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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Ceva, Giovanni</author> <title>Geometria motus</title> <date>1692</date> <place>Bologna</place> <translator></translator> <lang>la</lang> <cvs_file>cevag_geome_022_la_1692.xml</cvs_file> <cvs_version></cvs_version> <locator>022.xml</locator> </info> <text> <front> <section> <pb xlink:href="022/01/001.jpg"></pb> <figure id="id.022.01.001.1.jpg" xlink:href="022/01/001/1.jpg"></figure> <pb xlink:href="022/01/002.jpg"></pb> <pb xlink:href="022/01/003.jpg"></pb> <p type="main"> <s id="s.000001"><emph type="center"></emph>GEOMETRIA<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000002"><emph type="center"></emph>MOTUS<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000003"><emph type="center"></emph>OPVSCVLVM GEOMETRICVM<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000004"><emph type="center"></emph>A'<emph.end type="center"></emph.end><emph type="center"></emph><emph type="italics"></emph>IOANNE CEVA MEDIOLANENSI<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000005"><emph type="center"></emph>In gratiam Aquarum excogitatum.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000006"><emph type="center"></emph>CONTINET DVOS LIBROS<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000007"><emph type="center"></emph>Primum de Simplici Motu, <lb></lb>Alterum de Compoſito.<emph.end type="center"></emph.end></s> </p> <figure id="id.022.01.003.1.jpg" xlink:href="022/01/003/1.jpg"></figure> <p type="main"> <s id="s.000008"><emph type="center"></emph>BONONIÆ, M. DC. XCII.<emph.end type="center"></emph.end><lb></lb></s> </p> <p type="main"> <s id="s.000009"><emph type="center"></emph>Typis HH. </s> <s id="s.000010">Antonij Piſarij Superiorum permiſſu.<emph.end type="center"></emph.end></s> </p> <pb xlink:href="022/01/004.jpg"></pb> <pb xlink:href="022/01/005.jpg"></pb> <p type="main"> <s id="s.000011"><emph type="center"></emph>SERENISSIMO<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000012"><emph type="center"></emph>MANTVÆ DUCI<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000013"><emph type="center"></emph>FERDINANDO <lb></lb>CAROLO.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000014"><emph type="italics"></emph>ITerum, Sereniſſime Princeps, tuis aduolutus <lb></lb>genibus opuſculum exhibeo, in quo naturam motuum, pleniori <lb></lb>methodo, quàm puto antea ſit actum, geometricè exequor. <lb></lb></s> <s id="s.000015">Neceße habui hæc præmittere, quò viam aperirem, & quo<lb></lb>dammodo alueum ſternerem aquarum doctrinæ, quarum <lb></lb>argumentum vtiliſſimum, & profundæ indaginis iam diu <lb></lb>meditor. </s> <s id="s.000016">Quam arduum ſit, & per quas ſalebras eun<lb></lb>dum, vt nouum aliquid luce dignum è latebris naturæ eruarur <lb></lb>vtinam Celſitudini tuæ aliquis veritatum non vulgarium <lb></lb>indagator fidem faceret; ſcio equidem, & laboris improbitas <lb></lb>tangeret benigniſſimum animum tuum, & ſimul naturæ inge<lb></lb>nium ſuſpiceres, quæ mentibus aliquorum vim inuentricem <lb></lb>inſeruit, vt eorum iugi cogitatione humanis vſibus prouide-<emph.end type="italics"></emph.end><pb xlink:href="022/01/006.jpg"></pb><emph type="italics"></emph>ret. </s> <s id="s.000017">Et verò (ſi in hoc genere de me quidquam confiteri decet) <lb></lb>niſi aduerſæ valetudinis experimento prudentior factus indo<lb></lb>lem meam huiuſcemodi ſtudijs intemperanter addictam ali<lb></lb>quot ab hinc annis compeſcuißem; nec non quotidie munus à <lb></lb>Celſitudine Tua ſummo cum honore & beneficentia demanda<lb></lb>tum (adeo vt hoc etiam nomine Teſeruatorem meum appella<lb></lb>re poſſim) inde me reuocaſſet; eorum, credo equidem, ponderi, <lb></lb>aſſiduæque contemplationi ſuccumbere neceſſe erat. </s> <s id="s.000018">Vnde au<lb></lb>tem, Celſiſſime dux, huic ſcientiæ tanta vis, vt quos ſibi ſemet <lb></lb>adiunxerit, nonniſi altiori ratione queat a ſe ipſa dimittere? <lb></lb></s> <s id="s.000019">An quod fortaſſe vbi animus publicæ vtilitati deſeruire cæpe<lb></lb>rit, veluti in naturæ concilium admiſſus, ſui quodammodo <lb></lb>oblitus, propriam humilioremque ſedem reuiſere dedignetur; an <lb></lb>quia, cùm inter cæteras ſcientias Geometria demonſtrationem, <lb></lb>hoc eſt veritatem ſinceram, & quandam primi veri particu<lb></lb>lam profiteatur, hinc neſcio quid diuinum habent ſibi <expan abbr="propoſitũ">propoſitum</expan>, <lb></lb>vnde nonniſi Deo impellente, vbi nimirum officia, potiorque <lb></lb>ratio id poſtulant, ab eius intuitu retrahatur. </s> <s id="s.000020">Hoc equidem <lb></lb>puto; atque hinc diuina Geometria iure optimo a doctiſſimis, & <lb></lb>clariſſimis viris paſſim nuncupatur. </s> <s id="s.000021">Quamobrem nemo non <lb></lb>eam ſuſpiciat, eiuſque cultores oppidò diligat; ob eamque <expan abbr="causã">causam</expan> <lb></lb>huic etiam qualicunque opuſculo benignè annuas ſpero, adeo <lb></lb>vt iam Te in terris Dominum, Altorem, Seruatorem, Patro<lb></lb>numque appellare non dubitem, quam vna cum Celſiſſima do<lb></lb>mo mihi, tot tibi nominibus deuincto, ſuperi vt ſeruent ſoſpi<lb></lb>tentque, enixè oro, ac omnibus votis exopto.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000022"><emph type="italics"></emph>Sereniſsimæ Celſitudinis Tuæ<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000023"><emph type="italics"></emph>Humillimus, & Obſequentiſſimus Seruus<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000024">Ioannes Ceua. </s> </p> </section> </front> <body> <chap> <pb pagenum="1" xlink:href="022/01/007.jpg"></pb> <p type="main"> <s id="s.000025"><emph type="center"></emph>GEOMETRIA<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000026"><emph type="center"></emph>MOTVS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000027"><emph type="center"></emph>DEF. I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000028">CVrrat mobile ab A in D ſecundùm rectam <arrow.to.target n="marg1"></arrow.to.target><lb></lb>AD, & linea BHI ſit naturæ illius, vt dedu<lb></lb>ctis ad AD perpendicularibus AB, CH, DI <lb></lb>ex punctis quibuſcunque A, C, D; veloci<lb></lb>tatum gradus, quos mobile ſortitur in ijſ<lb></lb>dem punctis A, C, D menſurentur ab ipſis <lb></lb>rectis AB, CH, CI. </s> <s id="s.000029">Figuram planam BADIHB apellabi<lb></lb>mus geneſim motus ab A in D. </s> </p> <p type="margin"> <s id="s.000030"><margin.target id="marg1"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="main"> <s id="s.000031"><emph type="center"></emph>DEF. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000032">IIſdem manentibus, ſit etiam alia linea EFG talis natu<lb></lb><arrow.to.target n="marg2"></arrow.to.target><lb></lb>ræ, vt protractis rectis BA in E, HC in F, & ID in G ha<lb></lb>beat DG ad CF eandem reciprocè rationem, quam HC <lb></lb>ad ID. </s> <s id="s.000033">Item ſit CF ad HE vt reciprocè BA ad HC, vo<lb></lb>cabimus figuram planam ADGIEA imaginem tempo<lb></lb>ris motus ab A in D iuxta geneſim prædictam. </s> </p> <p type="margin"> <s id="s.000034"><margin.target id="marg2"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="main"> <s id="s.000035"><emph type="center"></emph>DEF. III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000036">ADhuc poſita illa geneſi, intelligatur linea PON eius <lb></lb><arrow.to.target n="marg3"></arrow.to.target><lb></lb>naturæ, vt ſi ſit KL ad LM vt tempus lationis ab A <lb></lb>in C ad tempus ab eodem C in D, habeat ſemper KP ad <lb></lb>LO eandem rationem, quam AB ad CH; & LO ad NM <lb></lb>eandem, quam HC ad ID: Figuram planam PKMNOP <pb pagenum="2" xlink:href="022/01/008.jpg"></pb>vocabimus imaginem iuxta geneſim BADI motus ab <lb></lb>A in D. </s> </p> <p type="margin"> <s id="s.000037"><margin.target id="marg3"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000038"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000039"><emph type="italics"></emph>Patet, cum motus ſunt æquabiles, geneſes, & imagines figu<lb></lb>ras eße parallelogrammas.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000040"><emph type="center"></emph>DEF. IV.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg4"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000041"><margin.target id="marg4"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="main"> <s id="s.000042">SI ſint duæ geneſes, aut imagines ABCD, FEG, ita vt <lb></lb>cum geneſes ſint, habeat AB ad FE eandem rationem, <lb></lb>quam velocitas in A ad velocitatem in F, & cum imagines <lb></lb>velocitatum, quarum tempora AD, FG, velocitas, quam <lb></lb>habet mobile inſtanti A ad velocitatem alterius mobilis <lb></lb>inſtanti F, ſit vt AB ad FE, & demum ipſis figuris vt imagi<lb></lb>nibus temporum conſideratis habeat velocitas in A ad <lb></lb>velocitatem in F rationem eandem, quam AB ad FE, vo<lb></lb>cabuntur tum geneſes illæ, cum imagines inter ſe homo<lb></lb>geneæ. </s> </p> <p type="main"> <s id="s.000043"><emph type="center"></emph>DEF. V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000044">EAm planam Figuram, in qua ductæ quotcunque <lb></lb>ęquidiſtantes eò deinceps decreſcunt, quò ad idem <lb></lb>extremum propiores fiunt, acuminatam nuncupabimus. </s> </p> <p type="main"> <s id="s.000045"><emph type="center"></emph>DEF. VI. AX. I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000046">INter maximam, & minimam eiuſdem imaginis veloci<lb></lb>tatem cadit quædam media, qua tantùm velocitate, ſi <lb></lb>conciperetur motus æquabilis, nihilominùs eodem tem<lb></lb>pore idem ſpatium curreretur, ac iuxta imaginem propoſi<lb></lb>tam: eam ergo mediam velocitatem dicimus propoſitæ <lb></lb>imaginis æquatricem. </s> </p> <pb pagenum="3" xlink:href="022/01/009.jpg"></pb> <p type="main"> <s id="s.000047"><emph type="center"></emph>AX. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000048">SPatium iuxta imaginem velocitatum quamcunque <lb></lb>exactum, vel iuxta æquatricem imaginis eſt maius eo <lb></lb>ſpatio, quod curreretur eodem tempore minima eiuſdem <lb></lb>imaginis velocitate; ſed minus eo, quod velocitate ma<lb></lb>xima. </s> </p> <p type="main"> <s id="s.000049"><emph type="center"></emph>AX. III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000050">TEmpus, quo curritur ſpatium iuxta quamlibet tem<lb></lb>poris imaginem, maius eſt eo, quo idem ſpatium <lb></lb>curreretur maxima velocitate, ſed contra minus eo altero, <lb></lb>quo ipſum ſpatium minima velocitate exigeretur, earum <lb></lb>videlicet, quæ ſunt in geneſi, aut imagine velocitatum pro<lb></lb>poſiti motus, cuius nempe illa eſt imago temporis. </s> <s id="s.000051">Fit er<lb></lb>go, vt tempus æquale ei, quo illud ipſum ſpatium currere<lb></lb>tur iuxta propoſitam imaginem, ſit inter vtrumque dicto<lb></lb>rum temporum maximi, & minimi. </s> </p> <p type="main"> <s id="s.000052"><emph type="center"></emph>AX. IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000053">QVæcunque excogitetur figura plana, vel eſt paralle<lb></lb>logrammum, vel acuminata figura, aut ex his com<lb></lb>poſitum. </s> <s id="s.000054">Has tamen figuras inter binas volu<lb></lb>mus parallelas, ita vt vnum latus ſit ipſas nectens normali<lb></lb>ter parallelas, quanquam etiam loco parallelarum poſſint <lb></lb>eſſe puncta, nempè vbi deſinunt in acuminatas prorſus <lb></lb>figuras. </s> </p> <p type="main"> <s id="s.000055"><emph type="center"></emph>PROP. I. THEOR. I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000056">TEmpora, quibus duo motus complentur ſunt in ra<arrow.to.target n="marg5"></arrow.to.target><lb></lb>tione imaginum homogenearum ipſorum <expan abbr="temporũ">temporum</expan>. </s> </p> <pb pagenum="4" xlink:href="022/01/010.jpg"></pb> <p type="margin"> <s id="s.000058"><margin.target id="marg5"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 5.</s> </p> <p type="main"> <s id="s.000059">Motus ſint primò æquabiles, curratque mobile ſpatium <lb></lb>AB tempore, cuius imago CAB, curratur item ab alio mo<lb></lb>bili ſpatium DE tempore, cuius imago DEF, & ſint ipſæ <lb></lb><arrow.to.target n="marg6"></arrow.to.target><lb></lb>temporum imagines interſe homogeneæ, ſcilicet FD ad <lb></lb>AC eandem habeat rationem, quam velocitas in A ad <lb></lb>velocitatem in D. Dico, tempus per AB ad id per DE eſ<lb></lb><arrow.to.target n="marg7"></arrow.to.target><lb></lb>ſe vt figura ABC, ad DEF. </s> <s id="s.000060">Cum motus æquabiles ſint <lb></lb>erunt figuræ dictarum imaginum rectangula, propterea il<lb></lb>lorum ratio componetur ex rationibus altitudinum AB ad <lb></lb><arrow.to.target n="marg8"></arrow.to.target><lb></lb>DE, & baſium AC ad DF, ex ijſdem verò rationibus ſpa<lb></lb>tiorum ſcilicet, & reciproca velocitatum (ſunt enim ima<lb></lb>gines inter ſe homogeneæ) nectitur etiam ratio temporum, <lb></lb>quibus <expan abbr="percurrũtur">percurruntur</expan> ipſa ſpatia AB, DE iuxta geneſes ima<lb></lb>ginum ACB, DEF, ergo eſt eadem ratio inter illa tempo<lb></lb>ra, ac inter imagines ſuas. <lb></lb><arrow.to.target n="marg9"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000061"><margin.target id="marg6"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000062"><margin.target id="marg7"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000063">Def.<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000064"><margin.target id="marg8"></margin.target><emph type="italics"></emph>Gal. </s> <s id="s.000065">pr. S de <lb></lb>motu æquab. <lb></lb></s> <s id="s.000066">Def.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000067"><margin.target id="marg9"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 6. <lb></lb><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000068">2. Sit motus vnus æquabilis, alter verò quicunque; ſit <lb></lb>tamen imago huius temporis figura acuminata vt ALGE, <lb></lb>& alterius temporis prædicti motus æquabilis, ſit HFM, </s> </p> <p type="main"> <s id="s.000069"><arrow.to.target n="marg10"></arrow.to.target><lb></lb>quæ rectangulum erit: Dico, imaginibus homogeneis exi<lb></lb>ſtentibus, fore inter has eandem rationem, ac homologè <lb></lb>inter tempora decurſuum ab A in E, & ab F in M iuxtą <lb></lb>geneſes imaginum temporum propoſitarum. </s> <s id="s.000070">Si enim non <lb></lb>eſt ita, ſit quædam alia magnitudo Y, maior, vel minor <lb></lb>imagine acuminata ALGE, quæ ad imaginem FHM ha<lb></lb>beat eandem rationem, quam tempus per AE iuxta imagi<lb></lb>nem ALGE ad tempus per FM iuxta imaginem alteram <lb></lb>FHM; ſit verò magnitudinis Y differentia ab imagine ma<lb></lb>gnitudo Z. </s> <s id="s.000071">Secetur AE bifariam in C, pariterque ſeg<lb></lb>menta AC, CE bifariam in B, D, & ſic vlteriùs progredia<lb></lb>tur, donec, ſi compleatur rectangulum poſtremum, & ma<lb></lb>ximum DG, hoc minus exiſtat quam Z. </s> <s id="s.000072">Tum ductis reli<lb></lb>quis æquidiſtantibus CI, BK, & à punctis N, I, K, I alijs <lb></lb>etiam æquidiſtantibus rectæ AE, efficiatur ipſi ALGE cir<lb></lb>cumſcripta figura, conſtans ex rectangulis æquealtis AK <pb pagenum="5" xlink:href="022/01/011.jpg"></pb>BI, CN, DG, & inſcripta compoſita ex rectangulis inter ſe <lb></lb>pariter æquealtis BL, CR, DI, EN. </s> <s id="s.000073">Cum circumſcriptą <lb></lb>figura differat ab inſcripta exceſſu, quo rectangulum DG <lb></lb>ſuperat BL; (nam reliqua circumſcripta AK, BI, CN, re<lb></lb>liquis inſcriptis æqualia ſunt) ſequitur, exceſſum illum eſſe <lb></lb>minorem magnitudine Z. </s> <s id="s.000074">Si ergo magnitudo Y ponatur <lb></lb>maior magnitudine ALGE pro exceſſu Z, maior etiam erit <lb></lb>circumſcripta AK, BI, CN, DG. </s> <s id="s.000075">Quòd ſi contrà Y intelli<lb></lb>gatur minor ipſa ALGE ex defectu Z, erit quoque eadem <lb></lb>Y minor, quàm inſcripta figura BL, CK, DI, EN. </s> <s id="s.000076">Itaque <lb></lb>nunc, ſi fieri poteſt, ſit Y maior magnitudine ALGE per ip<lb></lb>ſum exceſſum Z, & intelligantur tot motus, quot ſunt re<lb></lb>ctangula in circumſcripta figura, ſcilicet ſint ipſi motus ab <lb></lb>A in B, à B in C, à C in D, & à D in E ſecundum deinceps, <lb></lb>temporum imagines AK, BI, CN, DG rectangula, quæ <lb></lb>ſint interſe, & propoſitis imaginibus homogeneæ, qui <lb></lb>motus erunt proptereà æquabiles. </s> <s id="s.000077">His poſitis, tempus <lb></lb><arrow.to.target n="marg11"></arrow.to.target><lb></lb>per FM iuxta imaginem MH ad tempus per AB iuxta ima<lb></lb>ginem rectangulum AK eandem habet rationem, quam re<lb></lb>ctangulum MH ad rectangulum AK, ſimiliter idem tem<lb></lb>pus per FM ſecundùm ipſam imaginem rectangulum MH <lb></lb><arrow.to.target n="marg12"></arrow.to.target><lb></lb>ad ſingula reliqua tempora per BC, CD, DE imaginibus <lb></lb>deinceps rectangulis BI, CN, DG habet eandem rationem, <lb></lb>quam rectangulum MH ad ſingula eodem ordine rectan<lb></lb>gula BI, CN, DG. </s> <s id="s.000078">Quo circa totidem rectangula ex MH, <lb></lb><arrow.to.target n="marg13"></arrow.to.target><lb></lb>quot ſunt illa, ex quibus conſtat circumſcripta figura, ha<lb></lb>bebunt ad ea ipſa circumſcripta rectangula, ſeu ad eandem <lb></lb>circumſcriptam figuram AK, BI, CN, DG eandem ratio<lb></lb>nem, quam totidem tempora eiuſdem imaginis MH ad ſi<lb></lb>mul tempora, quorum imagines ſunt illa ipſa circumſcripta <lb></lb>rectangula AK, BI, CN, DG. </s> <s id="s.000079">Quare etiam vnicum re<lb></lb>ctangulum MH ad circumſcriptam figuram AK, BI, CN, <lb></lb>DG erit in eadem ratione, in quo vnicum tempus per FM <lb></lb>iuxta imaginem MH ad omnia ſimul illa tempora iuxtą <pb pagenum="6" xlink:href="022/01/012.jpg"></pb>imagines, quæ ſunt dicta circumſcripta rectangula. </s> <s id="s.000080">Et <lb></lb>quoniam figura imaginis eſt acuminata, habetque vi def. <lb></lb>2. huius, applicatas, quæ ſunt in ratione reciproca veloci<lb></lb>tatum, quibus nempe mobile afficitur in punctis ſpatij, à <lb></lb>quibus deducuntur ipſæ applicatæ; hinc fit, vt earum ve<lb></lb>locitatum, quas mobile habet in decurſu rectæ AB, ea, quę <lb></lb>in A maxima ſit, & quæ in B minima. </s> <s id="s.000081">Eodem modo iuxta <lb></lb>reliquas imagines BKIC, CIND, DNGE, quæ itidem acu<lb></lb>minatæ ſunt, velocitates in fine decurſuum C, D, E (ſunt <lb></lb>enim omnes versùs A acuminatæ) minimæ erunt, & ma<lb></lb>ximæ initio dictorum ſpatiorum. </s> <s id="s.000082">Ideo tempora, quę im<lb></lb><arrow.to.target n="marg14"></arrow.to.target><lb></lb>penduntur iuxta illas imagines, ſeu ipſam <expan abbr="imaginẽ">imaginem</expan> ALGE, <lb></lb>cuius illæ ſunt omnes partes, minora erunt temporibus, <lb></lb>quæ decurrerent, ſi illi decurſus forent æquabiles ex mini<lb></lb>mis illis velocitatibus exacti, vel quod in idem recidit, ſi <lb></lb>illi decurſus eſſent iuxta imagines rectangulorum circum<lb></lb>ſcriptorum AK, BI, CN, DG; itaque rectangulum MH ad <lb></lb>figuram circumſcriptam AK, BI, CN, DG habebit mino<lb></lb>rem rationem, quàm tempus per FM imagine MH ad tem<lb></lb>pus per AE imagine ALGE, ſeu quàm rectangulum MH <lb></lb>habet ex hypotheſi ad magnitudinem Y; igitur circumſcri<lb></lb>pta figura, quæ priùs minor oſtenſa fuit magnitudine Y; <lb></lb>nunc maior concluditur; quod cum ſit abſurdum, ſequi<lb></lb>tur falsò nos poſuiſſe magnitudinem Y maiorem; quàm̨ <lb></lb>ALGE. </s> <s id="s.000083">At ſi Y minor ponatur, <expan abbr="quã">quam</expan> magnitudo ALGE de<lb></lb>fectu Z; inſcripta, vt ſupra, figura conſtante ex rectangulis <lb></lb>æquè altis BL, CK, DI, EN, vt ſcilicet differentia ab ima<lb></lb>gine ſit minor magnitudine Z, liquebit, magnitudinem Y <lb></lb>minorem eſſe inſcripta figura BL, CK, DI, EN; deindę <lb></lb>procedendo vt ſupra, inueniemus rectangulum MH ad in<lb></lb>ſcriptam figuram BL, CK, DI, EN in eadem ratione, iņ <lb></lb>quo tempus per FM imagine MH ad omnia ſimul decur<lb></lb>ſuum tempora per AB, BC, CD, DE iuxta imagines re<lb></lb>ctangula inſcripta BL, CH, DI, EN; Hæc verò temporą <pb pagenum="7" xlink:href="022/01/013.jpg"></pb>minora ſunt temporibus iuxta imagines ALKB, BKIC, <lb></lb>CIND, INGE (nam velocitates initio decurſuum per <lb></lb>dictas rectas diximus eſſe maximas, & quibus <expan abbr="conſiderã-tur">conſideran<lb></lb>tur</expan> illi motus æquabiles ſecundùm imagines ipſa illa re<lb></lb>ctangula inſcripta) ergo rectangulum MH ad inſcriptam̨ <lb></lb>figuram BL, CK, DI, EN habebit maiorem rationem, <expan abbr="quã">quam</expan> <lb></lb>tempus per FM iuxta imaginem MH ad tempora ſimul <lb></lb>imaginibus ALKB, BKIC, CIND, DNGE, ſiue ad tempus <lb></lb>iuxta imaginem ALGE ex illis compoſitam. </s> <s id="s.000084">Ideoque re<lb></lb>ctangulum MH ad ipſam inſcriptam figuram habebit ma<lb></lb>iorem rationem, quàm ad magnitudinem Y, idcirco Y, quæ <lb></lb>minor oſtenſa fuit inſcriptà figura BL, CK, DI, EN, nunc <lb></lb>hac alia via maiorem inuenimus; ergo cum rurſus hoc ſit <lb></lb>abſurdum, neceſſe eſt magnitudinem Y neque minorem̨ <lb></lb>eſſe magnitudine ALGE, propterea æquales inter ſe <expan abbr="erũt">erunt</expan>, <lb></lb>atque adeo tempus per FM imagine MN ad tempus per <lb></lb>AE imagine ALGE habebit eandem rationem, quam ima<lb></lb>go MH ad imaginem ALGE. </s> <s id="s.000085">Quod &c. </s> </p> <p type="margin"> <s id="s.000086"><margin.target id="marg10"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000087">Def.<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000088"><margin.target id="marg11"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000089">Def.<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000090"><margin.target id="marg12"></margin.target><emph type="italics"></emph>Ex pramißą <lb></lb>parte.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000091"><margin.target id="marg13"></margin.target><emph type="italics"></emph>Euang. Tor<lb></lb>ric. lem.<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>in <lb></lb>libro de dim. <lb></lb></s> <s id="s.000092">parabolæ.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000093"><margin.target id="marg14"></margin.target><emph type="italics"></emph>Ax.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000094">3. Imagines propoſitæ ſint duæ acuminatæ. </s> <s id="s.000095">Dico ni<lb></lb><arrow.to.target n="marg15"></arrow.to.target><lb></lb>hilominus, tempora iuxta illas imagines per AE, HI eſſe vt <lb></lb>ipſæ imagines ALGE ad HIK, quæ ſint inter ſe homoge<lb></lb>neæ vt ſemper ſupponetur. </s> <s id="s.000096">Nam ſi intelligatur alius mo<lb></lb>tus per MF iuxta imaginem rectangulum MFN, qui æqua<lb></lb><arrow.to.target n="marg16"></arrow.to.target><lb></lb>bilis erit, manifeſtum eſt ex ſecundo caſu, tempus per AE <lb></lb>iuxta imaginem ALGE ad tempus per FM iuxta <expan abbr="imaginẽ">imaginem</expan> <lb></lb>rectangulum MH, habere eandem rationem, quam imago <lb></lb>ALGE ad imaginem rectangulum MH; & ſimiliter tem<lb></lb>pus per FM imagine rectangulum MN ad tempus per HI <lb></lb>iuxta imaginem HKI habet eandem rationem, quam ima<lb></lb>go NM ad imaginem HKI, ergo ex æquali tempus per AE <lb></lb>ad tempus per HI ſecundùm imagines propoſitas erit vt <lb></lb>imago ipſa ALGE ad imaginem HKI. </s> <s id="s.000097">Quod &c. </s> </p> <p type="margin"> <s id="s.000098"><margin.target id="marg15"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="margin"> <s id="s.000099"><margin.target id="marg16"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000100">Def.<emph.end type="italics"></emph.end> 3 <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000101">4. Demum imagines ſint quæcunque, modò ſint ho<lb></lb><arrow.to.target n="marg17"></arrow.to.target><lb></lb>mogeneæ, ADFB, GHKL: Dico rurſus inter ſe eſſe vt tem-<pb pagenum="8" xlink:href="022/01/014.jpg"></pb>pora per AB, AK iuxta ipſa imagines. </s> <s id="s.000102">Vel enim hæ ima<lb></lb>gines ſunt ſimplices, hoc eſt tantùm parallelogrammę, aut <lb></lb>tantùm acuminatæ, & tunc ſupra oſtendimus propoſitum, <lb></lb>quemadmodum etiam ſi vna acuminata, altera parallelo<lb></lb>gramma; vel non ſunt huiuſmodi & componentur ex illis. <lb></lb><arrow.to.target n="marg18"></arrow.to.target><lb></lb>Sint ergo in imagine ADFB partes ab æquidiſtantibus di<lb></lb>ſtinctæ ADEN, OFB acuminatæ & NEFO paralellogram-<lb></lb><arrow.to.target n="marg19"></arrow.to.target><lb></lb>mum, erunt hæ procul dubio inter ſe, totique imagini ho<lb></lb>mogeneæ; ſint pariter in alia imagine partes GHCM, <lb></lb>MCKL, per æquidiſtantem MC diſtinctæ inter ſe acumi<lb></lb><arrow.to.target n="marg20"></arrow.to.target><lb></lb>natæ, quæ itidem inter ſe, & imagini, cuius ſunt partes, ho<lb></lb>mogeneæ erunt. </s> <s id="s.000103">His acceptis, quoniam tempus per AN <lb></lb><arrow.to.target n="marg21"></arrow.to.target><lb></lb>iuxta imaginem ADEN acuminatam ad tempus per HC <lb></lb>iuxta aliam imaginem item acuminatam HGMC, habet <lb></lb>eandem rationem, ac imago ADEN ad <expan abbr="imaginẽ">imaginem</expan> GHCM. <lb></lb>ſimiliter tempus per HC iuxta imaginem GHCM ad tem<lb></lb>pus per CK iuxta imaginem acuminatam MCKL eſt vt <lb></lb>illa ad hanc imaginem; componendo, inde per conuerſio<lb></lb>nem rationis, & conuertendo, tempus per HC ſecundùm <lb></lb>imaginem GHCM ad tempora ſimul per HC, CK, <expan abbr="quorũ">quorum</expan> <lb></lb>imagines GHCM, MCKL, hoc eſt ad tempus per HK iux<lb></lb>ta imaginem GHKL habebit <expan abbr="eãdem">eandem</expan> rationem, quam ima<lb></lb>go GHCM ad imaginem GHCL; & ideo ex æquali tem<lb></lb>pus per AN, cuius imago ADEN, ad tempus per HK, iux<lb></lb>ta imaginem GHKL, erit in eadem ratione, in qua eſt ima<lb></lb>go ADEN ad imaginem GHKL. </s> <s id="s.000104">Præterea tempus per <lb></lb>AN iuxta imaginem ADEN ad idem ipſum tempus habet <lb></lb>eandem rationem, quam imago ADEN ad eandem ipſam; <lb></lb>tempus per NO iuxta imaginem rectangulum NEPO ad <lb></lb><arrow.to.target n="marg22"></arrow.to.target><lb></lb>tempus prædictum per AN eſt in eadem ratione <expan abbr="imaginũ">imaginum</expan> <lb></lb>NEPO ad ADEN, & ſimiliter tempus per OB iuxta ima<lb></lb>ginem OPFB habet ad tempus per AN eandem rationem, <lb></lb>ac imago OPFB ad imaginem ſæpè dictam ADEN; <expan abbr="itaq;">itaque</expan> ex <lb></lb>lem. 18. Toric. in lib. de dim: parabolæ, erunt tria <expan abbr="tẽpora">tempora</expan> per <pb pagenum="9" xlink:href="022/01/015.jpg"></pb>AN, NO, OB iuxta imagines deinceps ADEN, NEPO, <lb></lb>OPFB, hoc eſt erit tempus per AB iuxta imaginem ADFB <lb></lb>ad ſimul tria tempora per AN iuxta eandem imaginem <lb></lb>ADEN, vt imago ADFB ad triplum imaginis ADEN, & <lb></lb>cum tria æqualia tempora per AN ad vnicum ex illis ſit <lb></lb>vt triplum imaginis ADEN ad vnicam imaginem; ſequi<lb></lb>tur ex æquali tempus per AB ad tempus per AN iuxtą <lb></lb>imaginem ADEN habere eandem rationem, quam imago <lb></lb>ADFB ad imaginem ADEN: & oſtenſum fuit tempus per <lb></lb>AN iuxta imaginem ADEN ad tempus per HK iuxta <lb></lb>imaginem GHKL habere eandem rationem, quam imago <lb></lb>ADEN ad imaginem GHKL, ergo rurſus, & tandem ex <lb></lb>æquali, tempus per AB iuxta imaginem ADFB ad <expan abbr="tẽpus">tempus</expan> <lb></lb>per HK iuxta imaginem GHKL habebit eandem <expan abbr="rationẽ">rationem</expan>, <lb></lb>quam imago ADFB ad imaginem GHKL. </s> <s id="s.000106">Quod &c. </s> </p> <p type="margin"> <s id="s.000107"><margin.target id="marg17"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>Fig. 9<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000108"><margin.target id="marg18"></margin.target><emph type="italics"></emph>Ax.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000109"><margin.target id="marg19"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000110"><margin.target id="marg20"></margin.target><emph type="italics"></emph>Def:<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000111"><margin.target id="marg21"></margin.target><emph type="italics"></emph>Ex tertia <lb></lb>parte huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000112"><margin.target id="marg22"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>partę <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000113"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000114"><emph type="italics"></emph>Hinc colligitur, ſi prima magnitudo ad ſecundam fuerit vt <lb></lb>tertia ad quartam, item alia prima ad aliam ſecundam vt <lb></lb>alia tertia ad aliam quartam, & ſic vlteriùs quoad viſum̨ <lb></lb>fuerit, ſint præterea omnes primæ, item omnes tertiæ interſe <lb></lb>æquales, conſtat, inquam, primarum vnam ad omnes ſecun<lb></lb>das habere eandem rationem, ac vna tertiarum ad omnes <lb></lb>quartas.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000115"><emph type="center"></emph>PROP. II. THEOR. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000116">Spatia, quæ curruntur iuxta quaſcunque homogeneas <lb></lb><expan abbr="velocitatũ">velocitatum</expan> imagines, ſunt interſe, vt eædem illæ ima<lb></lb>gines. </s> <s id="s.000117">Sint primùm motus æquabiles, curraturque ſpa<lb></lb><arrow.to.target n="marg23"></arrow.to.target><lb></lb>tium AB iuxta imaginem velocitatum, quæ rectangulum <lb></lb>erit ILMK, ſpatium verò DE tranſigatur iuxta imaginem̨ <lb></lb>prædictæ homogeneam rectangulum FHNG (nam erunt <pb pagenum="10" xlink:href="022/01/016.jpg"></pb>homogeneæ ipſæ imagines, ſi vt ex Def. 4. huius IL ad HF <lb></lb>erit vt velocitas inſtanti I ad velocitatem mobilis inſtanti <lb></lb>F) Dico ſpatium AB ad DE eſſe vt imago rectangulum̨ <lb></lb>ILMK ad imaginem rectangulum FHNG. </s> <s id="s.000118">Componuntur <lb></lb>ipſa illa rectangula ex ratione altitudinum IK ad FG, & ex <lb></lb>ea baſium IL ad FH; verùm ex ijſdem, ea nempe <expan abbr="temporũ">temporum</expan> <lb></lb><arrow.to.target n="marg24"></arrow.to.target><lb></lb>IK ad FG, atque ea velocitatum IL ad FH componitur <lb></lb>etiam ratio ſpatiorum AB ad DE, ergo ipſa ſpatia erunt vt <lb></lb>propoſitę imagines. <lb></lb><arrow.to.target n="marg25"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000119"><margin.target id="marg23"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 9. <lb></lb><emph type="italics"></emph>Cor. </s> <s id="s.000120">Dif.<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000121"><margin.target id="marg24"></margin.target><emph type="italics"></emph>Gil. de motu <lb></lb>æquabili.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000122"><margin.target id="marg25"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>fig<emph.end type="italics"></emph.end> 10.