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Yet another new version.
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 12:23:20 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Fabri, Honoré</author> <title>Tractatus physicus de motu locali</title> <date>1646</date> <place>Lyon</place> <translator/> <lang>la</lang> <cvs_file>fabri_tract_026_la_1646.xml</cvs_file> <cvs_version/> <locator>026.xml</locator> </info> <text> <front> <section> <pb xlink:href="026/01/001.jpg"/> <p id="N1001B" type="head"> <s id="N1001D"><emph type="center"/>TRACTATVS <lb/>PHYSICVS <lb/>DE MOTV LOCALI, <lb/><emph type="italics"/>IN QVO<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N1002E" type="main"> <s id="N10030"><emph type="center"/>EFFECTVS OMNES, QVI AD IMPETVM, <lb/>Motum naturalem, violentum, & mixtum pertinent, <lb/>explicantur, & ex principiis Phy&longs;icis <lb/>demon&longs;trantur.<emph.end type="center"/></s> </p> <p id="N1003D" type="main"> <s id="N1003F"><emph type="center"/><emph type="italics"/>Auctore<emph.end type="italics"/> PETRO MOVSNERIO <emph type="italics"/>Doctore Medico:<emph.end type="italics"/><lb/>CVNCTA EXCERPTA<emph.end type="center"/></s> </p> <p id="N10052" type="main"> <s id="N10054"><emph type="center"/><emph type="italics"/>Ex prælectionibus<emph.end type="italics"/> R. P. HONORATI FABRY, <lb/><emph type="italics"/>Societatis<emph.end type="italics"/> IESV.<emph.end type="center"/></s> </p> <figure id="id.026.01.001.1.jpg" xlink:href="026/01/001/1.jpg"/> <p id="N1006C" type="main"> <s id="N1006E"><emph type="center"/><emph type="italics"/>LVGDVNI,<emph.end type="italics"/><lb/>Apud IOANNEM CHAMPION, <lb/>in foro Cambij.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1007E" type="main"> <s id="N10080"><emph type="center"/><emph type="italics"/>M. </s> <s id="N10087">D C. XLVI.<emph.end type="italics"/><lb/>Cum Priuilegio Regis, & Approbatione Doctorum.<emph.end type="center"/></s> </p> <pb xlink:href="026/01/002.jpg"/> </section> <section> <pb xlink:href="026/01/003.jpg"/> <figure id="id.026.01.003.1.jpg" xlink:href="026/01/003/1.jpg"/> <p id="N1009C" type="head"> <s id="N1009E"><emph type="center"/>AMPLISSIMO, <lb/>NOBILISSIMOQVE DOMINO,<emph.end type="center"/></s> </p> <p id="N100A7" type="main"> <s id="N100A9"><emph type="center"/>D. PETRO DE SEVE, <lb/>DOMINO DE FLECHERES, <lb/>SANCTIORIS CONSILII REGIS <lb/>Con&longs;iliario, in Lugdunen&longs;i Curia Prætori prima­<lb/>rio, & &longs;ecundùm Mercatorum Præpo&longs;ito, &c.<emph.end type="center"/></s> </p> <p id="N100B8" type="main"> <s id="N100BA"><emph type="center"/>PETRVS MOVSNERIVS,<emph.end type="center"/></s> </p> <p id="N100C1" type="main"> <s id="N100C3"><!-- NEW --><emph type="italics"/>TIBI alterum no&longs;træ Philo&longs;o­<lb/>phiæ fœtum in&longs;cribo, cui iam <lb/>primum in&longs;crip&longs;i<emph.end type="italics"/> (PRÆTOR <lb/>AMPLISSIME) <emph type="italics"/>nempe idem <lb/>e&longs;&longs;e debeo, quia tu &longs;emper idem <lb/>es: </s> <s id="N100D9"><!-- NEW -->non muta&longs;ti merita, non mu­<lb/>tabo officia: </s> <s id="N100DF"><!-- NEW -->multos non expo&longs;cam Patronos, qui <lb/>iam omnium optimum, & meriti&longs;simum habeo; </s> <s id="N100E5"><!-- NEW -->neo <lb/>enim &longs;acra Philo&longs;ophiæ anathemata rudi, & ru­<lb/>&longs;tico muro appendam, quæ ex &longs;acro tholo templi <lb/>Themidos amœniter pendent: </s> <s id="N100EF"><!-- NEW -->Nec leuem toti rei li­<lb/>terariæ iniuriam inferrem, &longs;i alium illi, quàm li-<emph.end type="italics"/><pb xlink:href="026/01/004.jpg"/><emph type="italics"/>teratum Mecænatem accer&longs;erem: </s> <s id="N100FD"><!-- NEW -->& verò Tracta­<lb/>tum hunc de Motu Locali, alteri quàm tibi in&longs;cri­<lb/>bere non debui, cuius imperia Ludgunen&longs;is orbis, po­<lb/>tiùs quàm vrbis, componunt: </s> <s id="N10107"><!-- NEW -->Tu prudens Intelli­<lb/>gentia, huic orbi &longs;emper a&longs;si&longs;tis; </s> <s id="N1010D"><!-- NEW -->ita motibus in­<lb/>uigilas, vt quieti publicæ con&longs;ulas, remque ita pu­<lb/>blicam admini&longs;tras, vt &longs;ingulis commoda procures: <lb/>Cæterùm dubitare non po&longs;&longs;um, quin hunc meu&mtail; <lb/>quantulumcumque conatum, fidemque meam ia&mtail; <lb/>tibi &longs;emel oppigneratam, & nunc altero voto peni­<lb/>tus ob&longs;trictam, æqui bonique &longs;is con&longs;ulturus, Val&etail;.<emph.end type="italics"/><!-- KEEP S--></s> </p> </section> <section> <pb xlink:href="026/01/005.jpg"/> <figure id="id.026.01.005.1.jpg" xlink:href="026/01/005/1.jpg"/> <p id="N10129" type="head"> <s id="N1012B"><emph type="center"/>PRÆFATIO.<emph.end type="center"/></s> </p> <p id="N10132" type="main"> <s id="N10134"><!-- NEW -->NIHIL habeo præfari (Beneuole Lector) <lb/>in gratiam huius tractatus de Motu Locali, <lb/>cuius amœnitatem & vtilitatem, rerum co­<lb/>piam & &longs;yluam, tuo gu&longs;tui & iudicio re­<lb/>linquo: </s> <s id="N10140"><!-- NEW -->Multi &longs;anè hactenus in hac mate­<lb/>ria feliciter de&longs;udarunt; </s> <s id="N10146"><!-- NEW -->& quidem præ cæteris magnus <lb/>ille Galileus, qui mirificâ, & ferè diuinâ ingenij acie, <lb/>motum localem eò perduxit, quò mortalium nemo per­<lb/>duxerat; </s> <s id="N10150"><!-- NEW -->quia tamen multa omi&longs;it, quæ ad motum &longs;pe­<lb/>ctant, vt nemo ne&longs;cit; </s> <s id="N10156"><!-- NEW -->nec ex principijs Phy&longs;icis mira­<lb/>biles illos effectus demon&longs;trauit, &longs;ed tantùm certis qui­<lb/>bu&longs;dam proportionibus ex geometricis addixit; </s> <s id="N1015E"><!-- NEW -->vt Phy­<lb/>&longs;icæ con&longs;ulamus, aliam inimus viam: </s> <s id="N10164"><!-- NEW -->Geometriam qui­<lb/>dem adhibemus, ad explicandas, exponenda&longs;que præ­<lb/>dictas illas proportiones, quæ motibus in&longs;unt; </s> <s id="N1016C"><!-- NEW -->&longs;ed effe­<lb/>ctus illos prædictis proportionibus affixos ad principia <lb/>Phy&longs;ica reducimus; </s> <s id="N10174"><!-- NEW -->id e&longs;t, cùm &longs;upponamus quòd &longs;int, <lb/>propter quid &longs;int demon&longs;tramus: </s> <s id="N1017A"><!-- NEW -->in votis erat motus <lb/>omnes vno volumine complecti; </s> <s id="N10180"><!-- NEW -->id e&longs;t effectus omnes <lb/>cuiu&longs;uis potentiæ motricis; </s> <s id="N10186"><!-- NEW -->tres enim agno&longs;cimus hu­<lb/>iu&longs;modi potentias: </s> <s id="N1018C"><!-- NEW -->primam naturalem voco, quæ e&longs;t <lb/>grauium: </s> <s id="N10192"><!-- NEW -->alteram animalem, quæ e&longs;t animantium: </s> <s id="N10196"><!-- NEW -->ter­<lb/>tiam mediam, quæ ten&longs;orum e&longs;t vel compre&longs;&longs;orum: </s> <s id="N1019C"><!-- NEW -->In <lb/>hoc tractatu tùm à motu progre&longs;&longs;iuo animantium, tùm <lb/>ab alijs motibus, qui in animato corpore, neruorum & <pb xlink:href="026/01/006.jpg"/>mu&longs;culorum opera fiunt, penitus ab&longs;tinemus; </s> <s id="N101A8"><!-- NEW -->cùm &longs;ci­<lb/>licèt eas notiones &longs;upponant, quæ huius loci e&longs;&longs;e non <lb/>po&longs;&longs;unt, ab&longs;tinemus etiam à mirifica illa ten&longs;orum & <lb/>compre&longs;&longs;orum vi, quæ mediæ illius virtutis e&longs;t; </s> <s id="N101B2"><!-- NEW -->neque <lb/>adhuc eò rem Phy&longs;icam adduximus; Sed hîc tantùm na­<lb/>turam impetus con&longs;ideramus, motus naturalis affectio­<lb/>nes, violenti, mixti ex rectis, reflexi, circularis, mixti <lb/>ex circularibus, illius qui fit in planis inclinatis &longs;ur&longs;um <lb/>& deor&longs;um, vibrationum funependuli, diuer&longs;arum im­<lb/>pre&longs;&longs;ionum, centri percu&longs;&longs;ionis, &c. </s> <s id="N101C2"><!-- NEW -->Fortè aliquis poten­<lb/>tias mechanicas de&longs;ideraret, lineas, motus, & cæle&longs;tes <lb/>&longs;piras; </s> <s id="N101CA"><!-- NEW -->&longs;ed hæ quidquid phy&longs;icum habent, &longs;ingulari tra­<lb/>ctatui de corpore cæle&longs;ti, reliqua verò A&longs;tronomiæ con­<lb/>cedunt: potentiæ mechanicæ ad Staticam pertinent, qua­<lb/>re illarum tantùm phy&longs;icum principium in hoc tractatu <lb/>explicamus, lineæ motus nihil phy&longs;icum habent. </s> <s id="N101D6"><!-- NEW -->Quare <lb/>ad vitandam confu&longs;ionem ad Mathe&longs;im illas remittimus, <lb/>cuius non modicam facient acce&longs;&longs;ionem; igitur &longs;ecun­<lb/>dum Tomum de motu locali non expectabis, qui ne <lb/>cuncta quidem, quæ ad motum &longs;pectant comprehende­<lb/>ret, &longs;ed huic &longs;tatim Metaphy&longs;icam demon&longs;tratiuam &longs;ub­<lb/>necto. </s> <s id="N101E6"><!-- NEW -->Cæterùm de &longs;ubtili&longs;&longs;imo i&longs;torum omnium inuen­<lb/>torum auctore nihil dicam, qui cum ægrè tulerit paucula <lb/>illa quæ in prima tractatu præfatus &longs;um, os mihi peni­<lb/>tus ob&longs;truxit: </s> <s id="N101F0"><!-- NEW -->omitto etiam quæ in me quidam iniquè <lb/>certè rerum æ&longs;timatores iactarunt: </s> <s id="N101F6"><!-- NEW -->reponere po&longs;&longs;em cum <lb/>fænore; </s> <s id="N101FC"><!-- NEW -->&longs;ed nos talem con&longs;uetudinem non habemus; </s> <s id="N10200"><!-- NEW -->de­<lb/>dici hactenus pati iniurias, non inferre; quod non modò <lb/>moralis Philo&longs;ophia, &longs;ed præ&longs;ertim Chri&longs;tiana Religio me <lb/>docet. </s> </p> <pb xlink:href="026/01/007.jpg"/> <p id="N1020D" type="main"> <s id="N1020F"><!-- NEW -->Vnum e&longs;t, de quo te monitum velim (Amice Lector) <lb/>opu&longs;culum i&longs;tud non &longs;ine aliquot erratis edi potui&longs;&longs;e, <lb/>præ&longs;ertim cùm in a&longs;&longs;ignandis cuilibet figuræ &longs;uis chara­<lb/>cteribus &longs;æpiùs peccatum &longs;it; </s> <s id="N10219"><!-- NEW -->operas excu&longs;abis in rebus <lb/>Geometricis minimè ver&longs;atos: auctor tibi &longs;um, vt errata, <lb/>quæ fideliter adnotaui ca&longs;tiges, vt deinde cum maiore <lb/>gu&longs;tu Librum hunc perlegere po&longs;&longs;is. <lb/><gap desc="hr tag"/></s> </p> <p id="N10226" type="main"> <s id="N10228"><emph type="center"/><emph type="italics"/>SYNOPSIS LIBRORVM<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N10233" type="main"> <s id="N10235"><emph type="center"/><emph type="italics"/>huius tractatus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N10240" type="table"> <s id="N10242">TABELLE WAR HIER<!-- KEEP S--></s> </p> </section> <section> <pb xlink:href="026/01/008.jpg"/> <figure id="id.026.01.008.1.jpg" xlink:href="026/01/008/1.jpg"/> <p id="N1024F" type="head"> <s id="N10251"><emph type="center"/><emph type="italics"/>SYNOPSIS AMPLIOR.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1025D" type="main"> <s id="N1025F">BREVISSIMAM huius operis Epitomem hîc <lb/>habes (Amice Lector) quam ex The&longs;ibus no&longs;tri <lb/>Philo&longs;ophi huc traduxi, quæ tibi ampli&longs;&longs;imi <lb/>indicis loco erit. </s> </p> <figure id="id.026.01.008.2.jpg" xlink:href="026/01/008/2.jpg"/> <p id="N1026D" type="main"> <s id="N1026F"><emph type="center"/><emph type="italics"/>De Impetu.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N1027A" type="main"> <s id="N1027C"><!-- NEW -->1. IMPETVS e&longs;t qualitas exigens motum &longs;ui &longs;ubiecti: </s> <s id="N10280"><!-- NEW --><lb/>datur impetus; </s> <s id="N10285"><!-- NEW -->quia non pote&longs;t e&longs;&longs;e alia cau&longs;a exi­<lb/>gitiua motus: </s> <s id="N1028B"><!-- NEW -->adde quòd, potentia motrix e&longs;t acti­<lb/>ua; </s> <s id="N10291"><!-- NEW -->igitur aliquid producit, &longs;ed non aliud quàm <lb/>impetum, vt con&longs;tat ex dictis de motu: </s> <s id="N10297"><!-- NEW -->e&longs;t aliquid di&longs;tinctum à <lb/>&longs;ub&longs;tantia mobilis, quæ pote&longs;t e&longs;&longs;e &longs;ine impetu: </s> <s id="N1029D"><!-- NEW -->non e&longs;t modus, <lb/>quia di&longs;tinguitur ab effectu &longs;uo formali &longs;ecundario: </s> <s id="N102A3"><!-- NEW -->impetus non <lb/>producitur in eo mobili, quod moueri non pote&longs;t à potentia mo­<lb/>trice applicata: </s> <s id="N102AB"><!-- NEW -->& produci tantùm pote&longs;t, vel in omni parte, vel <lb/>in nulla; </s> <s id="N102B1"><!-- NEW -->alioquin e&longs;&longs;et fru&longs;trà; & gratis ponitur ne&longs;cio quis impe­<lb/>tus inefficax. </s> </p> <p id="N102B7" type="main"> <s id="N102B9">2. Primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, ne &longs;imul im­<lb/>petus &longs;it in duobus locis. </s> <s id="N102BE"><!-- NEW -->Impetus productus ad extra non produci­<lb/>tur à quantitate, nec virtute re&longs;i&longs;titiua, nec ab alio, quàm ab impe­<lb/>tu, qui maximè e&longs;t cau&longs;a connaturalis alterius impetus: </s> <s id="N102C6"><!-- NEW -->agit tan­<lb/>tùm ad extra, vt tollat impedimentum: </s> <s id="N102CC"><!-- NEW -->hinc, cùm pro diuer&longs;a <lb/>applicatione &longs;it diuer&longs;um impedimentum, modò plùs, modò minùs <lb/>agit; </s> <s id="N102D4"><!-- NEW -->maximè verò, cum maximum e&longs;t impedimentum: </s> <s id="N102D8"><!-- NEW -->hinc ictus <lb/>per lineam perpendicularem forti&longs;&longs;imus e&longs;t: portò omnes partes <lb/>impetus agunt ad extra actione communi. </s> </p> <p id="N102E0" type="main"> <s id="N102E2"><!-- NEW -->3. Impetus inten&longs;us producere pote&longs;t remi&longs;&longs;um, minoris mobi­<lb/>lis in maiore; </s> <s id="N102E8"><!-- NEW -->& remi&longs;&longs;us inten&longs;um, maioris mobilis in minore, vt <lb/>patet; æqualis æqualem, æqualis mobilis in æquali, modò &longs;it debi-<pb xlink:href="026/01/009.jpg"/>ta applicatio, cum maximo impedimento, quod reuerâ tunc e&longs;t, <lb/>cùm linea directionis connectit centra grauitatis vtriu&longs;que. </s> <s id="N102F4"><!-- NEW -->Datur <lb/>impetus alio impetu perfectior, & imperfectior, &longs;ine quo non po­<lb/>te&longs;t explicari natura vectis: </s> <s id="N102FC"><!-- NEW -->itaque dato quocunque dari pote&longs;t per­<lb/>fectior, & imperfectior: quia dato quocunque motu pote&longs;t dari ve­<lb/>locior, & tardior. </s> </p> <p id="N10304" type="main"> <s id="N10306"><!-- NEW -->4. Propagatur impetus vniformiter tantùm, cùm omnes partes <lb/>corporis mouentur motu recto æquali: </s> <s id="N1030C"><!-- NEW -->ibi enim e&longs;t æqualis cau&longs;a, <lb/>vbi e&longs;t æqualis effectus: </s> <s id="N10312"><!-- NEW -->in motu circulari applicata potentia cen­<lb/>tro vectis, producitur æqualis perfectionis versùs circunferentiam, <lb/>& inæqualis numerus; </s> <s id="N1031A"><!-- NEW -->applicata verò potentia circunferentiæ, pro­<lb/>ducitur æqualis numerus, &longs;ed inæqualis perfectionis versùs cen­<lb/>trum; </s> <s id="N10322"><!-- NEW -->quia potentia non pote&longs;t producere immediatè perfectiorem, <lb/>& imperfectiorem in infinitum: </s> <s id="N10328"><!-- NEW -->eadem potentia nece&longs;&longs;aria æquali­<lb/>bus temporibus, & ii&longs;dem circun&longs;tantiis, producit æqualem impe­<lb/>tum, & inæqualibus inæqualem: e&longs;t enim hæc ratio cau&longs;æ nece&longs;­<lb/>&longs;ariæ. </s> </p> <p id="N10332" type="main"> <s id="N10334"><!-- NEW -->5. Impetus innatus e&longs;t tantùm determinatus ad lineam perpen­<lb/>dicularem deor&longs;um; </s> <s id="N1033A"><!-- NEW -->alioquin &longs;i ad aliam determinari po&longs;&longs;et, primo <lb/>e&longs;&longs;et æqualis motus per inclinatam, & perpendicularem; </s> <s id="N10340"><!-- NEW -->corpus <lb/>graue mi&longs;&longs;um per lineam inclinatam ab eo non declinaret; </s> <s id="N10346"><!-- NEW -->imò im­<lb/>petus &longs;emel productus (&longs;i liberum e&longs;&longs;et medium) non de&longs;trueretur: </s> <s id="N1034C"><!-- NEW --><lb/>quæ omnia phy&longs;icis hypothe&longs;ibus repugnant: omnis alius impetus, <lb/>etiam acqui&longs;itus motu naturali deor&longs;um, e&longs;t indifferens ad omnem <lb/>lineam, ad vitanda infinita ferè naturæ incommoda. </s> </p> <p id="N10355" type="main"> <s id="N10357"><!-- NEW -->6. Impetus indifferens determinatur ad lineam multis modis: <lb/>primò, à potentia motrice: </s> <s id="N1035D"><!-- NEW -->&longs;ecundò, ab impetu: </s> <s id="N10361"><!-- NEW -->tertiò, ab alio impe­<lb/>tu concurrente; quartò, ab obice occurrente: </s> <s id="N10367"><!-- NEW -->quintò, ab ip&longs;o appli­<lb/>cationis diuer&longs;o modo: quæ omnia clara &longs;unt: hinc duo impetus ad <lb/>motum mixtum &longs;æpè concurrunt, quod &longs;emper fit, ni&longs;i determina­<lb/>tiones &longs;int oppo&longs;itæ ex diametro. </s> <s id="N10371"><!-- NEW -->Impetus e&longs;t capax inten&longs;ionis; </s> <s id="N10375"><!-- NEW --><lb/>quia aliquando de&longs;truitur ex parte: </s> <s id="N1037A"><!-- NEW -->eius exten&longs;io commen&longs;uratur <lb/>exten&longs;ioni mobilis; </s> <s id="N10380"><!-- NEW -->quod etiam cæteris qualitatibus commune e&longs;t: <lb/>impetus productus non con&longs;eruatur à cau&longs;a primò productiua, à <lb/>qua etiam &longs;eparatus exi&longs;tit. </s> </p> <p id="N10388" type="main"> <s id="N1038A"><!-- NEW -->7. Impetus non e&longs;t contrarius alteri ratione entitatis; </s> <s id="N1038E"><!-- NEW -->quia qui­<lb/>libet cum quolibet in eodem &longs;ubiecto coëxi&longs;tere pote&longs;t: </s> <s id="N10394"><!-- NEW -->pugnat <lb/>tamen vnus cum alio ratione determinationis: </s> <s id="N1039A"><!-- NEW -->hinc vnus impetus <lb/>pugnat cum alio ratione lineæ motus: </s> <s id="N103A0"><!-- NEW -->hinc vnus videtur de&longs;trui ab <pb xlink:href="026/01/010.jpg"/>alio; </s> <s id="N103A8"><!-- NEW -->quanquam impetus tantùm de&longs;truitur, cùm e&longs;t fru&longs;trà: </s> <s id="N103AC"><!-- NEW -->hinc, &longs;i <lb/>e&longs;&longs;et tantùm vnicus in eodem mobili, & liberum e&longs;&longs;et medium, <lb/>nunquam de&longs;trueretur nec vnquam dici po&longs;&longs;et functus &longs;uo mune­<lb/>re; quod omninò gratis dicitur. </s> </p> <p id="N103B6" type="main"> <s id="N103B8"><!-- NEW -->8. Hinc, &longs;i &longs;int tantùm duo impetus in eodem mobili æquales <lb/>verbi gratia, vel ad eandem lineam determinantur, vel ad diver&longs;as; </s> <s id="N103BE"><!-- NEW --><lb/>&longs;i ad eandem, nihil impetus de&longs;truitur, &longs;ed e&longs;t duplò velocior mo­<lb/>tus; </s> <s id="N103C5"><!-- NEW -->&longs;i ad diuer&longs;as, vel &longs;unt oppo&longs;itæ ex diametro, vel concurrentes <lb/>faciunt angulum; </s> <s id="N103CB"><!-- NEW -->&longs;i primum, vterque de&longs;truitur impetus; &longs;i &longs;e­<lb/>cundum, de&longs;truitur aliquid illius, quod determinabimus in­<lb/>frà. </s> <s id="N103D3"><!-- NEW -->Impetus innatus nunquam de&longs;truitur: </s> <s id="N103D7"><!-- NEW -->dici po&longs;&longs;et grauitas ab­<lb/>&longs;oluta; &longs;altem nihil e&longs;t, quod di&longs;tingui ab illa probare po&longs;&longs;it. </s> <s id="N103DD">Porrò <lb/>nunquam de&longs;truitur; </s> <s id="N103E2"><!-- NEW -->quia nunquam e&longs;t fru&longs;trà; quippe eius finis, <lb/>vel v&longs;us, non e&longs;t tantùm motus deor&longs;um, &longs;ed grauitatio, &longs;eu ni&longs;us <lb/>quidam deor&longs;um. </s> <s id="N103EA">Sed de grauitate aliàs. </s> </p> <figure id="id.026.01.010.1.jpg" xlink:href="026/01/010/1.jpg"/> <p id="N103F2" type="main"> <s id="N103F4"><emph type="center"/><emph type="italics"/>De motu naturali deor&longs;um.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N103FF" type="main"> <s id="N10401">1. DAtur motus naturalis grauium deor&longs;um ab intrin&longs;eco, <lb/>quippe non pote&longs;t e&longs;&longs;e, vel à vi tractrice terræ vel fila­<lb/>mentis quibu&longs;dam, vel materia quadam tenui expultrice. </s> <s id="N10408">Eius finis <lb/>e&longs;t globi terre&longs;tris compactio, &c. </s> <s id="N1040D"><!-- NEW -->E&longs;t autem motus naturalis ab <lb/>impetu: </s> <s id="N10413"><!-- NEW -->primò, quia eius acceleratio &longs;ine impetu explicari non po­<lb/>te&longs;t: </s> <s id="N10419"><!-- NEW -->&longs;ecundò, quia, cùm graue deor&longs;um cadens imprimat impetum <lb/>in corpore occurrente, certè debet habere impetum: nec alio ar­<lb/>gumento mihi probabis, Solem e&longs;&longs;e lucidum, ignem calidum. </s> </p> <p id="N10421" type="main"> <s id="N10423"><!-- NEW -->2. Motus hic e&longs;t naturaliter acceleratus, &longs;cilicet, ab intrin&longs;eco; <lb/>patet experientiâ. </s> <s id="N10429">Ratio e&longs;t: </s> <s id="N1042C"><!-- NEW -->quia, cùm in libero medio non impe­<lb/>diatur motus, & impetus productus primo in&longs;tanti non con&longs;erue­<lb/>tur &longs;ecundo à cau&longs;a primò productiua, &longs;ed ab alia, &longs;itque ip&longs;a mo­<lb/>bilis &longs;ub&longs;tantia cau&longs;a nece&longs;&longs;aria; </s> <s id="N10436"><!-- NEW -->certè &longs;ecundo in&longs;tanti producit <lb/>nouum impetum: idem dica de tertio, quarto, &c. </s> <s id="N1043C"><!-- NEW -->igitur cre&longs;cit <lb/>cau&longs;a motus; </s> <s id="N10442"><!-- NEW -->igitur & motus: quæ ratio clari&longs;&longs;ima e&longs;t: </s> <s id="N10446"><!-- NEW -->hinc æquali­<lb/>bus temporibus æqualia acquiruntur velocitatis momenta; </s> <s id="N1044C"><!-- NEW -->quia <lb/>cau&longs;a nece&longs;&longs;aria æqualibus temporibus, æqualem effectum produ­<lb/>cit: quid clarius? </s> </p> <p id="N10454" type="main"> <s id="N10456"><!-- NEW -->3. Hinc non pote&longs;t cre&longs;cere hic impetus &longs;ecundùm porportio-<pb xlink:href="026/01/011.jpg"/>nem duplicatam temporum, cùm cre&longs;cat &longs;ecundùm proportionem <lb/>temporum, etïam ex mente Galilei: </s> <s id="N10460"><!-- NEW -->cre&longs;cit autem velocitas, vt im­<lb/>petus; </s> <s id="N10466"><!-- NEW -->effectus, &longs;cilicet, vt cau&longs;a: </s> <s id="N1046A"><!-- NEW -->idem dico de motu, ratione velo­<lb/>citatis; </s> <s id="N10470"><!-- NEW -->quippe motus ip&longs;e e&longs;t &longs;ua velocitas: at verò ip&longs;a &longs;patia, <lb/>quæ decurruntur illo motu, &longs;i con&longs;ideretur crementum in in&longs;tan­<lb/>tibus, cre&longs;cunt iuxta progre&longs;&longs;ionem arithmeticam &longs;implicem, <lb/>id e&longs;t, &longs;i primo in&longs;tanti, acquiritur vnum &longs;patium, &longs;ecundo acquiri­<lb/>tur vnum &longs;patium, &longs;ecundo acquiruntur duo, tertio 3. quarto 4. at­<lb/>que ita deinceps. </s> </p> <p id="N1047E" type="main"> <s id="N10480"><!-- NEW -->4. Hoc autem facilè pote&longs;t <expan abbr="demõ&longs;trari">demon&longs;trari</expan>: </s> <s id="N10488"><!-- NEW -->quia, cùm velocitas cre&longs;­<lb/>cat iuxta proportionem temporum, &longs;i primo in&longs;tanti &longs;it vnus gradus <lb/>velocitatis, &longs;ecundo erunt duo, tertio tres, at que ita deinceps: </s> <s id="N10490"><!-- NEW -->igitur, <lb/>&longs;i mobile cum vno gradu velocitatis acquirit vnum &longs;patium, certè <lb/>cum duobus acquiret duo &longs;patia, cum tribus tria, atque ita dein­<lb/>ceps: debet autem vera progre&longs;&longs;io crementorum a&longs;&longs;umi in &longs;ingulis <lb/>in&longs;tantibus, quia reuerà &longs;ingulis in&longs;tantibus phy&longs;icis (nam de iis <lb/>loquor) noua fit huius crementi acce&longs;&longs;io. </s> </p> <p id="N1049E" type="main"> <s id="N104A0"><!-- NEW -->5. Quia tamen in&longs;tantia non &longs;unt &longs;en&longs;ibilia, vt Phy&longs;icæ con&longs;u­<lb/>latur, quæ res &longs;en&longs;ibiles con&longs;iderat, a&longs;&longs;umi debent partes temporis <lb/>&longs;en&longs;ibiles, in quibus reuerâ progre&longs;&longs;io &longs;patiorum non e&longs;t arithmeti­<lb/>ca &longs;implex; &longs;ed tam propè accedit ad hanc numerorum imparium, <lb/>1. 3. 5. 7. &c. </s> <s id="N104AC"><!-- NEW -->quam Galileus excogitauit, vt &longs;ine &longs;crupulo hæc a&longs;­<lb/>&longs;umi po&longs;&longs;it: </s> <s id="N104B2"><!-- NEW -->hinc &longs;patia &longs;unt ferè vt temporum quadrata: dixi, ferè: </s> <s id="N104B6"><!-- NEW --><lb/>nam e&longs;t paulò minor proportio, cùm tantùm finita &longs;int in&longs;tantia <lb/>phy&longs;ica, quæ reuerà &longs;i infinita e&longs;&longs;ent in qualibet temporis &longs;en&longs;ibilis <lb/>parte, haud dubiè &longs;patia e&longs;&longs;ent omninò in ratione duplicata tem­<lb/>porum: &longs;ed, quia parum pro nihilo computatur, hanc progre&longs;&longs;io­<lb/>nem Galilei deinceps v&longs;urpabimus in Phy&longs;ica. <!-- KEEP S--></s> </p> <p id="N104C4" type="main"> <s id="N104C6">6. Hinc ratio euidens maioris ictus inflicti à corpore graui, <lb/>cùm ex maiori altitudine cadit. </s> <s id="N104CB"><!-- NEW -->Sunt autem ictus, vt impetus; <lb/>impetus, vt tempora; hæc demum, vt radices &longs;patiorum &longs;en&longs;ibi­<lb/>liter quæ omnia con&longs;tant ex dictis. </s> <s id="N104D3"><!-- NEW -->Impetus acqui&longs;itus in de&longs;cen&longs;u <lb/>e&longs;t &longs;emper imperfectior, &longs;i a&longs;&longs;umantur &longs;ingula in&longs;tantia, quæ reuerâ <lb/>&longs;unt &longs;emper minora; </s> <s id="N104DB"><!-- NEW -->quia motus fit &longs;emper velocior: cùm graue <lb/>de&longs;cendit in medio, quod re&longs;i&longs;tit, minùs accuratè &longs;eruantur prædi­<lb/>ctæ proportiones, quæ in vacuo modico accurati&longs;&longs;imè &longs;eruaren­<lb/>tur. </s> </p> <p id="N104E5" type="main"> <s id="N104E7"><!-- NEW -->7. Re&longs;i&longs;tentia medij non e&longs;t propter vllam formam improportio­<lb/>natam, qua&longs;i verò impetus &longs;it forma improportionata aëri: </s> <s id="N104ED"><!-- NEW -->&longs;ed in <pb xlink:href="026/01/012.jpg"/>duobus præ&longs;ertim con&longs;i&longs;tit; </s> <s id="N104F5"><!-- NEW -->primò, eò quòd medium detrahat ali­<lb/>quid grauitationis corporis grauis; </s> <s id="N104FB"><!-- NEW -->&longs;ecundò, eò quòd partes medij <lb/>aliquam implicationem habeant, quæ &longs;olui non pote&longs;t &longs;ine aliqua <lb/>compre&longs;&longs;ione, vel ten&longs;ione; </s> <s id="N10503"><!-- NEW -->vtraque autem re&longs;i&longs;tit impetui: quod <lb/>&longs;pectat ad primum, &longs;i medium &longs;it æqualis grauitatis cum ip&longs;o cor­<lb/>pore, detrahitur tota grauitatio, &longs;i &longs;ubduplæ &longs;ubduplum, &c. </s> <s id="N1050B">de quo <lb/>aliàs. </s> </p> <p id="N10510" type="main"> <s id="N10512"><!-- NEW -->8. Hinc corpus graue per medium rarius, cæteris paribus, fa­<lb/>cilè de&longs;cendit; non tamen ex re&longs;i&longs;tentia medij cognita, pote&longs;t co­<lb/>gno&longs;ci proportio grauitatis vtriu&longs;que, propter &longs;ecundum caput, ex <lb/>quo etiam petitur re&longs;i&longs;tentia. </s> <s id="N1051C"><!-- NEW -->Idem corpus cum eodem medio <lb/>comparatum, habet tres coniugationes: nam, vel e&longs;t grauius, vel­<lb/>e&longs;t grauius, vel æquè graue, vel minùs. </s> <s id="N10524">Sunt etiam tres aliæ con­<lb/>iugationes, &longs;cilicet, eiu&longs;dem mobilis cum diuer&longs;is mediis, duorum <lb/>mobilium cum eodem medio, duorum mobilium cum duobus <lb/>mediis. </s> </p> <p id="N1052D" type="main"> <s id="N1052F"><!-- NEW -->9. Figura corporis grauis deor&longs;um cadentis motum vel retardat <lb/>vel accelerat; </s> <s id="N10535"><!-- NEW -->retardat quidem, &longs;i plures partes medij amouendæ <lb/>&longs;unt vel pauciores velociori motu; accelerat è contrario: </s> <s id="N1053B"><!-- NEW -->hinc idem <lb/>corpus <expan abbr="parallelipedũ">parallelipedum</expan> iuxta tres diuer&longs;os &longs;itus, triplici motu diuer­<lb/>&longs;o de&longs;cendere pote&longs;t: hinc ratio, cur acuminata tam facilè de&longs;cen­<lb/>dant. </s> <s id="N10549"><!-- NEW -->Cubus, qui de&longs;cendit, imprimit aëri velociorem motum, <lb/>quàm ip&longs;e habeat; & quò maior e&longs;t eius &longs;uperficies, eò velociorem. </s> </p> <p id="N1054F" type="main"> <s id="N10551"><!-- NEW -->10. Duo globi, vel cubi eiu&longs;dem materiæ æquè velociter de&longs;­<lb/>cendunt: </s> <s id="N10557"><!-- NEW -->ratio e&longs;t, quia, licèt maioris vires habeant maiorem pro­<lb/>portionem ad molem aëris re&longs;i&longs;tentis, quàm vires minoris ad alte­<lb/>ram aëris molem, quæ proprium illius motum retardat, cùm tamen <lb/>aër, qui re&longs;i&longs;tit maiori cubo, debeat amoueri velociori motu, quàm <lb/>aër, qui re&longs;i&longs;tit minori, &longs;itque eadem proportio re&longs;i&longs;tentiæ ratione <lb/>motus, minoris ad maiorem, quæ e&longs;t ratione molis, maioris ad mi­<lb/>norem; </s> <s id="N10567"><!-- NEW -->certè ratio compo&longs;ita vtriu&longs;que erit eadem in vtroque cu­<lb/>bo: igitur æqualiter de&longs;cendet vterque. </s> </p> <p id="N1056D" type="main"> <s id="N1056F"><!-- NEW -->11. Si tamen &longs;int diuer&longs;æ materiæ, haud dubiè, qui con&longs;tat leuio­<lb/>ri materia, tardiùs de&longs;cendet; quia eius vires habent minorem <lb/>proportionem ad re&longs;i&longs;tentiam. </s> <s id="N10577"><!-- NEW -->Corpu&longs;cula etiam ex graui&longs;&longs;ima ma­<lb/>teria tardi&longs;&longs;imè de&longs;cendunt: </s> <s id="N1057D"><!-- NEW -->tum, quia à filamentis illis, quibus par­<lb/>tes aëris implicantur, facilè detinentur; </s> <s id="N10583"><!-- NEW -->analogiam habes in lapil­<lb/>lo, qui ab araneæ tela intercipitur: </s> <s id="N10589"><!-- NEW -->tum, quia, cùm lati&longs;&longs;imam ali­<lb/>quando habeant &longs;uperficiem pro modica mole, minimam habent <pb xlink:href="026/01/013.jpg"/><expan abbr="proportion&etilde;">proportionem</expan> virium ad <expan abbr="re&longs;i&longs;tentiã">re&longs;i&longs;tentiam</expan>: tùm denique, quia, cùm modico <lb/>impetu agitari po&longs;&longs;int ab aëre mobili, vnus motus alium impedit. </s> </p> <p id="N1059C" type="main"> <s id="N1059E"><!-- NEW -->12. Singulis in&longs;tantibus motus naturaliter accelerati cre&longs;cit <lb/>re&longs;i&longs;tentia; </s> <s id="N105A4"><!-- NEW -->quia, cùm motus cre&longs;cat, æqualibus temporibus, plures <lb/>partes medij occurrunt; </s> <s id="N105AA"><!-- NEW -->cre&longs;cunt tamen vires in eadem proportio­<lb/>ne, &longs;cilicet, impetus: igitur non mutatur progre&longs;&longs;io motus. </s> <s id="N105B0"><!-- NEW -->Hinc <lb/>colligo, contra Galilæum, motum rectum ex naturaliter accelerato <lb/>nunquam fieri æquabilem: dixi motum rectum; quia motus corpo­<lb/>rum cœle&longs;tium ex accelerato factus e&longs;t æqualis. </s> </p> <figure id="id.026.01.013.1.jpg" xlink:href="026/01/013/1.jpg"/> <p id="N105BF" type="main"> <s id="N105C1"><emph type="center"/><emph type="italics"/>De motu violento &longs;ur&longs;um.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N105CC" type="main"> <s id="N105CE">1. MOtus violentus &longs;ur&longs;um vulgò dicitur e&longs;&longs;e à principio ex­<lb/>trin&longs;eco. </s> <s id="N105D3"><!-- NEW -->Triplici modo accidere pote&longs;t: </s> <s id="N105D7"><!-- NEW -->primò, &longs;i reuerà <lb/>imprimatur impetus ab extrin&longs;eco, vt, cùm mitto lapidem &longs;ur&longs;um: </s> <s id="N105DD"><!-- NEW --><lb/>&longs;ecundò, &longs;i corpus deor&longs;um cadens deinde reflectatur &longs;ur&longs;um; </s> <s id="N105E2"><!-- NEW -->tunc <lb/>autem nihil e&longs;t ab extrin&longs;eco, ni&longs;i determinatio noua, quæ e&longs;t à cor­<lb/>pore reflectente: </s> <s id="N105EA"><!-- NEW -->tertiò, &longs;i terra vtrinque e&longs;&longs;et peruia; </s> <s id="N105EE"><!-- NEW -->nam lapis haud <lb/>dubiè non &longs;i&longs;teret in centro, &longs;altem po&longs;t primum de&longs;cen&longs;um; </s> <s id="N105F4"><!-- NEW -->igitur <lb/>a&longs;cenderet per eandem lineam; </s> <s id="N105FA"><!-- NEW -->nullum tamen e&longs;t principium ex­<lb/>trin&longs;ecum; igitur motus violentus dicit tantùm motum &longs;ur&longs;um <lb/>corporis grauis. </s> </p> <p id="N10602" type="main"> <s id="N10604"><!-- NEW -->2. Dari autem motum violentum, dubium e&longs;&longs;e non pote&longs;t, qui <lb/>&longs;upponit impetum, vel impre&longs;&longs;um ab extrin&longs;eco, vel in de&longs;cen&longs;u <lb/>acqui&longs;itum, qui reuerâ ine&longs;t ip&longs;i mobili, cùm ip&longs;um medium hunc <lb/>motum potiùs impediat, quàm iuuet: </s> <s id="N1060E"><!-- NEW -->hinc, &longs;i nullus e&longs;&longs;et impetus <lb/>extrin&longs;ecus, vel acqui&longs;itus, nullus e&longs;&longs;et motus violentus; quia im­<lb/>petus innatus illius cau&longs;a e&longs;&longs;e non pote&longs;t. </s> <s id="N10616">Portò hic motus non e&longs;t <lb/>acceleratus, nec æqualis, alioquin <expan abbr="nunquã">nunquam</expan> rediret deor&longs;um mobile. </s> </p> <p id="N1061F" type="main"> <s id="N10621"><!-- NEW -->3. Hinc nece&longs;&longs;ariò e&longs;t retardatus: </s> <s id="N10625"><!-- NEW -->igitur de&longs;truitur impetus, non <lb/>quidem ab ip&longs;a medij re&longs;i&longs;tentia; </s> <s id="N1062B"><!-- NEW -->quippe idem medium non magis <lb/>re&longs;i&longs;tit motui &longs;ur&longs;um, quàm motui deor&longs;um, vt patet: </s> <s id="N10631"><!-- NEW -->igitur de&longs;trui­<lb/>tur ille impetus motus violenti ab impetu innato aliquo modo; </s> <s id="N10637"><!-- NEW -->non <lb/>quidem vt à contrario ratione entitatis, &longs;ed ratione determinatio­<lb/>nis: </s> <s id="N1063F"><!-- NEW -->cùm enim impetus innatus exigat motum deor&longs;um, & alius &longs;ur­<lb/>&longs;um: </s> <s id="N10645"><!-- NEW -->hic quidem præualet, attamen fru&longs;trà e&longs;t, ratione gradus <lb/>æqualis impetui innato: igitur de&longs;truitur ille gradus illo in&longs;tanti. </s> </p> <pb xlink:href="026/01/014.jpg"/> <p id="N1064E" type="main"> <s id="N10650"><!-- NEW -->4. Hinc &longs;ingulis temporibus æqualibus de&longs;truitur gradus impe­<lb/>tui innato; </s> <s id="N10656"><!-- NEW -->e&longs;t enim eadem ratio pro omnibus: </s> <s id="N1065A"><!-- NEW -->igitur temporibus <lb/>æqualibus de&longs;truitur æqualis impetus: </s> <s id="N10660"><!-- NEW -->igitur amittit ille motus <lb/>æqualia velocitatis momenta: </s> <s id="N10666"><!-- NEW -->igitur e&longs;t naturaliter retardatus: </s> <s id="N1066A"><!-- NEW -->igi­<lb/>tur iuxta eam proportionem decre&longs;cit motus violentus, iuxtaquam <lb/>cre&longs;cit naturalis: igitur dici debent de hac progre&longs;&longs;ione retardatio­<lb/>nis, quæ dicta &longs;unt de illa progre&longs;&longs;ione accelerationis. </s> </p> <p id="N10674" type="main"> <s id="N10676"><!-- NEW -->5. Hinc impetus imperfectior initio de&longs;truitur: </s> <s id="N1067A"><!-- NEW -->quia, cùm motus <lb/>ille &longs;it velocior initio, in&longs;tantia &longs;unt minora: </s> <s id="N10680"><!-- NEW -->atqui minori tempore <lb/>minùs retardatur: </s> <s id="N10686"><!-- NEW -->igitur inperfectior impetus de&longs;truitur; </s> <s id="N1068A"><!-- NEW -->cùm è <lb/>contrario in motu acceleratio initio acquiratur imperfectior, quia <lb/>in&longs;tantia &longs;unt maiora: vnde vides, gradus impetus e&longs;&longs;e heteroge­<lb/>neos, & principium illud etiam in impetu valere, &longs;cilicet, &longs;ubiectum <lb/>ita compleri ab vna forma, vt alterius homogeneæ non &longs;it ampliùs <lb/>capax, &longs;altem naturaliter. </s> </p> <p id="N10698" type="main"> <s id="N1069A">6. Hinc vltimus gradus impetus violenti e&longs;t omnium perfecti&longs;­<lb/>&longs;imus, vt con&longs;tat. </s> <s id="N1069F"><!-- NEW -->Quie&longs;ceret vno in&longs;tanti mobile iactum &longs;ur&longs;um, &longs;i <lb/>gradus vltimus violenti e&longs;&longs;et æqualis perfectionis, cum impetu in­<lb/>nato: </s> <s id="N106A7"><!-- NEW -->vbi enim ventum e&longs;&longs;et ad in&longs;tans æqualitatis, neutrum præ­<lb/>ualere po&longs;&longs;et: </s> <s id="N106AD"><!-- NEW -->igitur in&longs;tanti &longs;equenti e&longs;&longs;et quies: </s> <s id="N106B1"><!-- NEW -->cùm tamen &longs;int <lb/>diuer&longs;æ perfectionis, perfectior præualet: vter autem &longs;it perfectior, <lb/>dicemus infrà. </s> </p> <p id="N106B9" type="main"> <s id="N106BB"><!-- NEW -->7. Cum mobile &longs;ur&longs;um reflectitur, vel terra perforata &longs;uam lineam <lb/>motus &longs;ur&longs;um versus oppo&longs;itam cœli plagam promouet, vel aliud <lb/>æqualis ponderis, vel maioris, &longs;ur&longs;um mouet, tunc certum e&longs;t, inna­<lb/>tum e&longs;&longs;e perfectiorem: </s> <s id="N106C5"><!-- NEW -->&longs;i verò imprimitur ab alia potentia motrice, <lb/>tunc etiam imperfectior e&longs;t impetu innato; </s> <s id="N106CB"><!-- NEW -->nam inæqualis e&longs;t; </s> <s id="N106CF"><!-- NEW -->alio­<lb/>quin, &longs;i e&longs;&longs;et æqualis, &longs;imul e&longs;&longs;ent in eodem &longs;ubiecto duo gradus <lb/>homogenei: </s> <s id="N106D7"><!-- NEW -->præ&longs;tat autem e&longs;&longs;e imperfectiorem, quàm perfectio­<lb/>rem, vt plura impetus puncta à potentia imprimantur; </s> <s id="N106DD"><!-- NEW -->quòd mul­<lb/>tum facit ad mouenda maiora pondera: hinc nullo in&longs;tanti quie&longs;­<lb/>cunt proiecta &longs;ur&longs;um. </s> </p> <p id="N106E5" type="main"> <s id="N106E7"><!-- NEW -->8. Tandiu durat &longs;en&longs;ibiliter de&longs;cen&longs;us globi proiecti &longs;ur&longs;um, <lb/>quandiu durauit a&longs;cen&longs;us; </s> <s id="N106ED"><!-- NEW -->e&longs;t enim eadem ratio: &longs;agittæ verò mi­<lb/>nùs durat a&longs;cen&longs;us, quàm de&longs;cen&longs;us propter mixtionem materiæ. </s> <s id="N106F3"><!-- NEW --><lb/>Si motus violentus e&longs;&longs;et æquabilis, percurreret proiectum &longs;patium <lb/>ferè duplum eo tempore, quo retardato percurrit &longs;ubduplum: </s> <s id="N106FA"><!-- NEW -->hinc <lb/>&longs;onus tam citò auditur; </s> <s id="N10700"><!-- NEW -->quia propagatur cum particulis aëris æqua­<lb/>bili ferè motu: </s> <s id="N10706"><!-- NEW -->e&longs;&longs;e autem &longs;patium ferè duplum, probatur ex eo, <pb xlink:href="026/01/015.jpg"/>quòd &longs;patium motu æquabili decur&longs;um re&longs;pondet rectangulo; </s> <s id="N1070E"><!-- NEW -->de­<lb/>cur&longs;um verò motu retardato, re&longs;pondet triangulo, &longs;ubduplo rectan­<lb/>guli: a&longs;&longs;umpto &longs;cilicet, æquali tempore. </s> </p> <p id="N10716" type="main"> <s id="N10718"><!-- NEW -->9. Vites potentiæ proiicientis toto ni&longs;u re&longs;pondent velocitati <lb/>acqui&longs;itæ in toto de&longs;cen&longs;u corporis proiecti; <expan abbr="tantũdem">tantundem</expan> enim <lb/>impetus in de&longs;cen&longs;u acquiritur, quantùm in a&longs;cen&longs;u deperditur. </s> <s id="N10724"><!-- NEW --><lb/>Impetus primo in&longs;tanti, quo e&longs;t, agit, &longs;i e&longs;t aliquod impedimen­<lb/>tum; </s> <s id="N1072B"><!-- NEW -->e&longs;t enim cau&longs;a nece&longs;&longs;aria: </s> <s id="N1072F"><!-- NEW -->primo in&longs;tanti motus aliquid im­<lb/>petus de&longs;truitur: </s> <s id="N10735"><!-- NEW -->&longs;iue præce&longs;&longs;erit motus violentus, &longs;iue non præce&longs;­<lb/>&longs;erit, corpus graue æquali motu deor&longs;um cadit: </s> <s id="N1073B"><!-- NEW -->re&longs;i&longs;tentia aëris e&longs;t <lb/>quidem maior initio; &longs;ed etiam &longs;unt maiores vires. </s> </p> <figure id="id.026.01.015.1.jpg" xlink:href="026/01/015/1.jpg"/> <p id="N10746" type="main"> <s id="N10748"><emph type="center"/><emph type="italics"/>De motu in planis inclinatis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N10753" type="main"> <s id="N10755"><!-- NEW -->1. PLanum inclinatum e&longs;t &longs;ur&longs;um, vel deor&longs;um: </s> <s id="N10759"><!-- NEW -->in hoc de&longs;cen­<lb/>dit corpus graue, ni&longs;i fortè retineatur ab a&longs;peritate, vel pro­<lb/>pria, vel ip&longs;ius plani: </s> <s id="N10761"><!-- NEW -->impeditur autem motus naturalis in plano <lb/>prædicto, quia impeditur eius linea: </s> <s id="N10767"><!-- NEW -->ideò e&longs;t tardior hic motus in <lb/>plano inclinato, quàm in perpendiculari: </s> <s id="N1076D"><!-- NEW -->in ea porrò proportione <lb/>e&longs;t tardior, in qua perpendiculum e&longs;t minus linea inclinata, eiu&longs;dem <lb/>&longs;cilicet, altitudinis; </s> <s id="N10775"><!-- NEW -->quippe eò tardior e&longs;t, quò magis impeditur, & <lb/>magis impeditur, quò maius &longs;patium decurrendum e&longs;t, ad acqui­<lb/>rendam eandem altitudinem: igitur eadem e&longs;t proportio impe­<lb/>dimenti, quæ &longs;patij, &c. </s> </p> <p id="N1077F" type="main"> <s id="N10781"><!-- NEW -->2. Hinc motus &longs;unt vt lineæ permutando: </s> <s id="N10785"><!-- NEW -->hinc mobile de&longs;cendit <lb/>per &longs;e in prædicto plano: </s> <s id="N1078B"><!-- NEW -->licet enim motus impediatur, non tamen <lb/><expan abbr="tous">totus</expan>, impetus, qui acquiritur in eodem plano e&longs;t imperfectior ac­<lb/>qui&longs;ito in perpendiculari in eadem proportione; </s> <s id="N10796"><!-- NEW -->nam impetus &longs;unt <lb/>vt motus: </s> <s id="N1079C"><!-- NEW -->hinc pote&longs;t perfectio impetus imminui in infinitum, cùm <lb/>po&longs;&longs;it e&longs;&longs;e in infinitum linea magis, ac magis inclinata: igitur mo­<lb/>tum imminui po&longs;&longs;e in infinitum, non tantùm ex vecte, &longs;ed etiam <lb/>ex planis inclinatis haberi pote&longs;t. </s> </p> <p id="N107A6" type="main"> <s id="N107A8"><!-- NEW -->3. Hinc producit impetum imperfectiorem impetus acqui&longs;itus <lb/>in hoc eodem plano, quàm acqui&longs;itus in perpendiculari, æqualibus <lb/>&longs;cilicet temporibus, quia cau&longs;a imperfectior imperfectiorem pro­<lb/>ducit effectum: </s> <s id="N107B2"><!-- NEW -->motus in plano inclinato deor&longs;um e&longs;t acceleratus <lb/>iuxta eandem proportionem, iuxta quam acceleratur in perpendi-<pb xlink:href="026/01/016.jpg"/>culo: </s> <s id="N107BC"><!-- NEW -->tempora, quibus percurruntur perpendiculum, & linea plani <lb/>inclinati, &longs;unt vt lineæ; &longs;patia autem, quæ in prædictis lineis acqui­<lb/>runtur æqualibus temporibus, &longs;unt vt motus, id e&longs;t, vt lineæ per­<lb/>mutando, vt patet ex dictis. </s> </p> <p id="N107C6" type="main"> <s id="N107C8">4. Ex his concludo, nece&longs;&longs;ariò per plana omnia eiu&longs;dem altitu­<lb/>dinis acquiri eandem velocitatem, quantumuis a&longs;&longs;umantur longi&longs;­<lb/>&longs;ima, modò &longs;cilicet perpendicula &longs;int &longs;emper parallela. </s> <s id="N107CF">Hinc habes <lb/>apud Galileum, per omnes chordas circuli erecti de&longs;cen&longs;um fieri <lb/>æqualibus temporibus. </s> <s id="N107D6"><!-- NEW -->Vires, quæ &longs;u&longs;tinent pondus in plano in­<lb/>clinato per lineam plano <expan abbr="parallelã">parallelam</expan>, &longs;unt ad eas, quæ &longs;u&longs;tinent in per­<lb/>pendiculo, vt lineæ permutando; quia debent adæquare impetum, <lb/>qui producitur, tùm in plano inclinato, tùm in perpendiculo. </s> </p> <p id="N107E4" type="main"> <s id="N107E6"><!-- NEW -->5. Porrò minùs grauitat in ip&longs;um planum inclinatum corpus gra­<lb/>ue, quàm in planum horizontale: </s> <s id="N107EC"><!-- NEW -->e&longs;t autem grauitatio in horizonta­<lb/>li, &longs;eu Tangente, ad grauitationem in inclinata, &longs;eu &longs;ecante, vt ip&longs;æ <lb/>lineæ permutando: quod facilè demon&longs;tramus. </s> <s id="N107F4"><!-- NEW -->Proiicitur mobile <lb/>faciliùs per inclinatum planum &longs;ur&longs;um, quàm per ip&longs;am perpendi­<lb/>cularem: patet experientia: cuius ratio e&longs;t, quia minùs re&longs;i&longs;tit im­<lb/>petus innatus, cuius minor e&longs;t ni&longs;us per inclinatam, vt con&longs;tat ex <lb/>dictis. </s> </p> <p id="N10800" type="main"> <s id="N10802"><!-- NEW -->6. Illæ vires, quæ &longs;ufficiunt ad eum motum &longs;ur&longs;um in perpendi­<lb/>culo, &longs;ufficiunt ad motum &longs;ur&longs;um in plano inclinato eiu&longs;dem alti­<lb/>tudinis: </s> <s id="N1080A"><!-- NEW -->quia illæ vires &longs;ufficiunt ad a&longs;cen&longs;um, quæ acquiruntur in <lb/>toto de&longs;cen&longs;u: &longs;ed in de&longs;cen&longs;u inclinatæ, & perpendiculi acquirun­<lb/>tur vires æquales, id e&longs;t, velocitas æqualis, vt dictum e&longs;t &longs;uprà. </s> <s id="N10812"><!-- NEW -->Om­<lb/>nia puncta plani inclinati rectilinei, imò & horizontalis, &longs;unt di­<lb/>uer&longs;æ inclinationis: in iis tamen planis inclinatis quæ vulgò a&longs;&longs;u­<lb/>muntur, non mutatur &longs;en&longs;ibiliter inclinatio. </s> </p> <p id="N1081C" type="main"> <s id="N1081E"><!-- NEW -->7. Hinc minùs de&longs;truitur impetus in plano inclinato &longs;ur&longs;um, <lb/>quàm in perpendiculo; </s> <s id="N10824"><!-- NEW -->quia diutiùs durat: </s> <s id="N10828"><!-- NEW -->cùm enim minùs ac­<lb/>quiratur in de&longs;cen&longs;u, vt dictum e&longs;t, minùs etiam de&longs;truitur in a&longs;­<lb/>cen&longs;u: </s> <s id="N10830"><!-- NEW -->hinc accedit propriùs hic motus ad æquabilem: </s> <s id="N10834"><!-- NEW -->in eodem <lb/>plano rectilineo pote&longs;t e&longs;&longs;e a&longs;cen&longs;us, & de&longs;cen&longs;us, versùs eandem <lb/>partem: </s> <s id="N1083C"><!-- NEW -->tale e&longs;&longs;et planum horizontale, in cuius vnico tantùm pun­<lb/>cto nulla e&longs;t inclinatio: in quolibet puncto huius plani e&longs;t &longs;ingu­<lb/>laris inclinatio, vt patet, quæ e&longs;t ad perpendiculum, vt Tangens ad <lb/>&longs;ecantem é&longs;tque eadem proportio motuum. </s> </p> <p id="N10846" type="main"> <s id="N10848"><!-- NEW -->8. Corpus graue in &longs;uperficie quadrantis caua, deor&longs;um cadit <lb/>motu naturaliter accelerato; </s> <s id="N1084E"><!-- NEW -->quia &longs;ingulis in&longs;tantibus accedit nouus <pb xlink:href="026/01/017.jpg"/>impetus; </s> <s id="N10856"><!-- NEW -->non tamen æqualibus temporibus, acquiruntur æqualia <lb/>velocitatis momenta; </s> <s id="N1085C"><!-- NEW -->quia in &longs;ingulis punctis quadrantis, e&longs;t diuer­<lb/>&longs;a tangens; </s> <s id="N10862"><!-- NEW -->igitur mutatur progre&longs;&longs;io accelerationis, quæ certè ma­<lb/>jor e&longs;t initio, & &longs;ub finem minor; quia initio tangentes acce­<lb/>dunt propriùs ad perpendiculum, & &longs;ub finem ad horizonta<lb/>lem. </s> </p> <p id="N1086C" type="main"> <s id="N1086E"><!-- NEW -->9. De&longs;cendit etiam in &longs;uperficie conuexa globi erecti motu ac­<lb/>celerato; </s> <s id="N10874"><!-- NEW -->initio quidem, in minore proportione; </s> <s id="N10878"><!-- NEW -->&longs;ub finem, in maio­<lb/>re; </s> <s id="N1087E"><!-- NEW -->vnde e&longs;t inuer&longs;a prioris: </s> <s id="N10882"><!-- NEW -->pote&longs;t etiam de&longs;cendere corpus graue <lb/>v&longs;que ad centrum terræ motu accelerato, in &longs;uperficie conuexa &longs;e­<lb/>micirculi: </s> <s id="N1088A"><!-- NEW -->&longs;i &longs;uperficies terræ e&longs;&longs;et læuigati&longs;&longs;ima, corpus proje­<lb/>ctum moueretur in ea motu æquabili, nec de&longs;trueretur impetus im­<lb/>pre&longs;&longs;us, vt con&longs;tat; </s> <s id="N10892"><!-- NEW -->pote&longs;t quoque de&longs;cendere per &longs;piralem: &longs;unt in­<lb/>finita plana curua, in quibus faciliùs moueri pote&longs;t, quam in ho­<lb/>rizontali recta. </s> </p> <figure id="id.026.01.017.1.jpg" xlink:href="026/01/017/1.jpg"/> <p id="N1089F" type="main"> <s id="N108A1"><emph type="center"/><emph type="italics"/>De motu mixto ex rectis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N108AC" type="main"> <s id="N108AE">1. DAri motum mixtum ille non dubitat, qui di&longs;cum proiicit. </s> <s id="N108B1"><!-- NEW --><lb/>Mixtus ex duobus rectis æquabilibus e&longs;t rectus, e&longs;t que <lb/>diagonalis vtriu&longs;que: </s> <s id="N108B8"><!-- NEW -->hinc de&longs;truitur aliquid impetus, iuxta pro­<lb/>portionem differentiæ diagonalis, & vtriu&longs;que lateris &longs;imul &longs;ump­<lb/>ti; </s> <s id="N108C0"><!-- NEW -->quia, &longs;cilicet, e&longs;t fru&longs;trà: </s> <s id="N108C4"><!-- NEW -->quò maior e&longs;t angulus, quem faciunt li­<lb/>neæ determinationum, minor e&longs;t diagonalis; igitur plùs impetus <lb/>de&longs;truitur, donec tandem concurrant in oppo&longs;itas lineas, tunc enim <lb/>totius impetus de&longs;truitur. </s> </p> <p id="N108CE" type="main"> <s id="N108D0"><!-- NEW -->2. <expan abbr="Quũ">Quum</expan> minor e&longs;t, vel acutior prædictus angulus, minùs impetus <lb/>de&longs;truitur; </s> <s id="N108DA"><!-- NEW -->quia diagonalis maior e&longs;t; </s> <s id="N108DE"><!-- NEW -->donec tandem conueniant in <lb/>eandem lineam, tunc enim nihil de&longs;truitur: </s> <s id="N108E4"><!-- NEW -->datur de facto hic mo­<lb/>tus in rerum natura; </s> <s id="N108EA"><!-- NEW -->talis e&longs;t motus nauis à duobus ventis impre&longs;­<lb/>&longs;us; vel eiu&longs;dem partis aëris; imò & ip&longs;ius venti: </s> <s id="N108F0"><!-- NEW -->motus mixtus ex <lb/>duobus retardatis iuxta eandem progre&longs;&longs;ionem e&longs;t rectus; </s> <s id="N108F6"><!-- NEW -->quia fit <lb/>per hypothenu&longs;im triangulorum proportionalium: idem dico de <lb/>duobus acceleratis. </s> </p> <p id="N108FE" type="main"> <s id="N10900"><!-- NEW -->3. Si mixtus &longs;it ex æquali, & accelerato, vel ex duobus accelera­<lb/>tis in diuer&longs;a progre&longs;&longs;ione, vel ex duobus retardatis &longs;imiliter, fit per <lb/>lineam curuam, vt patet: </s> <s id="N10908"><!-- NEW -->dum proiicitur corpus graue per horizon-<pb xlink:href="026/01/018.jpg"/>talem in medio libero e&longs;t motus mixtus ex accelerato naturali, & <lb/>retardato violento: e&longs;t enim acceleratus naturalis, cùm deor&longs;um <lb/>deor&longs;um tendat qua&longs;i per gradus, &longs;eu diuer&longs;a plana inclinata. </s> </p> <p id="N10914" type="main"> <s id="N10916"><!-- NEW -->4. Non tamen impetus acqui&longs;itus in eo motu e&longs;t eiu&longs;dem perfe­<lb/>ctionis cum illo, qui acquireretur in perpendiculari eiu&longs;dem longi­<lb/>tudinis; </s> <s id="N1091E"><!-- NEW -->&longs;ed tantùm eiu&longs;dem altitudinis: </s> <s id="N10922"><!-- NEW -->nam perinde cre&longs;cit ille <lb/>impetus, atque cre&longs;ceret in diuer&longs;is planis inclinaris: </s> <s id="N10928"><!-- NEW -->impetus verò <lb/>violentus in hoc motu retardatur; </s> <s id="N1092E"><!-- NEW -->tùm, quia, &longs;i maneret idem, maior <lb/>e&longs;&longs;et ictus &longs;ub finem iactus, quod e&longs;t ridiculum; nec e&longs;t, quòd aliqui <lb/>dicant, ab aëre de&longs;trui, qui non minùs re&longs;i&longs;tit naturali, quàm vio­<lb/>lento. </s> </p> <p id="N10938" type="main"> <s id="N1093A"><!-- NEW -->5. Adde, quòd e&longs;t duplex determinatio: </s> <s id="N1093E"><!-- NEW -->igitur aliquid de&longs;trui de­<lb/>bet, non acqui&longs;iti; igitur impre&longs;&longs;i: </s> <s id="N10944"><!-- NEW -->de&longs;trui autem non dicitur acqui­<lb/>&longs;itus, quòd, &longs;cilicet, plùs de nouo accedat, quàm pereat; </s> <s id="N1094A"><!-- NEW -->e&longs;t enim ac­<lb/>celeratus: </s> <s id="N10950"><!-- NEW -->adde, quòd non infligitur tantus ictus &longs;ub finem; </s> <s id="N10954"><!-- NEW -->igitur <lb/>de&longs;truitur aliquid impetus, non acqui&longs;iti, eo modo, quo diximus; </s> <s id="N1095A"><!-- NEW --><lb/>igitur impre&longs;&longs;i: ita tamen &longs;en&longs;im de&longs;truitur, vt pro æquabili per ali­<lb/>quod &longs;patium qua&longs;i haberi po&longs;&longs;it. </s> </p> <p id="N10961" type="main"> <s id="N10963"><!-- NEW -->6. Hinc mobile proiectum per horizontalem, ne primo quidem <lb/>in&longs;tanti per horizontalem mouetur, alioqui non e&longs;&longs;et motus mix­<lb/>tus: </s> <s id="N1096B"><!-- NEW -->tardiùs cadit mobile ita proiectum in planùm horizontale &longs;ub­<lb/>iectum, quàm cum &longs;ua &longs;ponte, ex eadem altitudine de&longs;cendit: </s> <s id="N10971"><!-- NEW -->cuius <lb/>rei clari&longs;&longs;ima e&longs;t experientia: ratio e&longs;t; </s> <s id="N10977"><!-- NEW -->quia impetus acqui&longs;itus in <lb/>hoc iactu non e&longs;t eiu&longs;dem perfectionis, cùm acqui&longs;ito in perpendi­<lb/>culo: </s> <s id="N1097F"><!-- NEW -->cùm proiicitur mobile per inclinatam &longs;ur&longs;um, mouetur motu <lb/>mixto ex naturali æquabili, & violento retardato: patet prima pars; </s> <s id="N10985"><!-- NEW --><lb/>quia acceleratur tantùm naturalis deor&longs;um, &longs;altem in inclinata: </s> <s id="N1098A"><!-- NEW -->&longs;e­<lb/>cunda pars etiam patet; quia &longs;ub finem minor e&longs;t ictus. </s> </p> <p id="N10990" type="main"> <s id="N10992"><!-- NEW -->7. Hinc linea motus e&longs;t curua: </s> <s id="N10996"><!-- NEW -->iuxta diuer&longs;am progre&longs;&longs;ionem de­<lb/>&longs;truitur hic impetus impre&longs;&longs;us: </s> <s id="N1099C"><!-- NEW -->tùm pro diuer&longs;a inclinatione plani, <lb/>cuius etiam hîc habetur ratio; </s> <s id="N109A2"><!-- NEW -->nam &longs;ingulis in&longs;tantibus mutatur: </s> <s id="N109A6"><!-- NEW --><lb/>tùm, quia modò plùs impetus e&longs;t fru&longs;trà, modò minùs; </s> <s id="N109AB"><!-- NEW -->plùs <lb/>certè, cùm linea determinationis impetus impre&longs;&longs;i facit obtu­<lb/>&longs;iorem: </s> <s id="N109B3"><!-- NEW -->atqui initio e&longs;t obtu&longs;ior; &longs;ub finem verò a&longs;cen&longs;us acu­<lb/>tior. </s> </p> <p id="N109B9" type="main"> <s id="N109BB"><!-- NEW -->8. A&longs;cen&longs;us proiecti per inclinatam diutiùs durat, quàm de&longs;­<lb/>cen&longs;us, ratione eiu&longs;dem plani horizontalis; </s> <s id="N109C1"><!-- NEW -->quia, &longs;cilicet, a&longs;­<lb/>cen&longs;us longior e&longs;t, quàm de&longs;cen&longs;us: </s> <s id="N109C7"><!-- NEW -->e&longs;t autem longior; </s> <s id="N109CB"><!-- NEW -->quia, vt <lb/>e&longs;&longs;et æqualis, nihil impetus impre&longs;&longs;i deberet de&longs;trui in a&longs;cen&longs;u <pb xlink:href="026/01/019.jpg"/>porrò in de&longs;cen&longs;u e&longs;t motus mixtus ex accelerato naturali, <lb/>& retardato violento, vt con&longs;tat ex dictis: </s> <s id="N109D7"><!-- NEW -->iactus per incli­<lb/>natam ad angulum 45. e&longs;t omnium maximus, ratione eiu&longs;dem <lb/>plani horizontalis: clara e&longs;t experientia. </s> <s id="N109DF">Ratio e&longs;t: </s> <s id="N109E2"><!-- NEW -->quia per verti­<lb/>calem &longs;ur&longs;um, nihil acquiritur in plano horizontali, ex quo fit ia­<lb/>ctus; </s> <s id="N109EA"><!-- NEW -->nihil etiam per ip&longs;am horizontalem; igitur plùs acquiritur per <lb/>illam, quæ maximè ab vtraque &longs;imul recedit. </s> </p> <p id="N109F0" type="main"> <s id="N109F2"><!-- NEW -->9. Hæc ratio e&longs;t verè phy&longs;ica, geometrica nulla e&longs;t: hinc illi <lb/>iactus æquale &longs;patium acquirunt in prædicto plano horizontali, <lb/>qui fiunt per inclinatas æqualiter à prædicta inclinata ad ang. 45. <lb/>di&longs;tantes. </s> <s id="N109FC"><!-- NEW -->Cùm emittitur mobile per inclinatum deor&longs;um, in libero <lb/>medio, mouetur motu mixto ex naturali accelerato, & impre&longs;­<lb/>&longs;o retardato, vt con&longs;tat ex dictis; </s> <s id="N10A04"><!-- NEW -->ille autem primus accelera­<lb/>tur per acce&longs;&longs;ionem impetus perfectionis quàm in iactu per ho­<lb/>rizontalem; </s> <s id="N10A0C"><!-- NEW -->&longs;ed imperfectionis, quàm in perpendiculo: </s> <s id="N10A10"><!-- NEW -->retarda­<lb/>tur verò impetus minùs, quàm in iactu per horizontalem; plùs ve­<lb/>rò, quàm in iactu per ip&longs;um perpendiculum, in quo nihil impetus <lb/>de&longs;truitur. </s> </p> <p id="N10A1A" type="main"> <s id="N10A1C"><!-- NEW -->10. Cùm è naui mobili &longs;ur&longs;um mittitur corpus graue, e&longs;t motus <lb/>mixtus ex tribus, in a&longs;cen&longs;u, &longs;cilicet, ex naturali æquabili, ex verti­<lb/>cali retardato, & horizontali æquabili: </s> <s id="N10A24"><!-- NEW -->mouetur &longs;ur&longs;um per cur­<lb/>uam, &longs;empérque capiti iaculatoris imminet; </s> <s id="N10A2A"><!-- NEW -->quippe tantùm acqui­<lb/>rit in horizontali, quantùm nauis: </s> <s id="N10A30"><!-- NEW -->in de&longs;cen&longs;u verò e&longs;t motus <expan abbr="mix­">mixtus</expan> <lb/>ex horizontali retardato, & naturali accelerato: </s> <s id="N10A3A"><!-- NEW -->quia tamen bre­<lb/>ui&longs;&longs;imo illo tempore, retardatio illa horizontalis non e&longs;t &longs;en&longs;ibilis, <lb/>ferè in ip&longs;ius iaculatoris caput de&longs;cendit; quod certè phænomenon <lb/>ex no&longs;tris principiis euincitur. </s> </p> <p id="N10A44" type="main"> <s id="N10A46"><!-- NEW -->11. Parum cautè Vfanus vniuer&longs;im a&longs;&longs;erit, iaculationem pilæ ex <lb/>tormento, maiorem e&longs;&longs;e ex naui in continentem, & minorem vi­<lb/>ci&longs;&longs;im, cùm vtriu&longs;que differentia peti po&longs;&longs;it, vel à puluere tormen­<lb/>tario, vel ab eius compre&longs;&longs;ione, vel humiditate, vel tormenti fabri­<lb/>ca, vel ip&longs;ius demum nauigij motu, qui pilæ motum, vel accelerat, &longs;i <lb/>versùs eandem partem e&longs;t, vel retardat è contrario: in plano ho­<lb/>rizontali duro pote&longs;t e&longs;&longs;e motus mixtus ex duobus, tribus, qua­<lb/>tuor, & pluribus aliis. </s> </p> <p id="N10A58" type="main"> <s id="N10A5A"><!-- NEW -->12. Cùm è naui mobili emittitur &longs;agitta per horizontalem, quæ fa­<lb/>cit angelum rectum cum linea directionis nauis, fertur qua&longs;i per dia­<lb/>gonalem vtriu&longs;que, &longs;altem per aliquod &longs;patium: </s> <s id="N10A62"><!-- NEW -->cùm verò emitti-<pb xlink:href="026/01/020.jpg"/>tur per horizontalem, quæ conueniat cum eadem linea directionis, <lb/>iactus e&longs;t longior toto illo &longs;patio, quod nauis decurrit, dum iactus <lb/>durat; </s> <s id="N10A6E"><!-- NEW -->breuior tamen, &longs;i in partem oppo&longs;itam fiat iactus in hoc ca­<lb/>&longs;u, &longs;i nauis æqualem impetum imprimeret, deor&longs;um rectà ferretur <lb/>mobile motu naturali; </s> <s id="N10A76"><!-- NEW -->imò &longs;agitta po&longs;&longs;et retorqueri in iaculatorem: </s> <s id="N10A7A"><!-- NEW --><lb/>&longs;i terra e&longs;&longs;et vtrimque peruia, lapis demi&longs;&longs;us per multa annorum <lb/>millia libraretur; non tamen e&longs;&longs;et motuus perpetuus. </s> </p> <figure id="id.026.01.020.1.jpg" xlink:href="026/01/020/1.jpg"/> <p id="N10A86" type="main"> <s id="N10A88"><emph type="center"/><emph type="italics"/>De motu reflexo.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N10A93" type="main"> <s id="N10A95"><!-- NEW -->1. MOtus reflexi vera cau&longs;a e&longs;t impetus prior, ad nouam li­<lb/>neam determinatus ab occurrente obice; </s> <s id="N10A9B"><!-- NEW -->planum refle­<lb/>ctens e&longs;t cau&longs;a nouæ determinationis &longs;uo modo; </s> <s id="N10AA1"><!-- NEW -->cau&longs;am enim di­<lb/>co eam, ex qua aliquid &longs;equitur: </s> <s id="N10AA7"><!-- NEW -->ex gemina determinatione, noua, <lb/>&longs;cilicet, per ip&longs;am perpendicularem erectam in puncto contactus, <lb/>& priore per lineam incidentiæ, ab eodem puncto contactus pro­<lb/>pagatam, fit determinatio mixta per lineam reflexionis; </s> <s id="N10AB1"><!-- NEW -->quæ omnia <lb/>patent ex terminis: </s> <s id="N10AB7"><!-- NEW -->hinc nullus impetus producitur à plano refle­<lb/>ctente; </s> <s id="N10ABD"><!-- NEW -->quippe prior pote&longs;t determinari ad nouam lineam: adde, <lb/>quòd planum, quod caret impetu, impetum producere non pote&longs;t. </s> </p> <p id="N10AC3" type="main"> <s id="N10AC5"><!-- NEW -->2. Imò nihil impetus de&longs;truitur in reflexione pura per &longs;e; </s> <s id="N10AC9"><!-- NEW -->quia ni­<lb/>hil impetus e&longs;t fru&longs;trà per &longs;e in pura reflexione; </s> <s id="N10ACF"><!-- NEW -->multus tamen im­<lb/>petus de&longs;truitur per accidens, tùm ab ip&longs;o attritu tùm mollitie <lb/>& ce&longs;&longs;ione, tùm pre&longs;&longs;ione: </s> <s id="N10AD7"><!-- NEW -->hinc &longs;uppo&longs;ito eodem iactu, perpendi­<lb/>cularis reflexa e&longs;t omnium reflexarum minima; </s> <s id="N10ADD"><!-- NEW -->quia per eam li­<lb/>neam maximus ictus infligitur; </s> <s id="N10AE3"><!-- NEW -->igitur maxima e&longs;t partium colli&longs;io, <lb/>& pre&longs;&longs;io: hinc etiam corpora duriora longiùs reflectuntur, per ip&longs;am <lb/>quoque <expan abbr="perpendicular&etilde;">perpendicularem</expan>, dum planum reflectens &longs;it æquè durum. </s> </p> <p id="N10AEF" type="main"> <s id="N10AF1"><!-- NEW -->3. Determinatio noua dupla e&longs;t prioris, po&longs;ita linea incidentiæ <lb/>perpendiculari, & po&longs;ito etiam plano reflectente immobili; </s> <s id="N10AF7"><!-- NEW -->quia <lb/>alioquin anguli reflexionis non e&longs;&longs;ent æquales angulis incidentiæ: </s> <s id="N10AFD"><!-- NEW --><lb/>&longs;i globus reflectens &longs;it æqualis impacto, æqualis e&longs;t ce&longs;&longs;io re&longs;i&longs;tenciæ <lb/>cùm &longs;it æquale agens re&longs;i&longs;tenti, perid enim reflectens re&longs;i&longs;tit, per <lb/>quod e&longs;t: </s> <s id="N10B06"><!-- NEW -->igitur, &longs;i æqualis re&longs;i&longs;tit, & cedit, certè æqualiter ce­<lb/>dit, & re&longs;i&longs;tit: </s> <s id="N10B0C"><!-- NEW -->hinc noua determinatio æqualis e&longs;t priori: </s> <s id="N10B10"><!-- NEW -->hinc glo­<lb/>bus impactis &longs;i&longs;tit immobilis; quia ex duabus determinationibus <lb/>oppo&longs;itis neutra præualet. </s> </p> <pb xlink:href="026/01/021.jpg"/> <p id="N10B1B" type="main"> <s id="N10B1D"><!-- NEW -->4. Tantum e&longs;t ab æqualitate prædicta ce&longs;&longs;ionis, & re&longs;i&longs;tentiæ, ad <lb/>nullam ce&longs;&longs;ionem, & notam re&longs;i&longs;tentiam, quantum e&longs;t ad nullam <lb/><expan abbr="re&longs;i&longs;t&etilde;tiam">re&longs;i&longs;tentiam</expan>, & totam ce&longs;&longs;ionem: </s> <s id="N10B28"><!-- NEW -->hinc, cùm à tota ce&longs;&longs;ione ad æqua­<lb/>litatem prædictam acquiratur tantùm noua determinato æqualis <lb/>priori; </s> <s id="N10B30"><!-- NEW -->igitur ab eadem æqualitate ad nullam ce&longs;&longs;ionem tantun­<lb/>dem acquiritur; </s> <s id="N10B36"><!-- NEW -->igitur dupla prioris, vt iam &longs;uprà dictum e&longs;t; </s> <s id="N10B3A"><!-- NEW -->nulla <lb/>e&longs;&longs;et re&longs;i&longs;tentia in vacuo; nulla e&longs;t ce&longs;&longs;io, cùm ip&longs;um corpus refle­<lb/>ctens nullo modo mouetur ab ictu. </s> </p> <p id="N10B42" type="main"> <s id="N10B44"><!-- NEW -->5. Determinatio noua per lineam obliquam, e&longs;t ad nouam per <lb/>lineam perpendicularem, vt &longs;inus rectus anguli incidentiæ, ad &longs;i­<lb/>num totum, in qualibet hypothe&longs;i; </s> <s id="N10B4C"><!-- NEW -->quia &longs;unt hæ, vt ictus, per vtran­<lb/>que lineam; </s> <s id="N10B52"><!-- NEW -->ictus verò vt grauitationes in horizontale planum, & <lb/>in planum inclinatum, &longs;ub angulo complementi anguli incidentiæ: </s> <s id="N10B58"><!-- NEW --><lb/>hinc noua determinatio per lineam obliquam, e&longs;t vt dupla &longs;inus re­<lb/>cti anguli incidentiæ, ad &longs;inum totum: </s> <s id="N10B5F"><!-- NEW -->hinc &longs;upra angulum inci­<lb/>dentiæ 30, noua e&longs;t maior priore, infrà minor; in ip&longs;o angulo 30. <lb/>æqualis, &longs;uppo&longs;ita hypothe&longs;i plani reflectentis immobilis. </s> </p> <p id="N10B67" type="main"> <s id="N10B69"><!-- NEW -->6. Ex hoc po&longs;itiuo principio demon&longs;tratur accurati&longs;&longs;imè æqua­<lb/>litas anguli reflexionis, & incidentiæ, quod certè demon&longs;tratum <lb/>non fuit ab Ari&longs;t. in problematis, &longs;ect. 17. problem. 4. & 13. quibus <lb/>in locis fusè &longs;atis explicatur hoc Theorema, ducta comparatione, <lb/>tùm à grauibus, quæ cadunt, tùm ab orbibus, quæ rotantur, rùm à <lb/>&longs;peculis: &longs;ed minimè demon&longs;tratur ex certis principiis &longs;ine petitio­<lb/>ne principij. </s> <s id="N10B79"><!-- NEW -->In puncto reflexionis, po&longs;ita hypothe&longs;i plani immo­<lb/>bilis reflectentis, nulla datur quies; </s> <s id="N10B7F"><!-- NEW -->quia vnum tantùm e&longs;t conta­<lb/>ctus in&longs;tans; &longs;ed eo in&longs;tanti e&longs;t motus, quo primo acquiritur locus. </s> </p> <p id="N10B85" type="main"> <s id="N10B87"><!-- NEW -->7. Omnes lineæ reflexæ per &longs;e &longs;unt æqualis longitudinis, & ab <lb/>eodem puncto contactus, ad communem peripheriam terminan­<lb/>tur: </s> <s id="N10B8F"><!-- NEW -->&longs;i globus impactus &longs;it æqualis reflectenti, &longs;itque linea inciden­<lb/>tiæ obliqua quælibet terminata ad idem punctum contactus, re­<lb/>flectitur prædictus globus per lineam tangentem globum refle­<lb/>ctentem in eodem puncto; </s> <s id="N10B99"><!-- NEW -->quia hæc tangens e&longs;t diagonalis com­<lb/>munis, & determinatio mixta communis omnibus lineis inciden­<lb/>tiæ: e&longs;t tamen modò longior, modò breuior linea reflexa, é&longs;tque vt <lb/>vt &longs;inus complementi anguli incidentiæ, ad &longs;inum totum, qui &longs;it <lb/>determinatio prior, vt facilè demon&longs;tramus. </s> </p> <p id="N10BA5" type="main"> <s id="N10BA7">8. Si globus impactus &longs;it minor corpore reflectente, reflectitur <lb/>etiam per ip&longs;am perpendicularem, & determinatio noua e&longs;t dupla­<lb/>prioris, minùs ratione globorum v. g. &longs;i globus impactus &longs;it &longs;ubdu-<pb xlink:href="026/01/022.jpg"/>plus, determinatio noua e&longs;t dupla prioris, minùs vna quarta, <lb/>&c. </s> <s id="N10BB4"><!-- NEW -->ratio e&longs;t, quia in ea proportione globus reflectens cedit, in <lb/>qua mouetur, igitur tantùm detrahitur determinationis impacto <lb/>globo, quantùm additur motus reflectenti: at verò noua determina­<lb/>tio per lineam incidentiæ obliquam, e&longs;t ad nouam per ip&longs;am per­<lb/>pendicularem, vt &longs;inus rectus anguli incidentiæ ad &longs;inum totum. </s> </p> <p id="N10BC0" type="main"> <s id="N10BC2"><!-- NEW -->9. In hac hypothe&longs;i lineæ reflexæ omnes &longs;unt &longs;upra prædictam <lb/>tangentem, &longs;eu &longs;ectionem plani, maiores, vel minores, pro diuer&longs;a <lb/>men&longs;ura diagonalis: </s> <s id="N10BCA"><!-- NEW -->in &longs;uperiori verò hypothe&longs;i æqualium globo­<lb/>rum, &longs;unt omnes in ip&longs;a &longs;ectione plani: &longs;i denique globus impactus <lb/>&longs;it maior alio, omnes &longs;unt infra prædictam &longs;ectionem. </s> <s id="N10BD2"><!-- NEW -->Porrò in hac <lb/>hypothe&longs;i vltima, determinatio noua per ip&longs;am perpendicularem <lb/>e&longs;t minor priore: </s> <s id="N10BDA"><!-- NEW -->hinc non modò nulla fit reflexio in perpendicula­<lb/>ri, &longs;ed linea directa vlteriùs propagatur; quia prior determinatio <lb/>præualet. </s> </p> <p id="N10BE2" type="main"> <s id="N10BE4"><!-- NEW -->10. Detrahitur priori portio æqualis rationi globorum; </s> <s id="N10BE8"><!-- NEW -->v. g. glo­<lb/>bus reflectens e&longs;t &longs;ubduplus impacto de trahitur priori determina­<lb/>tioni vna &longs;ecunda; </s> <s id="N10BF0"><!-- NEW -->e&longs;t &longs;ubquadruplus, vna quarta; atque ita dein­<lb/>ceps: </s> <s id="N10BF6"><!-- NEW -->ratio patet ex dictis: </s> <s id="N10BFA"><!-- NEW -->in linea verò incidentiæ obliqua, deter­<lb/>minatio e&longs;t ad determinationem in perpendiculari, vt &longs;inus rectus <lb/>anguli incidentiæ ad &longs;inum totum: linea demum reflexa e&longs;t modò <lb/>maior, modò minor pro diuer&longs;a diagonali. </s> </p> <p id="N10C04" type="main"> <s id="N10C06"><!-- NEW -->11. Si duo globi æquales in &longs;e inuicem impingantur æquali mo­<lb/>tu, per lineam connectentem centra, vterque æquali motu priori re­<lb/>troagitur; </s> <s id="N10C0E"><!-- NEW -->quia æqualis in æqualis æqualem impetum imprimit: </s> <s id="N10C12"><!-- NEW -->non <lb/>e&longs;t tamen motus reflexus; </s> <s id="N10C18"><!-- NEW -->quia totus prior impetus de&longs;truitur, vt <lb/>patet ex dictis: </s> <s id="N10C1E"><!-- NEW -->&longs;i autem inæquali motu concurrant, retroaguntur <lb/>ii&longs;dem motibus, permutando; quod etiam clarum e&longs;t: hinc egre­<lb/>gium paradoxum, &longs;i quod aliud con&longs;equitur, &longs;cilicet, globum A, v. <lb/>g. æqualem motum imprimere globo B, &longs;iue hic moueatur, &longs;iue <lb/>quie&longs;cat. </s> </p> <p id="N10C2A" type="main"> <s id="N10C2C"><!-- NEW -->12. Si verò linea incidentiæ &longs;it obliqua, vterque globus reflecte­<lb/>tur pror&longs;us vt à plano immobili: </s> <s id="N10C32"><!-- NEW -->hinc reflexio &longs;it ad angulos æqua­<lb/>les, & lineæ omnes reflexionis &longs;unt æquales: ratio e&longs;t; </s> <s id="N10C38"><!-- NEW -->quia, quantùm <lb/>detrahit globus reflectens re&longs;i&longs;tendo, tantùm addit in partem op­<lb/>po&longs;itam repellendo, po&longs;itiuo ni&longs;u, vel impetu: quòd &longs;i alter globus <lb/>maiore, vel minore motu moueatur, vel &longs;i globi &longs;int inæquales, <lb/>cum æquali motu, vel inæquali, res etiam determinari pote&longs;t ex <lb/>præmi&longs;&longs;is. </s> </p> <pb xlink:href="026/01/023.jpg"/> <p id="N10C49" type="main"> <s id="N10C4B"><!-- NEW -->13. Cum duo globi in &longs;e&longs;e inuicem impinguntur æquali motu, <lb/>minor retroagitur velociore motu, quàm ante moueretur, vt clarum <lb/>e&longs;t: </s> <s id="N10C53"><!-- NEW -->maior verò, &longs;i duplus e&longs;t alterius, &longs;i&longs;tit immobilis in puncto <lb/>contactus; </s> <s id="N10C59"><!-- NEW -->&longs;i maior duplo &longs;uum iter pro&longs;equitur, &longs;ed tardiore mo­<lb/>tu; &longs;i minor duplo, retroagitur: quæ omnia facilè ex dictis demon­<lb/>&longs;trantur. </s> <s id="N10C61"><!-- NEW -->Pote&longs;t impetus e&longs;&longs;e æqualis alteri, & præualere; pote&longs;t <lb/>æqualem impetum producere hoc in&longs;tanti, & &longs;tatim in&longs;tanti, quod <lb/>&longs;equitur, totus de&longs;trui. </s> </p> <p id="N10C69" type="main"> <s id="N10C6B"><!-- NEW -->14. Pote&longs;t globus retroagi in plano horizontali, licèt in aliud cor­<lb/>pus non incidat, ita vt initio tendat in ortum, verbi gratia: </s> <s id="N10C71"><!-- NEW -->tùm <lb/>deinde, licèt nihil pror&longs;us addatur, versùs occa&longs;um; </s> <s id="N10C77"><!-- NEW -->quod accidit, <lb/>cum globus vtroque motu, centri, &longs;cilicet, & orbis, mouetur, &longs;ed <lb/>contrario; primùm enim motus centri præualet, &longs;ed facilè cedit <lb/>propter attritum maiorem partium. </s> <s id="N10C81"><!-- NEW -->Nullus datur propriè motus <lb/>refractus: </s> <s id="N10C87"><!-- NEW -->licèt enim incuruetur linea motus, dum per aquam &longs;u­<lb/>bit mobile; hæc tamen e&longs;t reflexionis &longs;pecies. </s> </p> <p id="N10C8D" type="main"> <s id="N10C8F"><!-- NEW -->15. Globus reflectens, qui ab ictu alterius mouetur, non mouetur <lb/>in&longs;tanti contactus; </s> <s id="N10C95"><!-- NEW -->quia impetus primo in&longs;tanti, quo e&longs;t, non mo­<lb/>uetur; </s> <s id="N10C9B"><!-- NEW -->producitur enim impetus primo in&longs;tanti contactus: </s> <s id="N10C9F"><!-- NEW -->&longs;i impe­<lb/>tus e&longs;&longs;et tantùm determinatus ad vnam lineam, nulla fieri po&longs;&longs;et <lb/>reflexio, &longs;ed tantùm repercu&longs;&longs;io; </s> <s id="N10CA7"><!-- NEW -->quia veri&longs;&longs;ima cau&longs;a reflexionis <lb/>con&longs;i&longs;tit in noua determinatione: </s> <s id="N10CAD"><!-- NEW -->per reflexionem po&longs;&longs;unt colligi <lb/>plures partes aëris &longs;onori ad Echometriam: </s> <s id="N10CB3"><!-- NEW -->&longs;agitta emi&longs;&longs;a per ho­<lb/>rizontalem &longs;ursùm, tantillùm a&longs;cendit per arcum; quia tantillùm <lb/>reflectitur ab aëre. </s> </p> <figure id="id.026.01.023.1.jpg" xlink:href="026/01/023/1.jpg"/> <p id="N10CC0" type="main"> <s id="N10CC2"><emph type="center"/><emph type="italics"/>De motu circulari.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N10CCD" type="main"> <s id="N10CCF"><!-- NEW -->1. DAri motum circularem, probatur infinitis ferè experimen­<lb/>tis: </s> <s id="N10CD5"><!-- NEW -->cuius ratio à priori e&longs;t, quòd po&longs;&longs;int extremitates eiu&longs;­<lb/>dem cylindri in partes oppo&longs;itas pelli; </s> <s id="N10CDB"><!-- NEW -->vnde &longs;equitur nece&longs;&longs;ariò <lb/>motus circularis; quem ij negare coguntur, qui ex punctis mathe­<lb/>maticis quantitatem componunt. </s> <s id="N10CE3"><!-- NEW -->Motus circularis in &longs;ublunaribus <lb/>oritur ex recto impedito; </s> <s id="N10CE9"><!-- NEW -->quia, &longs;cilicet, determinatur tantùm im­<lb/>petus ad lineam rectam: </s> <s id="N10CEF"><!-- NEW -->hinc quidam motus circularis e&longs;t merè <lb/>per accidens, vt cùm retinetur extremitas funependuli, &longs;eu <pb xlink:href="026/01/024.jpg"/>fundæ, quæ &longs;i demittatur, &longs;equitur motus rectus: </s> <s id="N10CF9"><!