:space="preserve">D. Corol. <lb/>4. Gener. <lb/>34, lib. 2.</note> <note position="right" xlink:label="note-0307-04" xlink:href="note-0307-04a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div694" type="section" level="1" n="408"> <head xml:id="echoid-head428" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s7001" xml:space="preserve">SI in parabola ducatur quædam recta linea ad eiuſdem <lb/>axim, vel diametrum ordinatim applicata; </s> <s xml:id="echoid-s7002" xml:space="preserve">agantur de-<lb/>inde ipſx<unsure/>axi, vel diametro æquidiſtantes rectæ lineævſque <lb/>ad curuam parabolicam, & </s> <s xml:id="echoid-s7003" xml:space="preserve">dictam ordinatim applicatam, <lb/>quæ baſis erit eiuſdem parabolæ; </s> <s xml:id="echoid-s7004" xml:space="preserve">Dictæ æquidiſtantes rectę <pb o="288" file="0308" n="308" rhead="GEOMETRIÆ"/> lineæ erunt interſe, vtrectangula ſub partibus baſis ab ei-<lb/>ſdem æquidiſtantibus conſtitutis.</s> <s xml:id="echoid-s7005" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7006" xml:space="preserve">Sit ergo parabola, FCH, circa axim, vel diametrum, CG, ad <lb/>quam ordinatim applicetur recta linea vtcunq; </s> <s xml:id="echoid-s7007" xml:space="preserve">FH, ducantur dein-<lb/>de intra parabolam axi, vel diametro, CG, parallelæ vtcunque, A <lb/>N, MO, baſim, FH, in punctis, N, O, diuidentes. </s> <s xml:id="echoid-s7008" xml:space="preserve">Dico igitur re-<lb/>ctam, AN, ad rectam, MO, eſſe vt rectangulum, FNH, ad re-<lb/>ctangulum, FOH; </s> <s xml:id="echoid-s7009" xml:space="preserve">ducatur per, M, ipſi, FH, parallela, MI; </s> <s xml:id="echoid-s7010" xml:space="preserve">eſt <lb/>ergo, GC, ad, CI, vt quadratum, GH, ad quadratum, IM, vel <lb/> <anchor type="note" xlink:label="note-0308-01a" xlink:href="note-0308-01"/> ad quadratum, GO, ergo, perconuerſionem rationis, GC, ad, G <lb/> <anchor type="figure" xlink:label="fig-0308-01a" xlink:href="fig-0308-01"/> I, vel ad, MO, erit vt quadratum, H <lb/>G, ad ſuireliquum, dempto quadrato, <lb/>GO, hoc autem reſiduum eſt rectan-<lb/>gulum ſub, GOH, bis, vna cum qua-<lb/>drato, OH, quod eſt æqualerectan-<lb/>gulo, FOH, nam rectangulum, GO <lb/>H, cum quadrato, OH, æquatur re-<lb/>ctangulo, GHO, .</s> <s xml:id="echoid-s7011" xml:space="preserve">i. </s> <s xml:id="echoid-s7012" xml:space="preserve">rectangulo ſub, <lb/>FG, OH, cui ſi iunxeris rectangulum <lb/>ſub, GO, & </s> <s xml:id="echoid-s7013" xml:space="preserve">eadem, OH, conſurget <lb/>integrum rectangulum, FOH, æqualerectangulis ſub, GOH, bis, <lb/> <anchor type="note" xlink:label="note-0308-02a" xlink:href="note-0308-02"/> vna cum quadrato, OH, ergo, CG, ad, MO, erit vt quadratum, <lb/> <anchor type="note" xlink:label="note-0308-03a" xlink:href="note-0308-03"/> GH, .</s> <s xml:id="echoid-s7014" xml:space="preserve">i. </s> <s xml:id="echoid-s7015" xml:space="preserve">vt rectangulum, FGH, ad rectangulum, FOH, & </s> <s xml:id="echoid-s7016" xml:space="preserve">con-<lb/> <anchor type="note" xlink:label="note-0308-04a" xlink:href="note-0308-04"/> uertendo, MO, ad, CG, erit vtrectang. </s> <s xml:id="echoid-s7017" xml:space="preserve">HOF, ad rectangulum, H <lb/>GF; </s> <s xml:id="echoid-s7018" xml:space="preserve">codem modo oſtendemus, CG, ad, AN, eſſe vt idem rectan-<lb/>gulum, HGF, ad rectangulum, FNH, ergo ex æquali, & </s> <s xml:id="echoid-s7019" xml:space="preserve">conuer-<lb/>tendo, AN, ad, MO, erit vt rectangulum, FNH, ad rectangulum, <lb/>FOH, quod oſtendere oportebat. </s> <s xml:id="echoid-s7020" xml:space="preserve">Poſſunt autem vocari &</s> <s xml:id="echoid-s7021" xml:space="preserve">, AN, <lb/>MO, ordinatim applicatæ ad baſim parabolæ, FCH, ſcilicet ad <lb/>ipſam, FH.</s> <s xml:id="echoid-s7022" xml:space="preserve"/> </p> <div xml:id="echoid-div694" type="float" level="2" n="1"> <note position="left" xlink:label="note-0308-01" xlink:href="note-0308-01a" xml:space="preserve">38. EtSch. <lb/>40. lib. 1.</note> <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a"> <image file="0308-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0308-01"/> </figure> <note position="left" xlink:label="note-0308-02" xlink:href="note-0308-02a" xml:space="preserve">4. 2. Elem.</note> <note position="left" xlink:label="note-0308-03" xlink:href="note-0308-03a" xml:space="preserve">3.2. Elem.</note> <note position="left" xlink:label="note-0308-04" xlink:href="note-0308-04a" xml:space="preserve">1. 2. Elem.</note> </div> </div> <div xml:id="echoid-div696" type="section" level="1" n="409"> <head xml:id="echoid-head429" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s7023" xml:space="preserve">SI ad baſim parabolæ ordinatim applicetur vtcunque re-<lb/>cta linea, ſiat autem parallelogrammum, & </s> <s xml:id="echoid-s7024" xml:space="preserve">triangulum <lb/>habentia circa communem angulum dictam applicatam, & </s> <s xml:id="echoid-s7025" xml:space="preserve"><lb/>abſciſſam à baſiab vtrauis extremitatum eiuſdem, vel ſint <lb/>duæ ad baſim vtcunque ordinatim applicatæ, ſub alterutra <lb/>quarum, & </s> <s xml:id="echoid-s7026" xml:space="preserve">ſub in cluſa ab ijſdem portione baſis ſiat paralle-<lb/>logrammum, & </s> <s xml:id="echoid-s7027" xml:space="preserve">triangulum; </s> <s xml:id="echoid-s7028" xml:space="preserve">dicti parallelogrammi, vel trian- <pb o="289" file="0309" n="309" rhead="LIBER IV."/> guli, ad portionem parabolæ dicto parallelogrammo inſcri-<lb/>ptam ratio nota erit.</s> <s xml:id="echoid-s7029" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7030" xml:space="preserve">Sit parabola, cuius baſis, FG, ad axim, vel diametrum, CX, or-<lb/>dinatim applicata; </s> <s xml:id="echoid-s7031" xml:space="preserve">ad baſim autem, FG, ſit etiam ordinatim appli-<lb/>cata, AN, vtcunq; </s> <s xml:id="echoid-s7032" xml:space="preserve">diuidens baſim, FG, in puncto, N, ſiat autem <lb/>parallelogrammum, RN, & </s> <s xml:id="echoid-s7033" xml:space="preserve">triangulum, AFN, ſublateribus, A <lb/>N, NF, vel ſub, AN, NG; </s> <s xml:id="echoid-s7034" xml:space="preserve">Vel ſint duæ vtcunque ad baſim, FG, <lb/>ordinatim applicatæ, AN, CX, ſiat autem parallelogrammum, & </s> <s xml:id="echoid-s7035" xml:space="preserve"><lb/>triangulum ſub, AN, NX, vel ſub, CX, XN. </s> <s xml:id="echoid-s7036" xml:space="preserve">Dico parallelogram-<lb/> <anchor type="figure" xlink:label="fig-0309-01a" xlink:href="fig-0309-01"/> mum, RN, veltrian-<lb/>gulum, FAN, ad <lb/>portionem, AFN, <lb/>parabole, FCG, pa-<lb/>rallelogrammo, RN, <lb/>inſcriptam, eſſe in ra-<lb/>tione nota. </s> <s xml:id="echoid-s7037" xml:space="preserve">Similiter <lb/>parallelogrammum, <lb/>ZX, & </s> <s xml:id="echoid-s7038" xml:space="preserve">triangulum, <lb/>NCX, ad portio-<lb/>nem, ACXN, habe-<lb/>re rationem notam. <lb/></s> <s xml:id="echoid-s7039" xml:space="preserve">Producatur, CX, vt-<lb/>cunque in, E, & </s> <s xml:id="echoid-s7040" xml:space="preserve">cir-<lb/>ca ſemiaxes, vel ſemi-<lb/>diametros coniuga-<lb/>tas, FX, XE, intel-<lb/>ligatur deſcriptus ſe-<lb/>micirculus, velſemi-<lb/>ellipſis, FEG, pro-<lb/>ducantur deinde, RF, ZN, indeſinitè, ſecetque, ZN, curuam ſe-<lb/>micirculi, vel ſemiellipſis, FEG, in puncto, O, & </s> <s xml:id="echoid-s7041" xml:space="preserve">compleantur pa-<lb/>rallelogramma, VN, RX, ſumatur deindein, FN, vtcunq; </s> <s xml:id="echoid-s7042" xml:space="preserve">pun-<lb/>ctum, S, per quodipſi, CE, parallela ducatur, YT, ſecans curnam <lb/>parabolæ in, I, curuam autem, FEG, in, M; </s> <s xml:id="echoid-s7043" xml:space="preserve">eſt ergo, AN, ad, I <lb/> <anchor type="note" xlink:label="note-0309-01a" xlink:href="note-0309-01"/> S, vt rectangulum, GNF, ad rectangulum, GSF, eſt autem etiam <lb/> <anchor type="note" xlink:label="note-0309-02a" xlink:href="note-0309-02"/> quadratum, ON, ad quadratum, SM, vt rectangulum, GNF, ad <lb/>rectangulum, GSF, ergo, AN, vel, YS, ad, SI, erit vt quadra <lb/>tum, NO, vel vt quadratum, TS, ad quadratum, SM, ſunt au-<lb/>tem, RN, NV, parallelogramma in eiſdem baſibus, & </s> <s xml:id="echoid-s7044" xml:space="preserve">altitudini-<lb/> <anchor type="note" xlink:label="note-0309-03a" xlink:href="note-0309-03"/> bus cum portionibus, AFN, NFO, & </s> <s xml:id="echoid-s7045" xml:space="preserve">punctum, S, ſumptum eſt <lb/>vtcunque, repertumque eſt, vt, YS, ad, SI, ita eſſe quadratum, T <pb o="290" file="0310" n="310" rhead="GEOMETRIÆ"/> S, ad quadratum, SM, ergo horum quatuor ordinum magnitudines <lb/>erunt proportionales collectæ, iuxta quatuor iam dictas magnitudi-<lb/>nes proportionales .</s> <s xml:id="echoid-s7046" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7047" xml:space="preserve">omnes lineæ ipſius, RN, (ſumpta pro omni-<lb/>bus communi regula, CE,) ad omnes lineas trilinei, FIAN, erunt <lb/>vt omnia quadrata, FO, ad omnia quadrata trilinei, FMON, ra-<lb/>tio autem, quam habent omnia quadrata, FO, ad omnia quadrata <lb/>trilinei, FMON, iam notiſicata eſt lib.</s> <s xml:id="echoid-s7048" xml:space="preserve">3. </s> <s xml:id="echoid-s7049" xml:space="preserve">de circulo, & </s> <s xml:id="echoid-s7050" xml:space="preserve">ellipſi Pro-<lb/> <anchor type="figure" xlink:label="fig-0310-01a" xlink:href="fig-0310-01"/> poſit. </s> <s xml:id="echoid-s7051" xml:space="preserve">1. </s> <s xml:id="echoid-s7052" xml:space="preserve">ergo & </s> <s xml:id="echoid-s7053" xml:space="preserve">ratio <lb/>omnium linearum, R <lb/>N, ad omnes lineas <lb/>trilinei, FIAN, & </s> <s xml:id="echoid-s7054" xml:space="preserve"><lb/>ſubinde ratio paral-<lb/>lelogrammi, RN, ad <lb/>portionem, FIAN, <lb/>nota erit, & </s> <s xml:id="echoid-s7055" xml:space="preserve">ſubinde <lb/>nota erit ratio trian-<lb/>guli, FAN, quod eſt <lb/>dimidium parallelo-<lb/>grammi, RN, ad <lb/> <anchor type="note" xlink:label="note-0310-01a" xlink:href="note-0310-01"/> portionem, FIAN; <lb/></s> <s xml:id="echoid-s7056" xml:space="preserve">eodem modo oſten-<lb/>demus parallelográ-<lb/>mum, ZX, ad qua-<lb/>drilineum, NACX, <lb/>eſſe vt omnia qua-<lb/>drata, RX, ad om-<lb/>nia quadrata quadri-<lb/>linei, ONXF, ratio autem, quam habent omnia quadrata, RX, <lb/>ad omnia quadrata quadrilinei, ONXE, iam notiſicata eſt in ſupra-<lb/>dicto Libro, Propoſit.</s> <s xml:id="echoid-s7057" xml:space="preserve">3. </s> <s xml:id="echoid-s7058" xml:space="preserve">& </s> <s xml:id="echoid-s7059" xml:space="preserve">4. </s> <s xml:id="echoid-s7060" xml:space="preserve">ergo ratio pamllelogrammi, ZX, ad <lb/>quadrilineum, ſiue portionem parabolæ, ANXC, nota erit, veluti <lb/>& </s> <s xml:id="echoid-s7061" xml:space="preserve">ratio trianguli, CNX, ad eandem portionem, ANXC, pariter <lb/>nota erit, quod erat oſtendendum.</s> <s xml:id="echoid-s7062" xml:space="preserve"/> </p> <div xml:id="echoid-div696" type="float" level="2" n="1"> <figure xlink:label="fig-0309-01" xlink:href="fig-0309-01a"> <image file="0309-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0309-01"/> </figure> <note position="right" xlink:label="note-0309-01" xlink:href="note-0309-01a" xml:space="preserve">Exantec.</note> <note position="right" xlink:label="note-0309-02" xlink:href="note-0309-02a" xml:space="preserve">40. Et <lb/>Sch. 1. 1.</note> <note position="right" xlink:label="note-0309-03" xlink:href="note-0309-03a" xml:space="preserve">Coroll.3. <lb/>26. lib. 2.</note> <figure xlink:label="fig-0310-01" xlink:href="fig-0310-01a"> <image file="0310-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0310-01"/> </figure> <note position="left" xlink:label="note-0310-01" xlink:href="note-0310-01a" xml:space="preserve">3. Lib. 2.</note> </div> </div> <div xml:id="echoid-div698" type="section" level="1" n="410"> <head xml:id="echoid-head430" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7063" xml:space="preserve">HInc colligitur dicta parallelogramma ad portiones parabolæ ſibi <lb/>inſcriptas, ordmatimque ad parabolæ baſim applicatis incluſas, <lb/>eſſe, bt omnia quadrata parallelogrammorum illis è regione reſponden-<lb/>tium, quibuſq; </s> <s xml:id="echoid-s7064" xml:space="preserve">inſcribuntur ſemiportiones circuli. </s> <s xml:id="echoid-s7065" xml:space="preserve">vel ellipſis iam di-<lb/>ctæ ad omnia quadrata dictarum ſemiportionum, regula communt axi, <lb/>vel diametro, CE, exiftente. </s> <s xml:id="echoid-s7066" xml:space="preserve">Oſtenſum .</s> <s xml:id="echoid-s7067" xml:space="preserve">n. </s> <s xml:id="echoid-s7068" xml:space="preserve">eſt, RN, ad portionera, FA <lb/>N, eſſe, bt omnia quadrata, FO, ad omnia quadrata trilmet, FMON;</s> <s xml:id="echoid-s7069" xml:space="preserve"> <pb o="291" file="0311" n="311" rhead="LIBER IV."/> &</s> <s xml:id="echoid-s7070" xml:space="preserve">, ZX, ad portionem, ACXN, eſſe, vt omnia quádratà, ℟ X, ad <lb/>omnia quadrata quadrilinei, NOEX, &</s> <s xml:id="echoid-s7071" xml:space="preserve">, AN, CX, ordinatim ad ba-<lb/>ſim, FG, applicatæ ſumptæ ſunt vtcunq; </s> <s xml:id="echoid-s7072" xml:space="preserve">vnde patet.</s> <s xml:id="echoid-s7073" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div699" type="section" level="1" n="411"> <head xml:id="echoid-head431" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s7074" xml:space="preserve">DVctis vtcunque ad baſim parabolæ ordinatim applica-<lb/>tis, parallelogramma ſub ipſis, & </s> <s xml:id="echoid-s7075" xml:space="preserve">portionibus baſis ab <lb/>ijſdem abſciſſis ad ſibi inſcriptas portiones parabolæ infra-<lb/>ſcriptam rationem habebunt.</s> <s xml:id="echoid-s7076" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7077" xml:space="preserve">Sit ergo parabola, HGA, in baſi, HA, circa axim, vel diame-<lb/>trum, GO, & </s> <s xml:id="echoid-s7078" xml:space="preserve">ſint ductæ ipſi, GO, parallelæ vtcunque, ST, EC, <lb/>compleantur autem parallelogramma, LT, BO, DC, deinde pro-<lb/>ducatur, GO, vtcunque in, M, & </s> <s xml:id="echoid-s7079" xml:space="preserve">circa ſemiaxes, vel ſemidiame-<lb/>tros, HO, OM, intelligatur, HMA, ſemicirculus, vel ſemiellipſis, <lb/> <anchor type="figure" xlink:label="fig-0311-01a" xlink:href="fig-0311-01"/> cuius curuam, ST, <lb/>EC, productæ ſe-<lb/>cent in, VN, com-<lb/>pleàtur pariter pa-<lb/>rallelogramma, H <lb/>V, HM, HN, pro-<lb/>ducantur inſuper, Y <lb/>M, BG, vſque in, <lb/>& </s> <s xml:id="echoid-s7080" xml:space="preserve">℟, &</s> <s xml:id="echoid-s7081" xml:space="preserve">, SV, EN, <lb/>vſq; </s> <s xml:id="echoid-s7082" xml:space="preserve">ad puncta, P, <lb/>Z, Q, I, quæ ſunt <lb/>in lateribus, B ℟, Y <lb/>&</s> <s xml:id="echoid-s7083" xml:space="preserve">. Igitur paralle-<lb/>logrammum, LT, <lb/>ad portionem, HS <lb/>T, erit vt omnia <lb/>quadrata, HV, ad <lb/>omnia quadrata ſe-<lb/>miportionis, HT <lb/>V, (regula, GM, pro hac Propoſ. </s> <s xml:id="echoid-s7084" xml:space="preserve">ſumpta). </s> <s xml:id="echoid-s7085" xml:space="preserve">i. </s> <s xml:id="echoid-s7086" xml:space="preserve">vt, TA, ad compó-<lb/>ſitam ex {1/2}, TA, &</s> <s xml:id="echoid-s7087" xml:space="preserve">, {1/6}, HT, vt patet in Libro de Circulo, & </s> <s xml:id="echoid-s7088" xml:space="preserve">Ellipſi <lb/>Propoſitione I.</s> <s xml:id="echoid-s7089" xml:space="preserve"/> </p> <div xml:id="echoid-div699" type="float" level="2" n="1"> <figure xlink:label="fig-0311-01" xlink:href="fig-0311-01a"> <image file="0311-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0311-01"/> </figure> </div> <p> <s xml:id="echoid-s7090" xml:space="preserve">Similiter oſtendemus, BO, ſemiparabolæ, HGO, eſſe ſexqui-<lb/>alterum, eſt enim vt omnia quadrata, HM, ad omnia quadrata, H <lb/>VMO, ideſt in ratione ſexquialtera, vt patet in eadem Propoſit. </s> <s xml:id="echoid-s7091" xml:space="preserve">I.</s> <s xml:id="echoid-s7092" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7093" xml:space="preserve">Pariter demonſtrabimus, DC, ad portionem, HGEC, eſſe vt, <lb/>AC, ad compoſitam ex, {1/2}, AC, &</s> <s xml:id="echoid-s7094" xml:space="preserve">, {1/6}, CH, ſicenim ſunt omnia <pb o="292" file="0312" n="312" rhead="GEOMETRIÆ"/> quadrata, HN, ad omnia quadrata ſemiportionis, HMNC, vt <lb/>patet in eiuſdem Lib. </s> <s xml:id="echoid-s7095" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s7096" xml:space="preserve">I.</s> <s xml:id="echoid-s7097" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7098" xml:space="preserve">Quod ſi velimus comparare parallelogramma, quæ ſunt in baſi-<lb/>bus æqualibus axi, vel diametro, inueniemus infraſcriptas rationes <lb/>ſcilicet parallelogrammum, BT, ad portionem, HST, eſſe vt re-<lb/>ctangulum ſub, HO, & </s> <s xml:id="echoid-s7099" xml:space="preserve">tripla, OA, ad rectangulum ſub, HT, & </s> <s xml:id="echoid-s7100" xml:space="preserve"><lb/>ſub compoſita ex, TA, &</s> <s xml:id="echoid-s7101" xml:space="preserve">, AO, ſicuti ſunt omnia quadrata, HZ, <lb/>ad omnia quadrata ſemiportionis, HTV. </s> <s xml:id="echoid-s7102" xml:space="preserve">Eadem ratione, BC, ad <lb/> <anchor type="figure" xlink:label="fig-0312-01a" xlink:href="fig-0312-01"/> portionem, HGE <lb/>C, erit vt rectangu-<lb/>lum ſub, HO, & </s> <s xml:id="echoid-s7103" xml:space="preserve"><lb/>tripla, OA, ad re-<lb/>ctangulum ſub, H <lb/>C, & </s> <s xml:id="echoid-s7104" xml:space="preserve">ſub compoſi-<lb/>ta ex, CA, &</s> <s xml:id="echoid-s7105" xml:space="preserve">, AO, <lb/>ſic enim ſunt om-<lb/>nia quadrata, HI, <lb/>ad omnia quadrata <lb/>ſemiportionis, HM <lb/>NC, vt patet in eo-<lb/>dem Lib. </s> <s xml:id="echoid-s7106" xml:space="preserve">3. </s> <s xml:id="echoid-s7107" xml:space="preserve">Prop. </s> <s xml:id="echoid-s7108" xml:space="preserve">2.</s> <s xml:id="echoid-s7109" xml:space="preserve"/> </p> <div xml:id="echoid-div700" type="float" level="2" n="2"> <figure xlink:label="fig-0312-01" xlink:href="fig-0312-01a"> <image file="0312-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0312-01"/> </figure> </div> <p> <s xml:id="echoid-s7110" xml:space="preserve">Sitandem ſuma-<lb/>musparallelogram-<lb/>mum, PC, cui in-<lb/>ſcripta eſt parabolę <lb/>portio, TSGEC, <lb/>incluſa duabus, ST, <lb/>EC, ad baſim, HA, vtcunq; </s> <s xml:id="echoid-s7111" xml:space="preserve">ordinatim applicatis, ſiue intercipiant <lb/>axem, vel diametrum, GO, ſiue non, ſiue axis, vel diameter, GO, <lb/>ſit altera harum duarum ad baſim, HA, ordinatim applicatarum, <lb/>ſiue non; </s> <s xml:id="echoid-s7112" xml:space="preserve">reperiemus parallelogrammum, PC, ad portionem, TS <lb/>GEC, eſſe vt rectangulum, HOA, ad rectangulum ſub, AC, & </s> <s xml:id="echoid-s7113" xml:space="preserve"><lb/>ſub compoſita ex, {1/2}, CT, & </s> <s xml:id="echoid-s7114" xml:space="preserve">tota, TH, vna cum rectangulo ſub, T <lb/>C, & </s> <s xml:id="echoid-s7115" xml:space="preserve">ſub compoſita ex, {1/6}, TC, &</s> <s xml:id="echoid-s7116" xml:space="preserve">, {1/2}, TH, ſic enim eſſe inuenie-<lb/>mus omnia quadrata, TI, ad omnia quadrata quadrilinei, TVM <lb/>NC, vt patet eodem Lib. </s> <s xml:id="echoid-s7117" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s7118" xml:space="preserve">4.</s> <s xml:id="echoid-s7119" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div702" type="section" level="1" n="412"> <head xml:id="echoid-head432" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s7120" xml:space="preserve">_H_Inc habetur ſi fiant triangulæ, ductis, SH, PH, GH, QT, hæc <lb/>ad portiones, quibus inſcribuntur habere eaſdem rationes, quas <lb/>habent dimidia antecedentium ad eadem conſequentia ſuperius expoſita, <lb/>ſunt enim & </s> <s xml:id="echoid-s7121" xml:space="preserve">ipſa triangula dictorum parallelogrammorum dimedia.</s> <s xml:id="echoid-s7122" xml:space="preserve"/> </p> <pb o="293" file="0313" n="313" rhead="LIBER IV."/> </div> <div xml:id="echoid-div703" type="section" level="1" n="413"> <head xml:id="echoid-head433" xml:space="preserve">THEOREMA VI. PROPOS. VI.</head> <p> <s xml:id="echoid-s7123" xml:space="preserve">SI ad baſim datæ parabolæ ordinatim applicetur recta li-<lb/>nea, tota parabola ad abſciſſam portionem per ipſam or-<lb/>dinatim applicatam erit, vt parallelepipedum ſub altitudine <lb/>dimidia baſi, ſub baſi autem quadrato totius baſis, ad paral-<lb/>lelepipedum ſub altitudine linea compoſita ex dimidia baſi, <lb/>& </s> <s xml:id="echoid-s7124" xml:space="preserve">reliquo baſis, dempta abſciſſa ab eadem extremitate ba-<lb/>ſis, à qua portio parabolæ abſcinditur, & </s> <s xml:id="echoid-s7125" xml:space="preserve">ſub baſi quadrato <lb/>eiuſdem abſciſſæ per dictam ordinatim applicatam: </s> <s xml:id="echoid-s7126" xml:space="preserve">Vel erit, <lb/>vt cubus totius baſis ad parallelepipedum ſub baſi quadrato <lb/>abſciſſæ, altitudine tripla reliquæ, cum cubo dictæ abſciſſæ.</s> <s xml:id="echoid-s7127" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7128" xml:space="preserve">Sit parabola, HG <lb/> <anchor type="figure" xlink:label="fig-0313-01a" xlink:href="fig-0313-01"/> A, cuius baſis, HA, <lb/>& </s> <s xml:id="echoid-s7129" xml:space="preserve">axis, vel diameter, <lb/>GO; </s> <s xml:id="echoid-s7130" xml:space="preserve">ducatur deinde <lb/>ipſi, GO, vtcunque <lb/>parallela, ST. </s> <s xml:id="echoid-s7131" xml:space="preserve">Dico <lb/>parabolam, AGH, <lb/>ad vtramuis portio-<lb/>num, SHT, TSG <lb/>A, vt ad, SHT, eſſe <lb/>vt parallelepip. </s> <s xml:id="echoid-s7132" xml:space="preserve">ſub al-<lb/>titudine dimidia, H <lb/>A, quæ ſit, AX, illi <lb/>in directum conſtitu-<lb/>ta, baſi quadrato, A <lb/>H, ad parallelepipe-<lb/>dum ſub altitudine, X <lb/>T, baſi quadrato, T <lb/>H. </s> <s xml:id="echoid-s7133" xml:space="preserve">Producatur, GO, <lb/>in, M, & </s> <s xml:id="echoid-s7134" xml:space="preserve">circa ſemia-<lb/>xes, vel ſemidiame-<lb/>tros, HO, OM, in-<lb/>telligatur deſcriptus <lb/>ſemicirculus, vel ſe-<lb/>miellipſis, HMA, <lb/>deinde per puncta, G, M, ducantur ipſi, HA, parallelæ, B ℟, Y <pb o="294" file="0314" n="314" rhead="GEOMETRIÆ"/> &</s> <s xml:id="echoid-s7135" xml:space="preserve">, & </s> <s xml:id="echoid-s7136" xml:space="preserve">per, HA, ipſi, GM, parallelę, BY, ℟ &</s> <s xml:id="echoid-s7137" xml:space="preserve">, producaturque, T <lb/>S, vſque ad, B ℟, Y &</s> <s xml:id="echoid-s7138" xml:space="preserve">, in, P, Z, & </s> <s xml:id="echoid-s7139" xml:space="preserve">per, SV, ducantur, VF, SL, <lb/>parallelæipſi, HA, ſunt igitur parallelogramma, BA, AY, LT, <lb/>TF, BT, TY, PA, AZ. </s> <s xml:id="echoid-s7140" xml:space="preserve">Igitur parabola, AGH, ad portionem, <lb/>HST, habetrationem compoſitam ex ea, quam habet parabola, H <lb/> <anchor type="note" xlink:label="note-0314-01a" xlink:href="note-0314-01"/> GA, ad parallelogrammum, BA, ideſt ex ea, quam habent omnia <lb/>quadrata, H &</s> <s xml:id="echoid-s7141" xml:space="preserve">, (regula ſumpta pro hoc Theor. </s> <s xml:id="echoid-s7142" xml:space="preserve">ipſa, GM,) ad om-<lb/> <anchor type="note" xlink:label="note-0314-02a" xlink:href="note-0314-02"/> nia quadrata ſemicirculi, vel ſemiellipſis, HMA; </s> <s xml:id="echoid-s7143" xml:space="preserve">& </s> <s xml:id="echoid-s7144" xml:space="preserve">ex ea, quam <lb/> <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/> habet, AB, ad, BT, <lb/> <anchor type="note" xlink:label="note-0314-03a" xlink:href="note-0314-03"/> ideſt, AH, ad, HT, <lb/>ideſt omnia quadra-<lb/> <anchor type="note" xlink:label="note-0314-04a" xlink:href="note-0314-04"/> ta, & </s> <s xml:id="echoid-s7145" xml:space="preserve">H, ad omnia <lb/>quadrata, HZ; </s> <s xml:id="echoid-s7146" xml:space="preserve">& </s> <s xml:id="echoid-s7147" xml:space="preserve">ex <lb/> <anchor type="note" xlink:label="note-0314-05a" xlink:href="note-0314-05"/> ea, quam habet, BT, <lb/> <anchor type="note" xlink:label="note-0314-06a" xlink:href="note-0314-06"/> ad portionem, HST, <lb/>ideſt omnia quadra-<lb/>ta, HZ, ad omnia <lb/>quadrata ſemiportio-<lb/>nis, HTV, ſed etiam <lb/>omnia quadrata ſe-<lb/>micirculi, vel ſemiel-<lb/>lipſis, HMA, ad on-<lb/>nia quadrata ſemipor <lb/>tionis, HTV, ha-<lb/>bent rationem com-<lb/>poſitam ex ea, quam <lb/>habent omnia qua-<lb/>drata ſemicirculi, vel <lb/>ſemiellipſis, HMA, <lb/>ad omnia quadrata, <lb/>H &</s> <s xml:id="echoid-s7148" xml:space="preserve">, & </s> <s xml:id="echoid-s7149" xml:space="preserve">ex ea, quam <lb/>habent hęc ad omnia <lb/>quadrata ſemiportio-<lb/>nis, HTV, ergo pa-<lb/>rabola, HGA, ad portionem, HST, eſt vt omnia quadrata, HM <lb/> <anchor type="note" xlink:label="note-0314-07a" xlink:href="note-0314-07"/> A, ad omnia quadrata ſemiportionis, HTV, ideſt vt parallelepipe-<lb/>dum ſub altitudine, XA, baſi quadrato, AH, ad parallelepipedum <lb/>ſub altitudine, XT, baſi quadrato, TH; </s> <s xml:id="echoid-s7150" xml:space="preserve">vel vt cubus, AH, ad pa-<lb/>rallelepipedum ſub altitudine tripla, AT, baſi quadrato, TH, cum <lb/>cubo, TH, ſic. </s> <s xml:id="echoid-s7151" xml:space="preserve">n. </s> <s xml:id="echoid-s7152" xml:space="preserve">eſſe omnia quadrata ſemicirculi, vel ſemiellipſis, <lb/>HMA, ad omnia quadrata ſemiportionis, HVT, oſtenſum eſt <lb/>Lib. </s> <s xml:id="echoid-s7153" xml:space="preserve">3. </s> <s xml:id="echoid-s7154" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s7155" xml:space="preserve">6.</s> <s xml:id="echoid-s7156" xml:space="preserve"/> </p> <div xml:id="echoid-div703" type="float" level="2" n="1"> <figure xlink:label="fig-0313-01" xlink:href="fig-0313-01a"> <image file="0313-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0313-01"/> </figure> <note position="left" xlink:label="note-0314-01" xlink:href="note-0314-01a" xml:space="preserve">Diff. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0314-02" xlink:href="note-0314-02a" xml:space="preserve">Exante.</note> <figure xlink:label="fig-0314-01" xlink:href="fig-0314-01a"> <image file="0314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0314-01"/> </figure> <note position="left" xlink:label="note-0314-03" xlink:href="note-0314-03a" xml:space="preserve">5. Lib. 2.</note> <note position="left" xlink:label="note-0314-04" xlink:href="note-0314-04a" xml:space="preserve">9. Lib. 2.</note> <note position="left" xlink:label="note-0314-05" xlink:href="note-0314-05a" xml:space="preserve">Ex antec.</note> <note position="left" xlink:label="note-0314-06" xlink:href="note-0314-06a" xml:space="preserve">Defin. 12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0314-07" xlink:href="note-0314-07a" xml:space="preserve">Lib. 3.</note> </div> <pb o="295" file="0315" n="315" rhead="LIBER IV."/> </div> <div xml:id="echoid-div705" type="section" level="1" n="414"> <head xml:id="echoid-head434" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s7157" xml:space="preserve">_H_Inc patet, quod, diuidendo, portio parabolæ, SGAT, ad por-<lb/>tionem, SHT, erit bt omnia quadrata ſemiportionis, AMVT, <lb/>ad omnia quadrata ſemiportionis, HVT, .</s> <s xml:id="echoid-s7158" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7159" xml:space="preserve">bt parallelepipedum ſub <lb/>altitudine linea compoſita ex, OH, HT, baſi quadrato, TA, ad paral-<lb/>lelepipedum ſub altitudine, XT, baſi quadrato, HT, bt patet in Coroll. <lb/></s> <s xml:id="echoid-s7160" xml:space="preserve">ſupradictæ Propoſ. </s> <s xml:id="echoid-s7161" xml:space="preserve">6. </s> <s xml:id="echoid-s7162" xml:space="preserve">eiuſdem Libri 3.</s> <s xml:id="echoid-s7163" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div706" type="section" level="1" n="415"> <head xml:id="echoid-head435" xml:space="preserve">THEOREMA VII. PROPOS. VII.</head> <p> <s xml:id="echoid-s7164" xml:space="preserve">SI duæ ad baſim parabolæ applicentur vtcunque rectæ li-<lb/>neæ, abſciſſæ portiones parabolæ eruntinterſe, vt pa-<lb/>rallelepipeda ſub baſibus quadratis abſciſſarum à baſi per <lb/>eaſdem applicatas ab eadem extremitate, à qua portiones <lb/>abſciſſæ intelliguntur, altitudinibus compoſitis ex reſiduis <lb/>dictæ baſis (demptis abſciſſis) & </s> <s xml:id="echoid-s7165" xml:space="preserve">dimidia totius.</s> <s xml:id="echoid-s7166" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7167" xml:space="preserve">Sit ergo parabola, HGA, in baſi, HA, ad quam ordinatim ap-<lb/>plicentur duæ vtcunque lineæ, ST, RV, abſcindentes portiones, R <lb/> <anchor type="figure" xlink:label="fig-0315-01a" xlink:href="fig-0315-01"/> HV, SHT. </s> <s xml:id="echoid-s7168" xml:space="preserve">Dico portionem, RHV, ad <lb/>portionem, SHT, eſſe (ſi producatur, AX, <lb/>æqualis ipſius baſis, AH, medietati) vt pa-<lb/>rallelepipedum ſub altitudine, XV, baſi qua-<lb/>drato, VH, ad parallelepipedum ſub altitudi-<lb/>ne, XT, baſi quadrato, TH. </s> <s xml:id="echoid-s7169" xml:space="preserve">Eſt enim por-<lb/>tio, RHV, ad parabolam, AGH, vt paral-<lb/>lelepipedum ſub altitudine, XV, baſi quadra-<lb/> <anchor type="note" xlink:label="note-0315-01a" xlink:href="note-0315-01"/> to, VH, ad parallelepipedum ſub altitudine, <lb/>XA, baſi quadrato, AH, item parabola, A <lb/>GH, ad portionem, HST, eſt vt parallele-<lb/>pipedum ſub altitudine, XA, baſi quadrato, <lb/>AH, ad parall elepipedum ſub altitudine, XT, <lb/>baſi quadrato, TH, ergo exæquali portio, R <lb/>HV, ad portionem, SHT, eſt vt parallele-<lb/>pipedum ſub altitudine, XV, baſi quadrato, <lb/>VH, ad parallelepipedum ſub altitudine, X <lb/>T, baſi quadrato, TH, quod oſtendere opor-<lb/>tebar.</s> <s xml:id="echoid-s7170" xml:space="preserve"/> </p> <div xml:id="echoid-div706" type="float" level="2" n="1"> <figure xlink:label="fig-0315-01" xlink:href="fig-0315-01a"> <image file="0315-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0315-01"/> </figure> <note position="right" xlink:label="note-0315-01" xlink:href="note-0315-01a" xml:space="preserve">Exantec.</note> </div> <pb o="296" file="0316" n="316" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div708" type="section" level="1" n="416"> <head xml:id="echoid-head436" xml:space="preserve">THEOREMA VIII. PROPOS. VIII.</head> <p> <s xml:id="echoid-s7171" xml:space="preserve">SI ad baſim datæ parabolæ ordinatim applicetur recta li-<lb/>nea, ſub qua, & </s> <s xml:id="echoid-s7172" xml:space="preserve">ſub portione baſis abſciſſa, ac earum ex-<lb/>trema iungente, fiattriangulum, portio parabolæ abſciſſa ad <lb/>triangulum ſibi inſcriptum erit, vt ad reliquam baſis, dempta <lb/>abſciſſa, eadem reliqua cum, {1/3}, ipſius abſciſſæ.</s> <s xml:id="echoid-s7173" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7174" xml:space="preserve">Sit parabola, HGA, in baſi, HA, ad quam ordinatim applicetur <lb/>vtcunque recta linea, ST, fiat autem triangulum ſub, ST, & </s> <s xml:id="echoid-s7175" xml:space="preserve">vtrauis <lb/> <anchor type="figure" xlink:label="fig-0316-01a" xlink:href="fig-0316-01"/> duarum, HT, TA, vt ſub, HT, & </s> <s xml:id="echoid-s7176" xml:space="preserve">ſub, SH, <lb/>quod ſit, HST. </s> <s xml:id="echoid-s7177" xml:space="preserve">Dico portionem, HST, ad <lb/>triangulum, HST, eſſe vt compoſitam ex, A <lb/>T, &</s> <s xml:id="echoid-s7178" xml:space="preserve">, {1/3}, TH; </s> <s xml:id="echoid-s7179" xml:space="preserve">ad, AT, compleatur paralle-<lb/>logrammum, CT, eſt ergo parallelogrammum, <lb/> <anchor type="note" xlink:label="note-0316-01a" xlink:href="note-0316-01"/> CT, ad portionem, HST, vt, AT, ad com-<lb/>poſitam ex, {1/2}, AT, &</s> <s xml:id="echoid-s7180" xml:space="preserve">, {1/6}, TH, & </s> <s xml:id="echoid-s7181" xml:space="preserve">anteceden-<lb/>tium dimidia ſcilicet triangulum, HST, ad <lb/>portionem, HST, erit vt dimidia, AT, ad <lb/>compoſitam ex, {1/2}, AT, &</s> <s xml:id="echoid-s7182" xml:space="preserve">, {1/6}, TH, ideſt vt, <lb/>AT, ad, AT, cum, {1/3}, HT, & </s> <s xml:id="echoid-s7183" xml:space="preserve">conuertendo, <lb/>portio, HST, ad triangulum, HST, erit vt <lb/>compoſita ex, {1/3}, HT, & </s> <s xml:id="echoid-s7184" xml:space="preserve">tota, TA, ad, TA, <lb/>quod oſtendendum nobis erat.</s> <s xml:id="echoid-s7185" xml:space="preserve"/> </p> <div xml:id="echoid-div708" type="float" level="2" n="1"> <figure xlink:label="fig-0316-01" xlink:href="fig-0316-01a"> <image file="0316-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0316-01"/> </figure> <note position="left" xlink:label="note-0316-01" xlink:href="note-0316-01a" xml:space="preserve">5. Huius.</note> </div> </div> <div xml:id="echoid-div710" type="section" level="1" n="417"> <head xml:id="echoid-head437" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s7186" xml:space="preserve">_P_Oteſt autem, & </s> <s xml:id="echoid-s7187" xml:space="preserve">dicta ratio ſic conſtitui, triplicatis terminis, ſcili-<lb/>cet, quod portio, HST, ad triangulum, HST, ſit bt bna, HA, <lb/>cum duabus, AT, ad tres, AT, bel ſic, quod ſit, bt dimidia, HA, cum, <lb/>AT, ad ipſam, AT.</s> <s xml:id="echoid-s7188" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div711" type="section" level="1" n="418"> <head xml:id="echoid-head438" xml:space="preserve">PROBLEMA I. PROPOS. IX.</head> <p> <s xml:id="echoid-s7189" xml:space="preserve">A Data parabola portionem abſcindere per lineam ab <lb/>eiuſdem baſim ordinatim ductam, quę ad triangulum <lb/>ſub eadem ordinatim ducta, & </s> <s xml:id="echoid-s7190" xml:space="preserve">abſciſſa per eandem à baſi pa-<lb/>rabolæ ad eandem partem, ad quam abſcinditur portio, ha-<lb/>beat datam rationem, dummodò hæc ſit maior ſexquialtera.</s> <s xml:id="echoid-s7191" xml:space="preserve"/> </p> <pb o="297" file="0317" n="317" rhead="LIBER IV."/> <p> <s xml:id="echoid-s7192" xml:space="preserve">Hoc Problema ſoluetur methodo Propoſ. </s> <s xml:id="echoid-s7193" xml:space="preserve">8. </s> <s xml:id="echoid-s7194" xml:space="preserve">Lib. </s> <s xml:id="echoid-s7195" xml:space="preserve">3. </s> <s xml:id="echoid-s7196" xml:space="preserve">propterea circa <lb/>ipſum non immoror.</s> <s xml:id="echoid-s7197" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div712" type="section" level="1" n="419"> <head xml:id="echoid-head439" xml:space="preserve">THEOREMAIX. PROPOS. X.</head> <p> <s xml:id="echoid-s7198" xml:space="preserve">SI ad baſim datæ parabolæ ordinatim applicentur vtcun-<lb/>que rectæ lineæ, triangula ſub ipſis, & </s> <s xml:id="echoid-s7199" xml:space="preserve">portionibus baſis <lb/>abijſdem abſciſſis, erunt vt parallelepipeda ſub baſibus qua-<lb/>dratis abſciſſarum à baſi, altitudinibus autem reſiduis ipſius <lb/>baſis demptis abſciſſis.</s> <s xml:id="echoid-s7200" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7201" xml:space="preserve">Sit parabola, HGA, cuius baſis, HA, axis, vel diameter, GO, <lb/>ſint autem ductæ duæ vtcunq; </s> <s xml:id="echoid-s7202" xml:space="preserve">ordinatim applicatæ adipſam baſim, <lb/>HA, ipſæ, ST, VX, & </s> <s xml:id="echoid-s7203" xml:space="preserve">iungantur, SH, VH. </s> <s xml:id="echoid-s7204" xml:space="preserve">Dico triangulum, <lb/>VHX, ad triangulum, HST, eſſe vt parallelepipedum ſub altitudi-<lb/>ne, AX, baſi quadrato, XH, ad parallelepipedum ſub altitudine, A <lb/>T, baſi quadrato, TH. </s> <s xml:id="echoid-s7205" xml:space="preserve">Quoniam enim triangula vnum angulum <lb/> <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> vni angulo æqualem habentia ratio-<lb/> <anchor type="note" xlink:label="note-0317-01a" xlink:href="note-0317-01"/> nem habent ex ratione laterum illis <lb/>angulis circumſtãtium compoſitam, <lb/>ideò triangulum, VHX, ad triangu-<lb/> <anchor type="note" xlink:label="note-0317-02a" xlink:href="note-0317-02"/> lum, SHT, habebit rationem com-<lb/>poſitam ex ea, quam habet, VX, ad, <lb/>ST, ideſt rectangulum, AXH, ad <lb/> <anchor type="note" xlink:label="note-0317-03a" xlink:href="note-0317-03"/> rectangulum, ATH, & </s> <s xml:id="echoid-s7206" xml:space="preserve">ex ea, quam <lb/>habet, XH, ad, HT, ſed iſtæ duæ <lb/>rationes componunt rationem paral-<lb/>lelepipedi ſub altitudine, HX, baſi <lb/>rectangulo, AXH, ad parallelepi-<lb/>pedum ſub altitudine, HT, baſi re-<lb/>ctangulo, HTA, . </s> <s xml:id="echoid-s7207" xml:space="preserve">i. </s> <s xml:id="echoid-s7208" xml:space="preserve">parallelepipedi <lb/> <anchor type="note" xlink:label="note-0317-04a" xlink:href="note-0317-04"/> ſub altitudine, AX, baſi quadrato, XH, ad parallelepipedum ſub <lb/>altitudine, AT, baſi quadrato, TH, ergo triangulum, VHX, ad <lb/>triangulum, SHT, erit vt parallelepipedum ſub altitudine, AX, baſi <lb/>quadrato, XH, ad parallelepipedum ſub altitudine, AT, baſi qua-<lb/>drato, TH, quod erat oſtendendum.</s> <s xml:id="echoid-s7209" xml:space="preserve"/> </p> <div xml:id="echoid-div712" type="float" level="2" n="1"> <figure xlink:label="fig-0317-01" xlink:href="fig-0317-01a"> <image file="0317-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0317-01"/> </figure> <note position="right" xlink:label="note-0317-01" xlink:href="note-0317-01a" xml:space="preserve">6. Lib. 2.</note> <note position="right" xlink:label="note-0317-02" xlink:href="note-0317-02a" xml:space="preserve">3. Huius.</note> <note position="right" xlink:label="note-0317-03" xlink:href="note-0317-03a" xml:space="preserve">G. D Cor. <lb/>4. Gen. 34, <lb/>lib. 2.</note> <note position="right" xlink:label="note-0317-04" xlink:href="note-0317-04a" xml:space="preserve">Schol. 35. <lb/>lib. 2.</note> </div> <pb o="298" file="0318" n="318" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div714" type="section" level="1" n="420"> <head xml:id="echoid-head440" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s7210" xml:space="preserve">_H_Inc apparet, ſi producatur, GO, btcunq; </s> <s xml:id="echoid-s7211" xml:space="preserve">in, E, & </s> <s xml:id="echoid-s7212" xml:space="preserve">circa ſemiaxes, <lb/>bel ſemidiametros, HO, OE, deſcribi intelligatur ſemicirculus, <lb/>vel ſemiellipſis, HEA, quod, ſi etiam producantur, ST, VX, in, N, <lb/>M, & </s> <s xml:id="echoid-s7213" xml:space="preserve">iungantur, HN, HM; </s> <s xml:id="echoid-s7214" xml:space="preserve">omnia quadrata trianguli, HXM, ad <lb/>omnia quadrata trianguli, HTN, regula, OE, erunt in ratione com-<lb/>poſita ex ea, quam habet quadratum, XM, ad quadratum, TN, . </s> <s xml:id="echoid-s7215" xml:space="preserve">i re-<lb/>ctangulum, AXH, ad rectangulum, ATH, & </s> <s xml:id="echoid-s7216" xml:space="preserve">ex ea, quam habet, <lb/>XH, ad, HT, . </s> <s xml:id="echoid-s7217" xml:space="preserve">i. </s> <s xml:id="echoid-s7218" xml:space="preserve">erunt, bt parallelepipedum ſub altitudine, AX, baſt <lb/>quadrato, XH, ad parallelepipedum ſub altitudine, AT, baſi qua-<lb/>drato, TH.</s> <s xml:id="echoid-s7219" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div715" type="section" level="1" n="421"> <head xml:id="echoid-head441" xml:space="preserve">THEOREMA X. PROPOS. XI.</head> <p> <s xml:id="echoid-s7220" xml:space="preserve">SI ad axim, vel diametrum datæ parabolæ ordinatim ap-<lb/>plicentur duę rectæ lineę eandem ſecantes, deinde ſum-<lb/>pto extremo puncto minoris dictarum ordinatim applicata-<lb/>rum, & </s> <s xml:id="echoid-s7221" xml:space="preserve">alio extremo puncto maioris dictarum, ſed non ad <lb/>eandem partem, iungantur dicta puncta recta linea; </s> <s xml:id="echoid-s7222" xml:space="preserve">hæc di-<lb/>uidet quadrilineum duabus ordinatim applicatis incluſum <lb/>in duo trilinea: </s> <s xml:id="echoid-s7223" xml:space="preserve">Trilineum igitur conſtitutum in maiori di-<lb/>ctarum linearum ad trilineum cõſtitutum in minori tanquam <lb/>in baſi erit, vt dicta maior ordinatim ductarum, ſimul cum <lb/>tertia proportionali duarum, quarum prima eſt tripla com-<lb/>poſitę ex minori, & </s> <s xml:id="echoid-s7224" xml:space="preserve">dimidia exceſſus maioris ſuper minorem, <lb/>ſecunda autem eſt dimidia dicti exceſſus, ad eandem mino-<lb/>rem, cum eadem tertia proportionali.</s> <s xml:id="echoid-s7225" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7226" xml:space="preserve">Sit ergo parabola, cuius baſis, BH, axis, vel diameter, NO, due <lb/>adipſam vtcunque ordinatim applicatæ ſint, BH, baſis, &</s> <s xml:id="echoid-s7227" xml:space="preserve">, AM, <lb/>minor ipſa, BH, abſcindens parabolam, ANM, ſumatur autem <lb/>vtcunque punctum, A, extremum minoris, AM, & </s> <s xml:id="echoid-s7228" xml:space="preserve">punctum, H, <lb/>ad aliam partem de duobus extremis maioris, BH, & </s> <s xml:id="echoid-s7229" xml:space="preserve">iungantur, A, <lb/>H, puncta recta linea, AH, deindeà punctis, A, M, demittantur <lb/>verſus, BH, parallelæipſi, NO; </s> <s xml:id="echoid-s7230" xml:space="preserve">AC, MG, erit ergo, BC, GH, <lb/>exceſſus, BH, ſuper, AM, &</s> <s xml:id="echoid-s7231" xml:space="preserve">, BC, æqualis ipſi, GH, dimidium <lb/>dicti exceſſus; </s> <s xml:id="echoid-s7232" xml:space="preserve">fiat etiam, vt tripla, HC, ad, BC, ita, BC, ad, C <pb o="299" file="0319" n="319" rhead="LIBER IV."/> E, & </s> <s xml:id="echoid-s7233" xml:space="preserve">iungatur, AG. </s> <s xml:id="echoid-s7234" xml:space="preserve">Dico trilineum, ABH, ad trilineum, AM <lb/>H, eſſe vt, BH, cum, CE, ad ipſam, AM, cum, CE: </s> <s xml:id="echoid-s7235" xml:space="preserve">Prius au-<lb/>tem dico portionculam, ASB, eſſe æqualem portionculæ, MIH, <lb/>& </s> <s xml:id="echoid-s7236" xml:space="preserve">enim trapezium, ABOR, æquatur trapezio, ROHM, & </s> <s xml:id="echoid-s7237" xml:space="preserve"><lb/>quadrilineum, RASBO, ipſi quadrilineo, RMIHO, cum, A <lb/>O, axis, vel diameter bifariam diuidat omnes æquidiſtantes ipſi, B <lb/>H, & </s> <s xml:id="echoid-s7238" xml:space="preserve">ideò omnes lineæ quadrilinei, RASBO, æquentur omni-<lb/>bus lineis quadrilinei, RMHO, vnde dicta quadrilinea etiam ſunt <lb/>ęqualia, & </s> <s xml:id="echoid-s7239" xml:space="preserve">ideo portionculæ, ASB, MIH, inter ſe ſunt æquales: <lb/></s> <s xml:id="echoid-s7240" xml:space="preserve"> <anchor type="note" xlink:label="note-0319-01a" xlink:href="note-0319-01"/> Quoniam vero portio, ASBC, ad triangulum, ABC, eſt vt com-<lb/>poſita ex {1/3}, BC, & </s> <s xml:id="echoid-s7241" xml:space="preserve">ex, CH, ad, CH, ideò, diuidendo, portion-<lb/> <anchor type="note" xlink:label="note-0319-02a" xlink:href="note-0319-02"/> cula, ASB, ad triangulum, ABC, erit vt {1/3}, BC, ad, CH, vel <lb/>vt, BC, ad triplam, CH, . </s> <s xml:id="echoid-s7242" xml:space="preserve">i. </s> <s xml:id="echoid-s7243" xml:space="preserve">ſumpta, BC, communi altitudine, vt <lb/>quadratum, BC, ad rectangulum ſub, BC, & </s> <s xml:id="echoid-s7244" xml:space="preserve">tripla, CH; </s> <s xml:id="echoid-s7245" xml:space="preserve">eſt au-<lb/> <anchor type="note" xlink:label="note-0319-03a" xlink:href="note-0319-03"/> tem triangulum, ABC, ad triangulum, ABH, vt, CB, ad, BH, <lb/>ideſt (ſumpta communi altitudine tripla, CH,) vt rectangulum <lb/>ſub, BC, & </s> <s xml:id="echoid-s7246" xml:space="preserve">tripla, CH, ad rectangulum ſub, BH, & </s> <s xml:id="echoid-s7247" xml:space="preserve">tripla, CH, <lb/> <anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/> ergo ex æquali portioncula, ASB, <lb/>ad triangulum, ABH, erit vt qua-<lb/>dratum, BC, ad rectangulum ſub, <lb/>BH, & </s> <s xml:id="echoid-s7248" xml:space="preserve">tripla, HC, quoniam vero, <lb/>BC, eſt media proportionalis inter <lb/>triplam, HC, & </s> <s xml:id="echoid-s7249" xml:space="preserve">ipſam, CE, ideò <lb/> <anchor type="note" xlink:label="note-0319-04a" xlink:href="note-0319-04"/> quadiatum, BC, æquatur rectan-<lb/>gulo ſub tripla, HC, & </s> <s xml:id="echoid-s7250" xml:space="preserve">ſub, CE, <lb/>vnde portioncula, ASB, ad triangulum, ABH, erit vt rectangu-<lb/>lum ſub, CE, & </s> <s xml:id="echoid-s7251" xml:space="preserve">tripla, CH, ad rectangulum ſub, BH, & </s> <s xml:id="echoid-s7252" xml:space="preserve">tripla, <lb/>CH, ideſt erit, vt baſis, CE, ad baſim, BH, ergo, componendo, <lb/>trilineum, ASBH, ad triangulum, ABH, erit vt, CE, cum, B <lb/>H, ad ipſam, BH, triangulum verò, ABH, ad triangulum, AC <lb/>G, vel ad triangulum, AGM, eſt vt, BH, ad, CG, vel ad, AM, <lb/>eſt vero triangulum, AGM, æquale triangulo, AHM, ergo tri-<lb/>lineum, ASBH, ad triangulum, AMH, erit vt, CE, cum, BH, <lb/>ad, AM, eſt verò trilineum, ASBH, ad portionculam, ASB, <lb/>vel, MIH, illi æqualem, per conuerſionem rationis, vt, BH, <lb/>cum, CE, ad ipſam, CE, ergo, colligendo, trilineum, ASBH, <lb/>ad triangulum, AHM, & </s> <s xml:id="echoid-s7253" xml:space="preserve">portionculam, MIH, . </s> <s xml:id="echoid-s7254" xml:space="preserve">i. </s> <s xml:id="echoid-s7255" xml:space="preserve">ad trilineum, <lb/>AMIH, erit vt, BH, cum, CE, ad ipſam, AM, cum, CE, <lb/>quod oſtendere oportebat.</s> <s xml:id="echoid-s7256" xml:space="preserve"/> </p> <div xml:id="echoid-div715" type="float" level="2" n="1"> <note position="right" xlink:label="note-0319-01" xlink:href="note-0319-01a" xml:space="preserve">3.2.</note> <note position="right" xlink:label="note-0319-02" xlink:href="note-0319-02a" xml:space="preserve">8. huius.</note> <note position="right" xlink:label="note-0319-03" xlink:href="note-0319-03a" xml:space="preserve">5. l. 2.</note> <figure xlink:label="fig-0319-01" xlink:href="fig-0319-01a"> <image file="0319-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0319-01"/> </figure> <note position="right" xlink:label="note-0319-04" xlink:href="note-0319-04a" xml:space="preserve">Elicitur <lb/>ex 12. 1. 2.</note> </div> <pb o="300" file="0320" n="320" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div717" type="section" level="1" n="422"> <head xml:id="echoid-head442" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s7257" xml:space="preserve">_H_Inc patet triangulum, ABH, ad portionculam, ASB, eſſe bt, <lb/>BH, ad, CE.</s> <s xml:id="echoid-s7258" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div718" type="section" level="1" n="423"> <head xml:id="echoid-head443" xml:space="preserve">THEOREMA XI. PROPOS. XII.</head> <p> <s xml:id="echoid-s7259" xml:space="preserve">ASſumpta figura Propoſ. </s> <s xml:id="echoid-s7260" xml:space="preserve">ant. </s> <s xml:id="echoid-s7261" xml:space="preserve">dimiſſa recta, AG, & </s> <s xml:id="echoid-s7262" xml:space="preserve">con-<lb/>ſtituto parallelogrammo ſuper, BH, circa axim, vel <lb/>diametrum, RO, quod ſit, PH, iunctiſque, BR, RH, o-<lb/>ſtendemus parallelogrammum, PH, ad fruſtum parabolæ, <lb/>ASBHIM, eſſe vt, BH, ad, HC, cum, CE; </s> <s xml:id="echoid-s7263" xml:space="preserve">& </s> <s xml:id="echoid-s7264" xml:space="preserve">trian-<lb/>gulum, RBH, ad idem fruſtum eſſe vt, BH, ad duplam, <lb/>HC, CE.</s> <s xml:id="echoid-s7265" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7266" xml:space="preserve">Parallelogrammum enim, PH, eſt ad triangulum, ABH, vt <lb/>dupla, BH, ad ipſam, BH, triangulum verò, ABH, ad ſection-<lb/>culam, ASB, eſt vt, BH, ad, CE, ergo, ex æquali, parallelo-<lb/>grammum, PH, ad ſectionculam, ASB, eſt vt dupia, BH, ad, <lb/> <anchor type="note" xlink:label="note-0320-01a" xlink:href="note-0320-01"/> <anchor type="figure" xlink:label="fig-0320-01a" xlink:href="fig-0320-01"/> CE, & </s> <s xml:id="echoid-s7267" xml:space="preserve">ad duas portionculas, AS <lb/>B, MIH, erit vt dupla, BH, ad <lb/>duplam, CE, ideſt vt, BH, ad, C <lb/> <anchor type="note" xlink:label="note-0320-02a" xlink:href="note-0320-02"/> E. </s> <s xml:id="echoid-s7268" xml:space="preserve">Item parallelogrammum, PH, <lb/>ad trapezium, ABHM, eſt vt, B <lb/>H, ad, AM, cum dimidio exceſſus, <lb/>BH, ſuper, AM, . </s> <s xml:id="echoid-s7269" xml:space="preserve">i. </s> <s xml:id="echoid-s7270" xml:space="preserve">ad, AM, vel, <lb/>CG, GH, ergo, colligendo, pa-<lb/>rallelogrammum, PH, ad ſectionculas, ASB, MIH, cum trape-<lb/>zio, ABHM, . </s> <s xml:id="echoid-s7271" xml:space="preserve">i. </s> <s xml:id="echoid-s7272" xml:space="preserve">ad fruſtum parabolæ, ASBHIM, erit vt, BH, <lb/>ad, HC, cum, CE. </s> <s xml:id="echoid-s7273" xml:space="preserve">Quia verò triangulum, RBH, eſt dimidium <lb/>parallelogrammi, PH, ideò ad fruſtum, ASBH'IM, erit vt di-<lb/>midia, BH, ad, HC, cum, CE, . </s> <s xml:id="echoid-s7274" xml:space="preserve">i. </s> <s xml:id="echoid-s7275" xml:space="preserve">vt, BH, ad duplam, HC, C <lb/>E, quod erat oſtendendum.</s> <s xml:id="echoid-s7276" xml:space="preserve"/> </p> <div xml:id="echoid-div718" type="float" level="2" n="1"> <note position="left" xlink:label="note-0320-01" xlink:href="note-0320-01a" xml:space="preserve">Gorol. 11 <lb/>huius.</note> <figure xlink:label="fig-0320-01" xlink:href="fig-0320-01a"> <image file="0320-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0320-01"/> </figure> <note position="left" xlink:label="note-0320-02" xlink:href="note-0320-02a" xml:space="preserve">O. 1. 2.</note> </div> <figure> <image file="0320-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0320-02"/> </figure> <pb o="301" file="0321" n="321" rhead="LIBER IV."/> </div> <div xml:id="echoid-div720" type="section" level="1" n="424"> <head xml:id="echoid-head444" xml:space="preserve">THEOREMA XII. PROPOS. XIII.</head> <p> <s xml:id="echoid-s7277" xml:space="preserve">SIab extremo puncto baſis datæ parabolæ ducatur vſq; </s> <s xml:id="echoid-s7278" xml:space="preserve">ad <lb/>curuam parabolæ ſupra, vel infra baſim (indefinitè <lb/>producta ipſa curua) recta linea: </s> <s xml:id="echoid-s7279" xml:space="preserve">Data parabola ad ſegmen-<lb/>ta ſub ductis lineis, & </s> <s xml:id="echoid-s7280" xml:space="preserve">curua ab ijſdem abſciſſa comprehen-<lb/>ſa, ſingillatim ſumpta, erit vt cubus baſis ipſius datæpara-<lb/>bolæ ad cubum rectæ lineæ dicto puncto interceptæ, & </s> <s xml:id="echoid-s7281" xml:space="preserve">alio <lb/>puncto eiuſdem baſis productæ, ſi opus ſit, in quod cadit <lb/>recta linea, quæ ducitur ab alio extremo puncto baſis re-<lb/>ſecti ſegmenti parallela axi, vel diametro ipſius datæ pa-<lb/>rabolæ.</s> <s xml:id="echoid-s7282" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7283" xml:space="preserve">Sit ergo data parabola, HNB, inbaſi, HB, ſumpto autem vno <lb/>extremorum punctorum, H, B, ipſius baſis, H B, vtipſum, H, ab <lb/>eo ducatur vtcunq; </s> <s xml:id="echoid-s7284" xml:space="preserve">recta linea, HA, ſupra baſim, HB, & </s> <s xml:id="echoid-s7285" xml:space="preserve">indefi-<lb/>nitè producta curua, NAB, alia, HV, ſubterbàſim, vt ſint con-<lb/>ſtituta ſegmenta, ANH, VBNH, ſit autem axis, vel diameter, <lb/>NO, cui parallelæ ducantur per puncta, AV, verſus baſim, HB, <lb/> <anchor type="figure" xlink:label="fig-0321-01a" xlink:href="fig-0321-01"/> productam, ſi opus ſit, occur-<lb/>rentes illi in punctis, X, C. <lb/></s> <s xml:id="echoid-s7286" xml:space="preserve">Dico ergo parabolam, HNB, <lb/>ad ſegmentum, HN.</s> <s xml:id="echoid-s7287" xml:space="preserve">A, eſſe vt <lb/>cubus, HB, ad cubum, HC. </s> <s xml:id="echoid-s7288" xml:space="preserve"><lb/>Eandem verò ad ſegmentum, <lb/>HNBV, eſſe vt cubum, BH, <lb/>ad cubum, HX, iungantur <lb/>puncta, B, A; </s> <s xml:id="echoid-s7289" xml:space="preserve">B, N; </s> <s xml:id="echoid-s7290" xml:space="preserve">N, H, <lb/>& </s> <s xml:id="echoid-s7291" xml:space="preserve">ſit, CE, tertia proportiona-<lb/>lis duarum, quarum prima eſt <lb/>tripla, CH, ſecunda autem ipſa, BC. </s> <s xml:id="echoid-s7292" xml:space="preserve">Quoniam ergo triangula, <lb/> <anchor type="note" xlink:label="note-0321-01a" xlink:href="note-0321-01"/> NBH, BAH, ſunt in eadem baſi, BH, erunt inter ſe, vt altitu-<lb/>dines, vel vt lineæ, quæ a verticibus, NA, ad baſes ductæ cum <lb/>eiſdem æqualiter inclinantur .</s> <s xml:id="echoid-s7293" xml:space="preserve">i. </s> <s xml:id="echoid-s7294" xml:space="preserve">triangulum, HNB, ad triangu-<lb/>lum, HAB, erit vt, NO, ad, AC, .</s> <s xml:id="echoid-s7295" xml:space="preserve">i. </s> <s xml:id="echoid-s7296" xml:space="preserve">vt rectangulum, HOB, <lb/>ad rectangulum, HCB. </s> <s xml:id="echoid-s7297" xml:space="preserve">Inſuper triangulum, HNB, ad portion-<lb/> <anchor type="note" xlink:label="note-0321-02a" xlink:href="note-0321-02"/> culam, ASB, habet rationem compoſitam ex ratione trianguli, <lb/>HNB, ad triangulum, HAB, .</s> <s xml:id="echoid-s7298" xml:space="preserve">i. </s> <s xml:id="echoid-s7299" xml:space="preserve">ex ratione rectanguli, HOB, <lb/>ad rectangulum, HCB, & </s> <s xml:id="echoid-s7300" xml:space="preserve">ex ratione trianguli, HAB, ad por- <pb o="302" file="0322" n="322" rhead="GEOMETRIÆ"/> tionculam, ASB, .</s> <s xml:id="echoid-s7301" xml:space="preserve">i. </s> <s xml:id="echoid-s7302" xml:space="preserve">exratione, BH, ad, CE, quæ duæ rationes <lb/> <anchor type="note" xlink:label="note-0322-01a" xlink:href="note-0322-01"/> componunt rationem parallelepipedi ſub altitudine, BH, baſi re-<lb/>ctangulo, HOB, vel quadrato, OH, ad parallelepipedum ſub <lb/>altitudine, CE, baſi rectangulo, HCB, ergo triangulum, HNB, <lb/>ad portionculam, ASB, eſt vt parallelepipedum ſub altitudine, <lb/>BH, baſi quadrato, HO, ad parallelepipedum ſub altitudine, C <lb/>E, baſi rectangulo, HCB, eſt autem, vt dicebatur, triangulum, <lb/>HNB, ad triangulum, HAB, vt rectangulum, HOB, vel qua-<lb/>dratum, HO, ad rectangulum, HCB, ideſt ſumpta, HB, com-<lb/>munialtitudine, vt parallelepipedum ſub altitudine, HB, baſi qua-<lb/>drato, HO, ad parallelepipedum ſub altitudine, HB baſi rectan-<lb/>gulo, HCB, ergo, colligendo, triangulum, HNB, ad portion-<lb/>culam, ASB, cum triangulo, ABH, ſilicet ad trilineum, HAS <lb/>B, erit vt parallelepipedum ſub altitudine, HB, baſi quadrato, H <lb/>O, ad parallelepipedum ſub altitudine compoſita ex, HB, CE, <lb/>baſi rectangulo, HCB; </s> <s xml:id="echoid-s7303" xml:space="preserve">vel vt iſtorum quadrupla ſilicet vt paralle-<lb/>lepipedum ſub eadem altitudine, HB, baſi quadruplo quadrati, H <lb/>O, ideſt quadrato, HB, ſilicet vt cubus, HB, ad parallelepipedum <lb/>ſub eadem altitudine compoſita ex, HB, CE, baſi quadruplo re-<lb/> <anchor type="note" xlink:label="note-0322-02a" xlink:href="note-0322-02"/> ctanguli, HCB. </s> <s xml:id="echoid-s7304" xml:space="preserve">Quia verò parabola, HNB, eſt ſexquitertia trian-<lb/>guli, HNB, ideò erit ad ipſum, vt ſolidum ſexquitertium cubi, H <lb/>B, ad cubum, HB, eſt autem triangulum, HNB, ad trilineum, <lb/>HASB, vt cubus, HB, ad parallelepipedum ſub altitudine com-<lb/>poſita ex, HB, CE, & </s> <s xml:id="echoid-s7305" xml:space="preserve">ſub baſi quadruplo rectanguli, HCB, ergo <lb/> <anchor type="figure" xlink:label="fig-0322-01a" xlink:href="fig-0322-01"/> ex æquali parabola, HNB, <lb/>ad trilineum, HASB, erit vt <lb/>ſolidum ſexquitertium cubi, H <lb/>B, ad parallelepipedum ſub al-<lb/>titudine compoſita ex, HB, C <lb/>E, baſi quadruplo rectanguli, <lb/>HCB; </s> <s xml:id="echoid-s7306" xml:space="preserve">vel vt iſtorum ſubſex-<lb/>quitertia ſilicet vt cubus, HB, <lb/>ab parallelepipedum ſub ea-<lb/>dem altitudine compoſita ex, <lb/>HB, CE, baſi triplo rectan-<lb/>guli, HCB, eſt enim quadruplum rectanguli, HCB, ſexquiter-<lb/>tium tripli eiuſdem rectanguli; </s> <s xml:id="echoid-s7307" xml:space="preserve">hoc autem conſequens parallelepi-<lb/> <anchor type="note" xlink:label="note-0322-03a" xlink:href="note-0322-03"/> pedum poteſt diuidi in parallelepipedum ſub altitudine, CE, baſi <lb/>triplo rectanguli, HCB, vel baſi rectangulo ſub, BC, & </s> <s xml:id="echoid-s7308" xml:space="preserve">tripla, <lb/>CH, & </s> <s xml:id="echoid-s7309" xml:space="preserve">in parallelepipedum ſub altitudine, HB, baſi etiam rectan-<lb/>gulo ſub, BCH, ter ſumpto, quoniam verò tripla, HC, &</s> <s xml:id="echoid-s7310" xml:space="preserve">, CB, <lb/>CE, ſunt deinceps proportionales, ideò parallelepipedum, quod <pb o="303" file="0323" n="323" rhead="LIBER IV."/> ſit ab illis tribus æquale eſt cubo mediæ ideſt parallelepipedum ſub <lb/> <anchor type="note" xlink:label="note-0323-01a" xlink:href="note-0323-01"/> altitudine, CE, & </s> <s xml:id="echoid-s7311" xml:space="preserve">ſub baſi rectangulo ipſius, BC, ductæ in tri-<lb/>plam, CH, æquabitur cubo, BC, remanet adhuc parallelepipe-<lb/>dum ſub altitudine, HB, baſi t@ibus rectangulis, BCH, quod (al-<lb/>titudinem, BH, diuidentes in duas ſilicet in, BC, CH,) diuidi-<lb/>mus in parallelepipedum ſub altitudine, HC, baſirectangulo, H <lb/> <anchor type="note" xlink:label="note-0323-02a" xlink:href="note-0323-02"/> CB, ter ſumpto ideſt in parallelepipedum ſub altitudine, BC, baſi <lb/>quadrato, CH, ter ſumpto, & </s> <s xml:id="echoid-s7312" xml:space="preserve">in parallelepipedum ſub altitudine, <lb/> <anchor type="note" xlink:label="note-0323-03a" xlink:href="note-0323-03"/> BC, baſi rectangulo, BCH, terſumpto ideſt in parallelep pedum <lb/>ſub altitudine, HC, baſi quadrato, BC, ter ſumpto; </s> <s xml:id="echoid-s7313" xml:space="preserve">parallepipe-<lb/> <anchor type="note" xlink:label="note-0323-04a" xlink:href="note-0323-04"/> dum ergo ſub altitudine compoſita ex, HB, CE, baſi rec@angu-<lb/>lo, HCB, ter ſumpto, æquatur parallelepipedis ter ſub, BC, & </s> <s xml:id="echoid-s7314" xml:space="preserve"><lb/>quadrato, CH, terſub, HC, & </s> <s xml:id="echoid-s7315" xml:space="preserve">quadrato, CB, cum cubo, CB, <lb/>ad hæc ergo ſimul ſumpta cubus, HB, erit vt parabola, HNB, <lb/>ad trilineum, HASB; </s> <s xml:id="echoid-s7316" xml:space="preserve">quia verò parallelepipedum ter ſub, BC, <lb/> <anchor type="note" xlink:label="note-0323-05a" xlink:href="note-0323-05"/> & </s> <s xml:id="echoid-s7317" xml:space="preserve">quadrato, CH, cum parallelepipedo ter ſub, HC, & </s> <s xml:id="echoid-s7318" xml:space="preserve">quadrato, <lb/>CB, cum cubo, CB, deficiunt à cubo, BH, quantitate cubi, HC, <lb/>ideo, per conuerſionem rationis, parabola, HNB, ad ſegmentum, <lb/>HNA, erit vt cubus, BH, ad cubum, HC.</s> <s xml:id="echoid-s7319" xml:space="preserve"/> </p> <div xml:id="echoid-div720" type="float" level="2" n="1"> <figure xlink:label="fig-0321-01" xlink:href="fig-0321-01a"> <image file="0321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0321-01"/> </figure> <note position="right" xlink:label="note-0321-01" xlink:href="note-0321-01a" xml:space="preserve">Coroll.1. <lb/>19.huius.</note> <note position="right" xlink:label="note-0321-02" xlink:href="note-0321-02a" xml:space="preserve">Defin.12. <lb/>l.1.</note> <note position="left" xlink:label="note-0322-01" xlink:href="note-0322-01a" xml:space="preserve">Ex Co-<lb/>tol.antec.</note> <note position="left" xlink:label="note-0322-02" xlink:href="note-0322-02a" xml:space="preserve">1.huius.</note> <figure xlink:label="fig-0322-01" xlink:href="fig-0322-01a"> <image file="0322-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0322-01"/> </figure> <note position="left" xlink:label="note-0322-03" xlink:href="note-0322-03a" xml:space="preserve">38.1.2.</note> <note position="right" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">45.1.2.</note> <note position="right" xlink:label="note-0323-02" xlink:href="note-0323-02a" xml:space="preserve">35.1.2.</note> <note position="right" xlink:label="note-0323-03" xlink:href="note-0323-03a" xml:space="preserve">36.1.2.</note> <note position="right" xlink:label="note-0323-04" xlink:href="note-0323-04a" xml:space="preserve">36.1.2.</note> <note position="right" xlink:label="note-0323-05" xlink:href="note-0323-05a" xml:space="preserve">38.1.2.</note> </div> <p> <s xml:id="echoid-s7320" xml:space="preserve">Nuncdico parabolam, HNB, ad ſegmentum, HNBV, eſſe <lb/>vt cubum, BH, ad cubum, HX; </s> <s xml:id="echoid-s7321" xml:space="preserve">ducatur per, V, ipſi, BH, pa-<lb/>rallela, VZ, ſecans curuam parabolæ productam in, Z, & </s> <s xml:id="echoid-s7322" xml:space="preserve">à <lb/> <anchor type="note" xlink:label="note-0323-06a" xlink:href="note-0323-06"/> puncto, H, ipſi, NO, vel, XV, demittatur parallela, HI, oc-<lb/>currens ipſi, VZ, in, I, eſt ergo parabola, BNH, ad parabo-<lb/>lam, VBNHZ, vt cubus, BH, ad cubum, VZ, item parabo-<lb/>la, VBNHZ, ad ſegmentum, VBNH, (quia, VH, eſt ſupra <lb/>baſim, VZ,) eſt vt cubus, ZV, ad cubum, VI, vel, XH; </s> <s xml:id="echoid-s7323" xml:space="preserve">æqua-<lb/>lis, VI, quia, XI, eſt parallelogrammum; </s> <s xml:id="echoid-s7324" xml:space="preserve">ergo, ex æquali, pa-<lb/>rabola, HNB, ad ſegmentum, HNBV, conſtitutum per lineam <lb/>ductam à puncto extremo, H, baſis, BH, properantem infra <lb/>eandem baſim, BH, erit vt cubus, BH, ad cubum, HX, quæ o-<lb/>ſtendenda erant.</s> <s xml:id="echoid-s7325" xml:space="preserve"/> </p> <div xml:id="echoid-div721" type="float" level="2" n="2"> <note position="right" xlink:label="note-0323-06" xlink:href="note-0323-06a" xml:space="preserve">@. huius.</note> </div> </div> <div xml:id="echoid-div723" type="section" level="1" n="425"> <head xml:id="echoid-head445" xml:space="preserve">THEOREMA XIII. PROPOS. XIV.</head> <p> <s xml:id="echoid-s7326" xml:space="preserve">SIintra curuam parabolæ ducantur vtcunque duæ rectæ <lb/>lineæ in eandem curuam terminantes, parabola ab vna <lb/>ductarum conſtituta ad parabolam ab alia conſtitutam erit, <lb/>vt cubus primò ductæ ad cubum rectæ lineæ, quæ, ducitur <pb o="304" file="0324" n="324" rhead="GEOMETRIÆ"/> per punctum extremum alterius ſecundò ductæ, parallela <lb/>primò ductæ, incluſæ dicto puncto, & </s> <s xml:id="echoid-s7327" xml:space="preserve">alio eiuſdem paralle-<lb/>læ productæ, ſi opus ſit; </s> <s xml:id="echoid-s7328" xml:space="preserve">in quod cadit, quæ ducitur per <lb/>aliud extremum punctum ſecundò ductæ, parallela axi, vel <lb/>diametro parabolæ per primò ductam conſtitutæ.</s> <s xml:id="echoid-s7329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7330" xml:space="preserve">Sit curua parabolæ, BAEC, intra quam ſint vtcumq; </s> <s xml:id="echoid-s7331" xml:space="preserve">ductæ in <lb/>eandem curuam hinc inde terminantes (.</s> <s xml:id="echoid-s7332" xml:space="preserve">i. </s> <s xml:id="echoid-s7333" xml:space="preserve">quod non ſint ductæ pa-<lb/>rallelæ axi) primò, BC, ſecundò, AD; </s> <s xml:id="echoid-s7334" xml:space="preserve">ducatur deinde per vtrum <lb/> <anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/> libet extremorum punctorum ſecundò <lb/>ductæ, vt per, A, ipſa, AF, parallela <lb/>ipſi, BC, in quam productam, ſi opus <lb/>ſit, incidat parallela axi, quæ ducitur <lb/>per punctum, D, aliud extremum <lb/>ipſius, AD, occurrat autem illi in, F. <lb/></s> <s xml:id="echoid-s7335" xml:space="preserve">Dico parabolam, BAEC, ad para-<lb/>bolam, AED, eſſe vt cubum, BC, <lb/>ad cubum, AF. </s> <s xml:id="echoid-s7336" xml:space="preserve">Eſt enim parabola, <lb/> <anchor type="note" xlink:label="note-0324-01a" xlink:href="note-0324-01"/> BNC, ad parabolam, ANE, vt cubus, BC, ad cubum, AE, <lb/>item parabola, ANE, ad parabolam, ANED, eſt vt cubus, A <lb/> <anchor type="note" xlink:label="note-0324-02a" xlink:href="note-0324-02"/> E, ad cubum, AF, ergo parabola, BNC, ad parabolam, AN <lb/>ED, eſt vt cubus, BC, ad cubum, AF, quod oſtendere opus <lb/>erat.</s> <s xml:id="echoid-s7337" xml:space="preserve"/> </p> <div xml:id="echoid-div723" type="float" level="2" n="1"> <figure xlink:label="fig-0324-01" xlink:href="fig-0324-01a"> <image file="0324-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0324-01"/> </figure> <note position="left" xlink:label="note-0324-01" xlink:href="note-0324-01a" xml:space="preserve">@. huius.</note> <note position="left" xlink:label="note-0324-02" xlink:href="note-0324-02a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div725" type="section" level="1" n="426"> <head xml:id="echoid-head446" xml:space="preserve">THEOREMA XIV. PROPOS. XV.</head> <p> <s xml:id="echoid-s7338" xml:space="preserve">IN eadem antecedentis figura, ſi ducatur intra parabo-<lb/>lam, BNC, à puncto, V, ſumpto vtcumque in curua, B <lb/>NC, verſus baſim, BC, ipſa, VX, incidens baſi in, X, pa-<lb/>rallela axi, vel diametro eiuſdem parabolæ. </s> <s xml:id="echoid-s7339" xml:space="preserve">Dico parabo-<lb/>lam, ANED, ad ſegmentum, VCX, eſſe vt cubum, AF, <lb/>ad parallelepipedum ter ſub, BX, & </s> <s xml:id="echoid-s7340" xml:space="preserve">quadrato, XC, cum <lb/>cubo, XC.</s> <s xml:id="echoid-s7341" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7342" xml:space="preserve">Nam parabola, ANED, ad parabolam, BNC, conuertendo, <lb/>eſt vt cubus, AF, ad cubum, BC, item parabola, BNC, ad <lb/>ſegmentum, VCX, eſt vt cubus, BC, ad parallelepipedum ter ſub <lb/> <anchor type="note" xlink:label="note-0324-03a" xlink:href="note-0324-03"/> altitudine, BX, baſi quadrato, XC, cum cubo, XC, ergo, ex æ-<lb/>quali, parabola, ANED, ad ſegmentum, VXC, erit vt cubus, <lb/>AF, ad parallelepipedum terſub, BX, & </s> <s xml:id="echoid-s7343" xml:space="preserve">quadrato, XC, cum cu-<lb/>bo, XC, quod oſtendere oportebat.</s> <s xml:id="echoid-s7344" xml:space="preserve"/> </p> <div xml:id="echoid-div725" type="float" level="2" n="1"> <note position="left" xlink:label="note-0324-03" xlink:href="note-0324-03a" xml:space="preserve">6.huius.</note> </div> <pb o="305" file="0325" n="325" rhead="LIBER IV."/> </div> <div xml:id="echoid-div727" type="section" level="1" n="427"> <head xml:id="echoid-head447" xml:space="preserve">THEOREMA XV. PROPOS. XVI.</head> <p> <s xml:id="echoid-s7345" xml:space="preserve">INeadem ſupradicti Theorematis figura oſtendemus tri-<lb/>lineum, VNAI, ad trilineum, VNABX, eſſe vt pa-<lb/>rallelepipedum ter ſub, EI, & </s> <s xml:id="echoid-s7346" xml:space="preserve">quadrato, IA, cum cubo, <lb/>IA, ad parallelepipedum ter ſub, CX, & </s> <s xml:id="echoid-s7347" xml:space="preserve">quadrato, XB, <lb/>cum cubo, XB. </s> <s xml:id="echoid-s7348" xml:space="preserve">Similiter trilineum, VEI, ad trilineum, <lb/>VECX, eſſe vt parallelepipedum ter ſub, AI, & </s> <s xml:id="echoid-s7349" xml:space="preserve">quadra-<lb/>to, IE, cum cubo, IE, ad parallelepipedum terſub, BX, & </s> <s xml:id="echoid-s7350" xml:space="preserve"><lb/>quadrato, XC, cum cubo, XC.</s> <s xml:id="echoid-s7351" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7352" xml:space="preserve">Trilineum enim, VNAI, ad parabolam, ANE, eſt vt paral-<lb/> <anchor type="note" xlink:label="note-0325-01a" xlink:href="note-0325-01"/> lelepipedum ter ſub, EI, & </s> <s xml:id="echoid-s7353" xml:space="preserve">quadrato, IA, cum cubo, IA, ad cu-<lb/>bum, AE, item parabola, ANE, ad parabolam, BNC, eſt vt <lb/>cubus, AE, ad cubum, BC, & </s> <s xml:id="echoid-s7354" xml:space="preserve">tandem parabola, BNC, ad trili-<lb/> <anchor type="note" xlink:label="note-0325-02a" xlink:href="note-0325-02"/> neum, VABX, eſt vt cubus, CB, ad parallelepipedum ter ſub, C <lb/>X, & </s> <s xml:id="echoid-s7355" xml:space="preserve">quadrato, XB, cum cubo, BX, ergo, ex æquali, trilineum, <lb/>VNAI, ad trilineum, VNBX, erit vt parallelepipedum ter ſub, <lb/> <anchor type="note" xlink:label="note-0325-03a" xlink:href="note-0325-03"/> EI, & </s> <s xml:id="echoid-s7356" xml:space="preserve">quadrato, IA, cum cubo, IA, ad parallelepipedum ter <lb/>ſub, CX, & </s> <s xml:id="echoid-s7357" xml:space="preserve">quadrato, XB, cum cubo, XB. </s> <s xml:id="echoid-s7358" xml:space="preserve">Eodem modo o-<lb/>ſtendemus trilineum, VIE, ad trilineum, VXC, eſſe vt paralle-<lb/>lepipedum ter ſub, AI, & </s> <s xml:id="echoid-s7359" xml:space="preserve">quadrato, IE, cum cubo, IE, ad pa-<lb/>rallelepipedum ter ſub, BX, & </s> <s xml:id="echoid-s7360" xml:space="preserve">quadrato, XC, cum cubo, XC, <lb/>quod, &</s> <s xml:id="echoid-s7361" xml:space="preserve">c.</s> <s xml:id="echoid-s7362" xml:space="preserve"/> </p> <div xml:id="echoid-div727" type="float" level="2" n="1"> <note position="right" xlink:label="note-0325-01" xlink:href="note-0325-01a" xml:space="preserve">6.huius.</note> <note position="right" xlink:label="note-0325-02" xlink:href="note-0325-02a" xml:space="preserve">2.huius.</note> <note position="right" xlink:label="note-0325-03" xlink:href="note-0325-03a" xml:space="preserve">6.huius.</note> </div> </div> <div xml:id="echoid-div729" type="section" level="1" n="428"> <head xml:id="echoid-head448" xml:space="preserve">THEOREMA XVI. PROPOS. XVII.</head> <p> <s xml:id="echoid-s7363" xml:space="preserve">SI duæ intra curuam parabolicam ducantur rectæ lineæ <lb/>axem ſecantes, fuerint autem conſtitua@um ab eiſdem <lb/>parabolarum diametri, vel axis, & </s> <s xml:id="echoid-s7364" xml:space="preserve">diameter æquales, & </s> <s xml:id="echoid-s7365" xml:space="preserve">ipſe <lb/>parabolæ erunt æquales.</s> <s xml:id="echoid-s7366" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7367" xml:space="preserve">Sit curua parabolica, BAC, intra quam ducantur vtcunque duę, <lb/>DF, MC, axem ſecantes, ideſt non parallelæ axi, ſint autem, A <lb/>R, HO, diametri, vel axis, & </s> <s xml:id="echoid-s7368" xml:space="preserve">diameter inter ſe æquales. </s> <s xml:id="echoid-s7369" xml:space="preserve">Dico <lb/>parabolam, DAF, eſſe æqualem parabolæ, MHFC; </s> <s xml:id="echoid-s7370" xml:space="preserve">ducatur <pb o="306" file="0326" n="326" rhead="GEOMETRIÆ"/> per, C, ipſi, DF, parallela, CB, & </s> <s xml:id="echoid-s7371" xml:space="preserve">producantur, AR, HO, vſq; <lb/></s> <s xml:id="echoid-s7372" xml:space="preserve">ad, BC, in, P, Q, iunganturque, AC, HC, & </s> <s xml:id="echoid-s7373" xml:space="preserve">à puncto, M, <lb/>ducatur, MX, parallela axi, vel diametro, AP; </s> <s xml:id="echoid-s7374" xml:space="preserve">quoniam ergo, O <lb/> <anchor type="note" xlink:label="note-0326-01a" xlink:href="note-0326-01"/> Q, eſt parallela ipſi, MX, & </s> <s xml:id="echoid-s7375" xml:space="preserve">ipſa ſecat, MC, bifariam in, O, ſe-<lb/>cabit etiam, XC, bifariam in Q; </s> <s xml:id="echoid-s7376" xml:space="preserve">& </s> <s xml:id="echoid-s7377" xml:space="preserve">quia parabola, ABC, ad pa-<lb/>rabolam, MHFC, eſt vt cubus, BC, ad cubum, CX, vel vt cu-<lb/>bus, PC, ad cubum, CQ, ideò ſemiparabola, APC, ad ſemipa-<lb/>rabolam, HOC, erit vt cubus, PC, ad cubum, CQ, & </s> <s xml:id="echoid-s7378" xml:space="preserve">eorun-<lb/> <anchor type="note" xlink:label="note-0326-02a" xlink:href="note-0326-02"/> dem ſubſexquitertia .</s> <s xml:id="echoid-s7379" xml:space="preserve">i. </s> <s xml:id="echoid-s7380" xml:space="preserve">triangulum, APC, ad triangulum, HOC, <lb/>erit vt cubus, PC, ad cubum, CQ: </s> <s xml:id="echoid-s7381" xml:space="preserve">quoniam verò triangula æqui-<lb/>angula habent interſerationem compoſitam exratione baſium, &</s> <s xml:id="echoid-s7382" xml:space="preserve">G <lb/>altitudinum, vel linearum à verticibus earundem ductarum æqua-<lb/> <anchor type="note" xlink:label="note-0326-03a" xlink:href="note-0326-03"/> liter baſibus inclinatarum; </s> <s xml:id="echoid-s7383" xml:space="preserve">ideò triangulum, APC, ad triangu-<lb/>lum, HOC, habebit rationem compoſitam ex ratione baſis, PA, <lb/>ad baſim, OH, vel, AR, illi æqualem, & </s> <s xml:id="echoid-s7384" xml:space="preserve">ex ratione, PC, ad, C <lb/>Q, quæ vel ſunt altitudines, vellineæ ductæ à communi vertice, C, <lb/>cum æquali inclinatione ad baſes, AP, &</s> <s xml:id="echoid-s7385" xml:space="preserve">, HO, productam, quia, <lb/> <anchor type="figure" xlink:label="fig-0326-01a" xlink:href="fig-0326-01"/> AP, HQ, ſunt parallelæ, eſt autem vt, <lb/>PA, ad AR, ita quadratum, PC, ad <lb/>quadratum, RF, ergo triangulum, A <lb/>PC, ad triangulum, HOC, habebit <lb/>rationem compoſitam ex ea, quam ha-<lb/>bet quadratum, PC, ad quadratum, R <lb/>F, & </s> <s xml:id="echoid-s7386" xml:space="preserve">ex ea, quam habet, PC, ad CQ, <lb/>quia verò triangulum, APC, ad trian-<lb/>gulum, HOC, eſt vt cubus, PC, ad <lb/>cubum, CQ, ideò ad illud habet etiam rationem compoſitam ex <lb/>ea, quam habet, PC, ad CQ, & </s> <s xml:id="echoid-s7387" xml:space="preserve">ex ratione quadrati, PC, ad qua-<lb/>dratum, CQ, ergo iſtæ duæ rationes, ſcilicet quam habet, PC, <lb/>ad, CQ, & </s> <s xml:id="echoid-s7388" xml:space="preserve">quadratum, PC, ad quadratum, RF, componunt <lb/>eandem rationem, quam iſtæ duæ, ſcilicet ratio, PC, ad, CQ, & </s> <s xml:id="echoid-s7389" xml:space="preserve"><lb/>quadrati, PC, ad quadratum, CQ, eſt autem in his communis <lb/>ratio, quam habet, PC, ad, CQ, ergo reliqua ratio, quam habet <lb/>quadratum, PC, ad quadratum, CQ, erit eadem ei, quam habet <lb/>quadratum idem, PC, ad quadratum, RF, ergo quadratum, C <lb/> <anchor type="note" xlink:label="note-0326-04a" xlink:href="note-0326-04"/> Q, erit æquale quadrato, RF, &</s> <s xml:id="echoid-s7390" xml:space="preserve">, CQ, erit æqualis ipſi, RF. <lb/></s> <s xml:id="echoid-s7391" xml:space="preserve">Quoniam autem parabola, BAC, ad parabolam, DAF, eſt vt <lb/> <anchor type="note" xlink:label="note-0326-05a" xlink:href="note-0326-05"/> cubus, BC, ad cubum, DF, .</s> <s xml:id="echoid-s7392" xml:space="preserve">i. </s> <s xml:id="echoid-s7393" xml:space="preserve">vt cubus, PC, ad cubum, RF, <lb/>item oſtenſum eſt parabolam eandem, BAC, ad parabolam, MH <lb/>FC, eſſe vt cubum, PC, ad cubum, CQ, ideò parabola, DAF, <lb/>ad parabolam, MHFC, erit vt cubus, RF, ad cubum, QC, ſunt <lb/>autem, QC, RF, inter ſe æquales, vtoſtenſum eſt, & </s> <s xml:id="echoid-s7394" xml:space="preserve">ideò et@am <pb o="307" file="0327" n="327" rhead="LIBER IV."/> eorundem cubi ſunt æquales, ergo parabola, DAF, erit æqualis <lb/>parabolæ, MHFC, quod oſtendere opuserat.</s> <s xml:id="echoid-s7395" xml:space="preserve"/> </p> <div xml:id="echoid-div729" type="float" level="2" n="1"> <note position="left" xlink:label="note-0326-01" xlink:href="note-0326-01a" xml:space="preserve">12.huius.</note> <note position="left" xlink:label="note-0326-02" xlink:href="note-0326-02a" xml:space="preserve">@.huius.</note> <note position="left" xlink:label="note-0326-03" xlink:href="note-0326-03a" xml:space="preserve">Coroll.1. <lb/>19. 1. 2.</note> <figure xlink:label="fig-0326-01" xlink:href="fig-0326-01a"> <image file="0326-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0326-01"/> </figure> <note position="left" xlink:label="note-0326-04" xlink:href="note-0326-04a" xml:space="preserve">2.huius.</note> <note position="left" xlink:label="note-0326-05" xlink:href="note-0326-05a" xml:space="preserve">12.huius.</note> </div> </div> <div xml:id="echoid-div731" type="section" level="1" n="429"> <head xml:id="echoid-head449" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7396" xml:space="preserve">_H_Inc patet, ſi diametri, AR, HO, vel axis, & </s> <s xml:id="echoid-s7397" xml:space="preserve">diameter ſint æquà-<lb/>les, etiam, DF, XC, eſſe ęquales, nam oſtenſum eſt, QC, eſſe æqualem <lb/>ipſi, RF, eſt autem, XC, dupla, CQ &</s> <s xml:id="echoid-s7398" xml:space="preserve">, DF, dupla, FR, ideò etiam, <lb/>XC, DF, ſunt, æquales.</s> <s xml:id="echoid-s7399" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div732" type="section" level="1" n="430"> <head xml:id="echoid-head450" xml:space="preserve">THEOREMA XVII. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s7400" xml:space="preserve">EXpoſita ſemiparabola cum dimidia baſi, & </s> <s xml:id="echoid-s7401" xml:space="preserve">axi, vel <lb/>diametro totius, & </s> <s xml:id="echoid-s7402" xml:space="preserve">completo parallelogrammo ſub <lb/>dicto axi, vel diametro. </s> <s xml:id="echoid-s7403" xml:space="preserve">& </s> <s xml:id="echoid-s7404" xml:space="preserve">ſemibaſi, deſcriptaque ellipſis <lb/>quarta, vel circuli circa axem vel diametrum, & </s> <s xml:id="echoid-s7405" xml:space="preserve">ſemi-<lb/>baſim dictam, tanquam circa ſemiaxes, vel ſemidiame-<lb/>tros coniugatas integræ ellipſis, vel circuli; </s> <s xml:id="echoid-s7406" xml:space="preserve">ſi deinde ſu-<lb/>matur vtcunque punctum in ſemibaſi, per quod ducatur <lb/>recta linea ad oppoſitum latus parallelogrammi paralle-<lb/>la dictæ axi, vel diametro, portio huius inter ſemibaſim, <lb/>& </s> <s xml:id="echoid-s7407" xml:space="preserve">curuam ellipſis, vel circuli incluſa, erit media propor-<lb/>tionalis inter incluſam oppoſitis lateribus parallelogram-<lb/>mi iam dicti, & </s> <s xml:id="echoid-s7408" xml:space="preserve">eadem ſemibaſi, ac curua parabolæ. </s> <s xml:id="echoid-s7409" xml:space="preserve">Si <lb/>verò ſumatur punctum in axi, vel diametro iam dicta, <lb/>& </s> <s xml:id="echoid-s7410" xml:space="preserve">per ipſum ducatur ſemibaſi parallela, producta vſq; </s> <s xml:id="echoid-s7411" xml:space="preserve">ad <lb/>latus oppoſitum parallelogrammi iam dicti, & </s> <s xml:id="echoid-s7412" xml:space="preserve">iungantur <lb/>extrema puncta curuæ parabolæ recta linea, huius portio <lb/>incluſa inter axim, vel diametrum dictam, & </s> <s xml:id="echoid-s7413" xml:space="preserve">curuam pa-<lb/>rabolæ, erit media proportionalis inter eam, quæ inclu-<lb/>ditur lateribus oppoſitis dicti parallelogrammi, & </s> <s xml:id="echoid-s7414" xml:space="preserve">eam, <lb/>quæ includitur lateribus trianguli ſub dicta axi, vel diame-<lb/>tro, & </s> <s xml:id="echoid-s7415" xml:space="preserve">dicta ſemibaſi conſtituti.</s> <s xml:id="echoid-s7416" xml:space="preserve"/> </p> <figure> <image file="0327-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0327-01"/> </figure> <pb o="308" file="0328" n="328" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s7417" xml:space="preserve">Sit ſemiparabola, AOCB, in baſi, BC, & </s> <s xml:id="echoid-s7418" xml:space="preserve">axis, vel diameter <lb/>integræ, AB, compleaturq; </s> <s xml:id="echoid-s7419" xml:space="preserve">parallelogrammum, DB, & </s> <s xml:id="echoid-s7420" xml:space="preserve">circa, A <lb/>B, BC, tanquam ſemiaxes, vel ſemidiametros coniugatas, deicri-<lb/>batur quarta circuli, vel ellipſis, AICB, deinde ſumatur in baſi, B <lb/>C, vtcunque punctum, P, & </s> <s xml:id="echoid-s7421" xml:space="preserve">per, P, ducaturipſi, AB, parallela, <lb/>PH, ſecans curuam parabolæ in, X, & </s> <s xml:id="echoid-s7422" xml:space="preserve">circuli, vel ellipſis, AIC, <lb/> <anchor type="figure" xlink:label="fig-0328-01a" xlink:href="fig-0328-01"/> in, I. </s> <s xml:id="echoid-s7423" xml:space="preserve">Dico ergo, IP, eſſe mediam <lb/>proportionalem inter, HP, PX, pro-<lb/>ducatur, CB, verſus, B, in, Z, ita <lb/>vt, BZ, ſit æqualis, BC, eſt ergo <lb/>quadratum, AB, vel quadratum, H <lb/>P, ad quadratum, PI, vt rectangu-<lb/>lum, ZBC, ad rectangulum, ZPC, <lb/>.</s> <s xml:id="echoid-s7424" xml:space="preserve">i. </s> <s xml:id="echoid-s7425" xml:space="preserve">vt, AB, vel, HP, ad, PX, ergo <lb/>vt, HP, ad, PI, ita erit, IP, ad, PX.</s> <s xml:id="echoid-s7426" xml:space="preserve"/> </p> <div xml:id="echoid-div732" type="float" level="2" n="1"> <figure xlink:label="fig-0328-01" xlink:href="fig-0328-01a"> <image file="0328-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0328-01"/> </figure> </div> <p> <s xml:id="echoid-s7427" xml:space="preserve">Iungantur puncta, A, C, & </s> <s xml:id="echoid-s7428" xml:space="preserve">ſum-<lb/> <anchor type="note" xlink:label="note-0328-01a" xlink:href="note-0328-01"/> pto vtcunq. </s> <s xml:id="echoid-s7429" xml:space="preserve">puncto, V, in, AB, per ipſum ducaturipſi, BC, pa-<lb/>rallela, VF, ſecans curuam parabolæ in, O, & </s> <s xml:id="echoid-s7430" xml:space="preserve">rectam, AC, in, N. <lb/></s> <s xml:id="echoid-s7431" xml:space="preserve">Dico ergo, vt, FV, ad, VO, ita eſſe, VO, ad, VN; </s> <s xml:id="echoid-s7432" xml:space="preserve">eſt enim <lb/>quadratum, BC, vel quadratum, FV, ad quadratum, VO, vt, B <lb/>A, ad, AV, .</s> <s xml:id="echoid-s7433" xml:space="preserve">i. </s> <s xml:id="echoid-s7434" xml:space="preserve">vt, BC, vel, FV, ad, VN, ergo erit, vt, FV, ad, <lb/>VO, ſic, VO, ad, VN, quæ oſtendere oportebat.</s> <s xml:id="echoid-s7435" xml:space="preserve"/> </p> <div xml:id="echoid-div733" type="float" level="2" n="2"> <note position="left" xlink:label="note-0328-01" xlink:href="note-0328-01a" xml:space="preserve">40.cum <lb/>Sch.l.1. <lb/>3.huius.</note> </div> </div> <div xml:id="echoid-div735" type="section" level="1" n="431"> <head xml:id="echoid-head451" xml:space="preserve">THEOREMA XVIII. PROPOS. XIX.</head> <p> <s xml:id="echoid-s7436" xml:space="preserve">PArabolæ ſunt inter ſe, vt parallelogramma illis circum-<lb/>ſcripta latera habentia baſibus, & </s> <s xml:id="echoid-s7437" xml:space="preserve">eorundem axibus, <lb/>vel diametris parallela.</s> <s xml:id="echoid-s7438" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7439" xml:space="preserve">Patet hæc propoſitio, nam dictæ parabolæ ſunt ſubſexquialteræ <lb/> <anchor type="note" xlink:label="note-0328-02a" xlink:href="note-0328-02"/> dictorum parallelogrammorum, & </s> <s xml:id="echoid-s7440" xml:space="preserve">ideò ſunt inter ſe, vt ipſa paral-<lb/>lelogramma.</s> <s xml:id="echoid-s7441" xml:space="preserve"/> </p> <div xml:id="echoid-div735" type="float" level="2" n="1"> <note position="left" xlink:label="note-0328-02" xlink:href="note-0328-02a" xml:space="preserve">1.huius.</note> </div> </div> <div xml:id="echoid-div737" type="section" level="1" n="432"> <head xml:id="echoid-head452" xml:space="preserve">A. COROLL. SECTIO I.</head> <p style="it"> <s xml:id="echoid-s7442" xml:space="preserve">HInc patet, concluſiones, quę de parallelogrammis collectæ ſun@ <lb/>in Propoſ 5.</s> <s xml:id="echoid-s7443" xml:space="preserve">6.</s> <s xml:id="echoid-s7444" xml:space="preserve">7.</s> <s xml:id="echoid-s7445" xml:space="preserve">8. </s> <s xml:id="echoid-s7446" xml:space="preserve">Lib.</s> <s xml:id="echoid-s7447" xml:space="preserve">2. </s> <s xml:id="echoid-s7448" xml:space="preserve">ſuppoſitis quibuſdam conditionibus in <lb/>lateribus, vel in altitudine, & </s> <s xml:id="echoid-s7449" xml:space="preserve">baſi dictorum parallelogrammorum, poſ-<lb/>ſe colligi etiam pro parabolis eaſdem conditiones in axibus, vel diame-<lb/>tris, vel altitudinibus, & </s> <s xml:id="echoid-s7450" xml:space="preserve">baſtbus habentes; </s> <s xml:id="echoid-s7451" xml:space="preserve">quia enim tunc dictæ con- <pb o="309" file="0329" n="329" rhead="LIBER IV."/> ditiones reperiuntur etiam in lateribus circumſcriptorum illis paral-<lb/>lelogrammorum, vel in altitudine, & </s> <s xml:id="echoid-s7452" xml:space="preserve">baſi eorundem, quia baſis eſt <lb/>communis, & </s> <s xml:id="echoid-s7453" xml:space="preserve">reliquum latus axi, vel diametro parabola æquidiſtans, <lb/>ideò ſequuntur illicò oſtenſæ concluſiones pro parallelogrammis; </s> <s xml:id="echoid-s7454" xml:space="preserve">& </s> <s xml:id="echoid-s7455" xml:space="preserve"><lb/>conſequenter etiam pro ipſis parabolis, quarum ipſa parallelogramma <lb/>ſunt ſexquialtera, recipi poſſunt.</s> <s xml:id="echoid-s7456" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div738" type="section" level="1" n="433"> <head xml:id="echoid-head453" xml:space="preserve">B. SECTIO II.</head> <note position="right" xml:space="preserve">B</note> <p style="it"> <s xml:id="echoid-s7457" xml:space="preserve">_Q_Via ergo oſtenſum eſt pàrallelogramma, quæ ſunt in eadem altitu-<lb/>dine, eſſe inter ſe, vt baſes, & </s> <s xml:id="echoid-s7458" xml:space="preserve">quæ in eadem baſi, vel æqualibus <lb/>baſibus, eſſe interſe, vt altitudines, vel vtlinea à verticibus ad baſes <lb/>cum æquali inclinatione ad eaſdem ductæ: </s> <s xml:id="echoid-s7459" xml:space="preserve">ideò colligemus etiam para-<lb/>bolas, quæ ſunt circa eundem axem, vel diametrum, eſſe inter ſe, vt ba-<lb/>ſes; </s> <s xml:id="echoid-s7460" xml:space="preserve">& </s> <s xml:id="echoid-s7461" xml:space="preserve">quæ ſunt in eadem, vel æqualibus baſibus, eſſe inter ſe, vt alti, <lb/>tudines, vel vt lineæ, quæ à verticibus eorundem ad baſes cumæquali <lb/>inclinatione ducuntur, ſiue illa ſint axes, ſiue diametri.</s> <s xml:id="echoid-s7462" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div739" type="section" level="1" n="434"> <head xml:id="echoid-head454" xml:space="preserve">C. SECTIO III.</head> <note position="right" xml:space="preserve">C</note> <p style="it"> <s xml:id="echoid-s7463" xml:space="preserve">_S_Imiliter colligemus parabolas habere rationem compoſitam ex ra-<lb/>tione baſium, & </s> <s xml:id="echoid-s7464" xml:space="preserve">altitudinum, vel linearum, quæ à verticibus du-<lb/>cuntur, æqualiter baſibus inclinatarum, ſiue ſint axes, ſiue diametri.</s> <s xml:id="echoid-s7465" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div740" type="section" level="1" n="435"> <head xml:id="echoid-head455" xml:space="preserve">D. SECTIO IV.</head> <note position="right" xml:space="preserve">D</note> <p style="it"> <s xml:id="echoid-s7466" xml:space="preserve">_I_Tem parabolæ habentes baſes altitudinibus, vel lineis à verticibus <lb/>ductis æqualiter inclinatis reciprocas erunt æquales; </s> <s xml:id="echoid-s7467" xml:space="preserve">& </s> <s xml:id="echoid-s7468" xml:space="preserve">parabolæ <lb/>æquales, quarum diametri æqualiter ab baſes ſint inclinatæ, habebunt <lb/>baſes altitudinibus, vel lineis ductis à verticibus ad baſes æquali@er in-<lb/>clinatis reciprocas.</s> <s xml:id="echoid-s7469" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div741" type="section" level="1" n="436"> <head xml:id="echoid-head456" xml:space="preserve">E. SECTIO V.</head> <note position="right" xml:space="preserve">E</note> <p style="it"> <s xml:id="echoid-s7470" xml:space="preserve">_D_Eniq; </s> <s xml:id="echoid-s7471" xml:space="preserve">parabolæ, quarum axes, vel diametri, ad haſes ęqualiter in-<lb/>clinati, ad eaſdem baſes habent eandem rationem, ſunt in dupla <lb/>ratione baſium, ſiue axium, vel diametrorum, vel vt quadrata eorun-<lb/>dem: </s> <s xml:id="echoid-s7472" xml:space="preserve">N am parallelogramma his parabolis circumſcripta ſunt ſimilia, <lb/>& </s> <s xml:id="echoid-s7473" xml:space="preserve">ideò ſunt, vt quadrata laterum homologorum, quæ vel ſunt axes, aut <lb/>diametri, vel baſes dictarum parabolarum, & </s> <s xml:id="echoid-s7474" xml:space="preserve">ideò etiam ipſæ parabolæ <lb/>ſunt, vt quadrata axium, vel diametrorum æqualiter baſibus inclina-<lb/>tarum, vel vt quadrata baſium, quæ omnia facilè patent.</s> <s xml:id="echoid-s7475" xml:space="preserve"/> </p> <pb o="310" file="0330" n="330" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div742" type="section" level="1" n="437"> <head xml:id="echoid-head457" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s7476" xml:space="preserve">_D_Eſiderari fortè tamen videtur, quod oſtendamus has varietates <lb/>parabolis contingere poſſe, nec eaſdem eſſe, exempligratia, vt <lb/>circulos, quibus tantum contingit ſe habere, vt diametrorum quadra-<lb/>ta, nec alia ijſdem accidit variatio, propterea ſubſequens Theorema, <lb/>ſubijciemus.</s> <s xml:id="echoid-s7477" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div743" type="section" level="1" n="438"> <head xml:id="echoid-head458" xml:space="preserve">THEOREMA XIX. PROPOS. XX.</head> <p> <s xml:id="echoid-s7478" xml:space="preserve">DAto quocunq; </s> <s xml:id="echoid-s7479" xml:space="preserve">parallelogrammo, circa eiuſdem duo <lb/>latera angulum continentia ſemiparabola deſcribi <lb/>poteſt, cuius alterum eorundem laterum ſit baſis, alterum <lb/>axis, vel diameter integræ parabolæ, ad quem dicta baſis <lb/>ordinatim applicatur.</s> <s xml:id="echoid-s7480" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7481" xml:space="preserve">Sit parallelogrammum quodcunque, AD, cuius ſumantur vt-<lb/>cunque duo latera, AC, CD, circa angulum, ACD. </s> <s xml:id="echoid-s7482" xml:space="preserve">Dico cir-<lb/>ca, AC, CD, ſemiparabolam de@cribi poſſe, ita vt alterum ipſo-<lb/>rum, AC, CD, ſit baſis dictæ ſemiparabolæ, alterum ſit axis, vel <lb/> <anchor type="figure" xlink:label="fig-0330-01a" xlink:href="fig-0330-01"/> diameter integræ parabolæ; </s> <s xml:id="echoid-s7483" xml:space="preserve">Eſto <lb/>quod velimus, CD, eſſe baſim, &</s> <s xml:id="echoid-s7484" xml:space="preserve">, <lb/>CA, axim, vel diametrum inte-<lb/>græ parabolæ; </s> <s xml:id="echoid-s7485" xml:space="preserve">applicetur ergo ad, <lb/>AC, rectangulum æquale quadra-<lb/>to, CD, quod latitudinem faciat <lb/>ipſam, XA, erit ergo quadratum, <lb/>CD, æquale rectangulo ſub, CA, <lb/>AX, &</s> <s xml:id="echoid-s7486" xml:space="preserve">, AX, erit linea, iuxta <lb/>quam poſſunt, quæ à curua para-<lb/>bolæ tranſeunte per puncta, D, A, <lb/> <anchor type="note" xlink:label="note-0330-01a" xlink:href="note-0330-01"/> vertice, A, ad axim, vel diametrum, AC, ordinatim applicari <lb/>poſſunt; </s> <s xml:id="echoid-s7487" xml:space="preserve">erit ergo quædam ſemiparabola, cuius curua tranſibit per <lb/>puncta, AD, in baſi, CD, exiſtente, AC, axi, vel diametro in-<lb/>tegræ parabolæ, ſit autem dicta ſemiparabola, ACD, quod oſten-<lb/>dere opus erat.</s> <s xml:id="echoid-s7488" xml:space="preserve"/> </p> <div xml:id="echoid-div743" type="float" level="2" n="1"> <figure xlink:label="fig-0330-01" xlink:href="fig-0330-01a"> <image file="0330-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0330-01"/> </figure> <note position="left" xlink:label="note-0330-01" xlink:href="note-0330-01a" xml:space="preserve">Schol.40. <lb/>lib.1.</note> </div> <pb o="311" file="0331" n="331" rhead="LIBER IV."/> </div> <div xml:id="echoid-div745" type="section" level="1" n="439"> <head xml:id="echoid-head459" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7489" xml:space="preserve">_H_Incliquet, ſi cuilibetparallelogrammo eſt inſcriptibilis ſemip. </s> <s xml:id="echoid-s7490" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s7491" xml:space="preserve">rabola talipacto, quo dictum eſt, quod parietates, quæ paralle-<lb/>logrammis contingunt, etiam ipſis parabolis competere poſſunt.</s> <s xml:id="echoid-s7492" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div746" type="section" level="1" n="440"> <head xml:id="echoid-head460" xml:space="preserve">THEOREMA XX. PROPOS. XXI.</head> <p> <s xml:id="echoid-s7493" xml:space="preserve">O Mnia quadrata parallelogrammi in eadem baſi, & </s> <s xml:id="echoid-s7494" xml:space="preserve">cir-<lb/>ca eundem axim, vel diametrum cum parabola, regu-<lb/>la baſi, ſunt dupla omnium quadratorum ipſius parabolæ: <lb/></s> <s xml:id="echoid-s7495" xml:space="preserve">Omnia verò quadrata parabolæ ſunt fexquialtera omnium <lb/>quadratorum trianguli in eadem baſi, & </s> <s xml:id="echoid-s7496" xml:space="preserve">circa eundem axim, <lb/>vel diametrum cum ipſa conſtituti.</s> <s xml:id="echoid-s7497" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7498" xml:space="preserve">Sit ergo parabola, cuius baſis, VF, axis, vel diameter, EM, <lb/> <anchor type="figure" xlink:label="fig-0331-01a" xlink:href="fig-0331-01"/> ſit etiam parallelogrammum, <lb/>AF, & </s> <s xml:id="echoid-s7499" xml:space="preserve">triangulum, EVF, in <lb/>eadem baſi, VF, & </s> <s xml:id="echoid-s7500" xml:space="preserve">circa eun-<lb/>dem axim, vel diametrum, EM. <lb/></s> <s xml:id="echoid-s7501" xml:space="preserve">Dico, omnia quadrata, AF, re-<lb/>gula, VF, omnium quadrato-<lb/>rum parabolæ, VEF, effe du-<lb/>pla: </s> <s xml:id="echoid-s7502" xml:space="preserve">Omnia verò quadrata para-<lb/>bolæ, VEF, omnium quadra-<lb/>torum trianguli, VEF, effe fex-<lb/>quialtera. </s> <s xml:id="echoid-s7503" xml:space="preserve">Sumaturintra, EM, <lb/>vtcunque punctum, N, per quodipſi, VF, agatur parallela, ND, <lb/>ſecans curuam parabolę; </s> <s xml:id="echoid-s7504" xml:space="preserve">in, O; </s> <s xml:id="echoid-s7505" xml:space="preserve">eſt ergo quadratum, MF, vel qua-<lb/>dratum, ND, ad quadratum, NO, vt, ME, ad, EN, eſt au-<lb/>tem, EF, parallelogrammum in eadem baſi, & </s> <s xml:id="echoid-s7506" xml:space="preserve">altitudine cumſe-<lb/>miparabola, EMF, regula eſt, MF, & </s> <s xml:id="echoid-s7507" xml:space="preserve">punctum, N, ſumptum vt-<lb/>cunque, per quod regulæ parallela ducta eſt, ND, repertumq; </s> <s xml:id="echoid-s7508" xml:space="preserve">eſt, <lb/> <anchor type="note" xlink:label="note-0331-01a" xlink:href="note-0331-01"/> vt quadratum, DN, ad quadratum, NO, ita eſte, ME, ad EN, <lb/>ergo horum quatuor ordinum magnitudines erunt proportionales <lb/>collectæ iuxta dictas quatuor magnitudines proportionales ſci-<lb/>licet omnia quadrata, EF, magnitudines primi ordinis collectæ <lb/>iuxta primam ſcilicet iuxta quadratum, ND, ad omnia quadrata <lb/>femiparabolæ, EMF, magnitudines fecundi ordims collectas <lb/>iuxta ſecundam ſcilicet iuxta quadratum, NO, erunt vt maxi- <pb o="312" file="0332" n="332" rhead="GEOMETRIÆ"/> mæ abſciſſarum, EM, magnitudines tertij ordinis collectæ iuxta <lb/>tertiam . </s> <s xml:id="echoid-s7509" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7510" xml:space="preserve">iuxta, ME, ad omnes abiciſſas ipſius, ME, magnitudi-<lb/>nes quarti ordinis collectas iuxta quartam . </s> <s xml:id="echoid-s7511" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7512" xml:space="preserve">iuxta, EN, ſumptis <lb/>maximisabſciſſarum, EM, & </s> <s xml:id="echoid-s7513" xml:space="preserve">eiuſdem omnibus abſciſſis, vel recti, <lb/> <anchor type="note" xlink:label="note-0332-01a" xlink:href="note-0332-01"/> vel eiuſdem obliqui tranſitus; </s> <s xml:id="echoid-s7514" xml:space="preserve">ſunt autem maximæ abſciſſarum, E <lb/>M, duplæ omnium abſciſſarum, EM, recti, vel eiuſdem obliqui <lb/>tranſitus, ergo & </s> <s xml:id="echoid-s7515" xml:space="preserve">omnia quadrata, EF, erunt dupla omnium qua-<lb/>dratorum ſemiparabolæ, EMF, & </s> <s xml:id="echoid-s7516" xml:space="preserve">eorum quadrupla . </s> <s xml:id="echoid-s7517" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7518" xml:space="preserve">omnia <lb/>quadrata, AF, erunt dupla omnium quadratorum parabolæ, VE <lb/>F; </s> <s xml:id="echoid-s7519" xml:space="preserve">Quarum ergo partium omnia quadrata, AF, erunt ſex, earum <lb/>omnia quadrata parabolæ, VEF, erunt tres, ſed quarum partium <lb/>omnia quadrata, AF, ſunt ſex, earum omnia quadrata trianguli, <lb/>EVF, iunt duæ, quia omnia quadrata, AF, iunt tripla omnium <lb/> <anchor type="note" xlink:label="note-0332-02a" xlink:href="note-0332-02"/> quadratorum trianguli, EVF, ergo quarum partium omnia qua-<lb/>drata parabolæ, VEF, funttres, earum omnia quadrata triangu-<lb/>li, EVF, erunt duæ, ergo omnia quadrata parabolæ, VEF, erunt <lb/>ſexquialtera omnium quadratorum trianguli, VEF, quæ oſtende-<lb/>re oportebat.</s> <s xml:id="echoid-s7520" xml:space="preserve"/> </p> <div xml:id="echoid-div746" type="float" level="2" n="1"> <figure xlink:label="fig-0331-01" xlink:href="fig-0331-01a"> <image file="0331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0331-01"/> </figure> <note position="right" xlink:label="note-0331-01" xlink:href="note-0331-01a" xml:space="preserve">Coroll. 3. <lb/>16. 1. 2.</note> <note position="left" xlink:label="note-0332-01" xlink:href="note-0332-01a" xml:space="preserve">Coroll. 2. <lb/>19. 1. 2.</note> <note position="left" xlink:label="note-0332-02" xlink:href="note-0332-02a" xml:space="preserve">34. 1. 2.</note> </div> </div> <div xml:id="echoid-div748" type="section" level="1" n="441"> <head xml:id="echoid-head461" xml:space="preserve">THEOREMA XXI. PROPOS. XXII.</head> <p> <s xml:id="echoid-s7521" xml:space="preserve">SI ad eundem axim, vel diametrum parabolæ ordinatim <lb/>applicentur duæ rectæ lineæ parabolas conſtituentes, <lb/>quarum altera ſumatur pro regula, harum parabolarum <lb/>omnia quadrata erunt interſe, vt quadrata axium, vel dia-<lb/>metrorum earundem.</s> <s xml:id="echoid-s7522" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7523" xml:space="preserve">Sinr ergo ad eundem axim, vel diametrum, CG, parabolæ, F <lb/> <anchor type="figure" xlink:label="fig-0332-01a" xlink:href="fig-0332-01"/> CH, duæ vtcunque ordinatim <lb/>applicatæ, FH, OM, parabo-<lb/>las, FCH, OCM, abſcinden-<lb/>tes, ſit autem regula altera ha-<lb/>rum ordinatim applicatarum, vt, <lb/>FH. </s> <s xml:id="echoid-s7524" xml:space="preserve">Dico omnia quadrata pa-<lb/>rabolæ, FCH, ad omnia qua-<lb/>drata parabolæ, OCM, eſſe vt <lb/>quadratum, GC, ad quadratum, <lb/>CI: </s> <s xml:id="echoid-s7525" xml:space="preserve">Conſtituantur parallelo-<lb/>grammum, AH, in baſi, FH, <lb/>& </s> <s xml:id="echoid-s7526" xml:space="preserve">circa axim, vel diametrum, CG, & </s> <s xml:id="echoid-s7527" xml:space="preserve">parallelogrammum, RM, <lb/>in baſi, OM, & </s> <s xml:id="echoid-s7528" xml:space="preserve">circa axim, vel diametrum, CI. </s> <s xml:id="echoid-s7529" xml:space="preserve">Quoniam ergo <pb o="313" file="0333" n="333" rhead="LIBER IV."/> omnia quadrata, AH, ſunt dupla omnium quadratorum parabo-<lb/> <anchor type="note" xlink:label="note-0333-01a" xlink:href="note-0333-01"/> læ, FCH, & </s> <s xml:id="echoid-s7530" xml:space="preserve">omnia quadrata, RM, ſunt dupla omnium quadra-<lb/>torum parabolę, OCM, ideò omnia quadrata parabolę, FCH, ad <lb/>omnia quadrata parabolę, OCM, erunt vt omnia quadrata; </s> <s xml:id="echoid-s7531" xml:space="preserve">AH, <lb/>ad omnia quadrata, RM: </s> <s xml:id="echoid-s7532" xml:space="preserve">Omnia vero quadrata, AH, ad omnia <lb/>quadrata, RM, habentrationem compoſitam ex ea, quam habet <lb/>quadratum, FH, ad quadratum, OM, ideſt ex ea, quam habet, <lb/>GC, ad, CI, & </s> <s xml:id="echoid-s7533" xml:space="preserve">ex ea, quam habet, HE, ad, NM, quiaillę cum <lb/> <anchor type="note" xlink:label="note-0333-02a" xlink:href="note-0333-02"/> baſibus, OM, FH, continent angulos ęquales, duę autem ratio-<lb/>nes, ſcilicet, quam habet, GC, ad, CI, &</s> <s xml:id="echoid-s7534" xml:space="preserve">, HE, ad, NM, . </s> <s xml:id="echoid-s7535" xml:space="preserve">1. </s> <s xml:id="echoid-s7536" xml:space="preserve">G <lb/>C, ad, CI, componuntrationem quadrati, GC, ad quadratum, C <lb/>I, ergo omnia quadrata, AH, ad omnia quadrata, RM, vel om-<lb/>nia quadrata parabolę, FCH, ad omnia quadrata parabolę, OC <lb/>M, erunt vt quadratum, GC, ad quadratum, CI, quod oſtende-<lb/>re opus erat.</s> <s xml:id="echoid-s7537" xml:space="preserve"/> </p> <div xml:id="echoid-div748" type="float" level="2" n="1"> <figure xlink:label="fig-0332-01" xlink:href="fig-0332-01a"> <image file="0332-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0332-01"/> </figure> <note position="right" xlink:label="note-0333-01" xlink:href="note-0333-01a" xml:space="preserve">Exa@. tec.</note> <note position="right" xlink:label="note-0333-02" xlink:href="note-0333-02a" xml:space="preserve">11. 1. 2.</note> </div> </div> <div xml:id="echoid-div750" type="section" level="1" n="442"> <head xml:id="echoid-head462" xml:space="preserve">THEOREMA XXII. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s7538" xml:space="preserve">IN figura Prop. </s> <s xml:id="echoid-s7539" xml:space="preserve">12. </s> <s xml:id="echoid-s7540" xml:space="preserve">ſumpta regula ipſa, BH, oſtendemus <lb/>omnia quadrata, PH, ad omnia quadrata fruſti, ABH <lb/>M, eſſe vt, ON, ad compoſitam ex, NR, & </s> <s xml:id="echoid-s7541" xml:space="preserve">@. </s> <s xml:id="echoid-s7542" xml:space="preserve">RO: </s> <s xml:id="echoid-s7543" xml:space="preserve">Omnia <lb/>verò quadrata fruſti, ABHM, ad omnia quadrata triangu-<lb/>li, RBH, eſie vt compoſitam ex, ON, dupla, NR, et @. </s> <s xml:id="echoid-s7544" xml:space="preserve">R <lb/>O, ad ipſam, NO.</s> <s xml:id="echoid-s7545" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7546" xml:space="preserve">Sumatur in, RO, vtcunq; </s> <s xml:id="echoid-s7547" xml:space="preserve">punctum, X, per quod regulę, BH, <lb/>paralleia ducatur, XT, ſecans curuam parabolę in, I, eſt ergo qua-<lb/>dratum, OH, vel quadratum, TX, ad quadratum, XI, vt, ON, <lb/> <anchor type="figure" xlink:label="fig-0333-01a" xlink:href="fig-0333-01"/> ad, NX, eſt autem parallelogram-<lb/>mum, RH, in eadem bafi, & </s> <s xml:id="echoid-s7548" xml:space="preserve">alti-<lb/>tudine cum quadrilineo, ROHM, <lb/>& </s> <s xml:id="echoid-s7549" xml:space="preserve">punctum, X, ſumptum eſt vt <lb/>cunque, ductaque, XT, regulæ <lb/>parallela, repertum eſt quadratum, <lb/>TX, ad quadratum, XI, eſſe vt, <lb/>ON, ad, NX, ergo horum quatuor ordinum magnitudines erunt <lb/> <anchor type="note" xlink:label="note-0333-03a" xlink:href="note-0333-03"/> proportionales. </s> <s xml:id="echoid-s7550" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7551" xml:space="preserve">omnia quadrata, RH, magnitudines primi ordinis <lb/>collectę iuxta primam. </s> <s xml:id="echoid-s7552" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7553" xml:space="preserve">iuxta quadratum, TX, ad omnia quadrata <lb/>quadrilinei, RMHO, magnitudines ſecundi ordinis collectas iuxta fe-<lb/>cundã. </s> <s xml:id="echoid-s7554" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7555" xml:space="preserve">iuxta quadratum, XI, erunt vt maximę abſciſlarum, OR, <lb/>adiunctal, RN, ad omnes abiciſſas, OR, adiuncta, RN, quę ſunt <pb o="314" file="0334" n="334" rhead="GEOMETRIE"/> magnitudines collectæ iuxta tertiam, & </s> <s xml:id="echoid-s7556" xml:space="preserve">quartam. </s> <s xml:id="echoid-s7557" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7558" xml:space="preserve">iuxta, ON, ter-<lb/>tiam, &</s> <s xml:id="echoid-s7559" xml:space="preserve">, NX, quartam, ijſdem recti, vel eiuſdem obliqui tranſitus <lb/>ſumptis: </s> <s xml:id="echoid-s7560" xml:space="preserve">Quia verò datæ rectæ lineæ, OR, adiungitur, RN, ideò <lb/>maximæ abſciſſarum, OR, adiuncta, RN, ad omnes abſciſſas, OR, <lb/>adiuncta, RN, recti, vel eiuſdem obliqui tranſitus, ſunt vt, ON, ad <lb/>compoſitam ex, NR, & </s> <s xml:id="echoid-s7561" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s7562" xml:space="preserve">RO, ideò omnia quadrata, RH, ad om. <lb/></s> <s xml:id="echoid-s7563" xml:space="preserve"> <anchor type="note" xlink:label="note-0334-01a" xlink:href="note-0334-01"/> nia quadrata quadrilinei, RMHO, vel eorum quadrupla. </s> <s xml:id="echoid-s7564" xml:space="preserve">. </s> <s xml:id="echoid-s7565" xml:space="preserve">omnia <lb/>quadrata. </s> <s xml:id="echoid-s7566" xml:space="preserve">PH, ad omnia quadrata fruſt, ABHM, erunt vt, ON, <lb/>ad compoſitamex, NR, & </s> <s xml:id="echoid-s7567" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s7568" xml:space="preserve">RO; </s> <s xml:id="echoid-s7569" xml:space="preserve">Et conuertendo omnia quadrata <lb/>fruſti, ABHM, ad omnia quadrata, PH, erunt vt compoſita ex, <lb/>NR, & </s> <s xml:id="echoid-s7570" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s7571" xml:space="preserve">RO, ad, NO, omnia verò quadrata, PH, omnium qua-<lb/>dratorumtrianguli, RBH, ſunt tripla. </s> <s xml:id="echoid-s7572" xml:space="preserve">1. </s> <s xml:id="echoid-s7573" xml:space="preserve">ſunt vt, NO, ad {1/3}. </s> <s xml:id="echoid-s7574" xml:space="preserve">eiuſ-<lb/> <anchor type="note" xlink:label="note-0334-02a" xlink:href="note-0334-02"/> dem, NO, ergo, ex æquali, omnia quadrata fruſti, ABHM, ad <lb/>omnia quadrata trianguli, BRH, erunt vt compoſita ex, NR, & </s> <s xml:id="echoid-s7575" xml:space="preserve">{1/2}. <lb/></s> <s xml:id="echoid-s7576" xml:space="preserve">RO, ad {1/3}. </s> <s xml:id="echoid-s7577" xml:space="preserve">NO, vel vt horum tripla. </s> <s xml:id="echoid-s7578" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7579" xml:space="preserve">vt compoſita ex tribus, NR, <lb/>& </s> <s xml:id="echoid-s7580" xml:space="preserve">ſexquialtera, RO, adipſam, NO, porrò ſi iunxerimus vnam, <lb/>NR, cum, RO, fiet integra, ON, cum duabus, NR, & </s> <s xml:id="echoid-s7581" xml:space="preserve">dimidia, <lb/>RO, æqualistriplæ, NR, & </s> <s xml:id="echoid-s7582" xml:space="preserve">ſexqualteræ, RO; </s> <s xml:id="echoid-s7583" xml:space="preserve">ergo omnia qua-<lb/>drata fruſti, ABHM, ad omnia quadrata trianguli, RBH, erunt <lb/>vt compoſita ex dupla, NR, & </s> <s xml:id="echoid-s7584" xml:space="preserve">dimidia, RO, cum, NO; </s> <s xml:id="echoid-s7585" xml:space="preserve">ad ipſam, <lb/>NO; </s> <s xml:id="echoid-s7586" xml:space="preserve">quæ oſtendere oportebat.</s> <s xml:id="echoid-s7587" xml:space="preserve"/> </p> <div xml:id="echoid-div750" type="float" level="2" n="1"> <figure xlink:label="fig-0333-01" xlink:href="fig-0333-01a"> <image file="0333-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0333-01"/> </figure> <note position="right" xlink:label="note-0333-03" xlink:href="note-0333-03a" xml:space="preserve">Coroll. @. <lb/>26. 1. 2.</note> <note position="left" xlink:label="note-0334-01" xlink:href="note-0334-01a" xml:space="preserve">Corol. <lb/>20. 1. 2.</note> <note position="left" xlink:label="note-0334-02" xlink:href="note-0334-02a" xml:space="preserve">24. 1. 2.</note> </div> </div> <div xml:id="echoid-div752" type="section" level="1" n="443"> <head xml:id="echoid-head463" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7588" xml:space="preserve">_O_Via autem probatum fuit omnia quadrata, PH, ad omnia quã-<lb/>drata fruſti, ABHM, eſſe vt, NO, ad dimidiam, OR, cum, <lb/>RN, ſunt autem omnia quadrata, PH, ad omnia quadrata parallelo-<lb/>grammi, AG, vt quadratam, HO, ad quadratam, RM, . </s> <s xml:id="echoid-s7589" xml:space="preserve">i. </s> <s xml:id="echoid-s7590" xml:space="preserve">vt, ON, <lb/>ad NR, ideò omnia quadrata, PH, ad omnia quadrata fruſti, AB <lb/>HM, ab ijſdem demptis omnibus quadratis, AG, erunt vt, NO, <lb/>ad dimidiam ipſius, OR.</s> <s xml:id="echoid-s7591" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div753" type="section" level="1" n="444"> <head xml:id="echoid-head464" xml:space="preserve">THEOREMA XXIII. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s7592" xml:space="preserve">SI intra curuam parabolæ ducatur vtcunq; </s> <s xml:id="echoid-s7593" xml:space="preserve">recta linea in <lb/>eandem terminata, & </s> <s xml:id="echoid-s7594" xml:space="preserve">ad axem obliqua, deinde intra <lb/>portionem ab ipſa reſectam ducatur alia vtcunq; </s> <s xml:id="echoid-s7595" xml:space="preserve">prædictæ <lb/>parallela, agantur autem ab extremitate harum parallela-<lb/>rum lineæ axi æquidiſtantes: </s> <s xml:id="echoid-s7596" xml:space="preserve">Vt baſis reſectæ portionis ad <lb/>diſtantiam parallelarum ab eiuſdem extremitate ductarum, <pb o="315" file="0335" n="335" rhead="LIBER IV."/> ita erit alia prædictæ parallela ad diſtantiam parallelarum <lb/>ductarum ab eiuſdem extremitate ſecundò dictæ.</s> <s xml:id="echoid-s7597" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7598" xml:space="preserve">Sit ergo intra curuam parabolicam, ABCDF, ducta vtcunque, <lb/>BF, obliquè ſecans axem, NR, in eandem curuam terminata, <lb/>agatur deinde intra portionem, BNF, reſectam à, BF, recta, C <lb/>D, parallela ipſi, BF; </s> <s xml:id="echoid-s7599" xml:space="preserve">ducantur inſuperà punctis, B, C, D, F, axi, <lb/>NR, parallelæ, BO, CV, DG, FH, & </s> <s xml:id="echoid-s7600" xml:space="preserve">à puncto, F, cadat <lb/> <anchor type="figure" xlink:label="fig-0335-01a" xlink:href="fig-0335-01"/> ipſi, NR, perpendicularis, F <lb/>A, ſecans parallelas, DG <lb/>R, CV, BO, in punctis, G, <lb/>R, V, O, poterunt ergo dicta-<lb/>rum parallelarum diſtantiæ iumi <lb/>in ipſamet, AF, nam ipſa per-<lb/>pendiculariter dictas parallelas <lb/>ſecat, erit ergo, OF, diſtantia <lb/>parallelarum, BO, FH, ab <lb/>extremis punctis rectæ, BF, ductarum; </s> <s xml:id="echoid-s7601" xml:space="preserve">pariter, VG, erit diſtan-<lb/>tia parallelarum, CV, DG, ab extremis punctis, CD, ducta-<lb/>rum. </s> <s xml:id="echoid-s7602" xml:space="preserve">Dico ergo, BF, ad, FO, eſſe vt, CD, ad, VG: </s> <s xml:id="echoid-s7603" xml:space="preserve">Ducan-<lb/>tur a puncto, D, ipſi, CV, perpendicularis, DX, ſecans, BF, in, <lb/>M, quoniam ergo anguli, BOF, CXD, ſunt recti, ideò iuntin-<lb/>terſe æ quales, item anguius, OBF, eſt æqualis angulo, VIF, & </s> <s xml:id="echoid-s7604" xml:space="preserve">V <lb/>IF, ipſi angulo, XCD, ergo angulus, OBF, erit æqualis angulo, X <lb/> <anchor type="note" xlink:label="note-0335-01a" xlink:href="note-0335-01"/> CD, & </s> <s xml:id="echoid-s7605" xml:space="preserve">ideò reliquus, OFB, reliquo, XDC, æqualis erit, & </s> <s xml:id="echoid-s7606" xml:space="preserve">trian-<lb/>guli, BOF, CXD, ſimiles erunt, vnde, BF, ad, FO, erit vt, C <lb/>D, ad, DX, . </s> <s xml:id="echoid-s7607" xml:space="preserve">i. </s> <s xml:id="echoid-s7608" xml:space="preserve">ad, VG, quod oſtendere opus erat.</s> <s xml:id="echoid-s7609" xml:space="preserve"/> </p> <div xml:id="echoid-div753" type="float" level="2" n="1"> <figure xlink:label="fig-0335-01" xlink:href="fig-0335-01a"> <image file="0335-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0335-01"/> </figure> <note position="right" xlink:label="note-0335-01" xlink:href="note-0335-01a" xml:space="preserve">46. Elem.</note> </div> </div> <div xml:id="echoid-div755" type="section" level="1" n="445"> <head xml:id="echoid-head465" xml:space="preserve">PROBLEMA II. PROPOS. XXV.</head> <p> <s xml:id="echoid-s7610" xml:space="preserve">A Sſumpta iterum ſuperioris figura, dimiſſa axi, & </s> <s xml:id="echoid-s7611" xml:space="preserve">ei-<lb/>dem parallelis, BO, CV, DG, FH, & </s> <s xml:id="echoid-s7612" xml:space="preserve">ipſa, DX, <lb/>ſiguram plènam deſcribere cum portione, BCDF, com-<lb/>munem habens angulum mixtum ſub, BF, & </s> <s xml:id="echoid-s7613" xml:space="preserve">curua, FD <lb/>C, quifit ad punctum, F, ita vt quælibet in deſctipta figu-<lb/>ra recta linea ipſi, BF, æquidiſtanter ducta, ſit diſtantia pa-<lb/>rallelarum axi, quæ ab extremis punctis eiuſdem rectæ li-<lb/>neæ, productæ vſque ad curuam parabolicam, duci poſſunt: <lb/></s> <s xml:id="echoid-s7614" xml:space="preserve">Vocetur autem hæc deſcripta figura; </s> <s xml:id="echoid-s7615" xml:space="preserve">figura diſtantiarum <lb/>portionis, ſiue parabolæ, BCDF.</s> <s xml:id="echoid-s7616" xml:space="preserve"/> </p> <pb o="316" file="0336" n="336" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s7617" xml:space="preserve">Quoniam ergo, OF, eſt diſtantia parallelarum axi ductarum à <lb/>punctis, BF, abſcindatur à, BF, recta, FE, æqualis diſtantiæ, F <lb/>O, inſuper intelligatur adhuc ipſa, CD, ducta vtcunque parallela <lb/>rectæ, BF, terminans in puncta, CD, curuæ parabolæ, & </s> <s xml:id="echoid-s7618" xml:space="preserve">cum <lb/>ſit, VG, diſtantia parallelarumaxi, quæ à punctis, CD, ducun-<lb/>tur, abſcindatur ab ipſa, CD, verſus, D, ipſa, DZ, æqualis di-<lb/>ſtantiæ, VG; </s> <s xml:id="echoid-s7619" xml:space="preserve">ſic ductis in portione, BCDF, omnibus lineis, regu-<lb/>la, BF, in earundem ſingulis intell gantur ſumptæ diſtantiæ, ſicut <lb/>acceptæ ſuerunt, EF, ZD, quarum extrema puncta ſint in curua <lb/>parabolica, FDCB, ſint autem in huius curuæ ea parte, in qua <lb/>ſunt puncta, DF, patet ergo ſi fumamus punctum, S, verticem <lb/>portionis, BSF, quod dictarum omnium linearum extrema puncta <lb/>erunt in curua parabolica, quæ incipit a vertice, S, & </s> <s xml:id="echoid-s7620" xml:space="preserve">deſinit in, F; <lb/></s> <s xml:id="echoid-s7621" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0336-01a" xlink:href="fig-0336-01"/> per alia ergo extrema puncta earundem <lb/>diſtantiarum intelligatur ducta linea, S <lb/>ZE. </s> <s xml:id="echoid-s7622" xml:space="preserve">Dico figuram, SFE, compre-<lb/>henſam recta, EF, curua parabolica, <lb/>SDF, & </s> <s xml:id="echoid-s7623" xml:space="preserve">linea, SZE, eſſe huiuſmo-<lb/>di, quod, ſi duxerimus intra ipſam vt-<lb/>cunq; </s> <s xml:id="echoid-s7624" xml:space="preserve">ipſi, BF, parallelam, quæ pro-<lb/>ducatur vſq; </s> <s xml:id="echoid-s7625" xml:space="preserve">ad curuam parabolicam, <lb/>huius portio manens in figura, SEF, erit diſtantia parallelarum <lb/>axi, quæ ducuntur ab extremis punctis ab eadem producta in curua <lb/>parabolica ſignatis. </s> <s xml:id="echoid-s7626" xml:space="preserve">Intelligatur ergo ducta vtcunque, DZ, ipſi, B <lb/>F, parallela, & </s> <s xml:id="echoid-s7627" xml:space="preserve">producta vſq; </s> <s xml:id="echoid-s7628" xml:space="preserve">ad curuam parabolicam incidens illi <lb/>in puncto, C, quoniam ergo, CD, eſt vna earum, quæ dicuntur <lb/>omnes lineæ figurę, BSF, portio eiuſdem manens intra figuram, <lb/>SEF, erit diſtantia parallelarum axi, quę ab eiuſdem extremis pun-<lb/>ctis ductæ intelliguntur, & </s> <s xml:id="echoid-s7629" xml:space="preserve">hoc per conſtructionem patet, quoniam <lb/>abipſa, CD, abſciſſa eſt, DZ, quę terminat in lineam, SZE, æ-<lb/>qualis dictę diſtantię, ergo figura, SEF, deſcripta eſt, qualem pro-<lb/>blema poſtulabat; </s> <s xml:id="echoid-s7630" xml:space="preserve">quę vocetur figura diſtantiarum portionis, ſiue pa-<lb/>rabolę, BSF.</s> <s xml:id="echoid-s7631" xml:space="preserve"/> </p> <div xml:id="echoid-div755" type="float" level="2" n="1"> <figure xlink:label="fig-0336-01" xlink:href="fig-0336-01a"> <image file="0336-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0336-01"/> </figure> </div> </div> <div xml:id="echoid-div757" type="section" level="1" n="446"> <head xml:id="echoid-head466" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7632" xml:space="preserve">_Q_Via verò oſtenſum eſt, BF, ad diſtantiam parallelarum axià, B, <lb/>F, ductarum, eſſe vt, CD, ad diſtantiam parallelarum axi à <lb/>punctis, C, D, ductarum, ſunt autem, EF, ZD, æquales dictis diſtan-<lb/>tijs, ideò erit, BF, ad, FE, vt, CD, ad, DZ, & </s> <s xml:id="echoid-s7633" xml:space="preserve">ſic erit quælibet du-<lb/>cta in portione, BSF, parallelaipſi, BF, adeiuſdem partemincluſam@ <lb/>figura, SEF, vt, BF, ad, FE.</s> <s xml:id="echoid-s7634" xml:space="preserve"/> </p> <pb o="317" file="0337" n="337" rhead="LIBER IV."/> </div> <div xml:id="echoid-div758" type="section" level="1" n="447"> <head xml:id="echoid-head467" xml:space="preserve">THEOREMA XXIV. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s7635" xml:space="preserve">INeadem antecedentis figura oſtendemus omnia quadra-<lb/>ta portionis, BSF, ad rectangula ſub eadem portione, <lb/>BSF, & </s> <s xml:id="echoid-s7636" xml:space="preserve">ſub figura, SEF, regula communi, BF, eſſe vt, B <lb/>F, ad, FE.</s> <s xml:id="echoid-s7637" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7638" xml:space="preserve">Eſt enim quadratum, BF, ad rectangulum ſub, BF, FE, vt, B <lb/>F, ad, FE; </s> <s xml:id="echoid-s7639" xml:space="preserve">ſimiliter ducta vtcunque, CD, parallela regulę, BF, <lb/>oſtendemus quadratum, CD, ad rectangulum, ſub, CD, DZ, <lb/>eſſe vt, CD, ad, DZ, eſt autem vt, BF, ad, FE, ita, CD, ad, <lb/>DZ, ergo quadratum, BF, ad rectangulum, BFE, erit vt qua-<lb/>dratum, CD, ad rectangulum, CDZ, ſic oſtendemus quamlibet <lb/>ductam intra portionem, BSF, parallelam regulę, BF, ad eiuſdem <lb/> <anchor type="note" xlink:label="note-0337-01a" xlink:href="note-0337-01"/> portionem incluſam figura, SFE, eſſe vt quadratum, BF, ad re-<lb/>ctangulum ſub, BF, FE, ergo quadratum, BF, ad rectangulum <lb/>ſub, BF, FE, erit vt omnia quadrata portionis, BSF, ad rectan-<lb/>gula ſub portione, BSF, & </s> <s xml:id="echoid-s7640" xml:space="preserve">ſub figura, SEF, vt autem quadratum, <lb/>BF, ad rectangulum ſub, BF, FE, ita, BF, ad, FE, ergo omnia <lb/>quadrata portionis, BSF, ad rectangula ſub portione, BSF, & </s> <s xml:id="echoid-s7641" xml:space="preserve">fi-<lb/>gura, ESF, erunt vt, BF, ad, FE, quod oſtendere opus erat.</s> <s xml:id="echoid-s7642" xml:space="preserve"/> </p> <div xml:id="echoid-div758" type="float" level="2" n="1"> <note position="right" xlink:label="note-0337-01" xlink:href="note-0337-01a" xml:space="preserve">Corol. <lb/>4. 1. 2.</note> </div> </div> <div xml:id="echoid-div760" type="section" level="1" n="448"> <head xml:id="echoid-head468" xml:space="preserve">THEOREMA XXV. PROPOS. XXVII.</head> <p> <s xml:id="echoid-s7643" xml:space="preserve">SI intra curuam parabolæ duæ vtcunque ducantur rectæ <lb/>lineę in eandem @terminatæ, quarum vna rectè, altera <lb/>obliquè axim ſecet, ſint autem conſtitutarum ab eiſdem pa-<lb/>rabolarum diametri inter ſe æquales: </s> <s xml:id="echoid-s7644" xml:space="preserve">Omnia quadrata pa-<lb/>rabolæ per eam, quæ rectè axim ſecat, conſtitutæ, regula <lb/>eadem, erunt æqualia rectangulis ſub parabola per obli-<lb/>quam ad axem conſtituta, regula eadem, & </s> <s xml:id="echoid-s7645" xml:space="preserve">ſub figura di-<lb/>ſtantiarum eiuſdem parabolæ per obliquam ad axem con-<lb/>ſtitutæ.</s> <s xml:id="echoid-s7646" xml:space="preserve"/> </p> <figure> <image file="0337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0337-01"/> </figure> <pb o="318" file="0338" n="338" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s7647" xml:space="preserve">Sint intra curuam parabolicam, BAC, duæ vtcunquæ ductæ in <lb/>eandem terminatæ, DF, MC, quarum, DF, rectè, altera, MC, <lb/>obliquè ſecet axem, AP, ſit autem deſcripta linea, HR, vt ſit con-<lb/>ſtituta, HRC, figura diſtantiarum portionis, MFC, & </s> <s xml:id="echoid-s7648" xml:space="preserve">ab eodem <lb/>vertice, H, à quo ducitur linea, HR, ducatur, HQ, parallela <lb/>axi, AP, & </s> <s xml:id="echoid-s7649" xml:space="preserve">ſint diametri, AZ, HO, parabolarum, DAF, M <lb/>HC, inter ſeæquales. </s> <s xml:id="echoid-s7650" xml:space="preserve">Dico ergo omnia quadrata parabolæ, DA <lb/>F, regula, DF, eſſe æqualia rectangulis ſub parabola, MHC, re-<lb/>gula, MC, & </s> <s xml:id="echoid-s7651" xml:space="preserve">ſub, HRC, figura diſtantiarum eiuſdem parabolæ, <lb/>MHC. </s> <s xml:id="echoid-s7652" xml:space="preserve">Iungantur ergo, DA, AF, MH, HC, & </s> <s xml:id="echoid-s7653" xml:space="preserve">à puncto, M, <lb/>ducatur, MX, axi, AP, æquidiſtans, à puncto verò, C, perpendi-<lb/>cularis axi, AP, producta vſq; </s> <s xml:id="echoid-s7654" xml:space="preserve">in, B, tandem à puncto, H, ipſa, H <lb/>I, perpendicularis ipſi, MC: </s> <s xml:id="echoid-s7655" xml:space="preserve">Omnia ergo quadrata, DAF, para-<lb/>bolæ, regula, DF, adrectangula ſub parabola, MHC, regula, M <lb/>C, & </s> <s xml:id="echoid-s7656" xml:space="preserve">ſub trilineo, HRC, habent rationem compoſitam ex ea, <lb/>quam habent omnia quadrata parabolæ, DAF, regula, DF, ad <lb/> <anchor type="note" xlink:label="note-0338-01a" xlink:href="note-0338-01"/> omnia quadrata parabolæ, MHC, regula, MC, & </s> <s xml:id="echoid-s7657" xml:space="preserve">ex ea, quam <lb/>habent omnia quadrata parabolę, MHC, regula, MC, adrectan-<lb/>gula ſub parabola, MHC, & </s> <s xml:id="echoid-s7658" xml:space="preserve">ſub trilineo, HRC, regula eadem, <lb/> <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> MC: </s> <s xml:id="echoid-s7659" xml:space="preserve">Omnia verò quadrata para-<lb/>bolæ, DMF, regula, DF, ad om-<lb/>nia quadrata parabolæ, MHC, re-<lb/>gula, MC, ſunt vt omnia quadrata <lb/>trianguli, DAF, regula, DF, ad <lb/>omnia quadrata trianguli, MHC, <lb/>regula, MC, nam omnia quadrata <lb/>parabolarum ſunt ſexquialtera om-<lb/>nium quadratorum triangulorum in <lb/>eiſdem baſibus, & </s> <s xml:id="echoid-s7660" xml:space="preserve">circa eoſdem axes cum ipſis conſtitutorum, regu-<lb/> <anchor type="note" xlink:label="note-0338-02a" xlink:href="note-0338-02"/> lis baſibus: </s> <s xml:id="echoid-s7661" xml:space="preserve">Omnia inſuper quadrata trianguli, DAF, regula, DF, <lb/>ad omnia quadrata trianguli, MHC, regula, MC, habent ratio-<lb/> <anchor type="note" xlink:label="note-0338-03a" xlink:href="note-0338-03"/> nem compoſitam ex ratione altitudinum, & </s> <s xml:id="echoid-s7662" xml:space="preserve">quadratorum baſium <lb/>. </s> <s xml:id="echoid-s7663" xml:space="preserve">i. </s> <s xml:id="echoid-s7664" xml:space="preserve">ex ratione, quam habet, AZ, ad, HI, & </s> <s xml:id="echoid-s7665" xml:space="preserve">ex rationẽ, quam ha-<lb/>bet quadratum, DF, ad quadratum, MC, vel quadratum, ZF, <lb/>ad quadratum, OC, eſt autem, AZ, æqualis ipſi, HO, ex hypo-<lb/>teſi, &</s> <s xml:id="echoid-s7666" xml:space="preserve">, ZF, ipſi, QC, ergo omnia quadrata trianguli, DAF, ad <lb/> <anchor type="note" xlink:label="note-0338-04a" xlink:href="note-0338-04"/> omnia quadrata trianguli, MHC, regulis iam dictis, habebunt ra-<lb/>tionem compoſitam ex ea, quam habet, OH, ad HI, & </s> <s xml:id="echoid-s7667" xml:space="preserve">ex ea, <lb/>quam habet quadratum, QC, ad qu adratum, CO, quia verò trian-<lb/>guli, HIO, OQC, ſunt æquianguli, ideò, OH, ad, HI, erit vt, <lb/>OC, ad, CQ, ergo illa habebunt rationem compoſitam ex ea, quam <lb/>habet, OC, ad, CQ, & </s> <s xml:id="echoid-s7668" xml:space="preserve">quadratum QC, ad quadratum, CO, eſt <pb o="319" file="0339" n="339" rhead="LIBER IV."/> autem vt, OC, ad, CQ, ita, ſumpta, QC, communi altitudine, <lb/>rectangulum ſub OC, CQ, ad quadratum, QC, ergo ratio com-<lb/>poſita ex ea, quam habet, OC, ad, CQ, & </s> <s xml:id="echoid-s7669" xml:space="preserve">quadratum, QC, ad <lb/>quadratum, CO, eſt eadem compoſitæ ex ea, quam habet rectan-<lb/>gulum ſub, OC, CQ, ad quadratum, CQ, & </s> <s xml:id="echoid-s7670" xml:space="preserve">quadratum, CQ, <lb/>ad quadratum, CO, . </s> <s xml:id="echoid-s7671" xml:space="preserve">i. </s> <s xml:id="echoid-s7672" xml:space="preserve">eademei, quam habet rectangulum ſub, <lb/>QC, CO, ad quadratum, CO, . </s> <s xml:id="echoid-s7673" xml:space="preserve">i. </s> <s xml:id="echoid-s7674" xml:space="preserve">eadem ei, quam habet, QC, ad, <lb/>CO; </s> <s xml:id="echoid-s7675" xml:space="preserve">ergo omnia quadrata trianguli, DAF, ad omnia quadrata <lb/>trianguli, MHC, vel omnia quadrata parabolæ, DAF, ad om-<lb/>nia quadrata parabolæ, MHC, regulis iam dictis, erunt vt, QC, <lb/>ad, CO, quod ſerua.</s> <s xml:id="echoid-s7676" xml:space="preserve"/> </p> <div xml:id="echoid-div760" type="float" level="2" n="1"> <note position="left" xlink:label="note-0338-01" xlink:href="note-0338-01a" xml:space="preserve">Defin. <lb/>12. 1. 1.</note> <figure xlink:label="fig-0338-01" xlink:href="fig-0338-01a"> <image file="0338-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0338-01"/> </figure> <note position="left" xlink:label="note-0338-02" xlink:href="note-0338-02a" xml:space="preserve">21. huius.</note> <note position="left" xlink:label="note-0338-03" xlink:href="note-0338-03a" xml:space="preserve">D. Corol. <lb/>22. 1. 2.</note> <note position="left" xlink:label="note-0338-04" xlink:href="note-0338-04a" xml:space="preserve">Corol. <lb/>17. huius.</note> </div> <p> <s xml:id="echoid-s7677" xml:space="preserve">Vlterius omnia quadrata parabolæ, MHC, ad rectangula ſub <lb/> <anchor type="note" xlink:label="note-0339-01a" xlink:href="note-0339-01"/> parabola, MHC, & </s> <s xml:id="echoid-s7678" xml:space="preserve">trilineo, HRC, regula, MC, ſunt vt, M <lb/>C, ad, CR, vel ad, CX, . </s> <s xml:id="echoid-s7679" xml:space="preserve">i. </s> <s xml:id="echoid-s7680" xml:space="preserve">vt, OC, ad, CQ, ergo omnia qua-<lb/>drata parabolæ, DAF, regula, DF, ad rectangula ſub parabola, <lb/>MHC, & </s> <s xml:id="echoid-s7681" xml:space="preserve">trilineo, HRC, regula, MC, habebunt rationem <lb/>compoſitam ex ea, quam habet, QC, ad, CO, & </s> <s xml:id="echoid-s7682" xml:space="preserve">ex ea quam <lb/>habet, CO, ad, QC, ideſt habebunt eandem rationem, quam <lb/>habet, QC, ad, QC, ideſt eruntillis æqualia, quod oſtende-<lb/>re opus erat.</s> <s xml:id="echoid-s7683" xml:space="preserve"/> </p> <div xml:id="echoid-div761" type="float" level="2" n="2"> <note position="right" xlink:label="note-0339-01" xlink:href="note-0339-01a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div763" type="section" level="1" n="449"> <head xml:id="echoid-head469" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s7684" xml:space="preserve">_H_Inc patet omnia quadrata parabolæ, DAF, regula, DF, ad om-<lb/>nta quadrata parabolæ, MHC, regula, MC, eſſe vt QC, ad, <lb/>CO, vel, XC, ad CM, vel, DF, (quæ eſt æqualis ipſi, XC,) ad, MC, <lb/>dumdiametri, AZ, HO, ſunt æquales, vt in Theoremate oſtenſum eſt.</s> <s xml:id="echoid-s7685" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div764" type="section" level="1" n="450"> <head xml:id="echoid-head470" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s7686" xml:space="preserve">_P_Atet vlterius, ſi intra cùruam parabolicam duæ vtcunq; </s> <s xml:id="echoid-s7687" xml:space="preserve">rectæ li-<lb/>neæ obliquè axem ſecantes, & </s> <s xml:id="echoid-s7688" xml:space="preserve">in ipſam terminantes, ductæ fue-<lb/>rint, regula pro qualibetparabola ſumpta earum baſi, quod rectangula <lb/>ſub dictis parabolis per eaſdem conſtitutis, & </s> <s xml:id="echoid-s7689" xml:space="preserve">ſub figura diſtantiarum <lb/>earundem parabolarum, inter ſe erunt æqualia, quotieſcunq diametri <lb/>earundem ſint æquales, vtraq; </s> <s xml:id="echoid-s7690" xml:space="preserve">enim ſingillatim æquabuntur omnibus <lb/>qu dratis parabolæ, cuius baſis ſecet perpendeculariter axem eiuſdem <lb/>qui ſit æqualis diametris dictarum parabolarum, & </s> <s xml:id="echoid-s7691" xml:space="preserve">pro, qua ſit regula <lb/>eiuſdem baſis.</s> <s xml:id="echoid-s7692" xml:space="preserve"/> </p> <pb o="320" file="0340" n="340" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div765" type="section" level="1" n="451"> <head xml:id="echoid-head471" xml:space="preserve">THEOREMA XXVI. PROPOS. XXVIII.</head> <p> <s xml:id="echoid-s7693" xml:space="preserve">SI intra curuam parabolicam duæ vtcunque ductæ fue-<lb/>rint rectæ lineæ in eandem terminantes, quarum vna <lb/>rectè, altera obliquè ſecet axim; </s> <s xml:id="echoid-s7694" xml:space="preserve">omnia quadrata conſtitu-<lb/>tæ parabolæ per eam, quæ axim rectè ſecat, regula eadem, <lb/>ad rectangula ſub parabola conſtituta per obliquè ſecantem <lb/>axem, regula huius baſi, & </s> <s xml:id="echoid-s7695" xml:space="preserve">ſub ſigura diſtantiarum eiuſ-<lb/>dem parabolæ, erunt vt quadratum axis primò dictæ para. <lb/></s> <s xml:id="echoid-s7696" xml:space="preserve">bolæ ad quadratum diametriſecundò dictæ parabolæ.</s> <s xml:id="echoid-s7697" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7698" xml:space="preserve">Sintigitur intra curuam parabolicam, ADH, duæ ductæ rectæ <lb/>lineæ in eadem terminantes, quarum vna rectè, altera obliquè ſecet <lb/>axim, ſi ergo conſtitutarum ab ijſdem parabolarum diametri ſunt <lb/>æquales, pater veritas Propoſitionis ex antecedenti Theor. </s> <s xml:id="echoid-s7699" xml:space="preserve">non ſint <lb/>autem conſtitutarum parabolarum diametri æquales, ſint autem <lb/>duæ parabolas conſtituentes, AH, rectè ſecans axem, DO, & </s> <s xml:id="echoid-s7700" xml:space="preserve">C <lb/> <anchor type="figure" xlink:label="fig-0340-01a" xlink:href="fig-0340-01"/> G, obliquè ipſum diuidens, exiſtatq; <lb/></s> <s xml:id="echoid-s7701" xml:space="preserve">axis, DO, maior diametro parabo-<lb/>læ, CEG, quæ ſit, EM, & </s> <s xml:id="echoid-s7702" xml:space="preserve">ſit du-<lb/>cta linea, ER, & </s> <s xml:id="echoid-s7703" xml:space="preserve">conſtituta, ER <lb/>G, figura diſtantiarum parabolæ, C <lb/>EG. </s> <s xml:id="echoid-s7704" xml:space="preserve">Dico ergo omnia quadrata <lb/>parabolæ, ADH, regula, AH, <lb/>ad rectangula ſub parabola, CEG, <lb/>& </s> <s xml:id="echoid-s7705" xml:space="preserve">trilineo, ERG regula, CG, <lb/>eſſe vt quadratum, DO, ad quadratum, EM, abſcindatur ergo <lb/>ab, OD, DN, æqualis ipſi, EM, & </s> <s xml:id="echoid-s7706" xml:space="preserve">per, N, ducatur ipſi, AH, <lb/>parallela, BF. </s> <s xml:id="echoid-s7707" xml:space="preserve">Omnia ergo quadrata parabolæ, ADH, ad omnia <lb/>quadrata parabolæ, BDF, regula communi, AH, vel, BF, ſunt <lb/>vt qúadratum, OD, ad quadratum, DN, vel ad quadratum, E <lb/>M, ſedomnia quadrata parabolæ, BDF, regula, BF, ſunt æqua-<lb/> <anchor type="note" xlink:label="note-0340-01a" xlink:href="note-0340-01"/> lia rectangulis ſub parabola, CEG, & </s> <s xml:id="echoid-s7708" xml:space="preserve">trilineo, ERG, regula, C <lb/>G, ergo omnia quadrata parabolæ, ADH, regula, AH ad re-<lb/>ctangula ſub parabola, CEG, & </s> <s xml:id="echoid-s7709" xml:space="preserve">trilineo, ERG, regula, CG, <lb/> <anchor type="note" xlink:label="note-0340-02a" xlink:href="note-0340-02"/> erunt vt quadratum, OD, ad quadratum, EM, quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s7710" xml:space="preserve"/> </p> <div xml:id="echoid-div765" type="float" level="2" n="1"> <figure xlink:label="fig-0340-01" xlink:href="fig-0340-01a"> <image file="0340-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0340-01"/> </figure> <note position="left" xlink:label="note-0340-01" xlink:href="note-0340-01a" xml:space="preserve">22. huius.</note> <note position="left" xlink:label="note-0340-02" xlink:href="note-0340-02a" xml:space="preserve">Ex antec.</note> </div> <pb o="321" file="0341" n="341" rhead="LIBER IV."/> </div> <div xml:id="echoid-div767" type="section" level="1" n="452"> <head xml:id="echoid-head472" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7711" xml:space="preserve">_H_Inc patet, ſi àuæ rectæ lineæ ad axem obliquæ parabolas conſtitue-<lb/>rint, ſumpta pro regula conſtitutæ parabolæ recta eam conſtituen-<lb/>te, quoniam rectangula ſub dictis parabolis, & </s> <s xml:id="echoid-s7712" xml:space="preserve">figuris diſtantiarum ea-<lb/>rundem ad omnia quadrata parabolæ, cuius baſis ſit ad axim recta (quæ <lb/>pro eadem ſumatur pro regula) ſunt, vt quadrata diametrorum earun-<lb/>dem ad quadratum axis illius tertia parabolæ; </s> <s xml:id="echoid-s7713" xml:space="preserve">quod ideò illa rectangula <lb/>erunt inter ſe, vt diametrorum earundem parabolarum quadrata fuerint <lb/>quoq; </s> <s xml:id="echoid-s7714" xml:space="preserve">inter ſe.</s> <s xml:id="echoid-s7715" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div768" type="section" level="1" n="453"> <head xml:id="echoid-head473" xml:space="preserve">THEOREMA XXVII. PROPOS. XXIX:</head> <p> <s xml:id="echoid-s7716" xml:space="preserve">OMnia quadrata parabolarum, regulis baſibus ſuntin-<lb/>terſe, vt omnia quadrata parallelogrammorum, in <lb/>eiſdem baſibus, & </s> <s xml:id="echoid-s7717" xml:space="preserve">circa eoſdem axes, veldiametros exi-<lb/>ſtentium, regulis eiſdem baſibus.</s> <s xml:id="echoid-s7718" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7719" xml:space="preserve">Manifeſta eſt hæc propoſitio, nam omnia quadrata dictarum <lb/>parabolarum ſunt ſubdupla omnium quadratorum eorundem pa-<lb/> <anchor type="note" xlink:label="note-0341-01a" xlink:href="note-0341-01"/> rallelogrammorum, endemregulis aſſumptis, ſcilicet parabolarum <lb/>baſibus iam dictis.</s> <s xml:id="echoid-s7720" xml:space="preserve"/> </p> <div xml:id="echoid-div768" type="float" level="2" n="1"> <note position="right" xlink:label="note-0341-01" xlink:href="note-0341-01a" xml:space="preserve">21. huius</note> </div> </div> <div xml:id="echoid-div770" type="section" level="1" n="454"> <head xml:id="echoid-head474" xml:space="preserve">A. COROLL. SECTIO I.</head> <note position="right" xml:space="preserve">A</note> <p style="it"> <s xml:id="echoid-s7721" xml:space="preserve">_H_Inc colligimus concluſiones, quæ de omnibus quadratis parallelo-<lb/>grammorum collectæ ſuntin Theorematibus _9. </s> <s xml:id="echoid-s7722" xml:space="preserve">10. </s> <s xml:id="echoid-s7723" xml:space="preserve">11. </s> <s xml:id="echoid-s7724" xml:space="preserve">12. </s> <s xml:id="echoid-s7725" xml:space="preserve">13._ <lb/></s> <s xml:id="echoid-s7726" xml:space="preserve">Lib. </s> <s xml:id="echoid-s7727" xml:space="preserve">_2._ </s> <s xml:id="echoid-s7728" xml:space="preserve">regulis ibidem aſſumptis, ſuppoſitis quibuſdam conditionibus <lb/>circa altitudines, vel latera æqualiter baſibus inclinata, & </s> <s xml:id="echoid-s7729" xml:space="preserve">quadrata <lb/>baſium, vel ipſas baſes, veriſicarietiam de omnibus quadratis parabo-<lb/>larum, ſuppoſitis eiſdem conditionibus circa axes, vel altitudines, vel <lb/>circa diametros æqualiter baſibus inclinatas, & </s> <s xml:id="echoid-s7730" xml:space="preserve">circa quadrata baſium, <lb/>vel eaſdem baſes; </s> <s xml:id="echoid-s7731" xml:space="preserve">nam his conditionibus axibus, vel altitudmibus, vel <lb/>diametris, & </s> <s xml:id="echoid-s7732" xml:space="preserve">quadratis baſium, vel ipſis baſibus competentibus, etiam <lb/>altitudinibus, vel lateribus parallelogrammorum, æqualiter baſibus <lb/>inclinatis, & </s> <s xml:id="echoid-s7733" xml:space="preserve">quadratis baſium, vel eiſdem baſibus, pariter conueniunt, <lb/>quæ quidem parallelogramma ſint in eiſdem baſibus, & </s> <s xml:id="echoid-s7734" xml:space="preserve">circa eoſdem <lb/>axes vel diametros cum parabolis; </s> <s xml:id="echoid-s7735" xml:space="preserve">& </s> <s xml:id="echoid-s7736" xml:space="preserve">ideò dictæ concluſiones, quæ tunc <pb o="322" file="0342" n="342" rhead="GEOMETRIÆ"/> colliguntur pro omnibus quadratis dictorum parallelogrammorum, pro <lb/>omnibus quadratis etiam parabolarum eiſdem inſcriptiarum, tamquam <lb/>pro earundem partibus proportionalibus, ſcilicet dimidijs, pariter vt <lb/>vera recipi poſſunt.</s> <s xml:id="echoid-s7737" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div771" type="section" level="1" n="455"> <head xml:id="echoid-head475" xml:space="preserve">B. SECTIO II.</head> <note position="left" xml:space="preserve">B</note> <p style="it"> <s xml:id="echoid-s7738" xml:space="preserve">_E_T quia oſtenſum eſt omnia quadrata parallelogrammorum in ea-<lb/> <anchor type="note" xlink:label="note-0342-02a" xlink:href="note-0342-02"/> dem altitudine ſtantium, regulis baſibus, eſſe interſe, vt qua-<lb/>drata baſium; </s> <s xml:id="echoid-s7739" xml:space="preserve">& </s> <s xml:id="echoid-s7740" xml:space="preserve">exiſtentium in eadem baſi eſſe, vt altitudines, vel <lb/>etiam, vt latera eorundem æqualiter baſibus inclinata, ideò pariter hic <lb/>colligemus omnia quadrata parabolarum in eadam altitudine exiſten-<lb/>tium, regulis baſibus, eſſe vt quadrata baſium, & </s> <s xml:id="echoid-s7741" xml:space="preserve">exiſtentium in ea-<lb/>dem baſis eſſe interſe, vt altitudines, vel vt diametros æqualiter ba-<lb/>ſibus inclinatas.</s> <s xml:id="echoid-s7742" xml:space="preserve"/> </p> <div xml:id="echoid-div771" type="float" level="2" n="1"> <note position="left" xlink:label="note-0342-02" xlink:href="note-0342-02a" xml:space="preserve">_9. l. 2._</note> </div> </div> <div xml:id="echoid-div773" type="section" level="1" n="456"> <head xml:id="echoid-head476" xml:space="preserve">C. SECTIO III.</head> <note position="left" xml:space="preserve">C</note> <p style="it"> <s xml:id="echoid-s7743" xml:space="preserve">_S_Imiliter quia oſtenſum eſt omnia quadrata parallelogrammorum, re-<lb/> <anchor type="note" xlink:label="note-0342-04a" xlink:href="note-0342-04"/> gulis baſibus, babere inter ſe rationem compoſitam ex ratione qua-<lb/>dratorum baſium, & </s> <s xml:id="echoid-s7744" xml:space="preserve">altitudinum, vel laterum æqualiter baſibus in-<lb/>clinatorum; </s> <s xml:id="echoid-s7745" xml:space="preserve">ideò colligemus, hic, omnia quadrata parabolarum regu-<lb/>lis baſibus, babere inter ſe rationem compoſitam ex ratione quadrato-<lb/>rum baſium, & </s> <s xml:id="echoid-s7746" xml:space="preserve">altitudinum, vel diametrorum æqualiter baſibus in-<lb/>clinatorum.</s> <s xml:id="echoid-s7747" xml:space="preserve"/> </p> <div xml:id="echoid-div773" type="float" level="2" n="1"> <note position="left" xlink:label="note-0342-04" xlink:href="note-0342-04a" xml:space="preserve">_10. l. 2._</note> </div> </div> <div xml:id="echoid-div775" type="section" level="1" n="457"> <head xml:id="echoid-head477" xml:space="preserve">D. SECTIO IV.</head> <note position="left" xml:space="preserve">D</note> <p style="it"> <s xml:id="echoid-s7748" xml:space="preserve">_C_Onſimili metbodo colligemus, omnia quadrata parabolarum, regu-<lb/>lis baſibus, quarum baſium quadrata alt itudinibus, vel diametris <lb/>æqualiter baſibus inclinatis reciprocantur, eſſe æqualia, & </s> <s xml:id="echoid-s7749" xml:space="preserve">quæ ſunt <lb/>æqualia, eſſe parabolarum, quarum altitudines, vel diametri æquali-<lb/>ter baſibus inclinatæ, baſium quadratis reciprocantur.