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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:PVNDER1Y.xml</dcterms:identifier> <dcterms:creator identifier="GND:10001271X">Angeli, Stefano</dcterms:creator> <dcterms:title xml:lang="la">Miscellaneum hyperbolicum et parabolicum</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1659</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"/> <pb file="0002" n="2"/> <handwritten/> <handwritten/> <pb file="0003" n="3"/> <pb file="0004" n="4"/> <pb file="0005" n="5"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">MISCELL ANEVM <lb/>HYPERBOLICVM, <lb/>ET PARABOLICVM.</head> <head xml:id="echoid-head2" style="it" xml:space="preserve">IN QVO PRÆCIPVE AGITVR DE CENTRIS <lb/>Grauitatis Hyperbolæ, partium eiuſdem,</head> <head xml:id="echoid-head3" style="it" xml:space="preserve">Atque nonnullorum ſolidorum, de quibus nunquam Geometria locuta eſt. <lb/>Parabola nouiter quadratur dupliciter. <lb/>Ducuntur infinitarum parabolarum tangentes. <lb/>Aſſignantur maxima inſcriptibilia, minimaque circumſcriptibilia <lb/>Infinitis Parabolis, Conoidibus, ac ſemifuſis parabolicis. <lb/>Aliaque Geometrica noua exponuntur ſcitu digna.</head> <head xml:id="echoid-head4" xml:space="preserve">AVTHORE <lb/>F. STEPHANODE ANGELIS <lb/>VENETO,</head> <head xml:id="echoid-head5" style="it" xml:space="preserve">Ordinis Ieſuatorum S. HIERONY MI, in Veneta <lb/>Prouincia Definitore Prouinciali.</head> <head xml:id="echoid-head6" xml:space="preserve">AD ILLVSTRISSIMOS, ET SAPIENTISSIMOS <lb/>SENATVS BONONIENSIS <lb/>QVINQVAGINTA VIROS.</head> <figure> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0005-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head7" xml:space="preserve">VENETIIS, MD CLIX.</head> <head xml:id="echoid-head8" xml:space="preserve">Apud Ioannem La Noù.</head> <head xml:id="echoid-head9" style="it" xml:space="preserve">SVPERIORVM PERMISSV.</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/PVNDER1Y/figures/0007-01"/> </figure> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head10" xml:space="preserve">Illuſtriſſimis, & Sapientiſſimis</head> <head xml:id="echoid-head11" xml:space="preserve">BONONIENSIS SENATVS <lb/>QVINQVAGINTA VIRIS</head> <head xml:id="echoid-head12" xml:space="preserve">Dominis Colendiſſimis.</head> <head xml:id="echoid-head13" xml:space="preserve">F. STEPHANVS ANGELI VENETVS</head> <head xml:id="echoid-head14" xml:space="preserve">Ord. leſuatorum S. Hieronymi, ac in Prouincia <lb/>Veneta Prouincialis Definitor P.P.P.</head> <p style="it"> <s xml:id="echoid-s1" xml:space="preserve">EA Virtutis est vis (Illustri ſsimi & </s> <s xml:id="echoid-s2" xml:space="preserve"><lb/>Saptentiſſimi DD.)</s> <s xml:id="echoid-s3" xml:space="preserve">, ac ſolertiſſima <lb/>indo´es, vt an mum ſuauitèr imbuat, <lb/>diſcipliniſq; </s> <s xml:id="echoid-s4" xml:space="preserve">velutitemper amento per-<lb/>optimo, iucundè componat, & </s> <s xml:id="echoid-s5" xml:space="preserve">inſtruat. <lb/></s> <s xml:id="echoid-s6" xml:space="preserve">Quod viuere eſt corpori, id menti prę ſtat <lb/>ſcire excellentiùs; </s> <s xml:id="echoid-s7" xml:space="preserve">namq̀; </s> <s xml:id="echoid-s8" xml:space="preserve">veluti Prome-<lb/>thei inanis ſtatua homo degeret, ſi à ſcientiarum radio fęlici-<lb/>tèr non excitaretur ad vitam. </s> <s xml:id="echoid-s9" xml:space="preserve">Id docuit Apollinis lyra, quę <lb/>lapidem quondam dulciſona fecit carmina reddentem, vitales <lb/>indidit auras, & </s> <s xml:id="echoid-s10" xml:space="preserve">voces, cum in reliquis grauit aret inanimis, <lb/>at q́; </s> <s xml:id="echoid-s11" xml:space="preserve">imè tenderet in centrum. </s> <s xml:id="echoid-s12" xml:space="preserve">Explicet proſperè plumas <lb/>Dedalus, iungat bumeris alas, ſe ſelibret in aera, caſus fu-<lb/>giat crudelitatis deludens ingenium; </s> <s xml:id="echoid-s13" xml:space="preserve">animus verè tunc petit <lb/>æthera, cum ſapientiæ adiumento fulcitur, ſcientiarumq́;</s> <s xml:id="echoid-s14" xml:space="preserve"> <pb file="0008" n="8"/> acumine euadit nuperus Phęnix, vt vires ſumat ad ten-<lb/>tanda ſydera. </s> <s xml:id="echoid-s15" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s16" xml:space="preserve">volitabit mens incunctanter vbi <lb/>ſtudij artificium acceßerit, idq́; </s> <s xml:id="echoid-s17" xml:space="preserve">robur mutuabit à ſcientia, <lb/>quod ab Archytę curaretulit lignea olim columba, cui pennas <lb/>fabrefacere ad volatum, opificis ſors fuit, & </s> <s xml:id="echoid-s18" xml:space="preserve">elucubratio <lb/>valdè diligens. </s> <s xml:id="echoid-s19" xml:space="preserve">Ita est; </s> <s xml:id="echoid-s20" xml:space="preserve">ſi viuat corpus, àt rude extet in <lb/>genium, minimè dicendum, quod viuat homo, qui ſolum vt <lb/>intelligat viuit, opuſq́; </s> <s xml:id="echoid-s21" xml:space="preserve">intelligentiæ exercendo ab animan-<lb/>tibus cęteris ſecernitur. </s> <s xml:id="echoid-s22" xml:space="preserve">Natura greſſum dat pedibus vt cir-<lb/>cumcurſent per orbem; </s> <s xml:id="echoid-s23" xml:space="preserve">verùm, vt mens euebatur, virtus <lb/>eſt, quæ capiti iungit adminicula; </s> <s xml:id="echoid-s24" xml:space="preserve">ideo Mercurius Scientia-<lb/>rum Numen, & </s> <s xml:id="echoid-s25" xml:space="preserve">Pręſes, ceruicem, at q́; </s> <s xml:id="echoid-s26" xml:space="preserve">plant as iurè implicat <lb/>alis. </s> <s xml:id="echoid-s27" xml:space="preserve">Ergo ſi maxima debemus naturæ, cuius ope morituri <lb/>viuimus, potiora ſcientiæ inſcribenda, qua rectè, qua ſa-<lb/>pientèr, qua vtilitèr, qua decorè, qua perennitèr viuimus. <lb/></s> <s xml:id="echoid-s28" xml:space="preserve">Flla nos incunabulis, veluti carceri faſcijs adſtrictos, addicit; </s> <s xml:id="echoid-s29" xml:space="preserve"><lb/>hęc perennitati generosè fouet. </s> <s xml:id="echoid-s30" xml:space="preserve">Flla ab vtero in ærumnoſam <lb/>vitam; </s> <s xml:id="echoid-s31" xml:space="preserve">hęc in gloriæ Capitoliumeducit. </s> <s xml:id="echoid-s32" xml:space="preserve">Flla lacte, quo ſa-<lb/>ginamur infantes, ad corruptionem enutrit; </s> <s xml:id="echoid-s33" xml:space="preserve">bæc nos immor-<lb/>talitati parit, ac posthumos ſeruat. </s> <s xml:id="echoid-s34" xml:space="preserve">Illa demùm parentibus <lb/>emancipat, & </s> <s xml:id="echoid-s35" xml:space="preserve">Patriæ; </s> <s xml:id="echoid-s36" xml:space="preserve">hæc quidquid ſumus Lyceis, & </s> <s xml:id="echoid-s37" xml:space="preserve">præ-<lb/>ceptoribus in ſcribit; </s> <s xml:id="echoid-s38" xml:space="preserve">indeq́; </s> <s xml:id="echoid-s39" xml:space="preserve">profitetur Achilles, pluradebere <lb/>Chyrom, qui ab animo ruditatem eliminauit, quam Thety-<lb/>di, quæ corpus dedit, stygijſq́; </s> <s xml:id="echoid-s40" xml:space="preserve">vndis lotum ictibus expoſuit <lb/>in ffenſum. </s> <s xml:id="echoid-s41" xml:space="preserve">Bononia Glorioſa studiorum Mater, quæ Athe-<lb/>narum reparat vetuſtatem, quæ ſcientijs gymnaſia diſertiſ-<lb/>ſima aperit, quæ Virtuti ſola struit thronum, & </s> <s xml:id="echoid-s42" xml:space="preserve">domicilium, <lb/>quæ postremò Męce ates parat ſapientibus, ad Matheſis me <lb/>accendit Amorem, opportunitatem contulit, Archimedemq́;</s> <s xml:id="echoid-s43" xml:space="preserve"> <pb file="0009" n="9"/> exhibuit, Excellentiſſimum nempè Bonauenturam Cauale-<lb/>rium, qui Geometriæ gloriam perfecit, buiuſce preclariſſimæ <lb/>Vrbis auxit nitorem, Ieſuatorum cętum ampliſſime decora-<lb/>uit, vt puriori Geometricarum dulcedinum lacte, luculenter <lb/>nutrirer. </s> <s xml:id="echoid-s44" xml:space="preserve">Hauſi, quæ nunquam ad ſaturitatem deguſtabo <lb/>alimenta. </s> <s xml:id="echoid-s45" xml:space="preserve">Vestrum Filuſtriſſimi, & </s> <s xml:id="echoid-s46" xml:space="preserve">Sapientiſſimi D D. <lb/></s> <s xml:id="echoid-s47" xml:space="preserve">vrbanitatileniſſimæ, quæ Pręceptorem Caualerium fouit im-<lb/>pensè, iurè ſe ſtatuit diſcipulus, quò fidenter deditiſſima Vo-<lb/>bis hęc libet attramenta, quibus claritatem iungere, vt in-<lb/>occidua ſplendeſcant, veſtræ Nobilitatis, & </s> <s xml:id="echoid-s48" xml:space="preserve">laudis, opus erit, <lb/>as facinus pręſtantiſſimum. </s> <s xml:id="echoid-s49" xml:space="preserve">Tenuis manuſculi inopiam com-<lb/>mendet quapromitur obſequentiſſima vouentis deuotio; </s> <s xml:id="echoid-s50" xml:space="preserve">hęc <lb/>me vobis valdè ſpondet deuinctum, hęc conſulit, & </s> <s xml:id="echoid-s51" xml:space="preserve">iubet, <lb/>vt tandem, forſan cum fę nore, reddam, quę iam Geometri-<lb/>ca ab hoc Lyceo iucundiſſimè ebibi rudimenta. </s> <s xml:id="echoid-s52" xml:space="preserve">Primitiarum <lb/>titulis gloriantur bi labores, namq́; </s> <s xml:id="echoid-s53" xml:space="preserve">centrum grauitatis by-<lb/>perbolæ me primò fu ße perſcrutatum profiteor. </s> <s xml:id="echoid-s54" xml:space="preserve">Vos binc eli-<lb/>go Numina, quibus ęquiſſimè dicem, Vos operis optimè ſtæ-<lb/>tuo Patronos. </s> <s xml:id="echoid-s55" xml:space="preserve">Ioannes della Faille, qui primus centrum gra-<lb/>uitatis partium circuli, & </s> <s xml:id="echoid-s56" xml:space="preserve">Ell pſis est nactus, voluminis <lb/>verticem Philippi Quarti Hiſpaniarum Potentiſsimi Regis, <lb/>nomine, & </s> <s xml:id="echoid-s57" xml:space="preserve">maiestate coronauit. </s> <s xml:id="echoid-s58" xml:space="preserve">Quò gaudet communi ti-<lb/>tulo, hæc opella, eò præclariſsimis Viris ſe nouit fore ſacr an-<lb/>dam. </s> <s xml:id="echoid-s59" xml:space="preserve">Excipiatis hęc vota, ideo à Vobis omnibus numeris <lb/>maximis, cum exigua ſint, & </s> <s xml:id="echoid-s60" xml:space="preserve">penè minima, tuenda. </s> <s xml:id="echoid-s61" xml:space="preserve">Cæte-<lb/>rum ſi Palladis ortum ditauit irriguè pluens aurum, Vos pari-<lb/>tèr Sapientiſsimæ Vrbis Præſides, quiq́; </s> <s xml:id="echoid-s62" xml:space="preserve">ideò Mineruæ mu-<lb/>nus impletis, Aſtra ditent, ac proſperè tribuant ad gloriam <lb/>ſeneſcere. </s> <s xml:id="echoid-s63" xml:space="preserve">Valete.</s> <s xml:id="echoid-s64" xml:space="preserve"/> </p> <pb file="0010" n="10"/> <figure> <image file="0010-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0010-01"/> </figure> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head15" xml:space="preserve">LECTORI <lb/>BENEVOLO.</head> <p> <s xml:id="echoid-s65" xml:space="preserve">ELapſo Menſe Iulij exierunt è Typo-<lb/>graphi manibus quatuor noſtrilibri <lb/>circa Infinitas Parabolas verſantes. <lb/></s> <s xml:id="echoid-s66" xml:space="preserve">Subiectum equidem vetus, quum de <lb/>ipſo Caualerius antè annum 1640, <lb/>in problemate vltimo centuriæ ſuo-<lb/>rum problematum; </s> <s xml:id="echoid-s67" xml:space="preserve">& </s> <s xml:id="echoid-s68" xml:space="preserve">anno 1647. </s> <s xml:id="echoid-s69" xml:space="preserve">in exercitatio-<lb/>nibus geometricis; </s> <s xml:id="echoid-s70" xml:space="preserve">pertractauerit. </s> <s xml:id="echoid-s71" xml:space="preserve">Sed circa illud, <lb/>non modica vel totaliter ab ipſo intacta, vel pro-<lb/>prijs medijs oſtenſa, & </s> <s xml:id="echoid-s72" xml:space="preserve">roborata, manifeſtauimus. </s> <s xml:id="echoid-s73" xml:space="preserve"><lb/>Verum dum tertius illorum ſub prælo eſſet, ſuccurrit <lb/>modus centra grauitatis hyperbolæ, eiuſque partium <lb/>indagandi, ſuppoſita tamen ipſarum quadratura. </s> <s xml:id="echoid-s74" xml:space="preserve"><lb/>Aſt tunc noſtra intererat opus de infinitis parabolis <lb/>quam primum abſoluere; </s> <s xml:id="echoid-s75" xml:space="preserve">quapropter & </s> <s xml:id="echoid-s76" xml:space="preserve">in epiſtola <lb/>ad lectorem, & </s> <s xml:id="echoid-s77" xml:space="preserve">in calce quarti libri polliciti ſumus, <lb/>& </s> <s xml:id="echoid-s78" xml:space="preserve">argumentum illud, & </s> <s xml:id="echoid-s79" xml:space="preserve">tractatum de infinitis ſpira-<lb/>libus, ſequenti anno, explicare. </s> <s xml:id="echoid-s80" xml:space="preserve">Incępimus conſcri-<lb/>bere propoſitiones ad centrum grauitatis hyperbolæ <lb/>attinentes; </s> <s xml:id="echoid-s81" xml:space="preserve">quando tot nouæ cognitiones geometri- <pb file="0011" n="11"/> cæoccurrerunt, vt nos coegĕrint (neſcimus quo fa-<lb/>to) ſententiam mutare, impullerintque Miſcellaneum <lb/>præſens citiſſimè edere, opuſculum de infinitis ſpi-<lb/>ralibus ad aliud tempus reſeruantes. </s> <s xml:id="echoid-s82" xml:space="preserve">Etenim neſci-<lb/>mus an hoc primum futurum ſitillorum, quæforſan <lb/>elaboraturi ſumus. </s> <s xml:id="echoid-s83" xml:space="preserve">Modò namque phantaſiam occu-<lb/>pat argumentum quodam leuiter ab eximio Torri-<lb/>cellio tactum; </s> <s xml:id="echoid-s84" xml:space="preserve">circa quod, doctrinas tùm in Miſcel-<lb/>laneo præſenti, tùm in opere de infinitis parabolis <lb/>expoſitas, inſequentes, arbitramur nobis licitum fo-<lb/>re futurum explicare quamplurima noua, tam circa <lb/>menſuram, quam circa centra grauitatis infinitorum <lb/>ſolidorum, infinitiſque modis variatorum. </s> <s xml:id="echoid-s85" xml:space="preserve">Accipe <lb/>ergo, benignè Lector, in præſentiarum Miſcella-<lb/>neum hocce, in quo quas principaliter enucleauimus <lb/>doctrinas, habes in eius fronte. </s> <s xml:id="echoid-s86" xml:space="preserve">Porrò cupimus ad-<lb/>moneri, nos in ipſo aliqua indiuiſibilium methodo <lb/>dumtaxat confirmaſſe, Namque illaomittendo, pu-<lb/>tabamus, non modicè ingenium tuum labefactare. <lb/></s> <s xml:id="echoid-s87" xml:space="preserve">Haud enim indiuiſibilium methodo roboratis aſſen-<lb/>tiri, leuiterque circa regalem illum arguendi modum <lb/>hæſitare, aliud proculdubio non indicat, quam eius <lb/>vim, & </s> <s xml:id="echoid-s88" xml:space="preserve">energiam intimè, ac medulitùs minimè per-<lb/>cipi. </s> <s xml:id="echoid-s89" xml:space="preserve">Perlege ergo ſequentia ſi tibi placet, & </s> <s xml:id="echoid-s90" xml:space="preserve">Vale.</s> <s xml:id="echoid-s91" xml:space="preserve"/> </p> <pb file="0012" n="12"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head16" xml:lang="it" xml:space="preserve">Noi Reformatori dello Studio di Padoa.</head> <p xml:lang="it"> <s xml:id="echoid-s92" xml:space="preserve">HAuendo oſſeruato per fede del Padre Inquiſitore non <lb/>eſſerui, nel Libro di Materie Matematiche del Pad. </s> <s xml:id="echoid-s93" xml:space="preserve">F. <lb/></s> <s xml:id="echoid-s94" xml:space="preserve">Steffano Angeli dell´ Ordine de Geſuati, coſa contraria <lb/>alla Santa Fede, eparimente per atteſtato del Segreta-<lb/>rio noſtro niente contro Prencipi, è buoni coſtumi, per-<lb/>mettemo, che poſſi eſſere ſtampato, douendo oſſeruarſi <lb/>gl´Ordini, & </s> <s xml:id="echoid-s95" xml:space="preserve">eſſerne preſentate due Copie, vna per la Li-<lb/>braria di Padoa, e l´altra di queſta Città &</s> <s xml:id="echoid-s96" xml:space="preserve">c.</s> <s xml:id="echoid-s97" xml:space="preserve"/> </p> <p xml:lang="it"> <s xml:id="echoid-s98" xml:space="preserve">Dat. </s> <s xml:id="echoid-s99" xml:space="preserve">dal Magiſtr, noſtro li 8. </s> <s xml:id="echoid-s100" xml:space="preserve">Ottobre 1659.</s> <s xml:id="echoid-s101" xml:space="preserve"/> </p> <p xml:lang="it"> <s xml:id="echoid-s102" xml:space="preserve">{Nicolò Sagredo Cau. </s> <s xml:id="echoid-s103" xml:space="preserve">Proc, Ref.</s> <s xml:id="echoid-s104" xml:space="preserve"/> </p> <p xml:lang="it"> <s xml:id="echoid-s105" xml:space="preserve">Alemante Angelo Donini Segr.</s> <s xml:id="echoid-s106" xml:space="preserve"/> </p> <pb o="1" file="0013" n="13"/> <figure> <image file="0013-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0013-01"/> </figure> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head17" xml:space="preserve">MISCELLANEVM <lb/>HYPERBOLICVM, <lb/>PARABOLICVMQVE.</head> <p> <s xml:id="echoid-s107" xml:space="preserve">FÆCVNDITAS trium propoſi-<lb/>tionum initio tertij libri eorum, <lb/>quos de infinitis conſcripſimus pa-<lb/>rabolis, explicatarum, luculenter ex <lb/>pronunciatis ijſdem in libris fuit <lb/>omnibus patefacta. </s> <s xml:id="echoid-s108" xml:space="preserve">Hæc autem <lb/>eluceſcet magis, magiſque perluſtrantibus in præ-<lb/>ſentilibro à nobis aperienda. </s> <s xml:id="echoid-s109" xml:space="preserve">Centra grauitatis cir-<lb/>culi, & </s> <s xml:id="echoid-s110" xml:space="preserve">Ellipſis, aliquarumque ipſorum partium ad <lb/>noſtra tempora vſque incognita fuere. </s> <s xml:id="echoid-s111" xml:space="preserve">Noftro dum-<lb/>taxat ſeculo Ioannes della Failla, Guldinus, alijque <lb/>hæc detexere. </s> <s xml:id="echoid-s112" xml:space="preserve">Hæc & </s> <s xml:id="echoid-s113" xml:space="preserve">nos manifeſtauimus in 3. </s> <s xml:id="echoid-s114" xml:space="preserve">& </s> <s xml:id="echoid-s115" xml:space="preserve"><lb/>4. </s> <s xml:id="echoid-s116" xml:space="preserve">præcitatis libris, at methodo abomnibus diuer-<lb/>ſa. </s> <s xml:id="echoid-s117" xml:space="preserve">Aſt hæc centra inquirerentur fruſtra niſi circuli <lb/>quadratura ſupponeretur. </s> <s xml:id="echoid-s118" xml:space="preserve">Semidiameter etenim ad <lb/>interceptam inter centrum circuli, & </s> <s xml:id="echoid-s119" xml:space="preserve">centrum gra-<lb/>uitatis ſectoris eiuſdem eam dicitur habere ratio-<lb/>nem, quæ inter partem circumferentiæ, rectamque <pb o="2" file="0014" n="14"/> lineam cadit. </s> <s xml:id="echoid-s120" xml:space="preserve">Ratio verò inter rectum, & </s> <s xml:id="echoid-s121" xml:space="preserve">curuum <lb/>exprimenda, ſemota circuli quadratura, habetur nè <lb/>forſitan? </s> <s xml:id="echoid-s122" xml:space="preserve">Nequaquam. </s> <s xml:id="echoid-s123" xml:space="preserve">Igitur prædicta centra mi-<lb/>nimè reperirentur, niſi circuli quadratura ſuppone-<lb/>retur. </s> <s xml:id="echoid-s124" xml:space="preserve">Tres in geometria extant inſignes figuræ, <lb/>quarum deſideratur quadratura, Circulus, Ellipſis, <lb/>ac Hyperbola. </s> <s xml:id="echoid-s125" xml:space="preserve">Circuli & </s> <s xml:id="echoid-s126" xml:space="preserve">Ellipſis, ac eorum partium <lb/>(ſuppoſita talium figurarum quadratura) centra gra-<lb/>uitatis reperta fuere; </s> <s xml:id="echoid-s127" xml:space="preserve">curnon etiam ipſius hyperbo-<lb/>læ? </s> <s xml:id="echoid-s128" xml:space="preserve">Centrum grauitatis hyperbolæ ſub ſilentio re-<lb/>linquere quotquot de centro grauitatis figurarum <lb/>ſcripſere. </s> <s xml:id="echoid-s129" xml:space="preserve">Saltem neſcimus aliquem de ipſo verba <lb/>feciſſe. </s> <s xml:id="echoid-s130" xml:space="preserve">Imò Guldinus lib. </s> <s xml:id="echoid-s131" xml:space="preserve">pri. </s> <s xml:id="echoid-s132" xml:space="preserve">centrobarycæin cal-<lb/>ce pag. </s> <s xml:id="echoid-s133" xml:space="preserve">9. </s> <s xml:id="echoid-s134" xml:space="preserve">liberè pronunciat. </s> <s xml:id="echoid-s135" xml:space="preserve">_Deeſt hoc loco hyperbolæ,_ <lb/>_eiuſque partium centri grauitatis inueſtigatio._ </s> <s xml:id="echoid-s136" xml:space="preserve">Curabimus <lb/>ergo nos, hoc centrum, ſeù potius hæc centra, ma-<lb/>nifeſtare, at non niſihyperbolæ ſuppoſita quadratu-<lb/>ra; </s> <s xml:id="echoid-s137" xml:space="preserve">in primiſque oſtendemus in qua linea diametro <lb/>parallela ſit centrum grauitatis ſemihyperbolæ. </s> <s xml:id="echoid-s138" xml:space="preserve">Aſt <lb/>quoniam hoc inquirimus media ratione, quam ha-<lb/>bet cylindrus conoidi hyperbolico circumſcriptus, <lb/>ad ipſum conoides; </s> <s xml:id="echoid-s139" xml:space="preserve">licet hanc nos docuerit Archi-<lb/>medes lib. </s> <s xml:id="echoid-s140" xml:space="preserve">de conoid. </s> <s xml:id="echoid-s141" xml:space="preserve">& </s> <s xml:id="echoid-s142" xml:space="preserve">ſphæroid. </s> <s xml:id="echoid-s143" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s144" xml:space="preserve">27. </s> <s xml:id="echoid-s145" xml:space="preserve">atta-<lb/>men & </s> <s xml:id="echoid-s146" xml:space="preserve">nos prius hanc aſſignabimus pluribus mo-<lb/>dis, interſeque diuerſis, ac nunquam excogitatis; </s> <s xml:id="echoid-s147" xml:space="preserve">& </s> <s xml:id="echoid-s148" xml:space="preserve"><lb/>hoc eò libentius, quia data occaſione, aliqua nouæ <lb/>geometrica exponemus. </s> <s xml:id="echoid-s149" xml:space="preserve">Sit ergo.</s> <s xml:id="echoid-s150" xml:space="preserve"/> </p> <pb o="3" file="0015" n="15"/> </div> <div xml:id="echoid-div8" type="section" level="1" n="8"> <head xml:id="echoid-head18" xml:space="preserve">PROPOSITIO PRIMA.</head> <p style="it"> <s xml:id="echoid-s151" xml:space="preserve">Si circa diametrum hyperbolæ ſit etiam parabola ita diui-<lb/>dens baſim byperbolæ, vt quadratum ſemibaſis, ſit ad <lb/>quadratum ſemibaſis parabolæ, vt compoſita ex latere <lb/>tranſuerſo hyperbolæ, & </s> <s xml:id="echoid-s152" xml:space="preserve">ex diametro, ad tranſuerſam <lb/>latus. </s> <s xml:id="echoid-s153" xml:space="preserve">Tota parabola cadet intra hyperbolam.</s> <s xml:id="echoid-s154" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s155" xml:space="preserve">TRes ſequentes propoſit. </s> <s xml:id="echoid-s156" xml:space="preserve">probantur ferè ijſdem <lb/>terminis à Luca Valerio in append. </s> <s xml:id="echoid-s157" xml:space="preserve">ad lib. </s> <s xml:id="echoid-s158" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s159" xml:space="preserve">de cent grauit. </s> <s xml:id="echoid-s160" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s161" xml:space="preserve">pri & </s> <s xml:id="echoid-s162" xml:space="preserve">2. </s> <s xml:id="echoid-s163" xml:space="preserve">Efto ergo hyper-<lb/>bola A B C, cuius latus tranſuerſum G B, diame-<lb/>ter B D, circa quam ſit etiam parabola E B F, ſic <lb/>fecans A C, vt quadratum A D, ſit ad quadra-<lb/>tum D E, vt D G, ad G B. </s> <s xml:id="echoid-s164" xml:space="preserve">Dico totam para-<lb/>bolam E B F, cadereintra hyperbolam. </s> <s xml:id="echoid-s165" xml:space="preserve">Accipia-<lb/>tur arbitrariè punctum L, per quod ducatur ordi-<lb/>natim applicata H K L. </s> <s xml:id="echoid-s166" xml:space="preserve">Quoniam ex propoſit. </s> <s xml:id="echoid-s167" xml:space="preserve">21. </s> <s xml:id="echoid-s168" xml:space="preserve"><lb/>prim. </s> <s xml:id="echoid-s169" xml:space="preserve">conic. </s> <s xml:id="echoid-s170" xml:space="preserve">quadratum H L, eſt ad quadratum <lb/>A D, vt rectangulum G L B, ad rectangulum <lb/>G D B; </s> <s xml:id="echoid-s171" xml:space="preserve">& </s> <s xml:id="echoid-s172" xml:space="preserve">ex hypotheſi, eſt quadratum A D, ad <lb/>quadratum D E, vt D G, ad G B; </s> <s xml:id="echoid-s173" xml:space="preserve">nempeſum-<lb/>pta communi altitudine D B, vt rectangulum <lb/>G D B, ad rectangulum G B D. </s> <s xml:id="echoid-s174" xml:space="preserve">Ergo ex æquali, <lb/>erit quadratum H L, ad quadratum E D, vt re-<lb/>ctangulum G L B, ad rectangulum G B D. </s> <s xml:id="echoid-s175" xml:space="preserve">Rur-<lb/>ſum; </s> <s xml:id="echoid-s176" xml:space="preserve">quoniam in parabola eſt ex propoſit. </s> <s xml:id="echoid-s177" xml:space="preserve">20. </s> <s xml:id="echoid-s178" xml:space="preserve">lib. </s> <s xml:id="echoid-s179" xml:space="preserve"><lb/>eit. </s> <s xml:id="echoid-s180" xml:space="preserve">quadratum E D, ad quadratum K L, vt D B, <pb o="4" file="0016" n="16"/> <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> ad BL, nempe ſumpta communi altitudine G B, <lb/>vt rectangulum D B G, ad rectangulum L B G. <lb/></s> <s xml:id="echoid-s181" xml:space="preserve">Ergo ex æquali, erit quadratum H L, ad quadra-<lb/>tum K L, vt rectangulum G L B, ad rectangulum <lb/>G B L. </s> <s xml:id="echoid-s182" xml:space="preserve">At rectangulum G L B, maius eſt rectan-<lb/>gulo G B L. </s> <s xml:id="echoid-s183" xml:space="preserve">Ergo etiam quadratum H L, maius <lb/>erit quadrato K L. </s> <s xml:id="echoid-s184" xml:space="preserve">Sed punctum L, ſumptum eſt <lb/>arbitrariè. </s> <s xml:id="echoid-s185" xml:space="preserve">Ergo omnes lineæ ordinatim applica-<lb/>tæ in pa abola erunt minores ſingulis ordinatim ap-<lb/>plicatis in hyperbola. </s> <s xml:id="echoid-s186" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s187" xml:space="preserve"/> </p> <div xml:id="echoid-div8" type="float" level="2" n="1"> <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a"> <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0016-01"/> </figure> </div> <pb o="5" file="0017" n="17"/> </div> <div xml:id="echoid-div10" type="section" level="1" n="9"> <head xml:id="echoid-head19" xml:space="preserve">PROPOSITIO II.</head> <p style="it"> <s xml:id="echoid-s188" xml:space="preserve">Si quatuor magnitudinum ſit prima, ad ſecundam, vt tertia, <lb/>ad quartam; </s> <s xml:id="echoid-s189" xml:space="preserve">ſitque ablata pars primæ ad ablatam par-<lb/>tem ſecundæ, vt ablata pars tertiæ ad ablatam partem <lb/>quartæ et ſint partes primæ proportionales partibus ſecun-<lb/>dæ. </s> <s xml:id="echoid-s190" xml:space="preserve">Erit reliqua pars primæ ad reliquam partem ſecun-<lb/>dæ, vt reliqua pars tertiæ ad reliquam partem quartæ.</s> <s xml:id="echoid-s191" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s192" xml:space="preserve">SIT vt prima <lb/> <anchor type="figure" xlink:label="fig-0017-01a" xlink:href="fig-0017-01"/> A B, ad ſe-<lb/>cundam C D, ſic <lb/>tertia E F, ad <lb/>quartam G H; <lb/></s> <s xml:id="echoid-s193" xml:space="preserve">ſitque k B, ad <lb/>L D, vt MF, ad <lb/>N H: </s> <s xml:id="echoid-s194" xml:space="preserve">pariter ſit vt Ak, ad k B, ſic E M, ad M F. </s> <s xml:id="echoid-s195" xml:space="preserve"><lb/>Dico etiam A K, eſſe ad C L, vt E M, ad G N. </s> <s xml:id="echoid-s196" xml:space="preserve"><lb/>Quoniam ex hypotheſi componendo, eſt A B, ad <lb/>B k, vt E F, ad F M; </s> <s xml:id="echoid-s197" xml:space="preserve">& </s> <s xml:id="echoid-s198" xml:space="preserve">vt k B, ad L D, ſic M F, <lb/>ad N H; </s> <s xml:id="echoid-s199" xml:space="preserve">ergo ex æquali, vt A B, ad L D, ſic E F, <lb/>ad N H. </s> <s xml:id="echoid-s200" xml:space="preserve">At pariter eſt vt A B, ad totam C D, ſic <lb/>E F, ad totam G H. </s> <s xml:id="echoid-s201" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s202" xml:space="preserve">A B, erit ad reliquam <lb/>C L, vt E F, ad reliquam G N. </s> <s xml:id="echoid-s203" xml:space="preserve">Rurſum, quoniam <lb/>conuertendo, eſt B K, ad k A, vt F M, ad M E. </s> <s xml:id="echoid-s204" xml:space="preserve"><lb/>Ergo componendo, & </s> <s xml:id="echoid-s205" xml:space="preserve">conuertendo, erit Ak, ad A B, <lb/>vt EM, ad EF. </s> <s xml:id="echoid-s206" xml:space="preserve">Erat autem vt AB, ad CL, ſic EF, ad <pb o="6" file="0018" n="18"/> G N. </s> <s xml:id="echoid-s207" xml:space="preserve">Ergo ex æquali, erit A k, ad C L, vt E M, ad <lb/>G N. </s> <s xml:id="echoid-s208" xml:space="preserve">Quod &</s> <s xml:id="echoid-s209" xml:space="preserve">c.</s> <s xml:id="echoid-s210" xml:space="preserve"/> </p> <div xml:id="echoid-div10" type="float" level="2" n="1"> <figure xlink:label="fig-0017-01" xlink:href="fig-0017-01a"> <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0017-01"/> </figure> </div> </div> <div xml:id="echoid-div12" type="section" level="1" n="10"> <head xml:id="echoid-head20" xml:space="preserve">PROPOSITIO III.</head> <p style="it"> <s xml:id="echoid-s211" xml:space="preserve">Factis ijſdem quæ in prima propoſit. </s> <s xml:id="echoid-s212" xml:space="preserve">exceſſus quadratorum <lb/>ordinatim applicatarum in byperbola ſupra quadrata or-<lb/>dinatim applicatarum in parab la, erunt ad inuicem, vt <lb/>quadrata partium diametri interceptarum inter ipſas, & </s> <s xml:id="echoid-s213" xml:space="preserve"><lb/>verticem figurarum.</s> <s xml:id="echoid-s214" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s215" xml:space="preserve">IN eodem ſchemate, ſint ordinatim applicatæ ad <lb/>diametrum A E D C, H K L O. </s> <s xml:id="echoid-s216" xml:space="preserve">Dico exceſ-<lb/>ſum quadrati A D, ſupra quadratum E D, eſſe <lb/>ad exceſſum quadrati H L, ſupra quadratum k L, <lb/>vt quadratum D B, ad quadratum B L. </s> <s xml:id="echoid-s217" xml:space="preserve">Quo-<lb/>niam enim quadratum totum A D, eſt ad totum <lb/>quadratum H L, vt totum rectangulum G D B, <lb/>ad totum rectangulum G L B: </s> <s xml:id="echoid-s218" xml:space="preserve">& </s> <s xml:id="echoid-s219" xml:space="preserve">ablatum quadra-<lb/>tum E D, probatum eſt eſſe ad ablatum quadra-<lb/>tum K L, vt ablatum rectangulum D B G, ad <lb/>ablatum rectangulum L B G: </s> <s xml:id="echoid-s220" xml:space="preserve">eſtque ablatum qua-<lb/>dratum D E, ad reliquum rectangulum A E C, <lb/>vt ablatum quadratum L k, ad ablatum rectangu-<lb/>lum H k O (quiacum ex hypotheſi, ſit quadratum <lb/>A D, ad quadratum D E, vt D G, ad G B; <lb/></s> <s xml:id="echoid-s221" xml:space="preserve">nempe vt rectangulum G D B, ad rectangulum <lb/>G B D; </s> <s xml:id="echoid-s222" xml:space="preserve">erit diuidendo, & </s> <s xml:id="echoid-s223" xml:space="preserve">conuertendo, quadra-<lb/>tum D E, ad rectangulum A E C, vt rectangu- <pb o="7" file="0019" n="19"/> <anchor type="figure" xlink:label="fig-0019-01a" xlink:href="fig-0019-01"/> lum G B D, ad quadratum B D). </s> <s xml:id="echoid-s224" xml:space="preserve">Ergo ex pro-<lb/>poſit. </s> <s xml:id="echoid-s225" xml:space="preserve">anteced. </s> <s xml:id="echoid-s226" xml:space="preserve">erit & </s> <s xml:id="echoid-s227" xml:space="preserve">vt reliquum rectangulum <lb/>A E C, ad reliquum rectangulum H k O, vt reli-<lb/>quum quadratum D B, ad reliquum quadratum <lb/>B L. </s> <s xml:id="echoid-s228" xml:space="preserve">Quod &</s> <s xml:id="echoid-s229" xml:space="preserve">c.</s> <s xml:id="echoid-s230" xml:space="preserve"/> </p> <div xml:id="echoid-div12" type="float" level="2" n="1"> <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a"> <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0019-01"/> </figure> </div> </div> <div xml:id="echoid-div14" type="section" level="1" n="11"> <head xml:id="echoid-head21" xml:space="preserve">PROPOSITIO IV.</head> <p style="it"> <s xml:id="echoid-s231" xml:space="preserve">Si ex figuris antecedentium propoſitionum intelligantur ge-<lb/>nerari conoidea, in quibus inſcribentur coni ſuper ijſ-<lb/>dem baſibus, & </s> <s xml:id="echoid-s232" xml:space="preserve">circa eandem diametrum. </s> <s xml:id="echoid-s233" xml:space="preserve">Differen-<lb/>tia conoideorum tam ſecundum totum, quam ſecundum <pb o="8" file="0020" n="20"/> partes proportionales, erit æqualis differentiæ cono-<lb/>rum.</s> <s xml:id="echoid-s234" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s235" xml:space="preserve">SEd ex hyperbola A B C, & </s> <s xml:id="echoid-s236" xml:space="preserve">parabola E B F, <lb/>intelligantur genita conoidea, in quibus ſint <lb/>inſcripti pariter coni A B C, E B F. </s> <s xml:id="echoid-s237" xml:space="preserve">Dico diffe-<lb/>rentiam conoideorum, nempe exceſſum conoidis <lb/>hyperbolici ſupra conoides parabolicum, æqualem <lb/>fore differentiæ conorum. </s> <s xml:id="echoid-s238" xml:space="preserve">Sumatur in diametro <lb/>B D, arbitrariè punctum L, per quod agatur pla-<lb/>num H O, plano A C, parallelum, ſecans om-<lb/>nia dicta ſolida, vt in ſchemate. </s> <s xml:id="echoid-s239" xml:space="preserve">Quoniam enim vt <lb/>quadratum D B, ad quadratum B L, ſic eſt tam <lb/>quadratum totius A D, ad quadratum totius P L, <lb/>quam ablatum quadratum E D, ad ablatum qua-<lb/>dratum M L: </s> <s xml:id="echoid-s240" xml:space="preserve">& </s> <s xml:id="echoid-s241" xml:space="preserve">quadratum D E, eſt ad rectan-<lb/>gulum A E C, vt quadratum L M, ad rectangu-<lb/>lum P M R (quia proportiones horum quadra-<lb/>torum ad hæc rectangula componuntur ex ijſdem <lb/>proportionibus, vt facile quilibet modicè in geo-<lb/>metria expertus poteſt agnoſcere). </s> <s xml:id="echoid-s242" xml:space="preserve">Ergo ex propoſ. <lb/></s> <s xml:id="echoid-s243" xml:space="preserve">2. </s> <s xml:id="echoid-s244" xml:space="preserve">erit vt quadratum D B, ad quadratum B L, ſic <lb/>rectangulum A E C, ad rectangulum P M R. </s> <s xml:id="echoid-s245" xml:space="preserve">Sed <lb/>etiam ex propoſit. </s> <s xml:id="echoid-s246" xml:space="preserve">antec. </s> <s xml:id="echoid-s247" xml:space="preserve">eſt vt quadratum D B, ad <lb/>quadratum B L, ſic rectangulum A E C, ad rectan-<lb/>gulum H k O. </s> <s xml:id="echoid-s248" xml:space="preserve">Ergo vt rectangulum A E C, ad re-<lb/>ctangulum P M R, ſic idem rectangulum A E C, ad <lb/>rectangulum H k O. </s> <s xml:id="echoid-s249" xml:space="preserve">Ergo rectangulum P M R, <lb/>erit æquale rectangulo H k O. </s> <s xml:id="echoid-s250" xml:space="preserve">Quare etiam armilla <pb o="9" file="0021" n="21"/> <anchor type="figure" xlink:label="fig-0021-01a" xlink:href="fig-0021-01"/> circularis P M R, erit æqualis armillæ circulari <lb/>Hk O. </s> <s xml:id="echoid-s251" xml:space="preserve">Cum verò punctum L, ſumptum ſit arbi-<lb/>trariè, ſequitur omnes armillas differentiæ cono-<lb/>rum, æquales eſſe omnibus armillis differentiæ co-<lb/>noideorum. </s> <s xml:id="echoid-s252" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s253" xml:space="preserve">differentia conorum erit æqua-<lb/>lis differentiæ conoideorum.</s> <s xml:id="echoid-s254" xml:space="preserve"/> </p> <div xml:id="echoid-div14" type="float" level="2" n="1"> <figure xlink:label="fig-0021-01" xlink:href="fig-0021-01a"> <image file="0021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0021-01"/> </figure> </div> <p> <s xml:id="echoid-s255" xml:space="preserve">Sicuti autem probatum eſt totasillas differentias <lb/>æquales eſſe, ſic probari poteſt quaslibet ipſarum <lb/>partes proportionales item fore æquales. </s> <s xml:id="echoid-s256" xml:space="preserve">v. </s> <s xml:id="echoid-s257" xml:space="preserve">g. </s> <s xml:id="echoid-s258" xml:space="preserve">ſi in-<lb/>telligatur ductum planum H O, probari poteſt eo-<lb/>dem modo, partem differentiæ conoideorum con- <pb o="10" file="0022" n="22"/> tentam inter plana HO, AC, æqualem eſſe parti dif-<lb/>ferentię conorum inter eadem plana contentæ; </s> <s xml:id="echoid-s259" xml:space="preserve">quod <lb/>cum ſit de sè euidens, omittitur. </s> <s xml:id="echoid-s260" xml:space="preserve">Patet ergo diffe-<lb/>rentias conoideorum & </s> <s xml:id="echoid-s261" xml:space="preserve">conorum, æquales eſſe inter <lb/>ſe, tam ſecundum totum, quam ſecundum partes <lb/>proportionales. </s> <s xml:id="echoid-s262" xml:space="preserve">Quod &</s> <s xml:id="echoid-s263" xml:space="preserve">c.</s> <s xml:id="echoid-s264" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div16" type="section" level="1" n="12"> <head xml:id="echoid-head22" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s265" xml:space="preserve">Non turbetur autem lector videns præſentem <lb/>propoſitionem probari per indiuiſibilium metho-<lb/>dum, imo admiretur excellentiam, & </s> <s xml:id="echoid-s266" xml:space="preserve">vniuerſalita-<lb/>tem illius methodi veritatem prodientis etiam illis <lb/>modis, quibus nequit manifeſtari methodo antiquo-<lb/>rum. </s> <s xml:id="echoid-s267" xml:space="preserve">Nam in ſuperiori conſtructione neſcimus an <lb/>methodus antiquorum poſſit adhiberi, quia in diffe-<lb/>rentijs prædictis nequeunt inſcribi cylindri. </s> <s xml:id="echoid-s268" xml:space="preserve">Quid <lb/>ergo? </s> <s xml:id="echoid-s269" xml:space="preserve">Concluſio demonſtrata falſa erit, quia per in-<lb/>diuiſibilia fuit roborata? </s> <s xml:id="echoid-s270" xml:space="preserve">Nequaquam. </s> <s xml:id="echoid-s271" xml:space="preserve">Nam etiam <lb/>eadem concluſio probari poteſt methodo antiquo-<lb/>rum, ſed alia præparatione adhibita, vt patebit ſuo <lb/>loco.</s> <s xml:id="echoid-s272" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div17" type="section" level="1" n="13"> <head xml:id="echoid-head23" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s273" xml:space="preserve">Sed antequam nos expediamus à præſenti propo-<lb/>ſitione, opere pretium ducimus manifeſtare eas no-<lb/>titias, quas ex ipſa, & </s> <s xml:id="echoid-s274" xml:space="preserve">ex dictis in noſtro lib. </s> <s xml:id="echoid-s275" xml:space="preserve">4. </s> <s xml:id="echoid-s276" xml:space="preserve">de <lb/>infinitis parabolis poſſumus eruere. </s> <s xml:id="echoid-s277" xml:space="preserve">Cum enim ex- <pb o="11" file="0023" n="23"/> <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> ceſſus ſæpe dicti ſint æquales inter ſe tam ſecundum <lb/>totum, quam ſecundum partes proportionales, ſe-<lb/>quitur conſequenter iuxta doctrinam præcit. </s> <s xml:id="echoid-s278" xml:space="preserve">4. </s> <s xml:id="echoid-s279" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s280" xml:space="preserve">eſſe quantitates proportionaliter analogas tam ſe-<lb/>cundum magnitudinem, quam ſecundum grauita-<lb/>tem. </s> <s xml:id="echoid-s281" xml:space="preserve">Quare ex propoſit. </s> <s xml:id="echoid-s282" xml:space="preserve">13. </s> <s xml:id="echoid-s283" xml:space="preserve">eiuſdem libri, centra <lb/>grauitatis horum exceſſuum ſecabunt B D, eodem <lb/>pacto. </s> <s xml:id="echoid-s284" xml:space="preserve">Cum ergo centrum grauitatis differentiæ co-<lb/>norum, quodſit v. </s> <s xml:id="echoid-s285" xml:space="preserve">g. </s> <s xml:id="echoid-s286" xml:space="preserve">L, ſic ſecet B D, vt B L, fit <lb/>tripla L D (nam idem eſt centrum grauitatis ex-<lb/>ceſſus prædicti, & </s> <s xml:id="echoid-s287" xml:space="preserve">conorum A B C, E B F). </s> <s xml:id="echoid-s288" xml:space="preserve">Ergo <pb o="12" file="0024" n="24"/> etiam centrum grauitatis differẽtiæ conoideorum ſic <lb/>ſecabit B D, in L, vt B L, ſit tripla L D. </s> <s xml:id="echoid-s289" xml:space="preserve">Imo cum <lb/>traiecto quolibet plano H O, parallelo A C, pars <lb/>differentiæ conoideorum contenta inter plana H O, <lb/>A C, ſit proportionaliter analoga cum parte diffe-<lb/>rentiæ conorum contenta inter eadem plana; </s> <s xml:id="echoid-s290" xml:space="preserve">& </s> <s xml:id="echoid-s291" xml:space="preserve">cum <lb/>in illo lib. </s> <s xml:id="echoid-s292" xml:space="preserve">4. </s> <s xml:id="echoid-s293" xml:space="preserve">pluribus modis ſit aſſignatum centrum <lb/>grauitatis prædictæ partis differentiæ conorum, quia <lb/>centrum grauitatis illius ſic diuidit L D, ſicuti ip-<lb/>ſam diuidit centrum grauitatis fruſtorum conorum <lb/>E M N F, A P R C, vt conſideranti patebit: </s> <s xml:id="echoid-s294" xml:space="preserve">ſequi-<lb/>tur etiam pluribus modis haberi centrum grauitatis <lb/>differentiæ conoideorum contentæ inter plana H O, <lb/>A C. </s> <s xml:id="echoid-s295" xml:space="preserve">Notetur etiam nos in hoc opere citaturos eſ-<lb/>ſe antecedentia huius operis, & </s> <s xml:id="echoid-s296" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s297" xml:space="preserve">librorum <lb/>noſtrorum de infinitis parabolis. </s> <s xml:id="echoid-s298" xml:space="preserve">Dum ergo citabi-<lb/>mus propoſ. </s> <s xml:id="echoid-s299" xml:space="preserve">huius operis, dicemus, ex tali propoſit. <lb/></s> <s xml:id="echoid-s300" xml:space="preserve">vel ex ſchol. </s> <s xml:id="echoid-s301" xml:space="preserve">talis propoſit. </s> <s xml:id="echoid-s302" xml:space="preserve">Dum vero citabimus li-<lb/>bros de infinitis parabolis, dicemus ex prop. </s> <s xml:id="echoid-s303" xml:space="preserve">talilibri <lb/>talis. </s> <s xml:id="echoid-s304" xml:space="preserve">v.</s> <s xml:id="echoid-s305" xml:space="preserve">g. </s> <s xml:id="echoid-s306" xml:space="preserve">ex propoſ. </s> <s xml:id="echoid-s307" xml:space="preserve">4. </s> <s xml:id="echoid-s308" xml:space="preserve">lib. </s> <s xml:id="echoid-s309" xml:space="preserve">3. </s> <s xml:id="echoid-s310" xml:space="preserve">intelligendo ſemper <lb/>noſtri operis.</s> <s xml:id="echoid-s311" xml:space="preserve"/> </p> <div xml:id="echoid-div17" type="float" level="2" n="1"> <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0023-01"/> </figure> </div> </div> <div xml:id="echoid-div19" type="section" level="1" n="14"> <head xml:id="echoid-head24" xml:space="preserve">PROPOSITIO V.</head> <p style="it"> <s xml:id="echoid-s312" xml:space="preserve">Cylindrus circumſcriptus conoidi byperbolico eſt ad ipſum, <lb/>vt compoſita ex axi, ſeù diametro, & </s> <s xml:id="echoid-s313" xml:space="preserve">ex latere tranſ-<lb/>uerſo conoidis, ad dimidium lateris tranſuerſi, vna cum <lb/>tertia parte axis, ſeù diametri.</s> <s xml:id="echoid-s314" xml:space="preserve"/> </p> <pb o="13" file="0025" n="25"/> <figure> <image file="0025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0025-01"/> </figure> <p> <s xml:id="echoid-s315" xml:space="preserve">INtelligantur omnia ſolida antecedentis propo-<lb/>ſit. </s> <s xml:id="echoid-s316" xml:space="preserve">& </s> <s xml:id="echoid-s317" xml:space="preserve">ipſis conoidibus ſint circumſcripti cylindri <lb/>QC, TF. </s> <s xml:id="echoid-s318" xml:space="preserve">Quoniam conoides hyperbolicum con-<lb/>ftatex differentia conoideorum, & </s> <s xml:id="echoid-s319" xml:space="preserve">ex conoide para-<lb/>bolico; </s> <s xml:id="echoid-s320" xml:space="preserve">& </s> <s xml:id="echoid-s321" xml:space="preserve">differentia conoideorum eſt æqualis dif-<lb/>ferentiæ conorum; </s> <s xml:id="echoid-s322" xml:space="preserve">ergo ratio cylindri Q C, ad co-<lb/>noides A B C, erit eadem cum ratione eiuſdem cy-<lb/>lindri ad differentiam conorum, & </s> <s xml:id="echoid-s323" xml:space="preserve">ad conoides pa-<lb/>rabolicum E B F. </s> <s xml:id="echoid-s324" xml:space="preserve">At ratio cylindri QC, ad dif-<lb/>ferentiam conorum eſt eadem cum ratione quadrati <lb/>A D, ad tertiam partem rectanguli A E C, vt con-<lb/>ſideranti patebit; </s> <s xml:id="echoid-s325" xml:space="preserve">quia cum ſit ad conum A B C, vt <pb o="14" file="0026" n="26"/> quadratum A D, ad tertiam partem ſui; </s> <s xml:id="echoid-s326" xml:space="preserve">& </s> <s xml:id="echoid-s327" xml:space="preserve">ad co-<lb/>num E B F, vtidem quadratum A D, ad tertiam <lb/>partem quadrati E D; </s> <s xml:id="echoid-s328" xml:space="preserve">ſequitur eſſe ad differentiam <lb/>conorum vt idem quadratum A D, ad tertiam par-<lb/>tem differentiæ quadratorum A D, D E, nempe <lb/>ad tertiam partem rectanguli A E C. </s> <s xml:id="echoid-s329" xml:space="preserve">Cum verò ex <lb/>hypotheſi, ſit quadratum A D, ad quadratum E D, <lb/>vt D G, ad G B; </s> <s xml:id="echoid-s330" xml:space="preserve">ergo per conuerſionem rationis, <lb/>erit quadratum A D, ad rectangulum A E C, vt <lb/>G D, ad D B. </s> <s xml:id="echoid-s331" xml:space="preserve">Et quadratum A D, erit ad ter-<lb/>tiam partem rectanguli A E C, vt G D, ad ter-<lb/>tiam partem D B. </s> <s xml:id="echoid-s332" xml:space="preserve">Quare etiam cylindrus Q C, erit <lb/>ad differentiam conorum, & </s> <s xml:id="echoid-s333" xml:space="preserve">conſequenter ad diffe-<lb/>rentiam conoideorum, vt G D, ad tertiam partem <lb/>D B. </s> <s xml:id="echoid-s334" xml:space="preserve">Pariter ratio cylindri Q C, ad conoides E B F, <lb/>eſt eadem cum ratione quadrati A D, ad dimidium <lb/>quadrati E D. </s> <s xml:id="echoid-s335" xml:space="preserve">Quia cum ſit ad cylindrum T F, vt <lb/>quadratum A D, ad quadratum E D; </s> <s xml:id="echoid-s336" xml:space="preserve">& </s> <s xml:id="echoid-s337" xml:space="preserve">cum co-<lb/>noides E B F, ſit dimidium cylindri T F, vt ſæpe <lb/>probatum eſt in noſtris lib. </s> <s xml:id="echoid-s338" xml:space="preserve">de inſinit. </s> <s xml:id="echoid-s339" xml:space="preserve">parab. </s> <s xml:id="echoid-s340" xml:space="preserve">Ergo <lb/>cylindrus Q C, erit ad conoides E B F, vt quadra-<lb/>tum A D, ad dimidium quadrati E D; </s> <s xml:id="echoid-s341" xml:space="preserve">nempe ex <lb/>hypotheſi, vt D G, ad dimidiam G B. </s> <s xml:id="echoid-s342" xml:space="preserve">Ergo colli-<lb/>gendo conſequentia, erit cylindrus Q C, ad conoi-<lb/>des, & </s> <s xml:id="echoid-s343" xml:space="preserve">ad differentiam conoideorum, nempe ad co-<lb/>noides hyperbolicum A B C, vt G D, ad dimi-<lb/>diam G B, cum tertia parte B D. </s> <s xml:id="echoid-s344" xml:space="preserve">Quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s345" xml:space="preserve"/> </p> <pb o="15" file="0027" n="27"/> <figure> <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0027-01"/> </figure> </div> <div xml:id="echoid-div20" type="section" level="1" n="15"> <head xml:id="echoid-head25" xml:space="preserve">PROPOSITIO VI.</head> <p style="it"> <s xml:id="echoid-s346" xml:space="preserve">Fn ſolidis ſæpe dictis, exceßus conoidis hyperbolici ſupra <lb/>conum ſibi inſcriptum est æqualis exceſſui conoidis pa-<lb/>rabolici illi inſcripti ſupra conum illi inſcriptum, tam <lb/>ſecundum totum, quam ſecundam partes proportio-<lb/>nales.</s> <s xml:id="echoid-s347" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s348" xml:space="preserve">QVantum ad totos exceſſus ſic patebit. </s> <s xml:id="echoid-s349" xml:space="preserve">Cum <lb/>enim ex propoſit. </s> <s xml:id="echoid-s350" xml:space="preserve">4. </s> <s xml:id="echoid-s351" xml:space="preserve">exceſſus conoideorum ſit <lb/>æqualis exceſſui conorum, ſi communis auferatur <lb/>illa pars, quæ generatur ex reuolutione trilinei mixti <pb o="16" file="0028" n="28"/> A O E, & </s> <s xml:id="echoid-s352" xml:space="preserve">communis addatur pars genita ex figura <lb/>contenta à recta, & </s> <s xml:id="echoid-s353" xml:space="preserve">curua O B, patebit propo-<lb/>ſitum.</s> <s xml:id="echoid-s354" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s355" xml:space="preserve">Quantum verò ad partes proportionales, non erit <lb/>diſſimilis demonſtratio ab antecedenti, addendo, & </s> <s xml:id="echoid-s356" xml:space="preserve"><lb/>auferendo partes communes ſecundum quod pla-<lb/>num ſecans parallelum plano A C, tranſit vel <lb/>per puncta O, I, vel ſuprà, vel infrà ipſa. </s> <s xml:id="echoid-s357" xml:space="preserve">Qua-<lb/>re &</s> <s xml:id="echoid-s358" xml:space="preserve">c.</s> <s xml:id="echoid-s359" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div21" type="section" level="1" n="16"> <head xml:id="echoid-head26" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s360" xml:space="preserve">Ergo exceſſus prædicti conoideorum ſupra ſuos <lb/>conos erunt quantitates proportionaliter analogæ, <lb/>tam in magnitudine, quam in grauitate. </s> <s xml:id="echoid-s361" xml:space="preserve">Cum er-<lb/>go exceſſus conoidis parabolici E B F, ſupra ſuum <lb/>conum ſit dimidium talis coni, quia conoides eſt ſeſ-<lb/>quialterum coni. </s> <s xml:id="echoid-s362" xml:space="preserve">Ergo etiam exceſſus conoidis hy-<lb/>perbolici A B C, ſupra ſuum conum erit dimidium <lb/>coni inſcripti in conoide E B F. </s> <s xml:id="echoid-s363" xml:space="preserve">Quare cylindrus <lb/>Q C, quieſt ad conum inſcriptum in conoide para-<lb/>bolico, vt quadratum A D, ad tertiam partem qua-<lb/>drati E D, erit ad exceſſum conoidis A B C, ſupra <lb/>conum A B C, vt idem quadratum A D, ad ſextam <lb/>partem quadrati D E. </s> <s xml:id="echoid-s364" xml:space="preserve">Quod notetur.</s> <s xml:id="echoid-s365" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s366" xml:space="preserve">Item, quoniam exceſſus prædicti ſunt magnitudi-<lb/>nes proportionaliter analogæ in grauitate. </s> <s xml:id="echoid-s367" xml:space="preserve">Ergo <lb/>idem punctumin B D, erit centrum grauitatis cu-<lb/>iuslibet talium exceſſuum. </s> <s xml:id="echoid-s368" xml:space="preserve">Cum ergo punctum me- <pb o="17" file="0029" n="29"/> <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> dium ipſius B D, ſit centrum grauitatis exceſſus co-<lb/>noidis parabolici E B F, ſupra conum E B F; </s> <s xml:id="echoid-s369" xml:space="preserve">ſe-<lb/>quitur etiam centrum grauitatis exceſſus conoidis <lb/>A B C, ſupra ſuum conum eſſe in medio ipſius <lb/>B D.</s> <s xml:id="echoid-s370" xml:space="preserve"/> </p> <div xml:id="echoid-div21" type="float" level="2" n="1"> <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a"> <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0029-01"/> </figure> </div> <p> <s xml:id="echoid-s371" xml:space="preserve">Quod vero centrum grauitatis exceſſus conoidis <lb/>parabolici E B F, ſupra ſuum conum ſit medium <lb/>punctum ipſius B D, patet. </s> <s xml:id="echoid-s372" xml:space="preserve">Quia P, centrum <lb/>grauitatis conoidis diuidit B D, vt B P, ſit ad <lb/>P D, vt 2, ad 1, ſeù vt 8. </s> <s xml:id="echoid-s373" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s374" xml:space="preserve">N, verò cen-<lb/>trum grauitatis coni diuidit B D, ſic, vt B N, ſit <lb/>ad N D, vt 3. </s> <s xml:id="echoid-s375" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s376" xml:space="preserve">ſeù vt 9. </s> <s xml:id="echoid-s377" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s378" xml:space="preserve">Ergo qualium <pb o="18" file="0030" n="30"/> B D, eſt 12, talium P N, erit 1. </s> <s xml:id="echoid-s379" xml:space="preserve">Cum verò ſi ſiat <lb/>vt exceſſus conoidis ſupra conum ad conum, nem-<lb/>pe vt 1, ad 2, ſic reciprocè N P, ad P M, ſit M, <lb/>centrum grauitatis exceſſus prædicti. </s> <s xml:id="echoid-s380" xml:space="preserve">Sequitur qua-<lb/>lium B D, erat 12, P N, 1, & </s> <s xml:id="echoid-s381" xml:space="preserve">B P, 8, talium P M, <lb/>eſſe 2, & </s> <s xml:id="echoid-s382" xml:space="preserve">B M, 6. </s> <s xml:id="echoid-s383" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s384" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div23" type="section" level="1" n="17"> <head xml:id="echoid-head27" xml:space="preserve">PROPOSITIO VII.</head> <p style="it"> <s xml:id="echoid-s385" xml:space="preserve">Cylindrus circumſcriptus conoidi hyperbolico eſt ad ipſum, <lb/>vt compoſita ex axi, ſeù diametro, & </s> <s xml:id="echoid-s386" xml:space="preserve">ex latere tran-<lb/>ſuerſo conoidis, ad dimidium lateris tranſuerſi, vna <lb/>cum tertia parte axis, ſeù diametri.</s> <s xml:id="echoid-s387" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s388" xml:space="preserve">PRopoſitio ergo quinta probatur alio modo. </s> <s xml:id="echoid-s389" xml:space="preserve">Sint <lb/>ſolida prædicta, &</s> <s xml:id="echoid-s390" xml:space="preserve">c. </s> <s xml:id="echoid-s391" xml:space="preserve">Dico cylindrum Q C, eſ-<lb/>ſe ad conoides hyperbolicum A B C, vt G D, ad <lb/>dimidiam G B, cum tertia parte D B. </s> <s xml:id="echoid-s392" xml:space="preserve">Cum enim <lb/>conoides A B C, diuidatur in conum A B C, & </s> <s xml:id="echoid-s393" xml:space="preserve">in <lb/>exceſſum ipſius ſupraipſum; </s> <s xml:id="echoid-s394" xml:space="preserve">ſequitur Q C, cylin-<lb/>drum eſſe ad conoides A B C, vt eſt etiam ad co-<lb/>num A B C, & </s> <s xml:id="echoid-s395" xml:space="preserve">ad exceſſum conoidis ſupra conum. <lb/></s> <s xml:id="echoid-s396" xml:space="preserve">Cylindrus Q C, eſt ad conum A B C, vt quadra-<lb/>tum A D, ad ſui tertiam partem: </s> <s xml:id="echoid-s397" xml:space="preserve">& </s> <s xml:id="echoid-s398" xml:space="preserve">ex ſchol. </s> <s xml:id="echoid-s399" xml:space="preserve">ant. </s> <s xml:id="echoid-s400" xml:space="preserve"><lb/>eſt ad exceſſum conoidis A B C, ſupra ſuum co-<lb/>num vt quadratum A D, ad ſextam partem quadra-<lb/>ti D E. </s> <s xml:id="echoid-s401" xml:space="preserve">Ergo colligendo ambo conſequentia, erit <lb/>QC, ad conum, & </s> <s xml:id="echoid-s402" xml:space="preserve">ad exceſſum, nempe ad conoides <lb/>A B C, vt quadratum A D, ad ſui tertiam partem, <pb o="19" file="0031" n="31"/> <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/> vna cum ſexta parte quadrati E D. </s> <s xml:id="echoid-s403" xml:space="preserve">Cum autem ex <lb/>hypotheſi, ſit vt quadratum A D, ad quadratum <lb/>D E, ſic D G, ad G B; </s> <s xml:id="echoid-s404" xml:space="preserve">erit & </s> <s xml:id="echoid-s405" xml:space="preserve">vt quadratum A D, <lb/>ad ſui tertiam partem, cum ſexta parte quadrati E D, <lb/>ſic G D, ad fui tertiam partem cum ſexta parte <lb/>G B. </s> <s xml:id="echoid-s406" xml:space="preserve">Ergo etiam cylindrus Q C, erit ad conoides <lb/>A B C, vt D G, ad ſui tertiam partem (nempe ad <lb/>tertiam partem ipſarum G B, B D) vna cum ſexta <lb/>parte G B. </s> <s xml:id="echoid-s407" xml:space="preserve">At tertia pars G B, vna cum ſexta par-<lb/>te eiuſdem facit dimidiam G B. </s> <s xml:id="echoid-s408" xml:space="preserve">Ergo Q C, erit <lb/>ad conoides hyperbolicum A B C, vt G D, ad <pb o="20" file="0032" n="32"/> dimidiam G B, cum tertia parte B D. </s> <s xml:id="echoid-s409" xml:space="preserve">Quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s410" xml:space="preserve"/> </p> <div xml:id="echoid-div23" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0031-01"/> </figure> </div> </div> <div xml:id="echoid-div25" type="section" level="1" n="18"> <head xml:id="echoid-head28" xml:space="preserve">PROPOSITIO VIII.</head> <p style="it"> <s xml:id="echoid-s411" xml:space="preserve">Si ſruſto coni cuìus oppoſita plana parallela, circumſcri-<lb/>batur cylindrus, & </s> <s xml:id="echoid-s412" xml:space="preserve">alter inſcribatur, cuius baſis mi-<lb/>nor baſis frusti, & </s> <s xml:id="echoid-s413" xml:space="preserve">latera trapezij genitoris fruſti pro-<lb/>ducantur vſque ad concurſum cum diametro. </s> <s xml:id="echoid-s414" xml:space="preserve">Tubus <lb/>cylindricus, qui est exceſſus cylindri circumſcripti ſupra <lb/>cylindrum inſcriptum, erit ad exceſſum frusti ſupra <lb/>cylindrum inſcriptum, vt compoſita ex diametro fru-<lb/>sti, & </s> <s xml:id="echoid-s415" xml:space="preserve">ex dupla intercepta inter minorem baſim, & </s> <s xml:id="echoid-s416" xml:space="preserve"><lb/>punctum concurſus laterum trapezij, ad compoſitam ex <lb/>tali intercepta, & </s> <s xml:id="echoid-s417" xml:space="preserve">ex tertia parte diametri fruſti.</s> <s xml:id="echoid-s418" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s419" xml:space="preserve">FRuſto coni A B C D, cuius diameter ET, & </s> <s xml:id="echoid-s420" xml:space="preserve"><lb/>oppoſita plana parallela ad inuicem ſint B C, <lb/>A D, circumſcribatur cylindrus G D, & </s> <s xml:id="echoid-s421" xml:space="preserve">inſcri-<lb/>batur H C; </s> <s xml:id="echoid-s422" xml:space="preserve">& </s> <s xml:id="echoid-s423" xml:space="preserve">latera A B, D C, producantur vſ-<lb/>que dum occurrant T E, productæ in I. </s> <s xml:id="echoid-s424" xml:space="preserve">Dico tu-<lb/>bum cylindricum G H C D, eſſe ad exceſſum fruſti <lb/>A B C D, ſupra cylindrum B L, nempe ad ſolidum <lb/>genitum ex triangulo A B H, reuoluto circa E T, <lb/>vt compoſita ex T E, & </s> <s xml:id="echoid-s425" xml:space="preserve">ex dupla I E, ad I E, vna <lb/>cum tertia parte T E. </s> <s xml:id="echoid-s426" xml:space="preserve">Cum cnim cylindrus G D, <lb/>ſit ad cylindrum B L, vt quadratum A T, ad qua-<lb/>dratum T H, ſeù B E; </s> <s xml:id="echoid-s427" xml:space="preserve">nempe vt quadratum T I, <lb/>ad quadratum I E. </s> <s xml:id="echoid-s428" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s429" xml:space="preserve">per conuerſione ratio- <pb o="21" file="0033" n="33"/> <anchor type="figure" xlink:label="fig-0033-01a" xlink:href="fig-0033-01"/> nis, erit G D, ad tubum G H C D, vt quadratum <lb/>I T, ad exceſſum ipſius ſupra quadratum I E; </s> <s xml:id="echoid-s430" xml:space="preserve">nem-<lb/>pe ad duplum rectangulum I E T, cum quadrato <lb/>T E; </s> <s xml:id="echoid-s431" xml:space="preserve">nempe ad rectangulum ſub compoſita ex dupla <lb/>I E, & </s> <s xml:id="echoid-s432" xml:space="preserve">E T, & </s> <s xml:id="echoid-s433" xml:space="preserve">ſub E T. </s> <s xml:id="echoid-s434" xml:space="preserve">Quare & </s> <s xml:id="echoid-s435" xml:space="preserve">conuertendo, <lb/>erit tubus G H K, ad G D, vt prædictum rectan-<lb/>gulum ad quadratum I T. </s> <s xml:id="echoid-s436" xml:space="preserve">Cylindrus G D, eſt ex <lb/>dictis in ſchol. </s> <s xml:id="echoid-s437" xml:space="preserve">2. </s> <s xml:id="echoid-s438" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s439" xml:space="preserve">15. </s> <s xml:id="echoid-s440" xml:space="preserve">lib. </s> <s xml:id="echoid-s441" xml:space="preserve">2. </s> <s xml:id="echoid-s442" xml:space="preserve">ad fruſtum <lb/>A B C D, vt tripla T I, ad T I, I E, & </s> <s xml:id="echoid-s443" xml:space="preserve">harum ter-<lb/>tiam minorem proportionalem; </s> <s xml:id="echoid-s444" xml:space="preserve">nempe ducendo has <lb/>in I T, vt triplum quadratum I T, ad quadratum <lb/>I T, rectangulum T1 E, & </s> <s xml:id="echoid-s445" xml:space="preserve">rectangulum ſub T I, &</s> <s xml:id="echoid-s446" xml:space="preserve"> <pb o="22" file="0034" n="34"/> ſub tertia proportionali (quod rectangulum eſt æ-<lb/>quale quadrato I E): </s> <s xml:id="echoid-s447" xml:space="preserve">nempe ſubtriplando terminos, <lb/>eſt G D, ad A B C D, vt quadratum TI, ad ter-<lb/>tiam partem quadratorum T1, I E, & </s> <s xml:id="echoid-s448" xml:space="preserve">rectanguli <lb/>T I E, quæ tertia pars eſt æqualis quadrato I E, re-<lb/>ctangulo I E T, & </s> <s xml:id="echoid-s449" xml:space="preserve">tertiæ parti quadrati T E. </s> <s xml:id="echoid-s450" xml:space="preserve">At <lb/>idem cylindrus G D, eſt ad cylindrum B L, vt <lb/>quadratum A T, ad quadratum HT, feù B E; </s> <s xml:id="echoid-s451" xml:space="preserve">hoc <lb/>eſt vt quadratum T I, ad quadratum I E. </s> <s xml:id="echoid-s452" xml:space="preserve">Ergo <lb/>idem cylindrus G D, erit ad exceſſum fruſti A B C D, <lb/>ſupra cylindrum B L, vt quadratum T I, ad re-<lb/>ctangulum I E T, vna cum tertia parte quadrati <lb/>T E; </s> <s xml:id="echoid-s453" xml:space="preserve">nempe vna cum rectangulo contento ſub <lb/>T E, & </s> <s xml:id="echoid-s454" xml:space="preserve">ſub tertia parte T E. </s> <s xml:id="echoid-s455" xml:space="preserve">Aſt erat ſupra <lb/>tubus G H K, ad cylindrum G D, vt rectangulum <lb/>ſub compoſita ex dupla I E, & </s> <s xml:id="echoid-s456" xml:space="preserve">ex E T, & </s> <s xml:id="echoid-s457" xml:space="preserve">ſub T E, <lb/>ad quadratum I T. </s> <s xml:id="echoid-s458" xml:space="preserve">Ergo ex æquali, erit tubus GHk, <lb/>ad exceſſum fruſti A B C D, ſupra cylindrum B L, <lb/>vt prædictum rectangulum, ad rectangulum I E T, <lb/>vna cum rectangulo ſub T E, & </s> <s xml:id="echoid-s459" xml:space="preserve">ſub tertia parte E T. <lb/></s> <s xml:id="echoid-s460" xml:space="preserve">Quæ duo rectangula cum ſint idem ac rectangulum <lb/>ſub compoſita ex I E, & </s> <s xml:id="echoid-s461" xml:space="preserve">ex tertia parte E T, & </s> <s xml:id="echoid-s462" xml:space="preserve">ſub <lb/>T E. </s> <s xml:id="echoid-s463" xml:space="preserve">Sequitur G H k, eſſe ad exceſſum prædictum, <lb/>vt rectangulum ſub compoſita ex dupla I E, & </s> <s xml:id="echoid-s464" xml:space="preserve">ex <lb/>E T, & </s> <s xml:id="echoid-s465" xml:space="preserve">ſub E T, ad rectangulum ſub eadem E T, <lb/>& </s> <s xml:id="echoid-s466" xml:space="preserve">ſub compoſita ex I E, & </s> <s xml:id="echoid-s467" xml:space="preserve">ex tertia parte E T; </s> <s xml:id="echoid-s468" xml:space="preserve">nem-<lb/>pepropter commune latus E T, vt compoſita ex du-<lb/>pla I E, & </s> <s xml:id="echoid-s469" xml:space="preserve">ex E T, ad I E, cum tertia parte E T. </s> <s xml:id="echoid-s470" xml:space="preserve"><lb/>Quod erat oſtendendum.