</s> </p> <p type="main"> <s id="s.000123">2. Sint nunc motus iuxta imagines, quarum altera acu<lb></lb>minata, altera rectangulum ſit. </s> <s id="s.000124">Dico rurſus ſpatium AB, <lb></lb>quod curritur iuxta imaginem ABCD ad ſpatium DE, <lb></lb>quod curritur iuxta alteram imaginem, eſſe vt imago <lb></lb>ABCD ad imaginem PHNG. </s> <s id="s.000125">Niſi ita ſit, erit alia magni<lb></lb>tudo Y maior, vel minor imagine ABCD, quæ quidem ad <lb></lb>alteram imaginem HPGN habebit eandem rationem, <expan abbr="quã">quam</expan> <lb></lb>ſpatium AB ad DE. </s> <s id="s.000126">Sit primùm maior exceſſu Z. Cir<lb></lb>cumſcribatur; vt egimus in ſecunda parte primæ huius, fi<lb></lb>gura imagini ABCD conſtans ex rectangulis æquè altis, <lb></lb>excedatque imaginem ABCD exceſſu minori, quam Z; ſit <lb></lb>ergo circumſcripta illa AE, HF, IG, KG, quam primò fa<lb></lb>cilè oſtendemus minorem magnitudine Y; nam hæc exceſ<lb></lb>ſu magis diſtat ab imagine, quàm circumſcripta illa. </s> <s id="s.000127">Præ<lb></lb>terea ſi intelligantur tot motus æquabiles, quot ſunt <expan abbr="rectã-gula">rectan<lb></lb>gula</expan> circumſcripta, ij nempe, qui fierent temporibus AH, <lb></lb>HI, IK, KD iuxta deinceps imagines ipſa rectangula AE, <lb></lb>HF, IG, KC interſe, & propoſitis imaginibus homogeneas, <lb></lb>velocitates, quibus ijdem motus conſiderarentur, forent <lb></lb>HE, IF, KG, DC, nimirum maximæ imaginum ABEH, <lb></lb>HEFI, IFGK, KGCD; Cumque ita ſit, longiora ſpatia cur</s> </p> <p type="main"> <s id="s.000128"><arrow.to.target n="marg26"></arrow.to.target><lb></lb>rerentur iuxta imagines rectangula circumſcripta, quam <lb></lb>ijſdem temporibus, imaginibuſque poſtremis, hoc eſt <expan abbr="quã">quam</expan> <lb></lb>tempore AD iuxta imaginem ABCD; obidque ſpatium <lb></lb>AB ad DE, ſeu magnitudo Y ad imaginem HPGN habe-<pb pagenum="11" xlink:href="022/01/017.jpg"></pb>bit minorem rationem, quàm omnes illæ ſimul imagines, <lb></lb><arrow.to.target n="marg27"></arrow.to.target><lb></lb>ſeu quam circumſcripta figura AE, HF, IG, KC ad ean<lb></lb>dem imaginem HPGN; quare Y, quæ priùs oſtenſa fuit <lb></lb>maior, nunc reperitur minor eadem circumſcripta, quod <lb></lb>cum fieri nequeat, impoſſibile etiam eſt magnitudinem Y <lb></lb>maiorem eſſe magnitudine imaginis ABCD. </s> <s id="s.000129">Sit ergo mi<lb></lb>nor, ſi etiam fieri poteſt, & defectus ipſius Y ſupra ABCD <lb></lb>ſit Z. </s> <s id="s.000130">Inſcribatur imagini figura ex rectangulis æquealtis, vt <lb></lb>nempe deficiat ab imagine defectu minori Z; ſic enim ipſa <lb></lb>inſcripta, quæ ſit AB, IE, KF, DG erit magnitudine pro<lb></lb>pinquior imagini ABCD, quàm Y, ideoque Y minor erit <lb></lb>dicta inſcripta figura. </s> <s id="s.000131">Deinde, quoniam, ſi ponantur mo<lb></lb>tus æquabiles, quorum imagines rect angula inſcripta HB, <lb></lb>IE, KF, DG, quæque inter ſe, & propoſitis imaginibus ſint <lb></lb>homogeneæ; velocitates, quibus efficerentur dicti motus, <lb></lb>eſſent AB, IE, KF, DG, minimæ ſcilicet imaginum ABEH <lb></lb>HEFI, IFGK. KGCD, & ideo ſpatia, quæ percurrerentur <lb></lb>temporibus HA, HI, IK, KD imaginibus illis, maiora eſ<lb></lb><arrow.to.target n="marg28"></arrow.to.target><lb></lb>ſent, quàm quæ ijſdem temporibus tranſigerentur iuxtą <lb></lb>imagines prædictas rectangula circumſcripta, hinc fit vt <lb></lb>ſpatium AB ad DE, ſeu magnitudo Y ad imagine HPGN <lb></lb>habeat maiorem rationem, quàm inſcripta figura ad ean<lb></lb>dem imaginem HPGN; quare Y, quæ minor erat inſcripta <lb></lb>figura, modò reſultat maior, non ergo Y minor eſſe poteſt <lb></lb>imagine ABCD, ſed neque maior vt oſtendimus, ergo ſpa<lb></lb>tium AB ad DE erit, vt imago ABCD ad imaginem <lb></lb>PHNG. </s> <s id="s.000132">Quod &c. </s> </p> <p type="margin"> <s id="s.000133"><margin.target id="marg26"></margin.target><emph type="italics"></emph>Ax.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000134"><margin.target id="marg27"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000135">pr.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000136"><margin.target id="marg28"></margin.target><emph type="italics"></emph>Ex.<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000137">3. & 4. Si verò imagines acuminatæ ſint, aut demum <lb></lb>quæ cumque, eodem prorsùs modo, quo prima propoſitio<lb></lb>ne, oſtendemus hoc etiam propoſitum, ergo patet omne <lb></lb>intentum. </s> </p> <pb pagenum="12" xlink:href="022/01/018.jpg"></pb> <p type="main"> <s id="s.000138"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000139"><emph type="italics"></emph>Cum prorsùs geometricè oſtenderimus ſuperiores duas pro<lb></lb>poſitiones, vtiliſſimum eſt obſeruare, quomodo liceat vti tem<lb></lb>poris inſtantibus, non vt punctis prorsùs geometricis, ſed vt <lb></lb>quantitatibus dicam minoribus quibuſcunque datis. </s> <s id="s.000140">Hinc <lb></lb>oritur indiuiſibilium methodus, quæ intelligentiam affert <lb></lb>faciliorem, ac ſi rigori geometrico penitus inſiſteremus, quam<lb></lb>quam eæ tamen difficiliores Geometras mihi magis decerę <lb></lb>videantur.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000141"><emph type="center"></emph>PROP. III. THEOR. III.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg29"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000142"><margin.target id="marg29"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="main"> <s id="s.000143">SPatia, quæ curruntur iuxta quaslibet homogeneas ve<lb></lb>locitatum imagines, nectuntur ex rationibus tempo<lb></lb>rum, ac æquatricum. </s> </p> <p type="main"> <s id="s.000144">Velocitates æquatrices duorum motuum, quorum ima<lb></lb>gines velocitatum ſint ABCD, EFHI ponantur AG, EL. <lb></lb></s> <s id="s.000145">Dico ſpatia, ſeu ipſas imagines componi ex ratione tem<lb></lb>porum AD ad EI; & ex ea æquatricum AE ad EL. </s> <s id="s.000146">Nam <lb></lb>ſi motus, qui eſt iuxta imaginem ABCD perſeueret velo<lb></lb>citate AG, eſſet quidem æquabilis, idemque ſpatium illa </s> </p> <p type="main"> <s id="s.000147"><arrow.to.target n="marg30"></arrow.to.target><lb></lb>velocitate, & tempore AD percurreretur, ac ſecundùm̨ <lb></lb>imaginem ABCD; Itaque exiſtente rectangulo DE, quod <lb></lb><arrow.to.target n="marg31"></arrow.to.target><lb></lb>eſset imago velocitatum illius motus æquabilis, foret idem <lb></lb><arrow.to.target n="marg32"></arrow.to.target><lb></lb>æquale imagini ABCD (nam imagines ABCD, & DG <lb></lb>homogeneæ ſunt) eodem modo imago rectangulum VL <lb></lb>æquale eſset imagini EFHI. </s> <s id="s.000148">Cum ergo duæ imagines re<lb></lb>ctangula DE, IL componantur ex rationibus temporum <lb></lb>AD ad EI, & ex ea æquatricum AG ad EL; ex ijſdem̨ <lb></lb>prorsùs rationibus etiam imagines propoſitæ prædictis re<lb></lb>ctangulis æquales nectentur. </s> <s id="s.000149">Et ideo ſpatia, quæ propo<lb></lb>ſitis imaginibus tranſiguntur, quæque ipſis proportionalia <pb pagenum="13" xlink:href="022/01/019.jpg"></pb>ſunt, componentur ex rationibus temporum, & ex rationi<lb></lb>bus æquatricum. </s> </p> <p type="margin"> <s id="s.000150"><margin.target id="marg30"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>Ax.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="margin"> <s id="s.000151"><margin.target id="marg31"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000152"><margin.target id="marg32"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000153"><emph type="center"></emph><emph type="italics"></emph>Corollarium I.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000154"><emph type="italics"></emph>Hinc patet ſi lineæ, quæ in imagine velocitatum tempus <lb></lb>exhibet, aplicetur rectangulum æquale propoſitæ imagini ve<lb></lb>locitatum, fore vt latitudo eiuſdem rectanguli, ſit velocitas <lb></lb>æquatrix propoſitæ imaginis.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000155"><emph type="center"></emph><emph type="italics"></emph>Corollarium II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000156"><emph type="italics"></emph>Item constat, vbi tempora, vel æquatrices velocitates fue<lb></lb>rint æquales, rationem ſpatiorum eſſe eandem, quæ æquatri<lb></lb>cum, vel quæ temporum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000157"><emph type="center"></emph><emph type="italics"></emph>LEMMA.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000158"><emph type="italics"></emph>Si quælibet ratio compoſita ſit ex quotcumque rationibus, <lb></lb>harum quælibet nectetur ex propoſita, & ex reliquis contra<lb></lb>riò ſumptis rationibus. </s> <s id="s.000159">Sit A ad B compoſita ex rationibus E <lb></lb>æd F; G ad H; & I ad K. </s> <s id="s.000160">Dico quamlibet ist arum puta G ad <lb></lb>K conſtare ex rationibus A ad B, & ex reliquis reciprocè ſum<lb></lb>ptis F ad E, & I ad K. </s> <s id="s.000161">Vt E ad F, ita ſit A ad C, & vt D ad B <lb></lb>ſic I ad K; erit C ad D, vt G ad H; <expan abbr="ideoq;">ideoque</expan> C ad D, hoc eſt G ad<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="table1"></arrow.to.target><lb></lb><emph type="italics"></emph>H nectetur ex C ad A, ſeu F ad G, & ex rationibus A ad B, <lb></lb>B ad D, ſiue K ad I. </s> <s id="s.000162">Quod &c.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="14" xlink:href="022/01/020.jpg"></pb> <table> <table.target id="table1"></table.target> <row> <cell><emph type="italics"></emph>A<emph.end type="italics"></emph.end></cell> <cell><emph type="italics"></emph>E<emph.end type="italics"></emph.end></cell> <cell></cell> <cell></cell> </row> <row> <cell><emph type="italics"></emph>C<emph.end type="italics"></emph.end></cell> <cell><emph type="italics"></emph>F<emph.end type="italics"></emph.end></cell> <cell><emph type="italics"></emph>I.<emph.end type="italics"></emph.end></cell> <cell><emph type="italics"></emph>K<emph.end type="italics"></emph.end></cell> </row> <row> <cell><emph type="italics"></emph>D<emph.end type="italics"></emph.end></cell> <cell><emph type="italics"></emph>G<emph.end type="italics"></emph.end></cell> <cell></cell> <cell></cell> </row> <row> <cell><emph type="italics"></emph>B<emph.end type="italics"></emph.end></cell> <cell><emph type="italics"></emph>H<emph.end type="italics"></emph.end></cell> <cell></cell> <cell></cell> </row> </table> <p type="main"> <s id="s.000163"><emph type="center"></emph>PROP. IV. THEOR. IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000164">TEmpora, quibus abſoluuntur duo motus componun<lb></lb>tur ex ratione ſpatiorum, & ex reciproca æquatri<lb></lb>cum. </s> <s id="s.000165">Cum enim ſpatia <expan abbr="componãtur">componantur</expan> ex ratione temporum, <lb></lb><arrow.to.target n="marg33"></arrow.to.target><lb></lb>& ex ea velocitatum æquatricum, ſequitur per prædictum <lb></lb>Lemma, quòd tempora nectantur ex rationibus ſpatiorum, <lb></lb>& reciproca æquatricum. </s> </p> <p type="margin"> <s id="s.000166"><margin.target id="marg33"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000167"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000168"><emph type="italics"></emph>Manifeſtum eſt ſpatia, vel æquatrices velocitates, ſi ſint <lb></lb>æquales, eſſe tempora in reliqua ratione reciproca æquatri<lb></lb>cum, vel ſpatiorum non reciproca.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000169"><emph type="center"></emph>PROP. V. THEOR. V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000170">ÆQuatrices velocitates componuntur ex rationibus <lb></lb>ſpatiorum, & reciproca temporum. </s> </p> <p type="main"> <s id="s.000171">Cum ſpatia componantur ex rationibus temporum, & <lb></lb>velocitatum æquatricum, manifeſtum eſt ex eodem Lem<lb></lb>mate, velocitates ipſas necti ex rationibus ſpatiorum, & <lb></lb>reciproca temporum. </s> </p> <p type="main"> <s id="s.000172"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000173"><emph type="italics"></emph>Deducitur, æquatrices velocitates eſſe vt tempora reciprocè <lb></lb>ſumpta, vel vt ſpatia, ſi altera ratio fuerit æqualitatis.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000174"><emph type="center"></emph>D. </s> <s id="s.000175">E F. VII.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg34"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000176"><margin.target id="marg34"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="main"> <s id="s.000177">SI in geneſibus homogeneis AEC, GFK exiſtente AB <lb></lb>ad BC ſicut GI ad IK, habeat AE ad BD eandem ra-<pb pagenum="15" xlink:href="022/01/021.jpg"></pb>tionem, ac GF ad IH, motus, qui fiunt iuxta illas geneſes, <lb></lb>vocentur inter ſe ſimiles, & ipſæ geneſes dicentur ſimilium <lb></lb>motuum; quod verò attinet ad rectas AE, BD, GF, IH apel<lb></lb>labimus applicatas ad homologa puncta A, B, G, I propor<lb></lb>tionales. </s> </p> <p type="main"> <s id="s.000178"><emph type="center"></emph>PROP. VI. THEOR. VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000179">SI in imaginibus temporum homogeneis, applicatæ v<lb></lb>nius fuerint ad homologa puncta, proportionales ap<lb></lb>plicatis alterius imaginis, motus, quorum ſunt ipſæ imagi<lb></lb>nes, ſimiles erunt. </s> </p> <p type="main"> <s id="s.000180">Imagines temporum ſint &MLABC, &ONGIK, quæ </s> </p> <p type="main"> <s id="s.000181"><arrow.to.target n="marg35"></arrow.to.target><lb></lb>ſint homogeneæ, & cum GI ad IK ſit vt AB ad BC, habeat <lb></lb>quoque AL ad BM eandem rationem, ac GN ad IO. Di<lb></lb>co, motus, quorum ſunt illæ imagines temporum inter ſe ſi<lb></lb>miles eſſe. </s> </p> <p type="margin"> <s id="s.000182"><margin.target id="marg35"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000183">Sint apud ipſas imagines eorundem motuum geneſes, <lb></lb>ſcilicet EAC, FGK interſe homogeneæ. </s> <s id="s.000184">Exiſtente AL ad <lb></lb>BM, vt GN ad IO, erit conuertendo BM ad AL vt IO ad <lb></lb>GN; ſed vt BM ad AL ita ob geneſim EA ad DB, & vt IO <lb></lb><arrow.to.target n="marg36"></arrow.to.target><lb></lb>ad GN, ſic FG ad HI. ergo EA ad DB eſt vt FG ad HI, erat <lb></lb>autem vt AB ad BC ita etiam GI ad IK, ergo motus ſunt <lb></lb><arrow.to.target n="marg37"></arrow.to.target><lb></lb>ſimiles, & ipſæ imagines ſimilium motuum. </s> </p> <p type="margin"> <s id="s.000185"><margin.target id="marg36"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000186"><margin.target id="marg37"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000187"><emph type="center"></emph>PROP. VII. THEOR. VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000188">SI in imaginibus velocitatum vnius, applicate fuerint ex <lb></lb><arrow.to.target n="marg38"></arrow.to.target><lb></lb>punctis homologè ſumptis proportionales applicatis <lb></lb>alterius imaginis, motus iuxta ipſas imagines erunt ſimi<lb></lb>les, ideoque ipſæ imagines ſimilium motuum. </s> </p> <p type="margin"> <s id="s.000189"><margin.target id="marg38"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="main"> <s id="s.000190">Velocitatum imagines ſint ABCD, NPRT, ſitque AB <lb></lb>ad EF in eadem ratione, in qua NP ad TR; Dico exiſtenti<lb></lb>bus etiam BF ad FC, vt PQ ad QR eſſe propoſitas imagi<lb></lb>nes ſimilium motuum. </s> <s id="s.000191">Intelligantur eorundem motuum <pb pagenum="16" xlink:href="022/01/022.jpg"></pb>geneſes GHKL, YZ 43. & ſit pariter HI ad IK, vt ſegmen<lb></lb>tum ABFE ad EFCD. </s> <s id="s.000192">Sit ſimiliter Z <gap></gap> ad <gap></gap> 4 vt ſeg<lb></lb>mentum NPQV ad VQRT, ductiſque applicatis IM, QV, <lb></lb>manifeſtum eſt, vt velocitas AB æqualis eſt velocitati GH, <lb></lb>ſic EF æqualem fore ipſi IM; nam quia ſpatium <expan abbr="tranſactũ">tranſactum</expan> <lb></lb>iuxta imaginem ABFE ad ſpatium tranſactum imagine <lb></lb><arrow.to.target n="marg39"></arrow.to.target><lb></lb>EFCD eſt vt illa ad hanc imaginem, nempe vt HI ad IK, <lb></lb>erit mobile inſtanti F in puncto I, & ideo inibi erit veloci<lb></lb>tas eadem, quam habet mobile inſtanti F, ſcilicet æquales <lb></lb>erunt EF, IM. </s> <s id="s.000193">Eodem modo erunt æquales QV, <gap></gap> 2, & <lb></lb>ſunt etiam æquales NP, YZ, ergo ſicut ſe habet AB ad EF, <lb></lb>ita erit GH ad MI, & vt eſt NP ad <expan abbr="Vq.">Vque</expan> ita erit YZ ad 2 <gap></gap><lb></lb>Præterea concipiatur figura OPRSXO ſimilis ipſi ABCD, <lb></lb>ſcilicet ſit CB ad PR vt AB ad OP, vel (cum ſint BF ad <lb></lb>FC ita PQ ad QR, vt EF ad homologam XQ, erit ſeg<lb></lb>mentum ABFE ad ſibi ſimile ſegmentum OPQX in dupli<lb></lb>cata ratione laterum homologorum EF ad XQ, & item in <lb></lb><expan abbr="eadẽ">eadem</expan> duplicata ratione erunt interſe ſimilia <expan abbr="ſegmẽta">ſegmenta</expan> EFCD <lb></lb>ad XQRS, ſed cum etiam OPQX ſegmentum ad NPQV, <lb></lb>& XQRS ad ſegmentum VQRT ſint in eadem ratione <lb></lb>eiuſdem QX ad QV, erit ex æquali ſegmentum ABFE ad <lb></lb>ſegmentum NPQV, vt ſegmentum EFCD ad VQRT, & <lb></lb>permutando, ſegmentum ABFE ad ſegmentum EFCD ha<lb></lb>bebit eandem rationem, ac ſegmentum NPQV ad VQRT <lb></lb>ſcilicet erit HI ad IK vt Z <gap></gap> ad <gap></gap> 4, ob idque conſtat ge<lb></lb>neſium applicatas vnius proportionales eſſe applicatis al<lb></lb>terius, quare ſimiles motus erunt, qui fiunt iuxta imagines <lb></lb>velocitatum propoſitas. </s> </p> <p type="margin"> <s id="s.000194"><margin.target id="marg39"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000195"><emph type="center"></emph>PROP. VIII. THEOR. VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000196">SPatia, quæ curruntur ſimilibus motibus ſunt in ratione <lb></lb>compoſita temporum, & homologarum velocitatum, <lb></lb>inter quas ſunt extremæ, aut primæ. </s> </p> <pb pagenum="17" xlink:href="022/01/023.jpg"></pb> <p type="main"> <s id="s.000197">Imagines velocitatum ſimilium motuum ſint BCDE, <lb></lb><arrow.to.target n="marg40"></arrow.to.target><lb></lb>GMKI, & iuxta eas percurrantur ſpatia A, F. </s> <s id="s.000198">Dico iſta com<lb></lb>poni ex rationibus temporum BE ad GI, & ex ea veloci<lb></lb>tatem extremarum ED ad IK. </s> <s id="s.000199">Fiat vt BE ad GI, ita BC <lb></lb>ad GH, intelligatur que GHLI figura ſimilis ipſi BDE. Quo<lb></lb><arrow.to.target n="marg41"></arrow.to.target><lb></lb>niam ſpatium A ad F, hoc eſt imago BCDE ad imaginem <lb></lb>GMKI componitur ex ratione imaginis BCDE ad figu<lb></lb>ram ſibi ſimilem GHLI, & ex ratione huius ad imaginem <lb></lb>GMKI: prior ratio eſt duplicata homologorum laterum̨ <lb></lb>BE ad GI, ſeu eſt compoſita ex BE ad GI, & ex huic ſimi<lb></lb>li ratione ED ad IL, & ratio altera, imaginis ſcilicet GHLI <lb></lb>ad imaginem GMKI eſt, vt LI ad IK; ergo ex æquali ima<lb></lb>go BCDE ad imaginem GMKI, hoc eſt ſpatium A ad ſpa<lb></lb>tium F, componetur ex ratione temporum BE ad GI, & ex <lb></lb>rationibus ED ad LI, & IL ad IK, ſcilicet nectetur ex ra<lb></lb>tione BE ad GI, & ED ad IK, quæ poſtrema cum ſit ratio <lb></lb>velocitatum extremarum ED ad IK; conſtat, quod propo<lb></lb>ſuimus, ſpatia ſimilium motuum componi ex ratione tem<lb></lb>porum, & ex ratione homologarum velocitatum, hoc eſt <lb></lb>extremarum. </s> </p> <p type="margin"> <s id="s.000200"><margin.target id="marg40"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 5</s> </p> <p type="margin"> <s id="s.000201"><margin.target id="marg41"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huiu.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000202"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000203"><emph type="italics"></emph>Si tempora fuerint æqualia, ſimilium motuum ſpatia <expan abbr="erũt">erunt</expan> <lb></lb>vt extremæ, vel ſummæ velocitates, & contra, ſi iſtæ æquales <lb></lb>ſint, erunt ſpatia vt tempora.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000204"><emph type="center"></emph><emph type="italics"></emph>Corollarium II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000205"><emph type="italics"></emph>Cum ſpatia ſimilium motuum nectantur ex ratione tem<lb></lb>porum & ex ea velocitatum ſummarum, ſeu earum, quæ <expan abbr="sũt">sunt</expan> <lb></lb>ad inſtantia ſimiliter ſumpta in rectis BE, GI, constat ex <lb></lb>lem: infra cor.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius tempora componi ex rationi<lb></lb>bus ſpatiorum ſimilium motuum, & ex recìproca dictarum <emph.end type="italics"></emph.end><pb pagenum="18" xlink:href="022/01/024.jpg"></pb><emph type="italics"></emph>velocitatum. </s> <s id="s.000206">Ex eadem ratione patet eſſe velocitates ſum<lb></lb>mas, vel homologas vti diximus in ratione compoſita dicto<lb></lb>rum ſpatiorum, & ipſorum temporum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000207"><emph type="center"></emph><emph type="italics"></emph>Corollarium III.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000208"><emph type="italics"></emph>Quare ſi alteræ de dua<gap></gap>bus componentibus æqualis fuerit, <lb></lb>reliqua tantùm computanda erit.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000209"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000210"><emph type="italics"></emph>Hinc emergit omnis ferè doctrina grauium cum <expan abbr="deſcendũt">deſcendunt</expan> <lb></lb>prorſus libera, aut ſuper planis inclinatis ad horizontem̨: <lb></lb>nec accidit veritates iam patefactas huc rurſus lectoris taedio <lb></lb>afferre, ſed libeat potius, rationem metiendarum imaginum, <lb></lb>quamuis longitudine immenſarum, noſtra methodo exponere.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000211"><emph type="center"></emph>DEF. VIII.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg42"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000212"><margin.target id="marg42"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 6.</s> </p> <p type="main"> <s id="s.000213">SInt inter binas parallelas AB, GH, et IK, PQ planæ fi<lb></lb>guræ ABHG, IKQP, & in altera earum ducta altitudi<lb></lb>ne RV, ſint inter ſe ipſæ figuræ talis naturæ, vt cum ſit <lb></lb>GABH ad ſegmentum EABF factum per æquidiſtantem <lb></lb>ipſi GH ſicut VR ad RT, verificetur ſemper (ducta æqui<lb></lb>diſtanti NTO ipſi PQ) eſſe GH ad EF vt reciprocè NO ad <lb></lb>PQ tunc huiuſmodi figuras vocabimus inter ſe auuerſas. </s> </p> <p type="main"> <s id="s.000214"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000215"><emph type="italics"></emph>Sequitur ex vi nunc allatæ deffin., lineam IK tunc eſſe in<lb></lb>finitam, cum AB fuerit punctum, & ideo ſimul conſtat figu<lb></lb>ram IPQK immenſam eſſe longitudine versùs K aut I, aut <lb></lb>vtrinque, ſi nempe producerentur nunquam coituræ lineæ <lb></lb>QP, IK.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="19" xlink:href="022/01/025.jpg"></pb> <p type="main"> <s id="s.000216"><emph type="center"></emph>PROP. IX. THEOR. IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000217">REctangulum ſub altitudine, & baſi vnius auuerſarum <lb></lb>ad ipſam auuerſam figuram, eandem habet <expan abbr="rationẽ">rationem</expan>, <lb></lb>ac altera auuerſa figura ad rectangulum ex baſi in altitudi<arrow.to.target n="marg43"></arrow.to.target><lb></lb>nem eiuſdem huius figuræ. </s> </p> <p type="margin"> <s id="s.000219"><margin.target id="marg43"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> <gap></gap>. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="main"> <s id="s.000220">Sint auuerſæ figuræ ACB, GFDEG. </s> <s id="s.000221">Dico rectangu<lb></lb>lum DF in DE ad figuram GFDEG, eandem habere ratio<lb></lb>nem ac figura ACBA ad rectangulum AB in BC. </s> <s id="s.000222">Sint pri<lb></lb>mùm ABC, FDE anguli recti, & ducta qualibet HI paral<lb></lb><arrow.to.target n="marg44"></arrow.to.target><lb></lb>lela BC, ſit BAC ad HIA vt DF ad KF, erit ob naturam <lb></lb>auuerſarum KL ad DE vt BC ad HI; itaque ſi ponatur eſſe <lb></lb>quidam motus ab F in D iuxta imaginem <expan abbr="velocitatũ">velocitatum</expan> BAC, <lb></lb><arrow.to.target n="marg45"></arrow.to.target><lb></lb>erit GFDEG imago temporis eiuſdem motus; nam imago <lb></lb><arrow.to.target n="marg46"></arrow.to.target><lb></lb>BAC ad imaginem HIA eſt vt ſpatium DF ad ſpatium FK <lb></lb>& velocitas BC ad <expan abbr="velocitatẽ">velocitatem</expan> HI vt reciprocè KL ad DE. <lb></lb></s> <s id="s.000223">Sit etiam alius motus, ſed æquabilis, cuius imago velocita<lb></lb>tum æqualis ſit, & homogenea ipſi BAC, rectangulum <expan abbr="nẽ-pe">nen<lb></lb>pe</expan> AB in BM, & ideo ſi fiat BM ad BC ſicut DE ad DN, <lb></lb>concipiaturque rectangulum FD in DN, erit hoc imago <lb></lb><arrow.to.target n="marg47"></arrow.to.target><lb></lb>temporis dicti motus æquabilis, homogenea, & æqualis <lb></lb>imagini GFDEG; nam <expan abbr="tẽpora">tempora</expan>, ſcilicet imagines GFDEG, <lb></lb><arrow.to.target n="marg48"></arrow.to.target><lb></lb>FD in DN rectangulum componuntur ex rationibus ſpa<lb></lb><arrow.to.target n="marg49"></arrow.to.target><lb></lb>tiorum, hoc eſt imaginum velocitatum interſe æqualium, <lb></lb>ABM, ACB, & reciproca æquatricum pariter æqualium <lb></lb>BM, BM. </s> <s id="s.000224">Cum igitur rectangulum FD in DN æquale ſit <lb></lb><arrow.to.target n="marg50"></arrow.to.target><lb></lb>imagini, ſeu figuræ GFDEG, habebit eadem figurą <lb></lb>GFDEG ad rectangulum FD in DE eandem rationem, <lb></lb>quam DN ad DE, hoc eſt quam BC ad BM, ſeu quam re<lb></lb>ctangulum AB in BC ad rectangulum AB in BM, aut ad ei <lb></lb>æqualem figuram ABC; & conuertendo, manifeſtum eſt <lb></lb>quod propoſuimus, nempe rectangulum FD in DE ad fi<lb></lb>guram GFDEG habere eandem <expan abbr="rationẽ">rationem</expan>, ac figura ACBA <pb pagenum="20" xlink:href="022/01/026.jpg"></pb>ad rectangulum AB in BC. quod erat demonſtrandum <lb></lb>primo loco. </s> </p> <p type="margin"> <s id="s.000225"><margin.target id="marg44"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000226"><margin.target id="marg45"></margin.target><emph type="italics"></emph>Def:<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000227"><margin.target id="marg46"></margin.target><emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000228"><margin.target id="marg47"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000229"><margin.target id="marg48"></margin.target><emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000230"><margin.target id="marg49"></margin.target><emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000231"><margin.target id="marg50"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000232">pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000233">2. Si verò propoſitæ figuræ ſint quæcunque auuerſæ <lb></lb><arrow.to.target n="marg51"></arrow.to.target><lb></lb>DAE, QPLMQ poterunt hæ reuocari ad quaſdam alias <lb></lb>FKG, RSZX, quæ ſint inter eaſdem parallelas, queis com<lb></lb>prehenduntur propoſitæ figuræ, ad eo vt exiſtentibus re<lb></lb>ctis angulis KFG, RXZ ſint ipſæ binæ figuræ ab ijſdem pa<lb></lb>rallelis interceptæ. </s> <s id="s.000234">inter ſe æqualiter analogæ hoc eſt du<lb></lb>ctis æquidiſtantibus, vt viſum fuerit IHBC, VTNO, ſint <lb></lb>ſemper interiectæ lineæ IH, BC, & VT, NO æquales: hoc <lb></lb>modo non tantùm liquet figuras FKG, DAE, nec noņ <lb></lb>RSZX, PQML æquales inter ſe eſſe, verùm etiam FKG ad <lb></lb>IKH eſſe in eadem ratione, in qua QPLMQ ad QPNOQ, <lb></lb>quamobrem ex prima parte, rectangulum ZX in RM ad <lb></lb>figuram SRXZS, hoc eſt rectangulum LM in altitudinem <lb></lb>figuræ QPLMQ ad hanc ipſam figuram habebit eandem <lb></lb>rationem, quam figura FKG ad rectangulum KF in FG, <lb></lb>vel quam figura DAE ad rectangulum DE in altitudinem <lb></lb>eiuſdem huius figuræ DAE; quo circa conſtat omne pro<lb></lb>poſitum. </s> </p> <p type="margin"> <s id="s.000235"><margin.target id="marg51"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 8.</s> </p> <p type="main"> <s id="s.000236"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000237"><emph type="italics"></emph>Patet in prima parte repertum eſſe rectangulum FD iņ<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg52"></arrow.to.target><lb></lb><emph type="italics"></emph>DN æquale figuræ GFDEG, licèt hæc immenſe longitudinis <lb></lb>ſit versùs G, & ob id manifeſtum eſt, quòd quamuis aliquą <lb></lb>figura ſit ſinè fiue longa, non ideo ſemper magnitudinem ha<lb></lb>bet infinitam. </s> <s id="s.000238">Et ſimul illud conſtat, vbi vna auuerſarum, ſeu <lb></lb>vbi imago velocitatum, aut temporis ſit magnitudine termi<lb></lb>nata, etiam altera auuerſarum, vel imaginum erit huiuſ<lb></lb>modi &c.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="21" xlink:href="022/01/027.jpg"></pb> <p type="margin"> <s id="s.000239"><margin.target id="marg52"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000240">pr.<emph.end type="italics"></emph.end> 18. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000241"><emph type="center"></emph>PROP. X. THEOR. X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000242">IN quouis parallelogrammo BD ſint deinceps diagona<lb></lb><arrow.to.target n="marg53"></arrow.to.target><lb></lb>les AGC, AHC, AIC, ALC, aliæque numerò infinitæ, <lb></lb>ita vt acta quælibet recta EF parallela BA <expan abbr="ãs">ſecans</expan> ipſas dia<lb></lb>gonales in punctis G, L, H, I, ſit ſemper DA ad AF, vt CD, <lb></lb>aut EF ad FG; quadratum ex DA ad quadratum AF vt <lb></lb>EF ad FH; cubus ex DA ad cubum ex AF vt EF ad FI; <lb></lb>quadroquadratum ex DA ad quadroquadratum ex AF <lb></lb>vt EF ad FL; & ſic continuò procedendo per infinitas ex <lb></lb>ordine poteſtates: Stephanus de Angelis Author ſubtilis, <lb></lb>ac celeberrimus, libro ſuo infin. parabolarum vocat trian<lb></lb>gulum rectilineum ABC parabolam primam, BAHC ſe<lb></lb>cundam; tertiam BAIC, quartam BALC, & ita in infini<lb></lb>tum: His definitis docet ex Cauallerio parallelogrammum <lb></lb>BD ad quancunque dictarum parabolarum ſibi inſcripta<lb></lb>rum eſſe vt numerus, vel exponens parabolæ vnitate au<lb></lb>ctus ad ipſum exponentem, ſiue numerum parabolę, qua<lb></lb>re ad primam habebit ipſum parallelogrammum eandem <lb></lb>rationem, ac 2 ad 1; ad ſecundam vt 3 ad 2; ad tertiam vt <lb></lb>4 ad 3, & ita deinceps de reliquis; itaque per conuerſio<lb></lb>nem rationis habebit ipſum parallelogrammum ad exceſ<lb></lb>ſum illius ſupra quancunque parabolarum dictarum, ſcili<lb></lb>cet ad trilineum primum AGCD eandem rationem, quam <lb></lb>2 ad 1, ad ſecundum quam 3 ad 1, & ſic deinceps quam <lb></lb>numerus trilinei vnitate auctus ad ipſam vnitatem. </s> <s id="s.000244">Sed <lb></lb>eſt etiam admonendum verticem dictarum parabolarum <lb></lb>eſſe punctum A, & per conſequens AB diametrum, & BC <lb></lb>ordinatim aplicatam, ſeu baſim. </s> </p> <pb pagenum="22" xlink:href="022/01/028.jpg"></pb> <p type="margin"> <s id="s.000245"><margin.target id="marg53"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 9.</s> </p> <p type="main"> <s id="s.000246"><emph type="center"></emph>PROP. XI. THEOR. XI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000247">IIſdem adhuc manentibus, idem de Angelis monſtrat eo<lb></lb>dem illo tractatu pr. 3. ſi quæcunque ex dictis parabo<lb></lb>lis ſecta ſit qualibet recta parallela baſi BC, eſſe parabolam <lb></lb>ad reſectam portionem verſus verticem, vt poteſtas baſis, <lb></lb>cuius exponens eſt numerus parabolæ vnitate auctus ad <lb></lb>ſimilem poteſtatem ex baſi reſectæ portionis; itaque iņ <lb></lb>prima parabola eſt vt quadratum ad quadratum, in ſecun<lb></lb>da vt cubus ad cubum, & ſic de cæteris. </s> <s id="s.000248">Similiter ſi ſece<lb></lb>tur quodlibet ex infinitis trilineis linea GF baſi CD paral<lb></lb>lela, erit trilineum ad ſuperius ſui ſegmentum vt poteſtas <lb></lb>ex DA, cuius exponens eſt numerus trilinei vnitate auctus <lb></lb>ad ſimilem poteſtatem ex AF. quare trilineum primum̨ <lb></lb>CAD ad GAF erit vt quadratum ex DA ad quadratum <lb></lb>ex FA, ſecundum CHAD ad ſegmentum HAF vt cubus <lb></lb>ad cubum, & ita in cæteris eodem ordine. </s> </p> <p type="main"> <s id="s.000249"><emph type="center"></emph>PROP. XII. THEOR. XII.