</s> <s xml:id="echoid-s7750" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div776" type="section" level="1" n="458"> <head xml:id="echoid-head478" xml:space="preserve">E. SECTIO V.</head> <note position="left" xml:space="preserve">E</note> <p style="it"> <s xml:id="echoid-s7751" xml:space="preserve">_D_Eniq; </s> <s xml:id="echoid-s7752" xml:space="preserve">& </s> <s xml:id="echoid-s7753" xml:space="preserve">hoc obtinemus. </s> <s xml:id="echoid-s7754" xml:space="preserve">nempè omnia quadrata parabolarum, <lb/>regulis baſibus, quarum altitudines, veldiametri, baſibus æ-<lb/>qualiter inclinatæ, ad eaſdem baſes eandem rationem habeant, eſſe in-<lb/>ter ſe in tripla ratione baſium, vel altitudinum, vel diametrorum <lb/>æqualiter baſibus inclinatarum; </s> <s xml:id="echoid-s7755" xml:space="preserve">quæ omnia clarè, & </s> <s xml:id="echoid-s7756" xml:space="preserve">facilè patent.</s> <s xml:id="echoid-s7757" xml:space="preserve"/> </p> <pb o="323" file="0343" n="343" rhead="LIBER IV."/> </div> <div xml:id="echoid-div777" type="section" level="1" n="459"> <head xml:id="echoid-head479" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXX.</head> <p> <s xml:id="echoid-s7758" xml:space="preserve">SI duæ rectæ lineæ ducantur, quarum altera parabolam <lb/>tangat, altera verò axi, vel diametro parabolæ æqui-<lb/>diſtanter ducta eandem ſecet, in idem punctum concurren-<lb/>tes: </s> <s xml:id="echoid-s7759" xml:space="preserve">Omnia quadrata parallelogrammi, regula tangente, <lb/>erunt ſexcupla omnium quadratorum trilinei ſub dictis tan-<lb/>gente, & </s> <s xml:id="echoid-s7760" xml:space="preserve">ſecante, & </s> <s xml:id="echoid-s7761" xml:space="preserve">curua parabolæ ab ijſdem incluſa, <lb/>comprehenſi.</s> <s xml:id="echoid-s7762" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7763" xml:space="preserve">Sit ſemiparabola, ACB, quam tangat lineæ, AD, & </s> <s xml:id="echoid-s7764" xml:space="preserve">à pun-<lb/>cto, A, ducta, AB, axis, vel diameter, integræ parabolæ, dein-<lb/>de agatur vtcunque, DC, parallela axi, vel diametro, AB, ſecans <lb/>curuam parabolæ in, C, & </s> <s xml:id="echoid-s7765" xml:space="preserve">occurrens tangenti, AD, in, D, duca-<lb/>tur tandem à puncto, C, ipſi, AD, æquidiſtans, CB, ſecans, AB, <lb/>in, B, regula autem ſit, AD. </s> <s xml:id="echoid-s7766" xml:space="preserve">Dico ergo omnia quadrata parallelo-<lb/> <anchor type="figure" xlink:label="fig-0343-01a" xlink:href="fig-0343-01"/> grammi, AC, eſſe omnium quadratorum <lb/>trilinei, ADC, ſexcupla: </s> <s xml:id="echoid-s7767" xml:space="preserve">Omnia enim <lb/>quadrata, AC, ad rectangula ſub, AC, <lb/> <anchor type="note" xlink:label="note-0343-01a" xlink:href="note-0343-01"/> & </s> <s xml:id="echoid-s7768" xml:space="preserve">ſemiparabola, ABC, ſunt vt, AC, ad <lb/>ſemiparabolam, ABC, .</s> <s xml:id="echoid-s7769" xml:space="preserve">i. </s> <s xml:id="echoid-s7770" xml:space="preserve">ſexquialtera <lb/>.</s> <s xml:id="echoid-s7771" xml:space="preserve">i. </s> <s xml:id="echoid-s7772" xml:space="preserve">vt 6. </s> <s xml:id="echoid-s7773" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s7774" xml:space="preserve">omnia autem quadrata, AC, <lb/> <anchor type="note" xlink:label="note-0343-02a" xlink:href="note-0343-02"/> ad omnia quadrata ſemiparabolę ABC, <lb/>ſunt dupla .</s> <s xml:id="echoid-s7775" xml:space="preserve">i. </s> <s xml:id="echoid-s7776" xml:space="preserve">vt 6. </s> <s xml:id="echoid-s7777" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s7778" xml:space="preserve">ergo omnia qua-<lb/> <anchor type="note" xlink:label="note-0343-03a" xlink:href="note-0343-03"/> drata, AC, ad reſiduum demptis omni-<lb/>bus quadratis ſemiparabolæ, ABC, à re-<lb/>ctangulis ſub, AC, & </s> <s xml:id="echoid-s7779" xml:space="preserve">ſemiparabola, ABC, .</s> <s xml:id="echoid-s7780" xml:space="preserve">i. </s> <s xml:id="echoid-s7781" xml:space="preserve">ad rectangula ſub <lb/>ſemiparabola, ABC, & </s> <s xml:id="echoid-s7782" xml:space="preserve">trilineo, ADC, erunt in ratione ſex-<lb/>cuplaideſt vt 6. </s> <s xml:id="echoid-s7783" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s7784" xml:space="preserve">ad eadem verò bis ſumpta, vt 6. </s> <s xml:id="echoid-s7785" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s7786" xml:space="preserve">quo-<lb/>niam verò omnia quadrata, AC, ad omnia quadrata ſemipara-<lb/>bolæ, ABC, ſum vt 6. </s> <s xml:id="echoid-s7787" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s7788" xml:space="preserve">vt dictum eſt, ideò omnia quadrata, <lb/>AC, ad omnia quadrata ſemiparabolæ, ABC, cum rectangulis <lb/>bis ſub ſemiparabola, ABC, & </s> <s xml:id="echoid-s7789" xml:space="preserve">trilineo, ADC, erunt vt 6. </s> <s xml:id="echoid-s7790" xml:space="preserve">ad 5. <lb/></s> <s xml:id="echoid-s7791" xml:space="preserve">ergo ad reliquum ſcilicet ad omnia quadrata trilinei, ADC, omnia <lb/> <anchor type="note" xlink:label="note-0343-04a" xlink:href="note-0343-04"/> quadrata, AC, erunt vt 6. </s> <s xml:id="echoid-s7792" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s7793" xml:space="preserve">ideſt erunt eorundem ſexcupla, <lb/>quod oſtendere oportebat.</s> <s xml:id="echoid-s7794" xml:space="preserve"/> </p> <div xml:id="echoid-div777" type="float" level="2" n="1"> <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a"> <image file="0343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0343-01"/> </figure> <note position="right" xlink:label="note-0343-01" xlink:href="note-0343-01a" xml:space="preserve">Coroll. 1. <lb/>26. l. 2.</note> <note position="right" xlink:label="note-0343-02" xlink:href="note-0343-02a" xml:space="preserve">21. huius.</note> <note position="right" xlink:label="note-0343-03" xlink:href="note-0343-03a" xml:space="preserve">Coroll. <lb/>23. l. 2.</note> <note position="right" xlink:label="note-0343-04" xlink:href="note-0343-04a" xml:space="preserve">D. 23. l. 2.</note> </div> <pb o="324" file="0344" n="344" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div779" type="section" level="1" n="460"> <head xml:id="echoid-head480" xml:space="preserve">A. COROLL. SECT IO I.</head> <note position="left" xml:space="preserve">A</note> <p style="it"> <s xml:id="echoid-s7795" xml:space="preserve">_H_Inc habetur omnia quadrata trilineorum ſub tangentibus, & </s> <s xml:id="echoid-s7796" xml:space="preserve">ſe-<lb/>cantibus, veluti ſunt, AD, DC, regulis tangentibus, eſſe <lb/>inter ſe, vt omnia quadrata parallelogrammorum ſub eiſdem tangen-<lb/>tibus, & </s> <s xml:id="echoid-s7797" xml:space="preserve">ſecantibus, regulis ijſdem tangentibus, quoniam dictorum <lb/>trilineorum omnia quadrata ſunt ſextæ partes omnium quadratorum di-<lb/>ctorum parallelogram norum; </s> <s xml:id="echoid-s7798" xml:space="preserve">Etideò proipſis etiam has concluſiones <lb/>colligemus, ſcilicet.</s> <s xml:id="echoid-s7799" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div780" type="section" level="1" n="461"> <head xml:id="echoid-head481" xml:space="preserve">B. SECTIO II.</head> <note position="left" xml:space="preserve">B</note> <p style="it"> <s xml:id="echoid-s7800" xml:space="preserve">_S_I dicti triliaei fuerint in eadem altitudine, quòd omnia quadrata <lb/>earundem erunt inter ſe, vt baſium quadrata @ſ. </s> <s xml:id="echoid-s7801" xml:space="preserve">tangentium; </s> <s xml:id="echoid-s7802" xml:space="preserve">Et <lb/>ſi fuerint dictitrilinei in eadem baſi ſcilicet tangente, dicta omnia qua-<lb/>drata erunt inter ſe, vt altitudines, vel, vt ſecantes æqualiter baſibus <lb/>ſcilicettangentibus, inclinata.</s> <s xml:id="echoid-s7803" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div781" type="section" level="1" n="462"> <head xml:id="echoid-head482" xml:space="preserve">C. SECTIO III.</head> <note position="left" xml:space="preserve">C</note> <p style="it"> <s xml:id="echoid-s7804" xml:space="preserve">_I_Tem quod omnia quadrata dictorum trilineorum habebunt inter ſe <lb/>rationem compoſitam ex ratione quadratorum baſium, & </s> <s xml:id="echoid-s7805" xml:space="preserve">ex r atio-<lb/>ne altitudinum, vel ſecantium æqualiter baſibus, ſcilicet tangenti-<lb/>bus, inclinatarum.</s> <s xml:id="echoid-s7806" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div782" type="section" level="1" n="463"> <head xml:id="echoid-head483" xml:space="preserve">D. SECTIO IV.</head> <note position="left" xml:space="preserve">D</note> <p style="it"> <s xml:id="echoid-s7807" xml:space="preserve">_P_Ariter quod omnia quadrata dictorum trilineorum, quorum tan-<lb/>gentium quadrata altitudinibus, vel ſecantibus æqualiter tangen-<lb/>t<unsure/>ibus inclinatis reciprocantur, eſſe æqualia; </s> <s xml:id="echoid-s7808" xml:space="preserve">& </s> <s xml:id="echoid-s7809" xml:space="preserve">quæ ſunt æqualia, eſſe <lb/>trilineorum, quorum baſium, vel tangentium quadrata altitudinibus, <lb/>vel ſecantibus equaliter tangentibus inclinatis, reciprocantur.</s> <s xml:id="echoid-s7810" xml:space="preserve"/> </p> <figure> <image file="0344-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0344-01"/> </figure> <pb o="325" file="0345" n="345" rhead="LIBER IV."/> </div> <div xml:id="echoid-div783" type="section" level="1" n="464"> <head xml:id="echoid-head484" xml:space="preserve">E. SECTIO V.</head> <p style="it"> <s xml:id="echoid-s7811" xml:space="preserve">_T_Andem, quòd, ſi dictorum trilineorum ſecantes ád tángentes eán-<lb/>demrationem habuerint, omnia quadrata eorundem erunt in tri-<lb/>plaratione tangentium, vel ſecantium.</s> <s xml:id="echoid-s7812" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div784" type="section" level="1" n="465"> <head xml:id="echoid-head485" xml:space="preserve">THEOREMA XXIX. PROPOS. XXXI.</head> <p> <s xml:id="echoid-s7813" xml:space="preserve">EXponatur figura Theor. </s> <s xml:id="echoid-s7814" xml:space="preserve">antecedentis, & </s> <s xml:id="echoid-s7815" xml:space="preserve">intra paralle-<lb/>logrammum, AC, ducatur vtcunq; </s> <s xml:id="echoid-s7816" xml:space="preserve">recta, EF, pa-<lb/>rallelaipſi, BC, quæ ſumatur pro regula: </s> <s xml:id="echoid-s7817" xml:space="preserve">Oſtendemus. </s> <s xml:id="echoid-s7818" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s7819" xml:space="preserve">om@ia quadrata, AC, demptis omnibus quadratis ſemipa-<lb/>rabolæ, ABC, ad omnia quadrata, EC, demptis omni-<lb/>bus quadratis quadrilinei, MEBC, eſſe vt quadratum, A <lb/>B, ad quadratum, BE.</s> <s xml:id="echoid-s7820" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7821" xml:space="preserve">Omnia. </s> <s xml:id="echoid-s7822" xml:space="preserve">n. </s> <s xml:id="echoid-s7823" xml:space="preserve">quadrata, AC, ad omnia quadrata, EC, demptis <lb/>omnibus quadratis quadrilinei, EMCB, habent rationem compo-<lb/>ſitam ex ea, quam habent omnia quadrata, AC, ad omnia qua-<lb/>drata, EC, ideſt ex ea, quam habet, AB, ad, BE, & </s> <s xml:id="echoid-s7824" xml:space="preserve">ex ea, <lb/> <anchor type="note" xlink:label="note-0345-01a" xlink:href="note-0345-01"/> quam habent omnia quadrata, EC, ad reſiduum, demptis ab ijſdem <lb/> <anchor type="figure" xlink:label="fig-0345-01a" xlink:href="fig-0345-01"/> omnibus quadratis quadrilinei, MEBC, <lb/>.</s> <s xml:id="echoid-s7825" xml:space="preserve">i. </s> <s xml:id="echoid-s7826" xml:space="preserve">ex ea, quam habet, AB, ad {1/2}, BE, <lb/> <anchor type="note" xlink:label="note-0345-02a" xlink:href="note-0345-02"/> duæ autem hæ rationes .</s> <s xml:id="echoid-s7827" xml:space="preserve">ſ. </s> <s xml:id="echoid-s7828" xml:space="preserve">quàm habet, <lb/>AB, ad, BE, &</s> <s xml:id="echoid-s7829" xml:space="preserve">, AB, ad {1/2}. </s> <s xml:id="echoid-s7830" xml:space="preserve">BE, com-<lb/>ponunt rationem quadrati, AB, ad re-<lb/>ctangulum ſub, EB, & </s> <s xml:id="echoid-s7831" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s7832" xml:space="preserve">BE, .</s> <s xml:id="echoid-s7833" xml:space="preserve">i. </s> <s xml:id="echoid-s7834" xml:space="preserve">ad <lb/>dimidium quadrati, BE, ergo omnia <lb/> <anchor type="note" xlink:label="note-0345-03a" xlink:href="note-0345-03"/> quadrata, AC, ad omnia quadrata, E <lb/>C, demptis omnibus quadratis quadrili-<lb/>nei, MEBC, erunt vt quadratum, AB, <lb/>ad dimidium quadrati, BE, ſunt autem omnia quadrata, AC, <lb/>demptis omnibus quadratis ſemiparabolæ, ABC, dimidium <lb/>omnium quadratorum, AC, quia omnia quadrata, AC, ſunt du-<lb/>pla omnium quadratorum ſemiparabolæ, ABC, ergo omnia qua-<lb/>drata, AC, demptis omnibus quadratis ſemiparabolæ, ABC, ad <lb/>omnia quadrata, EC, demptis omnibus quadratis quadrilinei, E <lb/>MCB, erunt vt dimidium quadrati, AB, ad dimidium quadrati, <lb/>BE, ideſt vt quadratum, AB, ad quadratum, BE, quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s7835" xml:space="preserve"/> </p> <div xml:id="echoid-div784" type="float" level="2" n="1"> <note position="right" xlink:label="note-0345-01" xlink:href="note-0345-01a" xml:space="preserve">10. l. 2.</note> <figure xlink:label="fig-0345-01" xlink:href="fig-0345-01a"> <image file="0345-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0345-01"/> </figure> <note position="right" xlink:label="note-0345-02" xlink:href="note-0345-02a" xml:space="preserve">23. huius.</note> <note position="right" xlink:label="note-0345-03" xlink:href="note-0345-03a" xml:space="preserve">21. huius.</note> </div> <pb o="326" file="0346" n="346" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div786" type="section" level="1" n="466"> <head xml:id="echoid-head486" xml:space="preserve">THEOREMA XXX. PROPOS. XXXII.</head> <p> <s xml:id="echoid-s7836" xml:space="preserve">SI parallelogrammum, & </s> <s xml:id="echoid-s7837" xml:space="preserve">parabola fuerint in eadem ba-<lb/>ſi, & </s> <s xml:id="echoid-s7838" xml:space="preserve">circa eundem axim, vel diametrum conſtituta, <lb/>baſiſque ſumatur pro regula: </s> <s xml:id="echoid-s7839" xml:space="preserve">Omnia quadrata dicti paral-<lb/>lelogrammi ad omnia quadrata figuræ compoſitæ ex para-<lb/>bola, & </s> <s xml:id="echoid-s7840" xml:space="preserve">alterutro trilineorum, qui fiunt extra parabolam, <lb/>demptis omnibus quadratis eiuſdem trilinei, erunt vt di-<lb/>ctum parallelogrammum ad dictam parabolam; </s> <s xml:id="echoid-s7841" xml:space="preserve">ad eadem <lb/>verò cum omnibus quadratis illius trilinci erunt, vt dictum <lb/>parallelogrammum ad dictam parabolam ſimul cum {@/2} {1/4}. </s> <s xml:id="echoid-s7842" xml:space="preserve">di-<lb/>ctiparallelogrammi .</s> <s xml:id="echoid-s7843" xml:space="preserve">i. </s> <s xml:id="echoid-s7844" xml:space="preserve">vt 24. </s> <s xml:id="echoid-s7845" xml:space="preserve">ad 17.</s> <s xml:id="echoid-s7846" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7847" xml:space="preserve">Sit ergo parallelogrammum, AF, in eadem baſi, DF, & </s> <s xml:id="echoid-s7848" xml:space="preserve">circa <lb/>eundem axim, vel diametrum, BE, cum parabola, DBF, regula <lb/>ſit, DF. </s> <s xml:id="echoid-s7849" xml:space="preserve">Dico omnia quadrata, AF, ad omnia quadrat<unsure/>a figuræ, <lb/>CBDF, demptis omnibus quadratis trilinei, BCF, eſſe, vt, AF, <lb/>ad parabolam, DBF, eadem verò ad omnia quadrata fig. </s> <s xml:id="echoid-s7850" xml:space="preserve">CBD <lb/> <anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/> F, eſſe vt, AF, ad parabo-<lb/>lam, DBF, cum {@/2} {1/4}. </s> <s xml:id="echoid-s7851" xml:space="preserve">paral-<lb/>lelogrammi, AF; </s> <s xml:id="echoid-s7852" xml:space="preserve">quoniam <lb/>enim, BE, eſt axis, vel dia-<lb/>meter tum parabolæ, DBF, <lb/>tum parallelogrammi, AF, <lb/>ideò ſi ducatur intra paralle-<lb/>logrammum, AF, vtcunq-<lb/>recta linea parallelaipſi, D <lb/>F, portiones eiuſdem inclu-<lb/>ſæ trilineis, ADB, CFB, erunt inter ſe æquales, & </s> <s xml:id="echoid-s7853" xml:space="preserve">ideò para-<lb/>bola, DBF, erit figura, qualem poſtulat Prop. </s> <s xml:id="echoid-s7854" xml:space="preserve">29. </s> <s xml:id="echoid-s7855" xml:space="preserve">Lib. </s> <s xml:id="echoid-s7856" xml:space="preserve">3. </s> <s xml:id="echoid-s7857" xml:space="preserve">quapro-<lb/>pter omnia quadrata, AF, ad omnia quadrata figuræ, CBDF, <lb/>demptis omnibus quadratis trilinei, BCF, erunt vt, AF, ad para-<lb/>bolam, DBF.</s> <s xml:id="echoid-s7858" xml:space="preserve"/> </p> <div xml:id="echoid-div786" type="float" level="2" n="1"> <figure xlink:label="fig-0346-01" xlink:href="fig-0346-01a"> <image file="0346-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0346-01"/> </figure> </div> <p> <s xml:id="echoid-s7859" xml:space="preserve">Quoniam verò omnia quadrata, AF, ad omnia quadrata, BF, <lb/>ſunt vt quadratum, DF, ad quadratum, FE, .</s> <s xml:id="echoid-s7860" xml:space="preserve">i. </s> <s xml:id="echoid-s7861" xml:space="preserve">quadrupla .</s> <s xml:id="echoid-s7862" xml:space="preserve">i. </s> <s xml:id="echoid-s7863" xml:space="preserve">vt <lb/>24. </s> <s xml:id="echoid-s7864" xml:space="preserve">ad 6. </s> <s xml:id="echoid-s7865" xml:space="preserve">omnia verò quadrata, BF, ſunt ſexcupla omnium qua-<lb/>dratorum trilinei, BCF, .</s> <s xml:id="echoid-s7866" xml:space="preserve">i. </s> <s xml:id="echoid-s7867" xml:space="preserve">vt 6. </s> <s xml:id="echoid-s7868" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s7869" xml:space="preserve">igitur omnia quadrata, AF, <lb/> <anchor type="note" xlink:label="note-0346-01a" xlink:href="note-0346-01"/> ad omnia quadrata trilinei, BCF, erunt vt 24. </s> <s xml:id="echoid-s7870" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s7871" xml:space="preserve">ideſt vt, AF, <lb/>ad ſui ipſius {@/2} {1/4}. </s> <s xml:id="echoid-s7872" xml:space="preserve">ergo omnia quadrata, AF, ad omnia quadrata fi- <pb o="327" file="0347" n="347" rhead="LIBER IV."/> guræ, CBDF, erunt vt, AF, ad parabolam, DBF, cum {@/2} {1/4}. <lb/></s> <s xml:id="echoid-s7873" xml:space="preserve">parallelogrammi, AF; </s> <s xml:id="echoid-s7874" xml:space="preserve">parallelogrammum autem, AF, eſt ſex-<lb/>quialterum parabolæ, DBF, .</s> <s xml:id="echoid-s7875" xml:space="preserve">i. </s> <s xml:id="echoid-s7876" xml:space="preserve">ad illam, vt 24. </s> <s xml:id="echoid-s7877" xml:space="preserve">ad 16. </s> <s xml:id="echoid-s7878" xml:space="preserve">ergo ſi nu-<lb/>mero 16. </s> <s xml:id="echoid-s7879" xml:space="preserve">iungatur {1/4}. </s> <s xml:id="echoid-s7880" xml:space="preserve">eiuſdem parallelogrammi, AF, fient 17. </s> <s xml:id="echoid-s7881" xml:space="preserve"><lb/>igitur omnia quadrata, AF, ad omnia quadrata figuræ, CBDF, <lb/>erunt vt 24. </s> <s xml:id="echoid-s7882" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s7883" xml:space="preserve">quod oſtendendum erat.</s> <s xml:id="echoid-s7884" xml:space="preserve"/> </p> <div xml:id="echoid-div787" type="float" level="2" n="2"> <note position="left" xlink:label="note-0346-01" xlink:href="note-0346-01a" xml:space="preserve">30. huius.</note> </div> </div> <div xml:id="echoid-div789" type="section" level="1" n="467"> <head xml:id="echoid-head487" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7885" xml:space="preserve">_H_Inc patet omnia quadrata, AF, eſſe ſexquialtera omnium qua-<lb/>dratorum figuræ, CBDF, demptis omnibus quadratis trilinei, B <lb/> <anchor type="note" xlink:label="note-0347-01a" xlink:href="note-0347-01"/> CF, nam ſunt ad illa, vt, AF, ad parabolam, BDF, cuius parallelo-<lb/>grammum, AF, eſt ſexquialterum.</s> <s xml:id="echoid-s7886" xml:space="preserve"/> </p> <div xml:id="echoid-div789" type="float" level="2" n="1"> <note position="right" xlink:label="note-0347-01" xlink:href="note-0347-01a" xml:space="preserve">_1. huius._</note> </div> </div> <div xml:id="echoid-div791" type="section" level="1" n="468"> <head xml:id="echoid-head488" xml:space="preserve">THEOREMA XXXI. PROPOS. XXXIII.</head> <p> <s xml:id="echoid-s7887" xml:space="preserve">IN eodem antec. </s> <s xml:id="echoid-s7888" xml:space="preserve">Propoſit. </s> <s xml:id="echoid-s7889" xml:space="preserve">Schemate oſten demus omnia <lb/>quadrata figuræ, CBDF, demptis omnibus quadratis, <lb/>BF, ad omnia quadrata, BF, demptis omnibus quadratis <lb/>trilinei, BCF, eſſe vt 11. </s> <s xml:id="echoid-s7890" xml:space="preserve">ad 5.</s> <s xml:id="echoid-s7891" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7892" xml:space="preserve">Nam omnia quadrata, AF, ad omnia quadrata figuræ, CBD <lb/>F, oſtenſa ſunt eſſe, vt 24. </s> <s xml:id="echoid-s7893" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s7894" xml:space="preserve">eadem verò ad omnia quadrata, <lb/>BF, ſunt vt 24. </s> <s xml:id="echoid-s7895" xml:space="preserve">ad 6. </s> <s xml:id="echoid-s7896" xml:space="preserve">quia ſunt eorum quadrupla, ergo ad reſiduum <lb/>.</s> <s xml:id="echoid-s7897" xml:space="preserve">f. </s> <s xml:id="echoid-s7898" xml:space="preserve">ad omnia quadrata figuræ, CBDF, demptis omnib. </s> <s xml:id="echoid-s7899" xml:space="preserve">quadratis, <lb/> <anchor type="figure" xlink:label="fig-0347-01a" xlink:href="fig-0347-01"/> BF, erunt vt 24. </s> <s xml:id="echoid-s7900" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s7901" xml:space="preserve">con-<lb/>uertendo omnia quadrata fi-<lb/>gurę, CBDF, demptis om-<lb/>nibus quadratis, BF, ad om-<lb/>nia quadrata, AF, erunt vt <lb/>11. </s> <s xml:id="echoid-s7902" xml:space="preserve">ad 24. </s> <s xml:id="echoid-s7903" xml:space="preserve">Item omnia qua-<lb/>drata, AF, ad omnia qua-<lb/>drata, BF, ſunt vt 24. </s> <s xml:id="echoid-s7904" xml:space="preserve">ad 6. <lb/></s> <s xml:id="echoid-s7905" xml:space="preserve">omnia verò quadrata, BF, <lb/>ad omnia quadrata trilinei, <lb/>BCF, ſunt vt 6. </s> <s xml:id="echoid-s7906" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s7907" xml:space="preserve">ergo omnia quadrata, AF, ad omnia qua-<lb/>drata trilinei, BCF, ſunt vt 24. </s> <s xml:id="echoid-s7908" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s7909" xml:space="preserve">eadem verò ad omnia quadra-<lb/>ta, BF, ſunt vt 24. </s> <s xml:id="echoid-s7910" xml:space="preserve">ad 6. </s> <s xml:id="echoid-s7911" xml:space="preserve">ergo omnia quadrata, AF, ad omnia qua-<lb/>drata, BF, demptis omnibus quadratis trilinei, BCF, erunt vt 24. </s> <s xml:id="echoid-s7912" xml:space="preserve"><lb/>ad 5. </s> <s xml:id="echoid-s7913" xml:space="preserve">erant autem omnia quadrata figuræ, CBDF, demptis omni- <pb o="328" file="0348" n="348" rhead="GEOMETRIÆ"/> bus quadratis, BF, ad omnia quadrata, AF, vt 11. </s> <s xml:id="echoid-s7914" xml:space="preserve">ad 24. </s> <s xml:id="echoid-s7915" xml:space="preserve">ergo, <lb/>exæquali, omnia quadrata figuræ, CBDF, demptis omnibus qua-<lb/>dratis BF, ad omnia quadrata, BF, demptis omnibus quadratis tri-<lb/>linei, BCF, erunt vt 11. </s> <s xml:id="echoid-s7916" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s7917" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s7918" xml:space="preserve"/> </p> <div xml:id="echoid-div791" type="float" level="2" n="1"> <figure xlink:label="fig-0347-01" xlink:href="fig-0347-01a"> <image file="0347-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0347-01"/> </figure> </div> </div> <div xml:id="echoid-div793" type="section" level="1" n="469"> <head xml:id="echoid-head489" xml:space="preserve">THEOREMA XXXII. PROPOS. XXXIV.</head> <p> <s xml:id="echoid-s7919" xml:space="preserve">ASſumpta eadem anteced. </s> <s xml:id="echoid-s7920" xml:space="preserve">Theor. </s> <s xml:id="echoid-s7921" xml:space="preserve">figura, ſiproducatur <lb/>baſis, DF, (quæ retineatur pro regula) vtcunq; </s> <s xml:id="echoid-s7922" xml:space="preserve">in, <lb/>M, & </s> <s xml:id="echoid-s7923" xml:space="preserve">per M, ipſi, BE, parallela ducatur, MH, cui occur-<lb/>rat, AC, producta, in ipſo, H. </s> <s xml:id="echoid-s7924" xml:space="preserve">Omnia quadrata, AM, <lb/>demptis omnibus quadratis, CM, ad omnia quadrata figu-<lb/>ræ, HBDM, demptis omnibus quadratis quadrilinei, H <lb/>BFM, erunt vt, AF, ad parabolam, DBF, ideſt erunt <lb/>eorum ſexquialtera: </s> <s xml:id="echoid-s7925" xml:space="preserve">Quod facilè patebit, quia parabola, D <lb/>BF, inſcripta parallegrammo, AF, eſt figura, qualem po-<lb/>ſtulat Propoſit. </s> <s xml:id="echoid-s7926" xml:space="preserve">30. </s> <s xml:id="echoid-s7927" xml:space="preserve">Lib. </s> <s xml:id="echoid-s7928" xml:space="preserve">3. </s> <s xml:id="echoid-s7929" xml:space="preserve">Vlterius autem dico omnia qua-<lb/>drat’a, AM, ad omnia quadrata figuræ, BDMH, eſſe vt <lb/>quadratum, DM, ad quadratum, ME, dimidium qua-<lb/>drati, ED, cum rectangulo ſub ſexquitértia, DE, & </s> <s xml:id="echoid-s7930" xml:space="preserve"><lb/>ſub, EM.</s> <s xml:id="echoid-s7931" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7932" xml:space="preserve">In conſtructa figura igitur omnia quadrata figuræ, HBDM, per <lb/>rectam, BE, diuiduntur in omnia quadrata ſemiparabolæ, BDC, <lb/>in omnia quadrata, BM, & </s> <s xml:id="echoid-s7933" xml:space="preserve">in rectangula bis ſub ſemiparabola, B <lb/> <anchor type="note" xlink:label="note-0348-01a" xlink:href="note-0348-01"/> DE, & </s> <s xml:id="echoid-s7934" xml:space="preserve">ſub EH, nunc ad horum ſingula comparemus omnia qua-<lb/> <anchor type="figure" xlink:label="fig-0348-01a" xlink:href="fig-0348-01"/> drata, AM: </s> <s xml:id="echoid-s7935" xml:space="preserve">Om-<lb/>nia igitur quadr@@-<lb/>ta, AM, ad om-<lb/>nia quadrata, BM, <lb/>ſunt vt quadra-<lb/>tum, DM, ad <lb/>quadratum, ME, <lb/>quod ſerua. </s> <s xml:id="echoid-s7936" xml:space="preserve">Item <lb/>omnia quadrata, <lb/>AM, ad omnia quadrata, AE, ſunt vt quadratum, MD, ad qua-<lb/>dratum, DE, omnia verò quadrata, AE, ſunt dupla omnium qua-<lb/>dratorum ſemiparabolæ, BDE, ergo omnia quadrata, AM, ad <lb/>omnia quadrata ſemiparabolæ, BDE, ſunt vt quadratum, MD, <lb/>ad dimidium quadrati, DE, quod etiam ſerua. </s> <s xml:id="echoid-s7937" xml:space="preserve">Tandem omnia <pb o="329" file="0349" n="349" rhead="LIB ER IV."/> quadrata, AM, ad rectangula ſub, AE, EH, ſunt vt quadratum, <lb/>DM, ad rectangulum, DEM, rectangula verò ſub, AE, EH, ad <lb/>rectangula ſub ſemiparabola, BDE, & </s> <s xml:id="echoid-s7938" xml:space="preserve">ſub, EH, ſunt vt, AE, ad <lb/>ſemiparabolam, BDE, (quia, EH, eſt parallelogrammum) idelt <lb/>ſexquialtera .</s> <s xml:id="echoid-s7939" xml:space="preserve">i. </s> <s xml:id="echoid-s7940" xml:space="preserve">vt, DE, ad {2/3}. </s> <s xml:id="echoid-s7941" xml:space="preserve">DE, .</s> <s xml:id="echoid-s7942" xml:space="preserve">i. </s> <s xml:id="echoid-s7943" xml:space="preserve">vt rectangulum ſub, DEM, <lb/>(ſumpta, EM, communi altitudine) ad rectangulum ſub {2/3}, DE, & </s> <s xml:id="echoid-s7944" xml:space="preserve"><lb/>ſub, EM, ergo, ex æquali, omnia quadrata, AM, ad rectangula <lb/>ſub ſemiparabola, BDE, & </s> <s xml:id="echoid-s7945" xml:space="preserve">ſub, BM, erunt vt quadratum, DM, <lb/> <anchor type="note" xlink:label="note-0349-01a" xlink:href="note-0349-01"/> ad rectangulum ſub {2/3}. </s> <s xml:id="echoid-s7946" xml:space="preserve">DE, & </s> <s xml:id="echoid-s7947" xml:space="preserve">ſub, EM; </s> <s xml:id="echoid-s7948" xml:space="preserve">ad eadem verò bis ſumpta <lb/>erunt, vt idem quadratum, DM, ad rectangulum bis ſub {2/3}. </s> <s xml:id="echoid-s7949" xml:space="preserve">DE, .</s> <s xml:id="echoid-s7950" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s7951" xml:space="preserve">ſub ſexquitertia, DE, ſemel, & </s> <s xml:id="echoid-s7952" xml:space="preserve">ſub, EM, ergó, colligendo, omnia <lb/>quadrata, AM, ad omnia quadrata, BM, & </s> <s xml:id="echoid-s7953" xml:space="preserve">ad omnia quadrata ſe-<lb/>miparabolæ, BDE, cum rectangulis bis ſub, HE, & </s> <s xml:id="echoid-s7954" xml:space="preserve">ſemiparabo-<lb/>la, BDE, ideſt ad omnia quadrata figuræ, HBDM, erunt vt qua-<lb/>dratum, DM, ad quadratum, ME, & </s> <s xml:id="echoid-s7955" xml:space="preserve">dimidium quadrati, ED, <lb/>cum rectangulo ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s7956" xml:space="preserve">ſub, EM, ſimul iuncta quæ <lb/>nobis erant demonſtranda.</s> <s xml:id="echoid-s7957" xml:space="preserve"/> </p> <div xml:id="echoid-div793" type="float" level="2" n="1"> <note position="left" xlink:label="note-0348-01" xlink:href="note-0348-01a" xml:space="preserve">D. 23. l. 2.</note> <figure xlink:label="fig-0348-01" xlink:href="fig-0348-01a"> <image file="0348-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0348-01"/> </figure> <note position="right" xlink:label="note-0349-01" xlink:href="note-0349-01a" xml:space="preserve">Coroll. 1. <lb/>26. l. 2.</note> </div> </div> <div xml:id="echoid-div795" type="section" level="1" n="470"> <head xml:id="echoid-head490" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s7958" xml:space="preserve">_H_Inc apparet, quod methodo huius in Propoſ. </s> <s xml:id="echoid-s7959" xml:space="preserve">32. </s> <s xml:id="echoid-s7960" xml:space="preserve">oſtendi poterat <lb/>omnia quadrata, AF, ad omnia quadrata figuræ, CBDF, eſſe <lb/>vt 24. </s> <s xml:id="echoid-s7961" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s7962" xml:space="preserve">prius demonſtrando omnia quadrata, AF, ad omnia qua-<lb/>drata figuræ, CBDF, eſſe vt quadratum, DF, ad quadratum, FE, {1/2}. </s> <s xml:id="echoid-s7963" xml:space="preserve">qua-<lb/>drati, ED, & </s> <s xml:id="echoid-s7964" xml:space="preserve">rectangulum ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s7965" xml:space="preserve">ſub, EF, vt nem-<lb/>pè 24. </s> <s xml:id="echoid-s7966" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s7967" xml:space="preserve">veluti calculanti patebit, quod bic appoſui, vt eam ratio-<lb/>nem etiam hoc pacto teneamus.</s> <s xml:id="echoid-s7968" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div796" type="section" level="1" n="471"> <head xml:id="echoid-head491" xml:space="preserve">THEOREMA XXXIII. PROPOS. XXXV.</head> <p> <s xml:id="echoid-s7969" xml:space="preserve">IN eadem anteced. </s> <s xml:id="echoid-s7970" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s7971" xml:space="preserve">figura oſtendemus omnia <lb/>quadrata, BM, ad omnia quadrata figurę, BFMH, eſſe <lb/>vt quadratum, EM, ad quadratum, MF, cum rectangulo ſub <lb/>{2/3}. </s> <s xml:id="echoid-s7972" xml:space="preserve">EF, & </s> <s xml:id="echoid-s7973" xml:space="preserve">ſub, FM, vna cum {1/6}. </s> <s xml:id="echoid-s7974" xml:space="preserve">quadrati, EF, regula eadem <lb/>retenta.</s> <s xml:id="echoid-s7975" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7976" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s7977" xml:space="preserve">n. </s> <s xml:id="echoid-s7978" xml:space="preserve">quadrata figuræ, BFMH, per rectam, CF, di<unsure/>uidun-<lb/>tur in omnia quadrata, CM, in omnia quadrata trilinei, BCF, & </s> <s xml:id="echoid-s7979" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0349-02a" xlink:href="note-0349-02"/> in rectangula bis ſub trilineo, BCF, & </s> <s xml:id="echoid-s7980" xml:space="preserve">ſub, CM; </s> <s xml:id="echoid-s7981" xml:space="preserve">ad horum ergo <lb/>fingula comparemus omnia quadrata, BM; </s> <s xml:id="echoid-s7982" xml:space="preserve">hæc igitur ad omnia <pb o="330" file="0350" n="350" rhead="GEOMETRIÆ"/> quadrata, CM, ſunt vt quadratum, EM, ad quadratum, MF, quod <lb/>ſerua. </s> <s xml:id="echoid-s7983" xml:space="preserve">Item omnia quadrata, BM, ad omnia quadrata, BF, ſunt <lb/> <anchor type="figure" xlink:label="fig-0350-01a" xlink:href="fig-0350-01"/> vt quadratum, ME, <lb/>ad quadratum, EF, <lb/> <anchor type="note" xlink:label="note-0350-01a" xlink:href="note-0350-01"/> omnia verò quadra-<lb/>ta, BF, ſunt ſexcupla <lb/>omnium quadratorú <lb/>trilinei, BCF, .</s> <s xml:id="echoid-s7984" xml:space="preserve">i. </s> <s xml:id="echoid-s7985" xml:space="preserve">ſunt <lb/>ad illa, vt quadratum, <lb/>EF, ad eiuſdem {1/6}. </s> <s xml:id="echoid-s7986" xml:space="preserve">er-<lb/>go, ex æquali, om-<lb/>nia quadrata, BM, ad <lb/>omnia quadrata trili-<lb/>nei, BCF, ſunt vt quadratum, EM, ad {1/6}. </s> <s xml:id="echoid-s7987" xml:space="preserve">quadrati, EF, quod etiá <lb/>ſerua. </s> <s xml:id="echoid-s7988" xml:space="preserve">Tandem omnia quadrata, BM, ad rectangula ſub, BF, FH, <lb/>ſunt vt quadratum, EM, ad rectangulum ſub, EF, FM, rectangu-<lb/> <anchor type="note" xlink:label="note-0350-02a" xlink:href="note-0350-02"/> la verò ſub, BF, FH, ad rectangula ſub trilineo, BFC, & </s> <s xml:id="echoid-s7989" xml:space="preserve">ſub, C <lb/>M, ſunt vt, BF, ad trilineum, BCF, (nam, CM, eſt parallelogram-<lb/>mum) ideſt ſunt eorum tripla .</s> <s xml:id="echoid-s7990" xml:space="preserve">i. </s> <s xml:id="echoid-s7991" xml:space="preserve">ſunt vt, EF, ad {1/3}. </s> <s xml:id="echoid-s7992" xml:space="preserve">EF, .</s> <s xml:id="echoid-s7993" xml:space="preserve">i. </s> <s xml:id="echoid-s7994" xml:space="preserve">vt rectan-<lb/>gulum, EF/M, ad rectangulum ſub {1/3}. </s> <s xml:id="echoid-s7995" xml:space="preserve">EF, & </s> <s xml:id="echoid-s7996" xml:space="preserve">ſub, FM, ergo, ex æ-<lb/>quali, omnia quadrata, BM, ad rectangula ſub trilineo, BCF, & </s> <s xml:id="echoid-s7997" xml:space="preserve"><lb/>ſub, FH, erunt vt quadratum, EM, ad rectangulum ſub {1/3}. </s> <s xml:id="echoid-s7998" xml:space="preserve">EF, & </s> <s xml:id="echoid-s7999" xml:space="preserve"><lb/>ſub, FM, ad eadem verò bis ſumpta, vt quadratum, EM, ad re-<lb/>ctangulum bis ſub {1/3}. </s> <s xml:id="echoid-s8000" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8001" xml:space="preserve">ſub, FM, ideſt ſemel ſub {2/3}. </s> <s xml:id="echoid-s8002" xml:space="preserve">EF, ſub, F <lb/>M, ergo, colligendo, omnia quadrata, BM, ad omnia quadrata; </s> <s xml:id="echoid-s8003" xml:space="preserve">C <lb/>M, ad omnia quadrata trilinei, BCF, & </s> <s xml:id="echoid-s8004" xml:space="preserve">ad rectangula bis ſub tri-<lb/>lineo, BCF, & </s> <s xml:id="echoid-s8005" xml:space="preserve">ſub, FH, .</s> <s xml:id="echoid-s8006" xml:space="preserve">i. </s> <s xml:id="echoid-s8007" xml:space="preserve">ad omnia quadrata figuræ, BFMH, <lb/>erunt vt quadratum, EM, ad quadratum, MF, rectangulum ſub <lb/>{2/3}. </s> <s xml:id="echoid-s8008" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8009" xml:space="preserve">ſub, FM, vna cum {1/6}. </s> <s xml:id="echoid-s8010" xml:space="preserve">quadrati, EF, quod oſtendere opus <lb/>erat.</s> <s xml:id="echoid-s8011" xml:space="preserve"/> </p> <div xml:id="echoid-div796" type="float" level="2" n="1"> <note position="right" xlink:label="note-0349-02" xlink:href="note-0349-02a" xml:space="preserve">D. Corol. <lb/>23. l. 2.</note> <figure xlink:label="fig-0350-01" xlink:href="fig-0350-01a"> <image file="0350-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0350-01"/> </figure> <note position="left" xlink:label="note-0350-01" xlink:href="note-0350-01a" xml:space="preserve">30. huius.</note> <note position="left" xlink:label="note-0350-02" xlink:href="note-0350-02a" xml:space="preserve">Coroll. 1. <lb/>26. l. 2.</note> </div> </div> <div xml:id="echoid-div798" type="section" level="1" n="472"> <head xml:id="echoid-head492" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s8012" xml:space="preserve">_H_Inc colligimus omnia quadrata, BM, ad reſiduum, demptis ab <lb/>eiſdem omnibus quadratis figuræ, BFMH, eſſe vt quadratum, <lb/>EM, ad reſiduum, demptis à quadrato, EM, his omnibus .</s> <s xml:id="echoid-s8013" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8014" xml:space="preserve">quadrato, <lb/>FM, rectangulo ſub, MF, & </s> <s xml:id="echoid-s8015" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s8016" xml:space="preserve">FE, cum {1/6}. </s> <s xml:id="echoid-s8017" xml:space="preserve">quadrati, EF, hoc autem <lb/>reſiduum eſt rectangulum ſub, MF, & </s> <s xml:id="echoid-s8018" xml:space="preserve">ſexquitertia, FE, cum {5/@}. </s> <s xml:id="echoid-s8019" xml:space="preserve">qua-<lb/>drati, EF; </s> <s xml:id="echoid-s8020" xml:space="preserve">nam ſi à quadrato, EM, dempſeris quadratum, FM, rema-<lb/>nebunt duo rectangula ſub, MF, FE, cum quadratum, FE, vlterius ſi à <lb/>rectangulo bis ſub, MF, FE, dempſeris rectangulum ſub, MF, & </s> <s xml:id="echoid-s8021" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s8022" xml:space="preserve">F <pb o="331" file="0351" n="351" rhead="LIBER IV."/> E, ideſt rectangulum bis ſub, MF, & </s> <s xml:id="echoid-s8023" xml:space="preserve">ſub {1/3}. </s> <s xml:id="echoid-s8024" xml:space="preserve">FE, remanebitrectangu-<lb/>lum bis ſub, MF, & </s> <s xml:id="echoid-s8025" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s8026" xml:space="preserve">FE, idtſt ſemel ſub, MF, & </s> <s xml:id="echoid-s8027" xml:space="preserve">ſexquitertia, FE: <lb/></s> <s xml:id="echoid-s8028" xml:space="preserve">Tandem ablato {1/6}. </s> <s xml:id="echoid-s8029" xml:space="preserve">aquadrato, FE, remanent {5/6}. </s> <s xml:id="echoid-s8030" xml:space="preserve">eiuſdem quadrati, vnde <lb/>omnia quadrata, BM, ad reſiduum, demptis omnibus quadratis figuræ, <lb/>BFMH, erunt vt quadratum, EM, ad rectangulum ſub, MF, & </s> <s xml:id="echoid-s8031" xml:space="preserve">ſex-<lb/>quitertia, FE, cum {5/6}. </s> <s xml:id="echoid-s8032" xml:space="preserve">quadrati, FE.</s> <s xml:id="echoid-s8033" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div799" type="section" level="1" n="473"> <head xml:id="echoid-head493" xml:space="preserve">THEOREMA XXXIV. PROPOS. XXXVI.</head> <p> <s xml:id="echoid-s8034" xml:space="preserve">INeodem Schemate Theor. </s> <s xml:id="echoid-s8035" xml:space="preserve">Prop. </s> <s xml:id="echoid-s8036" xml:space="preserve">34. </s> <s xml:id="echoid-s8037" xml:space="preserve">oſtendemus omnia <lb/>quadrata figuræ, BDMH, demptis omnibus quadra-<lb/>tis, BM, ad omnia quadrata, BM, demptis omnibus qua-<lb/>dratis figuræ, BFMH, eſſe vt, EM, cum {1/2}. </s> <s xml:id="echoid-s8038" xml:space="preserve">EM, & </s> <s xml:id="echoid-s8039" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8040" xml:space="preserve">ED, <lb/>ad, MF, cum {1/3}. </s> <s xml:id="echoid-s8041" xml:space="preserve">MF, & </s> <s xml:id="echoid-s8042" xml:space="preserve">{5/6}. </s> <s xml:id="echoid-s8043" xml:space="preserve">FE.</s> <s xml:id="echoid-s8044" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8045" xml:space="preserve">Omnia enim quadrata, AM, ad omnia quadrata figuræ, DBH <lb/> <anchor type="figure" xlink:label="fig-0351-01a" xlink:href="fig-0351-01"/> M, oſtendimus eſſe, <lb/> <anchor type="note" xlink:label="note-0351-01a" xlink:href="note-0351-01"/> vt quadratum, D <lb/>M, ad quadratum, <lb/>ME, rectangulum <lb/>ſub, ME, & </s> <s xml:id="echoid-s8046" xml:space="preserve">ſexqui-<lb/>tertia, ED, cum {1/2}. <lb/></s> <s xml:id="echoid-s8047" xml:space="preserve">quadrati, ED, ea-<lb/>dem verò ad omnia <lb/>quadrata, BM, ſunt <lb/>vt quadratum, DM, ad quadratum, ME, ergo omnia quadrata, A <lb/>M, ad omnia quadrata figuræ, HBDM, demptis omnibus qua-<lb/>Matis, BM, erunt vt quadratum, DM, ad rectangulum ſub, ME, <lb/>& </s> <s xml:id="echoid-s8048" xml:space="preserve">ſexquitertia, ED, cum {1/2}. </s> <s xml:id="echoid-s8049" xml:space="preserve">quadrati, ED, &</s> <s xml:id="echoid-s8050" xml:space="preserve">, conuertendo, hæc <lb/>ad illa erunt, vt rectangulum ſub ſexquitertia, ED, cum {1/2}. </s> <s xml:id="echoid-s8051" xml:space="preserve">quadra-<lb/>ti, ED, ad quadratum, DM: </s> <s xml:id="echoid-s8052" xml:space="preserve">Omnia verò quadrata, AM, ad om-<lb/>nia quadrata. </s> <s xml:id="echoid-s8053" xml:space="preserve">BM, ſunt vt quadratum, DM, ad quadratum, ME, <lb/>& </s> <s xml:id="echoid-s8054" xml:space="preserve">tandem omnia quadrata, BM, adeorum reſiduum, demptis <lb/>omnibus quadratis figuræ, BHMF, ſunt vt quadratum, EM, <lb/> <anchor type="note" xlink:label="note-0351-02a" xlink:href="note-0351-02"/> ad rectangulum ſub, MF, & </s> <s xml:id="echoid-s8055" xml:space="preserve">ſexquitertia, FE, cum {5/6}. </s> <s xml:id="echoid-s8056" xml:space="preserve">quadrati, <lb/>FE, ergo, ex æquali, omnia quadrata figuræ, BDMH, <lb/>demptis omnibus quadratis, BM, ad omnia quadrata, BM, <lb/>demptis omnibus quadratis figuræ, BFMH, erunt vt rectangu-<lb/>lum ſub, ME, & </s> <s xml:id="echoid-s8057" xml:space="preserve">ſexquitertia, ED, cum {1/2}. </s> <s xml:id="echoid-s8058" xml:space="preserve">quadrati, ED, ad re-<lb/>ctangulum ſub, MF, & </s> <s xml:id="echoid-s8059" xml:space="preserve">ſexquitertia, FE, cum {5/6}. </s> <s xml:id="echoid-s8060" xml:space="preserve">quadrati, FE; <lb/></s> <s xml:id="echoid-s8061" xml:space="preserve">quia verò rectangulum ſub, ME, & </s> <s xml:id="echoid-s8062" xml:space="preserve">ſexquitertia, ED, æquaturre- <pb o="332" file="0352" n="352" rhead="GEOMETRIÆ"/> ctangulo ſub ſexquitertia, ME, & </s> <s xml:id="echoid-s8063" xml:space="preserve">ſub, ED, quia baſes eorum <lb/>ſunt altitudinibus reciprocę, & </s> <s xml:id="echoid-s8064" xml:space="preserve">eadem ratione rectangulum ſub ſex-<lb/>quitertia, EF, & </s> <s xml:id="echoid-s8065" xml:space="preserve">ſub, FM, æquatur rectãgulo ſub EF, & </s> <s xml:id="echoid-s8066" xml:space="preserve">ſexquitertia, <lb/>FM, ideò ſupradicta ratio erit eadem ei, quam habet rectangulũ ſub, <lb/>DE, vel, EF, & </s> <s xml:id="echoid-s8067" xml:space="preserve">ſub ſexquitertia, EM, cum {1/2}. </s> <s xml:id="echoid-s8068" xml:space="preserve">quadrati, DE, ideſt <lb/>cum rectangulo ſub, EF, & </s> <s xml:id="echoid-s8069" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8070" xml:space="preserve">EF, ad rectangulum ſub, EF, & </s> <s xml:id="echoid-s8071" xml:space="preserve">ſub <lb/>ſexquitertia, FM, cum {5/6}. </s> <s xml:id="echoid-s8072" xml:space="preserve">quadrati, EF, ideſt cum rectangulo ſub, <lb/>EF, & </s> <s xml:id="echoid-s8073" xml:space="preserve">{5/6}. </s> <s xml:id="echoid-s8074" xml:space="preserve">EF, duo autem rectangula ſub, EF, & </s> <s xml:id="echoid-s8075" xml:space="preserve">ſub ſexquitertia, <lb/>EM, & </s> <s xml:id="echoid-s8076" xml:space="preserve">ſub, EF, & </s> <s xml:id="echoid-s8077" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8078" xml:space="preserve">EF, conficiunt rectangulum ſub, EF, & </s> <s xml:id="echoid-s8079" xml:space="preserve">ſub <lb/>compoſita ex {1/2}. </s> <s xml:id="echoid-s8080" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8081" xml:space="preserve">ſexquitertia, EM; </s> <s xml:id="echoid-s8082" xml:space="preserve">pariter alia duo rectan-<lb/>gula ſub, EF, & </s> <s xml:id="echoid-s8083" xml:space="preserve">{5/6}. </s> <s xml:id="echoid-s8084" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8085" xml:space="preserve">ſub, EF, & </s> <s xml:id="echoid-s8086" xml:space="preserve">ſexquitertia, FM, conficiunt <lb/>rectangulum ſub, EF, & </s> <s xml:id="echoid-s8087" xml:space="preserve">compoſita ex {5/6}. </s> <s xml:id="echoid-s8088" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8089" xml:space="preserve">ſexquitertia, FM, <lb/>ergo omnia quadrata figuræ, BDMH, demptis omnibus quadra-<lb/>tis, BM, ad omnia quadrata, BM, demptis omnibus quadratis figu-<lb/> <anchor type="note" xlink:label="note-0352-01a" xlink:href="note-0352-01"/> ræ, BFMH; </s> <s xml:id="echoid-s8090" xml:space="preserve">erunt vt rectangulum ſub, EF, & </s> <s xml:id="echoid-s8091" xml:space="preserve">compoſita ex {1/2}. </s> <s xml:id="echoid-s8092" xml:space="preserve">E <lb/>F, & </s> <s xml:id="echoid-s8093" xml:space="preserve">ſexquitertia, EM, ad rectangulum ſub eadem altitudine, EF, <lb/>& </s> <s xml:id="echoid-s8094" xml:space="preserve">ſub compoſita ex {5/6}. </s> <s xml:id="echoid-s8095" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8096" xml:space="preserve">ſexquitertia, FM, .</s> <s xml:id="echoid-s8097" xml:space="preserve">i. </s> <s xml:id="echoid-s8098" xml:space="preserve">vt compoſita <lb/>ex {1/2}. </s> <s xml:id="echoid-s8099" xml:space="preserve">EF, vel {1/2}. </s> <s xml:id="echoid-s8100" xml:space="preserve">ED, & </s> <s xml:id="echoid-s8101" xml:space="preserve">ſexquitertia, EM, ad compoſitam ex {5/6}. </s> <s xml:id="echoid-s8102" xml:space="preserve">E <lb/>F, & </s> <s xml:id="echoid-s8103" xml:space="preserve">ſexquitertia, FM, ideſt vt, EM, cum {1/3}. </s> <s xml:id="echoid-s8104" xml:space="preserve">ME, & </s> <s xml:id="echoid-s8105" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8106" xml:space="preserve">ED, ad, M <lb/>F, cum {1/2}. </s> <s xml:id="echoid-s8107" xml:space="preserve">MF, & </s> <s xml:id="echoid-s8108" xml:space="preserve">{5/6}. </s> <s xml:id="echoid-s8109" xml:space="preserve">FE, quod oſtendere oportebat.</s> <s xml:id="echoid-s8110" xml:space="preserve"/> </p> <div xml:id="echoid-div799" type="float" level="2" n="1"> <figure xlink:label="fig-0351-01" xlink:href="fig-0351-01a"> <image file="0351-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0351-01"/> </figure> <note position="right" xlink:label="note-0351-01" xlink:href="note-0351-01a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-0351-02" xlink:href="note-0351-02a" xml:space="preserve">Ex corol. <lb/>autec.</note> <note position="left" xlink:label="note-0352-01" xlink:href="note-0352-01a" xml:space="preserve">1.2,elem.</note> </div> </div> <div xml:id="echoid-div801" type="section" level="1" n="474"> <head xml:id="echoid-head494" xml:space="preserve">THEOREMA XXXV. PROPOS. XXXVII.</head> <p> <s xml:id="echoid-s8111" xml:space="preserve">IN figura Prop. </s> <s xml:id="echoid-s8112" xml:space="preserve">32. </s> <s xml:id="echoid-s8113" xml:space="preserve">oſtendemus, regula, DF, omnia qua-<lb/>drata ſemiparabolæ, DBE, ad omnia quadrata ſiguræ, <lb/>CBDF, demptis omnibus quadratis trilinei, BCF, eſſe vt <lb/>octaua pars, DF, ad duas tertias eiuſdem, DF, .</s> <s xml:id="echoid-s8114" xml:space="preserve">i. </s> <s xml:id="echoid-s8115" xml:space="preserve">vt 3. </s> <s xml:id="echoid-s8116" xml:space="preserve">ad 16.</s> <s xml:id="echoid-s8117" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8118" xml:space="preserve">Nam omnia quadrata ſemiparabolæ, BDE, ſunt dimidium om. <lb/></s> <s xml:id="echoid-s8119" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0352-01a" xlink:href="fig-0352-01"/> nium quadratorum, AE, ideſt <lb/>ſunt ad illa, vt {1/2}. </s> <s xml:id="echoid-s8120" xml:space="preserve">quadrati, D <lb/> <anchor type="note" xlink:label="note-0352-02a" xlink:href="note-0352-02"/> E, ad quadratum, DE, item <lb/>omnia quadrata, AE, ad om-<lb/>nia quadrata, AF, ſunt vt qua-<lb/>dratum, DE, ad quadratum, <lb/>DF; </s> <s xml:id="echoid-s8121" xml:space="preserve">tandem omnia quadrata, <lb/>DF, ad omnia quadrata figurę, <lb/> <anchor type="note" xlink:label="note-0352-03a" xlink:href="note-0352-03"/> CBDF, demptis omnibus <lb/>quadratis trilinei, BCF, ſunt ſexquialtera, ideſt ſunt vt quadratũ, <lb/>DF, ad rectangulum ſub, DF, & </s> <s xml:id="echoid-s8122" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s8123" xml:space="preserve">DF, ergo, exæquali, omnia <pb o="333" file="0353" n="353" rhead="LIBER IV."/> quadrata ſemiparabolæ, BDC, ad omnia quadrata figuræ, CBD <lb/>F, demptis omnibus quadratis trilinei, BCF, erunt vt dimidium <lb/>quadrati, DE, .</s> <s xml:id="echoid-s8124" xml:space="preserve">i. </s> <s xml:id="echoid-s8125" xml:space="preserve">vt rectangulum ſub {1/8}. </s> <s xml:id="echoid-s8126" xml:space="preserve">DF, & </s> <s xml:id="echoid-s8127" xml:space="preserve">ſub, DF, ad rectã-<lb/>gulum ſub {2/3}. </s> <s xml:id="echoid-s8128" xml:space="preserve">DF, & </s> <s xml:id="echoid-s8129" xml:space="preserve">ſub, DF, .</s> <s xml:id="echoid-s8130" xml:space="preserve">i. </s> <s xml:id="echoid-s8131" xml:space="preserve">vt {1/8}. </s> <s xml:id="echoid-s8132" xml:space="preserve">DF, ad {1/3}. </s> <s xml:id="echoid-s8133" xml:space="preserve">DF, .</s> <s xml:id="echoid-s8134" xml:space="preserve">i. </s> <s xml:id="echoid-s8135" xml:space="preserve">vt {1/2} {3/4}. </s> <s xml:id="echoid-s8136" xml:space="preserve">D <lb/>F, ad {1/2} {6/4}. </s> <s xml:id="echoid-s8137" xml:space="preserve">DF, .</s> <s xml:id="echoid-s8138" xml:space="preserve">i. </s> <s xml:id="echoid-s8139" xml:space="preserve">vt 3. </s> <s xml:id="echoid-s8140" xml:space="preserve">ad 16. </s> <s xml:id="echoid-s8141" xml:space="preserve">quod oſtendere opus erat.</s> <s xml:id="echoid-s8142" xml:space="preserve"/> </p> <div xml:id="echoid-div801" type="float" level="2" n="1"> <figure xlink:label="fig-0352-01" xlink:href="fig-0352-01a"> <image file="0352-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0352-01"/> </figure> <note position="left" xlink:label="note-0352-02" xlink:href="note-0352-02a" xml:space="preserve">20. huius.</note> <note position="left" xlink:label="note-0352-03" xlink:href="note-0352-03a" xml:space="preserve">Corol. 32. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div803" type="section" level="1" n="475"> <head xml:id="echoid-head495" xml:space="preserve">THEOREMA XXXVI. PROP. XXXVIII.</head> <p> <s xml:id="echoid-s8143" xml:space="preserve">IN figura Prop. </s> <s xml:id="echoid-s8144" xml:space="preserve">34. </s> <s xml:id="echoid-s8145" xml:space="preserve">adhuc oſtendemus omnia quadrata <lb/>figuræ, HBDM, demptis omnibus quadratis figuræ, B <lb/>HMF, ad omnia quadrata figuræ, CBDF, demptis omni-<lb/>bus quadratis trilinei, BCF, eſſe vt, D/M, MF, ad, FD.</s> <s xml:id="echoid-s8146" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8147" xml:space="preserve">Quoniam .</s> <s xml:id="echoid-s8148" xml:space="preserve">n. </s> <s xml:id="echoid-s8149" xml:space="preserve">omnia quadrata, AM, demptis omnibus quadra-<lb/>tis, CM, ſunt ad omnia quadrata figuræ, HBDM, demptis om-<lb/> <anchor type="figure" xlink:label="fig-0353-01a" xlink:href="fig-0353-01"/> nibus quadratis figu-<lb/>ræ, BHMF, vt, AF, <lb/>ad parabolam, DBF, <lb/> <anchor type="note" xlink:label="note-0353-01a" xlink:href="note-0353-01"/> .</s> <s xml:id="echoid-s8150" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8151" xml:space="preserve">vt omnia quadrata, <lb/>A F, ad omnia quadra. <lb/></s> <s xml:id="echoid-s8152" xml:space="preserve">ta figuræ, CBDF, <lb/>demptis omnibus qua <lb/>dratis trilinei, BCF, <lb/>ergo, permutando, <lb/>omnia quadrata, A <lb/>M, demptis omnibus <lb/>quadratis, CM, ad omnia quadrata, AF, erunt vt omnia quadra-<lb/>ta figuræ, HBDM, demptis omnibus quadratis figuræ, HBFM, <lb/>ad omnia quadrata figuræ, CBDF, dem ptis omnibus quadratis <lb/>trilinei, BCF, ſunt autem omnia quadrata, AM, demptis omnibus <lb/>quadratis, CM, ad omnia quadrata, AF, vt rectangulum bis ſub, <lb/>MF, FD, cum quadrato, FD, .</s> <s xml:id="echoid-s8153" xml:space="preserve">i. </s> <s xml:id="echoid-s8154" xml:space="preserve">rectangulum ſub compoſita ex, <lb/>DM, MF, & </s> <s xml:id="echoid-s8155" xml:space="preserve">ſub; </s> <s xml:id="echoid-s8156" xml:space="preserve">FD, ad quadratum, FD, .</s> <s xml:id="echoid-s8157" xml:space="preserve">i. </s> <s xml:id="echoid-s8158" xml:space="preserve">vt compoſita ex, <lb/>DM, MF, ad, FD, ergo omnia quadrata figuræ, DM, HB, dem-<lb/>ptis omnibus quadratis figuræ CBDF, demptis omnibus quadra. </s> <s xml:id="echoid-s8159" xml:space="preserve"><lb/>tis trilinei, BCF, erunt vt, DM, MF, ad, FD, quod oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s8160" xml:space="preserve"/> </p> <div xml:id="echoid-div803" type="float" level="2" n="1"> <figure xlink:label="fig-0353-01" xlink:href="fig-0353-01a"> <image file="0353-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0353-01"/> </figure> <note position="right" xlink:label="note-0353-01" xlink:href="note-0353-01a" xml:space="preserve">34. huius.</note> </div> <pb o="334" file="0354" n="354" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div805" type="section" level="1" n="476"> <head xml:id="echoid-head496" xml:space="preserve">THEOREMA XXXVII. PROP. XXXIX.</head> <p> <s xml:id="echoid-s8161" xml:space="preserve">IN Schemate adhuc Prop. </s> <s xml:id="echoid-s8162" xml:space="preserve">antec. </s> <s xml:id="echoid-s8163" xml:space="preserve">oſtendemus omnia qua-<lb/>quadrata figuræ, HBDM, demptis omnibus quadratis <lb/>figuræ, BHMF, eſſe ad omnia quadrata ſemiparabolæ, B <lb/>DE, vt, DM, MF, ad @ {3/6}. </s> <s xml:id="echoid-s8164" xml:space="preserve">ipſius, FD.</s> <s xml:id="echoid-s8165" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8166" xml:space="preserve">Nam omnia quadrata figuræ, HBDM, demptis omnibus qua-<lb/>dratis figuræ, BHMF, ad omnia quadrata figuræ, CBDF, dem-<lb/> <anchor type="note" xlink:label="note-0354-01a" xlink:href="note-0354-01"/> ptis omnibus quadratis trilinei, BCF, oſtenſa ſunt eſſe, vt, DM, M <lb/>F, ad, FD, hæc autem ad omnia quadrata ſemiparabolæ, BDE, <lb/>ſunt vt {2/3}. </s> <s xml:id="echoid-s8167" xml:space="preserve">FD, ad {1/8}. </s> <s xml:id="echoid-s8168" xml:space="preserve">ipſius, FD, .</s> <s xml:id="echoid-s8169" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8170" xml:space="preserve">vt, FD, ad @ {3/6}. </s> <s xml:id="echoid-s8171" xml:space="preserve">FD, ergo, ex æ-<lb/>quali, omnia quadrata figuræ, HBDM, demptis omnibus qua-<lb/>dratis figuræ, BHMF, ad omnia quadrata ſemiparabolæ, BDE. <lb/></s> <s xml:id="echoid-s8172" xml:space="preserve">erunt vt, DM, MF, ad @ {3/6}. </s> <s xml:id="echoid-s8173" xml:space="preserve">ipſius, FD, quod oſtendere opus erat.</s> <s xml:id="echoid-s8174" xml:space="preserve"/> </p> <div xml:id="echoid-div805" type="float" level="2" n="1"> <note position="left" xlink:label="note-0354-01" xlink:href="note-0354-01a" xml:space="preserve">37. huius 1</note> </div> </div> <div xml:id="echoid-div807" type="section" level="1" n="477"> <head xml:id="echoid-head497" xml:space="preserve">THEOREMA XXXVIII. PROP. XL.</head> <p> <s xml:id="echoid-s8175" xml:space="preserve">SI in figuris Propoſ. </s> <s xml:id="echoid-s8176" xml:space="preserve">32. </s> <s xml:id="echoid-s8177" xml:space="preserve">& </s> <s xml:id="echoid-s8178" xml:space="preserve">34. </s> <s xml:id="echoid-s8179" xml:space="preserve">ducantur, GP, GV, regu-<lb/>lis, DF, DM, parallelæ, oſtendemus (ſi ipſę ſecauerint <lb/>parabolam, DBF,) omnia quadrata figuræ, CBDF, dem <lb/>ptis omnibus quadratis trilinei, BCF, ad omnia quadrata <lb/>figuræ, CBNP, demptis omnibus quadratis quadrilinei, <lb/>BCPO. </s> <s xml:id="echoid-s8180" xml:space="preserve">Vel omnia quadrata figuræ, HBDM, demptis <lb/>omnibus quadratis figuræ, BHMF, ad omnia quadrata fi-<lb/>guræ, HBNV, demptis omnibus quadratis figuræ, HVO <lb/>B, eſſe vt parabolam, DBF, ad parabolam, NBO.</s> <s xml:id="echoid-s8181" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8182" xml:space="preserve">Demonſtratio præſentis, Theor. </s> <s xml:id="echoid-s8183" xml:space="preserve">erit conformis demonſtratio. <lb/></s> <s xml:id="echoid-s8184" xml:space="preserve">nibus Prop. </s> <s xml:id="echoid-s8185" xml:space="preserve">19. </s> <s xml:id="echoid-s8186" xml:space="preserve">20. </s> <s xml:id="echoid-s8187" xml:space="preserve">Lib. </s> <s xml:id="echoid-s8188" xml:space="preserve">3. </s> <s xml:id="echoid-s8189" xml:space="preserve">quapropter inde petatur.</s> <s xml:id="echoid-s8190" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div808" type="section" level="1" n="478"> <head xml:id="echoid-head498" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s8191" xml:space="preserve">_H_Inc colligemus omnia quadrata ſiguræ, HBDM, demptis omni-<lb/>bus quadratis figuræ, BHMF, ad omnia quadrata figuræ, HBN <lb/>V, demptis omnibus quadratis figuræ, BHVO, eſſe vt omnia quadra-<lb/>ta figuræ, CBDF, demptis omnibus quadratis trilinei, BCF, ad omnia <pb o="335" file="0355" n="355" rhead="LIBER IV."/> quadrata figuræ, CBNP, demptis omnibus quadratis quadrilinei, B <lb/>CPO; </s> <s xml:id="echoid-s8192" xml:space="preserve">& </s> <s xml:id="echoid-s8193" xml:space="preserve">vtraque eſſe, vt cubum, DF. </s> <s xml:id="echoid-s8194" xml:space="preserve">ad cubum, NO.</s> <s xml:id="echoid-s8195" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">_Ex 2. hu-_ <lb/>_ius._</note> </div> <div xml:id="echoid-div809" type="section" level="1" n="479"> <head xml:id="echoid-head499" xml:space="preserve">THEOREMA XXXIX. PROPOS. XLI</head> <p> <s xml:id="echoid-s8196" xml:space="preserve">INeiſdem figuris oſtendemus, regulis adhuc ipſis, DM, <lb/>DF, omnia quadrata figuræ, CBDF, ad omnia quadra-<lb/>ta figuræ, CBNP, eſſe vt parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s8197" xml:space="preserve"><lb/>{1/2} {1/2}. </s> <s xml:id="echoid-s8198" xml:space="preserve">quadrati ipſius, DF, ad parallelepipedum ſub, BX, & </s> <s xml:id="echoid-s8199" xml:space="preserve"><lb/>his ſpatijs ſimul compoſitis .</s> <s xml:id="echoid-s8200" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8201" xml:space="preserve">quadrato, XP: </s> <s xml:id="echoid-s8202" xml:space="preserve">1. </s> <s xml:id="echoid-s8203" xml:space="preserve">quadrati, <lb/>NX, & </s> <s xml:id="echoid-s8204" xml:space="preserve">rectangulo ſub ſexquitertia, NX, & </s> <s xml:id="echoid-s8205" xml:space="preserve">ſub, XP; </s> <s xml:id="echoid-s8206" xml:space="preserve">Om-<lb/>nia verò quadrata figuræ, HBDM, ad omnia quadrata fi-<lb/>guræ, HBNV, eſſe vt parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s8207" xml:space="preserve">his <lb/>ſpatijs .</s> <s xml:id="echoid-s8208" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8209" xml:space="preserve">quadrato, ME, 1. </s> <s xml:id="echoid-s8210" xml:space="preserve">quadrati, ED, & </s> <s xml:id="echoid-s8211" xml:space="preserve">rectangulo <lb/>ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s8212" xml:space="preserve">ſub, EM, ad parallelepipedum ſub, <lb/>BX, & </s> <s xml:id="echoid-s8213" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8214" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8215" xml:space="preserve">quadrato, VX, 1. </s> <s xml:id="echoid-s8216" xml:space="preserve">quadrati, XN, & </s> <s xml:id="echoid-s8217" xml:space="preserve">re-<lb/>ctangulo, ſub ſexquitertia, NX, & </s> <s xml:id="echoid-s8218" xml:space="preserve">ſub, XV.</s> <s xml:id="echoid-s8219" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8220" xml:space="preserve">Ducatur per, N, ipſi, BE, parallela, NQ, in vtraq; </s> <s xml:id="echoid-s8221" xml:space="preserve">figura, igi-<lb/> <anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/> tur omnia quadrata <lb/>figuræ, CBDF, ad <lb/>omnia quadrata figu-<lb/>ræ, CBNP, habẽt ra-<lb/>tionem compoſitam <lb/>ex ea, quam habent <lb/>omnia quadrata figu-<lb/>ræ, CBDF, ad om-<lb/>nia quadrata, AF, .</s> <s xml:id="echoid-s8222" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s8223" xml:space="preserve">ex ea, quam habent <lb/>{3/2} {7/4}. </s> <s xml:id="echoid-s8224" xml:space="preserve">quadrati, DF, ad <lb/>quadratum, DF, & </s> <s xml:id="echoid-s8225" xml:space="preserve">ex ratione omnium quadratorum, AF, ad om-<lb/> <anchor type="note" xlink:label="note-0355-02a" xlink:href="note-0355-02"/> nia quadrata, AP, .</s> <s xml:id="echoid-s8226" xml:space="preserve">i. </s> <s xml:id="echoid-s8227" xml:space="preserve">ex ratione, EB, ad, BX, & </s> <s xml:id="echoid-s8228" xml:space="preserve">ex ratione om-<lb/>nium quadratorum, AP, ad omnia quadrata, QP, .</s> <s xml:id="echoid-s8229" xml:space="preserve">i. </s> <s xml:id="echoid-s8230" xml:space="preserve">ex ratione <lb/>quadrati, GP, vel quadrati, DF, ad quadratum, PN, & </s> <s xml:id="echoid-s8231" xml:space="preserve">tandem <lb/>ex ratione omnium quadratorum, QP, ad omnia quadrata figurę, C <lb/> <anchor type="note" xlink:label="note-0355-03a" xlink:href="note-0355-03"/> BNP, .</s> <s xml:id="echoid-s8232" xml:space="preserve">i. </s> <s xml:id="echoid-s8233" xml:space="preserve">ex ratione quadrati, NP, ad quadratum, PX, cum {1/2}. </s> <s xml:id="echoid-s8234" xml:space="preserve">quadra-<lb/>ti, XN, & </s> <s xml:id="echoid-s8235" xml:space="preserve">cum rectâgulo ſub ſexquitertia, NX, & </s> <s xml:id="echoid-s8236" xml:space="preserve">ſub, XP, harum au-<lb/>tem rat onum iſtæ .</s> <s xml:id="echoid-s8237" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8238" xml:space="preserve">quam habent {1/2} {7/4}. </s> <s xml:id="echoid-s8239" xml:space="preserve">quadrati, DF, ad quadratum, <lb/>DF, quadatum, DF, ad quadratum, NP, & </s> <s xml:id="echoid-s8240" xml:space="preserve">quadratum, NP, ad hęc <lb/>ſimul .</s> <s xml:id="echoid-s8241" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8242" xml:space="preserve">quadratum, PX, @. </s> <s xml:id="echoid-s8243" xml:space="preserve">quadrati, NX, & </s> <s xml:id="echoid-s8244" xml:space="preserve">rectangulum ſub ſex.</s> <s xml:id="echoid-s8245" xml:space="preserve"> <pb o="336" file="0356" n="356" rhead="GEOMETRIÆ"/> quitertia, NX, & </s> <s xml:id="echoid-s8246" xml:space="preserve">ſub, XP, conficiunt rationem @ {7/4}. </s> <s xml:id="echoid-s8247" xml:space="preserve">quadrati, DF, <lb/>ad hæc ſpatia vltimo dicta, hæc vero ratio, cum ea, quam habet, E <lb/>B, ad BX, conficit rationem parallelepidi ſub, BE, & </s> <s xml:id="echoid-s8248" xml:space="preserve">{1/2} {7/4}. </s> <s xml:id="echoid-s8249" xml:space="preserve">quadra-<lb/>ti, DF, ad parallelepipedum ſub, BX, & </s> <s xml:id="echoid-s8250" xml:space="preserve">dictis ſpatijs vltimò dictis, <lb/>ſcilicet quadrato, PX, {1/2}, quadrati, NX, & </s> <s xml:id="echoid-s8251" xml:space="preserve">rectangulo ſub ſexquiter-<lb/>nia, NX, & </s> <s xml:id="echoid-s8252" xml:space="preserve">ſub, XP, ergo omnia quadrata figurę, CBDF, ad om-<lb/>nia quadrata figuræ, CBNP, erunt vt parallelepipedum ſub, BE, & </s> <s xml:id="echoid-s8253" xml:space="preserve"><lb/>{1/2} {7/8}. </s> <s xml:id="echoid-s8254" xml:space="preserve">quadrati, DF, ad parallelepipedum ſub, BX, & </s> <s xml:id="echoid-s8255" xml:space="preserve">dictis ſpatijs <lb/>vltimo dictis.</s> <s xml:id="echoid-s8256" xml:space="preserve"/> </p> <div xml:id="echoid-div809" type="float" level="2" n="1"> <figure xlink:label="fig-0355-01" xlink:href="fig-0355-01a"> <image file="0355-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0355-01"/> </figure> <note position="right" xlink:label="note-0355-02" xlink:href="note-0355-02a" xml:space="preserve">32. huius</note> <note position="right" xlink:label="note-0355-03" xlink:href="note-0355-03a" xml:space="preserve">34. huius.</note> </div> <p> <s xml:id="echoid-s8257" xml:space="preserve">Eadem methodo compoſitionis proportionum, ſumptis medijs <lb/>omnibus quadratis, AM, AV, QV, inter omnia quadrata figura-<lb/>rum, HBDM, HBNV, oſtendemus parjter omnia quadrata fi-<lb/>guræ, HBDM, ad omnia quadrata figuræ, HBNV, eſſe vt pa-<lb/>rallelepipedum ſub, BE, & </s> <s xml:id="echoid-s8258" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8259" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8260" xml:space="preserve">quadrato, ME, {1/2}, quadra-<lb/>ti, ED, & </s> <s xml:id="echoid-s8261" xml:space="preserve">rectangulo ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s8262" xml:space="preserve">ſub, EM, ad paral-<lb/>lelepipedum ſub, BX, & </s> <s xml:id="echoid-s8263" xml:space="preserve">ſub his ſpatijs, .</s> <s xml:id="echoid-s8264" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8265" xml:space="preserve">quadrato, VX, {1/2}. </s> <s xml:id="echoid-s8266" xml:space="preserve">qua-<lb/>drati, XN, & </s> <s xml:id="echoid-s8267" xml:space="preserve">rectangulo ſub ſexquitertia, XN, & </s> <s xml:id="echoid-s8268" xml:space="preserve">ſub, XV, quæ <lb/>erant nobis oſtendenda.</s> <s xml:id="echoid-s8269" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div811" type="section" level="1" n="480"> <head xml:id="echoid-head500" xml:space="preserve">THEOREMA XL. PROPOS. XLII.</head> <p> <s xml:id="echoid-s8270" xml:space="preserve">SI intra parabolam axi, vel diametro eiuſdem parallela <lb/>ducatur recta linea in curuam, & </s> <s xml:id="echoid-s8271" xml:space="preserve">baſim parabolæ ter-<lb/>minata, quæ baſis ſumatur pro regula, ducta verò tangente <lb/>parabolam intermino dicti axis, vel diametri, & </s> <s xml:id="echoid-s8272" xml:space="preserve">producta <lb/>dicta parallela vſque ad ipſam, compleatur parallelogram-<lb/>mum ſub ipſa, & </s> <s xml:id="echoid-s8273" xml:space="preserve">baſis maiori portione: </s> <s xml:id="echoid-s8274" xml:space="preserve">Omnia quadrata <lb/>conſtituti parallelogrammi ad omnia quadrata reſiduæ fi-<lb/>guræ eodem iincluſæ parallelogrammo, ab eodem dempto <lb/>trilineo extra ſemiparabolam facto, erunt vt quadratum ba-<lb/>ſis dicti fruſti ad quadratum reſidui eiuſdem baſi, dempta <lb/>ab eadem dimidia baſis totius parabolæ, ſimul cum {1/2}. <lb/></s> <s xml:id="echoid-s8275" xml:space="preserve">quadrati huius dimidiæ, & </s> <s xml:id="echoid-s8276" xml:space="preserve">rectangulo ſub ſexquitertia ta-<lb/>lis dimidiæ, & </s> <s xml:id="echoid-s8277" xml:space="preserve">eodem baſis reſiduo iam dicto.</s> <s xml:id="echoid-s8278" xml:space="preserve"/> </p> <pb o="337" file="0357" n="357" rhead="LIBER IV."/> <p> <s xml:id="echoid-s8279" xml:space="preserve">Sit ergo parabola, HBM, cuius axis, vel diameter, BG, baſis, <lb/> <anchor type="figure" xlink:label="fig-0357-01a" xlink:href="fig-0357-01"/> HM, ducatur autem intra ipſam <lb/>eidem, BG, parallela, EF, ducta <lb/>verò tangente, AC, in termino, <lb/>B, quæ erit parallela baſi, HF, pro-<lb/>ducatur verſus, FE, illi productæ <lb/>occurrens in, C, & </s> <s xml:id="echoid-s8280" xml:space="preserve">compleatur pa-<lb/>rallelogrammum, AF, regula ve-<lb/>rò ſit, HM. </s> <s xml:id="echoid-s8281" xml:space="preserve">Dico ergo omnia qua-<lb/>drata, AF, ad omnia quadrata fi-<lb/>guræ, CBHF, eſſe vt quadratum, HF, ad quadratum, FG, {1/2}. </s> <s xml:id="echoid-s8282" xml:space="preserve">qua-<lb/>dtati, GH, & </s> <s xml:id="echoid-s8283" xml:space="preserve">rectangulum ſub ſexquitertia, HG, & </s> <s xml:id="echoid-s8284" xml:space="preserve">ſub, GF. </s> <s xml:id="echoid-s8285" xml:space="preserve">Hęc <lb/>autem erit conſimilis demonſtrationi ſecundæ partis Theor. </s> <s xml:id="echoid-s8286" xml:space="preserve">32. <lb/></s> <s xml:id="echoid-s8287" xml:space="preserve">ideo inde colligatur.</s> <s xml:id="echoid-s8288" xml:space="preserve"/> </p> <div xml:id="echoid-div811" type="float" level="2" n="1"> <figure xlink:label="fig-0357-01" xlink:href="fig-0357-01a"> <image file="0357-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0357-01"/> </figure> </div> </div> <div xml:id="echoid-div813" type="section" level="1" n="481"> <head xml:id="echoid-head501" xml:space="preserve">THEOREMA XLI. PROPOS. XLIII.</head> <p> <s xml:id="echoid-s8289" xml:space="preserve">IN eadem anteced. </s> <s xml:id="echoid-s8290" xml:space="preserve">Propoſit. </s> <s xml:id="echoid-s8291" xml:space="preserve">figura oſtendemus omnia <lb/>quadrata, AF, ad omnia quadrata figuræ, CBHF, dem-<lb/>ptis omnibus quadratis trilinei, BCE, eſſe vt parallelepi-<lb/>peduw ſub, BG, & </s> <s xml:id="echoid-s8292" xml:space="preserve">quadrato, HF, ad reliquum parallelepi-<lb/>pedi ſub, BG, & </s> <s xml:id="echoid-s8293" xml:space="preserve">his ſpatis .</s> <s xml:id="echoid-s8294" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8295" xml:space="preserve">quadrato, FG, {1/2}. </s> <s xml:id="echoid-s8296" xml:space="preserve">quadrati, <lb/>GH, & </s> <s xml:id="echoid-s8297" xml:space="preserve">rectangulo ſub ſexquitertia, HG, & </s> <s xml:id="echoid-s8298" xml:space="preserve">ſub, GF, ab eo-<lb/>dem dempto {1/3}. </s> <s xml:id="echoid-s8299" xml:space="preserve">parallelepipedi ſub, CE, & </s> <s xml:id="echoid-s8300" xml:space="preserve">quadrato, FG.</s> <s xml:id="echoid-s8301" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8302" xml:space="preserve">Nam omnia quadrata, AF, ad omnia quadrata, BF, ducta per, <lb/> <anchor type="note" xlink:label="note-0357-01a" xlink:href="note-0357-01"/> E, ipſa, EI, æquidiſtans, HM, ſunt vt parallelepipedum ſub, AH, <lb/>& </s> <s xml:id="echoid-s8303" xml:space="preserve">quadrato, HF, ad parallelepipedum ſub, BI, & </s> <s xml:id="echoid-s8304" xml:space="preserve">quadrato, IE, <lb/>ſuut autem omnia quadrata, BE, ſexcupla ommum quadratorum <lb/> <anchor type="note" xlink:label="note-0357-02a" xlink:href="note-0357-02"/> trilinei, BCE, ideò omnia quadrata, AF, ad omnia quadrata tri-<lb/> <anchor type="figure" xlink:label="fig-0357-02a" xlink:href="fig-0357-02"/> linei, BCE, erunt vt parallelepi-<lb/>pedum ſub, AH, vel, BG, & </s> <s xml:id="echoid-s8305" xml:space="preserve">ſub <lb/>quadrato, HF, ad parallelepipedi <lb/>ſub, BI, & </s> <s xml:id="echoid-s8306" xml:space="preserve">quadrato, IE, ſextam <lb/>partem: </s> <s xml:id="echoid-s8307" xml:space="preserve">Quia verò omnia quadra-<lb/>ta, AF, ad omnia quadrata figuræ, <lb/> <anchor type="note" xlink:label="note-0357-03a" xlink:href="note-0357-03"/> CBHF, ſunt vt quadratum, HF, <lb/>ad hæc ſpatia .</s> <s xml:id="echoid-s8308" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8309" xml:space="preserve">quadratum FG, <lb/>{1/2}. </s> <s xml:id="echoid-s8310" xml:space="preserve">quadrati, HG, & </s> <s xml:id="echoid-s8311" xml:space="preserve">rectangulum <lb/>ſub ſexquitertia, HG, & </s> <s xml:id="echoid-s8312" xml:space="preserve">ſub, GF, .</s> <s xml:id="echoid-s8313" xml:space="preserve">i. </s> <s xml:id="echoid-s8314" xml:space="preserve">ſumpta, BG, communi alti-<lb/>tudine, vt parallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8315" xml:space="preserve">quadrato, HF, ad paral- <pb o="338" file="0358" n="358" rhead="GEOMETRIÆ"/> lelepipedum ſub, BG, & </s> <s xml:id="echoid-s8316" xml:space="preserve">dicti, ſpatijs, ideò omnia quadrata, AF, <lb/>ad omnia quadrata figuræ, CBHF, demptis omnibus quadratis <lb/>trilinei, BGE, erunt vt parallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8317" xml:space="preserve">quadrato, H <lb/>F, ad reſiduum, dempta ſexta parte parallelepipedi ſub, BI, vel, C <lb/>E, exceſſu, BG, ſuper, EF, & </s> <s xml:id="echoid-s8318" xml:space="preserve">ſub quadrato, IE, à parallelepipedo <lb/>ſub, BG, & </s> <s xml:id="echoid-s8319" xml:space="preserve">dictis ſpatijs .</s> <s xml:id="echoid-s8320" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8321" xml:space="preserve">quadrato, FG, {1/2}. </s> <s xml:id="echoid-s8322" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s8323" xml:space="preserve">re-<lb/>ctangulo ſub ſexquitertia, HG, & </s> <s xml:id="echoid-s8324" xml:space="preserve">ſub, GF.</s> <s xml:id="echoid-s8325" xml:space="preserve"/> </p> <div xml:id="echoid-div813" type="float" level="2" n="1"> <note position="right" xlink:label="note-0357-01" xlink:href="note-0357-01a" xml:space="preserve">11.l. 2.</note> <note position="right" xlink:label="note-0357-02" xlink:href="note-0357-02a" xml:space="preserve">30. huius.</note> <figure xlink:label="fig-0357-02" xlink:href="fig-0357-02a"> <image file="0357-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0357-02"/> </figure> <note position="right" xlink:label="note-0357-03" xlink:href="note-0357-03a" xml:space="preserve">24. huius.</note> </div> </div> <div xml:id="echoid-div815" type="section" level="1" n="482"> <head xml:id="echoid-head502" xml:space="preserve">THEOREMA XLII. PROPOS. XLIV.</head> <p> <s xml:id="echoid-s8326" xml:space="preserve">INeadem figura Prop. </s> <s xml:id="echoid-s8327" xml:space="preserve">42. </s> <s xml:id="echoid-s8328" xml:space="preserve">ducta intra fruſtum parabolæ, <lb/>EBHF, recta, VR, parallela baſi, HM, oſtendemus <lb/>omnia quadrata figuræ, CBHF, ad omnia quadrata figu-<lb/>ræ, CBRV, eſſe vt parallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8329" xml:space="preserve">his ſpa-<lb/>tijs .</s> <s xml:id="echoid-s8330" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8331" xml:space="preserve">quadrato, FG, {1/2}. </s> <s xml:id="echoid-s8332" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s8333" xml:space="preserve">rectangulo ſub <lb/>ſexquitertia, HG, & </s> <s xml:id="echoid-s8334" xml:space="preserve">ſub, GF, ad parallelepipedum ſub, B <lb/>S, & </s> <s xml:id="echoid-s8335" xml:space="preserve">ſub his ſpatijs, ſcilicet quadrato, VS, 1. </s> <s xml:id="echoid-s8336" xml:space="preserve">quadrati, SR, <lb/>& </s> <s xml:id="echoid-s8337" xml:space="preserve">rectangulo ſub ſexquitertia, RS, & </s> <s xml:id="echoid-s8338" xml:space="preserve">ſub, SV.</s> <s xml:id="echoid-s8339" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8340" xml:space="preserve">Huius demonſtratio non eſt alia à demonſtratione Propoſ. </s> <s xml:id="echoid-s8341" xml:space="preserve">41. <lb/></s> <s xml:id="echoid-s8342" xml:space="preserve">ideò ibi in ſecunda eiuſdem parte recolatur.</s> <s xml:id="echoid-s8343" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div816" type="section" level="1" n="483"> <head xml:id="echoid-head503" xml:space="preserve">THEOREMA XLIII. PROP. XLV.</head> <p> <s xml:id="echoid-s8344" xml:space="preserve">INeodem Propoſ. </s> <s xml:id="echoid-s8345" xml:space="preserve">42. </s> <s xml:id="echoid-s8346" xml:space="preserve">Schemate oſtendemus omnia qua-<lb/>drata figuræ, CBHF, demptis omnibus qua dratis trili-<lb/>nei, BCE, ad omnia quadrata ſemiparabolæ, BHG, eſſe <lb/>vt reliquum parallelepipedi ſub, BG, & </s> <s xml:id="echoid-s8347" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8348" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8349" xml:space="preserve">qua-<lb/>drato, FG, {1/2}. </s> <s xml:id="echoid-s8350" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s8351" xml:space="preserve">rectangulo ſub, FG, & </s> <s xml:id="echoid-s8352" xml:space="preserve">ſex-<lb/>quitertia, GH, ab eodem dempta ſexta parte parallelepi-<lb/>pediſub, CE, & </s> <s xml:id="echoid-s8353" xml:space="preserve">quadrato, FG, ad dimidium parallelepi-<lb/>pediſub, BG, & </s> <s xml:id="echoid-s8354" xml:space="preserve">quadrato, GH.</s> <s xml:id="echoid-s8355" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8356" xml:space="preserve">Etenim omnia quadrata figuræ, CBHF, demptis omnibus <lb/>quadratis trilinei, BCE, ad omnia quadrata, AF, conuertendo, <lb/>ſunt vt parallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8357" xml:space="preserve">his ſpatijs, ſcilicet quadrato. <lb/></s> <s xml:id="echoid-s8358" xml:space="preserve">FG, {1/2}. </s> <s xml:id="echoid-s8359" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s8360" xml:space="preserve">rectangulo ſub, FG, & </s> <s xml:id="echoid-s8361" xml:space="preserve">ſexquitertia, G <lb/>H, ab eodem dempto {1/6}. </s> <s xml:id="echoid-s8362" xml:space="preserve">parallelepipedi ſub, CE, & </s> <s xml:id="echoid-s8363" xml:space="preserve">quadrato, F <pb o="339" file="0359" n="359" rhead="LIBER IV."/> G, ad parallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8364" xml:space="preserve">quadrato, HF; </s> <s xml:id="echoid-s8365" xml:space="preserve">item omnia <lb/>quadrata, AF, ad omnia quadrata, AG, ſunt vt quadratum, FH, <lb/>ad quadratum, HG, .</s> <s xml:id="echoid-s8366" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8367" xml:space="preserve">ſumpta, BG, communi altitudine, vt pa-<lb/>rallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8368" xml:space="preserve">quadrato, FH, ad parallelepipedum <lb/>ſub, BG, & </s> <s xml:id="echoid-s8369" xml:space="preserve">quadrato, HG: </s> <s xml:id="echoid-s8370" xml:space="preserve">_Tandem omnia quadrata,_ AG, dupla <lb/>ſunt omnium quadratorum ſemiparabolæ, BHG, ergo, ex æquali, <lb/>omnia quadrata figuræ, CBHF, demptis omnibus quadratis trili-<lb/>nei, BCE, ad omnia quadrata ſemiparabolæ, BHG, erunt vt pa-<lb/>rallelepipedum ſub, BG, & </s> <s xml:id="echoid-s8371" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8372" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8373" xml:space="preserve">quadrato, FG, {1/2}. </s> <s xml:id="echoid-s8374" xml:space="preserve">quadra-<lb/>ti, GH, & </s> <s xml:id="echoid-s8375" xml:space="preserve">rectangulo ſub, FG, & </s> <s xml:id="echoid-s8376" xml:space="preserve">ſexquitertia, GH, ab eodem <lb/>dempto {1/6}. </s> <s xml:id="echoid-s8377" xml:space="preserve">parallelepipedi ſub, CE, & </s> <s xml:id="echoid-s8378" xml:space="preserve">quadrato, FG, ad dimidium <lb/>parallelepipedi ſub, BG, & </s> <s xml:id="echoid-s8379" xml:space="preserve">quadrato, GH, quod erat demonſtran-<lb/>dum.</s> <s xml:id="echoid-s8380" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div817" type="section" level="1" n="484"> <head xml:id="echoid-head504" xml:space="preserve">THEOREMA XLIV. PROP. XLVI.</head> <p> <s xml:id="echoid-s8381" xml:space="preserve">IN parabola ducta axi, vel diametro æquidiſtanter rect@ <lb/>linea, ſi deinde fiat parallelogrammum ſub eadem du-<lb/>cta, & </s> <s xml:id="echoid-s8382" xml:space="preserve">ſub baſi, angulum habens æqualem angulo i<gap/> <lb/>tionis eiuſdem ductæ ad baſim, regula ſumpta baſi. </s> <s xml:id="echoid-s8383" xml:space="preserve">Re<gap/> <lb/>gula ſub parallelogram<gap/>, in quæ dictum parallelogram-<lb/>mum diuiditur à ducta linea, ſunt dupla rectangulorum <lb/>ſub portionibus fruſti parabolæ, dicto parallelogrammo in-<lb/>cluſæ, per eandem ductam conſtituris.</s> <s xml:id="echoid-s8384" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8385" xml:space="preserve">Sit parabola, AZG, in baſi, ZG, circa axim, vel diametrum, <lb/> <anchor type="figure" xlink:label="fig-0359-01a" xlink:href="fig-0359-01"/> AQ, cui parallela ducatur vt-<lb/>cumque recta, DP, fiat autem <lb/>parallelogrammum ſub, ZQ, <lb/>DP, angulum habens æqualẽ <lb/>angulo inclinationis, DP, ad <lb/>ZG, .</s> <s xml:id="echoid-s8386" xml:space="preserve">i. </s> <s xml:id="echoid-s8387" xml:space="preserve">angulo, qui ſit, DPG, <lb/>vtcunque exduobus, DPG, <lb/>DPZ, ſit autem hoc paralle. <lb/></s> <s xml:id="echoid-s8388" xml:space="preserve">logrammum, HG, regula ve. </s> <s xml:id="echoid-s8389" xml:space="preserve"><lb/>ro, HG. </s> <s xml:id="echoid-s8390" xml:space="preserve">Dico ergo, rectãgula <lb/>ſub, HP, PE, dupla eſſe rectãgulorũ ſub portionibus, BDPZ, DGP. </s> <s xml:id="echoid-s8391" xml:space="preserve"><lb/>Sumpto ergo vtcunq; </s> <s xml:id="echoid-s8392" xml:space="preserve">in, DP, puncto, T, per, T, ducatur, RF, ipſi, <lb/>ZG, æquidiſtans ſecanſq; </s> <s xml:id="echoid-s8393" xml:space="preserve">curuam parabolæ in, SI, &</s> <s xml:id="echoid-s8394" xml:space="preserve">, AQ, in, O. </s> <s xml:id="echoid-s8395" xml:space="preserve"><lb/>Rectangulum ergo, ZPQ, ad rectangulum, STI, habet rationem <pb o="340" file="0360" n="360" rhead="GEOMETRIÆ"/> compoſitam ex ea, quam habet rectangulum, ZPG, ad rectangu-<lb/>lum, ZQG, ideſt ex ea, quam habet, DP, ad, AQ, & </s> <s xml:id="echoid-s8396" xml:space="preserve">ex ra-<lb/>tione rectanguli, ZQG, ad rectangulum, SOI, vel quadra-<lb/>ti, QG, ad quadratum, OI, ideſt ex ea, quam habet, QA, ad, <lb/> <anchor type="note" xlink:label="note-0360-01a" xlink:href="note-0360-01"/> AO, & </s> <s xml:id="echoid-s8397" xml:space="preserve">ex ratione rectanguli, SOI, ad rectangulum, STI, ideſt <lb/>ex ratione, AO, ad, DT, ergo rectangulum, ZPG, vel, RTF, ad <lb/>rectangulum, STI, erit vt, PD, ad DT, abſciſſam. </s> <s xml:id="echoid-s8398" xml:space="preserve">Et quoniam, <lb/> <anchor type="note" xlink:label="note-0360-02a" xlink:href="note-0360-02"/> HG, eſt parallelogrammum in eadem baſi, & </s> <s xml:id="echoid-s8399" xml:space="preserve">altitudine cum fruſto, <lb/>BZGD, & </s> <s xml:id="echoid-s8400" xml:space="preserve">per punctum, T, vtcunq. </s> <s xml:id="echoid-s8401" xml:space="preserve">ſumptum ducta, BP, regulæ <lb/>parallela, quę eſt baſis, ZG, inuentũ eſt rectangulũ BTP, ad rectan-<lb/>gulũ, STI, eſſe vt, PD, ad DT; </s> <s xml:id="echoid-s8402" xml:space="preserve">quatuor ergo horum magnitudinum <lb/> <anchor type="note" xlink:label="note-0360-03a" xlink:href="note-0360-03"/> ordinibus conſtructis, iuxta has quatuor magnitudines, quę inuentę <lb/>ſunt eſſe proportionales, & </s> <s xml:id="echoid-s8403" xml:space="preserve">hoc modo ſolito, reperimus rectangula <lb/>ſub, HP, PE, ad rectangula ſub portionibus, BZPD, DGP, eſſe vt <lb/>maximę abſciſſarum, DP, ad omnes abſciſſas, DP, recti, vel eiuſdẽ <lb/> <anchor type="note" xlink:label="note-0360-04a" xlink:href="note-0360-04"/> obliqui tranſitus .</s> <s xml:id="echoid-s8404" xml:space="preserve">i. </s> <s xml:id="echoid-s8405" xml:space="preserve">eſſe eorum dupla, quod oſtendere opus erat.</s> <s xml:id="echoid-s8406" xml:space="preserve"/> </p> <div xml:id="echoid-div817" type="float" level="2" n="1"> <figure xlink:label="fig-0359-01" xlink:href="fig-0359-01a"> <image file="0359-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0359-01"/> </figure> <note position="left" xlink:label="note-0360-01" xlink:href="note-0360-01a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0360-02" xlink:href="note-0360-02a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0360-03" xlink:href="note-0360-03a" xml:space="preserve">Yux. Cor. <lb/>3. 26.l. 2.</note> <note position="left" xlink:label="note-0360-04" xlink:href="note-0360-04a" xml:space="preserve">Corol. 2. <lb/><gap/>2.</note> </div> </div> <div xml:id="echoid-div819" type="section" level="1" n="485"> <head xml:id="echoid-head505" xml:space="preserve">THEOREMA XLV. PROP. XLVII.</head> <p> <s xml:id="echoid-s8407" xml:space="preserve">IN anteced. </s> <s xml:id="echoid-s8408" xml:space="preserve">figura oſtendemus, regula eadem, ZG, omnia <lb/>quadrata, DG, ad omnia quadrata, DPG, eſſe vt, ZP, <lb/>ad compoſitam ex {1/3}. </s> <s xml:id="echoid-s8409" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8410" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8411" xml:space="preserve">PCOmnia verò quadrata, <lb/>DC, ad omnia quadrata trilinei, DGE, eſſe vt, ZP, ad ſui <lb/>reliquum, demptis ab eadem {2/3}. </s> <s xml:id="echoid-s8412" xml:space="preserve">ZP, cum {1/6}, PG.</s> <s xml:id="echoid-s8413" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8414" xml:space="preserve">Rectangula enim ſub, HP, PE, adrectangula ſub, HP, & </s> <s xml:id="echoid-s8415" xml:space="preserve">por-<lb/> <anchor type="note" xlink:label="note-0360-05a" xlink:href="note-0360-05"/> tione, DPG, ſunt vt, EP, ad portionem, DPG, .</s> <s xml:id="echoid-s8416" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8417" xml:space="preserve">vt, ZP, ad <lb/>compoſitam ex {1/4}. </s> <s xml:id="echoid-s8418" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8419" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8420" xml:space="preserve">PG; </s> <s xml:id="echoid-s8421" xml:space="preserve">eadem autem rectangula ſub, H <lb/>P, PE, ſunt dupla rectangulorum ſub portionibus, DBZP, DP <lb/> <anchor type="note" xlink:label="note-0360-06a" xlink:href="note-0360-06"/> G, .</s> <s xml:id="echoid-s8422" xml:space="preserve">i. </s> <s xml:id="echoid-s8423" xml:space="preserve">ſunt ad illa, vt, ZP, ad {1/2}. </s> <s xml:id="echoid-s8424" xml:space="preserve">ZP, ergo ad reſiduum rectangulo-<lb/> <anchor type="note" xlink:label="note-0360-07a" xlink:href="note-0360-07"/> <anchor type="figure" xlink:label="fig-0360-01a" xlink:href="fig-0360-01"/> rum ſub, HP, &</s> <s xml:id="echoid-s8425" xml:space="preserve">, DPG, dem-<lb/>ptis rectangulis ſub portioni-<lb/>bus, DBZP, DGP, ideſt <lb/> <anchor type="note" xlink:label="note-0360-08a" xlink:href="note-0360-08"/> ad rectangula ſub trilineo, D <lb/>P G, & </s> <s xml:id="echoid-s8426" xml:space="preserve">trilineo, BHZ, .</s> <s xml:id="echoid-s8427" xml:space="preserve">i. </s> <s xml:id="echoid-s8428" xml:space="preserve">tri-<lb/>lineo, DEG, erunt vt, ZP, <lb/>ad {1/6}. </s> <s xml:id="echoid-s8429" xml:space="preserve">PG, .</s> <s xml:id="echoid-s8430" xml:space="preserve">i. </s> <s xml:id="echoid-s8431" xml:space="preserve">ſumpta, PG, cõ-<lb/>muni altitudine, vt rectangu-<lb/>lum, ZPG, ad rectangulum <lb/>ſub, PG, & </s> <s xml:id="echoid-s8432" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8433" xml:space="preserve">PG, .</s> <s xml:id="echoid-s8434" xml:space="preserve">i. </s> <s xml:id="echoid-s8435" xml:space="preserve">ad {1/6}. <lb/></s> <s xml:id="echoid-s8436" xml:space="preserve">quadrati, PG, ſunt autem omnia quadrata, DG, ad rectangula ſub, <pb o="341" file="0361" n="361" rhead="LIBER IV."/> EP, PH, vt quadratum, GP, ad rectangulum, GPZ, ergo, exæ. <lb/></s> <s xml:id="echoid-s8437" xml:space="preserve">quali, omnia quadrata, DG, ad rectangul a ſub trilineis, DPG, D <lb/>EG, erunt vt quadratum, PG, ad {1/6}. </s> <s xml:id="echoid-s8438" xml:space="preserve">quadrati, PG, .</s> <s xml:id="echoid-s8439" xml:space="preserve">i. </s> <s xml:id="echoid-s8440" xml:space="preserve">erunt eorum <lb/>fexcupla: </s> <s xml:id="echoid-s8441" xml:space="preserve">Quoniam ergo omnia quadrata, DG, ad rectangula, ſub, <lb/>DG, &</s> <s xml:id="echoid-s8442" xml:space="preserve">?? </s> <s xml:id="echoid-s8443" xml:space="preserve">trilineo, DGP, ſunt vt, DG, ad, DGP, .</s> <s xml:id="echoid-s8444" xml:space="preserve">i. </s> <s xml:id="echoid-s8445" xml:space="preserve">vt, ZP, ad <lb/>compofitam ex {1/2}. </s> <s xml:id="echoid-s8446" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8447" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8448" xml:space="preserve">PG, ſunt autem omnia quadrata, D <lb/> <anchor type="note" xlink:label="note-0361-01a" xlink:href="note-0361-01"/> G, ſexcupla rectangulorum ſub trilineis, DPG, DEG, .</s> <s xml:id="echoid-s8449" xml:space="preserve">i. </s> <s xml:id="echoid-s8450" xml:space="preserve">ad ea, <lb/>vt, ZP, ad {1/6}, ZP, ergo omnia quadrata, DG, ad omnia quadrata, <lb/>D GP, erunt vt, ZG, ad reſiduum, dempto {1/6}. </s> <s xml:id="echoid-s8451" xml:space="preserve">ZP, à compoſita ex <lb/>{1/2}. </s> <s xml:id="echoid-s8452" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8453" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8454" xml:space="preserve">PG, quia verò ſi ab {1/2}. </s> <s xml:id="echoid-s8455" xml:space="preserve">ZP, dematur, {1/6}. </s> <s xml:id="echoid-s8456" xml:space="preserve">ZP, remanent <lb/>1. </s> <s xml:id="echoid-s8457" xml:space="preserve">ZP, ideò omnia quadrata, DG, ad omnia quadiata, DPG, erunt <lb/>vt, ZP, ad compoſitam ex {1/3}. </s> <s xml:id="echoid-s8458" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8459" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8460" xml:space="preserve">PG, vt dictum eſt.</s> <s xml:id="echoid-s8461" xml:space="preserve"/> </p> <div xml:id="echoid-div819" type="float" level="2" n="1"> <note position="left" xlink:label="note-0360-05" xlink:href="note-0360-05a" xml:space="preserve">Coroll. 1. <lb/>26.I. 2.</note> <note position="left" xlink:label="note-0360-06" xlink:href="note-0360-06a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0360-07" xlink:href="note-0360-07a" xml:space="preserve">Ex antec.</note> <figure xlink:label="fig-0360-01" xlink:href="fig-0360-01a"> <image file="0360-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0360-01"/> </figure> <note position="left" xlink:label="note-0360-08" xlink:href="note-0360-08a" xml:space="preserve">Jux. A. 23. <lb/>l. 2.</note> <note position="right" xlink:label="note-0361-01" xlink:href="note-0361-01a" xml:space="preserve">5. huius@</note> </div> <p> <s xml:id="echoid-s8462" xml:space="preserve">Quia verò nunc oſtenſum eſt omnia quadrata, DG, ad omnia <lb/>quadrata, DPG, eſſe vt, ZP, ad compoſitam ex {1/3}. </s> <s xml:id="echoid-s8463" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8464" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8465" xml:space="preserve">PG, <lb/>omnia autem quadrata, DG, ad rectangula ſub trilineis, DPG, D <lb/>E G, ſunt vt, ZP, ad {1/6}. </s> <s xml:id="echoid-s8466" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8467" xml:space="preserve">ad eadem bis ſumpta, vt, ZP, ad {1/3}. <lb/></s> <s xml:id="echoid-s8468" xml:space="preserve">Z P, ideò omnia quadrata, DG, ad omnia quadrata, DPG, & </s> <s xml:id="echoid-s8469" xml:space="preserve">ad <lb/>rectangula bis ſub, DPG, DEG, erunt vt, ZP, ad compoſitam ex <lb/>{2/3}. </s> <s xml:id="echoid-s8470" xml:space="preserve">ZP. </s> <s xml:id="echoid-s8471" xml:space="preserve">& </s> <s xml:id="echoid-s8472" xml:space="preserve">{1/6}: </s> <s xml:id="echoid-s8473" xml:space="preserve">PG, ergo omnia quadrata. </s> <s xml:id="echoid-s8474" xml:space="preserve">DG, ad reſiduum, demptis <lb/>omnibus quadratis, DPG, & </s> <s xml:id="echoid-s8475" xml:space="preserve">rectangulis bis ſub, DPG, DEG, .</s> <s xml:id="echoid-s8476" xml:space="preserve">i. </s> <s xml:id="echoid-s8477" xml:space="preserve"><lb/>ad omnia quadrata trilinei, DEG, erunt vt, ZP, ad reſiduum, dem-<lb/>ptis {2/3}. </s> <s xml:id="echoid-s8478" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8479" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8480" xml:space="preserve">PG, ab eadem, ZP, quæ nobis oſtendenda erat.</s> <s xml:id="echoid-s8481" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div821" type="section" level="1" n="486"> <head xml:id="echoid-head506" xml:space="preserve">THEOREMA XLVI. PROPOS. XLVIII.</head> <p> <s xml:id="echoid-s8482" xml:space="preserve">IN ſupradictæ Propoſ. </s> <s xml:id="echoid-s8483" xml:space="preserve">figura, ducta, AX, parallela baſi, <lb/>Z G, quæ tanget parabolam in, A, cui occurrat, GE, <lb/>producta, in puncto, X, oſtendemus omnia quadrata trili-<lb/>nei, DPG, ad omnia quadrata ſemiparabolæ, AQG, ha-<lb/>bere rationem compoſitam ex ea, quam habet compoſita ex <lb/>{1/3}. </s> <s xml:id="echoid-s8484" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8485" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8486" xml:space="preserve">PG, ad, ZP, & </s> <s xml:id="echoid-s8487" xml:space="preserve">ex ratione parallelepipedi ſub, <lb/>D P, & </s> <s xml:id="echoid-s8488" xml:space="preserve">quadrato, PG, ad dimidium parallelepipedi ſub, <lb/>A Q, & </s> <s xml:id="echoid-s8489" xml:space="preserve">quadrato, QG; </s> <s xml:id="echoid-s8490" xml:space="preserve">Omnia vero quadrata trilinei, AX <lb/>G. </s> <s xml:id="echoid-s8491" xml:space="preserve">ad omnia quadrata trilinei, DEG, habere rationem cõ-<lb/>poſitam ex ea, quam habet parallelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8492" xml:space="preserve">qua-<lb/>drato, QG, ſexta pars, ad parallelepipedum ſub, DP, & </s> <s xml:id="echoid-s8493" xml:space="preserve"><lb/>quadrato, PG, & </s> <s xml:id="echoid-s8494" xml:space="preserve">ex ea, quam habet, ZP, ad reſidunm, dem-<lb/>ptis ab eadem, ZP, {2/3}. </s> <s xml:id="echoid-s8495" xml:space="preserve">ZP, cum {1/6}. </s> <s xml:id="echoid-s8496" xml:space="preserve">PG.</s> <s xml:id="echoid-s8497" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8498" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s8499" xml:space="preserve">n. </s> <s xml:id="echoid-s8500" xml:space="preserve">quadrata trilinei, DPG, ad omnia quadrata ſemipa- <pb o="342" file="0362" n="362" rhead="GEO METRIÆ"/> ra bolæ, AQG, habent rationem compoſitam ex ea, quam habent <lb/>omnia quadrata, DPG, ad omnia quadrata, DG, ideſt ex ratione <lb/>compoſitę ex {1/2}. </s> <s xml:id="echoid-s8501" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8502" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8503" xml:space="preserve">PG, ad, ZP, & </s> <s xml:id="echoid-s8504" xml:space="preserve">ex ea, quam habent om-<lb/> <anchor type="note" xlink:label="note-0362-01a" xlink:href="note-0362-01"/> nia quadrata, DG, ad omnia quadrata, AG, .</s> <s xml:id="echoid-s8505" xml:space="preserve">i. </s> <s xml:id="echoid-s8506" xml:space="preserve">ex ratione paralle-<lb/>lepipedi ſub, DP, & </s> <s xml:id="echoid-s8507" xml:space="preserve">quadrato, PG, ad parallel epipedum ſub, AQ, <lb/>& </s> <s xml:id="echoid-s8508" xml:space="preserve">quadrato, QG, & </s> <s xml:id="echoid-s8509" xml:space="preserve">tandem ex ea, quam habent omnia quadrata, <lb/>A G, ad omnia quadrata ſemiparabolæ, AQG, .</s> <s xml:id="echoid-s8510" xml:space="preserve">i. </s> <s xml:id="echoid-s8511" xml:space="preserve">ex ratione paral-<lb/>lelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8512" xml:space="preserve">quadrato, QG, ad eiuſdem dimidium: </s> <s xml:id="echoid-s8513" xml:space="preserve">Duæ <lb/> <anchor type="note" xlink:label="note-0362-02a" xlink:href="note-0362-02"/> autem rationes parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s8514" xml:space="preserve">quadrato, PG, ad paral-<lb/>lelepipedum ſub, AQ, & </s> <s xml:id="echoid-s8515" xml:space="preserve">quadrato, QG, & </s> <s xml:id="echoid-s8516" xml:space="preserve">ratio huius ad eiuſdem <lb/> <anchor type="note" xlink:label="note-0362-03a" xlink:href="note-0362-03"/> dimidium, conſiciunt rationem parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s8517" xml:space="preserve">quadra-<lb/>to, PG, ad {1/2}. </s> <s xml:id="echoid-s8518" xml:space="preserve">parallelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8519" xml:space="preserve">quadrato, QG, ergo om-<lb/> <anchor type="note" xlink:label="note-0362-04a" xlink:href="note-0362-04"/> nia quadrata, D & </s> <s xml:id="echoid-s8520" xml:space="preserve">G, ad omnia quadrata ſemiparabolæ, AQG, ha-<lb/>bent rationem compoſitam ex ratione rectæ compoſitæ ex {1/3}. </s> <s xml:id="echoid-s8521" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8522" xml:space="preserve"><lb/>{1/2}. </s> <s xml:id="echoid-s8523" xml:space="preserve">PG, ad, ZP, & </s> <s xml:id="echoid-s8524" xml:space="preserve">ex ratione parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s8525" xml:space="preserve">quadrato, <lb/>P G, ad {1/2}. </s> <s xml:id="echoid-s8526" xml:space="preserve">parallelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8527" xml:space="preserve">quadrato, QG, vt dictum eſt.</s> <s xml:id="echoid-s8528" xml:space="preserve"/> </p> <div xml:id="echoid-div821" type="float" level="2" n="1"> <note position="left" xlink:label="note-0362-01" xlink:href="note-0362-01a" xml:space="preserve">Exantec.</note> <note position="left" xlink:label="note-0362-02" xlink:href="note-0362-02a" xml:space="preserve">Effcitur ex <lb/>x1. l. 2.</note> <note position="left" xlink:label="note-0362-03" xlink:href="note-0362-03a" xml:space="preserve">21. huius.</note> <note position="left" xlink:label="note-0362-04" xlink:href="note-0362-04a" xml:space="preserve">Defin. 12, <lb/>l. 1.</note> </div> <p> <s xml:id="echoid-s8529" xml:space="preserve">Inſuper omnia quadrata trilinei, AXG, ad omnia quadrata trili-<lb/>nei, DEG, habent rationem compoſitam ex ratione omnium qua-<lb/> <anchor type="note" xlink:label="note-0362-05a" xlink:href="note-0362-05"/> dratorum, AXG, ad omnia quadrata, AG, .</s> <s xml:id="echoid-s8530" xml:space="preserve">i. </s> <s xml:id="echoid-s8531" xml:space="preserve">ſubſexcupla .</s> <s xml:id="echoid-s8532" xml:space="preserve">i. </s> <s xml:id="echoid-s8533" xml:space="preserve">ex ra-<lb/>tione {1/6}. </s> <s xml:id="echoid-s8534" xml:space="preserve">parallelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8535" xml:space="preserve">quadrato, QG, ad idem paral-<lb/>lelepipedum, & </s> <s xml:id="echoid-s8536" xml:space="preserve">ex ratione omnium quadratorum, AG, ad omnia <lb/>quadrata, DG, .</s> <s xml:id="echoid-s8537" xml:space="preserve">i. </s> <s xml:id="echoid-s8538" xml:space="preserve">parallelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8539" xml:space="preserve">quadrato, QG, ad <lb/>parallelepipedum ſub, DP, & </s> <s xml:id="echoid-s8540" xml:space="preserve">quadrato, PG, quæ duæ rationes cõ-<lb/>ficiunt rationem {1/6}. </s> <s xml:id="echoid-s8541" xml:space="preserve">parallepipedi ſub, AQ, & </s> <s xml:id="echoid-s8542" xml:space="preserve">quadrato, QG, ad pa-<lb/>rallelepipedum ſub, DP, & </s> <s xml:id="echoid-s8543" xml:space="preserve">quadrato, PG, & </s> <s xml:id="echoid-s8544" xml:space="preserve">tandem ex ratione <lb/>omnium quadratorum, DG, ad omnia quadrata trilinei, DEG, .</s> <s xml:id="echoid-s8545" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s8546" xml:space="preserve"> <anchor type="note" xlink:label="note-0362-06a" xlink:href="note-0362-06"/> ex ea, quam habet, ZP, ad reſiduum, ab eadem, ZP, demptis {2/3}. </s> <s xml:id="echoid-s8547" xml:space="preserve">ZP, <lb/>cum {1/6}. </s> <s xml:id="echoid-s8548" xml:space="preserve">PG, ergo omnia quadrata trilinei, AXG, ad omnia quadra-<lb/>ta trilinei, DEG, habent rationem compoſitam ex ea, quam habet <lb/>{1/6}. </s> <s xml:id="echoid-s8549" xml:space="preserve">parallelepipedi, ſub, AQ, & </s> <s xml:id="echoid-s8550" xml:space="preserve">quadrato, QG, ad parallelepipedum <lb/>ſub, DP, & </s> <s xml:id="echoid-s8551" xml:space="preserve">quadrato, PG, & </s> <s xml:id="echoid-s8552" xml:space="preserve">ex ea, quam habet, ZP, ad ſui reſi-<lb/>duum, demptis ab ea {2/3}. </s> <s xml:id="echoid-s8553" xml:space="preserve">ZP, cum {1/6}. </s> <s xml:id="echoid-s8554" xml:space="preserve">PG, quæ oſtendere oportebat.</s> <s xml:id="echoid-s8555" xml:space="preserve"/> </p> <div xml:id="echoid-div822" type="float" level="2" n="2"> <note position="left" xlink:label="note-0362-05" xlink:href="note-0362-05a" xml:space="preserve">30. huius.</note> <note position="left" xlink:label="note-0362-06" xlink:href="note-0362-06a" xml:space="preserve">47. huius.</note> </div> </div> <div xml:id="echoid-div824" type="section" level="1" n="487"> <head xml:id="echoid-head507" xml:space="preserve">THEOREMA XLVII. PROPOS. XLIX.</head> <p> <s xml:id="echoid-s8556" xml:space="preserve">IN eadem figura Propoſ. </s> <s xml:id="echoid-s8557" xml:space="preserve">46. </s> <s xml:id="echoid-s8558" xml:space="preserve">oſtendemus, producta, PD, <lb/>verſus, AX, cui occurrat in, C, omnia quadrata trilinei, <lb/>D GP, ad omnia quadrata figuræ, CAZP, demptis omni-<lb/>bus quadratis trilinei, ACD, habere rationem compoſitam <lb/>ex ea, quam habet compoſita ex {1/3}. </s> <s xml:id="echoid-s8559" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8560" xml:space="preserve">{1/6}, PG, ad ZP, & </s> <s xml:id="echoid-s8561" xml:space="preserve"><lb/>ex ratione parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s8562" xml:space="preserve">quadrato, PG, ad pa- <pb o="343" file="0363" n="363" rhead="LIBER IV."/> rallelepipedum ſub, AQ, & </s> <s xml:id="echoid-s8563" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8564" xml:space="preserve">f. </s> <s xml:id="echoid-s8565" xml:space="preserve">quadrato, PQ {1/2}. <lb/></s> <s xml:id="echoid-s8566" xml:space="preserve">quadrati, QZ, & </s> <s xml:id="echoid-s8567" xml:space="preserve">rectangulo ſub ſexquitertia, ZQ, & </s> <s xml:id="echoid-s8568" xml:space="preserve">ſub, Q <lb/>P, ab eodem dempta {1/6}. </s> <s xml:id="echoid-s8569" xml:space="preserve">para llelepipedi ſub, CD, & </s> <s xml:id="echoid-s8570" xml:space="preserve">quadra-<lb/>to, QP,</s> </p> <p> <s xml:id="echoid-s8571" xml:space="preserve">Completo parallelogrammo, KP, omnia igitur quadrata trili-<lb/> <anchor type="figure" xlink:label="fig-0363-01a" xlink:href="fig-0363-01"/> nei, DPG, ad omnia qua-<lb/>drata figurę, CAZP, demp tis <lb/>omnibus quadratis trilinei, A <lb/>CD, habent rationem com-<lb/>poſitam ex ea, quam habent <lb/>omnia quadrata, DPG, ad <lb/>omnia quadrata, DG, .</s> <s xml:id="echoid-s8572" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8573" xml:space="preserve">ex <lb/>ratione compoſitæ ex {1/3}. </s> <s xml:id="echoid-s8574" xml:space="preserve">ZP, <lb/>& </s> <s xml:id="echoid-s8575" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8576" xml:space="preserve">PG, ad, ZP, & </s> <s xml:id="echoid-s8577" xml:space="preserve">ex ratio-<lb/> <anchor type="note" xlink:label="note-0363-01a" xlink:href="note-0363-01"/> ne omnium quadratorum, D <lb/>G, ad omnia quadrata, KP, .</s> <s xml:id="echoid-s8578" xml:space="preserve">i. </s> <s xml:id="echoid-s8579" xml:space="preserve">ex ratione parallelepipedi ſub, D <lb/>P, & </s> <s xml:id="echoid-s8580" xml:space="preserve">quadrato, PG, ad parallel epipedum ſub, AQ, & </s> <s xml:id="echoid-s8581" xml:space="preserve">quadrato, Z <lb/>P, & </s> <s xml:id="echoid-s8582" xml:space="preserve">tandem ex ratione omnium quadratorum, KP, ad omnia qua-<lb/>drata figuræ, CAZP, demptis omnibus quadratis trilinei, ACD, <lb/>.</s> <s xml:id="echoid-s8583" xml:space="preserve">f. </s> <s xml:id="echoid-s8584" xml:space="preserve">ex ratione parallelepipedi ſub, AQ, & </s> <s xml:id="echoid-s8585" xml:space="preserve">quadrato, ZP, ad paralle-<lb/>pipedum ſub, AQ, & </s> <s xml:id="echoid-s8586" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8587" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8588" xml:space="preserve">quadrato, PQ, {1/2}. </s> <s xml:id="echoid-s8589" xml:space="preserve">quadrati, <lb/> <anchor type="note" xlink:label="note-0363-02a" xlink:href="note-0363-02"/> QZ, & </s> <s xml:id="echoid-s8590" xml:space="preserve">rectangulo ſub, PQ, & </s> <s xml:id="echoid-s8591" xml:space="preserve">ſexquitertia, QZ, ab eodem dem-<lb/>pta {1/6}. </s> <s xml:id="echoid-s8592" xml:space="preserve">parallelepipedi ſub, CD, & </s> <s xml:id="echoid-s8593" xml:space="preserve">quadrato, PQ; </s> <s xml:id="echoid-s8594" xml:space="preserve">duæ autem ra-<lb/>tiones parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s8595" xml:space="preserve">quadrato PG, ad parallelepipe-<lb/>dum ſub, AQ, & </s> <s xml:id="echoid-s8596" xml:space="preserve">quadrato, ZP, & </s> <s xml:id="echoid-s8597" xml:space="preserve">huius parallelepipedi ad paral-<lb/>lelepipedum ſub, AQ, & </s> <s xml:id="echoid-s8598" xml:space="preserve">ſpatijs iam dictis, ab eodem dempta {1/6}. </s> <s xml:id="echoid-s8599" xml:space="preserve">pa-<lb/>rallelepipedi ſub, CD, & </s> <s xml:id="echoid-s8600" xml:space="preserve">quadrato, PQ, componunt rationem pa-<lb/>rallelepipedi ſub, DP, & </s> <s xml:id="echoid-s8601" xml:space="preserve">quadrato, PG, ad parallelepipedum ſub, <lb/> <anchor type="note" xlink:label="note-0363-03a" xlink:href="note-0363-03"/> AQ, & </s> <s xml:id="echoid-s8602" xml:space="preserve">dictis ſpatijs ab eodem dempta {1/6}. </s> <s xml:id="echoid-s8603" xml:space="preserve">parallelepipedi ſub, CD, <lb/>& </s> <s xml:id="echoid-s8604" xml:space="preserve">quadrato, PQ, ergo omnia quadrata trilinei, DGP, ad omnia, <lb/>quadrata figuræ, CAZP, demptis omnibus quadratis trilinei, AC <lb/>D, erunt in ratione compoſita ex ea, quam habet {1/3}. </s> <s xml:id="echoid-s8605" xml:space="preserve">ZP, cum {1/6}. </s> <s xml:id="echoid-s8606" xml:space="preserve">P <lb/>G, ad, ZP, & </s> <s xml:id="echoid-s8607" xml:space="preserve">ex ea, quam habet parallelepipedum ſub, DP, & </s> <s xml:id="echoid-s8608" xml:space="preserve">qua-<lb/>drato, PG, ad parallelepipedum ſub, AQ, & </s> <s xml:id="echoid-s8609" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s8610" xml:space="preserve">f. </s> <s xml:id="echoid-s8611" xml:space="preserve">quadra-<lb/>to, PQ, {1/2}. </s> <s xml:id="echoid-s8612" xml:space="preserve">quadrati, QZ, cum rectangulo ſub, PQ, & </s> <s xml:id="echoid-s8613" xml:space="preserve">ſexquitertia, <lb/>QZ, ab eodem parallelepipedo dempta {1/6}. </s> <s xml:id="echoid-s8614" xml:space="preserve">parallelepipedi ſub, CD, <lb/>& </s> <s xml:id="echoid-s8615" xml:space="preserve">quadrato, PQ, quod oſtendere opus erat.</s> <s xml:id="echoid-s8616" xml:space="preserve"/> </p> <div xml:id="echoid-div824" type="float" level="2" n="1"> <figure xlink:label="fig-0363-01" xlink:href="fig-0363-01a"> <image file="0363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0363-01"/> </figure> <note position="right" xlink:label="note-0363-01" xlink:href="note-0363-01a" xml:space="preserve">47. huius.</note> <note position="right" xlink:label="note-0363-02" xlink:href="note-0363-02a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-0363-03" xlink:href="note-0363-03a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> </div> <pb o="344" file="0364" n="364" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div826" type="section" level="1" n="488"> <head xml:id="echoid-head508" xml:space="preserve">THEOREMA XLVIII. PROPOS. L.</head> <p> <s xml:id="echoid-s8617" xml:space="preserve">IN eadem figura, ducta per, I, IL, æquidiſtante ipſi, A <lb/>Q, adhuc oſtendemus omnia quadrata trilinei, DGP, <lb/>ad omnia quadrata trilinei, DTI, habere rationem compo-<lb/>ſitam ex ea, quam habet rectangulum, ZPG, cum. </s> <s xml:id="echoid-s8618" xml:space="preserve">qua-<lb/>drati, PG, ad rectangulum, STI, cum quadrati, TI, & </s> <s xml:id="echoid-s8619" xml:space="preserve">ex ea, <lb/>quam habet quadratum, PG, ad quadratum TI.</s> <s xml:id="echoid-s8620" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8621" xml:space="preserve">Nam omnia quadrata, DGP, ad omnia quadrata, DIT, ha-<lb/>bent rationem compoſitam ex ea, quam habent omnia quadrata, <lb/>DGP, ad omnia quadrata, DG, .</s> <s xml:id="echoid-s8622" xml:space="preserve">i. </s> <s xml:id="echoid-s8623" xml:space="preserve">ex ea, quam habet {1/3}. </s> <s xml:id="echoid-s8624" xml:space="preserve">ZP, cũ <lb/> <anchor type="note" xlink:label="note-0364-01a" xlink:href="note-0364-01"/> {1/6}. </s> <s xml:id="echoid-s8625" xml:space="preserve">PG, ad, ZP, .</s> <s xml:id="echoid-s8626" xml:space="preserve">i. </s> <s xml:id="echoid-s8627" xml:space="preserve">ſumpta, PG, communi altitudine, ex ea, quam <lb/>habet rectangulum ſub {1/3}. </s> <s xml:id="echoid-s8628" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8629" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8630" xml:space="preserve">PG, & </s> <s xml:id="echoid-s8631" xml:space="preserve">ſub, PG, ad rectangu-<lb/>lum, ZPG, item ex ratione omnium quadratorum, DG, ad om-<lb/>nia quadrata, DI, ſcilicet compoſita ex ea, quam habet, PD, ad, D <lb/>T, & </s> <s xml:id="echoid-s8632" xml:space="preserve">quadratum, PG, ad quadratum, TI, eſt autem, vt, PD, ad, <lb/>DT, ita rectangulum, ZPG, ad rectangulum, STI; </s> <s xml:id="echoid-s8633" xml:space="preserve">Tandem <lb/>verò componitur ex ea, quam habent omnia quadrata, DI, <lb/> <anchor type="figure" xlink:label="fig-0364-01a" xlink:href="fig-0364-01"/> ad omnia quadrata, DIT, <lb/>ideſt ex ratione, ST, ad {1/3}. <lb/></s> <s xml:id="echoid-s8634" xml:space="preserve">ST, cum {1/6}. </s> <s xml:id="echoid-s8635" xml:space="preserve">TI, ideſt ſum-<lb/>pta, TI, communi altitudi-<lb/>ne, ex ea, quam habet rectan-<lb/>gulum, STI, ad rectangulum <lb/>fub, TI, & </s> <s xml:id="echoid-s8636" xml:space="preserve">compoſita ex {1/3}. </s> <s xml:id="echoid-s8637" xml:space="preserve">S <lb/>T, & </s> <s xml:id="echoid-s8638" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8639" xml:space="preserve">TI, iſtæ autem ratio-<lb/> <anchor type="note" xlink:label="note-0364-02a" xlink:href="note-0364-02"/> nes .</s> <s xml:id="echoid-s8640" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8641" xml:space="preserve">quam habet rectangu-<lb/>lum ſub {1/3}. </s> <s xml:id="echoid-s8642" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8643" xml:space="preserve">{1/6}, PG, & </s> <s xml:id="echoid-s8644" xml:space="preserve"><lb/>ſub, PG, ad rectangulum, ZPG, & </s> <s xml:id="echoid-s8645" xml:space="preserve">huius ad rectangulum, STI, <lb/>& </s> <s xml:id="echoid-s8646" xml:space="preserve">taudem rectanguli, STI, ad rectangulum ſub, TI, & </s> <s xml:id="echoid-s8647" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s8648" xml:space="preserve">ST, <lb/>cum {1/6}. </s> <s xml:id="echoid-s8649" xml:space="preserve">TI, componunt rationem rectanguli fub {1/3}. </s> <s xml:id="echoid-s8650" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s8651" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8652" xml:space="preserve">PG, <lb/>& </s> <s xml:id="echoid-s8653" xml:space="preserve">ſub, PG, ad rectangulum ſub {1/3}. </s> <s xml:id="echoid-s8654" xml:space="preserve">ST, & </s> <s xml:id="echoid-s8655" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s8656" xml:space="preserve">TI, & </s> <s xml:id="echoid-s8657" xml:space="preserve">ſub, TI<emph style="sub">6</emph> .</s> <s xml:id="echoid-s8658" xml:space="preserve">i, tri-<lb/>plicatis terminis, componunt rationem rectanguli ſub, ZP, PG, <lb/>cum rectangulo, ſub {3/6}. </s> <s xml:id="echoid-s8659" xml:space="preserve">PG, & </s> <s xml:id="echoid-s8660" xml:space="preserve">ſub, PG, .</s> <s xml:id="echoid-s8661" xml:space="preserve">i. </s> <s xml:id="echoid-s8662" xml:space="preserve">cum {1/2}. </s> <s xml:id="echoid-s8663" xml:space="preserve">quadrati, PG, <lb/>ad rectangulum ſub, ST, TI, cum rectanguio ſub {3/6}. </s> <s xml:id="echoid-s8664" xml:space="preserve">TI, & </s> <s xml:id="echoid-s8665" xml:space="preserve">ſub, T <lb/>I, .</s> <s xml:id="echoid-s8666" xml:space="preserve">i. </s> <s xml:id="echoid-s8667" xml:space="preserve">cum {1/2}. </s> <s xml:id="echoid-s8668" xml:space="preserve">quadrati, TI, & </s> <s xml:id="echoid-s8669" xml:space="preserve">remanſit ſola ratio quadrati, PG, ad <lb/>quadratum, TI, ergo omnia quadrata trilinei, DGP, ad om-<lb/>nia quadrata trilinei, DIT, habebunt rationem compoſitam ex <pb o="345" file="0365" n="365" rhead="LIBER IV."/> ea, quam habet rectangulum, ZPG, cum {1/2}. </s> <s xml:id="echoid-s8670" xml:space="preserve">quadrati, PG, ad re-<lb/>ctangulum, STI, cum {1/2}. </s> <s xml:id="echoid-s8671" xml:space="preserve">quadrati, TI, & </s> <s xml:id="echoid-s8672" xml:space="preserve">ex ea, quam habet qua-<lb/>dratum, PG, ad quadratum, TI, quod, &</s> <s xml:id="echoid-s8673" xml:space="preserve">c.</s> <s xml:id="echoid-s8674" xml:space="preserve"/> </p> <div xml:id="echoid-div826" type="float" level="2" n="1"> <note position="left" xlink:label="note-0364-01" xlink:href="note-0364-01a" xml:space="preserve">41. huius.</note> <figure xlink:label="fig-0364-01" xlink:href="fig-0364-01a"> <image file="0364-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0364-01"/> </figure> <note position="left" xlink:label="note-0364-02" xlink:href="note-0364-02a" xml:space="preserve">47. huius.</note> </div> </div> <div xml:id="echoid-div828" type="section" level="1" n="489"> <head xml:id="echoid-head509" xml:space="preserve">THEOREMA XLIX. PROPOS. LI.</head> <p> <s xml:id="echoid-s8675" xml:space="preserve">IN omnibus huius Libri 4. </s> <s xml:id="echoid-s8676" xml:space="preserve">Propoſitionibus, in quibus <lb/>duarum quarumcunque fi grarum notificata fuit ratio <lb/>omnium quadratorum, iuxta regulas in eiſdem aſſumptas, <lb/>nota etiam euadit ratio ſimilarium ſolidorum, quæ ex illis <lb/>gignuntur figuris, iuxta eaſdem regulas.</s> <s xml:id="echoid-s8677" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8678" xml:space="preserve">Quoniam enim oſtenſum eſt Lib. </s> <s xml:id="echoid-s8679" xml:space="preserve">2. </s> <s xml:id="echoid-s8680" xml:space="preserve">Prop. </s> <s xml:id="echoid-s8681" xml:space="preserve">33. </s> <s xml:id="echoid-s8682" xml:space="preserve">vt omnia quadra-<lb/>ta duarum figurarum inter ſe ſumpta cum datis regulis, ita effe ſo-<lb/>lida ſimilaria genita ex ijſdem figuris iuxta eaſdem regulas, ideò <lb/>cum in Propoſitionibus huius Libri inuenta eſt ratio omnium qua-<lb/>dratorum duarum figurarum cum quibuſdam regulis, colligemus <lb/>etiam nunc eandem eſſe rationem duorum ſimilarium ſolidorum, <lb/>quæ ex illis figuris iuxta eaſdem regulas, genita dicuntur. </s> <s xml:id="echoid-s8683" xml:space="preserve">Vtex. </s> <s xml:id="echoid-s8684" xml:space="preserve">g. <lb/></s> <s xml:id="echoid-s8685" xml:space="preserve">in Prop. </s> <s xml:id="echoid-s8686" xml:space="preserve">21. </s> <s xml:id="echoid-s8687" xml:space="preserve">conſpecta denuò illius figura, cum oſtenſum eſt omnia <lb/>quadrata, AF, eſſe dupla omnium quadratorum parabolæ, VEF, <lb/>regula ſumpta, VF; </s> <s xml:id="echoid-s8688" xml:space="preserve">& </s> <s xml:id="echoid-s8689" xml:space="preserve">item omnia quadrata parabolæ, VEF, eſ-<lb/>ſe ſexquialtera omnium quadratorum trianguli, VEF, conclude-<lb/>mus pariter ſolidum fimilare genitum ex, AF, ad ſibi ſimilare geni-<lb/>tum ex parabola, VEF, duplam habere rationem; </s> <s xml:id="echoid-s8690" xml:space="preserve">hoc verò ad ſo-<lb/>lidum ſibiſimilare genitum ex triangulo, VEF, habere rationem <lb/>ſexquialteram, genita autem dicta ſolida intellige iuxta dictam re-<lb/>gulam, VF; </s> <s xml:id="echoid-s8691" xml:space="preserve">pater ergo propoſitum.</s> <s xml:id="echoid-s8692" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div829" type="section" level="1" n="490"> <head xml:id="echoid-head510" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s8693" xml:space="preserve">_Q_V oniam autem apertè colligitur ex Lib. </s> <s xml:id="echoid-s8694" xml:space="preserve">I. </s> <s xml:id="echoid-s8695" xml:space="preserve">Prop. </s> <s xml:id="echoid-s8696" xml:space="preserve">46. </s> <s xml:id="echoid-s8697" xml:space="preserve">& </s> <s xml:id="echoid-s8698" xml:space="preserve">47 ſi onæ-<lb/>nes figuræ ſimiles parabolæ, quæ ſumantur regula eiuſdem baſi, <lb/>ſint circuli, diame tros in eadem parabola ſitos babentes, cui ſint ere-<lb/>cti, ſolidum ſimilare genitum ex dicta parabola eſſe conoides parabo-<lb/>licum, cuius baſis rectè ſecat axim; </s> <s xml:id="echoid-s8699" xml:space="preserve">ſi verò ſint ellipſes homologas <lb/>diametros in eadem parabola ſitos babentes eidem erectæ, quarum ſe-<lb/>cunda diametri ſint æquales diſtantia parallelarum, qua ducuntur ab <lb/>extremis primæ diametri æquidiſtanter axi, eſſe pariter conoides pa-<lb/>rabolicum, cuius baſis tunc obliquè axim ſecat. </s> <s xml:id="echoid-s8700" xml:space="preserve">Ideò ex bis infra- <pb o="346" file="0366" n="366" rhead="GEOMETRIÆ"/> ſcripta ſequuntur Corollaria, in quibus exempla adbibebimus, veluti <lb/>Lib. </s> <s xml:id="echoid-s8701" xml:space="preserve">3. </s> <s xml:id="echoid-s8702" xml:space="preserve">effectum eſt, aſſumptis nempè omnibus figuris ſimilibus genitri-<lb/>cium figurarum, quæ ſint circuli, diametros in ipſis genitricibus figu-<lb/>ris, quibus ſunt erecti, ſitos babentes, quæ per reuolutionem figuraram <lb/>circa ſuos axes deſcribi facilè appræbendi poſsũt, propter quod in exẽ-<lb/>plis tantũmodò axes aſſumemus congruenter ipſarũ genitrium figurarũ <lb/>reuolutioni, licet exempla etiam aſſumptis diametris confiici poſſent <lb/>per deſcriptionem omnium ſimilium figurarum haud tamen per reuolu-<lb/>tionem factam. </s> <s xml:id="echoid-s8703" xml:space="preserve">Liceat autem Prop. </s> <s xml:id="echoid-s8704" xml:space="preserve">antecedentium reaſſamptas figu-<lb/>ras ſub ampliori forma quandoque proponere, bel ſub auguſtiori, pro-<lb/>ut expedire comperietur, ſeruata ſemper earundem ſimilitudine.</s> <s xml:id="echoid-s8705" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div830" type="section" level="1" n="491"> <head xml:id="echoid-head511" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s8706" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8707" xml:space="preserve">21. </s> <s xml:id="echoid-s8708" xml:space="preserve">ergo ſi intelligantur tres figuræ, nempè parallelo-<lb/>grammum, AF, triangulus, EVF, & </s> <s xml:id="echoid-s8709" xml:space="preserve">parabola, VEF, circa com-<lb/> <anchor type="figure" xlink:label="fig-0366-01a" xlink:href="fig-0366-01"/> munem axem reuolui, qui ſuppona-<lb/>tur eſſe, EM, fiet ex, AF, cylindrus, <lb/>AF. </s> <s xml:id="echoid-s8710" xml:space="preserve">extriangulo, VEF, conus, VEF, <lb/>& </s> <s xml:id="echoid-s8711" xml:space="preserve">ex parabola, VEF, conoides para-<lb/>bolicum, VEF, vnde patebit cylindrũ, <lb/>AF, eſſe duplum conoidis, VEF, & </s> <s xml:id="echoid-s8712" xml:space="preserve"><lb/>hoc eſſe ſexquialterum coni, VEF; </s> <s xml:id="echoid-s8713" xml:space="preserve">& </s> <s xml:id="echoid-s8714" xml:space="preserve"><lb/>vniuerſaliſsimè, vt dictum eſt, ſoli-<lb/>dum ſimilare genitum ex, AF, ad ſibi <lb/>ſimilare genitum ex parabola, VEF, habere duplam rationem, hoc <lb/>verò ad ſibi ſimilare gentium ex triangulo, VEF, rationem ſexqui-<lb/>alteram, quod tamen, ne figuræ multiplicentur, ſeu nimis confun-<lb/>dantur (quod etiã impofteru obſeruabimus) vno tãtũ adhibito exẽ-<lb/>plo, reuolutionis figurarũ<unsure/> genitriciũ circa ſuos axes, explicare volui.</s> <s xml:id="echoid-s8715" xml:space="preserve"/> </p> <div xml:id="echoid-div830" type="float" level="2" n="1"> <figure xlink:label="fig-0366-01" xlink:href="fig-0366-01a"> <image file="0366-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0366-01"/> </figure> </div> </div> <div xml:id="echoid-div832" type="section" level="1" n="492"> <head xml:id="echoid-head512" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s8716" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8717" xml:space="preserve">22. </s> <s xml:id="echoid-s8718" xml:space="preserve">aſſumpta eius figura, fiat exemplum per reuolutio-<lb/> <anchor type="figure" xlink:label="fig-0366-02a" xlink:href="fig-0366-02"/> nem parabolæ, FCH, circa axẽ, <lb/>CG, dimiſsis parallelogrammis, AH, <lb/>RM, fient igitur in hac reuolutione <lb/>conoidea parabolica ex, FCH, OCM, <lb/>parabolis, quæ fint, FCH, OCM; </s> <s xml:id="echoid-s8719" xml:space="preserve">vn-<lb/>de patebit conoides parabolicum, <lb/>FCH, ad conoides parabolicũ, OCM, <lb/>eſſe, vt quadratũ, GG, ad quadratũ, <pb o="347" file="0367" n="367" rhead="LIBER IV."/> CI, & </s> <s xml:id="echoid-s8720" xml:space="preserve">ſic eſſe quodlibet ſolidum ſimilare genitũ ex parabola, FCH, <lb/>ad ſibi ſimilare genitum ex parabola OCM, iuxta communem re-<lb/>gulam, FH, ſiue CG, ſit axis, ſiue tantum diameter, quod iuxta an-<lb/>tecedentis explicationem facilè intelligi poteſt.</s> <s xml:id="echoid-s8721" xml:space="preserve"/> </p> <div xml:id="echoid-div832" type="float" level="2" n="1"> <figure xlink:label="fig-0366-02" xlink:href="fig-0366-02a"> <image file="0366-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0366-02"/> </figure> </div> </div> <div xml:id="echoid-div834" type="section" level="1" n="493"> <head xml:id="echoid-head513" xml:space="preserve">COROLLARIVM III.</head> <p> <s xml:id="echoid-s8722" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8723" xml:space="preserve">23. </s> <s xml:id="echoid-s8724" xml:space="preserve">aſſumpta figura Prop. </s> <s xml:id="echoid-s8725" xml:space="preserve">12. </s> <s xml:id="echoid-s8726" xml:space="preserve">ſcilicet parabola, BNH, <lb/>parallelogrammis, PH, AG, & </s> <s xml:id="echoid-s8727" xml:space="preserve">triangulo, BRH, reuoluatur pa-<lb/> <anchor type="figure" xlink:label="fig-0367-01a" xlink:href="fig-0367-01"/> rabola, BNH, vt fiat noſſrum exem-<lb/>plum, circa axem, NO, & </s> <s xml:id="echoid-s8728" xml:space="preserve">inſimul, PH, <lb/>AG, & </s> <s xml:id="echoid-s8729" xml:space="preserve">triangulus, BRH, circa RO, <lb/>patebit ergo cylindrum ex, PO, ad <lb/>fruſtum conoidis ex, ABOR, in reuo-<lb/>lutione genitum, eſſe vt, ON, ad com-<lb/>poſitam ex, NR, & </s> <s xml:id="echoid-s8730" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8731" xml:space="preserve">RO, ipſum ve-<lb/>rò ad idem dempto cylindro ex, AO, <lb/>.</s> <s xml:id="echoid-s8732" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8733" xml:space="preserve">AG, vt, NO, ad {1/2}. </s> <s xml:id="echoid-s8734" xml:space="preserve">RO, ex Corollario huic Propoſitioni ſubiecto, <lb/>hoc fruſtum tandem ad conum gentium ex triangulo, RBO, vt cõ-<lb/>poſita ex, ON, dupla, MR, & </s> <s xml:id="echoid-s8735" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s8736" xml:space="preserve">RO, adipſam, NO; </s> <s xml:id="echoid-s8737" xml:space="preserve">& </s> <s xml:id="echoid-s8738" xml:space="preserve">vniuerſa-<lb/>liter quæcunque ſolida ſimilaria ex eiſdem figuris genitricibus geni-<lb/>ta, iuxta communem regulam, BH, eaſdem rationes habere, vt ſu-<lb/>pradicta ad inuicem comparata, ſiue, NO, ſit axis, ſiue tantum dia-<lb/>meter; </s> <s xml:id="echoid-s8739" xml:space="preserve">quod ex Propol. </s> <s xml:id="echoid-s8740" xml:space="preserve">51. </s> <s xml:id="echoid-s8741" xml:space="preserve">clarè patet. </s> <s xml:id="echoid-s8742" xml:space="preserve">Intelligatur autem in ſe-<lb/>quentibus, licet ſemper aſſumatur axis, tamen pro ſolidis ſimilari-<lb/>bus etiam aſſu mptis diametris eadem ibi appoſita verificari.</s> <s xml:id="echoid-s8743" xml:space="preserve"/> </p> <div xml:id="echoid-div834" type="float" level="2" n="1"> <figure xlink:label="fig-0367-01" xlink:href="fig-0367-01a"> <image file="0367-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0367-01"/> </figure> </div> </div> <div xml:id="echoid-div836" type="section" level="1" n="494"> <head xml:id="echoid-head514" xml:space="preserve">COROLLARIVM IV.</head> <p> <s xml:id="echoid-s8744" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s8745" xml:space="preserve">26. </s> <s xml:id="echoid-s8746" xml:space="preserve">veluti oſtendimus in eiuſdem figura hic appoſita <lb/> <anchor type="figure" xlink:label="fig-0367-02a" xlink:href="fig-0367-02"/> omnia quadrata portionis, BSF, <lb/>ad rectangula ſub portione, BSF, & </s> <s xml:id="echoid-s8747" xml:space="preserve"><lb/>figura diſtantiarum, SEF, eſſe vt, BF, <lb/>ad FE, ſic oſtenſum fuiſſet (aſſumptis <lb/>vice quadratorum alijs figuris ſimili-<lb/>bus, & </s> <s xml:id="echoid-s8748" xml:space="preserve">vice rectangulorum, aſſumptis <lb/>alijs ſimilibus figuris eius generis, vt <lb/>veluti eſt vnum quoduis dictorum <lb/>omnium quadratorum ad rectãgulum adiacens lateri, à quo deſcri-<lb/>bitur, ita ſit figura ab eodem latere deſcripta vice quadrati ſumpta, <lb/>ad figuram deſcriptam eodem latere vice rectanguli ſumptam, fiet <pb o="348" file="0368" n="368" rhead="GEOMETRI Æ."/> enim eodem modo demonſtratio his figuris aſſumptis) omnes fi-<lb/>guras ſimiles portionis, BSF, ad figuras vice rectangulorum ſum-<lb/>ptas eſſe pariter, vt, BF, ad, EF, & </s> <s xml:id="echoid-s8749" xml:space="preserve">pariter ſoliudm, quorum omnes <lb/>dictæ figuræ ſimiles vice quadratorum ſumptæ ſunt omnia plana, <lb/>ad ſolidum, quorum figuræ vice rectangulorum ſumptæ ſunt om-<lb/>nia plana, eſſe, vt, BF, ad, FE; </s> <s xml:id="echoid-s8750" xml:space="preserve">quæ quidem ſolida non ſunt ſolida <lb/>ad inuicem ſimilarià, quia vtriuſque ſolidi figuræ non ſunt inter ſe <lb/>fimiles, ſed tantum ſunt ſimiles inter ſe, quæ ſunt in vnoquoque <lb/>horum ſolidorum ſingillatim ſumpto.</s> <s xml:id="echoid-s8751" xml:space="preserve"/> </p> <div xml:id="echoid-div836" type="float" level="2" n="1"> <figure xlink:label="fig-0367-02" xlink:href="fig-0367-02a"> <image file="0367-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0367-02"/> </figure> </div> </div> <div xml:id="echoid-div838" type="section" level="1" n="495"> <head xml:id="echoid-head515" xml:space="preserve">COROLL. V. SECTIO I.</head> <p> <s xml:id="echoid-s8752" xml:space="preserve">IN Prop 27. </s> <s xml:id="echoid-s8753" xml:space="preserve">ſimiliter aſſumpta eiuſdem ſigura, vt fiat noſtrum <lb/>exemplum reuoluatur parabola, BAC, circa AP, axem, vt fiat <lb/> <anchor type="figure" xlink:label="fig-0368-01a" xlink:href="fig-0368-01"/> cono des parabolicum, BAC, à quo <lb/>per planum à, DZ, deſcriptum in re-<lb/>uolutione abſcindetur conoides para-<lb/>bolicum, DAF, cuius baſis rectè axim, <lb/>AP, ſecat, & </s> <s xml:id="echoid-s8754" xml:space="preserve">eſt circulus, intelligatur <lb/>autem etiam per, MC, planum ex-<lb/>tendi rectũ ad planũ parabolæ, BAC, <lb/>per hoc igitur abſcindetur pariter co <lb/>noides parabolicum, cuius baſis erit ellipſis, cuius maior diameter, <lb/>MC, minor autem erit, CR. </s> <s xml:id="echoid-s8755" xml:space="preserve">Dico nunc hæc duo conoidea eſſe <lb/>inter ſe æqualia, cum diametri eorundem, AZ, HO, ſint æquales: <lb/></s> <s xml:id="echoid-s8756" xml:space="preserve">ſi enim intellexerimus conoides, DAF, planis parallelis baſi ſecari, <lb/>& </s> <s xml:id="echoid-s8757" xml:space="preserve">pariter conoides, MHC, ſecari planis parallelis ſuæ baſi, fient, <lb/> <anchor type="note" xlink:label="note-0368-01a" xlink:href="note-0368-01"/> ductis omnibus eorundem planis, in conoide, DAF, dicta omnia <lb/>plana, omnes figuræ ſimiles inter ſe .</s> <s xml:id="echoid-s8758" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8759" xml:space="preserve">omnes circuli figuræ geni-<lb/>tricis, quæ eſt parabola, DAF; </s> <s xml:id="echoid-s8760" xml:space="preserve">in conoide verò, MHC, dicta omnia <lb/>plana fient omnes figuræ ſimiles genitricis, MHC, .</s> <s xml:id="echoid-s8761" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8762" xml:space="preserve">omnes ellipſes <lb/>eiuſdem, quarum coniugatæ diametri erunt inter ſe, vt, MC, ad, C <lb/>R, maiores diametros in figura genitrice, MHC, ſitas habentes. <lb/></s> <s xml:id="echoid-s8763" xml:space="preserve">Intelligantur nunc circa illas maiores diametros deſcribi circuli in <lb/>planis ellipſium iacentes, erit ergo quilibet circulus ad ellipſim ab <lb/>eo comprehenſam, vt maior diameter ad minorem, & </s> <s xml:id="echoid-s8764" xml:space="preserve">quia iſtę con-<lb/> <anchor type="note" xlink:label="note-0368-02a" xlink:href="note-0368-02"/> iugatæ diametri ſunt omnes inter ſe, maiores .</s> <s xml:id="echoid-s8765" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8766" xml:space="preserve">ad minores, vt M <lb/>C, ad, CR, .</s> <s xml:id="echoid-s8767" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8768" xml:space="preserve">vt quadratum, MC, ad rectangulum, MCR, & </s> <s xml:id="echoid-s8769" xml:space="preserve">vt <lb/>vnum ad vnum, ſic omnia ad omnia .</s> <s xml:id="echoid-s8770" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8771" xml:space="preserve">vt omnes circuli figuræ ge-<lb/>nitricis, MHC, ad omnes eiuſdem ſimiles ellipſes, ita circulus circa, <lb/>MC, ad ellipſim circa, MC, .</s> <s xml:id="echoid-s8772" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8773" xml:space="preserve">ſic quadratum, MC, ad rectangulum, <pb o="349" file="0369" n="369" rhead="LIBER IV."/> MCR, .</s> <s xml:id="echoid-s8774" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8775" xml:space="preserve">ita omnia quadrata figuræ genitricis, vel parabolæ, MH <lb/>C, ad rectangula, ſub parabola, MHC, & </s> <s xml:id="echoid-s8776" xml:space="preserve">figura diſtantiarum, H <lb/> <anchor type="note" xlink:label="note-0369-01a" xlink:href="note-0369-01"/> RC, .</s> <s xml:id="echoid-s8777" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8778" xml:space="preserve">ita ſolidum, cuius omnes circuli figuræ genitricis, MHC, <lb/>iuxta regulam baſim, MC, ſumpti, ſunt omnia plana, ad ſolidum, <lb/>cuius omnia plana ſunt omnes ſimiles ellipſes iam dictæ figuræ ge-<lb/>nitricis, MHC, ſumptæ iuxta eandem regulam, ſcilicet ad conoides <lb/>parabolicũ, MHC; </s> <s xml:id="echoid-s8779" xml:space="preserve">ſunt verò omnes circuli parabolæ, DAF, iuxta <lb/>regulam, DF, ad omnes circulos parabolæ, MHC, iuxta regulam, <lb/> <anchor type="note" xlink:label="note-0369-02a" xlink:href="note-0369-02"/> MC, ita omnia quadrata, DAF, ad omnia quadrata, MHC, iuxta <lb/>eaſdem regulas: </s> <s xml:id="echoid-s8780" xml:space="preserve">ideo ex æquali omnes circuli parabolæ, DAF, ad <lb/>omnes ellipſes ſimiles iam dictas parabolæ, MHC, erunt vt omnia <lb/>quadrata, DAF, retentis ſemper eiſdem regulis, ad rectangula ſub, <lb/>MHC, & </s> <s xml:id="echoid-s8781" xml:space="preserve">figura diſtantiarum, HRC, .</s> <s xml:id="echoid-s8782" xml:space="preserve">i. </s> <s xml:id="echoid-s8783" xml:space="preserve">omnes circuli, DAF, erunt <lb/>æquales omnibus ſimilibus ellipſibus iam dictis figuræ, MHC, ve-<lb/>rum omnes circuli parabolæ, DAF, ſumpti iu xta regulam, DF, <lb/>quorum diametri ſunt in figura genitrice, DAF, funt omnia plana <lb/>conoidis geniti in reuolutione ex ſemiparabola, DAZ, omnes verò <lb/>ellipſes ſimiles iam dictæ parabolæ, MHC, ſunt omnia plana co-<lb/> <anchor type="note" xlink:label="note-0369-03a" xlink:href="note-0369-03"/> noidis parabolici reſecti a plano ducto per, MC, ergo conoides pa-<lb/>rabolicum, DAF, eſt æquale conoidi parabolico, MHC. </s> <s xml:id="echoid-s8784" xml:space="preserve">Sed vni-<lb/>uerſaliter ſolidum genitum ex, DAF, habens omnia plana, quæ ſint <lb/>omnes figuræ ſimiles inter ſe eiuſdem genitricis, DAF, erit æquale <lb/>ſolido genito ex, MHC, habenti omnia plana, quæ ſint omnes fi-<lb/>guræ ſimiles inter ſe eiuſdem genitricis figuræ, MHC, ad quas om-<lb/>nes figuræ ſimiles figuræ genitricis, DAF, ſint vt omnia quadrata, <lb/>DAF, ad rectangula ſub figura genitrice, MHC, & </s> <s xml:id="echoid-s8785" xml:space="preserve">figura diſtantia-<lb/>rum, HRC, dummodo diametri, AZ, HO, inter ſe ſint æquales, <lb/>quæ hic nobis erant colligenda.</s> <s xml:id="echoid-s8786" xml:space="preserve"/> </p> <div xml:id="echoid-div838" type="float" level="2" n="1"> <figure xlink:label="fig-0368-01" xlink:href="fig-0368-01a"> <image file="0368-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0368-01"/> </figure> <note position="left" xlink:label="note-0368-01" xlink:href="note-0368-01a" xml:space="preserve">45. l. 1.</note> <note position="left" xlink:label="note-0368-02" xlink:href="note-0368-02a" xml:space="preserve">10. l. 3.</note> <note position="right" xlink:label="note-0369-01" xlink:href="note-0369-01a" xml:space="preserve">3. l. 2.</note> <note position="right" xlink:label="note-0369-02" xlink:href="note-0369-02a" xml:space="preserve">Iuxta reg. <lb/>DF.</note> <note position="right" xlink:label="note-0369-03" xlink:href="note-0369-03a" xml:space="preserve">Elicitur <lb/>ex Corol. <lb/>47. l. 1.</note> </div> </div> <div xml:id="echoid-div840" type="section" level="1" n="496"> <head xml:id="echoid-head516" xml:space="preserve">SECTIO II.</head> <p> <s xml:id="echoid-s8787" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s8788" xml:space="preserve">v. </s> <s xml:id="echoid-s8789" xml:space="preserve">colligitur ſolidum ſimilare genitum ex parabola, D <lb/>AF, ad ſibi ſimilare genitum ex parabola, MHC, eſſe vt, DF, <lb/>ad MC, dum, AZ, HO, diametri fuerint æquales.</s> <s xml:id="echoid-s8790" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div841" type="section" level="1" n="497"> <head xml:id="echoid-head517" xml:space="preserve">SECTIO III.</head> <p> <s xml:id="echoid-s8791" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s8792" xml:space="preserve">2. </s> <s xml:id="echoid-s8793" xml:space="preserve">colligitur, ſi fuerint duo plana axem conoidis pa-<lb/>rabolicæ obliquè ſecantia, ſint autem abſciſſarum conoidum <lb/>diametri inter ſe æquales, quod abſciſsæ conoides erunt inter ſe <lb/>æquales; </s> <s xml:id="echoid-s8794" xml:space="preserve">ſed vniuerſaliter, vice ſimilium ellipſium, quæ ſunt om- <pb o="350" file="0370" n="370" rhead="GEOMETRIÆ"/> nia plana dictarum conoidum, alijs figuris ſimilibus ſeorſim in <lb/>vnoquoque ſolido aſsumptis, inter ſe eandem rationem, quam prę-<lb/>dictæ ſimiles ellipſes habentibus, quod ea ſolida, quorum aſsum-<lb/>ptæ fimiles figuræ ſunt omnia plana, erunt inter ſe æ qualia, dum <lb/>diametri genitricium eorundem figurarum, quæ ſunt abſciſsæ pa-<lb/>rabolæ, inter ſe quoq; </s> <s xml:id="echoid-s8795" xml:space="preserve">æquales fuerint.</s> <s xml:id="echoid-s8796" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div842" type="section" level="1" n="498"> <head xml:id="echoid-head518" xml:space="preserve">COROLLARIVM VI.</head> <p> <s xml:id="echoid-s8797" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s8798" xml:space="preserve">28. </s> <s xml:id="echoid-s8799" xml:space="preserve">& </s> <s xml:id="echoid-s8800" xml:space="preserve">eius Coroll. </s> <s xml:id="echoid-s8801" xml:space="preserve">aſsum pta illius figura, & </s> <s xml:id="echoid-s8802" xml:space="preserve">facto ſo-<lb/>lito exemplo per reuolutionem, ADH, parabolæ circa axim, <lb/> <anchor type="figure" xlink:label="fig-0370-01a" xlink:href="fig-0370-01"/> DO, habetur, quod ſi conois paraboli-<lb/>ca, ADH, in reuolutione deſcripta <lb/>ſecetur quomodocunque planis ſiue <lb/>ad axem rectis, ſiue obliquis, quod ab-<lb/>fcifsæ conoides erunt inter ſe, vt qua-<lb/>drata diametrorum eorundem, Nam <lb/>vt omnia quadrata, BDF, regula, BF, <lb/>quæ axim, DO, rectè ſecat, ad rectan-<lb/>gula ſub parabola, CEG, & </s> <s xml:id="echoid-s8803" xml:space="preserve">figura di-<lb/>ftantiarum, ERG, ita eſse omnes circulos, BDF, diametros in ea ſi-<lb/>tas habentes, ſumptos iuxta regulam, BF, ad omnes ſimiles elli-<lb/>pſes figuræ genitricis, CEG, ſumptas iuxta regulam, CG, quarum <lb/>diametri maiores ſunt in figura, CEG, minores verò in figura di-<lb/>ſtantiarum, REG, oſtendemus, methodo antecedentis, ergo dicti <lb/>omnes circuli parabolæ, BDF, ad dictas omnes ellipſes parabolæ, <lb/>CEG, erunt vt quadratum, DN, ad quadratum, EM, ergo & </s> <s xml:id="echoid-s8804" xml:space="preserve">co-<lb/>nois parabolica, BDF, ad conoidem parabolicam, CEG, erit vt <lb/>quadratum, DN, ad quadratum, EM, vnde, conuertendo, conois <lb/>parabolica, GEC, ad conoidem parabolicam, FDB, erit vt qua-<lb/>dratum, EM, ad quadratum, DN, ſi ergo aliud planum, vtcunq; <lb/></s> <s xml:id="echoid-s8805" xml:space="preserve">obliquè axem, DO, ſecauerit, erit conois parabolica, BDF, ad <lb/>hanc conoidem vltimò reſectam, vt quadratum, DN, ad quadra-<lb/>tum diametri huius reſectæ conodis, ergo ex æquali conois pa-<lb/>rabolica, CEG, ad hanc conoidem vltimò reſectam, cuius baſis <lb/>pariter obliquè ſecat axim, DO, erit vt quadratum, EM, ad huius <lb/>diametri quadratum, quomodocunque igitur reſecetur conois pla-<lb/>nis axem ſecantibus, reſecta ſegmenta ſunt, vt diametrorum qua-<lb/>drata. </s> <s xml:id="echoid-s8806" xml:space="preserve">Sed vniuerſaliter, ſi, vice circulorum, vel dictarum ellipſium, <lb/>ſummamus alias figuras ſimiles in vnoquoq; </s> <s xml:id="echoid-s8807" xml:space="preserve">ſolido ſeorſim, quo-<lb/>rum ſunt omnia plana, ijs exiſtentibus omnibus figuris ſimilibus <pb o="351" file="0371" n="371" rhead="LIBER IV."/> genitricium figurarum, quales ſunt parabolæ, BDF, CEG, dicta ex <lb/>ijſdem genita iolida iuxta regulas baſes abſciſſarum parabolarum, <lb/>ſi dictæ figuræ ſimiles fuerint inter ſe, vt prædicti circuli, vel ſimi-<lb/>les e lipſes, vel vt omnia quadrata, & </s> <s xml:id="echoid-s8808" xml:space="preserve">rectangula ſub abſciſſis pa-<lb/>rabolis, & </s> <s xml:id="echoid-s8809" xml:space="preserve">figuris diſtantiarum earundem, regulis ſemper pro vna-<lb/>quaque earundem parabolarum baſibus ſumptis, erunt inter ſe, vt <lb/>quadrata diametrorum abſciſſarum per ducta plana parabolarum, <lb/>intellige tamen reſecantia plana ſemper in ſupradictis eſſe erecta <lb/>plano genitricium figurarum, vt planum per, CG, erectum para-<lb/>bolæ, ADH, plano, ſimiſiter & </s> <s xml:id="echoid-s8810" xml:space="preserve">quod per, BF, ſiue in conoide, ſiue <lb/>in alijsiam dictis ſolidis, vt ſupradictum eſt genitis.