</s> <s xml:id="echoid-s471" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="1"> <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a"> <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0033-01"/> </figure> </div> <pb o="23" file="0035" n="35"/> </div> <div xml:id="echoid-div27" type="section" level="1" n="19"> <head xml:id="echoid-head29" xml:space="preserve">PROPOSITIO IX.</head> <p style="it"> <s xml:id="echoid-s472" xml:space="preserve">Si recta A B, ſit ſecta bifariam in C, & </s> <s xml:id="echoid-s473" xml:space="preserve">in D, E, æque <lb/>remotè à C, & </s> <s xml:id="echoid-s474" xml:space="preserve">pariter in F, G, æque remotè à C; </s> <s xml:id="echoid-s475" xml:space="preserve">ſit-<lb/>que rectangulum A F B, æquale quadrato D C. </s> <s xml:id="echoid-s476" xml:space="preserve">Erit <lb/>etiam rectangulum A D B, æquale quadrato F C.</s> <s xml:id="echoid-s477" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s478" xml:space="preserve">CVm enim rectangulum A F B, diuidatur in re-<lb/>ctangulum ſub A F, in D B, & </s> <s xml:id="echoid-s479" xml:space="preserve">in rectangulum <lb/>A F D, nempe in rectangulum ſub F D, in G B. </s> <s xml:id="echoid-s480" xml:space="preserve">Er-<lb/>go rectangula A F, D B; </s> <s xml:id="echoid-s481" xml:space="preserve">F D, G B, erunt æqualia <lb/>quadrato D C. </s> <s xml:id="echoid-s482" xml:space="preserve">Quare addito communi rectangu-<lb/>lo F D G. </s> <s xml:id="echoid-s483" xml:space="preserve">Ergo rectangula A F, D B; </s> <s xml:id="echoid-s484" xml:space="preserve">F D, G B; <lb/></s> <s xml:id="echoid-s485" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> F D G, erunt æqualia quadrato D C, & </s> <s xml:id="echoid-s486" xml:space="preserve">rectangulo <lb/>F D G; </s> <s xml:id="echoid-s487" xml:space="preserve">nempe quadrato F C. </s> <s xml:id="echoid-s488" xml:space="preserve">At rectangula F D G, <lb/>& </s> <s xml:id="echoid-s489" xml:space="preserve">F D, G B, faciunt rectangulum F D B. </s> <s xml:id="echoid-s490" xml:space="preserve">Quod cum <lb/>rectangulo A F, D B, facit rectangulum A D B. <lb/></s> <s xml:id="echoid-s491" xml:space="preserve">Quare etiam rectangulum A D B, erit æquale qua-<lb/>drato F C. </s> <s xml:id="echoid-s492" xml:space="preserve">Quod &</s> <s xml:id="echoid-s493" xml:space="preserve">c.</s> <s xml:id="echoid-s494" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="1"> <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a"> <image file="0035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0035-01"/> </figure> </div> </div> <div xml:id="echoid-div29" type="section" level="1" n="20"> <head xml:id="echoid-head30" xml:space="preserve">PROPOSITIO X.</head> <p style="it"> <s xml:id="echoid-s495" xml:space="preserve">Si conoides byperbolicum includatur intra fruſtum conicum <lb/>habens oppoſitas baſes parallelas, & </s> <s xml:id="echoid-s496" xml:space="preserve">latera trapezij geni-<lb/>toris frusti ſint partes aſymptoton hyperbolæ genitricis <pb o="24" file="0036" n="36"/> conoidis; </s> <s xml:id="echoid-s497" xml:space="preserve">intraque fruſtum conicum, & </s> <s xml:id="echoid-s498" xml:space="preserve">ſupra minori ba-<lb/>ſi ipſius inſcribatur cylindrus. </s> <s xml:id="echoid-s499" xml:space="preserve">Erit exceſſus fruſti coni-<lb/>ci ſupra cylindrum ſibi inſcriptum æqualis conoidi hy-<lb/>perbolico, tam ſeeundumtotum, quam ſecundum partes <lb/>proportionales.</s> <s xml:id="echoid-s500" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s501" xml:space="preserve">COnoides hyperbolicum A B C, cuius diame-<lb/>ter D B, latus tranſuerſum E B, centrum F, <lb/>aſymptoti hyperbolæ genitricis F G, F H, intelli-<lb/>gatur incluſum intra fruſtum conicum G I K H, cu-<lb/>ius oppoſita plana parallela ſint I k, G H, & </s> <s xml:id="echoid-s502" xml:space="preserve">in ipſo <lb/>ſit inſcriptus cylindrus I M. </s> <s xml:id="echoid-s503" xml:space="preserve">Dico exceſſum fruſti <lb/>G I k H, ſupra cylindrum I M, æqualem eſſe conoi-<lb/>di A B C, tam ſecundum totum, quam ſecundum <lb/>partes proportionales. </s> <s xml:id="echoid-s504" xml:space="preserve">Sumatur enim in diametro <lb/>arbitrariè punctum O, per quod agatur planum <lb/>N O P, G H, parallelum, ſecans omnia ſolida, vt in <lb/>ſchemate. </s> <s xml:id="echoid-s505" xml:space="preserve">Quoniam enim quadratum N O, eſt æ-<lb/>quale tam rectangulo N Q P, cum quadrato Q O, <lb/>quam rectangulo N R P, cum quadrato R O. </s> <s xml:id="echoid-s506" xml:space="preserve">Ergo <lb/>rectangulum N Q P, cum quadrato Q O, erit æ-<lb/>quale rectangulo N R P, cum quadrato R O. </s> <s xml:id="echoid-s507" xml:space="preserve">At <lb/>ex 2. </s> <s xml:id="echoid-s508" xml:space="preserve">conic. </s> <s xml:id="echoid-s509" xml:space="preserve">propoſit, 10. </s> <s xml:id="echoid-s510" xml:space="preserve">rectangulum N Q P, eſt æ-<lb/>quale quadrato I B, ſeù quadrato R O. </s> <s xml:id="echoid-s511" xml:space="preserve">Ergo reli-<lb/>quum rectangulum N R P, erit æquale quadrato <lb/>Q O. </s> <s xml:id="echoid-s512" xml:space="preserve">Quare etiam armilla circularis N R P, erit æ-<lb/>qualis circulo Q T. </s> <s xml:id="echoid-s513" xml:space="preserve">Punctum autem O, ſumptum <lb/>eſt arbitrariè; </s> <s xml:id="echoid-s514" xml:space="preserve">ergo omnes Armillæ genitæ ex reuo-<lb/>lutione trianguli G I L, circa B D, erunt æquales <pb o="25" file="0037" n="37"/> <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> omnibus circulis conoidis A B C, A C, parallelis. <lb/></s> <s xml:id="echoid-s515" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s516" xml:space="preserve">ſolidum genitum ex triangulo, nempe ex-<lb/>ceſſus fruſti G I K H, ſupra cylindrum I M, erit <lb/>æqualis ipſi conoidi A B C. </s> <s xml:id="echoid-s517" xml:space="preserve">Quod verò oſtenſum <lb/>eſt de totis iſtis ſolidis, probaretur etiam de partibus <lb/>proportionalibus; </s> <s xml:id="echoid-s518" xml:space="preserve">quia eodem modo probaretur v. </s> <s xml:id="echoid-s519" xml:space="preserve"><lb/>g. </s> <s xml:id="echoid-s520" xml:space="preserve">partem exceſſus contentam inter plana N P, <lb/>G H, æqualem eſſe fruſto hyperbolico A Q T C. </s> <s xml:id="echoid-s521" xml:space="preserve"><lb/>Quare patet prædicta ſolida æqualia eſſetam ſecun- <pb o="26" file="0038" n="38"/> d@in totum, quam ſecundum partes proportiona-<lb/>les. </s> <s xml:id="echoid-s522" xml:space="preserve">Quod &</s> <s xml:id="echoid-s523" xml:space="preserve">c.</s> <s xml:id="echoid-s524" xml:space="preserve"/> </p> <div xml:id="echoid-div29" 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/PVNDER1Y/figures/0037-01"/> </figure> </div> </div> <div xml:id="echoid-div31" type="section" level="1" n="21"> <head xml:id="echoid-head31" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s525" xml:space="preserve">Licet hæc propoſitio oſtenſa ſit per indiuiſibilia, <lb/>poteſt tamen probari modo Archimedeo. </s> <s xml:id="echoid-s526" xml:space="preserve">Cum e-<lb/>nim probatum ſit armillam circularem N R P, æ-<lb/>qualem eſſe circulo Q T, etiam (ſi inſcribantur) <lb/>tubus cylindricus N L P, inſcriptus in exceſſu fruſti <lb/>coni ſupra cylindrum, erit æqualis cylindro Q V, <lb/>inſcripto in conoide. </s> <s xml:id="echoid-s527" xml:space="preserve">Si ergo diuidatur B D, in <lb/>quibuſcunque punctis, & </s> <s xml:id="echoid-s528" xml:space="preserve">per hæc agantur plana vt <lb/>ſupra, & </s> <s xml:id="echoid-s529" xml:space="preserve">fiant tubi, & </s> <s xml:id="echoid-s530" xml:space="preserve">cylindri modo antedicto, fa-<lb/>cile patebit omnes tubos cylindricos inſcriptos in <lb/>exceſſu fruſti coni ſupra cylindrum, æquales fore <lb/>omnibus cylindris in conoide inſcriptis. </s> <s xml:id="echoid-s531" xml:space="preserve">Quare ſi <lb/>hæc diuiſio fiat per continuam biſlectionem D B, <lb/>partiumque eiuſdem; </s> <s xml:id="echoid-s532" xml:space="preserve">quia tam in exceſſu fruſti ſu-<lb/>pra cylindrum, quam in conoide inſcribemus ſolida <lb/>ab ipſis deficientibus defectu minori quacunque <lb/>data magnitudine; </s> <s xml:id="echoid-s533" xml:space="preserve">tandem concludemus exceſſum <lb/>prædictum, & </s> <s xml:id="echoid-s534" xml:space="preserve">conoides eſſe magnitudines æqua-<lb/>les. </s> <s xml:id="echoid-s535" xml:space="preserve">Hæc autem viris Euclideis, Archimedeiſque <lb/>ſunt nimis obuia.</s> <s xml:id="echoid-s536" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div32" type="section" level="1" n="22"> <head xml:id="echoid-head32" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s537" xml:space="preserve">Poteſt ergo conſequenter ad ſuperius ſæpe dicta, <pb o="27" file="0039" n="39"/> <anchor type="figure" xlink:label="fig-0039-01a" xlink:href="fig-0039-01"/> deduciex his, exceſſum prædictum, & </s> <s xml:id="echoid-s538" xml:space="preserve">conoides hy-<lb/>perbolicum, eſſe quantitates proportionaliter ana-<lb/>logas tam in magnitudine, quam in grauitate, tam <lb/>ſecundum totum, quam ſecundum partes proportio-<lb/>nales. </s> <s xml:id="echoid-s539" xml:space="preserve">Vnde ſi aliquo pacto inuenietur centrum <lb/>grauitatis, vel totius exceſſus prædicti, vel partis e-<lb/>ius in B D; </s> <s xml:id="echoid-s540" xml:space="preserve">idem crit centrum grauitatis conoidis <lb/>hyperbolici A B C, vel ſegmenti eiuſdem, &</s> <s xml:id="echoid-s541" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s542" xml:space="preserve">Idem intelligaturè contra.</s> <s xml:id="echoid-s543" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0039-01"/> </figure> </div> <pb o="28" file="0040" n="40"/> </div> <div xml:id="echoid-div34" type="section" level="1" n="23"> <head xml:id="echoid-head33" xml:space="preserve">SCHOLIVM III.</head> <p> <s xml:id="echoid-s544" xml:space="preserve">Galileus in poſtremis dialogis pag. </s> <s xml:id="echoid-s545" xml:space="preserve">apud nos, 28, <lb/>oſtendit paradoxum quodam; </s> <s xml:id="echoid-s546" xml:space="preserve">nimirum, circuli cir-<lb/>cumferentiam æqualem eſſe puncto. </s> <s xml:id="echoid-s547" xml:space="preserve">Vt hoc oſten-<lb/>dat vtitur exceſſu cylindri ſupra hemiſphærium, & </s> <s xml:id="echoid-s548" xml:space="preserve"><lb/>cono, vt ibidem poteſt conſpici. </s> <s xml:id="echoid-s549" xml:space="preserve">Sed ſicuti vſus fuit <lb/>exceſlu cylindri ſupra hemiſphærium, ſic etiam po-<lb/>terat vti exceſſu cylindri ſupra hemiſphæroides; </s> <s xml:id="echoid-s550" xml:space="preserve">ea-<lb/>dem enim fuiſſet demonſtratio. </s> <s xml:id="echoid-s551" xml:space="preserve">Paradoxum Galilei <lb/>oſtendimus & </s> <s xml:id="echoid-s552" xml:space="preserve">nos in appendice noſtri libelli ſexa-<lb/>ginta problematum geometricorum, adhibendo ex-<lb/>ceſſum cylindri ſupra conoides parabolicum, & </s> <s xml:id="echoid-s553" xml:space="preserve">ip-<lb/>ſum conoides. </s> <s xml:id="echoid-s554" xml:space="preserve">Hoc idem paradoxum facile ex præ-<lb/>ſenti propoſit. </s> <s xml:id="echoid-s555" xml:space="preserve">patebit confirmari poſſe, adhibendo <lb/>exceſſum prædictum fruſticoni G I K H, ſupra cy-<lb/>lindrum I M, & </s> <s xml:id="echoid-s556" xml:space="preserve">conoides hyperbolicum A B C. <lb/></s> <s xml:id="echoid-s557" xml:space="preserve">Probatum eſt enim, vbicunque traiciatur planum <lb/>N P, plano G H, parallelum, ſemper armillam <lb/>N R P, æqualem eſſe circulo Q T; </s> <s xml:id="echoid-s558" xml:space="preserve">ſicuti quamli-<lb/>bet partem exceſſus æqualem eſſe proportionali par-<lb/>ti conoidis. </s> <s xml:id="echoid-s559" xml:space="preserve">Cum ergo exceſſus prædictus deſinat <lb/>in circumferentia circuli cuius diameter l k, ſicuti <lb/>conoides deſinit in puncto B; </s> <s xml:id="echoid-s560" xml:space="preserve">videtur ergo colligi <lb/>circumferentiam æqualem eſſe vertici B.</s> <s xml:id="echoid-s561" xml:space="preserve"/> </p> <pb o="29" file="0041" n="41"/> </div> <div xml:id="echoid-div35" type="section" level="1" n="24"> <head xml:id="echoid-head34" xml:space="preserve">PROPOSITIO XI.</head> <p style="it"> <s xml:id="echoid-s562" xml:space="preserve">Cylindrus circumſcriptus conoidi hyperbolico eſt ad ipſum, <lb/>vt compoſita ex axi, ſeù diametro, & </s> <s xml:id="echoid-s563" xml:space="preserve">exlatere tranſ-<lb/>uerſo conoidis, ad dimidium lateris tranſuerſi, vna cum <lb/>tertia parte axis, ſeù diametri.</s> <s xml:id="echoid-s564" xml:space="preserve"/> </p> <figure> <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0041-01"/> </figure> <p> <s xml:id="echoid-s565" xml:space="preserve">COnoidi hyperbolico A B C, cuius diameter <lb/>D B, latus tranſuerſum E B, ſit circumſcri- <pb o="30" file="0042" n="42"/> ptus cylindrus O C. </s> <s xml:id="echoid-s566" xml:space="preserve">Dico hunc eſſe ad illud vt E D, <lb/>ad dimidiam E B, cum tertia parte B D. </s> <s xml:id="echoid-s567" xml:space="preserve">Sit F, <lb/>centrum hyperbolæ genitricis, & </s> <s xml:id="echoid-s568" xml:space="preserve">F G, F H, ſint <lb/>eius aſymptoti, & </s> <s xml:id="echoid-s569" xml:space="preserve">per B, ſit ducta I B, parallela <lb/>G D; </s> <s xml:id="echoid-s570" xml:space="preserve">intelligamuſque ex reuolutione trapezij <lb/>G I B D, circa B D, genitum eſſe fruſtum conicum <lb/>G I K H, cui ſit circumſcriptus cylindrus N H, & </s> <s xml:id="echoid-s571" xml:space="preserve"><lb/>inſcriptus I M. </s> <s xml:id="echoid-s572" xml:space="preserve">Quoniam linea G H, diuiſa eſt ſe-<lb/>cundum conditiones propoſit. </s> <s xml:id="echoid-s573" xml:space="preserve">9. </s> <s xml:id="echoid-s574" xml:space="preserve">nam ex propoſit. <lb/></s> <s xml:id="echoid-s575" xml:space="preserve">10. </s> <s xml:id="echoid-s576" xml:space="preserve">2. </s> <s xml:id="echoid-s577" xml:space="preserve">conic. </s> <s xml:id="echoid-s578" xml:space="preserve">rectangulum G A H, eſt æquale qua-<lb/>drato I B, ſeù quadrato L D. </s> <s xml:id="echoid-s579" xml:space="preserve">Ergo rectangulum <lb/>G L H, erit æquale quadrato A D. </s> <s xml:id="echoid-s580" xml:space="preserve">Ergo etiam ar-<lb/>milla circularis G L H, quæ eſt baſis tubi cylindrici <lb/>N L P, erit æqualis circulo A C, baſi cylindri O C. </s> <s xml:id="echoid-s581" xml:space="preserve"><lb/>Cum ergo ex propoſit. </s> <s xml:id="echoid-s582" xml:space="preserve">anteced. </s> <s xml:id="echoid-s583" xml:space="preserve">exceſſus fruſti coni <lb/>G I k H, ſupra cylindrum I M, ſit æqualis conoidi <lb/>hyperbolico A B C. </s> <s xml:id="echoid-s584" xml:space="preserve">Ergo tubus cylindricus N L P, <lb/>ad illum exceſſum, & </s> <s xml:id="echoid-s585" xml:space="preserve">cylindrus O C, ad conoides <lb/>erunt in eadem ratione. </s> <s xml:id="echoid-s586" xml:space="preserve">At ex propoſit. </s> <s xml:id="echoid-s587" xml:space="preserve">8. </s> <s xml:id="echoid-s588" xml:space="preserve">tubus eſt <lb/>ad exceſſum vt E D, ad F B, cum tertia parte D B. </s> <s xml:id="echoid-s589" xml:space="preserve"><lb/>Quare patet propoſitum.</s> <s xml:id="echoid-s590" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s591" xml:space="preserve">Oſten ſa ergo proportione cylindri circumſcripti <lb/>conoidi hyperbolico ad ipſum, facile docebimus in <lb/>qua linea diametro parallela ſit centrum grauitatis <lb/>ſemihyperbolæ. </s> <s xml:id="echoid-s592" xml:space="preserve">Sit ergo.</s> <s xml:id="echoid-s593" xml:space="preserve"/> </p> <pb o="31" file="0043" n="43"/> </div> <div xml:id="echoid-div36" type="section" level="1" n="25"> <head xml:id="echoid-head35" xml:space="preserve">PROPOSITIO XII.</head> <p style="it"> <s xml:id="echoid-s594" xml:space="preserve">Si fiat vt ſemihyperbola ad dimidium parallelogrammi ſibi <lb/>circumſcripti, ſic compoſita ex ſemilatere tranſuerſo hy-<lb/>perbolæ, & </s> <s xml:id="echoid-s595" xml:space="preserve">ex tertia parte axis eiuſdem, ad aliam: </s> <s xml:id="echoid-s596" xml:space="preserve">dein-<lb/>de fiat vt compoſita ex latere tranſuerſo & </s> <s xml:id="echoid-s597" xml:space="preserve">ex axi, ad <lb/>inuentam, ſic baſis ſemihyperbolæ ad ſui partem abſcin-<lb/>dendam incipiendo ab axi. </s> <s xml:id="echoid-s598" xml:space="preserve">Centrum grauitatis ſemihy-<lb/>perbolæ erit in line a per punctum ducta axi parallela.</s> <s xml:id="echoid-s599" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s600" xml:space="preserve">ESto hyperbola A B C, cuius axis B E; </s> <s xml:id="echoid-s601" xml:space="preserve">centrum <lb/>G; </s> <s xml:id="echoid-s602" xml:space="preserve">latus tranſuerſum F B; </s> <s xml:id="echoid-s603" xml:space="preserve">parallelogrammum <lb/>ei circumſcriptum ſit D C; </s> <s xml:id="echoid-s604" xml:space="preserve">ſitque B H, tertia pars <lb/>B E; </s> <s xml:id="echoid-s605" xml:space="preserve">& </s> <s xml:id="echoid-s606" xml:space="preserve">fiat vt A B E, ad dimidium D E, ſic G H, <lb/>ad E k; </s> <s xml:id="echoid-s607" xml:space="preserve">& </s> <s xml:id="echoid-s608" xml:space="preserve">pariter fiat vt F E, ad E k, ſic A E, ad <lb/>E L; </s> <s xml:id="echoid-s609" xml:space="preserve">ac per L, ducatur L M, parallela B E. </s> <s xml:id="echoid-s610" xml:space="preserve">Dico <lb/>in M L, eſſe centrum grauitatis ſemihyperbolæ <lb/>A B E. </s> <s xml:id="echoid-s611" xml:space="preserve">Intelligamus D E, cum ſemihyperbola. <lb/></s> <s xml:id="echoid-s612" xml:space="preserve">A B E, rotari circa B E. </s> <s xml:id="echoid-s613" xml:space="preserve">Quoniam ex propoſit. </s> <s xml:id="echoid-s614" xml:space="preserve">5. </s> <s xml:id="echoid-s615" xml:space="preserve">7. </s> <s xml:id="echoid-s616" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s617" xml:space="preserve">11. </s> <s xml:id="echoid-s618" xml:space="preserve">cylindrus D C, eſt ad conoides A B C, vt <lb/>F E, ad G H; </s> <s xml:id="echoid-s619" xml:space="preserve">& </s> <s xml:id="echoid-s620" xml:space="preserve">ratio F E, ad GH (de foris ſumpta <lb/>E k) componitur ex rationibus F E, ad E k, & </s> <s xml:id="echoid-s621" xml:space="preserve">hu-<lb/>ius ad G H. </s> <s xml:id="echoid-s622" xml:space="preserve">Ergo etiam ratio cylindri ad conoides <lb/>componetur ex ijſdem rationibus. </s> <s xml:id="echoid-s623" xml:space="preserve">Sed ex ſchol. </s> <s xml:id="echoid-s624" xml:space="preserve">1. </s> <s xml:id="echoid-s625" xml:space="preserve"><lb/>propoſit. </s> <s xml:id="echoid-s626" xml:space="preserve">3. </s> <s xml:id="echoid-s627" xml:space="preserve">lib. </s> <s xml:id="echoid-s628" xml:space="preserve">3. </s> <s xml:id="echoid-s629" xml:space="preserve">ratio cylindri ad conoides compo-<lb/>nitur etiam ex ratione dimidij D E, ad A B E, & </s> <s xml:id="echoid-s630" xml:space="preserve">ex <lb/>ratione A E, ad interceptam inter E B, & </s> <s xml:id="echoid-s631" xml:space="preserve">centrum <lb/>æquilibrij A B E, ſeù grauitatis duplicatæ A B E, <pb o="32" file="0044" n="44"/> <anchor type="figure" xlink:label="fig-0044-01a" xlink:href="fig-0044-01"/> ad partes A E; </s> <s xml:id="echoid-s632" xml:space="preserve">& </s> <s xml:id="echoid-s633" xml:space="preserve">ſupra factum eſt conuertendo, vt <lb/>dimidium D E, ad A B E, ſic k E, ad G H. </s> <s xml:id="echoid-s634" xml:space="preserve">Er-<lb/>go rationes F E, ad E k, & </s> <s xml:id="echoid-s635" xml:space="preserve">E k, ad G H, æquales <lb/>erunt rationibus E k, ad G H, & </s> <s xml:id="echoid-s636" xml:space="preserve">A E, ad prædi-<lb/>ctam interceptam. </s> <s xml:id="echoid-s637" xml:space="preserve">Ergo ſi auferatur communis ra-<lb/>tio k E, ad G H; </s> <s xml:id="echoid-s638" xml:space="preserve">F E, ad E k, erit vt A E, ad il-<lb/>lam interceptam. </s> <s xml:id="echoid-s639" xml:space="preserve">Sed ex conſtructione, vt F E, ad <lb/>E k, ſic A E, ad E L. </s> <s xml:id="echoid-s640" xml:space="preserve">Ergo L, erit centrum æqui-<lb/>librij ſemihyperbolæ. </s> <s xml:id="echoid-s641" xml:space="preserve">Et conſequenter in L M, <lb/>erit centrum grauitatis ſemihyperbolæ. </s> <s xml:id="echoid-s642" xml:space="preserve">Q od &</s> <s xml:id="echoid-s643" xml:space="preserve">c.</s> <s xml:id="echoid-s644" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0044-01"/> </figure> </div> <pb o="33" file="0045" n="45"/> </div> <div xml:id="echoid-div38" type="section" level="1" n="26"> <head xml:id="echoid-head36" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s645" xml:space="preserve">Tria autem, quæ collecta ſunt in quamplurimis <lb/>propoſitionibus lib. </s> <s xml:id="echoid-s646" xml:space="preserve">3. </s> <s xml:id="echoid-s647" xml:space="preserve">colligentur etiam nunc. </s> <s xml:id="echoid-s648" xml:space="preserve">Nam <lb/>primò, tam ſuper D E, quam ſupra A B E, intelle-<lb/>ctis cylindricis rectis æquealtis reſectis diagonaliter <lb/>plano tranſeunte per E B, & </s> <s xml:id="echoid-s649" xml:space="preserve">per latus oppoſitum ip-<lb/>ſi D A, colligentur cubationes amborum truncorum <lb/>cylindrici ſuper ſemihyperbola exiſtentis, cumhac <lb/>tamen diuerſitate; </s> <s xml:id="echoid-s650" xml:space="preserve">quod cubatio trunci ſiniſtri dabi-<lb/>tur ſemota hyperbolæ quadratura; </s> <s xml:id="echoid-s651" xml:space="preserve">quia ſine tali qua-<lb/>dratura datur ratio D C, cylindri ad conoides <lb/>A B C; </s> <s xml:id="echoid-s652" xml:space="preserve">ſecùs dicendum de cubatione trunci dexte-<lb/>ri, quæ non habetur niſi ſuppoſita quadratura. </s> <s xml:id="echoid-s653" xml:space="preserve">Se-<lb/>cundum eſt (quadratura ſuppoſita) ratio cylindri ex <lb/>D E, circa D A, ad annulum ſtrictum ex ſemihyper-<lb/>bola A B E, circa D A. </s> <s xml:id="echoid-s654" xml:space="preserve">Tertium eſt ratio conoi-<lb/>dis, & </s> <s xml:id="echoid-s655" xml:space="preserve">prædicti ſolidi ad inuicem, pariter ſuppoſita <lb/>quadratura.</s> <s xml:id="echoid-s656" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s657" xml:space="preserve">Sed antequam vlterius progrediamur, ſicuti plu-<lb/>ribus modis patefacta eſt ratio cylindri circumſcri-<lb/>pti ad conoides, ſic non erit inutile aſſignare centrum <lb/>grauitatis conoidis. </s> <s xml:id="echoid-s658" xml:space="preserve">Sit ergo.</s> <s xml:id="echoid-s659" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div39" type="section" level="1" n="27"> <head xml:id="echoid-head37" xml:space="preserve">PROPOSITIO XIII.</head> <p style="it"> <s xml:id="echoid-s660" xml:space="preserve">Centrum grauitatis conoidis hyperbolici ſic diuidit d uode <lb/>cimam partem diametri eiuſdem ordine quartam à ba- <pb o="34" file="0046" n="46"/> ſi, vt pars propinquior baſi, ſit ad reliquam, vt di-<lb/>midium lateris tranſuerſi conoidis, ad tertiam partem <lb/>ſuæ diametri.</s> <s xml:id="echoid-s661" xml:space="preserve"/> </p> <figure> <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0046-01"/> </figure> <p> <s xml:id="echoid-s662" xml:space="preserve">Esto conoides hyperbolicum quodcunque <lb/>A B C, cuius axis, ſeù diameter B D, ſic ſe-<lb/>cetur in L, vt B L, ſit dupla L D, & </s> <s xml:id="echoid-s663" xml:space="preserve">ſic in Q, vt <lb/>B Q, ſit tripla Q D. </s> <s xml:id="echoid-s664" xml:space="preserve">Ergo ſic L Q, erit duodecima <lb/>pars totius B D, & </s> <s xml:id="echoid-s665" xml:space="preserve">ordine quarta incipiendo à D. <lb/></s> <s xml:id="echoid-s666" xml:space="preserve">Sit G B, latus tranſuerſum conoidis, & </s> <s xml:id="echoid-s667" xml:space="preserve">L Q, ſic <pb o="35" file="0047" n="47"/> ſecetur in P, vt Q P, ſit ad P L, vt dimidia G B, ad <lb/>tertiam partem B D. </s> <s xml:id="echoid-s668" xml:space="preserve">Dico P, eſſe centrum graui-<lb/>tatis conoidis hyperbolici A B C. </s> <s xml:id="echoid-s669" xml:space="preserve">Inſcribantur co-<lb/>noides parabolicum E B F, & </s> <s xml:id="echoid-s670" xml:space="preserve">coni, vt factum eſt ſu-<lb/>pra. </s> <s xml:id="echoid-s671" xml:space="preserve">Quoniam ex ſchol. </s> <s xml:id="echoid-s672" xml:space="preserve">2. </s> <s xml:id="echoid-s673" xml:space="preserve">propoſit 4. </s> <s xml:id="echoid-s674" xml:space="preserve">Q, eſt cen-<lb/>trum grauitatis tam differentiæ conorum, quam dif-<lb/>ferentiæ conoideorum, & </s> <s xml:id="echoid-s675" xml:space="preserve">vt oſtenditur à multis, & </s> <s xml:id="echoid-s676" xml:space="preserve"><lb/>etiam à nobis lib. </s> <s xml:id="echoid-s677" xml:space="preserve">4. </s> <s xml:id="echoid-s678" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s679" xml:space="preserve">14, L, eſt centrum <lb/>grauitatis conoidis parabolici E B F; </s> <s xml:id="echoid-s680" xml:space="preserve">ergo ſi L Q, ſic <lb/>diuidatur in P, vt ſit reciprocè Q P, ad P L, vt co-<lb/>noides E B F, ad differentiam conoideorum, erit P, <lb/>centrũ grauitatistotius conoidis hyperbolici A B C. <lb/></s> <s xml:id="echoid-s681" xml:space="preserve">Sed vt conoides E B F, ad differentiam conoi-<lb/>deorum, ſic dimidia G B, ad tertiam partem D B, <lb/>vt ſtatim patebit. </s> <s xml:id="echoid-s682" xml:space="preserve">Ergo patet propoſitum.</s> <s xml:id="echoid-s683" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s684" xml:space="preserve">Aſſumptum vero patet ex dictis. </s> <s xml:id="echoid-s685" xml:space="preserve">Quia facile pa-<lb/>tebit conoides E B F, eſſe ad differentiam conoi-<lb/>deorum, ſeù ad differentiam conorum, vt dimidium <lb/>quadrati D E, ad tertiam partem rectanguli A E C. <lb/></s> <s xml:id="echoid-s686" xml:space="preserve">Sed cum ex data hypotheſi, ſit diuidendo, & </s> <s xml:id="echoid-s687" xml:space="preserve">con-<lb/>uertendo, quadratum D E, ad rectangulum A E C, <lb/>vt G B, ad B D. </s> <s xml:id="echoid-s688" xml:space="preserve">Erit & </s> <s xml:id="echoid-s689" xml:space="preserve">vt dimidium quadrati D E, <lb/>ad tertiam partem rectanguli A E C, ſic dimidia <lb/>G B, ad tertiam partem B D.</s> <s xml:id="echoid-s690" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div40" type="section" level="1" n="28"> <head xml:id="echoid-head38" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s691" xml:space="preserve">Siquis verò ſcire cupiat, in qua proportione ſece-<lb/>tur tota B D, à centro grauitatis P, hoc tali diſcur- <pb o="36" file="0048" n="48"/> <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/> fu obtinebit. </s> <s xml:id="echoid-s692" xml:space="preserve">Quoniam enim conuertendo L P, eſt <lb/>ad P Q, vt tertia pars B D, ad dimidiam G B; <lb/></s> <s xml:id="echoid-s693" xml:space="preserve">ergo cum B L, ſit octupla L Q, B P, erit ad P Q, <lb/>vt 9. </s> <s xml:id="echoid-s694" xml:space="preserve">tertiæ partes B D (nempe vt tripla B D) cum <lb/>8. </s> <s xml:id="echoid-s695" xml:space="preserve">dimidijs G B (nempe cum quadrupla G B) ad <lb/>dimidiam G B. </s> <s xml:id="echoid-s696" xml:space="preserve">Pariter cum D Q, ſit tripla Q L; </s> <s xml:id="echoid-s697" xml:space="preserve"><lb/>erit P Q, ad P D, vt dimidia G B, ad quadruplam <lb/>dimidiam G B (nempe ad duplam G B) vna cum <lb/>tribus tertijs partibus B D (nempe cum B D). </s> <s xml:id="echoid-s698" xml:space="preserve">Er-<lb/>go ex æquali, erit B P, ad P D, vt quadrupla G B, <pb o="37" file="0049" n="49"/> vna cum tripla B D, ad duplam G B, cum B D. <lb/></s> <s xml:id="echoid-s699" xml:space="preserve">Et ſubquadruplando terminos, erit B P, ad P D, <lb/>vt G B, cumſubſeſquitertia B D, ad dimidiam G B, <lb/>cum quarta parte B D.</s> <s xml:id="echoid-s700" xml:space="preserve"/> </p> <div xml:id="echoid-div40" 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/PVNDER1Y/figures/0048-01"/> </figure> </div> </div> <div xml:id="echoid-div42" type="section" level="1" n="29"> <head xml:id="echoid-head39" xml:space="preserve">PROPOSITIO XIV.</head> <p style="it"> <s xml:id="echoid-s701" xml:space="preserve">Centrum grauitatis conoidis hyperbolici ſic diuidit quartam <lb/>partem diametri eiuſdem ordine ſecundam à baſi, vt <lb/>pars propinquior baſi ſit adreliquam, vt ſexta pars la-<lb/>teris tranſuerſi, ad tertiam partem compoſitæ ex latere <lb/>tranſuerſo, & </s> <s xml:id="echoid-s702" xml:space="preserve">ex diametro.</s> <s xml:id="echoid-s703" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s704" xml:space="preserve">SEd in ſchem. </s> <s xml:id="echoid-s705" xml:space="preserve">anteced. </s> <s xml:id="echoid-s706" xml:space="preserve">ſupponat prudens geome-<lb/>tra diametrum B D, ſecari bifariam in L, & </s> <s xml:id="echoid-s707" xml:space="preserve"><lb/>L D, bifariam in Q; </s> <s xml:id="echoid-s708" xml:space="preserve">deinde L Q, ſic ſecari in P, <lb/>vt Q P, ſit ad P L, vt ſexta pars G B, ad tertiam <lb/>partem G D. </s> <s xml:id="echoid-s709" xml:space="preserve">Dico P, eſſe centrum grauitatis <lb/>conoidis A B C. </s> <s xml:id="echoid-s710" xml:space="preserve">Cum enim Q, ſit centrum graui-<lb/>tatis coni A B C, & </s> <s xml:id="echoid-s711" xml:space="preserve">ex ſchol. </s> <s xml:id="echoid-s712" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s713" xml:space="preserve">6. </s> <s xml:id="echoid-s714" xml:space="preserve">L, ſit <lb/>centrum exceſſus conoidis ſupra conum; </s> <s xml:id="echoid-s715" xml:space="preserve">& </s> <s xml:id="echoid-s716" xml:space="preserve">cum ſit <lb/>Q P, ad P L, vt ſexta pars G B, ad tertiam par-<lb/>tem G D, nempe exhypotheſi, vt ſexta pars qua-<lb/>drati D E, ad tertiam partem quadrati A D; </s> <s xml:id="echoid-s717" xml:space="preserve">nem-<lb/>pe ex ſchol. </s> <s xml:id="echoid-s718" xml:space="preserve">cit. </s> <s xml:id="echoid-s719" xml:space="preserve">vt exceſſus conoidis ſupra conum ad <lb/>ipſum conum. </s> <s xml:id="echoid-s720" xml:space="preserve">Ergo ex Archimede in æqueponde-<lb/>rantibus, erit P, centrum grauitatis totius co-<lb/>noidis.</s> <s xml:id="echoid-s721" xml:space="preserve"/> </p> <pb o="38" file="0050" n="50"/> </div> <div xml:id="echoid-div43" type="section" level="1" n="30"> <head xml:id="echoid-head40" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s722" xml:space="preserve">Modus præſens aſſignandi centrum grauitatis <lb/>conuenit cum antecedenti, vt attentè conſideranti <lb/>patebit. </s> <s xml:id="echoid-s723" xml:space="preserve">Eſſet etiam alius modus inueniendi tale <lb/>centrum grauitatis, inuento prius centro grauitatis <lb/>exceſſus fruſti conici ſupra cylindrum ſibi inſcri-<lb/>ptum. </s> <s xml:id="echoid-s724" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s725" xml:space="preserve">enim 3. </s> <s xml:id="echoid-s726" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s727" xml:space="preserve">10. </s> <s xml:id="echoid-s728" xml:space="preserve">patet talem <lb/>exceſſum, & </s> <s xml:id="echoid-s729" xml:space="preserve">conoides hyperbolicum, eſſe quantita-<lb/>tes proportionaliter analogas. </s> <s xml:id="echoid-s730" xml:space="preserve">Centrum verò gra-<lb/>uitatis prædicti exceſſus facile habebitur. </s> <s xml:id="echoid-s731" xml:space="preserve">Nam ex <lb/>dictis in lib. </s> <s xml:id="echoid-s732" xml:space="preserve">4. </s> <s xml:id="echoid-s733" xml:space="preserve">totius fruſti coni habetur pluribus <lb/>modis centrum grauitatis. </s> <s xml:id="echoid-s734" xml:space="preserve">Sed habetur etiam cen-<lb/>trum grauitatis cylindri in fruſto inſcripti; </s> <s xml:id="echoid-s735" xml:space="preserve">habetur-<lb/>que ratio talis cylindri ad exceſſum fruſti ſupra ip-<lb/>ſum. </s> <s xml:id="echoid-s736" xml:space="preserve">Quare centrum prædicti exceſſus non ignora-<lb/>bitur. </s> <s xml:id="echoid-s737" xml:space="preserve">Vice verſa tamen, modi reperiendi centrum <lb/>grauitatis conoidis aſſignati in dua bus propoſit. </s> <s xml:id="echoid-s738" xml:space="preserve">an-<lb/>teced quadrabunt etiam prædicto exceſſui.</s> <s xml:id="echoid-s739" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s740" xml:space="preserve">Sed ſicuti in ſuperioribus docuimus in qua linea <lb/>diametro parallela ſit centrum grauitatis ſemihy-<lb/>perbolæ, ſic videtur conueniens docere in qua linea <lb/>dian etro parallela ſit centrum grauitatis ſegmenti <lb/>ſemihy perbolæ contenti inter duas lineas baſi paral-<lb/>lelas. </s> <s xml:id="echoid-s741" xml:space="preserve">Sed cum inuentioni talis lineæ præmiſſa ſit ra-<lb/>tio, cylindri circumſcripti conoidi ad ipſum conoi-<lb/>des, ſic in præſentiarum anteponenda videtur atio <lb/>cylindri circumſcripti ſegmento conoidis hyper- <pb o="39" file="0051" n="51"/> bolici contento inter duo plana baſi parallela, ad <lb/>ipſum.</s> <s xml:id="echoid-s742" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div44" type="section" level="1" n="31"> <head xml:id="echoid-head41" xml:space="preserve">PROPOSITIO XV.</head> <p style="it"> <s xml:id="echoid-s743" xml:space="preserve">Si ſegmento conoidis hyperbolici reſecti plano baſi parallelo, <lb/>ſit circumſcriptus cylindrus. </s> <s xml:id="echoid-s744" xml:space="preserve">Erit bic ad ipſum ſegmen-<lb/>tum, vt rectangulum ſub compoſita ex latere tranſuer-<lb/>ſo, & </s> <s xml:id="echoid-s745" xml:space="preserve">ex diametro conoidis, & </s> <s xml:id="echoid-s746" xml:space="preserve">ſub diametro, ad re-<lb/>ctangulum ſub eadem compoſita, & </s> <s xml:id="echoid-s747" xml:space="preserve">ſub diametro co-<lb/>noidis ad verticem, vna cum rectangulo ſub compoſi-<lb/>ta ex dimidio lateris tranſuerſi, & </s> <s xml:id="echoid-s748" xml:space="preserve">ex tertia parte dia-<lb/>metri fruſti, & </s> <s xml:id="echoid-s749" xml:space="preserve">ſub eadem tertia parte.</s> <s xml:id="echoid-s750" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s751" xml:space="preserve">COnoides hyperbolicum cuius baſis A C, ver-<lb/>tex B, diameter D B, latus tranſuerſum. <lb/></s> <s xml:id="echoid-s752" xml:space="preserve">G B, intelligatur ſectum plano H K I, A C, pa-<lb/>rallelo, & </s> <s xml:id="echoid-s753" xml:space="preserve">ipſi ſit circumſcriptus cylindricus L C. </s> <s xml:id="echoid-s754" xml:space="preserve">Di-<lb/>co hunc eſſe ad ſegmentum conoidis, vt rectangu-<lb/>lum G D B, ad rectangulum ſub G D, in B k, <lb/>vna cum rectangulo ſub compoſita ex dimidia G B, <lb/>& </s> <s xml:id="echoid-s755" xml:space="preserve">tertia parte D k, & </s> <s xml:id="echoid-s756" xml:space="preserve">ſub tertia parte D k.</s> <s xml:id="echoid-s757" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s758" xml:space="preserve">Segmento A H I C, intelligatur inſcriptum ſeg. <lb/></s> <s xml:id="echoid-s759" xml:space="preserve">mentum E N O F, conoidis parabolici cuius ver-<lb/>tex B, conditionis ſupra ſæpe expoſitæ; </s> <s xml:id="echoid-s760" xml:space="preserve">& </s> <s xml:id="echoid-s761" xml:space="preserve">in talibus <lb/>ſegmentis intelligantur ſegmenta conorum inſcri-<lb/>ptorum in integris conoidibus, quæ ſint A P Q C, <lb/>E R S F. </s> <s xml:id="echoid-s762" xml:space="preserve">Quoniam fruſtum A H I C, conſtat ex <lb/>fruſto parabolico, & </s> <s xml:id="echoid-s763" xml:space="preserve">ex differentia fruſtorum conoi- <pb o="40" file="0052" n="52"/> <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> deorum; </s> <s xml:id="echoid-s764" xml:space="preserve">& </s> <s xml:id="echoid-s765" xml:space="preserve">ex propoſit. </s> <s xml:id="echoid-s766" xml:space="preserve">4, differentia fruſtorum co-<lb/>noideorum eſt æqualis differentiæ conorum; </s> <s xml:id="echoid-s767" xml:space="preserve">ergo <lb/>L C, erit ad fruſtum A H I C, vt eſt ad fruſtum <lb/>parabolicum, vna cum differentia fruſtorum cono-<lb/>rum. </s> <s xml:id="echoid-s768" xml:space="preserve">Hanc verò rationem ſic venabimur. </s> <s xml:id="echoid-s769" xml:space="preserve">Cylin-<lb/>drus L C, ad fruſtum parabolicum E N O F, ha-<lb/>bet rationem compoſitam ex ratione cylindri L C, <lb/>ad cylindrum T F, tali fruſto parabolico circum-<lb/>ſcriptum, & </s> <s xml:id="echoid-s770" xml:space="preserve">huius ad ipſum fruſtum: </s> <s xml:id="echoid-s771" xml:space="preserve">L C, ad T F, <lb/>eſt vt quadratum A D, ad quadratum E D; </s> <s xml:id="echoid-s772" xml:space="preserve">nem-<lb/>pe ex hypotheſi, vt D G, ad G B. </s> <s xml:id="echoid-s773" xml:space="preserve">Cum autem ex <pb o="41" file="0053" n="53"/> propoſit. </s> <s xml:id="echoid-s774" xml:space="preserve">3. </s> <s xml:id="echoid-s775" xml:space="preserve">lib. </s> <s xml:id="echoid-s776" xml:space="preserve">4. </s> <s xml:id="echoid-s777" xml:space="preserve">ſit T F, ad E N O F, vt paralle-<lb/>logrammum T F, ad trapezium E R S F; </s> <s xml:id="echoid-s778" xml:space="preserve">& </s> <s xml:id="echoid-s779" xml:space="preserve">cum ex <lb/>propoſit. </s> <s xml:id="echoid-s780" xml:space="preserve">8. </s> <s xml:id="echoid-s781" xml:space="preserve">& </s> <s xml:id="echoid-s782" xml:space="preserve">9. </s> <s xml:id="echoid-s783" xml:space="preserve">lib. </s> <s xml:id="echoid-s784" xml:space="preserve">prim. </s> <s xml:id="echoid-s785" xml:space="preserve">ſit T F, parallelogram-<lb/>mum ad trapezium E R S F, vt dupla E D, ad E D, <lb/>cum R K, vel vt dupla D B, ad D B, cum Bk; </s> <s xml:id="echoid-s786" xml:space="preserve">ſe-<lb/>quitur cylindrum L C, ad ſegmentum parabolicum <lb/>E N O F, habere rationem compoſitam ex ratione <lb/>D G, ad G B, & </s> <s xml:id="echoid-s787" xml:space="preserve">ex ratione duplæ D B, ad D B, <lb/>cum B k. </s> <s xml:id="echoid-s788" xml:space="preserve">Sed ex dictis rationibus componitur quo-<lb/>que ratio dupli rectanguli G D B, ad rectangulum <lb/>G B D, cum rectangulo G B k. </s> <s xml:id="echoid-s789" xml:space="preserve">Et vt duplum re-<lb/>ctangulum G D B, ad prædicta conſequentia, ſic <lb/>triplum rectangulum G D B, ad ſexquialterum re-<lb/>ctangulorum G B D, G B k. </s> <s xml:id="echoid-s790" xml:space="preserve">Ergo L C, erit ad <lb/>ſegmentum E N O F, vt triplum rectangulum <lb/>G D B, ad ſeſquialterum rectangulorum G B D; <lb/></s> <s xml:id="echoid-s791" xml:space="preserve">G B k. </s> <s xml:id="echoid-s792" xml:space="preserve">Quod ſeruetur.</s> <s xml:id="echoid-s793" xml:space="preserve"/> </p> <div xml:id="echoid-div44" 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/PVNDER1Y/figures/0052-01"/> </figure> </div> <p> <s xml:id="echoid-s794" xml:space="preserve">Ex propoſit. </s> <s xml:id="echoid-s795" xml:space="preserve">14, & </s> <s xml:id="echoid-s796" xml:space="preserve">15, lib. </s> <s xml:id="echoid-s797" xml:space="preserve">2. </s> <s xml:id="echoid-s798" xml:space="preserve">habemus tam totum <lb/>cylindrum L C, quam ablatum T F, eſſe illum ad <lb/>fruſtum conicum A P Q C, hunc verò ad fruſtum <lb/>conicum E R S F, vt tripla D B, ad D B, B R, & </s> <s xml:id="echoid-s799" xml:space="preserve"><lb/>harum tertiam minorem continuè proportionalem. <lb/></s> <s xml:id="echoid-s800" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s801" xml:space="preserve">reliquum ad reliquum erit vt totum ad to-<lb/>tum: </s> <s xml:id="echoid-s802" xml:space="preserve">nempetubus cylindricus L E M, erit ad diffe-<lb/>rentiam fruſtorum conorum, vt tripla D B, ad D B, <lb/>B k, & </s> <s xml:id="echoid-s803" xml:space="preserve">illam tertiam proportionalem. </s> <s xml:id="echoid-s804" xml:space="preserve">Tunc argu-<lb/>mentetur ſic. </s> <s xml:id="echoid-s805" xml:space="preserve">Ratio cylindri L C, ad differentiam <lb/>ſegmentorum conorum componitur ex ratione L C, <lb/>ad tubum L E M, & </s> <s xml:id="echoid-s806" xml:space="preserve">huius ad differentiam ſegmen- <pb o="42" file="0054" n="54"/> <anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/> torum conorum: </s> <s xml:id="echoid-s807" xml:space="preserve">at L C, ad tubum eſt vt quadra-<lb/>tum A D, ad rectangulum A E C, nempe ex hy-<lb/>potheſi ſuppoſita per conuerſionem rationis, vt <lb/>G D, ad D B: </s> <s xml:id="echoid-s808" xml:space="preserve">tubus autem eſt ad differentiam fru-<lb/>ſtorum conorum vt tripla D B, ad D B, B k, & </s> <s xml:id="echoid-s809" xml:space="preserve">il-<lb/>lam tertiam proportionalem. </s> <s xml:id="echoid-s810" xml:space="preserve">Ergo ratio L C, ad <lb/>differentiam ſegmentorum conorum componetur <lb/>quoque ex rationibus G D, ad D B, & </s> <s xml:id="echoid-s811" xml:space="preserve">triplæ D B, <lb/>ad D B, B K, & </s> <s xml:id="echoid-s812" xml:space="preserve">illam tertiam proportionalem. </s> <s xml:id="echoid-s813" xml:space="preserve">Sed <lb/>ex dictis rationibus componitur etiam ratio tripli <lb/>rectanguli G D B, ad quadratum D B, rectangulum <pb o="43" file="0055" n="55"/> D B k, & </s> <s xml:id="echoid-s814" xml:space="preserve">rectangulum ſub D B, & </s> <s xml:id="echoid-s815" xml:space="preserve">ſub illa tertia <lb/>proportionali (quod eſt æquale quadrato mediæ <lb/>B k). </s> <s xml:id="echoid-s816" xml:space="preserve">Ergo L C, erit ad differentiam fruſtorum co-<lb/>norum, vt triplum rectangulum G D B, ad quadra-<lb/>ta D B, B k, cum rectangulo D B K; </s> <s xml:id="echoid-s817" xml:space="preserve">nempe ad tria <lb/>quadrata B k, cum triplo rectangulo B k D, & </s> <s xml:id="echoid-s818" xml:space="preserve">cum <lb/>quadrato D k (, quia quadratum D B, diuiditur <lb/>in quadrata B k, k D, & </s> <s xml:id="echoid-s819" xml:space="preserve">in duo rectangula B k D; </s> <s xml:id="echoid-s820" xml:space="preserve">& </s> <s xml:id="echoid-s821" xml:space="preserve"><lb/>pariter rectangulum D B k, diuiditur in quadratum <lb/>B k, & </s> <s xml:id="echoid-s822" xml:space="preserve">in rectangulum B k D). </s> <s xml:id="echoid-s823" xml:space="preserve">Cum autem ſupra <lb/>probatum ſit, eſſe L C, ad fruſtum E N O F, vt <lb/>idem triplum rectangulum G D B, ad ſeſquialterum <lb/>rectangulorum G B D, G B k. </s> <s xml:id="echoid-s824" xml:space="preserve">Ergo colligendo am-<lb/>boconſe quentia, erit L C, ad fruſtum, & </s> <s xml:id="echoid-s825" xml:space="preserve">ad diffe-<lb/>rentiam fruſtorum conorum ſimul, nempe ad fru-<lb/>ſtum A H I C, vt triplum rectangulum G D B, ad <lb/>triplum quadratum B k, cum triplo rectangulo <lb/>B k D, cum quadrato K D, & </s> <s xml:id="echoid-s826" xml:space="preserve">cum ſeſquialtero re-<lb/>ctangulorum G B D, G B k. </s> <s xml:id="echoid-s827" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s828" xml:space="preserve">vt horum pla-<lb/>norum tertiæ partes: </s> <s xml:id="echoid-s829" xml:space="preserve">nempe L C, erit ad A H I C, <lb/>vt rectangulum G D B, ad quadratum B K, cum <lb/>rectangulo B k D, & </s> <s xml:id="echoid-s830" xml:space="preserve">cum tertia parte quadrati D k, <lb/>vna cum dimidio rectangulorum G B D, G B K. <lb/></s> <s xml:id="echoid-s831" xml:space="preserve">Cum verò dimidium rectanguli G B D, diuidatur <lb/>in dimidium G B K, & </s> <s xml:id="echoid-s832" xml:space="preserve">in dimidium G B, K D. </s> <s xml:id="echoid-s833" xml:space="preserve"><lb/>Ergo dimidium rectangulorum G B D, G B K, erit <lb/>rectangulum G B k, cum dimidio rectanguli G B, <lb/>K D. </s> <s xml:id="echoid-s834" xml:space="preserve">Si ergo ſimul iunxerimus rectangulum G B K, <lb/>cum quadrato B K, & </s> <s xml:id="echoid-s835" xml:space="preserve">cum rectangulo B K D, habe- <pb o="44" file="0056" n="56"/> <anchor type="figure" xlink:label="fig-0056-01a" xlink:href="fig-0056-01"/> bimus rectangulum G D, B k. </s> <s xml:id="echoid-s836" xml:space="preserve">Pariter ſi ſimul iun-<lb/>xerimus rectangulum ſub dimidia G B, & </s> <s xml:id="echoid-s837" xml:space="preserve">ſub D K, <lb/>cum tertia parte quadrati D K, nempe cum rectan-<lb/>gulo ſub D K, & </s> <s xml:id="echoid-s838" xml:space="preserve">ſub tertia parte D k, habebimus <lb/>rectangulum ſub compoſita ex dimidia G B, & </s> <s xml:id="echoid-s839" xml:space="preserve">ex <lb/>tertia parte D k, & </s> <s xml:id="echoid-s840" xml:space="preserve">ſub D K. </s> <s xml:id="echoid-s841" xml:space="preserve">Ergo à primo ad vlti-<lb/>mum concludemus, eſſe L C, ad fruſtum conoidis <lb/>hyperbolici A H I C, vt rectangulum G D B, ad re-<lb/>ctangulum G D, B K, cum rectangulo ſub compo-<lb/>ſita ex dimidia G B, & </s> <s xml:id="echoid-s842" xml:space="preserve">ex tertia parte D k, & </s> <s xml:id="echoid-s843" xml:space="preserve">ſub <lb/>D K. </s> <s xml:id="echoid-s844" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s845" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="2"> <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/PVNDER1Y/figures/0054-01"/> </figure> <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/PVNDER1Y/figures/0056-01"/> </figure> </div> <pb o="45" file="0057" n="57"/> </div> <div xml:id="echoid-div47" type="section" level="1" n="32"> <head xml:id="echoid-head42" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s846" xml:space="preserve">Proportionem prædicti cylindri ad illud ſegmen-<lb/>tum hyperbolicum, etiam duobus alijs modis, con-<lb/>ſequenter ad ſuperius dicta, liceret colligere. </s> <s xml:id="echoid-s847" xml:space="preserve">Cum <lb/>enim tale ſegmentum conſter ex ſegmento coniſibi <lb/>inſcripto, & </s> <s xml:id="echoid-s848" xml:space="preserve">ex exceſſu ſupra ipſum; </s> <s xml:id="echoid-s849" xml:space="preserve">& </s> <s xml:id="echoid-s850" xml:space="preserve">cum talis ex-<lb/>ceſſus ſit æqualis exceſſui ſegmenti conoidis para-<lb/>bolici ſupra ſuum ſegmentum conicum; </s> <s xml:id="echoid-s851" xml:space="preserve">& </s> <s xml:id="echoid-s852" xml:space="preserve">cum ex <lb/>dictis in ijs, quæ de infinitis parabolis conſcripſi-<lb/>mus, facile liceat colligere rationem L C, & </s> <s xml:id="echoid-s853" xml:space="preserve">ad ſeg-<lb/>mentum conicum A P Q C, & </s> <s xml:id="echoid-s854" xml:space="preserve">ad exceſlum ſegmen-<lb/>ti conoidis parabolici ENOF, ſupra ſegmentum <lb/>conicum E R S F: </s> <s xml:id="echoid-s855" xml:space="preserve">ſequitur facile etiam nos obtine-<lb/>re rationem LC, ad ſegmentum AHIC. </s> <s xml:id="echoid-s856" xml:space="preserve">Pari-<lb/>ter ſi in ſchemat. </s> <s xml:id="echoid-s857" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s858" xml:space="preserve">10. </s> <s xml:id="echoid-s859" xml:space="preserve">tam ſegmento v. </s> <s xml:id="echoid-s860" xml:space="preserve">g. <lb/></s> <s xml:id="echoid-s861" xml:space="preserve">A Q T C, quam ſegmento exceſſus fruſti conici <lb/>G N P H, ſupra cylindrum R M, mente concipia-<lb/>mus circumſcribi cylindros; </s> <s xml:id="echoid-s862" xml:space="preserve">patet ex dictis in eadem <lb/>propoſitione, tubum cylindricum cuius baſis armil-<lb/>la circularis G L H, altitudo OD, æqualem eſſe <lb/>cylindro circumſcripto ſegmento A Q T C. </s> <s xml:id="echoid-s863" xml:space="preserve">Pari-<lb/>terque patet exceſſum fruſti G N P H, ſupra cylin-<lb/>drum R M, æqualem eſſe ſegmento A Q T C. </s> <s xml:id="echoid-s864" xml:space="preserve">Cum <lb/>ergo ex dictis in opere ſupra citato, faciliſſime <lb/>poſſimus habere rationem prædicti tubi ad illum ex-<lb/>ceſſum ſupra cylindrum; </s> <s xml:id="echoid-s865" xml:space="preserve">faciliter etiam habebimus <lb/>rationem cylindri circum ſcripti ſegmento hyperbo- <pb o="46" file="0058" n="58"/> lico A Q T C, ad ipſum ſegmentum. </s> <s xml:id="echoid-s866" xml:space="preserve">Hæc non <lb/>continent multum difficultatis, quapropter ſufficiat <lb/>ea lectoribus indicaſſe.</s> <s xml:id="echoid-s867" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s868" xml:space="preserve">Sicuti ſufficiat ex antecedentibus indicare mo-<lb/>dum reperiendi in quà linea parallela D k, ſit cen-<lb/>trum grauitatis ſuppoſiti ſegmenti ſemihyperbolæ <lb/>A H k D. </s> <s xml:id="echoid-s869" xml:space="preserve">Hoc autem reperietur ex dictis, ſi ſuppo-<lb/>natur ſegmenti A H K D, quadratura, nempe ratio, <lb/>quam habet ad ipſum parallelogrammum L D. </s> <s xml:id="echoid-s870" xml:space="preserve">Cum <lb/>enim cylindrus L C, habeat ad ſegmentum conoi-<lb/>dis A H I C, ex ſchol. </s> <s xml:id="echoid-s871" xml:space="preserve">pri. </s> <s xml:id="echoid-s872" xml:space="preserve">prop. </s> <s xml:id="echoid-s873" xml:space="preserve">3. </s> <s xml:id="echoid-s874" xml:space="preserve">lib. </s> <s xml:id="echoid-s875" xml:space="preserve">3. </s> <s xml:id="echoid-s876" xml:space="preserve">rationem <lb/>compoſitam ex ratione dimidij parallelogrammi <lb/>L D, ad ſegmentum A H k D, & </s> <s xml:id="echoid-s877" xml:space="preserve">ex ratione A D, <lb/>ad interceptam inter D, & </s> <s xml:id="echoid-s878" xml:space="preserve">centrum æquilibrij ſeg-<lb/>menti acceptum in A D, hoc eſt centrum grauitatis <lb/>duplicati ſegmenti A H k D, ad partes A D; </s> <s xml:id="echoid-s879" xml:space="preserve">ſequi-<lb/>tur, quod ſi ex proportione cylindri L C, ad ſeg-<lb/>mentum conoidis A H I C; </s> <s xml:id="echoid-s880" xml:space="preserve">nempe ex ratione ex-<lb/>preſſa in pręſenti propoſitione, ſubtrahatur ſuppoſita <lb/>ratio dimidij parallelogrammi L D, ad ſegmentum <lb/>parabolæ A H K D, remanebit ratio A D, ad inter-<lb/>ceptam inter D, & </s> <s xml:id="echoid-s881" xml:space="preserve">centrum quæſitum.</s> <s xml:id="echoid-s882" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s883" xml:space="preserve">Hocpuncto inuento, non ignora bimus tria ſolita, <lb/>quæ ſæpe ſæpius deduximus in non paucis propoſi-<lb/>tionib is lib. </s> <s xml:id="echoid-s884" xml:space="preserve">3. </s> <s xml:id="echoid-s885" xml:space="preserve">Nam primo non ignorabimus ratio-<lb/>nem cylindri ex L D, ad ſolidum ex ſegmento <lb/>A H K D, circa L A. </s> <s xml:id="echoid-s886" xml:space="preserve">Secundo non ignorabimus <lb/>rationem ſegmenti A H I C, ad ſolidum prædictum <lb/>circa A L. </s> <s xml:id="echoid-s887" xml:space="preserve">Tertio tam ſupra L D, quam ſupra.</s> <s xml:id="echoid-s888" xml:space="preserve"> <pb o="47" file="0059" n="59"/> A H k D, intellectis cylindricis rectis æquealtis ſe-<lb/>ctis diagonaliter plano tranſeunte per D k, & </s> <s xml:id="echoid-s889" xml:space="preserve">per <lb/>latus oppoſitum ipſi L A, minimè ignorabimus cu-<lb/>bationes truncorum cylindrici ſuper A H k D, exi-<lb/>ſtentis. </s> <s xml:id="echoid-s890" xml:space="preserve">Hac tamen differentia, quod cubationem <lb/>trunci ſiniſtri habebimus ſine ſuppoſitione alicu-<lb/>ius quadraturæ; </s> <s xml:id="echoid-s891" xml:space="preserve">non ſic cubationem trunci dex-<lb/>teri.</s> <s xml:id="echoid-s892" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s893" xml:space="preserve">His oſtenſis non erit inutile oſtendere modum. <lb/></s> <s xml:id="echoid-s894" xml:space="preserve">inueniendi centrum grauitatis ſegmenti conoidis <lb/>hyperbolici A H I C. </s> <s xml:id="echoid-s895" xml:space="preserve">Sed prius oſtendatur ſequens <lb/>propoſitio.</s> <s xml:id="echoid-s896" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div48" type="section" level="1" n="33"> <head xml:id="echoid-head43" xml:space="preserve">PROPOSITIO XVI.</head> <p style="it"> <s xml:id="echoid-s897" xml:space="preserve">Differentia ſupradictorum fruſtorum conoideorum eſt ad <lb/>ſegmentum conoidis parabolici, vt quadrata axium to-<lb/>tius conoidis, & </s> <s xml:id="echoid-s898" xml:space="preserve">conoidis ad verticem, vna cum re-<lb/>ctangulo contento ſub his axibus, ad ſeſquialterum re-<lb/>ctangulorum contentorum ſub latere tranſuerſo, & </s> <s xml:id="echoid-s899" xml:space="preserve">ſub <lb/>prædictis axibus.</s> <s xml:id="echoid-s900" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s901" xml:space="preserve">SInt ergo ſegmenta anteced, propoſit. </s> <s xml:id="echoid-s902" xml:space="preserve">Dico dif-<lb/>ferentiam fruſtorum A H I C, E N O F, eſſe <lb/>ad ſegmentum parabolicum E N O F, vt quadrata <lb/>D B, B k, cum rectangulo D B k, ad ſeſquialterum <lb/>rectangulorum G B D, G B K. </s> <s xml:id="echoid-s903" xml:space="preserve">Differentia enim. <lb/></s> <s xml:id="echoid-s904" xml:space="preserve">prædicta ad ſegmentum E N O F, habet rationem <lb/>compoſitam ex ratione differentiæ ad tubum cylin- <pb o="48" file="0060" n="60"/> dricum LEM; </s> <s xml:id="echoid-s905" xml:space="preserve">huius ad cylindrum T F; </s> <s xml:id="echoid-s906" xml:space="preserve">& </s> <s xml:id="echoid-s907" xml:space="preserve">hu-<lb/>ius ad ſegmentum E N O F. </s> <s xml:id="echoid-s908" xml:space="preserve">Cum autem differen-<lb/>tia fruſtorum conoideorum ſit, ex ſupradictis, æqua-<lb/>lis differentiæ fruſtorum conorum inſcriptorum in <lb/>ipſis; </s> <s xml:id="echoid-s909" xml:space="preserve">& </s> <s xml:id="echoid-s910" xml:space="preserve">cum differentia fruſtorum conorum ſit ad <lb/>tubum L E M, vt facile poteſt deduci ex dictis in <lb/>ſchol. </s> <s xml:id="echoid-s911" xml:space="preserve">4. </s> <s xml:id="echoid-s912" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s913" xml:space="preserve">14. </s> <s xml:id="echoid-s914" xml:space="preserve">lib. </s> <s xml:id="echoid-s915" xml:space="preserve">2. </s> <s xml:id="echoid-s916" xml:space="preserve">vt D B, cum B K, & </s> <s xml:id="echoid-s917" xml:space="preserve"><lb/>cum harum tertia minori proportionali ad tres D B. <lb/></s> <s xml:id="echoid-s918" xml:space="preserve">Sequitur etiam differentiam ſegmentorum conoi-<lb/>deorum, eſſe ad tubum cylindricum L E M, vt D B, <lb/>B K, & </s> <s xml:id="echoid-s919" xml:space="preserve">illa tertia proportionalis ad tres D B. </s> <s xml:id="echoid-s920" xml:space="preserve">Cum <lb/>verò L E M, tubus ſit ad cylindrum T F, vt re-<lb/>ctangulum A E C, ad quadratum E D, nempe <lb/>diuidendo, ex hypotheſi frequenter vſa, vt D B, <lb/>ad B G, ſeù vt tripla D B, ad triplam G B. </s> <s xml:id="echoid-s921" xml:space="preserve">Ergo <lb/>ex æquali, erit differentia ſegmentorum conoideo-<lb/>rum ad cylindrum T F, vt D B, B k, cum illa ter-<lb/>tia proportionali ad triplam G B. </s> <s xml:id="echoid-s922" xml:space="preserve">Cylindrus T F, <lb/>eſt ad ſegmentum E N O F, vt dicetur inferius, vt <lb/>dupla D B, ad D B, cum B K. </s> <s xml:id="echoid-s923" xml:space="preserve">Ergo à primo ad <lb/>vltimum, differentia ſegmentorum conoideorum. </s> <s xml:id="echoid-s924" xml:space="preserve"><lb/>ad ſegmentum E N O F, habebit rationem com-<lb/>poſitam ex ratione D B, B k, & </s> <s xml:id="echoid-s925" xml:space="preserve">harum tertiæ pro-<lb/>portionalis ad triplam B G, & </s> <s xml:id="echoid-s926" xml:space="preserve">ex ratione duplæ D B, <lb/>ad D B, B k. </s> <s xml:id="echoid-s927" xml:space="preserve">Sed ex dictis rationibus componitur <lb/>quoque ratio duorum quadratorum B D, duorum <lb/>rectangulorum D B K, & </s> <s xml:id="echoid-s928" xml:space="preserve">duorum rectangulorum. </s> <s xml:id="echoid-s929" xml:space="preserve"><lb/>ſub D B, & </s> <s xml:id="echoid-s930" xml:space="preserve">ſub illa tertia proportionali (quæ duo <lb/>vltima rectangula ſunt æqualia duobus quadratis <pb o="49" file="0061" n="61"/> <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> mediæ B K), ad tria rectangula G B D, cum tribus <lb/>rectangulis G B k. </s> <s xml:id="echoid-s931" xml:space="preserve">Ergo differentia fruſtorum co-<lb/>noideorum, erit ad ſegmentum E N O F, vt duo <lb/>quadrata D B, cum duobus rectangulis D B k, & </s> <s xml:id="echoid-s932" xml:space="preserve"><lb/>cum duobus quadratis B K, ad tria rectangula, <lb/>G B k, cum tribus rectangulis G B D. </s> <s xml:id="echoid-s933" xml:space="preserve">Et vt ho-<lb/>rum terminorum dimidia. </s> <s xml:id="echoid-s934" xml:space="preserve">Nempe differentia præ-<lb/>dicta, erit ad prædictum ſegmentum, vt quadrata <lb/>D B, B k, cum rectangulo D B k, ad ſeſquialte-<lb/>rum rectangulorum G B D, G B k. </s> <s xml:id="echoid-s935" xml:space="preserve">Quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s936" xml:space="preserve"/> </p> <div xml:id="echoid-div48" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0061-01"/> </figure> </div> <pb o="50" file="0062" n="62"/> <p> <s xml:id="echoid-s937" xml:space="preserve">Quod verò T F, cylindrus ſit ad ſegmentum. <lb/></s> <s xml:id="echoid-s938" xml:space="preserve">E N O F, vt dupla D B, ad D B, B k, patet. </s> <s xml:id="echoid-s939" xml:space="preserve">Quía <lb/>ex propoſit. </s> <s xml:id="echoid-s940" xml:space="preserve">3. </s> <s xml:id="echoid-s941" xml:space="preserve">lib. </s> <s xml:id="echoid-s942" xml:space="preserve">4. </s> <s xml:id="echoid-s943" xml:space="preserve">cylindrus T F, eſt ad ſegmen-<lb/>tum conoidis parabolici E N O F, vt parallelo-<lb/>grammum T F, ad trapezium lineare E R S F, At <lb/>ex propoſit. </s> <s xml:id="echoid-s944" xml:space="preserve">9. </s> <s xml:id="echoid-s945" xml:space="preserve">lib. </s> <s xml:id="echoid-s946" xml:space="preserve">prim. </s> <s xml:id="echoid-s947" xml:space="preserve">eſt parallelogrammum ad <lb/>trapezium vt dupla D B, ad D B, & </s> <s xml:id="echoid-s948" xml:space="preserve">B k. </s> <s xml:id="echoid-s949" xml:space="preserve">Qua-<lb/>re patet propoſitum.</s> <s xml:id="echoid-s950" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div50" type="section" level="1" n="34"> <head xml:id="echoid-head44" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s951" xml:space="preserve">Ratio autem prædictorum ſolidorum collecta in <lb/>ſupradicta propoſitione, poteſt etiam reduci ad mi-<lb/>nora plana; </s> <s xml:id="echoid-s952" xml:space="preserve">quia poteſt reduci ad eam, quam habet <lb/>rectangulum D B k, cum tertia parte quadrati D k, <lb/>ad rectangulum G B K, cum dimidio rectanguli <lb/>G B, K D. </s> <s xml:id="echoid-s953" xml:space="preserve">Patet quia hæc plana ſunt tertiæ partes <lb/>priorum planorum.</s> <s xml:id="echoid-s954" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div51" type="section" level="1" n="35"> <head xml:id="echoid-head45" xml:space="preserve">PROPOSITIO XVII.</head> <head xml:id="echoid-head46" style="it" xml:space="preserve">Segmenti fupradicti conoidis hyperbolici centrum <lb/>grauitatis reperire.