-- NEW -->quidam tamen <lb/>non e&longs;t merè peraccidens, vt cùm pellitur extremitas cylindri in <lb/>plano horizontali; e&longs;t enim, iuxta in&longs;titutionem naturæ, ad facili­<lb/>tatem motus. </s> </p> <p id="N10D03" type="main"> <s id="N10D05"><!-- NEW -->2. Quippe tale e&longs;t naturæ in&longs;titutum, vt eo motu corpora mo­<lb/>ueantur, quo faciliùs moueri po&longs;&longs;unt: </s> <s id="N10D0B"><!-- NEW -->atqui cùm pellitur altera cy­<lb/>lindri extremitas, in plano horizontali putà innatantis, faciliùs <lb/>mouetur, quàm recto, & qua&longs;i minore &longs;umptu, cùm minùs &longs;patij <lb/>acquirat: æquali tempore: </s> <s id="N10D15"><!-- NEW -->pote&longs;t dari motus circularis mixtus ex <lb/>duobus rectis, quorum vnus &longs;it, vt &longs;inus recti, alius vt ver&longs;i; vix <lb/>tamen hoc accidit vnquàm, &longs;ed tantùm oritur hic motus ex <lb/>determinatione per tangentem impedita, ratione alicuius puncti <lb/>immobilis. </s> </p> <p id="N10D21" type="main"> <s id="N10D23"><!-- NEW -->3. Hinc, &longs;i tollatur impedimentum, &longs;tatim per tangentem or­<lb/>bis fit motus, vt patet in funda: </s> <s id="N10D29"><!-- NEW -->inæqualiter partes radij prædicti <lb/>orbis mouentur, iuxta proportionem di&longs;tantiæ maioris, & minoris <lb/>à centro: </s> <s id="N10D31"><!-- NEW -->hinc propagatio impetus inæqualis, de qua iam &longs;uprà, <lb/>&longs;ingulis in&longs;tantibus & punctis e&longs;t noua determinatio; </s> <s id="N10D37"><!-- NEW -->quia, &longs;cilicet, <lb/>&longs;ingulis punctis &longs;ua tangens re&longs;pondet: </s> <s id="N10D3D"><!-- NEW -->hinc, &longs;i imponatur rotæ <lb/>aliud corpus, &longs;tatim abigitur, &longs;ine &longs;it in &longs;itu verticali, &longs;iue in &longs;itu ho­<lb/>rizontali; hinc dum turbo rotatur, &longs;i vel aquæ guttula eius &longs;uper­<lb/>ficies a&longs;pergitur, & &longs;tatim di&longs;pergitur. </s> </p> <p id="N10D47" type="main"> <s id="N10D49"><!-- NEW -->4 Dari impetum in motu circulari certi&longs;&longs;imum e&longs;t: </s> <s id="N10D4D"><!-- NEW -->punctum phy­<lb/>&longs;icum e&longs;t capax huius motus; cuius finis multiplex e&longs;t; </s> <s id="N10D53"><!-- NEW -->corpus mo­<lb/>uetur motu circulari circa centrum immobile cum motus centri <lb/>impeditur non tamen motus orbis, ad quem impetus facilè deter­<lb/>minatur, cùm &longs;it ad omnes lineas indifferens: </s> <s id="N10D5D"><!-- NEW -->adde v&longs;um vectis, <lb/>trochleæ, aliorúmque organorum, qui &longs;ine motu circulari e&longs;&longs;e non <lb/>pote&longs;t: omitto motum progre&longs;&longs;iuum, ipsúmque brachiorum, & ti­<lb/>biarum v&longs;um, qui motu circulari carere non pote&longs;t. </s> </p> <p id="N10D67" type="main"> <s id="N10D69"><!-- NEW -->5. Motus circularis rotæ in plano verticali e&longs;t æquabilis per &longs;e; </s> <s id="N10D6D"><!-- NEW --><lb/>quia nihil e&longs;t, quod impetum &longs;emel impre&longs;&longs;um de&longs;truat: </s> <s id="N10D72"><!-- NEW -->licèt enim <lb/>&longs;ingulis in&longs;tantibus &longs;it noua determinatio, nullus tamen impetus <lb/>e&longs;t fru&longs;trà; </s> <s id="N10D7A"><!-- NEW -->quippe illud &longs;patium acquiritur in linea curua, quod in <lb/>recta, &longs;i nullum e&longs;&longs;et impedimentum, percurreret: </s> <s id="N10D80"><!-- NEW -->quemadmodum <lb/>enim in reflexione, quæ fit à plano immobili, nullus de&longs;truitur im­<lb/>petus; </s> <s id="N10D88"><!-- NEW -->ita nullus hîc de&longs;truitur; </s> <s id="N10D8C"><!-- NEW -->tam enim centrum illud immobile <lb/>ad &longs;e qua&longs;i trahit mobile, quàm planum immobile à &longs;e repellit; in <lb/>quo e&longs;t perfectè analogia. </s> </p> <pb xlink:href="026/01/025.jpg"/> <p id="N10D97" type="main"> <s id="N10D99"><!-- NEW -->6. Hinc per &longs;e motus circularis integri orbis e&longs;t perpetuus; </s> <s id="N10D9D"><!-- NEW -->de­<lb/>&longs;truitur tamen per accidens, &longs;cilicet, propter attritum axis: </s> <s id="N10DA3"><!-- NEW -->hinc <lb/>tam diu durat hic motus: </s> <s id="N10DA9"><!-- NEW -->clari&longs;&longs;imum experimentum habes in tur­<lb/>bine, cuius cu&longs;pis læuigati&longs;&longs;ima in plano læuigati&longs;&longs;imo rotatur; </s> <s id="N10DAF"><!-- NEW -->nec <lb/>vnquam ce&longs;&longs;aret hic motus &longs;ine prædicto attritu, & partium a&longs;peri­<lb/>tate: </s> <s id="N10DB7"><!-- NEW -->nec quidquam ob&longs;tat, quòd aliquæ partes rotæ, quæ in circu­<lb/>lo verticali voluitur, a&longs;cendant; </s> <s id="N10DBD"><!-- NEW -->quia etiam aliquæ de&longs;cendunt: qua­<lb/>re &longs;emper remanet perfectum æquilibrium, & harum de&longs;cen&longs;us, il­<lb/>larum a&longs;cen&longs;um compen&longs;at. </s> <s id="N10DC5"><!-- NEW -->Quò diutiùs potentia motrix manet <lb/>applicata manubrio axis rotæ, ita vt nouum &longs;emper producat im­<lb/>petum, rotæ motus velocior e&longs;t, atque diutiùs durat: idem pror&longs;us <lb/>dico de rota circulo horizontali parallela. </s> </p> <p id="N10DCF" type="main"> <s id="N10DD1"><!-- NEW -->7. Cùm mouetur æquali ni&longs;u acus circa immobile centrum, tùm <lb/>in plano <expan abbr="horizõtali">horizontali</expan>, tùm in verticali, &longs;iue &longs;it <expan abbr="lõgior">longior</expan> vna, &longs;iue breuior <lb/>alia, per &longs;e plures gyros non de&longs;cribit vna, quàm alia; </s> <s id="N10DE1"><!-- NEW -->quia per &longs;e <lb/>mouetur motu æquabili: </s> <s id="N10DE7"><!-- NEW -->per accidens tamen &longs;ecus accidit; </s> <s id="N10DEB"><!-- NEW -->quippe <lb/>maior e&longs;t maioris attritus: </s> <s id="N10DF1"><!-- NEW -->dixi, cùm mouetur æquali ni&longs;u; </s> <s id="N10DF5"><!-- NEW -->nam &longs;æpè <lb/>contingit, maiore ni&longs;u potentiam motricem agere circa maiorem; </s> <s id="N10DFB"><!-- NEW --><lb/>æquali tamen tempore numerus circuitionum minoris, e&longs;t ad nu­<lb/>merum circuitionum maioris per &longs;e vt acuum quadrata permu­<lb/>tando; &longs;unt enim motus vt &longs;patia, &longs;pacia vt quadrata. </s> </p> <p id="N10E04" type="main"> <s id="N10E06"><!-- NEW -->8. Verbi gratia, &longs;it acus maior 2. minor 1. certè cùm tota area or­<lb/>bis maioris &longs;it quadrupla minoris, &longs;itque area maioris, &longs;patium ma­<lb/>ioris, & area minoris &longs;patium minoris, haud dubiè de&longs;cribet minor <lb/>quatuor circuitiones, eo tempore, quo maior decurret vnicam: </s> <s id="N10E10"><!-- NEW -->li­<lb/>cèt enim extremitas minoris, quæ impellitur, habeat tantùm du­<lb/>plum impetum extremitatis maioris, &longs;itque impetus inten&longs;io in <lb/>minore, dupla inten&longs;ionis impetus in maiore; </s> <s id="N10E1A"><!-- NEW -->e&longs;t tamen quadrupla <lb/>illius, quæ e&longs;t in &longs;egmento maioris versùs centrum æquali minori <lb/>acui: porrò motus circulares æquabiles in vtraque cum eodem <lb/>impetu, &longs;unt vt motus recti. </s> </p> <p id="N10E24" type="main"> <s id="N10E26"><!-- NEW -->9. Rota in plano verticali faciliùs mouetur, quàm in horizonta­<lb/>li; </s> <s id="N10E2C"><!-- NEW -->quia in illo mouetur per minimam impetus, vel potentiæ acce&longs;­<lb/>&longs;ionem; </s> <s id="N10E32"><!-- NEW -->&longs;ecùs in i&longs;to; </s> <s id="N10E36"><!-- NEW -->quippe per minimam acce&longs;&longs;ionem tollitur <lb/>æquilibrium; </s> <s id="N10E3C"><!-- NEW -->imò moueri pote&longs;t in plano verticali, licèt nullus im­<lb/>primatur impetus rotæ, v. <!-- REMOVE S-->g. <!-- REMOVE S-->per additionem minimi ponderis, vel <lb/>momenti, vt patet; cùm tamen in plano horizontali moueri non <lb/>po&longs;&longs;it, ni&longs;i impetus imprimatur. </s> </p> <p id="N10E4A" type="main"> <s id="N10E4C"><!-- NEW -->10. Si cylindrus in plano horizontali læuigato in altera extremi­<lb/>tate per tangentem impellatur, mouebitur motu circulati, &longs;cilicet, <pb xlink:href="026/01/026.jpg"/>faciliori, circa centrum, quod di&longs;tet ab altera extremitate vna <lb/>quarta totius cylindri: ratio e&longs;t: quia faciliùs mouetur circa illud <lb/>centrum, quàm circa alia puncta, quòd, &longs;cilicet, minùs &longs;patij decur­<lb/>ratur, po&longs;ito eodem &longs;emper motu alterius extremitatis, cui appli­<lb/>catur immediatè potentia motrix. </s> </p> <p id="N10E5E" type="main"> <s id="N10E60"><!-- NEW -->11. Cùm rota mouetur in verticali, atque præponderat alter &longs;emi­<lb/>circulus, haud dubiè hic præponderans producit impetum in alio <lb/>&longs;emicirculo: </s> <s id="N10E68"><!-- NEW -->hinc fortè e&longs;t, quòd mirere, impetus determinatus <lb/>deor&longs;um producit alium &longs;ur&longs;um: </s> <s id="N10E6E"><!-- NEW -->hinc impetus vnius partis mobi­<lb/>lis pote&longs;t producere &longs;imilem in alia parte continua; </s> <s id="N10E74"><!-- NEW -->quod tantùm in <lb/>hoc ca&longs;u locum habet: </s> <s id="N10E7A"><!-- NEW -->quando corpus incumbit plano, quod mo­<lb/>uetur motu recto æquabili, ab eo non &longs;eparatur; &longs;ecùs verò, &longs;i in­<lb/>cumbat plano, quod mouetur motu circulari. </s> </p> <figure id="id.026.01.026.1.jpg" xlink:href="026/01/026/1.jpg"/> <p id="N10E87" type="main"> <s id="N10E89"><emph type="center"/><emph type="italics"/>De motu funependuli.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N10E94" type="main"> <s id="N10E96"><!-- NEW -->1. FVnependulum de&longs;cendit per arcum motu naturaliter acce­<lb/>lerato: </s> <s id="N10E9C"><!-- NEW -->experientia clari&longs;&longs;ima e&longs;t: cùm enim ex maiori &longs;ubli­<lb/>mitate de&longs;cendit, maiorem ictum infligit. </s> <s id="N10EA2"><!-- NEW -->Ratio à priori e&longs;t quia <lb/>priori impetui acqui&longs;ito nouus accedit: </s> <s id="N10EA8"><!-- NEW -->non acceleratur in eadem <lb/>proportione, in qua &longs;uprà dictum e&longs;t accelerari in linea recta; </s> <s id="N10EAE"><!-- NEW -->quia <lb/>in hac acceleratur vniformiter, id e&longs;t, æqualibus temporibus, <lb/>æqualia acquiruntur velocitatis momenta; </s> <s id="N10EB6"><!-- NEW -->quia vel e&longs;t &longs;emper ea­<lb/>dem inclinatio plani, vel idem perpendiculum: </s> <s id="N10EBC"><!-- NEW -->at verò in fune­<lb/>pendulo in &longs;ingulis punctis e&longs;t noua tangens; </s> <s id="N10EC2"><!-- NEW -->igitur noua inclina­<lb/>tio plani; igitur noua ratio motus. </s> </p> <p id="N10EC8" type="main"> <s id="N10ECA"><!-- NEW -->2. Initio acceleratur motus per maiora crementa, &longs;ub finem per mi­<lb/>nora; </s> <s id="N10ED0"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;i dato tempore acqui&longs;iuit vnum gradum impetus initio, <lb/>æquali deinde tempore acquiret minùs: ratio clara e&longs;t: </s> <s id="N10ED8"><!-- NEW -->quia, vt ac­<lb/>quireret æqualem, deberet e&longs;&longs;e eadem plani inclinatio; </s> <s id="N10EDE"><!-- NEW -->&longs;ed &longs;emper <lb/>cre&longs;cit Inclinatio; </s> <s id="N10EE4"><!-- NEW -->igitur &longs;emper imminuitur impetus æquali <expan abbr="t&etilde;pore">tempore</expan> <lb/>acqui&longs;itus: </s> <s id="N10EEE"><!-- NEW -->acquiritur tamen æqualis velocitas in arcu, & in chor­<lb/>da, &longs;eu plano inclinato, eiu&longs;dem altitudinis; igitur &longs;emper cre&longs;cit <lb/>motus funependuli in de&longs;cen&longs;u, &longs;ed minoribus incrementis. </s> </p> <p id="N10EF6" type="main"> <s id="N10EF8"><!-- NEW -->3. Hinc breuiore tempore de&longs;cendit per radium perpendicula­<lb/>rem, quàm per quadrantis arcum eiu&longs;dem radij; </s> <s id="N10EFE"><!-- NEW -->tùm quia breuior <lb/>e&longs;t linea; tùm, quia in perpendiculari acceleratur motus per maiora <lb/>crementa. </s> <s id="N10F06"><!-- NEW -->Vibratio maior eiu&longs;dem funependuli æquali ferè tem-<pb xlink:href="026/01/027.jpg"/>pore cum minore perficitur: ratio e&longs;t: </s> <s id="N10F0E"><!-- NEW -->quia, cùm ferè decurrantur <lb/>arcus iuxta &longs;ubten&longs;arum proportionem, certè cùm &longs;ubten&longs;æ om­<lb/>nes æquali tempore decurrantur, idem ferè fit in ip&longs;is arcubus: </s> <s id="N10F16"><!-- NEW -->dixi <lb/>ferè: </s> <s id="N10F1C"><!-- NEW -->nam reuerà minor vibratio citiùs, maior tardiùs perficitur, vt <lb/><expan abbr="cõ&longs;tat">con&longs;tat</expan> <expan abbr="experi&etilde;tia">experientia</expan>: neque dee&longs;t ratio, quam in <expan abbr="analyticcã">analyticam</expan> remittimus. </s> </p> <p id="N10F2D" type="main"> <s id="N10F2F"><!-- NEW -->4. Non a&longs;cendit funependulum ad eam altitudinem, ex qua priùs <lb/>de&longs;cenderat: </s> <s id="N10F35"><!-- NEW -->clara e&longs;t experientia: </s> <s id="N10F39"><!-- NEW -->neque ratio tantùm petitur ab <lb/>aëris re&longs;i&longs;tentia; </s> <s id="N10F3F"><!-- NEW -->tam enim re&longs;i&longs;tit de&longs;cen&longs;ui, quàm a&longs;cen&longs;ui; </s> <s id="N10F43"><!-- NEW -->&longs;ed ex <lb/>eo, quòd &longs;ingulis in&longs;tantibus &longs;it quædam pugna, inter impetum in­<lb/>natum, & alium determinatum ad arcum &longs;ur&longs;um: </s> <s id="N10F4B"><!-- NEW -->quippe impetus <lb/>innatus ad totum de&longs;cen&longs;um, &longs;ed nullo modo ad a&longs;cen&longs;um con­<lb/>currit: </s> <s id="N10F53"><!-- NEW -->hinc in maiori vibratione imminuitur motus, & &longs;patium in <lb/>maiori proportione, quàm in minori; </s> <s id="N10F59"><!-- NEW -->quia in hac lineæ &longs;ingulæ a&longs;­<lb/>cen&longs;us qua&longs;i <expan abbr="totid&etilde;">totidem</expan> inclinatæ &longs;unt inclinatiores; in illa verò minùs. </s> </p> <p id="N10F63" type="main"> <s id="N10F65"><!-- NEW -->5. Hinc diu vibratur funependulum per minores arcus, quippe <lb/>facilis e&longs;t a&longs;cen&longs;us per planum proximè ad horizontale accedens: </s> <s id="N10F6B"><!-- NEW --><lb/>hinc etiam in funependulo maiori diutiùs durant huiu&longs;modi vi­<lb/>brationes, idque in arcubus paulò maioribus; </s> <s id="N10F72"><!-- NEW -->quia &longs;ubten&longs;æ his <lb/>arcubus &longs;unt inclinatiores: </s> <s id="N10F78"><!-- NEW -->hinc refutabis eos, qui dicunt, vibra­<lb/>tiones funependuli in vacuo fore perpetuas: </s> <s id="N10F7E"><!-- NEW -->arcus vibratio­<lb/>nis a&longs;cen&longs;us fit motu naturaliter retardato, &longs;ed per imminu­<lb/>tiones inæquales; quia pro diuer&longs;a inclinatione plani diuer&longs;imodè <lb/>retardatur. </s> </p> <p id="N10F88" type="main"> <s id="N10F8A"><!-- NEW -->6. Vltimum punctum impetus acqui&longs;itus acqui&longs;itum in de&longs;cen&longs;u, <lb/>nullo modo ad de&longs;cen&longs;um concurrit, &longs;ed ad a&longs;cen&longs;um, vnico tan­<lb/>tùm in&longs;tanti; </s> <s id="N10F92"><!-- NEW -->quippe e&longs;t omnium imperfecti&longs;&longs;imum; </s> <s id="N10F96"><!-- NEW -->quod reuerà &longs;i <lb/>e&longs;&longs;et eiu&longs;dem perfectionis cum innato, a&longs;cen&longs;us æqualis e&longs;t de&longs;cen­<lb/>&longs;ui: </s> <s id="N10F9E"><!-- NEW -->&longs;i &longs;int funependula inæqualia, vibrationes non &longs;unt æquè diu­<lb/>turnæ: ratio e&longs;t: </s> <s id="N10FA4"><!-- NEW -->quia, &longs;i a&longs;&longs;umantur, v.g. duo quadrantes inæquales, <lb/>&longs;unt eju&longs;dem inclinationis; igitur minor citiùs percurritur. </s> </p> <p id="N10FAA" type="main"> <s id="N10FAC"><!-- NEW -->7. Porrò tempora vibrationum &longs;unt in ratione &longs;ubduplicata ar­<lb/>cuum &longs;imilium, vel chordarum &longs;imilium, vel radiorum; </s> <s id="N10FB2"><!-- NEW -->id e&longs;t, vt <lb/>radices &longs;patiorum &longs;imilium: </s> <s id="N10FB8"><!-- NEW -->verbi gratia, &longs;it quadruplus alterius, <lb/>tempus vibrationis maioris e&longs;t duplum temporis vibrationis mino­<lb/>ris; </s> <s id="N10FC0"><!-- NEW -->quod ita intelligendum e&longs;t, vt hæc proportio con&longs;ideretur in <lb/>partibus temporis &longs;en&longs;ibilibus, vt iam dictum e&longs;t de motu natura­<lb/>liter accelerato deor&longs;um in perpendiculo, & in planis inclinatis; <lb/>nam progre&longs;&longs;io arithmetica; a&longs;&longs;umpta in &longs;ingulis in&longs;tantibus, tran­<lb/>&longs;it in hanc, &longs;i a&longs;&longs;umantur partes temporis &longs;en&longs;ibiles, quarum &longs;ingu­<lb/>læ infinitis ferè con&longs;tent in&longs;tantibus. </s> </p> <pb xlink:href="026/01/028.jpg"/> <p id="N10FD1" type="main"> <s id="N10FD3"><!-- NEW -->8. In maiori quadrante, circa &longs;upremam extremitatem, e&longs;t minor <lb/>inclinatio, quàm in minore; </s> <s id="N10FD9"><!-- NEW -->hic enim &longs;tatim detorquetur à perpen­<lb/>diculo, cum quo facit angulum maiorem: </s> <s id="N10FDF"><!-- NEW -->at verò circa infirmam <lb/>extremitatem, e&longs;t maior inclinatio in maiore, quàm in minore: </s> <s id="N10FE5"><!-- NEW -->hinc, <lb/>&longs;i comparetur vibratio maioris, cum vibratione minoris in modico <lb/>arcu, tempus illius e&longs;t paulò maius duplo, temporis huius; in maxi­<lb/>mo arcu paulò minùs duplo, dum, &longs;cilicet, longitudinum ratio <lb/>&longs;it quadrupla. </s> </p> <p id="N10FF1" type="main"> <s id="N10FF3"><!-- NEW -->9. In de&longs;cen&longs;u funependuli velocitas acqui&longs;ita e&longs;t eadem cum ea, <lb/>quæ in &longs;ubten&longs;a eiu&longs;dem arcus acquiritur: </s> <s id="N10FF9"><!-- NEW -->hinc &longs;unt ijdem ictus: </s> <s id="N10FFD"><!-- NEW --><lb/>numerus, vibrationum non e&longs;t infinitus, licèt in vacuo vibraretur <lb/>funependulum; </s> <s id="N11004"><!-- NEW -->quia, cùm &longs;ingulæ imminuantur, & infinitis pun­<lb/>ctis non con&longs;tent; </s> <s id="N1100A"><!-- NEW -->tandem ad vltimam peruenitur: </s> <s id="N1100E"><!-- NEW -->illa autem e&longs;t vl­<lb/>tima, in cuius de&longs;cen&longs;u acquiritur tantùm vnum punctum impetus <lb/>&longs;upra innatum; in ea tamen &longs;ententia, quæ vel infinitas partes actu, <lb/>vel infinita puncta cogno&longs;cit, certè nunquam quie&longs;ceret funepen­<lb/>dulum in vacuo vibratum. </s> </p> <p id="N1101A" type="main"> <s id="N1101C"><!-- NEW -->10. Funependulum in fine a&longs;cen&longs;us non quie&longs;cit vno in&longs;tanti; </s> <s id="N11020"><!-- NEW --><lb/>quia impetui innato <expan abbr="nũquam">nunquam</expan> redditur æqualis acqui&longs;itus; </s> <s id="N11029"><!-- NEW -->po&longs;ita ta­<lb/>men illa æqualitate, in&longs;tanti &longs;equenti e&longs;&longs;et quies: </s> <s id="N1102F"><!-- NEW -->funependulum <lb/>grauius citiùs de&longs;cendit; </s> <s id="N11035"><!-- NEW -->e&longs;t enim eadem ratio, quæ fuit pro mo­<lb/>tu naturali; </s> <s id="N1103B"><!-- NEW -->corpus oblongum &longs;olidum circa punctum immobile <lb/>in circulo verticali rotatum vibratur adin&longs;tat funependuli; de&longs;­<lb/>cendit tamen citiùs, quàm funependulum eiu&longs;dem longitudinis. </s> </p> <p id="N11043" type="main"> <s id="N11045">11. Ratio facilis e&longs;t; </s> <s id="N11048"><!-- NEW -->quia partes &longs;olidæ, quæ accedunt propiùs <lb/>ad extremitatem immobilem, accelerant motum aliarum, quæ <lb/>ad mobilem extremitatem accedunt; </s> <s id="N11050"><!-- NEW -->faciunt enim arcum mino­<lb/>rem: </s> <s id="N11056"><!-- NEW -->hinc a&longs;cen&longs;us non peruenit ad tantam &longs;ublimitatem; </s> <s id="N1105A"><!-- NEW -->quia, vt <lb/>prædictæ partes accelerant motum aliarum in de&longs;cen&longs;u, ita retar­<lb/>dant in de&longs;cen&longs;u: </s> <s id="N11062"><!-- NEW -->hinc citiùs quie&longs;cit hoc penduli genus, quàm <lb/>aliud: </s> <s id="N11068"><!-- NEW -->ex hoc colligo paradoxon, &longs;cilicet, corpus moueri po&longs;&longs;e &longs;ua <lb/>&longs;ponte velociùs in arcu deor&longs;um, quàm in perpendiculo; v.g. <!-- REMOVE S-->&longs;i iuxta <lb/>extremitatem immobilem &longs;it nodus plumbeus, cuius vi, altera ex­<lb/>tremitas longiùs di&longs;tans deor&longs;um rapiatur. </s> </p> <figure id="id.026.01.028.1.jpg" xlink:href="026/01/028/1.jpg"/> <p id="N11079" type="main"> <s id="N1107B"><emph type="center"/><emph type="italics"/>De motu mixto ex circulari.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N11086" type="main"> <s id="N11088"><!-- NEW -->1. ROta, quæ mouetur in &longs;uperficie plana, mouetur motu mixto <lb/>ex recto centri, & circulari orbis: </s> <s id="N1108E"><!-- NEW -->axis tantùm rotæ mouetur <lb/>motu recto: </s> <s id="N11094"><!-- NEW -->punctum contactus rotæ mouetur motu tardi&longs;&longs;imo, <pb xlink:href="026/01/029.jpg"/>quando motus centri, & &longs;uprema rotæ pars in eandem partem &longs;e­<lb/>runtur; </s> <s id="N1109E"><!-- NEW -->punctum verò oppo&longs;itum veloci&longs;&longs;imo, quia in motu huius <lb/>rotus motus orbis additur motui centri; </s> <s id="N110A4"><!-- NEW -->in motu verò illius, to­<lb/>tus motus orbis, motui centri detrahitur: quod autem detrahit mo­<lb/>tus orbis, nunquam æquale e&longs;t toti motui centri. </s> </p> <p id="N110AC" type="main"> <s id="N110AE"><!-- NEW -->2. Hinc omnia puncta eiu&longs;dem circuli rotæ mobilis in plano <lb/>hoc motu mixto mouentur in æquali motu: </s> <s id="N110B4"><!-- NEW -->hoc etiam motu mo­<lb/>uetur globus de&longs;cendens in plano inclinato, in quo reuerâ motu <lb/>hæc habes: </s> <s id="N110BC"><!-- NEW -->primò, non modò accelerari <expan abbr="motũ">motum</expan> centri, verùm etiam <lb/>motum orbis; <expan abbr="&longs;ecũdò">&longs;ecundò</expan>, ita <expan abbr="impetũ">impetum</expan> propagari ab intrin&longs;eco, vt &longs;ingu­<lb/>lis partibus eiu&longs;dem circuli, & plani in æqualiter di&longs;tribuatur, tertiò <lb/>hoc motu motum rectum non impediri à circulari, & &longs;ed iuuari. </s> </p> <p id="N110D2" type="main"> <s id="N110D4"><!-- NEW -->3. Cùm rota voluitur in &longs;uperficie connexa, mouetur motu mix­<lb/>to ex duobus circularibus: &longs;imilis e&longs;t hic motus motui epicycli. </s> <s id="N110DA"><!-- NEW -->Ca­<lb/>lamus volatilis, cuius mi&longs;&longs;io frequens, & repercu&longs;&longs;io, ludi non in­<lb/>grati copiam facit: </s> <s id="N110E2"><!-- NEW -->mouetur motu mixto ex recto, & circulari: </s> <s id="N110E6"><!-- NEW -->in <lb/>hoc porrò motu præit calami caput, & &longs;equuntur pennæ; </s> <s id="N110EC"><!-- NEW -->quia aër <lb/>fortiùs re&longs;i&longs;tit pennis, quàm thecæ: hinc pennarum motum theca <lb/>grauior accelerat, cuius motum pennæ retardant. </s> </p> <p id="N110F4" type="main"> <s id="N110F6"><!-- NEW -->4. Hinc, &longs;i quando accidat, penas educi ex theca in libero medio; </s> <s id="N110FA"><!-- NEW --><lb/>&longs;tatim theca velociori motu mouetur, cùm tamen pennæ ip&longs;æ &longs;i­<lb/>&longs;tant: </s> <s id="N11101"><!-- NEW -->ex hac inæqualitate, ne impetus &longs;it fru&longs;trà, propter detortas <lb/>in alteram partem pennas ab aëre re&longs;i&longs;tente totum iaculum defle­<lb/>ctitur, agitúr que in orbem; hinc motus orbis traducitur ex theca in <lb/>pennas, non contrà, vt aliquis fortè exi&longs;timaret, licèt pennarum tar­<lb/>ditas, & obliqua deflexio, ratione cuius ab aëre re&longs;tante, in alteram <lb/>partem qua&longs;i reflectentur, &longs;int nece&longs;&longs;aria conditio huius traductio­<lb/>nis. </s> </p> <p id="N11111" type="main"> <s id="N11113"><!-- NEW -->5. Hinc motu recto prædictum iaculum in vacuo tantùm mo­<lb/>ueretur, vt patet: hinc: </s> <s id="N11119"><!-- NEW -->cùm pennæ &longs;unt explicatiores, tardiùs; </s> <s id="N1111D"><!-- NEW -->cùm <lb/>verò contractiores, velociùs mouetur, etiam motu orbis; </s> <s id="N11123"><!-- NEW -->cui non <lb/>minùs aër re&longs;i&longs;tit, in pennis, &longs;cilicet, quàm motui axis: </s> <s id="N11129"><!-- NEW -->hinc, &longs;i theca <lb/>&longs;it grauior, velociùs; </s> <s id="N1112F"><!-- NEW -->&longs;i leuior, tardiùs iaculum fertur; </s> <s id="N11133"><!-- NEW -->etiam tenera <lb/>plumarum lanugo tarditatem conciliat: </s> <s id="N11139"><!-- NEW -->porrò, &longs;i axis mouetur mo­<lb/>tu recto, quod reuerà fit, cùm iaculum deor&longs;um demittitur in per­<lb/>pendiculo, hic motus e&longs;t &longs;piralis cylindricus: ex his infinita ferè <lb/>phænomena explicari po&longs;&longs;unt. </s> </p> <p id="N11143" type="main"> <s id="N11145"><!-- NEW -->6. Sunt infiniti propemodum motus mixti; </s> <s id="N11149"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->cylindri ab alte­<lb/>ra extremitate rotata emi&longs;&longs;i; </s> <s id="N11153"><!-- NEW -->longioris ha&longs;tæ, quæ &longs;ur&longs;um facta cir­<lb/>cuitione emittitur; </s> <s id="N11159"><!-- NEW -->brachij, gladij, &c. &longs;ed poti&longs;&longs;imùm turbinis, qui <pb xlink:href="026/01/030.jpg"/>vel &longs;cutica, vel funiculo in torto circumagitur, in quo clari&longs;&longs;i­<lb/>mè apparet motus centri, & orbis: </s> <s id="N11163"><!-- NEW -->ratio motus orbis e&longs;t impe­<lb/>tus impre&longs;&longs;us vtrique extremitati diametri va&longs;is in partes contra­<lb/>rias; </s> <s id="N1116B"><!-- NEW -->ratio verò motus centri e&longs;t, quia adducitur funiculo vel ex­<lb/>ploditur, &longs;eu expellitur &longs;cutica: </s> <s id="N11171"><!-- NEW -->huius motus phænomena &longs;unt ferè <lb/>infinita: &longs;ingula ex no&longs;tris principiis facilè explicantur. </s> </p> <figure id="id.026.01.030.1.jpg" xlink:href="026/01/030/1.jpg"/> <p id="N1117C" type="main"> <s id="N1117E"><emph type="center"/><emph type="italics"/>De diuer&longs;is impre&longs;&longs;ionibus motus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N11189" type="main"> <s id="N1118B"><!-- NEW -->1. CVm &longs;u&longs;tinetur manus, &longs;eu brachium, in &longs;itu horizontali im­<lb/>mobile, producitur nece&longs;&longs;ariò impetus æqualis impetui gra­<lb/>uitationis; </s> <s id="N11193"><!-- NEW -->alioquin, &longs;i maior e&longs;&longs;et, &longs;ur&longs;um ferretur brachium; &longs;i verò <lb/>minor, deor&longs;um: </s> <s id="N11199"><!-- NEW -->quia præualeret grauitatio, porrò hic impetus pro­<lb/>ducitur tantùm à potentia motrice animantis, in &longs;ingulari organo; </s> <s id="N1119F"><!-- NEW --><lb/>non verò in aliis partibus, etiam animatis, ni&longs;i quando mouentur; </s> <s id="N111A4"><!-- NEW --><lb/>nec in ip&longs;o pondere, &longs;i aliquod &longs;u&longs;tinetur: &longs;ic men&longs;a in pondere &longs;u­<lb/>per po&longs;ito impetum nullum producit. </s> <s id="N111AB">Si anima immediatè in toto <lb/>corpore po&longs;&longs;et producere impetum, homo facilè volare po&longs;&longs;et. </s> </p> <p id="N111B0" type="main"> <s id="N111B2"><!-- NEW -->2. Cùm &longs;u&longs;tinetur funependulum, nullus impetus producitur à <lb/>&longs;u&longs;tinente in ip&longs;o globo, ne &longs;cilicet, &longs;it fru&longs;trà; </s> <s id="N111B8"><!-- NEW -->&longs;ecùs verò, &longs;i attolla­<lb/>tur: </s> <s id="N111BE"><!-- NEW -->&longs;ic per quamlibet lineam corpus retineri pote&longs;t &longs;ine impetu in <lb/>eo corpore producto per &longs;e: </s> <s id="N111C4"><!-- NEW -->hinc, cùm duo &longs;e&longs;e inuicem trahunt ad­<lb/>uer&longs;o ni&longs;u, neuter in altero producit impetum per &longs;e; </s> <s id="N111CA"><!-- NEW -->&longs;ed per acci­<lb/>dens, propter mollitiem, & ten&longs;ionem partium: </s> <s id="N111D0"><!-- NEW -->cùm verò defertur <lb/>aliquid coniunctum, producitur haud dubiè æqualis impetus; </s> <s id="N111D6"><!-- NEW -->hinc <lb/>&longs;eparari non pote&longs;t; </s> <s id="N111DC"><!-- NEW -->quia æqualis e&longs;t motus latoris, & delati: exem­<lb/>plum habes in naui. </s> </p> <p id="N111E2" type="main"> <s id="N111E4"><!-- NEW -->3. Si verò nauis illicò &longs;i&longs;tat, vel tardiùs moueri pergat, tunc fit &longs;e­<lb/>paratio: hinc liquida effunduntur, &longs;i dum feruntur, breuior quietis <lb/>in va&longs;e intercedat morula. </s> <s id="N111EC"><!-- NEW -->Vt feratur cylindrus humeris <expan abbr="cõmodiùs">commodiùs</expan> <lb/>debet &longs;u&longs;tineri in <expan abbr="c&etilde;tro">centro</expan> grauitatis, ad eleuationem anguli 49. quia <lb/><expan abbr="tũc">tunc</expan> manui, & humero æqualiter <expan abbr="põdus">pondus</expan> di&longs;tribuitur: </s> <s id="N11203"><!-- NEW -->ideò in circulo <lb/>voluitur &longs;cyphus aqua plenus &longs;ine effu&longs;ione; quia impetus determi­<lb/>natus per tangentem circuli aquam ip&longs;am à centro circuli remouet. </s> </p> <p id="N1120B" type="main"> <s id="N1120D"><!-- NEW -->4. Cùm trahitur aliquod corpus impetus impre&longs;&longs;us in vna parte <lb/>non producit impetum in alia, alioquin daretur proce&longs;&longs;us in infi­<lb/>nitum; </s> <s id="N11215"><!-- NEW -->&longs;i chorda vtrinque trahatur, rumpetur in medio: </s> <s id="N11219"><!-- NEW -->&longs;i affixa <lb/>extremitati immobili, trahatur à potentia applicata alteri extremi-<pb xlink:href="026/01/031.jpg"/>tati, rumpetur iuxta primam illam extremitatem: &longs;i denique pon­<lb/>ticulo &longs;uppo&longs;ito tendatur, vel pondere deprimente, in eo puncto <lb/>rumpetur. </s> <s id="N11227"><!-- NEW -->Ratio communis i&longs;torum omnium e&longs;t: </s> <s id="N1122B"><!-- NEW -->quia inter illas <lb/>duas partes fieri debet diui&longs;io per &longs;e, quarum vna mouetur, &longs;ecùs <lb/>alia; vel quarum vtraque in partes oppo&longs;itas mouetur. </s> </p> <p id="N11233" type="main"> <s id="N11235"><!-- NEW -->5. Vt quodlibet pondus faciliùs trahatur, &longs;inguli equi trahere <lb/>debent fune communi, potiùs quàm bigati; </s> <s id="N1123B"><!-- NEW -->quia tunc nihil ferè pe­<lb/>rit impetus: </s> <s id="N11241"><!-- NEW -->cùm plures idem pondus trahunt, agunt actione com­<lb/>muni, alioqui &longs;inguli in toto pondere &longs;uum impetum producerent; <lb/>igitur &longs;inguli &longs;eor&longs;um trahere? </s> <s id="N11249"><!-- NEW -->e&longs;&longs;ent, quod fal&longs;um e&longs;t: </s> <s id="N1124D"><!-- NEW -->ideò currus <lb/>paulò po&longs;t initium motus faciliùs mouetur; </s> <s id="N11253"><!-- NEW -->quia aliquid impetus <lb/>priùs producti remanet: hinc etiam rupto fune, quo trahitur currus, <lb/>currus ip&longs;e modicum tempus adhuc mouetur. </s> </p> <p id="N1125B" type="main"> <s id="N1125D"><!-- NEW -->6. Si, dum quis trahit toto ni&longs;u magnum aliquod pondus, funis <lb/>rumpatur, pronùs corruit: quia maiorem impetum in &longs;e producit, <lb/>totum, &longs;cilicet, illum, quem in toto pondere produxi&longs;&longs;et eo in&longs;tan­<lb/>ti, quo rumpitur finis, qui reuerà maior e&longs;t, propter impedimen­<lb/>tum, ex præmi&longs;&longs;is principiis, maiorique applicatione potentiæ, ner­<lb/>uorum ten&longs;ione, &c. </s> <s id="N1126B"><!-- NEW -->dum trahitur vnco an nullus immobilis ver­<lb/>sùs nauim, nauis fertur versùs littus; dum pellitur aduersùm littus, <lb/>recedit à littore, quia pede, vel genu, imprimitur naui impetus in <lb/>contrariam pattem. </s> </p> <p id="N11275" type="main"> <s id="N11277"><!-- NEW -->7. Cùm trahitur cylindrus vtrinque æqualiter, qui neque flecti, <lb/>neque tendi pote&longs;t, nullum impetum accipit; </s> <s id="N1127D"><!-- NEW -->imò in tractione nul­<lb/>lus impetus e&longs;t inutilis: </s> <s id="N11283"><!-- NEW -->brachium infligit maiorem ictum, cùm ma­<lb/>iorem <expan abbr="arcũ">arcum</expan> de&longs;cribit &longs;uo motu; </s> <s id="N1128D"><!-- NEW -->quia, &longs;cilicet, mouetur motu natu­<lb/>raliter accelerato: </s> <s id="N11293"><!-- NEW -->hinc auer&longs;a manu validior impingitur colaphus, <lb/>quàm aduer&longs;a; </s> <s id="N11299"><!-- NEW -->quia illa maiorem arcum de&longs;cribit: </s> <s id="N1129D"><!-- NEW -->hinc longius bra­<lb/>chium cæteris paribus grauiùs ferit: hinc diu qua&longs;i rotatur bra­<lb/>chium, vt longiùs mittatur lapis. </s> </p> <p id="N112A5" type="main"> <s id="N112A7"><!-- NEW -->8. Maiore fu&longs;te maior ictus infligitur; </s> <s id="N112AB"><!-- NEW -->quia potentia toto ni&longs;u <lb/>agens, diutiùs manet applicata maiori, quàm minori; </s> <s id="N112B1"><!-- NEW -->&longs;untque ictus <lb/>in ratione &longs;ubduplicata vtriu&longs;que fu&longs;tis; </s> <s id="N112B7"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->fu&longs;tis pendens vnam <lb/>libram per maximum arcum impactus, infligit &longs;ubduplum ictum <lb/>alterius, quem infligit fu&longs;tis quatuor pendens libras per eundem <lb/>arcum impactus: </s> <s id="N112C5"><!-- NEW -->idem dicatur de mi&longs;&longs;o lapide: principium huius <lb/>veritatis pendet ex iis, quæ diximus lib. 2. de motu naturali­<lb/>ter accelerate, iuxta progre&longs;&longs;ionem numerorum imparium, <lb/>1. 3. 5. &c. </s> </p> <p id="N112CF" type="main"> <s id="N112D1"><!-- NEW -->9. Fu&longs;tis circa centrum immobile vibratus, maximum ictum in-<pb xlink:href="026/01/032.jpg"/>fligit, non quidem in centro grauitatis, id e&longs;t, in medio, &longs;i &longs;it cy­<lb/>lindrus, vel parallelipedum; </s> <s id="N112DB"><!-- NEW -->nec in extremitate mobili; </s> <s id="N112DF"><!-- NEW -->&longs;ed in eo <lb/>puncto, in quo e&longs;t centrum impetus impre&longs;&longs;i, id e&longs;t, quod æqualem <lb/>vtrinque dirimit impetum: ratio e&longs;t; </s> <s id="N112E7"><!-- NEW -->quia tunc totus impetus agit, <lb/>quantùm pote&longs;t; </s> <s id="N112ED"><!-- NEW -->illud autem punctum Geometria demon&longs;trat e&longs;&longs;e <lb/>terminum mediæ proportionalis, inter totum cylindrum, & &longs;ub­<lb/>duplum; modò nulla ratio vectis habeatur alioquin centrum pro­<lb/>cu&longs;&longs;ionis di&longs;tat 2/3 ab extremitate immobili. </s> </p> <p id="N112F7" type="main"> <s id="N112F9"><!-- NEW -->10. Cùm fu&longs;tis inflectitur, reditque ad pri&longs;tinum &longs;tatum, vt <lb/>videre e&longs;t in tudicula maiore, maior ictus imprimitur: </s> <s id="N112FF"><!-- NEW -->quia non <lb/>tantùm agit impetus extrin&longs;ecùs adueniens; </s> <s id="N11305"><!-- NEW -->verùm etiam potentia <lb/>quædam media, quæ corpora compre&longs;&longs;a, vel ten&longs;a, ad pri&longs;tinum <lb/>&longs;tatum reducit: hinc maximus e&longs;t ictus tudiculæ, cùm eo in&longs;tanti, <lb/>quo reductum e&longs;t omninò manubrium priori rectitudini, infligitur <lb/>ictus, quia tunc vis potentiæ mediæ e&longs;t maxima. </s> </p> <p id="N11311" type="main"> <s id="N11313"><!-- NEW -->11. Rotato flagello ideò maxima vis ine&longs;t, quia diutiùs potentia <lb/>manet applicata: </s> <s id="N11319"><!-- NEW -->hinc vides hoc principium e&longs;&longs;e vniuer&longs;ali&longs;&longs;imum, <lb/>quod iactis, pul&longs;is, & impactis competit; </s> <s id="N1131F"><!-- NEW -->de malleorum ictu idem <lb/>pror&longs;us dicendum e&longs;t, quod de fu&longs;te; </s> <s id="N11325"><!-- NEW -->&longs;i autem mallei cadant <lb/>ex eadem altitudine, motu naturali accelerato, ictus &longs;unt vt <lb/>mallei, quia duplus malleus, v. <!-- REMOVE S-->g. <!-- REMOVE S-->duplum impetum acquirit: nam <lb/>&longs;ingulæ partes &longs;eor&longs;im æqualem impetum acquirunt. </s> </p> <p id="N11333" type="main"> <s id="N11335"><!-- NEW -->12. Si verò ex diuer&longs;a altitudine cadant, vel &longs;unt æquales, vel <lb/>inæquales: </s> <s id="N1133B"><!-- NEW -->&longs;i primum, ictus &longs;unt vt tempora, quibus cadunt: </s> <s id="N1133F"><!-- NEW -->&longs;i <lb/>&longs;ecundum, ictus &longs;unt in ratione compo&longs;ita temporum, & mal­<lb/>leorum: </s> <s id="N11347"><!-- NEW -->&longs;i &longs;unt infinitæ, partes actu, nulla e&longs;t proportio percu&longs;&longs;ionis <lb/>granuli cadentis, & rupis ingentis grauitantis; </s> <s id="N1134D"><!-- NEW -->&longs;ed hoc vltimum fal­<lb/>&longs;um e&longs;&longs;e con&longs;tat; </s> <s id="N11353"><!-- NEW -->non pote&longs;t tamen determinari proportio vitium <lb/>grauitationis, & percu&longs;&longs;ionis, ni&longs;i numerus in&longs;tantium: quibus durat <lb/>motus deor&longs;um cogno&longs;catur. </s> </p> <p id="N1135B" type="main"> <s id="N1135D"><!-- NEW -->13. Leui&longs;&longs;imi lapides vix emittuntur ad modicam di&longs;tantiam; </s> <s id="N11361"><!-- NEW --><lb/>quia &longs;tatim &longs;eparantur à potentia: </s> <s id="N11366"><!-- NEW -->parallelipedum cadens de or­<lb/>&longs;um in &longs;itu horizontali maximum ictum infligit in centro grauita­<lb/>tis, id e&longs;t, in medio; </s> <s id="N1136E"><!-- NEW -->quia tunc totus impetus agit, totus enim impe­<lb/>ditur: </s> <s id="N11374"><!-- NEW -->in aliis punctis minor e&longs;t ictus, iuxta proportionem maioris <lb/>di&longs;tantiæ à prædicto centro: &longs;i verò percutiatur cylindrus innatans, <lb/>maxima erit vis, vel effectus ictus in centro grauitatis propter ean­<lb/>dem rationem. </s> </p> </section> </front> <body> <chap id="N1137F"> <pb pagenum="1" xlink:href="026/01/033.jpg"/> <figure id="id.026.01.033.1.jpg" xlink:href="026/01/033/1.jpg"/> <p id="N11389" type="head"> <s id="N1138B"><emph type="center"/>LIBER PRIMVS, <lb/><emph type="italics"/>DE IMPETV.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N11398" type="main"> <s id="N1139A"><!-- NEW -->TRACTATVM hunc de motu locali <lb/>ab ip&longs;o impetu au&longs;picamur, ex cuius <lb/>profectò cognitione tota res i&longs;ta de­<lb/>pendet; </s> <s id="N113A4"><!-- NEW -->cum enim impetus &longs;it cau&longs;a <lb/>immediata motus, vt fusè demon&longs;tra­<lb/>bimus infrà; </s> <s id="N113AC"><!-- NEW -->& cum propter quid &longs;it res cogno&longs;ci <lb/>non po&longs;&longs;it, ni&longs;i eius cau&longs;a cogno&longs;catur; </s> <s id="N113B2"><!-- NEW -->dubium e&longs;&longs;e <lb/>non pote&longs;t, quin præmittenda &longs;it tractatio illa, quæ <lb/>e&longs;t de impetu, vt deinde affectiones ip&longs;ius motus <lb/>per cau&longs;am eiu&longs;dem demon&longs;trentur; immò au&longs;im <lb/>dicere ex vnius impetus cognitione, non modò mo­<lb/>tum ip&longs;um, verùm etiam totam rem Phy&longs;icam pen­<lb/>dere. <lb/><gap desc="hr tag"/></s> </p> <p id="N113C5" type="main"> <s id="N113C7"><emph type="center"/><emph type="italics"/>DEFINITIO I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N113D3" type="main"> <s id="N113D5">MOTVS <emph type="italics"/>localis e&longs;t tran&longs;itus mobilis è loco in locum continuo fluxu.<emph.end type="italics"/><lb/>Huius definitionis explicationem habebis in Metaphy&longs;icâ, <lb/>quæ &longs;anè explicatio ad rem præ&longs;entem non facit. </s> </p> <p id="N113E1" type="main"> <s id="N113E3"><emph type="center"/><emph type="italics"/>Definitio II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N113EF" type="main"> <s id="N113F1"><!-- NEW --><emph type="italics"/>Motus velox e&longs;t quo percurritur maius &longs;patium æquali tempore, vel <lb/>æquale &longs;patium minori tempore; contrà verò motus tardus.<emph.end type="italics"/></s> </p> <pb pagenum="2" xlink:href="026/01/034.jpg"/> <p id="N113FF" type="main"> <s id="N11401"><emph type="center"/><emph type="italics"/>Definitio III.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1140D" type="main"> <s id="N1140F"><emph type="italics"/>Impetus e&longs;t qualitas exigens motum, &longs;eu fluxum localem &longs;ui &longs;ubiecti, vel <lb/>qua est cau&longs;a proxima motus illius mobilis, cui ine&longs;t, eo &longs;cilicet modo, quo <lb/>pote&longs;t e&longs;&longs;e cau&longs;a motus.<emph.end type="italics"/></s> </p> <p id="N1141A" type="main"> <s id="N1141C"><!-- NEW -->Dico e&longs;&longs;e qualitatem &longs;iue di&longs;tincta &longs;it, &longs;iue non di&longs;tincta; </s> <s id="N11420"><!-- NEW -->quod hîc <lb/>certè non di&longs;cutio; </s> <s id="N11426"><!-- NEW -->nec enim affirmo in hac definitione dari impetum; </s> <s id="N1142A"><!-- NEW --><lb/>&longs;ed definio tantùm quid &longs;it impetus; </s> <s id="N1142F"><!-- NEW -->qui reuera aliud non e&longs;t, &longs;i e&longs;t: </s> <s id="N11433"><!-- NEW --><lb/>quippe id tantùm concipio, cum impetum appello; </s> <s id="N11438"><!-- NEW -->&longs;iue &longs;it, &longs;iue non &longs;it, <lb/>ne quis fortè initio &longs;tatim mihi litem intendat; </s> <s id="N1143E"><!-- NEW -->quemadmodum definit <lb/>circulum Geometra; </s> <s id="N11444"><!-- NEW -->licèt non a&longs;&longs;erat dari perfectum circulum; </s> <s id="N11448"><!-- NEW -->ita Phy­<lb/>&longs;icus definit impetum, quamuis non affirmet dari impetum; </s> <s id="N1144E"><!-- NEW -->quod tamen <lb/>in &longs;exto Theoremate demon&longs;trabimus; </s> <s id="N11454"><!-- NEW -->itaque &longs;i e&longs;t impetus, haud dubiè <lb/>nihil omninò præ&longs;tat in &longs;uo &longs;ubiecto ni&longs;i motum; quod quomodò fiat, <lb/>explicabimus intrà in Theorematis. <!-- KEEP S--></s> </p> <p id="N1145D" type="main"> <s id="N1145F"><emph type="center"/><emph type="italics"/>Hypothe&longs;is I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1146B" type="main"> <s id="N1146D"><!-- NEW --><emph type="italics"/>Datur motus localis<emph.end type="italics"/>; </s> <s id="N11476"><!-- NEW -->quis enim non videt volantem auem, natantem <lb/>pi&longs;cem; currentem equum, rotatum globum; denique vnum corpus mi­<lb/>grans è loco in locum? </s> <s id="N1147E"><!-- NEW -->&longs;ed hoc e&longs;t moueri per Def. <!-- REMOVE S-->1. igitur infinitis fe­<lb/>rè experimentis nititur hæc hypothe&longs;is, quam veram e&longs;&longs;e nece&longs;&longs;e e&longs;t, &longs;i <lb/>illa vera &longs;unt; &longs;ed illa certa &longs;unt phy&longs;icè, neque citra miraculum fallere <lb/>po&longs;&longs;unt. </s> </p> <p id="N1148A" type="main"> <s id="N1148C"><!-- NEW -->Diceret fortè aliquis etiam motum &longs;ube&longs;&longs;e oculorum fallaciæ; cùm è <lb/>naui mobili littus ip&longs;um moueri, ip&longs;umque nauigium non moueri iudi­<lb/>cemus. </s> <s id="N11494">Quis enim oculos in Solem intendens, primo intuitu Solem &longs;ta­<lb/>re non iudicet? </s> <s id="N11499"><!-- NEW -->cum tamen deinde pernici&longs;&longs;imo cur&longs;u rotari demon&longs;tre­<lb/>mus; </s> <s id="N1149F"><!-- NEW -->adde alias oculorum fallacias circa motum; </s> <s id="N114A3"><!-- NEW -->&longs;ic rotata &longs;cintilla, vel <lb/>carbo accen&longs;us immotum orbem de&longs;cribere videtur; </s> <s id="N114A9"><!-- NEW -->&longs;ic nota inu&longs;ta <lb/>trocho, dum celerrimè rotatur, orbem etiam immobilem de&longs;cribere iu­<lb/>dicatur; </s> <s id="N114B1"><!-- NEW -->&longs;ic &longs;tella cadens, vel exhalatio continenti &longs;ucce&longs;&longs;ione accen&longs;a <lb/>moueri videtur; </s> <s id="N114B7"><!-- NEW -->licet minimè moueatur; </s> <s id="N114BB"><!-- NEW -->idem dicendum de puluere <lb/>tormentario, vel alia qualibet materia; quæ continuata con&longs;ecutione <lb/>accenditur; </s> <s id="N114C3"><!-- NEW -->immò trochus ip&longs;e in orbem celerrimè agitatus, quie&longs;cere <lb/>videtur; </s> <s id="N114C9"><!-- NEW -->&longs;ic qui vertigine laborant, ea moueri exi&longs;timant, quæ quie&longs;cunt; </s> <s id="N114CD"><!-- NEW --><lb/>idem exemplum habemus in ebrio&longs;is, iracundis, in iis qui ex graui febris <lb/>ardore delirant, & in pueris qui diu in gyros eunt, vbi verti de&longs;ierint; </s> <s id="N114D4"><!-- NEW --><lb/>&longs;ic eorum quæ motu æquali feruntur, remotiora tardiùs moueri viden­<lb/>tur; </s> <s id="N114DB"><!-- NEW -->immò &longs;i per eandem lineam oculus, & mobile pari velocitate ince­<lb/>dant, ip&longs;um mobile quie&longs;cere videtur, plura leges apud Opticos, de <lb/>quibus agemus &longs;uo loco: Igitur ex his omnibus con&longs;tat minimè con&longs;ta­<lb/>re dari motum, ex eo quòd oculis aliquid moueri videatur. </s> </p> <p id="N114E5" type="main"> <s id="N114E7"><!-- NEW -->Re&longs;pondeo equidem fateri me, vi&longs;um ip&longs;um plurimis &longs;ube&longs;&longs;e fraudi­<lb/>bus; </s> <s id="N114ED"><!-- NEW -->attamen &longs;i rectè oculus admoueatur, iu&longs;ta di&longs;tantià, nec vllum &longs;it <lb/>impedimentum exterius nec interius; </s> <s id="N114F3"><!-- NEW -->fieri non pote&longs;t, quin oculus mo­<lb/>tum ob&longs;eruet; an fortè currentis calami motus oculum meum fallere po-<pb pagenum="3" xlink:href="026/01/035.jpg"/>te&longs;t? </s> <s id="N114FE"><!-- NEW -->quidquid &longs;it, fateor vltrò hanc hypothe&longs;im in eo tantùm certitudi­<lb/>nis gradu e&longs;&longs;e reponendam, in quo reponitur hæc cognitio, quâ modo <lb/>cogno&longs;co me &longs;cribere, manu&longs;que, & calami motum ob&longs;eruo; </s> <s id="N11506"><!-- NEW -->&longs;iue id tan­<lb/>tùm oculis fiat, &longs;iue intellectu ex oculis; quod aliàs di&longs;cutiemus; &longs;i quis <lb/>fortè in Phy&longs;ica maiorem certitudinem po&longs;tularet, cum eo certè conue­<lb/>nire non po&longs;&longs;um. </s> </p> <p id="N11510" type="main"> <s id="N11512"><!-- NEW -->Porrò quod &longs;pectat ad fallacias illas quæ &longs;upra adductæ &longs;unt; </s> <s id="N11516"><!-- NEW -->certum <lb/>e&longs;t vel obiectum e&longs;&longs;e remotius, quam par &longs;it; </s> <s id="N1151C"><!-- NEW -->vel moueri celeriùs, vel <lb/>e&longs;&longs;e aliquod impedimentum interius; </s> <s id="N11522"><!-- NEW -->præ&longs;ertim in iis, qui &longs;eu vertigine, <lb/>vel alio capitis morbo laborant; &longs;ed ne hîc opticum agere videar, harum <lb/>fallaciarum certi&longs;&longs;imas cau&longs;as in &longs;uum locum remittimus. </s> </p> <p id="N1152A" type="main"> <s id="N1152C"><!-- NEW -->Cæterùm licèt ad &longs;tatuendam, firmandamque hanc hypote&longs;im, Phy­<lb/>&longs;ica experimenta rectè applicato &longs;en&longs;u comprobata &longs;ufficere po&longs;&longs;int; <lb/>non de&longs;unt tamen rationes multæ à priori, vt vulgò aiunt, quibus euin­<lb/>citur, non modò quid &longs;it motus, verùm etiam propter quid &longs;it. </s> </p> <p id="N11536" type="main"> <s id="N11538"><!-- NEW -->Prima duci pote&longs;t à fine motus; </s> <s id="N1153C"><!-- NEW -->cum enim res creatæ vbique &longs;imul <lb/>e&longs;&longs;e non po&longs;&longs;int, certè, vt illo bono gaudeant, quo fortè carent, & vt <lb/>coniungantur &longs;uo fini, motu locali opus e&longs;t; </s> <s id="N11544"><!-- NEW -->&longs;itit equus, abe&longs;t aqua, <lb/>certè, ni&longs;i vel hæc propinetur, vel ille accedat, &longs;itim leuare non pote­<lb/>rit; </s> <s id="N1154C"><!-- NEW -->at neutrum &longs;ine motu haberi pote&longs;t: Lapis remouetur à &longs;uo centro, <lb/>à &longs;uo globo, à &longs;uo fine, vt &longs;e&longs;e illi re&longs;tituat, deor&longs;um cadat nece&longs;&longs;e e&longs;t. </s> <s id="N11552"><!-- NEW --><lb/>Itaque ad cum finem res omnes creatæ in&longs;titutæ &longs;unt, quem &longs;ine motu <lb/>a&longs;&longs;equi non po&longs;&longs;unt; </s> <s id="N11559"><!-- NEW -->igitur dari motum nece&longs;&longs;e e&longs;t, vt res creatæ cum lo­<lb/>cum acquirant, in quo &longs;uo bono, &longs;uo fini, &longs;uæ perfectioni coniungan­<lb/>tur; vel &longs;altem id muneris obeant, cui ab ipsâ naturâ de&longs;tinantur. </s> </p> <p id="N11561" type="main"> <s id="N11563"><!-- NEW -->Secunda ratio ducitur à cau&longs;a efficiente; ni&longs;i enim daretur motus, <lb/>fru&longs;trà daretur potentia motrix, tùm in animantibus, tùm in grauibus, <lb/>de quâ aliàs. </s> </p> <p id="N1156B" type="main"> <s id="N1156D"><!-- NEW -->Tertia petitur à cau&longs;a formali; cum enim detur impetus, vt demon­<lb/>&longs;trabimus infrà, nece&longs;&longs;e e&longs;t dari motum. </s> </p> <p id="N11573" type="main"> <s id="N11575"><!-- NEW -->Quarta petitur à termino motus; </s> <s id="N11579"><!-- NEW -->cum enim globus proiectus &longs;it in <lb/>nouo loco in quo ante non erat; </s> <s id="N1157F"><!-- NEW -->certè nouus locus qui &longs;uccedit alteri <lb/>relicto, e&longs;t terminus motus citra miraculum; igitur &longs;i e&longs;t nouus locus, <lb/>e&longs;t quoque motus. </s> </p> <p id="N11587" type="main"> <s id="N11589">Quinta ab v&longs;u; nec enim &longs;ine motu flueret aqua, caderet lapis, gyros <lb/>agerent a&longs;tra, flaret ventus, volarent nubes, &c. </s> </p> <p id="N1158E" type="main"> <s id="N11590"><!-- NEW -->Sexta ab ip&longs;a Mechanica, quæ organa motui mini&longs;trat: </s> <s id="N11594"><!-- NEW -->quis enim ne­<lb/>garet maius momentum e&longs;&longs;e cum maiori di&longs;tantiâ coniunctum; </s> <s id="N1159A"><!-- NEW -->&longs;i verò <lb/>maius momentum e&longs;t, nunquid præualebit; igitur deor&longs;um cadet, immò <lb/>&longs;euerior Geometria, vt omittam A&longs;tronomiam, motum &longs;upponit, cum ex <lb/>fluxu &longs;eu motu puncti infinitas fere lineas de&longs;cribat. </s> <s id="N115A4">Igitur certum e&longs;t <lb/>dari motum localem. </s> </p> <p id="N115A9" type="main"> <s id="N115AB"><emph type="center"/><emph type="italics"/>Hypothe&longs;is II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N115B7" type="main"> <s id="N115B9"><!-- NEW --><emph type="italics"/>Datur quies, id e&longs;t priuatio motus.<emph.end type="italics"/> Hæc hypothe&longs;is etiam certa e&longs;t, <pb pagenum="4" xlink:href="026/01/036.jpg"/>Quis enim neget &longs;edentem humi, vel decumbentem in lecto quie&longs;ceret <lb/>con&longs;ule &longs;en&longs;us rectè applicatos; </s> <s id="N115C9"><!-- NEW -->tam enim certus &longs;um me iam in cathe­<lb/>dra quie&longs;cere, quam &longs;um certus Solem lucere; igitur ex certis experi­<lb/>mentis certa hypothe&longs;is con&longs;equitur. </s> <s id="N115D1"><!-- NEW -->Non de&longs;unt rationes à priori; nam <lb/>primò res aliqua &longs;uo bono, &longs;eu fini coniuncta ab eo &longs;eparari non po&longs;tu­<lb/>lat, igitur nec moueri. </s> <s id="N115D9">Secundò maximum incommodum e&longs;&longs;et, &longs;i res &longs;e­<lb/>mel mota perpetuò moueretur. </s> <s id="N115DE"><!-- NEW -->Tertiò, finis, &longs;eu terminus motus recti, <lb/>e&longs;t quies; nam ideo lapis deor&longs;um cadit, vt in &longs;uo centro &longs;eu globo <lb/>quie&longs;cat, id e&longs;t vt cum aliis partibus totum illud, &longs;eu globum componat, <lb/>vt dicemus aliàs. </s> </p> <p id="N115E8" type="main"> <s id="N115EA"><!-- NEW -->Diceret fortè aliquis &longs;ententias prædictas non valere in &longs;ententiâ <lb/>Copernici, quæ terræ motum ad&longs;truit; præterea non modò falli &longs;en&longs;us <lb/>circa motum, verùm etiam circa quietem. </s> </p> <p id="N115F2" type="main"> <s id="N115F4"><!-- NEW -->Re&longs;pondeo primò illam Copernici &longs;ententiam e&longs;&longs;e fal&longs;i&longs;&longs;imam, vt &longs;uo <lb/>loco o&longs;tendemus: &longs;ecundò, licèt terra moueretur &longs;ecundum Coperni­<lb/>cum, Sol, & &longs;tellæ quie&longs;cerent. </s> </p> <p id="N115FC" type="main"> <s id="N115FE">Dices iuxta hypothe&longs;im Heraclidis Pontici, terra ip&longs;a, Sol etiam, & <lb/>&longs;tellæ mouentur. </s> <s id="N11603"><!-- NEW -->Re&longs;pondeo primò hypothe&longs;im illam e&longs;&longs;e fal&longs;am, vt &longs;uo <lb/>loco videbimus; </s> <s id="N11609"><!-- NEW -->&longs;ecundò etiam data illa hypothe&longs;i po&longs;&longs;et dari quies; </s> <s id="N1160D"><!-- NEW -->&longs;i <lb/>enim globus eodem ver&longs;us occa&longs;um impetu proiiceretur, quò ver&longs;us or­<lb/>tum à terra ip&longs;a rapitur, haùd dubiè quie&longs;ceret: præterea iuxta hanc hy­<lb/>pothe&longs;im, quietem appellarem vnius partis cum alia connexionem in ip­<lb/>&longs;o toto &longs;eu globo, & quie&longs;cere dicerem lapidem, qui tantùm totius glo­<lb/>bi motu mouetur, ex quo profectò tota &longs;oluitur difficultas. </s> </p> <p id="N1161B" type="main"> <s id="N1161D"><!-- NEW -->Quod verò &longs;pectat ad fallacias oculi circa quietem; </s> <s id="N11621"><!-- NEW -->eodem pror&longs;us <lb/>modo &longs;oluendæ &longs;unt, quo iam &longs;upra &longs;olutæ &longs;unt aliæ circa motum: <lb/>vtrùm verò motus, & quies dicant aliquid di&longs;tinctum à mobili, dice­<lb/>mus infrà. </s> </p> <p id="N1162B" type="main"> <s id="N1162D"><emph type="center"/><emph type="italics"/>Hypothe&longs;is III.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11639" type="main"> <s id="N1163B"><emph type="italics"/>Aliquid mouetur quod incœpit moueri.<emph.end type="italics"/></s> <s id="N11642"><!-- NEW --> Video lapidem quie&longs;centem, <lb/>qui deinde proiectus mouetur; </s> <s id="N11648"><!-- NEW -->igitur ante non mouebatur, igitur cum <lb/>deinde mouetur, cœpit moueri; mille aliis experimentis hæc hypothe­<lb/>&longs;is confirmari pote&longs;t. </s> </p> <p id="N11650" type="main"> <s id="N11652"><emph type="center"/><emph type="italics"/>Hypothe&longs;is IV.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1165E" type="main"> <s id="N11660"><emph type="italics"/>Aliquid mouetur quod tandem de&longs;init moueri, vel incipit quie&longs;cere.<emph.end type="italics"/></s> <s id="N11667"> Vi­<lb/>deo rotatam pilam, quæ tandem quie&longs;cit, cadentem lapidem, qui tan­<lb/>dem &longs;i&longs;tit, &c. </s> <s id="N1166E">igitur certa e&longs;t hæc hypothe&longs;is. </s> </p> <p id="N11671" type="main"> <s id="N11673"><emph type="center"/><emph type="italics"/>Hypothe&longs;is V.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1167F" type="main"> <s id="N11681"><emph type="italics"/>Idem mouetur modò tardiùs, modò velociùs.<emph.end type="italics"/></s> <s id="N11688"><!-- NEW --> Video rotatum globum, <lb/>qui &longs;en&longs;im quie&longs;cit: &longs;entio ab eodem globo modò maiorem, modò mi­<lb/>norem ictum infligi, &c. </s> <s id="N11690">igitur e&longs;t certa hypothe&longs;is. </s> </p> <pb pagenum="5" xlink:href="026/01/037.jpg"/> <p id="N11697" type="main"> <s id="N11699"><emph type="center"/><emph type="italics"/>Hypothe&longs;is VI.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N116A5" type="main"> <s id="N116A7"><emph type="italics"/>Corpus proiectum etiam à potentiâ motrice &longs;eiunctum adhuc mouetur.<emph.end type="italics"/><lb/>Oculos omnium te&longs;tes appello. </s> </p> <p id="N116B0" type="main"> <s id="N116B2"><emph type="center"/><emph type="italics"/>Hypothe&longs;is VII.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N116BE" type="main"> <s id="N116C0"><emph type="italics"/>Corpus proiectum, & in aliud impactum illud ip&longs;um impellit, & mouet.<emph.end type="italics"/></s> </p> <p id="N116C7" type="main"> <s id="N116C9"><emph type="center"/><emph type="italics"/>Hypothe&longs;is VIII.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N116D5" type="main"> <s id="N116D7"><emph type="italics"/>Ignis applicatus &longs;ubiectum aptum, cui rectè applicatur nece&longs;&longs;ariò calefa­<lb/>cit, nix frigefacit, Sol illuminat, corpus in aliud impactum illud ip&longs;um im­<lb/>pellit.<emph.end type="italics"/></s> <s id="N116E2"> Prædictæ omnes Hypothe&longs;es certi&longs;&longs;imis nixæ experimentis certi­<lb/>tudinem phy&longs;icam habent, & citra miraculum fallere non po&longs;&longs;unt. </s> </p> <p id="N116E7" type="main"> <s id="N116E9"><emph type="center"/><emph type="italics"/>Axioma I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N116F5" type="main"> <s id="N116F7"><emph type="italics"/>Contradictoria &longs;imul e&longs;&longs;e non po&longs;&longs;unt, vel non e&longs;&longs;e.<emph.end type="italics"/></s> <s id="N116FE"> Hoc ip&longs;um iam præ­<lb/>mi&longs;imus Logicæ no&longs;træ demon&longs;tratiuæ, complectiturque prima illa <lb/>principia Metaphy&longs;icæ. </s> </p> <p id="N11705" type="main"> <s id="N11707">1. <emph type="italics"/>Impo&longs;&longs;ibile est idem &longs;imul e&longs;&longs;e, & non e&longs;&longs;e.<emph.end type="italics"/></s> </p> <p id="N1170F" type="main"> <s id="N11711">2. <emph type="italics"/>Quodlibet e&longs;t, vel non est.<emph.end type="italics"/></s> </p> <p id="N11719" type="main"> <s id="N1171B">3. <emph type="italics"/>De eodem alterum contradictoriorum verè affirmatur, & alterum verè <lb/>negatur, non &longs;imul vtrumque.<emph.end type="italics"/></s> </p> <p id="N11725" type="main"> <s id="N11727"><emph type="center"/><emph type="italics"/>Axioma II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11733" type="main"> <s id="N11735"><emph type="italics"/>Maximum &longs;ignum di&longs;tinctionis realis in phy&longs;icis est &longs;eparabilitas, vel op­<lb/>po&longs;itio.<emph.end type="italics"/></s> <s id="N1173E"><!-- NEW --> Nihil enim a &longs;e ip&longs;o &longs;eparari po&longs;t; </s> <s id="N11742"><!-- NEW -->quippe, vbi e&longs;t &longs;eparatio, &longs;eu <lb/>diui&longs;io, e&longs;t pluralitas; cur enim nummus A & nummus B eiu&longs;dem ma­<lb/>teriæ, formæ, ponderis, realiter di&longs;tinguuntur? </s> <s id="N1174A"><!-- NEW -->quia &longs;cilicet vnus <lb/>non e&longs;t alius inquies; & quare vnus non e&longs;t alius? </s> <s id="N11750">quia vnus e&longs;t hic & <lb/>alius non e&longs;t hic, vnum tango, & alium non tango, vnus e&longs;t meus, & <lb/>alius non e&longs;t meus, &c. </s> <s id="N11757">vides prædicata contradictoria, quæ cum eidem <lb/>&longs;imul ine&longs;&longs;e non po&longs;&longs;int per Ax. 1. diuer&longs;is, & di&longs;tinctis ine&longs;&longs;e nece&longs;&longs;e <lb/>e&longs;t. </s> </p> <p id="N1175E" type="main"> <s id="N11760"><!-- NEW -->Diceret fortè aliquis hominem reproductum in duobus locis e&longs;&longs;e po&longs;­<lb/>&longs;e, & dum Romæ e&longs;t à &longs;e ip&longs;o Lugduni exi&longs;tente &longs;eiunctum e&longs;&longs;e; hoc <lb/>ip&longs;um aliàs examinabimus, dum con&longs;tet modò id totum, &longs;i fiat, mira­<lb/>culo tribuendum e&longs;&longs;e, cum tamen res phy&longs;icas citra miraculum con&longs;ide­<lb/>remus. </s> </p> <p id="N1176C" type="main"> <s id="N1176E"><emph type="center"/><emph type="italics"/>Axioma III.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1177A" type="main"> <s id="N1177C"><emph type="italics"/>Vt dicatur aliquid exi&longs;tere, vel debet &longs;en&longs;u percipi, vel aliqua ratione <lb/>probari.<emph.end type="italics"/></s> <s id="N11785"><!-- NEW --> Qui enim a&longs;&longs;erit rem aliquam po&longs;itiuam exi&longs;tere, certè po&longs;i­<lb/>tiuo argumento demon&longs;trare debet quod &longs;it; </s> <s id="N1178B"><!-- NEW -->illud porrò argumentum <lb/>duci pote&longs;t vel ab experimento certo; </s> <s id="N11791"><!-- NEW -->&longs;ic probo exi&longs;tere rem aliquam, <lb/>quam video; vel ab aliqua ratione; </s> <s id="N11797"><!-- NEW -->&longs;ic ex eo quòd cau&longs;a &longs;it nece&longs;&longs;aria <lb/>applicata &longs;ubiecto apto, probo effectum ip&longs;um produci; </s> <s id="N1179D"><!-- NEW -->vel eo quòd &longs;it <lb/>effectus probo cau&longs;am e&longs;&longs;e vel ex nece&longs;&longs;itate, quâ aliquid e&longs;t nece&longs;&longs;a­<lb/>rium ad aliquem finem à natura in&longs;titutum, quo natura ip&longs;a &longs;ine ab&longs;ur-<pb pagenum="6" xlink:href="026/01/038.jpg"/>do, vel graui&longs;&longs;imo incommodo carere non pote&longs;t, probo illud ip&longs;um <lb/>e&longs;&longs;e; </s> <s id="N117AC"><!-- NEW -->vel demùm ex aliqua reuelatione certa in rebus fidei; </s> <s id="N117B0"><!-- NEW -->igitur hoc <lb/>Axioma certum e&longs;t phy&longs;icè; </s> <s id="N117B6"><!-- NEW -->quod ni&longs;i recipiatur à Philo&longs;ophis; </s> <s id="N117BA"><!-- NEW -->cuique <lb/>licebit impunè mentiri; &longs;i enim dicam extra mundi huius fines e&longs;&longs;e <lb/>alios orbes, intra tuum mu&longs;æum, in quo &longs;olus fortè degis, e&longs;&longs;e quin­<lb/>quaginta homines, e&longs;&longs;e mille Soles, & totidem Lunas in cœlo, &c. </s> <s id="N117C4"><!-- NEW --><lb/>numquid &longs;tatim oppones Axioma i&longs;tud, <emph type="italics"/>qua ratio, qua experientia, qua <lb/>nece&longs;&longs;itas, qua reuelatio?<emph.end type="italics"/> Quæ&longs;tio facti e&longs;t, producendi &longs;unt te&longs;tes: huc <lb/>reuoca principium illud commune. </s> </p> <p id="N117D3" type="main"> <s id="N117D5">1. <emph type="italics"/>Non &longs;unt multiplicanda entia &longs;ine nece&longs;&longs;itate, quod certè non valet ni&longs;i <lb/>addas, vel &longs;ine ratione, vel &longs;ine experientia.<emph.end type="italics"/></s> </p> <p id="N117DF" type="main"> <s id="N117E1">2. <emph type="italics"/>Qui a&longs;&longs;erit aliquid po&longs;itiuè, debet argumento po&longs;itiuo probare.<emph.end type="italics"/></s> </p> <p id="N117E9" type="main"> <s id="N117EB"><emph type="center"/><emph type="italics"/>Axioma IV.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N117F7" type="main"> <s id="N117F9"><emph type="italics"/>Quidquid exi&longs;tit phy&longs;icè extra &longs;uas cau&longs;as ab omni alio &longs;eparatum, de­<lb/>terminatum e&longs;t.<emph.end type="italics"/></s> </p> <p id="N11802" type="main"> <s id="N11804"><!-- NEW -->Hoc Axioma explicatione modicâ indiget: </s> <s id="N11808"><!-- NEW -->Determinatum illud <lb/>apello, quod illud ip&longs;um e&longs;t, quod e&longs;t, & nihil aliud; </s> <s id="N1180E"><!-- NEW -->quod e&longs;t hoc, id <lb/>e&longs;t ab omni alio di&longs;tinctum; </s> <s id="N11814"><!-- NEW -->atqui quidquid productum e&longs;t, &longs;ingulare <lb/>e&longs;t, id e&longs;t, e&longs;t hoc; </s> <s id="N1181A"><!-- NEW -->&longs;i enim producitur, alicubi producitur, & ali­<lb/>quando, ergo dici pote&longs;t, e&longs;t hîc, e&longs;t nunc; igitur determinatum e&longs;t. </s> <s id="N11820"><!-- NEW --><lb/>Aliquis fortè &longs;tatim opponet mihi partes indeterminatas quantitatis: </s> <s id="N11825"><!-- NEW -->&longs;ed <lb/>pro&longs;ectò nulla pars actu e&longs;t quæ non &longs;it hæc, & non alia; </s> <s id="N1182B"><!-- NEW -->igitur quæ <lb/>non &longs;it determinata, de quo aliàs; quidquid &longs;it, &longs;altem partes illæ fa­<lb/>ciunt aliquod totum quod e&longs;t determinatum, quod mihi &longs;atis e&longs;t modò <lb/>ad veritatem huius Axiomatis. <!-- KEEP S--></s> <s id="N11836"><!-- NEW -->Dices aliquid po&longs;&longs;e e&longs;&longs;e nullibi; </s> <s id="N1183A"><!-- NEW -->has <lb/>nugas refutabimus in Metaphy&longs;ica, quæ in mentem &longs;apientis viri ca­<lb/>dere non po&longs;&longs;unt; nunc &longs;altem con&longs;tat id naturali modo fieri non <lb/>po&longs;&longs;e. </s> </p> <p id="N11844" type="main"> <s id="N11846"><emph type="center"/><emph type="italics"/>Axioma V.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11852" type="main"> <s id="N11854"><emph type="italics"/>Quod vnum e&longs;t, determinatum e&longs;t.<emph.end type="italics"/></s> <s id="N1185B"><!-- NEW --> Quia quod vnum e&longs;t, e&longs;t hoc, & <lb/>nihil aliud; </s> <s id="N11861"><!-- NEW -->nihil enim aliud e&longs;t vnum, ni&longs;i indiui&longs;um in &longs;e, & diui­<lb/>&longs;um à quolibet alio: </s> <s id="N11867"><!-- NEW -->quippè indifferentia, vel indeterminatio ibi tan­<lb/>tum e&longs;t, vbi &longs;unt plura; &longs;i enim tantum vnum e&longs;t, certè non datur op­<lb/>tio, &longs;i aliqua cau&longs;a e&longs;t indifferens ad effectum A & B, id e&longs;t &longs;i non e&longs;t, <lb/>cur vnum potius quàm alium producat? </s> <s id="N11871"><!-- NEW -->plures e&longs;&longs;e nece&longs;&longs;e e&longs;t; &longs;i enim <lb/>tantùm vnus e&longs;t, certè indifferens non e&longs;t. </s> </p> <p id="N11877" type="main"> <s id="N11879"><emph type="center"/><emph type="italics"/>Axioma VI.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11885" type="main"> <s id="N11887"><emph type="italics"/>Quidquid e&longs;t, fru&longs;trà non e&longs;t.<emph.end type="italics"/></s> <s id="N1188E"><!-- NEW --> Quidquid e&longs;t, id e&longs;t exi&longs;tit naturaliter <lb/>&longs;cilicet, & citra miraculum, fru&longs;trà non e&longs;t, id e&longs;t propter aliquem fi­<lb/>nem e&longs;t ab ip&longs;a natura in&longs;titutum; </s> <s id="N11896"><!-- NEW -->finem autem rei ex ip&longs;o v&longs;u cogno­<lb/>&longs;cimus; </s> <s id="N1189C"><!-- NEW -->v&longs;um verò ip&longs;o ferè &longs;en&longs;u: </s> <s id="N118A0"><!-- NEW -->quod vt breui inductione confirme­<lb/>mus, quidquid exi&longs;tit vel e&longs;t &longs;ub&longs;tantia, vel accidens; </s> <s id="N118A6"><!-- NEW -->&longs;i &longs;ub&longs;tantia, vel <lb/>incorporea, vel corporea; </s> <s id="N118AC"><!-- NEW -->&longs;i incorporea, vel e&longs;t Deus, vel Angelus, vel <pb pagenum="7" xlink:href="026/01/039.jpg"/>Anima rationalis; </s> <s id="N118B5"><!-- NEW -->atqui nihil horum fru&longs;trà e&longs;t, vt con&longs;tat; </s> <s id="N118B9"><!-- NEW -->&longs;i corporea, <lb/>vel e&longs;t corpus, vel forma; </s> <s id="N118BF"><!-- NEW -->&longs;i corpus, vel elementum, vel mixtum; </s> <s id="N118C3"><!-- NEW --><lb/>vtrumque &longs;uum finem habet, & con&longs;tantem v&longs;um; </s> <s id="N118C8"><!-- NEW -->&longs;i forma quamdiu <lb/>e&longs;t principium actionum compo&longs;iti fru&longs;trà non e&longs;t; </s> <s id="N118CE"><!-- NEW -->quippe ad cum finem <lb/>e&longs;t in&longs;tituta; </s> <s id="N118D4"><!-- NEW -->hinc optima ratio ducitur, cur forma materialis &longs;eparata <lb/>exi&longs;tere non po&longs;&longs;it citra miraculum, quia &longs;cilicet fru&longs;trà e&longs;&longs;et; </s> <s id="N118DA"><!-- NEW -->cum enim <lb/>non po&longs;&longs;it agere ni&longs;i in &longs;ubiecto, &longs;i &longs;ubiectum non e&longs;t, fru&longs;trà e&longs;t; </s> <s id="N118E0"><!-- NEW -->at verò <lb/>anima rationalis, quæ aliquas actiones in organicas habet, fru&longs;trà non <lb/>e&longs;t etiam &longs;eparata, igitur immortalis e&longs;t: </s> <s id="N118E8"><!-- NEW -->vtramque rationem &longs;uo loco fu­<lb/>sè demon&longs;trabimus; </s> <s id="N118EE"><!-- NEW -->&longs;i verò accidens e&longs;t, haud dubiè alteri ine&longs;&longs;e debet <lb/>propter &longs;uum finem intrin&longs;ecum, quem alibi effectum formalem &longs;ecun­<lb/>darium appellamus; </s> <s id="N118F6"><!-- NEW -->quem &longs;cilicet præ&longs;tat in &longs;uo &longs;ubiecto, cui certè &longs;i ni­<lb/>hil præ&longs;taret, in eo fru&longs;trà e&longs;&longs;et; </s> <s id="N118FC"><!-- NEW -->&longs;ic caloris effectus &longs;ecundarius e&longs;t rare­<lb/>factio, vel re&longs;olutio partium &longs;ui &longs;ubiecti, vel aliquid aliud; impetus, <lb/>motus &c. </s> <s id="N11904"><!-- NEW -->Igitur tunc effet fru&longs;trà accidens, cum &longs;uo illo effectu careret; </s> <s id="N11908"><!-- NEW --><lb/>hinc rationem contrarietatis aliquando petemus, certi&longs;&longs;imam quidem, <lb/>licet nouam, & inde clari&longs;&longs;imè con&longs;tabit, cur, & quomodo vnum contra­<lb/>rium ab alio de&longs;trui dicatur; </s> <s id="N11911"><!-- NEW -->&longs;ed non e&longs;t huius loci: cùm verò audis fi­<lb/>nem: </s> <s id="N11917"><!-- NEW -->ne quæ&longs;o cogites aliquid morale, nec enim illum finem intelligo, ad <lb/>quem ab agente rationabili de&longs;tinatur: &longs;ed eum dumtaxat, ad quem na­<lb/>tura ip&longs;a, vel e&longs;&longs;entia rei &longs;pectat, &longs;ed de his &longs;atis. </s> </p> <p id="N1191F" type="main"> <s id="N11921">Huc reuoca Principium illud, <emph type="italics"/>Deus & Natura nihil faciunt fru&longs;trà,<emph.end type="italics"/><lb/>id e&longs;t quod &longs;uo fine careat intrin&longs;eco. </s> </p> <p id="N1192B" type="main"> <s id="N1192D"><!-- NEW -->Dices fortè, multa videri e&longs;&longs;e fru&longs;trà, quæ tamen exi&longs;tunt; ad quid <lb/>enim vel tanta aquarum copia, vel tantus &longs;tellarum numerus, vel tot are­<lb/>næ puncta? </s> <s id="N11935">tot fluitantes atomi? </s> <s id="N11938">tot in&longs;ecta? </s> <s id="N1193B">& vermiculi: </s> <s id="N1193E"><!-- NEW -->Re&longs;pondeo <lb/>quamlibet &longs;tellam, quodlibet in&longs;ectum, &longs;eu vermiculum &longs;uis pollere pro­<lb/>prietatibus; </s> <s id="N11946"><!-- NEW -->igitur fru&longs;trà non e&longs;t, & quodlibet punctum, quamlibet ato­<lb/>mum, & quamlibet guttulam aquæ e&longs;&longs;e partem huius vniuer&longs;itatis: </s> <s id="N1194C"><!-- NEW -->quod <lb/>enim dices de vna, dicam de omnibus; </s> <s id="N11952"><!-- NEW -->equidem pauciores e&longs;&longs;e po&longs;&longs;ent; <lb/>attamen nulla e&longs;t fru&longs;trà, cum quælibet &longs;imul cum aliis totum hoc com­<lb/>ponat. </s> </p> <p id="N1195A" type="main"> <s id="N1195C"><emph type="center"/><emph type="italics"/>Axioma VII.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11968" type="main"> <s id="N1196A"><emph type="italics"/>Tunc ponenda e&longs;t forma distincta &longs;ub&longs;tantialis vel accidentalis, dum e&longs;t ali­<lb/>qua proprietas &longs;en&longs;ibilis, quæ non pote&longs;t tribui ip&longs;i materiæ,<emph.end type="italics"/> hîc res tantùm <lb/>naturales con&longs;idero, nec &longs;uper naturales attingo, quæ &longs;uas regulas diui­<lb/>næ fidei debent, non &longs;en&longs;ibus. </s> </p> <p id="N11978" type="main"> <s id="N1197A"><!-- NEW -->Hoc Axioma omninò certum e&longs;t, & per Ax. 3. confirmatur, vt enim <lb/>dicas aliquid di&longs;tinctum ab omni alio exi&longs;tere, vel debet id &longs;en&longs;u percipi, <lb/>vel aliqua ratione probari quod &longs;it; </s> <s id="N11982"><!-- NEW -->atqui formam &longs;ub&longs;tantialem &longs;en&longs;u <lb/>non percipis immediatè; </s> <s id="N11988"><!-- NEW -->igitur aliquem eius effectum &longs;en&longs;ibilem vel me­<lb/>diatè, vel immediatè; </s> <s id="N1198E"><!-- NEW -->qui certè &longs;i tribui po&longs;&longs;it materiæ, haud dubiè per il­<lb/>lum formam non probabis, ni&longs;i formæ ip&longs;ius e&longs;&longs;e antè demon&longs;tres; </s> <s id="N11994"><!-- NEW -->&longs;i ve­<lb/>to e&longs;t forma accidentalis, quam &longs;en&longs;u percipis; certè id tantùm accidit ex <pb pagenum="8" xlink:href="026/01/040.jpg"/>aliqua affectione, quâ &longs;en&longs;um ip&longs;um afficit hæc forma, igitur ex effectu il­<lb/>lo illam percipis, quod clarum e&longs;t. </s> </p> <p id="N119A1" type="main"> <s id="N119A3"><!-- NEW -->Huc reuoca vulgare illud principium, <emph type="italics"/>Frustrà fit per plura, quod po­<lb/>test fieri per pauciora,<emph.end type="italics"/> quod ad Tertium etiam reuocatur; </s> <s id="N119AF"><!-- NEW -->quod ita in­<lb/>telligi non debet, vt &longs;ine gutta aquæ Oceanus, &longs;ine &longs;tella cœlum, &longs;ine gra­<lb/>nulo arenæ terra, &longs;ine altero oculorum homo &longs;tare non po&longs;&longs;int; </s> <s id="N119B7"><!-- NEW -->quæ <lb/>omnia fal&longs;i&longs;&longs;ima e&longs;&longs;e con&longs;tat; &longs;ed tantùm quod illud dicatur exi&longs;tere &longs;iue <lb/>&longs;it &longs;ub&longs;tantia, &longs;iue accidens, quod vel experientia certa euincit, vel nece&longs;­<lb/>&longs;itas, vel ratio, vel diuina fides (immò & humana in rebus humanis, non <lb/>tamen in &longs;cientiis.) </s> </p> <p id="N119C3" type="main"> <s id="N119C5">Igitur nunquam claudicat hic equus Okami, vt vulgò dicitur, &longs;i hoc <lb/>fræno regatur, & præ&longs;cripto ambulet pa&longs;&longs;u. </s> </p> <p id="N119CA" type="main"> <s id="N119CC"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N119D8" type="main"> <s id="N119DA">Ob&longs;eruabis &longs;eptem præmi&longs;&longs;a Axiomata, licet metaphy&longs;ica &longs;altem ali­<lb/>qua ex parte e&longs;&longs;e videantur, ita pertinere ad Phy&longs;icam, vt plurimæ phy­<lb/>&longs;icæ affectiones &longs;ine illis explicari, & demon&longs;trari non po&longs;&longs;int. </s> </p> <p id="N119E1" type="main"> <s id="N119E3">Primum certum e&longs;t etiam certitudine metaphy&longs;ica, &longs;eu geometrica. </s> <s id="N119E6"><lb/>Secundum, Quartum, & Quintum per Primum demon&longs;trari po&longs;&longs;unt. </s> <s id="N119EA"><!-- NEW --><lb/>Tertium e&longs;t veluti communis po&longs;itio, &longs;eu commune po&longs;tulatum, in quo <lb/>docti omnes conunciunt; </s> <s id="N119F1"><!-- NEW -->quippe nihil &longs;ine ratione dici debet à philo&longs;o­<lb/>pho; </s> <s id="N119F7"><!-- NEW -->Sextum & Septimum probari po&longs;&longs;unt per Tertium; &longs;ed iam ad <lb/>alia, quæ propiùs ad phy&longs;icam accedunt, veniamus. </s> </p> <p id="N119FD" type="main"> <s id="N119FF"><emph type="center"/><emph type="italics"/>Axioma VIII.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11A0B" type="main"> <s id="N11A0D"><emph type="italics"/>Quidquid primò e&longs;t, & antè non erat, habet cau&longs;am di&longs;tinctam.