</s> <s xml:id="echoid-s8811" xml:space="preserve"/> </p> <div xml:id="echoid-div842" type="float" level="2" n="1"> <figure xlink:label="fig-0370-01" xlink:href="fig-0370-01a"> <image file="0370-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0370-01"/> </figure> </div> </div> <div xml:id="echoid-div844" type="section" level="1" n="499"> <head xml:id="echoid-head519" xml:space="preserve">APPENDIX.</head> <p> <s xml:id="echoid-s8812" xml:space="preserve">EXponatur parabola, ACE, circa axim, CM, in baſi, AE, cui <lb/>paraliela ducatur vtcunque, BD, intra ipſam, & </s> <s xml:id="echoid-s8813" xml:space="preserve">iungatur, BE, <lb/> <anchor type="figure" xlink:label="fig-0371-01a" xlink:href="fig-0371-01"/> ducaturque, RS, diameter parabolæ, <lb/>BRE, & </s> <s xml:id="echoid-s8814" xml:space="preserve">vt fiat noſtrum exemplum <lb/>reuo uatur parabola, ACE, circa axim <lb/>manentem, CM, vt fiant conoides <lb/>parabolicæ, ACE, BCD, & </s> <s xml:id="echoid-s8815" xml:space="preserve">per BE, <lb/>ducatur planum erectum plano para-<lb/>bolæ, ACE, ſcindens fruſtum conoi-<lb/>dis, BAED, in duas portiones, ſcilicer, <lb/>BAE, BDE. </s> <s xml:id="echoid-s8816" xml:space="preserve">Dico ergo portionem, BAE, ad portionem, BDE, <lb/>(reſecta, CO, æquali ipſi, RS,) eſſe vt quadratũ, MO, cũ rectangulo <lb/>bis ſub, MOC, ad quadratum, ON, cum rectangulo bis ſub, ONC.</s> <s xml:id="echoid-s8817" xml:space="preserve"/> </p> <div xml:id="echoid-div844" type="float" level="2" n="1"> <figure xlink:label="fig-0371-01" xlink:href="fig-0371-01a"> <image file="0371-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0371-01"/> </figure> </div> <p> <s xml:id="echoid-s8818" xml:space="preserve">Nam conois, ACE, ad conoidem, BRE, eſt vt quadratum, MC, <lb/>ad quadratum, SR, vel ad quadratum, OC, ergo, per conuerſionem <lb/>rationis, & </s> <s xml:id="echoid-s8819" xml:space="preserve">conuertendo, portio ſolida, BAE, ad conoidem para-<lb/>bolicam, ACE, erit vt reſiduum quadrati, MC, dempto quadrato, <lb/>OC, ad quadratum, MC, .</s> <s xml:id="echoid-s8820" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8821" xml:space="preserve">vt quadratum, MO, cum rectangulo <lb/>bis ſub, MOC, ad quadratum, MC, quod ſerua. </s> <s xml:id="echoid-s8822" xml:space="preserve">Item quia conoi-<lb/>dem, ACE, ad conoidem, BRE, diximus eſſe vt quadratum, MC, <lb/>ad quadratum, CO, eadem autem conois, ACE, ad conoidem, BCD, <lb/>eſt vt quadratum, MC, ad quadratum. </s> <s xml:id="echoid-s8823" xml:space="preserve">CN, ergo conois, ACE, ad <lb/>reliquum dempta conoide, BCD, à conoide, BRE, erit vt idem qua-<lb/>dratum, MC, ad reliquum, dempto quadrato, CN, à quadrato, CO, <lb/>.</s> <s xml:id="echoid-s8824" xml:space="preserve">i. </s> <s xml:id="echoid-s8825" xml:space="preserve">ad quadratum, ON, cum rectangulo bis ſub, ONC, eſt ergo co-<lb/>nois, ACE, ad portionem ſolidam, BDE, vt quadratum, MC, ad <lb/>quadratum, ON, cum rectangulo bis ſub, ONC, erat autem portio <pb o="352" file="0372" n="372" rhead="GEOMETRIÆ"/> ſolida, BAE, ad conoidem parabolicam, ACE, vt quadratum, MO, <lb/>cum rectangulo bis ſub, MOC, ad quadratum, MC, ergo, ex æqua-<lb/>li, portio ſolida, ABE, ad portionem ſolidam, BDE, erit vt qua-<lb/>dratum, MO, cum rectangulo bis ſub, MOC, ad quad. </s> <s xml:id="echoid-s8826" xml:space="preserve">ON, cum re-<lb/>ctang. </s> <s xml:id="echoid-s8827" xml:space="preserve">bis ſub, ONC, quod &</s> <s xml:id="echoid-s8828" xml:space="preserve">c.</s> <s xml:id="echoid-s8829" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8830" xml:space="preserve">Sed vniuerſaliter ſi ſint ſolida ſimilaria genita ex parabolis, ACE, <lb/>BCD, iuxta communem regulam, AE, & </s> <s xml:id="echoid-s8831" xml:space="preserve">ducatur planum per, BE, <lb/>rectum plano parabolæ, ACE, ſcindens ſolidum ſimilare genitum <lb/>ex, BDEA, in duas portiones ſolidas, BAE, BDE, adhuc, conſe-<lb/>quenter ſupradictis, inueniemus has duas portiones ſolidas eſſe in <lb/>eadem ratione, vt portiones ſolidæ productæ ex ſectione fruſti co-<lb/>noidis parabolicæ, BAED, .</s> <s xml:id="echoid-s8832" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8833" xml:space="preserve">eſſe vt quadratum, MO, cum rectan-<lb/>gulo bis ſub, MOC, ad quadrat, ũON, cum rectangulo bis ſub, ONC, <lb/>quod ex ſupradictis erui facilè poteſt; </s> <s xml:id="echoid-s8834" xml:space="preserve">quæ demonſtratio currit etiã, <lb/>ſi, CM, non ſit axis, ſed tantum diameter, vt confideranti clarè <lb/>patebit.</s> <s xml:id="echoid-s8835" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div846" type="section" level="1" n="500"> <head xml:id="echoid-head520" xml:space="preserve">A. COROLL. VII. SECTIO I.</head> <note position="left" xml:space="preserve">A.</note> <p> <s xml:id="echoid-s8836" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8837" xml:space="preserve">29. </s> <s xml:id="echoid-s8838" xml:space="preserve">& </s> <s xml:id="echoid-s8839" xml:space="preserve">Cor. </s> <s xml:id="echoid-s8840" xml:space="preserve">Sect. </s> <s xml:id="echoid-s8841" xml:space="preserve">1. </s> <s xml:id="echoid-s8842" xml:space="preserve">& </s> <s xml:id="echoid-s8843" xml:space="preserve">2. </s> <s xml:id="echoid-s8844" xml:space="preserve">colligimus ſolida ſimilaria geni-<lb/>ta ex parabolis in eadem altitudine conſtitutis, genita inquam <lb/>iuxta regulas ipſarum baſes, eſſe inter ſe, vt quadrata baſium, & </s> <s xml:id="echoid-s8845" xml:space="preserve">in <lb/>ijſdem baſibus conſtitutis, vt earum altitudines, vel vt diametros <lb/>æqualiter baſibus inclinatas; </s> <s xml:id="echoid-s8846" xml:space="preserve">hoc igitur nedum concluditur de co-<lb/>noidibus parabolicis in eadem altitudine ſtantibus, quod ſit, vt qua-<lb/>drata baſium, vel in eadem baſi exiſtentium, quod ſint, vt altitu-<lb/>dines, ſed de cæteris ſimilaribus ſolidis ex ipſis parabolis genitis <lb/>iuxta regulas baſes, vt dictum eſt.</s> <s xml:id="echoid-s8847" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div847" type="section" level="1" n="501"> <head xml:id="echoid-head521" xml:space="preserve">B. SECTIO II.</head> <note position="left" xml:space="preserve">B.</note> <p> <s xml:id="echoid-s8848" xml:space="preserve">ITem habemus conoides parabolicas, & </s> <s xml:id="echoid-s8849" xml:space="preserve">cætera ſolida ſimilaria <lb/>ex parabolis genita iuxta regulas baſes, habere inter ſe ratio-<lb/>nem eompoſitam ex ratione quadratorum baſium, & </s> <s xml:id="echoid-s8850" xml:space="preserve">altitudi-<lb/>num, vel diametrorum æqualiter baſibus inclinatarum.</s> <s xml:id="echoid-s8851" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div848" type="section" level="1" n="502"> <head xml:id="echoid-head522" xml:space="preserve">C. SECTIO III.</head> <note position="left" xml:space="preserve">C.</note> <p> <s xml:id="echoid-s8852" xml:space="preserve">ITem eadem ſolida, quarum baſes altitudinibus, vel diametris <lb/>æqualiter baſibus inclinatis reciprocantur, eſſe æqualia, & </s> <s xml:id="echoid-s8853" xml:space="preserve"><lb/>quæ ſunt æqualia habere baſes altitudinibus, vel diametris æqua-<lb/>liter baſibus inclinatis, reciprocas.</s> <s xml:id="echoid-s8854" xml:space="preserve"/> </p> <pb o="353" file="0373" n="373" rhead="LIBER IV."/> </div> <div xml:id="echoid-div849" type="section" level="1" n="503"> <head xml:id="echoid-head523" xml:space="preserve">D. SECTIO IV.</head> <note position="right" xml:space="preserve">D</note> <p> <s xml:id="echoid-s8855" xml:space="preserve">TAndem colligemus conoides parabolicas, & </s> <s xml:id="echoid-s8856" xml:space="preserve">cætera ſolida ſi-<lb/>milaria ex parabolis genita iuxta regulas ipſarum baſes, qua-<lb/> <anchor type="note" xlink:label="note-0373-02a" xlink:href="note-0373-02"/> rum axes, vel diametri ad homologas baſium diametros, vel late-<lb/>ra habeant eandem ratione .</s> <s xml:id="echoid-s8857" xml:space="preserve">i. </s> <s xml:id="echoid-s8858" xml:space="preserve">ſimiles conoides parabolicas, & </s> <s xml:id="echoid-s8859" xml:space="preserve">ſi-<lb/>milia ſolida ſimilaria genita ex parabolis iam dictis, eſſe in tripla <lb/>ratione dictarum homologarum linearum.</s> <s xml:id="echoid-s8860" xml:space="preserve"/> </p> <div xml:id="echoid-div849" type="float" level="2" n="1"> <note position="right" xlink:label="note-0373-02" xlink:href="note-0373-02a" xml:space="preserve">46. l. 1.</note> </div> </div> <div xml:id="echoid-div851" type="section" level="1" n="504"> <head xml:id="echoid-head524" xml:space="preserve">+ COROLL. VIII. SECTIO I.</head> <note position="right" xml:space="preserve">+</note> <p> <s xml:id="echoid-s8861" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8862" xml:space="preserve">30. </s> <s xml:id="echoid-s8863" xml:space="preserve">expoſita figura, vt fiat ſolitum exemplum, reuo-<lb/>luatur, ACD, circa manentem axim, DC, patebit ergo cylin-<lb/> <anchor type="figure" xlink:label="fig-0373-01a" xlink:href="fig-0373-01"/> drum genitum ex, BD, in reuolutione <lb/>.</s> <s xml:id="echoid-s8864" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8865" xml:space="preserve">BF, eſſe ſexcuplum ſolidi geniti ex <lb/>trilineo, CDA, .</s> <s xml:id="echoid-s8866" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8867" xml:space="preserve">ſolidi, CAF. </s> <s xml:id="echoid-s8868" xml:space="preserve">Sed <lb/>vniuerſaliter ſolidum ſimilare geni-<lb/>tum ex, BD, ad ſibi ſimilare genitum <lb/>ex, CDA, ſexcuplam rationem habe-<lb/>re, ſiue CD, ſit perpendicularis ipſi, <lb/>DA, ſiue non; </s> <s xml:id="echoid-s8869" xml:space="preserve">vocetur autem ſoli <lb/>dum genitum per reuolutionem ex, C <lb/>DA, Apex parabolicus.</s> <s xml:id="echoid-s8870" xml:space="preserve"/> </p> <div xml:id="echoid-div851" type="float" level="2" n="1"> <figure xlink:label="fig-0373-01" xlink:href="fig-0373-01a"> <image file="0373-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0373-01"/> </figure> </div> </div> <div xml:id="echoid-div853" type="section" level="1" n="505"> <head xml:id="echoid-head525" xml:space="preserve">A. SECTIO II.</head> <note position="right" xml:space="preserve">A</note> <p> <s xml:id="echoid-s8871" xml:space="preserve">IN Corollario autem colligimusapices parabolicos in eadem <lb/>altitudine exiſtentes, eſſe vt baſium quadrata, & </s> <s xml:id="echoid-s8872" xml:space="preserve">in eiſdem ba-<lb/>ſibus eſſe, vt altitudines, ſic etiam eſſe ſolida ſimilaria genita ex <lb/>trilineis in eadem altitudine, vel in eadem baſi exiſtentibus, geni-<lb/>ta inquam iuxta regulas tangentes ipſas parabolas.</s> <s xml:id="echoid-s8873" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div854" type="section" level="1" n="506"> <head xml:id="echoid-head526" xml:space="preserve">B. SECTIO III.</head> <note position="right" xml:space="preserve">B</note> <p> <s xml:id="echoid-s8874" xml:space="preserve">ITem, quod eadem ſolida quomodocunque ſint, habeant inter <lb/>ſe rationem compoſitam ex ratione baſium, & </s> <s xml:id="echoid-s8875" xml:space="preserve">altitudinum, <lb/>vel ſecantium æqualiter tangentibus inclinatarum.</s> <s xml:id="echoid-s8876" xml:space="preserve"/> </p> <pb o="354" file="0374" n="374" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div855" type="section" level="1" n="507"> <head xml:id="echoid-head527" xml:space="preserve">C. SECTIO IV.</head> <note position="left" xml:space="preserve">C</note> <p> <s xml:id="echoid-s8877" xml:space="preserve">ITem, quod eadem ſolida baſes habentia altitudinibus; </s> <s xml:id="echoid-s8878" xml:space="preserve">vel ſe-<lb/>cantibus æqualiter tangentibus inclinatis reciprocas, ſint æ-<lb/>qualia; </s> <s xml:id="echoid-s8879" xml:space="preserve">& </s> <s xml:id="echoid-s8880" xml:space="preserve">quæ ſunt æqualia, baſes habeant altitudinibus, vel ſe-<lb/>cantibus æqualiter tangentibus inclinatis reciprocas.</s> <s xml:id="echoid-s8881" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div856" type="section" level="1" n="508"> <head xml:id="echoid-head528" xml:space="preserve">D. SECTIO V.</head> <note position="left" xml:space="preserve">D</note> <p> <s xml:id="echoid-s8882" xml:space="preserve">TAndem, quod eadem ſolida ſint in tripla ratione tangen-<lb/>tium, vel ſecantium parabolas; </s> <s xml:id="echoid-s8883" xml:space="preserve">ſi tangentes ad ſecantes ſe-<lb/>miparabolas, ex quibus in reuolutione generantur, habeant ean-<lb/>dem rationem.</s> <s xml:id="echoid-s8884" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div857" type="section" level="1" n="509"> <head xml:id="echoid-head529" xml:space="preserve">COROLLARIVM IX.</head> <p> <s xml:id="echoid-s8885" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s8886" xml:space="preserve">31. </s> <s xml:id="echoid-s8887" xml:space="preserve">expoſiea figura, & </s> <s xml:id="echoid-s8888" xml:space="preserve">vt fiat noſtrum exemplum re-<lb/>uoluto, AC, circa manentem axim, AB, patet ſolidum, quod <lb/> <anchor type="figure" xlink:label="fig-0374-01a" xlink:href="fig-0374-01"/> in reuolutione fit ex trilineo, AD <lb/>C, ad ſolidum, quod fit ex trilineo, <lb/>MFC, eſſe vt quadratum, AB, ad <lb/>quadratum, BE, & </s> <s xml:id="echoid-s8889" xml:space="preserve">vniuerſaliter, ſo-<lb/>lidum ſimilare genitum ex, AC, <lb/>dempto ſolido ſimilari genito ex ſe-<lb/>miparabola, ACB, ad ſibi ſimilare <lb/>genitum ex, EC, dempto ſolido ſi-<lb/>milari genito ex fruſto, EMCB, <lb/>eſſe vt quadratum, AB, ad quadratum, BE, genita, in quam in-<lb/>tellige iuxta communem regulam, BC.</s> <s xml:id="echoid-s8890" xml:space="preserve"/> </p> <div xml:id="echoid-div857" type="float" level="2" n="1"> <figure xlink:label="fig-0374-01" xlink:href="fig-0374-01a"> <image file="0374-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0374-01"/> </figure> </div> </div> <div xml:id="echoid-div859" type="section" level="1" n="510"> <head xml:id="echoid-head530" xml:space="preserve">COROLL X. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s8891" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8892" xml:space="preserve">32. </s> <s xml:id="echoid-s8893" xml:space="preserve">expoſita figura, & </s> <s xml:id="echoid-s8894" xml:space="preserve">vt fiat noſtrum exemplum re-<lb/>uoluto, AF, circa manentem axim, CF, patebit cylindrum <lb/>in reuolutione genitum ex AF, ad ſolidum genitum ex parabola, <lb/>DBF, eſſe vt, AF, ad parabolam, DBF, & </s> <s xml:id="echoid-s8895" xml:space="preserve">ita eſſe quodlibet ſo-<lb/>lidum ſimilare genitum ex, AF, ad ſibi ſimilare genitum ex figu-<lb/>ra, CBDF, dempto ſolido ſimilari genito ex trilineo, BCF; </s> <s xml:id="echoid-s8896" xml:space="preserve">cy-<lb/>lindrum verò genitum ex, AF, .</s> <s xml:id="echoid-s8897" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8898" xml:space="preserve">AM, ad ſolidum in reuolutione <lb/>genitum ex figura, CBDF, eſſe vt, AF, ad parabolam, DBF, <pb o="355" file="0375" n="375" rhead="LIBER IV."/> cum {1/24}. </s> <s xml:id="echoid-s8899" xml:space="preserve">parallelogrammi, AF, .</s> <s xml:id="echoid-s8900" xml:space="preserve">1. </s> <s xml:id="echoid-s8901" xml:space="preserve">vt 24. </s> <s xml:id="echoid-s8902" xml:space="preserve">ad 17. </s> <s xml:id="echoid-s8903" xml:space="preserve">& </s> <s xml:id="echoid-s8904" xml:space="preserve">ita eſſe ſolidum <lb/> <anchor type="figure" xlink:label="fig-0375-01a" xlink:href="fig-0375-01"/> ſimilare ge-<lb/>nitum ex, A <lb/>F, ad ſibi ſi-<lb/>milare geni <lb/>tum ex figu-<lb/>ra, CBDF, <lb/>genita in-<lb/>quam iuxta <lb/>communem <lb/>regulam, D <lb/>F. </s> <s xml:id="echoid-s8905" xml:space="preserve">Vocetur <lb/>autem ſolidum, quod in reuolutione generatur ex parabola DB <lb/>F. </s> <s xml:id="echoid-s8906" xml:space="preserve">Semianulus ſtrictus parabolicus; </s> <s xml:id="echoid-s8907" xml:space="preserve">quod verò gignitur ex, figu-<lb/>ra, CBDF; </s> <s xml:id="echoid-s8908" xml:space="preserve">Semibaſis columnaris parabolica ſtricta.</s> <s xml:id="echoid-s8909" xml:space="preserve"/> </p> <div xml:id="echoid-div859" type="float" level="2" n="1"> <figure xlink:label="fig-0375-01" xlink:href="fig-0375-01a"> <image file="0375-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0375-01"/> </figure> </div> </div> <div xml:id="echoid-div861" type="section" level="1" n="511"> <head xml:id="echoid-head531" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s8910" xml:space="preserve">IN Corollario colligitur cylindrum, AM, eſſe ſexquialterum <lb/>ſemianuli ſtricti parabolici, DBFXM, vnde colligi poteſt pro-<lb/>prietates, quæ conoidibus, vel apicibus parabolicis in Corollarijs <lb/>7. </s> <s xml:id="echoid-s8911" xml:space="preserve">& </s> <s xml:id="echoid-s8912" xml:space="preserve">8. </s> <s xml:id="echoid-s8913" xml:space="preserve">Propoſit. </s> <s xml:id="echoid-s8914" xml:space="preserve">51. </s> <s xml:id="echoid-s8915" xml:space="preserve">huius in eſſe oſtenſa ſunt, & </s> <s xml:id="echoid-s8916" xml:space="preserve">de ſemianulis ſtri-<lb/>ctis parabolicis pariter concludi.</s> <s xml:id="echoid-s8917" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div862" type="section" level="1" n="512"> <head xml:id="echoid-head532" xml:space="preserve">COROLLARIVM XI.</head> <p> <s xml:id="echoid-s8918" xml:space="preserve">IN Propoſit. </s> <s xml:id="echoid-s8919" xml:space="preserve">33. </s> <s xml:id="echoid-s8920" xml:space="preserve">habemus partem interiorem ſemianuli ſtricti <lb/>parabolici, ad exteriorem (quæ partes diſſeparantur per ſuper-<lb/>ficiem in reuolutione deſcriptam in ſuperioris figura per lineam, ſi-<lb/>ue axim, BE,) eſse vt 5. </s> <s xml:id="echoid-s8921" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8922" xml:space="preserve">& </s> <s xml:id="echoid-s8923" xml:space="preserve">ſic eſse quodlibet ſolidum ſimila-<lb/>re genitum ex, BF, dempto ſolido ſimilari genito ex trilineo, BC <lb/>F, ad ſibi ſimilare genitum ex figura, CBDF, dempto ſolido ſi-<lb/>milari genito ex, BF, genita, inquam, iuxta communem regu-<lb/>lam, DF.</s> <s xml:id="echoid-s8924" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div863" type="section" level="1" n="513"> <head xml:id="echoid-head533" xml:space="preserve">COROLL. XII. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s8925" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8926" xml:space="preserve">34. </s> <s xml:id="echoid-s8927" xml:space="preserve">aſſumpta eiuſdem figura, vt fiat exemplum reuo-<lb/>luatur, AM, circa manentem axim, HM, fiat autem ex, AM, <lb/>in reuolutione cylindrus, AL; </s> <s xml:id="echoid-s8928" xml:space="preserve">patet igitur cylindrum, AL, ad ſo-<lb/>lidum in reuolutione genitum ex parabola, DBF, eſſe vt, AF, ad <pb o="356" file="0376" n="376" rhead="GEOMETRIÆ."/> parabolam, DBF, (hoc autem vocetur Semianulus latus parabot <lb/>licus) & </s> <s xml:id="echoid-s8929" xml:space="preserve">ad ſolidum genitum ex figura, HBDM, eſſe vt quadra. <lb/></s> <s xml:id="echoid-s8930" xml:space="preserve">tum, DM, ad quadratum, ME, {1/2}. </s> <s xml:id="echoid-s8931" xml:space="preserve">quadrati, ED, cum rectangu-<lb/>lo ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s8932" xml:space="preserve">ſub, EM, quod vocetur Semibaſis co-<lb/>lumnaris parabolica lata; </s> <s xml:id="echoid-s8933" xml:space="preserve">Et vniuerſaliter ſolidum ſimilare geni-<lb/> <anchor type="figure" xlink:label="fig-0376-01a" xlink:href="fig-0376-01"/> tum ex, AM, ad ſibi ſimilare genitum ex figura, HBDM, habe-<lb/>re eandem rationem proximè dictæ ad idem verò dempto ſolido <lb/>ſimilari genito ex quadrilineo, BFMH, eſse vt, AF, ad parabo-<lb/>lam, DBF,.</s> <s xml:id="echoid-s8934" xml:space="preserve">. in ratione ſexq uialtera.</s> <s xml:id="echoid-s8935" xml:space="preserve"/> </p> <div xml:id="echoid-div863" type="float" level="2" n="1"> <figure xlink:label="fig-0376-01" xlink:href="fig-0376-01a"> <image file="0376-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0376-01"/> </figure> </div> </div> <div xml:id="echoid-div865" type="section" level="1" n="514"> <head xml:id="echoid-head534" xml:space="preserve">SECTIO POSTERIOR,</head> <p> <s xml:id="echoid-s8936" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s8937" xml:space="preserve">poteſt colligi etiam in Cor. </s> <s xml:id="echoid-s8938" xml:space="preserve">10. </s> <s xml:id="echoid-s8939" xml:space="preserve">Prop. </s> <s xml:id="echoid-s8940" xml:space="preserve">51. </s> <s xml:id="echoid-s8941" xml:space="preserve">Sect. </s> <s xml:id="echoid-s8942" xml:space="preserve">po-<lb/>ſter. </s> <s xml:id="echoid-s8943" xml:space="preserve">concludi poſse cylindrum in reuolutione genitum ex, AF, <lb/>ad ſemibaſim columnarem ſtrictam parabolicam genitam ex figu-<lb/>ra, CBDF, eſse vt quadratum, DF, ad quadratum, FE, {1/2}, qua <lb/>drati, ED, & </s> <s xml:id="echoid-s8944" xml:space="preserve">rectangulum ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s8945" xml:space="preserve">ſub, EF, & </s> <s xml:id="echoid-s8946" xml:space="preserve"><lb/>ſic eſſe ſolida ſimilaria ex eiſdem genita iuxta communem regu-<lb/>lam DF.</s> <s xml:id="echoid-s8947" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div866" type="section" level="1" n="515"> <head xml:id="echoid-head535" xml:space="preserve">COROLL. XIII. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s8948" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8949" xml:space="preserve">35. </s> <s xml:id="echoid-s8950" xml:space="preserve">iterum aſsumpta antecedentis figura, patet cylin-<lb/>drum genitum in reuolutione ex, BM, .</s> <s xml:id="echoid-s8951" xml:space="preserve">i. </s> <s xml:id="echoid-s8952" xml:space="preserve">cylindrum, BY, ad <lb/>ſolidum genitum ex reuolutione figuræ, BHMF, .</s> <s xml:id="echoid-s8953" xml:space="preserve">ſ. </s> <s xml:id="echoid-s8954" xml:space="preserve">ad ſolidum, <lb/>BFKR, quod vocetur Semitympanum parabolicum, eſse vt qua-<lb/>dratum, EM, ad quadratum, MF, cum rectangulo ſub {2/3}. </s> <s xml:id="echoid-s8955" xml:space="preserve">EF, & </s> <s xml:id="echoid-s8956" xml:space="preserve"><lb/>FM, vna cum {1/6}. </s> <s xml:id="echoid-s8957" xml:space="preserve">quadrati, EF; </s> <s xml:id="echoid-s8958" xml:space="preserve">& </s> <s xml:id="echoid-s8959" xml:space="preserve">ſic eſse ſolidum ſimilare geni-<lb/>tum ex, BM, ad ſibi ſimilare genitum ex figura, BHMF, iuxta <lb/>communem regulam, DM.</s> <s xml:id="echoid-s8960" xml:space="preserve"/> </p> <pb o="357" file="0377" n="377" rhead="LIBER IV."/> </div> <div xml:id="echoid-div867" type="section" level="1" n="516"> <head xml:id="echoid-head536" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s8961" xml:space="preserve">EX Coroll. </s> <s xml:id="echoid-s8962" xml:space="preserve">habetur cylindrum, BY, ad reſiduum, dempto ſe-<lb/>mitympano parabolico, BFKR, ab eodem, eſse vt quadra-<lb/>tum, EM, ad rectangulum ſub, MF, & </s> <s xml:id="echoid-s8963" xml:space="preserve">lub ſexquitertia, FE, cum <lb/>{5/6}. </s> <s xml:id="echoid-s8964" xml:space="preserve">quadrati, FE, & </s> <s xml:id="echoid-s8965" xml:space="preserve">ſic eſse ſolidum ſimilare genitum ex, BM, ad <lb/>reſiduum, dempto ad eodem ſolido ſimilari genito ex figura, BH <lb/>MF, iuxta communem regulam, DM.</s> <s xml:id="echoid-s8966" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div868" type="section" level="1" n="517"> <head xml:id="echoid-head537" xml:space="preserve">COROLLARIVM XIV.</head> <p> <s xml:id="echoid-s8967" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8968" xml:space="preserve">36. </s> <s xml:id="echoid-s8969" xml:space="preserve">vila adhuc eadem figura, patet portionum ſemia-<lb/>nul@ lati parabolici ex, DBF, parabola in reuolutione geni-<lb/>ti, quæ ſepatantur a ſuperſicie deicripta ab axi, BE, exteriorem <lb/>ad interiorem .</s> <s xml:id="echoid-s8970" xml:space="preserve">1. </s> <s xml:id="echoid-s8971" xml:space="preserve">quæ gignitur a ſemiparabola, BDE, ad eam, <lb/>quæ gignitura ſemiparabola, BFE, eſse vt, EM, cum {1/3}. </s> <s xml:id="echoid-s8972" xml:space="preserve">EM, & </s> <s xml:id="echoid-s8973" xml:space="preserve"><lb/>{1/2}. </s> <s xml:id="echoid-s8974" xml:space="preserve">ED, ad MF, cum {1/3}. </s> <s xml:id="echoid-s8975" xml:space="preserve">MF, & </s> <s xml:id="echoid-s8976" xml:space="preserve">{5/6}. </s> <s xml:id="echoid-s8977" xml:space="preserve">FE, & </s> <s xml:id="echoid-s8978" xml:space="preserve">ſic eſse ſolidum ſimilare <lb/>genitum ex figura, DBHM, dempto ſolido ſimilari genito ex, B <lb/>M, ad ſolidum ſimilare genitum ex, BM, dempto ſolido ſimilari <lb/>genito ex figura, BFMH, iuxta communem regulam, DM.</s> <s xml:id="echoid-s8979" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div869" type="section" level="1" n="518"> <head xml:id="echoid-head538" xml:space="preserve">COROLLARIVM XV.</head> <p> <s xml:id="echoid-s8980" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s8981" xml:space="preserve">37. </s> <s xml:id="echoid-s8982" xml:space="preserve">viſa fig. </s> <s xml:id="echoid-s8983" xml:space="preserve">Cor. </s> <s xml:id="echoid-s8984" xml:space="preserve">10. </s> <s xml:id="echoid-s8985" xml:space="preserve">P. </s> <s xml:id="echoid-s8986" xml:space="preserve">51. </s> <s xml:id="echoid-s8987" xml:space="preserve">huius, patet conoidem pa-<lb/> <anchor type="figure" xlink:label="fig-0377-01a" xlink:href="fig-0377-01"/> rabol. </s> <s xml:id="echoid-s8988" xml:space="preserve">ge-<lb/>nitam in re-<lb/>uolutione ex <lb/>ſemiparabo-<lb/>la, BDE, ad <lb/>ſemianulum, <lb/>ſtrictum para <lb/>bolicum ge-<lb/>nitum ex pa-<lb/>rabola, DBF, <lb/>eſse vt {1/8}. </s> <s xml:id="echoid-s8989" xml:space="preserve">DF, <lb/>ad {2/3}. </s> <s xml:id="echoid-s8990" xml:space="preserve">DF,.</s> <s xml:id="echoid-s8991" xml:space="preserve">. <lb/>vt 3. </s> <s xml:id="echoid-s8992" xml:space="preserve">ad 16. </s> <s xml:id="echoid-s8993" xml:space="preserve">& </s> <s xml:id="echoid-s8994" xml:space="preserve">ſic eſſe ſolidum ſimilare genitum ex, DBE, ſemipa-<lb/>rabola, ad ſibi ſimilare genitum ex figura, CBDF, dempto ſolido <lb/>ſimilari genito ex trilineo, BCF, iuxta communem regulam, DF.</s> <s xml:id="echoid-s8995" xml:space="preserve"/> </p> <div xml:id="echoid-div869" type="float" level="2" n="1"> <figure xlink:label="fig-0377-01" xlink:href="fig-0377-01a"> <image file="0377-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0377-01"/> </figure> </div> <pb o="358" file="0378" n="378" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div871" type="section" level="1" n="519"> <head xml:id="echoid-head539" xml:space="preserve">COROLLARIVM XVI.</head> <p> <s xml:id="echoid-s8996" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s8997" xml:space="preserve">38. </s> <s xml:id="echoid-s8998" xml:space="preserve">conſpecta adhuc eadem ſuperiori figura, habe-<lb/>tur ſemianulum latum parabolicum genitum in reuolutione <lb/>ex parabola, DBF, ad ſemianu um ſtrictum pararabolicum geni-<lb/>tum ex eadem, eſse vt, DM, MF, ad FD, & </s> <s xml:id="echoid-s8999" xml:space="preserve">ſic eſse ſolidum ſimi-<lb/>lare quodcunq; </s> <s xml:id="echoid-s9000" xml:space="preserve">genitum ex figura, HBDM, dempto ſolido ſimi-<lb/>lari genito ex figura, BHMF, ad ſolidum ſibi ſimilari genitum ex <lb/>figura, CBDF, dempto ſolido ſimilari genito ex trilineo, BCF, <lb/>iuxta communem regulam, DM.</s> <s xml:id="echoid-s9001" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div872" type="section" level="1" n="520"> <head xml:id="echoid-head540" xml:space="preserve">COROLLARIVM XVII.</head> <p> <s xml:id="echoid-s9002" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9003" xml:space="preserve">39. </s> <s xml:id="echoid-s9004" xml:space="preserve">viſa eadem ſuperioris figura, habemus ſemianu-<lb/>lum latum parabolicum genitum ex parabola, DBF, ad co-<lb/>noidem parabolicam genitam ex eadem per reuolutionem, eſse vt, <lb/>DM, MF, ad {3/16}. </s> <s xml:id="echoid-s9005" xml:space="preserve">ipſius, FD, & </s> <s xml:id="echoid-s9006" xml:space="preserve">ſic eſse quodlibet ſolidum ſimila-<lb/>re genitum ex, HBDM, dempto ſolido ſimilari genito ex figura, <lb/>BHMF, ad ſolidum ſibi ſimilare genitum ex ſemiparabola, BDE, <lb/>iuxta communem regulam, DM.</s> <s xml:id="echoid-s9007" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div873" type="section" level="1" n="521"> <head xml:id="echoid-head541" xml:space="preserve">COROLL XVIII. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s9008" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9009" xml:space="preserve">40. </s> <s xml:id="echoid-s9010" xml:space="preserve">viſis fig. </s> <s xml:id="echoid-s9011" xml:space="preserve">Cor. </s> <s xml:id="echoid-s9012" xml:space="preserve">10, & </s> <s xml:id="echoid-s9013" xml:space="preserve">12. </s> <s xml:id="echoid-s9014" xml:space="preserve">ſuperiorum, & </s> <s xml:id="echoid-s9015" xml:space="preserve">ductis vt-<lb/>cumque baſi parabolæ, DBF, æquidiſtantibus intra ipſam, quę <lb/>ſint, GP, GPV, patet ſemianulos parabolicos ex parabolis, DBF, <lb/>NBO, in vtraq; </s> <s xml:id="echoid-s9016" xml:space="preserve">figura per reuolutionem genitos, eſse inter ſe, vt <lb/>ipſas parabolas, DBF, NBO, & </s> <s xml:id="echoid-s9017" xml:space="preserve">ſic eſse quodlibet ſolidum ſimi-<lb/>lare genitum ex figura, HBDM, dempto ſolido ſimilari genito <lb/>ex figura, BFMH, ad ſibi ſimilare genitum ex figura, HBNV, <lb/>dempto ſolido ſimilari genito ex figura, HBOV. </s> <s xml:id="echoid-s9018" xml:space="preserve">Et ſic etiam <lb/>ſolidum ſimilare genitum ex figura, CBDF, dempto ſolido ſimi-<lb/>lari genito ex trilineo, CBF, ad ſolidum ſibi ſimilare genitum ex <lb/>figura, CBNP, dempto ſolido ſimilari genito ex figura, BCPO, <lb/>genita inquam iuxta communes regulas, DF.</s> <s xml:id="echoid-s9019" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div874" type="section" level="1" n="522"> <head xml:id="echoid-head542" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s9020" xml:space="preserve">EX Coroll. </s> <s xml:id="echoid-s9021" xml:space="preserve">habetur ſemianulos latos parabolicos ex parabolis, <lb/>DBF, NBO, in reuolutione circa, HM, genitos eſse ad in- <pb o="359" file="0379" n="379" rhead="LIBER IV."/> uicẽ, vt ſemianulos ſtrictos parabolicos ex parabolis, DBF, NBC, <lb/>genitos in reuolut. </s> <s xml:id="echoid-s9022" xml:space="preserve">circa, CF, & </s> <s xml:id="echoid-s9023" xml:space="preserve">ſic ſolida ſimiliaria, &</s> <s xml:id="echoid-s9024" xml:space="preserve">c. </s> <s xml:id="echoid-s9025" xml:space="preserve">& </s> <s xml:id="echoid-s9026" xml:space="preserve">vtroſq; <lb/></s> <s xml:id="echoid-s9027" xml:space="preserve">ſemianulos parabolicos, ſiue latos, ſiue ſtrictos eſſe ad inuicom, vt <lb/>cubi dictarum parabolarum baſium, DF, NO, & </s> <s xml:id="echoid-s9028" xml:space="preserve">ſic etiam ſolida <lb/>ſimilaria, &</s> <s xml:id="echoid-s9029" xml:space="preserve">c.</s> <s xml:id="echoid-s9030" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div875" type="section" level="1" n="523"> <head xml:id="echoid-head543" xml:space="preserve">COROLLARIVM XIX.</head> <p> <s xml:id="echoid-s9031" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9032" xml:space="preserve">41. </s> <s xml:id="echoid-s9033" xml:space="preserve">viſis adhuc fig. </s> <s xml:id="echoid-s9034" xml:space="preserve">Cor. </s> <s xml:id="echoid-s9035" xml:space="preserve">10. </s> <s xml:id="echoid-s9036" xml:space="preserve">& </s> <s xml:id="echoid-s9037" xml:space="preserve">12. </s> <s xml:id="echoid-s9038" xml:space="preserve">ſuperiorum, patet <lb/>ſemibaſim columnarem ſtrictam parabolicam genitam in re-<lb/>uolutione ex figura, CBDF, ad ſemibaſim columnarem latam pa-<lb/>rabol cam genitam ex figura, CBNP, eſſe vt parallelepipedum <lb/>ſub, BE, & </s> <s xml:id="echoid-s9039" xml:space="preserve">{1/2} {7/4}. </s> <s xml:id="echoid-s9040" xml:space="preserve">quadrati ipſius, DF, ad parallelepipedum ſub, BX, <lb/>& </s> <s xml:id="echoid-s9041" xml:space="preserve">his ſpatijs. </s> <s xml:id="echoid-s9042" xml:space="preserve">ſ@ quadrato, XP, {1/2}. </s> <s xml:id="echoid-s9043" xml:space="preserve">quadrati, NX, & </s> <s xml:id="echoid-s9044" xml:space="preserve">rectangulo ſub <lb/>ſexquitertia, NX, & </s> <s xml:id="echoid-s9045" xml:space="preserve">ſub, XP; </s> <s xml:id="echoid-s9046" xml:space="preserve">& </s> <s xml:id="echoid-s9047" xml:space="preserve">ſic etiam eſse quodlibet ſolidum <lb/>ſimilare genitum ex figura, CBDF, ad ſolidum ſibi ſimilare genitũ <lb/>ex figura, CBNP, iuxta communem regulam, DF, patet inſuper <lb/>ſemibaſim columnarem latam parabolicam genitam in reuolutio-<lb/>ne circa, HM, ex figura, DBHM, ad ſemibaſim columnarem latã <lb/>parabolicam genitam ex figura, HBNV, eſse vt parallelepipedum <lb/>ſub, BE, & </s> <s xml:id="echoid-s9048" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s9049" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9050" xml:space="preserve">quadrato. </s> <s xml:id="echoid-s9051" xml:space="preserve">ME, {1/2}. </s> <s xml:id="echoid-s9052" xml:space="preserve">quadrati, ED, & </s> <s xml:id="echoid-s9053" xml:space="preserve">rectã-<lb/>gulo ſub ſexquitertia, DE, & </s> <s xml:id="echoid-s9054" xml:space="preserve">ſub, EM, ad parallelepipedum ſub, B <lb/>X, & </s> <s xml:id="echoid-s9055" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s9056" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9057" xml:space="preserve">quadrato, VX, {1/2}. </s> <s xml:id="echoid-s9058" xml:space="preserve">quadrati, XN, & </s> <s xml:id="echoid-s9059" xml:space="preserve">rectangulo <lb/>ſub ſexquitertia, NX, & </s> <s xml:id="echoid-s9060" xml:space="preserve">ſub, XV; </s> <s xml:id="echoid-s9061" xml:space="preserve">& </s> <s xml:id="echoid-s9062" xml:space="preserve">ſic etiam eſſe ſolidum ſimila-<lb/>re quodcunq; </s> <s xml:id="echoid-s9063" xml:space="preserve">genitum ex figura, HBDM, ad ſolidum ſibi ſimila-<lb/>re genitum ex figura, HBNV, iuxta communem regulam, DM.</s> <s xml:id="echoid-s9064" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div876" type="section" level="1" n="524"> <head xml:id="echoid-head544" xml:space="preserve">COROLLARIVM XX.</head> <p> <s xml:id="echoid-s9065" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9066" xml:space="preserve">42. </s> <s xml:id="echoid-s9067" xml:space="preserve">aſſumpta figura, quæ ad ipſum pertinet .</s> <s xml:id="echoid-s9068" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9069" xml:space="preserve">paralle-<lb/>logrãmo, AF, & </s> <s xml:id="echoid-s9070" xml:space="preserve">ſruſto parab læ maiori illi incluſo .</s> <s xml:id="echoid-s9071" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9072" xml:space="preserve">EBHF, <lb/> <anchor type="figure" xlink:label="fig-0379-01a" xlink:href="fig-0379-01"/> vt fiat noſtrum exẽ <lb/>plũ reuoluatur, AF, <lb/>eirca manẽtẽ axim, <lb/>CF, fiat autem ex, <lb/>AF, cylindrus, AN, <lb/>& </s> <s xml:id="echoid-s9073" xml:space="preserve">ex figura, HBCF, <lb/>ſolidum, HBON, <lb/>quod vocetur: </s> <s xml:id="echoid-s9074" xml:space="preserve">ſemi-<lb/>baſis collumnaris <lb/>media parabolica, & </s> <s xml:id="echoid-s9075" xml:space="preserve">extenſo plano, AF, macſimiè producatur in <pb o="360" file="0380" n="380" rhead="GEOMETRIÆ"/> cylndro parallelogrãmum, AN, & </s> <s xml:id="echoid-s9076" xml:space="preserve">in ſemibaſi columnari figura, <lb/>HBON, quæ erit circa exem, CF, compoſita ex duabus figuris, <lb/>FBHF, FEON, ſimilibus, & </s> <s xml:id="echoid-s9077" xml:space="preserve">æqualibus ei, quæ per reuolutionem <lb/>ſemibaſim columnarem, HBON, generat; </s> <s xml:id="echoid-s9078" xml:space="preserve">patet ergo in hac Pro-<lb/>poſit. </s> <s xml:id="echoid-s9079" xml:space="preserve">cylindrum, AN, ad ſemibaſem columnarem mediam para-<lb/>bo icam, HBON, eſſe vt quadratum baſis, HF, ad quadratũ, FG, <lb/>{1/2}. </s> <s xml:id="echoid-s9080" xml:space="preserve">quadrati, GH, cum rectangulo ſub ſex quitertia, HG, & </s> <s xml:id="echoid-s9081" xml:space="preserve">ſub, G <lb/>F; </s> <s xml:id="echoid-s9082" xml:space="preserve">ſic verò etiam erit quodlibet ſolidum ſimilare genitum ex, AF, <lb/>ad ſibi ſimilare genitum ex figura, CBHF, iuxta communem re-<lb/>gulam, HF.</s> <s xml:id="echoid-s9083" xml:space="preserve"/> </p> <div xml:id="echoid-div876" type="float" level="2" n="1"> <figure xlink:label="fig-0379-01" xlink:href="fig-0379-01a"> <image file="0379-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0379-01"/> </figure> </div> </div> <div xml:id="echoid-div878" type="section" level="1" n="525"> <head xml:id="echoid-head545" xml:space="preserve">COROLLARIVM XXI.</head> <p> <s xml:id="echoid-s9084" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9085" xml:space="preserve">43. </s> <s xml:id="echoid-s9086" xml:space="preserve">viſa ſuperioris figura, patebit cylindrum, AN, ad <lb/>ſolidum genitum in reuolutione ex fruſto maiori parabolæ, <lb/>EBHF, (quod vocetur Aceruus maior parabolicus) .</s> <s xml:id="echoid-s9087" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9088" xml:space="preserve">ad Aceruũ, <lb/>HBEON, eſſe vt parallelepipedum ſub, BG, & </s> <s xml:id="echoid-s9089" xml:space="preserve">quadrato, HF, ad <lb/>reliquum parallelepipedi ſub, BG, & </s> <s xml:id="echoid-s9090" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s9091" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9092" xml:space="preserve">quadrato, FG, <lb/>{1/2}. </s> <s xml:id="echoid-s9093" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s9094" xml:space="preserve">rectangulo ſub ſexquitertia, HG, & </s> <s xml:id="echoid-s9095" xml:space="preserve">ſub, GF, <lb/>ab eodem dempto <emph style="sub">1</emph>. </s> <s xml:id="echoid-s9096" xml:space="preserve">parallelepipediſub, CE, & </s> <s xml:id="echoid-s9097" xml:space="preserve">quadrato, FG; <lb/></s> <s xml:id="echoid-s9098" xml:space="preserve">Sic etiam erit ſolidum ſimilare quodcunq; </s> <s xml:id="echoid-s9099" xml:space="preserve">genitum ex, AF, ad ſibi <lb/>ſimilare genitum ex figura, CBHF, dempto ſolido ſimilari genito <lb/>ex trilineo, BCE, iuxta communem regulam, HF.</s> <s xml:id="echoid-s9100" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div879" type="section" level="1" n="526"> <head xml:id="echoid-head546" xml:space="preserve">COROLLARIVM XXII.</head> <p> <s xml:id="echoid-s9101" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9102" xml:space="preserve">44. </s> <s xml:id="echoid-s9103" xml:space="preserve">adiuncta ſuperioris figuræ linea, RV, parallela ipſi, <lb/>HF, quæ, RV, ſit producta vſq; </s> <s xml:id="echoid-s9104" xml:space="preserve">in, X, per ſpſam ducatur pla-<lb/>num æquidiſtans baſi, HN, quod faciet in ſemibaſi columnari, HB <lb/>ON, communem factionem circulum, RX, habetur ergo hinc ſe-<lb/>mibaſim columnarem mediam parabolicam, HBON, ad abſciſſum <lb/>per circulum, RX, fruſtum, RBOX, eſſe vt parallelepipedum ſub, <lb/>BG, & </s> <s xml:id="echoid-s9105" xml:space="preserve">his ſpatijs .</s> <s xml:id="echoid-s9106" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9107" xml:space="preserve">quadrato, FG, {1/2}. </s> <s xml:id="echoid-s9108" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s9109" xml:space="preserve">rectangulo <lb/>ſub ſexquitertia, HG, & </s> <s xml:id="echoid-s9110" xml:space="preserve">ſub, GF, ad parallelepipedum ſub, BS, & </s> <s xml:id="echoid-s9111" xml:space="preserve"><lb/>ſexquitertia, RS, & </s> <s xml:id="echoid-s9112" xml:space="preserve">ſub, SV. </s> <s xml:id="echoid-s9113" xml:space="preserve">Veluti etiam erit quodlibet ſolidum <lb/>ſimilare genitum ex figura, CBHF, ad ſibi ſimilare genitum ex fi-<lb/>gura, CBRV, iuxca communem regulam, HF.</s> <s xml:id="echoid-s9114" xml:space="preserve"/> </p> <pb o="361" file="0381" n="381" rhead="LIBER IV."/> </div> <div xml:id="echoid-div880" type="section" level="1" n="527"> <head xml:id="echoid-head547" xml:space="preserve">COROLLARIVM XXIII.</head> <p> <s xml:id="echoid-s9115" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9116" xml:space="preserve">45. </s> <s xml:id="echoid-s9117" xml:space="preserve">viſa adhuc anteced. </s> <s xml:id="echoid-s9118" xml:space="preserve">figura, patet aceruum maio-<lb/>rem parabolicum, HBEON, ad conoidem parabolicam geni-<lb/>tam ex ſemiparabola, BHG, eſſe vt reliquum parallelepiped@ſub, <lb/>GB, & </s> <s xml:id="echoid-s9119" xml:space="preserve">his ſpatis ſ. </s> <s xml:id="echoid-s9120" xml:space="preserve">quadrato, FG, {1/2}. </s> <s xml:id="echoid-s9121" xml:space="preserve">quadrati, GH, & </s> <s xml:id="echoid-s9122" xml:space="preserve">rectãgulo ſub, <lb/>FG, & </s> <s xml:id="echoid-s9123" xml:space="preserve">ſexquitertia, GH, ab eedẽ dẽpto {1/6}. </s> <s xml:id="echoid-s9124" xml:space="preserve">parallelepipediſub, CE, & </s> <s xml:id="echoid-s9125" xml:space="preserve"><lb/>quadrato, FG, ad dimidiũ parallepepidiſub, @BG, & </s> <s xml:id="echoid-s9126" xml:space="preserve">quadrato, GH; <lb/></s> <s xml:id="echoid-s9127" xml:space="preserve">vt etiã erit quodlibet ſolidũ ſimilare genitũ ex figura, CBHF, dem-<lb/>pto ſolido ſimilari genito ex trilineo, BCE, ad ſolidum ſibi ſimilare <lb/>genitum ex ſemiparabola, BHG, iuxta communem regulam, HF.</s> <s xml:id="echoid-s9128" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div881" type="section" level="1" n="528"> <head xml:id="echoid-head548" xml:space="preserve">COROLLARIVM XXIV.</head> <p> <s xml:id="echoid-s9129" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s9130" xml:space="preserve">47. </s> <s xml:id="echoid-s9131" xml:space="preserve">ſumatur ex figura Prop. </s> <s xml:id="echoid-s9132" xml:space="preserve">46. </s> <s xml:id="echoid-s9133" xml:space="preserve">fruſtum minus parabo-<lb/>læ, quod eſt, DPG, cum Parallelogrammo, DG, & </s> <s xml:id="echoid-s9134" xml:space="preserve">integra <lb/> <anchor type="figure" xlink:label="fig-0381-01a" xlink:href="fig-0381-01"/> baſi parabolæ, <lb/>ZAG, quæ eſt, <lb/>ZG, &</s> <s xml:id="echoid-s9135" xml:space="preserve">, vt fiat <lb/>ſolitum exem-<lb/>plum, reuolua-<lb/>tur, DG, circa <lb/>manentẽ axim, <lb/>DP, & </s> <s xml:id="echoid-s9136" xml:space="preserve">iterum <lb/>circa manentẽ <lb/>axim, EG, fiet ergo ex reuolutione circa, DP, à parallelogrammo, <lb/>DG, cylindrus, RG, & </s> <s xml:id="echoid-s9137" xml:space="preserve">à fruſto parabolæ minori, PDG, ſolidum, <lb/>quod ſit, HDG, quodque vocetur, Aceruus minor parabolicus; </s> <s xml:id="echoid-s9138" xml:space="preserve">& </s> <s xml:id="echoid-s9139" xml:space="preserve"><lb/>ex reuolutione circa; </s> <s xml:id="echoid-s9140" xml:space="preserve">EG, à parallelogrammo, DG, in alia figura <lb/>cylindrus, DV, & </s> <s xml:id="echoid-s9141" xml:space="preserve">à trilineo extra fruſtum minus parabolæ conſti-<lb/>tuto ſolidum, DGX@, quod eſt fruſtum apicis parabolici refectum <lb/>per circulum. </s> <s xml:id="echoid-s9142" xml:space="preserve">DX, quodque vocetur Fruſtum apicis parabolici. </s> <s xml:id="echoid-s9143" xml:space="preserve">Pa-<lb/>tet ergo cylindrum, RG, ad aceruum minorem parabolicu, HDG, <lb/>eſſe vt, ZP, ad compoſitam ex {1/3}. </s> <s xml:id="echoid-s9144" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s9145" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s9146" xml:space="preserve">PG, ac cylindrum, DV, <lb/>ad fruſtum apicis parabolici, DGX, eſſe vt, ZP, ad ſui reliquum, <lb/>demptis ab ea {2/3}. </s> <s xml:id="echoid-s9147" xml:space="preserve">ZP, cum {1/6}. </s> <s xml:id="echoid-s9148" xml:space="preserve">PG; </s> <s xml:id="echoid-s9149" xml:space="preserve">Sic autem etiam erit quodlibet <lb/>ſolidum ſimilare genitum ex, DG, ad ſolidum ſibi ſimilare genitum <lb/>ex fruſto minori, DGP, vt, inquam, in priori parte huius Theor. <lb/></s> <s xml:id="echoid-s9150" xml:space="preserve">dictum eſt; </s> <s xml:id="echoid-s9151" xml:space="preserve">& </s> <s xml:id="echoid-s9152" xml:space="preserve">ſic etiam ſolidum quodlibet ſimilare genitum ex, <lb/>DG, ad ſibi ſimilare genitum ex trilineo, DEG, iuxta communem <pb o="362" file="0382" n="382" rhead="GEOMETRIÆ"/> regulam, ZG, vt in poſteriore dicti Theor. </s> <s xml:id="echoid-s9153" xml:space="preserve">parte dictum eſt.</s> <s xml:id="echoid-s9154" xml:space="preserve"/> </p> <div xml:id="echoid-div881" type="float" level="2" n="1"> <figure xlink:label="fig-0381-01" xlink:href="fig-0381-01a"> <image file="0381-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0381-01"/> </figure> </div> </div> <div xml:id="echoid-div883" type="section" level="1" n="529"> <head xml:id="echoid-head549" xml:space="preserve">COROLLARIVM XXV.</head> <p> <s xml:id="echoid-s9155" xml:space="preserve">IN Propoſitione 48: </s> <s xml:id="echoid-s9156" xml:space="preserve">ſumatur de figura Propoſit. </s> <s xml:id="echoid-s9157" xml:space="preserve">46. </s> <s xml:id="echoid-s9158" xml:space="preserve">parabola. <lb/></s> <s xml:id="echoid-s9159" xml:space="preserve">ZAG, cum baſi, ZG, & </s> <s xml:id="echoid-s9160" xml:space="preserve">axi, AQ, & </s> <s xml:id="echoid-s9161" xml:space="preserve">axi, AQ, & </s> <s xml:id="echoid-s9162" xml:space="preserve">reſecto eius minori fruſto, <lb/>DPG, de eiuſdem figura adnuc ſumatur, AXG, trilineum, in quo <lb/>ducitur, DE, æquidiſtans ipſi, AX, & </s> <s xml:id="echoid-s9163" xml:space="preserve">ſeorſim ponatur, vt autem fiat <lb/> <anchor type="figure" xlink:label="fig-0382-01a" xlink:href="fig-0382-01"/> ſolitum exemplum, reuoluatur parabola, ZAG, circa manentem <lb/>axim, AQ, & </s> <s xml:id="echoid-s9164" xml:space="preserve">fruſtum minus eiuſdem, quod eſt, DPG, circa, DP, <lb/>ex quo fiat aceruus minor, RDG. </s> <s xml:id="echoid-s9165" xml:space="preserve">Inſuper trilineum, AXG, reuo-<lb/>luatur circa, GX, vt fiat apex parabolicus, AGZ, & </s> <s xml:id="echoid-s9166" xml:space="preserve">ex GDE, eius <lb/>fruſtum, GDY; </s> <s xml:id="echoid-s9167" xml:space="preserve">patet igitur ex ipſa Prop. </s> <s xml:id="echoid-s9168" xml:space="preserve">48. </s> <s xml:id="echoid-s9169" xml:space="preserve">aceruum minorem, <lb/>RDG, ad conoidem parabolicam, ZAG, habere rationem compo-<lb/>ſitam ex ea, quam habet compoſita ex {1/3}. </s> <s xml:id="echoid-s9170" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s9171" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s9172" xml:space="preserve">PG, ad, ZP, & </s> <s xml:id="echoid-s9173" xml:space="preserve"><lb/>ex ratione parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s9174" xml:space="preserve">quadrato, PG, ad dimidium <lb/>parallelepipedi ſub, AQ & </s> <s xml:id="echoid-s9175" xml:space="preserve">quadrato, QG; </s> <s xml:id="echoid-s9176" xml:space="preserve">& </s> <s xml:id="echoid-s9177" xml:space="preserve">ſic etiã eſſe quodlibet <lb/>ſolidum ſimilare genitum ex fruſto parabolæ, DGP, ad ſibi ſimila-<lb/>re genitum ex ſemiparabola, AQG; </s> <s xml:id="echoid-s9178" xml:space="preserve">iuxta communem regulam, <lb/>ZG. </s> <s xml:id="echoid-s9179" xml:space="preserve">Item ex eadem Prop. </s> <s xml:id="echoid-s9180" xml:space="preserve">patet apicem parabolicum, AGZ, ad <lb/>eius fruſtum, DGY, habere rationem compoſitam ex ea, quam <lb/>habet ſexta pars parallelepipedi ſub AQ, & </s> <s xml:id="echoid-s9181" xml:space="preserve">quadrato, QG, ad pa-<lb/>rallelepipedum ſub, DP, & </s> <s xml:id="echoid-s9182" xml:space="preserve">quadrato, PG, & </s> <s xml:id="echoid-s9183" xml:space="preserve">ex ea, quam habet, <lb/>ZP, ad reſiduum, demptis ab eadem, ZP, {2/3}. </s> <s xml:id="echoid-s9184" xml:space="preserve">ZP, cum {1/6}. </s> <s xml:id="echoid-s9185" xml:space="preserve">PG: </s> <s xml:id="echoid-s9186" xml:space="preserve">Sic <lb/>autem quoque erit quodcunque ſolidum ſimilare genitum ex trili-<lb/>neo, AXG, ad ſibi ſimilare genitum ex trilineo, GDE, iuxta com-<lb/>munem regulam, AX.</s> <s xml:id="echoid-s9187" xml:space="preserve"/> </p> <div xml:id="echoid-div883" type="float" level="2" n="1"> <figure xlink:label="fig-0382-01" xlink:href="fig-0382-01a"> <image file="0382-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0382-01"/> </figure> </div> <pb o="363" file="0383" n="383" rhead="LIBER IV."/> </div> <div xml:id="echoid-div885" type="section" level="1" n="530"> <head xml:id="echoid-head550" xml:space="preserve">COROLLARIVM XXVI.</head> <p> <s xml:id="echoid-s9188" xml:space="preserve">N Propoſitione 49. </s> <s xml:id="echoid-s9189" xml:space="preserve">aſſumpta de figura Propoſit. </s> <s xml:id="echoid-s9190" xml:space="preserve">46. </s> <s xml:id="echoid-s9191" xml:space="preserve">parabola,</s> </p> <p> <s xml:id="echoid-s9192" xml:space="preserve">ZAG, cum axi, AQ, & </s> <s xml:id="echoid-s9193" xml:space="preserve">illi parallela, DP, abſcindatur ab axi, <lb/> <anchor type="figure" xlink:label="fig-0383-01a" xlink:href="fig-0383-01"/> AQ, ipſa, AV, exceſſus, AQ, <lb/>ſuper, DP, & </s> <s xml:id="echoid-s9194" xml:space="preserve">vt fiat ſolitum <lb/>exemplum, reuoluantur tũ <lb/>fruſtum maius, ZADP, tum <lb/>fruſtum minus, DPG, circa <lb/>communem axem, DP, vt <lb/>ex fruſto maiori, DAZP, <lb/>fiat aceruus maior parabo <lb/>licus, ZADON, & </s> <s xml:id="echoid-s9195" xml:space="preserve">ex fruſto <lb/>minori, DPG, fiat aceruus <lb/>minor parabolicus, RDG: </s> <s xml:id="echoid-s9196" xml:space="preserve">Patet ergo ex hac Propoſ. </s> <s xml:id="echoid-s9197" xml:space="preserve">aceruum <lb/>minorem, RDG, ad aceruum maiorem, ZADON, habere ratio-<lb/>nem compoſitam ex ea, quam habet compoſita ex {1/3}. </s> <s xml:id="echoid-s9198" xml:space="preserve">ZP, & </s> <s xml:id="echoid-s9199" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s9200" xml:space="preserve">PG, <lb/>ad ZP, & </s> <s xml:id="echoid-s9201" xml:space="preserve">ex ratione parallelepipedi ſub, DP, & </s> <s xml:id="echoid-s9202" xml:space="preserve">quadrato, PG, ad <lb/>parallelepipedum ſub, AQ, & </s> <s xml:id="echoid-s9203" xml:space="preserve">his ſpatijs. </s> <s xml:id="echoid-s9204" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9205" xml:space="preserve">quadrato, PQ, {1/2}. </s> <s xml:id="echoid-s9206" xml:space="preserve">qua-<lb/>drati, QZ, & </s> <s xml:id="echoid-s9207" xml:space="preserve">rectangulo ſub ſexquitertia, ZQ, & </s> <s xml:id="echoid-s9208" xml:space="preserve">ſub, QP, dempto <lb/>ab eodem {1/6}. </s> <s xml:id="echoid-s9209" xml:space="preserve">parallelepipedi ſub, AV, & </s> <s xml:id="echoid-s9210" xml:space="preserve">quadrato, QP: </s> <s xml:id="echoid-s9211" xml:space="preserve">Sic autem <lb/>erit etiam quodlibet ſolidum ſimilare genitum ex fruſto minori, <lb/>DPG, ad ſibi ſimilare genitum ex figura, CAZP, Prop. </s> <s xml:id="echoid-s9212" xml:space="preserve">46. </s> <s xml:id="echoid-s9213" xml:space="preserve">dem-<lb/>pto ſolido ſimilari genit o ex trilineo, ACD, iuxta communem re-<lb/>gulam, ZG.</s> <s xml:id="echoid-s9214" xml:space="preserve"/> </p> <div xml:id="echoid-div885" type="float" level="2" n="1"> <figure xlink:label="fig-0383-01" xlink:href="fig-0383-01a"> <image file="0383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0383-01"/> </figure> </div> </div> <div xml:id="echoid-div887" type="section" level="1" n="531"> <head xml:id="echoid-head551" xml:space="preserve">COROLLARIVM XXVII.</head> <p> <s xml:id="echoid-s9215" xml:space="preserve">N Propoſitione 50. </s> <s xml:id="echoid-s9216" xml:space="preserve">de figura Propoſit. </s> <s xml:id="echoid-s9217" xml:space="preserve">46. </s> <s xml:id="echoid-s9218" xml:space="preserve">ſumatur fruſtum</s> </p> <p> <s xml:id="echoid-s9219" xml:space="preserve">minus parabolæ, quod eſt, DGP, quodque ſecatur per rectam, <lb/> <anchor type="figure" xlink:label="fig-0383-02a" xlink:href="fig-0383-02"/> TI, æquidiſtantem ipſi, PG, acci-<lb/>piantur inſuper duæ integræ, ZG, <lb/>SI, & </s> <s xml:id="echoid-s9220" xml:space="preserve">vt fiat noſtrum exemplum, <lb/>reuoluatur, DPG, fruſtum circa <lb/>manentem axim, DP, vt ex, DPG, <lb/>ſiat aceruus minor, RDG, & </s> <s xml:id="echoid-s9221" xml:space="preserve">ex, <lb/>DTI, eius fruſtum, YDI, patet er-<lb/>go ex hoc Theorem. </s> <s xml:id="echoid-s9222" xml:space="preserve">aceruum mi-<lb/>norem, RDG, ad reſectum fruſtum, YDI, habere rationem com-<lb/>poſitam ex ea, quam habet rectangulum, ZPG, cum {1/2}. </s> <s xml:id="echoid-s9223" xml:space="preserve">quadrati, <pb o="364" file="0384" n="384" rhead="GEOMETRIÆ."/> PG, ad rectangulum, STI, cu n {1/2}. </s> <s xml:id="echoid-s9224" xml:space="preserve">quadrati, TI, & </s> <s xml:id="echoid-s9225" xml:space="preserve">ex ea, quam <lb/>habet quadratum, PG, ad quadratum, TI. </s> <s xml:id="echoid-s9226" xml:space="preserve">Vt etiam erit quod-<lb/>libet ſolidum ſimilare genitum ex fruſto minori parabolæ, quod <lb/>eſt, DPG, ad ſibi ſimilare genitum ex trilineo, DTI, iuxta com-<lb/>munemregulam, PG.</s> <s xml:id="echoid-s9227" xml:space="preserve"/> </p> <div xml:id="echoid-div887" type="float" level="2" n="1"> <figure xlink:label="fig-0383-02" xlink:href="fig-0383-02a"> <image file="0383-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0383-02"/> </figure> </div> </div> <div xml:id="echoid-div889" type="section" level="1" n="532"> <head xml:id="echoid-head552" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s9228" xml:space="preserve">_P_Lurã alia poſſemus adbuc circa bac examinare, præcipuè ſolidita-<lb/>tem eius, quod produceretur, reuoluta parabola circ a baſim, vel <lb/>illi parallelam, ſine tangentem p arabolam, ſiue extra ipſam conſtitu-<lb/>tam, & </s> <s xml:id="echoid-s9229" xml:space="preserve">proportionem eiuſdem ſegmentorum; </s> <s xml:id="echoid-s9230" xml:space="preserve">necnon, & </s> <s xml:id="echoid-s9231" xml:space="preserve">aliorum <lb/>corporum, quorun notitia tum ob ſpeculationem iucunda, tum in or-<lb/>dine ad praxim conſiderata, non inutilis etiam eſſe videtur, ſed bæe <lb/>alij; </s> <s xml:id="echoid-s9232" xml:space="preserve">indaganda relinquam. </s> <s xml:id="echoid-s9233" xml:space="preserve">Hæc autem nunc delibaſſe ſufficiat.</s> <s xml:id="echoid-s9234" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div890" type="section" level="1" n="533"> <head xml:id="echoid-head553" xml:space="preserve">Finis quarti Libri.</head> <pb o="365" file="0385" n="385"/> </div> <div xml:id="echoid-div891" type="section" level="1" n="534"> <head xml:id="echoid-head554" xml:space="preserve">GEOMETRIÆ</head> <head xml:id="echoid-head555" xml:space="preserve">CAVALERII.</head> <head xml:id="echoid-head556" xml:space="preserve">LIBER QVINTVS.</head> <head xml:id="echoid-head557" style="it" xml:space="preserve">In quo de Hyperbola, Oppoſitis Sectionib us, <lb/>ac ſolidis ab eiſdem genitis, babetur <lb/>contemplatio.</head> <head xml:id="echoid-head558" xml:space="preserve">THEOREMA I. PROPOS. I.</head> <p> <s xml:id="echoid-s9235" xml:space="preserve">OMNIA quadrata Hyperbol@, regu-<lb/>la ſumpta baſi ſcilicet vna ex ordi-<lb/>natim applicatis ad axim, vel diame-<lb/>trum eiuſdem, ad omnia quadrata <lb/>parallelogrammi in eadem baſi, & </s> <s xml:id="echoid-s9236" xml:space="preserve"><lb/>altitudine cum ipſa, erunt vt linea <lb/>compoſita ex dimidia tranſuerſi la-<lb/>teris hyperbolæ, & </s> <s xml:id="echoid-s9237" xml:space="preserve">diametri, vel <lb/>axis eiuſdem, ad compoſita n ex tranſucrſo latere, & </s> <s xml:id="echoid-s9238" xml:space="preserve">axi, <lb/>vel dia netro eiuſdem: </s> <s xml:id="echoid-s9239" xml:space="preserve">Eade n verò ad omnia quadrata <lb/>trianguli in eadem baſi, & </s> <s xml:id="echoid-s9240" xml:space="preserve">altitudine cum ipſa erunt, vt cõ-<lb/>poſit@ ex ſexquialtera tranſuerſi lateris, & </s> <s xml:id="echoid-s9241" xml:space="preserve">axi, vel diame-<lb/>tro eiuſdem, ad compoſitam ex tranſuerſo latere, & </s> <s xml:id="echoid-s9242" xml:space="preserve">axi, <lb/>vel diametro eiuſdem.</s> <s xml:id="echoid-s9243" xml:space="preserve"/> </p> <pb o="366" file="0386" n="386" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s9244" xml:space="preserve">Sitigitur hyperbola, DBF, in baſi, DF, cuius axis, vel diameter, <lb/>EB, & </s> <s xml:id="echoid-s9245" xml:space="preserve">tranſuerſum latus, BO, bifariam diuiſum in, N, deſcribatur <lb/>vero paralielogrammũ, AF, in eadem altitudine, & </s> <s xml:id="echoid-s9246" xml:space="preserve">baſi cum hy-<lb/>perbola, DBF, & </s> <s xml:id="echoid-s9247" xml:space="preserve">nunc circa axim, vel diametrum, BE, circa quam <lb/> <anchor type="figure" xlink:label="fig-0386-01a" xlink:href="fig-0386-01"/> tit etiam triangulum, BDF. </s> <s xml:id="echoid-s9248" xml:space="preserve">Dico ergo <lb/>omnia quadrata hyperbolæ, DBF, regu-<lb/>la, DF, ad omnia quadrata, AF, eſſe vt <lb/>compoſitam ex, NB, & </s> <s xml:id="echoid-s9249" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9250" xml:space="preserve">BE, ad, OE; <lb/></s> <s xml:id="echoid-s9251" xml:space="preserve">ad omnia verò quadrata trianguli, DBF, <lb/>eſſe vt compoſitam ex ſexquialtera, OB, <lb/>& </s> <s xml:id="echoid-s9252" xml:space="preserve">ipſa, BE, ad, OE, ſumatur in, BE, vt-<lb/>cunq; </s> <s xml:id="echoid-s9253" xml:space="preserve">punctum, M, & </s> <s xml:id="echoid-s9254" xml:space="preserve">per, M, ducatur, <lb/>MG, parallela ipſi, DF, ſecans curuam <lb/>hyperbolæ in, H. </s> <s xml:id="echoid-s9255" xml:space="preserve">Eſt ergo quadratum, <lb/> <anchor type="note" xlink:label="note-0386-01a" xlink:href="note-0386-01"/> EF, vel quadratum, GM, ad quadratum, <lb/>MH, vt rectangulum, OEB, ad rectangu-<lb/>lum, OMB, eſt autem, BF, parallelogrã@ <lb/>mum in eadem altitudine, & </s> <s xml:id="echoid-s9256" xml:space="preserve">baſi cum ſemihyperbola, BEF, & </s> <s xml:id="echoid-s9257" xml:space="preserve"><lb/>punctum, M, vtcunq; </s> <s xml:id="echoid-s9258" xml:space="preserve">ſumptum, per quod acta eſtipſi, DF, pa-<lb/>rallela, MG, regula, DF, repertumque eſt, vt quadratum, GM, ad <lb/>quadratum, MH, ita eſſe rectangulum, OEB, ad rectangulum, O <lb/> <anchor type="note" xlink:label="note-0386-02a" xlink:href="note-0386-02"/> MB, ergo horum quatuor ordinum magnitudines erunt propor-<lb/>tionales .</s> <s xml:id="echoid-s9259" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9260" xml:space="preserve">omnia quadrata, BF, magnitudines primi ordinis col-<lb/>lectæ iuxta primam .</s> <s xml:id="echoid-s9261" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9262" xml:space="preserve">iuxta quadratum, GM, ad omnia quadra-<lb/>ta ſernihy perbolæ, BEF, magnitudines ſecundi ordinis collectas <lb/>iuxta ſecundam .</s> <s xml:id="echoid-s9263" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9264" xml:space="preserve">iuxta quadratum, MH, erunt vt rectangula ſub <lb/>maximis abſciſſarum, BE, magnitudines tertij ordinis collectę iux-<lb/>ta tertjam .</s> <s xml:id="echoid-s9265" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9266" xml:space="preserve">iuxta rectangulum ſub, OE, EB, ad rectangula ſub <lb/>omnibus abſciſsis, EB, adiuncta, BO, & </s> <s xml:id="echoid-s9267" xml:space="preserve">ſub omnibus abſciſsis, EB, <lb/>magnitudines quarti ordinis collectas iuxta primam, .</s> <s xml:id="echoid-s9268" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9269" xml:space="preserve">iuxta re-<lb/>ctangulum, OMB; </s> <s xml:id="echoid-s9270" xml:space="preserve">verum rectangula ſub maximis abſciſſarum, <lb/>EB, adiuncta, BO, & </s> <s xml:id="echoid-s9271" xml:space="preserve">ſub maximis abiciſſarum, EB, ad rectangula <lb/>ſub omnibus abſeiſsis, EB, adiuncta, BO, & </s> <s xml:id="echoid-s9272" xml:space="preserve">ſub omnibus abſciſsis, <lb/>EB, recti, vel eiuſdem obliqui tranſitus, ſunt vt, OE, ad compoſi-<lb/>tam ex, NB, & </s> <s xml:id="echoid-s9273" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9274" xml:space="preserve">BE, ergo, conuertendo, omnia quadrara ſemi-<lb/> <anchor type="note" xlink:label="note-0386-03a" xlink:href="note-0386-03"/> hyperbolæ, BEF, ad omnia quadrata, BF, vel eorum quadrupla .</s> <s xml:id="echoid-s9275" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s9276" xml:space="preserve">omnia quadrata hyperbolæ, DBF, ad omnia quadrata, AF, etiam <lb/>ſi, AF, non eſſet circa axim, vel diametrum, BE, ſed tantum in ea-<lb/>dem altitudine cum hyperbola, DBF, erunt, vt compoſita ex {1/2}. </s> <s xml:id="echoid-s9277" xml:space="preserve">O <lb/>B, & </s> <s xml:id="echoid-s9278" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9279" xml:space="preserve">BE, ad, OE.</s> <s xml:id="echoid-s9280" xml:space="preserve"/> </p> <div xml:id="echoid-div891" type="float" level="2" n="1"> <figure xlink:label="fig-0386-01" xlink:href="fig-0386-01a"> <image file="0386-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0386-01"/> </figure> <note position="left" xlink:label="note-0386-01" xlink:href="note-0386-01a" xml:space="preserve">39. l. 3. & <lb/>Scho. 40.</note> <note position="left" xlink:label="note-0386-02" xlink:href="note-0386-02a" xml:space="preserve">Coroll. 3. <lb/>26. l. 2.</note> <note position="left" xlink:label="note-0386-03" xlink:href="note-0386-03a" xml:space="preserve">Corol. 30. <lb/>l. 2.</note> </div> <note position="left" xml:space="preserve">24. l. 2.</note> <p> <s xml:id="echoid-s9281" xml:space="preserve">Quoniam verò omnia quadrata, AF, ſunt tripla omnium qua- <pb o="367" file="0387" n="387" rhead="LIBER V."/> dratorum trianguli, DBF, ideò ſunt ad illa, vt, OE, ad {1/3}. </s> <s xml:id="echoid-s9282" xml:space="preserve">OE, oſten-<lb/>ſum autem eſt omnia quadrata hyperbolæ, DBF, ad omnia qua-<lb/>drata, AF, eſſe vt compoſitam ex {1/2}. </s> <s xml:id="echoid-s9283" xml:space="preserve">OB, & </s> <s xml:id="echoid-s9284" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9285" xml:space="preserve">BE, ad, OE, ergo, <lb/>ex æquali, omnia quadrata hyperbolæ, DBF, ad omnia quadrata <lb/>trianguli, DBF, etiam ſi non eſſet circa axim, vel diametrum, BE, <lb/>ſed tantum in eadem altitudine cum hyperbola, DBF, erunt vt cõ. <lb/></s> <s xml:id="echoid-s9286" xml:space="preserve">poſita ex {1/2}. </s> <s xml:id="echoid-s9287" xml:space="preserve">OB. </s> <s xml:id="echoid-s9288" xml:space="preserve">& </s> <s xml:id="echoid-s9289" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9290" xml:space="preserve">BE, ad {1/3}. </s> <s xml:id="echoid-s9291" xml:space="preserve">OE, vel vt horum tripla .</s> <s xml:id="echoid-s9292" xml:space="preserve">i. </s> <s xml:id="echoid-s9293" xml:space="preserve">vt com-<lb/>poſita ex ſexquialtera, OB, & </s> <s xml:id="echoid-s9294" xml:space="preserve">ipſa, BE, ad, OE, quod oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s9295" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div893" type="section" level="1" n="535"> <head xml:id="echoid-head559" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s9296" xml:space="preserve">SI duæ ad axim, vel diametrum hyperbolæ ordinatim <lb/>applicatæ fuerint rectæ lineæ, hyperbolas conſtituen-<lb/>tes, ſit autem earum altera regula: </s> <s xml:id="echoid-s9297" xml:space="preserve">omnia quadrata hyper-<lb/>bolæ ab vna earundem conſtitutæ ad omnia quadrata hy-<lb/>perbolæ per aliam conſtitutæ, erunt vt parallelepipedum <lb/>ſub compoſita ex ſexquialtera tranſuerſi lateris hyparbola-<lb/>rum dictarum, & </s> <s xml:id="echoid-s9298" xml:space="preserve">ſub axi, vel diametro hyperbolæ primò <lb/>dictæ, & </s> <s xml:id="echoid-s9299" xml:space="preserve">ſub quadrato eiuſdem axis, vel diametri ad pa-<lb/>rallelepipedum ſub compoſita ex eiuſdem tranſuerſi lateris <lb/>ſexquialtera, & </s> <s xml:id="echoid-s9300" xml:space="preserve">axi, vel diametro hyperbolæ ſecundò di-<lb/>ctæ, & </s> <s xml:id="echoid-s9301" xml:space="preserve">ſub quadrato eiuſdern axis, vel diametri.</s> <s xml:id="echoid-s9302" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9303" xml:space="preserve">Sint intra curuam hyperbolæ duæ vtcunq; </s> <s xml:id="echoid-s9304" xml:space="preserve">ad axim, vel diame-<lb/>trum, NE, ordinatim ductæ rectæ lineæ, HG, DF, hyperbolas, N <lb/>HG, NDF, conſtituentes, ſit autem earum altera, vt, DF, ſum-<lb/>pta pro regula, & </s> <s xml:id="echoid-s9305" xml:space="preserve">tranſuerſum eorundem latus, NO, bifariam di-<lb/>uiſum in, B, cui in directum ſit adiecta, OX, æqualis dimidiæ, ON. <lb/></s> <s xml:id="echoid-s9306" xml:space="preserve">Dico ergo omnia quadrata hyperbolæ, DNF, ad omnia quadra-<lb/> <anchor type="note" xlink:label="note-0387-01a" xlink:href="note-0387-01"/> ta hyperbolæ, HNG, eſſe vt parallelepipedum ſub, XE, & </s> <s xml:id="echoid-s9307" xml:space="preserve">quadra-<lb/>to, EN, ad parallelepipedum ſub, XM, & </s> <s xml:id="echoid-s9308" xml:space="preserve">quadrato, MN. </s> <s xml:id="echoid-s9309" xml:space="preserve">Fiant <lb/>ergo in baſibus, DF, HG, & </s> <s xml:id="echoid-s9310" xml:space="preserve">circa axes, vel diametros, NM, ME, <lb/>parallelogramma, AF, CG: </s> <s xml:id="echoid-s9311" xml:space="preserve">Omnia ergo quadrata hyperbolæ, <lb/>DNF, ad omnia quadrata hyperbolæ, HNG, habent rationem <lb/>compoſitam ex ea, quam habent omnia quadrata hyperbolæ, D <lb/>NF, ad omnia quadrata, AF, omnia quadrata, AF, ad omnia qua-<lb/>drata, CG, & </s> <s xml:id="echoid-s9312" xml:space="preserve">omnia quadrata, CG, ad omnia quadrata hyperbo-<lb/>læ, HNG; </s> <s xml:id="echoid-s9313" xml:space="preserve">ſed omnia quadrata hyperbolæ, DNF, ad omnia qua-<lb/> <anchor type="note" xlink:label="note-0387-02a" xlink:href="note-0387-02"/> <pb o="368" file="0388" n="388" rhead="GEOMETRIÆ"/> drata, AF, ſunt vt compoſita ex {1/2}. </s> <s xml:id="echoid-s9314" xml:space="preserve">ON. </s> <s xml:id="echoid-s9315" xml:space="preserve">.</s> <s xml:id="echoid-s9316" xml:space="preserve">i. </s> <s xml:id="echoid-s9317" xml:space="preserve">ex, BN, & </s> <s xml:id="echoid-s9318" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9319" xml:space="preserve">NE, ad, <lb/> <anchor type="note" xlink:label="note-0388-01a" xlink:href="note-0388-01"/> OE, vel vt iſtorum tripla .</s> <s xml:id="echoid-s9320" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9321" xml:space="preserve">vt, XE, ad trip lam, OE. </s> <s xml:id="echoid-s9322" xml:space="preserve">Inſuper omnia <lb/> <anchor type="note" xlink:label="note-0388-02a" xlink:href="note-0388-02"/> quadrata, AF, ad omnia quadrata, CG, habent rationem compoſitã <lb/> <anchor type="figure" xlink:label="fig-0388-01a" xlink:href="fig-0388-01"/> ex ea, quã habet quadratu, DF, ad quadratũ, <lb/>HG, ideſt rectangulum, OEN, ad rectagulũ, <lb/>OMN, .</s> <s xml:id="echoid-s9323" xml:space="preserve">i. </s> <s xml:id="echoid-s9324" xml:space="preserve">horũ tripla, .</s> <s xml:id="echoid-s9325" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9326" xml:space="preserve">rectangulum ſubtri-<lb/> <anchor type="note" xlink:label="note-0388-03a" xlink:href="note-0388-03"/> pla, OE, &</s> <s xml:id="echoid-s9327" xml:space="preserve">, EN, ſola, ad rectã gulũ ſub tripla, <lb/>OM, & </s> <s xml:id="echoid-s9328" xml:space="preserve">ſola, MN, & </s> <s xml:id="echoid-s9329" xml:space="preserve">ex rñe, EN, ad, NM; </s> <s xml:id="echoid-s9330" xml:space="preserve">tã-<lb/>dem omnia quadrata, CG, ad omnia quadra-<lb/>ta hyperbolæ, HNG, ſunt vt, OM, ad cõpo-<lb/>ſitam ex, BN, & </s> <s xml:id="echoid-s9331" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9332" xml:space="preserve">NM, .</s> <s xml:id="echoid-s9333" xml:space="preserve">i. </s> <s xml:id="echoid-s9334" xml:space="preserve">vt tripla, OM, <lb/>ad, MX, ideſt ſumpta, MN, communi alti-<lb/>tudine, vt rectangulũ ſub tripla, OM, & </s> <s xml:id="echoid-s9335" xml:space="preserve">ſub, <lb/>MN, ad rectãgulũ ſub, XM, MN, ergo omnia <lb/>quadrata hyperbolæ, DNF, ad omnia qua-<lb/>drata hyperbolæ, HNG, habent rationem <lb/>compoſitam ex ea, quam habet, XE, ad tri-<lb/>plam, EO, .</s> <s xml:id="echoid-s9336" xml:space="preserve">i. </s> <s xml:id="echoid-s9337" xml:space="preserve">ſumpta, EN, communi altitudine, ex ea, quam ha-<lb/>bet rectangulum, XEN, ad rectangulum ſub, NE, & </s> <s xml:id="echoid-s9338" xml:space="preserve">tripla, EO, & </s> <s xml:id="echoid-s9339" xml:space="preserve"><lb/>ex ea, quam habet rectangulum ſub tripla; </s> <s xml:id="echoid-s9340" xml:space="preserve">OE, & </s> <s xml:id="echoid-s9341" xml:space="preserve">ſub, EN, ad re-<lb/>ctangulum ſub tripla, OM, & </s> <s xml:id="echoid-s9342" xml:space="preserve">ſub, MN, & </s> <s xml:id="echoid-s9343" xml:space="preserve">rectangulum ſub tripla, <lb/>OM, & </s> <s xml:id="echoid-s9344" xml:space="preserve">ſub, MN, ad rectangulum ſub, MN, & </s> <s xml:id="echoid-s9345" xml:space="preserve">MX, & </s> <s xml:id="echoid-s9346" xml:space="preserve">tandem ex <lb/>ea, quam habet, EN, ad, NM; </s> <s xml:id="echoid-s9347" xml:space="preserve">porrò iſtæ rationes .</s> <s xml:id="echoid-s9348" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9349" xml:space="preserve">quam habet <lb/>rectangulum ſub, XE, &</s> <s xml:id="echoid-s9350" xml:space="preserve">, EN, ad rectangulum ſub tripla, OE, &</s> <s xml:id="echoid-s9351" xml:space="preserve">, <lb/>EN, item quam habet rectangulum ſub tripla, OE, &</s> <s xml:id="echoid-s9352" xml:space="preserve">, EN, ad re-<lb/>ctangulum ſub tripla, OM, & </s> <s xml:id="echoid-s9353" xml:space="preserve">MN, & </s> <s xml:id="echoid-s9354" xml:space="preserve">quam habet rectangulum <lb/>ſubtripla, OM, &</s> <s xml:id="echoid-s9355" xml:space="preserve">, MN, ad rectangulum, XMN, conficiunt ratio-<lb/>nem rectanguli, XEN, ad rectangulum, XMN, quæ ſimul cum ra-<lb/>tione: </s> <s xml:id="echoid-s9356" xml:space="preserve">quam habet, EN, ad, NM, conficit rationem parallelepipe-<lb/>di ſub, NE, & </s> <s xml:id="echoid-s9357" xml:space="preserve">rectangulo, NEX, .</s> <s xml:id="echoid-s9358" xml:space="preserve">i. </s> <s xml:id="echoid-s9359" xml:space="preserve">ſub, XE, & </s> <s xml:id="echoid-s9360" xml:space="preserve">quadrato, EN, ad <lb/> <anchor type="note" xlink:label="note-0388-04a" xlink:href="note-0388-04"/> parallelepipedum ſub, NM, & </s> <s xml:id="echoid-s9361" xml:space="preserve">rectangu o, NMX, .</s> <s xml:id="echoid-s9362" xml:space="preserve">i. </s> <s xml:id="echoid-s9363" xml:space="preserve">ſub, XM & </s> <s xml:id="echoid-s9364" xml:space="preserve"><lb/>quadrato, MN, ergo omnia quadrata hyperbolæ, DNF, ad omnia <lb/>quadrata hyperbolæ, HNG, erunt vt parallelepipedum ſub, XE, <lb/>& </s> <s xml:id="echoid-s9365" xml:space="preserve">quadrato, EN, ad parallelepipedum ſub, XM, & </s> <s xml:id="echoid-s9366" xml:space="preserve">quadrato, M <lb/>N, quod oſtendere oportebat.</s> <s xml:id="echoid-s9367" xml:space="preserve"/> </p> <div xml:id="echoid-div893" type="float" level="2" n="1"> <note position="right" xlink:label="note-0387-01" xlink:href="note-0387-01a" xml:space="preserve">Defim. 12. <lb/>l.</note> <note position="right" xlink:label="note-0387-02" xlink:href="note-0387-02a" xml:space="preserve">Exantec.</note> <note position="left" xlink:label="note-0388-01" xlink:href="note-0388-01a" xml:space="preserve">11. l. 1.</note> <note position="left" xlink:label="note-0388-02" xlink:href="note-0388-02a" xml:space="preserve">Corol. 39. <lb/>& Sch. 40. <lb/>l. 1.</note> <figure xlink:label="fig-0388-01" xlink:href="fig-0388-01a"> <image file="0388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0388-01"/> </figure> <note position="left" xlink:label="note-0388-03" xlink:href="note-0388-03a" xml:space="preserve">EX antec.</note> <note position="left" xlink:label="note-0388-04" xlink:href="note-0388-04a" xml:space="preserve">36. l. 2.</note> </div> </div> <div xml:id="echoid-div895" type="section" level="1" n="536"> <head xml:id="echoid-head560" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s9368" xml:space="preserve">IN eadem antecedentis figura, ſi producatur, HG, hinc <lb/>inde vſque ad curuam hyperbolicam, cui incidat in, <pb o="369" file="0389" n="389" rhead="LIBER V."/> NX, & </s> <s xml:id="echoid-s9369" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9370" xml:space="preserve">{2/3}- NE, regula eadem, DF, retenta; </s> <s xml:id="echoid-s9371" xml:space="preserve">oſten-<lb/>demus?</s> <s xml:id="echoid-s9372" xml:space="preserve">? omnia quadrata parallelogrammi, SF, ad om-<lb/>nia quadrata fruſti hyperbolæ, HDFG, eſſe vt rectangu-<lb/>lum, OEN, ad rectangulum ſub, OE, &</s> <s xml:id="echoid-s9373" xml:space="preserve">, NM, vna cum re-<lb/>ctangulo ſub compoſita ex. </s> <s xml:id="echoid-s9374" xml:space="preserve">No, &</s> <s xml:id="echoid-s9375" xml:space="preserve">. ME, & </s> <s xml:id="echoid-s9376" xml:space="preserve">ſub, ME; <lb/></s> <s xml:id="echoid-s9377" xml:space="preserve">Omnia verò quadrata trianguli, DMF, ad omnia quadra-<lb/>ta eiuſdem fruſti, HDFG, eſſe vt rectangulum, OEN, ad <lb/>rectangulum ſub, OE, & </s> <s xml:id="echoid-s9378" xml:space="preserve">tripla, NM, vna cum rectangulo <lb/>ſub compoſita ex, NX, & </s> <s xml:id="echoid-s9379" xml:space="preserve">ME, ſub, ME.</s> <s xml:id="echoid-s9380" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9381" xml:space="preserve">Sumatur in, ME, vtcunq; </s> <s xml:id="echoid-s9382" xml:space="preserve">punctum, L, & </s> <s xml:id="echoid-s9383" xml:space="preserve">per ipſum regulæ, <lb/>DF, parallela ducatur, LK, curuam hyp rbolicam in, I, ſecans; <lb/></s> <s xml:id="echoid-s9384" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0389-01a" xlink:href="fig-0389-01"/> Eſt ergo quadratum, EF, vel quadratum, <lb/>LK, ad quadratum, LI, vt rectangulum, <lb/>OEN, adrectangulum, OLN; </s> <s xml:id="echoid-s9385" xml:space="preserve">eſt autem <lb/>parallelogrammum, MF, in eadem baſi, & </s> <s xml:id="echoid-s9386" xml:space="preserve"><lb/>altitudine cum figura, MGFE, punctum, <lb/>L, ſumptum eſt vt cunque, perq; </s> <s xml:id="echoid-s9387" xml:space="preserve">ipſum re-<lb/>gulæ, DF, ducta parallela, LK, repertum <lb/>eſt, vt quadratum, KL, ad quadratum, LI, <lb/>ita eſſe rectangulum, OEN, ad rectangulũ, <lb/>OLN; </s> <s xml:id="echoid-s9388" xml:space="preserve">quatuor ergo ordinum magnitudi-<lb/>nes conſtructæ iuxta has quatuor inuentas <lb/>magnitudines proportionales, erunt quoq; <lb/></s> <s xml:id="echoid-s9389" xml:space="preserve">proportionales .</s> <s xml:id="echoid-s9390" xml:space="preserve">i. </s> <s xml:id="echoid-s9391" xml:space="preserve">omnia quadrata: </s> <s xml:id="echoid-s9392" xml:space="preserve">MF, ad <lb/>omnia quadrata figuræ, MGFE, quæ ſunt <lb/>magnitudines primi, & </s> <s xml:id="echoid-s9393" xml:space="preserve">ſecundi ordinis conſtructæ iuxta primã, & </s> <s xml:id="echoid-s9394" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0389-01a" xlink:href="note-0389-01"/> ſecundam .</s> <s xml:id="echoid-s9395" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9396" xml:space="preserve">iuxta quadratum, KL, & </s> <s xml:id="echoid-s9397" xml:space="preserve">quadratum, LI, erunt, vt <lb/>rectangula ſub maximis abſciſſarum, EM, adiuncta, MO, & </s> <s xml:id="echoid-s9398" xml:space="preserve">ſub <lb/>maximis abſciſſarum, EM, adiuncta, MN, ad rectangula ſub om-<lb/>nibus abſciſsis, EM, adiuncta, MO, & </s> <s xml:id="echoid-s9399" xml:space="preserve">ſub omnibus abſciſsis, EM, <lb/>adiuncta, MN, quæ ſunt magnitudines tertij, & </s> <s xml:id="echoid-s9400" xml:space="preserve">quarti ordinis <lb/>collectæ iuxta tertiam, & </s> <s xml:id="echoid-s9401" xml:space="preserve">quartam .</s> <s xml:id="echoid-s9402" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9403" xml:space="preserve">iuxta rectangulum, OEN, <lb/>OLN; </s> <s xml:id="echoid-s9404" xml:space="preserve">verum rectangula ſub maximis abſciſlarum, EM, adiuncta, <lb/>MO, & </s> <s xml:id="echoid-s9405" xml:space="preserve">ſub eiſdem adiuncta, MN, ad rectangula ſub omnibus ab-<lb/>ſciſsis, EM, adiuncta, MO, & </s> <s xml:id="echoid-s9406" xml:space="preserve">ſub ijſdem adiuncta, MN, omnibus <lb/> <anchor type="note" xlink:label="note-0389-02a" xlink:href="note-0389-02"/> recti vel eiuſdem obliqui tranſitus ſumptis, ſunt, vt rectangulum, <lb/>OEN, ad rectangulum ſub, OE, &</s> <s xml:id="echoid-s9407" xml:space="preserve">, NM, vna cum rectangulo ſub <lb/>compoſita ex {1/2}. </s> <s xml:id="echoid-s9408" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9409" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9410" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9411" xml:space="preserve">ſub, ME, ergo omnia quadra-<lb/>ta, MF, ad omnia quadrata figuræ, GMEF, vel horum quadru-<lb/>pla .</s> <s xml:id="echoid-s9412" xml:space="preserve">i. </s> <s xml:id="echoid-s9413" xml:space="preserve">omnia quadrata, SF, ad omnia quadrata fruſti, HDFG, <pb o="370" file="0390" n="390" rhead="GEOMETRIÆ"/> erunt, vt rectangulum, OEN, ad rectangulum ſub, OE, &</s> <s xml:id="echoid-s9414" xml:space="preserve">, NM, <lb/>vna cum rectang. </s> <s xml:id="echoid-s9415" xml:space="preserve">ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9416" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9417" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9418" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9419" xml:space="preserve">ſub, ME.</s> <s xml:id="echoid-s9420" xml:space="preserve"/> </p> <div xml:id="echoid-div895" type="float" level="2" n="1"> <figure xlink:label="fig-0389-01" xlink:href="fig-0389-01a"> <image file="0389-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0389-01"/> </figure> <note position="right" xlink:label="note-0389-01" xlink:href="note-0389-01a" xml:space="preserve">Coroll. 3. <lb/>26. l. 2.</note> <note position="right" xlink:label="note-0389-02" xlink:href="note-0389-02a" xml:space="preserve">Corol. 21. <lb/>l. 2.</note> </div> <p> <s xml:id="echoid-s9421" xml:space="preserve">Quia verò omnia quadrata trianguli, DMF, ſunt {1/3}. </s> <s xml:id="echoid-s9422" xml:space="preserve">omnium <lb/> <anchor type="note" xlink:label="note-0390-01a" xlink:href="note-0390-01"/> quadratorum, SF, ideò ad omnia quadrata fruſti, HDFG, erunt <lb/>vt {1/3}. </s> <s xml:id="echoid-s9423" xml:space="preserve">rectanguli, OEN, ad rectangulum ſub, OE, &</s> <s xml:id="echoid-s9424" xml:space="preserve">, NM, vna cũ <lb/>rectangulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9425" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9426" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9427" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9428" xml:space="preserve">ſub, ME, vel vt <lb/>horum tripla .</s> <s xml:id="echoid-s9429" xml:space="preserve">i. </s> <s xml:id="echoid-s9430" xml:space="preserve">vt rectangulum, OEN, ad rectangulum ſub, OE, <lb/>& </s> <s xml:id="echoid-s9431" xml:space="preserve">tripla, NM, vna cum rectangulo ſub compoſita ex ſexquialte-<lb/>ra, ON, .</s> <s xml:id="echoid-s9432" xml:space="preserve">i. </s> <s xml:id="echoid-s9433" xml:space="preserve">ex, NX, & </s> <s xml:id="echoid-s9434" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9435" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9436" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9437" xml:space="preserve">ſub, ME, quę oſtendere opus <lb/>erat.</s> <s xml:id="echoid-s9438" xml:space="preserve"/> </p> <div xml:id="echoid-div896" type="float" level="2" n="2"> <note position="left" xlink:label="note-0390-01" xlink:href="note-0390-01a" xml:space="preserve">24. l. 2.</note> </div> </div> <div xml:id="echoid-div898" type="section" level="1" n="537"> <head xml:id="echoid-head561" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s9439" xml:space="preserve">IN eadem antecedentis figura productis, CH, RG, ver-<lb/>ſus baſem, DF, cui incidant in, PQ, regula eadem re-<lb/>tenta, oſtendemus omnia quadrata, SF, ad omnia quadra-<lb/>ta fruſti, HDFG, demptis omnibus quadratis, HQ, eſſe vt <lb/>rectangulum, OEN, ad rectangulum ſub, EM, & </s> <s xml:id="echoid-s9440" xml:space="preserve">ſub com-<lb/>poſita ex {1/3}. </s> <s xml:id="echoid-s9441" xml:space="preserve">EM, integra, MN, & </s> <s xml:id="echoid-s9442" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s9443" xml:space="preserve">NO: </s> <s xml:id="echoid-s9444" xml:space="preserve">Omnia verò quadra-<lb/>ta trianguli, DMF, ad eadem eſſe oſtendemus, vt rectang. <lb/></s> <s xml:id="echoid-s9445" xml:space="preserve">OEN, ad rect. </s> <s xml:id="echoid-s9446" xml:space="preserve">ſub compoſita ex, EX, dupla, NM, & </s> <s xml:id="echoid-s9447" xml:space="preserve">ſub, ME.</s> <s xml:id="echoid-s9448" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9449" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s9450" xml:space="preserve">n. </s> <s xml:id="echoid-s9451" xml:space="preserve">quadrata, SF, ad omnia qu adrata fruſti, HDFG <lb/> <anchor type="note" xlink:label="note-0390-02a" xlink:href="note-0390-02"/> <anchor type="figure" xlink:label="fig-0390-01a" xlink:href="fig-0390-01"/> oſtenſa ſunt eſſe, vt rectangulum, OEN, ad <lb/>rectangulum ſub, OE, & </s> <s xml:id="echoid-s9452" xml:space="preserve">NM, vna cum, <lb/>rectangulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9453" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9454" xml:space="preserve">{1/3}. <lb/></s> <s xml:id="echoid-s9455" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9456" xml:space="preserve">ſub, ME: </s> <s xml:id="echoid-s9457" xml:space="preserve">Omnia verò quadrata-<lb/>SF, ad omnia quadrata, HQ, ſunt vt qua <lb/>dratum, DF, ad quadratum, PQ, vel ad <lb/> <anchor type="note" xlink:label="note-0390-03a" xlink:href="note-0390-03"/> quadratum, HG, .</s> <s xml:id="echoid-s9458" xml:space="preserve">i. </s> <s xml:id="echoid-s9459" xml:space="preserve">vt rectangulum, OEN, <lb/>ad rectangulum, OMN, ergo eadem ad re-<lb/> <anchor type="note" xlink:label="note-0390-04a" xlink:href="note-0390-04"/> liquum omnium quadratorum fruſti, DH <lb/>GF, demptis omnibus quadratis, HQ, e-<lb/>runt vt rectangulum, OEN, ad reliquum, <lb/>dempto rectangulo, OMN, à rectangulis <lb/>ſub, OE, MN, & </s> <s xml:id="echoid-s9460" xml:space="preserve">ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9461" xml:space="preserve">ON, <lb/>& </s> <s xml:id="echoid-s9462" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9463" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9464" xml:space="preserve">ſub, ME, eſt autem rectangu-<lb/> <anchor type="note" xlink:label="note-0390-05a" xlink:href="note-0390-05"/> lum ſub, OE, MN, æquale rectangulis ſub, OM, MN, & </s> <s xml:id="echoid-s9465" xml:space="preserve">ſub, EM, <lb/>MN, ergo dempto rectangulo, OMN, à rectangulo ſub, OE, MN, <lb/>remanet rectangulum, EMN, ad quod vna cum rectangulo ſub <lb/>compoſita ex {1/2}. </s> <s xml:id="echoid-s9466" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9467" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9468" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9469" xml:space="preserve">ſub, ME, ipſum rectangulum <lb/>OEN, erit vt omnia quad. </s> <s xml:id="echoid-s9470" xml:space="preserve">SF, ad omnia quad. </s> <s xml:id="echoid-s9471" xml:space="preserve">fruſti, HDFG, dẽ- <pb o="371" file="0391" n="391" rhead="LIBER V."/> ptis omnibus quadratis, HQ, æquatur autem rectangulum, EM <lb/>N, cum rectangulo ſub, EM, & </s> <s xml:id="echoid-s9472" xml:space="preserve">iub compoſita ex {1/3}. </s> <s xml:id="echoid-s9473" xml:space="preserve">EM, & </s> <s xml:id="echoid-s9474" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s9475" xml:space="preserve">N <lb/>O, rectangulo ſub, EM, & </s> <s xml:id="echoid-s9476" xml:space="preserve">ſub compoſita ex {1/3}. </s> <s xml:id="echoid-s9477" xml:space="preserve">EM. </s> <s xml:id="echoid-s9478" xml:space="preserve">integra, MN, <lb/>& </s> <s xml:id="echoid-s9479" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s9480" xml:space="preserve">NO, ergo omnia quadrata, SF, ad omnia quadrata fruſti, D <lb/>HGF, demptis omnibus quadratis, HQ, erunt vt rectangulum, <lb/>OEN, ad rectangulum ſub, EM, & </s> <s xml:id="echoid-s9481" xml:space="preserve">ſub compoſita ex {1/3}. </s> <s xml:id="echoid-s9482" xml:space="preserve">EM, in-<lb/>tegra, MN, & </s> <s xml:id="echoid-s9483" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s9484" xml:space="preserve">NO.</s> <s xml:id="echoid-s9485" xml:space="preserve"/> </p> <div xml:id="echoid-div898" type="float" level="2" n="1"> <note position="left" xlink:label="note-0390-02" xlink:href="note-0390-02a" xml:space="preserve">In antec.</note> <figure xlink:label="fig-0390-01" xlink:href="fig-0390-01a"> <image file="0390-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0390-01"/> </figure> <note position="left" xlink:label="note-0390-03" xlink:href="note-0390-03a" xml:space="preserve">9. l. 2.</note> <note position="left" xlink:label="note-0390-04" xlink:href="note-0390-04a" xml:space="preserve">39. & Sch. <lb/>40. l. 1.</note> <note position="left" xlink:label="note-0390-05" xlink:href="note-0390-05a" xml:space="preserve">1. 2. Elem.</note> </div> <p> <s xml:id="echoid-s9486" xml:space="preserve">Omnia verò quadrata trianguli, DMF, ad eadem erunt, vt {1/3}. <lb/></s> <s xml:id="echoid-s9487" xml:space="preserve">rectanguli, OEN, ad rectangulum ſub, EM, & </s> <s xml:id="echoid-s9488" xml:space="preserve">ſub compoſita ex <lb/>{1/3}. </s> <s xml:id="echoid-s9489" xml:space="preserve">EM, integra, MN, & </s> <s xml:id="echoid-s9490" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s9491" xml:space="preserve">NO, .</s> <s xml:id="echoid-s9492" xml:space="preserve">i. </s> <s xml:id="echoid-s9493" xml:space="preserve">vt totum rectangulum ſub, O <lb/>EN, ad rectangulum ſub, EM, & </s> <s xml:id="echoid-s9494" xml:space="preserve">ſub compoſita ex, EM, tripla, <lb/>MN, &</s> <s xml:id="echoid-s9495" xml:space="preserve">, NX, .</s> <s xml:id="echoid-s9496" xml:space="preserve">i. </s> <s xml:id="echoid-s9497" xml:space="preserve">ſub, EM, & </s> <s xml:id="echoid-s9498" xml:space="preserve">ſub compoſita ex, EX, & </s> <s xml:id="echoid-s9499" xml:space="preserve">dupla, MN, <lb/>quæ oſtendenda erant.</s> <s xml:id="echoid-s9500" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div900" type="section" level="1" n="538"> <head xml:id="echoid-head562" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s9501" xml:space="preserve">IN eadem figura, regula eadem retenta, oſtendemus om-<lb/>mnia quadrata, AF, demptis omnibus quadratis hyper-<lb/>bolæ, DNF, ad omnia quadrata, SF, demptis omnibus <lb/>quadratis fruſti, HDFG, eſſe vt parallelepipedum ſub cõ-<lb/>poſita ex ipſa, XE, EN, & </s> <s xml:id="echoid-s9502" xml:space="preserve">ſub quadrato, NE, ad parallele-<lb/>pipedum ſub compoſita ex eadem, XE, & </s> <s xml:id="echoid-s9503" xml:space="preserve">cum, EN, NM, <lb/>& </s> <s xml:id="echoid-s9504" xml:space="preserve">ſub quadrato, ME.</s> <s xml:id="echoid-s9505" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9506" xml:space="preserve">Quia enim omnia quadrata, AF, ad omnia quadrata hyperbo-<lb/> <anchor type="note" xlink:label="note-0391-01a" xlink:href="note-0391-01"/> læ, DNF, ſunt vt, OE, ad compoſitam ex {1/2}. </s> <s xml:id="echoid-s9507" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9508" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9509" xml:space="preserve">NE, ideò <lb/>per conuerſionem rationis, & </s> <s xml:id="echoid-s9510" xml:space="preserve">conuertendo omnia quadrata, AF, <lb/>demptis omnibus quadratis hyperbolæ, DNF, ad omnia quadra-<lb/>ta, AF, erunt vt compoſita ex {1/2}. </s> <s xml:id="echoid-s9511" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9512" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9513" xml:space="preserve">NE, ad, OE, .</s> <s xml:id="echoid-s9514" xml:space="preserve">i. </s> <s xml:id="echoid-s9515" xml:space="preserve">ſum-<lb/>pta, NE, communialtitudine, vt rectangulum ſub compoſita ex <lb/>{1/2}. </s> <s xml:id="echoid-s9516" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9517" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9518" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9519" xml:space="preserve">ſub, NE, ad rectangulum, OEN. </s> <s xml:id="echoid-s9520" xml:space="preserve">Quoniam <lb/>verò omnia quadrata, AF, demptis omnibus quadratis hyperbo-<lb/>læ, DNF, ad omnia quadrata, SF, demptis omnibus quadratis <lb/>fruſti, DHGF, habent rationem compoſitam ex ea, quam habent <lb/>omnia quadrata, AF, demptis omnibus quadratis hyperbolæ, D <lb/> <anchor type="note" xlink:label="note-0391-02a" xlink:href="note-0391-02"/> NF, ad omnia quadrata, AF, .</s> <s xml:id="echoid-s9521" xml:space="preserve">i. </s> <s xml:id="echoid-s9522" xml:space="preserve">ex ea, quam habet rectangulum <lb/>ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9523" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9524" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9525" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9526" xml:space="preserve">ſub, N E, ad rectangu-<lb/>lum, NEO; </s> <s xml:id="echoid-s9527" xml:space="preserve">& </s> <s xml:id="echoid-s9528" xml:space="preserve">ex ratione, quam habent omnia quadrata, <lb/>AF, ad omnia quadrata, SF, ideſt ex ea, quam habet, NE, <lb/>ad, EM, & </s> <s xml:id="echoid-s9529" xml:space="preserve">tandem ex ea, quam habent omnia quadrata, SF, <lb/> <anchor type="note" xlink:label="note-0391-03a" xlink:href="note-0391-03"/> ad omnia quadrata, SF, demptis omnibus quadratis fruſti, HDF <pb o="372" file="0392" n="392" rhead="GEOMETRIE"/> G, ideò omnia quadrata, AF, demptis omnibus quadratis hyper-<lb/>bolæ, DNF, ad omnia quadrata, SF, demptis omnibus quadra-<lb/>tis fruſti, HDFG, habebunt rationem compoſitam ex ea, quam <lb/>habet rectangulum ſub compoſita ex {1/2}: </s> <s xml:id="echoid-s9530" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9531" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9532" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9533" xml:space="preserve">ſub, N <lb/>E, ad rectangulum, NEO, & </s> <s xml:id="echoid-s9534" xml:space="preserve">ex ea, quam habet, NE, ad, EM, <lb/>& </s> <s xml:id="echoid-s9535" xml:space="preserve">ex ea, quam habent omnia quadrata, SF, ad omnia quadrata, <lb/> <anchor type="figure" xlink:label="fig-0392-01a" xlink:href="fig-0392-01"/> SF, demptis omnibus quadratis fruſti, HD <lb/>FG. </s> <s xml:id="echoid-s9536" xml:space="preserve">Quoniam autem omnia quadrata, <lb/>SF, ad omnia quadrata fruſti, HDFG, sũt <lb/>vt rectã gulum, OEN, ad rect. </s> <s xml:id="echoid-s9537" xml:space="preserve">ſub OE, NM, <lb/>cum'rectãg. </s> <s xml:id="echoid-s9538" xml:space="preserve">ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9539" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9540" xml:space="preserve">{1/3}. <lb/></s> <s xml:id="echoid-s9541" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9542" xml:space="preserve">ſub, ME, ideò oĩa quadrata, SF, ad <lb/>reſiduum, daptis omnibus quadratis fruſti, <lb/>HDFG, erunt vt rectangulum, OEN, ad <lb/>reſiduum, demptis à rectangulo, OEN, re-<lb/> <anchor type="note" xlink:label="note-0392-01a" xlink:href="note-0392-01"/> ctangulo, ſub, OE, NM, vna eum rectan-<lb/>gulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9543" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9544" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9545" xml:space="preserve">ME, <lb/>& </s> <s xml:id="echoid-s9546" xml:space="preserve">ſub, ME; </s> <s xml:id="echoid-s9547" xml:space="preserve">ſi igitur à rectangulo, OEN, <lb/>dempſeris rectangulum ſub, OE, MN, re-<lb/>manebit rectangulum ſub, OE, EM, rur-<lb/>ſus ſi à rectangulo ſub, OE, EM, dempſeris rectangulum ſub com-<lb/>poſita ex {1/2}. </s> <s xml:id="echoid-s9548" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9549" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9550" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9551" xml:space="preserve">ſub, ME, .</s> <s xml:id="echoid-s9552" xml:space="preserve">i. </s> <s xml:id="echoid-s9553" xml:space="preserve">ſi dempſeris rectangu-<lb/>lum ſub, OB, &</s> <s xml:id="echoid-s9554" xml:space="preserve">, ME, remanebit rectangulum ſub, BE, EM, à quo <lb/>ſi adhuc auferas rectangulum ſub {1/3}. </s> <s xml:id="echoid-s9555" xml:space="preserve">ME, & </s> <s xml:id="echoid-s9556" xml:space="preserve">ſub, ME, .</s> <s xml:id="echoid-s9557" xml:space="preserve">i. </s> <s xml:id="echoid-s9558" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9559" xml:space="preserve">qua-<lb/>drati, ME, habebimus rectangulum, BEM, dempto {1/3}. </s> <s xml:id="echoid-s9560" xml:space="preserve">quadrati, <lb/>ME, ad quod rectangulum, OEN, erit vt omnia quadrata, SF, ad <lb/>ſui reliquum, demptis omnibus quadratis fruſti, HDFG, ergo om-<lb/>nia quadrata, AF, demptis omnibus quadratis hyperbolæ, DNF, <lb/>ad omnia quadrata, SF, demptis omnibus quadratis fruſti, HDF <lb/>G, habebunt rationem compoſitam ex his rationibus .</s> <s xml:id="echoid-s9561" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9562" xml:space="preserve">ex ea, quã <lb/>habet rectangulum ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9563" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9564" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9565" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9566" xml:space="preserve">ſub, NE, <lb/>ad rectangulum, OEN, & </s> <s xml:id="echoid-s9567" xml:space="preserve">ex ratione, NE, ad, EM, & </s> <s xml:id="echoid-s9568" xml:space="preserve">ex ea, quam <lb/>habet rectangulum, OEN, ad rectangulum, BEM, dempto {1/3}. </s> <s xml:id="echoid-s9569" xml:space="preserve">qua-<lb/>drati, ME; </s> <s xml:id="echoid-s9570" xml:space="preserve">harum autem iſtæ duæ, quam .</s> <s xml:id="echoid-s9571" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9572" xml:space="preserve">habet rectangulum <lb/>ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9573" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9574" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9575" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9576" xml:space="preserve">ſub, NE, ad rectangulum, <lb/>OEN, & </s> <s xml:id="echoid-s9577" xml:space="preserve">quam habet rectangulum, OEN, ad rectangulum, BEM, <lb/>dempto {1/3}. </s> <s xml:id="echoid-s9578" xml:space="preserve">quadrati, ME, conficiunt rationem rectanguli ſub com. <lb/></s> <s xml:id="echoid-s9579" xml:space="preserve">poſita ex {1/2}. </s> <s xml:id="echoid-s9580" xml:space="preserve">ON, & </s> <s xml:id="echoid-s9581" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s9582" xml:space="preserve">NE, & </s> <s xml:id="echoid-s9583" xml:space="preserve">ſub, NE, ad rectangulum, BEM <lb/>dempto {1/3}. </s> <s xml:id="echoid-s9584" xml:space="preserve">quadrati, ME, vel, his triplicatis, conficiunt rationem' <lb/>rectanguli ſub compoſita ex tribus, BN, .</s> <s xml:id="echoid-s9585" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9586" xml:space="preserve">ex, NX, & </s> <s xml:id="echoid-s9587" xml:space="preserve">ter {2/3}. </s> <s xml:id="echoid-s9588" xml:space="preserve">NE, <lb/>.</s> <s xml:id="echoid-s9589" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9590" xml:space="preserve">dupla, NE, .</s> <s xml:id="echoid-s9591" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9592" xml:space="preserve">ſub compoſita ex, NE, &</s> <s xml:id="echoid-s9593" xml:space="preserve">, EX, & </s> <s xml:id="echoid-s9594" xml:space="preserve">ſub, NE, ad <lb/>rectangulum ſub tripla, BE, & </s> <s xml:id="echoid-s9595" xml:space="preserve">ſub, EM, demptis {3/3}. </s> <s xml:id="echoid-s9596" xml:space="preserve">ideſt integro <pb o="373" file="0393" n="393" rhead="LIBER V."/> dempto quadrato, ME, quia verò tripla, BE, eſt compoſita ex, E <lb/>X, & </s> <s xml:id="echoid-s9597" xml:space="preserve">dupla, EN, ſi a rectangulo ſub compoſita ex, EX, & </s> <s xml:id="echoid-s9598" xml:space="preserve">dupla, <lb/>EN, & </s> <s xml:id="echoid-s9599" xml:space="preserve">ſub, EM, abſtuleris quadratum, ME, .</s> <s xml:id="echoid-s9600" xml:space="preserve">i. </s> <s xml:id="echoid-s9601" xml:space="preserve">rectangulum ſub, <lb/>MF, &</s> <s xml:id="echoid-s9602" xml:space="preserve">, ME, remanebit rectangulum ſub compoſita ex ipſa, XE, <lb/>EN, NM, & </s> <s xml:id="echoid-s9603" xml:space="preserve">ſub, EM, illas ergo tres componentes rationes in has <lb/>duas reſolutas habemus, ſcilicet in eam, quam habet rectangulũ <lb/>ſub, XEN, integra, & </s> <s xml:id="echoid-s9604" xml:space="preserve">ſub, EN, ad rectangulum ſub integra, XE, <lb/>EN, NM, & </s> <s xml:id="echoid-s9605" xml:space="preserve">ſub, ME, & </s> <s xml:id="echoid-s9606" xml:space="preserve">in eam, quam habet, NE, ad, EM, quæ <lb/>duæ rationes componunt rationem parallelepipedi ſub, NE, & </s> <s xml:id="echoid-s9607" xml:space="preserve"><lb/>ſub rectangulo integræ, XEN, ductæ in, EN, ideſt parallelepipe-<lb/> <anchor type="note" xlink:label="note-0393-01a" xlink:href="note-0393-01"/> di ſub integra, XEN, & </s> <s xml:id="echoid-s9608" xml:space="preserve">quadrato, NE, ad parallelepipedum ſub, <lb/>ME, & </s> <s xml:id="echoid-s9609" xml:space="preserve">rectangulo integræ, XE, EN, NM, ductæ in, ME, .</s> <s xml:id="echoid-s9610" xml:space="preserve">i. </s> <s xml:id="echoid-s9611" xml:space="preserve">ad <lb/>parallelepipedum ſub integra, XE, EN, NM, & </s> <s xml:id="echoid-s9612" xml:space="preserve">quadrato, ME, <lb/>ergo omnia quadrata, AF, demptis omnibus quadratis hyperbo-<lb/>læ, DNF, ad omnia quadrata, SF, demptis omnibus quadratis <lb/>fruſti, HDFG, erunt vt parallelepipedum ſub integra, XEN, & </s> <s xml:id="echoid-s9613" xml:space="preserve"><lb/>quadrato, NE, ad parallelepipedum ſub integra, XE, EN, NM, <lb/>& </s> <s xml:id="echoid-s9614" xml:space="preserve">quadrato, ME, quod erat oſtendendum.</s> <s xml:id="echoid-s9615" xml:space="preserve"/> </p> <div xml:id="echoid-div900" type="float" level="2" n="1"> <note position="right" xlink:label="note-0391-01" xlink:href="note-0391-01a" xml:space="preserve">Is huius.</note> <note position="right" xlink:label="note-0391-02" xlink:href="note-0391-02a" xml:space="preserve">Defin. 12. <lb/>1. I.</note> <note position="right" xlink:label="note-0391-03" xlink:href="note-0391-03a" xml:space="preserve">10, 1.2.</note> <figure xlink:label="fig-0392-01" xlink:href="fig-0392-01a"> <image file="0392-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0392-01"/> </figure> <note position="left" xlink:label="note-0392-01" xlink:href="note-0392-01a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-0393-01" xlink:href="note-0393-01a" xml:space="preserve">3.6. .1.</note> </div> </div> <div xml:id="echoid-div902" type="section" level="1" n="539"> <head xml:id="echoid-head563" xml:space="preserve">PROBLEMA I. PROPOS. VI.</head> <p> <s xml:id="echoid-s9616" xml:space="preserve">A Data hyperbola portionem abſcindere per lineam <lb/>ad eiuſdem axim, vel diametrum ordinatim appli-<lb/>catam, cuius omnia quadrata, regula propoſitæ hyperbo-<lb/>læ baſi, ad omnia quadrata trianguli in eadem baſi, & </s> <s xml:id="echoid-s9617" xml:space="preserve">cir-<lb/>ca eundem axim, vel diametrum cum portione, ſiue hyper-<lb/>bola abſciſſa, exiſtentis, habeant datam rationem, quam <lb/>oportet eſſe quidem maioris inæqualitatis, ſed tamen mi-<lb/>norem ſexquialtera.</s> <s xml:id="echoid-s9618" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9619" xml:space="preserve">Sit ergo data hyperbola, FEG, cuius axis, vel diameter, E M & </s> <s xml:id="echoid-s9620" xml:space="preserve"><lb/>larus tranſuerſum, CE, cuius ſit, AE, ſexquialtera, baſis, & </s> <s xml:id="echoid-s9621" xml:space="preserve">regu-<lb/>la, FG, data ratio, quam habet, HR, ad, RL, maioris inæquali-<lb/>tatis, ſed minor ſexquialtera, oportet ergo ab hyperbola, FEG, <lb/>per lineam ad, EM, ordinatim applicatam .</s> <s xml:id="echoid-s9622" xml:space="preserve">i. </s> <s xml:id="echoid-s9623" xml:space="preserve">baſi, fiue regulæ, <lb/>FG, parallelam, portionem, ſiue hyperbolam abſcindele, cuius <lb/>omnia quadrata ad omnia quadrata trianguli in eadem baſi, & </s> <s xml:id="echoid-s9624" xml:space="preserve"><lb/>circa eundem axim, vel diametrum cum ipſa habeant rationem, <lb/>quam habet, HR, ad, RL; </s> <s xml:id="echoid-s9625" xml:space="preserve">quia ergo ratio, HR, ad, RL, eſt mi-<lb/>nor ſexquialtera, erit minor ea, quam habet, AE, ad, EC, & </s> <s xml:id="echoid-s9626" xml:space="preserve">etiã <pb o="374" file="0394" n="394" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0394-01a" xlink:href="fig-0394-01"/> diuidendo minor ea, quam habet, <lb/>AC, ad, CE, eandem ergo, quam <lb/>habet, HL, ad, LR, habebit, AC, ad <lb/>maiorem, CE, ſit illa, CO, & </s> <s xml:id="echoid-s9627" xml:space="preserve">per, <lb/>O, ducatur, SV, parallela ipſi regu-<lb/>læ, FG, iunganturque, SE, EV: </s> <s xml:id="echoid-s9628" xml:space="preserve">Om-<lb/>nia ergo quadrata hyperbolæ, SEV, <lb/>ad omnia quadrata trianguli, SEV, <lb/>ſunt vt, AO, ad, OC, quia verò, AC, <lb/>ad, CO, eſt vt, HL, ad, LR, compo-<lb/>nendo, AO, ad, OC, erit vt, HR, ad, <lb/>RL, ergo omnia quadrata hyperbo-<lb/>læ, SEV, ad omnia quadrata triangu-<lb/>li, SEV, erunt vt, HR, ad, RL, .</s> <s xml:id="echoid-s9629" xml:space="preserve">i. </s> <s xml:id="echoid-s9630" xml:space="preserve">in <lb/>ratione data, quod facere opus erat.</s> <s xml:id="echoid-s9631" xml:space="preserve"/> </p> <div xml:id="echoid-div902" type="float" level="2" n="1"> <figure xlink:label="fig-0394-01" xlink:href="fig-0394-01a"> <image file="0394-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0394-01"/> </figure> </div> </div> <div xml:id="echoid-div904" type="section" level="1" n="540"> <head xml:id="echoid-head564" xml:space="preserve">THEOREMA VI. PROPOS. VII.</head> <p> <s xml:id="echoid-s9632" xml:space="preserve">SI circa datam hyperbolam deſcribantur aſymptoti, <lb/>eiuſdem autem baſis vſq; </s> <s xml:id="echoid-s9633" xml:space="preserve">ad aſymptotos producatur, <lb/>quæ ſumatur pro regula: </s> <s xml:id="echoid-s9634" xml:space="preserve">O nnia quadrata hyperbolæ ad <lb/>omnia quadrata trianguli aſymptotis, & </s> <s xml:id="echoid-s9635" xml:space="preserve">baſi comprchen-<lb/>ſi, habebunt rationem compoſitam ex ea, quam habet <lb/>quadratum baſis hyperbolæ ad quadratum baſis trianguli, <lb/>& </s> <s xml:id="echoid-s9636" xml:space="preserve">ex ea, quam habet rectangulum ſub compoſita ex ſex-<lb/>quialtera tranſuerſi lateris, & </s> <s xml:id="echoid-s9637" xml:space="preserve">axi, vel diametro datæ hy-<lb/>perbolæ, ſub eodem axi, vel diametro, ad rectangulum <lb/>ſub compoſita ex tranſuerſo latere, & </s> <s xml:id="echoid-s9638" xml:space="preserve">axi, vel diametro <lb/>eiuſdem hyperbolæ; </s> <s xml:id="echoid-s9639" xml:space="preserve">& </s> <s xml:id="echoid-s9640" xml:space="preserve">ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9641" xml:space="preserve">tranſuerſi late-<lb/>ris, & </s> <s xml:id="echoid-s9642" xml:space="preserve">eodem axi, vel diametro.</s> <s xml:id="echoid-s9643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9644" xml:space="preserve">Sit igitar data hyperbola, cuius baſis, SX, circa axim, vel dia-<lb/>metrum, OV, cuius tranſuerſum latus ſit, BO, bifariam in C, di-<lb/>uiſum, ſit autem illi in directum adiuncta, AB, æqualis, BC, de-<lb/>inde ducta per, O, tangente hyperbolam, quæ ſit, ED, cui erit <lb/>parallela baſis, SX, abicindantur, EO, OD, ita vt quadratum, E <lb/>O, & </s> <s xml:id="echoid-s9645" xml:space="preserve">quadratum, OD, ſeorſim ſint æqualia quartæ parti rectan-<lb/>guli ſub, BO, latere tranſuerſo, & </s> <s xml:id="echoid-s9646" xml:space="preserve">ſub eiuſdem recto latere, ſi ergo <lb/>iunctis, CE, CD, ipsæ producantur indefinitè verſus baſim, SX, <lb/> <anchor type="note" xlink:label="note-0394-01a" xlink:href="note-0394-01"/> cui productæ occurant in punctis, H, R, erunt, CH, CR, afym- <pb o="375" file="0395" n="395" rhead="LIBER V."/> ptoti datæ hyperbolæ. </s> <s xml:id="echoid-s9647" xml:space="preserve">Dico igitur omnia quadrata hyperbolę, <lb/>SOX, ad omnia quadrata trianguli, HCR, habere rationem com-<lb/> <anchor type="figure" xlink:label="fig-0395-01a" xlink:href="fig-0395-01"/> poſitam ex ea, quam habet quadratum, <lb/>SX, ad quadratum, HR, & </s> <s xml:id="echoid-s9648" xml:space="preserve">rectangulũ, <lb/>AVO, ad rectangulum, BVC, inngan-<lb/>tur, OS, OX: </s> <s xml:id="echoid-s9649" xml:space="preserve">Omnia ergo quadrata hy-<lb/> <anchor type="note" xlink:label="note-0395-01a" xlink:href="note-0395-01"/> perbolæ, SOX, ad omnia quadrata triã-<lb/>guli, HCR, habent rationem compoſi-<lb/>tam ex ea, quam habent omnia quadra-<lb/>ta hyperbolæ, SOX, ad omnia quadra-<lb/>ta trianguli, SOX, .</s> <s xml:id="echoid-s9650" xml:space="preserve">i. </s> <s xml:id="echoid-s9651" xml:space="preserve">ex ea, quam Habet, <lb/>AV, ad, VB, & </s> <s xml:id="echoid-s9652" xml:space="preserve">ex ea, quam habent om-<lb/>nia quadrata trianguli, SOX, ad omnia <lb/>quadrata trianguli, HCR, quæ eſt com-<lb/>poſita ex ea, quam habet quadratum, S <lb/> <anchor type="note" xlink:label="note-0395-02a" xlink:href="note-0395-02"/> X, ad quadratum, HR, & </s> <s xml:id="echoid-s9653" xml:space="preserve">ex ea, quam <lb/>habet, OV, ad, VC, habemus ergo has tres rationes componen-<lb/>tes rationem, quam habent omnia quadrata hyperbolæ, SOX, ad <lb/>omnia quadrata trianguli, HCR, ſcilicet eam, quam habet qua-<lb/>dratum, SX, ad quadratum, HR, & </s> <s xml:id="echoid-s9654" xml:space="preserve">quam habet, AV, ad, VB, & </s> <s xml:id="echoid-s9655" xml:space="preserve"><lb/>tandem, quam habet, OV, ad, VC, harum autem iſtæ duæ .</s> <s xml:id="echoid-s9656" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9657" xml:space="preserve">quã <lb/>habet, AV, ad, VB, &</s> <s xml:id="echoid-s9658" xml:space="preserve">, OV, ad; </s> <s xml:id="echoid-s9659" xml:space="preserve">VC, componunt rationem rectã-<lb/>guli, AVO, ad rectangulum, BVC, ergo omnia quadrata hyper-<lb/>bolæ, SOX, ad omnia quadrata trianguli, HCR, habent rationẽ <lb/>compoſitam ex ea, quam habet quadratum, SX, ad quadratum, <lb/>HR, & </s> <s xml:id="echoid-s9660" xml:space="preserve">rectangulum, AVO, ad rectangulum, BVC, quod oſten-<lb/>dere opus erat.</s> <s xml:id="echoid-s9661" xml:space="preserve"/> </p> <div xml:id="echoid-div904" type="float" level="2" n="1"> <note position="left" xlink:label="note-0394-01" xlink:href="note-0394-01a" xml:space="preserve">1.2. Con.</note> <figure xlink:label="fig-0395-01" xlink:href="fig-0395-01a"> <image file="0395-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0395-01"/> </figure> <note position="right" xlink:label="note-0395-01" xlink:href="note-0395-01a" xml:space="preserve">Defin .12. <lb/>l. 1. <lb/>1. huius. <lb/>D. Cor. <lb/>22. l. 2.</note> <note position="right" xlink:label="note-0395-02" xlink:href="note-0395-02a" xml:space="preserve">6 ſec.</note> </div> </div> <div xml:id="echoid-div906" type="section" level="1" n="541"> <head xml:id="echoid-head565" xml:space="preserve">THEOREMA VII. PROPOS. VIII.</head> <p> <s xml:id="echoid-s9662" xml:space="preserve">IN eadem anteced. </s> <s xml:id="echoid-s9663" xml:space="preserve">figura, regula eadem, retenta, oſten-<lb/>demus (ducta intra hyperbolam, SOX ipſa, IY, occur-<lb/>rente aſymptotis, CH, CR, in, T, P,) omnia quadrata tra-<lb/>pezij, THRP, ad omnia quadrata fruſti hyperbolæ, ISXY, <lb/>eſſe in ratione compoſita ex ea, quam habet rectangulum <lb/>ſub, GP, VR, cum .</s> <s xml:id="echoid-s9664" xml:space="preserve">quadrati, PM, ad quadratum, VX, & </s> <s xml:id="echoid-s9665" xml:space="preserve"><lb/>ex ea, quam habet rectangulum, BVO, ad rectangulum <lb/>ſub, BV, OG, vna cum rectangulo ſub compoſita ex .</s> <s xml:id="echoid-s9666" xml:space="preserve">BO, <lb/>& </s> <s xml:id="echoid-s9667" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9668" xml:space="preserve">GV, & </s> <s xml:id="echoid-s9669" xml:space="preserve">ſub, GV.</s> <s xml:id="echoid-s9670" xml:space="preserve"/> </p> <pb o="376" file="0396" n="396" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s9671" xml:space="preserve">Ducantur per puncta, X, R, XN, RM, rectæ lineæ parallelæ <lb/>axi, vel diametro hyperbolæ, OV, occurrentes, TP, productæ, in, <lb/>N, M: </s> <s xml:id="echoid-s9672" xml:space="preserve">Omnia ergo quadr. </s> <s xml:id="echoid-s9673" xml:space="preserve">trapezij, GPRV, ad omnia quadrata <lb/>quadlilinei, GVXY, habent rationem compoſitam ex ea, quam <lb/>habent omnia quadrata trapezij, PGVR, ad omnia quadrata, <lb/>GR, .</s> <s xml:id="echoid-s9674" xml:space="preserve">i. </s> <s xml:id="echoid-s9675" xml:space="preserve">ex ea, quam habet rectangulum ſub, PG, VR, cum {1/3}. </s> <s xml:id="echoid-s9676" xml:space="preserve">qua-<lb/>drati, PM, ad quadratum, VR, & </s> <s xml:id="echoid-s9677" xml:space="preserve">ex ea, quam habent omnia qua-<lb/>drata, GR, ad omnia quadrata, GX, ideſt ex ea, quam habet qua-<lb/>dratum, RV, ad quadratum, VX; </s> <s xml:id="echoid-s9678" xml:space="preserve">quæ duæ rationes componunt <lb/> <anchor type="figure" xlink:label="fig-0396-01a" xlink:href="fig-0396-01"/> rationem, quam habet rectangulum ſub, <lb/>GP, VR, cum {1/3}. </s> <s xml:id="echoid-s9679" xml:space="preserve">quadrati, PM, ad qua <lb/> <anchor type="note" xlink:label="note-0396-01a" xlink:href="note-0396-01"/> dratum, VX; </s> <s xml:id="echoid-s9680" xml:space="preserve">& </s> <s xml:id="echoid-s9681" xml:space="preserve">tandem ex ea, quam ha-<lb/>bent omnia quadrata, GX, ad omnia <lb/>quadrata, GYXV, .</s> <s xml:id="echoid-s9682" xml:space="preserve">i. </s> <s xml:id="echoid-s9683" xml:space="preserve">ex ea, quam habet <lb/> <anchor type="note" xlink:label="note-0396-02a" xlink:href="note-0396-02"/> rectangulum, BVO, ad rectangulum ſub, <lb/>BV, GO, vna cum rectangulo ſub com-<lb/>poſita ex {1/2}. </s> <s xml:id="echoid-s9684" xml:space="preserve">BO, & </s> <s xml:id="echoid-s9685" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9686" xml:space="preserve">GV, & </s> <s xml:id="echoid-s9687" xml:space="preserve">ſub, GV, <lb/> <anchor type="note" xlink:label="note-0396-03a" xlink:href="note-0396-03"/> ergo omnia quadrata trapezij, PGVR, <lb/>ad omnia quadrata quadrilinei, YGVX, <lb/>vel eorum quadrupla .</s> <s xml:id="echoid-s9688" xml:space="preserve">i. </s> <s xml:id="echoid-s9689" xml:space="preserve">omnia quadrata <lb/>trapezij, THRP, ad omnia quadrata fru-<lb/>ſti, ISXY, habebunt rationem compoſi <lb/> <anchor type="note" xlink:label="note-0396-04a" xlink:href="note-0396-04"/> tam ex ea; </s> <s xml:id="echoid-s9690" xml:space="preserve">quam habet rectangulum ſub, <lb/>GP, VR, cum {1/3}. </s> <s xml:id="echoid-s9691" xml:space="preserve">quadrati, PM, ad quadratum, VX, & </s> <s xml:id="echoid-s9692" xml:space="preserve">ex ea, quã <lb/>habet rectangulum, BVO, ad rectangulum ſub, BV, GO, vna cum <lb/>rectangulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s9693" xml:space="preserve">BO, & </s> <s xml:id="echoid-s9694" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9695" xml:space="preserve">GV, & </s> <s xml:id="echoid-s9696" xml:space="preserve">ſub, GV, quod <lb/>oſtendere opus erat.</s> <s xml:id="echoid-s9697" xml:space="preserve"/> </p> <div xml:id="echoid-div906" type="float" level="2" n="1"> <figure xlink:label="fig-0396-01" xlink:href="fig-0396-01a"> <image file="0396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0396-01"/> </figure> <note position="left" xlink:label="note-0396-01" xlink:href="note-0396-01a" xml:space="preserve">28.2.</note> <note position="left" xlink:label="note-0396-02" xlink:href="note-0396-02a" xml:space="preserve">9 2.</note> <note position="left" xlink:label="note-0396-03" xlink:href="note-0396-03a" xml:space="preserve">6.2.</note> <note position="left" xlink:label="note-0396-04" xlink:href="note-0396-04a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div908" type="section" level="1" n="542"> <head xml:id="echoid-head566" xml:space="preserve">THEOREMA VIII. PROPOS. IX.</head> <p> <s xml:id="echoid-s9698" xml:space="preserve">VIſa adhuc anteced. </s> <s xml:id="echoid-s9699" xml:space="preserve">figura, exponemus aliter rationẽ <lb/>ibi adinuentam tantummodo compoſitam ex dua-<lb/>bus, ad vnam ſolum eandem reducentes, probando .</s> <s xml:id="echoid-s9700" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9701" xml:space="preserve">om-<lb/>nia quadrata trianguli, HCR, regula eadem, HR, retenta <lb/>ad omnia quadrata hyperbolæ, SOX, eſſe vt cubus, CV, <lb/>eſt ad parallelepipedum ter ſub, CO, & </s> <s xml:id="echoid-s9702" xml:space="preserve">quadrato, OV, <lb/>cum cubo, OV.</s> <s xml:id="echoid-s9703" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9704" xml:space="preserve">Nam vt in ſupradicta Propoſit. </s> <s xml:id="echoid-s9705" xml:space="preserve">oſtenſum eſt, omnia quadrata <lb/> <anchor type="note" xlink:label="note-0396-05a" xlink:href="note-0396-05"/> trianguli, CHR, ad omnia quadrata hyperbolæ, SOX, conuertẽ. <lb/></s> <s xml:id="echoid-s9706" xml:space="preserve">do, habent rationem compoſitam ex ea, quam habet quadratũ, <pb o="377" file="0397" n="397" rhead="LIBER V."/> HR, ad quadratum, SX, & </s> <s xml:id="echoid-s9707" xml:space="preserve">rectangulum, BVC, ad rectangulum: <lb/></s> <s xml:id="echoid-s9708" xml:space="preserve">AVO, quæ eſt compoſita pariter ex duabus .</s> <s xml:id="echoid-s9709" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9710" xml:space="preserve">ex ea, quam habet, <lb/> <anchor type="note" xlink:label="note-0397-01a" xlink:href="note-0397-01"/> CV, ad, VO, & </s> <s xml:id="echoid-s9711" xml:space="preserve">ex ea, quam habet, BV, ad, VA, vt autem, BV, <lb/>ad, VA, ſic eſt, ſumpta, OV, communi altitudine, rectangulum, <lb/>BVO, ad rectangulum, AVO, quodſerua. </s> <s xml:id="echoid-s9712" xml:space="preserve">Sumatur nunc harum <lb/>rationum componentium ea, quam habet quadratu, HR, ad qua-<lb/>dratum, SX, quæ eſt eadem el, quam habet quadratum, HV, ad <lb/> <anchor type="note" xlink:label="note-0397-02a" xlink:href="note-0397-02"/> quadratum, VS, quia verò rectangulum, HSR, cum quadrato, SV, <lb/> <anchor type="note" xlink:label="note-0397-03a" xlink:href="note-0397-03"/> eſt æquale quadrato, HV, ideò quadratum, VS, eſt exceſſus, quo <lb/>quadratũ, HV, ſuperat rectang. </s> <s xml:id="echoid-s9713" xml:space="preserve">HSR, & </s> <s xml:id="echoid-s9714" xml:space="preserve">quia rectangul. </s> <s xml:id="echoid-s9715" xml:space="preserve">HSR, eſt <lb/>æquale quadrato, EO, ideò, vt quadratum, HV, ad rectaugulum, <lb/> <anchor type="note" xlink:label="note-0397-04a" xlink:href="note-0397-04"/> HSR, ita erit idem quadratum, HV, ad quadratum, EO, & </s> <s xml:id="echoid-s9716" xml:space="preserve">ita <lb/>erit quadratum, VC, ad quadratum, CO, quia triangula, CEO, <lb/>CHV, ſunt ſimilia, ergo, per conuerſionem rationis, quadratum, <lb/>HV, ad exceſſum ſui ſuper quadratum, EO, .</s> <s xml:id="echoid-s9717" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9718" xml:space="preserve">ad quadratum, VS, <lb/>erit vt quadratum, VC, ad exceſſum ſui ſuper quadratum, CO, .</s> <s xml:id="echoid-s9719" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s9720" xml:space="preserve">ad rectangulũ bis ſub, CO, OV, cum quadrato, OV, .</s> <s xml:id="echoid-s9721" xml:space="preserve">i. </s> <s xml:id="echoid-s9722" xml:space="preserve">ad rectan-<lb/>gulum ſemel ſub, BO, OV, cum quadrato, OV, .</s> <s xml:id="echoid-s9723" xml:space="preserve">i. </s> <s xml:id="echoid-s9724" xml:space="preserve">ad integrum <lb/>rectangul@, BVO; </s> <s xml:id="echoid-s9725" xml:space="preserve">erit ergo, vt quadratum, HV, ad quadratum, <lb/>VS, ita quadratum, CV, ad rectangulum, BVO, hæc ergo ratio, <lb/>quam nempè habet quadratum, CV, ad rectangulum, BVO, ſum-<lb/>pta vice eius, quam habet quadratum, HV, ad quadratum, VS, <lb/>vel quadratum, HR, ad quadratum, SX, (quæ erat vna rationum <lb/>componentium) componit rationem omnium quadratorum triã-<lb/>guli, HCR, ad omnia quadrata hyperbolæ, SOX, ſimul ſumpta <lb/>cum ea, quam habet rectangulum, BVO, ad rectangulum, AVO, <lb/>& </s> <s xml:id="echoid-s9726" xml:space="preserve">cum ea, quam habet, CV, ad, VO; </s> <s xml:id="echoid-s9727" xml:space="preserve">harum autem trium rationũ <lb/>componentium ea, quam habet quadiatum, CV, ad rectangulum, <lb/>BVO, & </s> <s xml:id="echoid-s9728" xml:space="preserve">quam habet hoc rectangulum, BVO, ad rectangulum, <lb/>AVO, componunt rationem quadrat, CV, ad rectangulum, AV <lb/>O, illas ergo tres in has duas rationes reiolutas habemus .</s> <s xml:id="echoid-s9729" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9730" xml:space="preserve">in eam, <lb/>quam habea quadratum, CV, ad rectangulum, AVO, & </s> <s xml:id="echoid-s9731" xml:space="preserve">in eam, <lb/>quam habet, CV, ad, VO, porro iſtæ duæ rationes componunt <lb/>rationem parallelepipedi ſub, CV, & </s> <s xml:id="echoid-s9732" xml:space="preserve">quadrato, CV, .</s> <s xml:id="echoid-s9733" xml:space="preserve">i. </s> <s xml:id="echoid-s9734" xml:space="preserve">cubi, CV, <lb/>ad parallelepipedum ſub, OV, & </s> <s xml:id="echoid-s9735" xml:space="preserve">rectangulo, AVO, .</s> <s xml:id="echoid-s9736" xml:space="preserve">i. </s> <s xml:id="echoid-s9737" xml:space="preserve">parallele-<lb/>pipedi ſub, AV, & </s> <s xml:id="echoid-s9738" xml:space="preserve">quadiato, VO, .</s> <s xml:id="echoid-s9739" xml:space="preserve">i. </s> <s xml:id="echoid-s9740" xml:space="preserve">parallelepiped ſub, AO, & </s> <s xml:id="echoid-s9741" xml:space="preserve"><lb/>quadrato, OV, cum cubo, OV, i. </s> <s xml:id="echoid-s9742" xml:space="preserve">parallelepipedi ter ſub, CO, & </s> <s xml:id="echoid-s9743" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0397-05a" xlink:href="note-0397-05"/> quadrato, OV, cum cubo, OV; </s> <s xml:id="echoid-s9744" xml:space="preserve">ergo omnia quadrata trianguli, <lb/>HCR, ad omnia quadrata hyperbolæ, SOX, erunt vt cubus, CV, <lb/>ad parallelepipedum ter ſub, CO, & </s> <s xml:id="echoid-s9745" xml:space="preserve">quadrato, OV, cum cubo, O <lb/>V, quod oſtendere opus erat.</s> <s xml:id="echoid-s9746" xml:space="preserve"/> </p> <div xml:id="echoid-div908" type="float" level="2" n="1"> <note position="left" xlink:label="note-0396-05" xlink:href="note-0396-05a" xml:space="preserve">6. huius</note> <note position="right" xlink:label="note-0397-01" xlink:href="note-0397-01a" xml:space="preserve">5. 1. 2.</note> <note position="right" xlink:label="note-0397-02" xlink:href="note-0397-02a" xml:space="preserve">5. 2. Elem.</note> <note position="right" xlink:label="note-0397-03" xlink:href="note-0397-03a" xml:space="preserve">10. 2. Cõ.</note> <note position="right" xlink:label="note-0397-04" xlink:href="note-0397-04a" xml:space="preserve">4. 6. Elem.</note> <note position="right" xlink:label="note-0397-05" xlink:href="note-0397-05a" xml:space="preserve">36. 1. 1.</note> </div> <pb o="378" file="0398" n="398" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div910" type="section" level="1" n="543"> <head xml:id="echoid-head567" xml:space="preserve">THEOREMA IX. PROPOS. X.</head> <p> <s xml:id="echoid-s9747" xml:space="preserve">SI à centro hyperbolæ duæ intra aſymptotos eiuſdem <lb/>ductæ fuerint rectæ lineæ indefinitè productæ, agan-<lb/>tur autem intra curuam hyperbolicam parallelæ tangenti-<lb/>bus in punctis concurius ductarum linearum, & </s> <s xml:id="echoid-s9748" xml:space="preserve">curuæ hy-<lb/>perbolicæ hinc inde ad eandem productæ, erunt iſtæ ba-<lb/>ſes hyperbolarum, quarum diametri, vel axes erunt por-<lb/>tiones ductarum à centro interceptæ inter ipſas, & </s> <s xml:id="echoid-s9749" xml:space="preserve">earun-<lb/>dem hyperbolarum vertices: </s> <s xml:id="echoid-s9750" xml:space="preserve">Dico autem omnia quadra-<lb/>ta vnius dictarum hyperbolarum, regula eiuſdem baſi, ad <lb/>omnia quadrata alterius regula quoq; </s> <s xml:id="echoid-s9751" xml:space="preserve">huius baſi, habere <lb/>rationem compoſitam ex ratione rectanguli ſub compoſita <lb/>ex ſexquialtera tranſuerſi lateris hyperbolæ primò dictæ, <lb/>& </s> <s xml:id="echoid-s9752" xml:space="preserve">axi, vel diametro eiuſdem, & </s> <s xml:id="echoid-s9753" xml:space="preserve">ſub compoſita extran-<lb/>ſuerſo latere, & </s> <s xml:id="echoid-s9754" xml:space="preserve">axi, vel diametro hyperbolæ ſecundò di-<lb/>ctæ, ad rectangulum ſub compoſita ex ſexquialtera tran-<lb/>ſuerſi lateris hyperbolæ ſecundò dictæ, & </s> <s xml:id="echoid-s9755" xml:space="preserve">axi, vel diame-<lb/>tro eiuſdem, & </s> <s xml:id="echoid-s9756" xml:space="preserve">ſub compoſita ex tranſuerſo latere, & </s> <s xml:id="echoid-s9757" xml:space="preserve">axi, <lb/>vel diametro hyperbolæ primò dictæ; </s> <s xml:id="echoid-s9758" xml:space="preserve">& </s> <s xml:id="echoid-s9759" xml:space="preserve">ex ratione paral-<lb/>lelepipedi ſub altitudine hyperbolæ primò dictæ, baſi ve-<lb/>rò, baſis eiuſdem quadrato ad parallelepipedum ſub alti-<lb/>tudine hyperbolæ ſecundò dictæ, baſiq; </s> <s xml:id="echoid-s9760" xml:space="preserve">pariter eiuſdem <lb/>baſis quadrato.</s> <s xml:id="echoid-s9761" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9762" xml:space="preserve">Sit ergo hyperbolæ, ADC, vtcunq: </s> <s xml:id="echoid-s9763" xml:space="preserve">baſis, AC, centrum, E, per <lb/>quod intra eiuſdem aſymptotos, EY, EZ, ductæ ſint, FEDB, HE <lb/>VI, vtcunq; </s> <s xml:id="echoid-s9764" xml:space="preserve">indefinitè productæ, ſit tamen altera earum diameter <lb/>iam expoſite hyperbolæ, pro alia hyperbola autem conſtituenda, <lb/>ducta pariter ſit vtcunq: </s> <s xml:id="echoid-s9765" xml:space="preserve">intra curuam hyperbolicam, & </s> <s xml:id="echoid-s9766" xml:space="preserve">in eandẽ <lb/>hinc inde producta ipſa, OX, parallela tangenti curuam hyperbo-<lb/>licam in puncto, V, in quo ipſam, HI, ſecat. </s> <s xml:id="echoid-s9767" xml:space="preserve">Dico ergo omnia <lb/>quadrata hyperbolæ, ADC, regula, AC, ad omnia quadrata hy-<lb/>perbolæ, OVX, regula, OX, habere rationem compoſitam (ſum-<lb/>ptis, EF, FM, æqualibus ipſi, ED, &</s> <s xml:id="echoid-s9768" xml:space="preserve">, EH, HR, æqualibus ipſi, <lb/>EV,) ex ratione rectanguli ſub, MB, HI, ad rectangulum ſub, RI, <lb/>FB; </s> <s xml:id="echoid-s9769" xml:space="preserve">& </s> <s xml:id="echoid-s9770" xml:space="preserve">ex ratione parallelepipedi ſub altitudine hyperbolæ, ADC, <pb o="379" file="0399" n="399" rhead="LIBER V."/> & </s> <s xml:id="echoid-s9771" xml:space="preserve">baſi quadrato, AC, ad parallelepipedum ſub altitudine hyper <lb/>bolæ, OVX, baſi autem quadrato, OX. </s> <s xml:id="echoid-s9772" xml:space="preserve">Nam omnia quadrata <lb/>hyperbolæ, ADC, regula, AC, ad omnia quadrata hyperbolæ, <lb/>OVX, regula, OX, (iunctis, AD, DC, OV, VX,) ſumptis medijs <lb/> <anchor type="note" xlink:label="note-0399-01a" xlink:href="note-0399-01"/> <anchor type="figure" xlink:label="fig-0399-01a" xlink:href="fig-0399-01"/> omnibus quadratis triangulorum, AD <lb/>C, OVX, habent rationem compoſitã <lb/>ex ratione omnium quadratorum hy-<lb/>perbolæ, ADC, ad omnia quadrata <lb/> <anchor type="note" xlink:label="note-0399-02a" xlink:href="note-0399-02"/> trianguli, ADC, .</s> <s xml:id="echoid-s9773" xml:space="preserve">i. </s> <s xml:id="echoid-s9774" xml:space="preserve">ex ratione, MB, ad, <lb/>BF, & </s> <s xml:id="echoid-s9775" xml:space="preserve">ex ratione omnium quadratorũ <lb/>trianguli, ADC, ad omnia quadrata <lb/> <anchor type="note" xlink:label="note-0399-03a" xlink:href="note-0399-03"/> trianguli, OVX, quæ eſt compoſita ex <lb/>ratione altitudinis trianguli, ADC, vel <lb/>hyperbolæ, ADC, ad altitudinem triã-<lb/>guli, OVX, vel hyperbolæ, OVX, & </s> <s xml:id="echoid-s9776" xml:space="preserve">ex <lb/>ratione quadrati, AC, ad quadratum, <lb/>OX, & </s> <s xml:id="echoid-s9777" xml:space="preserve">tandem eſt compoſita ex ratio-<lb/> <anchor type="note" xlink:label="note-0399-04a" xlink:href="note-0399-04"/> ne omnium quadratorum trianguli, O <lb/>VX, ad omnia quadrata hyperbolæ, O <lb/>VX, .</s> <s xml:id="echoid-s9778" xml:space="preserve">i. </s> <s xml:id="echoid-s9779" xml:space="preserve">ex ea, quam habet, HI, ad, IR, harum autem rationum <lb/> <anchor type="note" xlink:label="note-0399-05a" xlink:href="note-0399-05"/> componentium iſtæ duæ .</s> <s xml:id="echoid-s9780" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9781" xml:space="preserve">quam habet, MB, ad, BF, &</s> <s xml:id="echoid-s9782" xml:space="preserve">, HI, ad, <lb/>IR, componunt rationem rectanguli ſub, MB, HI, ad rectangulũ <lb/>ſub, RI, FB; </s> <s xml:id="echoid-s9783" xml:space="preserve">aliæ autem duæ rationes componentes .</s> <s xml:id="echoid-s9784" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9785" xml:space="preserve">quam ha-<lb/>bet altitudo hyperbolæ, ADC, ad altitudinem hyperbolæ, OVX, <lb/>& </s> <s xml:id="echoid-s9786" xml:space="preserve">quam habet quadratum, AC, ad quadratum, OX, componunt <lb/>rationem parallelepipedi ſub altitudine hyperbolæ, ADC, baſi <lb/>quadrato, AC, ad parallelepipedum ſub altitudine hyperbolæ, O <lb/>VX, baſi quadrato, OX, ergo omnia quadrata hyperbolæ, ADC, <lb/>regula, AC, ad omnia quadrata hyperbolæ, OVX, regula, OX, <lb/>habent rationem compoſitam ex ratione rectanguli ſub, MB, HI, <lb/>ad rectangulum ſub, RI, FB, & </s> <s xml:id="echoid-s9787" xml:space="preserve">ex ratione parallelepipedi ſub al-<lb/>titudine hyperbolę, ADC, baſi quadrato, AC, ad parallelepipe-<lb/>dum ſub altitudine hyperbolæ, OVX, baſi verò quadrato, OX, <lb/>quod oſtendere opus erat.</s> <s xml:id="echoid-s9788" xml:space="preserve"/> </p> <div xml:id="echoid-div910" type="float" level="2" n="1"> <note position="right" xlink:label="note-0399-01" xlink:href="note-0399-01a" xml:space="preserve">Defin. 12. <lb/>1. 1.</note> <figure xlink:label="fig-0399-01" xlink:href="fig-0399-01a"> <image file="0399-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0399-01"/> </figure> <note position="right" xlink:label="note-0399-02" xlink:href="note-0399-02a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-0399-03" xlink:href="note-0399-03a" xml:space="preserve">C. Col. 22. <lb/>1. 2.</note> <note position="right" xlink:label="note-0399-04" xlink:href="note-0399-04a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-0399-05" xlink:href="note-0399-05a" xml:space="preserve">6, 1. 2.</note> </div> </div> <div xml:id="echoid-div912" type="section" level="1" n="544"> <head xml:id="echoid-head568" xml:space="preserve">THEOREMA X. PROPOS. XI.</head> <p> <s xml:id="echoid-s9789" xml:space="preserve">IN eadem antec. </s> <s xml:id="echoid-s9790" xml:space="preserve">figura, iuncta, DV, & </s> <s xml:id="echoid-s9791" xml:space="preserve">à puncto, X, ducta, <lb/>XP, parallela ipſi, DV, indefinitè producta, à puncto <lb/>autem, O, ipſa, OP, parallela ei, quæ tangeret hypeibo- <pb o="380" file="0400" n="400" rhead="GEOMETRIÆ"/> lam, ADC, in puncto, D, quæ indefinitè quoq; </s> <s xml:id="echoid-s9792" xml:space="preserve">producta <lb/>occurrat ipſi, XP, in puncto, P, ſuppoſitoque, BD, eſſe <lb/>axim, oſtendemus omnia quadrata hyperbolæ, ADC, ad <lb/>rectangula omnia hyperbolæ, OVX, ſimilia rectangulo <lb/>ſub, XO, OP, habere rationem compoſit am ex ratione re-<lb/>ctanguli ſub, MB, HI, ad rectangulum ſub, RI, FB, & </s> <s xml:id="echoid-s9793" xml:space="preserve">ex <lb/>tatione parallelepipedi ſub altitudine hyperbolæ, ADC, <lb/>& </s> <s xml:id="echoid-s9794" xml:space="preserve">baſi quadrato, AC, ad parallelepipedum ſub altitudine <lb/>hyperbolæ, OVX, baſi aute m rectangulo ſub, XO, OP.</s> <s xml:id="echoid-s9795" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9796" xml:space="preserve">Nam omnia quadrata hyperbolæ, ADC, regula eadẽ, AC, ad oĩa <lb/>quadrata hyperbolę, OVX, regula, OX, oſtenſa ſunt habere ratio-<lb/>nẽ cõpoſitam ex ratione rectang. </s> <s xml:id="echoid-s9797" xml:space="preserve">ſub, MB, HI, ad rectang. </s> <s xml:id="echoid-s9798" xml:space="preserve">ſub, RI, <lb/>FB, & </s> <s xml:id="echoid-s9799" xml:space="preserve">parallelepipedi ſub altitudine hyperbolę, ADC, baſi quadr. <lb/></s> <s xml:id="echoid-s9800" xml:space="preserve"> <anchor type="note" xlink:label="note-0400-01a" xlink:href="note-0400-01"/> AC, ad parallelepipedũ ſub altitudine hyperbolę, OVX, baſi autẽ <lb/> <anchor type="figure" xlink:label="fig-0400-01a" xlink:href="fig-0400-01"/> quadrato, OX; </s> <s xml:id="echoid-s9801" xml:space="preserve">inſuper omnia quadra-<lb/>ta hyperbolę, OVX, ad rectangula <lb/>omnia eiuſdem hyperbolę ſimilia re-<lb/>ctangulo, XOP, regula, XO, ſunt vt <lb/>vnum ad vnum, ſcilicet vt quadratũ, <lb/>XO, ad rectangulum, XOP, .</s> <s xml:id="echoid-s9802" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9803" xml:space="preserve">ſumpta <lb/>communi altitudine eiuſdem hyperbo-<lb/>læ, OVX, altitudine, vt parallelepi-<lb/>pedum ſub altitudine hyperbolæ, O <lb/>VX, baſi quadrato, OX, ad parallele-<lb/>pipedum ſub eadem altitudine, baſi <lb/>autem rectangulo, XOP, ergo omnia <lb/>quadrata hyperbolę, ADC, regula, <lb/>AC, ad omnia rectangula hyperbolę, <lb/>OVX, ſimilia rectangulo, XOP, regu-<lb/>la, OX, erunt in ratione compoſita ex ratione rectanguli ſub, MB, <lb/>HI, ad rectangulum ſub, RI, FB, & </s> <s xml:id="echoid-s9804" xml:space="preserve">parallelepipedi ſub altitudine <lb/>hyperbolę, ADC, & </s> <s xml:id="echoid-s9805" xml:space="preserve">ſub quadrato, AC, ad parallelepipedum ſub <lb/>altitudine hyperbolę, OVX, baſi quadrato, OX, & </s> <s xml:id="echoid-s9806" xml:space="preserve">ex ratione hui-<lb/>us parallelepipedi ad parallelepipedum ſub eiuſdem hyperbolę, O <lb/>VX, altitudine baſi rectangulo, XOP, quę duę vltimò dictę racio-<lb/>nes componunt rationem parallelepipedi ſub altitudine hyperbo-<lb/>lę, ADC, baſi quadrato, AC, ad parallelepipedum ſub altitudine <lb/>hyperbolę, OVX, baſi rectangulo, XOP, ergo omnia quadrata <lb/>hyperbolę, ADC, regula, AC, ad omnia rectangula hyperbolę, <pb o="381" file="0401" n="401" rhead="LIBER V."/> OVX, ſimilia rectangulo, XOP, regula, OX, habebunt rationem <lb/>compoſitam ex ea, quam habetrectangulum ſub, MB, HI, adre-<lb/>ctangulum ſub, RI, FB, & </s> <s xml:id="echoid-s9807" xml:space="preserve">ex ea, quam habet parallelepipedum <lb/>ſub altitudine hyperbole, ADC, baſi quadrato, AC, ad parallele-<lb/>bipedum ſub altitudine hyperbole, OVX, baſi rectangulo, XOP, <lb/>quod erat demonſtrandum.</s> <s xml:id="echoid-s9808" xml:space="preserve"/> </p> <div xml:id="echoid-div912" type="float" level="2" n="1"> <note position="left" xlink:label="note-0400-01" xlink:href="note-0400-01a" xml:space="preserve">Iu antec.</note> <figure xlink:label="fig-0400-01" xlink:href="fig-0400-01a"> <image file="0400-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0400-01"/> </figure> </div> </div> <div xml:id="echoid-div914" type="section" level="1" n="545"> <head xml:id="echoid-head569" xml:space="preserve">THEOREMA XI. PROPOS. XII.</head> <p> <s xml:id="echoid-s9809" xml:space="preserve">ASſumptis quibuſcunq; </s> <s xml:id="echoid-s9810" xml:space="preserve">hyperbolis, in vnaquaq; </s> <s xml:id="echoid-s9811" xml:space="preserve">re-<lb/>gula baſi, oſtendemus omnia quadrata vnius ad om-<lb/>nia quadrata alterius, habere rationem compoſitam ex ra-<lb/>tione rectanguli ſub compoſita ex ſexquialtera tranſuerſi <lb/>lateris, & </s> <s xml:id="echoid-s9812" xml:space="preserve">axi, vel diametro hyperbolæ primò dictæ, & </s> <s xml:id="echoid-s9813" xml:space="preserve">ſub <lb/>compoſita ex tranſuerſo latere, & </s> <s xml:id="echoid-s9814" xml:space="preserve">axi, vel diametro hyper-<lb/>bolæ ſecundò dictæ ad re ctangulum ſub compoſita ex trã-<lb/>ſuerſi lateris ſexquialtera, & </s> <s xml:id="echoid-s9815" xml:space="preserve">axi, vel diametro hyperbolæ <lb/>ſecundò dictæ, & </s> <s xml:id="echoid-s9816" xml:space="preserve">ſub compoſita ex tranſuerſo latere, & </s> <s xml:id="echoid-s9817" xml:space="preserve">axi <lb/>vel diametro hyperbolæ primò dictæ, & </s> <s xml:id="echoid-s9818" xml:space="preserve">ex ratione paral-<lb/>lelepipediſub altitudine hyperbolæ primò dictæ, baſiau-<lb/>tem quadrato baſis eiuſdem, ad parallelepipedum ſub al-<lb/>t tudine hyp rbolæ ſecundò dictæ, baſi pariter quadrato <lb/>b ſis eiuſdem. </s> <s xml:id="echoid-s9819" xml:space="preserve">Velſi comparentur omnia quadrata hy-<lb/>perbolæ primò dictæ, ad omnia rectangula hyperbolæ fe-<lb/>cundò dictæ ſimilia cuidam rectangulo, illa ad hæchabe-<lb/>buntrationem compoſitam exratione prædictorum rectã-<lb/>gulorum, & </s> <s xml:id="echoid-s9820" xml:space="preserve">exratione parallelepipedi primò dictiad pa-<lb/>rallelepipedum ſub altitudine hyperbolæ ſecundò, dictæ <lb/>baſirectangulo, cuiomnia dicta rectangula ſunt ſimilia. <lb/></s> <s xml:id="echoid-s9821" xml:space="preserve">Vel tandem ſi comparentur omnia rectangula primæ hy-<lb/>perbolæ ſimilia cuidam rectangulo ad omnia rectangula <lb/>ſecundæ hyperbolæ ſimilia pariter cuidam rectangulo, il-<lb/>la ad hæchabebunt rationem compoſitam ex ratione pa-<lb/>rallelepipedi ſub altitudine hyperbolæ primò dictæ baſi <lb/>rectangulo, cuiomnia eiuſdem rectangula ſunt ſimilia, ad <lb/>parallelepipedum ſub altitudine ecundæ hyperbolæ baſi <lb/>rectangulo, cuiomnia eiuſdem rectangula iam dicta ſunt <pb o="382" file="0402" n="402" rhead="GEOMETRIÆ"/> ſimilia, & </s> <s xml:id="echoid-s9822" xml:space="preserve">ex ratione, quæ in huius Theorematis ſupradi-<lb/>ctis caſibusinter illa duorectangula primò loco expoſita <lb/>fuit.</s> <s xml:id="echoid-s9823" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9824" xml:space="preserve">Sint aſſumptę quęcunq; </s> <s xml:id="echoid-s9825" xml:space="preserve">hyperbolę, BAD, HMQ, circa axes, <lb/>vel diametros, AC, MP, circa quas ſint quoq; </s> <s xml:id="echoid-s9826" xml:space="preserve">triangula, BAD, H <lb/>MQ, & </s> <s xml:id="echoid-s9827" xml:space="preserve">in baſibus, BD, HQ, latus autem tranſuerſum hyperbolę, <lb/>BAD, ſit, GA, cuius ſexquialtera, VA; </s> <s xml:id="echoid-s9828" xml:space="preserve">& </s> <s xml:id="echoid-s9829" xml:space="preserve">larus tranſuerſum hy-<lb/>perbolę, HMQ, ſit, MX, cuius ſexquialtera, MR, ſint autem ex-<lb/>poſitæ duæ vtcunque rectæ lineæ, FY, EN. </s> <s xml:id="echoid-s9830" xml:space="preserve">Dico omnia qua-<lb/>drata hyperbolę, BAD, regula, BD, ad omnia quadrata hyper. <lb/></s> <s xml:id="echoid-s9831" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0402-01a" xlink:href="fig-0402-01"/> perbolę, HMQ, regu-<lb/>la, HQ, habereratio-<lb/>nem compoſitã ex ea, <lb/>quam habet rectangu-<lb/>lum ſub, VC, XP, adre-<lb/>ctangulum ſub RP, C <lb/>G, & </s> <s xml:id="echoid-s9832" xml:space="preserve">ex ea, quam ha-<lb/>bet parallelepipedum <lb/>ſub altitudinehyperbo-<lb/>lę, BAD, & </s> <s xml:id="echoid-s9833" xml:space="preserve">ſub qua-<lb/>drato, BD, ad paralle-<lb/>lepipedum ſub altitu-<lb/>dine hyperbolę, HMQ, <lb/>baſi quadrato, HQ; <lb/></s> <s xml:id="echoid-s9834" xml:space="preserve">quod oſtendemus ad <lb/>modum Propoſ. </s> <s xml:id="echoid-s9835" xml:space="preserve">10. </s> <s xml:id="echoid-s9836" xml:space="preserve">Si <lb/>verò comparentur om-<lb/>nia quadrata hyperbolæ, BAD, ad omnia rectangula hyperbolæ, <lb/>HMQ, ſimilia rectangulo ſub, HQ, EN, oſtendemus illa ad hæc <lb/>habere rationem compoſitam ex ratione primò dicta inter illa re-<lb/>ctangula, & </s> <s xml:id="echoid-s9837" xml:space="preserve">ex ratione parallelepipedi ſub altitud ne hyperpolæ, <lb/>BAD, baſiquad ato, BD, ad parallelepipedum ſub altitudine hy-<lb/>perbolæ, HMQ, baſi rectangulo ſub, HQ, EN; </s> <s xml:id="echoid-s9838" xml:space="preserve">hocq; </s> <s xml:id="echoid-s9839" xml:space="preserve">oſtende-<lb/>mus iuxta methodum Propol. </s> <s xml:id="echoid-s9840" xml:space="preserve">antecedentis. </s> <s xml:id="echoid-s9841" xml:space="preserve">Sitandem compa. </s> <s xml:id="echoid-s9842" xml:space="preserve"><lb/>rentur omnia rectangula hyperbolæ, BAD, ſimilia rectangulo ſub, <lb/>BD, FY, ad omnia rectangula hy perbolæ, HMQ, ſimilia rectan-<lb/>gulo ſub, HQ, EN, oſten demus propoſitum de his hoc pacto: </s> <s xml:id="echoid-s9843" xml:space="preserve">Nã <lb/>omnia rectangula hyperbolæ, BAD, ſimilia rectangulo ſub, BD, <lb/>FY, ad omnia quadrata eiuſdem, BAD, ſunt vt rectangulum ſub, <lb/>BD, FY, ad?