</head> <p> <s xml:id="echoid-s955" xml:space="preserve">SEgmenti conoidis hyperbolici A H I C, cen-<lb/>trum grauitatis reperietur ſic. </s> <s xml:id="echoid-s956" xml:space="preserve">Inſcriptis ſoli-<lb/>dis vt ſupra, ſecetur K D, ſic in X, vt K X, ſit ad <lb/>X D, vt duplum quadratum E D, cum quadrato <lb/>N K, ad duplum quadratum N K, cum quadrato <pb o="51" file="0063" n="63"/> <anchor type="figure" xlink:label="fig-0063-01a" xlink:href="fig-0063-01"/> E D, ſeù vt dupla D B, cum B K, ad duplam B K, <lb/>cum B D. </s> <s xml:id="echoid-s957" xml:space="preserve">Ergo ex ſchol. </s> <s xml:id="echoid-s958" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s959" xml:space="preserve">15. </s> <s xml:id="echoid-s960" xml:space="preserve">lib 4. </s> <s xml:id="echoid-s961" xml:space="preserve">erit <lb/>X, centrum grauitatis fruſti conoidis parabolici <lb/>E N O F. </s> <s xml:id="echoid-s962" xml:space="preserve">B D, & </s> <s xml:id="echoid-s963" xml:space="preserve">B K, ſic ſecentur in Y, +, vt <lb/>B Y, ſit tripla ipſius Y K, & </s> <s xml:id="echoid-s964" xml:space="preserve">pariter B +, tripla ſit <lb/>ipſius + D: </s> <s xml:id="echoid-s965" xml:space="preserve">& </s> <s xml:id="echoid-s966" xml:space="preserve">fiat vt exceſſus cubi D B, ſupra cu-<lb/>bum B K, ad cubum B K, ſic Y +, ad + ℟. </s> <s xml:id="echoid-s967" xml:space="preserve">Ergo <lb/>exſchol. </s> <s xml:id="echoid-s968" xml:space="preserve">propoſit, 18. </s> <s xml:id="echoid-s969" xml:space="preserve">eiuſdem libri erit ℟, centrum <lb/>grauitatis differentiæ fruſtorum conorum; </s> <s xml:id="echoid-s970" xml:space="preserve">& </s> <s xml:id="echoid-s971" xml:space="preserve">conſe-<lb/>quenter exſchol. </s> <s xml:id="echoid-s972" xml:space="preserve">2. </s> <s xml:id="echoid-s973" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s974" xml:space="preserve">4. </s> <s xml:id="echoid-s975" xml:space="preserve">huius, erit centrum <lb/>grauitatis differentiæ fruſtorum conoideorum. </s> <s xml:id="echoid-s976" xml:space="preserve">Di- <pb o="52" file="0064" n="64"/> uidatur ergo X ℟, in Z, vt ſit X Z, ad Z ℟, vt qua-<lb/>drata D B, B K, cum rectangulo D B K, ad ſeſqui-<lb/>alterum rectangulorum G B D, G B K; </s> <s xml:id="echoid-s977" xml:space="preserve">ſeù vt rectan-<lb/>gulum D B K, cum tertia parte quadrati D K, ad re-<lb/>ctangulum G B K, cum dimidio rectanguli G B, K D; <lb/></s> <s xml:id="echoid-s978" xml:space="preserve">nenipe ex propoſit. </s> <s xml:id="echoid-s979" xml:space="preserve">anteced. </s> <s xml:id="echoid-s980" xml:space="preserve">vt eſt differentia fruſto-<lb/>rum conoideo rum ad fruſtum conoidis parabolici <lb/>E N O F. </s> <s xml:id="echoid-s981" xml:space="preserve">Dico inuentum eſſe Z, centrum grauita-<lb/>tis fruſti conoidis hyperbolici A H I C. </s> <s xml:id="echoid-s982" xml:space="preserve">Cum au-<lb/>tem res ſit de sè euidens ex doctrinis Archimedis in <lb/>æqueponderantibus, relinquitur conſiderationi le-<lb/>ctoris.</s> <s xml:id="echoid-s983" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="1"> <figure xlink:label="fig-0063-01" xlink:href="fig-0063-01a"> <image file="0063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0063-01"/> </figure> </div> </div> <div xml:id="echoid-div53" type="section" level="1" n="36"> <head xml:id="echoid-head47" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s984" xml:space="preserve">Alij modi ex ſuperioribus non deſunt reperiendi <lb/>tale centrum grauitatis; </s> <s xml:id="echoid-s985" xml:space="preserve">ſed nè lectorem nimis quam <lb/>par ſit defatigemus, ad alia, & </s> <s xml:id="echoid-s986" xml:space="preserve">noua tranſeamus; </s> <s xml:id="echoid-s987" xml:space="preserve">præ-<lb/>cipuè ad centrum grauitatis hyperbolæ reperien-<lb/>dum. </s> <s xml:id="echoid-s988" xml:space="preserve">Quod tamen non reperietur niſi præmiſſis qui-<lb/>buſdam demonſtrationibus.</s> <s xml:id="echoid-s989" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div54" type="section" level="1" n="37"> <head xml:id="echoid-head48" xml:space="preserve">PROPOSITIO XVIII.</head> <p style="it"> <s xml:id="echoid-s990" xml:space="preserve">Si ſemihyperbola cum ſibi circumſcripto parallelogrammo <lb/>rotetur circa ſecundam coniugatam diametrum. </s> <s xml:id="echoid-s991" xml:space="preserve">An-<lb/>nulus latus ortus ex rotatione exceſſus parallelogram-<lb/>mi ſupra ſemihyperbolam, erit æqualis cono ex triangu-<lb/>lo, cuius vnum latus dimidia ſecundæ diametri, aliud <pb o="53" file="0065" n="65"/> intercepta inter ſecundam diametrum, & </s> <s xml:id="echoid-s992" xml:space="preserve">aſymptotum, <lb/>reuoluto cicca ſecundam diametrum; </s> <s xml:id="echoid-s993" xml:space="preserve">& </s> <s xml:id="echoid-s994" xml:space="preserve">hoc tam ſecun-<lb/>dum totum, quam ſecundum partes proportionales.</s> <s xml:id="echoid-s995" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s996" xml:space="preserve">ESto ſemihyperbola A B C, cuius diameter A B; <lb/></s> <s xml:id="echoid-s997" xml:space="preserve">E B dimidium lateris tranſuerſi; </s> <s xml:id="echoid-s998" xml:space="preserve">centrum E; </s> <s xml:id="echoid-s999" xml:space="preserve"><lb/>aſymptotus E G; </s> <s xml:id="echoid-s1000" xml:space="preserve">ſecunda diameter E F; </s> <s xml:id="echoid-s1001" xml:space="preserve">& </s> <s xml:id="echoid-s1002" xml:space="preserve">pa-<lb/>rallelogrammum A D, ſemihy perbolæ circumſcri-<lb/>ptum cum triangulo E F G, rotentur circa E F. </s> <s xml:id="echoid-s1003" xml:space="preserve">Di-<lb/>co annulum latum ortum ex rotatione trilinei mixti <lb/>C B D, circa E F, æqualem eſſe cono G E M, & </s> <s xml:id="echoid-s1004" xml:space="preserve"><lb/>hoc tam ſecundum totum, quam ſecundum partes <lb/>proportionales. </s> <s xml:id="echoid-s1005" xml:space="preserve">Intelligantul oppoſitæ ſectiones vt <lb/>in ſchemate, & </s> <s xml:id="echoid-s1006" xml:space="preserve">ſumatur a bitrariè in E F, quodli-<lb/>bet punctum I, per quod ducatur O I N, paralle-<lb/>la L C, ſecans aſymptotum E G, in P. </s> <s xml:id="echoid-s1007" xml:space="preserve">Quadra-<lb/>tum I O, eſt æquale tam rectangulo O P N, cum <lb/>quadrato P I, quam rectangulo O Q N, cum. </s> <s xml:id="echoid-s1008" xml:space="preserve"><lb/>quadrato Q I. </s> <s xml:id="echoid-s1009" xml:space="preserve">Ergo rectangulum O P N, cum <lb/>quadrato P I, erit æquale rectangulo O Q N, cum <lb/>quadrato Q I. </s> <s xml:id="echoid-s1010" xml:space="preserve">Sed ex propoſit. </s> <s xml:id="echoid-s1011" xml:space="preserve">11. </s> <s xml:id="echoid-s1012" xml:space="preserve">ſec. </s> <s xml:id="echoid-s1013" xml:space="preserve">conic. </s> <s xml:id="echoid-s1014" xml:space="preserve">re-<lb/>ctangulum O P N, eſt aquale quadrato B E, ſeù <lb/>quadrato Q I. </s> <s xml:id="echoid-s1015" xml:space="preserve">Ergo reliquum rectangulum O Q N, <lb/>erit æquale reliquo quadrato P I. </s> <s xml:id="echoid-s1016" xml:space="preserve">Quare & </s> <s xml:id="echoid-s1017" xml:space="preserve">armil-<lb/>la circularis O Q N, erit æqualis circulo P R. </s> <s xml:id="echoid-s1018" xml:space="preserve">Cum <lb/>vero punctum I, ſumptum ſit arbitrariè ergo om-<lb/>nes armillæ circulares parallelæ armillæ C D L, or-<lb/>tæ ex rotatione trilinei C B D, circa E F, erunt <lb/>æquales omnibus circulis coni G E M. </s> <s xml:id="echoid-s1019" xml:space="preserve">Et conſe- <pb o="54" file="0066" n="66"/> <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> quenter annulus latus ortus ex rotatione illius trili-<lb/>nei circa E F, erit æqualis cono G E M. </s> <s xml:id="echoid-s1020" xml:space="preserve">Quod <lb/>vero probatum eſt de totis, patet eodem modo poſſe <lb/>probari de partibus proportionalibus; </s> <s xml:id="echoid-s1021" xml:space="preserve">v. </s> <s xml:id="echoid-s1022" xml:space="preserve">g. </s> <s xml:id="echoid-s1023" xml:space="preserve">eodem <lb/>modo probabimus partem annuli lati ortam ex rota-<lb/>tione trapezij mixti C O Q D, æqualem eſſe ſeg-<lb/>mento com G P R M. </s> <s xml:id="echoid-s1024" xml:space="preserve">Quare patet ſolida prædi-<lb/>cta æqualia eſſe inter ſetam ſecundum totum, quam <lb/>ſecundum partes proportionales.</s> <s xml:id="echoid-s1025" xml:space="preserve"/> </p> <div xml:id="echoid-div54" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0066-01"/> </figure> </div> <pb o="55" file="0067" n="67"/> </div> <div xml:id="echoid-div56" type="section" level="1" n="38"> <head xml:id="echoid-head49" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s1026" xml:space="preserve">Licet autem præſens propoſitio probata fit per <lb/>indiuiſibilia, poteſt tamen probari etiam modo ar-<lb/>chimedeo; </s> <s xml:id="echoid-s1027" xml:space="preserve">quia facta conſtructione vt in ſchemate, <lb/>facile patebit tubum cylindricum O D N, inſcri-<lb/>ptum in annulo, æqualem eſſe cylindro in cono in-<lb/>ſcripto. </s> <s xml:id="echoid-s1028" xml:space="preserve">Si ergo diuidatur E F, bifariam, & </s> <s xml:id="echoid-s1029" xml:space="preserve">partes <lb/>bifariam, & </s> <s xml:id="echoid-s1030" xml:space="preserve">hocſemper, & </s> <s xml:id="echoid-s1031" xml:space="preserve">per puncta diuiſionum <lb/>fiant conſtructiones ſimiles factæ; </s> <s xml:id="echoid-s1032" xml:space="preserve">patebit faciliter <lb/>omnes tubos cylindricos inſcriptos in annulo, æqua-<lb/>les fore omnibus cylindris in cono inſcriptis. </s> <s xml:id="echoid-s1033" xml:space="preserve">Qua-<lb/>re cum facta hac inſcriptione, tam cylindri in cono <lb/>inſcripti, quam tubi in annulo poſſint deficere à ma-<lb/>gnitu dinibus in quibus inſcribuntur magnitudine <lb/>quacumque data minore; </s> <s xml:id="echoid-s1034" xml:space="preserve">modo archimedeo dedu-<lb/>cetur, annulum æqualem eſſe cono.</s> <s xml:id="echoid-s1035" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div57" type="section" level="1" n="39"> <head xml:id="echoid-head50" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s1036" xml:space="preserve">Ex dictis ergo in præſenti propoſit. </s> <s xml:id="echoid-s1037" xml:space="preserve">& </s> <s xml:id="echoid-s1038" xml:space="preserve">in lib. </s> <s xml:id="echoid-s1039" xml:space="preserve">4. </s> <s xml:id="echoid-s1040" xml:space="preserve">de <lb/>Infin. </s> <s xml:id="echoid-s1041" xml:space="preserve">Parab. </s> <s xml:id="echoid-s1042" xml:space="preserve">poſſumus deducere, annulum prædi-<lb/>ctum, & </s> <s xml:id="echoid-s1043" xml:space="preserve">conum G E M, eſſe quantitates proportio-<lb/>naliter annalogas tam in magnitudine, quam in gra-<lb/>uitate, tam fecundum totum, quam ſecundum par-<lb/>tes proportionales. </s> <s xml:id="echoid-s1044" xml:space="preserve">Quare cum ex dictis in ſchol-<lb/>prim. </s> <s xml:id="echoid-s1045" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1046" xml:space="preserve">8. </s> <s xml:id="echoid-s1047" xml:space="preserve">eiuſdem libri, conus, trilineum pa-<lb/>rabolicum quadraticum, & </s> <s xml:id="echoid-s1048" xml:space="preserve">exceſſus cylindri cir- <pb o="56" file="0068" n="68"/> <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> cumſcripti hemiſphærio, ſeù hemiſphæroidi ſint <lb/>quatuor magnitudines proportionaliter analogæ: </s> <s xml:id="echoid-s1049" xml:space="preserve">ſe-<lb/>quitur his etiam aſſociari pro quinta magnitudine <lb/>annulum latum prædictum. </s> <s xml:id="echoid-s1050" xml:space="preserve">Ex dictis ergo in lib cit. <lb/></s> <s xml:id="echoid-s1051" xml:space="preserve">habebimus, quod centrum grauitatis talis annuli ſic <lb/>ſecabit E F, vt pars terminata ad E, ſit ad par-<lb/>tem terminatam ad F, vt 3. </s> <s xml:id="echoid-s1052" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s1053" xml:space="preserve">Pariter ſi con-<lb/>ſiderabimus quamlibet partem eiuſdem annuli re-<lb/>ſectiplano C L, parallelo, & </s> <s xml:id="echoid-s1054" xml:space="preserve">terminatam ad circu- <pb o="57" file="0069" n="69"/> lum B E K, v.</s> <s xml:id="echoid-s1055" xml:space="preserve">g. </s> <s xml:id="echoid-s1056" xml:space="preserve">illam, quæ oritur ex rotatione tri-<lb/>linei B O Q ci ca E F; </s> <s xml:id="echoid-s1057" xml:space="preserve">agnoſcemus eius centrum <lb/>grauitatis ſecare E I, in eadem ratione. </s> <s xml:id="echoid-s1058" xml:space="preserve">Quia ta-<lb/>lis pars eſt proportionaliter an aloga cum cono P E R. <lb/></s> <s xml:id="echoid-s1059" xml:space="preserve">Cum vero etiam pars annuli orta ex rotatione trape-<lb/>zij mixti C O Q D, ſit probata proportionaliter <lb/>analoga ſegmento conico G P R M, & </s> <s xml:id="echoid-s1060" xml:space="preserve">cum talis <lb/>ſegmenti conici ſit in libro cit. </s> <s xml:id="echoid-s1061" xml:space="preserve">pluribus modis inuen-<lb/>tum centrum grauitatis; </s> <s xml:id="echoid-s1062" xml:space="preserve">ex dictis ibidem reperie nus <lb/>in quo puncto I F, ſit centrum grauitatis prædicti <lb/>ſegmenti annuli.</s> <s xml:id="echoid-s1063" xml:space="preserve"/> </p> <div xml:id="echoid-div57" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0068-01"/> </figure> </div> </div> <div xml:id="echoid-div59" type="section" level="1" n="40"> <head xml:id="echoid-head51" xml:space="preserve">SCHOLIVM III.</head> <p> <s xml:id="echoid-s1064" xml:space="preserve">Sed paradoxum Galilei, de quo locuti ſumus ſu-<lb/>pra ſchol. </s> <s xml:id="echoid-s1065" xml:space="preserve">2. </s> <s xml:id="echoid-s1066" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1067" xml:space="preserve">10. </s> <s xml:id="echoid-s1068" xml:space="preserve">poſlumus etiam deducere <lb/>ex præſenti propoſitione. </s> <s xml:id="echoid-s1069" xml:space="preserve">Nam etiam ex hac facto <lb/>concinno diſcurſu, tandem concludemus, circumfe-<lb/>rentiam B E k, extremitatem annuli, æqualem fore <lb/>E, vertici coni.</s> <s xml:id="echoid-s1070" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div60" type="section" level="1" n="41"> <head xml:id="echoid-head52" xml:space="preserve">PROPOSITIO XIX.</head> <p style="it"> <s xml:id="echoid-s1071" xml:space="preserve">In ſchem. </s> <s xml:id="echoid-s1072" xml:space="preserve">anteced. </s> <s xml:id="echoid-s1073" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1074" xml:space="preserve">annulus ſtrictus ex quadrila-<lb/>tero mixto C B E G, circa E F, eſt æqualis cylindro <lb/>D K, tam ſecundum totum, quam ſecundum partes <lb/>proportionales.</s> <s xml:id="echoid-s1075" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1076" xml:space="preserve">PAtet faciliter. </s> <s xml:id="echoid-s1077" xml:space="preserve">Cum enim in anteced. </s> <s xml:id="echoid-s1078" xml:space="preserve">propoſit. <lb/></s> <s xml:id="echoid-s1079" xml:space="preserve">oſtenſum ſit, annulum latum ex trilineo CBD, <pb o="58" file="0070" n="70"/> circa E F, æqualem eſſe cono G E M; </s> <s xml:id="echoid-s1080" xml:space="preserve">ergo com-<lb/>muni addito cylindro K D, erit ſo idum C B k L, <lb/>æquale cylindro D K, & </s> <s xml:id="echoid-s1081" xml:space="preserve">cono G E M. </s> <s xml:id="echoid-s1082" xml:space="preserve">Quò hinc <lb/>inde ablato. </s> <s xml:id="echoid-s1083" xml:space="preserve">Ergo ſolidum G C B E k L M, erit æ-<lb/>quale cylindro k D.</s> <s xml:id="echoid-s1084" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1085" xml:space="preserve">Eodem modo oſtendemus æqualitatem partium <lb/>proportionalium, v. </s> <s xml:id="echoid-s1086" xml:space="preserve">g. </s> <s xml:id="echoid-s1087" xml:space="preserve">partem annuli ortam ex rota-<lb/>tione quadrilateri mixti C O P G, æqualem eſſe <lb/>cylindro Q S. </s> <s xml:id="echoid-s1088" xml:space="preserve">Addendo enim cylindrum Q S, & </s> <s xml:id="echoid-s1089" xml:space="preserve"><lb/>auferrendo G P R M, fruſtum conicum, patebit <lb/>propoſitum.</s> <s xml:id="echoid-s1090" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div61" type="section" level="1" n="42"> <head xml:id="echoid-head53" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s1091" xml:space="preserve">Præſens propoſitio potuiſſet immediate probari <lb/>per indiuiſibilia independenter ab anteced. </s> <s xml:id="echoid-s1092" xml:space="preserve">propo-<lb/>ſit. </s> <s xml:id="echoid-s1093" xml:space="preserve">Quia facta conſtructione vt in anteced propoſit. <lb/></s> <s xml:id="echoid-s1094" xml:space="preserve">ſtatim patebit ex propoſit. </s> <s xml:id="echoid-s1095" xml:space="preserve">11. </s> <s xml:id="echoid-s1096" xml:space="preserve">2. </s> <s xml:id="echoid-s1097" xml:space="preserve">Conic. </s> <s xml:id="echoid-s1098" xml:space="preserve">& </s> <s xml:id="echoid-s1099" xml:space="preserve">rectangu-<lb/>lum O P N, æquale eſſe quadrato B E, ſeù Q I; </s> <s xml:id="echoid-s1100" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s1101" xml:space="preserve">armillam circularem O P N, æqualem pariter <lb/>fore circulo cuius radius Q I. </s> <s xml:id="echoid-s1102" xml:space="preserve">Quare facile patebit <lb/>& </s> <s xml:id="echoid-s1103" xml:space="preserve">omnes armillas ſolidi ex quadrilatero mixto <lb/>C B E G, æquales eſſe omnibus circulis cylindri k D, <lb/>& </s> <s xml:id="echoid-s1104" xml:space="preserve">ipſum annulum ex quadrilatero mixto, æqualem <lb/>eſſe cylindro k D. </s> <s xml:id="echoid-s1105" xml:space="preserve">Maluimus tamen hanc ex ante-<lb/>cedenti deducere, vt pauidis geometris non relin-<lb/>quamus vllum locum hæſitandi de certitudine præ-<lb/>ſentis propoſitionis; </s> <s xml:id="echoid-s1106" xml:space="preserve">nam adhibita præſenti conſtru-<lb/>ctione propoſitio non probatur niſi per indiuiſibi- <pb o="59" file="0071" n="71"/> <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/> lia; </s> <s xml:id="echoid-s1107" xml:space="preserve">quia in annulo ex quadrilatero mixto C B E G, <lb/>nequit fieri inſcriptio tuborum cylindricorum, quæ <lb/>patuit poſſe fieri in annulo ex trilineo mixto <lb/>C B D.</s> <s xml:id="echoid-s1108" xml:space="preserve"/> </p> <div xml:id="echoid-div61" type="float" level="2" n="1"> <figure xlink:label="fig-0071-01" xlink:href="fig-0071-01a"> <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0071-01"/> </figure> </div> </div> <div xml:id="echoid-div63" type="section" level="1" n="43"> <head xml:id="echoid-head54" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s1109" xml:space="preserve">Pater ergo conſequenter ad ſæpeſæpius repetita, <lb/>annulum præſatum G C B E k L M, & </s> <s xml:id="echoid-s1110" xml:space="preserve">cylindrum <pb o="60" file="0072" n="72"/> K D, eſſe quantitates proportionaliter analogas om-<lb/>niquaque: </s> <s xml:id="echoid-s1111" xml:space="preserve">quod etiam intelligẽdum eſtſi ſemihyper-<lb/>bola cum omnibus duplicetur. </s> <s xml:id="echoid-s1112" xml:space="preserve">Annulus ergo præ-<lb/>dictus etiam duplicatus ad partes K B, erit corpus <lb/>ſibi ſimilare, ad modum quo cylindrus K D, ſic du-<lb/>plicatus eſt corpus ſibi ſimilare. </s> <s xml:id="echoid-s1113" xml:space="preserve">Hoc eſt, quod ſicut <lb/>cylindrus ſectus planis baſibus parallelis, ſemper ſe-<lb/>catur in proportione partium axis, ſic etiam in tali <lb/>proportione ſecabitur talis annulus. </s> <s xml:id="echoid-s1114" xml:space="preserve">Sicuti ergo <lb/>centrum grauitatis cylindri, cuiuslibetque eius par-<lb/>tis contentæ inter plana baſibus parallela eſt in me-<lb/>dio axis; </s> <s xml:id="echoid-s1115" xml:space="preserve">ſic etiam centrum grauitatis talis annuli, & </s> <s xml:id="echoid-s1116" xml:space="preserve"><lb/>cuiuslibet eiuſdem ſegmenti reſecti plano C L, pa-<lb/>rallelo, erit vel in medio E F, vel in medio partis <lb/>E F, correſpondentis parti annuli, vel quæ ſit al-<lb/>titudo partis annuli. </s> <s xml:id="echoid-s1117" xml:space="preserve">Quæ omnia vtique nobis vi-<lb/>dentur admitabilia, & </s> <s xml:id="echoid-s1118" xml:space="preserve">neicimus an fortè corpus huic <lb/>ſimile in tota geometria adinueniatur, præter vni-<lb/>cum, quod antequam ad vlteriora progrediamur, <lb/>intelligimus in propoſitione ſequenti explicare.</s> <s xml:id="echoid-s1119" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div64" type="section" level="1" n="44"> <head xml:id="echoid-head55" xml:space="preserve">PROPOSITIO XX.</head> <p style="it"> <s xml:id="echoid-s1120" xml:space="preserve">Exceſſus fruſti crnici propoſit. </s> <s xml:id="echoid-s1121" xml:space="preserve">10. </s> <s xml:id="echoid-s1122" xml:space="preserve">ſupra conoides hyper-<lb/>bolicum, eſt æqualis cylindro ſuper minore baſi frusti, <lb/>& </s> <s xml:id="echoid-s1123" xml:space="preserve">circa diametrum cum ipſo: </s> <s xml:id="echoid-s1124" xml:space="preserve">& </s> <s xml:id="echoid-s1125" xml:space="preserve">hoc tam ſecundum to-<lb/>tum, quam ſecundum partes proportionales.</s> <s xml:id="echoid-s1126" xml:space="preserve"/> </p> <pb o="61" file="0073" n="73"/> <figure> <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0073-01"/> </figure> <p> <s xml:id="echoid-s1127" xml:space="preserve">ESto ergo in ſchem propoſit. </s> <s xml:id="echoid-s1128" xml:space="preserve">10. </s> <s xml:id="echoid-s1129" xml:space="preserve">fruſtum coni-<lb/>cum G I K H, conoides hyperbolicum ſit <lb/>A B C, cuius aſymptoti G F, F H, & </s> <s xml:id="echoid-s1130" xml:space="preserve">ſit cylin-<lb/>drus I M, cuius baſis I B K, minor baſis fruſti. <lb/></s> <s xml:id="echoid-s1131" xml:space="preserve">Dico exceſſum ſruſti conici G I k H, ſupra conoi-<lb/>des A B C, æqua´em eſſe cylindro I M, ram ſe-<lb/>cundum totum, quam ſecundum partes proportio-<lb/>nales. </s> <s xml:id="echoid-s1132" xml:space="preserve">De totis patet. </s> <s xml:id="echoid-s1133" xml:space="preserve">Quia cum ex cit. </s> <s xml:id="echoid-s1134" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1135" xml:space="preserve"><lb/>10. </s> <s xml:id="echoid-s1136" xml:space="preserve">exceſſus G I k H, ſupra cylindrum I M, ſit <pb o="62" file="0074" n="74"/> æqualis conoidi A B C; </s> <s xml:id="echoid-s1137" xml:space="preserve">ſi cylindrus I M, adda-<lb/>tur. </s> <s xml:id="echoid-s1138" xml:space="preserve">Ergo exceſſus cum cylindro, nempe fruſtum <lb/>G I k H, erit æquale cylindro, & </s> <s xml:id="echoid-s1139" xml:space="preserve">conoidi ſimul. <lb/></s> <s xml:id="echoid-s1140" xml:space="preserve">Ablato ergo conoide, exceſſus fruſti ſupra conoides <lb/>remanebit æqualis cylindro.</s> <s xml:id="echoid-s1141" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1142" xml:space="preserve">Non alio modo oſtendetur æqualitas partium, <lb/>proportionalium, v. </s> <s xml:id="echoid-s1143" xml:space="preserve">g. </s> <s xml:id="echoid-s1144" xml:space="preserve">exceſſum fruſti G N P H, <lb/>ſupra fruſtum conoidis A Q T C, æqualem eſſe <lb/>cylindro R M. </s> <s xml:id="echoid-s1145" xml:space="preserve">Quia ex dictis in præcitata propo-<lb/>ſit. </s> <s xml:id="echoid-s1146" xml:space="preserve">10. </s> <s xml:id="echoid-s1147" xml:space="preserve">exceſſus fruſti G N P H, ſupra cylindrum <lb/>R M, eſt æqualis ſegmento A Q T C; </s> <s xml:id="echoid-s1148" xml:space="preserve">addito ergo, <lb/>vt prius, cylindro R M, & </s> <s xml:id="echoid-s1149" xml:space="preserve">ablato ſegmento A Q T C, <lb/>intentum probabitur. </s> <s xml:id="echoid-s1150" xml:space="preserve">Quare patuit talia ſolida æ-<lb/>qualia fore tam ſecundum totum, quam ſecundum <lb/>partes.</s> <s xml:id="echoid-s1151" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div65" type="section" level="1" n="45"> <head xml:id="echoid-head56" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s1152" xml:space="preserve">Sed etiam præſens propoſitio poſſet immediate <lb/>per indiuiſibilia oſtendi. </s> <s xml:id="echoid-s1153" xml:space="preserve">Sumpto enim arbitrariè <lb/>puncto O, & </s> <s xml:id="echoid-s1154" xml:space="preserve">acto plano N O P, G H, paralle-<lb/>lo. </s> <s xml:id="echoid-s1155" xml:space="preserve">Ex propoſit. </s> <s xml:id="echoid-s1156" xml:space="preserve">10. </s> <s xml:id="echoid-s1157" xml:space="preserve">ſec. </s> <s xml:id="echoid-s1158" xml:space="preserve">conic. </s> <s xml:id="echoid-s1159" xml:space="preserve">rectangulum N Q P, <lb/>eſt æquale quadrato I B, ſeù quadrato R O. </s> <s xml:id="echoid-s1160" xml:space="preserve">Et <lb/>conſequenter armilla circularis N Q P, eſt æqua-<lb/>lis circulo R O S: </s> <s xml:id="echoid-s1161" xml:space="preserve">& </s> <s xml:id="echoid-s1162" xml:space="preserve">omnes armillæ ęqualis omni-<lb/>b s irculis; </s> <s xml:id="echoid-s1163" xml:space="preserve">& </s> <s xml:id="echoid-s1164" xml:space="preserve">exceſſus prędictus ęqualis cylindro <lb/>I M. </s> <s xml:id="echoid-s1165" xml:space="preserve">Sed hac conſtructione adhibita, demonſtratio <lb/>non reducitur ad modum Archimedeum, quia in prę-<lb/>dicto exceſſu nequeunt inſcribi tubi cylindrici.</s> <s xml:id="echoid-s1166" xml:space="preserve"/> </p> <pb o="63" file="0075" n="75"/> <figure> <image file="0075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0075-01"/> </figure> <p> <s xml:id="echoid-s1167" xml:space="preserve">Patet ergo exceſſum prędictum, & </s> <s xml:id="echoid-s1168" xml:space="preserve">cylindrum. <lb/></s> <s xml:id="echoid-s1169" xml:space="preserve">I M, eſſe quantitates proportionaliter analogas tam <lb/>ſecundum totum, quam ſecundum partes, tam in <lb/>magnitudine, quam in grauitate. </s> <s xml:id="echoid-s1170" xml:space="preserve">Inſuper patet ex-<lb/>ceſſum A G I B k H C, prędictum eſſe corpus ſibi <lb/>ſimilare vt explicatum eſt in ſchol. </s> <s xml:id="echoid-s1171" xml:space="preserve">2. </s> <s xml:id="echoid-s1172" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1173" xml:space="preserve">ant. </s> <s xml:id="echoid-s1174" xml:space="preserve"><lb/>Hoc eſt quod ſi ſecetur plano N P, quocunque, G H, <lb/>parallelo, ſemper ſecabitur in ratione partium axis <lb/>D B. </s> <s xml:id="echoid-s1175" xml:space="preserve">Item centrum grauitatis eius erit in medio <pb o="64" file="0076" n="76"/> D B; </s> <s xml:id="echoid-s1176" xml:space="preserve">ſicutietiam centrum grauitatis cuiuslibet eius <lb/>partis erit in medio partis B D, quæ erit altitudo <lb/>partis exceſſus.</s> <s xml:id="echoid-s1177" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div66" type="section" level="1" n="46"> <head xml:id="echoid-head57" xml:space="preserve">PROPOSITIO XXI.</head> <p style="it"> <s xml:id="echoid-s1178" xml:space="preserve">In ſchemate prop. </s> <s xml:id="echoid-s1179" xml:space="preserve">19. </s> <s xml:id="echoid-s1180" xml:space="preserve">cylindrus ex parallelogrammo A F, <lb/>circa E F, eſt ad ſchdum ex figur a mixta C B E F, circa <lb/>candem E F, vt quadratum E A, ad quadratum E B, <lb/>cum tertia parte rect anguli K A B.</s> <s xml:id="echoid-s1181" xml:space="preserve"/> </p> <figure> <image file="0076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0076-01"/> </figure> <pb o="65" file="0077" n="77"/> <p> <s xml:id="echoid-s1182" xml:space="preserve">QVoniam enim probatum eſt in propoſit. </s> <s xml:id="echoid-s1183" xml:space="preserve">19. </s> <s xml:id="echoid-s1184" xml:space="preserve">ſo-<lb/>lidum C B k L, æquari cylindro B S, & </s> <s xml:id="echoid-s1185" xml:space="preserve"><lb/>cono G E M; </s> <s xml:id="echoid-s1186" xml:space="preserve">ergo cylindrus A L, ad hæc ſoli-<lb/>da habebit eandem rationem. </s> <s xml:id="echoid-s1187" xml:space="preserve">At cylindrus A L, <lb/>ad cylindrum B S, & </s> <s xml:id="echoid-s1188" xml:space="preserve">ad conum G E M, eſt vt qua-<lb/>dratum E A, ad quadratum E B, cum tertia parte <lb/>rectanguli K A B. </s> <s xml:id="echoid-s1189" xml:space="preserve">Quare &</s> <s xml:id="echoid-s1190" xml:space="preserve">c.</s> <s xml:id="echoid-s1191" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1192" xml:space="preserve">Aſſumptum patebit ſic. </s> <s xml:id="echoid-s1193" xml:space="preserve">Cylindrus A L, ad cy-<lb/>lindrum B S, eſt vt quadratum A F, ad quadratum <lb/>E B. </s> <s xml:id="echoid-s1194" xml:space="preserve">Pariter idem cylindrus A L, ad conum GEM, <lb/>eſt vt quadratum C F, ſeù vt idem quadratum A E, <lb/>ad tertiam partem quadrati G F. </s> <s xml:id="echoid-s1195" xml:space="preserve">Ergo colligendo <lb/>ambo conſequentia, erit cylindrus A L, ad cylin-<lb/>drum B S, cum cono G E M, nempe ad ſolidum <lb/>C B k L, vt quadratum A E, ad quadratum E B, <lb/>cum tertia parte quadrati F G. </s> <s xml:id="echoid-s1196" xml:space="preserve">At tertia pars qua-<lb/>drati F G, eſt æqualis tertiæ parti rectanguli k A B. <lb/></s> <s xml:id="echoid-s1197" xml:space="preserve">Nam quadratum E A, diuiditur in quadratum E B, <lb/>& </s> <s xml:id="echoid-s1198" xml:space="preserve">in rectangulum k A B: </s> <s xml:id="echoid-s1199" xml:space="preserve">pariter quadratum idem <lb/>E A, ſeù F C, diuiditur in quadratum F G, & </s> <s xml:id="echoid-s1200" xml:space="preserve">in <lb/>rectangulum C G L, ſeù M C G. </s> <s xml:id="echoid-s1201" xml:space="preserve">Ergo quadra-<lb/>tum E B, cum rectangulo K A B, erit æquale qua-<lb/>drato F G, & </s> <s xml:id="echoid-s1202" xml:space="preserve">rectangulo M C G. </s> <s xml:id="echoid-s1203" xml:space="preserve">Sed ex ſec. </s> <s xml:id="echoid-s1204" xml:space="preserve">co-<lb/>nic. </s> <s xml:id="echoid-s1205" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1206" xml:space="preserve">11. </s> <s xml:id="echoid-s1207" xml:space="preserve">rectangulum M C G, eſt æquale <lb/>quadrato B E. </s> <s xml:id="echoid-s1208" xml:space="preserve">Quare reliquum rectangulum k A B, <lb/>erit æquale reliquo quadrato F G. </s> <s xml:id="echoid-s1209" xml:space="preserve">Quare etiam il-<lb/>lorum tertiæ partes erunt æquales. </s> <s xml:id="echoid-s1210" xml:space="preserve">Ergo cylindrus <lb/>A L, erit ad ſolidum C B k L, vt quadratum E A, <pb o="66" file="0078" n="78"/> ad quadratum EB, cum tertia parte rectanguli k A B. <lb/></s> <s xml:id="echoid-s1211" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1212" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1213" xml:space="preserve">His oſtenſis adinuenietur centrum grauitatis hy-<lb/>perbolæ ſic.</s> <s xml:id="echoid-s1214" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div67" type="section" level="1" n="47"> <head xml:id="echoid-head58" xml:space="preserve">PROPOSITIO XXII.</head> <p style="it"> <s xml:id="echoid-s1215" xml:space="preserve">Si hyperbolæ circumſcriptum par allelogrammum intelliga-<lb/>tur productum vſque ad ſecundam diametrum, & </s> <s xml:id="echoid-s1216" xml:space="preserve">fiat <lb/>vt quadratum compoſitæ ex axi hyperbolæ, & </s> <s xml:id="echoid-s1217" xml:space="preserve">ex di-<lb/>midia lateris tranſuerſi, ad quadratum dimidiæ lateris <lb/>tranſuerſi cum rectangulo ſub axi, & </s> <s xml:id="echoid-s1218" xml:space="preserve">ſub compoſita <lb/>ex axi, & </s> <s xml:id="echoid-s1219" xml:space="preserve">ex latere tranſuerſo, ſic compoſita ex di-<lb/>midia lateris tranſuerſi, & </s> <s xml:id="echoid-s1220" xml:space="preserve">ex axi, ad aliam: </s> <s xml:id="echoid-s1221" xml:space="preserve">item <lb/>fiat vt dimidium prædicti parallelogrammi ad exceſſum <lb/>totius parallelogrammi ſupra hyperbolam, ſic compoſita <lb/>ex axi, & </s> <s xml:id="echoid-s1222" xml:space="preserve">ex dimidia lateris tranſuerſi, ad aliam: <lb/></s> <s xml:id="echoid-s1223" xml:space="preserve">tandem fiat vt ſecunda inuenta ad primam inuentam, <lb/>ſic compoſita ex axi, & </s> <s xml:id="echoid-s1224" xml:space="preserve">ex dimidia lateris tranſuerſi <lb/>ad ſui partem abſcindendam incipiendo à ſecunda dia-<lb/>metro. </s> <s xml:id="echoid-s1225" xml:space="preserve">Erit punctum quod est alter terminus huius ab-<lb/>ſciſſæ centrum grauitatis exceſſus parallelogrammi ſupra <lb/>hyperbolam.</s> <s xml:id="echoid-s1226" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1227" xml:space="preserve">ESto hyperbola A B C, cuius axis B D; </s> <s xml:id="echoid-s1228" xml:space="preserve">latus <lb/>tranſuerſum B E; </s> <s xml:id="echoid-s1229" xml:space="preserve">centrum F; </s> <s xml:id="echoid-s1230" xml:space="preserve">ſecunda dia-<lb/>meter G H; </s> <s xml:id="echoid-s1231" xml:space="preserve">& </s> <s xml:id="echoid-s1232" xml:space="preserve">G C, ſit parallclogrammum: </s> <s xml:id="echoid-s1233" xml:space="preserve">fiat <lb/>vt quadratum F D, ad quadratum F B, cum tertia <pb o="67" file="0079" n="79"/> <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/> parte rectanguli E D B, ſic D F, ad F O: </s> <s xml:id="echoid-s1234" xml:space="preserve">item fiat <lb/>vt parallelogrammum G D, ad exceſſum parallelo-<lb/>grammi G C, ſupra hyperbolam A B C, ſic D F, <lb/>ad F L: </s> <s xml:id="echoid-s1235" xml:space="preserve">tandem fiat vt L F, ad F O, ſic D F, ad <lb/>F k. </s> <s xml:id="echoid-s1236" xml:space="preserve">Dico punctum k, eſſe centrum grauitatis fi-<lb/>guræ A G H C B.</s> <s xml:id="echoid-s1237" xml:space="preserve"/> </p> <div xml:id="echoid-div67" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0079-01"/> </figure> </div> <p> <s xml:id="echoid-s1238" xml:space="preserve">Quoniam enim ex propoſit. </s> <s xml:id="echoid-s1239" xml:space="preserve">anteced. </s> <s xml:id="echoid-s1240" xml:space="preserve">cylindrus ex <lb/>G C, circa G H, eſt ad ſolidum ex figura A G H C B, <lb/>circa eandem G H, vt quadratum F D, ad quadra-<lb/>tum F B, cum tertia parte rectanguli E D B; </s> <s xml:id="echoid-s1241" xml:space="preserve">nem- <pb o="68" file="0080" n="80"/> pe ex conſtructionē, vt D F, ad F O; </s> <s xml:id="echoid-s1242" xml:space="preserve">& </s> <s xml:id="echoid-s1243" xml:space="preserve">ratio D F, <lb/>ad F O (de foris ſumpta F L) componitur ex ratio-<lb/>ne D F, ad F L, & </s> <s xml:id="echoid-s1244" xml:space="preserve">huius ad F O. </s> <s xml:id="echoid-s1245" xml:space="preserve">Ergo etiam ra-<lb/>tio cylindri prædicti ex G C, ad ſolidum ex exceſſu <lb/>G C, ſupra hyperbolam componetur ex ijſdem ra-<lb/>tionibus. </s> <s xml:id="echoid-s1246" xml:space="preserve">At ex ſchol. </s> <s xml:id="echoid-s1247" xml:space="preserve">prim. </s> <s xml:id="echoid-s1248" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1249" xml:space="preserve">3. </s> <s xml:id="echoid-s1250" xml:space="preserve">lib. </s> <s xml:id="echoid-s1251" xml:space="preserve">3. </s> <s xml:id="echoid-s1252" xml:space="preserve">ratio <lb/>prædicti cylindri ad antedictum ſolidum componi-<lb/>tur etiam ex ratione parallelogrammi G D, ad figu-<lb/>ram A G H C B, & </s> <s xml:id="echoid-s1253" xml:space="preserve">ex ratione D F, ad interceptam <lb/>inter F, & </s> <s xml:id="echoid-s1254" xml:space="preserve">centrum grauitatis figuræ A G H C B. <lb/></s> <s xml:id="echoid-s1255" xml:space="preserve">Ergo etiam rationes D F, ad F L, & </s> <s xml:id="echoid-s1256" xml:space="preserve">F L, ad FO, <lb/>erunt æquales rationibus G D, ad A G H C B, & </s> <s xml:id="echoid-s1257" xml:space="preserve"><lb/>D F, ad prædictam interceptam. </s> <s xml:id="echoid-s1258" xml:space="preserve">Sed ex conſtru-<lb/>ctione, rationes G D, ad A G H C B, & </s> <s xml:id="echoid-s1259" xml:space="preserve">D F, ad <lb/>F L, ſunt æquales. </s> <s xml:id="echoid-s1260" xml:space="preserve">Ergo ſi hæ rationes auferantur à <lb/>prædictis, etiam reliquæ erunt æquales. </s> <s xml:id="echoid-s1261" xml:space="preserve">Ergo ratio <lb/>L F, ad F O, erit æqualis rationi D F, ad interce-<lb/>ptam prædictam. </s> <s xml:id="echoid-s1262" xml:space="preserve">Sed factum fuit ſupra vt L F, ad <lb/>F O, ſic D F, ad F k. </s> <s xml:id="echoid-s1263" xml:space="preserve">Ergo k, erit centrum gra-<lb/>uitatis figuræ A G H C B. </s> <s xml:id="echoid-s1264" xml:space="preserve">Quod erat oſtenden-<lb/>dum.</s> <s xml:id="echoid-s1265" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div69" type="section" level="1" n="48"> <head xml:id="echoid-head59" xml:space="preserve">SCHOLIVMI.</head> <p> <s xml:id="echoid-s1266" xml:space="preserve">Inuento autem centro prædicto, facile erit etiam <lb/>centrum grauitatis hyperbolæ reperire. </s> <s xml:id="echoid-s1267" xml:space="preserve">Si enim <lb/>ſupponamus F D, ſectam bifariam in O, & </s> <s xml:id="echoid-s1268" xml:space="preserve">ſuppo-<lb/>namus k, eſſe centrum grauitatis figuræ A G H C B, <lb/>ſi fiat vt A B C, ad A G H C B, ſic reciprocè k O, <pb o="69" file="0081" n="81"/> <anchor type="figure" xlink:label="fig-0081-01a" xlink:href="fig-0081-01"/> ad O L. </s> <s xml:id="echoid-s1269" xml:space="preserve">Erit ex doctrinis Archimedis, L, centrum <lb/>grauitatis hyperbolæ.</s> <s xml:id="echoid-s1270" xml:space="preserve"/> </p> <div xml:id="echoid-div69" type="float" level="2" n="1"> <figure xlink:label="fig-0081-01" xlink:href="fig-0081-01a"> <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0081-01"/> </figure> </div> <p> <s xml:id="echoid-s1271" xml:space="preserve">Sed etiam in præſenti eſt adnotandum, poſſe <lb/>colligi tria ſolita. </s> <s xml:id="echoid-s1272" xml:space="preserve">Nempe rationem ſolidorum ex <lb/>A G H C B, ſigura reuoluta & </s> <s xml:id="echoid-s1273" xml:space="preserve">circa G H, & </s> <s xml:id="echoid-s1274" xml:space="preserve">circa <lb/>A C, ad inuicem. </s> <s xml:id="echoid-s1275" xml:space="preserve">Cubationem truncorum cylindrici <lb/>recti ſuperipſa ſigura exiſtentis reſecti plano diago-<lb/>naliter tranſeunte per G H, & </s> <s xml:id="echoid-s1276" xml:space="preserve">per A C, parallelam. </s> <s xml:id="echoid-s1277" xml:space="preserve">Aſt <lb/>cubatio trunci ſiniſtri habetur ſine ſuppoſitione qua-<lb/>draturæ hyperbolæ, ſed cubatio trunci dexteri non <pb o="70" file="0082" n="82"/> habetur ſine tali quadratura; </s> <s xml:id="echoid-s1278" xml:space="preserve">ſine quanon habemus <lb/>nec etiam tertium, nempe rationem cylindri ex <lb/>G C, circa A C, ad ſolidum ex figura A G H C B, <lb/>circa eandem A C.</s> <s xml:id="echoid-s1279" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1280" xml:space="preserve">Sed hyperbolæ A B C, intellecto circumſcripto <lb/>parallelogrammo, cum hyperbolæ inuentum ſit cen-<lb/>trum grauitatis, tria ordinatia colligentur etiam in <lb/>ſolidis genitis ex hyperbola. </s> <s xml:id="echoid-s1281" xml:space="preserve">Sed hæc non colligen-<lb/>turniſi ſuppoſita ipſiu, quadratura. </s> <s xml:id="echoid-s1282" xml:space="preserve">Hac ergo ſup-<lb/>poſita habebimus rationem cylindri ex parallelo-<lb/>grammo hyperbolæ circumſcripto ad alterutrum ſo-<lb/>lidorum ex pſa reuoluta ſiue circa A C, ſiue circa <lb/>latus parallelogrammi tranſiens per B. </s> <s xml:id="echoid-s1283" xml:space="preserve">Item habebi-<lb/>mus rationem horum ſolidorum ad inuicem. </s> <s xml:id="echoid-s1284" xml:space="preserve">Ft cu-<lb/>bationem truncorum cylindrici recti ſupra ipſa exi-<lb/>ſtentis, reſectique plano conſueto modo diagonali-<lb/>ter tranſennte. </s> <s xml:id="echoid-s1285" xml:space="preserve">Ex quibus pater ſuppoſita hyperbo-<lb/>læ quadratura, nos aſſignaſſe rationem cylindri cir-<lb/>cumſcripti ſuſo hyperbolico, ad ipſum; </s> <s xml:id="echoid-s1286" xml:space="preserve">quod pari-<lb/>ter alio modo præſtitit Bonauentura Caualerius in <lb/>exercit. </s> <s xml:id="echoid-s1287" xml:space="preserve">4. </s> <s xml:id="echoid-s1288" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1289" xml:space="preserve">35.</s> <s xml:id="echoid-s1290" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div71" type="section" level="1" n="49"> <head xml:id="echoid-head60" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s1291" xml:space="preserve">Repertum eſt ergo centrum grauitatis hyperbo-<lb/>læ, ſuppoſita ipſius quadratura, quod nullus (quod <lb/>ſciamus) ante nos tentauit. </s> <s xml:id="echoid-s1292" xml:space="preserve">Sed non modo licet re-<lb/>perire hoc, ſed etiam poſſumus aſſignare centrum-<lb/>æquilibrij cuiuſcunque eius partis conſtitutæ ex ſe- <pb o="71" file="0083" n="83"/> ctione hypèrbolæ linea, vellineis diametro paralle-<lb/>lis; </s> <s xml:id="echoid-s1293" xml:space="preserve">& </s> <s xml:id="echoid-s1294" xml:space="preserve">conſequenter centrum grauitatis talis partis <lb/>duplicatæ. </s> <s xml:id="echoid-s1295" xml:space="preserve">Explicabimus hoc in vna, ex huiuſque <lb/>explicatione lector adnotabit modum in alijs exer-<lb/>cendum. </s> <s xml:id="echoid-s1296" xml:space="preserve">Intelligamus in ſequenti figura reperire <lb/>centrum grauitatis portionis T O C, reſectæ linea <lb/>T O, diametro B A, parallela. </s> <s xml:id="echoid-s1297" xml:space="preserve">Quoniam ſupia in <lb/>propoſit. </s> <s xml:id="echoid-s1298" xml:space="preserve">19. </s> <s xml:id="echoid-s1299" xml:space="preserve">probatum fuit annulum ex figura mix-<lb/>ta C O P G, æqualem fore cylindro Q S; </s> <s xml:id="echoid-s1300" xml:space="preserve">commu-<lb/>ai addito fruſto conico G P R M, totum ſolidum <lb/>C O N L, erit æquale cylindro Q S, & </s> <s xml:id="echoid-s1301" xml:space="preserve">fruſto <lb/>G P R M. </s> <s xml:id="echoid-s1302" xml:space="preserve">Cum ergo ad modum ſuperiorum poſſi-<lb/>mus reperire rationem, quam habet cylindrus T L, <lb/>ad cylindrum Q S, & </s> <s xml:id="echoid-s1303" xml:space="preserve">ad ſegmentum conicum-<lb/>G P R M, ſimul; </s> <s xml:id="echoid-s1304" xml:space="preserve">habebimus etiam rationem, quam <lb/>habet cylindrus T L, ad ſolidum C O N L. </s> <s xml:id="echoid-s1305" xml:space="preserve">Hac <lb/>habita, ſi ex ipſa ſubtrahamus rationem, quam <lb/>habet dimidium I C, ſuppoſitam, ad figu-<lb/>ram C O I F; </s> <s xml:id="echoid-s1306" xml:space="preserve">habebimus rationem, quam habet <lb/>T I, ad interceptam inter I, & </s> <s xml:id="echoid-s1307" xml:space="preserve">centrum æquilibrij <lb/>figuræ C O I F, in I T. </s> <s xml:id="echoid-s1308" xml:space="preserve">Et conſequenter facile re-<lb/>periemus centrum æquilibrij talis figuræ. </s> <s xml:id="echoid-s1309" xml:space="preserve">Hoc in-<lb/>uento reperietur etiam centrum ęquilibrij portionis <lb/>hyperbolę T O C, in T O; </s> <s xml:id="echoid-s1310" xml:space="preserve">& </s> <s xml:id="echoid-s1311" xml:space="preserve">conſequenter cen-<lb/>trum grauitatis duplicatę T O C, ad partes T O. <lb/></s> <s xml:id="echoid-s1312" xml:space="preserve">Ex quibus poſtea reliqua ſolita deduci, colligeren-<lb/>tur. </s> <s xml:id="echoid-s1313" xml:space="preserve">Hęcergo, & </s> <s xml:id="echoid-s1314" xml:space="preserve">ſimilia liceret reperire. </s> <s xml:id="echoid-s1315" xml:space="preserve">Ex qui-<lb/>bus paterent ea omnia, quę oſtendit Caualerius in <lb/>loc. </s> <s xml:id="echoid-s1316" xml:space="preserve">cit. </s> <s xml:id="echoid-s1317" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1318" xml:space="preserve">36. </s> <s xml:id="echoid-s1319" xml:space="preserve">& </s> <s xml:id="echoid-s1320" xml:space="preserve">multo plura. </s> <s xml:id="echoid-s1321" xml:space="preserve">Sed quia hęc <pb o="72" file="0084" n="84"/> <anchor type="figure" xlink:label="fig-0084-01a" xlink:href="fig-0084-01"/> non reperiuntur niſi ex ſuppoſita quadratura, ideo <lb/>reliquuntur. </s> <s xml:id="echoid-s1322" xml:space="preserve">Sufficit enim nobis lectori indicare. <lb/></s> <s xml:id="echoid-s1323" xml:space="preserve">hęc nequaquam ignorari à nobis. </s> <s xml:id="echoid-s1324" xml:space="preserve">Sicuti ſufficiet ip-<lb/>ſi indicare nos poſſe habere centra grauitatis om-<lb/>nium cylindricorum exiſtentium ſuper hyperbola, & </s> <s xml:id="echoid-s1325" xml:space="preserve"><lb/>ſuper omnibus ipſius partibus, quarum inuenitur <lb/>centrum grauitatis. </s> <s xml:id="echoid-s1326" xml:space="preserve">Erit enim in medio lineæ iun-<lb/>gentis centra grauitatis oppoſitarum baſium. </s> <s xml:id="echoid-s1327" xml:space="preserve">Reli- <pb o="73" file="0085" n="85"/> ctis ergo his, tranſeamus ad quadrandam parabolam <lb/>duobus nouis modis.</s> <s xml:id="echoid-s1328" xml:space="preserve"/> </p> <div xml:id="echoid-div71" 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/PVNDER1Y/figures/0084-01"/> </figure> </div> </div> <div xml:id="echoid-div73" type="section" level="1" n="50"> <head xml:id="echoid-head61" xml:space="preserve">PROPOSITIO XXIII.</head> <p style="it"> <s xml:id="echoid-s1329" xml:space="preserve">Si ſemihyperbola cum ſibi circumſ ripto parallelogrammo ro-<lb/>tetur circa ſecundan. </s> <s xml:id="echoid-s1330" xml:space="preserve">diametrum. </s> <s xml:id="echoid-s1331" xml:space="preserve">Tubus cylindruus <lb/>ex parallelogrammo, erit ſeſquialter annuli lati ex ſe-<lb/>mibyperbola.</s> <s xml:id="echoid-s1332" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1333" xml:space="preserve">SEmihyperbola A B C, cum ſibi circumſcripto <lb/>parallogrammo A D, rotetur circa E F, ſe-<lb/>cundam dametrum. </s> <s xml:id="echoid-s1334" xml:space="preserve">Dico tubum cylindricum. <lb/></s> <s xml:id="echoid-s1335" xml:space="preserve">A D H, eſſe ſeſquialterum annuli lati ex ſemihy-<lb/>perbola A B C, circa E F, reuoluta. </s> <s xml:id="echoid-s1336" xml:space="preserve">Quoniam <lb/>tubus C B S H, eſt ad cylindrum A L, vt rectan-<lb/>gulum H B A, ad quadratum E A; </s> <s xml:id="echoid-s1337" xml:space="preserve">nempe vt re-<lb/>ctangulum k A B, ad idem quadratum E A; </s> <s xml:id="echoid-s1338" xml:space="preserve">& </s> <s xml:id="echoid-s1339" xml:space="preserve">cy-<lb/>lindrus A L, probatus eſt eſſe in propoſit. </s> <s xml:id="echoid-s1340" xml:space="preserve">21. </s> <s xml:id="echoid-s1341" xml:space="preserve">ad ſo-<lb/>lidum C B k L, vt quadratum E A, ad quadratum <lb/>E B, cum tertia parte rectanguli K A B; </s> <s xml:id="echoid-s1342" xml:space="preserve">vnde per <lb/>conuerſionem rationis, eſt idem cylindrus A L, ad <lb/>annulum ex ſemihyperbola A B C, circa E F, vt <lb/>idem quadratum E A, ad exceſſum ipſius ſupra <lb/>quadratum E B, & </s> <s xml:id="echoid-s1343" xml:space="preserve">ſupra tertiam partem rectanguli <lb/>k A B; </s> <s xml:id="echoid-s1344" xml:space="preserve">ergo ex æquali, erit tubus cylindricus A D k L, <lb/>ad talem annulum latum, vt rect angulum A B H, ad <lb/>prædictum exceſſum. </s> <s xml:id="echoid-s1345" xml:space="preserve">Sed quadratum E A, cum ſit <lb/>æquale quadrato E B, & </s> <s xml:id="echoid-s1346" xml:space="preserve">rectangulo k A B, excedit <pb o="74" file="0086" n="86"/> illa plana duobus tertijs rectanguli k A B. </s> <s xml:id="echoid-s1347" xml:space="preserve">Ergo tu-<lb/>bus cylindricus A D K L, erit ad prædictum annu-<lb/>lum, vt rectangulum K A B, ad duotertia eiuſdem <lb/>rectanguli; </s> <s xml:id="echoid-s1348" xml:space="preserve">nempe in ratione ſeſquialtera. </s> <s xml:id="echoid-s1349" xml:space="preserve">Quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s1350" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div74" type="section" level="1" n="51"> <head xml:id="echoid-head62" xml:space="preserve">PROPOSITIO XXIV.</head> <p style="it"> <s xml:id="echoid-s1351" xml:space="preserve">Si recta linea A B, ſecetur in C, bifariam, & </s> <s xml:id="echoid-s1352" xml:space="preserve">in D, <lb/>E, æque remotè à C, eodemque modo in F, G. </s> <s xml:id="echoid-s1353" xml:space="preserve">Re-<lb/>ctangulum A G B, erit exceſſus rectanguli A E B, ſu-<lb/>pra rectangulum F E G.</s> <s xml:id="echoid-s1354" xml:space="preserve"/> </p> <figure> <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0086-01"/> </figure> <p> <s xml:id="echoid-s1355" xml:space="preserve">NAm rectangulum A E B, diuiditur in rectan-<lb/>gulum A E G, & </s> <s xml:id="echoid-s1356" xml:space="preserve">in rectangulum A F, G B. <lb/></s> <s xml:id="echoid-s1357" xml:space="preserve">Pariter rectangulum A E G, diuiditur in rectangu-<lb/>lum F E G, & </s> <s xml:id="echoid-s1358" xml:space="preserve">in rectangulum A F, E G, ſeù B G F, <lb/>quia A F, @xhypotheſi, @ſt æqualis G B. </s> <s xml:id="echoid-s1359" xml:space="preserve">Ergo ex-<lb/>ceſſus rectanguli A E B, ſupra rectangulum F E G, <lb/>eſt rectangulum A E, G B, cum rectangulo E G B; </s> <s xml:id="echoid-s1360" xml:space="preserve"><lb/>quæ duo rectangula ſunt æqualia rectangulo A G B. </s> <s xml:id="echoid-s1361" xml:space="preserve"><lb/>Quare patet propoſitum.</s> <s xml:id="echoid-s1362" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div75" type="section" level="1" n="52"> <head xml:id="echoid-head63" xml:space="preserve">PROPOSITIO XXV.</head> <p style="it"> <s xml:id="echoid-s1363" xml:space="preserve">Si in oppoſitis ſection bus, quæ hyperb læ appellantur du-<lb/>cantur lineæ lateri tranſuerſo parallelæ, occurrentes <pb o="75" file="0087" n="87"/> æqualibus ad diametros applicatis in ambabus hyper-<lb/>bolis. </s> <s xml:id="echoid-s1364" xml:space="preserve">Rectangula ſub partibus ipſarum reſectarum ab <lb/>eadem curua hyperbolæ erunt ad inuicem, vt rectan-<lb/>gula ſub partibus ordinatim applicatæ ab ipſis ſectæ.</s> <s xml:id="echoid-s1365" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1366" xml:space="preserve">SInt oppoſitæ ſe-<lb/> <anchor type="figure" xlink:label="fig-0087-01a" xlink:href="fig-0087-01"/> ctiones hyper-<lb/>bolæ A B C, D E F, <lb/>quarum latus tranſ-<lb/>uerſum E B, & </s> <s xml:id="echoid-s1367" xml:space="preserve">D F, <lb/>A C, ſint æquales or-<lb/>dinatim applicatæ ad <lb/>æquales diametros <lb/>K E, B H, & </s> <s xml:id="echoid-s1368" xml:space="preserve">ſint du-<lb/>ctæ L O, P S, paral-<lb/>lelæ k H. </s> <s xml:id="echoid-s1369" xml:space="preserve">Dico re-<lb/>ctangulum L N O, eſ-<lb/>ſe ad rectangulum. <lb/></s> <s xml:id="echoid-s1370" xml:space="preserve">P R S, vt rectangu-<lb/>lum A O C, ad re-<lb/>ctangulum A S C. </s> <s xml:id="echoid-s1371" xml:space="preserve"><lb/>Applicentur à punctis <lb/>N, R, N T, R I, ordi-<lb/>nation ad diametrum; </s> <s xml:id="echoid-s1372" xml:space="preserve"><lb/>item à punctis M, Q <lb/>ordinatim applicen-<lb/>tur ad k E, M V, Q X. </s> <s xml:id="echoid-s1373" xml:space="preserve"><lb/>Q oniam enim ex <lb/>prim. </s> <s xml:id="echoid-s1374" xml:space="preserve">conic. </s> <s xml:id="echoid-s1375" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1376" xml:space="preserve">21. </s> <s xml:id="echoid-s1377" xml:space="preserve">rectangulum E H B, ad <pb o="76" file="0088" n="88"/> rectangulum E T B, eſt vt quadratum A H, ad qua-<lb/>dratum N T, ſeù O H; </s> <s xml:id="echoid-s1378" xml:space="preserve">& </s> <s xml:id="echoid-s1379" xml:space="preserve">rectangulis E H B, E T B, <lb/>ſunt æqualia rectangula K B H, V B T, quia k E, <lb/>B H, & </s> <s xml:id="echoid-s1380" xml:space="preserve">V E, B T, ſunt æquales; </s> <s xml:id="echoid-s1381" xml:space="preserve">ergo erit vt rectan-<lb/>gulum K B H, ad rectangulum V B T, ſic quadra-<lb/>tum A H, ad quadratum H O. </s> <s xml:id="echoid-s1382" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1383" xml:space="preserve">per con-<lb/>uerſionem rationis, erit rectangulum K B H, ad ex-<lb/>ceſſum ipſius ſupra rectangulum V B T; </s> <s xml:id="echoid-s1384" xml:space="preserve">nempe ex <lb/>propoſit. </s> <s xml:id="echoid-s1385" xml:space="preserve">anteced. </s> <s xml:id="echoid-s1386" xml:space="preserve">ad rectangulum k T H, ſeù ad ei <lb/>æquale L N O, vt quadratum A H, ad rectangu-<lb/>lum A O C. </s> <s xml:id="echoid-s1387" xml:space="preserve">Et conuertendo, erit rectangulum. <lb/></s> <s xml:id="echoid-s1388" xml:space="preserve">A O C, ad quadratum A H, vt rectangulum L N O, <lb/>ad rectangulum K B H. </s> <s xml:id="echoid-s1389" xml:space="preserve">Eodem modo oſtendetur <lb/>eſſe rectangulum K B H, ad rectangulum P R S, <lb/>vt quadratum A H, ſeù H C, ad rectangulum. </s> <s xml:id="echoid-s1390" xml:space="preserve"><lb/>A S C. </s> <s xml:id="echoid-s1391" xml:space="preserve">Quare ex æquali, erit rectangulum L N O, <lb/>ad rectangulum P R S, vt rectangulum A O C, ad <lb/>rectangulum A S C. </s> <s xml:id="echoid-s1392" xml:space="preserve">Quod &</s> <s xml:id="echoid-s1393" xml:space="preserve">c.</s> <s xml:id="echoid-s1394" xml:space="preserve"/> </p> <div xml:id="echoid-div75" type="float" level="2" n="1"> <figure xlink:label="fig-0087-01" xlink:href="fig-0087-01a"> <image file="0087-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0087-01"/> </figure> </div> </div> <div xml:id="echoid-div77" type="section" level="1" n="53"> <head xml:id="echoid-head64" xml:space="preserve">PROPOSITIO XXVI.</head> <p style="it"> <s xml:id="echoid-s1395" xml:space="preserve">Parallelogrammum circum ſcriptum parabolæ quadraticæ, eſt <lb/>ad ipſam, vt tubus @ylindricus ex gyratione parallelo-<lb/>gramm@ circurnſcripti hyperbolæ circa ſecundam coniuga-<lb/>tam diametrum, ad annulum latum ex reuolutione hyper-<lb/>bo´æ circa eandem diametrum; </s> <s xml:id="echoid-s1396" xml:space="preserve">& </s> <s xml:id="echoid-s1397" xml:space="preserve">hoc tam ſecundum to-<lb/>tum, quam ſecundum partes proportionales; </s> <s xml:id="echoid-s1398" xml:space="preserve">dummodo ba-<lb/>ſes pa abolæ, & </s> <s xml:id="echoid-s1399" xml:space="preserve">hyperbolæ genitricis annuli proportiona-<lb/>liter ſecentur.</s> <s xml:id="echoid-s1400" xml:space="preserve"/> </p> <pb o="77" file="0089" n="89"/> <p> <s xml:id="echoid-s1401" xml:space="preserve">ESto hyperbola A B C, cuius axis B N, diame-<lb/>ter tranſuerſa E B, centrum L, ſecunda dia-<lb/>meter k M, parallelogrammum ei circumſcriptum <lb/>ſit G C: </s> <s xml:id="echoid-s1402" xml:space="preserve">pariter ſit parabola quadratica A O C, <lb/>cum ſibi circumſcripto parallelogrammo P C. </s> <s xml:id="echoid-s1403" xml:space="preserve">Di-<lb/>co tubum cylindricum ex reuolutione C G, circa <lb/>k M, eſſe ad annulum latum ex reuolutione A B C, <lb/>circa eandem K M, vt parallelogrammum P C, ad <lb/>A O C, parabolam. </s> <s xml:id="echoid-s1404" xml:space="preserve">In A C, communi baſi para-<lb/>bolæ, & </s> <s xml:id="echoid-s1405" xml:space="preserve">hyperbolæ accipiatur arbitrariè punctum I, <lb/>per quod agatur F I T, parallela O E, ſecans om-<lb/>nia vt in ſchemate. </s> <s xml:id="echoid-s1406" xml:space="preserve">Quoniam ex propoſit. </s> <s xml:id="echoid-s1407" xml:space="preserve">anteced. <lb/></s> <s xml:id="echoid-s1408" xml:space="preserve">rectangulum A N C, e@t ad rectangulum A I C, vt <lb/>rectangulum V B N, ad rectangulum T H I; </s> <s xml:id="echoid-s1409" xml:space="preserve">& </s> <s xml:id="echoid-s1410" xml:space="preserve">vt <lb/>rectangulum V B N, ad rectangulum T H I, ſic <lb/>armilla circularis ex B N, reuoluta circa K M, ad <lb/>armillam circularem ex H I, reuoluta circa eandem <lb/>K M; </s> <s xml:id="echoid-s1411" xml:space="preserve">ergo vt rectangulum A N C, ad rectangulum <lb/>A I C, ſic armilla circula is V B N, ſeù T S I, ad ar-<lb/>millam circularem T H I. </s> <s xml:id="echoid-s1412" xml:space="preserve">Sed vt rectangulum. </s> <s xml:id="echoid-s1413" xml:space="preserve"><lb/>A N C, ad rectangulum A I C, ſic ex ſchol. </s> <s xml:id="echoid-s1414" xml:space="preserve">propo-<lb/>ſitionis 22. </s> <s xml:id="echoid-s1415" xml:space="preserve">libri primi N O, ſeù F I, ad I R. </s> <s xml:id="echoid-s1416" xml:space="preserve"><lb/>Ergo vt armilla circularis T S I, ad armillam circu-<lb/>larem T H I, ſic F I, ad I R. </s> <s xml:id="echoid-s1417" xml:space="preserve">Sed punctum I, ſum-<lb/>ptum fuit vt cunque. </s> <s xml:id="echoid-s1418" xml:space="preserve">Ergo vt omnes armillæ circula-<lb/>res parallelæ armillæ V B N, ex parallelogrammo <lb/>G C, re@oluto circa k M, ad omnes armillas circu-<lb/>lares parallelas eidem V B N, ex hype bola A B C, <lb/>reuoluta circa eandem k M, ſic omnes lineæ paralle- <pb o="78" file="0090" n="90"/> <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> logrammi P C, parallelæ N O, ad omnes lineas pa-<lb/>rabolæ A O C, parallelas eidem O N. </s> <s xml:id="echoid-s1419" xml:space="preserve">Nempe vt <pb o="79" file="0091" n="91"/> tubus cylindricus ad annulum ex hyperbola, ſic pa-<lb/>rallelogrammum P C, ad parabolam A O C.</s> <s xml:id="echoid-s1420" xml:space="preserve"/> </p> <div xml:id="echoid-div77" 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/PVNDER1Y/figures/0090-01"/> </figure> </div> <p> <s xml:id="echoid-s1421" xml:space="preserve">Quod autem probatum fuitdetotis, patet eodem <lb/>modo probari poſſe de partibus proportionalibus; <lb/></s> <s xml:id="echoid-s1422" xml:space="preserve">nimirum eodem modo poteſt probari eſſe v. </s> <s xml:id="echoid-s1423" xml:space="preserve">g. </s> <s xml:id="echoid-s1424" xml:space="preserve">tu-<lb/>bum cylindricum ex parallelogrammo I B, circa. </s> <s xml:id="echoid-s1425" xml:space="preserve"><lb/>K M, ad partem annuli ex ſegmento hyperbolæ <lb/>I H B N, circa eandem K M, vt parallelogrammum <lb/>F N, ad ſegmentum parabolæ I R O N. </s> <s xml:id="echoid-s1426" xml:space="preserve">Quare pa-<lb/>tet propoſitum in omnibus, & </s> <s xml:id="echoid-s1427" xml:space="preserve">peromnia.</s> <s xml:id="echoid-s1428" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div79" type="section" level="1" n="54"> <head xml:id="echoid-head65" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s1429" xml:space="preserve">Præſens propoſitio, quæ probata fuit perindiuiſi-<lb/>bilium methodum breuiorem, probari quoque po-<lb/>teſt per methodum antiquam prolixiorem. </s> <s xml:id="echoid-s1430" xml:space="preserve">Nam <lb/>cum probatum ſit eſſe armillam circularem T S I. <lb/></s> <s xml:id="echoid-s1431" xml:space="preserve">ad armillam circularem T H I, vt F I, ad I R; </s> <s xml:id="echoid-s1432" xml:space="preserve">& </s> <s xml:id="echoid-s1433" xml:space="preserve">cum <lb/>ſit armilla circularis T S I, ad armillam circularem <lb/>T H, ſictubus cylindricus ex parallelogrammo S N, <lb/>circa K M, ad tubum cylindricum ex parallelogram-<lb/>mo H N, circa eandem K M, quitubus eſt inſcriptus <lb/>in annulo ex hyperbola; </s> <s xml:id="echoid-s1434" xml:space="preserve">& </s> <s xml:id="echoid-s1435" xml:space="preserve">cum pariter ſit vt F I, ad <lb/>I R, ſic parallelogrammum F N, ad parallelogram-<lb/>mum R N, inſcriptum in parabola: </s> <s xml:id="echoid-s1436" xml:space="preserve">ſequitur vt tu-<lb/>bus ex parallelogrammo S N, ad tubum ex paralle-<lb/>logrammo H N, ſic eſſe parallelogrammum F N, <lb/>ad parallelogrammum R N. </s> <s xml:id="echoid-s1437" xml:space="preserve">Quare ſi A N, v. </s> <s xml:id="echoid-s1438" xml:space="preserve">g. </s> <s xml:id="echoid-s1439" xml:space="preserve"><lb/>b@ſſ@caretur, & </s> <s xml:id="echoid-s1440" xml:space="preserve">hocidem fieret de eiuſdem partibus, <pb o="80" file="0092" n="92"/> <anchor type="figure" xlink:label="fig-0092-01a" xlink:href="fig-0092-01"/> & </s> <s xml:id="echoid-s1441" xml:space="preserve">in hyperbola, & </s> <s xml:id="echoid-s1442" xml:space="preserve">parabola inſcriberentur paralle-<lb/>logramma; </s> <s xml:id="echoid-s1443" xml:space="preserve">eodem modo probaremus partes tubi <pb o="81" file="0093" n="93"/> cylindrici ex G C, eſſe ad omnes tubos ex paralle-<lb/>logrammis inſcriptis in hyperbola, qui tubi inſcri-<lb/>buntur in annulo ex hyperbola, vt partes parallelo-<lb/>grammi P C, ad omnia parallelogramma inſcripta <lb/>in parabola. </s> <s xml:id="echoid-s1444" xml:space="preserve">Cumque, tubi inſcripti in annulo ex <lb/>hyperbola, ſicuti parallelogramma inſcripta in pa-<lb/>rabola, per continuatam talem biſſectionem poſſint <lb/>tandem deficere à magnitudinibus in quibus inſcri-<lb/>buntur, defectu, quacunque data magnitudine mi-<lb/>nori: </s> <s xml:id="echoid-s1445" xml:space="preserve">ſequitur tandem modo archimedeo per dedu-<lb/>ctionem ad impoſſibile poſſe concludi, tubum cy-<lb/>lindricum ex parallelogrammo eſſe ad annulum la-<lb/>tum ex hyperbola, vt parallelogrammum ad para-<lb/>bolam.