<emph.end type="italics"/></s> <s id="N11A14"><!-- NEW --> Id e&longs;t quid­<lb/>quid incipit e&longs;&longs;e ab alio e&longs;t; </s> <s id="N11A1A"><!-- NEW -->quippe à &longs;e e&longs;&longs;e non pote&longs;t; </s> <s id="N11A1E"><!-- NEW -->nihil enim à &longs;e <lb/>ip&longs;o dependere pote&longs;t &longs;eu produci; </s> <s id="N11A24"><!-- NEW -->quia quod à &longs;e e&longs;t, nece&longs;&longs;ariò e&longs;t, <lb/>quod verò nece&longs;&longs;ariò e&longs;t, non e&longs;&longs;e non pote&longs;t, alioquin priùs e&longs;&longs;et, & <lb/>po&longs;terius, priùs vt cau&longs;a, po&longs;teriùs vt effectus: </s> <s id="N11A2C"><!-- NEW -->præterea quidquid produci­<lb/>tur aliquando producitur, & alicubi, vt certi&longs;&longs;imum e&longs;t; </s> <s id="N11A32"><!-- NEW -->&longs;ed quia hoc ali­<lb/>qui negant, contendo tantùm in hoc rerum ordine, & naturaliter lo­<lb/>quendo, quidquid producitur alicubi produci, & aliquando, quod nemo <lb/>negabit; </s> <s id="N11A3C"><!-- NEW -->Igitur &longs;i aliquid &longs;e producit; cur hîc potiùs quam illîc? </s> <s id="N11A40">cur <lb/>nunc potius quam antè? </s> <s id="N11A45">cum enim antè nullibi e&longs;&longs;et, cur de&longs;init non <lb/>e&longs;&longs;e hîc & non illîc, nunc & non antè? </s> <s id="N11A4A"><!-- NEW -->hinc quod à &longs;e e&longs;t, vbique, & <lb/>&longs;emper e&longs;t, &longs;ed ne quis mihi litem intendat, licet hoc Axioma certitudi­<lb/>nem geometricam habeat; </s> <s id="N11A52"><!-- NEW -->&longs;ufficit modò habere phy&longs;icam, quod ex om­<lb/>nibus hypothe&longs;ibus demon&longs;tratur; </s> <s id="N11A58"><!-- NEW -->&longs;i enim aliquid de nouo produci­<lb/>tur, quod certum e&longs;t, ab alio produci video: </s> <s id="N11A5E"><!-- NEW -->calor ab igne mediatè <lb/>vel immediatè, impetus à potentia motrice, vel ab alio impetu: </s> <s id="N11A64"><!-- NEW -->cuncta <lb/>hæc &longs;i reuera producuntur de quo alibi, ab alio produci con&longs;tat; </s> <s id="N11A6A"><!-- NEW -->in Me­<lb/>taphy&longs;ica hoc ip&longs;um geometricè demon&longs;trabimus; </s> <s id="N11A70"><!-- NEW -->cum enim agere &longs;up­<lb/>ponat e&longs;&longs;e; </s> <s id="N11A76"><!-- NEW -->quippe omnis actio alicuius agentis e&longs;t; </s> <s id="N11A7A"><!-- NEW -->& cum agere termi­<lb/>netur ad effectum, nam fieri e&longs;t alicuius fieri; certè agens, & terminus, <lb/>cau&longs;a, & effectus di&longs;tinguuntur, igitur. <emph type="italics"/>Quidquid primo e&longs;t, &c.<emph.end type="italics"/></s> </p> <pb pagenum="9" xlink:href="026/01/041.jpg"/> <p id="N11A8B" type="main"> <s id="N11A8D"><emph type="center"/><emph type="italics"/>Axioma IX.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11A99" type="main"> <s id="N11A9B"><emph type="italics"/>Cau&longs;a debet exi&longs;tere vt immediatè agat.<emph.end type="italics"/></s> <s id="N11AA2"> Hoc certum e&longs;t; </s> <s id="N11AA5"><!-- NEW -->quia agere <lb/>&longs;upponit e&longs;&longs;e; </s> <s id="N11AAB"><!-- NEW -->quippe agere e&longs;t perfectio realis actu exi&longs;tens; igitur ali­<lb/>cuius actu exi&longs;tentis; igitur certum e&longs;t etiam Geometricè, de quo in <lb/>Metaph. <!-- KEEP S--></s> <s id="N11AB4">Iam vero &longs;ufficiat certum e&longs;&longs;e phi&longs;icè, vt con&longs;tat ex omnibus <lb/>hypoth. </s> <s id="N11AB9">phy&longs;icis; </s> <s id="N11ABC"><!-- NEW -->nihil enim videmus agere, ni&longs;i quod e&longs;t; </s> <s id="N11AC0"><!-- NEW -->&longs;i enim age­<lb/>ret quod non e&longs;t; cur potius hîc, & nunc quam alibi, & aliàs? </s> <s id="N11AC6">cur in <lb/>hoc &longs;ubiecto potius quàm in alio? </s> </p> <p id="N11ACB" type="main"> <s id="N11ACD"><!-- NEW -->Dices, finis qui non e&longs;t influit; </s> <s id="N11AD1"><!-- NEW -->igitur agit; </s> <s id="N11AD5"><!-- NEW -->Re&longs;pondeo finem non <lb/>agere, nec influere ni&longs;i obiectiuè; </s> <s id="N11ADB"><!-- NEW -->atqui quod non exi&longs;tit actu, id e&longs;t in <lb/>&longs;tatu entatiuo, & reali, pote&longs;t e&longs;&longs;e in &longs;tatu obiectiuo; </s> <s id="N11AE1"><!-- NEW -->id e&longs;t quod non <lb/>habet actum rei, pote&longs;t habere actum obiecti, id e&longs;t e&longs;&longs;e cognitum, & <lb/>volitum, de quo aliàs; porrò hîc tantùm intelligimus cau&longs;am efficien­<lb/>tem, &c. </s> </p> <p id="N11AEB" type="main"> <s id="N11AED"><!-- NEW -->Dices, cau&longs;a principalis pulli exclu&longs;i pote&longs;t non e&longs;&longs;e; </s> <s id="N11AF1"><!-- NEW -->hæc omnia di­<lb/>&longs;cutiemus &longs;uo loco cum de generatione animalium; </s> <s id="N11AF7"><!-- NEW -->&longs;ufficiat dixi&longs;&longs;e non <lb/>e&longs;&longs;e cau&longs;am immediatam, de qua hîc tantum loquimur; idem re&longs;pon&longs;um <lb/>e&longs;to de rana vaga. </s> </p> <p id="N11AFF" type="main"> <s id="N11B01"><emph type="center"/><emph type="italics"/>Axioma X.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11B0D" type="main"> <s id="N11B0F"><emph type="italics"/>Cau&longs;a debet e&longs;&longs;e applicata vt immediatè agat.<emph.end type="italics"/></s> <s id="N11B16"><!-- NEW --> Cur enim potiùs hîc <lb/>quam illîc; in hoc &longs;ubiecto potiùs, quam in alio, in hac di&longs;tantia potiùs, <lb/>quam in alia? </s> <s id="N11B1E"><!-- NEW -->quidquid &longs;it, certum e&longs;t phy&longs;icè; nec enim ignis, qui e&longs;t <lb/>Romæ, calefacit Lugduni. </s> </p> <p id="N11B24" type="main"> <s id="N11B26"><!-- NEW -->Dices dari fortè actionem in di&longs;tans; </s> <s id="N11B2A"><!-- NEW -->Re&longs;pondeo negando, quod de­<lb/>mon&longs;trabimus in Metaph. præterea, licet daretur in productione quali­<lb/>tatum occultarum, & &longs;impathicorum quorundam effectuum, quos exa­<lb/>minabimus &longs;uo loco; </s> <s id="N11B34"><!-- NEW -->nemo tamen dubitat quin productio caloris, lu­<lb/>minis, impetus; de quibus hic tantùm agimus, debeat e&longs;&longs;e ab applicata <lb/>cau&longs;a. </s> </p> <p id="N11B3C" type="main"> <s id="N11B3E"><!-- NEW -->Dices impetum produci in extremitate perticæ, quæ non e&longs;t applica­<lb/>ta, vel in globo tudiculario etiam non applicato; calorem & lucem <lb/>produci à Sole in terra non applicata. </s> <s id="N11B46"><!-- NEW -->Re&longs;pondeo, e&longs;&longs;e applicationem <lb/>mediatam; nam &longs;i reuera hæ qualitates producuntur continuata propa­<lb/>gatione, diffunduntur per medium, in quo non e&longs;t difficultas. </s> </p> <p id="N11B4E" type="main"> <s id="N11B50"><!-- NEW -->Dices etiam partes interiores cau&longs;æ v. <!-- REMOVE S-->g. <!-- REMOVE S-->Solis agunt, &longs;ed non agunt <lb/>per totum medium; alioquin agerent in alias partes Solis, à quibus <lb/>obteguntur. </s> <s id="N11B5C"><!-- NEW -->Re&longs;pondeo, diffu&longs;ionem vel propagationem actionis in­<lb/>choari tantum ab ipsâ &longs;uperficie Solis; </s> <s id="N11B62"><!-- NEW -->quippe omnes partes agunt <lb/>actione communi, de quo infrà; atqui actio communis à communi me­<lb/>dio incipit. </s> </p> <p id="N11B6A" type="main"> <s id="N11B6C">Dices ignem produci in parte medij remota interrupta propagatio­<lb/>ne, vt con&longs;tat, &longs;i vitro per refractionem, vel &longs;peculo per reflectionem <lb/>radios Solares colligas. </s> </p> <p id="N11B73" type="main"> <s id="N11B75"><!-- NEW -->Re&longs;pondeo, ignem quidem accendi in data di&longs;tantia; </s> <s id="N11B79"><!-- NEW -->at non &longs;ine <pb pagenum="10" xlink:href="026/01/042.jpg"/>aliqua applicatione, &longs;altem virtutis, in quo non e&longs;t difficultas; </s> <s id="N11B82"><!-- NEW -->quomo­<lb/>do vero ignis accendatur, & quid &longs;it ignem accendi, explicabimus &longs;uo <lb/>loco; quidquid &longs;it, certum e&longs;t ad productionem impetus requiri ali­<lb/>quam applicationem, vt patet etiam in magnete. </s> </p> <p id="N11B8F" type="main"> <s id="N11B91"><emph type="center"/><emph type="italics"/>Axioma XI.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11B9D" type="main"> <s id="N11B9F"><emph type="italics"/>Si cau&longs;a vniuoca applicata, & non impedita est &longs;ufficiens ad productionem <lb/>effectus, non e&longs;t ponenda alia &longs;cilicet æquiuoca.<emph.end type="italics"/></s> <s id="N11BA8"><!-- NEW --> Non dico omnem cau&longs;am <lb/>e&longs;&longs;e vniuocam, &longs;ed tantùm vniuocam &longs;ufficientem, & applicatam e&longs;&longs;e <lb/>cau&longs;am, v. <!-- REMOVE S-->g. <!-- REMOVE S-->calor e&longs;t cau&longs;a &longs;ufficiens caloris, vt con&longs;tat in aqua calida; </s> <s id="N11BB4"><!-- NEW --><lb/>igitur &longs;i calor e&longs;t applicatus &longs;ubiecto, in quo producitur calor non &longs;upe­<lb/>rans vires caloris applicati; </s> <s id="N11BBB"><!-- NEW -->dicendum e&longs;t calorem illum ab hoc produ­<lb/>ci; </s> <s id="N11BC1"><!-- NEW -->cum calor &longs;it cau&longs;a nece&longs;&longs;aria; </s> <s id="N11BC5"><!-- NEW -->igitur &longs;i &longs;it applicatus &longs;ubjecto apto, <lb/>nece&longs;&longs;ariò agit; </s> <s id="N11BCB"><!-- NEW -->igitur quantum pote&longs;t; igitur effectus non e&longs;t tribuen­<lb/>dus alteri cau&longs;æ, quam &longs;ufficientem e&longs;&longs;e ignoramus. </s> </p> <p id="N11BD1" type="main"> <s id="N11BD3"><!-- NEW -->Ad hoc Axioma aliud reuoca. <emph type="italics"/>Si ex applicatione alicuius &longs;equitur &longs;em­<lb/>per effectus aliquis, illud ip&longs;um cau&longs;a dici debet huius effectus; </s> <s id="N11BDC"><!-- NEW -->licet aliud &longs;it <lb/>coniunctum, ex quo &longs;eor&longs;im &longs;umpto applicato non &longs;equitur effectus<emph.end type="italics"/>; </s> <s id="N11BE5"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->ex <lb/>applicatione aquæ calidæ &longs;equitur productio caloris; </s> <s id="N11BEF"><!-- NEW -->ex applicatione &longs;o­<lb/>lius aquæ non &longs;equitur; </s> <s id="N11BF5"><!-- NEW -->igitur dicendum e&longs;t calorem hunc produci ab <lb/>ip&longs;o calore, qui aquæ ine&longs;t, non verò ab ip&longs;a aquæ &longs;ub&longs;tantia; idem dico <lb/>de ferro frigido, &c. </s> </p> <p id="N11BFD" type="main"> <s id="N11BFF"><!-- NEW -->Dices non e&longs;&longs;e certum calorem produci; Re&longs;pondeo, negando; </s> <s id="N11C03"><!-- NEW -->&longs;ed, <lb/>quidquid &longs;it, loquor tantùm hypotheticè; dixi enim &longs;i producatur, à <lb/>calore aquæ inhærente producitur. </s> </p> <p id="N11C0B" type="main"> <s id="N11C0D">Dices produci po&longs;&longs;e ab aliqua cau&longs;a ignota po&longs;ita dumtaxat tali, vel <lb/>tali conditione. </s> <s id="N11C12"><!-- NEW -->Re&longs;pondeo, hoc reuera geometricè non probari, &longs;ed <lb/>tantùm phy&longs;icè; </s> <s id="N11C18"><!-- NEW -->quidquid &longs;it, voco cau&longs;am id, ex cuius applicatione <lb/>&longs;equitur &longs;emper effectus, & nunquam aliàs; </s> <s id="N11C1E"><!-- NEW -->nam phy&longs;icè loquendo, &longs;iue <lb/>&longs;it alia cau&longs;a, &longs;iue non, eodem modo &longs;e habet, ac &longs;i e&longs;&longs;et cau&longs;a; quippe <lb/>certum e&longs;t phy&longs;icè ignem calefacere, Solem illuminare, quod &longs;atis e&longs;t. </s> </p> <p id="N11C26" type="main"> <s id="N11C28"><emph type="center"/><emph type="italics"/>Axioma XII.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11C34" type="main"> <s id="N11C36"><emph type="italics"/>Cau&longs;a nece&longs;&longs;aria &longs;ubiecto apto applicata, & non impedita &longs;emper agit, & <lb/>quantum pote&longs;t.<emph.end type="italics"/></s> <s id="N11C3F"><!-- NEW --> Hoc Axioma duas partes habet; </s> <s id="N11C43"><!-- NEW -->prima certa e&longs;t per hy­<lb/>poth. 8. & per definitionem cau&longs;æ nece&longs;&longs;ariæ, quæ in hoc differt à libe­<lb/>râ: Secunda pars probatur; </s> <s id="N11C4B"><!-- NEW -->quia &longs;i partem effectus omitteret, quam ta­<lb/>men ponere po&longs;&longs;et; haud dubiè non e&longs;&longs;et cau&longs;a nece&longs;&longs;aria contra hypoth. </s> <s id="N11C51"><!-- NEW --><lb/>nam &longs;i vnam partem effectus omittat; cur vnam potiùs quam aliam? </s> <s id="N11C56"><lb/>cur non duas? </s> <s id="N11C5A">cur non omnes? </s> <s id="N11C5D">denique video cau&longs;am eandem eidem <lb/>&longs;ubiecto eodem modo applicatam, eundem &longs;emper effectum producere <lb/>per Hyp. <!-- REMOVE S-->8. </s> </p> <p id="N11C66" type="main"> <s id="N11C68"><emph type="center"/><emph type="italics"/>Axioma XIII<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11C74" type="main"> <s id="N11C76"><emph type="italics"/>Exten&longs;io cau&longs;a non intendit effectum ad intra.<emph.end type="italics"/></s> <s id="N11C7D"><!-- NEW --> Quælibet pars maioris <lb/>ignis non habet calorem inten&longs;iorem, quàm quælibet pars minoris; idem <pb pagenum="11" xlink:href="026/01/043.jpg"/>dico de grauitate plumbi, &c. </s> <s id="N11C88">nec enim libra plumbi coniuncta cum <lb/>alia habet diuer&longs;am grauitatem ab eâ, quam habet &longs;eparata. </s> </p> <p id="N11C8D" type="main"> <s id="N11C8F">Dixi ad intra; </s> <s id="N11C92"><!-- NEW -->quia ad extra multum iuuat exten&longs;io; </s> <s id="N11C96"><!-- NEW -->&longs;ic maior ignis <lb/>longiùs diffundit &longs;uum calorem; </s> <s id="N11C9C"><!-- NEW -->corpus grauiùs cadens majorem ictum <lb/>infligit; Ad hoc Axioma reuocatur i&longs;tud. </s> </p> <p id="N11CA2" type="main"> <s id="N11CA4"><!-- NEW -->1. <emph type="italics"/>Omnes partes eiu&longs;dem cau&longs;æ agunt ad extra actione communi,<emph.end type="italics"/> iuxta <lb/>eum modum quo illam explicabimus in Metaph. nec punctum Solis &longs;e­<lb/>paratum ad eandem di&longs;tantiam &longs;uam lucem, caloremque &longs;uum diffunde­<lb/>ret; </s> <s id="N11CB4"><!-- NEW -->ad quam diffundit coniunctum cum aliis; </s> <s id="N11CB8"><!-- NEW -->idem dico de igne maiori, <lb/>& minori; de quibus omnibus &longs;uo loco. </s> <s id="N11CBE">Huc etiam reuoca dicta illa <lb/>communia. </s> </p> <p id="N11CC3" type="main"> <s id="N11CC5">2. <emph type="italics"/>Plures partes cau&longs;a plures partes effectus producunt, & vici&longs;&longs;im.<emph.end type="italics"/></s> </p> <p id="N11CCD" type="main"> <s id="N11CCF">3. <emph type="italics"/>Maior, & perfectior cau&longs;a maiorem effectum producit, & perfectiorem, <lb/>& vici&longs;&longs;im.<emph.end type="italics"/></s> </p> <p id="N11CD9" type="main"> <s id="N11CDB"><!-- NEW -->4. <emph type="italics"/>Perfectior effectus, vel imperfectior arguit cau&longs;am perfectiorem, vel im­<lb/>perfectiorem, &longs;uppo&longs;itâ eâdem applicatione; </s> <s id="N11CE4"><!-- NEW -->&longs;i enim maior e&longs;t applicatio &longs;ine <lb/>ratione loci, &longs;iue ratione temporis; haud dubiè maior erit effectus, vt con&longs;tat.<emph.end type="italics"/></s> </p> <p id="N11CEC" type="main"> <s id="N11CEE"><emph type="center"/><emph type="italics"/>Axioma XIV.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11CFA" type="main"> <s id="N11CFC"><emph type="italics"/>Quidquid de&longs;truitur non e&longs;t à &longs;e.<emph.end type="italics"/></s> <s id="N11D03"><!-- NEW --> Hoc Axioma geometricum e&longs;t; </s> <s id="N11D07"><!-- NEW -->Quod <lb/>enim e&longs;t à &longs;e, nece&longs;&longs;ariò e&longs;t; </s> <s id="N11D0D"><!-- NEW -->cùm à libertate &longs;eu voluntate alterius non <lb/>pendeat; </s> <s id="N11D13"><!-- NEW -->cum enim primo in&longs;tanti quo res e&longs;t, non &longs;it à &longs;e per Axiom. <!-- REMOVE S-->8. <lb/>de &longs;ecundo idem dici debet, quod de primo, vt patet: </s> <s id="N11D1B"><!-- NEW -->quippe id eo <lb/>primo in&longs;tanti non e&longs;t nece&longs;&longs;ariò, quia ita e&longs;t illo in&longs;tanti, vt po&longs;&longs;it non <lb/>e&longs;&longs;e; </s> <s id="N11D23"><!-- NEW -->&longs;ed etiam &longs;ecundo in&longs;tanti ita e&longs;t vt po&longs;&longs;it non e&longs;&longs;e; igitur non e&longs;t <lb/>nece&longs;&longs;ariò, igitur pendet ab alio, quod pote&longs;t facere vt non &longs;it. </s> </p> <p id="N11D29" type="main"> <s id="N11D2B">Dices po&longs;&longs;e de&longs;trui &longs;ecundo in&longs;tanti ab aliquo contrario, à quo tamen <lb/>non pendet per po&longs;itiuum influxum. </s> <s id="N11D30"><!-- NEW -->Re&longs;pondeo, non videri quomo­<lb/>do de&longs;trui po&longs;&longs;it, quod influxu po&longs;itiuo non indiget, vt &longs;it; quid enim <lb/>faceret contrarium, quod tantùm exigere pote&longs;t contrarij de&longs;tructio­<lb/>nem, quid e&longs;t porro de&longs;trui, ni&longs;i de&longs;inere con&longs;eruari? </s> <s id="N11D3A"><!-- NEW -->quæ omnia fusè <lb/>in Metaphy&longs;ica demon&longs;trabimus; </s> <s id="N11D40"><!-- NEW -->quidquid enim e&longs;t aliquo in&longs;tanti vel <lb/>e&longs;t à &longs;e, vel non à &longs;e; &longs;i primùm Deus e&longs;t; </s> <s id="N11D46"><!-- NEW -->&longs;i &longs;ecundum ab alio e&longs;t: <lb/>quidquid &longs;it, hoc Axioma certum e&longs;t phy&longs;icè. </s> </p> <p id="N11D4C" type="main"> <s id="N11D4E">Huc reuoca Axiomata &longs;equentia, quæ ex hoc vno deducuntur. </s> </p> <p id="N11D51" type="main"> <s id="N11D53">1. <emph type="italics"/>Quidquid e&longs;t, & non e&longs;t à &longs;e, e&longs;t, &longs;eu pendet, &longs;eu con&longs;eruatur ab alio.<emph.end type="italics"/><lb/>Hæc enim &longs;unt idem, vt con&longs;tat. </s> </p> <p id="N11D5D" type="main"> <s id="N11D5F">2. <emph type="italics"/>Quidquid destruitur, ad exigentiam alicuius de&longs;truitur, &longs;altem totius <lb/>natura, ne aliquid &longs;it fru&longs;trà.<emph.end type="italics"/></s> <s id="N11D69"><!-- NEW --> Hoc etiam ex hypothe&longs;ibus &longs;equitur; </s> <s id="N11D6D"><!-- NEW -->cum <lb/>enim de&longs;trui &longs;it idem ac de&longs;inere con&longs;eruari; </s> <s id="N11D73"><!-- NEW -->certè qui de&longs;init con&longs;er­<lb/>uare in&longs;tanti A potiùs quam in&longs;tanti B, hoc facere non pote&longs;t ni&longs;i ali­<lb/>quid hoc exigat; &longs;cilicet iuxta leges naturæ. </s> </p> <p id="N11D7B" type="main"> <s id="N11D7D">3. <emph type="italics"/>Tandiu aliquid con&longs;eruatur, quandiu nihil exigit eius de&longs;tructionem.<emph.end type="italics"/><lb/>Hoc &longs;equitur ex priori, id e&longs;t quandiu e&longs;t eadem ratio, cur &longs;it, & con­<lb/>&longs;eruetur, quæ erat antè. </s> </p> <pb pagenum="12" xlink:href="026/01/044.jpg"/> <p id="N11D8D" type="main"> <s id="N11D8F"><emph type="center"/><emph type="italics"/>Axioma XV.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11D9B" type="main"> <s id="N11D9D"><emph type="italics"/>Contraria pugnant pro rata.<emph.end type="italics"/></s> <s id="N11DA4"><!-- NEW --> Nec enim alia regula e&longs;&longs;e pote&longs;t; </s> <s id="N11DA8"><!-- NEW -->&longs;ic minor <lb/>calor minùs de&longs;truit frigoris; minor impetus minùs de&longs;truit impetus <lb/>contrarij (&longs;i contrarium habet) quæ omnia con&longs;tant ex hypothe&longs;ibus. </s> <s id="N11DB0"><lb/>Ratio e&longs;t, quia plùs vel minùs contrarij de&longs;truere, multam habet ex­<lb/>ten&longs;ionem. </s> <s id="N11DB6">v.g. <!-- REMOVE S-->&longs;int duo contraria A & B, &longs;it A vt 20. &longs;it B vt 5. certè &longs;i <lb/>B de&longs;truat A &longs;upra ratam, vel &longs;upra id, quod &longs;ibi ex æquo re&longs;pondet, id <lb/>e&longs;t &longs;upra 5. cur potius 6. quam 7. 8. &c. </s> <s id="N11DBF">Si infra, cur potius 4. quam 3. <lb/>2. &c. </s> <s id="N11DC4">Igitur cum plures &longs;int termini tùm infra, tùm &longs;upra 5. cur potius <lb/>vnus quàm alius? </s> <s id="N11DC9">atqui vnus tantùm ex æquo re&longs;pondet, &longs;cilicet 5. &longs;ed <lb/>quod vnum e&longs;t determinatum e&longs;t, per Axioma 5. igitur pugnant pro <lb/>rata. </s> <s id="N11DD0"><!-- NEW -->Nec dicas A totum de&longs;trui à B, quòd e&longs;t contra hypothe&longs;im, nam <lb/>modicum caloris non de&longs;truit totum frigus: </s> <s id="N11DD6"><!-- NEW -->in impetu res e&longs;t clari&longs;&longs;ima; <lb/>adde quod minor cau&longs;a minùs agit per Ax. 13. num. </s> <s id="N11DDC">3. igitur minùs exi­<lb/>git; porrò cum dico vnum ab alio de&longs;trui, intelligo tantùm ex applica­<lb/>tione vnius &longs;equi de&longs;tructionem alterius &longs;altem ex parte. </s> </p> <p id="N11DE3" type="main"> <s id="N11DE5">Ob&longs;eruabis hæc Axiomata &longs;altem maiori ex parte e&longs;&longs;e metaph. </s> <s id="N11DE8">quæ <lb/>nos fusè in Theorematis metaph. </s> <s id="N11DED"><!-- NEW -->explicabimus, & demon&longs;trabimus; </s> <s id="N11DF1"><!-- NEW -->&longs;ed <lb/>nobis hoc loco &longs;atis e&longs;t, &longs;i parem cum phy&longs;icis &longs;upponas habere cer­<lb/>titudinem, quod nemo negabit; con&longs;tátque ex hypothe&longs;ibus, licèt ma­<lb/>iorem etiam habeant, de qua &longs;uo loco. </s> </p> <p id="N11DFB" type="main"> <s id="N11DFD">Ob&longs;eruabis prætereà nos diutiùs hæ&longs;i&longs;&longs;e in præmittendis huic libro <lb/>Axiomatis, quod tamen in aliis libris non faciemus. </s> </p> <p id="N11E02" type="main"> <s id="N11E04"><emph type="center"/><emph type="italics"/>Postulatum,<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11E10" type="main"> <s id="N11E12"><emph type="italics"/>Liceat datum corpus impellere, proiicere, deor&longs;um cadens excipere, motus <lb/>durationem &longs;en&longs;ibilem, &longs;patiumque &longs;en&longs;ibile, metiri, comparare, &c.<emph.end type="italics"/></s> </p> <p id="N11E1B" type="main"> <s id="N11E1D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11E2A" type="main"> <s id="N11E2C"><emph type="italics"/>Motus e&longs;t aliquid realiter di&longs;tinctum à mobili.<emph.end type="italics"/></s> <s id="N11E33"> Demon&longs;tratur; Motus <lb/>e&longs;t in mobili, in quo antè non erat per hypoth. </s> <s id="N11E38"><!-- NEW -->3. & de&longs;init e&longs;&longs;e in mobili, <lb/>in quo antè erat per hypoth.4. igitur mobile e&longs;t, & non e&longs;t motus; </s> <s id="N11E3E"><!-- NEW -->igi­<lb/>tur à motu &longs;eparatum; </s> <s id="N11E44"><!-- NEW -->igitur realiter di&longs;tinctum per Ax. 2. præterea <lb/>moueri, & non moueri &longs;unt prædicata contradictoria, vt con&longs;tat; </s> <s id="N11E4A"><!-- NEW -->igi­<lb/>tur eidem &longs;imul ine&longs;&longs;e non po&longs;&longs;unt per Ax. 1. igitur cum eo non &longs;unt <lb/>idem; </s> <s id="N11E52"><!-- NEW -->alioquin &longs;imul e&longs;&longs;ent; </s> <s id="N11E56"><!-- NEW -->igitur alterum illorum e&longs;t di&longs;tinctum à <lb/>mobili; </s> <s id="N11E5C"><!-- NEW -->non quies, vt con&longs;tat, quæ e&longs;t tantùm negatio motus, &longs;eu per­<lb/>&longs;euerantia in eodem loco; </s> <s id="N11E62"><!-- NEW -->igitur nullam dicit mutationem; at verò <lb/>motus mutationem dicit, per Def. <!-- REMOVE S-->1. hoc Theorema fusè demon&longs;trabo <lb/>in Metaph. <!-- KEEP S--></s> </p> <p id="N11E6D" type="main"> <s id="N11E6F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11E7C" type="main"> <s id="N11E7E"><emph type="italics"/>Motus non pote&longs;t dici propriè productus immediatè, vel effectus immedia­<lb/>tus cau&longs;æ efficientis.<emph.end type="italics"/></s> <s id="N11E87"> Demon&longs;t. </s> <s id="N11E8A"><!-- NEW -->Motus e&longs;t mutatio, &longs;eu tran&longs;itus ex loco <lb/>in locum per Def. <!-- REMOVE S-->1. &longs;ed mutatio propriè non producitur; </s> <s id="N11E92"><!-- NEW -->quippè pro­<lb/>ductio tantùm terminatur ad ens; </s> <s id="N11E98"><!-- NEW -->nihil enim ni&longs;i ens produci pote&longs;t; </s> <s id="N11E9C"><!-- NEW --><pb pagenum="13" xlink:href="026/01/045.jpg"/>atqui nulla mutatio dicit tantùm ens; </s> <s id="N11EA4"><!-- NEW -->præ&longs;ertim hæc, quæ tantùm dicit <lb/>terminum à quo, ide&longs;t locum relictum; </s> <s id="N11EAA"><!-- NEW -->& terminum ad quem, id e&longs;t lo­<lb/>cum immediatum acqui&longs;itum; </s> <s id="N11EB0"><!-- NEW -->nam &longs;eparato quocunque alio ab ip&longs;o <lb/>mobili; </s> <s id="N11EB6"><!-- NEW -->modo &longs;imul, id e&longs;t eodem in&longs;tanti relinquat primum locum, & <lb/>nouum acquirat, omninò mouetur, &longs;ed concretum illud ex loco relicto, <lb/>& acqui&longs;ito produci non pote&longs;t; </s> <s id="N11EBE"><!-- NEW -->illud autem e&longs;t motus, qui certè non <lb/>dicit tantùm locum relictum &longs;ine acqui&longs;ito; </s> <s id="N11EC4"><!-- NEW -->alioqui &longs;i mobile de&longs;true­<lb/>retur, diceretur moueri; </s> <s id="N11ECA"><!-- NEW -->nec etiam locum acqui&longs;itum &longs;ine priori relicto: </s> <s id="N11ECE"><!-- NEW --><lb/>alioqui &longs;i mobile primò produceretur, diceretur moueri localiter; </s> <s id="N11ED3"><!-- NEW -->igitur <lb/>motus neutrum dicit &longs;eor&longs;im; &longs;i primum, diceretur de&longs;tructus; </s> <s id="N11ED9"><!-- NEW -->&longs;i &longs;ecun­<lb/>dum, diceretur aliquo modo productus, vel potiùs acqui&longs;itus; </s> <s id="N11EDF"><!-- NEW -->at vtrum­<lb/>que coniunctim, &longs;imulque e&longs;&longs;entialiter dicit motus; </s> <s id="N11EE5"><!-- NEW -->nec enim conci­<lb/>pio aliud, dum concipio motum: </s> <s id="N11EEB"><!-- NEW -->porrò vtrumque &longs;imul &longs;umptum indi­<lb/>ui&longs;ibiliter non pote&longs;t dici, vel de&longs;tructum propriè, vel productum; Di­<lb/>xi propriè; nam impropriè dici pote&longs;t motus productus. </s> </p> <p id="N11EF4" type="main"> <s id="N11EF6"><!-- NEW -->Dices Motus e&longs;t ens, non à &longs;e; igitur ab alio; igitur motus e&longs;t pro­<lb/>ductus. </s> <s id="N11EFC"><!-- NEW -->Re&longs;pondeo Motum non e&longs;&longs;e ens ab&longs;olutum, &longs;ed e&longs;&longs;e mutatio­<lb/>nem entis, quæ mutatio e&longs;t concretum quoddam ex ente & non ente; </s> <s id="N11F02"><!-- NEW --><lb/>quòd certè non pote&longs;t dici propriè productum, &longs;ed re&longs;ultans, vt relatio; </s> <s id="N11F07"><!-- NEW --><lb/>nam producatur, &longs;i fieri pote&longs;t; </s> <s id="N11F0C"><!-- NEW -->certè e&longs;t aliquid, quod tam facilè de­<lb/>&longs;trui pote&longs;t, quam produci; </s> <s id="N11F12"><!-- NEW -->igitur de&longs;truatur, & remaneat tantùm en­<lb/>titas mobilis, quæ, quo in&longs;tanti priorem locum relinquit, nouum acqui­<lb/>rat; certè dicitur adhuc moueri, & tamen non erit motus ex &longs;uppo&longs;itio­<lb/>ne, quod ab&longs;urdum e&longs;t. </s> </p> <p id="N11F1C" type="main"> <s id="N11F1E"><!-- NEW -->Dices potentia motrix e&longs;t actiua; </s> <s id="N11F22"><!-- NEW -->igitur agit; igitur producit, &longs;ed ni­<lb/>hil ni&longs;i motum. </s> <s id="N11F28">Re&longs;p. potentiam motricem e&longs;&longs;e actiuam vt dicemus, <lb/>& ab eâ produci impetum, qui deinde exigit motum, vt dicemus <lb/>infrà. </s> </p> <p id="N11F2F" type="main"> <s id="N11F31"><!-- NEW -->Nec e&longs;t quod aliqui ita mirentur hæc à me dici; </s> <s id="N11F35"><!-- NEW -->cum certum &longs;it effe­<lb/>ctus formales &longs;ecundarios principum ferè qualitatum tales e&longs;&longs;e, vt mini­<lb/>mè producantur; </s> <s id="N11F3D"><!-- NEW -->&longs;ed qua&longs;i re&longs;ultent ab exigentia; v. <!-- REMOVE S-->g. <!-- REMOVE S-->effectus calo­<lb/>ris in &longs;uo &longs;ubiecto e&longs;t eiu&longs;dem &longs;ubiecti rarefactio, quæ reuerâ non <lb/>producitur, vt con&longs;tat. </s> </p> <p id="N11F49" type="main"> <s id="N11F4B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11F58" type="main"> <s id="N11F5A"><emph type="italics"/>Motus e&longs;t ab alio di&longs;tincto in aliquo genere cau&longs;æ.<emph.end type="italics"/></s> <s id="N11F61"><!-- NEW --> Demon&longs;tratur, quia <lb/>motus, qui non erat, incipit e&longs;&longs;e per hypothe&longs;im tertiam; &longs;ed quod <lb/>huiu&longs;modi e&longs;t, habet cau&longs;am di&longs;tinctam per Ax.8. <!-- KEEP S--></s> </p> <p id="N11F6A" type="main"> <s id="N11F6C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11F78" type="main"> <s id="N11F7A"><!-- NEW -->Ob&longs;eruabis motum localem e&longs;&longs;e duplicis generis; </s> <s id="N11F7E"><!-- NEW -->primum genus mo­<lb/>tus e&longs;t actio potentiæ motricis, quæ reuerà mouet, & cuius exercitium <lb/>dicitur motus, &longs;eu latio, &longs;eu motio, &longs;eu actio, qua reuerâ agit, produ­<lb/>citque impetum, non motum; </s> <s id="N11F88"><!-- NEW -->cum etiam &longs;ine motu defatigetur, vt cum <lb/>quis alium pellit, à quo pellitur æquali ni&longs;u; </s> <s id="N11F8E"><!-- NEW -->patet etiam in manu &longs;u­<lb/>&longs;tinente aliquod pondus, quæ non mouetur; </s> <s id="N11F94"><!-- NEW -->licet reuerâ etiam &longs;ummo <pb pagenum="14" xlink:href="026/01/046.jpg"/>conatu agat: </s> <s id="N11F9D"><!-- NEW -->immò &longs;i potentia motrix produceret motum primum, non <lb/>impetum in corpore proiecto; </s> <s id="N11FA3"><!-- NEW -->nulla deinde e&longs;&longs;et cau&longs;a applicata ad pro­<lb/>ducendum impetum: </s> <s id="N11FA9"><!-- NEW -->Itaque hic motus primi generis, &longs;i comparetur <lb/>cum potentia motrice, e&longs;t verè influxus, vel actio; </s> <s id="N11FAF"><!-- NEW -->&longs;i cum termino, e&longs;t <lb/>eius fieri, &longs;eu dependentia; </s> <s id="N11FB5"><!-- NEW -->&longs;i cum &longs;ubiecto, &longs;eu mobili e&longs;t pa&longs;&longs;io; </s> <s id="N11FB9"><!-- NEW -->nec <lb/>propriè dicitur produci, ni&longs;i vt quo (vt vulgò loquuntur) nec enim <lb/>actio e&longs;t terminus, vel effectus, in quo &longs;i&longs;tat cau&longs;a; &longs;ed e&longs;t via, qua ten­<lb/>dit ad terminum. </s> <s id="N11FC3"><!-- NEW -->Motus &longs;ecundi generis e&longs;t mutatio, &longs;eu tran&longs;itus ex <lb/>vno loco in alium; </s> <s id="N11FC9"><!-- NEW -->hoc e&longs;t finis, vel effectus formalis &longs;ecundarius, <lb/>quem exigit impetus; </s> <s id="N11FCF"><!-- NEW -->& fru&longs;trà ponitur alia entitas, quæ tantùm e&longs;&longs;et <lb/>in&longs;tituta ad exigendam i&longs;tam loci mutationem; Igitur &longs;i &longs;ufficienter <lb/>exigatur ab ip&longs;o impetu, de quo infrà, certè fru&longs;tra ponitur quodcun­<lb/>que aliud per Ax.3. & 7. </s> </p> <p id="N11FD9" type="main"> <s id="N11FDB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N11FE8" type="main"> <s id="N11FEA"><emph type="italics"/>Cau&longs;a illa immediata motus, quæ non est efficiens, potest tantùm e&longs;&longs;e exi­<lb/>gens, quæ reducitur ad formalem, quæ &longs;uum effectum formalem &longs;ecundarium, <lb/>id est &longs;uum finem intrin&longs;ecum exigit.<emph.end type="italics"/></s> <s id="N11FF5"><!-- NEW --> Sic calor exigit rarefactionem, vel <lb/>re&longs;olutionem, impetus motum; </s> <s id="N11FFB"><!-- NEW -->cum enim non &longs;it cau&longs;a efficiens per Th. <!-- REMOVE S--><lb/>2. &longs;it tamen cau&longs;a per Th.3. nec &longs;it materialis, nec finalis, vt con&longs;tat, de­<lb/>bet e&longs;&longs;e formalis, vel exigens, &longs;eu exigitiua; </s> <s id="N12004"><!-- NEW -->vt patet ex ip&longs;a cau&longs;arum <lb/>enumeratione; </s> <s id="N1200A"><!-- NEW -->non e&longs;t materialis, quia non recipit motum, ni&longs;i ab alio; </s> <s id="N1200E"><!-- NEW --><lb/>nec finalis, quæ &longs;upponit alias; cum ip&longs;a non &longs;it dum ponitur <lb/>effectus. </s> </p> <p id="N12015" type="main"> <s id="N12017"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12024" type="main"> <s id="N12026"><!-- NEW --><emph type="italics"/>Entitas &longs;eu &longs;ubstantia mobilis non e&longs;t cau&longs;a immediata motus,<emph.end type="italics"/> Sit enim <lb/>lapis proiectus per Po&longs;tul. <!-- REMOVE S-->haud dubiè &longs;ub&longs;tantia lapidis non e&longs;t cau&longs;a <lb/>huius motus; </s> <s id="N12035"><!-- NEW -->quia lapis tandem &longs;i&longs;tit per hypoth.4. igitur non e&longs;t cau&longs;a <lb/>motus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; </s> <s id="N1203B"><!-- NEW -->igitur &longs;emper cau&longs;aret per Ax.12. præ­<lb/>terea potentia motrix proiicientis verè agit, cum etiam defatigetur; </s> <s id="N12041"><!-- NEW -->igi­<lb/>tur aliquid producit, non motum immediatè, qui produci non pote&longs;t pro<lb/>prièper Th. 2. Adde quod motus &longs;ecundi generis habet tantùm cau&longs;am <lb/>immediatam exigentem, &longs;ed potentia motrix non exigit; </s> <s id="N1204B"><!-- NEW -->quia primò <lb/>non defatigaretur exigendo; </s> <s id="N12051"><!-- NEW -->&longs;ecundò quia lapis &longs;eparatus à manu etiam <lb/>mouetur, &longs;ed non ad exigentiam potentiæ motricis, vt patet; </s> <s id="N12057"><!-- NEW -->quia &longs;tatim <lb/>po&longs;t &longs;eparationem pote&longs;t illa potentia de&longs;trui, licèt lapis longo pò&longs;t <lb/>tempore moueatur; &longs;ed quod non e&longs;t, nihil exigit. </s> </p> <p id="N1205F" type="main"> <s id="N12061">Aliquis fortè diceret potentiam motricem exigere primam partem <lb/>motus, quæ deinde &longs;ecundam exigit, & &longs;ecunda tertiam, tertia quar­<lb/>tam, &c. </s> <s id="N12068">Sed contra; </s> <s id="N1206B"><!-- NEW -->quæro quid &longs;it prima illa pars motus; </s> <s id="N1206F"><!-- NEW -->nec enim <lb/>aliud agno&longs;co ni&longs;i primam mutationem loci, quæ mutatio non pote&longs;t <lb/>exigere ni&longs;i quando e&longs;t; </s> <s id="N12077"><!-- NEW -->atqui quando e&longs;t, nihil reale e&longs;t actu ni&longs;i mo­<lb/>bile, & nouus locus acqui&longs;itus, mobile ip&longs;um non exigit, vt demon&longs;tra­<lb/>tum e&longs;t, & conce&longs;&longs;um, nec etiam locus de nouo acqui&longs;itus, in quo <lb/>&longs;cilicet mobile &longs;i&longs;tere pote&longs;t: quidquid pones aliud, impetum appellabo. </s> </p> <pb pagenum="15" xlink:href="026/01/047.jpg"/> <p id="N12085" type="main"> <s id="N12087">Dices cum graue aliquod mouetur deor&longs;um, vel leue &longs;ur&longs;um, vel <lb/>corpus animatum &longs;e ip&longs;um mouet, dici pote&longs;t &longs;ub&longs;tantia corporis cau&longs;a <lb/>immediata motus. </s> <s id="N1208E"><!-- NEW -->Re&longs;p. negando, tùm quia omnis potentia motrix <lb/>agit; </s> <s id="N12094"><!-- NEW -->igitur producit aliquid aliud, quod e&longs;t cau&longs;a motus: præterea po­<lb/>tentia motrix corporis animati, agit v&longs;que ad defatigationem, &longs;udorem, <lb/>licèt non &longs;it motus, igitur aliud producit, de corpore graui probabi­<lb/>mus infrà. </s> </p> <p id="N1209E" type="main"> <s id="N120A0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N120AD" type="main"> <s id="N120AF"><emph type="italics"/>Datur impetus.<emph.end type="italics"/></s> <s id="N120B6"><!-- NEW --> Demon&longs;tro, Sub&longs;tantia mobilis non e&longs;t cau&longs;a imme­<lb/>diata motus, per Th.5. ergo aliquid aliud; igitur impetus, nam quod di­<lb/>&longs;tinctum e&longs;t à &longs;ub&longs;tantia mobilis, & exigit motum, e&longs;t impetus per <lb/>Def.3. &longs;ed quia hoc Theorema e&longs;t veluti princeps huius tractatus cardo, <lb/>in eo paulò diutius hærendum e&longs;t, igitur. </s> </p> <p id="N120C2" type="main"> <s id="N120C4"><!-- NEW -->Demon&longs;tro primò dari impetum: </s> <s id="N120C8"><!-- NEW -->Quidquid e&longs;t, & antè non erat, non <lb/>e&longs;t à &longs;e, &longs;ed habet cau&longs;am per Ax.8. Motus de nouo e&longs;t per hypothe&longs;im <lb/>tertiam; </s> <s id="N120D0"><!-- NEW -->igitur habet cau&longs;am, &longs;ed non aliam, quam impetum, quod pro­<lb/>bo: </s> <s id="N120D6"><!-- NEW -->Lapis cadens, vel impactus in alium lapidem mouet illum per hy­<lb/>poth.7. &longs;ed &longs;ub&longs;tantia lapidis in alium impacti non e&longs;t cau&longs;a huius mo­<lb/>tus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt patet; </s> <s id="N120DE"><!-- NEW -->igitur applicata eundem effe­<lb/>ctum produceret per Ax.12. &longs;ed etiam applicata immediata non agit, vt <lb/>con&longs;tat experientia; igitur per idem Axioma non e&longs;t cau&longs;a. </s> </p> <p id="N120E6" type="main"> <s id="N120E8"><!-- NEW -->Scio e&longs;&longs;e aliquas re&longs;pon&longs;iones, quas infrà refellemus; nunc &longs;ufficiat <lb/>dixi&longs;&longs;e lapidem impactum non producere motum, qui propriè non pro­<lb/>ducitur per Th.2. nec exigere, vt con&longs;tat ex &longs;ecunda probatione Th. 5. <lb/>igitur &longs;i aliquid exigit, vel producit, voco impetum. </s> </p> <p id="N120F2" type="main"> <s id="N120F4">Secundò probatur; potentia motrix e&longs;t actiua, quia defatigatur, quis <lb/>hoc neget? </s> <s id="N120F9"><!-- NEW -->igitur aliquid producit; </s> <s id="N120FD"><!-- NEW -->non motum, qui propriè non pro­<lb/>ducitur per Th.2. igitur aliquid aliud; voco impetum; </s> <s id="N12103"><!-- NEW -->adde quod etiam <lb/>&longs;ine motu agit, & defatigatur vt iam dictum e&longs;t; </s> <s id="N12109"><!-- NEW -->igitur habet alium effe­<lb/>ctum immediatum; denique mouere, pellere, trahere, proiicere, percu­<lb/>tere, nihil ni&longs;i actionem &longs;onant. </s> </p> <p id="N12111" type="main"> <s id="N12113">Tertiò probatur; pila di&longs;iuncta à manu proiicientis diu adhuc mo­<lb/>uetur per hypoth.6. igitur hic motus habet cau&longs;am per Ax. 8. quælibet <lb/>enim pars motus de nouo e&longs;t, neque duæ illius partes &longs;imul e&longs;&longs;e po&longs;&longs;unt. </s> <s id="N1211A"><!-- NEW --><lb/>atqui potentia motrix non e&longs;t cau&longs;a per Ax.10. immò pote&longs;t e&longs;&longs;e de&longs;tru­<lb/>cta; igitur non e&longs;t cau&longs;a per Ax. 9. <!--neuer Satz-->Non e&longs;t etiam cau&longs;a &longs;ub&longs;tantia pilæ <lb/>mobilis per Th.5.5. nec priores pattes motus per re&longs;p. </s> <s id="N12125"><!-- NEW -->ad primam in­<lb/>&longs;tantiam Th 5. igitur aliquid aliud; voco impetum. </s> </p> <p id="N1212B" type="main"> <s id="N1212D">Quartò probatur; </s> <s id="N12130"><!-- NEW -->pila proiecta &longs;en&longs;im &longs;ine &longs;en&longs;u tardiore motu <lb/>mouetur; donec tandem moueri omnino de&longs;inat per hypoth. </s> <s id="N12136"><!-- NEW -->5. igitur <lb/>non e&longs;t &longs;emper æqualis, & eadem cau&longs;a huius motus per Ax. 12. & 13. <lb/>num.3. igitur cau&longs;a huius motus eodem modo debilitatur, &longs;eu remitti­<lb/>tur, quo ip&longs;e motus; </s> <s id="N12140"><!-- NEW -->&longs;ed decre&longs;cit &longs;ub&longs;tantia mobilis, nec potentia mo-<pb pagenum="16" xlink:href="026/01/048.jpg"/>trix, vel corpus prius impactum; </s> <s id="N12149"><!-- NEW -->ergo e&longs;t alia cau&longs;a præ&longs;ens, quæ mi­<lb/>nuitur; voco impetum. </s> </p> <p id="N1214F" type="main"> <s id="N12151"><!-- NEW -->Quintò corpus graue deor&longs;um cadens accelerat &longs;uum motum, vt patet <lb/>experientia; </s> <s id="N12157"><!-- NEW -->quæ maximè clara e&longs;t in funependulis, de qua in &longs;equen­<lb/>tibus libris; </s> <s id="N1215D"><!-- NEW -->igitur debet e&longs;&longs;e cau&longs;a huius motus velocioris; </s> <s id="N12161"><!-- NEW -->non e&longs;t au­<lb/>tem &longs;ub&longs;tantia lapidis, nec grauitas per Ax. 12. nec aliud quidpiam ex­<lb/>trin&longs;ecum, vt videbimus &longs;uo loco; igitur aliquid aliquid intrin&longs;ecum, <lb/>voco impetum. </s> <s id="N1216B"><!-- NEW -->Igitur certum e&longs;t dari impetum; qui certè tribui non <lb/>pote&longs;t, vel vlli connotationi, vel alteri exigentiæ, vt con&longs;tat ex <lb/>dictis. </s> </p> <p id="N12173" type="main"> <s id="N12175"><!-- NEW -->Diceret fortè alius hæc omnia e&longs;&longs;e dubia; </s> <s id="N12179"><!-- NEW -->nam fieri pote&longs;t vt Deus <lb/>tantùm moueat; </s> <s id="N1217F"><!-- NEW -->quod &longs;ine impetu fieri po&longs;&longs;e certum e&longs;t; </s> <s id="N12183"><!-- NEW -->Re&longs;p. equi­<lb/>dem per miraculum hoc fieri po&longs;&longs;e; &longs;ed quemadmadum certum e&longs;t phy­<lb/>&longs;icè ignem applicatum calefacere, niuem frigefacere, & modò calamum <lb/>à me hæc &longs;cribente moueri, ita certum o&longs;t phy&longs;icè &longs;agittam à &longs;agittario <lb/>emitti, & pilam à proiiciente, &c. </s> <s id="N1218F"><!-- NEW -->adde quod Deus, vt auctor naturæ <lb/>e&longs;t, agit tantùm; </s> <s id="N12195"><!-- NEW -->vel de&longs;init agere iuxta exigentiam cau&longs;arum &longs;ecunda­<lb/>rum; denique cau&longs;am phy&longs;icè appello, ex cuius applicatione nunquam <lb/>non &longs;equitur effectus per Ax.11. num.1. </s> </p> <p id="N1219D" type="main"> <s id="N1219F"><!-- NEW -->Dicerent alij hoc totum prouenire à corpu&longs;culis; </s> <s id="N121A3"><!-- NEW -->vel atomis, vel fila­<lb/>mentis &longs;ine vlla actione; </s> <s id="N121A9"><!-- NEW -->equidem non reiicio corpu&longs;cula, & perennia <lb/>corporum effluuia: </s> <s id="N121AF"><!-- NEW -->Dico tamen primò globum quie&longs;centem humi ha­<lb/>bere &longs;altem aliquas partes quie&longs;centes, vel immobiles; quis hoc neget? </s> <s id="N121B5"><!-- NEW --><lb/>immò maximam &longs;uarum partium partem; </s> <s id="N121BA"><!-- NEW -->igitur cum deinde proiicitur <lb/>idem globus, illæ partes mouentur; </s> <s id="N121C0"><!-- NEW -->dari igitur debet cau&longs;a huius motus <lb/>per Ax.8, igitur impetus: </s> <s id="N121C6"><!-- NEW -->nec dicas moueri illas partes à corpu&longs;culis; </s> <s id="N121CA"><!-- NEW -->quia <lb/>antè erant eadem, immò plura corpu&longs;cula; </s> <s id="N121D0"><!-- NEW -->& tamen non mouebant: </s> <s id="N121D4"><!-- NEW -->igi­<lb/>tur non &longs;unt cau&longs;a huius motus per Ax.12. Dices excitari; &longs;ed quid hoc <lb/>e&longs;t excitari? </s> <s id="N121DC"><!-- NEW -->vel enim mutantur, vel non mutantur; </s> <s id="N121E0"><!-- NEW -->&longs;ecundum dici <lb/>non pote&longs;t; </s> <s id="N121E6"><!-- NEW -->quia vt excitentur, ex non excitatis mutari debent; igitur <lb/>per aliquid: </s> <s id="N121EC"><!-- NEW -->deinde quid e&longs;t illa excitatio, ni&longs;i impul&longs;io; igitur &longs;i mouen­<lb/>tur illa corpu&longs;cula, & excitantur à potentia motrice, etiam partes prius <lb/>immobiles mouebuntur, & excitabuntur per Ax.12. quia &longs;unt applicatæ <lb/>cau&longs;æ nece&longs;&longs;ariæ. </s> </p> <p id="N121F6" type="main"> <s id="N121F8"><!-- NEW -->Dico &longs;ecundò minimum ex his corpu&longs;culis non &longs;emper moueri; </s> <s id="N121FC"><!-- NEW -->po­<lb/>te&longs;t enim &longs;i&longs;tere; quis hoc neget? </s> <s id="N12202">igitur &longs;i modò mouetur, modò quie&longs;­<lb/>cit, motus ab eo di&longs;tinguitur per Th.1. igitur mouetur per impetum, de <lb/>quo infrà. </s> </p> <p id="N12209" type="main"> <s id="N1220B">Igitur datur nece&longs;&longs;ariò impetus, &longs;ine quo non po&longs;&longs;unt explicari prædi­<lb/>ctæ omnes hypothe&longs;es, contra quem &longs;unt quidem graui&longs;&longs;imæ difficultates, <lb/>quas &longs;en&longs;im in &longs;equentibus Theorematis, in quibus explicantur pro­<lb/>prietates huius impetus, di&longs;cutiemus. </s> </p> <p id="N12214" type="main"> <s id="N12216"><!-- NEW -->Diceret aliquis lapidem impul&longs;um ab aëre deinde propelli; </s> <s id="N1221A"><!-- NEW -->&longs;ed aër po­<lb/>tius re&longs;i&longs;tit motui; vt con&longs;tat experientiâ; &longs;ed hoc &longs;oluemus infrà. </s> </p> <pb pagenum="17" xlink:href="026/01/049.jpg"/> <p id="N12224" type="main"> <s id="N12226"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N12232" type="main"> <s id="N12234"><emph type="italics"/>Impetus est aliquid distinctum à &longs;ubstantiâ mobilis.<emph.end type="italics"/></s> <s id="N1223B"> Demon&longs;tratur. </s> <s id="N1223E"><lb/>Quia &longs;ub&longs;tantia mobilis non e&longs;t cau&longs;a exigens motum per Th. 5. Impe­<lb/>tus e&longs;t cau&longs;a exigens per Def. <!-- REMOVE S-->3. & Th. 6. de eodem contradictoria dici <lb/>non po&longs;&longs;unt per Ax. 1. n. </s> <s id="N12248"><!-- NEW -->3. Igitur impetus non e&longs;t idem cum &longs;ub&longs;tantià <lb/>mobilis; igitur di&longs;tinctus; deinde &longs;eparari pote&longs;t à &longs;ub&longs;tantia mobilis <lb/>per Hypoth. <!-- KEEP S--></s> <s id="N12251">4. igitur e&longs;t di&longs;tinctus per Ax. 2. <!-- KEEP S--></s> </p> <p id="N12255" type="main"> <s id="N12257"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N12263" type="main"> <s id="N12265"><!-- NEW --><emph type="italics"/>Impetus est accidens<emph.end type="italics"/>; </s> <s id="N1226E"><!-- NEW -->Quippe non e&longs;t corpus, nec forma &longs;ub&longs;tantia­<lb/>lis; quia omne corpus, & omnis forma &longs;ub&longs;tantialis moueri pote&longs;t, & <lb/>non moueri, vt con&longs;tat ex po&longs;t. </s> <s id="N12276">& ex Hypoth. <!-- KEEP S--></s> <s id="N1227A"><!-- NEW -->3. & 4. igitur di&longs;tingui­<lb/>tur à motu; </s> <s id="N12280"><!-- NEW -->igitur & ab impetu per Ax. 2. igitur impetus non e&longs;t &longs;ub­<lb/>&longs;tantia; igitur accidens. </s> </p> <p id="N12286" type="main"> <s id="N12288"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N12294" type="main"> <s id="N12296"><emph type="italics"/>Impetus non e&longs;t modus.<emph.end type="italics"/></s> <s id="N1229D"><!-- NEW --> Modus duplicis generis e&longs;&longs;e pote&longs;t: </s> <s id="N122A1"><!-- NEW -->Modus <lb/>primi generis e&longs;t entitas quædam diminuta, vt vulgò loquuntur, di&longs;tin­<lb/>cta quidem modaliter, vt aiunt, à re, cui adhæret; ac proinde ab ca &longs;e­<lb/>parari pote&longs;t, non tamen exi&longs;tere &longs;eparata. </s> <s id="N122AB"><!-- NEW -->Modus &longs;ecundi generis non <lb/>e&longs;t entitas quidem di&longs;tincta; </s> <s id="N122B1"><!-- NEW -->e&longs;t tamen &longs;tatus quidam corporis; &longs;ic &longs;e&longs;&longs;io <lb/>e&longs;t modus, conden&longs;atio, compre&longs;&longs;io, &c. </s> <s id="N122B7"><!-- NEW -->His po&longs;itis Impetus non e&longs;t mo­<lb/>dus primi generis; </s> <s id="N122BD"><!-- NEW -->nihil enim probat impetum e&longs;&longs;e modum, quod etiam <lb/>non probet calorem, & lucem e&longs;&longs;e modos; </s> <s id="N122C3"><!-- NEW -->dicere autem omnia acci­<lb/>dentia e&longs;&longs;e modos non debemus, de quo &longs;uo loco; </s> <s id="N122C9"><!-- NEW -->modus enim ita à na­<lb/>turâ comparatus e&longs;t, vt &longs;ine &longs;ubiecto actuali &longs;eu fulcro non exi&longs;tere mo­<lb/>dò, &longs;ed ne concipi quidem po&longs;&longs;it; </s> <s id="N122D1"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->actio non pote&longs;t concipi ni&longs;i &longs;it <lb/>alicuius actio; </s> <s id="N122DB"><!-- NEW -->nec fieri &longs;ine facto; </s> <s id="N122DF"><!-- NEW -->nec via &longs;ine termino; </s> <s id="N122E3"><!-- NEW -->nec dependen­<lb/>tia &longs;ine dependente; </s> <s id="N122E9"><!-- NEW -->at verò po&longs;&longs;um concipere calorem, & impetum <lb/>&longs;ine alio, quod &longs;it actu; </s> <s id="N122EF"><!-- NEW -->licèt enim calor exigat re&longs;olutionem partium <lb/>&longs;ui &longs;ubiecti, &longs;eu rarefactionem, & impetus motum; nihil tamen impe­<lb/>dit, quin per miraculum calor, & impetus con&longs;eruari po&longs;&longs;int &longs;ine eo. </s> <s id="N122F7"><!-- NEW --><lb/>quod exigunt, hoc e&longs;t &longs;ine &longs;uo &longs;ine; </s> <s id="N122FC"><!-- NEW -->igitur &longs;ine &longs;ubiecto; </s> <s id="N12300"><!-- NEW -->non e&longs;t etiam <lb/>modus &longs;ecundi generis vt patet, &longs;ed de modis in Metaphy&longs;ica; vix enim <lb/>hoc Theorema ad rem Phy&longs;icam quicquam facit. </s> </p> <p id="N12308" type="main"> <s id="N1230A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N12316" type="main"> <s id="N12318"><emph type="italics"/>Impetus e&longs;t qualitas Phy&longs;ica.<emph.end type="italics"/><!-- KEEP S--></s> <s id="N12320"> Sequitur ex dictis; cum nec &longs;it motus. </s> <s id="N12323"><!-- NEW --><lb/>nec &longs;ub&longs;tantia, nec modus, nec quidquam negatiuum, alioquin exige­<lb/>ret; </s> <s id="N1232A"><!-- NEW -->igitur e&longs;t aliud accidens; vocetur qualitas. </s> </p> <p id="N1232E" type="main"> <s id="N12330"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N1233C" type="main"> <s id="N1233E"><emph type="italics"/>Impetus est qualitas Phy&longs;ica.<emph.end type="italics"/><!-- KEEP S--></s> <s id="N12346"> Quia impetus e&longs;t di&longs;tinctus realiter à &longs;ue <lb/>&longs;ubiecto per Th. 7. E&longs;t enim &longs;eparabilis per Hypoth. <!-- KEEP S--></s> <s id="N1234C"><!-- NEW -->3. & 4. igitur di­<lb/>&longs;tinctus per Ax. 2. &longs;ed qualitatem realiter di&longs;tinctam apello Phy&longs;icam; <lb/>præ&longs;ertim cum nec moralis &longs;it, nec Logica, &c. </s> </p> <pb pagenum="18" xlink:href="026/01/050.jpg"/> <p id="N12358" type="main"> <s id="N1235A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N12366" type="main"> <s id="N12368"><emph type="italics"/>Impetus est qualitas permanens.<emph.end type="italics"/></s> <s id="N1236F"><!-- NEW --> Quia lapis proiectus etiam &longs;eparatus <lb/>mouetur aliquandiu per Hyp. <!-- REMOVE S-->6. igitur durat eius cau&longs;a, &longs;cilicet impe­<lb/>tus; igitur e&longs;t qualitas permanens. </s> </p> <p id="N12379" type="main"> <s id="N1237B"><!-- NEW -->Diceret fortè aliquis lapidem proiectum pelli ab aëre à tergo in&longs;tan­<lb/>te, vt voluit Ari&longs;toteles pluribus in locis; </s> <s id="N12381"><!-- NEW -->&longs;ed præ&longs;ertim 8. Ph.c.vlt.& 7. <lb/>cap.2. 3.de Cœlo, cap. 3. Re&longs;pondeo hoc dici non po&longs;&longs;e; </s> <s id="N12387"><!-- NEW -->Primò quia non <lb/>modò non iuuat aër; </s> <s id="N1238D"><!-- NEW -->&longs;ed etiam impedit motum proiecti, quod de omni <lb/>medio nece&longs;&longs;ariò dicendum e&longs;t, vt patet experientiâ; </s> <s id="N12393"><!-- NEW -->vnde quo cra&longs;&longs;ius, <lb/>&longs;eu den&longs;ius e&longs;t <expan abbr="mediũ">medium</expan>, motum potentiùs retardat, vt videmus in proiectis <lb/>per aquam; </s> <s id="N1239F"><!-- NEW -->rationem à priori afferemus infrà, cum de re&longs;i&longs;tentia medij: </s> <s id="N123A3"><!-- NEW --><lb/>Secundò, quis dicat pilam rotatam in &longs;olo moueri aëris appul&longs;u? cum <lb/>alia corpora, quæ pila rotata præterlambendo qua&longs;i allambit, nullo mo­<lb/>do moueantur; præ&longs;ertim granula pulueris. </s> <s id="N123AC"><!-- NEW -->Tertiò, an fortè aër id præ­<lb/>&longs;tare pote&longs;t &longs;ine vi impre&longs;&longs;a; </s> <s id="N123B2"><!-- NEW -->igitur non minus ip&longs;i pilæ proiectæ, quam <lb/>aëri ambienti imprimi poterit: </s> <s id="N123B8"><!-- NEW -->Quartò, nullus aër à tergo pellitur; </s> <s id="N123BC"><!-- NEW -->&longs;ed <lb/>potius ip&longs;a pila aduer&longs;us aëra pellit, dum emittitur manu; igitur &longs;i aër <lb/>&longs;uccedit à tergo, id totum accidit, vel metu vacui, vel ne aër compri­<lb/>matur, vt videbimus infrà. </s> <s id="N123C6"><!-- NEW -->Quintò denique, cum diu moueatur eadem <lb/>pars aëris, haud dubiè in ca manet vis impre&longs;&longs;a; igitur impetus erit ad­<lb/>huc qualitas permanens. </s> </p> <p id="N123CE" type="main"> <s id="N123D0"><!-- NEW -->Ad id quod obiicitur ex Ari&longs;totele; </s> <s id="N123D4"><!-- NEW -->aliqui putant inclina&longs;&longs;e in cam &longs;en­<lb/>tentiam; </s> <s id="N123DA"><!-- NEW -->cùm tam en no&longs;tram teneant illu&longs;tres Peripatetici, quorum no­<lb/>minibus parco, ne tot citationes paginas impleant; vide apud Conim­<lb/>bric. </s> <s id="N123E2"><!-- NEW -->l. <!-- REMOVE S-->7. Phy&longs;. cap. 2. Aliqui excu&longs;ant ip&longs;um Ari&longs;totelem, putantque <lb/>non e&longs;&longs;e locutum ex propriâ &longs;ententiâ: </s> <s id="N123EA"><!-- NEW -->Alij dicunt Ari&longs;totelem quidem <lb/>tribui&longs;&longs;e aliquam vim extrin&longs;ecam aëri; </s> <s id="N123F0"><!-- NEW -->non tamen nega&longs;&longs;e intrin&longs;ecam <lb/>impetus; </s> <s id="N123F6"><!-- NEW -->quidquid &longs;it, ip&longs;a verba Ari&longs;totelis demon&longs;trant ip&longs;um agno­<lb/>ui&longs;&longs;e vim motricem impre&longs;&longs;am aëri, hoc e&longs;t impetum (<emph type="italics"/>potentia enim<emph.end type="italics"/> (in­<lb/>quit) &longs;cilicet motrix, <emph type="italics"/>quâ pollet proijciens qua&longs;i vim impre&longs;&longs;am tradit vtrique<emph.end type="italics"/>) <lb/>id e&longs;t aëri &longs;ur&longs;um, & deor&longs;um; quid porrò e&longs;t illa vis motrix, ni&longs;i impetus. </s> </p> <p id="N1240C" type="main"> <s id="N1240E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N1241A" type="main"> <s id="N1241C"><emph type="italics"/>Impetus non producit motum.<emph.end type="italics"/></s> <s id="N12423"><!-- NEW --> Probatur, quia motus non dicitur pro­<lb/>ductus per Th. 2. Adde &longs;i vis rationem metaphy&longs;icam; </s> <s id="N12429"><!-- NEW -->quia nihil cogit <lb/>dicere accidens aliquod, ex iis &longs;cilicet, quæ &longs;en&longs;u percipimus, agere ad <lb/>intra; </s> <s id="N12431"><!-- NEW -->quod videtur e&longs;&longs;e proprium &longs;ub&longs;tantiæ, &longs;altem naturaliter; vt <lb/>demon&longs;trabimus in Metaph. <!-- KEEP S--></s> </p> <p id="N12438" type="main"> <s id="N1243A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N12446" type="main"> <s id="N12448"><!-- NEW --><emph type="italics"/>Impetus exigit motum, id est fluxum mobilis in loco<emph.end type="italics"/>; </s> <s id="N12451"><!-- NEW -->quia cau&longs;a imme­<lb/>diata motus e&longs;t tantum exigens, per Th. 4. &longs;ed impetus e&longs;t cau&longs;a motus <lb/>immediata per Th. 5. & 6. igitur e&longs;t cau&longs;a exigens, adde quod id tantùm <lb/>accidens &longs;en&longs;ibile præ&longs;tare pote&longs;t in &longs;uo &longs;ubiecto, vt aliquam illius mu­<lb/>tationem præ&longs;tet, vel exigat; </s> <s id="N1245D"><!-- NEW -->quæ vel e&longs;t localis, hoc e&longs;t fluxus quidam: </s> <s id="N12461"><!-- NEW --><pb pagenum="19" xlink:href="026/01/051.jpg"/>per &longs;patium loci; </s> <s id="N12469"><!-- NEW -->vel alteratiua, vt vulgò vocatur; quà &longs;cilicet vel re­<lb/>&longs;oluuntur partes, vel rarefiunt, vel lique&longs;cunt, vel concre&longs;cunt &c. </s> <s id="N1246F"><!-- NEW -->vel <lb/>demùm mutant &longs;en&longs;ibilem &longs;tatum; </s> <s id="N12475"><!-- NEW -->vel e&longs;t perfectiua aliquo modo, qua­<lb/>tenus &longs;ubiectum nouam aliquam habitudinem acquirit ad &longs;en&longs;us; &longs;ic <lb/>lumen illuminando obiectum reddit illud vi&longs;ibile. </s> <s id="N1247D">&c. </s> <s id="N12480">de quibus aliàs. </s> </p> <p id="N12483" type="main"> <s id="N12485"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N12491" type="main"> <s id="N12493"><emph type="italics"/>Motus e&longs;t effectus formalis &longs;ecundarius impetus.<emph.end type="italics"/></s> <s id="N1249A"><!-- NEW --> Cum enim &longs;it cau&longs;a <lb/>exigens per Th. 121. Voco effectum formalem &longs;ecundarium, quem in <lb/>mobili exigit impetus; </s> <s id="N124A2"><!-- NEW -->quippe, vt iam dictum e&longs;t, cau&longs;a exigens redu­<lb/>citur ad formalem; </s> <s id="N124A8"><!-- NEW -->nec enim cau&longs;at aliquid producendo, quod &longs;pectat ad <lb/>efficientem; </s> <s id="N124AE"><!-- NEW -->nec mouendo, quod &longs;pectat ad finalem; </s> <s id="N124B2"><!-- NEW -->nec determinando, <lb/>quod &longs;pectat ad obiectiuam; </s> <s id="N124B8"><!-- NEW -->nec recipiendo, quod &longs;pectat ad materia­<lb/>lem; </s> <s id="N124BE"><!-- NEW -->nec dirigendo, quod &longs;pectat ad idæalem, vel exemplarem; &longs;ed <lb/>exigendo; </s> <s id="N124C4"><!-- NEW -->quatenus &longs;cilicet ad id à natura e&longs;t in&longs;tituta, vt ex eius in <lb/>&longs;ubiecto præ&longs;entia talis affectio, vel mutatio con&longs;equatur; </s> <s id="N124CA"><!-- NEW -->vocatur au­<lb/>tem effectus formalis &longs;ecundarius; non verò primarius, qui e&longs;t tantùm <lb/>concretum ex ip&longs;a formâ, & &longs;ubiecto. </s> </p> <p id="N124D2" type="main"> <s id="N124D4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N124E0" type="main"> <s id="N124E2"><emph type="italics"/>Motus e&longs;t finis intrin&longs;ecus impetus.<emph.end type="italics"/></s> <s id="N124E9"><!-- NEW --> Dum finem audis intrin&longs;ecum, <lb/>cogita quæ&longs;o aliquid phy&longs;icum; </s> <s id="N124EF"><!-- NEW -->e&longs;t enim id, propter quod talis, vel ta­<lb/>lis forma in&longs;tituta e&longs;t: </s> <s id="N124F5"><!-- NEW -->quid enim aliud e&longs;&longs;e pote&longs;t; </s> <s id="N124F9"><!-- NEW -->finem enim rerum <lb/>naturalium ex ip&longs;o v&longs;u cogno&longs;cimus; </s> <s id="N124FF"><!-- NEW -->immò idem e&longs;t finis cum ip&longs;o v&longs;u; </s> <s id="N12503"><!-- NEW --><lb/>cum igitur impetus illum tantùm v&longs;um habeat, quem in ip&longs;o mobili <lb/>præ&longs;tare cernimus, &longs;cilicet motum; </s> <s id="N1250A"><!-- NEW -->dicendum e&longs;t motum e&longs;&longs;e finem in­<lb/>trin&longs;ecum impetus; </s> <s id="N12510"><!-- NEW -->adde quod cum fru&longs;trà &longs;it impetus ille, qui non præ­<lb/>&longs;tat motum mediatè &longs;altem in &longs;uo &longs;ubiecto; quid enim aliud in &longs;uo &longs;ub­<lb/>iecto præ&longs;taret, quem effectum, quam mutationem? </s> <s id="N12518"><!-- NEW -->certè &longs;i fru&longs;trà e&longs;t, non <lb/>e&longs;t, per Ax.6.igitur vt &longs;it, debet habere id, &longs;ine quo e&longs;&longs;e non pote&longs;t; igitur <lb/>maximum eius bonum e&longs;t, igitur finis, quem natiuâ vel innatâ velut <lb/>appetentiâ concupi&longs;cit, vel exigit. </s> <s id="N12522">Dixi mediatè, vel immediatè; </s> <s id="N12525"><!-- NEW -->num <lb/>reuera datur fortè aliquis impetus, vt dicemus infrà; </s> <s id="N1252B"><!-- NEW -->&longs;cilicet primus na­<lb/>turalis, qui &longs;cilicet duos fines habet di&longs;iunctiuè; quorum alter e&longs;t gra­<lb/>uitatio, alter motus deor&longs;um. </s> </p> <p id="N12533" type="main"> <s id="N12535"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N12541" type="main"> <s id="N12543"><emph type="italics"/>Ni&longs;i e&longs;&longs;et motus non e&longs;&longs;et impetus.<emph.end type="italics"/></s> <s id="N1254A"><!-- NEW --> Probatur quia motus e&longs;t finis intrin­<lb/>&longs;ecus impetus per Th. 16. igitur &longs;i nullus motus e&longs;&longs;e po&longs;&longs;et, &longs;uo fine ca­<lb/>reret impetus; </s> <s id="N12552"><!-- NEW -->igitur non e&longs;&longs;et, vt patet, igitur non e&longs;&longs;et; </s> <s id="N12556"><!-- NEW -->quia quod <lb/>fru&longs;trà e&longs;t, non e&longs;t per Ax. 6. nec ob&longs;tat quod &longs;uprà indicatum e&longs;t de im <lb/>petu naturali primo vel innato (&longs;ic enim deinceps appellabimus vt recti <lb/>di&longs;tinguamus ab acqui&longs;ito quem vocabimus impetum accelerationis) <lb/>qui &longs;ine motu con&longs;eruatur in corpore grauitante; </s> <s id="N12562"><!-- NEW -->quia ni&longs;i po&longs;&longs;ibilis e&longs;­<lb/>&longs;et motus deor&longs;um nulla e&longs;&longs;et grauitatio; </s> <s id="N12568"><!-- NEW -->quippe grauitare e&longs;t deor­<lb/>&longs;um inclinari, motumque inclinationis impediri; </s> <s id="N1256E"><!-- NEW -->hinc dicemus <pb pagenum="20" xlink:href="026/01/052.jpg"/>in &longs;ecundo libro impetum innatum &longs;æpiùs e&longs;&longs;e &longs;ine motu; cum &longs;cilicet <lb/>impeditur à corpore &longs;u&longs;tinente? </s> <s id="N12579">immò dicemus infrà primo in&longs;tanti, <lb/>quo e&longs;t impetus, nondum e&longs;&longs;e motum. </s> </p> <p id="N1257E" type="main"> <s id="N12580"><!-- NEW -->Ob&longs;eruabis autem certi&longs;&longs;imam regulam; &longs;cilicet ex impo&longs;&longs;ibilitate <lb/>effectus formalis, &longs;equi impo&longs;&longs;ibilitatem cau&longs;æ formalis, huiu&longs;que po&longs;&longs;i­<lb/>bilitatem ex illius po&longs;&longs;ibilitate. </s> </p> <p id="N12588" type="main"> <s id="N1258A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N12596" type="main"> <s id="N12598"><emph type="italics"/>Ni&longs;i e&longs;&longs;et impetus, non e&longs;&longs;et naturaliter motus.<emph.end type="italics"/></s> <s id="N1259F"><!-- NEW --> Quia ni&longs;i e&longs;&longs;et cau&longs;a, non <lb/>e&longs;&longs;et naturaliter effectus per Ax. 8. Impetus enim e&longs;t cau&longs;a motus per <lb/>Th.15. Deinde omnis motus e&longs;t ab aliqua potentia motrice, vt patet ex <lb/>omni hypothe&longs;i; </s> <s id="N125A9"><!-- NEW -->&longs;iue &longs;it naturalis in grauibus, & leuibus, &longs;iue &longs;it vitalis <lb/>in viuentibus; </s> <s id="N125AF"><!-- NEW -->&longs;iue &longs;it media in compre&longs;&longs;is, & dilatatis; &longs;iue alia quæli­<lb/>bet: </s> <s id="N125B5"><!-- NEW -->&longs;ed omnis potentia motrix e&longs;t actiua, quia mouet; </s> <s id="N125B9"><!-- NEW -->ergo agit, &longs;ed <lb/>motum non producit per Th. 2. Igitur impetum, qui deinde exigit mo­<lb/>tum per Th. 14. Dixi naturaliter; </s> <s id="N125C1"><!-- NEW -->quia non e&longs;t dubium, quin Deus &longs;ine <lb/>impetu aliquo modo mouere po&longs;&longs;it; </s> <s id="N125C7"><!-- NEW -->ide&longs;t, facere &longs;ine impetu, vt corpus <lb/>mutet locum: </s> <s id="N125CD"><!-- NEW -->nec dicas Deum non po&longs;&longs;e &longs;upplere vices cau&longs;æ formalis; </s> <s id="N125D1"><!-- NEW --><lb/>nam concedo id quidem pro effectu formali primario; </s> <s id="N125D6"><!-- NEW -->nec enim Deus <lb/>pote&longs;t facere, vt aliquid &longs;it calidum &longs;ine calore; </s> <s id="N125DC"><!-- NEW -->cum e&longs;&longs;e calidum &longs;it <lb/>idem, ac e&longs;&longs;e habens calorem; </s> <s id="N125E2"><!-- NEW -->id tamen nego pro effectu &longs;ecundario, <lb/>quem &longs;cilicet cau&longs;a formalis exigit: </s> <s id="N125E8"><!-- NEW -->Etenim &longs;icut pote&longs;t &longs;ummo iure non <lb/>&longs;atisfacere exigentiæ; </s> <s id="N125EE"><!-- NEW -->ita pote&longs;t id <expan abbr="cõferre">conferre</expan> &longs;ine exigentiâ, quòd cum exi­<lb/>gentia conferre pote&longs;t; &longs;ic pote&longs;t corpus re&longs;oluere &longs;ine calore, mouere <lb/>fine impetu &c. </s> <s id="N125FA"><!-- NEW -->quanquam vt verum fatear non e&longs;&longs;et propriè motus, &longs;ed <lb/>qua&longs;i continuæ reproductionis modus; </s> <s id="N12600"><!-- NEW -->nam motus dicit aliquam pa&longs;­<lb/>&longs;ionem; &longs;cilicet actum entis in potentiâ, vt aiunt. </s> </p> <p id="N12606" type="main"> <s id="N12608"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N12614" type="main"> <s id="N12616"><!-- NEW --><emph type="italics"/>Si e&longs;&longs;et motus naturaliter &longs;ine impetu, corpus per &longs;e ip&longs;um moueretur,<emph.end type="italics"/> id e&longs;t, <lb/>exigeret motum per &longs;uam entitatem; </s> <s id="N12621"><!-- NEW -->quia nullus impetus exigeret; </s> <s id="N12625"><!-- NEW -->ergo <lb/>aliquid aliud, nihil di&longs;tinctum, alioquin e&longs;&longs;et impetus; ergo ip&longs;a corpo­<lb/>ris entitas; quanquam non e&longs;&longs;et motus, vt iam dictum e&longs;t, quia non e&longs;­<lb/>&longs;et pa&longs;&longs;io. </s> </p> <p id="N1262F" type="main"> <s id="N12631"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N1263D" type="main"> <s id="N1263F"><!-- NEW --><emph type="italics"/>Corpus illud æquali &longs;emper motu ferretur per &longs;e<emph.end type="italics"/>; </s> <s id="N12648"><!-- NEW -->Quia e&longs;&longs;et &longs;emper ea­<lb/>dem cau&longs;a nece&longs;&longs;aria motus, id e&longs;t, ip&longs;a entitas corporis; </s> <s id="N1264E"><!-- NEW -->igitur idem <lb/>effectus per Axioma 12. igitur idem, vel æqualis motus: dixi per &longs;e pro­<lb/>pter diuer&longs;um medium. </s> </p> <p id="N12656" type="main"> <s id="N12658"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N12664" type="main"> <s id="N12666"><emph type="italics"/>Si e&longs;&longs;et aliquod corpus e&longs;&longs;entialiter mobile, impetu non indigeret.<emph.end type="italics"/></s> <s id="N1266D"> Probatur; </s> <s id="N12670"><!-- NEW --><lb/>quia in tantum indiget mobile impetu vt impetus exigat motum; </s> <s id="N12675"><!-- NEW -->&longs;ed <lb/>corpus illud per &longs;uam e&longs;&longs;entiam exigeret motum; </s> <s id="N1267B"><!-- NEW -->igitur non indigeret <lb/>impetu; </s> <s id="N12681"><!-- NEW -->po&longs;&longs;et tamen impediri eius motus, vt patet; immò e&longs;&longs;et capax <lb/>recipiendi impetus., &longs;iue quem in ip&longs;o produceret, &longs;iue quem ab alia <pb pagenum="21" xlink:href="026/01/053.jpg"/>cau&longs;a extrin&longs;eca acciperet. </s> </p> <p id="N1268C" type="main"> <s id="N1268E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N1269A" type="main"> <s id="N1269C"><emph type="italics"/>Si e&longs;&longs;et aliquod corpus e&longs;&longs;entialiter immobile, e&longs;&longs;et incapax impetus.<emph.end type="italics"/></s> <s id="N126A3"> Pro­<lb/>batur; quia, ni&longs;i e&longs;&longs;et motus, non e&longs;&longs;et impetus per Th. 17. igitur &longs;ubie­<lb/>ctum incapax motus e&longs;t incapax impetus. </s> </p> <p id="N126AA" type="main"> <s id="N126AC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N126B8" type="main"> <s id="N126BA"><emph type="italics"/>Si e&longs;&longs;et aliquod &longs;ubiectum incapax impetus, e&longs;&longs;et incapax motus.<emph.end type="italics"/></s> <s id="N126C1"> Quia <lb/>vbi non pote&longs;t e&longs;&longs;e cau&longs;a formalis, ibi non pote&longs;t e&longs;&longs;e effectus forma­<lb/>lis, quod certum e&longs;t. </s> </p> <p id="N126C8" type="main"> <s id="N126CA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N126D6" type="main"> <s id="N126D8"><!-- NEW --><emph type="italics"/>Omne corpus, quod e&longs;t capax motus, e&longs;t capax impetus, & vici&longs;&longs;im.<emph.end type="italics"/><lb/>Probatur 1. pars; </s> <s id="N126E2"><!-- NEW -->quia impetus in eo non e&longs;&longs;et fru&longs;trà; haberet enim <lb/>&longs;uum effectum formalem, & finem intrin&longs;ecum. </s> <s id="N126E8">Probatur 2.pars; </s> <s id="N126EB"><!-- NEW -->quia in <lb/>eo impetus non e&longs;&longs;et fru&longs;trà per Ax. 6. igitur haberet &longs;uum effectum; <lb/>igitur motum. </s> </p> <p id="N126F3" type="main"> <s id="N126F5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N12701" type="main"> <s id="N12703"><emph type="italics"/>Omne corpus finitum e&longs;t capax motus, & impetus.<emph.end type="italics"/></s> <s id="N1270A"> Probatur 1. pars; </s> <s id="N1270D"><!-- NEW --><lb/>quia non e&longs;t vbique, igitur pote&longs;t transferri è loco in locum; cur enim <lb/>non po&longs;&longs;et? </s> <s id="N12714"><!-- NEW -->Dices fortè quia affixum e&longs;&longs;et e&longs;&longs;entialiter tali, vel tali lo­<lb/>co, &longs;ed contra; </s> <s id="N1271A"><!-- NEW -->quia de&longs;truantur omnia, præter ip&longs;um corpus; certè <lb/>nulli affixum manet. </s> <s id="N12720">Dices &longs;patio imaginario; apage i&longs;tas nugas: <lb/>de i&longs;to &longs;patio plura demon&longs;trabimus in Metaphy. <!-- KEEP S--></s> <s id="N12726">Probatur 2. pars; </s> <s id="N12729"><!-- NEW -->quia <lb/>&longs;i e&longs;t capax motus, e&longs;t capax impetus per Th. 24. Quod dixi de corpo­<lb/>re; dicendum e&longs;t de omni re creata finita permanente. </s> </p> <p id="N12731" type="main"> <s id="N12733"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N1273F" type="main"> <s id="N12741"><emph type="italics"/>Quod durat tantùm vno in&longs;tanti, e&longs;t incapax motus, & impetus.<emph.end type="italics"/></s> <s id="N12748"><!-- NEW --> Pro­<lb/>batur, quia non e&longs;t moueri, ni&longs;i relinquat locum, & acquirat alium; </s> <s id="N1274E"><!-- NEW -->&longs;ed <lb/>1. acquirere locum, e&longs;t 1. e&longs;&longs;e in illo loco; </s> <s id="N12754"><!-- NEW -->& relinquere locum e&longs;t, <lb/>1. non e&longs;&longs;e in eo loco; </s> <s id="N1275A"><!-- NEW -->nec &longs;imul e&longs;t in vtroque, quia in duobus locis <lb/>idem &longs;imul e&longs;&longs;e non pote&longs;t; vt demon&longs;tramus in Metaphy&longs;ica; </s> <s id="N12760"><!-- NEW -->& phy­<lb/>&longs;icè certum e&longs;t ex omni hypothe&longs;i; </s> <s id="N12766"><!-- NEW -->igitur moueri nunc, id e&longs;t, hoc in­<lb/>&longs;tanti, id e&longs;t, 1. acquirere nouum locum, & 1. relinquere priorem, <lb/>&longs;upponit nece&longs;&longs;ariò antè fui&longs;&longs;e in loco nunc relicto; </s> <s id="N1276E"><!-- NEW -->&longs;ed quod durat <lb/>tantùm in in&longs;tanti, non habet antè, neque po&longs;t; </s> <s id="N12774"><!-- NEW -->igitur quod durat tan­<lb/>tùm vno in&longs;tanti, moueri non pote&longs;t; </s> <s id="N1277A"><!-- NEW -->igitur e&longs;t incapax motus; igitur <lb/>& impetus. </s> </p> <p id="N12780" type="main"> <s id="N12782"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N1278E" type="main"> <s id="N12790"><!-- NEW --><emph type="italics"/>Deus e&longs;t incapax motus, & impetus<emph.end type="italics"/>: </s> <s id="N12799"><!-- NEW -->Tum quia vbique, e&longs;t igitur <lb/>nouum locum acquirere non pote&longs;t; </s> <s id="N1279F"><!-- NEW -->igitur nec moueri per Definitio­<lb/>nem 1. tùm quia æternitas Dei tota &longs;imul e&longs;t; </s> <s id="N127A5"><!-- NEW -->igitur nec fuit antè, ne­<lb/>que po&longs;t in ca; </s> <s id="N127AB"><!-- NEW -->igitur non pote&longs;t dici antè habui&longs;&longs;e locum, quo nunc <lb/>caret: </s> <s id="N127B1"><!-- NEW -->& nunc non habere illum quo caret; </s> <s id="N127B5"><!-- NEW -->tùm quia immutabilitas <pb pagenum="22" xlink:href="026/01/054.jpg"/>Dei hoc prohibet; </s> <s id="N127BE"><!-- NEW -->nam moueri, e&longs;t affici intrin&longs;ecè; </s> <s id="N127C2"><!-- NEW -->quia etiam de­<lb/>&longs;tructis omnibus extrin&longs;ecis creatis moueri po&longs;&longs;em, & fru&longs;trà recurres <lb/>ad partes virtuales immen&longs;itatis Dei, quas ferè animus abhorret; apa­<lb/>ge partes in Deo: quis hoc ferre po&longs;&longs;it? </s> <s id="N127CC"><!-- NEW -->præterea &longs;i &longs;unt, &longs;unt e&longs;&longs;entia­<lb/>liter immobiles; </s> <s id="N127D2"><!-- NEW -->igitur valet &longs;emper ratio allata; </s> <s id="N127D6"><!-- NEW -->igitur Deus e&longs;t inca­<lb/>pax motus; igitur & impetus. </s> </p> <p id="N127DC" type="main"> <s id="N127DE"><!-- NEW -->Diceret aliquis Deum quantumuis Immen&longs;um in orbem conuolui <lb/>po&longs;&longs;e; igitur 1. ratio non probat de omni motu. </s> <s id="N127E4"><!-- NEW -->Re&longs;pondeo adhuc va­<lb/>lere, quia etiam in orbem conuolui non pote&longs;t, ni&longs;i mutetur intrin&longs;e­<lb/>cè; </s> <s id="N127EC"><!-- NEW -->atqui &longs;i e&longs;t immen&longs;us, non pote&longs;t mutari intrin&longs;ecè per motum; </s> <s id="N127F0"><!-- NEW --><lb/>quia nullum locum de nouo acquireret; </s> <s id="N127F5"><!-- NEW -->&longs;ed de hoc motu aliàs, cum de <lb/>infinito; </s> <s id="N127FB"><!-- NEW -->vel de puncto phy&longs;ico mobili; quidquid &longs;it. </s> <s id="N127FF">valet &longs;altem <lb/>1. ratio pro motu recto, & aliæ duæ pro omni motu. </s> </p> <p id="N12804" type="main"> <s id="N12806"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N12812" type="main"> <s id="N12814"><emph type="italics"/>Motus ip&longs;e moueri non pote&longs;t.<emph.end type="italics"/></s> <s id="N1281B"><!-- NEW --> Quia cum tantùm dicat mutationem <lb/>loci; </s> <s id="N12821"><!-- NEW -->certè mutatio non e&longs;t in loco; dicit enim tantùm locum relictum <lb/>eo in&longs;tanti, quo nouus acquiritur. </s> <s id="N12827"><!-- NEW -->Præterea quod e&longs;t in loco dicit tan­<lb/>tùm ens phy&longs;icum; </s> <s id="N1282D"><!-- NEW -->&longs;ed mutatio dicit etiam non ens; <emph type="italics"/>Hinc egregium pa­<lb/>radoxum; illud non mouetur per quod cuncta mouentur, quæ mouentur.<emph.end type="italics"/></s> </p> <p id="N12838" type="main"> <s id="N1283A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N12846" type="main"> <s id="N12848"><emph type="italics"/>Duratio moueri non pote&longs;t.<emph.end type="italics"/></s> <s id="N1284F"> Cum enim &longs;it &longs;ucce&longs;&longs;iua, fluit per partes, <lb/>igitur quælibet illius pars, &longs;eu quod durat vna in&longs;tanti tantùm e&longs;t inca­<lb/>pax motus, per Th. 26. </s> </p> <p id="N12856" type="main"> <s id="N12858"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N12864" type="main"> <s id="N12866"><!