</s> <s xml:id="echoid-s9844" xml:space="preserve">? quadratum, BD, .</s> <s xml:id="echoid-s9845" xml:space="preserve">i. </s> <s xml:id="echoid-s9846" xml:space="preserve">vt parallelepipedum ſub altitudi- <pb o="383" file="0403" n="403" rhead="LIBER V."/> ne hyperbolæ, BAD, bafi rectangulo ſub, BD, FY, adparallele-<lb/>pipedum ſub eadem altitudine baſi quadrato, BD: </s> <s xml:id="echoid-s9847" xml:space="preserve">pariter omnia <lb/>quadrata hyperbolæ, BAD, ad omnia rectangula hyperbolę, HM <lb/>Q, ſimilia rectangulo ſub, HQ, EN, habent rationem compofitã <lb/>ex ratione rectanguli ſub, VC, XP, ad rectangulum ſub, RP, GC, <lb/>& </s> <s xml:id="echoid-s9848" xml:space="preserve">parallelepipedi ſub altitudine hyperbolæ, BAD, & </s> <s xml:id="echoid-s9849" xml:space="preserve">ſub quadra-<lb/>to, BD, ad parallelepipedum ſub altitudine hyperbolæ, HMQ, <lb/>bafi rectangulo ſub, HQ, EN, ergo, ex æquo, omnia rectangula <lb/>hyperbolæ, BAD, ſimilia rectangulo ſub, BD, FY, regula, BD, ad <lb/>omnia rectangula hyperbolæ, HMQ, ſimilia rectangulo ſub, HQ, <lb/>EN, regula, HQ, habebunt rationem compoſitam ex ratione re-<lb/>ctanguli, ſub, VC, XP, ad rectangulum ſub, RP, GC, & </s> <s xml:id="echoid-s9850" xml:space="preserve">ex ratio-<lb/>ne parallelepipedi ſub altitudine hyperbolæ, BAD, baſi rectangu-<lb/>lo ſub, BD, FY, ad parallelepipedum ſub eadem altitudine, & </s> <s xml:id="echoid-s9851" xml:space="preserve">baſi <lb/>quadrato, BD, & </s> <s xml:id="echoid-s9852" xml:space="preserve">ex ratione huius parallelepipedi ad parallelepi-<lb/>pedum ſub altitudine hyperbolę, HMQ, baſi rectangulo ſub, HQ, <lb/>EN; </s> <s xml:id="echoid-s9853" xml:space="preserve">.</s> <s xml:id="echoid-s9854" xml:space="preserve">i. </s> <s xml:id="echoid-s9855" xml:space="preserve">compoſitã ex ratione parallelepipedi ſub altitudine hy-<lb/>perbolę, ABD, baſi rectangulo ſub, BD, FY, ad parallelepipedum <lb/>ſub altitudine hyperbolę, HMQ, baſi rectangulo ſub, HQ, EN, <lb/>quę erant oſtend.</s> <s xml:id="echoid-s9856" xml:space="preserve"/> </p> <div xml:id="echoid-div914" type="float" level="2" n="1"> <figure xlink:label="fig-0402-01" xlink:href="fig-0402-01a"> <image file="0402-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0402-01"/> </figure> </div> </div> <div xml:id="echoid-div916" type="section" level="1" n="546"> <head xml:id="echoid-head570" xml:space="preserve">THEOREMA XII. PROPOS. XIII.</head> <p> <s xml:id="echoid-s9857" xml:space="preserve">SImilium hyperbolarum omnia quadrata, regulis ea-<lb/>rum baſibus, ſunt in tripla ratione axium, vel diame-<lb/>trorum earundem.</s> <s xml:id="echoid-s9858" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9859" xml:space="preserve">Sint ſimiles hyperbolæ, BAD, HMQ, earum latera tranſuerſa, <lb/>GA, XM, quorum ſint ſexquialteræ, AV, MR, in directum axi-<lb/>bus, vel diametris, AC, MP, baſes, & </s> <s xml:id="echoid-s9860" xml:space="preserve">regulæ ſint, BD, HQ. </s> <s xml:id="echoid-s9861" xml:space="preserve">Di-<lb/>co omnia quadrata hyperbolæ, BAD, ad omnia quadrata hyper-<lb/>bolæ, HMQ, eſſe in tripla ratione eius, quam habet, AC, ad, M <lb/>P, iungantur, BA, AD, HM, MQ. </s> <s xml:id="echoid-s9862" xml:space="preserve">Quoniam ergo hyperbolæ <lb/> <anchor type="note" xlink:label="note-0403-01a" xlink:href="note-0403-01"/> ſunt ſimiles baſis, BD, ad, CA, erit vt baſis, HQ, ad, PM, & </s> <s xml:id="echoid-s9863" xml:space="preserve">ſunt <lb/>anguli in clinationis, AC, ad, BD, & </s> <s xml:id="echoid-s9864" xml:space="preserve">MP, ad, HQ, inter ſe æqua-<lb/>les, ergo triangula, BAD, HMQ, ſunt ſimilia, & </s> <s xml:id="echoid-s9865" xml:space="preserve">ideo omnia qua. <lb/></s> <s xml:id="echoid-s9866" xml:space="preserve">drata eorundem, regulis ijſdem, erunt inter ſe in triplaratione la-<lb/>terum homologorum .</s> <s xml:id="echoid-s9867" xml:space="preserve">i. </s> <s xml:id="echoid-s9868" xml:space="preserve">eius, quam habet, BD, ad, HQ, vel, AC, <lb/>ad, MP; </s> <s xml:id="echoid-s9869" xml:space="preserve">quia verò quadratum, BC, ad rectangulum, GCA, eſt vt <lb/>hyperbolæ, BAD, rectum latus ad tranſuerſum .</s> <s xml:id="echoid-s9870" xml:space="preserve">I. </s> <s xml:id="echoid-s9871" xml:space="preserve">vt rectum latus <lb/> <anchor type="note" xlink:label="note-0403-02a" xlink:href="note-0403-02"/> ad tranſuerſum hyperbolæ, HMQ, quia ille ſunt ſimiles .</s> <s xml:id="echoid-s9872" xml:space="preserve">I. </s> <s xml:id="echoid-s9873" xml:space="preserve">vt qua- <pb o="384" file="0404" n="404" rhead="GEOMETRIÆ"/> dratum, HP, ad rectangulum, MPX, ideò quadratum, BC, ad re-<lb/> <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/> ctangulum, ACG, erit vt quadratum, HP, ad rectangulum, MPX; <lb/></s> <s xml:id="echoid-s9874" xml:space="preserve">quia autem ratio, quam habet, BC, ad, CA, &</s> <s xml:id="echoid-s9875" xml:space="preserve">, BC, ad, CG, com-<lb/> <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/> ponit rationem quadrati, BC, ad rectangulum, ACG, & </s> <s xml:id="echoid-s9876" xml:space="preserve">item ra-<lb/>tio, quam habet, HP, ad, PM, &</s> <s xml:id="echoid-s9877" xml:space="preserve">, HP, ad PX, componit rationem <lb/> <anchor type="figure" xlink:label="fig-0404-01a" xlink:href="fig-0404-01"/> quadrati, HP, ad rectã-<lb/>gulum, MPX, harum <lb/>autem rationum com-<lb/>ponentium ea, quam <lb/>habet, BC, ad, CA, eſt <lb/>eadem, ei, quam ha-<lb/>bet, HP, ad, PM, ideò <lb/>reliquæ componentiũ <lb/>erunt eædem .</s> <s xml:id="echoid-s9878" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9879" xml:space="preserve">BC, ad <lb/>CG, erit vt, HP, ad, P <lb/>X, eſt autem etiam, A <lb/>C, ad, CB, conuerten. <lb/></s> <s xml:id="echoid-s9880" xml:space="preserve">do, vt, MP, ad PH, <lb/>ergo, ex æquali, & </s> <s xml:id="echoid-s9881" xml:space="preserve">con-<lb/>uertendo, GC, ad CA, <lb/>erit vt, XP, ad PM, & </s> <s xml:id="echoid-s9882" xml:space="preserve"><lb/>diuidendo, GA, ad, A <lb/>C, erit vt, XM, ad MP, & </s> <s xml:id="echoid-s9883" xml:space="preserve">antecedentium dimidia .</s> <s xml:id="echoid-s9884" xml:space="preserve">ſ. </s> <s xml:id="echoid-s9885" xml:space="preserve">VG, ad AC, <lb/>erit vt, RX, ad, MP, eſt autem eadem, VG, ad, GA, vt eadem, R <lb/>X, ad, XM, ergo, VG, ad, GC, erit vt, RX, ad, XP, & </s> <s xml:id="echoid-s9886" xml:space="preserve">componen-<lb/>do, VC, ad, CG, erit vt, RP, ad PX, eſt autem, VC, ad, CG, vt om-<lb/>nia quadrata hyperbolæ, BAD, ad omnia quadrata trianguli, B <lb/>AD, &</s> <s xml:id="echoid-s9887" xml:space="preserve">, RP, ad PX, vt omnia quadrata hyperbolę, HMQ, ad om-<lb/>nia quadrata trianguli, HMQ, ergo omnia quadrata hyperbolę, <lb/> <anchor type="note" xlink:label="note-0404-03a" xlink:href="note-0404-03"/> BAD, ad omnia quadrata trianguli, BAD, erunt vt omnia qua-<lb/>drata hyperbolæ, HMQ, ad omnia quadrata trianguli, HMQ, & </s> <s xml:id="echoid-s9888" xml:space="preserve"><lb/>permutando, omnia quadrata hyperbolę, BAD, ad omnia gua-<lb/>drata hyperbolę, HMQ, erunt vt omnia quadrata trianguli, BA <lb/>D, ad omnia quadrata trianguli, HMQ, .</s> <s xml:id="echoid-s9889" xml:space="preserve">@. </s> <s xml:id="echoid-s9890" xml:space="preserve">in tripla ratione eius, <lb/> <anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/> quam habet, AC, ad, MP, quod oſtendere opus erat.</s> <s xml:id="echoid-s9891" xml:space="preserve"/> </p> <div xml:id="echoid-div916" type="float" level="2" n="1"> <note position="right" xlink:label="note-0403-01" xlink:href="note-0403-01a" xml:space="preserve">Iuxta def. <lb/>Apoll. 6. <lb/>Con.</note> <note position="right" xlink:label="note-0403-02" xlink:href="note-0403-02a" xml:space="preserve">F Cor. 22 <lb/>l. 2.</note> <note position="left" xlink:label="note-0404-01" xlink:href="note-0404-01a" xml:space="preserve">21 primi <lb/>Co 1.</note> <note position="left" xlink:label="note-0404-02" xlink:href="note-0404-02a" xml:space="preserve">Iuxta def. <lb/>Cõmand. <lb/>& dicta <lb/>ad Schol. <lb/>28. l. 1.</note> <figure xlink:label="fig-0404-01" xlink:href="fig-0404-01a"> <image file="0404-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0404-01"/> </figure> <note position="left" xlink:label="note-0404-03" xlink:href="note-0404-03a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-0404-04" xlink:href="note-0404-04a" xml:space="preserve">F Cor. 22. <lb/>l. 2.</note> </div> </div> <div xml:id="echoid-div918" type="section" level="1" n="547"> <head xml:id="echoid-head571" xml:space="preserve">THEOREMA XIII, PROPOS. XIV.</head> <p> <s xml:id="echoid-s9892" xml:space="preserve">SIexponatur ſemiperbola, quæ per axem, vel diametrũ <lb/>integrę ſit abſciſſa, habens pro baſi dimidiam baſis <pb o="385" file="0405" n="405" rhead="LIBER V."/> integræ hyperbolæ, ſiat autem parallelogram mum ſub di-<lb/>cta baſi, & </s> <s xml:id="echoid-s9893" xml:space="preserve">axi, vel diametio, in angulo ab eiſdon comen-<lb/>to, ſumpta baſi pro regula: </s> <s xml:id="echoid-s9894" xml:space="preserve">Omnia quadrata dicti paralle. <lb/></s> <s xml:id="echoid-s9895" xml:space="preserve">logrammi ad omnia quadrata trilinei extia hypeibolam <lb/>conſtituti, erunt vt idem parallelogran n@un ad ſuiieli-<lb/>quum ab eodem dempta ſemil yperbola, vna cum exceſſu, <lb/>quo dicta ſemihyperbola ſuperat. </s> <s xml:id="echoid-s9896" xml:space="preserve">dicti parallel@ gram-<lb/>mi, cum {1/6}. </s> <s xml:id="echoid-s9897" xml:space="preserve">parallelogrammi ſub targente hyperbolan, & </s> <s xml:id="echoid-s9898" xml:space="preserve"><lb/>axis, vel diametri hyperbolæ ea poitione, ad quam teli-<lb/>qua ſit, vt integra axis, vel dian eter ad eiuſdem latus <lb/>tranſuerſum.</s> <s xml:id="echoid-s9899" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9900" xml:space="preserve">Sit ergo axis, vel diameter hyperbolę, BE, cuius <lb/> <anchor type="figure" xlink:label="fig-0405-01a" xlink:href="fig-0405-01"/> dimidia, BED, latus tranſuerſum, AB, & </s> <s xml:id="echoid-s9901" xml:space="preserve">in angu-<lb/>lo, BED, ſub, BE, ED, conſticutum parallelogrã-<lb/>mum, GE, ſit autem, vt, EB, ad, BA, ita, EH, ad, H <lb/>B, & </s> <s xml:id="echoid-s9902" xml:space="preserve">per, H, ducta, HM, parallela ipſi, ED, quę ſu-<lb/>matur, pro regula, ita vt ſit conſtitutum parallelo-<lb/>grammum ſub, HB, & </s> <s xml:id="echoid-s9903" xml:space="preserve">ſub, BG, quæ erit tangens <lb/>hyperbolam in puncto, B. </s> <s xml:id="echoid-s9904" xml:space="preserve">Dico gitur omnia qua-<lb/>drata, BD, ad omnia quadrata trilinei, BGD, eſſe <lb/>ut, BD, ad ſui reliquum, dempto ab eodem ſemihyperbola, BE <lb/>D, vna cum exceſſu, quo ipſa ſuperat {1/3}. </s> <s xml:id="echoid-s9905" xml:space="preserve">dicti paralle ogran mi, B <lb/>D, cum {1/6}. </s> <s xml:id="echoid-s9906" xml:space="preserve">B M. </s> <s xml:id="echoid-s9907" xml:space="preserve">Nam omnia quadrata, BD, ad rectangula ſub, B <lb/>D, & </s> <s xml:id="echoid-s9908" xml:space="preserve">ſemihyperbola, BED, ſunt vt, BD, adiplam, BED, rectan-<lb/> <anchor type="note" xlink:label="note-0405-01a" xlink:href="note-0405-01"/> gula verò ſub, BD,, & </s> <s xml:id="echoid-s9909" xml:space="preserve">BED, æquantur rectangulis ſub, BOD, B <lb/>ED, ſimul cum omnibus quadratis, BED, ergo omnia quadrata, <lb/>BD, ad rectangula ſub, BGD, BED, cum ommbus quadratis, BE <lb/> <anchor type="note" xlink:label="note-0405-02a" xlink:href="note-0405-02"/> D, erunt vt, BD, ad, BED; </s> <s xml:id="echoid-s9910" xml:space="preserve">ſunt autem omnia quadrata, BD, ad <lb/>omnia quadrata, BED, vt, AE, ad compoſitam ex {1/2}. </s> <s xml:id="echoid-s9911" xml:space="preserve">AB, & </s> <s xml:id="echoid-s9912" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9913" xml:space="preserve">B <lb/>E, .</s> <s xml:id="echoid-s9914" xml:space="preserve">i. </s> <s xml:id="echoid-s9915" xml:space="preserve">vt, BE, ad compoſitam ex {1/2}. </s> <s xml:id="echoid-s9916" xml:space="preserve">BH, & </s> <s xml:id="echoid-s9917" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9918" xml:space="preserve">HE, quia, AE, BE, <lb/>proportionaliter diuiduntur in punctis, B, H, .</s> <s xml:id="echoid-s9919" xml:space="preserve">i. </s> <s xml:id="echoid-s9920" xml:space="preserve">vt parallelogram-<lb/> <anchor type="note" xlink:label="note-0405-03a" xlink:href="note-0405-03"/> mum, BD, ad compofitum ex {1/2}. </s> <s xml:id="echoid-s9921" xml:space="preserve">BM, & </s> <s xml:id="echoid-s9922" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s9923" xml:space="preserve">HD, .</s> <s xml:id="echoid-s9924" xml:space="preserve">i. </s> <s xml:id="echoid-s9925" xml:space="preserve">vt, BD, ad <lb/>compoſitum ex {1/3}. </s> <s xml:id="echoid-s9926" xml:space="preserve">BD, & </s> <s xml:id="echoid-s9927" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s9928" xml:space="preserve">BM, ergo omnia quadrata, BD, ad <lb/>rectangula ſub, BGD, BED, erunt vt, BD, ad exceſſum, quo ſe-<lb/> <anchor type="note" xlink:label="note-0405-04a" xlink:href="note-0405-04"/> mihyperbola ſuperat {1/3}. </s> <s xml:id="echoid-s9929" xml:space="preserve">BD, cum {1/6}. </s> <s xml:id="echoid-s9930" xml:space="preserve">BM, erant autem omnia qua-<lb/>drata, BD, ad rectangula ſub, IGD, BED, vna cum ommb. </s> <s xml:id="echoid-s9931" xml:space="preserve">qua-<lb/>dratis, BED, vt, BD, ad, BED, ergo omnia quadrata, BD, ad <lb/>rectangula bis ſub, BGD, BED, vna cum omnibus quadratis, BE <pb o="386" file="0406" n="406" rhead="GEOMETRIÆ"/> D, erunt vt, BD, ad, BED, vna cum exceſſu, quo, BED, ſuperat, <lb/>{1/3}. </s> <s xml:id="echoid-s9932" xml:space="preserve">BD, cu-n {1/6}. </s> <s xml:id="echoid-s9933" xml:space="preserve">BM, ergo per conuerſionem rationis omnia qua-<lb/>drata, BD, ad omnia quadrata, BGD, erunt vt, BD, ad iui reli-<lb/>quum, ab eodem dempta ſemihyperbola, BED, & </s> <s xml:id="echoid-s9934" xml:space="preserve">exceſſu, quo <lb/>eadem ſuperat {1/3}. </s> <s xml:id="echoid-s9935" xml:space="preserve">BD, & </s> <s xml:id="echoid-s9936" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s9937" xml:space="preserve">BM, quod erat oſtendendum.</s> <s xml:id="echoid-s9938" xml:space="preserve"/> </p> <div xml:id="echoid-div918" type="float" level="2" n="1"> <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a"> <image file="0405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0405-01"/> </figure> <note position="right" xlink:label="note-0405-01" xlink:href="note-0405-01a" xml:space="preserve">Coroll. 1. <lb/>26. 2.</note> <note position="right" xlink:label="note-0405-02" xlink:href="note-0405-02a" xml:space="preserve">C Co. 23. <lb/>l. 2.</note> <note position="right" xlink:label="note-0405-03" xlink:href="note-0405-03a" xml:space="preserve">I. huius</note> <note position="right" xlink:label="note-0405-04" xlink:href="note-0405-04a" xml:space="preserve">5. l. 2.</note> </div> </div> <div xml:id="echoid-div920" type="section" level="1" n="548"> <head xml:id="echoid-head572" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s9939" xml:space="preserve">HAnc Propoſi@ionem appoſui, vt & </s> <s xml:id="echoid-s9940" xml:space="preserve">nonnullas alias inferius <lb/>quæ licet ſupponant quadraturam byperbolæ iam notam, vt & </s> <s xml:id="echoid-s9941" xml:space="preserve"><lb/>ipſæ completè inteligantur, non inutiliter tamen aliqualiter ſcrriexi-<lb/>ſtimaui, vt ſi alicuius ind uſtria illius quadratura in lucem prodeat, <lb/>illico & </s> <s xml:id="echoid-s9942" xml:space="preserve">bic appoſita nota fiant; </s> <s xml:id="echoid-s9943" xml:space="preserve">vel è conuersò, vt per bæc aliquan-<lb/>do adinuenta ſtacim illius quadr atura nobis inoteſcat; </s> <s xml:id="echoid-s9944" xml:space="preserve">vade cumſcie-<lb/>mus, quam rationem habeat, BD, ad ſemihyperbolam, BED, appreben-<lb/>demus ſtatim, quam rationem babeant omnia quadrata, BD, ad omnia <lb/>quadrata trilinei, BGD: </s> <s xml:id="echoid-s9945" xml:space="preserve">Vel è contra, ſi quando notificabimus, quam <lb/>rationem babeant omnia quadrata, BD, ad omnia quadrata trilinei, <lb/>BGD, ſtatim compertum babebimus, quam rationembabeat, BD, ad <lb/>ſemibyperbolam, BED, & </s> <s xml:id="echoid-s9946" xml:space="preserve">eius quadratura notareddetur.</s> <s xml:id="echoid-s9947" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div921" type="section" level="1" n="549"> <head xml:id="echoid-head573" xml:space="preserve">THEOREMA XIV. PROPOS. XV.</head> <p> <s xml:id="echoid-s9948" xml:space="preserve">SI parallelogrammum, & </s> <s xml:id="echoid-s9949" xml:space="preserve">hyperbola fuerint in eadem <lb/>baſi, & </s> <s xml:id="echoid-s9950" xml:space="preserve">circa eundem axim, vel diametrum, regula <lb/>baſi. </s> <s xml:id="echoid-s9951" xml:space="preserve">Omnia quadrata dicti parallelogrammi ad omnia <lb/>quadrata figuræ compoſitæ ex hyperbola, & </s> <s xml:id="echoid-s9952" xml:space="preserve">alterutro tri-<lb/>lineorum extra hyperbolam conſtitutorum, demptis om-<lb/>nibus quadratis aſſumpti trilinei, eruut vt dictum paral-<lb/>lelogramum ad inſcriptam hyperbolam.</s> <s xml:id="echoid-s9953" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9954" xml:space="preserve">Sit hyperbola, CBD, in baſi, CD, <lb/> <anchor type="figure" xlink:label="fig-0406-01a" xlink:href="fig-0406-01"/> circa axim, vel diametrum, BE, eius <lb/>latus tranſuerſum, AB, in eadem au-<lb/>tem baſi, CD, & </s> <s xml:id="echoid-s9955" xml:space="preserve">circa eundem axim, <lb/>vel diametrum, BE, ſit parallelogrã-<lb/>mum, FD, regula verò, CD. </s> <s xml:id="echoid-s9956" xml:space="preserve">Dico <lb/>ergo omnia quadrata, FD, ad omnia <lb/>quadrata ſiguræ, GBCD, demptis <lb/>omnibus quadratis trilinei, BGD, al-<lb/>terutrius ex duobus, BFC, BGD, ef- <pb o="387" file="0407" n="407" rhead="LIBER V."/> fe vt, FD, ad hyperbolã, CBD, quod patet nam, CBD, eſt ſigura <lb/>qualem poſtulat Prop. </s> <s xml:id="echoid-s9957" xml:space="preserve">29. </s> <s xml:id="echoid-s9958" xml:space="preserve">Lib. </s> <s xml:id="echoid-s9959" xml:space="preserve">3. </s> <s xml:id="echoid-s9960" xml:space="preserve">eſt enim, BE, communis axis, <lb/>vel diameter, FD, parallelogrammi, & </s> <s xml:id="echoid-s9961" xml:space="preserve">hyperbolæ, CBD, vnde <lb/>patet propoſitum.</s> <s xml:id="echoid-s9962" xml:space="preserve"/> </p> <div xml:id="echoid-div921" type="float" level="2" n="1"> <figure xlink:label="fig-0406-01" xlink:href="fig-0406-01a"> <image file="0406-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0406-01"/> </figure> </div> </div> <div xml:id="echoid-div923" type="section" level="1" n="550"> <head xml:id="echoid-head574" xml:space="preserve">THEOREMA XV. PROPOS. XVI.</head> <p> <s xml:id="echoid-s9963" xml:space="preserve">IN eadem anteced. </s> <s xml:id="echoid-s9964" xml:space="preserve">Prepoſ. </s> <s xml:id="echoid-s9965" xml:space="preserve">figura, ſi producatur, CD, <lb/>vtcunq; </s> <s xml:id="echoid-s9966" xml:space="preserve">in, M, & </s> <s xml:id="echoid-s9967" xml:space="preserve">compleatur parallelogrammum, HC, <lb/>regula, CM: </s> <s xml:id="echoid-s9968" xml:space="preserve">Omnia quadrata, FM. </s> <s xml:id="echoid-s9969" xml:space="preserve">demptis omnibus qua-<lb/>dratis, GM, ad omnia quadrata figuræ, HBCM, dem-<lb/>ptis omnibus quadratis figuræ, HBDM, erunt vt, FD, ad <lb/>hyperbolam, CBD.</s> <s xml:id="echoid-s9970" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9971" xml:space="preserve">Patet hoc Theor. </s> <s xml:id="echoid-s9972" xml:space="preserve">nam, CBD, eſt ſigura, qualem poſtulat Prop. <lb/></s> <s xml:id="echoid-s9973" xml:space="preserve">30. </s> <s xml:id="echoid-s9974" xml:space="preserve">Lib. </s> <s xml:id="echoid-s9975" xml:space="preserve">3. </s> <s xml:id="echoid-s9976" xml:space="preserve">quia, BE, eſt communis ax@s, vel diameter, parallelo-<lb/>grammi, FD, & </s> <s xml:id="echoid-s9977" xml:space="preserve">hyperbolæ, CBD, vnde, &</s> <s xml:id="echoid-s9978" xml:space="preserve">c.</s> <s xml:id="echoid-s9979" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div924" type="section" level="1" n="551"> <head xml:id="echoid-head575" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s9980" xml:space="preserve">_H_Inc babetur omnia quadrata, FD ad omnia quad ſigura, GBCD, <lb/>demptis omnibus quadratis trilmet, BGD, eſſe vt omnia qua-<lb/>drata FM, demptis omnibus quadratis, GM, ad omnia quadrata figu-<lb/>ræ, HBCM, demptis omnibus quadratis ſig. </s> <s xml:id="echoid-s9981" xml:space="preserve">HBDM, quia vtraq; </s> <s xml:id="echoid-s9982" xml:space="preserve">ſunt, <lb/>vt, FD, ad byperbolam, CBD.</s> <s xml:id="echoid-s9983" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div925" type="section" level="1" n="552"> <head xml:id="echoid-head576" xml:space="preserve">THEOREMA XVI. PROPOS. XVII.</head> <p> <s xml:id="echoid-s9984" xml:space="preserve">IN eadem Prop. </s> <s xml:id="echoid-s9985" xml:space="preserve">15. </s> <s xml:id="echoid-s9986" xml:space="preserve">figura ſi intelligamus ductam vt-<lb/>cunq; </s> <s xml:id="echoid-s9987" xml:space="preserve">axi, vel diametro, BE, parallelam, RS, fiat au-<lb/>tẽ, vt oia quad FE, ad oia q uad. </s> <s xml:id="echoid-s9988" xml:space="preserve">ſemihyperb BCE, regula, <lb/>CD, .</s> <s xml:id="echoid-s9989" xml:space="preserve">i. </s> <s xml:id="echoid-s9990" xml:space="preserve">vt, AE, ad compoſitam ex. </s> <s xml:id="echoid-s9991" xml:space="preserve">AB, & </s> <s xml:id="echoid-s9992" xml:space="preserve">BE, ita quadra-<lb/>tum, CE, ad quadratum, EI, & </s> <s xml:id="echoid-s9993" xml:space="preserve">vt FE, ad ſemihyperbolã, <lb/>BCE, ita eſſe ſupponatur, CE, ad, EV, vbicunq; </s> <s xml:id="echoid-s9994" xml:space="preserve">cadatpũ-<lb/>ctum, V. </s> <s xml:id="echoid-s9995" xml:space="preserve">Dico omnia quadrata, FS, ad omnia quadrata <lb/>figuræ, RBCS, regula, CD, eſſe vt quadratum, CD, ad <lb/>quadratum, SE, quadratum, EI, & </s> <s xml:id="echoid-s9996" xml:space="preserve">rectangulum bis ſub, <lb/>VE, ES.</s> <s xml:id="echoid-s9997" xml:space="preserve"/> </p> <pb o="388" file="0408" n="408" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s9998" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s9999" xml:space="preserve">n. </s> <s xml:id="echoid-s10000" xml:space="preserve">quadrata figuræ, RBCS, ſecantur per, BE, in omnia <lb/> <anchor type="note" xlink:label="note-0408-01a" xlink:href="note-0408-01"/> quadrata, BS, in omnia quadrata ſemihyperbolæ, BCE, & </s> <s xml:id="echoid-s10001" xml:space="preserve">inre-<lb/>ctangula bis ſub, BCE, & </s> <s xml:id="echoid-s10002" xml:space="preserve">ſub, BS, ad horum ergo ſingula compa-<lb/>remus omnia quadrata, FS; </s> <s xml:id="echoid-s10003" xml:space="preserve">hæcigitur ad omnia quadrata, BS, <lb/>ſunt vt quadratum, CS, ad quadratum, SE, pariter omnia qua-<lb/>drata, FS, ad omnia quadrata, FE, ſunt vt quadratum, SC, ad <lb/>quadratum, CE, omnia verò quadrata, FE, ad omnia quadrata, <lb/> <anchor type="figure" xlink:label="fig-0408-01a" xlink:href="fig-0408-01"/> BCE, ſunt vt quadratum, CE, ad <lb/> <anchor type="figure" xlink:label="fig-0408-02a" xlink:href="fig-0408-02"/> quadratum, EI, ergo, ex æquali, om-<lb/>nia quadrata, FS, ad omnia quadra-<lb/>ta, BCE, erunt vt quadratum, CS, ad <lb/>quadratum, EI; </s> <s xml:id="echoid-s10004" xml:space="preserve">quodſerua. </s> <s xml:id="echoid-s10005" xml:space="preserve">Item <lb/>omnia quadrata, FS, ad rectangula <lb/>ſub, FE, ER, ſunt vt quadratum, C <lb/>S, ad rectangulum, CES, rectangula <lb/> <anchor type="note" xlink:label="note-0408-02a" xlink:href="note-0408-02"/> verò ſub, FE, ER, adrectang. </s> <s xml:id="echoid-s10006" xml:space="preserve">ſub, BC <lb/>E, ER, ſunt vt, FE, ad, BCE, .</s> <s xml:id="echoid-s10007" xml:space="preserve">i. </s> <s xml:id="echoid-s10008" xml:space="preserve">vt, C <lb/>E, ad, VE, .</s> <s xml:id="echoid-s10009" xml:space="preserve">i. </s> <s xml:id="echoid-s10010" xml:space="preserve">ſumpta, ES, communi altitudine, vt rectangulum, <lb/>CES, ad rectangulum, VES, ergo, ex æquo, omnia quadrata, FS, <lb/> <anchor type="note" xlink:label="note-0408-03a" xlink:href="note-0408-03"/> ad rectangula ſub, BCE, ER, erunt vt quadratum, CS, ad rectan-<lb/>gulum, VES, ad eadem verò bis ſumpta, vt quadratum, CS, ad <lb/>rectangulum bis ſub, VES; </s> <s xml:id="echoid-s10011" xml:space="preserve">ergo, conſequentibus ſimul collectis, <lb/>omnia quadrata, FS, ad omnia quadrata, BS, ad omnia quadra-<lb/>ta, BCE, & </s> <s xml:id="echoid-s10012" xml:space="preserve">ad rectangula bis ſub, BCE, ER, ideſt ad omnia qua-<lb/> <anchor type="note" xlink:label="note-0408-04a" xlink:href="note-0408-04"/> dratà figuræ, RBCS, erunt vt quadratum, CS, ad quadra-<lb/>SE, quadratum, EI, & </s> <s xml:id="echoid-s10013" xml:space="preserve">rectangulum bis ſub, VE, ES; </s> <s xml:id="echoid-s10014" xml:space="preserve">qua me-<lb/>thodo ſimiliter oſtendemus omnia quadrata, FD, ad omnia <lb/>quadrata figurę, GBCD, eſſe vt quadratum, CD, ad quadra-<lb/>tum, DE, quadratum, EI, & </s> <s xml:id="echoid-s10015" xml:space="preserve">rectangulum bis ſub, VE, ED; <lb/></s> <s xml:id="echoid-s10016" xml:space="preserve">& </s> <s xml:id="echoid-s10017" xml:space="preserve">ſimiliter omnia quadrata, FM, ad omnia quadrata figuræ, HB <lb/>CM, eſſe vt quadratum, CM, ad quadratum, ME, quadratum, <lb/>EI, & </s> <s xml:id="echoid-s10018" xml:space="preserve">rectangulum bis ſub, VE, EM, quod oſtendere opus erat.</s> <s xml:id="echoid-s10019" xml:space="preserve"/> </p> <div xml:id="echoid-div925" type="float" level="2" n="1"> <note position="left" xlink:label="note-0408-01" xlink:href="note-0408-01a" xml:space="preserve">D.Co.23. <lb/>l. 2.</note> <figure xlink:label="fig-0408-01" xlink:href="fig-0408-01a"> <image file="0408-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0408-01"/> </figure> <figure xlink:label="fig-0408-02" xlink:href="fig-0408-02a"> <image file="0408-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0408-02"/> </figure> <note position="left" xlink:label="note-0408-02" xlink:href="note-0408-02a" xml:space="preserve">14. @ 2.</note> <note position="left" xlink:label="note-0408-03" xlink:href="note-0408-03a" xml:space="preserve">Coroll 1. <lb/>26. l. 2.</note> <note position="left" xlink:label="note-0408-04" xlink:href="note-0408-04a" xml:space="preserve">D. Co 3. 2. <lb/>l. 2.</note> </div> </div> <div xml:id="echoid-div927" type="section" level="1" n="553"> <head xml:id="echoid-head577" xml:space="preserve">THE OREMA XVII. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s10020" xml:space="preserve">IN eadem Prop. </s> <s xml:id="echoid-s10021" xml:space="preserve">15. </s> <s xml:id="echoid-s10022" xml:space="preserve">figura oſtendemus omnia quadrata <lb/>figuræ, HBCM, dempti omnibus quadratis figuræ, H <lb/>BDM regula eadem retenta, ad omnia quadrata figuræ, <lb/>GBCD, demptis omnibus quadratis trilinei, BGD, eſſe <lb/>vt compoſita ex, CM, MD, ad, DC.</s> <s xml:id="echoid-s10023" xml:space="preserve"/> </p> <pb o="389" file="0409" n="409" rhead="LIBER V."/> <p> <s xml:id="echoid-s10024" xml:space="preserve">Hoc Theorema demonſtrabitur methodo Sect. </s> <s xml:id="echoid-s10025" xml:space="preserve">2. </s> <s xml:id="echoid-s10026" xml:space="preserve">Collorarij <lb/>29. </s> <s xml:id="echoid-s10027" xml:space="preserve">Prop. </s> <s xml:id="echoid-s10028" xml:space="preserve">33. </s> <s xml:id="echoid-s10029" xml:space="preserve">Lib. </s> <s xml:id="echoid-s10030" xml:space="preserve">3. </s> <s xml:id="echoid-s10031" xml:space="preserve">quod ſimiliter quacunq; </s> <s xml:id="echoid-s10032" xml:space="preserve">figura exiſtente, CB <lb/>D, dummodo, BE, ſit communis axis eius, &</s> <s xml:id="echoid-s10033" xml:space="preserve">, FD, facilè collige-<lb/>mus.</s> <s xml:id="echoid-s10034" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div928" type="section" level="1" n="554"> <head xml:id="echoid-head578" xml:space="preserve">THEOREMA XVIII. PROPOS. XIX.</head> <p> <s xml:id="echoid-s10035" xml:space="preserve">IN eodem Prop. </s> <s xml:id="echoid-s10036" xml:space="preserve">15. </s> <s xml:id="echoid-s10037" xml:space="preserve">figura oſtendemus omnia quadrata <lb/>BCE, regula, CD, ad omnia quadrata figuræ, GBCD, <lb/>demptis omnibus quadratis trilinei, BGD, eſſe vt quadra-<lb/>tum, IE, ad rectangulum ſub, CD, & </s> <s xml:id="echoid-s10038" xml:space="preserve">dupla, VE.</s> <s xml:id="echoid-s10039" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10040" xml:space="preserve">Nam omnia quadrata, BCE, ad omnia quadrata, FE, ſunt vt <lb/>quadratum, IE, ad quadratum, EC, item omnia quadrata, FE, <lb/>ad omnia quadrata, FD, ſunt vt quadratum, EC, ad quadratum, <lb/> <anchor type="note" xlink:label="note-0409-01a" xlink:href="note-0409-01"/> CD, & </s> <s xml:id="echoid-s10041" xml:space="preserve">tandem omnia quadrata, FD, ad omnia quad. </s> <s xml:id="echoid-s10042" xml:space="preserve">figuræ, GBC <lb/>D: </s> <s xml:id="echoid-s10043" xml:space="preserve">demptis omnib. </s> <s xml:id="echoid-s10044" xml:space="preserve">quadratis trilinei, BGD, ſunt vt, FD, ad hyper-<lb/>bolã, CBD, .</s> <s xml:id="echoid-s10045" xml:space="preserve">i. </s> <s xml:id="echoid-s10046" xml:space="preserve">vt, CE, ad, EV, velvt, CD, ad duplã, VE, vel vt qua-<lb/> <anchor type="note" xlink:label="note-0409-02a" xlink:href="note-0409-02"/> dratum, CD, ad rectangulũ ſub, CD, & </s> <s xml:id="echoid-s10047" xml:space="preserve">dupla, VE, ergo, ex æqua-<lb/>li, omnia quadrata ſemihyperbolæ, BCE, ad omnia quadrata figu-<lb/>ræ, GBCD, dẽptis omnibus quadratis trilinei, BGD, eruntvt qua-<lb/>dratum, EL, ad rectangulum ſub, CD, & </s> <s xml:id="echoid-s10048" xml:space="preserve">dupla, VE, quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s10049" xml:space="preserve"/> </p> <div xml:id="echoid-div928" type="float" level="2" n="1"> <note position="right" xlink:label="note-0409-01" xlink:href="note-0409-01a" xml:space="preserve">9. l. 2.</note> <note position="right" xlink:label="note-0409-02" xlink:href="note-0409-02a" xml:space="preserve">15. huius.</note> </div> </div> <div xml:id="echoid-div930" type="section" level="1" n="555"> <head xml:id="echoid-head579" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s10050" xml:space="preserve">_O_Via ve ò omnia quadrata figuræ, GBCD, demptis omnibus qua-<lb/>d atis trilinei, BGD, ad mnia quadrata fig. </s> <s xml:id="echoid-s10051" xml:space="preserve">HBCM, demptis om <lb/>nibus quadratis figuræ, BHMD, oſtenſa ſunt eſſe, vt, CD, ad, DMC, .</s> <s xml:id="echoid-s10052" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s10053" xml:space="preserve">ſumpta communi altitudine dupla, VE, vt rectangulum ſub, CD, & </s> <s xml:id="echoid-s10054" xml:space="preserve"><lb/>dupla, VE, ad rectangulum ſub, CMD, & </s> <s xml:id="echoid-s10055" xml:space="preserve">dupla, VE, ideò etiam, ex <lb/> <anchor type="note" xlink:label="note-0409-03a" xlink:href="note-0409-03"/> æquali, omnia quadrata, B E, ad omnia quadrata figuræ HBCM, d@m-<lb/>ptis omnibus quadratis figuræ, BHMD, erunt vt quadratum, EI, ad <lb/>vectungulum ſub, CMD, & </s> <s xml:id="echoid-s10056" xml:space="preserve">dupla, VE.</s> <s xml:id="echoid-s10057" xml:space="preserve"/> </p> <div xml:id="echoid-div930" type="float" level="2" n="1"> <note position="right" xlink:label="note-0409-03" xlink:href="note-0409-03a" xml:space="preserve">_18. huius._</note> </div> </div> <div xml:id="echoid-div932" type="section" level="1" n="556"> <head xml:id="echoid-head580" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s10058" xml:space="preserve">_H_Aec, & </s> <s xml:id="echoid-s10059" xml:space="preserve">ſimilia poſſumus circa byperbolam, eiuſque portiones <lb/>contemplari, quorum plurima Lectoris induſtriæ ex aminanda <lb/>relinquo, tum ad nimiam prolixitatem euitandam, tum etiam; </s> <s xml:id="echoid-s10060" xml:space="preserve">quia <lb/>bæc Tbeoremata minus fortè reliquis iucunda erunt, tum completa <pb o="390" file="0410" n="410" rhead="GEOMETRIÆ"/> eorum notitia in ſuppoſitione eiuſdem byperbolæ qudaraturæ deficiat; <lb/></s> <s xml:id="echoid-s10061" xml:space="preserve">ſi quis tamen adbuc voluerit aliacirca eandem contemplari, metbo-<lb/>dum tenere poterit Lib. </s> <s xml:id="echoid-s10062" xml:space="preserve">2. </s> <s xml:id="echoid-s10063" xml:space="preserve">& </s> <s xml:id="echoid-s10064" xml:space="preserve">3. </s> <s xml:id="echoid-s10065" xml:space="preserve">à me proſequutam, mibi verò poſt <lb/>byperbolarum ſpeculationem ad oppoſitas ſectiones, & </s> <s xml:id="echoid-s10066" xml:space="preserve">coniugatas <lb/>Appolonij opportunè videtur tranſeundum.</s> <s xml:id="echoid-s10067" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div933" type="section" level="1" n="557"> <head xml:id="echoid-head581" xml:space="preserve">THEOREMA XIX. PROPOS. XX.</head> <p> <s xml:id="echoid-s10068" xml:space="preserve">SI ad axim, vel diametrum vtriuſq; </s> <s xml:id="echoid-s10069" xml:space="preserve">oppoſitarum ſectio-<lb/>num ordinatim applicentur rectæ lineæ in eaſdem <lb/>terminatæ, ita vt abſciſſæ per eaſdem ab axibu, vel dia-<lb/>metris verſus vertices ſint æquales erũt iſtæ applicatæ pa-<lb/>rallelogrammi oppoſita latera, quod parallelogrãmum ſi <lb/>compleatur, regula applicatarum altera ſumpta, omnia <lb/>quadrata parallelogrammi conſtituti ad reliquum, dem-<lb/>ptis ab ijſdem omnibus quadratis oppoſitarum hyperbo-<lb/>larum iam ſer dictas ordinatim applicatas conſtitutarum, <lb/>erunt vt rectangulum ſub compoſita ex tranſuerſo latere, <lb/>& </s> <s xml:id="echoid-s10070" xml:space="preserve">axi, vel diametro alterutrius oppoſitarum hyperbola-<lb/>rum, & </s> <s xml:id="echoid-s10071" xml:space="preserve">ſub compoſita ex hoc axi, vel diametro, & </s> <s xml:id="echoid-s10072" xml:space="preserve">. </s> <s xml:id="echoid-s10073" xml:space="preserve">tran-<lb/>ſuerſi lateris, ad rectangulum bis ſub . </s> <s xml:id="echoid-s10074" xml:space="preserve">tranſuerſi lateris, <lb/>& </s> <s xml:id="echoid-s10075" xml:space="preserve">ſum compoſita ex. </s> <s xml:id="echoid-s10076" xml:space="preserve">eiuſdem tranſuerſi lateris, & </s> <s xml:id="echoid-s10077" xml:space="preserve">axi, <lb/>vel diamerro alrerutrius oppoſitarum hyperbolarum, cum <lb/>{2/3}. </s> <s xml:id="echoid-s10078" xml:space="preserve">quadrati eiuſdem axis, vel diametri.</s> <s xml:id="echoid-s10079" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10080" xml:space="preserve">Sint oppoſitæ ſectiones, AMC, BND, quarum latus tranſuer-<lb/>ſum ſit, NM, communis axis, vel diameter earundem, ad quam <lb/>hincinde productam ordinatim applicẽtur, BD, AC, in fectiones <lb/>terminatæ, abicindentes verſus vertices, NM, axes, vel diame-<lb/>tros, FN, ME, hyperbolarum, BND, AMC, (quas pariter oppo-<lb/>ſitas voco) quæ ſint inter ſe æquales, iunganturque, BA, DC, & </s> <s xml:id="echoid-s10081" xml:space="preserve"><lb/>ſit, O, centrum oppoſitarum ſectionum, BND, AMC: </s> <s xml:id="echoid-s10082" xml:space="preserve">Quoniam <lb/>ergo, FN, ME, ſunt æquales, erunt etiam æquales, BD, AC, & </s> <s xml:id="echoid-s10083" xml:space="preserve"><lb/>ſunt equidiſtantes, quia ad eandem diametrum, velaxim, FE, ſunt <lb/> <anchor type="note" xlink:label="note-0410-01a" xlink:href="note-0410-01"/> ordinatim applicate, ergo, BA, DC, erunt ęquidiſtantes, &</s> <s xml:id="echoid-s10084" xml:space="preserve">, BC, <lb/>parallelogrammum. </s> <s xml:id="echoid-s10085" xml:space="preserve">Dico ergo (regula ſumpta altera applica-<lb/>tarum, AC, BD, vt, AC,) omnia quadrata, BC, ad reliquum eo-<lb/>rundem demptis omnibus quadratis oppoſitarum hyperbolarum, <lb/>BND, AMC, eſſe vt rectangulum, NEO, ad rectangulum, NOE, <pb o="391" file="0411" n="411" rhead="LIBER V."/> bis, cum {2/3}. </s> <s xml:id="echoid-s10086" xml:space="preserve">quadrati, EM. </s> <s xml:id="echoid-s10087" xml:space="preserve">Ducanturper, M, O, puncta paralie <lb/> <anchor type="note" xlink:label="note-0411-01a" xlink:href="note-0411-01"/> lę, AC, ipſę, VS, TR, igitur, TR, tanget ſectionem, AMC, & </s> <s xml:id="echoid-s10088" xml:space="preserve"><lb/>ſunt parallelogramma, TC, VC, VD: </s> <s xml:id="echoid-s10089" xml:space="preserve">Omnia ergo quadrata pa. <lb/></s> <s xml:id="echoid-s10090" xml:space="preserve">rallelogrammi, VC, ad omnia quadrata hyperbolæ, AMC, ha <lb/>bent ratiíonem compoſtionem ex ea, quam habent omnia quadrata, <lb/> <anchor type="note" xlink:label="note-0411-02a" xlink:href="note-0411-02"/> VC, ad omnia quadrata, TC, .</s> <s xml:id="echoid-s10091" xml:space="preserve">i. </s> <s xml:id="echoid-s10092" xml:space="preserve">exratione, OE, ad, EM, & </s> <s xml:id="echoid-s10093" xml:space="preserve">ex <lb/> <anchor type="figure" xlink:label="fig-0411-01a" xlink:href="fig-0411-01"/> ratione omnium quadratorum, TC, <lb/>ad omnia quadrata hyperbolę, AMC, <lb/>ideſt ex ea, quam habet, NE, ad cõ-<lb/>poſitam ex, OM, & </s> <s xml:id="echoid-s10094" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10095" xml:space="preserve">ME, iſtę duę <lb/> <anchor type="note" xlink:label="note-0411-03a" xlink:href="note-0411-03"/> rationes autem .</s> <s xml:id="echoid-s10096" xml:space="preserve">i. </s> <s xml:id="echoid-s10097" xml:space="preserve">quam habet, OE, <lb/>ad, EM, &</s> <s xml:id="echoid-s10098" xml:space="preserve">, NE, ad compoſitam ex, <lb/>OM, & </s> <s xml:id="echoid-s10099" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10100" xml:space="preserve">ME, componuntrationem <lb/>rectanguli ſub, NE, EO, ad rectangu-<lb/>lum ſub, EM, & </s> <s xml:id="echoid-s10101" xml:space="preserve">ſub compoſita ex, O <lb/>M, & </s> <s xml:id="echoid-s10102" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10103" xml:space="preserve">ME, ergo omnia quadrata, <lb/>VC, ad omnia quadrata hyperbolę, <lb/>AMC, ſunt vt rectangulum, NEO, ad <lb/> <anchor type="note" xlink:label="note-0411-04a" xlink:href="note-0411-04"/> rectangulum ſub, EM, & </s> <s xml:id="echoid-s10104" xml:space="preserve">ſub compo-<lb/>ſita ex, OM, & </s> <s xml:id="echoid-s10105" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10106" xml:space="preserve">ME, quod eſt ęqua <lb/>le rectangulis ſub, OM, &</s> <s xml:id="echoid-s10107" xml:space="preserve">, ME, & </s> <s xml:id="echoid-s10108" xml:space="preserve">ſub <lb/>{1/3}. </s> <s xml:id="echoid-s10109" xml:space="preserve">ME, & </s> <s xml:id="echoid-s10110" xml:space="preserve">ſub, ME, .</s> <s xml:id="echoid-s10111" xml:space="preserve">i. </s> <s xml:id="echoid-s10112" xml:space="preserve">rectangulo, OME, cum {1/3}. </s> <s xml:id="echoid-s10113" xml:space="preserve">quadrati, ME; <lb/></s> <s xml:id="echoid-s10114" xml:space="preserve">qu<unsure/>a verò rectangulum, NEO, ęquatur rectangulo, NEO, cum <lb/> <anchor type="note" xlink:label="note-0411-05a" xlink:href="note-0411-05"/> quadrato, OE, quadratum verò, OE, ęquatur quadratis, EM, M <lb/>O, cum rectangulis bis ſub, EMO, ideò ſi ab his dempſeris ſemel <lb/>rectangulum, EMO, remanebit de quadrato, OE, rectangulum, <lb/>EMO, cum quadratis, EM, MO, rurſus ſi dempſeris {1/3}. </s> <s xml:id="echoid-s10115" xml:space="preserve">quadrati, <lb/>EM, à quadrato, EM, remanebunt {2/3}. </s> <s xml:id="echoid-s10116" xml:space="preserve">quadrati, EM, rectangulũ, <lb/>EMO, cum quadrato, MO, rectangulum verò, EMO, cum qua-<lb/>drato, OM, ęquatur rectangulo, EOM, vel, EON, quod collectũ <lb/>ſimul cum {2/3}. </s> <s xml:id="echoid-s10117" xml:space="preserve">quadrati, EM, eſt reſiduum, quod remanet detracto <lb/>rectangulo, EMO, cum {1/3}. </s> <s xml:id="echoid-s10118" xml:space="preserve">quadrati, EM, a quadrato, EO, ergo <lb/>detracto rectangulo, EMO, cum {1/3}. </s> <s xml:id="echoid-s10119" xml:space="preserve">quadrati, EM, à quadrato, E <lb/>O, iuncto rectangulo, EON, .</s> <s xml:id="echoid-s10120" xml:space="preserve">i. </s> <s xml:id="echoid-s10121" xml:space="preserve">à rectangulo, NEO, remanent <lb/>duo rectangula, NOE, cum {2/3}. </s> <s xml:id="echoid-s10122" xml:space="preserve">quadrati, ME; </s> <s xml:id="echoid-s10123" xml:space="preserve">quia ergo oſtenſum <lb/>eſt omnia quadrata, VC, ad omnia quadrata hyperbolę, AMC, <lb/>eiſe vt rectangulum, NEO, ad rectangulum, OME, cum {1/3} qua-<lb/>drati, ME, ideò, per conuerſionem rationis, omnia quadrata, V <lb/>O, ad reliquum, demptis ab ijſdem omnibus, quadratis hyperbo-<lb/>lę, AMC, erunt vt rectangulum, NEO, ad rectangulum bisſub, <lb/>NOE, cum {2/3}. </s> <s xml:id="echoid-s10124" xml:space="preserve">quadrati, EM. </s> <s xml:id="echoid-s10125" xml:space="preserve">Eodem pacto, ſi ducamus per, N, <pb o="392" file="0412" n="412" rhead="GEOMETRIÆ"/> ipſam, QP, parallelam ipſi, BD, quæerit tangens fectionem, BN <lb/>D, in puncto, N, oſtendemus omnia quadrata, BS, ad reliquum, <lb/>demptis omnibus quadratis. </s> <s xml:id="echoid-s10126" xml:space="preserve">hyperbolæ, BND, (ſumptis medijs <lb/>omnibus quadratis, BP,) eſſe vt rectangulum, MFO, ad rectangu, <lb/>lum bis ſub, MOF, cum {2/3}. </s> <s xml:id="echoid-s10127" xml:space="preserve">quadrati, FN, .</s> <s xml:id="echoid-s10128" xml:space="preserve">i. </s> <s xml:id="echoid-s10129" xml:space="preserve">vt rectangulum, NE <lb/>O, ad rectangulum bis ſub, NOE, cum {2/3}. </s> <s xml:id="echoid-s10130" xml:space="preserve">quadrati, EM, nam, E <lb/>M, eſt æqualis, NF, & </s> <s xml:id="echoid-s10131" xml:space="preserve">ideò etiam, EN, ęqualis, MF, &</s> <s xml:id="echoid-s10132" xml:space="preserve">, EO, pa-<lb/>riter eſt æqualis ipſi, OF. </s> <s xml:id="echoid-s10133" xml:space="preserve">Tandem vt vnum ad vnum, ita omnia <lb/>ad omnia .</s> <s xml:id="echoid-s10134" xml:space="preserve">i. </s> <s xml:id="echoid-s10135" xml:space="preserve">vt omnia quadrata, BS, ad reliquum, demptis omni. <lb/></s> <s xml:id="echoid-s10136" xml:space="preserve">bus quadratis hyperbolæ, BND, .</s> <s xml:id="echoid-s10137" xml:space="preserve">i. </s> <s xml:id="echoid-s10138" xml:space="preserve">vt rectangulum, MFO, ad re-<lb/>ctangulum bis ſub, MOF, cum {2/3}. </s> <s xml:id="echoid-s10139" xml:space="preserve">quadrati, FN, ita omnia qua-<lb/>drata, BC, adreliquum. </s> <s xml:id="echoid-s10140" xml:space="preserve">demptis ab eiſdem omnibus quadratis hy-<lb/>perbolarum oppolitarum, AMC, BND, eſt autem, vt rectangu-<lb/>lum, MFO, ad rectangulum bis ſub, MOF, cum {2/3} quadrati, FN, <lb/>ita rectangulum, NOE, ad rectangulum bis ſub, NOE, cum {2/3}. </s> <s xml:id="echoid-s10141" xml:space="preserve"><lb/>quadra @, EM, ergo omnia quadrata, BC, ad reliquum demptis <lb/>ab j@d@m omnibus quadratis oppoſitarum hyperbolarum, AMC, <lb/>BND, erunt vt rectangulum ſub, NEO, ad rectangulum bis ſub, <lb/>NOE, cum {2/3}. </s> <s xml:id="echoid-s10142" xml:space="preserve">quadrati, ME, quod oſtendereopus erat.</s> <s xml:id="echoid-s10143" xml:space="preserve"/> </p> <div xml:id="echoid-div933" type="float" level="2" n="1"> <note position="left" xlink:label="note-0410-01" xlink:href="note-0410-01a" xml:space="preserve">Blicitur <lb/>ex 29. Pri. <lb/>Con.</note> <note position="right" xlink:label="note-0411-01" xlink:href="note-0411-01a" xml:space="preserve">17. P<unsure/>imi <lb/>Con.</note> <note position="right" xlink:label="note-0411-02" xlink:href="note-0411-02a" xml:space="preserve">Defin. 12 <lb/>l. 1. <lb/>10. l.2.</note> <figure xlink:label="fig-0411-01" xlink:href="fig-0411-01a"> <image file="0411-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0411-01"/> </figure> <note position="right" xlink:label="note-0411-03" xlink:href="note-0411-03a" xml:space="preserve">.huius.</note> <note position="right" xlink:label="note-0411-04" xlink:href="note-0411-04a" xml:space="preserve">6.‘.2.</note> <note position="right" xlink:label="note-0411-05" xlink:href="note-0411-05a" xml:space="preserve">4. Sec. E. <lb/>lem.</note> </div> </div> <div xml:id="echoid-div935" type="section" level="1" n="558"> <head xml:id="echoid-head582" xml:space="preserve">THEOREMA XX. PROPOS. XXI.</head> <p> <s xml:id="echoid-s10144" xml:space="preserve">SI, veluti in anteced. </s> <s xml:id="echoid-s10145" xml:space="preserve">ſit parallelogrammum habens op-<lb/>poſita latera, quæ ſint ad diametrum tranſuerſam op-<lb/>poſitaruin ſectionem ordinatim applicata, quæq; </s> <s xml:id="echoid-s10146" xml:space="preserve">oppoſi-<lb/>tarum hyperbolarum ſint baſes, inſuper deſcribantur earũ <lb/>aſymptoti, & </s> <s xml:id="echoid-s10147" xml:space="preserve">regula ſit latus tranſuerſum, conſtituti paral-<lb/>lelogra nmi omnia quadrata ad omnia quadrata figuræ, <lb/>quæ continetur lateribus parallelogrammi iam dicti, late-<lb/>ritranſuerſo parallelis, & </s> <s xml:id="echoid-s10148" xml:space="preserve">portionibus oppoſitarum ſectio-<lb/>num inter eadem latera comprehenſis, erunt vt quadratũ <lb/>vniuſcuiuſuis laterum dicti para llelogrammi lateri tran-<lb/>ſuerſo æquidiſtantium ad quadratum lateris tranſuerſi, <lb/>vna cum. </s> <s xml:id="echoid-s10149" xml:space="preserve">quadrati portionis dicti lateris eiuſdem paral-<lb/>lelogrammi, quæ inter aſymptotos incluſa manet.</s> <s xml:id="echoid-s10150" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10151" xml:space="preserve">Sint oppoſitæ ſectiones, FAD, EVC, quarum latus tranſuer-<lb/>ſum, AV, centrum, O, per quod tranſeant earum aſymptoti, YO <pb o="393" file="0413" n="413" rhead="LIBER V."/> H, NOS, ſit autem, veluti in anteced. </s> <s xml:id="echoid-s10152" xml:space="preserve">Prop. </s> <s xml:id="echoid-s10153" xml:space="preserve">conſſitutum paralle-<lb/>logrammum, FC, cuius oppoſita latera, FD, EC, ſint ad axim, <lb/>vel diametrum, AV, in eadem productam, órdinatim applica@a, <lb/> <anchor type="figure" xlink:label="fig-0413-01a" xlink:href="fig-0413-01"/> erunt, DC, FE, ipſi, AV, ęqui <lb/>diftantes, ſint earum portiones <lb/>inter aſymptotos concluſæ, H <lb/>S, NY, regula ſit, AV. </s> <s xml:id="echoid-s10154" xml:space="preserve">Dico <lb/>ergo omnia quadrata, FC, ad <lb/>omnia quadrata figuræ, FAD <lb/>CVE, ideſt figuræ concluſæ in-<lb/>ter latera, FE, DC, & </s> <s xml:id="echoid-s10155" xml:space="preserve">oppoſi-<lb/>tarum ſectionum portiones in-<lb/>ter eadem manentes, quę ſunt, <lb/>FAD, EVC, eſſe, vt quadratũ, <lb/>DC, vel, FE, ad quadratum, <lb/>AV, cum {1/3}. </s> <s xml:id="echoid-s10156" xml:space="preserve">quadrati, HS, vel, <lb/>NY. </s> <s xml:id="echoid-s10157" xml:space="preserve">Per puncta ergo, O, V, <lb/>ducantur, XL, VP, ad ipſam, <lb/>AV, ordinatim applicatæ, erit <lb/>igitur, XL, ſecunda diameter, <lb/>&</s> <s xml:id="echoid-s10158" xml:space="preserve">, VP, tanget ſectionem, EV <lb/>C. </s> <s xml:id="echoid-s10159" xml:space="preserve">Quoniam ergo rectangulum, HCS, æquatur quadrato. </s> <s xml:id="echoid-s10160" xml:space="preserve">OV, <lb/>ideſt quadrato, LP, rectangulum verò, HCS, æquatur rectangu-<lb/> <anchor type="note" xlink:label="note-0413-01a" xlink:href="note-0413-01"/> lo, LSC, bis vna cum quadrato, SC, ideò, rectangulum, LSC, bis <lb/>vna cum quadrato, SC, erit æquale quadrato, LP; </s> <s xml:id="echoid-s10161" xml:space="preserve">eodem pacto <lb/>ſi intelligamus ipſi, LC, æquidiſſantem vtcunq; </s> <s xml:id="echoid-s10162" xml:space="preserve">ductam intra pa-<lb/>rallelogrammum, OP, viq; </s> <s xml:id="echoid-s10163" xml:space="preserve">ad ſectionem, VC, productam, oſten-<lb/>demus rectangulum bis ſub eius portionibus inter, OL, OS, & </s> <s xml:id="echoid-s10164" xml:space="preserve">in-<lb/>ter, OS, & </s> <s xml:id="echoid-s10165" xml:space="preserve">ſectionem, VC, concluſis, vna cum quadrato eius, quę <lb/>inter, OS, & </s> <s xml:id="echoid-s10166" xml:space="preserve">ſectionem, VC, clauditur, æquari quadrato eius, quę <lb/>manet inter, OL, VP, & </s> <s xml:id="echoid-s10167" xml:space="preserve">ſic de reliquis conſimihter ſumptis; </s> <s xml:id="echoid-s10168" xml:space="preserve">vnde <lb/>patebit tandem rectangula ſub trianguio, LOS, & </s> <s xml:id="echoid-s10169" xml:space="preserve">figura, OVCS, <lb/>bis ſumpta, vna cum omnibus quadratis figuræ, CVCS, æquari <lb/>omnibus quadratis, OP, regula, AV, iam ſuppoſita, quia ergo <lb/> <anchor type="note" xlink:label="note-0413-02a" xlink:href="note-0413-02"/> omnia quadrata, CO, ad omnia quadrata, OP, ſunt vt quadratũ, <lb/>ZO, ad quadratum, OV, ideò pam<unsure/>teron @ a quadrata, OC, ad <lb/>rectangula ſub triangulo, LOS, & </s> <s xml:id="echoid-s10170" xml:space="preserve">figura, OVCS, bis, vna cum om-<lb/>nibus quadratis figuræ, OVCS, erunt vt quadratum, ZO, ad qua-<lb/>dratum, OV; </s> <s xml:id="echoid-s10171" xml:space="preserve">quod ſerua.</s> <s xml:id="echoid-s10172" xml:space="preserve"/> </p> <div xml:id="echoid-div935" type="float" level="2" n="1"> <figure xlink:label="fig-0413-01" xlink:href="fig-0413-01a"> <image file="0413-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0413-01"/> </figure> <note position="right" xlink:label="note-0413-01" xlink:href="note-0413-01a" xml:space="preserve">II. Secun. <lb/>Con.</note> <note position="right" xlink:label="note-0413-02" xlink:href="note-0413-02a" xml:space="preserve">9. lib. 2.</note> </div> <p> <s xml:id="echoid-s10173" xml:space="preserve">Inſuper omnia quadrata, CO, ad omnia quadrata parallelogrã-<lb/>mi, SO, ſi compleretur, eſſent vt quadiatum, CL, ad quadratum, <pb o="394" file="0414" n="414" rhead="GEOMETRIÆ"/> LS, ſunt autem omnia quadrata trianguli, OLS, {1/3}. </s> <s xml:id="echoid-s10174" xml:space="preserve">omnium qua-<lb/>dratorum parallelogrammi, SO, ergo omnia quadrata, CO, ad <lb/>omnia quadrata trianguli, LOS, erunt vt quadratum, CL, ad {1/3}. <lb/></s> <s xml:id="echoid-s10175" xml:space="preserve">quadrati, LS; </s> <s xml:id="echoid-s10176" xml:space="preserve">& </s> <s xml:id="echoid-s10177" xml:space="preserve">quontam oſtenſum eſt omnia quadrata, CO, ad <lb/>rectangula bis ſub triangulo, LOS, & </s> <s xml:id="echoid-s10178" xml:space="preserve">figura, OVCS, vna cum <lb/> <anchor type="figure" xlink:label="fig-0414-01a" xlink:href="fig-0414-01"/> omnibus quadratis figuræ, O <lb/>VCS, eſſe vt quadratum, ZO, <lb/>vel, CL, ad quadratum, OV, <lb/>ideò omnia quadrata, CO, ad <lb/>rectangula bis ſub triangulo, <lb/>LOS, & </s> <s xml:id="echoid-s10179" xml:space="preserve">figura, OVCS, vna <lb/>com omnibus quadratis tum <lb/>figuræ, OVCS, tum trianguli, <lb/> <anchor type="note" xlink:label="note-0414-01a" xlink:href="note-0414-01"/> OLS, .</s> <s xml:id="echoid-s10180" xml:space="preserve">i. </s> <s xml:id="echoid-s10181" xml:space="preserve">omnia quadrata, CO, <lb/>ad omnia quadrata figuræ, O <lb/>LCV, erunt vt quadratum, C <lb/>L, ad quadratum, OV, vna cũ <lb/> <anchor type="note" xlink:label="note-0414-02a" xlink:href="note-0414-02"/> {1/3}. </s> <s xml:id="echoid-s10182" xml:space="preserve">quadrati, LS, & </s> <s xml:id="echoid-s10183" xml:space="preserve">anteceden-<lb/>tium dupla .</s> <s xml:id="echoid-s10184" xml:space="preserve">i. </s> <s xml:id="echoid-s10185" xml:space="preserve">omnia quadra-<lb/>ta, XC, ad omnia quadrata fi-<lb/>guræ, EVCLX, erunt vt qua-<lb/>dratum, CL, ad quadratum, <lb/>OV, cum {1/3}. </s> <s xml:id="echoid-s10186" xml:space="preserve">quadrati, LS, & </s> <s xml:id="echoid-s10187" xml:space="preserve"><lb/>adhuciſtorum quadrupla .</s> <s xml:id="echoid-s10188" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10189" xml:space="preserve">omnia quadrata, FC, ad omnia qua-<lb/> <anchor type="note" xlink:label="note-0414-03a" xlink:href="note-0414-03"/> drata figuræ, FADCVE, erunt vt quadratum, CL, ad quadratum, <lb/>OV, vna cum {1/3}. </s> <s xml:id="echoid-s10190" xml:space="preserve">quadrati, LS, vel vt iſſorum quadrup a, ſedicet. <lb/></s> <s xml:id="echoid-s10191" xml:space="preserve">vt quadratum, CD, ad quadratum, AV, vna cum {1/3}. </s> <s xml:id="echoid-s10192" xml:space="preserve">quadrati, H <lb/>S, vel, NY, quod oſtendere opus erat.</s> <s xml:id="echoid-s10193" xml:space="preserve"/> </p> <div xml:id="echoid-div936" type="float" level="2" n="2"> <figure xlink:label="fig-0414-01" xlink:href="fig-0414-01a"> <image file="0414-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0414-01"/> </figure> <note position="left" xlink:label="note-0414-01" xlink:href="note-0414-01a" xml:space="preserve">D. Cor. <lb/>23. l. 2.</note> <note position="left" xlink:label="note-0414-02" xlink:href="note-0414-02a" xml:space="preserve">10. l. 2.</note> <note position="left" xlink:label="note-0414-03" xlink:href="note-0414-03a" xml:space="preserve">9. l. 2.</note> </div> </div> <div xml:id="echoid-div938" type="section" level="1" n="559"> <head xml:id="echoid-head583" style="it" xml:space="preserve">A@@ter ſupradictam rationem explicare.</head> <p> <s xml:id="echoid-s10194" xml:space="preserve">Dico omnia quadrata, FC, ad omnia quadrata figuræ, FADC <lb/>VE, eſſe vt quadratum, RZ, compoſitæ ex tranſuerſo latere, AV, <lb/>& </s> <s xml:id="echoid-s10195" xml:space="preserve">axibus, vel diametris oppoſitarum hyperbolarum, FAD, EVC, <lb/>ad quadratum, AV, vna cum rectangulo ſub, AZ, & </s> <s xml:id="echoid-s10196" xml:space="preserve">ſexquitertia, <lb/>ZV, Nam oſtenſum eſt eadem eſſe, vt quadratum, CL, vel, ZO, <lb/>ad quadratum, OV, cum {1/3}. </s> <s xml:id="echoid-s10197" xml:space="preserve">quadrat, LS, & </s> <s xml:id="echoid-s10198" xml:space="preserve">quoniam rectangu-<lb/>lum, CSD, cum quadrato, SL, eſt æquale quadrato, CL, vel, ZO, <lb/> <anchor type="note" xlink:label="note-0414-04a" xlink:href="note-0414-04"/> rectangulu n auten, CSD, eſſ æquale quadrato, OV, ideò qua-<lb/>dratum, LS, erit æquale reliquo quadrati, ZO, dempto quadra-<lb/>to, DV, .</s> <s xml:id="echoid-s10199" xml:space="preserve">i. </s> <s xml:id="echoid-s10200" xml:space="preserve">erit ælquale rectangulo ſub, OVZ, bis, vna cum qua-<lb/>drato, VZ, .</s> <s xml:id="echoid-s10201" xml:space="preserve">i. </s> <s xml:id="echoid-s10202" xml:space="preserve">rectangulo ſub, AVZ, ſemel cum quadrato, VZ, .</s> <s xml:id="echoid-s10203" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s10204" xml:space="preserve">rectangulo ſub, AZV, & </s> <s xml:id="echoid-s10205" xml:space="preserve">ideò {1/3}. </s> <s xml:id="echoid-s10206" xml:space="preserve">quadrati, LS, erit æquale {1/3}. </s> <s xml:id="echoid-s10207" xml:space="preserve">re- <pb o="395" file="0415" n="415" rhead="LIBER V."/> ctanguli fub, AZ, ZV, .</s> <s xml:id="echoid-s10208" xml:space="preserve">i. </s> <s xml:id="echoid-s10209" xml:space="preserve">erit æquale rectangulo ſub, AZ, & </s> <s xml:id="echoid-s10210" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10211" xml:space="preserve">Z <lb/>V, & </s> <s xml:id="echoid-s10212" xml:space="preserve">ideò omnia quadrata, FC, ad omnia quadrata figuræ, FAD <lb/>CVE, erunt vt quadratum, ZO, ad quadratum, OV, cum rectan. <lb/></s> <s xml:id="echoid-s10213" xml:space="preserve">gulo ſub, AZ, & </s> <s xml:id="echoid-s10214" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10215" xml:space="preserve">ZV, .</s> <s xml:id="echoid-s10216" xml:space="preserve">i. </s> <s xml:id="echoid-s10217" xml:space="preserve">vt horum quadrupla, nempè, vt qua-<lb/>dratum, RZ, ad quadratum, AV, cum rectangulo ſub, AZ, & </s> <s xml:id="echoid-s10218" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10219" xml:space="preserve"><lb/>ZV, .</s> <s xml:id="echoid-s10220" xml:space="preserve">i. </s> <s xml:id="echoid-s10221" xml:space="preserve">ſub, AZ, & </s> <s xml:id="echoid-s10222" xml:space="preserve">ſexquitertia, ZV, quæ ratio ſic proponebatur <lb/>explicanda, quæque, vt libet, retineri poterit.</s> <s xml:id="echoid-s10223" xml:space="preserve"/> </p> <div xml:id="echoid-div938" type="float" level="2" n="1"> <note position="left" xlink:label="note-0414-04" xlink:href="note-0414-04a" xml:space="preserve">5 Sec. El. <lb/>Elicitur <lb/>exl@. fe@. <lb/>Con. au-<lb/>xilto 16. <lb/>pri. Con.</note> </div> </div> <div xml:id="echoid-div940" type="section" level="1" n="560"> <head xml:id="echoid-head584" xml:space="preserve">COROLLARIVM:</head> <p style="it"> <s xml:id="echoid-s10224" xml:space="preserve">_H_Inc patet quadratum dimidiæ eius, quælateri tranſuerſo op-<lb/>poſitarum ſectionum æquidiſtanter ducitur, ſubtenditurq; </s> <s xml:id="echoid-s10225" xml:space="preserve">an-<lb/>gulo, qui deinceps eſt angulo ſub aſymptotis comprebenſo, ſectiones <lb/>continenti æquale eſſe rectangulo ſub compoſita ex latere tranſuerſo, <lb/>& </s> <s xml:id="echoid-s10226" xml:space="preserve">axi, vel diametro alterutrius conſtitutarum hyperbolarum per or-<lb/>d natim applicatas à punctis, quibus dicta ſubtenſa incidit, producta, <lb/>ipſis oppoſitis ſectionibus, & </s> <s xml:id="echoid-s10227" xml:space="preserve">ſub eodem axi, vel diametro, quod pa-<lb/>tet, veluti oſtenſum eſt quadratum, SL, æquari rectangulo, .</s> <s xml:id="echoid-s10228" xml:space="preserve">AZV.</s> <s xml:id="echoid-s10229" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div941" type="section" level="1" n="561"> <head xml:id="echoid-head585" xml:space="preserve">THEOREMA XXI. PROPOS. XXII.</head> <p> <s xml:id="echoid-s10230" xml:space="preserve">SI per vertices oppoſitarum ſectionum rectæ lineæ or-<lb/>dinatim ad eorum axim, vel diametrum applicentur <lb/>vſque ad aſymptotos productæ, quarum extrema ad eaſdé <lb/>partes ſumpta iungantur rectis lineis, iungenteſq; </s> <s xml:id="echoid-s10231" xml:space="preserve">vſq; </s> <s xml:id="echoid-s10232" xml:space="preserve">ad <lb/>oppoſitas ſectiones producantur, erunt iſtæ parallelogrã, <lb/>mi oppoſita latera, quod parallelogrammum ſi complea-<lb/>tur, regula exiſtente latere tranſuerſo: </s> <s xml:id="echoid-s10233" xml:space="preserve">Omnia quadrata <lb/>conſtituti parallelogrammi erunt ſexquialtera omnium <lb/>quadratorum figuræ comprehenſæ ſub lateribus dictipa-<lb/>rallelogrammi lateri tranſuerſo æquidiſtantibus, & </s> <s xml:id="echoid-s10234" xml:space="preserve">ſub <lb/>oppoſitarum ſectionum portionibus inter eadem latera cõ-<lb/>cluſis: </s> <s xml:id="echoid-s10235" xml:space="preserve">Omnia verò quadrata dictæ figuræ erunt quadru-<lb/>pla omnium quadratorum triangulorum, quiſub aſympto-<lb/>tis & </s> <s xml:id="echoid-s10236" xml:space="preserve">ijſdem incluſis portionibus laterum parallelogram-<lb/>mi, tranſuerſo lateri æquidiſtantium continentur.</s> <s xml:id="echoid-s10237" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10238" xml:space="preserve">Sint oppofitæ fectiones, FAD, EVC, quarum latus tranſuer- <pb o="396" file="0416" n="416" rhead="GEOMETRIÆ"/> ſum ſit, AV, centrum, O, aſymptoti indefinitè producti, NP, HY, <lb/>per puncta autem, AV, ſint ductæ ordinatim, NH, YP, productæ <lb/>vſq; </s> <s xml:id="echoid-s10239" xml:space="preserve">ad aſymptotos, in punctis, N, H, Y, P, ipſis incidentes, quæ <lb/>fectiones tangent, deinde iunctis, NY, HP, producantur ipſæ tun-<lb/> <anchor type="figure" xlink:label="fig-0416-01a" xlink:href="fig-0416-01"/> gentes vſq; </s> <s xml:id="echoid-s10240" xml:space="preserve">ad ſectiones illis in punctis, F, E, C, D, <lb/>occurrentes, iunganturque, FD, EC, & </s> <s xml:id="echoid-s10241" xml:space="preserve">per, O, ad <lb/> <anchor type="note" xlink:label="note-0416-01a" xlink:href="note-0416-01"/> ipfam, AV, ordinatim applicetur, XL, incidens, F <lb/>E, in, X, &</s> <s xml:id="echoid-s10242" xml:space="preserve">, DC, in, L, quæ erit ſecunda diameter, <lb/>& </s> <s xml:id="echoid-s10243" xml:space="preserve">producatur, AV, indeſinitè incidens ipſis, FD, <lb/>EC, in punctis, R, Z, erit ergo, FC, parailelogrã-<lb/> <anchor type="note" xlink:label="note-0416-02a" xlink:href="note-0416-02"/> mum, nam rectangulum, YFN, .</s> <s xml:id="echoid-s10244" xml:space="preserve">i. </s> <s xml:id="echoid-s10245" xml:space="preserve">rectangulum, E <lb/>NF, eſt æquale quadrato, AO, ideſt rectangulo, C <lb/>HD, item quadratum; </s> <s xml:id="echoid-s10246" xml:space="preserve">NX, eſt æquale quadrato, <lb/>HL, & </s> <s xml:id="echoid-s10247" xml:space="preserve">ideo rectangulum, ENF, cum quadrato, NX, .</s> <s xml:id="echoid-s10248" xml:space="preserve">i. </s> <s xml:id="echoid-s10249" xml:space="preserve">quadra-<lb/> <anchor type="note" xlink:label="note-0416-03a" xlink:href="note-0416-03"/> tum, XF, erit æquale rectangulo, CHD, cum quadrato, LH, .</s> <s xml:id="echoid-s10250" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s10251" xml:space="preserve">quadrato, LD, & </s> <s xml:id="echoid-s10252" xml:space="preserve">ideò, XF, erit æqualis ipſi, LD, & </s> <s xml:id="echoid-s10253" xml:space="preserve">eius dupla, F <lb/>E, æqualis duplæ, CD, & </s> <s xml:id="echoid-s10254" xml:space="preserve">eidem parallela, vnde, FD, erit, EC, <lb/>parallela, & </s> <s xml:id="echoid-s10255" xml:space="preserve">ambæ ordinatim ad axim, vel diametrum, RZ, or-<lb/>dinatim applicatæ, & </s> <s xml:id="echoid-s10256" xml:space="preserve">ideò in, R, Z, bifatiam ſectæ, &</s> <s xml:id="echoid-s10257" xml:space="preserve">, FC, erit <lb/>parallelogrammum, ſitregula latus tranſuerſum, AV, Dico nũc <lb/>omnia quadrata parallelogrammi, FC, eſſe ſexquialtera omnium <lb/> <anchor type="note" xlink:label="note-0416-04a" xlink:href="note-0416-04"/> quadratorum figuræ, FADCVE; </s> <s xml:id="echoid-s10258" xml:space="preserve">& </s> <s xml:id="echoid-s10259" xml:space="preserve">hęc eſſe quadrupla omnium <lb/>quadratorum triangulorum, NOY, HOP; </s> <s xml:id="echoid-s10260" xml:space="preserve">nam omnia quadrata, <lb/>FC, ad omnia quadrata figuræ, FADCVE, oſtenſa ſunt eſſe, vt <lb/>quadratum, DC, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10261" xml:space="preserve">quadrati, HP, eſt <lb/>autem quadratum, HP, æquale quadrato, AV, & </s> <s xml:id="echoid-s10262" xml:space="preserve">ideò ſunt, vt <lb/>quadratum, DC, ad quadratum, HP, cum {1/3}. </s> <s xml:id="echoid-s10263" xml:space="preserve">quadrati, HP, vel vt <lb/>quadratum, RZ, ad quadratum, AV, cum, {1/3}. </s> <s xml:id="echoid-s10264" xml:space="preserve">quadrati, AV. </s> <s xml:id="echoid-s10265" xml:space="preserve">Pro-<lb/>ducantur nunc aſymptoti, NP, HY, verſus, EC, cui productæ in-<lb/>cidant in S, I, eſt ergo rectangulum, SEI, æquale quadrato, YV, <lb/> <anchor type="note" xlink:label="note-0416-05a" xlink:href="note-0416-05"/> .</s> <s xml:id="echoid-s10266" xml:space="preserve">i. </s> <s xml:id="echoid-s10267" xml:space="preserve">quadrato, EZ, & </s> <s xml:id="echoid-s10268" xml:space="preserve">ideò rectangulum, SEI, cum quadrato, EZ, <lb/>duplum eſt quadrati, EZ, vel quadrati, YV, eſt autem rectangulũ, <lb/>SEI, cum quadrato, EZ, ęquale quadrato, SZ, & </s> <s xml:id="echoid-s10269" xml:space="preserve">ideò quadratũ, <lb/>SZ, duplum eſt quadrati, YV, eſt autem, vt quadratum, SZ, ad <lb/>quadratum, YV, ita quadratum, ZO, ad quadratum, OV, ergo <lb/>quadratum, ZO, erit duplum quadrati, OV, vel eorum quadru pla <lb/>.</s> <s xml:id="echoid-s10270" xml:space="preserve">f. </s> <s xml:id="echoid-s10271" xml:space="preserve">quadratum, ZR, duplum quadrati, AV; </s> <s xml:id="echoid-s10272" xml:space="preserve">quia verò dictum eſt <lb/>omnia quadrata, FC, ad omnia quadrata figuræ, FADCVE, eſſe <lb/>vt quadratum, RZ, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10273" xml:space="preserve">quadrati, AV, .</s> <s xml:id="echoid-s10274" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s10275" xml:space="preserve">vt quadratum, RZ, ad {1/3}. </s> <s xml:id="echoid-s10276" xml:space="preserve">quadrati, AV, & </s> <s xml:id="echoid-s10277" xml:space="preserve">eſt quadratum, RZ, <lb/>duplum quadrati, AV, ideò quadratum, RZ, erit {6/3}. </s> <s xml:id="echoid-s10278" xml:space="preserve">quadrati, AV, <pb o="397" file="0417" n="417" rhead="LIBER V."/> ergo quadratum, RZ, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10279" xml:space="preserve">quadrati, AV, <lb/>erit vt {6/3}. </s> <s xml:id="echoid-s10280" xml:space="preserve">ad {4/3}. </s> <s xml:id="echoid-s10281" xml:space="preserve">.</s> <s xml:id="echoid-s10282" xml:space="preserve">i. </s> <s xml:id="echoid-s10283" xml:space="preserve">vt 6. </s> <s xml:id="echoid-s10284" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s10285" xml:space="preserve">.</s> <s xml:id="echoid-s10286" xml:space="preserve">i. </s> <s xml:id="echoid-s10287" xml:space="preserve">in ratione ſexquialtera, ergo omnia <lb/>quadriata, FC, ad omnia quadrata figurę, FADCVE, erunt in ra-<lb/>tione ſexquialtera.</s> <s xml:id="echoid-s10288" xml:space="preserve"/> </p> <div xml:id="echoid-div941" type="float" level="2" n="1"> <figure xlink:label="fig-0416-01" xlink:href="fig-0416-01a"> <image file="0416-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0416-01"/> </figure> <note position="left" xlink:label="note-0416-01" xlink:href="note-0416-01a" xml:space="preserve">37. Secun. <lb/>Con.</note> <note position="left" xlink:label="note-0416-02" xlink:href="note-0416-02a" xml:space="preserve">3. 1. Secun. <lb/>Con.</note> <note position="left" xlink:label="note-0416-03" xlink:href="note-0416-03a" xml:space="preserve">4. Sec. Hle,</note> <note position="left" xlink:label="note-0416-04" xlink:href="note-0416-04a" xml:space="preserve">31, huius.</note> <note position="left" xlink:label="note-0416-05" xlink:href="note-0416-05a" xml:space="preserve">11. Secun. <lb/>Con.</note> </div> <p> <s xml:id="echoid-s10289" xml:space="preserve">Igitur conuertendo omnia quadrata figurę, FADCVE, ad om-<lb/>nia quadrata, FC, eruntin ratione ſubiexquialtera, .</s> <s xml:id="echoid-s10290" xml:space="preserve">i. </s> <s xml:id="echoid-s10291" xml:space="preserve">vt 4. </s> <s xml:id="echoid-s10292" xml:space="preserve">ad 6. <lb/></s> <s xml:id="echoid-s10293" xml:space="preserve">ſunt autem omnia quadrata, FC, ad omnia quadrata, NP, vt qua. </s> <s xml:id="echoid-s10294" xml:space="preserve"><lb/>dratum, ZR, ad quadratum, AV, ideſt dupla .</s> <s xml:id="echoid-s10295" xml:space="preserve">i. </s> <s xml:id="echoid-s10296" xml:space="preserve">vt 6. </s> <s xml:id="echoid-s10297" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s10298" xml:space="preserve">& </s> <s xml:id="echoid-s10299" xml:space="preserve">om-<lb/>nia quadrata, NP, ſunt tripla omnium quadratorum triangulo-<lb/>rum, NYO, OHP, .</s> <s xml:id="echoid-s10300" xml:space="preserve">i. </s> <s xml:id="echoid-s10301" xml:space="preserve">ſunt ad illa, vt 3. </s> <s xml:id="echoid-s10302" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s10303" xml:space="preserve">ergo ex ęquali, omnia <lb/>quadrata figurę, FADCVE, ad omnia quad. </s> <s xml:id="echoid-s10304" xml:space="preserve">triangulorum, NY <lb/>O, OHP, erunt vt 4. </s> <s xml:id="echoid-s10305" xml:space="preserve">ad I. </s> <s xml:id="echoid-s10306" xml:space="preserve">.</s> <s xml:id="echoid-s10307" xml:space="preserve">1. </s> <s xml:id="echoid-s10308" xml:space="preserve">eorum quadrupla, quę erant oſten-<lb/>denda.</s> <s xml:id="echoid-s10309" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div943" type="section" level="1" n="562"> <head xml:id="echoid-head586" xml:space="preserve">COROLLARIVM. I.</head> <p style="it"> <s xml:id="echoid-s10310" xml:space="preserve">_Q_V oniam Verò in Propoſ. </s> <s xml:id="echoid-s10311" xml:space="preserve">antec. </s> <s xml:id="echoid-s10312" xml:space="preserve">oſtenſum eſt, ac in eius figura, <lb/>omnia quadrata, FC, ad omnia quadrata figuræ, F ADCVE, eſſe <lb/>Vt quadratum, DC, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10313" xml:space="preserve">quadrati, HS, & </s> <s xml:id="echoid-s10314" xml:space="preserve"><lb/>quia omnia quadrata, ZL, ad omnia quadratatrianguli, OSL, vel eo-<lb/>rum quadrupla .</s> <s xml:id="echoid-s10315" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10316" xml:space="preserve">omnia quadrata, RC, ad omnia quadrata trianguli, <lb/>SOH, oſtenſa ſunt eſſe, Vt quadratum, CL, ad {1/3}. </s> <s xml:id="echoid-s10317" xml:space="preserve">quadrati, LS, vel Vt <lb/>quadratum, CD, ad @. </s> <s xml:id="echoid-s10318" xml:space="preserve">quadrati, HS, & </s> <s xml:id="echoid-s10319" xml:space="preserve">ſic eorum dupla. </s> <s xml:id="echoid-s10320" xml:space="preserve">.</s> <s xml:id="echoid-s10321" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10322" xml:space="preserve">Vt quadra. <lb/></s> <s xml:id="echoid-s10323" xml:space="preserve">tum, LC, ad {1/3}. </s> <s xml:id="echoid-s10324" xml:space="preserve">quadrati, SH ita omnia quadrata, FC, ad omnia qua-<lb/>drata triangulorum, NOR HOS, erant autem omnia quadrata, FC, ad <lb/>omnia quadrata figuræ, FADCVE, vt quadratum, DC, ad quadratum, <lb/>AV, cum {1/3}. </s> <s xml:id="echoid-s10325" xml:space="preserve">quadrati, HS, ergo omnia quadrata, FC, ad reliquum, <lb/>demptis omnibus quadratis triangulorum, NOR, HOS, abomnibus <lb/>quadratis figuræ, FADCVE, erunt, vt quadratum, DC, ad quadratum, <lb/>AV; </s> <s xml:id="echoid-s10326" xml:space="preserve">& </s> <s xml:id="echoid-s10327" xml:space="preserve">ideò in præſenti Propoſ. </s> <s xml:id="echoid-s10328" xml:space="preserve">omnia quadrata, FC, ad omnia qua-<lb/>dratà figuræ FAD, CVE, demptis ab ijſdem omnibus quadratis triã-<lb/>gulorum NOR, HOP, erunt vt quadratum, RZ, ad quadratum, AV, <lb/>ideſt dupla.</s> <s xml:id="echoid-s10329" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div944" type="section" level="1" n="563"> <head xml:id="echoid-head587" xml:space="preserve">COROLLARIVM. II.</head> <head xml:id="echoid-head588" xml:space="preserve">Ex præcedenti deductum.</head> <p style="it"> <s xml:id="echoid-s10330" xml:space="preserve">_P_Atet etiam nos poſsè inuenire parallelogrammum circumſcri-<lb/>ptum ſectionibus oppoſitis, veluti, FC, ideſt ita quod eius duo <lb/>oppoſita latera ſint baſes oppoſitarum hyperbolarum, & </s> <s xml:id="echoid-s10331" xml:space="preserve">reliqua duo <pb o="398" file="0418" n="418" rhead="GEOMETRIÆ"/> lateri tranſ uerſo parallela, quod ſumabur pro regula, ita inquám, vt <lb/>omnia quadrata deſeripti parallelogram ni ad omnia quadrata figuræ <lb/>dictis lateribus, quæ tranſuerſo lateri æ quidiſtant, & </s> <s xml:id="echoid-s10332" xml:space="preserve">ab ijſdem ſe-<lb/>ctionum oppoſitarum in cluſis portionibus compræbenſæ, demptis om-<lb/>bus quadratis triangulorum ſub aſymptotis, & </s> <s xml:id="echoid-s10333" xml:space="preserve">ab ijs incluſis portio-<lb/>nibus laterum, parallelogrammi tranſuerſo lateri æquidiſta ntium, <lb/>babeant datam rationem, dummodo ea ſit maioris inæqualitatis: </s> <s xml:id="echoid-s10334" xml:space="preserve">Sit <lb/>in figura Propos. </s> <s xml:id="echoid-s10335" xml:space="preserve">21. </s> <s xml:id="echoid-s10336" xml:space="preserve">data ratio maioris inæ quadlitatis, quam babet, <lb/>KB, ad, GM, & </s> <s xml:id="echoid-s10337" xml:space="preserve">ſupponatur ductam ſuiſſe, FE, æqudiſtantem lateri <lb/>tranſuerſo, AV, ita vt quadratum, FE, ad quadratum, Av, ſit vt, K <lb/>B, ad, GM, & </s> <s xml:id="echoid-s10338" xml:space="preserve">conſtructam fuiſſe figuram, velutibi factum eſt, patet <lb/>igitur, quia omnia quadrata, FC, ad omnia quadrata figurę, FADCV <lb/>E, ſunt vt quadratum, FE, ad quadratum, AV, ex Coroll, antec. </s> <s xml:id="echoid-s10339" xml:space="preserve">dem-<lb/>ptis tamen ab omnibus quadratis dictæ figuræ, omnibus quadratis <lb/>triangulorum, NOR, HOS, quod ideò ad eadom erunt in ratione da-<lb/>ta .</s> <s xml:id="echoid-s10340" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10341" xml:space="preserve">m ea quam babet, KB, ad, GM.</s> <s xml:id="echoid-s10342" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div945" type="section" level="1" n="564"> <head xml:id="echoid-head589" xml:space="preserve">THEOREMA XXII. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s10343" xml:space="preserve">SI duo parallelogramma vtcunq; </s> <s xml:id="echoid-s10344" xml:space="preserve">fectionibus oppoſitis <lb/>circumſcripta fuerint modo ſolito, habentia ſcilicet <lb/>duo oppoſita latera, quæ ſint oppoſitarum hyperbolarum <lb/>baſes, & </s> <s xml:id="echoid-s10345" xml:space="preserve">reliqua duo lateri rianſuerſo æ quidiſtantia, regu-<lb/>la vna dictarum baſium: </s> <s xml:id="echoid-s10346" xml:space="preserve">Omnia quadrata vnius paralle-<lb/>logrammi, demptis omnibus quadratis oppoſitarum hy-<lb/>perbolarum communes cum eo baſes habentium, ad om-<lb/>nia quadrata alterius parallelogrammi, demptis omnibus <lb/>quadratis oppoſitarum hyperbolarum communes cum eo <lb/>baſes habentium, erunt vt parallelepipedum ſub altitudi-<lb/>ne axi, vel diametro vnius hyperbolarum, cuius eſt com-<lb/>munis baſis cum parallelogrammo primò dicto, baſirectã-<lb/>gulo ſub dimidia tranſuerſi lateris, & </s> <s xml:id="echoid-s10347" xml:space="preserve">ſub compoſita ex ea-<lb/>dem dimidia, & </s> <s xml:id="echoid-s10348" xml:space="preserve">axi, vel diametro dictæ hyperbolæ, vna <lb/>cum. </s> <s xml:id="echoid-s10349" xml:space="preserve">quadrati eiuſdem axis, vel diametri, ad parallele-<lb/>pipedum ſub altitudine axi, vel diametro hyperbolæ, cui-<lb/>us eſt communis baſis cum parallelogrammo ſecundò di-<lb/>cto, baſirectangulo ſub dimidia tranſuerſi lateris, & </s> <s xml:id="echoid-s10350" xml:space="preserve">ſub <lb/>compoſita ex eadem dimidia, & </s> <s xml:id="echoid-s10351" xml:space="preserve">axi, vel diametro hyper- <pb o="399" file="0419" n="419" rhead="LIBER V."/> bolæ poſtremò dictæ, vna cum {1/3}. </s> <s xml:id="echoid-s10352" xml:space="preserve">quadrati eiuſdem axis, <lb/>vel diametri.</s> <s xml:id="echoid-s10353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10354" xml:space="preserve">Sint oppoſitis ſectionibus, FAD, EVC, quorum latus tranſuer-<lb/>ſum, AV, centrum, O, circumſcripta parallelogramma vtcunque' <lb/> <anchor type="figure" xlink:label="fig-0419-01a" xlink:href="fig-0419-01"/> FC, TN, quorum duo oppoſita latera <lb/>ſint baſes oppoſitarum hyperbolarum, F <lb/>D, EC, nempè hyperbolarum, FAD, EV <lb/>C, &</s> <s xml:id="echoid-s10355" xml:space="preserve">, TY, MN, hyperbolarum, TAY, M <lb/>VN, nempè ſint ad axim, vel diametrum <lb/>tranſuerſam, AV, ordinatim applicata, & </s> <s xml:id="echoid-s10356" xml:space="preserve"><lb/>reliqua latera, ad ſecundum axim, vel dia-<lb/>metrum, quæ ſit, XL, pariter ordinatim <lb/>applicata, regula autem vna dictarum ba-<lb/>ſinum, vt, EC. </s> <s xml:id="echoid-s10357" xml:space="preserve">Dico ergo omnia quadra. <lb/></s> <s xml:id="echoid-s10358" xml:space="preserve">ta, FC, demptis omnibus quadratis oppo. </s> <s xml:id="echoid-s10359" xml:space="preserve"><lb/>ſitarum hyperbolarum, FAD, EVC, ad <lb/>omnia quadrata, FN, demptis omnibus <lb/>quadratis oppotitarum hyperbolarum, T <lb/>AY, MVN, eſſe vt parallelepipedum ſub <lb/>alcicudine, ZV, baſi rectangulo VOZ, cũ <lb/>{1/3}. </s> <s xml:id="echoid-s10360" xml:space="preserve">quadrati, ZV, ad parallelpipedu ſub altitudine, SV, baſi re-<lb/>ctangulo, VOS, cum {1/3}. </s> <s xml:id="echoid-s10361" xml:space="preserve">quadrati, SV: </s> <s xml:id="echoid-s10362" xml:space="preserve">Omnia .</s> <s xml:id="echoid-s10363" xml:space="preserve">o. </s> <s xml:id="echoid-s10364" xml:space="preserve">quadrata, FC, <lb/>dempt somnibus quadratis oppoſitarum hyperbolarum, FAD, E <lb/>VC, ad omnia quadrata, TN, demp@ sommbus quadratis oppo-<lb/>ſitarum hyperbolarum, TAY, MVN, habentrationem compo-<lb/>fitam ex ea, quain habent omnia quadrata, FC, demptis omnibus <lb/>quadratis oppoſitarum hyperbolarum, FAD, EVC, ad omnia <lb/>quadrata, FC, & </s> <s xml:id="echoid-s10365" xml:space="preserve">ex ratione horum ad omnia quadrata, TN, & </s> <s xml:id="echoid-s10366" xml:space="preserve"><lb/>ex ratione iſtorum ad omnia eorundem quadrata, demptis omni-<lb/>bus quadratis oppoſitarum hyperbolarum, TAY, MVN; </s> <s xml:id="echoid-s10367" xml:space="preserve">verum <lb/>omnia quadrata, FC, demptis omnibus quadratis oppoſitarum <lb/>hyperbolarum, FAD, EVC, ad omnia quadrata, FC, ſunt vt re-<lb/>ctangulum, AOZ, b@s, cum {2/3}. </s> <s xml:id="echoid-s10368" xml:space="preserve">quadrati, ZV, ad rectangulum, A <lb/> <anchor type="note" xlink:label="note-0419-01a" xlink:href="note-0419-01"/> ZO: </s> <s xml:id="echoid-s10369" xml:space="preserve">Omnia item quadrata, FC, ad omnia quadrata, TN, habẽt <lb/>rationem compoſitam ex ratione, FE, ad, TM, vel, EX, ad, MH, <lb/>ſiue, ZO, ad, OS, & </s> <s xml:id="echoid-s10370" xml:space="preserve">ex ratione quadrati, EC, ad quadratum, MN, <lb/> <anchor type="note" xlink:label="note-0419-02a" xlink:href="note-0419-02"/> ſiue rectanguli, AZV, ad rectangulum, ASV: </s> <s xml:id="echoid-s10371" xml:space="preserve">@andem omnia <lb/>quadrata, TN ad eadem demptis omnibus quadratis oppoſitarũ <lb/>hyperbolarum, TAY, MVN, ſunt vt rectangulum, ASO, ad re-<lb/>ctangulum, AOS, bis, cum {2/3}. </s> <s xml:id="echoid-s10372" xml:space="preserve">quadrati, SV, habemus ergo has <lb/> <anchor type="note" xlink:label="note-0419-03a" xlink:href="note-0419-03"/> <pb o="400" file="0420" n="420" rhead="GEOMETRIÆ"/> quatuor rationes primò dictam rationem componentes .</s> <s xml:id="echoid-s10373" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10374" xml:space="preserve">ratione <lb/>rectanguli, AOZ, bis, cum {2/3}. </s> <s xml:id="echoid-s10375" xml:space="preserve">quadrati, VZ, ad rectangulum, AZ <lb/>O, rationem, ZO, ad OS, rationem rectanguli, AZV, ad rectan-<lb/>gulum, ASV, & </s> <s xml:id="echoid-s10376" xml:space="preserve">tandem rationem rectanguli, ASO, ad rectangu-<lb/>lum, AOS, bis cum {2/3}. </s> <s xml:id="echoid-s10377" xml:space="preserve">quadrati, SV, harum autem rationum illa, <lb/> <anchor type="figure" xlink:label="fig-0420-01a" xlink:href="fig-0420-01"/> quam habet rectangulum, AZV, adrectã, <lb/>gulum, ASV, componitur ex ratione, Z <lb/>V, ad, VS, & </s> <s xml:id="echoid-s10378" xml:space="preserve">ex ratione, ZA, ad, AS, ha-<lb/>bemus ergo quinque rationes componen-<lb/>tes rationem primò dictam, ſitigitur pri-<lb/>mo loco diſpoſita ratio, quam habet re-<lb/>ctangulum bis ſub, AOZ, cum {2/3} quadra-<lb/>ti, ZV, ad rectangulum, AZO; </s> <s xml:id="echoid-s10379" xml:space="preserve">ſi rurlus <lb/>aſſumamus ex cæteris quatuor rationibus <lb/>eam, quam habet, ZA, ad, AS, vel (rum <lb/>pta, ZO, communi altitudine) quam ha <lb/>bet rectangulum, AZO, ad rectangulu <lb/>ſub, ZO, AS, quæ habeatur ſecundo roco; <lb/></s> <s xml:id="echoid-s10380" xml:space="preserve">& </s> <s xml:id="echoid-s10381" xml:space="preserve">inſuper ſiex cæteris tribus ratiombus <lb/>ſumamus, quam habet, ZO, ad, OS, vel <lb/>(ſumpta, AS, communi altitudine) quam <lb/>habet rectangulum ſub, ZO, AS, ad rectangulum, ASO, quæ ſit <lb/>poſita tertio loco, & </s> <s xml:id="echoid-s10382" xml:space="preserve">tandem ſiteneamus quartò loco eam, quam <lb/>habet rectangulum, ASO, ad rectangulum ſub, AOS, bis, {2/3}. </s> <s xml:id="echoid-s10383" xml:space="preserve">qua-<lb/>drati, SV, habebimus has quatuor hoc ordine diſpoſitas conſequẽ-<lb/>ter rationes .</s> <s xml:id="echoid-s10384" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10385" xml:space="preserve">rationem rectanguli ſub, AOZ, bis cum {2/3}, quadra-<lb/>ti, ZV, ad rectangulum, AZO, rationem huius ad rectangulum <lb/>ſub, AS, OZ, rationem huius ad rectangulum, ASO, & </s> <s xml:id="echoid-s10386" xml:space="preserve">tandem <lb/>rationem huius ad rectangulum, AOS, bis, vna cum {2/3}. </s> <s xml:id="echoid-s10387" xml:space="preserve">quadrati, <lb/>SV, quæ component rationem primæ ad vitimam .</s> <s xml:id="echoid-s10388" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10389" xml:space="preserve">eam, quam <lb/> <anchor type="note" xlink:label="note-0420-01a" xlink:href="note-0420-01"/> habet rectangulum, AOZ, bis vna cum {2/3}. </s> <s xml:id="echoid-s10390" xml:space="preserve">quadrati, ZV, ad rectã-<lb/>gulum, AOS, bis, vna cum {2/3}. </s> <s xml:id="echoid-s10391" xml:space="preserve">quadrati, SV, vel quam habent ho-<lb/>rum dimidia .</s> <s xml:id="echoid-s10392" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10393" xml:space="preserve">rectangulum, AOZ, cum {1/3}. </s> <s xml:id="echoid-s10394" xml:space="preserve">quadrat V, Z, ad rectã-<lb/>gulum, AOS, cum {1/3}. </s> <s xml:id="echoid-s10395" xml:space="preserve">quadrati, SV, @llas ergo quatuor rationes in <lb/>ſolam hane redegimus; </s> <s xml:id="echoid-s10396" xml:space="preserve">huicſi iungamus eam, quam habet, ZV, <lb/>ad, VS, quæ erat quinta ratio, quæreſidua erat, componetur ex <lb/>dictis quinque rationibus hæcſola .</s> <s xml:id="echoid-s10397" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10398" xml:space="preserve">quam habet parallelepipedũ <lb/>ſub altitudine, ZV, baſi rectangulo, AOZ, vel, VOZ, cum {1/3}. </s> <s xml:id="echoid-s10399" xml:space="preserve">qua-<lb/>drati, ZV, ad parallelepipedum ſub altitudine, SV, baſi rectangu-<lb/>lo, AOS, vel, VOS, cum {1/3}. </s> <s xml:id="echoid-s10400" xml:space="preserve">quadrati, SV, quę erit ea, quam habe-<lb/>bunt omnia quadrata, FC, demptis omnibus quadratis oppoſita <pb o="401" file="0421" n="421" rhead="LIBER V."/> rum hyperbol arum, FAD, EVC, ad omnia quadrata, TN, dem-<lb/>ptis omnibus quadratis oppoſitarum hyperbolarum, TAY, MVN, <lb/>quod, &</s> <s xml:id="echoid-s10401" xml:space="preserve">c.</s> <s xml:id="echoid-s10402" xml:space="preserve"/> </p> <div xml:id="echoid-div945" type="float" level="2" n="1"> <figure xlink:label="fig-0419-01" xlink:href="fig-0419-01a"> <image file="0419-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0419-01"/> </figure> <note position="right" xlink:label="note-0419-01" xlink:href="note-0419-01a" xml:space="preserve">20. huius,</note> <note position="right" xlink:label="note-0419-02" xlink:href="note-0419-02a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> <note position="right" xlink:label="note-0419-03" xlink:href="note-0419-03a" xml:space="preserve">20: huius.</note> <figure xlink:label="fig-0420-01" xlink:href="fig-0420-01a"> <image file="0420-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0420-01"/> </figure> <note position="left" xlink:label="note-0420-01" xlink:href="note-0420-01a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> </div> </div> <div xml:id="echoid-div947" type="section" level="1" n="565"> <head xml:id="echoid-head590" xml:space="preserve">THEOREMA XXIII. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s10403" xml:space="preserve">IN eadem antecedentis figura, regula ſumpta, DC, oſtẽ-<lb/>demus omnia quad. </s> <s xml:id="echoid-s10404" xml:space="preserve">fig. </s> <s xml:id="echoid-s10405" xml:space="preserve">FADCVE, ad omnia quadra-<lb/>ta figuræ, TAYNVM, eſſe vt paralle lepipedum ſub, XL, & </s> <s xml:id="echoid-s10406" xml:space="preserve"><lb/>quadrato, RZ, cum duplo quadrati, AV, ad parallelepipe-<lb/>dum ſub, HG, & </s> <s xml:id="echoid-s10407" xml:space="preserve">quadrato, BS, cum duplo quadrati, AV.</s> <s xml:id="echoid-s10408" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10409" xml:space="preserve">Omnia namque quadrata figuræ, FADCVE, ad omnia quadra-<lb/>ta figuræ, TAYNVM, habent rationem compoſitam ex ea, quam <lb/>habent omnia quadrata figuræ, FADCVE, ad omnia quadrata, <lb/>FC, .</s> <s xml:id="echoid-s10410" xml:space="preserve">i. </s> <s xml:id="echoid-s10411" xml:space="preserve">ex ratione quadrati, AV, cum {1/3}. </s> <s xml:id="echoid-s10412" xml:space="preserve">quadrati, KI, (quæ ſit <lb/>portio, DC, capta inter aſymptotos, qui ſint, PI, KQ, ducti per, <lb/> <anchor type="note" xlink:label="note-0421-01a" xlink:href="note-0421-01"/> O, ſecantes, YN, in, &</s> <s xml:id="echoid-s10413" xml:space="preserve">, ℟, FE, in, P, Q, &</s> <s xml:id="echoid-s10414" xml:space="preserve">, TM, in, Ω, Π) ad <lb/>quadratum, DC, & </s> <s xml:id="echoid-s10415" xml:space="preserve">ex ratione omnium quadratorum, FC, ad om-<lb/>nia quadrata, TN, quæ eſt compoſita ex ea, quam habet quadra-<lb/>tum, DC, ad quadratum, YN, & </s> <s xml:id="echoid-s10416" xml:space="preserve">ex ea, quam habet, EC, ad, MN, <lb/> <anchor type="note" xlink:label="note-0421-02a" xlink:href="note-0421-02"/> & </s> <s xml:id="echoid-s10417" xml:space="preserve">tandem ex ratione omnium quadratorum, TN, ad omnia qua-<lb/>drata figuræ, TAYNVM, .</s> <s xml:id="echoid-s10418" xml:space="preserve">i. </s> <s xml:id="echoid-s10419" xml:space="preserve">ex ratione quadrati, YN, ad quadra-<lb/>tum, AV, cum {1/3}. </s> <s xml:id="echoid-s10420" xml:space="preserve">quadrati, & </s> <s xml:id="echoid-s10421" xml:space="preserve">℟, porrò ex his rationibus compo-<lb/>nentibus ea, quam habet quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10422" xml:space="preserve">quadrati, KI, ad <lb/> <anchor type="note" xlink:label="note-0421-03a" xlink:href="note-0421-03"/> quadratum, DC, item quadratum, DC, ad quadratum, YN, & </s> <s xml:id="echoid-s10423" xml:space="preserve"><lb/>quadratum, YN, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10424" xml:space="preserve">quadrati, & </s> <s xml:id="echoid-s10425" xml:space="preserve">℟, com-<lb/>ponunt rationem quadrati, AV, cum {1/3}. </s> <s xml:id="echoid-s10426" xml:space="preserve">quadrati, KI, ad quadra-<lb/>tum, AV, cum {1/3}. </s> <s xml:id="echoid-s10427" xml:space="preserve">quadrati, & </s> <s xml:id="echoid-s10428" xml:space="preserve">℟, vel triplicatis terminis, compo-<lb/>nunt rationem trium quadratorum, AV, cum quadrato, KI, ad <lb/>tria quadrata, AV, cum quadrato, & </s> <s xml:id="echoid-s10429" xml:space="preserve">℟, vel componunt rationem <lb/> <anchor type="note" xlink:label="note-0421-04a" xlink:href="note-0421-04"/> trium quadratorum, OV, cum quadrato, LI, ad tria quadrata, O <lb/>V, cum quadrato, G ℟; </s> <s xml:id="echoid-s10430" xml:space="preserve">quadratum autem, LI, eſt æquale rectan-<lb/>gulo, OVZ, bis cum quadrato, VZ, & </s> <s xml:id="echoid-s10431" xml:space="preserve">quadratum, G℟, æquale <lb/>rectangulo, OVZ, bis cum quadrato, VS; </s> <s xml:id="echoid-s10432" xml:space="preserve">nam rectangulum, KC <lb/>I, ex prop. </s> <s xml:id="echoid-s10433" xml:space="preserve">11. </s> <s xml:id="echoid-s10434" xml:space="preserve">Lib. </s> <s xml:id="echoid-s10435" xml:space="preserve">2. </s> <s xml:id="echoid-s10436" xml:space="preserve">Conicorum æquatur quadrato, OV, & </s> <s xml:id="echoid-s10437" xml:space="preserve">idẽ <lb/>rectangulum, KCI, cum quadrato, IL, æquatur quadrato. </s> <s xml:id="echoid-s10438" xml:space="preserve">LC, vel <lb/>quadrato, OZ, vnde quadratum, LI, remanet æquale rectangulo <lb/>ſub, OVZ, bis cum quadrato, VZ; </s> <s xml:id="echoid-s10439" xml:space="preserve">& </s> <s xml:id="echoid-s10440" xml:space="preserve">ſic etiam quadratum, G℟, <lb/>concludetur æquale eſſe rectangulo bis ſub, OVS, cum quadrato, <pb o="402" file="0422" n="422" rhead="GEOMETRIÆ"/> VS; </s> <s xml:id="echoid-s10441" xml:space="preserve">componunt ergo rationem trium quadratorum, OV, cum <lb/>rectangulo, OVZ, bis, & </s> <s xml:id="echoid-s10442" xml:space="preserve">quadrato, VZ, .</s> <s xml:id="echoid-s10443" xml:space="preserve">i. </s> <s xml:id="echoid-s10444" xml:space="preserve">duorum quadratorum, <lb/>OV, cum quadrato, OZ, ad tria quadrata, OV, cum rectangulo, <lb/>OVS, bis & </s> <s xml:id="echoid-s10445" xml:space="preserve">quadrato, VS, .</s> <s xml:id="echoid-s10446" xml:space="preserve">i. </s> <s xml:id="echoid-s10447" xml:space="preserve">ad duo quadrata, OV, cum quadra-<lb/>to, OS, hæc autem ratio ſimul cum ea, quæ remanſit .</s> <s xml:id="echoid-s10448" xml:space="preserve">i. </s> <s xml:id="echoid-s10449" xml:space="preserve">cum ra-<lb/>tione, EC, ad, MN, componit rationem parallelepipedi ſub, EC, <lb/>& </s> <s xml:id="echoid-s10450" xml:space="preserve">baſi quadrato, ZO, cum duplo quadrati, OV, ad parallelepipe-<lb/>dum ſub, MN, & </s> <s xml:id="echoid-s10451" xml:space="preserve">baſi quadrato, SO, cum duplo quadrati, OV; <lb/></s> <s xml:id="echoid-s10452" xml:space="preserve">vel parallelepipedi ſub, XL, baſi quadrato, RZ, cum duplo qua-<lb/>drati, AV, ad parallelepipedum ſub, HG, baſi quadrato, BS, cum <lb/>duplo quadrati, AV, quod nobis erat oſtendendum.</s> <s xml:id="echoid-s10453" xml:space="preserve"/> </p> <div xml:id="echoid-div947" type="float" level="2" n="1"> <note position="right" xlink:label="note-0421-01" xlink:href="note-0421-01a" xml:space="preserve">21. huius.</note> <note position="right" xlink:label="note-0421-02" xlink:href="note-0421-02a" xml:space="preserve">11. l. 2.</note> <note position="right" xlink:label="note-0421-03" xlink:href="note-0421-03a" xml:space="preserve">21. huius</note> <note position="right" xlink:label="note-0421-04" xlink:href="note-0421-04a" xml:space="preserve">Def. 12. <lb/>l. 1.</note> </div> </div> <div xml:id="echoid-div949" type="section" level="1" n="566"> <head xml:id="echoid-head591" xml:space="preserve">THEOREMA XXIV. PROPOS. XXV.</head> <p> <s xml:id="echoid-s10454" xml:space="preserve">IN eadem figura Prop. </s> <s xml:id="echoid-s10455" xml:space="preserve">23. </s> <s xml:id="echoid-s10456" xml:space="preserve">oſtendemus omnia quadrata <lb/>figuræ, FADCVE, (regula eadem, AV,) demptis omni-<lb/>bus quadratis triangulorum kOI, POQ, ad omnia quadra-<lb/>ta figuræ, TAYNVM, demptis omnibus quadratis trian-<lb/>gulorum, &</s> <s xml:id="echoid-s10457" xml:space="preserve">O, ΩΠ, eſſe vt, EC, ad, MN, vel, XL, ad, H <lb/>G, qui ſuntſecundiaxes, vel diametri.</s> <s xml:id="echoid-s10458" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10459" xml:space="preserve">Nam omnia quadrata figuræ, FADCVE, demptis omnibus <lb/> <anchor type="figure" xlink:label="fig-0422-01a" xlink:href="fig-0422-01"/> quadratis triangulorum, kOI, POQ, ad <lb/>omnia quadrata figuræ, TAYNVM, dẽ <lb/>ptis omnibus quadratis triangulorum, & </s> <s xml:id="echoid-s10460" xml:space="preserve"><lb/>O℟, ΩΟΠ, habent rationem compoſitã <lb/>ex ratione omnium quadratorum figuræ, <lb/>FADCVE, demptis omnibus quadratis <lb/>triangulorum, KOI, POQ, ad omnia qua-<lb/>drata, FC, .</s> <s xml:id="echoid-s10461" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10462" xml:space="preserve">ex ratione quadrati, AV, ad <lb/> <anchor type="note" xlink:label="note-0422-01a" xlink:href="note-0422-01"/> quadratum, DC, item ex ratione omniũ <lb/>quadratorum, FC, ad omnia quadrata, T <lb/>N, quæ eſt compoſita ex ratione quadra-<lb/>ti, DC, ad quadratum, YN, & </s> <s xml:id="echoid-s10463" xml:space="preserve">ex ratione, <lb/>CE, ad, NM, & </s> <s xml:id="echoid-s10464" xml:space="preserve">tandem componitur ex <lb/>ratione omnium quadratorum, TN, ad <lb/>omnia quadrata figurę, TAYNVM, dẽ-<lb/>ptis omnibus quadratis triangulorum, & </s> <s xml:id="echoid-s10465" xml:space="preserve"><lb/>O℟, ΩΟΠ, .</s> <s xml:id="echoid-s10466" xml:space="preserve">i. </s> <s xml:id="echoid-s10467" xml:space="preserve">ex ea, quam habet quadratum, YN, ad quadratũ <lb/> <anchor type="note" xlink:label="note-0422-02a" xlink:href="note-0422-02"/> AV, ex his autem rationibus illa, quam habet quadratum, AV, ad <pb o="403" file="0423" n="423" rhead="LIBER V."/> quadratum, DC, quadratum, DC, ad quadratum, YN, & </s> <s xml:id="echoid-s10468" xml:space="preserve">qua-<lb/> <anchor type="note" xlink:label="note-0423-01a" xlink:href="note-0423-01"/> dratum, YN, ad quadratum, AV, componunt rationem quadra-<lb/>ti, AV, ad quadratum, AV, quę ſimul cum ratione ipſius, EC, ad <lb/>MN, componit rationem parallelepipedi lub, EC, & </s> <s xml:id="echoid-s10469" xml:space="preserve">quadrato, <lb/>AV, ad parallelepipedum ſub, MN, & </s> <s xml:id="echoid-s10470" xml:space="preserve">quadrato, AV, quæ tandẽ <lb/>eſt eadem ei, quam habet, EC, ad, MN, quia illa ſunt parallelepi-<lb/>peda in eadem baſi, & </s> <s xml:id="echoid-s10471" xml:space="preserve">ideò omnia quadrata figuræ, FADCVE, <lb/>demptis omnibus quadratis triangulorum, KOI, POQ, ad om-<lb/>nia quadrata figuræ, TAYNVM, demptis omnibus quadratis <lb/>triangulorum, & </s> <s xml:id="echoid-s10472" xml:space="preserve">O℟, ΩΟΠ, erum vt, EC, ad, MN, vel, XL, ad, <lb/>HG, quod demonſtrare opus erat.</s> <s xml:id="echoid-s10473" xml:space="preserve"/> </p> <div xml:id="echoid-div949" type="float" level="2" n="1"> <figure xlink:label="fig-0422-01" xlink:href="fig-0422-01a"> <image file="0422-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0422-01"/> </figure> <note position="left" xlink:label="note-0422-01" xlink:href="note-0422-01a" xml:space="preserve">Coroll. 1. <lb/>22. huius.</note> <note position="left" xlink:label="note-0422-02" xlink:href="note-0422-02a" xml:space="preserve">Coroll. 1. <lb/>22. huius.</note> <note position="right" xlink:label="note-0423-01" xlink:href="note-0423-01a" xml:space="preserve">Defin. 13. <lb/>lib .1.</note> </div> </div> <div xml:id="echoid-div951" type="section" level="1" n="567"> <head xml:id="echoid-head592" xml:space="preserve">THEOREMA XXV. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s10474" xml:space="preserve">ASumpta iterum figura Propoſ. </s> <s xml:id="echoid-s10475" xml:space="preserve">23. </s> <s xml:id="echoid-s10476" xml:space="preserve">dimiſſo quouis pa-<lb/>rallelogrammorum, FC, TN, vt dimiſſo, TN, cæte-<lb/>ris ijſdem manentibus, oſtendemus omnia quadrata, FC, <lb/>demptis omnibus quadratis oppoſitarum hyperbolarum, <lb/>FAD, EVC, regula, EC, ad omnia quadrata figuræ, FAD <lb/>CVE, regula, DC, vel, AV, habere rationem compoſitam <lb/>ex ratione rectanguli, AOZ, bis cum @. </s> <s xml:id="echoid-s10477" xml:space="preserve">quadrati, VZ, ad <lb/>rectangulum, AZO, & </s> <s xml:id="echoid-s10478" xml:space="preserve">ex ratione rectanguli ſub, DC, vel, <lb/>RZ, & </s> <s xml:id="echoid-s10479" xml:space="preserve">ſub EC, ad quadratum, AV, cum. </s> <s xml:id="echoid-s10480" xml:space="preserve">quadrati, KI, <lb/>vel cum rectangulo ſub, AZ, & </s> <s xml:id="echoid-s10481" xml:space="preserve">ſexquitertia, ZV.</s> <s xml:id="echoid-s10482" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10483" xml:space="preserve">Omnia namq; </s> <s xml:id="echoid-s10484" xml:space="preserve">quadrata, FC, demptis omnibus quadratis oppo-<lb/>ſitarum hyperbolarum, FAD, EVC, regula, EC, ad omnia qua-<lb/>drata figuræ, FADCVE, regula, AV, habent rationem compoſi-<lb/>tam ex ratione omnium quadratorum, FC, demptis omnious qua-<lb/>dratis oppoſitarum hyperbolarum, FAD, EVC, ad omnia qua-<lb/>drata, FC, communiregula, EC, .</s> <s xml:id="echoid-s10485" xml:space="preserve">f. </s> <s xml:id="echoid-s10486" xml:space="preserve">ex ratione rectanguli, AOZ, <lb/> <anchor type="note" xlink:label="note-0423-02a" xlink:href="note-0423-02"/> bis cum {2/3}. </s> <s xml:id="echoid-s10487" xml:space="preserve">quadrati, VZ, ad rectangulum, AZO, & </s> <s xml:id="echoid-s10488" xml:space="preserve">ex ratione <lb/>omnium quadratorum, FC, regula, EC, ad omnia quadrata, FC, <lb/>regula, CD, vel, AV, .</s> <s xml:id="echoid-s10489" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10490" xml:space="preserve">ex ratione, EC, ad, CD, vel rectanguli <lb/> <anchor type="note" xlink:label="note-0423-03a" xlink:href="note-0423-03"/> ſub, EC, CD, ad quadratum, CD, & </s> <s xml:id="echoid-s10491" xml:space="preserve">tandem componitur ex ra-<lb/>tione omnium quadratorum, FC, regula, DC, ad omnia quadra-<lb/>ta figuræ, FADCVE, regula eadem, DC, .</s> <s xml:id="echoid-s10492" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10493" xml:space="preserve">ex ratione quadrati, <lb/> <anchor type="note" xlink:label="note-0423-04a" xlink:href="note-0423-04"/> DC, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10494" xml:space="preserve">quadrati KI, duæ verò rationes, <lb/>ſcilicet rectanguli ſub, ED, CD, ad quadratum, CD, & </s> <s xml:id="echoid-s10495" xml:space="preserve">quadrati, <lb/>CD, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10496" xml:space="preserve">quadrati, KI, componunt ratio- <pb o="404" file="0424" n="424" rhead="GEOMETRIÆ"/> nem rectanguli, DC, CE, vel ſub, RZ, EC, ad quadratum, AV, <lb/>cum {1/3}. </s> <s xml:id="echoid-s10497" xml:space="preserve">quadrati, kI, ergo omnia quadrata, FC, demptis omnibus <lb/> <anchor type="figure" xlink:label="fig-0424-01a" xlink:href="fig-0424-01"/> quadratis oppoſitarum hyperbola-<lb/>rum, FAD, EVC, regula, EC, ad <lb/>omnia quadrata figuræ, FADCVE, <lb/>regula, DC, vel, AV, habebunt ra-<lb/>tionem compoſitam ex ractione re-<lb/>ctanguli, AOZ, bis cum {2/3}. </s> <s xml:id="echoid-s10498" xml:space="preserve">quadra <lb/>ti, KI, ad rectangulum, AZO, & </s> <s xml:id="echoid-s10499" xml:space="preserve">ex <lb/>ratione rectanguli ſub, RZ, EC, ad <lb/>quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10500" xml:space="preserve">quadrati, <lb/>KI, .</s> <s xml:id="echoid-s10501" xml:space="preserve">i. </s> <s xml:id="echoid-s10502" xml:space="preserve">cum {4/12}. </s> <s xml:id="echoid-s10503" xml:space="preserve">quadrati, KI, quæ <lb/>ſunt {4/3}. </s> <s xml:id="echoid-s10504" xml:space="preserve">quadrati, LI, .</s> <s xml:id="echoid-s10505" xml:space="preserve">i. </s> <s xml:id="echoid-s10506" xml:space="preserve">rectanguli, <lb/> <anchor type="note" xlink:label="note-0424-01a" xlink:href="note-0424-01"/> AZV, vnde rectangulum ſub, AZ, <lb/>& </s> <s xml:id="echoid-s10507" xml:space="preserve">ſexquitertia, ZV, erit æquale ter-<lb/>tiæ parti quadrati, kI, erit igitur di-<lb/>cta ratio compoſita ex ratione pri-<lb/>mò dicta, & </s> <s xml:id="echoid-s10508" xml:space="preserve">ex ratione fectanguli <lb/>ſub, RZ, EC, ad quadratum, AV, <lb/>cum {1/3}. </s> <s xml:id="echoid-s10509" xml:space="preserve">quadrati, kI, ſiue cum rectangulo ſub, AZ, & </s> <s xml:id="echoid-s10510" xml:space="preserve">ſexquitertia, <lb/>ZV, quod oſtendere propoſitum erat.</s> <s xml:id="echoid-s10511" xml:space="preserve"/> </p> <div xml:id="echoid-div951" type="float" level="2" n="1"> <note position="right" xlink:label="note-0423-02" xlink:href="note-0423-02a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-0423-03" xlink:href="note-0423-03a" xml:space="preserve">29. l. 2.</note> <note position="right" xlink:label="note-0423-04" xlink:href="note-0423-04a" xml:space="preserve">21. huius.</note> <figure xlink:label="fig-0424-01" xlink:href="fig-0424-01a"> <image file="0424-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0424-01"/> </figure> <note position="left" xlink:label="note-0424-01" xlink:href="note-0424-01a" xml:space="preserve">Corol, 21. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div953" type="section" level="1" n="568"> <head xml:id="echoid-head593" xml:space="preserve">THEOREMA XXVI. PROPOS. XXVII.</head> <p> <s xml:id="echoid-s10512" xml:space="preserve">SI in eadem anteced. </s> <s xml:id="echoid-s10513" xml:space="preserve">Propoſit. </s> <s xml:id="echoid-s10514" xml:space="preserve">figura intelligantur de-<lb/>ſcriptæ ſectiones, quæ ab Apollonio coniugatæ vo-<lb/>cantur, quæ ſint, Y & </s> <s xml:id="echoid-s10515" xml:space="preserve">B, HTN, coniugatæ prædictis, FAD, <lb/>EVC, habentes ſcilicet quadratum tranſuerſi lateris, & </s> <s xml:id="echoid-s10516" xml:space="preserve">T, <lb/>æquale rectingulo ſub alio tr anſuerſo latere, AV, & </s> <s xml:id="echoid-s10517" xml:space="preserve">linea <lb/>iuxta qua n poſſunt, ſiue latere recto oppoſitarum ſectionũ, <lb/>FAD, EVC, & </s> <s xml:id="echoid-s10518" xml:space="preserve">regula ſit DC, latus parallelogrammi, FC, <lb/>expoſitis primò ſectionibus oppoſiti, FAD, EVC, circum-<lb/>ſcriptum, æquidiſtans earu nlateri tranſuerſo, AV: </s> <s xml:id="echoid-s10519" xml:space="preserve">Om-<lb/>nia quadrata, FC, ad omnia quadrata figuræ, FADCVE, <lb/>demptis omnibus quadratis oppoſitarum hyperbolarum, <lb/>Y & </s> <s xml:id="echoid-s10520" xml:space="preserve">B HTN, quæ portionibus laterum, FE, DC, inter op-<lb/>poſitas ſectiones, Y & </s> <s xml:id="echoid-s10521" xml:space="preserve">B HTN, exiſtentium conſtituuntur, <lb/>erunt vt parallelepipedum ſub dimidia baſis primò expoſi-<lb/>tarum alterutrius, hyperbolarum, vt ſub, ZC, & </s> <s xml:id="echoid-s10522" xml:space="preserve">ſub qua- <pb o="405" file="0425" n="425" rhead="LIBER V."/> drato, ZS, (quæ habetur, productis, ZC, OI, donec ſibi <lb/>occurrant, vt in, S,) ad parallelepipedum bis ſub, LT, & </s> <s xml:id="echoid-s10523" xml:space="preserve"><lb/>quadrato, TO, cum cubo, TO, & </s> <s xml:id="echoid-s10524" xml:space="preserve">amplius @. </s> <s xml:id="echoid-s10525" xml:space="preserve">eiuſdem cubi.</s> <s xml:id="echoid-s10526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10527" xml:space="preserve">Producatur, OL, indefinitè, cui occurrat, SG, ducta per, S, ipſi, <lb/>ZO, æquidiſtans, & </s> <s xml:id="echoid-s10528" xml:space="preserve">occurſus ſit in puncto, G, & </s> <s xml:id="echoid-s10529" xml:space="preserve">per, T, ipſa, MT, <lb/>æquidiſtans ducatur ipſi, AV, & </s> <s xml:id="echoid-s10530" xml:space="preserve">per, V, VM, æquidiſtans ipſi, V <lb/>T, quæ tangent ſectiones in punctis, VT, & </s> <s xml:id="echoid-s10531" xml:space="preserve">conuenient interſe <lb/>in aſymptoto, OS, vt in, M, vt ex pri. </s> <s xml:id="echoid-s10532" xml:space="preserve">Secundi Conicorum elici <lb/>poteſt: </s> <s xml:id="echoid-s10533" xml:space="preserve">Omnia ergo quadrata, FC, ad omnia quadrata figuræ, <lb/>FADCVE, vel omnia quadrata, RC, ad omnia quadrata figuræ, <lb/> <anchor type="note" xlink:label="note-0425-01a" xlink:href="note-0425-01"/> DAVC, ſunt vt quadratum, DC, ad quadratum, AV, cum {1/3}. </s> <s xml:id="echoid-s10534" xml:space="preserve">qua-<lb/>drati, kI, ſiue vt quadratum, CL, ad quadratum, OV, vel, TM, <lb/>cum {1/3}. </s> <s xml:id="echoid-s10535" xml:space="preserve">quadrati, LI, quia verò quadratum, CL, vel, SG, ad qua-<lb/>dratum, MT, eſt vt quadratum, GO, ad quadratum, OT, & </s> <s xml:id="echoid-s10536" xml:space="preserve">qua-<lb/>dratum, GS, ad quadratum, LI, eſt vt quadratum, GO, ad quadra-<lb/>tum, OL, ideo quadratum, SG, ad quadratum, TM, vel, OV, cum <lb/>{1/3}. </s> <s xml:id="echoid-s10537" xml:space="preserve">quadrati, LI, erit vt quadratum, GO, ad quadratum, OT, cum <lb/>{1/3}. </s> <s xml:id="echoid-s10538" xml:space="preserve">quadrati, OL, ſiue vt triplum quadrat, GO, ad quadratum, LO, <lb/>cum tribus quadratis, OT, vel ſumpta, ’LO, communi altitudine, <lb/>vt parallelepipedum ſub, LO, & </s> <s xml:id="echoid-s10539" xml:space="preserve">triplo quadrati, OG, ad paralle-<lb/>le ipedum ſub, LO, & </s> <s xml:id="echoid-s10540" xml:space="preserve">quadrato, OL, cum triplo quadrati, OT, <lb/>ſic igitur erunt omnia quadrata, RC, ad omnia quadrata figuræ, <lb/>DAVC, quod ſerua.</s> <s xml:id="echoid-s10541" xml:space="preserve"/> </p> <div xml:id="echoid-div953" type="float" level="2" n="1"> <note position="right" xlink:label="note-0425-01" xlink:href="note-0425-01a" xml:space="preserve">21. huius.</note> </div> <p> <s xml:id="echoid-s10542" xml:space="preserve">Inſuper omnia quadrata, RC, ad omnia quadrata trianguli, K <lb/>OI, ſunt vt quadratum, DC, ad {1/3}. </s> <s xml:id="echoid-s10543" xml:space="preserve">quadrati, kI, vel vt quadratum, <lb/>CL, vel quadratum, GS, ad {1/3}. </s> <s xml:id="echoid-s10544" xml:space="preserve">quadrati, LI, vel vt quadratum, G <lb/>O, ad {1/3}. </s> <s xml:id="echoid-s10545" xml:space="preserve">quadrati, OL, vel vt trip um quadrati, GO, ad quadra-<lb/>tum, OL, Vel, ſump@@, OL, communi altitudine, vt parallelepipe-<lb/>dum ſub, LO, & </s> <s xml:id="echoid-s10546" xml:space="preserve">triplo quadrati, CG, ad parallelepipedum ſub L <lb/>O, & </s> <s xml:id="echoid-s10547" xml:space="preserve">quadrato, LO, .</s> <s xml:id="echoid-s10548" xml:space="preserve">i. </s> <s xml:id="echoid-s10549" xml:space="preserve">ad cubum, LO. </s> <s xml:id="echoid-s10550" xml:space="preserve">Vlterius omnia quadra <lb/> <anchor type="note" xlink:label="note-0425-02a" xlink:href="note-0425-02"/> ta trianguli, KOI, ad omnia quadrata hyperbolæ, HTN, ſunt vt <lb/>cubus, LO, ad parallelepipedum ter ſub, OT, & </s> <s xml:id="echoid-s10551" xml:space="preserve">quadrato, TL, cũ <lb/>cubo, TL, ergo, ex æquali, omnia quadrata, RC, ad omnia qua-<lb/>drata hyperbolæ, HTN, erunt vt parallelepipedum ſub, LO, & </s> <s xml:id="echoid-s10552" xml:space="preserve"><lb/>triplo quadrati, OG, ad parallelepipedum ter ſub, OT, & </s> <s xml:id="echoid-s10553" xml:space="preserve">qua-<lb/>drato, TL, cum cubo, TL, erant autem omnia quadrata, RC, ad <lb/>omnia quadrata figurę, DAVC, vt idem parallelepipedum ſub, L <lb/>O, & </s> <s xml:id="echoid-s10554" xml:space="preserve">triplo quadrati, OG, ad parallelepipedum ſub, LO, & </s> <s xml:id="echoid-s10555" xml:space="preserve">qua-<lb/>drato, OL, cuin triplo quadrati, OT, ergo omnia quadrata, RC, <lb/>ad omnia quadrata figuræ, DAVC, demptis omnibus quadratis <pb o="406" file="0426" n="426" rhead="GEOMETRIÆ"/> hyperbolæ, HTN, erunt vt parallelepipedum ſub, LO, & </s> <s xml:id="echoid-s10556" xml:space="preserve">triplo <lb/>quadrati, OG, ad reliquum, quod habetur, dempto parallelepipe-<lb/> <anchor type="figure" xlink:label="fig-0426-01a" xlink:href="fig-0426-01"/> do ter ſub, OT, & </s> <s xml:id="echoid-s10557" xml:space="preserve">quadrato, TL, <lb/>cum cubo, TL, à parallelepipedo <lb/>ſub, LO, & </s> <s xml:id="echoid-s10558" xml:space="preserve">quadrato, LO, .</s> <s xml:id="echoid-s10559" xml:space="preserve">i. </s> <s xml:id="echoid-s10560" xml:space="preserve">à cu-<lb/>bo, LO, & </s> <s xml:id="echoid-s10561" xml:space="preserve">parallelepipedo ſub, LO, <lb/>& </s> <s xml:id="echoid-s10562" xml:space="preserve">triplo quadrati, OT, verum, quia <lb/>cubus, LO, æquatur parallelepipe-<lb/>dis ter ſub, OT, & </s> <s xml:id="echoid-s10563" xml:space="preserve">quadrato, TL, <lb/>ter ſub, TL, & </s> <s xml:id="echoid-s10564" xml:space="preserve">quadrato, TO, cum <lb/>cubis, OT, TL, ideò ſi à cubo, OL, <lb/>dematur parallelepipedum ter ſub, <lb/>OT, & </s> <s xml:id="echoid-s10565" xml:space="preserve">quadrato, TL, cum cubo, <lb/> <anchor type="note" xlink:label="note-0426-01a" xlink:href="note-0426-01"/> TL, remanebit parallelepipedum <lb/>ter ſub, LT, & </s> <s xml:id="echoid-s10566" xml:space="preserve">quadrato, TO, cum <lb/>cubo, TO, quod iungendum eſt pa-<lb/>rallelepipedo ſub, LO, & </s> <s xml:id="echoid-s10567" xml:space="preserve">triplo qua-<lb/>drati, TO, habebimus ergo pro <lb/>quæſito reſiduo parallelepipedum <lb/>ſub, LO, & </s> <s xml:id="echoid-s10568" xml:space="preserve">quadrato, OT, ter .</s> <s xml:id="echoid-s10569" xml:space="preserve">i. </s> <s xml:id="echoid-s10570" xml:space="preserve">ſub, LT, & </s> <s xml:id="echoid-s10571" xml:space="preserve">quadrato, TO, <lb/>ter, cum tribus cubis, TO, & </s> <s xml:id="echoid-s10572" xml:space="preserve">adhuc parallelepipedum ſub, LT, & </s> <s xml:id="echoid-s10573" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0426-02a" xlink:href="note-0426-02"/> quadrato, TO, ter cum cubo, TO, .</s> <s xml:id="echoid-s10574" xml:space="preserve">i. </s> <s xml:id="echoid-s10575" xml:space="preserve">habebimus parallelepipe-<lb/>dum ſub, LT, & </s> <s xml:id="echoid-s10576" xml:space="preserve">quadrato, TO, ſexies, cum quatuor cubis, TO, <lb/>pro quæſito reſiduo, igitur omnia quadrata, RC, ad omnia qua-<lb/>drata figuræ, DAVC, demptis omnibus quadratis hyperbolæ, HT <lb/>N, vel omnia quadrata, FC, ad omnia quadrata figuræ, FADCV <lb/>E, demptis omnibus quadratis oppoſitarum hyperbolarum, Y & </s> <s xml:id="echoid-s10577" xml:space="preserve"><lb/>B, HTN, erunt vt parallelepipedum ſub, LO, & </s> <s xml:id="echoid-s10578" xml:space="preserve">triplo quadrati, <lb/>OG, ad parallelepipedum ſexies ſub, LT, & </s> <s xml:id="echoid-s10579" xml:space="preserve">quadrato, TO, cum <lb/>quatuor cubis, TO, .</s> <s xml:id="echoid-s10580" xml:space="preserve">i. </s> <s xml:id="echoid-s10581" xml:space="preserve">vt parallelepipedum ſub, LO, vel, ZC, & </s> <s xml:id="echoid-s10582" xml:space="preserve"><lb/>quadrato, OG, vel, ZS, ad parallelepipedum bis ſub, LT, & </s> <s xml:id="echoid-s10583" xml:space="preserve">qua-<lb/>drato, TO, cum cubo, TO, & </s> <s xml:id="echoid-s10584" xml:space="preserve">amplius eiuſdem cubi, TO, nam <lb/>hæc ſunt eorundem ſubtripla, vt conſideranti facilè patebit, quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s10585" xml:space="preserve"/> </p> <div xml:id="echoid-div954" type="float" level="2" n="2"> <note position="right" xlink:label="note-0425-02" xlink:href="note-0425-02a" xml:space="preserve">9. huius.</note> <figure xlink:label="fig-0426-01" xlink:href="fig-0426-01a"> <image file="0426-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0426-01"/> </figure> <note position="left" xlink:label="note-0426-01" xlink:href="note-0426-01a" xml:space="preserve">38. l. 2.</note> <note position="left" xlink:label="note-0426-02" xlink:href="note-0426-02a" xml:space="preserve">3. 6. l. 2,</note> </div> </div> <div xml:id="echoid-div956" type="section" level="1" n="569"> <head xml:id="echoid-head594" xml:space="preserve">THEOREMA XXVII. PROPOS. XXVIII.</head> <p> <s xml:id="echoid-s10586" xml:space="preserve">SI, expoſitis ſectionibus coniugatis, parallelogrammũ <lb/>deſcribatur, habens latera earundem axibus, vel dia-<lb/>metris coniugatis parallela, in earum aſymptotis conue- <pb o="407" file="0427" n="427" rhead="LIBER V."/> nientia, eaſdemq; </s> <s xml:id="echoid-s10587" xml:space="preserve">oppoſitas ſectiones diuidentia’ alterutro <lb/>axium, vel diametrorum, ſumpto pro regula. </s> <s xml:id="echoid-s10588" xml:space="preserve">Omnia qua-<lb/>drata deſcripti parallelogrammi ad omnia quadrata figurę <lb/>duobus oppoſitis lateribus parallelogrammi regulæ æqui-<lb/>diſtantibus, & </s> <s xml:id="echoid-s10589" xml:space="preserve">reliquorum laterum portionibus inter ſe-<lb/>ctiones coniugatas, & </s> <s xml:id="echoid-s10590" xml:space="preserve">prædicta latera concluſis, & </s> <s xml:id="echoid-s10591" xml:space="preserve">ipſis <lb/>coniugatis ſectionibus, comprehenſæ, demptis ab ijſdem <lb/>omnibus quadratis oppoſitarum hyperbolarum, quarum <lb/>latus tranſuerſum non fuit ſumptum pro regula, erunt vt <lb/>cubus dimidij lateris parallelogrammi regulæ non æqui-<lb/>diſtantis, ad duo parallelepipeda, quorum vnum contine-<lb/>tur ſub dimidio exceſſus dicti lateris ſuper baſim hyperbo-<lb/>læ, quam idem latus abſcindit, & </s> <s xml:id="echoid-s10592" xml:space="preserve">ſub quadrato dimidij <lb/>eiuſdem lateris, aliud verò ſub dimidio baſis dictæ hyper-<lb/>bolæ, & </s> <s xml:id="echoid-s10593" xml:space="preserve">ſub @. </s> <s xml:id="echoid-s10594" xml:space="preserve">quadrati eiuſdem, cum quadrato dimidij <lb/>lateristranſuerſi, quod non eſt regula, ab his tamen dem-<lb/>pto paralleledipedo ſub dimidio lateris tranſuerſi, quod <lb/>non eſt regu a, & </s> <s xml:id="echoid-s10595" xml:space="preserve">ſub quadrato axis, vel diametri alteru-<lb/>trius hyperbolarum, quarum eſt latus tranſuerſum, vna cũ <lb/>{1/3}. </s> <s xml:id="echoid-s10596" xml:space="preserve">cubi eiuſdem axis, vel diametri.</s> <s xml:id="echoid-s10597" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10598" xml:space="preserve">Sintigitur expoſitæ ſectiones coniugatæ, AEC, MON, PIQ, <lb/>BFH, quarum communes aſymptoti indefinitè cum ſectionibus <lb/>fint producti, qui ſint, TSV, RSX, ſint autem earum axes, vel dia-<lb/>metri coniugatæ, EO, FI, quarum alterutra ſit ſumpta pro regu-<lb/>la, vt, FI, ſit vlterius deſcriptum parallelogrammum, TV, latera <lb/>habens æquidiſtantia ipſis, EO, FI, & </s> <s xml:id="echoid-s10599" xml:space="preserve">in aſymptotis, TV, XR, cõ-<lb/>uenientia in punctis, T, R, V, X, ipſaſq; </s> <s xml:id="echoid-s10600" xml:space="preserve">ſectiones diuidentia, ita <lb/>vt, quæ inter ſectiones manent, fiuntq; </s> <s xml:id="echoid-s10601" xml:space="preserve">hyperbolarum baſes ſint, <lb/>PQ, NM, HB, AC, quorum æquidiſtantia erunt æqualia. </s> <s xml:id="echoid-s10602" xml:space="preserve">Dico <lb/>ergo omnia quadrata parallelogrammi, TV, ad omnia quadrata <lb/>figuræ inte, TX, RV, TB, HR, VQ, PX, & </s> <s xml:id="echoid-s10603" xml:space="preserve">ſectiones, BFH, PIQ, <lb/>cõcluſæ, demptis ab ijſdë omnibus quadratis oppoſitarum hyper-<lb/>bolarum, AEC, MON, eſſe vt cubus dimidij, XV, ad parallelepi-<lb/>pedum ſub, QV, & </s> <s xml:id="echoid-s10604" xml:space="preserve">quadrato dimidij lateris, XV, vna cum paral-<lb/>lelepipedo ſub dimidio, PQ, & </s> <s xml:id="echoid-s10605" xml:space="preserve">ſub compoſito ex {1/3}. </s> <s xml:id="echoid-s10606" xml:space="preserve">quadrati eiuſ-<lb/>dem dimidij, PQ, & </s> <s xml:id="echoid-s10607" xml:space="preserve">quadrato, SO, ab his tamen dempto paralle-<lb/>lepipedo ſub, SO, & </s> <s xml:id="echoid-s10608" xml:space="preserve">quadrato reliquæ ad medietatem, XV, cum <pb o="408" file="0428" n="428" rhead="GEOMETRIÆ"/> {1/3}. </s> <s xml:id="echoid-s10609" xml:space="preserve">cubi eiuſdem reliquæ. </s> <s xml:id="echoid-s10610" xml:space="preserve">Producantur, FI, EO, hinc inde vſq; </s> <s xml:id="echoid-s10611" xml:space="preserve">ad <lb/>latera, TX, XV, VR, RT, quibus occurrant in punctis, &</s> <s xml:id="echoid-s10612" xml:space="preserve">, Z, Y, <lb/> <anchor type="figure" xlink:label="fig-0428-01a" xlink:href="fig-0428-01"/> G, in quibus illa bifariam diui-<lb/>duntur, & </s> <s xml:id="echoid-s10613" xml:space="preserve">per, Q, ducatur, QK, <lb/>ęequidiſtans ipſi, RV: </s> <s xml:id="echoid-s10614" xml:space="preserve">Omnia <lb/>igitur quadrata parallelogram-<lb/>mi, SV, ad omnia quadrata ſigu <lb/>rę, SIQK, habent rationem com-<lb/>poſitam ex ea, quam habent om-<lb/>nia quadrata, SV, ad omnia qua-<lb/> <anchor type="note" xlink:label="note-0428-01a" xlink:href="note-0428-01"/> drata, SQ, .</s> <s xml:id="echoid-s10615" xml:space="preserve">i. </s> <s xml:id="echoid-s10616" xml:space="preserve">ex ratione, YS, ad, <lb/>Sk, & </s> <s xml:id="echoid-s10617" xml:space="preserve">ex ratione omnium qua-<lb/>dratorum, SQ, ad omnia qua-<lb/>drata figurę, SIQk, .</s> <s xml:id="echoid-s10618" xml:space="preserve">i. </s> <s xml:id="echoid-s10619" xml:space="preserve">ex ratione <lb/> <anchor type="note" xlink:label="note-0428-02a" xlink:href="note-0428-02"/> quadrati, KQ, ad quadratum, SI, <lb/>cum {1/3}. </s> <s xml:id="echoid-s10620" xml:space="preserve">quadrati, kD, .</s> <s xml:id="echoid-s10621" xml:space="preserve">i. </s> <s xml:id="echoid-s10622" xml:space="preserve">exra-<lb/>tione quadrati, YS, ad quadratũ <lb/>SO, cum {1/3}. </s> <s xml:id="echoid-s10623" xml:space="preserve">quadrati, SK, duę <lb/>autem rationes, YS, ad, Sk, & </s> <s xml:id="echoid-s10624" xml:space="preserve"><lb/>quadrati, YS, ad quadratum, SO, Cum {1/3}. </s> <s xml:id="echoid-s10625" xml:space="preserve">quadrati, Sk, componũt <lb/>rationem cubi, YS, ad parallelepipedum ſub, KS, & </s> <s xml:id="echoid-s10626" xml:space="preserve">compoſito ex <lb/>quadrato, SO, & </s> <s xml:id="echoid-s10627" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10628" xml:space="preserve">quadrati, Sk, ergo omnia quadrata, SV, ad <lb/>omnia quadrata figurę SIQk, erunt vt cubus, YS, ad parallelepi-<lb/>pedum ſub, kS, & </s> <s xml:id="echoid-s10629" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10630" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10631" xml:space="preserve">quadrati, Sk: <lb/></s> <s xml:id="echoid-s10632" xml:space="preserve">Omnia item quadrata, SV, ad omnia quadrata, kV, ſunt vt, SY, <lb/> <anchor type="note" xlink:label="note-0428-03a" xlink:href="note-0428-03"/> ad, Yk, .</s> <s xml:id="echoid-s10633" xml:space="preserve">i. </s> <s xml:id="echoid-s10634" xml:space="preserve">ſumpta communi baſi quadrato, SY, vt cubus, SY, ad <lb/>parallelepipedum ſub, Yk, & </s> <s xml:id="echoid-s10635" xml:space="preserve">quadrato, YS, ergo omnia quadrata’ <lb/>SV, ad omnia quadrata figuræ, SIQk, & </s> <s xml:id="echoid-s10636" xml:space="preserve">parallelogrammi, kV, .</s> <s xml:id="echoid-s10637" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s10638" xml:space="preserve">ad omnia quadrata figuræ, SIQVY, erunt vt cubus, YS, ad paral-<lb/>lelepipedum ſub, kY, & </s> <s xml:id="echoid-s10639" xml:space="preserve">quadrato, YS, vna cum parallelepipedo <lb/>ſub, kS, & </s> <s xml:id="echoid-s10640" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10641" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10642" xml:space="preserve">quadrati, Sk: </s> <s xml:id="echoid-s10643" xml:space="preserve">Quo-<lb/>niam verò omnia quadrata, SV, ſunt tripla omnium quadratorũ <lb/> <anchor type="note" xlink:label="note-0428-04a" xlink:href="note-0428-04"/> trianguli, SYV, hæc verò ad omnia quadrata ſemihyperbolæ, OY <lb/>N, ſunt vt cubus, SY, ad parallelepipedum ter ſub, SQ, & </s> <s xml:id="echoid-s10644" xml:space="preserve">quadra-<lb/>to, OY, cum cubo, OY, ideò omnia quadrata, SV, ad omnia qua-<lb/> <anchor type="note" xlink:label="note-0428-05a" xlink:href="note-0428-05"/> drata ſemihyperbolæ, YON, erunt vt tres cubi, SY, ad parallele-<lb/>pipedum ter ſub, SO, & </s> <s xml:id="echoid-s10645" xml:space="preserve">quadrato, OY, cum cubo, OY, .</s> <s xml:id="echoid-s10646" xml:space="preserve">i. </s> <s xml:id="echoid-s10647" xml:space="preserve">vt cu-<lb/>bus, SY, ad parallelepipedum ſub, SO, & </s> <s xml:id="echoid-s10648" xml:space="preserve">quadrato, OY, cum {1/3}. <lb/></s> <s xml:id="echoid-s10649" xml:space="preserve">cubi, OY; </s> <s xml:id="echoid-s10650" xml:space="preserve">erant autem omnia quadrata, SV, ad omnia quadrata <lb/>figuræ, SIQVY, vt cubus, SY, ad parallelepipedum ſub, kY, & </s> <s xml:id="echoid-s10651" xml:space="preserve"><lb/>quadrato, YS, vna cum parallelepipedo ſub, kS, & </s> <s xml:id="echoid-s10652" xml:space="preserve">compoſito ex <pb o="409" file="0429" n="429" rhead="LIBER V."/> quadrato, SO, & </s> <s xml:id="echoid-s10653" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10654" xml:space="preserve">quadrati, Sk, ergo omnia quadrata, SV, ad <lb/>omnia quadrata figuræ, SIQVY, demptis omnibus quadratis ſe-<lb/>mihyperbolæ, YON, vel horũ quadrupla .</s> <s xml:id="echoid-s10655" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10656" xml:space="preserve">omnia quadrata, GV, <lb/>ad omnia quadrata figuræ, FIQVRH, demptis omnibus quadratis <lb/>hyperbolę, MON, vel horum dupla .</s> <s xml:id="echoid-s10657" xml:space="preserve">i. </s> <s xml:id="echoid-s10658" xml:space="preserve">omnia quadrata, TV, ad om-<lb/>nia quad. </s> <s xml:id="echoid-s10659" xml:space="preserve">rigurę, XPIQVRHFBT, dẽptis omnibus quadratis op-<lb/>poſitarum hyperbolarũ, AEC, MON, erunt vt cubus, YS, vel, ZV, <lb/>ad parallelepipedum ſub, kY, & </s> <s xml:id="echoid-s10660" xml:space="preserve">quadrato, YS, vel ſub, QV, & </s> <s xml:id="echoid-s10661" xml:space="preserve">qua-<lb/>drato, VZ, vna cum parallelepipedo ſub, kS, & </s> <s xml:id="echoid-s10662" xml:space="preserve">compoſito ex qua-<lb/>drato, SO, & </s> <s xml:id="echoid-s10663" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10664" xml:space="preserve">quadrati, Sk, vel vna cum parallelepipedo ſub, Z <lb/>Q, & </s> <s xml:id="echoid-s10665" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10666" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10667" xml:space="preserve">quadrati, ZQ, ab his ta-<lb/>men dempto parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s10668" xml:space="preserve">quadrato, OY, quę, eſt <lb/>reliqua ad ipſam, SY, vel, ZV, vna cum {1/3}. </s> <s xml:id="echoid-s10669" xml:space="preserve">cubi eiuſdem reliquę, N <lb/>Y, quę eſt diameter alterutrius hyperbolarum dictarum, quod, &</s> <s xml:id="echoid-s10670" xml:space="preserve">c.</s> <s xml:id="echoid-s10671" xml:space="preserve"/> </p> <div xml:id="echoid-div956" type="float" level="2" n="1"> <figure xlink:label="fig-0428-01" xlink:href="fig-0428-01a"> <image file="0428-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0428-01"/> </figure> <note position="left" xlink:label="note-0428-01" xlink:href="note-0428-01a" xml:space="preserve">Defin, 12. <lb/>11.</note> <note position="left" xlink:label="note-0428-02" xlink:href="note-0428-02a" xml:space="preserve">10. l. 2, <lb/>21. huius.</note> <note position="left" xlink:label="note-0428-03" xlink:href="note-0428-03a" xml:space="preserve">10. l. 2.</note> <note position="left" xlink:label="note-0428-04" xlink:href="note-0428-04a" xml:space="preserve">24. l. 2.</note> <note position="left" xlink:label="note-0428-05" xlink:href="note-0428-05a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div958" type="section" level="1" n="570"> <head xml:id="echoid-head595" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s10672" xml:space="preserve">_H_Inc habetur omnia quadrata, TV, ad omnia quadrata fig. </s> <s xml:id="echoid-s10673" xml:space="preserve">ianz <lb/>dictæ, quæ comprehenditur terminis, quiſunt, TX, RV, TB, <lb/>HR, VQ, PX, & </s> <s xml:id="echoid-s10674" xml:space="preserve">ſectionibus oppoſitis, BFH, PIQ eſſe vt cubus RS, <lb/>ad parallelepipedum, ſub, KY, & </s> <s xml:id="echoid-s10675" xml:space="preserve">quadrato. </s> <s xml:id="echoid-s10676" xml:space="preserve">YS, Vna cum parallele-<lb/>pipedo ſub, KS, & </s> <s xml:id="echoid-s10677" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10678" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10679" xml:space="preserve">quadrati, SK, Vt <lb/>ſuperius oſtenſum eſt.</s> <s xml:id="echoid-s10680" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div959" type="section" level="1" n="571"> <head xml:id="echoid-head596" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXIX.</head> <p> <s xml:id="echoid-s10681" xml:space="preserve">INeadem anteced. </s> <s xml:id="echoid-s10682" xml:space="preserve">figura ſi aliud parallelogrammum de-<lb/>ſcribatur vtcunque, conditionibus tamen, quo, TV, de-<lb/>ſcriptum eſt, cuius latera ſectiones coniugatas diuidant, <lb/>quod ſit parallelogrammum, βΩ, cuius latera ſectiones cõ-<lb/>iugatas diuidant in punctis, L, Γ, Φ, Λ, 3, Σ, 7, 9, & </s> <s xml:id="echoid-s10683" xml:space="preserve">axes, <lb/>vel diametros coniugatas, &</s> <s xml:id="echoid-s10684" xml:space="preserve">, Y, GZ, in punctis, ℟, 8, 6, 4, <lb/>regula alterutro axium, vel diametrorum coniugatarum, V <lb/>T, FI: </s> <s xml:id="echoid-s10685" xml:space="preserve">Oſtendemus omnia quadrata figuræ, quæ remanet <lb/>demptis oppoſitis hyperbolis, BFH, PIQ à parallelogram-<lb/>mo, TV, ablatis ab ijſdem omnibus quadratis oppoſitarum <lb/>hyperbolarum, AEC, MON, (quæ figura breuitatis cau-<lb/>ſa dicatur, figura parallelogrammi, TV,) ad omnia qua-<lb/>drata figuræ, quæ remanet, demptis oppoſitis hyperbolis, <pb o="410" file="0430" n="430" rhead="GEOMETRIÆ"/> ΦΙΛ, 9F7, à parallelogrammo, βΩ, ablatis ab ijſdem om-<lb/>nibus quadratis oppoſitarum hyperbolarum, LΕΓ, ΣΟ3, <lb/>quæ dicatur figura parallelogrammi, βΩ, eſſe vt paralle-<lb/>lepipedum ſub, QV, & </s> <s xml:id="echoid-s10686" xml:space="preserve">quadrato. </s> <s xml:id="echoid-s10687" xml:space="preserve">VZ, vna cum parallelepi-<lb/>pedo ſub, QZ, & </s> <s xml:id="echoid-s10688" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10689" xml:space="preserve">@. </s> <s xml:id="echoid-s10690" xml:space="preserve">quadra-<lb/>ti, QZ, ab his dempto parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s10691" xml:space="preserve">quadra-<lb/>to, OY, & </s> <s xml:id="echoid-s10692" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10693" xml:space="preserve">cubi, OY, ad parallelepipedum ſub, ΛΩ, & </s> <s xml:id="echoid-s10694" xml:space="preserve"><lb/>quadrato, Ω4, vna cum parallelepipedo ſub, Λ4, & </s> <s xml:id="echoid-s10695" xml:space="preserve">com-<lb/>poſito ex quadrato, SO, & </s> <s xml:id="echoid-s10696" xml:space="preserve">@. </s> <s xml:id="echoid-s10697" xml:space="preserve">quadrati, Λ4, dempto paral-<lb/>lelepipedo ſub, SO, & </s> <s xml:id="echoid-s10698" xml:space="preserve">quadrato, O6, cum {1/3}. </s> <s xml:id="echoid-s10699" xml:space="preserve">cubi, O6.</s> <s xml:id="echoid-s10700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10701" xml:space="preserve">Nam omnia quadrata figuræ parallelogrammi, TV, demptis <lb/>iam dictis, ad omnia quadrata figuræ parallelogrammi, βΩ, dem-<lb/>ptis iam dictis, habent rationem compoſitam ex ratione omniũ <lb/>quadratorum primo dictæ figuræ, demptis, &</s> <s xml:id="echoid-s10702" xml:space="preserve">c. </s> <s xml:id="echoid-s10703" xml:space="preserve">ad omnia qua-<lb/>drata, TV, .</s> <s xml:id="echoid-s10704" xml:space="preserve">i. </s> <s xml:id="echoid-s10705" xml:space="preserve">ex ea, quam habet parallelepipedum ſub, QV, & </s> <s xml:id="echoid-s10706" xml:space="preserve"><lb/>quadrato, VZ, vna cum parallelepipedo ſub, QZ, & </s> <s xml:id="echoid-s10707" xml:space="preserve">compoſita <lb/> <anchor type="note" xlink:label="note-0430-01a" xlink:href="note-0430-01"/> ex quadrato, OS, & </s> <s xml:id="echoid-s10708" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10709" xml:space="preserve">quadrati, QZ, dempto ab his parallelepi-<lb/>pedo ſub, SO, & </s> <s xml:id="echoid-s10710" xml:space="preserve">quadrato, OY, & </s> <s xml:id="echoid-s10711" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10712" xml:space="preserve">cubi, OY, ad cubum, ZV, itẽ <lb/>ex ratione omnium quadratorum, TV, ad omnia quadrata, βΩ, <lb/>ideſt ex ratione cubi, VZ, ad cubum, Ω4, quia parallelogramma, <lb/> <anchor type="note" xlink:label="note-0430-02a" xlink:href="note-0430-02"/> TV, βΩ, ſunt ſimilia, cum ſint circa eandem diametrum, & </s> <s xml:id="echoid-s10713" xml:space="preserve">tandẽ <lb/>ex ratione omnium quadratorum, βΩ, ad omnia quadrata figuræ <lb/>parallelogrammi, βΩ, demptis iam dictis, .</s> <s xml:id="echoid-s10714" xml:space="preserve">i. </s> <s xml:id="echoid-s10715" xml:space="preserve">ex ratione cubi, Ω4, <lb/> <anchor type="note" xlink:label="note-0430-03a" xlink:href="note-0430-03"/> ad parallelepipedum ſub, ΛΩ, & </s> <s xml:id="echoid-s10716" xml:space="preserve">quadrato, Ω4, vna cum paralle-<lb/>lepipedo ſub, Λ4, & </s> <s xml:id="echoid-s10717" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10718" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10719" xml:space="preserve">quadrati, <lb/>Λ4, ab his dempto parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s10720" xml:space="preserve">quadrato, O6, vna <lb/>cum {1/3}. </s> <s xml:id="echoid-s10721" xml:space="preserve">cubi, O6, rationes autem parallelepipedorum primò dicto-<lb/>rum, dempto parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s10722" xml:space="preserve">quadrato, OY, cum {1/3}. <lb/></s> <s xml:id="echoid-s10723" xml:space="preserve">cubi, OY, ad cubum, ZV, cubi, ZV, ad cubum, Ω4, & </s> <s xml:id="echoid-s10724" xml:space="preserve">cubi, Ω4, <lb/>ad parallelepipeda poſtremò dicta, dempto parallerepipedo ſub, S <lb/>O, & </s> <s xml:id="echoid-s10725" xml:space="preserve">quadrato, O6, cum {1/3}. </s> <s xml:id="echoid-s10726" xml:space="preserve">cubi, O6, componunt rationem pa-<lb/>rallelepipedorum primò dictorum, dempto iam dicto ad parallele-<lb/> <anchor type="note" xlink:label="note-0430-04a" xlink:href="note-0430-04"/> pipeda poſtremò dicta, dempto iam dicto, ergo omnia quadrata <lb/>figuræ parallelogrammi, TV, demptis omnibus quadratis oppoſi-<lb/>tarum hyperbolarum, AEC, MON, ad omnia quadrata figuræ <lb/>parallelogrammi, βΩ, demptis omnibus quadratis oppoſitarum <lb/>hyperbolarum, LEI, ΣΟ3, erunt vt parallelepipedum ſub, QV, <lb/>& </s> <s xml:id="echoid-s10727" xml:space="preserve">quadrato, VZ, vna cum parallelepipedo ſub, QZ, & </s> <s xml:id="echoid-s10728" xml:space="preserve">compoſi-<lb/>to ex quadrato, SO, & </s> <s xml:id="echoid-s10729" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10730" xml:space="preserve">quadrati, QZ, ab his dempto parallele- <pb o="411" file="0431" n="431" rhead="LIBER V."/> pipedo ſub, SO, & </s> <s xml:id="echoid-s10731" xml:space="preserve">quadrato, OY, cum {1/3}. </s> <s xml:id="echoid-s10732" xml:space="preserve">cubi, OY, ad parallele. <lb/></s> <s xml:id="echoid-s10733" xml:space="preserve">pipedum ſub, ΛΩ, & </s> <s xml:id="echoid-s10734" xml:space="preserve">quadrato, Ω4, vna cum parallelepipedo ſub, <lb/>Λ4, & </s> <s xml:id="echoid-s10735" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s10736" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10737" xml:space="preserve">quadrati, Λ4, ab his dẽ-<lb/>pto parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s10738" xml:space="preserve">quadrato, Ο6, cum {1/3}. </s> <s xml:id="echoid-s10739" xml:space="preserve">cubi, Ο6, <lb/>quod oſtendere opus erat.</s> <s xml:id="echoid-s10740" xml:space="preserve"/> </p> <div xml:id="echoid-div959" type="float" level="2" n="1"> <note position="left" xlink:label="note-0430-01" xlink:href="note-0430-01a" xml:space="preserve">Exantec.</note> <note position="left" xlink:label="note-0430-02" xlink:href="note-0430-02a" xml:space="preserve">3. l. 2.</note> <note position="left" xlink:label="note-0430-03" xlink:href="note-0430-03a" xml:space="preserve">Exantec.</note> <note position="left" xlink:label="note-0430-04" xlink:href="note-0430-04a" xml:space="preserve">Deſin. 12. <lb/>l. 1.</note> </div> </div> <div xml:id="echoid-div961" type="section" level="1" n="572"> <head xml:id="echoid-head597" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s10741" xml:space="preserve">_H_Inc patet, quod eadem methodo oſtendemus omnia quadrátá fi-<lb/>guræ parallelogrammi, TV, nibil ab eis dempto, ad omnia <lb/>quadrata figuræ parallelogrammi βΩ, nibil pariter ab eis dempto, eſſe <lb/>Vt parallelepipeda primò dicta ad parallelepipeda ſecundò dicta.</s> <s xml:id="echoid-s10742" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div962" type="section" level="1" n="573"> <head xml:id="echoid-head598" xml:space="preserve">THEOREMA XXIX. PROPOS. XXX.</head> <p> <s xml:id="echoid-s10743" xml:space="preserve">IN omnibus huius Lib. </s> <s xml:id="echoid-s10744" xml:space="preserve">5. </s> <s xml:id="echoid-s10745" xml:space="preserve">Propoſitionibus, in quibus <lb/>duarum quarumcunq; </s> <s xml:id="echoid-s10746" xml:space="preserve">figurarum notificata fuit ratio <lb/>omn ium quadratorum, iuxta regulas in eiſdem aſſumptas, <lb/>nota etiam euadit ratio ſimiliarium ſolidorum, quæ exillis <lb/>gignuntur figuris, iuxta eaſdem regulas.</s> <s xml:id="echoid-s10747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10748" xml:space="preserve">Quoniam enim oſtenſum eſt Lib. </s> <s xml:id="echoid-s10749" xml:space="preserve">2. </s> <s xml:id="echoid-s10750" xml:space="preserve">Prop. </s> <s xml:id="echoid-s10751" xml:space="preserve">23. </s> <s xml:id="echoid-s10752" xml:space="preserve">vt omnia qua-<lb/>drata duarum figurarum inter ſe ſumpta cum datis regulis, ita eſſe <lb/>ſolida ſimilaria genita ex ijſdem figuris iuxta eaſdem regulas, ideò <lb/>cum in huius Libri Propoſitionibus inuẽta eſt ratio omnium qua-<lb/>dratorum duarum figurarum cum talibus regulis, colligemus etiã <lb/>nunc eandem eſſe rationem duorum ſimilarium ſolidorum, quæ <lb/>ex illis figuris iuxta eaſdem regulas genita dicuntur, quæ amplius <lb/>in ſequentibus dilucidabimus ſingulas Propoſitiones, quæ oppor-<lb/>tunæ fuerint, denuò aſſumentes.</s> <s xml:id="echoid-s10753" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10754" xml:space="preserve">Vnde cum in prima Propoſ. </s> <s xml:id="echoid-s10755" xml:space="preserve">exempli gratia oſtenſum eſt (con-<lb/>ſpecta denuò eiuſdem figura) omnia quadrata hyperbolæ, DBF, <lb/>regula, DF, ad omnia quadrata, AF, eſſe vt compoſitam ex, NB, <lb/>& </s> <s xml:id="echoid-s10756" xml:space="preserve">{1/3}, BE, ad, OE, eandem comperiemus habere rationem ſolidum <lb/>ſimilare genitum ex hyperbola, DBF, ad ſolidum ſimilare genitũ <lb/>ex, AF, iuxta communem regulam, DF; </s> <s xml:id="echoid-s10757" xml:space="preserve">& </s> <s xml:id="echoid-s10758" xml:space="preserve">eodem pacto collige-<lb/>mus, veluti omnia quadrata hyperbolæ, DBF, ad omnia quadra-<lb/>ta trianguli, DBF, ſunt vt compoſita ex ſexquialtera, OB, & </s> <s xml:id="echoid-s10759" xml:space="preserve">ex, <lb/>BE, ad, OE, ita eſſe ſolidum ſimilare genitum ex hyperbola, DBF, <pb o="412" file="0432" n="432" rhead="GEOMETRIÆ"/> ad ſibi ſimilare genitum ex triangulo, DBF, iuxta communem re-<lb/>gulam, DF.</s> <s xml:id="echoid-s10760" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div963" type="section" level="1" n="574"> <head xml:id="echoid-head599" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s10761" xml:space="preserve">_Q_V oniam verò ex Propoſ. </s> <s xml:id="echoid-s10762" xml:space="preserve">45. </s> <s xml:id="echoid-s10763" xml:space="preserve">Lib. </s> <s xml:id="echoid-s10764" xml:space="preserve">Primi habetur, quod ſi quæ <lb/>cunque conois byperbolica, im cuius baſi ſit cylindrus, & </s> <s xml:id="echoid-s10765" xml:space="preserve">conus <lb/>& </s> <s xml:id="echoid-s10766" xml:space="preserve">circa eundem axem, vel diametrum ſecetur planis baſi æquidiſtan-<lb/>tibus, quibus pariter ſecentur cylindrus, & </s> <s xml:id="echoid-s10767" xml:space="preserve">conus, fient conceptæ in <lb/>ſolidis figuræ ſimiles baſi, ideò omnia plana eorundem regula baſi erũt <lb/> <anchor type="note" xlink:label="note-0432-01a" xlink:href="note-0432-01"/> omnes figuræ ſimiles dictorum ſolidorum, in quibus ſi ducatur pla-<lb/>num per axem, producet in ipſis figuras genitrices earundem, nempè <lb/>parallelogrammum in cylindro, hyperbolam in conoide, & </s> <s xml:id="echoid-s10768" xml:space="preserve">trian-<lb/>gulum in cono, dicta autem ſolida erunt ſimilaria genita ex his figu. <lb/></s> <s xml:id="echoid-s10769" xml:space="preserve">ris genitricibus iuxta communem regulam ipſam baſim, & </s> <s xml:id="echoid-s10770" xml:space="preserve">ideò eorũ <lb/>ratio nota erit, quia ſcimus quam rationem habeant inter ſe omnia <lb/>quadrata dictarum genitricium figurarum, regula baſi. </s> <s xml:id="echoid-s10771" xml:space="preserve">Hæc autem <lb/>ſimiliter pro ſequentibus memoria teneantùr, in quibus ſiet noſtrum <lb/>ſolitum exemplum per reuolutionem ſigurarum circa ſuos axes, Vt <lb/>habeamus omnes figuras ſimiles genitorum ſolidorum, quæ ſint circuli <lb/>diametros in figuris genitricibus, quibus ſint erecti, ſitas habentes, <lb/>licet eadem Verificentur ſſumptis non axibus, ſed tantum diametris, <lb/>Vt alibi pluriés repetitum eſt, ex dictis autem infraſcripta habentur <lb/>Corollaria.