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg54"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000250"><margin.target id="marg54"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="main"> <s id="s.000251">SIt modò ACD angulus rectus, & linea FE talis naturæ, <lb></lb>vt deductis ad libitum rectis AF, BE parallelis ipſi <lb></lb>CD, poteſtas ex CA ad ſimilem poteſtatem ex CB ſit reci<lb></lb>procè vt alia quædam poteſtas ex BE ad ſimilem huic po<lb></lb>teſtatem ex AF; patet rectas CA, CD nondum iungi cum <lb></lb>EF, quamuis in immenſum vnà producerentur. </s> <s id="s.000252">Ab hoc <lb></lb>proprietate VValliſius & Fermatius ſubtiliſſimi authores <lb></lb>vocauerunt curuam FE nouam hyperbolam, & eius aſ<lb></lb>ſymptotos AC, CD. </s> <s id="s.000253">Omnes huiuſmodi hyperbolæ, quæ <lb></lb>infinitæ numero ſunt, terminantur ad vnam partem ma<lb></lb>gnitudine, cum hyperbola <expan abbr="cõmunis">communis</expan>, ſeu Apolloniaca ſit in <lb></lb>vtranque partem magnitudine infinita. </s> <s id="s.000254">Quod ergo exi<lb></lb>mium eſt, oſtenderunt ipſi authores rectangulum FA iņ <pb pagenum="23" xlink:href="022/01/029.jpg"></pb>AC ad ſpatium hyperbolicum quà finitum eſt, licèt ſinè <lb></lb>fine longum, eandem habere rationem, quam differentia <lb></lb>exponentium poteſtatum hyperbolæ ad exponentem po<lb></lb>teſtatis minoris. </s> <s id="s.000255">Quare ſi in hyperbola ſit vt cubus CB <lb></lb>ad cubum CA ita quadratum AF ad quadratum BE, erit <lb></lb>prædictum rectangulum CA in AF dimidium Spatij ſinè <lb></lb>fine producti A & FA; at ſi quadratum CB ad quadratum <lb></lb>CA ſit vt recta AF ad rectam BE, rectangulum ipſum CA <lb></lb>in AF æquale erit ſpatio A & FA, quòd ſi poteſtas CA vel <lb></lb>CB non fuerit altior poteſtate ex BE, vel AF, tunc ipſum <lb></lb>illud ſpatium, infinitum quoque erit magnitudine, etenim <lb></lb>nullus exceſſus exponentis prædictæ poteſtatis ex CA ſu<lb></lb>pra exponentem poteſtatis BE, habet ad numerum expo<lb></lb>nentis poteſtatis BE rationem infinitam. </s> </p> <p type="main"> <s id="s.000256"><emph type="center"></emph>DEMONSTRATIO.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000257">SVpradictum propoſitum habetur in commercio epi<lb></lb>ſtolico Ioannis Valliſij Epiſtola quarta, quem libellum <lb></lb>vnà cum alijs doctiſſimis ſuis operibus Vincentius Viuia<lb></lb>nus ingens æui noſtri Geometra, antequam ſumma cum̨ <lb></lb>humanitate miſiſſet, eidem ipſi quadraturam vnius ex di<lb></lb>ctis hyperbolis ex noſtris principijs deductam, ac excogi<lb></lb>tatam, indicauimus. </s> <s id="s.000258">Cum verò poſtea nobis eueniſſet <lb></lb>vniuerſaliorem ad alias hyperbolas (ſemper communi ex<lb></lb>cepta) accomodatam reperijſſe, huc debemus afferre, pri<lb></lb>mùm vt quendam fructum ſcientiæ huius; deinde cum di<lb></lb>ctorum authorum ipſam propoſitionis demonſtrationem <lb></lb>non habuerimus, & demum quia ipſarum hyperbolarum <lb></lb>menſura, ac quadratura in aquarum rationibus erunt po<lb></lb>tiſſimum ex vſu. </s> <s id="s.000259">Sit igitur BC vna ex infinitis hyperbolis, <arrow.to.target n="marg55"></arrow.to.target><lb></lb>quarum aſſymptoti AE, EL; Sint etiam quæcunque apli<lb></lb>catæ AB, DC aſsymptoto EL æquidiſtantes, & habeat <lb></lb>DE ad EA eandem rationem v. g. </s> <s id="s.000261">quam cubus ex AB ad <pb pagenum="24" xlink:href="022/01/030.jpg"></pb><arrow.to.target n="marg56"></arrow.to.target><lb></lb>cubum DC. </s> <s id="s.000262">Patet ſi proponeretur illi auuerſa figurą <lb></lb><arrow.to.target n="marg57"></arrow.to.target><lb></lb>FGK, eſſetque AE ad DE vt figura GFK ad figuram IHK <lb></lb>eſſe etiam FG ad IH vt DC ad AB, eſt autem cubus ex <lb></lb>DC ad cubum ex AB vt AE ad ED; ergo etiam figurą <lb></lb>FGK ad IHK (ſunt enim FG, IH parallelę) habebit ean<lb></lb>dem rationem, ac cubus ex FG ad cubum ex IH: Itaquę <lb></lb>GFK erit comunis parabola, hoc eſt quadratica, ſeu <expan abbr="ſecũ-">ſecun<lb></lb></expan><arrow.to.target n="marg58"></arrow.to.target><lb></lb>da in ſerie infinitarum parabolarum, & ob id eadem GFK <lb></lb><arrow.to.target n="marg59"></arrow.to.target><lb></lb>parabola ad rectangulum GF in FK erit vt 2 ad 3, in qua <lb></lb>ratione ſe habebit quoque rectangulum BA in AE ad ſpa<lb></lb>tium infinitè longum & BM, et erit vt 2 ad 1; ſcilicet vt ex<lb></lb>ceſſus exponentis maioris poteſtatis, quæ cubica eſt, ſuper <lb></lb>numerum exponentis, qui hoc caſu eſt tantùm vnitas ra<lb></lb>dicis, eſt ad hunc ipſum exponentem, ſeu vnitatem lineæ <lb></lb>indicantem, quod concordat cum propoſita dictorum̨ <lb></lb>authorum. </s> </p> <p type="margin"> <s id="s.000263"><margin.target id="marg55"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="margin"> <s id="s.000264"><margin.target id="marg56"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>fig<emph.end type="italics"></emph.end> 2.</s> </p> <p type="margin"> <s id="s.000265"><margin.target id="marg57"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000266"><margin.target id="marg58"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000267"><margin.target id="marg59"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000268"><emph type="center"></emph><emph type="italics"></emph>Exemplum aliud.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg60"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000269"><margin.target id="marg60"></margin.target><emph type="italics"></emph>In eadem fi<lb></lb>guræ.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000270">SIt etiam cubus ex DE ad cubum ex AE, ſicut quadra<lb></lb>to quadratum AB ad quadroquadratum DC, & rur<lb></lb>ſus propoſita GKF auerſa huius hyperbolæ: patet ſi ſit AE <lb></lb>ad DE vt figura GFK ad figuram IKH, eſſe etiam FG ad </s> </p> <p type="main"> <s id="s.000271"><arrow.to.target n="marg61"></arrow.to.target><lb></lb>IH vt DC ad AB; cumque ſit cubus ex AE ad cubum ex <lb></lb>DE ſicut quadroquadratum ex DC ad <expan abbr="quadroquadratũ">quadroquadratum</expan> <lb></lb>ex AB, erit etiam quadroquadratum ex FG ad quadro<lb></lb>quadratum ex IH, vt cubus ex AE ad cubum ex DE; ſi <lb></lb>igitur intelligatur quædam ratio, quæ ſit ſubduodecupla <lb></lb>tam rationis quadroquadratorum quàm huic ſimilis cu<lb></lb>borum prædictorum, erit porrò FG ad IH triplicata, & <lb></lb>AE ad ED quadruplicata eiuſdem dictæ ſubduodecuplæ; <lb></lb>quamobrem etiam ratio figuræ GFK ad <expan abbr="figurã">figuram</expan> IHK, quæ <lb></lb>eſſe debet vt AE ad ED, erit quadruplicata eiuſdem ſub<lb></lb>duodecuplæ: & ideò ſi ponamus IK ad KI in ratione <pb pagenum="25" xlink:href="022/01/031.jpg"></pb>eiuſdem ſubduodecuplæ, erit figura GFK illius naturæ, vt <lb></lb><arrow.to.target n="marg62"></arrow.to.target><lb></lb>ſit ſemper cubus ex FK ad cubum ex KI ſicut GF ad IH, & <lb></lb>hoc modo eadem illa figura erit trilineum tertium, ſeu cu<lb></lb>bicum, ex quo ergo ſequitur, GFK ad HIK ſit in eadem ra<lb></lb>tione, in qua quadroquadratum ex FK ad quadroqua<lb></lb>dratum ex KI, hoc eſt ſit vt AE ad ED; ſequiturque etiam <lb></lb><arrow.to.target n="marg63"></arrow.to.target><lb></lb>ob hoc figuram GFK ſubquadruplam eſle circumſcripti <lb></lb>rectanguli GF in FK; eſt autem vt trilineum GFK ad <expan abbr="rectã-">rectan<lb></lb></expan><arrow.to.target n="marg64"></arrow.to.target><lb></lb>gulum GF in FK circumſcriptum, ſic rectangulum ABME <lb></lb>ad auuerſam eidem trilineo figuram AB & EA, ergo re<lb></lb>ctangulum ABME ſubquadruplum erit eiuſdem figuræ <lb></lb>AB & EA longitudinis infinitæ, quare ipſum rectangulum <lb></lb>erit ſubtriplum portionis & BM & longitudinis pariter im<lb></lb>menſæ. </s> <s id="s.000272">Cum ita ſit, conſtat exemplo hoc quoque, <expan abbr="eandẽ">eandem</expan> <lb></lb>illam rationem eſſe exceſſum maioris exponentis ſuprą <lb></lb>minorem exponentem ad hoc ipſum, dictarum <expan abbr="poteſtatũ">poteſtatum</expan> <lb></lb>hyperbolæ. </s> </p> <p type="margin"> <s id="s.000273"><margin.target id="marg61"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000274"><margin.target id="marg62"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000275"><margin.target id="marg63"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000276"><margin.target id="marg64"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000277"><emph type="center"></emph>PROP. XIII. THEOR. XIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000278">SVperior demonſtratio effecta fuiſſet ampliſſima, ſi prę<lb></lb>ponere voluiſſemus <expan abbr="quadraturã">quadraturam</expan> vt datam omnis ge<lb></lb>neris parabolarum, & trilineorum, verùm cum iſta pars <expan abbr="nõ">non</expan> <lb></lb>ſit plenè tradita, vt videre eſt quinto libro infinitarum pa<lb></lb>rabolarum eiuſdem de Angelis, ſatius ideo duximus qua<lb></lb>draturam hyperbolarum à VValiſio, & Fermatio acutiſſi<lb></lb>mis illis viris propoſitam omnino veram admittere, vt indè <lb></lb>eam parabolarum & trilineorum vniuerſalem, quam adhuc <lb></lb>ab alijs non habemus, facillimè, compendiosèque depro<lb></lb>meremus. </s> <s id="s.000279">Hanc igitur ita proponimus vt ſubinde oſten<lb></lb>damus. </s> </p> <p type="main"> <s id="s.000280">Si ſimiles poteſtates applicatarum fuerint in eadem ra<lb></lb>tione, ac ſunt interſe poteſtates quædam aliæ, & eiuſdem <lb></lb>gradus diametrorum ab ipſis applicatis abſciſſarum vſque <pb pagenum="26" xlink:href="022/01/032.jpg"></pb>ad verticem parabolarum, vel trilineorum; erit rectangu<lb></lb>lum ad parabolam ſibi inſcriptam vt aggregatum <expan abbr="exponẽ-tium">exponen<lb></lb>tium</expan> vtriuſque poteſtatis ad exponentem altioris ipſarum <lb></lb>poteſtatum parabolæ; & ad trilineum vt aggregatum ex<lb></lb>ponentium poteſtatum trilinei ad exponentem inferioris <lb></lb>poteſtatis eiuſdemmet trilinei. </s> <s id="s.000281">Sic enim in expoſita figu<lb></lb>ra prædicta, ſi eſſet quadratum ex FG ad quadratum ex <lb></lb>IH, ſicut cubus ex FK ad cubum ex IH, eſſet rectangulum <lb></lb>GF in FK ad figuram GFK (quæ tunc foret trilineum, vt <lb></lb>5 ad 2; nam vbi poteſtas abſciſſarum maior eſt illa applica. <lb></lb></s> <s id="s.000282">tarum eſt ſemper GF trilineum. </s> <s id="s.000283">Simili modo, ſi ſit vt qua<lb></lb>dratum ex FK ad quadratum ex KI ita cubocubus ex FG <lb></lb>ad cubocubum ex IH; hoc eſt ſi ſit cubus ex FG ad <expan abbr="cubũ">cubum</expan> <lb></lb>ex IH, vt linea FK ad KI (tolluntur enim vtrinque ex ſimi<lb></lb>libus ſimiles rationes) erit ſigura GFK parabola, ad quam <lb></lb>ſibi circumſcriptum rectangulum eandem habebit <expan abbr="rationẽ">rationem</expan>, <lb></lb>quam 4 ad 3, & ſic dicendum erit de omnibus alijs para<lb></lb>bolis atque trilineis. </s> </p> <p type="main"> <s id="s.000284"><emph type="center"></emph>DEMONSTRATIO.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000285">VErùm vt propoſitum oſtendamus, eſto quælibet ex <lb></lb>parabolis GFK, nimirum quadratocubus ex FG ad <lb></lb>quadratocubum ex IH habeat eandem rationem, quam̨ <lb></lb>cubus ex FK ad cubum ex IK. Demonſtro, rectangulum <lb></lb>GF in FK habere eandem rationem ad parabolam GFK, <lb></lb>quam aggregatum exponentium 8 ad maiorem exponen<lb></lb>tem 5. Primùm, quam rationem habet rectangulum GF in <lb></lb>FK ad parabolam GFK, eandem habebit rectangulum HI <lb></lb>in IK ad parabolam HIK (hoc enim demonſtrabimus in<lb></lb>frà) permutandoque, erit rectangulum GF in FK ad re<lb></lb>ctangulum HI in IK, vt parabola GFK ad parabolam HIK; <lb></lb>componuntur verò illa rectangula ex rationibus GF ad <lb></lb>IH, & FK ad IK, ergo etiam parabola ad parabolam com-<pb pagenum="27" xlink:href="022/01/033.jpg"></pb>ponetur ex ijſdem rationibus; & quoniam ductis inuicem <lb></lb>exponentibus poſſunt conſiderari quindecim rationes in<lb></lb>ter ſe ſimiles, ex quibus conſtet tam ratio dictorum cubo<lb></lb>rum, quàm huic ſimilis altera quadratocuborum, & tunc <lb></lb>GF ad IH erit triplicata, et FK ad KI quintuplicata <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb>ſubquindecuplæ rationis, quæ ſit A ad B; ergo ſimul ad<lb></lb>ditis ijſdem rationibus, quintuplicata ſcilicet, & triplicata <lb></lb>exiliet ratio octuplicata ipſius A ad B; proptereaque pa<lb></lb>rabola GFK ad HIK, ſeu ſi conſideremus figuram & BAEL <lb></lb>auuerſam parabolæ GFK, ita vt AE ad ED ſit vt para<lb></lb><arrow.to.target n="marg65"></arrow.to.target><lb></lb>bola GFK ad <expan abbr="parabolã">parabolam</expan> HIK; AE ad ED erit pariter octu<lb></lb>plicata eiuſdem A ad B; & cum ſit ob naturam <expan abbr="auuerſarũ">auuerſarum</expan> <lb></lb>FG ad HI vt DC ad AB; erit DC ad AB triplicata <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb>rationis A ad B, qnare vt cubus AE ad cubum DE, itą <lb></lb>quadratocubocubus DC ad quadratocubocubum ex <lb></lb>AB: rectangulum igitur ABME ad ſpatium hyperbolicum <lb></lb>infin<gap></gap> è longum & BM & erit vt quinque ad tria, & ad vni<lb></lb><arrow.to.target n="marg66"></arrow.to.target><lb></lb>uerſum ſpatium & BAE & vt 5 ad 8, in qua nempe ratio<lb></lb>ne debet eſſe parabola GF<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ad rectangulum GF in FK. <lb></lb><arrow.to.target n="marg67"></arrow.to.target><lb></lb>Quod &c. </s> </p> <p type="margin"> <s id="s.000286"><margin.target id="marg65"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000287"><margin.target id="marg66"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000288"><margin.target id="marg67"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000289"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000290"><emph type="italics"></emph>Conſtat ſi fuerit ratio A ad B eò ſubmultiplicata rationis <lb></lb>applicatarum, quoties eſt numerus exponentis poteſtatis ab<lb></lb>ſciſſarum eiuſdem parabolæ, eſſe ipſam parabolam ad ſui por<lb></lb>tionem in tam multiplicata ratione A ad B, ac eſt numerus <lb></lb>aggregati exponentium ambarum poteſtatum parabola. </s> <s id="s.000291">Nam <lb></lb>cum eſſet quadratocubus ex FG ad quadratocubum ex IH, ſi<lb></lb>cut cubus ex FK ad cubum ex IK, propoſita inſuper eſſet A ad <lb></lb>B. ſubquindecupla alterius dictarum ſimilium rationum ex <lb></lb>poteſt atibus parabola, oſtenſum fuit rationem A ad B ſubtri<lb></lb>plicatam ipſius GF ad IH, & ſubquintuplicatam alterius FK <lb></lb>ad KI, & tandem oſtendimus parabolam GFK ad portionem <emph.end type="italics"></emph.end><pb pagenum="28" xlink:href="022/01/034.jpg"></pb><emph type="italics"></emph>eius HIK eße in octuplicata ratione eiuſdem A ad B; quod <lb></lb>idem omnino diceretur ſi figura GFK trilineum eſſet. </s> <s id="s.000292">Ratio <lb></lb>autem A ad B dicetur impoſterum logarithmica poteſtatum <lb></lb>parabolæ, ſeu trilinei, aut hyperbolæ.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000293"><emph type="center"></emph>ASSVMPTVM.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000294">REliquum eſt vt oſtendamus, parabolam GFK ad <lb></lb>portionem HIK eſſe vt rectangulum GF ad rectan<lb></lb>gulum HI in IK, ſcilicet eſſe in ratione compoſita baſium, <lb></lb>& altitudinum parabolarum, quod nempe ſic oſtendetur, <lb></lb>Sit vt ſupra FGK parabola, eiuſque portio IHK; exiſtenti<lb></lb>bus verò applicatis FG, IH, fiat EG ad IE vt FK ad KI, ſit<lb></lb><arrow.to.target n="marg68"></arrow.to.target><lb></lb>que IE baſis, et K vertex parabolę IEK ſimilis ipſi GFK pa<lb></lb>tet propter ſimilitudinem figurarum, eſſe parabolam GFK <lb></lb>ad parabolam IEK in eadem duplicata ratione FG ad IE, <lb></lb>in qua nempe eſt rectangulum GF in FK ad ſibi ſimile re<lb></lb>ctangulum EI in IK, ob idque rectangulum GF in FK ad <lb></lb>rectangulum EI in IK, cum ſint interſe vt parabola GFK ad <lb></lb>parabolam EIK, hæc verò parabola ad ipſam IHK habeat <lb></lb>eandem rationem, ac IE ad IH; ſeu ob eandem altitudinem <lb></lb>IK vt rectangulum EI in IK ad rectangulum HI in IK, erit <lb></lb>ex æquali parabola GFK ad parabolam HIK vt rectangu<lb></lb>lum GF in FK ad rectangulum HI in IK. </s> <s id="s.000295">Quod &c. </s> </p> <p type="margin"> <s id="s.000296"><margin.target id="marg68"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="main"> <s id="s.000297"><emph type="center"></emph>PROP. XIV. THEOR. XIV.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg69"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000298"><margin.target id="marg69"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000299">IN quacunque hyperbola (excepta ſemper conica) cu<lb></lb>ius aſſymptoti EA, EM, ſi ſit poteſtas applicatarum DC <lb></lb>AB altior poteſtate abſciſſarum AE, ED (ſic enim finitą <lb></lb>erit magnitudine ſecundum eam aſſymptoton, quæ appli<lb></lb>catis parallela eſt) ſpatium ipſum hyperbolæ & BAE & <lb></lb>ad ſui portionem & CDE & habebit eandem rationem, ac <lb></lb>rectangulum BAE ad rectangulum CDE, ſeu (aſſumpta <pb pagenum="29" xlink:href="022/01/035.jpg"></pb>ratione logarithmica A ad B poteſtatum hyperbolæ) <expan abbr="quã">quam</expan> <lb></lb>poteſtas ex A, cuius exponens eſt differentia <expan abbr="exponentiũ">exponentium</expan> <lb></lb>poteſtatum hyperbolæ ad ſimilem poteſtatem ex B. </s> </p> <p type="main"> <s id="s.000300"><emph type="center"></emph>DEMONSTRATIO.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000301">QVam rationem habet rectangulum BAE ad ſpatium <lb></lb>& BAE &, eandem habet rectangulum CDE ad </s> </p> <p type="main"> <s id="s.000302"><arrow.to.target n="marg70"></arrow.to.target><lb></lb>ſpatium & CDE, & permutando erit rectangu<lb></lb>lum BAE ad CDE, ſicut ſpatium & BAE & ad ſpatium̨ <lb></lb>& CDE &; ſi igitur in eadem propoſita hyperbola ſit po<lb></lb>teſtas applicatarum DC, AB quintuplicata ipſius A ad B, <lb></lb>& AE ad ED ſeptuplicata ſit eiuſdem; erit ſeptuplicatą <lb></lb>applicatarum in eadem ratione, ac quintuplicata abſciſſa<lb></lb>rum; ſcilicet quadratoquadratocubus ex DC ad ſimilem <lb></lb>poteſtatem ex AB erit vt quadratocubus ex AE ad qua<lb></lb>dratocubum ex DE, eritque ſic maior poteſtas applicata<lb></lb>rum, atque adeo componetur rectangulum EAB ad EDC <lb></lb>ex ſeptuplicata ipſius A ad B, qualis eſt AE ad ED, & ſub<lb></lb>quintuplicata eiuſdem A ad B, quæ eſt AB ad DC; nimi<lb></lb>rùm erit rectangulum EAB ad EDC in duplicata tantum <lb></lb>ratione ipſius A ad B: quare ſpatium & BAE & ad id <lb></lb>& CDE &, quæ ſunt inter ſe, vt ipſa rectangula, erit vt po<lb></lb>teſtas ex A, cuius exponens eſt differentia exponentium & <lb></lb>S poteſtatum hyperbolæ ad ſimilem poteſtatem ex B. <lb></lb></s> <s id="s.000303">Quod &c. </s> </p> <p type="margin"> <s id="s.000304"><margin.target id="marg70"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000305"><emph type="center"></emph>PROP. XV. THEOR. XV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000306">SI ab exponente poteſtatis applicatarum hyperbolę de<lb></lb>trahatur exponens minoris poteſtatis abſciſſarum, po<lb></lb>teſtas reliqui exponetis erit applicatarum auuerſæ figuræ, <lb></lb>in abſciſſis verò adeſt vtrobique eadem poteſtas. </s> <s id="s.000307">Itaque <lb></lb>cum in ſuperiori hyperbola reſidui exponentis poteſtas <pb pagenum="30" xlink:href="022/01/036.jpg"></pb>quadratum eſſet, porrò in eius auuerſa eſſet poteſtas appli<lb></lb>catarum quadratica, & abſciſſarum quadratocubica. </s> </p> <p type="main"> <s id="s.000308"><emph type="center"></emph>DEMONSTRATIO.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg71"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000309"><margin.target id="marg71"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000310">ESto rurſus hyperbola & BAE &, et ſicut dictum eſt <lb></lb>AE ad ED ſit in ſeptuplicata ratione logarithmicæ <lb></lb>rationis A ad B, at DC ad AB in quintuplicata, videlicet <lb></lb>quadratocubus ex AE ad quadratocubum ex DE eandem <lb></lb>habeat rationem, ac quadratoquadratocubus ex DC ad <lb></lb>ſimilem poteſtatem ex AB; Dico in auuerſa figura poteſta<lb></lb>tem aplicatarum eſſe quadratum, cuius <expan abbr="exponẽs">exponens</expan> 2 eſt dif<lb></lb>ferentia exponentium poteſtatum hyperbolæ; poteſtatem <lb></lb>verò abſciſſarum eandem eſſe, abſciſſarum eiuſdem hyper<lb></lb>bolæ. </s> <s id="s.000311">Sit vt ſupra FK ad KI vt hyperbola & BAE & ad <lb></lb>& CDE &, hoc eſt, ſit vt poteſtas ex A, cuius exponens </s> </p> <p type="main"> <s id="s.000312"><arrow.to.target n="marg72"></arrow.to.target><lb></lb>eſt differentia exponentium poteſtatum hyperbolæ ad ſi<lb></lb>milem poteſtatem ex B, & ideo FK ad KI erit duplicata ip<lb></lb>ſius A ad B, ſed DC ad AB eiuſdem illius logarithmicæ <lb></lb>quintuplicata; eſtque in hac eadem ratione etiam GF ad <lb></lb>IH; ergo cum duplicata huius ſit ſimilis quintuplicatæ KF <lb></lb>ad KI (nam vtraque ratio continet decies A ad B) pater, <lb></lb>quadratum ex FG ad quadratum ex IH eſſe eam poteſta<lb></lb>tem, quam propoſuimus euenire in applicatis auuerſæ, cum <lb></lb>aliàs in abſciſſis ſit vtrobique poteſtas eadem, nempe qua<lb></lb>dratocubi. </s> <s id="s.000313">Quod &c. </s> </p> <p type="margin"> <s id="s.000314"><margin.target id="marg72"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 14. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000315"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000316"><emph type="italics"></emph>Patet ex noto trilineo, vel parabola FGK eſſe in auuerſa, <lb></lb>ſcilicet in hyperbola & BAE & (quæ tunc eſt ſemper magnitu<lb></lb>dine finita iuxta aſsymptoton EM &) poteſtatem <expan abbr="applicatarũ">applicatarum</expan>, <lb></lb>qua pro exponente habet ſummam exponentium poteſtatum <lb></lb>parabolæ, aut trilinei; nam cum eßet in trilineo pracedenti<emph.end type="italics"></emph.end><pb pagenum="31" xlink:href="022/01/037.jpg"></pb><emph type="italics"></emph>quadratum ex FG ad quadratum ex IH vt quadratocubus <lb></lb>ex FK ad quadratocubum ex IK, fuit equidem in hyperbolą <lb></lb>quadratoquadratocubus ex DC<gap></gap> quadratoquadratocubum <lb></lb>ex AB ſicut quadratocubus ex AE ad ſimilem poteſtatem ex <lb></lb>DE, ſcilicet inuariata poteſtate abſerſarum in ambabus au<lb></lb>uerſis. </s> <s id="s.000317">Quare ex poteſtatibus notis vnius auuerſarum fa<lb></lb>cilè inoteſcent poteſtates alterius, atque etiam illius magnitu<lb></lb>do. </s> <s id="s.000318">Nunc redeamus ad motus, nouamque adhuc methodum, <lb></lb>quam hoc loco reſeruauimus, afferamus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000319"><emph type="center"></emph>DEF. IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000320">SIt quædam Geneſis ACBH, cuius imago temporis <lb></lb>& DCB &; item ſit FCBK geneſis alterius motus ab <lb></lb><arrow.to.target n="marg73"></arrow.to.target><lb></lb>eodem C in B; & actà rectà OIGE ipſi AFCD parallelą, <lb></lb>ponantur CD, GE loco minimorum temporum, ita vt <expan abbr="tẽ-pore">ten<lb></lb>pore</expan> CD, dum mobile ex C affectum velocitate CA, <lb></lb>currat minimum ſpatiolum indicatum per C, cui eſt æqua<lb></lb>le ſpatiolum aliud indicatum per G, quodque tranſigitur <lb></lb>tempore GE velocitate GD (nam vt eſſent illa ſpatia iņ <lb></lb>C, G æqualia, effectum fuit vt velocitas AC ad GD ean<lb></lb>dem reciprocè rationem haberet, ac tempus GE ad CD, <lb></lb>id quod patet ex natura geneſis ACBH, & imaginis & <lb></lb>DCB &) et hic rurſus notatu digniſſimum eſt nulli errori <lb></lb>obnoxium eſſe, quòd æquabiles in illis minimis ſpatiolis <lb></lb>intellexerimus motus, quamuis potius deberet videri, in <lb></lb>ijſdem interuallis reperiri innumeras, ac inæquales veloci<lb></lb>tates, queis nempe efficerentur motus inæquabiles, quòd <lb></lb>geneſes inæquabiles ſint. </s> <s id="s.000321">Cur iſta ſe ita habeant, hic non <lb></lb>eſt nobis diſputandum, ego enim puto, non ex indiuiſibili <lb></lb>velocitates alijs ſuccedere, ſed reuera minutulum tempo<lb></lb>ris conſiderari debere antequam motus diuerſimodè pro<lb></lb>cedat, nempe ac ſi velocitas, quæ ſuccedere debet priori, <lb></lb>non ita ſit in promptu, aut non ita ſtatim mobile afficiat ad <pb pagenum="32" xlink:href="022/01/038.jpg"></pb>motum ſibi proportionatum. </s> <s id="s.000322">Sed linquamus hæc alijs diſ<lb></lb>putanda: ſatis nobis ſit, methodum noſtram, quoad <expan abbr="noſtrũ">noſtrum</expan> <lb></lb>eſt, demonſtrare. </s> <s id="s.000323">Ijs igitur vt ſupra propoſitis, concipia<lb></lb>tur adhuc tempore CD velocitate FC <expan abbr="ſpatiũ">ſpatium</expan> exigi quod<lb></lb>dam, item aliud tempore EG, velocitateque GI, & ſic per <lb></lb>omnes quaſcunque applicatas: quæritur, quod ſpatium̨ <lb></lb>vltimò exactum eſſet, hoc eſt quam rationem id haberet ad <lb></lb>illud alterum ſpatium, quod eodem tempore tranſigitur <lb></lb>iuxta geneſim HACB, cuius imago temporis CD & B. <lb></lb></s> <s id="s.000324">Iſti duo motus in exemplo eſſent, ſi in quodam plano mo<lb></lb>ueretur formica, dum ipſum planum vna eius extremitate <lb></lb>immobili circumduceretur, Sic formica difficiliùs <expan abbr="aſcẽde-ret">aſcende<lb></lb>ret</expan> prout ipſum planum magis ad horizontem erigeretur. <lb></lb></s> <s id="s.000325">Iam motus extremitatis plani circumactæ habet geneſim <lb></lb>ACBH, cuius temporis imago & DCB &, et altera geneſis <lb></lb>FCBK tribueretur motui formicæ, nam vt <expan abbr="dictũ">dictum</expan> eſt varius <lb></lb>motus formicæ pendet ex latione plani, ideò velocitates <lb></lb>eiuſdem (nam in plano immobili ponimus æquabiliter fer<lb></lb>ri) durant ijſdem temporibus, quibus velocitates præcipuæ <lb></lb>geneſis ACBH. </s> <s id="s.000326">Sit denique LMSR imago velocitatum <lb></lb>iuxta geneſim ACBH, cuius temporis imago CD & B; pa<lb></lb>tet ſi ſit MP ad PS ſicut imago temporis CDEG ad ima<lb></lb>ginem & BGE &, fore LM ad PQ vt AC ad OG, & con<lb></lb>cepta etiam figura MNOTS inter parallelas LMN, RST <lb></lb>ita vt ſit ſemper MN ad PO ſicut FC ad GI, nec non LM <lb></lb>ad MN vt AC ad FC. (ſunt enim initio motuum in C, aut <lb></lb>inſtanti M, velocitates geneſium AC, CF, ſcilicet LM, MN; <lb></lb>& in G, hoc eſt inſtanti P ſunt velocitates OC, GI; nimi<lb></lb>rum QP, PO) vocetur proinde geneſis FCBK ſpuria, ac <lb></lb>adſtricta imagini temporis & DCB &, cuius imago veloci<lb></lb>tatum MNTS pariter ſpuria, homogenea tamen ipſi legiti<lb></lb>mæ LMSR. </s> </p> <pb pagenum="33" xlink:href="022/01/039.jpg"></pb> <p type="margin"> <s id="s.000327"><margin.target id="marg73"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="main"> <s id="s.000328"><emph type="center"></emph>PROP. XVI. THEOR. XVI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000329">SI ſint duo motus iuxta geneſes legitimam, & ſpuriam, <lb></lb>erunt mobilium exacta ſpatia, vt imagines interſe <lb></lb>homogeneæ velocitatum, legitima ad ſpuriam. </s> </p> <p type="main"> <s id="s.000330">Eſto geneſis legitima ACBH, cuius imago temporis <lb></lb><arrow.to.target n="marg74"></arrow.to.target><lb></lb>& DCA &, & imago velocitatum MLRS. </s> <s id="s.000331">Sit etiam gene<lb></lb>ſis altera illi homogenea, ſed ſpuria, & adſtricta imagini <lb></lb>temporis & DCB &, cuius imago velocitatum ſpuria, prio<lb></lb>rique legitimæ homogenea NMST. Dico, ſpatia iuxta has <lb></lb>imagines tranſacta eſſe vt ipſæ imagines legitima LMSR <lb></lb>ad ſpuriam NMST. </s> <s id="s.000332">Cum temporis momenta M, P in<lb></lb>telligantur ex minimis temporibus, quæ proponi poſſunt, <lb></lb>interſe æqualibus, & quibus æquabiliter perdurant ve<lb></lb>locitates, quas mobile ſortitur in aduentu ſuo in punctis <lb></lb>C, G, erit vt velocitas FC ad velocitatem GI ſic interſe <lb></lb><arrow.to.target n="marg75"></arrow.to.target><lb></lb>ſpatia, quæ iſtis velocitatibus, temporibuſque illis æqua<lb></lb>libus percurrerentur, in qua ratione eſt etiam NM ad OP. <lb></lb></s> <s id="s.000333">Deinde momento M peragerentur ſpatia proportionalia <lb></lb>velocitatibus FC, AC, ſeu rectis NM, ML, momento <lb></lb>autem P ſpatia proportionalia velocitatibus GI, GD, <lb></lb>in qua ratione eſt etiam OP ad PQ, & ſic deinceps <lb></lb>procedendo per ſingula temporis MR momenta, adeo <lb></lb>vt, cum ſpatium velocitate FC exactum ad id veloci<lb></lb>tate CA, ſit vt NM ad ML, ſpatium velocitate IG ad id <lb></lb>exactum velocitate GD ſit vt OP ad PQ, & ſint præterea <lb></lb>primæ interſe, hoc eſt ſpatia velocitatibus FC, GI tran<lb></lb>ſacta, proportionalia tertijs, ſpatijs videlicet tranſactis <lb></lb>velocitatibus ML, PQ ergo vt omnes primæ ad omnes <lb></lb>tertias quantitates, hoc eſt omnia ſpatia tranſacta iuxta <lb></lb>geneſim FCBK ad omnia ſpatia iuxta geneſim ACB, ita <lb></lb>erit ſumma ſecundarum ad omnes quartas, ſcilicet iſta <lb></lb>erit imago NMST ad imaginem LMSR. </s> <s id="s.000334">Quod & c. </s> </p> <pb pagenum="34" xlink:href="022/01/040.jpg"></pb> <p type="margin"> <s id="s.000335"><margin.target id="marg74"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="margin"> <s id="s.000336"><margin.target id="marg75"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000337"><emph type="center"></emph>LIBER ALTER<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000338"><emph type="center"></emph>DE<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000339"><emph type="center"></emph>Motu Compoſito.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000340">MOtum appellamus compoſitum, vbi dum fer<lb></lb>tur mobile, conſideratur habere plures iņ <lb></lb>diuerſas partes, vel <expan abbr="etiã">etiam</expan> in eandem partem <lb></lb>conatus, ex quibus oriatur tertia vis diſtin<lb></lb>cta ab illis. </s> <s id="s.000341">Hunc librum, cum expleueri<lb></lb>mus, non pauca vnà cum priori, dicta erunt de motu, erit<lb></lb>que ea methodus, qua ſimul geometrica quædam, difficil<lb></lb>lima ſcitu ſatis breuiter oſtendemus. </s> <s id="s.000342">Nam vibrationes <lb></lb>pendulorum exigi temporibus; quæ ſint in ſubduplicatą <lb></lb>ratione longitudinum eorundem, planè tandem conſtabit <lb></lb>aliàs nobis diſſentientibus: aperiemus etiam, qua arte in<lb></lb>telligi queant anguli rectilinei curuilineis æquales; nec non <lb></lb>exponemus parabolas quibuſdam ſpiralibus æquales, vt <lb></lb>eſt vulgata ſpirali Archimedeæ, cùm videlicet baſis para<lb></lb>bolæ radio circuli ſpiralem continentis, & dimidium huius <lb></lb>circumferentiæ circuli altitudini eiuſdem parabolæ, æqua<lb></lb>les ſint. </s> </p> <p type="main"> <s id="s.000343"><emph type="center"></emph>PROP. I. THEOR. I.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg76"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000344"><margin.target id="marg76"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="main"> <s id="s.000345">SI in eadem recta linea currantur ſpatia temporibus <lb></lb>æqualibus, & ſint motus ſimplices, ac ad eaſdem par<lb></lb>tes tendentes, eadem illa ſpatia ſimul motu compoſito, ab <lb></lb>eodemque mobili duabus illis geneſibus affecto, vnicoque <lb></lb>ex dictis temporibus æqualibus, excurrentur. </s> </p> <pb pagenum="35" xlink:href="022/01/041.jpg"></pb> <p type="main"> <s id="s.000346">Curratur LI iuxta imaginem velocitatum HAEF, et IO <lb></lb>iuxta aliam dictæ homogeneam BAED. </s> <s id="s.000347">Dico LO ſum<lb></lb>mam dictorum ſpatiorum LI, IO exactum iri vnico tem<lb></lb>pore AE, ſi nempe mobile feratur <expan abbr="ſecũdum">ſecundum</expan> vtranque ima<lb></lb>ginem. </s> </p> <p type="main"> <s id="s.000348">Per quodlibet punctum, ſeu temporis momentum M <lb></lb>agatur recta GMC parallela HB, vel FD. </s> <s id="s.000349">Habebit mobi<lb></lb>le momento A, <expan abbr="dũ">dum</expan> ſcilicet mouetur motu compoſito duas <lb></lb>ſimul velocitates AH, AB, ideſt vnicam HB. </s> <s id="s.000350">Similiter mo<lb></lb>mento M habebit GC, & momento E ipſam FD. </s> <s id="s.000351">Itaque </s> </p> <p type="main"> <s id="s.000352"><arrow.to.target n="marg77"></arrow.to.target><lb></lb>erit HBDF imago velocitatum compoſiti motus, qui fiet <lb></lb>tempore AE iuxta imaginem, quæ aggregatum eſt <expan abbr="dictarũ">dictarum</expan> <lb></lb>HAEF, ABDE. </s> <s id="s.000353">Eſt verò LI ad IO vt imago HAEF ad <lb></lb>imaginem ABDE; ergo conuertendo, componendoquę <lb></lb>erit vt LI ad LO, ſic imago HAEF ad imaginem HBDF; <lb></lb>propterea quemadmodum ſpatium LI currebatur iuxtą <lb></lb>imaginem HAEF, ſic LO percurretur imagine HBDF ſolo, <lb></lb>eodemque tempore AE. </s> <s id="s.000354">Quod &c. </s> </p> <p type="margin"> <s id="s.000355"><margin.target id="marg77"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>prima.