-- NEW --><emph type="italics"/>Hinc actio moueri non pote&longs;t<emph.end type="italics"/>; </s> <s id="N1286F"><!-- NEW -->cum enim actio per quam res con&longs;erua­<lb/>tur, &longs;it eius duratio; vt con&longs;tabit ex iis, quæ demon&longs;trabimus in Me­<lb/>taphy&longs;ica, & cum duratio moueri non po&longs;&longs;it, per Th. 29. certè neque <lb/>actio moueri pote&longs;t. </s> </p> <p id="N12879" type="main"> <s id="N1287B"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12888" type="main"> <s id="N1288A"><!-- NEW -->Hinc in tanta rerum creatarum multitudine &longs;unt tantùm duæ, quæ <lb/>&longs;unt e&longs;&longs;entialiter immobiles; &longs;cilicet motus, & actio; </s> <s id="N12890"><!-- NEW -->quorum ille cum <lb/>&longs;it mutatio non e&longs;t adæquatè aliquid po&longs;itiuum; &longs;ecus actio. </s> </p> <p id="N12896" type="main"> <s id="N12898"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N128A5" type="main"> <s id="N128A7"><!-- NEW -->Hinc &longs;unt tantùm duo adæquatè po&longs;itiua, quæ moueri non po&longs;&longs;unt; </s> <s id="N128AB"><!-- NEW --><lb/>&longs;cilicet Deus, & actio; Deus, qui &longs;emper e&longs;t; </s> <s id="N128B0"><!-- NEW -->actio, quæ tantùm vno <lb/>in&longs;tanti e&longs;t; Deus vbique e&longs;&longs;entialiter; actio hic tantum e&longs;&longs;entialiter; </s> <s id="N128B6"><!-- NEW --><lb/>Deus primum ens; actio infinitum ens; </s> <s id="N128BB"><!-- NEW -->e&longs;t enim modus; </s> <s id="N128BF"><!-- NEW -->Deus primum <lb/>mouens; actio ip&longs;e motus; &longs;cilicet primi generis, de quo in &longs;ect. </s> <s id="N128C5">Th.3. </s> </p> <p id="N128C8" type="main"> <s id="N128CA"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N128D7" type="main"> <s id="N128D9"><!-- NEW -->Hinc &longs;i res aliqua creata per actionem tantæ perfectionis, quæ mille <lb/>annis e&longs;&longs;entialiter re&longs;ponderet, con&longs;eruaretur; </s> <s id="N128DF"><!-- NEW -->certè per totum illud <lb/>tempus moueri non po&longs;&longs;et; </s> <s id="N128E5"><!-- NEW -->e&longs;&longs;et enim vnicum in&longs;tans, hoc e&longs;t duratio <pb pagenum="23" xlink:href="026/01/055.jpg"/>tota &longs;imul; </s> <s id="N128EE"><!-- NEW -->&longs;ed eodem in&longs;tanti in pluribus locis e&longs;&longs;e non pote&longs;t; igitur <lb/>nec moueri; </s> <s id="N128F4"><!-- NEW -->adde quod per cam actionem &longs;um in loco, per quam &longs;um <lb/>in tempore; </s> <s id="N128FA"><!-- NEW -->igitur &longs;i hæc e&longs;t &longs;emper eadem, illam eandem e&longs;&longs;e nece&longs;&longs;e <lb/>e&longs;t; &longs;ed hæc &longs;unt metaphy&longs;ica, quæ obiter tantùm attingo, aliàs fusè <lb/>de mon&longs;trabo. </s> </p> <p id="N12902" type="main"> <s id="N12904"><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12910" type="main"> <s id="N12912"><!-- NEW -->Ob&longs;eruabis primò ex dictis præclarum naturæ in&longs;titutum; </s> <s id="N12916"><!-- NEW -->cum enim <lb/>corpus moueri &longs;emper non debeat, (quippe hoc e&longs;&longs;et maximè incom­<lb/>modum) certè per &longs;uam entitatem moueri non exigit; </s> <s id="N1291E"><!-- NEW -->alioquin &longs;emper <lb/>moueretur; </s> <s id="N12924"><!-- NEW -->igitur per aliud ab entitate di&longs;tinctum, id e&longs;t per impetum; </s> <s id="N12928"><!-- NEW --><lb/>itaque licet per &longs;uam entitatem exigat fluxum in tempore, id e&longs;t con&longs;er­<lb/>uari, & durare; </s> <s id="N1292F"><!-- NEW -->id e&longs;t nouam &longs;emper actionem con&longs;eruatiuam; </s> <s id="N12933"><!-- NEW -->quia <lb/>maximum eius bonum e&longs;t durare vel exi&longs;tere; </s> <s id="N12939"><!-- NEW -->Igitur per &longs;e ip&longs;um illud <lb/>exigit; </s> <s id="N1293F"><!-- NEW -->quia &longs;emper exigit, non tamen per &longs;e ip&longs;um exigit fluxum in <lb/>loco, id e&longs;t motum; quia moueri non &longs;emper e&longs;t bonum. </s> </p> <p id="N12945" type="main"> <s id="N12947"><!-- NEW -->Ob&longs;eruabis &longs;ecundò, cum idem corpus aliquando velociùs, tardiùs <lb/>aliquando moueri exigat; </s> <s id="N1294D"><!-- NEW -->&longs;i per &longs;uam entitatem moueri exigeret, eo­<lb/>dem &longs;emper ferretur motu; </s> <s id="N12953"><!-- NEW -->quia eadem &longs;emper e&longs;&longs;et exigentia; </s> <s id="N12957"><!-- NEW -->igitur <lb/>debet e&longs;&longs;e aliquid aliud; </s> <s id="N1295D"><!-- NEW -->illud autem e&longs;t impetus, qui aliquando maior <lb/>&longs;eu perfectior, aliquando verò minor e&longs;t; </s> <s id="N12963"><!-- NEW -->igitur maiorem &longs;eu <expan abbr="velcio­rem">velocio­<lb/>rem</expan> motum aliquando exigit, aliquando minorem, &longs;eu tardiorem; </s> <s id="N1296D"><!-- NEW --><lb/>cum enim motus &longs;it eius finis intrin&longs;ecus, vt re&longs;olutio e&longs;t finis caloris <lb/>vel rarefactio; </s> <s id="N12974"><!-- NEW -->quemadmodum maior calor maiorem exigit, &longs;eu præ­<lb/>&longs;tat re&longs;olutionem; ita & maior, &longs;eu perfectior impetus maiorem, &longs;eu <lb/>velociorem motum exigit. </s> </p> <p id="N1297C" type="main"> <s id="N1297E"><!-- NEW -->Ob&longs;eruabis tertiò aliud naturæ in&longs;titutum, quo &longs;cilicet in eo tan­<lb/>tùm &longs;ubiecto recipi pote&longs;t cau&longs;a formalis, in quo recipi pote&longs;t eius effe­<lb/>ctus formalis &longs;ecundarius: </s> <s id="N12986"><!-- NEW -->nec alia regula, præter eam excogitari pote&longs;t; </s> <s id="N1298A"><!-- NEW --><lb/>cum enim aliqua forma ad talem, vel talem finem à natura in&longs;tituta e&longs;t; </s> <s id="N1298F"><!-- NEW --><lb/>certè propter illum finem e&longs;t, igitur in eo non e&longs;t, in quo &longs;uum finem <lb/>con&longs;equi non pote&longs;t; </s> <s id="N12996"><!-- NEW -->alioquin fru&longs;trà e&longs;&longs;et; </s> <s id="N1299A"><!-- NEW -->& contra in eo e&longs;&longs;e pote&longs;t, <lb/>in quo fru&longs;trà non e&longs;t; </s> <s id="N129A0"><!-- NEW -->cum &longs;cilicet in eo &longs;uum finem con&longs;equatur; </s> <s id="N129A4"><!-- NEW -->ad­<lb/>de quod finis ille intrin&longs;ecus phy&longs;icus &longs;cilicet, non moralis, aliquis no­<lb/>uus effectus e&longs;t; </s> <s id="N129AC"><!-- NEW -->atqui nouus effectus &longs;ine &longs;ua cau&longs;a e&longs;&longs;e non pote&longs;t, neque <lb/>cau&longs;a nece&longs;&longs;aria &longs;ine effectu; igitur ibi, &longs;cilicet in hoc &longs;ubiecto, in quo <lb/>e&longs;t, vel e&longs;&longs;e pote&longs;t effectus formalis, cau&longs;a formalis e&longs;t, vel e&longs;&longs;e pote&longs;t, <lb/>e&longs;t inquam citra miraculum. </s> </p> <p id="N129B6" type="main"> <s id="N129B8">Ob&longs;eruabis quartò egregiam rationem; </s> <s id="N129BB"><!-- NEW -->propter quam res eadem in <lb/>pluribus locis naturaliter e&longs;&longs;e non pote&longs;t; </s> <s id="N129C1"><!-- NEW -->quippe cum res fuerit primo <lb/>producta in aliquo loco, illa certè nouum locum acquirere non pote&longs;t <lb/>naturaliter; </s> <s id="N129C9"><!-- NEW -->ni&longs;i per motum, atqui motus dicit nece&longs;&longs;ario priorem lo­<lb/>tum relictum, & nouum acqui&longs;itum; </s> <s id="N129CF"><!-- NEW -->igitur cum tot acquirantur loca <lb/>per motum, quot relinquuntur; </s> <s id="N129D5"><!-- NEW -->&longs;i ante motum vnus tantùm erat eiu&longs;­<lb/>dem rei locus, po&longs;t motum etiam vnus e&longs;t: </s> <s id="N129DB"><!-- NEW -->quod autem producatur tan-<pb pagenum="24" xlink:href="026/01/056.jpg"/>tùm res in vno loco patet; </s> <s id="N129E4"><!-- NEW -->vel enim à cau&longs;a prima vel ab aliqua 2. pro­<lb/>ducitur; </s> <s id="N129EA"><!-- NEW -->&longs;i à 2. ergo ab aliqua aplicata; </s> <s id="N129EE"><!-- NEW -->igitur ex &longs;uppo&longs;itione quòd il­<lb/>la cau&longs;a 2. in vno tantùm loco producta &longs;it, vni tantum applicari po­<lb/>te&longs;t; </s> <s id="N129F6"><!-- NEW -->quod autem cau&longs;a 1. in pluribus locis naturaliter eundem effectum <lb/>non producat, certum e&longs;t, & demon&longs;trabimus in Metaphy&longs;. quia &longs;in­<lb/>gulis effectibus &longs;ingulæ &longs;ufficiunt actiones; </s> <s id="N129FE"><!-- NEW -->&longs;ingulis terminis &longs;ingulæ <lb/>viæ; </s> <s id="N12A04"><!-- NEW -->immò hoc requiri videtur, &longs;eu &longs;pectare ad huius vniuer&longs;itatis or­<lb/>dinem; </s> <s id="N12A0A"><!-- NEW -->quippe &longs;i res eadem in pluribus locis e&longs;&longs;et; cur potius in duo­<lb/>bus quam in tribus? </s> <s id="N12A10"><!-- NEW -->deinde multiplex iure po&longs;&longs;et exi&longs;timari; </s> <s id="N12A14"><!-- NEW -->denique <lb/>quod vnum e&longs;t in entitate creata, &longs;eu dependente ab eadem cau&longs;a, vnum <lb/>e&longs;t etiam in dependentia; </s> <s id="N12A1C"><!-- NEW -->quæ e&longs;t actio, per quam dependet; &longs;ed de his <lb/>aliàs. </s> </p> <p id="N12A22" type="main"> <s id="N12A24"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N12A30" type="main"> <s id="N12A32"><emph type="italics"/>Impetus non producitur in eo mobili, quod moueri non pote&longs;t à potentia <lb/>motrice applicata, licèt à fortiori moueri po&longs;&longs;it.<emph.end type="italics"/></s> <s id="N12A3B"><!-- NEW --> Probatur, quia impetus <lb/>e&longs;t tantùm propter motum, qui eius effectus e&longs;t, & finis, per Th. 15. <lb/>& 16. Igitur vbi non e&longs;t motus, fru&longs;trà e&longs;t impetus; </s> <s id="N12A43"><!-- NEW -->&longs;ed quod fru&longs;trà <lb/>e&longs;t, non e&longs;t; id e&longs;t non e&longs;t, quod fru&longs;trà e&longs;&longs;et, &longs;i e&longs;&longs;et per Ax. 6. Exci­<lb/>pio tamen impetum naturalem innatum, qui nunquam e&longs;t fru&longs;trà, vt <lb/>dictum e&longs;t &longs;uprà in Theorem. <!-- KEEP S--></s> <s id="N12A4E"><!-- NEW -->17. adde quod non pote&longs;t cogno&longs;ci <lb/>impetus, ni&longs;i vel ex motu, vel ex ictu, vel ex contrario ni&longs;u, vel <lb/>impul&longs;u; </s> <s id="N12A56"><!-- NEW -->&longs;ed nihil horum cernitur in rupe quam ferio; </s> <s id="N12A5A"><!-- NEW -->Igitur non <lb/>e&longs;t dicendum in ea produci impetum, cuius rationem afferemus infrà; </s> <s id="N12A60"><!-- NEW --><lb/>nunc &longs;atis e&longs;t Ax. 3. id manife&longs;tè probari; </s> <s id="N12A65"><!-- NEW -->nam qui diceret in rupe im­<lb/>mobili impetum imprimi; </s> <s id="N12A6B"><!-- NEW -->certè po&longs;itiuo argumento probare tenere­<lb/>tur, quod tantùm duci pote&longs;t, vel ab experimento; </s> <s id="N12A71"><!-- NEW -->atqui hîc nullum e&longs;t; </s> <s id="N12A75"><!-- NEW --><lb/>vel à nece&longs;&longs;itate, quæ nulla e&longs;t; </s> <s id="N12A7A"><!-- NEW -->vel ex alio quocumque capite, quod <lb/>nullum excogitari pote&longs;t; </s> <s id="N12A80"><!-- NEW -->&longs;ed maiorem lucem huic Th. 3. ex proximè <lb/>&longs;equentibus accer&longs;emus; </s> <s id="N12A86"><!-- NEW -->nec e&longs;t quòd aliqui dicant produci impetum <lb/>inefficacem; </s> <s id="N12A8C"><!-- NEW -->qui cum fru&longs;trà &longs;it, &longs;i e&longs;t, ex nullo capite probari pote&longs;t: </s> <s id="N12A90"><!-- NEW -->ad­<lb/>de quòd de&longs;truitur impetus, ne &longs;it fru&longs;trà; </s> <s id="N12A96"><!-- NEW -->Igitur non producitur, ne &longs;it <lb/>fru&longs;trà; </s> <s id="N12A9C"><!-- NEW -->nam con&longs;eruatio e&longs;t vera actio, vt dicemus &longs;uo loco; </s> <s id="N12AA0"><!-- NEW -->Igitur &longs;i <lb/>hæc non ponitur, ne aliquid &longs;it fru&longs;trà; </s> <s id="N12AA6"><!-- NEW -->etiam 1. productio poni non <lb/>debet; vnde commentum illud impetus inefficacis pror&longs;us inefficax e&longs;t. </s> </p> <p id="N12AAC" type="main"> <s id="N12AAE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N12ABA" type="main"> <s id="N12ABC"><emph type="italics"/>Ideo potentia motrix non producit impetum in prædicta rupe.<emph.end type="italics"/> v.g. <emph type="italics"/>quia de­<lb/>bilior e&longs;t.<emph.end type="italics"/></s> <s id="N12ACB"> Probatur, & explicatur; quippe debilior potentia minorem ef­<lb/>fectum producit per. </s> <s id="N12AD0"><!-- NEW -->Ax. 13. <emph type="italics"/>num.<emph.end type="italics"/> 2. igitur pauciores partes impetus <lb/>æquales vni certæ per idem <emph type="italics"/>num.<emph.end type="italics"/> 1. igitur &longs;i &longs;int plures partes &longs;ubiecti <lb/>mobilis, &longs;eu rupis, quàm impetus; </s> <s id="N12AE4"><!-- NEW -->cum vna pars impetus duobus parti­<lb/>bus &longs;ubiecti ine&longs;&longs;e non po&longs;&longs;it; </s> <s id="N12AEA"><!-- NEW -->licet plures vni &longs;imul in e&longs;&longs;e po&longs;&longs;int; </s> <s id="N12AEE"><!-- NEW --><lb/>non e&longs;t mirum &longs;i nullus impetus producatur; </s> <s id="N12AF3"><!-- NEW -->cum non po&longs;&longs;int tot partes <lb/>illius produci, quot e&longs;&longs;ent nece&longs;&longs;ariæ; vt &longs;altem &longs;ingulæ &longs;ingulis &longs;ubie­<lb/>cti, &longs;eu rupis partibus di&longs;tribuerentur. </s> </p> <pb pagenum="25" xlink:href="026/01/057.jpg"/> <p id="N12AFF" type="main"> <s id="N12B01"><!-- NEW -->Ob&longs;eruabis autem nouum quoddam genús re&longs;i&longs;tentiæ; </s> <s id="N12B05"><!-- NEW -->nam &longs;ingulæ <lb/>partes rupis ab applicata potentiâ aptæ &longs;unt loco moueri per impre&longs;­<lb/>&longs;um impetum, & maior potentia &longs;imul omnes loco moueret; </s> <s id="N12B0D"><!-- NEW -->at verò <lb/>omnes &longs;imul, & coniunctim con&longs;ideratæ; </s> <s id="N12B13"><!-- NEW -->quatenus &longs;cilicet vna pars <lb/>non pote&longs;t moueri &longs;ine alia, & comparatæ cum illa potentia debili di­<lb/>cuntur habere prædictam re&longs;i&longs;tentiam, quæ &longs;uperat potentiæ vires; </s> <s id="N12B1B"><!-- NEW --><lb/>quòd &longs;cilicet à maiori moueri tantùm po&longs;&longs;int; quia plures partes im­<lb/>petus po&longs;tulantur, quam &longs;int eæ, quæ à prædictâ potentiâ po&longs;&longs;unt pro­<lb/>duci. </s> </p> <p id="N12B24" type="main"> <s id="N12B26"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N12B32" type="main"> <s id="N12B34"><!-- NEW --><emph type="italics"/>Vel producitur impetus in omnibus &longs;ubiecti partibus vnitis, vel in nulla; </s> <s id="N12B3A"><!-- NEW --><lb/>modò nulla fiat &longs;eparatio, neque compre&longs;&longs;io<emph.end type="italics"/>: Certum e&longs;t enim in ijs omni­<lb/>bus partibus, quæ auolant ab ictu, produci impetum. </s> <s id="N12B44"><!-- NEW -->Probatur igitur <lb/>1. quia &longs;i non producatur in omnibus partibus, & nulla &longs;eparetur ab <lb/>alijs; </s> <s id="N12B4C"><!-- NEW -->certè nulla mouetur, vt certum e&longs;t; </s> <s id="N12B50"><!-- NEW -->igitur nulla habet impetum; </s> <s id="N12B54"><!-- NEW --><lb/>quia ibi non e&longs;t cau&longs;a formalis, vbi non e&longs;t effectus formalis; alioquin <lb/>e&longs;&longs;et fru&longs;trà, contra Ax. 6.2. </s> <s id="N12B5B"><!-- NEW -->Tu dicis produci impetum in aliquot parti­<lb/>bus; hoc dicis, hoc proba? </s> <s id="N12B61">an potes digno&longs;cere impetum ni&longs;i ex motu? </s> <s id="N12B64"><lb/>vel con&longs;eruaretur hîc impetus &longs;equentibus in&longs;tantibus, vel &longs;tatim &longs;ecun­<lb/>do in&longs;tanti de&longs;trueretur. </s> <s id="N12B6A">Primum dicere ab&longs;urdum e&longs;t; </s> <s id="N12B6D"><!-- NEW -->quia &longs;i hoc e&longs;&longs;et <lb/>multisictibus repetitis tandem moueretur totum mobile; &longs;i verò de­<lb/>&longs;trui dicatur. </s> <s id="N12B75">Secundo in&longs;tanti; eadem ratio probat non produci. </s> <s id="N12B78">Pri­<lb/>mo in&longs;tanti, quæ probat de&longs;trui. </s> <s id="N12B7D">Secundo nam ideo de&longs;truitur. </s> <s id="N12B80">Secun­<lb/>do quia e&longs;t fru&longs;trà, &longs;ed non minus e&longs;t fru&longs;trà. </s> <s id="N12B85">Primo igitur non produ­<lb/>citur. </s> <s id="N12B8A">Primo 4. probatur; </s> <s id="N12B8D"><!-- NEW -->quia cum non &longs;ufficiant partes impetus, quas <lb/>dixi produci, vt omnibus partibus &longs;ubiecti di&longs;tribuantur; </s> <s id="N12B93"><!-- NEW -->certè non e&longs;t <lb/>vlla ratio, cur potiùs his quàm illis di&longs;tribui dicantur; cum vna &longs;it tan­<lb/>tùm immediatè applicata. </s> <s id="N12B9B">Igitur certum e&longs;t vel produci in omnibus, vel <lb/>in nulla, ni&longs;i forte aliquæ auolent, &longs;ed tunc &longs;eparantur. </s> </p> <p id="N12BA0" type="main"> <s id="N12BA2"><!-- NEW -->Obiiciet aliquis 1. e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam applicatam &longs;ubiecto ap­<lb/>to: </s> <s id="N12BA8"><!-- NEW -->igitur agit per Ax. 12. Re&longs;pondeo e&longs;&longs;e impeditam; </s> <s id="N12BAC"><!-- NEW -->nam re&longs;i&longs;tentia <lb/>&longs;ubiecti &longs;uperat vires potentiæ vt dictum e&longs;t; </s> <s id="N12BB2"><!-- NEW -->immò in ip&longs;o motu re­<lb/>torqueo argumentum; licèt enim &longs;it applicata cau&longs;a nece&longs;&longs;aria mouens, <lb/>non tamen mouet. </s> </p> <p id="N12BBA" type="main"> <s id="N12BBC"><!-- NEW -->Obiiciet 2. Ignis applicatus agit in nonnullas partes &longs;ubiecti, licèt <lb/>non agat in omnes; igitur & potentia motrix. </s> <s id="N12BC2"><!-- NEW -->Re&longs;pondeo non e&longs;&longs;e pa­<lb/>ritatem; </s> <s id="N12BC8"><!-- NEW -->quia vna pars pote&longs;t calefieri, & re&longs;olui &longs;ine alia, vt con&longs;tat <lb/>non tamen vna moueri &longs;ine alia, cui coniuncta e&longs;t, ni&longs;i &longs;eparetur; igi­<lb/>tur nec recipere impetum &longs;ine alia. </s> </p> <p id="N12BD0" type="main"> <s id="N12BD2">Obiiciet. </s> <s id="N12BD5"><!-- NEW -->3. &longs;int duo trahentes idem mobile; </s> <s id="N12BD9"><!-- NEW -->ita vt &longs;eor&longs;im neuter <lb/>trahere po&longs;&longs;it, coniunctim verò vterque po&longs;&longs;it; </s> <s id="N12BDF"><!-- NEW -->certè &longs;i alter non pro­<lb/>ducit impetum &longs;eor&longs;im, nec etiam coniunctim producet; </s> <s id="N12BE5"><!-- NEW -->nec enim au­<lb/>gentur eius vires ab altero: </s> <s id="N12BEB"><!-- NEW -->Re&longs;pondeo vtrunque agere actione com­<lb/>muni; igitur non e&longs;t mirum &longs;i effectus maior e&longs;t, quem tamen neuter <pb pagenum="26" xlink:href="026/01/058.jpg"/>&longs;eor&longs;im producere pote&longs;t. </s> </p> <p id="N12BF6" type="main"> <s id="N12BF8"><!-- NEW -->Dices &longs;i vterque coniunctim producit effectum: </s> <s id="N12BFC"><!-- NEW -->&longs;int v. <!-- REMOVE S-->g. <!-- REMOVE S-->100. par­<lb/>tes impetus; Igitur &longs;inguli producunt tantùm 50. Igitur cur potiùs in <lb/>in his partibus &longs;ubiecti, quàm in alijs, cum vtriu&longs;que potentia eidem <lb/>&longs;ubiecti parti po&longs;&longs;et e&longs;&longs;e applicata? </s> <s id="N12C0A"><!-- NEW -->Re&longs;pondeo &longs;ingulos producere 100. <lb/>actione &longs;cilicet communi indiui&longs;ibiliter; </s> <s id="N12C10"><!-- NEW -->&longs;int enim duo trahentes A. & <lb/>B. A. producit 100. &longs;ed non &longs;olus; </s> <s id="N12C16"><!-- NEW -->B. producit ea&longs;dem 100. &longs;ed non &longs;o­<lb/>lus; &longs;ed explicabimus hunc modum actionis communis in Metaphys. <!-- REMOVE S--><lb/>quod autem agant actione communi patet per Ax. 13. </s> </p> <p id="N12C1F" type="main"> <s id="N12C21"><!-- NEW -->Obiicies 4. producitur &longs;onus &longs;i ferias rupem; </s> <s id="N12C25"><!-- NEW -->igitur & impetus; </s> <s id="N12C29"><!-- NEW -->Re&longs;­<lb/>pondeo ad &longs;onum &longs;olam aëris colli&longs;ionem &longs;ufficere, quam fieri certum <lb/>e&longs;t à prædicto ictu; </s> <s id="N12C31"><!-- NEW -->deinde mallej motus impacti in rupem facit &longs;onum; </s> <s id="N12C35"><!-- NEW --><lb/>quidquid tandem &longs;it &longs;onus, de quo hîc non di&longs;puto: </s> <s id="N12C3A"><!-- NEW -->adde quod in ru­<lb/>pe &longs;unt &longs;emper aliquæ partes tremulæ, quæ modico tantùm, eoque flexi­<lb/>bili nexu cum alijs partibus copulantur; adde aliquam compre&longs;&longs;ionem, <lb/>ex qua modicæ vibrationes &longs;equuntur. </s> </p> <p id="N12C44" type="main"> <s id="N12C46"><!-- NEW -->Obiicies 5. Quando aliquæ partes auolant ab ictu, haud dubiè auo­<lb/>lant propter impetum impre&longs;&longs;um: </s> <s id="N12C4C"><!-- NEW -->Igitur prius e&longs;t imprimi impetum, <lb/>quàm auolare; igitur productus e&longs;t impetus in nonnullis partibus, & <lb/>non in aliis, cum quibus illæ &longs;unt coniunctæ. </s> <s id="N12C54"><!-- NEW -->Re&longs;pondeo equidem im­<lb/>petum produci in illis partibus antequam auolent; </s> <s id="N12C5A"><!-- NEW -->&longs;ed ideo produci vt <lb/>deinde auolent nam tota ratio cur non producatur, e&longs;t ne &longs;it fru&longs;trà; </s> <s id="N12C60"><!-- NEW -->&longs;ed <lb/>&longs;i auolent aliquæ partes: certè in ijs non e&longs;t fru&longs;trà, in quibus habet <lb/>&longs;uum effectum, id e&longs;t, motum. </s> </p> <p id="N12C68" type="main"> <s id="N12C6A">Dices; </s> <s id="N12C6D"><!-- NEW -->igitur primo in&longs;tanti impetus ille e&longs;t fru&longs;trà; </s> <s id="N12C71"><!-- NEW -->in quo non <lb/>habet &longs;uum effectum; </s> <s id="N12C77"><!-- NEW -->Re&longs;pondeo nunquam primo in&longs;tanti e&longs;&longs;e fru&longs;trà, <lb/>modò &longs;it motus &longs;ecundo cum etiam primo in&longs;tanti, quo e&longs;t impetus, <lb/>non po&longs;&longs;it e&longs;&longs;e motus, vt demon&longs;trabo infrà; immò ideo ponitur im­<lb/>petus primo vt &longs;it motus &longs;ecundo exigendo pro in&longs;tant &longs;equenti, de <lb/>cum impetus ponat tantùm motum quo aliàs. </s> </p> <p id="N12C83" type="main"> <s id="N12C85">Dices; </s> <s id="N12C88"><!-- NEW -->&longs;ed potentia motrix ne&longs;cit an po&longs;&longs;it pars aliqua mobilis &longs;epa­<lb/>rari; igitur non e&longs;t quòd aliquando producat impetum, aliquando <lb/>non producat. </s> <s id="N12C90"><!-- NEW -->Re&longs;pondeo non &longs;tare per cau&longs;am nece&longs;&longs;ariam, quin &longs;em­<lb/>per agat; </s> <s id="N12C96"><!-- NEW -->&longs;ed per &longs;ubiectum, quod &longs;i aptum e&longs;t, & capax effectus; </s> <s id="N12C9A"><!-- NEW -->haud <lb/>dubiè eo ip&longs;o cau&longs;a nece&longs;&longs;aria applicata in ip&longs;um aget; &longs;i verò ineptum. </s> <s id="N12CA0"><lb/>haud dubiè non aget; </s> <s id="N12CA4"><!-- NEW -->nam ad hoc vt producatur effectus in &longs;ubiecto; </s> <s id="N12CA8"><!-- NEW --><lb/>non &longs;atis e&longs;t cau&longs;am po&longs;&longs;e producere, ni&longs;i etiam &longs;ubiectum po&longs;&longs;it recipe­<lb/>re; </s> <s id="N12CAF"><!-- NEW -->igitur cum &longs;it talis ordo à natura in&longs;titutus, ne aliquid &longs;it fru&longs;trà; </s> <s id="N12CB3"><!-- NEW --><lb/>certè &longs;i impetus producibilis &longs;it futurus fru&longs;trà, hauddubiè non produ­<lb/>cetur; &longs;ecus verò &longs;i fru&longs;trà non &longs;it futurus, in quo non e&longs;t difficultas. </s> </p> <p id="N12CBA" type="main"> <s id="N12CBC"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12CC8" type="main"> <s id="N12CCA">Ob&longs;eruabis 1. vix fieri po&longs;&longs;e quin &longs;emper aliquæ partes &longs;eparentur, <lb/>comprimantur, vel dilatentur, vt patet experientiâ. </s> </p> <p id="N12CCF" type="main"> <s id="N12CD1"><!-- NEW -->Ob&longs;eruabis 2. etiam maximam corporis molem à debili potentia mi-<pb pagenum="27" xlink:href="026/01/059.jpg"/>nimo etiam ictu moueri; </s> <s id="N12CDA"><!-- NEW -->quod etiam ob&longs;eruauit Galileus in &longs;uis dialo­<lb/>gis de motu; </s> <s id="N12CE0"><!-- NEW -->quem certè motum ob&longs;eruabis etiam in&longs;en&longs;ibilem, tùm <lb/>operâ radij luminis repercu&longs;&longs;i, & ad aliquod interuallum proiecti; </s> <s id="N12CE6"><!-- NEW -->tùm <lb/>operâ &longs;eu pi&longs;orum in tympani membranâ tremulo qua&longs;i motu &longs;ub&longs;ul­<lb/>tantium; quâ etiam arte deprehenditur in arce ob&longs;e&longs;&longs;a, &longs;ub quam muri <lb/>partem cuniculi agantur. </s> </p> <p id="N12CF0" type="main"> <s id="N12CF2"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12CFF" type="main"> <s id="N12D01"><!-- NEW -->Hinc egregia ratio erui pote&longs;t, cur ingens corporis moles à debili po­<lb/>tentia loco moueri non po&longs;&longs;it; </s> <s id="N12D07"><!-- NEW -->cum enim tot &longs;altem requirantur partes <lb/>impetus, quot &longs;unt partes &longs;ubiecti: </s> <s id="N12D0D"><!-- NEW -->quia vel in omnibus, vel in nulla <lb/>producitur; </s> <s id="N12D13"><!-- NEW -->certè cum &longs;int plures partes &longs;ubiecti, quàm vt in &longs;ingulis <lb/>ab ea dumtaxat potentiâ impetus produci po&longs;&longs;it; quid mirum e&longs;t, &longs;i mo­<lb/>ueri non po&longs;&longs;it. </s> </p> <p id="N12D1B" type="main"> <s id="N12D1D"><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12D2A" type="main"> <s id="N12D2C"><!-- NEW -->Hinc certa ratio alterius vulgaris effectus potentiæ motricis, quæ lapi­<lb/>dem 40. librarum tardo tantùm motu impellit, etiam cum &longs;ummo ni&longs;u, <lb/>cum tamen &longs;axo vnius libræ velociorem motum imprimat; </s> <s id="N12D34"><!-- NEW -->quia &longs;cilicet <lb/>partes impetus producti di&longs;tribuuntur pluribus partibus &longs;ubiecti in ma­<lb/>iori lapide, & paucioribus in minori; </s> <s id="N12D3C"><!-- NEW -->igitur &longs;ingulæ partes minoris <lb/>habent plures partes impetus, vt manife&longs;tè con&longs;tat; </s> <s id="N12D42"><!-- NEW -->ergo ille impetus <lb/>inten&longs;ior e&longs;t; igitur maiorem exigit &longs;eu perfectiorem motum per Ax. <!-- REMOVE S--><lb/>13. num.2. </s> </p> <p id="N12D4B" type="main"> <s id="N12D4D"><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12D5A" type="main"> <s id="N12D5C"><!-- NEW -->Hinc &longs;ublata ratione diuer&longs;æ re&longs;i&longs;tentiæ medij, dato pondere <lb/>mobilis vtriu&longs;que, datoque ni&longs;u communi potentiæ, pote&longs;t de­<lb/>terminari certus velocitatis gradus vtriu&longs;que; </s> <s id="N12D64"><!-- NEW -->nam ratio velocitatum <lb/>e&longs;t inuer&longs;a ponderum v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it pondùs 4. librarum; </s> <s id="N12D6E"><!-- NEW -->fit etiam 2. librarum <lb/>&longs;it impetus impre&longs;&longs;us vtrique &longs;uppo&longs;ito communi, & æquali ni&longs;u <lb/>potentiæ, & æquali tempore; </s> <s id="N12D76"><!-- NEW -->haud dubiè velocitas mobilis 2. libra­<lb/>rum erit dupla velocitatis mobilis 4. librarum; </s> <s id="N12D7C"><!-- NEW -->quia cum &longs;int duplo <lb/>plures partes &longs;ubiecti in hoc mobili quàm in illo (accipio enim vtrum­<lb/>que eiu&longs;dem materiæ, vt omnes lites fugiam) igitur in minori e&longs;t duplo <lb/>inten&longs;ior impetus: Igitur duplo velocior motus; </s> <s id="N12D86"><!-- NEW -->dixi, &longs;i fiat æquali <lb/>ni&longs;u, & æquali tempore; </s> <s id="N12D8C"><!-- NEW -->quia reuerâ non fit in tempore æquali, &longs;ed <lb/>inæquali, &longs;i &longs;upponatur idem arcus brachij v. <!-- REMOVE S-->g. <!-- REMOVE S-->iacientis; </s> <s id="N12D96"><!-- NEW -->nam tempo­<lb/>ra &longs;unt in ratione &longs;ubduplicata ponderum; vt demon&longs;trabimus lib. 10. <lb/>& velocitates &longs;unt vt tempora permutando. </s> </p> <p id="N12D9E" type="main"> <s id="N12DA0"><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12DAD" type="main"> <s id="N12DAF"><!-- NEW -->Hinc facilè determinari pote&longs;t proportio impetus impre&longs;&longs;i cognitâ <lb/>grauitate mobilium; </s> <s id="N12DB5"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it mobile graue vt4. & aliud graue vt 2. haud <lb/>dubiè vt moueatur æquali gradu velocitatis, debet produci duplo <lb/>maior impetus in maiori mobili, hoc e&longs;t, iuxta rationem maioris ad mi­<lb/>nus, quod clari&longs;&longs;imè &longs;equitur ex dictis; </s> <s id="N12DC3"><!-- NEW -->vt enim tot &longs;int gradus impetus <pb pagenum="28" xlink:href="026/01/060.jpg"/>in qualibet parte minoris, quot &longs;unt in qualibet parte minoris; </s> <s id="N12DCC"><!-- NEW -->haud <lb/>dubiè impetus maioris habet eandem rationem ad impetum minoris; <lb/>quam habet maius ad minus. </s> </p> <p id="N12DD4" type="main"> <s id="N12DD6"><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12DE3" type="main"> <s id="N12DE5"><!-- NEW -->Hinc quoque ducitur manife&longs;ta ratio &longs;eu re&longs;pon&longs;io ad illud præcla­<lb/>rum certè quorundam philo&longs;ophorum <expan abbr="comm&etilde;tum">commentum</expan>, qui volunt ex mini­<lb/>ma ponderis acce&longs;&longs;ione totam terræ molem inclinari, vt in nouo æqui­<lb/>librio &longs;tatuatur; quod omninò fal&longs;um e&longs;t; </s> <s id="N12DF3"><!-- NEW -->nam ex &longs;uppotione quòd <lb/>terra non grauitet (vt vulgò dicitur, & aliàs à nobis <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan>) illa <lb/>certè moueri non pote&longs;t ni&longs;i producantur tot partes impetus quot &longs;unt <lb/>partes &longs;ubiecti in tota terra; quæ certè maximas <expan abbr="pot&etilde;tiæ">potentiæ</expan> vires po&longs;tulant. </s> </p> <p id="N12E05" type="main"> <s id="N12E07"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N12E13" type="main"> <s id="N12E15"><emph type="italics"/>Primo in&longs;tanti, quo est impetus, non est ille motus, cuius hic impetus e&longs;t <lb/>cau&longs;a.<emph.end type="italics"/></s> <s id="N12E1E"> Probatur; </s> <s id="N12E21"><!-- NEW -->quia non pote&longs;t e&longs;&longs;e motus, ni&longs;i &longs;it locus prior reli­<lb/>ctus, & nouus acqui&longs;itus, igitur &longs;i eodem in&longs;tanti, quo e&longs;t impetus, <lb/>haberet motum, eodem in&longs;tanti e&longs;&longs;et in duobus locis, quod dici non <lb/>pote&longs;t; & iam diximus in Th. 26. igitur impetus primo in&longs;tanti quo <lb/>e&longs;t non habet &longs;uum motum. </s> </p> <p id="N12E2D" type="main"> <s id="N12E2F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N12E3B" type="main"> <s id="N12E3D"><emph type="italics"/>Immò nihil e&longs;t, quod primo in&longs;tanti, quo e&longs;t, moueri po&longs;&longs;it.<emph.end type="italics"/></s> <s id="N12E44"><!-- NEW --> Quia non pote&longs;t <lb/>moueri, ni&longs;i acquirat nouum locum, & priorem relinquat; </s> <s id="N12E4A"><!-- NEW -->igitur, vel &longs;i­<lb/>mul in vtroque e&longs;t, quod dici non pote&longs;t; </s> <s id="N12E50"><!-- NEW -->vel in relicto antè fuit; igitur <lb/>non e&longs;t primum in&longs;tans, contra &longs;uppo&longs;itionem. </s> </p> <p id="N12E56" type="main"> <s id="N12E58"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N12E64" type="main"> <s id="N12E66"><emph type="italics"/>Potest impetus aliquo in&longs;tanti non moueri quo mouetur ip&longs;um mobile, in <lb/>quo est.<emph.end type="italics"/></s> <s id="N12E6F"><!-- NEW --> Nam moueatur mobile quodlibet; </s> <s id="N12E73"><!-- NEW -->& dum mouetur, impella­<lb/>tur, factâ &longs;cilicet acce&longs;&longs;ione noui impetus; haud dubiè hoc primo in­<lb/>&longs;tanti, quo producitur impetus in dato mobili non mouetur per Th. <!-- REMOVE S--><lb/>35. quo tamen in&longs;tanti mouetur prædictum mobile. </s> </p> <p id="N12E7E" type="main"> <s id="N12E80"><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12E8D" type="main"> <s id="N12E8F"><!-- NEW -->Hinc egregium paradoxon; <emph type="italics"/>Pote&longs;t alique in&longs;tanti moueri &longs;ubiectum, licèt <lb/>non moueantur illa omnia, que eidem &longs;ubiecto reuerâ in&longs;unt.<emph.end type="italics"/></s> </p> <p id="N12E9A" type="main"> <s id="N12E9C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12EA9" type="main"> <s id="N12EAB"><!-- NEW -->Hinc etiam aliud paradoxon; </s> <s id="N12EAF"><!-- NEW --><emph type="italics"/>Impetus primo in&longs;tanti, quo e&longs;t, non habet <lb/>&longs;uum finem, nec habere pote&longs;t<emph.end type="italics"/>; patet, quia primo in&longs;tanti non habet <expan abbr="motũ">motum</expan>. </s> </p> <p id="N12EBE" type="main"> <s id="N12EC0"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12ECD" type="main"> <s id="N12ECF"><!-- NEW -->Hinc pote&longs;t aliquid dato in&longs;tanti carere &longs;uo fine; </s> <s id="N12ED3"><!-- NEW -->licèt non &longs;it fru&longs;trà; <lb/>fru&longs;trâ enim tantùm dicitur ille impetus, qui pro in&longs;tanti &longs;equenti <lb/>non pote&longs;t habere motum. </s> </p> <p id="N12EDB" type="main"> <s id="N12EDD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N12EE9" type="main"> <s id="N12EEB"><emph type="italics"/>Impetus pars recepta in parte &longs;ubiecti non exigit motum aliarum partium<emph.end type="italics"/><pb pagenum="29" xlink:href="026/01/061.jpg"/><emph type="italics"/>eiu&longs;dem &longs;ubiecti, licèt coniunctarum.<emph.end type="italics"/></s> <s id="N12EFB"><!-- NEW --> Probatur 1. quia alioquin vna pars <lb/>impetus &longs;ufficeret ad mouendam ingentem rupem; quod ab&longs;urdum e&longs;t. </s> <s id="N12F01"><!-- NEW --><lb/>2. &longs;icut vna pars caloris non re&longs;oluit alias partes &longs;ubiecti; ita nec im­<lb/>petus. </s> <s id="N12F08"><!-- NEW -->3. Ratio à priori e&longs;t; quia impetus non e&longs;t cau&longs;a efficiens motus <lb/>per Th. 13. &longs;ed tantùm cau&longs;a formalis per Th. 15. Igitur præ&longs;tat tantùm <lb/>&longs;uum effectum formalem in eo &longs;ubiecto, in quo e&longs;t. </s> </p> <p id="N12F10" type="main"> <s id="N12F12"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12F1F" type="main"> <s id="N12F21"><!-- NEW -->Hinc partes impetus non cau&longs;ant motum in &longs;uo &longs;ubiecto actione, vel <lb/>exigentia communi; </s> <s id="N12F27"><!-- NEW -->quia quælibet pars impetus exigit tantùm motum <lb/>&longs;ui &longs;ubiecti; </s> <s id="N12F2D"><!-- NEW -->id e&longs;t illius partis, quàm afficit; quod etiam probatur per <lb/>Ax. 13. </s> </p> <p id="N12F33" type="main"> <s id="N12F35"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N12F42" type="main"> <s id="N12F44"><!-- NEW -->Hinc corpus grauius per&longs;e, &longs;altem eiu&longs;dem materiæ, non cadit velo­<lb/>ciùs, quàm leuius, vti globus plumbeus 100. librarum, quàm globus <lb/>vnius libræ plumbeus; </s> <s id="N12F4C"><!-- NEW -->quia &longs;cilicet impetus vnius partis non iuuat mo­<lb/>tum alterius: </s> <s id="N12F52"><!-- NEW -->præterea tam facilè 2, partes impetus in 2. partibus &longs;ubie­<lb/>cti receptæ ea&longs;dem mouent, quàm 100. alias 100. dixi per &longs;e; </s> <s id="N12F58"><!-- NEW -->nam di­<lb/>uer&longs;a e&longs;&longs;e pote&longs;t medij re&longs;i&longs;tentia; &longs;ed de his fu&longs;e in 2. lib. <!-- REMOVE S--><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N12F69" type="main"> <s id="N12F6B"><!-- NEW --><emph type="italics"/>Impetus recipitur tantùm in ip&longs;a &longs;ub&longs;tantia &longs;ubiecti naturaliter.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i <lb/>mobile &longs;it ferrum calidum, recipitur in ip&longs;a &longs;ub&longs;tantia ferri; </s> <s id="N12F7A"><!-- NEW -->non verò <lb/>in ip&longs;o calore (ex &longs;uppo&longs;itione quod calor &longs;it accidens, vt aliàs demon­<lb/>&longs;trabimus; </s> <s id="N12F82"><!-- NEW -->nec in alijs accidentibus, &longs;i quæ &longs;unt, in eodem &longs;ubiecto; </s> <s id="N12F86"><!-- NEW -->pro­<lb/>batur 1. quia &longs;i produceretur etiam impetus in accidentibus, quo plu­<lb/>ra e&longs;&longs;ent accidentia in aliquo &longs;ubiecto; </s> <s id="N12F8E"><!-- NEW -->plures quoque partes impetus <lb/>producendæ e&longs;&longs;ent; igitur maiori potentiâ opus e&longs;&longs;et per Ax. 13. n. </s> <s id="N12F94">4. <lb/>Igitur difficiliùs mouerentur, quod e&longs;t ab&longs;urdum. </s> <s id="N12F99"><!-- NEW -->Diceret fortè ali­<lb/>quis eundem impetum recipi &longs;imul in &longs;ub&longs;tantia & in ip&longs;is accidenti­<lb/>bus; </s> <s id="N12FA1"><!-- NEW -->&longs;ed contra, nam reuera, &longs;i hoc e&longs;&longs;et, dum proijcitur ferrum cali­<lb/>dum, & &longs;tatim frigefit, de&longs;trueretur totus impetus, de&longs;tructo &longs;cilicet <lb/>eius &longs;ubiecto: </s> <s id="N12FA9"><!-- NEW -->2. qui hoc diceret, deberet probare; </s> <s id="N12FAD"><!-- NEW -->nam eodem modo <lb/>mouetur corpus &longs;iue afficiatur pluribus accidentibus, &longs;iue paucioribus; <lb/>igitur non euincit experientia recipi in illis impetum, nec etiam ratio, <lb/>vt dicam paulò po&longs;t. </s> <s id="N12FB7"><!-- NEW -->Ratio à priori e&longs;&longs;e pote&longs;t; </s> <s id="N12FBB"><!-- NEW -->quia accidens cum &longs;uo <lb/>&longs;ubiecto coniunctum exigit &longs;emper e&longs;&longs;e præ&longs;ens &longs;ubiecto, cum natura­<lb/>liter extra &longs;ubiectum exi&longs;tere non po&longs;&longs;it; </s> <s id="N12FC3"><!-- NEW -->igitur cum exigat con&longs;erua­<lb/>ri, & exi&longs;tere; </s> <s id="N12FC9"><!-- NEW -->eo tantùm modo, quo pote&longs;t naturaliter con&longs;eruari & <lb/>exi&longs;tere; </s> <s id="N12FCF"><!-- NEW -->certè exigit con&longs;eruari, & ine&longs;&longs;e &longs;ubiecto; </s> <s id="N12FD3"><!-- NEW -->igitur exi&longs;tere in <lb/>eo loco, in quo exi&longs;tit &longs;ubiectum, vt patet; igitur, &longs;i &longs;ubiectum mutet <lb/>locum etiam accidens cum eo coniunctum mutare debet. </s> </p> <p id="N12FDB" type="main"> <s id="N12FDD"><!-- NEW -->Dices, igitur &longs;imiliter dici pote&longs;t non recipi impetum in omni­<lb/>bus partibus &longs;ubiecti mobilis, &longs;ed in vnâ dumtaxat; cui cum <lb/>aliæ &longs;int vnitæ, exigunt moueri &longs;ine impetu ad illius motum? </s> <s id="N12FE5"><!-- NEW -->cum <lb/>hoc ip&longs;um ad omnem vnionem &longs;pectare videatur; </s> <s id="N12FEB"><!-- NEW -->Re&longs;pondeo vnam <pb pagenum="30" xlink:href="026/01/062.jpg"/>partem plumbi ita coniungi cum alia, vt etiam &longs;eparata naturaliter <lb/>exi&longs;tere po&longs;&longs;it; </s> <s id="N12FF6"><!-- NEW -->igitur non e&longs;t par ratio; </s> <s id="N12FFA"><!-- NEW -->præterea vna pars plumbi non <lb/>e&longs;t in loco alterius; </s> <s id="N13000"><!-- NEW -->nec enim inuicem penetrantur cum &longs;it compene­<lb/>tratio accidentium cum &longs;ubiecto; </s> <s id="N13006"><!-- NEW -->deinde, quò plures &longs;unt partes vnitæ, <lb/>maior e&longs;t re&longs;i&longs;tentia, quæ ip&longs;o etiam &longs;en&longs;u percipitur; </s> <s id="N1300C"><!-- NEW -->denique non vide­<lb/>tur cur potius produceretur in vna parte, quam in alia; quæ omnia <lb/>iam &longs;uprà Th. 33. demon&longs;trauimus. </s> </p> <p id="N13014" type="main"> <s id="N13016"><!-- NEW -->Adde quod &longs;i impetus produceretur in ip&longs;is accidentibus, etiam in <lb/>ip&longs;o impetu prius producto alius impetus produceretur; </s> <s id="N1301C"><!-- NEW -->cum &longs;cilicet <lb/>noua fit impetus acce&longs;&longs;io; quod &longs;atis ridiculum e&longs;t; qua&longs;i verò impetus <lb/>indigeat impetu &c. </s> <s id="N13024"><!-- NEW -->hîc loquor tantùm de accidentibus in &longs;ubiecto; <lb/>non verò de Euchari&longs;ticis, quæ à &longs;ubiecto per miraculum &longs;eparata etiam <lb/>moueri po&longs;&longs;unt per impre&longs;&longs;um impetum. </s> </p> <p id="N1302C" type="main"> <s id="N1302E"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1303B" type="main"> <s id="N1303D"><!-- NEW -->Hinc manife&longs;tè patet, quid dicendum &longs;it de anima bruti, quæ moue­<lb/>tur etiam &longs;ine impetu; </s> <s id="N13043"><!-- NEW -->quia exigit &longs;emper e&longs;&longs;e coniuncta corpori, à <lb/>quo di&longs;iuncta naturaliter exi&longs;tere non pote&longs;t, vt &longs;uo loco dicemus; igi­<lb/>tur ad motum corporis, &longs;eu &longs;ubiecti moueri deber. </s> </p> <p id="N1304B" type="main"> <s id="N1304D"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1305A" type="main"> <s id="N1305C"><!-- NEW -->Idem quoque de Anima rationali dicendum e&longs;&longs;e videtur; </s> <s id="N13060"><!-- NEW -->licèt <lb/>enim à corpore &longs;eparata naturaliter exi&longs;tere po&longs;&longs;it; </s> <s id="N13066"><!-- NEW -->tandiù tamen cum <lb/>corpore manet coniuncta, quandiu agere pote&longs;t in organis corporeis; <lb/>ac proinde exigit con&longs;eruari in corpore ip&longs;o, quandiu &longs;uas operatio­<lb/>nes organicas in eo exercere pote&longs;t. </s> </p> <p id="N13070" type="main"> <s id="N13072"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1307F" type="main"> <s id="N13081"><!-- NEW -->Hinc patet ratio manife&longs;ta ad quæ&longs;itum illud; </s> <s id="N13085"><!-- NEW -->quomodo &longs;cilicet po­<lb/>tentia motrix materialis v.g. <!-- REMOVE S-->Taurus &longs;uo cornu hominem ventilare po&longs;­<lb/>&longs;it; nec vlla &longs;upere&longs;t difficultas, dum dicas impetum non produci in <lb/>anima. </s> </p> <p id="N13091" type="main"> <s id="N13093"><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1309F" type="main"> <s id="N130A1"><!-- NEW -->Ob&longs;eruabis primò In hoc Theoremate dictum e&longs;&longs;e naturaliter; quia <lb/>per miraculum accidens &longs;eparatum ab omni &longs;ub&longs;tantia, dum &longs;it impe­<lb/>netrabile, per impetum &longs;ibi impre&longs;&longs;um moueri pote&longs;t. </s> </p> <p id="N130A9" type="main"> <s id="N130AB">Ob&longs;eruabis &longs;ecundò de anima bruti per miraculum &longs;eparatâ, idem <lb/>pror&longs;us dicendum e&longs;&longs;e. </s> </p> <p id="N130B0" type="main"> <s id="N130B2"><!-- NEW -->Ob&longs;eruabis tertiò etiam Animam rationalem &longs;eparatam, modò &longs;it <lb/>cum impenetrabilitate coniuncta, capacem e&longs;&longs;e impetus; </s> <s id="N130B8"><!-- NEW -->quem etiam <lb/>à potentia motrice corporea recipere pote&longs;t; </s> <s id="N130BE"><!-- NEW -->idem dictum e&longs;to de An­<lb/>gelo; &longs;ed de vtroque aliàs. </s> </p> <p id="N130C4" type="main"> <s id="N130C6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N130D2" type="main"> <s id="N130D4"><!-- NEW --><emph type="italics"/>Quando corpus pellitur ab alio corpore per impetum impre&longs;&longs;um; </s> <s id="N130DA"><!-- NEW -->haud du­<lb/>biè impetus ille impre&longs;&longs;us ab aliqua cau&longs;a efficiente producitur<emph.end type="italics"/>; patet <lb/>per Ax. 8. </s> </p> <pb pagenum="31" xlink:href="026/01/063.jpg"/> <p id="N130E9" type="main"> <s id="N130EB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N130F7" type="main"> <s id="N130F9"><emph type="italics"/>Ille impetus non producitur à &longs;ub&longs;tantia corporis in aliud impacti.<emph.end type="italics"/></s> <s id="N13100"> Proba­<lb/>tur; </s> <s id="N13105"><!-- NEW -->quia &longs;i produceretur, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt <expan abbr="clarũ">clarum</expan> e&longs;t; igitur appli­<lb/>cata, & non impedita ageret per Ax. 32. quod e&longs;t contra experientiam. </s> <s id="N1310F"><!-- NEW --><lb/>Dicunt aliqui requiri <expan abbr="motũ">motum</expan> præuium, vt agat; &longs;ed contra; </s> <s id="N13118"><!-- NEW -->nam motus <lb/>præuius non requiritur vt cau&longs;a, vt patet; </s> <s id="N1311E"><!-- NEW -->quia cau&longs;a vt agat debet exi­<lb/>&longs;tere per Ax. 9. Igitur requiritur, vt conditio, quod dici non pote&longs;t; </s> <s id="N13124"><!-- NEW --><lb/>quia primo etiam conditio debet e&longs;&longs;e præ&longs;ens; </s> <s id="N13129"><!-- NEW -->&longs;ed motus præuius de <lb/>nihil pre&longs;enti e&longs;t &longs;ecundo quia non pote&longs;t excogitari aliud munus con­<lb/>ditionis; </s> <s id="N13131"><!-- NEW -->ni&longs;i vel vt tollat impedimentum, vel vt applicet cau&longs;am &longs;ubie­<lb/>cto apto; </s> <s id="N13137"><!-- NEW -->præterea motus præuius non e&longs;t; </s> <s id="N1313B"><!-- NEW -->igitur eodem modo &longs;e <lb/>habet, ac &longs;i nunquam extiti&longs;&longs;et; & &longs;i eo in&longs;tanti quo corpus impa­<lb/>ctum primo tangit, amitteret totum impetum, ita vt expræterito motu <lb/>nihil reliquum e&longs;&longs;et, haud dubiè corpus aliud non pelleret. </s> </p> <p id="N13145" type="main"> <s id="N13147"><!-- NEW -->Diceret alius impetum e&longs;&longs;e tantùm conditionem, quæ &longs;emper e&longs;t <lb/>de præ&longs;enti: </s> <s id="N1314D"><!-- NEW -->ad hanc in&longs;tantiam non valet &longs;uperior re&longs;pon&longs;io; </s> <s id="N13151"><!-- NEW -->& certè <lb/>&longs;i eo ip&longs;o in&longs;tanti contactus noua fieret impetus acce&longs;&longs;io; </s> <s id="N13157"><!-- NEW -->haud dubiè <lb/>maior e&longs;&longs;et ictus; </s> <s id="N1315D"><!-- NEW -->licèt cum eodem motu præuio, & tamen idem e&longs;&longs;et <lb/>corpus <expan abbr="impactũ">impactum</expan>, Igitur ad hanc <expan abbr="in&longs;tantiã">in&longs;tantiam</expan> alio modo re&longs;pondeo, ex appli­<lb/>catione impetus &longs;emper &longs;equitur productio alterius impetus; </s> <s id="N1316D"><!-- NEW -->dum &longs;cili­<lb/>cet &longs;ubiectum, cui applicatur &longs;it capax motus; </s> <s id="N13173"><!-- NEW -->ex applicatione corporis <lb/>&longs;eu &longs;ubiecti ip&longs;ius non &longs;emper &longs;equitur; igitur dicendum e&longs;t impetum <lb/>ip&longs;um e&longs;&longs;e cau&longs;am alterius per Ax. 11. n. </s> <s id="N1317B"><!-- NEW -->1. voco enim illud cau&longs;am, <lb/>ex cuius applicatione &longs;emper &longs;equitur &longs;imilis effectus; alioquin &longs;i hoc <lb/>neges; </s> <s id="N13183"><!-- NEW -->proba mihi aliter ignem accendi ab alio igne; </s> <s id="N13187"><!-- NEW -->dicam enim tibi <lb/>ignem applicatum e&longs;&longs;e tantùm conditionem, & produci à cœlo; proba <lb/>mihi aliter calorem produci à calore? </s> <s id="N1318F">quo enim medio, vel argu­<lb/>mento id euinces? </s> <s id="N13194"><!-- NEW -->quo etiam non euincam impetum produci ab im­<lb/>petu: Deinde affer rationem à priori, propter quam &longs;ub&longs;tantia <lb/>corporis producat impetum &longs;ur&longs;um? </s> <s id="N1319C"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->cum non exigat à &longs;e ip&longs;a mo­<lb/>tum &longs;ursùm, qui violentus e&longs;t corpori graui; numquid certum e&longs;t, vt <lb/>dicemus infrà impetum produci ad extra, vt tollatur impedimentum <lb/>motus? </s> <s id="N131AA"><!-- NEW -->igitur illius e&longs;t tollere impedimentum, cuius e&longs;t exigere motum, <lb/>corpus ip&longs;um graue non exigit motum &longs;ur&longs;um, &longs;ed impetus; </s> <s id="N131B0"><!-- NEW -->igitur im­<lb/>petus e&longs;t tollere impedimentum &longs;ui effectus; </s> <s id="N131B6"><!-- NEW -->igitur producere impetum, <lb/>quo vno tolli tantùm pote&longs;t: </s> <s id="N131BC"><!-- NEW -->En tibi rationem à priori, cutum nullam <lb/>habeas: Præterea, cur negas impetum e&longs;&longs;e cau&longs;am &longs;ufficientem alterius <lb/>impetus, cum ex eius applicatione ip&longs;o &longs;en&longs;u percipiamus produci alium <lb/>impetum? </s> <s id="N131C6">quæ ratio? </s> <s id="N131C9"><!-- NEW -->quid inde ab&longs;urdi, quid incommodi: Igitur tàm <lb/>certum e&longs;t, immò certius impetum produci ab alio impetu, quàm calo­<lb/>rem à calore. </s> <s id="N131D1"><!-- NEW -->Dices impetum iam habere alium effectum &longs;cilicet mo­<lb/>tum; bella profecto ratio! &longs;ed numquid motus e&longs;t effectus formalis im­<lb/>petus? </s> <s id="N131D9">prætereà e&longs;t-ne effectus ad extra? </s> <s id="N131DC"><!-- NEW -->deinde idem dico de calore; </s> <s id="N131E0"><!-- NEW --><pb pagenum="32" xlink:href="026/01/064.jpg"/>qui reuera habet effectum formalem &longs;ecundarium ad intra, &longs;cilicet rare­<lb/>factionem, quæ e&longs;t mutatio exten&longs;ionis; </s> <s id="N131EA"><!-- NEW -->quemadmodum motus e&longs;t mu­<lb/>tatio loci, vel vbicationis; </s> <s id="N131F0"><!-- NEW -->igitur cum hoc | non ob&longs;tante, calor pro­<lb/>ducat calorem ad extra; cur impetus non producit impetum? </s> <s id="N131F6"><!-- NEW -->cuius pro­<lb/>ductionem concedis virtuti corporum re&longs;i&longs;titiuæ, id e&longs;t vnioni, impe­<lb/>netrabilitati, & cæteris huiu&longs;modi modorum &longs;uperfluorum qui&longs;quiliis; <lb/>de quibus plurimi tecum contendunt. </s> </p> <p id="N13200" type="main"> <s id="N13202"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1320E" type="main"> <s id="N13210"><!-- NEW -->Ob&longs;eruabis nonnullas e&longs;&longs;e difficultates, quæ communes &longs;unt etiam <lb/>illi &longs;ententiæ, quam &longs;equuntur ij, qui exi&longs;timant impetum ad extra <lb/>produci à corpore impacto; quas tamen facilè &longs;oluemus infrà in conti­<lb/>nuata no&longs;trorum Theorematum &longs;erie. </s> </p> <p id="N1321A" type="main"> <s id="N1321C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N13228" type="main"> <s id="N1322A"><emph type="italics"/>Aliquis impetus non producitur ab alio impetu.<emph.end type="italics"/></s> <s id="N13231"> Probatur, quia aliquis <lb/>impetus producitur ad intra à potentia motrice, vt patet. </s> <s id="N13236">2. cum non <lb/>detur progre&longs;&longs;us in infinitum, nec impetus idem producatur à &longs;e ip&longs;o, ad <lb/>aliquem tandem vltimum &longs;eu primum deueniendum e&longs;t, qui ab alio im­<lb/>petu non producatur. </s> </p> <p id="N1323F" type="main"> <s id="N13241"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N1324D" type="main"> <s id="N1324F"><emph type="italics"/>Impetus producitur &longs;emper ad extra ab alio impetu.<emph.end type="italics"/></s> <s id="N13256"><!-- NEW --> Quia cum &longs;emper <lb/>ad illius productionem requiratur applicatio alterius impetus; certè <lb/>non e&longs;t ponenda alia cau&longs;a per Ax. 11. </s> </p> <p id="N1325E" type="main"> <s id="N13260"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N1326C" type="main"> <s id="N1326E"><!-- NEW --><emph type="italics"/>Hinc impetus habet duplex munus cau&longs;æ; </s> <s id="N13274"><!-- NEW -->&longs;cilicet cau&longs;æ exigentis ad intra <lb/>& efficientis ad extra<emph.end type="italics"/>; vtrumque patet ex dictis. </s> </p> <p id="N1327D" type="main"> <s id="N1327F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N1328B" type="main"> <s id="N1328D"><!-- NEW --><emph type="italics"/>Impetus agit tantùm ad extra, vt tollat impedimentum motus<emph.end type="italics"/>; </s> <s id="N13296"><!-- NEW -->cum enim <lb/>motus &longs;it finis intrin&longs;ecus impetus; </s> <s id="N1329C"><!-- NEW -->certè &longs;i nihil impediret motum, <lb/>haud dubiè gauderet impetus &longs;uo fine; </s> <s id="N132A2"><!-- NEW -->igitur fru&longs;trà quidquam aliud <lb/>de&longs;ideraret; </s> <s id="N132A8"><!-- NEW -->præterea licèt applicetur à tergo aliud mobile; </s> <s id="N132AC"><!-- NEW -->non tamen <lb/>propterea in eo producit, vt con&longs;tat experientiâ; </s> <s id="N132B2"><!-- NEW -->denique cum tan­<lb/>tùm impetum cogno&longs;camus per motum; </s> <s id="N132B8"><!-- NEW -->cum nequidem e&longs;&longs;et impetus, <lb/>&longs;i non e&longs;&longs;et motus, per Th. 17. certè totus e&longs;t impetus propter motum <lb/>qui e&longs;t eius finis; </s> <s id="N132C0"><!-- NEW -->igitur non agit ni&longs;i propter motum: </s> <s id="N132C4"><!-- NEW -->&longs;ed non pote&longs;t <lb/>excogitari, quid faciat propter motum, dum agit, ni&longs;i dicamus ideo <lb/>tantùm agere, vt tollatur impedimentum; cum certum &longs;it corpus im­<lb/>mobile, in quod impingitur aliud mobile, impedire eius motum. </s> </p> <p id="N132CE" type="main"> <s id="N132D0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N132DC" type="main"> <s id="N132DE"><!-- NEW --><emph type="italics"/>Hinc non &longs;imul agit impetus in orbem &longs;ed tantùm per lineam <lb/>&longs;ui motus; </s> <s id="N132E6"><!-- NEW -->cui &longs;i nullum corpus occurrit reuerà non agit,<emph.end type="italics"/> Ratio e&longs;t; </s> <s id="N132ED"><!-- NEW -->quia li­<lb/>cèt aliud corpus mobili admoueatur in alia linea; </s> <s id="N132F3"><!-- NEW -->cum non impediat <lb/>eius motum, vt &longs;uppono; </s> <s id="N132F9"><!-- NEW -->cum agat tantùm impetus ad extra, vt tollat, <pb pagenum="33" xlink:href="026/01/065.jpg"/>impedimentum motu &longs;ui &longs;ubiecti, in eo non agit, quod non impedit; </s> <s id="N13302"><!-- NEW -->& <lb/>cum impediatur tantùm in vna linea, in ca tantùm agit; igitur non <lb/>agit in orbem. </s> </p> <p id="N1330A" type="main"> <s id="N1330C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13318" type="main"> <s id="N1331A"><!-- NEW -->Ob&longs;eruabis primò, hanc primam e&longs;&longs;e difficultatem; cum in hoc im­<lb/>petus maximè differat ab alijs qualitatibus &longs;i quæ &longs;unt, quæ agunt in or­<lb/>bem, vt dicemus &longs;uo loco. </s> </p> <p id="N13322" type="main"> <s id="N13324"><!-- NEW -->Ob&longs;eruabis &longs;ecundò, hanc etiam e&longs;&longs;e communem illorum &longs;ententiam, <lb/>qui dicunt impetum ad extrà produci ab ip&longs;o mobili, &longs;ed ita vt ab illis <lb/>vix &longs;olui po&longs;&longs;it; cum tamen à nobis facilè &longs;oluatur. </s> </p> <p id="N1332C" type="main"> <s id="N1332E"><!-- NEW -->Ob&longs;eruabis tertiò, impetum in vtroque munere cau&longs;æ &longs;ube&longs;&longs;e tantùm <lb/>vni lineæ; </s> <s id="N13334"><!-- NEW -->&longs;cilicet exigit motum per vnam lineam; </s> <s id="N13338"><!-- NEW -->cum per plures &longs;i­<lb/>mul motus e&longs;&longs;e non po&longs;&longs;it; </s> <s id="N1333E"><!-- NEW -->ne idem mobile &longs;imul e&longs;&longs;et in pluribus lo­<lb/>cis; </s> <s id="N13344"><!-- NEW -->& producit impetum per vnam lineam; cum producat tantùm pro­<lb/>pter motum. </s> </p> <p id="N1334A" type="main"> <s id="N1334C"><!-- NEW -->Ob&longs;eruabis quartò, alias qualitates, &longs;i quæ &longs;unt, non agere ad extra, <lb/>vt tollant impedimentum &longs;ui effectus ad intra; </s> <s id="N13352"><!-- NEW -->qui &longs;cilicet ab impedi­<lb/>mento extrin&longs;eco impediri non pote&longs;t; </s> <s id="N13358"><!-- NEW -->vt accidit in ip&longs;o impetu; </s> <s id="N1335C"><!-- NEW -->etenim <lb/>corpus non pote&longs;t moueri ni&longs;i nouum locum acquirat: neque nouum <lb/>locum acquirere ab alio corpore occupatum, ni&longs;i corpus hoc loco ce­<lb/>dat, neque hoc loco cedere pote&longs;t &longs;ine motu, vel moueri &longs;ine impetu, <lb/>igitur cum impediat motum amoueri debet, accepto dumtaxat impetu <lb/>ab alio mobili. </s> </p> <p id="N1336A" type="main"> <s id="N1336C"><!-- NEW -->Ob&longs;eruabis quintò nonnullos e&longs;&longs;e, qui volunt motum vnius corporis <lb/>transferri in aliud corpus; </s> <s id="N13372"><!-- NEW -->&longs;ed mera e&longs;t metaphora; </s> <s id="N13376"><!-- NEW -->nihil enim pror&longs;us <lb/>e&longs;t quod ab vno in aliud tran&longs;eat, &longs;eu transferatur; nec aliud dici po­<lb/>te&longs;t, ni&longs;i quod dictum e&longs;t, impetum &longs;cilicet nouum produci. </s> </p> <p id="N1337E" type="main"> <s id="N13380"><!-- NEW -->Hinc etiam reiicies commentum illorum, qui dicunt ideo vnum <lb/>corpus ab alio moueri, quia ab vno in aliud deriuantur corpu&longs;cula illa, <lb/>quæ faciunt lumen, & calorem; </s> <s id="N13388"><!-- NEW -->quia lumen, & calor &longs;unt veræ qualita­<lb/>tes, non corpu&longs;cula, vt demon&longs;trabimus in 5. tractatu: </s> <s id="N1338E"><!-- NEW -->Adde quod li­<lb/>cet ferrum candens aliud frigidum impellat, etiam veloci&longs;&longs;imè; </s> <s id="N13394"><!-- NEW -->hoc ip­<lb/>&longs;um æquè frigidum manet; </s> <s id="N1339A"><!-- NEW -->denique in cra&longs;&longs;is tenebris nix &longs;eu glacies <lb/>frigidi&longs;&longs;ima pernici&longs;&longs;imè moueri pote&longs;t: &longs;ed apage i&longs;ta commenta. </s> </p> <p id="N133A0" type="main"> <s id="N133A2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N133AE" type="main"> <s id="N133B0"><emph type="italics"/>Omnes partes impetus mobilis agunt ad extra actione communi.<emph.end type="italics"/></s> <s id="N133B7"> Probatur <lb/>per Ax. 13. n. </s> <s id="N133BC"><!-- NEW -->1. ni&longs;i enim agerent actione communi &longs;ed quælibet &longs;uam <lb/>produceret; </s> <s id="N133C2"><!-- NEW -->cur potius in hac parte &longs;ubiecti, quam in alia, deinde ap­<lb/>plicatur tantùm vna immediatè; </s> <s id="N133C8"><!-- NEW -->Igitur agunt omnes actione commu­<lb/>ni; </s> <s id="N133CE"><!-- NEW -->omnes inquam illæ, quæ impediuntur; </s> <s id="N133D2"><!-- NEW -->cum enim impetus agat <lb/>tantùm ad extrà vt tollat impedimentum &longs;ui motus; ille pro&longs;ectò age­<lb/>re non debet, cuius motus vel effectus non impeditur. </s> </p> <pb pagenum="34" xlink:href="026/01/066.jpg"/> <p id="N133DE" type="main"> <s id="N133E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N133EC" type="main"> <s id="N133EE"><emph type="italics"/>Hinc maiora corpora putà onerariæ naues, licèt tardi&longs;&longs;imo motu ferantur, <lb/>cum in aliud corpus impinguntur maxima vi illud impellunt.<emph.end type="italics"/></s> <s id="N133F7"> Ratio e&longs;t; <lb/>quia cum &longs;int plures partes impetus in pluribus partibus &longs;ubiecti, & <lb/>omnes agant actione communi, non mirum e&longs;t &longs;i maiorem effectum <lb/>producant, per Ax. 13. n. </s> <s id="N13400">2. <!-- KEEP S--></s> </p> <p id="N13404" type="main"> <s id="N13406"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13412" type="main"> <s id="N13414"><!-- NEW -->Vides primò in hoc ca&longs;u compen&longs;ari inten&longs;ionem ab exten&longs;ione; </s> <s id="N13418"><!-- NEW --><lb/>quippe quod præ&longs;tarent plures partes impetus in minore corporis mole <lb/>inten&longs;æ; hoc idem præ&longs;tare po&longs;&longs;unt exten&longs;æ in maiore mole. </s> </p> <p id="N1341F" type="main"> <s id="N13421">Secundò &longs;icut maior moles aptior e&longs;t ad motum imprimendum, & mi­<lb/>nùs apta ad recipiendum ita minor contrà aptior e&longs;t ad recipiendum, & <lb/>minùs apta ad imprimendum. </s> </p> <p id="N13428" type="main"> <s id="N1342A"><!-- NEW -->Tertiò, Hinc corpora illa, quorum partes vel nullo vel modico nexu <lb/>copulantur, minimo ferè impul&longs;u commouentur; </s> <s id="N13430"><!-- NEW -->&longs;ic aër & aqua mini­<lb/>mo flante vento agitantur, nubes pelluntur; </s> <s id="N13436"><!-- NEW -->hinc tot procellæ tempe­<lb/>&longs;tate&longs;que cientur; nec vlla e&longs;t alia ratio, cur minima ferè venti vis, cui <lb/>modicum &longs;axum re&longs;i&longs;tit, tantam aquæ, vel aëris molem commoueat, ni­<lb/>&longs;i quia cum partes illorum corporum nullo ferè nexu coniunctæ &longs;int vna <lb/>&longs;ine alia moueri pote&longs;t, quod in aqua gelu concreta minimè accidit. </s> </p> <p id="N13442" type="main"> <s id="N13444">Quartò, Hinc &longs;i maxima rupes ita comminueretur vt tota in pulue­<lb/>rem &longs;eu &longs;abulum abiret, minima vis impre&longs;&longs;a particulas illas moueret. </s> </p> <p id="N13449" type="main"> <s id="N1344B"><!-- NEW -->Quintò, Hinc diuino penè con&longs;ilio factum e&longs;t, vt partes terre&longs;tris <lb/>globi arctiore fibula copulentur; </s> <s id="N13451"><!-- NEW -->ne, &longs;i di&longs;iunctæ e&longs;&longs;ent, minimo flatu <lb/>di&longs;pergerentur: vt videre e&longs;t in puluere etiam graui&longs;&longs;imo, qui ab aura <lb/>flant e di&longs;pergitur. </s> </p> <p id="N13459" type="main"> <s id="N1345B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N13467" type="main"> <s id="N13469"><emph type="italics"/>Impetus, cuius motus non impeditur, non agit ad extrà.<emph.end type="italics"/></s> <s id="N13470"><!-- NEW --> Probatur per <lb/>Th. 44. hinc &longs;i aliud corpus affigas mobili à tergo, nullum impetum in <lb/>eo producet, cuius effectus, qui certè impetui &longs;ingularis e&longs;t, alia ratio <lb/>e&longs;&longs;e non pote&longs;t; </s> <s id="N1347A"><!-- NEW -->tam enim corpus e&longs;t applicatum à tergo, quam in <lb/>ip&longs;a fronte; & nihil e&longs;t in vno, quod non &longs;it in alio, ni&longs;i quod in fronte <lb/>impedit motum, à tergo verò non impedit. </s> </p> <p id="N13482" type="main"> <s id="N13484"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13491" type="main"> <s id="N13493"><!-- NEW -->Hinc egregium paradoxon erui pote&longs;t; </s> <s id="N13497"><!-- NEW -->quod &longs;cilicet cau&longs;a nece&longs;&longs;aria <lb/>etiam immediatè applicata, & non impedita in &longs;ubiecto apto non agit; </s> <s id="N1349D"><!-- NEW --><lb/>quod videtur e&longs;&longs;e contra Ax. 12. vnde vt agat cau&longs;a nece&longs;&longs;aria, debet <lb/>applicari debito modo; </s> <s id="N134A4"><!-- NEW -->&longs;i agat in orbem, omnis applicatio &longs;ufficiens <lb/>e&longs;t: </s> <s id="N134AA"><!-- NEW -->&longs;i verò agat tantùm per vnam lineam; </s> <s id="N134AE"><!-- NEW -->certè applicari debet in ca <lb/>linea; alioquin non aget defectu debitæ applicationis. </s> </p> <p id="N134B4" type="main"> <s id="N134B6"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N134C3" type="main"> <s id="N134C5"><!-- NEW -->Hinc etiam aliud paradoxon non minus iucundum; </s> <s id="N134C9"><!-- NEW -->cau&longs;a nece&longs;&longs;aria <pb pagenum="35" xlink:href="026/01/067.jpg"/>applicata, & non impedita non agit; </s> <s id="N134D2"><!-- NEW -->at verò agit impedita; </s> <s id="N134D6"><!-- NEW -->&longs;cilicet <lb/>impetus qui tantùm agit, vt tollat impedimentum; igitur, &longs;i non <lb/>impediatur non agit. </s> </p> <p id="N134DE" type="main"> <s id="N134E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N134EC" type="main"> <s id="N134EE"><!-- NEW --><emph type="italics"/>Quo minùs impeditur impetus, minùs agit ad extra, & contrà; quo plùs <lb/>impeditur, plùs agit.<emph.end type="italics"/></s> <s id="N134F8"><!-- NEW --> Cum enim ideò agat ad extra, vt tollat impedi­<lb/>mentum; </s> <s id="N134FE"><!-- NEW -->certè &longs;i nullum e&longs;t, nihil agit, &longs;i minùs, minùs agit; igitur <lb/>agit pro rata, id e&longs;t, pro diuer&longs;a impedimenti ratione. </s> </p> <p id="N13504" type="main"> <s id="N13506"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s> </p> <p id="N13512" type="main"> <s id="N13514"><emph type="italics"/>Si linea motus, quam directionis appellant, ducatur per centrum vtriu&longs;que <lb/>corporis, maximum est impedimentum,<emph.end type="italics"/> vt patet. </s> <s id="N1351E"><!-- NEW -->&longs;int enim duo globi, <lb/>A mobilis, & B. occurrens ip&longs;i A, &longs;itque linea directionis DE ducta <lb/>per centrum vtriu&longs;que AB, & punctum contactus &longs;it C; </s> <s id="N13526"><!-- NEW -->certè glo­<lb/>bus B maximum ponit impedimentum, quod ab eo poni po&longs;&longs;it; </s> <s id="N1352C"><!-- NEW -->Igitur <lb/>impetus globi A agit quantùm pote&longs;t in globum B; vt &longs;cilicet maxi­<lb/>mum impedimentum remoueat. </s> </p> <p id="N13534" type="main"> <s id="N13536"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N13542" type="main"> <s id="N13544"><!-- NEW --><emph type="italics"/>Si linea motus vel ip&longs;ius parallela cadat perpendiculariter in extremam <lb/>diametrum globi immobilis: </s> <s id="N1354C"><!-- NEW -->haud dubiè nihil impedit<emph.end type="italics"/>; </s> <s id="N13553"><!-- NEW -->&longs;it enim globus <lb/>mobilis A, Immobilis B, linea directionis &longs;it GA, ip&longs;i parallela FC; </s> <s id="N13559"><!-- NEW --><lb/>certè globus B. non impedit motum globi A. cum nihil loci globi B <lb/>occupari debeat à globo A; Igitur impetus A non agit in globum B per <lb/>Th. 48. </s> </p> <p id="N13562" type="main"> <s id="N13564"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N13570" type="main"> <s id="N13572"><!-- NEW --><emph type="italics"/>Si linea motus &longs;it inter vtramque; </s> <s id="N13578"><!-- NEW -->est minus impedimentum.<emph.end type="italics"/> &longs;it globus <lb/>immobilis BA; </s> <s id="N13581"><!-- NEW -->&longs;it linea motus GC cum impedimento, de qua in Th. 50. <lb/>&longs;it alia KB cum nullo impedimento, de qua in Th. 51. &longs;int aliæ HD, <lb/>IE; </s> <s id="N13589"><!-- NEW -->certè minus e&longs;t impedimentum in contactu D, quàm in C; </s> <s id="N1358D"><!-- NEW -->quia ca­<lb/>dit obliquè in D, perinde atque &longs;i caderet in tangentem NO; Igitur <lb/>minus impeditur; in qua vero proportione, dicemus aliàs, cum de re­<lb/>flexione, & de motu mixto. </s> </p> <p id="N13597" type="main"> <s id="N13599"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N135A5" type="main"> <s id="N135A7"><!-- NEW --><emph type="italics"/>Hinc producitur in contactu<emph.end type="italics"/> C, <emph type="italics"/>totus impetus; </s> <s id="N135B3"><!-- NEW -->in contactu<emph.end type="italics"/> D, <emph type="italics"/>minùs; </s> <s id="N135BD"><!-- NEW -->in <lb/>contactu<emph.end type="italics"/> E <emph type="italics"/>adhuc minùs; </s> <s id="N135C9"><!-- NEW -->in<emph.end type="italics"/> B <emph type="italics"/>nihil<emph.end type="italics"/>; </s> <s id="N135D6"><!-- NEW -->quia in ea proportione producitur <lb/>plùs vel minùs impetus, quo plùs e&longs;t, vel minùs impedimenti per <lb/>Th. 49. &longs;ed minùs e&longs;t impedimentum in E, quàm in C; </s> <s id="N135DE"><!-- NEW -->& in E, quàm <lb/>in D, per Th. 52; Igitur in D producitur minùs impetus, quàm in C, <lb/>& minùs in E, quàm in D. <!-- KEEP S--></s> </p> <p id="N135E7" type="main"> <s id="N135E9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N135F5" type="main"> <s id="N135F7"><!-- NEW --><emph type="italics"/>Hinc eadem cau&longs;a nece&longs;&longs;aria etiam immediate applicata diuer&longs;um impe<emph.end type="italics"/><pb pagenum="36" xlink:href="026/01/068.jpg"/><emph type="italics"/>tum producit; vt patet in impetu, non tamen est eodem modo applicata, <lb/>id e&longs;t in eadem linea.<emph.end type="italics"/></s> </p> <p id="N1360A" type="main"> <s id="N1360C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N13618" type="main"> <s id="N1361A"><emph type="italics"/>Hinc ratio multorum effectuum phy&longs;icorum e. </s> <s id="N1361F">ui potest<emph.end type="italics"/>; </s> <s id="N13625"><!-- NEW -->cur &longs;cilicet cor­<lb/>pus incidens in aliud perpendiculariter maximum ictum infligat; </s> <s id="N1362B"><!-- NEW -->quia <lb/>&longs;cilicet maximum impetum producit, qui po&longs;&longs;it ab eo produci; </s> <s id="N13631"><!-- NEW -->cur <lb/>idem corpus obliquè incidens in aliud minorem ictum infligat; cuius <lb/>rei alia ratio e&longs;&longs;e non pote&longs;t. </s> <s id="N13639"><!-- NEW -->Huc etiam reuoca tormenta bellica, quæ <lb/>vel directo, vel obliquo ictu muros verberant; </s> <s id="N1363F"><!-- NEW -->hinc perpendicularis <lb/>forti&longs;&longs;ima e&longs;t; licèt eadem ratio pro motu corporum non valeat, quæ <lb/>valet pro diffu&longs;ione, &longs;eu propagatione qualitatum. </s> </p> <p id="N13647" type="main"> <s id="N13649"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N13655" type="main"> <s id="N13657"><!-- NEW -->Hinc pote&longs;t determinari quota pars impetus producatur, & quantus <lb/>&longs;it ictus; </s> <s id="N1365D"><!-- NEW -->cognito &longs;cilicet & &longs;uppo&longs;ito eo impetus gradu, qui producitur, <lb/>cum totus producitur, vt fit in perpendiculari; </s> <s id="N13663"><!-- NEW -->quippe tota men&longs;ura <lb/>impetus continetur in arcu CB; quam proportionem nos infrà demon­<lb/>&longs;trabimus. </s> </p> <p id="N1366B" type="main"> <s id="N1366D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N13679" type="main"> <s id="N1367B"><!-- NEW --><emph type="italics"/>Si linea directionis ducatur per centrum vtriu&longs;que globi, mobilis &longs;cilicet <lb/>& immobilis, impetus producit totum impetum quem pote&longs;t producere &longs;iue in <lb/>maiori globo, &longs;iue in minori, &longs;iue in æquali<emph.end type="italics"/>; patet experientia; cuius ratio <lb/>e&longs;t; </s> <s id="N1368A"><!-- NEW -->quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria; </s> <s id="N1368E"><!-- NEW -->Igitur idem impetus eodem mo­<lb/>do applicatus æquali tempore, æqualem &longs;emper effectum producit, per <lb/>Ax. 12. igitur cum impetus agat tantùm, vt tollat impedimentum per <lb/>Th. 44. & cum in prædicta linea agat quantum pote&longs;t per Th. 50. cer­<lb/>tè æqualem effectum producat nece&longs;&longs;e e&longs;t; &longs;iue in maiori &longs;iue in mino­<lb/>ri, &longs;iue in æquali globo immobili. </s> </p> <p id="N1369C" type="main"> <s id="N1369E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s> </p> <p id="N136AA" type="main"> <s id="N136AC"><!-- NEW --><emph type="italics"/>Hinc impetus remi&longs;&longs;us potest producere inten&longs;um; </s> <s id="N136B2"><!-- NEW -->& hæc e&longs;t altera difficul­<lb/>tas; </s> <s id="N136B8"><!-- NEW -->cum &longs;cilicet maior globus in minorem impingitur<emph.end type="italics"/>; </s> <s id="N136BF"><!-- NEW -->cum enim omnes <lb/>partes impetus maioris globi agant actione communi per Th. 46. & <lb/>cum agant quantùm maximè po&longs;&longs;unt; </s> <s id="N136C7"><!-- NEW -->in minore globo, tot partes pro­<lb/>ducunt impetus, quot in maiore, vt patet; </s> <s id="N136CD"><!-- NEW -->igitur in minore globo pau­<lb/>cioribus partibus &longs;ubiecti di&longs;tribuuntur plures partes impetus; </s> <s id="N136D3"><!-- NEW -->ergo in <lb/>qualibet parte &longs;ubiecti &longs;unt plures; </s> <s id="N136D9"><!-- NEW -->&longs;ed hoc e&longs;t e&longs;&longs;e inten&longs;um, vt con&longs;tat, <lb/>igitur impetus remi&longs;&longs;us producit inten&longs;um; quod e&longs;t paradoxon egre­<lb/>gium. </s> </p> <p id="N136E1" type="main"> <s id="N136E3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N136EF" type="main"> <s id="N136F1"><!-- NEW --><emph type="italics"/>Hinc etiam impetus inten&longs;us producit remi&longs;&longs;um, cum &longs;cilicet minor globus <lb/>in maiorem incidit<emph.end type="italics"/>; </s> <s id="N136FC"><!-- NEW -->quia &longs;cilicet pauciores partes impetus di&longs;tribuun­<lb/>tur pluribus partibus &longs;ubiecti; </s> <s id="N13702"><!-- NEW -->igitur quælibet &longs;ubiecti pauciores impe­<lb/>tus habet; quæ omnia con&longs;tant ex dictis. </s> </p> <pb pagenum="37" xlink:href="026/01/069.jpg"/> <p id="N1370C" type="main"> <s id="N1370E"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1371A" type="main"> <s id="N1371C"><!-- NEW -->Ob&longs;eruabis primò, &longs;ingularem impetus proprietatem, quæ alijs qua­<lb/>litatibus minimè competit; </s> <s id="N13722"><!-- NEW -->nam aliæ qualitates v. <!-- REMOVE S-->g. <!-- REMOVE S-->calor; </s> <s id="N1372A"><!-- NEW -->lumen in <lb/>eadem di&longs;tantia effectum &longs;emper æquè inten&longs;um producunt; </s> <s id="N13730"><!-- NEW -->&longs;ecus verò <lb/>impetus, qui pro maiori vel minori obice maiorem, vel minorem, hoc <lb/>e&longs;t inten&longs;iorem, vel remi&longs;&longs;iorem impetum in eadem di&longs;tantia producit; </s> <s id="N13738"><!-- NEW --><lb/>cuius ratio ex eo capite petitur; </s> <s id="N1373D"><!-- NEW -->quòd impetus agat tantùm ad extra <lb/>propter &longs;uum effectum ad intra, vt &longs;cilicet tollat impedimentum; </s> <s id="N13743"><!-- NEW -->igi­<lb/>tur in totum, quod impedit, agit; </s> <s id="N13749"><!-- NEW -->igitur non habet certam, & deter­<lb/>minatam &longs;phæram; </s> <s id="N1374F"><!-- NEW -->cum tantùm agat in obicem, &longs;iue &longs;it maior, &longs;iue <lb/>minor: </s> <s id="N13755"><!-- NEW -->Quia verò e&longs;t cau&longs;a nece&longs;&longs;aria, æqualem effectum producit, id <lb/>e&longs;t tot partes impetus in maiore, quot in minore, ergo, cum in mino­<lb/>re &longs;int pauciores partes &longs;ubiecti, & plures in maiore; </s> <s id="N1375D"><!-- NEW -->haud dubiè quæli­<lb/>bet pars minoris habebit plures partes effectus, & quælibet pars maio­<lb/>ris pauciores; igitur effectus erit inten&longs;ior in minore, & remi&longs;&longs;ior in <lb/>maiore. </s> </p> <p id="N13767" type="main"> <s id="N13769"><!-- NEW -->Prætereà, cum dixi omnes partes mobilis actione communi agere ad <lb/>extra; </s> <s id="N1376F"><!