</s> <s xml:id="echoid-s10772" xml:space="preserve"/> </p> <div xml:id="echoid-div963" type="float" level="2" n="1"> <note position="left" xlink:label="note-0432-01" xlink:href="note-0432-01a" xml:space="preserve">_45. l. 1._ <lb/>_Elicitur ex_ <lb/>_45. l. 1. pſo_ <lb/>_Conoide_ <lb/>_Hyperb. &_ <lb/>_ex Cor. 3_ <lb/>_34. l. 2. pro_ <lb/>_Cylindro._ <lb/>_& Cono-_</note> </div> </div> <div xml:id="echoid-div965" type="section" level="1" n="575"> <head xml:id="echoid-head600" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s10773" xml:space="preserve">VTigitur fiat noſtrum exemplum in Prop. </s> <s xml:id="echoid-s10774" xml:space="preserve">1. </s> <s xml:id="echoid-s10775" xml:space="preserve">reuoluatur pa-<lb/>rallelogrammum, AF, circam manentem axim, BE, vt fiat <lb/> <anchor type="figure" xlink:label="fig-0432-01a" xlink:href="fig-0432-01"/> ex parallelogrammo, AF, cylindrus, A <lb/>F, ex hyperbola, DBF, conois, DBF, <lb/>& </s> <s xml:id="echoid-s10776" xml:space="preserve">ex triangulo, DBF, conus, DBF, col-<lb/>ligitur ergo cylindrum, AF, ad conoidé, <lb/>DBF, eſſe vt, OE, ad compoſitam ex, <lb/>NB, & </s> <s xml:id="echoid-s10777" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10778" xml:space="preserve">BE, conoidem autem, DBF, <lb/>ad conum, DBF, eſſe vt compoſitam <lb/>ex ſexquialtera, OB, & </s> <s xml:id="echoid-s10779" xml:space="preserve">ex, BE, ad OE; <lb/></s> <s xml:id="echoid-s10780" xml:space="preserve">& </s> <s xml:id="echoid-s10781" xml:space="preserve">ita eſſe ſolida quæcunque ſimilaria <lb/>genita ex eiſdem figuris .</s> <s xml:id="echoid-s10782" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10783" xml:space="preserve">ex parallelo-<lb/>grammo, AF, hyperbola, DBF, & </s> <s xml:id="echoid-s10784" xml:space="preserve">triã-<lb/>gulo, DBF, iuxta communem regulam, <lb/>DF, vt ſupra dictum eſt, quę declarare oportebat.</s> <s xml:id="echoid-s10785" xml:space="preserve"/> </p> <div xml:id="echoid-div965" type="float" level="2" n="1"> <figure xlink:label="fig-0432-01" xlink:href="fig-0432-01a"> <image file="0432-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0432-01"/> </figure> </div> <pb o="413" file="0433" n="433" rhead="LIBER V."/> </div> <div xml:id="echoid-div967" type="section" level="1" n="576"> <head xml:id="echoid-head601" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s10786" xml:space="preserve">INProp. </s> <s xml:id="echoid-s10787" xml:space="preserve">2. </s> <s xml:id="echoid-s10788" xml:space="preserve">aſſumpta eius figura, dimiſſis parallelogrammis, A <lb/>Z, CG, & </s> <s xml:id="echoid-s10789" xml:space="preserve">rectis, CH, RG, LK, vt fiat noſtrum exemplum re-<lb/> <anchor type="figure" xlink:label="fig-0433-01a" xlink:href="fig-0433-01"/> uoluatur figura circa manentem axim, <lb/>NE, vt fiat ex parallelogrammo, SF, cy, <lb/>lindrus, SF, ex triangulo, DMF, conus-<lb/>DMF, & </s> <s xml:id="echoid-s10790" xml:space="preserve">ex hyperbolis, DNF, HNG, <lb/>conoides hyperbolicæ, DNF, HNG, <lb/>patet ergo ex hac Propoſ. </s> <s xml:id="echoid-s10791" xml:space="preserve">conoidem, D <lb/>NF, ad conoidem, HNG, abſciſſam pla-<lb/>no, HG, æquidiſtanteipſi plano, DF, <lb/>eſſe vt parallelepipedũ ſub, XE, & </s> <s xml:id="echoid-s10792" xml:space="preserve">qua-<lb/>drato, EN, ad parallelepipedum ſub, X <lb/>M, & </s> <s xml:id="echoid-s10793" xml:space="preserve">quadrato, MN, & </s> <s xml:id="echoid-s10794" xml:space="preserve">ſic eſſe quod-<lb/>libet ſolidum ſimilare genitum ex hy-<lb/>perbola, DNF, ad ſibi ſimilare genitũ <lb/>ex hyperbola, HNG, iuxta communem <lb/>regulam, DF.</s> <s xml:id="echoid-s10795" xml:space="preserve"/> </p> <div xml:id="echoid-div967" type="float" level="2" n="1"> <figure xlink:label="fig-0433-01" xlink:href="fig-0433-01a"> <image file="0433-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0433-01"/> </figure> </div> </div> <div xml:id="echoid-div969" type="section" level="1" n="577"> <head xml:id="echoid-head602" xml:space="preserve">COROLLARIVM III.</head> <p> <s xml:id="echoid-s10796" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10797" xml:space="preserve">3. </s> <s xml:id="echoid-s10798" xml:space="preserve">patet in ſuperioris figura, in qua eius exemplum <lb/>conſtructum eſt, cylindrum, SF, ad fruſtum conoidis, HDFG, <lb/>eſſe vt rectangulum, OEN, ad rectangulum ſub, OE, &</s> <s xml:id="echoid-s10799" xml:space="preserve">, NM, <lb/>vna cum rectangulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s10800" xml:space="preserve">NO, & </s> <s xml:id="echoid-s10801" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10802" xml:space="preserve">ME, & </s> <s xml:id="echoid-s10803" xml:space="preserve">ſub, M <lb/>E. </s> <s xml:id="echoid-s10804" xml:space="preserve">Et conum, DMF, ad idem fruſtum eſſe, vt rectangulum, OE <lb/>N, ad rectangulum ſub, OE, & </s> <s xml:id="echoid-s10805" xml:space="preserve">tripla, NM, vna cum rectangulo <lb/>ſub compoſita ex, NX, &</s> <s xml:id="echoid-s10806" xml:space="preserve">, ME, & </s> <s xml:id="echoid-s10807" xml:space="preserve">ſub, ME; </s> <s xml:id="echoid-s10808" xml:space="preserve">& </s> <s xml:id="echoid-s10809" xml:space="preserve">ſic eſſe ſolida ſi-<lb/>milaria quæcunq; </s> <s xml:id="echoid-s10810" xml:space="preserve">genita ex eiſdem figuris, parallelogrammo nẽ-<lb/>pè, SF, fruſto hyperbolæ, HDFG, & </s> <s xml:id="echoid-s10811" xml:space="preserve">triangulo, DMF, iuxta cõ. <lb/></s> <s xml:id="echoid-s10812" xml:space="preserve">munem regulam, DF.</s> <s xml:id="echoid-s10813" xml:space="preserve"/> </p> <figure> <image file="0433-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0433-02"/> </figure> <pb o="414" file="0434" n="434" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div970" type="section" level="1" n="578"> <head xml:id="echoid-head603" xml:space="preserve">COROLLARIVM IV.</head> <p> <s xml:id="echoid-s10814" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10815" xml:space="preserve">4. </s> <s xml:id="echoid-s10816" xml:space="preserve">iterum aſſumpta figura Coroll. </s> <s xml:id="echoid-s10817" xml:space="preserve">2. </s> <s xml:id="echoid-s10818" xml:space="preserve">patebit cylindrũ, <lb/> <anchor type="figure" xlink:label="fig-0434-01a" xlink:href="fig-0434-01"/> SF, ad fruſtum hyperbolicum, HD <lb/>FG, ab eo dempto cylindro, HQ, eſſe <lb/>vt rectangulum, OEN, ad rectangulum <lb/>ſub compoſita ex {1/3}. </s> <s xml:id="echoid-s10819" xml:space="preserve">EM, integra, MN, <lb/>& </s> <s xml:id="echoid-s10820" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s10821" xml:space="preserve">NO. </s> <s xml:id="echoid-s10822" xml:space="preserve">Conum verò, DMF, ad idẽ <lb/>fruſtum, dempto cylindro, HQ, eſſe vt <lb/>rectangulum, OEN, ad rectangulum ſub <lb/>compoſita ex, EX, & </s> <s xml:id="echoid-s10823" xml:space="preserve">dupla, NM: </s> <s xml:id="echoid-s10824" xml:space="preserve">& </s> <s xml:id="echoid-s10825" xml:space="preserve">ſic <lb/>eſſe quæcunq; </s> <s xml:id="echoid-s10826" xml:space="preserve">ſolida ſimilaria genita ex <lb/>eiſdem figuris .</s> <s xml:id="echoid-s10827" xml:space="preserve">ſ. </s> <s xml:id="echoid-s10828" xml:space="preserve">parallelogrammo, SF, <lb/>fruſto hyperbolæ, HDFG, dempto ſo-<lb/>lido ſimilari genito ex, HQ, & </s> <s xml:id="echoid-s10829" xml:space="preserve">triangu-<lb/>lo, DMF, iuxta communem regulam, <lb/>DF.</s> <s xml:id="echoid-s10830" xml:space="preserve"/> </p> <div xml:id="echoid-div970" type="float" level="2" n="1"> <figure xlink:label="fig-0434-01" xlink:href="fig-0434-01a"> <image file="0434-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0434-01"/> </figure> </div> </div> <div xml:id="echoid-div972" type="section" level="1" n="579"> <head xml:id="echoid-head604" xml:space="preserve">COROLLARIV M V.</head> <p> <s xml:id="echoid-s10831" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10832" xml:space="preserve">5. </s> <s xml:id="echoid-s10833" xml:space="preserve">aſſumpta iterum figura Prop. </s> <s xml:id="echoid-s10834" xml:space="preserve">2. </s> <s xml:id="echoid-s10835" xml:space="preserve">dimiſsis rectis, CP, <lb/>RQ, Lk, & </s> <s xml:id="echoid-s10836" xml:space="preserve">triangulo, DM, &</s> <s xml:id="echoid-s10837" xml:space="preserve">, vt fiat ſolitum exemplum, ea <lb/> <anchor type="figure" xlink:label="fig-0434-02a" xlink:href="fig-0434-02"/> reuoluta circa axem, NE, vt ex, A <lb/>F, fiat cylindrus, AF, ex hyperbola, <lb/>DNF, conois, DNF, quæ ſolida <lb/>ſint ſecta plano, SZ, baſi, DF, ęqui-<lb/>diſtante, patet cylindrum, AF, dẽ-<lb/>pta conoide, DNF, ad cylindrum, <lb/>SF, dempto fruſto conoidis, DHG <lb/>F, eſſe vt parallelepipedum ſub cõ-<lb/>poſita ex, XE, EN, & </s> <s xml:id="echoid-s10838" xml:space="preserve">ſub quadrato, <lb/>NE, ad parallelepipedum ſub com-<lb/>poſita ex, XE, EN, NM, & </s> <s xml:id="echoid-s10839" xml:space="preserve">ſub qua-<lb/>drato, ME; </s> <s xml:id="echoid-s10840" xml:space="preserve">& </s> <s xml:id="echoid-s10841" xml:space="preserve">ſic eſſe quodlibet <lb/>ſolidum ſimilare genitum ex, AF, <lb/>dempto ſolido ſimilari genito ex <lb/>hyperbola, DNF, ad ſibi ſimilare <lb/>genitum ex, SF, dempto ſolido ſimilari genito ex fruſto hyperbo-<lb/>læ, DHFG, iuxta communem regulam, DF.</s> <s xml:id="echoid-s10842" xml:space="preserve"/> </p> <div xml:id="echoid-div972" type="float" level="2" n="1"> <figure xlink:label="fig-0434-02" xlink:href="fig-0434-02a"> <image file="0434-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0434-02"/> </figure> </div> <pb o="415" file="0435" n="435" rhead="LIBER V."/> </div> <div xml:id="echoid-div974" type="section" level="1" n="580"> <head xml:id="echoid-head605" xml:space="preserve">COROLLARIVM VI.</head> <p> <s xml:id="echoid-s10843" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10844" xml:space="preserve">6. </s> <s xml:id="echoid-s10845" xml:space="preserve">expoſita eius figura, & </s> <s xml:id="echoid-s10846" xml:space="preserve">vt fiat noſtrum exemplum ea-<lb/> <anchor type="figure" xlink:label="fig-0435-01a" xlink:href="fig-0435-01"/> dein circa, EM, reuoluta, patet pla-<lb/>num tranſiens per, SV, æquid ſtans ba-<lb/>ſi, FG, a cono de, FEG, reſecare conoi-<lb/>dem, SEV, quæ ad conum, SEV, habet <lb/>rationem datam, quam nempè habet H <lb/>R, ad, RL, idq; </s> <s xml:id="echoid-s10847" xml:space="preserve">diſcimus efficere quo-<lb/>cunque ſolido ſimilari exiſtente, FEG, <lb/>cuius figura genitrix ſit, FEG, à quo .</s> <s xml:id="echoid-s10848" xml:space="preserve">ſ. <lb/></s> <s xml:id="echoid-s10849" xml:space="preserve">ſciemus abſcindere per planũ baſi ęqui-<lb/>diſtans ſolidum ſibi ſimilare, quod nem-<lb/>pè erit genitũ ex hyperbola, SEV, quod <lb/>ad ſolidum ſibi ſimilare genitum ex triã-<lb/>gulo, SEV, habeat rationem datam, <lb/>dummodo data ratio ſit quidem maioris <lb/>in æqualitatis, ſed minor ſexquialtera.</s> <s xml:id="echoid-s10850" xml:space="preserve"/> </p> <div xml:id="echoid-div974" type="float" level="2" n="1"> <figure xlink:label="fig-0435-01" xlink:href="fig-0435-01a"> <image file="0435-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0435-01"/> </figure> </div> </div> <div xml:id="echoid-div976" type="section" level="1" n="581"> <head xml:id="echoid-head606" xml:space="preserve">COROLLARIVM VII.</head> <p> <s xml:id="echoid-s10851" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10852" xml:space="preserve">7. </s> <s xml:id="echoid-s10853" xml:space="preserve">expoſita eius figura, dimiſsis tamen rectis, ED, SO <lb/> <anchor type="figure" xlink:label="fig-0435-02a" xlink:href="fig-0435-02"/> XO, & </s> <s xml:id="echoid-s10854" xml:space="preserve">parallelogrammis, <lb/>GX, GR, eadem reuoluatur cir-<lb/>ca manentem axim, CV, vt ex <lb/>triangulo, HCR, fiat conus, H <lb/>CR, & </s> <s xml:id="echoid-s10855" xml:space="preserve">ex hyperbola, SOX, co-<lb/>nois, SOX, patet ergo conum, <lb/>HCR, ad conoidem, SOX, eſſe <lb/>in ratione compoſita ex ea, quã <lb/>habet quadratum, SX, ad qua-<lb/>dratum, HR, & </s> <s xml:id="echoid-s10856" xml:space="preserve">rectangulum, <lb/>AVO, ad rectangulum, BVC.</s> <s xml:id="echoid-s10857" xml:space="preserve"/> </p> <div xml:id="echoid-div976" type="float" level="2" n="1"> <figure xlink:label="fig-0435-02" xlink:href="fig-0435-02a"> <image file="0435-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0435-02"/> </figure> </div> <pb o="416" file="0436" n="436" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div978" type="section" level="1" n="582"> <head xml:id="echoid-head607" xml:space="preserve">COROLLARIVM VIII.</head> <p> <s xml:id="echoid-s10858" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10859" xml:space="preserve">8. </s> <s xml:id="echoid-s10860" xml:space="preserve">ſumpto exemplo ex anteced. </s> <s xml:id="echoid-s10861" xml:space="preserve">figura, in qu a trap <lb/>zium, THRP, in reuolutione genuit fruſtum coni, THRP, & </s> <s xml:id="echoid-s10862" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0436-01a" xlink:href="fig-0436-01"/> fruſtum hyperbolæ, IYXS, ge-<lb/>nuit fruſtum conoidis, IYXS; </s> <s xml:id="echoid-s10863" xml:space="preserve">pa-<lb/>tet fruſtum coni, THRP, ad fru-<lb/>ſtum conoidis, IYXS, habere ra-<lb/>tionem compoſitam ex ea, quã <lb/>habet rectangulum ſub, GP, VR, <lb/>cum {1/3}. </s> <s xml:id="echoid-s10864" xml:space="preserve">quadratiearum differen-<lb/>tiæ ad quadratum, VX, & </s> <s xml:id="echoid-s10865" xml:space="preserve">ex ea, <lb/>quam habet rectãgulum, BVO, <lb/>ad rectangulum ſub, BV, OG, <lb/>vna cum rectangulo ſub compo. <lb/></s> <s xml:id="echoid-s10866" xml:space="preserve">ſita ex {1/2}. </s> <s xml:id="echoid-s10867" xml:space="preserve">BO, & </s> <s xml:id="echoid-s10868" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s10869" xml:space="preserve">GV, & </s> <s xml:id="echoid-s10870" xml:space="preserve">ſub, <lb/>GV; </s> <s xml:id="echoid-s10871" xml:space="preserve">& </s> <s xml:id="echoid-s10872" xml:space="preserve">ſic eſſe quodlibet ſolidũ <lb/>ſimilare genitum ex trapezio, T <lb/>HRP, ad ſibi ſimilare genitum ex fruſto hyperbolæ, ISXY, iux-<lb/>ta communem regulam, HR.</s> <s xml:id="echoid-s10873" xml:space="preserve"/> </p> <div xml:id="echoid-div978" type="float" level="2" n="1"> <figure xlink:label="fig-0436-01" xlink:href="fig-0436-01a"> <image file="0436-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0436-01"/> </figure> </div> </div> <div xml:id="echoid-div980" type="section" level="1" n="583"> <head xml:id="echoid-head608" xml:space="preserve">COROLLARIVM IX.</head> <p> <s xml:id="echoid-s10874" xml:space="preserve">EX Propoſ. </s> <s xml:id="echoid-s10875" xml:space="preserve">9. </s> <s xml:id="echoid-s10876" xml:space="preserve">conſpecta figura Corollarij 7. </s> <s xml:id="echoid-s10877" xml:space="preserve">manifeſtò colli-<lb/>gitur Conum, HCR, ad conoidem, SOX, eſſe vt cubus, CV, <lb/>ad parallelepipedum ter ſub, CO. </s> <s xml:id="echoid-s10878" xml:space="preserve">& </s> <s xml:id="echoid-s10879" xml:space="preserve">quadrato, OV, cumcubo, O <lb/>V, & </s> <s xml:id="echoid-s10880" xml:space="preserve">ſic etiam eſſe quæcunq; </s> <s xml:id="echoid-s10881" xml:space="preserve">ſolida ſimilaria genita ex triangulo, <lb/>HCR, ad ſibi ſimilaria genita ex hyperbola, SOX, iuxta commu-<lb/>nem regulam, HR.</s> <s xml:id="echoid-s10882" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div981" type="section" level="1" n="584"> <head xml:id="echoid-head609" xml:space="preserve">COROLLARIVM X.</head> <p> <s xml:id="echoid-s10883" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10884" xml:space="preserve">10. </s> <s xml:id="echoid-s10885" xml:space="preserve">colligimus ſolida ſim laria genita ex hyperbolis, <lb/>AOC, OVX, habere inter ſe rationem compoſi@am ex ratio-<lb/>nibus ibi appoſitis, quæ breuitatis gratia inibi recola@tur.</s> <s xml:id="echoid-s10886" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div982" type="section" level="1" n="585"> <head xml:id="echoid-head610" xml:space="preserve">COROLLARIVM XI.</head> <p> <s xml:id="echoid-s10887" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s10888" xml:space="preserve">11. </s> <s xml:id="echoid-s10889" xml:space="preserve">expoſita eius figura, habemus conoides hyper-<lb/>bolicas ab eadem conoide diremptas. </s> <s xml:id="echoid-s10890" xml:space="preserve">habere inter ſe rationẽ <pb o="417" file="0437" n="437" rhead="LIBER V."/> compoſitam ex duabus rationibus ibidem appoſitis. </s> <s xml:id="echoid-s10891" xml:space="preserve">Vt autem <lb/>fiat noſtrum exemplum, intelligatur in ipia (in qua dimittantur <lb/>aſymptoti, & </s> <s xml:id="echoid-s10892" xml:space="preserve">rectæ, ad, DC, OV, VX, PO, PX,) BD, eſſe axem, <lb/>circa quam reuoluatur figura, vt ex hypeibola, ADC, fiat conois <lb/> <anchor type="figure" xlink:label="fig-0437-01a" xlink:href="fig-0437-01"/> hyperbolica, ADC; </s> <s xml:id="echoid-s10893" xml:space="preserve">vlterius per, <lb/>OX, traducatur planum, OX, ere-<lb/>ctum plano genitricis hyperbolæ, <lb/>ADC, cuius pars in conoide con-<lb/>cepta erit ellipſis, OX, cuius maior <lb/>diameter, OX, minor autem in fi-<lb/>gura propoſitionis linea, PO, ha-<lb/>bemus igitur ex Prop. </s> <s xml:id="echoid-s10894" xml:space="preserve">11. </s> <s xml:id="echoid-s10895" xml:space="preserve">conoi-<lb/>dem, ADC, ad conoidem, OVX, <lb/>habere rationem compoſitam ex <lb/>ratione rectanguli ſub, MB, HI, <lb/>ad rectangulum ſub, RI, FB, & </s> <s xml:id="echoid-s10896" xml:space="preserve">ex <lb/> <anchor type="note" xlink:label="note-0437-01a" xlink:href="note-0437-01"/> ratione parallelepipedi ſub altitu-<lb/>dine hyperbolæ, ADC, baſi qua-<lb/>drato, AC, ad parallelepipedum ſub altitudine hyperbolæ, OVX, <lb/>baſi autem rectangulo ſub, XO, OP, veluti ſunt omnia quadrata <lb/>hyperbolæ, ADC, regula, AC, ad omnia rectangula hyperbolæ, <lb/>OVX, (regula, OX,) ſimilia rectangulo ſub, XO, OP, ſiue omnes <lb/>circuli eiuſdem ad omnes ellipſes hyperbolæ, OVX, ſimiles ellipſi, <lb/> <anchor type="note" xlink:label="note-0437-02a" xlink:href="note-0437-02"/> cuius coniugati axes, vel diametri ſunt, XO, OP, XO, maior, OP, <lb/>minor, nam omnes dicti circuli ſunt omnia plana conoidis, ADC, <lb/>regula, AC, & </s> <s xml:id="echoid-s10897" xml:space="preserve">dictæ omnes ellipſes ſunt omnia plana conoidis, O <lb/>VX, eandem autem rationem ſupradictæ comperiemus habere <lb/>quæcunq; </s> <s xml:id="echoid-s10898" xml:space="preserve">lolida non quidem ſimilaria inter ſe, ſed quorum om-<lb/>nia plana ſint omnes figuræ ſimiles genitricium figurarum, ADC, <lb/>OVX, a quibus genita dicuntur, quæ habeant inter ſeeandem ra-<lb/>tionem ei, quam habet quadratum, AC, ad rectangulum, XOP.</s> <s xml:id="echoid-s10899" xml:space="preserve"/> </p> <div xml:id="echoid-div982" type="float" level="2" n="1"> <figure xlink:label="fig-0437-01" xlink:href="fig-0437-01a"> <image file="0437-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0437-01"/> </figure> <note position="right" xlink:label="note-0437-01" xlink:href="note-0437-01a" xml:space="preserve">43. l. 1. <lb/>Coro. 44. <lb/>l. 1.</note> <note position="right" xlink:label="note-0437-02" xlink:href="note-0437-02a" xml:space="preserve">Corol. 2. <lb/>33. l. 2.</note> </div> </div> <div xml:id="echoid-div984" type="section" level="1" n="586"> <head xml:id="echoid-head611" xml:space="preserve">COR OLLARIVM XII.</head> <p> <s xml:id="echoid-s10900" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s10901" xml:space="preserve">12. </s> <s xml:id="echoid-s10902" xml:space="preserve">conſpecta illius figura, & </s> <s xml:id="echoid-s10903" xml:space="preserve">completis conoidibus, <lb/>BAD, HMQ, patet eorum rationem eſſe compoſitam ex ra-<lb/>tionibus ibi explicatis, vbi videri poterunt. </s> <s xml:id="echoid-s10904" xml:space="preserve">Quas quidem ratio-<lb/>nes comperiemus etiam habere quæcunq; </s> <s xml:id="echoid-s10905" xml:space="preserve">ſolida, licet etiam non <lb/>ſimilaria ad inuicem, genita tamen ex eildem figuris, quarum om-<lb/>nes figuræ ſimiles (inter ſe, quę ſunt vnius, vtriuſq; </s> <s xml:id="echoid-s10906" xml:space="preserve">tamen figuræ <lb/>genitricis diſſimiles) habeant eandem rationem, quam habent <pb o="418" file="0438" n="438" rhead="GEOMETRIÆ"/> prædicta omnia quadrata, vel rectangula, vt ſupra ad inuicem <lb/>comparata.</s> <s xml:id="echoid-s10907" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div985" type="section" level="1" n="587"> <head xml:id="echoid-head612" xml:space="preserve">COROLLARIVM XIII.</head> <p> <s xml:id="echoid-s10908" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10909" xml:space="preserve">13. </s> <s xml:id="echoid-s10910" xml:space="preserve">habetur ſimiles conoides hyperbolicas eſſe in tri-<lb/> <anchor type="note" xlink:label="note-0438-01a" xlink:href="note-0438-01"/> pla ratione axium, vel diametrorum earundem, quippe quæ <lb/>ex ſimilibus hyperbolis naſcuntur: </s> <s xml:id="echoid-s10911" xml:space="preserve">lgitur in anteced. </s> <s xml:id="echoid-s10912" xml:space="preserve">Corollarij <lb/>figura, ſi ſupponantur ſimiles hyperbolæ, BAD, HMQ, vt fiant <lb/>ex illis ſimiles conoides hyperbolicæ, FEG, HTS, iſtæ erunt inter <lb/>ſe in tripla ratione axium, AC, MP, & </s> <s xml:id="echoid-s10913" xml:space="preserve">ſic erit quodlibet ſolidum <lb/> <anchor type="note" xlink:label="note-0438-02a" xlink:href="note-0438-02"/> ſimilare genitum ex hyperbola, FEG, ad ſibi ſimilare genitum ex <lb/>hyperbola, HTS, iuxta regulas, FG, HS.</s> <s xml:id="echoid-s10914" xml:space="preserve"/> </p> <div xml:id="echoid-div985" type="float" level="2" n="1"> <note position="left" xlink:label="note-0438-01" xlink:href="note-0438-01a" xml:space="preserve">Eliciture 1 <lb/>Coro. 50. <lb/>l. 1.</note> <note position="left" xlink:label="note-0438-02" xlink:href="note-0438-02a" xml:space="preserve">Coro. 50. <lb/>l. 1.</note> </div> </div> <div xml:id="echoid-div987" type="section" level="1" n="588"> <head xml:id="echoid-head613" xml:space="preserve">COROLLARIV M XIV.</head> <p> <s xml:id="echoid-s10915" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10916" xml:space="preserve">14. </s> <s xml:id="echoid-s10917" xml:space="preserve">expoſita eius figura, vt fiat exem plum, reuolua-<lb/> <anchor type="figure" xlink:label="fig-0438-01a" xlink:href="fig-0438-01"/> tur circa axim, DG, vt ex, EG, fiat <lb/>cylindrus, EF, & </s> <s xml:id="echoid-s10918" xml:space="preserve">ex trilineo, DGB, ſo-<lb/>lidum, DBGF, quod vocetur: </s> <s xml:id="echoid-s10919" xml:space="preserve">Apex <lb/>hyperbolicus; </s> <s xml:id="echoid-s10920" xml:space="preserve">patet ergo cylindrum, E <lb/>F, ad apicem, BDF, eſſe vt, BD, ad ſui <lb/>reliquum, dempta ab eodem ſemihy-<lb/>perbola, BED, vna cum exceſſu, quo <lb/>ipſa ſuperat {1/3}. </s> <s xml:id="echoid-s10921" xml:space="preserve">parall: </s> <s xml:id="echoid-s10922" xml:space="preserve">logrammi, BD, & </s> <s xml:id="echoid-s10923" xml:space="preserve"><lb/>{1/6}. </s> <s xml:id="echoid-s10924" xml:space="preserve">BM; </s> <s xml:id="echoid-s10925" xml:space="preserve">& </s> <s xml:id="echoid-s10926" xml:space="preserve">ſic eſſe patet, quodlibet ſoli-<lb/>dum ſimilate genitum ex, BD, ad ſibi <lb/>ſimilare genitum ex ſemihyperbola, BE <lb/>D, iuxta communem regulam, ED.</s> <s xml:id="echoid-s10927" xml:space="preserve"/> </p> <div xml:id="echoid-div987" type="float" level="2" n="1"> <figure xlink:label="fig-0438-01" xlink:href="fig-0438-01a"> <image file="0438-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0438-01"/> </figure> </div> </div> <div xml:id="echoid-div989" type="section" level="1" n="589"> <head xml:id="echoid-head614" xml:space="preserve">COROLLARIVM XV.</head> <p> <s xml:id="echoid-s10928" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s10929" xml:space="preserve">15. </s> <s xml:id="echoid-s10930" xml:space="preserve">expoſita <lb/> <anchor type="figure" xlink:label="fig-0438-02a" xlink:href="fig-0438-02"/> eius figura, vt fiat exẽ-<lb/>p@u@n, eadem voluatur cir-<lb/>ca axim, GD, vt ex, FD, <lb/>fiat cylindrus, FH, & </s> <s xml:id="echoid-s10931" xml:space="preserve">ex <lb/>hyperbola, CBD, ſolidum, <lb/>CBDNH, quod voeetur: <lb/></s> <s xml:id="echoid-s10932" xml:space="preserve">Sem anulus ſtr@ctus hyper-<lb/>bolicus: </s> <s xml:id="echoid-s10933" xml:space="preserve">@ntelligantur autẽ <lb/>ſe@nper hæc ſolida ſecari <lb/>per axem, vt ijs producan-<lb/>tur figuræ, quæ in reuolu- <pb o="419" file="0439" n="439" rhead="LIBER V."/> tione eadem generant, nempè ex tenſo plano, FD, per axem, G <lb/>D, produci figuram, FH, compoſitam ex duobus parallelogram-<lb/>mis, FD, DM, & </s> <s xml:id="echoid-s10934" xml:space="preserve">figuram, CBDNH, compoſitam ex duabus hy-<lb/>perbolis, CBD, DNH; </s> <s xml:id="echoid-s10935" xml:space="preserve">patet ergo cylindrum, FH, ad ſolidum, C <lb/>BDNH, eſſe vt, FD, ad hyperbolam, CBD, & </s> <s xml:id="echoid-s10936" xml:space="preserve">ſic eſſe quodlibet <lb/>ſolidum ſimilare genitum ex, FD, ad ſibi ſimilare genitum ex hy-<lb/>perbola, CBD, iuxta communem regulam, CD.</s> <s xml:id="echoid-s10937" xml:space="preserve"/> </p> <div xml:id="echoid-div989" type="float" level="2" n="1"> <figure xlink:label="fig-0438-02" xlink:href="fig-0438-02a"> <image file="0438-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0438-02"/> </figure> </div> </div> <div xml:id="echoid-div991" type="section" level="1" n="590"> <head xml:id="echoid-head615" xml:space="preserve">COROLL. XVI. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s10938" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10939" xml:space="preserve">16. </s> <s xml:id="echoid-s10940" xml:space="preserve">vt fiat exemplum, reuoluatur eius figura circa <lb/>axim, HM, vt ex, FM, fiat cylindrus, FQ, & </s> <s xml:id="echoid-s10941" xml:space="preserve">ex hyperbola, C <lb/> <anchor type="figure" xlink:label="fig-0439-01a" xlink:href="fig-0439-01"/> BD, ſolidum, <lb/>CBD, SOQ, <lb/>quod voce-<lb/>tur: </s> <s xml:id="echoid-s10942" xml:space="preserve">Semia-<lb/>nulus latus <lb/>hyperbolicus <lb/>patet ergo cy <lb/>lindrum, FQ, <lb/>ad ſemianulũ <lb/>latum hyper-<lb/>bolicum, CB <lb/>DSOQ, eſſe <lb/>vt, FD, ad <lb/>hyperbolam, CBD, & </s> <s xml:id="echoid-s10943" xml:space="preserve">ſic eſſe quodlibet ſolidum ſimilare genitum <lb/>ex, FD, ad ſibi ſimilare genitu@n ex hyperbola, CBD, iuxta com-<lb/>munem regulam, CD.</s> <s xml:id="echoid-s10944" xml:space="preserve"/> </p> <div xml:id="echoid-div991" type="float" level="2" n="1"> <figure xlink:label="fig-0439-01" xlink:href="fig-0439-01a"> <image file="0439-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0439-01"/> </figure> </div> </div> <div xml:id="echoid-div993" type="section" level="1" n="591"> <head xml:id="echoid-head616" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s10945" xml:space="preserve">VNde habetur ex Corollario cylindrum, FH, in figura Co-<lb/>rollarij antecedentis ad ſemianulum ſtrictum hyperbolicu, <lb/>CBDNH, eſſe vt cylindrum, FQ, in ſigura huius Corollar jad <lb/>ſemianulum latum hyperbolicum, CBDSOQ, & </s> <s xml:id="echoid-s10946" xml:space="preserve">ſic ſolida ſimi-<lb/>laria, &</s> <s xml:id="echoid-s10947" xml:space="preserve">c.</s> <s xml:id="echoid-s10948" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div994" type="section" level="1" n="592"> <head xml:id="echoid-head617" xml:space="preserve">COROLLARIVM XVII.</head> <p> <s xml:id="echoid-s10949" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10950" xml:space="preserve">17. </s> <s xml:id="echoid-s10951" xml:space="preserve">ſi, ducta parallela axi, vel diametro hyperbolæ, B <lb/>E, ſit, GD, patet in ſigura Corollarij 15. </s> <s xml:id="echoid-s10952" xml:space="preserve">quam ra@onem ha- <pb o="420" file="0440" n="440" rhead="GEOMETRIÆ"/> beat cylindrus, FH, ad ſolidum, CBNH, quod vocetur: </s> <s xml:id="echoid-s10953" xml:space="preserve">Semiba-<lb/>ſis columnaris ſtricta hyperbolica: </s> <s xml:id="echoid-s10954" xml:space="preserve">Si verò dicta parallela ſit, HM, <lb/> <anchor type="figure" xlink:label="fig-0440-01a" xlink:href="fig-0440-01"/> patet in figura Corollarij <lb/>anteced. </s> <s xml:id="echoid-s10955" xml:space="preserve">quam rationem <lb/>habeat cylindrus, FQ, ad <lb/>ſolidum, CBOQ, quod vo-<lb/>cetur: </s> <s xml:id="echoid-s10956" xml:space="preserve">Semibaſis colum-<lb/>naris lata hyperbolica: </s> <s xml:id="echoid-s10957" xml:space="preserve">ſi <lb/>tandem ſit, RS, voluto, FS, <lb/>circa axim, RS, vt ex, FS, <lb/>fiat cylindrus, FH, & </s> <s xml:id="echoid-s10958" xml:space="preserve">ex fi-<lb/>gura, CBRS, ſolidum, CB <lb/>DH, quod vocetur: </s> <s xml:id="echoid-s10959" xml:space="preserve">Se-<lb/>mibaſis columnaris media <lb/>hyperbolica: </s> <s xml:id="echoid-s10960" xml:space="preserve">patet cylin-<lb/>drum, FH, ad ſemibaſim, CBDH, eſſe vt quadratum, CS, ad qua-<lb/>dratum, SE, quadratum, EI, & </s> <s xml:id="echoid-s10961" xml:space="preserve">rectangulum bis ſub, VE, ES, vt <lb/>& </s> <s xml:id="echoid-s10962" xml:space="preserve">ſolida ſimilaria ex eiſdem genita iuxta communẽ regulam, CS.</s> <s xml:id="echoid-s10963" xml:space="preserve"/> </p> <div xml:id="echoid-div994" type="float" level="2" n="1"> <figure xlink:label="fig-0440-01" xlink:href="fig-0440-01a"> <image file="0440-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0440-01"/> </figure> </div> </div> <div xml:id="echoid-div996" type="section" level="1" n="593"> <head xml:id="echoid-head618" xml:space="preserve">COROLLARIVM XVIII.</head> <p> <s xml:id="echoid-s10964" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10965" xml:space="preserve">18. </s> <s xml:id="echoid-s10966" xml:space="preserve">habetur, viſis proximis antecedentibus figuris, ſe-<lb/>m@anulum latuin hyperb@l@cum, CBDSOQ, ad ſemianulum <lb/>ſtrictum hyperbo icum, CBDNH, eſſe vt, CM, MD, ad, DC; </s> <s xml:id="echoid-s10967" xml:space="preserve">& </s> <s xml:id="echoid-s10968" xml:space="preserve"><lb/>ſic ſolida ſimilaria, &</s> <s xml:id="echoid-s10969" xml:space="preserve">c.</s> <s xml:id="echoid-s10970" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div997" type="section" level="1" n="594"> <head xml:id="echoid-head619" xml:space="preserve">COROLL. XIX. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s10971" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10972" xml:space="preserve">19. </s> <s xml:id="echoid-s10973" xml:space="preserve">habetur, viſa figura Corollarij 15. </s> <s xml:id="echoid-s10974" xml:space="preserve">conoidem hy-<lb/>perbo@icam genitam ex ſem@hyperbola, CBE, ad ſem@anulũ <lb/>@tr@ctum hyperbolicum, CBDNH, eſſe vt quadiatum, IE, ad re-<lb/>ctangulum ſub, CD, & </s> <s xml:id="echoid-s10975" xml:space="preserve">dupla, VE.</s> <s xml:id="echoid-s10976" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div998" type="section" level="1" n="595"> <head xml:id="echoid-head620" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s10977" xml:space="preserve">VNde in Corollario colligitur eandem conoidem ad ſemianu-<lb/>lum latum hyperbolicum eſſe vt quadratum, EI, ad rectan-<lb/>gulum ſub compoſitæex, CM, MD, & </s> <s xml:id="echoid-s10978" xml:space="preserve">ſub dupla, VE, & </s> <s xml:id="echoid-s10979" xml:space="preserve">ſic ſoli-<lb/>da ſim@laria ex eiſdem figuris genita iuxta communem regulam, <lb/>CD.</s> <s xml:id="echoid-s10980" xml:space="preserve"/> </p> <pb o="421" file="0441" n="441" rhead="LIBER V."/> </div> <div xml:id="echoid-div999" type="section" level="1" n="596"> <head xml:id="echoid-head621" xml:space="preserve">COROLLARIVM XX.</head> <p> <s xml:id="echoid-s10981" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10982" xml:space="preserve">20. </s> <s xml:id="echoid-s10983" xml:space="preserve">expoſita eius figura, & </s> <s xml:id="echoid-s10984" xml:space="preserve">vt fiatſolitum exemplum; <lb/></s> <s xml:id="echoid-s10985" xml:space="preserve">ea circa axim, FE, reuoluta, vt ex, BE, fiat cylindrus, BC, & </s> <s xml:id="echoid-s10986" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0441-01a" xlink:href="fig-0441-01"/> ex oppoſitis hyperbolis, BND, AM <lb/>C, fiant conoides, BND, AMC, quę <lb/>pariter dicantur; </s> <s xml:id="echoid-s10987" xml:space="preserve">Conoides oppoſitę, <lb/>Pitet cylindrum, BC, ad reliquum, <lb/>demptis ab eodem oppoſitis cono@di-<lb/>bus, AMC, BND, eſſe vt rectangulũ, <lb/>NEO, ad rectangulum, NOE, bis, <lb/>cum {2/3}. </s> <s xml:id="echoid-s10988" xml:space="preserve">quadrati, ME; </s> <s xml:id="echoid-s10989" xml:space="preserve">& </s> <s xml:id="echoid-s10990" xml:space="preserve">ſic eſſe quod-<lb/>libet lolidum ſimilare genitum ex, B <lb/>E, ad reliquum, demptis ab e@dem <lb/>ſolid@s ſimilaribus gen tis ex ſemihy-<lb/>perbolis, BNF, AME, vel ſic ſolidum <lb/>quodlibet ſimilare genitum ex, BC, ad <lb/>relliquum ab eodem, dempt@s ſolidis <lb/>ſimilaribus genitis ex hyperbolis op-<lb/>poſitis, BND, AMC, iuxta commu-<lb/>nem regulam, AC.</s> <s xml:id="echoid-s10991" xml:space="preserve"/> </p> <div xml:id="echoid-div999" type="float" level="2" n="1"> <figure xlink:label="fig-0441-01" xlink:href="fig-0441-01a"> <image file="0441-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0441-01"/> </figure> </div> </div> <div xml:id="echoid-div1001" type="section" level="1" n="597"> <head xml:id="echoid-head622" xml:space="preserve">COROLLARIVM XXL</head> <p> <s xml:id="echoid-s10992" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s10993" xml:space="preserve">21. </s> <s xml:id="echoid-s10994" xml:space="preserve">expoſita eius figura, & </s> <s xml:id="echoid-s10995" xml:space="preserve">ea circa axim, LX, reuo-<lb/>luta, vt ex, DX, fiat cylindrus, DE, & </s> <s xml:id="echoid-s10996" xml:space="preserve">ex figura, DAFEVC, <lb/> <anchor type="figure" xlink:label="fig-0441-02a" xlink:href="fig-0441-02"/> ſolidum, DAFEVC, <lb/>quod vocetur: </s> <s xml:id="echoid-s10997" xml:space="preserve">@ym-<lb/>panum hyperbolicũ: <lb/></s> <s xml:id="echoid-s10998" xml:space="preserve">& </s> <s xml:id="echoid-s10999" xml:space="preserve">ex triangulis, HL <lb/>O, OXN, oppoſit <lb/>coni, HOS, NOY, <lb/>patet cylindrum, D <lb/>E, adtympanum, D <lb/>AFEVC, eſſe vt qua-<lb/>dratum, FE, ad qua <lb/>dratum, AV, cum<emph style="sub">1</emph>. </s> <s xml:id="echoid-s11000" xml:space="preserve"><lb/>quadrat@ HS,. </s> <s xml:id="echoid-s11001" xml:space="preserve">Ve (vt aliter ibiexplicatur) vt quadratum, RZ, <lb/>ad quadratum, AV, vna cum rectangulo ſub, AZ, & </s> <s xml:id="echoid-s11002" xml:space="preserve">rexquitertia, <lb/>ZV; </s> <s xml:id="echoid-s11003" xml:space="preserve">& </s> <s xml:id="echoid-s11004" xml:space="preserve">ſic eſſe quod@bet ſolidum ſimilare genitum ex, DE, ad ſibi <pb o="422" file="0442" n="442" rhead="GEOMETRIÆ"/> ſimilare genitum ex figura, DAFEVC, iuxta communem regu-<lb/>lam, FE.</s> <s xml:id="echoid-s11005" xml:space="preserve"/> </p> <div xml:id="echoid-div1001" type="float" level="2" n="1"> <figure xlink:label="fig-0441-02" xlink:href="fig-0441-02a"> <image file="0441-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0441-02"/> </figure> </div> </div> <div xml:id="echoid-div1003" type="section" level="1" n="598"> <head xml:id="echoid-head623" xml:space="preserve">COROLL. XXII. SECTIO PRIMA.</head> <p> <s xml:id="echoid-s11006" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s11007" xml:space="preserve">22. </s> <s xml:id="echoid-s11008" xml:space="preserve">ſi in ſuperioris figura ſupponamus, FR, eſſe æqua-<lb/>lem ei, quæ tangens ſectionem, DAF, in, A, concluditur in-<lb/>ter, A, & </s> <s xml:id="echoid-s11009" xml:space="preserve">aſymptoton, ON, habetur cylindrum, DE, eſſe ſexquial-<lb/>terum tympani hyperbol ci, DAFEVC, & </s> <s xml:id="echoid-s11010" xml:space="preserve">hoc tympanũ eſſe qua-<lb/>druplum conorum, NOY, HOS, & </s> <s xml:id="echoid-s11011" xml:space="preserve">ſic eſſe ſolida ſimilaria ex <lb/>eiſdem figuris genita iuxta communem regulam, DC.</s> <s xml:id="echoid-s11012" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1004" type="section" level="1" n="599"> <head xml:id="echoid-head624" xml:space="preserve">SECTIO II.</head> <p> <s xml:id="echoid-s11013" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s11014" xml:space="preserve">1. </s> <s xml:id="echoid-s11015" xml:space="preserve">habetur cylindrum, DE, ad tympanum hyperbo-<lb/>licum, DAFEVC, demptis conis oppoſitis, HOS, NOY, eſſe <lb/>vt quadratum, DC, ad quadratum, AV, & </s> <s xml:id="echoid-s11016" xml:space="preserve">in caſu præſentis Prop. <lb/></s> <s xml:id="echoid-s11017" xml:space="preserve">eſſe eorum dupla, & </s> <s xml:id="echoid-s11018" xml:space="preserve">ſic ſolida ſimilaria, &</s> <s xml:id="echoid-s11019" xml:space="preserve">c.</s> <s xml:id="echoid-s11020" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1005" type="section" level="1" n="600"> <head xml:id="echoid-head625" xml:space="preserve">SECTIO III.</head> <p> <s xml:id="echoid-s11021" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s11022" xml:space="preserve">2. </s> <s xml:id="echoid-s11023" xml:space="preserve">diſcimus inuenire cylindrum deſcriptum à paral-<lb/>lelogrammo ſectionibus oppoſitis circumſcripto, vt ibi dicitur, <lb/>quod ad reliquum tympani hyperbolici, demptis oppoſitis conis, <lb/>habeatrationem datam, dummodo ea ſit maioris inæqualit. </s> <s xml:id="echoid-s11024" xml:space="preserve">idem <lb/>intellige de ſol dis ſimilaribus, &</s> <s xml:id="echoid-s11025" xml:space="preserve">c.</s> <s xml:id="echoid-s11026" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1006" type="section" level="1" n="601"> <head xml:id="echoid-head626" xml:space="preserve">COROLLARIVM XXIII.</head> <p> <s xml:id="echoid-s11027" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s11028" xml:space="preserve">23. </s> <s xml:id="echoid-s11029" xml:space="preserve">aſſumpta eius figura, &</s> <s xml:id="echoid-s11030" xml:space="preserve">, <lb/> <anchor type="figure" xlink:label="fig-0442-01a" xlink:href="fig-0442-01"/> vt fiat exemplum, ea circa axim, RZ, <lb/>reuoluta, vt ex, FC, fiat cylindrus, FC, & </s> <s xml:id="echoid-s11031" xml:space="preserve"><lb/>ex, TN, cylindrus, TN, & </s> <s xml:id="echoid-s11032" xml:space="preserve">ex oppoſitis, <lb/>hyperbolis, FAD, EVC, oppoſitæ conoi-<lb/>des, FAD, EVC, & </s> <s xml:id="echoid-s11033" xml:space="preserve">ex, TAY, MVN, op-<lb/>poſitæ, conoides, TAY, MVN, patet er-<lb/>go cylindrum, FC, demotis oppoſitis co-<lb/>noidibus, FAD, EVC, ad cylindrum, TN, <lb/>demptis oppoſitis conoidibus, TAX, MV <lb/>N, eſſe vt parallelepipedum ſub, ZV, & </s> <s xml:id="echoid-s11034" xml:space="preserve"><lb/>baſirectangulo, VOZ, cum {1/3}. </s> <s xml:id="echoid-s11035" xml:space="preserve">quadrati, <lb/>ZV, ad parallelepipedum ſub, ZV, baſi <lb/>rectangulo, VOS, cum {1/3}. </s> <s xml:id="echoid-s11036" xml:space="preserve">quadrati, SV, &</s> <s xml:id="echoid-s11037" xml:space="preserve"> <pb o="423" file="0443" n="443" rhead="LIBER V."/> ſic eſſe quodlibet ſolidum ſimilare genitum ex, FC, demptis ſoli-<lb/>dis ſimilaribus genitis ex oppoſitis hypeibolis, FAD, EVC, ad @o-<lb/>lidum ſibi ſimilare genitum ex, TN, demptis ſolidis ſimilaribus <lb/>genitis ex oppoſitis hyperbolis, TAY, MVN, iuxta communem <lb/>regulam, EC.</s> <s xml:id="echoid-s11038" xml:space="preserve"/> </p> <div xml:id="echoid-div1006" type="float" level="2" n="1"> <figure xlink:label="fig-0442-01" xlink:href="fig-0442-01a"> <image file="0442-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0442-01"/> </figure> </div> </div> <div xml:id="echoid-div1008" type="section" level="1" n="602"> <head xml:id="echoid-head627" xml:space="preserve">COROLLARIVM XXIV.</head> <p> <s xml:id="echoid-s11039" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s11040" xml:space="preserve">24. </s> <s xml:id="echoid-s11041" xml:space="preserve">expoſita eius figura, &</s> <s xml:id="echoid-s11042" xml:space="preserve">, vt fiat exemplum, ea <lb/>circa axem, XL, reuoluta, vt ex figura, EV@DAF, fiat tyin-<lb/>panum hyperbolicum, EVCDAF, & </s> <s xml:id="echoid-s11043" xml:space="preserve">ex figura, MVNYAT, fiat <lb/> <anchor type="figure" xlink:label="fig-0443-01a" xlink:href="fig-0443-01"/> tympanum hyperbolicum, <lb/>MVNYAT, patet tympa-<lb/>num, EVCDAF, ad tympa-<lb/>num, MVNYAT, eſſe vt <lb/>parallelep pedum ſub, XL, <lb/>& </s> <s xml:id="echoid-s11044" xml:space="preserve">quadrato, RZ, cum duplo <lb/>quadrati, AV, ad parallele <lb/>pipedum ſub, HG, & </s> <s xml:id="echoid-s11045" xml:space="preserve">qua <lb/>drato, B@, cum duplo qua-<lb/>drati, AV; </s> <s xml:id="echoid-s11046" xml:space="preserve">& </s> <s xml:id="echoid-s11047" xml:space="preserve">ſic eſſe quod-<lb/>libet ſolidum ſim@lare genitũ <lb/>ex figura, EVCDAF, ad ſibi ſimilare genitum ex figura, MVNY <lb/>AT, iuxta communem regulam, CD.</s> <s xml:id="echoid-s11048" xml:space="preserve"/> </p> <div xml:id="echoid-div1008" type="float" level="2" n="1"> <figure xlink:label="fig-0443-01" xlink:href="fig-0443-01a"> <image file="0443-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0443-01"/> </figure> </div> </div> <div xml:id="echoid-div1010" type="section" level="1" n="603"> <head xml:id="echoid-head628" xml:space="preserve">COROLLARIVM XXV.</head> <p> <s xml:id="echoid-s11049" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s11050" xml:space="preserve">25. </s> <s xml:id="echoid-s11051" xml:space="preserve">viſa figura anteced. </s> <s xml:id="echoid-s11052" xml:space="preserve">Coroll in qua ex triangulis, <lb/>QOP, IOK, geniti ſint coni opooſiti, QOP, IOK, & </s> <s xml:id="echoid-s11053" xml:space="preserve">ex trian-<lb/>gulis, ΠΟΩ, ℟O&</s> <s xml:id="echoid-s11054" xml:space="preserve">, coni oppoſiti. </s> <s xml:id="echoid-s11055" xml:space="preserve">ΠΟΩ, ℟O&</s> <s xml:id="echoid-s11056" xml:space="preserve">, patet tympanu, <lb/>EVCDAF, demptis con@s, QOP, IOK, ad tympanum, MVNYA <lb/>T, demptis conis, ΠΟΩ ℟O&</s> <s xml:id="echoid-s11057" xml:space="preserve">, eſſe vt, XL, ad, HG; </s> <s xml:id="echoid-s11058" xml:space="preserve">& </s> <s xml:id="echoid-s11059" xml:space="preserve">ſic eſſe ſo-<lb/>lidum ſimilare genitum ex figura, EV@DAF, demptis ſolidis ſimi-<lb/>laribus genitis ex triangulis, QOP, IOk, ad ſolidum ſimilare geni-<lb/>tum ex figura, TAYNVM, demptis ſolidis ſimilaribus genitis ex <lb/>triangulis, &</s> <s xml:id="echoid-s11060" xml:space="preserve">O℟, Ω@Π, @uxta communem regulam, AV.</s> <s xml:id="echoid-s11061" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1011" type="section" level="1" n="604"> <head xml:id="echoid-head629" xml:space="preserve">COROLLARIVM XXVI.</head> <p> <s xml:id="echoid-s11062" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s11063" xml:space="preserve">26. </s> <s xml:id="echoid-s11064" xml:space="preserve">viſis figuris Corollarij 23. </s> <s xml:id="echoid-s11065" xml:space="preserve">24. </s> <s xml:id="echoid-s11066" xml:space="preserve">& </s> <s xml:id="echoid-s11067" xml:space="preserve">ſuppoſito, FC, <lb/>eſſe idem parallelogrammum, in vunq; </s> <s xml:id="echoid-s11068" xml:space="preserve">figuris pate@ cylin- <pb o="424" file="0444" n="444" rhead="GEOMETRIÆ"/> drum, FC, in figura Coroll. </s> <s xml:id="echoid-s11069" xml:space="preserve">23. </s> <s xml:id="echoid-s11070" xml:space="preserve">demptis oppoſitis conoidibus, F <lb/>AD, EVC, ad tympanum hyperbolicum genitum ex figura, EVC <lb/>DAF, in figura Coroll. </s> <s xml:id="echoid-s11071" xml:space="preserve">24. </s> <s xml:id="echoid-s11072" xml:space="preserve">ſcilicet ad tympanum hyperbolicum, <lb/>EVCDAF, habere rationem compoſitam ex ratione rectanguli, <lb/>AOZ, bis cum {2/3}. </s> <s xml:id="echoid-s11073" xml:space="preserve">quadrati, VZ, ad rectangulum, AZO, & </s> <s xml:id="echoid-s11074" xml:space="preserve">ex ra-<lb/>tione rectanguli ſub, DC, vel, RZ, & </s> <s xml:id="echoid-s11075" xml:space="preserve">ſub, EC, ad quadratum, A <lb/>V, cum {1/3}. </s> <s xml:id="echoid-s11076" xml:space="preserve">quadrati, KI, vel cum rectangulo ſub, AZ, & </s> <s xml:id="echoid-s11077" xml:space="preserve">ſexqui. <lb/></s> <s xml:id="echoid-s11078" xml:space="preserve">tertia, ZV, & </s> <s xml:id="echoid-s11079" xml:space="preserve">ſic eſſe quodlibet ſolidum ſimilare genitum ex, FC, <lb/>de@nptis ſolidis ſimilaribus genitis ex oppoſitis hyperbolis, FAD, <lb/>EVC, iuxta communem regulam, EC, ad ſolidum ſimilare ſibi <lb/>genitum ex figura, EVCDAF, iuxta regulam, CD.</s> <s xml:id="echoid-s11080" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1012" type="section" level="1" n="605"> <head xml:id="echoid-head630" xml:space="preserve">COROLLARIVM XXVII.</head> <p> <s xml:id="echoid-s11081" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s11082" xml:space="preserve">27. </s> <s xml:id="echoid-s11083" xml:space="preserve">conſpecta figura Coroll. </s> <s xml:id="echoid-s11084" xml:space="preserve">21. </s> <s xml:id="echoid-s11085" xml:space="preserve">intelligantur deſcri-<lb/>ptæ ſectiones, BIG, kMP, quæ dicuntur coniugatæ ſectioni-<lb/>bus, DAF, CVE, ex quibus in reuolutione genitæ fuerint oppoſi-<lb/>tæ conoides, B G, kMP, patet igitur cylindrum, DE, ad tympa-<lb/>num hyperbolicum, DAF, EVC, demptis oppoſitis conoidibus, <lb/>BIG, KMP, eſſe vt parallelepipedum ſub, ZC, & </s> <s xml:id="echoid-s11086" xml:space="preserve">ſub quadrato, Z <lb/>Q (quæ habetur extenſa, ZC, ad aſymptoton producta .</s> <s xml:id="echoid-s11087" xml:space="preserve">ſ. </s> <s xml:id="echoid-s11088" xml:space="preserve">ad, OS, <lb/>cu@occurrat in, Q,) a@ parallele ipedum bis ſub, XM, & </s> <s xml:id="echoid-s11089" xml:space="preserve">quadra-<lb/>to, MO, cum cubo, MO, & </s> <s xml:id="echoid-s11090" xml:space="preserve">amplius {1/3}. </s> <s xml:id="echoid-s11091" xml:space="preserve">eiuſdem cubi; </s> <s xml:id="echoid-s11092" xml:space="preserve">& </s> <s xml:id="echoid-s11093" xml:space="preserve">ſic eſſe <lb/>quodlibet ſolidum ſimilare genitum ex, FC, ad ſibi ſimlare g@nitũ <lb/>ex figura, DAFEVC, demptis ſolidis ſimilaribus genitis ex oppo-<lb/>ſitis hyperbolis, BIG, KMP, iuxta cõmunẽ regulam, FE, vel, AV.</s> <s xml:id="echoid-s11094" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1013" type="section" level="1" n="606"> <head xml:id="echoid-head631" xml:space="preserve">COROLL. XXVIII. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s11095" xml:space="preserve">IN Prop. </s> <s xml:id="echoid-s11096" xml:space="preserve">28. </s> <s xml:id="echoid-s11097" xml:space="preserve">illius aſſumpta figu-<lb/> <anchor type="figure" xlink:label="fig-0444-01a" xlink:href="fig-0444-01"/> ra, eadem reuoluatur circa axẽ, <lb/>&</s> <s xml:id="echoid-s11098" xml:space="preserve">Y, vel, GZ, ſit autem reuolutio <lb/>circa, &</s> <s xml:id="echoid-s11099" xml:space="preserve">Y; </s> <s xml:id="echoid-s11100" xml:space="preserve">patet ergo cylindrum ge-<lb/>nitum ex, TV, nempè, TV, ad reli-<lb/>quum, ab eodem demptis ſolidis ge <lb/>nitis ex quatuor hyperbolis coniu-<lb/>gatis, BFH, PIQ, AEC, MON, eſſe <lb/>vt cubus, ZV, vel, SY, ad parallele-<lb/>pipedum ſub, QV, & </s> <s xml:id="echoid-s11101" xml:space="preserve">quad. </s> <s xml:id="echoid-s11102" xml:space="preserve">ZV, vna <lb/>cum parallelepipedo ſub, ZQ, & </s> <s xml:id="echoid-s11103" xml:space="preserve">ſub <lb/>compoſito ex {1/3}. </s> <s xml:id="echoid-s11104" xml:space="preserve">quadrati, ZQ, & </s> <s xml:id="echoid-s11105" xml:space="preserve"><lb/>qua@rato, SO, ab his tamen dempto <lb/>parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s11106" xml:space="preserve">quadra <lb/>to, OY, cum {1/3}. </s> <s xml:id="echoid-s11107" xml:space="preserve">cubi, OY, & </s> <s xml:id="echoid-s11108" xml:space="preserve">ſic eſſe <pb o="425" file="0445" n="445" rhead="LIBER V."/> ſolidum ſimilare quodcunque genitum ex parallelogrammo, TV, <lb/>ad ſibi ſimilare genitum ex figura, TBFHRVQIPX, demptis ſo-<lb/>lidis ſimilaribus genitis ex oppoſitis hyperbolis, AEC, MON, iux-<lb/>ta communem regulam, RV; </s> <s xml:id="echoid-s11109" xml:space="preserve">eadem verò eſſe oſtendemus ſum-<lb/>pta pro regula ipſa, VX, & </s> <s xml:id="echoid-s11110" xml:space="preserve">reuolutione facta circa axem, GZ.</s> <s xml:id="echoid-s11111" xml:space="preserve"/> </p> <div xml:id="echoid-div1013" type="float" level="2" n="1"> <figure xlink:label="fig-0444-01" xlink:href="fig-0444-01a"> <image file="0444-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0444-01"/> </figure> </div> </div> <div xml:id="echoid-div1015" type="section" level="1" n="607"> <head xml:id="echoid-head632" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s11112" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s11113" xml:space="preserve">colligitur cylindrum, TV, ad cylindrum, TP, &</s> <s xml:id="echoid-s11114" xml:space="preserve">, HV, <lb/>cum tympano, BFHQIP, eſſe vt cubus, YS, ad parallelepipedũ <lb/>ſub, KY, & </s> <s xml:id="echoid-s11115" xml:space="preserve">quadrato, YS, vna cum parallelepido ſub, KS, & </s> <s xml:id="echoid-s11116" xml:space="preserve">com-<lb/>poſito ex quadrato, SO, & </s> <s xml:id="echoid-s11117" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s11118" xml:space="preserve">quadrati, SK; </s> <s xml:id="echoid-s11119" xml:space="preserve">& </s> <s xml:id="echoid-s11120" xml:space="preserve">ſic ſolida ſimilaria ex <lb/>elſdem figuris genita iuxta ibi aſſumptam regulam, RV.</s> <s xml:id="echoid-s11121" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1016" type="section" level="1" n="608"> <head xml:id="echoid-head633" xml:space="preserve">COROLL. XXIX. SECTIO PRIOR.</head> <p> <s xml:id="echoid-s11122" xml:space="preserve">IN Propoſ. </s> <s xml:id="echoid-s11123" xml:space="preserve">29. </s> <s xml:id="echoid-s11124" xml:space="preserve">viſa eius figura, eaque reuoluta circa axem, &</s> <s xml:id="echoid-s11125" xml:space="preserve">Y, <lb/>vt in anteced. </s> <s xml:id="echoid-s11126" xml:space="preserve">conſpicitur, patet ſolidum in reuolutione de-<lb/>ſcriptum a figura reſidua, demptis à parallelogrammo, T V, qua-<lb/>tuor hyperbolis, BFH, PIQ, AEC, MON, ad ſolidum deſ@riptum <lb/>in reuolutione ex figura reſidua, demptis à parallelogrammo, βΩ, <lb/>quatuor hyperbolis, 9F7, ΦΙΛ, L E ſ, ΣΟ3, eſſe vt parallelepipe-<lb/>dum ſub QV, & </s> <s xml:id="echoid-s11127" xml:space="preserve">quadrato, VZ, vna cum parallelepipedo ſub, QZ, <lb/>& </s> <s xml:id="echoid-s11128" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s11129" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s11130" xml:space="preserve">quadrati, QZ, ab his dempto <lb/>parallelepipedo ſub, SO, & </s> <s xml:id="echoid-s11131" xml:space="preserve">quadrato, OY, & </s> <s xml:id="echoid-s11132" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s11133" xml:space="preserve">cubi, OY, ad paral-<lb/>lelepipedum ſub, ΛΩ, & </s> <s xml:id="echoid-s11134" xml:space="preserve">quadrato, Ω4, vna cum parallelepipedo <lb/>ſub, Λ4, & </s> <s xml:id="echoid-s11135" xml:space="preserve">compoſito ex quadrato, SO, & </s> <s xml:id="echoid-s11136" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s11137" xml:space="preserve">quadrati, Λ4, dempto <lb/>parallelepido ſub, SO, & </s> <s xml:id="echoid-s11138" xml:space="preserve">quadrato, O6, cum {1/3}. </s> <s xml:id="echoid-s11139" xml:space="preserve">cubi, O6: </s> <s xml:id="echoid-s11140" xml:space="preserve">Sic etiam <lb/>patet eſſe quodlibet ſolidum ſimilare genitum ex figura, TBFHR <lb/>VQIPX, demptis ſolidis ſimilaribus genitis ex opoſitis hyperbolis, <lb/>AEC, MON, ad ſibi ſimilare genitũ ex figura, B9F72ΩΛΙΦΛ, dem-<lb/>ptis ſolidis ſimilaribus genitis ex oppoſitis hyperbolis, LEF, ΣΟ3, <lb/>iuxta communem regulam, RV.</s> <s xml:id="echoid-s11141" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1017" type="section" level="1" n="609"> <head xml:id="echoid-head634" xml:space="preserve">SECTIO POSTERIOR.</head> <p> <s xml:id="echoid-s11142" xml:space="preserve">IN Coroll. </s> <s xml:id="echoid-s11143" xml:space="preserve">colligitur eadem ſolida, genita nempè ex figuris, <lb/>TBFHRVQIPX β9F72ΩΛ ΙΦΔ, iuxta communem regulam, <lb/>RV, nihil ab eis dempto, eſſe vt dicta parallelepipeda, nihil pari-<lb/>@er ab eiſdem dempto.</s> <s xml:id="echoid-s11144" xml:space="preserve"/> </p> <pb o="426" file="0446" n="446" rhead="GEOMETRIÆ LIB. V."/> </div> <div xml:id="echoid-div1018" type="section" level="1" n="610"> <head xml:id="echoid-head635" xml:space="preserve">SCHOLIV M.</head> <p style="it"> <s xml:id="echoid-s11145" xml:space="preserve">_S_I verò intelligeremus, reuolutionem parallelogrammi, TV, non <lb/>fieri circa, &</s> <s xml:id="echoid-s11146" xml:space="preserve">γ, ſed circa, XV, vel illi parallelam, ſeu circa, TX, <lb/>aut illi parallelam, quoniam figura, T BFHRV QIPX, cxempli gratia, <lb/>talis eſt, qualem poſtulant Propoſ. </s> <s xml:id="echoid-s11147" xml:space="preserve">29. </s> <s xml:id="echoid-s11148" xml:space="preserve">& </s> <s xml:id="echoid-s11149" xml:space="preserve">30. </s> <s xml:id="echoid-s11150" xml:space="preserve">lib. </s> <s xml:id="echoid-s11151" xml:space="preserve">3. </s> <s xml:id="echoid-s11152" xml:space="preserve">vt facilè patet, <lb/>ideò cylindrus, vel faſcia cylindrica genita ex parallelogrammo, TV, <lb/>ad ſolidũ genitũ in tali reuolutione ex dicta figura, TBFHRV QIPX, <lb/>erit vt dictum parallelogrammum, TV, ad dictam figuram, T BFHR <lb/>VQIPX. </s> <s xml:id="echoid-s11153" xml:space="preserve">Mitto autem hic pariter quamplurima, quæ adbuc circa hæc <lb/>indaganda ſuperſunt, vt Lectori in his laborandi locus relinqu atur-<lb/>Hæc verò circa hyperbolam, & </s> <s xml:id="echoid-s11154" xml:space="preserve">oppoſitas ſectiones pro nunc adinue-<lb/>niſſe fit ſatis.</s> <s xml:id="echoid-s11155" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1019" type="section" level="1" n="611"> <head xml:id="echoid-head636" xml:space="preserve">Finis Quinti Libri.</head> <figure> <image file="0446-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0446-01"/> </figure> <pb o="427" file="0447" n="447"/> </div> <div xml:id="echoid-div1020" type="section" level="1" n="612"> <head xml:id="echoid-head637" xml:space="preserve">GEOMETRIAE</head> <head xml:id="echoid-head638" xml:space="preserve">CAVALER II</head> <head xml:id="echoid-head639" style="it" xml:space="preserve">LIBER SEXTVS.</head> <head xml:id="echoid-head640" xml:space="preserve">In quo de Spatijs Helicis, & Solidis in-<lb/>de genitis, ac alijs quibuſdam ex <lb/>ſuperioribus deductis, ſpecula-<lb/>tio inſtituitur.</head> <head xml:id="echoid-head641" xml:space="preserve">DEFINITIONES,</head> <head xml:id="echoid-head642" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s11156" xml:space="preserve">SI, dato quocumque circulo, ſuper eiuſdem <lb/>centro, ad diſtantiam omnium punctorum <lb/> <anchor type="note" xlink:label="note-0447-01a" xlink:href="note-0447-01"/> recti tranſitus ipſius ſemidiametri, circulo-<lb/>rum circumferentiæ deſcribi intelligan-<lb/>tur; </s> <s xml:id="echoid-s11157" xml:space="preserve">prædictæ circumferentiæ ſimul ſum-<lb/>ptæ dicantur. </s> <s xml:id="echoid-s11158" xml:space="preserve">Omnes circumferentiæ dati circuli.</s> <s xml:id="echoid-s11159" xml:space="preserve"/> </p> <div xml:id="echoid-div1020" type="float" level="2" n="1"> <note position="right" xlink:label="note-0447-01" xlink:href="note-0447-01a" xml:space="preserve">Deffin. 3. <lb/>l. 2.</note> </div> </div> <div xml:id="echoid-div1022" type="section" level="1" n="613"> <head xml:id="echoid-head643" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s11160" xml:space="preserve">ET ſi à præfato circulo quęcumque figura abſciſſa in-<lb/>telligatur; </s> <s xml:id="echoid-s11161" xml:space="preserve">portiones omnium circumferentiarum <lb/>dicti circuli, conceptæ in abſciſſa figura, dicentur. </s> <s xml:id="echoid-s11162" xml:space="preserve">Om-<lb/>nes circumferentiæ eiuſdem abſciſſæ figuræ.</s> <s xml:id="echoid-s11163" xml:space="preserve"/> </p> <pb o="428" file="0448" n="448" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1023" type="section" level="1" n="614"> <head xml:id="echoid-head644" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s11164" xml:space="preserve">SPatium Helicum voco, quod copræhenditur ſub ſpira-<lb/>li, vel eius quacumque portione, & </s> <s xml:id="echoid-s11165" xml:space="preserve">rectis, quæ à ter-<lb/>minis eiuſdem ſpiralis, ſeu illius portionis, ad initium re-<lb/>uolutionis ducuntur.</s> <s xml:id="echoid-s11166" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1024" type="section" level="1" n="615"> <head xml:id="echoid-head645" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s11167" xml:space="preserve">SPiralem verò intelligo iuxta diffinitionem Archimedis <lb/>lib. </s> <s xml:id="echoid-s11168" xml:space="preserve">de Spiralibus, nempè, ſi cuiuſcumque circuli ra-<lb/>dius æquali celeritate moueatur circa ipſius centrum (cu-<lb/>ius aliud extremum punctum periphæriam deſcribet) ini-<lb/>tio autem circulationis diſcedat à centro punctum æque-<lb/>uelociter motum ſuper radio, taliter vt eodem tempore <lb/>prædictum punctum percurrat circumferentiam, & </s> <s xml:id="echoid-s11169" xml:space="preserve">hoc ip-<lb/>ſum radium, quod ex compoſitione duorum motuum de-<lb/>ſcripta à puncto, quod radium percurrit, ipſa linea, ſit ea, <lb/>quam voco ſpiralem, cuius initium dicitur ipſum centrum, <lb/>terminus verò aliud extremum punctum ipſius radij; </s> <s xml:id="echoid-s11170" xml:space="preserve">& </s> <s xml:id="echoid-s11171" xml:space="preserve">ini-<lb/>tium circulationis, ſiue voluta ipſe radius: </s> <s xml:id="echoid-s11172" xml:space="preserve">Appellatur au-<lb/>tem hæc, ſpiralis in prima reuolutione genita, ſicuti aliæ <lb/>etiam ſunt in alijs reuolutionibus deſcriptibiles, produ-<lb/>cto radio, & </s> <s xml:id="echoid-s11173" xml:space="preserve">continuato motu, vt in ſecunda, in tertia, in <lb/>quarta reuolutione, & </s> <s xml:id="echoid-s11174" xml:space="preserve">ſic deinceps, vnde & </s> <s xml:id="echoid-s11175" xml:space="preserve">deſcripti cir-<lb/>culi dicuntur primi, ſecundi, tertij, &</s> <s xml:id="echoid-s11176" xml:space="preserve">c. </s> <s xml:id="echoid-s11177" xml:space="preserve">quæ Archimedem <lb/>lib. </s> <s xml:id="echoid-s11178" xml:space="preserve">de Spiralibus recolenti melius innoteſcent, eiuſdem <lb/>enim terminos in hoc Libro paſſim vſurpabimus</s> </p> </div> <div xml:id="echoid-div1025" type="section" level="1" n="616"> <head xml:id="echoid-head646" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s11179" xml:space="preserve">_A_Spice Schema Prop. </s> <s xml:id="echoid-s11180" xml:space="preserve">9. </s> <s xml:id="echoid-s11181" xml:space="preserve">huius, in quo eſt circuliradius, AE, qui <lb/>æqueuelociter motus circa, A, deſcribit circulum, SME, ipſum <lb/>verò, E, circumferentiam, MSE, initio autem reuolutionis diſcedat <lb/>ab, A, punctum motum æqueuelociter ſuper, AE, quam percurrat eo <lb/>tempore, quo punctum, E, pertranſit circumferentiam, MSE, deſi-<lb/>gnans curuam, AIE, hæc igitur dic itur ſpiralis in prima reuolutione <lb/>orta, cuius initium, A, terminus, E, &</s> <s xml:id="echoid-s11182" xml:space="preserve">, AE, vocatur circulationis <lb/>initium: </s> <s xml:id="echoid-s11183" xml:space="preserve">Exempla autem ſpir alm̃ in alijs reuolutionibus geni@ arum <lb/>babes in Schemate Cor. </s> <s xml:id="echoid-s11184" xml:space="preserve">Prop. </s> <s xml:id="echoid-s11185" xml:space="preserve">20. </s> <s xml:id="echoid-s11186" xml:space="preserve">huius, etenim, LSO, in ſecunda, <lb/>OTP, in tertia, PVG, autem in quarta reuolutione genitæ dicuntur.</s> <s xml:id="echoid-s11187" xml:space="preserve"/> </p> <pb o="429" file="0449" n="449" rhead="LIBER VI."/> </div> <div xml:id="echoid-div1026" type="section" level="1" n="617"> <head xml:id="echoid-head647" xml:space="preserve">THEOREMA I. PROPOS. I.</head> <p> <s xml:id="echoid-s11188" xml:space="preserve">CIrculorum æqualium, necnon ſectorum æqualium, <lb/>& </s> <s xml:id="echoid-s11189" xml:space="preserve">ab eodem, vel æqualibus circulis abſciſſorum, <lb/>omnes circumferentiæ ſunt æquales.</s> <s xml:id="echoid-s11190" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11191" xml:space="preserve">Hæc Propoſitio facilè per ſuperpoſitionem oſtendetur. </s> <s xml:id="echoid-s11192" xml:space="preserve">Si <lb/>enim circuli æquales ad inuicem ſuperponantur, ita vt centrum <lb/>centro congruat, etiam ipſi circuli congruent, cum ſupponantur <lb/>æquales, vnde & </s> <s xml:id="echoid-s11193" xml:space="preserve">eorum radij ſint æquales, congruentibus autem <lb/>circulis, etiam omnes vnius circumferentiæ congruent omnibus <lb/>alterius circumferentijs, & </s> <s xml:id="echoid-s11194" xml:space="preserve">ideò inter ſe æquales erunt. </s> <s xml:id="echoid-s11195" xml:space="preserve">Eadem <lb/>pariter ſuperpoſitionis adhibita via, oſtendemus ſectorum æqua-<lb/>lium, ab eodem, vel æqualibus circulis abſciſſorum omnes circum-<lb/>ferentias inter ſe æquales eſſe, quod erat demonſtrandum.</s> <s xml:id="echoid-s11196" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1027" type="section" level="1" n="618"> <head xml:id="echoid-head648" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s11197" xml:space="preserve">OMnis circulus æqualis eſt triangulo rectangulo, cu-<lb/>ius radius eſt par vni eorum, quæ ſunt circa rectum <lb/>angulum, circumferentia verò baſi.</s> <s xml:id="echoid-s11198" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11199" xml:space="preserve">Hæc oſtenditur ab Archimede lib. </s> <s xml:id="echoid-s11200" xml:space="preserve">de Dimenſione Circuli, Pro-<lb/>poſ. </s> <s xml:id="echoid-s11201" xml:space="preserve">1. </s> <s xml:id="echoid-s11202" xml:space="preserve">propterea ibirecolatur.</s> <s xml:id="echoid-s11203" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1028" type="section" level="1" n="619"> <head xml:id="echoid-head649" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s11204" xml:space="preserve">OMnis ſector circuli æqualis eſt triangulo rectangu-<lb/>lo, cuius circuli radius eſt par vni eorum, quæ ſunt <lb/>circa rectum, circumferentia verò baſi illius ſectoris.</s> <s xml:id="echoid-s11205" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11206" xml:space="preserve">Si circulus, ABCD, cuius radius, ED, & </s> <s xml:id="echoid-s11207" xml:space="preserve">ſector, EDC, expo <lb/>ſito vero triangulo, HOM, cuius angulus, HMO, ſit rectus, & </s> <s xml:id="echoid-s11208" xml:space="preserve"><lb/>letus, HM, æquale ipſi, ED, &</s> <s xml:id="echoid-s11209" xml:space="preserve">, MO, circumferentiæ, ABCD, <lb/> <anchor type="note" xlink:label="note-0449-01a" xlink:href="note-0449-01"/> ſit, MN, æqualis circumferentiæ, CD; </s> <s xml:id="echoid-s11210" xml:space="preserve">& </s> <s xml:id="echoid-s11211" xml:space="preserve">iungatur, HN. </s> <s xml:id="echoid-s11212" xml:space="preserve">Dico <lb/>ergo ſectorem, ECD, æquari triangulo, HNM,. </s> <s xml:id="echoid-s11213" xml:space="preserve">Nam circulus, <lb/>ABCD, ad ſectorem, CED, eſt vt circumferentia, ABCD, ad cir- <pb o="428" file="0450" n="450" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0450-01a" xlink:href="fig-0450-01"/> cumferentiam, CD, eſt <lb/>autem circulus, ABCD, <lb/>æqualis triang. </s> <s xml:id="echoid-s11214" xml:space="preserve">HOM, <lb/>ergo triangulus, HOM, <lb/>ad ſectorem, ECD, eſt <lb/>vt circumferẽtia, ABCD, <lb/>ad circumferentiam, CD, <lb/>ideſt vt, OM, ad, MN, id-<lb/>eſt vt triangulus, HOM, <lb/>ad, HNM, igitur idem <lb/>triangulus, HOM, ad ſe-<lb/>ctorem, ECD, ad trian-<lb/>gulum, HNM, eandem <lb/>habet rationem, ergo ſector, ECD, eſt æqualis triangulo, <lb/>HNM, quod oſtendere opus erat.</s> <s xml:id="echoid-s11215" xml:space="preserve"/> </p> <div xml:id="echoid-div1028" type="float" level="2" n="1"> <note position="right" xlink:label="note-0449-01" xlink:href="note-0449-01a" xml:space="preserve">33. Sexti <lb/>Blem. <lb/>Exantec.</note> <figure xlink:label="fig-0450-01" xlink:href="fig-0450-01a"> <image file="0450-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0450-01"/> </figure> </div> </div> <div xml:id="echoid-div1030" type="section" level="1" n="620"> <head xml:id="echoid-head650" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s11216" xml:space="preserve">_H_Inc patet, ſi ſumpto vtcumque puncto in, ED, vt, X, centro, <lb/>E, ad diſtantiam, X, circuferentia, FZX, deſcripta fuerit in-<lb/>ſuper abſciſſa, HR, æquali ipſi, EX, per, R, ducta fuerit, SR, parllela <lb/>ipſi, OM, ſecans, HN, in, I, trape zium, OSRM, æquari reſiduo circu-<lb/>li, ABCD, ab eo dempto circulo, FZX, quod reſiduum dicatur faſcia <lb/>circulorum, BD, FX, nam circulus, BD, ad circulum, FX, eſt vt qua-<lb/>dratum, DE ad quadratum, EX, ideſt vt quadratum, MH, ad qua-<lb/>dratum, HR, ideſt vt triangulus, HOM, ad triangulum, HSR, vnde, <lb/> <anchor type="note" xlink:label="note-0450-01a" xlink:href="note-0450-01"/> quia circulus, BD, æquatur triangulo, HOM, etiam circulus, FZX, <lb/>æquatur triangulo, HSR, vnde faſcia, BF, æquatur trapezio, OSRM; <lb/></s> <s xml:id="echoid-s11217" xml:space="preserve">eoden modo colligemns reſiduum ſectoris DEC, ab eo dempto ſectore, <lb/> <anchor type="note" xlink:label="note-0450-02a" xlink:href="note-0450-02"/> XEZ, quod dicatur eorundem ſectorum faſcia, ſcilicet ipſum, ZXDC, <lb/>æquart trapezio, IRMN.</s> <s xml:id="echoid-s11218" xml:space="preserve"/> </p> <div xml:id="echoid-div1030" type="float" level="2" n="1"> <note position="left" xlink:label="note-0450-01" xlink:href="note-0450-01a" xml:space="preserve">_Coroll. 1._ <lb/>_11. l. 3._</note> <note position="left" xlink:label="note-0450-02" xlink:href="note-0450-02a" xml:space="preserve">_Coroll. 1._ <lb/>_19. l. 2._</note> </div> </div> <div xml:id="echoid-div1032" type="section" level="1" n="621"> <head xml:id="echoid-head651" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s11219" xml:space="preserve">_P_Atet inſuper, quia circulus, CDB, æquatur triangulo, HOM, & </s> <s xml:id="echoid-s11220" xml:space="preserve"><lb/>circulus, FZX, triang. </s> <s xml:id="echoid-s11221" xml:space="preserve">HSR, item circumferentia, ABCD, ipſi <lb/>OM, &</s> <s xml:id="echoid-s11222" xml:space="preserve">, FZX, ipſi, SR, (nam, DE, æquatur ipſi, MH, &</s> <s xml:id="echoid-s11223" xml:space="preserve">, EX, ipſi, RH,) <lb/>quod veluti, OM, ad, SR, eſt vt, MH, ad, HR ita circumferentia, ABCD, <lb/>ad, FZX, erit vt, DE, ad, EX. </s> <s xml:id="echoid-s11224" xml:space="preserve">Sic etiam oſtendemus ſimilium ſectorum, <lb/>CED, ZEX, circumferentias, CD, ZX, eſſe vt ſemidiametri, DE EX, & </s> <s xml:id="echoid-s11225" xml:space="preserve"><lb/>ipſos ſimiles ſectores eſſe vt quadrata ſemidiametrorũ, DE, EX, quoniã <pb o="429" file="0451" n="451" rhead="LIBER VI."/> ſunt circulorum, à quibus abſcinduntur partes proportionales, ipſi au-<lb/>tem circuli ſunt, vt diametrorum quadrata.</s> <s xml:id="echoid-s11226" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1033" type="section" level="1" n="622"> <head xml:id="echoid-head652" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s11227" xml:space="preserve">DAti circuli, necnon ſimiles fectores inter ſe ſunt, vt <lb/>omnes eorundem circumferentiæ.</s> <s xml:id="echoid-s11228" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11229" xml:space="preserve">Sint in eadem antecedentis figura circuli quinque, BADC, <lb/>FXZ, deſcripti ſuper eodem centro, E, & </s> <s xml:id="echoid-s11230" xml:space="preserve">abijſdem intelligantur <lb/> <anchor type="figure" xlink:label="fig-0451-01a" xlink:href="fig-0451-01"/> abſciſſi ſimiles ſectores, D <lb/>ED, XEZ. </s> <s xml:id="echoid-s11231" xml:space="preserve">Dico circulos, <lb/>DABC, FZX, necnon ſe-<lb/>ctores, DEC, XEZ, inter ſe <lb/>eſſe, vt omnes iplorum cir-<lb/>cumferentiæ. </s> <s xml:id="echoid-s11232" xml:space="preserve">Sit denuò <lb/>expoſitũ triãgulum, HOM, <lb/>cuius ſit angulus rectus, H <lb/>MO, latus, HM, æquale ra-<lb/>dio, ED, &</s> <s xml:id="echoid-s11233" xml:space="preserve">, MO, circum-<lb/>ferentiæ, DCBA, abſciſſa <lb/>autem, HR, æqualr ipſi, <lb/>EX, & </s> <s xml:id="echoid-s11234" xml:space="preserve">per, R, ducta paral-<lb/>lela ipſi, OM, quæ ſit, SR, intercepta lateribus, HO, HM, patet, <lb/>vt dicebatur in Corol. </s> <s xml:id="echoid-s11235" xml:space="preserve">2. </s> <s xml:id="echoid-s11236" xml:space="preserve">ant. </s> <s xml:id="echoid-s11237" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s11238" xml:space="preserve">quod circumferentia, FZX, <lb/>æquatur ipſi, SR, eodem modo abſcindentes ab ipſis, HM, ED, <lb/>verſus, H, E, puncta æquales quaſcunque rectas lineas, & </s> <s xml:id="echoid-s11239" xml:space="preserve">per ea-<lb/>rum terminos ducentes parallelam quidem ipſi, OM, in triangulo, <lb/>& </s> <s xml:id="echoid-s11240" xml:space="preserve">circumfer entiam iuper cenrro, E, in circulo, ABCD, manife-<lb/>ſtum erit prædictam circumferentiam æquari prædictæ parallelæ, <lb/>lateribus, HO, HM, interceptæ, & </s> <s xml:id="echoid-s11241" xml:space="preserve">vnicuique circumferentiæ in <lb/>circulo, ABCD, fic deſcriptæ reſpondere ſuam parallelam in triã-<lb/>gulo, HOM, cum ſint rectę, HM, ED, æquales, igitur conclude-<lb/>mus omnes circumferentias circuli, DABC, æquari omnibus li-<lb/>neis trianguli, HOM, regula, OM, ſicut etiam omnes circumfe-<lb/>rentias circuli, FZX, æquari omnibus lineis trianguli, HSR, regu-<lb/>la eadem, OM, quapropter, vt omnes lineæ trianguli, HOM, ad <lb/>omnes lineas trianguli, HSR, ideſi vt triãgulum, HOM, ad, HSR, <lb/> <anchor type="note" xlink:label="note-0451-01a" xlink:href="note-0451-01"/> ideſt vt circulus, DABC, ad circulum, FZX, ita omnes circumfe-<lb/>rentiæ circuli, ABCD, erunt ad omnes circumfarentias circuli <lb/>ciuidem, FZX; </s> <s xml:id="echoid-s11242" xml:space="preserve">quod & </s> <s xml:id="echoid-s11243" xml:space="preserve">fimuli methodo de ſectoribus ex. </s> <s xml:id="echoid-s11244" xml:space="preserve">g. </s> <s xml:id="echoid-s11245" xml:space="preserve">DEC, <pb o="432" file="0452" n="452" rhead="GEOMETRIÆ"/> XEZ, ducta, HN, quæ abſcindat, NM, æqualem circumferentię, <lb/>CD, facilè oſtendemus, hæc autem erant demonſtranda.</s> <s xml:id="echoid-s11246" xml:space="preserve"/> </p> <div xml:id="echoid-div1033" type="float" level="2" n="1"> <figure xlink:label="fig-0451-01" xlink:href="fig-0451-01a"> <image file="0451-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0451-01"/> </figure> <note position="right" xlink:label="note-0451-01" xlink:href="note-0451-01a" xml:space="preserve">3. l. 2.</note> </div> </div> <div xml:id="echoid-div1035" type="section" level="1" n="623"> <head xml:id="echoid-head653" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s11247" xml:space="preserve">QVicumque ſectores inter ſe comparati, ſeu quæcum-<lb/>que figuræ ex ſectoribus compoſitæ ad ſectores, vel <lb/>ad figuras ex ſectoribus compoſitas comparatæ, habent <lb/>eandem rationem, quam omnes ipſarum circumferentiæ <lb/>ad omnes illarum circumferentias.</s> <s xml:id="echoid-s11248" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11249" xml:space="preserve">Sint quicunque circuli ſuper centro, A, nempè, ECD, maior, &</s> <s xml:id="echoid-s11250" xml:space="preserve">, <lb/>VIO, minor, & </s> <s xml:id="echoid-s11251" xml:space="preserve">in, ECD, fit ſector quicunque, CAD, & </s> <s xml:id="echoid-s11252" xml:space="preserve">ſimiliter <lb/>in, VIO, quilibet ſector, OAB. </s> <s xml:id="echoid-s11253" xml:space="preserve">Dico ſectorem, CAD, ad ſectorẽ, <lb/>OAB, eſſe vt omnes circumferentias, CAD, ad omnes circumfe-<lb/>rentias, OAB. </s> <s xml:id="echoid-s11254" xml:space="preserve">Secent radij, CA, AD, circumferentiam, VIO, in <lb/> <anchor type="figure" xlink:label="fig-0452-01a" xlink:href="fig-0452-01"/> punctis, I, O; </s> <s xml:id="echoid-s11255" xml:space="preserve">Eſt ergo ſector, CAD, ſi-<lb/>milis ſectori, IAO, & </s> <s xml:id="echoid-s11256" xml:space="preserve">ideo eſt ad illum, <lb/>vt omnes circumferentiæ ad omnes <lb/> <anchor type="note" xlink:label="note-0452-01a" xlink:href="note-0452-01"/> circumferentias, ſed & </s> <s xml:id="echoid-s11257" xml:space="preserve">vt ſector, IAO, <lb/>ad ſectorem, OAB, ita omnes cir-<lb/>cumferentiæ ad omnes circumferen-<lb/>tias, nam ſector, IAO, ad, OAB, <lb/>eſt vt circumferentia, IO, ad, OB, <lb/>vt verò, IO, ab, OB, ſic, deſcripta cir-<lb/>cumferentia, STR, vtcumque, ipſa, ST, <lb/>ad, TR, eſt enim, IO, ad, ST, vt OA, ad, <lb/> <anchor type="note" xlink:label="note-0452-02a" xlink:href="note-0452-02"/> AT, ideſt vt, OB, ad, TR, vnde, permu-<lb/>tando, vt, IO, 3d, OB, ſic, ST, ad, SR, & </s> <s xml:id="echoid-s11258" xml:space="preserve">vt vnum ad vnum, ita <lb/>omnia ad omnia, ideſt vt, IO, ad, OB, ita omnes circumferentiæ, <lb/>IAO, ſectoris ad omnes circumferentias ſectoris, OAB, ſed vt, IO, <lb/>ad, OB, ſic, vt dectũ eſt, ſe habet ſector, IAO, ad, OAB, ergo, IAO, <lb/>ad, OAB, eſt vt omnes circum ferentiæ, IAO, ad omnes circumfe-<lb/>rentias, OAB, ſed & </s> <s xml:id="echoid-s11259" xml:space="preserve">ſectorem, CAD, ad, IAO, eſſe oſtenſum eſt, <lb/>vt omnes circumferentiæ, CAD, ad omnes circumferentias, IAO, <lb/>ergo ex æquali ſector, CAD, ad ſectorem OAB, eſt vt omnes cir-<lb/>cumferentiæ, CAD, ad omnes circumferentias, OAB. </s> <s xml:id="echoid-s11260" xml:space="preserve">Et com-<lb/>ponendo figura compoſita ex ſectoribus, CAD, OAB, ad ſectorẽ, <lb/>OAB, erit vt omnes circumferentiæ figuę eiuſdem, ad omnes cir-<lb/>cumferentias ſectoris, OAB, veiuti etiam ſi prædicta figura non <pb o="433" file="0453" n="453" rhead="LIBER VI."/> ad ſectorem, ſed ad aliam quamcunq; </s> <s xml:id="echoid-s11261" xml:space="preserve">figuram ex ſectoribus com-<lb/>poſitam compararetur, oſtenderemus, eaſdem figuras eſſe inter ſe, <lb/>vt omnes earundem circumferentiæ, quod demonſtrare opus <lb/>erat.</s> <s xml:id="echoid-s11262" xml:space="preserve"/> </p> <div xml:id="echoid-div1035" type="float" level="2" n="1"> <figure xlink:label="fig-0452-01" xlink:href="fig-0452-01a"> <image file="0452-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0452-01"/> </figure> <note position="left" xlink:label="note-0452-01" xlink:href="note-0452-01a" xml:space="preserve">Exantec.</note> <note position="left" xlink:label="note-0452-02" xlink:href="note-0452-02a" xml:space="preserve">Coroll. 2. <lb/>3 huius.</note> </div> </div> <div xml:id="echoid-div1037" type="section" level="1" n="624"> <head xml:id="echoid-head654" xml:space="preserve">COROLL ARIVM.</head> <p style="it"> <s xml:id="echoid-s11263" xml:space="preserve">_P_Atet aùtem, veluti oſtenſum eſt ſectores, AIO, AOB, eſſe vt om-<lb/>nes eorum circumferentiæ eodem modo demonſtrari poſſe, circu-<lb/>lum, VOB, & </s> <s xml:id="echoid-s11264" xml:space="preserve">ſectorem, AOB, & </s> <s xml:id="echoid-s11265" xml:space="preserve">in uniuerſam circulos, & </s> <s xml:id="echoid-s11266" xml:space="preserve">ſuos ſe-<lb/>ctores inter ſe eſſe, vt omnes eorum circumferentiæ.</s> <s xml:id="echoid-s11267" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1038" type="section" level="1" n="625"> <head xml:id="echoid-head655" xml:space="preserve">THEOREMA VI. PROPOS. VI.</head> <p> <s xml:id="echoid-s11268" xml:space="preserve">SI in circulo ab eiuſdem centro ad circumferentiam <lb/>curuam quædam linea illius conditionis producatur, <lb/>vt quæcunq; </s> <s xml:id="echoid-s11269" xml:space="preserve">rectæ lineæ à centro ad ipſam pertingentes <lb/>(præter illius extrema iungentem) intra illud ſpatium ca-<lb/>dant, quod comprehenditur ducta curua, & </s> <s xml:id="echoid-s11270" xml:space="preserve">illius extrema <lb/>iungente: </s> <s xml:id="echoid-s11271" xml:space="preserve">Erit dictum ſpatium ad propoſitum circulum, <lb/>vel quemcunq; </s> <s xml:id="echoid-s11272" xml:space="preserve">ſectorem, vt omnes eiuſdem circumferen-<lb/>tiæ ad omnes illius circumferentias.</s> <s xml:id="echoid-s11273" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11274" xml:space="preserve">Sit quicunque circulus, NOQT, & </s> <s xml:id="echoid-s11275" xml:space="preserve">centrum, A, curua, AFN, <lb/>ducta à centro, A, ad periphæriam, cui incidat in, N, & </s> <s xml:id="echoid-s11276" xml:space="preserve">fit eius <lb/>conditionis, qualis ſuppoſitum eſt, ſitq; </s> <s xml:id="echoid-s11277" xml:space="preserve">iuncta, AN, Dico igitur <lb/> <anchor type="figure" xlink:label="fig-0453-01a" xlink:href="fig-0453-01"/> ſpatium, ſeu figuram, AFN, ad <lb/>circulum, NOQT, vel ad quem-<lb/>cunque ſectorem, eſſe vt omnes <lb/>eiuſdem circumferentiæ ad om-<lb/>nes illius circumferentias. </s> <s xml:id="echoid-s11278" xml:space="preserve">Fiat <lb/>vt circulus, NOQT, ad figuram, <lb/>NFA, ita circumferentia, NOQ <lb/>T, ad circumferentiam, QR, ita <lb/>enim erit, & </s> <s xml:id="echoid-s11279" xml:space="preserve">circulus, NOQT, <lb/>ad ſectorem, QAR, iunctis, QA, <lb/>AR, vnde ſector, QAR, erit æ-<lb/>qualis figuræ, AFN, vel ergo <lb/>omnes circumferentiæ, QAR, æquantur etiam omnibus circum-<lb/>ferentijs figuræ, AFN, & </s> <s xml:id="echoid-s11280" xml:space="preserve">ſic quia fector, QAR, ad circulum, NO <pb o="434" file="0454" n="454" rhead="GEOMETRIÆ"/> QT, eſt vt omnes eiſdem circumferentiæ ad omnes illius circumf <lb/>etiam figura, AFN, ad circulum, NOQT, & </s> <s xml:id="echoid-s11281" xml:space="preserve">conſequenter etiam <lb/>ad quẽcumq; </s> <s xml:id="echoid-s11282" xml:space="preserve">illius ſectorem per anteced. </s> <s xml:id="echoid-s11283" xml:space="preserve">Prop. </s> <s xml:id="echoid-s11284" xml:space="preserve">& </s> <s xml:id="echoid-s11285" xml:space="preserve">Cor. </s> <s xml:id="echoid-s11286" xml:space="preserve">erit vt om-<lb/>nes circumfer. </s> <s xml:id="echoid-s11287" xml:space="preserve">ad omnes circumferentias: </s> <s xml:id="echoid-s11288" xml:space="preserve">Vel, niſi omnes circũ. <lb/></s> <s xml:id="echoid-s11289" xml:space="preserve">ferentiæ, QAR, æquantur omnibus circumferentijs figuræ, AFN, <lb/>erunt eiſdem maiores, vel minores, ſint primò maiores, qua ntita-<lb/>te omnium circumferentiarum ſectoris, AST, intellecta autem à <lb/> <anchor type="figure" xlink:label="fig-0454-01a" xlink:href="fig-0454-01"/> centro, A, ducta ipſa, AO, tan-<lb/>gente curuam, AFN, in puncto, <lb/>A, quæ circumferentiæ incidat <lb/>in, O, ſecetur circumferentia, O <lb/>N, bifariam in, L, & </s> <s xml:id="echoid-s11290" xml:space="preserve">rurſus par-<lb/>tes, OL, LN, bifariam in pun-<lb/>ctis, P, M, & </s> <s xml:id="echoid-s11291" xml:space="preserve">hoc ſemper fiat <lb/>donec ad circumferentias deuẽ-<lb/> <anchor type="note" xlink:label="note-0454-01a" xlink:href="note-0454-01"/> tum ſit, quarum vna quæque ſit <lb/>minor, ST, ſcilicet iplæ, OP, P <lb/>L, LM, MN, & </s> <s xml:id="echoid-s11292" xml:space="preserve">à centro, A, ad <lb/>puncta, P, L, M, extendantur re-<lb/>ctæ, AP, AL, AM, quæ ſecaburt curuam, AFN, earum enim por-<lb/>tiones inter centrum, & </s> <s xml:id="echoid-s11293" xml:space="preserve">curuam interceptæ, ex hypoteſi cadunt <lb/>intra ſpatium, ANFA, ſecent in, C, F, I, & </s> <s xml:id="echoid-s11294" xml:space="preserve">centro, A, interuallis, <lb/>AC, AF, AI, arcus deſcribantur, BCD, EFG, HIK, incidẽtes pro-<lb/>ximis rectis lineis, à centro eductis, in punctis, B, D; </s> <s xml:id="echoid-s11295" xml:space="preserve">E, G; </s> <s xml:id="echoid-s11296" xml:space="preserve">H, k. <lb/></s> <s xml:id="echoid-s11297" xml:space="preserve">Quoniam ergo omnes circumferentiæ ſectoris, QAR, ſuperant <lb/>omnes circumferentias figuræ, AFN, quantitate omnium circum. </s> <s xml:id="echoid-s11298" xml:space="preserve"><lb/>ferentiarum ſectoris, AST, omnes autem circumferentiæ figuræ <lb/>compoſitæ ex ſectoribus, NAM, IAH, FAE, CAB, ideſt figuræ <lb/>ſpatio, AFN, circumſcriptæ, ſuperant omnes circumferentias fi-<lb/>guræ compoſitæ ex ſectoribus, KAI, GAF, DAC, ideſt figuræ ei-<lb/>dem ſpatio inſcriptæ, quantitate omnium circumferentiarum fe-<lb/>ſectoris, BAC, & </s> <s xml:id="echoid-s11299" xml:space="preserve">quadrilineorum, ECDF, HFGI, MIkN, quæ ſi-<lb/>mul adæquantur omnibus circumfèrentijs ſectoris, MAN, vt fa-<lb/>cilè oſtendi poteſt, propterea omnes circumferentiæ figuræ circũ-<lb/>ſcriptæ ſuperant omnes circumferentias inſcriptæ quantitate om-<lb/>nium circumferentiarum ſectoris, MAN, quæ cum ſint minores <lb/>omnibus circumferentijs ſectoris, TAS, ideò omnes circumferen-<lb/>tiæ figuræ, circumſcriptæ ſuperabunt omnes circumferentias in-<lb/>ſcriptæ minori quantitate, & </s> <s xml:id="echoid-s11300" xml:space="preserve">eædem multò minori quantitate ſu-<lb/>perabunt omnes circumferentias ſpatij, AFN, quam omnes circũ-<lb/>ferentiæ ſectoris, AQR, ſuperent omnes circumferentias ſpatij, A <pb o="435" file="0455" n="455" rhead="LIBER VI."/> FN, ergo omnes circumferentiæ figuræ circumſcriptæ minores <lb/> <anchor type="note" xlink:label="note-0455-01a" xlink:href="note-0455-01"/> erunt omnibus circumferentijs ſectoris, QAR, cum verò figura ex <lb/>ſectoribus compoſita ad ſectorem, ſit vt omnes circumferentiæ ad <lb/>omnes circumferentias, ideò etiam figura circumſcripta minor erit <lb/>ſectore, QAR, & </s> <s xml:id="echoid-s11301" xml:space="preserve">multò minor eri<unsure/>t fig ira, AFN, ſectore, QAR, ſed <lb/>& </s> <s xml:id="echoid-s11302" xml:space="preserve">æqualis illi oſtenſa fuit, quod eſt abſurdum, igitur abſurdum etiã <lb/>eſt dicere omnes circumferentias ſectoris, QAR, maiores eſſe om-<lb/>nibus circumferentijs ſpatij, AFN. </s> <s xml:id="echoid-s11303" xml:space="preserve">Dico nunc neque eſſe mino-<lb/>res, ſi hoc verum eſt, ſint minores omnibus circumferentijs ſecto-<lb/>ris, SAT, & </s> <s xml:id="echoid-s11304" xml:space="preserve">repetita eadem conſtructione, ſit ſpatio, AFN, circũ-<lb/>ſcripta figura ex ſectoribus compoſita, & </s> <s xml:id="echoid-s11305" xml:space="preserve">alia inſcripta, ita vt cir-<lb/>cumſcriptæ figurę omnes circumferentiæ ſuperent omnes circum-<lb/>ferentias inſcriptæ minori quantitate, quam ſint omnes circum-<lb/>ferentiæ ſectoris, SAT, ergo omnes circumferentiæ figuræ, AFN, <lb/>ſuperabunt omnes circumferentias figuræ inſcriptæ multò minori <lb/>quantitate, quam eædem ſuperent omnes circumferentias, QAR, <lb/>ergo omnes circumferentię inſcriptæ figuræ maiores erunt omni-<lb/>bus circumferentijs ſectoris, QAR, ergo figura inſcripta maior e-<lb/>tiam erit ſectore, QAR, & </s> <s xml:id="echoid-s11306" xml:space="preserve">eodem multò maior erit figura, AFN, <lb/>contra hypoteſim, eſt enim illi æ qualis, quod eſt obſurdum, igitur <lb/>abſurdum etiam eſt omnes circumferentias ſectoris, QAR, mino-<lb/>res eſſe omnibus circumferentijs figuræ, AFN, ſed neq; </s> <s xml:id="echoid-s11307" xml:space="preserve">ſunt illis <lb/>maiores, vt oſtenſum eſt, ergo ſunt eiſdem æquales, ſed omnes cir-<lb/>cumferentiæ ſectoris, AQR, ad circulum, OQSN, vel quemcunq; <lb/></s> <s xml:id="echoid-s11308" xml:space="preserve">ſectorem comparatæ ſunt, vt ſpatium ad ſpatium, ergo ſpatium <lb/> <anchor type="note" xlink:label="note-0455-02a" xlink:href="note-0455-02"/> quoque, AFN, ad circulum, OQSN, vel ad quemcunque ſectorem, <lb/>erit, vt omnes illius circumfer. </s> <s xml:id="echoid-s11309" xml:space="preserve">ad omnes iſtius circumferentias, <lb/>quod, &</s> <s xml:id="echoid-s11310" xml:space="preserve">c.</s> <s xml:id="echoid-s11311" xml:space="preserve"/> </p> <div xml:id="echoid-div1038" type="float" level="2" n="1"> <figure xlink:label="fig-0453-01" xlink:href="fig-0453-01a"> <image file="0453-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0453-01"/> </figure> <figure xlink:label="fig-0454-01" xlink:href="fig-0454-01a"> <image file="0454-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0454-01"/> </figure> <note position="left" xlink:label="note-0454-01" xlink:href="note-0454-01a" xml:space="preserve">1. Decimi <lb/>Elem.</note> <note position="right" xlink:label="note-0455-01" xlink:href="note-0455-01a" xml:space="preserve">Exantec.</note> <note position="right" xlink:label="note-0455-02" xlink:href="note-0455-02a" xml:space="preserve">Exantec,</note> </div> </div> <div xml:id="echoid-div1040" type="section" level="1" n="626"> <head xml:id="echoid-head656" xml:space="preserve">THEOREMA VII. PROPOS. VII.</head> <p> <s xml:id="echoid-s11312" xml:space="preserve">SI in ſpiralem ex prima reuolutione ortam incidant duę <lb/>lineæ à puncto, quod eſt initium ſpiralis, & </s> <s xml:id="echoid-s11313" xml:space="preserve">producã-<lb/>tur vſq; </s> <s xml:id="echoid-s11314" xml:space="preserve">ad circumferentiam primi circuli, eandem rationẽ <lb/>inter ſe habebunt iſtæ in ſpiralem incidentes, quam arcus <lb/>circuli, medij inter terminum ſpiralis, & </s> <s xml:id="echoid-s11315" xml:space="preserve">limites linearum <lb/>productarum in citcumferentia factos, ſumptis in conſe-<lb/>quentia arcubus à fine ſpiralis.</s> <s xml:id="echoid-s11316" xml:space="preserve"/> </p> <pb o="436" file="0456" n="456" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1041" type="section" level="1" n="627"> <head xml:id="echoid-head657" xml:space="preserve">THEOREMA VIII. PROPOS. VIII.</head> <p> <s xml:id="echoid-s11317" xml:space="preserve">SI in ſpirales in alijs reuolutionibus genitas, quam in <lb/> <anchor type="figure" xlink:label="fig-0456-01a" xlink:href="fig-0456-01"/> prima incidant duę <lb/>lineæ ab initio ſpira-<lb/>lis, habebunt illæ in-<lb/>ter ſe eandem rationẽ, <lb/>quam arcus circuli pri-<lb/>mi, intercepti, veluti <lb/>dicitur in anteceden-<lb/>te, cum integra circũ-<lb/>ferentia toties aſſum-<lb/>pta, quotus eſt vnitate <lb/>minor reuolut ionum <lb/>numerus.</s> <s xml:id="echoid-s11318" xml:space="preserve"/> </p> <div xml:id="echoid-div1041" type="float" level="2" n="1"> <figure xlink:label="fig-0456-01" xlink:href="fig-0456-01a"> <image file="0456-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0456-01"/> </figure> </div> <p> <s xml:id="echoid-s11319" xml:space="preserve">Hę duę Propoſitiones oſtenduntur ab Archimede lib. </s> <s xml:id="echoid-s11320" xml:space="preserve">de pir. <lb/></s> <s xml:id="echoid-s11321" xml:space="preserve">Prop. </s> <s xml:id="echoid-s11322" xml:space="preserve">14. </s> <s xml:id="echoid-s11323" xml:space="preserve">& </s> <s xml:id="echoid-s11324" xml:space="preserve">15.</s> <s xml:id="echoid-s11325" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1043" type="section" level="1" n="628"> <head xml:id="echoid-head658" xml:space="preserve">SCOLIVM.</head> <p style="it"> <s xml:id="echoid-s11326" xml:space="preserve">_I_N prima reuolutione orta ſit ſpiralis, ACER, & </s> <s xml:id="echoid-s11327" xml:space="preserve">RTV MG, in ſe-<lb/>cunda, &</s> <s xml:id="echoid-s11328" xml:space="preserve">, AC, AE, pertingant ad primam, AV, AM, ad ſecun-<lb/>dam, erit, AC, ad, AE, vt circumferentia, RSO, ad, RSN, AV, verò <lb/>ad, AM, erit vt circumferentia tota, RNOS, cum, RSO, ad, RNOS, <lb/>totam, cum, RSON, &</s> <s xml:id="echoid-s11329" xml:space="preserve">, ſic in cæteris.</s> <s xml:id="echoid-s11330" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1044" type="section" level="1" n="629"> <head xml:id="echoid-head659" xml:space="preserve">THEOREMA IX. PROPOS. IX.</head> <p> <s xml:id="echoid-s11331" xml:space="preserve">SPatium compręhenſum à ſpirali ex prima reuolutione <lb/>orta, & </s> <s xml:id="echoid-s11332" xml:space="preserve">prima linea, quæ initium eſt reuolutionis, eſt <lb/>tertia pars primi circuli.</s> <s xml:id="echoid-s11333" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11334" xml:space="preserve">Sit ſpiralis in prima reuolutione genita ipſa, AIE, AE, verò re-<lb/>uolutionis initium, & </s> <s xml:id="echoid-s11335" xml:space="preserve">centro, A, interuallo, AE, ſit primus circu-<lb/>lus deſcriptus, ESM. </s> <s xml:id="echoid-s11336" xml:space="preserve">Dico ſpatium, AIE, tertiam partem eſſe cir-<lb/>culi, EMS. </s> <s xml:id="echoid-s11337" xml:space="preserve">Sumpto itaq; </s> <s xml:id="echoid-s11338" xml:space="preserve">vtcunq; </s> <s xml:id="echoid-s11339" xml:space="preserve">puncto, vt, V, in, AE, centro, <lb/>A, interuallo, AV, circulus deſcribatur, VIT, & </s> <s xml:id="echoid-s11340" xml:space="preserve">iuncta, AI, prò- <pb o="437" file="0457" n="457" rhead="LIBER VI."/> <anchor type="figure" xlink:label="fig-0457-01a" xlink:href="fig-0457-01"/> ducatur ad, S, deinde exponatur triangulum rectangulum, OQR, <lb/>cuius latus, OQ, circa rectum, OQR, ſit æquale ipſi, AE, &</s> <s xml:id="echoid-s11341" xml:space="preserve">, QR, <lb/>circumferentiæ, SME, & </s> <s xml:id="echoid-s11342" xml:space="preserve">compleatur rectangulum, QZ, abſcin-<lb/>datur autem, OX, æqualis, AV, & </s> <s xml:id="echoid-s11343" xml:space="preserve">per, X, ducatur, XY, parallela, <lb/>RE, ſecans, ZR, in, Y, &</s> <s xml:id="echoid-s11344" xml:space="preserve">, OR, in, N, & </s> <s xml:id="echoid-s11345" xml:space="preserve">vertice, O, per punctum, <lb/> <anchor type="note" xlink:label="note-0457-01a" xlink:href="note-0457-01"/> R, deſcribatur ſemiparabola, RGO, circa axem, OZ, quam ſecet, <lb/>YX, in, G, & </s> <s xml:id="echoid-s11346" xml:space="preserve">per, G, agatur, GB, parallela, OQ, incidens ipſi, ZO, <lb/>in, B. </s> <s xml:id="echoid-s11347" xml:space="preserve">Quoniam ergo quadratum, ZR, ad quadratum, BG, eſt <lb/> <anchor type="note" xlink:label="note-0457-02a" xlink:href="note-0457-02"/> vt, ZO, ad, OB, ideò, RQ, ad, GX, erit vt quadratum, QO, ad <lb/>quadratum, OX, ideſt vt quadratum, EA, ad quadratum, AV, ſed <lb/>ſic etiam eſt circumferentia, ESM, ad circumferentiam, ITV, ete-<lb/>nim ad eam habet rationem compoſitam ex ratione circumferẽ-<lb/>tiæ, ESM, ad circumferentiam, IVT, ideſt ex ea, quam habet, EA, <lb/> <anchor type="note" xlink:label="note-0457-03a" xlink:href="note-0457-03"/> ad, AV, & </s> <s xml:id="echoid-s11348" xml:space="preserve">ex ratione circumferentiæ, IVT, ad circumferentiam, <lb/>ITV, ideſt circumferentiæ, MSE, ad circumferentiam, SME, ideſt <lb/>ex ratione, EA, ad, AI, vel ad, AV, duæ verò rationes, EA, ad, A <lb/> <anchor type="note" xlink:label="note-0457-04a" xlink:href="note-0457-04"/> V, componunt rationem quadrati, EA, ad quadratum, AV, ergo <lb/> <anchor type="note" xlink:label="note-0457-05a" xlink:href="note-0457-05"/> <pb o="438" file="0458" n="458" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0458-01a" xlink:href="fig-0458-01"/> circumferentia, MSE, ad circumferentiam, ITV, eſt vt quadratũ <lb/>EA, ad quadratum, AV, ideſt vt, RQ, ad, XG, eſt autem, RQ, æ, <lb/>qualis circumferentiæ, MSE, ergo &</s> <s xml:id="echoid-s11349" xml:space="preserve">, GX, circumferentiæ, ITV, <lb/>æqualis erit, & </s> <s xml:id="echoid-s11350" xml:space="preserve">ſic oſtendemus quamlibet circumferentiam ipſi <lb/>A, concentricam, & </s> <s xml:id="echoid-s11351" xml:space="preserve">interceptam inter ſpiralem, AIE, & </s> <s xml:id="echoid-s11352" xml:space="preserve">rectam <lb/>AE, tamen extra ſpatium helicum, AIE, adæquari ductæ in trili-<lb/>neo, OGRQ, ipſi, RQ, ductæ parallelæ, quæ nempè abſcindunt <lb/>verſus puncta, O, A, ipſarum, OQ, AE, partes ęquales, & </s> <s xml:id="echoid-s11353" xml:space="preserve">quia, <lb/>OQ, AE, ſupponuntur æquales, ideò omnes lineę trilinei, OGR <lb/>Q, regula, RQ, omnibus circumferentijs trilinei recta, AE, ſpira-<lb/>li, AIE, & </s> <s xml:id="echoid-s11354" xml:space="preserve">circũferentia, MSE, cõpręhenſi æquales erunt. </s> <s xml:id="echoid-s11355" xml:space="preserve">Similiter, <lb/>quia eſt, RQ, ad, NX, vt, QO, ad, OX, vel, EA, ad, AV, vel circũ-<lb/>ferentia, MSE, ad, TIV, æquatur autem, RQ, ipſi, MSE, ergo, N <lb/>X, æquatur circumferentiæ, TIV, & </s> <s xml:id="echoid-s11356" xml:space="preserve">ſic oſtendemus omnes lineas <lb/>trianguli, ORQ, adæquari omnibus circumferentijs circuli, MSE, <lb/>ergo vt trianguli, ORQ, omnes lineæ ad omnes lineas trilinei, OG <lb/>RQ, vel vt triangulum, ORQ, ad trilineum, OGRQ, ita omnes <lb/> <anchor type="note" xlink:label="note-0458-01a" xlink:href="note-0458-01"/> <pb o="439" file="0459" n="459" rhead="LIBER V."/> circumferentiæ circuli, MSE, erunt ad omnes circumferentias fi-<lb/>guræ ſpirali, AIE, recta, AE, & </s> <s xml:id="echoid-s11357" xml:space="preserve">circumferentia, MSE, concluſæ, <lb/>& </s> <s xml:id="echoid-s11358" xml:space="preserve">per conuerſionem rationis triangulum, ORQ, vel, OZR, ad fi-<lb/>guram, OGR, erit vt omnes circumferentiæ circuli, MSE, ad om-<lb/>nes circumferentias ſpatij helici, AIE, ideſt vt circulus ad ſpatium, <lb/>AIE, (quia curua, AIE, eſt talis conditionis, qualem poſtulat Prop. <lb/></s> <s xml:id="echoid-s11359" xml:space="preserve"> <anchor type="note" xlink:label="note-0459-01a" xlink:href="note-0459-01"/> 6. </s> <s xml:id="echoid-s11360" xml:space="preserve">vt elicitur ex Prop. </s> <s xml:id="echoid-s11361" xml:space="preserve">7. </s> <s xml:id="echoid-s11362" xml:space="preserve">huius) cum verò ſemiparabola, OGRZ, <lb/>ſit ſexquitertia trianguli, OZR, vnde diuidendo figura, OGR, ſit <lb/>tertia pars trianguli, OZR, ideò, & </s> <s xml:id="echoid-s11363" xml:space="preserve">ſpatium helicum, AIE, tertia <lb/>pars erit circuli, MSE, quod demonſtrare oportebat.</s> <s xml:id="echoid-s11364" xml:space="preserve"/> </p> <div xml:id="echoid-div1044" type="float" level="2" n="1"> <figure xlink:label="fig-0457-01" xlink:href="fig-0457-01a"> <image file="0457-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0457-01"/> </figure> <note position="right" xlink:label="note-0457-01" xlink:href="note-0457-01a" xml:space="preserve">20.l.4.,</note> <note position="right" xlink:label="note-0457-02" xlink:href="note-0457-02a" xml:space="preserve">38. & Sc. <lb/>40.l.1.</note> <note position="right" xlink:label="note-0457-03" xlink:href="note-0457-03a" xml:space="preserve">C. Cor. 2. <lb/>3. huius.</note> <note position="right" xlink:label="note-0457-04" xlink:href="note-0457-04a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-0457-05" xlink:href="note-0457-05a" xml:space="preserve">E 23. Sex. <lb/>Elem.</note> <figure xlink:label="fig-0458-01" xlink:href="fig-0458-01a"> <image file="0458-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0458-01"/> </figure> <note position="left" xlink:label="note-0458-01" xlink:href="note-0458-01a" xml:space="preserve">3. l. 2.</note> <note position="right" xlink:label="note-0459-01" xlink:href="note-0459-01a" xml:space="preserve">1. l. 4.</note> </div> </div> <div xml:id="echoid-div1046" type="section" level="1" n="630"> <head xml:id="echoid-head660" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s11365" xml:space="preserve">_H_Vcuſq; </s> <s xml:id="echoid-s11366" xml:space="preserve">per methodum indiuiſibilium etiam in boc Libro libuit <lb/>procedere, vt innoteſceret nos poſſe, quæ Archimedes oſtendit <lb/>Lib. </s> <s xml:id="echoid-s11367" xml:space="preserve">de Spiralibus, circa ſpatiorum menſuram, etiam tali artificio de-<lb/>monſtrare, etenim ſi quis hoc attentauerit circa ſequentes Propoſitio-<lb/>nes, idipſum obtineri poſſe facilè animaduertet, veruntamen hoc ar-<lb/>bitrio, ac iudicio Lectoris relinquendo, placuit etiam ſtylo veteri, <lb/>aliter tamen ab Archimede, eaſdem propoſitiones demonſtrare.</s> <s xml:id="echoid-s11368" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1047" type="section" level="1" n="631"> <head xml:id="echoid-head661" style="it" xml:space="preserve">Præfatæ Propoſ. alia demonſtratio.</head> <p> <s xml:id="echoid-s11369" xml:space="preserve">SIt alia ſpiralis ex prima reuolutione orta, ASRMB, AB, verò <lb/>initium reuolutionis, & </s> <s xml:id="echoid-s11370" xml:space="preserve">centro, A, interuallo, AB, ſit primus <lb/>circulus deſcriptus, ECDB, deinde exponatur triangulus, FHG, <lb/>rectum habens angulum ad, G, cuius latus, FG, ſit æquale ipſi, A <lb/>B, & </s> <s xml:id="echoid-s11371" xml:space="preserve">HG, circumferentiæ, ECDB, erit ergo triangulus, FHG, æ-<lb/>qualis circulo, ECDB, intelligatur deinde in eiuſdem trianguli <lb/> <anchor type="note" xlink:label="note-0459-02a" xlink:href="note-0459-02"/> plano tranſire parabolam, HLF, cuius vertex ſit, F, &</s> <s xml:id="echoid-s11372" xml:space="preserve">, HG, pa-<lb/> <anchor type="note" xlink:label="note-0459-03a" xlink:href="note-0459-03"/> rallela eiuſdem axi, ad quemipſa, GF, ſit ordinatim applicata, quę <lb/>tanget ſectionem in puncto, F. </s> <s xml:id="echoid-s11373" xml:space="preserve">Dico igitur, FLHG, trilineum <lb/> <anchor type="note" xlink:label="note-0459-04a" xlink:href="note-0459-04"/> æquari ſpatio reſiduo, dempto à circulo, ECDB, ſpatio helico ſub <lb/>ſpirali, ASRMB, &</s> <s xml:id="echoid-s11374" xml:space="preserve">, AB, ſi enim non eſt illi æquale, erit eodem, <lb/> <anchor type="note" xlink:label="note-0459-05a" xlink:href="note-0459-05"/> vel maius, vel minus, ſit primò maius quantitate ſpatij, quod vo-<lb/>cetur, Ω, rurſus diuidatur, HG, bifariam in, Π, & </s> <s xml:id="echoid-s11375" xml:space="preserve">iungantur, FΠ, <lb/>& </s> <s xml:id="echoid-s11376" xml:space="preserve">ſic ipſæ, AΠ, ΠG, diuidantur bifariam in, P, Γ, & </s> <s xml:id="echoid-s11377" xml:space="preserve">iungantur, <lb/>PF, ΓF, ſicque ſemper fiat donec deuentum ſit, vt ad triangulum, <lb/>FΓG, quod ſit minus ſpatio, Ω, deueniemus autem, nam à ma-<lb/>gnitudine propoſita, & </s> <s xml:id="echoid-s11378" xml:space="preserve">his, quæ relinquuntur, ſemper aufertur <lb/> <anchor type="note" xlink:label="note-0459-06a" xlink:href="note-0459-06"/> dimidium, ſecent autem iungentes, F, cum diuiſionum punctis cur- <pb o="440" file="0460" n="460" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0460-01a" xlink:href="fig-0460-01"/> uam parabolæ in punctis, I, K, L, per quæ ipſi, HG, parallelæ <lb/>ducantur, XQ, YKT, ZLV, ſecantes, FG, in punctis, Q, T, V, di-<lb/>co, FG, per hæc ſecari in partes æquales, nam, HG, ad, GT, ha-<lb/>betrationem compoſitam ex ea, quam habet, HG, ad, IQ, &</s> <s xml:id="echoid-s11379" xml:space="preserve">, IQ, <lb/>ad, ΓG, ſed, AG, ad, IQ, eſt vt quadratum, GF, ad quadratum, F <lb/>Q, &</s> <s xml:id="echoid-s11380" xml:space="preserve">, IQ, ad, ΓG, vt, QF, ad, FG, ideſt vt quadratum, QF, ad <lb/>rectangulum, QFG, ergo, HG, ad, GT, habebit rationem com-<lb/> <anchor type="note" xlink:label="note-0460-01a" xlink:href="note-0460-01"/> poſitam ex ea, quam habet quadratum, GF, ad quadratum, FQ, & </s> <s xml:id="echoid-s11381" xml:space="preserve"><lb/>quadratum, FQ, ad rectangulum, QFG, quæ erit eadem ei, quam <lb/> <anchor type="note" xlink:label="note-0460-02a" xlink:href="note-0460-02"/> habet quadratum, GF, ad rectangulum, GFQ, ideſt ei, quam ha-<lb/>bet, GF, ad, FQ, igitur, HG, ad, GT, erit vt, GF, ad, FQ, eodem <lb/> <anchor type="note" xlink:label="note-0460-03a" xlink:href="note-0460-03"/> modo oſtendemus, HG, ad, GΠ. </s> <s xml:id="echoid-s11382" xml:space="preserve">eſſe vt, GF, ad, FT, & </s> <s xml:id="echoid-s11383" xml:space="preserve">HG, ad, <lb/>GP, vt, GF, ad, FV, vnde, FG, diuiſa erit in partes æquales; </s> <s xml:id="echoid-s11384" xml:space="preserve">ha-<lb/> <anchor type="note" xlink:label="note-0460-04a" xlink:href="note-0460-04"/> bemus ergo ſpatio, FLHG, circumſcriptam figuram ex triangu-<lb/>lo, FIQ, & </s> <s xml:id="echoid-s11385" xml:space="preserve">ex trapezijs, KQ, LT, HV; </s> <s xml:id="echoid-s11386" xml:space="preserve">compoſitam, & </s> <s xml:id="echoid-s11387" xml:space="preserve">aliam in-<lb/> <anchor type="note" xlink:label="note-0460-05a" xlink:href="note-0460-05"/> ſcriptam ex trapezijs, PV, OT, NQ, compoſitam, & </s> <s xml:id="echoid-s11388" xml:space="preserve">exceſlus cir- <pb o="441" file="0461" n="461" rhead="LIBER VI."/> eumſcriptæ ſuper inſcriptam ſunttrapezia, HL, LK, KI, cum tri-<lb/>angulo, IFQ, quę, quia ęquantur trapezijs, ΓV, VN, NQ, & </s> <s xml:id="echoid-s11389" xml:space="preserve"><lb/>triangulo, IFQ, (nam dicta trapezia ſunt reſidua triangulorum in <lb/>ęqualibus baſibus, & </s> <s xml:id="echoid-s11390" xml:space="preserve">altitudinibus conſtitutorum) ideſt triangulo, <lb/>FΓG, ſubinde ſunt minora ſpatio, Ω, & </s> <s xml:id="echoid-s11391" xml:space="preserve">ideo circumſcriptã ſuperat <lb/>inſcriptaminori ſpatio, quam ſit Ω, ergo trilineum, FLHG, excedit <lb/>inſcriptã multò minori ſpatio, excedit autem ſpatium reſiduum cir-<lb/>culi, ECDB, iam dictum ſpatio Ω, ergo ſigura inſcriptaerit maior <lb/>dicto ſpatio reſiduo; </s> <s xml:id="echoid-s11392" xml:space="preserve">quod ſerua.</s> <s xml:id="echoid-s11393" xml:space="preserve"/> </p> <div xml:id="echoid-div1047" type="float" level="2" n="1"> <note position="right" xlink:label="note-0459-02" xlink:href="note-0459-02a" xml:space="preserve">z. huius.</note> <note position="right" xlink:label="note-0459-03" xlink:href="note-0459-03a" xml:space="preserve">20. l. 4.</note> <note position="right" xlink:label="note-0459-04" xlink:href="note-0459-04a" xml:space="preserve">17. Primi <lb/>Conic.</note> <note position="right" xlink:label="note-0459-05" xlink:href="note-0459-05a" xml:space="preserve">Defi. 3. <lb/>huius.</note> <note position="right" xlink:label="note-0459-06" xlink:href="note-0459-06a" xml:space="preserve">1. Decimè <lb/>Elem.</note> <figure xlink:label="fig-0460-01" xlink:href="fig-0460-01a"> <image file="0460-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0460-01"/> </figure> <note position="left" xlink:label="note-0460-01" xlink:href="note-0460-01a" xml:space="preserve">Defin, 12. <lb/>l. 1.</note> <note position="left" xlink:label="note-0460-02" xlink:href="note-0460-02a" xml:space="preserve">Coroll. 1. <lb/>l. 4.</note> <note position="left" xlink:label="note-0460-03" xlink:href="note-0460-03a" xml:space="preserve">4. Sexti <lb/>Elem. <lb/>5. l. 2.</note> <note position="left" xlink:label="note-0460-04" xlink:href="note-0460-04a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> <note position="left" xlink:label="note-0460-05" xlink:href="note-0460-05a" xml:space="preserve">5. l. 2.</note> </div> <p> <s xml:id="echoid-s11394" xml:space="preserve">Diuidatut nunc, AB, ſim liter, ac diuiditut, FG, in punctis, 3, <lb/>4, 7, centro autem communi, A, ad diſtantiam punctorum, 3, 4, <lb/>7, deſcribantur circumferentiæ, 35Δ, 4ΣRβ, 789M, ſecantes ſpi-<lb/>ralem in punctis, S, R, M, per quæ tranſeant eductæ à centro, A, <lb/>productæque vſque ad circumferentiam, ECDB, rectæ, AD, AC, <lb/>AE, vt igitur in præhabita demonſtratione oſtendemus circumfer. <lb/></s> <s xml:id="echoid-s11395" xml:space="preserve">53, & </s> <s xml:id="echoid-s11396" xml:space="preserve">rectam, IQ, inter ſe æquales eſſe, & </s> <s xml:id="echoid-s11397" xml:space="preserve">ſimiliter circumferen-<lb/>tiam, RΣ4, æquari rectæ, KT, &</s> <s xml:id="echoid-s11398" xml:space="preserve">, M987, ipſi, LV, & </s> <s xml:id="echoid-s11399" xml:space="preserve">quia, 53, <lb/> <anchor type="note" xlink:label="note-0461-01a" xlink:href="note-0461-01"/> circumferentia ad circumfer. </s> <s xml:id="echoid-s11400" xml:space="preserve">Σ4, eſt, vt, 3 A, ad, A4, ideſt vt, QF, <lb/>ad, FT, ideſt vt, IQ, ad, NT, eſt autem ęqualis, 53, ipſi, IQ, ergo, <lb/>Σ4, erit ęqualis ipſi, NT, & </s> <s xml:id="echoid-s11401" xml:space="preserve">eſt, 34, ęqualis ipſi, QT, ergo faſcia, 53 <lb/>4Σ, erit ęqualis trapezio, IQTN; </s> <s xml:id="echoid-s11402" xml:space="preserve">eodem modo oſtendemus faſciã, <lb/>R9874, æquari trapezio, KV, & </s> <s xml:id="echoid-s11403" xml:space="preserve">faſciam, MECDB7, ęquari tra-<lb/>pexio, PLVG, & </s> <s xml:id="echoid-s11404" xml:space="preserve">ideo figura compoſita ex dictis faſcijs ęqualis <lb/>erit figurę compoſitę ex his trapezijs inſcriptę trilineo, FLHG, eſt <lb/>autem hæc figura inſcripta maior ſpatio reſiduo circuli, ECDB, ab <lb/>eo dempto ſpatio ſub ſpirali, & </s> <s xml:id="echoid-s11405" xml:space="preserve">voluta, AB, ergo figura compo-<lb/>ſita ex dictis ſpatijs erit maior ſpatio dicto reſiduo, cui tamen eſt <lb/>inſcripta, quod eſt abſurdum, non ergo trilineum, FLHG, maius <lb/>eſt dicto reſiduo.</s> <s xml:id="echoid-s11406" xml:space="preserve"/> </p> <div xml:id="echoid-div1048" type="float" level="2" n="2"> <note position="right" xlink:label="note-0461-01" xlink:href="note-0461-01a" xml:space="preserve">Corol. 1. <lb/>3.huius.</note> </div> <p> <s xml:id="echoid-s11407" xml:space="preserve">Dico neq; </s> <s xml:id="echoid-s11408" xml:space="preserve">eſſe minus. </s> <s xml:id="echoid-s11409" xml:space="preserve">Sit, ſi fieri poteſt, minus ſpatio eodem, <lb/>Ω, ſit autem vt ſupra trilineo, FLHG, circumſcripta figura, ex <lb/>trapezijs, KQ, LT, HV, & </s> <s xml:id="echoid-s11410" xml:space="preserve">triangulo, IFk, compoſita, & </s> <s xml:id="echoid-s11411" xml:space="preserve">alia ei-<lb/>dem in cripta ex trapezijs, PO, OT, NQ, ita vt earum differentia <lb/>ſit minor ſpatio, Ω, igitur circumſcripta excedet trilineum, FLH <lb/>G, multò minori ſpatio, ergo circumſcripta figura minor erit ſpa-<lb/>tio reſiduo jam dicto circuli, ECDB, quod excedit trilineum, FLH <lb/>G, ſpatio, Ω, quod tamem eſt abſurdum, nam ſectorem, AS3, pa-<lb/>ret ęqualem eſſe triangulo, FIQ, faſciamque, ΑRΣ43, æquari oſtẽ <lb/>dem us trapezio, kQ, modo ſupra adhibito, & </s> <s xml:id="echoid-s11412" xml:space="preserve">faſciam, βΜ874, ip-<lb/>ſi trapezio, LT, & </s> <s xml:id="echoid-s11413" xml:space="preserve">totam faſciam, 679C, trapezio, HV, vnde fi-<lb/>gura compoſita ex dictis faſcijs, & </s> <s xml:id="echoid-s11414" xml:space="preserve">ſectore, A53, erit ęqualis com- <pb o="442" file="0462" n="462" rhead="GEOMETRIÆ"/> poſitæ ex dictis trapezijs, & </s> <s xml:id="echoid-s11415" xml:space="preserve">triangulo, FIQ, quæ oſtenſa eſt eſſe <lb/>minor ſpatio reſiduo iam dicto circuli, ECDB, & </s> <s xml:id="echoid-s11416" xml:space="preserve">ideò figura com-<lb/>poſita ex dictis faſcijs erit minor ſpatio reſiduo iam dicto, cuita-<lb/>men circumſcribitur, quod eſt abſurdum, non eſt ergo trilineum, <lb/>FLHG, minus dicto ſpatio reſiduo circuli, ECDB, & </s> <s xml:id="echoid-s11417" xml:space="preserve">oſtenſum eſt <lb/>neq; </s> <s xml:id="echoid-s11418" xml:space="preserve">eſſe illo maius, ergo erit illi æquale, & </s> <s xml:id="echoid-s11419" xml:space="preserve">triangulus, FHG, eſt <lb/>æqualis circulo, ECDB, ergo triangulus, FHG, ad trilineum, <lb/>FLHG, erit vt circulus, ECDB, ad reſiduum ſpatium ab eo dem-<lb/>pto ſpatio ſub ſpirali, A53 MB, & </s> <s xml:id="echoid-s11420" xml:space="preserve">voluta, AB, ſed triangulus, FH <lb/>G, eſt ſexquialter trilinei, FLHG, ergo circulus, ECDB, erit ſex-<lb/>quialter ſpatij reſidui iam dicti, & </s> <s xml:id="echoid-s11421" xml:space="preserve">conſequenter erit triplus ſpatij, <lb/>quod comprehenditur ſub ſpirali, A53MB, & </s> <s xml:id="echoid-s11422" xml:space="preserve">voluta, AB, quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s11423" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1050" type="section" level="1" n="632"> <head xml:id="echoid-head662" xml:space="preserve">COROLLARIVM</head> <p style="it"> <s xml:id="echoid-s11424" xml:space="preserve">_H_Inc patet eductas à vertice parabolæ ad ſecantem quamcunq; <lb/></s> <s xml:id="echoid-s11425" xml:space="preserve">diametro eiuſdem parallelam, parabola, ac tangente ibidem <lb/>interceptam, ſimiliter ſecare eandem, actranſiens per punctum cur-<lb/>uæ parabolæ, in quo prædicta eam diuidit, eidemq; </s> <s xml:id="echoid-s11426" xml:space="preserve">parallela, ſecatip-<lb/>ſam tangentem, eſtenſum enim eſt, ex. </s> <s xml:id="echoid-s11427" xml:space="preserve">g. </s> <s xml:id="echoid-s11428" xml:space="preserve">HG, ad GΓ, eſſe vt, GF, ad, <lb/>FQ. </s> <s xml:id="echoid-s11429" xml:space="preserve">ex quo nouus, ni fallor, ac pulcberrimus deſcribendi parabolam <lb/>elicitur modus.</s> <s xml:id="echoid-s11430" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1051" type="section" level="1" n="633"> <head xml:id="echoid-head663" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s11431" xml:space="preserve">_S_It deſcribendæ parabolæ diameter, A2, baſis, QX, cuiper, A, ſit <lb/>ducta parallela, LF, ſintque, AF, AL, æquales ipſis, 2X, 2Q, æ-<lb/>qualibus, ſecta autem, AF, in quotcunq; </s> <s xml:id="echoid-s11432" xml:space="preserve">partes æquales, vt in quin-<lb/>que, velutietiam, LA, in punctis, K, I, H, G, B, C, D, E, per ipſa du-<lb/>cantur diametro, A2, æquidiſtantes, ΚΣ, 19, H8, G7, B3, C4, D5, E6, <lb/>ſecantes ſi niliter baſim, QX, in æquas partes in punctis, Σ, 9, 8, 7, 3, <lb/>4, 5, 6, tandem iunctis,, LQ, FX, ipſæ ſimiliter ſecentur ac, AF, vel, <lb/>AL, ſcilicet in quinq; </s> <s xml:id="echoid-s11433" xml:space="preserve">partes æquales in punctis, R, S, T, V, M, N, <lb/>O, P, & </s> <s xml:id="echoid-s11434" xml:space="preserve">ad bæc puncta ducantur ab, A, rectæ lineæ, AR, AS, AT, <lb/>AV, AM, AN, AO, AP, necnon, AQ, AX, notentur autem pun-<lb/>cta, in quibus eductæ ab, A, ſecant parallelas diametro, A2, eatamẽ, <lb/>in quibus eductæ diuidunt eas parallelas, quæ viciſſim abſcindunt de <lb/>ipſis, AL, AF, verſus, A, eandem partem, quam ab ipſis, QL, XF, ab-<lb/>ſcindunt eductę, verſus tamen puncta, L, F, vt ex. </s> <s xml:id="echoid-s11435" xml:space="preserve">g. </s> <s xml:id="echoid-s11436" xml:space="preserve">notabimus pun-<lb/>ctum, γ, in quo educta, AP, abſcindit {4/5}. </s> <s xml:id="echoid-s11437" xml:space="preserve">ipſius, QL, verſus, L, ſicut <pb o="443" file="0463" n="463" rhead="LIBER VI."/> <anchor type="figure" xlink:label="fig-0463-01a" xlink:href="fig-0463-01"/> etiam parallela, ΚΣ, <lb/>abſcinditab, LA, ver <lb/>ſus, A, {4/5}. </s> <s xml:id="echoid-s11438" xml:space="preserve">ipſius, LA, <lb/>ſic ergo puncta notata <lb/>erunt, Q, γ Ζ, &</s> <s xml:id="echoid-s11439" xml:space="preserve">, Φ, <lb/>Δ, Γ, Π, ℟, Χ, per <lb/>quæ ſi extend itur cur-<lb/>ua linea, dico propin-<lb/>quiſſimè ſic Parabolã <lb/>delineari, prædictanẽ-<lb/>pè puncta eſſe in Pa-<lb/>rabola, cuius diame-<lb/>ter, A2, & </s> <s xml:id="echoid-s11440" xml:space="preserve">baſis, QX, <lb/>etenim babet bæc pro-<lb/>prietatem in præbabito Corollario declaratam, vel, vt clarius loquar, <lb/>XF, ad, E℟, exempligratia babetrationem compoſitam ex ratione, X <lb/>F, ad, FV, ideſt, propter conſtructionem, ex ratione, FA, ad, AE, & </s> <s xml:id="echoid-s11441" xml:space="preserve">ex <lb/>ratione, VF, ad, E℟, boc eſt adbuc ex ratione, FA, ad, AE, duæ autẽ <lb/>rationes, F A, ad, AE, componunt ratione quadrati, FA, ad quadra-<lb/>tum, AE, ergo, XF, ad, E℟, eſt Vt quadratum, FA, ad quadratum, A <lb/> <anchor type="note" xlink:label="note-0463-01a" xlink:href="note-0463-01"/> E, ſedſic etiam eſt, FX, ad parallelam ipſi, A2, interiectam inter, A <lb/>F, & </s> <s xml:id="echoid-s11442" xml:space="preserve">Parabolam circa diametrum, A2, in baſi, QX, ergo punctum, <lb/>℟, eſt in tali parabola: </s> <s xml:id="echoid-s11443" xml:space="preserve">Hoc idem oſtendemus eodem modo de cęteris <lb/>punctis, Π Γ, Δ, Φ, &</s> <s xml:id="echoid-s11444" xml:space="preserve">, Z γ, ergo dicta puncta ſunt omnia in dicta pa-<lb/>rabola. </s> <s xml:id="echoid-s11445" xml:space="preserve">Hic quidẽ modus debuiſſet poni Lib. </s> <s xml:id="echoid-s11446" xml:space="preserve">4. </s> <s xml:id="echoid-s11447" xml:space="preserve">ſiue in meo Tractatu de <lb/>Specuio V ſtorio iam in lucem edito, ſed quia oritur bic ex proprietate <lb/>proximè demonſtrata, nec illud prius menti ſubuenit, propterea idip. <lb/></s> <s xml:id="echoid-s11448" xml:space="preserve">ſum bic ſubiungere libuit.</s> <s xml:id="echoid-s11449" xml:space="preserve"/> </p> <div xml:id="echoid-div1051" type="float" level="2" n="1"> <figure xlink:label="fig-0463-01" xlink:href="fig-0463-01a"> <image file="0463-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0463-01"/> </figure> <note position="right" xlink:label="note-0463-01" xlink:href="note-0463-01a" xml:space="preserve">Corol. 1. <lb/>1. 4.</note> </div> </div> <div xml:id="echoid-div1053" type="section" level="1" n="634"> <head xml:id="echoid-head664" xml:space="preserve">THEOREMA X. PROPOS. X.</head> <p> <s xml:id="echoid-s11450" xml:space="preserve">SI in ſpirali ex prima reuolutione orta ſumatur punctũ, <lb/>quod non ſit initium, nec terminus eiuſdem ſpiralis, <lb/>ab initio autem ſpiralis ad dictum punctum agatur recta <lb/>linea, & </s> <s xml:id="echoid-s11451" xml:space="preserve">ſuper initio ſpiralis centro ad diſtantiam dicti pũ-<lb/>cti deſctibatur circulus, eiuſdem portio comprehenſa du-<lb/>cta linea, & </s> <s xml:id="echoid-s11452" xml:space="preserve">portione eius, quæ dicitur reuolutionis initia-<lb/>tiua, quam abſcindit circumferentia dicti circuli, & </s> <s xml:id="echoid-s11453" xml:space="preserve">circũ-<lb/>ferentia eiuſdem, quæ eſt ad conſequentia, tripla eſt figu-<lb/>ræ comprehenſæ ducta linea, & </s> <s xml:id="echoid-s11454" xml:space="preserve">portione ſpiralis, quæ eſt <lb/>ad conſequentia vſquc ad initium ſpiralis.</s> <s xml:id="echoid-s11455" xml:space="preserve"/> </p> <pb o="444" file="0464" n="464" rhead="GEOMETRIÆ"/> <figure> <image file="0464-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0464-01"/> </figure> <p> <s xml:id="echoid-s11456" xml:space="preserve">Sit ſpiralis ex prima reuolutione orta, AOVE, primus circulus, <lb/>EYG, ſumptum in ſpirali vtcumq; </s> <s xml:id="echoid-s11457" xml:space="preserve">punctum, V, & </s> <s xml:id="echoid-s11458" xml:space="preserve">centro, A, in-<lb/>teruallo autem, AV, circulus deſcriptus, VHX. </s> <s xml:id="echoid-s11459" xml:space="preserve">Dico portionẽ, <lb/>AOVA, comprehenſam ſpiralis portione, AOV, & </s> <s xml:id="echoid-s11460" xml:space="preserve">recta, AV, eſ-<lb/>ſe {1/3}. </s> <s xml:id="echoid-s11461" xml:space="preserve">portionis eiuſdem circuli comprehenſæ rectis, AV, AC, & </s> <s xml:id="echoid-s11462" xml:space="preserve"><lb/>circumferentia, VHXC. </s> <s xml:id="echoid-s11463" xml:space="preserve">Exponatur triangulus rectangulus, HkF, <lb/>rectum habens angulum, FKH, cuius latus, HK, æquale ſit ipſi, <lb/>AC, & </s> <s xml:id="echoid-s11464" xml:space="preserve">kF, circumferentiæ, CXHV, erit ergo triangulus, HFk, <lb/>æqualis portioni circuli, cuius baſis eſt circumferentia, CXHV; <lb/></s> <s xml:id="echoid-s11465" xml:space="preserve"> <anchor type="note" xlink:label="note-0464-01a" xlink:href="note-0464-01"/> deſcripta deinde intelligatur parabola, F℟H, cuius vertex, H, <lb/>quam tangat, KH, in, H, &</s> <s xml:id="echoid-s11466" xml:space="preserve">, FK, ſit axi eiuſdem æquidiſtans. <lb/></s> <s xml:id="echoid-s11467" xml:space="preserve">Dico trilineum, H℟Fk, eſſe æqualem ſpatio circumferentia, VHX <lb/>C, ſpirali, VOA, & </s> <s xml:id="echoid-s11468" xml:space="preserve">recta, AC, contento (quod ſpatium breuitatis <lb/>cauſa dicatur reſiduum portionis circuli, VHC,) ſi enim non, erit <lb/>co maius, vel minus, ſit primò maius, & </s> <s xml:id="echoid-s11469" xml:space="preserve">vt in antecedenti trilineo, <lb/>H℟Fk, figura circumſcripta intelligatur ex triangulo, HM3, & </s> <s xml:id="echoid-s11470" xml:space="preserve"><lb/>ex trapexijs, P3, ℟4, F6, compoſita, & </s> <s xml:id="echoid-s11471" xml:space="preserve">alia inſcripta ex trapezijs, <pb o="445" file="0465" n="465" rhead="LIBER VI."/> M4, P6, ℟k, pariter compoſita, ita vt circumſcripta ſuperet inſcri-<lb/>ptam minori ſpatio, quam ſit differentia dictarum figurarum (quę <lb/>differentia ſit ſpatium, Ω,) igitur trilineum, H℟FK, minori quã-<lb/>titate ſuperabit figuram inſcriptam, quam ſpatium reſiduum por-<lb/>tionis circuli, VHC, ergo figura inſcripta erit maior dicto reſiduo, <lb/>quod eſt abſurdum, nam ſi, AC, diuidamus ſimiliter, vt, KH, <lb/>in punctis, IBD, & </s> <s xml:id="echoid-s11472" xml:space="preserve">deſcripſerimus per eadem puncta ſuper centro, <lb/>A, circumferentias, INS, BRZ, DΠΟΣ, oſtendemus, vt in ante-<lb/>cedenti figuram compoſitam ex faſcijs, ΙΒβ, ΒDΔ, DCΧΦ, eſſe <lb/>æqualem figuræ inſcriptæ trilineo, H℟Fk, & </s> <s xml:id="echoid-s11473" xml:space="preserve">conſequenter eſſe <lb/>maiorem ſpatio reſiduo portionis circuli, VHC, cui tamen inſcri-<lb/>bitur, quodeſt abſurdum.</s> <s xml:id="echoid-s11474" xml:space="preserve"/> </p> <div xml:id="echoid-div1053" type="float" level="2" n="1"> <note position="left" xlink:label="note-0464-01" xlink:href="note-0464-01a" xml:space="preserve">2. huius. <lb/>10. l. 4.</note> </div> <p> <s xml:id="echoid-s11475" xml:space="preserve">Sit nunc trilineum, H℟Fk, minus eodem, Ω, dicto reſiduo, & </s> <s xml:id="echoid-s11476" xml:space="preserve"><lb/>cætera, vt prius conſtructa, quia ergo circumſcripta figura ſuperat <lb/>inſcriptã minorr quantitate, quam ſit, Ω, ſuperabit ipſum trilineũ, <lb/>H℟FK, multò minori quantitate, ergo figura circumſcripta mi-<lb/>nor erit ſpatio reſiduo portionis circuli, VHC, oſtendemus autem, <lb/>vt ſupra figuram compoſitam ex ſectore, ANI, & </s> <s xml:id="echoid-s11477" xml:space="preserve">ex faſcijs, IBR, <lb/>BDO, DCV, eſſe æqualem figuræ circumſcriptæ trilineo, H℟Fk, <lb/>ergo erit minor ſpatio reſiduò iam dicto, cui tamen circunſcribitur <lb/>quod eſt abſurdum, trilineum ergo, H℟Fk, neq; </s> <s xml:id="echoid-s11478" xml:space="preserve">maius, neq; </s> <s xml:id="echoid-s11479" xml:space="preserve">mi-<lb/>nus eſt ſpatio reſiduo iam dicto, ergo illi æquale, ſicut triangulus, <lb/> <anchor type="note" xlink:label="note-0465-01a" xlink:href="note-0465-01"/> HFK, eſt æqualis portioni circuli, cuius baſis eſt circumferentia, <lb/>CHV, ſed triangulus, HFk, eſt ſexquialter trilinei, H℟FK, ergo <lb/>talis portio eſt ſexquialtera ſpatij reſidui iam dicti, ergo eſt tripla <lb/>ſpatij, quod ſpirali, AROV, & </s> <s xml:id="echoid-s11480" xml:space="preserve">recta, AV, continetur, quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s11481" xml:space="preserve"/> </p> <div xml:id="echoid-div1054" type="float" level="2" n="2"> <note position="right" xlink:label="note-0465-01" xlink:href="note-0465-01a" xml:space="preserve">Elicitur <lb/>ex prima <lb/>1. 4.</note> </div> </div> <div xml:id="echoid-div1056" type="section" level="1" n="635"> <head xml:id="echoid-head665" xml:space="preserve">THEOREMA XI. PROPOS. XI.</head> <p> <s xml:id="echoid-s11482" xml:space="preserve">SI ab initio ſpiralis in prima reuolutione ortæ educan-<lb/>tur rectæ lineæ vtcumque ad ipſam ſpiralem terminã-<lb/>tes, ſpatia ſub portionibus ſpiralis abſciſsis per eductas <lb/>verſus initium, erunt vt cubi earundem eductarum.</s> <s xml:id="echoid-s11483" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11484" xml:space="preserve">Sit ſpiralis in prima reuolutione orta, ACDB, ipſa, AB, reuo-<lb/>luta, & </s> <s xml:id="echoid-s11485" xml:space="preserve">ſpiralis initium, A, à quo ad ipſam ſpiralem terminantes <lb/>ſint eductæ vtcumq; </s> <s xml:id="echoid-s11486" xml:space="preserve">AC, AD. </s> <s xml:id="echoid-s11487" xml:space="preserve">Dico ſpatium ſub portione ſpira-<lb/>lis, AXC, & </s> <s xml:id="echoid-s11488" xml:space="preserve">educta, AC, ad ſpatium ſub portione ſpiralis, AXCD, <lb/>& </s> <s xml:id="echoid-s11489" xml:space="preserve">educta, AD, eſſe vt cubum, AC, ad cubum, AD. </s> <s xml:id="echoid-s11490" xml:space="preserve">Centro igitur, <lb/>A, interuallis, C, D, ſint deſcripti circuli, CMVN, DGE, & </s> <s xml:id="echoid-s11491" xml:space="preserve">ſit <pb o="446" file="0466" n="466" rhead="GEOMETRIÆ"/> producta, AC, vſq; </s> <s xml:id="echoid-s11492" xml:space="preserve">ad circumferentiam circuli, DG, cui incidat <lb/> <anchor type="figure" xlink:label="fig-0466-01a" xlink:href="fig-0466-01"/> in, O, portio igi-<lb/>tur circuli, CAV <lb/>N, ad portionem <lb/>circuli, DAEGO, <lb/>habet rationem <lb/>compoſitá ex ea, <lb/>quam habet por-<lb/> <anchor type="note" xlink:label="note-0466-01a" xlink:href="note-0466-01"/> tio, CAVN, ad <lb/>portionem, OAE <lb/> <anchor type="note" xlink:label="note-0466-02a" xlink:href="note-0466-02"/> G, ideſt ex ratio-<lb/>ne quadrati, VA, <lb/> <anchor type="note" xlink:label="note-0466-03a" xlink:href="note-0466-03"/> ad quadratum, A <lb/>E, & </s> <s xml:id="echoid-s11493" xml:space="preserve">ex ratione <lb/>portionis, OAEG, ad portionem, DAEGO, ideſt ex r@tione cir-<lb/>cumferentiæ, EGO, ad circumferentiam, EGD, ideſt ex ratione, <lb/>VA, ad, AE, duæ autem rationes quadrati, VA, ad qua lratum, A <lb/>E, & </s> <s xml:id="echoid-s11494" xml:space="preserve">ipſius, VA, ad, AE, componunt rationem cubi, VA, ad cu-<lb/>bum, AE, ergo portio, CAVN, ad portionem, DAEGO, erit vt <lb/>cubus, VA, ad cubum, AE, ſunt autem ſpatia, AXC, AXCD, ter-<lb/>tiæ partes dictarum portionum, ergo ſpacium, AXC, ad ſpatium, <lb/>AXCD, erit vt cubus, VA, ad cubum, AE, quoderat oſtenden-<lb/> <anchor type="note" xlink:label="note-0466-04a" xlink:href="note-0466-04"/> dum.</s> <s xml:id="echoid-s11495" xml:space="preserve"/> </p> <div xml:id="echoid-div1056" type="float" level="2" n="1"> <figure xlink:label="fig-0466-01" xlink:href="fig-0466-01a"> <image file="0466-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0466-01"/> </figure> <note position="left" xlink:label="note-0466-01" xlink:href="note-0466-01a" xml:space="preserve">Deſin. 12. <lb/>1. 1.</note> <note position="left" xlink:label="note-0466-02" xlink:href="note-0466-02a" xml:space="preserve">Coroll. 2. <lb/>3. huius.</note> <note position="left" xlink:label="note-0466-03" xlink:href="note-0466-03a" xml:space="preserve">33. Sexti. <lb/>Elem. <lb/>7. huius.</note> <note position="left" xlink:label="note-0466-04" xlink:href="note-0466-04a" xml:space="preserve">Exantec.</note> </div> </div> <div xml:id="echoid-div1058" type="section" level="1" n="636"> <head xml:id="echoid-head666" xml:space="preserve">THEOREMA XII. PROPOS. XII.</head> <p> <s xml:id="echoid-s11496" xml:space="preserve">COmpræhenſum ſpatium ſub ſpirali, q æ eſt minor <lb/>ea, quæ ſub prima reuolutione fit, nec abet termi-<lb/>num initium ſpiralis, & </s> <s xml:id="echoid-s11497" xml:space="preserve">rectis, quæ à terminis ipſius in ſpi-<lb/>ralis initium ducuntur, ad ſectorem habentem radium æ-<lb/>qualem maiori earum, quæ à termino ad initium ſpiralis <lb/>ducuntur, arcum verò, qui intercipitur inter duas rectas <lb/>ſecundum eaſdem partes ſpiralis, habet eandem rationem, <lb/>quam rectangulum compræhenſum ſub rectis à terminis <lb/>in principium ſpiralis ductis, vna cum. </s> <s xml:id="echoid-s11498" xml:space="preserve">quadrati exceſſus, <lb/>quo maior dictarum linearum ſuperat minotẽ, ad quadra-<lb/>tum maioris linearum à terminis ad initium ſpiralis coniũ-<lb/>ctarum.</s> <s xml:id="echoid-s11499" xml:space="preserve"/> </p> <pb o="447" file="0467" n="467" rhead="LIBER VI."/> <figure> <image file="0467-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0467-01"/> </figure> <p> <s xml:id="echoid-s11500" xml:space="preserve">Sit ſpiralis ex prima reuolutione@@o, OP℟QX, primus circu-<lb/>lus, ΩkXF, cuius, radius, & </s> <s xml:id="echoid-s11501" xml:space="preserve">voluta ſit, OX, ſpiralis, ℟Q, minor <lb/>ea, quæ ſub prima reuolutione fit, nec habet terminum initium <lb/>ſpiralis, iunctis autem, OA, OQ, & </s> <s xml:id="echoid-s11502" xml:space="preserve">ijs vſque ad circumferentiam, <lb/>FΩΚΧ, productis, cui incidant in, Ω, F. </s> <s xml:id="echoid-s11503" xml:space="preserve">Dico trilineum, ℟OQ, <lb/>ad ſectorem, AOQ eſſe vt rectangulum, AO℟, cum {1/3}. </s> <s xml:id="echoid-s11504" xml:space="preserve">quadrati, <lb/>A℟, ad quadratum, AO. </s> <s xml:id="echoid-s11505" xml:space="preserve">Exponatur parallelogrammum rectan-<lb/>gulum, ED, cuius latus, CD, ſit æqualeipſi, OX, &</s> <s xml:id="echoid-s11506" xml:space="preserve">, BD, circum-<lb/>ferentiæ, FΩΚΧ, & </s> <s xml:id="echoid-s11507" xml:space="preserve">ſit iuncta, BC, &</s> <s xml:id="echoid-s11508" xml:space="preserve">, CT, ſit æqualis circumferẽ-<lb/>tiæ, XkΩ, TM, circumferentiæ, ΩF, &</s> <s xml:id="echoid-s11509" xml:space="preserve">, ME, circumferentiæ, FX, <lb/>& </s> <s xml:id="echoid-s11510" xml:space="preserve">per puncta, M, T, ducanturipſi, CD, parallelæ, MH, TN, <lb/>quarum, MH, ſecet, BC, in, I, & </s> <s xml:id="echoid-s11511" xml:space="preserve">per, I, ipſi, EC, parallela duca-<lb/>tur, IG, erit ergo, MC, æqualis circumferentiæ, XkΩF, & </s> <s xml:id="echoid-s11512" xml:space="preserve">quia <lb/>circumferentia, ΧFΩκ, ad circumferentiã, FΩkX, eſt vt, XO, ad, <lb/> <anchor type="note" xlink:label="note-0467-01a" xlink:href="note-0467-01"/> OQ, ideſt, EC, ad, CM, eſt vt, XO, ad, OQ, eſt autem, EC, ad, <lb/>CM, vt, EB, ad, MI, ergo, EB, ad, MI, erit vt, XO, ad, OQ, ſunt <lb/>autem ipſæ, XO, EB, æquales, ergo etiam æquales eruntiplæ, M <pb o="448" file="0468" n="468" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0468-01a" xlink:href="fig-0468-01"/> I, QO, ſic oſtendemus eſſe æquales ipſas, O℟, TL, quia ergo ſe-<lb/> <anchor type="note" xlink:label="note-0468-01a" xlink:href="note-0468-01"/> ctor, AOQ, ad ſectorem, ΩΟF, eſt vt quadratum, QO, ad quadra-<lb/>tum, OF, ideſt vt quadratum, IM, ad quadratum, MH, ideſt vt <lb/>omnia quadrata, MG, regula, EB, ad omnia quadrata, MN, & </s> <s xml:id="echoid-s11513" xml:space="preserve"><lb/>ſector, ΩΟF, ad circulum, FK, eſt vt circumferentia, ΩF, ad cir-<lb/> <anchor type="note" xlink:label="note-0468-02a" xlink:href="note-0468-02"/> cumferentiam, FΩΚΧ, ideſt vt, MT, ad, EC, ideſt vt omnia qua-<lb/>drata, MN, regula, EB, ad omnia quadrata, ED, & </s> <s xml:id="echoid-s11514" xml:space="preserve">circulus, ΩΚΧ <lb/>F, ſpatij, OXQ℟PO, triplus eſt, ideſt, ſe habet ad illud, vt omnia <lb/> <anchor type="note" xlink:label="note-0468-03a" xlink:href="note-0468-03"/> quadrata, ED, ad omnia quadrata trianguli, EBC, regula, EB, <lb/>item ſpatium, OXQ℟P, ad ſpatium, OQ℟PO, eſt vt cubus, OX, <lb/> <anchor type="note" xlink:label="note-0468-04a" xlink:href="note-0468-04"/> ab cubum, OQ, ideſt vt cubus, EB, ad cubum, MI, ideſt vt omnia <lb/>quadrata trianguli, EBC, ad omnia quadrata trianguli, MIC, er-<lb/> <anchor type="note" xlink:label="note-0468-05a" xlink:href="note-0468-05"/> go ex æquali ſector, AOQ, ad ſpatium, OQ℟PO, erit vt omnia <lb/>quadrata, MG, ad omnia quadrata trianguli, MIC, & </s> <s xml:id="echoid-s11515" xml:space="preserve">quia ſpatiũ, <lb/>OQ℟PO, ad ſpatium, ℟PO, eſt vt cubus, OQ, ad cubum, O℟, id-<lb/>eſt vt cubus, MI, ad cubum, TL, ideſt vt omnia quadrata triangu-<lb/> <anchor type="note" xlink:label="note-0468-06a" xlink:href="note-0468-06"/> li, MIC, ad omnia quadrata trianguli, TLC, ergo ſector, AOQ ad <pb o="449" file="0469" n="469" rhead="LIBER VI."/> ſpatium, OP℟, erit vt omnia quadrata, MG, regula, MI, ad om-<lb/>nia quadrata trianguli, TLC, eſt autem idem ſector, AOQ, ad ſpa-<lb/>tium, OP℟Q, vt omnia quadrata, MG, ad omnia quadratatrian-<lb/>guli, MIC, regula eadem, ergoſector, AOQ, ad reliquum ſpa-<lb/>tium, dempto ſpatio, OP℟, à ſpatio, OP℟Q, erit vt omnia qua-<lb/>drata, MG, regula, MI, ad omnia quadrata trapezij, MILT, ſed <lb/> <anchor type="note" xlink:label="note-0469-01a" xlink:href="note-0469-01"/> hæcſunt, vt quadratum, GT, adrectangulum, GTL, cum {1/3}. </s> <s xml:id="echoid-s11516" xml:space="preserve">qua-<lb/>drati, LG, ergo, conuertendo, ſpatium, ℟OQ, ad ſectorem, AOQ, <lb/>erit vtrectangulum, AO℟, cum tertia parte quadrati, A℟, ad qua-<lb/>dratum, AO, quod erat oſtendendum.</s> <s xml:id="echoid-s11517" xml:space="preserve"/> </p> <div xml:id="echoid-div1058" type="float" level="2" n="1"> <note position="right" xlink:label="note-0467-01" xlink:href="note-0467-01a" xml:space="preserve">7. huius. <lb/>4. Sexti <lb/>Elem.</note> <figure xlink:label="fig-0468-01" xlink:href="fig-0468-01a"> <image file="0468-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0468-01"/> </figure> <note position="left" xlink:label="note-0468-01" xlink:href="note-0468-01a" xml:space="preserve">Corol. 2. <lb/>3. huius.</note> <note position="left" xlink:label="note-0468-02" xlink:href="note-0468-02a" xml:space="preserve">Io. 1. 2,</note> <note position="left" xlink:label="note-0468-03" xlink:href="note-0468-03a" xml:space="preserve">9. huius. <lb/>24. 1. 2.</note> <note position="left" xlink:label="note-0468-04" xlink:href="note-0468-04a" xml:space="preserve">Ex ant.</note> <note position="left" xlink:label="note-0468-05" xlink:href="note-0468-05a" xml:space="preserve">Coro. 22, <lb/>1. 2.</note> <note position="left" xlink:label="note-0468-06" xlink:href="note-0468-06a" xml:space="preserve">F. Cor. 22. <lb/>1, 2.</note> <note position="right" xlink:label="note-0469-01" xlink:href="note-0469-01a" xml:space="preserve">28. 1. 2.</note> </div> </div> <div xml:id="echoid-div1060" type="section" level="1" n="637"> <head xml:id="echoid-head667" xml:space="preserve">THEOREMA XIII. PROPOS. XIII.</head> <p> <s xml:id="echoid-s11518" xml:space="preserve">IN eadem antecedentis ſigura centro, O, diſtantia, O℟, <lb/>deſcripta circumferentia, ℟Y, oſtendemus trilineum, <lb/>A℟Q, ad trilineum, ℟QY, eſſe vt, ℟O, cum {2/3}. </s> <s xml:id="echoid-s11519" xml:space="preserve">℟A, ad, ℟ <lb/>O, cum tertia parte ipſius, ℟A.</s> <s xml:id="echoid-s11520" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11521" xml:space="preserve">Quia enim ex antecedenteſector, AOQ, ad ſpatium, Q℟O, eſt <lb/>vt quadratum, AO, ad rectangulum, AO℟, cum {1/3}. </s> <s xml:id="echoid-s11522" xml:space="preserve">quadrati, A ℟, <lb/>per conuerſionem rationis, idem ſector ad trilineum, A℟Q, erit vt <lb/>quadratum, AO, ad rectangulum, O℟A, cum {2/3}. </s> <s xml:id="echoid-s11523" xml:space="preserve">quadrati, ℟A, nã <lb/>dempto rectangulo, AO℟, à quadrato, AO, remanet rectangulũ, <lb/>OA℟,. </s> <s xml:id="echoid-s11524" xml:space="preserve">i. </s> <s xml:id="echoid-s11525" xml:space="preserve">rectangulũ, O℟A, cum quadrato, ℟A, à quo ablato {1/3}. </s> <s xml:id="echoid-s11526" xml:space="preserve">re-<lb/>manet rectangulum, O℟A, cum {2/3}. </s> <s xml:id="echoid-s11527" xml:space="preserve">quadrati, ℟A, ideſt cum re-<lb/> <anchor type="note" xlink:label="note-0469-02a" xlink:href="note-0469-02"/> ctanguloſub {2/3}. </s> <s xml:id="echoid-s11528" xml:space="preserve">℟ A, & </s> <s xml:id="echoid-s11529" xml:space="preserve">ſub, ℟A, quod cum rectangulo, O℟A, cõ-<lb/>ficit rectangulum ſub compoſita ex, O℟, & </s> <s xml:id="echoid-s11530" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s11531" xml:space="preserve">℟A, & </s> <s xml:id="echoid-s11532" xml:space="preserve">ſub, ℟A, <lb/>conuertendo igitur trilineum, A℟Q, ad ſectorem, AOQ, erit vt <lb/> <anchor type="note" xlink:label="note-0469-03a" xlink:href="note-0469-03"/> rectangulum ſub compoſita ex, O℟, & </s> <s xml:id="echoid-s11533" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s11534" xml:space="preserve">℟A, & </s> <s xml:id="echoid-s11535" xml:space="preserve">ſub, ℟A, ad qua-<lb/>dratum, OA, inſuper ſector, AOQ, ad ſectorem, ℟OY, eſt vt qua-<lb/>dratum, AO, ad quadratum, O℟, & </s> <s xml:id="echoid-s11536" xml:space="preserve">quia idem ſector, AOQ, ad <lb/>ſpatium, Q℟O, eſt vt quadratum, AO, ad rectangulum, AO℟, cũ <lb/>{1/3}. </s> <s xml:id="echoid-s11537" xml:space="preserve">quadrati, ℟A, ideò ſector, AOQ, ad reliquum dempto à ſpatio, <lb/>℟OQ, ſectore, ℟OY, ideſt ad trilineum, Q℟Y, erit vt quadratu, <lb/> <anchor type="note" xlink:label="note-0469-04a" xlink:href="note-0469-04"/> AO, ad reliquum rectanguli, AO℟, cum {1/3}. </s> <s xml:id="echoid-s11538" xml:space="preserve">quadrati, ℟A, ab eo <lb/>dempto quadrato, ℟O, ideſt ad rectangulum, O℟A, cum {1/3}. </s> <s xml:id="echoid-s11539" xml:space="preserve">qua-<lb/>drati, ℟A, erat autem trilineum, A℟Q, ad rectorem, AOQ, vt re-<lb/>ctangulum ſub compoſita ex, O℟, & </s> <s xml:id="echoid-s11540" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s11541" xml:space="preserve">℟A, ad quadratum, AO, <lb/>ergo ex æquali trilineum, A℟Q, ad trilineum, ℟QY, erit vt rectã-<lb/>gulum ſub compoſita ex, O℟, & </s> <s xml:id="echoid-s11542" xml:space="preserve">{2/3}. </s> <s xml:id="echoid-s11543" xml:space="preserve">℟A, & </s> <s xml:id="echoid-s11544" xml:space="preserve">ſub, ℟A, ad rectangu-<lb/>lum, O℟A, cum 3. </s> <s xml:id="echoid-s11545" xml:space="preserve">parte quadrati, ℟A, ideſt ad rectang. </s> <s xml:id="echoid-s11546" xml:space="preserve">ſub com- <pb o="450" file="0470" n="470" rhead="GEOMETRIÆ"/> poſita ex, O℟, & </s> <s xml:id="echoid-s11547" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s11548" xml:space="preserve">℟A, & </s> <s xml:id="echoid-s11549" xml:space="preserve">ſub, ℟A, & </s> <s xml:id="echoid-s11550" xml:space="preserve">quia horum rectangulo-<lb/> <anchor type="note" xlink:label="note-0470-01a" xlink:href="note-0470-01"/> rum altitudines ſunt æquales, ideò trilineum, A℟Q, ad trilineum, <lb/>℟QY, erit vt, O℟, cum {2/3}. </s> <s xml:id="echoid-s11551" xml:space="preserve">℟A, ad, O℟, cum tertia parte, ℟A, <lb/>quod oſtendcre opus erat.</s> <s xml:id="echoid-s11552" xml:space="preserve"/> </p> <div xml:id="echoid-div1060" type="float" level="2" n="1"> <note position="right" xlink:label="note-0469-02" xlink:href="note-0469-02a" xml:space="preserve">1. Secundi <lb/>Elem.</note> <note position="right" xlink:label="note-0469-03" xlink:href="note-0469-03a" xml:space="preserve">Coroll. 2. <lb/>3. huius.</note> <note position="right" xlink:label="note-0469-04" xlink:href="note-0469-04a" xml:space="preserve">3. Secundi <lb/>Elem.</note> <note position="left" xlink:label="note-0470-01" xlink:href="note-0470-01a" xml:space="preserve">5. 1. 2.</note> </div> </div> <div xml:id="echoid-div1062" type="section" level="1" n="638"> <head xml:id="echoid-head668" xml:space="preserve">THEOREMA XIV. PROPOS. XIV.</head> <p> <s xml:id="echoid-s11553" xml:space="preserve">SI duæ rectæ lineę ducantur, quarum altera parabolam <lb/>tangat, altera verò ducta axi, vel diametro eiuſdem <lb/>æquidiſtans, eandem ſecet, iuncto verò puncto contactus <lb/>cum hoc ſectionis puncto, rurſus ab hoc puncto ad latus <lb/>illi oppoſitum in facto triangulo recta producatur, quæ <lb/>curuam ſecabit parabolæ, à quo ſectionis puncto ducatur <lb/>axi, vel diametro parallela quouſq; </s> <s xml:id="echoid-s11554" xml:space="preserve">incidat in tangentem: <lb/></s> <s xml:id="echoid-s11555" xml:space="preserve">Triangulum ſub eductis ad ſecantem à puncto contactus, <lb/>ad portionem parabolæ eiſdem interceptam erit, vt qua-<lb/>dratum totius tangentis ad rectangulum ſub eadem, & </s> <s xml:id="echoid-s11556" xml:space="preserve">ſub <lb/>illius abſciſſa per eam verſus punctum conta ctus per ſecũ-<lb/>dò ductam axi, vel diametro parallelam, vna cum {1/3}. </s> <s xml:id="echoid-s11557" xml:space="preserve">qua-<lb/>drati differentiæ dictarum tangentium.</s> <s xml:id="echoid-s11558" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11559" xml:space="preserve">Sit parabola curua, BIA, quam tangat, DA, in puncto, ADB, <lb/>vero axi, vel diametro eiuſdem parallel a eandem ſecet in puncto, <lb/>B, iunctis verò, BA, à puncto, A, ducatur intra triangulum, ABD, <lb/>adlatus oppoſitum, BD, vtcumq; </s> <s xml:id="echoid-s11560" xml:space="preserve">AC, ſecans curuam, AIB, in, I, <lb/>à quo verſus tangentem, AD, ducatur, IE, axi, vel diametro iam <lb/>dicto æquidiſtans. </s> <s xml:id="echoid-s11561" xml:space="preserve">Dico igitur triangulum, ABC, ad trilineum, <lb/>ABI, eſſe vt quadratum, DA, ad rectangulum, DAE, vna cum <lb/>{1/3}. </s> <s xml:id="echoid-s11562" xml:space="preserve">quadrati, DE. </s> <s xml:id="echoid-s11563" xml:space="preserve">Exponatur parallelogrammum, FP, cuius an-<lb/>gulus, OPH, ſit æqualis angulo, ADB, &</s> <s xml:id="echoid-s11564" xml:space="preserve">, OP, æqualis ipſi, AD, <lb/>&</s> <s xml:id="echoid-s11565" xml:space="preserve">, HP, ipſi, BD, abſcindatur deinde ab, OP, verſus, O, ipſa, ON, <lb/>æqualis ipſi, AE, & </s> <s xml:id="echoid-s11566" xml:space="preserve">per, N, ducatur, GN, parallela ipſi, HP, ſe-<lb/>cans iungentem, HO, in, M, (ſint enim iuncta, H, O, puncta re-<lb/>cta, HO,) ſit verò regula, HP. </s> <s xml:id="echoid-s11567" xml:space="preserve">Quia ergo, BD, ad, DC, eſt vt, D <lb/> <anchor type="note" xlink:label="note-0470-02a" xlink:href="note-0470-02"/> A, ad, AE, per conuerſionem rationis, & </s> <s xml:id="echoid-s11568" xml:space="preserve">conuertendo, CB, ad, B <lb/>D, erit vt, ED, ad, DA, ideſt vt, NP, ad, PO, ideſt vt omnia qua-<lb/>drata, GP, ad omnia quadrata, FP, regula, HP, ſed vt, CB, ad, B <lb/>D, ſic triangulus, ABC, ad triangulum, ABD, ergo vt omnia qua-<lb/>drata, GP, ad omnia quadrata, FP, ſic erit triangulus, ABC, ad <lb/>triangulum, ABD, quod ſerua,</s> </p> <div xml:id="echoid-div1062" type="float" level="2" n="1"> <note position="left" xlink:label="note-0470-02" xlink:href="note-0470-02a" xml:space="preserve">Corol. 