</s> <s xml:id="echoid-s1446" xml:space="preserve"/> </p> <div xml:id="echoid-div79" type="float" level="2" n="1"> <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a"> <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0092-01"/> </figure> </div> <p> <s xml:id="echoid-s1447" xml:space="preserve">Patet ergo ex dictis haberi nouo modo parabo-<lb/>læ quadraticæ quadraturam; </s> <s xml:id="echoid-s1448" xml:space="preserve">nimirum parallelo-<lb/>grammum ei circumſcriptum, eſſe ipſius ſeſquial-<lb/>terum. </s> <s xml:id="echoid-s1449" xml:space="preserve">Probatum fuit enim in anteced. </s> <s xml:id="echoid-s1450" xml:space="preserve">propoſit. <lb/></s> <s xml:id="echoid-s1451" xml:space="preserve">tubum cylindricum ex parallelogrammo G C, cir-<lb/>ca k M, eſſe ſeſquialterum annuli lati ex hyperbola <lb/>circa eandem k M. </s> <s xml:id="echoid-s1452" xml:space="preserve">Sed infra adhibendo aliud ſoli-<lb/>dum hyperbolicum, parabolam alio nouo modo <lb/>quadrabimus; </s> <s xml:id="echoid-s1453" xml:space="preserve">nunc ſuggerendæ ſunt lectori quam-<lb/>plurimæ nouæ notitiæ geometricæ, quæ ex hac pro-<lb/>poſitione, & </s> <s xml:id="echoid-s1454" xml:space="preserve">ex dictis in lib. </s> <s xml:id="echoid-s1455" xml:space="preserve">de Infin. </s> <s xml:id="echoid-s1456" xml:space="preserve">Par. </s> <s xml:id="echoid-s1457" xml:space="preserve">deducuntur.</s> <s xml:id="echoid-s1458" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div81" type="section" level="1" n="55"> <head xml:id="echoid-head66" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s1459" xml:space="preserve">Dedu@itur ergo ex dictis, & </s> <s xml:id="echoid-s1460" xml:space="preserve">ad modum ſuperio- <pb o="82" file="0094" n="94"/> <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> rum, parabolam A O C, & </s> <s xml:id="echoid-s1461" xml:space="preserve">annulum latum prædi-<lb/>ctum ex hyperbola A B C, eſſe quantitates propor- <pb o="83" file="0095" n="95"/> tionaliter analogas tam in magnitudine, quam in <lb/>grauitate; </s> <s xml:id="echoid-s1462" xml:space="preserve">tam ſecundum totum, quam ſecundum <lb/>partes proportionales. </s> <s xml:id="echoid-s1463" xml:space="preserve">Quot ergo nouæ notitiæ de-<lb/>ducantur ex hac doctrina tam circa magnitudinem, <lb/>quam circa grauitatem talis annulilati, ex noſtro ope-<lb/>re cit. </s> <s xml:id="echoid-s1464" xml:space="preserve">vniſquiſque poteſt agnoſcere.</s> <s xml:id="echoid-s1465" xml:space="preserve"/> </p> <div xml:id="echoid-div81" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0094-01"/> </figure> </div> <p> <s xml:id="echoid-s1466" xml:space="preserve">Ex propoſit. </s> <s xml:id="echoid-s1467" xml:space="preserve">enim 9, lib. </s> <s xml:id="echoid-s1468" xml:space="preserve">prim. </s> <s xml:id="echoid-s1469" xml:space="preserve">agnoſcet quænam <lb/>ſit ratio, quam habet tubus cylindricus ex G I, ad <lb/>portionem annuli lati ex portione minori hyperbo-<lb/>læ A H I; </s> <s xml:id="echoid-s1470" xml:space="preserve">nempe eſſe ad ipſum vt tres A N, ad ex-<lb/>ceſſum ipſarum ſupra A N, N I, & </s> <s xml:id="echoid-s1471" xml:space="preserve">harum tertiam <lb/>minorem proportionalem. </s> <s xml:id="echoid-s1472" xml:space="preserve">Vel ſubtriplandotermi-<lb/>nos, eſſe vt A N, ad ſubſeſquialteram A I, cum <lb/>tertia parte exceſſus I N, ſupra illam tertiam pro-<lb/>portionalem.</s> <s xml:id="echoid-s1473" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1474" xml:space="preserve">Ex ſchol prim. </s> <s xml:id="echoid-s1475" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1476" xml:space="preserve">10. </s> <s xml:id="echoid-s1477" xml:space="preserve">agnoſcet, tubum cy-<lb/>lindricum ex parallelogrammo S N, eſſe ad portio-<lb/>nem annuli ex ſegmento hyperbolæ I H B N, vt tri-<lb/>pla A N, ad duplam A N, vna cum exceſſu ipſius <lb/>ſupra prædictam tertiam proportionalem. </s> <s xml:id="echoid-s1478" xml:space="preserve">Et ſub-<lb/>triplando terminos, eſſe vt A N, ad A I, cum duo-<lb/>bus tertijs I N, & </s> <s xml:id="echoid-s1479" xml:space="preserve">cum tertia parte exceſſus I N, <lb/>ſupraillam tertiam proportionalem. </s> <s xml:id="echoid-s1480" xml:space="preserve">Imo ex ſchol. <lb/></s> <s xml:id="echoid-s1481" xml:space="preserve">3. </s> <s xml:id="echoid-s1482" xml:space="preserve">cit. </s> <s xml:id="echoid-s1483" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1484" xml:space="preserve">agnoſcet, eſſe eundem tubum cylin-<lb/>dricum ad eandem portionem annuli, vt triplum <lb/>recta gulum T S I, ad duplum rectangulum T S I, <lb/>cum rectangulo T H I. </s> <s xml:id="echoid-s1485" xml:space="preserve">Et ſubtriplando terminos, <lb/>vt rectangulum T S I, ad ſubſeſquialterum ipſius, <lb/>cum tertia parte rectanguli T H I.</s> <s xml:id="echoid-s1486" xml:space="preserve"/> </p> <pb o="84" file="0096" n="96"/> <figure> <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0096-01"/> </figure> <p> <s xml:id="echoid-s1487" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1488" xml:space="preserve">prim. </s> <s xml:id="echoid-s1489" xml:space="preserve">propofit. </s> <s xml:id="echoid-s1490" xml:space="preserve">12. </s> <s xml:id="echoid-s1491" xml:space="preserve">agnoſcet rationem <lb/>tubi cylindrici ex parallelogrammo S Q, ad ſeg- <pb o="85" file="0097" n="97"/> mentum annuli ex ſegmento intermedio ſemihy-<lb/>perbolæ Q X H I.</s> <s xml:id="echoid-s1492" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1493" xml:space="preserve">Ex ſchol prim. </s> <s xml:id="echoid-s1494" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1495" xml:space="preserve">13. </s> <s xml:id="echoid-s1496" xml:space="preserve">agnoſcet rationem <lb/>tubi ex parallelogrammo S C, ad portionem annuli <lb/>ex portione maiori hyperbolæ I H B C.</s> <s xml:id="echoid-s1497" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1498" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1499" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1500" xml:space="preserve">14. </s> <s xml:id="echoid-s1501" xml:space="preserve">agnoſcet rationem, quam <lb/>habet tubus cylindricus ex parallelogrammo S Y, <lb/>ad ſegmentum annuli ex ſegmento intermedio <lb/>I H B Z Y, intercipiente axim B N.</s> <s xml:id="echoid-s1502" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1503" xml:space="preserve">Sed portioni minori hyperbolæ A H I, intellecto <lb/>circumſcripto parallelogrammo H A, agnoſcet ex <lb/>propoſit. </s> <s xml:id="echoid-s1504" xml:space="preserve">15. </s> <s xml:id="echoid-s1505" xml:space="preserve">tubum cylindricum ex parallelogram-<lb/>mo H A, eſſe ad portionem annuli ex portione <lb/>A H I, vt tripla A N, cum tripla N I, ad duplam <lb/>A N, cum vnica N I. </s> <s xml:id="echoid-s1506" xml:space="preserve">Imo ex ſchol. </s> <s xml:id="echoid-s1507" xml:space="preserve">eiuſdem pro-<lb/>poſit. </s> <s xml:id="echoid-s1508" xml:space="preserve">agnoſcet, tubum prædictum eſſe ad prædictam <lb/>annuli portionem, vt I C ad dimidiam I C, cum <lb/>ſexta parte I A.</s> <s xml:id="echoid-s1509" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1510" xml:space="preserve">Ex ſcholio propoſit. </s> <s xml:id="echoid-s1511" xml:space="preserve">17. </s> <s xml:id="echoid-s1512" xml:space="preserve">agnoſcet rationem tubi <lb/>cylindrici ex parallelogrammo H C, ad portionem <lb/>annuli ex portione maiori I H B C. </s> <s xml:id="echoid-s1513" xml:space="preserve">Ex eodem ſchol. <lb/></s> <s xml:id="echoid-s1514" xml:space="preserve">etiam agnoſcet talem rationem eſſe, vt eſt A I, ad <lb/>dimidiam A I, cum ſexta parte I C. </s> <s xml:id="echoid-s1515" xml:space="preserve">Quare agno-<lb/>ſcet vniuerſaliter, quod tubus cylindricus ex altero <lb/>parallelogrammorum H A, H C, ad portionem an-<lb/>nuli ſibi correſpondentem eſſe, vt baſis reliquæ por-<lb/>tionis hyperbolæ, ad ſui dimidiam, cum ſexta parte <lb/>baſis portionis reuolutæ.</s> <s xml:id="echoid-s1516" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1517" xml:space="preserve">Ex propoſit, 18. </s> <s xml:id="echoid-s1518" xml:space="preserve">aguoſect rationem tubi ex paral- <pb o="86" file="0098" n="98"/> lelogrammo H Q, circumſcripto ſegmento inter-<lb/>medio Q X H I, ad ſegmentum annuli ex tali ſeg-<lb/>mento intermedio.</s> <s xml:id="echoid-s1519" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1520" xml:space="preserve">Tandem ex ſchol. </s> <s xml:id="echoid-s1521" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1522" xml:space="preserve">20. </s> <s xml:id="echoid-s1523" xml:space="preserve">agnoſcet rationem <lb/>ſegmenti annuli ex ſegmento I H B N, ad portio-<lb/>nem annuli ex portione I A H. </s> <s xml:id="echoid-s1524" xml:space="preserve">Qua agnita, non <lb/>ignorabit rationem portionis annuli ex portione <lb/>I H B C, ad prædictam portionem annuli ex por-<lb/>tione A H I.</s> <s xml:id="echoid-s1525" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div83" type="section" level="1" n="56"> <head xml:id="echoid-head67" xml:space="preserve">SCHOLIVM III.</head> <p> <s xml:id="echoid-s1526" xml:space="preserve">Pariter, cum vt diximus, prædictus annulus latus <lb/>ex hyperbola ſit quantitas proportionaliter analoga <lb/>etiam in grauitate cum parabola quadratica; </s> <s xml:id="echoid-s1527" xml:space="preserve">ex lib. <lb/></s> <s xml:id="echoid-s1528" xml:space="preserve">3. </s> <s xml:id="echoid-s1529" xml:space="preserve">de In fin. </s> <s xml:id="echoid-s1530" xml:space="preserve">Parab agnoſcet lector centrum grauita-<lb/>tis quamplurium ſegmentorum prædicti annuli lati.</s> <s xml:id="echoid-s1531" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1532" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1533" xml:space="preserve">ergo 2. </s> <s xml:id="echoid-s1534" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1535" xml:space="preserve">2. </s> <s xml:id="echoid-s1536" xml:space="preserve">agnoſcet centrum <lb/>grauitatis annuli ex ſemihy perbola A B N, ſic ſe-<lb/>care k L, vt pars terminata ad k, ſit ad partem ter-<lb/>minatam ad L, vt 5. </s> <s xml:id="echoid-s1537" xml:space="preserve">ad 3.</s> <s xml:id="echoid-s1538" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1539" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1540" xml:space="preserve">pri. </s> <s xml:id="echoid-s1541" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1542" xml:space="preserve">14. </s> <s xml:id="echoid-s1543" xml:space="preserve">agnoſcet centrum gra-<lb/>uitatis in K L, portionis annuli ex portione mino-<lb/>ri A H I.</s> <s xml:id="echoid-s1544" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1545" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1546" xml:space="preserve">prim. </s> <s xml:id="echoid-s1547" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1548" xml:space="preserve">16. </s> <s xml:id="echoid-s1549" xml:space="preserve">agnoſcet centrum <lb/>grauitatis ſegmenti annuli ex ſegmento I H B N. <lb/></s> <s xml:id="echoid-s1550" xml:space="preserve">Hoc autem centrum etiam alio modo agnoſcet ex di-<lb/>ctis in calce eiuſdem ſcholij.</s> <s xml:id="echoid-s1551" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1552" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1553" xml:space="preserve">prim. </s> <s xml:id="echoid-s1554" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1555" xml:space="preserve">17. </s> <s xml:id="echoid-s1556" xml:space="preserve">agnoſcet modum re- <pb o="87" file="0099" n="99"/> periendi centrum grauitatis ſegmenti annuli ex ſeg-<lb/>mento intermedio Q X H I. </s> <s xml:id="echoid-s1557" xml:space="preserve">Quod etiam inueniet <lb/>alio modo expreſſo in eodem ſchol o.</s> <s xml:id="echoid-s1558" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1559" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s1560" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1561" xml:space="preserve">19. </s> <s xml:id="echoid-s1562" xml:space="preserve">agnoſcet modum reperien-<lb/>di centrum grauitatis portionis annuli ex portione <lb/>maiori I H B C.</s> <s xml:id="echoid-s1563" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1564" xml:space="preserve">Tandem ex ſchol. </s> <s xml:id="echoid-s1565" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1566" xml:space="preserve">21. </s> <s xml:id="echoid-s1567" xml:space="preserve">agnoſcet modum <lb/>reperiendi centrum grauitatis ſegmenti intermedij <lb/>annuli ex ſegmento intermedio I H B Z Y, interci-<lb/>piente axim B N.</s> <s xml:id="echoid-s1568" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1569" xml:space="preserve">Hæ ergo ſunt notitiæ geometricæ, quæ deducun-<lb/>tur ex anteced. </s> <s xml:id="echoid-s1570" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1571" xml:space="preserve">Quibus addenda eſt. </s> <s xml:id="echoid-s1572" xml:space="preserve">Quod <lb/>cum notatum ſit in ſchol. </s> <s xml:id="echoid-s1573" xml:space="preserve">prim propoſit. </s> <s xml:id="echoid-s1574" xml:space="preserve">8. </s> <s xml:id="echoid-s1575" xml:space="preserve">lib. </s> <s xml:id="echoid-s1576" xml:space="preserve">4. </s> <s xml:id="echoid-s1577" xml:space="preserve">Pa-<lb/>rabolam, ſphæram, ſphæroides, & </s> <s xml:id="echoid-s1578" xml:space="preserve">exceſſum cylin-<lb/>dri ſupra duos conos inuersè poſitos, quorum baſes <lb/>oppoſitæ baſes cylindri, vertex verò medium pun-<lb/>ctum axis, eſſe magnitudines proportionaliter ana-<lb/>logas tam in magnitudine, quam in grauitate; </s> <s xml:id="echoid-s1579" xml:space="preserve">ſe qui <lb/>ex dictis, his aſſociari annulum prædictum ex hy-<lb/>perbola.</s> <s xml:id="echoid-s1580" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div84" type="section" level="1" n="57"> <head xml:id="echoid-head68" xml:space="preserve">PROPOSITIO XXVII.</head> <p style="it"> <s xml:id="echoid-s1581" xml:space="preserve">In ſchematæ propoſit. </s> <s xml:id="echoid-s1582" xml:space="preserve">quintæ, exceſſus cylindri circumſcri-<lb/>pti conoidi hyperbolico ſupra cylindrum circumſcriptum <lb/>conoidi parabolico, erit triplus exceſſus conoidis hyperbo-<lb/>lici ſupra conoides parabolicum.</s> <s xml:id="echoid-s1583" xml:space="preserve"/> </p> <pb o="88" file="0100" n="100"/> <p> <s xml:id="echoid-s1584" xml:space="preserve">COnoidibus hyperbolico A B C, & </s> <s xml:id="echoid-s1585" xml:space="preserve">parabolico <lb/>E B F, ſint circumſcripti cylindri Q C, T F. <lb/></s> <s xml:id="echoid-s1586" xml:space="preserve">Dico tubum cylindricum Q E L C, triplum eſſe ex-<lb/>ceſſus conoidis A B C, ſupra conoides E B F. </s> <s xml:id="echoid-s1587" xml:space="preserve">Quo-<lb/>niam enim cylindrus Q C, eſt ad cylindrum T F, <lb/>vt quadratum A D, ad quadratum D E; </s> <s xml:id="echoid-s1588" xml:space="preserve">nempe <lb/>ex hypotheſi, vt D G, ad G B, ergo per conuer-<lb/>ſionem rationis & </s> <s xml:id="echoid-s1589" xml:space="preserve">conuertendo, erit tubus cylin-<lb/>dricus Q E L C, ad cylindrum Q C, vt B D, ad <lb/>D G. </s> <s xml:id="echoid-s1590" xml:space="preserve">Sed ex propoſit. </s> <s xml:id="echoid-s1591" xml:space="preserve">5. </s> <s xml:id="echoid-s1592" xml:space="preserve">7. </s> <s xml:id="echoid-s1593" xml:space="preserve">& </s> <s xml:id="echoid-s1594" xml:space="preserve">11. </s> <s xml:id="echoid-s1595" xml:space="preserve">cylindrus Q C, <lb/>eſt ad conoides A B C, vt D G, ad dimidium B G, <lb/>cum tertia parte D B: </s> <s xml:id="echoid-s1596" xml:space="preserve">ergo ex æquali, erit tubus <lb/>Q E L C, ad conoides A B C, vt D B, ad dimi-<lb/>diam G B, cum tertia parte D B. </s> <s xml:id="echoid-s1597" xml:space="preserve">Rurſum, quoniam <lb/>diuidendo, eſt tubus Q E L C, ad cylindrum T F, <lb/>vt rectangulum A E C, ad quadratum E D, nem-<lb/>pe ex hypotheſi, vt D B, ad B G, & </s> <s xml:id="echoid-s1598" xml:space="preserve">conoides <lb/>E B F, eſt dimidium cylindri T F, vt oſtendimus <lb/>præcipuè in lib. </s> <s xml:id="echoid-s1599" xml:space="preserve">2. </s> <s xml:id="echoid-s1600" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1601" xml:space="preserve">15. </s> <s xml:id="echoid-s1602" xml:space="preserve">Ergo tubus Q E L C, <lb/>erit ad conoides E B F, vt D B, ad dimidiam G B. </s> <s xml:id="echoid-s1603" xml:space="preserve"><lb/>Sed erat ad totum conoides A B C, vt eadem D B, <lb/>ad dimidiam G B, cum tertia parte D B. </s> <s xml:id="echoid-s1604" xml:space="preserve">Ergo <lb/>Q E L C, erit ad reliquum, nempe ad differentiam <lb/>conoideorum, vt D B, ad ſui tertiam partem; </s> <s xml:id="echoid-s1605" xml:space="preserve"><lb/>nempe erit triplus talis exceſſus. </s> <s xml:id="echoid-s1606" xml:space="preserve">Quod e@@@ oſten-<lb/>dendum.</s> <s xml:id="echoid-s1607" xml:space="preserve"/> </p> <pb o="89" file="0101" n="101"/> <figure> <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0101-01"/> </figure> </div> <div xml:id="echoid-div85" type="section" level="1" n="58"> <head xml:id="echoid-head69" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s1608" xml:space="preserve">Quoniam tam totus cylindrus Q C, eſt triplus <lb/>totius coni A B C, quam ablatus cylindrus T F, eſt <lb/>triplus ablati coni E B F (inſcriptis prius conis in <lb/>conoidibus); </s> <s xml:id="echoid-s1609" xml:space="preserve">ergo & </s> <s xml:id="echoid-s1610" xml:space="preserve">reliquus tubus Q E L C, tri-<lb/>plus erit reliqui; </s> <s xml:id="echoid-s1611" xml:space="preserve">nempe differentiæ conorum. </s> <s xml:id="echoid-s1612" xml:space="preserve">Sed <lb/>ex propoſit. </s> <s xml:id="echoid-s1613" xml:space="preserve">4. </s> <s xml:id="echoid-s1614" xml:space="preserve">differentia conorum eſt æqualis diffe-<lb/>rentiæ conoideorum. </s> <s xml:id="echoid-s1615" xml:space="preserve">Ergo tubus erit etiam triplus <lb/>differentiæ conoideorum. </s> <s xml:id="echoid-s1616" xml:space="preserve">Quod&</s> <s xml:id="echoid-s1617" xml:space="preserve">c.</s> <s xml:id="echoid-s1618" xml:space="preserve"/> </p> <pb o="90" file="0102" n="102"/> </div> <div xml:id="echoid-div86" type="section" level="1" n="59"> <head xml:id="echoid-head70" xml:space="preserve">PROPOSITIO XXVIII.</head> <p style="it"> <s xml:id="echoid-s1619" xml:space="preserve">Exceſſus cylindri circumſ@ripti conoidi hyperbolico ſupra <lb/>cylindrum circumſcriptum conoidi parabolico ſæpe ex-<lb/>plicato, est ad differentiam conoideorum, vt paralle-<lb/>logrammum circumſcriptum trilineo quadratico ad ip-<lb/>ſum, tam ſecundum totum, quam ſecundum partes <lb/>proportionales; </s> <s xml:id="echoid-s1620" xml:space="preserve">ſi diametri trilinei, & </s> <s xml:id="echoid-s1621" xml:space="preserve">conoidis ſecentur <lb/>proportionaliter.</s> <s xml:id="echoid-s1622" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1623" xml:space="preserve">SInt ergo conoidea hyperbolicum A B C, & </s> <s xml:id="echoid-s1624" xml:space="preserve">pa-<lb/>rabolicum E B F, vt ſæpe dictum eſt, cum cir-<lb/>cumſcriptis cylindris Q C, T F, & </s> <s xml:id="echoid-s1625" xml:space="preserve">inſuper ſit ſe-<lb/>miparabola B C O, cuius diameter O B, baſis <lb/>O C, & </s> <s xml:id="echoid-s1626" xml:space="preserve">parallelogrammum ei circumſcriptum ſit <lb/>D O, adeovt D B C, ſit trilineum quadraticum, cu-<lb/>ius diameter D B. </s> <s xml:id="echoid-s1627" xml:space="preserve">Dico tubum cylindricum <lb/>Q E L C, eſſe ad differentiam conoideorum, vt pa-<lb/>rallelogrammum D O, ad trilineum B D C, tam <lb/>ſecundum totum, quam fecundum partes propor-<lb/>tionales. </s> <s xml:id="echoid-s1628" xml:space="preserve">Sumatur in D B, diametro arbitrariè pun-<lb/>ctum G, per quod in ſolidis intelligatur tranfire pla-<lb/>num H K, plano A C, parallelum, ſecans tubum <lb/>in P, conoides hyperbolicum in M, & </s> <s xml:id="echoid-s1629" xml:space="preserve">paraboli-<lb/>cum in R: </s> <s xml:id="echoid-s1630" xml:space="preserve">item in parallelogrammo ducatur GK, <lb/>parallela D C, ſecans curuam parabolicam in S. <lb/></s> <s xml:id="echoid-s1631" xml:space="preserve">Quoniam ex propoſit. </s> <s xml:id="echoid-s1632" xml:space="preserve">3. </s> <s xml:id="echoid-s1633" xml:space="preserve">rectangulum A E C, eſt <lb/>ad rectangulum M R V, vt quadratum D B, ad <pb o="91" file="0103" n="103"/> <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/> quadratum B G; </s> <s xml:id="echoid-s1634" xml:space="preserve">& </s> <s xml:id="echoid-s1635" xml:space="preserve">vt rectangulum A E C, hoc eſt <lb/>rectangulum H P k, ad rectangulum M R V, ſic <lb/>armilia circularis H P k, ad armillam circularem <lb/>M R V: </s> <s xml:id="echoid-s1636" xml:space="preserve">ergo vt armilla circularis H P k, ad armil-<lb/>lam circularem M R V, ſic quadratum D B, ad <lb/>quadratum B G. </s> <s xml:id="echoid-s1637" xml:space="preserve">Sed ex natura parabolæ quadrati-<lb/>cæ, eſt etiam vt quadratum D B, ad quadratum <lb/>B G, ſic D C, ſeù K G, ad G S. </s> <s xml:id="echoid-s1638" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1639" xml:space="preserve">vt ar-<lb/>milla H P k, ad armillam M R V, ſic k G, ad G S. <lb/></s> <s xml:id="echoid-s1640" xml:space="preserve">Cum verò punctum G, ſumptum ſit ad libitum; </s> <s xml:id="echoid-s1641" xml:space="preserve">er-<lb/>go vt omnes armillæ tubi cylindrici Q E L C, pa-<lb/>rallelæ armillæ A E C, ad omnes armillas differen-<lb/>tiæ conoideorum, parallelas A E C, ſic omnes li-<lb/>meæ parallelogrammi D O, parallelæ D C, ad om- <pb o="92" file="0104" n="104"/> nes lineas trilinei C D B, parallelas itidem D C; <lb/></s> <s xml:id="echoid-s1642" xml:space="preserve">nempe vt tubus ad differentiam, ſic parallelogram-<lb/>mum ad trilineum.</s> <s xml:id="echoid-s1643" xml:space="preserve"/> </p> <div xml:id="echoid-div86" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0103-01"/> </figure> </div> <p> <s xml:id="echoid-s1644" xml:space="preserve">Cum vero quod oſtenſum eſt de totis, pateat poſ-<lb/>ſe eodem modo probari de partibus proportionali-<lb/>bus, ideo patet propoſitum.</s> <s xml:id="echoid-s1645" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div88" type="section" level="1" n="60"> <head xml:id="echoid-head71" xml:space="preserve">SCHOLIVMI.</head> <p> <s xml:id="echoid-s1646" xml:space="preserve">Patet ergo quomodo adhibito etiam alio ſolido <lb/>hyperbolico, nempe differentia conoideorum, poſſi-<lb/>mus quadrare parabolam. </s> <s xml:id="echoid-s1647" xml:space="preserve">Cum enim ex propoſit. <lb/></s> <s xml:id="echoid-s1648" xml:space="preserve">anteced. </s> <s xml:id="echoid-s1649" xml:space="preserve">tubus cylindricus Q E L C, ſit triplus dif-<lb/>ferentiæ conoideorum; </s> <s xml:id="echoid-s1650" xml:space="preserve">etiam parallelogrammum <lb/>triplum erit trilinei; </s> <s xml:id="echoid-s1651" xml:space="preserve">& </s> <s xml:id="echoid-s1652" xml:space="preserve">conſequenter ſeſquialterum <lb/>femiparabolæ.</s> <s xml:id="echoid-s1653" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1654" xml:space="preserve">Inſuper patet, quod cum in ſchol. </s> <s xml:id="echoid-s1655" xml:space="preserve">2. </s> <s xml:id="echoid-s1656" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1657" xml:space="preserve">18. <lb/></s> <s xml:id="echoid-s1658" xml:space="preserve">probatum ſit, conum, trilineum quadraticum, exceſ-<lb/>ſum cylindri circumſcripti hemiſphærio, & </s> <s xml:id="echoid-s1659" xml:space="preserve">hemiſ-<lb/>phæroidi, & </s> <s xml:id="echoid-s1660" xml:space="preserve">exceſſum tubi cylindrici ſuper annulum <lb/>latum ex hyperbola circa ſecundam diametrum, eſſe <lb/>quantitates proportion aliter analogas, patet in-<lb/>quam, his pro ſexta addi differentiam conoideorum <lb/>prædictam.</s> <s xml:id="echoid-s1661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div89" type="section" level="1" n="61"> <head xml:id="echoid-head72" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s1662" xml:space="preserve">In propoſit. </s> <s xml:id="echoid-s1663" xml:space="preserve">11. </s> <s xml:id="echoid-s1664" xml:space="preserve">lib. </s> <s xml:id="echoid-s1665" xml:space="preserve">2. </s> <s xml:id="echoid-s1666" xml:space="preserve">de Infinit. </s> <s xml:id="echoid-s1667" xml:space="preserve">Parab. </s> <s xml:id="echoid-s1668" xml:space="preserve">cuius <lb/>ſchema hic apponimus, probauimus, quod ſi ſint <pb o="93" file="0105" n="105"/> <anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/> duæ quælibet figuræ A B C, A E F C, ſupra ea-<lb/>dem baſi A C, & </s> <s xml:id="echoid-s1669" xml:space="preserve">circa communem axim B D; </s> <s xml:id="echoid-s1670" xml:space="preserve">ſint-<lb/>que hæ talis naturæ, vt ipſis duplicatis ad partes <lb/>A C, hæc euadat communis axis ambarum figura-<lb/>rum; </s> <s xml:id="echoid-s1671" xml:space="preserve">probauimus inquam, intellectis ambabus figu- <pb o="94" file="0106" n="106"/> ris gyrari circa parallelam ipſi B D, ductam per <lb/>punctum C, quæ ſit v.</s> <s xml:id="echoid-s1672" xml:space="preserve">g. </s> <s xml:id="echoid-s1673" xml:space="preserve">C F, ſolidum rotundum <lb/>ortum ex figura A E F C, eſſe ad ſolidum rotundum <lb/>ex figura A B C, vt figura A E F C, ad figuram <lb/>A B C. </s> <s xml:id="echoid-s1674" xml:space="preserve">Hoc probauimus medijs truncis ſiniſtris cy-<lb/>lindricorum rectorum ſupra figuris exiſtentium, vt <lb/>loco cit. </s> <s xml:id="echoid-s1675" xml:space="preserve">poteſt conſpici. </s> <s xml:id="echoid-s1676" xml:space="preserve">Ex hac vniuerſali propoſi-<lb/>tione deduximus ibidem quamplurima corollaria; <lb/></s> <s xml:id="echoid-s1677" xml:space="preserve">quibus poteſt aggregari, quod ſi A B C, eſſet hy-<lb/>perbola, & </s> <s xml:id="echoid-s1678" xml:space="preserve">E C, eſſet parallelogrammum ipſam <lb/>circumſcribens, & </s> <s xml:id="echoid-s1679" xml:space="preserve">haberetur quadratura hyperbo-<lb/>læ, nequaquam ignoraretur ratio cylindriex E C, cir-<lb/>ca C F, ad annulum ſtrictum ex hyperbola A B C, <lb/>circa C F. </s> <s xml:id="echoid-s1680" xml:space="preserve">Verum illa propoſitio poteſt vniuerſa-<lb/>lius proponi; </s> <s xml:id="echoid-s1681" xml:space="preserve">nonſolum enim illud verum eſt; </s> <s xml:id="echoid-s1682" xml:space="preserve">ſed <lb/>etiam veriſicatur, quod ſi illæ duæ figuræ rotentur <lb/>circa parall lamipſi C F, ſed extra figuras ductam, <lb/>adeovt ex figuris cratis generentur annuli lati: </s> <s xml:id="echoid-s1683" xml:space="preserve">ni-<lb/>hilominus annulum larum ex A E F C, ad annulum <lb/>latum ex A B C, eſſe vt figura A E F C, ad figuram <lb/>A B C. </s> <s xml:id="echoid-s1684" xml:space="preserve">Hoc peſiet probari medijs ijſdem truncis, <lb/>& </s> <s xml:id="echoid-s1685" xml:space="preserve">hoc pacto liceret ampliare doctrinas de truncis in <lb/>illo opere expoſitas; </s> <s xml:id="echoid-s1686" xml:space="preserve">fed de his forſan aliquando. </s> <s xml:id="echoid-s1687" xml:space="preserve">In <lb/>præſenti probabimus medijs ad noſtrum inſtitutum <lb/>magis accomodatis, ſequentem propoſitionem vt ex <lb/>huius cognitione inquiramus centra grauitatis infi-<lb/>nitorum annulorum, vt infià patebit.</s> <s xml:id="echoid-s1688" xml:space="preserve"/> </p> <div xml:id="echoid-div89" type="float" level="2" n="1"> <figure xlink:label="fig-0105-01" xlink:href="fig-0105-01a"> <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0105-01"/> </figure> </div> <pb o="95" file="0107" n="107"/> </div> <div xml:id="echoid-div91" type="section" level="1" n="62"> <head xml:id="echoid-head73" xml:space="preserve">PROPOSITIO XXIX.</head> <p style="it"> <s xml:id="echoid-s1689" xml:space="preserve">Si ſuper eadem baſi & </s> <s xml:id="echoid-s1690" xml:space="preserve">circa eandem diametrum ſint quælibet <lb/>figura & </s> <s xml:id="echoid-s1691" xml:space="preserve">parallelogrammum ipſam circumſcribens. </s> <s xml:id="echoid-s1692" xml:space="preserve">Cy-<lb/>lindrus ex parallelogrammo ad ſolidum ex figura, reuolutis <lb/>ambobus circa parallelam diametro ductam velper extre-<lb/>mitatem baſis, vel extra baſim, erit vt parallelogram-<lb/>mum ad figur am.</s> <s xml:id="echoid-s1693" xml:space="preserve"/> </p> <figure> <image file="0107-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0107-01"/> </figure> <p> <s xml:id="echoid-s1694" xml:space="preserve">SVper eadem baſi A C, & </s> <s xml:id="echoid-s1695" xml:space="preserve">circa eandem dia-<lb/>metrum B D, ſint quælibet figura A B C, & </s> <s xml:id="echoid-s1696" xml:space="preserve"><lb/>parallelogrammum E C, ipſam circumſcribens <lb/>& </s> <s xml:id="echoid-s1697" xml:space="preserve">intelligamus ambas figuras prius rotari circa F C. <lb/></s> <s xml:id="echoid-s1698" xml:space="preserve">Dico cylindrum E G, eſſe ad ſolidum ex figura, <lb/>A B C, circa eandem F C, quod ſit A B C H G, vt <lb/>E C, ad A B C. </s> <s xml:id="echoid-s1699" xml:space="preserve">Accipiatur in B D, arbitrariè <lb/>punctum 1, per quod intelligantur tranſire in <lb/>figuris linea k N, A C, parallela, in ſolidis verò <pb o="96" file="0108" n="108"/> planum K N, item A G, parallelum. </s> <s xml:id="echoid-s1700" xml:space="preserve">Quoniam <lb/>enim vt k N, ad L M, ſic (ſumpta N R, com-<lb/>muni altitudine) rectangulum k N R, ad rectan-<lb/>gulum ſub L M, & </s> <s xml:id="echoid-s1701" xml:space="preserve">ſub N R; </s> <s xml:id="echoid-s1702" xml:space="preserve">& </s> <s xml:id="echoid-s1703" xml:space="preserve">N R, eſt æ-<lb/>qualis M Q, quia M N, eſt æqualis, tam N O, <lb/>quam Q R, vndè etiam rectangulum ſub L M, <lb/>& </s> <s xml:id="echoid-s1704" xml:space="preserve">ſub N R, eſt æquale rectangulo L M Q. </s> <s xml:id="echoid-s1705" xml:space="preserve">Ergo <lb/>etiam vt k N, ad L M, ſic rectangulum k N R, <lb/>ad rectangulum L M Q. </s> <s xml:id="echoid-s1706" xml:space="preserve">Sed vt rectangulum <lb/>k N R, ad rectangulum L M Q, ſic circulus, <lb/>k N R, ad armillam circulatem L M Q. </s> <s xml:id="echoid-s1707" xml:space="preserve">Ergo <lb/>& </s> <s xml:id="echoid-s1708" xml:space="preserve">vt K N, ad L M, ſic circulus K N R, ad ar-<lb/>millam circularem L M Q. </s> <s xml:id="echoid-s1709" xml:space="preserve">At punctum I, ſum-<lb/>ptum eſt vtcunque. </s> <s xml:id="echoid-s1710" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1711" xml:space="preserve">vt vnum ad vnum, ita <lb/>omnia ad omnia. </s> <s xml:id="echoid-s1712" xml:space="preserve">Ergo vt omnes lineæ figuræ E C, <lb/>A C, parallelæ ad omnes lineas figuræ A B C, item <lb/>A C, parallelas, ſic omnes circuli ſolidi E G, circulo <lb/>A G, paralleli ad omnes armillas ſolidi A B C H G. <lb/></s> <s xml:id="echoid-s1713" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1714" xml:space="preserve">vt figura ad figuram, ſic ſolidum ad ſoli-<lb/>dum.</s> <s xml:id="echoid-s1715" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1716" xml:space="preserve">Sed ſupponamus figuras prædictas rotari circa <lb/>S T, poſitam vltra C, ipſi B D, parallelam, adeo-<lb/>vt ex figuris generentur tubus cylindricus, & </s> <s xml:id="echoid-s1717" xml:space="preserve">annu-<lb/>lus latus vt in ſequenti ſchemate. </s> <s xml:id="echoid-s1718" xml:space="preserve">Dico nihilomi-<lb/>nus eſſe E C, ad figuram A B C, vt tubus E C Y, ad <lb/>annulum ex figura A B C. </s> <s xml:id="echoid-s1719" xml:space="preserve">Nam accepto vt prius, <lb/>puncto I, arbitrariè, factiſque ijſdem, conclu-<lb/>demus eodem modo eſſe vt K N, ad L M, ſic re-<lb/>ctangulum K N R, ad rectangulum L M Q; </s> <s xml:id="echoid-s1720" xml:space="preserve">nem- <pb o="97" file="0109" n="109"/> <anchor type="figure" xlink:label="fig-0109-01a" xlink:href="fig-0109-01"/> pe ſic armillam circularem k N R, ad armillam cir-<lb/>cularem L M Q. </s> <s xml:id="echoid-s1721" xml:space="preserve">Quare eodem modo concludemus <lb/>eſſe figuram E C, ad figuram A B C, vt ſolidum <lb/>ex E C, circa S T, ad ſolidum ex figura A B C, cir-<lb/>ca eandem T S. </s> <s xml:id="echoid-s1722" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1723" xml:space="preserve"/> </p> <div xml:id="echoid-div91" type="float" level="2" n="1"> <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a"> <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0109-01"/> </figure> </div> </div> <div xml:id="echoid-div93" type="section" level="1" n="63"> <head xml:id="echoid-head74" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s1724" xml:space="preserve">Cum præſens propoſitio ſit propoſita in tanta vni-<lb/>uerſalitate, adeovt comprehendat infinitas figuras <lb/>circa diametrum, & </s> <s xml:id="echoid-s1725" xml:space="preserve">infinitis modis diuerſificatas, <lb/>impoſſibile videtur poſſe ipſam oſtendi in tali vni-<lb/>uerſalitate vnica conſtructione niſi per indiuiſibilia. <lb/></s> <s xml:id="echoid-s1726" xml:space="preserve">Modo etiam archimedeo probari poteſt, ſed in caſi-<lb/>bus particularibus, & </s> <s xml:id="echoid-s1727" xml:space="preserve">conſtructionibus proprijs, vt <lb/>quilibet poterit experiri.</s> <s xml:id="echoid-s1728" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1729" xml:space="preserve">Ex hac autem vniuerſaliſſima propoſitione, ea om-<lb/>nia, quæ ſunt deducta in corollarijs propoſit. </s> <s xml:id="echoid-s1730" xml:space="preserve">cit. </s> <s xml:id="echoid-s1731" xml:space="preserve">in <lb/>opere de in finit. </s> <s xml:id="echoid-s1732" xml:space="preserve">parab circa varia ſolida annulorum <pb o="98" file="0110" n="110"/> ſtrictorum ex varijs figuris genitorum, poſſunt dedu-<lb/>ci etiam in infinitis ſolidis annulorum latorum; </s> <s xml:id="echoid-s1733" xml:space="preserve">quæ <lb/>autem ea ſint, inſpiciatur ibidem. </s> <s xml:id="echoid-s1734" xml:space="preserve">Nos enim in præ-<lb/>ſenti non manifeſtabimus niſi inſinitorum annulo-<lb/>rum tam ſtrictorum, quam latorum centra grauita-<lb/>tis. </s> <s xml:id="echoid-s1735" xml:space="preserve">Nam facili negotio ex dictis in lib. </s> <s xml:id="echoid-s1736" xml:space="preserve">4. </s> <s xml:id="echoid-s1737" xml:space="preserve">infinit. </s> <s xml:id="echoid-s1738" xml:space="preserve">pa-<lb/>rab. </s> <s xml:id="echoid-s1739" xml:space="preserve">agnoſcemus figuras prædictas eſſe quantitates <lb/>proportionaliter analogas cum ſuis annulis, tam ſtri-<lb/>ctis, quam latis. </s> <s xml:id="echoid-s1740" xml:space="preserve">V. </s> <s xml:id="echoid-s1741" xml:space="preserve">g. </s> <s xml:id="echoid-s1742" xml:space="preserve">facile agnoſcemus figuram <lb/>A B C, eſſe quantitatem proportionaliter analogam <lb/>tam cum annulo ſtricto A B C H G, in prima figu-<lb/>ra, quam cum annulo lato ex eadem A B C, in ſe-<lb/>cunda figura. </s> <s xml:id="echoid-s1743" xml:space="preserve">Quare etiam duo annuli ex eadem <lb/>figura, nempe & </s> <s xml:id="echoid-s1744" xml:space="preserve">ſtrictus, & </s> <s xml:id="echoid-s1745" xml:space="preserve">latus erunt quantitates <lb/>proportionaliter analogæ tam in magnitudine, quam <lb/>in grauitate. </s> <s xml:id="echoid-s1746" xml:space="preserve">Sequitur ergo nos habere centra gra-<lb/>uitatis omnium illorum annulorum tam ſtrictorum, <lb/>quam latorum, quorum figurarum genitricium ſupra <lb/>explicatarum, habemus centrum grauitatis.</s> <s xml:id="echoid-s1747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1748" xml:space="preserve">Si ergo ſupponamus A B C, eſſe parallelogram-<lb/>mum veluti E C, quod rotetur vel circa ſuum latus <lb/>F C, vel circa T S, ei parallelum (quod ſemper intelli-<lb/>gendum erit in dicendis impoſterum, ne cogamur <lb/>idem cum lectorum tedio repetere) centrum grauita-<lb/>tis cylindri, vel tubi cylindrici, ſecabit F C, vel T S, <lb/>in ea ratione, in qua ſecat B D, centrum grauitatis <lb/>parallelogrammi.</s> <s xml:id="echoid-s1749" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1750" xml:space="preserve">Si verò ſupponamus A B C, nobis repræſentare <lb/>infinitas parabolas, habebimus centrum grauitatis <pb o="99" file="0111" n="111"/> <anchor type="figure" xlink:label="fig-0111-01a" xlink:href="fig-0111-01"/> infinitorum annulorum ex ipſis ſic ſecare F C, vt <lb/>pars terminata ad F, ſit ad partem terminatam ad <lb/>C, in primo annulo ex prima parabola vt 2. </s> <s xml:id="echoid-s1751" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s1752" xml:space="preserve">In <lb/>ſec. </s> <s xml:id="echoid-s1753" xml:space="preserve">vt 3. </s> <s xml:id="echoid-s1754" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s1755" xml:space="preserve">in tertio vt 4. </s> <s xml:id="echoid-s1756" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1757" xml:space="preserve">& </s> <s xml:id="echoid-s1758" xml:space="preserve">ſic in infinitum. <lb/></s> <s xml:id="echoid-s1759" xml:space="preserve">Ratio eſt, quia ex ſchol. </s> <s xml:id="echoid-s1760" xml:space="preserve">prim. </s> <s xml:id="echoid-s1761" xml:space="preserve">propoſit 2. </s> <s xml:id="echoid-s1762" xml:space="preserve">lib. </s> <s xml:id="echoid-s1763" xml:space="preserve">2. </s> <s xml:id="echoid-s1764" xml:space="preserve">ha-<lb/>bemus centrum grauitatis infinitarum parabolarum <lb/>ſic ſecare B D.</s> <s xml:id="echoid-s1765" xml:space="preserve"/> </p> <div xml:id="echoid-div93" type="float" level="2" n="1"> <figure xlink:label="fig-0111-01" xlink:href="fig-0111-01a"> <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0111-01"/> </figure> </div> <p> <s xml:id="echoid-s1766" xml:space="preserve">Si autem ſupponamus A B C, eſſe quamlibet <lb/>infinitarum parabolarum, & </s> <s xml:id="echoid-s1767" xml:space="preserve">E C, eſſe parallelo-<lb/>grammum infinitis parabolis circumſcriptum. </s> <s xml:id="echoid-s1768" xml:space="preserve">Ha-<lb/>bebimus centrum grauitatis infinitorum annulorum <lb/>ortorum ex reuolutione exceſſuum infinitorum pa-<lb/>rallelogrammorum ſupra infinitas parabolas. </s> <s xml:id="echoid-s1769" xml:space="preserve">Hoc <lb/>autem centrum grauitatis ſic ſecabit F C, vt pars <lb/>terminata ad F, ſit ad partem terminatam ad C, vt <lb/>numerus annuli vnitate auctus, ad triplum nume-<lb/>rum annuli vnitate auctum. </s> <s xml:id="echoid-s1770" xml:space="preserve">V. </s> <s xml:id="echoid-s1771" xml:space="preserve">g. </s> <s xml:id="echoid-s1772" xml:space="preserve">in primo annulo <lb/>vt 2. </s> <s xml:id="echoid-s1773" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s1774" xml:space="preserve">In ſecundo, vt 3. </s> <s xml:id="echoid-s1775" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s1776" xml:space="preserve">In tertio vt 4. </s> <s xml:id="echoid-s1777" xml:space="preserve">ad <pb o="100" file="0112" n="112"/> 10. </s> <s xml:id="echoid-s1778" xml:space="preserve">& </s> <s xml:id="echoid-s1779" xml:space="preserve">ſic in infinitum. </s> <s xml:id="echoid-s1780" xml:space="preserve">Ratio eſt, quia ex ſchol. <lb/></s> <s xml:id="echoid-s1781" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1782" xml:space="preserve">8. </s> <s xml:id="echoid-s1783" xml:space="preserve">eiuſdem libri centrum grauitatis exceſ-<lb/>ſus parallelogrammi E C, ſupra parabolam ſic ſecat <lb/>ipſam B D.</s> <s xml:id="echoid-s1784" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1785" xml:space="preserve">Sed ſupponentes A B C, eſſe vel ſemicirculum, <lb/>vel ſemiellipſim, vel circuli, aut ellipſis portionem, <lb/>vel etiam hyperbolam. </s> <s xml:id="echoid-s1786" xml:space="preserve">Habebimus centrum gra-<lb/>uitatis annulorum talium figurarum, ſed ſuppoſita <lb/>figurarum quadratura. </s> <s xml:id="echoid-s1787" xml:space="preserve">Hæcautem patent vera eſſe <lb/>partim ex dictis in lib. </s> <s xml:id="echoid-s1788" xml:space="preserve">3. </s> <s xml:id="echoid-s1789" xml:space="preserve">vbi in propoſit. </s> <s xml:id="echoid-s1790" xml:space="preserve">24. </s> <s xml:id="echoid-s1791" xml:space="preserve">aſſigna-<lb/>uimus centrum grauitatis ſemicirculi; </s> <s xml:id="echoid-s1792" xml:space="preserve">& </s> <s xml:id="echoid-s1793" xml:space="preserve">in ſchol. <lb/></s> <s xml:id="echoid-s1794" xml:space="preserve">prim. </s> <s xml:id="echoid-s1795" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1796" xml:space="preserve">25. </s> <s xml:id="echoid-s1797" xml:space="preserve">omnium ipſius portionum; </s> <s xml:id="echoid-s1798" xml:space="preserve">& </s> <s xml:id="echoid-s1799" xml:space="preserve">in <lb/>propoſit. </s> <s xml:id="echoid-s1800" xml:space="preserve">vltima lib. </s> <s xml:id="echoid-s1801" xml:space="preserve">4. </s> <s xml:id="echoid-s1802" xml:space="preserve">in qua aſſignauimus centrum <lb/>grauitatis omnium partium ellipſis; </s> <s xml:id="echoid-s1803" xml:space="preserve">partim ex dictis <lb/>in propoſit. </s> <s xml:id="echoid-s1804" xml:space="preserve">22. </s> <s xml:id="echoid-s1805" xml:space="preserve">huius, & </s> <s xml:id="echoid-s1806" xml:space="preserve">in ſcholio eiuſdem, vbiaſ-<lb/>ſignauimus centrum grauitatis hyperbolæ. </s> <s xml:id="echoid-s1807" xml:space="preserve">Imo ſi <lb/>in ſchemate illius propoſitionis, intelligamus exceſ-<lb/>ſum parallelogrammi G C, ſupra hyperbolam <lb/>A B C, rotari vel circa H C, vel circa ipſi paralle-<lb/>lam extra parallelogrammum: </s> <s xml:id="echoid-s1808" xml:space="preserve">ex dictis ibidem, agno-<lb/>ſcetur centrum grauitatis annulorum genitorum.</s> <s xml:id="echoid-s1809" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1810" xml:space="preserve">Exiſtimantes autem A B C, eſſe cycloidem pri-<lb/>mariam; </s> <s xml:id="echoid-s1811" xml:space="preserve">placitis Torricellij in lib. </s> <s xml:id="echoid-s1812" xml:space="preserve">1. </s> <s xml:id="echoid-s1813" xml:space="preserve">de motu grau. <lb/></s> <s xml:id="echoid-s1814" xml:space="preserve">ſchol. </s> <s xml:id="echoid-s1815" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1816" xml:space="preserve">18. </s> <s xml:id="echoid-s1817" xml:space="preserve">annuentes, intelligemus centrum <lb/>grauitatis annuli ex cycloide ſic ſecare F C, vt pars <lb/>terminata ad F, ſit ad partem terminatam ad C, vt <lb/>7. </s> <s xml:id="echoid-s1818" xml:space="preserve">ad 5.</s> <s xml:id="echoid-s1819" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1820" xml:space="preserve">Sed accipiamus ſchema ſequens, in quo intelli-<lb/>gamus ſemiparabolam B A D, duplicari ad partes <pb o="101" file="0113" n="113"/> <anchor type="figure" xlink:label="fig-0113-01a" xlink:href="fig-0113-01"/> baſis A D, adeo vt hæc euadat communis axis dua-<lb/>rum ſemiparabolarum ſimul coniunctarum, hanc-<lb/>que figuram intelligamus rotari vel circa O N, vel <lb/>circa parallelam A D, extra figuram: </s> <s xml:id="echoid-s1821" xml:space="preserve">centrum gra-<lb/>uitatis productorum annulorum ita ſecabit O N, <lb/>vel illi parallelam &</s> <s xml:id="echoid-s1822" xml:space="preserve">c. </s> <s xml:id="echoid-s1823" xml:space="preserve">vt pars terminata ad O, ſit ad <pb o="102" file="0114" n="114"/> pattem terminatam ad N, vt numerus annuli au-<lb/>ctus ternario ad numerum annuli auctum vnitate. <lb/></s> <s xml:id="echoid-s1824" xml:space="preserve">Nimirum in primo vt 4. </s> <s xml:id="echoid-s1825" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s1826" xml:space="preserve">Inſec. </s> <s xml:id="echoid-s1827" xml:space="preserve">vt 5. </s> <s xml:id="echoid-s1828" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1829" xml:space="preserve">In <lb/>tertio vt 6. </s> <s xml:id="echoid-s1830" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s1831" xml:space="preserve">& </s> <s xml:id="echoid-s1832" xml:space="preserve">ſic in infinitum. </s> <s xml:id="echoid-s1833" xml:space="preserve">Ita enim ex <lb/>ſchol. </s> <s xml:id="echoid-s1834" xml:space="preserve">2. </s> <s xml:id="echoid-s1835" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1836" xml:space="preserve">2. </s> <s xml:id="echoid-s1837" xml:space="preserve">lib. </s> <s xml:id="echoid-s1838" xml:space="preserve">3. </s> <s xml:id="echoid-s1839" xml:space="preserve">centrum æquilibrij ſemi-<lb/>parabolæ A B D, ſeù centrum grauitatis figuræ <lb/>N A B, diuidit A D.</s> <s xml:id="echoid-s1840" xml:space="preserve"/> </p> <div xml:id="echoid-div94" type="float" level="2" n="2"> <figure xlink:label="fig-0113-01" xlink:href="fig-0113-01a"> <image file="0113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0113-01"/> </figure> </div> <p> <s xml:id="echoid-s1841" xml:space="preserve">Prædictæ autem figuræ circumſcripto parallelo-<lb/>grammo E N, & </s> <s xml:id="echoid-s1842" xml:space="preserve">figura conſtante ex duobus trili-<lb/>neis N O A B E, reuoluta prædicto modo: </s> <s xml:id="echoid-s1843" xml:space="preserve">centrum <lb/>grauitatis ſolidi geniti ſic fecabit O N, vt pars ter-<lb/>minata ad O, ſit ad partem terminatam ad N, vt <lb/>vnitas ad numerum annuli vnitate auctum. </s> <s xml:id="echoid-s1844" xml:space="preserve">Nempe <lb/>in primo vt 1. </s> <s xml:id="echoid-s1845" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s1846" xml:space="preserve">In ſec: </s> <s xml:id="echoid-s1847" xml:space="preserve">vt 1. </s> <s xml:id="echoid-s1848" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1849" xml:space="preserve">In tertio vt 1. <lb/></s> <s xml:id="echoid-s1850" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s1851" xml:space="preserve">Et ſic in infinitum. </s> <s xml:id="echoid-s1852" xml:space="preserve">Ratio eſt quia centrum <lb/>grauitatis talium trilincorum ſimul coniunctorum <lb/>ſic diuidit A D, vt centrum æquilibrij vnius v. </s> <s xml:id="echoid-s1853" xml:space="preserve">g. </s> <s xml:id="echoid-s1854" xml:space="preserve"><lb/>A E B, diuidit E B. </s> <s xml:id="echoid-s1855" xml:space="preserve">Atex ſchol. </s> <s xml:id="echoid-s1856" xml:space="preserve">prim. </s> <s xml:id="echoid-s1857" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s1858" xml:space="preserve">2. </s> <s xml:id="echoid-s1859" xml:space="preserve"><lb/>lib. </s> <s xml:id="echoid-s1860" xml:space="preserve">3. </s> <s xml:id="echoid-s1861" xml:space="preserve">E B, in prædicta ratione ſecatur à tali centro <lb/>æquilibrij. </s> <s xml:id="echoid-s1862" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s1863" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1864" xml:space="preserve">At ſi ſemiparabola quælibet intelligatur duplicari <lb/>ad partes B F, vt figura conſtans ſit C D B Q P, & </s> <s xml:id="echoid-s1865" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s1866" xml:space="preserve">hæc rotetur vel circa D C, vel circa ipſi paralle-<lb/>lam. </s> <s xml:id="echoid-s1867" xml:space="preserve">Centrum grauitatis ſolidi geniti ſecabit pari-<lb/>ter D C, vt pars terminata ad C, ſit ad partem ter-<lb/>minatam ad D, vt numerus annuli ternario auctus, <lb/>ad numerum annuli vnitate auctum. </s> <s xml:id="echoid-s1868" xml:space="preserve">Nempe vt 4, <lb/>ad 2. </s> <s xml:id="echoid-s1869" xml:space="preserve">vt 5. </s> <s xml:id="echoid-s1870" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1871" xml:space="preserve">&</s> <s xml:id="echoid-s1872" xml:space="preserve">c. </s> <s xml:id="echoid-s1873" xml:space="preserve">Item ſi trilineum C B Q, ſic ro-<lb/>tetur; </s> <s xml:id="echoid-s1874" xml:space="preserve">D C, ſic ſecabitur vt pars terminata ad D, <pb o="103" file="0115" n="115"/> <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> ſit ad partem terminatam ad C, vt numèrus annu-<lb/>li vnitate auctus, ad vnitatem. </s> <s xml:id="echoid-s1875" xml:space="preserve">Ratio eſt quia eodem <lb/>modo ſecatur A D, à centro grauitatis figuræ <lb/>N A B, ſicuti ſecatur B F, à centro grauitatis fi-<lb/>guræ D C B Q P; </s> <s xml:id="echoid-s1876" xml:space="preserve">ita tamen vt homologi termini <lb/>extremi ſint A, & </s> <s xml:id="echoid-s1877" xml:space="preserve">F; </s> <s xml:id="echoid-s1878" xml:space="preserve">D, & </s> <s xml:id="echoid-s1879" xml:space="preserve">B. </s> <s xml:id="echoid-s1880" xml:space="preserve">Item eodem <pb o="104" file="0116" n="116"/> modo ſecatur A D, à centro grauitatis figuræ <lb/>O N A B E, ſicuti ſecatur B F, à centro grauitatis <lb/>figuræ C B Q; </s> <s xml:id="echoid-s1881" xml:space="preserve">exiſtentibus pariter homologis pun-<lb/>ctis extremis A, F; </s> <s xml:id="echoid-s1882" xml:space="preserve">D, B.</s> <s xml:id="echoid-s1883" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="3"> <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/PVNDER1Y/figures/0115-01"/> </figure> </div> <p> <s xml:id="echoid-s1884" xml:space="preserve">Cum verò eodem etiam modo ſecetur B D, à <lb/>centro grauitatis figuræ A B C, ſicuti ſecatur F C, <lb/>à centro grauitatis duplicatæ ſemiparabolæ D B C, <lb/>in B D C R G: </s> <s xml:id="echoid-s1885" xml:space="preserve">pariter cum eodem modo ſecetur <lb/>B D, à centro grauitatis trilineorum A E B F C, <lb/>ſicuti ſecatur F C, à centro grauitatis ipſius B C R; <lb/></s> <s xml:id="echoid-s1886" xml:space="preserve">ſequitur quod ſi intelligamus figuram B D C R G, <lb/>rotari circa R G, &</s> <s xml:id="echoid-s1887" xml:space="preserve">c. </s> <s xml:id="echoid-s1888" xml:space="preserve">intelligemus pariter R G, <lb/>ſic diuidi à centro grauitatis geniti ſolidi, vt pars <lb/>terminata ad R, ſit ad partem terminatam ad G, <lb/>vt numerus annuli vnitate auctus, ad numerum an-<lb/>nuli. </s> <s xml:id="echoid-s1889" xml:space="preserve">Nempe vt 2. </s> <s xml:id="echoid-s1890" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s1891" xml:space="preserve">vt 3. </s> <s xml:id="echoid-s1892" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s1893" xml:space="preserve">&</s> <s xml:id="echoid-s1894" xml:space="preserve">c. </s> <s xml:id="echoid-s1895" xml:space="preserve">Item ſi in-<lb/>telliganius ſic rotari figuram B C R; </s> <s xml:id="echoid-s1896" xml:space="preserve">R G, ſic ſe-<lb/>cabitur vt pars terminata ad R, ſit ad partem termi-<lb/>natam ad G, vt numerus annuli vnitate auctus ad <lb/>triplum numerum annuli vnitate auctum. </s> <s xml:id="echoid-s1897" xml:space="preserve">Nempe <lb/>vt 2 ad 4. </s> <s xml:id="echoid-s1898" xml:space="preserve">vt 3. </s> <s xml:id="echoid-s1899" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s1900" xml:space="preserve">vt 4. </s> <s xml:id="echoid-s1901" xml:space="preserve">ad 10. </s> <s xml:id="echoid-s1902" xml:space="preserve">Et ſic in in-<lb/>finitum.</s> <s xml:id="echoid-s1903" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1904" xml:space="preserve">Quæ autem dicta ſunt ſupra de parabola quatuor <lb/>modis diſpoſita, quantum ad aſſignationem centro-<lb/>rum grauita is ſolidorum rotundorum ex ipſa geni-<lb/>torum, paret poſſe eriam applicari ſuo modo ſoli-<lb/>dis genitis ex reuo utione portionum circuli, & </s> <s xml:id="echoid-s1905" xml:space="preserve">el-<lb/>lipſis, item ſemihyperbolæ ſic diſpoſitarum. </s> <s xml:id="echoid-s1906" xml:space="preserve">Sed <lb/>quodnam ſit tale centrum relinquimus lectori conſi- <pb o="105" file="0117" n="117"/> <anchor type="figure" xlink:label="fig-0117-01a" xlink:href="fig-0117-01"/> derandum. </s> <s xml:id="echoid-s1907" xml:space="preserve">Præcipu è quia centra grauitatis figura-<lb/>rum genitricium non habentur niſi ſuppoſita ipſa-<lb/>rum figurarum quadratura. </s> <s xml:id="echoid-s1908" xml:space="preserve">Non ſic relinquemus <lb/>conſiderandum lectori, in quo puncto ip ſius F C, <lb/>vel ipſi parallelæ, ſit centrum grauitatis ſo lidi geniti <lb/>ex exceſſu parallelogrammi E C, ſupra ſuppoſitam <pb o="106" file="0118" n="118"/> cycloidem primariam A B C, reuoluto vel cir-<lb/>ca F C, vel circa dictam parallelam: </s> <s xml:id="echoid-s1909" xml:space="preserve">Item in <lb/>quo puncto ipſius R G, vel ipſi parallelæ ſit cen-<lb/>trum grauitatis duplicatæ ſemicycloidis B D C R G, <lb/>ad partes F C: </s> <s xml:id="echoid-s1910" xml:space="preserve">ſed admonebimus, centrum graui-<lb/>tatis ſolidi orti ex reuolutione figuræ B D C R G, ſic <lb/>ſecare dictam R G, vt pars terminata ad R, ſit ad <lb/>partem terminatam ad G, vt 7. </s> <s xml:id="echoid-s1911" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s1912" xml:space="preserve">Ratio eſt, <lb/>quia ita diuidit B D, centrum grauitatis cycloidis <lb/>A B C, ſicuti diuidit FC, centrum figuræ B D C R G. <lb/></s> <s xml:id="echoid-s1913" xml:space="preserve">Item admonebimus, centrum grauitatis ſolidi orti <lb/>ex gyratione figuræ A E B F C, circa F C, ſic ſeca-<lb/>re F C, vt pars terminata ad F, ſit ad partem ter-<lb/>minatam ad C, vt 1. </s> <s xml:id="echoid-s1914" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1915" xml:space="preserve">Ratio eſt quia ſic di-<lb/>uidit B D, centrum grauitatis prædictæ figuræ re-<lb/>uolutæ. </s> <s xml:id="echoid-s1916" xml:space="preserve">Nam cum ex Torricellio de dimenſione cy-<lb/>cloidis, & </s> <s xml:id="echoid-s1917" xml:space="preserve">ex Tacquet in diſlertatione de circulorum <lb/>volutationibus propoſit. </s> <s xml:id="echoid-s1918" xml:space="preserve">20. </s> <s xml:id="echoid-s1919" xml:space="preserve">demonſtratione nun-<lb/>quam ſatis laudata, conſtet, A E B F C, eſſe tertiam <lb/>partem cycloidis A B C; </s> <s xml:id="echoid-s1920" xml:space="preserve">& </s> <s xml:id="echoid-s1921" xml:space="preserve">cum ex eodem Torri-<lb/>cellio ſupra citato, ſupponamus centrum grauitatis <lb/>cycloidis ſic ſecare B D, vt pars terminata ad B, ſit <lb/>ad partem terminatam ad D, vt 7. </s> <s xml:id="echoid-s1922" xml:space="preserve">ad 5; </s> <s xml:id="echoid-s1923" xml:space="preserve">& </s> <s xml:id="echoid-s1924" xml:space="preserve">pariter <lb/>cum medium punctum B D, ſit centrum grauitatis <lb/>torius parallelogrammi E C, nempe centrum gra-<lb/>uitatis parallelogramn irelinquat hinc inde 6, par-<lb/>tes, quarum B D, ſupponitur 12; </s> <s xml:id="echoid-s1925" xml:space="preserve">lector in doctri-<lb/>nis A chimed s exercitatus facile agnoſcet, centrum <lb/>grauitatis prædicti exceſſus ſic ſecare B D, vt pars <pb o="107" file="0119" n="119"/> <anchor type="figure" xlink:label="fig-0119-01a" xlink:href="fig-0119-01"/> terminata ad B, ſit ad partem terminatam ad D, <lb/>vt 3. </s> <s xml:id="echoid-s1926" xml:space="preserve">ad 9. </s> <s xml:id="echoid-s1927" xml:space="preserve">ſeù vt 1. </s> <s xml:id="echoid-s1928" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1929" xml:space="preserve">Lector autem ſic edo-<lb/>ctus facile agnoſcet etiam centrum grauitatis figuræ <lb/>B C R, reuolutæ circa R G, &</s> <s xml:id="echoid-s1930" xml:space="preserve">c. </s> <s xml:id="echoid-s1931" xml:space="preserve">ſic ſecare R G, <lb/>vt pars terminata ad R, fit ad partem terminatam <lb/>ad G, vt 1. </s> <s xml:id="echoid-s1932" xml:space="preserve">ad 3.</s> <s xml:id="echoid-s1933" xml:space="preserve"/> </p> <div xml:id="echoid-div96" type="float" level="2" n="4"> <figure xlink:label="fig-0117-01" xlink:href="fig-0117-01a"> <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0117-01"/> </figure> <figure xlink:label="fig-0119-01" xlink:href="fig-0119-01a"> <image file="0119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0119-01"/> </figure> </div> <pb o="108" file="0120" n="120"/> <p> <s xml:id="echoid-s1934" xml:space="preserve">Supponamus autem A B D, eſſe portionem mi-<lb/>norem parabolæ cuiuſcunque reſectæ linea B D, <lb/>diametro parallela, adeovt A D, ſit baſis talis por-<lb/>tionis; </s> <s xml:id="echoid-s1935" xml:space="preserve">& </s> <s xml:id="echoid-s1936" xml:space="preserve">intelligamus portionem A B D, duplicari <lb/>ad partes B D, adeo vt B D, diametro parallela <lb/>euadat axis figuræ A B C; </s> <s xml:id="echoid-s1937" xml:space="preserve">& </s> <s xml:id="echoid-s1938" xml:space="preserve">intelligamus con-<lb/>ſueto modo figuram A B C, rotari circa F C, &</s> <s xml:id="echoid-s1939" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1940" xml:space="preserve">Ex propoſit. </s> <s xml:id="echoid-s1941" xml:space="preserve">15. </s> <s xml:id="echoid-s1942" xml:space="preserve">lib. </s> <s xml:id="echoid-s1943" xml:space="preserve">3. </s> <s xml:id="echoid-s1944" xml:space="preserve">in qua aſſignatur centrum æ-<lb/>quilibrij portionis A B D, in B D, diametro pa-<lb/>rallela, & </s> <s xml:id="echoid-s1945" xml:space="preserve">conſequenter centrum grauitatis figuræ <lb/>A B C, habebimus centrum grauitatis talis ſolidi. </s> <s xml:id="echoid-s1946" xml:space="preserve"><lb/>Si vero intelligamus figuræ A B C, circumſcriptum <lb/>parallelogrammum E C; </s> <s xml:id="echoid-s1947" xml:space="preserve">cum exceſſus ipſius habea-<lb/>mus centrum grauitatis, quia habemus centrum gra-<lb/>uitatis & </s> <s xml:id="echoid-s1948" xml:space="preserve">parallelogrammi, & </s> <s xml:id="echoid-s1949" xml:space="preserve">portionis, & </s> <s xml:id="echoid-s1950" xml:space="preserve">ex pro-<lb/>poſit. </s> <s xml:id="echoid-s1951" xml:space="preserve">15. </s> <s xml:id="echoid-s1952" xml:space="preserve">lib. </s> <s xml:id="echoid-s1953" xml:space="preserve">pri. </s> <s xml:id="echoid-s1954" xml:space="preserve">habemus rationem parallelogram-<lb/>mi ad ſiguram, & </s> <s xml:id="echoid-s1955" xml:space="preserve">conſequenter illius exceſſus ad fi-<lb/>guram; </s> <s xml:id="echoid-s1956" xml:space="preserve">habebimus etiam centrum grauitatis ſolidi <lb/>ex illo exceſſu circa F C, vel illis parallelam. </s> <s xml:id="echoid-s1957" xml:space="preserve">Quod <lb/>vero dictum eſt de figura A B C, patet ex ſupradi-<lb/>ctis intelligendum etiam fore de figura B D C R G. </s> <s xml:id="echoid-s1958" xml:space="preserve"><lb/>Sed ſi talis figura intelligeretur duplicata ad partes <lb/>A D, adeovt baſis D A, euadat axis figuræ N A B. </s> <s xml:id="echoid-s1959" xml:space="preserve"><lb/>Ex propoſit. </s> <s xml:id="echoid-s1960" xml:space="preserve">14. </s> <s xml:id="echoid-s1961" xml:space="preserve">lib. </s> <s xml:id="echoid-s1962" xml:space="preserve">3. </s> <s xml:id="echoid-s1963" xml:space="preserve">habebimus centrum grauita-<lb/>tis annulorum ex N A B, circa O N, vel illi pa-<lb/>rallelam. </s> <s xml:id="echoid-s1964" xml:space="preserve">Idemque intelligendum eſt ſi figura intel-<lb/>ligeretur duplicata vt C D B Q P.