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000356"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000357"><emph type="italics"></emph>Hinc patet graue perpendiculariter, violenterque deiectum <lb></lb>minimè ad terram venturum aggregato virium, quarum vna <lb></lb>eſt ab impellente impreßa, altera verò à grauitate <expan abbr="dependẽs">dependens</expan>. <lb></lb></s> <s id="s.000358">Nam ex impartita vt celerior fit caſus, quam vt graue in de<lb></lb>curſu ſuo poſſit ex acceleratione naturali eum gradum acqui<lb></lb>rere, quem certè ſponte ſua tantùm deſcendens in fine eiuſdem <lb></lb>altitudinis adeptum eſſet. </s> <s id="s.000359">Hoc ita verum eſt, vt aliquando <lb></lb>minimum interſit, inter impetum ab ambabus cauſis proue<lb></lb>nientem, & eum, qui a ſola oritur grauitate, quamobrem pa<lb></lb>rum is proficeret, qui conaretur maiorem impetum componere <lb></lb>in caſu grauis, illi nempe adiecta vi, mobile idem in decurſu <lb></lb>impellente, vltra natiuam grauitatem, quod tamen fieri haud <lb></lb>dubiè poſſet, ſi caſus obliquus eßet.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="36" xlink:href="022/01/042.jpg"></pb> <p type="main"> <s id="s.000360"><emph type="italics"></emph>Illud quoque hac occaſione aperiendum eſt, graue naturali<lb></lb>ter deſcendens eò concitatiùs ferri, quoad potentia reſiſtentis <lb></lb>aeris (validior namque iſta fit, vbi mobilis caſus eſt celerior) <lb></lb>vi grauitatis mobili inhærenti exaquatur, tunc enim cauſą <lb></lb>vlterioris accelerationis adempta eſt, conſumiturque in lucta<lb></lb>tione aeris contranitentis: quare tunc grane progrederetur <lb></lb>æquabili motu, id quòd citiùs euenire deberet ſi grane intrą <lb></lb>aquam deſcendat.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000361"><emph type="center"></emph>PROP. II. THEOR. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000362">SI in eadem recta duos motus ſibi contrarios, ſimplices, <lb></lb>ac eodem tempore peractos intelligamus, mobile di<lb></lb>ferentiam illorum ſpatiorum, ſi vtroque motu eſſet affe<lb></lb>ctum, percurreret. <lb></lb><arrow.to.target n="marg78"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000363"><margin.target id="marg78"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="main"> <s id="s.000364">Curratur à puncto L ſpatium LO imagine velocitatum <lb></lb>ABFG, & codem tempore curratur etiam recta OM ex <lb></lb>puncto altero O, ſcilicet contrario motu, & iuxta <expan abbr="imaginẽ">imaginem</expan> <lb></lb>AHIG prædictę homogeneam. </s> <s id="s.000365">Dico mobile, <expan abbr="cõpoſito">compoſito</expan> ex <lb></lb>vtriſque motu, & tempore ipſo AG curſurum differentiam <lb></lb>LM dictorum ſpatiorum LO, OM. </s> </p> <p type="main"> <s id="s.000366">Primùm intra parallelas AB, GF non ſe ſecent lineæ <arrow.to.target n="marg79"></arrow.to.target><lb></lb>BF, HI, & ducatur quælibet DC æquidiſtans AB, vel GF, <lb></lb>quæ fecet HI in E. </s> <s id="s.000368">Manifeſtum eſt, mobile, compoſito <lb></lb>motu feratur habere duplicem velocitatem, vnam AB al<lb></lb>teram illi oppoſitam AH, ob idque moueri verſus O ſolą <lb></lb>velocitate HB differentia dictarum interſe pugnantium <lb></lb>velocitatum: pariter momento D feretur mobile veloci<lb></lb>tate EC differentia duarum DE, DC, & inſtanti G habebit <lb></lb><arrow.to.target n="marg80"></arrow.to.target><lb></lb>differentialem IF; ex quo ſequitur figuram BHEIFCB, dif<lb></lb>ferentiam imaginum ABFG, HAGI, aptatam tempori AC <lb></lb>imaginem eſſe velocitatum compoſiti motus. </s> <s id="s.000369">Hoc po<lb></lb><arrow.to.target n="marg81"></arrow.to.target><lb></lb>ſito habebit LM ad LO eandem rationem, ac BHIF ad <lb></lb>ABFG; Propterea LM, quæ eſt differentia ſpatiorum LO, <pb pagenum="37" xlink:href="022/01/043.jpg"></pb>MO curretur iuxta imaginem BHIF, nempe compoſito <lb></lb>motu, & tempore AG. </s> </p> <p type="margin"> <s id="s.000370"><margin.target id="marg79"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="margin"> <s id="s.000371"><margin.target id="marg80"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>prima.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000372"><margin.target id="marg81"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primą <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000373">2. Se nunc ſecent lineæ BF, HI in C. </s> <s id="s.000374">Ducatur CD pa<lb></lb><arrow.to.target n="marg82"></arrow.to.target><lb></lb>rallela alteri æquidiſtantium AB, GF. </s> <s id="s.000375">Conſtat ex prima <lb></lb>parte, quòd mobile compoſito motu, & iuxta imaginem <lb></lb>HBC feretur verſus O tempore AD; ſit ergo ſpatium, quod <lb></lb>curreretur illa imagine, PR, & ob id LO ad PR eandem̨ <lb></lb><arrow.to.target n="marg83"></arrow.to.target><lb></lb>habebit rationem quam imago ABFG ad imaginem̨ <lb></lb>HBC. </s> </p> <p type="margin"> <s id="s.000376"><margin.target id="marg82"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="margin"> <s id="s.000377"><margin.target id="marg83"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>prima<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000378">Similiter dum mobile mouetur tempore DG iuxta ima<lb></lb>gines DCIG, DCFG, feretur verè ſecundùm imaginem̨ <lb></lb><arrow.to.target n="marg84"></arrow.to.target><lb></lb>FCI verſus L, quamobrem ſi ſpatium, quod exigeretur <lb></lb>hac imagine ſit RQ, habebit iſtud ad LO eandem rationem, <lb></lb><arrow.to.target n="marg85"></arrow.to.target><lb></lb>quam imago CFI ad imaginem ABFG, & ideo ex æquali <lb></lb>QR ad PR ſe habebit vt imago CFI ad imaginem HBC; ſi <lb></lb>igitur ponatur ABFG maior imagine AHIG, demptà co<lb></lb>muniter AHCFG relinquetur HBC maior imagine CEI, & <lb></lb>ideo etiam PR maior QR: curritur verò PR versùs R tem<lb></lb>pore AD, & RQ versùs P tempore DG, ergo toto tempo<lb></lb>re AG curretur PQ differentia ſpatiorum PR, RQ Cum <lb></lb>verò HBC ad CFI, ſit vt PR ad RQ, erit diuidendo vt ex<lb></lb>ceſſus imaginis HBC ſupra imaginem FCI ad imaginem̨ <lb></lb>iſtam, ita PQ ad QR, & oſtenſum eſt QR ad LO, ſicut ima<lb></lb>go FCI ad imaginem ABFG, ergo ex æquali exceſſus ima<lb></lb>ginis HBC ſupra imaginem AHIG habebit eandem ratio<lb></lb>nem ad imaginem AHIG, ac PQ ad LO, at eſt in illa <expan abbr="eadẽ">eadem</expan> <lb></lb>ratione etiam LM ad LO (eſt enim LO ad MO vt imago <lb></lb>ABFG ad imaginem AHIG) ergo PQ erit æqualis LM, <lb></lb>atque adeo mobile dum currit vtroque motu, hoc eſt iux<lb></lb>ta ſimul duas imagines propoſitas contrariorum motuum, <lb></lb>peraget ſpatium LM versùs O ſecundùm imaginem, quæ <lb></lb>differentia eſt propoſitarum ABFG, AHIG, tempore AG. <lb></lb></s> <s id="s.000379">Quod &c. </s> </p> <pb pagenum="38" xlink:href="022/01/044.jpg"></pb> <p type="margin"> <s id="s.000380"><margin.target id="marg84"></margin.target><emph type="italics"></emph>Ex primą <lb></lb>parte.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000381"><margin.target id="marg85"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>prima.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000382"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000383"><emph type="italics"></emph>Deducìtur, mobile nullum ſpatium emenſurum, vbi ima<lb></lb>gines ſimplicium motuum fuerint aquales.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000384"><emph type="center"></emph>PROP. III. THEOR. III.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg86"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000385"><margin.target id="marg86"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="main"> <s id="s.000386">REperire eam velocitatem, eamque directionem, quæ <lb></lb>orirentur, ſi mobile pluribus eodem momento velo<lb></lb>citatibus, ſeu conatibus affectum eſſet. </s> <s id="s.000387">Opportet autem <lb></lb>non ſolum has velocitates, verùm etiam earum directio<lb></lb>nes manifeſtas eſſe. </s> </p> <p type="main"> <s id="s.000388">Habeat mobile A, eodem momento conatum AB, quo <lb></lb>tendat in R; AC; quo in C; & AD, quo in D. </s> <s id="s.000389">Quæritur ve<lb></lb>locitas, & directio, quas mobile habiturum eſſet in multi<lb></lb>plici illa affectione (Nam actu vnam velocitatem, vnam<lb></lb>que tantùm directionem ſortiri debet) Ex duabus qui<lb></lb>buſque AD, AC intelligatur perfici parallelogrammum <lb></lb>ACED, & ducta diametro AE fiat itidem aliud parallelo<lb></lb>grammum ABFE, cuius agatur diameter AF. </s> <s id="s.000390">Dico AF <lb></lb>eſſe quæſitam velocitatem, ac directionem, quibus mobile <lb></lb>ex illis pluribus conatibus motum ſuum inſtitueret. </s> </p> <p type="main"> <s id="s.000391">Si mobili A currendum eſſet æquabili motu ſpatium <lb></lb>AE, pertranſiret eodem tempore tam rectam AD, quàm </s> </p> <p type="main"> <s id="s.000392"><arrow.to.target n="marg87"></arrow.to.target><lb></lb>ipſam AC; nam cum fertur ab A in E verè deſcendit ab A <lb></lb>in C, & ab A in D motu pariter æquabili; ergo AD ad <lb></lb>AC, erit vt velocitas, qua curritur per AD ad velocitatem, <lb></lb>qua curritur per AC. </s> <s id="s.000393">Itaque ſi mobile dum eſt in A in<lb></lb>telligatur affectum velocitatibus AD, AC habentibus di<lb></lb>rectiones ipſas rectas AD, AC, perinde eſſet, ac ſi ſola fo<lb></lb>ret mobili velocitas vnâ cum directione AE. </s> <s id="s.000394">Eadem ra<lb></lb>tione AF velocitas habens directionem AF, æquipollebit <lb></lb>duabus velocitatibus AB, AE iuxta directiones rectas eaſ-<pb pagenum="39" xlink:href="022/01/045.jpg"></pb>dem ABAE; hoc æquiualebit tribus AB, AC, AD. </s> <s id="s.000395">Mo<lb></lb>bile igitur ex affectione trium illorum conatuum, vt ſup<lb></lb>poſitum fuit, nitetur ſecundùm AF velocitate ipſa AF <lb></lb>Quod &c. </s> </p> <p type="margin"> <s id="s.000396"><margin.target id="marg87"></margin.target><emph type="italics"></emph>Gal. </s> <s id="s.000397">pr. de mo<lb></lb>tu aquab.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000398"><emph type="center"></emph>DEF. I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000399">ACcelerationem alicuius motus, tunc intelligimus, <expan abbr="cũ">cum</expan> <lb></lb>velocitates, quæ ſubinde mobili adueniunt, non de<lb></lb>lentur, ſed prorſus integræ, atque indelebiles mobili in ipſo <lb></lb>motu perſeuerant. </s> <s id="s.000400">Ex quo ſequitur motum ſimplicem di<lb></lb>ci, cum præteritæ velocitates protinus euaneſcunt, illæ<lb></lb>que tantum conſiderantur, quæ mobili ſubinde oriun<lb></lb>tur. </s> </p> <p type="main"> <s id="s.000401"><emph type="center"></emph>PROP. IV. PROB. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000402">IMaginem accelerationis cuiuſcunque ſimplicis motus <lb></lb>exhibere. </s> </p> <p type="main"> <s id="s.000403">Imago velocitatum ſimplicis motus eſto rectangulum <lb></lb><arrow.to.target n="marg88"></arrow.to.target><lb></lb>AFDC: ſic motus eſt æquabilis, vt acceleretur debent in<lb></lb><arrow.to.target n="marg89"></arrow.to.target><lb></lb>ſtanti C vigere omnes velocitates in imagine AFDC <expan abbr="cõ-prehenſæ">con<lb></lb>prehenſæ</expan>, & item ducta quacunque BE parallela AF, vel <lb></lb><arrow.to.target n="marg90"></arrow.to.target><lb></lb>CD, erit mobile momento B affectum omnibus antece<lb></lb>dentibus velocitatibus, comprehenſis nempe ab imaginis <lb></lb>portione AFEB; quare ſi ponamus HLG imaginem eſſę <lb></lb>accelerationis, itaut nempe tempus GL æquale ſit tempo<lb></lb>ri AC; item KL æquale tempori AB, erit vt figura CAFD <lb></lb>ad figuram BAFE, ſic velocitas, qua mobile fertur <expan abbr="momẽ-to">momen<lb></lb>to</expan> G ad velocitatem, quam habet inſtanti K; & ideo quia <lb></lb>ponitur imago ſimplicis motus rectangulum AFDC, erit <lb></lb>rectangulum CF ad BF, hoc eſt recta CA ad AB immò <lb></lb>LG ad LK, vt GH ad KI; quamobrem GLH imago velo<lb></lb><arrow.to.target n="marg91"></arrow.to.target><lb></lb>citatum huiuſmodi motus, erit triangulum. </s> <s id="s.000404">Quod ſi ima-<pb pagenum="40" xlink:href="022/01/046.jpg"></pb>go ſimplicis motus fuiſſet triangulum, imago velocitatum <lb></lb>accelerationis foret trilineum ſecundum, & ita pro<lb></lb>portionaliter de infinitis numero accelerationibus. </s> </p> <p type="margin"> <s id="s.000405"><margin.target id="marg88"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> <gap></gap>.</s> </p> <p type="margin"> <s id="s.000406"><margin.target id="marg89"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000407">def.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>pri<lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000408"><margin.target id="marg90"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000409"><margin.target id="marg91"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000410"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000411"><emph type="italics"></emph>Hinc obiter habemus, quo pacto imago velocitatum corpo<lb></lb>rum naturaliter deſcendentium triangulum ſit. </s> <s id="s.000412">Nam quo<lb></lb>libet momento ſui caſus habet graue idem inſe principium̨ <lb></lb>motus, ſeu grauitas, ex qua concipitur imago ſimplicis motus <lb></lb>ſi nempe priores gradus velocitatis ſubinde deperirent, at <lb></lb>quia in eius deſcenſu prorſus perſeuerant (id enim ſupponi<lb></lb>tur abſtrahendo ab aere) inde motus concitatur, & fit vti di<lb></lb>ximus imago accelerationis triangulum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000413"><emph type="center"></emph>AXIOMA<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000414">QVælibet linea, vt fluxus puncti concipi po<lb></lb>teſt. </s> </p> <p type="main"> <s id="s.000415"><emph type="center"></emph>AX. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000416">VT propoſita linea ex fluxu puncti exarètur, duò tan<lb></lb>tùm neceſſaria ſunt, ſcilicet motus, & puncti di<lb></lb>rectio. </s> </p> <p type="main"> <s id="s.000417"><emph type="center"></emph>PROP. V. THEOR. III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000418">REcta, quæ priùs deſcripta eſt, poteſt alijs à primis <lb></lb>velocitatibus, rurſus exarari. </s> </p> <p type="main"> <s id="s.000419">Nam punctum poteſt fluere ſecundum quamcunque <lb></lb>rectam, quocunque motu, ergo illam poteſt etiam quibuſ<lb></lb>cunque velocitatibus affectum rurſus exarare. </s> </p> <pb pagenum="41" xlink:href="022/01/047.jpg"></pb> <p type="main"> <s id="s.000420"><emph type="center"></emph>PROP. VI. THEOR. IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000421">VT eadem recta ex fluxu puncti renouetur, opportet in <lb></lb>quocunque illius puncto ſeruari priſtinas directio<lb></lb>nes, </s> </p> <p type="main"> <s id="s.000422">Cum, vti diximus, ad deſcriptionem lineæ duo tantùm <lb></lb><arrow.to.target n="marg92"></arrow.to.target><lb></lb>exigantur, nempe motus, & puncti directio; motus verò po<lb></lb>teſt eſſe quilibet, ſequitur ergo directionem, alteram de <lb></lb>duobus, ſeruari debere. </s> </p> <p type="margin"> <s id="s.000423"><margin.target id="marg92"></margin.target><emph type="italics"></emph>Ax.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>buius. <lb></lb></s> <s id="s.000424">pr.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000425"><emph type="center"></emph>DEF. II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000426">LIneam dicimus curuam, in qua ſumptis duobus ad<lb></lb>libitum punctis, recta, quæ ipſa puncta coniunge<lb></lb>ret, nullam cum propoſita linea partem ſit habitura com<lb></lb>munem. </s> </p> <p type="main"> <s id="s.000427"><emph type="center"></emph>PROP. VII. THEOR. V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000428">DIrectiones puncti deſcribentis lineam, iuxta rectas <lb></lb>lineas concipi debent. </s> </p> <p type="main"> <s id="s.000429">Dum punctum fluere intelligimus, ineſt in eo ſingulis <lb></lb>momentis certus, ac præfixus gradus velocitatis, quo tan<lb></lb>tùm attento, rectà, <expan abbr="æquabiliq;">æquabilique</expan> motu in certam partem con<lb></lb>tenderet; at huiuſmodi iter, aliud non eſt, quàm directio <lb></lb>puncti, qua eius temporis momento proficiſcitur; ergo iux<lb></lb>ta rectas lineas, directiones omnes conſiderari opportet. </s> </p> <p type="main"> <s id="s.000430"><emph type="center"></emph>PROP. VIII. THEOR. VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000431">TAngens, & directio motus in quouis curuæ puncto <lb></lb>eſt vna, <expan abbr="atq;">atque</expan> eadem recta. </s> </p> <p type="main"> <s id="s.000432">Nam in deſcriptione <expan abbr="cuiuſcunq;">cuiuſcunque</expan> rectæ procedit pun<lb></lb><arrow.to.target n="marg93"></arrow.to.target><pb pagenum="42" xlink:href="022/01/048.jpg"></pb>ctum iuxta tendentias rectas, obliquatur tamen ob ſubſe<lb></lb>quentes, aliò tendentes niſus, & ob id diſtrahitur punctum <lb></lb>ipſum à priori tendentia, idem accidit ex alia parte ſi re<lb></lb>flaxiſſet idem punctum, nempe hinc inde vnicam rectam <lb></lb>eandemque, continuantibus oppoſitis ad idem punctum <lb></lb>directionibus, ergo directio, & tangens vna, & eadem eſt <lb></lb>recta. </s> </p> <p type="margin"> <s id="s.000433"><margin.target id="marg93"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000434"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000435"><emph type="italics"></emph>Hinc ſequitur, vnicam lineam dicendam eſſe, cum à quo<lb></lb>cunque illius puncto vnica tantùm ex vtraque parte egre<lb></lb>ditur tangens.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000436"><emph type="center"></emph>DEF. III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000437">QVòd ſi ex aliquo puncto duæ tangentes hinc inde <lb></lb>egredientes angulum efficiant; tunc propoſitam li<lb></lb>neam inflexam dicemus, & punctum, in quo ſunt <lb></lb>contactus, inflexionis appellabitur. </s> </p> <p type="main"> <s id="s.000438"><emph type="center"></emph><emph type="italics"></emph>Corollarium I.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000439"><emph type="italics"></emph>Ab hiſce deffinitionibus, & priori coroll. </s> <s id="s.000440">manat artificium <lb></lb>componendi duas curuas, vel curuam & rectam, adeout vni<lb></lb>cam lineam efforment, nullumque angulum; nempe cum ſic <lb></lb>inuicem iungamus, vt tangentes ad punctum connexus, vnam <lb></lb>tantùm rectam efficiant.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000441"><emph type="center"></emph><emph type="italics"></emph>Corollarium II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000442"><emph type="italics"></emph>Sed & illud patet, quibus angulis inflectantur lineæ inui<lb></lb>cem compoſitæ, ſi ad punctum inflexionis angulum tangen<lb></lb>tium obſeruauerimus, ſunt enim interſe æquales, licèt diuer<lb></lb>ſa ſpeciei, cum vnus ſit curuilineus, & rectilineus alter.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="43" xlink:href="022/01/049.jpg"></pb> <p type="main"> <s id="s.000443"><emph type="center"></emph>PROP. IX. THEOR. VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000444">TAngens, ſeu directio motus in quocunque curuæ <lb></lb>puncto eſt illa recta, quæ vtrinque ſtatim cadens <lb></lb>extra curuæ conuexum ad eandem, quàm fieri poteſt ex <lb></lb>vtraque parte accedit. </s> </p> <p type="main"> <s id="s.000445">Nam alia quæque recta tranſiens per punctum conta<lb></lb>ctus ad ſectionem magis accedere nequit, quin ipſam illinc <lb></lb>ſecet, ob id extra conuexum eius non cadet, ab altera ve<lb></lb>rò parte magis à propoſita curua ſeparabitur, quamobrem <lb></lb>nulla alia recta, quàm tangens poterit ſimul extra curuam <lb></lb>eſſe, & quàm fieri poteſt ad ipſam accedere. </s> </p> <p type="main"> <s id="s.000446"><emph type="center"></emph>DEF. IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000447">LIneæ AC, AD occurrant ſibi in A, quod punctum in<lb></lb><arrow.to.target n="marg94"></arrow.to.target><lb></lb>telligatur transferri ab A in C vnà cum linea AD <lb></lb>ſemper ſibi parallela, quo tempore punctum A currat ip<lb></lb>ſam latam lineam ex A in D. </s> <s id="s.000448">Manifeſtum eſt idipſum <lb></lb>punctum A deſcripturum eſſe motu compoſito lineam <lb></lb>quandam AB diagonalem ſuperficiei parallelogrammæ <lb></lb>ABCD. </s> <s id="s.000449">Vocamus ergo diagonalem illam ſemitam com<lb></lb>poſiti motus, & AC, AD latera illius. </s> </p> <p type="margin"> <s id="s.000450"><margin.target id="marg94"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 6.</s> </p> <p type="main"> <s id="s.000451"><emph type="center"></emph><emph type="italics"></emph>Corollarium I.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000452"><emph type="italics"></emph>Manifeſtum eſt mobile dum currit AB tranſire etiam AC, <lb></lb>AD, licèt curuæ ſint, nam verè transfertur illo tempore, tam <lb></lb>ad lineam CB quam ad DB.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000453"><emph type="center"></emph><emph type="italics"></emph>Corollarium II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000454"><emph type="italics"></emph>Præterea ſi ducerentur, autſint AC, CB, DA, DB, AB<emph.end type="italics"></emph.end><pb pagenum="44" xlink:href="022/01/050.jpg"></pb><emph type="italics"></emph>rectæ lineæ, efficeretur ex ijs parallelogrammum ACBD, cu<lb></lb>ius diameter AB; quamobrem ex datis punctis C, A, D repe<lb></lb>riretur ſtatim punctum B, ſcilicet extremum ſemitæ compo<lb></lb>ſiti motus, cuius latera ipſæ curuæ, aut rectæ AC, AD<emph.end type="italics"></emph.end> —. </s> </p> <p type="main"> <s id="s.000455"><emph type="center"></emph>PROP. X. PROB. III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000456">EX datis <expan abbr="quotcunq;">quotcunque</expan> lateribus compoſiti motus, huius <lb></lb><arrow.to.target n="marg95"></arrow.to.target><lb></lb>ſemitæ terminum exhibere. </s> </p> <p type="margin"> <s id="s.000457"><margin.target id="marg95"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="main"> <s id="s.000458">Si latera compoſiti motus eſſent duo tantùm AB, AC. <lb></lb></s> <s id="s.000459">Facto parallelogrammo vt dictum eſt, inueniretur pun<lb></lb>ctum E extremum motus: & <expan abbr="quæcunq;">quæcunque</expan> ſit ſemita, ſeu mo<lb></lb>tus, poteſt idem E ſupponi tanquam extremum alterius la<lb></lb>teris, adeoque, ſi motus conſtet ex tribus lateribus AC, <lb></lb>AB, AD, perinde ſit ac ſi foret duorum laterum AE, AD; <lb></lb>nam AC, AD valent ſimul ac ſolum AE; cum ita ſit, facto <lb></lb>etiam parallelogrammo EADF ex datis punctis E, A, D, <lb></lb>habebitur F extremum ſemitæ, cuius ſunt tria latera CA, <lb></lb>AD, AB — </s> </p> <p type="main"> <s id="s.000460"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000461"><emph type="italics"></emph>Deducitur artificium deſcribendæ ſemitæ AE, vel AF, ſi <lb></lb>nempe aſſumptis partibus AG, AH, AI in dictis lateribus, <lb></lb>quæ quidem ſciantur percurri temporibus æqualibus, ſi per <lb></lb>ipſas ſingulas mobile punctum ferretur eo modo, quo in com<lb></lb>poſito motu nititur per eaſdem directiones; reperietur in<lb></lb>quam punctum K in ſemita AE, atque L in ſemita AF: qua<lb></lb>re hoc modo ſumptis alijs, atque alijs partibus in ipſis lateri<lb></lb>bus, reperientur alia, atque alia puncta ad ipſam ſemitam̨ <lb></lb>pertinentia, quorum tandem beneficio, facile erit quaſitam <lb></lb>fermè ſemitam exarare.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="45" xlink:href="022/01/051.jpg"></pb> <p type="main"> <s id="s.000462"><emph type="center"></emph>PROP. XI. PROB. IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000463">EX datis imaginibus velocitatum, iuxta quas ſimplici <lb></lb><arrow.to.target n="marg96"></arrow.to.target><lb></lb>motu currantur latera compoſiti motus; datis item <lb></lb>tangentibus ad quæcunque puncta ipſorum laterum, repe<lb></lb>rire ſemitam compoſiti motus, nec non directiones, <expan abbr="veloci-tateſq;">veloci<lb></lb>tateſque</expan> puncti deſcribentis ipſam ſemitam. </s> </p> <p type="margin"> <s id="s.000464"><margin.target id="marg96"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 8.</s> </p> <p type="main"> <s id="s.000465">Opportet tamen latera ipſa, <expan abbr="itemq;">itemque</expan> imagines prædictas, <lb></lb>in imperatas ſecari poſſe rationes, quamquam nos non la<lb></lb>teat, in lateribus curuis hoc effici non poſſe, præterquam̨ <lb></lb>aliquatenus in periphærijs circulorum. </s> </p> <p type="main"> <s id="s.000466">Sint AB, AF latera compoſiti motus, quæ quidem ſeor<lb></lb>ſim currantur eodem tempore QM, ſcilicet AB iuxta ima<lb></lb>ginem MNPQ, et AF iuxta imaginem alteram ei homoge<lb></lb>neam TMQR. </s> <s id="s.000467">Ponatur AB circuli arcus, quem tangat re<lb></lb>cta BC æqualis QB, at AF lineam', quæ parabola ſit, con<lb></lb>tingat recta FG æqualis RQ Reperiemus illicò punctum <lb></lb><arrow.to.target n="marg97"></arrow.to.target><lb></lb>H extremum ſemitæ compoſiti motus; ſunt enim data pun<lb></lb>cta A, F, B. </s> <s id="s.000468">Cum igitur mobile venerit in H. Dico, eo <lb></lb>temporis momento velocitatem, ac directionem HL, quæ <lb></lb>recta diameter eſt parallelogrammi, cuius duo latera ſunt <lb></lb>dictæ lineæ HI, HK; Iam vti diximus punctum H eſt ex<lb></lb>tremum compoſiti motus, quare eo momento, quo pun<lb></lb>ctum mobile eſt in H, habet inibi eaſdem illas velocitates, <lb></lb>quas haberet in B, et F, dum ſeorſim illa latera excurriſſet; <lb></lb>ſcilicet conſideratur ipſum mobile habens ſimul velocita<lb></lb>tem HI æqualem, ac æquedirectam, ſeu æquidiſtantem <lb></lb>ipſi CB, cui eſt æqualis alia QP; & velocitatem HK æqua<lb></lb>lem, ſimiliterque directam, ipſi GF æquali RQ Cum ita <lb></lb><arrow.to.target n="marg98"></arrow.to.target><lb></lb>ſit erit HL velocitas, & directio quæſita momento <expan abbr="q.">que</expan> Eo<lb></lb>dem modo, ſi ſit, vel fiat vt imago PNMQ ad ONMV <lb></lb>(ducta ſcilicet applicata SVO) ita BA ad AX, et ONMV <lb></lb>ad imaginem VMTS, vt XA ad AI, percurrentur AX, AI <lb></lb><arrow.to.target n="marg99"></arrow.to.target><pb pagenum="46" xlink:href="022/01/052.jpg"></pb>eodem tempore MV, eritque ob id in X velocitas, & dire<lb></lb>ctio, tangens ipſa ZX æqualis VO, & in I velocitas, & di<lb></lb>rectio, tangens 2 I æqualis VS; Itaque datis punctis X, I, A <lb></lb><arrow.to.target n="marg100"></arrow.to.target><lb></lb>dabitur etiam Y extremum ſemitæ compoſiti motus, cuius <lb></lb>latera AX, AI, & ideo mobile dum eſt in Y momento V <lb></lb>affectum erit duplici velocitate, hoc eſt Y 4 æquali ve<lb></lb>locitati ZX, ſeu VO, ac æquidiſtante eidem ZX, et veloci<lb></lb>tate altera Y 3 æquali, & æquèdirecta ipſi 2 I: quare ex <lb></lb>datis punctis 4, Y, 3 inuenietur punctum S quartus angu<lb></lb>lus parallelogrammi habentis diametrum YI, quæ quidem <lb></lb><arrow.to.target n="marg101"></arrow.to.target><lb></lb>erit directio, & velocitas mobilis currentis compoſito mo<lb></lb>tu inſtanti V. </s> <s id="s.000469">Cumque alia quotcunque puncta eadem <lb></lb>methodo reperire queamus, per quæ duci poſſit linea ferè <lb></lb>quæſitam ſemitam repræſentans, <expan abbr="atq;">atque</expan> emulans, patet idcir<lb></lb>co, quod propoſuimus. </s> </p> <p type="margin"> <s id="s.000470"><margin.target id="marg97"></margin.target><emph type="italics"></emph>Pr,<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000471"><margin.target id="marg98"></margin.target><emph type="italics"></emph>Ex pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>hu.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000472"><margin.target id="marg99"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000473"><margin.target id="marg100"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>def.<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000474"><margin.target id="marg101"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000475"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000476"><emph type="italics"></emph>Cum verò directiones ſint idem, ac tangentes, liquet HL<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg102"></arrow.to.target><lb></lb><emph type="italics"></emph>VS tangentes eſſe compoſiti motus.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000477"><margin.target id="marg102"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 8 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000478"><emph type="center"></emph>PROP. XII. THEOR. VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000479">CVm imagines velocitatum, iuxta quas curruntur duę <lb></lb><arrow.to.target n="marg103"></arrow.to.target><lb></lb>rectæ, quæ ſint latera compoſiti motus, ſunt paral-<lb></lb>lelogrammum, & triangulum; tunc ſemita compoſiti motus <lb></lb>erit communis parabola. </s> </p> <p type="margin"> <s id="s.000481"><margin.target id="marg103"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="main"> <s id="s.000482">Tempore HM curratur latus AC iuxta imaginem velo<lb></lb>citatum HILM rectangulum, & latus AB iuxta imaginem <lb></lb><arrow.to.target n="marg104"></arrow.to.target><lb></lb>triangulum HMN; erit CA ad AB, vt imago <expan abbr="parallelogrã-mum">parallelogram<lb></lb>mum</expan> HILM ad aliam imaginem triangulum NHM. </s> <s id="s.000483">Fiat <lb></lb><arrow.to.target n="marg105"></arrow.to.target><lb></lb><expan abbr="parellogrãmum">parellogrammum</expan> ACDB erit in D extremum ſemitæ com<lb></lb>poſiti motus, quæ ſi ponatur AFC; Dico eſſe parabolam. <lb></lb></s> <s id="s.000484">Sumatur in ipſa linea quoduis punctum F, ab ipſo dedu-<pb pagenum="47" xlink:href="022/01/053.jpg"></pb>cta FE parallela AB, vti etiam FG parallela AC, erunt <lb></lb><arrow.to.target n="marg106"></arrow.to.target><lb></lb>AE, AG latera compoſiti motus, cuius ſemita AF: Con<lb></lb>cipiatur modò P momentum, quo mobile adeſt in F, & <lb></lb>ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad <lb></lb><arrow.to.target n="marg107"></arrow.to.target><lb></lb><expan abbr="imaginẽ">imaginem</expan> PHIK, hoc eſt MH ad HP, vt CA ad AE, ſeu vt BD <lb></lb>ad GF. </s> <s id="s.000485">Pariter erit imago NHM ad <expan abbr="imaginẽ">imaginem</expan> OHP, hoc eſt <lb></lb>quadratum ex MH ad <expan abbr="quadratũ">quadratum</expan> ex PH; immò id ex BO ad <lb></lb>illud ex GF, vt BA ad AG; quamobrem punctum F cadet <lb></lb>in curuam parabolicam communem, cuius diameter AB, <lb></lb>& baſis, ſeu ordinatim applicata BD, ſcilicet AFD erit ipſa <lb></lb>curua parabolica. </s> <s id="s.000486">Quod &c. </s> </p> <p type="margin"> <s id="s.000487"><margin.target id="marg104"></margin.target><emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primum <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000488"><margin.target id="marg105"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000489"><margin.target id="marg106"></margin.target><emph type="italics"></emph>Ex eadem.