9. <lb/>huius ad <lb/>poſteriorẽ <lb/>demonſt. <lb/>10. 1. 2.</note> </div> <pb o="451" file="0471" n="471" rhead="LIBER VI."/> <figure> <image file="0471-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0471-01"/> </figure> <p> <s xml:id="echoid-s11569" xml:space="preserve">Inſuper omnia quadrata, FP, ſunt tripla omnium quadratorum <lb/> <anchor type="note" xlink:label="note-0471-01a" xlink:href="note-0471-01"/> trianguli, OHP, & </s> <s xml:id="echoid-s11570" xml:space="preserve">ideo ſunt ad illa, vt triangulus, ABD, ad ſe-<lb/>ctionem, AIB, cuius eſt triplus, quod etiam ſerua. </s> <s xml:id="echoid-s11571" xml:space="preserve">Vlterius om-<lb/> <anchor type="note" xlink:label="note-0471-02a" xlink:href="note-0471-02"/> nia quadrata trianguli, OHP, ad omnia quadrata trianguli, OMN, <lb/>ſunt vt cubus, PO, ad cubum, ON, ideſt vt cubus, DA, ad cubum, <lb/>AE, ideſt vt ſectio, AIB, ad ſectionem, AXI, (ſunt enim tertiæ par-<lb/> <anchor type="note" xlink:label="note-0471-03a" xlink:href="note-0471-03"/> tes triangulorum, ABD, AIE, qui inter ſe ſunt, vt cubi, DA, AE,) <lb/>ergo ex &</s> <s xml:id="echoid-s11572" xml:space="preserve">quali omnia quadrata, GP, ad omnia quadrata trianguli, <lb/>OMN, erunt vt triangulus, ABC, ad ſectionem, AXI, ſed omnia <lb/>quadrata, GP, ad omnia quadrata trianguli, OHP, erant vt idem <lb/>triangulum, ABC, ad ſectionem, AIB, ergo omnia quadrata, G <lb/>P, ad reliquum, demptis omnibus quadratis trianguli, OMN, ab <lb/> <anchor type="note" xlink:label="note-0471-04a" xlink:href="note-0471-04"/> omnibus quadratis trianguli, OHP, ſcilicet ad omnia quadrati tra-<lb/>pezij, MHPN, erunt vt triangulus, ABC, ad reliquum, dempta <lb/>ſectione, AXI, a ſectione, AIB, ſcilicet ad trilineum, AIB, ſed om-<lb/>nia quadrata, GP, ad omnia quadrata trapezij, MHPN, ſunt vt <lb/>quadratum, HP, ad rectangulum, ſub, HP, MN, vna cum {1/5}. </s> <s xml:id="echoid-s11573" xml:space="preserve">qua-<lb/>drati, GM, ideſt vt quadratum, PO, ad rectangulum ſub, PO, ON, <lb/>vna cum {1/3}. </s> <s xml:id="echoid-s11574" xml:space="preserve">quadrati, PN, ergo triangulus, ABC, ad trilineum, A <lb/>BI, eric vt quadratum, PO, ad rectangulum, PON, vna cum {1/3}. </s> <s xml:id="echoid-s11575" xml:space="preserve">qua-<lb/>drati, PN, ideſt vt quadratum, DA, ad rectangulum, DAE, vna <lb/>cum {1/3}. </s> <s xml:id="echoid-s11576" xml:space="preserve">quadrati, DE, quod erat oſtendendum.</s> <s xml:id="echoid-s11577" xml:space="preserve"/> </p> <div xml:id="echoid-div1063" type="float" level="2" n="2"> <note position="right" xlink:label="note-0471-01" xlink:href="note-0471-01a" xml:space="preserve">24. l. 2.</note> <note position="right" xlink:label="note-0471-02" xlink:href="note-0471-02a" xml:space="preserve">Elicitur <lb/>ex prima <lb/>l. 4.</note> <note position="right" xlink:label="note-0471-03" xlink:href="note-0471-03a" xml:space="preserve">F. Cor. 22. <lb/>l. 2.</note> <note position="right" xlink:label="note-0471-04" xlink:href="note-0471-04a" xml:space="preserve">28. l. 2.</note> </div> </div> <div xml:id="echoid-div1065" type="section" level="1" n="639"> <head xml:id="echoid-head669" xml:space="preserve">THEOREMA XV. PROPOS. XV.</head> <p> <s xml:id="echoid-s11578" xml:space="preserve">SPatium ſub ſpirali ex quacunq; </s> <s xml:id="echoid-s11579" xml:space="preserve">reuolutione genita, <lb/>præterquam ex prima, & </s> <s xml:id="echoid-s11580" xml:space="preserve">recta eiuſdem numeri cum <pb o="452" file="0472" n="472" rhead="GEOMETRIÆ"/> ſpatio, ad circulum eiuſdem numeri, eſt vt compoſitum ex <lb/>rectangulo ſub radio eiuſdem circuli, & </s> <s xml:id="echoid-s11581" xml:space="preserve">ſub radio circuli <lb/>vnitate minoris, vna cum 3. </s> <s xml:id="echoid-s11582" xml:space="preserve">parte quad. </s> <s xml:id="echoid-s11583" xml:space="preserve">differentiæ vtri-<lb/>uſq; </s> <s xml:id="echoid-s11584" xml:space="preserve">radij, ad quadratum maioris radij prædictorum.</s> <s xml:id="echoid-s11585" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11586" xml:space="preserve">Sit quicunq; </s> <s xml:id="echoid-s11587" xml:space="preserve">circulus, CDFB, ſpatium eiuſdem numeri cum eo, <lb/>quod cõtinetur ſub ſpirali, GMSIB, & </s> <s xml:id="echoid-s11588" xml:space="preserve">voluta, GB; </s> <s xml:id="echoid-s11589" xml:space="preserve">circulus vnita-<lb/>te minor ipſæ, <emph style="sc">Ya</emph>HG. </s> <s xml:id="echoid-s11590" xml:space="preserve">Dico ſpatium dictum ad circulum, BCD <lb/>F, eſſe vt rectangulum, BAG, cum tertia parte quadrati, GB, ad <lb/>quadratum, AB. </s> <s xml:id="echoid-s11591" xml:space="preserve">Exponatur triangulus, LPQ, rectum habens an-<lb/>gulum ad, P, cuius latus, LP, ſit æquale ipſi, AB, & </s> <s xml:id="echoid-s11592" xml:space="preserve">latus, PQ, <lb/>æquale compoſito ex tot circumferentijs circuli, CDFB, quot ra-<lb/>dij primi circuii ſunt in, AB, deinde intra triangulum, LPQ, ver-<lb/>tice, L, deſcripta ſit parabola, cuius curua tranſeat per, Q, quæ <lb/> <anchor type="note" xlink:label="note-0472-01a" xlink:href="note-0472-01"/> ſit, LΩQ, ita vt, LP, ſit eandem tangens in, L, & </s> <s xml:id="echoid-s11593" xml:space="preserve">ſecans paralle-<lb/>la axi ipſa, QP, abſcindatur deinde ab, LP, recta, Lβ, æqualis ipſi-<lb/> <anchor type="figure" xlink:label="fig-0472-01a" xlink:href="fig-0472-01"/> <pb o="453" file="0473" n="473" rhead="LIBER VI."/> AG, & </s> <s xml:id="echoid-s11594" xml:space="preserve">per, β, ducatur, βΔ, parallela ipſi, QP, ſecans curuam pa-<lb/>rabolæ in, Ω, & </s> <s xml:id="echoid-s11595" xml:space="preserve">iunctis, LΩ, producatur, LΩ, vſq; </s> <s xml:id="echoid-s11596" xml:space="preserve">ad, QP, cui in-<lb/> <anchor type="note" xlink:label="note-0473-01a" xlink:href="note-0473-01"/> cidat in, Σ. </s> <s xml:id="echoid-s11597" xml:space="preserve">Quia igitur eſt, QP, ad, ΡΣ, vt, PL, ad, Lβ, per con-<lb/>uerſionem rationis, PQ, ad, QΣ, erit vt, LP, ad, Ρβ, quotuplex <lb/>ergo eſt, LP, ipſius, Ρβ, radio primi circuli æqualis, totuplex erit, <lb/>QP, ipſius, QΣ, eſt autem etiam totuplex, QP, circumferentiæ, C <lb/>DFB, ergo, QΣ, erit æqualis circumferentiæ, CDFB, eſt autem, L <lb/>P, æqualis ipſi, AB, ergo triangulus, LQΣ, circulo, CDFB, æqua-<lb/>lis erit. </s> <s xml:id="echoid-s11598" xml:space="preserve">Dico vlterius trilineum, LQΩ, æquari ſpatio circuli, CD <lb/> <anchor type="note" xlink:label="note-0473-02a" xlink:href="note-0473-02"/> FB, nempè contento ſub ſpirali, GSIB, & </s> <s xml:id="echoid-s11599" xml:space="preserve">voluta, GB, ſi enim nõ, <lb/>erit eo maior, vel minor, ſit primò, maior quantitate ſpatij ſeorſim <lb/>expoſiti, 8, diuiſa autem bifariam, QΣ, in, ℟, iungatur, L℟, rur-<lb/>ſus bifariam diuidantur, Q℟, RΣ, in punctis, &</s> <s xml:id="echoid-s11600" xml:space="preserve">, Π, & </s> <s xml:id="echoid-s11601" xml:space="preserve">iungantur, <lb/>&</s> <s xml:id="echoid-s11602" xml:space="preserve">L, ΠL, & </s> <s xml:id="echoid-s11603" xml:space="preserve">ſic ſemper fiat donec deuentum ſit ad triangulum mi-<lb/>norem ſpatio, 8, ſit is triangulus, LΠΣ, per puncta autem, in qui-<lb/> <anchor type="note" xlink:label="note-0473-03a" xlink:href="note-0473-03"/> bus, LΠ, L℟, L&</s> <s xml:id="echoid-s11604" xml:space="preserve">, ſecant curuam, QΩ, ſcilicet per, O, V, Z, du-<lb/>cantur, QP, parallelæ, 7Γ, 69, Φ+, quæ ſi producantur ſecent, β <lb/>P, in punctis, 2, 3, 4, quia ergo, Q&</s> <s xml:id="echoid-s11605" xml:space="preserve">, &</s> <s xml:id="echoid-s11606" xml:space="preserve">℟, ℟Π, ΠΣ, ſunt æqua-<lb/>les facilè oſtendemus per Coroll. </s> <s xml:id="echoid-s11607" xml:space="preserve">Prop. </s> <s xml:id="echoid-s11608" xml:space="preserve">9. </s> <s xml:id="echoid-s11609" xml:space="preserve">huius, etiam, P4, 43, 32, <lb/> <anchor type="note" xlink:label="note-0473-04a" xlink:href="note-0473-04"/> 2β, eſſe æquales, ſimiliter facilè oſtendemus, trapezia, QZ, ZV, V <lb/>O, & </s> <s xml:id="echoid-s11610" xml:space="preserve">triangulum, LΟΓ, ſimul collecta æquari triangulo, LΠΣ, .</s> <s xml:id="echoid-s11611" xml:space="preserve">i. <lb/></s> <s xml:id="echoid-s11612" xml:space="preserve">eſſe minora ſpatio, 8, habemus ergo ſpatio, LQΩ, circum ſcriptam <lb/>figuram ex triangulis, LQ&</s> <s xml:id="echoid-s11613" xml:space="preserve">, L+Z, L9V, LΓΟ, & </s> <s xml:id="echoid-s11614" xml:space="preserve">aliam eidem in-<lb/>ſcriptam ex triangulis, LZ+, LV6, LO7, LΩΣ, compoſitam, <lb/>quam circumſcripta excedit minori ſpatio, quam ſit, 8, ergo tri-<lb/>lineum, LQΩ, excedet inſcriptam multò minori ſpatio, ergo in-<lb/>ſcripta erit manior ſpatio, GMSIB, quod eſt abſurdum, nam ſi cen-<lb/>tro, A, ſemidiametris æqualibus ipſis, L4, L3, L2, deſcribantur ſe-<lb/>ctores, vel ſectorum reſidua. </s> <s xml:id="echoid-s11615" xml:space="preserve">AkIR, XSN, TME, habebimus ſpa-<lb/>tio, BISMG, inſcriptam figuram ex ſectoribus, vel ſectorum reſi-<lb/>duis iam dictis compoſitam, & </s> <s xml:id="echoid-s11616" xml:space="preserve">aliam circumſcriptam ex ſectori-<lb/>bus, vel ſectorum reſidius, BAC, IAR, SAN, MAE, compoſitam, <lb/> <anchor type="note" xlink:label="note-0473-05a" xlink:href="note-0473-05"/> & </s> <s xml:id="echoid-s11617" xml:space="preserve">quia, ΣQ, ad, QP, eſt vt, βΡ, ad, PL, &</s> <s xml:id="echoid-s11618" xml:space="preserve">, PQ. </s> <s xml:id="echoid-s11619" xml:space="preserve">ad, Q&</s> <s xml:id="echoid-s11620" xml:space="preserve">, eſt vt, L <lb/>P, ad, P4, ex æquali, ΣQ, ad, Q&</s> <s xml:id="echoid-s11621" xml:space="preserve">, erit vt, βΡ, ad, P4, ideſt vt GB, <lb/>ad, BK, ideſt vt circumferentia, CDFB, ad circumferentiam, CB, <lb/>(nam dum punctus, B, deſcribit totam circumferentiam, CDFB, <lb/>punctus deſcribens ſpiralem percurrit ipſam, GB, & </s> <s xml:id="echoid-s11622" xml:space="preserve">dum, B, deſcri-<lb/>pſit circumferentiam, CB, idem punctus percurrit ipſam, Bκ,) eſt <lb/> <anchor type="note" xlink:label="note-0473-06a" xlink:href="note-0473-06"/> autem, QΣ, æqualis circumferentiæ, CDFB, ergo, Q&</s> <s xml:id="echoid-s11623" xml:space="preserve">, æqualis <lb/>erit circumfer. </s> <s xml:id="echoid-s11624" xml:space="preserve">CB, eſt verò, Q&</s> <s xml:id="echoid-s11625" xml:space="preserve">, ad, ΦΖ, vt, PL, ad, L4, ideſt vt, <lb/>BA, ad, Ak, ideſt vt circumferentia, CB, ad circumferentiam, IK, <pb o="454" file="0474" n="474" rhead="GEOMETRIÆ"/> ergo, ΦΖ, erit æqualis circumferentiæ, IK, & </s> <s xml:id="echoid-s11626" xml:space="preserve">eſt altitudo triangu-<lb/> <anchor type="note" xlink:label="note-0474-01a" xlink:href="note-0474-01"/> li, LΦ Ζ, ideſt L4, æqualis ipſi, kA, ergo triangulus, LΦΖ, ſectori, <lb/>KAI, æqualis erit. </s> <s xml:id="echoid-s11627" xml:space="preserve">Eodem modo oſtendemus triangulum, LVB, <lb/> <anchor type="note" xlink:label="note-0474-02a" xlink:href="note-0474-02"/> æquari ſectori, AXS, & </s> <s xml:id="echoid-s11628" xml:space="preserve">triangulum, LO7, ſectori, ATM, & </s> <s xml:id="echoid-s11629" xml:space="preserve">tan-<lb/>dem triangulum, Ab<emph style="sub">9</emph>Ω, ſectori, AHG, ergo figura inſcripta trili-<lb/>neo, LQΩ, æqualis erit inſcriptæ ſpatio, GMSIB, eſt autem illa <lb/>maior ſpatio, GMSIB, èrgo figura inſcripta ſpatio, GMSIB, erit <lb/>eodem ſpatio, GMSIB, maior, quod eſt abſurdum, non ergo tri-<lb/>lineus, LQΩ, maior eſt ſpatio, GMSIB.</s> <s xml:id="echoid-s11630" xml:space="preserve"/> </p> <div xml:id="echoid-div1065" type="float" level="2" n="1"> <note position="left" xlink:label="note-0472-01" xlink:href="note-0472-01a" xml:space="preserve">20. l. 4.</note> <figure xlink:label="fig-0472-01" xlink:href="fig-0472-01a"> <image file="0472-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0472-01"/> </figure> <note position="right" xlink:label="note-0473-01" xlink:href="note-0473-01a" xml:space="preserve">‘Corol’ .9. <lb/>huius, ad <lb/>poſter de. <lb/>moaſtr.</note> <note position="right" xlink:label="note-0473-02" xlink:href="note-0473-02a" xml:space="preserve">Iuxta 2. <lb/>huius</note> <note position="right" xlink:label="note-0473-03" xlink:href="note-0473-03a" xml:space="preserve">Iuxta prè <lb/>10. Elem.</note> <note position="right" xlink:label="note-0473-04" xlink:href="note-0473-04a" xml:space="preserve">Iux. Cor. <lb/>1. tertiæ <lb/>huius.</note> <note position="right" xlink:label="note-0473-05" xlink:href="note-0473-05a" xml:space="preserve">Corol. 9. <lb/>huius, ad <lb/>poſteric-<lb/>rem dc-<lb/>monſt.</note> <note position="right" xlink:label="note-0473-06" xlink:href="note-0473-06a" xml:space="preserve">Elicitur <lb/>ex 4 Sex. <lb/>ti Elem.</note> <note position="left" xlink:label="note-0474-01" xlink:href="note-0474-01a" xml:space="preserve">Corol. 2. <lb/>3. huius.</note> <note position="left" xlink:label="note-0474-02" xlink:href="note-0474-02a" xml:space="preserve">Elici tur <lb/>ex Cor. 1. <lb/>3. huius.</note> </div> <p> <s xml:id="echoid-s11631" xml:space="preserve">Sed dico neq; </s> <s xml:id="echoid-s11632" xml:space="preserve">eſſe minorem eodem ſpatio, GMSIB, ſi enim eſt <lb/>ſit adhuc defectus ſpatium, 8, modo autem ſupra adhibito circum-<lb/>ſcribatur trilineo, ΙΩQ, figura, & </s> <s xml:id="echoid-s11633" xml:space="preserve">alia inſcribatur ex triangulis <lb/>compoſita, ita vt circumſcripta fuperet inſcriptam minori ſpatio, <lb/>quam ſit, 8, deſeruiant autem nobis iam in prima parte deſcriptæ <lb/>figuræ, tum intra, & </s> <s xml:id="echoid-s11634" xml:space="preserve">extra trilineum, LΩQ, tum intra, vel extra <lb/>ſpatium, GMSIB. </s> <s xml:id="echoid-s11635" xml:space="preserve">Igitur figura circumſcripta trilineo, LΩQ, ſu-<lb/>perabit eundem trilineum multò minori ſpatio, quam ſit, 8, nem-<lb/>pè quam ſpatium, GMSIB, excedat trilineum, LΩQ, ergo figura <lb/>huic trilineo circumſcripta erit minor ſpatio, GMSIB, oſtendemus <lb/>autem eandem ęquari figurę circumſcriptæ eidem ſpatio, GMSIB, <lb/>modo ſuprapoſito, ergo figura circumſcripta ſpatio, GMSIB, erit <lb/>eodem minor, quod eſt abſurdum, igitur trilineus, LΩQ, neq; </s> <s xml:id="echoid-s11636" xml:space="preserve">eſt <lb/>maior, neq; </s> <s xml:id="echoid-s11637" xml:space="preserve">minor ſpatio, GMSIB, ergo eſt eidem æqualis, & </s> <s xml:id="echoid-s11638" xml:space="preserve">eſt <lb/>triangulus, LQΣ, æqualis circulo, CDFB, ergo circulus, CDFB, <lb/> <anchor type="note" xlink:label="note-0474-03a" xlink:href="note-0474-03"/> ad ſpatium, GMSIB, erit vt triangulus, LQΣ, ad tr lineum, LQΩ, <lb/>eſt autem triangulus, LQΣ, ad trilineum, LQΩ, vt quadratum, P <lb/>Lad rectangulum, ΡLβ, vnam {1/3}. </s> <s xml:id="echoid-s11639" xml:space="preserve">quadrati, Ρβ, ergo circulus, CD <lb/>FB, ad ſpatium, GMSIB, erit vt quadratum, PL, ad rectangulum, <lb/>ΡLβ, vna cum {1/3}. </s> <s xml:id="echoid-s11640" xml:space="preserve">quadrati, Ρβ, ideſt vt quadratum, BA, ad rectan-<lb/>gulum, BAG, vna cum {1/3}. </s> <s xml:id="echoid-s11641" xml:space="preserve">quadrati, GB, quod erat nobis oſtendẽ-<lb/>dum.</s> <s xml:id="echoid-s11642" xml:space="preserve"/> </p> <div xml:id="echoid-div1066" type="float" level="2" n="2"> <note position="left" xlink:label="note-0474-03" xlink:href="note-0474-03a" xml:space="preserve">Ex ant.</note> </div> </div> <div xml:id="echoid-div1068" type="section" level="1" n="640"> <head xml:id="echoid-head670" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s11643" xml:space="preserve">_P_Oterant autem, vt in Prop. </s> <s xml:id="echoid-s11644" xml:space="preserve">5. </s> <s xml:id="echoid-s11645" xml:space="preserve">& </s> <s xml:id="echoid-s11646" xml:space="preserve">6. </s> <s xml:id="echoid-s11647" xml:space="preserve">huius, componi figuræ, quæ <lb/>circumſcrihuntur, & </s> <s xml:id="echoid-s11648" xml:space="preserve">inſcrihuntur, ex trapezijs, in quo caſu, <lb/>circumſcriptio, & </s> <s xml:id="echoid-s11649" xml:space="preserve">inſcriptio intelligi dehuiſſet cir ca trilineum, Q <lb/>ΩΣ, vel in ſupra demonſtratis propoſitionibus poterant dicta figuræ <lb/>ex triangulis componi, veluti in hac effectum eſt, & </s> <s xml:id="echoid-s11650" xml:space="preserve">tunc circumſcri-<lb/>ptio, & </s> <s xml:id="echoid-s11651" xml:space="preserve">inſcriptio ſectionibus, FLH, in Schem ate poſterioris demõſtra-<lb/>t ionis Prop. </s> <s xml:id="echoid-s11652" xml:space="preserve">9. </s> <s xml:id="echoid-s11653" xml:space="preserve">& </s> <s xml:id="echoid-s11654" xml:space="preserve">H℟F, in Propoſ. </s> <s xml:id="echoid-s11655" xml:space="preserve">10. </s> <s xml:id="echoid-s11656" xml:space="preserve">fieridebuiſſet intelligi, banc <lb/>tamen varietatem proſequutus ſum, vt pateat vtroq; </s> <s xml:id="echoid-s11657" xml:space="preserve">modo nos, quod <lb/>inquirimus, obtinere poſſe.</s> <s xml:id="echoid-s11658" xml:space="preserve"/> </p> <pb o="455" file="0475" n="475" rhead="LIBER VI."/> </div> <div xml:id="echoid-div1069" type="section" level="1" n="641"> <head xml:id="echoid-head671" xml:space="preserve">THEOREMA XVI. PROPOS. XVI.</head> <p> <s xml:id="echoid-s11659" xml:space="preserve">SI in ſpirali ex quacunq; </s> <s xml:id="echoid-s11660" xml:space="preserve">reuolutione genita ſumatur <lb/>punctum, quod non ſit initium, nec terminus eiuſdem <lb/>ſpiralis, & </s> <s xml:id="echoid-s11661" xml:space="preserve">iungantur cum puncto, quod eſt initium reuo-<lb/>lutionis, quo tanquam centro ad diſtantiam ſumpti puncti <lb/>circulus ſit deſcriptus, huius ſector, vel ſectoris reſiduum, <lb/>cuius baſis ſit circumferentia inter hoc punctum, & </s> <s xml:id="echoid-s11662" xml:space="preserve">princi-<lb/>pium circulationis ad partes conſequentes incluſa, ad ſpa-<lb/>tium helicum ab eodem ſectore, vel ſectoris reſiduo, ap-<lb/>prehenſum, erit vt quadratum ſemidiametri deſcripti cir-<lb/>culi, ad rectangulum ſub eodem, & </s> <s xml:id="echoid-s11663" xml:space="preserve">ſub radio circuli eiuf-<lb/>dem numeri cum ſpirali vnitate prædicta minoris, vna cũ <lb/>tertia parte quadrati exceſſus vtriuſq; </s> <s xml:id="echoid-s11664" xml:space="preserve">radij.</s> <s xml:id="echoid-s11665" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11666" xml:space="preserve">Conſpiciatur antecedentis figura, in qua ſumpto vtcunq; </s> <s xml:id="echoid-s11667" xml:space="preserve">pun-<lb/>cto in ſpirali, GMSIB, quod ſit, I, intelligatur deſcriptus circulus, <lb/>IRεK. </s> <s xml:id="echoid-s11668" xml:space="preserve">Dico igitur ſectorem, vel eius reſiduum, cuius baſis eſt <lb/>circumferentia, IRεk, ad rectas, IA, Ak, terminata, ad ſpatium <lb/>ſub ſpiralis portiones, ISMG, & </s> <s xml:id="echoid-s11669" xml:space="preserve">rectis, IA, AG, eſſe vt quadratũ, <lb/>IA, ad rectangulum ſub, IA, AG, vna cum {1/3}. </s> <s xml:id="echoid-s11670" xml:space="preserve">quadrati, Gk; </s> <s xml:id="echoid-s11671" xml:space="preserve">in ip-<lb/>ſa enim, QΣ, iam habebus, & </s> <s xml:id="echoid-s11672" xml:space="preserve">Σ, æqualem circumferentiæ, CDFB, <lb/>terminanti ad, C, B, producatur, Φ+, quouſque ſecet ambas, LΠ, L <lb/>Σ, vt in, {12/ }, {13/ }, & </s> <s xml:id="echoid-s11673" xml:space="preserve">quia, &</s> <s xml:id="echoid-s11674" xml:space="preserve">Σ, ad, Z, {13/ }, eſt vt, ΣL, ad, L, {13/ }, vel vt, PL, ad, <lb/> <anchor type="note" xlink:label="note-0475-01a" xlink:href="note-0475-01"/> L4, ſiue, BA, ad, AK, ſiue circumferentia, CDFB, ad circumferẽtiam, <lb/>IRεK, ideò circumferentia, IRεk, erit æqualis ipſi, Z {13/ }, ſi ergo diui-<lb/>damus, Z, {13/ }, bifariam, & </s> <s xml:id="echoid-s11675" xml:space="preserve">factas portiones adhuc bifariam, & </s> <s xml:id="echoid-s11676" xml:space="preserve">ſic sẽ-<lb/>per fiat, iungẽtes diuiſionum pũcta cum, L, & </s> <s xml:id="echoid-s11677" xml:space="preserve">per puncta, in quibus <lb/>iſtę iungẽtes ſecant curuã parabolę, ΖΩ, ductis ipſi, Z {13/ }, parallelis, <lb/>vt in antecedenti circumſcripſerimus trilineo, LΖΩ, figuram, & </s> <s xml:id="echoid-s11678" xml:space="preserve">aliã <lb/>inſcripſerimus, ex triangulis compoſitam, & </s> <s xml:id="echoid-s11679" xml:space="preserve">ſimiliter ſpatio, AIS <lb/>MGA, figuram ex ſectoribus, vel eorum reſiduis compoſitam cir-<lb/>cumſcripſerimus, velut in antecedenti (quam quia antecedentis <lb/>propoſitionis methodo ſimilis eſt, hic explanare mitto) & </s> <s xml:id="echoid-s11680" xml:space="preserve">aliam <lb/>inſcripſerimus, tandem oſtendemus trilineum, LΖΩ, neq; </s> <s xml:id="echoid-s11681" xml:space="preserve">maius, <lb/>neq; </s> <s xml:id="echoid-s11682" xml:space="preserve">minus eſſe ſpatio, AISMGA, & </s> <s xml:id="echoid-s11683" xml:space="preserve">ideò illi eſſe æquale; </s> <s xml:id="echoid-s11684" xml:space="preserve">ſimili-<lb/>ter oſten demus triangulum, LZ {13/ }, ſectori, IPεK, vel ſectoris reſi-<lb/> <anchor type="note" xlink:label="note-0475-02a" xlink:href="note-0475-02"/> duo, æqualem eſſe, nam triangulus, LQΣ, ad triangulum, LZ {13/ }, <pb o="456" file="0476" n="476" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0476-01a" xlink:href="fig-0476-01"/> habet rationem compoſitam ex ratione trianguli, LQΣ, ad trian-<lb/>gulum, L&</s> <s xml:id="echoid-s11685" xml:space="preserve">Σ, ideſt ex ratione, QΣ, ad, Σ&</s> <s xml:id="echoid-s11686" xml:space="preserve">, vel ex ratione circum-<lb/> <anchor type="note" xlink:label="note-0476-01a" xlink:href="note-0476-01"/> ferentiæ, CDFBC, ad circumferentiam, CDFB, quia prædictis æ-<lb/>quatur ideſt ex ratione circuli, CDFB, ad ſectorem, vel eius reſi-<lb/>duum, ACDFBA, & </s> <s xml:id="echoid-s11687" xml:space="preserve">ex ratione trianguli, L&</s> <s xml:id="echoid-s11688" xml:space="preserve">Σ, ad triangulum, <lb/>LZ {13/ }, ideſt ex ratione quadrati, PL, ad quadratum, L4, ideſt ex ra-<lb/> <anchor type="note" xlink:label="note-0476-02a" xlink:href="note-0476-02"/> tione quadrati, BA, ad quadratum, AK, ideſt ex ratione ſectoris <lb/>(dicatur ſic breuitatis cauſa, ſiue ſit ſector, ſiue eius reſiduum) AC <lb/>DFB, ad ſectorem, AIR εkA, quæ duæ rationes componunt ratio-<lb/> <anchor type="note" xlink:label="note-0476-03a" xlink:href="note-0476-03"/> nem circuli, CDFB, ad ſectorem, AIR εKA, ergo triangulus, LQ <lb/>Σ, ad triangulum, LZ {13/ }, erit vt circulus, CDFB, ad ſectorem, AI <lb/>RεKA, ſed triangulus, LQΣ, eſt æqualis circulo, CDFB, ergo triã-<lb/>gulus, LZ {13/ }, ſectori, AIR εKA, æqualis erit, & </s> <s xml:id="echoid-s11689" xml:space="preserve">eſt trilineus, LΖΩ, <lb/> <anchor type="note" xlink:label="note-0476-04a" xlink:href="note-0476-04"/> æqualis ſpatio, AISMGA, ergo ſector, AIR εKA, ad ſpatium, AIS <lb/>MGA, erit vt triangulus, LZ {13/ }, ad trilineum, LΖΩ, .</s> <s xml:id="echoid-s11690" xml:space="preserve">i. </s> <s xml:id="echoid-s11691" xml:space="preserve">vt quadra-<lb/>tum, 4L, ad rectangulum ſub, 4L, Lβ, cum {1/3}. </s> <s xml:id="echoid-s11692" xml:space="preserve">quadrati, 4β, .</s> <s xml:id="echoid-s11693" xml:space="preserve">i. </s> <s xml:id="echoid-s11694" xml:space="preserve">vt qua-<lb/> <anchor type="note" xlink:label="note-0476-05a" xlink:href="note-0476-05"/> dratum, IA, ad rectangulum, ſub, IA, AG, cum {1/3}. </s> <s xml:id="echoid-s11695" xml:space="preserve">quadrati, GK, <lb/>quod erat oſtendendum.</s> <s xml:id="echoid-s11696" xml:space="preserve"/> </p> <div xml:id="echoid-div1069" type="float" level="2" n="1"> <note position="right" xlink:label="note-0475-01" xlink:href="note-0475-01a" xml:space="preserve">Iuxta 4. <lb/>Sexti Ele.</note> <note position="right" xlink:label="note-0475-02" xlink:href="note-0475-02a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> <figure xlink:label="fig-0476-01" xlink:href="fig-0476-01a"> <image file="0476-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0476-01"/> </figure> <note position="left" xlink:label="note-0476-01" xlink:href="note-0476-01a" xml:space="preserve">Coroll. 1. <lb/>19. l. 2. <lb/>33. Sexti <lb/>Elem.</note> <note position="left" xlink:label="note-0476-02" xlink:href="note-0476-02a" xml:space="preserve">Coroll. 1. <lb/>19. l. 2.</note> <note position="left" xlink:label="note-0476-03" xlink:href="note-0476-03a" xml:space="preserve">Coroll. 2. <lb/>3. huius. <lb/>Defin. 12. <lb/>l. 1.</note> <note position="left" xlink:label="note-0476-04" xlink:href="note-0476-04a" xml:space="preserve">Iuxta 2. <lb/>huius.</note> <note position="left" xlink:label="note-0476-05" xlink:href="note-0476-05a" xml:space="preserve">14. huius.</note> </div> <pb o="457" file="0477" n="477" rhead="LIBER VI."/> </div> <div xml:id="echoid-div1071" type="section" level="1" n="642"> <head xml:id="echoid-head672" xml:space="preserve">THEOREMA XVII. PROPOS. XVII.</head> <p> <s xml:id="echoid-s11697" xml:space="preserve">COmpræhenſ@m ſpatium ſub ſpirali, quæ eſt minor ea, <lb/>quæ ſub vna reuolutione fit, nec habet terminum <lb/>initium ſpiralis, & </s> <s xml:id="echoid-s11698" xml:space="preserve">rectis, quæ à terminis ipſius in reuolu-<lb/>tionis initium ducuntur ad ſectorem habentem radium æ-<lb/>qualem maiori earum, quę à termino ad initium reuolutio-<lb/>nis ducitur, arcum verò, qui intercipitur inter duas rectas <lb/>ſecundum eaſdem partem ſpiralis; </s> <s xml:id="echoid-s11699" xml:space="preserve">habet eandem rationẽ, <lb/>quam rectangulum compræhenſum ſub rectis à terminis <lb/>ad initium reuolutionis ductis, vna cum tertia parte qua-<lb/>drati exceſſus, quo maior dictarum linearum ſuperat mi-<lb/>norem, ad quadratum maioris earundem.</s> <s xml:id="echoid-s11700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11701" xml:space="preserve">In eadem antecedentis figura ſupponamus arſumptam, IS, por-<lb/>tionem ſpiralis in vna reuolutione genitæ, quæ non habcat termi-<lb/>num initium talis ſpiralis, a cuius extremis punctis, I, S, ſint du-<lb/>ctæ ad, A, initium reuolutionis ipſæ, SA, IA, & </s> <s xml:id="echoid-s11702" xml:space="preserve">ſit ſector, IAR, <lb/>cuius ſemidiameter ſit æqual s maiori ductarum, IA, AS, nempè <lb/>ipſi, IA. </s> <s xml:id="echoid-s11703" xml:space="preserve">Dico ſectorem, IAR, ad trilineum, IAS, eſſe vt quadra-<lb/>tum, RA, ad rectangulum, RAS, vna cum {1/3}. </s> <s xml:id="echoid-s11704" xml:space="preserve">quadrati, RS, (vta-<lb/>mur conſtructis in eadem figura) Sector igitur, AIRεKA, eſt æqua-<lb/>lis triangulo, LZ {12/ }, vt in antecedenti oſtenſum eſt, eodem modo <lb/>probabimus triangulum, L+{13/ }, eſſe æqualem ſectori, ARεKA, <lb/>ergo reliquus triangulus, LZ+, erit æqualis reliquo ſectori, IAR; <lb/></s> <s xml:id="echoid-s11705" xml:space="preserve">ſimiliter iuxta antecedentem oſtendemus ſpatium, AISMGA, eſſe <lb/>æqualem trilineo, LΖΩ, & </s> <s xml:id="echoid-s11706" xml:space="preserve">ſpatium, ASMGA, eſſe æqualem tri-<lb/> <anchor type="note" xlink:label="note-0477-01a" xlink:href="note-0477-01"/> lineo, LVΩ ergo reliquum ſpatium, IAS, erit æquale trilineo, LZ <lb/>V, ergo ſector, IAR, ad trilineum, LZV, erit vt triangulus LZ+, <lb/>ad trilineum, LZV, ideſt vt quadratum, L4, ad rectangulum ſub, <lb/>4L3, cum {1/3}, quadrati, 34, ideſt vt quadratum, IA, vel, RA, ad re-<lb/>ctangulum ſub, RA, A<emph style="sub">c</emph>, vna cum {1/3}. </s> <s xml:id="echoid-s11707" xml:space="preserve">quadrati, RS, quod oſtende-<lb/>re opus erat.</s> <s xml:id="echoid-s11708" xml:space="preserve"/> </p> <div xml:id="echoid-div1071" type="float" level="2" n="1"> <note position="right" xlink:label="note-0477-01" xlink:href="note-0477-01a" xml:space="preserve">14. huius.</note> </div> </div> <div xml:id="echoid-div1073" type="section" level="1" n="643"> <head xml:id="echoid-head673" xml:space="preserve">THEOREMA XVIII. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s11709" xml:space="preserve">TRilineum, IRS, ad trilineum, ISX, erit vt, SA, cum <lb/>{2/3}. </s> <s xml:id="echoid-s11710" xml:space="preserve">SR, ad, SA, cum {1/3}. </s> <s xml:id="echoid-s11711" xml:space="preserve">SR.</s> <s xml:id="echoid-s11712" xml:space="preserve"/> </p> <pb o="458" file="0478" n="478" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s11713" xml:space="preserve">Huius demoſtratio non erit alia à demoſtratione 13. </s> <s xml:id="echoid-s11714" xml:space="preserve">huius, <lb/>propterea ibi recolatur.</s> <s xml:id="echoid-s11715" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1074" type="section" level="1" n="644"> <head xml:id="echoid-head674" xml:space="preserve">THEOREMA XIX. PROPOS. XIX.</head> <p> <s xml:id="echoid-s11716" xml:space="preserve">PRimi circuli ſpatium helicum ad ſpatium helicum ſe-<lb/>cundi circuli erit, vt tertia pars quadrati radij primi <lb/>circuli ad rectangulum ſub rad io primi, & </s> <s xml:id="echoid-s11717" xml:space="preserve">ſecundi circuli, <lb/>vna cum tertia parte quadrati exceſſus radij ſecundi circu-<lb/>li ſuper radium primi. </s> <s xml:id="echoid-s11718" xml:space="preserve">Spatium verò ſecundi circuli ad <lb/>ſpatium tertij erit, vt rectangulum ſub radio eiuſdem, & </s> <s xml:id="echoid-s11719" xml:space="preserve"><lb/>ſub radio circuli vnitate minoris, ideſt primi, vna cum ter-<lb/>tia parte quadrati differentiæ horum radiorũ, ad rectangu-<lb/>lum ſub radio eiuſdem, & </s> <s xml:id="echoid-s11720" xml:space="preserve">ſub radio circuli vnitate maio-<lb/>ris, ideſt tertij, vna cum tertia parte quadrati differentiæ <lb/>iſtorum radiorum, & </s> <s xml:id="echoid-s11721" xml:space="preserve">ſic deinceps in reliquis.</s> <s xml:id="echoid-s11722" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11723" xml:space="preserve">Exponantur ſuper eodem centro, A, circuli, primus, HRVO, <lb/>ſecundus, kLNM, tertius autem, CDFB, cum ſpatijs ſub ſpirali-<lb/>bus eiuſdem numeri cum circulis, primo, AGHA, ſecundo, HPK <lb/>MH, tertio autem, MZSBM. </s> <s xml:id="echoid-s11724" xml:space="preserve">Dico ſpatium primum ad ſecundũ <lb/>eſſe vt {1/3}. </s> <s xml:id="echoid-s11725" xml:space="preserve">quadrati HA, ad rectangulum ſub, HA, AM, vna cum {1/3}. <lb/></s> <s xml:id="echoid-s11726" xml:space="preserve">quadrati, HM, ſecundum verò ad tertium eſſe vt rectangulum ſub, <lb/>HA, AM, cum {1/3}. </s> <s xml:id="echoid-s11727" xml:space="preserve">quadrati, HM, ad rectangulum ſub, MA, AB, <lb/>vna cum {1/5}. </s> <s xml:id="echoid-s11728" xml:space="preserve">quadrati, MB. </s> <s xml:id="echoid-s11729" xml:space="preserve">Nam ſpatium, AGH, ad ſpatium, HP <lb/>kM, habet rationem compoſita ex ratione ſpatij, AGH, ad cir-<lb/>culum, OVRH, ideſt ex ratione {1/3}. </s> <s xml:id="echoid-s11730" xml:space="preserve">quadrati, HA, ad quadratum, <lb/> <anchor type="note" xlink:label="note-0478-01a" xlink:href="note-0478-01"/> HA, & </s> <s xml:id="echoid-s11731" xml:space="preserve">ex ratione circuli, OVRH, ad circulum, MkLN, ideſt ex <lb/>ratione quadrati, HA, ad quadratum, AM, & </s> <s xml:id="echoid-s11732" xml:space="preserve">ex ratione circuli, C <lb/> <anchor type="note" xlink:label="note-0478-02a" xlink:href="note-0478-02"/> DFB, ad ſpatium, HPMH, ideſt ex ratione quadrati, MA, ad re-<lb/>ctangulum, MAH, vna cum {1/3}. </s> <s xml:id="echoid-s11733" xml:space="preserve">quadrati, MH, quæ rationes com-<lb/>ponunt rationem {1/3}. </s> <s xml:id="echoid-s11734" xml:space="preserve">quadrati, AH, ad rectangulum, MAH, cum <lb/>{1/3}. </s> <s xml:id="echoid-s11735" xml:space="preserve">quadrati, HM. </s> <s xml:id="echoid-s11736" xml:space="preserve">Item ſpatium, HPMH, ad ſpatium, MZSBM, <lb/>habet rationem compoſitam ex ratione ſpatij, HPMH, ad circu-<lb/>lum, kLNM, ideſt ex ratione rectanguli, HAM, cum {1/3}. </s> <s xml:id="echoid-s11737" xml:space="preserve">quadrati, <lb/> <anchor type="note" xlink:label="note-0478-03a" xlink:href="note-0478-03"/> HM, ad quadratum, AM, & </s> <s xml:id="echoid-s11738" xml:space="preserve">ex ratione circuli, kLNM, ad circu-<lb/>lum, CDFB, ideſt quadrati, MA, ad quadratum, AB, & </s> <s xml:id="echoid-s11739" xml:space="preserve">tandem <lb/> <anchor type="note" xlink:label="note-0478-04a" xlink:href="note-0478-04"/> ex ratione circuli, CDFB, ad ſpatium, MZSBM, ideſt ex ratione <lb/>quadrati, BA, ad rectangulum, BAM, cum {1/3}. </s> <s xml:id="echoid-s11740" xml:space="preserve">quadrati, MB, quæ <pb o="459" file="0479" n="479" rhead="LIBER VI."/> <anchor type="figure" xlink:label="fig-0479-01a" xlink:href="fig-0479-01"/> rationes componunt rationem rectanguli, HAM, cum {1/3}. </s> <s xml:id="echoid-s11741" xml:space="preserve">quadrati, <lb/>HM, ad rectangulum, MAB, cum {1/3}. </s> <s xml:id="echoid-s11742" xml:space="preserve">quadrati, MB. </s> <s xml:id="echoid-s11743" xml:space="preserve">Et ſic dein-<lb/>ceps oſtendemus tertium ſpatium ad quartum eſſe, vt rectangulũ, <lb/>MAB, cum {1/3}. </s> <s xml:id="echoid-s11744" xml:space="preserve">quadrati, MB, ad rectangulum ſub, BA, & </s> <s xml:id="echoid-s11745" xml:space="preserve">radio <lb/>circuli vnitate maioris, vna cum {1/3}. </s> <s xml:id="echoid-s11746" xml:space="preserve">quadrati differentiæ horum <lb/>radiorum, quæ differentia ſemper eſt æqualis radio primi circuli, <lb/>quod oſtendere opus erat.</s> <s xml:id="echoid-s11747" xml:space="preserve"/> </p> <div xml:id="echoid-div1074" type="float" level="2" n="1"> <note position="left" xlink:label="note-0478-01" xlink:href="note-0478-01a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-0478-02" xlink:href="note-0478-02a" xml:space="preserve">Coroll. 2. <lb/>11. l. 3. <lb/>15. huius. <lb/>Defin. 12. <lb/>l. 1.</note> <note position="left" xlink:label="note-0478-03" xlink:href="note-0478-03a" xml:space="preserve">15. huiu.</note> <note position="left" xlink:label="note-0478-04" xlink:href="note-0478-04a" xml:space="preserve">Coroll. 2. <lb/>11. l. 3. <lb/>15. huius.</note> <figure xlink:label="fig-0479-01" xlink:href="fig-0479-01a"> <image file="0479-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0479-01"/> </figure> </div> </div> <div xml:id="echoid-div1076" type="section" level="1" n="645"> <head xml:id="echoid-head675" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s11748" xml:space="preserve">EXponatur triangulus, ETI, habens rectum angulum ad, T, <lb/>cuius latus, ET, ſit æquale radio primi circuli, &</s> <s xml:id="echoid-s11749" xml:space="preserve">, TI, eiuſdẽ <lb/>circumferentiæ, & </s> <s xml:id="echoid-s11750" xml:space="preserve">per, EI, tranſeat parabolæ curua quam tangat, <lb/>TE, in, E, vertice, ſecet verò, TI, in, I, eiuſdem axi æquidiſtans, <lb/>deinde indefinitè producta, ET, verſus, T, in ea ſumantur tot par-<lb/> <anchor type="note" xlink:label="note-0479-01a" xlink:href="note-0479-01"/> tes æquales ipſi, ET, quot radij primi circuli ſunt in radio, AB, quę <pb o="460" file="0480" n="480" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0480-01a" xlink:href="fig-0480-01"/> ſint, ET, TY, YZ, & </s> <s xml:id="echoid-s11751" xml:space="preserve">per puncta, YZ, ducantur parabolæ axi æ-<lb/>quidiſtantes, YX, ZQ, curuæ eiuſdem indefinitè productæ occur-<lb/>rentes in punctis, X, Q, & </s> <s xml:id="echoid-s11752" xml:space="preserve">iungantur, EX, EQ. </s> <s xml:id="echoid-s11753" xml:space="preserve">Erit igitur ſectio, <lb/>EIX, ad ſectionem, E℟I, vt cubus, YE, ad cubum, ET, ſic enim <lb/>ſunt eorum tripla ſcilicet triangula, EIT, EXY, quod elicitur ex <lb/>prima Lib. </s> <s xml:id="echoid-s11754" xml:space="preserve">4. </s> <s xml:id="echoid-s11755" xml:space="preserve">&</s> <s xml:id="echoid-s11756" xml:space="preserve">, diuidendo, trilineum, EXI, ad ſectionem, E℟I, <lb/>erit vt parallelepipedum ter ſub, ET, ac quadrato, TY, & </s> <s xml:id="echoid-s11757" xml:space="preserve">ter ſub, <lb/> <anchor type="note" xlink:label="note-0480-01a" xlink:href="note-0480-01"/> YT, & </s> <s xml:id="echoid-s11758" xml:space="preserve">quadrato, TE, cum cubo, TY, ad cubum, TE, vel vt horũ <lb/>ſubtripla, ſcilicet, vt parallelepipedum ſemel ſub, YT, & </s> <s xml:id="echoid-s11759" xml:space="preserve">quadra-<lb/>to, TE, & </s> <s xml:id="echoid-s11760" xml:space="preserve">ſub, ET, & </s> <s xml:id="echoid-s11761" xml:space="preserve">quadrato, TY, ſcilicet ſub, YT, & </s> <s xml:id="echoid-s11762" xml:space="preserve">rectan-<lb/> <anchor type="note" xlink:label="note-0480-02a" xlink:href="note-0480-02"/> gulo, YTE, cum {1/3}. </s> <s xml:id="echoid-s11763" xml:space="preserve">cubi, TY, ideſt cum parallelepipedo ſub, TY, & </s> <s xml:id="echoid-s11764" xml:space="preserve"><lb/>{1/3}. </s> <s xml:id="echoid-s11765" xml:space="preserve">quadrati, TY, ad {1/ }. </s> <s xml:id="echoid-s11766" xml:space="preserve">cubi, TE, ideſt ad parallelepipedum ſub, T <lb/>E, vel, TY, & </s> <s xml:id="echoid-s11767" xml:space="preserve">tertia parte quadrati, TE, nempé vt parallelepipe-<lb/>dum ſub, TY, & </s> <s xml:id="echoid-s11768" xml:space="preserve">quadr. </s> <s xml:id="echoid-s11769" xml:space="preserve">ET, & </s> <s xml:id="echoid-s11770" xml:space="preserve">rectangulo, YTE, & </s> <s xml:id="echoid-s11771" xml:space="preserve">tertia parte <lb/>quadrati, YT, quod conficit parallelepipedum ſub, YT, & </s> <s xml:id="echoid-s11772" xml:space="preserve">rectan-<lb/>gulo ſub, YET, & </s> <s xml:id="echoid-s11773" xml:space="preserve">ſub tertia parte quadrati, YT, ad parallelepi-<lb/> <anchor type="note" xlink:label="note-0480-03a" xlink:href="note-0480-03"/> pedum ſub, YT, & </s> <s xml:id="echoid-s11774" xml:space="preserve">ſub t ertia parte quadrati, TE, & </s> <s xml:id="echoid-s11775" xml:space="preserve">quia horum <lb/>parallelepipedorum altitudines ſunt eædem, ideò erunt, vt baſes <lb/>ſcilicet, vt rectangulum ſub, YET, cum tertia parte quadrati, TY, <lb/> <anchor type="note" xlink:label="note-0480-04a" xlink:href="note-0480-04"/> ad, {1/ }. </s> <s xml:id="echoid-s11776" xml:space="preserve">quadrati, ET. </s> <s xml:id="echoid-s11777" xml:space="preserve">Eodem modo oſtendemus trilineum, EQX, <lb/>adtrilineum, EXI, eſſe vt exceſſus cubi, ZE, ſuper cubum, YE, ad <lb/>exceſſum cubi, YE, ſuper cubum, TE, .</s> <s xml:id="echoid-s11778" xml:space="preserve">i. </s> <s xml:id="echoid-s11779" xml:space="preserve">vt parallelepipedum ter <lb/>ſub, ZY, & </s> <s xml:id="echoid-s11780" xml:space="preserve">quadrato, YE, ter ſub EY, & </s> <s xml:id="echoid-s11781" xml:space="preserve">quadrato, YZ, cum cu-<lb/> <anchor type="note" xlink:label="note-0480-05a" xlink:href="note-0480-05"/> bo, YZ, ad parallelepipedum ter ſub, ET, & </s> <s xml:id="echoid-s11782" xml:space="preserve">quadrato, TY, ter <pb o="461" file="0481" n="481" rhead="LIBER VI."/> ſub, YT, & </s> <s xml:id="echoid-s11783" xml:space="preserve">quadrato, TE, cum cubo, TY, vel vt horum ſub tri-<lb/>pla .</s> <s xml:id="echoid-s11784" xml:space="preserve">i. </s> <s xml:id="echoid-s11785" xml:space="preserve">vt parallelepipedum ſub, ZY, & </s> <s xml:id="echoid-s11786" xml:space="preserve">quadrato, YE, ſub, EY, & </s> <s xml:id="echoid-s11787" xml:space="preserve"><lb/>quadrato, YZ; </s> <s xml:id="echoid-s11788" xml:space="preserve">.</s> <s xml:id="echoid-s11789" xml:space="preserve">i. </s> <s xml:id="echoid-s11790" xml:space="preserve">ſub, ZY, & </s> <s xml:id="echoid-s11791" xml:space="preserve">rectangulo, ZYE, cum tertia parte <lb/>cubi, ZY, quæ conficiunt parallelepipedum ſub, ZY, & </s> <s xml:id="echoid-s11792" xml:space="preserve">his iunctis <lb/> <anchor type="note" xlink:label="note-0481-01a" xlink:href="note-0481-01"/> ideſt rectangulo, ZYE, quadrato, YE, cum tertia parte quadrati, <lb/>ZY, ideſt ſub, ZY, & </s> <s xml:id="echoid-s11793" xml:space="preserve">rectangulo, ZEY, cum tertia parte quadrati, <lb/>ZY, ad parallelepipedum ſub, YT, & </s> <s xml:id="echoid-s11794" xml:space="preserve">quadrato, TE, ſub, ET, & </s> <s xml:id="echoid-s11795" xml:space="preserve"><lb/>quadrato, TY, cum tertia parte cubi, TY, quæ eſſe æqualia oſtẽ-<lb/>demus parallelepipedo ſub, YT, & </s> <s xml:id="echoid-s11796" xml:space="preserve">rectangulo YET, cum tertia <lb/>parte quadrati, YT, igitur trilineum, EQX, ad trilineum, EXI, <lb/>erit vt parallelepipedum ſub, ZY, & </s> <s xml:id="echoid-s11797" xml:space="preserve">rectangulo, ZEY, cum tertia <lb/>@arte quadrati, ZY, ad parallelepipedum ſub, YT, ideſt ſub, ZY, <lb/> <anchor type="note" xlink:label="note-0481-02a" xlink:href="note-0481-02"/> & </s> <s xml:id="echoid-s11798" xml:space="preserve">ſub rectangulo, YET, cum tertia parte quadrati, TY, & </s> <s xml:id="echoid-s11799" xml:space="preserve">quia <lb/>hæc parallelepipeda ſunt in eadem altitudine, ideo ſunt vt baſes, <lb/>igitur trilineum, EQX, ad trilineum, EXI, erit vt rectangulum, ZE <lb/> <anchor type="note" xlink:label="note-0481-03a" xlink:href="note-0481-03"/> Y, cum tertia parte quadrati, YZ, ad rectangulum, YET, cum ter-<lb/>tia parte quadrati, YT, eſt autem ſectio, E℟I, æqualis ſpatio, AG <lb/>H, & </s> <s xml:id="echoid-s11800" xml:space="preserve">trilineum, EXI, ſpatio, HPMH, & </s> <s xml:id="echoid-s11801" xml:space="preserve">trilineum, EQX, ſpatio, <lb/>MZSBM, ergo ſpatium, AGH, ad ſpatium, HPMH, erit vt ter-<lb/>tia pars quadrati, TE, ad rectangulum, TEY, cum tertia parte <lb/>quadrati, TY, ideſt vt tertia pars quadrati, HA, ad rectangulum, <lb/>HAM, cum tertia parte quadrati, HM. </s> <s xml:id="echoid-s11802" xml:space="preserve">Similiter concludemus <lb/>ſpatium, HPMH, ad ſpatium, MZSBM, eſſe vt rectangulum, HA <lb/>M, cum tertia parte quadrati, HM, ad rectangulum, MAB, cum <lb/>tertia parte quadrati, MB, quod oſtendere opus erat.</s> <s xml:id="echoid-s11803" xml:space="preserve"/> </p> <div xml:id="echoid-div1076" type="float" level="2" n="1"> <note position="right" xlink:label="note-0479-01" xlink:href="note-0479-01a" xml:space="preserve">20. l. 4.</note> <figure xlink:label="fig-0480-01" xlink:href="fig-0480-01a"> <image file="0480-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0480-01"/> </figure> <note position="left" xlink:label="note-0480-01" xlink:href="note-0480-01a" xml:space="preserve">38. l. 2.</note> <note position="left" xlink:label="note-0480-02" xlink:href="note-0480-02a" xml:space="preserve">36. l. 2.</note> <note position="left" xlink:label="note-0480-03" xlink:href="note-0480-03a" xml:space="preserve">35. l. 2.</note> <note position="left" xlink:label="note-0480-04" xlink:href="note-0480-04a" xml:space="preserve">R. G. Cor. <lb/>4. gener. <lb/>34. l. 2.</note> <note position="left" xlink:label="note-0480-05" xlink:href="note-0480-05a" xml:space="preserve">38. l. 2.</note> <note position="right" xlink:label="note-0481-01" xlink:href="note-0481-01a" xml:space="preserve">35 l. 2. <lb/>33. l. 2.</note> <note position="right" xlink:label="note-0481-02" xlink:href="note-0481-02a" xml:space="preserve">B. G. Cor. <lb/>4. gener. <lb/>34. l. 2.</note> <note position="right" xlink:label="note-0481-03" xlink:href="note-0481-03a" xml:space="preserve">Blicitur ex <lb/>9. huius. <lb/>Elicitur <lb/>ex 15. hui <lb/>us.</note> </div> </div> <div xml:id="echoid-div1078" type="section" level="1" n="646"> <head xml:id="echoid-head676" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s11804" xml:space="preserve">_H_Inc patet ſi ducatur quædam tangens parabolam, quæ in partes <lb/>quotcunq; </s> <s xml:id="echoid-s11805" xml:space="preserve">æquales diuidatur, & </s> <s xml:id="echoid-s11806" xml:space="preserve">per puncta diuiſionum du-<lb/>cantur recta linea diametro parallelæ, quouſq; </s> <s xml:id="echoid-s11807" xml:space="preserve">incidant in curuam <lb/>parabolæ, his incidentiæ punctis cum contactus puncto iunctis, ſpa-<lb/>tium ſub prima iungente, & </s> <s xml:id="echoid-s11808" xml:space="preserve">ſubtenſa curua parabola ad trilineum <lb/>ſub prima, & </s> <s xml:id="echoid-s11809" xml:space="preserve">ſecunda iungente, & </s> <s xml:id="echoid-s11810" xml:space="preserve">ab illis appræhenſa curua, eſſe vt <lb/>tertia pars quadrati primæ partis tangentis eſt ad rectangulum ſub <lb/>prima parte, & </s> <s xml:id="echoid-s11811" xml:space="preserve">compoſito ex prima, & </s> <s xml:id="echoid-s11812" xml:space="preserve">ſecunda cum tertia parte qua-<lb/>dratis ſecundæ. </s> <s xml:id="echoid-s11813" xml:space="preserve">Similiter hoc trilineum ad trilineum ſub ſecunda, <lb/>& </s> <s xml:id="echoid-s11814" xml:space="preserve">tertia iungente, & </s> <s xml:id="echoid-s11815" xml:space="preserve">ab illis appræhenſa curua parabolæ, eſſe vt re-<lb/>ctangulum ſub prima, & </s> <s xml:id="echoid-s11816" xml:space="preserve">ſub compoſito ex prima, & </s> <s xml:id="echoid-s11817" xml:space="preserve">ſecunda parte <lb/>tangentis (enumeratione ſemper à puncto contactus incepta) vna cum <lb/>tertia parte quadrati ſecunda ad rectangulum ſub compoſita ex prima, <pb o="462" file="0482" n="482" rhead="GEOMETRIÆ"/> & </s> <s xml:id="echoid-s11818" xml:space="preserve">ſecunda, & </s> <s xml:id="echoid-s11819" xml:space="preserve">ſub compoſito ex prima, ſecunda, & </s> <s xml:id="echoid-s11820" xml:space="preserve">tertia parte, <lb/>vna cum tertia parte quadrati tertiæ partis, & </s> <s xml:id="echoid-s11821" xml:space="preserve">ſic trilinea deinceps <lb/>ſequentia eſſe, vt hæc rectangula deinceps ſequentia cum tertia parte <lb/>dictorum quadratorum, eodem enim modo ſupra adhibito hoc oſtende-<lb/>tur. </s> <s xml:id="echoid-s11822" xml:space="preserve">Quotieſcunq; </s> <s xml:id="echoid-s11823" xml:space="preserve">autem tangens ſit æqualis radio circuli ſpiralium <lb/>alicuius numeri veluti fuit, EZ, æqualis ipſi, AB, & </s> <s xml:id="echoid-s11824" xml:space="preserve">diuidatur in tot <lb/>partes æquales, in quot radius talis circuli diuiditur à circumferen-<lb/>tijs infertorum circulorum, tunc nedum in parabola dicta ſpatia ſe <lb/>habent, vt dictum eſt, ſed etiam ſunt æqualia ſpatijs dictorum circu-<lb/>lorum, primum nempè primo, ſecundum ſecundo, & </s> <s xml:id="echoid-s11825" xml:space="preserve">ſic deinceps, à <lb/>puncto contactus parabolæ dictorum ſpatiorum enumeratione facta, <lb/>quod eſt admirabile, hęc autem ex ſupradictis manifeſta ſunt.</s> <s xml:id="echoid-s11826" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1079" type="section" level="1" n="647"> <head xml:id="echoid-head677" xml:space="preserve">THEOREMA XX. PROPOS. XX.</head> <p> <s xml:id="echoid-s11827" xml:space="preserve">SI parabolam tetigerit recta linea, quæ diuidatur in <lb/>quotcunq; </s> <s xml:id="echoid-s11828" xml:space="preserve">partes æquales, per puncta autem diuiſio-<lb/>num, & </s> <s xml:id="echoid-s11829" xml:space="preserve">extremum ducantur rectæ lineæ diametro para-<lb/>bolæ, æquidiſtautes, quouſq; </s> <s xml:id="echoid-s11830" xml:space="preserve">in eiuſdem curuam incidant, <lb/>iungantur autem puncta incidentiæ cum puncto cõtactus. <lb/></s> <s xml:id="echoid-s11831" xml:space="preserve">Spatium ſub prima iungente, & </s> <s xml:id="echoid-s11832" xml:space="preserve">ſubtenſa ab eadem curua <lb/>erit ſeptima pars ſpatij ſub prima, & </s> <s xml:id="echoid-s11833" xml:space="preserve">ſecunda iungente, & </s> <s xml:id="echoid-s11834" xml:space="preserve"><lb/>ab ijs appræhenſa curua compræhenſi. </s> <s xml:id="echoid-s11835" xml:space="preserve">Hoc verò ad ſpa-<lb/>tium ſub ſecunda, & </s> <s xml:id="echoid-s11836" xml:space="preserve">tertia iungente, & </s> <s xml:id="echoid-s11837" xml:space="preserve">appræhenſa curua, <lb/>erit vt 7. </s> <s xml:id="echoid-s11838" xml:space="preserve">ad 19 Hoc autem ad ſpatium ſub tertia, & </s> <s xml:id="echoid-s11839" xml:space="preserve">quar-<lb/>ta iungente & </s> <s xml:id="echoid-s11840" xml:space="preserve">ab ijs incluſa curua, vt 19. </s> <s xml:id="echoid-s11841" xml:space="preserve">ad 37. </s> <s xml:id="echoid-s11842" xml:space="preserve">& </s> <s xml:id="echoid-s11843" xml:space="preserve">ſic de-<lb/>i@ceps, prout indicat appoſita numerorum ſeries.</s> <s xml:id="echoid-s11844" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11845" xml:space="preserve">Sit tangens parabolam, AHF, ipſa, AE, diuiſa in quotcumq; <lb/></s> <s xml:id="echoid-s11846" xml:space="preserve">partes æquales, AB, BC, CD, DE, ductis autem à punctis, B, C, <lb/>D, E, diametro parallelis, quouſq; </s> <s xml:id="echoid-s11847" xml:space="preserve">incidant curuæ, AHF, ipſæ, B <lb/>N, CM, DH, EF, iungantur puncta incidèntiæ, quæ ſint, F, H, M, <lb/>N, cum puncto, A, &</s> <s xml:id="echoid-s11848" xml:space="preserve">, AN, dicatur prima iungens, AM, tecunda, <lb/>AH, tertia, & </s> <s xml:id="echoid-s11849" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s11850" xml:space="preserve">Dico ipatium ſub, AN, & </s> <s xml:id="echoid-s11851" xml:space="preserve">ab ea ſub-<lb/>tenſa curua, eſſe ad ſpatium ſub, NA, AM, & </s> <s xml:id="echoid-s11852" xml:space="preserve">curua, MN, ideſt <lb/>ad trilineum, AMN, vt 1. </s> <s xml:id="echoid-s11853" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s11854" xml:space="preserve">hoc verò ad trilineum, AHM, vt <lb/>7. </s> <s xml:id="echoid-s11855" xml:space="preserve">ad 19. </s> <s xml:id="echoid-s11856" xml:space="preserve">& </s> <s xml:id="echoid-s11857" xml:space="preserve">ſic deinceps, prout indicat oppoſita numerorum ſeries <lb/> <anchor type="note" xlink:label="note-0482-01a" xlink:href="note-0482-01"/> ſe habere trilinea deinceps ſubſequentia. </s> <s xml:id="echoid-s11858" xml:space="preserve">Eſt enim ſpatium, AIN, <lb/>ad trilineum, AMN, vt {1/3}. </s> <s xml:id="echoid-s11859" xml:space="preserve">quadrati, AB, ad rectangulum, CAB,</s> </p> <div xml:id="echoid-div1079" type="float" level="2" n="1"> <note position="left" xlink:label="note-0482-01" xlink:href="note-0482-01a" xml:space="preserve">Ex Coro. <lb/>antec.</note> </div> <pb o="463" file="0483" n="483" rhead="LIBER VI."/> <p> <s xml:id="echoid-s11860" xml:space="preserve">Series ſpatiorum 1. </s> <s xml:id="echoid-s11861" xml:space="preserve">2. </s> <s xml:id="echoid-s11862" xml:space="preserve">3. </s> <s xml:id="echoid-s11863" xml:space="preserve">4. </s> <s xml:id="echoid-s11864" xml:space="preserve">5. </s> <s xml:id="echoid-s11865" xml:space="preserve">6. </s> <s xml:id="echoid-s11866" xml:space="preserve">7.</s> <s xml:id="echoid-s11867" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11868" xml:space="preserve">Series numerorum 1. </s> <s xml:id="echoid-s11869" xml:space="preserve">7. </s> <s xml:id="echoid-s11870" xml:space="preserve">19. </s> <s xml:id="echoid-s11871" xml:space="preserve">37. </s> <s xml:id="echoid-s11872" xml:space="preserve">61. </s> <s xml:id="echoid-s11873" xml:space="preserve">91. </s> <s xml:id="echoid-s11874" xml:space="preserve">127.</s> <s xml:id="echoid-s11875" xml:space="preserve"/> </p> <figure> <image file="0483-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0483-01"/> </figure> <p> <s xml:id="echoid-s11876" xml:space="preserve">cum {1/3}. </s> <s xml:id="echoid-s11877" xml:space="preserve">quadrati, CB, ſi ergo, AB, ſtatuatur 3. </s> <s xml:id="echoid-s11878" xml:space="preserve">erit, AC, 6. </s> <s xml:id="echoid-s11879" xml:space="preserve">rectan-<lb/>gulum, CAB, 18. </s> <s xml:id="echoid-s11880" xml:space="preserve">tertia pars quadrati, BC, erit 3. </s> <s xml:id="echoid-s11881" xml:space="preserve">quæ iuncta ipſi <lb/>18. </s> <s xml:id="echoid-s11882" xml:space="preserve">efficit 21. </s> <s xml:id="echoid-s11883" xml:space="preserve">erit ergo qualium partium quadratum, AB, eſt 9. </s> <s xml:id="echoid-s11884" xml:space="preserve">re-<lb/>ctangulum, CAB, cum tertia parte quadrati, BC, 21. </s> <s xml:id="echoid-s11885" xml:space="preserve">& </s> <s xml:id="echoid-s11886" xml:space="preserve">tertia pars <lb/>quadrati, AB, eſt 3. </s> <s xml:id="echoid-s11887" xml:space="preserve">eſt igitur ſpatium, AIN, ad trilineum, AMN, <lb/>vt 3. </s> <s xml:id="echoid-s11888" xml:space="preserve">ad 21. </s> <s xml:id="echoid-s11889" xml:space="preserve">ideſt vt 1. </s> <s xml:id="echoid-s11890" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s11891" xml:space="preserve">Eodem modo reperiemus trilineum, A <lb/>NM, ad, AMH, eſſe vt 7. </s> <s xml:id="echoid-s11892" xml:space="preserve">ad 19. </s> <s xml:id="echoid-s11893" xml:space="preserve">& </s> <s xml:id="echoid-s11894" xml:space="preserve">hoc ad trilineum, AHF, vt 19. <lb/></s> <s xml:id="echoid-s11895" xml:space="preserve">ad 37. </s> <s xml:id="echoid-s11896" xml:space="preserve">& </s> <s xml:id="echoid-s11897" xml:space="preserve">ſic deinceps, prout indicat ſeries numerorum ſupra poſi-<lb/>ta, quod demonſtrandum erat.</s> <s xml:id="echoid-s11898" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1081" type="section" level="1" n="648"> <head xml:id="echoid-head678" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s11899" xml:space="preserve">_H_Inc patet ſi expoſita ſint ſpirales in quotcunque reuolutionibus <lb/>genitæ, initio circulationis exiſtente in, K, ſint autem volutæ <lb/>ipſæ, KL, LO, OP, PG, & </s> <s xml:id="echoid-s11900" xml:space="preserve">ſpirales eodem ordine procedentes, KRL, L <lb/>SO, OTP, PVG, quod ſi, KG, fuerit æqualis ipſi AE, & </s> <s xml:id="echoid-s11901" xml:space="preserve">diuiſa in pun-<lb/>ctis, L, O, P, prout diuiditur, AE, in punctis, B, C, D, ſpatium, KRL, <lb/> <anchor type="note" xlink:label="note-0483-01a" xlink:href="note-0483-01"/> erit æquale ſpatio, AIN, &</s> <s xml:id="echoid-s11902" xml:space="preserve">, LSO, trilineo, AMN, &</s> <s xml:id="echoid-s11903" xml:space="preserve">, OTP, trili-<lb/>neo, AMH, & </s> <s xml:id="echoid-s11904" xml:space="preserve">tãdem, PVG, trilineo, AHF, & </s> <s xml:id="echoid-s11905" xml:space="preserve">ſic deinceps, vnde <lb/>etiam hæc ſpatia ſe habebunt, prout indicat ſuprapoſita ſeries nume-</s> </p> <div xml:id="echoid-div1081" type="float" level="2" n="1"> <note position="right" xlink:label="note-0483-01" xlink:href="note-0483-01a" xml:space="preserve">_Elicitur e@_ <lb/>_9. huius._ <lb/>_Elicitur_ <lb/>_15. huiur._</note> </div> <p> <s xml:id="echoid-s11906" xml:space="preserve">Secunda ſeries num. </s> <s xml:id="echoid-s11907" xml:space="preserve">1. </s> <s xml:id="echoid-s11908" xml:space="preserve">6. </s> <s xml:id="echoid-s11909" xml:space="preserve">12. </s> <s xml:id="echoid-s11910" xml:space="preserve">18. </s> <s xml:id="echoid-s11911" xml:space="preserve">24. </s> <s xml:id="echoid-s11912" xml:space="preserve">30. </s> <s xml:id="echoid-s11913" xml:space="preserve">36.</s> <s xml:id="echoid-s11914" xml:space="preserve"/> </p> <pb o="464" file="0484" n="484" rhead="GEOMETRIÆ"/> <p style="it"> <s xml:id="echoid-s11915" xml:space="preserve">rorum. </s> <s xml:id="echoid-s11916" xml:space="preserve">Si autem primum ſpatium ſubtrahatur à ſecundo, ſecundum <lb/>à tertio, tertium à quarto, & </s> <s xml:id="echoid-s11917" xml:space="preserve">ſic deinceps, habebimus hanc numero-<lb/>rum ſecundum ſerieum indic antem rationem primi ſpatij ad faſciam ſe-<lb/>quentem, & </s> <s xml:id="echoid-s11918" xml:space="preserve">huius ad faſciam ſequentem, & </s> <s xml:id="echoid-s11919" xml:space="preserve">ſic deinceps, in qua pa-<lb/>tet primum ſpatium eſſe _{1/6}_. </s> <s xml:id="echoid-s11920" xml:space="preserve">faſcia ſequentis, ſecundam verò faſciam <lb/>prima eſſe duplam, tertiam eiuſdem triplam, quartam quadruplam, & </s> <s xml:id="echoid-s11921" xml:space="preserve"><lb/>ſic deinceps, ſec undum numenorum continuum incrementum, qua in-<lb/>uentis ab Archimede eſſe conformia eiuſdem de ſpiralibus librum <lb/>perlegenti compertum fiet.</s> <s xml:id="echoid-s11922" xml:space="preserve"/> </p> <figure> <image file="0484-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0484-01"/> </figure> </div> <div xml:id="echoid-div1083" type="section" level="1" n="649"> <head xml:id="echoid-head679" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s11923" xml:space="preserve">_H_Aec libuit apponere, tum quia adhibita methodus ab Archime-<lb/>dea diuerſa eſt, tum etiam, vt admirabilem connexionem, &</s> <s xml:id="echoid-s11924" xml:space="preserve">, <lb/>vt ita dicam parabolici, ac helici ſpatij, affinitatem, talia ſpeculanti, <lb/>puto, non aſpernendam, ob oculos ponerem; </s> <s xml:id="echoid-s11925" xml:space="preserve">quibus, & </s> <s xml:id="echoid-s11926" xml:space="preserve">ſequentia <pb o="467" file="0485" n="485" rhead="LIBER VI."/> ſubnectere non inutile mihi viſum fuit. </s> <s xml:id="echoid-s11927" xml:space="preserve">Hoc autem tántum circa prę-<lb/>fatas demonſtrationes dicam, quod licet in Prop. </s> <s xml:id="echoid-s11928" xml:space="preserve">_12._ </s> <s xml:id="echoid-s11929" xml:space="preserve">& </s> <s xml:id="echoid-s11930" xml:space="preserve">_14._ </s> <s xml:id="echoid-s11931" xml:space="preserve">indiuiſi-<lb/>bilibus, nempè omnibus qu adratis parallelogrammorum, quæ ibi de-<lb/>ſcribuntur, vſus fuerim, tamen etiam modo conſueto potuiſsent de-<lb/>monſtrari, ſi ex .</s> <s xml:id="echoid-s11932" xml:space="preserve">g. </s> <s xml:id="echoid-s11933" xml:space="preserve">vice omnium quadratorum parallelogrammi, ED, re-<lb/>gula, EB, ibi aſſumpta, Vſus fuiſsem parallelepipedo ſub altitudine, <lb/>DB, baſi autem quadrato, EB, vel pro omnibus quadratis trianguli, <lb/>CBE, regula eadem, EB, vſus eſſem pyramide ſub altitudine, CE, ba-<lb/>ſi eodem quadrato, EB, etenim ſimiliter demonſtratio abſolui potuiſ-<lb/>ſet, hac omnium quadratorum parallelogrammorum ibidem conſide-<lb/>ratorum dimiſſa congerie, & </s> <s xml:id="echoid-s11934" xml:space="preserve">ſubſtitutis parallelepipedis, vel pyrami-<lb/>dibus, aut earum fruſtis, vbi opus erat. </s> <s xml:id="echoid-s11935" xml:space="preserve">Hæc inuenire volui, vt præ-<lb/>dicta omnia ſtylo veteri demonſtrabilia eſſe, etiam aliter ab Archi-<lb/>mede patefiat.</s> <s xml:id="echoid-s11936" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1084" type="section" level="1" n="650"> <head xml:id="echoid-head680" xml:space="preserve">THEOREMA XXI. PROPOS. XXI.</head> <p> <s xml:id="echoid-s11937" xml:space="preserve">SI exponatur ſeries ſpiralium, & </s> <s xml:id="echoid-s11938" xml:space="preserve">circulorum deinceps à <lb/>primis, in ſpatijs verò ſub ſpiralibus, & </s> <s xml:id="echoid-s11939" xml:space="preserve">volutis, cylin-<lb/>drici, & </s> <s xml:id="echoid-s11940" xml:space="preserve">conici in eadem altitudiue ſtantes intelligantur <lb/>conſtituti tamquam in baſibus, ſimiliter & </s> <s xml:id="echoid-s11941" xml:space="preserve">in circulis con-<lb/>ſtituti eſſe cylindri, & </s> <s xml:id="echoid-s11942" xml:space="preserve">coni inte lligantur. </s> <s xml:id="echoid-s11943" xml:space="preserve">Cylindri inter <lb/>ſe, & </s> <s xml:id="echoid-s11944" xml:space="preserve">cylindrici pariter inter ſe, ſiue ad cylindros compa-<lb/>rati, ſiue coni inter ſe, & </s> <s xml:id="echoid-s11945" xml:space="preserve">conici inter ſe ſiue ad conos com-<lb/>parati eandem rationem, quam baſes habebunt.</s> <s xml:id="echoid-s11946" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11947" xml:space="preserve">Patet hæc propoſitio, nam cylindrici, & </s> <s xml:id="echoid-s11948" xml:space="preserve">conici in eadem alti-<lb/> <anchor type="note" xlink:label="note-0485-01a" xlink:href="note-0485-01"/> tudine conſtituti ſunt inter ſe, vt baſes; </s> <s xml:id="echoid-s11949" xml:space="preserve">ſunt autem prædicta ſoli-<lb/>da per conſtructionem in eadem altitudine poſita, ergo erunt in-<lb/>ter ſe, vt ipſæ baſes; </s> <s xml:id="echoid-s11950" xml:space="preserve">Vocentur autem Cylindri, & </s> <s xml:id="echoid-s11951" xml:space="preserve">Cylindrici, nec-<lb/>non Conici eiuſdem numeri cum ſpatijs, quibus inſiſtunt .</s> <s xml:id="echoid-s11952" xml:space="preserve">i. </s> <s xml:id="echoid-s11953" xml:space="preserve">pri-<lb/>mus cylindrus, vel conus, qui eſt in primo circulo, ſecundus cylin-<lb/>drus, vel conus, qui eſt in ſecundo circulo tamquam in baſi; </s> <s xml:id="echoid-s11954" xml:space="preserve">pri-<lb/>mus cylindricus, vel conicus, qui eſt in ſpatio helico primi circuli <lb/>tamquam in baſi, ſecundus cylindricus, vel conicus, qui eſt in ſpa-<lb/>tio ſecundi circuli, & </s> <s xml:id="echoid-s11955" xml:space="preserve">ſic deinceps.</s> <s xml:id="echoid-s11956" xml:space="preserve"/> </p> <div xml:id="echoid-div1084" type="float" level="2" n="1"> <note position="right" xlink:label="note-0485-01" xlink:href="note-0485-01a" xml:space="preserve">B. G. H. <lb/>Coroll. 4. <lb/>gener. 34. <lb/>l. 2.</note> </div> <pb o="468" file="0486" n="486" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1086" type="section" level="1" n="651"> <head xml:id="echoid-head681" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s11957" xml:space="preserve">_E_T quia in ſuprápoſitis Propoſitionibus baſium prædictorum ſoli-<lb/>dorum ratio fuit adinuenta, ideò eandem pro dictis ſolidis ratio-<lb/>nem inde colligemus.</s> <s xml:id="echoid-s11958" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1087" type="section" level="1" n="652"> <head xml:id="echoid-head682" xml:space="preserve">THEOREMA XXII. PROPOS. XXII.</head> <p> <s xml:id="echoid-s11959" xml:space="preserve">PRimus cylindrus nonuplus eſt primi conici.</s> <s xml:id="echoid-s11960" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11961" xml:space="preserve">Hæc Propoſitio pariter manifeſta eſt, nam primus cylindrus ad <lb/>primum cylindricum eſt, vt primus circulus ad ſuum ſpatium <lb/>ideſt in ratione tripla, primus verò cylindricus ad primum conicũ <lb/>eſt in ratione tripla, quia ſunt in eadem baſi, quod eſt ſpatium pri-<lb/> <anchor type="note" xlink:label="note-0486-01a" xlink:href="note-0486-01"/> mi circuli, & </s> <s xml:id="echoid-s11962" xml:space="preserve">in eadem altitudine, & </s> <s xml:id="echoid-s11963" xml:space="preserve">ideò primus cylindrus ad <lb/>primum conicum eſt in ratione nonupla, quæ ex duabus triplis <lb/>conflatur.</s> <s xml:id="echoid-s11964" xml:space="preserve"/> </p> <div xml:id="echoid-div1087" type="float" level="2" n="1"> <note position="left" xlink:label="note-0486-01" xlink:href="note-0486-01a" xml:space="preserve">1. Cor. 4. <lb/>gener. 34. <lb/>l. 2.</note> </div> </div> <div xml:id="echoid-div1089" type="section" level="1" n="653"> <head xml:id="echoid-head683" xml:space="preserve">THEOREMA XXIII. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s11965" xml:space="preserve">SEcundus cylindrus ad ſecundum conicum eſt, vt tri-<lb/>plum quadrati radij ſecundi circuli, ad rectangulum <lb/>ſub radio eiuſdem ſecundi, & </s> <s xml:id="echoid-s11966" xml:space="preserve">radio primi circuli, vna cũ <lb/>tertia parte quadrati differentiæ eorundem radiorum.</s> <s xml:id="echoid-s11967" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11968" xml:space="preserve">Secundus enim cylindrus ad ſecundum cylindricum eſt, vt ſecũ-<lb/> <anchor type="note" xlink:label="note-0486-02a" xlink:href="note-0486-02"/> dus circulus ad ſuum ſpatium ideſt vt quadratum radij ſecundi cir-<lb/>culi ad rectangulum ſub radio eiuſdem, & </s> <s xml:id="echoid-s11969" xml:space="preserve">ſub radio primi vna cũ <lb/>tertia parce quadrati differentiæ eorundem radiorum, ſecundus ve-<lb/> <anchor type="note" xlink:label="note-0486-03a" xlink:href="note-0486-03"/> rò cylindricus triplus eſt conici ſecundi, quoniam in eadem baſi, <lb/>& </s> <s xml:id="echoid-s11970" xml:space="preserve">altitudine cum eo conſtituitur, ergo eſt ad illum, vt dictum re-<lb/>ctangulum ſub radijs primi, & </s> <s xml:id="echoid-s11971" xml:space="preserve">ſecundi circuli, vna cum tertia par-<lb/>te quadrati differentiæ eorundem ad horum coniunctorum tertiã <lb/>partem, & </s> <s xml:id="echoid-s11972" xml:space="preserve">ex æquali ſecundus cylindrus ad ſecundum conicum <lb/>erit, vt quadratum radij primi circuli ad tertiam partem rectangu-<lb/>li ſub radijs primi, & </s> <s xml:id="echoid-s11973" xml:space="preserve">ſecundi circuli, cum nona parte quadrati dif-<lb/>ferentiæ eorundem radiorum, ideſt, vt triplum quadrati radij ſe-<lb/>cundi circuli ad rectangulum ſub radijs primi, & </s> <s xml:id="echoid-s11974" xml:space="preserve">ſecundi circuli, <lb/>vna cum tertia parte quadrati differentiæ eorundem radiorum.</s> <s xml:id="echoid-s11975" xml:space="preserve"/> </p> <div xml:id="echoid-div1089" type="float" level="2" n="1"> <note position="left" xlink:label="note-0486-02" xlink:href="note-0486-02a" xml:space="preserve">15, huius.</note> <note position="left" xlink:label="note-0486-03" xlink:href="note-0486-03a" xml:space="preserve">1. Coro. 4. <lb/>gener. 14. <lb/>l. 2.</note> </div> <pb o="469" file="0487" n="487" rhead="LIBER VI."/> </div> <div xml:id="echoid-div1091" type="section" level="1" n="654"> <head xml:id="echoid-head684" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s11976" xml:space="preserve">_H_Inc patet reliquorum cylindrorum ad conicos eiuſdem numeri <lb/>rationem eandem eſſe illi, quam habet triplum quadrati radij <lb/>circuli, qui eſt baſis talis cylindri, ad rectangulum ſub eodem radio, & </s> <s xml:id="echoid-s11977" xml:space="preserve"><lb/>radio circuli vnitate minoris, vna cum tertia parte quadrati differen-<lb/>tiæ vtriuſq; </s> <s xml:id="echoid-s11978" xml:space="preserve">radij, quod eod. </s> <s xml:id="echoid-s11979" xml:space="preserve">modo oſtendetur.</s> <s xml:id="echoid-s11980" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1092" type="section" level="1" n="655"> <head xml:id="echoid-head685" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s11981" xml:space="preserve">_P_Atet inſuper, quod eadem methodo facilè inueniemus rationem <lb/>cuiuſcunq; </s> <s xml:id="echoid-s11982" xml:space="preserve">cylindri, vel fruſti cylindri, & </s> <s xml:id="echoid-s11983" xml:space="preserve">conici, vel fruſti co-<lb/>nici, in baſibus aliquibus ex iam conſideratis ſpatijs conſtituti, quæ ob <lb/>facilitatem dimitto; </s> <s xml:id="echoid-s11984" xml:space="preserve">vt ad aliqua ex antecedentium librorum, & </s> <s xml:id="echoid-s11985" xml:space="preserve">hui-<lb/>us propoſitionibus conſtructa Problemat a, ſiue Theoremata, ſpecula-<lb/>tionem noſtram conuertentes, vtilitatis eximiæ, quam ſuperius tra-<lb/>dita doctrina, etiam ad praxim deducta, afferre poteſt, illuſtriora quæ-<lb/>dam præbeamus argumenta.</s> <s xml:id="echoid-s11986" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1093" type="section" level="1" n="656"> <head xml:id="echoid-head686" xml:space="preserve">PROBLEMA I. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s11987" xml:space="preserve">CYlindrum, vel conum conſtituere æqualem datæ <lb/>ſphæræ, vel ſphæroidi, vel eiuſdem portioni.</s> <s xml:id="echoid-s11988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11989" xml:space="preserve">Sit ſphæra, vel ſphæroides, ACEG, circa diametrum, AE, opor-<lb/> <anchor type="note" xlink:label="note-0487-01a" xlink:href="note-0487-01"/> tet illi cylindrum, vel conum æqualem conſtituere. </s> <s xml:id="echoid-s11990" xml:space="preserve">Exponatur <lb/>cylindrus, RQ, & </s> <s xml:id="echoid-s11991" xml:space="preserve">conus, SPQ, quorum altitudo, vt, SV, ſit æqualis <lb/>ipſi, AE, & </s> <s xml:id="echoid-s11992" xml:space="preserve">baſis æqualis circulo tranſeunti per centrum, N, qui ſit, <lb/>CG, recté axem ſecans, ſeu pro ſphæroide, ſi, AE, non ſit axis, RQ, <lb/>altitudinẽ hab eat æqualem altitudini ſphæroidis @uxta planũ, CG, <lb/>aſſumptę, & </s> <s xml:id="echoid-s11993" xml:space="preserve">ſit in baſi æquali ellipſi, CG. </s> <s xml:id="echoid-s11994" xml:space="preserve">Erit ergo cylindrus, RQ, <lb/>ſexquialter ſphæræ, vel ſphæroidis, ACEG, & </s> <s xml:id="echoid-s11995" xml:space="preserve">conus ſubduplus <lb/>eiuſdẽ, ſi igitur in eadẽ baſi fiat cylindrus, cuius altitudo ſit {2/3}. </s> <s xml:id="echoid-s11996" xml:space="preserve">ipſius, <lb/>SV, hic erit æqualis datæ ſphæræ, vel ſpæroid@, ACEG, ſi verò, fiat <lb/>conus altitudinis duplæ ipſius, VS, in eadem pariter baſi, i@@e eidẽ <lb/>ſphæræ, vel ſphæroidi æqualis erit, coni enim, & </s> <s xml:id="echoid-s11997" xml:space="preserve">cylindri in eadẽ <lb/>baſi conſtituti ſunt, vt altitudines.</s> <s xml:id="echoid-s11998" xml:space="preserve"/> </p> <div xml:id="echoid-div1093" type="float" level="2" n="1"> <note position="right" xlink:label="note-0487-01" xlink:href="note-0487-01a" xml:space="preserve">Coroll, 1. <lb/>34. l. 3.</note> </div> <p> <s xml:id="echoid-s11999" xml:space="preserve">Sit rurſus conſtituendus cylindrus, vel conus, æqualis eiuſdem <lb/> <anchor type="note" xlink:label="note-0487-02a" xlink:href="note-0487-02"/> ſphæræ, vel ſpræroidis, portioni, BAH, vel, DAF, @upponatur nũc <lb/>ergo cylindrus, RQ, cuius baſis ſit æqualis círculo, vel ellipſi, DF, <lb/>altitudo verò, SV, æqualis ipſi, AO, ſeu altitudini portionis, ADF, <pb o="470" file="0488" n="488" rhead="GEOMETRI Æ"/> <anchor type="figure" xlink:label="fig-0488-01a" xlink:href="fig-0488-01"/> iuxta planum, DF, aſſum-<lb/>ptæ, erit igitur hic cylin-<lb/> <anchor type="note" xlink:label="note-0488-01a" xlink:href="note-0488-01"/> drus ad portionem, DAF, <lb/>vt, OE, ad compoſitam ex <lb/>{1/2}. </s> <s xml:id="echoid-s12000" xml:space="preserve">OE, & </s> <s xml:id="echoid-s12001" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s12002" xml:space="preserve">OA, hanc au-<lb/>tem rationem habeat, SV, <lb/>ad aliam altitudinem, erit <lb/>ergo cylindrus, RQ, ad cy-<lb/>lindrum altitudinis inuen-<lb/>tæ, & </s> <s xml:id="echoid-s12003" xml:space="preserve">in eadem baſi, PQ, <lb/>conſtitutum, vt, OE, ad <lb/>compoſitam ex {1/2}. </s> <s xml:id="echoid-s12004" xml:space="preserve">OE, & </s> <s xml:id="echoid-s12005" xml:space="preserve"><lb/>{1/6}. </s> <s xml:id="echoid-s12006" xml:space="preserve">OA, .</s> <s xml:id="echoid-s12007" xml:space="preserve">i. </s> <s xml:id="echoid-s12008" xml:space="preserve">vt cylindrus, R <lb/>Q, ad portionem, DAF, <lb/>igitur inuentus cylindrus <lb/>erit æqualis portioni, DA <lb/>F. </s> <s xml:id="echoid-s12009" xml:space="preserve">Triplicetur nunc alti-<lb/>tudo inuenti cylindri, & </s> <s xml:id="echoid-s12010" xml:space="preserve"><lb/>fiat conus talis altitudinis, <lb/>in eadem cum eo baſi, hic <lb/>igitur conus erit æqualis <lb/>inuento cylindro, & </s> <s xml:id="echoid-s12011" xml:space="preserve">ſub-<lb/>inde portioni, DAF. </s> <s xml:id="echoid-s12012" xml:space="preserve">Eo-<lb/>dem modo inueniemus cy-<lb/>lindrum, vel conum æ-<lb/>qualem portioni, BAH.</s> <s xml:id="echoid-s12013" xml:space="preserve"/> </p> <div xml:id="echoid-div1094" type="float" level="2" n="2"> <note position="right" xlink:label="note-0487-02" xlink:href="note-0487-02a" xml:space="preserve">C. G. H. <lb/>Coroll. 4. <lb/>gener. 34. <lb/>l. 2.</note> <figure xlink:label="fig-0488-01" xlink:href="fig-0488-01a"> <image file="0488-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0488-01"/> </figure> <note position="left" xlink:label="note-0488-01" xlink:href="note-0488-01a" xml:space="preserve">Coroll. 1. <lb/>34. l. 3.</note> </div> </div> <div xml:id="echoid-div1096" type="section" level="1" n="657"> <head xml:id="echoid-head687" xml:space="preserve">PROBLEMA II. PROPOS. XXV.</head> <p> <s xml:id="echoid-s12014" xml:space="preserve">SOlido quocunq; </s> <s xml:id="echoid-s12015" xml:space="preserve">in eadem baſi, & </s> <s xml:id="echoid-s12016" xml:space="preserve">altitudine cum cy-<lb/>lindro conſtituto, ad quod cylindrus notam rationẽ <lb/>habeat, cylindrum, & </s> <s xml:id="echoid-s12017" xml:space="preserve">conum, inuenire, æqualem dato <lb/>ſolido.</s> <s xml:id="echoid-s12018" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12019" xml:space="preserve">Sit ſolidum quodcunque, DAF, ad quod cylindrus, BF, in eadem <lb/>baſi, DF, & </s> <s xml:id="echoid-s12020" xml:space="preserve">eadẽ altitudine cum eodẽ cõſtitutus, habeat notam ra-<lb/>tionem. </s> <s xml:id="echoid-s12021" xml:space="preserve">Oportet cylindrum inuenire, & </s> <s xml:id="echoid-s12022" xml:space="preserve">conum, æqualem dato ſo-<lb/>lido. </s> <s xml:id="echoid-s12023" xml:space="preserve">Fiat ergo, vt cylindrus, BF, ad ſolidum, DAF, ſic altitudo, quæ <lb/>ſit, AE, ad altitudinem, EI, & </s> <s xml:id="echoid-s12024" xml:space="preserve">per, I, ducatur planum producẽs <lb/>in cylindro, BF, circulum, GK, conſtituenſque cylindrum, GF, <lb/>igitur, vt, AE, ad, EI, ſic erit cylindrus, BF, ad cylindrum, GF, &</s> <s xml:id="echoid-s12025" xml:space="preserve"> <pb o="471" file="0489" n="489" rhead="LIBER VI."/> <anchor type="figure" xlink:label="fig-0489-01a" xlink:href="fig-0489-01"/> ſic cylindrus, BF, ad ſolidum, DAF, <lb/>vnde cylindrus, GF, erit æqualis ſo-<lb/>lido, DAF. </s> <s xml:id="echoid-s12026" xml:space="preserve">Rurſus triplicetur alti-<lb/>tudo, EI, & </s> <s xml:id="echoid-s12027" xml:space="preserve">fiat conus eiuſdem al-<lb/>titudinis in baſi, DF, hic igitur co-<lb/>nus erit æqualis cylindro, GF, & </s> <s xml:id="echoid-s12028" xml:space="preserve"><lb/>ſubinde ſolido, DAF, quod inueni-<lb/>re opus erat.</s> <s xml:id="echoid-s12029" xml:space="preserve"/> </p> <div xml:id="echoid-div1096" type="float" level="2" n="1"> <figure xlink:label="fig-0489-01" xlink:href="fig-0489-01a"> <image file="0489-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0489-01"/> </figure> </div> </div> <div xml:id="echoid-div1098" type="section" level="1" n="658"> <head xml:id="echoid-head688" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s12030" xml:space="preserve">_H_Inc patet nos etiam poſſe inuenire cylindrum, & </s> <s xml:id="echoid-s12031" xml:space="preserve">conum, nedũ <lb/>æqualem dicto ſolido, ſed qui ad ipſum babeat datam rationem, <lb/>ſi enim altitudo inuenti cylindri, vel coni æqualis dicto ſolido, fiat <lb/>id ali am altitudinem in data ratione, tamen conuerſa, & </s> <s xml:id="echoid-s12032" xml:space="preserve">harum al-<lb/>tatudinum vltimò inuentarum in eiſdem baſibus cum prædictis fiant <lb/>cylindrus, & </s> <s xml:id="echoid-s12033" xml:space="preserve">conus, habebunt iſti ad dictum ſolidum datam ratio-<lb/>nem, vt facilè apparet.</s> <s xml:id="echoid-s12034" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1099" type="section" level="1" n="659"> <head xml:id="echoid-head689" xml:space="preserve">COROLL. II. SECTIO I.</head> <note position="right" xml:space="preserve">A</note> <p style="it"> <s xml:id="echoid-s12035" xml:space="preserve">_H_Inc etiam patet cylindrum in baſi apicis ſphæralis, vel ſphæ-<lb/> <anchor type="note" xlink:label="note-0489-02a" xlink:href="note-0489-02"/> roidalis, conſtitutum cuius altitudo ad altitudinem eiuſdem <lb/>apicis ſit, vt, _2._ </s> <s xml:id="echoid-s12036" xml:space="preserve">ad _21._ </s> <s xml:id="echoid-s12037" xml:space="preserve">eſſe æqualem eidem apici.</s> <s xml:id="echoid-s12038" xml:space="preserve"/> </p> <div xml:id="echoid-div1099" type="float" level="2" n="1"> <note position="right" xlink:label="note-0489-02" xlink:href="note-0489-02a" xml:space="preserve">_Coro. 21._ <lb/>_34. l. 3._</note> </div> </div> <div xml:id="echoid-div1101" type="section" level="1" n="660"> <head xml:id="echoid-head690" xml:space="preserve">SECTIO II.</head> <note position="right" xml:space="preserve">B</note> <p style="it"> <s xml:id="echoid-s12039" xml:space="preserve">_V_Lterius habetur quoq; </s> <s xml:id="echoid-s12040" xml:space="preserve">cylindrum, ad cuius altitudinem altitu-<lb/>do tympani ſphæralis, vel ſphæroidalis ſit, vt ſemidiameter <lb/>baſis tympani ad reliquum, dempta ab eadem recta linea, ad quam di-<lb/>midia ſecundæ diametri circuli, vel ellipſis ſit, vt circulus ad qua-<lb/>dratum, cui circumſcribitur ſimul cum exceſſu, quo dicta linea exce-<lb/>dit _{2/3}_. </s> <s xml:id="echoid-s12041" xml:space="preserve">tertiæ proportionalis ſemidiametri baſis tympani, & </s> <s xml:id="echoid-s12042" xml:space="preserve">dimidiæ <lb/>ſecundæ diametri dicti circuli, vel ellipſis, eſſe æqualem dato tym-<lb/> <anchor type="note" xlink:label="note-0489-04a" xlink:href="note-0489-04"/> pano ſphærali, vel ſphæroidali, ſi ſit in baſi eiuſdcm tympani.</s> <s xml:id="echoid-s12043" xml:space="preserve"/> </p> <div xml:id="echoid-div1101" type="float" level="2" n="1"> <note position="right" xlink:label="note-0489-04" xlink:href="note-0489-04a" xml:space="preserve">Corò. 12 <lb/>34. l. 3.</note> </div> </div> <div xml:id="echoid-div1103" type="section" level="1" n="661"> <head xml:id="echoid-head691" xml:space="preserve">SECTIO III.</head> <note position="right" xml:space="preserve">C</note> <p style="it"> <s xml:id="echoid-s12044" xml:space="preserve">_E_T cylindrum, ad cuius altitudinem, altitudo anuli ſtricti circu-<lb/>laris, vel elliptici, ſit vt quadratum ad circulum, cui circum-<lb/>ſcribitur, in baſi exiſtentem circulo, cuius radius ſit æqualis ſecundæ <pb o="472" file="0490" n="490" rhead="GEOMETRIÆ"/> diametro circuli, vel ellipſis, quæ reuoluitur, eſſe æqualem dicto anu-<lb/> <anchor type="note" xlink:label="note-0490-01a" xlink:href="note-0490-01"/> lo ſtricto. </s> <s xml:id="echoid-s12045" xml:space="preserve">Conſimiliter autem inueniemus cylindrum æqualem anulo <lb/>lato circulari, vel elliptico; </s> <s xml:id="echoid-s12046" xml:space="preserve">& </s> <s xml:id="echoid-s12047" xml:space="preserve">eius portionibus, abſciſſis planis ad <lb/>axem reuolutionis rectis; </s> <s xml:id="echoid-s12048" xml:space="preserve">ſiue cuicunq; </s> <s xml:id="echoid-s12049" xml:space="preserve">ex figuris Corollariorum 26. <lb/></s> <s xml:id="echoid-s12050" xml:space="preserve">27. </s> <s xml:id="echoid-s12051" xml:space="preserve">28. </s> <s xml:id="echoid-s12052" xml:space="preserve">29. </s> <s xml:id="echoid-s12053" xml:space="preserve">Lib. </s> <s xml:id="echoid-s12054" xml:space="preserve">3. </s> <s xml:id="echoid-s12055" xml:space="preserve">Similiter inueniemus cylindrum æqualem Malo <lb/>Roſeo, vel Cotoneo, vel Citri@, vel Oliuæ; </s> <s xml:id="echoid-s12056" xml:space="preserve">Conoidi Parabolico, Vel <lb/>hyperbolico, eiuſdem fruſto, Apici parabolico, Semianulo ſtricto, vel <lb/>lato, & </s> <s xml:id="echoid-s12057" xml:space="preserve">ſemibaſibus ſtrictis, medijs, vel latis, Aceruis minori, vel <lb/>maiori parabolicis. </s> <s xml:id="echoid-s12058" xml:space="preserve">Tympano hyperbolico, & </s> <s xml:id="echoid-s12059" xml:space="preserve">portionibus eorundẽ <lb/>ſupra conſideratis, & </s> <s xml:id="echoid-s12060" xml:space="preserve">cylindricis, vel conicis, qui in baſibus ſpatijs <lb/>ſub ſpiralibus, & </s> <s xml:id="echoid-s12061" xml:space="preserve">volutis conſtituuntur. </s> <s xml:id="echoid-s12062" xml:space="preserve">Triplicatis autem altitu-<lb/>dinibus inuentorum cylindrorum, in quibus, & </s> <s xml:id="echoid-s12063" xml:space="preserve">eiſdem baſibus cum <lb/>cylindris, conſtituantur coni, iſti prædictis ſolidis æquales erunt, & </s> <s xml:id="echoid-s12064" xml:space="preserve"><lb/>iuxta Coroll. </s> <s xml:id="echoid-s12065" xml:space="preserve">_1_ antecedentis inueniemus pariter cylindrum, vel co-<lb/>num, qui ad quoduis ex prædictis ſolidis datam ratione mhabeat.</s> <s xml:id="echoid-s12066" xml:space="preserve"/> </p> <div xml:id="echoid-div1103" type="float" level="2" n="1"> <note position="left" xlink:label="note-0490-01" xlink:href="note-0490-01a" xml:space="preserve">_Coro. 13._ <lb/>_34. l. 3._</note> </div> </div> <div xml:id="echoid-div1105" type="section" level="1" n="662"> <head xml:id="echoid-head692" xml:space="preserve">PROBLEMA III. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s12067" xml:space="preserve">SPhæram inuenire æqualem dato cylindro. </s> <s xml:id="echoid-s12068" xml:space="preserve">Similiter, & </s> <s xml:id="echoid-s12069" xml:space="preserve"><lb/>ſphæroidem circa datum axim æqualem dato cyin-<lb/>dro -</s> </p> <p> <s xml:id="echoid-s12070" xml:space="preserve">Sit cylindrus datus, A, oportet illi æqualem ſphæram inuenire. <lb/></s> <s xml:id="echoid-s12071" xml:space="preserve">Fiat cylindrus rectus, CFD, ſexquialter cylindri, A, deinde inter, <lb/> <anchor type="note" xlink:label="note-0490-02a" xlink:href="note-0490-02"/> <anchor type="figure" xlink:label="fig-0490-01a" xlink:href="fig-0490-01"/> altitudinem, FE, & </s> <s xml:id="echoid-s12072" xml:space="preserve"><lb/>baſis diametrũ, CD, <lb/>duæ mediæ continuè <lb/>proportionales, iuxta <lb/>methodũ ab alijs tra-<lb/>ditam, inueniantur, <lb/>quę ſint, M, GH, de-<lb/>ſcripto autem circu-<lb/>lo circa alterà dictarũ <lb/>mediarum tanquam <lb/>diametrum, vt circa, <lb/>GH, fiat is baſis cu-<lb/>uſdam cylindri altitu-<lb/>dinis æqualis ipſi, G <lb/>H, & </s> <s xml:id="echoid-s12073" xml:space="preserve">ſit tandem ſphæra, B, circa diametrum æqualem ipſi, GH, <lb/>conſt tuta. </s> <s xml:id="echoid-s12074" xml:space="preserve">Dico ſphæram, B, eſſe æqualem cylindro, A. </s> <s xml:id="echoid-s12075" xml:space="preserve">Eſt en@m <lb/>CD, ad, GH, vt, M, ad, FF, permutando, CD, ad, M, eſt vt, G <pb o="473" file="0491" n="491" rhead="LIBER VI."/> H, vel, LK, altituto, ad, FE, vt verò, CD, ad, M, ita quadratum <lb/>CD, ad quadratum, GH, vel circulus, CD, ad circulum, GH, er-<lb/>go vt, LK, ad, FE, ſic circulus, CD, ad circulum, GH, ergo cy-<lb/> <anchor type="note" xlink:label="note-0491-01a" xlink:href="note-0491-01"/> lindri, CFD, GLH, ſunt æquales, eſt autem cylindrus, CFD, ſex-<lb/>quialter cylindri, A, ergo cylindrus, GLH, erit ſexquialter cylin-<lb/>dri, A, eſt autem cylindrus, GLH, etiam ſexquialter ſphæræ circa <lb/>diametrum, GH, vel illi æqualem, NO, deſcriptæ ideſt ſphæræ, B, <lb/> <anchor type="note" xlink:label="note-0491-02a" xlink:href="note-0491-02"/> ergo ſphæra, B, erit æqualis dato cylindro, A.</s> <s xml:id="echoid-s12076" xml:space="preserve"/> </p> <div xml:id="echoid-div1105" type="float" level="2" n="1"> <note position="left" xlink:label="note-0490-02" xlink:href="note-0490-02a" xml:space="preserve">Vide Da-<lb/>uidem Ri <lb/>ualtum in <lb/>Commé. <lb/>in Arch. <lb/>ad prop. <lb/>1. ſecundi <lb/>de Sphæ-<lb/>ra, & Cy-<lb/>lindro.</note> <figure xlink:label="fig-0490-01" xlink:href="fig-0490-01a"> <image file="0490-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0490-01"/> </figure> <note position="right" xlink:label="note-0491-01" xlink:href="note-0491-01a" xml:space="preserve">E. G. <lb/>Coroll. 4. <lb/>gene. 34. <lb/>l. 2.</note> <note position="right" xlink:label="note-0491-02" xlink:href="note-0491-02a" xml:space="preserve">Coroll. 1. <lb/>34. l. 3.</note> </div> <p> <s xml:id="echoid-s12077" xml:space="preserve">Sit nunc datus axis, NO, circa quem ſit conſtituenda ſphærois <lb/>æqualis dato cylindro, A, ſi igitur ſphæra circa diametrum, NO, <lb/>eſſet æqualis dato cylindro, non poſſet circa hanc diametrum fie-<lb/>ri alia ſphærois ęqualis dato cylindro, ſed talis ſphęrois eſſet eadẽ <lb/>ſphęra. </s> <s xml:id="echoid-s12078" xml:space="preserve">Non ſit autem ęqualis ſphęra, B, cylindro, A, tunc fiat <lb/>ſphęra ęqualis cylindro, A, quę ſit circa diametrum, ST, deinde <lb/>fiat, vt, NO, ad, ST, ſic quadratum, ST, ad, X1, bifariam diuiſam <lb/>in, B, centro, & </s> <s xml:id="echoid-s12079" xml:space="preserve">fiat ſphęrois circa diametros, NO, XI, igitur pri. <lb/></s> <s xml:id="echoid-s12080" xml:space="preserve">miaxes, NO, ST, reciprocè reſpondent ſecundorum axium, ST, <lb/>vel, 34, XI, quadratis ergo ſphęra, ST, erit ęqualis ſphęroidis, NX <lb/> <anchor type="note" xlink:label="note-0491-03a" xlink:href="note-0491-03"/> OI, ergo ſphęrois, NXOI, circa datum axim, erit ęqualis dato cy-<lb/>lindro, A, quod erat inueniendum.</s> <s xml:id="echoid-s12081" xml:space="preserve"/> </p> <div xml:id="echoid-div1106" type="float" level="2" n="2"> <note position="right" xlink:label="note-0491-03" xlink:href="note-0491-03a" xml:space="preserve">Corol. 10. <lb/>Prop. 34. <lb/>l. 3. ſect. 4.</note> </div> </div> <div xml:id="echoid-div1108" type="section" level="1" n="663"> <head xml:id="echoid-head693" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s12082" xml:space="preserve">_H_Inc colligitur cuicunq; </s> <s xml:id="echoid-s12083" xml:space="preserve">ex ſolidis in antecedenti, & </s> <s xml:id="echoid-s12084" xml:space="preserve">Corolla. <lb/></s> <s xml:id="echoid-s12085" xml:space="preserve">rijs eiuſdem nominatis ſphæram æqualem nos ſcire conſtitue-<lb/>re, necnon ſphæroidem æqualem circa datum axem, ſphæramque, ac <lb/>ſphæoidem, quæ ad quodcunq; </s> <s xml:id="echoid-s12086" xml:space="preserve">ex ipſis datam rationem habeat. </s> <s xml:id="echoid-s12087" xml:space="preserve">Pro-<lb/> <anchor type="note" xlink:label="note-0491-04a" xlink:href="note-0491-04"/> poſito enim ex illis quocunque ſolido, inuenietur primò cylindrus, <lb/>qui ad ipſum datam rationem habeat, deinde fiet ſphæra, vel ſphærois <lb/>circa datum axim, æqualis inuento c ylindro, quæ ſubinde ad datum <lb/>ſolidum datam rationem habebit: </s> <s xml:id="echoid-s12088" xml:space="preserve">Et vniuerſaliter patet ſi diſcamus, <lb/>dato cylindro æquale ſolidum ex iam @onſideratorum genere construe-<lb/>re, conſequenter eiuſmodi ſolidum nos ſcire conſtruere, quod ad ali-<lb/>quod ex nominatis in antecedenti Propoſitione, & </s> <s xml:id="echoid-s12089" xml:space="preserve">eiuſdem Corolla-<lb/>rijs, datam rationem habeat.</s> <s xml:id="echoid-s12090" xml:space="preserve"/> </p> <div xml:id="echoid-div1108" type="float" level="2" n="1"> <note position="right" xlink:label="note-0491-04" xlink:href="note-0491-04a" xml:space="preserve">25. huius.</note> </div> </div> <div xml:id="echoid-div1110" type="section" level="1" n="664"> <head xml:id="echoid-head694" xml:space="preserve">PROBLEMA IV. PROPOS. XXVII.</head> <p> <s xml:id="echoid-s12091" xml:space="preserve">DAto cylindro apicem ſphæralem æqualem conſtitue-<lb/>re, vel ſphæroidalem, & </s> <s xml:id="echoid-s12092" xml:space="preserve">hunc circa datum axem.</s> <s xml:id="echoid-s12093" xml:space="preserve"/> </p> <pb o="474" file="0492" n="492" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s12094" xml:space="preserve">Vtamur antecedentis figura, in qua ſupponamus dato cylin-<lb/>dro, A, conſticuendum eſſe æqualem apicem ſphæralem, vel ſphę-<lb/>roidalem, & </s> <s xml:id="echoid-s12095" xml:space="preserve">hunc circa datum axem. </s> <s xml:id="echoid-s12096" xml:space="preserve">Exponatur autem cylin-<lb/>drus, FCD, qui ad cylindrum, A, ſit, vt 21. </s> <s xml:id="echoid-s12097" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s12098" xml:space="preserve">deinde inter, C <lb/>D, FE, ſumantur duæ inediæ continuè proportionales, GH, M, & </s> <s xml:id="echoid-s12099" xml:space="preserve"><lb/>fiat cylindrus altitudinis, GH, qui ſit, GLH, ac ſupponatur ipſi, <lb/>LK, aſſumptam eſſe æqualem ipſam, NO, igitur ductis tangenti-<lb/>bus circulum circa, NO, in punctis, O, R, N, quæ ſint, OZ, Z℟, <lb/>℟N, concurrentibus in Z, ℟, patet, OZ, eſſe æqualem ipſi, GK, <lb/>&</s> <s xml:id="echoid-s12100" xml:space="preserve">, ℟Z, æquatur ipſi, Lk, crgo cylindrus, qui naſceretur ex reuò-<lb/>lutione parallelogrammi, NZ, circa manentem axem, ℟Z, eſſet <lb/>æqualis cylindro, GLH, oſtendemus autem, vt in antecedenti cy-<lb/>lindrum, GLH, eſſe æqualem cylindro, CFD, vnde patebit cylin-<lb/>drum genitum ex, NZ, ad cylindrum, A, eſſe vt 21. </s> <s xml:id="echoid-s12101" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s12102" xml:space="preserve">ſedidem <lb/>ad apicem, qui naſceretur ex reuolutione trilinei, OZR, circa, RZ, <lb/> <anchor type="note" xlink:label="note-0492-01a" xlink:href="note-0492-01"/> eſt vt 21. </s> <s xml:id="echoid-s12103" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s12104" xml:space="preserve">nam cylindrus ex, NZ, duplus eſt cylindriex, BZ, <lb/>ergo apex genitus ex trilineo, OZR, æqualis erit cylindro, A.</s> <s xml:id="echoid-s12105" xml:space="preserve"/> </p> <div xml:id="echoid-div1110" type="float" level="2" n="1"> <note position="left" xlink:label="note-0492-01" xlink:href="note-0492-01a" xml:space="preserve">Corol. 11. <lb/>34. l. 3.</note> </div> <p> <s xml:id="echoid-s12106" xml:space="preserve">Sit nunc inueniendus apex ſphæroidalis circa datum axem, RZ, <lb/>vel illi æqualem, qui ſit æqualis cylindro, A, ſi ergo talis eſſet apex <lb/>ſphæralis, qui fit ex, OZR, non eſſet alius apex ſphæroidalis circa, <lb/>RZ, vel illi æqualem, qui eſſet æqualis cylindro, A; </s> <s xml:id="echoid-s12107" xml:space="preserve">ſi verò non <lb/>ſit, inueniatur apex ſphæralis, vt, ΤΔ4, æqualis cylindro, A, dein-<lb/>de vt, RZ, ad huius facti apicis axim, 4Δ, ita fiat dimidij diametri <lb/>baſis eiuſdem, ideſt, ΤΔ, quadratum ad quadratum, OY, ſiue, BI, <lb/>& </s> <s xml:id="echoid-s12108" xml:space="preserve">per, 1, tranſeat elliplis, NIO, & </s> <s xml:id="echoid-s12109" xml:space="preserve">ducatur eandem tangens in, I, <lb/>quę fit, IY, igitur quia, RZ, ad, 4Δ, axim facti apicis ſphæralis eſt, <lb/>vt quadratum, ΤΔ, dimidij diametri baſis, ad quadratum, OY, <lb/>ideſt, vt circulus, qui eſt baſis facti apicis ſphæralis, ΤΔ4, ad cir-<lb/>culum, qui eſt baſis alterius, ideò iſti apices erunt æquales: </s> <s xml:id="echoid-s12110" xml:space="preserve">nam <lb/>ſe habebunt, vt cylindri in eiſdem cum illis baſibus, & </s> <s xml:id="echoid-s12111" xml:space="preserve">circa eoſ-<lb/>dem axes exiſtentes, qui cylindrici erunt æquales, nam axes baſi-<lb/>bus reciprocè reſpondet; </s> <s xml:id="echoid-s12112" xml:space="preserve">ergo apex ſuphæroidalis, qui fiet ex, O <lb/>YI, & </s> <s xml:id="echoid-s12113" xml:space="preserve">eſt circa axim, IY, æqualem ipſi, RZ, datæ, erit æqualis cy-<lb/>lindro, A, quæ inuenienda erant.</s> <s xml:id="echoid-s12114" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1112" type="section" level="1" n="665"> <head xml:id="echoid-head695" xml:space="preserve">COR OLL ARIVM.</head> <p style="it"> <s xml:id="echoid-s12115" xml:space="preserve">_P_Atet autem, quod iuxta Corollarium antec@dentis poterimus etiã <lb/>inuenire apices ſphęrales, vel ſphæroidales circa datum axim, ad <lb/>datum quodcunq; </s> <s xml:id="echoid-s12116" xml:space="preserve">ſolidum ex enumeratis in dicto Corollario datam <lb/>ratoinem habentes.</s> <s xml:id="echoid-s12117" xml:space="preserve"/> </p> <pb o="473" file="0493" n="493" rhead="LIBER VI."/> </div> <div xml:id="echoid-div1113" type="section" level="1" n="666"> <head xml:id="echoid-head696" xml:space="preserve">PROBLEMA V. PROPOS. XXVIII.</head> <p> <s xml:id="echoid-s12118" xml:space="preserve">DAto cylindro tympanum ſphærale eidem æquale cõ-<lb/>ſtituere, cuius axis ſemidiametro baſis ſit æqualis.</s> <s xml:id="echoid-s12119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12120" xml:space="preserve">Sit datus cylindrus, <lb/> <anchor type="figure" xlink:label="fig-0493-01a" xlink:href="fig-0493-01"/> ABC, cuius axis, AD, <lb/>baſis, BC, oportet illi <lb/>æquale tympanu ſphę-<lb/>rale conſtituere, cuius <lb/>axis ſemidiametro ba-<lb/>ſis ſit æqualis. </s> <s xml:id="echoid-s12121" xml:space="preserve">Vt hoc <lb/>fiat exponatur recta li-<lb/>nea terminata quæcũ-<lb/>que, EI, quæ ſit bifariã <lb/>ſecta in, G, & </s> <s xml:id="echoid-s12122" xml:space="preserve">vt quad. <lb/></s> <s xml:id="echoid-s12123" xml:space="preserve">circulo cuicunq; </s> <s xml:id="echoid-s12124" xml:space="preserve">circũ. </s> <s xml:id="echoid-s12125" xml:space="preserve"><lb/>ſcriptum ad eundem <lb/>circulum, ita fiat, GE, <lb/>ad, EF, ſumatur dein-<lb/>de, FH, quæ ſit exceſ-<lb/>ſus, quo, FE, ſuperat <lb/>{2/3}. </s> <s xml:id="echoid-s12126" xml:space="preserve">tertiæ proportiona-<lb/>lis duarum, IE, EG, <lb/>deinde vt, HI, ad, IE, ita fiat, AD, ad, LO, altitudinem alterius <lb/>cylindri in baſi, NP, æquali ipſi, BC, exiſtentis, & </s> <s xml:id="echoid-s12127" xml:space="preserve">tandem inter <lb/>LO, &</s> <s xml:id="echoid-s12128" xml:space="preserve">, ON, baſis ſemidiametrum, ſumantur duæ mediæ conti-<lb/>nuò proportionales, QO, OR, & </s> <s xml:id="echoid-s12129" xml:space="preserve">in altitudine æquali, OR, nem-<lb/>pè, T8, baſi, ℟8, æquali eidem, OR, fiat parallelogrammum re-<lb/>ctangulum, 58, in cuius plano deſcribatur ſemicirculus, SY℟, & </s> <s xml:id="echoid-s12130" xml:space="preserve"><lb/>ipſum cum parallelogrammo reuo uatur circa manentem axim, <lb/>T8, donec redeat vnde diſceſſit, patet autem, quod ex parallelo-<lb/>grammo fiet cylindrus, vt, Sφ, & </s> <s xml:id="echoid-s12131" xml:space="preserve">ex figura, SY℟8T, tympanum <lb/>ſphærale, 5Yφ. </s> <s xml:id="echoid-s12132" xml:space="preserve">Dico igitur hoc tympanum eſſe æquale dato cy-<lb/>lindro, ABC, &</s> <s xml:id="echoid-s12133" xml:space="preserve">, T8, æqualem ipſi, 8℟, ſemidiametro baſis, ℟φ. </s> <s xml:id="echoid-s12134" xml:space="preserve"><lb/>Sint parallelogrammum, Sφ, & </s> <s xml:id="echoid-s12135" xml:space="preserve">figura, SYφ, per axem tranſeun-<lb/>tia, & </s> <s xml:id="echoid-s12136" xml:space="preserve">X&</s> <s xml:id="echoid-s12137" xml:space="preserve">, per centra, X, &</s> <s xml:id="echoid-s12138" xml:space="preserve">, circulorum ducta, ſecans, T8, in, Z, <lb/>manifeſtum eſt igitur, quod, XZ, bifariam ſecabitur à circumfe-<lb/>rentia, SY℟, vt in, Y, cum, 8℟, ſit æqualis, ℟S, & </s> <s xml:id="echoid-s12139" xml:space="preserve">ipſi, XZ, S℟, <lb/>autem ſit dupla, XY, vnde ſi ſecetur, ℟8, bifariam in, 4@erit, ℟4, <lb/>æqualis ipſi, XY, ſit autem ab ea dempta, ℟@, ad qua ?</s> <s xml:id="echoid-s12140" xml:space="preserve">? 4℟, ſit vt <pb o="474" file="0494" n="494" rhead="GEOMETRIÆ"/> quadratum ad inſcriptum circulum, & </s> <s xml:id="echoid-s12141" xml:space="preserve">in, ℟8, ſumpta, 37, æqua <lb/>lis exceſſui, quo, 3℟, ſuperat {2/3}. </s> <s xml:id="echoid-s12142" xml:space="preserve">tertiæ proportionalis duarum, 8℟, <lb/>℟4, patet ergo, quod cylindrus, Sφ, ad tympanum, SYφ, eſt vt, ℟ <lb/>8, ad, 87. </s> <s xml:id="echoid-s12143" xml:space="preserve">Quoniam vero, vt, LO, ad, OQ, ſic eſt, QO, ad, OR, <lb/> <anchor type="note" xlink:label="note-0494-01a" xlink:href="note-0494-01"/> ideò vt, LO, ad, OR, vel, T8, ipſi æqualem, ita quadratum, LO, <lb/>ad quadratum, OQ, vel ita quadratum, RO, ſeu quadratum, ℟8, <lb/>ad quadratum, NO, vel ita circulus, ℟φ, ad circulum, NP, ergo <lb/>duo cylindri, Sφ, KP, quorum axes reciprocè baſibus reſpondent, <lb/>erunt æquales, quod ſerua. </s> <s xml:id="echoid-s12144" xml:space="preserve">Vlterius, quia vt, IE, ad, EG, ſic eſt, <lb/> <anchor type="note" xlink:label="note-0494-02a" xlink:href="note-0494-02"/> 8℟, ad, ℟4, & </s> <s xml:id="echoid-s12145" xml:space="preserve">vt, EG, ad, EF, ſic quadratum ad inſcriptum circu-<lb/>lum, & </s> <s xml:id="echoid-s12146" xml:space="preserve">ita etiam, ℟4, ad, ℟3, ergo ex æquali, IE, ad, EF, erit vt, <lb/>8℟, ad, ℟3. </s> <s xml:id="echoid-s12147" xml:space="preserve">Similiter quia, IE, ad, EG, eſt vt, 8℟, ad, ℟4, &</s> <s xml:id="echoid-s12148" xml:space="preserve">, E <lb/>G, ad {2/3}. </s> <s xml:id="echoid-s12149" xml:space="preserve">tertiæ proportionalis duarum, IE, EG, eſt vt, 4℟, ad {2/3}. <lb/></s> <s xml:id="echoid-s12150" xml:space="preserve">tertiæ proportionalis duarum, 8℟, ℟4, ideò ex æquali, vt, IE, ad, <lb/>{2/3}. </s> <s xml:id="echoid-s12151" xml:space="preserve">tertiæ proportionalis duarum, IE, EG, ita, 8℟, erit ad {2/3}. </s> <s xml:id="echoid-s12152" xml:space="preserve">tertię <lb/>proportionalis duarum, 8℟, ℟4, eædem autem, IE, 8℟, ad, FE, 3 <lb/>℟, erant in eadem ratione, ergo ad exceſſus duarum, EF, ℟3, ſu-<lb/>per {2/3}. </s> <s xml:id="echoid-s12153" xml:space="preserve">tertiarum proportionalium, IE, EG, ex vna parte, &</s> <s xml:id="echoid-s12154" xml:space="preserve">, 8℟, <lb/>℟4, ex alia, erunt in eadem ratione .</s> <s xml:id="echoid-s12155" xml:space="preserve">i. </s> <s xml:id="echoid-s12156" xml:space="preserve">vt, IE, ad, FH, ita erit, 8℟, <lb/>ad, 37, ſed etiam, vt, IE, ad, EF, ſic eſſe oſtenſum eſt, 8℟, ad, ℟3, <lb/>ergo, coll@gendo, vt, IE, ad, EH, ita, 8℟, ad, ℟7, & </s> <s xml:id="echoid-s12157" xml:space="preserve">per conuer-<lb/>ſionem rationis, & </s> <s xml:id="echoid-s12158" xml:space="preserve">conuertendo, vt, Hl, ad, IE, ideſt vt, AD, ad, <lb/>LO, ideſt vt cylindrus, BAC, ad cylindrum, NLP, vel illi æqualẽ, <lb/>Sφ, (vt oſtenſum eſt) ita, 78, ad, 8℟, ſed vt, 78, ad, 8℟, ita tym-<lb/>panum, SYφ, ad cylindrum, Sφ, ergo, vt cylindrus, BAC, ad cy-<lb/> <anchor type="note" xlink:label="note-0494-03a" xlink:href="note-0494-03"/> lindrum, Sφ, ita tympanum, SYφ, ad cylindrum, Sφ, ergo cylin-<lb/>drus, BAC, æquaturtympano, SYφ, cuius axis, T8, ſemidiametro <lb/>baſis, ℟8, eſt æqualis, quod, &</s> <s xml:id="echoid-s12159" xml:space="preserve">c.</s> <s xml:id="echoid-s12160" xml:space="preserve"/> </p> <div xml:id="echoid-div1113" type="float" level="2" n="1"> <figure xlink:label="fig-0493-01" xlink:href="fig-0493-01a"> <image file="0493-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0493-01"/> </figure> <note position="left" xlink:label="note-0494-01" xlink:href="note-0494-01a" xml:space="preserve">Corol. 12. <lb/>34. l. 3.</note> <note position="left" xlink:label="note-0494-02" xlink:href="note-0494-02a" xml:space="preserve">E. Cor. 4. <lb/>gener. 34. <lb/>l. 2.</note> <note position="left" xlink:label="note-0494-03" xlink:href="note-0494-03a" xml:space="preserve">Corol. 12. <lb/>34. l. 3.</note> </div> </div> <div xml:id="echoid-div1115" type="section" level="1" n="667"> <head xml:id="echoid-head697" xml:space="preserve">COROLL ARIVM.</head> <p style="it"> <s xml:id="echoid-s12161" xml:space="preserve">_C_Olligitur autemiuxta Corollarium Propoſit. </s> <s xml:id="echoid-s12162" xml:space="preserve">26. </s> <s xml:id="echoid-s12163" xml:space="preserve">huius, nos poſ-<lb/>ſe in uenire tympana ſphæralia, quorum axes ſemidiametris <lb/>baſium ſint æquales, quæ ad datum quodcunq; </s> <s xml:id="echoid-s12164" xml:space="preserve">ex ſolidis in ſectione 3. <lb/></s> <s xml:id="echoid-s12165" xml:space="preserve">Coroll. </s> <s xml:id="echoid-s12166" xml:space="preserve">2. </s> <s xml:id="echoid-s12167" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s12168" xml:space="preserve">25. </s> <s xml:id="echoid-s12169" xml:space="preserve">huius enumeratis, datam rationem habeant.</s> <s xml:id="echoid-s12170" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1116" type="section" level="1" n="668"> <head xml:id="echoid-head698" xml:space="preserve">PROBLEMA VI. PROPOS. XIX.</head> <p> <s xml:id="echoid-s12171" xml:space="preserve">DAto cylindro anulum ſtrictum circularem æqualem <lb/>inuenire.</s> <s xml:id="echoid-s12172" xml:space="preserve"/> </p> <pb o="475" file="0495" n="495" rhead="LIBER VI."/> <p> <s xml:id="echoid-s12173" xml:space="preserve">Sit datus cylindrus, A, oportet illi anulum ſtrictum circularem <lb/>æqualem inuenire. </s> <s xml:id="echoid-s12174" xml:space="preserve">Reperiamus ergo cilindrum, quiad cylindrũ <lb/>A, ſit, vt duplum cuiuiſuis quadrati ad circulum dicto quadrato <lb/> <anchor type="note" xlink:label="note-0495-01a" xlink:href="note-0495-01"/> inſcriptum, & </s> <s xml:id="echoid-s12175" xml:space="preserve">dein de huic inuento cylindro alius inueniatur ęqua-<lb/> <anchor type="figure" xlink:label="fig-0495-01a" xlink:href="fig-0495-01"/> lis, BC, cuius axis ſit <lb/>ęqualis diametro ba. <lb/></s> <s xml:id="echoid-s12176" xml:space="preserve">ſis .</s> <s xml:id="echoid-s12177" xml:space="preserve">@. </s> <s xml:id="echoid-s12178" xml:space="preserve">MN, ipſi, OC, <lb/>qui diuidatur bifariã <lb/>in, R, & </s> <s xml:id="echoid-s12179" xml:space="preserve">per, R, du-<lb/>cto plano oppoſitis <lb/>baſibus æquidiſtan-<lb/>te, ſit conſtitutus cy <lb/>lindrus, DC, in quo <lb/>planum per axem <lb/>ductum produxerit <lb/>parallelogrammum, <lb/>DC, quod in duo ſe. </s> <s xml:id="echoid-s12180" xml:space="preserve"><lb/>parab tur quadrata per ipſam, RN, ſint illis inſcripti æquales cir-<lb/>culi, E, F, ex quorum reuolutione circa, RN, intelligatur effectus <lb/>anulus ſtrictus circularis, EF. </s> <s xml:id="echoid-s12181" xml:space="preserve">Dico hunc eſſe æqualem cylindro, <lb/>A. </s> <s xml:id="echoid-s12182" xml:space="preserve">Nam, BC, ad, A, eſt vt parallelogrammum, BN, .</s> <s xml:id="echoid-s12183" xml:space="preserve">i. </s> <s xml:id="echoid-s12184" xml:space="preserve">vt duplũ <lb/> <anchor type="note" xlink:label="note-0495-02a" xlink:href="note-0495-02"/> quadrati, DN, ad circulum, E, ſic autem eſt, BC, ad anulum ſtri-<lb/>ctum genitum ex, E, igitur hic anulus cylindro, A, æqualis erit, <lb/>quod inuenire opus erat.</s> <s xml:id="echoid-s12185" xml:space="preserve"/> </p> <div xml:id="echoid-div1116" type="float" level="2" n="1"> <note position="right" xlink:label="note-0495-01" xlink:href="note-0495-01a" xml:space="preserve">Vtin pro. <lb/>pol. 26, <lb/>huius.</note> <figure xlink:label="fig-0495-01" xlink:href="fig-0495-01a"> <image file="0495-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0495-01"/> </figure> <note position="right" xlink:label="note-0495-02" xlink:href="note-0495-02a" xml:space="preserve">Elicitu@ <lb/>ex Corol. <lb/>13. 34. l. 3.</note> </div> </div> <div xml:id="echoid-div1118" type="section" level="1" n="669"> <head xml:id="echoid-head699" xml:space="preserve">PROBLEMA VII- PROPOS. XXX.</head> <p> <s xml:id="echoid-s12186" xml:space="preserve">DAto cylindro anulo latum circularem ęqualem inue-<lb/>nire, dato circulo, qui per reuolutionem ipſum ge-<lb/>nerat; </s> <s xml:id="echoid-s12187" xml:space="preserve">oportet autem datum cylin drum maiorem eſſe anu-<lb/>lo ſtricto ab eodem circulo per reuolutionem genito.</s> <s xml:id="echoid-s12188" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12189" xml:space="preserve">Sit datus cylindrus, E, datus circulus, CD, ſit autem datus cy-<lb/>lindrus, E, maior anulo ſtricto per reuolutionem dati circuli circa <lb/>ipſum rectam tangentem genito. </s> <s xml:id="echoid-s12190" xml:space="preserve">Oportet anulum latum circula-<lb/>rem inuenire ab eodem circulo per reuolutionem genitum, æqua-<lb/>lem dato cylindro, E. </s> <s xml:id="echoid-s12191" xml:space="preserve">Sit tangens circulum, CD, in puncto, D, <lb/>ipſa, MN, circa quam fieri intelligatur reuolutio, vt deſcribatur <lb/>anulus ſtrictus circularis ex, CD, fiat deinde, vt anulus ſtrictus ab <lb/>eo genitus ad cylindrum, E, ita, DC, diameter eiuldem ad aliam, <lb/>FH, (quæ erit eadem maior, quia etiam c@lindrus, E, eſt maior <pb o="476" file="0496" n="496" rhead="GEO METRIÆ"/> dicto anulo @tricto) cui adijciatur in directum, HI, ipſi, DC, æqua <lb/>lis, deinde tota, FI, bifariam diuidatur in, G, & </s> <s xml:id="echoid-s12192" xml:space="preserve">producta, DC, <lb/>verſus, C, indefinitè in ea ſumatur, AD, æqualis ipſi, GI, & </s> <s xml:id="echoid-s12193" xml:space="preserve">ab-<lb/>ſciſla ab eadem ad punctum, A, ipſa, AR, eidem, CD, ęquali, in-<lb/>telligatur circa, AR, diametrum deſcriptus circulus, AR, ęqualis, <lb/> <anchor type="figure" xlink:label="fig-0496-01a" xlink:href="fig-0496-01"/> CD. </s> <s xml:id="echoid-s12194" xml:space="preserve">Dico anulũ <lb/>latum circularem <lb/>deſcriptum per, A <lb/>R, reuolutum cir-<lb/>ca, MN, in tali ſi-<lb/>tu, cylindro, E, ę-<lb/>qualem eſſe. </s> <s xml:id="echoid-s12195" xml:space="preserve">Nam <lb/>ſtrictus anulus de-<lb/>ſeriptus à, CD, ad <lb/>cylindrum, E, eſt <lb/>vt, DC, ad, FH, & </s> <s xml:id="echoid-s12196" xml:space="preserve"><lb/>quia, GI, eſt æqualis ipſi, AD, &</s> <s xml:id="echoid-s12197" xml:space="preserve">, CD, ipſi, HI, erit, GH, æqualis <lb/>ipſi, AC, ergo vt, DC, ad, FH, ita eſt eadem, DC, ad, IGH, vel ad, <lb/>DAC, ſiue, AR, ad, ADR, (nam compoſita ex, AD, DR, eſt æ-<lb/> <anchor type="note" xlink:label="note-0496-01a" xlink:href="note-0496-01"/> qualis compoſitæ ex, DA, AC,) eſt autem vt, AR, ad compoſitã <lb/>ex, AD, DR, ita anulus ſtrictus genitus ex circulo, CD, ad anu lũ <lb/>latum genitum ex circulo, AR, ergo anulus ſtrictus genitus ex cir-<lb/>culo, CD, ad anulum latum genitum ex circulo, AR, erit, vt idem <lb/>anulus ſtrictus ad cylindrum, E, ergo anulus latus genitus ex circu-<lb/>lo dato, AR, ſiue, CD, in tali ſitu, æqualis erit cylindro, E. </s> <s xml:id="echoid-s12198" xml:space="preserve">In-<lb/>uentum ergo eſt, quod opus erat.</s> <s xml:id="echoid-s12199" xml:space="preserve"/> </p> <div xml:id="echoid-div1118" type="float" level="2" n="1"> <figure xlink:label="fig-0496-01" xlink:href="fig-0496-01a"> <image file="0496-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0496-01"/> </figure> <note position="left" xlink:label="note-0496-01" xlink:href="note-0496-01a" xml:space="preserve">Elicitur ex <lb/>dictis in <lb/>Corol. 29. <lb/>34. lib. 3. <lb/>Sect. 2. fi <lb/>ea circulis <lb/>applicẽtur.</note> </div> </div> <div xml:id="echoid-div1120" type="section" level="1" n="670"> <head xml:id="echoid-head700" xml:space="preserve">COROLL ARIVM I.</head> <p style="it"> <s xml:id="echoid-s12200" xml:space="preserve">_I_Vxta Coroll. </s> <s xml:id="echoid-s12201" xml:space="preserve">autem Prop. </s> <s xml:id="echoid-s12202" xml:space="preserve">26. </s> <s xml:id="echoid-s12203" xml:space="preserve">huius, manifeſtum eſt nos etiam di-<lb/>ctos anulos in dataratione ad datum cylindrum inuenire poſſe, & </s> <s xml:id="echoid-s12204" xml:space="preserve"><lb/>ſubinde etiam in data ratione ad quodcunq; </s> <s xml:id="echoid-s12205" xml:space="preserve">ex ſolidis in Sect. </s> <s xml:id="echoid-s12206" xml:space="preserve">3. </s> <s xml:id="echoid-s12207" xml:space="preserve">Cor. </s> <s xml:id="echoid-s12208" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s12209" xml:space="preserve">Prop. </s> <s xml:id="echoid-s12210" xml:space="preserve">25. </s> <s xml:id="echoid-s12211" xml:space="preserve">huius enumeratis.</s> <s xml:id="echoid-s12212" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1121" type="section" level="1" n="671"> <head xml:id="echoid-head701" xml:space="preserve">COROLL ARIVM II.</head> <p style="it"> <s xml:id="echoid-s12213" xml:space="preserve">_H_Abetur injuper ſi in recta, DA, indefinitè producta, continuen-<lb/>tur à puncto, D, æquales circulorum diametri; </s> <s xml:id="echoid-s12214" xml:space="preserve">ab eiſdem cir-<lb/>@ulis per reuolutionem circa, MN, deinceps genitos anulos ſeſe ha-<lb/>bere, vt numeros impares ab vnitate continuò progredientes. </s> <s xml:id="echoid-s12215" xml:space="preserve">Quod <lb/>ſi in eadem recta linea perpendiculari ipſi, MN, Vt in eadem, DA, in@ <pb o="477" file="0497" n="497" rhead="LIBER VI."/> definitè producta, continuentur à puncto, D, parallelogrammor@m re-<lb/>ctangulorum, in eademq; </s> <s xml:id="echoid-s12216" xml:space="preserve">altitudine exiſtentium, æquales baſes, ijſq; <lb/></s> <s xml:id="echoid-s12217" xml:space="preserve">bifariam ſectis, ab effectis punctis educantur parallelogrammorum di-<lb/>ctorum diametri, circa quas exiſtant aliæ planæ figuræ eius conditio-<lb/>nis, vt ducta quacunq; </s> <s xml:id="echoid-s12218" xml:space="preserve">parallela. </s> <s xml:id="echoid-s12219" xml:space="preserve">AD, illius portiones in his figuris <lb/>conceptæſint æquales, tum anuli deſoripti à dictis parallelogrammis <lb/>ſe habebunt vt numeri impares ab vnitate deinceps expoſiti, tum etiã <lb/>anuli geniti à prædictis figuris: </s> <s xml:id="echoid-s12220" xml:space="preserve">Etenim iſti anuli deinceps ſe habe-<lb/>bunt, vt quedratum primæ æqualium rectarum linearum, in ipſa, D <lb/>A, aſſumptarum, & </s> <s xml:id="echoid-s12221" xml:space="preserve">exceſſus quadratorum deinceps ſubſequentium <lb/>æqualium linearum, vt facilè innoteſcet, ſi in memoriam reuocentur, <lb/>quæ dicta ſunt in Coroll. </s> <s xml:id="echoid-s12222" xml:space="preserve">29. </s> <s xml:id="echoid-s12223" xml:space="preserve">34. </s> <s xml:id="echoid-s12224" xml:space="preserve">Lib. </s> <s xml:id="echoid-s12225" xml:space="preserve">3. </s> <s xml:id="echoid-s12226" xml:space="preserve">pro ibi conſideratis figuris, <lb/>quibus hæc quoq; </s> <s xml:id="echoid-s12227" xml:space="preserve">adaptantur.</s> <s xml:id="echoid-s12228" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1122" type="section" level="1" n="672"> <head xml:id="echoid-head702" xml:space="preserve">COROLLARIVM III.</head> <p style="it"> <s xml:id="echoid-s12229" xml:space="preserve">_M_Anifeſtum etiam eſt nos poſſe iuxta ſupradictam metbodum <lb/>cætera ſolida attentare, vt e adem dato cylindro tum æqualia, <lb/>tum etiam in data ratione inueniamus, veluti ex. </s> <s xml:id="echoid-s12230" xml:space="preserve">gr. </s> <s xml:id="echoid-s12231" xml:space="preserve">baſim columna-<lb/>rem ſtr ictam, latam, ac med am, Malum Roſeum, Citrium, & </s> <s xml:id="echoid-s12232" xml:space="preserve">reliqua, <lb/>quæ in Sect. </s> <s xml:id="echoid-s12233" xml:space="preserve">3. </s> <s xml:id="echoid-s12234" xml:space="preserve">Cor. </s> <s xml:id="echoid-s12235" xml:space="preserve">2. </s> <s xml:id="echoid-s12236" xml:space="preserve">Prop. </s> <s xml:id="echoid-s12237" xml:space="preserve">25. </s> <s xml:id="echoid-s12238" xml:space="preserve">huius enumerantur, vt ſubinde cui-<lb/>libet ex conſideratis in hoc volumine ſolidis inueniamus ex genere <lb/>cuiuslibet nedum æquale, ſed etiam in data ratione, quæ omnia ſingil. <lb/></s> <s xml:id="echoid-s12239" xml:space="preserve">latim proſequi minimè volui, tum ad vitandam prolixitatem, tum <lb/>etiam, vt alijs iucundi exercitij occaſionem non eripiam, veluti, & </s> <s xml:id="echoid-s12240" xml:space="preserve"><lb/>centri grauitatis nouorum ſolidorum inuentionem, nemini, quod ſciã <lb/>adhuc tentatam, alijs pro nunc relinquam, ſufficiat enim in præſenti <lb/>prædicta ſolida inueniendi rationem aliqualiter declaraſſe, centriq; </s> <s xml:id="echoid-s12241" xml:space="preserve"><lb/>grauitatis dictorum ſolidorum inueſtigandi materiam præbuiſſe.</s> <s xml:id="echoid-s12242" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1123" type="section" level="1" n="673"> <head xml:id="echoid-head703" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s12243" xml:space="preserve">_A_Duertendum eſt autem circa ſupradicta ſolida, quorum menſurã <lb/>præcisè non inuenimus, vt ex. </s> <s xml:id="echoid-s12244" xml:space="preserve">g. </s> <s xml:id="echoid-s12245" xml:space="preserve">patet de apicibus ſphærali-<lb/>bus, tympanis, anulis, & </s> <s xml:id="echoid-s12246" xml:space="preserve">alijs plurimis, neq; </s> <s xml:id="echoid-s12247" xml:space="preserve">inuentionem prædictam <lb/>eſſe, vel fore præciſam, non tamen aſpernendam, cum proximè ad ve-<lb/>ritatem accedat.</s> <s xml:id="echoid-s12248" xml:space="preserve"/> </p> <pb o="478" file="0498" n="498" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1124" type="section" level="1" n="674"> <head xml:id="echoid-head704" xml:space="preserve">THEOREMA XXIV. PROPOS. XXXI.</head> <p> <s xml:id="echoid-s12249" xml:space="preserve">SI in ſpatio helico primi circuli ſpiralium conicus in <lb/>eadem altitudine cum apice parabolico, in baſi dicto <lb/>circulo exiſtente, ſit conſtitutus; </s> <s xml:id="echoid-s12250" xml:space="preserve">apex parabolicus erit <lb/>ſexquialter dicti conici.</s> <s xml:id="echoid-s12251" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12252" xml:space="preserve">Patet hæc Propoſitio, nam ſi in dicto circulo, vt in baſi, & </s> <s xml:id="echoid-s12253" xml:space="preserve">cir-<lb/> <anchor type="note" xlink:label="note-0498-01a" xlink:href="note-0498-01"/> ca eundem axim cum dictis ſolidis ſit cylindrus conſtitutus, hic <lb/>erit ſexcuplus apicis parabolici, & </s> <s xml:id="echoid-s12254" xml:space="preserve">nonuplus dicti primi conici, er-<lb/>go apex parabolicus ad cylindrum erit, vt 3. </s> <s xml:id="echoid-s12255" xml:space="preserve">ad 18. </s> <s xml:id="echoid-s12256" xml:space="preserve">& </s> <s xml:id="echoid-s12257" xml:space="preserve">conicus ad <lb/>ipſum, vt 2. </s> <s xml:id="echoid-s12258" xml:space="preserve">ad 18. </s> <s xml:id="echoid-s12259" xml:space="preserve">vnde apex adconicum erit, vt 3. </s> <s xml:id="echoid-s12260" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s12261" xml:space="preserve">ideſt in <lb/>ratione ſexquialtera, quod erat oſtendendum.</s> <s xml:id="echoid-s12262" xml:space="preserve"/> </p> <div xml:id="echoid-div1124" type="float" level="2" n="1"> <note position="left" xlink:label="note-0498-01" xlink:href="note-0498-01a" xml:space="preserve">Coroll. 8 <lb/>Prop. 51. <lb/>l. 4. ſect. 1. <lb/>22. huius.</note> </div> </div> <div xml:id="echoid-div1126" type="section" level="1" n="675"> <head xml:id="echoid-head705" xml:space="preserve">THEOREMA XXV. PROPOS. XXXII.</head> <p> <s xml:id="echoid-s12263" xml:space="preserve">SI circa diametrum baſis ſemianuli ſtricti parabolici <lb/>tanquam circa propriam diametrum ſphæra, vel ſphę-<lb/>rois, fuerit conſtituta, cuius ſecunda diamet@er ſit æqualis <lb/>altitudine, ſiue axi, eiuſdem ſemianuli; </s> <s xml:id="echoid-s12264" xml:space="preserve">dicta ſphæra, vel <lb/>ſphærois ipſi ſemianulo æqualis erit.</s> <s xml:id="echoid-s12265" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12266" xml:space="preserve">Hæc etiam patet, nam cylindrus in eadem baſi cum ſemianulo <lb/>dicto, & </s> <s xml:id="echoid-s12267" xml:space="preserve">eadem altitudine, eſt eiuſdem ſexquialter, eſt autem etiã <lb/> <anchor type="note" xlink:label="note-0498-02a" xlink:href="note-0498-02"/> ſexquialter d ctæ ſphæræ, vel ſphæroidis, & </s> <s xml:id="echoid-s12268" xml:space="preserve">ideò dicta ſphæra, vel <lb/>ſphærois, erit æqualis dicto ſemianulo, quod oſtendendum erat.</s> <s xml:id="echoid-s12269" xml:space="preserve"/> </p> <div xml:id="echoid-div1126" type="float" level="2" n="1"> <note position="left" xlink:label="note-0498-02" xlink:href="note-0498-02a" xml:space="preserve">Corol. 10. <lb/>51. lib. 4 <lb/>ſect. poiſe <lb/>rior.</note> </div> <note position="left" xml:space="preserve">Coroll. 1. <lb/>34. l. 3.</note> </div> <div xml:id="echoid-div1128" type="section" level="1" n="676"> <head xml:id="echoid-head706" xml:space="preserve">THEOREMA XXVI. PROPOS. XXXIII.</head> <p> <s xml:id="echoid-s12270" xml:space="preserve">SI cylindrus, & </s> <s xml:id="echoid-s12271" xml:space="preserve">conus, hæmiſphærium, vel hæmiſphę-<lb/>roides, conoides parabolicum, apex parabolicus, & </s> <s xml:id="echoid-s12272" xml:space="preserve"><lb/>ſphæralis, fuerint in baſi eodem circul<unsure/>o, & </s> <s xml:id="echoid-s12273" xml:space="preserve">circa eundem <lb/>axim, infraſcriptam rationem inter ſe habebunt -</s> </p> <p> <s xml:id="echoid-s12274" xml:space="preserve">Sit cylindrus, BE, in baſi circulo, CE, circa axem, FD, in qui-<lb/>bus ſint etiam hæmiſphærium, vel hæmiſphæroides, CFE, conoi-<lb/>d@s parabolicum, CRFkE, conus, CFE, apex parabolicus, CVF <lb/>ZE, & </s> <s xml:id="echoid-s12275" xml:space="preserve">apex ſphæralis, vel ſphæroidalis, CXFYE, qualium igitur <lb/>partium cylindrus, BE, eſt 126. </s> <s xml:id="echoid-s12276" xml:space="preserve">talium hæmilphærium eſt 84. </s> <s xml:id="echoid-s12277" xml:space="preserve">co- <pb o="479" file="0499" n="499" rhead="LIBER VI."/> <anchor type="figure" xlink:label="fig-0499-01a" xlink:href="fig-0499-01"/> noides 63: </s> <s xml:id="echoid-s12278" xml:space="preserve">conus 42. </s> <s xml:id="echoid-s12279" xml:space="preserve">apex para-<lb/> <anchor type="note" xlink:label="note-0499-01a" xlink:href="note-0499-01"/> bolicus 21. </s> <s xml:id="echoid-s12280" xml:space="preserve">apex ſphæralis 12. <lb/></s> <s xml:id="echoid-s12281" xml:space="preserve">vnde patet hæmiſphęrium, vel <lb/>hæmiſphæroides ſexquitertium <lb/> <anchor type="note" xlink:label="note-0499-02a" xlink:href="note-0499-02"/> eſſe conoidis parabolici, quadru-<lb/> <anchor type="note" xlink:label="note-0499-03a" xlink:href="note-0499-03"/> plum apicis parabolici, & </s> <s xml:id="echoid-s12282" xml:space="preserve">ſe-<lb/>ptuplum apicis ſphæralis. </s> <s xml:id="echoid-s12283" xml:space="preserve">Co-<lb/>noides verò parabolicum tri-<lb/> <anchor type="note" xlink:label="note-0499-04a" xlink:href="note-0499-04"/> plum eſſe apicis parabolici, & </s> <s xml:id="echoid-s12284" xml:space="preserve"><lb/>quintuplum ſexquiquartum apicis ſphæralis, quæ ex ipſis nume-<lb/> <anchor type="note" xlink:label="note-0499-05a" xlink:href="note-0499-05"/> ris colliguntur, ſimiliter conum, FCE, duplum eſſe apicis para <lb/>bolici, triplum ſexquialterum proximè apicis ſpæralis, quoad api-<lb/>cem ſphæralem enim ſemper proximam dictam rationẽ intellige, <lb/>& </s> <s xml:id="echoid-s12285" xml:space="preserve">tandem apex parabolicus ad ſphæralem erit ſexquiſupertripar-<lb/>tiens quartas.</s> <s xml:id="echoid-s12286" xml:space="preserve"/> </p> <div xml:id="echoid-div1128" type="float" level="2" n="1"> <figure xlink:label="fig-0499-01" xlink:href="fig-0499-01a"> <image file="0499-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0499-01"/> </figure> <note position="right" xlink:label="note-0499-01" xlink:href="note-0499-01a" xml:space="preserve">Corol. 10. <lb/>51. l. 4. ſec. <lb/>poſterior.</note> <note position="right" xlink:label="note-0499-02" xlink:href="note-0499-02a" xml:space="preserve">Coroll. 1. <lb/>51. l. 4.</note> <note position="right" xlink:label="note-0499-03" xlink:href="note-0499-03a" xml:space="preserve">I. Corol. 4. <lb/>gener. 34. <lb/>l. 2.</note> <note position="right" xlink:label="note-0499-04" xlink:href="note-0499-04a" xml:space="preserve">Coroll. 8. <lb/>51. l. 4. ſe-<lb/>ctio 1.</note> <note position="right" xlink:label="note-0499-05" xlink:href="note-0499-05a" xml:space="preserve">Coroll. 11. <lb/>34. l. 3.</note> </div> </div> <div xml:id="echoid-div1130" type="section" level="1" n="677"> <head xml:id="echoid-head707" xml:space="preserve">THEOREMA XXVII. PROPOS. XXXIV.</head> <p> <s xml:id="echoid-s12287" xml:space="preserve">SI in baſi cylindri, & </s> <s xml:id="echoid-s12288" xml:space="preserve">circa eundem axim, fuerint hæ-<lb/>miſphærium, vel hæmiſphæroides, conoides parabo-<lb/>licum, hyperbolicum, & </s> <s xml:id="echoid-s12289" xml:space="preserve">conus, ſecto verò axi vtcunque, <lb/>ducatur planum per punctum ſectionis baſi æquidiſtans. <lb/></s> <s xml:id="echoid-s12290" xml:space="preserve">Abſciſſæ per ductum planum à dictis ſolidis portiones erũt <lb/>ad ſolida, à quibus abſcinduntur in ratione infraſcripta. </s> <s xml:id="echoid-s12291" xml:space="preserve"><lb/>Similiter demptis dictis ſolidis ſingillatim à cylindro, ab-<lb/>ſciſſæ per ductum planum portiones ad reſiduum cylindri, <lb/>demptis ſolidis iam dictis, erunt in ratione infraſcripta.</s> <s xml:id="echoid-s12292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12293" xml:space="preserve">Sit cylindrus, BF, in baſi circulo, DF, & </s> <s xml:id="echoid-s12294" xml:space="preserve">circa axim, AE, circa <lb/>quem in eadem baſi ſit hæmiſphærium, vel hæmiſphæroides, DV <lb/>ATF, conoides parabolicum DOARF, hyperbolicum, DNASF, <lb/>& </s> <s xml:id="echoid-s12295" xml:space="preserve">conus, DMAIF, ſumpto autẽ vtcunq; </s> <s xml:id="echoid-s12296" xml:space="preserve">puncto in, AE, quod ſit, k, <lb/>per, k, ducatur planũ, CG, baſi, DF, æquidiſtans. </s> <s xml:id="echoid-s12297" xml:space="preserve">Igitur hæmiſphę-<lb/> <anchor type="note" xlink:label="note-0499-06a" xlink:href="note-0499-06"/> riũ, vel hæmiſphæroides, DVATF, ad portionẽ, VAT, erit vt pa-<lb/>ralle lepipedũ ſub dupla, AE, & </s> <s xml:id="echoid-s12298" xml:space="preserve">quadrato, AE, ad parallelepipedum <lb/>ſub compoſita ex dupla, AE, & </s> <s xml:id="echoid-s12299" xml:space="preserve">ex, EK, & </s> <s xml:id="echoid-s12300" xml:space="preserve">@ub quadrato, kA. </s> <s xml:id="echoid-s12301" xml:space="preserve">Co-<lb/>noides parabolicum, DOARF, ad conoides, OAR, erit vt qua-<lb/> <anchor type="note" xlink:label="note-0499-07a" xlink:href="note-0499-07"/> dratum, EA, ad quadratum, AK. </s> <s xml:id="echoid-s12302" xml:space="preserve">Conoides hyperbolicum, DN <lb/>ASF, ad conoides, NAS, vt parallelepipedum ſub compoſita ex <lb/> <anchor type="note" xlink:label="note-0499-08a" xlink:href="note-0499-08"/> ſexquialtera tranſuerſi eiuldem lateris, &</s> <s xml:id="echoid-s12303" xml:space="preserve">, EA, & </s> <s xml:id="echoid-s12304" xml:space="preserve">ſub quadrato, E <pb o="480" file="0500" n="500" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0500-01a" xlink:href="fig-0500-01"/> A, ad parallelepipe-<lb/>dum ſub compoſita <lb/>ex ſexquialteraeiuſ-<lb/>dem tranſuerſi late-<lb/>ris, &</s> <s xml:id="echoid-s12305" xml:space="preserve">, KA, & </s> <s xml:id="echoid-s12306" xml:space="preserve">qua-<lb/>drato, KA. </s> <s xml:id="echoid-s12307" xml:space="preserve">Conus <lb/>verò, DAF, ad co-<lb/>num, MAI, vt cu-<lb/>bus, EA, ad cubum, <lb/>AK.</s> <s xml:id="echoid-s12308" xml:space="preserve"/> </p> <div xml:id="echoid-div1130" type="float" level="2" n="1"> <note position="right" xlink:label="note-0499-06" xlink:href="note-0499-06a" xml:space="preserve">Coroll. 7. <lb/>34. l. 3.</note> <note position="right" xlink:label="note-0499-07" xlink:href="note-0499-07a" xml:space="preserve">Coroll. 3. <lb/>51. l. 4.</note> <note position="right" xlink:label="note-0499-08" xlink:href="note-0499-08a" xml:space="preserve">Coroll. 2. <lb/>30. l. 5.</note> <figure xlink:label="fig-0500-01" xlink:href="fig-0500-01a"> <image file="0500-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0500-01"/> </figure> </div> <p> <s xml:id="echoid-s12309" xml:space="preserve">Nunc intelliga-<lb/> <anchor type="note" xlink:label="note-0500-01a" xlink:href="note-0500-01"/> tur demptum à cy-<lb/>lindro, BF, hæmi-<lb/>ſphærium, vel hæ-<lb/>miſphæroides, DVATF. </s> <s xml:id="echoid-s12310" xml:space="preserve">Igitur per demonſtrata patet reliquum <lb/>cylindri ab abſciſſam ab eo portionem per ductum planum eſſe, vt <lb/> <anchor type="note" xlink:label="note-0500-02a" xlink:href="note-0500-02"/> cubus, AE, eſt ad cubum, EK. </s> <s xml:id="echoid-s12311" xml:space="preserve">Dempto autem conoide parabo-<lb/>lico ab eodem cylindro; </s> <s xml:id="echoid-s12312" xml:space="preserve">reliquum cylindri ab abſciſſam portionẽ <lb/>erit, vt quadratum, AE, ad quadratum, EK. </s> <s xml:id="echoid-s12313" xml:space="preserve">Dempto verò co-<lb/> <anchor type="note" xlink:label="note-0500-03a" xlink:href="note-0500-03"/> noide hyperbolico ab eodem cylindro, reliquum cylindri ad ab-<lb/>ſciſſam portionem erit, vt parallelepipedum ſub compoſita ex <lb/>ſexquialtera tranſuerſi lateris, & </s> <s xml:id="echoid-s12314" xml:space="preserve">dupla axis eiuſdem, & </s> <s xml:id="echoid-s12315" xml:space="preserve">ſub qua-<lb/>drato eiuſdem axis, ad parallelepipedum ſub compoſita ex ſex-<lb/> <anchor type="note" xlink:label="note-0500-04a" xlink:href="note-0500-04"/> quialtera eiuſdem tranſuerſi lateris, & </s> <s xml:id="echoid-s12316" xml:space="preserve">axibus vtriuſq; </s> <s xml:id="echoid-s12317" xml:space="preserve">portionis, <lb/>& </s> <s xml:id="echoid-s12318" xml:space="preserve">ſub quadrato exceſſus maioris axis ſuper minorem. </s> <s xml:id="echoid-s12319" xml:space="preserve">Tandem <lb/>dempto cono, DAF, à cylindro, BF, reſiduum cylindri ad abſciſsã <lb/>portionem erit, vt cubus, AE, ad parallelepipedum ſub ſexquialte-<lb/> <anchor type="note" xlink:label="note-0500-05a" xlink:href="note-0500-05"/> ra, kE, & </s> <s xml:id="echoid-s12320" xml:space="preserve">ſub rectangulo, AKE, cum {2/3}. </s> <s xml:id="echoid-s12321" xml:space="preserve">quadrati, KE. </s> <s xml:id="echoid-s12322" xml:space="preserve">Nam cy-<lb/>lindrus, BF, ad reliquum cylindri, CF, dempto fruſto coni, DMIF, <lb/> <anchor type="note" xlink:label="note-0500-06a" xlink:href="note-0500-06"/> habet rationem compoſitam ex ea, quam habet cylindrus, BF, ad <lb/>cylindrum, CF, ideſt ex ea, quam habet, AE, ad, EK, & </s> <s xml:id="echoid-s12323" xml:space="preserve">ex ratio-<lb/> <anchor type="note" xlink:label="note-0500-07a" xlink:href="note-0500-07"/> ne cylindri, CF, ad reliquum, dempto à cylindro, CF, fruſto, DM <lb/>IF, quæ eſt ea, quam habet quadratum, DE, ad rectangulum, CM <lb/>K, cum {2/3}. </s> <s xml:id="echoid-s12324" xml:space="preserve">quadrati, CM, vel quadratum, EA, ad rectangulum, E <lb/>KA, cum {2/3}. </s> <s xml:id="echoid-s12325" xml:space="preserve">quadrati, EK, eſt autem reliquum cylindri, BF, dem-<lb/>pto cono, DAF, {2/3}. </s> <s xml:id="echoid-s12326" xml:space="preserve">eiuſdem cylindri, ergo reliquum cylindri, BF, <lb/>dempto cono, DAF, ad reliquum cylindri, CF, dempto fruſto, D <lb/>MIF, erit in ratione compoſita ex ea, quam habẽt {2/3}. </s> <s xml:id="echoid-s12327" xml:space="preserve">AE, ad, EK, <lb/>ideſt, AE, ad ſexquialteram, EK, & </s> <s xml:id="echoid-s12328" xml:space="preserve">quadratum, AE, ad rectangu-<lb/> <anchor type="note" xlink:label="note-0500-08a" xlink:href="note-0500-08"/> lum, AKE, cum {2/3}. </s> <s xml:id="echoid-s12329" xml:space="preserve">quadrati, kE, quæ duæ rationes componunt <lb/>rationem cubi, AE, ad parallelepipedum ſub ſexquialtera, Ek, &</s> <s xml:id="echoid-s12330" xml:space="preserve"> <pb o="481" file="0501" n="501" rhead="LIBER VI"/> ſub rectangulo, AkE, cum {2/3}. </s> <s xml:id="echoid-s12331" xml:space="preserve">quadrati, kE, ſic igitur erit reſiduu@ <lb/>cylindri, BF, dempto cono, DAF, ad reſiduum cylindri, CF, dem-<lb/>pto fruſto, DMIF; </s> <s xml:id="echoid-s12332" xml:space="preserve">cætera autem ex ſuis Propoſitionibus patent, <lb/>quæ explicanda proponebantur.</s> <s xml:id="echoid-s12333" xml:space="preserve"/> </p> <div xml:id="echoid-div1131" type="float" level="2" n="2"> <note position="left" xlink:label="note-0500-01" xlink:href="note-0500-01a" xml:space="preserve">F. H. <lb/>Cor. gen. <lb/>34. l. 2.</note> <note position="left" xlink:label="note-0500-02" xlink:href="note-0500-02a" xml:space="preserve">Coroll. 5. <lb/>34. l. 3.</note> <note position="left" xlink:label="note-0500-03" xlink:href="note-0500-03a" xml:space="preserve">Coroll. 9. <lb/>51. l. 4.</note> <note position="left" xlink:label="note-0500-04" xlink:href="note-0500-04a" xml:space="preserve">Coroll. 5. <lb/>30. l. 5.</note> <note position="left" xlink:label="note-0500-05" xlink:href="note-0500-05a" xml:space="preserve">Defin. 12. <lb/>l. 1.</note> <note position="left" xlink:label="note-0500-06" xlink:href="note-0500-06a" xml:space="preserve">C. Cor. 4. <lb/>gener. 34. <lb/>l. 2.</note> <note position="left" xlink:label="note-0500-07" xlink:href="note-0500-07a" xml:space="preserve">Collig. ex <lb/>L. Coroll. <lb/>4. gener. <lb/>34. l. 2.</note> <note position="left" xlink:label="note-0500-08" xlink:href="note-0500-08a" xml:space="preserve">D. G. <lb/>Cor. gen. <lb/>34. l. 2.</note> </div> </div> <div xml:id="echoid-div1133" type="section" level="1" n="678"> <head xml:id="echoid-head708" xml:space="preserve">COROLL. GENERALE.</head> <p style="it"> <s xml:id="echoid-s12334" xml:space="preserve">_L_Icet autem in ſuperioribus bu ius Libri Propoſitionibus tantum. <lb/></s> <s xml:id="echoid-s12335" xml:space="preserve">modo cylindros, conos, ſphæras, ſphæroides, conoides paraboli-<lb/>cas, & </s> <s xml:id="echoid-s12336" xml:space="preserve">byperbolicas, apices ſphærales, atque anulos, apices parabo-<lb/>licos, & </s> <s xml:id="echoid-s12337" xml:space="preserve">ſemianulos, ac cætera conſimilia ſolida fuerimus contempla-<lb/>ti, quorum omnia planà ſunt omnes figuræ ſimiles figurarum, quæ eo-<lb/>rundem genitrices appellantur, ſcilicet in his ſolidis, aſſumptis tan-<lb/>tum ſimilibus figuris, quæ ſint circuli, vel ellipſes; </s> <s xml:id="echoid-s12338" xml:space="preserve">tamen manifeſtum <lb/>eſt, ſi vice circulorum, vel ellipſium alia fuiſſent aſſumptæ ſimiles fi-<lb/>guræ, quod eadem circa talia ſolida Theoremata, vel Problemata ſi-<lb/>milia præpoſitis conſtruere potuiſſemus. </s> <s xml:id="echoid-s12339" xml:space="preserve">Vnde ex .</s> <s xml:id="echoid-s12340" xml:space="preserve">g. </s> <s xml:id="echoid-s12341" xml:space="preserve">veluti in Prop. </s> <s xml:id="echoid-s12342" xml:space="preserve"><lb/>26. </s> <s xml:id="echoid-s12343" xml:space="preserve">buius inuenimus ſphæram æqualem dato cylindro, ita ſi vice cy-<lb/>lindri habuiſſemus cylindricum, cuius baſis fuiſſet triangulum æqui-<lb/>laterum, poteramus vice ſphæræ inuenire ſolidum, cylindrico dato ſi-<lb/>milare, genitum ex circulo, cuius nempè omnia plana ſuiſſent omnes <lb/>figuræ ſimiles, ideſt omnia triangula æquilatera, circuli, qui erat ſphæ-<lb/>ræ, & </s> <s xml:id="echoid-s12344" xml:space="preserve">eſt buius ſolidi genitrix figura, & </s> <s xml:id="echoid-s12345" xml:space="preserve">eodem modo in cæteris banc <lb/>commutationem proſequi, aſſumptis quibuſcumque ſimilibus figuris <lb/>genitricium figurarum, ex quibus dicta ſolida ad inuicem ſimilaria <lb/>genita dicuntur, quam varietatem, vt & </s> <s xml:id="echoid-s12346" xml:space="preserve">alia quamplurima tum Pro-<lb/>blemata, tum Theoremata, quæ ex bactenus oftenſis deduci poſſent, <lb/>quaq; </s> <s xml:id="echoid-s12347" xml:space="preserve">Lectoris induſtria relinquuntur, cuiq; </s> <s xml:id="echoid-s12348" xml:space="preserve">licebit iuxta propoſit am <lb/>methodum facilè meditari, & </s> <s xml:id="echoid-s12349" xml:space="preserve">propterea circa bæc non amplius im-<lb/>morandum mibieſſe cenſui.</s> <s xml:id="echoid-s12350" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1134" type="section" level="1" n="679"> <head xml:id="echoid-head709" xml:space="preserve">Finis Sexti Libri.</head> <pb o="482" file="0502" n="502"/> </div> <div xml:id="echoid-div1135" type="section" level="1" n="680"> <head xml:id="echoid-head710" xml:space="preserve">GEOMETRIÆ <lb/>CAVALERII</head> <head xml:id="echoid-head711" xml:space="preserve">LIBER SEPTIMVS.</head> <head xml:id="echoid-head712" style="it" xml:space="preserve">In quo quæcumque in antecedentibus Libris me-<lb/>thodo indiuiſibilium demonſtrata fuere, <lb/>alia ratione, ab eadem independen-<lb/>te, breuiter oſtenduntur.</head> <head xml:id="echoid-head713" xml:space="preserve">PRÆFATIO.</head> <p style="it"> <s xml:id="echoid-s12351" xml:space="preserve">_G_EOMETRIAE, in ſex prioribus Libris <lb/>per eam, quam indiuiſibilium methodum <lb/>non incongruè appellamus, bactenus pro-<lb/>motę, talis fuit, qualis bucuſq; </s> <s xml:id="echoid-s12352" xml:space="preserve">videri po-<lb/>tuit, ſtructura, necnon talia, qualia iacta <lb/>ſunt fundamenta. </s> <s xml:id="echoid-s12353" xml:space="preserve">Illa quidem adeò firma, <lb/>atq; </s> <s xml:id="echoid-s12354" xml:space="preserve">inconcuſſa, eſſe decuit, vt velut ada-<lb/>mantina ſummorum ingeniorum tamquam <lb/>arietum ictrbus pulfata ne minimum quidẽ <lb/>nutantia agnoſcerentur: </s> <s xml:id="echoid-s12355" xml:space="preserve">Hoc enim Mathematicarum dignitati, ac <lb/>ſummæ certitudini, quam præ omnibus alijs humanis ſcientijs, nemi-<lb/>ne philoſopborum reclamante, ipſæ ſibi vindicarunt, maximè conue-<lb/>nire manifeſtum eſt. </s> <s xml:id="echoid-s12356" xml:space="preserve">An id ego ſufficienter præſtiterim a iorum iudi-<lb/>cio relinquam; </s> <s xml:id="echoid-s12357" xml:space="preserve">vnicuique enim hæc perlegenti ex animi ſui ſenten-<lb/>tia iudicare licebit. </s> <s xml:id="echoid-s12358" xml:space="preserve">Haud quidem me latet circa continui compoſi-<lb/>tionem, necnon circa inſinitum, plurma à philoſophis diſputari, quæ <lb/>meis principijs obeſſe non paucis ſortaſſe videbuntur, propterea nem-<lb/>pè hæſitantes i quod omnium linearum, ſeu omnium planorum, con-<lb/>ceptus cimerijs veluti obſeurior tenebris inapprehenſibilis videatur: <lb/></s> <s xml:id="echoid-s12359" xml:space="preserve">Vel quod in continui ex indiuiſibilibus compoſitionem mea ſententia <lb/>prolabatur: </s> <s xml:id="echoid-s12360" xml:space="preserve">Vel tandem quod vnum infinitum alio maius dari poſſe <lb/>pro ſirmiſſimo Geometriæ ſternere auſerimfundamento, circa quæ mil-<lb/>libus, quæ paſſim in ſcholis circumferuntur argumeutis, ne Achil- <pb o="483" file="0503" n="503" rhead="LIBER VII"/> lea quidem armareſiſtere poſſe exiſtimantur. </s> <s xml:id="echoid-s12361" xml:space="preserve">His tamen ego per ea, <lb/>quæ Lib. </s> <s xml:id="echoid-s12362" xml:space="preserve">2. </s> <s xml:id="echoid-s12363" xml:space="preserve">Prop. </s> <s xml:id="echoid-s12364" xml:space="preserve">1. </s> <s xml:id="echoid-s12365" xml:space="preserve">ac illius Scholio præcipuè declarata, ac demon-<lb/>ſtrataſunt, ſatisfieri poſſe dijudicaui: </s> <s xml:id="echoid-s12366" xml:space="preserve">quoad conceptum enim omniã <lb/>linearum, ſeu omnium planorum efformandum, facilè hoc per nega-<lb/>tionem nos conſequi poſſe exiſtimaui, ita nempè vt nulla linearum, <lb/>ſeu planorum, excludi intelligatur. </s> <s xml:id="echoid-s12367" xml:space="preserve">Quoad continui autem compo. <lb/></s> <s xml:id="echoid-s12368" xml:space="preserve">ſitionem manifeſtum eſt ex præoſtenſis ad ipſum ex indiuiſibilibus cõ-<lb/>ponendum nos minimè cogi, ſolum enim continua ſequi indiuiſibilium <lb/>proportionem, & </s> <s xml:id="echoid-s12369" xml:space="preserve">è conuersò, probare intentum ſuit, quod quidem cũ <lb/>vtraq; </s> <s xml:id="echoid-s12370" xml:space="preserve">poſitione ſtare poteſt. </s> <s xml:id="echoid-s12371" xml:space="preserve">Tandem verò dicta indiuiſibilium aggre-<lb/>gata non ita pertractauimus vt infinitatis rationem, propter infinitas <lb/>lineas, ſeu plana, ſubire videntur, ſed quatenus finitatis quandam <lb/>conditionem, & </s> <s xml:id="echoid-s12372" xml:space="preserve">naturam ſortiuntur, vt propterea, & </s> <s xml:id="echoid-s12373" xml:space="preserve">augeri, & </s> <s xml:id="echoid-s12374" xml:space="preserve"><lb/>diminui poſſint, vt ibidem oſtenſum fuit, ſi ipſa prout diffinita ſunt <lb/>accipiantur. </s> <s xml:id="echoid-s12375" xml:space="preserve">Sed his nihilominus fortè obſtrepent Philoſophi, vecla-<lb/>mabuntq; </s> <s xml:id="echoid-s12376" xml:space="preserve">Geometræ, qui puriſſimos veritatis latices ex clariſſimis <lb/>baurire fontibus conſueſcunt ſic obijcientes. </s> <s xml:id="echoid-s12377" xml:space="preserve">Hic dicendi modus ad. </s> <s xml:id="echoid-s12378" xml:space="preserve"><lb/>buc videtur ſubobſcurus, durior quam par eſt euadit hic omnium li-<lb/>nearum, ſeu omnium planorum conceptus, quapropter bun@ tuæ <lb/>Geometriæ ceu Gordium nodum aut auſeras, aut ſaltem frangas, niſi <lb/>diſſoluas. </s> <s xml:id="echoid-s12379" xml:space="preserve">Fregiſſem qu idem fateor, ò Geometræ, vel emninò à prio-<lb/>ribus Libris ſuſtuli ſſem, niſi indignum facinus mihi viſum fuiſſet no-<lb/>ua hæc Geometriæ veluti myſteria ſapientiſſimis abſcondere viris; </s> <s xml:id="echoid-s12380" xml:space="preserve">vt, <lb/>his fundamentis, quibus tot concluſionum ab alijs quoq; </s> <s xml:id="echoid-s12381" xml:space="preserve">oſtenſarum <lb/>veritates adeò mirè concordant, alicuius induſtria melius fortè con-<lb/>cinnatis, huiuſce nodi exoptatam illis diſſolutionem aliquando præ-<lb/>ſtare poſſint. </s> <s xml:id="echoid-s12382" xml:space="preserve">Interim qualiſcumq; </s> <s xml:id="echoid-s12383" xml:space="preserve">mea ſuerit illius tentata diſſolu-<lb/>tio, ipſum tamen in præſenti Libro, nouis alijs denuò ſtratis funda-<lb/>mentis, quibus ea omnia, quæ indiuiſibilium methodo in antecedenti-<lb/>bus Libris iam oſtenſa ſunt, alia ratione ab infinitatis exempta conce-<lb/>ptu comprobantur, omninò è medio tollendum eſſe cenſui. </s> <s xml:id="echoid-s12384" xml:space="preserve">Hoc verò <lb/>præcipuè à nobis factum eſt, tum vt apud eos, quibus noſtra hæc indi-<lb/>uiſibilium methodus minus probabitur, non indignè noſtram banc de <lb/>Continuis doctrinam Geometriæ titulo inſigniri clarius eluceſcat; </s> <s xml:id="echoid-s12385" xml:space="preserve">tum <lb/>etiam vt appareat, quod non leui ratione ducti, cum poſſemus cuncta <lb/>per indiuiſibilium methodum præoſtenſa, tantum per huius Libri fun-<lb/>damenta demonſtrare, illam quoque methodum tanquam nouam, ac <lb/>conſideratione dignam, fuimus preſequuti. </s> <s xml:id="echoid-s12386" xml:space="preserve">Nodum verò ipſum, cui <lb/>negotium faceſſeret, non inaniter in præcedentibus Libris relictum eſ-<lb/>ſe, quinimmo nos ipſum alicui Alexandro aut frangendum, aut iuxta <lb/>ſcrupoliſiſsimi cuiuſq; </s> <s xml:id="echoid-s12387" xml:space="preserve">Geometræ vota diſſoluendum, meritò reſer-<lb/>uaſſe, nõ ineptè quiſpiam iudicabit.