</s> <s xml:id="echoid-s1965" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1966" xml:space="preserve">Si vero in ſequenti figura, portio maior A I B D, <lb/>parabolæ cuiuſcunque, cuius baſis A D, intelliga- <pb o="109" file="0121" n="121"/> <anchor type="figure" xlink:label="fig-0121-01a" xlink:href="fig-0121-01"/> tur duplicata quatuor modis ſupra dictis, & </s> <s xml:id="echoid-s1967" xml:space="preserve">intelliga-<lb/>mus generari ſolida prædicta; </s> <s xml:id="echoid-s1968" xml:space="preserve">nihilominus ipſorum <lb/>ſolidorum habebimus centra grauitatis. </s> <s xml:id="echoid-s1969" xml:space="preserve">Ratio eſt <lb/>quia in propoſit. </s> <s xml:id="echoid-s1970" xml:space="preserve">19. </s> <s xml:id="echoid-s1971" xml:space="preserve">& </s> <s xml:id="echoid-s1972" xml:space="preserve">20. </s> <s xml:id="echoid-s1973" xml:space="preserve">lib. </s> <s xml:id="echoid-s1974" xml:space="preserve">3. </s> <s xml:id="echoid-s1975" xml:space="preserve">habemus centra <lb/>æquilibrij maioris portionis parabolæ cuiuſcunque <lb/>reſectæ linea diametro parallela, tam in prædicta li-<lb/>nea diametro parallela, quam in baſi. </s> <s xml:id="echoid-s1976" xml:space="preserve">Vndè etiam <lb/>habemus centra grauitatis duplicatæ portionis qua- <pb o="110" file="0122" n="122"/> tuor illis modis; </s> <s xml:id="echoid-s1977" xml:space="preserve">& </s> <s xml:id="echoid-s1978" xml:space="preserve">conſequenter centra grauitatis il-<lb/>lorum annulorum.</s> <s xml:id="echoid-s1979" xml:space="preserve"/> </p> <div xml:id="echoid-div97" type="float" level="2" n="5"> <figure xlink:label="fig-0121-01" xlink:href="fig-0121-01a"> <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0121-01"/> </figure> </div> <p> <s xml:id="echoid-s1980" xml:space="preserve">Sed ſi in eodem ſchemate, portionem LMIBD, <lb/>parabolæ cuiuſcunque reſectæ duabus lineis L M, <lb/>B D, diametro I K, inter ipſas interceptæ, paral-<lb/>lelis, intelligamus diſponi quatuor prædictis modis, <lb/>& </s> <s xml:id="echoid-s1981" xml:space="preserve">intelligamus conſueto modo, generari quatuor <lb/>ſpecies annulorum, vtſæpe dictum eſt: </s> <s xml:id="echoid-s1982" xml:space="preserve">illorum om-<lb/>nium ſciemus centra grauitatis; </s> <s xml:id="echoid-s1983" xml:space="preserve">hæcque nos docent <lb/>propoſit. </s> <s xml:id="echoid-s1984" xml:space="preserve">21. </s> <s xml:id="echoid-s1985" xml:space="preserve">& </s> <s xml:id="echoid-s1986" xml:space="preserve">22. </s> <s xml:id="echoid-s1987" xml:space="preserve">lib. </s> <s xml:id="echoid-s1988" xml:space="preserve">3. </s> <s xml:id="echoid-s1989" xml:space="preserve">in quibus aſſignantur cen-<lb/>tra æquilibrij illorum ſegmentorum tam in baſi, <lb/>quam in lineis diametro parallelis.</s> <s xml:id="echoid-s1990" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1991" xml:space="preserve">Sed ſi in ſequenti ſchemate ſupponamus A B E F, <lb/>eſſe ſegmentum ſemiparabolæ cuiuſcunque reſectæ <lb/>linea B E, baſi A F, parallela, intelligamuſque <lb/>hoc aptari quatuor conſuetis modis, & </s> <s xml:id="echoid-s1992" xml:space="preserve">vt in ſche-<lb/>mate. </s> <s xml:id="echoid-s1993" xml:space="preserve">Habebimus centra grauitatis ſolidorum ge-<lb/>nitorum modis ſupra explicatis. </s> <s xml:id="echoid-s1994" xml:space="preserve">Videat lector pro-<lb/>poſit. </s> <s xml:id="echoid-s1995" xml:space="preserve">10. </s> <s xml:id="echoid-s1996" xml:space="preserve">lib. </s> <s xml:id="echoid-s1997" xml:space="preserve">3. </s> <s xml:id="echoid-s1998" xml:space="preserve">in qua aſſignatur in E F, centrum <lb/>grauitatis ſegmenti A B C D; </s> <s xml:id="echoid-s1999" xml:space="preserve">& </s> <s xml:id="echoid-s2000" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2001" xml:space="preserve">11. </s> <s xml:id="echoid-s2002" xml:space="preserve">in <lb/>qua aſſignatur centrum æquilibrij ſegmenti ABEF, <lb/>in baſi A F.</s> <s xml:id="echoid-s2003" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2004" xml:space="preserve">Sed ſupponamus F A B E, eſſe vtique ſegmen-<lb/>tum ſemiparabolæ cuiuſcunque, ſed ſic diſpoſitæ vt <lb/>A F, ſit diameter, & </s> <s xml:id="echoid-s2005" xml:space="preserve">B E, parallela diametro, <lb/>adeovt F A B E, ſit ſegmentum ad diametrum, quod <lb/>intelligatur duplicatum quatuor modis vt in ſchema-<lb/>te. </s> <s xml:id="echoid-s2006" xml:space="preserve">Solidorum genitorum conſueto modo ex figuris <lb/>ſic diſpoſitis habebimus centra grauitatis. </s> <s xml:id="echoid-s2007" xml:space="preserve">Quia in <pb o="111" file="0123" n="123"/> <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> propoſit. </s> <s xml:id="echoid-s2008" xml:space="preserve">15. </s> <s xml:id="echoid-s2009" xml:space="preserve">& </s> <s xml:id="echoid-s2010" xml:space="preserve">16. </s> <s xml:id="echoid-s2011" xml:space="preserve">libri 3. </s> <s xml:id="echoid-s2012" xml:space="preserve">habemus centra æqui-<lb/>libiij ſegmenti ad diametrum parabolæ cuiuſcun-<lb/>que, tam in baſi, quam in linea diametro parallela. <lb/></s> <s xml:id="echoid-s2013" xml:space="preserve">Solum videtur nobis lectorem admonendum, cir-<lb/>cumſcriptis figuris parallelogrammis; </s> <s xml:id="echoid-s2014" xml:space="preserve">ſolidum ex <lb/>exceſſu parallelogrammi G D, ſupra figuram <lb/>A B C D, haberetale centrum grauitatis, quod ſic <lb/>ſecet D H, F E, parallelam, vt pars terminata <lb/>ad D, ſit ad partem terminatam ad H, vt nume-<lb/>rus annuli vnitate auctus ad vnitatem. </s> <s xml:id="echoid-s2015" xml:space="preserve">V.</s> <s xml:id="echoid-s2016" xml:space="preserve">g. </s> <s xml:id="echoid-s2017" xml:space="preserve">in pri-<lb/>m, vt 2. </s> <s xml:id="echoid-s2018" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s2019" xml:space="preserve">In ſecundo vt 3. </s> <s xml:id="echoid-s2020" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s2021" xml:space="preserve">Et ſic in <lb/>infinitum. </s> <s xml:id="echoid-s2022" xml:space="preserve">Ratio eſt quia A G B, eſt trilineum <pb o="112" file="0124" n="124"/> ſimile toti trilineo totius ſemiparabolæ, in quo pari-<lb/>ter centrum æquilibrij ſic diuidit A G; </s> <s xml:id="echoid-s2023" xml:space="preserve">& </s> <s xml:id="echoid-s2024" xml:space="preserve">conſe-<lb/>quenter centrum grauitatis duorum trilineorum <lb/>A G B, C D H, ſimul ſic diuidit F E, vt pars ter-<lb/>minata ad F, ſit ad partem terminatam ad E, vt <lb/>numerus trilinei vnitate auctus, ad vnitatem. </s> <s xml:id="echoid-s2025" xml:space="preserve">Idem <lb/>propter eandem rationem, intelligendum eſt de tri-<lb/>lineo C D N, reuoluto vel circa ductam per N, <lb/>ſeù C, ipſi E F, parallelam, vel circa alias paral-<lb/>Ielas E F, extra trilineum ductas.</s> <s xml:id="echoid-s2026" xml:space="preserve"/> </p> <div xml:id="echoid-div98" type="float" level="2" n="6"> <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a"> <image file="0123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0123-01"/> </figure> </div> <p> <s xml:id="echoid-s2027" xml:space="preserve">Sed tandem ſupponamus A B E F, eſſe ſegmen-<lb/>tum intermedium ſemiparabolæ cuiuſcunque reſe-<lb/>ctæ duabus lineis B E, A F, diametro parallelis, <lb/>quod ſegmentum intelligatur diſpoſitum quatuor <lb/>modis. </s> <s xml:id="echoid-s2028" xml:space="preserve">Omnium ſolidorum genitorum conſueto <lb/>modo nobis innoteſcent centra grauitatis ex propo-<lb/>ſit. </s> <s xml:id="echoid-s2029" xml:space="preserve">17. </s> <s xml:id="echoid-s2030" xml:space="preserve">& </s> <s xml:id="echoid-s2031" xml:space="preserve">18. </s> <s xml:id="echoid-s2032" xml:space="preserve">lib. </s> <s xml:id="echoid-s2033" xml:space="preserve">3.</s> <s xml:id="echoid-s2034" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2035" xml:space="preserve">Quot igitur ſolidorum habeantur ex antedicta <lb/>propoſit. </s> <s xml:id="echoid-s2036" xml:space="preserve">centra grauitatis, de quibus neutiquam co-<lb/>gnitio tenebatur, potuit lector animaduertere. </s> <s xml:id="echoid-s2037" xml:space="preserve">Sed <lb/>non minorem vtilitatem capiemus ex ſequenti pro-<lb/>poſitione, quæ, modo ad noſtrum inſtitutum apto, <lb/>explicata, ducet nos in cognitionem centrorum gra-<lb/>uitatis quorundam ſolidorum, quæ vſque nunc geo-<lb/>metria ignorauit. </s> <s xml:id="echoid-s2038" xml:space="preserve">Præcipuè exipſa venabimur cen-<lb/>tra grauitatis omnium ſemifuſorum parabolicorum; <lb/></s> <s xml:id="echoid-s2039" xml:space="preserve">nempe docebimus in quo puncto baſis ſit centrum <lb/>grauitatis ſolidi ex ſemipa@abola quacunque reuo-<lb/>luta circa baſim.</s> <s xml:id="echoid-s2040" xml:space="preserve"/> </p> <pb o="113" file="0125" n="125"/> </div> <div xml:id="echoid-div100" type="section" level="1" n="64"> <head xml:id="echoid-head75" xml:space="preserve">PROPOSITIO XXX.</head> <p style="it"> <s xml:id="echoid-s2041" xml:space="preserve">Annulus ſtrictus figuræ antecedentis propoſitionis æquatur <lb/>quatuor ſolidis, quorum duo ſint, qui oriuntur ex re-<lb/>uolutione ſemifiguræ circa diametrum, alia duo ex re-<lb/>uolutione ſemifiguræ circa, parallelam diametro per ex-<lb/>tremitatem baſis; </s> <s xml:id="echoid-s2042" xml:space="preserve">& </s> <s xml:id="echoid-s2043" xml:space="preserve">hoc tam ſecundum totum, quam <lb/>ſecundum partes proportionales. </s> <s xml:id="echoid-s2044" xml:space="preserve">Item annulus latus ex <lb/>eadem figura æquatur duobus primis ſolidis, & </s> <s xml:id="echoid-s2045" xml:space="preserve">duobus <lb/>annulis latis ex ſemifigura circa parallelam diametro <lb/>extra ipſam.</s> <s xml:id="echoid-s2046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2047" xml:space="preserve">ESto ergo figura A B C, in primis, quæ reuo-<lb/>luatur circa C F, diametro B D, parallelam <lb/>ductam per extremitatem baſis C. </s> <s xml:id="echoid-s2048" xml:space="preserve">Dico annulum <lb/>A B C H G, æqualem eſſe duobus ſolidis ex ſemifi-<lb/>gura D B C, circa B D, & </s> <s xml:id="echoid-s2049" xml:space="preserve">duobus ſolidis ex ea-<lb/>dem D B C, circa C F. </s> <s xml:id="echoid-s2050" xml:space="preserve">Diſponantur iſta ſolida, <lb/>vt in ſchemate, ſec. </s> <s xml:id="echoid-s2051" xml:space="preserve">adeovt contineantur omnia in-<lb/>ter duo plana A B, C D, parallela. </s> <s xml:id="echoid-s2052" xml:space="preserve">Sicuti autem <lb/>taliter ſunt diſpoſita vt duo genita ex reuolutione <lb/>D B C, circa diametrum occupent medium locum, <lb/>ita potuiſſent diſponi quocunque alio modo; </s> <s xml:id="echoid-s2053" xml:space="preserve">& </s> <s xml:id="echoid-s2054" xml:space="preserve">ſicu-<lb/>ti diſponuntur vt vnum aliud tangat, ità potuiſſent <lb/>diſponi vt eſſent ab inuicem diſſita quocunque inter-<lb/>uallo. </s> <s xml:id="echoid-s2055" xml:space="preserve">Diſpoſita autem fuerunt ſic tanquam concin-<lb/>no modo ad inferrenda pulcherrima, quæ ex tali pro-<lb/>poſitione deducentur. </s> <s xml:id="echoid-s2056" xml:space="preserve">Accipiatur in diametro B D, <pb o="114" file="0126" n="126"/> <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> primæ figuræ, quodhbet punctum I, per quod du-<lb/>catur planum L Q, plano A G, parallelum. </s> <s xml:id="echoid-s2057" xml:space="preserve">Cum <lb/>autem C A, in ſecunda figura ſupponatur æqualis <lb/>ipſi B D, in prima, fiat C F, æqualis B I, & </s> <s xml:id="echoid-s2058" xml:space="preserve">per <lb/>E, agatur planum E F, A B, C D, planis paralle-<lb/>lum. </s> <s xml:id="echoid-s2059" xml:space="preserve">Rectangult m L M Q, primæ figuræ, diui-<lb/>ditur in rectangula I M Q, & </s> <s xml:id="echoid-s2060" xml:space="preserve">L I, M Q. </s> <s xml:id="echoid-s2061" xml:space="preserve">Re-<lb/>ctangulum I M Q, eſt æquale rectangulis I M P;</s> <s xml:id="echoid-s2062" xml:space="preserve"> <pb o="115" file="0127" n="127"/> I M, P Q, ſeù MIL. </s> <s xml:id="echoid-s2063" xml:space="preserve">Pariter rectangulum L I, <lb/>M Q, cum ſit æquale rectangulo I M Q, diuiditur <lb/>in eadem rectangula. </s> <s xml:id="echoid-s2064" xml:space="preserve">Quare colligemus, rectan-<lb/>gulum L M Q, æquale eſſe duobus rectangulis <lb/>I M P, & </s> <s xml:id="echoid-s2065" xml:space="preserve">duobus rectangulis M I L. </s> <s xml:id="echoid-s2066" xml:space="preserve">Rectangulum <lb/>I M P, in prima figura, æquatur rectangulo EGK, <lb/>in ſecunda; </s> <s xml:id="echoid-s2067" xml:space="preserve">vnde duo rectangula I M P, primæ, <lb/>æquantur duobus rectangulis E G k, R S F, ſe-<lb/>cundæ: </s> <s xml:id="echoid-s2068" xml:space="preserve">item duo rectangula M I L, primæ, æquan-<lb/>tur duobus rectangulis L O M, N P Q, ſecundæ; <lb/></s> <s xml:id="echoid-s2069" xml:space="preserve">vnde omnia quatuor rectangula primæ, æquantur <lb/>quatuor rectangulis ſecundæ. </s> <s xml:id="echoid-s2070" xml:space="preserve">Ergo etiam rectangu-<lb/>lum L M Q, primæ, æquabitur rectangulis E G k; </s> <s xml:id="echoid-s2071" xml:space="preserve"><lb/>L O M; </s> <s xml:id="echoid-s2072" xml:space="preserve">N P Q; </s> <s xml:id="echoid-s2073" xml:space="preserve">R S F, ſecundæ. </s> <s xml:id="echoid-s2074" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2075" xml:space="preserve">aimilla <lb/>circularis L M Q, ſolidi primæ figuræ, æquabitur <lb/>armillis circularibus E G k; </s> <s xml:id="echoid-s2076" xml:space="preserve">R S F, & </s> <s xml:id="echoid-s2077" xml:space="preserve">circulis <lb/>L O M, N P Q, ſecundæ. </s> <s xml:id="echoid-s2078" xml:space="preserve">Cumautem puncta I, <lb/>& </s> <s xml:id="echoid-s2079" xml:space="preserve">E, ſumpta ſint ad libitum, inuentaque ſit æqua-<lb/>litas inter plana prædicta; </s> <s xml:id="echoid-s2080" xml:space="preserve">rectè deducemus, necdum <lb/>omnes armillas circulares ſolidi primæ figuræ plano <lb/>A G, parallelas, ęquales eſſe omnibus armillis cir-<lb/>cularibus, & </s> <s xml:id="echoid-s2081" xml:space="preserve">omnibus circulis ſolidorum ſecundæ; </s> <s xml:id="echoid-s2082" xml:space="preserve"><lb/>ſed etiam ſolidum primæ ęquari omnibus ſolidis ſe-<lb/>cundæ.</s> <s xml:id="echoid-s2083" xml:space="preserve"/> </p> <div xml:id="echoid-div100" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0126-01"/> </figure> </div> <p> <s xml:id="echoid-s2084" xml:space="preserve">Quod autem probatum fuit de totis, patet eo-<lb/>dem modo probari poſſe de partibus proportionali-<lb/>bus; </s> <s xml:id="echoid-s2085" xml:space="preserve">quia non diſſimili modo probabimus partem ſo-<lb/>lidi primæ contentam inter plana parallela L Q, <lb/>A G, ęquari parti ſolidorum ſecundæ, conten- <pb o="116" file="0128" n="128"/> tæ inter plana A B, E F, parallela. </s> <s xml:id="echoid-s2086" xml:space="preserve">Quare patet <lb/>propoſitum.</s> <s xml:id="echoid-s2087" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2088" xml:space="preserve">Secunda pars propoſitionis; </s> <s xml:id="echoid-s2089" xml:space="preserve">nempe quod in ſe-<lb/>quenti figura, annulus latus ex figura A B C, eirca <lb/>T S, reuoluta ſit ęqualis duobus ſolidis ex D B C, <lb/> <anchor type="figure" xlink:label="fig-0128-01a" xlink:href="fig-0128-01"/> reuoluta circa B D, & </s> <s xml:id="echoid-s2090" xml:space="preserve">duobus ex eadem reuolu-<lb/>ta circa T S; </s> <s xml:id="echoid-s2091" xml:space="preserve">facta pręparatione ſimili anteceden-<lb/>ti, lector facile proprio Marte cognoſcet, diſcur-<lb/>rendo vt nos ſupra fecimus. </s> <s xml:id="echoid-s2092" xml:space="preserve">Quare patet propo-<lb/>ſitum.</s> <s xml:id="echoid-s2093" xml:space="preserve"/> </p> <div xml:id="echoid-div101" type="float" level="2" n="2"> <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/PVNDER1Y/figures/0128-01"/> </figure> </div> </div> <div xml:id="echoid-div103" type="section" level="1" n="65"> <head xml:id="echoid-head76" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s2094" xml:space="preserve">Nec etiam pręſens propoſitio in tanta vniuerſa-<lb/>litate propoſita, videtur vnica conſtructione proba-<lb/>ri poſſe niſi methodo indiuiſibilium. </s> <s xml:id="echoid-s2095" xml:space="preserve">In figuris vero <lb/>particularibus, factis particularibus præparationi-<lb/>bus, probari etiam poterit modo Archimedeo. </s> <s xml:id="echoid-s2096" xml:space="preserve">Si <lb/>enim ſupponamus A B C, eſſe figuram ad partes <pb o="117" file="0129" n="129"/> B, deficientem, lector in geometricis peritus fa-<lb/>cile agnoſcet probari poſſe modo Archimedeo.</s> <s xml:id="echoid-s2097" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2098" xml:space="preserve">Ex his ergo, & </s> <s xml:id="echoid-s2099" xml:space="preserve">ex dictis in lib. </s> <s xml:id="echoid-s2100" xml:space="preserve">4. </s> <s xml:id="echoid-s2101" xml:space="preserve">de Infinit. </s> <s xml:id="echoid-s2102" xml:space="preserve">Parab. <lb/></s> <s xml:id="echoid-s2103" xml:space="preserve">colligemus ſæpe replicatam doctrinam; </s> <s xml:id="echoid-s2104" xml:space="preserve">nimirum. </s> <s xml:id="echoid-s2105" xml:space="preserve"><lb/>annulum primę figurę, & </s> <s xml:id="echoid-s2106" xml:space="preserve">ſolida ſimul ſecundę, eſſe <lb/>quantitates proportionaliter analogas tam in ma-<lb/>gnitudine, quam in grauitate. </s> <s xml:id="echoid-s2107" xml:space="preserve">Vnde cum ſolidum <lb/>primę ſit magnitudo ſic analoga cum figura A B C. </s> <s xml:id="echoid-s2108" xml:space="preserve"><lb/>Sequitur etiam omnia ſolida ſecundę figurę ſimul, eſ-<lb/>ſe analoga cum figura A B C, tam in magnitudi-<lb/>ne, quam in grauitate. </s> <s xml:id="echoid-s2109" xml:space="preserve">Cum autem facile etiam ſic <lb/>cognoſcere prędictorum ſolidorum ſimul ſecundę <lb/>figurę eſſe centrum grauitatis in V X (vt hoc enim <lb/>ſequatur ſic ex induſtria diſpoſita fuerunt;) </s> <s xml:id="echoid-s2110" xml:space="preserve">ergo <lb/>centrum grauitatis prędictorum ſolidorum ſimul ita <lb/>ſecabit V X, vt centrum grauitatis figurę A B C, <lb/>ſecat B D. </s> <s xml:id="echoid-s2111" xml:space="preserve">Ex hac doctrina adinueniemus centrum <lb/>grauitatis nonnullorum ſolidorum. </s> <s xml:id="echoid-s2112" xml:space="preserve">Sed prius adno-<lb/>tabim s vnum particulare in ſequenti ſcholio, quod <lb/>exiſtimamus P. </s> <s xml:id="echoid-s2113" xml:space="preserve">Marium Bettinum Societatis lesù ſi <lb/>viueret, libenter excepiſſe.</s> <s xml:id="echoid-s2114" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div104" type="section" level="1" n="66"> <head xml:id="echoid-head77" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s2115" xml:space="preserve">Galileus, in poſtremis Dialogis pag. </s> <s xml:id="echoid-s2116" xml:space="preserve">apud nos 28, <lb/>loquitur de paradoxo quodam geometrico, in quo <lb/>intelligit demonſtrare circuli circumferentiam. <lb/></s> <s xml:id="echoid-s2117" xml:space="preserve">ęqualem eſſe puncto. </s> <s xml:id="echoid-s2118" xml:space="preserve">De hoc paradoxo veſtigia Ga-<lb/>lilei ſequentes, locuti ſumus & </s> <s xml:id="echoid-s2119" xml:space="preserve">in appendicula ſexa- <pb o="118" file="0130" n="130"/> ginta problematum geometricorum, & </s> <s xml:id="echoid-s2120" xml:space="preserve">in hoc ope-<lb/>re in ſchol. </s> <s xml:id="echoid-s2121" xml:space="preserve">3. </s> <s xml:id="echoid-s2122" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2123" xml:space="preserve">10, & </s> <s xml:id="echoid-s2124" xml:space="preserve">in ſchol. </s> <s xml:id="echoid-s2125" xml:space="preserve">3 propoſit. <lb/></s> <s xml:id="echoid-s2126" xml:space="preserve">18. </s> <s xml:id="echoid-s2127" xml:space="preserve">At P. </s> <s xml:id="echoid-s2128" xml:space="preserve">Bettinus ſupradictus in tom. </s> <s xml:id="echoid-s2129" xml:space="preserve">3. </s> <s xml:id="echoid-s2130" xml:space="preserve">ſui Ærarij <lb/>pareg. </s> <s xml:id="echoid-s2131" xml:space="preserve">geom. </s> <s xml:id="echoid-s2132" xml:space="preserve">ſchol. </s> <s xml:id="echoid-s2133" xml:space="preserve">prim. </s> <s xml:id="echoid-s2134" xml:space="preserve">& </s> <s xml:id="echoid-s2135" xml:space="preserve">alibi, admonet parado-<lb/>xum pręſens nequaquam intelligendum eſſe geome-<lb/>tricè, ſed phyſicè: </s> <s xml:id="echoid-s2136" xml:space="preserve">nam geometricè loquendo, Eucli-<lb/>des, doctrinaque eius tradita in defin. </s> <s xml:id="echoid-s2137" xml:space="preserve">3. </s> <s xml:id="echoid-s2138" xml:space="preserve">lib. </s> <s xml:id="echoid-s2139" xml:space="preserve">5. </s> <s xml:id="echoid-s2140" xml:space="preserve">Ele-<lb/>ment. </s> <s xml:id="echoid-s2141" xml:space="preserve">ab omnibuſque paſſim recepta huic aſſerto ad-<lb/>uerſatur. </s> <s xml:id="echoid-s2142" xml:space="preserve">Proportio enim eſt duarum magnitudi-<lb/>num eiuſdem generis, quatenus ad quantitatem per-<lb/>tinet, mutua quædam habitudo. </s> <s xml:id="echoid-s2143" xml:space="preserve">Quando ergo com-<lb/>paratur circumferentia cum puncto, & </s> <s xml:id="echoid-s2144" xml:space="preserve">colligitur æ-<lb/>qualitas, fit comparatio impropria, & </s> <s xml:id="echoid-s2145" xml:space="preserve">quæ non eſt, <lb/>cum ſint quantitates diuerſorum generum. </s> <s xml:id="echoid-s2146" xml:space="preserve">At non <lb/>deeſt alius medius terminus geometricus oſtendens <lb/>Galilei Parallogiſmum ſi intelligat geometrice lo-<lb/>qui, non phyſicè. </s> <s xml:id="echoid-s2147" xml:space="preserve">Hicque nobis ſuppeditatur ab an-<lb/>tecedenti propoſitione, antecedentibuſque ſolidis. </s> <s xml:id="echoid-s2148" xml:space="preserve"><lb/>Nam ad modum Galilei diſcurrentes, in maximum <lb/>abſurdum incideremus: </s> <s xml:id="echoid-s2149" xml:space="preserve">oſtenderemus enim circuli <lb/>circumferentiam æqualem eſſe duabus circuli cir-<lb/>cumferentijs, quarum vnaquæque priori eſſet ęqua-<lb/>lis, & </s> <s xml:id="echoid-s2150" xml:space="preserve">inſuper duobus punctis. </s> <s xml:id="echoid-s2151" xml:space="preserve">Cum enim proba-<lb/>tum ſit, ſolidum ex A B C, in prima figura, æqua-<lb/>le eſſe quatuor ſolidis in ſecunda figura tam ſecun-<lb/>dum totum, quamſecendum partes proportionales; </s> <s xml:id="echoid-s2152" xml:space="preserve"><lb/>ſequeretur ex doctrina Galilei, quod cum tandem. </s> <s xml:id="echoid-s2153" xml:space="preserve"><lb/>ſolidum A B C H G, in prima figura deſinat in cir-<lb/>cumferentia circuli, cuius diameter B H; </s> <s xml:id="echoid-s2154" xml:space="preserve">item <pb o="119" file="0131" n="131"/> quatuor ſolidorum in ſecunda figura, duo extrema <lb/>deſinant in circumferentijs, quarum diametri C H, <lb/>T D, media verò in punctis Y, Z; </s> <s xml:id="echoid-s2155" xml:space="preserve">ſequeretur in-<lb/>quam, circun ferentiam B H, æqualem eſſe cir-<lb/>cumferentijs C H, T D, & </s> <s xml:id="echoid-s2156" xml:space="preserve">punctis Y, Z. </s> <s xml:id="echoid-s2157" xml:space="preserve">Quod <lb/>eſt abſurdiſſimum. </s> <s xml:id="echoid-s2158" xml:space="preserve">Nam cum circumferentiæ ſint vt <lb/>diametri, & </s> <s xml:id="echoid-s2159" xml:space="preserve">cum B H, C H, & </s> <s xml:id="echoid-s2160" xml:space="preserve">T D, ſint ęqua-<lb/>les; </s> <s xml:id="echoid-s2161" xml:space="preserve">ſequitur etiam circumferentias circulorum, quo-<lb/>rum diametri C H, T D, duplas eſſe circumfe-<lb/>rentiæ, cuius diameter B H. </s> <s xml:id="echoid-s2162" xml:space="preserve">Erroneus ergo eſt di-<lb/>ſcurſus, ex quo hauritur circumferentiam B H, <lb/>ęquari circumferentijs C H, T D, & </s> <s xml:id="echoid-s2163" xml:space="preserve">punctis <lb/>Y, Z; </s> <s xml:id="echoid-s2164" xml:space="preserve">& </s> <s xml:id="echoid-s2165" xml:space="preserve">conſequenter erroneus eſt Galilei diſ-<lb/>curſus.</s> <s xml:id="echoid-s2166" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div105" type="section" level="1" n="67"> <head xml:id="echoid-head78" xml:space="preserve">PROPOSITIO XXXI.</head> <head xml:id="echoid-head79" style="it" xml:space="preserve">Semifuſi parabolici cuiuſcunque, centrum grauitatis <lb/>reperire.</head> <p> <s xml:id="echoid-s2167" xml:space="preserve">ESto A B D, ſemiparabola quæcunque in prima <lb/>figura, cuius diameter A D, baſis B D, quæ <lb/>reuoluta circa baſim B D, generet ſemifuſum para-<lb/>bolicum; </s> <s xml:id="echoid-s2168" xml:space="preserve">huius oportet centrum grauitatis aſſigna-<lb/>re. </s> <s xml:id="echoid-s2169" xml:space="preserve">Semiparabola A B D, intelligatur duplicata <lb/>ad partes baſis B D, & </s> <s xml:id="echoid-s2170" xml:space="preserve">figura A B C, ex duabus <lb/>ſemiparabolis conſtans intelligatur rotari circa F C, <lb/>B D, parallelam. </s> <s xml:id="echoid-s2171" xml:space="preserve">Item in ſecunda figura intelligan-<lb/>tur quatuor ſolida ſic diſpoſita, vt duo extrema A H, <pb o="120" file="0132" n="132"/> <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> T B, ſint illa, quæ otiuntur ex ſemipar abola D B C, <lb/>reuoluta circa C F, duo vero media ſint illa, quæ <lb/>oriuntur ex reuolutione ſemiparabolæ A B D, cir-<lb/>ca baſim B D, nempe ſint duo ſemifuſi parabolici <lb/>ex data ſemiparabola. </s> <s xml:id="echoid-s2172" xml:space="preserve">Ex propoſit. </s> <s xml:id="echoid-s2173" xml:space="preserve">anteced. </s> <s xml:id="echoid-s2174" xml:space="preserve">con-<lb/>ſtat quatuor ſolida ſecundæ figuræ eſſe proportiona-<lb/>liter analoga cum ſolido A B C H G, primæ. </s> <s xml:id="echoid-s2175" xml:space="preserve">Sed <lb/>ſolidum A B C H G, primæ eſt proportionalit er <pb o="121" file="0133" n="133"/> analogum cum figura A B C, conſtante ex duabus <lb/>ſemiparabolis. </s> <s xml:id="echoid-s2176" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2177" xml:space="preserve">quatuor ſolida ſecundæ fi-<lb/>guræ ſimul erunt proportionaliter analoga cum fi-<lb/>gura A B C. </s> <s xml:id="echoid-s2178" xml:space="preserve">Sed ex ſchol. </s> <s xml:id="echoid-s2179" xml:space="preserve">2. </s> <s xml:id="echoid-s2180" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2181" xml:space="preserve">2. </s> <s xml:id="echoid-s2182" xml:space="preserve">lib. </s> <s xml:id="echoid-s2183" xml:space="preserve">3. </s> <s xml:id="echoid-s2184" xml:space="preserve">cen-<lb/>trum grauitatis figuræ A B C, ſic diuidit B D, vt <lb/>pars terminata ad B, ſit ad partem terminatam ad <lb/>D, vt numerus parabolæ ternario auctus ad nume-<lb/>rum parabolæ vnitate auctum. </s> <s xml:id="echoid-s2185" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2186" xml:space="preserve">centrum gra-<lb/>uitatis quatuor ſolidorum ſecundæ figuræ ſimul ſic <lb/>ſecabit V X, vt pars terminata ad V, ſit ad partem <lb/>terminatam ad X, vt numerus parabolæ ternario <lb/>auctus ad numerum parabolæ vnitate auctum. </s> <s xml:id="echoid-s2187" xml:space="preserve">Sup-<lb/>ponatur à perito geometra, ſic diuiſa in ℟. </s> <s xml:id="echoid-s2188" xml:space="preserve">Item ex <lb/>propoſit. </s> <s xml:id="echoid-s2189" xml:space="preserve">18. </s> <s xml:id="echoid-s2190" xml:space="preserve">lib. </s> <s xml:id="echoid-s2191" xml:space="preserve">4. </s> <s xml:id="echoid-s2192" xml:space="preserve">de infin. </s> <s xml:id="echoid-s2193" xml:space="preserve">parab. </s> <s xml:id="echoid-s2194" xml:space="preserve">conſtat centrum <lb/>grauitatis ſolidi ex ſemiparabola D B C, in prima <lb/>figura circa C F, ſic diuidere F C, vt pars termi-<lb/>nata ad F, ſit ad partem terminatam ad C, vt <lb/>duplus numerus parabolæ ternario auctus, ad du-<lb/>plum numerum vnitate auctum. </s> <s xml:id="echoid-s2195" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2196" xml:space="preserve">centrum <lb/>grauitatis ſolidorum extremorum in ſecunda figura, <lb/>ſic ſecabunt lineas circa quas ſemiparabolæ intelli-<lb/>guntur reuolutæ. </s> <s xml:id="echoid-s2197" xml:space="preserve">Cum ergo talia ſolida ſint ex inſti-<lb/>tuto ſic diſpoſita, vt commune amborum centrum <lb/>grauitatis cadat in V X: </s> <s xml:id="echoid-s2198" xml:space="preserve">ſi ergo V X, ſic diuida-<lb/>tur in +, vt V +, ſit ad + X, vt duplus nume-<lb/>rus parabolæ ternario auctus, ad duplum numerum <lb/>parabolæ vnitate auctum; </s> <s xml:id="echoid-s2199" xml:space="preserve">+ erit centrum grauita-<lb/>tis illorum ſolidorum ſimul. </s> <s xml:id="echoid-s2200" xml:space="preserve">Cum ergo in VX, ſit <lb/>centrum grauitatis tam quatuor ſolidorum ſimul, <pb o="122" file="0134" n="134"/> <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> quam duorum extremorum; </s> <s xml:id="echoid-s2201" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2202" xml:space="preserve">reliquorum <lb/>duorum mediorum ſimul erit in V X, centrum gra-<lb/>uitatis. </s> <s xml:id="echoid-s2203" xml:space="preserve">Hoc autem reperictur ſi fiat reciprocè vt <lb/>duo media ad duo extrema, ſic +℟, ad ℟ 2. </s> <s xml:id="echoid-s2204" xml:space="preserve">Cum <lb/>eigo ex corollar. </s> <s xml:id="echoid-s2205" xml:space="preserve">prim. </s> <s xml:id="echoid-s2206" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2207" xml:space="preserve">4. </s> <s xml:id="echoid-s2208" xml:space="preserve">lib. </s> <s xml:id="echoid-s2209" xml:space="preserve">3. </s> <s xml:id="echoid-s2210" xml:space="preserve">ſit ſolidum <lb/>vnum medium ad vnum ſolidorum extremorum, <lb/>nompe duo media ad duo extrema, vt numerus para-<lb/>bolæ ad numerum parabolæ vnitate auctum; </s> <s xml:id="echoid-s2211" xml:space="preserve">ſi fiat <pb o="123" file="0135" n="135"/> vt numerus parabolæ ad numerum parabolæ vnitate <lb/>auctum, ſic +℟, ad ℟ 2. </s> <s xml:id="echoid-s2212" xml:space="preserve">Erit 2, centrum gra-<lb/>uitatis duorum ſolidorum mediorum ſimul. </s> <s xml:id="echoid-s2213" xml:space="preserve">Sed cum <lb/>hæc fuerint ſic diſpoſita vt centrum grauitatis vniuſ-<lb/>cuiuſque ipſorum ſic ſecet illorumaxim; </s> <s xml:id="echoid-s2214" xml:space="preserve">ſi ergo axis <lb/>B D, ſemifufi in prima figura, ſic ſecetur in T, vt <lb/>B T, ſit ad T D, vt V 2, ad 2 +: </s> <s xml:id="echoid-s2215" xml:space="preserve">erit T, cen-<lb/>trum grauitatis ſemifuſi A B C, orti ex reuolutione <lb/>ſemiparabolæ A B D, circa baſim B D. </s> <s xml:id="echoid-s2216" xml:space="preserve">Quod <lb/>erat reperiendum.</s> <s xml:id="echoid-s2217" xml:space="preserve"/> </p> <div xml:id="echoid-div105" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0132-01"/> </figure> <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/PVNDER1Y/figures/0134-01"/> </figure> </div> </div> <div xml:id="echoid-div107" type="section" level="1" n="68"> <head xml:id="echoid-head80" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2218" xml:space="preserve">Inuentio huius centri grauitatis non continet ali-<lb/>quam ſeriem ordinatam. </s> <s xml:id="echoid-s2219" xml:space="preserve">Verum tamen eſt, quod <lb/>quilibet numero potert exprimere rationem in qua <lb/>ſecetur B D, à centrograuitatis tais ſemifuſi, ſi or-<lb/>dinem obſeruauerit, quem nostenemus in inuentio-<lb/>ne talis centri in ſemifuſo parabolico quadratico. </s> <s xml:id="echoid-s2220" xml:space="preserve">In <lb/>primo enim ſemifuſo, cum ſit conus, iam patet B D, <lb/>ſic ſecari vt pars ad B, ſit ad partem ad D, vt 3. <lb/></s> <s xml:id="echoid-s2221" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s2222" xml:space="preserve">In quadratico verò, conſequenter ad ſupra <lb/>dicta, ſi B D, ſic ſecetur in S, vt B S, ſit ad <lb/>S D, vt numerus parabolæ ternario auctus ad nu-<lb/>merum parabolæ vnitate auctum; </s> <s xml:id="echoid-s2223" xml:space="preserve">quarum B D, erit <lb/>8, talium B S, erit 5, & </s> <s xml:id="echoid-s2224" xml:space="preserve">quarum B D, erit 12, <lb/>talium B S, erit 7, cum dimidia. </s> <s xml:id="echoid-s2225" xml:space="preserve">Item ſi ſecetur <lb/>in I, vt B I, ſit ad I D, vt duplus numerus ter-<lb/>nario auctus, ad duplum numerum vnitate auctum, <pb o="124" file="0136" n="136"/> <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> quarum B D, erit 12, B I, erit 7. </s> <s xml:id="echoid-s2226" xml:space="preserve">Ergo quarum <lb/>B D, erit 12, talium B I, erit 7; </s> <s xml:id="echoid-s2227" xml:space="preserve">B S, 7, cum <lb/>dimidio IS, dimidium; </s> <s xml:id="echoid-s2228" xml:space="preserve">I D, 5; </s> <s xml:id="echoid-s2229" xml:space="preserve">D S, 4. </s> <s xml:id="echoid-s2230" xml:space="preserve">cum <lb/>dimidio. </s> <s xml:id="echoid-s2231" xml:space="preserve">Fiat ergo vt numerus parabolæ ad nume-<lb/>rum parabolæ vnitate auctum ſic I S, ad S T. </s> <s xml:id="echoid-s2232" xml:space="preserve">Er-<lb/>go<unsure/> quarum partium I S, eſt 2, talium S T, erit tria. <lb/></s> <s xml:id="echoid-s2233" xml:space="preserve">Cum ergo quarum BD, erat 12, talium B S, eſſet 7, <lb/>cum dimidio, & </s> <s xml:id="echoid-s2234" xml:space="preserve">IS, dimidium. </s> <s xml:id="echoid-s2235" xml:space="preserve">Ergo quarum B D, <pb o="125" file="0137" n="137"/> erit 24; </s> <s xml:id="echoid-s2236" xml:space="preserve">IS, erit 1; </s> <s xml:id="echoid-s2237" xml:space="preserve">& </s> <s xml:id="echoid-s2238" xml:space="preserve">B S, 15. </s> <s xml:id="echoid-s2239" xml:space="preserve">Et qualium B D, <lb/>erit 48, talium I S, erit 2, & </s> <s xml:id="echoid-s2240" xml:space="preserve">BS, 30. </s> <s xml:id="echoid-s2241" xml:space="preserve">Sed qualium <lb/>IS, erat 2, talium S T, erat 3. </s> <s xml:id="echoid-s2242" xml:space="preserve">Ergo qualium B D, <lb/>erit 48, talium B T, erit 33, & </s> <s xml:id="echoid-s2243" xml:space="preserve">T D, 15. </s> <s xml:id="echoid-s2244" xml:space="preserve">Ergo <lb/>centrum grauitatis ſemifuſi parabolici quadratici ſic <lb/>diuidit B D, in T, vt B T, ſit ad T D, vt 33, ad <lb/>15; </s> <s xml:id="echoid-s2245" xml:space="preserve">& </s> <s xml:id="echoid-s2246" xml:space="preserve">ſubtriplando terminos, vt 11, ad 5.</s> <s xml:id="echoid-s2247" xml:space="preserve"/> </p> <div xml:id="echoid-div107" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0136-01"/> </figure> </div> <p> <s xml:id="echoid-s2248" xml:space="preserve">Sed non ſolum ſupradicta methodo reperiemus <lb/>centrum grauitatis ſemifuſi parabolici, ſed etiam ex-<lb/>ceſſus cylindri ipſi circumſcripti ſupra ipſum; </s> <s xml:id="echoid-s2249" xml:space="preserve">nem-<lb/>pe centrum grauitatis ſolidi ex trilineo E B A, in pri-<lb/>ma figura, reuoluto circa baſim ſemiparabolæ B D. <lb/></s> <s xml:id="echoid-s2250" xml:space="preserve">Cum autem tale centrum facilius inuen@atur<unsure/> alio mo-<lb/>do, ideo hunc experiemur in parabola quadratica in <lb/>numeris. </s> <s xml:id="echoid-s2251" xml:space="preserve">Supponamus ergo BD, ſectam bifariam <lb/>in S, & </s> <s xml:id="echoid-s2252" xml:space="preserve">in T, ſic vt BT, ſit ad T D, vt 11, ad 5. </s> <s xml:id="echoid-s2253" xml:space="preserve"><lb/>adeo vt T, ſit centrum grauitatis ſemifuſi A B C. </s> <s xml:id="echoid-s2254" xml:space="preserve">Er-<lb/>go quarum BD, erit 16, talium ST, erit 3, & </s> <s xml:id="echoid-s2255" xml:space="preserve"><lb/>B S, 8. </s> <s xml:id="echoid-s2256" xml:space="preserve">Ergo qualium B D, erit 37, cum tertia par-<lb/>te, talium ST, erit 7, & </s> <s xml:id="echoid-s2257" xml:space="preserve">BS, 18, cum duobus ter-<lb/>tijs. </s> <s xml:id="echoid-s2258" xml:space="preserve">Cum autem ex ſchol. </s> <s xml:id="echoid-s2259" xml:space="preserve">prim. </s> <s xml:id="echoid-s2260" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2261" xml:space="preserve">14. </s> <s xml:id="echoid-s2262" xml:space="preserve">lib. </s> <s xml:id="echoid-s2263" xml:space="preserve">2. </s> <s xml:id="echoid-s2264" xml:space="preserve"><lb/>ſit exceſſus cylindri circumſcripti ſemifuſo ad ipſum <lb/>vt 7, ad 8, & </s> <s xml:id="echoid-s2265" xml:space="preserve">ſi fiat vt talis exceſſus ad ſemifuſum, <lb/>ſic reriprocè T S, ad S I, ſit 1, centrum grauitatis <lb/>prædicti exceſlus; </s> <s xml:id="echoid-s2266" xml:space="preserve">erit SI, 8, qualium BS, eſt 18, <lb/>cum duobus tertijs. </s> <s xml:id="echoid-s2267" xml:space="preserve">Ergo talium reliqua BI, erit <lb/>10, cum duobus tertijs. </s> <s xml:id="echoid-s2268" xml:space="preserve">Qualium ergo BD, eſt <lb/>37, cum tertia parte, erit BI, 10, cum duabus <lb/>tertijs partibus, & </s> <s xml:id="echoid-s2269" xml:space="preserve">reliqua DI, 26, cum duo bus ter- <pb o="126" file="0138" n="138"/> tijs. </s> <s xml:id="echoid-s2270" xml:space="preserve">Ergo centrum grauitatis prædicti exceſſus ſecat <lb/>BD, in I, in prædicta ratione.</s> <s xml:id="echoid-s2271" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div109" type="section" level="1" n="69"> <head xml:id="echoid-head81" xml:space="preserve">PROPOSITIO XXXII.</head> <p style="it"> <s xml:id="echoid-s2272" xml:space="preserve">Semifuſi hyperbolici cuiuſcunque, ſuppoſita hyperbolœ quæ-<lb/>dratura, poſſumus centrum grauit atis reperire.</s> <s xml:id="echoid-s2273" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2274" xml:space="preserve">SVpponamus in ſeq. </s> <s xml:id="echoid-s2275" xml:space="preserve">figura D B C, eſſe ſemi-<lb/>hyperbolam, cuius diameter C D, baſis B D, <lb/>latus tranſuerſum CZ, centrum S. </s> <s xml:id="echoid-s2276" xml:space="preserve">Dico, ſuppo-<lb/>ſita hyperbolæ quadratura, nos poſſe reperire cen-<lb/>trum grauitatis ſemifuſi hyperbolici A B C. </s> <s xml:id="echoid-s2277" xml:space="preserve">Diſpo-<lb/>nantur quatuor ſolida vt ſupra, & </s> <s xml:id="echoid-s2278" xml:space="preserve">vt in ſecunda figu-<lb/>ra, ſed duo extrema A H, T B, intelligantur eſſe <lb/>annulos non ſtrictos, vt ſchema exprimit, ſed latos, <lb/>ortos ex rotatione ſemihyperbolæ D B C, ſeq. </s> <s xml:id="echoid-s2279" xml:space="preserve">fi-<lb/>guræ circa ſecundam diametrum T S. </s> <s xml:id="echoid-s2280" xml:space="preserve">Ergo horum <lb/>quatuor ſolidorum ſic diſpoſitorum vt in illa fi-<lb/>gura habemus centrum grauitatis in V X, quia ha-<lb/>bemus centrum grauitatis ſolidi A B C Z H G, ſeq. <lb/></s> <s xml:id="echoid-s2281" xml:space="preserve">figuræ, quod ex propoſit. </s> <s xml:id="echoid-s2282" xml:space="preserve">30. </s> <s xml:id="echoid-s2283" xml:space="preserve">eſt proportionali-<lb/>ter analogum cum quatuor ſolidis ſecundæ figu-<lb/>ræ. </s> <s xml:id="echoid-s2284" xml:space="preserve">Habemus autem centrum grauitatis ſolidi <lb/>A B C Z H G, quia habemus in baſi B D, centrum <lb/>grauitatis figuræ A B C, conſtantis ex duabus ſe-<lb/>mihyperbolis, ex propoſit. </s> <s xml:id="echoid-s2285" xml:space="preserve">12. </s> <s xml:id="echoid-s2286" xml:space="preserve">in qua, ſuppoſita hy-<lb/>perbol quadratura, inuentum fuit centrum æqui-<lb/>librij ſemihype bolæ D B C, in baſi B D, & </s> <s xml:id="echoid-s2287" xml:space="preserve">con- <pb o="127" file="0139" n="139"/> <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> ſequenter centrum grauitatis in B D, ipſius A B C. <lb/></s> <s xml:id="echoid-s2288" xml:space="preserve">Pariter, cum ex ſchol. </s> <s xml:id="echoid-s2289" xml:space="preserve">3. </s> <s xml:id="echoid-s2290" xml:space="preserve">prop. </s> <s xml:id="echoid-s2291" xml:space="preserve">26. </s> <s xml:id="echoid-s2292" xml:space="preserve">habeamus centrum <lb/>grauitatis, ſine ſuppoſitione quadraturæ hyperbolæ, <lb/>annuli lati ex ſemihy perbola D B C, in hac figu-<lb/>ra reuoluta circa ſecundam diametrum T S; </s> <s xml:id="echoid-s2293" xml:space="preserve">habe-<lb/>bimus conſequenter ad ſupra dicta, in ſecunda figu-<lb/>ra, in V X, centrum grauitatis duorum ſolidorum <lb/>extremorum, nempe duorum annulorum latorum <lb/>A H, T B. </s> <s xml:id="echoid-s2294" xml:space="preserve">Inſuper ex ſchol. </s> <s xml:id="echoid-s2295" xml:space="preserve">2. </s> <s xml:id="echoid-s2296" xml:space="preserve">prop. </s> <s xml:id="echoid-s2297" xml:space="preserve">32. </s> <s xml:id="echoid-s2298" xml:space="preserve">ſuppoſita hy-<lb/>perbolæ quadratura, habemus in hac figura ra-<lb/>tionem, quam habet annulus latus D B C Z H ℟, <lb/>ad ſemifuſum A B C; </s> <s xml:id="echoid-s2299" xml:space="preserve">& </s> <s xml:id="echoid-s2300" xml:space="preserve">conſequenter in ſecunda <lb/>figura, habemus rationem duorum ſolidorum extre-<lb/>morum ſimul ad duo ſolida media. </s> <s xml:id="echoid-s2301" xml:space="preserve">Ergo conſequen-<lb/>ter habebimus in V X, ſecundæ figuræ centrum gra-<lb/>uitatis duorum ſolidorum mediorum ſimul. </s> <s xml:id="echoid-s2302" xml:space="preserve">Et pari-<lb/>ter in hac figura, habebimus centrum in B D, ſe-<lb/>mifuſi A B C. </s> <s xml:id="echoid-s2303" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2304" xml:space="preserve">c.</s> <s xml:id="echoid-s2305" xml:space="preserve"/> </p> <div xml:id="echoid-div109" 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/PVNDER1Y/figures/0139-01"/> </figure> </div> <pb o="128" file="0140" n="140"/> </div> <div xml:id="echoid-div111" type="section" level="1" n="70"> <head xml:id="echoid-head82" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s2306" xml:space="preserve">Sed non ſolum habebimus tale centrum grauita-<lb/>tis, ſed etiam centrum grauitatis exceſſus cylindri <lb/>E C, ſupra ipſum.</s> <s xml:id="echoid-s2307" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div112" type="section" level="1" n="71"> <head xml:id="echoid-head83" xml:space="preserve">PROPOSITIO XXXIII.</head> <p style="it"> <s xml:id="echoid-s2308" xml:space="preserve">Annuli stricti ex ſemiparabola quacunque, cuius expe-<lb/>nens ſit numerus par, reuoluta circa parallelam dia-<lb/>metro ductam per extremitatem baſis, centrum grauita-<lb/>tis aſſignare.</s> <s xml:id="echoid-s2309" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2310" xml:space="preserve">ESto ſemiparabola quæcunque D B C, cuius ex-<lb/>ponens ſit numerus par, ſitque eius diameter <lb/>B D, baſis D C, & </s> <s xml:id="echoid-s2311" xml:space="preserve">intelligamus D B C, rotari cir-<lb/>ca C F, parallelam diametro B D, ductam per C: <lb/></s> <s xml:id="echoid-s2312" xml:space="preserve">oporteat annuli producti centrum grauitatis reperi-<lb/>re. </s> <s xml:id="echoid-s2313" xml:space="preserve">Intelligamus ſemiparabolam duplicari ad partes <lb/>B D, vt fiat tota parabola A B C, & </s> <s xml:id="echoid-s2314" xml:space="preserve">intelligamus <lb/>hanc totam rotari circa F C, vt fiat annulus <lb/>A B C H G. </s> <s xml:id="echoid-s2315" xml:space="preserve">Cum hic annulus ex propoſit. </s> <s xml:id="echoid-s2316" xml:space="preserve">30. </s> <s xml:id="echoid-s2317" xml:space="preserve">ſit <lb/>æqualis quatuor ſolidis dictis in illa propoſitione, di-<lb/>ſponantur hęc ſolida vt in ſecunda figura. </s> <s xml:id="echoid-s2318" xml:space="preserve">Ergo ho-<lb/>rum quatuor folidorum ſimul centrum grauitatis ita <lb/>ſecabit V X, vt ſecat B D, centrum grauitatis pa-<lb/>rabolæ A B C. </s> <s xml:id="echoid-s2319" xml:space="preserve">Sed ex ſchol. </s> <s xml:id="echoid-s2320" xml:space="preserve">prim. </s> <s xml:id="echoid-s2321" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2322" xml:space="preserve">2. </s> <s xml:id="echoid-s2323" xml:space="preserve">lib. </s> <s xml:id="echoid-s2324" xml:space="preserve">3. </s> <s xml:id="echoid-s2325" xml:space="preserve"><lb/>hoc centrum ita ſecat B D, vt pars terminata ad B, <pb o="129" file="0141" n="141"/> <anchor type="figure" xlink:label="fig-0141-01a" xlink:href="fig-0141-01"/> ſit ad partem terminatam ad D, vt numerus para-<lb/>bolæ vnitate auctus ad numerum parabolæ. </s> <s xml:id="echoid-s2326" xml:space="preserve">Ergo ſi <lb/>V X, ſic ſecetur in ℟, vt ſit V ℟, ad ℟ X, vt nu-<lb/>merus parabolæ, ſeù annuli vnitate auctus, ad nu-<lb/>merum parabolæ erit ℟, centrum grauitatis ſoli-<lb/>dorum quatuor ſimul ſumptorum. </s> <s xml:id="echoid-s2327" xml:space="preserve">Pariter, quo-<lb/>niam ex propoſit. </s> <s xml:id="echoid-s2328" xml:space="preserve">14. </s> <s xml:id="echoid-s2329" xml:space="preserve">lib. </s> <s xml:id="echoid-s2330" xml:space="preserve">4. </s> <s xml:id="echoid-s2331" xml:space="preserve">centrum grauitatis co-<lb/>noidis A B C, ſic in prima figura diuidit B D, vt <pb o="130" file="0142" n="142"/> pars terminata ad B, ſit ad partem terminatam ad <lb/>D, vt dimidium numeri conoidis vnitate aucti, ad <lb/>dimidium numeri conoidis; </s> <s xml:id="echoid-s2332" xml:space="preserve">& </s> <s xml:id="echoid-s2333" xml:space="preserve">cum ſic in ſecunda fi-<lb/>gura ſint diſpoſita ex induſtria duo conoidea me-<lb/>dia, vt centrum grauitatis amborum ſimul ſit in <lb/>V X; </s> <s xml:id="echoid-s2334" xml:space="preserve">ſi hæc ſic ſecetur in 2, vt ſit V 2, ad 2 X, vt <lb/>dimidium numeri conoidis aucti vnitate ad dimi-<lb/>dium numeri conoidis; </s> <s xml:id="echoid-s2335" xml:space="preserve">erit 2, centrum grauitatis <lb/>duorum conoideorum ſimul. </s> <s xml:id="echoid-s2336" xml:space="preserve">Cum ergo in V X, ſit <lb/>centrum grauitatis tam quatuor ſolidorum ſimul, <lb/>quam duorum conoideorum; </s> <s xml:id="echoid-s2337" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2338" xml:space="preserve">in V X, erit <lb/>centrum grauitatis duorum annulorum extremo-<lb/>rum. </s> <s xml:id="echoid-s2339" xml:space="preserve">Si ergo fiat vt duos annulos ſimul, ad duo co-<lb/>noidea ſimul, vel vt vnus annulus ad vnum conoi-<lb/>des, nempe ex coroll. </s> <s xml:id="echoid-s2340" xml:space="preserve">3. </s> <s xml:id="echoid-s2341" xml:space="preserve">lib. </s> <s xml:id="echoid-s2342" xml:space="preserve">3. </s> <s xml:id="echoid-s2343" xml:space="preserve">vt numerus conoidis <lb/>ternario auctus ad numerum conoidis vnitate au-<lb/>ctum, ſic reciprocè 2 ℟, ad ℟ +. </s> <s xml:id="echoid-s2344" xml:space="preserve">Erit + centrum <lb/>grauitatis duorum annulorum ſimul. </s> <s xml:id="echoid-s2345" xml:space="preserve">Et ſi in prima fi-<lb/>gura ſecetur F C, in puncto in ratione F +, ad <lb/>+ X. </s> <s xml:id="echoid-s2346" xml:space="preserve">Erit illud inuentum centrum grauitatis illius <lb/>annuli. </s> <s xml:id="echoid-s2347" xml:space="preserve">Res de ſe patet. </s> <s xml:id="echoid-s2348" xml:space="preserve">Quare &</s> <s xml:id="echoid-s2349" xml:space="preserve">c.</s> <s xml:id="echoid-s2350" xml:space="preserve"/> </p> <div xml:id="echoid-div112" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0141-01"/> </figure> </div> </div> <div xml:id="echoid-div114" type="section" level="1" n="72"> <head xml:id="echoid-head84" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2351" xml:space="preserve">Sed nec etiam inuentio huius centri continet ali-<lb/>quam pulchram ſeriem; </s> <s xml:id="echoid-s2352" xml:space="preserve">quilibet tamen aſſignabit in <lb/>numeris rationem ſecundum quam diuiditur F C, <lb/>à centro grauitatis prædicti annuli, ſi notabit ſequen-<lb/>tem ordinem quem tenemus in annulo ſemiparabolæ <pb o="131" file="0143" n="143"/> <anchor type="figure" xlink:label="fig-0143-01a" xlink:href="fig-0143-01"/> quadraticæ. </s> <s xml:id="echoid-s2353" xml:space="preserve">In illa enim V X, ſic ſecatur in ℟, <lb/>centro grauitatis quatuor ſolidorum ſimul, vt V ℟, <lb/>ſit ad ℟ X, vt 3. </s> <s xml:id="echoid-s2354" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s2355" xml:space="preserve">In 2. </s> <s xml:id="echoid-s2356" xml:space="preserve">vero vt V 2, ſit <lb/>ad 2 X, vt 2, ad 1, nempe vt 3, cum tertia par-<lb/>te, ad 1, cum duobustertijs. </s> <s xml:id="echoid-s2357" xml:space="preserve">Ergo qualium V X, <lb/>eſt 5, talium V ℟, eſt 3, & </s> <s xml:id="echoid-s2358" xml:space="preserve">V 2, eſt 3, cum ter-<lb/>sia parte; </s> <s xml:id="echoid-s2359" xml:space="preserve">℟ 2, tertia pars; </s> <s xml:id="echoid-s2360" xml:space="preserve">& </s> <s xml:id="echoid-s2361" xml:space="preserve">qualium V X, eſt <lb/>15, talium V ℟, eſt 9; </s> <s xml:id="echoid-s2362" xml:space="preserve">V 2, 10; </s> <s xml:id="echoid-s2363" xml:space="preserve">& </s> <s xml:id="echoid-s2364" xml:space="preserve">℟ 2, vnitas.</s> <s xml:id="echoid-s2365" xml:space="preserve"> <pb o="132" file="0144" n="144"/> Qualium ergo ℟ 2, eſt 5, talium V X, eſt 75, <lb/>V ℟, 45, & </s> <s xml:id="echoid-s2366" xml:space="preserve">V 2, 50. </s> <s xml:id="echoid-s2367" xml:space="preserve">Cum ergo qualium ℟ 2, eſt <lb/>5, talium ℟ +, ſit 3. </s> <s xml:id="echoid-s2368" xml:space="preserve">Ergo qualium V X, eſt 75, <lb/>talium V +, erit 42. </s> <s xml:id="echoid-s2369" xml:space="preserve">V X, ergo centrum grauita-<lb/>tis duorum annulorum ſecabitur in +, & </s> <s xml:id="echoid-s2370" xml:space="preserve">conſe-<lb/>quenter F C, ſic ſecabitur à centro grauitatis præ-<lb/>dicti annuli quadratici v. </s> <s xml:id="echoid-s2371" xml:space="preserve">g. </s> <s xml:id="echoid-s2372" xml:space="preserve">in N, vt F N, ſit ad <lb/>N C, vt 42, ad 33; </s> <s xml:id="echoid-s2373" xml:space="preserve">nempe ſubtriplando termi-<lb/>nos, vt 14, ad 11.</s> <s xml:id="echoid-s2374" xml:space="preserve"/> </p> <div xml:id="echoid-div114" 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/PVNDER1Y/figures/0143-01"/> </figure> </div> <p> <s xml:id="echoid-s2375" xml:space="preserve">Habito autem centro grauitatis talis annuli, non <lb/>ignorabitur centrum grauitatis conici B C H, orti <lb/>ex rotatione trilinei B F C, circa baſim F C. </s> <s xml:id="echoid-s2376" xml:space="preserve">Quod <lb/>licet poſſit haberi independenter ab inuento centro <lb/>grauitatis annuli, vt patet ex ſuperioribus, conſide-<lb/>rando perſe, ſolidum ortum ex reuolutione exceſſus <lb/>parallelogrammi E C, ſupra parabolam A B C, <lb/>circa F C, faciendo diſpoſtionem vt ſupra; </s> <s xml:id="echoid-s2377" xml:space="preserve">faci-<lb/>lius tamen inuenietur ex centro annuli ex ſemipara-<lb/>bola prius inuento. </s> <s xml:id="echoid-s2378" xml:space="preserve">Nam habetur etiam centrum <lb/>grauitatis totius cylindri D H; </s> <s xml:id="echoid-s2379" xml:space="preserve">& </s> <s xml:id="echoid-s2380" xml:space="preserve">ex propoſit. </s> <s xml:id="echoid-s2381" xml:space="preserve">15. <lb/></s> <s xml:id="echoid-s2382" xml:space="preserve">lib. </s> <s xml:id="echoid-s2383" xml:space="preserve">2. </s> <s xml:id="echoid-s2384" xml:space="preserve">habetur ratio prædicti annuli ad conicum. </s> <s xml:id="echoid-s2385" xml:space="preserve"><lb/>B C H. </s> <s xml:id="echoid-s2386" xml:space="preserve">Hoc autem ſic in numeris inuenietur in co-<lb/>nico quadratico: </s> <s xml:id="echoid-s2387" xml:space="preserve">ſupponamus in ſecunda figura (in <lb/>qua faciemus operationem in V X, & </s> <s xml:id="echoid-s2388" xml:space="preserve">quam in ip-<lb/>ſa faciemus intelligemus factam in F C) V X, eſſe <lb/>ſectam bifariam in ℟, & </s> <s xml:id="echoid-s2389" xml:space="preserve">in 2, vt V 2, ſit ad 2 X, <lb/>vt 14, ad 11. </s> <s xml:id="echoid-s2390" xml:space="preserve">Ergo ℟, erit centrum grauitatis to-<lb/>tius cylindri annulo circumſcripti, & </s> <s xml:id="echoid-s2391" xml:space="preserve">2, erit ex di-<lb/>ctis, centrum grauitatis annuli. </s> <s xml:id="echoid-s2392" xml:space="preserve">Ergo qualium to- <pb o="133" file="0145" n="145"/> ta V X, eſt 25; </s> <s xml:id="echoid-s2393" xml:space="preserve">V 2, 14; </s> <s xml:id="echoid-s2394" xml:space="preserve">& </s> <s xml:id="echoid-s2395" xml:space="preserve">2 X, 11; </s> <s xml:id="echoid-s2396" xml:space="preserve">talium V ℟, <lb/>erit 12, cum dimidia; </s> <s xml:id="echoid-s2397" xml:space="preserve">& </s> <s xml:id="echoid-s2398" xml:space="preserve">℟ 2, 1, cum dimidia. </s> <s xml:id="echoid-s2399" xml:space="preserve">Cum <lb/>ergo ex ſecunda parte propoſit. </s> <s xml:id="echoid-s2400" xml:space="preserve">15, lib. </s> <s xml:id="echoid-s2401" xml:space="preserve">ſecun. </s> <s xml:id="echoid-s2402" xml:space="preserve">ſit di-<lb/>uidendo conicus B C H, ad annulum vt 2, ad 10, <lb/>ſeù vt 1, ad 5; </s> <s xml:id="echoid-s2403" xml:space="preserve">& </s> <s xml:id="echoid-s2404" xml:space="preserve">ſi fiat reciprocè vt conicus, <lb/>ad annulum, nempe vt 1, ad 5, ſic 2 ℟, ad ℟ +, ſit <lb/>+, centrum grauitatis conici; </s> <s xml:id="echoid-s2405" xml:space="preserve">& </s> <s xml:id="echoid-s2406" xml:space="preserve">cum ſit vt 1, ad 5, <lb/>ſic vnum cum dimidio ad 7, cum dimidio. </s> <s xml:id="echoid-s2407" xml:space="preserve">Ergo <lb/>+ ℟, erit 7, cum dimidio. </s> <s xml:id="echoid-s2408" xml:space="preserve">Quare reliqua V +, erit <lb/>5, & </s> <s xml:id="echoid-s2409" xml:space="preserve">+ X, 20. </s> <s xml:id="echoid-s2410" xml:space="preserve">Ergo V X, ſic ſecatur in +, & </s> <s xml:id="echoid-s2411" xml:space="preserve">F C, <lb/>v. </s> <s xml:id="echoid-s2412" xml:space="preserve">g. </s> <s xml:id="echoid-s2413" xml:space="preserve">in N, à centro grauitatis conici B C H, vt <lb/>C N, ſit ad N F, vt 20, ad 5, ſeù vt 4. </s> <s xml:id="echoid-s2414" xml:space="preserve">ad 1.</s> <s xml:id="echoid-s2415" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div116" type="section" level="1" n="73"> <head xml:id="echoid-head85" xml:space="preserve">PROPOSITIO XXXIV.</head> <p style="it"> <s xml:id="echoid-s2416" xml:space="preserve">Annuli stricti orti ex reuolutione ſemihyperbolæ, vt in an-<lb/>teced. </s> <s xml:id="echoid-s2417" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2418" xml:space="preserve">ſuppoſita hyperbolæ quadratura, poſſumus <lb/>centrum grauitatis aſſignare.</s> <s xml:id="echoid-s2419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2420" xml:space="preserve">SEd ſupponamus D B C, eſſe ſemihyperbolam, <lb/>&</s> <s xml:id="echoid-s2421" xml:space="preserve">c. </s> <s xml:id="echoid-s2422" xml:space="preserve">Dico etiam nos poſſe aſſignare centrum <lb/>grauitatis annuli ſtricti ex ſemihyperbola D B C, <lb/>circa F C. </s> <s xml:id="echoid-s2423" xml:space="preserve">Reuoluta enim hyperbola A B C, tota <lb/>circa F C, vt fiat annulus A B C H G, cum hic ſit <lb/>æqualis quatuor ſolidis diſpoſicis vt in ſecunda figu-<lb/>ra, vt ſæpe dictum eſt; </s> <s xml:id="echoid-s2424" xml:space="preserve">ergo ex propoſit. </s> <s xml:id="echoid-s2425" xml:space="preserve">22. </s> <s xml:id="echoid-s2426" xml:space="preserve">in qua <lb/>aſſignatur centrum grauitatis in B D, hyperbolæ <lb/>A B C, habebimus etiam centrum grauitatis qua-<lb/>tuor illorum ſolidorum ſimul diſpoſitorum. </s> <s xml:id="echoid-s2427" xml:space="preserve">Sit hoc <pb o="134" file="0146" n="146"/> centrum ℟. </s> <s xml:id="echoid-s2428" xml:space="preserve">Item ex prop. </s> <s xml:id="echoid-s2429" xml:space="preserve">13. </s> <s xml:id="echoid-s2430" xml:space="preserve">& </s> <s xml:id="echoid-s2431" xml:space="preserve">14. </s> <s xml:id="echoid-s2432" xml:space="preserve">habemus centrum <lb/>grauitatis conoidis hyperbolici, & </s> <s xml:id="echoid-s2433" xml:space="preserve">conſequenter <lb/>duorum conoideorum diſpoſitorum vt in ſecunda fi-<lb/>gura. </s> <s xml:id="echoid-s2434" xml:space="preserve">Sit hoc 2. </s> <s xml:id="echoid-s2435" xml:space="preserve">Pariter, quoniam ex propoſit. </s> <s xml:id="echoid-s2436" xml:space="preserve">12. <lb/></s> <s xml:id="echoid-s2437" xml:space="preserve">habemus centrum æquilibrij ſemihyperbolæ D B C, <lb/>in D C; </s> <s xml:id="echoid-s2438" xml:space="preserve">habebimus etiam ex propoſit. </s> <s xml:id="echoid-s2439" xml:space="preserve">4 lib 3. </s> <s xml:id="echoid-s2440" xml:space="preserve">ra-<lb/>tionem quam habent ſolida ex ſemihyperbola D B C, <lb/>reuoluta circa B D, & </s> <s xml:id="echoid-s2441" xml:space="preserve">F C, ad inuicem; </s> <s xml:id="echoid-s2442" xml:space="preserve">& </s> <s xml:id="echoid-s2443" xml:space="preserve">conſe-<lb/>quenter habebimus rationem, quam habent in ſe-<lb/>cunda figura duo ſolida extrema ad duo media. </s> <s xml:id="echoid-s2444" xml:space="preserve">Si er-<lb/>go fiat vt duo ſolida extrema ad duo media ſic reci-<lb/>procè 2 ℟, ad ℟ +. </s> <s xml:id="echoid-s2445" xml:space="preserve">Erit +, centrum grauitatis <lb/>duorum annulorum ſimul. </s> <s xml:id="echoid-s2446" xml:space="preserve">Vndè patet quomodo <lb/>poſſimus habere centrum grauitatis vnius annuli ſoli <lb/>ex ſemihyperbola. </s> <s xml:id="echoid-s2447" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2448" xml:space="preserve">c.</s> <s xml:id="echoid-s2449" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div117" type="section" level="1" n="74"> <head xml:id="echoid-head86" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2450" xml:space="preserve">Habito centro grauitatis annuli, non ignorabitur <lb/>centrum grauitatis conici hyperbolici B C H; </s> <s xml:id="echoid-s2451" xml:space="preserve">pro <lb/>quare conſideretur ſcholium antecedentis propoſi-<lb/>tionis, diſcurſuſque in ipſo expoſitus imitetur.</s> <s xml:id="echoid-s2452" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2453" xml:space="preserve">Q oniam autem ex doctrinis ſuperius traditis li-<lb/>cet nobis colligere centra grauitatis aliquorum ſoli-<lb/>dorum, de quibus nunquam geometria locuta eſt; <lb/></s> <s xml:id="echoid-s2454" xml:space="preserve">ideo vt hoc expeditius fiat, opere pretium ducimus <lb/>doctrinas ſuperius traditas aptius ordinare, regulam <lb/>quandam generalem exponendo. </s> <s xml:id="echoid-s2455" xml:space="preserve">Sciendum ergo <lb/>eſt, quatuor eſſe centra grauitatis, quorum tribus <pb o="135" file="0147" n="147"/> datis, licet quartum colligere. </s> <s xml:id="echoid-s2456" xml:space="preserve">Nempe cèntrum <lb/>grauitatis figuræ A B C, circa diametrum: </s> <s xml:id="echoid-s2457" xml:space="preserve">centrum <lb/>æquilibrij ſemifiguræ D B C, in D C: </s> <s xml:id="echoid-s2458" xml:space="preserve">centrum <lb/>grauitatis ſolidi A B C, orti ex reuolutione ſemi-<lb/>figuræ A B D, circa B D: </s> <s xml:id="echoid-s2459" xml:space="preserve">& </s> <s xml:id="echoid-s2460" xml:space="preserve">centrum grauitatis ſe-<lb/>miſiguræ D B C, reuolutæ circa F C. </s> <s xml:id="echoid-s2461" xml:space="preserve">Nam datis <lb/>tribus primis, patebit dari quartum ſic. </s> <s xml:id="echoid-s2462" xml:space="preserve">Dato cen-<lb/>tro grauitatis figuræ A B C, datur centrum graui-<lb/>tatis ſolidi orti ex gyratione A B C, circa C F; </s> <s xml:id="echoid-s2463" xml:space="preserve">& </s> <s xml:id="echoid-s2464" xml:space="preserve"><lb/>conſequenter centrum grauitatis quatuor ſolidorum <lb/>diſpoſitorum in ſecunda figura. </s> <s xml:id="echoid-s2465" xml:space="preserve">Secundo dato cen-<lb/>tro æquilibrij ſemifiguræ D B C, in D C, dabitur <lb/>ratio ſolidi ex ſemifigura D B C, reuoluta circa D B, <lb/>ad ſolidum ex eadem reuoluta circa C F; </s> <s xml:id="echoid-s2466" xml:space="preserve">ex propo-<lb/>ſit. </s> <s xml:id="echoid-s2467" xml:space="preserve">4. </s> <s xml:id="echoid-s2468" xml:space="preserve">lib. </s> <s xml:id="echoid-s2469" xml:space="preserve">3. </s> <s xml:id="echoid-s2470" xml:space="preserve">& </s> <s xml:id="echoid-s2471" xml:space="preserve">conſequenter in ſecunda figura dabi-<lb/>tur ratio duorum ſolidorum mediorum ad duo extre-<lb/>ma. </s> <s xml:id="echoid-s2472" xml:space="preserve">Tertio dato centro grauitatis ſolidi A B C, da-<lb/>bitur etiam in ſecunda figura centrum duorum ſoli-<lb/>dorum mediorum ſimul. </s> <s xml:id="echoid-s2473" xml:space="preserve">Si ergo ℟, ſit centrum <lb/>quatuor ſimul, iam datum, & </s> <s xml:id="echoid-s2474" xml:space="preserve">2, ſit centrum duo-<lb/>rum mediorum etiam datum, ſi fiat 2 ℟, ad ℟ +, in <lb/>ratione data, nempe vt duo ſolida extrema, ad duo <lb/>media, vel vt vnum ad vnum; </s> <s xml:id="echoid-s2475" xml:space="preserve">erit + centrum gra-<lb/>uitatis duorum extremorum, vel vnius extremi, quod <lb/>eſt quartum, quod quærebatur. </s> <s xml:id="echoid-s2476" xml:space="preserve">Ita ſuppoſitis dari <lb/>tribus quibuſuis quatuor iam dictorum, patebit ſimi-<lb/>li diſcurſu, dari quartum. </s> <s xml:id="echoid-s2477" xml:space="preserve">His animaduerſis.