-- NEW -->ita primò intelligi debet, vt omnes illæ partes moueantur: </s> <s id="N13773"><!-- NEW -->&longs;ecun­<lb/>dò, vt linea motus, &longs;eu directionis per centra grauitatis vtriu&longs;que glo­<lb/>bi v, g. <!-- REMOVE S-->ducatur; </s> <s id="N1377D"><!-- NEW -->alioquin, vel omnes actione communi non agunt, vel <lb/>minus agunt, de quo infrà; </s> <s id="N13783"><!-- NEW -->&longs;ufficit verò iuxta præ&longs;ens in&longs;titutum, vt <lb/>globus ita impellat alium vel æqualem, vel inæqualem, vt linea dire­<lb/>ctionis ducatur per centrum grauitatis alterius; vide figuram. </s> <s id="N1378B">in qua <lb/>linea directionis e&longs;t DE. </s> </p> <p id="N13790" type="main"> <s id="N13792"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N1379E" type="main"> <s id="N137A0"><!-- NEW --><emph type="italics"/>Impetus globi impacti in alium globum eo modo, quo diximus, id est, linea <lb/>directionis ducta per centra grauitatis vtriu&longs;que producit in eo æqualem<emph.end type="italics"/>; </s> <s id="N137AB"><!-- NEW -->Pro­<lb/>batur, quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria, quæ tunc agit quantum pote&longs;t <lb/>per Th. 57. &longs;ed æqualis pote&longs;t producere æqualem: </s> <s id="N137B3"><!-- NEW -->Probatur primò, <lb/>exemplo aliarum qualitatum; </s> <s id="N137B9"><!-- NEW -->&longs;ecundò, quia ideo agit vt tollat impedi­<lb/>mentum, hoc e&longs;t vt corpus illud amoueat loco; </s> <s id="N137BF"><!-- NEW -->igitur æquali motu per <lb/>&longs;e; </s> <s id="N137C5"><!-- NEW -->alioquin ni&longs;i æquali motu amoueret, non tolleret impedimentum, <lb/>vt pater; </s> <s id="N137CB"><!-- NEW -->tertiò &longs;int 30. partes impetus, certè vel producent plures vel <lb/>pauciores, vel totidem, non plures; </s> <s id="N137D1"><!-- NEW -->cur enim potius 31. quam 32. <lb/>nec etiam pauciores; cur enim potius 20. quam 18, &c. </s> <s id="N137D7"><!-- NEW -->Igitur totidem; </s> <s id="N137DB"><!-- NEW --><lb/>quia cum &longs;int plures numeri plurium partium &longs;upra 30. & pauciorum <lb/>infra vt patet; </s> <s id="N137E2"><!-- NEW -->&longs;itque tantùm vnicus numerus æqualium; </s> <s id="N137E6"><!-- NEW -->certè quod <lb/>vnum e&longs;t, determinatum e&longs;t, per Ax. 5. hæc ratio licèt videatur negati­<lb/>ua e&longs;t tamen potenti&longs;&longs;ima: </s> <s id="N137EE"><!-- NEW -->quartò, quia actus &longs;ecundus, re&longs;pondet actui <lb/>primo, id e&longs;t, effectus productus virtuti cau&longs;æ producentis; </s> <s id="N137F4"><!-- NEW -->itaque cum <lb/>virtus agendi impetus &longs;it eius entitas, vt patet, certè impetus productus <lb/>e&longs;t per &longs;e æqualis impetui producenti per &longs;e; id e&longs;t remoto omni <lb/>impedimento, & facto eo contactu iuxta modum prædictum, ea quo-<pb pagenum="38" xlink:href="026/01/070.jpg"/>que lege, vt impetus agat quantum pote&longs;t, & omnes partes mobilis <lb/>moueantur æquali motu. </s> </p> <p id="N13805" type="main"> <s id="N13807"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13814" type="main"> <s id="N13816"><!-- NEW -->Hinc reijcis illos, qui volunt à globo æquali produci in æquali &longs;ub­<lb/>duplum impetum; in &longs;ubduplo &longs;ubtriplum; in &longs;ubquadruplo &longs;ubquin­<lb/>tuplum; ratio illorum e&longs;t; </s> <s id="N1381E"><!-- NEW -->quia duo globi æquales in&longs;tanti contactus <lb/>perinde &longs;e habent, atque &longs;i conflarent vnum corpus; </s> <s id="N13824"><!-- NEW -->&longs;ed &longs;i conflarent <lb/>vnum corpus quilibet &longs;ubduplum impetum haberet; </s> <s id="N1382A"><!-- NEW -->&longs;i verò globus cum <lb/>alio &longs;ubduplo faceret vnum mobile; </s> <s id="N13830"><!-- NEW -->haud dubiè minor, id e&longs;t, &longs;ubduplus <lb/>haberet tantùm &longs;ubtriplum impetum; atque ita deinceps; </s> <s id="N13836"><!-- NEW -->hoc totum <lb/>fal&longs;i&longs;&longs;imum e&longs;t; </s> <s id="N1383C"><!-- NEW -->nam primò &longs;i globus æqualis acciperet tantùm &longs;ubdu­<lb/>plum impetum ab alio, &longs;ubduplo tantùm motu ferretur; </s> <s id="N13842"><!-- NEW -->igitur &longs;ubdu­<lb/>plum &longs;patium decurreret, quod e&longs;t contra experientiam, & Th. 47. Se­<lb/>cundò, ratio propo&longs;ita nulla e&longs;t; </s> <s id="N1384A"><!-- NEW -->quia quando globus impactus impellit <lb/>alium, e&longs;t veluti potentiâ, quæ cum tota &longs;ua vi, & cum impetu agit, <lb/>cuius nulla pars transfertur in alium globum; </s> <s id="N13852"><!-- NEW -->nec enim migrat de <lb/>de &longs;ubiecto in &longs;ubiectum, &longs;ed producit &longs;ibi æqualem: </s> <s id="N13858"><!-- NEW -->equidem &longs;i duo <lb/>globi æquales e&longs;&longs;ent vel coniuncti, vel contigui in linea directionis, <lb/>quilibet pro rata acciperet impetus producti partem à potentia applica­<lb/>ta; </s> <s id="N13862"><!-- NEW -->&longs;i e&longs;&longs;ent æquales, qui&longs;que &longs;ubduplum: &longs;i alter &longs;ubduplus &longs;ubtri­<lb/>plum, &c. </s> <s id="N13868">&longs;ed hæc &longs;unt &longs;atis facilia. </s> </p> <p id="N1386B" type="main"> <s id="N1386D"><!-- NEW -->Obijci fortè po&longs;&longs;et ab aliquo primò experientia; </s> <s id="N13871"><!-- NEW -->videmus enim &longs;æpè <lb/>globum impul&longs;um in ludo Tudiculario moueri tardiùs globo impellen­<lb/>te; </s> <s id="N13879"><!-- NEW -->re&longs;pondeo id &longs;æpè accidere; </s> <s id="N1387D"><!-- NEW -->tùm quia linea directionis non connec­<lb/>tit centra vtriu&longs;que globi; </s> <s id="N13883"><!-- NEW -->igitur minor e&longs;t ictus per Th 52. tùm quia <lb/>globus impellens, vel impul&longs;us deficiunt à perfecta &longs;phæra; </s> <s id="N13889"><!-- NEW -->tùm quia <lb/>non e&longs;t perfecta æqualitas globorum; adde quod quò accuratiùs prædi­<lb/>ctæ leges ob&longs;eruantur, ip&longs;i motus ad æqualitatem propiùs accedunt, vt <lb/>con&longs;tat experientia. </s> </p> <p id="N13893" type="main"> <s id="N13895"><!-- NEW -->Obiici po&longs;&longs;et &longs;ecundò de&longs;trui aliquid impetus globi impellentis ab ip&longs;o <lb/>ictu, vt con&longs;tat experientia; </s> <s id="N1389B"><!-- NEW -->igitur illa pars impetus, quæ de&longs;truitur, non <lb/>producit nouum impetum in globo impul&longs;o; </s> <s id="N138A1"><!-- NEW -->Re&longs;pondeo de&longs;trui quidem <lb/>aliquid impetus in globo impacto, vt videbimus infrà; </s> <s id="N138A7"><!-- NEW -->cum tamen de­<lb/>&longs;truatur tantùm &longs;equenti po&longs;t ictum in&longs;tanti; </s> <s id="N138AD"><!-- NEW -->certè cum exi&longs;tat adhuc <lb/>ip&longs;o in&longs;tanti contactus, nece&longs;&longs;ariò agit, quippe aliquid vltimo in&longs;tanti <lb/>pote&longs;t agere; </s> <s id="N138B5"><!-- NEW -->adde quod illud ip&longs;um repugnat manife&longs;tæ experientiæ; </s> <s id="N138B9"><!-- NEW --><lb/>licèt enim aliquando de&longs;truatur totus impetus in globo impacto, quod <lb/>&longs;æpè accidit in ludo Tudiculario, nam illicò &longs;i&longs;tit pila eburnea; alius <lb/>tamen globus velociter mouetur, cuius effectus rationem infrà addu­<lb/>cemus. </s> </p> <p id="N138C4" type="main"> <s id="N138C6"><!-- NEW -->Obijci po&longs;&longs;et tertiò inde &longs;equi progre&longs;&longs;um in infinitum, nam globus <lb/>A impactus in globum B impellet cum æquali motu, & B in C etiam <lb/>æquali, C in D, atque ita deinceps; </s> <s id="N138CE"><!-- NEW -->modò illi globi ita &longs;tatuantur, vt <lb/>linea directionis per omnium centra rectà ducatur; </s> <s id="N138D4"><!-- NEW -->Re&longs;pondeo, vel il-<pb pagenum="39" xlink:href="026/01/071.jpg"/>los omnes globos ita e&longs;&longs;e contiguos, vt mutuo contactu &longs;e inuicem tan­<lb/>gant; </s> <s id="N138DF"><!-- NEW -->vel aliquod &longs;patium inter &longs;ingulos intercipi; </s> <s id="N138E3"><!-- NEW -->&longs;i primum, produci­<lb/>tur impetus à potentia motrice in omnibus, &longs;i &longs;ufficiens e&longs;t; </s> <s id="N138E9"><!-- NEW -->non verò <lb/>vnus globus in alio, vt con&longs;tat; </s> <s id="N138EF"><!-- NEW -->&longs;icut duo pondera &longs;imul attollo, quorum <lb/>vnum alteri incumbit: </s> <s id="N138F5"><!-- NEW -->&longs;i verò non &longs;e tangant, dico antequam A im­<lb/>pingatur in B, dum &longs;patium illud interiectum percurrit, amittere aliquid <lb/>impetus: </s> <s id="N138FD"><!-- NEW -->idem dico de B, & C, vnde &longs;i nihil impetus in eo primo motu <lb/>periret & linea directionis omnium centra perfectè connecteret; </s> <s id="N13903"><!-- NEW -->ita vt <lb/>omnium ictus illi omnino &longs;ine vlla deflexione re&longs;ponderent; </s> <s id="N13909"><!-- NEW -->haud du­<lb/>biè non po&longs;&longs;ent e&longs;&longs;e tot globi, quin po&longs;&longs;et alius addi, qui ab vltimo <lb/>pelleretur; </s> <s id="N13911"><!-- NEW -->&longs;ed vix illa omnia de quibus &longs;uprà po&longs;&longs;unt ob&longs;eruari; </s> <s id="N13915"><!-- NEW -->Hinc <lb/>tamen facilè vna pars aëris aliam pellit, quod di&longs;tinctè videmus in <lb/>aqua; &longs;ed de his aliàs, &longs;ufficiat modò propo&longs;itam obiectionem inde <lb/>manere &longs;olutam. </s> </p> <p id="N1391F" type="main"> <s id="N13921"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N1392D" type="main"> <s id="N1392F"><!-- NEW --><emph type="italics"/>Globus maior impactus in minorem imprimit illi inten&longs;iorem impetum, & <lb/>velociorem motum per Th.<emph.end type="italics"/> 48. <emph type="italics"/>&<emph.end type="italics"/> 47. Nec e&longs;t quod aliqui opponant Prin­<lb/>cipium illud mechanicum; </s> <s id="N13942"><!-- NEW -->id e&longs;t, nullum corpus po&longs;&longs;e maiorem veloci­<lb/>tatis gradum alteri corpori imprimere; </s> <s id="N13948"><!-- NEW -->eo &longs;cilicet gradu, quem ip&longs;um <lb/>habet; </s> <s id="N1394E"><!-- NEW -->nec enim inuenio Principium illud apud eos Mechanicos, qui <lb/>mechanica momenta &longs;uarum demon&longs;trationum momentis confirmant; <lb/>quî porro fieri pote&longs;t, vt principium illud admittatur, quod manife&longs;tæ <lb/>experientiæ repugnat? </s> <s id="N13958">Quis enim non vidit vel maius &longs;axum in aliud <lb/>etiam tardo motu impactum maiorem motum, & impetum imprimere? </s> <s id="N1395D"><lb/>quis non vidit maiores illas onerarias naues etiam pigro, & tardo motu <lb/>labentes maximum impetum minori occurrenti cymbæ etiam impri­<lb/>mere? </s> <s id="N13965"><!-- NEW -->Rationem habes in Th. 47. &longs;ed dices; </s> <s id="N13969"><!-- NEW -->igitur aliquis velocitatis <lb/>gradus nullam habet cau&longs;am; igitur e&longs;t à nihilo, quod dici non pote&longs;t. </s> <s id="N1396F"><!-- NEW --><lb/>Re&longs;pondeo, plures partes impetus non produci in minore globo, quàm <lb/>&longs;int in maiore; </s> <s id="N13976"><!-- NEW -->igitur nulla pars e&longs;t impetus minoris globi, quæ &longs;ui <lb/>cau&longs;am &longs;ufficientem non habeat; </s> <s id="N1397C"><!-- NEW -->&longs;ed cum partes impetus maioris globi <lb/>di&longs;tribuantur pluribus partibus &longs;ubiecti, faciunt remi&longs;&longs;um impetum, igi­<lb/>tur & tardum; </s> <s id="N13984"><!-- NEW -->cum &longs;cilicet impetus vnius partis non iuuet motum alte­<lb/>rius per Th. 37. at verò cum partes impetus producti in minore globo <lb/>di&longs;tribuantur paucioribus partibus &longs;ubiecti, faciunt inten&longs;iorem im­<lb/>petum; igitur velociorem motum, quippe omnes producuntur ab <lb/>omnibus illis actione communi per Ax. 17. num. </s> <s id="N13990">1. quid clarius. </s> </p> <p id="N13993" type="main"> <s id="N13995"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N139A1" type="main"> <s id="N139A3"><!-- NEW --><emph type="italics"/>Globus minor imprimit maiori remi&longs;&longs;iorem impetum & tardiorem motum <lb/>& æqualis, æquali æqualem<emph.end type="italics"/>; hæc omnia probantur per Th. 60. & præ-, <lb/>cedentia. </s> </p> <pb pagenum="40" xlink:href="026/01/072.jpg"/> <p id="N139B4" type="main"> <s id="N139B6"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N139C2" type="main"> <s id="N139C4"><!-- NEW -->Ob&longs;eruabis primò, vtrumque globum e&longs;&longs;e eiu&longs;dem materiæ; </s> <s id="N139C8"><!-- NEW -->&longs;i enim <lb/>&longs;int diuer&longs;æ materiæ, &longs;ecùs accidit, quàm diximus; </s> <s id="N139CE"><!-- NEW -->&longs;i v. <!-- REMOVE S-->g. <!-- REMOVE S-->æneus mi­<lb/>nor pellatur ab eburneo maiore, maiorem motum hic illi non impri­<lb/>met; </s> <s id="N139DA"><!-- NEW -->licèt enim &longs;it maior exten&longs;io eburnei; </s> <s id="N139DE"><!-- NEW -->e&longs;t tamen minus pondus; <lb/>igitur pauciores partes. </s> </p> <p id="N139E4" type="main"> <s id="N139E6"><!-- NEW -->Secundò, eos globos accipiendos e&longs;&longs;e, quorum partes, vel non auo­<lb/>lent ab ictu, vel non comprimantur; </s> <s id="N139EC"><!-- NEW -->comprimuntur in plumbeis, <lb/>æneis, & auolant in vitreis; cum enim &longs;it compre&longs;&longs;io, vel partium di­<lb/>ui&longs;io, de&longs;truitur multùm impetus. </s> </p> <p id="N139F4" type="main"> <s id="N139F6"><!-- NEW -->Tertiò reiice commentum illorum, qui dicunt corpus illud e&longs;&longs;e ma­<lb/>joris velocitatis capax, quod plures habet partes materiæ &longs;ub eadem <lb/>quantitate; </s> <s id="N139FE"><!-- NEW -->nam &longs;uppo&longs;ita eadem re&longs;i&longs;tentiæ ratione, omne corpus e&longs;t <lb/>capax illius velocitatis, cuius aliud e&longs;t capax; </s> <s id="N13A04"><!-- NEW -->cum nullus &longs;it motus, quo <lb/>non po&longs;&longs;it dari velocior, & tardior, vt dicemus infrà; </s> <s id="N13A0A"><!-- NEW -->immò &longs;it glo­<lb/>bus plumbeus 12. librarum, &longs;it eburneus eiu&longs;dem diametri 2. librarum, <lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->haud dubiè eadem potentia producet inten&longs;iorem impetum in <lb/>eburneo, vt patet experientia, & ratio con&longs;tat ex dictis; </s> <s id="N13A18"><!-- NEW -->qua&longs;i verò &longs;it <lb/>aliqua materiæ inertia, quæ motum re&longs;puat; </s> <s id="N13A1E"><!-- NEW -->licèt fortè maior &longs;it pro­<lb/>portio re&longs;i&longs;tentiæ medij comparatæ cum globo eburneo, quàm compa­<lb/>ratæ cum plumbeo; &longs;ed de re&longs;i&longs;tentia de percu&longs;&longs;ione, & de &longs;patio age­<lb/>mus infra. </s> </p> <p id="N13A28" type="main"> <s id="N13A2A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s> </p> <p id="N13A36" type="main"> <s id="N13A38"><emph type="italics"/>Omnis globus, qui in alium, qui mouetur impingitur, dum hic mouetur, ve­<lb/>lociùs mouetur eo &c. </s> <s id="N13A3F"><!-- NEW -->in quem impingitur <emph.end type="italics"/> patet; alioquin numquam a&longs;&longs;equi <lb/>po&longs;&longs;et, quod ex ip&longs;is terminis con&longs;tat. </s> </p> <p id="N13A48" type="main"> <s id="N13A4A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s> </p> <p id="N13A56" type="main"> <s id="N13A58"><!-- NEW --><emph type="italics"/>Ex hac hypothe&longs;i globus impactus producit in alie nouas partes impetus<emph.end type="italics"/>; <lb/>quia impeditur eius motus, igitur vt tollat impedimentum, agit ad <lb/>extra per Th. 44. </s> </p> <p id="N13A65" type="main"> <s id="N13A67"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s> </p> <p id="N13A73" type="main"> <s id="N13A75"><!-- NEW --><emph type="italics"/>Hic impetus nouus productus minor e&longs;t eo qui produceretur in eodem globo <lb/>immobili<emph.end type="italics"/>: ratio e&longs;t; </s> <s id="N13A80"><!-- NEW -->quia &longs;i &longs;i&longs;teret, maius e&longs;&longs;et impedimentum, quia <lb/>totum motum impediret, cuius tantùm partem impedit, dum mouetur , <lb/>licèt paulò tardius; igitur minus agit ad extra per Th. 49. </s> </p> <p id="N13A88" type="main"> <s id="N13A8A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s> </p> <p id="N13A96" type="main"> <s id="N13A98"><!-- NEW --><emph type="italics"/>Mobile adhærens alteri mobili à tergo; dum vtrumque æque velociter <lb/>feratur nullum producit in eo impetum.<emph.end type="italics"/></s> <s id="N13AA2"><!-- NEW --> Probatur, quia mobile quod præit, <lb/>non impedit motum &longs;ub&longs;equentis; igitur nullum impetum ab eo acci<lb/>pit per Th. 48. </s> </p> <pb pagenum="41" xlink:href="026/01/073.jpg"/> <p id="N13AAE" type="main"> <s id="N13AB0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s> </p> <p id="N13ABC" type="main"> <s id="N13ABE"><!-- NEW --><emph type="italics"/>Hinc paradoxon egregium &longs;i quod aliud; </s> <s id="N13AC4"><!-- NEW -->globus percu&longs;&longs;us ab alio eadem <lb/>&longs;emper velocitate mouetur, &longs;iue moueretur in&longs;tanti percu&longs;&longs;ionis, &longs;iue &longs;i­<lb/>&longs;teret.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it globus A quie&longs;cens, cui imprimantur ab alio B 40. gra­<lb/>dus velocitatis: id e&longs;t æqualis impetus impetui percutientis, iam verò <lb/>moueatur A, cum 20. grad. <!-- REMOVE S-->velocitatis, & B, qui mouetur cum 40. <lb/>impingatur, certè cum impediatur tantùm &longs;ubduplum motus, produce­<lb/>tur tantùm &longs;ubduplum impetus, id e&longs;t 20. qui &longs;i addantur 20. grad. <!-- REMOVE S-->erunt <lb/>40. quæ omnia con&longs;tant per Th.49.48.&c. </s> </p> <p id="N13AE1" type="main"> <s id="N13AE3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13AF0" type="main"> <s id="N13AF2">Hinc æquale &longs;emper &longs;patium percu&longs;&longs;us globus conficit, &longs;iue ante per­<lb/>cu&longs;&longs;ionem moueretur, &longs;iue quie&longs;ceret. </s> </p> <p id="N13AF7" type="main"> <s id="N13AF9"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13B06" type="main"> <s id="N13B08"><!-- NEW -->Hinc &longs;i &longs;ecundò percutiatur idem globus, &longs;patium totum, quod per­<lb/>currit tùm à primò, tùm à &longs;ecundo ictu e&longs;t maius eo, quod à primo ictu <lb/>confeci&longs;&longs;et, &longs;i non fui&longs;&longs;et &longs;ecundò percu&longs;&longs;us; maius inquam &longs;egmento &longs;pa­<lb/>tij interiecto inter primum & &longs;ecundum ictum. </s> </p> <p id="N13B12" type="main"> <s id="N13B14"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13B21" type="main"> <s id="N13B23"><!-- NEW -->Hinc reiicies aliquos, quorum &longs;ententiam habes apud Doctum Mer­<lb/>&longs;ennium, <emph type="italics"/>in prop.<emph.end type="italics"/> 20. <emph type="italics"/>phæn. mech. quorum &longs;unt hæc verba; </s> <s id="N13B32"><!-- NEW -->&longs;i malleus pilam <lb/>currentem eodem, ac anteà modo percutiat, nonam &longs;ui motus partem; &longs;i verò <lb/>currentem tertia vice percutiat, vnam vige&longs;imam &longs;eptimam &longs;ui motus par­<lb/>tem ei tribuet, atque ita deinceps.<emph.end type="italics"/></s> <s id="N13B3E"> Supponit primò hæc &longs;ententia mal­<lb/>leum e&longs;&longs;e duplum pilæ percu&longs;&longs;æ. </s> <s id="N13B43"><!-- NEW -->Secundò, malleum imprimere pilæ &longs;ub­<lb/>duplæ &longs;ubtriplum motum; quod fal&longs;um e&longs;t, vt con&longs;tat ex Th 6. & Co­<lb/>roll. </s> <s id="N13B4B"><!-- NEW -->1. Præterea, licètin primà percu&longs;&longs;ione imprimeret tantùm prædi­<lb/>ctæ pilæ &longs;ubtriplum impetum, in &longs;ecunda percu&longs;&longs;ione maiorem impri­<lb/>meret po&longs;t longiorem motum, vbi iam ad quietem propiùs accedit; </s> <s id="N13B53"><!-- NEW -->mi­<lb/>norem verò paulò po&longs;t initium motus, vt con&longs;tat ex dictis, & ex ip&longs;a ex­<lb/>perientia; </s> <s id="N13B5B"><!-- NEW -->pote&longs;t quidem in aliquo puncto &longs;ui motus &longs;ecunda vice per­<lb/>cuti, in quo &longs;ubtriplum tantùm motum imprimet; </s> <s id="N13B61"><!-- NEW -->hoc e&longs;t eo in&longs;tanti­<lb/>quo tantùm ami&longs;it tertiam fui impetus partem; </s> <s id="N13B67"><!-- NEW -->tum deinde in tertia <lb/>percu&longs;&longs;ione pote&longs;t tantùm (1/27) motus partem illi tribuere; </s> <s id="N13B6D"><!-- NEW -->eo &longs;cilicet in­<lb/>&longs;tanti, quo tantùm ami&longs;it (1/27) &longs;ui impetus partem; </s> <s id="N13B73"><!-- NEW -->&longs;ed in alijs temporis <lb/>punctis longè alia erit impetus producti ratio; Igitur tota hæc progre&longs;­<lb/>&longs;io gratis omninò fuit excogitata. </s> </p> <p id="N13B7B" type="main"> <s id="N13B7D"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13B8A" type="main"> <s id="N13B8C"><!-- NEW -->Hinc etiam po&longs;t &longs;ecundam percu&longs;&longs;ionem æquale &longs;patium conficiet al­<lb/>teri, quod iam confecit po&longs;t primam æqualibus temporibus; </s> <s id="N13B92"><!-- NEW -->igitur æqua­<lb/>lis e&longs;t velocitas vtriu&longs;que motus; </s> <s id="N13B98"><!-- NEW -->quia &longs;cilicet, &longs;i e&longs;t æqualis impetus, e&longs;t <lb/>qualis motus: Ex his maximam carum dubitationum partem &longs;oluere po­<lb/>teris quæ in eadem Mer&longs;enni propo&longs;itione courinentur reliquas vero ex <lb/>dicendis infrà. </s> </p> <pb pagenum="42" xlink:href="026/01/074.jpg"/> <p id="N13BA6" type="main"> <s id="N13BA8"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13BB5" type="main"> <s id="N13BB7"><!-- NEW -->Ex dictis etiam colliges diuer&longs;as percu&longs;&longs;ionum rationes &longs;uppo&longs;ita di­<lb/>uer&longs;a ratione ponderum globi percutientis, & percu&longs;&longs;i; </s> <s id="N13BBD"><!-- NEW -->cum enim impe­<lb/>tus productus &longs;it æqualis per &longs;e impetui producenti, per Th.60. modò <lb/>debita fiat applicatio, de qua in Th.50. &longs;i percutiens &longs;it duplus percu&longs;&longs;i, <lb/>&longs;uppo&longs;ita eadem materia, motus percu&longs;&longs;i erit duplò velocior; quia im­<lb/>petus erit duplò inten&longs;ior, vt con&longs;tat ex Th. 61. &longs;i verò &longs;it quadruplus, <lb/>quadruplo, &c. </s> <s id="N13BCB">Igitur velocitates motuum &longs;unt in ratiòne ponderum <lb/>permutando. </s> </p> <p id="N13BD0" type="main"> <s id="N13BD2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s> </p> <p id="N13BDE" type="main"> <s id="N13BE0"><!-- NEW --><emph type="italics"/>Si corpus percu&longs;&longs;um &longs;it oblongum, & percu&longs;&longs;io fiat in centro grauitatis eiu&longs;­<lb/>dem corporis; </s> <s id="N13BE8"><!-- NEW -->producitur impetus in percu&longs;&longs;io æqualis impetui percutientis<emph.end type="italics"/>; </s> <s id="N13BEF"><!-- NEW -->&longs;ed <lb/>opus e&longs;t aliqua figura: </s> <s id="N13BF5"><!-- NEW -->Sit corpus AD, parallelipedum; </s> <s id="N13BF9"><!-- NEW -->diuidatur æqua­<lb/>liter in E ita vt E &longs;it centrum grauitatis; </s> <s id="N13BFF"><!-- NEW -->&longs;i percu&longs;&longs;io fiatin E per lineam <lb/>perpendicularem HE, producetur impetus in corpore AD æqualis im­<lb/>petui corporis percutientis; </s> <s id="N13C07"><!-- NEW -->quia &longs;cilicet à corpore AD non pote&longs;t maius <lb/>e&longs;&longs;e impedimentum; igitur agit quantùm pote&longs;t impetus corporis per­<lb/>cutientis per Th.50. igitur producit æqualem per Th.69. </s> </p> <p id="N13C0F" type="main"> <s id="N13C11"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s> </p> <p id="N13C1D" type="main"> <s id="N13C1F"><!-- NEW --><emph type="italics"/>Si percu&longs;&longs;io fiat in<emph.end type="italics"/> F <emph type="italics"/>per lineam perpendicularem<emph.end type="italics"/> IF, <emph type="italics"/>minus erit impedi­<lb/>mentum, quàm per<emph.end type="italics"/> HE, Quia &longs;i per HE, moueri tantùm pote&longs;t motu <lb/>recto, &longs;i per IF, etiam motu circulari circa aliquod centrum; </s> <s id="N13C38"><!-- NEW -->&longs;ed hic <lb/>motus e&longs;t facilior quam ille; </s> <s id="N13C3E"><!-- NEW -->igitur minus e&longs;t impedimentum; </s> <s id="N13C42"><!-- NEW -->(&longs;uppono <lb/>autem cylindrum BC vtroque modo moueri po&longs;&longs;e ab applicata potentia) <lb/>igitur minùs impetus producitur, &longs;i percu&longs;&longs;io fiat per IF, quàm &longs;i fiat <lb/>per LK: </s> <s id="N13C4C"><!-- NEW -->In qua verò proportione &longs;it minus impedimentum, & minori <lb/>opus impetu, po&longs;ito eodem potentiæ ni&longs;u, determinabimus facilè aliàs; <lb/>vt etiam demon&longs;trabimus circa quod centrum hic circularis motus fieri <lb/>debeat. </s> </p> <p id="N13C56" type="main"> <s id="N13C58"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13C64" type="main"> <s id="N13C66"><!-- NEW -->Ex duobus capitibus minus e&longs;&longs;e pote&longs;t impedimentum; </s> <s id="N13C6A"><!-- NEW -->primum e&longs;t, <lb/>quod petitur à puncto contactus, &longs;ecundum à linea incidentiæ; </s> <s id="N13C70"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i <lb/>accipiatur punctum E, in quo e&longs;t centrum grauitatis corporis AD, & in <lb/>eo fiat percu&longs;&longs;io; </s> <s id="N13C7C"><!-- NEW -->maximum e&longs;t impedimentum ratione puncti conta­<lb/>ctus, in quo fit percu&longs;&longs;io; </s> <s id="N13C82"><!-- NEW -->&longs;i verò percu&longs;&longs;io fiat per lineam perpendicu­<lb/>larem HE, maximum e&longs;t impedimentum, ratione lineæ; </s> <s id="N13C88"><!-- NEW -->&longs;i autem ex <lb/>vtroque capite &longs;imul accidat impedimentum, maximum e&longs;t omnium; </s> <s id="N13C8E"><!-- NEW --><lb/>iam verò &longs;i accipiatur punctum E, & linea percu&longs;sionis ME; </s> <s id="N13C93"><!-- NEW -->minor e&longs;t <lb/>percu&longs;sio ratione lineæ non puncti; </s> <s id="N13C99"><!-- NEW -->accipiatur punctum N, & linea <lb/>percu&longs;sionis MN, minor e&longs;t percu&longs;sio ratione puncti non lineæ, acci­<lb/>piatur punctum N, & linea IN, minor e&longs;t percu&longs;sio ratione vtriu&longs;que; </s> <s id="N13CA1"><!-- NEW --><lb/>&longs;i demum accipiatur punctum E, & linea ME, minor e&longs;t percu&longs;sio ra­<pb pagenum="43" xlink:href="026/01/075.jpg"/>tione lineæ non puncti; </s> <s id="N13CAB"><!-- NEW -->accipiatur punctum N linea percu&longs;&longs;ionis MN, <lb/>minor e&longs;t percu&longs;&longs;io ratione puncti non lineæ; </s> <s id="N13CB1"><!-- NEW -->&longs;i accipiatur punctum N, <lb/>& linea IN, minor e&longs;t percu&longs;&longs;io ratione vtriu&longs;que: </s> <s id="N13CB7"><!-- NEW -->&longs;i demum accipia­<lb/>tur punctum E & linea HE, maior e&longs;t percu&longs;&longs;io ratione vtriu&longs;que; </s> <s id="N13CBD"><!-- NEW -->igi­<lb/>tur &longs;unt quatuor coniugationes; &longs;eu quatuor cla&longs;&longs;es diuer&longs;arum percu&longs;­<lb/>&longs;ionum. </s> </p> <p id="N13CC5" type="main"> <s id="N13CC7"><!-- NEW -->Hinc compen&longs;ari pote&longs;t ratione vnius quod dee&longs;t ratione alterius, <lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i fiat percu&longs;&longs;io in puncto E per lineam ME, pote&longs;t &longs;ciri punctum <lb/>inter ED, in quo percu&longs;&longs;io per lineam perpendicularem &longs;it æqualis <lb/>percu&longs;&longs;ioni per lineam ME; &longs;ed de his infrà in lib. 10. cum de percu&longs;­<lb/>&longs;ione, determinabimus enim vnde proportiones i&longs;tæ petendæ &longs;int, & <lb/>demon&longs;trabimus totam i&longs;tam rem, quæ multùm curio&longs;itatis habet, & <lb/>vtilitatis. </s> </p> <p id="N13CDD" type="main"> <s id="N13CDF">Determinabimus etiam dato puncto percu&longs;&longs;ionis F v.g. <!-- REMOVE S-->cum &longs;equatur <lb/>motus vectis, quodnam &longs;it centrum vectis &longs;eu huius motus. </s> </p> <p id="N13CE6" type="main"> <s id="N13CE8"><!-- NEW -->Hinc demum &longs;equitur, ne hoc omittam, data minimâ percu&longs;&longs;ione per <lb/>lineam MN dari po&longs;&longs;e adhuc minorem per lineam IN, & alias incli­<lb/>natas; </s> <s id="N13CF0"><!-- NEW -->& data percu&longs;&longs;ione per lineam quantumuis inclinatam, po&longs;&longs;e da­<lb/>ri æqualem per lineam perpendicularem; </s> <s id="N13CF6"><!-- NEW -->& data per lineam perpendi­<lb/>cularem extra centrum grauitatis E, po&longs;&longs;e dari æqualem; & in qualibet <lb/>data ratione per aliquam inclinatam, quæ cadat in E, &longs;ed de his fusè <lb/>&longs;uo loco. </s> </p> <p id="N13D00" type="main"> <s id="N13D02"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s> </p> <p id="N13D0E" type="main"> <s id="N13D10"><!-- NEW --><emph type="italics"/>Corpus oblongum parallelipedum percutiens aliud corpus, putà globu&mtail;, <lb/>motu recto per lineam directionis, quæ producta à puncto contactus ducitur per <lb/>centrum globi, dum fiat contactus in centro grauitatis parallelipedi, maximum <lb/>ictum infligit, &longs;eu agit quantùm pote&longs;t.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it parallelipedum EB; quod <lb/>moueatur motu recto parallelo, lineis CD, HG, &c. </s> <s id="N13D25"><!-- NEW -->&longs;itque globus in <lb/>D; </s> <s id="N13D2B"><!-- NEW -->haud dubiè agit quantùm pote&longs;t, quia &longs;cilicet e&longs;t maximum impedi­<lb/>mentum per Th.68. Tam enim globus in D impedit motum paralleli­<lb/>pedi, quàm parallelipedum motum globi impacti per lineam ID; </s> <s id="N13D33"><!-- NEW -->impedit <lb/>inquam ratione oppo&longs;itionis; </s> <s id="N13D39"><!-- NEW -->quia centra grauitatis vtriu&longs;que con­<lb/>currunt in eadem linea; igitur &longs;i maximum e&longs;t impedimentum, agit <lb/>quantùm pote&longs;t Th. 50. hinc producitur impetus æqualis per Th.60. </s> </p> <p id="N13D41" type="main"> <s id="N13D43"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s> </p> <p id="N13D4F" type="main"> <s id="N13D51"><!-- NEW --><emph type="italics"/>Si percu&longs;&longs;io fiat in G, id e&longs;t &longs;i globus e&longs;&longs;et in G, producetur minor impetus, <lb/>& in<emph.end type="italics"/> M <emph type="italics"/>adhuc minor<emph.end type="italics"/>; </s> <s id="N13D62"><!-- NEW -->vt con&longs;tat ex dictis in &longs;uperioribus Theorematis; <lb/>in qua vero proportione determinabimus aliàs. </s> </p> <p id="N13D68" type="main"> <s id="N13D6A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s> </p> <p id="N13D76" type="main"> <s id="N13D78"><!-- NEW --><emph type="italics"/>Si corpus percutiens non &longs;it parallelipedum, &longs;ed alterius figuræ v.g.<emph.end type="italics"/> <emph type="italics"/>trigo­<lb/>non,<emph.end type="italics"/> ADE, &longs;itque maioris facilitatis gratia Orthonium; </s> <s id="N13D89"><!-- NEW -->eiu&longs;que motus <lb/>&longs;it parallelus lineis ED, BC: </s> <s id="N13D8F"><!-- NEW -->&longs;it autem DA dupla DE; </s> <s id="N13D93"><!-- NEW -->&longs;itque diui&longs;a to­<lb/>ta DA æqualiter in C, in C non erit maximus ictus; </s> <s id="N13D99"><!-- NEW -->quia in C non <pb pagenum="44" xlink:href="026/01/076.jpg"/>e&longs;t centrum grauitatis, vt patet; </s> <s id="N13DA2"><!-- NEW -->vt autem habeatur centrum impre&longs;&longs;io­<lb/>nis; </s> <s id="N13DA8"><!-- NEW -->a&longs;&longs;umatur AN media proportionalis inter totam AD, & &longs;ubdu­<lb/>plum AC; </s> <s id="N13DAE"><!-- NEW -->certè cum triangulum ANO &longs;it &longs;ubduplum totius ADE, <lb/>vt con&longs;tat ex Geometria, & æquale trapezo ND EO; </s> <s id="N13DB4"><!-- NEW -->erit impetus in <lb/>vtroque æqualis; </s> <s id="N13DBA"><!-- NEW -->igitur in N erit centrum impre&longs;&longs;ionis, vel impetus; </s> <s id="N13DBE"><!-- NEW -->vt <lb/>autem habeatur centrum percu&longs;&longs;ionis; </s> <s id="N13DC4"><!-- NEW -->in quo &longs;cilicet maximus ictus in­<lb/>fligitur, inueniatur centrum grauitatis H, ducaturque KHI parallela <lb/>DE, centrum percu&longs;&longs;ionis erit in I; </s> <s id="N13DCC"><!-- NEW -->quippe in I totus impeditur impetus <lb/>grauitatis vtrimque, cum &longs;it in æquilibrio; </s> <s id="N13DD2"><!-- NEW -->quomodo verò inueniatur <lb/>punctum H facilè habetur ex Archimede, ductis &longs;cilicet AF, DB, quæ <lb/>diuidant bifariam æqualiter DE, EA; vel a&longs;&longs;umpta AI dupla ID, quod <lb/>demon&longs;trabimus in Mechan. <!-- KEEP S--></s> </p> <p id="N13DDD" type="main"> <s id="N13DDF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N13DEB" type="main"> <s id="N13DED"><!-- NEW --><emph type="italics"/>Si circa centrum immobile rotetur corpus parallelipedum<emph.end type="italics"/> CA, <emph type="italics"/>diuer&longs;a e&longs;t <lb/>ratio percu&longs;&longs;ionum ab ea, quàm &longs;uprà propo&longs;uimus<emph.end type="italics"/>; </s> <s id="N13DFE"><!-- NEW -->moueatur enim circa <lb/>centrum C, fitque CA diui&longs;a bifariam in B, haud dubiè punctum A <lb/>faciet arcum AE eo tempore, quò punctum B faciet BD &longs;ubduplum <lb/>AE; </s> <s id="N13E08"><!-- NEW -->igitur punctum A duplò velociùs mouetur quàm B, vt con&longs;tat; </s> <s id="N13E0C"><!-- NEW -->igi­<lb/>tur habet duplò maiorem impetum; cum effectum habeat duplò maio­<lb/>rem per Ax. 13. n. </s> <s id="N13E14"><!-- NEW -->4. igitur cum totus motus &longs;egmenti AB &longs;it ad to­<lb/>tum motum &longs;egmenti BC, vt &longs;patia acqui&longs;ita; </s> <s id="N13E1A"><!-- NEW -->certè &longs;patia acqui&longs;ita <lb/>&longs;unt vt arcus; </s> <s id="N13E20"><!-- NEW -->igitur & trapezus BAED, continet 3/4 totius CAE, vt <lb/>con&longs;tat; </s> <s id="N13E26"><!-- NEW -->&longs;unt enim &longs;ectores &longs;imilis in ratione duplicata radiorum; </s> <s id="N13E2A"><!-- NEW -->igi­<lb/>tur totus motus &longs;egmenti BC &longs;ubquadruplus motus totius CA; igitur <lb/>& impetus; </s> <s id="N13E32"><!-- NEW -->vt autem habeatur centrum impre&longs;&longs;ionis, vel impetus; </s> <s id="N13E36"><!-- NEW -->&longs;it &longs;e­<lb/>ctor CHI, &longs;ubduplus totius CAE quod quomodo fiat, patet ex Geo­<lb/>metria; </s> <s id="N13E3E"><!-- NEW -->accipiatur tantùm &longs;ubdupla diagonalis quadrati lateris CA, igi­<lb/>tur in puncto H e&longs;t centrum impre&longs;&longs;ionis, &longs;eu media proportionalis in­<lb/>ter totam CA, & &longs;ubduplam CB: </s> <s id="N13E46"><!-- NEW -->vt autem habeatur percu&longs;&longs;ionis, a&longs;­<lb/>&longs;umatur CY dupla YA; </s> <s id="N13E4C"><!-- NEW -->Dico punctum Y e&longs;&longs;e centrum percu&longs;&longs;ionis; <lb/>quia perinde &longs;e habet, atque &longs;i e&longs;&longs;et trianguli cadentis ictus, vt demon­<lb/>&longs;trabimus aliàs nunc tantùm indica&longs;&longs;e &longs;ufficiat. </s> </p> <p id="N13E54" type="main"> <s id="N13E56"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13E63" type="main"> <s id="N13E65"><!-- NEW -->Hinc etiam &longs;oluetur, quod proponunt aliqui; &longs;eu potiùs quærunt; </s> <s id="N13E69"><!-- NEW --><lb/>in quà &longs;cilicet parte maiorem ictum infligat en&longs;is; </s> <s id="N13E6E"><!-- NEW -->&longs;i enim &longs;it eiu&longs;dem <lb/>cra&longs;&longs;itiei in omnibus &longs;uis partibus, idem dicendum e&longs;t quod de cylin­<lb/>dro CA; &longs;i verò in mucronem de&longs;inat, inueniemus etiam centrum <lb/>percu&longs;&longs;ionis. </s> </p> <p id="N13E78" type="main"> <s id="N13E7A"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13E87" type="main"> <s id="N13E89"><!-- NEW -->Huc etiam reuoca clauarum ictus, vel aliorum corporum, quæ ad in­<lb/>&longs;tar &longs;eu conorum, &longs;eu pyramidum ver&longs;us mucronem maiora &longs;unt, vel <lb/>den&longs;iora; quippe ex iacto &longs;uprà principio i&longs;torum omnium effectuum <lb/>rationes demon&longs;trabimus. </s> </p> <pb pagenum="45" xlink:href="026/01/077.jpg"/> <p id="N13E97" type="main"> <s id="N13E99"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13EA6" type="main"> <s id="N13EA8"><!-- NEW -->Colligemus etiam quid dicendum &longs;it de malleorum ictu; </s> <s id="N13EAC"><!-- NEW -->&longs;it enim <lb/>malleus F æqualis malleo G (in his vna fere manubrij longitudinis ha­<lb/>betur ratio) ducatur arcus NM, itemque OG; </s> <s id="N13EB4"><!-- NEW -->ictus mallei G e&longs;t ferè <lb/>&longs;ubduplus alterius, dum vterque malleus &longs;it æqualis; </s> <s id="N13EBA"><!-- NEW -->dixi ferè, quia <lb/>motus totius mallei G non e&longs;t omninò &longs;ubduplus motus mallei F, quia <lb/>&longs;cilicet trapezus OD e&longs;t minor &longs;ubduplo alterius NE; </s> <s id="N13EC2"><!-- NEW -->quotâ vero parte <lb/>&longs;it minor facilè pote&longs;t &longs;ciri opera Geometriæ: &longs;ed hæc omnia determi­<lb/>nabimus. </s> </p> <p id="N13ECA" type="main"> <s id="N13ECC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s> </p> <p id="N13ED8" type="main"> <s id="N13EDA"><!-- NEW --><emph type="italics"/>Si daretur potentia motrix, quæ &longs;emper agere po&longs;&longs;et, impetus po&longs;&longs;et intendi <lb/>in infinitum<emph.end type="italics"/>; </s> <s id="N13EE5"><!-- NEW -->pater, quia quocumque dato motu pote&longs;t dari velocior in <lb/>infinitum; igitur pote&longs;t dari impetus inten&longs;ior, & inten&longs;ior in infinitum. </s> </p> <p id="N13EEB" type="main"> <s id="N13EED"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13EF9" type="main"> <s id="N13EFB"><!-- NEW -->Hîc ob&longs;erua nouum di&longs;crimen, quod intercedit inter impetum, & <lb/>alias qualitates; </s> <s id="N13F01"><!-- NEW -->quæ fortè non po&longs;&longs;unt intendi in infinitum, ratio di&longs;­<lb/>criminis e&longs;t, quia totus calor exten&longs;us in maiore &longs;ubiecto non pote&longs;t <lb/>produci in minore, in quo eadem cau&longs;a eumdem &longs;emper effectum pro­<lb/>ducit; </s> <s id="N13F0B"><!-- NEW -->quia &longs;cilicet agit vniformiter difformiter; at verò impetus exten­<lb/>&longs;us in magno <expan abbr="den&longs;o&qacute;ue">den&longs;oque</expan> malleo pote&longs;t producere æqualem in maximâ <lb/>ferè pilâ. </s> </p> <p id="N13F17" type="main"> <s id="N13F19"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s> </p> <p id="N13F25" type="main"> <s id="N13F27"><!-- NEW --><emph type="italics"/>Impetus &longs;imilis, id e&longs;t, ad <expan abbr="eãdem">eandem</expan> lineam determinatus, & æqualis in in­<lb/>ten&longs;ione, non pote&longs;t intendere alium &longs;imilem<emph.end type="italics"/>; </s> <s id="N13F36"><!-- NEW -->Probatur, quia agit tantùm ad <lb/>extra, vt tollat impedimentum per Th. 44. &longs;ed eorum mobilium, quæ <lb/>ver&longs;us <expan abbr="eãdem">eandem</expan> partem pari velocitate mouentur, neutrum impedit al­<lb/>terius motum, vt con&longs;tat; igitur impetus &longs;imilis, &c. </s> </p> <p id="N13F44" type="main"> <s id="N13F46"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13F52" type="main"> <s id="N13F54"><!-- NEW -->Ob&longs;erua de impetu &longs;imili id tantùm dici; </s> <s id="N13F58"><!-- NEW -->&longs;imili inquam id e&longs;t non <lb/>modò eiu&longs;dem inten&longs;ionis; </s> <s id="N13F5E"><!-- NEW -->&longs;ed etiam eiu&longs;dem lineæ: </s> <s id="N13F62"><!-- NEW -->&longs;i enim alterum <lb/>de&longs;it, haud dubiè &longs;imilis impetus non e&longs;t; </s> <s id="N13F68"><!-- NEW -->&longs;ic impetus quatuor grad. <!-- REMOVE S-->in­<lb/>tendere pote&longs;t impetum duorum graduum; </s> <s id="N13F70"><!-- NEW -->licèt vterque ad <expan abbr="eãdem">eandem</expan> li­<lb/>neam &longs;it determinatus; </s> <s id="N13F7A"><!-- NEW -->&longs;i verò ad diuer&longs;as lineas determinentur; etiam <lb/>impetus vt duo pote&longs;t intendere impetum vt quatuor. </s> </p> <p id="N13F80" type="main"> <s id="N13F82"><!-- NEW -->Ob&longs;eruabis præterea hoc Theorema ita e&longs;&longs;e intelligendum, vt impe­<lb/>tus mobilis præeuntis nullo modo impediatur; alioquin mobile &longs;ucce­<lb/>dens omninò aliud vrgeret, vt con&longs;tat. </s> </p> <p id="N13F8A" type="main"> <s id="N13F8C"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13F98" type="main"> <s id="N13F9A"><!-- NEW -->Hinc &longs;imile pote&longs;t in aliquo ca&longs;u agere in &longs;imile; </s> <s id="N13F9E"><!-- NEW -->vnde rectè colligo <lb/>id tantùm dictum e&longs;&longs;e ab Ari&longs;totele de qualitatibus alteratiuis; </s> <s id="N13FA4"><!-- NEW -->quid <lb/>verò accidat, cum mobile graue mobili alteri &longs;uperponitur; dicemus <lb/>infrà. </s> </p> <pb pagenum="46" xlink:href="026/01/078.jpg"/> <p id="N13FB0" type="main"> <s id="N13FB2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s> </p> <p id="N13FBE" type="main"> <s id="N13FC0"><!-- NEW --><emph type="italics"/>Exten&longs;io impetus respondet extentioni &longs;ui &longs;ubiecti, &longs;cilicet mobilis<emph.end type="italics"/>; </s> <s id="N13FC9"><!-- NEW -->cum <lb/>enim extra &longs;ubjectum e&longs;&longs;e non po&longs;&longs;it, cum &longs;it qualitas; </s> <s id="N13FCF"><!-- NEW -->certè ibi e&longs;t, vbi <lb/>&longs;ubjectum e&longs;t; nam penetratur accidens cum ip&longs;o <expan abbr="&longs;ujecto">&longs;ubjecto</expan>. </s> </p> <p id="N13FD9" type="main"> <s id="N13FDB"><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N13FE7" type="main"> <s id="N13FE9"><!-- NEW -->Ob&longs;eruabis qualitatem omnem ita &longs;uo &longs;ubjecto coëxtendi, vt æqua­<lb/>lem omnino quodlibet eius punctum, &longs;eu pars extentionem habeat ex­<lb/>tentioni puncti, &longs;eu partis &longs;ui &longs;ubjecti; </s> <s id="N13FF1"><!-- NEW -->nec enim aliud e&longs;t, vnde po&longs;&longs;it <lb/>determinari extentio qualitatum, præter ip&longs;am exten&longs;ionem &longs;ubjecti; </s> <s id="N13FF7"><!-- NEW --><lb/>quod maximè in impetu videre e&longs;t, cuius partes in mobili den&longs;o minori <lb/>extentioni &longs;ubjacent, quàm in mobili raro; </s> <s id="N13FFE"><!-- NEW -->cum ex maiore ictu &longs;eu per­<lb/>cu&longs;&longs;ione in mobili den&longs;o plures impetus agentis partes e&longs;&longs;e con&longs;tet; quia <lb/>&longs;cilicet &longs;unt plures partes &longs;ubiecti. </s> </p> <p id="N14006" type="main"> <s id="N14008"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s> </p> <p id="N14014" type="main"> <s id="N14016"><!-- NEW --><emph type="italics"/>Datur impetus altero impetu perfectior &longs;ecundum entitatem<emph.end type="italics"/>; dixi &longs;ecun­<lb/>dum entitatem; </s> <s id="N14021"><!-- NEW -->quia iam dictum e&longs;t &longs;uprà dari perfectiorem &longs;ecundum <lb/>inten&longs;ionem; </s> <s id="N14027"><!-- NEW -->huius Theorematis veritas mihi maximè demon&longs;tranda <lb/>e&longs;t, ex quo tàm multa infrà deducemus; </s> <s id="N1402D"><!-- NEW -->&longs;ic autem probamus; </s> <s id="N14031"><!-- NEW -->Quotie&longs;­<lb/>cunque mouetur corpus, producuntur &longs;altem tot partes impetus quot <lb/>&longs;unt partes mobilis per Th. 33. Quotie&longs;cunque producuntur in mobili <lb/>tot partes impetus quot &longs;unt in mobili partes &longs;ubjecti, mouetur mobile, <lb/>modó non impediatur; </s> <s id="N1403D"><!-- NEW -->quia po&longs;ita cau&longs;a nece&longs;&longs;aria, & non impedita per <lb/>Ax. 11. ponitur effectus, quod de omni cau&longs;a, &longs;ed de formali poti&longs;&longs;imum <lb/>dici debet; </s> <s id="N14045"><!-- NEW -->præterea datur aliquod pondus, quod data potentia &longs;ine me­<lb/>chanico organo mouere non pote&longs;t, licèt cum organo facilè moueat; </s> <s id="N1404B"><!-- NEW -->hæc <lb/>hypothe&longs;is certa e&longs;t; </s> <s id="N14051"><!-- NEW -->igitur cum mouet, producit tot partes impetus quot <lb/>&longs;unt nece&longs;&longs;ariæ, vt omnibus partibus mobilis di&longs;tribuantur per idem Th. <!-- REMOVE S--><lb/>33. cum verò non mouet, non producit tot partes impetus vt con&longs;tat ex <lb/>dictis; </s> <s id="N1405C"><!-- NEW -->igitur producit plures cum organo in mobili, quàm &longs;ine organo; <lb/>igitur imperfectiores, quod demon&longs;tro: </s> <s id="N14062"><!-- NEW -->&longs;it enim vectis BF, cuius cen­<lb/>trum &longs;eu fulcrum &longs;it in A, potentia in B, pondus G, quod attollitur in F; </s> <s id="N14068"><!-- NEW --><lb/>plures partes impetus produci po&longs;&longs;unt in F, vel in E, quàm in B, &longs;cilicet <lb/>in ip&longs;o pondere; </s> <s id="N1406F"><!-- NEW -->quia pondus quod non pote&longs;t attolli in B, attollitur in <lb/>E, vel in F, vt patet ex dictis; </s> <s id="N14075"><!-- NEW -->præterea punctum F mouetur tardius, quàm <lb/>B; </s> <s id="N1407B"><!-- NEW -->quia motus &longs;unt vt arcus, arcus vt &longs;emidiametri, hæ demum vt AF, <lb/>ad AB; </s> <s id="N14081"><!-- NEW -->igitur motus puncti F, e&longs;t tardior, vel imperfectior; </s> <s id="N14085"><!-- NEW -->igitur im­<lb/>petus puncti F, e&longs;t imperfectior impetu puncti B, per Ax. 13 num.4. atqui <lb/>non e&longs;t imperfectior ratione numeri partium, igitur ratione entitatis, <lb/>quæ imperfectior e&longs;t; igitur datur impetus altero impetu imperfectior. </s> </p> <p id="N1408F" type="main"> <s id="N14091"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1409D" type="main"> <s id="N1409F"><!-- NEW -->Ob&longs;eruabis primò multa hîc &longs;upponi &longs;eu de&longs;iderari, quæ pertinent <lb/>ad propagationem impetus, de quibus infrà; Secundò hoc Theorema <pb pagenum="47" xlink:href="026/01/079.jpg"/>per Axioma illud Metaph. probari, <emph type="italics"/>Data quacumque creatura dari potest <lb/>perfectior, vel imperfectior.<emph.end type="italics"/></s> </p> <p id="N140B1" type="main"> <s id="N140B3"><!-- NEW -->Tertiò, &longs;i dato quocunque motu pote&longs;t dari tardior: igitur dato quo­<lb/>cunque impetu pote&longs;t dari imperfectior. </s> </p> <p id="N140B9" type="main"> <s id="N140BB"><!-- NEW -->Quartò, &longs;i daretur punctum impetus in inten&longs;ione: non po&longs;&longs;et dari <lb/>motus tardior in infinitum &longs;ine diuer&longs;is gradibus perfectionis. </s> </p> <p id="N140C1" type="main"> <s id="N140C3"><!-- NEW -->Quintò, &longs;ine hac diuer&longs;a impetus perfectione non po&longs;&longs;et explicari <lb/>productio continua impetus, quæ &longs;it temporibus inæqualibus, neque de­<lb/>&longs;tructio eiu&longs;dem impetus; nec motus in diuer&longs;is planis inclinatis, vel in­<lb/>diuer&longs;is lineis citra perpendicularem, &longs;ed de his omnibus &longs;uo loco. </s> </p> <p id="N140CD" type="main"> <s id="N140CF"><!-- NEW -->Sextò, Denique ratio propo&longs;ita rem i&longs;tam euincit; </s> <s id="N140D3"><!-- NEW -->cum enim in motu <lb/>vectis plures partes producantur ver&longs;us centrum, &longs;cilicet, in maiori pon­<lb/>dere, quod attollitur; & cum hæ habeant motum tardiorem, &longs;equitur ne­<lb/>ce&longs;&longs;ariò e&longs;&longs;e imperfectiores. </s> </p> <p id="N140DD" type="main"> <s id="N140DF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s> </p> <p id="N140EB" type="main"> <s id="N140ED"><emph type="italics"/>Dato quocumque impetu dari pote&longs;t imperfectior, & imperfectior,<emph.end type="italics"/> quia da­<lb/>to quocumque motu dari pote&longs;t tardior, ergo dato quocumque impetu <lb/>imperfectior. </s> </p> <p id="N140F9" type="main"> <s id="N140FB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s> </p> <p id="N14107" type="main"> <s id="N14109"><emph type="italics"/>Non pote&longs;t explicari tarditas motus &longs;ine diuer&longs;a perfectione impetus, per <lb/>pauciores &longs;cilicet eiu&longs;dem impetus partes.<emph.end type="italics"/></s> <s id="N14112"><!-- NEW --> Primò, quia cum retardari po&longs;&longs;it <lb/>hic motus, & de&longs;trui &longs;ucce&longs;&longs;inè hic impetus; </s> <s id="N14118"><!-- NEW -->cumque in&longs;tantia motus <lb/>velocioris &longs;int breuiora; </s> <s id="N1411E"><!-- NEW -->certè initio motus, breuiori &longs;cilicet tempore <lb/>imperfectior impetus de&longs;trui tantùm pote&longs;t; </s> <s id="N14124"><!-- NEW -->cum enim æqualis æquali­<lb/>bus temporibus; certè inæqualis inæqualibus. </s> <s id="N1412A">Secundò quia vix explica­<lb/>ri pore&longs;t quomodo duæ formæ homogeneæ in eodem &longs;ubiecti puncto <lb/>exi&longs;tere po&longs;&longs;int, quod etiam in commune e&longs;t calori, lumini, &c. </s> </p> <p id="N14131" type="main"> <s id="N14133"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s> </p> <p id="N1413F" type="main"> <s id="N14141"><emph type="italics"/>Cum applicatur potentia centro vectis, non producitur æqualis impetus ver­<lb/>&longs;us circumferentiam in omnibus partibus, &longs;ed maior ver&longs;us eandem circumfe­<lb/>rentiam,<emph.end type="italics"/> quia e&longs;t maior motus. </s> </p> <p id="N1414D" type="main"> <s id="N1414F"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1415C" type="main"> <s id="N1415E"><!-- NEW -->Hinc difficiliùs attollitur pertica CA ex puncto C motu circulari, <lb/>quàm ex puncto B motu recto; </s> <s id="N14164"><!-- NEW -->quia &longs;cilicet, cum motu recto ex puncto B <lb/>attollitur, omnes partes mouentur motu æquali; </s> <s id="N1416A"><!-- NEW -->igitur impetus æqualiter <lb/>omnibus di&longs;tribuitur; </s> <s id="N14170"><!-- NEW -->igitur modò producantur tot partes impetus, quot <lb/>&longs;unt partes in mobili; </s> <s id="N14176"><!-- NEW -->haud dubiè attolletur: </s> <s id="N1417A"><!-- NEW -->at verò, cum motu circulari <lb/>ex puncto C attollitur, omnes partes inæquali motu attolluntur; </s> <s id="N14180"><!-- NEW -->igitur <lb/>plures &longs;unt nece&longs;&longs;ariæ, vt attollatur motu circulari; </s> <s id="N14186"><!-- NEW -->igitur difficiliùs iuxta <lb/>experimentum; adde quod cum applicatur potentia in C, punctum A, <lb/>maius momentum habet, de quo aùàs. </s> </p> <p id="N1418E" type="main"> <s id="N14190"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1419D" type="main"> <s id="N1419F"><!-- NEW -->Hinc ratio euidens illius experimenti, quo manife&longs;tè con&longs;tat perti-<pb pagenum="48" xlink:href="026/01/080.jpg"/>cam CA, ex A, facilius attolli motu recto, quàm circulari; cum &longs;ci­<lb/>licet cuiu&longs;dam qua&longs;i reflexionis opera eodem tempore vtraque extremi­<lb/>tas æquali motu attollitur. </s> </p> <p id="N141AC" type="main"> <s id="N141AE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s> </p> <p id="N141BA" type="main"> <s id="N141BC"><arrow.to.target n="note1"/></s> </p> <p id="N141C1" type="margin"> <s id="N141C3"><margin.target id="note1"/><emph type="italics"/>Fig.<emph.end type="italics"/>7. <lb/><emph type="italics"/>Tab.<emph.end type="italics"/>1.<!-- KEEP S--></s> </p> <p id="N141D5" type="main"> <s id="N141D7"><!-- NEW --><emph type="italics"/>Si verò applicetur potentia extra centrum vectis v. <!-- REMOVE S-->g. <!-- REMOVE S-->in<emph.end type="italics"/> F, <emph type="italics"/>po&longs;ito centro in<emph.end type="italics"/><lb/>A, <emph type="italics"/>producitur impetus minor ab<emph.end type="italics"/> F, <emph type="italics"/>ver&longs;us<emph.end type="italics"/> A; </s> <s id="N141F7"><!-- NEW --><emph type="italics"/>ab verò ver&longs;us<emph.end type="italics"/> E, <emph type="italics"/>producitur <lb/>eiu&longs;dem perfectionis proportionaliter, cuius e&longs;t ab<emph.end type="italics"/> F, <emph type="italics"/>ver&longs;us<emph.end type="italics"/> A; denique ab E, <lb/>ver&longs;us B, producitur quidem vnum punctum, vel vnus gradus impetus <lb/>eiu&longs;dem perfectionis cum eo, qui productus e&longs;t in F, & in E (&longs;upponi­<lb/>tur enim ex. gr. <!-- REMOVE S-->vnus tantùm gradus in F, & in E, productus) at verò <lb/>producuntur alij imperfectiones. </s> <s id="N14218"><!-- NEW -->v.g. <!-- REMOVE S-->in D, præter æquè perfectum pro­<lb/>ducuntur 3. alij adæquantes perfectionem prioris; </s> <s id="N14220"><!-- NEW -->in C verò, præter 4. <lb/>&longs;imiles ijs, qui &longs;unt in D, producuntur 5. alij adæquantes prioris perfe­<lb/>ctionem in B7; atque ita deinceps per numeros impares, & quadrata, <lb/>nullus tamen producitur perfectioris entitatis. </s> </p> <p id="N1422A" type="main"> <s id="N1422C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s> </p> <p id="N14238" type="main"> <s id="N1423A"><!-- NEW --><emph type="italics"/>Determinatur hæc diuer&longs;a perfectio impetus à diuer&longs;a perfectione motus, <lb/>quatenus fit tali modo<emph.end type="italics"/>; </s> <s id="N14245"><!-- NEW -->quæ non pote&longs;t explicari per impetum remi&longs;&longs;io­<lb/>rem, vel inten&longs;iorem; </s> <s id="N1424B"><!-- NEW -->nam cum &longs;it tantùm impetus in&longs;titutus propter <lb/>motum; </s> <s id="N14251"><!-- NEW -->certè ille tantùm impetus produci pote&longs;t, ex quo pote&longs;t &longs;equi <lb/>motus; </s> <s id="N14257"><!-- NEW -->igitur &longs;i tali tantùm motu data pars mobilis moueri pote&longs;t; haud <lb/>dubiè talis tantùm impetus, ex quo &longs;equitur talis motus, in ea produ­<lb/>cetur, & tali modo. </s> </p> <p id="N1425F" type="main"> <s id="N14261"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s> </p> <p id="N1426D" type="main"> <s id="N1426F"><!-- NEW --><emph type="italics"/>Perfectio impetus non petitur tantùm à perfectione motus &longs;i con&longs;ideretur <lb/>&longs;eor&longs;im entitas eiu&longs;dem impetus; </s> <s id="N14277"><!-- NEW -->&longs;ed debet comparari tota collectio omniu&mtail; <lb/>partium impetus, quæ in&longs;unt datæ parti &longs;ubiecti, cum tota collectione partium <lb/>quæ alteri parti mobilis in&longs;unt<emph.end type="italics"/>; </s> <s id="N14282"><!-- NEW -->quippe plures partes impetus po&longs;&longs;unt ha­<lb/>bere eum motum, vel potius eam motus perfectionem, quam pauciores <lb/>haberent; igitur perfectio illarum e&longs;t ab ip&longs;o motu, quatenus cum ip&longs;o <lb/>partium numero comparatur. </s> </p> <p id="N1428C" type="main"> <s id="N1428E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s> </p> <p id="N1429A" type="main"> <s id="N1429C"><!-- NEW --><emph type="italics"/>Impetus perfectus producere pote&longs;t imperfectum<emph.end type="italics"/>; patet in vecte; </s> <s id="N142A5"><!-- NEW -->nam po­<lb/>tentia, &longs;en pondus extremitati appen&longs;um producit in &longs;e impetum, à quo <lb/>deinde impetus in toto vecte producitur per Th.42. &longs;ed impetus pon­<lb/>deris appen&longs;i e&longs;t eiu&longs;dem perfectionis cum impetu producto in ip&longs;a ve­<lb/>ctis extremitate, ex qua pendet; </s> <s id="N142B1"><!-- NEW -->cum &longs;it vtriu&longs;que æqualis motus; &longs;ed <lb/>ver&longs;us centrum eiu&longs;dem vectis producitur impetus imperfectior per <lb/>Th.82. igitur imperfectus à perfecto producitur. </s> </p> <p id="N142B9" type="main"> <s id="N142BB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s> </p> <p id="N142C7" type="main"> <s id="N142C9"><!-- NEW --><emph type="italics"/>Impetus perfectus nunquam producitur ab imperfecto, per Ax.<emph.end type="italics"/> 3. <emph type="italics"/>num.<emph.end type="italics"/> 2. <lb/>adde quod nunquam effectus perfectio &longs;uperat perfectionem cau&longs;æ; </s> <s id="N142DA"><!-- NEW -->dixi <lb/>perfectum ab imperfecto; </s> <s id="N142E0"><!-- NEW -->&longs;cilicet &longs;i con&longs;ideretur perfectio ratione en-<pb pagenum="49" xlink:href="026/01/081.jpg"/>titatis; </s> <s id="N142E9"><!-- NEW -->cum reuerâ, vt dictum e&longs;t &longs;uprà, remi&longs;&longs;us producat inten&longs;um, <lb/>quod in vecte clari&longs;&longs;imum e&longs;t; </s> <s id="N142EF"><!-- NEW -->quippe momentum applicatum in F, quod <lb/>tardiùs mouetur deor&longs;um, quàm B, &longs;ur&longs;um, vt patet, habet impetum re­<lb/>mi&longs;&longs;iorem, qui tamen producit in B, inten&longs;iorem: </s> <s id="N142F7"><!-- NEW -->Pro quo, ob&longs;eruabis <lb/>impetum imperfectum cum alio perfecto actione communi agentem <lb/>po&longs;&longs;e concurrere ad producendum perfectum, vt patet; </s> <s id="N142FF"><!-- NEW -->non tamen in <lb/>ratione cau&longs;æ totalis: </s> <s id="N14305"><!-- NEW -->&longs;imiliter plures imperfecti &longs;imul concurrentes <lb/>po&longs;&longs;unt producere perfectum; quia plures imperfecti conjunctim adæ­<lb/>quant perfectionem alterius perfectioris &longs;inguli &longs;eor&longs;im. </s> </p> <p id="N1430D" type="main"> <s id="N1430F"><!-- NEW -->Ob&longs;eruabis &longs;ecundò præclarum naturæ in&longs;titutum, quo factum e&longs;t; </s> <s id="N14313"><!-- NEW --><lb/>vt cum vires hominum maiora pondera leuare non po&longs;&longs;int, &longs;i &longs;eor&longs;un <lb/>con&longs;iderentur; </s> <s id="N1431A"><!-- NEW -->cum organis tamen mechanicis conjunctæ nullum pon­<lb/>dus quantumuis immane leuare non po&longs;&longs;int; </s> <s id="N14320"><!-- NEW -->quod certè nullo modo ac­<lb/>cideret, ni&longs;i plures partes impetus producerent neque plures producere <lb/>po&longs;&longs;ent, ni&longs;i minoris perfectionis e&longs;&longs;ent; quia faciliùs producitur effe­<lb/>ctus imperfectus, quam perfectus per Ax. 13.num.4. </s> </p> <p id="N1432A" type="main"> <s id="N1432C"><!-- NEW -->Tertiò hinc optimè à natura proui&longs;um e&longs;t, vt motus tardior in infi­<lb/>nitum e&longs;&longs;e po&longs;&longs;it; quod reuerâ fieri non po&longs;&longs;et, ni&longs;i dari po&longs;&longs;et impetus <lb/>alio imperfectior. </s> </p> <p id="N14334" type="main"> <s id="N14336"><!-- NEW -->Quartò, hinc quoque benè explicatur diuer&longs;itas impetus, quæ oritur <lb/>tum à diuer&longs;o medio, tùm à plano inclinato, tùm ab aliis impedimentis, <lb/>tùm à diuer&longs;o ni&longs;u eiu&longs;dem potentiæ, tùm maximè à diuer&longs;o applicatio­<lb/>nis modo; de quibus aliàs. </s> </p> <p id="N14340" type="main"> <s id="N14342"><!-- NEW -->Quintò, &longs;i potentia applicata mobili immediatè illud moueat motu <lb/>recto, vel in &longs;ingulis punctis mobilis producitur vnum punctum impe­<lb/>tus, vel plura; </s> <s id="N1434A"><!-- NEW -->&longs;i primum, erit primus tantùm gradus maximæ perfectio­<lb/>nis; </s> <s id="N14350"><!-- NEW -->ita vt perfectiorem producere non po&longs;&longs;it, ad quem e&longs;t determinata <lb/>potentia; </s> <s id="N14356"><!-- NEW -->imperfectiorem tamen impetu innato, de quo infrà; &longs;i verò <lb/>&longs;ecundum, producet in &longs;ingulis partibus <expan abbr="eũdem">eundem</expan> gradum perfecti&longs;&longs;i­<lb/>mum cum aliis pluribus, vel paucioribus heterogeneis, & imperfectio­<lb/>ribus. </s> </p> <p id="N14364" type="main"> <s id="N14366"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s> </p> <p id="N14372" type="main"> <s id="N14374"><!-- NEW --><emph type="italics"/>Potentia naturalis grauium producit tantùm vno in&longs;tanti ad intra vnicum <lb/>punctum impetus in quolibet puncto &longs;ubiecti; </s> <s id="N1437C"><!-- NEW -->&longs;i tamen impetum producit, quod <lb/>definiam lib.<emph.end type="italics"/> 20. <emph type="italics"/>& &longs;i dentur puncta &longs;ubiecti, quod ad præ&longs;ens in&longs;titutum non <lb/>pertinet<emph.end type="italics"/>; </s> <s id="N1438D"><!-- NEW -->Probatur, quia fru&longs;trà e&longs;&longs;ent plura puncta impetus; nec enim <lb/>&longs;unt multiplicandæ formæ &longs;ine nece&longs;&longs;itate, ratione &c. </s> <s id="N14393">per Ax. 7. & 3. <lb/>n. </s> <s id="N14398">1. Præterea non e&longs;t, cur potius produceret 2. quàm 3. 4. &c. </s> <s id="N1439B">atqui <lb/>quod vnum e&longs;t, determinatum e&longs;t per Ax. 5. <!-- KEEP S--></s> </p> <p id="N143A1" type="main"> <s id="N143A3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s> </p> <p id="N143AF" type="main"> <s id="N143B1"><!-- NEW --><emph type="italics"/>Potentia motrix animantium etiam vno in&longs;tanti plura puncta, &longs;en partes <lb/>impetus in eadem parte &longs;ubiecti producere potest<emph.end type="italics"/>; </s> <s id="N143BC"><!-- NEW -->Probatur in proiectis, <lb/>quorum impetus aliquando plùs, aliquando minùs durat licèt &longs;en&longs;im <lb/>&longs;ingulis in&longs;tantibus aliquid illius de&longs;truatur; </s> <s id="N143C4"><!-- NEW -->determinatur autem <pb pagenum="50" xlink:href="026/01/082.jpg"/>numerus punctorum, &longs;eu partium ab ea potentia, cui &longs;ube&longs;t potentia <lb/>motrix; quia modò maior e&longs;t ni&longs;us, modò minor. </s> </p> <p id="N143CF" type="main"> <s id="N143D1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s> </p> <p id="N143DD" type="main"> <s id="N143DF"><!-- NEW --><emph type="italics"/>Eadem potentia inæqualibus temporibus impetum inæqualem in perfectio­<lb/>ne producit<emph.end type="italics"/>; </s> <s id="N143EA"><!-- NEW -->accipiatur enim totum illud tempus, quo vnicum tantùm <lb/>punctum impetus producit (vocetur in&longs;tans) de quo in Th. 86; certè <lb/>&longs;i in minori tempore agat, minùs aget, per Ax. 13. num. </s> <s id="N143F2"><!-- NEW -->4. &longs;ed non <lb/>pote&longs;t minùs agere ratione numeri, vt patet; igitur ratione perfectio­<lb/>nis. </s> </p> <p id="N143FA" type="main"> <s id="N143FC"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14408" type="main"> <s id="N1440A">Ob&longs;eruabis &longs;ine hoc Theoremate explicari non po&longs;&longs;e accelerationem <lb/>motus naturalis, vel augmentum impetus, vt videbimus. </s> </p> <p id="N1440F" type="main"> <s id="N14411"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s> </p> <p id="N1441D" type="main"> <s id="N1441F"><!-- NEW --><emph type="italics"/>Impetus violenti, qui &longs;en&longs;im de&longs;truitur in proiectis, po&longs;itis ij&longs;dem circum­<lb/>&longs;tantiis medij, & re&longs;i&longs;tentiæ, minori tempore minùs de&longs;truitur; </s> <s id="N14427"><!-- NEW -->plus verò ma­<lb/>jori:<emph.end type="italics"/> Quia hæc de&longs;tructio habet cau&longs;am; nam quidquid de&longs;truitur, ad <lb/>exigentiam alicuius de&longs;truitur, per Ax. 14. num. </s> <s id="N14432">2. igitur minori <lb/>tempore minùs de&longs;truitur per Ax. <!-- REMOVE S-->13. 4. alioquin totus &longs;imul debe­<lb/>ret de&longs;trui. </s> </p> <p id="N1443B" type="main"> <s id="N1443D"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14449" type="main"> <s id="N1444B">Ob&longs;eruabis etiam &longs;ine hoc Theoremate non po&longs;&longs;e explicari de&longs;tru­<lb/>ctionem impetus violenti, vt videbimus infrà. </s> </p> <p id="N14450" type="main"> <s id="N14452"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1445F" type="main"> <s id="N14461">Hinc, quò potentia diutiùs manet applicata (putà malleo) percu&longs;&longs;io ma­<lb/>ior e&longs;t. </s> </p> <p id="N14466" type="main"> <s id="N14468"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14475" type="main"> <s id="N14477">Hinc, quò impedimentum diutiùs manet applicatum, illa de&longs;tructio <lb/>e&longs;t maior. </s> </p> <p id="N1447C" type="main"> <s id="N1447E"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1448B" type="main"> <s id="N1448D"><!-- NEW -->Hinc præclara eruitur ratio, cur maior lapis, quàm minor impactus <lb/>maiorem ictum infligat; </s> <s id="N14493"><!-- NEW -->licèt tot partes impetus eodem in&longs;tanti produ­<lb/>cantur in vno, quot in alio: </s> <s id="N14499"><!-- NEW -->quia &longs;cilicet diutiùs manet applicatus po­<lb/>tentiæ; &longs;ed hanc rationem explicabimus fusè lib. 10. cum de percu&longs;­<lb/>&longs;ione. </s> </p> <p id="N144A1" type="main"> <s id="N144A3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s> </p> <p id="N144AF" type="main"> <s id="N144B1"><emph type="italics"/>Impetus propagatur nece&longs;&longs;ariò per totum corpus impul&longs;um, &longs;eu proiectum.<emph.end type="italics"/></s> </p> <p id="N144B8" type="main"> <s id="N144BA">Probatur; </s> <s id="N144BD"><!-- NEW -->quia cum omnes eius partes moueantur, nec vlla &longs;ine im­<lb/>petu moueri po&longs;&longs;it per Th. 18. & 33. cum etiam potentia motrix non <lb/>&longs;it omnibus immediatè applicata, vt con&longs;tat; certè &longs;ine propagatione, <lb/>vel diffu&longs;ione non pote&longs;t explicari productio huius motus. </s> </p> <pb pagenum="51" xlink:href="026/01/083.jpg"/> <p id="N144CB" type="main"> <s id="N144CD"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N144D9" type="main"> <s id="N144DB">Ob&longs;eruabis propagationem impetus, vel alterius qualitatis e&longs;&longs;e tan­<lb/>tùm continuatam eiu&longs;dem productionem, quæ incipit ab ea parte, cui <lb/>potentia e&longs;t immediatè applicata, & propagatur, &longs;eu diffunditur per <lb/>omnes alias donec ad vltimam perueniat eo modo, quo iam definio. </s> </p> <p id="N144E4" type="main"> <s id="N144E6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s> </p> <p id="N144F2" type="main"> <s id="N144F4"><!-- NEW --><emph type="italics"/>Illa progatio non fit per motum localem, ita vt pars impetus producta in <lb/>prima parte &longs;ubiecti tran&longs;eat ad &longs;ecundam,<emph.end type="italics"/> patet; quia cum impetus &longs;it ac­<lb/>cidens per Th. 8. de &longs;ubiecto in &longs;ubiectum tran&longs;ire non pote&longs;t per deff. </s> <s id="N14501"><lb/>accidentis; de qua in Metaphy&longs;icâ; </s> <s id="N14505"><!-- NEW -->nec e&longs;t quod aliqui dicant &longs;e <expan abbr="nõ">non</expan> po&longs;&longs;e <lb/>concipere, quomodo id fiat &longs;ine motu locali; </s> <s id="N1450F"><!-- NEW -->cum ip&longs;is etiam oculis <lb/>qua&longs;i cernatur; </s> <s id="N14515"><!-- NEW -->cum enim percutis corpus oblongum AE, & cadit ictus <lb/>in extremitatem A, corpus ip&longs;um totum &longs;imul moues; igitur pars impe­<lb/>tus, quæ recipitur in A, non migrat in E, &longs;ed hæc producitur in A, & <lb/>alia in B, alia in C, atque ita deinceps. </s> </p> <p id="N1451F" type="main"> <s id="N14521"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1452D" type="main"> <s id="N1452F">Ob&longs;eruabis ex hac propagatione impetus per analogiam rectè om­<lb/>ninò explicari propagationem luminis, & aliarum qualitatum, de qui­<lb/>bus &longs;uo loco. </s> </p> <p id="N14536" type="main"> <s id="N14538"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s> </p> <p id="N14544" type="main"> <s id="N14546"><!-- NEW --><emph type="italics"/>In propagatione impetus prima pars<emph.end type="italics"/> A v. <!-- REMOVE S-->g. <emph type="italics"/>non producit partem<emph.end type="italics"/> B, <emph type="italics"/>& <lb/>hæc<emph.end type="italics"/> C; </s> <s id="N1455F"><!-- NEW --><emph type="italics"/>hæc verò<emph.end type="italics"/> D, <emph type="italics"/>atque ita deinceps<emph.end type="italics"/>; Probatur. <!-- KEEP S--></s> <s id="N1456F"><!-- NEW -->Primò, quia &longs;i hoc e&longs;&longs;et, <lb/>omne corpus po&longs;&longs;et moueri à qualibet potentia; nam modò po&longs;&longs;et pro­<lb/>duci vnum punctum impetus, hoc etiam aliud produceret, & hoc aliud, <lb/>atque ita deinceps. </s> <s id="N14579">Secundò, Minimum granum &longs;uperpo&longs;itum rupi, to­<lb/>tam ip&longs;am rupem mouere po&longs;&longs;et. </s> <s id="N1457E"><!-- NEW -->Tertio, Quia vel in omnibus, vel in <lb/>nulla parte impetus producitur per Th.33. Quartò, quia impetus mobi­<lb/>lis projecti intenderetur; nam impetus vnius partis impetum alterius <lb/>intenderet. </s> <s id="N14588"><!-- NEW -->Quintò, quia impetus partis B, tàm ageret in A, trahendo, <lb/>quàm in C pellendo; cum impetus vtroque modo propagetur. </s> <s id="N1458E"><!-- NEW -->Sextò, &longs;i <lb/>applicaretur potentia in C, non video, cur impetus partis C, ageret po­<lb/>tius versùs E, quàm versùs A? alioquin eadem pars impetus plures pro­<lb/>ducere po&longs;&longs;et; igitur impetus potentiæ motricis &longs;ufficiens erit cau&longs;a ad <lb/>producendum totum alium. </s> <s id="N1459A"><!-- NEW -->Septimò, tractionis impetus explicari non <lb/>pote&longs;t, &longs;i impetus vnius partis producat in alia impetum; alioquin dare­<lb/>tur mutua actio infinities repetita, vt con&longs;ideranti patebit. </s> <s id="N145A2"><!-- NEW -->Octauò, &longs;i <lb/>impetus vnius partis producit in alia; </s> <s id="N145A8"><!-- NEW -->&longs;int duo globi contigui; igitur il­<lb/>le, qui impellit alium, reflecti po&longs;&longs;et, quod nunquam accidit quando <lb/>&longs;unt contigui. </s> </p> <p id="N145B0" type="main"> <s id="N145B2"><!-- NEW -->Ob&longs;eruabis illud quidem verum e&longs;&longs;e in motu recto, &longs;ecus in circulari; </s> <s id="N145B6"><!-- NEW --><lb/>nam cum cylindrus circa alteram extremitatem vibratus deor&longs;um cadit; <lb/>partes, quæ propiùs ad extremitatem immobilem accedunt iuuant mo­<lb/>tum aliarum, quæ longiùs ab eadem recedunt. </s> </p> <pb pagenum="52" xlink:href="026/01/084.jpg"/> <p id="N145C3" type="main"> <s id="N145C5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s> </p> <p id="N145D1" type="main"> <s id="N145D3"><emph type="italics"/>Impetus propagatur eodem in&longs;tanti, id e&longs;t, &longs;ine temporis &longs;ucce&longs;&longs;ione.<emph.end type="italics"/></s> <s id="N145DA"> Proba­<lb/>tur; </s> <s id="N145DF"><!-- NEW -->&longs;it enim applicata potentia in A, dico &longs;imul produci impetum in <lb/>BCDE; </s> <s id="N145E5"><!-- NEW -->quia &longs;i primo in&longs;tanti produceretur in A, & &longs;ecundo in B, vel <lb/>A moueretur ante B, vel impetus in A e&longs;&longs;et fru&longs;trà; </s> <s id="N145EB"><!-- NEW -->vtrumque e&longs;t ab&longs;ur­<lb/>dum; nam totum AE, &longs;imul mouetur. </s> </p> <p id="N145F1" type="main"> <s id="N145F3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s> </p> <p id="N145FF" type="main"> <s id="N14601"><emph type="italics"/>Tribus tantùm modis propagari pote&longs;t impetus ratione inten&longs;ionis.<emph.end type="italics"/></s> <s id="N14608"><!-- NEW --> Primò <lb/>&longs;i æqualiter omnibus partibus &longs;ubjecti di&longs;tribuatur; id e&longs;t vniformiter. </s> <s id="N1460E"><lb/>Secundò, &longs;i plùs partibus propioribus, & minùs remotioribus. </s> <s id="N14612"><!-- NEW -->Tertiò, è <lb/>contra, &longs;i plùs remotioribus, & minùs propioribus; </s> <s id="N14618"><!-- NEW -->tribus etiam ratione <lb/>perfectionis eo modo, quo diximus de inten&longs;ione; </s> <s id="N1461E"><!-- NEW -->at verò nouem mo­<lb/>dis propagari pote&longs;t ratione vtriu&longs;que; patet ex regula combinationum; </s> <s id="N14624"><!-- NEW --><lb/>&longs;i enim 3. ducantur in 3. habebis 9. Iam &longs;upere&longs;t, vt videamus, an reue­<lb/>rà omnibus i&longs;tis modis impetus re ip&longs;a propagetur; </s> <s id="N1462B"><!-- NEW -->quod licèt difficile <lb/>&longs;it, & vix hactenus explicatum: Audeo tamen polliceri meum &longs;uper hac <lb/>re conatum non pror&longs;us inutilem fore. </s> </p> <p id="N14633" type="main"> <s id="N14635"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s> </p> <p id="N14641" type="main"> <s id="N14643"><!-- NEW --><emph type="italics"/>Impetus propagatur vniformiter in mobili, cuius omnes partes mouentur <lb/>æquali motu<emph.end type="italics"/>; </s> <s id="N1464E"><!-- NEW -->probatur, quia impetus non cogno&longs;citur ni&longs;i per motum; <lb/>igitur vbi e&longs;t æqualis motus, debet e&longs;&longs;e æqualis impetus in omnibus par­<lb/>tibus, id e&longs;t æqualis graduum heterogeneorum collectio, in quo non <lb/>e&longs;t difficultas. </s> </p> <p id="N14658" type="main"> <s id="N1465A"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14666" type="main"> <s id="N14668"><!-- NEW -->Ob&longs;eruabis illud mobile moueri motu æquali &longs;ecundum omnes &longs;ui <lb/>partes, quod mouetur motu recto; quippe fieri non pote&longs;t, quin omnes <lb/>partes, quæ mouentur motu recto &longs;implici, motu etiam æquali mouean­<lb/>tur. </s> </p> <p id="N14672" type="main"> <s id="N14674"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s> </p> <p id="N14680" type="main"> <s id="N14682"><!-- NEW --><emph type="italics"/>Cum duo corpora &longs;e&longs;e mutuò tangunt, impetus in vtroque propagatur<emph.end type="italics"/> &longs;int <lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->globi A & B, æquales &longs;ibi inuicem contigui in C, &longs;it applicata po­<lb/>tentia in D, non modò producet impetum in globo A, &longs;ed etiam in B: </s> <s id="N14693"><!-- NEW --><lb/>probatur primò, quia &longs;e habent per modum vnius, vt patet ex re&longs;i&longs;ten­<lb/>tia, nec enim A moueri pote&longs;t &longs;ine B per lineam DE, quod certè cla­<lb/>ri&longs;&longs;imum e&longs;t; probatur &longs;ecundò quia &longs;i A produceret impetum in B, duo <lb/>globi, vel 3. vel 5. vel infiniti tantùm re&longs;i&longs;terent, quantùm vnicus glo­<lb/>bus, quod fal&longs;um & ab&longs;urdum e&longs;t. </s> <s id="N146A0"><!