</s> <s xml:id="echoid-s12388" xml:space="preserve"/> </p> <pb o="484" file="0504" n="504" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1136" type="section" level="1" n="681"> <head xml:id="echoid-head714" xml:space="preserve">THEOREMA I. PROPOS. I.</head> <p> <s xml:id="echoid-s12389" xml:space="preserve">FIguræ planæ quæcunq; </s> <s xml:id="echoid-s12390" xml:space="preserve">in eiſdem parallelis conſtitutę, <lb/>in quibus, ductis quibuſcunq; </s> <s xml:id="echoid-s12391" xml:space="preserve">eiſdem parallelis ęqui-<lb/>diſtantibus rectis lineis, conceptæ cuiuſcumq; </s> <s xml:id="echoid-s12392" xml:space="preserve">rectæ lineæ <lb/>portiones ſunt æquales, etiam inter ſe æquales erunt: </s> <s xml:id="echoid-s12393" xml:space="preserve">Et <lb/>figuræ ſolidæ quæcumq; </s> <s xml:id="echoid-s12394" xml:space="preserve">in eiſdem planis parallelis conſti-<lb/>tutæ, in quibus, ductis quibuſcunq; </s> <s xml:id="echoid-s12395" xml:space="preserve">planis eiſdem planis <lb/>parallelis æquidiſtantibus, conceptæ cuiuſcunq; </s> <s xml:id="echoid-s12396" xml:space="preserve">ſic ducti <lb/>plani in ipſis ſolidis figuræ planæ ſunt æquales, pariter in-<lb/>terſe æquales erunt. </s> <s xml:id="echoid-s12397" xml:space="preserve">Dicantur autem figuræ æqualiter <lb/>analogæ, tum planæ, tum ipſæ ſolidæ interſe comparatæ, <lb/>ac etiam iuxta regulas lineas, ſeu plana parallela, in qui-<lb/>bus eſse ſupponuntur, cum hoc fuerit opus explicare.</s> <s xml:id="echoid-s12398" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12399" xml:space="preserve">Sint quæcunq; </s> <s xml:id="echoid-s12400" xml:space="preserve">planę figuræ, BZ&</s> <s xml:id="echoid-s12401" xml:space="preserve">, CβΛ, in eiſdem parallelis, <lb/>AD, Y4, conſtitutæ, ductis autem ipſis, AD, Y4, quibuſcunque <lb/>parallelis, E6, LΣ, portione ex. </s> <s xml:id="echoid-s12402" xml:space="preserve">g. </s> <s xml:id="echoid-s12403" xml:space="preserve">ipſius, E6, in figuris conceptę, <lb/>nempè, FG, HI, inter ſe ſint æquales, necnon ipſius, LΣ, portio-<lb/>nes, MN, OP, ſimul ſumptæ (ſit enim figura, BZ&</s> <s xml:id="echoid-s12404" xml:space="preserve">, ex. </s> <s xml:id="echoid-s12405" xml:space="preserve">g. </s> <s xml:id="echoid-s12406" xml:space="preserve">intus ca-<lb/>ua ſecundum ambitum, {12/ }, N, {13/ }, O,) ipſi, SV, ſint pariter ęqua-<lb/>les, & </s> <s xml:id="echoid-s12407" xml:space="preserve">hoc contingat in quibuſcunq; </s> <s xml:id="echoid-s12408" xml:space="preserve">alijs ipſi, AD, æquidiſtanti-<lb/>bus. </s> <s xml:id="echoid-s12409" xml:space="preserve">Dico figuras, BZ&</s> <s xml:id="echoid-s12410" xml:space="preserve">, CβΛ, inter ſe æquales eſſe. </s> <s xml:id="echoid-s12411" xml:space="preserve">Aſſumpta <lb/>ergo alterutra figurarum, BZ&</s> <s xml:id="echoid-s12412" xml:space="preserve">, CβΛ, vt ipſa, BZ&</s> <s xml:id="echoid-s12413" xml:space="preserve">, cum paralle-<lb/>larum, AD, Y4, portionibus ipſi conterminantibus, nempè cum, <lb/>AB, Y&</s> <s xml:id="echoid-s12414" xml:space="preserve">, ſuperponatur reliquæ figuræ, CβΛ, ita tamen vt ipſæ, A <lb/>B, Y&</s> <s xml:id="echoid-s12415" xml:space="preserve">, cadant ſuper, BD, &</s> <s xml:id="echoid-s12416" xml:space="preserve">4, vel ergo tota, BZ& </s> <s xml:id="echoid-s12417" xml:space="preserve">congruit toti, <lb/>CβΛ, & </s> <s xml:id="echoid-s12418" xml:space="preserve">ita cum ſibi congruant æquales erunt, vel non, aliqua <lb/>tamen pars eſto, quod congruerit alicui parti, vt, CΙγβ587, pars <lb/>figuræ, BZ&</s> <s xml:id="echoid-s12419" xml:space="preserve">, ipſi, CΙγβ587, parti figuræ, CβΛ. </s> <s xml:id="echoid-s12420" xml:space="preserve">Manifeſtum eſt <lb/>autem ſuperpoſitione figurarum taliter effecta, vt portiones pa-<lb/>rallelarum, AD, Y4, ipſius figuris conterminantes ſint inuicem <lb/>ſuperpoſitæ, quod quæcumq; </s> <s xml:id="echoid-s12421" xml:space="preserve">rectæ lineæ in figuris conceptæ erãt <lb/>ſibi in Birectum, manent etiam ſibi in directum, vt ex. </s> <s xml:id="echoid-s12422" xml:space="preserve">g. </s> <s xml:id="echoid-s12423" xml:space="preserve">cum, M <lb/>N, OP, eſſent in directum ipſi, SV, dicta ſuperpoſitione facta, ma-<lb/>nent etiam ſibi in directum, nempè, QR, ST, in directum ipſi, SV, <lb/>eſt enim diſtantia ipſarum, MN, OP, ab, AD, æqualis diſtantiæ, <lb/>SV, ab eadem, AD, vnde quotieſcunque, AB, extendatur ſuper, B <lb/>D, vbicunque hoc fiat, ſemper, MN, OP, manebunt in directum <pb o="485" file="0505" n="505" rhead="LIBER VII."/> <anchor type="figure" xlink:label="fig-0505-01a" xlink:href="fig-0505-01"/> ipſi, SV, quod, & </s> <s xml:id="echoid-s12424" xml:space="preserve">de cæteris quibuſcunq; </s> <s xml:id="echoid-s12425" xml:space="preserve">ipſi, AD, parallelis in <lb/>vtraque figura liquidò apparet. </s> <s xml:id="echoid-s12426" xml:space="preserve">Quod verò pars vnius figuræ, vt, <lb/>BZ&</s> <s xml:id="echoid-s12427" xml:space="preserve">, congruat neceſſariò parti figuræ, @βΛ, & </s> <s xml:id="echoid-s12428" xml:space="preserve">non toti, dum fit <lb/>ſuperpoſitio tali lege, quali dictum eſt, ſic demonſtrabitur. </s> <s xml:id="echoid-s12429" xml:space="preserve">Cum <lb/>enim ductis quibuſcunq; </s> <s xml:id="echoid-s12430" xml:space="preserve">ipſi, AD, parallelis conceptæ in figuris <lb/>ipſarum portiones, quæ erant ſibi in directum, adhuc poſt ſuper-<lb/>poſitionem maneant ſibi in directum, illæ vero ante ſuperpoſitio-<lb/>nem effent ex hypoteſi æquales, ergo poſt ſuperpoſitionem por-<lb/>tiones parallelarum ipſi, AD, in figuris ſuperpoſitis conceptæ erũt <lb/>pariter æquales, vt ex.</s> <s xml:id="echoid-s12431" xml:space="preserve">g. </s> <s xml:id="echoid-s12432" xml:space="preserve">QR, ST, ſimul ſumptæ æquabuntur ipſi, <lb/>SV, ergo niſi vtræque, QR, ST, congruant toti, SV, congruente <lb/>parte alicui parti, vt, ST, ipſi, ST, erit, QR, æqualis ipſi, TV, &</s> <s xml:id="echoid-s12433" xml:space="preserve">, <lb/>QR, quidem erit in reſiduo figuræ, BZ&</s> <s xml:id="echoid-s12434" xml:space="preserve">, ſuperpoſitæ, TV, verò <lb/>in reſiduo ſiguræ, @βΛ, cu@ fit ſuperpoſitio. </s> <s xml:id="echoid-s12435" xml:space="preserve">Eodem modo oſten-<lb/>demus cuicunq; </s> <s xml:id="echoid-s12436" xml:space="preserve">parallelæ ipſi, AD, conceptæ in reſiduo, figuræ, B <lb/>Z&</s> <s xml:id="echoid-s12437" xml:space="preserve">, ſuperpoſitæ, quod ſit, H℟597, reſpondere in directum ęqua-<lb/>lem rectam lineam, quę erit in reſiduo figuræ, @βΛ, cui fit ſuper-<lb/>poſitio, ergo ſuperpoſitione hac lege facta, cum ſupereſt aliquid <lb/>de figura uperpoſita, quod non cadatſuper figuram, cui fitſu-<lb/>perpoſitio, neceſſe eſt reliquæ figuræ aliquid etiam ſupereſſe, ſuper <lb/>quod nihil ſit ſuperpoſitum. </s> <s xml:id="echoid-s12438" xml:space="preserve">Cum autem vnicuiq; </s> <s xml:id="echoid-s12439" xml:space="preserve">rectæ lineæ <lb/>parallelę, AD, conceptæ in reſiduo, vel reſiduis (quia poſſunt eſſe <lb/>plures figuræ reſiduæ) figuræ, BZ&</s> <s xml:id="echoid-s12440" xml:space="preserve">, ſiue, C℟γ, ſuperpoſitæ, re-<lb/>ſpondeat in directum in reſiduo, vel reſiduis ſiguræ; </s> <s xml:id="echoid-s12441" xml:space="preserve">CβΛ, alia re-<lb/>cta linea, manifeſtum eſt has reſiduas figuras, ſiue reſiduarum ag-<lb/>gregata, eſſe in eiſdem parallelis, cum ergo reſidua figura, H℟597, <pb o="486" file="0506" n="506" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0506-01a" xlink:href="fig-0506-01"/> ſit in parallelis, E6, Y4, etiam reſidua figura, vel reſiduarum ag-<lb/>gregatum, ipſius, CβΛ, (quod ſit ipſi fruſta, ΙΓΛ, 785,) erit in eiſdẽ <lb/>parallelis; </s> <s xml:id="echoid-s12442" xml:space="preserve">E6, Y4, ſi enim non pertingeret hinc inde ad parallelas, <lb/>E6, Y4, vt ex. </s> <s xml:id="echoid-s12443" xml:space="preserve">g. </s> <s xml:id="echoid-s12444" xml:space="preserve">ſi pertingeret quidem vſq; </s> <s xml:id="echoid-s12445" xml:space="preserve">ad, E6, non tamen vſq; <lb/></s> <s xml:id="echoid-s12446" xml:space="preserve">ad, Y4, ſed tantum vſque ad, LΣ, conceptis rectis lineis in fruſto, <lb/>Q℟β59R, ipſi, AD, parallelis non reſponderent in reſiduo figuræ, <lb/>CβΛ, ſeu ex reſiduis aggregato, aliæ rectæ lineæ, vt ſuperius neceſ-<lb/>ſe eſſe probatum eſt, ſunt ergo hæc reſidua, vel reſiduorum aggre-<lb/>gata in eiſdem parallelis, & </s> <s xml:id="echoid-s12447" xml:space="preserve">in illis conceptæ parallelarum ipſis, A <lb/>D, Y4, portiones inter ſe ſunt æquales, vt ſupra oſtendimus, ergo <lb/>reſidua, ſeu reſiduorum aggregata, ſunt eius conditionis, cuius ip-<lb/>ſas, BZ&</s> <s xml:id="echoid-s12448" xml:space="preserve">, CβΛ, figuras iam eſſe ſuppoſitum fuit, ideſt æqualiter <lb/>analoga. </s> <s xml:id="echoid-s12449" xml:space="preserve">Fiat ergo denuo reſiduorum ſuperpoſitio, ita tamen vt <lb/>parallelæ, GH, & </s> <s xml:id="echoid-s12450" xml:space="preserve">β, ſuper parallelas, HK, β4, ſint conſtitutæ, & </s> <s xml:id="echoid-s12451" xml:space="preserve"><lb/>congruat pars, VΔΛ, fruſti, H℟597, parti, VΔΛ, fruſti, ΙΓΛ, oſten-<lb/>demus ergo vt ſupra, dum vnius habetur reſiduum haberi etiam al-<lb/>terius, & </s> <s xml:id="echoid-s12452" xml:space="preserve">hæc reſidua, ſiue reſiduorum aggregata, eſſe in eiſdem <lb/>parallelis, ſit autem ad figuram, BZ&</s> <s xml:id="echoid-s12453" xml:space="preserve">, ſpectans reſiduum, ΚVΛ3 <lb/>ΠΧ, ad figuram autem, CβΛ, ſint pertinentia reſidua, ΙΓΔV, 785, <lb/>quorum aggregatum eſt in eiſdem parallelis cum reſiduo, ΚVΛ3 <lb/>ΠΧ, nem pè in parallelis, E6, Y4, ſi ergo horum reſiduorum fiat <lb/>denuò ſuperpoſitio, ita tamen vt parallelæ, in quibus exiſtunt, ſint <lb/>ſemper ad inuicem ſuperpoſitę, & </s> <s xml:id="echoid-s12454" xml:space="preserve">hoc ſemper fieri intelligatur, do-<lb/>nec tota figura, BZ&</s> <s xml:id="echoid-s12455" xml:space="preserve">, fuerit ſuperpoſita, dico totam debere ipſi, <lb/>CβΛ, congruere, alioquin ſi eſſet aliquod reſiduum vt figurę, CβΛ, <lb/>cui nihil eſſet ſuperpoſitum, eſſet etiam aliquod reſiduum figurę, <pb o="487" file="0507" n="507" rhead="LIBER VII."/> BZ&</s> <s xml:id="echoid-s12456" xml:space="preserve">, quod non eſſet ſuperpoſitum, vt ſupra oſtendimus neceſſe <lb/>eſſe, ponitur autem totam, BZ&</s> <s xml:id="echoid-s12457" xml:space="preserve">, eſſe ſuperpoſitam ipſi, CβΛ, er-<lb/>go ita ſunt ad inuicem ſuperpoſitę, vt neutrius reſidua habeantur, <lb/>ergo ita ſunt ſuperpoſitę, vt ſibi congruant, ergo figuræ, BZ&</s> <s xml:id="echoid-s12458" xml:space="preserve">, C <lb/>βΛ, inter ſe ſunt æquales.</s> <s xml:id="echoid-s12459" xml:space="preserve"/> </p> <div xml:id="echoid-div1136" type="float" level="2" n="1"> <figure xlink:label="fig-0505-01" xlink:href="fig-0505-01a"> <image file="0505-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0505-01"/> </figure> <figure xlink:label="fig-0506-01" xlink:href="fig-0506-01a"> <image file="0506-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0506-01"/> </figure> </div> <p> <s xml:id="echoid-s12460" xml:space="preserve">Sint nunc in eodem ſchemate duæ figuræ ſolidæ quæeunque, <lb/>BZ&</s> <s xml:id="echoid-s12461" xml:space="preserve">, CβΛ, in eiſdem planis parallelis, AD, Y4, conſſitutæ, ductis <lb/>autem quibuſcunq; </s> <s xml:id="echoid-s12462" xml:space="preserve">planis, E6, LΣ, præfatis æquidiſtantibus, ſint <lb/>conceptæ in ſolidis figuræ, quæ iacent in eodem plano, ſemper in-<lb/>ter ſe æquales, vt, FG, æqualis, HI, &</s> <s xml:id="echoid-s12463" xml:space="preserve">, MN, OP, ſimul ſumptæ <lb/>(ſit enim ſolida figura ex. </s> <s xml:id="echoid-s12464" xml:space="preserve">g. </s> <s xml:id="echoid-s12465" xml:space="preserve">BZ&</s> <s xml:id="echoid-s12466" xml:space="preserve">, intus vtcunq; </s> <s xml:id="echoid-s12467" xml:space="preserve">caua ſecundum <lb/>ſuperficiem, {12/ }, N, {13/ }, O,) æquales ipſi, SV. </s> <s xml:id="echoid-s12468" xml:space="preserve">Dico eaſdem ſolidas <lb/>figuras æquales eſſe. </s> <s xml:id="echoid-s12469" xml:space="preserve">Si enim ſolidum, BZ&</s> <s xml:id="echoid-s12470" xml:space="preserve">, cum portionibus, <lb/>ABY, & </s> <s xml:id="echoid-s12471" xml:space="preserve">planorũ, AD, Y4, ipſi conterminantibus, ſolido, CβΛ, ita <lb/>ſuperpoſuerimus, vt planum, AB, ſit in plano, AD, &</s> <s xml:id="echoid-s12472" xml:space="preserve">, Y&</s> <s xml:id="echoid-s12473" xml:space="preserve">, in pla-<lb/>no, Y4, oſtendemus (vt fecimus ſuperius circa parallelarum ipſi, A <lb/>D, conceptas in figuris planis, BZ&</s> <s xml:id="echoid-s12474" xml:space="preserve">, CβΛ, portiones) figuras in <lb/>ſolidis, BZ&</s> <s xml:id="echoid-s12475" xml:space="preserve">, CβΛ, conceptas, quæ erant in eodem plano, etiam <lb/>poſt ſuperpoſitionem manere in eode plano, & </s> <s xml:id="echoid-s12476" xml:space="preserve">ideò adhuc ęqua-<lb/>les eſſe figuras in ſuperpoſitis ſolidis conceptas, & </s> <s xml:id="echoid-s12477" xml:space="preserve">ipſis, AD, Y4, <lb/>parallelas. </s> <s xml:id="echoid-s12478" xml:space="preserve">Niſi ergo totum ſolidum toti congruat in prima ſu-<lb/>perpoſitione, relinquentur reſidua ſolida, vel ex reſiduis compoſi-<lb/>ta in vtroq; </s> <s xml:id="echoid-s12479" xml:space="preserve">ſolido, quæ non erunt ad inuicem ſuperpoſita, cum <lb/>enim ex. </s> <s xml:id="echoid-s12480" xml:space="preserve">g. </s> <s xml:id="echoid-s12481" xml:space="preserve">figuræ, QR, ST, æquentur figuræ, SV, dempta com-<lb/>muni figura, ST, reliqua, QR, æquabitur reliquæ, TV, hocq; </s> <s xml:id="echoid-s12482" xml:space="preserve">cõ-<lb/>tinget in quouis alio plano ipſi, AD, parallelo occurrenteſolidis, <lb/>C℟Γ, CβΛ, ergo ſemper habentes reſiduum vnius ſolidi, habebimus <lb/>etiam reſiduum alterius, & </s> <s xml:id="echoid-s12483" xml:space="preserve">patebit, iuxta methodum adhibitam in <lb/>priori parte huius Propoſitionis circa figuras planas, reſidua ſoli-<lb/>da, vel reſiduorum aggregata ſemper eſſe in eiſdem parallelis pla-<lb/>nis, vt reſidua, H℟597, ΙΓΛ, 785, eſſe in planis parallelis, E6, Y4, <lb/>ac æqualiter analoga: </s> <s xml:id="echoid-s12484" xml:space="preserve">ſi ergo hæc reſidua adhuc ſuperponantur, <lb/>ita vt planum, EH, locetur in plano, H6, &</s> <s xml:id="echoid-s12485" xml:space="preserve">, Υβ, in, β4, & </s> <s xml:id="echoid-s12486" xml:space="preserve">hoc <lb/>ſempei fieri intelligatur, donec quod ſuperponitur, vt, BZ&</s> <s xml:id="echoid-s12487" xml:space="preserve">, totũ <lb/>fuerit aſſumptum, tandem ipſum totum, BZ&</s> <s xml:id="echoid-s12488" xml:space="preserve">, congruet toti, Cβ <lb/>Λ, niſi enim toto ſolido, BZ&</s> <s xml:id="echoid-s12489" xml:space="preserve">, ipſi, CβΛ, ſuperpoſito, @pſa ſibi cõ-<lb/>gruerent, eſſet aliquod reſiduum vnius, vt ſolidi, CβΛ, ergo etiam <lb/>eſſet aliquod reſiduum ſolidi, C℟Γ, ſeu, BZ&</s> <s xml:id="echoid-s12490" xml:space="preserve">, illudq; </s> <s xml:id="echoid-s12491" xml:space="preserve">non eſſet ſu-<lb/>perpoſitum, quod eſt abſurdum, ponitur enim iam totum ſolidũ, <lb/>BZ&</s> <s xml:id="echoid-s12492" xml:space="preserve">, eſſe ipſi, CβΛ, ſuperpoſitum, non ergo erit aliquod reſiduũ <lb/>in ipſisſolidis, ergo ſibi congruent, ergo dictæ figuræ ſolidæ, BZ&</s> <s xml:id="echoid-s12493" xml:space="preserve">, <lb/>CβΛ, inter ſe æquales erunt, quæ fuerunt demonſtranda. </s> <s xml:id="echoid-s12494" xml:space="preserve">Præfatę <pb o="488" file="0508" n="508" rhead="GEOMETRIÆ"/> autem figuræ, vt ſupra innuimus, dicatur æqualiter analogæ, & </s> <s xml:id="echoid-s12495" xml:space="preserve">ſi <lb/>opus erit, iuxta regulas lineas parallelas, ſeu plana parallela, AD, <lb/>Y4.</s> <s xml:id="echoid-s12496" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1138" type="section" level="1" n="682"> <head xml:id="echoid-head715" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s12497" xml:space="preserve">_C_Vm antecedens Prop. </s> <s xml:id="echoid-s12498" xml:space="preserve">maximi ſit momenti, vt in ſequentibus <lb/>apparebit, aliuſq; </s> <s xml:id="echoid-s12499" xml:space="preserve">modus priorem partem demonſtrandi, ſtylo <lb/>Archimedeo haud abſimilis, menti ſuccurrerit, idipſum ne pereatin <lb/>Lemmata diſtributum hic ſubiungere placuit.</s> <s xml:id="echoid-s12500" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1139" type="section" level="1" n="683"> <head xml:id="echoid-head716" xml:space="preserve">LEMMA PRIMVM.</head> <p> <s xml:id="echoid-s12501" xml:space="preserve">SI in eadem, vel æqualibus baſibus, & </s> <s xml:id="echoid-s12502" xml:space="preserve">in eiſdem paral-<lb/>lelis figuræ planę æqualiter analogę iuxta eaſdem ba-<lb/>ſes fuerint conſtitutæ, itatamen, vt quæcunq; </s> <s xml:id="echoid-s12503" xml:space="preserve">æquidiſtã-<lb/>tium baſibus linearum portiones in eiſdem conceptæ figu-<lb/>ris integræ ſint, ac eidem baſi, vel baſibus æquales, ipſæ <lb/>pariter figuræ inter ſe æquales erunt.</s> <s xml:id="echoid-s12504" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12505" xml:space="preserve">Sint in eadem baſi, GH, ſeu in æqualibus baſibus, & </s> <s xml:id="echoid-s12506" xml:space="preserve">in eiſdem <lb/>parallelis, AF, PQ, figuræ planæ, AGHB, EGHF, æqualiter ana-<lb/> <anchor type="figure" xlink:label="fig-0508-01a" xlink:href="fig-0508-01"/> logę iux-<lb/>ta eandẽ <lb/>baſem, G <lb/>H, ſeu ba <lb/>ſes ęqua-<lb/>les iã di-<lb/>ctas, ex-<lb/>tenſa ve-<lb/>ro qua-<lb/>cunq; </s> <s xml:id="echoid-s12507" xml:space="preserve">ip-<lb/>ſis, PQ, A <lb/>F, parallela, SR, eiuſdem portiones captæ in præfatis figuris, vt, <lb/>ST, NO, integræ ſint, ac æquales baſi, GH, ſeu dictis æqualibus <lb/>baſibus. </s> <s xml:id="echoid-s12508" xml:space="preserve">Dico etiam præfatas figuras inter ſe æquales eſſe. </s> <s xml:id="echoid-s12509" xml:space="preserve">In ea-<lb/>dem enim baſi, GH, ſeu in altera dictarum æqualium baſium ſit <lb/>conſtitutum, & </s> <s xml:id="echoid-s12510" xml:space="preserve">in eiſdem parallelis, AF, PQ, quodcunq; </s> <s xml:id="echoid-s12511" xml:space="preserve">paralle-<lb/>logrammum, CH, in quo portio concepta ipſius, SR, ſit, LM, quę <lb/>erit æqualis ipſi, GH, & </s> <s xml:id="echoid-s12512" xml:space="preserve">conſequenter ipſi, NO, vnde addita cõ-<lb/>muni, MN, fiet, LN, æqualis, MO. </s> <s xml:id="echoid-s12513" xml:space="preserve">Eodem modo autem oſten-<lb/>demus, CE, eſſe æqualem, DE, & </s> <s xml:id="echoid-s12514" xml:space="preserve">reliquas huiu ſmodi ſimiliter <pb o="489" file="0509" n="509" rhead="LIBER VII."/> adæquari. </s> <s xml:id="echoid-s12515" xml:space="preserve">Nunc aſſumpto trilineo, ECG, & </s> <s xml:id="echoid-s12516" xml:space="preserve">poſito, C, in, D, &</s> <s xml:id="echoid-s12517" xml:space="preserve">, <lb/>CG, in, DH, cadet, G, in, H, quia, CG, DH, ſunt æquales, ca-<lb/>dente verò trilineo, ECG, ſuper, FDH, extendetur, CE, ſuper, D <lb/>F, cum angulus, FDH, exterior fit æqualis interiori, ECG; </s> <s xml:id="echoid-s12518" xml:space="preserve">paral-<lb/>lelarum, DH, CG, & </s> <s xml:id="echoid-s12519" xml:space="preserve">punctum, E, erit in, F, ambituſque, ENG, <lb/>cadet ſuper ambitum, FOH, ſi enim non, eſto quod aliquod pun-<lb/>ctum ambitus, ENG, non cadat ſuper, FOH, cadet ergo, vel extra <lb/>trilineum, FDH, vel intra, cadat extra, vt in, R, ita vt ambitus, E <lb/>NGH, cadat vt, FRH, erit ergo, MR, maior, MO, ſed, MR, eſt <lb/>æqualis, LN, ergo, LN, erit maior, MO, ſed eſt etiam æqualis ei-<lb/>dem, MO, ex demonſtratis, ergo eſſet æqualis, & </s> <s xml:id="echoid-s12520" xml:space="preserve">maior eadem, <lb/>MO, quod eſt abſurdum, non ergo aliquod punctum ambitus, EN <lb/>G, cadit extra trilineum, FDH, eodem modo probabitur, nec ca-<lb/>dere intra eundem trilineum, ergo ambitus, ENG, cadet ſuper am-<lb/>bitum, FOH, congruens totus toti, & </s> <s xml:id="echoid-s12521" xml:space="preserve">conſequenter etiam trili-<lb/>neus, ECG, congruet trilineo, FDH, & </s> <s xml:id="echoid-s12522" xml:space="preserve">illi æqualis erit, vnde abla-<lb/>to communi trilineo, DIE, & </s> <s xml:id="echoid-s12523" xml:space="preserve">addito communi trilineo, GIH, fiet, <lb/>EGHF, figura æqualis parallelogrammo, CH. </s> <s xml:id="echoid-s12524" xml:space="preserve">Eodem modo <lb/>oſtendemus figuram, AGHB, æquari eidem, CH, ergo figuræ, A <lb/>GHB, EGHF, inter ſe æquales erunt. </s> <s xml:id="echoid-s12525" xml:space="preserve">Cum autem dictæ figuræ <lb/>fuerint in æqualibus baſibus, tum conſtituentes ſuper vnamquãq; <lb/></s> <s xml:id="echoid-s12526" xml:space="preserve">parallelogrammum in eiſdem parallelis cum ijidem poſitum, con-<lb/>cludemus etiam dictas figuras æquales eſſe, probantes eodem mo-<lb/>do deſcriptis parallelogrammis adæquari, quę quidem inter ſe erũt <lb/>æqualia, quod demonſtrare opus erat. </s> <s xml:id="echoid-s12527" xml:space="preserve">Hæcautem vocentur pa-<lb/>rallelogramma curuilinea, cum, AG, BH, EG, FH, fuerint curuæ <lb/>lineæ, cum verò fuerint rectæ lineæ, parallelogramma rectilinea <lb/>ad illorum differentiam eadem appeliabimus, ſed vtraq; </s> <s xml:id="echoid-s12528" xml:space="preserve">in gene-<lb/>re, ſi libuerit, nomine parallelogrammi tantum ctiam nuncupa-<lb/>bimus.</s> <s xml:id="echoid-s12529" xml:space="preserve"/> </p> <div xml:id="echoid-div1139" type="float" level="2" n="1"> <figure xlink:label="fig-0508-01" xlink:href="fig-0508-01a"> <image file="0508-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0508-01"/> </figure> </div> </div> <div xml:id="echoid-div1141" type="section" level="1" n="684"> <head xml:id="echoid-head717" xml:space="preserve">LEMMA II.</head> <p> <s xml:id="echoid-s12530" xml:space="preserve">SI in æqualibus rectis lineis, tamqũam in baſibus, & </s> <s xml:id="echoid-s12531" xml:space="preserve">iu <lb/>eiſdem parallelis, fuerint quæcunq; </s> <s xml:id="echoid-s12532" xml:space="preserve">planæ figuræ, æ-<lb/>qualiter analogæ iuxta dictas baſes; </s> <s xml:id="echoid-s12533" xml:space="preserve">portiones autem æ-<lb/>quidiſtantium quotcunq; </s> <s xml:id="echoid-s12534" xml:space="preserve">ipſis baſibus linearum in figuris <lb/>conceptæ integræ fuerint, ac in altera dictarum figurarum <lb/>ſic ſe habentes, vt quælibet propinquior baſi ſit maior re-<lb/>motiori, dictæ figuræ interſe æquales erunt.</s> <s xml:id="echoid-s12535" xml:space="preserve"/> </p> <pb o="490" file="0510" n="510" rhead="GEOMETRIÆ"/> <figure> <image file="0510-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0510-01"/> </figure> <p> <s xml:id="echoid-s12536" xml:space="preserve">+</s> </p> <p> <s xml:id="echoid-s12537" xml:space="preserve">Sint in æqualibus rectis lineis, QP, TY, tamquam in baſibus, <lb/>& </s> <s xml:id="echoid-s12538" xml:space="preserve">in eiſdem parallelis, AL, QY, quæcunq; </s> <s xml:id="echoid-s12539" xml:space="preserve">planæ figuræ CQPD, <lb/>HTYL, æqualiter analogæ iuxta dictas baſes, QP, ΓΥ, ductis au-<lb/>tem quotcunq; </s> <s xml:id="echoid-s12540" xml:space="preserve">baſibus parallelis, vt, ΛΧ, ΓV, βΚ, earum in figu-<lb/>ris conceptæ portiones integræ ſint, ac in altera figurarum pro-<lb/>pinquior baſi maior remotiori, vt ſi conceptæ in, CQPD, ſint, N <lb/>℟, I&</s> <s xml:id="echoid-s12541" xml:space="preserve">, EZ, & </s> <s xml:id="echoid-s12542" xml:space="preserve">in, HTYL, ipſæ, SX, RV, OK, iſtæ quidem integræ <lb/>ſint necnon ex. </s> <s xml:id="echoid-s12543" xml:space="preserve">g. </s> <s xml:id="echoid-s12544" xml:space="preserve">in figura, CQPD, N℟, maior, I&</s> <s xml:id="echoid-s12545" xml:space="preserve">; I&</s> <s xml:id="echoid-s12546" xml:space="preserve">, maior, <lb/>EZ, & </s> <s xml:id="echoid-s12547" xml:space="preserve">ſic in cæteris (erit enim etiam, SX, maior, RV, &</s> <s xml:id="echoid-s12548" xml:space="preserve">, RV, ma-<lb/>ior, OK, & </s> <s xml:id="echoid-s12549" xml:space="preserve">ſic in cæteris, cum ſint æqualiter analogæ iuxta ba-<lb/>ſes, QP, TY.) </s> <s xml:id="echoid-s12550" xml:space="preserve">Dico figuras, CQPD, HTYL, inter ſe æquales eſ-<lb/>ſe. </s> <s xml:id="echoid-s12551" xml:space="preserve">Si enim non ſint æquales, altera earum maior erit, ſit maior, <lb/>HTYL, ipſa, CQPD, ſpatio, +, tunc minoris figuræ baſis, QP, <lb/> <anchor type="note" xlink:label="note-0510-01a" xlink:href="note-0510-01"/> moueatur verſus, AD, ſemper ipſi, AD, æquidiſtanter, ac manẽ-<lb/>te iugiter puncto, P, in linea, PD, donec congruat ipſi, AD, igitur <lb/>punctum, Q, deſcribet lineam, QΓΑ, &</s> <s xml:id="echoid-s12552" xml:space="preserve">, QP, deſcribet parallelo-<lb/>grammum, AP, rectilineum, ſeu curuilineum, prout, AQ, DP, <lb/>fuerint rectæ, vel curuæ, erit autem, CΓΑ, tota extra figuram, CQ <lb/>PD, cum parallelæ, QP, in figura, CQPD, ipſi, QP, propinquio-<lb/>res remotioribus ſint ſemper maiores (quo pacto data baſi, & </s> <s xml:id="echoid-s12553" xml:space="preserve">cur-<lb/>ua linea, tota in eodem plano cum ipſa baſi, ac vni extremorum <lb/>eiuſdem conterminante, parallelogrammum curuilineum, ab ij@dẽ <lb/>apprehenium, deſcribere docemur) ſimiliter compleatur paralle-<lb/>logrammum, FY, ducaturquæ, ΓV, parallela, QY, b fariam diui-<lb/>dens altitud nem figurari m, CQPD, HTYL, reſpectu, QY, aſſum-<lb/>ptam, ſecanſque, AQ, in, Γ, CQ, in, I, DP, in, &</s> <s xml:id="echoid-s12554" xml:space="preserve">, FT, in, Π, HT, <lb/>in, R, &</s> <s xml:id="echoid-s12555" xml:space="preserve">, LY, in, V, per, ΓV, igitur diuidetur parallelogrammum, <pb o="491" file="0511" n="511" rhead="LIBER VII."/> AP, in æqualia parallelogramma, A &</s> <s xml:id="echoid-s12556" xml:space="preserve">, &</s> <s xml:id="echoid-s12557" xml:space="preserve">Q: </s> <s xml:id="echoid-s12558" xml:space="preserve">rurſus autem per <lb/> <anchor type="note" xlink:label="note-0511-01a" xlink:href="note-0511-01"/> alias ipſi, QY, parallelas diuidantur dictæ altitudinis portiones bi-<lb/>fariam, & </s> <s xml:id="echoid-s12559" xml:space="preserve">ſic ſemper ſiat (ſectis inſimul conſtitutis parallelogram-<lb/>mi@, quæ idcircò etiam bifariam diuidentur) donec ad parallelo-<lb/>grammum, vt ad, ℟Q, deueniatur minus ſpatio, +, ſit igitur ſe-<lb/> <anchor type="note" xlink:label="note-0511-02a" xlink:href="note-0511-02"/> ctum, AP, in parallelogramma, æquè alta, AZ, β&</s> <s xml:id="echoid-s12560" xml:space="preserve">, γ℟, Δ Ρ, per <lb/>ęquidiſtantes lineas, βκ, γV, ΔΧ, quæ ſecent lineas, AQ, in pun-<lb/>ctis, β, Γ, Δ, CQ, in, E, I, N, DP, in, Z, &</s> <s xml:id="echoid-s12561" xml:space="preserve">, ℟, FT, in, ΛΠΣ, HT, <lb/>in, O, R, S, & </s> <s xml:id="echoid-s12562" xml:space="preserve">tandem, LY, in, K, V, X, compleanturq; </s> <s xml:id="echoid-s12563" xml:space="preserve">paralle-<lb/>logramma, BZ, 2&</s> <s xml:id="echoid-s12564" xml:space="preserve">, 3℟, ΔΡ, iuxta deſcriptionem ſuperius tradi-<lb/>tam, erunt enim lineæ, BE.</s> <s xml:id="echoid-s12565" xml:space="preserve">, 2I, 3N, ΔQ, extra figuram, CQPD, <lb/>quod patebit, veluti, AQ, extra, CQPD, fimiliter cadere oſten-<lb/>ſa eſt, & </s> <s xml:id="echoid-s12566" xml:space="preserve">conſequeuter figura ex parallelogrammis, BZ, 2&</s> <s xml:id="echoid-s12567" xml:space="preserve">, 3℟, Δ <lb/>P, compoſita comprehendet ſpatium, CQPD, ſint autem etiam <lb/>completa parallelogramma, E&</s> <s xml:id="echoid-s12568" xml:space="preserve">, Ι℟, NP, quorum deſcriptæ li-<lb/>neæ, ΕΦ, ΙΩ, NM, intra figuram, CQPD, quidem cadere oſtende-<lb/>mus ex eadem ratione, quod dictæ parallelæ ipſi, PQ, propinquio-<lb/>res remotioribus ſint ſemper maiores, & </s> <s xml:id="echoid-s12569" xml:space="preserve">ſubinde patebit figuram <lb/>ex parallelogrammis, E&</s> <s xml:id="echoid-s12570" xml:space="preserve">, Ι℟, NP, compoſitam comprehendi à <lb/>figura, CQPD. </s> <s xml:id="echoid-s12571" xml:space="preserve">Tandem compleantur parallelogramma quoque, <lb/>Gκ, 6V, 9X, ΣΥ, ex quibus compoſitam figuram ſpatium, HTYL, <lb/>eadem methodo comprehendere demonſtrabimus. </s> <s xml:id="echoid-s12572" xml:space="preserve">Cum ergo fi-<lb/>gura comprehendens ſpatium, CQPD, ſuperet ab eo compreh en-<lb/>ſam parallelogrammis, BZ, 2Φ, 3Ω, ΔΜ, hoc eſt parallelogram-<lb/>mo, ΔΡ, quod eſt minus ſpatio, +, dicta comprehendens figura <lb/>ſuperabit, CQPD, muitò minori ſpatio, quam ſit, +, ſed, HTY <lb/>L, ſuperat, CQPD, ex hypoteſi ſpatio, +, ergo figura compre-<lb/>hendens, CQPD, minor eſt, HTYL, & </s> <s xml:id="echoid-s12573" xml:space="preserve">multò minor figura ipſum, <lb/>HTYL, comprehendente, quæ iam deſcripta fuit, hoc autem eſt <lb/> <anchor type="note" xlink:label="note-0511-03a" xlink:href="note-0511-03"/> abſurdum, cum enim paralſelogràmmum, BZ, æquetur ipſi, GK, 2 <lb/>&</s> <s xml:id="echoid-s12574" xml:space="preserve">, 6V, 3℟, 9X, &</s> <s xml:id="echoid-s12575" xml:space="preserve">, ΔΡ, ΣΥ, tota toti adæquatur contra præde-<lb/>monſtrata, non ergo figura, HTYL, maior eſt, CQPD.</s> <s xml:id="echoid-s12576" xml:space="preserve"/> </p> <div xml:id="echoid-div1141" type="float" level="2" n="1"> <note position="left" xlink:label="note-0510-01" xlink:href="note-0510-01a" xml:space="preserve">Poſt. 1. <lb/>l. 2.</note> <note position="right" xlink:label="note-0511-01" xlink:href="note-0511-01a" xml:space="preserve">Elicit. ex <lb/>ant. Lem.</note> <note position="right" xlink:label="note-0511-02" xlink:href="note-0511-02a" xml:space="preserve">1. Deoim@ <lb/>Elem.</note> <note position="right" xlink:label="note-0511-03" xlink:href="note-0511-03a" xml:space="preserve">Ex antec. <lb/>Lem.</note> </div> <p> <s xml:id="echoid-s12577" xml:space="preserve">Sit nunc eadem minor, ſi poſſibile eſt, eodem ſpatio, +, igitur <lb/>deſcriptis circa, CQPD, eiſdem figuris, ita vt comprehendens, CQ <lb/>PD, ſuperet ab eo comprehenſam minori ſpatio, quam ſit, +, cõ-<lb/>pleantur parallelogramma, OV, RX, SY, ex quibus compoſitam <lb/>figuram, vt ſupra à ſpatio, HTYL, comprehendi oſtendemus. </s> <s xml:id="echoid-s12578" xml:space="preserve">Igi-<lb/>tur ſi comprehendens, CQPD, ſuperat figuram comprehenſam <lb/>minori ſpatio, quam ſit, +, ipſum ſpatium, CQPD, ſuperabit ab <lb/>eo comprehenſam figuram multò minori ſpatio, quam ſit, +, idẽ <lb/>autem ſuperat, HTYL, ſpatio, +, ergo figura comprehenſa à <pb o="492" file="0512" n="512" rhead="GEOMETRIÆ"/> ſpatio, CQPD, maior erit ſpatio, HTYL, & </s> <s xml:id="echoid-s12579" xml:space="preserve">multò maior erit figu-<lb/>ra iam deſcripta, ab eodem ſpatio, HTYL, compi ehenſa, quod eſt <lb/> <anchor type="note" xlink:label="note-0512-01a" xlink:href="note-0512-01"/> abſurdum, cum enim parallelogiammum, E&</s> <s xml:id="echoid-s12580" xml:space="preserve">, æquetur, OV, Ι℟, <lb/>ipſi, RX, necnon, NP, ipſi, SY, tota toti adæquator contra præ-<lb/>demonſtrata, nec ergo figura, HTYL, minor eſſe poteſt figura, C <lb/>QPD, ſed neque eadem maior, vt oſtenſum eſt, ergo eidem æqua-<lb/>lis etit, quod demonſtrare oportebat. </s> <s xml:id="echoid-s12581" xml:space="preserve">Vnamquamque autem di-<lb/>ctarum figurarum, CQPD, HTYL, pręfatas conditiones haben-<lb/>tium, figuram in alteram partem deficientem appellabimus, regu-<lb/>la baſi, ſeu quacunq; </s> <s xml:id="echoid-s12582" xml:space="preserve">illi ęquidiſtante.</s> <s xml:id="echoid-s12583" xml:space="preserve"/> </p> <div xml:id="echoid-div1142" type="float" level="2" n="2"> <note position="left" xlink:label="note-0512-01" xlink:href="note-0512-01a" xml:space="preserve">Ex antec. <lb/>Lem.</note> </div> </div> <div xml:id="echoid-div1144" type="section" level="1" n="685"> <head xml:id="echoid-head718" xml:space="preserve">LEMMA III.</head> <p> <s xml:id="echoid-s12584" xml:space="preserve">SI curua linea quæcunq; </s> <s xml:id="echoid-s12585" xml:space="preserve">tota ſit in eodem plano, cuioc-<lb/>currat recta in duobus punctis, aut rectis lineis, vel in <lb/>recta, & </s> <s xml:id="echoid-s12586" xml:space="preserve">puncto, poterimus aliam rectam lineam præfatæ <lb/>æquidiſtantem ducere, quæ tangat portionem curuæ lineæ <lb/>inter duos predictos occurſus continuatam.</s> <s xml:id="echoid-s12587" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1145" type="section" level="1" n="686"> <head xml:id="echoid-head719" xml:space="preserve">DEFINITIO. +</head> <p style="it"> <s xml:id="echoid-s12588" xml:space="preserve">_T_Angere autem dico rectam lineam aliam quamcunque curuam <lb/>totam in eodem plano cum ea exiſtantem, cum ipſa recta linea <lb/>ſiue in puncto, ſiue in recta linea, euruæ, occurrente, eadem curua vel <lb/>tota eſt ad eandem partem, vel illius nihil eſt ad alteram partem illi <lb/>occurrentis rectæ lineæ.</s> <s xml:id="echoid-s12589" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12590" xml:space="preserve">Sit curua linea, BAC, tota in eodem exiſtens plano, cui recta, B <lb/>C, occurrat in duobus punctis, ſeu rectis lineis, vel in recta, & </s> <s xml:id="echoid-s12591" xml:space="preserve">pun-<lb/> <anchor type="figure" xlink:label="fig-0512-01a" xlink:href="fig-0512-01"/> cto, B, C. </s> <s xml:id="echoid-s12592" xml:space="preserve">Dico nos aliam rectam <lb/>ipſi, BC, æquidiſtantem ducere poſ-<lb/>ſe, quæ tangat portionem curuæ li-<lb/>neæ inter duos occurſus, B, C, con-<lb/>tinuatam. </s> <s xml:id="echoid-s12593" xml:space="preserve">Quoniam ergo recta eſt, <lb/>BC, & </s> <s xml:id="echoid-s12594" xml:space="preserve">curua, BAC, ideò inter ſe ſpa-<lb/>tium comprehendent, figuramque, <lb/>vt, BAC, conſtituent, ergo poſſibi-<lb/>le erit figuræ, BAC, reſpectu rectæ, BC, verticem inuenire, ſit is <lb/> <anchor type="note" xlink:label="note-0512-02a" xlink:href="note-0512-02"/> punctum, A, per quod ducatur, DF, parallela, BC, igitur, BF, tan-<lb/>get figuram, BAC, ergo totus ambitus, BAC, eſt ad eandem par- <pb o="493" file="0513" n="513" rhead="LIBER VII."/> tem rectæ, DF, vel nihil eſt ſaltem ad alteram partem, ſi enim ali-<lb/>qua illius portio eſſet ad alteram partem rectæ, DF, iam recta, D <lb/>F, ſecaret figuram, BAC, quod eſt abſurdum, ergorecta, DF, tan-<lb/>git curuam, BAC, igitur poſſibile eſt, &</s> <s xml:id="echoid-s12595" xml:space="preserve">c.</s> <s xml:id="echoid-s12596" xml:space="preserve"/> </p> <div xml:id="echoid-div1145" type="float" level="2" n="1"> <figure xlink:label="fig-0512-01" xlink:href="fig-0512-01a"> <image file="0512-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0512-01"/> </figure> <note position="left" xlink:label="note-0512-02" xlink:href="note-0512-02a" xml:space="preserve">1. lib. 7.</note> </div> </div> <div xml:id="echoid-div1147" type="section" level="1" n="687"> <head xml:id="echoid-head720" xml:space="preserve">COROLL ARIV M.</head> <p style="it"> <s xml:id="echoid-s12597" xml:space="preserve">_H_Inc manifeſtum eſt quomodo ducenda ſit recta linea datam cur-<lb/>uam totam in eodem plano cum ea exiſtentem contingens, quæ <lb/>quidem data recta linea ſit æquidiſtans.</s> <s xml:id="echoid-s12598" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1148" type="section" level="1" n="688"> <head xml:id="echoid-head721" xml:space="preserve">LEMMA IV.</head> <p> <s xml:id="echoid-s12599" xml:space="preserve">SI propoſita quæcumque figura plana vni regulæ paral-<lb/>lelis quotcumque lineis ita ſecari poſſit, vt conceptæ <lb/>in figura rectæ lineæ integræ ſemper exiſtant: </s> <s xml:id="echoid-s12600" xml:space="preserve">Ipſa ex pa-<lb/>rallelogrammis rectilineis, aut curuilineis, ſeu ex figuris <lb/>in alteram partem deficientibus, regula eadem, compo-<lb/>netur.</s> <s xml:id="echoid-s12601" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12602" xml:space="preserve">Sit quæcumque figura plana, SPFR, talis tamen, vt ſecta quot-<lb/>cumq; </s> <s xml:id="echoid-s12603" xml:space="preserve">vni regulæ, vt, FE, parallelis, conceptæ in ipſa rectæ lineę <lb/>integræ ſint. </s> <s xml:id="echoid-s12604" xml:space="preserve">Dico ipſam, velex parallelogrammis rectilineis, aut <lb/> <anchor type="figure" xlink:label="fig-0513-01a" xlink:href="fig-0513-01"/> curuilineis, vel ex figuris in al-<lb/>teram partem deficientibus, <lb/>reg. </s> <s xml:id="echoid-s12605" xml:space="preserve">eadem, FE, componi. <lb/></s> <s xml:id="echoid-s12606" xml:space="preserve">Sint enim ductæ, SA, FE, op-<lb/>poſitæ tangentes figuræ, SPF <lb/>R, regula eadem, FE, quibus <lb/>incidat quomodocumq; </s> <s xml:id="echoid-s12607" xml:space="preserve">recta <lb/>linea, AE, moueatur autem, F <lb/> <anchor type="note" xlink:label="note-0513-01a" xlink:href="note-0513-01"/> E, verſus, SA, ſemperæquidi-<lb/>ſtanter eidem, SA, donec illi <lb/>congruat, interim verò punctum, E, ita in ipſa feratur, vt deſcri-<lb/>bat lineam, ENA, cum, AE, figuram, ANE, comprehenden@em, <lb/>quæ eidem, SPFR, ſit æqualiter analoga iuxta regulam, FE; </s> <s xml:id="echoid-s12608" xml:space="preserve">in <lb/>eadem integris exiſtentibus parallelis ipſi, FE, ad ambitum, ANE, <lb/>terminantibus: </s> <s xml:id="echoid-s12609" xml:space="preserve">rurſus feratur recta linea, AE, verſus ambitum, A <lb/>NE, ſemper ipſi, AE, æquidiſtanter donec totam pertranſierit fi-<lb/>guram, ANE, adnotentur autem contactus lineæ ſic decurrentis <pb o="494" file="0514" n="514" rhead="GEOMETRIÆ"/> in ambitu, ANE, vel enim tanget in linea, aut lineis, vel in pun-<lb/>ctis, & </s> <s xml:id="echoid-s12610" xml:space="preserve">lineis, vel tantum in punctis, eſto quod fiat contactus in re-<lb/>cta, LM, & </s> <s xml:id="echoid-s12611" xml:space="preserve">in puncto, N, tranſeantq; </s> <s xml:id="echoid-s12612" xml:space="preserve">per puncta, L, M, N, rectæ <lb/>lineæ regulæ, FE, parallelæ, HD, QC, PB, ſecantes ambitum fi-<lb/>guræ, SPFR, in punctis, P, Q, H, I, O, R, & </s> <s xml:id="echoid-s12613" xml:space="preserve">rectam, AE, in pun-<lb/> <anchor type="figure" xlink:label="fig-0514-01a" xlink:href="fig-0514-01"/> ctis, B, C, D, nulluſque alius <lb/>factus fuerit contactus in am-<lb/>bitu, ANMLE. </s> <s xml:id="echoid-s12614" xml:space="preserve">Quoniam er-<lb/>go a puncto, N, ad, A, nullus <lb/>datur contactus, erit, ANB, fi-<lb/>gura in alteram partem, nem-<lb/>pè verſus, A, deficiens, hoc eſt <lb/>quælibet in figura, ANB, pa-<lb/>rallela, NB, erit maior remo-<lb/>tiori, ſi enim non, eſto quod <lb/>aliqua vt, YT, non ſit maior remotiori, XV, ad ambitum termina-<lb/>ta, vel ergo erit illi æqualis, vel eadem maior, ſit illi æqualis, & </s> <s xml:id="echoid-s12615" xml:space="preserve"><lb/>iungatur, YX, hæc ergo erit paralſela, AE, & </s> <s xml:id="echoid-s12616" xml:space="preserve">occurrit ambitui in <lb/>duobus punctis, Y, X, ergo poſſibile erit ducere rectam lineam ipſi, <lb/>YX, ſeu, AE, parallelam, tangentem portionem curuæ lineæ, hoc <lb/> <anchor type="note" xlink:label="note-0514-01a" xlink:href="note-0514-01"/> eſt ambitus, AN, inter duos occurſus, Y, X, continuatam, quod <lb/>eſt contra ſuppoſitum: </s> <s xml:id="echoid-s12617" xml:space="preserve">quod ſi dicatur, YT, eſſe minorem, XV, <lb/>multò magis conuincetur præfatum abſurdum, ergo, YT, erit ma-<lb/>ior, XV, & </s> <s xml:id="echoid-s12618" xml:space="preserve">quælibet, NB, propinquior remotiore maior, ergo fi-<lb/>gura, ANB, erit in alteram partem deficiens: </s> <s xml:id="echoid-s12619" xml:space="preserve">eodem modo autem <lb/>oſtendemus etiam, NMCB, LED, eſſe figuras in alteram partem <lb/>deficientes, LMCD, autem manifeſtum eſt eſſe parallelogrammũ <lb/>rectilineum, ergo in figura, SPFR, ipſa, SPR, quæ eſt æqualiter <lb/>analoga ipſi, ANB, erit figura in alteram partem deficiens, ſic etiã, <lb/>PQOR, HFI, QHIO, verò erit paralle@ @grammum rectilineum, <lb/>ſeu curuilineum, prout, QH, OI, rectæ, vel curuæ, eſſe poſſunt, <lb/>ergo figura, SPFR, componitur ex figuris in alteram partem defi-<lb/>cientibus, ac ex parallelogrammo rectilineo, ſeu curuilineo, re-<lb/>gula, FE, quod oſtendere opus erat.</s> <s xml:id="echoid-s12620" xml:space="preserve"/> </p> <div xml:id="echoid-div1148" type="float" level="2" n="1"> <figure xlink:label="fig-0513-01" xlink:href="fig-0513-01a"> <image file="0513-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0513-01"/> </figure> <note position="right" xlink:label="note-0513-01" xlink:href="note-0513-01a" xml:space="preserve">1. lib. 1</note> <figure xlink:label="fig-0514-01" xlink:href="fig-0514-01a"> <image file="0514-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0514-01"/> </figure> <note position="left" xlink:label="note-0514-01" xlink:href="note-0514-01a" xml:space="preserve">Ex antec. <lb/>Lem.</note> </div> </div> <div xml:id="echoid-div1150" type="section" level="1" n="689"> <head xml:id="echoid-head722" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s12621" xml:space="preserve">_H_Inc habetur figuram, SPFR, ipſi, ANE, æqualem eſſe, & </s> <s xml:id="echoid-s12622" xml:space="preserve">vni-<lb/>uerſaliter figuras planas æqualit er analogas, in quibus earum <lb/>regula æquidiſtantium quotcunq; </s> <s xml:id="echoid-s12623" xml:space="preserve">linearum concepta portiones inte-<lb/>graſunt, inter ſe æquales eſſe.</s> <s xml:id="echoid-s12624" xml:space="preserve"/> </p> <pb o="495" file="0515" n="515" rhead="LIBER VII."/> <p> <s xml:id="echoid-s12625" xml:space="preserve">PRopoſit. </s> <s xml:id="echoid-s12626" xml:space="preserve">antacedens, aliter, quoad priorem partem, <lb/>oſtenſa.</s> <s xml:id="echoid-s12627" xml:space="preserve"/> </p> <figure> <image file="0515-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0515-01"/> </figure> <p> <s xml:id="echoid-s12628" xml:space="preserve">Sint quęcũq; </s> <s xml:id="echoid-s12629" xml:space="preserve">figurę planę ęqualiter analogę iuxta regulã, GM, <lb/>ipſę, BHIC, DQK, quarum oppoſitę tangentes, AF, GM, regula pa-<lb/>riter, GM, parallelarũ, autẽ ipſi, GM, quotcumque portiones in vna-<lb/>que dictarum figurarum integrę ſint, ſiue non. </s> <s xml:id="echoid-s12630" xml:space="preserve">Dico eaſdem ęqua-<lb/>les eſſe. </s> <s xml:id="echoid-s12631" xml:space="preserve">Incidat ergo parallelis, AF, GM, quomodocumque recta <lb/>linea, EL, in eiſdem terminata, moueatur autem, GM, verſus, A <lb/>F, ſemper eidem, AF, ęquidiſtanter donec illi congruerit, interim <lb/>autem vnum punctum moueatur in eadem, GM, ſic mota, deſcri-<lb/>bens ambitum, I βΕ, figurę ęqualiter analogę ipſi, DQK, & </s> <s xml:id="echoid-s12632" xml:space="preserve">aliud <lb/>punctum in eadem motum ad aliam partem, EL, deſcribat ambi-<lb/>turn figurę, EYL, ęqualiter a@ alogę ipſi, BHIC, in quibus quidem <lb/>ſic deſcriptis figuris conceptę ipſi, GM, parallelarum portiones <lb/>quęcumque integrę ſint. </s> <s xml:id="echoid-s12633" xml:space="preserve">Erit ergo figura, @ βL, ęqualis figurę, E <lb/>YL, eſto autem quod in figura, DQK, conceptę portiones paral-<lb/>lelarum ipſi, GM, non omnes ſint integrę, ſed aliquę fractę per in-<lb/>teriorem ambitum, nempè, quę intercipiuntur parallelis, Q6, ΦΩ, <lb/> <anchor type="note" xlink:label="note-0515-01a" xlink:href="note-0515-01"/> in quibus habeantur duo figurę fruſta, 67RΩ, CΦR, in quorum <lb/>tamen vnoquoque dictę parallelarum port ones integrę habean-<lb/>tur, ſit autem in motu, GM, a quodam runcto deſcripta linea, <lb/>& </s> <s xml:id="echoid-s12634" xml:space="preserve">γ5, nempè ambitus figurę, 5 & </s> <s xml:id="echoid-s12635" xml:space="preserve">ΣΤ, eodem modo, quo delcripti <lb/>fuerunt ambitus, @β@, EYL, g@ręinquam, 5&</s> <s xml:id="echoid-s12636" xml:space="preserve">Σ @, ęqualiter ana-<lb/>logę fruſto, 7R Ω@; </s> <s xml:id="echoid-s12637" xml:space="preserve">er tergorel@qu@ figura, 5Δ&</s> <s xml:id="echoid-s12638" xml:space="preserve">, qualiter analo-<lb/>ga ſrulto, QΦ@, cumtota, Τ ΔΣ ſit toti compoſito ex fruſti, QΦ <lb/>R, 7R Ω6, ęqualiter analoga, & </s> <s xml:id="echoid-s12639" xml:space="preserve">ſunt portiones ipſi, GM, paralle- <pb o="496" file="0516" n="516" rhead="GEOMETRIE"/> <anchor type="figure" xlink:label="fig-0516-01a" xlink:href="fig-0516-01"/> laru n in vnaquaq; </s> <s xml:id="echoid-s12640" xml:space="preserve">figura, 5Δ&</s> <s xml:id="echoid-s12641" xml:space="preserve">, 5&</s> <s xml:id="echoid-s12642" xml:space="preserve">ΣΤ, integræ omnes, ſicut <lb/>contingere ſuppoſuimus in fruſtis, QΦR, 7R Ω6, ergo cum, QΦR, <lb/> <anchor type="note" xlink:label="note-0516-01a" xlink:href="note-0516-01"/> 5Δ&</s> <s xml:id="echoid-s12643" xml:space="preserve">, ſint figuræ etiam æqualiter analogæ, inter ſe æquales erunt: <lb/></s> <s xml:id="echoid-s12644" xml:space="preserve">Eadem ratione patebit fruſtum, 7RΩ6, æquari figuræ, S&</s> <s xml:id="echoid-s12645" xml:space="preserve">ΣΤ, er-<lb/>go fruſta, QΦR, 7RΩ6, ſimul ſumpta æquabuntur figuræ, Τ5βΔΣ, <lb/>ſed & </s> <s xml:id="echoid-s12646" xml:space="preserve">figuram, 76D, ipſi, EST, adæquari, necnon, ΦΚΩ, ipſi, ΔL <lb/> <anchor type="note" xlink:label="note-0516-02a" xlink:href="note-0516-02"/> Σ, pariter adęquari manifeſtum eſt, cum ſint figuræ æqualiter ana-<lb/>logæ, & </s> <s xml:id="echoid-s12647" xml:space="preserve">portiones parallelarum ipſi, GM, in eiſdem conceptarũ <lb/>integræ ſint, ergo tota figura, DQK, toti, ΕβL, æqualis erit. </s> <s xml:id="echoid-s12648" xml:space="preserve">Cõ-<lb/>ſimili modo in figura, BHIC, ducentes rectas lineas ipſi, GM, pa-<lb/>rallelas, nempè, O2, P3, quibusipſa diſtinguatur in fruſta, capien-<lb/>tia dictas parallelarum portiones integras ſcilicet in fruſta, BON, <lb/>CN2, PH4, 4I3, OP32, PH4, 4I3, eaſdem, O2, P3, producentes <lb/>vt ſecent ambitum figuræ, EYL, velutin, T, X, ℟Υ, deſcriptiſq; <lb/></s> <s xml:id="echoid-s12649" xml:space="preserve">lineis, EV, ZL, vt fuit deſcripta, 5Γ&</s> <s xml:id="echoid-s12650" xml:space="preserve">, vt conſtituatur figura@, ET <lb/> <anchor type="note" xlink:label="note-0516-03a" xlink:href="note-0516-03"/> V, æqualiter analoga fruſto, CN2, (ex quo remanet, EVX, æqua-<lb/>liter analoga ipſi, BON,) & </s> <s xml:id="echoid-s12651" xml:space="preserve">figura, Ζ℟L, ęqualiter analoga ipſi, <lb/>4I3. </s> <s xml:id="echoid-s12652" xml:space="preserve">(ex quo, ZLY, remanet etiam æqualiter analoga ipſi, PH4,) <lb/>cum in his captę parallelarum dictæ portiones integræ ſint, mani-<lb/>feſtum erit fig. </s> <s xml:id="echoid-s12653" xml:space="preserve">ETV, æquari ipſi, CN2, EVX, ipſi, BON, Ζ℟L, <lb/>ipſi, 4I3, ZLY, ipſi, PH4, & </s> <s xml:id="echoid-s12654" xml:space="preserve">tandem, ΧΤ℟Υ, ipſi, OP32, ex quo <lb/>concludemus figuram, BHIC, æquari ipſi, EYL, hoc eſt ipſi, ΕβL, <lb/>ſed eidem,, ΕβL, oſtenſa eſt æqualis etiam, DQK, ergo figuræ, B <lb/>HIC, DQK, inter ſe æquales erunt, igitur quæcumq; </s> <s xml:id="echoid-s12655" xml:space="preserve">planæ figu-<lb/>ræ æqualiter analogæ inter ſe æquales erunt, quod oſtendendum <lb/>erat. </s> <s xml:id="echoid-s12656" xml:space="preserve">Per hæc autem priori parti Propoſ. </s> <s xml:id="echoid-s12657" xml:space="preserve">1. </s> <s xml:id="echoid-s12658" xml:space="preserve">huius iam ſatisfactum <lb/>eſſe manifeſtum eſt.</s> <s xml:id="echoid-s12659" xml:space="preserve"/> </p> <div xml:id="echoid-div1150" type="float" level="2" n="1"> <note position="right" xlink:label="note-0515-01" xlink:href="note-0515-01a" xml:space="preserve">Ex antec. <lb/>Lem.</note> <figure xlink:label="fig-0516-01" xlink:href="fig-0516-01a"> <image file="0516-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0516-01"/> </figure> <note position="left" xlink:label="note-0516-01" xlink:href="note-0516-01a" xml:space="preserve">En antec. <lb/>Lem.</note> <note position="left" xlink:label="note-0516-02" xlink:href="note-0516-02a" xml:space="preserve">Ex antec. <lb/>Lem.</note> <note position="left" xlink:label="note-0516-03" xlink:href="note-0516-03a" xml:space="preserve">Ex autec. <lb/>Lem.</note> </div> <pb o="497" file="0517" n="517" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1152" type="section" level="1" n="690"> <head xml:id="echoid-head723" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s12660" xml:space="preserve">FIguræ planæ quæcumq; </s> <s xml:id="echoid-s12661" xml:space="preserve">in eiſdem parallelis conſtitu-<lb/>tæ, in quibus, ductis quibuſcumq; </s> <s xml:id="echoid-s12662" xml:space="preserve">eiſdem parallelis <lb/>æquidiſta t@bus rectis lineis, conceptæ cuiuſcumq; </s> <s xml:id="echoid-s12663" xml:space="preserve">rectæ <lb/>lineæ portiones ſunt inter ſe, vt cuiuſlibet alterius in eiſdẽ <lb/>figuris conceptæ portiones (homologis tamen in eadem <lb/>figura ſemper exiſtentibus) eandem inter ſe proportionem <lb/>habebunt, quam dictæ portiones. </s> <s xml:id="echoid-s12664" xml:space="preserve">Dicantur autem pro-<lb/>portionaliter analogæ, ac etiam, ſi libuerit, iuxta regulas <lb/>ipſas parallelas, in quibus exiſtunt.</s> <s xml:id="echoid-s12665" xml:space="preserve"/> </p> <figure> <image file="0517-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0517-01"/> </figure> <p> <s xml:id="echoid-s12666" xml:space="preserve">Sint duæ quælibet figuræ planæ, Β&</s> <s xml:id="echoid-s12667" xml:space="preserve">℟ΚΓΔ, CΦλ, inter paralle-<lb/>las AD, ΧΩ, conſtitutæ, ducta vero vtcumq; </s> <s xml:id="echoid-s12668" xml:space="preserve">EQ, prædictis paral-<lb/>lela, eiuſdem portiones in figura, Β&</s> <s xml:id="echoid-s12669" xml:space="preserve">Δ, conceptæ, quæ ſint, HI, L <lb/>M, ſimul ſumptæ ſint ad eam, ſeu ad eas, quæ concipiuntur in fi-<lb/>gura, CΦλ, vt aliæ quælibet ſimiliter ſumptæ, nempè ex. </s> <s xml:id="echoid-s12670" xml:space="preserve">g. </s> <s xml:id="echoid-s12671" xml:space="preserve">vt, & </s> <s xml:id="echoid-s12672" xml:space="preserve"><lb/>℟ΓΔ, ad, Φλ. </s> <s xml:id="echoid-s12673" xml:space="preserve">Dica figuram, Β&</s> <s xml:id="echoid-s12674" xml:space="preserve">℟κΓΔ, ad figuram, CΦλ, eſſe vt, <lb/>HI, LM, ad, NO, velvt, &</s> <s xml:id="echoid-s12675" xml:space="preserve">℟, ΓΔ, ad, Φλ, vel vt quęlibet aliæ ſi-<lb/>militer ſumptæ. </s> <s xml:id="echoid-s12676" xml:space="preserve">Accipiantur in, Φλ, producta verſus, λ, quotcũq; <lb/></s> <s xml:id="echoid-s12677" xml:space="preserve">eidem, Φλ, æquales, vt; </s> <s xml:id="echoid-s12678" xml:space="preserve">λ2, ſimiliter quælibet linearum figuræ, CΦ <lb/>λ, producatur, & </s> <s xml:id="echoid-s12679" xml:space="preserve">in ipſa intelligantur tot aſſumptæ æquales vni-<lb/>cuiq; </s> <s xml:id="echoid-s12680" xml:space="preserve">productarum, quot aſſumptæ ſunt æquales ipſi, Φλ, ex.</s> <s xml:id="echoid-s12681" xml:space="preserve">g. </s> <s xml:id="echoid-s12682" xml:space="preserve">vni- <pb o="498" file="0518" n="518" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0518-01a" xlink:href="fig-0518-01"/> ca tantum, & </s> <s xml:id="echoid-s12683" xml:space="preserve">per omnium terminos ex parte, 2, tranſeat linea, C <lb/>P2, ſimiliter in alia figura, Β&</s> <s xml:id="echoid-s12684" xml:space="preserve">Δ, ſumantur quotcumq; </s> <s xml:id="echoid-s12685" xml:space="preserve">in ipſa, Δ&</s> <s xml:id="echoid-s12686" xml:space="preserve">, <lb/>producta verſus, &</s> <s xml:id="echoid-s12687" xml:space="preserve">, æquales ipſis, &</s> <s xml:id="echoid-s12688" xml:space="preserve">℟ΓΔ, fimul ſumptis, & </s> <s xml:id="echoid-s12689" xml:space="preserve">pro-<lb/>ductis reliquis in fig. </s> <s xml:id="echoid-s12690" xml:space="preserve">Β&</s> <s xml:id="echoid-s12691" xml:space="preserve">Δ, ipſi, &</s> <s xml:id="echoid-s12692" xml:space="preserve">Δ, parallelis, aliæ tot æquales <lb/>ſuis productis in directum capiantur, per quorum omnium termi-<lb/>nos tranſeant lineæ, BGZ, BFY. </s> <s xml:id="echoid-s12693" xml:space="preserve">Quoniam ergo figurę, BFYZG, <lb/>BGZ&</s> <s xml:id="echoid-s12694" xml:space="preserve">H, Β&</s> <s xml:id="echoid-s12695" xml:space="preserve">℟κΓΔ, ſunt in eiſdem parallelis, AD, ΧΩ; </s> <s xml:id="echoid-s12696" xml:space="preserve">& </s> <s xml:id="echoid-s12697" xml:space="preserve">ductis <lb/>in eiſdem quomodocumq; </s> <s xml:id="echoid-s12698" xml:space="preserve">ipſis, AD, ΧΩ, parallelis, interceptæ in <lb/>figuris portiones ſunt æquales, ideò ip@æ figuræ, BYZ, BZ&</s> <s xml:id="echoid-s12699" xml:space="preserve">, Β& </s> <s xml:id="echoid-s12700" xml:space="preserve"><lb/>℟κΓΔ, æqualiter analogæ, & </s> <s xml:id="echoid-s12701" xml:space="preserve">ſubinde æquales, erunt: </s> <s xml:id="echoid-s12702" xml:space="preserve">Quo pa-<lb/> <anchor type="note" xlink:label="note-0518-01a" xlink:href="note-0518-01"/> cto etiam oſtendemus figuras, Φ λ, λC2, æquales eſſe: </s> <s xml:id="echoid-s12703" xml:space="preserve">Quotu-<lb/>plex ergo eſt aggregatum ex, Υ℟, ΓΔ, aggregati ex, &</s> <s xml:id="echoid-s12704" xml:space="preserve">℟, ΓΔ, to-<lb/>tuplex erit aggregatum ex figuris, BYZ, BZ&</s> <s xml:id="echoid-s12705" xml:space="preserve">, Β&</s> <s xml:id="echoid-s12706" xml:space="preserve">℟κΓΔ, ſeu figu-<lb/>ra, ΒΥ℟κ @Δ, figuræ, Β&</s> <s xml:id="echoid-s12707" xml:space="preserve">℟κΓΔ; </s> <s xml:id="echoid-s12708" xml:space="preserve">ſimiliter quotuplex erit, Φ2, ipſi-<lb/>us, φλ, totuplex erit aggregatum ex figuris, Cφλ, Cλ2, hoc eſt figu-<lb/>ra, CΦ2, ipſius figuræ, CΦλ, habemus ergo æquè multiplices pri-<lb/>mæ, & </s> <s xml:id="echoid-s12709" xml:space="preserve">tertiæ vtcumq; </s> <s xml:id="echoid-s12710" xml:space="preserve">aſſumptas, ſimiliter & </s> <s xml:id="echoid-s12711" xml:space="preserve">æquè multiplices ſe-<lb/>cundæ, & </s> <s xml:id="echoid-s12712" xml:space="preserve">quartæ. </s> <s xml:id="echoid-s12713" xml:space="preserve">Quoniam verò ex. </s> <s xml:id="echoid-s12714" xml:space="preserve">g. </s> <s xml:id="echoid-s12715" xml:space="preserve">Υ℟, ΓΔ; </s> <s xml:id="echoid-s12716" xml:space="preserve">FI, LM, ſunt <lb/>æquè multiplices ipſarum, &</s> <s xml:id="echoid-s12717" xml:space="preserve">℟, ΓΔ; </s> <s xml:id="echoid-s12718" xml:space="preserve">HI, LM, ſimiliter, 2Φ, PN, <lb/>ſunt æquè multiplices ipſarum, Φλ, NO, ipſę verò, &</s> <s xml:id="echoid-s12719" xml:space="preserve">℟, ΓΔ, HI, <lb/>LM, φλ, NO, ſunt proportionales, ideò ſi aggregatum ex, Υ℟, ΓΔ, <lb/>adæquabitur ipſi, φ2, etiam aggregatum ex, FI, LM, adæquabitur <lb/>ipſi, NP, vt & </s> <s xml:id="echoid-s12720" xml:space="preserve">reliquæ omnes ſimiliter ſumptæ, & </s> <s xml:id="echoid-s12721" xml:space="preserve">conſequenter <lb/> <anchor type="note" xlink:label="note-0518-02a" xlink:href="note-0518-02"/> etiam figura, ΒΥ℟κΓΔ, adæquabitur figuræ, Cφ2, ſi verò aggre-<lb/>gutum ex, Υ℟, ΓΔ; </s> <s xml:id="echoid-s12722" xml:space="preserve">ſuperet, φ2, eodem modo patebit figuram, ΒΥ <lb/>℟κΓΔ, ſuperare figuram, Cφ2, vel ſuperari ab eadem, ſi, Υ℟, ΓΔ;</s> <s xml:id="echoid-s12723" xml:space="preserve"> <pb o="499" file="0519" n="519" rhead="LIBER VII."/> ſupereretur à, φ2, ergo prima ad ſecundam erit, vt tertia ad quar-<lb/>tam. </s> <s xml:id="echoid-s12724" xml:space="preserve">f. </s> <s xml:id="echoid-s12725" xml:space="preserve">figura, Β&</s> <s xml:id="echoid-s12726" xml:space="preserve">℟ΚΓΔ, ad figuram, Cφλ, erit, vt aggregatum <lb/>ex, &</s> <s xml:id="echoid-s12727" xml:space="preserve">℟, ΓΔ; </s> <s xml:id="echoid-s12728" xml:space="preserve">ad, φλ, vel vt aggregatum ex, HI, LM, ad, NO, ſeu <lb/>vt quælibet aliæ duæ ſimiliter ſumptę, quod erat oſtendendum, <lb/>Dicantur autem dictę figuræ proportionaliter analogę iuxta regu-<lb/>lam, AD, vel, ΧΩ.</s> <s xml:id="echoid-s12729" xml:space="preserve"/> </p> <div xml:id="echoid-div1152" type="float" level="2" n="1"> <figure xlink:label="fig-0518-01" xlink:href="fig-0518-01a"> <image file="0518-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0518-01"/> </figure> <note position="left" xlink:label="note-0518-01" xlink:href="note-0518-01a" xml:space="preserve">Per ant.</note> <note position="left" xlink:label="note-0518-02" xlink:href="note-0518-02a" xml:space="preserve">Ex ant.</note> </div> </div> <div xml:id="echoid-div1154" type="section" level="1" n="691"> <head xml:id="echoid-head724" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s12730" xml:space="preserve">FIguræ ſolidæ quæcumq; </s> <s xml:id="echoid-s12731" xml:space="preserve">in eiſdem planis parallelis <lb/>conſtitutæ, in quibus ductis quibuſcumque planis di-<lb/>ctis parallelis æquidiſtantibus, coneeptæ cuiuſcumq; </s> <s xml:id="echoid-s12732" xml:space="preserve">ſic <lb/>ducti plani in ipſis ſolidis figurę planę ſunt inter ſe, vt eiuſ-<lb/>modi cuiuſlibet alterius plani in eiſdem ſolis conceptæ ſi-<lb/>guræ (homologis tamen in eodem ſolido ſemper exiſter ti-<lb/>bus) eandem inter ſe, quam dictæ iam-conceptæ cuiuſcũq; <lb/></s> <s xml:id="echoid-s12733" xml:space="preserve">plani figuræ, rationem habebunt. </s> <s xml:id="echoid-s12734" xml:space="preserve">Dicanrur autem figurę <lb/>proportionaliter analogæ, iuxta regulas ipſa plana paral-<lb/>lela, in quibus exiſtunt.</s> <s xml:id="echoid-s12735" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12736" xml:space="preserve">Sint duę quelibet fig. </s> <s xml:id="echoid-s12737" xml:space="preserve">ſolidę, AMEGF, PQRY, in eiſdem planis <lb/>parallelis conſtitutę; </s> <s xml:id="echoid-s12738" xml:space="preserve">ductis verò quibuſcumq; </s> <s xml:id="echoid-s12739" xml:space="preserve">planis præfatis pa-<lb/>rallelis ęquidiſtantibus, eorum conceptę, in ſolidis figurę ſint vnius <lb/>plani ex. </s> <s xml:id="echoid-s12740" xml:space="preserve">g. </s> <s xml:id="echoid-s12741" xml:space="preserve">figuræ, NSTV, ΖΩΔ, alteriusautem, MEGF, QRY, <lb/>vel contingat has eſſe ſolidorum baſes, ac in altero planorum pa-<lb/>rallelorum, ſolida, AMEGF, PQRY, contingentium, ſit verò figu-<lb/>ra, MEGF, ad figuram, QRY, vt figura, NSTV, ad figuram, ΖΩ <lb/>Δ, homologis nempè in eodem ſolido exiſtentibus. </s> <s xml:id="echoid-s12742" xml:space="preserve">Dico ſolidum, <lb/>AMEGF, ad ſolidum, PQRY, eſſe vt, NSTV, figura, ad figuram, <lb/>ΖΩΔ, vel vt figura, MEGF, ad figuram, QRY. </s> <s xml:id="echoid-s12743" xml:space="preserve">Ducatur enim in <lb/>figura, MEGF, vtcumq; </s> <s xml:id="echoid-s12744" xml:space="preserve">recta, EF, ad illius ambitum terminata, <lb/>cui ducta parallela, SV, in figura, NSTV, producantur ambæ in-<lb/>definitè verſus puncta, S, E, in quibus ſumantur vtcũq; </s> <s xml:id="echoid-s12745" xml:space="preserve">ęquè mul-<lb/>tiplices, BS, CE, ſimiliter in eiſdem figuris ductis ali js eiſ dem, SV. <lb/></s> <s xml:id="echoid-s12746" xml:space="preserve">EF, ęquidiſtantibus, ſumãtur earum pariter ęquè, multiplices iux-<lb/>ta prędictarum multiplicitatem, & </s> <s xml:id="echoid-s12747" xml:space="preserve">omnium termini ſint in lineis, <lb/>NBT, MICHG, ſicut ipſarum partium termini ſint in lineis, NST, <lb/>NOT, NBT, MEG, MDG, MCG, traductis verò alijs quotcumq; </s> <s xml:id="echoid-s12748" xml:space="preserve"><lb/>planis pręfatis parallelis, ac ipſa ſolida ſecantibus, hoc idem fiat <lb/>circa ipſorum figuras in ipſis ſolidis conceptas, omnium verò ita <pb o="500" file="0520" n="520" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0520-01a" xlink:href="fig-0520-01"/> reſultantium figurarum termini ſint in ſuperficiebus, AMCG, A <lb/>MDG; </s> <s xml:id="echoid-s12749" xml:space="preserve">AMEG ſimiliter in alio ſolido eſto quod plana, quę produ-<lb/>xerũt in ſolido, AMEGF, figur. </s> <s xml:id="echoid-s12750" xml:space="preserve">MEGF, NSTV, genuerint figuras, <lb/>QRY, ΖλΩ, ad quas illæ habent eandem rationem, ductis autem, <lb/>vel aſſumptis rectis, QY, ΖΔ, inter ſe parallelis, illæ producantur <lb/>verſus eandem partem, ΔΥ, in ijſq; </s> <s xml:id="echoid-s12751" xml:space="preserve">productis accipiantur quæcũq; <lb/></s> <s xml:id="echoid-s12752" xml:space="preserve">æquè multiplices, vel æquales, YX, Δ℟, & </s> <s xml:id="echoid-s12753" xml:space="preserve">idem fiat in cæteris <lb/>ipſis parallelis in figuris, QRY, ΖΩΔ, ſic productis, & </s> <s xml:id="echoid-s12754" xml:space="preserve">omnium <lb/>termini ſint in lineis, YXR, Δ℟Ω, hæ verò lineæ, ſicut & </s> <s xml:id="echoid-s12755" xml:space="preserve">reliqua-<lb/>rum figurarum eodem modo producibilium, ſint in ſuperficiebus, <lb/> <anchor type="note" xlink:label="note-0520-01a" xlink:href="note-0520-01"/> PYR, PYXR. </s> <s xml:id="echoid-s12756" xml:space="preserve">Manifeſtum eſt autem figuras, MEGF, MDGE, M <lb/>CGD, eſſe æqualiter analogas, & </s> <s xml:id="echoid-s12757" xml:space="preserve">ideò inter ſe æquales, ſicut etiã <lb/>figuræ, NSTV, NOTS, NBTO, pariter inter ſe ſunt æquales, & </s> <s xml:id="echoid-s12758" xml:space="preserve"><lb/>quecunq; </s> <s xml:id="echoid-s12759" xml:space="preserve">aliæ ſunt in eodem plano, ex quo habemus etiam ſoli-<lb/>da, AMEGF, AMDGE, AMCGD, eſſe æqualiter analoga, & </s> <s xml:id="echoid-s12760" xml:space="preserve">ideò <lb/>interſe æqualia. </s> <s xml:id="echoid-s12761" xml:space="preserve">Eodem modo oſtendemus ſolida, PQRY, PRX <lb/>Y, pariter inter ſe æqualia eſſe. </s> <s xml:id="echoid-s12762" xml:space="preserve">Quotupiex eſt ergo ſolidum, AM <lb/>CGF, extribus, AMCGD, AMDGE, AMEGF, compoſitum, to-<lb/>tuplex eſt figura, MCGF, ex tribus, MCGD, MDGE, MEGF, cõ-<lb/>poſita, figuræ, MEGF. </s> <s xml:id="echoid-s12763" xml:space="preserve">Similiter quotuplex eſt ſolidum, PQRX, <lb/>ex duobus, PQRY, PYRX, compoſitum ipſius, PQRY, totuplex <lb/>eſt baſis, QRX, ex duabus, QRY, YRX, compoſita, fig. </s> <s xml:id="echoid-s12764" xml:space="preserve">QRY; </s> <s xml:id="echoid-s12765" xml:space="preserve">ita <lb/>vt habeamus æquè multiplices primæ, & </s> <s xml:id="echoid-s12766" xml:space="preserve">tertiæ, necnon ſecundę, <lb/>& </s> <s xml:id="echoid-s12767" xml:space="preserve">quartę magnitudinis. </s> <s xml:id="echoid-s12768" xml:space="preserve">Cum autem figuræ, FMCG, VNBT, <lb/>ſint æquè multiplices figurarum, MEGF, NSTV, & </s> <s xml:id="echoid-s12769" xml:space="preserve">pariter figu-<lb/>ræ, QRX, ΖΩ℟, ſint æquè multiplices figurarum, QRY, ΖΩΔ, <pb o="501" file="0521" n="521" rhead="LIBER VII."/> ipſæ verò figuræ, MEGF, QRY, NSTV, ΖΩΔ, ſint proportiona-<lb/> <anchor type="note" xlink:label="note-0521-01a" xlink:href="note-0521-01"/> les, & </s> <s xml:id="echoid-s12770" xml:space="preserve">homologæ, MEGF, NSTV, ideò ſi figura, MCGF, fuerit <lb/>æqualis figuræ, QRX, etiam figura, NBTV, erit æqualis figuræ, <lb/>ΖΩ℟, & </s> <s xml:id="echoid-s12771" xml:space="preserve">quælibet alia in ſolido, AMCGF, ſibi reſpondenti in alio <lb/>ſolido, PQRX, vnde & </s> <s xml:id="echoid-s12772" xml:space="preserve">ſolidum, AMCGF, æquabitur ſolido, PQ <lb/>RX. </s> <s xml:id="echoid-s12773" xml:space="preserve">Et ſi figura, MCGF, ſuperauerit figuram, QRX, eodem <lb/> <anchor type="note" xlink:label="note-0521-02a" xlink:href="note-0521-02"/> modo oſtendemus, quod ſolidum, AMCGF, ſuperabit ſolidum, P <lb/>QRX, & </s> <s xml:id="echoid-s12774" xml:space="preserve">ſi illa ſuperabitur, etiam hoc ſuperabitur, ergo prima ad <lb/>ſecundam erit, vt tertia ad quartam, hoc eſt ſolidum, AMEGF, ad <lb/>ſolidum, PQRY, erit vt figura, MEGF, ad figuram, QRY, vel vt <lb/>figura, NSTV, ad figuram, ΖΩΔ, vel vt alia quælibet eiuſmodi <lb/> <anchor type="note" xlink:label="note-0521-03a" xlink:href="note-0521-03"/> in ſolido, AMEGF, ad ſibi reipo identem in alio ſolido, PQRY, <lb/>hoc eſt ad exiſtentem in eodem cum ipſa plano quod oſtendere o-<lb/>erat. </s> <s xml:id="echoid-s12775" xml:space="preserve">Dicantur autem figuræ proportionaliter analogæ, iuxta re-<lb/>gulas, MEGF, QRY.</s> <s xml:id="echoid-s12776" xml:space="preserve"/> </p> <div xml:id="echoid-div1154" type="float" level="2" n="1"> <figure xlink:label="fig-0520-01" xlink:href="fig-0520-01a"> <image file="0520-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0520-01"/> </figure> <note position="left" xlink:label="note-0520-01" xlink:href="note-0520-01a" xml:space="preserve">Ex antec.</note> <note position="right" xlink:label="note-0521-01" xlink:href="note-0521-01a" xml:space="preserve">Conuerſ. <lb/>Deſin. 4. <lb/>Qui, El.</note> <note position="right" xlink:label="note-0521-02" xlink:href="note-0521-02a" xml:space="preserve">Ex 1. hu-<lb/>ius.</note> <note position="right" xlink:label="note-0521-03" xlink:href="note-0521-03a" xml:space="preserve">Defi.5. <lb/>Qui. El.</note> </div> </div> <div xml:id="echoid-div1156" type="section" level="1" n="692"> <head xml:id="echoid-head725" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s12777" xml:space="preserve">HÆc, & </s> <s xml:id="echoid-s12778" xml:space="preserve">antecedens methodo Indiuiſibilium oſtenſæ quoq; <lb/></s> <s xml:id="echoid-s12779" xml:space="preserve">fuerunt Lib. </s> <s xml:id="echoid-s12780" xml:space="preserve">2. </s> <s xml:id="echoid-s12781" xml:space="preserve">Prop. </s> <s xml:id="echoid-s12782" xml:space="preserve">4. </s> <s xml:id="echoid-s12783" xml:space="preserve">cum verò prima, ſecunda, & </s> <s xml:id="echoid-s12784" xml:space="preserve">tertia <lb/>Prop. </s> <s xml:id="echoid-s12785" xml:space="preserve">eiuſdem libri ſint illius methodi fundamenta, hinc opus erit <lb/>in præſenti Lib. </s> <s xml:id="echoid-s12786" xml:space="preserve">quaſcumq; </s> <s xml:id="echoid-s12787" xml:space="preserve">illas ſubſequentes, & </s> <s xml:id="echoid-s12788" xml:space="preserve">ex dicta indiui-<lb/>ſibilium methodo Propoſitiones dependentes, aliter demonſtra-<lb/>re, vt vel ſcrupoloſo cuiq; </s> <s xml:id="echoid-s12789" xml:space="preserve">Geometrę ſatisfiat. </s> <s xml:id="echoid-s12790" xml:space="preserve">Igitur ab hac Lib. </s> <s xml:id="echoid-s12791" xml:space="preserve"><lb/>2. </s> <s xml:id="echoid-s12792" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s12793" xml:space="preserve">4. </s> <s xml:id="echoid-s12794" xml:space="preserve">incipientes, curabimus, vt, quę per illam methodum <lb/>vera eſſe demonſtrata ſunt, etiam per noua hæc fundamenta con-<lb/>firmentur. </s> <s xml:id="echoid-s12795" xml:space="preserve">Primi Lib. </s> <s xml:id="echoid-s12796" xml:space="preserve">autem Prop. </s> <s xml:id="echoid-s12797" xml:space="preserve">nullatenus à dicta methodo <lb/>pendere manifeſtum eſt circa nonnullas tamen obiter prius hæc <lb/>pauca maioris facilitatis gratia libuit declarare.</s> <s xml:id="echoid-s12798" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12799" xml:space="preserve">In Prop. </s> <s xml:id="echoid-s12800" xml:space="preserve">4. </s> <s xml:id="echoid-s12801" xml:space="preserve">igitur Lib. </s> <s xml:id="echoid-s12802" xml:space="preserve">primi ſciat Lector tacitè ſupponi omnes <lb/>vertices datę figuræ, reſpectu eiuſdem regulę aſſumptos, eſſe in ea-<lb/>dem recta linea regulæ parallela; </s> <s xml:id="echoid-s12803" xml:space="preserve">ſeu, pro figuris ſolidis, in eodem <lb/>plano regulæ æquidiſtante, diffinitionibus conformiter; </s> <s xml:id="echoid-s12804" xml:space="preserve">quod ob <lb/>ſui claritatem inter axiomata poterat recenſeri.</s> <s xml:id="echoid-s12805" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12806" xml:space="preserve">In Prop. </s> <s xml:id="echoid-s12807" xml:space="preserve">26. </s> <s xml:id="echoid-s12808" xml:space="preserve">prætermiſſa fuit demonſtratio pręſentis caſus, cum <lb/>nempe, AG, contingit eſſe perpendicularem, GV, & </s> <s xml:id="echoid-s12809" xml:space="preserve">hoc cum fa-<lb/>cile, intellecto difficiliori caſu (qui ibidem explicatur) hoc probari <lb/>poſſet; </s> <s xml:id="echoid-s12810" xml:space="preserve">concludetur autem hoc modo, quod prætendimus, nempe <lb/>in tali caſu etiam, KY, eſſe perpendicularem ipſi, ΥΔ, & </s> <s xml:id="echoid-s12811" xml:space="preserve">ſecunda <lb/>plana, AV, ΚΔ, ad plana, HV, &</s> <s xml:id="echoid-s12812" xml:space="preserve">Δ, æquè ad eandem partem in-<lb/>clinari. </s> <s xml:id="echoid-s12813" xml:space="preserve">Sit, AG, ad, GP, vt, KY, ad, YX, iunctis, AP, PE, KX, X <pb o="502" file="0522" n="522" rhead="GEOMETRIÆ"/> T, & </s> <s xml:id="echoid-s12814" xml:space="preserve">cęteris vt ibidem conſtructis, eodem modo prius oſtende-<lb/>mus vt ibitriangula, AFE, KZT, necnon, AFG, KZY, EFG, TZ <lb/>Y, & </s> <s xml:id="echoid-s12815" xml:space="preserve">AGE, KYT, eſſe inter ſe ſimilia, & </s> <s xml:id="echoid-s12816" xml:space="preserve">angulum, PGE, æquari <lb/>angulo, XYT. </s> <s xml:id="echoid-s12817" xml:space="preserve">Hoc ſuppoſito, cum, PG, ad, GA, ſit vt, XY, ad, <lb/>YK, &</s> <s xml:id="echoid-s12818" xml:space="preserve">, AG, ad GE, vt, KY, ad, YT, exæquali, PG, ad GE, erit vt, X <lb/>Y, ad, YT, & </s> <s xml:id="echoid-s12819" xml:space="preserve">ſunt circa æquales angulos, PGE, XYT, ergotrian-<lb/> <anchor type="note" xlink:label="note-0522-01a" xlink:href="note-0522-01"/> gula, PGE, XYT, ſunt ſimilia, ergo, PE, ad, EG, eſt vt, XT, ad, <lb/>TY, &</s> <s xml:id="echoid-s12820" xml:space="preserve">, GE, ad, EA, vt, YT, ad, Tk, ergo, PE, ad, EA, eſt vt, X <lb/>T, ad, TH, & </s> <s xml:id="echoid-s12821" xml:space="preserve">ſunt circa rectos, PEA, XTK, ergo triangula, PEA, <lb/> <anchor type="note" xlink:label="note-0522-02a" xlink:href="note-0522-02"/> XTk, ſunt ſimilia, ergo, AP, ad, PE, erit vt, KX, ad, XT, ſed &</s> <s xml:id="echoid-s12822" xml:space="preserve">, <lb/>PE, ad, PG, eſt vt, XT, ad, XY, ergo, AP, ad, PG, erit vt, KX, ad, <lb/>XY, &</s> <s xml:id="echoid-s12823" xml:space="preserve">, PG, ad, GA, eſt vt, XY, ad, Yk, ergo triangula, APG, kXY, <lb/> <anchor type="note" xlink:label="note-0522-03a" xlink:href="note-0522-03"/> ſunt ſimilia, rectus autem eſt angulus, AGP, cum rectus ponatur, <lb/>AGV, ergo, kYX, &</s> <s xml:id="echoid-s12824" xml:space="preserve">, ΚΥΔ, rectus erit, vnde anguli, AGV, κΥΔ ę-<lb/>quales erunt. </s> <s xml:id="echoid-s12825" xml:space="preserve">Cum verò quadratum, PA, ęquetur quadratis, PG, <lb/> <anchor type="note" xlink:label="note-0522-04a" xlink:href="note-0522-04"/> GA, ſeu quadratis, PG, GE, EA, & </s> <s xml:id="echoid-s12826" xml:space="preserve">quadratum PA, ęquetur etiam <lb/>quadratis, PE, EA, duo quadrata, PE, EA, æ quabuntur tribus <lb/>quadratis, PG, GE, EA, & </s> <s xml:id="echoid-s12827" xml:space="preserve">ablato communi quadrato, EA, erit <lb/>quadratum, PE, æquale quadratis, PG, GE, vnde angulus, PGE, <lb/> <anchor type="note" xlink:label="note-0522-05a" xlink:href="note-0522-05"/> rectus erit, & </s> <s xml:id="echoid-s12828" xml:space="preserve">conſequenter etiam rectus ipſe, XYT, vnde anguli, <lb/>AGE, kYT, erunt inclinationes ſecundorum planorum, AV, ΚΛ, <lb/>cum ſubiectis planis, HV, &</s> <s xml:id="echoid-s12829" xml:space="preserve">Δ, & </s> <s xml:id="echoid-s12830" xml:space="preserve">inter ſe ęquales, per quę ſuppo-<lb/>ſito caſui ſatisfieri maniſeſtum eſt.</s> <s xml:id="echoid-s12831" xml:space="preserve"/> </p> <div xml:id="echoid-div1156" type="float" level="2" n="1"> <note position="left" xlink:label="note-0522-01" xlink:href="note-0522-01a" xml:space="preserve">6. Sex. Ele.</note> <note position="left" xlink:label="note-0522-02" xlink:href="note-0522-02a" xml:space="preserve">6. Sex. Ele.</note> <note position="left" xlink:label="note-0522-03" xlink:href="note-0522-03a" xml:space="preserve">5. Sex. Ele.</note> <note position="left" xlink:label="note-0522-04" xlink:href="note-0522-04a" xml:space="preserve">47. Primi <lb/>Elem.</note> <note position="left" xlink:label="note-0522-05" xlink:href="note-0522-05a" xml:space="preserve">48. Primi <lb/>Elem. <lb/>Deſin. 6. <lb/>Vnd, Ele.</note> </div> <p> <s xml:id="echoid-s12832" xml:space="preserve">In Lemmate 5. </s> <s xml:id="echoid-s12833" xml:space="preserve">poſt Prop. </s> <s xml:id="echoid-s12834" xml:space="preserve">8. </s> <s xml:id="echoid-s12835" xml:space="preserve">prętermiſſa fui demonſtratio prę-<lb/>ſentis caſus, cum eadem facilis exiſtimaretur, nempè quando, FE, <lb/>FG, cum, AE, AG, &</s> <s xml:id="echoid-s12836" xml:space="preserve">, LI, LM, cum, HI, HM, concurrere mini-<lb/>me poſſe contingat, vt cum angulos, EAF, GAF, IHL, MHL, re-<lb/>ctos, vel recto maiores acciderit eſſe: </s> <s xml:id="echoid-s12837" xml:space="preserve">Sic autem tum hic, tum ſup-<lb/>poſitus ibi caſus poterit vniuerſaliter demonſtrari. </s> <s xml:id="echoid-s12838" xml:space="preserve">Intelligantur <lb/> <anchor type="note" xlink:label="note-0522-06a" xlink:href="note-0522-06"/> ipſę, AE, AF, AG, HI, HL, HM, inter ſe ęquales, & </s> <s xml:id="echoid-s12839" xml:space="preserve">iungantur, <lb/>EF, FG, EG, IL, LM, IM: </s> <s xml:id="echoid-s12840" xml:space="preserve">Cum ergo anguli, FAG, LHM, ſup-<lb/>ponantur ęquales, & </s> <s xml:id="echoid-s12841" xml:space="preserve">latera, FA, LH, &</s> <s xml:id="echoid-s12842" xml:space="preserve">, AG, HM, ęqualia, erunt <lb/>pariter baſes, FG, LM, æquales: </s> <s xml:id="echoid-s12843" xml:space="preserve">Sic autem probabimus tum, EF, <lb/> <anchor type="note" xlink:label="note-0522-07a" xlink:href="note-0522-07"/> IL, tum, EG, IM, inter ſe æquales eſſe. </s> <s xml:id="echoid-s12844" xml:space="preserve">Rurſus ſuſpenſa pyrami-<lb/>de, AEFG, ponatur, F, in, L, demittaturq; </s> <s xml:id="echoid-s12845" xml:space="preserve">FG, ſuper, LM, cui cõ-<lb/>gruet, & </s> <s xml:id="echoid-s12846" xml:space="preserve">triangulo, EFG, cadente ſuper, ILM, punctum, E, pari-<lb/>ter erit in, I; </s> <s xml:id="echoid-s12847" xml:space="preserve">Sed & </s> <s xml:id="echoid-s12848" xml:space="preserve">punctum, A, dico fore in, H, tres enim ſphæ-<lb/>ricæ ſuperficies ſuper centris, I, L, M, radijs inuicem ſe ſecantibus <lb/>deſcriptæ, nempè radijs, HI, HL, HM, ſeu, AE, AF, AG, in duo-<lb/>bus tantum punctis ſeſe decuſſare poſſunt, vt facile oſtendi poteſt, <lb/>duę enim quęlibet ſphæricæ ſuperficies in circuli periphæria ſe ſe- <pb o="503" file="0523" n="523" rhead="LIBER VII."/> cabunt, tertia verò hanc periphęriam diuidet in duobus punctis, <lb/>quę ſunt ab ambas partes plani, ILM, nempè vnum ſupra alterum <lb/>infra ipſum, quare non ad aliud punctum, quam ad, H, concurrẽt <lb/>tres rectæ lineæ, AE, AF, AG, ad eandem partem plani, ILM, cum <lb/>ipſis, HI, HL, HM, conſtitutæ, ergo, AF, cadet in, HL, AE, in, <lb/>HI, &</s> <s xml:id="echoid-s12849" xml:space="preserve">, AG, in, HM, quibus pręoſtenſis, reliquum demonſtratio-<lb/>nis, vt ibi, proſequemur.</s> <s xml:id="echoid-s12850" xml:space="preserve"/> </p> <div xml:id="echoid-div1157" type="float" level="2" n="2"> <note position="left" xlink:label="note-0522-06" xlink:href="note-0522-06a" xml:space="preserve">4. Primi <lb/>Elem.</note> <note position="left" xlink:label="note-0522-07" xlink:href="note-0522-07a" xml:space="preserve">7. Primi <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div1159" type="section" level="1" n="693"> <head xml:id="echoid-head726" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s12851" xml:space="preserve">PArallelogramma in eadem altitudine exiſtentia inter <lb/>ſe ſunt vt baſes.</s> <s xml:id="echoid-s12852" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12853" xml:space="preserve">Sin in ſigura Prop. </s> <s xml:id="echoid-s12854" xml:space="preserve">5. </s> <s xml:id="echoid-s12855" xml:space="preserve">lib. </s> <s xml:id="echoid-s12856" xml:space="preserve">2. </s> <s xml:id="echoid-s12857" xml:space="preserve">parallelogramma, AM, MC, in ea-<lb/>dem altitudine. </s> <s xml:id="echoid-s12858" xml:space="preserve">Dico eadem eſſe inter ſe, vt baſes, GM, MH. <lb/></s> <s xml:id="echoid-s12859" xml:space="preserve"> <anchor type="note" xlink:label="note-0523-01a" xlink:href="note-0523-01"/> Hoc autem manifeſtum eſt, ſunt enim dicta parallelogramma figu-<lb/>rę proportionaliter analogę, iuxta ipſas baſes, cum ſit, GM, ad, <lb/>MH, vt, DE, ad, EI, &</s> <s xml:id="echoid-s12860" xml:space="preserve">, DI, ducta ſit vtcumque, vnde patet pro-<lb/>poſitum etiam independenter à methodo Indiuiſibilium.</s> <s xml:id="echoid-s12861" xml:space="preserve"/> </p> <div xml:id="echoid-div1159" type="float" level="2" n="1"> <note position="right" xlink:label="note-0523-01" xlink:href="note-0523-01a" xml:space="preserve">Ex 2. hu-<lb/>ius.</note> </div> </div> <div xml:id="echoid-div1161" type="section" level="1" n="694"> <head xml:id="echoid-head727" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s12862" xml:space="preserve">PRopofitionis 5. </s> <s xml:id="echoid-s12863" xml:space="preserve">Lib. </s> <s xml:id="echoid-s12864" xml:space="preserve">2. </s> <s xml:id="echoid-s12865" xml:space="preserve">prior pars pendet quidem ab Indiuiſi-<lb/>bilium methodo, verum pars poſterior, necnon Prop. </s> <s xml:id="echoid-s12866" xml:space="preserve">6.</s> <s xml:id="echoid-s12867" xml:space="preserve">7. </s> <s xml:id="echoid-s12868" xml:space="preserve">& </s> <s xml:id="echoid-s12869" xml:space="preserve"><lb/>8. </s> <s xml:id="echoid-s12870" xml:space="preserve">abſq; </s> <s xml:id="echoid-s12871" xml:space="preserve">illa methodo, vt intuenti apparebit, oſtenduntur, qua-<lb/>propter, cum ab eadem exemptę ſint, non indigent vt reſtauren-<lb/>tur, ſed illas tamquam ſtylo veteri demonſtratas, vt veras in hoc <lb/>libro quoq; </s> <s xml:id="echoid-s12872" xml:space="preserve">vſurpabimus, quod etiam de alijs Propoſitionibus fiet, <lb/>quę a methodo Indiuiſibilium immediatę pendere non conſpicien-<lb/>tur, etiamſi mediatè ab eadem vtiq; </s> <s xml:id="echoid-s12873" xml:space="preserve">dependere competiantur, ſuffi-<lb/>ciet enim illas Prop. </s> <s xml:id="echoid-s12874" xml:space="preserve">de nouo oſtendere, quę immediatè ab ipſa <lb/>methodo Indiuiſibilium fidem ſumpſiſſe videbuntur. </s> <s xml:id="echoid-s12875" xml:space="preserve">Cum verò <lb/>ſubſequentes Propoſitiones, in quibus parallelogrammorum om-<lb/>nia quadrata, ſeu omnes figurę ſimiles, regulis baſibus, examinan-<lb/>tur, ſint in gratiam cylindricorum, prima verò tantum pendeat ex <lb/>methodo Indiuiſibilium, propterea illa erit denuo oſtendenda, <lb/>quam nunc ſubiungo.</s> <s xml:id="echoid-s12876" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1162" type="section" level="1" n="695"> <head xml:id="echoid-head728" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s12877" xml:space="preserve">CYlindrici in eadem altitudine exiſtentes inter ſe ſunt <lb/>vt baſes.</s> <s xml:id="echoid-s12878" xml:space="preserve"/> </p> <pb o="504" file="0524" n="524" rhead="GEOMETRIÆ"/> <p> <s xml:id="echoid-s12879" xml:space="preserve">Manifeſla eſt ſimiliter hæc Prop. </s> <s xml:id="echoid-s12880" xml:space="preserve">cum enim ſecto quolibet cy-<lb/> <anchor type="note" xlink:label="note-0524-01a" xlink:href="note-0524-01"/> lindrico plano æquidiſtanter baſi, producatur in eo figura æqualis <lb/>ipſi baſi, propterea vt baſis ad baſim, ſic erit figura ad ſiguram ab <lb/>eodem plano baſibus æquidiſtante etcumq; </s> <s xml:id="echoid-s12881" xml:space="preserve">productam, ergo hi <lb/>cylindrici erunt figurę proportionaliter analogæ, iuxta ipſas baſes, <lb/> <anchor type="note" xlink:label="note-0524-02a" xlink:href="note-0524-02"/> ergo cylindrici æquè alti erunt inter ſe vt baſes.</s> <s xml:id="echoid-s12882" xml:space="preserve"/> </p> <div xml:id="echoid-div1162" type="float" level="2" n="1"> <note position="left" xlink:label="note-0524-01" xlink:href="note-0524-01a" xml:space="preserve">Corol.12. <lb/>lib. 1.</note> <note position="left" xlink:label="note-0524-02" xlink:href="note-0524-02a" xml:space="preserve">3.huius.</note> </div> </div> <div xml:id="echoid-div1164" type="section" level="1" n="696"> <head xml:id="echoid-head729" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s12883" xml:space="preserve">HOc demonſtrato haud difficile erit ſtylo veteri oſtendere cy-<lb/>lindricos exiſtentes in eadem baſi eſſe inter ſe vt altitudines, <lb/>vel vt latera æqualiter baſibus inclinata.</s> <s xml:id="echoid-s12884" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12885" xml:space="preserve">Similiter eoſdem habere inter ſe rationem compoſitam ex ra-<lb/>tione baſium, & </s> <s xml:id="echoid-s12886" xml:space="preserve">altitudinum, vel laterum æqualiter baſibus incli-<lb/>natorum. </s> <s xml:id="echoid-s12887" xml:space="preserve">Et eos qui habent baſes altitudinibus, vel lateribus æ-<lb/>qualiter baſibus inclinatis, reciprocas æquales eſſe; </s> <s xml:id="echoid-s12888" xml:space="preserve">Vel æquales, <lb/>baſes haberet altitudinibus, ſeu lateribus æqualiter baſibus incli-<lb/>natis, reciprocas atq; </s> <s xml:id="echoid-s12889" xml:space="preserve">ſimiles cylindricos eſſe in tripla ratione late-<lb/>rum hom ologorum. </s> <s xml:id="echoid-s12890" xml:space="preserve">Sufficiet namq; </s> <s xml:id="echoid-s12891" xml:space="preserve">nos methodum imitari, qua <lb/>dem onſtrata Prop. </s> <s xml:id="echoid-s12892" xml:space="preserve">9. </s> <s xml:id="echoid-s12893" xml:space="preserve">lib. </s> <s xml:id="echoid-s12894" xml:space="preserve">2. </s> <s xml:id="echoid-s12895" xml:space="preserve">poſtmodum reliquæ vſq; </s> <s xml:id="echoid-s12896" xml:space="preserve">ad Prop. </s> <s xml:id="echoid-s12897" xml:space="preserve">14. <lb/></s> <s xml:id="echoid-s12898" xml:space="preserve">oſtenſæ fuerunt, probando circa cylindricos, quod ibi circa omnia <lb/>quadrata datorum parallelogram. </s> <s xml:id="echoid-s12899" xml:space="preserve">oſtendebatur. </s> <s xml:id="echoid-s12900" xml:space="preserve">Hæc autem pro <lb/>cylindricis poſtea collecta ſunt in eodem lib. </s> <s xml:id="echoid-s12901" xml:space="preserve">2. </s> <s xml:id="echoid-s12902" xml:space="preserve">Prop. </s> <s xml:id="echoid-s12903" xml:space="preserve">34. </s> <s xml:id="echoid-s12904" xml:space="preserve">Cor. </s> <s xml:id="echoid-s12905" xml:space="preserve">4. </s> <s xml:id="echoid-s12906" xml:space="preserve"><lb/>generali a ſec. </s> <s xml:id="echoid-s12907" xml:space="preserve">B. </s> <s xml:id="echoid-s12908" xml:space="preserve">vſq; </s> <s xml:id="echoid-s12909" xml:space="preserve">ad ſec. </s> <s xml:id="echoid-s12910" xml:space="preserve">G. </s> <s xml:id="echoid-s12911" xml:space="preserve">quæ quidem animaduertere opus <lb/>erat.</s> <s xml:id="echoid-s12912" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12913" xml:space="preserve">In Prop. </s> <s xml:id="echoid-s12914" xml:space="preserve">15. </s> <s xml:id="echoid-s12915" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s12916" xml:space="preserve">2. </s> <s xml:id="echoid-s12917" xml:space="preserve">hæc ſupplenda videntur. </s> <s xml:id="echoid-s12918" xml:space="preserve">In Sec. </s> <s xml:id="echoid-s12919" xml:space="preserve">A. <lb/></s> <s xml:id="echoid-s12920" xml:space="preserve">probatur figuram, KQM, ipſi, ABD, &</s> <s xml:id="echoid-s12921" xml:space="preserve">, ΠΤΩ, ipſi, φΣΛ, æqualem <lb/>eſſe ex Prop. </s> <s xml:id="echoid-s12922" xml:space="preserve">3. </s> <s xml:id="echoid-s12923" xml:space="preserve">eiuſdem, nempè ex methodo Indiuiſibilium, hoc <lb/>autem patebit etiam ex prop. </s> <s xml:id="echoid-s12924" xml:space="preserve">prima huius, ſunt enim dictæ figuræ <lb/>æqualiter analogæ. </s> <s xml:id="echoid-s12925" xml:space="preserve">In ſec. </s> <s xml:id="echoid-s12926" xml:space="preserve">B. </s> <s xml:id="echoid-s12927" xml:space="preserve">figuram, MZP, adæquari ipſi, KQ <lb/>M, &</s> <s xml:id="echoid-s12928" xml:space="preserve">, Ω℟&</s> <s xml:id="echoid-s12929" xml:space="preserve">, ipſi, ΠΤΩ, eodem modo deducetur ex prima huius. </s> <s xml:id="echoid-s12930" xml:space="preserve"><lb/>In ſec. </s> <s xml:id="echoid-s12931" xml:space="preserve">C. </s> <s xml:id="echoid-s12932" xml:space="preserve">probabitur vt, MP, ad, PO, ita eſſe figuram, MZP, ad, O <lb/>ZP, ex prop. </s> <s xml:id="echoid-s12933" xml:space="preserve">2. </s> <s xml:id="echoid-s12934" xml:space="preserve">huius. </s> <s xml:id="echoid-s12935" xml:space="preserve">In ſec. </s> <s xml:id="echoid-s12936" xml:space="preserve">D. </s> <s xml:id="echoid-s12937" xml:space="preserve">ſimiliter oſtendemus figuram, O <lb/>ZP, ad figuram, Ω℟&</s> <s xml:id="echoid-s12938" xml:space="preserve">, eſſe vt, ZP, ad, ℟&</s> <s xml:id="echoid-s12939" xml:space="preserve">, ſimiliter ex prop. </s> <s xml:id="echoid-s12940" xml:space="preserve">2. </s> <s xml:id="echoid-s12941" xml:space="preserve"><lb/>huius. </s> <s xml:id="echoid-s12942" xml:space="preserve">Cętera verò abſq; </s> <s xml:id="echoid-s12943" xml:space="preserve">methodo indiuiſibilium ſubſiſtunt; </s> <s xml:id="echoid-s12944" xml:space="preserve">vt & </s> <s xml:id="echoid-s12945" xml:space="preserve"><lb/>Corollaria, & </s> <s xml:id="echoid-s12946" xml:space="preserve">prop. </s> <s xml:id="echoid-s12947" xml:space="preserve">16.</s> <s xml:id="echoid-s12948" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12949" xml:space="preserve">In Prop. </s> <s xml:id="echoid-s12950" xml:space="preserve">17. </s> <s xml:id="echoid-s12951" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s12952" xml:space="preserve">2. </s> <s xml:id="echoid-s12953" xml:space="preserve">hæc pariter ſupplenda ſunt. </s> <s xml:id="echoid-s12954" xml:space="preserve">In ſec. <lb/></s> <s xml:id="echoid-s12955" xml:space="preserve">A. </s> <s xml:id="echoid-s12956" xml:space="preserve">elicitur ex 3. </s> <s xml:id="echoid-s12957" xml:space="preserve">pariter lib. </s> <s xml:id="echoid-s12958" xml:space="preserve">2. </s> <s xml:id="echoid-s12959" xml:space="preserve">ſolidum, HZ {00/ }, æquari ſolido, ABP <lb/>C, &</s> <s xml:id="echoid-s12960" xml:space="preserve">, ΣΓ2, ſolido, VΠ&</s> <s xml:id="echoid-s12961" xml:space="preserve">Ω, cum verò hæc ſolida ſint figuræ æqua-<lb/>liter analogæ vt eorum conditiones expendenti patebit, ideò quod <lb/>ibi ex 3. </s> <s xml:id="echoid-s12962" xml:space="preserve">lib. </s> <s xml:id="echoid-s12963" xml:space="preserve">2. </s> <s xml:id="echoid-s12964" xml:space="preserve">hic ex prima huius deducemus. </s> <s xml:id="echoid-s12965" xml:space="preserve">In ſec. </s> <s xml:id="echoid-s12966" xml:space="preserve">B. </s> <s xml:id="echoid-s12967" xml:space="preserve">ſolidum, <pb o="505" file="0525" n="525" rhead="LIBER VII."/> LDGF, æquari ipſi, HZ {00/ }, &</s> <s xml:id="echoid-s12968" xml:space="preserve">, 3687, ſolido, ΣΓ2, pariter ex prima <lb/>huius colligemus. </s> <s xml:id="echoid-s12969" xml:space="preserve">In ſec. </s> <s xml:id="echoid-s12970" xml:space="preserve">D. </s> <s xml:id="echoid-s12971" xml:space="preserve">quod figura, LED, ad, OED, ſit vt, <lb/>LE, ad, EO, ſeu quod figura, QAMY, ad, TIMY, ſit vt, QY, ad, Y <lb/>T, ideſt vt, LE, ad, EO, vel quod figura, LFE, ad, OFE, ſfit vt, LE, <lb/>ad, EO, patet, ex prop. </s> <s xml:id="echoid-s12972" xml:space="preserve">2. </s> <s xml:id="echoid-s12973" xml:space="preserve">huius: </s> <s xml:id="echoid-s12974" xml:space="preserve">Quod verò ſolidum, LDFE, ad <lb/>ſolidum, ODFE, ſit vt figura, LEF, ad figuram, OEF, ideſi vt, LE, <lb/>ad, EO, manifeſtum eſt pariter ex 3. </s> <s xml:id="echoid-s12975" xml:space="preserve">huius. </s> <s xml:id="echoid-s12976" xml:space="preserve">In ſec. </s> <s xml:id="echoid-s12977" xml:space="preserve">F. </s> <s xml:id="echoid-s12978" xml:space="preserve">ſolidum, O <lb/>DFE, ad ſolidum, 3674, eſſe vt figura, EDF, ad figuram, 467, <lb/>pateb@t ex 3. </s> <s xml:id="echoid-s12979" xml:space="preserve">huius, ſunt enim dicta ſolida figuræ proportionali-<lb/>ter analogæ vt conſideranti manifeſtum erit. </s> <s xml:id="echoid-s12980" xml:space="preserve">Cætera huius prop. <lb/></s> <s xml:id="echoid-s12981" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s12982" xml:space="preserve">& </s> <s xml:id="echoid-s12983" xml:space="preserve">prop. </s> <s xml:id="echoid-s12984" xml:space="preserve">18. </s> <s xml:id="echoid-s12985" xml:space="preserve">abſq; </s> <s xml:id="echoid-s12986" xml:space="preserve">methodo Indiuiſibilium ſubſiſtunt, <lb/>vt examinantifacilè apparebit.</s> <s xml:id="echoid-s12987" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1165" type="section" level="1" n="697"> <head xml:id="echoid-head730" xml:space="preserve">THEOREMA VI. PROPOS. VI.</head> <p> <s xml:id="echoid-s12988" xml:space="preserve">QVæcunq; </s> <s xml:id="echoid-s12989" xml:space="preserve">de parallelogrammis oſtenduntur in Prop. <lb/></s> <s xml:id="echoid-s12990" xml:space="preserve">5. </s> <s xml:id="echoid-s12991" xml:space="preserve">6. </s> <s xml:id="echoid-s12992" xml:space="preserve">7. </s> <s xml:id="echoid-s12993" xml:space="preserve">& </s> <s xml:id="echoid-s12994" xml:space="preserve">8. </s> <s xml:id="echoid-s12995" xml:space="preserve">Lib. </s> <s xml:id="echoid-s12996" xml:space="preserve">2. </s> <s xml:id="echoid-s12997" xml:space="preserve">eadem etiam de triangulis, con-<lb/>ditiones ibi ſuppoſitas circa ſuas baſes, & </s> <s xml:id="echoid-s12998" xml:space="preserve">altitudines, ſeu <lb/>latera æqualiter baſibus inclinata, habentibus, verifi-<lb/>cantur.</s> <s xml:id="echoid-s12999" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13000" xml:space="preserve">Hæe Propoſitio maniſeſta eſt, cum enim expoſito quocunq; <lb/></s> <s xml:id="echoid-s13001" xml:space="preserve">triangulo, & </s> <s xml:id="echoid-s13002" xml:space="preserve">aſſumptis duobus quibuſuis lateribus angulum quē-<lb/>libet continentibus parallelogrammum compleri poſſit in illo an-<lb/>gulo, cuius triangulum erit dimidium, ideò quæcunq; </s> <s xml:id="echoid-s13003" xml:space="preserve">triangula <lb/> <anchor type="note" xlink:label="note-0525-01a" xlink:href="note-0525-01"/> erunt, vt eorum completa paral clogramma, habentibus autem <lb/>triangulis circa baſes, & </s> <s xml:id="echoid-s13004" xml:space="preserve">altitudines, ſeu latera æqualiter baſibus <lb/>inclinata, præfatas conditiones, eam pariter habent completa <lb/>parallelogramma, & </s> <s xml:id="echoid-s13005" xml:space="preserve">de illis verificantur ea, quæ in dictis propo-<lb/> <anchor type="note" xlink:label="note-0525-02a" xlink:href="note-0525-02"/> ſitionibus fuerunt propoſita, ergo eadem de eorum medietatibus, <lb/>hoc eſt de dictis triangulis verificabuntur. </s> <s xml:id="echoid-s13006" xml:space="preserve">Triangula ergo, quæ <lb/>ſunt in eadem altitudine inter ſe ſunt, vt baſes; </s> <s xml:id="echoid-s13007" xml:space="preserve">Et quæ ſunt in <lb/>ea dem, vel æqualibus baſibus, vt altitudines, vel vt latera, quæ <lb/>æqualiter baſi, ſeu baſibus, inclinantur. </s> <s xml:id="echoid-s13008" xml:space="preserve">Habent inter ſe rationem <lb/>compoſitam ex ratione baſium, & </s> <s xml:id="echoid-s13009" xml:space="preserve">altitudinum, vel laterum ęqua-<lb/>liter baſibus inclinatorum. </s> <s xml:id="echoid-s13010" xml:space="preserve">Habentia baſes altitudinibus, vel la-<lb/>teribus baſibus æqual ter incl natis, reciprocas, ſunt æqualia; </s> <s xml:id="echoid-s13011" xml:space="preserve">Et <lb/>quæ ſunt æqualia baſes habent altitudinibus, vel lateribus æqua-<lb/>liter baſibus inclinatis, reciprocas. </s> <s xml:id="echoid-s13012" xml:space="preserve">Et tandem ſimil a triangula <lb/>ſunt in dupla ratione laterum homologorum; </s> <s xml:id="echoid-s13013" xml:space="preserve">Quæ omnia etiam <lb/>Lib. </s> <s xml:id="echoid-s13014" xml:space="preserve">2. </s> <s xml:id="echoid-s13015" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13016" xml:space="preserve">19. </s> <s xml:id="echoid-s13017" xml:space="preserve">Coroll. </s> <s xml:id="echoid-s13018" xml:space="preserve">1. </s> <s xml:id="echoid-s13019" xml:space="preserve">ex methodo Indiuiſibilium collige ban- <pb o="506" file="0526" n="526" rhead="GEOMETRIE"/> tur. </s> <s xml:id="echoid-s13020" xml:space="preserve">Coroll. </s> <s xml:id="echoid-s13021" xml:space="preserve">2. </s> <s xml:id="echoid-s13022" xml:space="preserve">autem ſpectat ad dictam methodum pertractan-<lb/>dam, propterea non opus eſt, quod aliter oſtendatur: </s> <s xml:id="echoid-s13023" xml:space="preserve">Lemm2 <lb/>verò antecedens Propoſ. </s> <s xml:id="echoid-s13024" xml:space="preserve">20. </s> <s xml:id="echoid-s13025" xml:space="preserve">ſtylo veteri demonſtratur, ſicut & </s> <s xml:id="echoid-s13026" xml:space="preserve"><lb/>ipſa Propoſ. </s> <s xml:id="echoid-s13027" xml:space="preserve">20. </s> <s xml:id="echoid-s13028" xml:space="preserve">& </s> <s xml:id="echoid-s13029" xml:space="preserve">21. </s> <s xml:id="echoid-s13030" xml:space="preserve">quorum Corollaria haud nobis opus eſt ali-<lb/>ter demonſtrare, cum eorum vſus non ſit, niſi pro methodo Indi-<lb/>uiſibilium.</s> <s xml:id="echoid-s13031" xml:space="preserve"/> </p> <div xml:id="echoid-div1165" type="float" level="2" n="1"> <note position="right" xlink:label="note-0525-01" xlink:href="note-0525-01a" xml:space="preserve">34. Primi <lb/>Elem.</note> <note position="right" xlink:label="note-0525-02" xlink:href="note-0525-02a" xml:space="preserve">4. huius, <lb/>cum An-<lb/>not.</note> </div> </div> <div xml:id="echoid-div1167" type="section" level="1" n="698"> <head xml:id="echoid-head731" xml:space="preserve">THEOREMA VII. PROPOS. VII.</head> <p> <s xml:id="echoid-s13032" xml:space="preserve">COnici in eadem, vel æqualibus altitudinibus exiſtẽ-<lb/>tes inter ſe ſunt vt baſes.</s> <s xml:id="echoid-s13033" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13034" xml:space="preserve">Sint quicunq; </s> <s xml:id="echoid-s13035" xml:space="preserve">conici in eadem, vel æqualibus altitudinibus, A <lb/>E, BF, exiſtentes, AKLM, BSQTR. </s> <s xml:id="echoid-s13036" xml:space="preserve">Dico hos eſſe inter ie, vt ip-<lb/>ſorum baſes, KLM, SQTR. </s> <s xml:id="echoid-s13037" xml:space="preserve">Abſciſſis enim ab altitudinibus, AE, <lb/>BF, vtcunq; </s> <s xml:id="echoid-s13038" xml:space="preserve">partibus æqualibus verſus, A, B, ipſis, AC, BD, per, <lb/> <anchor type="figure" xlink:label="fig-0526-01a" xlink:href="fig-0526-01"/> C, ducatur planum baſi, K <lb/>LM, æquidiſtans, & </s> <s xml:id="echoid-s13039" xml:space="preserve">per, <lb/>D, ſimiliter planum baſi, <lb/>SQTR, æquidiſtans, qui-<lb/>bus in conicis producan-<lb/> <anchor type="note" xlink:label="note-0526-01a" xlink:href="note-0526-01"/> tur figuræ, GIO, XNVP, <lb/> <anchor type="note" xlink:label="note-0526-02a" xlink:href="note-0526-02"/> erit ergo, GIO, ſimilis ipſi, <lb/>kLM, quarum lat era ho-<lb/>mologa, IO, LM, ſimili-<lb/>ter, XNVP, erit ſimilis baſi, SQTR, ducto autem plano tranſeun-<lb/>te per altitudinem, BF, ſecetur baſis in recta, QR, vtcunque, & </s> <s xml:id="echoid-s13040" xml:space="preserve"><lb/>figura, XNVP, in recta, NP, ſuperficies verò conicularis in rectis, <lb/>BQ, BR, erunt ergo hæ ſimiliter ſecta in punctis, N, P, ac, BF, in, <lb/> <anchor type="note" xlink:label="note-0526-03a" xlink:href="note-0526-03"/> D, ſicut etiam, Ak, AL, AM, erunt ſimiliter ſecta in punctis, G, I, <lb/>O, ac, AE, in, C, &</s> <s xml:id="echoid-s13041" xml:space="preserve">, QR, NP, latera homologa ſimilium figura-<lb/> <anchor type="note" xlink:label="note-0526-04a" xlink:href="note-0526-04"/> rum, SQTR, XNVP, ſunt autem etiam, AE, BF, altitudines æqua-<lb/>les ſimiliter ſectæ in punctis, C, D. </s> <s xml:id="echoid-s13042" xml:space="preserve">Cum ergo figura, KLM, ſimi-<lb/>lis ſit ipſi, GIO, habebit, KLM, ad, GIO, duplam proportionem <lb/> <anchor type="note" xlink:label="note-0526-05a" xlink:href="note-0526-05"/> eius, quam, LM, ad, IO, vel, MA, ad, AO, vel, EA, ad, AC, ſeu, <lb/>FB, ad, BD, vel, RB, ad, BP, vel tandem eius, quam habet, QR, <lb/>ad, NP, ſed etiam ſigura, SQTR, ad, XNVP, habet duplam ratio-<lb/>nem eius, quam habet, QR, ad, NP, ergo figura, KLM, ad, GIO, <lb/>eſt vt, SQTR, ad, XNVP, & </s> <s xml:id="echoid-s13043" xml:space="preserve">permutando figura, kLM, ad, SQT <lb/>R, erit vt figura, GIO, ad figuram, XNVP, & </s> <s xml:id="echoid-s13044" xml:space="preserve">puncta, C, D, ſum-<lb/>pta ſunt vtcunque, ac conici, AKLM, BSQTR, ſunt in æquali-<lb/>bus altitudinibus, AE, BF, reſpectu baſium, KLM, SQTR, aſſum- <pb o="507" file="0527" n="527" rhead="LIBER VII."/> ptis,, ergo ſunt figuræ proportionaliter analogæ, ergo dicti cy-<lb/>lindrici erunt inter ſe, vt baſes, KLM, SQTR, quod erat demon-<lb/> <anchor type="note" xlink:label="note-0527-01a" xlink:href="note-0527-01"/> ſtrandum.</s> <s xml:id="echoid-s13045" xml:space="preserve"/> </p> <div xml:id="echoid-div1167" type="float" level="2" n="1"> <figure xlink:label="fig-0526-01" xlink:href="fig-0526-01a"> <image file="0526-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0526-01"/> </figure> <note position="left" xlink:label="note-0526-01" xlink:href="note-0526-01a" xml:space="preserve">19. lib. 1.</note> <note position="left" xlink:label="note-0526-02" xlink:href="note-0526-02a" xml:space="preserve">21. lib. 1.</note> <note position="left" xlink:label="note-0526-03" xlink:href="note-0526-03a" xml:space="preserve">17. Vnde, <lb/>Elem.</note> <note position="left" xlink:label="note-0526-04" xlink:href="note-0526-04a" xml:space="preserve">21. lib. 1.</note> <note position="left" xlink:label="note-0526-05" xlink:href="note-0526-05a" xml:space="preserve">15. l.b.2.</note> <note position="right" xlink:label="note-0527-01" xlink:href="note-0527-01a" xml:space="preserve">3.huius.</note> </div> </div> <div xml:id="echoid-div1169" type="section" level="1" n="699"> <head xml:id="echoid-head732" xml:space="preserve">COROLLARIV M.</head> <p style="it"> <s xml:id="echoid-s13046" xml:space="preserve">_C_Vm verò etiam cylindrici in eiſdem baſibus, & </s> <s xml:id="echoid-s13047" xml:space="preserve">altitudinibus <lb/>prædictis æqualibus, ſint inter ſe, vt ipſæ baſes, propterea erũt <lb/>etiam inter ſe, vt ipſi conici, vnde ſi in vna ſpecie cylindricorum, & </s> <s xml:id="echoid-s13048" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0527-02a" xlink:href="note-0527-02"/> conicorum oſtenſum fuerit, cylindricum triplum eſſe conici in eadem <lb/>baſi, & </s> <s xml:id="echoid-s13049" xml:space="preserve">altitudine cum eo exiſtentis, illicò hoc etiam de reliquis ſpe-<lb/>crebus cylindricorum, & </s> <s xml:id="echoid-s13050" xml:space="preserve">conicorum facilè colligemus.</s> <s xml:id="echoid-s13051" xml:space="preserve"/> </p> <div xml:id="echoid-div1169" type="float" level="2" n="1"> <note position="right" xlink:label="note-0527-02" xlink:href="note-0527-02a" xml:space="preserve">_5. huius._</note> </div> </div> <div xml:id="echoid-div1171" type="section" level="1" n="700"> <head xml:id="echoid-head733" xml:space="preserve">THEOREMA VIII. PROPOS. VIII.</head> <p> <s xml:id="echoid-s13052" xml:space="preserve">QVilibet Cylindricus triplus eſt Conici in eadem ba-<lb/>ſi, & </s> <s xml:id="echoid-s13053" xml:space="preserve">altitudine, cum eo exiſtentis.</s> <s xml:id="echoid-s13054" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13055" xml:space="preserve">Sit quicunq; </s> <s xml:id="echoid-s13056" xml:space="preserve">cylindricus, GO, & </s> <s xml:id="echoid-s13057" xml:space="preserve">conicus in eadem baſi, IMNO, <lb/>& </s> <s xml:id="echoid-s13058" xml:space="preserve">eadem altitudine cum ipſo. </s> <s xml:id="echoid-s13059" xml:space="preserve">Dico cylindricum, GO, triplum <lb/> <anchor type="figure" xlink:label="fig-0527-01a" xlink:href="fig-0527-01"/> eſſe conici, HIMNO. <lb/></s> <s xml:id="echoid-s13060" xml:space="preserve">Exponatur enim pri-<lb/>ſma, AFDE, triangu-<lb/>lares habens baſes, A <lb/>BC, FDE; </s> <s xml:id="echoid-s13061" xml:space="preserve">altitudinis <lb/>æqualis altitudini cy-<lb/>lindrici, GO, in baſi <lb/>verò, FDE, ſit pyra-<lb/>mis, CDFE; </s> <s xml:id="echoid-s13062" xml:space="preserve">erit er-<lb/>go priſma, ADEF, <lb/>triplum pyramidis, C <lb/>DEF, cum reſoluatur in tres pyramides æquales, FDBC, FDEC, <lb/>FBAC, vt oſtendit Euclides Vnd. </s> <s xml:id="echoid-s13063" xml:space="preserve">Element. </s> <s xml:id="echoid-s13064" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13065" xml:space="preserve">7. </s> <s xml:id="echoid-s13066" xml:space="preserve">vt autem ſe <lb/> <anchor type="note" xlink:label="note-0527-03a" xlink:href="note-0527-03"/> habet priſma, ADEF, ad pyramidem, CDEF, ita ſe habet cylin-<lb/>dricus, GO, ad conicum, HIMNO, ergo, GO, triplus eſt conici, H <lb/>MO, vnde omnis cylindricus triplus eſt conici in eadem baſi, & </s> <s xml:id="echoid-s13067" xml:space="preserve">al-<lb/>titudine cum eo conſtituti, illi enim conici, qui ſunt in eadem ba-<lb/>ſi, & </s> <s xml:id="echoid-s13068" xml:space="preserve">altitudine ex ant. </s> <s xml:id="echoid-s13069" xml:space="preserve">omnes inter ſe ſunt æquales, quod oſten-<lb/>dendum erat.</s> <s xml:id="echoid-s13070" xml:space="preserve"/> </p> <div xml:id="echoid-div1171" type="float" level="2" n="1"> <figure xlink:label="fig-0527-01" xlink:href="fig-0527-01a"> <image file="0527-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0527-01"/> </figure> <note position="right" xlink:label="note-0527-03" xlink:href="note-0527-03a" xml:space="preserve">EX ant.</note> </div> <pb o="508" file="0528" n="528" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1173" type="section" level="1" n="701"> <head xml:id="echoid-head734" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13071" xml:space="preserve">PErant. </s> <s xml:id="echoid-s13072" xml:space="preserve">prop. </s> <s xml:id="echoid-s13073" xml:space="preserve">ſatisfit prop. </s> <s xml:id="echoid-s13074" xml:space="preserve">22. </s> <s xml:id="echoid-s13075" xml:space="preserve">lib. </s> <s xml:id="echoid-s13076" xml:space="preserve">2. </s> <s xml:id="echoid-s13077" xml:space="preserve">ex ea enim pariter habe-<lb/>tur omnes cylindricos eandem rationem habere ad conicos <lb/>in eadem baſi, & </s> <s xml:id="echoid-s13078" xml:space="preserve">altitudine cum ipſis exiſtentes, cum eorum eſſe <lb/>triplos fuerit demonſtratum, & </s> <s xml:id="echoid-s13079" xml:space="preserve">eadem, quæ ex ipſa deduceban-<lb/>tur, hic pariter colliguntur, proprietates inquam illæ, quas cylin-<lb/>dricis competere dictum eſt in Annot. </s> <s xml:id="echoid-s13080" xml:space="preserve">prop. </s> <s xml:id="echoid-s13081" xml:space="preserve">5. </s> <s xml:id="echoid-s13082" xml:space="preserve">huius. </s> <s xml:id="echoid-s13083" xml:space="preserve">Sic ergo <lb/>ratum, ac firmum eſt, Conicos in eadem, vel æqualibus baſibus <lb/>exiſtentes, eſſe inter ſe vt altitudines. </s> <s xml:id="echoid-s13084" xml:space="preserve">Habereq; </s> <s xml:id="echoid-s13085" xml:space="preserve">rationem compo-<lb/>ſitam ex ratione baſium, & </s> <s xml:id="echoid-s13086" xml:space="preserve">altitudinum. </s> <s xml:id="echoid-s13087" xml:space="preserve">Eos verò, quorum ba-<lb/>ſes altitudinibus reciprocantur, æquales eſſe, & </s> <s xml:id="echoid-s13088" xml:space="preserve">æqualium baſe-<lb/>altitudinibus reciprocari. </s> <s xml:id="echoid-s13089" xml:space="preserve">Ac tandem ſimiles conicos eſſe in tri-<lb/>pla ratione linearum, vel laterum homologorum eorundem ba-<lb/>ſium, ſeu ſimilium triangulorum per verticem traſeuntium, quæ <lb/>in ipſius prop. </s> <s xml:id="echoid-s13090" xml:space="preserve">22. </s> <s xml:id="echoid-s13091" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13092" xml:space="preserve">Sectionibus, in gratiam Conicorum pari-<lb/>ter colligebantur. </s> <s xml:id="echoid-s13093" xml:space="preserve">Per hanc etiam ſatisſit prop. </s> <s xml:id="echoid-s13094" xml:space="preserve">24. </s> <s xml:id="echoid-s13095" xml:space="preserve">eiuſdem lib. <lb/></s> <s xml:id="echoid-s13096" xml:space="preserve">2. </s> <s xml:id="echoid-s13097" xml:space="preserve">cum per eam ibi demonſtrati intendatur cylindricum quemcũq; </s> <s xml:id="echoid-s13098" xml:space="preserve"><lb/>triplum eſſe conici in eadem baſi, & </s> <s xml:id="echoid-s13099" xml:space="preserve">altitudine cum eo exiſtentis, <lb/>vt in Sec. </s> <s xml:id="echoid-s13100" xml:space="preserve">1. </s> <s xml:id="echoid-s13101" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13102" xml:space="preserve">4. </s> <s xml:id="echoid-s13103" xml:space="preserve">gen. </s> <s xml:id="echoid-s13104" xml:space="preserve">prop. </s> <s xml:id="echoid-s13105" xml:space="preserve">34. </s> <s xml:id="echoid-s13106" xml:space="preserve">poſtea declaratur. </s> <s xml:id="echoid-s13107" xml:space="preserve">Aduerte au-<lb/>tem, quod pag. </s> <s xml:id="echoid-s13108" xml:space="preserve">79. </s> <s xml:id="echoid-s13109" xml:space="preserve">lin. </s> <s xml:id="echoid-s13110" xml:space="preserve">15. </s> <s xml:id="echoid-s13111" xml:space="preserve">hæc verba, _& </s> <s xml:id="echoid-s13112" xml:space="preserve">cum omnibus quadratis_ <lb/>_duorum triangulorun CBM, EMF_, ponenda ſunt poſt hæc verba, <lb/>_dupla erunt omnium quadratorum, AF_.</s> <s xml:id="echoid-s13113" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1174" type="section" level="1" n="702"> <head xml:id="echoid-head735" xml:space="preserve">THEOREMA IX. PROPOS. IX.</head> <p> <s xml:id="echoid-s13114" xml:space="preserve">COnicorum fruſta æquè alta, & </s> <s xml:id="echoid-s13115" xml:space="preserve">in baſibus æquè alto- <lb/>rum conicorum, à quibus abſcinduntur, conſtituta;</s> <s xml:id="echoid-s13116" xml:space="preserve"> <lb/>inter ſe ſunt vt baſes.</s> <s xml:id="echoid-s13117" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13118" xml:space="preserve">Videatur ſchema prop. </s> <s xml:id="echoid-s13119" xml:space="preserve">7. </s> <s xml:id="echoid-s13120" xml:space="preserve">huius, in quo ſint conicorum æquè <lb/>altorum, AkLM, BSQTR, fruſta, GIOLKM, XVTS, in eiſdem <lb/>cum illis baſibus, kLM, SQTR, & </s> <s xml:id="echoid-s13121" xml:space="preserve">in æqualibus altitudinibus, C <lb/>E, DF, exiſtentia, igitur abſciſſis verſus puncta, C, D, altitudinum <lb/>partibus æqualibus, & </s> <s xml:id="echoid-s13122" xml:space="preserve">per earum terminos ductis planis baſibus <lb/>parallelis, oſtendemus ab ijſdem productas in fruſtis figuras eſſe <lb/> <anchor type="note" xlink:label="note-0528-01a" xlink:href="note-0528-01"/> inter ſe vt ipſæ baſes eodem modo, quo ibi factum eſt, vnde pate-<lb/>bit dicta ſruſta eſſe figuras proportionaliter analogas, quapropter <lb/>ipſa eſſe inter ſe vt baſes pariter concludemus, quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s13123" xml:space="preserve"/> </p> <div xml:id="echoid-div1174" type="float" level="2" n="1"> <note position="left" xlink:label="note-0528-01" xlink:href="note-0528-01a" xml:space="preserve">3. huius.</note> </div> <pb o="509" file="0529" n="529" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1176" type="section" level="1" n="703"> <head xml:id="echoid-head736" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s13124" xml:space="preserve">_C_Vm verò etiam cylindrici in baſibus dictorum fruſtorum, & </s> <s xml:id="echoid-s13125" xml:space="preserve">in <lb/>æqualibus cum eiſdem altituainibus conſtituti, ſint inter ſe vt <lb/>baſes, erunt etiam inter ſe vt dicta fruſta, & </s> <s xml:id="echoid-s13126" xml:space="preserve">permutando habebunt <lb/> <anchor type="note" xlink:label="note-0529-01a" xlink:href="note-0529-01"/> eandem rationem ad dicta fruſta, vnde propoſito quocunq; </s> <s xml:id="echoid-s13127" xml:space="preserve">fruſto coni-<lb/>co, & </s> <s xml:id="echoid-s13128" xml:space="preserve">cylindrico in eadem baſi, & </s> <s xml:id="echoid-s13129" xml:space="preserve">altitudine, cum eo exiſtente, vt <lb/>rationem cylindrici ad fruſtum conicum inueniamus, ſufficiet alicuius <lb/>cylindrici præfatæ altitudinis rationem ad fruſtum conicum in eadem <lb/>baſi, & </s> <s xml:id="echoid-s13130" xml:space="preserve">altitudine cum eo exiſtens inuenire, ex ea enim propoſiti cylin <lb/>drici, & </s> <s xml:id="echoid-s13131" xml:space="preserve">fruſti conici ratio illicò apparebit. </s> <s xml:id="echoid-s13132" xml:space="preserve">Per hanc autem Propoſ. <lb/></s> <s xml:id="echoid-s13133" xml:space="preserve">ſatisfit etiam Prop. </s> <s xml:id="echoid-s13134" xml:space="preserve">27. </s> <s xml:id="echoid-s13135" xml:space="preserve">Lib. </s> <s xml:id="echoid-s13136" xml:space="preserve">2. </s> <s xml:id="echoid-s13137" xml:space="preserve">& </s> <s xml:id="echoid-s13138" xml:space="preserve">Sect. </s> <s xml:id="echoid-s13139" xml:space="preserve">K. </s> <s xml:id="echoid-s13140" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13141" xml:space="preserve">4. </s> <s xml:id="echoid-s13142" xml:space="preserve">gen. </s> <s xml:id="echoid-s13143" xml:space="preserve">34. </s> <s xml:id="echoid-s13144" xml:space="preserve">eiuſdem Lib. </s> <s xml:id="echoid-s13145" xml:space="preserve">2. </s> <s xml:id="echoid-s13146" xml:space="preserve"><lb/>vbi contenditur probare, conicorum fruſta in eadem baſi, & </s> <s xml:id="echoid-s13147" xml:space="preserve">altitudi-<lb/>ne exiſtentia, eſſe inuicem æqualia, hoc enim per hanc Prop maniſe-<lb/>ſtum eſt.</s> <s xml:id="echoid-s13148" xml:space="preserve"/> </p> <div xml:id="echoid-div1176" type="float" level="2" n="1"> <note position="right" xlink:label="note-0529-01" xlink:href="note-0529-01a" xml:space="preserve">_5. huius._</note> </div> </div> <div xml:id="echoid-div1178" type="section" level="1" n="704"> <head xml:id="echoid-head737" xml:space="preserve">THEOREMA X PROPOS. X.</head> <p> <s xml:id="echoid-s13149" xml:space="preserve">CYlindri cus ad fruſtum conicum quodcunq; </s> <s xml:id="echoid-s13150" xml:space="preserve">in eadem <lb/>baſi, & </s> <s xml:id="echoid-s13151" xml:space="preserve">altitudine cum eo conſtitutum (ſumptis dua- <lb/><emph style="bf">bus homologis in oppoſitis baſibus fruſti conici) eam ha-</emph> <lb/><emph style="bf">bet rationem, quam quadratum maioris homologarum ad</emph> <lb/><emph style="bf">rectangulum ſub ambabus homologis, vna cum tertia</emph> <lb/>parte quadrati differentiæ earundem. </s> <s xml:id="echoid-s13152" xml:space="preserve">Idem verò fruſtum <lb/>ad conicum in eadem baſi, & </s> <s xml:id="echoid-s13153" xml:space="preserve">altitudine, cum eo exiſten- <lb/>tem, erit vt rectangulum ſub maiori, & </s> <s xml:id="echoid-s13154" xml:space="preserve">tripla minoris, vna <lb/><emph style="bf">cum quadrato diferentiæ earundem homologarum, ad</emph> <lb/>maioris quadratum.</s> <s xml:id="echoid-s13155" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13156" xml:space="preserve">Sint in quacunque baſi, PRQS, & </s> <s xml:id="echoid-s13157" xml:space="preserve">eadem altitudine cylindri-<lb/>cus, FQ, fi uſtum conici, BPRQS, nempè, LNMISR, in baſi mi-<lb/>noriquoque, LNMI, & </s> <s xml:id="echoid-s13158" xml:space="preserve">conicus, OPRQS, ſecto autem quomodo-<lb/>cunq; </s> <s xml:id="echoid-s13159" xml:space="preserve">conico plano per verticem acto, producatur triangulum, B <lb/> <anchor type="note" xlink:label="note-0529-02a" xlink:href="note-0529-02"/> PQ, ſecans oppoſitas baſes fruſti conici in rectis, LM, PQ, quæ <lb/>erunt homologæ ſimilium ſigurarum, LNMI, PRQS, ſimiliter, <lb/>codem extenſo plano, ac completo cylindrico in eadem altitudi-<lb/>ne cum conico, BRS, ſecentur eius oppoſitæ baſes, necnon ſigu-<lb/>ra, FVHG, ab eodem plano in rectis, AC, FH, PQ. </s> <s xml:id="echoid-s13160" xml:space="preserve">Dico ergo <pb o="510" file="0530" n="530" rhead="GEOMETRIÆ"/> <anchor type="figure" xlink:label="fig-0530-01a" xlink:href="fig-0530-01"/> cylindricum, FQ, ad fruſtum conici, <lb/>NISR, eandem rationem habere, <lb/>quam quadratum, PQ, ad rectan <lb/>gulem, ſub, PQ, LM, vna cum {1/3}. <lb/></s> <s xml:id="echoid-s13161" xml:space="preserve">quadradifferentiæ earundem. </s> <s xml:id="echoid-s13162" xml:space="preserve">Idem <lb/>verò fruſtum ad conicum, OPQ, eſſe <lb/>vt rectangulum ſub, PQ, & </s> <s xml:id="echoid-s13163" xml:space="preserve">tripla, <lb/>LM, vna cum quadrato differentiæ <lb/>earundem, ad idem quadratum, P <lb/>Q. </s> <s xml:id="echoid-s13164" xml:space="preserve">Etenim cylindricus, FQ, ad fru-<lb/>ſtum conici, NISR, habet rationem <lb/>compoſitam ex ratione cylindrici, <lb/>FQ, ad cylindricum, AQ, ideſt ex <lb/>ratione, FP, ad, PA, vel, LP, ad, BP, <lb/> <anchor type="note" xlink:label="note-0530-01a" xlink:href="note-0530-01"/> vel exceſius, PQ, ſuper, LM, (qui <lb/>ſit, FX,) ad, PQ, & </s> <s xml:id="echoid-s13165" xml:space="preserve">ex ratione cylindri-<lb/>ci, AQ, ad conicum, BSR, ideſt ex <lb/>ea, quam habet, PQ, ad {1/3}. </s> <s xml:id="echoid-s13166" xml:space="preserve">PQ, & </s> <s xml:id="echoid-s13167" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0530-02a" xlink:href="note-0530-02"/> tandem ex ratione conici, BSR, ad <lb/>fruſtum, ISRN, quæ eſt eadem ei, quam habet cubus, PQ, vel, F <lb/>H, ad parallelepipedum ter ſub, HX, & </s> <s xml:id="echoid-s13168" xml:space="preserve">quadrato, XF, ter ſub, F <lb/>X, & </s> <s xml:id="echoid-s13169" xml:space="preserve">quadrato, XH, cum cubo, FX, eſt enim conicus, BSR, ſimi-<lb/> <anchor type="note" xlink:label="note-0530-03a" xlink:href="note-0530-03"/> lis conico, BIN, & </s> <s xml:id="echoid-s13170" xml:space="preserve">ideò, BSR, ad, BIN, eſt vt cubus, PQ, vel, FH, <lb/>ad cubum, LM, ſeu ad cubum, XH, vnde cum cubus, FH, æquæ-<lb/>tur cubis, FX, XH, cum parallelepipedis ter ſub, FX, & </s> <s xml:id="echoid-s13171" xml:space="preserve">quadrato, <lb/>XH, & </s> <s xml:id="echoid-s13172" xml:space="preserve">ter ſub, HX, & </s> <s xml:id="echoid-s13173" xml:space="preserve">quadrato, XF, ideò per conuerſionem ra-<lb/>tionis conicus, BSR, ad fruſtum, ISRN, erit vt cubus, FH, ad <lb/>parallelepipedum ter ſub, FX, & </s> <s xml:id="echoid-s13174" xml:space="preserve">quadrato, XF, ter ſub, XF, & </s> <s xml:id="echoid-s13175" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0530-04a" xlink:href="note-0530-04"/> quadrato, HX, cum cubo, HX. </s> <s xml:id="echoid-s13176" xml:space="preserve">Duæ rationes autem nempè, quã <lb/>habet, FX, ad, PQ, &</s> <s xml:id="echoid-s13177" xml:space="preserve">, PQ, ad ſui {1/3}. </s> <s xml:id="echoid-s13178" xml:space="preserve">componunt rationem, FX, ad <lb/>{1/3}. </s> <s xml:id="echoid-s13179" xml:space="preserve">PQ, vel triplæ, FX, ad, PQ, ſeu, FH, vel, ſumpto pro communi <lb/>baſi quadrato, FH, componunt rationem parallelepipedi ſub tri-<lb/>pla, FX, & </s> <s xml:id="echoid-s13180" xml:space="preserve">ſub quadrato, FH, ad cubum, FH, quæ proportio cum <lb/>ea, quam d ximus habere cubum, FH, ad paralleiepipedum ter ſub <lb/>HX, & </s> <s xml:id="echoid-s13181" xml:space="preserve">quadrato, XF, ter ſub, XF, & </s> <s xml:id="echoid-s13182" xml:space="preserve">quadrato, HX, cum, cubo, <lb/>FX, componit rationem parallelepipedi ſub tripla, FX, & </s> <s xml:id="echoid-s13183" xml:space="preserve">quadra-<lb/>to, FH, ad parallelepipedum ter ſub, HX, & </s> <s xml:id="echoid-s13184" xml:space="preserve">quadrato, XF, ter <lb/>ſub, XF, & </s> <s xml:id="echoid-s13185" xml:space="preserve">quadrato, HX, cum cubo, XF, ergo cylindricus, FQ, <lb/>ad fruſtum, ISRN, erit vt parallelepipedum ſub tripla, FX, & </s> <s xml:id="echoid-s13186" xml:space="preserve">qua-<lb/>drato, FH, ad dicta ſex parailelepipeda cum cubo, FX, vel vt eo-<lb/>rum ſub tripla, idéſt vt parallelepipedum ſub, FX, & </s> <s xml:id="echoid-s13187" xml:space="preserve">quadrato, F <pb o="511" file="0531" n="531" rhead="LIBER VII."/> H, ad parallelepipedum ſub, FX, & </s> <s xml:id="echoid-s13188" xml:space="preserve">quadrato, XH, ſub, HX, & </s> <s xml:id="echoid-s13189" xml:space="preserve"><lb/>quadrato, XF, cum {1/5}. </s> <s xml:id="echoid-s13190" xml:space="preserve">cubi, XF, hæc tria verò æquantur paralle-<lb/>lepipedo ſub, FX, & </s> <s xml:id="echoid-s13191" xml:space="preserve">rectangulo, FHX, cum {3/3}. </s> <s xml:id="echoid-s13192" xml:space="preserve">quadrati, FX, nam <lb/>parallelepipedum ſub, HX, & </s> <s xml:id="echoid-s13193" xml:space="preserve">quadrato, XF, idem eſt cum paral-<lb/>lelepipedo ſub, FX, & </s> <s xml:id="echoid-s13194" xml:space="preserve">rectangulo, FXH, quod ſi ipſum iunxeris <lb/>pcrallelepipedo ſub, FX, & </s> <s xml:id="echoid-s13195" xml:space="preserve">quadrato, XH, ſimul cum {1/3}. </s> <s xml:id="echoid-s13196" xml:space="preserve">cubi, FX, <lb/>ideſt vna cum parallelepipedo ſub, FX, & </s> <s xml:id="echoid-s13197" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s13198" xml:space="preserve">quadrati, FX, (cum <lb/>ſit communis altitudo) fiet parallelepipedum ſub, FX, & </s> <s xml:id="echoid-s13199" xml:space="preserve">rectan-<lb/>gulo, FXH, cum quadrato, XH, ideſt ſub, FX, & </s> <s xml:id="echoid-s13200" xml:space="preserve">rectangulo, FHX, <lb/>& </s> <s xml:id="echoid-s13201" xml:space="preserve">ſub {1/3}. </s> <s xml:id="echoid-s13202" xml:space="preserve">quadrati, FX, igitur cylindricus, FQ, ad fruſtum, ISRN, <lb/>erit vt parallelepipedum ſub, XF, & </s> <s xml:id="echoid-s13203" xml:space="preserve">quadrato, FH, ad parallelepi-<lb/>pedum ſub, XF, & </s> <s xml:id="echoid-s13204" xml:space="preserve">rectangulo, FHX, cum {1/3}. </s> <s xml:id="echoid-s13205" xml:space="preserve">quadrati, FX, ideſt vt <lb/>quadratum, FH, vel quadratum, PQ, ad rectangulum ſub, FH, HX, <lb/>vel ſub, PQ, LM, vna cum {1/3}. </s> <s xml:id="echoid-s13206" xml:space="preserve">quadrati, FX, differentiæ ipſarum <lb/>homologarum, PQ, LM. </s> <s xml:id="echoid-s13207" xml:space="preserve">Quoniam verò conicus, OSR, eſt {1/3}. </s> <s xml:id="echoid-s13208" xml:space="preserve">cy-<lb/> <anchor type="note" xlink:label="note-0531-01a" xlink:href="note-0531-01"/> lindrici, FQ, idcircò adidem fruſtum, ISRN, conicus, OSR, erit <lb/>vt {1/3}. </s> <s xml:id="echoid-s13209" xml:space="preserve">quadrati, PQ, ad rectangulum ſub, PQ, LM, cum {1/3}. </s> <s xml:id="echoid-s13210" xml:space="preserve">quadra-<lb/>ti, FX, vel vt quadratum, PQ, ad rectangulum ſub, PQ, & </s> <s xml:id="echoid-s13211" xml:space="preserve">tripla, <lb/>LM, cum quadrato, FX, & </s> <s xml:id="echoid-s13212" xml:space="preserve">conuertendo fruſtum, ISRN, ad coni-<lb/>cum, OSR, erit vt rectangulum ſub, PQ, & </s> <s xml:id="echoid-s13213" xml:space="preserve">tripla, LM, cum {1/3}. <lb/></s> <s xml:id="echoid-s13214" xml:space="preserve">quadrati, FX, differentiæ earundem homologarum, ad quadra-<lb/>tum, PQ quæ oſtendere opus erat.</s> <s xml:id="echoid-s13215" xml:space="preserve"/> </p> <div xml:id="echoid-div1178" type="float" level="2" n="1"> <note position="right" xlink:label="note-0529-02" xlink:href="note-0529-02a" xml:space="preserve">Coro. 21. <lb/>l. 1.</note> <figure xlink:label="fig-0530-01" xlink:href="fig-0530-01a"> <image file="0530-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0530-01"/> </figure> <note position="left" xlink:label="note-0530-01" xlink:href="note-0530-01a" xml:space="preserve">Annot. p. <lb/>5. huius.</note> <note position="left" xlink:label="note-0530-02" xlink:href="note-0530-02a" xml:space="preserve">8. huius.</note> <note position="left" xlink:label="note-0530-03" xlink:href="note-0530-03a" xml:space="preserve">Ex diff. 7. <lb/>l, 1. <lb/>Annot.p. <lb/>8. huius.</note> <note position="left" xlink:label="note-0530-04" xlink:href="note-0530-04a" xml:space="preserve">38. lib. 2.</note> <note position="right" xlink:label="note-0531-01" xlink:href="note-0531-01a" xml:space="preserve">8. huius.</note> </div> </div> <div xml:id="echoid-div1180" type="section" level="1" n="705"> <head xml:id="echoid-head738" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13216" xml:space="preserve">PEr ſuperiorem autem demonſtrationem ſuppletur prop. </s> <s xml:id="echoid-s13217" xml:space="preserve">28. <lb/></s> <s xml:id="echoid-s13218" xml:space="preserve">1, 2. </s> <s xml:id="echoid-s13219" xml:space="preserve">necnon ei, quod colligitur in ſec. </s> <s xml:id="echoid-s13220" xml:space="preserve">L. </s> <s xml:id="echoid-s13221" xml:space="preserve">& </s> <s xml:id="echoid-s13222" xml:space="preserve">M. </s> <s xml:id="echoid-s13223" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13224" xml:space="preserve">4. </s> <s xml:id="echoid-s13225" xml:space="preserve">gen. </s> <s xml:id="echoid-s13226" xml:space="preserve">24. </s> <s xml:id="echoid-s13227" xml:space="preserve"><lb/>eiuſdem 1. </s> <s xml:id="echoid-s13228" xml:space="preserve">2. </s> <s xml:id="echoid-s13229" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13230" xml:space="preserve">autem prop. </s> <s xml:id="echoid-s13231" xml:space="preserve">28. </s> <s xml:id="echoid-s13232" xml:space="preserve">eſt in gratiam methodi indiui-<lb/>ſibilium. </s> <s xml:id="echoid-s13233" xml:space="preserve">Quod prop. </s> <s xml:id="echoid-s13234" xml:space="preserve">29. </s> <s xml:id="echoid-s13235" xml:space="preserve">eiuſdem 1. </s> <s xml:id="echoid-s13236" xml:space="preserve">2. </s> <s xml:id="echoid-s13237" xml:space="preserve">ſi intelligamus in eius figu-<lb/>ra latera, CD, DB, deſcribere ſimiles figuras planas, in quibus tã-<lb/>quam in baſibus cylindrici conſiſtant, quorum latera ſint, CD, <lb/>pro figura, DB, & </s> <s xml:id="echoid-s13238" xml:space="preserve">DB, pro figura, CD, oſtendemus conſimili ibi <lb/>traditæ demonſtrationi cylindricum ſublateræ, DB, baſi figura, D <lb/>C, ad cylindricum ſub latere, DC, baſi figura, BD, prædictæ ſimili <lb/>eſſe vt, DC, ad, DB, & </s> <s xml:id="echoid-s13239" xml:space="preserve">ſic etiam eſſe conicum ſub lateribus, CB, B <lb/>D, baſi figura, CD, ad conicum ſub lateribus, BC, CD, baſi figura <lb/>ipſius, DB, habent enim cylindrici inter ſe, necnon & </s> <s xml:id="echoid-s13240" xml:space="preserve">conici, ra-<lb/>tionem, compoſitam ex ratione baſium, & </s> <s xml:id="echoid-s13241" xml:space="preserve">altitudinum, leu late-<lb/>rum æqualiter baſibus inclinatorum, vt ſuperius denuò animadu-<lb/> <anchor type="note" xlink:label="note-0531-02a" xlink:href="note-0531-02"/> crſum eſt: </s> <s xml:id="echoid-s13242" xml:space="preserve">Per hæc autem ſatisfit ctiam Sec. </s> <s xml:id="echoid-s13243" xml:space="preserve">N. </s> <s xml:id="echoid-s13244" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13245" xml:space="preserve">4. </s> <s xml:id="echoid-s13246" xml:space="preserve">gen. </s> <s xml:id="echoid-s13247" xml:space="preserve">34.</s> <s xml:id="echoid-s13248" xml:space="preserve"> <pb o="512" file="0532" n="532" rhead="GEOMETRIÆ"/> eiuſdem lib. </s> <s xml:id="echoid-s13249" xml:space="preserve">2. </s> <s xml:id="echoid-s13250" xml:space="preserve">Circa verò prop. </s> <s xml:id="echoid-s13251" xml:space="preserve">25. </s> <s xml:id="echoid-s13252" xml:space="preserve">& </s> <s xml:id="echoid-s13253" xml:space="preserve">26. </s> <s xml:id="echoid-s13254" xml:space="preserve">cum Corollarijs nihil <lb/>dictum fuit, cum ſintlemmaticæ pro methodo indiuiſibilium, qua <lb/>propter reſtauratione minimè indigere viſæ fuerunt; </s> <s xml:id="echoid-s13255" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13256" xml:space="preserve">33. </s> <s xml:id="echoid-s13257" xml:space="preserve">au. <lb/></s> <s xml:id="echoid-s13258" xml:space="preserve">tem recoletur in examine lib. </s> <s xml:id="echoid-s13259" xml:space="preserve">3. </s> <s xml:id="echoid-s13260" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13261" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13262" xml:space="preserve">34. </s> <s xml:id="echoid-s13263" xml:space="preserve">conſiſtit inde-<lb/>pendenter à methodo indiuiſibilium, vt illius etiam Corollaria, vn. </s> <s xml:id="echoid-s13264" xml:space="preserve"><lb/>de necipſa reſtauranda viſa ſunt. </s> <s xml:id="echoid-s13265" xml:space="preserve">Veruntamen circa Cor. </s> <s xml:id="echoid-s13266" xml:space="preserve">4. </s> <s xml:id="echoid-s13267" xml:space="preserve">gene-<lb/>rale eiuſdẽ prop. </s> <s xml:id="echoid-s13268" xml:space="preserve">34. </s> <s xml:id="echoid-s13269" xml:space="preserve">ſuperius ſuis locis adnotata fuerunt, quæ ani-<lb/>maduertenda erant. </s> <s xml:id="echoid-s13270" xml:space="preserve">Reliquæ tandem propoſitiones à 35. </s> <s xml:id="echoid-s13271" xml:space="preserve">vſq; </s> <s xml:id="echoid-s13272" xml:space="preserve">ad <lb/>finem lib. </s> <s xml:id="echoid-s13273" xml:space="preserve">2. </s> <s xml:id="echoid-s13274" xml:space="preserve">non pendent ab indiuiſibilium methodo, & </s> <s xml:id="echoid-s13275" xml:space="preserve">propte-<lb/>rea circa illas nihil nobis dicendum occurrit. </s> <s xml:id="echoid-s13276" xml:space="preserve">Relicta deniq; </s> <s xml:id="echoid-s13277" xml:space="preserve">fuit <lb/>vltimoloco prop. </s> <s xml:id="echoid-s13278" xml:space="preserve">23. </s> <s xml:id="echoid-s13279" xml:space="preserve">cum Corollarij ſectionibus, ac prop. </s> <s xml:id="echoid-s13280" xml:space="preserve">30. </s> <s xml:id="echoid-s13281" xml:space="preserve">31. </s> <s xml:id="echoid-s13282" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s13283" xml:space="preserve">32. </s> <s xml:id="echoid-s13284" xml:space="preserve">a 23. </s> <s xml:id="echoid-s13285" xml:space="preserve">præcipuè dependentibus cum paulò diligentiorem ani-<lb/>maduerſionem popoſcere viderentur, præſertim verò cum propo-<lb/>ſitione 23. </s> <s xml:id="echoid-s13286" xml:space="preserve">reſtaurata, aliæ quædam propoſitiones lemmaticæ, ad <lb/>rem noſtram pertinentes, forent fuperexſtruendæ, vt in ſequenti-<lb/>bus manifeſtum crit.</s> <s xml:id="echoid-s13287" xml:space="preserve"/> </p> <div xml:id="echoid-div1180" type="float" level="2" n="1"> <note position="right" xlink:label="note-0531-02" xlink:href="note-0531-02a" xml:space="preserve">Annot. p-<lb/>5. & 8. hu. <lb/>ius.</note> </div> </div> <div xml:id="echoid-div1182" type="section" level="1" n="706"> <head xml:id="echoid-head739" xml:space="preserve">THEOREMA XI. PROPOS. XI.</head> <p> <s xml:id="echoid-s13288" xml:space="preserve">SI propoſitum quodcumq; </s> <s xml:id="echoid-s13289" xml:space="preserve">ſolidum parallelis quotcumq; <lb/></s> <s xml:id="echoid-s13290" xml:space="preserve">planis ita ſecari poſſit, vt conceptæ ex ſecantibus pla-<lb/>nis in eo figuræ ſint ſemper parallelogramma rectangula, <lb/>latera verò eadem deſcribentia ſint omnia vni cuidam la-<lb/>teri, vt regulæ æquidiſtantia: </s> <s xml:id="echoid-s13291" xml:space="preserve">Illud ſuperficiebus cylin-<lb/>draceis comprehenſum erit.</s> <s xml:id="echoid-s13292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13293" xml:space="preserve">Sit propoſitum quodcunq; </s> <s xml:id="echoid-s13294" xml:space="preserve">ſolidum, ASOC, quod quidem pa-<lb/>rallelis quotcunq; </s> <s xml:id="echoid-s13295" xml:space="preserve">pianis fectum eſſe ſupponatur, efficientibus in <lb/>eo parallelogramma rectangula, EH, IM, latera verò hæc deſcri-<lb/>bentia, GH, LM, vt & </s> <s xml:id="echoid-s13296" xml:space="preserve">reliqua omnia præfata parallelogramma <lb/> <anchor type="note" xlink:label="note-0532-01a" xlink:href="note-0532-01"/> pariter deſcribentia, ſint vni cuidam regulæ, PQ, æquidiſtantia. <lb/></s> <s xml:id="echoid-s13297" xml:space="preserve">Dicoſolidum, ASOC, ſuperficiebus cylindraceis comprehendi. </s> <s xml:id="echoid-s13298" xml:space="preserve"><lb/>Quod enim ſuperficies, in qua iacent omnia prædicta latera, quæ <lb/>rectangula deſcribunt (quæ ſit, CNOD,) ſit cylindracea, manife. </s> <s xml:id="echoid-s13299" xml:space="preserve"><lb/>ſtum eſt ex eo, quod omnia vni regulæ, PQ, ſint parallela, & </s> <s xml:id="echoid-s13300" xml:space="preserve">eadẽ <lb/>ratione ſuperficies, in qua iacent latera rectangulorum prædictis <lb/>oppoſita (quæ ſit, ARSB,) erit cylindracea. </s> <s xml:id="echoid-s13301" xml:space="preserve">Similiter cum planũ, <lb/>EH, æquidiſtet plano, IM, &</s> <s xml:id="echoid-s13302" xml:space="preserve">, GH, ipſi, LM, etiam, EG, ipſi, LI, <lb/>æquidiſtabit, eodem modo autem etiam oſtendemus reliqua late-<lb/>ra, quæ præfatis rectangula deſcribentibus lateribus perpendicola- <pb o="513" file="0533" n="533" rhead="LIBER VII."/> <anchor type="figure" xlink:label="fig-0533-01a" xlink:href="fig-0533-01"/> riter inſiſtunt, eidẽ, <lb/>LI, æquidiſtare, ex <lb/>quo concludemus <lb/>hæc omnia pariter <lb/>in ſuperficie cylin-<lb/>dracea coextendi, <lb/>quæ ſit, ACNR, <lb/>qua methodo pate <lb/>bit etiam ſuperficiẽ, <lb/>BSOD, eſſe cylin-<lb/>draceam, in qua <lb/>quidem iacent late-<lb/>ra rectangulorũ prę-<lb/>dictis oppoſita. </s> <s xml:id="echoid-s13303" xml:space="preserve">Nũc <lb/>ſi ducta intelligan-<lb/>tur oppoſita plana <lb/>ſolidum, AO, tan-<lb/>gentia, ac præfatis <lb/>ſecantibus planis æ-<lb/>quidiſtantia, contin-<lb/>gere poteſt, vt ipſo-<lb/>rũ planorũ cõtactus ſit ex vtraq; </s> <s xml:id="echoid-s13304" xml:space="preserve">parte, vel in puncto, vel in linea, vel <lb/>in plano, vel ex vna parte contactus in vno iſtorum, ex altera verò <lb/>in alio promiſcuè, vt conſideranti facilè innoteſcet, attamen quo-<lb/>modocunq; </s> <s xml:id="echoid-s13305" xml:space="preserve">res ſe habeat etiam ratione iſtorum contactuum fiet, <lb/>vt dictum ſolidum cylindraceis ſuperficiebus comprehendatur, ſi <lb/>enim contactus ex neutra parte fiat in plano, dictum ſolidum non <lb/>alijs ſuperficiebus cylindraceis, quam ijs, quæ dictæ ſunt compre-<lb/>hendetur, vt manifeſtum eſt, ſi vero contactus ſit in plano, illud <lb/>erit parallelogrammum rectangulum, vt, AD, RO, cum enim <lb/>hæc tangentia plana æquidiſtent planis ſecantibus, quę tranſeunt <lb/>per latera cylindrici, cuius, ACNR, BDOS, ſunt ſuperficies, etiam <lb/>ipſa per eiuſdem latera tranſibunt, ergo, AC, BD, ſicut etiam, R <lb/>N, SO, per quæ tranſeunt dicta tangentia plana, ipſis, EG, FH, æ-<lb/> <anchor type="note" xlink:label="note-0533-01a" xlink:href="note-0533-01"/> quidiſtabunt, quo pacto oſtendemus etiam, AB, CD, RS, NO, ipſi, <lb/>EF, GH, pariter æquidiſtare, ergo plana contactuum, AD, RO, <lb/>erunt parallelogramma rectangula, ergo & </s> <s xml:id="echoid-s13306" xml:space="preserve">ipſa erunt ſuperficies <lb/>cylindraceæ, ergo etiam ratione contingentium planorum ſecan-<lb/>tibus planis æquidiſtantium præfatũ ſolidũ ſuperficiebus cylindra-<lb/>ceis comprehendi manifeſtum eſt, quod oſtendere opus erat.</s> <s xml:id="echoid-s13307" xml:space="preserve"/> </p> <div xml:id="echoid-div1182" type="float" level="2" n="1"> <note position="left" xlink:label="note-0532-01" xlink:href="note-0532-01a" xml:space="preserve">Defin. 3. <lb/>lib. 1.</note> <figure xlink:label="fig-0533-01" xlink:href="fig-0533-01a"> <image file="0533-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0533-01"/> </figure> <note position="right" xlink:label="note-0533-01" xlink:href="note-0533-01a" xml:space="preserve">9. l. 1.</note> </div> <pb o="514" file="0534" n="534" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1184" type="section" level="1" n="707"> <head xml:id="echoid-head740" xml:space="preserve">DEFINITIO. A.</head> <p> <s xml:id="echoid-s13308" xml:space="preserve">HViuſmodi ergo ſolida appellabimus nomine com-<lb/>muniſolid a rectangula. </s> <s xml:id="echoid-s13309" xml:space="preserve">Cum verò vnumquodque <lb/>in eiſdem ſolidis ex ſecantibus planis productorum paral-<lb/>lelogrammorum rectangulorum fuerit quadratum, etiam <lb/>ſolida quadrata vocabuntur. </s> <s xml:id="echoid-s13310" xml:space="preserve">Et ipſorum regulæ, quibus <lb/>latera plana rectangula continentia, æquidiſtant.</s> <s xml:id="echoid-s13311" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1185" type="section" level="1" n="708"> <head xml:id="echoid-head741" xml:space="preserve">DEFINITIO. B.</head> <p> <s xml:id="echoid-s13312" xml:space="preserve">INſuper ſolidum quodcunque rectangulum ſub duabus <lb/>quibuſcunque ſuperficiebus dicetur contineri (regulis <lb/>ijſdem ſupradictis) in quibus vnumquodq; </s> <s xml:id="echoid-s13313" xml:space="preserve">ęquidiſtantium <lb/>planorum, ipſum ſolidum rectangulum ita ſecantium, vt <lb/>dictum fuit, æqualia latera per ſectionem iſdem deſigna-<lb/>uerit, ſub quibus parallelogrammum rectangulum, ab eo-<lb/>dem plano ſecante in ſolido productum, continetur. </s> <s xml:id="echoid-s13314" xml:space="preserve">Et <lb/>cum fuerit ſolidum quadratum porerit etiam appellari, ſo-<lb/>lidum quadratum alterutrius dictarum ſuperficierũ ipſum <lb/>continentium. </s> <s xml:id="echoid-s13315" xml:space="preserve">Ipſas verò ſuperficies, æqualia rectangu-<lb/>lorum planorum latera capientes, homologas pariter nun-<lb/>cupabimus, regula quocunq; </s> <s xml:id="echoid-s13316" xml:space="preserve">dictorum eaſdem ſecantium <lb/>planorum.</s> <s xml:id="echoid-s13317" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1186" type="section" level="1" n="709"> <head xml:id="echoid-head742" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13318" xml:space="preserve">IVxta ergo ſuprapoſitas definitiones manifeſtum eſt, quaſnam <lb/>conditiones habere debeant ea ſolida, quæ vocantur ſolida re-<lb/>ctangula: </s> <s xml:id="echoid-s13319" xml:space="preserve">Erit igitur, ASOC, rectangulum ſolidum: </s> <s xml:id="echoid-s13320" xml:space="preserve">quod ſi, A <lb/>D, EH, IM, RO, & </s> <s xml:id="echoid-s13321" xml:space="preserve">cætera huiuſmodi plana fuerint quadrata, <lb/>poterit etiam dici, ASOC, quadratum ſolidum: </s> <s xml:id="echoid-s13322" xml:space="preserve">Ipſius autem re-<lb/>gulæ erunt ex. </s> <s xml:id="echoid-s13323" xml:space="preserve">g. </s> <s xml:id="echoid-s13324" xml:space="preserve">NO, OS, quibus latera rectangula continentia <lb/>æquidiſtant. </s> <s xml:id="echoid-s13325" xml:space="preserve">Eſto nunc, quod parallela plana, quæ in ſolido, A <lb/>O, rectangula, AD, EH, IM, RO, genuerunt, indefinitè produ-<lb/>cta occurrerint ex g. </s> <s xml:id="echoid-s13326" xml:space="preserve">tribus ſuperficiebus, TX℟Y, DOrZ, φΝΟΛ8, <lb/>in quibus per ſectionem deſignauerint, TY, æqualem ipſi, BD, &</s> <s xml:id="echoid-s13327" xml:space="preserve"> <pb o="515" file="0535" n="535" rhead="LIBER VII."