</s> <s xml:id="echoid-s2478" xml:space="preserve"/> </p> <pb o="136" file="0148" n="148"/> </div> <div xml:id="echoid-div118" type="section" level="1" n="75"> <head xml:id="echoid-head87" xml:space="preserve">PROPOSITIO XXXV.</head> <p style="it"> <s xml:id="echoid-s2479" xml:space="preserve">Annuli stricti orti ex reuolutione ſegmenti ſemiparabolæ <lb/>cuiuſcunque, cuius exponens ſit numerus par, reſectæ <lb/>linea baſi parallela, circa lineam ductam parallelam dia-<lb/>metro per extremitatem baſis poſſumus centrum graui-<lb/>tatis aſſignare.</s> <s xml:id="echoid-s2480" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2481" xml:space="preserve">PArabola quæcunque A B C, cuius numerus <lb/>par, ſit ſecta L M, A C, parallela, & </s> <s xml:id="echoid-s2482" xml:space="preserve">intelli-<lb/>gamus D I M C, rotari circa C F. </s> <s xml:id="echoid-s2483" xml:space="preserve">Dico annuli <lb/>orti nos poſſe aſſignare centrum grauitatis. </s> <s xml:id="echoid-s2484" xml:space="preserve">Nam <lb/>cum ex propoſit. </s> <s xml:id="echoid-s2485" xml:space="preserve">10. </s> <s xml:id="echoid-s2486" xml:space="preserve">lib. </s> <s xml:id="echoid-s2487" xml:space="preserve">3. </s> <s xml:id="echoid-s2488" xml:space="preserve">habeamus centrum gra-<lb/>uitatis ſegmenti parabolæ A L M C, habebimus <lb/>etiam ex ſupra dictis, centrum grauitatis annuli <lb/>A L M C O Q G; </s> <s xml:id="echoid-s2489" xml:space="preserve">& </s> <s xml:id="echoid-s2490" xml:space="preserve">conſequenter quatuor ſolido-<lb/>rum diſpoſitorum vt in ſecunda figura. </s> <s xml:id="echoid-s2491" xml:space="preserve">Ex propo-<lb/>ſit. </s> <s xml:id="echoid-s2492" xml:space="preserve">11. </s> <s xml:id="echoid-s2493" xml:space="preserve">eiuſdem libri habemus centrum æquilibtij fi-<lb/>guræ D I M C, in baſi D C. </s> <s xml:id="echoid-s2494" xml:space="preserve">Ex ſchol. </s> <s xml:id="echoid-s2495" xml:space="preserve">propoſit. <lb/></s> <s xml:id="echoid-s2496" xml:space="preserve">15. </s> <s xml:id="echoid-s2497" xml:space="preserve">lib. </s> <s xml:id="echoid-s2498" xml:space="preserve">3. </s> <s xml:id="echoid-s2499" xml:space="preserve">habemus centrum grauitatis ſolidi <lb/>A L M C. </s> <s xml:id="echoid-s2500" xml:space="preserve">Ergo quartum non ignorabitur; </s> <s xml:id="echoid-s2501" xml:space="preserve">nempe <lb/>centrum grauitatis ſolidi orti ex rotatione D I M C, <lb/>circa N C. </s> <s xml:id="echoid-s2502" xml:space="preserve">Quod & </s> <s xml:id="echoid-s2503" xml:space="preserve">c.</s> <s xml:id="echoid-s2504" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div119" type="section" level="1" n="76"> <head xml:id="echoid-head88" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2505" xml:space="preserve">Cum autem habeamus centrum grauitatis cylin-<lb/>dri IV; </s> <s xml:id="echoid-s2506" xml:space="preserve">& </s> <s xml:id="echoid-s2507" xml:space="preserve">rationem ex ſchol. </s> <s xml:id="echoid-s2508" xml:space="preserve">2. </s> <s xml:id="echoid-s2509" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2510" xml:space="preserve">11. </s> <s xml:id="echoid-s2511" xml:space="preserve">lib.</s> <s xml:id="echoid-s2512" xml:space="preserve"> <pb o="137" file="0149" n="149"/> <anchor type="figure" xlink:label="fig-0149-01a" xlink:href="fig-0149-01"/> 3. </s> <s xml:id="echoid-s2513" xml:space="preserve">quam habet cylindrus IV, ad conicum M C O; <lb/></s> <s xml:id="echoid-s2514" xml:space="preserve">habemus etiam in N C, centrum grauitatis talis <lb/>conici.</s> <s xml:id="echoid-s2515" xml:space="preserve"/> </p> <div xml:id="echoid-div119" type="float" level="2" n="1"> <figure xlink:label="fig-0149-01" xlink:href="fig-0149-01a"> <image file="0149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0149-01"/> </figure> </div> </div> <div xml:id="echoid-div121" type="section" level="1" n="77"> <head xml:id="echoid-head89" xml:space="preserve">PROPOSITIO XXXVI.</head> <p style="it"> <s xml:id="echoid-s2516" xml:space="preserve">Annuli ſtricti orti ex rotatione ſegmenti ſemihyperbolæ re-<lb/>ſectæ linea baſi parallela (ſuppoſita ſegmenti quadratu- <pb o="138" file="0150" n="150"/> ra) modo in propoſit. </s> <s xml:id="echoid-s2517" xml:space="preserve">anteced. </s> <s xml:id="echoid-s2518" xml:space="preserve">explicato, poſſumus cen-<lb/>trum grauitatis aſſignare.</s> <s xml:id="echoid-s2519" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2520" xml:space="preserve">VIce parabolæ propoſit. </s> <s xml:id="echoid-s2521" xml:space="preserve">antèced. </s> <s xml:id="echoid-s2522" xml:space="preserve">ſit hyperbola. <lb/></s> <s xml:id="echoid-s2523" xml:space="preserve">Dico nos poſſe aſſignare centrum grauitatis <lb/>annuli ſtricti D I M C O P V. </s> <s xml:id="echoid-s2524" xml:space="preserve">Nam cum ex propo-<lb/>ſit. </s> <s xml:id="echoid-s2525" xml:space="preserve">22, habeamus centrum grauitatis tam hyperbolæ <lb/>A B C, quam hyperbolæ L B M, & </s> <s xml:id="echoid-s2526" xml:space="preserve">cum ex ſup-<lb/>poſitione quadraturæ facile poſſimus elicere ratio-<lb/>nem ſegmenti A L M C, ad hyperbolam L B M; </s> <s xml:id="echoid-s2527" xml:space="preserve"><lb/>habebimus centrum grauitatis ſegmenti hyperbolæ <lb/>A L M C; </s> <s xml:id="echoid-s2528" xml:space="preserve">& </s> <s xml:id="echoid-s2529" xml:space="preserve">conſequenter ſolidi A L M C O Q G: </s> <s xml:id="echoid-s2530" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s2531" xml:space="preserve">conſequenter quatuor ſolidorum diſpoſitorum vt <lb/>in ſecunda figura. </s> <s xml:id="echoid-s2532" xml:space="preserve">Item ex ſchol. </s> <s xml:id="echoid-s2533" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2534" xml:space="preserve">15. </s> <s xml:id="echoid-s2535" xml:space="preserve">habe-<lb/>mus centrum æquilibrij in D C, ſegmenti D I M C. </s> <s xml:id="echoid-s2536" xml:space="preserve"><lb/>Ex propoſit. </s> <s xml:id="echoid-s2537" xml:space="preserve">17, habemus centrum grauitatis ſolidi <lb/>A L M C. </s> <s xml:id="echoid-s2538" xml:space="preserve">Ergo nec etiam in præſenti quartum <lb/>ignorabitur; </s> <s xml:id="echoid-s2539" xml:space="preserve">nempe centrum grauitatis annuli <lb/>D I M C O P V. </s> <s xml:id="echoid-s2540" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2541" xml:space="preserve">c.</s> <s xml:id="echoid-s2542" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div122" type="section" level="1" n="78"> <head xml:id="echoid-head90" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2543" xml:space="preserve">Ex prædicto centro inuento, & </s> <s xml:id="echoid-s2544" xml:space="preserve">ex ratione cylin-<lb/>dri IV, reperta in citato ſchol. </s> <s xml:id="echoid-s2545" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2546" xml:space="preserve">15. </s> <s xml:id="echoid-s2547" xml:space="preserve">per <lb/>conuerſionem rationis, ad conicum M C O, re-<lb/>periemus in N C, centrum grauitatis conici M C O, <lb/>prædicti.</s> <s xml:id="echoid-s2548" xml:space="preserve"/> </p> <pb o="139" file="0151" n="151"/> </div> <div xml:id="echoid-div123" type="section" level="1" n="79"> <head xml:id="echoid-head91" xml:space="preserve">PROPOSITIO XXXVII.</head> <p style="it"> <s xml:id="echoid-s2549" xml:space="preserve">Variorum ſegmentorum infinitorum fuſorum par abolicorum, <lb/>poſſumus centra grauitatis aſſignare.</s> <s xml:id="echoid-s2550" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2551" xml:space="preserve">ESto parabola quæcunque R B A, quam intelli-<lb/>gamus rotari circa R A, adeo vt generetur <lb/>quilibet fuſus parabolicus. </s> <s xml:id="echoid-s2552" xml:space="preserve">Dico variorum ſegmen-<lb/>torum huius fuſi nos poſſe centra granitatis aſſi-<lb/>gnare.</s> <s xml:id="echoid-s2553" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2554" xml:space="preserve">In primis parabola ſecetur linea I T, diametro <lb/>E B, parallela, poſſumus aſſignare centrum graui-<lb/>tatis partis fuſi ortæ ex reuolutione ſegmenti ad dia-<lb/>metrum I T B E, circa I E. </s> <s xml:id="echoid-s2555" xml:space="preserve">Nam in primis ex pro-<lb/>poſit. </s> <s xml:id="echoid-s2556" xml:space="preserve">16. </s> <s xml:id="echoid-s2557" xml:space="preserve">lib. </s> <s xml:id="echoid-s2558" xml:space="preserve">3. </s> <s xml:id="echoid-s2559" xml:space="preserve">habemus centrum æquilibrij in I E, <lb/>baſi ſegmenti I T B E, nempe centrum grauitatis <lb/>duplicatæ figuræ I T B E, ad partes I E. </s> <s xml:id="echoid-s2560" xml:space="preserve">Secundo, <lb/>ex propoſit. </s> <s xml:id="echoid-s2561" xml:space="preserve">18. </s> <s xml:id="echoid-s2562" xml:space="preserve">lib. </s> <s xml:id="echoid-s2563" xml:space="preserve">4. </s> <s xml:id="echoid-s2564" xml:space="preserve">habemus centrum grauita. <lb/></s> <s xml:id="echoid-s2565" xml:space="preserve">tis portionis annuli orti ex reuolutione ſegmenti <lb/>I T B E, circa B V. </s> <s xml:id="echoid-s2566" xml:space="preserve">Tertio ex ſchol. </s> <s xml:id="echoid-s2567" xml:space="preserve">3. </s> <s xml:id="echoid-s2568" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2569" xml:space="preserve"><lb/>15. </s> <s xml:id="echoid-s2570" xml:space="preserve">lib. </s> <s xml:id="echoid-s2571" xml:space="preserve">3. </s> <s xml:id="echoid-s2572" xml:space="preserve">habemus centrum ſegmenti I T B E, in <lb/>E B, nempe habemus rationem, quam habet ſoli-<lb/>dum ex I T B E, ſegmento reuoluto circa V B, ad <lb/>ſolidum ex eodem ſegmento reuoluto circa I E. </s> <s xml:id="echoid-s2573" xml:space="preserve">Ex <lb/>iftis tribus centris datis, ad modum ſuperiorum de-<lb/>ducemus quartum, nempe centrum grauitatis ſeg-<lb/>menti fuſi ex I T B E, ſegmento reuoluto circa I E.</s> <s xml:id="echoid-s2574" xml:space="preserve"/> </p> <pb o="140" file="0152" n="152"/> <figure> <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0152-01"/> </figure> <p> <s xml:id="echoid-s2575" xml:space="preserve">Secundo ſecetur parabola ctiam L P, E B, dia-<lb/>metro parallela, adeovt I T, L P, intercipiant dia-<lb/>metrum, poſlumus aſſignare centrum grauitatis ſeg-<lb/>menti intermedij fuſi orti ex reuolucione ſegmenti <lb/>intermedij I T B P L, reuoluti circa I L. </s> <s xml:id="echoid-s2576" xml:space="preserve">Nam ex <lb/>propoſit. </s> <s xml:id="echoid-s2577" xml:space="preserve">21. </s> <s xml:id="echoid-s2578" xml:space="preserve">lib. </s> <s xml:id="echoid-s2579" xml:space="preserve">3. </s> <s xml:id="echoid-s2580" xml:space="preserve">habemus centrum grauitatis du-<lb/>plicatæ figuræ I T B P L, ad partes I L. </s> <s xml:id="echoid-s2581" xml:space="preserve">Secundo <lb/>ex propoſit. </s> <s xml:id="echoid-s2582" xml:space="preserve">22. </s> <s xml:id="echoid-s2583" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s2584" xml:space="preserve">habemus centrum æ-<lb/>quilibrij ſegmenti in L G; </s> <s xml:id="echoid-s2585" xml:space="preserve">nempe rationem ſoli- <pb o="141" file="0153" n="153"/> dorum reuolutorum circa V G, & </s> <s xml:id="echoid-s2586" xml:space="preserve">I L. </s> <s xml:id="echoid-s2587" xml:space="preserve">Tertio ex <lb/>propoſit. </s> <s xml:id="echoid-s2588" xml:space="preserve">18. </s> <s xml:id="echoid-s2589" xml:space="preserve">lib. </s> <s xml:id="echoid-s2590" xml:space="preserve">4. </s> <s xml:id="echoid-s2591" xml:space="preserve">habemus centrum grauitatis ſeg-<lb/>menti annuli ex ſegmento I T B P L, reuoluto cir-<lb/>ca V G. </s> <s xml:id="echoid-s2592" xml:space="preserve">Ergo quartum, nempe centrum ſegmen-<lb/>ti fuſi ex eodem ſegmento circa L I, non ignora-<lb/>bitur.</s> <s xml:id="echoid-s2593" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2594" xml:space="preserve">Sic cognoſcemus centrum grauitatis portionis <lb/>fuſi ex portione maiori I T B A. </s> <s xml:id="echoid-s2595" xml:space="preserve">Nam centrum <lb/>grauitatis duplicatæ portionis habetur ex propoſit. <lb/></s> <s xml:id="echoid-s2596" xml:space="preserve">19. </s> <s xml:id="echoid-s2597" xml:space="preserve">lib. </s> <s xml:id="echoid-s2598" xml:space="preserve">3. </s> <s xml:id="echoid-s2599" xml:space="preserve">Ex propoſit. </s> <s xml:id="echoid-s2600" xml:space="preserve">20. </s> <s xml:id="echoid-s2601" xml:space="preserve">eiuſdem libri, habemus <lb/>rationem ſolidorum ex portione reuoluta circa V B, <lb/>& </s> <s xml:id="echoid-s2602" xml:space="preserve">circa I A. </s> <s xml:id="echoid-s2603" xml:space="preserve">Tertio ex citata propoſit. </s> <s xml:id="echoid-s2604" xml:space="preserve">18. </s> <s xml:id="echoid-s2605" xml:space="preserve">lib 4. </s> <s xml:id="echoid-s2606" xml:space="preserve"><lb/>habemus centrum portionis annuli ex portione <lb/>I T B A, reuoluta circa V B. </s> <s xml:id="echoid-s2607" xml:space="preserve">Quare &</s> <s xml:id="echoid-s2608" xml:space="preserve">c.</s> <s xml:id="echoid-s2609" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2610" xml:space="preserve">Pariter cognoſcemus centrum grauitatis portio-<lb/>nis fuſi ex portione minori R T I, quia ex propoſit. <lb/></s> <s xml:id="echoid-s2611" xml:space="preserve">14. </s> <s xml:id="echoid-s2612" xml:space="preserve">lib. </s> <s xml:id="echoid-s2613" xml:space="preserve">3. </s> <s xml:id="echoid-s2614" xml:space="preserve">habemus centrum grauitatis in R I, du-<lb/>plicatæ portionis R T I. </s> <s xml:id="echoid-s2615" xml:space="preserve">Secundo habemus ratio-<lb/>nem, quam habet prædict portio fuſi, ad portio-<lb/>nem annuli ex portione I R T, reuoluta circa S V. </s> <s xml:id="echoid-s2616" xml:space="preserve"><lb/>Quia mente portioni intellecto circumſcripto paral-<lb/>lelogrammo, habemus ex ſchol 2 propoſit 15. </s> <s xml:id="echoid-s2617" xml:space="preserve">eiuſ-<lb/>dem libri, rationem portionis f ſi, ad cylindrum ſi-<lb/>bi circumſcriptum: </s> <s xml:id="echoid-s2618" xml:space="preserve">pariter habemus rationem præ-<lb/>dicti cylindri ad cylindrum R X, quia habemus, ex <lb/>data portione, rationem I T, ad I V; </s> <s xml:id="echoid-s2619" xml:space="preserve">& </s> <s xml:id="echoid-s2620" xml:space="preserve">conſe-<lb/>quenter quadrati I T, ad quadratum I V: </s> <s xml:id="echoid-s2621" xml:space="preserve">item <lb/>habemus exſchol. </s> <s xml:id="echoid-s2622" xml:space="preserve">2. </s> <s xml:id="echoid-s2623" xml:space="preserve">propoſit 4. </s> <s xml:id="echoid-s2624" xml:space="preserve">lib. </s> <s xml:id="echoid-s2625" xml:space="preserve">4 rationem cy-<lb/>lindri R X, ad portionem annuli ex portione R T I, <pb o="142" file="0154" n="154"/> circa S V. </s> <s xml:id="echoid-s2626" xml:space="preserve">Vnde ex æquali, habemus rationem por-<lb/>tionis fuſi ad portionem annuli. </s> <s xml:id="echoid-s2627" xml:space="preserve">Tertio habemus <lb/>centrum grauitatis prædictæ portionis annuli ex cit. <lb/></s> <s xml:id="echoid-s2628" xml:space="preserve">prop. </s> <s xml:id="echoid-s2629" xml:space="preserve">18. </s> <s xml:id="echoid-s2630" xml:space="preserve">lib. </s> <s xml:id="echoid-s2631" xml:space="preserve">4. </s> <s xml:id="echoid-s2632" xml:space="preserve">Ergo quartum, nempe centrum graui-<lb/>tatis portionis fuſi non ignorabitur.</s> <s xml:id="echoid-s2633" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2634" xml:space="preserve">Sed nec in ſequenti figura, ſuppoſita ſemiparabo-<lb/>la E B A, ſecta duabus lineis H N, L P, diame-<lb/>tro E B, parallelis, ignorabimus centrum grauit atis <lb/>ſegmenti fuſi ex ſegmento intermedio H N P L. <lb/></s> <s xml:id="echoid-s2635" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> <pb o="143" file="0155" n="155"/> Nam centrum grauitatis in H L, duplicati ſegmen-<lb/>ti ad partes H L, habetur ex propoſit. </s> <s xml:id="echoid-s2636" xml:space="preserve">17. </s> <s xml:id="echoid-s2637" xml:space="preserve">libri 3. <lb/></s> <s xml:id="echoid-s2638" xml:space="preserve">Item ex præcitata propoſit. </s> <s xml:id="echoid-s2639" xml:space="preserve">18. </s> <s xml:id="echoid-s2640" xml:space="preserve">lib. </s> <s xml:id="echoid-s2641" xml:space="preserve">4. </s> <s xml:id="echoid-s2642" xml:space="preserve">habemus cen-<lb/>trum grauitatis ſegmenti annuli ex ſegmento <lb/>H N P L, circa B D. </s> <s xml:id="echoid-s2643" xml:space="preserve">Tertium nempe ratio ſeg-<lb/>menti fuſi ad ſegmentum annuli patebit haberi. </s> <s xml:id="echoid-s2644" xml:space="preserve"><lb/>Quia habemus ex ſchol. </s> <s xml:id="echoid-s2645" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2646" xml:space="preserve">18. </s> <s xml:id="echoid-s2647" xml:space="preserve">lib. </s> <s xml:id="echoid-s2648" xml:space="preserve">3. </s> <s xml:id="echoid-s2649" xml:space="preserve">rationem <lb/>ſegmenti fuſi ad cylindrum ex parallelogrammo <lb/>L N, ſibi circumſcripto; </s> <s xml:id="echoid-s2650" xml:space="preserve">ſed habemus rationem ta-<lb/>lis cylindri ad cylindrum H M, & </s> <s xml:id="echoid-s2651" xml:space="preserve">huius ex præcit. </s> <s xml:id="echoid-s2652" xml:space="preserve"><lb/>ſchol. </s> <s xml:id="echoid-s2653" xml:space="preserve">2. </s> <s xml:id="echoid-s2654" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2655" xml:space="preserve">4, lib. </s> <s xml:id="echoid-s2656" xml:space="preserve">4. </s> <s xml:id="echoid-s2657" xml:space="preserve">ad ſegmentum annuli. </s> <s xml:id="echoid-s2658" xml:space="preserve"><lb/>Quare ex æquali, patet propoſitum. </s> <s xml:id="echoid-s2659" xml:space="preserve">Cognitis ve-<lb/>rò tribus præcedentibus, quartum centrum quæſi-<lb/>tum innoteſcet. </s> <s xml:id="echoid-s2660" xml:space="preserve">Patuit ergo propoſitum in omni-<lb/>bus prædictis partibus.</s> <s xml:id="echoid-s2661" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0154-01"/> </figure> </div> </div> <div xml:id="echoid-div125" type="section" level="1" n="80"> <head xml:id="echoid-head92" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2662" xml:space="preserve">Sicuti autem in antecedentibus reperta ſunt cen-<lb/>tra grauitatis variorum ſegmentorum infinitorum <lb/>fuſorum parabolicorum, ſic ex ſuppoſita quadratura <lb/>hyperbolæ, eiuſque ſegmentorum, liceret reperire <lb/>tam centra grauitatis variorum ſegmentorum hy-<lb/>perbol quam variorum ſegmentorum fufi ex hy-<lb/>perbola, quod indicaſſe lectori ſufficiat.</s> <s xml:id="echoid-s2663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2664" xml:space="preserve">Ex ſuperius ergo dictis patuit quot ſint ea, quæ <lb/>deducuntur ex propoſit. </s> <s xml:id="echoid-s2665" xml:space="preserve">30. </s> <s xml:id="echoid-s2666" xml:space="preserve">ſuperiori, ſed inſuper <lb/>alia poſſunt deduci nempe tres regulæ vniuerſales in <lb/>tribus ſequentibus propoſic. </s> <s xml:id="echoid-s2667" xml:space="preserve">exprimendæ.</s> <s xml:id="echoid-s2668" xml:space="preserve"/> </p> <pb o="144" file="0156" n="156"/> </div> <div xml:id="echoid-div126" type="section" level="1" n="81"> <head xml:id="echoid-head93" xml:space="preserve">PROPOSITIO XXXVIII.</head> <p style="it"> <s xml:id="echoid-s2669" xml:space="preserve">Data cuiuſcunque ſemifigurœ circa diametrum quadratu-<lb/>ra, dataque ratione cylindri circumſcripti ſolido ex ſe-<lb/>mifigura reuoluta ſiue circa diametrum, ſiue circa du-<lb/>ctam diametro parallelam, vel per extremitatem baſis, <lb/>vel extra figuram. </s> <s xml:id="echoid-s2670" xml:space="preserve">Datur ratio cylindri circumſcripti <lb/>altero dictorum ſolidorum adipſum.</s> <s xml:id="echoid-s2671" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2672" xml:space="preserve">SIt data quælibet ſemifigura D B C, circa dia-<lb/>metrum B D, & </s> <s xml:id="echoid-s2673" xml:space="preserve">data ſitratio quam habet pa-<lb/>rallelogrammum B C, ad ipſam figuram; </s> <s xml:id="echoid-s2674" xml:space="preserve">inſuper <lb/>detur ratio, quam habet cylindrus ex B C, in pri-<lb/>ma figura, reuoluto ſiue circa D B, ſiue circa F C, <lb/>ad alterum ſolidorum ex ſemifigura D B C, ſiue cir-<lb/>ca B D, ſiue circa F C: </s> <s xml:id="echoid-s2675" xml:space="preserve">vel in ſecunda figura detur <lb/>vel ratio cylindri E C, ad ſolidum A B C, vel cy-<lb/>lindri D H, ad ſolidum ex D B C, reuoluta cir-<lb/>ca T S. </s> <s xml:id="echoid-s2676" xml:space="preserve">Dico dari etiam rationem alterius cylindri, <lb/>ad alterum ſolidum exſemifigura.</s> <s xml:id="echoid-s2677" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2678" xml:space="preserve">Probetur prius in prima figura, in qua intelliga-<lb/>mus parallelogrammum E C, cum figura integra <lb/>A B C, rotari circa F C. </s> <s xml:id="echoid-s2679" xml:space="preserve">Frgo ex propoſit. </s> <s xml:id="echoid-s2680" xml:space="preserve">29. <lb/></s> <s xml:id="echoid-s2681" xml:space="preserve">cum data ſit ratio ex hypotheſi, parallelogrammi <lb/>E C, ad figuram A B C, dabitur quoque ratio cy-<lb/>lindri E G, ad ſohdum A B C H G. </s> <s xml:id="echoid-s2682" xml:space="preserve">Sed tale ſoli-<lb/>dum ex propoſit. </s> <s xml:id="echoid-s2683" xml:space="preserve">30. </s> <s xml:id="echoid-s2684" xml:space="preserve">æquatur duobus ſolidis ex <lb/>D B C, circa D B, & </s> <s xml:id="echoid-s2685" xml:space="preserve">duobus, ex eadem circa F C.</s> <s xml:id="echoid-s2686" xml:space="preserve"> <pb o="145" file="0157" n="157"/> <anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/> Ergo dabitur quoque iatio cylindri E G, ad hæc <lb/>quatuor ſolida. </s> <s xml:id="echoid-s2687" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2688" xml:space="preserve">cylindri E C, qui eſt <lb/>quarta pars cylindri E G, ad eadem quatuor ſoli-<lb/>da. </s> <s xml:id="echoid-s2689" xml:space="preserve">Ergo dabitur quoque ratio cylindri E C, ſeù <lb/>ei æqualis, D H, ad duo tantum illorum ſolido-<lb/>rum, ſcilicet ad vnum, & </s> <s xml:id="echoid-s2690" xml:space="preserve">vnum, nempe ad vnum <lb/>circa D B, & </s> <s xml:id="echoid-s2691" xml:space="preserve">ad vnum circa F C. </s> <s xml:id="echoid-s2692" xml:space="preserve">Sed ex hypothe-<lb/>ſi, datur quoque ratio cylindri E C, ſeù D H, ad <lb/>alterum tantum ſolidorum ex D B C, reuoluta ſiue <pb o="146" file="0158" n="158"/> circa D B, ſiue circa F C. </s> <s xml:id="echoid-s2693" xml:space="preserve">Ergo quacunque data, <lb/>dabitur etiam altera; </s> <s xml:id="echoid-s2694" xml:space="preserve">nempe data ratione cylindri <lb/>E C, ad ſolidum A B C, dabitur quoque ratio cy-<lb/>lindri D H, ad ſolidum ex D B C, circa F C, & </s> <s xml:id="echoid-s2695" xml:space="preserve"><lb/>è contra.</s> <s xml:id="echoid-s2696" xml:space="preserve"/> </p> <div xml:id="echoid-div126" type="float" level="2" n="1"> <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a"> <image file="0157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0157-01"/> </figure> </div> <p> <s xml:id="echoid-s2697" xml:space="preserve">Pariter in ſecunda figura. </s> <s xml:id="echoid-s2698" xml:space="preserve">Quoniam datur ratio <lb/>parallelogrammi D F, ad ſemifiguram D B C, ſi-<lb/>ue parallelogrammi E C, ad integram figuram <lb/>A B C, dabitur ex propoſit. </s> <s xml:id="echoid-s2699" xml:space="preserve">29. </s> <s xml:id="echoid-s2700" xml:space="preserve">ratio tubi cylindri-<lb/>ci E C Y, ad annulum latum A B C Z H G. </s> <s xml:id="echoid-s2701" xml:space="preserve">Ergo <lb/>ex propoſit. </s> <s xml:id="echoid-s2702" xml:space="preserve">30. </s> <s xml:id="echoid-s2703" xml:space="preserve">dabitur quoque ratio prædicti tubi <lb/>ad quatuor ſolida ex D B C, duabus vicibus reuo-<lb/>luta circa B D, & </s> <s xml:id="echoid-s2704" xml:space="preserve">duabus circa T S. </s> <s xml:id="echoid-s2705" xml:space="preserve">Ergo dabi-<lb/>tur quoque ratio talis tubiad vnum ſolidum A B C, <lb/>& </s> <s xml:id="echoid-s2706" xml:space="preserve">ad vnum D B C Z H ℟. </s> <s xml:id="echoid-s2707" xml:space="preserve">Cum autem detur ratio <lb/>D S, tam ad A C, quam ad C G (hoc enim eſt <lb/>ſupponendum, quia danda eſt C S, qua data dantur <lb/>prædicta) dabitur etiam ratio quadrati D S, ad re-<lb/>ctangulum A C G; </s> <s xml:id="echoid-s2708" xml:space="preserve">nempe dabitur ratio cylindri <lb/>D H, ad tubum cylindricum E C Y. </s> <s xml:id="echoid-s2709" xml:space="preserve">Ergo ex æqua-<lb/>li, dabitur quoque ratio cylindri B ℟, ad ſolidum <lb/>A B C, ſimul cum ſolido D B C Z H ℟. </s> <s xml:id="echoid-s2710" xml:space="preserve">Siergo de-<lb/>tur etiam ex hypotheſi, ratio cylindri E C, ad ſoli-<lb/>dum A B C, quiacum detur ratio cylindri D H, ad <lb/>cylindrum E C, datur etiam ratio cylindri D H, <lb/>ad ſolidum A B C. </s> <s xml:id="echoid-s2711" xml:space="preserve">Ergo dabitur quoque ratio eiuſ-<lb/>dem cylindri D H, ad ſolidum D B C Z H ℟. </s> <s xml:id="echoid-s2712" xml:space="preserve">Si <lb/>vero detur ratio ex hypotheſi, cylindri D H, ad ſo-<lb/>lidum D B C Z H ℟; </s> <s xml:id="echoid-s2713" xml:space="preserve">ergo dabitur quoque ratio e- <pb o="147" file="0159" n="159"/> <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> iuſdem cylindri ad ſolidum A B C. </s> <s xml:id="echoid-s2714" xml:space="preserve">Sed etiam datur <lb/>ratio cylindri E C, ad cylindrum D H. </s> <s xml:id="echoid-s2715" xml:space="preserve">Ergo <lb/>quoque ex æquali, dabitur ratio cylindri E C, <lb/>ad ſolidum A B C. </s> <s xml:id="echoid-s2716" xml:space="preserve">Ergo in omnibus patuit pro-<lb/>poſitio.</s> <s xml:id="echoid-s2717" xml:space="preserve"/> </p> <div xml:id="echoid-div127" type="float" level="2" n="2"> <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0159-01"/> </figure> </div> <pb o="148" file="0160" n="160"/> </div> <div xml:id="echoid-div129" type="section" level="1" n="82"> <head xml:id="echoid-head94" xml:space="preserve">PROPOSITIO XXXIX.</head> <p style="it"> <s xml:id="echoid-s2718" xml:space="preserve">Datis ijſdem, quœ in antecedenti propoſitione in primo <lb/>ſehemate, datur centrum æquilibrij figuræ in <lb/>linea, quæ eſt radius rotationis.</s> <s xml:id="echoid-s2719" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2720" xml:space="preserve">SEd dentur eadem, quæ ſupra in primo ſchema-<lb/>te. </s> <s xml:id="echoid-s2721" xml:space="preserve">Dico dari in D C, quæ eſt radius rotatio-<lb/>nis, centrum æquilibrij ſemifiguræ D B C. </s> <s xml:id="echoid-s2722" xml:space="preserve">Cum <lb/>enim exanteced. </s> <s xml:id="echoid-s2723" xml:space="preserve">propoſit datis ijs, detur etiam ra-<lb/>tio cylindri ad alterum ſolidorum. </s> <s xml:id="echoid-s2724" xml:space="preserve">Ergo dabitur e-<lb/>tiam ratio ſolidorum ad inuicem; </s> <s xml:id="echoid-s2725" xml:space="preserve">nempe dabitur ra-<lb/>tio ſolidi A B C, ad ſolidum D B C H V. </s> <s xml:id="echoid-s2726" xml:space="preserve">Sed ex <lb/>propoſit. </s> <s xml:id="echoid-s2727" xml:space="preserve">4. </s> <s xml:id="echoid-s2728" xml:space="preserve">lib. </s> <s xml:id="echoid-s2729" xml:space="preserve">3. </s> <s xml:id="echoid-s2730" xml:space="preserve">ſolidum ad ſolidum eſt vt pars D C, <lb/>terminata à D, & </s> <s xml:id="echoid-s2731" xml:space="preserve">àcentro æquilibrij figuræ D B C, <lb/>ad reliquam partem D C. </s> <s xml:id="echoid-s2732" xml:space="preserve">Quare patet propo-<lb/>ſitum.</s> <s xml:id="echoid-s2733" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div130" type="section" level="1" n="83"> <head xml:id="echoid-head95" xml:space="preserve">PROPOSITIO XL.</head> <p style="it"> <s xml:id="echoid-s2734" xml:space="preserve">Fi ſecundo ſchemate datis ijſdem, & </s> <s xml:id="echoid-s2735" xml:space="preserve">dataratione annu-<lb/>li lati ex ſernifigura ad annulum ſtrictum eiuſdem, <lb/>dabitur prœdictum centrum.</s> <s xml:id="echoid-s2736" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2737" xml:space="preserve">SEd in ſecundo ſchemate, vltra data in ante-<lb/>cedenti, detur etiam ratio annuli lati <lb/>D B C Z H ℟, ad annulum ſtrictum exeadem D B C, <lb/>reuoluta circa F C. </s> <s xml:id="echoid-s2738" xml:space="preserve">Dico dari eius centrum æqui- <pb o="149" file="0161" n="161"/> librij in D C. </s> <s xml:id="echoid-s2739" xml:space="preserve">Nam eodem modo patebit, dari ra-<lb/>tionem ſolidi A B C, ad ſolidum D B C Z H ℟. <lb/></s> <s xml:id="echoid-s2740" xml:space="preserve">Sed etiam datur ratio ex hypotheſi, D B C Z H ℟, <lb/>ad annulum ſtrictum ex D B C, circa C F. </s> <s xml:id="echoid-s2741" xml:space="preserve">Ergo <lb/>ex æquali, dabitur ratio A B C, ſolidi ad prædi-<lb/>ctum annulum ſtrictum. </s> <s xml:id="echoid-s2742" xml:space="preserve">Quare ex cit propoſit. </s> <s xml:id="echoid-s2743" xml:space="preserve">3. </s> <s xml:id="echoid-s2744" xml:space="preserve"><lb/>dabitur quoque in D C, centrum æquilibrij quæſi-<lb/>tum. </s> <s xml:id="echoid-s2745" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2746" xml:space="preserve">c.</s> <s xml:id="echoid-s2747" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div131" type="section" level="1" n="84"> <head xml:id="echoid-head96" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2748" xml:space="preserve">Ex his tribus propoſitionibus poſſumus necdum <lb/>ex ſola quadratura in finitarum parabolarum inuenire <lb/>rationem cylindrorum circumſcripto ũ ad infinitos <lb/>fuſos parabolicos; </s> <s xml:id="echoid-s2749" xml:space="preserve">ſed etiam centrum grauitatis in-<lb/>finitarum parabolarum. </s> <s xml:id="echoid-s2750" xml:space="preserve">Nam cum in propoſit. </s> <s xml:id="echoid-s2751" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s2752" xml:space="preserve">lib. </s> <s xml:id="echoid-s2753" xml:space="preserve">4. </s> <s xml:id="echoid-s2754" xml:space="preserve">& </s> <s xml:id="echoid-s2755" xml:space="preserve">in ſcholijs eiuſdem, oſtenſum ſit in ſchema-<lb/>te illius propoſit. </s> <s xml:id="echoid-s2756" xml:space="preserve">data qualibet ſemiparabola R B E, <lb/>cuius baſis R E, diameter B E, quæ reuoluatur <lb/>cum ſibi circumſcripto parallelogrammo R B, cir-<lb/>ca B S: </s> <s xml:id="echoid-s2757" xml:space="preserve">cylindrum R K, eſſe ad ſolidum E R B Z k, <lb/>vt parallelogrammum R B, ad ſemiparabolam <lb/>E R B, cuius baſis E R, diameter E B, quæ ſit <lb/>gradus dupli, gradus ſemiparabolæ reuolutæ circa <lb/>S B; </s> <s xml:id="echoid-s2758" xml:space="preserve">patet ex data quadratura infinitarum parabola-<lb/>rum, dari rationem cylindri R K, ad annuIum <lb/>E R B Z k. </s> <s xml:id="echoid-s2759" xml:space="preserve">Data hac ratione, dabitur etiam ex pro-<lb/>poſit. </s> <s xml:id="echoid-s2760" xml:space="preserve">anteced. </s> <s xml:id="echoid-s2761" xml:space="preserve">ratio cylindri R k, velei æqualis or-<lb/>ti ex R B, circa R E, ad ſolidum ex E R B, circa <pb o="150" file="0162" n="162"/> <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> R E; </s> <s xml:id="echoid-s2762" xml:space="preserve">nempe ad ſemifuſum parabolicum. </s> <s xml:id="echoid-s2763" xml:space="preserve">His datis <lb/>dabitur etiam ratio illorum ſolidorum ad inuicem; <lb/></s> <s xml:id="echoid-s2764" xml:space="preserve">& </s> <s xml:id="echoid-s2765" xml:space="preserve">conſequenter centrum æquilibrij ſemiparabolæ <lb/>E R B, in E B; </s> <s xml:id="echoid-s2766" xml:space="preserve">& </s> <s xml:id="echoid-s2767" xml:space="preserve">conſequenter centrum grauitatis <lb/>parabolæ R B A, in diametro B E.</s> <s xml:id="echoid-s2768" xml:space="preserve"/> </p> <div xml:id="echoid-div131" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0162-01"/> </figure> </div> <p> <s xml:id="echoid-s2769" xml:space="preserve">Sed hic notetur, parabolas inſeruientes inuentio-<lb/>ni centri grauitatis infinitarum parabolarum, non <lb/>eſſe omnes, ſed illas dumtaxat, quarum exponentes <lb/>ſunt numeri pares; </s> <s xml:id="echoid-s2770" xml:space="preserve">quia hæ dumtaxat inſeruiunt in- <pb o="151" file="0163" n="163"/> uentioni rationis infinitorum cylindrorum R K, ad <lb/>infinitos annulos E R B Z k, vt luculenter explica-<lb/>tum fuit in admirabili ſcholio 4. </s> <s xml:id="echoid-s2771" xml:space="preserve">citat. </s> <s xml:id="echoid-s2772" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2773" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s2774" xml:space="preserve">lib. </s> <s xml:id="echoid-s2775" xml:space="preserve">4.</s> <s xml:id="echoid-s2776" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2777" xml:space="preserve">Inſuper cum in varijs propoſitionibus lib. </s> <s xml:id="echoid-s2778" xml:space="preserve">prim. <lb/></s> <s xml:id="echoid-s2779" xml:space="preserve">aſſignata fuerit ratio, quam habet quælibet pars pa-<lb/>rallelogrammi A S, ad quamlibet partem parabolæ <lb/>R B A, quam pars parallelogrammi includit, & </s> <s xml:id="echoid-s2780" xml:space="preserve">cum <lb/>in cit. </s> <s xml:id="echoid-s2781" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2782" xml:space="preserve">4. </s> <s xml:id="echoid-s2783" xml:space="preserve">lib. </s> <s xml:id="echoid-s2784" xml:space="preserve">4. </s> <s xml:id="echoid-s2785" xml:space="preserve">& </s> <s xml:id="echoid-s2786" xml:space="preserve">in eiuſdem ſcholijs, aſſi-<lb/>gnata fuerit ratio ex illa ſimplici analogia, quam ha-<lb/>bet quælibet pars cylindri R C, ad quamlibet par-<lb/>tem annuli A R B Z C; </s> <s xml:id="echoid-s2787" xml:space="preserve">v. </s> <s xml:id="echoid-s2788" xml:space="preserve">g. </s> <s xml:id="echoid-s2789" xml:space="preserve">oſtenſa ſit ratio, quam <lb/>habet cylindrus I K, ad partem annuli ex E I T B, <lb/>circa V B; </s> <s xml:id="echoid-s2790" xml:space="preserve">patet ex propoſit. </s> <s xml:id="echoid-s2791" xml:space="preserve">antecedentibus, nec-<lb/>dum dari rationem cuiuslibet partis cylindri R C, <lb/>v. </s> <s xml:id="echoid-s2792" xml:space="preserve">g. </s> <s xml:id="echoid-s2793" xml:space="preserve">I k, vel ei æqualis ex I B, circa I E, ad par-<lb/>tem fuſi ex I T B E, circa I E: </s> <s xml:id="echoid-s2794" xml:space="preserve">ſed etiam dari in <lb/>B E, vel in V I, centrum æquilibrij ſegmenti <lb/>I T B E, vel grauitatis duplicati ſegmenti ad par-<lb/>tes B E, vel I V.</s> <s xml:id="echoid-s2795" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2796" xml:space="preserve">In propoſit. </s> <s xml:id="echoid-s2797" xml:space="preserve">autem 3. </s> <s xml:id="echoid-s2798" xml:space="preserve">lib. </s> <s xml:id="echoid-s2799" xml:space="preserve">4. </s> <s xml:id="echoid-s2800" xml:space="preserve">patuit cylindrum <lb/>E C, eſſe ad quodlibet conoides parabolicum ABC, <lb/>cuius exponens ſit numerus par, vt parallelogram-<lb/>mum E C, ad parabolam A B C, cuius exponens <lb/>ſit ſubduplus exponentis conoidis. </s> <s xml:id="echoid-s2801" xml:space="preserve">Quare, vt ibi-<lb/>dem patuit, infinitæ parabolæ non inſeruierunt in-<lb/>uentioni rationi infinitorum cylindrorum ad in finita <lb/>conoidea, ſed tantum ad ea, quorum exponentes <lb/>ſunt numeri pares. </s> <s xml:id="echoid-s2802" xml:space="preserve">Eliciemus crgo ex antecedenti- <pb o="152" file="0164" n="164"/> <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> bus propoſitionibus, inſeruire infinitas parabolas <lb/>inuentioni rationi cylindrorum E C, vel eis æqua-<lb/>lium factorum ex E D, circa E A, ad annulos ex <lb/>A B D, circa A E, quorum exponentes ſint numeri <lb/>pares. </s> <s xml:id="echoid-s2803" xml:space="preserve">Pariter eliciemus nos ex his habere centrum <lb/>æquilibrij in baſi A D, ſemiparabolarum A B D, <lb/>quarum exponentes ſunt numeri pares, & </s> <s xml:id="echoid-s2804" xml:space="preserve">non om-<lb/>nium.</s> <s xml:id="echoid-s2805" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="2"> <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a"> <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0164-01"/> </figure> </div> <p> <s xml:id="echoid-s2806" xml:space="preserve">Patet ergo ex dictis, aliquod admirabile, & </s> <s xml:id="echoid-s2807" xml:space="preserve">non <lb/>minus eo, quod expoſitum fuit in prædicto ſchol. </s> <s xml:id="echoid-s2808" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s2809" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2810" xml:space="preserve">4. </s> <s xml:id="echoid-s2811" xml:space="preserve">lib. </s> <s xml:id="echoid-s2812" xml:space="preserve">4. </s> <s xml:id="echoid-s2813" xml:space="preserve">Hoc autem eſt, quod infinitæ para-<lb/>bolæ inſeruiunt tam inuentioni centri grauitatis in- <pb o="153" file="0165" n="165"/> finitarum parabolarum in diametro, quam inuentio-<lb/>ni centri æquilibrij infinitarum ſemiparabolarum in <lb/>baſi. </s> <s xml:id="echoid-s2814" xml:space="preserve">At inuenimus centra grauitatis infinitarum. <lb/></s> <s xml:id="echoid-s2815" xml:space="preserve">parabolarum in diamctro non adhibendo infinitas <lb/>parabolas, ſed illas tantum, quarum exponentes <lb/>ſunt numeri pares. </s> <s xml:id="echoid-s2816" xml:space="preserve">E contra verò adhibendo infi-<lb/>nitas parabolas, non inuenimus centra æquilibrij in <lb/>baſi infinitarum ſemiparabolarum, ſed illarum tan-<lb/>tum, quarum exponentes ſunt numeri pares.</s> <s xml:id="echoid-s2817" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2818" xml:space="preserve">Ex cit. </s> <s xml:id="echoid-s2819" xml:space="preserve">autem propoſit. </s> <s xml:id="echoid-s2820" xml:space="preserve">3. </s> <s xml:id="echoid-s2821" xml:space="preserve">lib. </s> <s xml:id="echoid-s2822" xml:space="preserve">4. </s> <s xml:id="echoid-s2823" xml:space="preserve">& </s> <s xml:id="echoid-s2824" xml:space="preserve">ex ſchol eiuſ-<lb/>dem, poſſumus ex propoſit. </s> <s xml:id="echoid-s2825" xml:space="preserve">anteced. </s> <s xml:id="echoid-s2826" xml:space="preserve">elicere ratio-<lb/>nem, quam habet cylindrus ex AM, circa E A, ad <lb/>partem annuli ex APMD, circa E A, cuius expo-<lb/>nens ſit numerus par. </s> <s xml:id="echoid-s2827" xml:space="preserve">Et inſuper centrum æquili-<lb/>brij in A D, ſegmenti APMD, ſemiparabolæ <lb/>A B D, cuius exponens itidem ſit numerus par. <lb/></s> <s xml:id="echoid-s2828" xml:space="preserve">Hæc autem facile patent ex dictis.</s> <s xml:id="echoid-s2829" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2830" xml:space="preserve">Quot igitur ſolidorum manifeſtata ſint centra <lb/>grauitatis, potuit lector ex dictis cognoſcere. </s> <s xml:id="echoid-s2831" xml:space="preserve">Sed <lb/>nolumus ſub ſilentio relinquere aliqua, quæ nobis <lb/>ſcitu digna videntur.</s> <s xml:id="echoid-s2832" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div134" type="section" level="1" n="85"> <head xml:id="echoid-head97" xml:space="preserve">PROPOSITIO XLI.</head> <p style="it"> <s xml:id="echoid-s2833" xml:space="preserve">Si ſuper eadem baſi, & </s> <s xml:id="echoid-s2834" xml:space="preserve">circa eandem diametrum ſint ſe-<lb/>mihyperbola, & </s> <s xml:id="echoid-s2835" xml:space="preserve">ſemiparabola. </s> <s xml:id="echoid-s2836" xml:space="preserve">Tota ſemihy-<lb/>perbola cadet intra ſemipar abolam.</s> <s xml:id="echoid-s2837" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2838" xml:space="preserve">SInt ſemihy perbola A E B D, & </s> <s xml:id="echoid-s2839" xml:space="preserve">ſemiparabola <lb/>A F B D, quarum eadem baſis A D, eadem- <pb o="154" file="0166" n="166"/> <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> que diameter B D. </s> <s xml:id="echoid-s2840" xml:space="preserve">Dico totam femihyperbolam <lb/>cadere intra ſemiparabolam. </s> <s xml:id="echoid-s2841" xml:space="preserve">Sit G B, latus tranſ-<lb/>uerſum hyperbolæ, & </s> <s xml:id="echoid-s2842" xml:space="preserve">accepto in B D, arbitrariè <lb/>puncto H, ordinatim applicetur H E F. </s> <s xml:id="echoid-s2843" xml:space="preserve">Quo-<lb/>niam enim in hyperbola eſt ex primo conic propo-<lb/>ſit. </s> <s xml:id="echoid-s2844" xml:space="preserve">21. </s> <s xml:id="echoid-s2845" xml:space="preserve">vt quadratum E H, ad quadratum A D, ſic <lb/>rectangulum G H B, ad rectangulum G D B: </s> <s xml:id="echoid-s2846" xml:space="preserve">& </s> <s xml:id="echoid-s2847" xml:space="preserve"><lb/>in parabola eſt ex propoſit. </s> <s xml:id="echoid-s2848" xml:space="preserve">20. </s> <s xml:id="echoid-s2849" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s2850" xml:space="preserve">quadra-<lb/>tum A D, ad quadratum F H, vt D B, ad B H; <lb/></s> <s xml:id="echoid-s2851" xml:space="preserve">nempe vt rectangulum G D B, ad rectangulum- <pb o="155" file="0167" n="167"/> ſub G D, in B H: </s> <s xml:id="echoid-s2852" xml:space="preserve">ergo ex æquali, erit quadratum <lb/>E H, ad quadratum F H, vt rectangulum G H B, <lb/>ad rectangulum ſub G D, in B H. </s> <s xml:id="echoid-s2853" xml:space="preserve">Sed rectangu-<lb/>lum G H B, minus eſt rectangulo ſub G D, in B H. <lb/></s> <s xml:id="echoid-s2854" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2855" xml:space="preserve">quadratum E H, minus erit quadrato F H. </s> <s xml:id="echoid-s2856" xml:space="preserve"><lb/>Ergo & </s> <s xml:id="echoid-s2857" xml:space="preserve">E H, minor erit F H. </s> <s xml:id="echoid-s2858" xml:space="preserve">Punctum autem H, <lb/>ſumptum fuit arbitrariè. </s> <s xml:id="echoid-s2859" xml:space="preserve">Ergo omnes lineæ hyper-<lb/>bolæ minores erunt ſingulis lineis parabolæ. </s> <s xml:id="echoid-s2860" xml:space="preserve">Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s2861" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0166-01"/> </figure> </div> </div> <div xml:id="echoid-div136" type="section" level="1" n="86"> <head xml:id="echoid-head98" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2862" xml:space="preserve">Patet ergo, quod ſiex prędictis figuris in telligantur <lb/>genita conoidea hyperbolicum A E B C, & </s> <s xml:id="echoid-s2863" xml:space="preserve">para-<lb/>bolicum A F B C, conoides hyperbolicum cadet <lb/>intra parabolicum.</s> <s xml:id="echoid-s2864" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div137" type="section" level="1" n="87"> <head xml:id="echoid-head99" xml:space="preserve">PROPOSITIO XLII.</head> <p style="it"> <s xml:id="echoid-s2865" xml:space="preserve">Differentiæ ſupradictorum conoideorum centrum grauitatis <lb/>eſt medium punctum diametri.</s> <s xml:id="echoid-s2866" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2867" xml:space="preserve">SInt ergo vt in propoſit. </s> <s xml:id="echoid-s2868" xml:space="preserve">anteced. </s> <s xml:id="echoid-s2869" xml:space="preserve">conoidea hy-<lb/>perbolicum A E B C, & </s> <s xml:id="echoid-s2870" xml:space="preserve">parabolicum A F B C. <lb/></s> <s xml:id="echoid-s2871" xml:space="preserve">Dico cent um grauitatis exceſſus conoidis paraboli-<lb/>ci ſupra conoides hyperbolicum eſſe in medio B D. </s> <s xml:id="echoid-s2872" xml:space="preserve"><lb/>In conoidibus inſcribatur conus A B C. </s> <s xml:id="echoid-s2873" xml:space="preserve">Cum ergo <lb/>ex ſchol. </s> <s xml:id="echoid-s2874" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2875" xml:space="preserve">4. </s> <s xml:id="echoid-s2876" xml:space="preserve">ſit in medio B D, centrum <lb/>grauitatis tam totius, nempe excefſus conoidis pa- <pb o="156" file="0168" n="168"/> <anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/> rabolici ſupra conum A B C, quam partis; </s> <s xml:id="echoid-s2877" xml:space="preserve">nempe <lb/>exceſſus conoidis hyperbolici ſupra eundem conum. <lb/></s> <s xml:id="echoid-s2878" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2879" xml:space="preserve">reliquæ partis, nempe exceſſus conoidis pa-<lb/>rabolier ſupra conoides hyperbolicum erit centrum <lb/>grauitatis in medio B D. </s> <s xml:id="echoid-s2880" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2881" xml:space="preserve">c.</s> <s xml:id="echoid-s2882" xml:space="preserve"/> </p> <div xml:id="echoid-div137" 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/PVNDER1Y/figures/0168-01"/> </figure> </div> </div> <div xml:id="echoid-div139" type="section" level="1" n="88"> <head xml:id="echoid-head100" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2883" xml:space="preserve">Sed cum in præfenti occurrerit modus alius com-<lb/>pendioſus aſſignandi centrum grauitatis conoidis <pb o="157" file="0169" n="169"/> hyperbolici diuerſus ab illis, quos tradidimus ſupra <lb/>in propoſit. </s> <s xml:id="echoid-s2884" xml:space="preserve">13. </s> <s xml:id="echoid-s2885" xml:space="preserve">& </s> <s xml:id="echoid-s2886" xml:space="preserve">14. </s> <s xml:id="echoid-s2887" xml:space="preserve">nolumus ipſum omittere, ſed <lb/>præmittenda eſt ſequens propoſitio eius manifeſta-<lb/>tioni.</s> <s xml:id="echoid-s2888" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div140" type="section" level="1" n="89"> <head xml:id="echoid-head101" xml:space="preserve">PROPOSITIO XLIII.</head> <p style="it"> <s xml:id="echoid-s2889" xml:space="preserve">Differentia ſupradictorum conoideorum, est ad conoides hy-<lb/>perboluum vt ſexta pars diametri ad tertiam partem <lb/>ciuſdem, vna cum dimidio lateris tranſuerſi.</s> <s xml:id="echoid-s2890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2891" xml:space="preserve">IN ſchemate ſuperiori. </s> <s xml:id="echoid-s2892" xml:space="preserve">Dico exceſſum conoidis <lb/>parabolici A F B C, ſupra conoides hyperboli-<lb/>cum A E B C, eſſe vt ſexta pars D B, ad tertiam <lb/>partem D B, cum dimidio G B. </s> <s xml:id="echoid-s2893" xml:space="preserve">Quoniam enim <lb/>vt elicitur ex propoſit. </s> <s xml:id="echoid-s2894" xml:space="preserve">15. </s> <s xml:id="echoid-s2895" xml:space="preserve">lib. </s> <s xml:id="echoid-s2896" xml:space="preserve">2. </s> <s xml:id="echoid-s2897" xml:space="preserve">conoides paraboli-<lb/>cum eſt ſeſquialterum coni A B C; </s> <s xml:id="echoid-s2898" xml:space="preserve">ergo erit ad ip-<lb/>ſum vt G D, ad duo tertia G D; </s> <s xml:id="echoid-s2899" xml:space="preserve">nempe vt dimi-<lb/>dium G D, ad tertiam partem G D. </s> <s xml:id="echoid-s2900" xml:space="preserve">Rurſum cum <lb/>ex propoſit.</s> <s xml:id="echoid-s2901" xml:space="preserve">, 5 7. </s> <s xml:id="echoid-s2902" xml:space="preserve">& </s> <s xml:id="echoid-s2903" xml:space="preserve">11. </s> <s xml:id="echoid-s2904" xml:space="preserve">ſit cylindrus conoidi hyper-<lb/>bolico circumſcriptus, ad ipſum, vt G D, ad dimi-<lb/>diam G B, cum tertia parte D B; </s> <s xml:id="echoid-s2905" xml:space="preserve">erit conus A B C, <lb/>tertia pars cylindri, ad conoides hyperbolicum, vt <lb/>tertia pars G D, ad dimidiam G B cum tertia par-<lb/>te D B Quare ex quali, erit conoides paraboli-<lb/>cum ad conoides hyperbolicum vt dimidium G D, <lb/>ad dimidium G B, cum tertia parte B D. </s> <s xml:id="echoid-s2906" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2907" xml:space="preserve"><lb/>diuidendo, erit differentia conoideorum ad conoi- <pb o="158" file="0170" n="170"/> des hyperbolicum vt ſexta pars D B, ad dimidium <lb/>G B, cum tertia parte D B. </s> <s xml:id="echoid-s2908" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2909" xml:space="preserve">c.</s> <s xml:id="echoid-s2910" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div141" type="section" level="1" n="90"> <head xml:id="echoid-head102" xml:space="preserve">PROPOSITIO XLIV.</head> <p style="it"> <s xml:id="echoid-s2911" xml:space="preserve">Centrum grauitatis conoidis hyperbolici ſic diuidit ipſius <lb/>diametrum vt pars ad verticem ſit ad reliquam, vt <lb/>latus tranſuerſum cum ſubſeſquitertia diametri, ad di-<lb/>midium lateris tranſuer ſicum quarta parte diametri.</s> <s xml:id="echoid-s2912" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2913" xml:space="preserve">ESto in ſchemate antecedenti conoides hyper-<lb/>bolicum A E B C, cuius diameter D B, latus <lb/>tranſuerſum G B, & </s> <s xml:id="echoid-s2914" xml:space="preserve">ſit k, eius centium grauitatis. <lb/></s> <s xml:id="echoid-s2915" xml:space="preserve">Dico B K, ad k D, eſſe vt G B, cum ſubſeſquiter-<lb/>tia B D, ad dimidiam G B, cum quarta parte D B. </s> <s xml:id="echoid-s2916" xml:space="preserve"><lb/>Eſto conoides parabolicum A F B C; </s> <s xml:id="echoid-s2917" xml:space="preserve">& </s> <s xml:id="echoid-s2918" xml:space="preserve">ſit H, me-<lb/>dium punctum B D, adeo vt ſicuti elicitur ex pro-<lb/>poſ. </s> <s xml:id="echoid-s2919" xml:space="preserve">42. </s> <s xml:id="echoid-s2920" xml:space="preserve">ſit centrum grauitatis differentiæ conoideo-<lb/>rum: </s> <s xml:id="echoid-s2921" xml:space="preserve">pariter B I, ſit dupla I D, adeo vt ſit 1, ex <lb/>propoſit. </s> <s xml:id="echoid-s2922" xml:space="preserve">14. </s> <s xml:id="echoid-s2923" xml:space="preserve">lib. </s> <s xml:id="echoid-s2924" xml:space="preserve">4. </s> <s xml:id="echoid-s2925" xml:space="preserve">centrum grauitatis conoidis pa-<lb/>rabolici. </s> <s xml:id="echoid-s2926" xml:space="preserve">Siergo fiat H I, ad I k, vt dimidium G B, <lb/>cum tertia parte B D, ad ſextam partem B D, nem-<lb/>pe ex prop ſit anteced. </s> <s xml:id="echoid-s2927" xml:space="preserve">reciprocè vt conoides hy-<lb/>perbolicum ad exceſſum conoidis parabolici ſupra <lb/>ipſum, erit k, centrum conoidis hyperbolici. </s> <s xml:id="echoid-s2928" xml:space="preserve">Tunc <lb/>argumente@ur ſic. </s> <s xml:id="echoid-s2929" xml:space="preserve">Quoniam B I, quadrupla eſt <lb/>I H, ergo B I, erit ad I k, vt dupla G B, vna cum <lb/>ſeſquite tia B D, ad ſextam partem B D. </s> <s xml:id="echoid-s2930" xml:space="preserve">Et com-<lb/>ponendo erit B K, ad k I, vt dupla G B, vna cum <pb o="159" file="0171" n="171"/> ſeſquitertia B D, & </s> <s xml:id="echoid-s2931" xml:space="preserve">cum ſexta parte eiuſdem, ad <lb/>ſextam partem eiuſdem. </s> <s xml:id="echoid-s2932" xml:space="preserve">Cum autem D I, ſit dupla <lb/>IH, erit k I, ad I D, vt ſexta pars B D, ad G B, <lb/>cum duabus tertijs partibus B D. </s> <s xml:id="echoid-s2933" xml:space="preserve">Et diuidendo, <lb/>erit Ik, ad k D, vt ſexta pars B D, ad G B, cum <lb/>dimidia B D. </s> <s xml:id="echoid-s2934" xml:space="preserve">Quare ex æquali, erit B k, ad k D, <lb/>vt dupla G B, cum ſeſquitertia B D, & </s> <s xml:id="echoid-s2935" xml:space="preserve">cum ſexta <lb/>parte eiuſdem, ad G B, cum dimidia B D. </s> <s xml:id="echoid-s2936" xml:space="preserve">Et vt <lb/>horum terminorum dimidia. </s> <s xml:id="echoid-s2937" xml:space="preserve">Ergo B k, erit <lb/>ad k D, vt G B, cum ſubſeſquitertia D B, ad di-<lb/>midiam G B, cumquarta parte B D. </s> <s xml:id="echoid-s2938" xml:space="preserve">Quod &</s> <s xml:id="echoid-s2939" xml:space="preserve">c.</s> <s xml:id="echoid-s2940" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div142" type="section" level="1" n="91"> <head xml:id="echoid-head103" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2941" xml:space="preserve">In noſtro libello 60, problematum geomètrico-<lb/>rum oſtendimus in propoſit. </s> <s xml:id="echoid-s2942" xml:space="preserve">53. </s> <s xml:id="echoid-s2943" xml:space="preserve">quandam proprieta-<lb/>tem communem conoidibus parabolico, & </s> <s xml:id="echoid-s2944" xml:space="preserve">hyper-<lb/>bolico, portionibus ſphæræ, & </s> <s xml:id="echoid-s2945" xml:space="preserve">ſphæroidis, & </s> <s xml:id="echoid-s2946" xml:space="preserve">etiam <lb/>cono. </s> <s xml:id="echoid-s2947" xml:space="preserve">Alia proprietas communis omnibus prædictis <lb/>ſolidis reperitur circa illorum grauitatis centrum. <lb/></s> <s xml:id="echoid-s2948" xml:space="preserve">Hanc in ſequentibus patefaciemus, ſed prius oſten-<lb/>demus aliqua, quæ vtique non videntur turpiora, & </s> <s xml:id="echoid-s2949" xml:space="preserve"><lb/>ſunt præmitenda.</s> <s xml:id="echoid-s2950" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div143" type="section" level="1" n="92"> <head xml:id="echoid-head104" xml:space="preserve">PROPOSITIO XLV.</head> <p style="it"> <s xml:id="echoid-s2951" xml:space="preserve">Si in qualibet ſphæræ, portione inſcribatur conus, quæ por-<lb/>tio cum cono ſecetur plano baſi parallelo ſecante axim bi-<lb/>fariam, & </s> <s xml:id="echoid-s2952" xml:space="preserve">intelligatur tubus cylindricus circa eundem <pb o="160" file="0172" n="172"/> axim cum portionè, cuíus baſis ſit armilla exceſſus cìrcu-<lb/>li facti in portione, ſupra circulum factum in cono à pla-<lb/>no ſecante. </s> <s xml:id="echoid-s2953" xml:space="preserve">Hic erit ad exceſſum portionis ſupra conum <lb/>tam ſecundum totum, quam ſecundum partes propor-<lb/>tionales, vt parallelogrammum circum ſcriptum parabolæ <lb/>quadraticæ ad ipſam; </s> <s xml:id="echoid-s2954" xml:space="preserve">dummodo hæc ſecetur ſecundum <lb/>diametro parallelas.</s> <s xml:id="echoid-s2955" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2956" xml:space="preserve">SIt A B C, quælibet portio ſphæræ, in qua in-<lb/>telligatur inſcriptus conus A B C, ſectoque axi <lb/>B D, bifariam in E, ducatur per E, planum F E G, <lb/>plano A D C, parallelum, faciens in cono circulum <lb/>H E I; </s> <s xml:id="echoid-s2957" xml:space="preserve">intelligamus tubum cylindricum k L M P, <lb/>circa eundem axim B D, cuius baſis armilla N L P, <lb/>æqualis armillæ F H G: </s> <s xml:id="echoid-s2958" xml:space="preserve">pariter in ſecunda figura <lb/>intelligamus parabolam quadraticam A B C, cuius <lb/>axis B D, baſis vero A C, ſit æqualis axi B D, por-<lb/>tionis, & </s> <s xml:id="echoid-s2959" xml:space="preserve">ei ſit circumſcriptum parallelogrammum. <lb/></s> <s xml:id="echoid-s2960" xml:space="preserve">Dicotubum cylindricum k L M C, eſſe ad exceſſum <lb/>portionis A B C, ſupra conum A B C, vt paralle-<lb/>logrammum E C, ad parabolam A B C. </s> <s xml:id="echoid-s2961" xml:space="preserve">Sumatur <lb/>in B D, axi portionis arbitrariè punctum V, per <lb/>quod tiaiciatur planum Q Z, plano A C, paral-<lb/>lelum ſecans omnia ſolida vt in ſchemate; </s> <s xml:id="echoid-s2962" xml:space="preserve">& </s> <s xml:id="echoid-s2963" xml:space="preserve">pariter <lb/>in parabola facta A F, æquali B V, per F, ducatur <lb/>F G H, parallela D B. </s> <s xml:id="echoid-s2964" xml:space="preserve">Quoniam enim rectangu-<lb/>lum D E B, eſt ad rectangulum D V B, vt rectan-<lb/>gulum A H B, ad rectangulum A I B, quia propor-<lb/>tiones horum rectangulorum componuntur ex ijſ- <pb o="161" file="0173" n="173"/> <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> dem proportionibus; </s> <s xml:id="echoid-s2965" xml:space="preserve">& </s> <s xml:id="echoid-s2966" xml:space="preserve">rectangulis in circulo A H B, <lb/>A T B, ſunt æqualia rectangula F H G, R T Y; </s> <s xml:id="echoid-s2967" xml:space="preserve">ergo <lb/>vt rectangulum D E B, ad rectanguium D V B, ſic <lb/>rectangulum F H G, ſeù Q S Z, ad rectangulum <lb/>R T Y. </s> <s xml:id="echoid-s2968" xml:space="preserve">Sed vt rectangulum Q S Z, ad rectangu-<lb/>lum R T Y, ſic armilla circularis Q S Z, ad armil-<lb/>lam circularem R T Y. </s> <s xml:id="echoid-s2969" xml:space="preserve">Ergo vt armilla ad armil-<lb/>lam, ſic rectangulum D E B, ad rectangulum D V B. <lb/></s> <s xml:id="echoid-s2970" xml:space="preserve">Sed vt rectangulum D E B, in portione ad rect an-<lb/>gulum D V B, ſic rectangulum C D A, in parabo-<lb/>la ad rectangulum C F A; </s> <s xml:id="echoid-s2971" xml:space="preserve">& </s> <s xml:id="echoid-s2972" xml:space="preserve">vt rectangulum C D A, <lb/>ad rectangulum C F A, ſic D B, ſeù F H, ad F G, <lb/>ex ſchol. </s> <s xml:id="echoid-s2973" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2974" xml:space="preserve">22. </s> <s xml:id="echoid-s2975" xml:space="preserve">lib. </s> <s xml:id="echoid-s2976" xml:space="preserve">prim. </s> <s xml:id="echoid-s2977" xml:space="preserve">Ergo vt armilla cir-<lb/>cularis Q S Z, ad armillam circularem R T Y, ſic <lb/>H F, ad F G. </s> <s xml:id="echoid-s2978" xml:space="preserve">Cum vero puncta V, F, ſumpta ſint <lb/>arbitrariè; </s> <s xml:id="echoid-s2979" xml:space="preserve">ergo concludemus omnes armillas circu-<lb/>lares tubi parallelas armillæ N L P, eſſe ad omnes <lb/>armillas circulares exceſſus portionis ſupra conum, <pb o="162" file="0174" n="174"/> parallelas eidem armillæ N L P, vt omnès lineæ pa-<lb/>rallelogrammi C E, parallelæ D B, ad omnes lineas <lb/>parabolæ itidem parallelas D B. </s> <s xml:id="echoid-s2980" xml:space="preserve">Quare etiam tu-<lb/>bus ad exceſſum, erit vt parallelogrammum ad pa-<lb/>rabolam.