-- NEW -->Tertiò, Ratio à priori e&longs;t; </s> <s id="N146A4"><!-- NEW -->quia ideo <lb/>producitur, & propagatur impetus in toto A; </s> <s id="N146AA"><!-- NEW -->quia vna pars non pote&longs;t <lb/>moueri &longs;ine alia per Th. 33. &longs;ed non pote&longs;t A moueri ni&longs;i moueatur B; <lb/>igitur in vtroque &longs;imul, & æqualiter propagatur impetus. </s> </p> <p id="N146B2" type="main"> <s id="N146B4"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N146C1" type="main"> <s id="N146C3">Hinc ratio manife&longs;ta cur maior &longs;it re&longs;i&longs;tentia duorum quàm vnius. </s> </p> <pb pagenum="53" xlink:href="026/01/085.jpg"/> <p id="N146CA" type="main"> <s id="N146CC"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N146D9" type="main"> <s id="N146DB"><!-- NEW -->Hinc eadem vis requiritur ad &longs;u&longs;tinenda duo pondera; &longs;iue vtrum­<lb/>que &longs;eor&longs;im humeris incubet, &longs;iue alterum alteri &longs;uperponatur. </s> </p> <p id="N146E1" type="main"> <s id="N146E3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N146F0" type="main"> <s id="N146F2">Hinc percu&longs;&longs;io vel ictus globi B, cui alter A à tergo immediatè in­<lb/>&longs;i&longs;tit maior e&longs;t. </s> </p> <p id="N146F7" type="main"> <s id="N146F9"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14706" type="main"> <s id="N14708">Hinc pondus alteri &longs;uperpo&longs;itum actione communi cum alio graui­<lb/>tat in &longs;uppo&longs;itam manum. </s> <s id="N1470D">v. <!-- REMOVE S-->g. <!-- REMOVE S--><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1471E" type="main"> <s id="N14720">Hinc potentia applicata in D, minùs impetus &longs;ingulis imprimit. </s> </p> <p id="N14723" type="main"> <s id="N14725"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14732" type="main"> <s id="N14734"><!-- NEW -->Hinc demum licèt impetus ratione inten&longs;ionis &longs;it æqualis in vtroque <lb/>globo; </s> <s id="N1473A"><!-- NEW -->attamen, &longs;i accipiatur numerus partium vtriu&longs;que impetus, im­<lb/>petus &longs;unt vt globi v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i B e&longs;t æqualis A impetus productus in B e&longs;t <lb/>æqualis producto in A, &longs;i B &longs;it &longs;ubduplus, vel &longs;ubtriplus, impetus e&longs;t <lb/>&longs;ubtriplus, vel &longs;ubduplus; quorum omnium rationes patent ex Th.96. </s> </p> <p id="N14748" type="main"> <s id="N1474A"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N14756" type="main"> <s id="N14758">Hinc etiam colligi pote&longs;t manife&longs;tum di&longs;crimen, quod intercedit inter <lb/>propagationem impetus, & aliarum qualitatum, quæ (vt vulgò dicitur) <lb/>vniformiter difformiter propagantur, id e&longs;t, æqualiter in æquali <lb/>di&longs;tantia, & inæqualiter inæquali. </s> </p> <p id="N14761" type="main"> <s id="N14763"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1476F" type="main"> <s id="N14771"><!-- NEW -->Hinc demum colligi pote&longs;t non modò impetum produci in globo B <lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->verùm etiam in aëre ambiente, cui &longs;cilicet globus contiguus e&longs;t; </s> <s id="N1477B"><!-- NEW --><lb/>qui reuera aër facilè amouetur; </s> <s id="N14780"><!-- NEW -->tùm quia propter raritatem pauci&longs;&longs;imæ <lb/>partes mouendæ &longs;unt; </s> <s id="N14786"><!-- NEW -->tùm quia facilè diuiduntur, de quibus alias; </s> <s id="N1478A"><!-- NEW -->tùm <lb/>quia, ne detur vaçuum, &longs;patium à tergo relictum occupare debet, quod <lb/>reuerà præ&longs;tat breui peracto circuitu, vt videre e&longs;t in aqua; </s> <s id="N14792"><!-- NEW -->nec enim <lb/>totus aër agitari debet; </s> <s id="N14798"><!-- NEW -->quis enim id con&longs;equi po&longs;&longs;et; tum denique, quia <lb/>aër non grauitat in aëre, igitur cum non re&longs;i&longs;tat vlla grauitatio, facilè <lb/>moueri pote&longs;t. </s> </p> <p id="N147A0" type="main"> <s id="N147A2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s> </p> <p id="N147AE" type="main"> <s id="N147B0"><!-- NEW --><emph type="italics"/>Cum applicatur potentia centro motus circularis, ita propagatur impetus, vt <lb/>plures partes impetus continuò producantur ver&longs;us <expan abbr="circumferentiã">circumferentiam</expan><emph.end type="italics"/>; &longs;it enim <lb/>cylindrus CA, fig. </s> <s id="N147C0"><!-- NEW -->Th. 73. &longs;it centrum motus C; </s> <s id="N147C4"><!-- NEW -->haud dubiè plures <lb/>partes impetus producuntur in B, quàm in C, & plures in A, quam in B; </s> <s id="N147CA"><!-- NEW --><lb/>quia, cum pars B moueatur velociùs, quàm C, & A quàm B; certè, vbi e&longs;t <lb/>maior motus, vel effectus, ibi debet e&longs;&longs;e maior impetus, vel cau&longs;a per <lb/>Ax. 13. n. </s> <s id="N147D3"><!-- NEW -->4. quod autem &longs;it maior motus, con&longs;tat ex maioribus &longs;patiis, <lb/>vel arcubus æquali tempore confectis; quod verò &longs;it impetus inten&longs;ior <pb pagenum="54" xlink:href="026/01/086.jpg"/>versùs circumferentiam, non perfectior, patet per Th. 8. </s> </p> <p id="N147DE" type="main"> <s id="N147E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s> </p> <p id="N147EC" type="main"> <s id="N147EE"><!-- NEW --><emph type="italics"/>Inten&longs;io impetus propagati iuxta hunc modum &longs;e habet, vt distantia à cen­<lb/>tro motus<emph.end type="italics"/>; </s> <s id="N147F9"><!-- NEW -->&longs;int enim punctum B, & punctum A: </s> <s id="N147FD"><!-- NEW -->ita &longs;e habet inten&longs;io <lb/>impetus puncti A ad inten&longs;ionem impetus puncti B, vt di&longs;tantia AC <lb/>ad BC. Probatur, quia cum impetus &longs;int vt motus, motus vt &longs;patia, &longs;patia <lb/>verò &longs;int arcus AE. BD; </s> <s id="N14807"><!-- NEW -->arcus &longs;unt, vt &longs;emidiametri AC, BC; igitur vt <lb/>di&longs;tantiæ quòd erat demon&longs;trandum. </s> </p> <p id="N1480D" type="main"> <s id="N1480F"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1481C" type="main"> <s id="N1481E"><!-- NEW -->Hinc &longs;i di&longs;tantia CA e&longs;t dupla di&longs;tantiæ CB, impetus in A e&longs;t du­<lb/>plus impetus in B: at verò impetus &longs;egmenti e&longs;t ad impetum alterius, <lb/>vt diximus in Th. 73. </s> </p> <p id="N14826" type="main"> <s id="N14828"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14835" type="main"> <s id="N14837"><!-- NEW -->Hinc hæc propagatio fit iuxta progre&longs;&longs;ionem arithmeticam id e&longs;t, &longs;i <lb/>in primâ parte ver&longs;us centrum producitur impetus vt 1. in &longs;ecunda pro­<lb/>ducitur vt duo, in tertiâ vt tria, atque ita deinceps; quia proportio <lb/>arithmetica e&longs;t laterum, &longs;eu linearum. </s> </p> <p id="N14841" type="main"> <s id="N14843"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14850" type="main"> <s id="N14852">Hinc hæc propagatio e&longs;t omninò inuer&longs;a illius, quæ aliis qualitatibus <lb/>competit, vt patet. </s> </p> <p id="N14857" type="main"> <s id="N14859"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14866" type="main"> <s id="N14868">Hinc etiam manife&longs;ta ratio &longs;equitur illius experimenti, quod propo­<lb/>&longs;uimus corol. </s> <s id="N1486D">2. Th. 80. </s> </p> <p id="N14870" type="main"> <s id="N14872"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1487F" type="main"> <s id="N14881"><!-- NEW -->Hinc &longs;i tantùm habeatur ratio impetus, facilè pote&longs;t determinari in <lb/>qua proportione cylindrus faciliùs moueatur motu recto, quàm motu <lb/>circulari; </s> <s id="N14889"><!-- NEW -->po&longs;ito &longs;cilicet centro motus in altera extremitate, cui applica­<lb/>tur potentia; </s> <s id="N1488F"><!-- NEW -->quippe impetus propagatus in motu circulari e&longs;t &longs;umma <lb/>terminorum; </s> <s id="N14895"><!-- NEW -->propagatus verò in motu recto e&longs;t vltimus terminorum, <lb/>v.g. <!-- REMOVE S-->&longs;int &longs;ex puncta &longs;ubiecti; </s> <s id="N1489D"><!-- NEW -->in quolibet producatur impetus vt vnum; </s> <s id="N148A1"><!-- NEW --><lb/>haud dubiè erit motus rectus; </s> <s id="N148A6"><!-- NEW -->vt verò &longs;it motus circularis in primo <lb/>puncto; </s> <s id="N148AC"><!-- NEW -->producatur vt 1. in &longs;ecundo vt 2. in tertio, vt 3. atque ita dein­<lb/>ceps; &longs;umma erit 21. cum tamen in motu recto e&longs;&longs;ent tantùm 6. igitur <lb/>vt &longs;e habent 21. ad 6. ita &longs;e habet facilitas motus recti ad facilitatem <lb/>motus circularis. </s> </p> <p id="N148B6" type="main"> <s id="N148B8"><!-- NEW -->Dixi, &longs;i tantùm habeatur ratio impetus; </s> <s id="N148BC"><!-- NEW -->quia &longs;i addatur ratio graui­<lb/>tationis, &longs;eu momenti; haud dubiè maior erit adhuc difficultas, de <lb/>quo infrà in Schol. <!-- REMOVE S--><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N148D0" type="main"> <s id="N148D2"><!-- NEW -->Hinc quò longior e&longs;t cylindrus, v. <!-- REMOVE S-->g. <!-- REMOVE S-->cre&longs;cit proportio maioris illius <lb/>facilitatis, vt patet inductione; </s> <s id="N148DC"><!-- NEW -->nam &longs;i &longs;int tantùm 2. puncta, proportio <lb/>erit 3. ad 2.; </s> <s id="N148E2"><!-- NEW -->&longs;it tria 6. ad 3.; </s> <s id="N148E6"><!-- NEW -->&longs;i 4. 10. ad 4. &longs;i 5. 15. ad 5.; </s> <s id="N148EA"><!-- NEW -->&longs;i 6. 21. ad 6. <pb pagenum="55" xlink:href="026/01/087.jpg"/>&longs;i 7. 28. ad 7; </s> <s id="N148F3"><!-- NEW -->&longs;i 8. 36. ad 8; </s> <s id="N148F7"><!-- NEW -->&longs;i 9. 45. ad 9; atque ita deinceps; ex quibus primò <lb/>vides cre&longs;cere &longs;emper proportionem. </s> <s id="N148FD"><!-- NEW -->Secundò inter duplam, & triplam <lb/>rationem, &longs;cilicet 6. ad 3. & 15. ad 5. intercedere 2 1/2; </s> <s id="N14903"><!-- NEW -->inter triplam & <lb/>quadruplam intercedere 3. 1/2; </s> <s id="N14909"><!-- NEW -->inter quadruplam & quintuplam inter­<lb/>cedere 4 1/2; atque ita deinceps. </s> </p> <p id="N1490F" type="main"> <s id="N14911"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1491D" type="main"> <s id="N1491F"><!-- NEW -->Colligo denique po&longs;&longs;e in motu recto cum maiore ni&longs;u produci inten­<lb/>&longs;iorem impetum in data ratione; </s> <s id="N14925"><!-- NEW -->&longs;it enim cylindrus AB, qui moueatur <lb/>circa centrum A, percurrátque B, arcum BD; </s> <s id="N1492B"><!-- NEW -->qui accipiatur vt recta, <lb/>quæ à minimis arcubus &longs;en&longs;u di&longs;tingui non pote&longs;t; </s> <s id="N14931"><!-- NEW -->haud dubiè &longs;i eo <lb/>tempore, vel æquali, quo AB tran&longs;it in AD; </s> <s id="N14937"><!-- NEW -->eadem AB, vel æqualis <lb/>motu recto tran&longs;eat in FD, Dico impetum huius motus e&longs;&longs;e duplò in­<lb/>ten&longs;iorem impetu illius; </s> <s id="N1493F"><!-- NEW -->quia impetus &longs;unt vt motus; </s> <s id="N14943"><!-- NEW -->motus verò vt <lb/>&longs;patia, quæ percurruntur æqualibus temporibus; </s> <s id="N14949"><!-- NEW -->&longs;ed &longs;patium rectanguli <lb/>AD, e&longs;t duplum trianguli ADB; </s> <s id="N1494F"><!-- NEW -->igitur & motus; </s> <s id="N14953"><!-- NEW -->igitur & impetus; </s> <s id="N14957"><!-- NEW -->&longs;i <lb/>verò AB tran&longs;eat in EL, ita vt AF, &longs;it dupla AE; </s> <s id="N1495D"><!-- NEW -->impetus erunt <lb/>æquales; quia rectangulum AC, e&longs;t æquale triangulo ABD. </s> </p> <p id="N14963" type="main"> <s id="N14965"><!-- NEW -->Dixi arcum BD, accipi vt lineam rectam; </s> <s id="N14969"><!-- NEW -->Si enim accipiatur vt ar­<lb/>cus; haud dubiè motus cylindri AB, dum transfertur in FD, e&longs;t ad mo­<lb/>tum eiu&longs;dem AB, dum transfertur in AD, vt rectangulum AD, ad &longs;e­<lb/>ctorem, cuius arcus &longs;it æqualis rectæ BD, & radius ip&longs;i AB. <!-- KEEP S--></s> </p> <p id="N14974" type="main"> <s id="N14976"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14982" type="main"> <s id="N14984">Ob&longs;eruabis primò, id quod &longs;uprà dictum e&longs;t ita e&longs;&longs;e intelligendum, <lb/>vt momentum grauitationis nullo modo con&longs;ideretur, & prædictus <lb/>cylindrus cen&longs;eatur potiùs moueri in plano horizontali, à quo &longs;u&longs;tinea­<lb/>tur, quàm in circulo verticali, in quo libera &longs;it eius libratio, &longs;eu gra­<lb/>uitatio. </s> </p> <p id="N1498F" type="main"> <s id="N14991"><!-- NEW -->Secundò, non po&longs;&longs;e &longs;u&longs;tineri cylindrum horizonti parallelum, ni&longs;i <lb/>aliqua eius portio &longs;eu manu, &longs;eu forcipe, vel alio quouis modo accipia­<lb/>tur, v.g. <!-- REMOVE S-->&longs;it cylindrus AG horizonti parallelus; vt in hoc &longs;itu reti­<lb/>neatur, debet aliqua eius portio putà AB, manu teneri, alioqui ne à po­<lb/>tentiâ quidem infinita &longs;u&longs;tineri po&longs;&longs;et. </s> </p> <p id="N1499F" type="main"> <s id="N149A1"><!-- NEW -->Tertiò, &longs;i &longs;upponatur fulcitus in B; </s> <s id="N149A5"><!-- NEW -->vt retineatur in æquilibrio, debet <lb/>addi momentum in A; &longs;eu debet retineri ab ip&longs;a potentiâ applicata <lb/>in A. <!-- KEEP S--></s> </p> <p id="N149AE" type="main"> <s id="N149B0">Quartò, pondus in G &longs;e habet ad idem pondus in A, &longs;tatuto centro in <lb/>B, vt &longs;egmentum GB, ad BA, id e&longs;t, vt 5. ad 1. <!-- KEEP S--></s> </p> <p id="N149B6" type="main"> <s id="N149B8"><!-- NEW -->Quintò, &longs;i proprio pondere frangeretur BG, haud dubiè in B frange­<lb/>retur; </s> <s id="N149BE"><!-- NEW -->e&longs;t autem momentum ponderis BG, vt &longs;ubduplum eiu&longs;dem BG <lb/>po&longs;itum in G, vt demon&longs;trat Galileus prop.1.de re&longs;i&longs;tentia corp.&longs;it enim <lb/>BG, duarum librarum, &longs;itque BG, diui&longs;a bifariam in H; </s> <s id="N149C6"><!-- NEW -->haud dubiè <lb/>pondus in H, facit momentum &longs;ubduplum eiu&longs;dem in G, vt patet; </s> <s id="N149CC"><!-- NEW -->&longs;unt <lb/>enim vt di&longs;tantiæ; </s> <s id="N149D2"><!-- NEW -->igitur cum &longs;egmentum HG tantùm addat momenti <lb/>&longs;upra H, quantùm detrahit HB; </s> <s id="N149D8"><!-- NEW -->certè momentum totius ponderis BG, <pb pagenum="56" xlink:href="026/01/088.jpg"/>e&longs;t tantùm &longs;ubduplum eiu&longs;dem po&longs;iti in G; </s> <s id="N149E1"><!-- NEW -->itaque &longs;it BG, 10. librarum, <lb/>æquiualet 5. libris &longs;tatutis in G, & AB, vni libræ po&longs;itæ in A; </s> <s id="N149E7"><!-- NEW -->&longs;ed hæc <lb/>libra in A, habet tantùm &longs;ubquintuplum momentum eiu&longs;dem in G, igi­<lb/>tur 5. libræ in A, æquiualent vni in G; </s> <s id="N149EF"><!-- NEW -->igitur vt &longs;tatuatur æquilibrium, <lb/>debent e&longs;&longs;e 24. libræ in A, &longs;eu vires æquiualentes; </s> <s id="N149F5"><!-- NEW -->quibus adde pondus <lb/>ab&longs;olutum 12. librarum; erunt 36. igitur re&longs;i&longs;tentia ad motum circula­<lb/>rem verticalem ex triplici capite oritur. </s> <s id="N149FD">Primò ex ip&longs;o pondere ab&longs;olutè <lb/>&longs;umpto, quæ communis e&longs;t motui propagationis. </s> <s id="N14A02"><!-- NEW -->Secundò, ex momento <lb/>eiu&longs;dem ponderis; </s> <s id="N14A08"><!-- NEW -->Tertiò, ex tali genere propagationis, de quo &longs;uprà; <lb/>quæ omnia &longs;unt apprimè tenenda, ne quis error &longs;ubrepat. </s> </p> <p id="N14A0E" type="main"> <s id="N14A10"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 99.<emph.end type="center"/></s> </p> <p id="N14A1C" type="main"> <s id="N14A1E"><!-- NEW --><emph type="italics"/>Cum applicatur potentia circumferentiæ motus circularis; </s> <s id="N14A24"><!-- NEW -->ita propagatur <lb/>impetus, vt plures partes ver&longs;us centrum motus producantur in pondere, quod <lb/>attollitur<emph.end type="italics"/>; </s> <s id="N14A2F"><!-- NEW -->&longs;it enim idem cylindrus CA; </s> <s id="N14A33"><!-- NEW -->&longs;itque applicata potentia in <lb/>A, dico ver&longs;us C, plures partes produci in pondere, Probatur, quia attol­<lb/>litur pondus in C, quod moueri non pote&longs;tin A, operâ vectis AC, vt con­<lb/>&longs;tat ex certa hypothe&longs;i; igitur plures partes impetus producuntur per <lb/>rationem 6. & 7. Th.77, </s> </p> <p id="N14A3F" type="main"> <s id="N14A41"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14A4D" type="main"> <s id="N14A4F"><!-- NEW -->Scio quidem hoc ip&longs;um à nemine hactenus, quod &longs;ciam, explicatum <lb/>e&longs;&longs;e; </s> <s id="N14A55"><!-- NEW -->atque fore vt à multis tanquam nouum, & in&longs;olens minùs fortè <lb/>probetur: </s> <s id="N14A5B"><!-- NEW -->quamquam illa hypothe&longs;is hoc ip&longs;um euincit, vulgaris certè, <lb/>& nemini qua&longs;i non nota; </s> <s id="N14A61"><!-- NEW -->qua nempè dicimus in omnibus partibus mo­<lb/>bilis, quod actu mouetur, impetum produci; </s> <s id="N14A67"><!-- NEW -->& &longs;i quando accidat corpo­<lb/>ris ingentem molem ab applicata potentia non po&longs;&longs;e moueri, illud e&longs;&longs;e <lb/>tantùm, quòd non po&longs;&longs;int produci tot partes impetus, quot &longs;unt nece&longs;&longs;a­<lb/>riæ, vt omnibus partibus &longs;ubjecti di&longs;tribuantur; igitur ex hac hypothe­<lb/>&longs;i, quæ ex manife&longs;tis ducitur experimentis, nece&longs;&longs;ariò dicendum e&longs;t plu­<lb/>res partes impetus versùs centrum vectis produci in pondere, quod at­<lb/>tollitur, cuius propagationis proportionem infrà demon&longs;trabimus. </s> </p> <p id="N14A77" type="main"> <s id="N14A79"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s> </p> <p id="N14A85" type="main"> <s id="N14A87"><emph type="italics"/>Impetus, qui producitur ver&longs;us centrum vectis in pondere, licèt cre&longs;cat nu­<lb/>mero, decre&longs;cit tamen in perfectione.<emph.end type="italics"/></s> <s id="N14A90"> Probatur per Th.81. ex motu imper­<lb/>fectiore, cui re&longs;pondet impetus imperfectior per Ax. 17.num.4. non ratio­<lb/>ne numeri, qui maior e&longs;t per Th.99. igitur ratione entitatis, &longs;eu perfe­<lb/>ctionis entitatiuæ. </s> </p> <p id="N14A99" type="main"> <s id="N14A9B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 101.<emph.end type="center"/></s> </p> <p id="N14AA7" type="main"> <s id="N14AA9"><!-- NEW --><emph type="italics"/>Tota collectio impetus, quæ in pondere ex dato puncto vectis producitur, e&longs;t <lb/>ad aliam collectionem alterius puncti in perfectione, vt distantia illius puncti <lb/>à centro, ad di&longs;tantiam huius<emph.end type="italics"/>: </s> <s id="N14AB6"><!-- NEW -->probatur, quia perfectio vnius collectionis <lb/>e&longs;t ad perfectionem alterius, vt motus ad motum; motus verò &longs;unt vt <lb/>&longs;patia, &longs;patia vt arcus, arcus vt &longs;emediametri, hæ demum, vt di&longs;tantiæ. </s> </p> <pb pagenum="57" xlink:href="026/01/089.jpg"/> <p id="N14AC2" type="main"> <s id="N14AC4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 102.<emph.end type="center"/></s> </p> <p id="N14AD0" type="main"> <s id="N14AD2"><!-- NEW --><emph type="italics"/>Impetus in ip&longs;o vecte &longs;ine pondere addito ita propagatur, vt &longs;it imperfectior <lb/>ver&longs;us centrum vectis<emph.end type="italics"/>; </s> <s id="N14ADD"><!-- NEW -->probatur, quia pondus ver&longs;us centrum mouetur <lb/>minore motu, vt con&longs;tat; igitur ab imperfectiore impetu; </s> <s id="N14AE3"><!-- NEW -->&longs;ed non e&longs;t <lb/>imperfectior tantùm ratione numeri, id e&longs;t, pauciorum partium impe­<lb/>tus; </s> <s id="N14AEB"><!-- NEW -->quia &longs;i hoc e&longs;&longs;et, &longs;it vectis AC, motus B, e&longs;t &longs;ubduplus motus <lb/>A; </s> <s id="N14AF1"><!-- NEW -->igitur &longs;i e&longs;t impetus eiu&longs;dem perfectionis entitatiuæ, vt &longs;ic loquar; </s> <s id="N14AF5"><!-- NEW --><lb/>ita &longs;e habet numerus partium impetus in B, ad numerum partium in A, <lb/>vt motus B, ad motum A; </s> <s id="N14AFC"><!-- NEW -->& hic vt arcus BD, ad arcum AE; </s> <s id="N14B00"><!-- NEW -->& hic vt <lb/>BC, ad AC; </s> <s id="N14B06"><!-- NEW -->igitur e&longs;t &longs;ubduplus; </s> <s id="N14B0A"><!-- NEW -->igitur æqualis omninò producitur <lb/>impetus ab eadem potentia in vecte AC, &longs;iue applicetur centro C, &longs;iue <lb/>circumferentiæ A; </s> <s id="N14B12"><!-- NEW -->igitur æquè facilè; quod e&longs;t contra experientiam; </s> <s id="N14B16"><!-- NEW --><lb/>probatur &longs;ecundò, quia &longs;i hoc e&longs;&longs;et, pondus idem tàm facilè attolleretur <lb/>in A, quàm in B; quia idem impetus produceretur, quod e&longs;t contra ex­<lb/>perientiam. </s> </p> <p id="N14B1F" type="main"> <s id="N14B21"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 103.<emph.end type="center"/></s> </p> <p id="N14B2D" type="main"> <s id="N14B2F"><emph type="italics"/>Ex hoc facilè intelligitur, cur impetus propagetur faciliùs à circumferen­<lb/>tia ad centrum, quàm à centro ad circumferentiam, & cur longior vectis ab <lb/>eadem potentia moueri po&longs;&longs;it primo modo, non &longs;ecundo, quod clarum est.<emph.end type="italics"/></s> </p> <p id="N14B3A" type="main"> <s id="N14B3C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 104.<emph.end type="center"/></s> </p> <p id="N14B48" type="main"> <s id="N14B4A"><!-- NEW --><emph type="italics"/>Decre&longs;cit impetus ver&longs;us centrum iuxta rationem distantiarum<emph.end type="italics"/>; </s> <s id="N14B53"><!-- NEW -->probatur <lb/>quia decre&longs;cit iuxta rationem motuum; & hæc iuxta rationem di&longs;tan­<lb/>tiarum. </s> </p> <p id="N14B5B" type="main"> <s id="N14B5D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 105.<emph.end type="center"/></s> </p> <p id="N14B69" type="main"> <s id="N14B6B"><!-- NEW --><emph type="italics"/>Non decre&longs;cit numerus partium impetus à circumferentia ad centrum<emph.end type="italics"/>; </s> <s id="N14B74"><!-- NEW --><lb/>probatur, quia cum à circumferentia ad centrum ita propagetur impe­<lb/>tus, vt vnicum tantùm punctum producatur in ip&longs;a extremitate mobilis; </s> <s id="N14B7B"><!-- NEW --><lb/>certè non pote&longs;t minùs impetus produci ver&longs;us centrum ratione nume­<lb/>ri; </s> <s id="N14B82"><!-- NEW -->igitur non decre&longs;cit numerus; hinc producitur nece&longs;&longs;ariò imperfe­<lb/>ctior ver&longs;us centrum. </s> </p> <p id="N14B88" type="main"> <s id="N14B8A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 106.<emph.end type="center"/></s> </p> <p id="N14B96" type="main"> <s id="N14B98"><!-- NEW --><emph type="italics"/>Non producuntur plures partes impetus in vecte ver&longs;us centrum, id est, non <lb/>&longs;unt plures in puncto vectis propiùs ad centrum accedente, quàm in co; quod <lb/>longiùs distat:<emph.end type="italics"/> Probatur primò, quia fru&longs;trà e&longs;&longs;ent plures. </s> <s id="N14BA5">Secundò, cur <lb/>potiùs in vna proportione, quàm in alia? </s> </p> <p id="N14BAA" type="main"> <s id="N14BAC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 107.<emph.end type="center"/></s> </p> <p id="N14BB8" type="main"> <s id="N14BBA"><!-- NEW --><emph type="italics"/>Ex his constat produci impetum æqualem numero in omnibus punctis vectis <lb/>a circumferentia ad centrum, cum &longs;cilicet applicatur potentia circumferentiæ<emph.end type="italics"/>; </s> <s id="N14BC5"><!-- NEW --><lb/>probatur, quia non producitur numerus minor per Th.105. neque maior <lb/>per Th. 106. igitur æqualis; </s> <s id="N14BCC"><!-- NEW -->adde quod res explicari non pote&longs;t per ma­<lb/>iorem, neque per minorem; ita vt &longs;cilicet pondera, quæ à data potentia <lb/>leuantur, &longs;int vt di&longs;tantiæ, de quo &longs;uprà. </s> </p> <pb pagenum="58" xlink:href="026/01/090.jpg"/> <p id="N14BD8" type="main"> <s id="N14BDA"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14BE6" type="main"> <s id="N14BE8"><!-- NEW -->Ob&longs;eruabis, quod aliquando in mentem venerat; </s> <s id="N14BEC"><!-- NEW -->&longs;cilicet, ver&longs;us cen­<lb/>trum produci maiorem numerum in ratione di&longs;tantiarum permutando; </s> <s id="N14BF2"><!-- NEW --><lb/>& imperfectiorem in ratione duplicata earumdem di&longs;tantiarum, etiam <lb/>permutando, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it idem vectis AC &longs;ectus bifariam in B; </s> <s id="N14BFD"><!-- NEW -->in puncto <lb/>B producitur numerus duplus producti in A; </s> <s id="N14C03"><!-- NEW -->at verò perfectio impetus <lb/>in B e&longs;t ad perfectionem impetus in A, vt quadratum BC ad quadra­<lb/>tum AC; </s> <s id="N14C0B"><!-- NEW -->vel in ratione &longs;ubquadrupla, licèt tota collectio impetus B <lb/>&longs;it tantùm &longs;ubdupla perfectione collectionis impetus A; </s> <s id="N14C11"><!-- NEW -->&longs;ed hoc profe­<lb/>ctò dici non pote&longs;t; </s> <s id="N14C17"><!-- NEW -->nam &longs;int in A 4. partes impetus; igitur in B erunt <lb/>8. applicetur autem pondus in B. </s> <s id="N14C1D"><!-- NEW -->Primò producentur in eo partes 8. <lb/>impetus perfectionis &longs;ubquadruplæ; </s> <s id="N14C23"><!-- NEW -->&longs;i comparentur cum partibus A, <lb/>tum producentur 16. quæ æquiualent 4 A; </s> <s id="N14C29"><!-- NEW -->igitur 24. at verò in A pro­<lb/>ducentur primò 4. tum deinde 2. quæ æquiualent 8. productis in B; igitur <lb/>6. igitur pondus, quod leuari pote&longs;t in B, e&longs;t ad pondus, quod leuari pote&longs;t <lb/>in A, vt 24. ad 6.id e&longs;t, in ratione quadrupla quod omninò fal&longs;um e&longs;t. </s> </p> <p id="N14C33" type="main"> <s id="N14C35"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 108.<emph.end type="center"/></s> </p> <p id="N14C41" type="main"> <s id="N14C43"><!-- NEW --><emph type="italics"/>Iam facilè explicatur ex dictis, quomodo, & cuius rationis pondera attol­<lb/>lantur ex diuer&longs;is punctis vectis<emph.end type="italics"/>; &longs;it enim idem vectis AC, & producan­<lb/>tur.v.g. </s> <s id="N14C50"><!-- NEW -->in &longs;ingulis punctis vectis &longs;ingula puncta impetus, &longs;ed diuer&longs;æ <lb/>perfectionis; </s> <s id="N14C56"><!-- NEW -->haud dubiè plures partes impetus imperfecti po&longs;&longs;unt face­<lb/>re impetum æqualem in perfectione alteri, qui con&longs;tat paucioribus, &longs;ed <lb/>perfectioribus; </s> <s id="N14C5E"><!-- NEW -->igitur cum impetus B &longs;it imperfectior duplò quàm im­<lb/>petus in A, duplò plures partes impetus producentur in B, quàm in A, er­<lb/>go duplò maius pondus mouebitur; atque ita deinceps; </s> <s id="N14C66"><!-- NEW -->eum enim ap­<lb/>ponitur pondus in B, producuntur in eo partes impetus omnes eiu&longs;dem <lb/>perfectionis; </s> <s id="N14C6E"><!-- NEW -->quæ &longs;cilicet re&longs;pondet B, id e&longs;t, quæ e&longs;t &longs;ubdupla perfectio­<lb/>nis impetus A; </s> <s id="N14C74"><!-- NEW -->igitur plures partes producuntur, quàm &longs;i e&longs;&longs;ent perfe­<lb/>ctionis A; </s> <s id="N14C7A"><!-- NEW -->&longs;ed pauciores quàm &longs;i e&longs;&longs;ent perfectionis O, quæ minor e&longs;t; <lb/>quippe eadem potentia, &longs;eu cau&longs;a, quæ agit quantum pote&longs;t (quod &longs;up­<lb/>pono modò) producit æqualem effectum in perfectione, per Ax. 13. n. </s> <s id="N14C82"><!-- NEW --><lb/>4. &longs;ed æqualis perfectio pote&longs;t con&longs;tare pluribus, vel paucioribus parti­<lb/>bus perfectionis, nam 4. pattes perfectionis vt 4. faciunt æqualem effe­<lb/>ctum alteri qui con&longs;tat 8. partibus perfectionis vt 2. quod certum e&longs;t; &longs;ed <lb/>de his plura aliàs. </s> </p> <p id="N14C8D" type="main"> <s id="N14C8F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 109.<emph.end type="center"/></s> </p> <p id="N14C9B" type="main"> <s id="N14C9D"><!-- NEW --><emph type="italics"/>Perfectio decre&longs;cit ver&longs;us centrum iuxta diuer&longs;am rationem longitudinum <lb/>vectis, &longs;eu distantiarum.<emph.end type="italics"/> v.g.&longs;it idem vectis AC, ita decre&longs;cit ab A ver&longs;us <lb/>centrum C; </s> <s id="N14CAA"><!-- NEW -->vt impetus puncti B &longs;it &longs;ubduplus in perfectione, puncti R <lb/>&longs;ubtriplus: </s> <s id="N14CB0"><!-- NEW -->iam verò &longs;it vectis &longs;ubduplus prioris BC, &longs;ectus bifariam in <lb/>Z; </s> <s id="N14CB6"><!-- NEW -->&longs;i impetus productus in B, qu&etail; e&longs;t extremitas minoris vectis B &longs;it æqua­<lb/>lis perfectionis cum impetu producto in A (& reuera &longs;unt æquales) &longs;i <lb/>æquali tempore percurrant arcus æquales, &longs;cilicet AV, & BD) certè im-<pb pagenum="59" xlink:href="026/01/091.jpg"/>petus productus in Z e&longs;t æqualis producto in B, cum B pertinet ad ma­<lb/>iorem vectem; </s> <s id="N14CC5"><!-- NEW -->quia vt AC totus maior vectis e&longs;t ad BC ita BC ad <lb/>ZC: igitur decre&longs;cit perfectio versùs centrum iuxta rationem longi­<lb/>tudinum. </s> </p> <p id="N14CCD" type="main"> <s id="N14CCF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 110.<emph.end type="center"/></s> </p> <p id="N14CDB" type="main"> <s id="N14CDD"><!-- NEW --><emph type="italics"/>Minima potentia est illa, quæ in extremitate vectis, quæ procul recedit à <lb/>centro, vnam tantùm partem, vel vnum punctum impetus producit<emph.end type="italics"/>; nihil <lb/>enim minùs produci pote&longs;t, po&longs;ito quod potentia applicata ad talem gra­<lb/>dum perfectionis &longs;it determinata, id e&longs;t ad producendum impetum talis <lb/>perfectionis in ea parte &longs;ubjecti, cui applicatur immediatè, vt &longs;uprà di­<lb/>ctum e&longs;t. </s> </p> <p id="N14CF0" type="main"> <s id="N14CF2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 111.<emph.end type="center"/></s> </p> <p id="N14CFE" type="main"> <s id="N14D00"><emph type="italics"/>Si &longs;int tantum duo puncta vel duæ partes vectis, illa potentia ad illum mo­<lb/>uendum &longs;ufficiens motu circulari est ad aliam &longs;ufficientem ad illum mouen­<lb/>dum motu recto, vt<emph.end type="italics"/> 1/2 <emph type="italics"/>ad<emph.end type="italics"/> 2. &longs;i &longs;int tria puncta vt 2. ad 3. &longs;i 4. vt 2. 1/2 ad 4. <lb/>&longs;i 5. vt 3. ad 5. &longs;i 6. vt 3. 1/2 ad 6. atque ita deinceps iuxta hanc propor­<lb/>tionem in quo non e&longs;t difficultas, cum hoc totum &longs;equatur ex Th. 109. </s> </p> <p id="N14D16" type="main"> <s id="N14D18"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14D24" type="main"> <s id="N14D26"><!-- NEW -->Ob&longs;erua tamen quacumque data potentia po&longs;&longs;e dari minorem; </s> <s id="N14D2A"><!-- NEW -->quia <lb/>quocumque dato motu, etiam recto, pote&longs;t dari tardior; </s> <s id="N14D30"><!-- NEW -->igitur quocum­<lb/>que impetu imperfectior; </s> <s id="N14D36"><!-- NEW -->igitur quando appellaui potentiam minimam; </s> <s id="N14D3A"><!-- NEW --><lb/>intellige illam quæ comparatur cum vnico puncto impetus talis perfe­<lb/>ctionis; hæc enim reuera minima e&longs;t illarum omnium, quæ po&longs;&longs;unt pro­<lb/>ducere impetum talis perfectionis, &longs;i verò comparetur cum impetu im­<lb/>perfectiore, haud dubiè minima non e&longs;t. </s> </p> <p id="N14D45" type="main"> <s id="N14D47"><!-- NEW -->Ob&longs;erua præterea &longs;uppo&longs;itum e&longs;&longs;e hactenus in extremitate vectis &longs;iue <lb/>maioris, &longs;iue minoris, produci impetum eiu&longs;dem perfectionis, eiu&longs;que <lb/>vnicum punctum, &longs;eu partem, vnde potentia quæ applicatur maiori vecti <lb/>conuenit quidem cum ea, quæ applicatur minori in eo, quòd vtraque in <lb/>extremitate &longs;ui vectis producat vnum punctum impetus eiu&longs;dem perfe­<lb/>ctionis; differt tamen in eo, quòd illa, quæ applicatur maiori vecti, &longs;it <lb/>maior iuxta rationes prædictas in Theoremate. <!-- KEEP S--></s> <s id="N14D58">v. <!-- REMOVE S-->g. <!-- REMOVE S-->illa, quæ applicatur <lb/>vecti. </s> <s id="N14D61"><!-- NEW -->2. punctorum e&longs;t ad eam, quæ applicatur vecti trium punctorum, <lb/>&longs;cu partium, vt 1. 1/2 ad 2. & &longs;i vectis &longs;it 4. punctorum ad 2. 1/2; </s> <s id="N14D67"><!-- NEW -->&longs;i 5. ad 3. <lb/>&longs;i 6. ad 3. 1/2; </s> <s id="N14D6D"><!-- NEW -->&longs;i 7. ad 4. &longs;i 8. ad 4. 1/2. Vides egregiam progre&longs;&longs;ionem; </s> <s id="N14D71"><!-- NEW -->&longs;it <lb/>enim vectis 2. punctorum AB, in puncto A, quod e&longs;t extremitas, produ­<lb/>catur punctum impetus datæ perfectionis, in B producetur aliud, cuius <lb/>perfectio e&longs;t &longs;ubdupla prioris per Th. 109. igitur caracter, &longs;eu momen­<lb/>tum totius impetus e&longs;t 1. 1/2. &longs;it porrò vectis 4. punctorum CDEF, in <lb/>C, quod e&longs;t extremitas; </s> <s id="N14D7F"><!-- NEW -->producatur vnum punctum impetus eiu&longs;dem <lb/>perfectionis cum eo, quod productum e&longs;t in A; </s> <s id="N14D85"><!-- NEW -->certè in D producetur <lb/>aliud cuius perfectio erit ad priorem vt 3.ad 4. per idem Th. &longs;ic autem <lb/>notetur 1/4, in E 2/4, in F 3/4, in C vero 4/4; </s> <s id="N14D8D"><!-- NEW -->perfectiones enim &longs;unt vt lon-<pb pagenum="60" xlink:href="026/01/092.jpg"/>gitudines; </s> <s id="N14D96"><!-- NEW -->quæ &longs;i colligantur, habebis characterem totius impetus, 2 1/2: </s> <s id="N14D9A"><!-- NEW --><lb/>igitur totus impetus productus in minore vecte, qui con&longs;tat 2. punctis, <lb/>e&longs;t ad impetum, qui producitur in maiore con&longs;tante 4.punctis, vt 1. 1/2 ad <lb/>2. 1/2; </s> <s id="N14DA3"><!-- NEW -->igitur vectis maior maiorem potentiam ad mouendum ip&longs;um ve­<lb/>ctem requirit; non certè in de&longs;cen&longs;u; </s> <s id="N14DA9"><!-- NEW -->quippe &longs;uo pondere de&longs;cendit, &longs;ed <lb/>in plano horizontali; </s> <s id="N14DAF"><!-- NEW -->ni&longs;i enim potentia po&longs;&longs;it mouere vectem; haud <lb/>dubiè nullum pondus vecte mouebit. </s> </p> <p id="N14DB5" type="main"> <s id="N14DB7"><!-- NEW -->At verò &longs;i potentia &longs;it tantùm dupla minimæ, quæ datum vectem mo­<lb/>uere po&longs;&longs;it; </s> <s id="N14DBD"><!-- NEW -->haud dubiè dato illo vecte datum ferè quodcumque pondus <lb/>mouere poterit; cum ip&longs;e vectis con&longs;tet ferè infinitis punctis in longi­<lb/>tudine, vt patet ex dictis, & con&longs;ideranti patebit. </s> </p> <p id="N14DC5" type="main"> <s id="N14DC7"><!-- NEW -->Ob&longs;eruabis demum in mechanicis nullam ferè haberi rationem pon­<lb/>deris ip&longs;ius vectis; </s> <s id="N14DCD"><!-- NEW -->parum enim pro nihilo computatur: </s> <s id="N14DD1"><!-- NEW -->Ex his tamen <lb/>erui po&longs;&longs;unt veri&longs;&longs;imæ rationes Phy&longs;icæ proportionum vectis AH; </s> <s id="N14DD7"><!-- NEW -->&longs;ia­<lb/>que A extremitas, H centrum; </s> <s id="N14DDD"><!-- NEW -->&longs;itque BH 1/2. CH 1/4, DH 1/2, EH (1/16), <lb/>FH (1/32), GH (1/64) pondus I applicetur in A, & moueatur; </s> <s id="N14DE3"><!-- NEW -->certè in B moue­<lb/>bitur pondus K duplum I; </s> <s id="N14DE9"><!-- NEW -->quia, cum impetus productus in B, &longs;it &longs;ubdu­<lb/>plus in perfectione illius, qui producitur in A; </s> <s id="N14DEF"><!-- NEW -->vt æqualis producatur in <lb/>B, & in A, debent produci in B duplò plures partes impetus; </s> <s id="N14DF5"><!-- NEW -->igitur du­<lb/>plò maius pondus mouebit; </s> <s id="N14DFB"><!-- NEW -->at verò in C mouebitur pondus L quadru­<lb/>plum I, in D octuplum, atque ita deinceps; donec tandem in G mouea­<lb/>tur pondus, quod &longs;it ad I vt 64. ad 1. & cum adhuc po&longs;&longs;int accipi inter <lb/>GH, partes aliquotæ minores, & minores ferè in infinitum, non mirum <lb/>e&longs;t &longs;i pondus maius po&longs;&longs;it adhuc moueri. </s> </p> <p id="N14E07" type="main"> <s id="N14E09"><!-- NEW -->Ob&longs;eruabis etiam in omni vecte ab&longs;trahendo ab eius pondere, & ap­<lb/>plicata eadem potentia, hoc e&longs;&longs;e commune; </s> <s id="N14E0F"><!-- NEW -->vt po&longs;&longs;it quodcumque pon­<lb/>dus attolli, licèt difficiliùs in minore; </s> <s id="N14E15"><!-- NEW -->quia hic non pote&longs;t in tam mul­<lb/>tas partes aliquotas &longs;en&longs;ibiliter diuidi, in medio tamen vecte duplum <lb/>&longs;emper pondus mouetur; &longs;iue ip&longs;e vectis &longs;it maior, &longs;iue minor. </s> </p> <p id="N14E1D" type="main"> <s id="N14E1F">Ob&longs;eruabis deinde, &longs;i centrum vectis non &longs;it in altera extremitate, <lb/>&longs;ed. </s> <s id="N14E24"><!-- NEW -->v.g. <!-- REMOVE S-->in C; </s> <s id="N14E2A"><!-- NEW -->haud dubiè producitur in H, & in B impetus æqualis; </s> <s id="N14E2E"><!-- NEW -->quia <lb/>æqualiter di&longs;tat vtrumque punctum à centro C; </s> <s id="N14E34"><!-- NEW -->igitur æquale pondus <lb/>mouebitur in B, & in H; propagatur tamen nouo modo à C ver&longs;us H, de <lb/>quo iam &longs;uprà dictum e&longs;t. </s> </p> <p id="N14E3C" type="main"> <s id="N14E3E">Ob&longs;eruabis denique triplicem propagationem impetus e&longs;&longs;e legiti­<lb/>mam. </s> <s id="N14E43">Prima e&longs;t in motu recto, cum propagatur per partes æquales, tùm <lb/>in perfectione, tùm in numero in &longs;ingulis partibus &longs;ubjecti per gradus, <lb/>&longs;cilicet heterogeneos. </s> <s id="N14E4A"><!-- NEW -->Secunda e&longs;t in motu circulari, applicata &longs;cilicet <lb/>potentia centro; cum propagatur per partes æquales in perfectione, & <lb/>inæquales in numero. </s> <s id="N14E52">Tertia e&longs;t in vecte, cum propagatur per partes <lb/>æquales in numero, & inæquales in perfectione. </s> </p> <p id="N14E57" type="main"> <s id="N14E59"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 112.<emph.end type="center"/></s> </p> <p id="N14E65" type="main"> <s id="N14E67"><!-- NEW --><emph type="italics"/>Impetus debet determinari ad aliquam lineam motus<emph.end type="italics"/>; </s> <s id="N14E70"><!-- NEW -->probatur, quia <lb/>non pote&longs;t e&longs;&longs;e impetus, ni&longs;i exigat motum per Th.14. nec exigere mo-<pb pagenum="61" xlink:href="026/01/093.jpg"/>tum, ni&longs;i per aliquam lineam, vt patet; </s> <s id="N14E7B"><!-- NEW -->&longs;ed hoc e&longs;t impetum e&longs;&longs;e de­<lb/>terminatum ad aliquam lineam motus; </s> <s id="N14E81"><!-- NEW -->præterea &longs;i non e&longs;t determina­<lb/>tus ad aliquam lineam; </s> <s id="N14E87"><!-- NEW -->igitur indeterminatus, & indifferens per Ax.1. <lb/>&longs;ed indifferens manere non pote&longs;t; cur enim potius haberet motum <lb/>per vnam lineam, quàm per aliam? </s> <s id="N14E8F">igitur debet determinari. </s> </p> <p id="N14E92" type="main"> <s id="N14E94"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 113.<emph.end type="center"/></s> </p> <p id="N14EA0" type="main"> <s id="N14EA2"><!-- NEW --><emph type="italics"/>Impetus ad plures lineas &longs;eor&longs;im indifferens e&longs;t:<emph.end type="italics"/> Probatur, quia idem im­<lb/>petus pilæ in aliam impactæ producit in ea impetum, qui pro diuer&longs;o <lb/>contactu ad diuer&longs;am lineam determinari pote&longs;t; </s> <s id="N14EAF"><!-- NEW -->præterea corpus graue <lb/>in diuer&longs;is planis inclinatis de&longs;cendit; </s> <s id="N14EB5"><!-- NEW -->igitur per diuer&longs;as lineas; </s> <s id="N14EB9"><!-- NEW -->deinde <lb/>pila reflectitur propter impetum priorem, qui tantùm mutat lineam, vt <lb/>dicemus infrà; </s> <s id="N14EC1"><!-- NEW -->adde quod funependuli vibrati impetus &longs;ine reflexione <lb/>mutat lineam motus; igitur idem impetus ad plures lineas &longs;eor&longs;im e&longs;t <lb/>indifferens. </s> </p> <p id="N14EC9" type="main"> <s id="N14ECB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 114.<emph.end type="center"/></s> </p> <p id="N14ED7" type="main"> <s id="N14ED9"><!-- NEW --><emph type="italics"/>Hinc idem impetus ad plures lineas potest determinari &longs;eor&longs;im<emph.end type="italics"/>; </s> <s id="N14EE2"><!-- NEW -->quia ad <lb/>eas pote&longs;t determinari, ad quas e&longs;t indifferens, vt patet; &longs;ed ad multas <lb/>e&longs;t indifferens per Theorema 113. igitur ad multas pote&longs;t determi­<lb/>nari. </s> </p> <p id="N14EEC" type="main"> <s id="N14EEE"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N14EFA" type="main"> <s id="N14EFC"><!-- NEW -->Ob&longs;eruabis primò determinationem hanc nihil e&longs;&longs;e aliud, ni&longs;i ip&longs;um <lb/>impetum cum tali linea comparatum, &longs;eu coniunctum; </s> <s id="N14F02"><!-- NEW -->vnam verò li­<lb/>neam differre ab alia ratione terminorum v. <!-- REMOVE S-->g. <!-- REMOVE S-->illa quæ tendit ver&longs;us <lb/>ortum differt ab ea, quæ tendit ver&longs;us au&longs;trum, vel occa&longs;um, &longs;cilicet <lb/>ratione terminorum, &longs;unt enim duo termini, nempè à quo, & ad quem; </s> <s id="N14F10"><!-- NEW --><lb/>4. autem modis differunt termini lineæ, vel enim neuter communis e&longs;t <lb/>vt AB. DC, vel terminus à quo vtrique lineæ communis e&longs;t, vt BA. <lb/>BE, vel terminus ad quem vt AB, EB; vel denique vici&longs;&longs;im commu­<lb/>tantur termini, vt BE, EB, & hæc terminorum coniugatio facit oppo­<lb/>&longs;itionem maximam, id e&longs;t diametralem. </s> </p> <p id="N14F1D" type="main"> <s id="N14F1F"><!-- NEW -->Secundò ob&longs;eruabis aliquando videri e&longs;&longs;e vtrumque terminum com­<lb/>munem licèt differant lineæ; </s> <s id="N14F25"><!-- NEW -->&longs;it linea recta BE, habet communes ter­<lb/>minos cum curua BFE, licèt omninò differat ab illa; </s> <s id="N14F2B"><!-- NEW -->at profectò licèt <lb/>BE videatur e&longs;&longs;e vnica &longs;implex linea duobus terminis clau&longs;a; </s> <s id="N14F31"><!-- NEW -->con&longs;tat <lb/>ramen ex pluribus aliis continuata, rectáque &longs;erie iunctis; </s> <s id="N14F37"><!-- NEW -->vnde, vt <lb/>linea dicatur eadem e&longs;&longs;e cum alia, debet vna cum aliâ conuenire; ita vt <lb/>alteri &longs;uperpo&longs;ita nec excedat, nec deficiat. </s> </p> <p id="N14F3F" type="main"> <s id="N14F41"><!-- NEW -->Tertiò linea motus non differt ab ip&longs;o motu continuo tractu, &longs;eu <lb/>fluxu qua&longs;i labenti: </s> <s id="N14F47"><!-- NEW -->Porrò vnus motus differt ab alio, vel ratione velo­<lb/>citatis, vel ratione terminorum; &longs;ed hæc parum difficultatis habent. </s> </p> <p id="N14F4D" type="main"> <s id="N14F4F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 115.<emph.end type="center"/></s> </p> <p id="N14F5B" type="main"> <s id="N14F5D"><!-- NEW --><emph type="italics"/>Impetus aliquis ad vnam tantùm lineam pote&longs;t e&longs;&longs;e determinatus<emph.end type="italics"/>; </s> <s id="N14F66"><!-- NEW -->v. <!-- REMOVE S-->g. <lb/><emph type="italics"/>impetus naturalis innatus, de quo in Th.<emph.end type="italics"/> 17. <emph type="italics"/>nam de acqui&longs;ito certum e&longs;t ad<emph.end type="italics"/><pb pagenum="62" xlink:href="026/01/094.jpg"/><emph type="italics"/>plures determinari po&longs;&longs;e, vt videbimus cum de motu reflexo<emph.end type="italics"/>; </s> <s id="N14F82"><!-- NEW -->probatur quia <lb/>motus deor&longs;um e&longs;t finis huius impetus; </s> <s id="N14F88"><!-- NEW -->quia ideo corpus graue produ­<lb/>cit in &longs;e impetum (&longs;i tamen producit) vt tendat deor&longs;um, vt certum e&longs;t; </s> <s id="N14F8E"><!-- NEW --><lb/>tàm enim omne graue non impeditum tendit deor&longs;um, quàm omnis <lb/>ignis e&longs;t calidus; </s> <s id="N14F95"><!-- NEW -->igitur &longs;i e&longs;t proprietas omnis ignis e&longs;&longs;e calidum, quia <lb/>omni competit; </s> <s id="N14F9B"><!-- NEW -->ita omni graui competit tendere infrà leuius, modò <lb/>non impediatur; </s> <s id="N14FA1"><!-- NEW -->igitur e&longs;t eius proprietas; </s> <s id="N14FA5"><!-- NEW -->igitur ille impetus e&longs;t de­<lb/>terminatus ad lineam quæ tendit deor&longs;um; </s> <s id="N14FAB"><!-- NEW -->&longs;ed de hoc impetu naturali <lb/>innato fusè agemus infrà in &longs;ecundò libro; nunc &longs;ufficiat dixi&longs;&longs;e po&longs;&longs;e <lb/>dari aliquem impetum ita determinatum ad certam lineam, vt ad aliam <lb/>determinari non po&longs;&longs;it naturaliter, nulla e&longs;t enim repugnantia. </s> </p> <p id="N14FB5" type="main"> <s id="N14FB7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 116.<emph.end type="center"/></s> </p> <p id="N14FC3" type="main"> <s id="N14FC5"><!-- NEW --><emph type="italics"/>Impetus determinatur aliquando ad lineam motus à potentia motrice<emph.end type="italics"/>; </s> <s id="N14FCE"><!-- NEW -->pro­<lb/>batur, quia primus impetus ab ip&longs;a potentia productus &longs;ine impedimen­<lb/>to ab alio determinari non pote&longs;t; potentia porrò motrix vel e&longs;t gra­<lb/>uium, vel leuium, vel animantium, vel proiectorum, vel compre&longs;&longs;o­<lb/>rum, &c. </s> </p> <p id="N14FDA" type="main"> <s id="N14FDC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 117.<emph.end type="center"/></s> </p> <p id="N14FE8" type="main"> <s id="N14FEA"><!-- NEW --><emph type="italics"/>Potentia verò motrix determinatur vel à &longs;uo fine intrin&longs;eco, vel potius ab <lb/>ip&longs;a &longs;ua natura<emph.end type="italics"/>; </s> <s id="N14FF5"><!-- NEW -->&longs;ic grauitas &longs;eu potentia motrix grauium determinata <lb/>e&longs;t ad motum deor&longs;um perpendicularem, dum in medio libero corpus <lb/>graue mouetur; vel à plano inclinato; </s> <s id="N14FFD"><!-- NEW -->pro cuius diuer&longs;a inclinatione <lb/>diuer&longs;a e&longs;t linea motus deor&longs;um; </s> <s id="N15003"><!-- NEW -->vel ab ip&longs;a via, &longs;eu exitu patefacto; <lb/>&longs;ic potentia motrix compre&longs;&longs;orum &longs;uas vires exerit, & mobile ip&longs;um <lb/>agit, quâ patet viâ, &longs;ur&longs;um, deor&longs;um &c. </s> <s id="N1500B"><!-- NEW -->vel ab appetitu &longs;eu libero, &longs;eu <lb/>&longs;en&longs;itiuo; </s> <s id="N15011"><!-- NEW -->&longs;ic potentia progre&longs;&longs;iua animantium cò corpus agit, quò iu­<lb/>bet appetitus, vel ab aliqua affectione intrin&longs;eca intrin&longs;ecùs vel extrin­<lb/>&longs;ecùs adueniente; </s> <s id="N15019"><!-- NEW -->&longs;ic dilatatur pupilla, vel contrahitur pro diuer&longs;a lu­<lb/>minis appul&longs;i vi, vel obiecti di&longs;tantia: Huc reuoca motus illos natura­<lb/>les, qui animalibus competunt v. <!-- REMOVE S-->g. <!-- REMOVE S-->tu&longs;&longs;is, &longs;ingultus, &longs;ternutationis, &c. </s> <s id="N15025"><lb/>de quibus fusè &longs;uo loco. </s> </p> <p id="N15029" type="main"> <s id="N1502B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 118.<emph.end type="center"/></s> </p> <p id="N15037" type="main"> <s id="N15039"><!-- NEW --><emph type="italics"/>Impetus determinatur aliquando ad lineam ab alio impetu producente<emph.end type="italics"/>; </s> <s id="N15042"><!-- NEW --><lb/>&longs;ic impetus corporis proiecti determinatur ab impetu vel organi vel <lb/>manus proiicientis; </s> <s id="N15049"><!-- NEW -->quia nihil e&longs;t aliud à quo determinari po&longs;&longs;it, vt <lb/>patet; </s> <s id="N1504F"><!-- NEW -->adde figuram organi, di&longs;po&longs;itionem &longs;eu &longs;itum mobilis, quod ma­<lb/>nu tenetur; </s> <s id="N15055"><!-- NEW -->impedimenti etiam habetur ratio v. <!-- REMOVE S-->g. <!-- REMOVE S-->corpus oblongum <lb/>proiici pote&longs;t, vel motu recto ad in&longs;tar teli, vel motu mixto ex recto <lb/>& circulari; cum &longs;cilicet diuer&longs;imodè vibratur: </s> <s id="N15061"><!-- NEW -->&longs;i enim altera extremi­<lb/>tas adhuc hæreat in manu, dum altera mouetur, vt cum quis baculo <lb/>ferit; </s> <s id="N15069"><!-- NEW -->tunc certè e&longs;t aliquòd impedimenti genus, ex quo oritur talis li­<lb/>nea motus; illud autem impedimentum emergit ex diuer&longs;a applicatione <lb/>diuer&longs;aque brachij vibratione, quæ omnia &longs;unt &longs;atis clara. </s> </p> <pb pagenum="63" xlink:href="026/01/095.jpg"/> <p id="N15075" type="main"> <s id="N15077"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 119.<emph.end type="center"/></s> </p> <p id="N15083" type="main"> <s id="N15085"><!-- NEW --><emph type="italics"/>Impetus determinatus ad vnam lineam pote&longs;t ad aliam in &longs;uo fluxu deter­<lb/>minatu<emph.end type="italics"/>; </s> <s id="N15090"><!-- NEW -->vt patet in corpore reflexo; nec enim dici pote&longs;t totum prio­<lb/>rem impetum in ip&longs;o reflexionis puncto de&longs;trui, vt demon&longs;trabimus <lb/>aliàs. </s> <s id="N15098">Probatur etiam ex impetu proiectorum, quæ mutant lineam mo­<lb/>tus manente adhuc priore impetu &longs;altem ex parte. </s> </p> <p id="N1509D" type="main"> <s id="N1509F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 120.<emph.end type="center"/></s> </p> <p id="N150AB" type="main"> <s id="N150AD"><!-- NEW --><emph type="italics"/>Corpus proiectum in aliud ita illud impellit, vt determinet lineam motus <lb/>ratione puncti contactus<emph.end type="italics"/>; </s> <s id="N150B8"><!-- NEW -->Sit enim, ne multiplicemus figuras, globus, <lb/>cuius linea directionis &longs;it DC, punctum contactus C, ita globus A im­<lb/>pellet globum B, vt linea motus, ad quam determinatur, &longs;it CB, id e&longs;t <lb/>ducta à puncto contactus ad centrum globi impul&longs;i; </s> <s id="N150C2"><!-- NEW -->&longs;it etiam globus <lb/>P impactus in globum A punctum contactus &longs;it D, linea motus, ad <lb/>quam determinatur, e&longs;t DA, quæ &longs;cilicet à puncto contactus ducitur <lb/>per centrum grauitatis corporis impul&longs;i: </s> <s id="N150CC"><!-- NEW -->experientia huius rei certa <lb/>e&longs;t, nec ignorant qui in ludo minoris tudiculæ ver&longs;ati &longs;unt; </s> <s id="N150D2"><!-- NEW -->ratio au­<lb/>tem inde tantùm duci pote&longs;t, quod &longs;cilicet ab ip&longs;o puncto contactus ita <lb/>diffunditur impetus, vt hinc inde æqualiter in vtroque hemi&longs;phærio <lb/>diffundatur; </s> <s id="N150DC"><!-- NEW -->coniungitur autem vtrumque hemi&longs;phærium circulo A, <lb/>vel B, in priore figura, e&longs;tque vtriu&longs;que communis &longs;ectio; </s> <s id="N150E2"><!-- NEW -->cum autem <lb/>vtrimque &longs;it æqualis impetus, nulla e&longs;t ratio, cur linea directionis in­<lb/>clinet potiùs in vnum hemi&longs;phærium, quàm in aliud: </s> <s id="N150EA"><!-- NEW -->præterea cum <lb/>motus orbis globi determinetur à motu centri; </s> <s id="N150F0"><!-- NEW -->cum &longs;cilicet globus in <lb/>globum impingitur; </s> <s id="N150F6"><!-- NEW -->haud dubiè non pote&longs;t e&longs;&longs;e alius motus centri, ni&longs;i <lb/>qui determinatur à puncto contactus, à quo vnica tantùm linea ad cen­<lb/>trum duci pote&longs;t, vt con&longs;tat; & hæc ratio veri&longs;&longs;ima e&longs;t, & totam rem <lb/>ip&longs;am euincit. </s> </p> <p id="N15100" type="main"> <s id="N15102"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 121.<emph.end type="center"/></s> </p> <p id="N1510E" type="main"> <s id="N15110"><emph type="italics"/>Hinc licèt diuer&longs;æ &longs;int linea motus globi impellentis, &longs;i tamen &longs;it idem pun­<lb/>ctum contactus ad <expan abbr="eãdem">eandem</expan> lineam globus impul&longs;us determinabitur,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->li­<lb/>cet globus P. eiu&longs;dem figuræ tangat globum A in D per lineam PD &longs;iue <lb/>per lineam HD &longs;iue per quamlibet aliam, globus A mouebitur &longs;emper <lb/>per lineam directionis DA propter rationem propo&longs;itam, quod etiam <lb/>mille experimentis conuincitur. </s> </p> <p id="N1512A" type="main"> <s id="N1512C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 122.<emph.end type="center"/></s> </p> <p id="N15138" type="main"> <s id="N1513A"><!-- NEW --><emph type="italics"/>Determinatur impetus corporis proiecti impacti in corpus reflectens ad no­<lb/>uam lineam<emph.end type="italics"/>; </s> <s id="N15145"><!-- NEW -->patet experientiâ in pilâ reflexâ; reflexionis autem ratio­<lb/>nem afferemus in lib. de motu reflexo. </s> </p> <p id="N1514B" type="main"> <s id="N1514D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 123.<emph.end type="center"/></s> </p> <p id="N15159" type="main"> <s id="N1515B"><emph type="italics"/>Non determinatur tantùm ratione puncti contactus.<emph.end type="italics"/></s> <s id="N15162"><!-- NEW --> Probatur, quia cum <lb/>eodem puncto contactus pote&longs;t e&longs;&longs;e determinatio ad diuer&longs;am lineam, <lb/>vt manife&longs;tum e&longs;t; &longs;it enim reflexio per angulum æqualem incidentiæ, <lb/>&longs;ed diuer&longs;i anguli po&longs;&longs;unt in idem punctum coire, vt patet. </s> </p> <pb pagenum="64" xlink:href="026/01/096.jpg"/> <p id="N15170" type="main"> <s id="N15172"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 124.<emph.end type="center"/></s> </p> <p id="N1517E" type="main"> <s id="N15180"><!-- NEW --><emph type="italics"/>Non determinatur noua linea in motu reflexo â priore tantùm linea <lb/>incidentiæ<emph.end type="italics"/>; probatur, quia pote&longs;t e&longs;&longs;e eadem linea incidentiæ cum di­<lb/>uer&longs;is lineis motus reflexi, vt patet. </s> </p> <p id="N1518D" type="main"> <s id="N1518F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 125.<emph.end type="center"/></s> </p> <p id="N1519B" type="main"> <s id="N1519D"><!-- NEW --><emph type="italics"/>Non determinatur noua linea motus reflexi ratione tantùm plani reflecten­<lb/>tis<emph.end type="italics"/>: Probatur, quia cum eodem plano reflectente diuer&longs;æ lineæ motus <lb/>reflexi e&longs;&longs;e po&longs;&longs;unt, vt con&longs;tat. </s> </p> <p id="N151AA" type="main"> <s id="N151AC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 126.<emph.end type="center"/></s> </p> <p id="N151B8" type="main"> <s id="N151BA"><!-- NEW --><emph type="italics"/>Determinatur noua linea motus reflexi ratione lineæ prioris incidentiæ com­<lb/>paratæ cum plano reflectente,<emph.end type="italics"/> e&longs;t enim angulus reflexionis æqualis angu­<lb/>lo incidentiæ, cuius effectus rationem aliàs afferemus, cum de motu <lb/>reflexo; </s> <s id="N151C9"><!-- NEW -->& verò multa hîc cur&longs;im tantùm per&longs;tringimus, quæ in libro <lb/>de motu reflexo accurati&longs;&longs;imè demon&longs;trabimus; Hìc tantùm dixi&longs;&longs;e &longs;uf­<lb/>ficiat determinari mobile in reflexionis puncto ad nouam lineam motus, <lb/>quod nemo in dubium reuocare pote&longs;t, & propter quid fiat loco citato <lb/>demon&longs;trabimus. </s> </p> <p id="N151D5" type="main"> <s id="N151D7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 127.<emph.end type="center"/></s> </p> <p id="N151E3" type="main"> <s id="N151E5"><!-- NEW --><emph type="italics"/>Quando globus in globum æqualem ita impingitur, vt linea directionis per <lb/>centra vtriu&longs;que ducatur, determinatio noua e&longs;t æqualis priori<emph.end type="italics"/>; </s> <s id="N151F0"><!-- NEW -->Patet ex­<lb/>perientia in pilis illis eburneis, quas de&longs;iderat ludus minoris tudiculæ; </s> <s id="N151F6"><!-- NEW --><lb/>nec e&longs;t vlla ratio, cur determinatio &longs;it maior potiùs, quàm minor, cum <lb/>vtraque pila &longs;it æqualis; </s> <s id="N151FD"><!-- NEW -->&longs;i enim maior e&longs;&longs;et, vel minor; cur potiùs vno <lb/>gradu, quàm duobus? </s> <s id="N15203">quàm tribus? </s> <s id="N15206"><!-- NEW -->Præterea, cum re&longs;i&longs;tens, vel im­<lb/>pediens e&longs;t æquale agenti; </s> <s id="N1520C"><!-- NEW -->certe &longs;icut agens refundit in pa&longs;&longs;um totum <lb/>id, quod habet, id e&longs;t æqualem impetum in inten&longs;ione, & æquè velo­<lb/>cem motum per Th. 60. <!--neuer Satz-->Ita re&longs;i&longs;tens, vel impediens refundit æquale <lb/>impedimentum, quod tantùm &longs;umi pote&longs;t ex æqualitate mobilium; </s> <s id="N15218"><!-- NEW -->&longs;ed <lb/>ex æquali impedimento duci tantùm pote&longs;t æqualis determinatio priori; <lb/>denique pote&longs;t dari determinatio noua æqualis priori, vt con&longs;tat, &longs;ed <lb/>aliunde duci non pote&longs;t quàm ex ip&longs;a mobilium æqualitate, modò fiat <lb/>contactus per lineam connectentem centra. </s> </p> <p id="N15224" type="main"> <s id="N15226"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 128.<emph.end type="center"/></s> </p> <p id="N15232" type="main"> <s id="N15234"><!-- NEW --><emph type="italics"/>Hinc ratio manife&longs;ta illius mirifici effectus, &longs;cilicet quietis pilæ impactæ<emph.end type="italics"/>; </s> <s id="N1523D"><!-- NEW --><lb/>quippe hæc quie&longs;cet illicò ab ictu; </s> <s id="N15242"><!-- NEW -->quia &longs;cilicet, cum noua determina­<lb/>tio &longs;it æqualis priori, non e&longs;t vlla ratio, cur alterutra præualeat; </s> <s id="N15248"><!-- NEW -->nec <lb/>etiam pote&longs;t e&longs;&longs;e determinatio communis, &longs;eu mixta; cur enim potius <lb/>dextror&longs;um quam &longs;ini&longs;tror&longs;um? </s> <s id="N15250">de quo infrà. </s> </p> <p id="N15253" type="main"> <s id="N15255"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 129.<emph.end type="center"/></s> </p> <p id="N15261" type="main"> <s id="N15263"><!-- NEW --><emph type="italics"/>Quando linea directionis globi impacti non connectit centra vtriu&longs;qu&etail; <lb/>globi, determinatur noua linea motus tùm à priore linea incidentiæ, tùm à <lb/>connectente centra, quæ &longs;cilicet per punctum contactus à centro impacti globi<emph.end type="italics"/><pb pagenum="65" xlink:href="026/01/097.jpg"/><emph type="italics"/>ad centrum alterius ducitur<emph.end type="italics"/>; quippe nihil e&longs;t aliud à quo determinari. </s> <s id="N15279"><lb/>po&longs;&longs;it, vt patet; </s> <s id="N1527D"><!-- NEW -->non determinatur etiam ab alterutra &longs;eor&longs;im, vt con­<lb/>&longs;tat, igitur ab vtraque conjunctim; </s> <s id="N15283"><!-- NEW -->in qua verò proportione dicemus, <lb/>& demon&longs;trabimus in libro de motu reflexo; &longs;unt enim mirificæ quæ­<lb/>dam reflexionum proportiones, quas ibidem explicabimus. </s> </p> <p id="N1528B" type="main"> <s id="N1528D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 130.<emph.end type="center"/></s> </p> <p id="N15299" type="main"> <s id="N1529B"><!-- NEW --><emph type="italics"/>Hinc globus &longs;ic impactus nunquam quie&longs;cit<emph.end type="italics"/>; </s> <s id="N152A4"><!-- NEW -->ratio e&longs;t, quia vtraque linea <lb/>determinationis cum angulum faciat, in communem lineam abit; </s> <s id="N152AA"><!-- NEW -->nam <lb/>ex duabus lineis motus minimè oppo&longs;itis ex diametro, fit alia tertia me­<lb/>dia pro rata; hîc etiam latent my&longs;teria, de quibus loco citato. </s> </p> <p id="N152B2" type="main"> <s id="N152B4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 131.<emph.end type="center"/></s> </p> <p id="N152C0" type="main"> <s id="N152C2"><!-- NEW --><emph type="italics"/>Si globus minor in maiorem impingatur per quamcumque lineam directio­<lb/>nis, determinatur ad nouam lineam motus reflexi<emph.end type="italics"/>; </s> <s id="N152CD"><!-- NEW -->experientia clara e&longs;t; ra­<lb/>tio e&longs;t, quia maior globus maius e&longs;t impedimentum, hinc nunquam <lb/>quie&longs;cit minor globus impactus. </s> </p> <p id="N152D5" type="main"> <s id="N152D7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 132.<emph.end type="center"/></s> </p> <p id="N152E3" type="main"> <s id="N152E5"><!-- NEW --><emph type="italics"/>Si globus major in minorem impingatur per lineam directionis, quæ conne­<lb/>ctat centra, &longs;eruat <expan abbr="eãdem">eandem</expan> lineam<emph.end type="italics"/>; </s> <s id="N152F4"><!-- NEW -->patet etiam experientiâ, cuius ratio e&longs;t <lb/>minor re&longs;i&longs;tentia minoris globi; </s> <s id="N152FA"><!-- NEW -->&longs;i verò &longs;it alia linea directionis, omni­<lb/>nò reflectitur &longs;uo modo; </s> <s id="N15300"><!-- NEW -->id e&longs;t mutat lineam; </s> <s id="N15304"><!-- NEW -->&longs;ed de his omnibus fusè <lb/>aliàs; </s> <s id="N1530A"><!-- NEW -->hîc tantùm &longs;ufficiat indica&longs;&longs;e; </s> <s id="N1530E"><!-- NEW -->(&longs;uppo&longs;ita linea directionis cen­<lb/>trali &longs;eu connectente centra, &longs;ic enim deinceps eam appellabimus, in <lb/>quo ca&longs;u duplex determinatio tertiam mediam conflare non pote&longs;t) in­<lb/>dica&longs;&longs;e inquam &longs;ufficiat nouam determinationem, vel e&longs;&longs;e æqualem prio­<lb/>ri, vel maiorem, vel minorem; </s> <s id="N1531A"><!-- NEW -->&longs;i æqualis e&longs;t, globus impactus &longs;i&longs;tit; &longs;i <lb/>maior, reflectitur; &longs;i minor, <expan abbr="eãdem">eandem</expan> lineam, &longs;ed lentiùs pro rata pro­<lb/>&longs;equitur. </s> </p> <p id="N15326" type="main"> <s id="N15328"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 133.<emph.end type="center"/></s> </p> <p id="N15334" type="main"> <s id="N15336"><!-- NEW --><emph type="italics"/>Si &longs;it duplex impetus æqualis ad diuer&longs;as lineas determinatus in eodem mo­<lb/>bili, &longs;ique illæ &longs;int ex diametro oppo&longs;itæ &longs;i&longs;tere debet mobile<emph.end type="italics"/>; patet; </s> <s id="N15341"><!-- NEW -->&longs;it enim <lb/>globus vtrimque gemino malleo percu&longs;&longs;us æquali ictu; </s> <s id="N15347"><!-- NEW -->haud dubiè &longs;i&longs;tit; <lb/>cur enim potiùs in vnam partem quam in aliam? </s> <s id="N1534D">cum &longs;imul in vtramque <lb/>moueri non po&longs;&longs;it. </s> </p> <p id="N15352" type="main"> <s id="N15354"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 134.<emph.end type="center"/></s> </p> <p id="N15360" type="main"> <s id="N15362"><!-- NEW --><emph type="italics"/>Si verò alter impetus &longs;it inten&longs;ior, po&longs;ito eodem ca&longs;u, haud dubiè eius de­<lb/>terminatio præualebit pro rata<emph.end type="italics"/>; patet etiam experientià; </s> <s id="N1536D"><!-- NEW -->ratio e&longs;t, quia im­<lb/>petus fortior debiliorem vincit; pugnant enim pro rata per Ax. 15. <lb/>hinc &longs;i &longs;it duplò inten&longs;ior, &longs;ubduplum &longs;uæ velocitatis amittet, &longs;i triplè <lb/>&longs;ubtriplum, &c. </s> <s id="N15377">de quo aliàs. </s> </p> <p id="N1537A" type="main"> <s id="N1537C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 135.<emph.end type="center"/></s> </p> <p id="N15388" type="main"> <s id="N1538A"><emph type="italics"/>Si duo globi projecti &longs;ibi inuicem occurrant in lineæ directionis connectente <lb/>centra, reflectitur vterque æquali motu, quo antè.<emph.end type="italics"/></s> <s id="N15393"> Probatur; </s> <s id="N15396"><!-- NEW -->&longs;unt enim globi <pb pagenum="66" xlink:href="026/01/098.jpg"/>A & B, & A feratur per lineam DE, & B per lineam ED, punctum con­<lb/>tactus &longs;it C, haud dubiè globus A impactus in B amittit totum &longs;uum im­<lb/>petum per Th.127. & 128. B, item impactus in A amittit totum &longs;uum per <lb/>eandem rationem; </s> <s id="N153A5"><!-- NEW -->globus A producit impetum in B æqualem &longs;uo per <lb/>Th.60. item B producit in A æqualem per idem Th. igitur tantùm perit <lb/>impetus quantùm accedit; </s> <s id="N153AD"><!-- NEW -->igitur in vtroque globo remanet æqualis im­<lb/>petus priori; igitur æquali motu vterque mouetur, quod erat dem. </s> <s id="N153B3">& hæc <lb/>e&longs;t ratio veri&longs;&longs;ima toties probatæ experientiæ. </s> </p> <p id="N153B8" type="main"> <s id="N153BA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 136.<emph.end type="center"/></s> </p> <p id="N153C6" type="main"> <s id="N153C8"><!-- NEW --><emph type="italics"/>Hinc æquale &longs;patium conficiet regrediendo po&longs;t reflexionem, quem confeci&longs;­<lb/>&longs;et motu directo, &longs;i propagatus fui&longs;&longs;et &longs;ine obice<emph.end type="italics"/>; </s> <s id="N153D3"><!-- NEW -->nam æquali motu æquali <lb/>tempore in eodem plano &longs;eu medio idem &longs;patium decurritur; quid verò <lb/>accidat in aliis punctis contactus dicemus infrà, cum de reflexione. </s> </p> <p id="N153DB" type="main"> <s id="N153DD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 137.<emph.end type="center"/></s> </p> <p id="N153E9" type="main"> <s id="N153EB"><!-- NEW --><emph type="italics"/>Si in eodem mobili duplex impetus producatur, quorum vterque &longs;eor&longs;im <lb/>ad duas lineas &longs;it determinatus quæ conjunctæ faciant angulum, determinatur <lb/>vterque ad tertiam lineam mediam<emph.end type="italics"/>; </s> <s id="N153F8"><!-- NEW -->&longs;it enim mobile in A. v. <!-- REMOVE S-->g. <!-- REMOVE S-->globus, <lb/>cui &longs;imul imprimatur impetus determinatus ad lineam AD, in plano <lb/>horizontali AF; </s> <s id="N15404"><!-- NEW -->&longs;i vterque &longs;it æqualis, ad nouam lineam determinabi­<lb/>tur AE; </s> <s id="N1540A"><!-- NEW -->quippe tantùm debet acquirere in horizontali AB, vel in eius <lb/>parallela DE, quantum acquirit in alia horizontali AD, vel in eius pa­<lb/>rallela BE; </s> <s id="N15412"><!-- NEW -->igitur debet ferri in E; </s> <s id="N15416"><!-- NEW -->igitur per diagonalem AE; </s> <s id="N1541A"><!-- NEW -->clara e&longs;t <lb/>omninò experientia; </s> <s id="N15420"><!-- NEW -->cuius ratio à priori hæc e&longs;t, quòd &longs;cilicet impetus <lb/>po&longs;&longs;it determinari ad quamlibet lineam ab alio impetu per Th.118.119. <lb/>igitur in eodem mobili pro rata quilibet alium determinat; </s> <s id="N15428"><!-- NEW -->igitur &longs;i <lb/>vterque æqualis e&longs;t, vterque æqualiter; igitur debet tantum &longs;patij acqui­<lb/>ri in linea vnius, quantum in linea alterius. </s> </p> <p id="N15430" type="main"> <s id="N15432"><!-- NEW -->Si verò impetus per AC &longs;it duplus impetus per AD; </s> <s id="N15436"><!-- NEW -->accipiatur AC <lb/>dupla AD, ducatur DF æqualis & parallela AC; </s> <s id="N1543C"><!-- NEW -->linea motus noua <lb/>erit diagonalis AF, quia vtraque determinatio concurrit ad nouam pro <lb/>rata; igitur debet &longs;patium acqui&longs;itum in AC e&longs;&longs;e duplum acqui&longs;iti <lb/>in AD. <!-- KEEP S--></s> </p> <p id="N15447" type="main"> <s id="N15449"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 138.<emph.end type="center"/></s> </p> <p id="N15455" type="main"> <s id="N15457"><!-- NEW --><emph type="italics"/>Si &longs;it duplex impetus in eodem mobili ad <expan abbr="eãdem">eandem</expan> lineam determinatus, non <lb/>mutabitur linea; </s> <s id="N15463"><!-- NEW -->&longs;ed cre&longs;cet motus & &longs;patium<emph.end type="italics"/> Imprimatur impetus in A, <lb/>per AB, quo dato tempore percurratur &longs;patium AB; </s> <s id="N1546C"><!-- NEW -->deinde produca­<lb/>tur &longs;imul alius impetus æqualis priori in eodem mobili per lineam AB; </s> <s id="N15472"><!-- NEW --><lb/>Dico quod eodem tempore percurretur tota AE, dupla &longs;cilicet AB; </s> <s id="N15477"><!-- NEW --><lb/>quia &longs;cilicet dupla cau&longs;a non impedita duplum effectum habet per Ax. <!-- REMOVE S--><lb/>13. num.1. duplus impetus duplum motum; igitur duplum &longs;patium; &longs;i <lb/>verò &longs;it triplus impetus, triplum erit &longs;patium, &c. </s> </p> <p id="N15481" type="main"> <s id="N15483"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 139.<emph.end type="center"/></s> </p> <p id="N1548F" type="main"> <s id="N15491"><!-- NEW --><emph type="italics"/>Si lineæ duplicis impetus, faciunt angulum acutiorem, longius erit &longs;patium<emph.end type="italics"/><pb pagenum="67" xlink:href="026/01/099.jpg"/><emph type="italics"/>acqui&longs;itum<emph.end type="italics"/>: </s> <s id="N154A3"><!-- NEW -->&longs;int duæ lineæ IK IL, mobili &longs;cilicet &longs;tatuto in I; </s> <s id="N154A7"><!-- NEW --><lb/>haud dubiè noua linea erit IM; </s> <s id="N154AC"><!-- NEW -->& quo angulus KIL, erit acutior (&longs;up­<lb/>po&longs;itis æqualibus &longs;emper lateribus IK IL) Diagonalis IM, erit ma­<lb/>ior; </s> <s id="N154B4"><!-- NEW -->donec tandem IL & IK coeant in eandem lineam; </s> <s id="N154B8"><!-- NEW -->tunc enim li­<lb/>nea erit dupla IK per Th. &longs;uperius: </s> <s id="N154BE"><!-- NEW -->quandiu verò e&longs;t aliquis angulus in <lb/>I quantumuis acutus, linea motus erit minor dupla IK, ad quam tamen <lb/>propiùs &longs;emper accedit; quæ omnia con&longs;tant ex elementis. </s> </p> <p id="N154C6" type="main"> <s id="N154C8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 140.<emph.end type="center"/></s> </p> <p id="N154D4" type="main"> <s id="N154D6"><!-- NEW --><emph type="italics"/>Si lineæ duplicis impetus faciunt angulum obtu&longs;um, &longs;patium acqui&longs;itum erit <lb/>breuius, & eò breuius quò angulus e&longs;t obtu&longs;ior<emph.end type="italics"/>; </s> <s id="N154E1"><!-- NEW -->&longs;int enim <emph type="sup"/>c<emph.end type="sup"/> duæ lineæ AD <lb/>AB mobili &longs;tatuto in A, noua linea erit AC per Th. 137. & &longs;i accipia­<lb/>tur angulus obtu&longs;ior HEF; </s> <s id="N154EF"><!-- NEW -->noua linea erit EG, eo rectè breuior, <lb/>quò angulus e&longs;t obtu&longs;ior, non tamen iuxta rationem angulorum; </s> <s id="N154F5"><!-- NEW -->donec <lb/>tandem de&longs;inat angulus, & ED EF coëant in vnam lineam; </s> <s id="N154FB"><!-- NEW -->tunc enim <lb/>nullum erit &longs;patium, quia &longs;i&longs;ter omninò mobile per Th.133.quæ omnia <lb/>ip&longs;a luce clariora e&longs;&longs;e con&longs;tat; </s> <s id="N15503"><!-- NEW -->quippe quæ cum certis experimentis, & <lb/>clari&longs;&longs;imis principiis con&longs;entiant; &longs;ed de his plura infrà. </s> </p> <p id="N15509" type="main"> <s id="N1550B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 141.<emph.end type="center"/></s> </p> <p id="N15517" type="main"> <s id="N15519"><!-- NEW --><emph type="italics"/>Ex his nece&longs;&longs;aria ducitur ratio, cur impetus duplus ad diuer&longs;as lineas de­<lb/>terminatus non habeat motum duplum, & con&longs;equenter &longs;patium duplum<emph.end type="italics"/>; </s> <s id="N15524"><!-- NEW -->nec <lb/>enim AE e&longs;t dupla AB, vt con&longs;tat; </s> <s id="N1552A"><!-- NEW -->nam &longs;i lineæ &longs;int oppo&longs;itæ ex <lb/>diametro vt BA BE totus de&longs;truitur impetus, per Th.133. &longs;i verò vna <lb/>in <expan abbr="eãdem">eandem</expan> lineam coëat cum aliâ, nihil impetus de&longs;truitur, nec impedi­<lb/>tur per Th.138. igitur quà proportione propiùs accedet ad oppo&longs;itas; </s> <s id="N15538"><!-- NEW --><lb/>plùs de&longs;truetur, & minus erit &longs;patium; & quâ proportione accedent <lb/>propiùs ad coëuntes, minùs de&longs;truetur, & maius erit &longs;patium, vt con&longs;tat <lb/>ex dictis. </s> </p> <p id="N15541" type="main"> <s id="N15543"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 142.<emph.end type="center"/></s> </p> <p id="N1554F" type="main"> <s id="N15551"><!-- NEW --><emph type="italics"/>Hinc impetus ad diuer&longs;as lineas determinati it a pugnant pro rata, vt mi­<lb/>nùs pugnent, quorum lineæ propiùs accedunt ad coëuntes; plùs verò, quorum <lb/>lineæ propiùs accedunt ad oppo&longs;itas, idque iuxta proportiones Diagonalium,<emph.end type="italics"/><lb/>quod totum &longs;equitur ex dictis. </s> </p> <p id="N1555F" type="main"> <s id="N15561"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1556D" type="main"> <s id="N1556F"><!-- NEW -->Ob&longs;eruabis vt faciliùs concipias duos impetus ad duas lineas deter­<lb/>minatos; </s> <s id="N15575"><!-- NEW -->finge tibi nauim à diuer&longs;is ventis impul&longs;am, &longs;eu lapidem pro­<lb/>jectum è naui mobili; &longs;ed de his plura in lib.4. cum de motu mixto. </s> </p> <p id="N1557B" type="main"> <s id="N1557D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 143.<emph.end type="center"/></s> </p> <p id="N15589" type="main"> <s id="N1558B"><emph type="italics"/>Impetus &longs;emel productus, quamdiu durat motus, con&longs;eruatur.<emph.end type="italics"/></s> <s id="N15592"> Probatur, <lb/>quia non pote&longs;t e&longs;&longs;e effectus, ni&longs;i &longs;it eius cau&longs;a per Ax. 8. igitur &longs;i e&longs;t mo­<lb/>tus, e&longs;t impetus. </s> </p> <p id="N15599" type="main"> <s id="N1559B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 144.