/> φ8, æqualem, CD, ſimiliter, & </s> <s xml:id="echoid-s13328" xml:space="preserve">Δ, VΩ, X℟, deinceps æquales ipſis, <lb/>FH, kM, SO, ſicut etiam, ΣΛ, ΠΛ, NO, deinceps æqualesipſis, <lb/>GH, LM, NO, & </s> <s xml:id="echoid-s13329" xml:space="preserve">in ſuperficie, DΖΓΟ, ipſas, DZ, H9, Μβ, CΓ, <lb/>deinceps æquales eiſdem, BD, FH, KM, SO, & </s> <s xml:id="echoid-s13330" xml:space="preserve">cætera plana pa-<lb/>rallela ſimiliter ſe habuerint (ipſę autem ſuperficies, BO, DΓ, T℟, <lb/>inter ſe, vti etiam, CO, φO, inter ſe, erunt homologæ, regula quo-<lb/>cunq; </s> <s xml:id="echoid-s13331" xml:space="preserve">dictorum eaſdem ſecantium planorum inter ſe æquidiſtan-<lb/>tium.) </s> <s xml:id="echoid-s13332" xml:space="preserve">Dicimus ergo ſolidum rectangulum, AO, nedum contine-<lb/>ri ex. </s> <s xml:id="echoid-s13333" xml:space="preserve">g. </s> <s xml:id="echoid-s13334" xml:space="preserve">ſub ſuperficiebus, BDOS, CDON, in quibus iacent latera <lb/>præfata rectangula continentia, ſed etiam ſub ſuperficiebus, T℟, <lb/>CO, vel, T℟, φO, vel ſub ſuperficiebus, ΓΖDΟ, ODCN, vel ſub, <lb/>ΓΖDΟ, φΝΟΛ8, in his enim plana parallela produxerunt latera <lb/>ijs æqualia, ſub quibus parallelogramma rectangula, AD, EH, I <lb/>M, RO, & </s> <s xml:id="echoid-s13335" xml:space="preserve">cætera huiuſmodi continentur, vt dictum fuit, in quo <lb/>non nihil à modo loquendi in planis diſcedere videmur, dicitur. </s> <s xml:id="echoid-s13336" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s13337" xml:space="preserve">ex. </s> <s xml:id="echoid-s13338" xml:space="preserve">g. </s> <s xml:id="echoid-s13339" xml:space="preserve">rectangulum planum, AD, contineri ſub, BD, DC, quæ re-<lb/>ctum angulum conſtituunt, & </s> <s xml:id="echoid-s13340" xml:space="preserve">non ſub, TY, φ8, quæ ipſius rectũ <lb/> <anchor type="note" xlink:label="note-0535-01a" xlink:href="note-0535-01"/> angulum non conſtituunt, hoc tamen loquendimodo vſus ſum, <lb/>potius ſoliditatis deterrninationẽ reſpiciens, quam continentiam, <lb/>quæ fit à ſuperficiebus in ambitu contentorum ſolidorum exiſten-<lb/>tibus, cum enim cernerem non omnes ſuperficies ſolidum rectan-<lb/>gulum vt ſic continentes poſſe in ipſius contenti ſolidi ambitu re-<lb/>periri (vt ex. </s> <s xml:id="echoid-s13341" xml:space="preserve">g. </s> <s xml:id="echoid-s13342" xml:space="preserve">cum contineretur duabus ſuperficiebus planis in il-<lb/>lius ambitu exiſtentibus, aliæ autem illis homologæ eſſent curuæ) <lb/>& </s> <s xml:id="echoid-s13343" xml:space="preserve">tamen latera ‘in his concepta viderem adæquari lateribus rectā-<lb/>gula plana continentibus, & </s> <s xml:id="echoid-s13344" xml:space="preserve">conſequenter eorundem areæ quan-<lb/>titatem præſcribere, vnde & </s> <s xml:id="echoid-s13345" xml:space="preserve">iſtæ prædictis homologæ ſuperficies <lb/>viderentur ipſius contenti ſoliditatẽ determinare (quęcumq; </s> <s xml:id="echoid-s13346" xml:space="preserve">enim <lb/>ſolida ſub ip ſius contineantur inter ſe erunt æqualia, vt infra oſtẽ-<lb/>demus) ideò volui præfata ſolida rectangula dici ſub omnibus his <lb/>ſuper ficiebus homologis ſecundum eandem regulam contineri. <lb/></s> <s xml:id="echoid-s13347" xml:space="preserve">Quemadmodum ſi quis aliter ab Euclide diceret parallelogrammũ <lb/>rectangulum nedum ſub lateribus ipſius angulum rectum conſti-<lb/>tuentibus, ſed etiam ſub quibuſcunq; </s> <s xml:id="echoid-s13348" xml:space="preserve">alijs lateribus prædictis æ-<lb/>qualibus contineri, ſubintelligendo non hoc parallelogrammũ <lb/>in ipſius ambitu neceſlariò ipſa latera continentia habere, ſed per <lb/>ea ſiue ſint in ambitu, ſiue non, ipſius areæ quantitatem determi-<lb/>nari, patallelogrammum enim rectangulum contentum ſub duo-<lb/>bus lateribus, iuxta modum loquendi Euclidianum, æquatur cui-<lb/>cumq; </s> <s xml:id="echoid-s13349" xml:space="preserve">parallelogrammo rectangulo ſub alijs duobus prædictis æ-<lb/>qualibus contento. </s> <s xml:id="echoid-s13350" xml:space="preserve">Quod ſi quis attendat demonſtrationes ſec.</s> <s xml:id="echoid-s13351" xml:space="preserve"> <pb o="516" file="0536" n="536" rhead="GEOMETRIÆ"/> Elem. </s> <s xml:id="echoid-s13352" xml:space="preserve">à prima illius def. </s> <s xml:id="echoid-s13353" xml:space="preserve">dependentes, animaduertet ſuam ſortiri <lb/>veritatem ſiue ſecundum hanc, ſiue ſecundum adductam defini-<lb/>tionem intelligantur; </s> <s xml:id="echoid-s13354" xml:space="preserve">conſimilem autem demonſtrationum ſeriẽ <lb/>exſuperioribus definitionibus emanantem, inferius & </s> <s xml:id="echoid-s13355" xml:space="preserve">ipſæ ſubiun-<lb/>gam.</s> <s xml:id="echoid-s13356" xml:space="preserve"/> </p> <div xml:id="echoid-div1186" type="float" level="2" n="1"> <note position="right" xlink:label="note-0535-01" xlink:href="note-0535-01a" xml:space="preserve">Pri. Def. <lb/>Sec. Elem.</note> </div> </div> <div xml:id="echoid-div1188" type="section" level="1" n="710"> <head xml:id="echoid-head743" xml:space="preserve">THEOREMA XII. PROPOS. XII.</head> <p> <s xml:id="echoid-s13357" xml:space="preserve">PRopoſito quocunq; </s> <s xml:id="echoid-s13358" xml:space="preserve">ſolido rectangulo iuxta datas re-<lb/>gulas, ac ſub duabus quibuſdam ſuperficiebus con-<lb/>tento; </s> <s xml:id="echoid-s13359" xml:space="preserve">indefinita numero ſolida rectangula pariter dari <lb/>poſſunt, iuxta eaſdem regulas, quorum vnumquodq; </s> <s xml:id="echoid-s13360" xml:space="preserve">pro-<lb/>poſito ſolido æquale erit, ac ſub eiſdem ſuperficiebus <lb/>continebitur.</s> <s xml:id="echoid-s13361" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13362" xml:space="preserve">Sit propoſitum quodcunq; </s> <s xml:id="echoid-s13363" xml:space="preserve">ſolidum rectangulum, POIS, ſub <lb/>duabus ſuperficiebus, QSIB, OBIH, contentum, cuius regulæ <lb/>ſint, HI, IS. </s> <s xml:id="echoid-s13364" xml:space="preserve">Dico indefinita numero ſolida rectangula regulis <lb/>eiſdem pariter dari poſſe, quorum vnumquodq; </s> <s xml:id="echoid-s13365" xml:space="preserve">ipſi, POIS, æqua-<lb/>le erit, ac ſub eiſdem ſuperficiebus, QSIB, OBIH, continebitur. <lb/></s> <s xml:id="echoid-s13366" xml:space="preserve">Igitur rectangulum ſolidum, POIS, ſuperficiebus cylindraceis cõ-<lb/>prehendetur, illæ ergo ſuperficies indefinitè hincinde produci in-<lb/>telligantur, in quibus latera fignata per plana parallela, in ſolido <lb/>parallelogramma rectangula gignentia, vni regulæ, vt ipſi, HI, æ-<lb/>quidiſtant, tales autem ſunt ſuperficies, PS, SH, HB, BP, ſicut <lb/> <anchor type="note" xlink:label="note-0536-01a" xlink:href="note-0536-01"/> etiam, PH, HS, SB, BP, quarum eſt pariter regula, SI, cum enim, <lb/>RI, PB, fuerint parallelogramma rectangula, tam iuxta regulam, <lb/>HI, quam iuxta, SI, poſſunt in ipſis rectę lineæ vni cuidam paral-<lb/>lelæ deſignari: </s> <s xml:id="echoid-s13367" xml:space="preserve">Producatur autem ipſæ, PS, SH, HB, BP, hinc <lb/>inde inderinitè, intelligaturq; </s> <s xml:id="echoid-s13368" xml:space="preserve">ſimiliter in quacunq; </s> <s xml:id="echoid-s13369" xml:space="preserve">productarum <lb/>ſuperficierum, vt in, OI, producta, exiſtere figura quæcunque, ΔΚ. <lb/></s> <s xml:id="echoid-s13370" xml:space="preserve">Μλ, homologa, iuxta regulam, RI, ipſi, OHIB, in eadem ſuperfi-<lb/>cie exi@tenti, deinde per illus ambitum, ΔΚΜλ, feratur quædam <lb/>recta linea indeficitè hinc inde producta, temper ipſi, SI, æquidi-<lb/>ſtanter, donec omnem illius percurrerit ambitum, gignens ſuper-<lb/>ficies cylindraceas, CΔΚΝ, NM, GMλD, DΔ, abſcindenſq; </s> <s xml:id="echoid-s13371" xml:space="preserve">a fu-<lb/>perficie, QR, indefinitè producta ſuperficiem cylindraceam, DCN <lb/>G. </s> <s xml:id="echoid-s13372" xml:space="preserve">Eſto igitur, quod vnum parallelorum planorum in ſolido, PI, <lb/>rectangula plana gignentium, vt, quod genuit, XV, indefinitè pro-<lb/>ductum, ita vt fecet ſolidum, CM, in eo produxerit figuram, E℟, <lb/>quoniam ergo, EF, eſt parallela ipſi, & </s> <s xml:id="echoid-s13373" xml:space="preserve">℟, nam eſt portio, EY, quę, <lb/>eſt parallela ipſi, & </s> <s xml:id="echoid-s13374" xml:space="preserve">V, ſimiliter, E&</s> <s xml:id="echoid-s13375" xml:space="preserve">, eſt parallela ipſi, F℟, erit, E℟, <pb o="517" file="0537" n="537" rhead="LIBER VII."/> parallelogrammum, &</s> <s xml:id="echoid-s13376" xml:space="preserve">, F℟&</s> <s xml:id="echoid-s13377" xml:space="preserve">, eſt angulus rectus, eſt enim exterior <lb/>parallelarum, F℟, XT, & </s> <s xml:id="echoid-s13378" xml:space="preserve">ideòipſi interiori, XT&</s> <s xml:id="echoid-s13379" xml:space="preserve">, æqualis, erit, <lb/>E℟, etiam rectangulum, & </s> <s xml:id="echoid-s13380" xml:space="preserve">quia, & </s> <s xml:id="echoid-s13381" xml:space="preserve">℟, æquatur ipſi, TV, ſunt. </s> <s xml:id="echoid-s13382" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s13383" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0537-01a" xlink:href="fig-0537-01"/> ΔΜ, OI, figuræ homologæ, <lb/>ſicut etiam, F℟, æquatur ip-<lb/>ſi, YV, ideò rectangulum, E <lb/>℟, erit æquale rectangulo, X <lb/>V. </s> <s xml:id="echoid-s13384" xml:space="preserve">Eadem ratione oſtende-<lb/>mus, quæcunq; </s> <s xml:id="echoid-s13385" xml:space="preserve">alia duo re-<lb/>ctangula ab eodem dictorum <lb/>ęquidiſtantium plano in ipſis <lb/>ſolidis producta ęqualia eſie, <lb/>ergo cum ſolida, CM, PI, ſint <lb/>in eadem altitudine ſumpta <lb/>regulis eiſdem æqualibus re-<lb/>ctangulis, cõcluduntur enim <lb/>inter extrema plana parallela, quorum contactus eſt in planis, N <lb/>M, RI; </s> <s xml:id="echoid-s13386" xml:space="preserve">Cλ, PB, ideò dicta ſolida erunt æqualiter analoga iuxta di-<lb/> <anchor type="note" xlink:label="note-0537-01a" xlink:href="note-0537-01"/> ctas regulas, ergo inter ſe æqualia erunt; </s> <s xml:id="echoid-s13387" xml:space="preserve">& </s> <s xml:id="echoid-s13388" xml:space="preserve">cum ſuperficies, ΔΜ, <lb/>ſit homologa ipſi, OI, &</s> <s xml:id="echoid-s13389" xml:space="preserve">, DM, ipſi, QI, regula plano, RI, propte-<lb/>rea & </s> <s xml:id="echoid-s13390" xml:space="preserve">erit, CM, ſolidum rectangulum æquale ipſi, PI, & </s> <s xml:id="echoid-s13391" xml:space="preserve">ſub eiſdẽ <lb/>ſuperficiebus, QI, IO, continebitur, & </s> <s xml:id="echoid-s13392" xml:space="preserve">eius regulæ erunt pariter <lb/>ipſæ, HI, IS. </s> <s xml:id="echoid-s13393" xml:space="preserve">Cum verò in ſuperficie, OI, indefinitè producta, in-<lb/>definitæ numero figuræ ipſi, OI, homologæ, regula plano, RI, <lb/>ſupponi poſſint, vt facillimè apparet, ideò ſupradicta methodo tot <lb/>ſolida rectangula ijſdem ſuperexſtrui poterunt, regulis eiſdẽ, quot <lb/>erunt figuræ ipſi, HP, homologæ, iuxta dictas regulas, ideſt nu-<lb/>mero indefinita, quorum vnumquodq; </s> <s xml:id="echoid-s13394" xml:space="preserve">ipſi, PI, adæquari, ac ſub <lb/>eiſdem ſuperficiebus, QI, IO, contineri, vt ſupra oſtendemus. <lb/></s> <s xml:id="echoid-s13395" xml:space="preserve">Quemadmodũ ſi etiã indefinitè ſuperficies, PH, HS, SB, BP, ſupra, <lb/>vel infra producerentur, alia indefinita numero ſolida rectangula <lb/>inueniri eodem modo poſſent, quorum vnumquodque ipſi, PI, <lb/>adæquari, ac ſub eiſdem ſuperficiebus, QI, IO, contineri, regulis <lb/>eiſdem, HI, IS, pari ratione probaremus. </s> <s xml:id="echoid-s13396" xml:space="preserve">Hæc autem oſtenden-<lb/>da proponebantur.</s> <s xml:id="echoid-s13397" xml:space="preserve"/> </p> <div xml:id="echoid-div1188" type="float" level="2" n="1"> <note position="left" xlink:label="note-0536-01" xlink:href="note-0536-01a" xml:space="preserve">11. huius.</note> <figure xlink:label="fig-0537-01" xlink:href="fig-0537-01a"> <image file="0537-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0537-01"/> </figure> <note position="right" xlink:label="note-0537-01" xlink:href="note-0537-01a" xml:space="preserve">1. huius.</note> </div> </div> <div xml:id="echoid-div1190" type="section" level="1" n="711"> <head xml:id="echoid-head744" xml:space="preserve">COROLL ARIVM I.</head> <p style="it"> <s xml:id="echoid-s13398" xml:space="preserve">_E_X ſupra demonſtratis manifeſtum eſt, quomodo ſolidum rectan-<lb/>gulum ſub duabus datis ſuperficiebus contentum, iuxta datas <lb/>regulas, in data ſuperficie cylindracea, quæ continentium altera fit <pb o="518" file="0538" n="538" rhead="GEOMETRIE"/> homologæ, deſcribi poſſit, ſuperficies enim CG, deſcribetur & </s> <s xml:id="echoid-s13399" xml:space="preserve">ipſa <lb/>lateræ, NG, moto per lineam, NEC, ſemper ipſi, HI, æquidiſtanter.</s> <s xml:id="echoid-s13400" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1191" type="section" level="1" n="712"> <head xml:id="echoid-head745" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s13401" xml:space="preserve">_I_N ſuper innoteſcit ſolidum rectangulum quodcunq; </s> <s xml:id="echoid-s13402" xml:space="preserve">eſſe ſemper <lb/>portionem ſolidam duobus cylindricis ſe ſe inuicem per ſuas ſu-<lb/>perficies cylindraceas decuſſantibus communem, quorum laterum re-<lb/>regulæ ſi ſimul ad vnum punctum componantur, ſibi inuicem perpen-<lb/>diculares erunt, vt regula, HI, cui æquidiſtant latera ſuperficiei cy-<lb/>lindraceæ, PSHBP, eſt ad angulum rectum cum regula, IS, cui æquidi-<lb/>ſtant latera ſuperficici cylindracea, PHSBP, quod quidem ſolidum, P <lb/>I, patet gigni ex concurſu dictarum ſuperficierum, ſicut, CM, ex con-<lb/>curſu earundem ‘PSHBP, indefinitè productarum, necnon ipſarum, <lb/>CκGΛC, hoc eſt vnumquodq; </s> <s xml:id="echoid-s13403" xml:space="preserve">ipſorum, CM, PI, eſſe portionem ſolidam <lb/>communem duobus cylindricis, quorum laterum regulæ ſunt ipſæ, HI, <lb/>IS, ad inuicem perpendiculares.</s> <s xml:id="echoid-s13404" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1192" type="section" level="1" n="713"> <head xml:id="echoid-head746" xml:space="preserve">COROLLARIVM III.</head> <p style="it"> <s xml:id="echoid-s13405" xml:space="preserve">_V_Lterius patet, quod ſolida rectangula ſub ſuperficiebus bomolo-<lb/>gis iu xta eaſdem regulas contenta, inter ſe ſunt æqualia: </s> <s xml:id="echoid-s13406" xml:space="preserve">Et <lb/>enim ſi propoſit@ ex. </s> <s xml:id="echoid-s13407" xml:space="preserve">g. </s> <s xml:id="echoid-s13408" xml:space="preserve">eſſent ſuperficies, QI, IO, homologæ ipſis, DM, <lb/>ΜΔ, regula plano, RI, & </s> <s xml:id="echoid-s13409" xml:space="preserve">completa fuiſſent ſolida rectangula, PI, CM, <lb/>ecdem modo oſtenſum fuiſſetipſa inter ſe æqualia eſſe.</s> <s xml:id="echoid-s13410" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1193" type="section" level="1" n="714"> <head xml:id="echoid-head747" xml:space="preserve">COROLLARIVM IV.</head> <p style="it"> <s xml:id="echoid-s13411" xml:space="preserve">_E_X hæc Prop. </s> <s xml:id="echoid-s13412" xml:space="preserve">& </s> <s xml:id="echoid-s13413" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13414" xml:space="preserve">ant. </s> <s xml:id="echoid-s13415" xml:space="preserve">deniq; </s> <s xml:id="echoid-s13416" xml:space="preserve">apparet, quam congruenter di-<lb/>ctum fuerit ſolidum rectangulum nedum ſub duabus ſuperficie-<lb/>bus in eiuſdem ambitu exiſtentibus contineri, ſed etiam ſub duabus <lb/>alijs quibuſcumq; </s> <s xml:id="echoid-s13417" xml:space="preserve">prædictis homologis, iuxta eaſdem regulas, licet <lb/>enim diuerſis ſuperficiébus ipſa ſolida comprehendatur, tamen eadem <lb/>ſemper ſolidatis quantitas conſeruatur, retentis eiſdem regulis, cuius <lb/>determinatio cum ex lateribus habeatur, vel rectangula plana dicto-<lb/>rum ſolidorum continentibus, vel æqualia ijs, quæ eadem continent, <lb/>iaceant verò hæc in dictis ſuperficiebus, propterea non incõgruè, puto, <lb/>dictum fuit præfata ſolida ſub talibus quibuſcumque homologis ſuper-<lb/>ficiebus, regulis eiſdem, contineri.</s> <s xml:id="echoid-s13418" xml:space="preserve"/> </p> <pb o="519" file="0539" n="539" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1194" type="section" level="1" n="715"> <head xml:id="echoid-head748" xml:space="preserve">THEOREMA XIII. PROPOS. XIII.</head> <p> <s xml:id="echoid-s13419" xml:space="preserve">SI, expoſitis duabus quibuſcumq; </s> <s xml:id="echoid-s13420" xml:space="preserve">ſolidorum rectangu-<lb/>lorum deſcriptibilium regulis, ad vnum punctum cõ-<lb/>poſitis, iuxta eaſdem ſolidum rectangulum contineatur <lb/>ſub parallelogrammo, & </s> <s xml:id="echoid-s13421" xml:space="preserve">alia quacumq; </s> <s xml:id="echoid-s13422" xml:space="preserve">figura plana in <lb/>ambitu contenti ſolidi exiſtente, ipſum ſolidum rectangu-<lb/>lum erit cylindricus, & </s> <s xml:id="echoid-s13423" xml:space="preserve">figura plana ſuperius dicta erit il-<lb/>lius baſis. </s> <s xml:id="echoid-s13424" xml:space="preserve">Quod ſi etiam prædicta figura fuerit parallelo-<lb/>grammum, & </s> <s xml:id="echoid-s13425" xml:space="preserve">ambo in illius ambitu, contentum ijſdem <lb/>ſolidum rectangulum erit parallelepipedum.</s> <s xml:id="echoid-s13426" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13427" xml:space="preserve">Exponantur duæ inuicem perpendiculares regulæ, BC, CD, ſo-<lb/>lidorũ deſcriptib liũ ſub parallelogrammo, AC, & </s> <s xml:id="echoid-s13428" xml:space="preserve">figura plana qua <lb/>cumque, HDC, ſit autem deſcriptum ſolidum rectangulum ſub eiſ-<lb/> <anchor type="figure" xlink:label="fig-0539-01a" xlink:href="fig-0539-01"/> dem contentum, AG <lb/>CH, iuxta regulas, B <lb/>C, CD, ita tamen vt <lb/>figura plana, HDC, <lb/>ſit in ambitu ipſius <lb/>contenti ſolidi. </s> <s xml:id="echoid-s13429" xml:space="preserve">Di-<lb/>co, AGCH, eſſe cy-<lb/>lindricum. </s> <s xml:id="echoid-s13430" xml:space="preserve">Quod. </s> <s xml:id="echoid-s13431" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s13432" xml:space="preserve">AC, CG, GH, ſint <lb/>ſuperficies cylindra-<lb/>ceæ, quarum regula, <lb/>BC, manifeſtum eſt, <lb/>quod verò latera per <lb/>ſecantia para lela plana in ipſis deſignata ſint æqualia ipſi, BC, la-<lb/>teri parallelogrammi, AC, ex dictis etiam cõſtare poteſt, ſed maio-<lb/>ris dilucidationis gratia ſit ab aliquo dictorum ſecantium planorũ, <lb/>in ſolido, AGHC, productum rectangulum, IMON, eſt ergo, IN, <lb/>æqualis, MO, hoc eſt ipſi, BC, quo pacto idem de cæteris oſten-<lb/>demus, in parallelogrammo autem, GC, eadem verificantur, & </s> <s xml:id="echoid-s13433" xml:space="preserve">in <lb/>illi oppoſito, ſi contactus plani ipſi, GC, oppoſiti eſſent in plano, <lb/>vt manifeſtum eſt, ergo perinde eſt ac ſi latus æquale, BC, ambitũ <lb/> <anchor type="note" xlink:label="note-0539-01a" xlink:href="note-0539-01"/> figuræ, HDC, extremo ſui puncto ſemper ipſi, BC, æquidiſtanter <lb/>percuriſſet ipſam ſuperficiem, ADBH, deſcribendo, erit ergo, A <lb/>GCH, cylindricus, cuius baſis eſt, HDC, figurá. </s> <s xml:id="echoid-s13434" xml:space="preserve">Præfatum qui-<lb/>dem ſolidum habet in ambitu figuras ipſum continentes, ſed ſi ve- <pb o="520" file="0540" n="540" rhead="GEOMETRIÆ"/> limus etiam caſum intelligere cum tantum figura planaeſt in illius <lb/>ambitu; </s> <s xml:id="echoid-s13435" xml:space="preserve">hoc in ſchemate ant. </s> <s xml:id="echoid-s13436" xml:space="preserve">prop. </s> <s xml:id="echoid-s13437" xml:space="preserve">facilè percipiemus, in qua <lb/>ſint regulæ, SI, IH, continentes verò figuræ, QI, ΔΜ, quarum, <lb/>QI, ſupponatur eſſe parallelogrammum, ſed non in ambitu con-<lb/>tenti eiſdem ſolidi, quod ſit, CM, ΔΜ, verò ſit figura plana, quæ <lb/>debet in ambitu ſolidi reperiri, igitur conſimili methodo oſtende-<lb/>mus etiam, CM, eſſe cylindricum, in baſi, ΔΜ, conſtitutum. </s> <s xml:id="echoid-s13438" xml:space="preserve">Quod <lb/>ſicontinentes figuræ, QI, IO, fuerint ambo parallelogramma, ac <lb/>in ambitu contenti ſolidi, quod ſit, PI, manifeſtum eſt nedum, PI, <lb/>eſſe cylindricum, ſed etiam eſſe parallelepipedum, ſunt enim pla-<lb/>na, RI, PB, parallela, necnon, PH, eſt ſuperficies plana ipſi, QI, <lb/>parallela, ac, PS, eſt plana, necnon ipſi, HB, ſimiliter parallela, <lb/>quod oſtendere oportebat.</s> <s xml:id="echoid-s13439" xml:space="preserve"/> </p> <div xml:id="echoid-div1194" type="float" level="2" n="1"> <figure xlink:label="fig-0539-01" xlink:href="fig-0539-01a"> <image file="0539-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0539-01"/> </figure> <note position="right" xlink:label="note-0539-01" xlink:href="note-0539-01a" xml:space="preserve">Def. 3. l. 1.</note> </div> </div> <div xml:id="echoid-div1196" type="section" level="1" n="716"> <head xml:id="echoid-head749" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s13440" xml:space="preserve">_E_X hoc colligitur, ſi, ducta, EH, per, H, parallela, DC, in paral-<lb/>lelis, EH, DC, ind finitè productis, reperiatur alia quæcunq; <lb/></s> <s xml:id="echoid-s13441" xml:space="preserve">plana figura, vt, EHC, ſolidum rectangulum ſub parallelogrammo <lb/>propoſito, AC, ſeu illi analoga ſuperficie ſecundum regulam planum, <lb/>GC, & </s> <s xml:id="echoid-s13442" xml:space="preserve">ſub figura, FHC, in ambitu contenti ſolidi exi lente, quod ſit, <lb/>AFCH, ad contentum ſub eodem parallelogrammo, AC, ſeu illi ana-<lb/>loga ſuperficie ſecundum dictam regulam, & </s> <s xml:id="echoid-s13443" xml:space="preserve">ſub figura, HDC, dummo-<lb/>do ea ſit in ambitu pariter contenti ſolidii, eſſe vt figura, EHC, ad figu-<lb/>ram, HDC, ſunt cnim hæc ſolida, ABFHC, ABGHC, cylindrici in ea-<lb/> <anchor type="note" xlink:label="note-0540-01a" xlink:href="note-0540-01"/> dem altitudine ſumpta reſpectu baſium, EHC, DHC, & </s> <s xml:id="echoid-s13444" xml:space="preserve">ideò ſunt inter <lb/>ſe vt ipſæ baſes, vnde cum ipſæ fuerint æquales etiam dicta ſolida re-<lb/>ctangula æqualia erunt.</s> <s xml:id="echoid-s13445" xml:space="preserve"/> </p> <div xml:id="echoid-div1196" type="float" level="2" n="1"> <note position="left" xlink:label="note-0540-01" xlink:href="note-0540-01a" xml:space="preserve">5. huius.</note> </div> </div> <div xml:id="echoid-div1198" type="section" level="1" n="717"> <head xml:id="echoid-head750" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s13446" xml:space="preserve">_H_Abetur inſuper ſi in eodem ſchemate ducatur in parallelogram-<lb/>mo, AC, quacumq; </s> <s xml:id="echoid-s13447" xml:space="preserve">parallela, HC, vt, RS, conſtituens paralle-<lb/>logrammum, RG, rectangulum ſolidu n ſub, AC, & </s> <s xml:id="echoid-s13448" xml:space="preserve">figura plana ex. </s> <s xml:id="echoid-s13449" xml:space="preserve">g. <lb/></s> <s xml:id="echoid-s13450" xml:space="preserve">HDC, contentum, dummodo hæc ſit in ipſius ambitu, ad rectangulum <lb/>ſolidum ſub, RC, & </s> <s xml:id="echoid-s13451" xml:space="preserve">eadem figura, HDC, in huius etiam ambitu exi-<lb/>ſtente, ſeu ſub quacumq; </s> <s xml:id="echoid-s13452" xml:space="preserve">alia plana figura in eiſdem parallelis cum, H <lb/>DC, exiſtent, dummodo ſit in ipſius ambitu, regulis ijſdem, BC, CD, eſſe <lb/>vt parallelogrammum, AC, ad par allelogrammum, CR, ſeu vt, BC, ad, <lb/>CS; </s> <s xml:id="echoid-s13453" xml:space="preserve">Et ſi ſint etiam parallelogramma, HV, HD, habetur etiam rectaã-<lb/>gulum ſolidum ſub, AC, CE, ad rectangulum ſolidum ſub, RC, CT, eſſe <pb o="521" file="0541" n="541" rhead="LIBER VII."/> vt rect angulum, BCD, ad rectangulum, SCV, ſunt enim hæc plana ré <lb/>ctangula baſes dictorum rectangulorum ſolidorum, quæ ex dictis ſunt <lb/>parallelepipeda, ſeu cylindrici eiuſdem altitudinis ſumptæ reſpectu <lb/>dict arum baſium, & </s> <s xml:id="echoid-s13454" xml:space="preserve">ideò ſunt vt ipſæ baſes, hoc eſt vt dicta rectangu-<lb/>la, ſuppoſito tamen quod continentia parallelogramma ſint in ambitu <lb/>contentorum ſolidorum.</s> <s xml:id="echoid-s13455" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1199" type="section" level="1" n="718"> <head xml:id="echoid-head751" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13456" xml:space="preserve">POterant quidem exhiberi parallelogramma, AC, RC, in eodē <lb/>plano cum figuris, EHC, CHD, & </s> <s xml:id="echoid-s13457" xml:space="preserve">in eiſdem cum ipſis paral-<lb/>lelis, vt, HY, proipſo, AC, &</s> <s xml:id="echoid-s13458" xml:space="preserve">, HR, proipſo, RC, & </s> <s xml:id="echoid-s13459" xml:space="preserve">intelligi me-<lb/>taliter deſcripta ſolida rectang. </s> <s xml:id="echoid-s13460" xml:space="preserve">iam dicta ſub iſtis in eodem plano <lb/>iacentibus fig. </s> <s xml:id="echoid-s13461" xml:space="preserve">prout dictum eſt, quo pacto eadem intelligi potuiſ-<lb/>ſent, ſed cum nonnihil difficile captu initio huius nouæ doctrinæ <lb/>hoc mihi fore videretur, eadem vt ſupra exhibere malui, verunta-<lb/>men valde expediet pro ſequentibus aſſuefieri dictorum ſolidorum <lb/>mentali deſcriptioni, exhibitis continentibus eadem fig. </s> <s xml:id="echoid-s13462" xml:space="preserve">(quæ, pu-<lb/>to, ſemper planæ erunt ) in eiſdem parallelis conſtitutis, quemad-<lb/>modum duabus quibuſcung; </s> <s xml:id="echoid-s13463" xml:space="preserve">rectis lineis exhibitis, illico rectangu-<lb/>lum ſub ipſis mentaliter deſcribere ſolemus, ſicuti & </s> <s xml:id="echoid-s13464" xml:space="preserve">quadratum <lb/>datæ rectæ lineæ cuiuſcumq; </s> <s xml:id="echoid-s13465" xml:space="preserve">abſque eo, quod ſemper in ſchema-<lb/>tibus ipſa deſcripta exhibeantur, ſic ergo & </s> <s xml:id="echoid-s13466" xml:space="preserve">ſolida rectang. </s> <s xml:id="echoid-s13467" xml:space="preserve">& </s> <s xml:id="echoid-s13468" xml:space="preserve">ſolida <lb/>quadrata, ſub duabus planis figuris in eiſdem parallelis exiſtentibus <lb/>iuxta datas regulas contenta, ad figurarum confuſionem euitan-<lb/>dam & </s> <s xml:id="echoid-s13469" xml:space="preserve">nos quoq; </s> <s xml:id="echoid-s13470" xml:space="preserve">mentaliter vt plurimum deſcribemus.</s> <s xml:id="echoid-s13471" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1200" type="section" level="1" n="719"> <head xml:id="echoid-head752" xml:space="preserve">THEOREMA XIV. PROPOS. XIV.</head> <p> <s xml:id="echoid-s13472" xml:space="preserve">SI duo triangula fuerint in eiſdem parallelis conſtituta. <lb/></s> <s xml:id="echoid-s13473" xml:space="preserve">Solidum rectangolum ſub eiſdem contentum, regula <lb/>altera dictarum parallelarum, ac alia quadam illi in ſubli-<lb/>mi perpendiculari, erit pyramis, habens in baſi parallelo-<lb/>grammum rectangulum, ſub dictorum triangulorum baſi-<lb/>bus contentum, dummodo alterum dictorum triangulorũ <lb/>ſit in ambitu contentiſolidi.</s> <s xml:id="echoid-s13474" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13475" xml:space="preserve">Sint duo triangula in eiſdem parallelis conſtituta, LK, ND, nẽ <lb/>pè, ABC, ACD, in baſibus, BC, CD, in parallela, ND, diſpoſitis, <lb/>eleuetur autem à puncto, C, quædam, CF, perpendicularis ipſi, C <pb o="522" file="0542" n="542" rhead="GEOMETRIÆ"/> B. </s> <s xml:id="echoid-s13476" xml:space="preserve">Dico ſolidum rectangulum ſub duobus triangulis, ABC, ACD, <lb/>contentum, regulis, BC, CF, eſſe pyramidem, cuius baſis erit pa-<lb/>rallelogrammum rectangulum ſub prædictis baſibus, BC, CD, pa-<lb/>riter contentum, dummodo alterum dictorum triangulorum ſit in <lb/>ambitu ipſius contenti ſolidi. </s> <s xml:id="echoid-s13477" xml:space="preserve">Sit enim deſcriptum ipſum ſolidum <lb/>rectangulum ſub triangulis, ABC, ACD, contentum, nempè, AE <lb/> <anchor type="figure" xlink:label="fig-0542-01a" xlink:href="fig-0542-01"/> BCF, ſit tamen alterum ip-<lb/>ſorum, vt, ABC, in ambitu <lb/>ipſius contenti, ſolidi, &</s> <s xml:id="echoid-s13478" xml:space="preserve">, AF <lb/>C, ſuperficies homologa ipſi, <lb/>ACD, iuxta regulam planũ, <lb/>BCF, erit ergo, ACF, trian-<lb/>gulum, eſto enim, quod vnũ <lb/>parallelorum ipſi, BF, plano-<lb/>rum, ſolidum, AEC, ſecanti-<lb/>um, in eo effecerit parallelo <lb/>grammum rectangulũ, GMIH, & </s> <s xml:id="echoid-s13479" xml:space="preserve">intriangulo, ACD, rectam, IY, <lb/>iam ſcimus, quod, HI, eſt in eodem plano cum, FC, cui eſt paral-<lb/>lela, & </s> <s xml:id="echoid-s13480" xml:space="preserve">ambo ſunt in eodem plano cum, AC, quod etiam de reli-<lb/>quis in ſuperficie, ACF, ipſi, FC, parallelis exiſtentibus eodem mo-<lb/>do oſtendetur, ergo iacent omnes in plano ipſarum, AC, CF, ergo, <lb/>ACF, eſt ſuperficies plana cum vero vt, CD, ad, IY, ita ſit, CA, ad, <lb/>AI, & </s> <s xml:id="echoid-s13481" xml:space="preserve">ita etiam, CF, ad, IH, erit, CF, ad, IH, vt, CA, ad, AI, er-<lb/>gotria puncta, FHA, erunt in recta linea, in eadem autem eſſe <lb/>oſtendemus etiam reliquarum ipſi, CF, parallelarum extrema pun-<lb/>cta ex hac parte, ergo, ACF, erit triangulum: </s> <s xml:id="echoid-s13482" xml:space="preserve">Conſimili autem <lb/> <anchor type="note" xlink:label="note-0542-01a" xlink:href="note-0542-01"/> modo pariter demonſtrabimus, ABE, AEF, eſſe triangula, & </s> <s xml:id="echoid-s13483" xml:space="preserve">eſt, <lb/>BF, parallelogrammum rectangulum, ergo ſolidum, ABF, eſt py-<lb/>ramis, & </s> <s xml:id="echoid-s13484" xml:space="preserve">eius baſis parallelogrammum, BF, quod oſtendere opus <lb/>erat.</s> <s xml:id="echoid-s13485" xml:space="preserve"/> </p> <div xml:id="echoid-div1200" type="float" level="2" n="1"> <figure xlink:label="fig-0542-01" xlink:href="fig-0542-01a"> <image file="0542-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0542-01"/> </figure> <note position="left" xlink:label="note-0542-01" xlink:href="note-0542-01a" xml:space="preserve">Lemwa 1. <lb/>22. l. 1.</note> </div> </div> <div xml:id="echoid-div1202" type="section" level="1" n="720"> <head xml:id="echoid-head753" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s13486" xml:space="preserve">_E_X hoc pariter intelligipoteſt, quod ſolidum rectang. </s> <s xml:id="echoid-s13487" xml:space="preserve">contentum <lb/>ſub trapezijs ex.</s> <s xml:id="echoid-s13488" xml:space="preserve">g.</s> <s xml:id="echoid-s13489" xml:space="preserve">MBCI, ICDγ, in eiſdem parallelis, Sγ, ND, <lb/>exiſtentibus, regulis ijſdem, BC, CF, eſt fruſtum pyramidis abſciſſæ per <lb/>planum baſi, BF, æquidiſtans, vt, GECI, dummodo alterum dictorum <lb/>trapeziorum in ambitu contenti ſolidi conſiſtat.</s> <s xml:id="echoid-s13490" xml:space="preserve"/> </p> <pb o="523" file="0543" n="543" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1203" type="section" level="1" n="721"> <head xml:id="echoid-head754" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s13491" xml:space="preserve">SImiliter ſi compleautur parallelogramma, PC, CR, CO, ſolidum <lb/>rectangulum ſub, RC, CP, ſeu ſub, OC, CP, contentum, quod eſt <lb/>parallelepipedum, triplum erit contenti ſub triangulis prædictis ideſt <lb/>pyramidis, AEC. </s> <s xml:id="echoid-s13492" xml:space="preserve">Contentum verò ſub parallelogrammis, TC, &</s> <s xml:id="echoid-s13493" xml:space="preserve">, C <lb/>X, ad contentum ſub dictis trapezijs hoc eſt ad fruſtum pyramidis, GE <lb/>CI, erit vt quadratum, BC, rectangulum ſub, XI, IM, vna cum {1/3}. </s> <s xml:id="echoid-s13494" xml:space="preserve">qua. <lb/></s> <s xml:id="echoid-s13495" xml:space="preserve"> <anchor type="note" xlink:label="note-0543-01a" xlink:href="note-0543-01"/> drat. </s> <s xml:id="echoid-s13496" xml:space="preserve">XM, retentis ſemper ijſdem reg. </s> <s xml:id="echoid-s13497" xml:space="preserve">BC, CF. </s> <s xml:id="echoid-s13498" xml:space="preserve">Hæc autem V<unsure/>era ſunt <lb/>ſiue latus, AC, ſit commune præfatis triangulis, ſeu parallelogram -<lb/>mis, ſiue non, ac ſine latus, IC, ſit cõmune predictis trapezijs, ſeu paral <lb/>lelogrammis, ſiue nõvt facilè intuenti innoteſcet.</s> <s xml:id="echoid-s13499" xml:space="preserve"/> </p> <div xml:id="echoid-div1203" type="float" level="2" n="1"> <note position="right" xlink:label="note-0543-01" xlink:href="note-0543-01a" xml:space="preserve">_10.huius._</note> </div> </div> <div xml:id="echoid-div1205" type="section" level="1" n="722"> <head xml:id="echoid-head755" xml:space="preserve">COROLLARIVM III.</head> <p style="it"> <s xml:id="echoid-s13500" xml:space="preserve">_P_Atet vltimo ſolida rectangula ſub dictis triangulis, regulis iam <lb/>dictis, contenta, ſe babereinter ſe, vt ipſæ pyramides, nempè <lb/>æquè alta eſſe in pro portione baſium, & </s> <s xml:id="echoid-s13501" xml:space="preserve">in eadem, vel æqualiqus ba-<lb/>ſibus exiſtentia eſſe in proportione altitudinem reſpectu baſium aſſum-<lb/>ptarum, quod eſt ſimile illi, quod animaduerſum eſt in Cor. </s> <s xml:id="echoid-s13502" xml:space="preserve">I. </s> <s xml:id="echoid-s13503" xml:space="preserve">& </s> <s xml:id="echoid-s13504" xml:space="preserve">2, <lb/>prop. </s> <s xml:id="echoid-s13505" xml:space="preserve">ant. </s> <s xml:id="echoid-s13506" xml:space="preserve">circa parallelogramma ſolida rectangula continentia.</s> <s xml:id="echoid-s13507" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1206" type="section" level="1" n="723"> <head xml:id="echoid-head756" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13508" xml:space="preserve">ADuerte autem cum ſolidum rectangulum fuerit quadratum, <lb/>tunc vnam ſufficere exponi figuram, vt ex. </s> <s xml:id="echoid-s13509" xml:space="preserve">g. </s> <s xml:id="echoid-s13510" xml:space="preserve">triangulum, A <lb/>BC, quod tunc æquipollet duobus expoſitis, ABC, ACD; </s> <s xml:id="echoid-s13511" xml:space="preserve">& </s> <s xml:id="echoid-s13512" xml:space="preserve">con-<lb/>tentum ſolidum ſub, ABC, ACD, tunc etiam dicimus quadratum <lb/>ſolidum ipſius, ABC, regulis, BC, CF, hæc autem planarum figu-<lb/>rarum quadrata ſolida mentaliter quoque vt plurimum deſcripta <lb/>eſſe intelligemus, vt etiam ſuperius animaduerſum fuit. </s> <s xml:id="echoid-s13513" xml:space="preserve">His au-<lb/>tem præpoſitis, nunc illa ſubiungemus, quæ aſſimilantur Prop. <lb/></s> <s xml:id="echoid-s13514" xml:space="preserve">Sec. </s> <s xml:id="echoid-s13515" xml:space="preserve">Elem. </s> <s xml:id="echoid-s13516" xml:space="preserve">ac iuxta methodum indiuiſibilium lib. </s> <s xml:id="echoid-s13517" xml:space="preserve">2. </s> <s xml:id="echoid-s13518" xml:space="preserve">prop. </s> <s xml:id="echoid-s13519" xml:space="preserve">23. </s> <s xml:id="echoid-s13520" xml:space="preserve">oſtẽ-<lb/>ſa fuere.</s> <s xml:id="echoid-s13521" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1207" type="section" level="1" n="724"> <head xml:id="echoid-head757" xml:space="preserve">THEOREMA XV. PROPOS. XV.</head> <p> <s xml:id="echoid-s13522" xml:space="preserve">SI duæ expoſitæ fuerint ſuperficies ſolidum rectangulũ <lb/>iuxta datas regulas continentes, altera autem earum <lb/>fuerit in quotcunq; </s> <s xml:id="echoid-s13523" xml:space="preserve">partes diuiſa per lineas ſecantes quaſ-<lb/>cunq; </s> <s xml:id="echoid-s13524" xml:space="preserve">ſuæ regulæ intra dictam ſuperficiem parallelas, alte- <pb o="524" file="0544" n="544" rhead="GEOMETRIE"/> ra autem fuerit indiuiſa: </s> <s xml:id="echoid-s13525" xml:space="preserve">Solidum rectangulum ſub indi-<lb/>uiſa, & </s> <s xml:id="echoid-s13526" xml:space="preserve">ſub diuiſa contentum, æquabitur ſolidis rectangu-<lb/>lis ſub eadem indiuiſa, & </s> <s xml:id="echoid-s13527" xml:space="preserve">ſub partibus diuiſæ, regulis ijſ-<lb/>dem, contentis.</s> <s xml:id="echoid-s13528" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13529" xml:space="preserve">Sint duæ expoſitæ ſuperficies, AC, CH, ſolidum recrangulum, <lb/>FC, iuxca regulas, kC, CB, continentes, earum autem altera, vt, <lb/>AC, ſit diuila in quotcumq; </s> <s xml:id="echoid-s13530" xml:space="preserve">partes, vt per lineam, DEC, ſecan-<lb/>tem quaſcumq; </s> <s xml:id="echoid-s13531" xml:space="preserve">intra ſuperficiem, AC, ipſiregulæ, BC, parallelas, <lb/> <anchor type="figure" xlink:label="fig-0544-01a" xlink:href="fig-0544-01"/> in duas partes, DEC, ADECB, ipſa <lb/>verò, HC, ſit indiuiſa. </s> <s xml:id="echoid-s13532" xml:space="preserve">Dico ſolidum <lb/>contentum ſub indiuiſa, HC, & </s> <s xml:id="echoid-s13533" xml:space="preserve">ſub <lb/>diuiſa, AC, ideſt, FC, æquari ſolidis <lb/>contentis ſub, DEC, CH, & </s> <s xml:id="echoid-s13534" xml:space="preserve">ſub, DE <lb/>CBA, & </s> <s xml:id="echoid-s13535" xml:space="preserve">ſub eadem, CH. </s> <s xml:id="echoid-s13536" xml:space="preserve">Intelliga-<lb/>tur ergo quandam rectam lineam fer-<lb/>ri peripſam, CED, indefinitè produ <lb/>ctam, donec totam percurrerit, ac <lb/>ſemper moueri ipſi regulæ, KC, æqui-<lb/>diſtanter, deſcribet ergo ſuperficiem <lb/>cylindraceam, quæ ſit, KEH, & </s> <s xml:id="echoid-s13537" xml:space="preserve">ab-<lb/>ſcindet à ſuperſiciebus, FK, AC, ſu-<lb/>perficies cylindraceas, HIK, DEC, &</s> <s xml:id="echoid-s13538" xml:space="preserve">, HC, eſt cylindracea, & </s> <s xml:id="echoid-s13539" xml:space="preserve">hoc <lb/>ſiue ſit in ambitu contenti ſolidi, ſiue non, alioquin non poſſent <lb/>latera, quæ per ſolidum, FC, ſecantia plana, ipſi, GC, æquidiſtã-<lb/>tia ſignantur in ipſa ſuperficie, HC, omnia vni regulæ, kC, æqui-<lb/>diſtare, ergo ſolidum, HIKCED, ſuperficiebus cylindraceis com-<lb/>prehenditur, quarum regulæ ſunt, kC, CB, inuicem perpendicula-<lb/>res ergo ſi ſolidum, HIKCED, ſecetur planis ipſi, kB, parallelis fiẽt <lb/>in ſolido parallelogramina ipſi, kB, æquiangula, hoc eſt rectangu-<lb/>la, & </s> <s xml:id="echoid-s13540" xml:space="preserve">ideò dictum ſolidum erit ſolidum rectangulum contentum <lb/>ſub, HC, CED, ſuperficiebus: </s> <s xml:id="echoid-s13541" xml:space="preserve">Eodem modo oſtendemus, HIKC <lb/>EDAG, eſſe ſolidum rectangulum contentum ſub ſuperficie, DIC, <lb/>hoe eſt, Dk, il i homologa iuxta planum, BK, ac ſub, DECBA, eſt <lb/>autem ſolidum, FC, æquale duobus ſolidis, HIC, CIHFB, ſimul <lb/>ſumptis, ergo ſolidum rectangulum contentum ſub indiuiſa ſuperö <lb/>ficie, HC, & </s> <s xml:id="echoid-s13542" xml:space="preserve">ſub diuiſa, AC, æquale eſt ſolidis rectangulis conten-<lb/>tis ſub eadem indiuiſa, HC, & </s> <s xml:id="echoid-s13543" xml:space="preserve">ſub partibus diuiſæ, DEC, DECB <lb/>A, regulis ſemper ijſdem, BC, CK, retentis, quod oſtendere opus <lb/>erat.</s> <s xml:id="echoid-s13544" xml:space="preserve"/> </p> <div xml:id="echoid-div1207" type="float" level="2" n="1"> <figure xlink:label="fig-0544-01" xlink:href="fig-0544-01a"> <image file="0544-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0544-01"/> </figure> </div> <pb o="525" file="0545" n="545" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1209" type="section" level="1" n="725"> <head xml:id="echoid-head758" xml:space="preserve">COROLLARIVM I.</head> <p style="it"> <s xml:id="echoid-s13545" xml:space="preserve">_E_Xpoſita figura plana quacumq; </s> <s xml:id="echoid-s13546" xml:space="preserve">BGEO, in parallelis, AC, DF, & </s> <s xml:id="echoid-s13547" xml:space="preserve"><lb/>aſſumptis pro regulis, DF, FH, inuicem perpendicularibus, ita <lb/> <anchor type="figure" xlink:label="fig-0545-01a" xlink:href="fig-0545-01"/> tamen vt, FH, ſit extra planum parallelarum, <lb/>AC, DF, primò colligitur, ſi ipſa figura per <lb/>ſolam lineam, BE, (ſecantem quaſcumq; </s> <s xml:id="echoid-s13548" xml:space="preserve">in-<lb/>tra eandem figuram, ipſiregulæ, DF, para l-<lb/>lelas deſcriptibiles) diutdatur vtcunq; </s> <s xml:id="echoid-s13549" xml:space="preserve">qua-<lb/>dratum ſolidum ſubindiuiſa, BGEO, & </s> <s xml:id="echoid-s13550" xml:space="preserve">ſub <lb/>eadem, BGEO, quatenus diuiſa, æquari rectangulis ſolidis ſub eadem <lb/>indiuiſa, BGEO, & </s> <s xml:id="echoid-s13551" xml:space="preserve">ſub partibus, BGE, BOE.</s> <s xml:id="echoid-s13552" xml:space="preserve"/> </p> <div xml:id="echoid-div1209" type="float" level="2" n="1"> <figure xlink:label="fig-0545-01" xlink:href="fig-0545-01a"> <image file="0545-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0545-01"/> </figure> </div> </div> <div xml:id="echoid-div1211" type="section" level="1" n="726"> <head xml:id="echoid-head759" xml:space="preserve">COROLLARIVM II.</head> <p style="it"> <s xml:id="echoid-s13553" xml:space="preserve">_C_olligitur ſecundò rectangulum ſolidum ſub indiuiſa, BEO, & </s> <s xml:id="echoid-s13554" xml:space="preserve">ſub <lb/>diuiſa, BGEO, æquarirectangulis ſolidis ſub eadem indiuiſa, BE <lb/>O, & </s> <s xml:id="echoid-s13555" xml:space="preserve">ſub parte, BEO, hoc eſt quadrato ſolido, BEO, & </s> <s xml:id="echoid-s13556" xml:space="preserve">rectangulo ſo-<lb/>lido ſub, BEO, BEG.</s> <s xml:id="echoid-s13557" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1212" type="section" level="1" n="727"> <head xml:id="echoid-head760" xml:space="preserve">COROLLARIVM III.</head> <p style="it"> <s xml:id="echoid-s13558" xml:space="preserve">_C_olligitur tertiò quadratum ſolidum ipſius, BGEO, æquari rectã-<lb/>gulis ſolidis ſub, BGEO, ac, viriuſq; </s> <s xml:id="echoid-s13559" xml:space="preserve">partibus, BEG, BEO, per <lb/>Cor. </s> <s xml:id="echoid-s13560" xml:space="preserve">prim. </s> <s xml:id="echoid-s13561" xml:space="preserve">& </s> <s xml:id="echoid-s13562" xml:space="preserve">ſubinde æquari quadratis partium, BEG, BEO, vnacum <lb/>rectangulobis ſub eiſdem partibus, BEG, BEO, per Cor. </s> <s xml:id="echoid-s13563" xml:space="preserve">ant.</s> <s xml:id="echoid-s13564" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1213" type="section" level="1" n="728"> <head xml:id="echoid-head761" xml:space="preserve">COROLLARIVM. IV.</head> <p style="it"> <s xml:id="echoid-s13565" xml:space="preserve">_C_olligitur quartò, ſi linea, BIE, bifariam, BNE, verò non bifari@ <lb/>ſecent dictas ipſi, DF, parallelas: </s> <s xml:id="echoid-s13566" xml:space="preserve">Rectangulum ſolidum ſub in <lb/>diuiſa, BNEO, & </s> <s xml:id="echoid-s13567" xml:space="preserve">ſub diuiſa, BGEN, per ipſam, BIE, æquari rectan-<lb/>gulo ſolido ſub eadem indiuiſa, BNEO, & </s> <s xml:id="echoid-s13568" xml:space="preserve">ſub partibus, BIEN, BGEI, <lb/>diuiſæ, hoc eſt æquari rectangulo ſub eadem, BNEO, & </s> <s xml:id="echoid-s13569" xml:space="preserve">ſub, BIEO, cum <lb/>ſolido rectangulo ſub, BOEN, BNEI, cui (i addatur quad. </s> <s xml:id="echoid-s13570" xml:space="preserve">ſolidum, BI <lb/>EN, (ex quibus integratur rectang ſolidum ſub, BIEO, BIEN, per <lb/>Cor. </s> <s xml:id="echoid-s13571" xml:space="preserve">primum) fiet quadratum ſolidum, BEO, cui æquabitur rectangu-<lb/>lum ſolidum ſub, BGEN, BNEO, cum quadrato ſolido figuræ, BIEN, <lb/>intermediæ ſec intibus lineis, BIE, BNE, liceat autem, curn dicimus <lb/>v<unsure/>ectangulum ſolidum ſub duabus figuris, ſubintelligere ſemper con-<lb/>tentum, breuitatis gratia, etiam ſi non exprimatur, vt in planis fieri <lb/>conſacuit.</s> <s xml:id="echoid-s13572" xml:space="preserve"/> </p> <pb o="526" file="0546" n="546" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1214" type="section" level="1" n="729"> <head xml:id="echoid-head762" xml:space="preserve">COROLLARIVM V.</head> <p style="it"> <s xml:id="echoid-s13573" xml:space="preserve">_C_olligitur quintò, ſi ſupponamus, BIE, bifariam ſecare dictas <lb/>ipſi, DF, parallelas in figura, BGEN, & </s> <s xml:id="echoid-s13574" xml:space="preserve">deinde illis adiungi, <lb/>BNEO, figuram in eiſdem parallelis cum, BGEN, conſtitutam; </s> <s xml:id="echoid-s13575" xml:space="preserve">rectan-<lb/>gulum ſolidum ſub, BGEO, & </s> <s xml:id="echoid-s13576" xml:space="preserve">ſub, BNEO, hoc eſt vnum ſub, BGEL, <lb/>ſeu, BIEN, & </s> <s xml:id="echoid-s13577" xml:space="preserve">ſub, BNEO, indiuiſa, aliud ſub, BIEO, & </s> <s xml:id="echoid-s13578" xml:space="preserve">ſub, BNE <lb/>O, cum quadrato ſolido, BIEN, ( quod iunctum rectangulo ſolido ſub, <lb/>BIEN, BNEO, facit rectangulum ſolidum ſub, BIEN, BIEO, per Cor. <lb/></s> <s xml:id="echoid-s13579" xml:space="preserve">2.) </s> <s xml:id="echoid-s13580" xml:space="preserve">æquari quadrato ſolido, BIEO, per Cor. </s> <s xml:id="echoid-s13581" xml:space="preserve">I. </s> <s xml:id="echoid-s13582" xml:space="preserve">hoc eft rectangulum ſo-<lb/>lidam ſub figura compoſita ex propoſita, BGEN, & </s> <s xml:id="echoid-s13583" xml:space="preserve">adiecta, BNEO, & </s> <s xml:id="echoid-s13584" xml:space="preserve"><lb/>ſub adiecta, BNEO, cum quadrato ſolido, BIEN, dimidiæ ipſius propo-<lb/>ſitæ, æquari quadrato ſolido, BIEO, compoſitæ ex dimidia, BIEN, & </s> <s xml:id="echoid-s13585" xml:space="preserve"><lb/>adiecta, BNEO.</s> <s xml:id="echoid-s13586" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1215" type="section" level="1" n="730"> <head xml:id="echoid-head763" xml:space="preserve">COROLLARIVM VI.</head> <p style="it"> <s xml:id="echoid-s13587" xml:space="preserve">_C_olligitur ſextò, in eadem fig. </s> <s xml:id="echoid-s13588" xml:space="preserve">BGEO, poſito, quod per lineam tan-<lb/>tum, BNE, ſecentur dictæ parallelæ ipſi, DF, quad. </s> <s xml:id="echoid-s13589" xml:space="preserve">ſolidum fi-<lb/> <anchor type="figure" xlink:label="fig-0546-01a" xlink:href="fig-0546-01"/> guræ, BGEO, cum quadrato ſolido figuræ, BN <lb/>EO, æquari rectangulo ſolido bis ſub, BGEO, &</s> <s xml:id="echoid-s13590" xml:space="preserve">, <lb/>BNEO, figuris contento, cum quadrato ſolido <lb/>reliquæ figuræ, BGEN. </s> <s xml:id="echoid-s13591" xml:space="preserve">Nam quadratum <lb/>ſolidum, BGEO, æquatur quadratis ſolidis, BG <lb/>EN, BNEO, cum duobus rectangulis ſolidis <lb/>ſub eiſdem figuris, addito ergo quadrato ſolido communi, BNEO, fient <lb/>quadrata ſolida figuraxum, BGEO, BNEO, æqualia duobus rectangulis <lb/>ſolidis ſub figuris, BGEN, BNEO, cum duobus quadratis ſolidis, BN <lb/>EO, hoc eſt duobus rectangulis ſolidis ſub, BGEO, BNEO, cum qua-<lb/>drato ſolido, BGEN.</s> <s xml:id="echoid-s13592" xml:space="preserve"/> </p> <div xml:id="echoid-div1215" type="float" level="2" n="1"> <figure xlink:label="fig-0546-01" xlink:href="fig-0546-01a"> <image file="0546-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0546-01"/> </figure> </div> </div> <div xml:id="echoid-div1217" type="section" level="1" n="731"> <head xml:id="echoid-head764" xml:space="preserve">COROLLARIVM VII.</head> <p style="it"> <s xml:id="echoid-s13593" xml:space="preserve">_C_olligitur ſeptimò, ſi propoſitæ figuræ, B G E N, diuidatur per li-<lb/>neam, B I E, dictas quoq; </s> <s xml:id="echoid-s13594" xml:space="preserve">parallelas ipſi, D F, ſecantem, rectan-<lb/>gulum ſolidum quater ſub, B G E N, B I E N, cum quadrato ſolido, <lb/>B G E I, æquari quadrato ſolido figuræ compoſitæ ex, B G E N, & </s> <s xml:id="echoid-s13595" xml:space="preserve"><lb/>figura, B I E N, ſeu illi homologa, quæ ſit, B N E O. </s> <s xml:id="echoid-s13596" xml:space="preserve">Duo enim <lb/>rectangula ſolida ſub, B G E N, B I E N, cum quadrato ſolido, B GEI, <lb/>æquantur duobus quadratis ſolidis, B G E N, B I E N, ex Cor. </s> <s xml:id="echoid-s13597" xml:space="preserve">ant.</s> <s xml:id="echoid-s13598" xml:space="preserve"> <pb o="527" file="0547" n="547" rhead="LIBER VII."/> hoc eſt quadratis ſolidis, BGEN, BNEO, additis communibus duobus <lb/>adbuc rectangulis ſub, BGEN, BIEN, ſeu, BNEO, quæ ſuperſunt, fi-<lb/>eut quatuor rectangula ſolida ſub, BGEN, BIEN, cum quadrato ſo-<lb/>lido, BGEI, æqualia duobus quadratis ſolidis, BGEN, BNEO, cum <lb/>duobus rectangulis ſolidis ſub, BGEN, BNEO, hoc eſt quadrato ſoli-<lb/>do, BGEO, per Cor. </s> <s xml:id="echoid-s13599" xml:space="preserve">Tertium.</s> <s xml:id="echoid-s13600" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1218" type="section" level="1" n="732"> <head xml:id="echoid-head765" xml:space="preserve">COROLLARIVM VIII.</head> <p style="it"> <s xml:id="echoid-s13601" xml:space="preserve">_C_olligitur octauò, ſi figuræ, BGEO, ſecetur vt in Cor. </s> <s xml:id="echoid-s13602" xml:space="preserve">4. </s> <s xml:id="echoid-s13603" xml:space="preserve">quadrata <lb/>ſolida figurarum, BGEN, BNEO, dupla eſſe quadratorum ſoli-<lb/>dorum, BGEI, BIEN. </s> <s xml:id="echoid-s13604" xml:space="preserve">Nam quad. </s> <s xml:id="echoid-s13605" xml:space="preserve">ſolidum, BGEN, æquatur quadra-<lb/>tis ſolidis, BGEI, BIEN cum duobus rectangulis ſolidis ſub, BGEI, BI <lb/>EN, per Cor. </s> <s xml:id="echoid-s13606" xml:space="preserve">Tertium, ideſt cum duobus rectangulis ſolidis ſub, BIEO, <lb/>(homologa ipſi, BGEI,) &</s> <s xml:id="echoid-s13607" xml:space="preserve">, BIEN, quibus ſi addatur reſiduum qua-<lb/>dratum ſolidum, BNEO fiunt duo rect angula ſolida ſub, BIEO, BIE <lb/>N, cum quadrato ſolido, BNEO, æqualia quadrato ſolido, BIEO, ſeu, <lb/>BGEI, cum quadrato ſolido, BIEN, igitur quadrata ſolida, BGEN, B <lb/>NEO, dupla ſunt quadratorum ſolidorum; </s> <s xml:id="echoid-s13608" xml:space="preserve">BGEI, BIEN.</s> <s xml:id="echoid-s13609" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1219" type="section" level="1" n="733"> <head xml:id="echoid-head766" xml:space="preserve">COROLLARIVM IX.</head> <p style="it"> <s xml:id="echoid-s13610" xml:space="preserve">_C_olligitur nonò, ſuppoſitis in figura ſectionibus ipſius Cor. </s> <s xml:id="echoid-s13611" xml:space="preserve">5. </s> <s xml:id="echoid-s13612" xml:space="preserve">qua-<lb/>drata ſolida, BGEO, BNEO, dupla eſſe quadratorum ſolidorum, <lb/>BGEI, BIEO. </s> <s xml:id="echoid-s13613" xml:space="preserve">Etenim quadratum ſolidum, BGEO, æquatur per Cor 3. <lb/></s> <s xml:id="echoid-s13614" xml:space="preserve">quadratis ſolidis, BGEI, BIEO, cum duobus rectangulis ſolidis ſub, BG <lb/>EI, ſeu, BIEN, illi homologa, &</s> <s xml:id="echoid-s13615" xml:space="preserve">, BIEO, quæ duo rectangula ſolida fa-<lb/>ciunt cum quadrato ſolido, BNEO, reſiduo, quadrata ſolida, BIEO, BI <lb/>EN, ſeu, BGEI, BIEO, ergo quadrata ſolida, BGEO, BNEO, dupla <lb/>ſunt quadratorum ſolidorum, BGEI, BIEO.</s> <s xml:id="echoid-s13616" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1220" type="section" level="1" n="734"> <head xml:id="echoid-head767" xml:space="preserve">COROLLARIVM X.</head> <p style="it"> <s xml:id="echoid-s13617" xml:space="preserve">_C_olligitur decimò, & </s> <s xml:id="echoid-s13618" xml:space="preserve">vltimò, ſi tandem ex. </s> <s xml:id="echoid-s13619" xml:space="preserve">g. </s> <s xml:id="echoid-s13620" xml:space="preserve">lineæ, BNE, ſecet <lb/>quaſcumq; </s> <s xml:id="echoid-s13621" xml:space="preserve">intra figuram, BNE, ipſi, DF, æquidiſtantes, ſecun-<lb/>dum extremam, ac mediam rationem, ita vt maior portio cuiuſcumq; <lb/></s> <s xml:id="echoid-s13622" xml:space="preserve">ſectæ lineæ ſit ex. </s> <s xml:id="echoid-s13623" xml:space="preserve">g. </s> <s xml:id="echoid-s13624" xml:space="preserve">in figura, BGEN, rectangulum ſolidum ſub, BGE <lb/>O, BNEO, æquari quadrato ſolido, BGEN, bæc enim ſolida erunt <lb/>æqualiter analogaiuxta regulam planum, DFH, ex eo quod in vnoquo-<lb/>que eidem parallelorũ planorum ipſa ſolida ſecantiũ, ac capientiũ vnũ <lb/>rectangulum, & </s> <s xml:id="echoid-s13625" xml:space="preserve">vnum quadratum, ſemper rectangulum eſt æquale, <lb/>quadrato in eodem plano exiſtenti.</s> <s xml:id="echoid-s13626" xml:space="preserve"/> </p> <pb o="528" file="0548" n="548" rhead="GEOMETRIÆ"/> </div> <div xml:id="echoid-div1221" type="section" level="1" n="735"> <head xml:id="echoid-head768" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13627" xml:space="preserve">DDuertatur autem me in omnibus ſupra poſitis Corollarijs <lb/>ſupponere ſecantes lineas, parallelas ipſi, DF, in dictis figu-<lb/>ris, non niſi ſemel occurrere eidem rectæ lineæ, vt, BIE, ſemel, ac, <lb/>BNE, ſeorſim ſemel tantum; </s> <s xml:id="echoid-s13628" xml:space="preserve">ipſas verò parallelas ad ambitum fi-<lb/>guræ terminari, ac ſingulas integras eſſe, quódetiam ſuppono in <lb/>prop. </s> <s xml:id="echoid-s13629" xml:space="preserve">2 3. </s> <s xml:id="echoid-s13630" xml:space="preserve">lib. </s> <s xml:id="echoid-s13631" xml:space="preserve">2. </s> <s xml:id="echoid-s13632" xml:space="preserve">integras autem eſſe ſubintelligo; </s> <s xml:id="echoid-s13633" xml:space="preserve">cum in plures re-<lb/>ctas lineas, aliquo interuallo ſeparatas, per ambitum figuræ, quæ <lb/>ab eadem regulæ parallela efficiuntur, diſiungi minimè comperien-<lb/>tur, in quo ſenſu ſciat lector (ne quis circa hoc hæſitaret) me ſem-<lb/>per in his libris hunc terminũ vſurpare, ſciat inſuper eaſdẽ regulas, <lb/>DF, FH, pro omnibus ſemperretineri. </s> <s xml:id="echoid-s13634" xml:space="preserve">Hæc autẽ ſegnius, quam for-<lb/>tè par erat, à me nunc explicata ſunt, ſed cum Propoſitiones Lib. <lb/></s> <s xml:id="echoid-s13635" xml:space="preserve">Sec. </s> <s xml:id="echoid-s13636" xml:space="preserve">Elem. </s> <s xml:id="echoid-s13637" xml:space="preserve">hæc imitarentur, & </s> <s xml:id="echoid-s13638" xml:space="preserve">inſuper conſimilis doctrina, adhi-<lb/>bita tamen indiuiſibilium methodo, tradita iam fuiſſet Lib. </s> <s xml:id="echoid-s13639" xml:space="preserve">2. </s> <s xml:id="echoid-s13640" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13641" xml:space="preserve"><lb/>23. </s> <s xml:id="echoid-s13642" xml:space="preserve">ideò ne rerum ſimilitudo faſtidium pareret, currenti, vtita di-<lb/>cam, calamo adnotata ſunt. </s> <s xml:id="echoid-s13643" xml:space="preserve">Ex ſupradictis autem facile eſt intel-<lb/>ligere nomen quadrati ſolidi alicuius figuræ planæ æquipollere <lb/>nomini omnium quadratorum eiuſdem figuræ, & </s> <s xml:id="echoid-s13644" xml:space="preserve">nomen rectan-<lb/>guli ſolidi ſub duabus figuris æquipollere nomini rectangulorum <lb/>ſub eiſdem figuris, quibus quidem in methodo indiuiſibilium vte-<lb/>bamur, ex quo patet, vt ſic nos indefinitum planorum numerum <lb/>euitare, cui ipſorum, quæ rectangula ſolida appellauimus, ſolidita-<lb/>tem ſatis concinne puto ſubſtituimus. </s> <s xml:id="echoid-s13645" xml:space="preserve">His autem paratis, ſequen-<lb/>tium propoſitionum demonſtrationes tum quæ ſuperſunt 1. </s> <s xml:id="echoid-s13646" xml:space="preserve">2. </s> <s xml:id="echoid-s13647" xml:space="preserve">tum <lb/>lib. </s> <s xml:id="echoid-s13648" xml:space="preserve">3. </s> <s xml:id="echoid-s13649" xml:space="preserve">4. </s> <s xml:id="echoid-s13650" xml:space="preserve">ac 5. </s> <s xml:id="echoid-s13651" xml:space="preserve">paucis mutatis compendioſiſſimè per hanc nouam <lb/>methodum, abſq; </s> <s xml:id="echoid-s13652" xml:space="preserve">ſolidarum figurarum circumſcriptione, & </s> <s xml:id="echoid-s13653" xml:space="preserve">inſcri-<lb/>ptione, vt alij conſueuerunt, necnon facile, oſtendemus, per hæc <lb/>verò Prop. </s> <s xml:id="echoid-s13654" xml:space="preserve">23. </s> <s xml:id="echoid-s13655" xml:space="preserve">Lib. </s> <s xml:id="echoid-s13656" xml:space="preserve">2. </s> <s xml:id="echoid-s13657" xml:space="preserve">iam ſatisfactum eſſe manifeſtò apparet.</s> <s xml:id="echoid-s13658" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1222" type="section" level="1" n="736"> <head xml:id="echoid-head769" xml:space="preserve">THEOREMA XVI. PROPOS. XVI.</head> <p> <s xml:id="echoid-s13659" xml:space="preserve">COnſpecta denuò figura Prop. </s> <s xml:id="echoid-s13660" xml:space="preserve">30. </s> <s xml:id="echoid-s13661" xml:space="preserve">lib. </s> <s xml:id="echoid-s13662" xml:space="preserve">2. </s> <s xml:id="echoid-s13663" xml:space="preserve">& </s> <s xml:id="echoid-s13664" xml:space="preserve">aſſumpta <lb/>regula, FD, & </s> <s xml:id="echoid-s13665" xml:space="preserve">alia, quæ à puncto, F, quomodocunq; <lb/></s> <s xml:id="echoid-s13666" xml:space="preserve">intelligatur eleuata ſuper planum, AF, perpendiculariter <lb/>ipſi, FD. </s> <s xml:id="echoid-s13667" xml:space="preserve">Rectangulum ſolidum ſub, AE, EC, ad rectan-<lb/>gulum ſolidum ſub, ADEC, trapezio, & </s> <s xml:id="echoid-s13668" xml:space="preserve">triangulo, CEF, re-<lb/>gulis iam dictis, contentum, erit vt, DE, ad compoſitam ex @. </s> <s xml:id="echoid-s13669" xml:space="preserve"><lb/>DE, & </s> <s xml:id="echoid-s13670" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13671" xml:space="preserve">EF.</s> <s xml:id="echoid-s13672" xml:space="preserve"/> </p> <pb o="529" file="0549" n="549" rhead="LIBER VII."/> <p> <s xml:id="echoid-s13673" xml:space="preserve">Hoc oſtendetur eodem modo, ac inſupradicta prop. </s> <s xml:id="echoid-s13674" xml:space="preserve">30. </s> <s xml:id="echoid-s13675" xml:space="preserve">lib. </s> <s xml:id="echoid-s13676" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s13677" xml:space="preserve">mutatis tantum ſupradictis nominibus, nempe ſi vbi dicimus re-<lb/>ctangula ſub duabus quibuſdam figuris, hic dicamus rectangulum <lb/>ſolidum ſub eiſdem figuris, ſicuti etiam cumdicuntur omnia qua-<lb/> <anchor type="figure" xlink:label="fig-0549-01a" xlink:href="fig-0549-01"/> drata cuiuſdam figuræ, nos illius vice <lb/>nunc ſubſtituemus nomen quadrati ſo-<lb/>lidi eiuſdem figuræ, vt ſupra dicebatur. <lb/></s> <s xml:id="echoid-s13678" xml:space="preserve">Igitur cum rectangulum ſolidum ſub <lb/>trapezio, ADEC, diuiſo per lineam, B <lb/> <anchor type="note" xlink:label="note-0549-01a" xlink:href="note-0549-01"/> E, & </s> <s xml:id="echoid-s13679" xml:space="preserve">ſub triangulo, CEF, indiuiſo, æ-<lb/>quetur rectangulo ſolido ſub, AE, & </s> <s xml:id="echoid-s13680" xml:space="preserve"><lb/>triangulo, CEF, vel triangulo, BEC, & </s> <s xml:id="echoid-s13681" xml:space="preserve"><lb/>rectangulo ſolido ſub triangulo, BEC, <lb/>& </s> <s xml:id="echoid-s13682" xml:space="preserve">triangulo, CEF, primò patet rectan-<lb/>gulum ſolidum ſub, AE, EC, adrectang. </s> <s xml:id="echoid-s13683" xml:space="preserve">ſolidum ſub, AE, & </s> <s xml:id="echoid-s13684" xml:space="preserve">triã-<lb/>gulo, BEC, eſſe vt, BF, ad, BEC, ideſt vt, DE, ad {1/2}. </s> <s xml:id="echoid-s13685" xml:space="preserve">DE, eſt enim, <lb/>BF, duplum trianguli, BEC. </s> <s xml:id="echoid-s13686" xml:space="preserve">Similiter rectangulum ſolidum ſub, <lb/> <anchor type="note" xlink:label="note-0549-02a" xlink:href="note-0549-02"/> AE, EC, ad quadratum ſolidum, BF, eſt vt rectangulum, DEF, ad <lb/>quadratum, EF, ideſt vt, DE, ad, EF, quadratum verò ſolidum, B <lb/>F, cum ſit triplum quadrati ſolidi, CEF, & </s> <s xml:id="echoid-s13687" xml:space="preserve">quadrati ſolidi, BEC, <lb/>erit etiam triplum duorum rectangulorum ſolidorum ſub, BEC, CE <lb/>F, (quadratum ſolidum enim, BF, oſtenſum eſt æquari quadratis <lb/>ſolidis, BEC, CEF, cum duobus rectang. </s> <s xml:id="echoid-s13688" xml:space="preserve">ſolidis ſub, BEC, CEF,) <lb/> <anchor type="note" xlink:label="note-0549-03a" xlink:href="note-0549-03"/> & </s> <s xml:id="echoid-s13689" xml:space="preserve">ideò erit ſexcuplum rectanguli ſolidi, ſub, BEC, CEF, ideſt erit <lb/>ad illud vt, EF, ad ſui {1/6}. </s> <s xml:id="echoid-s13690" xml:space="preserve">ergo ex æquali rectangulum ſolidum ſub, <lb/>AE, EC, ad rectangulum ſolidum ſub, BEC, CEF, erit vt, DE, ad <lb/>{1/6}. </s> <s xml:id="echoid-s13691" xml:space="preserve">EF, & </s> <s xml:id="echoid-s13692" xml:space="preserve">ad rectangulum ſolidum ſub, AE, & </s> <s xml:id="echoid-s13693" xml:space="preserve">triangulo, BEC, ſeu, <lb/>CEF, oſtenſum eſt eſſe vt, DE, ad {1/2}. </s> <s xml:id="echoid-s13694" xml:space="preserve">DE, ergo colligendo rectan-<lb/>gulum ſolidum ſub, AE, EC, ad rectangulum ſolidum ſub, AE, CE <lb/>F, & </s> <s xml:id="echoid-s13695" xml:space="preserve">ſub, CBE, CEF, ideſt ſub trapezio, CADE, & </s> <s xml:id="echoid-s13696" xml:space="preserve">triangulo, CE <lb/>F, erit vt, DE, ad compoſitam ex {1/2}. </s> <s xml:id="echoid-s13697" xml:space="preserve">DE, & </s> <s xml:id="echoid-s13698" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s13699" xml:space="preserve">EF, quod oſtendere <lb/>opuserat.</s> <s xml:id="echoid-s13700" xml:space="preserve"/> </p> <div xml:id="echoid-div1222" type="float" level="2" n="1"> <figure xlink:label="fig-0549-01" xlink:href="fig-0549-01a"> <image file="0549-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0549-01"/> </figure> <note position="right" xlink:label="note-0549-01" xlink:href="note-0549-01a" xml:space="preserve">SI.huius@</note> <note position="right" xlink:label="note-0549-02" xlink:href="note-0549-02a" xml:space="preserve">Coroll.1. <lb/>13. huius. <lb/>Coroll.2. <lb/>13. huius. <lb/>Coroll.2, <lb/>14.huius.</note> <note position="right" xlink:label="note-0549-03" xlink:href="note-0549-03a" xml:space="preserve">Coro. 15. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1224" type="section" level="1" n="737"> <head xml:id="echoid-head770" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13701" xml:space="preserve">Præſentem propoſitionem denuò ſecun<gap/> <lb/>thodum oſtendere volui, vt <gap/>um hanc nouam me-<lb/><gap/> nuius imitationem, reliquæ <lb/>ſuppleri po<gap/>, non alia, quam ſupradictorum nomi-<lb/>num mutatione facta, demonſtratio ſimillima fit, cum ea pariter <lb/>fuerint ſtabilita principia, vt in antecedentibus potuit ſtudioſus <lb/>animaduertere, quæ principijs methodi indiuiſibilium ſimilia ap-<lb/>parebant, ſufficiet ergo tales propoſitiones, tantum innuere, cum <pb o="530" file="0550" n="550" rhead="GEOMETRIÆ"/> illæ non aliam mutationem, quam prædictam in ſuis demonſtra-<lb/>tionibus, popoſcere videbuntur. </s> <s xml:id="echoid-s13702" xml:space="preserve">Quoad regulas autem, iuxta <lb/>quas dicimus ſolida rectangula contineri, poterimus etiam vice <lb/>duarum vnam tantum retinere, pro vt in methodo indiuiſibilium <lb/>effectum eſt, vt ex. </s> <s xml:id="echoid-s13703" xml:space="preserve">g. </s> <s xml:id="echoid-s13704" xml:space="preserve">in fig. </s> <s xml:id="echoid-s13705" xml:space="preserve">huius prop. </s> <s xml:id="echoid-s13706" xml:space="preserve">poterat ſufficere ipſa, DF, <lb/>altera enim regula non alio fungitur offitio, quam determinandi <lb/>cum priori regula vnum planum, cui plana ſolida rectangula ſecã-<lb/>tia, ac in illis rectangula plana producentia, æquidiſtant, & </s> <s xml:id="echoid-s13707" xml:space="preserve">hoc in <lb/>antecedentibus effectum eſt, vt clarior ſolidorum rectangulorum <lb/>deſcripcio haberetur, in poſterum tamen vnam tantum regulam <lb/>innuemus, alteram tacitè ſubintelligentes, dum præfata vni cuidã <lb/>eſſe parallela ſemper ſupponere debeamus, erunt autem eædem <lb/>regulæ, quæ in propoſitionibus infra citandis adhibitæ fuerunt, niſi <lb/>alias regulas innuendi quandoq; </s> <s xml:id="echoid-s13708" xml:space="preserve">neceſſitatem habuerimus.</s> <s xml:id="echoid-s13709" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1225" type="section" level="1" n="738"> <head xml:id="echoid-head771" xml:space="preserve">THEOREMA XVII. PROPOS. XVII.</head> <p> <s xml:id="echoid-s13710" xml:space="preserve">IN eodem Prop. </s> <s xml:id="echoid-s13711" xml:space="preserve">30. </s> <s xml:id="echoid-s13712" xml:space="preserve">Lib. </s> <s xml:id="echoid-s13713" xml:space="preserve">2. </s> <s xml:id="echoid-s13714" xml:space="preserve">ſchemate, regula eadem ibi <lb/>aſſumpta, rectangulum ſolidum ſub, AF, FB, ad rectan-<lb/>gulum ſolidum ſub trapezio, ADEC, & </s> <s xml:id="echoid-s13715" xml:space="preserve">triangulo, BEC, <lb/>erit vt, DF, ad compoſitam ex, {1/2}. </s> <s xml:id="echoid-s13716" xml:space="preserve">DE, & </s> <s xml:id="echoid-s13717" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s13718" xml:space="preserve">EF.</s> <s xml:id="echoid-s13719" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13720" xml:space="preserve">Hæc oſtendetur vtibi, prædicta tantum nominum mutatione <lb/>facta, vt meditanti innoteſcet.</s> <s xml:id="echoid-s13721" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1226" type="section" level="1" n="739"> <head xml:id="echoid-head772" xml:space="preserve">THEOREMA XVIII PROPOS. XVIII.</head> <p> <s xml:id="echoid-s13722" xml:space="preserve">IN ſchemate Prop. </s> <s xml:id="echoid-s13723" xml:space="preserve">3 I. </s> <s xml:id="echoid-s13724" xml:space="preserve">eiuſdem Lib. </s> <s xml:id="echoid-s13725" xml:space="preserve">2. </s> <s xml:id="echoid-s13726" xml:space="preserve">regula eadem, <lb/>rectangulum ſolidum ſub, AO, OB, ad rectangulum ſo-<lb/>lidum ſub trapezijs, HACN, MBCN, eſt vt rectangulum, <lb/>HOM, ad rectangulum ſub, HO, MN, cum rectangulo ſub <lb/>compoſito ex {1/2}. </s> <s xml:id="echoid-s13727" xml:space="preserve">HM, & </s> <s xml:id="echoid-s13728" xml:space="preserve">@. </s> <s xml:id="echoid-s13729" xml:space="preserve">NO, & </s> <s xml:id="echoid-s13730" xml:space="preserve">ſub, NO.</s> <s xml:id="echoid-s13731" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13732" xml:space="preserve">Hæc ſimiliter vt antecedens expedietur.</s> <s xml:id="echoid-s13733" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1227" type="section" level="1" n="740"> <head xml:id="echoid-head773" xml:space="preserve">THEOREMA XIX. PROPOS. XIX.</head> <p> <s xml:id="echoid-s13734" xml:space="preserve">IN ſchemate Prop. </s> <s xml:id="echoid-s13735" xml:space="preserve">32. </s> <s xml:id="echoid-s13736" xml:space="preserve">Lib. </s> <s xml:id="echoid-s13737" xml:space="preserve">2. </s> <s xml:id="echoid-s13738" xml:space="preserve">ſimiliter regula eadem <lb/>retenta, rectangulum ſolidum ſub, AE, ER, ad rectan- <pb o="531" file="0551" n="551" rhead="LIBER VII."/> gulum ſolidum ſub trapezijs, ADEC, CESR, erit vt rectan-<lb/>gulum, DES, ad rectangulum ſub, DE, & </s> <s xml:id="echoid-s13739" xml:space="preserve">compoſita ex, SF, <lb/>& </s> <s xml:id="echoid-s13740" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13741" xml:space="preserve">FE, vna cum rectangulo ſub, EF, & </s> <s xml:id="echoid-s13742" xml:space="preserve">compoſita ex {1/6}. </s> <s xml:id="echoid-s13743" xml:space="preserve">E <lb/>F, & </s> <s xml:id="echoid-s13744" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13745" xml:space="preserve">FS.</s> <s xml:id="echoid-s13746" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13747" xml:space="preserve">Hæc etiam vt antecedentes abſoluetur.</s> <s xml:id="echoid-s13748" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1228" type="section" level="1" n="741"> <head xml:id="echoid-head774" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13749" xml:space="preserve">HVcuſq; </s> <s xml:id="echoid-s13750" xml:space="preserve">Propoſitionibus Lib. </s> <s xml:id="echoid-s13751" xml:space="preserve">2. </s> <s xml:id="echoid-s13752" xml:space="preserve">quæ reſtauratione indigere <lb/>videbantur ſatisfactum eſſe manifeſto apparet. </s> <s xml:id="echoid-s13753" xml:space="preserve">Reliquum <lb/>eſt, vt & </s> <s xml:id="echoid-s13754" xml:space="preserve">ſequentium Librarum Propoſitiones denuò perpenden-<lb/>tes, per hanc nouam methodum à nobis quoq; </s> <s xml:id="echoid-s13755" xml:space="preserve">& </s> <s xml:id="echoid-s13756" xml:space="preserve">ipſæ reſtauren-<lb/>tur, quod maiori, qua fieri poterit, breuitate, ac facilitate, nunc <lb/>præſtare conabimur.</s> <s xml:id="echoid-s13757" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1229" type="section" level="1" n="742"> <head xml:id="echoid-head775" xml:space="preserve">THEOREMA XX. PROPOS. XX.</head> <p> <s xml:id="echoid-s13758" xml:space="preserve">ASſumpto ex Schemate Prop. </s> <s xml:id="echoid-s13759" xml:space="preserve">1. </s> <s xml:id="echoid-s13760" xml:space="preserve">Lib. </s> <s xml:id="echoid-s13761" xml:space="preserve">3. </s> <s xml:id="echoid-s13762" xml:space="preserve">ſemicircolo, <lb/>vel ſemiellipſi, EPR, circa diametrum, ER, ſimul cũ <lb/>applicata, BP, quæ etiam ſit regula, & </s> <s xml:id="echoid-s13763" xml:space="preserve">parallelogrammo, H <lb/>B, iuxta quemlibet trium ibi allatorum caſuum nunc oſten-<lb/>demus, conſpecta etiam illa figura, quadratum ſolidum <lb/>portionis, DEP, ad quadratum ſolidum parallelogrammi, <lb/>FP, eſſe vt compoſita ex {1/6}. </s> <s xml:id="echoid-s13764" xml:space="preserve">EB, & </s> <s xml:id="echoid-s13765" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13766" xml:space="preserve">BR, ad ipſam, BR.</s> <s xml:id="echoid-s13767" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13768" xml:space="preserve">Producantur enim indefinitè verius, B, E, ipſę, PB, HE, & </s> <s xml:id="echoid-s13769" xml:space="preserve">fiant, <lb/>B℟, EG, ſingulæ æquales ipſi, RE, & </s> <s xml:id="echoid-s13770" xml:space="preserve">iungantur, G℟, capiaturq; <lb/></s> <s xml:id="echoid-s13771" xml:space="preserve">BX, æqualis ipſi, BE, & </s> <s xml:id="echoid-s13772" xml:space="preserve">per, X, agatur, XL, parallela, ER, & </s> <s xml:id="echoid-s13773" xml:space="preserve">iun-<lb/>gatur, XE, ac ſit quæcumque, CN, applicata in ſemiportione, EP <lb/> <anchor type="figure" xlink:label="fig-0551-01a" xlink:href="fig-0551-01"/> B, quæ producatur indefinitè <lb/>hinc inde vt ſecet, HP, vt in, M, <lb/>EX, vt in, D, LX, vt in, Z, & </s> <s xml:id="echoid-s13774" xml:space="preserve">G℟, <lb/>vt in, Q; </s> <s xml:id="echoid-s13775" xml:space="preserve">ſunt ergo, GB, LB, G <lb/>X, parallelogramma, &</s> <s xml:id="echoid-s13776" xml:space="preserve">, ℟X, eſt <lb/>æqualis, RB, XB, autem ipſi, B <lb/>E, vnderectangulum, ℟XB, eſt <lb/>æquale rectangulo, RBE, hoc <lb/>eſt, in circulo quadrato, BP: </s> <s xml:id="echoid-s13777" xml:space="preserve">ea-<lb/>dem ratione oſtendemus tum rectangulum, QZC, æquari quadra-<lb/>to, CM, tum rectangulnm, QDC, æquari quadrato, CN, & </s> <s xml:id="echoid-s13778" xml:space="preserve">hoc <pb o="532" file="0552" n="552" rhead="GEOMETRI Æ"/> idem probabimus circa alias quaſcumque applicatas. </s> <s xml:id="echoid-s13779" xml:space="preserve">In ellipſi <lb/>verò oſtendemus rectangula, ℟XB, QDC, eſſe vt quadrata, BP, C <lb/>N, ſicut rectangula, ℟XB, QZC, vt quadrata, BP, CM. </s> <s xml:id="echoid-s13780" xml:space="preserve">Ergo ſi <lb/>intelligamus ſolidum rectangulum fieri ſub parallelogrammis, GX, <lb/> <anchor type="figure" xlink:label="fig-0552-01a" xlink:href="fig-0552-01"/> XE, & </s> <s xml:id="echoid-s13781" xml:space="preserve">quadratum ſolidum, EP, <lb/>communi regula, BP, erunt hæc <lb/>ſolida inter ſe æqualiter, vel pro-<lb/>portionaliter, analoga, cum ſint <lb/>in eiſdem planis parallelis, nem-<lb/>pè tranſeuntibus per lineas, ℟P, <lb/>GH, & </s> <s xml:id="echoid-s13782" xml:space="preserve">quæcunq; </s> <s xml:id="echoid-s13783" xml:space="preserve">plana his pa-<lb/>rallela præfata ſolida ſecantia, <lb/>producant in ipſis æquales figu-<lb/>ras planas, vel ſaltem proportionales, ſicut patuit de rectangulo, Q <lb/>ZC, æquali quadrato, CM; </s> <s xml:id="echoid-s13784" xml:space="preserve">vel ad idem exiſtente, vt rectangulum, <lb/>℟XB, ad quadratum, BP. </s> <s xml:id="echoid-s13785" xml:space="preserve">Eadem ratione, quia probauimus re-<lb/>ctangulum, QDC, æquari quadrato, CN, vel ad idem eſſe vt re-<lb/>ctangulum, ℟XB, ad quadratum, BP, concludemus ſolidum rectã-<lb/>gulũ ſub trapezio, EG℟X, & </s> <s xml:id="echoid-s13786" xml:space="preserve">triangulo, EXB, eſſe ęqualiter, vel pro-<lb/>portionaliter, analogum quadr. </s> <s xml:id="echoid-s13787" xml:space="preserve">ſolido, EBP, iuxta communem re-<lb/>gulam, BP, igitur rectangulum ſolidum, ſub GX, XF, æquabitur qua-<lb/>drato ſolido, EP, & </s> <s xml:id="echoid-s13788" xml:space="preserve">rectangulum ſolidum ſub, EG℟X, EXB, ęqua-<lb/>bitur quadrato ſolido, EBP, vel ſaltem erunt proportionalia in el-<lb/>lipſi, ergo quadratum ſolidum, EP, ad quadr. </s> <s xml:id="echoid-s13789" xml:space="preserve">ſolidum, EBP, erit <lb/> <anchor type="note" xlink:label="note-0552-01a" xlink:href="note-0552-01"/> <anchor type="note" xlink:label="note-0552-02a" xlink:href="note-0552-02"/> vt rectangulum ſolidum ſub, GX, XE, ad rectangulum ſolidum <lb/>ſub, EG℟X, &</s> <s xml:id="echoid-s13790" xml:space="preserve">, EXB, hoc eſt, vt, ℟X, ad compoſitam ex {1/2}. </s> <s xml:id="echoid-s13791" xml:space="preserve">℟X, <lb/>& </s> <s xml:id="echoid-s13792" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s13793" xml:space="preserve">XB, ideſt, vt, RB, ad compoſitam ex {1/2}. </s> <s xml:id="echoid-s13794" xml:space="preserve">RB, & </s> <s xml:id="echoid-s13795" xml:space="preserve">{1/6}. </s> <s xml:id="echoid-s13796" xml:space="preserve">BE, ergo, <lb/>iterum conſpecta figura prop. </s> <s xml:id="echoid-s13797" xml:space="preserve">1. </s> <s xml:id="echoid-s13798" xml:space="preserve">lib. </s> <s xml:id="echoid-s13799" xml:space="preserve">3. </s> <s xml:id="echoid-s13800" xml:space="preserve">quadratum ſolidum portio-<lb/> <anchor type="note" xlink:label="note-0552-03a" xlink:href="note-0552-03"/> nis, DEP, ad quadratum ſolidum, EP, erit vt compoſita ex {1/6}. </s> <s xml:id="echoid-s13801" xml:space="preserve">BE, <lb/>& </s> <s xml:id="echoid-s13802" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13803" xml:space="preserve">BR, ad ipſam, BR, cum enim ſemiportiones, DEB, BEP, <lb/>ſint homologę ſecundum regulam planum tranſiens per regulam, <lb/>BP, cuiæquidiſtant plana ſolida ſecantia, ſicut etiam, FB, BH, & </s> <s xml:id="echoid-s13804" xml:space="preserve"><lb/>cum quadratum ſolidum figuræ, FP, diuiſæ per lineam, EB, æque-<lb/> <anchor type="note" xlink:label="note-0552-04a" xlink:href="note-0552-04"/> tur quadratis ſolidis, FB, BH, & </s> <s xml:id="echoid-s13805" xml:space="preserve">duobus rectangulis ſolidis ſub, FB, <lb/>BH, ideſt quatuor quadratis ſolidis, BH, ideò quadratum ſolidum, <lb/>FP, quadruplum erit quadrati ſolidi, BH, ſicut etiam patebit qua-<lb/>dratum ſolidum portionis, DEP, quadruplum eſſe quadrati ſolidi <lb/>ſemiportionis, EBP, ergo, vt quadratum ſolidum, EBP, ad qua-<lb/>dratum ſolidum, BH, ita eſt quadratum ſolidum portionis, DEP, ad <lb/>quadratum ſolidum, DH, ideſt vt compoſita ex {1/6}. </s> <s xml:id="echoid-s13806" xml:space="preserve">BE, & </s> <s xml:id="echoid-s13807" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13808" xml:space="preserve">BR, ad <lb/>ipſam, BR, quod oſtendendum erat.</s> <s xml:id="echoid-s13809" xml:space="preserve"/> </p> <div xml:id="echoid-div1229" type="float" level="2" n="1"> <figure xlink:label="fig-0551-01" xlink:href="fig-0551-01a"> <image file="0551-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0551-01"/> </figure> <figure xlink:label="fig-0552-01" xlink:href="fig-0552-01a"> <image file="0552-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0552-01"/> </figure> <note position="left" xlink:label="note-0552-01" xlink:href="note-0552-01a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-0552-02" xlink:href="note-0552-02a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0552-03" xlink:href="note-0552-03a" xml:space="preserve">16. huius.</note> <note position="left" xlink:label="note-0552-04" xlink:href="note-0552-04a" xml:space="preserve">Cor. 3. 15. <lb/>huius.</note> </div> <pb o="533" file="0553" n="553" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1231" type="section" level="1" n="743"> <head xml:id="echoid-head776" xml:space="preserve">COROLLARIVM.</head> <p style="it"> <s xml:id="echoid-s13810" xml:space="preserve">_E_x proximè dictis manifeſtum eſſe poteſt quadratum ſolidum cuiuſ-<lb/>cumq; </s> <s xml:id="echoid-s13811" xml:space="preserve">figuræ circa diametrum, regula baſi, quadruplum eſſe <lb/>quadrati ſolidi cuiuſuis eiuſdem portionum, quæ ab ipſa diametro ſe-<lb/>parantur.</s> <s xml:id="echoid-s13812" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1232" type="section" level="1" n="744"> <head xml:id="echoid-head777" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13813" xml:space="preserve">POſterior pars prop. </s> <s xml:id="echoid-s13814" xml:space="preserve">1. </s> <s xml:id="echoid-s13815" xml:space="preserve">lib. </s> <s xml:id="echoid-s13816" xml:space="preserve">3. </s> <s xml:id="echoid-s13817" xml:space="preserve">oſtendetur vt ibi dicta nominum <lb/>tantum mutatione facta cum Cor. </s> <s xml:id="echoid-s13818" xml:space="preserve">ſicut etiam prop. </s> <s xml:id="echoid-s13819" xml:space="preserve">2.</s> <s xml:id="echoid-s13820" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1233" type="section" level="1" n="745"> <head xml:id="echoid-head778" xml:space="preserve">THEOREMA XXI. PROPOS. XXI.</head> <p> <s xml:id="echoid-s13821" xml:space="preserve">ASſumpto ex ſchemate prop. </s> <s xml:id="echoid-s13822" xml:space="preserve">3. </s> <s xml:id="echoid-s13823" xml:space="preserve">ſemicirculo, vel ſemiel-<lb/>lipſi, ASFD, circa diametrum, AD, ſimul cum appli-<lb/>catis, RF, MS, quarum altera ſit regula, & </s> <s xml:id="echoid-s13824" xml:space="preserve">parallelogram-<lb/>mo, NR, oſtendemus in illius figura, quadratum ſolidum <lb/>FB, ad quadratum ſolidum portionis, ICFS, eſſe vt rectan-<lb/>gulum, DRA, ad rectangulom ſub, DR, & </s> <s xml:id="echoid-s13825" xml:space="preserve">ſub compoſita <lb/>ex {1/2}. </s> <s xml:id="echoid-s13826" xml:space="preserve">RM, & </s> <s xml:id="echoid-s13827" xml:space="preserve">ex, MA, vna cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s13828" xml:space="preserve">ſub <lb/>compoſita ex {1/6}. </s> <s xml:id="echoid-s13829" xml:space="preserve">RM, & </s> <s xml:id="echoid-s13830" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13831" xml:space="preserve">MA.</s> <s xml:id="echoid-s13832" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13833" xml:space="preserve">Producantur enim indefinitè ipſæ applicatæ, SM, FR, verſus, <lb/>MR, à quibus abſcindantur, CR, HM, ſingillatim ipſi, DA, æqua-<lb/> <anchor type="figure" xlink:label="fig-0553-01a" xlink:href="fig-0553-01"/> les, vt etiam, GQ, HK, ſin-<lb/>gillatim pariter æquales ipſi, <lb/>DR, &</s> <s xml:id="echoid-s13834" xml:space="preserve">, YR, LM, æquales <lb/>ipſi, MA, & </s> <s xml:id="echoid-s13835" xml:space="preserve">iungantur, HG, <lb/>kQ, LY, QL, & </s> <s xml:id="echoid-s13836" xml:space="preserve">ſit, TX, quę <lb/>cumq; </s> <s xml:id="echoid-s13837" xml:space="preserve">inter, RF, MS, diame-<lb/>tro, AD, ſimiliter applicata, <lb/>quæ indefinitè hincinde ex-<lb/>tendatur ſecans, NF, in, V, A <lb/>D, in, T, LY, in, I, LQ, in, O, KQ, in, Z, &</s> <s xml:id="echoid-s13838" xml:space="preserve">, HG, in, P. </s> <s xml:id="echoid-s13839" xml:space="preserve">Erunt <lb/>ergo, HQ, KY, LR, parallelogramma, & </s> <s xml:id="echoid-s13840" xml:space="preserve">rectangulum, GQR, æ-<lb/>quabitur rectangulo, DRA, cum autem, GQ, ſit æqualis, DR, &</s> <s xml:id="echoid-s13841" xml:space="preserve">, <lb/>YR, ipſi, MA, erit, QY, æqualis, RM, hoc eſt ipſi, YL, eſt autem, <lb/>QY, ad, YL, vt, OI, ad, L, ergo, OI, æquatur, IL, ideſt, TM, &</s> <s xml:id="echoid-s13842" xml:space="preserve">, Z <lb/>I, ipſi, RM, ergo, ZO, æquatur, RT, ergo rectangulum, POT, æ- <pb o="534" file="0554" n="554" rhead="GEOMETR I Æ"/> quatur quoq; </s> <s xml:id="echoid-s13843" xml:space="preserve">rectangulo, DTA, ergo vt rectangulum, GQR, ad <lb/>rectangulum, POT, ita rectangulum, DRA, erit ad rectangulum, <lb/>DTA, hoc eſt ita quadratum, RF, ad quadratum, TX, ergo per-<lb/>mutando rectangulum, GQR, ad quadratum, RF, erit vt rectan-<lb/>gulum, POT, ad quadratum, TX, quod & </s> <s xml:id="echoid-s13844" xml:space="preserve">in reliquis huiuſmodi <lb/> <anchor type="note" xlink:label="note-0554-01a" xlink:href="note-0554-01"/> oſtendetur ſpatijs ergo rectangulum ſolidum ſub trapezijs, LHG <lb/> <anchor type="note" xlink:label="note-0554-02a" xlink:href="note-0554-02"/> Q, LMRQ, & </s> <s xml:id="echoid-s13845" xml:space="preserve">quadratum ſolidum, MSXFR, erunt, vel æqualiter <lb/>in circulo, vel proportionaliter analoga in ellipſi, ergo erunt inter <lb/>ſe vt rectangulum, GQR, & </s> <s xml:id="echoid-s13846" xml:space="preserve">quadratum, RF, ſunt quoq; </s> <s xml:id="echoid-s13847" xml:space="preserve">inter ſe, <lb/>ſed vt rectangulum, GQR, ad quadratum, RF, ſic etiam eſſe oſtẽ-<lb/>demus rectangulum ſolidum ſub, HQ, QM, ad quadratum ſolidũ, <lb/>NR, ergo rectangulum ſolidum ſub, HQ, QM, ad quadratum ſo-<lb/>lidum, NR, erit vt rectang. </s> <s xml:id="echoid-s13848" xml:space="preserve">ſolidum ſub, LHGQ, LMRQ, ad qua-<lb/>dratum ſolidum, MSXFR, & </s> <s xml:id="echoid-s13849" xml:space="preserve">permutando rectangulum iolidum <lb/>ſub, HQ, QM, ad rectangulum ſolidum ſub, LHGQ, LMRQ, erit <lb/>vt quadratum ſolidum, NR, ad quadratum ſolidum, MSXFR, eſt <lb/>autem rectangulum ſolidum ſub, HQ, QM, ad rectangulum ſoli-<lb/>dum ſub, HGQL, LQRM, vt rectangulum ſub, GQR, ad rectan-<lb/> <anchor type="note" xlink:label="note-0554-03a" xlink:href="note-0554-03"/> gulum ſub, GQ; </s> <s xml:id="echoid-s13850" xml:space="preserve">& </s> <s xml:id="echoid-s13851" xml:space="preserve">ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s13852" xml:space="preserve">QY, & </s> <s xml:id="echoid-s13853" xml:space="preserve">ex, YR; </s> <s xml:id="echoid-s13854" xml:space="preserve">vna cum <lb/>rectangulo ſub, QY, & </s> <s xml:id="echoid-s13855" xml:space="preserve">ſub compoſita ex {1/6}. </s> <s xml:id="echoid-s13856" xml:space="preserve">QY, & </s> <s xml:id="echoid-s13857" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13858" xml:space="preserve">YR, hoc eſt <lb/>vt rectangulum, DRA, ad rectangulum ſub, DR, & </s> <s xml:id="echoid-s13859" xml:space="preserve">ſub compoſi-<lb/>ta ex {1/2}. </s> <s xml:id="echoid-s13860" xml:space="preserve">RM, & </s> <s xml:id="echoid-s13861" xml:space="preserve">ex, MA, vna cum rectangulo ſub, RM, & </s> <s xml:id="echoid-s13862" xml:space="preserve">ſub cõ-<lb/>poſita ex {1/6}. </s> <s xml:id="echoid-s13863" xml:space="preserve">RM, & </s> <s xml:id="echoid-s13864" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s13865" xml:space="preserve">MA, ergo ſic etiam erit quadratum ſolidum <lb/>NR, ad quadratum ſolidum ſemiportionis, MSXFR, & </s> <s xml:id="echoid-s13866" xml:space="preserve">ita etiam <lb/>quadratum ſolidum, BF, ad quadratum ſolidum ipſius, ICFS, con-<lb/>ſpecta figura dìctæ prop. </s> <s xml:id="echoid-s13867" xml:space="preserve">3. </s> <s xml:id="echoid-s13868" xml:space="preserve">quod oſtendere opus erat.</s> <s xml:id="echoid-s13869" xml:space="preserve"/> </p> <div xml:id="echoid-div1233" type="float" level="2" n="1"> <figure xlink:label="fig-0553-01" xlink:href="fig-0553-01a"> <image file="0553-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0553-01"/> </figure> <note position="left" xlink:label="note-0554-01" xlink:href="note-0554-01a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-0554-02" xlink:href="note-0554-02a" xml:space="preserve">Coroll. 2. <lb/>13. huius.</note> <note position="left" xlink:label="note-0554-03" xlink:href="note-0554-03a" xml:space="preserve">19. huius.</note> </div> </div> <div xml:id="echoid-div1235" type="section" level="1" n="746"> <head xml:id="echoid-head779" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13870" xml:space="preserve">POſterior pars dictæ Prop. </s> <s xml:id="echoid-s13871" xml:space="preserve">3. </s> <s xml:id="echoid-s13872" xml:space="preserve">oſtendetur vt ibi, ſolita nominum <lb/>mutatione facta, ſicut etiam Prop. </s> <s xml:id="echoid-s13873" xml:space="preserve">4. </s> <s xml:id="echoid-s13874" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13875" xml:space="preserve">5. </s> <s xml:id="echoid-s13876" xml:space="preserve">reſtauratione <lb/>non indiget; </s> <s xml:id="echoid-s13877" xml:space="preserve">Cor. </s> <s xml:id="echoid-s13878" xml:space="preserve">autem deducetur eodem modo, vt ibi mutatis <lb/>tantum dictis nominibus, fiunt enim quadrata ſolida figurarum <lb/>ijſdem parallelis in eiuſdem ſchemate interceptarum, figuræ ſolidę <lb/> <anchor type="note" xlink:label="note-0554-04a" xlink:href="note-0554-04"/> æqualiter analogæ, vnde etiam ſunt æquales, ex quo concluditur <lb/>deinde Corollarium eodem modo, quo ibi factum eſt. </s> <s xml:id="echoid-s13879" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13880" xml:space="preserve">6. <lb/></s> <s xml:id="echoid-s13881" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13882" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13883" xml:space="preserve">7. </s> <s xml:id="echoid-s13884" xml:space="preserve">8. </s> <s xml:id="echoid-s13885" xml:space="preserve">9. </s> <s xml:id="echoid-s13886" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13887" xml:space="preserve">pariter vt ibi oſtendentur, muta-<lb/>tis nominibus, vt ſupra Prop. </s> <s xml:id="echoid-s13888" xml:space="preserve">10. </s> <s xml:id="echoid-s13889" xml:space="preserve">ſic patebit probabuntur enim fi-<lb/>guræ, AFH, AGH, eſſe proportionaliter analogæ, & </s> <s xml:id="echoid-s13890" xml:space="preserve">ideò eſſe inter <lb/>ſe, vt, FH, HG, eodem modo, quo ibi factum eſt, ex quo ſimiliterr <lb/>concludetur, AFVT, ad, AGVS, eſſe vt, FT, ad, GS; </s> <s xml:id="echoid-s13891" xml:space="preserve">& </s> <s xml:id="echoid-s13892" xml:space="preserve">non diſ- <pb o="535" file="0555" n="555" rhead="LI R VII."/> ſimiliter in Cor. </s> <s xml:id="echoid-s13893" xml:space="preserve">colligemus quadratum ſolidum, AFVT, ad quad <lb/>ſolidum, AGVS, eſſe vt quadratum, FT, ad quadratum, GS, ſubau-<lb/>di tamen in illius ſchemate ſecundas diametros, FT, GS, eſſe in eadẽ <lb/>recta linea. </s> <s xml:id="echoid-s13894" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13895" xml:space="preserve">11. </s> <s xml:id="echoid-s13896" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13897" xml:space="preserve">demonſtrantur, vt ibi, Prop. </s> <s xml:id="echoid-s13898" xml:space="preserve">verò <lb/>12. </s> <s xml:id="echoid-s13899" xml:space="preserve">ſimiliter, ſolita tantum nominum mutatione facta. </s> <s xml:id="echoid-s13900" xml:space="preserve">Prop. </s> <s xml:id="echoid-s13901" xml:space="preserve">13. <lb/></s> <s xml:id="echoid-s13902" xml:space="preserve">oſtendetur quoque mutatis nominibus, & </s> <s xml:id="echoid-s13903" xml:space="preserve">c in qua aduerte pag. </s> <s xml:id="echoid-s13904" xml:space="preserve">27. </s> <s xml:id="echoid-s13905" xml:space="preserve"><lb/>lin. </s> <s xml:id="echoid-s13906" xml:space="preserve">22. </s> <s xml:id="echoid-s13907" xml:space="preserve">ſuperfluè dici in, EF, quadratum, EI, detractum à rectan-<lb/>gulo ſub, IE, EF, relinquere rectangulum ſub, EI, IF, vt concluda-<lb/>tur detractis omnibus quadratis ſemiportionis, OCD, à rectangu-<lb/>lis ſub parallelogrammo, OV, & </s> <s xml:id="echoid-s13908" xml:space="preserve">ſemiportione, OCD, relinqui re-<lb/>ctangula ſub, OCD, DCV, hoc enim conſtat ex C. </s> <s xml:id="echoid-s13909" xml:space="preserve">23. </s> <s xml:id="echoid-s13910" xml:space="preserve">1. </s> <s xml:id="echoid-s13911" xml:space="preserve">2. </s> <s xml:id="echoid-s13912" xml:space="preserve">vt ci-<lb/>tatur in margine, illud tamen ad maiorem declarationem appoſi-<lb/>tum erat. </s> <s xml:id="echoid-s13913" xml:space="preserve">Corollarium eiuſdem pariter declarabitur mutatis, &</s> <s xml:id="echoid-s13914" xml:space="preserve">c. </s> <s xml:id="echoid-s13915" xml:space="preserve"><lb/>Prop. </s> <s xml:id="echoid-s13916" xml:space="preserve">14. </s> <s xml:id="echoid-s13917" xml:space="preserve">ſimiliter probabitur, cum Cor. </s> <s xml:id="echoid-s13918" xml:space="preserve">mutatis nominibus, &</s> <s xml:id="echoid-s13919" xml:space="preserve">c. </s> <s xml:id="echoid-s13920" xml:space="preserve"><lb/>Sic etiam Prop. </s> <s xml:id="echoid-s13921" xml:space="preserve">15. </s> <s xml:id="echoid-s13922" xml:space="preserve">16. </s> <s xml:id="echoid-s13923" xml:space="preserve">17. </s> <s xml:id="echoid-s13924" xml:space="preserve">18. </s> <s xml:id="echoid-s13925" xml:space="preserve">19. </s> <s xml:id="echoid-s13926" xml:space="preserve">20. </s> <s xml:id="echoid-s13927" xml:space="preserve">21. </s> <s xml:id="echoid-s13928" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13929" xml:space="preserve">in qua patebit <lb/>rectangula ſub, ASB, AHTFB, æquari rectangulis ſub triangulis, <lb/>ABD, AVD, cum ſint ſolida æqualiter analoga, & </s> <s xml:id="echoid-s13930" xml:space="preserve">hoc in figura <lb/>circuli, in figura autem ellipſis dicta ſolida oſtendentur eſſe propor-<lb/>tion aliter analoga, ac inter ſe vt coniugatarum diametrorum qua-<lb/>drata. </s> <s xml:id="echoid-s13931" xml:space="preserve">Sic etiam Prop. </s> <s xml:id="echoid-s13932" xml:space="preserve">22. </s> <s xml:id="echoid-s13933" xml:space="preserve">23. </s> <s xml:id="echoid-s13934" xml:space="preserve">24. </s> <s xml:id="echoid-s13935" xml:space="preserve">25. </s> <s xml:id="echoid-s13936" xml:space="preserve">26. </s> <s xml:id="echoid-s13937" xml:space="preserve">in qua ſchema ante-<lb/>cedentis reponendum eſt. </s> <s xml:id="echoid-s13938" xml:space="preserve">Propoſ. </s> <s xml:id="echoid-s13939" xml:space="preserve">27. </s> <s xml:id="echoid-s13940" xml:space="preserve">28. </s> <s xml:id="echoid-s13941" xml:space="preserve">29. </s> <s xml:id="echoid-s13942" xml:space="preserve">30. </s> <s xml:id="echoid-s13943" xml:space="preserve">31. </s> <s xml:id="echoid-s13944" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13945" xml:space="preserve"><lb/>Prop. </s> <s xml:id="echoid-s13946" xml:space="preserve">32. </s> <s xml:id="echoid-s13947" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s13948" xml:space="preserve">ac tandem Prop. </s> <s xml:id="echoid-s13949" xml:space="preserve">33. </s> <s xml:id="echoid-s13950" xml:space="preserve">pariter cum Cor.</s> <s xml:id="echoid-s13951" xml:space="preserve"/> </p> <div xml:id="echoid-div1235" type="float" level="2" n="1"> <note position="left" xlink:label="note-0554-04" xlink:href="note-0554-04a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div1237" type="section" level="1" n="747"> <head xml:id="echoid-head780" xml:space="preserve">THEOREMA XXII. PROPOS. XXII.</head> <p> <s xml:id="echoid-s13952" xml:space="preserve">EXpoſitis duabus quibuſcumq; </s> <s xml:id="echoid-s13953" xml:space="preserve">figuris planis, & </s> <s xml:id="echoid-s13954" xml:space="preserve">in ea-<lb/>rum vnaquaq; </s> <s xml:id="echoid-s13955" xml:space="preserve">ſumpta vtcumq; </s> <s xml:id="echoid-s13956" xml:space="preserve">regula, vt quadrata <lb/>ſolida earundem figurarum iuxta dictas regulas, ita erunt <lb/>ſolida quæcumq; </s> <s xml:id="echoid-s13957" xml:space="preserve">ad inuicem ſimilaria ex eiſdem genita fi-<lb/>guris, iuxta eaſdem regulas.</s> <s xml:id="echoid-s13958" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13959" xml:space="preserve">Sint duæ quæcumq; </s> <s xml:id="echoid-s13960" xml:space="preserve">figuræ planæ, ABC, DEF, in qnibus duæ <lb/>vtcumq; </s> <s xml:id="echoid-s13961" xml:space="preserve">ſint ſumptæ, BC, EF, rectælineæ. </s> <s xml:id="echoid-s13962" xml:space="preserve">Dico igitur vt qua-<lb/>dratum ſolidum figuræ, ABC, ad quadratum ſolidum figuræ, DE <lb/>F, regulis iam dictis ita eſſe quodcunq; </s> <s xml:id="echoid-s13963" xml:space="preserve">ſolidum ſimilare genitum <lb/>ex, ABC, ad ſibi ſimilare genitum ex, DEF, iuxta eaſdem regulas. <lb/></s> <s xml:id="echoid-s13964" xml:space="preserve">Ducatur in altera figurarum, vt in, DEF, vtcumq; </s> <s xml:id="echoid-s13965" xml:space="preserve">regulæ, EF, pa-<lb/> <anchor type="note" xlink:label="note-0555-01a" xlink:href="note-0555-01"/> rallela, HM. </s> <s xml:id="echoid-s13966" xml:space="preserve">Igitur quadratum, EF, ad quadratum, HM, habet <lb/>duplicatam rationem eius, quam habet, EF, ad, HM, ſed etiam <lb/> <anchor type="note" xlink:label="note-0555-02a" xlink:href="note-0555-02"/> alia quælibet figura plana deſcripta ab, EF, ad ſibi ſimilem deſcri- <pb o="536" file="0556" n="556" rhead="GEOMETRIE"/> ptam ab, HM, prædictæ homologa, habet duplicatam rationem <lb/>eius, quam, EF, habet ad, HM, ergo vt quadratum, EF, ad qua-<lb/>dratum, HM, ita eſt figura, EF, ad ſibi ſimilem figuram deſcriptã <lb/>ab, HM, & </s> <s xml:id="echoid-s13967" xml:space="preserve">permutando vt quadratum, EF, ad figuram quamcuq; <lb/></s> <s xml:id="echoid-s13968" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0556-01a" xlink:href="fig-0556-01"/> aliam deſcriptam ab, EF, ita erit quadratum, <lb/>HM, ad figuram prędictæ ſimilem deſcriptam <lb/>ab, HM, prædictæ homologa, ergo quadratũ <lb/>ſolidum figuræ, DEF, & </s> <s xml:id="echoid-s13969" xml:space="preserve">ſolidum ſimilare quo-<lb/>dcumq; </s> <s xml:id="echoid-s13970" xml:space="preserve">genitum ex figura, DEF, luxta comu <lb/>nem regulam, EF, ſunt ſolida proportionaliter <lb/>analoga ſecundum communem regulã, EF, ergo erunt inter ſe vt fi-<lb/> <anchor type="note" xlink:label="note-0556-01a" xlink:href="note-0556-01"/> guræ planæ ab eodem latere, vt ab, EF, deſcriptę. </s> <s xml:id="echoid-s13971" xml:space="preserve">Eodẽ modo oſtẽ-<lb/>demus quadratum ſolidũ, ABC, & </s> <s xml:id="echoid-s13972" xml:space="preserve">ſolidũ aliud quodcumq; </s> <s xml:id="echoid-s13973" xml:space="preserve">ſimilare <lb/>genitum ex figura, ABC, iuxta cõmunẽ regulam, BC, eſſe inter ſe, vt <lb/>figuræ à, BC, deſcriptæ, ſunt autem duo quadrata, BC, EF, & </s> <s xml:id="echoid-s13974" xml:space="preserve">duæ <lb/>aliæ quæcumq; </s> <s xml:id="echoid-s13975" xml:space="preserve">ſimiles figuræ planæ deſcriptæ ab homologis, BC, <lb/>EF, proportionales, ergo & </s> <s xml:id="echoid-s13976" xml:space="preserve">dicta ſolida proportionalia erunt, nẽpè <lb/>vt quadratũ, BC, ad figuram, BC, ſic erat quadratum ſolidũ, ABC, <lb/>ad ſolidum ſimilare genitũ ex, ABC, ſed vt quadratũ, BC, ad figurã, <lb/>BC, ita eſt qua dratum, EF, ad figuram, EF, prædictæ ſimilem, & </s> <s xml:id="echoid-s13977" xml:space="preserve">ita <lb/>etiam quadratum ſolidum, DEF, ad ſolidum prædicto ſimilare ge-<lb/>nitum ex, DEF, ergo quadratum ſolidum, ABC, ad ſolidum ſimi-<lb/>lare, ABC, eſt vt quadratum ſolidum, DEF, ad ſolidum præ dicto <lb/>ſimilare genitũ ex, DEF, & </s> <s xml:id="echoid-s13978" xml:space="preserve">permutando quad. </s> <s xml:id="echoid-s13979" xml:space="preserve">ſolidũ, ABC, ad qua-<lb/>dratũ ſolidũ, DEF, erit vt ſolidũ quodcũq; </s> <s xml:id="echoid-s13980" xml:space="preserve">ſimilare genitũ ex, ABC, <lb/>ad ſibi ſimilare gen tũ ex, DEF, luxta dictas regulas, quod oſtendere <lb/>opus erat.</s> <s xml:id="echoid-s13981" xml:space="preserve"/> </p> <div xml:id="echoid-div1237" type="float" level="2" n="1"> <note position="right" xlink:label="note-0555-01" xlink:href="note-0555-01a" xml:space="preserve">15. 1. 2.</note> <note position="right" xlink:label="note-0555-02" xlink:href="note-0555-02a" xml:space="preserve">15. l. 2.</note> <figure xlink:label="fig-0556-01" xlink:href="fig-0556-01a"> <image file="0556-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0556-01"/> </figure> <note position="left" xlink:label="note-0556-01" xlink:href="note-0556-01a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div1239" type="section" level="1" n="748"> <head xml:id="echoid-head781" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s13982" xml:space="preserve">HVius demonſtratio ſimilis eſt demonſtrationi Prop. </s> <s xml:id="echoid-s13983" xml:space="preserve">33. </s> <s xml:id="echoid-s13984" xml:space="preserve">1. </s> <s xml:id="echoid-s13985" xml:space="preserve">2. </s> <s xml:id="echoid-s13986" xml:space="preserve">cui <lb/>per hanc ſuppletur, Coro laria autem iuxta methodum ibi <lb/>adhibitam facilè quoq; </s> <s xml:id="echoid-s13987" xml:space="preserve">deducentur, illam vero huc reſeruaui, vt <lb/>promptiorem pro colligendis ſequentibus Corollarijs lib. </s> <s xml:id="echoid-s13988" xml:space="preserve">3. </s> <s xml:id="echoid-s13989" xml:space="preserve">ex hac <lb/>pendentibus eam haberemus. </s> <s xml:id="echoid-s13990" xml:space="preserve">Adhibuit quidem nomen ſolidi ſi-<lb/>milaris, quod per indefinitum numerum parallelorum planorum <lb/>fuit pariter explicatum lib. </s> <s xml:id="echoid-s13991" xml:space="preserve">2. </s> <s xml:id="echoid-s13992" xml:space="preserve">ad B. </s> <s xml:id="echoid-s13993" xml:space="preserve">Defin. </s> <s xml:id="echoid-s13994" xml:space="preserve">8. </s> <s xml:id="echoid-s13995" xml:space="preserve">attamen ſi vice om-<lb/>nium planorum, ſeu deſcriptarum figurarum, ſubſtituamus quot-<lb/>cumq; </s> <s xml:id="echoid-s13996" xml:space="preserve">plana, ſeu deſcriptas figuras, ita vt perimetri deſcriptarum <lb/>figurarum iacere intelligantur in ſuperficie ipſum ſolidum ambiẽ-<lb/>te, intelligemus nihilominus, licet nonnihil diuerſo modo, eſſe idẽ <lb/>ſolidum, quod dicitur ſimilare, ac à propria genitrice deſcriptum, <pb o="537" file="0557" n="557" rhead="LIBER VII."/> iuxta datam regulam, ſiue ſecundam illam definitionem abſolutè, <lb/>ſiue per eandem ſic modificatam, vt hæc ſimilaria ſolida ab infini-<lb/>tatis conceptu, ſeu ab indiuiſibilium methodo, eximerentur; </s> <s xml:id="echoid-s13997" xml:space="preserve">Non <lb/>eſt autem difficile inſuper intelligere quadrata ſolida quarumcũq; <lb/></s> <s xml:id="echoid-s13998" xml:space="preserve">planarum figurarum, in ambitu eorundem exiſtentium, eſſe etiam <lb/>ſolida ſimilaria, genita ex eiſdem figuris, quarum dicuntur qua-<lb/>drata ſolida, iuxta eaſdem regulas, iuxta quas quadrata ſolida di-<lb/>cebantur: </s> <s xml:id="echoid-s13999" xml:space="preserve">& </s> <s xml:id="echoid-s14000" xml:space="preserve">è conuerſo ſolida ſimilaria, genita ex quibuſcumq; </s> <s xml:id="echoid-s14001" xml:space="preserve"><lb/>figuris iuxta quaſuis regulas, quarum figuræ, à genitricium lineis <lb/>homologis deſcriptæ tamquam à lateribus, ſint quadrata, eſſe pa-<lb/>riter quadrata ſolida earundem figurarum iuxta eaſdem regulas. </s> <s xml:id="echoid-s14002" xml:space="preserve"><lb/>Igitur ad rem noſtram manifeſtum eſt, quod quæcumq; </s> <s xml:id="echoid-s14003" xml:space="preserve">ſolida ad <lb/>inuicem ſimilaria, genita ex figuris lib. </s> <s xml:id="echoid-s14004" xml:space="preserve">3. </s> <s xml:id="echoid-s14005" xml:space="preserve">hic denuò conſideratis, <lb/>iuxta aſſumptas regulas (quarum patefacta eſt ratio quadratorum <lb/>ſolidorum) habebunt rationem notam, per quod ſuppletur Propo-<lb/>ſit. </s> <s xml:id="echoid-s14006" xml:space="preserve">34. </s> <s xml:id="echoid-s14007" xml:space="preserve">lib. </s> <s xml:id="echoid-s14008" xml:space="preserve">3. </s> <s xml:id="echoid-s14009" xml:space="preserve">colligentur autem vt ibi factum eſt ſequentia Corol-<lb/>laria vſq; </s> <s xml:id="echoid-s14010" xml:space="preserve">ad finem eiuſdem lib. </s> <s xml:id="echoid-s14011" xml:space="preserve">3. </s> <s xml:id="echoid-s14012" xml:space="preserve">mutatis tantum ſæpè dictis no-<lb/>minibus, vbi neceſſe fuerit, quod enim ibi per omnia quadrata hic <lb/>per quadrata ſolida conſideratarum figurarum colligetur. </s> <s xml:id="echoid-s14013" xml:space="preserve">Do-<lb/>ctrina autem ſcholij ſubſequentis etiam pro hac noua methodo <lb/>ſubſiſtit, ſi tamen vice omnium figurarum, ſeu omnium planorum, <lb/>ſubſtitutas intelligamus quotcumque figuras, ſeu quotcumque <lb/>plana, cætera cnim à methodo indiuiſibilium exempta ſunt, & </s> <s xml:id="echoid-s14014" xml:space="preserve">hæc <lb/>ſufficiant circa examen lib. </s> <s xml:id="echoid-s14015" xml:space="preserve">3. </s> <s xml:id="echoid-s14016" xml:space="preserve">nunc autem Prop. </s> <s xml:id="echoid-s14017" xml:space="preserve">lib. </s> <s xml:id="echoid-s14018" xml:space="preserve">4. </s> <s xml:id="echoid-s14019" xml:space="preserve">ſimiliter <lb/>perluſtrabimus.</s> <s xml:id="echoid-s14020" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1240" type="section" level="1" n="749"> <head xml:id="echoid-head782" xml:space="preserve">THEOREMA XXIII. PROPOS. XXIII.</head> <p> <s xml:id="echoid-s14021" xml:space="preserve">ASſumpta ex ſchemate Prop. </s> <s xml:id="echoid-s14022" xml:space="preserve">1. </s> <s xml:id="echoid-s14023" xml:space="preserve">Lib. </s> <s xml:id="echoid-s14024" xml:space="preserve">4. </s> <s xml:id="echoid-s14025" xml:space="preserve">ſemiparabola, <lb/>CHG, cum parallelogrammo, EG, viſa tamen etiam <lb/>illa figura, oſtendemus parallelogrammum, EF, ſexquial-<lb/>terum eſſe parabolæ, HCF.</s> <s xml:id="echoid-s14026" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14027" xml:space="preserve">Producta enim diametro, CG, vtcumq; </s> <s xml:id="echoid-s14028" xml:space="preserve">in, V, deſcribatur qua-<lb/>drans circuli, vel ellipſis, HGV, iuxta duas ſemidiametros coniu-<lb/>gatas, HG, GV, & </s> <s xml:id="echoid-s14029" xml:space="preserve">per, H, ducta, EP, parallela, CV, & </s> <s xml:id="echoid-s14030" xml:space="preserve">indefinitè <lb/>extenſa, agantur ſimiliter à punctis, CV, parallele, HG, ipſæ, EC, <lb/>PV, erunt ergo parallelogramma, EG, GP, EG, quidem circum-<lb/>ſcriptum ſemiparabolæ, HCG, &</s> <s xml:id="echoid-s14031" xml:space="preserve">, PG, quadrati, HGV, ſit inſu-<lb/>per quæcumq; </s> <s xml:id="echoid-s14032" xml:space="preserve">MO, ordinatim ad, HG, applicata, regula, CG, quę <pb o="538" file="0558" n="558" rhead="GEOMETRI Æ"/> pro alis in hac propoſitione ſit pariter regula, extendaturq; </s> <s xml:id="echoid-s14033" xml:space="preserve">hinc <lb/>indevt ſecet EC, vt in, N, HV, vt in, R, & </s> <s xml:id="echoid-s14034" xml:space="preserve">PV, vt in, Q, ergo, CG, <lb/> <anchor type="figure" xlink:label="fig-0558-01a" xlink:href="fig-0558-01"/> ad, MO, erit vt quadratum, GH, <lb/>ad rectangulum ſub compoſita <lb/>ex, HG, GO, & </s> <s xml:id="echoid-s14035" xml:space="preserve">ſub, OH, (hoc. </s> <s xml:id="echoid-s14036" xml:space="preserve">n. <lb/></s> <s xml:id="echoid-s14037" xml:space="preserve">deducitur ex prop. </s> <s xml:id="echoid-s14038" xml:space="preserve">3. </s> <s xml:id="echoid-s14039" xml:space="preserve">lib. </s> <s xml:id="echoid-s14040" xml:space="preserve">4. </s> <s xml:id="echoid-s14041" xml:space="preserve">quæ <lb/>non dependet à prop. </s> <s xml:id="echoid-s14042" xml:space="preserve">1. </s> <s xml:id="echoid-s14043" xml:space="preserve">neq; </s> <s xml:id="echoid-s14044" xml:space="preserve">indi-<lb/>get, quod denuò demonſtretur) <lb/>ideſt vt quadratum, GV, ad qua-<lb/>dratum, OR, ſed vt, CG, ad MO, <lb/>ita eſt quadratum, CG, ad rectan-<lb/>gulum ſub, CG, ſeu, NO, & </s> <s xml:id="echoid-s14045" xml:space="preserve">OM, ergo quadratum, CG, ad rectan-<lb/>gulum, NOM, erit vt quadratum, GV, ad quadratum, OR, & </s> <s xml:id="echoid-s14046" xml:space="preserve">per-<lb/>mutando quadratum, CG, ad quadratum, GV, erit vt rectan-<lb/>gulum, NOM, ad quadratum, OR, ſic ductis alijs parallelis <lb/>euenire oſtendemus; </s> <s xml:id="echoid-s14047" xml:space="preserve">ergo ſolidum rectangulum ſub, EG, paral-<lb/>lelogrammo, & </s> <s xml:id="echoid-s14048" xml:space="preserve">ſemiparabola, CHG, erit proportionaliter ana-<lb/>logum quadrato ſolido quadrantis, HVG, ſecundum regulam, <lb/>C V, ſecundum eandem autem oſtendemus etiam quadratum <lb/>ſolidum, EG, eſſe proportionaliter analogum quadrato ſolido, H <lb/>V, etenim quadratum, CG, ad quadratum, GV, eſt vt quadratum, <lb/>NO, ad quadratum, OQ, vnde vt quadratum, CG, ad quadratum, <lb/>GV, ſic erit quadratum ſolidum, EG, ad quadratum ſolidum, G <lb/>P, & </s> <s xml:id="echoid-s14049" xml:space="preserve">ſic etiam rectangulum ſolidum ſub, EG, HCG, ad quadratum <lb/>ſolidum, HGV, ergo quadratum ſolidum, EG, ad quadratum ſoli-<lb/>dum, HV, erit vt rectangulum ſolidum ſub, EG, HCG, ad quadra-<lb/>tum ſolidum, HGV, ergo permutando quadratum ſolidum, EG, ad <lb/>rectangulum ſolidum ſub, EG, HCG, erit vt quadratum ſolidum, <lb/>HV, ad quadratum ſolidum, HGV, ſed quadratum ſolidum, HV, <lb/> <anchor type="note" xlink:label="note-0558-01a" xlink:href="note-0558-01"/> ſequialterum eſt quadrati ſolidi, HGV, cum, VG, tranſeat per cen-<lb/>trum, G, ergo quadratum ſolidum, EG, ſexquialterum erit rectan-<lb/>guli ſolidi ſub, EG, HCG, ſed vt quadratum ſolidum, EG, ad rectan-<lb/>gulum ſolidum ſub, EG, HCG, ita baſis, EG, ad baſim, HCG, ergo, <lb/>EG, erit ſexquialtera, HCG, & </s> <s xml:id="echoid-s14050" xml:space="preserve">conſequenter, viſa figura prop. </s> <s xml:id="echoid-s14051" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s14052" xml:space="preserve"> <anchor type="note" xlink:label="note-0558-02a" xlink:href="note-0558-02"/> lib. </s> <s xml:id="echoid-s14053" xml:space="preserve">4. </s> <s xml:id="echoid-s14054" xml:space="preserve">erit parallelogrammum, AH, ſequialterum parabolæ, FCH, <lb/>quod oſtendendum erat.</s> <s xml:id="echoid-s14055" xml:space="preserve"/> </p> <div xml:id="echoid-div1240" type="float" level="2" n="1"> <figure xlink:label="fig-0558-01" xlink:href="fig-0558-01a"> <image file="0558-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0558-01"/> </figure> <note position="left" xlink:label="note-0558-01" xlink:href="note-0558-01a" xml:space="preserve">Elicitur <lb/>ex 20. hu-<lb/>ius.</note> <note position="left" xlink:label="note-0558-02" xlink:href="note-0558-02a" xml:space="preserve">Cor. 1. 13. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1242" type="section" level="1" n="750"> <head xml:id="echoid-head783" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s14056" xml:space="preserve">PEr hanc autem ſuppletur prop. </s> <s xml:id="echoid-s14057" xml:space="preserve">1. </s> <s xml:id="echoid-s14058" xml:space="preserve">lib. </s> <s xml:id="echoid-s14059" xml:space="preserve">4. </s> <s xml:id="echoid-s14060" xml:space="preserve">etenim illius poſterior <lb/>pars deducetur, vt ibi, hac vero demonſtrata reliqua nunc <pb o="539" file="0559" n="559" rhead="LIBER VII."/> percurremus. </s> <s xml:id="echoid-s14061" xml:space="preserve">Igitur circa Corollarium p. </s> <s xml:id="echoid-s14062" xml:space="preserve">1. </s> <s xml:id="echoid-s14063" xml:space="preserve">nihil dicendum eſt. <lb/></s> <s xml:id="echoid-s14064" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14065" xml:space="preserve">2. </s> <s xml:id="echoid-s14066" xml:space="preserve">autem reſtauratione non indiget. </s> <s xml:id="echoid-s14067" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14068" xml:space="preserve">3. </s> <s xml:id="echoid-s14069" xml:space="preserve">ſimiliter. </s> <s xml:id="echoid-s14070" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14071" xml:space="preserve">4. </s> <s xml:id="echoid-s14072" xml:space="preserve"><lb/>oſten detur eo modo, quo nos primam demonſtrauimus, Corolla-<lb/>riũ verò deducetur vt ibi, mutatis tamẽ ſepè dictis nominibus &</s> <s xml:id="echoid-s14073" xml:space="preserve">c. </s> <s xml:id="echoid-s14074" xml:space="preserve">ex <lb/>hac autem oſtenſa facilè deducetur prop. </s> <s xml:id="echoid-s14075" xml:space="preserve">5. </s> <s xml:id="echoid-s14076" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14077" xml:space="preserve">mutatis &</s> <s xml:id="echoid-s14078" xml:space="preserve">c. </s> <s xml:id="echoid-s14079" xml:space="preserve"><lb/>vt etiam prop. </s> <s xml:id="echoid-s14080" xml:space="preserve">6. </s> <s xml:id="echoid-s14081" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14082" xml:space="preserve">p. </s> <s xml:id="echoid-s14083" xml:space="preserve">7. </s> <s xml:id="echoid-s14084" xml:space="preserve">8. </s> <s xml:id="echoid-s14085" xml:space="preserve">cum dictis in Scholio. </s> <s xml:id="echoid-s14086" xml:space="preserve">Similiter <lb/>Prop. </s> <s xml:id="echoid-s14087" xml:space="preserve">9. </s> <s xml:id="echoid-s14088" xml:space="preserve">10. </s> <s xml:id="echoid-s14089" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14090" xml:space="preserve">mutatis &</s> <s xml:id="echoid-s14091" xml:space="preserve">c. </s> <s xml:id="echoid-s14092" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14093" xml:space="preserve">11. </s> <s xml:id="echoid-s14094" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14095" xml:space="preserve">p. </s> <s xml:id="echoid-s14096" xml:space="preserve">12. </s> <s xml:id="echoid-s14097" xml:space="preserve">13. </s> <s xml:id="echoid-s14098" xml:space="preserve">14. </s> <s xml:id="echoid-s14099" xml:space="preserve">15. </s> <s xml:id="echoid-s14100" xml:space="preserve"><lb/>16. </s> <s xml:id="echoid-s14101" xml:space="preserve">17. </s> <s xml:id="echoid-s14102" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14103" xml:space="preserve">18. </s> <s xml:id="echoid-s14104" xml:space="preserve">19. </s> <s xml:id="echoid-s14105" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14106" xml:space="preserve">20. </s> <s xml:id="echoid-s14107" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14108" xml:space="preserve">reſtaurationem mi-<lb/>nimè poſtulant, cum a methodo in diuiſibilium non dependeant.</s> <s xml:id="echoid-s14109" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1243" type="section" level="1" n="751"> <head xml:id="echoid-head784" xml:space="preserve">THEOREMA XXIV. PROPOS. XXIV.</head> <p> <s xml:id="echoid-s14110" xml:space="preserve">EXpoſito denuò Schemate prop. </s> <s xml:id="echoid-s14111" xml:space="preserve">21. </s> <s xml:id="echoid-s14112" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s14113" xml:space="preserve">4. </s> <s xml:id="echoid-s14114" xml:space="preserve">regu-<lb/>la eadem, VF, retenta, oſtendemus quadratum ſolidũ, <lb/>AF, duplum eſſe quadrati ſolidi parabolæ, VEF, & </s> <s xml:id="echoid-s14115" xml:space="preserve">hoc eſſe <lb/>ſexquialterum quadrati ſolidi trianguli, VEF.</s> <s xml:id="echoid-s14116" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14117" xml:space="preserve">Eſtò quòd, ND, ſecet, EF, in, I, igitur rectangulum, DNI, eſt æ-<lb/>quale quadrato, NO, quod & </s> <s xml:id="echoid-s14118" xml:space="preserve">circa quaſcumq; </s> <s xml:id="echoid-s14119" xml:space="preserve">applicatas con-<lb/> <anchor type="note" xlink:label="note-0559-01a" xlink:href="note-0559-01"/> <anchor type="figure" xlink:label="fig-0559-01a" xlink:href="fig-0559-01"/> tingere concludemus, ergo rectan-<lb/>gulum ſolidum ſub parallelogram-<lb/>mo, CM, & </s> <s xml:id="echoid-s14120" xml:space="preserve">triangulo, EMF, erit æ-<lb/>qualiter analogum quadrato ſolido <lb/>ſemiparabolæ, EMF, quadratum <lb/>ſolidum autem, CM, ad rectangulũ <lb/>ſolidum ſub eodem parallelogram-<lb/>mo, CM, & </s> <s xml:id="echoid-s14121" xml:space="preserve">ſub triangulo, EMF, eſt <lb/>vt, CM, ad EMF, ideſt duplum, ergo quadratum ſolidum, CM, du-<lb/> <anchor type="note" xlink:label="note-0559-02a" xlink:href="note-0559-02"/> plum erit quadrati ſolidi, EMF, & </s> <s xml:id="echoid-s14122" xml:space="preserve">conſequenter quadratum ſoli-<lb/>dum, AF, duplum etiam erit quadrati ſolidi parabolæ, VEF, vnde <lb/>& </s> <s xml:id="echoid-s14123" xml:space="preserve">quadratum ſolidum, VEF, ſexquialterum erit quadrati ſolidi, E <lb/>VF, quod &</s> <s xml:id="echoid-s14124" xml:space="preserve">c.</s> <s xml:id="echoid-s14125" xml:space="preserve"/> </p> <div xml:id="echoid-div1243" type="float" level="2" n="1"> <note position="right" xlink:label="note-0559-01" xlink:href="note-0559-01a" xml:space="preserve">13 l. 4.</note> <figure xlink:label="fig-0559-01" xlink:href="fig-0559-01a"> <image file="0559-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0559-01"/> </figure> <note position="right" xlink:label="note-0559-02" xlink:href="note-0559-02a" xml:space="preserve">Cor. 1. 13. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1245" type="section" level="1" n="752"> <head xml:id="echoid-head785" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s14126" xml:space="preserve">PEr ſuprapoſitam prop. </s> <s xml:id="echoid-s14127" xml:space="preserve">ſuppletur prop. </s> <s xml:id="echoid-s14128" xml:space="preserve">21. </s> <s xml:id="echoid-s14129" xml:space="preserve">prop. </s> <s xml:id="echoid-s14130" xml:space="preserve">22. </s> <s xml:id="echoid-s14131" xml:space="preserve">verò dedu-<lb/>cetur eodem modo mutatis nominibus &</s> <s xml:id="echoid-s14132" xml:space="preserve">c.</s> <s xml:id="echoid-s14133" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1246" type="section" level="1" n="753"> <head xml:id="echoid-head786" xml:space="preserve">THEOREMA XXV. PROPOS. XXV.</head> <p> <s xml:id="echoid-s14134" xml:space="preserve">ASſumpta ex Schemate prop. </s> <s xml:id="echoid-s14135" xml:space="preserve">23. </s> <s xml:id="echoid-s14136" xml:space="preserve">ſemiparabola, NOH, <lb/>cum fruſto, MROH, & </s> <s xml:id="echoid-s14137" xml:space="preserve">parallelogrammo, VO, ac re- <pb o="540" file="0560" n="560" rhead="GEOMETRI Æ"/> cta, TX, ſecante curuam, MH, in, I, regula, OH, oſtendemus <lb/>quadratum ſolidum, PH, viſa dicta figura, ad quadratum ſo-<lb/>lidum, ABHM, eſſe vt, ON, ad compoſitam ex, NR, & </s> <s xml:id="echoid-s14138" xml:space="preserve">@. </s> <s xml:id="echoid-s14139" xml:space="preserve">RO.</s> <s xml:id="echoid-s14140" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14141" xml:space="preserve">Extendantur .</s> <s xml:id="echoid-s14142" xml:space="preserve">n. </s> <s xml:id="echoid-s14143" xml:space="preserve">HO, VR, verſus, OR, & </s> <s xml:id="echoid-s14144" xml:space="preserve">fiant, OC, RA, ſingulæ <lb/>æquales ipſi, ON, &</s> <s xml:id="echoid-s14145" xml:space="preserve">, DO, BR, capiantur ſingulæ æquales ipſi, RN <lb/>& </s> <s xml:id="echoid-s14146" xml:space="preserve">iungantur, AC, BD, CB, quas extenſſa indefinitè, TX, ſecet in, F’ <lb/>P, E. </s> <s xml:id="echoid-s14147" xml:space="preserve">Etunt ergo, AO, BO, AD, parallelogramma. </s> <s xml:id="echoid-s14148" xml:space="preserve">Cum verò, CO’ <lb/>æquetur, ON, & </s> <s xml:id="echoid-s14149" xml:space="preserve">DO, ipſi, RN, erit, CD, æqualis, OR .</s> <s xml:id="echoid-s14150" xml:space="preserve">i. </s> <s xml:id="echoid-s14151" xml:space="preserve">ipſi, DB, <lb/>vnde etiam, EP, ipſi, PB, hoc eſt ipſi, RX, & </s> <s xml:id="echoid-s14152" xml:space="preserve">tota, EX, toti, NX, & </s> <s xml:id="echoid-s14153" xml:space="preserve"><lb/>reliqua, FE, reliquæ, OX, æqualis erit. </s> <s xml:id="echoid-s14154" xml:space="preserve">Quoniam vero quadratum, <lb/> <anchor type="figure" xlink:label="fig-0560-01a" xlink:href="fig-0560-01"/> OH, ad quadratum, XI, eſt <lb/>vt, ON, ad, NX, hoc eſt, FX, <lb/>ad, XE, hoc eſt quadratum, <lb/>FX, vel quadratum, CO, ad <lb/>rectangulum, FXE, ideo per-<lb/>mutando quadratum, HO, <lb/>ad quadratum, OC, erit vt <lb/>quadratum, IX, ad rectangu-<lb/>lum, FXE, ex quo conclude-<lb/>mus, vt in ſuperioribus rectangulum ſolidum ſub, AO, & </s> <s xml:id="echoid-s14155" xml:space="preserve">trapezio, <lb/>BCOR, eſſe proportionaliter analogum quadrato ſolido, RMHO. <lb/></s> <s xml:id="echoid-s14156" xml:space="preserve">Similiter oſtendemus quadrata ſolida, AO, OV, eſſe proportionali-<lb/>ter analoga, & </s> <s xml:id="echoid-s14157" xml:space="preserve">conſequenter prædictis duobus ſolidis eſſe propor-<lb/>tionalia colligemus, vnde permutando quadratum ſolidum, VO, <lb/>ad quadratum ſolidum, MORH, ſeu (conſpecta figura prop. </s> <s xml:id="echoid-s14158" xml:space="preserve">23. </s> <s xml:id="echoid-s14159" xml:space="preserve">lib. </s> <s xml:id="echoid-s14160" xml:space="preserve"><lb/>4.) </s> <s xml:id="echoid-s14161" xml:space="preserve">quadratum ſolidum, PH, ad quadratum ſolidum fruſti, ABHM, <lb/>erit vt quadratum ſolidum, AO, ad rectangulum ſolidum ſub, AO, <lb/> <anchor type="note" xlink:label="note-0560-01a" xlink:href="note-0560-01"/> & </s> <s xml:id="echoid-s14162" xml:space="preserve">trapezio, BCOR .</s> <s xml:id="echoid-s14163" xml:space="preserve">i. </s> <s xml:id="echoid-s14164" xml:space="preserve">vt, AO, ad, BCOR .</s> <s xml:id="echoid-s14165" xml:space="preserve">i. </s> <s xml:id="echoid-s14166" xml:space="preserve">vt, CO, ad compoſi-<lb/>tam ex, OD, & </s> <s xml:id="echoid-s14167" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s14168" xml:space="preserve">DC, .</s> <s xml:id="echoid-s14169" xml:space="preserve">i. </s> <s xml:id="echoid-s14170" xml:space="preserve">vt, ON, ad compoſitam ex, NR, & </s> <s xml:id="echoid-s14171" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s14172" xml:space="preserve">R <lb/>O, quod &</s> <s xml:id="echoid-s14173" xml:space="preserve">c.</s> <s xml:id="echoid-s14174" xml:space="preserve"/> </p> <div xml:id="echoid-div1246" type="float" level="2" n="1"> <figure xlink:label="fig-0560-01" xlink:href="fig-0560-01a"> <image file="0560-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0560-01"/> </figure> <note position="left" xlink:label="note-0560-01" xlink:href="note-0560-01a" xml:space="preserve">Cor. 1. 13. <lb/>huius 20. <lb/>l. 2.</note> </div> </div> <div xml:id="echoid-div1248" type="section" level="1" n="754"> <head xml:id="echoid-head787" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s14175" xml:space="preserve">PEr hanc ſimiliter ſuppletur prop. </s> <s xml:id="echoid-s14176" xml:space="preserve">23. </s> <s xml:id="echoid-s14177" xml:space="preserve">poſterior .</s> <s xml:id="echoid-s14178" xml:space="preserve">n. </s> <s xml:id="echoid-s14179" xml:space="preserve">pars, cum <lb/>Cor. </s> <s xml:id="echoid-s14180" xml:space="preserve">deducetur vt ibi, mutatis nominibus &</s> <s xml:id="echoid-s14181" xml:space="preserve">c. </s> <s xml:id="echoid-s14182" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14183" xml:space="preserve">24. </s> <s xml:id="echoid-s14184" xml:space="preserve">reſtau-<lb/>ratione non indiget, ſicut etiam p. </s> <s xml:id="echoid-s14185" xml:space="preserve">25. </s> <s xml:id="echoid-s14186" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14187" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14188" xml:space="preserve">26. </s> <s xml:id="echoid-s14189" xml:space="preserve">oſtende-<lb/>tur etiam vt ibi, mutatis, &</s> <s xml:id="echoid-s14190" xml:space="preserve">c. </s> <s xml:id="echoid-s14191" xml:space="preserve">ſicut & </s> <s xml:id="echoid-s14192" xml:space="preserve">p. </s> <s xml:id="echoid-s14193" xml:space="preserve">27. </s> <s xml:id="echoid-s14194" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14195" xml:space="preserve">ſimiliter p. <lb/></s> <s xml:id="echoid-s14196" xml:space="preserve">28. </s> <s xml:id="echoid-s14197" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14198" xml:space="preserve">p. </s> <s xml:id="echoid-s14199" xml:space="preserve">29. </s> <s xml:id="echoid-s14200" xml:space="preserve">cum Corollarijs, p. </s> <s xml:id="echoid-s14201" xml:space="preserve">30. </s> <s xml:id="echoid-s14202" xml:space="preserve">cum Corollarijs, p. </s> <s xml:id="echoid-s14203" xml:space="preserve"><lb/>31. </s> <s xml:id="echoid-s14204" xml:space="preserve">32. </s> <s xml:id="echoid-s14205" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14206" xml:space="preserve">p. </s> <s xml:id="echoid-s14207" xml:space="preserve">33. </s> <s xml:id="echoid-s14208" xml:space="preserve">34. </s> <s xml:id="echoid-s14209" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14210" xml:space="preserve">35. </s> <s xml:id="echoid-s14211" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14212" xml:space="preserve">p. </s> <s xml:id="echoid-s14213" xml:space="preserve">36. </s> <s xml:id="echoid-s14214" xml:space="preserve">37. </s> <s xml:id="echoid-s14215" xml:space="preserve">38. </s> <s xml:id="echoid-s14216" xml:space="preserve">39. </s> <s xml:id="echoid-s14217" xml:space="preserve">40. </s> <s xml:id="echoid-s14218" xml:space="preserve"><lb/>cum Cor. </s> <s xml:id="echoid-s14219" xml:space="preserve">41. </s> <s xml:id="echoid-s14220" xml:space="preserve">42. </s> <s xml:id="echoid-s14221" xml:space="preserve">43. </s> <s xml:id="echoid-s14222" xml:space="preserve">44. </s> <s xml:id="echoid-s14223" xml:space="preserve">45. </s> <s xml:id="echoid-s14224" xml:space="preserve">p. </s> <s xml:id="echoid-s14225" xml:space="preserve">46. </s> <s xml:id="echoid-s14226" xml:space="preserve">autem eſt nobis reſtauranda.</s> <s xml:id="echoid-s14227" xml:space="preserve"/> </p> <pb o="541" file="0561" n="561" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1249" type="section" level="1" n="755"> <head xml:id="echoid-head788" xml:space="preserve">THEOREMA XXVI. PROPOS. XXVI.</head> <p> <s xml:id="echoid-s14228" xml:space="preserve">IN figura prop.</s> <s xml:id="echoid-s14229" xml:space="preserve">46. </s> <s xml:id="echoid-s14230" xml:space="preserve">oſtendemus, regula eadem retenta, re-<lb/>ctangulum ſolidum ſub, HP, PE, duplum eſſe rectangu-<lb/>liſolidi ſub, BZPD, DPG.</s> <s xml:id="echoid-s14231" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14232" xml:space="preserve">Sumatur .</s> <s xml:id="echoid-s14233" xml:space="preserve">n. </s> <s xml:id="echoid-s14234" xml:space="preserve">de illius ſchemate <lb/> <anchor type="figure" xlink:label="fig-0561-01a" xlink:href="fig-0561-01"/> parallelogrammum, HG, cum fru-<lb/>ſto parabolæ, BZGD, & </s> <s xml:id="echoid-s14235" xml:space="preserve">rectis, R <lb/>F, DP, fiat autem inſuper, AP, <lb/>æqualis, PD, & </s> <s xml:id="echoid-s14236" xml:space="preserve">ducta, AM, paral-<lb/>lela, DP, iungatur, AD, ſecans, C <lb/>T, in, N. </s> <s xml:id="echoid-s14237" xml:space="preserve">Cum ergo in dicta prop. <lb/></s> <s xml:id="echoid-s14238" xml:space="preserve">independenter ab indiuiſibilium methodo, cõcludatur rectangulũ’, <lb/>RTF, ad, STI, eſſe vt, PD, ad, DT, idpſum & </s> <s xml:id="echoid-s14239" xml:space="preserve">hic tanquã demonſtra-<lb/>tũ reci piemus, ſed, PD, ad, DT, hoc eſt, CT, ad, TN, eſt vt quadra-<lb/>tum, CT, ad rectangulum, CTN, ergo rectangulum, RTF, ad, STI, <lb/>erit vt quadratum, CT, ad rectangulum, CTN, eſt autem, RF, vt-<lb/>cumq; </s> <s xml:id="echoid-s14240" xml:space="preserve">ducta parallela, ZG, ergo modo conſueto oſtendemus ſoli-<lb/>dum rectangulum ſub, HP, PE, eſſe proportionaliter analogum <lb/>quadrato ſolido, MP, ſicut rectangulum ſolidum ſub, BZPD, DP <lb/>G, eſſe proportionalites analogum rectangulo ſolidoſub, MP, PA <lb/>D, & </s> <s xml:id="echoid-s14241" xml:space="preserve">tandem concludemus hæc ſolida eſſe proportionalia, ideſt <lb/>rectangulum ſolidum ſub, HP, PE, ad rectang. </s> <s xml:id="echoid-s14242" xml:space="preserve">ſolidum ſub, BZPD, <lb/>DPG, eſſe vt quadratum ſolidum MP, ad rectangulum ſolidum ſub, <lb/>MP, PDA, ideſt vt, MP, ad, PDA, ideſt concludemus rectangulum <lb/>ſolidum ſub, HP, PE, duplum eſſe rectanguli ſolidi ſub, BZPD, PD <lb/> <anchor type="note" xlink:label="note-0561-01a" xlink:href="note-0561-01"/> G, quod oſtendendum erat.</s> <s xml:id="echoid-s14243" xml:space="preserve"/> </p> <div xml:id="echoid-div1249" type="float" level="2" n="1"> <figure xlink:label="fig-0561-01" xlink:href="fig-0561-01a"> <image file="0561-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0561-01"/> </figure> <note position="right" xlink:label="note-0561-01" xlink:href="note-0561-01a" xml:space="preserve">Cor. I. 13. <lb/>hu.u.s</note> </div> </div> <div xml:id="echoid-div1251" type="section" level="1" n="756"> <head xml:id="echoid-head789" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s14244" xml:space="preserve">PRop.</s> <s xml:id="echoid-s14245" xml:space="preserve">46. </s> <s xml:id="echoid-s14246" xml:space="preserve">igitur reſtaurata, ſtylo noſtro ſequentium propoſitio-<lb/>num demonſtrationes proſequemur ab hac vſque a d 51. </s> <s xml:id="echoid-s14247" xml:space="preserve">inclu-<lb/>fiuè, quæ quidem veritatem habere comperitur ex prop.</s> <s xml:id="echoid-s14248" xml:space="preserve">22. </s> <s xml:id="echoid-s14249" xml:space="preserve">hu us. <lb/></s> <s xml:id="echoid-s14250" xml:space="preserve">Scholium autem ſequens retineatur vt ib, ſubſtituendo tamen no-<lb/>mini omnium ſimilium figurarum hocaliud, nempe quotcunque <lb/>ſimiles figuras &</s> <s xml:id="echoid-s14251" xml:space="preserve">c. </s> <s xml:id="echoid-s14252" xml:space="preserve">vt in examine lib. </s> <s xml:id="echoid-s14253" xml:space="preserve">3. </s> <s xml:id="echoid-s14254" xml:space="preserve">animaduerſum eſt. </s> <s xml:id="echoid-s14255" xml:space="preserve">His ve-<lb/>rò prædemonſtratis ſubſequentia Corollaria vſq; </s> <s xml:id="echoid-s14256" xml:space="preserve">ad finem lib.</s> <s xml:id="echoid-s14257" xml:space="preserve">4. </s> <s xml:id="echoid-s14258" xml:space="preserve">ſo-<lb/>lita nominum mutatione facta, cuncta facillimè deducentur per <lb/>iam oſtenſa circa quadrata, ſeu rectangula ſolida ſub talibus, & </s> <s xml:id="echoid-s14259" xml:space="preserve">ta- <pb o="542" file="0562" n="562" rhead="GEOMETRI Æ"/> libus figuris, in antecedentibus prop. </s> <s xml:id="echoid-s14260" xml:space="preserve">conſideratis. </s> <s xml:id="echoid-s14261" xml:space="preserve">Appendix au-<lb/>tem Cor 6 reſtauratione minimè indigere manifeſtum eſt. </s> <s xml:id="echoid-s14262" xml:space="preserve">Et hæc <lb/>circa prop.</s> <s xml:id="echoid-s14263" xml:space="preserve">lib.</s> <s xml:id="echoid-s14264" xml:space="preserve">4. </s> <s xml:id="echoid-s14265" xml:space="preserve">adnotaſſe ſufficiat, reliquum eſt, vt ad lib.</s> <s xml:id="echoid-s14266" xml:space="preserve">5. </s> <s xml:id="echoid-s14267" xml:space="preserve">exami-<lb/>nandum nos conferamus.</s> <s xml:id="echoid-s14268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1252" type="section" level="1" n="757"> <head xml:id="echoid-head790" xml:space="preserve">THEOREMA XXVII. PROPOS. XXVII.</head> <p> <s xml:id="echoid-s14269" xml:space="preserve">IN Schemate prop. </s> <s xml:id="echoid-s14270" xml:space="preserve">1. </s> <s xml:id="echoid-s14271" xml:space="preserve">lib. </s> <s xml:id="echoid-s14272" xml:space="preserve">5. </s> <s xml:id="echoid-s14273" xml:space="preserve">regula eadem retenta, oſten-<lb/>demus quadratum ſolidum parallelogrammi, AF, ad <lb/>quadratum ſolidum hyperbolæ, DBF, eſſe vt, OE, ad com-<lb/>poſitam ex, NB, &</s> <s xml:id="echoid-s14274" xml:space="preserve">. {1/3}. </s> <s xml:id="echoid-s14275" xml:space="preserve">BE.</s> <s xml:id="echoid-s14276" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14277" xml:space="preserve">Aſſumatur .</s> <s xml:id="echoid-s14278" xml:space="preserve">n. </s> <s xml:id="echoid-s14279" xml:space="preserve">ex eo paral-<lb/> <anchor type="figure" xlink:label="fig-0562-01a" xlink:href="fig-0562-01"/> lelogrammum, CE, cum ſe-<lb/>mihyperbola, BEF, & </s> <s xml:id="echoid-s14280" xml:space="preserve">recta, <lb/>OE, necnon, MG, quæcum-<lb/>q; </s> <s xml:id="echoid-s14281" xml:space="preserve">ex ordinatim applicatis ad <lb/>diametrum, BE, extendantur <lb/>autem, CB, FE, & </s> <s xml:id="echoid-s14282" xml:space="preserve">fiant BD, <lb/>EQ, ſingulæ æquales ipſi, E <lb/>O, necnon, RE, AB, ſingulæ <lb/>æquales ipſi, EB, & </s> <s xml:id="echoid-s14283" xml:space="preserve">iungan-<lb/>tur, DQ, AR, AE, quas, GM, <lb/>indefinitè quoq; </s> <s xml:id="echoid-s14284" xml:space="preserve">producta ſecet in punctis, P, S, T. </s> <s xml:id="echoid-s14285" xml:space="preserve">Erunt ergo, D <lb/>R, DE, AE, parallelogramma. </s> <s xml:id="echoid-s14286" xml:space="preserve">Quoniam verò quad. </s> <s xml:id="echoid-s14287" xml:space="preserve">EF, ad quad. <lb/></s> <s xml:id="echoid-s14288" xml:space="preserve"> <anchor type="note" xlink:label="note-0562-01a" xlink:href="note-0562-01"/> MH, eſt vt rectang. </s> <s xml:id="echoid-s14289" xml:space="preserve">OEB, ad, OMB, hoc eſt vt rectangulum, QER, <lb/>ad rectangulum, PTS, permutando quadratum, FE, ad rectangu-<lb/>lum, QER, erit vt quadratum, HM, ad rectangulum, PTS, quod <lb/>& </s> <s xml:id="echoid-s14290" xml:space="preserve">in cæteris oſten demus, ergo quadratum ſolidum, BEF, & </s> <s xml:id="echoid-s14291" xml:space="preserve">re-<lb/>ctangulum ſolidum ſub trapezio, DQEA, & </s> <s xml:id="echoid-s14292" xml:space="preserve">triangulo, AER, erunt <lb/>proportionaliter analoga, ac in proportione quadrati, FE, & </s> <s xml:id="echoid-s14293" xml:space="preserve">re-<lb/>ctanguli, QER, Conſimili modo probabimus quadratum ſolidum, <lb/>CE, eſſe æqualiter analogum rectangulo ſolido ſub, OB, BR, & </s> <s xml:id="echoid-s14294" xml:space="preserve">ad <lb/>ipſum pariter eſſe in proportione quadrati, EF, ad rectangulum, Q <lb/>ER, ergo dicta ſolida proportionalia erunt, & </s> <s xml:id="echoid-s14295" xml:space="preserve">permutãdo quadratũ <lb/>ſolidumCE, ad quad, ſolidũ, BEF, erit vt rectangulum ſolidum ſub, <lb/>QB, BR, ad rectangulum ſolidum ſub, DQEA, ARE, hoc eſt vt, QE, <lb/> <anchor type="note" xlink:label="note-0562-02a" xlink:href="note-0562-02"/> ad cõpoſitam ex {1/2}. </s> <s xml:id="echoid-s14296" xml:space="preserve">QR, & </s> <s xml:id="echoid-s14297" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s14298" xml:space="preserve">RE, hoc eſt vt, OE; </s> <s xml:id="echoid-s14299" xml:space="preserve">ad compoſitãex, N <lb/>B, (quæ eſt dimidia, BO,) & </s> <s xml:id="echoid-s14300" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s14301" xml:space="preserve">BE, igitur, viſo ſchemate dictę prop. <lb/></s> <s xml:id="echoid-s14302" xml:space="preserve">1 quadratum ſolidum, AF, ad quadratum ſolidum, DBF, erit vt, O <lb/>E, ad compoſitam ex, NB, & </s> <s xml:id="echoid-s14303" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s14304" xml:space="preserve">BE, quod demonſtrare oportebat.</s> <s xml:id="echoid-s14305" xml:space="preserve"/> </p> <div xml:id="echoid-div1252" type="float" level="2" n="1"> <figure xlink:label="fig-0562-01" xlink:href="fig-0562-01a"> <image file="0562-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0562-01"/> </figure> <note position="left" xlink:label="note-0562-01" xlink:href="note-0562-01a" xml:space="preserve">39, & ſch. <lb/>40. l. 1.</note> <note position="left" xlink:label="note-0562-02" xlink:href="note-0562-02a" xml:space="preserve">17. huius.</note> </div> <pb o="543" file="0563" n="563" rhead="LIBER VII."/> </div> <div xml:id="echoid-div1254" type="section" level="1" n="758"> <head xml:id="echoid-head791" xml:space="preserve">ANNOTATIO.</head> <p> <s xml:id="echoid-s14306" xml:space="preserve">PEr hanc ſuppletur prior parti prop. </s> <s xml:id="echoid-s14307" xml:space="preserve">1. </s> <s xml:id="echoid-s14308" xml:space="preserve">poſterior verò oſtendetur <lb/>vt ibi, mutatis conſuetis nominibus &</s> <s xml:id="echoid-s14309" xml:space="preserve">c. </s> <s xml:id="echoid-s14310" xml:space="preserve">ſicut etiam prop. </s> <s xml:id="echoid-s14311" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s14312" xml:space="preserve">Conſimili autem methodo adhibita in præſenti prop. </s> <s xml:id="echoid-s14313" xml:space="preserve">oſtendemus <lb/>quad. </s> <s xml:id="echoid-s14314" xml:space="preserve">ſolidum, GE, ad quadratum ſolidum, HMEF, in ſuperiori fig. </s> <s xml:id="echoid-s14315" xml:space="preserve"><lb/>(hoc eſt in figura p.</s> <s xml:id="echoid-s14316" xml:space="preserve">3.</s> <s xml:id="echoid-s14317" xml:space="preserve">lib.</s> <s xml:id="echoid-s14318" xml:space="preserve">5. </s> <s xml:id="echoid-s14319" xml:space="preserve">quadratum ſolidum, SF, ad quadratum <lb/>ſolidum, HDFG,) eſſe vt rectangulum ſolidum ſub, QM, MR, ad <lb/>rectangulum ſolidum ſub trapezijs, PQET, SRET, hoc eſt vt re-<lb/>ctangulum, QER, ad rectangulum ſub, QE, ST, vna cum rectan-<lb/>gulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s14320" xml:space="preserve">PS, & </s> <s xml:id="echoid-s14321" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s14322" xml:space="preserve">TM, & </s> <s xml:id="echoid-s14323" xml:space="preserve">ſub, TM, ideſt viſo ſche-<lb/>mate p.</s> <s xml:id="echoid-s14324" xml:space="preserve">3. </s> <s xml:id="echoid-s14325" xml:space="preserve">vt rectangulum, OEN, ad rectangulum ſub, OE, & </s> <s xml:id="echoid-s14326" xml:space="preserve">NM, <lb/> <anchor type="note" xlink:label="note-0563-01a" xlink:href="note-0563-01"/> vna cum rectangulo ſub compoſita ex {1/2}. </s> <s xml:id="echoid-s14327" xml:space="preserve">NO, & </s> <s xml:id="echoid-s14328" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s14329" xml:space="preserve">ME, & </s> <s xml:id="echoid-s14330" xml:space="preserve">ſub, M <lb/>E; </s> <s xml:id="echoid-s14331" xml:space="preserve">poſterior pars autem eiuſdem prop.</s> <s xml:id="echoid-s14332" xml:space="preserve">3. </s> <s xml:id="echoid-s14333" xml:space="preserve">deducetur vt ibidem, mu-<lb/>tatis nominibus &</s> <s xml:id="echoid-s14334" xml:space="preserve">c. </s> <s xml:id="echoid-s14335" xml:space="preserve">Sicut & </s> <s xml:id="echoid-s14336" xml:space="preserve">omnes prop. </s> <s xml:id="echoid-s14337" xml:space="preserve">à 4. </s> <s xml:id="echoid-s14338" xml:space="preserve">vſque ad 20. </s> <s xml:id="echoid-s14339" xml:space="preserve">incluſi-<lb/>uè, cum earum Corollarijs. </s> <s xml:id="echoid-s14340" xml:space="preserve">In prop. </s> <s xml:id="echoid-s14341" xml:space="preserve">21. </s> <s xml:id="echoid-s14342" xml:space="preserve">verò patebit quadratum ſo-<lb/>lidum, OP, viſa illius figura æquari rectangulo ſolido ſub, O <lb/>LS, OVCS, figuris, regula, DC, etenim ex ibi demonſtratis liquidò <lb/>apparet hæc eſſe ſolida æqualiter analoga iuxta dictam reguiam, <lb/>ex quo de inde reliqua concludentur mutatis nominibus &</s> <s xml:id="echoid-s14343" xml:space="preserve">c. </s> <s xml:id="echoid-s14344" xml:space="preserve">ſicuti <lb/>& </s> <s xml:id="echoid-s14345" xml:space="preserve">Cor. </s> <s xml:id="echoid-s14346" xml:space="preserve">In prop: </s> <s xml:id="echoid-s14347" xml:space="preserve">22. </s> <s xml:id="echoid-s14348" xml:space="preserve">figura ſic eſt corrigenda, debet enim, EC, hinc <lb/>inde produci, vt incidat aſymptotis, OY, OP, verſus eam productis, <lb/>in, S, I, quælitterę deſunt, cæterum prop. </s> <s xml:id="echoid-s14349" xml:space="preserve">oſtendetur vt ibidem mu-<lb/>tatis &</s> <s xml:id="echoid-s14350" xml:space="preserve">c. </s> <s xml:id="echoid-s14351" xml:space="preserve">ſimul cum Corollarijs, necnon prop. </s> <s xml:id="echoid-s14352" xml:space="preserve">23. </s> <s xml:id="echoid-s14353" xml:space="preserve">24. </s> <s xml:id="echoid-s14354" xml:space="preserve">25. </s> <s xml:id="echoid-s14355" xml:space="preserve">26. </s> <s xml:id="echoid-s14356" xml:space="preserve">27. </s> <s xml:id="echoid-s14357" xml:space="preserve">28. <lb/></s> <s xml:id="echoid-s14358" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14359" xml:space="preserve">& </s> <s xml:id="echoid-s14360" xml:space="preserve">29. </s> <s xml:id="echoid-s14361" xml:space="preserve">cum Cor. </s> <s xml:id="echoid-s14362" xml:space="preserve">Prop. </s> <s xml:id="echoid-s14363" xml:space="preserve">30. </s> <s xml:id="echoid-s14364" xml:space="preserve">autem patet ex dictis. </s> <s xml:id="echoid-s14365" xml:space="preserve">His deniq; </s> <s xml:id="echoid-s14366" xml:space="preserve"><lb/>reſtauratis, acſequenti ſcholio modificato, iuxta quod dictum fuit <lb/> <anchor type="note" xlink:label="note-0563-02a" xlink:href="note-0563-02"/> in examine lib. </s> <s xml:id="echoid-s14367" xml:space="preserve">3. </s> <s xml:id="echoid-s14368" xml:space="preserve">& </s> <s xml:id="echoid-s14369" xml:space="preserve">4. </s> <s xml:id="echoid-s14370" xml:space="preserve">ſequentia Corollaria vſq; </s> <s xml:id="echoid-s14371" xml:space="preserve">ad finem eiuſdem l. <lb/></s> <s xml:id="echoid-s14372" xml:space="preserve">5. </s> <s xml:id="echoid-s14373" xml:space="preserve">per quadratorum ſolidorum prædemonſtrata, ſimiliter, vt in præ-<lb/>fatis libris, colligentur, hæc autem pro reſtauratione lib.</s> <s xml:id="echoid-s14374" xml:space="preserve">5. </s> <s xml:id="echoid-s14375" xml:space="preserve">dicta <lb/>ſint ſatis.</s> <s xml:id="echoid-s14376" xml:space="preserve"/> </p> <div xml:id="echoid-div1254" type="float" level="2" n="1"> <note position="right" xlink:label="note-0563-01" xlink:href="note-0563-01a" xml:space="preserve">18. huius.</note> <note position="right" xlink:label="note-0563-02" xlink:href="note-0563-02a" xml:space="preserve">22. huius. <lb/>Annot.22. <lb/>& 26. hu-<lb/>ius.</note> </div> <p> <s xml:id="echoid-s14377" xml:space="preserve">Quoad lib.</s> <s xml:id="echoid-s14378" xml:space="preserve">6. </s> <s xml:id="echoid-s14379" xml:space="preserve">verò patet in eo traditas demonſtrationes, quæ ex <lb/>methodo indiuiſibilium dependebant, ibidẽ fuiſſe reſtauratas. </s> <s xml:id="echoid-s14380" xml:space="preserve">Illæ <lb/>autem propoſitiones, in quibus adhibentur aliquando nomina om-<lb/>nium quadratorum talium, vel talium figurarum, adhuc ſubſiſtent, <lb/>ſi illis nomina quadratorum ſolidorum earundem figurarum ſub-<lb/>ſtituamus, hac.</s> <s xml:id="echoid-s14381" xml:space="preserve">n. </s> <s xml:id="echoid-s14382" xml:space="preserve">ſola mutatione facta, cætera omnia manent in <lb/>ſuo robore, vt in eo libro innuitur in ſcholio prop. </s> <s xml:id="echoid-s14383" xml:space="preserve">20. </s> <s xml:id="echoid-s14384" xml:space="preserve">ac ſuperius <lb/>ſæpè ſæpius repetitum fuit.</s> <s xml:id="echoid-s14385" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1256" type="section" level="1" n="759"> <head xml:id="echoid-head792" style="it" xml:space="preserve">Finis Septimi Libri.</head> <pb file="0564" n="564"/> <pb file="0565" n="565"/> <pb file="0566" n="566"/> <pb file="0567" n="567"/> <handwritten/> <handwritten/> <pb file="0568" n="568"/> <figure> <image file="0568-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0568-01"/> </figure> </div></text> </echo>