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000490"><margin.target id="marg107"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000491"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000492"><emph type="italics"></emph>Quoniam graue, quod iaculatur extræ perpendiculum, li<lb></lb>berum ab omni obice, niſi turbaretur eius motus à proprią <lb></lb>grauitate pergerct moueri æquabiliter iuxta directionem, ve<lb></lb>locitatemque ei traditam; habet verò coniunctam grauita<lb></lb>tem, qua, niſi ab impreſſo impetu flecteretur motus, deſcen<lb></lb>deret iuxta perpendiculum motu naturaliter concitato, cuius <lb></lb>imago velocitatum, triangulum eſt; Hinc propterea granę <lb></lb>vltra perpendiculum proiectum deſcribit in curſu ſuo, motu <lb></lb>ſcilicet compoſite, parabolam vulgatam. </s> <s id="s.000493">Verùm enim verò <lb></lb>deſcriptionem iſt am neceſſe aliquo pacto eſt ex duabus cauſis <lb></lb>vitiari, hoc est ab aeris reſiſtentia, & perpendiculis non in<lb></lb>terſe parallelis, quippe in idem, <expan abbr="vnumq;">vnumque</expan> punctum, vniuerſi <lb></lb>centrum, conuergentibus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000494"><emph type="center"></emph>PROP. XIII. THEOR. IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000495">SI ab aſſumpto hyperbolæ puncto, recta axi primo pa<lb></lb><arrow.to.target n="marg108"></arrow.to.target><lb></lb>rallela deducatur, quæ ad ſecundam diametrum per<lb></lb>tingat; Quadrilineum comprehenſum ab ipſa curua hy<lb></lb>perbolica. </s> <s id="s.000496">& dictis tribus rectis, erit imago velocitatis il-<pb pagenum="48" xlink:href="022/01/054.jpg"></pb>lius motus deſcribentis curuam parabolicam, cuius baſis <lb></lb>ad axem eius habet eandem rationem, quam duplus axis <lb></lb>propoſitæ hyperbolæ ad ductam illam <expan abbr="æquidiſtãtem">æquidiſtantem</expan> inter <lb></lb>eiuſdem hyperbolæ aſſymptotos interiectam. </s> </p> <p type="margin"> <s id="s.000497"><margin.target id="marg108"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="main"> <s id="s.000498">Hyperbolæ IRS ſit centrum H, ſemiaxis HI, aſſymptoti <lb></lb>HT, NH, et SN parallela HI; tùm ducta HM ſecunda dia<lb></lb>metro hyperbolæ, intelligatur deſcriptio parabolæ AFD; <lb></lb>itaut duplus axis hyperbolæ, hoc eſt quadruplum ipſius <lb></lb>HI ad NT eandem habeat rationem, quam DB baſis pa<lb></lb>rabolæ ad BA axim eiuſdem. </s> <s id="s.000499">Dico quadrilineum HISM <lb></lb>eſſe imaginem velocitatum, iuxta quam motu compoſito <lb></lb>deſcribitur parabola AFD; & cum ſit homogenea imagi<lb></lb><arrow.to.target n="marg109"></arrow.to.target><lb></lb>nibus HILM, HTM, eſſe quoque rectangulum HDLM ad <lb></lb>imaginem ipſam HISM vt recta CA ad curuam AFD. <lb></lb></s> <s id="s.000500">Fiat rectangulum ACDB, et HM ſit tempus, quo curritur <lb></lb><arrow.to.target n="marg110"></arrow.to.target><lb></lb>vtrunque latus AB, AC, nempe axis AB motu grauium <lb></lb>iuxta imaginem triangulum HTM, alterum verò latus AC <lb></lb><arrow.to.target n="marg111"></arrow.to.target><lb></lb>æquabili motu iuxta imaginem rectangulum HILM, quod <lb></lb>quidem erit HILM; etenim AB ad ſpatium AC eſt vt ima<lb></lb>go triangulum HMT ad imaginem rectangulum HILM, <lb></lb>ſcilicet eſt vt MT ad duplam HI, vel vt NT ad quadru<lb></lb>plam HI, quemadmodum poſuimus. </s> <s id="s.000501">Iam monſtrauimus <lb></lb>lineam, quæ curritur iuxta illas imagines motu compoſito <lb></lb>parabolam eſſe, cuius diameter AB, & baſis BD; & pro<lb></lb>pterea erit ipſa AFD (nam vnica tantum parabola ex <lb></lb>datis AB, BD poſitione, ac magnitudine, axi ſcilicet, ac <lb></lb>baſi dari poteſt) Ducatur nunc à quolibet puncto F dictæ <lb></lb>parabolæ rectæ FE, FG parallelogrammum conſtituentes <lb></lb>AEFG; & P ſit momentum, quo mobile punctum inueni<lb></lb><arrow.to.target n="marg112"></arrow.to.target><lb></lb>tur in F. </s> <s id="s.000502">Habebit inibi ipſo temporis momento P veloci<lb></lb>tatem PQ iuxta directionem GF, ſunt verò iſtæ directiones <lb></lb>ſibi ipſis perpendiculares; ergo recta, quæ diameter eſſet <lb></lb>rectanguli AEFG, & ob id potentiâ æqualis duabus PK, <lb></lb><arrow.to.target n="marg113"></arrow.to.target><lb></lb>PQ erit gradus velocitatis, quem mobile habet momen-<pb pagenum="49" xlink:href="022/01/055.jpg"></pb>to F motu compoſito currens; verùm quia quadratum ex <lb></lb>PR ęquatur rectangulo ORQ vnà cum quadrato ex PQ, & <lb></lb><arrow.to.target n="marg114"></arrow.to.target><lb></lb>eſt ob hyperbolam rectangulum ORQ æquale quadrato <lb></lb>ex HI, vel PK; ergo PR quadratum æquale erit duobus ſi<lb></lb>mul quadratis PQ, PK; itaque PR erit gradus velocitatis <lb></lb>prædicti mobilis in F momento P, compoſitoque motu <lb></lb>currentis iuxta curuam parabolicam. </s> <s id="s.000503">Pariter momento <lb></lb>M, cum mobile eſſet in D velocitas compoſiti motus foret <lb></lb>MS poteſtate æqualis duabus MT, ML, ac demum in A <lb></lb>initio motus velocitas eſt HI: quare HISM erit imago ve<lb></lb>locitatis motus compoſiti dum mobile punctum deſcripſe<lb></lb><arrow.to.target n="marg115"></arrow.to.target><lb></lb>rit curuam parabolicam AFD, eſtque illa imago imagini<lb></lb>bus diuiſorum, ſeu ſimplicium, motuum homogenea; ergo <lb></lb>conſtat baſim etiam BD ad parabolam AFD eandem ha<lb></lb>bere rationem, quam rectangulum HILM ad quadrili<lb></lb>neum HISM. </s> <s id="s.000504">Quod &c. </s> </p> <p type="margin"> <s id="s.000505"><margin.target id="marg109"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>primi <lb></lb>& pr.<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000506"><margin.target id="marg110"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000507">pr.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>hu.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000508"><margin.target id="marg111"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000509"><margin.target id="marg112"></margin.target><emph type="italics"></emph>Ex pr.<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>hu.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000510"><margin.target id="marg113"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000511"><margin.target id="marg114"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>l.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>co<lb></lb>nic.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000512"><margin.target id="marg115"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>prima <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000513"><emph type="center"></emph><emph type="italics"></emph>Corollarium. I.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000514"><emph type="italics"></emph>Patet, cum latera compoſiti motus ſint duo, & ſibi ipſis per<lb></lb>pendicularia, tunc gradum velocitatis eìuſdem motus compo<lb></lb>ſiti æqualem eſſe potentiâ duobus ſimul gradibus, quos habet <lb></lb>mobile eodem momento, ac ſi ſeorſim intelligatur in ipſis ferri <lb></lb>lateribus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000515"><emph type="center"></emph><emph type="italics"></emph>Corollarium. II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000516"><emph type="italics"></emph>Si verò conſiderentur imagines primi ſecundique Caſus <lb></lb>interſe homogenea, erit vt quadrilineum HISM primi ad<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg116"></arrow.to.target><lb></lb><emph type="italics"></emph>quadrilineum ijſdem literis notatum ſecundi caſus, vt cur<lb></lb>ua illa parabolica ad hanc ſecundi caſus parabolam.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="50" xlink:href="022/01/056.jpg"></pb> <p type="margin"> <s id="s.000517"><margin.target id="marg116"></margin.target><emph type="italics"></emph>Pr<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primą <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000518"><emph type="center"></emph><emph type="italics"></emph>Corollarium. III.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000519"><emph type="italics"></emph>Illud etiam conſtat, eſſe in vtroque caſu vt quadrilineum <lb></lb>HIRP ad ipſum PRSM, ita AF ad FD.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000520"><emph type="center"></emph>PROP. XIV. THEOR. X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000521">PRopoſitis Spirali Archimedea primæ circulationis <lb></lb><arrow.to.target n="marg117"></arrow.to.target><lb></lb>ABD, et AGF <expan abbr="cõmuni">communi</expan> parabola, ſit FG baſis huius <lb></lb>æqualis radio DA, et GA ſit dimidium circumferentię cir<lb></lb>culi AEG; erit parabola AGF axem habens GA æqualis <lb></lb>propoſitæ ſpirali. </s> </p> <p type="margin"> <s id="s.000522"><margin.target id="marg117"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000523">Sit PNK communis hyperbola, cuius coniugati ſemia<lb></lb><arrow.to.target n="marg118"></arrow.to.target><lb></lb>xes ſint IK, IH, & aſſymptotos IO. </s> <s id="s.000524">Eſto etiam axis hy<lb></lb>perbolæ huius, dupla ſcilicet IK, ad HO illi ęquidiſtantem <lb></lb>vt FG ad AG. </s> <s id="s.000525">Iam conſtat quadrilineum IHPK fore ima<lb></lb>ginem velocitatum, iuxta quam curreretur parabola AGF <lb></lb>tempore IH: ſi modo oſtendimus hoc ipſum <expan abbr="quadrilineũ">quadrilineum</expan> <lb></lb>eſſe pariter homogeneam imaginem alterius compoſiti <lb></lb>motus, quo videlicet deſcribitur ſpiralis propoſita ABD, <lb></lb><arrow.to.target n="marg119"></arrow.to.target><lb></lb>palam erit, ipſam parabolam eidem illi ſpirali æqualem fu<lb></lb>turam. </s> <s id="s.000526">Ducatur recta KL, quæ æquidiſtet IH; item ex <lb></lb>quouis puncto Q <expan abbr="tẽporis">temporis</expan> IH alia deducatur recta QRMN <lb></lb>parallela IK: erit parallelogrammum rectangulum HIKL <lb></lb>imago velocitatum, iuxta quam curritur FG, et HIO trian<lb></lb>gulum imago, qua curritur AG motu grauium deſcenden<lb></lb>tium: Verùm quia eodem tempore IH, ſi mobile currat <lb></lb>æquabili motu DA æqualem FG, eſt eius imago idem re<lb></lb>ctangulum IHKL, curriturque illo eodem tempore IH (ſpi<lb></lb>rali exigente) omnis circuli circunferentia AGEA æqua<lb></lb>bili etiam motu ab extremitate A radij AD circumducti in <lb></lb>deſcriptione ſpiralis; ob idque factum eſt, vt IK ad HO eſ<lb></lb>ſet vt DA ad circunferentiam ipſam AGEA; nam hoc mo-<pb pagenum="51" xlink:href="022/01/057.jpg"></pb>do rectangulum IH in HO eſt imago velocitatum eiuſ<lb></lb>dem motus per AGEA. </s> <s id="s.000527">Ducatur nunc ex quocun<lb></lb><arrow.to.target n="marg120"></arrow.to.target><lb></lb>que momento Q linea QRMN ipſi IK æquidiſtans, & au<lb></lb>ſpicato motu ex centro D momento I, vt nempe oriatur <lb></lb>ſpiralis, intelligatur momento Q ventum eſſe in B, quamo<lb></lb>brem ductâ DBE, erit rectangulum, ſeu imago QIKR ad <lb></lb>imaginem rectangulum HIKL, ita DB ad DE, in qua ra<lb></lb>tione, cum propter ſpiralem, ſit etiam circunferentia AGE <lb></lb>ad circunferentiam AGEA, erit rectangulum IQ in HO <lb></lb>imago velocitatis per AGE, eſtque velocitas iuxta tangen<lb></lb>tem in E ad velocitatem iuxta tangentem circulum BC in <lb></lb>B vt ED ad DB, ſeu vt HO ad QM; ergo cum iuxta <expan abbr="tangẽ-tem">tangen<lb></lb>tem</expan> in A, hoc eſt in E velocitas ſit HO, erit ſecundùm tan<lb></lb>gentem circulum BC in B, ipſa QM velocitas; propterea<lb></lb>que imago triangulum HIO, quæ in parabolæ deſcriptio<lb></lb>ne erat per AG, nunc erit per omnes tangentes circulos ſu<lb></lb>binde creſcentes ex D in E: ſcilicet momento I, erit mobi<lb></lb>li puncto ſecundùm DA, velocitas IK; momento Q dum̨ <lb></lb>adeſt in B, erit ſecundùm BE velocitas QR, & iuxta <expan abbr="tangẽ-tem">tangen<lb></lb>tem</expan> in B circuli BC velocitas QM; quæ ambæ, hoc eſt ve<lb></lb>locitates QR, QM cum ſint normaliter directæ, erit eidem <lb></lb><arrow.to.target n="marg121"></arrow.to.target><lb></lb>mobili in B iuxta ſpiralem velocitas QN potentia ipſis am<lb></lb>babus æqualis. </s> <s id="s.000528">Similiterque momento H cum mobilę <lb></lb>fuerit in A, erit velocitas iuxta ſpiralem, ipſa HP æqualis <lb></lb>potentiâ duabus velocitatibus HL iuxta radium, et HO <lb></lb>iuxta tangentem; & ſic omnino liquet, ipſum quadrilineum <lb></lb>HIKP eſſe imaginem velocitatum tam in deſcriptione pa<lb></lb>rabolæ AGF, quàm ſpiralis Archimedeæ DBA, & cum ſit <lb></lb>in ijſdem deſcriptionibus homogenea ſibi ipſi, conſtat ip<lb></lb><arrow.to.target n="marg122"></arrow.to.target><lb></lb>ſas curuis æquales eſſe. </s> <s id="s.000529">Nam vt imago illa ad ſe ipſam ita <lb></lb>parabola ad ſpiralem prædictam. </s> <s id="s.000530">Quod &c. </s> </p> <pb pagenum="52" xlink:href="022/01/058.jpg"></pb> <p type="margin"> <s id="s.000531"><margin.target id="marg118"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000532"><margin.target id="marg119"></margin.target><emph type="italics"></emph>Pr. <gap></gap>. </s> <s id="s.000533">prima.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000534"><margin.target id="marg120"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>prima.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000535"><margin.target id="marg121"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>huius & <lb></lb>Cor. </s> <s id="s.000536">pr.<emph.end type="italics"></emph.end> 13.</s> </p> <p type="margin"> <s id="s.000537"><margin.target id="marg122"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000538"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000539"><emph type="italics"></emph>Hinc aparet, ſpiralem DB ad ſpiralem DBG eandem habe<lb></lb>re rationem, quam quadrilineum QIKN ad quadrilineum <lb></lb>HIKP; pariterque rectam DA ad eandem ſpiralem DCB ha<lb></lb>bere ipſam rationem, ac rectangulum HIKL ad dictum qua<lb></lb>drilineum HIKP. </s> <s id="s.000540">Eodem ferè modo exhiberi pißet ratio ſpi<lb></lb>ralis ad ſpiralem, licèt plurium interſe circulationum, eritque <lb></lb>prorſus ea, quam habet vnum ad alterum eiuſdem illius na<lb></lb>turæ, quadrilineorum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000541"><emph type="center"></emph>PROP. XV. THEOR. XI.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg123"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000542"><margin.target id="marg123"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="main"> <s id="s.000543">SPiralis orta ex motu naturaliter accelerato per <expan abbr="radiũ">radium</expan> <lb></lb>circuli comprehendentis ſpiralem ipſam, & ex motu <lb></lb>æquabili circa <expan abbr="circumferentiã">circumferentiam</expan> eiuſdem circuli, æqualis eſt <lb></lb>ei curuæ parabolicæ natæ ex motu compoſito, cuius vnum <lb></lb>latus curritur iuxta imaginem trianguli, nempe motu gra<lb></lb>uium, alterum verò latus iuxta imaginem trilinei ſecundi, <lb></lb>habebitque parabola ipſa axim æqualem radio, & baſim̨ <lb></lb>tertiæ parti circunferentiæ eiuſdem circuli ſpiralem com<lb></lb>prehendentis. </s> </p> <p type="main"> <s id="s.000544">Eſto ſpiralis ACB, quæ ſignatur ex motu <expan abbr="pũcti">puncti</expan> A æqua<lb></lb>biliter lati circa circumferentiam ADA, dum nempe <expan abbr="eodẽ">eodem</expan> <lb></lb>tempore IF, punctum B currit à quiete lineam BA motu <lb></lb>grauium deſcendentium; ſit verò imago velocitatum dicti <lb></lb>motus æquabilis per ADA rectangulum HGFI, & alte<arrow.to.target n="marg124"></arrow.to.target><lb></lb>rius motus imago, (quæ triangulum erit) eſto FEIM. Pa<lb></lb><arrow.to.target n="marg125"></arrow.to.target><lb></lb>tet, quia ipſæ imagines ponuntur homogeneæ, eſſe rectan<lb></lb>gulum HGFI ad triangulum IFM vt ADA circumferentia <lb></lb>ad radium BA, & propterea IM ad IH erit vt BA ad dimi<lb></lb>dium circunferentiæ AEDA. </s> <s id="s.000546">Sumatur quodlibet <expan abbr="momẽ-tum">momen<lb></lb>tum</expan> K, & ducatur ONKL æquidiſtans HM, puteturque <pb pagenum="53" xlink:href="022/01/059.jpg"></pb>eodem illo momento mobile <expan abbr="vẽtum">ventum</expan> eſſe in C ſpiralis pro<lb></lb>poſitæ BCA: agatur per ipſum punctum radius BCD, & ſic <lb></lb>illo momento extremitas A currendo circa periphæriam <lb></lb>reperietur in D, eritque circunferentia AED ad ipſam <lb></lb>AEDA, vt imago rectangulum OGFK ad <expan abbr="imaginẽ">imaginem</expan> GHIF, <lb></lb>hoc eſt erit vt KF ad FI; at BC ad BD erit vt imago trian<lb></lb>gulum KFL ad triangulum FIM, nempe vt quadratum KF <lb></lb>ad quadratum FI, eſt autem vt BD ad BC ita velocitas <lb></lb>iuxta tangentem in D ad velocitatem iuxta tangentem in <lb></lb>C circulum, cuius radius BC; ſcilicet ita velocitas IH ad <lb></lb>velocitatem KN, quadrati nempe IF ad quadratum KF, & <lb></lb>ob id velocitates, quæ ſunt iuxta tangentes circulos ſubin<lb></lb>de <expan abbr="creſcẽtes">creſcentes</expan> ex centro B, <expan abbr="erũt">erunt</expan> expreſſæ in trilineo HNFIH <lb></lb>ſecundo, cuius ſcilicet indoles eſt vt abſciſſarum quadrata <lb></lb>ſint vt applicatæ. </s> <s id="s.000547">His compoſitis, intellectiſque erit in B, <lb></lb>momento F, nulla velocitas, in C momento K duæ velo<lb></lb>citates quarum vnà KI mobile iret iuxta CD, ſed cum al<lb></lb>tera ſit KN iuxta tangentem circulum, cuius radius CB, ne<lb></lb><arrow.to.target n="marg126"></arrow.to.target><lb></lb>ctitur vna ex duabus illis, quibus <expan abbr="eiſdẽ">eiſdem</expan> potentia eſt æqua<lb></lb><arrow.to.target n="marg127"></arrow.to.target><lb></lb>lis, & qua idem mobile mouetur iuxta ſpiralem illo mo<lb></lb>mento K. </s> <s id="s.000548">Similiter cum mobile eſt in D, ſcilicet momento <lb></lb>I, habebit velocitatem potentia æqualem HI, qua dirigitur <lb></lb>iuxta tangentem, & velocitati IM, qua ſecundùm radium, <lb></lb>Itaque imago velocitatum mobilis deſcribentis ſpiralem <lb></lb>propoſitis motibus tempore IF, ea erit, cuius applicatæ <lb></lb>ſunt vbique æquales potentia ijs applicatis, quæ ab <expan abbr="eodẽ">eodem</expan> <lb></lb>momento intelligi queunt in imaginibus ſimplicibus, nem<lb></lb>pe partialium motuum, HNFI, IFM. </s> <s id="s.000549">Cum præterea OT <lb></lb>ponatur tertia pars eſſe circumferentiæ AEDA, & eſt <expan abbr="etiã">etiam</expan> <lb></lb>trilineum HFI vtpote ſecundum tertia pars <expan abbr="parallelogrã-mi">parallelogram<lb></lb>mi</expan> HGFI, erit triangulum IFM ad trilineum ipſum HFI vt <lb></lb><arrow.to.target n="marg128"></arrow.to.target><lb></lb>BA, vel ei æqualis QO ad OT; curritur verò vt ſupponi<lb></lb>tur OQ tempore IF iuxta imaginem triangulum IFM, ergo <lb></lb><arrow.to.target n="marg129"></arrow.to.target><lb></lb>eodem tempore iuxta trilineum HNF curretur alterum la-<pb pagenum="54" xlink:href="022/01/060.jpg"></pb>tus OT, ſiue baſis parabolæ QI. </s> <s id="s.000550">Si itaque parabola ipſa <lb></lb>putetur eſſe ORI, in qua punctum R eſto vbi mobile adeſt <lb></lb>momento K, deducantur verò ab eodem illo puncto RS <lb></lb>parallela axi QO, et RP æquidiſtans QI, vel OT, profectò <lb></lb>in O, momento F, ſicuti in ſpirali, nulla erit mobili veloci<lb></lb>tas, ſed cum eſt in R momento K habebit geminam veloci<lb></lb>tatem, KL ſecundùm SR, et KN iuxta PR perpendicularem <lb></lb>ipſi SR, quæ duæ velocitates itidem component vnicam <lb></lb>potentia ſimul illis æqualem, & cum idem dicatur de qui<lb></lb>buſcunque alijs punctis parabolæ, momentis temporis FI <lb></lb>reſpondentibus, manifeſtum eſt ſpirali BCA, & parabolæ <lb></lb>ORI vnicam, eandemque eſſe imaginem velocitatum, pro<lb></lb>pterquam quòd ipſæ curuæ, quòd ſint vt imagines, erunt <lb></lb>interſe æquales. <lb></lb><arrow.to.target n="marg130"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000551"><margin.target id="marg124"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000552">pr.<emph.end type="italics"></emph.end> 4. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000553"><margin.target id="marg125"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primą<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000554"><margin.target id="marg126"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000555"><margin.target id="marg127"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000556">prop.<emph.end type="italics"></emph.end> 13. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000557"><margin.target id="marg128"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000558"><margin.target id="marg129"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000559"><margin.target id="marg130"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>prima.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000560"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000561"><emph type="italics"></emph>Exemplo traditarum curuarum, poſſunt innumeræ ſpira<lb></lb>les ſuis parabolis æquales excogitari, nec ideo res minùs de<lb></lb>monſtrabitur, ſi loco rectarum, ſeu laterum OT, OP compoſiti <lb></lb>motus, ſubſtituantur circuli, aut circulorum arcus, qui ad re<lb></lb>ctos angulos ſe ſecent, ſcilicet <expan abbr="cũ">cum</expan> tangentes ad punctum infle<lb></lb>xionis, ſeu occurſus ipſarum curuarum ſibi ipſis perpendicula<lb></lb>res fuerint. </s> <s id="s.000562">Quòd ſi ipſa curua latera ad rectos angulos non <lb></lb>ſe ſecent curuæ nihilominus ab ipſo compoſito motu naſcen<lb></lb>tes poterunt exhiberi curuas parabolicas exequantes, quarum <lb></lb>itidem latera ſint rectæ eundem angulum, quem prædictæ <expan abbr="tã-gentes">tan<lb></lb>gentes</expan>, comprehendentes. </s> <s id="s.000563">Sed de his ſatis, nunc dicamus ea <lb></lb>tempora, quibus duorum pendulorum ſimiles vibrationes ab<lb></lb>ſoluuntur, hoc eſt Galilei ſententiam demonſtrabimus, quam <lb></lb>quondam haud ruditer decepti falſam credidimus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000564"><emph type="italics"></emph>Vincentius Viuianus eximius noſtri æui Geometra vt tue<lb></lb>retur Galilei ſententiam, cuius digniſſimè ſe fuiſſe diſcipu<lb></lb>lum profitetur, tradidit mihi per admodum Reuerendum, at-<emph.end type="italics"></emph.end><pb pagenum="55" xlink:href="022/01/061.jpg"></pb><emph type="italics"></emph>que cultiſſimum Patrem Ioſeph Ferronum è Societate Ieſu, de<lb></lb>monſtrationem ſuam verè pulcherrimam, ac diſertiſſimè <lb></lb>exaratam, qua vna potuiſſem de Galilei aßerto ſatisfactus <lb></lb>eſſe; eam demonſtrationem, ijſdem prorſus verbis, ac figuris, <lb></lb>quibus ad me peruenit hic duxi reponendam, ne gloriam̨, <lb></lb>quam Vir tantus meretur, ipſi videremur noſtra, quam inde <lb></lb>ſubdemus, demonſtratione, ſubripere.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000565"><emph type="italics"></emph>Inquit ergo.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000566">TEmpora naturalium de curſuum ſphærarum grauium <lb></lb><arrow.to.target n="marg131"></arrow.to.target><lb></lb>per ſimiles, ſimiliterque ad horizontem inclinatos <lb></lb>arcus curuarum linearum in planis, aut verticalibus, aut <lb></lb>ad horizontem æqualiter inclinatis deſcriptarum, & quæ <lb></lb>totæ ſint ad eaſdem partes cauæ, interſe ſunt in ſubdupli<lb></lb>cata ratione chordarum eorundem arcuum homologè <lb></lb>ſumptarum. </s> </p> <p type="margin"> <s id="s.000567"><margin.target id="marg131"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 1. 2 <lb></lb>3. 4.</s> </p> <p type="main"> <s id="s.000568">Ex puncto A ad curuam lineam BCD extra ipſam iņ <lb></lb>plano poſitam, & in totum ad eaſdem partes cauam, quæ<lb></lb>cunque ea ſit (vel nimirum pars aliqua circumferentiæ <lb></lb>circuli, vel alicuius ex infinitis ellipſibus, aut parabolis, aut <lb></lb>hyperbolis, aut ſpiralibus, aut cycloidibus, vel concoidis, <lb></lb>vel ciſoidis, ſeu alterius cuiuſcumque ex notis, vel ignotis <lb></lb>curuis educantur omnes rectæ AB, AC, AD &c. </s> <s id="s.000569">quæ à <lb></lb>punctis E, F, C, vel intra, vel extra eas ſumptis proportio<lb></lb>nalibus ſecentur, ita vt ſit AB ad AE, ſicut AC ad AF, & <lb></lb>ſicut AD ad AG &c. </s> <s id="s.000570">& hoc ſemper. </s> <s id="s.000571">Sic enim dubio pro<lb></lb>cul apparet, prout facillimum eſt oſtendere, lineam EFG <lb></lb>tranſeuntem per ſingula puncta E, F, G ſic inuenta, cur<lb></lb>uam <expan abbr="quoq;">quoque</expan> eſſe, & eiuſdem penitus naturæ, ac data BCD <lb></lb>eique ſimilem, ſimiliterque cum ipſa poſitam, atque in to<lb></lb>tum cauam ad eaſdem partes, ad quas ponitur caua ipſą <lb></lb>BCD. </s> <s id="s.000572">Concipiatur modò planum, in quo manent huiuſ<lb></lb>modi ſimilium curuarum ſimiles arcus BCD, EFG, vel eſſe <lb></lb>ad horizontem erectum, nempè verticale, vel ad ipſum̨ <pb pagenum="56" xlink:href="022/01/062.jpg"></pb>horizontem inclinatum iuxta curuitates ipſorum arcuum <lb></lb>BCD, EFG inflexas eſſe ſuperficies eidem plano erectas, <lb></lb>ita tamen, vt ſuper has poſitis grauibus ſphæris in A, E per <lb></lb>ipſas ſic inflexas ſuperficies eædem ſphæræ naturaliter <lb></lb>decurrere queant; id quod ſanè accidet, cum arcus BCD <lb></lb>totus fuerit infra horizontalem IL ex arcus ſubli<lb></lb>miori puncto B ductam, fuerintque ab hac continuati re<lb></lb>ceſſus, ac totus ad vnam partem perpendiculi BH: nam ſic <lb></lb>talis quoque erit alter arcus EFG illi BCD ſimilis, ſimili<lb></lb>terque poſitus. </s> <s id="s.000573">His omnibus ſic manentibus: Dico tem<lb></lb>pus decurſus ſphæræ grauis E per ſimilem, ſimiliterque po<lb></lb>ſitum arcum EFG, eſſe in ſubduplicata ratione chordarum <lb></lb>BO, EG arcus ipſos ſubtendentium. </s> <s id="s.000574">Secto enim bifariam <lb></lb>angulo BAD per rectam AC arcum BD ſecantem in C, <lb></lb>atque arcum EFG in F, iungantur chordæ BC, CD, et EF, <lb></lb>FG, quæ ex huiuſmodi curuarum natura cadent totæ intra <lb></lb>ipſos arcus, ſed in prima, & ſecunda figura ad partes poli <lb></lb>A, in tertia verò, & quarta ad oppoſitas. </s> </p> <p type="main"> <s id="s.000575">Et quoniam, ex talium curuarum geneſi, eſt vt BA ad <lb></lb>AE, ita DA, ad AG, erit BD ipſi EG parallela, hoc eſt <lb></lb>vtraque ad horizontem æqualiter inclinata, atque in ra<lb></lb>tione BA ad AE. </s> <s id="s.000576">Similiter cum ſit, vt BA ad AE, ita CA <lb></lb>ad AF, etiam BC, EF interſe æquidiſtabunt, ſeu ad hori<lb></lb>zontem æqualiter inclinabuntur, eruntque in ratione ea<lb></lb>dem, ac BA ad AE. </s> <s id="s.000577">Idemque oſtenditur de chordis CD, <lb></lb>FG, quare ex magni Galilei ſententia de motu naturaliter <lb></lb>accelerato indubitanter ſequitur tempus decurſus ſphæræ <lb></lb>grauis ex B in D per binas chordas BC, CD ad tempus <lb></lb>decurſus per vnicam BD, eſſe vt tempus decurſus grauis <lb></lb>ſphæræ ex E in G per binas EF, FG ad tempus decurſus <lb></lb>per vnicam EG: eadem itidem ratione demonſtratur (an<lb></lb>gulis pariter BAC, CAD bifariam ſectis per rectas, quæ <lb></lb>ſimiles arcus BC, EF, ac CD, FG duas in partes diuidant) <lb></lb>ex quatuor vtrinque arcuum horum cordis, illas interſe <pb pagenum="57" xlink:href="022/01/063.jpg"></pb>homologas, ſimileſque arcus ſubtendentes ad horizonte m <lb></lb>eſſe æqualiter inclinatas, ac alteram alteri in ratione ea<lb></lb>dem, in qua ſunt rectæ AB, AE &c: ac propterea ex ea<lb></lb>dem Galilei ſcientia conſtabit vtique, tempus decurſus ex <lb></lb>B in C ſphæræ grauis B per quatuor chordas quatuor par<lb></lb>tes arcus BCD ſubtendentes ad tempus decurſus per vni<lb></lb>cam BD, eſſe vt tempus decurſus ſphæræ grauis E ex E in <lb></lb>G per quatuor illis homologas chordas quatuor partes <lb></lb>arcus EFG pariter ſubtendentes ad tempus decurſus per <lb></lb>vnicam chordam EG: & hoc ſemper ita euenire demon<lb></lb>ſtrabitur quantacunque, & maxima fuerit in perpetua an<lb></lb>gulorum biſectione æquèmultiplicitas in vtroque arcu <lb></lb>talium chordarum homologè ſumptarum, ac interſe pro<lb></lb>portionalium, æqualiterque ad horizontem inclinatarum: <lb></lb>Propterquam quòd ſemper decurſus ex B in D per aggre<lb></lb>gatum chordarum omnium in arcu BCD ad tempus de<lb></lb>curſus per ſolam chordam BD eſſe vt tempus decurſus ex <lb></lb>E in G per aggregatum totidem chordarum in arcu EFG <lb></lb>ad tempus decurſus per vnicam chordam EG; adeo vt de<lb></lb>nique iure optimo educi poſſe videatur, tempus decurſus <lb></lb>grauis ex B in D per aggregatum infinitarum chordarum <lb></lb>totum arcum BCD conſtituentium, ſeu tempus per ipſum <lb></lb>arcum BCD ad tempus decurſus per ſolam cordam BD <lb></lb>eſſe vt tempus decurſus grauis ex E in G per aggregatum <lb></lb>totidem infinitarum chordarum dictis homologè propor<lb></lb>tionalium, æqualiterque ſingulæ ſingulis ad horizontem̨ <lb></lb>inclinatarum, ac totum arcum EFG conformantium, ſiue <lb></lb>vt tempus per ipſum arcum EFG per ſolam chordam EG. <lb></lb></s> <s id="s.000578">Quocirca permutando, tempus, decurſus ſphæræ grauis B <lb></lb>per arcum BCD ad tempus decurſus ſphæræ grauis E per <lb></lb>arcum ſimilem, ſimiliterque poſitum EG erit vt tempus <lb></lb>decurſus per chordam BD ad tempus decurſus per chor<lb></lb>dam EG; ſed ex eadem Galilaica ſcientia de motu, tempus <lb></lb>decurſus per chordam BD ad tempus decurſus per æqua-<pb pagenum="58" xlink:href="022/01/064.jpg"></pb>iter inclinatam EG eſt in ſubduplicata ratione ipſarum̨ <lb></lb>chordarum BD, EG; ergo tempus quoque decurſus ex B <lb></lb>per arcum BCD ad tempus decurſus ex E per arcum EFG <lb></lb>eſt in eadem ſubduplicata ratione chordę BD ad chordam <lb></lb>EG, quod oſtendendum propoſuimus. </s> </p> <p type="main"> <s id="s.000579"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000580"><emph type="italics"></emph>Ex modò ostenſis ſuper prima, ac ſecunda figura, manife<lb></lb>ſtum fit celeberrimum illud magni Galilei pronuntiatum, <lb></lb>quòd videlicet, ratio temporum ſimilium vibrationum pen<lb></lb>dulorum ſit ſubduplicata rationis longitudinum filorum ho<lb></lb>mologè ſumptorum, non tantum verum eſſe de vibrationibus <lb></lb>pendulorum per arcus ſimiles, ſimiliterque poſitos, ſumptos <lb></lb>ex circulorum quadrantibus ad perpendiculum vſque termi<lb></lb>nantes, ſed etiam de vibrationibus per arcus quoſcumque ſi<lb></lb>miles quadrantum à perpendiculo ſeiunctos: dummodo ipſi <lb></lb>ſimiles arcus ſint quoque ſimiliter poſiti: quales nimirùm ap<lb></lb>parent in figuris prima, ac ſecunda arcus BCD, EFG, dum <lb></lb>grauia B, E ex filis, aut haſtulis AB, AE circa punctum A <lb></lb>conuertibilibus appenſa concipiantur.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000581"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000582"><emph type="italics"></emph>Si curua BCD, EFG in prima, & ſecunda figura fuerint <lb></lb>ſimiles arcus ex circulis commune centrum A habentibus; ac <lb></lb>in verticali plano poſitis, & in prima figura recta AB, AE <lb></lb>fuerint fila aut haſtulæ quædam circa clauum A conuertibi<lb></lb>les, in ſecunda verò recta AB, AE concipiantur, vt haſtulæ <lb></lb>inflexibiles, volubileſque circa imum punctum E, atque ex <lb></lb>huiuſmodi filorum, aut haſtularum terminis B, E pendeant <lb></lb>graues ſphæræ B, E (cum eadem ſint tempora prout aſſumi<lb></lb>tur quoque ab ipſo met Ceua) tempora inquam decurſuum <lb></lb>liberorum granium B, E per arcus BCD, EFG, ac temporą<emph.end type="italics"></emph.end><pb pagenum="59" xlink:href="022/01/065.jpg"></pb><emph type="italics"></emph>deſcenſuum ipſorum grauium per eoſdem arcus (vel hac à <lb></lb>filis pendeant, vel ab hastulis ſustineantur) erit quoque tem<lb></lb>pus deſcenſus, ſeu vibrationis penduli B per arcum BCD ad <lb></lb>tempus deſcenſus, ſeu vibrationis penduli E per arcum EFD <lb></lb>in ſubduplicata ratione chordæ BD ad chordam EG; ſed hæc <lb></lb>ratio chordarum BD, EG eadem eſt, ac ratio filorum, aut ha<lb></lb>ſtularum AB, AE; Ergo tempus vibrationis penduli AB per <lb></lb>arcum BCD ad tempus vibrationis penduli AE per arcum il<lb></lb>li ſimilem, ſimiliterque poſitum EFG est quoque in ſubdupli<lb></lb>cata ratione longitudinum, vel filorum, aut haſtularum, ex <lb></lb>quibus eadem grauia pendula ſimiles vibrationes abſoluunt <lb></lb>BCD, EFG.