<emph.end type="center"/></s> </p> <p id="N155A7" type="main"> <s id="N155A9"><emph type="italics"/>Impetus non con&longs;eruatur à cau&longs;a primò productiua.<emph.end type="italics"/></s> <s id="N155B0"> Probatur; quia proii-<pb pagenum="68" xlink:href="026/01/100.jpg"/>ciatur mobile per Po&longs;tulatum, etiam mouetur &longs;eparatum à potentia mo­<lb/>trice per hypoth. </s> <s id="N155BA">6. igitur non con&longs;eruatur à potentia motrice per Ax. <!-- REMOVE S--><lb/>10. igitur nec à causâ primò productiua. </s> </p> <p id="N155C0" type="main"> <s id="N155C2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 145.<emph.end type="center"/></s> </p> <p id="N155CE" type="main"> <s id="N155D0"><!-- NEW --><emph type="italics"/>Hinc ab alia causâ con&longs;eruari nece&longs;&longs;e e&longs;t impetum<emph.end type="italics"/>: Probatur, quia impe­<lb/>tus non e&longs;t à &longs;e, quia de&longs;truitur aliquando per Ax. 14. igitur con&longs;eruatur <lb/>ab alio per Ax.14. num. </s> <s id="N155DD"><!-- NEW -->1. non à cau&longs;a primò productiua per Th.144.igi­<lb/>tur ab alia, eaque applicata per Ax. <!-- REMOVE S-->10. quæcumque tandem illa &longs;it, ali­<lb/>quando cau&longs;am primam e&longs;&longs;e demon&longs;trabimus; </s> <s id="N155E7"><!-- NEW -->nunc verò &longs;ufficiat dixi&longs;­<lb/>&longs;e dari aliquam cau&longs;am reuerâ applicatam, quæ ip&longs;um con&longs;eruat impe­<lb/>tum; immò ex hac ip&longs;a rerum con&longs;eruatione argumentum aliquando <lb/>ducemus, quo Deum ip&longs;um exi&longs;tere demon&longs;trabimus. </s> </p> <p id="N155F1" type="main"> <s id="N155F3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 146.<emph.end type="center"/></s> </p> <p id="N155FF" type="main"> <s id="N15601"><emph type="italics"/>Si impetus con&longs;eruaretur à cau&longs;a primò productiua, nunquam de&longs;truere­<lb/>tur, quamdiu e&longs;&longs;et applicata.<emph.end type="italics"/></s> <s id="N1560A"><!-- NEW --> Demon&longs;tratur, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria <lb/>(nam de hac ip&longs;a loquor) igitur &longs;emper ageret, igitur &longs;emper con­<lb/>&longs;eruaret, quod e&longs;t contra experientiam; </s> <s id="N15612"><!-- NEW -->nam reuerâ impetus pro­<lb/>ductus deor&longs;um à corpore graui motu naturaliter accelerato de&longs;truitur, <lb/>vt patet; </s> <s id="N1561A"><!-- NEW -->præterea &longs;i corpus graue con&longs;eruaret impetum primò produ­<lb/>ctum, non produceret nouum contra experientiam; </s> <s id="N15620"><!-- NEW -->quippe cau&longs;a ne­<lb/>ce&longs;&longs;aria non plùs agit vno in&longs;tanti quàm alio, per Ax.12. adde quod im­<lb/>petus de&longs;truitur ad exigentiam alterius, quidquid tandem illud &longs;it per <lb/>Ax.14. num.2. & 3. &longs;ed cau&longs;a primò productiua impetus non nouit rerum <lb/>exigentiam; </s> <s id="N1562C"><!-- NEW -->igitur illi facere &longs;atis non pote&longs;t; ex hoc etiam capite cau­<lb/>&longs;æ primæ exi&longs;tentiam &longs;uo loco demon&longs;trabimus. </s> </p> <p id="N15632" type="main"> <s id="N15634"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15640" type="main"> <s id="N15642"><!-- NEW -->Ob&longs;eruabis primò rem quamlibet ideo de&longs;trui, quia ce&longs;&longs;at cau&longs;a con­<lb/>&longs;eruans illam con&longs;eruare; </s> <s id="N15648"><!-- NEW -->quippe quod de&longs;truitur eo in&longs;tanti dicitur de­<lb/>&longs;trui, quo primò non e&longs;t, &longs;eu quo incipit primò non e&longs;&longs;e; atqui incipit <lb/>primò non e&longs;&longs;e &longs;eu de&longs;init e&longs;&longs;e, cum de&longs;init con&longs;eruari. </s> </p> <p id="N15650" type="main"> <s id="N15652"><!-- NEW -->Secundò ob&longs;eruabis præclarum naturæ in&longs;titutum, quod etiam ex ip&longs;is <lb/>hypothe&longs;ibus con&longs;tat, quo fit vt qualitates quæ carent contrario à cau&longs;a <lb/>primò productiua con&longs;eruentur, vt lumen; </s> <s id="N1565A"><!-- NEW -->ne &longs;i ab alia con&longs;eruarentur, <lb/>de&longs;truerentur vmquam; </s> <s id="N15660"><!-- NEW -->cum earum de&longs;tructionem nihil exigeret per <lb/>Ax.14.n.2. & 3. at verò qualitates, quæ contrarias habent: </s> <s id="N15666"><!-- NEW -->&longs;i quæ &longs;unt, <lb/>à cau&longs;a primò productiua minimè con&longs;eruantur; </s> <s id="N1566C"><!-- NEW -->cum enim ideo con­<lb/>trarium dicatur de&longs;truere contrarium, quia exigit eius de&longs;tructionem, id <lb/>e&longs;t, ne con&longs;eruetur amplius; </s> <s id="N15674"><!-- NEW -->certè vt cau&longs;a con&longs;eruans ce&longs;&longs;et con&longs;eruare, <lb/>debet no&longs;&longs;e illam exigentiam; atqui nulla cognitione pollent cau&longs;æ illæ <lb/>motrices naturales, de quibus e&longs;t quæ&longs;tio. </s> </p> <p id="N1567C" type="main"> <s id="N1567E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 147.<emph.end type="center"/></s> </p> <p id="N1568A" type="main"> <s id="N1568C"><!-- NEW --><emph type="italics"/>Tamdiu con&longs;eruatur impetus, quamdiu nihil exigit eius destructionem<emph.end type="italics"/>; </s> <s id="N15695"><!-- NEW -->quia <lb/>de&longs;truitur tantùm ad exigentiam alicuius, quidquid tandem illud &longs;it, de <pb pagenum="69" xlink:href="026/01/101.jpg"/>quo infrà, per Ax.14.num.2. certè tamdiu non de&longs;truitur, quamdiu nihil <lb/>e&longs;t, quod exigat eius de&longs;tructionem; igitur tamdiu con&longs;eruatur per Ax. <!-- REMOVE S--><lb/>14.num.3. </s> </p> <p id="N156A5" type="main"> <s id="N156A7"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N156B4" type="main"> <s id="N156B6"><!-- NEW -->Inde certa ducitur ratio, cur mobile etiam &longs;eparatum à manu mouea­<lb/>tur; </s> <s id="N156BC"><!-- NEW -->quia &longs;cilicet ip&longs;i adhuc ine&longs;t impetus, qui e&longs;t cau&longs;a motus; </s> <s id="N156C0"><!-- NEW -->quippe <lb/>&longs;uppo&longs;ui iam antè de hac hypothe&longs;i quod &longs;it, non tamen propter quid &longs;it; <lb/>igitur hæc e&longs;t germana illius ratio & cau&longs;a. </s> </p> <p id="N156C8" type="main"> <s id="N156CA"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N156D7" type="main"> <s id="N156D9">Hinc etiam rationem ducemus æquè præclaram in lib.2. motus natu­<lb/>raliter accelerati. </s> </p> <p id="N156DE" type="main"> <s id="N156E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 148.<emph.end type="center"/></s> </p> <p id="N156EC" type="main"> <s id="N156EE"><!-- NEW --><emph type="italics"/>Impetus productus aliquando de&longs;truitur<emph.end type="italics"/>; Probatur, quia mobile, quod <lb/>antè mouebatur, de&longs;init tandem moueri per hyp. </s> <s id="N156F9"><!-- NEW -->4. igitur de&longs;truitur <lb/>impetus; alioqui &longs;i remaneret, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria &longs;ine effectu contra <lb/>Ax.12. ideo porrò de&longs;truitur, quia aliquid exigit eius de&longs;tructionem, <lb/>quippe hæc e&longs;t vnica de&longs;tructionis ratio per Ax.14. num.2. </s> </p> <p id="N15703" type="main"> <s id="N15705"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 149.<emph.end type="center"/></s> </p> <p id="N15711" type="main"> <s id="N15713"><!-- NEW --><emph type="italics"/>In lineis oppo&longs;itis impetus de&longs;truitur ab impetu &longs;uo modo<emph.end type="italics"/>; </s> <s id="N1571C"><!-- NEW -->&longs;it enim globus <lb/>proiectus ver&longs;us au&longs;trum; </s> <s id="N15722"><!-- NEW -->cui deinde imprimatur nouus impetus ver­<lb/>&longs;us Boream; </s> <s id="N15728"><!-- NEW -->de&longs;truitur prior vt con&longs;tat, igitur ad exigentiam alicuius, <lb/>&longs;ed nihil e&longs;t quod po&longs;&longs;it exigere, ni&longs;i nouus impetus, &longs;cilicet mediatè; <lb/>nihil enim aliud e&longs;t applicatum, igitur nihil aliud exigit per Ax. 10. <lb/>hæc porrò exigentia non e&longs;t immediata, &longs;ed mediata, vt dixi. </s> </p> <p id="N15732" type="main"> <s id="N15734"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 150.<emph.end type="center"/></s> </p> <p id="N15740" type="main"> <s id="N15742"><!-- NEW --><emph type="italics"/>Impetus naturalis innatus exigit de&longs;tructionem alterius, qui ab extrin&longs;eco <lb/>ad diuer&longs;am lineam corpori graui impre&longs;&longs;us e&longs;t &longs;cilicet mediatè,<emph.end type="italics"/> experientia <lb/>certa e&longs;t in proiectis, quæ tandem quie&longs;cunt; </s> <s id="N1574F"><!-- NEW -->igitur ad exigentiam ali­<lb/>cuius, &longs;ed illud tantùm e&longs;t impetus innatus; </s> <s id="N15755"><!-- NEW -->nec enim e&longs;t &longs;ub&longs;tantia <lb/>corporis; </s> <s id="N1575B"><!-- NEW -->tùm quia qualitas &longs;ub&longs;tantiæ non opponitur; </s> <s id="N1575F"><!-- NEW -->tùm quia nulla <lb/>e&longs;&longs;et ratio, cur &longs;ub&longs;tantia de&longs;trueret potiùs vno in&longs;tanti vnum gradum, <lb/>quàm duos, quàm tres; </s> <s id="N15767"><!-- NEW -->adde quod ex duobus violentis oppo&longs;itis alte­<lb/>rum de&longs;truit; igitur impetus e&longs;t cau&longs;a &longs;ufficiens de&longs;tructiua impetus, <lb/>igitur non e&longs;t ponenda alia, eo &longs;cilicet modo, quo diximus. </s> </p> <p id="N1576F" type="main"> <s id="N15771"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 151.<emph.end type="center"/></s> </p> <p id="N1577D" type="main"> <s id="N1577F"><!-- NEW --><emph type="italics"/>In reflexione de&longs;truitur aliquid impotus &longs;altem per accidens<emph.end type="italics"/>; patet expe­<lb/>rientia, &longs;iue propter nouam determinationem, &longs;iue propter attritum, <lb/>vel pre&longs;&longs;ionem partium, de quo infrà. </s> </p> <p id="N1578C" type="main"> <s id="N1578E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 152.<emph.end type="center"/></s> </p> <p id="N1579A" type="main"> <s id="N1579C"><!-- NEW --><emph type="italics"/>Hinc &longs;i excipias tantùm impetum naturalem innatum, qui per &longs;uam de­<lb/>terminationem nece&longs;&longs;ariam, & quam nunquam mutat, pugnat cum omni<emph.end type="italics"/><pb pagenum="70" xlink:href="026/01/102.jpg"/><emph type="italics"/>extrin&longs;eco ad aliam lineam determinato, & cum ip&longs;o acqui&longs;ito, quando mu­<lb/>tat lineam perpendicularem deor&longs;um, de quo infrà; &longs;i hunc igitur excipias, <lb/>omnes aly pugnant tantùm ratione diuer&longs;æ lineæ, &longs;eu determinationis, in eodem <lb/>mobili:<emph.end type="italics"/> Vnde ille idem, qui modo pugnat probè conueniet, &longs;i ad ean­<lb/>dem lineam determinetur. </s> </p> <p id="N157B8" type="main"> <s id="N157BA"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N157C6" type="main"> <s id="N157C8"><!-- NEW -->Ob&longs;eruabis primò, præclarum naturæ in&longs;titutum, quo fit, vt impe­<lb/>tus perennis non &longs;it; vnde certè infinita propemodum emergerent ab­<lb/>&longs;urda, & incommoda. </s> </p> <p id="N157D0" type="main"> <s id="N157D2"><!-- NEW -->Secundò, faciliorem modum de&longs;tructionis impetus in&longs;titui non po­<lb/>tui&longs;&longs;e, immò nec excogitari po&longs;&longs;e; quàm enim facilè, vel impetus op­<lb/>po&longs;itus in mobili producitur, vel corpus durum opponitur &c. </s> </p> <p id="N157DA" type="main"> <s id="N157DC">Tertiò, præcipuam rationem huius de&longs;tructionis ducendam e&longs;&longs;e ex <lb/>Ax.6. in quo dicimus nihil e&longs;&longs;e fru&longs;trà, cumque ordinem à natura e&longs;&longs;e <lb/>in&longs;titutum, vt potiùs aliquid de&longs;truatur, & de&longs;inat e&longs;&longs;e, quàm fru&longs;trà &longs;it, <lb/>& dicimus de&longs;trui ad exigentiam totius naturæ. </s> </p> <p id="N157E5" type="main"> <s id="N157E7"><!-- NEW -->Quartò, cum impetus &longs;uo fine caret, fru&longs;trà e&longs;t; </s> <s id="N157EB"><!-- NEW -->finis impetus e&longs;t mo­<lb/>tus, vt &longs;æpè diximus, &longs;ic cum globus impactus in alium æqualem &longs;tatim <lb/>ab ictu &longs;i&longs;tit immobilis; </s> <s id="N157F3"><!-- NEW -->certe ne fru&longs;trà &longs;it impetus, de&longs;truitur per Ax.6. <lb/>& per Ax. 14. num.2. cum verò determinatio altera maior e&longs;t, certè præ­<lb/>ualet tantùm pro rata; </s> <s id="N157FB"><!-- NEW -->igitur minor e&longs;t motus; </s> <s id="N157FF"><!-- NEW -->igitur, ne aliqui gradus <lb/>impetus &longs;int fru&longs;trà, de&longs;truuntur, cum verò &longs;unt duo impetus in eodem <lb/>mobili, vt in naui mobili ad lineas oppo&longs;itas determinati; </s> <s id="N15807"><!-- NEW -->haud dubiè <lb/>maior impetus præualet pro rata per Ax. 15. Igitur non modò totus <lb/>impetus minor perit, ne &longs;it fru&longs;trà; </s> <s id="N1580F"><!-- NEW -->&longs;ed etiam aliquot gradus maioris, ne <lb/>&longs;int etiam fru&longs;trà; nec enim in communem lineam coïre po&longs;&longs;unt. </s> </p> <p id="N15815" type="main"> <s id="N15817">Denique quando &longs;unt duo impetus ad lineas diuer&longs;as determinati, <lb/>&longs;ed non oppo&longs;itas ex diametro, pugnant pro diuer&longs;o oppo&longs;itionis gradu, <lb/>vt &longs;uprà fusè dictum e&longs;t. </s> <s id="N1581E"><!-- NEW -->Igitur cum totus impetus non habeat totum <lb/>motum, quod duplex illa determinatio impedit, ne aliqui gradus <lb/>&longs;int fru&longs;trà, de&longs;truuntur; </s> <s id="N15826"><!-- NEW -->igitur vides impetum impre&longs;&longs;um ab ex­<lb/>trin&longs;eco de&longs;trui tantùm ne &longs;it fru&longs;trà; faceret enim vt e&longs;&longs;et fru&longs;trà vel <lb/>nouus impetus, vel determinato noua, & in hoc &longs;en&longs;u dicitur impetus <lb/>de&longs;trui ab impetu. </s> </p> <p id="N15830" type="main"> <s id="N15832"><!-- NEW -->Quintò, &longs;i de&longs;trueretur mobile, etiam de&longs;trueretur impetus per idem <lb/>Ax. 6. quia e&longs;&longs;et fru&longs;trà &longs;eparatum; </s> <s id="N15838"><!-- NEW -->immò ex hoc vno principio demon­<lb/>&longs;tramus accidentia & formas &longs;ub&longs;tantiales materiales non po&longs;&longs;e natura­<lb/>liter con&longs;eruari extra &longs;uum &longs;ubiectum, quia &longs;cilicet e&longs;&longs;ent fru&longs;trà; quip­<lb/>pe finem &longs;uum habent in &longs;ubiecto. </s> </p> <p id="N15842" type="main"> <s id="N15844"><!-- NEW -->Sextò, Impetus naturalis innatus nunquam de&longs;truitur; </s> <s id="N15848"><!-- NEW -->quia nunquam <lb/>e&longs;t fru&longs;trà; quippe &longs;emper habet alterum &longs;uorum effectuum formalium, <lb/>id e&longs;t vel motum deor&longs;um, vel grauitationem, adde quod fru&longs;trà de­<lb/>&longs;trueretur, cum &longs;it &longs;emper applicata potentia, id e&longs;t ip&longs;a grauitas, &longs;ed de <lb/>his infrâ fusè. </s> </p> <pb pagenum="71" xlink:href="026/01/103.jpg"/> <p id="N15858" type="main"> <s id="N1585A"><!-- NEW -->Septimò, Impetus &longs;ur&longs;um de&longs;truitur etiam, quia e&longs;t fru&longs;trà; </s> <s id="N1585E"><!-- NEW -->quippe <lb/>naturalis detrahit aliquid &longs;patij pro rata; </s> <s id="N15864"><!-- NEW -->igitur ne aliquid impetus &longs;it <lb/>fru&longs;trà, de&longs;truitur; </s> <s id="N1586A"><!-- NEW -->idem dico de impetu per inclinatam &longs;ur&longs;um, licèt <lb/>minùs de&longs;truatur quàm in perpendiculari &longs;ur&longs;um; </s> <s id="N15870"><!-- NEW -->idem de impetu per <lb/>inclinatam deor&longs;um, &longs;ed minùs adhuc, &longs;ed hæc acuratiori meditationi <lb/>&longs;unt relinquenda; </s> <s id="N15878"><!-- NEW -->quod reuerâ præ&longs;tabimus in lib.4. de motu mixto; </s> <s id="N1587C"><!-- NEW --><lb/>quidquid &longs;it, con&longs;tat ex dictis per idem Principium probari po&longs;&longs;e de­<lb/>&longs;tructionem impetus, &longs;cilicet ne &longs;it fru&longs;trà; &longs;ed de his aliàs fusè. </s> </p> <p id="N15883" type="main"> <s id="N15885"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 153.<emph.end type="center"/></s> </p> <p id="N15891" type="main"> <s id="N15893"><!-- NEW --><emph type="italics"/>Impetus productus ab extrin&longs;eco e&longs;t tantùm contrarius ratione diuer&longs;æ de­<lb/>terminationis, &longs;eu diuer&longs;æ lineæ<emph.end type="italics"/>; </s> <s id="N1589E"><!-- NEW -->Probatur primò, quia vterque ad omnem <lb/>lineam e&longs;t indifferens per Th.113. igitur vnus non e&longs;t alteri contrarius <lb/>ratione entitatis; </s> <s id="N158A6"><!-- NEW -->cùm vterque &longs;imilem motum, immò <expan abbr="eũdem">eundem</expan> habere <lb/>po&longs;&longs;it, vt patet ex dictis: </s> <s id="N158B0"><!-- NEW -->Igitur ratione tantùm lineæ vnus alteri e&longs;t <lb/>contrarius; hinc minùs e&longs;t contrarietatis, quo minùs e&longs;t oppo&longs;itionis <lb/>inter lineas & contrà. </s> </p> <p id="N158B8" type="main"> <s id="N158BA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 154.<emph.end type="center"/></s> </p> <p id="N158C6" type="main"> <s id="N158C8"><emph type="italics"/>Impetus naturalis acqui&longs;itus e&longs;t tantùm contrarius alteri extrin&longs;eco ratio­<lb/>ne lineæ.<emph.end type="italics"/></s> <s id="N158D1"> Probatur eodem modo; quia determinari pote&longs;t ad omnem li­<lb/>neam, vt patet ex reflexione grauis cadentis. </s> </p> <p id="N158D6" type="main"> <s id="N158D8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 155.<emph.end type="center"/></s> </p> <p id="N158E4" type="main"> <s id="N158E6"><!-- NEW --><emph type="italics"/>Impetus naturalis innatus non e&longs;t tantùm contrarius ratione lineæ<emph.end type="italics"/>; quia <lb/>&longs;cilicet non pote&longs;t determinari ad omnem lineam, patet, alioquin cor­<lb/>pus graue, quod &longs;ur&longs;um po&longs;t ca&longs;um reflectitur non de&longs;cenderet amplius, <lb/>de quo aliàs, hæc enim cur&longs;im tantùm per&longs;tringo, ne quid aliis libris <lb/>detrahatur. </s> </p> <p id="N158F7" type="main"> <s id="N158F9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 156.<emph.end type="center"/></s> </p> <p id="N15905" type="main"> <s id="N15907"><!-- NEW --><emph type="italics"/>Impetus ex naturali acqui&longs;ito pote&longs;t fieri violentus<emph.end type="italics"/>; </s> <s id="N15910"><!-- NEW -->vt patet in motu re­<lb/>flexo grauium; ratio e&longs;t. </s> <s id="N15916">quia mutatur linea. </s> </p> <p id="N15919" type="main"> <s id="N1591B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 157.<emph.end type="center"/></s> </p> <p id="N15927" type="main"> <s id="N15929"><!-- NEW --><emph type="italics"/>Impetus ex non contrario eidem fit contrarius<emph.end type="italics"/>; </s> <s id="N15932"><!-- NEW -->vt patet in eodem ca&longs;u; <lb/>nam impetus naturalis innatus, qui in de&longs;cen&longs;u non erat contrarius <lb/>acqui&longs;ito, in motu &longs;ur&longs;um reflexo fit contrarius. </s> </p> <p id="N1593A" type="main"> <s id="N1593C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 158.<emph.end type="center"/></s> </p> <p id="N15948" type="main"> <s id="N1594A"><!-- NEW --><emph type="italics"/>Impetus deor&longs;um ab extrin&longs;eco non e&longs;t contrarius naturali innato ratione <lb/>lineæ,<emph.end type="italics"/> quia &longs;cilicet e&longs;t determinatus ad eandem lineam, &longs;i tamen e&longs;t con­<lb/>trarius, id tantùm e&longs;t ratione propagationis impetus acqui&longs;iti, vel ac <lb/>celerationis motus; quod reuerà multa, & benè longâ explicatione indi­<lb/>get, quam con&longs;ule in lib.4. </s> </p> <p id="N1595B" type="main"> <s id="N1595D"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15969" type="main"> <s id="N1596B"><!-- NEW -->Ob&longs;eruabis cogno&longs;ci tantùm contrarietatem qualitatum ex mutua de­<lb/>&longs;tructione; </s> <s id="N15971"><!-- NEW -->cur verò vna qualitas dicatur de&longs;truere aliam, & cur illam <pb pagenum="72" xlink:href="026/01/104.jpg"/>de&longs;tructionem exigat; </s> <s id="N1597A"><!-- NEW -->maximum my&longs;terium e&longs;t, quod alibi enucleabi­<lb/>mus; quàm multa enim &longs;uper hac re tacuere Philo&longs;ophi! <!-- KEEP S--></s> </p> <p id="N15981" type="main"> <s id="N15983"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 159.<emph.end type="center"/></s> </p> <p id="N1598F" type="main"> <s id="N15991"><!-- NEW --><emph type="italics"/>Impetus &longs;ibi ip&longs;i pote&longs;t reddi contrarius,<emph.end type="italics"/> vt reuerâ accidit in reflexione, <lb/>in qua de&longs;truitur impetus ex parte propter diuer&longs;as determinationes; </s> <s id="N1599C"><!-- NEW --><lb/>cum &longs;cilicet corpus reflectens mouetur; igitur impetus prout determina­<lb/>tus ad lineam incidentiæ e&longs;t aliquo modo &longs;ibi ip&longs;i contrarius, prout e&longs;t <lb/>determinatus ad lineam reflexionis. </s> </p> <p id="N159A5" type="main"> <s id="N159A7">Iam ferè tumultuatim, &longs;i quæ &longs;unt reliqua, Theoremata congeremus. </s> </p> <p id="N159AA" type="main"> <s id="N159AC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 160.<emph.end type="center"/></s> </p> <p id="N159B8" type="main"> <s id="N159BA"><!-- NEW --><emph type="italics"/>Impetus violentus intendi pote&longs;t à naturali, & vici&longs;&longs;im<emph.end type="italics"/>; patet in projectis <lb/>deor&longs;um. </s> </p> <p id="N159C5" type="main"> <s id="N159C7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 161.<emph.end type="center"/></s> </p> <p id="N159D3" type="main"> <s id="N159D5">Idem impetus pote&longs;t <expan abbr="eũdem">eundem</expan> alium aliquando plùs, aliquando minùs <lb/>intendere. </s> <s id="N159DE">v. <!-- REMOVE S-->g. <!-- REMOVE S-->4. gradus impetus additi aliis 4. per <expan abbr="eãdem">eandem</expan> lineam <lb/>iidem ei&longs;dem, minùs intendunt, vt iam &longs;uprà &longs;atis fusè dictum e&longs;t. </s> </p> <p id="N159EB" type="main"> <s id="N159ED"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 162.<emph.end type="center"/></s> </p> <p id="N159F9" type="main"> <s id="N159FB"><emph type="italics"/>Impetus dici pote&longs;t propriè de&longs;trui ad exigentiam totius naturæ<emph.end type="italics"/> per Ax.14. <lb/>num.2. vt con&longs;tat ex multis Theorematis &longs;uperioribus. </s> </p> <p id="N15A05" type="main"> <s id="N15A07"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 163.<emph.end type="center"/></s> </p> <p id="N15A13" type="main"> <s id="N15A15"><!-- NEW --><emph type="italics"/>Omnis dici debet incipere, & de&longs;inere intrin&longs;ecè, & extrin&longs;ecè<emph.end type="italics"/>; quod enim <lb/>hoc in&longs;tanti primo e&longs;t, immediatè antecedenti vltimo non fuit, & quod <lb/>primo non e&longs;t hoc in&longs;tanti, immediatè antè vltimo fuit, nec pote&longs;t e&longs;&longs;e <lb/>immediatè pò&longs;t, ni&longs;i &longs;it immediatè antè, & vici&longs;&longs;im. </s> </p> <p id="N15A24" type="main"> <s id="N15A26"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 164.<emph.end type="center"/></s> </p> <p id="N15A32" type="main"> <s id="N15A34"><!-- NEW --><emph type="italics"/>Ideo producitur hic impetus numero potiùs, quàm alius omninò &longs;imilis<emph.end type="italics"/>; </s> <s id="N15A3D"><!-- NEW -->quia <lb/>potentia motrix e&longs;t determinata ad tale indiuiduum &longs;iue à &longs;e, &longs;iue ab <lb/>alio; </s> <s id="N15A45"><!-- NEW -->idem enim de illa dicendum e&longs;t, quod de aliis cau&longs;is naturalibus; <lb/>porrò idem dici debet de de&longs;tructione, quod de productione. </s> </p> <p id="N15A4B" type="main"> <s id="N15A4D"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15A59" type="main"> <s id="N15A5B">Ob&longs;eruabis breuiter aliqua, quæ fortè in no&longs;tris Theorematis fuere <lb/>omi&longs;&longs;a. </s> </p> <p id="N15A60" type="main"> <s id="N15A62"><!-- NEW -->Primò qualitates, quæ à cau&longs;a primò productiua con&longs;eruantur, ab ea <lb/>intendi non po&longs;&longs;e; </s> <s id="N15A68"><!-- NEW -->quia &longs;ingulis in&longs;tantibus nouum effectum non pro­<lb/>ducit; </s> <s id="N15A6E"><!-- NEW -->exemplum habes in luce; &longs;ecus vero de iis dicendum e&longs;t, quæ à <lb/>cau&longs;a primò productiua non con&longs;eruantur. </s> </p> <p id="N15A74" type="main"> <s id="N15A76"><!-- NEW -->Secundò qualitates, quæ contrarias habent, etiam de&longs;trui po&longs;&longs;e ab <lb/>alio, quam ab iis, &longs;cilicet ad exigentiam totius naturæ; ne &longs;cilicet &longs;int <lb/>fru&longs;trà. </s> </p> <p id="N15A7E" type="main"> <s id="N15A80">Tertiò aliqua carere contrario, non tamen con&longs;eruari à cau&longs;a primò <lb/>productiua. </s> <s id="N15A85">v.g. <!-- REMOVE S-->anima bruti, quæ de&longs;truitur ad exigentiam totius natu­<lb/>ræ, nç &longs;it fru&longs;trà. </s> </p> <pb pagenum="73" xlink:href="026/01/105.jpg"/> <p id="N15A90" type="main"> <s id="N15A92"><!-- NEW -->Quartò, impetum inten&longs;iorem in projectis diutiùs durare; </s> <s id="N15A96"><!-- NEW -->quia cum <lb/>&longs;en&longs;im de&longs;truatur; certè plures partes maiori tempore de&longs;truuntur, quàm <lb/>pauciores. </s> </p> <p id="N15A9E" type="main"> <s id="N15AA0"><!-- NEW -->Quintò, &longs;i totus impetus de&longs;trueretur vno in&longs;tanti, minima re&longs;i&longs;tentia <lb/>&longs;ufficeret ad motum impediendum: adde quod contraria pugnant pro <lb/>rata per Ax.15. </s> </p> <p id="N15AA8" type="main"> <s id="N15AAA">Sextò, ob&longs;eruabis plurima in hoc libro qua&longs;i obiter e&longs;&longs;e indicata, quæ <lb/>in aliis fusè explicata maiorem lucem accipient. </s> </p> <p id="N15AAF" type="main"> <s id="N15AB1"><!-- NEW -->Septimò, denique totam rem i&longs;tam, quæ pertinet ad impetum paulò <lb/>fu&longs;ius pertractatam in hoc primo libro; </s> <s id="N15AB7"><!-- NEW -->quòd &longs;cilicet ab ea reliqua ferè <lb/>omnia pendeant, quæ in hoc tractatu habentur; &longs;ed de his &longs;atis. <lb/><figure id="id.026.01.105.1.jpg" xlink:href="026/01/105/1.jpg"/></s> </p> </chap> <chap id="N15AC3"> <pb pagenum="74" xlink:href="026/01/106.jpg"/> <figure id="id.026.01.106.1.jpg" xlink:href="026/01/106/1.jpg"/> <p id="N15ACD" type="head"> <s id="N15ACF"><emph type="center"/>LIBER SECVNDVS, <lb/><emph type="italics"/>DE MOTV NATVRALI.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N15ADC" type="main"> <s id="N15ADE"><!-- NEW -->MOtus localis naturalis latè &longs;umptus e&longs;t, <lb/>qui ab aliqua causâ naturali ponitur; </s> <s id="N15AE4"><!-- NEW --><lb/>&longs;trictè verò &longs;umitur pro motu grauium <lb/>deor&longs;um, à principio intrin&longs;eco &longs;altem <lb/>&longs;en&longs;ibiliter; </s> <s id="N15AED"><!-- NEW -->In hoc vltimo &longs;en&longs;u mo­<lb/>tum naturalem v&longs;urpabo; &longs;it ergo. <lb/><gap desc="hr tag"/></s> </p> <p id="N15AF6" type="main"> <s id="N15AF8"><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15B04" type="main"> <s id="N15B06"><!-- NEW --><emph type="italics"/>MOtus localis naturalis e&longs;t, qui e&longs;t à grauitate deor&longs;um.<emph.end type="italics"/> hæc defini­<lb/>tio vix aliqua explicatione indiget; dicitur e&longs;&longs;e à grauitate, <lb/>quidquid &longs;it grauitas, &longs;iue qualitas di&longs;tincta, &longs;iue non. </s> </p> <p id="N15B13" type="main"> <s id="N15B15"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15B22" type="main"> <s id="N15B24"><emph type="italics"/>Motus æquabilis e&longs;t, quo æqualibus quibu&longs;cumque temporibus æqualia per­<lb/>curruntur &longs;patia ab eodem mobili.<emph.end type="italics"/></s> </p> <p id="N15B2D" type="main"> <s id="N15B2F"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15B3C" type="main"> <s id="N15B3E"><!-- NEW --><emph type="italics"/>Motus naturaliter acceleratus e&longs;t, quo &longs;ecundo tempore æquali primo ma­<lb/>ius &longs;patium acquiritur, & tertio, quàm &longs;ecundo, & quarto quàm tertio, atque <lb/>ita deinceps; nulla &longs;cilicet addita vi ab extrin&longs;eco &longs;altem &longs;en&longs;ibiliter.<emph.end type="italics"/></s> </p> <p id="N15B4A" type="main"> <s id="N15B4C"><!-- NEW -->Definit aliter hunc motum Galileus; </s> <s id="N15B50"><!-- NEW -->dicit enim eum e&longs;&longs;e, qui æquali­<lb/>bus temporibus æqualia acquirit velocitatis momenta; </s> <s id="N15B56"><!-- NEW -->&longs;ed profectò non <lb/>conuenit hæc definitio omni motui naturaliter accelerato, v. <!-- REMOVE S-->g. <!-- REMOVE S-->motui <lb/>de&longs;cen&longs;us funependuli, vel in orbe cauo, vel etiam in plano decliui ma­<lb/>ximæ longitudinis; definitio no&longs;tra clarior e&longs;t. </s> </p> <p id="N15B64" type="main"> <s id="N15B66"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15B73" type="main"> <s id="N15B75"><!-- NEW --><emph type="italics"/>Corpus graue cadit deor&longs;um, & cadens ex maiori altitudine maiorem ictum <lb/>infligit quam &longs;i caderet ex minore<emph.end type="italics"/>; &longs;i quis hoc neget hoc probet, patet ma­<lb/>nife&longs;ta experientia. </s> </p> <pb pagenum="75" xlink:href="026/01/107.jpg"/> <p id="N15B86" type="main"> <s id="N15B88"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15B95" type="main"> <s id="N15B97"><!-- NEW --><emph type="italics"/>Arcus maior & minor eiu&longs;dem funependuli æqualibus ferè temporibus, <lb/>percurruntur<emph.end type="italics"/>; hæc etiam &longs;æpiùs probata e&longs;t, & &longs;i quis fidem detrectat, <lb/>probare conetur. </s> </p> <p id="N15BA4" type="main"> <s id="N15BA6"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15BB3" type="main"> <s id="N15BB5"><!-- NEW -->Globus per planum inclinatum læuigatum de&longs;cendens &longs;ecundum &longs;pa­<lb/>tium citiùs percurrit, quàm primum; quod etiam &longs;en&longs;u percipi pote&longs;t, <lb/>& tam &longs;æpè probatum e&longs;t, vt nemo iam negare audeat motus naturalis <lb/>accelerationem. </s> </p> <p id="N15BBF" type="main"> <s id="N15BC1"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15BCE" type="main"> <s id="N15BD0"><!-- NEW --><emph type="italics"/>Omne tempus &longs;en&longs;ibile non e&longs;t; </s> <s id="N15BD6"><!-- NEW -->idem dico de &longs;patio,<emph.end type="italics"/> quod nemo etiam <lb/>negare au&longs;it; alioquin &longs;i quis negaret, dicat mihi quæ&longs;o quot &longs;int in mi­<lb/>nuto horæ in&longs;tantia? </s> <s id="N15BE1">quot in apice acus puncta? </s> </p> <p id="N15BE4" type="main"> <s id="N15BE6"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15BF3" type="main"> <s id="N15BF5"><!-- NEW --><emph type="italics"/>Impetus additus alteri, & determinatus ad <expan abbr="eãdem">eandem</expan> lineam, facit maiorem <lb/>& inten&longs;iorem impetum<emph.end type="italics"/>; patet, & vici&longs;&longs;im, & detractus alteri minorem <lb/>facit, & vici&longs;&longs;im. </s> </p> <p id="N15C06" type="main"> <s id="N15C08"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15C15" type="main"> <s id="N15C17"><emph type="italics"/>Quâ proportione cre&longs;cit cau&longs;a, eâdem cre&longs;cit effectus, & vici&longs;&longs;im, &longs;i eodem <lb/>modo eidemque &longs;ubjecto &longs;it applicata,<emph.end type="italics"/> probatur per Ax.12. l. <!-- REMOVE S-->1. & quâ pro­<lb/>portione illa decre&longs;cit, hic decre&longs;cit, & vici&longs;&longs;im. </s> </p> <p id="N15C25" type="main"> <s id="N15C27"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15C34" type="main"> <s id="N15C36"><emph type="italics"/>Eadem cau&longs;a nece&longs;&longs;aria non impedita &longs;ubjecto apte applicata æqualibus <lb/>temporibus æqualem effectum producit, & contrà.<emph.end type="italics"/></s> <s id="N15C3F"> Probatur per Ax.12.l. </s> <s id="N15C42">1. & <lb/>vici&longs;&longs;im æqualis effectus &longs;upponit æqualem cau&longs;am. </s> </p> <p id="N15C47" type="main"> <s id="N15C49"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15C56" type="main"> <s id="N15C58"><!-- NEW --><emph type="italics"/>Ille effectus, qui non producitur à causâ primâ, & ad cuius productionem <lb/>nulla cau&longs;a extrin&longs;eca e&longs;t applicata, producitur ab intrin&longs;eco<emph.end type="italics"/>; probatur, quia <lb/>habere debet aliquam cau&longs;am per Ax.8. <!-- KEEP S--></s> </p> <p id="N15C66" type="main"> <s id="N15C68"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15C75" type="main"> <s id="N15C77"><!-- NEW --><emph type="italics"/>Illa cau&longs;a plus agit proportionaliter quæ habet minorem re&longs;istentiam; minùs <lb/>verò, quæ maiorem, quæ demum æqualem, æquali proportione agit.<emph.end type="italics"/> v.g. <!-- REMOVE S-->cau&longs;a, <lb/>cuius virtus, vel actiuitas e&longs;t vt 20. & re&longs;i&longs;tentia vt 10. agit in maiori <lb/>proportione, quàm illa cuius actiuitas e&longs;t 30. & re&longs;i&longs;tentia 20. in minori <lb/>verò quàm ea, cuius actiuitas e&longs;t vt 3. & re&longs;i&longs;tentia vt 1. in æquali de­<lb/>nique cum illa, cuius actiuitas e&longs;t vt 4. & re&longs;i&longs;tentia vt 2. <!-- KEEP S--></s> </p> <p id="N15C8D" type="main"> <s id="N15C8F"><!-- NEW -->Hoc Axioma certi&longs;&longs;imum e&longs;t; </s> <s id="N15C93"><!-- NEW -->quippe 20. faciliùs &longs;uperabunt 10. quàm <lb/>30. 20. & difficiliùs quam 3. 1. & æquè facilè, ac 4. 2. In motu locali <lb/>res e&longs;t clari&longs;&longs;ima; </s> <s id="N15C9B"><!-- NEW -->quippe vires vt 12. tam facilè mouebunt 12. libras, <lb/>quàm vires vt 4. 4.libras; </s> <s id="N15CA1"><!-- NEW -->&longs;ed faciliùs, quàm vires vt 20. 30.libras, & dif­<lb/>ficiliùs quàm vires vt 4. 3. libras; quid clarius? </s> <s id="N15CA7">Igitur illa cau&longs;a faciliùs <pb pagenum="76" xlink:href="026/01/108.jpg"/>&longs;uperat re&longs;i&longs;tentiam impedimenti, quæ habet maiorem proportionem <lb/>virium cum re&longs;i&longs;tentia, quàm quæ minorem. </s> </p> <p id="N15CB1" type="main"> <s id="N15CB3"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15CBF" type="main"> <s id="N15CC1">Si quando appellandum erit aliquod Axioma vel Theorema lib. 1.ci­<lb/>tabitur Liber. </s> </p> <p id="N15CC6" type="main"> <s id="N15CC8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15CD5" type="main"> <s id="N15CD7"><emph type="italics"/>Datur motus localis naturalis, i&longs;que ab intrin&longs;eco.<emph.end type="italics"/></s> <s id="N15CDE"> Probatur; corpus gra­<lb/>ue mouetur localiter deor&longs;um per hypoth. </s> <s id="N15CE3"><!-- NEW -->hic motus e&longs;t ab intrin&longs;eco, <lb/>quod probatur; </s> <s id="N15CE9"><!-- NEW -->non e&longs;t ab vllâ causâ extrin&longs;ecâ; igitur e&longs;t ab intrin&longs;eca <lb/>per Ax.4. antecedens probatur inductione factâ omnium extrin&longs;ecorum. </s> <s id="N15CEF"><!-- NEW --><lb/>Primò non e&longs;t à cau&longs;a prima, vt aliquis fortè minùs prudenter, & magis <lb/>piè, quàm par &longs;it, diceret; </s> <s id="N15CF6"><!-- NEW -->quia ille effectus tribui tantùm debet cau&longs;æ <lb/>primæ, qui nullam habere pote&longs;t cau&longs;am &longs;ecundam applicatam, vt patet; </s> <s id="N15CFC"><!-- NEW --><lb/>&longs;ed hic effectus pote&longs;t habere cau&longs;am &longs;ecundam applicatam, quam a&longs;&longs;i­<lb/>gnabimus infrà; </s> <s id="N15D03"><!-- NEW -->deinde cau&longs;a prima agit tantùm naturaliter iuxta exi­<lb/>gentiam cau&longs;arum &longs;ecundarum; </s> <s id="N15D09"><!-- NEW -->igitur ideo moueret corpus graue deor­<lb/>&longs;um; </s> <s id="N15D0F"><!-- NEW -->quia tunc motum corpus graue exigeret; </s> <s id="N15D13"><!-- NEW -->&longs;ed hoc mihi &longs;ufficit, vt <lb/>dicatur hic motus e&longs;&longs;e ab intrin&longs;eco; </s> <s id="N15D19"><!-- NEW -->præterea, &longs;i dicatur Deus mouere <lb/>corpus graue deor&longs;um iuxta illius exigentiam, dicetur etiam tùm cale­<lb/>facere, tùm illuminare, ad exigentiam ignis; </s> <s id="N15D21"><!-- NEW -->quippe tàm mihi &longs;en&longs;ibile <lb/>e&longs;t corpus graue de&longs;cendere &longs;ine vi impre&longs;&longs;a ab extrin&longs;eco, quàm ignem <lb/>calefacere, & &longs;olem lucere &longs;ine vi extrin&longs;eca; </s> <s id="N15D29"><!-- NEW -->adde quod illud &longs;olenne <lb/>e&longs;t naturæ in&longs;titutum, vt id, quod exigit res aliqua ad finem &longs;uum con&longs;e­<lb/>quendum, per virtutem intrin&longs;ecam po&longs;&longs;it ponere, &longs;i dumtaxat excipias <lb/>concur&longs;um diuinum, & ip&longs;am con&longs;eruationem; </s> <s id="N15D33"><!-- NEW -->&longs;ic animal exigit vide­<lb/>re, audire, &longs;entire, moueri; </s> <s id="N15D39"><!-- NEW -->igitur habet virtutem intrin&longs;ecam, per quam <lb/>videat, audiat, & moueatur; </s> <s id="N15D3F"><!-- NEW -->&longs;ic ignis exigit calefacere, lucere; </s> <s id="N15D43"><!-- NEW -->aër, vel aqua <lb/>frigefacere, quidquid tandem &longs;int i&longs;tæ qualitates, de quibus alibi; </s> <s id="N15D49"><!-- NEW -->&longs;ic <lb/>demum corpus graue exigit moueri deor&longs;um; quis enim neget corpori <lb/>graui tàm natiuum e&longs;&longs;e tendere deor&longs;um, cum &longs;cilicet corpus leuius &longs;ub­<lb/>e&longs;t, quàm &longs;it animali progredi, vrere igni, lucere, &c. </s> </p> <p id="N15D53" type="main"> <s id="N15D55"><!-- NEW -->Denique &longs;atis e&longs;t mihi, vt dicatur aliquid cau&longs;a, Phy&longs;icè loquendo, &longs;i <lb/>ex illius applicatione &longs;emper &longs;equatur effectus; </s> <s id="N15D5B"><!-- NEW -->nam non nego po&longs;&longs;e fie­<lb/>ri effectus omnes, qui no&longs;tris &longs;en&longs;ibus &longs;ubiiciuntur, &longs;altem extrin&longs;ecos, <lb/>e&longs;&longs;e à cau&longs;a prima, quippe &longs;i &longs;emper ex ignis applicatione Deus diffun­<lb/>deret lucem, & calorem, quem &longs;olus ip&longs;e produceret, igne ip&longs;o inerte re­<lb/>licto, nullam pror&longs;us mutationem perciperemus; </s> <s id="N15D67"><!-- NEW -->& nemo e&longs;&longs;et, qui non <lb/>exi&longs;timaret lucem hanc & calorem hunc e&longs;&longs;e ab igne; </s> <s id="N15D6D"><!-- NEW -->igitur Phy&longs;icè lo­<lb/>quendo cau&longs;am appellamus id, ex cuius applicatione &longs;emper &longs;equitur <lb/>effectus, vt iam diximus in Ax. 11.l.1. n.1. Igitur cum ex corpore graui <lb/>po&longs;ito in aëre libero &longs;equatur motus deor&longs;um; dicendum e&longs;t, Phy&longs;icè lo­<lb/>quendò, e&longs;&longs;e huius motus cau&longs;am, id e&longs;t in ordine ad Phy&longs;icam, perinde <lb/>omninò &longs;e habere, atque &longs;i e&longs;&longs;et cau&longs;a, licèt cau&longs;a non e&longs;&longs;et. </s> </p> <pb pagenum="77" xlink:href="026/01/109.jpg"/> <p id="N15D7F" type="main"> <s id="N15D81"><!-- NEW -->Secundò hic motus non e&longs;t ab aëre ambiente; </s> <s id="N15D85"><!-- NEW -->probatur, ruderet aër <lb/>deor&longs;um corpus graue, quia leuior e&longs;t, id e&longs;t ne &longs;uprà &longs;e corpus grauius <lb/>haberet; </s> <s id="N15D8D"><!-- NEW -->&longs;ed eâdem ratione corpus graue debet remouere &longs;ur&longs;um aëra, <lb/>id e&longs;t corpus leue, ne infrà &longs;e habeat corpus leuius; </s> <s id="N15D93"><!-- NEW -->e&longs;t enim par omni­<lb/>nò ratio: </s> <s id="N15D99"><!-- NEW -->Præterea &longs;i aër trudit deor&longs;inn corpus graue, quia ip&longs;i loco <lb/>cedit; </s> <s id="N15D9F"><!-- NEW -->certè ip&longs;e aër mouetur, igitur ab intrin&longs;eco; </s> <s id="N15DA3"><!-- NEW -->&longs;i enim vna pars aë­<lb/>ris pellit aliam, & hæc aliam, tandem ad aliquam peruenitur, quæ &longs;e ip­<lb/>&longs;am mouet; </s> <s id="N15DAB"><!-- NEW -->igitur motus illius e&longs;t ab intrin&longs;eco; </s> <s id="N15DAF"><!-- NEW -->igitur motus natura­<lb/>lis; </s> <s id="N15DB5"><!-- NEW -->deinde non modò lapis de&longs;cendit per aëra, &longs;ed per mediam aquam; </s> <s id="N15DB9"><!-- NEW --><lb/>igitur &longs;i ab aëre truditur deor&longs;um, idem dicendum e&longs;t de aquâ, a qui <lb/>haud dubiè maiore vi truderetur; </s> <s id="N15DC0"><!-- NEW -->nam corpus den&longs;um maiore vi pellit, <lb/>quàm rarum, vt con&longs;tat exprientiâ; </s> <s id="N15DC6"><!-- NEW -->cum tamen corpus graue per me­<lb/>dium den&longs;ius difficiliùs decendat; igitur medium ip&longs;um re&longs;i&longs;tit motui, <lb/>quis hoc neget? </s> <s id="N15DCE">igitur non e&longs;t cau&longs;a motus, quem impedit. </s> </p> <p id="N15DD1" type="main"> <s id="N15DD3"><!-- NEW -->Denique &longs;i corpus graue non tendit, fertur que deor&longs;um &longs;uá &longs;ponte, <lb/>&longs;ed ab aëre extru&longs;um; </s> <s id="N15DD9"><!-- NEW -->igitur dum vix &longs;u&longs;tineo manu; o. </s> <s id="N15DDD">libras ferri, &longs;eu <lb/>plumbi; </s> <s id="N15DE2"><!-- NEW -->hæc vis illata manui, quam probè &longs;entio, e&longs;t ab aëre impel­<lb/>lente plumbum, quod e&longs;t ridiculum, cum eadem quantitas aëris incu­<lb/>ber, & &longs;ub&longs;it manui, &longs;iue &longs;u&longs;tineat plumbum, &longs;iue &longs;it vacua; ex hoc, ni <lb/>fallor, euincitur pondus ip&longs;um &longs;ui &longs;ponte deor&longs;um tendere. </s> </p> <p id="N15DEC" type="main"> <s id="N15DEE"><!-- NEW -->Tertiò non de&longs;unt, qui dicant corpus graue trahi ab ip&longs;a vi quadam <lb/>magneticâ, quod triplici modo fieri pote&longs;t; </s> <s id="N15DF4"><!-- NEW -->Primò per qualitatem <lb/>quamdam diffu&longs;am, quod dici non pote&longs;t; </s> <s id="N15DFA"><!-- NEW -->quia capillus traheretur faci­<lb/>liùs, quàm ingens &longs;axum, quàm ma&longs;&longs;a, &longs;eu lamina; </s> <s id="N15E00"><!-- NEW -->& faciliùs eadem po­<lb/>tentia motrix minus pondus moueret quàm maius, cæteris paribus; </s> <s id="N15E06"><!-- NEW -->præ­<lb/>terea manum meam æqualiter traheret, &longs;iue &longs;it cum aliquo pondere con­<lb/>iuncta, &longs;iue &longs;it nuda &longs;ine pondere; </s> <s id="N15E0E"><!-- NEW -->deinde illa virtus tractrix ita diffun­<lb/>ditur, vt in maiori di&longs;tantia &longs;it infirmior, fortior in minori; </s> <s id="N15E14"><!-- NEW -->alioqui <lb/>diffunderetur in infinitum, quod dici non pote&longs;t; </s> <s id="N15E1A"><!-- NEW -->igitur &longs;i idem lapis <lb/>demittatur ex maiore altitudine, tum ex minore; </s> <s id="N15E20"><!-- NEW -->haud dubiè morus ille <lb/>primus initio e&longs;&longs;et tardior i&longs;to contra experientiam; </s> <s id="N15E26"><!-- NEW -->deinde in &longs;pecu al­<lb/>ti&longs;&longs;ima &longs;ubterranea trahi po&longs;&longs;et corpus vndequaque, &longs;icut in magnete; </s> <s id="N15E2C"><!-- NEW --><lb/>quæ omnia intelligi non po&longs;&longs;unt; denique virtutes illas &longs;eu qualitates <lb/>tractrices refellemus &longs;uo loco. </s> </p> <p id="N15E33" type="main"> <s id="N15E35"><!-- NEW -->Secundò, aliqui dicunt hoc totum fieri per vim quamdam &longs;ympathi­<lb/>cam, quod etiam fal&longs;i&longs;&longs;imum e&longs;t; </s> <s id="N15E3B"><!-- NEW -->tùm quia hæc &longs;ympathia explicari <lb/>non pote&longs;t; </s> <s id="N15E41"><!-- NEW -->tùm quia vel terra ip&longs;a producit aliquid in corpore graui, <lb/>quod in aëre libratur; </s> <s id="N15E47"><!-- NEW -->vel corpus in &longs;e ip&longs;o; &longs;i primum; </s> <s id="N15E4B"><!-- NEW -->refellitur ii&longs;­<lb/>dem omninò rationibus, quibus ip&longs;am vim terræ tractricem &longs;uprà expu­<lb/>gnauimus; &longs;i verò &longs;ecundum, hoc ip&longs;um e&longs;t, quod &longs;uprà diximus. </s> </p> <p id="N15E53" type="main"> <s id="N15E55"><!-- NEW -->Tertiò, Dixere aliqui &longs;ubtiliùs profectò quàm veriùs, corpus graue <lb/>trahi deor&longs;um, non vi quadam occultâ, vt &longs;uprà dictum e&longs;t; </s> <s id="N15E5B"><!-- NEW -->&longs;ed filamen­<lb/>tis quibu&longs;dam, &longs;eu ductili terræ profluuio, quod illius capillitium vo­<lb/>cant; </s> <s id="N15E63"><!-- NEW -->idque tantùm fieri probant ducta ab electro analogiâ, quod pa­<lb/>leam & minutiora corpu&longs;cula hac eâdem arte trahit; </s> <s id="N15E69"><!-- NEW -->&longs;ed profectò gra-<pb pagenum="78" xlink:href="026/01/110.jpg"/>uiores &longs;unt difficultates, quam vt illis fieri &longs;atis queat; </s> <s id="N15E72"><!-- NEW -->nam primò cor­<lb/>pus leuius ab his filamentis abripi faciliùs po&longs;&longs;et, vt con&longs;tat in electro; <lb/>igitur citiùs de&longs;cenderet. </s> </p> <p id="N15E7A" type="main"> <s id="N15E7C">Secundò, corpus vicinius etiam faciliùs abriperetur. </s> </p> <p id="N15E7F" type="main"> <s id="N15E81">Tertiò, numquid flante vento, vel imbre cadente di&longs;&longs;ipantur hæc fi­<lb/>lamenta? </s> <s id="N15E86">quod etiam videmus in electro. </s> </p> <p id="N15E89" type="main"> <s id="N15E8B">Quartò, manum meam æquè facilè traheret terra his funiculis &longs;eu <lb/>pondere grauatam, &longs;eu vacuam. </s> </p> <p id="N15E90" type="main"> <s id="N15E92"><!-- NEW -->Quintò, quemadmodum electrum ex omni parte trahit, ita terra ip&longs;a <lb/>per omnem lineam traheret; immò etiam &longs;ur&longs;um in &longs;ubterranea &longs;pecu, <lb/>quod e&longs;t ab&longs;urdum. </s> </p> <p id="N15E9A" type="main"> <s id="N15E9C"><!-- NEW -->Sextò, hæc filamenta, quæ deinde reducuntur, debent habere cau­<lb/>&longs;am huius reductionis non extrin&longs;ecam; </s> <s id="N15EA2"><!-- NEW -->igitur intrin&longs;ecam; igitur datur <lb/>motus naturalis. </s> </p> <p id="N15EA8" type="main"> <s id="N15EAA">Septimò, hæc filamenta per mediam flammam non traherent, quod <lb/>etiam fieri videmus in electro. </s> </p> <p id="N15EAF" type="main"> <s id="N15EB1"><!-- NEW -->Quartò, motus naturalis non e&longs;t à virtute quadam pellente, quam <lb/>cælo quidam affingunt; </s> <s id="N15EB7"><!-- NEW -->nam vel ab omni parte cæli deor&longs;um trudere­<lb/>tur, vel ab vnâ; &longs;i ab vna; </s> <s id="N15EBD"><!-- NEW -->igitur in omni cæli plaga corpus non fertur <lb/>deor&longs;um; </s> <s id="N15EC3"><!-- NEW -->&longs;i ab omni, ergo cum pellatur corpus per plures lineas etiam <lb/>oppo&longs;itas moueri non pote&longs;t: </s> <s id="N15EC9"><!-- NEW -->Præterea debilior e&longs;&longs;et hæc vis in maiori <lb/>di&longs;tantiâ; denique vapores, & alia minutiora corpu&longs;cula in aëre fluitan­<lb/>tia faciliùs deor&longs;um truderentur, contra experientiam. </s> </p> <p id="N15ED1" type="main"> <s id="N15ED3"><!-- NEW -->Sed non e&longs;t omittendum, quod aliqui putant ex illis filamentis con­<lb/>texi po&longs;&longs;e legitimam rationem, cur atomi etiam plumbeæ materiæ non <lb/>ita facilè de&longs;cendant; </s> <s id="N15EDB"><!-- NEW -->quòd &longs;cilicet propter &longs;uam tenuitatem ab illis fi­<lb/>lamentis non ita intercipi vel implicari po&longs;&longs;int; </s> <s id="N15EE1"><!-- NEW -->&longs;ed qua&longs;i pi&longs;ces per fo­<lb/>ramina retium euadant; &longs;ed profectò longè alia ratio e&longs;t, quàm &longs;uo loco <lb/>afferemus, nam etiam plumæ, fe&longs;tucæ, paleæ, & alia corpu&longs;cula longio­<lb/>ra, &longs;ed leui&longs;&longs;ima iis filamentis implicarentur, vt videre e&longs;t in electro. </s> </p> <p id="N15EEB" type="main"> <s id="N15EED"><!-- NEW -->Quintò, aliqui recentiores exi&longs;timant corpora deor&longs;um trudi ab <lb/>ip&longs;a luce, quæ nihil e&longs;t aliud, quam motio æthereæ cuiu&longs;dam &longs;ub&longs;tan­<lb/>tiæ per poros aëris traductæ, vt ip&longs;i volunt; &longs;ed neque hoc probari po­<lb/>te&longs;t. </s> <s id="N15EF7"><!-- NEW -->Primò quia de nocte corpora æquali motu deor&longs;um feruntur; pe­<lb/>rinde atque de die, nec minùs in ob&longs;curi&longs;&longs;imo conclaui, quàm &longs;ub dio, <lb/>vel aperto cælo. </s> <s id="N15EFF"><!-- NEW -->Secundò, in &longs;ubterraneis locis etiam grauia æquè veloci­<lb/>ter de&longs;cendunt; </s> <s id="N15F05"><!-- NEW -->licèt eò lumen non penetret; </s> <s id="N15F09"><!-- NEW -->quod &longs;i aliquis ob&longs;tinatè, <lb/>id a&longs;&longs;ereret; </s> <s id="N15F0F"><!-- NEW -->haud dubiè per medium aëra maior huius materiæ copia <lb/>diffunditur, quàm per medias rupes, quis hoc neget; igitur pauci&longs;&longs;imi <lb/>radij v&longs;que ad interius & inferius antrum perueniunt. </s> <s id="N15F17"><!-- NEW -->Tertiò, manum <lb/>meam &longs;iue ponderi coniunctam &longs;iue ab eo <expan abbr="&longs;eparatã">&longs;eparatam</expan> æqualis portio illius <lb/>materiæ deor&longs;um pelleret, vt patet; igitur æquali motus vi. </s> <s id="N15F23">Quartò, cor­<lb/>pus diaphanum, per cuius poros facilè traiicitur hæc materia, e&longs;&longs;et leuius <lb/>alio quod tamen fal&longs;um e&longs;t, vt videre e&longs;t in vitro, cry&longs;tallo, adamante, <lb/>glacie. </s> <s id="N15F2C"><!-- NEW -->Quintò maxima huius materiæ copia collecta &longs;eu &longs;peculi opera <pb pagenum="79" xlink:href="026/01/111.jpg"/>&longs;eu vitri, maiore vi corpora deor&longs;um truderet; </s> <s id="N15F35"><!-- NEW -->quia maior cau&longs;a maio­<lb/>rem effectum producit per Ax.2. Sextò po&longs;t refractionem lineam mutat <lb/>radius luminis; igitur deor&longs;um rectà non pelleret. </s> <s id="N15F3D"><!-- NEW -->Septimò radij traie­<lb/>cti per vitrum maiore vi deor&longs;um pellerent quàm per lignum, vel &longs;pon­<lb/>giam; quippè per hæc corpora traiecti &longs;ecundum authores huius &longs;enten­<lb/>tiæ di&longs;trahuntur propter obliquitatem pororum. </s> <s id="N15F47"><!-- NEW -->Octauò denique radij <lb/>profecti à Sole iuxta ortum, vel occa&longs;um &longs;unt valdè obliqui; igitur non <lb/>truderent deor&longs;um rectà. </s> </p> <p id="N15F4F" type="main"> <s id="N15F51"><!-- NEW -->Nec e&longs;t quod prædicti àuthores confugiant ad experientiam, qua <lb/>&longs;cilicet videmus tripudiantes atomos in radio &longs;olari immer&longs;as; igitur <lb/>agitantur ab ip&longs;o radio, quod maximè accidit in linea v&longs;toria, cuius <lb/>effectus veri&longs;&longs;imam rationem &longs;uo loco afferemus, cum de lumine. </s> </p> <p id="N15F5B" type="main"> <s id="N15F5D"><!-- NEW -->Sextò, &longs;unt denique multi, <expan abbr="ii&qacute;ue">iique</expan> ex &longs;euerioribus Peripateticis, qui <lb/>exi&longs;timant grauia moueri deor&longs;um à generante, quod expre&longs;&longs;is verbis <lb/>traditum e&longs;t ab <emph type="italics"/>Ari&longs;totele l.<emph.end type="italics"/>8. <emph type="italics"/>phy&longs;. cap.<emph.end type="italics"/>4. <emph type="italics"/>iuxta<emph.end type="italics"/> principium illud vniuer­<lb/>&longs;ali&longs;&longs;imum; <emph type="italics"/>Quidquid mouetur; </s> <s id="N15F80"><!-- NEW -->ab alio mouetur<emph.end type="italics"/>; </s> <s id="N15F87"><!-- NEW -->&longs;ed profectò ij ip&longs;i, qui <lb/>motum grauium generanti tribuunt, tanquam principi cau&longs;æ, non ne­<lb/>gant ine&longs;&longs;e grauibus grauitatem, quæ &longs;it principium actiuum minus <lb/>principale motus; </s> <s id="N15F91"><!-- NEW -->ad quem etiam, vt ip&longs;i exi&longs;timant, forma &longs;ub&longs;tantialis <lb/>concurrit; </s> <s id="N15F97"><!-- NEW -->In hoc quippe conueniunt omnes tùm &longs;ectarum Principes, <lb/>tùm recentiores: </s> <s id="N15F9D"><!-- NEW -->quidquid &longs;it etiam ex iis ip&longs;is datur motus naturalis, <lb/>qui e&longs;t à virtute proxima intrin&longs;eca; hoc ip&longs;um etiam &longs;en&longs;it Ari&longs;toteles <lb/>lib.4. de cælo cap. 3. t. </s> <s id="N15FA5"><!-- NEW -->25. vbi ait grauibus & leuibus ine&longs;&longs;e principium <lb/>actiuum &longs;uorum motuum; </s> <s id="N15FAB"><!-- NEW -->immò &longs;i totum cap.4. l.8. phy&longs;. attentè lega­<lb/>tur, vbi dicit moueri à generante, haud dubiè intelligetur nihil aliud in­<lb/>tendi&longs;&longs;e Ari&longs;totelem quàm grauia à generante, in&longs;tanti, quo generan­<lb/>tur, accipere actum primum huius motus; id e&longs;t virtutem, à qua po&longs;­<lb/>&longs;int reduci ad actum &longs;ecundum, id e&longs;t ad ip&longs;um motum, de cuius rei ve­<lb/>ritate iam mihi non e&longs;t laborandum. </s> </p> <p id="N15FB9" type="main"> <s id="N15FBB"><!-- NEW -->Igitur non mouetur corpus graue à cau&longs;a primâ, licèt hæc concurrat <lb/>cum aliâ ad eius motum, nec ab aëre, nec à virtute magnetica, quæ in­<lb/>&longs;it terræ, nec adductis, reducti&longs;que filamentis, nec à cælo pellente, nec <lb/>à vi &longs;ympathicâ, nec à generante proximè & immediatè; </s> <s id="N15FC5"><!-- NEW -->quia fortè iam <lb/>interiit, nec ab vllo alio extrin&longs;eco, vt con&longs;tat inductione; </s> <s id="N15FCB"><!-- NEW -->igitur ab ali­<lb/>quâ vi intrin&longs;ecâ, quidquid &longs;it, de qua alibi: hæc omnia paulò fu&longs;iùs <lb/>tractauimus, quia in hoc vno Theoremate totam motus naturalis rem <lb/>verti iudicamus. </s> </p> <p id="N15FD5" type="main"> <s id="N15FD7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N15FE4" type="main"> <s id="N15FE6"><!-- NEW --><emph type="italics"/>Motus naturalis est aliquid distinctum realiter à mobili:<emph.end type="italics"/> Probatur; </s> <s id="N15FEF"><!-- NEW --><lb/>mobile ip&longs;um aliquando quie&longs;cit per hypoth.4.lib.1. igitur e&longs;t &longs;ine mo­<lb/>tu; </s> <s id="N15FF6"><!-- NEW -->igitur &longs;eparatum à motu; </s> <s id="N15FFA"><!-- NEW -->igitur realiter di&longs;tinctum per Ax.2. lib.1. <lb/>hoc etiam probatus per Th. 1.lib. 1. Et certè mirari &longs;atis non po&longs;&longs;um <lb/>aliquos recentiores non po&longs;&longs;e concipere, vt ip&longs;i aiunt, motum e&longs;&longs;e ali­<lb/>quid ab ip&longs;o mobili di&longs;tinctum; </s> <s id="N16004"><!-- NEW -->nam quotie&longs;cunque duo prædicata, vel <pb pagenum="80" xlink:href="026/01/112.jpg"/>attributa contradictoria, quorum &longs;cilicet vnum negat aliud, eidem &longs;ub­<lb/>jecto diuer&longs;is temporibus ine&longs;&longs;e dicuntur, haud dubiè alterum &longs;altem ab <lb/>eo di&longs;tingui realiter nece&longs;&longs;e e&longs;t; </s> <s id="N16011"><!-- NEW -->alioqui &longs;i vtrumque idem e&longs;&longs;e cum vno <lb/>tertio vere dicitur; </s> <s id="N16017"><!-- NEW --><emph type="italics"/>mouetur, non monetur,<emph.end type="italics"/> quæ &longs;unt prædicata contradi­<lb/>ctoria; </s> <s id="N16022"><!-- NEW -->igitur vel moueri, vel non moueri dicit di&longs;tinctum realiter à mo­<lb/>bili; Secundum e&longs;t mera negatio; </s> <s id="N16028"><!-- NEW -->nam eo ip&longs;o, quod mobile e&longs;t &longs;ine vllo <lb/>addito, non mouetur; </s> <s id="N1602E"><!-- NEW -->igitur &longs;uprà ip&longs;um mobile dicit puram putam ne­<lb/>gationem motus; igitur moueri, dicit aliquid di&longs;tinctum. </s> </p> <p id="N16034" type="main"> <s id="N16036"><!-- NEW -->Præterea quotie&longs;cunque prædicatum aliquod tribuitur in propo&longs;i­<lb/>tione affirmatiua falsâ; </s> <s id="N1603C"><!-- NEW -->certè prædicatum illud non ine&longs;t &longs;ubiecto; </s> <s id="N16040"><!-- NEW -->alio­<lb/>quin e&longs;&longs;et vera, vt patet; </s> <s id="N16046"><!-- NEW -->igitur di&longs;tinguitur à &longs;ubiecto realiter; </s> <s id="N1604A"><!-- NEW -->&longs;ed hæc <lb/>propo&longs;itio, <emph type="italics"/>lapis mouetur,<emph.end type="italics"/> dum ip&longs;e quie&longs;cit, e&longs;t fal&longs;a; igitur motus non <lb/>ine&longs;t mobili, igitur ab eo di&longs;tinguitur realiter, &longs;eu modaliter, quæ e&longs;t <lb/>di&longs;tinctio realis minor. </s> </p> <p id="N1605A" type="main"> <s id="N1605C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16069" type="main"> <s id="N1606B"><!-- NEW --><emph type="italics"/>Metus naturalis non e&longs;t immediatè ab entitate mobilis, ita vt nihil &longs;it aliud <lb/>vnde &longs;it hic motus:<emph.end type="italics"/> Probatur; lapis cadens ex maiore altitudine maiorem <lb/>ictum infligit perhypoth. </s> <s id="N16078"><!-- NEW -->1. maior e&longs;t effectus, igitur maior cau&longs;a, id e&longs;t <lb/>motus; </s> <s id="N1607E"><!-- NEW -->igitur cau&longs;a motus per Ax.2. &longs;ed e&longs;t eadem entitas mobilis, vt <lb/>patet; </s> <s id="N16084"><!-- NEW -->igitur non e&longs;t cau&longs;a immediata motus; Præterea globus per pla­<lb/>num inclinatum deuolutus &longs;uum motum accelerat per hypotl. </s> <s id="N1608A">3. & fune­<lb/>pendulum &longs;uam vibrationem per hypoth. </s> <s id="N1608F">2. igitur debet e&longs;&longs;e cau&longs;a huius <lb/>maioris, &longs;eu velocioris motus per Ax.8. lib. 1. hæc porrò non e&longs;t &longs;ub­<lb/>&longs;tantia ip&longs;ius corporis, quæ &longs;emper eadem e&longs;t, tùm initio, tùm in fine <lb/>motus per Ax.2. </s> </p> <p id="N16098" type="main"> <s id="N1609A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N160A7" type="main"> <s id="N160A9"><emph type="italics"/>Motus naturalis non e&longs;t immediatè ab ip&longs;a grauitate.<emph.end type="italics"/></s> <s id="N160B0"><!-- NEW --> Probatur, &longs;int <lb/>enim eædem hypoth.1.2.3. igitur maior ictus in fine motus, & velocior <lb/>motus debent habere cau&longs;am; &longs;ed hæc grauitas non e&longs;t, quæ &longs;emper ea­<lb/>dem e&longs;t, vt patet, vtrum verò di&longs;tinguatur grauitas ab ip&longs;a corporis <lb/>&longs;ub&longs;tantia di&longs;cutiemus in tractatu &longs;equenti. </s> <s id="N160BC"><!-- NEW -->Fuit aliquis non infimæ no­<lb/>tæ Philo&longs;ophus, qui diceret maiorem illum ictum e&longs;&longs;e ab ipsâ corporis <lb/>&longs;ub&longs;tantiâ; &longs;ed hoc iam refellimus Theoremate 4. lib.1. Adde quod im­<lb/>petu, ad extra producitur ab alio impetu per Th.42.lib.1. Dicebat etiam <lb/>velociorem motum e&longs;&longs;e ab ipsâ grauitate connotante præuium motum, <lb/>quod etiam refellemus infrà. </s> </p> <p id="N160CA" type="main"> <s id="N160CC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N160D9" type="main"> <s id="N160DB"><emph type="italics"/>Hinc motus naturalis e&longs;t ab impetu.<emph.end type="italics"/></s> <s id="N160E2"> Probatur; e&longs;t ab aliqua cau&longs;a per <lb/>Ax.8. lib.1. ab aliqua intrin&longs;eca per Th. 1. non à &longs;ub&longs;tantia corporis <lb/>grauis per Th. 3. non à grauitate per Th. 4. igitur ab impetu, quia <lb/>nihil aliud e&longs;&longs;e pote&longs;t intrin&longs;ecum, à quo &longs;it motus per definitionem <lb/>3. lib. 1. <!-- KEEP S--></s> </p> <pb pagenum="81" xlink:href="026/01/113.jpg"/> <p id="N160F2" type="main"> <s id="N160F4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16101" type="main"> <s id="N16103"><emph type="italics"/>Ille impetus ab aliqua cau&longs;a producitur.<emph.end type="italics"/></s> <s id="N1610A"> Probatur, quia quidquid de no­<lb/>uo e&longs;t, habet cau&longs;am per Ax.8. lib. 1. <!-- KEEP S--></s> </p> <p id="N16110" type="main"> <s id="N16112"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1611E" type="main"> <s id="N16120"><!-- NEW -->Producitur ab aliqua cau&longs;a intrin&longs;eca, quia non producitur ab aliqua <lb/>extrin&longs;eca; alioquin motus naturalis e&longs;&longs;et ab extrin&longs;eco contra definitio­<lb/>nem primam, & Th.1. <!-- KEEP S--></s> </p> <p id="N16129" type="main"> <s id="N1612B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N16137" type="main"> <s id="N16139"><!-- NEW --><emph type="italics"/>Hinc produci tantùm pote&longs;t ab ip&longs;a &longs;ubstantia corporis grauis; </s> <s id="N1613F"><!-- NEW -->nam graui­<lb/>tas e&longs;t ip&longs;e impetus innatus, de qua infrà:<emph.end type="italics"/> probatur; </s> <s id="N16148"><!-- NEW -->quia nihil e&longs;t aliud in­<lb/>trin&longs;ecum, à quo produci po&longs;&longs;it; quòd autem non producatur ab alio im­<lb/>petu ad intra, patet per Th.41. lib. 1. <!-- KEEP S--></s> </p> <p id="N16151" type="main"> <s id="N16153"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N1615F" type="main"> <s id="N16161"><emph type="italics"/>Impetus productus primo instanti durat proximè &longs;equenti.<emph.end type="italics"/></s> <s id="N16168"> Probatur pri­<lb/>mò; </s> <s id="N1616D"><!-- NEW -->quia &longs;emper habet &longs;uum effectum formalem; vel grauitationis, &longs;i <lb/>impeditur; </s> <s id="N16173"><!-- NEW -->vel motus in medio libero; </s> <s id="N16177"><!-- NEW -->igitur non e&longs;t fru&longs;trà; </s> <s id="N1617B"><!-- NEW -->igitur <lb/>non de&longs;truitur per Th.162.lib.1. nihil enim exigit de&longs;tructionem; </s> <s id="N16181"><!-- NEW -->non <lb/>tota natura, quia non e&longs;t fru&longs;trà per Ax. 6. non à contrario impetu, qui <lb/>&longs;æpè abe&longs;t, vt cum liberè mouetur corpus graue in aëre, vel &longs;u&longs;tinetur, <lb/>v.g. <!-- REMOVE S-->glans plumbea ab ingenti rupe: </s> <s id="N1618D"><!-- NEW -->adde quod, licèt producatur in cor­<lb/>pore graui impetus violentus &longs;ur&longs;um, non de&longs;truitur, tamen innatus; alio­<lb/>quin nihil e&longs;&longs;et, quod de&longs;trueret violentum per Th.150. & Schol. <!-- REMOVE S-->Th. <!-- REMOVE S--><lb/>152.num.6.lib.1. <!-- KEEP S--></s> </p> <p id="N1619B" type="main"> <s id="N1619D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N161A9" type="main"> <s id="N161AB"><!-- NEW --><emph type="italics"/>Impetus ille innatus, qui durat &longs;ecundo instanti, con&longs;eruatur ab aliqua cau­<lb/>&longs;a<emph.end type="italics"/>; e&longs;t certum per Ax. 14.lib.1.num.1. <!-- KEEP S--></s> </p> <p id="N161B7" type="main"> <s id="N161B9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N161C5" type="main"> <s id="N161C7"><emph type="italics"/>Non con&longs;eruatur à cau&longs;a primò productiua.<emph.end type="italics"/></s> <s id="N161CE"><!-- NEW --> Probatur per Th.144. lib.1. <lb/>alioquin non po&longs;&longs;et intendi ab eadem cau&longs;a per Th. 146. lib 1. quippè <lb/>con&longs;eruatio nihil e&longs;t aliud, quàm repetita productio, vt con&longs;tat; </s> <s id="N161D6"><!-- NEW -->nam <lb/>cau&longs;a con&longs;eruans verè influit; </s> <s id="N161DC"><!-- NEW -->igitur &longs;i e&longs;t cau&longs;a nece&longs;&longs;aria primo, & &longs;e­<lb/>cundo in&longs;tanti æquali ni&longs;u influit; </s> <s id="N161E2"><!-- NEW -->influit enim quantum pote&longs;t per Ax. <!-- REMOVE S--><lb/>12.lib.1.quòd autem impetus intendatur, demon&longs;trabimus infrà; </s> <s id="N161E9"><!-- NEW -->con&longs;ule <lb/>Schol.Th.146.lib.1.in quo habes rationem præclari natura in&longs;tituti; </s> <s id="N161EF"><!-- NEW -->quo <lb/>&longs;cilicet factum e&longs;t, vt qualitates, quæ contrario carent à causâ primò pro­<lb/>ductiua; aliæ verò, quæ contrarium habent, ab alia causà con&longs;er­<lb/>uentur. </s> </p> <p id="N161F9" type="main"> <s id="N161FB"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16207" type="main"> <s id="N16209"><!-- NEW -->Hinc ab aliâ causâ con&longs;eruari nece&longs;&longs;e e&longs;t, vt patet, eáque aplicatâ per <lb/>Ax.10.lib.1. quæcumque tandem illa &longs;it; nos aliquando cau&longs;am primam <lb/>e&longs;&longs;e dicemus. </s> </p> <pb pagenum="82" xlink:href="026/01/114.jpg"/> <p id="N16215" type="main"> <s id="N16217"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N16223" type="main"> <s id="N16225"><emph type="italics"/>Quando graue e&longs;t in medio libero, per quod &longs;cilicet de&longs;cendere pote&longs;t, &longs;ecun­<lb/>do instanti producitur nouus impetus, itemque tertio, quarto, quinto. </s> <s id="N1622C">&c.<emph.end type="italics"/></s> <s id="N16231"> Pro­<lb/>batur primò; </s> <s id="N16236"><!-- NEW -->quia &longs;ecundo in&longs;tanti e&longs;t eadem cau&longs;a quæ primo non ma­<lb/>gis impedita, eáque nece&longs;&longs;aria; </s> <s id="N1623C"><!-- NEW -->igitur nece&longs;&longs;ariò agit per Ax. 12. lib.1. <lb/>igitur aliquem effectum producit; &longs;ed hic effectus non e&longs;t impetus pro­<lb/>ductus primo in&longs;tanti, quia non con&longs;eruatur à cau&longs;a primò productiua <lb/>per Th.11. igitur e&longs;t nouus. </s> <s id="N16246">Probatur &longs;ecundò; cre&longs;cit motus grauium in <lb/>libero medio per hypoth. </s> <s id="N1624B">1.2.3. igitur cre&longs;cit impetus; </s> <s id="N1624E"><!-- NEW -->quia cum motus <lb/>naturalis &longs;it ab impetu per Th.5. quâ proportione cre&longs;cit effectus, &longs;cilicet <lb/>formalis, & exigentiæ; </s> <s id="N16256"><!-- NEW -->&longs;ic enim motus e&longs;t effectus impetus per Th. 15. <lb/>lib.1.eàdem cre&longs;cit cau&longs;a per Ax.2. Probatur tertiò, quia corpus graue ex <lb/>maiore altitudine cadens maiorem quoque ictum infligit per hypoth.1. <lb/>igitur maior impetus imprimitur in corpore percu&longs;&longs;o; &longs;ed impetus ad ex­<lb/>tra producitur ab alio impetu per Th.42.lib.1. igitur &longs;i cre&longs;cit productus <lb/>inpetus, cre&longs;cit impetus producens. </s> </p> <p id="N16264" type="main"> <s id="N16266"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16272" type="main"> <s id="N16274"><!-- NEW -->Hinc reiicies quorumdam placitum, qui volunt cau&longs;am velocioris <lb/>motus e&longs;&longs;e grauitatem ip&longs;am, quatenus connotat motum præuium; </s> <s id="N1627A"><!-- NEW -->quia <lb/>&longs;cilicet grauitas non producit illum maiorem impetum ad extra, vt con­<lb/>&longs;tat; </s> <s id="N16282"><!-- NEW -->nec &longs;ub&longs;tantia ip&longs;ius corporis grauis per Th.40.lib.1.igitur ip&longs;e im­<lb/>petus: </s> <s id="N16288"><!-- NEW -->præterea &longs;i hoc e&longs;&longs;et, fru&longs;trà requireretur impetus contra Th. 5. <lb/>Denique motus præuius nihil e&longs;t amplius, cum alius &longs;uccedit: Vide Th. <!-- REMOVE S--><lb/>40.lib.1. vbi hæc fusè di&longs;cu&longs;&longs;imus. </s> </p> <p id="N16291" type="main"> <s id="N16293"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N1629F" type="main"> <s id="N162A1"><!-- NEW --><emph type="italics"/>Impetus productus &longs;ecundo instanti in medio libero con&longs;eruatur tertio, & <lb/>productus tertio con&longs;eruatur, quarto, atque ita deinceps<emph.end type="italics"/>; </s> <s id="N162AC"><!-- NEW -->quia &longs;cilicet nec con­<lb/>&longs;eruantur à cau&longs;a primo productiua per Th.144.libri: </s> <s id="N162B2"><!-- NEW -->nec aliquid exigit <lb/>de&longs;tructionem; </s> <s id="N162B8"><!-- NEW -->non contrarius impetus, quia nullus e&longs;t applicatus, vt <lb/>con&longs;tat; </s> <s id="N162BE"><!-- NEW -->non re&longs;i&longs;tentia medij, quæ quidem alicuius momenti e&longs;t; </s> <s id="N162C2"><!-- NEW -->&longs;ed <lb/>non tanti, vt impedire po&longs;&longs;it motum omninò, vt con&longs;tat; </s> <s id="N162C8"><!-- NEW -->nam &longs;uppono <lb/>liberum medium, igitur nec de&longs;truere impetum; </s> <s id="N162CE"><!-- NEW -->cum tamdiu duret cau­<lb/>&longs;a quamdiu durat effectus, vt patet; </s> <s id="N162D4"><!-- NEW -->igitur nihil e&longs;t quod exigat impe­<lb/>tus huius de&longs;tructionem; igitur non de&longs;truitur per Ax. 14. lib.1. <lb/><expan abbr="qūanta">quanta</expan> verò &longs;it, & quid &longs;it cuiu&longs;libet medij re&longs;i&longs;tentia, dicemus <lb/>infrà. </s> </p> <p id="N162E1" type="main"> <s id="N162E3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N162EF" type="main"> <s id="N162F1"><emph type="italics"/>Si impetus innatus impeditur, ita vt moueri non po&longs;&longs;it corpus graue, &longs;e­<lb/>cundo instanti non producitur nouus impetus.<emph.end type="italics"/></s> <s id="N162FA"><!-- NEW --> Probatur primò, non cre&longs;cit <lb/>corporis grauis &longs;eu grauitas, &longs;eu grauitatio, vt con&longs;tat experientiâ; </s> <s id="N16300"><!-- NEW -->igitur <lb/>non cre&longs;cit impetus; </s> <s id="N16306"><!-- NEW -->alioquin &longs;i cre&longs;ceret cau&longs;a, cre&longs;ceret effectus per <lb/>Ax.2. igitur de re, quòd &longs;it, certum e&longs;t, atque euidens; </s> <s id="N1630C"><!-- NEW -->iam demon&longs;tratur <lb/>propter quid &longs;it; </s> <s id="N16312"><!-- NEW -->impetus &longs;ecundo in&longs;tanti productus e&longs;&longs;et fru&longs;trà; </s> <s id="N16316"><!-- NEW -->careret <pb pagenum="83" xlink:href="026/01/115.jpg"/>enim &longs;uo fine, vel effectu formali, id e&longs;t motu; igitur e&longs;&longs;et fru&longs;trà, &longs;ed <lb/>quod fru&longs;trà e&longs;t, non e&longs;t per Ax.6.l.1. </s> </p> <p id="N16321" type="main"> <s id="N16323"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1632F" type="main"> <s id="N16331">Ob&longs;erua quæ&longs;o, quod iam &longs;uprà indicatum e&longs;t, e&longs;&longs;e tres veluti &longs;pecies <lb/>impetus. </s> <s id="N16336">Prima e&longs;t impetus naturalis innati. </s> <s id="N16339">Secunda naturalis acqui&longs;iti. </s> <s id="N1633C"><lb/>Tertia violenti; </s> <s id="N16340"><!-- NEW -->innatus e&longs;t qui vel à generante &longs;imul cum corpore <lb/>graui productus e&longs;t; </s> <s id="N16346"><!-- NEW -->qui&longs;quis tandem &longs;it generans, de quo aliàs; </s> <s id="N1634A"><!-- NEW -->vel ab <lb/>ip&longs;o graui qua&longs;i profunditur, id e&longs;t, producitur in &longs;e ip&longs;o &longs;tatim initio, <lb/>quo e&longs;t; </s> <s id="N16352"><!-- NEW -->porrò cum in corpore graui duplex qua&longs;i proprietas &longs;en&longs;ibilis <lb/>e&longs;&longs;e videatur, &longs;cilicet grauitas, &longs;eu pondus & motus deor&longs;um; </s> <s id="N16358"><!-- NEW -->certè de­<lb/>bet e&longs;&longs;e in eo aliquid per quod tùm cogno&longs;ci po&longs;&longs;it eius pondus, tùm in­<lb/>cipiat moueri deor&longs;um; </s> <s id="N16360"><!-- NEW -->quippe maximè corpora ex pondere cogno&longs;ci­<lb/>mus, vnumque ab alio di&longs;tinguimus; </s> <s id="N16366"><!-- NEW -->igitur debet e&longs;&longs;e aliquid, quod &longs;en­<lb/>&longs;um afficiat, vt cogno&longs;ci po&longs;&longs;it; </s> <s id="N1636C"><!-- NEW -->atqui illud ip&longs;um non e&longs;t &longs;ub&longs;tantia cor­<lb/>poris; </s> <s id="N16372"><!-- NEW -->nam corpus graue meæ manui &longs;u&longs;tinenti impetum imprimit; </s> <s id="N16376"><!-- NEW --><lb/>immò vim alterius impetus infringit; </s> <s id="N1637B"><!-- NEW -->igitur operâ alterius per Th. 40. <lb/>& 42.lib.1. Præterea illud ip&longs;um, quod agit, &longs;eu deor&longs;um pellit &longs;u&longs;tinen­<lb/>tem manum, e&longs;t illud ip&longs;um quod inclinat corpus graue deor&longs;um imme­<lb/>diatè, &longs;eu quod exigit motum naturalem deor&longs;um; </s> <s id="N16385"><!-- NEW -->illud autem quod <lb/>immediatè præ&longs;tat hunc motum, nec e&longs;t &longs;ub&longs;tantia corporis grauis per <lb/>Th.3. igitur ip&longs;e impetus per Th.5. adde quod primo in&longs;tanti, quo e&longs;t im­<lb/>petus, non e&longs;t motus ille, quem exigit per Th.34. lib.1. igitur præexi&longs;tit <lb/>&longs;emper impetus, qui ne &longs;it fru&longs;trà, habet primum effectum &longs;uum forma­<lb/>lem, id e&longs;t grauitationem: </s> <s id="N16393"><!-- NEW -->Ex his dicendum e&longs;t hunc impetum natiuum <lb/>nunquam de&longs;trui, quia nunquam e&longs;t fru&longs;trà, habet enim &longs;emper aliquem <lb/>effectum, primum quidem &longs;i caret &longs;ecundo; </s> <s id="N1639B"><!-- NEW -->&longs;ecundum verò &longs;i caret pri­<lb/>mo; </s> <s id="N163A1"><!-- NEW -->quippe vtrumque &longs;imul habere non pote&longs;t; nam corporis pondus <lb/>cogno&longs;ci non pote&longs;t, dum fertur deor&longs;um accelerato motu, quot verò, <lb/>& quanta commoda ex cognitione ponderis cuiu&longs;libet materiæ proce­<lb/>dant, vix explicari pote&longs;t. </s> </p> <p id="N163AB" type="main"> <s id="N163AD"><!-- NEW -->Ex his verò concludendum &longs;upere&longs;t impetum innatum e&longs;&longs;e proprie­<lb/>tatem quarto modo, vt vulgò aiunt, corporis grauis; </s> <s id="N163B3"><!-- NEW -->ac proinde ab illo <lb/>in&longs;eparabilem; </s> <s id="N163B9"><!-- NEW -->quid verò fiat de illo, cum corpus graue fit leue; &longs;i tamen <lb/>hoc aliquando accidit, dicemus cum de grauitate, & grauitatione, iam <lb/>verò &longs;atis e&longs;t ad præ&longs;ens in&longs;titutum impetum innatum ab ip&longs;o corpori <lb/>graui nunquam &longs;eparari, quandiu remanet graue. </s> </p> <p id="N163C3" type="main"> <s id="N163C5"><!-- NEW -->Impetus naturalis acqui&longs;itus producitur ab eodem principio intrin­<lb/>&longs;eco; </s> <s id="N163CB"><!-- NEW -->hinc dicitur naturalis: </s> <s id="N163CF"><!-- NEW -->dicitur verò acqui&longs;itus, quia non e&longs;t inna­<lb/>tus; </s> <s id="N163D5"><!