</s> <s xml:id="echoid-s2981" xml:space="preserve"/> </p> <div xml:id="echoid-div143" type="float" level="2" n="1"> <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a"> <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0173-01"/> </figure> </div> <p> <s xml:id="echoid-s2982" xml:space="preserve">Hoc autem quod probatum fuit de totis, patet eo-<lb/>dem modo probari poſſe de partibus proportionali-<lb/>bus. </s> <s xml:id="echoid-s2983" xml:space="preserve">V. </s> <s xml:id="echoid-s2984" xml:space="preserve">g. </s> <s xml:id="echoid-s2985" xml:space="preserve">codem modo probare poterimus, partem <lb/>tubi K Z, eſſe ad partem exceſſus inter plana k M, <lb/>Q Z, contentam, vt parallelogrammum A H, ad <lb/>portionem A G F. </s> <s xml:id="echoid-s2986" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s2987" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div145" type="section" level="1" n="93"> <head xml:id="echoid-head105" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s2988" xml:space="preserve">Cum ergo ex ſchol. </s> <s xml:id="echoid-s2989" xml:space="preserve">prim. </s> <s xml:id="echoid-s2990" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s2991" xml:space="preserve">1. </s> <s xml:id="echoid-s2992" xml:space="preserve">lib. </s> <s xml:id="echoid-s2993" xml:space="preserve">prim. </s> <s xml:id="echoid-s2994" xml:space="preserve">ſit <lb/>parallelogrammum E C, ſeſquialterum parabolæ, <lb/>etiam tubus erit ſeſquialter prædicti exceſſus. </s> <s xml:id="echoid-s2995" xml:space="preserve">l@mo ex <lb/>propoſitionibus varijs eiuſdem lib. </s> <s xml:id="echoid-s2996" xml:space="preserve">prim. </s> <s xml:id="echoid-s2997" xml:space="preserve">habebimus <lb/>varias rationes partium tubi contentarum inter pla-<lb/>na plano A C, parallela. </s> <s xml:id="echoid-s2998" xml:space="preserve">Quæ autem hæ ſint re-<lb/>linquimus lectori conſiderare ex illis propoſitioni-<lb/>bus, in quibus aſſignantur rationes variarum par-<lb/>tium parallelogrammi C E, ad varia ſegmenta pa-<lb/>rabolæ.</s> <s xml:id="echoid-s2999" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div146" type="section" level="1" n="94"> <head xml:id="echoid-head106" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s3000" xml:space="preserve">Ad modum ergo perſæpe rememoratorum, poſſu-<lb/>mus deducere, exceſſum portionis A B C, ſupra <pb o="163" file="0175" n="175"/> <anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/> ſuum conum, & </s> <s xml:id="echoid-s3001" xml:space="preserve">parabolam eſſe quantitates propor-<lb/>tionaliter analogas tam in magnitudine, quam in <lb/>grauitate, tam ſecundum totum, quam ſecundum <lb/>partes proportionales. </s> <s xml:id="echoid-s3002" xml:space="preserve">Vnde quantum ad magnitu-<lb/>dinem, patet illum exceſſum ſecari à plano F G, bi-<lb/>fariam, ſicuti etiam parabola ſecatur bifariam à dia-<lb/>metro, ſed ſic bifariam, vt partes ſupra, & </s> <s xml:id="echoid-s3003" xml:space="preserve">infrà pla-<lb/>num F G, ſint ſemper ſimiles, & </s> <s xml:id="echoid-s3004" xml:space="preserve">æquales tam ſe-<lb/>cundum totum, quam ſecundum partes proportio-<lb/>nales. </s> <s xml:id="echoid-s3005" xml:space="preserve">Quantum vero ad grauitatem, patet in pri-<lb/>mis centrum grauitatis prædicti exceſſus eſſe in me-<lb/>dio B D, ſicuti in medio A C, baſis parabolæ, eſt <lb/>centrum æquilibrij parabolæ. </s> <s xml:id="echoid-s3006" xml:space="preserve">Inſuper pater, dimi-<lb/>dij exceſſus ſuperioris centrum grauitatis ſic ſecare <lb/>B E, vt pars ad B, ſit ad partem ad E, vt 5, ad 3; <lb/></s> <s xml:id="echoid-s3007" xml:space="preserve">quod habetur ex ſchol. </s> <s xml:id="echoid-s3008" xml:space="preserve">2. </s> <s xml:id="echoid-s3009" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3010" xml:space="preserve">2. </s> <s xml:id="echoid-s3011" xml:space="preserve">lib. </s> <s xml:id="echoid-s3012" xml:space="preserve">3. </s> <s xml:id="echoid-s3013" xml:space="preserve">In eadem <lb/>ratione ſecatur D E, à centro grauitatis partis inſe-<lb/>rioris, adeovt pars ad D, terminata, ſit ad partem <pb o="164" file="0176" n="176"/> terminatamad E, vt 5, ad 3. </s> <s xml:id="echoid-s3014" xml:space="preserve">Patet etiam ex dict is <lb/>in varijs propoſitionibuslib. </s> <s xml:id="echoid-s3015" xml:space="preserve">3. </s> <s xml:id="echoid-s3016" xml:space="preserve">qualiter poſſimus ha-<lb/>bere centrum grauitatis variorum ſegmentorum di-<lb/>cti exceſſus, ſicuti habemus centrum æquilibrij in <lb/>baſi A C, variorum ſegmentorum parabolæ.</s> <s xml:id="echoid-s3017" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="1"> <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a"> <image file="0175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0175-01"/> </figure> </div> <p> <s xml:id="echoid-s3018" xml:space="preserve">Sed duo etiam adnotentur. </s> <s xml:id="echoid-s3019" xml:space="preserve">Primumeſt, magni-<lb/>tudinibus inſchol. </s> <s xml:id="echoid-s3020" xml:space="preserve">3. </s> <s xml:id="echoid-s3021" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s3022" xml:space="preserve">26. </s> <s xml:id="echoid-s3023" xml:space="preserve">oſtenſis proportio-<lb/>naliter analogis, aſſociarietiam exceſſum prædictum <lb/>ſupra conum. </s> <s xml:id="echoid-s3024" xml:space="preserve">Alterumeſt, quod quæ dicta ſunt de <lb/>exceſſu portionis ſphæræ ſupra ſuumconum, intelli-<lb/>genda etiam ſunt de exceſſu portionis ſphæroidis <lb/>ſupra ſuum conum. </s> <s xml:id="echoid-s3025" xml:space="preserve">Quia in lib. </s> <s xml:id="echoid-s3026" xml:space="preserve">4. </s> <s xml:id="echoid-s3027" xml:space="preserve">de infinit. </s> <s xml:id="echoid-s3028" xml:space="preserve">para-<lb/>bolis, probata eſt perpetua analogia reperta inter <lb/>proportionales partes ſphæræ, & </s> <s xml:id="echoid-s3029" xml:space="preserve">ſphæroidis.</s> <s xml:id="echoid-s3030" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div148" type="section" level="1" n="95"> <head xml:id="echoid-head107" xml:space="preserve">PROPOSITIO XLVI.</head> <p style="it"> <s xml:id="echoid-s3031" xml:space="preserve">Si in quolibet conoide hyperbolico, & </s> <s xml:id="echoid-s3032" xml:space="preserve">parabolico quadra-<lb/>tico; </s> <s xml:id="echoid-s3033" xml:space="preserve">item in qualibet ſphœiœ, vel ſphœroidis portione <lb/>inſcribatur conus. </s> <s xml:id="echoid-s3034" xml:space="preserve">Centrum grauitatis exceſſus prœdi-<lb/>ctorum ſolidorum ſupra ſuos conos erit in medio puncto <lb/>diametri ipſorum.</s> <s xml:id="echoid-s3035" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3036" xml:space="preserve">SIt conoides parabolicum quadraticum, vt in <lb/>prima figura in ſchem. </s> <s xml:id="echoid-s3037" xml:space="preserve">ſequent. </s> <s xml:id="echoid-s3038" xml:space="preserve">B A C, vel <lb/>hyperbolicum vt in ſecunda; </s> <s xml:id="echoid-s3039" xml:space="preserve">vel quælibet portio <lb/>ſphæræ, vel ſphæroidis vt in tertia, & </s> <s xml:id="echoid-s3040" xml:space="preserve">in iſtis ſolidis <lb/>intelligantur inſcripti coni B A C. </s> <s xml:id="echoid-s3041" xml:space="preserve">Dico centrum <lb/>grauitatis exceſſuum prædictorum ſolidorum ſupra <pb o="165" file="0177" n="177"/> conos eſſe in E, diuidente bifariam A D. </s> <s xml:id="echoid-s3042" xml:space="preserve">De ex-<lb/>ceſſu conoideorum ſupra conos, patuit in ſcholio <lb/>propoſit. </s> <s xml:id="echoid-s3043" xml:space="preserve">6. </s> <s xml:id="echoid-s3044" xml:space="preserve">De exceſſu portionis ſphæræ, vel ſphæ-<lb/>roidis patuit in anteced propoſit. </s> <s xml:id="echoid-s3045" xml:space="preserve">Quare quoadom-<lb/>nia patet propoſitum.</s> <s xml:id="echoid-s3046" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div149" type="section" level="1" n="96"> <head xml:id="echoid-head108" xml:space="preserve">PROPOSITIO XLVII.</head> <p style="it"> <s xml:id="echoid-s3047" xml:space="preserve">Si in ſolidis antecedentis propoſitionis inſcribantur coni vt <lb/>dictum eſt, & </s> <s xml:id="echoid-s3048" xml:space="preserve">ſect s diametris ipſorum bifariam ordi-<lb/>natim applicentur lineœ, ſecantes latus conorum inſcri-<lb/>ptorum. </s> <s xml:id="echoid-s3049" xml:space="preserve">Diametri prœdictorum ſolidorum, & </s> <s xml:id="echoid-s3050" xml:space="preserve">etiam <lb/>coni, ſic ſecabuntur ab ipſorum centris grauitatis, vt <lb/>partes terminatœ ad verticem ſint ad partes terminatas <lb/>ad baſim vt quadratum ordinatim applicatœ, vna cum <lb/>duobus quadratis ductœ in conis, ad quadratum ordi-<lb/>natim applicatœ.</s> <s xml:id="echoid-s3051" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3052" xml:space="preserve">SInt ergo ſolida vt in antecedenti propoſitione, & </s> <s xml:id="echoid-s3053" xml:space="preserve"><lb/>inſuper etiam conus, vt in quarta figura BAC, <lb/>quorum diametri A D, ſint ſectæ bifariamin E, & </s> <s xml:id="echoid-s3054" xml:space="preserve"><lb/>ord natim applicentur E G F, ſitque horum cen-<lb/>trum grauitatis punctum O. </s> <s xml:id="echoid-s3055" xml:space="preserve">Dico A O, eſſe ad <lb/>O D, vt quadratum F E, cum duobus quadratis <lb/>G E, ad quadratum F E. </s> <s xml:id="echoid-s3056" xml:space="preserve">In cono res eſt manifeſta, <lb/>quia ſicuti A O, eſt tripla O D, ſic tria quadrata <lb/>G E, ſunt tripla vnius quadrati G E. </s> <s xml:id="echoid-s3057" xml:space="preserve">In alijs ſic <lb/>patebit. </s> <s xml:id="echoid-s3058" xml:space="preserve">Fiat D P, quarta pars D A. </s> <s xml:id="echoid-s3059" xml:space="preserve">Ergo P, <lb/>erit centrum grauitatis conorum. </s> <s xml:id="echoid-s3060" xml:space="preserve">Cum ergo ex pro- <pb o="166" file="0178" n="178"/> <anchor type="figure" xlink:label="fig-0178-01a" xlink:href="fig-0178-01"/> poſit. </s> <s xml:id="echoid-s3061" xml:space="preserve">antëced. </s> <s xml:id="echoid-s3062" xml:space="preserve">ſit etiam F, centrum grauitatis ex-<lb/>ceſſus ſolidorum ſupra conos, & </s> <s xml:id="echoid-s3063" xml:space="preserve">ex ſuppoſito, ſit <lb/>O, centrum grauitatis ſolidorum; </s> <s xml:id="echoid-s3064" xml:space="preserve">ergo erit reci-<lb/>procè vt P O, ad OE, ſic exceſſus ſolidorum ſu-<lb/>pra conos ad ipſos conos. </s> <s xml:id="echoid-s3065" xml:space="preserve">Et componendo, vt P E, <pb o="167" file="0179" n="179"/> ad O E, ſic ſolida ad ipſos conos. </s> <s xml:id="echoid-s3066" xml:space="preserve">Sed ex propoſit. <lb/></s> <s xml:id="echoid-s3067" xml:space="preserve">53. </s> <s xml:id="echoid-s3068" xml:space="preserve">lib. </s> <s xml:id="echoid-s3069" xml:space="preserve">noſtri ſexaginta problematum geometrico-<lb/>rum, ſolida ſunt ad conos vt quadrata F E, E G, ad <lb/>duplum quadratum E G. </s> <s xml:id="echoid-s3070" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3071" xml:space="preserve">P E, erit ad E O, <lb/>vt quadrata F E, E G, ad duplum quadratum E G. </s> <s xml:id="echoid-s3072" xml:space="preserve"><lb/>Et antecedentium dupla. </s> <s xml:id="echoid-s3073" xml:space="preserve">Ergo vt D E, ad E O, <lb/>ſic duo quadrata F E, cum duobus quadratis E G, <lb/>ad duo quadrata E G. </s> <s xml:id="echoid-s3074" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3075" xml:space="preserve">per conuerſionem <lb/>rationis vt E D, ad D O, ſic duo quadrata F E, <lb/>cum duobus quadratis E G, ad duo quadrata F E; </s> <s xml:id="echoid-s3076" xml:space="preserve"><lb/>nempe ſic dimidium ad dimidium, ſcilicet ſic qua-<lb/>drata F E, E G, ad quadratum F E. </s> <s xml:id="echoid-s3077" xml:space="preserve">Et vt antece-<lb/>dentium dupla. </s> <s xml:id="echoid-s3078" xml:space="preserve">Ergo vt A D, ad D O, ſic duo <lb/>quadrata F E, cum duobus quadratis G E, ad qua-<lb/>dratum F E. </s> <s xml:id="echoid-s3079" xml:space="preserve">Et diuidendo vt A O, ad O D, ſic <lb/>quadratum F E, cum duobus quadratis G E, ad <lb/>quadratum F E. </s> <s xml:id="echoid-s3080" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s3081" xml:space="preserve"/> </p> <div xml:id="echoid-div149" type="float" level="2" n="1"> <figure xlink:label="fig-0178-01" xlink:href="fig-0178-01a"> <image file="0178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0178-01"/> </figure> </div> </div> <div xml:id="echoid-div151" type="section" level="1" n="97"> <head xml:id="echoid-head109" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3082" xml:space="preserve">Cum ergo in progreſſu demonſtrationis proba-<lb/>tum ſit, eſſe D F, ad E O, vt duo quadrata F E, <lb/>cum duobus quadratis G E, ad duo quadrata G E; <lb/></s> <s xml:id="echoid-s3083" xml:space="preserve">nempe vt quadrata F E, E G, ad quadratum E G; </s> <s xml:id="echoid-s3084" xml:space="preserve"><lb/>ergo etiam diuidendo, erit D O, ad O E, vt qua-<lb/>dratum F E, ad quadratum G E. </s> <s xml:id="echoid-s3085" xml:space="preserve">Quod etiam pa-<lb/>tet verificari in cono. </s> <s xml:id="echoid-s3086" xml:space="preserve">Sed ex hac propoſitione, & </s> <s xml:id="echoid-s3087" xml:space="preserve"><lb/>ex analogia, quæ reperitur inter parabolam qua-<lb/>draticam, & </s> <s xml:id="echoid-s3088" xml:space="preserve">ſphæram, poteſt colligi quædam pro- <pb o="168" file="0180" n="180"/> poſitio vniuerſalis in qualibet portione parabolæ <lb/>quadraticæ.</s> <s xml:id="echoid-s3089" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div152" type="section" level="1" n="98"> <head xml:id="echoid-head110" xml:space="preserve">PROPOSITIO XLVIII.</head> <p style="it"> <s xml:id="echoid-s3090" xml:space="preserve">Si in quacunque portione parabolœ quadraticœ reſectœ linea <lb/>diametro parallela inſcribatur triangulum, & </s> <s xml:id="echoid-s3091" xml:space="preserve">baſis por-<lb/>tionts parabolœ ſecetur bifariam, & </s> <s xml:id="echoid-s3092" xml:space="preserve">per punctum biſſe-<lb/>ctionis ducatur parallela diametro. </s> <s xml:id="echoid-s3093" xml:space="preserve">Centrum œquilibrij <lb/>ſecundum baſim prœdictœ portionis ſic ſecabit baſim, <lb/>vt pars ad curuam terminata ſit ad reliquam, vt pa-<lb/>rallela diametro ducta à puncto biſſectionis, vna cum tn-<lb/>tercepta inter punctum bißectionis, & </s> <s xml:id="echoid-s3094" xml:space="preserve">latus trianguli, <lb/>ad prœdictam parallelam diametro.</s> <s xml:id="echoid-s3095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3096" xml:space="preserve">ESto parabola A B C, quadratica, cuius baſis <lb/>A C, diameter B D, & </s> <s xml:id="echoid-s3097" xml:space="preserve">ſit quælibet eius por-<lb/>tio E F B C, reſecta F E, diametro B D, paralle-<lb/>la, & </s> <s xml:id="echoid-s3098" xml:space="preserve">in portione ſit in ſciiptum triangulum CFE; </s> <s xml:id="echoid-s3099" xml:space="preserve">ſit-<lb/>que C E, ſecta bifariam in G, & </s> <s xml:id="echoid-s3100" xml:space="preserve">per G, ducatur <lb/>GIH, parallela diametro, ſitque K, centrum æ-<lb/>quilibrij in baſi portionis E F B C. </s> <s xml:id="echoid-s3101" xml:space="preserve">Dico C k, eſſe <lb/>ad k E, vt H G, cum G I, ad H I. </s> <s xml:id="echoid-s3102" xml:space="preserve">In tertia figura <lb/>ſchematis anteced. </s> <s xml:id="echoid-s3103" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s3104" xml:space="preserve">intelligatur portio ſphæ-<lb/>ræ, vel ſphæroidis B A C, proportionalis E F B C, <lb/>portioni parabolæ, & </s> <s xml:id="echoid-s3105" xml:space="preserve">intelligantur in ea omnia, <lb/>quæ ſupra, Ergo C K, erit ad k E, in portione pa-<lb/>rabolæ, vt A O, ad O D, in portione ſphæræ; </s> <s xml:id="echoid-s3106" xml:space="preserve">nem-<lb/>pe ex propoſit. </s> <s xml:id="echoid-s3107" xml:space="preserve">anteced. </s> <s xml:id="echoid-s3108" xml:space="preserve">vt duplum quadratum G E, <pb o="169" file="0181" n="181"/> <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> cum quadrato F E, ad quadratum F E. </s> <s xml:id="echoid-s3109" xml:space="preserve">Sed cum <lb/>G E, ſit dimidia B D, eius quadratum erit quarta <lb/>pars quadrati B D; </s> <s xml:id="echoid-s3110" xml:space="preserve">& </s> <s xml:id="echoid-s3111" xml:space="preserve">duo quadrata G E, erunt di-<lb/>midium quadrati B D. </s> <s xml:id="echoid-s3112" xml:space="preserve">Ergo A O, ad O D, & </s> <s xml:id="echoid-s3113" xml:space="preserve">C K, <lb/>ad K E, in portione parabolæ, erunt vt quadratum. <lb/></s> <s xml:id="echoid-s3114" xml:space="preserve">F E, cum dimidio quadrati B D, ad quadratum <lb/>F E; </s> <s xml:id="echoid-s3115" xml:space="preserve">nempe vt dimidium rectanguli H D A, cum <lb/>rectangulo H E A, ad rectangulum H E A. </s> <s xml:id="echoid-s3116" xml:space="preserve">Sed vt <lb/>illa plana ad inuicem in portione ſphæræ, ſic in por-<lb/>tione parabolæ quadraticæ dimidium rectanguli <lb/>A E C, cum rectangulo A G C, ad rectangulum. </s> <s xml:id="echoid-s3117" xml:space="preserve"><lb/>A G C. </s> <s xml:id="echoid-s3118" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3119" xml:space="preserve">vt C K, ad k E, ſic dimidium re-<lb/>ctanguli A E C, cum rectangulo A G C, ad rectan- <pb o="170" file="0182" n="182"/> gulum A G C. </s> <s xml:id="echoid-s3120" xml:space="preserve">Sed vt hæc plana ad inuicem ſic di-<lb/>midia F E, nempe G I, cum H G, ad H G. </s> <s xml:id="echoid-s3121" xml:space="preserve">Qua-<lb/>re & </s> <s xml:id="echoid-s3122" xml:space="preserve">vt C k, ad K E, ſic G I, cum G H, ad G H. <lb/></s> <s xml:id="echoid-s3123" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s3124" xml:space="preserve"/> </p> <div xml:id="echoid-div152" 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/PVNDER1Y/figures/0181-01"/> </figure> </div> </div> <div xml:id="echoid-div154" type="section" level="1" n="99"> <head xml:id="echoid-head111" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s3125" xml:space="preserve">Sed ex progreſſu demonſtrationis poteſt etiam fa-<lb/>cile probarieſſe C k, ad k E, vt A E, cum A G, ad <lb/>A G. </s> <s xml:id="echoid-s3126" xml:space="preserve">Nam cum probatum ſit eſſe C k, ad k E, vt <lb/>dimidium rectanguli A E C (nempe vt rectangu-<lb/>lum A E, G C) ſimul cum rectangulo A G C, ad <lb/>rectangulum A G C. </s> <s xml:id="echoid-s3127" xml:space="preserve">Patet hæc rectangula ob com-<lb/>mune latus C G, eſſe vt A E, A G, ad A G. </s> <s xml:id="echoid-s3128" xml:space="preserve">Qua-<lb/>re & </s> <s xml:id="echoid-s3129" xml:space="preserve">ſic C k, ad k E.</s> <s xml:id="echoid-s3130" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3131" xml:space="preserve">Eliciet ergo lector facile, eſſe E k, ad k G, vt <lb/>H G, ad dimidiam G I; </s> <s xml:id="echoid-s3132" xml:space="preserve">vel vt G A, ad dimidiam <lb/>A E. </s> <s xml:id="echoid-s3133" xml:space="preserve">Ex quibus etiam patebit in portione B A C, <lb/>fphæræ, vel ſphæroidis eſſe A O, ad O D, vt D H, <lb/>H E, ad H E. </s> <s xml:id="echoid-s3134" xml:space="preserve">Et D O, eſſe ad O E, vt E H, ad <lb/>dimidiam H D.</s> <s xml:id="echoid-s3135" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3136" xml:space="preserve">Sed hæc, quæ probata fuerunt ex analogia reper-<lb/>ta inter portiones parabolæ, & </s> <s xml:id="echoid-s3137" xml:space="preserve">ſphæræ, poſlunt ab-<lb/>folutè probari ex proprijs ipſius paiabolic. </s> <s xml:id="echoid-s3138" xml:space="preserve">Nam <lb/>cum F B C, ſit verè parabola ex prim. </s> <s xml:id="echoid-s3139" xml:space="preserve">conic. </s> <s xml:id="echoid-s3140" xml:space="preserve">propo-<lb/>ſit. </s> <s xml:id="echoid-s3141" xml:space="preserve">47. </s> <s xml:id="echoid-s3142" xml:space="preserve">cuius diameter H I, erit in G, centrum æ-<lb/>quilibrij parabolæ F B C, appenſæ ſecundum C E. <lb/></s> <s xml:id="echoid-s3143" xml:space="preserve">Fiat C L, dupla L E. </s> <s xml:id="echoid-s3144" xml:space="preserve">Ergo L, erit centrum æqui-<lb/>librii trianguli E F C, appenſi ſecundum, C E. </s> <s xml:id="echoid-s3145" xml:space="preserve">Er- <pb o="171" file="0183" n="183"/> <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> go erit reciprocè vt L k, ad k G, ſic F B C, ad tri-<lb/>angulum F C E. </s> <s xml:id="echoid-s3146" xml:space="preserve">Et componendo, erit L G, ad <lb/>G k, vt portio E F B C, ad triangulum E F C. </s> <s xml:id="echoid-s3147" xml:space="preserve">Sed <lb/>cum exſchol. </s> <s xml:id="echoid-s3148" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3149" xml:space="preserve">17. </s> <s xml:id="echoid-s3150" xml:space="preserve">lib prim. </s> <s xml:id="echoid-s3151" xml:space="preserve">ſit conuerten-<lb/>do, portio ad parallelogrammum duplum trianguli, <lb/>vt dimidia A E, vna cum ſexta parte C E, ad A E; <lb/></s> <s xml:id="echoid-s3152" xml:space="preserve">& </s> <s xml:id="echoid-s3153" xml:space="preserve">ad ipſum triangulum, vt idem antecedens ad di-<lb/>midiam A E. </s> <s xml:id="echoid-s3154" xml:space="preserve">Ergo erit etiam, vt L G, ad G K, <lb/>fic dimidia A E, cum ſexta parte C E, ad dimidiam <lb/>A E. </s> <s xml:id="echoid-s3155" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3156" xml:space="preserve">vt antecedentium tripla. </s> <s xml:id="echoid-s3157" xml:space="preserve">Ergo vt <lb/>E G, tripla L G, ad G k, lic ſeſquialtera A E, <lb/>cum dimidia C E, ad dimidiam A E. </s> <s xml:id="echoid-s3158" xml:space="preserve">Et per con-<lb/>uerſionem rationis, vt G E, ad E K, ſic ſeſquialte- <pb o="172" file="0184" n="184"/> ra A E; </s> <s xml:id="echoid-s3159" xml:space="preserve">cum dimidia C E, ad dimidiam C E, cum <lb/>A E. </s> <s xml:id="echoid-s3160" xml:space="preserve">Et rurſum vt antecedentium dupla. </s> <s xml:id="echoid-s3161" xml:space="preserve">Ergo vt <lb/>C E, ad E K, ſic C E, cum tripla A E, ad dimi-<lb/>diam C E, cum A E. </s> <s xml:id="echoid-s3162" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3163" xml:space="preserve">diuidendo, vt dimi-<lb/>dia C E, cum dupla A E, ad dimidiam C E, cum <lb/>A E, ſic C K, ad K E. </s> <s xml:id="echoid-s3164" xml:space="preserve">Sed vt dimidia C E, cum <lb/>dupla A E, nempe vt G A, cum A E, ad dimi-<lb/>diam C E, cum A E, nempe ad G A, ſic ſumpta <lb/>communi altitudine C G, rectangulum A G C, cum <lb/>rectangulo ſub A E, in G C, ad rectangulum A G C: <lb/></s> <s xml:id="echoid-s3165" xml:space="preserve">Et vt rectangulum A G C, cum rectangulo A E, G C, <lb/>ad rectangulum A G C, ſic H G, cum dimidia F E, <lb/>nempe cum I G, ad H G. </s> <s xml:id="echoid-s3166" xml:space="preserve">Quare & </s> <s xml:id="echoid-s3167" xml:space="preserve">vt C K, ad <lb/>k E, ſic H G, cum G I, ad H G. </s> <s xml:id="echoid-s3168" xml:space="preserve">Quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s3169" xml:space="preserve"/> </p> <div xml:id="echoid-div154" 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/PVNDER1Y/figures/0183-01"/> </figure> </div> </div> <div xml:id="echoid-div156" type="section" level="1" n="100"> <head xml:id="echoid-head112" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s3170" xml:space="preserve">Sed cum in ſchol. </s> <s xml:id="echoid-s3171" xml:space="preserve">2. </s> <s xml:id="echoid-s3172" xml:space="preserve">prop. </s> <s xml:id="echoid-s3173" xml:space="preserve">45 probatum ſit parabo-<lb/>lam quadraticam, ſphæram, & </s> <s xml:id="echoid-s3174" xml:space="preserve">ſphæroides eſſe quan-<lb/>titates proportionaliter analogas cum tribus alijs <lb/>ſolidis, ſequitur etiam in illis currere ſupra explica-<lb/>tum compendium circa illorum centra grauitatis. <lb/></s> <s xml:id="echoid-s3175" xml:space="preserve">Quon am ergo exceſſus, in ſchemate ſequenti, por-<lb/>tionis A B C, ſphæræ, vel ſphæroidis ſupra conum <lb/>A B C, eſt proportionaliter analogus cum parabola <lb/>quadratica A B C; </s> <s xml:id="echoid-s3176" xml:space="preserve">ſequitur inquam, quod ſi prius <lb/>fecetur plano F E G, deinde plano R V Y, ſecante <lb/>B E, biſariam in V, quod centrum grauitatis partis <pb o="173" file="0185" n="185"/> <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> exceſſus ex F B H, reuoluta circa B E, ſic ſecabic <lb/>B E, vt pars terminata ad B, ſit ad partem termina-<lb/>tam ad E, vel vt rectangulum R T Y, cum dimi-<lb/>dio rectanguli F H G, ad rectangulum R T Y: </s> <s xml:id="echoid-s3177" xml:space="preserve">vel <lb/>vt rectangulum A T B, cum dimidio rectanguli <lb/>A H B, ad rectangulum A T B: </s> <s xml:id="echoid-s3178" xml:space="preserve">vel vt rectangulum <lb/>D V B, cum dimidio rectanguli D E B, ad rectan-<lb/>gulum D V B: </s> <s xml:id="echoid-s3179" xml:space="preserve">vel compendioſius, vt E D, D V, <lb/>ad D V: </s> <s xml:id="echoid-s3180" xml:space="preserve">ſeù, quod idem eſt, vt A H A T, ad A T. <lb/></s> <s xml:id="echoid-s3181" xml:space="preserve">Pariter ſequitur, quod E V, ſic ſecabitur à prædi-<lb/>cto centro, vt pars terminata ad E, ſit ad partem <lb/>terminatam ad V, vt V D, ad dimidiam D E: </s> <s xml:id="echoid-s3182" xml:space="preserve">ſeù <lb/>vt T A, ad dimidiam A H: </s> <s xml:id="echoid-s3183" xml:space="preserve">ſeù vt rectangulum <lb/>B V D, ad dimidium rectanguli B E D: </s> <s xml:id="echoid-s3184" xml:space="preserve">ſeù vt re-<lb/>ctangulum B T A, ad dimidium rectanguli B H A: </s> <s xml:id="echoid-s3185" xml:space="preserve"><lb/>feù tandem vt rectangulum R T Y, ad dimidium <lb/>rectanguli F H G.</s> <s xml:id="echoid-s3186" xml:space="preserve"/> </p> <div xml:id="echoid-div156" type="float" level="2" n="1"> <figure xlink:label="fig-0185-01" xlink:href="fig-0185-01a"> <image file="0185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0185-01"/> </figure> </div> <p> <s xml:id="echoid-s3187" xml:space="preserve">Item cumin ſchem. </s> <s xml:id="echoid-s3188" xml:space="preserve">poſito in ſchol. </s> <s xml:id="echoid-s3189" xml:space="preserve">prop. </s> <s xml:id="echoid-s3190" xml:space="preserve">40. </s> <s xml:id="echoid-s3191" xml:space="preserve">ſuppo- <pb o="174" file="0186" n="186"/> ſito R B Z, A B C, eſſe conos, probatŭ ſit ibidem exceſ-<lb/>ſum cylindri R C, ſupra illos conoseſſe proportio-<lb/>naliter analogum cum parabola quadratica; </s> <s xml:id="echoid-s3192" xml:space="preserve">ſequi-<lb/>tur, quod ſi prædictus exceſſus ſecetur plano L P M, <lb/>deinde ſupponamus rurſum ſecari plano I T X, ſe-<lb/>cante bifariam S G, in V: </s> <s xml:id="echoid-s3193" xml:space="preserve">ſequitur inquam S G, ſe-<lb/>cari à centro grauitatis partis exceſlus geniti ex re-<lb/>uolutione ſegmenti L P B T R, in prædictis ratio-<lb/>nibus.</s> <s xml:id="echoid-s3194" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3195" xml:space="preserve">Tandem inſpiciatur ſchema poſitum in propoſit. <lb/></s> <s xml:id="echoid-s3196" xml:space="preserve">26. </s> <s xml:id="echoid-s3197" xml:space="preserve">in quo ex cit. </s> <s xml:id="echoid-s3198" xml:space="preserve">ſchol. </s> <s xml:id="echoid-s3199" xml:space="preserve">annulus latus ex hyperbola <lb/>A B C, circa K M, probatus fuit proportionaliter <lb/>analogus cum parabola quadratica A O C. </s> <s xml:id="echoid-s3200" xml:space="preserve">Si ergo <lb/>illæ annulus ſecetur prius vbilibet plano N B V, de-<lb/>inde plano I S T, ſecante bifariam K L, in puncto, <lb/>in quo ipſam ſecat; </s> <s xml:id="echoid-s3201" xml:space="preserve">eadem compendia ſupra expoſita <lb/>colligemus circa centrum grauitatis portionis annu-<lb/>li ex portione hyperbolæ A B N. </s> <s xml:id="echoid-s3202" xml:space="preserve">Hæc enim omnia <lb/>patent ex dictis, & </s> <s xml:id="echoid-s3203" xml:space="preserve">lector memor ſupradictorum fa-<lb/>cile percipiet. </s> <s xml:id="echoid-s3204" xml:space="preserve">Nè ergo ipſi tædium afferamus ad <lb/>alia tranſeamus.</s> <s xml:id="echoid-s3205" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3206" xml:space="preserve">Parabola quadratica habet lineam quandam, <lb/>quæ appellatur parameter, ſeù latus rectum; </s> <s xml:id="echoid-s3207" xml:space="preserve">cuius <lb/>natura eſt, vt quadrata ordinatim applicatarum, æ-<lb/>qualia ſint rectangulis contentis ſub hac, & </s> <s xml:id="echoid-s3208" xml:space="preserve">ſub por-<lb/>tionibus axis abſciſſis verſus verticem ab ordinatim <lb/>applicatis. </s> <s xml:id="echoid-s3209" xml:space="preserve">Hanc proprietatem habent quoque aliæ <lb/>inſinitæ parabolæ, ſed ſuo modo: </s> <s xml:id="echoid-s3210" xml:space="preserve">adeovt in quali-<lb/>bet ſit aſſignabilis quędamlinea, vt poteſtates ordi- <pb o="175" file="0187" n="187"/> natim applicatarum parabolæ congruentes, ęquales <lb/>ſint poteſtatibus factis ſub prędictis abſciſſis ab or-<lb/>dinatim applicatis, & </s> <s xml:id="echoid-s3211" xml:space="preserve">ſub poteſtate talis lineæ vno <lb/>gradu depreſſiore poteſtate parabolę. </s> <s xml:id="echoid-s3212" xml:space="preserve">Sit ergo.</s> <s xml:id="echoid-s3213" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div158" type="section" level="1" n="101"> <head xml:id="echoid-head113" xml:space="preserve">PROPOSITIO XLIX.</head> <p style="it"> <s xml:id="echoid-s3214" xml:space="preserve">Si fiat vt diameter parabolæ ad ſemibaſim, ſic buius po-<lb/>testas vno gradu depreſſior poteſtate parabolæ ad ſimi-<lb/>lem poteſtatem lineæ inueniendæ. </s> <s xml:id="echoid-s3215" xml:space="preserve">Potestates applicata-<lb/>rum ordinatim in parabola eiuſdem gradus cum parabo-<lb/>la, æquales erunt factis ſub abſciſſis diametri verſus <lb/>verticem ab ordinatim applicatis, & </s> <s xml:id="echoid-s3216" xml:space="preserve">ſub poteſtate li-<lb/>neæ inuentæ, vno gradu depreſſiore poteſtate para-<lb/>bolæ.</s> <s xml:id="echoid-s3217" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3218" xml:space="preserve">Esto quælibet parabola B A C, in qua fiat vt dia-<lb/>meter A D, ad ſemibaſim D B, ſic poteſtas <lb/>huius vnogradu depreſſior poteſtate parabolæ, ad <lb/>fimilem poteſtatem A H: </s> <s xml:id="echoid-s3219" xml:space="preserve">v. </s> <s xml:id="echoid-s3220" xml:space="preserve">g. </s> <s xml:id="echoid-s3221" xml:space="preserve">ſi parabola eſt qua-<lb/>dratica, ſic D B, ad A H; </s> <s xml:id="echoid-s3222" xml:space="preserve">ſi eſt cubica, ſic quadra-<lb/>tum D B, ad quædratum A H: </s> <s xml:id="echoid-s3223" xml:space="preserve">ſi eſt quadratoqua-<lb/>dratica, ſic cubus D B, ad cubum A H. </s> <s xml:id="echoid-s3224" xml:space="preserve">Dico, quod <lb/>ſi ordinatim applicentur G L, E k, poteſtas G L, <lb/>eiuſdem gradus cum parabola ęqualis erit facto ſub <lb/>L A, & </s> <s xml:id="echoid-s3225" xml:space="preserve">ſub poteſtate A H, vno gradu depreſſiore <lb/>poteſtate parabolæ, & </s> <s xml:id="echoid-s3226" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s3227" xml:space="preserve">Quoniam e-<lb/>nim vt A D, ad D B, ſic poteſtas D B, vno gradu <lb/>depreſſior poteſtate parabolæ, ad ſimilem poteſta- <pb o="176" file="0188" n="188"/> <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> tem A H; </s> <s xml:id="echoid-s3228" xml:space="preserve">ergo factum ſub D A, & </s> <s xml:id="echoid-s3229" xml:space="preserve">ſub prædicta <lb/>poteſtate A H, erit ęquale poteſtati B D, eiuſdem <lb/>gradus cumparabola. </s> <s xml:id="echoid-s3230" xml:space="preserve">Cum autem ſit ex geneſi pa-<lb/>rabolæ, vt poteſtas B D, eiuſdem gradus cum para-<lb/>bola ad ſimilem poteſtatem G L, ſic D A, ad A L. <lb/></s> <s xml:id="echoid-s3231" xml:space="preserve">Ft vt D A, ad A L, ſic factum ſub D A, & </s> <s xml:id="echoid-s3232" xml:space="preserve">ſub po-<lb/>teſtate A H, vno gradu depreſſiore poteſtate para-<lb/>bolæ, ad factum ſub L A, & </s> <s xml:id="echoid-s3233" xml:space="preserve">ſub prędicta poteſtate <lb/>A H. </s> <s xml:id="echoid-s3234" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3235" xml:space="preserve">vt factum ſub D A, & </s> <s xml:id="echoid-s3236" xml:space="preserve">ſub tali pote-<lb/>ſtate A H, ad factum ſub L A, & </s> <s xml:id="echoid-s3237" xml:space="preserve">ſub poteſtate <lb/>A H, ſic poteſtas B D, eiuſdem gradus cum parabo-<lb/>la ad ſimilem poteſtatem G L. </s> <s xml:id="echoid-s3238" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3239" xml:space="preserve">permutan-<lb/>do, vt factum ſub D A, & </s> <s xml:id="echoid-s3240" xml:space="preserve">ſub tali poteſtate A H, <lb/>ad poteſtatem B D, eiuſdem gradus cum parabola, <lb/>ſic factum ſub L A, & </s> <s xml:id="echoid-s3241" xml:space="preserve">ſub poteſtate A H, ad pote- <pb o="177" file="0189" n="189"/> ſtàtem G L, eiuſdem gradus cum parabola. </s> <s xml:id="echoid-s3242" xml:space="preserve">Cum <lb/>autem factum ſub D A, & </s> <s xml:id="echoid-s3243" xml:space="preserve">ſub poteſtate A H, <lb/>oſtenſum fuerit ęquale poteſtati prędictę B D. </s> <s xml:id="echoid-s3244" xml:space="preserve">Ergo <lb/>& </s> <s xml:id="echoid-s3245" xml:space="preserve">factum ſub L A, & </s> <s xml:id="echoid-s3246" xml:space="preserve">ſub poteſtate A H, erit ęqua-<lb/>le poteſtati G L. </s> <s xml:id="echoid-s3247" xml:space="preserve">Idem patebit de reliquis. </s> <s xml:id="echoid-s3248" xml:space="preserve">Quare <lb/>etiam patebit propoſitum.</s> <s xml:id="echoid-s3249" xml:space="preserve"/> </p> <div xml:id="echoid-div158" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0188-01"/> </figure> </div> </div> <div xml:id="echoid-div160" type="section" level="1" n="102"> <head xml:id="echoid-head114" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3250" xml:space="preserve">Sed lubet huic tractatui finem imponere infinita-<lb/>rum parabolarum tangentibus, ac maximis inſcripti-<lb/>bilibus, minimiſque circumſcriptilibus infinitis para-<lb/>bolis, infinitis conoidibus, ac ſemifufis parabo-<lb/>licis. </s> <s xml:id="echoid-s3251" xml:space="preserve">Pro quibus reperien dis nobis neceſſaria eſt <lb/>doctrina quædam, quę cum ſit nimis prolixa, ex alijs <lb/>eſt petenda. </s> <s xml:id="echoid-s3252" xml:space="preserve">Euclides in 6. </s> <s xml:id="echoid-s3253" xml:space="preserve">Elementorum libro, pro-<lb/>poſit. </s> <s xml:id="echoid-s3254" xml:space="preserve">27. </s> <s xml:id="echoid-s3255" xml:space="preserve">oſtendit. </s> <s xml:id="echoid-s3256" xml:space="preserve">_Omnium parallelogrammorum ad_ <lb/>_eandem rectam lineam applicatorum, & </s> <s xml:id="echoid-s3257" xml:space="preserve">deficientium figu-_ <lb/>_ris parallelogrammis ſimilibus, & </s> <s xml:id="echoid-s3258" xml:space="preserve">ſimiliter poſitis ei, quæ_ <lb/>_à dimidia deſcribitur, maximum eſt quod ad dimidiam eſt_ <lb/>_applicatum, ſimile existens defectur_. </s> <s xml:id="echoid-s3259" xml:space="preserve">Quod Euclides de-<lb/>monſtrauit in planis, Eutocius de ſphæra, & </s> <s xml:id="echoid-s3260" xml:space="preserve">cylind. <lb/></s> <s xml:id="echoid-s3261" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3262" xml:space="preserve">3. </s> <s xml:id="echoid-s3263" xml:space="preserve">Bonauentura Caualerius, in exercit. </s> <s xml:id="echoid-s3264" xml:space="preserve">6. </s> <s xml:id="echoid-s3265" xml:space="preserve"><lb/>propoſit. </s> <s xml:id="echoid-s3266" xml:space="preserve">28. </s> <s xml:id="echoid-s3267" xml:space="preserve">Ricardus Albius in ſuo hemiſphę. </s> <s xml:id="echoid-s3268" xml:space="preserve">diſ-<lb/>fecto. </s> <s xml:id="echoid-s3269" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3270" xml:space="preserve">42. </s> <s xml:id="echoid-s3271" xml:space="preserve">extenderunt ſuo medo ad ſolida, <lb/>patefacientes. </s> <s xml:id="echoid-s3272" xml:space="preserve">_Omnium parallelepipedorum ad eandem_ <lb/>_rectam lineam applicatorum cubiſque deficientium, maxi-_ <lb/>_mum eſse, quod ad tertiam illius partem applicatur_. </s> <s xml:id="echoid-s3273" xml:space="preserve">Hanc <lb/>denique doctrinam Petrus Paulus Carauaggius Me- <pb o="178" file="0190" n="190"/> diolanenſis eruditiſſimus geometra in ſua geometria <lb/>applicationum, ampliauit ad altiores poteſtates, o-<lb/>ſtendendo applicationem aliarum poteſtatum ſerua-<lb/>re ſimilem ordinem partium ad quas fit applicatio; <lb/></s> <s xml:id="echoid-s3274" xml:space="preserve">adeo vt magnitudo ad quam fieri debet applicatio <lb/>ſit ſecanda in tot partes quota eſt magnitudo, quæ <lb/>debet applicari, in ordine graduum; </s> <s xml:id="echoid-s3275" xml:space="preserve">& </s> <s xml:id="echoid-s3276" xml:space="preserve">applicatio <lb/>ſit facienda ad illarum vnicam. </s> <s xml:id="echoid-s3277" xml:space="preserve">V.</s> <s xml:id="echoid-s3278" xml:space="preserve">g. </s> <s xml:id="echoid-s3279" xml:space="preserve">ſi ad partem <lb/>datæ A B, ſit applicandum parallelogrammum di-<lb/>ficiens, & </s> <s xml:id="echoid-s3280" xml:space="preserve">c. </s> <s xml:id="echoid-s3281" xml:space="preserve">hoc eſt <lb/> <anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/> ſi A B, ſit ſic ſe-<lb/>canda in C, vtre-<lb/>ctangulum A C B, <lb/>ſit omnium maxi-<lb/>mum illorum, quæ <lb/>poſſunt fieri ex <lb/>partibus A B; </s> <s xml:id="echoid-s3282" xml:space="preserve">pun-<lb/>ctum C, ſit illud <lb/>quod biſſecat A C. <lb/></s> <s xml:id="echoid-s3283" xml:space="preserve">Si veto ſit applicandum parallelepipedum, hoc eſt ſi <lb/>A B, taliter ſit ſecanda in C, vt ſolidum factum ſub <lb/>A C, in quadratum C B, ſit omnium maximum; </s> <s xml:id="echoid-s3284" xml:space="preserve"><lb/>A C, debet eſſe tertia pars A B. </s> <s xml:id="echoid-s3285" xml:space="preserve">Si vero ſit appli-<lb/>candum planoplanum, adeo vt factum ſub A C, in <lb/>cubum C B, ſit omnium maximum. </s> <s xml:id="echoid-s3286" xml:space="preserve">A C; </s> <s xml:id="echoid-s3287" xml:space="preserve">debet <lb/>eſſe quarta pars A B. </s> <s xml:id="echoid-s3288" xml:space="preserve">Et ſic in infinitum in altiori-<lb/>bus poteſtatibus. </s> <s xml:id="echoid-s3289" xml:space="preserve">Hæc ergo doctrina nobis eſt ne-<lb/>ceſſaria pro impoſterum dicendis. </s> <s xml:id="echoid-s3290" xml:space="preserve">Quam etiam le- <pb o="179" file="0191" n="191"/> ctor debet ſupponere, velin citat. </s> <s xml:id="echoid-s3291" xml:space="preserve">opere Carauaggij <lb/>inſpicere.</s> <s xml:id="echoid-s3292" xml:space="preserve"/> </p> <div xml:id="echoid-div160" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0190-01"/> </figure> </div> </div> <div xml:id="echoid-div162" type="section" level="1" n="103"> <head xml:id="echoid-head115" xml:space="preserve">PROPOSITIO L.</head> <p style="it"> <s xml:id="echoid-s3293" xml:space="preserve">Sì in qualibet infinitarum parabolarum ſumatur aliquod <lb/>punctum à quo ad diametrum recta linea ordinatim <lb/>applicetur, diameterque ità producatur vt pars extra <lb/>parabolam ſit ad partem diametri abſciſſam ab ordina-<lb/>tim applicata verſus verticem vt numerus parabolæ <lb/>vnitate minutus ad vnitatem. </s> <s xml:id="echoid-s3294" xml:space="preserve">Recta linea, quæ ab ex-<lb/>tremitate inuentæ lineæ ducitur ad illud punctum, quod <lb/>ſumptum fuer at, parabolam continget.</s> <s xml:id="echoid-s3295" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3296" xml:space="preserve">ESto quælibet ſemiparabola cuius vertex B, dia-<lb/>meter B D, & </s> <s xml:id="echoid-s3297" xml:space="preserve">in curua parabolica ſumatur <lb/>quodlibet punctum E, per quod ordinatim appli-<lb/>cetur E H, producaturque H B, in G, vt G B, ſit <lb/>ad B H, vt numerus parabolæ vnitate minutus ad <lb/>vnitatem: </s> <s xml:id="echoid-s3298" xml:space="preserve">v. </s> <s xml:id="echoid-s3299" xml:space="preserve">g ſi parabola ſit quadratica, fiat æqua-<lb/>lis B G, ipſi B H: </s> <s xml:id="echoid-s3300" xml:space="preserve">ſi ſit cubica ſit G B, dupla B H, <lb/>& </s> <s xml:id="echoid-s3301" xml:space="preserve">ſic in infinitum (ſupponatur in præſenti parabo-<lb/>lam eſſe cubicam) & </s> <s xml:id="echoid-s3302" xml:space="preserve">iungatur G E. </s> <s xml:id="echoid-s3303" xml:space="preserve">Dico hanc pa-<lb/>rabolam contingere. </s> <s xml:id="echoid-s3304" xml:space="preserve">Sinon, cadat intra; </s> <s xml:id="echoid-s3305" xml:space="preserve">& </s> <s xml:id="echoid-s3306" xml:space="preserve">intelli-<lb/>gatur ordinatim applicata A K D. </s> <s xml:id="echoid-s3307" xml:space="preserve">Quoniam A D, <lb/>maior eſt D K, ergo quælibet poteſtas A D, maior <lb/>erit qualibet poteſtate K D, eiuſdem gradus. </s> <s xml:id="echoid-s3308" xml:space="preserve">Ergo <lb/>quælibet poteſtas A D, eiuſdem gradus cum para-<lb/>bola ad poteſtatem E H, eiuſdem gradus, habebit <pb o="180" file="0192" n="192"/> maiorem rationem quam ſimilis poteſtas K D, ad <lb/>eandem poteſtatem E H. </s> <s xml:id="echoid-s3309" xml:space="preserve">V. </s> <s xml:id="echoid-s3310" xml:space="preserve">g. </s> <s xml:id="echoid-s3311" xml:space="preserve">maior erit ratio cubi <lb/>A D, ad cubum E H, quam cubi K D, ad eundem <lb/>cubum E H. </s> <s xml:id="echoid-s3312" xml:space="preserve">Sed vt <lb/> <anchor type="figure" xlink:label="fig-0192-01a" xlink:href="fig-0192-01"/> poteſtas A D, ad po-<lb/>teſtatem E H, ſic ex <lb/>natura parabolæ, D B, <lb/>ad B H; </s> <s xml:id="echoid-s3313" xml:space="preserve">& </s> <s xml:id="echoid-s3314" xml:space="preserve">vt D B, ad <lb/>B H, ſic factum ſub <lb/>D B, & </s> <s xml:id="echoid-s3315" xml:space="preserve">ſub poteſtate <lb/>B G, vno gradu inferio-<lb/>ri poteſtate parabolæ, <lb/>ad factum ſub eadem <lb/>poteſtate G B, & </s> <s xml:id="echoid-s3316" xml:space="preserve">ſub <lb/>B H. </s> <s xml:id="echoid-s3317" xml:space="preserve">Ergo maior erit <lb/>ratio facti ſub D B, & </s> <s xml:id="echoid-s3318" xml:space="preserve"><lb/>ſub tali poteſtate B G, <lb/>ad factum ſub H B, & </s> <s xml:id="echoid-s3319" xml:space="preserve"><lb/>ſub eadem poteſtate <lb/>B G, ratione poteſtatis <lb/>K D, eiuſdem gradus <lb/>cum parabola, ad ſimi-<lb/>lem poteſtatem E H. </s> <s xml:id="echoid-s3320" xml:space="preserve">V. </s> <s xml:id="echoid-s3321" xml:space="preserve">g. </s> <s xml:id="echoid-s3322" xml:space="preserve">maior crit ratio facti <lb/>ſub D B, & </s> <s xml:id="echoid-s3323" xml:space="preserve">ſub quadrato B G, ad factum ſub H B, <lb/>& </s> <s xml:id="echoid-s3324" xml:space="preserve">ſub quadrato B G, ratione cubi K D, ad cubum <lb/>E H. </s> <s xml:id="echoid-s3325" xml:space="preserve">Sed vt poteſtas K D, ad ſimilem poteſtatem <lb/>E H, ſic ſimilis poteſtas DG, ad ſimilem poteſtatem <lb/>G H. </s> <s xml:id="echoid-s3326" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3327" xml:space="preserve">factum ſub D B, & </s> <s xml:id="echoid-s3328" xml:space="preserve">ſub poteſtate <lb/>B G, vno gradu depreſſiori poteſtate parabolæ, ad <pb o="181" file="0193" n="193"/> ſimile factum ſub H B, & </s> <s xml:id="echoid-s3329" xml:space="preserve">ſub eadem poteſtate B G, <lb/>erit in maiori rationc quam poteſtas D G, eiuſdem <lb/>gradus cum parabola ad ſimilem poteſtatem G H. <lb/></s> <s xml:id="echoid-s3330" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3331" xml:space="preserve">permutando primum factum ad poteſtatem <lb/>D G, crit in maiori ratione quam fecundum factum <lb/>ad poteſtatem G H. </s> <s xml:id="echoid-s3332" xml:space="preserve">V. </s> <s xml:id="echoid-s3333" xml:space="preserve">g. </s> <s xml:id="echoid-s3334" xml:space="preserve">factum ſub D B, in qua-<lb/>dratum B G, habebit ad cubum D G, maiorem ra-<lb/>tionem, quam factum ſub H B, & </s> <s xml:id="echoid-s3335" xml:space="preserve">ſub quadrato B G, <lb/>ad cubum H G. </s> <s xml:id="echoid-s3336" xml:space="preserve">Quod implicat, quia factum ſub <lb/>D B, & </s> <s xml:id="echoid-s3337" xml:space="preserve">ſub poteſtate B G, eſt in minori ratione ad <lb/>poteſtatem D G, & </s> <s xml:id="echoid-s3338" xml:space="preserve">non in maiori. </s> <s xml:id="echoid-s3339" xml:space="preserve">Quia ex doctri-<lb/>na ſcholij anteced. </s> <s xml:id="echoid-s3340" xml:space="preserve">factum ſub H B, & </s> <s xml:id="echoid-s3341" xml:space="preserve">ſub poteſtate <lb/>B G, eſt omnium maximum homogeneorum ſub par-<lb/>tibus H G, non ſic factum ſub D B, & </s> <s xml:id="echoid-s3342" xml:space="preserve">ſub poteſta-<lb/>te B G, eſt maximum homogeneorum ſub partibus <lb/>D G. </s> <s xml:id="echoid-s3343" xml:space="preserve">V. </s> <s xml:id="echoid-s3344" xml:space="preserve">g. </s> <s xml:id="echoid-s3345" xml:space="preserve">factum ſub H B, & </s> <s xml:id="echoid-s3346" xml:space="preserve">ſub quadrato B G, <lb/>eſt maximum omnium parallelepipedorum applica-<lb/>bilium ad partem H G, non ſic eſt maximum factum <lb/>ſub D B, & </s> <s xml:id="echoid-s3347" xml:space="preserve">ſub quadrato B G, applicabilium ad <lb/>partem D G. </s> <s xml:id="echoid-s3348" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s3349" xml:space="preserve"/> </p> <div xml:id="echoid-div162" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0192-01"/> </figure> </div> </div> <div xml:id="echoid-div164" type="section" level="1" n="104"> <head xml:id="echoid-head116" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s3350" xml:space="preserve">Ex dictis facile eliciemus, quod ſi circa diametrum <lb/>B D, & </s> <s xml:id="echoid-s3351" xml:space="preserve">ſuper eadem baſi A D, intelligamus infini-<lb/>tas ſemiparabolas, & </s> <s xml:id="echoid-s3352" xml:space="preserve">accepto in diametro B D, pun-<lb/>cto H, ducatur H C E F G, parallela A D, ſecans <lb/>omnes curuas parabolicas, & </s> <s xml:id="echoid-s3353" xml:space="preserve">pariter intelligamus <lb/>infinitas tangentes K E, L F, M G, &</s> <s xml:id="echoid-s3354" xml:space="preserve">c. </s> <s xml:id="echoid-s3355" xml:space="preserve">eliciemus <lb/>inquam, triangula infinita C B H, E k H, F L H, <pb o="182" file="0194" n="194"/> G M H, &</s> <s xml:id="echoid-s3356" xml:space="preserve">c. </s> <s xml:id="echoid-s3357" xml:space="preserve">eſſe talis <lb/> <anchor type="figure" xlink:label="fig-0194-01a" xlink:href="fig-0194-01"/> naturæ vt latera H B, <lb/>H K, H L, H M, &</s> <s xml:id="echoid-s3358" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s3359" xml:space="preserve">ſint in continua pro-<lb/>portione Arithmetica; </s> <s xml:id="echoid-s3360" xml:space="preserve"><lb/>baſes vero E H, F H, <lb/>G H, &</s> <s xml:id="echoid-s3361" xml:space="preserve">c. </s> <s xml:id="echoid-s3362" xml:space="preserve">ſint maiores <lb/>omnium mediarum pro-<lb/>portio nalium reperibi-<lb/>lium inter A D, C H. </s> <s xml:id="echoid-s3363" xml:space="preserve"><lb/>Primum patet, quia H B, <lb/>B k, K L, L M, &</s> <s xml:id="echoid-s3364" xml:space="preserve">c. </s> <s xml:id="echoid-s3365" xml:space="preserve">ſunt <lb/>omnes æquales. </s> <s xml:id="echoid-s3366" xml:space="preserve">Secun-<lb/>dum patet; </s> <s xml:id="echoid-s3367" xml:space="preserve">quia cum ſit <lb/>vt quadratum A D, ad <lb/>quadratum EH, ſic D B, <lb/>ad B H, ſeù A D, ad <lb/>C H; </s> <s xml:id="echoid-s3368" xml:space="preserve">E H, erit media <lb/>proportionalisinter A D, <lb/>C H. </s> <s xml:id="echoid-s3369" xml:space="preserve">Item cum ſit vt <lb/>cubus A D, ad cubum <lb/>F H, ſic D B, ad B H, ſeù A D, ad C H; </s> <s xml:id="echoid-s3370" xml:space="preserve">erit F H, <lb/>maior duarum mediarum inter A D, C H. </s> <s xml:id="echoid-s3371" xml:space="preserve">Et ſic di-<lb/>catur de cæteris.</s> <s xml:id="echoid-s3372" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0194-01"/> </figure> </div> <p> <s xml:id="echoid-s3373" xml:space="preserve">Notetur etiam, quod à ſupradicta regula inue-<lb/>niendi tangentem non excluditur prima parabola, <lb/>nempe triangulum. </s> <s xml:id="echoid-s3374" xml:space="preserve">Si enim in triangulo A B D, ſit <lb/>datum punctum C, ad quod debeat duci tangens; <lb/></s> <s xml:id="echoid-s3375" xml:space="preserve">ducta C H, imperat regula generalis producendam <pb o="183" file="0195" n="195"/> eſſe H B, vt pars vltra B, ſit ad B H, vt numerus <lb/>parabolæ vnitate minutus, nempe vt nihil, ad vnita-<lb/>tem. </s> <s xml:id="echoid-s3376" xml:space="preserve">Ergo H B, non eſt producenda, ſed à puncto <lb/>B, ad C, ducenda eſt linea, quæ vtique quodam-<lb/>modo poteſt dici tangere triangulum, quia ipſum <lb/>non ſecat.</s> <s xml:id="echoid-s3377" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div166" type="section" level="1" n="105"> <head xml:id="echoid-head117" xml:space="preserve">PROPOSITIO LI.</head> <p style="it"> <s xml:id="echoid-s3378" xml:space="preserve">Maximum triangulum inſcriptum in quolibet triangulo, eſt <lb/>cutus baſis bifariam diuidit diametrum <lb/>circum ſcripti.</s> <s xml:id="echoid-s3379" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3380" xml:space="preserve">ESto triangulum A B C, cuius diameter B D, <lb/>quæ ſecetur in F, bifariam à baſe E O, trian-<lb/>guli E D O. </s> <s xml:id="echoid-s3381" xml:space="preserve">Dico triangulum E D O, eſſe maxi-<lb/>mum omnium inſcriptibilium in triangulo A B C. <lb/></s> <s xml:id="echoid-s3382" xml:space="preserve">Quoniam enim triangulum A B C, ad triangulum <lb/>E D O, habet rationem compoſitam ex ratione <lb/>A C, ad E O (nempe ex ratione D B, ad B F) & </s> <s xml:id="echoid-s3383" xml:space="preserve"><lb/>ex ratione B D, ad D F; </s> <s xml:id="echoid-s3384" xml:space="preserve">& </s> <s xml:id="echoid-s3385" xml:space="preserve">hæ duæ rationes com-<lb/>ponunt rationem quadrati B D, ad rectangulum <lb/>B F D. </s> <s xml:id="echoid-s3386" xml:space="preserve">Ergo triangulum A B C, erit ad E D O, <lb/>vt quadratum D B, ad rectangulum B F D. </s> <s xml:id="echoid-s3387" xml:space="preserve">Sed <lb/>rectangulum B F D, eſt maximum omnium rectan-<lb/>gulorum factibilium ex partibus B D, in puncto di-<lb/>uifæ. </s> <s xml:id="echoid-s3388" xml:space="preserve">Ergo etiam triangulum E D O, erit ma-<lb/>ximum omnium inſcriptibilium intra A B C. </s> <s xml:id="echoid-s3389" xml:space="preserve">Quod <lb/>&</s> <s xml:id="echoid-s3390" xml:space="preserve">c.</s> <s xml:id="echoid-s3391" xml:space="preserve"/> </p> <pb o="184" file="0196" n="196"/> <figure> <image file="0196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0196-01"/> </figure> </div> <div xml:id="echoid-div167" type="section" level="1" n="106"> <head xml:id="echoid-head118" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3392" xml:space="preserve">Notetur obiter centrum grauitatis amborum. <lb/></s> <s xml:id="echoid-s3393" xml:space="preserve">triangulorum A B C, E D O, eſſe idem punctum. </s> <s xml:id="echoid-s3394" xml:space="preserve"><lb/>Sit enim H, centrum grauitatis trianguli A B C. </s> <s xml:id="echoid-s3395" xml:space="preserve"><lb/>Ergo qualium B D, eſt 6, & </s> <s xml:id="echoid-s3396" xml:space="preserve">D F, 3, B H, erit <lb/>4, D H, 2, & </s> <s xml:id="echoid-s3397" xml:space="preserve">H F, 1. </s> <s xml:id="echoid-s3398" xml:space="preserve">Ergo H, erit etiam centrum <lb/>grauitatis trianguli E D O.</s> <s xml:id="echoid-s3399" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div168" type="section" level="1" n="107"> <head xml:id="echoid-head119" xml:space="preserve">PROPOSITIO LII.</head> <p style="it"> <s xml:id="echoid-s3400" xml:space="preserve">Maximus conus inſcriptibilis in quolibet cono, eſt cuius dia-<lb/>meter est tertia pars circumſcripti.</s> <s xml:id="echoid-s3401" xml:space="preserve"/> </p> <pb o="185" file="0197" n="197"/> <p> <s xml:id="echoid-s3402" xml:space="preserve">HÆc propoſit. </s> <s xml:id="echoid-s3403" xml:space="preserve">oſtenditur etiam ab Albio in <lb/>hemiſphæ. </s> <s xml:id="echoid-s3404" xml:space="preserve">diſſec. </s> <s xml:id="echoid-s3405" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3406" xml:space="preserve">44. </s> <s xml:id="echoid-s3407" xml:space="preserve">Sed ſuppona-<lb/>mus A B C, E D O, eſſe conos, & </s> <s xml:id="echoid-s3408" xml:space="preserve">D F, eſſe tertiam <lb/>partem D B. </s> <s xml:id="echoid-s3409" xml:space="preserve">Dico conum E D O, eſſe maximum <lb/>omnium, &</s> <s xml:id="echoid-s3410" xml:space="preserve">c. </s> <s xml:id="echoid-s3411" xml:space="preserve">Nam, cum conus A B C, ad conum <lb/>E D O, habeat rationem compoſitam ex ratione <lb/>quadrati A D, ad quadratum E F (nempe quadra-<lb/>ti D B, ad quadratum B F) & </s> <s xml:id="echoid-s3412" xml:space="preserve">ex ratione D B, ad <lb/>D F; </s> <s xml:id="echoid-s3413" xml:space="preserve">& </s> <s xml:id="echoid-s3414" xml:space="preserve">cum hæ duæ rationes componant rationem <lb/>cubi B D, ad factum ſub quadrato B F, & </s> <s xml:id="echoid-s3415" xml:space="preserve">ſub F D; <lb/></s> <s xml:id="echoid-s3416" xml:space="preserve">ergo A B C, erit ad E D O, vt cubus B D, ad fa-<lb/>ctum ſub quadrato F B, & </s> <s xml:id="echoid-s3417" xml:space="preserve">ſub F D. </s> <s xml:id="echoid-s3418" xml:space="preserve">Cum ergo hoc <lb/>factum ſit maximum omnium homogeneorum ipſi <lb/>factorum ex partibus B D, in puncto diuiſæ. </s> <s xml:id="echoid-s3419" xml:space="preserve">Ergo <lb/>etiam conus E D O, erit maximus omnium inſcri-<lb/>ptibilium &</s> <s xml:id="echoid-s3420" xml:space="preserve">c. </s> <s xml:id="echoid-s3421" xml:space="preserve">Quod &</s> <s xml:id="echoid-s3422" xml:space="preserve">c.</s> <s xml:id="echoid-s3423" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div169" type="section" level="1" n="108"> <head xml:id="echoid-head120" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3424" xml:space="preserve">Sed hìc etiam obiter notetur centrum grauitatis <lb/>amborum conorum eſſe idem punctum. </s> <s xml:id="echoid-s3425" xml:space="preserve">Sit enim <lb/>rurſum H, centrum grauitatis coni A B C. </s> <s xml:id="echoid-s3426" xml:space="preserve">Ergo <lb/>qualium B D, eſt 12, D F, 4, & </s> <s xml:id="echoid-s3427" xml:space="preserve">D H, 3, talium <lb/>H F, eſt 1. </s> <s xml:id="echoid-s3428" xml:space="preserve">Ergo H, erit centrum grauitatis etiam <lb/>coni E D O.</s> <s xml:id="echoid-s3429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3430" xml:space="preserve">Pariter notetur, conum A B C, eſſe ad conum <lb/>E D O, vt 27, ad 4. </s> <s xml:id="echoid-s3431" xml:space="preserve">Nam ſic eſt cubus B D, ad <lb/>factum ſub quadrato B F, & </s> <s xml:id="echoid-s3432" xml:space="preserve">ſub F D.</s> <s xml:id="echoid-s3433" xml:space="preserve"/> </p> <pb o="186" file="0198" n="198"/> </div> <div xml:id="echoid-div170" type="section" level="1" n="109"> <head xml:id="echoid-head121" xml:space="preserve">PROPOSITIO LIII.</head> <p style="it"> <s xml:id="echoid-s3434" xml:space="preserve">Datam A D, taliter producere in B, vt B D, ſit ad <lb/>exceſſum D A, ſupra dimidiam A B, in <lb/>data proportione.</s> <s xml:id="echoid-s3435" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3436" xml:space="preserve">DAta ratio ſit, quam habet AD, ad H, & </s> <s xml:id="echoid-s3437" xml:space="preserve">ſic ſece-<lb/>tur A D, in E, vt ſit A E, ad E D, vt H, ad dimi-<lb/>diam A D, & </s> <s xml:id="echoid-s3438" xml:space="preserve">ipſi D E, fiat ęqualis D B, Ergo ſi A B, <lb/> <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> diuidatur bifariam in C, punctum C, cadet inter <lb/>A, D. </s> <s xml:id="echoid-s3439" xml:space="preserve">Sit ergo A B, diuiſa bifariam in C. </s> <s xml:id="echoid-s3440" xml:space="preserve">Quo-<lb/>niam A E, eſt æqualis A B, minus E B, ergo etiam <lb/>dimidia A E, erit æqualis dimidiæ A B, minus dimi-<lb/>dia E B. </s> <s xml:id="echoid-s3441" xml:space="preserve">Sed C B, eſt dimidia A B, & </s> <s xml:id="echoid-s3442" xml:space="preserve">B D, eſt <lb/>dimidia E B; </s> <s xml:id="echoid-s3443" xml:space="preserve">ergo dimidia A E, erit æqualis C B, <lb/>minus D B; </s> <s xml:id="echoid-s3444" xml:space="preserve">nempe C D. </s> <s xml:id="echoid-s3445" xml:space="preserve">Tunc, quoniam factum <lb/>fuit vt H, ad dimidiam A D, ſic A E, ad E D; <lb/></s> <s xml:id="echoid-s3446" xml:space="preserve">ergo & </s> <s xml:id="echoid-s3447" xml:space="preserve">ad conſequentium dupla. </s> <s xml:id="echoid-s3448" xml:space="preserve">Ergo vt H, ad <lb/>A D, ſic A E, ad E B. </s> <s xml:id="echoid-s3449" xml:space="preserve">Et conuertendo, vt A D, <lb/>ad H, ſic B E, ad E A. </s> <s xml:id="echoid-s3450" xml:space="preserve">Sed vt B E, ad E A, ita <lb/>B D, dimidia B E, ad dimidiam A E, nempe ad <lb/>C D, ei æqualem. </s> <s xml:id="echoid-s3451" xml:space="preserve">Ergo vt A D, ad H, ſic B D, <pb o="187" file="0199" n="199"/> ad D C, exceſſum D A, ſupra A C, dimidiam A B. <lb/></s> <s xml:id="echoid-s3452" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s3453" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0198-01"/> </figure> </div> </div> <div xml:id="echoid-div172" type="section" level="1" n="110"> <head xml:id="echoid-head122" xml:space="preserve">PROPOSITIO LIV.</head> <p style="it"> <s xml:id="echoid-s3454" xml:space="preserve">Sidiameter cuiuslibet infinitarum parabolarum ſic produca <lb/>tur vt pars exterior producta, ſit ad exceſſum diametrì <lb/>ſupra dimidiam compoſitæ ex diametro, & </s> <s xml:id="echoid-s3455" xml:space="preserve">ex producta <lb/>vt numerus parabolæ vnitate minor, ad vnitatem. <lb/></s> <s xml:id="echoid-s3456" xml:space="preserve">Triangulum inſcripium in parabold, cums baſis bißecet <lb/>illam compoſitam, erit omnium maximum in ipſa inſcri-<lb/>ptibilium.</s> <s xml:id="echoid-s3457" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3458" xml:space="preserve">DB, diameter parabolæ cuiuſcunque A B C, ſic <lb/>producatur in E, vt E B, ſit ad B F, exceſſum <lb/>B D, ſupra D F, medietatem D E, vt numerus pa-<lb/>rabolæ vnitate minutus, ad vnitatem, & </s> <s xml:id="echoid-s3459" xml:space="preserve">fiat triangu-<lb/>lum G D H. </s> <s xml:id="echoid-s3460" xml:space="preserve">Dico hoc eſſe maximum omnium in-<lb/>ſcriptibilium in A B C. </s> <s xml:id="echoid-s3461" xml:space="preserve">Ducantur E G K, E H L. <lb/></s> <s xml:id="echoid-s3462" xml:space="preserve">Ergo ex propoſit. </s> <s xml:id="echoid-s3463" xml:space="preserve">50. </s> <s xml:id="echoid-s3464" xml:space="preserve">erunt tangentes parabolam, & </s> <s xml:id="echoid-s3465" xml:space="preserve"><lb/>triangulum K E L, erit parabolæ circumſcriptum. </s> <s xml:id="echoid-s3466" xml:space="preserve"><lb/>Si ergo triangulum G D H, non eſt maximum para-<lb/>bolæ inſcrip um, ſit hoc triangulum, cuius baſis <lb/>O P, infra, velſupra G H, quæ producatur vſque <lb/>ad triangulum in M, & </s> <s xml:id="echoid-s3467" xml:space="preserve">N; </s> <s xml:id="echoid-s3468" xml:space="preserve">& </s> <s xml:id="echoid-s3469" xml:space="preserve">pariter intelligatur <lb/>triangulum M D N, cuius baſis M N. </s> <s xml:id="echoid-s3470" xml:space="preserve">Cum D E, <lb/>ſecta ſit bifariam in F; </s> <s xml:id="echoid-s3471" xml:space="preserve">ergo triangulum G D H, erit <lb/>maximum inſcriptibilium intra triangulum K E L. </s> <s xml:id="echoid-s3472" xml:space="preserve"><lb/>Ergo erit maius triangulo cuius baſis M N. </s> <s xml:id="echoid-s3473" xml:space="preserve">Ergo <pb o="188" file="0200" n="200"/> <anchor type="figure" xlink:label="fig-0200-01a" xlink:href="fig-0200-01"/> multo maius triangulo O D P, cuius baſis O P. <lb/></s> <s xml:id="echoid-s3474" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s3475" xml:space="preserve"/> </p> <div xml:id="echoid-div172" 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/PVNDER1Y/figures/0200-01"/> </figure> </div> </div> <div xml:id="echoid-div174" type="section" level="1" n="111"> <head xml:id="echoid-head123" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s3476" xml:space="preserve">Ab hac regula generali reperiendi triangulum <lb/>maximum inſcriptibilium in parabola non excludi-<lb/>tur prima parabola, nempe triangulum. </s> <s xml:id="echoid-s3477" xml:space="preserve">Cum enim <lb/>iubeat regula ſic eſſe producendam diametrum D B, <lb/>vt pars extra ſit ad exceſſum B D, ſupra medietatem <lb/>compoſitæ ex B D, & </s> <s xml:id="echoid-s3478" xml:space="preserve">ex producta, vt numerus pa-<lb/>rabolæ vnitate minutus ad vnitatem; </s> <s xml:id="echoid-s3479" xml:space="preserve">patet in prima <lb/>parabola, cuius numerus eſt vnitas, numerum vni- <pb o="189" file="0201" n="201"/> tate minutum eſſe nihil; </s> <s xml:id="echoid-s3480" xml:space="preserve">vnde D B, in triangulo non <lb/>eſt producenda; </s> <s xml:id="echoid-s3481" xml:space="preserve">ſed ſupponendo A B C, eſſe trian-<lb/>gulum, B D, eſt biſlecanda, & </s> <s xml:id="echoid-s3482" xml:space="preserve">triangulum G D H, <lb/>eſt maximum. </s> <s xml:id="echoid-s3483" xml:space="preserve">Quod ſic eſſe, probatum eſt ſupra <lb/>propoſit. </s> <s xml:id="echoid-s3484" xml:space="preserve">51.