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000583"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000584"><emph type="italics"></emph>Cæterùm non me latet conſtructionem, ac demonstratio-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg132"></arrow.to.target><lb></lb><emph type="italics"></emph>nem à nobis ſuperiùs allatam nonnullis euidentiorem fortaſſe <lb></lb>euaſuram, ſi ommiſſa illa continua biſectione angulorum ſi<lb></lb>miles, ſimiliterque poſitos arcus abſcindentium ex ſimilibus <lb></lb>curuis ibidem deſcriptis; atque ommiſſa pariter continua co<lb></lb>niunctione chordarum, vt ibi factum fuit, horum vice, vt in <lb></lb>quinta figura, ex punctis B, D binæ tangentes curuam BCD <lb></lb>ducantur BH, DH, quæ omninò mutuò ſe ſecabunt in puncto <lb></lb>H (ob conditiones in ipſa Theorematis expoſitione vltimo lo<lb></lb>co poſitas) atque ex E, G ipſis BH, DH agantur æquidistan<lb></lb>tes, quæ iunctæ, AH ſimul occurrent in I, curuamque EFG <lb></lb>contingent pariter ad E, G (quæ omnia ſi opus fuerit, facilè <lb></lb>demonſtrabuntur) ac inſuper, ſi à puncto C, in quo iunctą <lb></lb>AH ſecat arcum BCD, agatur tangens LM primas BH, DH <lb></lb>ſecans in LM; Per F verò, in quo AICH ſecat arcum EFG <lb></lb>agatur NO parallela tangenti LM, quæ curuam pariter EFG <lb></lb>tanget ad F, ac tangentes EI, GI ſecabit ad NO: & ſi iunctis <lb></lb>inſuper AL, AM, eadem, quam nunc explicauimus, continue<lb></lb>tur conſtructio per alias, atque alias tangentes, ac parallelas<emph.end type="italics"></emph.end><pb pagenum="60" xlink:href="022/01/066.jpg"></pb><emph type="italics"></emph>&c. </s> <s id="s.000585">ſic enim vnicuique harum curuarum circumſcribetur <lb></lb>rectilineum, primò ex binis tangentibus, ſecundò ex tribus, <lb></lb>tertiò ex quinque, quartò ex ſeptem, & ſic vlteriùs iuxta re<lb></lb>liquos impares numeros ſucceſſiuè ſumptos; atque omnia pa<lb></lb>ria talium æquidiſtantium tangentium eam ſemper inter ſe <lb></lb>rationem ſeruabunt, quam habent chorda BD, EG, ſen quam <lb></lb>habent rectæ BA, EA, <expan abbr="eruntq;">eruntque</expan> interſe æqualiter inclinatæ; <lb></lb><expan abbr="adeoq;">adeoque</expan> tempora decurſuum grauium B, E tam per ſummas <lb></lb>binarum tangentium BH, HD, EI, IG, quàm per minores <lb></lb>ſummas, ex quinque ſimul chordis vtrinque ſumptas, aut <lb></lb>quàm per alias ſemper minores ſummas huiuſmodi tangen<lb></lb>tium iuxta quantumuis maiorem numerum imparem æquè <lb></lb>multipliciter ſumptarum, erunt perpetuò proportionalia tem<lb></lb>poribus decurſuum per chordas BD, EG; & hoc ſemper; etiam<lb></lb>ſi per huiuſmodi decrementa aggregatorum ex tangentibus <lb></lb>vtrinque æquèmultipliciter ſumptis, deueniatur ad vltimus, <lb></lb>ac breuiſſimas ipſis arcubus circumſcriptiones polygonorum <lb></lb>ex lateribus numero innumerabiliter aquèmultiplicibus, hoc <lb></lb>eſt ad ipſos ſimiles, ſimiliterque poſitos arcus BCD, EFG, <lb></lb>quorum ſingula homologorum laterum, ſeu punctorum paria, <lb></lb>vt B, & E; C et F; D, et G &c. </s> <s id="s.000586">haberi poßunt tanquam tot <lb></lb>paria parallelarum, ac proportionalium tangentium ipſos ſi<lb></lb>miles, ac ſimiliter poſitos arcus conſtituentia. </s> <s id="s.000587">Quapropter <lb></lb>ratio <expan abbr="quoq;">quoque</expan> temporum decurſuum per ipſos arcus, ſimilis erit <lb></lb>rationi temporum decurſuum per chordas; ſed horum decur<lb></lb>ſuum ratio ſubdupla eſt rationis inter ipſas chordas. </s> <s id="s.000588">Quare, <lb></lb>& alia hac methodo conſtaret propoſitum.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000589"><margin.target id="marg132"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 6 <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 5.</s> </p> <p type="main"> <s id="s.000590"><emph type="italics"></emph>Hactenus grauiſſimus Vir; ſupereſt modò, vt quemadmo<lb></lb>dum annuimus, veritatem eandem noſtra quoque methodo, <lb></lb>confirmemus, vt ijs, quibus ſatis probat demonſtratio allata, <lb></lb>ſit nostra, quam afferemus, in experimentum traditarum hùc <lb></lb><expan abbr="vſq;">vſque</expan> rerum; & quibus ſecùs acciderit ex aliqua dubitatione, <lb></lb>hæc per demonſtrationes noſtras prorſus, ſtatimq tollatur. <lb></lb></s> <s id="s.000591">Illud etiam admoneo, eam rem non tantum me oſtenſurum,<emph.end type="italics"></emph.end><pb pagenum="61" xlink:href="022/01/067.jpg"></pb><emph type="italics"></emph>vt pulcherrima, <expan abbr="vtilimaq;">vtilimaque</expan> veritas pluribus demonſtrationi<lb></lb>bus aperiatur; verùm potius vt ampliſſima Methodus, qua tum <lb></lb>vtemur, aliorum motuum demonſtrandorum in exemplum <lb></lb>veniat.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000592"><emph type="center"></emph>PROP. XVI. THEOR. XII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000593">IN eadem recta CD coeant duæ planæ, <expan abbr="interſeq;">interſeque</expan> ſimiles, <lb></lb><arrow.to.target n="marg133"></arrow.to.target><lb></lb>ac prorſus æquales figuræ ADCA, BDCB, & quidem <lb></lb>ita, vt ab eodem puncto M ſi ducatur MH parallela CA, <lb></lb>et ML ipſi CB, ſit ſemper MH æqualis ML, quemadmo<lb></lb>dum æquales ſunt interſe CA, CB. </s> <s id="s.000594">Dico (ſi concipiatur <lb></lb>ſolidum eius indolis, vt ductis rectis BA, LH cadant iſtæ <lb></lb>omninò in ſolidi iſtius ſuperficie; ipſum verò ſolidum, quod <lb></lb>ſit BADC, ſecetur plano quolibet æquidiſtante figuræ <lb></lb>BCD) fore, vt ſectio iſta KFEIK, ſit prorſus ſimilis, æqua<lb></lb>liſque alteri conterminæ AEI; ſed opportet, vt palam eſt, <lb></lb>coeuntes illæ figuræ non in eodem plano reperiantur. </s> </p> <p type="margin"> <s id="s.000595"><margin.target id="marg133"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="main"> <s id="s.000596">Cum duo plana inuicem parallela KIE, BCD ſecent <lb></lb>alia duo interſe item parallela ACB, HML, erunt commu<lb></lb>nes ſectiones, interſe omnes æquidiſtantes rectæ lineæ KI, <lb></lb>GF, ML, CB. </s> <s id="s.000597">Cum verò ob naturam ſolidi, ſectiones <lb></lb>BAC, IHM triangula ſint rectilinea, erit vt BC ad CA, <lb></lb>ita KI ad IA. </s> <s id="s.000598">Sunt autem priores interſe æquales, ergo & <lb></lb>poſtremæ KI, AI interſe æquabuntur. </s> <s id="s.000599">Eademque ratione <lb></lb>ſunt æquales HG, GF: & quoniam ob ſimilitudinem figu<lb></lb>rarum angulus BCD æquatur angulo ACD, & angulus <lb></lb>BCD æqualis angulo KIE (nam etiam CD, IE ſunt rectæ <lb></lb>æquidiſtantes, cum nempe ſint communes ſectiones plani <lb></lb>DCA ſecantis duo æquidiſtantia KIE, BCD) ergo cum̨ <lb></lb>angulus pariter ACD æquet angulum AIE, erunt anguli <lb></lb>KIE, AIE, et FGE, HGF æquales. </s> <s id="s.000600">Quod &c. </s> </p> <pb pagenum="62" xlink:href="022/01/068.jpg"></pb> <p type="main"> <s id="s.000601"><emph type="center"></emph>PROP. XVII. THEOR. XIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000602">IIſdem manentibus. </s> <s id="s.000603">Dico triangula ACB, LHM eſſę <lb></lb>ſimilia. </s> <s id="s.000604">Sunt enim parallelæ &c. </s> <s id="s.000605">interſe tam rectæ CB, <lb></lb>ML, quàm CA, MH; ideo anguli ACB, HML interſe <lb></lb>æquabuntur, & ſunt circa eos proportionalia latera, nem. <lb></lb></s> <s id="s.000606">pe BC ad CA, vt LM, MH; ergo conſtat propoſitum. </s> </p> <p type="main"> <s id="s.000607"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000608"><emph type="center"></emph><emph type="italics"></emph>Simul conſtat rectas AB, LH interſe æquidiſtare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000609"><emph type="center"></emph>PROP. XVIII. THEOR. XIV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000610">IIſdem vt ſupra manentibus, ita tamen vt ACD ſit an<lb></lb>gulus rectus (ſic enim DC perpendicularis erit duabus <lb></lb>AC, CB) Dico ſolidum huiuſmodi ad priſma, cuius baſis <lb></lb>ABC, & altitudo CD eandem habere rationem, quam ſo<lb></lb>lidum rotundum ortum ex rotatione figuræ CAD circą <lb></lb>axem CD ad cylindrum genitum ex conuerſione rectan<lb></lb>guli AC in CD circa eundem axem. <lb></lb><arrow.to.target n="marg134"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000611"><margin.target id="marg134"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 8.</s> </p> <p type="main"> <s id="s.000612">Compleatur ipſum priſma, & ſit quidem AQDPBC, <lb></lb>quod ſecetur vnà cum propoſito ſolido per quoduis pla<lb></lb>num baſi ACB æquidiſtans: fiet in priſmate ſectio trian<lb></lb>gulum OMN ſimile, æqualeque ipſi ACB, & in altero ſo<lb></lb>lido triangulum LHM eidem ACB ſimile. </s> <s id="s.000613">Triangulum <lb></lb>ACB priſmatis ad <expan abbr="triãgulum">triangulum</expan> idem ſolido propoſito com<lb></lb>mune, eſt vt circulus radio CA deſcriptus ad circulum <lb></lb>eundem; Item triangulum NOM ſectio priſmatis eſt ad <lb></lb>triangulum LHM ſectionem propoſiti ſolidi, vt circulus ex <lb></lb>radio MO deſcriptus ad circulum radio MH. </s> <s id="s.000614">Cum dein<lb></lb>de idem dicatur de alijs omnibus ſectionibus priſmatis, & <pb pagenum="63" xlink:href="022/01/069.jpg"></pb>propoſiti ſolidi erunt omnes ſimul primæ, quæ interſę </s> </p> <p type="main"> <s id="s.000615"><arrow.to.target n="marg135"></arrow.to.target><lb></lb>æquales ſunt, ad omnes ſimul ſecundas vt omnes tertiæ, <lb></lb>his partibus interſe æqualibus, ad omnes quartas; ſcilicet <lb></lb>erunt omnia triangula priſmatis, ſeu ipſum priſma ad om<lb></lb>nia triangula propoſiti ſolidi, ſeu ad ipſum ſolidum, vt om<lb></lb>nes circuli eius cylindri, qui oritur ex conuerſione figuræ <lb></lb>ADCA circa axem CD, hoc eſt vt ipſum ſolidum rotun<lb></lb>dum, ſeu cylindrus ad omnes ſimul circulos ſolidi rotundi <lb></lb>geniti ex rotatione figuræ AHDCA circa axem <expan abbr="ipsũ">ipsum</expan> CD, <lb></lb>ſeu ad ipſum propoſitum ſolidum. </s> <s id="s.000616">Quod &c. </s> </p> <p type="margin"> <s id="s.000617"><margin.target id="marg135"></margin.target><emph type="italics"></emph>lemmæ<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>in <lb></lb>libro de dim. <lb></lb></s> <s id="s.000618">parab. </s> <s id="s.000619">Euang. <lb></lb>Tęrricel.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000620"><emph type="center"></emph>PROP. XIX. THEOR. XV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000621">ET rurſus ipſa manente figura patet, ſi ducantur HR, <lb></lb>LS parallelæ MD, fore non ſolum figuram AHDPA, <lb></lb>ſimilem, ac æqualem BLDQB; verùm etiam APRHA ipſi <lb></lb>BLSQB: Cum ita ſit, aio, eundem cylindrum ad ſoli<lb></lb>dum rotundum genitum, ex volutatione figuræ APD cir<lb></lb>ca eundem axem CD eandem rationem habere, ac priſma <lb></lb><expan abbr="prædictũ">prædictum</expan>, cuius baſis ACB, altitudo AP ad ſolidum, quod <lb></lb>ſupereſt ex ipſo priſmate, dempto ſolido ACBLDHA. </s> </p> <p type="main"> <s id="s.000622">Nam ex præterita propoſitione nouimus, dictum priſma <lb></lb>ad ſolidum eius partem ACBLDHA eſſe vt cylindrus or<lb></lb>tus ex conuerſione rectanguli CP circa axem CD ad par<lb></lb>tem eius rotundum circa axem eundem CD conuerſa fi<lb></lb>gura ADC, ergo per conuerſionem rationis, erit id quod <lb></lb>propoſuimus. </s> </p> <p type="main"> <s id="s.000623"><emph type="center"></emph>DEF. IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000624">QVodcunque ex dictis propoſitis ſolidis vocetur ab <lb></lb>ea figura, iuxta quam intelligitur ortum. </s> <s id="s.000625">Scilicet <lb></lb>ACBLDHA dicatur à figura AHDCA, & alte<lb></lb>rum, quod fuit reſiduum prædictum dicatur à figura AH<lb></lb>DPA. </s> </p> <pb pagenum="64" xlink:href="022/01/070.jpg"></pb> <p type="main"> <s id="s.000626"><emph type="center"></emph>PROP. XX. THEOR. XVI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000627">SI à quibuſcunque figuris fuerint duo ſolida, hæc inter<lb></lb><arrow.to.target n="marg136"></arrow.to.target><lb></lb>ſe erunt vt ſolida alia genita ex conuerſione illarum <lb></lb>figurarum circa communem ſectionem ſimilium, æqua<lb></lb>lium, ac interſe coeuntium figurarum. </s> </p> <p type="margin"> <s id="s.000628"><margin.target id="marg136"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 9.</s> </p> <p type="main"> <s id="s.000629">Solidum à figura ABC ſit CAFDBC, & quod eſt à fi<lb></lb>gura GLH eſto HGILH. </s> <s id="s.000630">Dico illud ad hoc ſolidum eſſe <lb></lb>vt rotundum natum ex conuerſione figuræ ABC circą <lb></lb>axem CE ad rotundum ortum ex <expan abbr="cõuerſione">conuerſione</expan> figuræ GLH <lb></lb>circa axem HL. </s> <s id="s.000631">Opportet tamen angulos ACF, GHI <lb></lb>æquales eſſe. </s> <s id="s.000632">Intelligantur priſmata triangularia, quorum <lb></lb>baſes ACF, GHI, & altitudines CE, HL; hoc eſt ſint ipſa <lb></lb>ſolida priſmatica AFCEBD, GIHLMK. </s> <s id="s.000633">Solidum à figu<lb></lb><arrow.to.target n="marg137"></arrow.to.target><lb></lb>ra ABC ad priſma AFCEBD habet eandem rationem, <lb></lb>quam ſolidum rotundum ortum ex conuerſione ſiguræ <lb></lb>ABC circa axem CE ad cylindrum natum ex rotatione <lb></lb>ABEC circa eundem axem CE; hic verò cylindrus ad cy<lb></lb>lindrum alium natum ex rotatione rectanguli GMLH cir<lb></lb>ca axem HL eſt vt priſma, cuius baſis ACF, altitudineque <lb></lb>CE ad alterum priſma baſem habens GHI ſimilem ipſi CF <lb></lb>(nam circa angulos æquales H, C ſunt latera etiam pro<lb></lb>portionalia, nempe æqualia) & altitudinem HL. </s> <s id="s.000634">Solidum <lb></lb>præterea, hoc eſt priſma GKHM ad ſolidum, quod eſt à <lb></lb><arrow.to.target n="marg138"></arrow.to.target><lb></lb>plano GLH habet eandem rationem, ac cylindrus, qui fit <lb></lb>ex conuerſione rectanguli HM circa axem HL ad ſolidum <lb></lb>rotundum ortum ex circumactione figuræ GLH circa ip<lb></lb>ſum axem HL, ergo ex æquali erit ſolidum à figura ABC <lb></lb>ad ſolidum à figura GLH, vt rotundum ex rotatione figu<lb></lb>ræ ABC circa axem CE ad rotundum alterum ex conuer<lb></lb>ſione alterius figuræ GLH circa axem HL. </s> <s id="s.000635">Quod &c. </s> </p> <pb pagenum="65" xlink:href="022/01/071.jpg"></pb> <p type="margin"> <s id="s.000636"><margin.target id="marg137"></margin.target>18. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000637"><margin.target id="marg138"></margin.target><emph type="italics"></emph>Ex eadem.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000638"><emph type="center"></emph>PROP. XXI. THEOR. XVI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000639">PRopoſitis ijſdem ſolidis, erunt inter ſe, vt momenta fi<lb></lb>gurarum a quibus ſunt, quæ tamen figuræ ſuſpenſæ <lb></lb>ſint ex longitudinibus deductis ab ipſarum grauitatum̨ <lb></lb>centris vſque ad coeuntium figurarum communes illas ſe<lb></lb>ctiones. </s> </p> <p type="main"> <s id="s.000640">Figuræ, à quibus ſunt ſolida, ponantur ABC, GLH, <expan abbr="cẽ-">cen<lb></lb></expan><arrow.to.target n="marg139"></arrow.to.target><lb></lb>tra grauitatum illarum M, N; axes, ſiue communes ſectio<lb></lb>nes coeuntium binarum interſe ſimilium, ac æqualium fi<lb></lb>gurarum à quibus dicuntur ipſa ſolida; & demum MO, NP <lb></lb>perpendiculares ſint ab ipſis centris ad illas communes ſe<lb></lb>ctiones deductæ CE, HL. Dico, ſolidum à plana figurą <lb></lb>ABC ad ſolidum a plana GHL eandem habere rationem, <lb></lb>ac momentum figuræ ABC pendentis ex MO ad momen<lb></lb><arrow.to.target n="marg140"></arrow.to.target><lb></lb>tum alterius figuræ ſuſpenſæ ex NP, ſunt enim hæc ſoli<lb></lb>da interſe, vt rotunda, quorum genetrices figuræ ABC, <lb></lb>GLH circa axes CE, HL, huiuſmodi verò ſolida ſunt vt <lb></lb><arrow.to.target n="marg141"></arrow.to.target><lb></lb>momenta propoſita; ergo ſolidum à plana figura ABC ad <lb></lb>ſolidum à plana GLH, erit vt momentum figuræ ABC <lb></lb>ſuſpenſæ ex MO ad momentum GLH pendentis ex NP. <lb></lb></s> <s id="s.000641">Quod &c. </s> </p> <p type="margin"> <s id="s.000642"><margin.target id="marg139"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 10.</s> </p> <p type="margin"> <s id="s.000643"><margin.target id="marg140"></margin.target><emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000644"><margin.target id="marg141"></margin.target><emph type="italics"></emph>Ter. lem.<emph.end type="italics"></emph.end> 31. <lb></lb><emph type="italics"></emph>in libro </s> <s id="s.000645">di<lb></lb>men. parabolæ. <emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000646"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000647"><emph type="italics"></emph>Cum ipſa illa momenta nectantur ex rationibus figurarum <lb></lb><arrow.to.target n="marg142"></arrow.to.target><lb></lb>ABC, GLH, & ex longitudinibus, ex quibus pendent ipſæ fi<lb></lb>gura (nam habentur vt grauia) ex ijſdem etiam rationibus <lb></lb>componentur ſolida, qua ſunt ab ipſis figuris—<emph.end type="italics"></emph.end></s> </p> <pb pagenum="66" xlink:href="022/01/072.jpg"></pb> <p type="margin"> <s id="s.000648"><margin.target id="marg142"></margin.target>Ex mechani<lb></lb>cis,</s> </p> <p type="main"> <s id="s.000649"><emph type="center"></emph>PROP. XXII. THEOR. XVII.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg143"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000650"><margin.target id="marg143"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="main"> <s id="s.000651">IMagines velocitatum, ſeu ſpatia, quæ curruntur accele<lb></lb>ratis motibus, ſunt vt ſolida ab imaginibus ſimplicium <lb></lb>motuum, ex quibus ipſi gignuntur accelerati. </s> </p> <p type="main"> <s id="s.000652">Sint imagines ſimplicium motuum ABC, GLH, & ſoli<lb></lb>da ab ipſis imaginibus (angulis ACQ, GHD ſemper re<lb></lb>ctis, aut ſaltem æqualibus) intelligantur ABCRQ, GLHD. <lb></lb>Dico, vt ſunt interſe iſta ſolida, ſic eſſe homologè ſpatium <lb></lb>exactum tempore AC motu accelerato ex ſimplici motu <lb></lb>imaginis ABC ad ſpatium tranſactum tempore GH motu <lb></lb>item accelerato ex ſimplici imagine priori homogeneą <lb></lb>GLH: ſecetur ſolidum ABCRQ plano æquidiſtanti QCR, </s> </p> <p type="main"> <s id="s.000653"><arrow.to.target n="marg144"></arrow.to.target><lb></lb>quod faciat in ſolido ipſo ſectionem TSVX: erit hæc figu<lb></lb>ra prorſus ſimilis, ac æqualis conterminæ ABVI; quare <lb></lb><arrow.to.target n="marg145"></arrow.to.target><lb></lb>cum in accelerato motu velocitas, quæ habetur momen<lb></lb>to C ad velocitatem momento S ſit vt imago ABC ſim<lb></lb><arrow.to.target n="marg146"></arrow.to.target><lb></lb>plex ad ſegmentum eius ABVS: erit etiam QCR æqualis <lb></lb>ABC ad ſectionem ſolidi TSVX, quæ æquatur ABVS, vt <lb></lb>illa eadem velocitas momento C mobili inhærens ad ve<lb></lb>locitatem momento S alterius accelerati motus. </s> <s id="s.000654">Eſt au<lb></lb>tem ſectio TSVX ad libitum ſumpta; ergo ſolidum ABC<lb></lb><arrow.to.target n="marg147"></arrow.to.target><lb></lb>QR poteſt ſumi merito vt imago velocitatum accelerati <lb></lb><arrow.to.target n="marg148"></arrow.to.target><lb></lb>motus, cuius ſimplex imago ABC: & eodem modo ſoli<lb></lb>dum alterum vicem geret imaginis velocitatum alterius <lb></lb>motus ex ſimplici imagine GLH, itaque erit ob homoge<lb></lb>neitatem ſpatium tranſactum motu accelerato iuxta ſim<lb></lb>plicem imaginem ABC ad ſpatium tranſactum motu ac<lb></lb>celerato iuxta ſimplicem imaginem GLH, <expan abbr="tẽporibus">temporibus</expan> AC, <lb></lb>GH, vt ſolidum ABCQR ad ALHD, </s> </p> <pb pagenum="67" xlink:href="022/01/073.jpg"></pb> <p type="margin"> <s id="s.000655"><margin.target id="marg144"></margin.target>16. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000656"><margin.target id="marg145"></margin.target>4. <emph type="italics"></emph>huius,<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000657"><margin.target id="marg146"></margin.target>16. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000658"><margin.target id="marg147"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> .3. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000659"><margin.target id="marg148"></margin.target><emph type="italics"></emph>& Def.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>hu<lb></lb>ius vnà cum <lb></lb>pr.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000660"><emph type="center"></emph>PROP. XXII. THEOR. XVIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000661">SInt nunc CE, HL communes ſectiones imaginum ſim<lb></lb><arrow.to.target n="marg149"></arrow.to.target><lb></lb>plicium ABC, GLH, ſi extenderentur cum ſujs æqua<lb></lb>libus, ac ſimilibus coeuntibus figuris. </s> <s id="s.000662">Eſto pariter M cen<lb></lb>trum grauitatis imaginis ABC, et N grauitatis alterius ima<lb></lb>ginis GLH; actis demùm MO, NP perpendicularibus ad <lb></lb>ipſas CE, HL. Dico, ſpatium accelerati motus ab imagine <lb></lb>ſimplici ABC ad <expan abbr="ſpatiũ">ſpatium</expan> accelerati alterius motus ab ima<lb></lb>gine ſimplici GLH componi ex ratione imaginis ABC ad <lb></lb>imaginem GLH, & ex ea perpendicularis MO ad perpen<lb></lb>dicularem NP. </s> <s id="s.000663">Cum hæc ipſa ſpatia ſint oſtenſa, vt ſoli<lb></lb><arrow.to.target n="marg150"></arrow.to.target><lb></lb>da à figuris ABC, GLH; hæc verò ſunt vt momenta ipſa<lb></lb><arrow.to.target n="marg151"></arrow.to.target><lb></lb>rum figurarum ſuſpenſarum ex MO, NP. </s> <s id="s.000664">Ergo quemad<lb></lb>modum momenta iſta nectuntur ex rationibus figurarum <lb></lb><arrow.to.target n="marg152"></arrow.to.target><lb></lb>tanquam magnitudinum ABC ad LGH, & diſtantiarum <lb></lb>MO ad NP, ita pariter ex his nectentur propoſita ſpatia. </s> </p> <p type="margin"> <s id="s.000665"><margin.target id="marg149"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="margin"> <s id="s.000666"><margin.target id="marg150"></margin.target>21. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000667"><margin.target id="marg151"></margin.target>20. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000668"><margin.target id="marg152"></margin.target><emph type="italics"></emph>Ex mechani<lb></lb>cis.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000669"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000670"><emph type="italics"></emph>Patet communes ſectiones CE, HL eſſe æquidiſtantes ap<lb></lb>plicatis AB, HL, quæ in imaginibus ſumuntur perpendicula<lb></lb>res rectis AC, GH. nam HL est recta, in quam coeunt figura <emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg153"></arrow.to.target><lb></lb><emph type="italics"></emph>planæ ſimiles, ac æquales.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000671"><margin.target id="marg153"></margin.target><emph type="italics"></emph>Pr<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>primą <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000672"><emph type="center"></emph>PROP. XXIV. THEOR. XIX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000673">SI imagines ſimplicium motuum fuerint ſimiles, ſimili<lb></lb>terque ſuſpenſæ, imagines velocitatum accelerato<lb></lb>rum motuum erunt in triplicata ratione temporum ſimpli<lb></lb>cium motuum, aut in triplicata homologarum, vel extre<lb></lb>marum velocitatum eorundem ſimplicium motuum. </s> </p> <p type="main"> <s id="s.000674">Cum centra grauitatum ſimilium imaginum, ſeu figu<lb></lb><arrow.to.target n="marg154"></arrow.to.target><pb pagenum="68" xlink:href="022/01/074.jpg"></pb>rarum, ſint puncta in ijſdem figuris ſimiliter poſita, ponun<lb></lb>tur verò imagines ſimiliter ſuſpenſæ, ergo ſequitur ipſas <lb></lb>longitudines eſſe vt latera homologa dictarum imaginum, <lb></lb>ſcilicet vt tempus AC ad tempus FG, vel vt extremæ ve<lb></lb>locitates BC ad KE. </s> <s id="s.000675">Quamobrem imagines ipſæ, cum ſint <lb></lb>in duplicata ratione laterum homologorum, ſi huic dupli<lb></lb>catæ addatur alia ratio ſimilis rationi longitudinum, fiet <lb></lb>ratio imaginum velocitatum, ſeu ſpatiorum acceleratorum <lb></lb>motuum ex ſimplicibus illis deriuantium triplicata tempo<lb></lb>rum, vel extremarum velocitatum ſimplicium motuum. </s> </p> <p type="margin"> <s id="s.000676"><margin.target id="marg154"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000677"><emph type="center"></emph>PROP. XXV. THEOR. XX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000678">SI verò ſimplices motus extiterint ſimiles, <expan abbr="æqualibuſq;">æqualibuſque</expan> <lb></lb>temporibus abſoluantur, imagines acceleratorum <lb></lb><arrow.to.target n="marg155"></arrow.to.target><lb></lb>motuum erunt in ſola ratione amplitudinum imaginum <lb></lb>ſimplicium. </s> </p> <p type="margin"> <s id="s.000679"><margin.target id="marg155"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 4.</s> </p> <p type="main"> <s id="s.000680">Sint imagines ſimilium, ac ſimplicium motuum BAC, <lb></lb>KFG, quarum grauitatis centra D, H, erunt ex hypotheſi <lb></lb><arrow.to.target n="marg156"></arrow.to.target><lb></lb>tempora AC, FG æqualia; & ideo ſpatia, ſcilicet imagines <lb></lb><arrow.to.target n="marg157"></arrow.to.target><lb></lb>velocitatum BAC, KFG habebunt eandem rationem, <lb></lb>quam ſummæ, aut extremæ motuum ſimplicium velocita<lb></lb>tes, ſcilicet, quam amplitudines imaginum, ſeu geneſum: <lb></lb>ſunt verò diſtantiæ DE, HI pariter æquales, quia AC, FG <lb></lb><arrow.to.target n="marg158"></arrow.to.target><lb></lb>æquales ſunt; ergo cum ſpatia acceleratorum motuum ne<lb></lb>ctantur ex imaginibus ſimplicium motuum ABC, KFG, & <lb></lb>ex diſtantijs DE ad HI, liquet ipſa ſpatia eſſe in vnica, ſo<lb></lb>laque ratione amplitudinum BC, KG, aut amplitudinum <lb></lb>geneſum. </s> </p> <p type="margin"> <s id="s.000681"><margin.target id="marg156"></margin.target>8 <emph type="italics"></emph>primi huius<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000682"><margin.target id="marg157"></margin.target>2 <emph type="italics"></emph>primi huius<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000683"><margin.target id="marg158"></margin.target>23. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000684"><emph type="center"></emph>PROP. XXVI. THEOR. XX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000685">AT ſi ſimplicium, ſimiliumque motuum fuerint imagi<lb></lb>nes æquè amplæ, imagines acceleratorum motuum, <pb pagenum="69" xlink:href="022/01/075.jpg"></pb>ſiue tempora erunt in duplicata ratione temporum iſto<lb></lb>rum, vel illorum motuum. </s> </p> <p type="main"> <s id="s.000686">Amplitudines imaginum ſimplicium, velocitatumque <lb></lb><arrow.to.target n="marg159"></arrow.to.target><lb></lb>BAC, KFG ſunto BC, KG, quæ æquales ſint. </s> <s id="s.000687">Dico ſpa<lb></lb>tia acceleratorum motuum ab illis ſimplicibus imaginibus <lb></lb>fore in duplicata ratione temporum AC ad FG (quę ſem<lb></lb>per in acceleratis ponuntur eadem, ac in ſimplicibus, nec <lb></lb>aliter eſſe poſſunt.) Vt FG ad GK, ita ſit AC ad CL, & <lb></lb>intelligatur LAC imago alterius motus ſimilis motui, cuius <lb></lb>imago BAC, vel KFG. </s> <s id="s.000688">Facilè demonſtrabitur ipſam fi<lb></lb><arrow.to.target n="marg160"></arrow.to.target><lb></lb>guram LAC ſimilem eſſe ipſi KFG, & ad BAC eandem̨ <lb></lb>habere rationem, quam LC ad BC. </s> <s id="s.000689">Cum ergo imago BAC <lb></lb>ad imaginem KFG componatur ex ratione imaginis BAC <lb></lb>ad LAC (quæ ſunt vt BC ad CL) & ex ratione imagi<lb></lb>nis ALC ad imaginem KFG, quæ ſunt in ratione compo<lb></lb>ſita LC ad KG, et AC ad FG: priores verò duæ rationes <lb></lb>componunt vnicam æqualitatis, ergo relinquitur, imagi<lb></lb>nem BAC ad imaginem KFG eſſe vt AC ad FG; ſpatium <lb></lb>verò accelerati motus ex ſimplici imagine BAC ad accele<lb></lb>ratum ex ſimplici KFG nectitur ex ratione imaginum ſim<lb></lb><arrow.to.target n="marg161"></arrow.to.target><lb></lb>plicium ipſarum, & ex ea diſtantiarum DE, HI à centris <lb></lb>grauitatum deductarum D, H, et ſunt hæ rectæ in eadem <lb></lb>ratione, ac altitudines AC, FG (nam in figuris, ſeu imagi<lb></lb>nibus ſimilium motuum BAC, LAC centra grauitatum <lb></lb>ſunt in eadem recta parallela ipſi BC, & in LAC, KFG <lb></lb>ſunt in punctis ſimiliter poſitis, adeo ut, ſicut poſitum eſt, <lb></lb>ratio ipſarum diſtantiarum in ipſis figuris LAC, KFG, ſeu <lb></lb>BAC, KEG eadem ſit, ac laterum homologorum LC ad <lb></lb>KG, vel AC ad FG) ergo ſpatium accelerati motus ex ſim<lb></lb>plici imagine KFG, erit vt quadratum ex AC ad quadra<lb></lb>tum ex FG, nempe in duplicata ratione temporum ſimpli<lb></lb>cium motuum. </s> </p> <pb pagenum="70" xlink:href="022/01/076.jpg"></pb> <p type="margin"> <s id="s.000690"><margin.target id="marg159"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 5.</s> </p> <p type="margin"> <s id="s.000691"><margin.target id="marg160"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000692"><margin.target id="marg161"></margin.target>23. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000693"><emph type="center"></emph>PROP. XXVII. THEOR. XXI.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg162"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000694"><margin.target id="marg162"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 6.</s> </p> <p type="main"> <s id="s.000695">DEmùm ſi ſint imagines, quæcunque velocitatum ſim<lb></lb>plicium, ſimiliumque motuum, imagines accelera<lb></lb>torum motuum, ſeu ſpatia ijs motibus exacta componen<lb></lb>tur ex duplicata temporum ratione, & ex ea amplitudi<lb></lb>num, vel applicatarum homologarum earundem imagi<lb></lb>num. </s> </p> <p type="main"> <s id="s.000696">Imagines ſimilium, ſimpliciumque motuum ſint BAC, <lb></lb>KFG. Dico, imagines acceleratorum motuum ab illis ſim<lb></lb>plicibus deriuantium habere rationem compoſitam ex du<lb></lb>plicata temporum AC ad FG, & amplitudinum imaginum <lb></lb>dictarum, vel geneſum. </s> <s id="s.000697">Intelligatur alius ſimilis motus, <lb></lb>cuius velocitatum imago ſit DFG æquèampla, ac homo<lb></lb>genea ipſi BCA; nimirum ſit DG æqualis BC. </s> <s id="s.000698">Quoniam <lb></lb>imago accelerati motus ex ſimplici imagine BA ad imagi<lb></lb>nem accelerati ex ſimplici imagine KFG componitur ex <lb></lb>ratione imaginis accelerati motus, cuius ſimplex imago <lb></lb>BAC ad imaginem accelerati motus ex ſimplici DFG, & <lb></lb>ex imagine huius accelerati motus ad accelerati imaginem <lb></lb>à ſimplici KFG; eſt autem prior ratio imaginum, ſeu ſpa<lb></lb>tiorum acceleratis motibus percurſorum ipſa temporum </s> </p> <p type="main"> <s id="s.000699"><arrow.to.target n="marg163"></arrow.to.target><lb></lb>duplicata AC ad FG, & altera dictarum imaginum, ſeu <lb></lb>ſpatiorum item acceleratis motibus confectorum, & quo<lb></lb><arrow.to.target n="marg164"></arrow.to.target><lb></lb>rum ſimplices imagines ſunt DFG, KFG, eſt eadem, ac ra<lb></lb>tio amplitudinum DG, ſeu BC ad KG. </s> <s id="s.000700">Ergo cum iſtæ <lb></lb>amplitudines ſint eædem, ac illæ geneſum, conſtat propo<lb></lb>ſitam rationem acceleratorum motuum ex ſimplicibus <lb></lb>imaginibus BAC, KFG habere rationem compoſitam ex <lb></lb>duplicata temporum AC ad FG, & ex ea amplitudinum <lb></lb>imaginum ſimplicium BC ad KG, ſeu amplitudinum gene<lb></lb>ſum. </s> <s id="s.000701">Quod &c. </s> </p> <pb pagenum="71" xlink:href="022/01/077.jpg"></pb> <p type="margin"> <s id="s.000702"><margin.target id="marg163"></margin.target><emph type="italics"></emph>Pr.<emph.end type="italics"></emph.end> 26 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000703"><margin.target id="marg164"></margin.target>25. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000704"><emph type="center"></emph>PROP. XXVIII. THEOR. XXII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000705">SI geneſes ſimilium, ſimpliciumque motuum fuerint <lb></lb>æquèamplæ, imagines acceleratorum motuum erunt <lb></lb>in duplicata ratione temporum, vel altitudinum ipſarum <lb></lb>geneſum. </s> </p> <p type="main"> <s id="s.000706">Geneſes ſimilium, ac ſimplicium motuum ſunto ABC, <lb></lb><arrow.to.target n="marg165"></arrow.to.target><lb></lb>DEF, quarum amplitudines æquales ſint AC, DF. Dico, <lb></lb>imagines, ſiue ſpatia acceleratorum motuum eſſe in dupli<lb></lb>cata ratione temporum, vel altitudinum BC ad EF. </s> <s id="s.000707">Cum <lb></lb>AC, DF ſint gradus velocitatum in extremitatibus ſimpli<lb></lb>cium decurſuum, etiam imagines velocitatum, iuxta ipſas <lb></lb>geneſes, quæ ſint interſe homogeneæ, erunt æquèamplæ, <lb></lb>& ſunt ſimilium motuum; ergo imagines acceleratorum <lb></lb><arrow.to.target n="marg166"></arrow.to.target><lb></lb>motuum, iuxta ſimplices illas geneſes, aut imagines æquè<lb></lb>amplas erunt in duplicata ratione temporum: ſunt autem <lb></lb>imagines velocitatum æquèamplæ, ſimiliumque motuum, <lb></lb><arrow.to.target n="marg167"></arrow.to.target><lb></lb>hoc eſt ſpatia BC ad EF vt ipſa tempora; ergo ſpatia acce<lb></lb>leratorum, propoſitorumque motuum erunt in ratione du<lb></lb>plicata altitudinum BC, EF ſimplicium geneſum, ABC, <lb></lb>DEF. </s> <s id="s.000708">Quod &c. </s> </p> <p type="margin"> <s id="s.000709"><margin.target id="marg165"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="margin"> <s id="s.000710"><margin.target id="marg166"></margin.target>26. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000711"><margin.target id="marg167"></margin.target>26. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000712"><emph type="center"></emph>PROP. XXIX. THEOR. XXIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000713">SI geneſes ſimilium, ſimpliciumque motuum fuerint <lb></lb>æquèaltæ, imagines, ſiue ſpatia, acceleratorum mo<lb></lb>tuum erunt vt tempora, vel reciprocè vt amplitudines ge<lb></lb>neſum ipſorum ſimplicium motuum. </s> </p> <p type="main"> <s id="s.000714">Geneſes ſimilium, ſimpliciumque motuum, ac interſe <lb></lb><arrow.to.target n="marg168"></arrow.to.target><lb></lb>homogeneæ ſint BAC, DEF, quæ habeant altitudines <lb></lb>AC, EF æquales. </s> <s id="s.000715">Dico, imagines acceleratorum motuum <lb></lb>eſſe inter ſe, vt tempora dictorum ſimplicium motuum, vel <lb></lb>reciprocè vt amplitudines ipſarum geneſum. </s> <s id="s.000716">Concipian-<pb pagenum="72" xlink:href="022/01/078.jpg"></pb>tur imagines velocitatum <expan abbr="ſimpliciũ">ſimplicium</expan> motuum, ſcilicet GHI <lb></lb>iuxta geneſim BAC, et MKL iuxta <expan abbr="alterã">alteram</expan> geneſim DEF, & <lb></lb>quia, vtpotè homogeneę, ſunt inter ſe vt ſpatia ęqualia AC <lb></lb>ad EF, <expan abbr="erũt">erunt</expan> ipſæ imagines ęquales inter ſe, <expan abbr="cũ">cum</expan> verò ob ſimili <lb></lb><expan abbr="tudinẽ">tudinem</expan> motuum eæ ipſæ imagines nectantur ex rationibus <lb></lb>GI ad ML, & ex ea, quam habet HI ad KL, ſequitur eſſe <lb></lb>GI ad ML, vt KL ad IH, & demum quia acceleratorum <lb></lb>motuum ſpatia à ſimplicibus imaginibus GHI, MKL ne<lb></lb>ctuntur ex duplicata temporum HI ad KL, & ex ea ampli<lb></lb><arrow.to.target n="marg169"></arrow.to.target><lb></lb>tudinum GI ad ML, ſiue ex ea, quam habet KL ad HI, re<lb></lb>linquitur, ſpatia acceleratis illis motibus confecta eſſe in <lb></lb>ſola, <expan abbr="vnicaq;">vnicaque</expan> ratione temporum HI ad KL, vel in ei ęqua<lb></lb>li ratione, reciproca amplitudinum imaginum ML ad GI, <lb></lb>vel geneſum DF ad BC. </s> <s id="s.000717">Quod &c, </s> </p> <p type="margin"> <s id="s.000718"><margin.target id="marg168"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 8.</s> </p> <p type="margin"> <s id="s.000719"><margin.target id="marg169"></margin.target>27. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000720"><emph type="center"></emph>PROP. XXX. THEOR. XXIV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000721">QVæcunque fuerint geneſes ſimilium, ſimpliciumque <lb></lb>motuum, dum interſe homogeneæ, ſpatia accelera<lb></lb>tis motibus ex illis ſimplicibus exacta nectentur <lb></lb>ex duplicata ratione altitudinum, & reciproca amplitudi<lb></lb>num earundem ſimplicium geneſum, </s> </p> <p type="main"> <s id="s.000722">Sint quæcunque ſimilium motuum geneſes BAC, KFG. <lb></lb><arrow.to.target n="marg170"></arrow.to.target><lb></lb>Dico, ſpatia acceleratorum motuum, ab ijs ſimplicibus de<lb></lb>riuantium, componi ex duplicata ratione altitudinum AC <lb></lb>ad FG, & ex ratione extremarum velocitatum, ſeu ampli<lb></lb>tudinum reciprocè ſumptarum ipſarum geneſum: eſto alia <lb></lb>geneſis DFG illis homogenea, & motu pariter ſimilis cum <lb></lb>ijſdem geneſibus. </s> <s id="s.000723">Eadem ſit amplitudine æqualis BAC, <lb></lb>& altitudo eius ſit FG, ſpatia acceleratorum motuum ex <lb></lb><arrow.to.target n="marg171"></arrow.to.target><lb></lb>ſimplicibus geneſibus æquales amplitudines habentibus, <lb></lb>& ſimilium motuum BAC, DFG ſunt in duplicata ratione <lb></lb>rectarum, ſeu altitudinum AC ad FG, & ſpatia accelera<lb></lb><arrow.to.target n="marg172"></arrow.to.target><pb pagenum="73" xlink:href="022/01/079.jpg"></pb>torum motuum ex ſimplicibus geneſibus, quæ ſint in ea<lb></lb>dem altitudine DFG, KFG, ſunt in reciproca ratione am<lb></lb>plitudinum, ſeu primarum velocitatum KG ad DG, vel <lb></lb>BC; ex æquali igitur ſpatia acceleratorum motuum ex <lb></lb>propoſitis ſimplicibus geneſibus BAC, KFG nectentur ex <lb></lb>ratione duplicata altitudinum AC ad FG, & reciproca <lb></lb>amplitudinum KG ad BC earundem geneſum BAC, <lb></lb>KFG. </s> <s id="s.000724">Quod &c. </s> </p> <p type="margin"> <s id="s.000725"><margin.target id="marg170"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 6.</s> </p> <p type="margin"> <s id="s.000726"><margin.target id="marg171"></margin.target>28. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000727"><margin.target id="marg172"></margin.target>29. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000728"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000729"><emph type="italics"></emph>At quia in ſpatijs, quæ accelerato motu peraguntur; non <lb></lb>ſeruatur ratio altitudinum geneſum ſimplicium, ex quo ori<lb></lb>tur in hac methodo quædam percipiendi difficultas; ideo ſe<lb></lb>quenti problemate, alijſque iam notis veritatibus, rem planè <lb></lb>illuſtrabimus, ac ſimul doctrina vſum trademus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000730"><emph type="center"></emph>PROP. XXXI. PROB. VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000731">EX datis ſpatijs accelerato motu confectis, cognitiſ<lb></lb>que primis, aut poſtremis ſimilium, ſimpliciumque <lb></lb>motuum velocitatibus, reperire tempora ipſorum de<lb></lb>curſuum. </s> </p> <p type="main"> <s id="s.000732">Spatia motibus acceleratis exacta ſunt C, D, & velo<lb></lb><arrow.to.target n="marg173"></arrow.to.target><lb></lb>tates, ſeu amplitudines geneſum ponantur eſſe A, B, ſcili<lb></lb>cet A principio motus per C, & B initio motus per D, quæ<lb></lb>ritur ratio temporum, quibus exiguntur propoſita ſpatia. <lb></lb></s> <s id="s.000733">Vt A ad B, ita fiat C ad E, & inter E, et D ſumatur F me<lb></lb>dia proportionalis. </s> <s id="s.000734">Dico ipſa tempora eſſe vt E ad F. <lb></lb></s> <s id="s.000735">Componuntur ſpatia acceleratis motibus exacta ex ratio<lb></lb><arrow.to.target n="marg174"></arrow.to.target><lb></lb>ne quadratorum temporum, & ex ea amplitudinum, ſeu <lb></lb>homologarum velocitatum in ſimplicibus motibus, ſimili<lb></lb><arrow.to.target n="marg175"></arrow.to.target><lb></lb>buſque ſumptarum; & ideo temporum quadrata necten<lb></lb>tur ex ratione ſpatiorum C ad D, & ex reciproca ampli-<pb pagenum="74" xlink:href="022/01/080.jpg"></pb>tudinum E ad C; temporum igitur quadrata erunt vt E ad <lb></lb>D, ipſa verò tempora vt E ad F. </s> <s id="s.000736">Quod &c. </s> </p> <p type="margin"> <s id="s.000737"><margin.target id="marg173"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 1.</s> </p> <p type="margin"> <s id="s.000738"><margin.target id="marg174"></margin.target>27. <emph type="italics"></emph>huiuij<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000739"><margin.target id="marg175"></margin.target><emph type="italics"></emph>lem. </s> <s id="s.000740">pr.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000741"><emph type="center"></emph>PROP. XXXII. PROB. VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000742">EXdatis ſpatijs accelerato motu tranſactis, datis item <lb></lb>primis velocitatibus ſimilium, ſimpliciumque mo<lb></lb>tuum, inuenire altitudines ſimplicium geneſum, ex quibus <lb></lb><arrow.to.target n="marg176"></arrow.to.target><lb></lb>propoſita ſpatia effecta ſunt. </s> </p> <p type="margin"> <s id="s.000743"><margin.target id="marg176"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 1. <lb></lb>30. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000744">Spatia ſint E, D reliquis, vt ſupra, manentibus: quoniam <lb></lb>ſpatia accelerato motu tranſacta componuntur ex ratio<lb></lb>nibus amplitudinum geneſum ſimplicium, ſimiliumquę <lb></lb>motuum reciprocè ſumptarum B ad A, ſiue E ad C, & ex <lb></lb>ea quadratorum altitudinum ipſarum geneſum; erit ratio <lb></lb>dictarum altitudinum duplicata C ad D; quare F, ſi ſit me<lb></lb>dia proportionalis, non inter E, & D (vt antea poſuimus) <lb></lb>ſed inter C ad D; erit ſanè C ad F ratio altitudinum gene<lb></lb>ſum ſimplicium, ſimiliumque motuum, quam quereba<lb></lb>mus. </s> </p> <p type="main"> <s id="s.000745"><emph type="center"></emph><emph type="italics"></emph>Exemplum primum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000746">SI idem graue naturaliter cadens percurrerit à quiete <lb></lb>duo ſpatia; tempora erunt in ratione ſubduplicatą <lb></lb>eorundem ſpatiorum. </s> </p> <p type="main"> <s id="s.000747">Ex Cor. </s> <s id="s.000748">pr: 4. huius conſtat rectangula eſſe geneſes ſim<lb></lb>plicium motuum grauium naturaliter deſcendentium, & <lb></lb>ex def. 7. primi liquet eaſdem geneſes eſſe motuum ſimi<lb></lb>lium. </s> <s id="s.000749">Cumque eiuſdem mobilis naturaliter cadentis ve<lb></lb>locitas à quiete ſit vna, eademque; ſimplices motus erunt <lb></lb>ij, vt geneſum ſimilium, ſimpliciumque motuum amplitu<lb></lb>dines æquales ſint, proptereaque, vt in figura præcedentis <lb></lb>propoſitionis æquales erunt C, E, atque adeo ſpatium̨ <lb></lb>C, ſiue E ad D erit in duplicata ratione temporum E ad F. </s> </p> <pb pagenum="75" xlink:href="022/01/081.jpg"></pb> <p type="main"> <s id="s.000750"><emph type="center"></emph><emph type="italics"></emph>Exemplum II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000751"><emph type="center"></emph>PROP. XXXIV. THEOR. XXVII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000752">TEmpora ſimilium vibrationum ſunt in ſubduplicata <lb></lb>ratione arcuum exactorum, ſeu longitudinum pen<lb></lb>dulorum, quorum ſunt vibrationes. </s> <s id="s.000753">Sint grauia pendula <lb></lb>LA, LF, quæ ab eadem recta LF diſcedentia currant ſuſ<lb></lb><arrow.to.target n="marg177"></arrow.to.target><lb></lb>penſa ex L duos ſimiles arcus circulares FI, AC. </s> <s id="s.000754">Dico <lb></lb>tempora horum deſcenſuum eſſe in ratione ſubduplicatą <lb></lb>arcuum FI, AC, ſeu longitudinum filorum, aut haſtularum <lb></lb>FA, LA. </s> <s id="s.000755">Ducamus quamcumque rectam LBG, erit AB <lb></lb>ad BC, vt FG ad GI, & cum præterea velocitates pendu<lb></lb>lorum a quiete in A, F ſint æquales, pariterque velocita<lb></lb>tes æquales a quiete in B, G; erit velocitas in A ad veloci<lb></lb>tatem in B, vt velocitas in F ad velocitatem in G, quare <lb></lb>conſideratis arcubus ABC, FGI, vt altitudines rectę, (quæ <lb></lb>item forent in B, G proportionaliter ſectę) geneſum ſimi<lb></lb><arrow.to.target n="marg178"></arrow.to.target><lb></lb>lium ſimpliciumque motuum, quarum amplitudines æqua <lb></lb>les ſunt, erunt ſpatia in acceleratis decurſubus per FI, AC <lb></lb>in ratione duplicata temporum, ſcilicet ipſi arcus, aut lon<lb></lb>gitudines LF, LA erunt in ratione duplicata temporum̨. <lb></lb></s> <s id="s.000756">Quod &c. </s> </p> <p type="margin"> <s id="s.000757"><margin.target id="marg177"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 2.</s> </p> <p type="margin"> <s id="s.000758"><margin.target id="marg178"></margin.target><emph type="italics"></emph>Def.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>primi.<gap></gap><emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000759">Idem demonſtratum eſſet beneficio imaginum, quæ vt<lb></lb>pote eorundem illorum motuum ſimplicium, forent etiam <lb></lb>ſimilium, & ſunt amplitudines æquales, etenim eædem̨ <lb></lb>ſunt, ac geneſum ergo rurſus ſpatia, hoc eſt arcus ABC, <lb></lb>FGI, nempe longitudines filorum IF, AC erunt in ratione <lb></lb>duplicata temporum. </s> <s id="s.000760">Quod &c. </s> </p> <pb pagenum="76" xlink:href="022/01/082.jpg"></pb> <p type="main"> <s id="s.000761"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000762"><emph type="italics"></emph>Vides, quàm breuiter rei diſficillimæ demonſtrationem at<lb></lb>tulimus, nec dubium, quin illa extendi queat ad quaſcum<lb></lb>que lineas decurſuum, dummodo ſimiles, ac ſimiliter poſitas in <lb></lb>ijſdem, vel æqualibus ab horizonte planis elenatis, quemad<lb></lb>modum Dominus Viuianus pulcherrimè propoſuit.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000763"><emph type="center"></emph><emph type="italics"></emph>Exemplum III.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000764"><emph type="center"></emph>PROP. XXXV. THEOR. XXVIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000765">TEmpora lationum à quiete per plana eandem eleua<lb></lb><arrow.to.target n="marg179"></arrow.to.target><lb></lb>tionem habentia ſunt homologè vt longitudines <lb></lb>planorum. </s> </p> <p type="margin"> <s id="s.000766"><margin.target id="marg179"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 3.</s> </p> <p type="main"> <s id="s.000767">Sint plana AB, AC eandem eleuationem AD habentia. <lb></lb></s> <s id="s.000768">Dico tempus lationis per AC ad id per AB eſſe vt AC ad <lb></lb>AB. (hæc Torricellij propoſitio, <expan abbr="expoſitioq;">expoſitioque</expan> eſt, hancque <lb></lb>eandem veritatem ex noſtris principijs demonſtrare <expan abbr="visũ">visum</expan> <lb></lb>eſt, non vt de re illa dubitemus, immò contrà, quòd de eą <lb></lb>plenè ſatisfacti ſimus, ex eo rurſus demonſtrandam ſuſce<lb></lb>pimus, vt exinde methodus noſtra, quàm vera ſit, eluceſ<lb></lb>cat) Momentum deſcenſus inplano AC ad id deſcenſus ſu<lb></lb><arrow.to.target n="marg180"></arrow.to.target><lb></lb>per plano AB eſt vt AB ad AC; ſunt autem <expan abbr="deſcendentiũ">deſcendentium</expan> <lb></lb>grauium, etiam ſuper planis inclinatis motus, quos ſimpli<lb></lb>ces appellamus, inter ſe ſimiles, nempe quorum geneſes <lb></lb><arrow.to.target n="marg181"></arrow.to.target><lb></lb>ſunt rectangula; ergo habebimus ſimplices geneſes, vnam, <lb></lb>cuius altitudo AC amplitudoque AB; alteram, cuius am<lb></lb>plitudo AC, altitudo autem AB; itaque propoſitis ſpatijs <lb></lb>AC, AB, primiſque velocitatibus AB, AC, ſi fiat AB ad AC <lb></lb>vt CA ad EA, erit EA ad AB duplicata <expan abbr="tẽporum">temporum</expan>, & ideo <lb></lb><arrow.to.target n="marg182"></arrow.to.target><lb></lb>ratio temporum per AC, AB erit CA ad AB. </s> <s id="s.000769">Quod &c. </s> </p> <pb pagenum="77" xlink:href="022/01/083.jpg"></pb> <p type="margin"> <s id="s.000770"><margin.target id="marg180"></margin.target><emph type="italics"></emph>Tor. pr.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>de <lb></lb>motu <expan abbr="grauiũ">grauium</expan>.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000771"><margin.target id="marg181"></margin.target><emph type="italics"></emph>Cor pr.<emph.end type="italics"></emph.end> 4. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000772"><margin.target id="marg182"></margin.target>31. <emph type="italics"></emph>vel<emph.end type="italics"></emph.end> 27. <emph type="italics"></emph>hu.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000773"><emph type="center"></emph><emph type="italics"></emph>Exemplum IV.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000774"><emph type="center"></emph>PROP. XXXVI. THEOR. XXIX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000775">IIſdem prorſus manentibus demonſtrarunt Gallileus, ac <lb></lb>Torricellius, gradus velocitatum acquiſitos in B, et C <lb></lb>eiuſdem mobilis deſcendentis à quiete in A pares eſſe; <lb></lb>idipſum nos oſtendemus. </s> </p> <p type="main"> <s id="s.000776">Cum tempora ſint vt AC ad AB, & velocitates à quie<lb></lb>te in ratione reciproca temporum, ſcilicet vt AB ad AC, <lb></lb><arrow.to.target n="marg183"></arrow.to.target><lb></lb>ſint deinde velocitates eæ vt amplitudines imaginum ſim<lb></lb>plicium, ſimiliumque illorum motuum (nam amplitudines <lb></lb>imaginum velocitatum ſunt prorſus eædem, ac illæ gene<lb></lb>ſum) erunt ipſæ imagines ſimplicium motuum æquales; <lb></lb>nam tempora, quæ ſummuntur vt altitudines imaginum <lb></lb>reciprocantur, vt dictum eſt, amplitudinibus, ſeu primis à <lb></lb><arrow.to.target n="marg184"></arrow.to.target><lb></lb>quiete velocitatibus, at in motibus acceleratis ipſæ inte<lb></lb>græ imagines ſimplicium motuum ſunt loco graduum ve<lb></lb>locitatum in extremo ſpatiorum acquiſitorum; ergo in B, et <lb></lb>C gradus velocitatum æquales erunt. </s> </p> <p type="margin"> <s id="s.000777"><margin.target id="marg183"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 4. <lb></lb>33. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000778"><margin.target id="marg184"></margin.target>4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000779"><emph type="center"></emph>PROP. XXXVII. THEOR. XXX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000780">SI æqualia pondera, ſuſpenſa ſint ex filis, quorum par<lb></lb>tes interſe æquales, præ tractione æqualiter elongen<lb></lb>ter tempora in reditu ipſorum filorum, cum ab ipſis graui<lb></lb>bus ſtatim liberantur, æqualia erunt. </s> <s id="s.000781">Hoc primùm <expan abbr="demõ-">demon<lb></lb></expan><arrow.to.target n="marg185"></arrow.to.target><lb></lb>ſtrabimus alia via, tum methodo noſtra, vt de ea aliud <lb></lb>exemplum tradamus. </s> <s id="s.000782">Sint funiculi AB, DC, & ex ijs <lb></lb>pendeant æqualia grauia B, C, adeo vt ſumptis hinc indè <lb></lb>partibus æqualibus eorundem funiculorum, conſtet ipſas <lb></lb>æqualiter ab ipſis grauibus trahi, atque produci. </s> <s id="s.000783">Dico, ſi <lb></lb>elongationes ſint HB, GC, & omnibus ſic ſtantibus pon-<pb pagenum="78" xlink:href="022/01/084.jpg"></pb>dera ſubmoueantur ex B, et C funiculis cæſis, fore vt eæ<lb></lb>dem extremitates reſtituantur in H, et G æqualibus tem<lb></lb>poribus. </s> <s id="s.000784">Sit AE æqualis DC, erit porrò elongatio facta <lb></lb>per idem graue B, quæ ſit EF, æqualis GC; propterea li<lb></lb>beratis funiculis ad B, et C, eodem tempore reſtituetur C <lb></lb>in G, ac E in F, quo tempore etiam B in H reſtitutum fue<lb></lb>rit; nam vno puncto in primum ſuum locum redito, etiam <lb></lb>alia ſingula in ſuum locum perueniſſe, opportebit. </s> </p> <p type="margin"> <s id="s.000785"><margin.target id="marg185"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 5.</s> </p> <p type="main"> <s id="s.000786"><emph type="center"></emph><emph type="italics"></emph>Exemplum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000787">HAc occaſione de funiculis erit non iniucunda diſer<lb></lb>tatio, remque ſic adhuc intactam promouebimus, <lb></lb>ſimulque demonſtrabimus. </s> </p> <p type="main"> <s id="s.000788">Idipſum propoſitum noſtris principijs ſic demonſtra<lb></lb>mus. </s> </p> <p type="main"> <s id="s.000789">Sint <expan abbr="eadẽ">eadem</expan>, quæ ſupra, ſcilicet conceptis in filo AB quot<lb></lb>libet partibus interſe æqualibus, <expan abbr="lõgitudinẽque">longitudinenque</expan> totam im<lb></lb>plentibus, hæ ſingulæ æqualiter à pondere B trahentur, <lb></lb>eritque BH ſumma omnium dictarum partium elongatio<lb></lb>num, & eodem pacto EF erit ſumma elongationum <expan abbr="partiũ">partium</expan> <lb></lb>omnium in AE contentarum, ab eodemque pondere effe<lb></lb>ctarum; propterea vt AB ad BH, ita erit AE ad EF; quamo <lb></lb>brem velocitas etiam puncti B ſublato pondere B erit ad <lb></lb>velocit atem puncti E ob eandem detractionem, vt BH ad <lb></lb>EF, vel BA ad EA (nam quot ſunt partes conceptę iņ <lb></lb>vtraque fili longitudine, totidem ſunt etiam impetus inter <lb></lb>ſe æquales) idem oſtenderemus ſi loco ponderis B, minus <lb></lb>quodcumque ſuſpenderemus, vt ſcilicet puncta B, et E ad <lb></lb>quemuis locum ſuperius remanerent, librarenturque cum <lb></lb>reſiſtentijs <expan abbr="partiũ">partium</expan> eò elongatarum, ergo tranſitus ex B in H, <lb></lb><arrow.to.target n="marg186"></arrow.to.target><lb></lb>& puncti E in F ſubducto pondere B erunt motus ſimilium <lb></lb>ſimpliciumque; ſed motus ex C in G exempto pondere C <lb></lb>eſt prorſus idem, ac motus E in F, ergo motus ſimiles, ac <pb pagenum="79" xlink:href="022/01/085.jpg"></pb>ſimplices ex B in H, & ex C in G, ex quibus fiunt accele<lb></lb>rati, geneſes habebunt, quarum primæ velocitates, ſeu am<lb></lb>plitudines proportionales ſunt altitudinibus earundem, <lb></lb>ſpatijs nimirum CG, BH accelerato motu exigendis; qua<lb></lb>mobrem componentur ex ratione ipſarum velocitatum, <lb></lb>ſeu amplitudinum CG ad BH, & ex ea quadratorum tem<lb></lb>porum, quæ proinde æqualitatis erit; itaque etiam huius <lb></lb>ſubduplicata; hoc eſt tempora in tranſitibus accelarato <lb></lb>motu exactis, erunt paria. </s> </p> <p type="margin"> <s id="s.000790"><margin.target id="marg186"></margin.target><emph type="italics"></emph>pr.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000791"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000792"><emph type="italics"></emph>Hinc patet, vbi æquè craſſis filis eiuſdemque materiei vel <lb></lb>cedentiæ ſuſpenſa ſint æqualia pondera, tunc primas velocita<lb></lb>tes, ſubductis ponderibus, fore in eadem ratione <expan abbr="elongationũ">elongationum</expan>, <lb></lb>vel longitudinum filorum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000793"><emph type="center"></emph>PROP. XXXVIII. THEOR. XXXI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000794">SI extremitatibus funiculorum ex vna parte <expan abbr="firmatorũ">firmatorum</expan>, <lb></lb>ac eandem craſſitiem habentium, nec non eiuſdem <lb></lb>cædentiæ exiſtentium, fuerint ſuſpenſa æqualia pondera, <lb></lb>quæ inde ijſdem longitudinibus ſeruatis, quomodo opor<lb></lb>tet tollantur, erunt ſpatia recurſuum, temporibus ſimpli<lb></lb>cium motuum exacta in ratione longitudinum pendulo<lb></lb>rum. </s> </p> <p type="main"> <s id="s.000795">Sit funiculus AC æquè craſſus ac BD, & ſuſpenſis <lb></lb>hinc inde ponderibus æqualibus, elongatio primi funiculi <lb></lb>ſit CE, & alterius ſit DF. </s> <s id="s.000796">Dico ſpatia temporibus ſimpli<lb></lb>cium imaginum, ab extremitatibus ſolutis exacta, fore iņ <lb></lb>ratione longitudinum ipſorum funiculorum. </s> </p> <p type="main"> <s id="s.000797">Iam conſtat CE ad DF eſſe, vt AC ad BD, in qua ratione <lb></lb>ſunt etiam velocitates à quiete, dum pondera ſubduceren<lb></lb>tur ex E, et F, vel ex alijs punctis quibuſcunque ſi æqualia <pb pagenum="80" xlink:href="022/01/086.jpg"></pb>pondera ſuſpenſa fuiſſent maioris, vel minoris ponderis, <lb></lb>ſic enim concipiuntur geneſes ſimilium, ſimpliciumque <lb></lb>motuum, quarum altitudines æquantur elongationibus <lb></lb>funiculorum; propterea ſpatia recurſuum temporibus ſim<lb></lb>plicium motuum exacta, nectentur ex rationibus duplicata <lb></lb>CE ad DF, hoc eſt AC ad BD, & ex reciproca filorum, <lb></lb>ſcilicet BD ad AC, quæ ratio, vti diximus, eſt reciprocą <lb></lb>primarum velocitatum, ſeu amplitudinum geneſum ſimpli<lb></lb>cium, ergo ipſa ſpatia in reditu filorum ab extremitatibus <lb></lb>ſolutis exacta, erunt vt AC ad BF, ſeu vt CE ad DF. <lb></lb></s> <s id="s.000798">Quod &c. </s> </p> <p type="main"> <s id="s.000799"><emph type="center"></emph>PROP. XXXIX. THEOR. XXXI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000800">TEmpora ſimplicium, ſimiliumque dictorum motuum <lb></lb>ſunt æqualia. </s> </p> <p type="main"> <s id="s.000801">Nam cor. </s> <s id="s.000802">2. pr. 8. huius primi demonſtratum eſt, tem<lb></lb>pora ſimplicium, ſimiliumque motuum componi ex ratio<lb></lb>ne ſpatiorum, ſeu altitudinum geneſum, & reciproca pri<lb></lb>marum, aut extremarum velocitatum, ſeu amplitudinum <lb></lb>geneſum: ſunt autem altitudines geneſum tractiones, ſeu <lb></lb>elongationes funiculorum, quæ ſunt vt longitudines funi<lb></lb>culorum, ergo tempora æqualia erunt. </s> </p> <p type="main"> <s id="s.000803"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000804"><emph type="italics"></emph>Conſtat, tempora a ſimplicium geneſum in tractionibus fu<lb></lb>niculorum, eſſe compoſita ex ratione elongationum funiculo<lb></lb>rum, & ex reciproca primarum velocitatum.<emph.end type="italics"></emph.end></s> </p> <pb pagenum="81" xlink:href="022/01/087.jpg"></pb> <p type="main"> <s id="s.000805"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000806"><emph type="italics"></emph>Superioris propoſitionis veritas concordat cum prop.<emph.end type="italics"></emph.end> 37. <emph type="italics"></emph>hu<lb></lb>ius, in eo tantùm variatur, quod ibi ponuntur data ſpatią <lb></lb>elongitiones funiculorum, hic verò tempora ſimplicium̨ <lb></lb>motuum, & quia elongationes oſtenſæ ſunt proportionales ſpa<lb></lb>tijs nunc exactis, manifeſtum eſt, noſtri iuris eſſe modò ſpatia <lb></lb>acceleratis motibus exact a ex temporibus ſimplicium <expan abbr="motuũ">motuum</expan> <lb></lb>datis concludere, modò contrà, ex ſpatijs altitudinibus gene<lb></lb>ſum proportionalibus, qua item data ſunt, tempora inuenire, <lb></lb>qua proinde methodus mihi videtur ampliſſima.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000807"><emph type="center"></emph>PROP. XXXX. THEOR. XXXIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000808">SI eiuſdem craſſitiei funiculis pondera dependeant, quę <lb></lb>ſint in ratione reciproca longitudinum ipſorum funi<lb></lb>culorum, ſpatia temporibus geneſum ſimplicium motuum <lb></lb>exacta erunt in ratione duplicata elongationum. <lb></lb><arrow.to.target n="marg187"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000809"><margin.target id="marg187"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 6.</s> </p> <p type="main"> <s id="s.000810"><expan abbr="Nã">Nam</expan> ſi ſit <expan abbr="põdus">pondus</expan> E ad F ſicuti <expan abbr="lõgitudo">longitudo</expan> DB ad CA, & ſint, <lb></lb>craſsities <expan abbr="funiculorũ">funiculorum</expan> æquales erit ſanè ratio, quæ <expan abbr="cõponi-tur">componi<lb></lb>tur</expan> ex ratione <expan abbr="funiculorũ">funiculorum</expan>, & ex ea <expan abbr="põderum">ponderum</expan>, æqualitatis; ob <lb></lb>idque geneſes <expan abbr="ſimpliciũ">ſimplicium</expan> <expan abbr="motuũ">motuum</expan>, <expan abbr="quarũ">quarum</expan> altitudines CE, DF <lb></lb><expan abbr="habebũt">habebunt</expan> amplitudines, <expan abbr="nẽpe">nempe</expan> primas velocitates interſe ęqua<lb></lb>les (nam cum pondera erant æqualia, primæ velocitates <lb></lb>proportionabantur longitudinibus <expan abbr="funiculorũ">funiculorum</expan>, ideo, cum </s> </p> <p type="main"> <s id="s.000811"><arrow.to.target n="marg188"></arrow.to.target><lb></lb>pondera reciprocantur longitudinibus ijſdem, ſeu viribus <lb></lb>funiculorum, fit vt primæ velocitates æquales reddantur) <lb></lb>cum ergo ita ſit, ſpatia recurſuum temporibus imaginum̨ <lb></lb>ſimplicium & accelerato motu confecta erunt in ratione <lb></lb>duplicata elongationum. </s> </p> <pb pagenum="82" xlink:href="022/01/088.jpg"></pb> <p type="margin"> <s id="s.000812"><margin.target id="marg188"></margin.target>28. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000813"><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000814"><emph type="italics"></emph>Cum ex eadem pr.<emph.end type="italics"></emph.end> 28. <emph type="italics"></emph>huius, eadem ſpatia ſint vt quadra<lb></lb>ta temporum, erunt ipſa tempera in ratione ſubduplicatą <lb></lb>elongationum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000815"><emph type="center"></emph>PROP. XXXXI THEOR. XXXIV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000816">SI funiculis æqualem craſſitiem habentibus fuerint ſuſ<lb></lb><arrow.to.target n="marg189"></arrow.to.target><lb></lb>penſa inæqualia pondera, ſpatia, quæ acceleratis mo<lb></lb>tibus, ac temporibus geneſum ſimplicium recurruntur ne<lb></lb>ctentur ex ratione duplicata elongationum, & ex duabus <lb></lb>reciprocè ſumptis rationibus, nempe longitudinum prima<lb></lb>rum funiculorum, antequam pondera ſuſpenderentur; & <lb></lb>ipſorum ponderum. </s> </p> <p type="margin"> <s id="s.000817"><margin.target id="marg189"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>Fig.<emph.end type="italics"></emph.end> 6.</s> </p> <p type="main"> <s id="s.000818">In antecedenti figura illud primum ſatis patet, quòd ſi <lb></lb>loco ponderis F ſuſpenſum fuiſſet pondus aliud grauius, <lb></lb>aut leuius, prior velocitas in aſcenſu fili, ſeu funiculi, aut <lb></lb>chordæ aucta, vel imminuta fuiſſet pro magnitudine pon<lb></lb>deris ſubſtituti; quamobrem priores velocitates ex inæqua <lb></lb>litate ponderum eidem chordæ ſuſpenſorum dependentes <lb></lb>forent, vt ipſa pondera; verùm cum ſuppoſitis funiculis <lb></lb>æqualia pondera ſuſpenſa veniunt, primæ velocitates ſunt <lb></lb><arrow.to.target n="marg190"></arrow.to.target><lb></lb>vt longitudines funiculorum, ergo velocitates primæ, cum <lb></lb>inæqualia ſunt pondera, quæ ſubtrahuntur, nectentur ex <lb></lb>ratione longitudinum funiculorum, & ex ea ponderum <lb></lb>inæqualium: quæcumque igitur ſit tractio DF, geneſes ha<lb></lb>bebimus ſimilium ſimpliciumque motuum, vnam, cuius al<lb></lb>titudo CE, & alteram habentem altitudinem DF, & ſunt <lb></lb>earundem geneſum amplitudines, ſeu primæ velocitates <lb></lb>in ratione compoſita funiculorum AC ad BD, & ponderis <lb></lb><arrow.to.target n="marg191"></arrow.to.target><lb></lb>pendentis ex E ad pondus ſuſpenſum in F; ergo ſpatia ac<lb></lb>celeratis motibus tranſacta temporibus geneſum <expan abbr="ſimpliciũ">ſimplicium</expan> <pb pagenum="83" xlink:href="022/01/089.jpg"></pb>nectentur ex ratione dublicata elongationum, ſiue altitu<lb></lb>dinum geneſum, & ex duabus rationibus reciprocè ſum<lb></lb>ptis funiculorum AC ad BD, & ponderum E ad F. <lb></lb></s> <s id="s.000819">Quod &c. </s> </p> <p type="margin"> <s id="s.000820"><margin.target id="marg190"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000821">pr.<emph.end type="italics"></emph.end> 37. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000822"><margin.target id="marg191"></margin.target>30. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000823"><emph type="center"></emph>PROP. XXXXII. THEOR. XXXV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000824">IIſdem poſitis, ſi ſpatia recurſuum erunt ipſæ elongatio<lb></lb>nes, tempora, quibus ab extremitatibus ſolutis recur<lb></lb>runtur, erunt in ratione ſubduplicata eorundem. </s> <s id="s.000825">Nam cum <lb></lb>geneſes ſimilium, ſimpliciumque motuum ſint æquè am<lb></lb>plæ, erunt, tempora in ratione ſubduplicata imaginum, <lb></lb>ſeu ſpatiorum acceleratorum motuum, ſunt verò ſpatia <lb></lb>ipſæ elongationes; ergo &c. </s> </p> <p type="main"> <s id="s.000826"><emph type="center"></emph>PROP. XXXXIII. THEOR. XXXVI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000827">CHordæ non eiuſdem craſſitiei, eiuſdem tamen mate<lb></lb>riæ, ac longitudinis, tunc æquè trahentur vbi <expan abbr="ſuſpẽ-ſa">ſuſpen<lb></lb>ſa</expan> pondera craſſitut inibus proportionalia fuerint. </s> <s id="s.000828">Nam <lb></lb>craſſior chorda poteſt concipi compoſita ex funiculis eiuſ <lb></lb>dem craſſitiei alterius chordæ, ſi illa huius fuerit multiplex, <lb></lb>& ſi partes exilior funiculus fuerit alterius craſſioris, erit <lb></lb>craſſities alicuius alterius funiculi, quæ pluries acceptą <lb></lb>conſtituere poterit vtranque craſſitiem funiculorum pro<lb></lb>poſitorum (hìc enim non accidit enumerare craſſities in<lb></lb>terſe irrationales, quippe quia, quod de iam dictis oſten<lb></lb>derimus, de his quoque facilè eſt iudicare, ſecùs eſſemus <lb></lb>longi, quam par eſt, potiſſimùm cum hæc præter <expan abbr="inſtitutũ">inſtitutum</expan> <lb></lb>adijciantur, & quidem vt conſtet, quomodo methodus iſta <lb></lb>noſtra facilis ſit, ac vtiliſſima) quapropter ſi cuique acce<lb></lb>ptarum æqualium chordarum, pondera æqualia ſuſpenſa <lb></lb>ſint, porrò hæc omnes æquè trahentur ab ipſis æqualibus <lb></lb>ponderibus, & ſic etiam compoſita, nempe choidæ pro-<pb pagenum="84" xlink:href="022/01/090.jpg"></pb>poſitæ; ſuntque ita pondera in eadem ratione craſſitierum, <lb></lb>ſicut propoſuimus; ergo patet propoſitum. </s> </p> <p type="main"> <s id="s.000829"><emph type="center"></emph>PROP. XXXXIV. THEOR. XXXVII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000830">SI fuerint eiuſdem materiæ funiculi, & ſint illis ſuſpenſa <lb></lb>pondera craſſitiebus proportionalia, ratio ſpatiorum <lb></lb>in reditibus accelerato motu exactorum, <expan abbr="tẽporibus">temporibus</expan> ſim<lb></lb><arrow.to.target n="marg192"></arrow.to.target><lb></lb>plicium geneſum, erit eadem ac funiculorum. </s> </p> <p type="margin"> <s id="s.000831"><margin.target id="marg192"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 6. <lb></lb>42. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000832">Nam, vt in præcedenti figura, erit tractio CE ad DF ita <lb></lb><arrow.to.target n="marg193"></arrow.to.target><lb></lb>AC ad BD, vel AE ad BF, ſunt autem primæ velocitates, <lb></lb>ſeu amplitudines geneſum ſimplicium, ſimiliumque <expan abbr="motuũ">motuum</expan> <lb></lb>in ratione funiculorum, ergo decurſuum ſpatia motibus <lb></lb><arrow.to.target n="marg194"></arrow.to.target><lb></lb>acceleratis exacta nectentur ex ratione duplicata altitu<lb></lb>dinum geneſum ſimplicium, nempe duplicata <expan abbr="funiculorũ">funiculorum</expan>, <lb></lb>& reciproca amplitudinum, ſuntque ipſæ amplitudines <lb></lb>homologè vt longitudines funiculorum, ergo relinquitur <lb></lb>vt ipſa ſpatia ſint in vnica ratione longitudinum funicu<lb></lb>lorum. </s> </p> <p type="margin"> <s id="s.000833"><margin.target id="marg193"></margin.target><emph type="italics"></emph>Cor. </s> <s id="s.000834">pr.<emph.end type="italics"></emph.end> 37. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000835"><margin.target id="marg194"></margin.target>27. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000836">Quòd ſi ſpatia recurſuum ponantur ipſæ tractiones, vel <lb></lb>longitudines funiculorum, oſtendetur tempora eſſe æqua<lb></lb>lia, quemadmodum æqualia ſunt tempora ſuperius pro<lb></lb>poſita ſimplicium geneſum. </s> </p> <p type="main"> <s id="s.000837"><emph type="center"></emph>PROP. XXXXV. THEOR. XXXVIII.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg195"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000838"><margin.target id="marg195"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="main"> <s id="s.000839">SI eiuſdem materiei quibuſcunque funiculis aligentur <lb></lb>quæcunque pondera, ijs ſublatis aſcenſuum ſpatia ab <lb></lb>extremitatibus ſolutis exacta temporibus geneſum ſimpli<lb></lb>cium, ijs nempe quæ impenderentur in motibus iuxta ſim<lb></lb>plices geneſes, erunt in ratione compoſita quadratorum <lb></lb>elongationum chordarum, ex ea craſſitierum, & ex duabus <lb></lb>reciprocè ſumptis rationibus, nempe longitudinum fu<lb></lb>niculorum antequam traherentur; & ſuſpenſorum ponde<lb></lb>rum. </s> </p> <pb pagenum="85" xlink:href="022/01/091.jpg"></pb> <p type="main"> <s id="s.000840">Funiculi AB, GH trahantur à ponderibus quibuſcunque <lb></lb>C, I in C, et I. </s> <s id="s.000841">Dico ſi exempta ſint pondera, fore, vt ſpatia <lb></lb>quæ acceleratis motibus exiguntur ab extremitatibus ſo<lb></lb>lutis C, I ſint in ratione compoſita ex duplicata IH ad BC, <lb></lb>craſſitudinis ad craſſitudinem funiculorum AB, GH; dein<lb></lb>de ex funiculi longitudine HG ad longitudinem AB, pon<lb></lb>deriſque I ad pondus C. </s> <s id="s.000842">Intelligatur funiculus, ſeu chor<lb></lb>da, æque craſſa, ac ſimiliter cedens, quàm GH (id quod <lb></lb>ſemper intelligimus quoties funiculi, interſe comparantur) <lb></lb>ſed æquè longa, ac AB, ſitque illi pondus F adiectum, ad <lb></lb>quod C eandem habeat rationem, ac craſſities AB ad craſ<lb></lb>ſitiem DE, conſtat elongationem EF æqualem fieri ipſi <lb></lb>CB, & cum primæ velocitates, ſeu amplitudines æquè al<lb></lb>tarum geneſum ſimilium, ſimpliciumque motuum ſint <expan abbr="etiã">etiam</expan> <lb></lb>æquales, ſpatia decurſuum acceleratis motibus exacta <expan abbr="erũt">erunt</expan> <lb></lb>prorſus æqualia; ſunt verò funiculi DE, GH eiuſdem craſ<lb></lb>ſitiei, eiſque ſunt ſuſpenſa duo'pondera inæqualia F, I; ergo <lb></lb>decurſuum ſpatia ab extremitatibus ſolutis exacta <expan abbr="nectẽ-tur">necten<lb></lb>tur</expan> ex ratione duplicata elongationum FE, ſeu CB ad IH, <lb></lb>ex ratione, quam habent longitudines funiculorum HG ad <lb></lb>DE, ſeu AB, & ex ea ponderum I ad F; verùm pondera I <lb></lb>ad F nectuntur ex rationibus ponderum I ad C et C ad F, <lb></lb>quæ poſtrema eſt ratio craſſitiei funiculi AB ad craſſitiem <lb></lb>funiculi DE, ſeu GH; ergo vt propoſuimus ſpatia accele<lb></lb>ratis motibus exacta, nectentur ex rationibus <expan abbr="quadratorũ">quadratorum</expan> <lb></lb>CB ad HI; craſſitudinum funiculorum AB, GH; <expan abbr="ponderũ">ponderum</expan> <lb></lb>I ad C, & longitudinum HG ad AB. </s> <s id="s.000843">Quod &c. </s> </p> <p type="main"> <s id="s.000844"><emph type="center"></emph>PROP. XXXXVI. THEOR. XXXIX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000845">TEmpora geneſum ſimplicium, dum chordis ſuſpen<lb></lb>duntur quæcunque grauia, nectuntur, ex ratione <lb></lb>elongationum funiculorum, & ex contrariè ſumptis ratio <lb></lb>nibus, craſſitudinum, longitudinumque funiculorum, nec <pb pagenum="86" xlink:href="022/01/092.jpg"></pb>non ponderum funiculis ſuſpenſorum. </s> </p> <p type="main"> <s id="s.000846">Nam Cor: 2. pr. 8. primi demonſtratum eſt, temporą <lb></lb>ſimplicium ſimiliumque motuum componi ex ratione ſpa<lb></lb>tiorum, ſeu altitudinum geneſum, & reciproca primarum <lb></lb>velocitatum, ſeu amplitudinum geneſum, ſunt autem alti<lb></lb>tudines geneſum tractiones, ſeu elongationes funiculorum; <lb></lb>velocitatesverò primæ nectuntur ex rationibus craſſitudi<lb></lb>num, & ex ea longitudinum funiculorum antequam tra<lb></lb>herentur (hoc enim ſubinde oſtendemus) ergo tempora <lb></lb>propoſita ſimplicium geneſum, dum chordis <expan abbr="alligãtur">alligantur</expan> quę<lb></lb>cunque inæqualia pondera, componentur ex rationibus <lb></lb>elongationum funiculorum, & ex contrariè ſumptis craſſi<lb></lb>tudinum, longitudinumque funiculorum, & ponderum. </s> </p> <p type="main"> <s id="s.000847"><emph type="center"></emph><emph type="italics"></emph>Aßumptum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000848"><arrow.to.target n="marg196"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000849"><margin.target id="marg196"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>fig.<emph.end type="italics"></emph.end> 7.</s> </p> <p type="main"> <s id="s.000850">VErùm primæ velocitates in ijſdem chordis componi <lb></lb>ex ratione craſſitudinum, longitudinum <expan abbr="funiculorũ">funiculorum</expan>, <lb></lb>& ſuſpenſorum ponderum, ſic oſtendemus, </s> </p> <p type="main"> <s id="s.000851">Quoniam in eadem poſtrema figura velocitas, quam̨ <lb></lb>haberet funiculus AB ex liberatione ponderis eſt æqualis <lb></lb>velocitati, quam haberet alius funiculus, vbi hic etiam li<arrow.to.target n="marg197"></arrow.to.target><lb></lb>beraretur à pondere, ſcilicet cum pondera craſſitiebus fu<lb></lb>niculorum proportionalia ſunt, & ipſi funiculi æquè longi; <lb></lb>velocitas funiculi DE à pondere F ad velocitatem <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb><arrow.to.target n="marg198"></arrow.to.target><lb></lb>funiculi, ſi loco ponderis F ſubſtitutum eſſet aliud æquale <lb></lb>ipſi I, eſſet vt pondus F ad ſubſtitutum, ſeu ad I, eſt autem <lb></lb>velocitas eiuſdem funiculi DE, dum fuiſſet pondus ei ſuſ<lb></lb>penſum æquale I ad velocitatem funiculi GH a pondere I <lb></lb>vt longitudo DE ad GH; ergo patet propoſitum. </s> </p> <pb pagenum="87" xlink:href="022/01/093.jpg"></pb> <p type="margin"> <s id="s.000853"><margin.target id="marg197"></margin.target>43. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="margin"> <s id="s.000854"><margin.target id="marg198"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 41. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000855"><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000856"><emph type="italics"></emph>Quod hucuſque ostendimus in funiculis ponderibus de<lb></lb>grauatis, non abſimili modo præst abimus in chordis ad <expan abbr="vtrã-que">vtran<lb></lb>que</expan> extremitatem firmatis, & adductis, hoc tantum diſcrimi<lb></lb>ne, vt ſi in ijs pondere ſublato, motus extremitatis ſolutæ at<lb></lb>tendebatur, hìc media parte attractâ chordâ, & ſubinde ſui <lb></lb>iuris relictâ, vibrationem eius obſeruamus, & equidem illa <lb></lb>omnia in hunc finem oſtendimus, quippe ab hac re, plurima <lb></lb>vtiliſſimæque veritates manere poſſunt. </s> <s id="s.000857">Nam de arcubus poſſes <lb></lb>pulcherrima inſtitui ratio, & qui vellet armonicorum ſono<lb></lb>rum, vel vocum per chordarum vibrationes editarum, tempo<lb></lb>ra, cum ſoni ad aures perueniunt, inueſtigare, reor non aliam <lb></lb>viam, quàm hanc ingredi nos debere, atque indè conſonantia<lb></lb>rum fortaſſe naturam percipere poſſe, vt primus <gap></gap>ui<gap></gap> Gal<lb></lb>lileus quamquam vibrationes tenſarum chordarum <expan abbr="differãt">differant</expan> <lb></lb>ab ijs pendulorum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000858"><emph type="center"></emph><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <pb pagenum="88" xlink:href="022/01/094.jpg"></pb> <p type="main"> <s id="s.000859"><emph type="center"></emph>SPIEGATIONE<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000860"><emph type="center"></emph>di vna nuoua ſpecie di Baleſtra.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="marg199"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000861"><margin.target id="marg199"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 9.</s> </p> <p type="main"> <s id="s.000862">IN queſta figura ſi eſprime vna nuoua <expan abbr="inuẽ-tione">inuen<lb></lb>tione</expan> di Baleſtra, la quale, maſſimamente <lb></lb>in grande, per tirar granate, ò ſaſſi può eſ<lb></lb>ſere di gran conſeguenze nella militare, co<lb></lb>me dimoſtreraſſi. </s> </p> <p type="main"> <s id="s.000863">Dalle ſue parti ſi verrà in cognitione del <lb></lb>modo di fabbricarta, e ſono le ſeguenti. </s> </p> <p type="main"> <s id="s.000864">AM, MN ſono amendue le braccia. </s> <s id="s.000865">Il punto M è il cen<lb></lb>tro della machina. </s> <s id="s.000866">Per la cauità M deue paſlar la pallą <lb></lb>ſcagliata dalla corda; e per di ſotto M ſi ferma & incaſtra <lb></lb>nel manico, al modo delle baleſtre communi. </s> <s id="s.000867">Ai due capi, <lb></lb>ò ſiano eſtremità A, N ſi annette la fune. </s> <s id="s.000868">I punti A, E, F, <lb></lb>G, I, K ſono in vna linea retta. </s> <s id="s.000869">Gl' interualli AE, EF, FG, <lb></lb>GI, IK, ſono, benche non di neceſſità, eguali. </s> <s id="s.000870">Le altezze, <lb></lb>ò commeſſure KL, IH, GD, FC, EB perpendiculari, nell' <lb></lb>incuruarſi dell' arco, ſi aprono intorno a' centri K, I, G, F, <lb></lb>E. </s> <s id="s.000871">Donde ne ſiegue, che prendendo la corda dal ſuo mez<lb></lb>zo, e tirandola verſo O; amendue le braccia ſi aprono nel<lb></lb>le predette commeſſure, come compare nell' vno d'eſſi ſe<lb></lb>gnato a punti con le lettere corriſpondenti. </s> <s id="s.000872">Ciaſcuna <lb></lb>delle predette commeſſure viene ſtrettamente rinſerratą <lb></lb>da vna molla, come ſi vede in L, H, D, C, B; e queſte mol<lb></lb>le, quanto più ſi auuicinano al centro M, deuono eſſere più <lb></lb>grandi e più maſſiccie, in modo che, per cagione della <expan abbr="grã-dezza">gran<lb></lb>dezza</expan> opportuna, vengano ad aprirſi con egual facilità <lb></lb>dell'altre, e per cagione della groſſezza, habbiano nel ſer<lb></lb>rarſi maggior forza, ò ſia momento, per la ragione, che ſot<lb></lb>to ſi dirà. </s> </p> <p type="main"> <s id="s.000873">Ciò preſuppoſto, è facil coſa dimoſtrare i vantaggi di <pb pagenum="89" xlink:href="022/01/095.jpg"></pb>queſta machina ſopra le ordinarie. </s> </p> <p type="main"> <s id="s.000874">Primieramente nel triangolo ALK, eſſendo le altezzę <lb></lb>EB, FC, GD, IH, KL perpendicolari, e perciò paralelle; <lb></lb>ne ſiegue che le proportioni di AE ad EB, di AF ad FC, di <lb></lb>AG a GD, di AI ad IH, di AK a KL ſieno tutte eguali; e <lb></lb>douendo eſſere parimente eguali le reſiſtenze delle molle <lb></lb>in B, C, D, H, L, che ſi ſuppongono di egual neruo nell' <lb></lb>aprirſi; ne ſiegue (ſecondo i principij della Meccanica) che <lb></lb>attraendoſi con la fune l'eſtremità A, nel medeſimo tempo <lb></lb>e con la medeſima facilità vinceraſſi l'equilibrio di tutte le <lb></lb>molle; la reſiſtenza delle quali ſi conſidera in ragione di <lb></lb>peſo, ſi come le linee AE, EB; AF, FC; AG, GD; &c. ſi con<lb></lb>ſiderano come vetti, ò lieue, che hanno i loro ippomoclij, ò <lb></lb>ſiano centri in E, F, G, I, K, e la potenza in A, la quale è <lb></lb>comune a tutte. </s> </p> <p type="main"> <s id="s.000875">In ſecondo luogo, hauendo il braccio AE al braccio EB <lb></lb>(il ſimile dicaſi degli altri) hauendo, dico, gran proportio<lb></lb>ne, reſterà molto ageuolato il moto. </s> </p> <p type="main"> <s id="s.000876">Terzo eſſendo molte le molle, e a prendoſi tutte, ne deue <lb></lb>ſeguire vn notabile incuruamento d'amendue le braccia; <lb></lb>onde laſciando l'arco in libertà, e chiudendoſi tutte le ſu<lb></lb>det te molle nel medeſimo tempo, cioè quaſi in vn'attimo; <lb></lb>dourà la corda, che era tirata verſo O, paſſare quaſi in <expan abbr="iſtã-te">iſtan<lb></lb>te</expan> verſo M; il che non potendoſi fare ſe non con ſomma ve<lb></lb>locità, per la grandezza dello ſpatio; e a queſta corriſpon<lb></lb>dendo la forza, ne ſeguirà vn colpo molto conſiderabile, e <lb></lb>vantaggioſo, come ciaſcuno può arguire. </s> </p> <p type="main"> <s id="s.000877">Reſtano hora a ſciorſi alcune difficoltà. </s> <s id="s.000878">La prima è, <lb></lb>che, quantunque ſia vero, che quella forza baſtante in A <lb></lb>per vincer l'equilibrio della molla B, quella medeſima al<lb></lb>treſi ſia ſufficiente a vincer l' equilibrio di tutte l'altre, per <lb></lb>eſſere eguali le proportioni delle vetti; ciò non oſtante, <expan abbr="cõ-ſiderandoſi">con<lb></lb>ſiderandoſi</expan> il braccio incuruato, come ſi vede nell' arco <lb></lb>KLA ſegnato a punti, le proportioni rieſcono alterate; do-<pb pagenum="90" xlink:href="022/01/096.jpg"></pb>uendoſi prendere per le lunghezze delle vetti ſudette, non <lb></lb>più le lunghezze di prima, ma bensi le applicate di detto <lb></lb>arco, cioe af, ag, ai, ak; delle quali aK, e l' altre a lei più <lb></lb>vicine ſi abbreuiano molto più quando l' arco è incurua<lb></lb>to, che quando non è: Onde per tal ragione dourebbero <lb></lb>le parti più vicine al centro M aprirſi meno dell'altre più <lb></lb>vicine alle eſtremità. </s> <s id="s.000879">A ciò ſi riſponde, che per eſſer la <lb></lb>corda a o più obliqua alla lunghezza a e di quel che ſia all' <lb></lb>altre più vicine al centro M, quindi ne ſiegue, che per quel' <lb></lb>altra cagione s' aprono più ageuolmente le parti vicine al <lb></lb>centro; onde, temperata vna ragione con l' altra (quando <lb></lb>l' arco non ſia eſtremamente incuruato) ſi conſeguiſce vno <lb></lb>ſtato d'apertura opportuna. </s> </p> <p type="main"> <s id="s.000880">La ſeconda difficoltà è che ciaſcuna molla nel ſuo re<lb></lb>ſtringerſi, par che cagioni qualche effetto contrario all'in<lb></lb>tento. </s> <s id="s.000881">Imperoche, per eſempio, nella molla B il mezzo <lb></lb>anello, che riſguarda l'eſtremità A, nello ſtringerſi fà benſi <lb></lb>il ſuo douere, perche il ſuo moto è verſo il centro M; ma l' <lb></lb>altra metà, che riſguarda il ſudetto centro M, nello ſtrin<lb></lb>gerſi, hauendo il ſuo moto verſo A, ſi oppone al chiudi<lb></lb>mento della molla ſeguente C; e il ſimile dicaſi dell' altre. <lb></lb></s> <s id="s.000882">A ciò ſi è poſto rimedio col far più grandi, e più maſſiccie <lb></lb>le molle più vicine al centro M, accreſcendole, e ingroſſan<lb></lb>dole di mano in mano opportunamente. </s> <s id="s.000883">Quindi ne ſegue <lb></lb>che per la maggior grandezza <expan abbr="cõſentono">conſentono</expan> egualmente all' <lb></lb>aprirſi con facilità; ma all' incontro nel ſerrarſi, per eſſere <lb></lb>più maſſiccie, e di maggior corpo, vengono ad hauere <lb></lb>maggior momento delle men corpulenti, ſuperando coņ <lb></lb>ciò non ſolo il detto moto oppoſto, ma etiandio impri<lb></lb>mendo maggior moto al ferro dell'arco, con cui ſi acco<lb></lb>muna il moto. </s> </p> <p type="main"> <s id="s.000884">Auuertaſi, che quanto ſaranno di maggior numero le <lb></lb><expan abbr="cõmeſlure">commeſture</expan>, le molle di maggior peſo, e l'arco più pouero di <lb></lb>corpo, tanto riuſcirà il colpo a diſmiſura maggiore, per l' <pb pagenum="91" xlink:href="022/01/097.jpg"></pb>incuruamento notabile delle braccia, e per il maggior mo<lb></lb>mento delle molle; e ciò con adoperare la medeſima <lb></lb>forza. </s> </p> <p type="main"> <s id="s.000885">Auuertaſi parimente, che il braccio AE, è il ſuo corriſ<lb></lb>pondente deuono eſſere alquanto più corti, cioè A vna <lb></lb>delle eſtremità dell'arco deue eſſere più verſo il centro di <lb></lb>quel che ſia il concorſo delle linee LB, KE, come pure dall' <lb></lb>altra parte; perche ſi vede che aprendoſi meno le parti vi<lb></lb>cine ad A, l'altre molle fanno miglior effetto. </s> </p> <p type="main"> <s id="s.000886">Finalmente la ſperienza ha moſtrato, che eſſendoſi la<lb></lb>uorata vna tal machina con pochiſſimi nodi, ageuoliſſima <lb></lb>ad aprirſi, e ſenza hauer ingrandite e ingroſlate le molle, <lb></lb>che più ſi vanno auuicinando al centro M, come ſi è det<lb></lb>to; con tutto ciò l' ordigno è riuſcito di forza molto ſupe<lb></lb>riore a vna baleſtra grande, e difficilifſima a inarcarſi. </s> <s id="s.000887">On<lb></lb>de non dubito, che, facendoſi con tutte le regole accenna<lb></lb>te, non debba riuſcire vna machina di effetto marauiglioſo <lb></lb>aggiungendo che per tirar granate dourebbero i bracci <lb></lb>eſſer di legno, armati di ferro ſol doue ſi richiede. </s> </p> <pb pagenum="92" xlink:href="022/01/098.jpg"></pb> <p type="main"> <s id="s.000888"><emph type="center"></emph>Nouum genus Baliſtæ <lb></lb>Explicatio.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000889"><arrow.to.target n="marg200"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000890"><margin.target id="marg200"></margin.target><emph type="italics"></emph>Tab.<emph.end type="italics"></emph.end> 9.</s> </p> <p type="main"> <s id="s.000891">IN hac figura exprimitur nouum genus Bali<lb></lb>ſtæ, quæ machina præſertim in mole maio<lb></lb>ri, non parum vtilitatis afferre poteſt rei mi<lb></lb>litari ad eiaculanda miſſilia, vt demonſtra<lb></lb>bitur. </s> <s id="s.000892">Ex eius verò partibus, quas ſubinde <lb></lb>recenſeo, etiam modus ſtructuræ apparebit. </s> </p> <p type="main"> <s id="s.000893">AM, MX ſunt brachia. </s> <s id="s.000894">Punctum M centrum machi<lb></lb>næ. </s> <s id="s.000895">Per cauitatem M tranſit telum emiſſum. </s> <s id="s.000896">Infra M in<lb></lb>ſeritur manubrium, vt in baliſtis vulgaribus. </s> <s id="s.000897">Extremis <lb></lb>capitibus A, N adnectitur funis. </s> <s id="s.000898">Puncta A, E, F, G, I, K <lb></lb>ſunt in linea recta. </s> <s id="s.000899">Interualla AE, EF, FG, GI, IK ſunt (li<lb></lb>cèt non neceſſariò) æqualia. </s> <s id="s.000900">Altitudinis, ſeu commiſſuræ <lb></lb>KL, IH, GD, FC, EB ſunt perpendiculares rectæ occultæ <lb></lb>KA. </s> <s id="s.000901">Singulæ autem, dum curuatur arcus, aperiuntur cer<lb></lb>ca centra K, I, G, F, E. </s> <s id="s.000902">Hinc ſequitur vt funis ex medio <lb></lb>dum attrahitur in O, aperiantur prædictæ commiſſuræ, ſeu <lb></lb>nodi, & curuentur vtraque brachia, vt in eorum altero ap<lb></lb>paret punctis notato. </s> <s id="s.000903">Quilibet ex his nodis arctiſſimè ſtrin<lb></lb>gitur ſupernè, a ſuo elaterio, vt videre eſt in L, H, D, C, B. <lb></lb></s> <s id="s.000904">Elateria autem quò propinquiora centro M tanto maiora, <lb></lb>& craſſiora debent eſſe remotioribus: Hinc fit vt, propter <lb></lb>molem opportunè auctam, æquè facilè aperiantur, ac cæ<lb></lb>tera; & vice verſa, propter craſſitiem maiorem, ſibi relicta <lb></lb>validiùs reſtringantur. </s> <s id="s.000905">Cuius rei paulo infra rationem <lb></lb>dabimus. </s> </p> <p type="main"> <s id="s.000906">His poſitis facile eſt oſtendere, quantum præſtet hu<lb></lb>iuſcemodi machina vulgaribus & communibus baliſtis. </s> </p> <p type="main"> <s id="s.000907">Primùm, in Triangulo ALK cùm altitudines EB, FC, <lb></lb>GD, IH, KL ſint perpendiculares, ideoque parallelæ, hinc <pb pagenum="93" xlink:href="022/01/099.jpg"></pb>ſit vtrationes AE ad EB, AF ad FC, AG ad GD, AI ad <lb></lb>IH, AK ad KL ſint æquales. </s> <s id="s.000908">Sunt pariter æquales reſi<lb></lb>ſtentiæ elateriorum in B, C, D, H, L (poſuimus enim ela<lb></lb>teria ita opportunè aucta vt æquè facile ſingula aperian<lb></lb>tur) ergo (ex primis principijs mechanicorum) dum attra<lb></lb>huntur fune extrema capita A, N, eodem tempore, eadem<lb></lb>que facili ate vincetur æquilibrium omnium elateriorum, <lb></lb>quorum reſiſtentia in ſingulis conſideratur in ratione pon<lb></lb>deris, quemadmodum lineæ AE, EB; AF, FC; AG, GD <lb></lb>&c. conſiderantur vt vectes, quorum hippomoclia ſeu <expan abbr="cẽ-tra">cen<lb></lb>tra</expan> ſunt in E, F, G, I, K, potentia autem conſideratur in A <lb></lb>communis omnibus. </s> </p> <p type="main"> <s id="s.000909">Secundò, cùm AE ad EB (idem die de cæteris) habeant <lb></lb>magnam proportionem, facilè aperientur nodi, & curuabi<lb></lb>tur arcus; quantumuis augeatur numerus nodorum. </s> </p> <p type="main"> <s id="s.000910">Tertiò Cum ſint plures nodi, atque omnes aperiantur, <lb></lb>neceſſe eſt vt brachia arcus valdè incuruentur; <expan abbr="quamobrẽ">quamobrem</expan> <lb></lb>ſi idem arcus ſibi relinquatur, prædicti nodi omnes, vi ela<lb></lb>teriorum, ictu oculi claudentur; eodemque puncto tempo<lb></lb>ris corda ex O percurret totum ſpatium vſque ad M: Quòd <lb></lb>cùm fieri nequeat niſi ſumma velocitate, propter magni<lb></lb>tudinem prædicti ſpatij, & velocitati reſpondent vis, atque <lb></lb>impetus, neceſſe eſt vt hinc ſequatur ictus valde notabilis, <lb></lb>vt facilè eſt vnicuique conijcere. </s> </p> <p type="main"> <s id="s.000911">Super ſunt nunc difficultates nonnullæ ſoluendæ. </s> <s id="s.000912">Prima <lb></lb>eſt, quòd licèt vis ſufficiens in A ad vincendum <expan abbr="æquilibriũ">æquilibrium</expan> <lb></lb>elaterij B, illa eadem quoque ſufficiat ad vincendum æqui<lb></lb>librium cæterorum, propter æquales proportiones <expan abbr="vectiũ">vectium</expan>; <lb></lb>his tamen non obſtantibus, ſi conſideretur brachium iam <lb></lb>incuruatum, vt apparet in KLA punctis notato, proportio<lb></lb>nes illæ cernuntur notabiliter variatæ. </s> <s id="s.000913">Neque enim pro <lb></lb>longitudinibus vectium ſumi poſſunt longitudines priores, <lb></lb>ſed loco ipſarum accipiendæ ſunt applicatæ arcus, videli<lb></lb>cet af, ag, ai, ak quarum ak, eidemque propinquiores, <expan abbr="quã-">quan-</expan><pb pagenum="94" xlink:href="022/01/100.jpg"></pb>do arcus incuruatur, breuiores fiunt, quàm eſſent anteą. <lb></lb></s> <s id="s.000914">Reſpondeo, quòd corda ao cùm ſit obliquior reſpectu <lb></lb>longitudinis ae, quàm reſpectu cæterarum centro propin<lb></lb>quiorum, hinc fit vt, quantùm eſt ex hac ratione, faciliùs <lb></lb>aperiantur partes propinquiores centro; quamobrem, vtra<lb></lb>que ratione inuicem temperata, dummodo arcus non ſit <lb></lb>ſummè incuruatus omnes partes aperientur, quantum ſa<lb></lb>tis eſt ad intentum. </s> </p> <p type="main"> <s id="s.000915">Altera difficultas eſt, quod elaterium quodlibet dum <lb></lb>reſtringitur videtur obſtare motui elaterij ſequentis. </s> <s id="s.000916">Nam, <lb></lb>exempli gratia, in elaterio B ſemiannulus reſpiciens extre<lb></lb>mum A, dum ſtringitur, optimè præſtat ſuum effectum, <expan abbr="cũ">cum</expan> <lb></lb>eius motus ſit versùs centrum M At è contrario reliqua <lb></lb>pars, ſeu ſemiannulus reſpiciens prædictum centrum M, <expan abbr="cũ">cum</expan> <lb></lb>habeat ſuum motum verſus A videtur obſtare, quo minus <lb></lb>liberè claudatur ſequens elaterium C. </s> <s id="s.000917">Aque idem de cæte<lb></lb>ris dicendum. </s> <s id="s.000918">Huic incommodo conſultum eſt augendo <lb></lb>magnitudinem, & craſſitiem elateriorum, quò magis acce<lb></lb>dunt ad centrum M. </s> <s id="s.000919">Hinc enim ſequitur vt propter ma<lb></lb>gnitudinem facilè conſentiant arcui, dum incuruatur; at <lb></lb>dum idem arcus liberè ſibi relinquitur, cum ſint corpulen<lb></lb>tiora & craſſiora habent maius momentum, quàm cætera <lb></lb>graciliora, ideoque non modo vincunt motum illum op<lb></lb>poſitum, ſed etiam imprimunt maiorem motum ferro ar<lb></lb>cus, cui ille motus communicatur. </s> </p> <p type="main"> <s id="s.000920">Aduerte quod commiſſuræ ſeu nodi, quò plures fuerint, <lb></lb>elateria autem maioris ponderis, arcus denique corporis <lb></lb>gracilioris equæ expeditioris, tanto ictus longat præſtan<lb></lb>tior ſequetur, tum propter notabilem curuaturam brachio<lb></lb>rum, tum propter momentum maius elateriorum, & <expan abbr="quidẽ">quidem</expan> <lb></lb>poſita eadem potentia, aut etiam minori, pro vt longitudi<lb></lb>nes vectium ſtatuuntur. </s> </p> <p type="main"> <s id="s.000921">Aduerte etiam, longitudinem brachij AE, eiuſdemquę <lb></lb>correſpondentis debere cæteris paribus nonnihil imminui, <pb pagenum="95" xlink:href="022/01/101.jpg"></pb>quod fiet ſi A, alterum extremum arcus, ſit propriùs cen<lb></lb>tro M, quàm ſit concurſus linearum LB, KE. </s> <s id="s.000922">Idem dicen<lb></lb>dum de altero extremo N. </s> <s id="s.000923">Nam cùm minus aperiantur <lb></lb>partes propinquiores puncto A, cætera elateria, vt com<lb></lb>pertum eſt, meliorem effectum præſtant. </s> </p> <p type="main"> <s id="s.000924">Fauet denique experientia. </s> <s id="s.000925">Nam huiuſcemodi machi<lb></lb>na pauciſſimis nodis conſtructa, facillimæ curuaturæ, cum <lb></lb>elaterijs eiuſdem prorſus molis & craſſitudinis; nihilomi<lb></lb>nus longè ſuperauit vim baliſtæ communis maximæ, & dif<lb></lb>ficillimæ flexionis. </s> <s id="s.000926">Quamobrem non dubito quin, ſi præ<lb></lb>cepta ſuperiùs data exactè ſeruentur, elaborari poſſit ma<lb></lb>china miræ vtilitatis. </s> <s id="s.000927">Adde poſtremo ad iacienda <expan abbr="quædã">quædam</expan> <lb></lb>miſſilia, vt eſt genus quoddam bolidum, vulgo <emph type="italics"></emph>granate,<emph.end type="italics"></emph.end> op<lb></lb>portuniora eſſe brachia lignea, tantummodo, vbi neceſſi<lb></lb>tas poſtulat, armata ferro. </s> </p> <p type="main"> <s id="s.000928"><emph type="center"></emph><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> </chap> </body> <back> <pb xlink:href="022/01/102.jpg"></pb> <section> <p type="main"> <s id="s.000929"><emph type="italics"></emph>Vid. D. Bernardus Marchellus Re<lb></lb>ctor Pœnitent. in Metropol. Bonon. <lb></lb>pro Illuſtriſs. & Reverendiſs. Domino <lb></lb>D. Iacobo Boncompagno Archiepiſ<lb></lb>copo, & Principe.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000930"><emph type="italics"></emph>Vid. Silueſter Bonfiliolus Inquiſitionis <lb></lb>reuiſor, & imprimi poſſe cenſuit.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000931"><emph type="italics"></emph>Stante atteſtatione.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000932"><emph type="center"></emph><emph type="italics"></emph>Imprimatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000933"><emph type="italics"></emph>F. Ioſeph Maria Agudius Vicarius <lb></lb>Sancti Offi c ij Bononiæ.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000934"><emph type="center"></emph>8 00 57<emph.end type="center"></emph.end></s> </p> <pb xlink:href="022/01/103.jpg"></pb> <p type="caption"> <s id="s.000935">TABVLA I.<lb></lb><figure id="id.022.01.103.1.jpg" xlink:href="022/01/103/1.jpg"></figure></s> </p> <pb xlink:href="022/01/104.jpg"></pb> <p type="caption"> <s id="s.000936">TABVLA II.<lb></lb><figure id="id.022.01.104.1.jpg" xlink:href="022/01/104/1.jpg"></figure></s> </p> <pb xlink:href="022/01/105.jpg"></pb> <p type="caption"> <s id="s.000937">TABVLA III.<lb></lb><figure id="id.022.01.105.1.jpg" xlink:href="022/01/105/1.jpg"></figure></s> </p> <pb xlink:href="022/01/106.jpg"></pb> <p type="caption"> <s id="s.000938">TABVLA VI.<lb></lb><figure id="id.022.01.106.1.jpg" xlink:href="022/01/106/1.jpg"></figure></s> </p> <pb xlink:href="022/01/107.jpg"></pb> <p type="caption"> <s id="s.000939">TABVLA V.<lb></lb><figure id="id.022.01.107.1.jpg" xlink:href="022/01/107/1.jpg"></figure></s> </p> <pb xlink:href="022/01/108.jpg"></pb> <p type="caption"> <s id="s.000940">TABVLA IV.<lb></lb><figure id="id.022.01.108.1.jpg" xlink:href="022/01/108/1.jpg"></figure></s> </p> <pb xlink:href="022/01/109.jpg"></pb> <p type="caption"> <s id="s.000941">TABVLA VII.<lb></lb><figure id="id.022.01.109.1.jpg" xlink:href="022/01/109/1.jpg"></figure></s> </p> <pb xlink:href="022/01/110.jpg"></pb> <p type="caption"> <s id="s.000942">TABVLA VIII.<lb></lb><figure id="id.022.01.110.1.jpg" xlink:href="022/01/110/1.jpg"></figure><pb xlink:href="022/01/111.jpg"></pb><figure id="id.022.01.111.1.jpg" xlink:href="022/01/111/1.jpg"></figure></s> </p> <pb xlink:href="022/01/112.jpg"></pb> <figure id="id.022.01.112.1.jpg" xlink:href="022/01/112/1.jpg"></figure> <p type="caption"> <s id="s.000943">TABVLA VIIII.</s> </p> </section> </back> </text> </archimedes>