</s> <s xml:id="echoid-s3485" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div175" type="section" level="1" n="112"> <head xml:id="echoid-head124" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s3486" xml:space="preserve">Triangulum ergo G D H, maximum inſcriptibi-<lb/>lium intra parabolam A B C, ſic diuidit D B, in F, <lb/>vt B F, ſit ad F D, vt vnitas adnumerum parabolæ. <lb/></s> <s xml:id="echoid-s3487" xml:space="preserve">V. </s> <s xml:id="echoid-s3488" xml:space="preserve">g. </s> <s xml:id="echoid-s3489" xml:space="preserve">in triangulo vt 1, ad 1. </s> <s xml:id="echoid-s3490" xml:space="preserve">In parabola quadrati-<lb/>ca vt 1, ad 2. </s> <s xml:id="echoid-s3491" xml:space="preserve">In cubica vt 1, ad 3. </s> <s xml:id="echoid-s3492" xml:space="preserve">Et ſic in infini-<lb/>tum. </s> <s xml:id="echoid-s3493" xml:space="preserve">In triangulo enim, patet ex dictis. </s> <s xml:id="echoid-s3494" xml:space="preserve">In alijs ſic <lb/>patebit. </s> <s xml:id="echoid-s3495" xml:space="preserve">Quum etenim ſit E B, ad B F, vt numerus <lb/>parabolæ vnitate minutus, ad vnitatem; </s> <s xml:id="echoid-s3496" xml:space="preserve">erit com-<lb/>ponendo, E F, ad F B, vt numerus parabolæ ad <lb/>vnitatem. </s> <s xml:id="echoid-s3497" xml:space="preserve">Sed F D, eſt æqualis E F. </s> <s xml:id="echoid-s3498" xml:space="preserve">Quare patet <lb/>propoſitum.</s> <s xml:id="echoid-s3499" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div176" type="section" level="1" n="113"> <head xml:id="echoid-head125" xml:space="preserve">PROPOSITIOLV.</head> <p style="it"> <s xml:id="echoid-s3500" xml:space="preserve">Maximum triangulum inſcriptibile in figura conſtante ex <lb/>duabus quibuſcunque ſemiparabolis ſic diſpoſitis, vt ſe-<lb/>mibaſis euadat diameter, eſt æquale maximo inſcripto in <lb/>parabola.</s> <s xml:id="echoid-s3501" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3502" xml:space="preserve">MEnte intelligamus ſemiparabolam A B D, du-<lb/>plicari ad partes A D. </s> <s xml:id="echoid-s3503" xml:space="preserve">Dico maximum trian- <pb o="190" file="0202" n="202"/> gulum inſcriptibile in tali figura, eſſe æquale trian-<lb/>gulo G D H. </s> <s xml:id="echoid-s3504" xml:space="preserve">Hoc oſtendetur in ſemiparabola, quod <lb/>enim probabitur de dimidia, patebit etiam detota. <lb/></s> <s xml:id="echoid-s3505" xml:space="preserve">Sit ergo G D H, maximum triangulum inſcriptibi-<lb/>le in parabola, & </s> <s xml:id="echoid-s3506" xml:space="preserve">ducatur G Q, B D, diametro paral-<lb/>lela: </s> <s xml:id="echoid-s3507" xml:space="preserve">patet triangulum G Q D, eſſe æquale triangu. </s> <s xml:id="echoid-s3508" xml:space="preserve"><lb/>lo G D F; </s> <s xml:id="echoid-s3509" xml:space="preserve">& </s> <s xml:id="echoid-s3510" xml:space="preserve">eius duplum, ipſi G D H. </s> <s xml:id="echoid-s3511" xml:space="preserve">Dico trian-<lb/>gulum G Q D, eſſe maximum &</s> <s xml:id="echoid-s3512" xml:space="preserve">c. </s> <s xml:id="echoid-s3513" xml:space="preserve">Etenim, cum <lb/>E D, ſit dupla D F, ſeù G Q, etiam D k, erit du-<lb/>pla D Q Ergo triangulum D Q G, erit maximum <lb/>inſcriptibilium intra triangulum k E D. </s> <s xml:id="echoid-s3514" xml:space="preserve">Si ergo <lb/>G Q D, non eſt maximum inſcriptibilium etiam in <lb/>ſemiparabola, ſit aliud, cuius baſis producta vſ-<lb/>que ad E k, ſecetipſam, & </s> <s xml:id="echoid-s3515" xml:space="preserve">curuam parabolicam in-<lb/>fra, vel ſupra G Q, vt ſupra dictum eſt de M N. </s> <s xml:id="echoid-s3516" xml:space="preserve"><lb/>Ergo triangulum cuius baſis ſecans k E, erit minus <lb/>triangulo G Q D. </s> <s xml:id="echoid-s3517" xml:space="preserve">Ergo triangulum cuius baſis per-<lb/>tingens tantum ad curuam parabolicam, erit multo <lb/>minus triangulo G Q D. </s> <s xml:id="echoid-s3518" xml:space="preserve">Quare patet propoſi-<lb/>tum.</s> <s xml:id="echoid-s3519" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div177" type="section" level="1" n="114"> <head xml:id="echoid-head126" xml:space="preserve">PROPOSITIOLVI.</head> <p style="it"> <s xml:id="echoid-s3520" xml:space="preserve">Si A B, ſit taliter ſecta in C, & </s> <s xml:id="echoid-s3521" xml:space="preserve">D, vt A C, ſit ter-<lb/>tia pars A B. </s> <s xml:id="echoid-s3522" xml:space="preserve">Erit C D, duo tertia A D, mi-<lb/>nus tertia parte D B.</s> <s xml:id="echoid-s3523" xml:space="preserve"/> </p> <pb o="191" file="0203" n="203"/> <figure> <image file="0203-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0203-01"/> </figure> <p> <s xml:id="echoid-s3524" xml:space="preserve">CVm enim A C, ſit tertia pars A B; </s> <s xml:id="echoid-s3525" xml:space="preserve">ergo C B, <lb/>erit duo tertia A B; </s> <s xml:id="echoid-s3526" xml:space="preserve">nempe duo tertia A D, <lb/>cum duobus tertijs D B. </s> <s xml:id="echoid-s3527" xml:space="preserve">Ergo C D, erit duo ter-<lb/>tia A D, minus tertia parte D B. </s> <s xml:id="echoid-s3528" xml:space="preserve">Quod &</s> <s xml:id="echoid-s3529" xml:space="preserve">c.</s> <s xml:id="echoid-s3530" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div178" type="section" level="1" n="115"> <head xml:id="echoid-head127" xml:space="preserve">PROPOSITIO LVII.</head> <p style="it"> <s xml:id="echoid-s3531" xml:space="preserve">Datam A D, taliter producere in B, vt B D, ſit ad ex-<lb/>ceſſum D A, ſupra tertiam partem A B, in <lb/>data proportione.</s> <s xml:id="echoid-s3532" xml:space="preserve"/> </p> <figure> <image file="0203-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0203-02"/> </figure> <p> <s xml:id="echoid-s3533" xml:space="preserve">DAta proportio ſit, quam habet A D, ad H; <lb/></s> <s xml:id="echoid-s3534" xml:space="preserve">& </s> <s xml:id="echoid-s3535" xml:space="preserve">fiat vt tripla H, cum A D, ad A D, ità <lb/>dupla A D, ad D B. </s> <s xml:id="echoid-s3536" xml:space="preserve">Patet B D, minorem eſſe <lb/>dupla A D. </s> <s xml:id="echoid-s3537" xml:space="preserve">Quare ſi fiat A C, tertia pars A B, <lb/>punctum C, cadet inter A, D. </s> <s xml:id="echoid-s3538" xml:space="preserve">Sit ergo A C, <lb/>tertia pars A B.</s> <s xml:id="echoid-s3539" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3540" xml:space="preserve">Quoniam vt tripla H, cum A D, ad A D, ſic <lb/>dupla A D, ad D B; </s> <s xml:id="echoid-s3541" xml:space="preserve">ergo diuidendo vt tripla H, <pb o="192" file="0204" n="204"/> ad A D, ità dupla A D, minus D B, ad D B. </s> <s xml:id="echoid-s3542" xml:space="preserve">Et <lb/>antecedentium ſubtripla. </s> <s xml:id="echoid-s3543" xml:space="preserve">Ergo vt H, ad A D, ita <lb/>duo tertia A D, minus tertia parte D B, ad D B. <lb/></s> <s xml:id="echoid-s3544" xml:space="preserve">Sed ex propoſit. </s> <s xml:id="echoid-s3545" xml:space="preserve">anteced. </s> <s xml:id="echoid-s3546" xml:space="preserve">C D, eſt duo tertia A D, <lb/>minus tertia parte D B. </s> <s xml:id="echoid-s3547" xml:space="preserve">Ergo vt H, ad A D, ſic <lb/>C D, ad D B. </s> <s xml:id="echoid-s3548" xml:space="preserve">Et conuertendo, vt A D, ad H, ſic <lb/>B D, ad D C, exceſſum D A, ſupra A C, tertiam <lb/>partem A B. </s> <s xml:id="echoid-s3549" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s3550" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div179" type="section" level="1" n="116"> <head xml:id="echoid-head128" xml:space="preserve">PROPOSITIO LVIII.</head> <p style="it"> <s xml:id="echoid-s3551" xml:space="preserve">Si diameter cuiuslibet infinitorum conoideorum ſic produ-<lb/>catur, vt pars exterior producta ſit ad exce ßum diame-<lb/>tri ſupra tertiam partem compoſitæ ex diametro, & </s> <s xml:id="echoid-s3552" xml:space="preserve">ex <lb/>producta, vt numerus parabolæ vnitate minutus ad <lb/>vnitatem. </s> <s xml:id="echoid-s3553" xml:space="preserve">Conus inſcriptus in conoide, cuius diameter <lb/>ſit tertia pars illius compoſitæ, erit maximus omnium in-<lb/>ſcriptibilium in conoide.</s> <s xml:id="echoid-s3554" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3555" xml:space="preserve">DB, diameter conoidis cuiuſcunque A B C, ſic <lb/>producatur in E, vt E B, ſit ad B F, exceſ-<lb/>ſum B D, ſupra D F, tertiam partem D E, vt nu-<lb/>merus parabolæ vnitate minutus, ad vnitatem; </s> <s xml:id="echoid-s3556" xml:space="preserve">& </s> <s xml:id="echoid-s3557" xml:space="preserve">in-<lb/>telligamus conum G D H, cuius diameter F D. </s> <s xml:id="echoid-s3558" xml:space="preserve">Di-<lb/>co hunc eſſe omnium maximum inſcriptibilium in <lb/>conoide. </s> <s xml:id="echoid-s3559" xml:space="preserve">Ductis enim tangentibus E G K, E H L, <lb/>intelligamus conum k E L, circumſcriptus conoi-<lb/>di. </s> <s xml:id="echoid-s3560" xml:space="preserve">Et ſi conns G D H, non eſt omnium maximus, <lb/>ſit alius cuius baſis O P, infrà, vel ſupra G H, quæ <pb o="193" file="0205" n="205"/> <anchor type="figure" xlink:label="fig-0205-01a" xlink:href="fig-0205-01"/> producatur in M N. </s> <s xml:id="echoid-s3561" xml:space="preserve">Ergo ex propoſit. </s> <s xml:id="echoid-s3562" xml:space="preserve">52. </s> <s xml:id="echoid-s3563" xml:space="preserve">conus <lb/>M D N, cuius baſis M N, erit minor cono G D H. <lb/></s> <s xml:id="echoid-s3564" xml:space="preserve">Ergo conus cuius baſis O P, erit multo minor cono <lb/>G D H. </s> <s xml:id="echoid-s3565" xml:space="preserve">Patet ergo propoſitum.</s> <s xml:id="echoid-s3566" xml:space="preserve"/> </p> <div xml:id="echoid-div179" 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/PVNDER1Y/figures/0205-01"/> </figure> </div> </div> <div xml:id="echoid-div181" type="section" level="1" n="117"> <head xml:id="echoid-head129" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3567" xml:space="preserve">Sicuti ergo ſupra diximus regulam generalem aſſi-<lb/>gnatam in parabolis, habere locum etiam in prima <lb/>parabola, ſic nunc animaduertimus præſentem ge-<lb/>neralem regulam habere locum etiam in primo co-<lb/>noide, nempe in cono. </s> <s xml:id="echoid-s3568" xml:space="preserve">Hoc autem facile quilibet <lb/>cognoſcet.</s> <s xml:id="echoid-s3569" xml:space="preserve"/> </p> <pb o="194" file="0206" n="206"/> <p> <s xml:id="echoid-s3570" xml:space="preserve">Sicuti facile agnoſcet D B, taliter ſecari in F, vt <lb/>B F, ſit ad F D, vt vnitas ad dimidium numeri co-<lb/>noidis. </s> <s xml:id="echoid-s3571" xml:space="preserve">Nempe in cono vt 1, ad dimidium, ſeù vt <lb/>2. </s> <s xml:id="echoid-s3572" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s3573" xml:space="preserve">In conoide quadratico, vt 1, ad 1. </s> <s xml:id="echoid-s3574" xml:space="preserve">In cu-<lb/>bico vt 1, ad 1, cum dimidio, & </s> <s xml:id="echoid-s3575" xml:space="preserve">ſic in infinitum. <lb/></s> <s xml:id="echoid-s3576" xml:space="preserve">In cono res ſupra patuit in propoſit. </s> <s xml:id="echoid-s3577" xml:space="preserve">52. </s> <s xml:id="echoid-s3578" xml:space="preserve">In alijs co-<lb/>noidibus ſic patebit. </s> <s xml:id="echoid-s3579" xml:space="preserve">Nam cum E B, ſit ad B F, <lb/>vt numerus conoidis vnitate minutus ad vnitatem, <lb/>erit componendo, E F, ad F B, vt numerus conoidis <lb/>ad vnitatem. </s> <s xml:id="echoid-s3580" xml:space="preserve">Cumautem D F, ſit dimidium F E, pa-<lb/>tet conuertendo, propoſitum.</s> <s xml:id="echoid-s3581" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div182" type="section" level="1" n="118"> <head xml:id="echoid-head130" xml:space="preserve">PROPOSITIO LIX.</head> <p style="it"> <s xml:id="echoid-s3582" xml:space="preserve">Si A B, taliter ſecetur in C, & </s> <s xml:id="echoid-s3583" xml:space="preserve">D, vt A C, ſit duo <lb/>tertia A B. </s> <s xml:id="echoid-s3584" xml:space="preserve">C D, erit tertia pars A D, minus <lb/>duobus tertijs D B.</s> <s xml:id="echoid-s3585" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3586" xml:space="preserve">CVm enim A C, ſit duo tertia A B, ergo C D, <lb/>erit tertia pars A B; </s> <s xml:id="echoid-s3587" xml:space="preserve">nempe tertia pars A D, <lb/>plus tertia parte D B. </s> <s xml:id="echoid-s3588" xml:space="preserve">Quare C D, ſola erit tertia <lb/>pars A D, minus duobustertijs D B. </s> <s xml:id="echoid-s3589" xml:space="preserve">Quod &</s> <s xml:id="echoid-s3590" xml:space="preserve">c.</s> <s xml:id="echoid-s3591" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div183" type="section" level="1" n="119"> <head xml:id="echoid-head131" xml:space="preserve">PROPOSITIO LX.</head> <p style="it"> <s xml:id="echoid-s3592" xml:space="preserve">Datam A D, taliter producere in B, vt B D, ſit ad <lb/>exceßum D A, ſupra duo tertia A B, in <lb/>data proportione.</s> <s xml:id="echoid-s3593" xml:space="preserve"/> </p> <pb o="195" file="0207" n="207"/> <figure> <image file="0207-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0207-01"/> </figure> <p> <s xml:id="echoid-s3594" xml:space="preserve">ITidem ratio data ſit quam habet A D, ad H; </s> <s xml:id="echoid-s3595" xml:space="preserve">& </s> <s xml:id="echoid-s3596" xml:space="preserve"><lb/>fiat vt tripla H, cum dupla A D, ad A D, ita <lb/>A D, ad D B. </s> <s xml:id="echoid-s3597" xml:space="preserve">Patet B D, minorem eſſe ſubdupla <lb/>A D; </s> <s xml:id="echoid-s3598" xml:space="preserve">& </s> <s xml:id="echoid-s3599" xml:space="preserve">conſequenter tertia parte totius A B. </s> <s xml:id="echoid-s3600" xml:space="preserve">Qua-<lb/>re A D, eſt maior duobus tertijs A B, quȩ ſit A C. </s> <s xml:id="echoid-s3601" xml:space="preserve">Di-<lb/>co A D, eſſe ſic productam in B, vt B D, ſit ad D C, <lb/>exceſſum A D, ſupra A C, dno tertia A B, vt A D, <lb/>ad H. </s> <s xml:id="echoid-s3602" xml:space="preserve">Quoniam enim factum eſt vt tripla H, cum <lb/>dupla A D, ad A D, ita A D, ad D B; </s> <s xml:id="echoid-s3603" xml:space="preserve">ergo & </s> <s xml:id="echoid-s3604" xml:space="preserve">dua-<lb/>bus vicibus diuidendo, erit tripla H, ad AD, vt A D, <lb/>minus dupla D B, ad D B. </s> <s xml:id="echoid-s3605" xml:space="preserve">Et antecedentium ſub-<lb/>tripla, nempe vt H, ad A D, ita tertia pars A D, mi-<lb/>nus duobus tertijs B D, ad B D. </s> <s xml:id="echoid-s3606" xml:space="preserve">Et conuertendo, <lb/>vt A D, ad H, ſic B D, ad tertiam partem A D, mi-<lb/>nus duobus tertijs D B; </s> <s xml:id="echoid-s3607" xml:space="preserve">nempe ex prop. </s> <s xml:id="echoid-s3608" xml:space="preserve">ant. </s> <s xml:id="echoid-s3609" xml:space="preserve">ad D C. <lb/></s> <s xml:id="echoid-s3610" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s3611" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div184" type="section" level="1" n="120"> <head xml:id="echoid-head132" xml:space="preserve">PROPOSITIO LXI.</head> <p style="it"> <s xml:id="echoid-s3612" xml:space="preserve">Si diameter cuiuſcunque parabolæ ſic producatur vt pars <lb/>exterior producta, ſit ad exceſſum diametri ſupra duo ter- <pb o="196" file="0208" n="208"/> tia compoſitæ ex diametro, & </s> <s xml:id="echoid-s3613" xml:space="preserve">ex producta, vt numerus <lb/>parabolæ vnitate minutus, ad vnitatem. </s> <s xml:id="echoid-s3614" xml:space="preserve">Conus cuius <lb/>radius baſis ſit æqualis duobus tertijs prædictæ compoſitæ, <lb/>erit maximus omnium inſcriptibilium in ſemifuſo ex ſemi-<lb/>parabola.</s> <s xml:id="echoid-s3615" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3616" xml:space="preserve">DIameter D B, in ſchem. </s> <s xml:id="echoid-s3617" xml:space="preserve">antec. </s> <s xml:id="echoid-s3618" xml:space="preserve">parabolæ cuiuſ-<lb/>cunque ſic producatur in E, vt E B, ſit ad B F, <lb/>exceſſum B D, ſupra D F, duo tertia D E, vt nu-<lb/>merus parabolæ vnitate minutus ad vnitatem, & </s> <s xml:id="echoid-s3619" xml:space="preserve">fiat <lb/>triangulum G Q D, vt G Q, ſit æqualis F D; </s> <s xml:id="echoid-s3620" xml:space="preserve">in-<lb/>telligamuſque ſemiparabolam A B D, cum triangu-<lb/>lo Q G D, rotari circa A D. </s> <s xml:id="echoid-s3621" xml:space="preserve">Dico conum ex Q G D, <lb/>eſſe maximum omnium inſcriptibilium in ſemifuſo. <lb/></s> <s xml:id="echoid-s3622" xml:space="preserve">Intelligatur tangens E G K, & </s> <s xml:id="echoid-s3623" xml:space="preserve">conus ex triangulo <lb/>k E D, circa k D. </s> <s xml:id="echoid-s3624" xml:space="preserve">Quoniam E F, eſt tertia pars <lb/>E D, nempe G E, eſt tertia pars E K, ergo & </s> <s xml:id="echoid-s3625" xml:space="preserve">Q D, <lb/>erit tertia pars D k. </s> <s xml:id="echoid-s3626" xml:space="preserve">Ergo conus ex triangulo Q G D, <lb/>erit ex propoſit 52. </s> <s xml:id="echoid-s3627" xml:space="preserve">@ aximus omnium inſcriptibi-<lb/>lium in cono ex triang lo k E D, reuolutis ambobus <lb/>circa k D. </s> <s xml:id="echoid-s3628" xml:space="preserve">Siau emconus non ſit maximus, ſit alius, <lb/>ſi eſt poſſibile; </s> <s xml:id="echoid-s3629" xml:space="preserve">& </s> <s xml:id="echoid-s3630" xml:space="preserve">deducetur ad abſurdum vt f@ctum <lb/>eſt prius. </s> <s xml:id="echoid-s3631" xml:space="preserve">Quare ex dictis, patebit propoſitum.</s> <s xml:id="echoid-s3632" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div185" type="section" level="1" n="121"> <head xml:id="echoid-head133" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3633" xml:space="preserve">Nec etiam in præſenti excluditur à regula gene-<lb/>rali primus ſemifuſus, nempe conus, vt conſideranti <lb/>patebit.</s> <s xml:id="echoid-s3634" xml:space="preserve"/> </p> <pb o="197" file="0209" n="209"/> <p> <s xml:id="echoid-s3635" xml:space="preserve">Sed notetur, in ſemifuſis, B D, ſecari in F, ali-<lb/>qua continuata ſerie, nempe ſic vt B F, ſit ad F D, <lb/>vt vnitas ad duplum numerum fuſi. </s> <s xml:id="echoid-s3636" xml:space="preserve">Nempe in pri-<lb/>mo vt 1, ad 2. </s> <s xml:id="echoid-s3637" xml:space="preserve">In ſecundo vt 1, ad 4. </s> <s xml:id="echoid-s3638" xml:space="preserve">In tertio vt 1, <lb/>ad 6. </s> <s xml:id="echoid-s3639" xml:space="preserve">& </s> <s xml:id="echoid-s3640" xml:space="preserve">ſic in infinitum. </s> <s xml:id="echoid-s3641" xml:space="preserve">Quod enim in primo ſe-<lb/>mifuſo, nempe in cono ſit vt 1, ad 2, patet ex dictis. <lb/></s> <s xml:id="echoid-s3642" xml:space="preserve">In alijs ſic patebit. </s> <s xml:id="echoid-s3643" xml:space="preserve">Nam cum ſit E F, ad F B, com-<lb/>ponendo, vt numerus parabolæ ad vnitatem; </s> <s xml:id="echoid-s3644" xml:space="preserve">erit <lb/>conuertendo F B, ad F E, vt vnitas ad numerum <lb/>parabolæ. </s> <s xml:id="echoid-s3645" xml:space="preserve">Et ad D F, duplam F E, vt vnitas ad <lb/>duplum numerum parabolæ, ſeù ſemifuſi.</s> <s xml:id="echoid-s3646" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div186" type="section" level="1" n="122"> <head xml:id="echoid-head134" xml:space="preserve">PROPOSITIO LXII.</head> <p style="it"> <s xml:id="echoid-s3647" xml:space="preserve">Minimum trianguium circumſcriptum cuilibet infinitarum <lb/>p@rabolarum, eſt illud cuius latera tangunt baſim maximi <lb/>triangu<gap/> in parabola in ſcripti.</s> <s xml:id="echoid-s3648" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3649" xml:space="preserve">ESto ſemiparabola quælibet A B C, cuius dia-<lb/>meter B C, & </s> <s xml:id="echoid-s3650" xml:space="preserve">in ipſa ſit in ſcriptum maximum <lb/>trianguium E C F (quod enim dicetur de dimidia <lb/>intelligetur etiam de tota) ſitque ei circumſcriptum <lb/>triangulum G E I C. </s> <s xml:id="echoid-s3651" xml:space="preserve">Dico hoc eſſe minimum om-<lb/>nium circumſcriptibilium ſemiparabolæ. </s> <s xml:id="echoid-s3652" xml:space="preserve">Si non, <lb/>ſit minimum H O k C, & </s> <s xml:id="echoid-s3653" xml:space="preserve">per punctum E, duca-<lb/>tur L E M, parallela K H. </s> <s xml:id="echoid-s3654" xml:space="preserve">Patet manifeſtè trian-<lb/>gulum L M C, minus eſſe triangulo k O H C, cum <lb/>L M, ſecet, k H, vero tangat parabolam. </s> <s xml:id="echoid-s3655" xml:space="preserve">Quoniam <lb/>autem ex ſuperioribus, triangulum E F C, eſt ma- <pb o="198" file="0210" n="210"/> <anchor type="figure" xlink:label="fig-0210-01a" xlink:href="fig-0210-01"/> ximum inſcriptibilium intra triangulum I G C, quia <lb/>ſupponitur ſecare G C, bifariam in F, ergo non erit <lb/>maximum inſcriptibilium intra triangulum L M C, <lb/>quia M C, non fecabitur bifariam in F. </s> <s xml:id="echoid-s3656" xml:space="preserve">Ergo trian-<lb/>gulum E F C, habebit ad triangulum I G C, ma-<lb/>iorem rationem, quam adtriangulum L M C. </s> <s xml:id="echoid-s3657" xml:space="preserve">Sed <lb/>idemtriangulum E F C, ad triangulum L M C, ha-<lb/>bet maiorem rationem quam ad triangulum k H C. <lb/></s> <s xml:id="echoid-s3658" xml:space="preserve">Ergo E F C, erit ad I G C, in multo maiori rationc <lb/>quam ad k H C. </s> <s xml:id="echoid-s3659" xml:space="preserve">Ergo I G C, minus erit k H C.</s> <s xml:id="echoid-s3660" xml:space="preserve"> <pb o="199" file="0211" n="211"/> Non ergo KHC, eſt minimum, ſed I G C. </s> <s xml:id="echoid-s3661" xml:space="preserve">Quod <lb/>&</s> <s xml:id="echoid-s3662" xml:space="preserve">c.</s> <s xml:id="echoid-s3663" xml:space="preserve"/> </p> <div xml:id="echoid-div186" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0210-01"/> </figure> </div> </div> <div xml:id="echoid-div188" type="section" level="1" n="123"> <head xml:id="echoid-head135" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s3664" xml:space="preserve">Cum autem in propoſit. </s> <s xml:id="echoid-s3665" xml:space="preserve">54. </s> <s xml:id="echoid-s3666" xml:space="preserve">aſſignatus ſit modus <lb/>reperiendi triangulum maximum E F C, fuit conſe-<lb/>quenter expoſitus etiam modus reperiendi triangu-<lb/>lum minimum G I C.</s> <s xml:id="echoid-s3667" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3668" xml:space="preserve">Inſuper notetur, triangulum minimum circum-<lb/>ſcriptum parabolæ, æquale eſſe triangulo minimo <lb/>circumſcripto figuræ conſtante ex duabus ſemipara-<lb/>bolis ſupra expoſitis. </s> <s xml:id="echoid-s3669" xml:space="preserve">Triangulum enim GIC, du-<lb/>plicatum ad partes G C, eſt æquale eidem G I C, <lb/>duplicato ad partes IC.</s> <s xml:id="echoid-s3670" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div189" type="section" level="1" n="124"> <head xml:id="echoid-head136" xml:space="preserve">PROPOSITIO LXIII.</head> <p style="it"> <s xml:id="echoid-s3671" xml:space="preserve">Conus minimus circum ſcriptus cuilibet infinitorum conoìdeo-<lb/>rum vel ſemifuſorum par abolicorum, eſt ille, qui tangit <lb/>baſim maximi coni in illis ſolidis inſcripti.</s> <s xml:id="echoid-s3672" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3673" xml:space="preserve">SEd ſupponamus conum ex triangulo EFC, eſſe <lb/>maximum inſcriptibilium intra conoides ex ſe-<lb/>miparabola A B C, circa B C, & </s> <s xml:id="echoid-s3674" xml:space="preserve">conum ex triangulo <lb/>G C, tangere baſim coniinſcripti. </s> <s xml:id="echoid-s3675" xml:space="preserve">Dico conum ex <lb/>triangulo G I C, eſſe minimum circumſcriptibilium <lb/>conoidi. </s> <s xml:id="echoid-s3676" xml:space="preserve">Si non, ſit minimus ille, qui oritur ex trian-<lb/>gulo H k C, & </s> <s xml:id="echoid-s3677" xml:space="preserve">ducta L E M, parallela KH, intelli- <pb o="200" file="0212" n="212"/> gamus conum ex triangulo L M C, qui vtique erit <lb/>minor cono ex triangulo K H C. </s> <s xml:id="echoid-s3678" xml:space="preserve">Conus ergo ex <lb/>triangulo E F C, cum ſit maximus inſcriptus in co-<lb/>noide, erit ex dictis, maximus inſcriptus in cono ex <lb/>triangulo I G C. </s> <s xml:id="echoid-s3679" xml:space="preserve">Non ergo erit maximus inſcriptus <lb/>in cono ex triangulo L M C. </s> <s xml:id="echoid-s3680" xml:space="preserve">Ergo conus ex triangu-<lb/>lo E F C, erit ad conum ex triangulo G I C, in ma-<lb/>iori ratione quam ad conum ex triangulo L C M. </s> <s xml:id="echoid-s3681" xml:space="preserve">Er-<lb/>go in multo maiori quam ad conum ex triangulo <lb/>H k C. </s> <s xml:id="echoid-s3682" xml:space="preserve">Non ergo erit minimus conus ex triangulo <lb/>k H C, ſed ille ex triangulo IGC.</s> <s xml:id="echoid-s3683" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3684" xml:space="preserve">Patiter ſi conus ex triangulo E N C, ſit maximus <lb/>inſcriptus in ſemifuſo ex ſemiparabola A B C, reuo-<lb/>luta circa A C, conus ex triangulo G I C, circa I C, <lb/>erit minimus circumſcriptus ſemifuſo; </s> <s xml:id="echoid-s3685" xml:space="preserve">quod, vt pa-<lb/>tet, probabitur eodem modo. </s> <s xml:id="echoid-s3686" xml:space="preserve">Quare pater propo-<lb/>ſitum.</s> <s xml:id="echoid-s3687" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div190" type="section" level="1" n="125"> <head xml:id="echoid-head137" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s3688" xml:space="preserve">Cum ergo in propoſitionibus 58, & </s> <s xml:id="echoid-s3689" xml:space="preserve">61, aſſigna-<lb/>uerimus conos maximos inſcriptos in conoidibus, & </s> <s xml:id="echoid-s3690" xml:space="preserve"><lb/>in ſemifuſis, pariter explicauimus vnica vice, conos <lb/>ctiam minimos prædictis ſolidis circumſcriptos. </s> <s xml:id="echoid-s3691" xml:space="preserve">No-<lb/>tandum tamen diuerſos eſſe conos minimos his ſoli-<lb/>dis circumſcriptos; </s> <s xml:id="echoid-s3692" xml:space="preserve">nam in cono circumſcripto co-<lb/>noidi, C F, eſt tertia pars G C; </s> <s xml:id="echoid-s3693" xml:space="preserve">in cono vero cir-<lb/>cumſcripto ſemifuſo, C F, eſt duæ tertiæ partes G C. <lb/></s> <s xml:id="echoid-s3694" xml:space="preserve">Quæ omnia cum ſint manifeſtiſſima ex ſupra dictis, <pb o="201" file="0213" n="213"/> ideo circa ipſa nequaquam immoramur. </s> <s xml:id="echoid-s3695" xml:space="preserve">Solum ani-<lb/>maduertendum eſt, quod cum ſupra in ſcholijs pro-<lb/>poſit. </s> <s xml:id="echoid-s3696" xml:space="preserve">51, & </s> <s xml:id="echoid-s3697" xml:space="preserve">52, oſtenſum ſit idem eſſe centrum gra-<lb/>uitatis maximi trianguli inſcripti in triangulo, & </s> <s xml:id="echoid-s3698" xml:space="preserve">ip-<lb/>ſius trianguli; </s> <s xml:id="echoid-s3699" xml:space="preserve">item maximi coni in cono inſcripti, & </s> <s xml:id="echoid-s3700" xml:space="preserve"><lb/>ipſius coni; </s> <s xml:id="echoid-s3701" xml:space="preserve">patet conſequenter idem eſſe centrum <lb/>grauitatis maximi trianguli inſcripti in parabola, & </s> <s xml:id="echoid-s3702" xml:space="preserve"><lb/>minimi circumſcripti: </s> <s xml:id="echoid-s3703" xml:space="preserve">item idem eſſe centrum gra-<lb/>uitatis maximi coni inſcripti in quolibet conoide, & </s> <s xml:id="echoid-s3704" xml:space="preserve"><lb/>in quolibet ſemifuſo para bolico, & </s> <s xml:id="echoid-s3705" xml:space="preserve">minimorum co-<lb/>norum ipſis circumſcriptorum.</s> <s xml:id="echoid-s3706" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div191" type="section" level="1" n="126"> <head xml:id="echoid-head138" xml:space="preserve">PROPOSITIO LXIV.</head> <p style="it"> <s xml:id="echoid-s3707" xml:space="preserve">Quælibet parabola est ad maximum triangulum ſibi inſcri-<lb/>ptum, vt pars ſemibaſis parabolæ, quæ ſe babeat ad ſemi-<lb/>baſim vt binarium ad numerum parabolæ vnitate au-<lb/>ctum, ad vltimam proportionalem proportionis ſemibaſis <lb/>parabolæ, ad ſemibaſim trianguli, continuatæ in tot termi-<lb/>nos, vt numerus eorum excedat numerum parabolæ bi-<lb/>nario.</s> <s xml:id="echoid-s3708" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3709" xml:space="preserve">ESto quælibet parabola A B C, ſitque maximum <lb/>triangulum in ea inſcriptum G D H, vt ſupra <lb/>dictum eſt. </s> <s xml:id="echoid-s3710" xml:space="preserve">Dico parabolam eſſe ad triangulum <lb/>G D H, vt talis pars A D, quæ sè habeat ad A D, <lb/>vt binarium ad numerum parabolæ vnitate auctum, <lb/>ad vltimum terminum proportionis A D, ad G F, <lb/>continuatæ in tot terminos, vt numerus eorum exce- <pb o="202" file="0214" n="214"/> <anchor type="figure" xlink:label="fig-0214-01a" xlink:href="fig-0214-01"/> dat numerum parabolæ binario. </s> <s xml:id="echoid-s3711" xml:space="preserve">V. </s> <s xml:id="echoid-s3712" xml:space="preserve">g. </s> <s xml:id="echoid-s3713" xml:space="preserve">in prima pa-<lb/>rabola, nempein triangulo vt A D, ad tertiam pro-<lb/>portionalem. </s> <s xml:id="echoid-s3714" xml:space="preserve">In quadratica vt duo tertia A D, ad <lb/>quartam. </s> <s xml:id="echoid-s3715" xml:space="preserve">In cubica vt duo quarta, feù dimidium <lb/>A D, ad quintam. </s> <s xml:id="echoid-s3716" xml:space="preserve">Etſic in infinitum. </s> <s xml:id="echoid-s3717" xml:space="preserve">Sit illa vltima <lb/>proportionalis A Q. </s> <s xml:id="echoid-s3718" xml:space="preserve">In prima parabola, nempe in <lb/>triangulo res eſt euidens, quia ſicuti triangulum <lb/>A B C, eſſet quadruplum trianguli G D H, maximi <lb/>ſibi inſcripti, ſic A D, quia A D, eſſet dupla G F, <lb/>eſſet quadrupla A Q, tertiæ proportionalis. </s> <s xml:id="echoid-s3719" xml:space="preserve">In alijs <lb/>parabolis nè ſchemata multiplicemus, intelligamus <lb/>inſcripta triangula etiam A B C, quorum baſes A C, <lb/>diametri D B. </s> <s xml:id="echoid-s3720" xml:space="preserve">Triangulum A B C, ad triangulum <pb o="203" file="0215" n="215"/> G D H, habet rationem compoſitam ex rationibus <lb/>A D, ad G F, & </s> <s xml:id="echoid-s3721" xml:space="preserve">B D, ad D F. </s> <s xml:id="echoid-s3722" xml:space="preserve">Sed B D, ad D F, <lb/>eſt ex ſchol. </s> <s xml:id="echoid-s3723" xml:space="preserve">2. </s> <s xml:id="echoid-s3724" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3725" xml:space="preserve">54. </s> <s xml:id="echoid-s3726" xml:space="preserve">componendo, vt nume-<lb/>rus parabolæ vnitate auctus ad numerum parabolæ, <lb/>& </s> <s xml:id="echoid-s3727" xml:space="preserve">pariter ex natura parabolæ, cum ſit B D, ad D F, <lb/>vt poteſtas A D, eiuſdem gradus cum parabola, ad <lb/>exceſſum ipſius ſupra ſimilem poteſtatem G F, nem-<lb/>pead tot tales poteſtates. </s> <s xml:id="echoid-s3728" xml:space="preserve">G F, quotus eſt numerus <lb/>parabolæ. </s> <s xml:id="echoid-s3729" xml:space="preserve">Ergo ratio trianguli A B C, ad G D H, <lb/>componetur ex ratione A D, ad G F, & </s> <s xml:id="echoid-s3730" xml:space="preserve">ex ratione <lb/>poteſtatis A D, eiuſdem gradus cum parabola ad <lb/>totſimiles poteſtates G F, quotus eſt numerus para-<lb/>bolæ Sed ex iſtis rationib s componitur quoque ra-<lb/>tio poteſtatis A D, vno gradu altioris poteſtate pa-<lb/>rabolæ, ad tot ſimiles poteſtates G F, quotus eſtnu-<lb/>merus parabolæ. </s> <s xml:id="echoid-s3731" xml:space="preserve">Ergo triangulum A B C, erit ad <lb/>triangulum G D H, vt illa poteſtas A D, ad illas <lb/>poteſtates G F. </s> <s xml:id="echoid-s3732" xml:space="preserve">Sedvt poteſtas A D, ad vnam po-<lb/>teſtatem G F, ſic D A, ad A Q: </s> <s xml:id="echoid-s3733" xml:space="preserve">ergo & </s> <s xml:id="echoid-s3734" xml:space="preserve">vt pote-<lb/>ſtas dicta A D, ad omnes illas poteſtates G F, ſic <lb/>D A, ad tot A Q. </s> <s xml:id="echoid-s3735" xml:space="preserve">Erit ergo triangulum A B C, ad <lb/>triangulum G D H, vt D A, ad tot A Q, quotus <lb/>eſt numerus parabolæ. </s> <s xml:id="echoid-s3736" xml:space="preserve">Quoniam vero ex propoſit. <lb/></s> <s xml:id="echoid-s3737" xml:space="preserve">1. </s> <s xml:id="echoid-s3738" xml:space="preserve">lib. </s> <s xml:id="echoid-s3739" xml:space="preserve">prim eſt conuertendo, parabola A B C, ad pa-<lb/>rallelogrammum ſibi circumſcriptum vt numerus <lb/>parabolæ ad numerum parabolæ vnitate auctum, <lb/>nempe vt duplus numerus parabolæ, ad duplum nu-<lb/>merum binario auctum; </s> <s xml:id="echoid-s3740" xml:space="preserve">ergo parabola A B C, erit <lb/>ad triangulum A B C, dimidium parallelogrammi <pb o="204" file="0216" n="216"/> <anchor type="figure" xlink:label="fig-0216-01a" xlink:href="fig-0216-01"/> ſibi circumſcripti vt duplus numerus parabolę ad nu-<lb/>merum parabolæ vnitate auctum; </s> <s xml:id="echoid-s3741" xml:space="preserve">nempe vt magni-<lb/>tudo, quæ ſe habeat ad A D, vt duplus numerus pa-<lb/>rabolæ, ad numerum parabolæ vnitate auctum, ad <lb/>A D. </s> <s xml:id="echoid-s3742" xml:space="preserve">Quare ex ęquali, erit parabola A B C, ad trian-<lb/>gulum G D H, vt dicta magnitudo, quæ ad A D, sè <lb/>habeat vt duplus numerus parabolæ ad numerum pa-<lb/>rabolæ vnitate auctum, ad tot A Q, quotus eſt nu-<lb/>merus parabolæ. </s> <s xml:id="echoid-s3743" xml:space="preserve">Cum verò antecedens huius pro-<lb/>portionis contineat duplum numerum parabolæ, & </s> <s xml:id="echoid-s3744" xml:space="preserve"><lb/>conſequens numerum parabolæſequitur antecedens <lb/>diuidi in tot binaria, in quot vnitates diuiditur con-<lb/>ſequens: </s> <s xml:id="echoid-s3745" xml:space="preserve">vnde erit vt præ dictum antecedens ad præ- <pb o="205" file="0217" n="217"/> dictum conſequens, ſic vnum binarium anteceden-<lb/>tis, ad vnitatem conſequentis. </s> <s xml:id="echoid-s3746" xml:space="preserve">Erit ergo vt duæ par-<lb/>tes illius magnitudinis diuiſæ in tot partes quotus eſt <lb/>numerus parabolę duplus, & </s> <s xml:id="echoid-s3747" xml:space="preserve">conſequenter ipſius A D, <lb/>diuiſæ in tot partes quotus eſt numerus parabolę vni-<lb/>tate auctus, ad A Q. </s> <s xml:id="echoid-s3748" xml:space="preserve">Quoderat oſtendendum.</s> <s xml:id="echoid-s3749" xml:space="preserve"/> </p> <div xml:id="echoid-div191" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0214-01"/> </figure> <figure xlink:label="fig-0216-01" xlink:href="fig-0216-01a"> <image file="0216-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0216-01"/> </figure> </div> </div> <div xml:id="echoid-div193" type="section" level="1" n="127"> <head xml:id="echoid-head139" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3750" xml:space="preserve">Cum autem in propoſit. </s> <s xml:id="echoid-s3751" xml:space="preserve">55, viſum ſit, triangulum <lb/>G Q D, eſſe dimidium trianguli maximi inſcripti in <lb/>figura conſtante ex duabus ſemiparabolis; </s> <s xml:id="echoid-s3752" xml:space="preserve">ſequitur <lb/>hoc eſſe ad triangulum maximum ſibi inſcriptum in <lb/>ſupra dicta ratione, continuata ratione A D, ad D Q, <lb/>diametrum trianguli æqualem G F, vt dictum eſt. <lb/></s> <s xml:id="echoid-s3753" xml:space="preserve">Pariter cum minima trian gula circum ſcripta tam in-<lb/>finitis parabolis, quam infinitis figuris conſtantibus <lb/>ex duabus ſemiparabolis, ſint quadrupla maximo-<lb/>rum triangulorum in ipſis inſcriptorum; </s> <s xml:id="echoid-s3754" xml:space="preserve">ſequitur <lb/>prædictas figuras eſſe ad minima triangula circum-<lb/>ſcripta, vt idem antecedens ad quadruplum conſe-<lb/>quentis: </s> <s xml:id="echoid-s3755" xml:space="preserve">vel vt quarta pars antecedentis ad idem <lb/>conſequens.</s> <s xml:id="echoid-s3756" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div194" type="section" level="1" n="128"> <head xml:id="echoid-head140" xml:space="preserve">PROPOSITIO LXV.</head> <p style="it"> <s xml:id="echoid-s3757" xml:space="preserve">Quodlibet conoides parabolicum eſt ad maximum conum ſibi <lb/>inſcriptum, vt pars radij baſis conoidis, quæ ſe habeat ad <lb/>totum radium vt vnitas ad numerum conoidis binario <pb o="206" file="0218" n="218"/> auctum, ad ſextam partem vltimæ proportionalis propor-<lb/>tionis dicti radij ad radium baſis coni, continuatæ in tot <lb/>terminos vt numerus eorum excedat numerum conoidis <lb/>ternario.</s> <s xml:id="echoid-s3758" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3759" xml:space="preserve">SEd ſupponamus A B C, eſſe conoides paraboli-<lb/>cum, & </s> <s xml:id="echoid-s3760" xml:space="preserve">D G H, maximum conum illi inſcri-<lb/>ptum, &</s> <s xml:id="echoid-s3761" xml:space="preserve">c. </s> <s xml:id="echoid-s3762" xml:space="preserve">& </s> <s xml:id="echoid-s3763" xml:space="preserve">ratio A D, ad G F, continuetur in tot <lb/>terminos vt numerus excedat numerum conoidis ter-<lb/>nario, ſitque vltimus terminus A Q. </s> <s xml:id="echoid-s3764" xml:space="preserve">Dico conoides <lb/>ad conum eſſe vt pars A D, quæ sè habeat ad dictam <lb/>A D, vt vnitas ad numerum conoidis binario au-<lb/>ctum, ad ſextam partem A Q. </s> <s xml:id="echoid-s3765" xml:space="preserve">V. </s> <s xml:id="echoid-s3766" xml:space="preserve">g. </s> <s xml:id="echoid-s3767" xml:space="preserve">in primo conoi-<lb/>de, nempe in cono, vt tertia pars A D, ad ſextam <lb/>partem A Q, quartæ proportionalis. </s> <s xml:id="echoid-s3768" xml:space="preserve">In ſecundo, <lb/>vt quarta pars A D, ad ſextam partem A Q, quin-<lb/>tæ proportionalis. </s> <s xml:id="echoid-s3769" xml:space="preserve">In cubico, vt quinta pars <lb/>A D, ad ſextam partem A Q, ſextæ. </s> <s xml:id="echoid-s3770" xml:space="preserve">Et ſic in in-<lb/>finitum.</s> <s xml:id="echoid-s3771" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3772" xml:space="preserve">In cono, patet. </s> <s xml:id="echoid-s3773" xml:space="preserve">Quia ſi A B C, eſt conus, B F, <lb/>eſt dupla F D. </s> <s xml:id="echoid-s3774" xml:space="preserve">Cumque pateat ex propoſ. </s> <s xml:id="echoid-s3775" xml:space="preserve">52, A B C, <lb/>eſſe ad G D H, vt cubus D B, ad factum ſub qua-<lb/>drato B F, in F D, nempe in medietatem B F; <lb/></s> <s xml:id="echoid-s3776" xml:space="preserve">nempe ad medietatem cubi B F; </s> <s xml:id="echoid-s3777" xml:space="preserve">& </s> <s xml:id="echoid-s3778" xml:space="preserve">cum ſit vt cubus <lb/>D B, ad medietatem cubi B F, ſic cubus A D, ad <lb/>medietatem cubi G F; </s> <s xml:id="echoid-s3779" xml:space="preserve">nempe tertia pars cubi A D, <lb/>ad ſextam partem cubi G F: </s> <s xml:id="echoid-s3780" xml:space="preserve">& </s> <s xml:id="echoid-s3781" xml:space="preserve">pariter cum ſit vt cu-<lb/>bus A D, ad cubum G F, ſic A D, ad A Q, & </s> <s xml:id="echoid-s3782" xml:space="preserve">vt <lb/>tertia pars cubi A D, ad ſextam partem cubi G F, <pb o="207" file="0219" n="219"/> ſic tertia pars A D, ad ſextam partem A Q; </s> <s xml:id="echoid-s3783" xml:space="preserve">ergo <lb/>patet propoſitum.</s> <s xml:id="echoid-s3784" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3785" xml:space="preserve">In alijs vero conoidibus, mente intelligamus co-<lb/>num A B C, inſcriptum in conoide: </s> <s xml:id="echoid-s3786" xml:space="preserve">ergo conus <lb/>A B C, ad conum G D H, habet rationem compo-<lb/>ſitam ex ratione quadrati A D, ad quadratum G F, <lb/>& </s> <s xml:id="echoid-s3787" xml:space="preserve">ex ratione B D, ad D F. </s> <s xml:id="echoid-s3788" xml:space="preserve">Sed ex natura conoidis, <lb/>B D, ad D F, eſt vt poteſtas A D, eiuſdem gradus <lb/>cum conoide, ad exceſſum eiuſdem ſupra ſimilem. <lb/></s> <s xml:id="echoid-s3789" xml:space="preserve">poteſtatem G F; </s> <s xml:id="echoid-s3790" xml:space="preserve">& </s> <s xml:id="echoid-s3791" xml:space="preserve">pariter ex ſchol. </s> <s xml:id="echoid-s3792" xml:space="preserve">propoſit. </s> <s xml:id="echoid-s3793" xml:space="preserve">58, <lb/>componendo, eſt B D, ad D F, vt dimidium nu-<lb/>meri conoidis vnitate auctum ad dimidium numeri <lb/>conoidis; </s> <s xml:id="echoid-s3794" xml:space="preserve">nempe vt numerus conoidis binario au-<lb/>ctus, ad numerum conoidis; </s> <s xml:id="echoid-s3795" xml:space="preserve">vnde exceſſus prædictæ <lb/>poteſtatis A D, ſupra ſimilem poteſtatem G F, <lb/>continet tot partes prædictæ poteſtatis A D, diuiſæ <lb/>in tot partes quotus eſt numerus conoidis binario <lb/>auctus, quotus eſt numerus conoidis; </s> <s xml:id="echoid-s3796" xml:space="preserve">nempe tot me-<lb/>dietates ſimilis poteſtatis G F, quotus eſt nume-<lb/>rus conoidis. </s> <s xml:id="echoid-s3797" xml:space="preserve">Ergo proportio coni A B C, ad co-<lb/>num G D H, componetur ex ratione quadrati A D, <lb/>ad quadratum G F, & </s> <s xml:id="echoid-s3798" xml:space="preserve">ex ratione poteſtatis A D, ad <lb/>tot medietates ſimilis poteſtatis G F, quotus eſt <lb/>numerus conoidis. </s> <s xml:id="echoid-s3799" xml:space="preserve">Ergo conus A B C, erit ad co-<lb/>num G D H, vt poteſtas A D, duplici gradu altior <lb/>poteſtate conoidis, ad factum ſub quadrato G F, & </s> <s xml:id="echoid-s3800" xml:space="preserve"><lb/>ſub prædictis medietatibus poteſtatis G F; </s> <s xml:id="echoid-s3801" xml:space="preserve">nempe <lb/>ad tot medietates ſimilis poteſtatis G F, quotus eſt <lb/>numerus conoidis; </s> <s xml:id="echoid-s3802" xml:space="preserve">nempe vt A D, ad tot medieta- <pb o="208" file="0220" n="220"/> tes A Q, quotus eſt numerus conoidis. </s> <s xml:id="echoid-s3803" xml:space="preserve">Aſt cum ex <lb/>propoſit. </s> <s xml:id="echoid-s3804" xml:space="preserve">15, lib. </s> <s xml:id="echoid-s3805" xml:space="preserve">3. </s> <s xml:id="echoid-s3806" xml:space="preserve">ſit conuertendo, conoides A B C, <lb/>ad cylindrum ſibi circum ſcriptum vt numerus co-<lb/>noidis ad numerum conoidis binario auctum; </s> <s xml:id="echoid-s3807" xml:space="preserve">nempe <lb/>vt triplus numerus conoidis, ad triplum numerum <lb/>conoidis ſenario auctum: </s> <s xml:id="echoid-s3808" xml:space="preserve">erit idem conoides ad co-<lb/>num A B C, tertiam partem talis cylindri, vt tri-<lb/>plus numerus conoidis, ad numerum conoidis bina-<lb/>rio auctum: </s> <s xml:id="echoid-s3809" xml:space="preserve">nempe vt tot partes A D, diuiſæ in tot <lb/>partes quotus eſt numerus conoidis binario auctus, <lb/>quotus eſt triplus numerus conoidis, ad A D. </s> <s xml:id="echoid-s3810" xml:space="preserve">Ergo <lb/>ex æquali, erit conoides A B C, ad conum G D H, <lb/>vt prædictæ partes A D, quotus eſt triplus numerus <lb/>conoidis, ad tot medietates A Q, quotus eſt nume-<lb/>rus conoidis. </s> <s xml:id="echoid-s3811" xml:space="preserve">Et diuiſis vtriſque terminis per 3, erit <lb/>conoides A B C, ad conum G D H, vt tres partes <lb/>A D, diuiſæ prædicto modo, ad dimidiam A Q. </s> <s xml:id="echoid-s3812" xml:space="preserve">Et <lb/>ſubtriplando hos terminos, vt vnica talium partium <lb/>A D, ad ſextam partem A Q. </s> <s xml:id="echoid-s3813" xml:space="preserve">Quod erat oſtenden-<lb/>dum.</s> <s xml:id="echoid-s3814" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div195" type="section" level="1" n="129"> <head xml:id="echoid-head141" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3815" xml:space="preserve">Cum ex ſupra dictis, conſtet, minimum conum. <lb/></s> <s xml:id="echoid-s3816" xml:space="preserve">k E L, conoidi circumſcriptum, eſſe maximum cir-<lb/>cumſcriptum cono G D H; </s> <s xml:id="echoid-s3817" xml:space="preserve">& </s> <s xml:id="echoid-s3818" xml:space="preserve">cum ex ſchol. </s> <s xml:id="echoid-s3819" xml:space="preserve">prop. </s> <s xml:id="echoid-s3820" xml:space="preserve"><lb/>52, conſtet conum G D H, eſſe ad conum k E L, vt <lb/>4, ad 27, ſequitur conoides eſſe ad conum K E L, vt <lb/>prædicta pars A D, ad A Q, cum eius octaua parte.</s> <s xml:id="echoid-s3821" xml:space="preserve"/> </p> <pb o="209" file="0221" n="221"/> </div> <div xml:id="echoid-div196" type="section" level="1" n="130"> <head xml:id="echoid-head142" xml:space="preserve">PROPOSITIO LXVI.</head> <p style="it"> <s xml:id="echoid-s3822" xml:space="preserve">Quilibet ſemifuſus parabolicus, eſt ad maximum conum ſibi <lb/>inſcriptum vt vnica pars quadrati ſemibaſis parabolæ di-<lb/>uiſi in tot partes quot vnitates continet tertia parsre-<lb/>ctanguli contenti ſub numero fuſi vnitate aucto, & </s> <s xml:id="echoid-s3823" xml:space="preserve">ſub <lb/>duplo numero fuſi vnitate aucto, ad duo rectangula con-<lb/>tenta ſub duobus vltimis terminis proportionis baſis ſemi-<lb/>parabolæ ad altitudinem coni, continuatæ in tot terminos, <lb/>vt numerus eorum excedat numerum fuſibinario.</s> <s xml:id="echoid-s3824" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3825" xml:space="preserve">SEd intelligamus ſemiparabolam A B D, cuius <lb/>baſis A D, diameter B D, cum triangulo <lb/>G Q D, rotari circa A D, adeovt conus genitus ſit <lb/>maximus in ſemiſuſo inſcriptus: </s> <s xml:id="echoid-s3826" xml:space="preserve">& </s> <s xml:id="echoid-s3827" xml:space="preserve">ratio A D, ad <lb/>D Q, ſit continuata ad totterminos, vt numerus co-<lb/>rum excedat numerum fuſi binario; </s> <s xml:id="echoid-s3828" xml:space="preserve">ſintque <lb/>duo vltimi minimi termini Q A, A k, Dico ſemi-<lb/>fuſum ex B A D, eſſe ad conum ex G Q D, vt vnica <lb/>pars quadrati A D, diuiſi in tot partes quot vnita-<lb/>tes continet tertia pars rectanguli ſub numero fuſi <lb/>vnitate aucto, & </s> <s xml:id="echoid-s3829" xml:space="preserve">ſub duplo numero fuſi vnitate au-<lb/>cto, ad duo rectangula Q A k. </s> <s xml:id="echoid-s3830" xml:space="preserve">V.</s> <s xml:id="echoid-s3831" xml:space="preserve">g. </s> <s xml:id="echoid-s3832" xml:space="preserve">in primo ſemi-<lb/>fuſo, vt dimidium quadrati A D, adilla duo rectan-<lb/>gula. </s> <s xml:id="echoid-s3833" xml:space="preserve">In ſecundo, vt quinta pars quadrati A D. </s> <s xml:id="echoid-s3834" xml:space="preserve">In <lb/>tertio vt vnica pars quadrati A D, diuiſi in 9, cum <lb/>tertia parte vnius. </s> <s xml:id="echoid-s3835" xml:space="preserve">Et ſic diſcurrendo.</s> <s xml:id="echoid-s3836" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3837" xml:space="preserve">Quod enim in cono ſicres ſehabeat, patet. </s> <s xml:id="echoid-s3838" xml:space="preserve">Quia <pb o="210" file="0222" n="222"/> <anchor type="figure" xlink:label="fig-0222-01a" xlink:href="fig-0222-01"/> in ipſo ratio A D, ad D Q, continuanda eſt tan-<lb/>tum ad tertium terminum; </s> <s xml:id="echoid-s3839" xml:space="preserve">hic ſit k A; </s> <s xml:id="echoid-s3840" xml:space="preserve">vnde duo vl-<lb/>timi minimi terminierunt D Q, k A. </s> <s xml:id="echoid-s3841" xml:space="preserve">Ergo eſt pro-<lb/>bandum conum ex B A D, eſſe ad conum ex G D Q, <lb/>vt dimidium quadrati A D, ad duo rectangula D Q, <lb/>K A. </s> <s xml:id="echoid-s3842" xml:space="preserve">Cum enim in tali caſu, ſit A Q, dupla Q D, <lb/>erit conus ad conum vt cubus A D, ad 4. </s> <s xml:id="echoid-s3843" xml:space="preserve">cubos Q D; <lb/></s> <s xml:id="echoid-s3844" xml:space="preserve">nempe vt dimidium cubi A D, ad duos cubos D Q. </s> <s xml:id="echoid-s3845" xml:space="preserve"><lb/>Sed vt dimidium cubi A D, ad duos cubos Q D, ſic <lb/>dimidium quadrati A D, ad duo rectangula D Q, <lb/>A k. </s> <s xml:id="echoid-s3846" xml:space="preserve">Quare patet propoſium.</s> <s xml:id="echoid-s3847" xml:space="preserve"/> </p> <div xml:id="echoid-div196" type="float" level="2" n="1"> <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/PVNDER1Y/figures/0222-01"/> </figure> </div> <p> <s xml:id="echoid-s3848" xml:space="preserve">Quod vero vt dimidium cubi ad duos cubos, ſic <lb/>dimidium quadrati ad duo rectangula, eſt maniſe- <pb o="211" file="0223" n="223"/> ſtum; </s> <s xml:id="echoid-s3849" xml:space="preserve">quia rationes antecedentium ad conſequentia <lb/>componuntur exijſdem rationibus. </s> <s xml:id="echoid-s3850" xml:space="preserve">Ratio enim di-<lb/>midij cubi A D, ad cubum D Q, componitur ex <lb/>ratione A D, ad D Q, & </s> <s xml:id="echoid-s3851" xml:space="preserve">ex ratione dimidij quadra-<lb/>ti A D, ad quadratum D Q, quæ ratio eſt æqualis <lb/>rationi dimidiæ A D, ad A K, ex quibus rationi-<lb/>bus componitur quoque ratio dimidij quadrati A D, <lb/>adrectangulum D Q, A k.</s> <s xml:id="echoid-s3852" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3853" xml:space="preserve">In alijs vero, intellecto triangulo B A D, reuolu-<lb/>toque ipſo circa A D, habet conus ex ipſo ad co-<lb/>num ex Q G D, rationem compoſitam ex ratio-<lb/>ne A D, ad D Q, & </s> <s xml:id="echoid-s3854" xml:space="preserve">ex ratione quadrati B D, <lb/>ad quadratum D F, nempe ex duplici ratione <lb/>B D, ad D F. </s> <s xml:id="echoid-s3855" xml:space="preserve">Cum autem ſit componendo, ex <lb/>ſchol, propoſit. </s> <s xml:id="echoid-s3856" xml:space="preserve">61, B D, ad D F, vt duplus nu-<lb/>merus fuſi vnitatc auctus ad duplum nume-<lb/>rum fuſi; </s> <s xml:id="echoid-s3857" xml:space="preserve">& </s> <s xml:id="echoid-s3858" xml:space="preserve">cum pariter ſit B D, ad D F, vt pote-<lb/>ſtas A D, eiuſdem gradus cum fuſo ad exceſſum ip-<lb/>ſius ſupra ſimilem poteſtatem G F, nempe ad tot <lb/>ſimiles poteſtates G F, quotus eſt duplus numerus <lb/>fuſi. </s> <s xml:id="echoid-s3859" xml:space="preserve">Ergo proportio coni ex triangulo B A D, ad <lb/>conum ex triangulo G Q D, componetur ex ratio-<lb/>ne A D, ad D Q, & </s> <s xml:id="echoid-s3860" xml:space="preserve">ex ratione poteſtatis A D, ad <lb/>tot ſimiles poteſtates G F, ſeù Q D, quotus eſt <lb/>duplus numerus fuſi, & </s> <s xml:id="echoid-s3861" xml:space="preserve">ex ratione B D, ad D F. <lb/></s> <s xml:id="echoid-s3862" xml:space="preserve">Sed ex rationibus A D, ad D Q, & </s> <s xml:id="echoid-s3863" xml:space="preserve">poteſtatis dictæ <lb/>A D, ad dictas poteſtates Q D, componitur ratio <lb/>poteſtatum vnius gradus altioris. </s> <s xml:id="echoid-s3864" xml:space="preserve">Ergo ratio coniad <lb/>conum componetur ex ratione poteſtatis A D, vno <pb o="212" file="0224" n="224"/> <anchor type="figure" xlink:label="fig-0224-01a" xlink:href="fig-0224-01"/> gradu altioris poteſtate fuſi ad tot ſimiles poteſtates <lb/>D Q, quotus eſt duplus numerus fuſi, & </s> <s xml:id="echoid-s3865" xml:space="preserve">ex ratione <lb/>B D, ad D F. </s> <s xml:id="echoid-s3866" xml:space="preserve">Sed cum ſit vt poteſtas A D, vno <lb/>gradu altior poteſtate fuſi ad ſimilem poteſtatem <lb/>D Q, ſic D A, ad A k; </s> <s xml:id="echoid-s3867" xml:space="preserve">vnde & </s> <s xml:id="echoid-s3868" xml:space="preserve">vt poteſtas A D, ad <lb/>tot poteſtates D Q, quotus eſt duplus numerus fuſt <lb/>ſic D A, ad tot numero A k. </s> <s xml:id="echoid-s3869" xml:space="preserve">Ergo ratio coni ex <lb/>triangulo B A D, ad conum ex triangulo G Q D, <lb/>componetur ex ratione A D, ad tot A k, quotus eſt <lb/>duplus numerus fuſi, & </s> <s xml:id="echoid-s3870" xml:space="preserve">ex ratione B D, ad D F. <lb/></s> <s xml:id="echoid-s3871" xml:space="preserve">Rurſum B D, ad D F, patuit ſupra, eſſe vt poteſtas <lb/>A D, eiuſdem gradus cum fuſo ad tot ſimiles poteſta-<lb/>tes Q D, quotus eſt duplus numerus fuſi; </s> <s xml:id="echoid-s3872" xml:space="preserve">& </s> <s xml:id="echoid-s3873" xml:space="preserve">vt talis <pb o="213" file="0225" n="225"/> poteſtas ad tales poteſtates ſic, D A, ad tot numero <lb/>A Q. </s> <s xml:id="echoid-s3874" xml:space="preserve">Ergo ratio coni ad conum componetur ex ra-<lb/>tionibus A D, ad tot A k, & </s> <s xml:id="echoid-s3875" xml:space="preserve">eiuſdem A D, ad <lb/>tot Q A, quotus eſt duplus numerus fuſi: </s> <s xml:id="echoid-s3876" xml:space="preserve">nimirum <lb/>crit conus ad conum vt quadratum A D, ad rectan-<lb/>gulum ſub illis tot k A, & </s> <s xml:id="echoid-s3877" xml:space="preserve">A Q, quotus eſt duplus <lb/>numerus fuſi. </s> <s xml:id="echoid-s3878" xml:space="preserve">Aſt quoniam ex propoſit. </s> <s xml:id="echoid-s3879" xml:space="preserve">16, lib 2. </s> <s xml:id="echoid-s3880" xml:space="preserve">eſt <lb/>conuertendo, ſemifuſus ex ſemiparabola B A D, ad <lb/>cylindrum ſibi circumſcriptum, vt quadratũ num eri <lb/>parabolę ad rectangulũ ſub dimidio numeri parabolę <lb/>vnitate aucti, & </s> <s xml:id="echoid-s3881" xml:space="preserve">ſub duplo numero parabolæ vnitate <lb/>aucto; </s> <s xml:id="echoid-s3882" xml:space="preserve">vel vt duplum ad duplum; </s> <s xml:id="echoid-s3883" xml:space="preserve">nempe vt duplum <lb/>quadratum numeri parabolæ ad rectangulum ſub nu-<lb/>mero vnitate aucto, & </s> <s xml:id="echoid-s3884" xml:space="preserve">ſub duplo numero vnitate au-<lb/>cto, vnde eſt ſemifuſus ad tertiam partem cylindri, <lb/>nempe ad conum ex triangulo B A D, vt antece-<lb/>dens, ad tertiam partem conſequentis; </s> <s xml:id="echoid-s3885" xml:space="preserve">& </s> <s xml:id="echoid-s3886" xml:space="preserve">vt antece-<lb/>dens ad tertiam partem conſequentis, ſic tot partes <lb/>quot vnitates continet duplum quadratum numeri <lb/>fuſi (hoc eft rectangulum ſub numero, & </s> <s xml:id="echoid-s3887" xml:space="preserve">ſub duplo <lb/>numero) quadrati A D, diuiſi in tot paites quot vni-<lb/>tates continet tertia pars rectanguli ſub numero fuſi <lb/>vnitate aucto, & </s> <s xml:id="echoid-s3888" xml:space="preserve">ſub duplo numero vnitate aucto, ad <lb/>quadratum A D. </s> <s xml:id="echoid-s3889" xml:space="preserve">Ergo ex æquali, erit ſemifuſus ad <lb/>conum ex G Q D, vt tot partes quadrati A D, diuiſi <lb/>vt dictum eſt, quot vnitates continet rectangulum <lb/>ſub numero fuſi, & </s> <s xml:id="echoid-s3890" xml:space="preserve">ſub duplo numero, ad tot rectan-<lb/>gula ſub tot K A, & </s> <s xml:id="echoid-s3891" xml:space="preserve">ſub tot A Q, quotus eſt du-<lb/>plus numerus fuſi. </s> <s xml:id="echoid-s3892" xml:space="preserve">Cum vero numerus antecedentis, <pb o="214" file="0226" n="226"/> nempe partium quadrati A D, ſit numerus ortus ex <lb/>numero fuſi, & </s> <s xml:id="echoid-s3893" xml:space="preserve">ex duplo numero; </s> <s xml:id="echoid-s3894" xml:space="preserve">& </s> <s xml:id="echoid-s3895" xml:space="preserve">numerus rectan-<lb/>gulorum ex k A, A Q, ſit numerus ortus ex duplo nu-<lb/>mero, & </s> <s xml:id="echoid-s3896" xml:space="preserve">ex duplo numero; </s> <s xml:id="echoid-s3897" xml:space="preserve">ſequitur primum nume-<lb/>rum, nempe quadratorum, eſle dimidium numeri <lb/>ſecundi, nempe rectangulorum K A Q. </s> <s xml:id="echoid-s3898" xml:space="preserve">Quare quot <lb/>vnitates continet numerus quadratorum, tot binaria <lb/>continet numerus rectangulorum. </s> <s xml:id="echoid-s3899" xml:space="preserve">Erit ergo vt om-<lb/>nia illa quadrata ad omnia rectangula, ſic vnicum <lb/>quadratum ad vnicum rectangulum. </s> <s xml:id="echoid-s3900" xml:space="preserve">Erit ergo ſemi-<lb/>fuſus ad conum ex GQD, maximum ſibi inſcriptum <lb/>vt vnica pars quadrati A D, diuiſi in tot partes quot <lb/>vnitates continet tertia pars rectanguli ſub numero <lb/>fuſi vnitate aucto, & </s> <s xml:id="echoid-s3901" xml:space="preserve">ſub duplo numero vnitate au-<lb/>cto, ad duo rectangula Q A k. </s> <s xml:id="echoid-s3902" xml:space="preserve">Quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s3903" xml:space="preserve"/> </p> <div xml:id="echoid-div197" type="float" level="2" n="2"> <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/PVNDER1Y/figures/0224-01"/> </figure> </div> </div> <div xml:id="echoid-div199" type="section" level="1" n="131"> <head xml:id="echoid-head143" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3904" xml:space="preserve">Cum ergo conus minimus circumſcriptus ſemifu-<lb/>ſo ſit ad maximum inſcriptum vt 27, ad 4; </s> <s xml:id="echoid-s3905" xml:space="preserve">ſequitur <lb/>ſemifuſum eſſe ad ipſum, vt prædictum antecedens <lb/>ad 13, rectangula Q A K, cum dimidio.</s> <s xml:id="echoid-s3906" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3907" xml:space="preserve">Hæc ergo ſunt benigne lector, quæ pro tertia hac <lb/>vice determinauimus tibi communicare. </s> <s xml:id="echoid-s3908" xml:space="preserve">Impreſſio <lb/>noſtri operis de Infinitis Parabolis abſoluta fuit die <lb/>quarta præteriti Menſis Iulij. </s> <s xml:id="echoid-s3909" xml:space="preserve">Compoſitio Miſcella-<lb/>nei præſentis terminata fuit die 26. </s> <s xml:id="echoid-s3910" xml:space="preserve">Auguſti. </s> <s xml:id="echoid-s3911" xml:space="preserve">Hæc <lb/>tibiexponimus vt habeas vnde colligas fauorabiles <pb o="215" file="0227" n="227"/> excuſationes pro imperfectionibus in ipſo contentis. <lb/></s> <s xml:id="echoid-s3912" xml:space="preserve">Sufficere enim arbitramur notificare compoſitum <lb/>fuiſſe tempore æſtiuo, & </s> <s xml:id="echoid-s3913" xml:space="preserve">dum Canicula, & </s> <s xml:id="echoid-s3914" xml:space="preserve">Leo ma-<lb/>gis, magiſque feruent. </s> <s xml:id="echoid-s3915" xml:space="preserve">Hæc etenim tempora potius <lb/>otio, & </s> <s xml:id="echoid-s3916" xml:space="preserve">quieti, quam ſpeculationibus geometricis, <lb/>hoceſt ſublimibus, videntur accomodata. </s> <s xml:id="echoid-s3917" xml:space="preserve">Verum <lb/>propemodum impoſſibile eſt cohibere intellectum <lb/>nè vagetur vbicunque eilibuerit. </s> <s xml:id="echoid-s3918" xml:space="preserve">Præter quamquod <lb/>in inuentionibus rerum geometricarum, expectandæ <lb/>ſunt illæ fauorabiles cæleſtes directiones, quæ influ-<lb/>unt non quando nos, ſed quando ipſæ volunt. </s> <s xml:id="echoid-s3919" xml:space="preserve">Tabel-<lb/>lam errorum non exhibemus; </s> <s xml:id="echoid-s3920" xml:space="preserve">relinquimus enim illos <lb/>tuæ diligentiæ, tuæque humanitati. </s> <s xml:id="echoid-s3921" xml:space="preserve">Diligentiæ vt <lb/>illos corrigas; </s> <s xml:id="echoid-s3922" xml:space="preserve">humanitati vt eos libenter ſuſtineas; </s> <s xml:id="echoid-s3923" xml:space="preserve"><lb/>memor impreſſionem librorum matrem eſſe erro-<lb/>rum; </s> <s xml:id="echoid-s3924" xml:space="preserve">atque in impreſſione ſpeculationum abſtracta-<lb/>rum, intellectum auctoris ſic incumbere ſubſtantiæ, <lb/>vt accidentia cogaturnegligere. </s> <s xml:id="echoid-s3925" xml:space="preserve">Vale.</s> <s xml:id="echoid-s3926" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div200" type="section" level="1" n="132"> <head xml:id="echoid-head144" xml:space="preserve">FINIS.</head> <pb file="0228" n="228"/> <pb file="0229" n="229"/> <pb file="0230" n="230"/> <pb file="0231" n="231"/> <pb file="0232" n="232"/> </div></text> </echo>