-- NEW -->&longs;ed &longs;eparatur à corpore graui; </s> <s id="N163D9"><!-- NEW -->quod &longs;emper eo caret, quandiu <lb/>quie&longs;cit: </s> <s id="N163DF"><!-- NEW -->&longs;ed innato tantùm accedit ad motus accelerationem, & ad alia <lb/>quamplurima, quæ ex ea &longs;equuntur; </s> <s id="N163E5"><!-- NEW -->putà maiorem percu&longs;&longs;ionem, re&longs;i­<lb/>&longs;tentiam, vim, & ad tollendum totius naturæ languidiorem; </s> <s id="N163EB"><!-- NEW -->quo certè af­<lb/>ficeretur, &longs;i corpus graue tardi&longs;&longs;imo motu deor&longs;um ferretur, de quo in­<lb/>frà; </s> <s id="N163F3"><!-- NEW -->Porrò impetus acqui&longs;itus in multis differt ab innato; primò quia <pb pagenum="84" xlink:href="026/01/116.jpg"/>de&longs;truitur à corpore re&longs;i&longs;tente eo modo, quo diximus, & dicemus infrà. </s> <s id="N163FC"><lb/>Secundò, quia determinari pote&longs;t ad omnem lineam. </s> </p> <p id="N16400" type="main"> <s id="N16402">Impetus violentus e&longs;t, qui e&longs;t ab extrin&longs;eco, de quo agemus infrà, & <lb/>iam &longs;uprà in lib.1. multa &longs;unt de eo demon&longs;trata. </s> </p> <p id="N16407" type="main"> <s id="N16409"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N16415" type="main"> <s id="N16417"><!-- NEW --><emph type="italics"/>Impetus naturalis corporis grauis intenditur dum hoc ip&longs;um de&longs;cendit in <lb/>medio libero<emph.end type="italics"/>; demon&longs;tratur, Impetus nouus producitur in &longs;ecundo, ter­<lb/>tio, quarto, &c. </s> <s id="N16424">in&longs;tantibus per Th.12. &longs;ed productus in primo con&longs;er­<lb/>uatur &longs;ecundo, per Th.9. productus &longs;ecundo con&longs;eruatur tertio, produ­<lb/>ctus tertio con&longs;eruatur quarto per Th.13. igitur &longs;ecundus additur tertio, <lb/>tertius primo, &longs;ecundo, quartus primo, &longs;ecundo, & tertio, &c.&longs;ed impetus <lb/>additus alteri facit inten&longs;iorem impetum per Ax.1. igitur impetus natu­<lb/>ralis intenditur, quod crat demon&longs;trandum. </s> </p> <p id="N16431" type="main"> <s id="N16433"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N1643F" type="main"> <s id="N16441"><!-- NEW --><emph type="italics"/>Hinc motus naturalis deor&longs;um acceleratur<emph.end type="italics"/>; </s> <s id="N1644A"><!-- NEW -->hoc ip&longs;um &longs;uppo&longs;ui &longs;uprà <lb/>Quod e&longs;&longs;et in hyp.1.2.3. iam verò demon&longs;tro propter quid e&longs;t; </s> <s id="N16450"><!-- NEW -->&longs;ie enim <lb/>hypothe&longs;is in Theorema conuerti pote&longs;t, vt &longs;æpè monuimus in metho­<lb/>do; </s> <s id="N16458"><!-- NEW -->igitur probatur hoc Theorema facilè; </s> <s id="N1645C"><!-- NEW -->cre&longs;cit impetus in corpore gra­<lb/>ui, quod tendit deor&longs;um in libero medio per T. 15. igitur cre&longs;cit cau&longs;a <lb/>motus; </s> <s id="N16464"><!-- NEW -->nam impetus e&longs;t cau&longs;a immediata motus naturalis per Th. 51. <lb/>&longs;ed quâ proportione cre&longs;cit cau&longs;a, debet cre&longs;cere effectus per Ax.2. igi­<lb/>tur motus naturalis deor&longs;um cre&longs;cit, id e&longs;t acceleratur, id e&longs;t fit velo­<lb/>cior, quod erat dem: </s> <s id="N1646E"><!-- NEW -->nec e&longs;t quod aliquis exi&longs;timet hic à me committi <lb/>vitio&longs;um argumentationis circulum; </s> <s id="N16474"><!-- NEW -->quippe probaui &longs;uprà cre&longs;cere im­<lb/>petum, quia cre&longs;cit motus; </s> <s id="N1647A"><!-- NEW -->iam verò probo cre&longs;cere motum, quia cre&longs;­<lb/>cit impetus; nam primò probaui produci nouum impetum in Th.12. eo <lb/>quod &longs;ecundo in&longs;tanti. </s> <s id="N16482"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it eadem cau&longs;a nece&longs;&longs;aria applicata non im­<lb/>pedita, igitur tàm debet agere &longs;ecundo quàm primo in&longs;tanti, hæc fuit <lb/>mea probatio à priori; </s> <s id="N1648C"><!-- NEW -->&longs;ecundò verò probaui ex hypothe&longs;i certa; </s> <s id="N16490"><!-- NEW -->quia <lb/>&longs;cilicet cre&longs;cit motus, cuius veritatem cogno&longs;co &longs;en&longs;ibiliter in &longs;e, vnde <lb/>&longs;uppono tantùm de illa quod &longs;it; </s> <s id="N16498"><!-- NEW -->igitur nullus committitur circulus; nam <lb/>diuer&longs;a e&longs;t omninò cognitio. </s> <s id="N1649E">Prima &longs;cilicet qua cogno&longs;co de motu na­<lb/>turaliter accelerato quod &longs;it, quæ mihi, & ru&longs;tico communis e&longs;t. </s> <s id="N164A3"><!-- NEW -->Secun­<lb/>da verò qua non modò cogno&longs;co de motu illo quod &longs;it acceleratus, ve­<lb/>rùm propter quid &longs;it acceleratus, id e&longs;t cau&longs;am huius accelerationis, id <lb/>e&longs;t propter quam attributum hoc ine&longs;t &longs;ubiecto, & hæc e&longs;t vera demon­<lb/>&longs;tratio à priori; porrò in Phy&longs;ica de effectu &longs;en&longs;ibili &longs;upponi debet quod <lb/>&longs;it, hoc enim percipitur &longs;en&longs;u. </s> <s id="N164B1">v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;upponam in Phy&longs;ica quod &longs;it motus <lb/>acceleratus, quod ignis &longs;it calidus, Sol lucidus, nix candida, vinum ru­<lb/>brum, &c. </s> <s id="N164BC">at verò demon&longs;trabo propter quid hæc &longs;int, &longs;ed de his <lb/>&longs;atis. </s> </p> <p id="N164C1" type="main"> <s id="N164C3"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N164CF" type="main"> <s id="N164D1"><!-- NEW -->Ob&longs;eruabis etiam aliud naturæ in&longs;titutum, quo &longs;cilicet factum e&longs;t, vt <pb pagenum="85" xlink:href="026/01/117.jpg"/>corpora grauia motu naturali accelerato deor&longs;um ferantur; </s> <s id="N164DA"><!-- NEW -->&longs;i enim motu <lb/>ferrentur æquabili, vel e&longs;&longs;et æqualis illi quem initio &longs;ui de&longs;cen&longs;us ha­<lb/>bent, qui e&longs;t tardi&longs;&longs;imus, vt con&longs;tat ex ip&longs;a ictuum differentia; </s> <s id="N164E2"><!-- NEW -->atque <lb/>ita infinitum ferè tempus ponerent grauia in minimo etiam de&longs;cen&longs;u, <lb/>quod e&longs;&longs;et maximè incommodum; &longs;i verò motus ille e&longs;&longs;et æqualis mo­<lb/>tui v.g. <!-- REMOVE S-->quem acqui&longs;iuit in &longs;patio 3. vel 4. perticarum, pondera corpo­<lb/>rum cre&longs;cerent in immen&longs;um, ide&longs;t in ea proportione, qua ictus, qui in­<lb/>fligitur à corpore graui confecto 4. perticarum &longs;patio maior e&longs;t ictu, qui <lb/>infligitur po&longs;t decur&longs;um minimum omnium &longs;patiorum, quod valdè in­<lb/>commodum e&longs;&longs;et. </s> </p> <p id="N164F6" type="main"> <s id="N164F8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N16504" type="main"> <s id="N16506"><!-- NEW --><emph type="italics"/>Æqualibus temporibus æqualis impetus producitur, &longs;i &longs;it eadem applica­<lb/>tio, idemque impedimentum<emph.end type="italics"/>; </s> <s id="N16511"><!-- NEW -->probatur, quia cau&longs;a huius impetus e&longs;t ne­<lb/>ce&longs;&longs;aria; &longs;ed eadem cau&longs;a nece&longs;&longs;aria æqualibus temporibus æqualem <lb/>impetum producit per Ax.3. </s> </p> <p id="N16519" type="main"> <s id="N1651B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N16527" type="main"> <s id="N16529"><!-- NEW --><emph type="italics"/>Qua proportione cre&longs;cit impetus acceleratur motus<emph.end type="italics"/>; quia quæ proportio­<lb/>ne cre&longs;cit cau&longs;a, etiam cre&longs;cit effectus per Ax.2. </s> </p> <p id="N16534" type="main"> <s id="N16536"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N16542" type="main"> <s id="N16544"><!-- NEW --><emph type="italics"/>Hinc æqualibus temporibus in de&longs;cen&longs;u corpus graue acquirit aqualia ve­<lb/>locitatis, vel accelerationis momenta<emph.end type="italics"/>; </s> <s id="N1654F"><!-- NEW -->hoc ip&longs;um e&longs;t quod definitionis lo­<lb/>co Galileus in dialogo tertio de motu naturali a&longs;&longs;umit; </s> <s id="N16555"><!-- NEW -->quod tamen <lb/>meo iudicio fuit antè demon&longs;trandum quàm &longs;upponendum; quare &longs;ic <lb/>demon&longs;tramus, quâ proportione cre&longs;cit impetus, cre&longs;cit motus per Th. <!-- REMOVE S--><lb/>18. &longs;ed temporibus æqualibus acquiruntur æquales impetus gradus per <lb/>Th.17. igitur æqualia velocitatis momenta, vel incrementa. </s> </p> <p id="N16562" type="main"> <s id="N16564"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N16570" type="main"> <s id="N16572"><!-- NEW --><emph type="italics"/>Spatia que per curruntur motu æquabili æqualibus temporibus &longs;unt æqualia<emph.end type="italics"/>; <lb/>Probatur per Def.2. </s> </p> <p id="N1657D" type="main"> <s id="N1657F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N1658B" type="main"> <s id="N1658D"><!-- NEW --><emph type="italics"/>Duo motus æquabiles, qui durant æqualibus temporibus, &longs;unt vt &longs;patia<emph.end type="italics"/>; <lb/>patet; </s> <s id="N16598"><!-- NEW -->cùm enim impetus &longs;int vt motus per Ax. 2. motus &longs;unt vt &longs;patia; <lb/>quippe vt ex impetu &longs;equitur motus, ita ex motu confectum &longs;pa­<lb/>tium. </s> </p> <p id="N165A0" type="main"> <s id="N165A2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N165AE" type="main"> <s id="N165B0"><emph type="italics"/>Duo motus æquabiles, quibus percurruntur &longs;patia æqualia &longs;unt vt tempora <lb/>permutande<emph.end type="italics"/>;, patet, quia velocior e&longs;t, quò percurritur &longs;patium æquale <lb/>minori tempore per Def.2. l. <!-- REMOVE S-->1. Igitur eò velocior, quò minori tem­<lb/>pore. </s> </p> <p id="N165C0" type="main"> <s id="N165C2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N165CE" type="main"> <s id="N165D0"><!-- NEW --><emph type="italics"/>Spatium, quod percurritur maiori tempore motu æquabili, est maius eo, <lb/>quod percurritur minori æquè veloci motu in ea ratione, qua vnum tempus<emph.end type="italics"/><pb pagenum="86" xlink:href="026/01/118.jpg"/><emph type="italics"/>est maius alio<emph.end type="italics"/>; </s> <s id="N165E4"><!-- NEW -->patet, quia æqualia &longs;unt æqualibus temporibus per Th. <!-- REMOVE S--><lb/>20. igitur inæqualibus inæqualia iuxta rationem temporum; item &longs;pa­<lb/>tium, quod idem percurritur minori tempore minus e&longs;t. </s> </p> <p id="N165ED" type="main"> <s id="N165EF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N165FB" type="main"> <s id="N165FD"><!-- NEW --><emph type="italics"/>Tempus quo maius &longs;patium percurritur eodem motu æquabili, e&longs;t maius eò <lb/>quò minus conficitur iuxta rationem &longs;patiorum:<emph.end type="italics"/> Si enim &longs;patia &longs;unt vt tem­<lb/>pora, igitur tempora &longs;unt vt &longs;patia; item tempus, quo minus &longs;patium <lb/>percurritur e&longs;t minus co, quo maius. </s> </p> <p id="N1660C" type="main"> <s id="N1660E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N1661A" type="main"> <s id="N1661C"><emph type="italics"/>Spatium, quod conficitur motu velociore, e&longs;t maius eo, quod percur­<lb/>ritur æquali certè tempore, &longs;ed tardiore motu,<emph.end type="italics"/> vt con&longs;tat per def. </s> <s id="N16626">2. l. <!-- REMOVE S-->1. <lb/>imò e&longs;t maius iuxta rationem velocitatis maioris, item e&longs;t minus iuxta <lb/>rationem tarditatis maioris. </s> </p> <p id="N1662F" type="main"> <s id="N16631"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N1663D" type="main"> <s id="N1663F"><!-- NEW --><emph type="italics"/>Tempus, quo conficitur &longs;patium æquale &longs;ed uelociore motu, est minus eo <lb/>quo conficitur tardiore<emph.end type="italics"/>; </s> <s id="N1664A"><!-- NEW -->Probatur per def.2. & per Th.22. idque in ratio­<lb/>ne velocitatum permutando; item tempus quo conficitur &longs;patium æqua­<lb/>le tardiore motu e&longs;t maius eo, quo conficitur velociore, patet. </s> </p> <p id="N16652" type="main"> <s id="N16654"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N16660" type="main"> <s id="N16662"><emph type="italics"/>Si datum mobile eodem motu æquabili duo percurrat &longs;patia, tempora mo­<lb/>tuum erunt vt &longs;patia, & vici&longs;&longs;im &longs;patia vt tempora.<emph.end type="italics"/></s> <s id="N1666B"> Probatur per Th. <!-- REMOVE S--><lb/>24. & 23. </s> </p> <p id="N16671" type="main"> <s id="N16673"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N1667F" type="main"> <s id="N16681"><!-- NEW --><emph type="italics"/>Si idem mobile temporibus æqualibus percurrat duo &longs;patia motu æquabili, <lb/>&longs;ed inæquali velocitate; </s> <s id="N16689"><!-- NEW -->&longs;patia erunt vt velocitates, & hæ vt illa; </s> <s id="N1668D"><!-- NEW -->imò &longs;i <lb/>&longs;patia &longs;unt vt velocitates, tempora erunt æqualia<emph.end type="italics"/>; pater etiam per <lb/>Th.25. </s> </p> <p id="N16698" type="main"> <s id="N1669A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N166A6" type="main"> <s id="N166A8"><!-- NEW --><emph type="italics"/>Si percurrantur à mobili æqualia &longs;patia, &longs;ed inæquali velocitate, ip&longs;æ ve­<lb/>locitates erunt in ratione permutata temporum, ide&longs;t maior velocitas re&longs;pon­<lb/>debit minori tempori, & minor maiori<emph.end type="italics"/>; Probatur per Th.23. </s> </p> <p id="N166B5" type="main"> <s id="N166B7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N166C3" type="main"> <s id="N166C5"><!-- NEW --><emph type="italics"/>Si duo mobilia mouentur motu æquabili, &longs;ed inæquali velocitate, & inæqua­<lb/>libus temporibus, &longs;patia &longs;unt in ratione compo&longs;ita ex ratione temporum, & ex <lb/>ratione velocitatum,<emph.end type="italics"/> &longs;i enim æqualia &longs;int tempora, &longs;patia erunt vt velo­<lb/>citates per Th.25. &longs;i æquales &longs;int velocitates, &longs;patia erunt vt tempora, per <lb/>Th.29. igitur &longs;i nec æquales velocitates, nec æqualia tempora, erit ratio <lb/>&longs;patiorum compo&longs;ita ex ratione temporum, & ex ratione velocitatum; <lb/>&longs;it ratio temporum 3/2 ratio velocitatum 2/3 compo&longs;ita ex vtraque erit 6/2 <lb/>&longs;eu 3. vt con&longs;tat ex ip&longs;is elementis. </s> </p> <pb pagenum="87" xlink:href="026/01/119.jpg"/> <p id="N166E0" type="main"> <s id="N166E2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N166EE" type="main"> <s id="N166F0"><!-- NEW --><emph type="italics"/>Si duo mobilia ferantur motu æquabili per diuer&longs;a &longs;patia, & diuer&longs;a velo­<lb/>citate, tempora erunt in ratione compo&longs;ita ex ratione &longs;patiorum & ratione <lb/>velocitatum permutata<emph.end type="italics"/>; </s> <s id="N166FD"><!-- NEW -->probatur eodem modo quo &longs;uperius Th. 30. &longs;it <lb/>ratio &longs;patiorum 4/1, velocitatum 4/2; </s> <s id="N16703"><!-- NEW -->permutetur hæc 1/4; componetur ex <lb/>vtraque 4/1, ide&longs;t 1/2, quæ e&longs;t ratio temporum. </s> </p> <p id="N16709" type="main"> <s id="N1670B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N16717" type="main"> <s id="N16719"><!-- NEW --><emph type="italics"/>Si duo mobilia æquabili motu ferantur per diuer&longs;a &longs;patia, & inæqualibus <lb/>temporibus; </s> <s id="N16721"><!-- NEW -->ratio velocitatum erit compo&longs;ita ex ratione &longs;patiorum, & ex ra­<lb/>tione temporum permutata<emph.end type="italics"/>; Probatur eodem modo; &longs;it ratio &longs;patiorum <lb/>4/2 temporum 1/2, permutetur 2/1, compo&longs;ita ex vtraque erit 2/2, ide&longs;t 4. <lb/>quæ e&longs;t ratio velocitatum. </s> </p> <p id="N1672E" type="main"> <s id="N16730"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1673C" type="main"> <s id="N1673E">Ob&longs;eruabis hæc omnia à vige&longs;imo Theoremate maiori ex parte tradi <lb/>à Galileo &longs;uo modo, optimo quidem, &longs;ed fortè longiore quàm par &longs;it, <lb/>nulla habita ratione cau&longs;arum phy&longs;icarum. </s> </p> <p id="N16745" type="main"> <s id="N16747"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N16753" type="main"> <s id="N16755"><!-- NEW --><emph type="italics"/>In motu naturaliter accelerato impetus nouus acquiritur &longs;ingulis in&longs;tanti­<lb/>bus<emph.end type="italics"/>; Probatur quia &longs;ingulis in&longs;tantibus e&longs;t eadem cau&longs;a nece&longs;&longs;aria, igi­<lb/>tur &longs;ingulis in&longs;tantibus aliquem effectum producit, per Ax. 12. l.1. &longs;ed <lb/>priorem non con&longs;eruat, vt dictum e&longs;t &longs;uprà, igitur nouum producit. </s> </p> <p id="N16764" type="main"> <s id="N16766"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N16772" type="main"> <s id="N16774"><emph type="italics"/>Hinc &longs;ingulis in&longs;tantibus æqualibus nouus impetus æqualis acquiritur,<emph.end type="italics"/> quip­<lb/>pe e&longs;t æqualis, imò eadem cau&longs;a, igitur æqualem effectum producit per <lb/>Ax.12. l.1. </s> </p> <p id="N16780" type="main"> <s id="N16782"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N1678E" type="main"> <s id="N16790"><!-- NEW --><emph type="italics"/>Hinc &longs;ingulis in&longs;tantibus intenditur impetus in hoc motu<emph.end type="italics"/>; cum &longs;ingulis <lb/>in&longs;tantibus producatur nouus, & prior con&longs;eruetur, cui cum addatur, <lb/>intenditur per Ax. 1. <!-- KEEP S--></s> </p> <p id="N1679E" type="main"> <s id="N167A0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N167AC" type="main"> <s id="N167AE"><emph type="italics"/>Hinc &longs;ingulis in&longs;tantibus æqualiter cre&longs;cit & intenditur impetus<emph.end type="italics"/> per Th. <!-- REMOVE S--><lb/>34. igitur æqualiter etiam &longs;ingulis in&longs;tantibus cre&longs;cit velocitas motus <lb/>per Ax.2. </s> </p> <p id="N167BB" type="main"> <s id="N167BD"><emph type="center"/><emph type="italics"/>Scholium<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N167C8" type="main"> <s id="N167CA"><!-- NEW -->Ob&longs;eruabis <expan abbr="dictū">dictum</expan> e&longs;&longs;e &longs;uprà <emph type="italics"/>instantibus æqualibus,<emph.end type="italics"/> quia temporis natura <lb/>aliter explicari non pote&longs;t, quàm per in&longs;tantia finita, vt demon&longs;trabimus <lb/>in Metaphy&longs;ica; </s> <s id="N167DC"><!-- NEW -->quid quid &longs;it, voco in&longs;tans totum illud tempus, quo res <lb/>aliqua &longs;imul producitur, &longs;iue &longs;it maius, &longs;iue minus, &longs;iue &longs;it pars maior, <lb/>vel minor, quod ad rem no&longs;tram nihil facit penitus; </s> <s id="N167E4"><!-- NEW -->nam dato quocun­<lb/>que tempore finito pote&longs;t dari maius & minus, quod certum e&longs;t; </s> <s id="N167EA"><!-- NEW -->igitur <lb/>totum illud tempus, quo producitur primus impetus acqui&longs;itus, vo-<pb pagenum="88" xlink:href="026/01/120.jpg"/>co in&longs;tans primum motus; cui æqualia deinde &longs;uccedunt tem­<lb/>pora. </s> </p> <p id="N167F7" type="main"> <s id="N167F9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N16805" type="main"> <s id="N16807"><!-- NEW --><emph type="italics"/>Hinc cre&longs;cit impetus iuxta progre&longs;&longs;ionem arithmeticam; </s> <s id="N1680D"><!-- NEW -->cum &longs;ingula in­<lb/>&longs;tantia æqualem impetum addant<emph.end type="italics"/>; </s> <s id="N16816"><!-- NEW -->&longs;i primo in&longs;tanti &longs;it vnus gradus, erunt <lb/>duo; productus &longs;cilicet alteri additus qui con&longs;eruatur, tertio erunt;. </s> <s id="N1681C"><lb/>quarto 4. quinto 5. &c. </s> <s id="N16820">igitur cre&longs;cit &longs;ecundum progre&longs;&longs;ionem arith­<lb/>meticam. </s> </p> <p id="N16825" type="main"> <s id="N16827"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N16833" type="main"> <s id="N16835"><emph type="italics"/>Eodem modo cre&longs;cit velocitas, quia &longs;ingulis in&longs;tantibus æqualia acquirun­<lb/>tur velocitatis momenta<emph.end type="italics"/> per Ax.2. & per Th.36. </s> </p> <p id="N1683F" type="main"> <s id="N16841"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N1684D" type="main"> <s id="N1684F"><emph type="italics"/>Maius &longs;patium acquiritur &longs;ecundo in&longs;tanti, quàm primo, quia &longs;ecundo<emph.end type="italics"/><lb/>in&longs;tanti motus e&longs;t velocior per Th.36. igitur maius conficitur &longs;patium, <lb/>tempore &longs;cilicet æquali per Def. <!-- REMOVE S-->2. l. <!-- REMOVE S-->1. idem dico de tertio, quar­<lb/>to, &c. </s> </p> <p id="N16860" type="main"> <s id="N16862"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N1686E" type="main"> <s id="N16870"><emph type="italics"/>Spatium quod acquiritur &longs;ecundò instanti e&longs;t ad &longs;patium quod acquiritur <lb/>primo vt velocitas, quæ e&longs;t &longs;ecundo ad velocitatem, quæ e&longs;t primo.<emph.end type="italics"/></s> <s id="N16879"><!-- NEW --> Patet per <lb/>Th.28. quia cum tempora illa &longs;int æqualia, &longs;patia &longs;unt nece&longs;&longs;ariò vt ve­<lb/>locitates; quippe æquali velocitati æquale &longs;patium re&longs;pondet tempore <lb/>æquali, igitur inæquale inæquali, igitur maius maiori, idem dico de <lb/>aliis in&longs;tantibus. </s> </p> <p id="N16885" type="main"> <s id="N16887"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N16893" type="main"> <s id="N16895"><emph type="italics"/>Hinc &longs;patium qucd acquiritur &longs;ecundo in&longs;tanti e&longs;t duplum illius, quod ac­<lb/>quiritur primo.<emph.end type="italics"/></s> <s id="N1689E"> Probatur, quia velocitas e&longs;t dupla per Th 38. igitur &longs;pa­<lb/>tium duplum, & triplum tertio, quadruplum quarto, &c. </s> </p> <p id="N168A3" type="main"> <s id="N168A5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N168B1" type="main"> <s id="N168B3"><!-- NEW --><emph type="italics"/>Hinc quodlibet &longs;patium cre&longs;cit æqualiter &longs;ingulis in&longs;tantibus æqualibus<emph.end type="italics"/>; </s> <s id="N168BC"><!-- NEW --><lb/>quia &longs;patia cre&longs;cunt vt motus, &longs;eu vt velocitates; hæ cre&longs;cunt æqualiter <lb/>&longs;ingulis in&longs;tantibus æqualibus per Th.36. igitur æqualiter cre&longs;cunt &longs;in­<lb/>gula &longs;patia per Th.40. </s> </p> <p id="N168C5" type="main"> <s id="N168C7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N168D3" type="main"> <s id="N168D5"><!-- NEW --><emph type="italics"/>Hinc &longs;patia cre&longs;cunt &longs;ingulis in&longs;tantibus æqualibus &longs;ecundùm progre&longs;&longs;io­<lb/>nem arithmeticam<emph.end type="italics"/>; quia cre&longs;cit vt velocitas per Th.40. hæc vt impetus <lb/>per Th.38. hic demum iuxta progre&longs;&longs;ionem arithmeticam per Th. 37. <lb/>igitur &longs;i &longs;patium acqui&longs;itum primo in&longs;tanti &longs;it 1. acqui&longs;itum &longs;ecundo erit <lb/>2. tertio 3. quarto 4. &c. </s> <s id="N168E6">hinc &longs;patia acqui&longs;ita &longs;ingulis in&longs;tantibus &longs;unt <lb/>vt &longs;eries numerorum, qui componunt progre&longs;&longs;ionem &longs;implicem, &longs;cilicet <lb/>1.2.3.4.5.6. &c. </s> <s id="N168ED"><!-- NEW -->dixi &longs;ingulis in&longs;tantibus æqualibus, quod e&longs;t apprimè <lb/>tenendum; &longs;i enim a&longs;&longs;umantur partes temporis maiores, perturbatur <lb/>hæc progre&longs;&longs;io, de quo infrà. </s> </p> <pb pagenum="89" xlink:href="026/01/121.jpg"/> <p id="N168F9" type="main"> <s id="N168FB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N16907" type="main"> <s id="N16909"><!-- NEW --><emph type="italics"/>Hinc pete&longs;t dici cre&longs;cere velocitatem quolibet in&longs;tanti iuxta rationem &longs;patij <lb/>quod illo in&longs;tanti decurritur<emph.end type="italics"/>; </s> <s id="N16914"><!-- NEW -->quod certè verum e&longs;t, dum intelligatur legi­<lb/>timus horum verborum &longs;en&longs;us; </s> <s id="N1691A"><!-- NEW -->quidquid reclamet Saluiatus apud <lb/>Galil. <!-- REMOVE S-->dialogo 3. modò a&longs;&longs;umatur progre&longs;&longs;io incrementi in &longs;ingulis in­<lb/>&longs;tantibus, in quibus reuerà fit; cur enim potiùs in vno quàm in alio? </s> <s id="N16924"><!-- NEW --><lb/>quippe &longs;i comparetur velocitas vnius in&longs;tantis cum velocitate alterius; </s> <s id="N16929"><!-- NEW --><lb/>haud dubiè erit eadem vtriu&longs;que ratio, quæ &longs;patiorum; </s> <s id="N1692E"><!-- NEW -->&longs;i enim vno in­<lb/>&longs;tanti percurritur vnum &longs;patium cum vno velocitatis gradu; </s> <s id="N16934"><!-- NEW -->certè in­<lb/>&longs;tanti æquali acquiritur duplum &longs;patium cum duobus velocitatis gradi­<lb/>bus, nec obe&longs;t, quod obiicit Galileus tunc motus e&longs;&longs;e æquabiles; </s> <s id="N1693C"><!-- NEW -->quia <lb/>motus qui fit in in&longs;tanti debet con&longs;iderari vt æquabilis; </s> <s id="N16942"><!-- NEW -->appello enim <lb/>in&longs;tans totum illud tempus, quo &longs;imul acquiritur aliquid impetus, ali­<lb/>quid enim &longs;imul acquiri nece&longs;&longs;e e&longs;t; </s> <s id="N1694A"><!-- NEW -->nec demum ob&longs;tat quod dicit, dari <lb/>non po&longs;&longs;e motum in&longs;tantaneum, quod multi haud dubiè negabunt; </s> <s id="N16950"><!-- NEW -->ego <lb/>in Metaphy&longs;ica explicabo quonam pacto dari po&longs;&longs;it motus in&longs;tanta­<lb/>neus, qui reuerà datur actu, non potentiâ; </s> <s id="N16958"><!-- NEW -->quia quacunque duratione <lb/>data pote&longs;t dari minor; igitur quocunque dato motu pote&longs;t dari minor. </s> </p> <p id="N1695E" type="main"> <s id="N16960"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1696C" type="main"> <s id="N1696E"><!-- NEW -->Ob&longs;eruabis primò hanc &longs;patiorum rationem, quæ e&longs;t eadem cum ra­<lb/>tione velocitatum a&longs;&longs;umendam tantùm e&longs;&longs;e in iis &longs;patiis, quæ acquirun­<lb/>tur &longs;ingulis in&longs;tantibus; </s> <s id="N16976"><!-- NEW -->&longs;i enim accipiantur partes temporis maiores, quæ <lb/>conflentur ex multis in&longs;tantibus; </s> <s id="N1697C"><!-- NEW -->haud dubiè maior erit ratio &longs;patio­<lb/>rum, quàm velocitatum.v.g.&longs;i primo in&longs;tanti acquiratur primo &longs;patium, <lb/>&longs;ecundo, 2.tertio, 3.quarto 4.igitur &longs;i <expan abbr="cõparetur">comparetur</expan> velocitas primi in&longs;tantis <lb/>cum velocitate quarti æqualis erit, vt ratio &longs;patiorum, id e&longs;t, vt 1. ad 4. <lb/>At verò &longs;i accipiatur pars temporis con&longs;tans duobus in&longs;tantibus, hæc 4. <lb/>in&longs;tantia conflabunt tantùm 2. partes temporis æquales; </s> <s id="N1698E"><!-- NEW -->in prima ac­<lb/>quirentur 3.&longs;patia, in &longs;ecunda 7.vt patet: </s> <s id="N16994"><!-- NEW -->&longs;ed quia velocitas primæ par­<lb/>tis temporis non e&longs;t æquabilis, nec etiam velocitas &longs;ecundæ; </s> <s id="N1699A"><!-- NEW -->addantur <lb/>velocitates primi & &longs;ecundi in&longs;tantis, itemque &longs;eor&longs;im velocitates tertij, <lb/>& quarti; </s> <s id="N169A2"><!-- NEW -->certè ratio collectorum erit vt ratio &longs;patiorum; &longs;i enim velo­<lb/>citas &longs;ecundi in&longs;tantis comparetur cum velocitate quarti e&longs;t tantùm <lb/>1/2 cum tamen primum &longs;patium &longs;it ad &longs;ecundum in ratione 3/7. </s> </p> <p id="N169AA" type="main"> <s id="N169AC">Secundò, &longs;i comparentur &longs;patia cum temporibus e&longs;t alia ratio v.g.&longs;pa­<lb/>tium acqui&longs;itum vno in&longs;tanti &longs;e habet ad &longs;patium acqui&longs;itum in duobus <lb/>in&longs;tantibus, vt 1, ad 3.in tribus vt 1.ad 6.in 4. vt 1. ad 10. </s> </p> <p id="N169B3" type="main"> <s id="N169B5"><!-- NEW -->Tertiò ob&longs;eruabis, non po&longs;&longs;e &longs;en&longs;u percipi in&longs;tans, imò neque tempo­<lb/>ris partem ex mille in&longs;tantibus conflatam; </s> <s id="N169BB"><!-- NEW -->nec etiam &longs;patium quod ac­<lb/>quiritur primo in&longs;tanti; </s> <s id="N169C1"><!-- NEW -->adhibenda &longs;unt tamen in&longs;tantia nece&longs;&longs;ariò ad <lb/>explicandam proportionem huius accelerationis, quæ fit in &longs;ingulis in­<lb/>&longs;tantibus; vt verò rem i&longs;tam reuocemus ad &longs;en&longs;ibilem praxim, a&longs;&longs;ume­<lb/>mus proportionem aliam &longs;en&longs;ibilem, quæ proximè ad veram accedit, nec <lb/>ferè &longs;en&longs;ibiliter fallere pote&longs;t, de qua infrà. </s> </p> <pb pagenum="90" xlink:href="026/01/122.jpg"/> <p id="N169D1" type="main"> <s id="N169D3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N169DF" type="main"> <s id="N169E1"><!-- NEW --><emph type="italics"/>Collectio &longs;patiorum e&longs;t &longs;umma terminorum huius progre&longs;&longs;ionis arithmeticæ<emph.end type="italics"/>; <lb/></s> <s id="N169EB"><!-- NEW -->Cùm enim ratio &longs;patiorum &longs;it vt ratio velocitatum; </s> <s id="N169EF"><!-- NEW -->dum &longs;cilicet hæc <lb/>progre&longs;&longs;io accipitur in in&longs;tantibus, & ratio velocitatum vt ratio incre­<lb/>menti impetuum; vt con&longs;tat ex dictis, & hæc &longs;equatur &longs;implicem <lb/>progre&longs;&longs;ionem 1. 2. 3. 4. &c. </s> <s id="N169F9">certè collectio &longs;patiorum e&longs;t &longs;umma ter­<lb/>minorum. </s> </p> <p id="N169FE" type="main"> <s id="N16A00"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N16A0C" type="main"> <s id="N16A0E"><emph type="italics"/>Hinc cognito primo termino, & vltimo, id e&longs;t &longs;patio quod per curritur primo <lb/>in&longs;tanti & &longs;patio quod percurritur vltimo instanti, cogno&longs;citur &longs;umma, id e&longs;t <lb/>collectio &longs;patiorum, id e&longs;t, totum &longs;patium confectum.<emph.end type="italics"/> v.g.&longs;i primus terminus, <lb/>&longs;ecundus S.igitur &longs;umma e&longs;t 36. quippe vltimus terminus indicat nume­<lb/>rum terminorum, quia primus e&longs;t &longs;emper vnitas, & progre&longs;&longs;iuus etiam <lb/>vnitas. </s> </p> <p id="N16A20" type="main"> <s id="N16A22"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N16A2E" type="main"> <s id="N16A30"><!-- NEW --><emph type="italics"/>Hinc cognita &longs;umma & vltimo termino cogno&longs;citur etiam numerus in&longs;tan­<lb/>tium æqualium, qui &longs;emper est idem cum numero terminorum, cogno&longs;citur <lb/>etiam primus terminus, id e&longs;t &longs;patium quod primo instanti percurritur, cogno­<lb/>&longs;cuntur etiam gradus velocitatis<emph.end type="italics"/>; </s> <s id="N16A3F"><!-- NEW -->quippe hæc omnia &longs;unt in eadem ratio­<lb/>ne; </s> <s id="N16A45"><!-- NEW -->quæ omnia con&longs;tant ex regulis arithmeticis præter alia multa data, <lb/>quæ lubens omitto; </s> <s id="N16A4B"><!-- NEW -->tùm quia Phy&longs;icam non &longs;apiunt, tùm quia hypothe­<lb/>&longs;is illa e&longs;t impo&longs;&longs;ibilis phy&longs;icè; quis enim &longs;en&longs;u percipere po&longs;&longs;it & di­<lb/>&longs;tinguere vnum temporis in&longs;tans, vel &longs;patij punctum? </s> <s id="N16A53">licèt recen&longs;enda <lb/>fuerit hæc accelerati motus proportio in in&longs;tantibus, vt ad &longs;ua phy&longs;ica <lb/>principia reduceretur. </s> </p> <p id="N16A5A" type="main"> <s id="N16A5C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N16A68" type="main"> <s id="N16A6A"><!-- NEW --><emph type="italics"/>Data &longs;umma progre&longs;&longs;ionis huius &longs;implicis, inuenietur numerus terminorum, <lb/>&longs;i inueniatur numerus, per quem diuidatur, qui &longs;uperet tantùm vnitate du­<lb/>plum quotientis<emph.end type="italics"/>; </s> <s id="N16A77"><!-- NEW -->quippe habebis in duplo quotientis numerum termino­<lb/>rum v.g. <!-- REMOVE S-->&longs;it &longs;umma 10. diui&longs;or &longs;it 5. quotiens 2. duplus 4. hic e&longs;t nume­<lb/>rus terminorum datæ &longs;ummæ; </s> <s id="N16A81"><!-- NEW -->&longs;it alia &longs;umma 21. diui&longs;or &longs;it 7.quotiens 3. <lb/>numerus terminorum 6. &longs;it alia &longs;umma 36. dini&longs;or &longs;it 9. quotiens 4. nu­<lb/>merus terminorum 8. &longs;it alia &longs;umma 45. partitor &longs;it 10. quotiens 4 1/2, <lb/>numerus terminorum 9. quomodo verò hic partitor inueniri po&longs;&longs;it, vi­<lb/>derint Arithmetici; </s> <s id="N16A8D"><!-- NEW -->nec enim e&longs;t huius loci, quamquam datâ &longs;ummâ <lb/>huius progre&longs;&longs;ionis &longs;implicis facilè cogno&longs;ci pote&longs;t numerus termino­<lb/>rum; duplicetur enim, & radix 9. neglecto re&longs;iduo dabit numerum ter­<lb/>minorum v.g. <!-- REMOVE S-->&longs;it &longs;umma 21. duplicetur, erit 42. rad. </s> <s id="N16A99"><!-- NEW -->9. 6. dat numerum <lb/>terminorum; &longs;it &longs;umma 36. duplicetur, erit 72.rad.9.8. dabit numerum <lb/>terminorum. </s> </p> <p id="N16AA1" type="main"> <s id="N16AA3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N16AAF" type="main"> <s id="N16AB1"><!-- NEW --><emph type="italics"/>Semper decre&longs;cit proportio incrementi velocitatis, id est maior est proportio <lb/>velocitatis &longs;ecundi in&longs;tantis ad primum quàm tertij ad &longs;ecundum, & maior<emph.end type="italics"/><pb pagenum="91" xlink:href="026/01/123.jpg"/><emph type="italics"/>tertij ad &longs;ecundum quàm quarti ad tertium, atque ita deinceps<emph.end type="italics"/>; </s> <s id="N16AC5"><!-- NEW -->&longs;it enim <lb/>primo in&longs;tanti velocitas vt 1.&longs;ecundo erit, vt 2.tertio, vt 3.quarto, vt 4. <lb/>&longs;ed maior e&longs;t proportio 2.ad 1.quàm 3.ad 2. & hæc maior quàm 4. ad 3. <lb/>atque ita deinceps; </s> <s id="N16ACF"><!-- NEW -->&longs;imiliter maior e&longs;t proportio &longs;patij quod percurritur <lb/>&longs;ecundo in&longs;tanti ad &longs;patium, quod percurritur primo, quàm &longs;patij, quod <lb/>percurritur &longs;ecundo in&longs;tanti ad &longs;patium, quod percurritur primo quàm <lb/>&longs;patij quod percurritur tertio ad &longs;patium, quod percurritur &longs;ecundo, at­<lb/>que ita deinceps; e&longs;t enim eadem ratio &longs;patiorum quæ &longs;ingulis in&longs;tanti­<lb/>bus re&longs;pondent, quæ velocitatum, vt demon&longs;tratum e&longs;t &longs;uprà. </s> </p> <p id="N16ADD" type="main"> <s id="N16ADF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N16AEB" type="main"> <s id="N16AED"><!-- NEW --><emph type="italics"/>Minor e&longs;t proportio totius &longs;patij, quod acquiritur duobus instantibus ad to<lb/>tum &longs;patium, quod acquiritur vno, quàm &longs;it illius, quod acquiritur quatuor in­<lb/>&longs;tantibus ad aliud, quod acquiritur duobus<emph.end type="italics"/>; patet ex dictis; </s> <s id="N16AFA"><!-- NEW -->&longs;i enim primo <lb/>in&longs;tanti acquiritur vnum &longs;patium, &longs;ecundo acquiruntur 2.igitur duobus <lb/>&longs;imul acquirantur 3. igitur proportio e&longs;t vt 3.ad 1.Sed &longs;i duobus acqui­<lb/>runtur 3. &longs;patia; </s> <s id="N16B04"><!-- NEW -->certè 4.in&longs;tantibus acquiruntur 10. igitur proportio e&longs;t <lb/>vt 10.ad 3. &longs;ed proportio 10/3 e&longs;t maior 3/1, erit adhuc maior proportio &longs;pa­<lb/>tij quod acquiretur 6. in&longs;tantibus ad illud quod acquiritur tribus; e&longs;t <lb/>enim (21/6) vt patet. </s> </p> <p id="N16B0E" type="main"> <s id="N16B10"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N16B1C" type="main"> <s id="N16B1E"><!-- NEW --><emph type="italics"/>Si componatur æquabilis motus ex &longs;ubdupla velocitate maxima, & mini­<lb/>ma, æquali tempore, idem &longs;patium percurretur hoc motu naturaliter accelera­<lb/>to<emph.end type="italics"/>; </s> <s id="N16B2B"><!-- NEW -->&longs;it enim maxima velocitas vt 6. minima vt 1. motu naturaliter acce­<lb/>lerato percurrentur &longs;patia 21. cuius &longs;ummæ termini &longs;unt 6.igitur 6. in­<lb/>&longs;tantibus con&longs;tat hic motus; </s> <s id="N16B33"><!-- NEW -->accipiatur &longs;ubduplum maximæ, & minimæ <lb/>velocitatis, &longs;cilicet 3 1/2. sítque velocitas motus æquabilis in&longs;tantium 6. <lb/>haud dubiè &longs;i ducantur 3 1/2 in 6 erunt 21.ratio ex eo petitur quod &longs;cili­<lb/>cet, vt habeatur &longs;umma progre&longs;&longs;ionis arithmeticæ, debet addi primus <lb/>terminus maximo, & a&longs;&longs;umi &longs;ubduplum totius; </s> <s id="N16B3F"><!-- NEW -->illudque ducere in nu­<lb/>merum terminorum per regulam arithmeticam; </s> <s id="N16B45"><!-- NEW -->atqui eadem e&longs;t ratio <lb/>velocitatum, quæ &longs;patiorum; vt dictum e&longs;t &longs;uprà; &longs;cilice, in &longs;ingulis <lb/>in&longs;tantibus. </s> </p> <p id="N16B4D" type="main"> <s id="N16B4F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N16B5B" type="main"> <s id="N16B5D"><!-- NEW --><emph type="italics"/>Si a&longs;&longs;umantur partes temporis majores; quæ &longs;cilicet pluribus in&longs;tantibus <lb/>constent, &longs;erueturque eadem accelerationis progre&longs;&longs;io arithmetica, &longs;patium <lb/>quod ex &longs;umma huius progre&longs;&longs;ionis re&longs;ultabit, erit minus vero,<emph.end type="italics"/> &longs;int enim 6.in­<lb/>&longs;tantia, & cuilibet iuxta progre&longs;&longs;ionem prædictam &longs;uum &longs;patium re&longs;pon­<lb/>deat, haud dubiè &longs;patium &longs;ecundi erit duplum &longs;patij primi, & tertium <lb/>triplum, &c. </s> <s id="N16B70">vt con&longs;tat ex dictis; </s> <s id="N16B73"><!-- NEW -->igitur erunt &longs;patia 21. iam verò a&longs;&longs;u­<lb/>mantur 3. partes temporis, quarum quælibet ex 2. con&longs;tet in&longs;tantibus; </s> <s id="N16B79"><!-- NEW --><lb/>primæ parti tria ex prædictis &longs;patiis re&longs;pondeant; </s> <s id="N16B7E"><!-- NEW -->certè &longs;i &longs;eruetur pro­<lb/>gre&longs;&longs;io arithmetica, &longs;ecundæ re&longs;pondebunt 6. & tertiæ 9. igitur totum <lb/>&longs;patium erit 18. minus vero quod erat 21. &longs;i verò a&longs;&longs;umantur tantùm 2. <lb/>partes, quarum quælibet tribus in&longs;tantibus con&longs;tet; </s> <s id="N16B88"><!-- NEW -->primæ parti re&longs;pon-<pb pagenum="92" xlink:href="026/01/124.jpg"/>debunt 6. &longs;ecundæ 12. igitur &longs;umma erit 18. minor vero &longs;patio &longs;cilicet <lb/>21.hinc vides &longs;uppo&longs;ito eodem in&longs;tantium numero &longs;patium e&longs;&longs;e &longs;emper <lb/>æquale, &longs;iue a&longs;&longs;umantur partes maiores temporis, &longs;iue minores, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;up­<lb/>po&longs;itis 6.in&longs;tantibus, ex quibus totum &longs;patium 21.con&longs;equitur, &longs;iue a&longs;&longs;u­<lb/>mantur tres partes, quarum quælibet con&longs;tet 2. in&longs;tantibus, &longs;iue duæ, <lb/>quarum quælibet con&longs;tet tribus, &longs;patium quod ex illis re&longs;ultat, e&longs;t &longs;em­<lb/>per idem &longs;cilicet 18. a&longs;&longs;umptis verò 8. in&longs;tantibus, & totali &longs;patio, quod <lb/>illis re&longs;pondet 36. &longs;patium quod ex partibus re&longs;ultabit erit 30. &longs;iue &longs;int <lb/>duæ partes, quarum quælibet con&longs;tet 4. in&longs;tantibus, &longs;iue &longs;int 4. quarum <lb/>quælibet con&longs;tet duobus: </s> <s id="N16BA7"><!-- NEW -->hinc rur&longs;us vides a&longs;&longs;umpto maiori in&longs;tantium <lb/>numero &longs;patium verum habere maiorem rationem ad non verum, quàm <lb/>a&longs;&longs;umpto minori in&longs;tantium numero, v.g.a&longs;&longs;umantur 4.in&longs;tantia, &longs;umma <lb/>&longs;patiorum erit 10. &longs;i verò a&longs;&longs;umantur 2.partes temporis, quarum quæli­<lb/>bet duobus in&longs;tantibus re&longs;pondeat; </s> <s id="N16BB3"><!-- NEW -->&longs;umma &longs;patij erit 9.igitur ratio ve­<lb/>ri &longs;patij ad non verum e&longs;t (10/9). a&longs;&longs;umantur 6. in&longs;tantia &longs;patij veri, &longs;umma <lb/>erit 21.non veri 18. igitur ratio (21/18) &longs;eu 7/6 quæ maior e&longs;t priori: denique <lb/>a&longs;&longs;umantur 8. in&longs;tantia &longs;patij veri, &longs;umma erit 36. non veri 30 igitur ra­<lb/>tio (36/30) &longs;eu 6/3 quæ maior e&longs;t prioribus, atque ita deinceps. </s> </p> <p id="N16BBF" type="main"> <s id="N16BC1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N16BCD" type="main"> <s id="N16BCF"><!-- NEW --><emph type="italics"/>Datis duabus partibus temporis, & cognito &longs;patio quod percurritur in prima, <lb/>matius &longs;patium re&longs;pondebit &longs;ecundæ quo vtraque in plures partes minores diui­<lb/>detur, &longs;uppo&longs;ita &longs;emper eadem progre&longs;&longs;ione arithmetica in ip&longs;o incremento<emph.end type="italics"/>; </s> <s id="N16BDC"><!-- NEW --><lb/>&longs;int enim duæ partes temporis &longs;en&longs;ibiles æquales AG. GH. & &longs;pa­<lb/>tium quod percurritur prima parte temporis AG &longs;it HI; </s> <s id="N16BE3"><!-- NEW -->in &longs;ecunda <lb/>percurretur IO, id e&longs;t, duplum HI; </s> <s id="N16BE9"><!-- NEW -->at verò diuidatur pars temporis <lb/>AG in duas æquales AF, FG, & con&longs;equenter totum tempus AH in 4. <lb/>æquales; </s> <s id="N16BF1"><!-- NEW -->haud dubiè in prima AF percurretur NP &longs;ubtripla HI, & in <lb/>&longs;ecunda FG percurretur PK dupla NP; </s> <s id="N16BF7"><!-- NEW -->igitur in 4. partibus temporis <lb/>AH percurretur &longs;patium decuplum PN, &longs;ed HO e&longs;t tantùm nonecupla <lb/>NP; </s> <s id="N16BFF"><!-- NEW -->igitur re&longs;ultabit maius &longs;patium in 4.partibus temporis, quam in dua­<lb/>bus; licèt duæ æquiualeant 4. iuxta progre&longs;&longs;ionem arithmeticam. </s> </p> <p id="N16C05" type="main"> <s id="N16C07"><!-- NEW -->Similiter AF diuidatur bifariam in E. & tota AH in 8. æquales AE; </s> <s id="N16C0B"><!-- NEW --><lb/>certè primis 4.percurretur idem &longs;patium ML æquale NK & HI; </s> <s id="N16C10"><!-- NEW -->igitur <lb/>in prima AE percurretur MR. cuius ML &longs;it decupla; </s> <s id="N16C16"><!-- NEW -->nam 4. terminis <lb/>re&longs;pondet &longs;umma 10. &longs;ed 8. terminis id e&longs;t 8.partibus temporis re&longs;pon­<lb/>det &longs;umma; </s> <s id="N16C1E"><!-- NEW -->6. æqualium RM; </s> <s id="N16C22"><!-- NEW -->&longs;ed HO tripla ML e&longs;t tantum 30. <lb/>æqualium MR; igitur in 8.partibus re&longs;ultabit maius &longs;patium, quàm in <lb/>4.quæ æquiualent 8. </s> </p> <p id="N16C2A" type="main"> <s id="N16C2C"><!-- NEW -->Ex quibus etiam con&longs;tat quo plures accipientur partes temporis ma­<lb/>ius &longs;patium re&longs;ultare, donec tandem perueniatur ad vltima in&longs;tantia, ex <lb/>quibus re&longs;ultat maximum; </s> <s id="N16C34"><!-- NEW -->& &longs;i accipias AG partes temporis AG. GH. <lb/>habebitur HO; </s> <s id="N16C3A"><!-- NEW -->&longs;i verò 4.æquales AF, cre&longs;cet &longs;patium &longs;eu &longs;umma 1/9 HO; </s> <s id="N16C3E"><!-- NEW --><lb/>&longs;i autem 8. æquales AE cre&longs;cet 1/5 HO; </s> <s id="N16C43"><!-- NEW -->&longs;i porrò 16. æquales AD cre&longs;­<lb/>cet (22/108) &longs;i 32. æquales AC cre&longs;cet (120/408); &longs;i 64. æquales AB cre&longs;cet (496/1584). </s> </p> <pb pagenum="93" xlink:href="026/01/125.jpg"/> <p id="N16C4D" type="main"> <s id="N16C4F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N16C5B" type="main"> <s id="N16C5D"><!-- NEW --><emph type="italics"/>In progre&longs;&longs;ione arithmetica &longs;i diuidatur numerus terminorum bifariam æ­<lb/>qualiter nunquam &longs;umma po&longs;terioris &longs;egmenti e&longs;t tripla prioris<emph.end type="italics"/>; &longs;ed &longs;i acci­<lb/>piantur duo termini e&longs;t tantùm 2/1, &longs;i 4. e&longs;t 7/3 &longs;i 6. e&longs;t (15/6), &longs;i 8. e&longs;t (26/10), &longs;i 10­<lb/>(40/15), &longs;i 12. (57/21), &longs;i 14. (77/28), atque ita deinceps. </s> </p> <p id="N16C6C" type="main"> <s id="N16C6E"><!-- NEW -->Ex quo ob&longs;erua mirabilem con&longs;equutionem; </s> <s id="N16C72"><!-- NEW -->quippe &longs;i a&longs;&longs;umantur <lb/>tantùm duo termini, & diuidantur bifariam, &longs;umma po&longs;terioris medie­<lb/>tatis e&longs;t tripla primæ minùs vnitate; </s> <s id="N16C7A"><!-- NEW -->&longs;i accipiantur 4. e&longs;t tripla minùs <lb/>2. &longs;i 6. minùs 3. &longs;i 8. minùs 4. &longs;i 10. minùs 5. &longs;i 12. minùs 6. &longs;i 14. mi­<lb/>nùs 7. atque ita deinceps; vnde &longs;umma po&longs;terioris medietatis e&longs;t &longs;emper <lb/>tripla minùs numero &longs;uorum terminorum, vel quod clarum e&longs;t minùs <lb/>&longs;ubduplo vltimi, &longs;eu maximi termini, vel numeri terminorum totius <lb/>progre&longs;&longs;ionis, quod probè omninò tenendum e&longs;t, vt omnes experientiæ <lb/>explica ri po&longs;&longs;int, quod infrà faciemus. </s> </p> <p id="N16C8A" type="main"> <s id="N16C8C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s> </p> <p id="N16C98" type="main"> <s id="N16C9A"><!-- NEW --><emph type="italics"/>Ex dictis hactenus facilè redditur ratio maioris ictus eiu&longs;dem corporis im­<lb/>pacti quod cadit ex maiori altitudine<emph.end type="italics"/>; fuit hyp. </s> <s id="N16CA5">1. &longs;ed ideò e&longs;t maior ictus, <lb/>quia maior imprimitur impetus, vt patet, at ideò maior impetus impri­<lb/>mitur, quia maior e&longs;t imprimens per Ax. 2. cre&longs;cit enim impetus, vt <lb/>con&longs;tat ex dictis. </s> </p> <p id="N16CAE" type="main"> <s id="N16CB0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N16CBC" type="main"> <s id="N16CBE"><!-- NEW --><emph type="italics"/>Hinc quoque ratio maximæ percu&longs;&longs;ionis ex &longs;olo pondere cadentis illius arie­<lb/>tis inflictæ<emph.end type="italics"/>; quâ &longs;cilicet altè infiguntur lignei pali, quibus in mediis <lb/>aquis tanquam iacto fundamini &longs;uperædificatur ingens &longs;æpè ædificij <lb/>moles. </s> </p> <p id="N16CCD" type="main"> <s id="N16CCF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N16CDB" type="main"> <s id="N16CDD"><!-- NEW --><emph type="italics"/>Hinc ex minima altitudine cadens corpus graue minimum ferè ictum in­<lb/>fligit<emph.end type="italics"/>; quia primus impetus valdè debilis e&longs;t, qui tamen deinde facta <lb/>acce&longs;&longs;ione maximus ferè euadit. </s> </p> <p id="N16CEA" type="main"> <s id="N16CEC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N16CF8" type="main"> <s id="N16CFA"><!-- NEW --><emph type="italics"/>Hinc ratio, cur tanta &longs;it differentia impetus grauitationis, & percu&longs;&longs;ionis <lb/>ab eodem mobili<emph.end type="italics"/>; </s> <s id="N16D05"><!-- NEW -->quia &longs;cilicet quantumuis tempore breui&longs;&longs;imo mouea­<lb/>tur, plurimis tamen eius motus durat in&longs;tantibus; atqui quolibet in&longs;tan­<lb/>ti motus acquiritur impetus æqualis primo impetui grauitationis, vt <lb/>con&longs;tat ex dictis. </s> <s id="N16D0F"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it mobile quod moueatur per mille in&longs;tantia <lb/>(modicum certè tempus & minimè &longs;en&longs;ibile) po&longs;t hunc motum impetus <lb/>erit millecuplus; </s> <s id="N16D1B"><!-- NEW -->igitur effectus etiam millecuplus; quæ omnia con&longs;tant <lb/>ex dictis. </s> </p> <p id="N16D21" type="main"> <s id="N16D23"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N16D2F" type="main"> <s id="N16D31"><!-- NEW --><emph type="italics"/>Hinc percu&longs;&longs;io quæ fit in primo in&longs;tanti contactus cre&longs;cit vt tempus<emph.end type="italics"/>; </s> <s id="N16D3A"><!-- NEW -->quia <lb/>cùm &longs;ingulis in&longs;tantibus cre&longs;cat impetus per partes æquales, & cùm per­<lb/>cu&longs;&longs;io &longs;it vt impetus; etiam erit vt tempus; </s> <s id="N16D42"><!-- NEW -->igitur percu&longs;&longs;io, quæ fit po&longs;t <lb/>duo in&longs;tantia motus eiu&longs;dem corporis grauis deor&longs;um cadentis e&longs;t du-<pb pagenum="94" xlink:href="026/01/126.jpg"/>pla illius, quæ &longs;it po&longs;t vnum in&longs;tans motus, & quæ fit po&longs;t tria tripla, po&longs;t <lb/>4. quadrupla, atque ita deinceps; cùm enim æqualibus temporibus æqua­<lb/>lia acquirantur velocitatis momenta, id e&longs;t æquales impetus, impetus <lb/>erunt vt tempora, percu&longs;&longs;iones vt impetus, igitur percu&longs;&longs;iones vt tem­<lb/>pora. </s> </p> <p id="N16D55" type="main"> <s id="N16D57"><!-- NEW -->Dixi in primo in&longs;tanti contactus; nam reuerâ &longs;ecundò in&longs;tanti con­<lb/>tactus, ni&longs;i fiat reflexio, augetur vis ictus, quia cau&longs;a nece&longs;&longs;aria e&longs;t ap­<lb/>plicata. </s> </p> <p id="N16D5F" type="main"> <s id="N16D61"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N16D6D" type="main"> <s id="N16D6F"><!-- NEW --><emph type="italics"/>Hinc po&longs;&longs;unt comparari duæ percu&longs;&longs;iones duorum grauium inæqualium <lb/>dum cadunt deor&longs;um<emph.end type="italics"/>; </s> <s id="N16D7A"><!-- NEW -->&longs;i enim cadunt æqualibus temporibus, percu&longs;&longs;io­<lb/>nes erunt vt corpora &longs;eu grauitates, vt patet v.g. <!-- REMOVE S-->corpus 2. librarum po&longs;t <lb/>2. in&longs;tantia motus infligit duplam percu&longs;&longs;ionem illius, quam infligit cor­<lb/>pus vnius libræ po&longs;t 2. in&longs;tantia motus; </s> <s id="N16D86"><!-- NEW -->&longs;i verò tempora motus &longs;unt inæ­<lb/>qualia, & grauitates æquales, percu&longs;&longs;iones erunt vt tempora; </s> <s id="N16D8C"><!-- NEW -->&longs;i demum <lb/>grauitates inæquales, & tempora motus inæqualia, percu&longs;&longs;iones erunt <lb/>in ratione compo&longs;ita ex ratione grauitatum & temporum, quæ omnia <lb/>patent ex dictis in Th. &longs;uperioribus, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it corpus duarum librarum, <lb/>& alterum trium librarum; </s> <s id="N16D9C"><!-- NEW -->primum moueatur per 5. in&longs;tantia, & &longs;ecun­<lb/>dum 2.per 5. ratio grauitatum e&longs;t 3/2; </s> <s id="N16DA2"><!-- NEW -->ratio temporum e&longs;t 7/5; </s> <s id="N16DA6"><!-- NEW -->compo&longs;ita <lb/>ex vtraque erit (21/10); & hæc e&longs;t ratio percu&longs;&longs;ionum. </s> </p> <p id="N16DAC" type="main"> <s id="N16DAE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N16DBA" type="main"> <s id="N16DBC"><emph type="italics"/>Hinc pote&longs;t &longs;ciri ratio percu&longs;&longs;ionis. </s> <s id="N16DC1"><!-- NEW -->& grauitationis eiu&longs;dem mobilis in pri­<lb/>mo in&longs;tanti vtriu&longs;que, &longs;i cogno&longs;catur numerus in&longs;tantium motus<emph.end type="italics"/>; </s> <s id="N16DCA"><!-- NEW -->cum enim <lb/>&longs;ingulis in&longs;tantibus æqualis impetus accedat, vt &longs;æpè dictum e&longs;t; </s> <s id="N16DD0"><!-- NEW -->certè <lb/>erit percu&longs;&longs;io ad grauitationem, vt numerus in&longs;tantium motus ad vnita­<lb/>tem, v.g. <!-- REMOVE S-->grauitatio &longs;it vt 4.&longs;it&qacute;ue motus eiu&longs;dem corporis per 8. in&longs;tan­<lb/>tia; percu&longs;&longs;io erit ad grauitationem, vt 32. ad 4.vel vt 8.ad 1.quæ om­<lb/>nia con&longs;tant ex dictis. </s> </p> <p id="N16DDE" type="main"> <s id="N16DE0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N16DEC" type="main"> <s id="N16DEE"><!-- NEW --><emph type="italics"/>Hinc data percu&longs;&longs;ione, &longs;i cogno&longs;ceretur probè numerus in&longs;tantium motus, <lb/>dari po&longs;&longs;et grauitatio ip&longs;i æqualis<emph.end type="italics"/>; </s> <s id="N16DF9"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it percu&longs;&longs;io dati corporis cadentis <lb/>per 8.in&longs;tantia, eius percu&longs;&longs;io e&longs;t octupla grauitationis eiu&longs;dem per Th. <!-- REMOVE S--><lb/>56. igitur &longs;i detur grauitatio octupla huius, erit æqualis datæ percu&longs;­<lb/>&longs;ioni; dabitur autem grauitatio octupla, &longs;i detur corpus eiu&longs;dem mate­<lb/>riæ octuplò grauius, vt con&longs;tat. </s> </p> <p id="N16E08" type="main"> <s id="N16E0A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N16E16" type="main"> <s id="N16E18"><emph type="italics"/>Hinc primo in&longs;tanti grauitationis nullum ferè &longs;entitur pondus,<emph.end type="italics"/> quia mini­<lb/>ma vis e&longs;t, quæ con&longs;equentibus in&longs;tantibus augetur, hinc licèt corpus <lb/>breui tempore quis &longs;u&longs;tineat, paulò po&longs;t tamen ponderi cedit, ratio e&longs;t <lb/>clara ex dictis. </s> </p> <pb pagenum="95" xlink:href="026/01/127.jpg"/> <p id="N16E2A" type="main"> <s id="N16E2C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16E38" type="main"> <s id="N16E3A"><!-- NEW -->Ob&longs;eruabis primò numerum in&longs;tantium non po&longs;&longs;e à quoquam &longs;en&longs;u <lb/>percipi, nec in calculos vocari, vt patet; </s> <s id="N16E40"><!-- NEW -->vnde Theoremata non po&longs;&longs;unt <lb/>ad praxim reduci defectu huius cognitionis; quam &longs;upra adhibui hypo­<lb/>the&longs;eos loco. </s> </p> <p id="N16E48" type="main"> <s id="N16E4A"><!-- NEW -->Secundò non pote&longs;t ad amu&longs;&longs;im tempus cum tempore componi ad <lb/>æqualitatem, vel aliam datam rationem; </s> <s id="N16E50"><!-- NEW -->licèt enim vnum tempus &longs;en&longs;i­<lb/>bile haberet mille in&longs;tantia &longs;upra aliud; </s> <s id="N16E56"><!-- NEW -->illa tamen inæqualitas &longs;en&longs;u <lb/>minimè perciperetur; idem dico de aliis rationibus, in quo, ni fallor, <lb/>maximè peccant, qui temporum æqualitatem perfectam ob&longs;eruari po&longs;&longs;e <lb/>contendunt. </s> </p> <p id="N16E60" type="main"> <s id="N16E62"><!-- NEW -->Tertiò, idem dico de percu&longs;&longs;ionum ratione; </s> <s id="N16E66"><!-- NEW -->quippe non pote&longs;t &longs;en&longs;u <lb/>percipi inæqualitas duarum percu&longs;&longs;ionum, licèt vires vnius præualeant <lb/>mille punctis &longs;eu gradibus in&longs;en&longs;ibilibus; </s> <s id="N16E6E"><!-- NEW -->quippe non pote&longs;t di&longs;tingui <lb/>ab alia ni&longs;i vel ex &longs;patio; </s> <s id="N16E74"><!-- NEW -->atqui di&longs;cerni non pote&longs;t, an vnum &longs;patium <lb/>&longs;uperet aliud mille punctis; vel ex &longs;ono; </s> <s id="N16E7A"><!-- NEW -->atqui &longs;onus pote&longs;t diuidi in in­<lb/>finitos ferè gradus &longs;en&longs;u minimè perceptibiles; </s> <s id="N16E80"><!-- NEW -->igitur nulla hypothe&longs;is <lb/>in his experimentis &longs;tatui pote&longs;t, quibus æqualitas vel temporum, vel <lb/>&longs;patiorum cogno&longs;ci dicatur; </s> <s id="N16E88"><!-- NEW -->nec dicas aliquot in&longs;tantia parùm di&longs;eri­<lb/>minis importare, nam cùm &longs;ingulis in&longs;tantibus fiat æqualis impetus ac­<lb/>ce&longs;&longs;io, mille in&longs;tantia reddunt percu&longs;&longs;ionem millecuplam grauitationis; </s> <s id="N16E90"><!-- NEW --><lb/>hinc certum e&longs;t ex numero in&longs;tantium cognito cogno&longs;ci tantùm po&longs;&longs;e <lb/>numerum punctorum, & vici&longs;&longs;im; </s> <s id="N16E97"><!-- NEW -->at certè neuter &longs;en&longs;u percipi pote&longs;t; ne­<lb/>que tanti e&longs;t hoc &longs;cire. </s> </p> <p id="N16E9D" type="main"> <s id="N16E9F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N16EAB" type="main"> <s id="N16EAD"><!-- NEW --><emph type="italics"/>Hinc &longs;i corpus graue de&longs;cenderet motu æquabili eoque æquali motui primi <lb/>in&longs;tantis; </s> <s id="N16EB5"><!-- NEW -->certè vix modicum &longs;patium post multos annos decurreret<emph.end type="italics"/>; </s> <s id="N16EBC"><!-- NEW -->&longs;uppo­<lb/>namus enim quod plures habent, licèt accuratè experimento &longs;ubii­<lb/>ci non po&longs;&longs;it, &longs;cilicet vno &longs;ecundo minuto temporis decurri à corpore <lb/>graui deor&longs;um 12. pedes &longs;patij; </s> <s id="N16EC6"><!-- NEW -->in &longs;ecundo minuto &longs;upponamus e&longs;&longs;e <lb/>mille in&longs;tantia, quamuis infinita penè contineat; </s> <s id="N16ECC"><!-- NEW -->&longs;itque in primo in­<lb/>&longs;tanti motus vnus gradus impetus; </s> <s id="N16ED2"><!-- NEW -->&longs;ic enim vocetur illa pars impetus, que <lb/>producitur primo in&longs;tanti; </s> <s id="N16ED8"><!-- NEW -->certè po&longs;t mille in&longs;tantia motus, erunt mille <lb/>gradus impetus; </s> <s id="N16EDE"><!-- NEW -->iam vcrò &longs;i accipiatur &longs;ubduplum maximæ & minimæ <lb/>velocitatis; </s> <s id="N16EE4"><!-- NEW -->id e&longs;t vnius gradus, & mille graduum, &longs;cilicet 500. 1/2 tri­<lb/>buaturque motui æquabili; </s> <s id="N16EEA"><!-- NEW -->haud dubiè vno fecundo minuto percur­<lb/>rentur 12. pedes &longs;patij per Th. 46. Igitur &longs;i cum velocitate vt 500, 1/2 <lb/>percurrentur 12. pedes 1.minuto, cum velocitate vt 1. percurrentur <lb/>12. pedes 500.&longs;ecundis minutis, &; 30. tertiis; </s> <s id="N16EF4"><!-- NEW -->&longs;i verò accipiantur plura <lb/>in&longs;tantia, v.g. <!-- REMOVE S-->1000000.in&longs;tantia, percurrentur 12. pedes 500000. &longs;e­<lb/>cundis minutis; </s> <s id="N16EFE"><!-- NEW -->&longs;i verò 1000000000000. percurremur 500000000000. <lb/>&longs;ecundis, id e&longs;t 8333333333. minutis, id e&longs;t 138888888. horis <pb pagenum="96" xlink:href="026/01/128.jpg"/>id e&longs;t 5787037. diebus id e&longs;t 89031. annis, omitto minutias; atqui lon­<lb/>gè adhuc plura in vno minuto continentur in&longs;tantia. </s> </p> <p id="N16F0B" type="main"> <s id="N16F0D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N16F19" type="main"> <s id="N16F1B"><!-- NEW --><emph type="italics"/>Si corpus graue de&longs;cenderet motu æquabili, eoque æquali motui vltimi in­<lb/>stantis, duplum ferè &longs;patium æquali tempore conficeret illius quod conficit <lb/>motu accelerato, duplum inquam ferè &longs;cilicet paulò minùs<emph.end type="italics"/>; </s> <s id="N16F28"><!-- NEW -->quia conficit <lb/>idem motu æquabili; </s> <s id="N16F2E"><!-- NEW -->cuius velocitas e&longs;t &longs;ubdupla maximæ & minimæ; </s> <s id="N16F32"><!-- NEW --><lb/>&longs;ed minima velocitas primi in&longs;tantis pro nihilo reputatur; </s> <s id="N16F37"><!-- NEW -->igitur acci­<lb/>piatur tantùm &longs;ubduplum maximæ, igitur cum velocitate æquali maxi­<lb/>mæ, eodem tempore duplum &longs;patium percurretur; </s> <s id="N16F3F"><!-- NEW -->igitur in vno minuto <lb/>&longs;ecundo, v.g. <!-- REMOVE S-->24. pedes; </s> <s id="N16F47"><!-- NEW -->igitur in vno minuto primo eodem motu æqua­<lb/>bili 1440. pedes percurrentur; </s> <s id="N16F4D"><!-- NEW -->igitur in vna hora 86400. pedes; hinc <lb/>non e&longs;t quod aliqui adeo mirentur, &longs;eu potiùs reiiciant hanc motus <lb/>accelerationem quod ex ea tùm tardi&longs;&longs;imus motus, tùm veloci&longs;&longs;imus <lb/>con&longs;equatur. </s> </p> <p id="N16F57" type="main"> <s id="N16F59"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N16F65" type="main"> <s id="N16F67"><!-- NEW --><emph type="italics"/>Motus naturaliter acceleratus non propagatur per omnes tarditatis gra­<lb/>dus<emph.end type="italics"/>; </s> <s id="N16F72"><!-- NEW -->quia tot &longs;unt huius propagationis gradus, quot &longs;unt in&longs;tantia, <lb/>quibus durat hic motus, cum &longs;ingulis in&longs;tantibus noua fiat impetus ac­<lb/>ce&longs;&longs;io, &longs;ed non &longs;unt infinita in&longs;tantia, vt demon&longs;trabimus in Metaphy­<lb/>&longs;ica; </s> <s id="N16F7C"><!-- NEW -->prætereà licèt e&longs;&longs;ent infinita in&longs;tantia, non fieret adhuc per omnes <lb/>tarditatis gradus hæc propagatio; </s> <s id="N16F82"><!-- NEW -->quia daretur aliquis gradus tarditatis, <lb/>quem non comprehenderet hæc graduum &longs;eries; </s> <s id="N16F88"><!-- NEW -->nam incipit moueri <lb/>tardiùs in plano inclinato quàm in libero medio rectà deor&longs;um, vt con­<lb/>&longs;tat, & in medio den&longs;o quàm in raro v.g. <!-- REMOVE S-->in aqua quàm in aëre; igitur <lb/>hic tarditatis gradus, quo incipit moueri in plano tantillùm inclinato, <lb/>non continetur inter illos, quibus mouetur rectà deor&longs;um. </s> </p> <p id="N16F96" type="main"> <s id="N16F98">Hinc duplici nomine reiice Galilæum qui hoc a&longs;&longs;erit. </s> <s id="N16F9B"><!-- NEW -->Primò, quia <lb/>fru&longs;trà ponit infinita in&longs;tantia &longs;ine nece&longs;&longs;itate; </s> <s id="N16FA1"><!-- NEW -->&longs;ecundò, quia ratio, quam <lb/>habet, non conuincit; </s> <s id="N16FA7"><!-- NEW -->vocat enim quietem tarditatem infinitam; </s> <s id="N16FAB"><!-- NEW -->à qua <lb/>dum recedit mobile, haud dubiè per omnes tarditatis gradus propagari <lb/>pote&longs;t eius motus; &longs;ed contrà primò, nam reuerà quies non e&longs;t tarditas, <lb/>quæ motui tantùm ine&longs;&longs;e pote&longs;t. </s> <s id="N16FB5">Secundò, quia tàm ex quiete &longs;equi po­<lb/>te&longs;t immediatè velox motus, quàm tardus, vt patet in proiectis. </s> <s id="N16FBA">Tertiò, <lb/>quia motus incipit; </s> <s id="N16FBF"><!-- NEW -->igitur per aliquid &longs;ui, igitur ille primus motus à <lb/>quiete infinitè non di&longs;tat; denique rationes &longs;uprà propo&longs;itæ rem i&longs;tam <lb/>euincunt. </s> </p> <p id="N16FC7" type="main"> <s id="N16FC9"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16FD5" type="main"> <s id="N16FD7">Ob&longs;eruabis con&longs;ideratum e&longs;&longs;e hactenus hunc motum nulla habita <lb/>ratione re&longs;i&longs;tentiæ medij, quæ haud dubiè hanc propo&longs;itionem motus <lb/>accelerati tantillùm impedit, &longs;ed de re&longs;i&longs;tentià medij agemus infrà. </s> </p> <p id="N16FDE" type="main"> <s id="N16FE0"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N16FED" type="main"> <s id="N16FEF">Ex dictis facilè reiicies primò &longs;ententiam illorum, qui negant mo-<pb pagenum="97" xlink:href="026/01/129.jpg"/>tum naturalem accelerari, quos non ratio modò euidenti&longs;&longs;ima, &longs;ed adeò <lb/>&longs;en&longs;ibile experimentum omninò conuincere pote&longs;t. </s> </p> <p id="N16FF9" type="main"> <s id="N16FFB"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N17008" type="main"> <s id="N1700A"><!-- NEW -->Secundò reiicies illos, qui volunt accelerationem motus e&longs;&longs;e, vel à vi <lb/>magnetica, quâ terra trahit ad &longs;e omnia grauia; </s> <s id="N17010"><!-- NEW -->vel ab alia vi occulta, <lb/>quâ cœlum pellit deor&longs;um; </s> <s id="N17016"><!-- NEW -->vel à cœle&longs;ti illa, imò potiùs fabulosâ mate­<lb/>riâ; </s> <s id="N1701C"><!-- NEW -->vel demum ab ip&longs;a vi &longs;ympathicâ, quâ corpus &longs;uo centro propiùs <lb/>factum totas &longs;uas vires exerit, vt ei &longs;e conjungat; quæ omnia gratis di­<lb/>cuntur, & ex dictis plu&longs;quam efficaciter refelli po&longs;&longs;unt, ne fru&longs;trà tempus <lb/>in iis iterum refellendis teramus. </s> </p> <p id="N17026" type="main"> <s id="N17028"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N17035" type="main"> <s id="N17037"><!-- NEW -->Tertiò reiicies, qui volunt motum accelerari ex aëris à tergo impel­<lb/>lentis appul&longs;u, quod ridiculum e&longs;t: </s> <s id="N1703D"><!-- NEW -->licèt enim Ari&longs;toteles videatur illud <lb/>&longs;en&longs;i&longs;&longs;e de projectis, quod examinabimus &longs;uo loco; </s> <s id="N17043"><!-- NEW -->nunquam tamen hoc <lb/>dixit de motu naturali; </s> <s id="N17049"><!-- NEW -->quin potiùs antiquorum fuit omnium hic &longs;en­<lb/>&longs;us, fieri <expan abbr="acce&longs;&longs;ion&etilde;">acce&longs;&longs;ionem</expan> mobili alicuius, vnde reddatur motus velocior; </s> <s id="N17053"><!-- NEW -->hinc <lb/>dictum illud vulgare, <emph type="italics"/>vire&longs;que acquirit eundo<emph.end type="italics"/>; </s> <s id="N1705F"><!-- NEW -->nihil porrò intelligi pote&longs;t <lb/>nomine virium, ni&longs;i id, ex quo maior ictus, &longs;eu percu&longs;&longs;io &longs;equitur; illud <lb/>autem e&longs;&longs;e impetum con&longs;tat. </s> </p> <p id="N17067" type="main"> <s id="N17069"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N17076" type="main"> <s id="N17078"><!-- NEW -->Quartò ex his &longs;ententia Ari&longs;totelica de motu accelerato optimè vin­<lb/>dicatur; </s> <s id="N1707E"><!-- NEW -->quòd &longs;cilicet grauia &longs;ub finem &longs;ui motus velociùs &longs;erantur ver­<lb/>sùs centrum; quod ex dictis, & &longs;implici&longs;&longs;imis, certi&longs;&longs;imi&longs;que principiis <lb/>demon&longs;tratum fuit. </s> </p> <p id="N17086" type="main"> <s id="N17088"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N17095" type="main"> <s id="N17097"><!-- NEW -->Quintò reiicies etiam illorum &longs;ententiam, qui hanc accelerationem <lb/>tribuunt vel medio minùs re&longs;i&longs;tenti, vel grauitatis augmento, vel impe­<lb/>tui violento priùs impre&longs;&longs;o dum corpus graue attollitur, quod meo iudi­<lb/>cio ridiculum e&longs;t; qua&longs;i verò fru&longs;tum rupis deci&longs;um, deor&longs;umque ruens <lb/>impetum violentum aliquando habuerit. </s> </p> <p id="N170A3" type="main"> <s id="N170A5"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N170B2" type="main"> <s id="N170B4">Sextò reiicies illorum &longs;ententiam, qui volunt accelerationem motus <lb/>naturalis ita fieri, vt &longs;patia temporibus æqualibus acqui&longs;ita &longs;equantur &longs;e­<lb/>riem numerorum imparium 1.3.5.7.9.11.13. &c. </s> <s id="N170BB"><!-- NEW -->& &longs;patia &longs;int vt <lb/>quadrata temporum v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i primo in&longs;tanti acquiritur 1.&longs;patium: &longs;ecundo <lb/>acquiruntur 3. tertio 5. quarto 7. &c. </s> <s id="N170C7"><!-- NEW -->fique vno in&longs;tanti acquiritur 1. <lb/>&longs;patium, duobus acquiruntur 4. tribus 9. quatuor 16. atque ita deinceps <lb/>per quadrata, quæ omnia ex dictis fal&longs;a e&longs;&longs;e con&longs;tat; </s> <s id="N170CF"><!-- NEW -->quippe &longs;i æqualibus <lb/>temporibus acquiruntur æqualia velocitatis momenta; igitur &longs;i primo <lb/>in&longs;tanti e&longs;t 1.gradus, &longs;ecundo erunt 2. igitur &longs;ecundo tempore cum duo­<lb/>bus gradibus velocitatis vel impetus percurrentur duo tantùm &longs;patia, &longs;i <lb/>primò in&longs;tanti æquali cum vno gradu percurritur vnus, &longs;ed de his fusè <lb/>infrà. </s> </p> <pb pagenum="98" xlink:href="026/01/130.jpg"/> <p id="N170E1" type="main"> <s id="N170E3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N170EF" type="main"> <s id="N170F1">Septimò reiicies etiam aliquos recentiores, qui volunt fieri hanc pro­<lb/>gre&longs;&longs;ionem &longs;patiorum æqualibus temporibus re&longs;pondentium &longs;ecundùm <lb/>progre&longs;&longs;ionem Geometricam, duplam, &longs;cilicet iuxta hos numeros 1. 2. 4. <lb/>8. 16. 32. &c. </s> <s id="N170FA"><!-- NEW -->quod etiam ex eadem ratione facilè confutatur: </s> <s id="N170FE"><!-- NEW -->reiicies <lb/>etiam alium recentiorem, qui vult hanc progre&longs;&longs;ionem &longs;umi ex linea <lb/>proportionaliter &longs;ectâ, id e&longs;t in mediam & extremam rationem; </s> <s id="N17106"><!-- NEW -->&longs;ed de <lb/>his omnibus in di&longs;&longs;ertatione &longs;equenti fusè di&longs;putamus; quippe rem hanc <lb/>tanti e&longs;&longs;e putamus, vt nihil omittendum &longs;it, quod ad eius pleni&longs;&longs;imam <lb/>confirmationem pertineat. <lb/><gap desc="hr tag"/></s> </p> <p id="N17113" type="main"> <s id="N17115"><emph type="center"/>DISSERTATIO<emph.end type="center"/></s> </p> <p id="N1711C" type="main"> <s id="N1711E"><emph type="center"/><emph type="italics"/>De Motu naturaliter accelerato.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N17129" type="main"> <s id="N1712B"><!-- NEW -->DVæ &longs;unt poti&longs;&longs;imùm in hac materia celebres &longs;ententiæ; </s> <s id="N1712F"><!-- NEW -->Prima e&longs;t <lb/>Galilei, & ferè omnium recentiorum, qui po&longs;t Galileum de motu <lb/>&longs;crip&longs;erunt; </s> <s id="N17137"><!-- NEW -->inter quos, ne omittam Genuen&longs;em Patricium, Balianum; </s> <s id="N1713B"><!-- NEW --><lb/>Doctus Mer&longs;ennus, & eruditus Ga&longs;&longs;endus primum locum obtinent; </s> <s id="N17140"><!-- NEW --><lb/>quorum ille hanc &longs;ententiam multis in locis, &longs;cilicet in &longs;uis quæ&longs;tioni­<lb/>bus Phy&longs;icis, in &longs;ua Galilei ver&longs;ione, in harmonia vniuer&longs;ali, & demum <lb/>in &longs;ua Bali&longs;tica pa&longs;&longs;im, tùm fusè proponit, & explicat, tùm etiam &longs;uis ra­<lb/>tionibus confirmat; Galileus verò illam habet tùm in gemino &longs;y&longs;tema­<lb/>te, tùm in dialogo tertio de motu locali. </s> </p> <p id="N1714D" type="main"> <s id="N1714F"><!-- NEW -->Secunda &longs;ententia no&longs;tra e&longs;t, de qua non &longs;emel di&longs;putandum fuit à <lb/>Magi&longs;tro, tùm verbis tùm etiam litteris &longs;criptis; & ne quid fortè di&longs;&longs;imu­<lb/>lem, illa e&longs;t &longs;ententia quam anonimo Philo&longs;ophe (quem non &longs;ine laude <lb/>appellat idem Mer&longs;ennus) tribuit. </s> <s id="N17159"><!-- NEW -->prop.18.&longs;uæ Bali&longs;ticæ &longs;ub finem; illa <lb/>e&longs;t inquam &longs;ententia, quam hactenus meo iudicio &longs;atis luculenter de­<lb/>mon&longs;trauimus. </s> </p> <p id="N17161" type="main"> <s id="N17163"><!-- NEW -->Sunt tres aliæ &longs;ententiæ, quæ ab eodem Mer&longs;enno referuntur; prima <lb/>e&longs;t quæ progre&longs;&longs;ionem &longs;patiorum <expan abbr="eãdem">eandem</expan> e&longs;&longs;e vult cum eâ, quæ e&longs;t &longs;i­<lb/>nuum ver&longs;orum, centro quadrantis po&longs;ito in centro terræ, & altero ex­<lb/>tremo &longs;inus totius in eo punctò, in quo incipit motus. </s> <s id="N17171">Secunda e&longs;t quo­<lb/>rumdam, qui volunt progre&longs;&longs;ionem &longs;patiorum, quæ &longs;ingulis temporibus <lb/>re&longs;pondent, e&longs;&longs;e in progre&longs;&longs;ione geometrica dupla iuxta hos numeros, <lb/>1.2.4.8.32. Tertia e&longs;t alicuius, qui voluit e&longs;&longs;e iuxta proportionem lineæ <lb/>&longs;ectæ in mediam, & extremam rationem. </s> </p> <p id="N1717C" type="main"> <s id="N1717E"><!-- NEW -->Tres vltimæ &longs;ententiæ nullo pror&longs;us nituntur fundamento; igitur vel <lb/>inde maximè confutantur, quòd gratis &longs;ine vllo pror&longs;us vel rationis vel <lb/>experimenti momento excogitatæ &longs;int. </s> <s id="N17186"><!-- NEW -->Igitur in hac di&longs;&longs;ertatione duæ <lb/>tantùm primæ di&longs;cutiendæ &longs;unt Sententiæ Galilei &longs;chema hic habes <lb/>in linea AF, in qua a&longs;&longs;umitur AB, &longs;patium &longs;cilicet, quod dato tempore <pb pagenum="99" xlink:href="026/01/131.jpg"/>corpus graue &longs;uo motu percurrit; </s> <s id="N17193"><!-- NEW -->& &longs;ecundo tempore æquali BC, quæ <lb/>tripla e&longs;t AB, tertio CD quintupla quarto DE &longs;eptupla, quinto EF <lb/>nonecupla; vides primò &longs;eriem numerorum imparium 1. 3. 5. 7. 9.atque <lb/>ita deinceps. </s> <s id="N1719D">Secundò vides &longs;patia e&longs;&longs;e in ratione duplicata temporum, <lb/>hoc e&longs;t vt temporum quadrata. </s> <s id="N171A2"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;i accipiatur &longs;patium AB primo tem­<lb/>pore peractum, & &longs;patium AC duobus temporibus confectum: ratio hu­<lb/>ius ad illud e&longs;t vt 4.ad 1.id e&longs;t vt quadratum 2.ad quadratum 1. &longs;imiliter, <lb/>&longs;i accipiatur &longs;patium AD confectum tribus temporibus, erit 9.id e&longs;t qua­<lb/>dratum 3, &longs;patium AE confectum 4.temporibus erit 16.id e&longs;t quadratum <lb/>4. & AF 25. quadratum 5. <!-- KEEP S--></s> </p> <p id="N171B3" type="main"> <s id="N171B5"><!-- NEW -->Hæc &longs;ententia ingeniosè à Galileo excogitata ex duplici capite à &longs;uis <lb/>auctoribus confirmatur; primò experientiâ, &longs;ecundò ratione. </s> <s id="N171BB"><!-- NEW -->Experien­<lb/>tia tribus poti&longs;&longs;imum experimentis fulcitur; primum e&longs;t in motu deor­<lb/>&longs;um per lineam perpendicularem. </s> <s id="N171C3"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->in linea AF; </s> <s id="N171CB"><!-- NEW -->nam reuerà multi <lb/>&longs;unt, iique graui&longs;&longs;imi auctores in rebus tùm philo&longs;ophicis, tùm mathe­<lb/>maticis ver&longs;ati&longs;&longs;imi, qui &longs;æpiùs &longs;en&longs;u ip&longs;o probarunt, repetitis v&longs;que ad <lb/>nau&longs;eam experimentis, tempore vnius &longs;ecundi minuti corpus graue in <lb/>libero aëre 12. pedes &longs;patij motu naturali deor&longs;um percurrere; in 2.ve­<lb/>rò &longs;ecundis 48. in 3.&longs;ecundis 108.&longs;ed &longs;patia i&longs;ta &longs;unt vt temporum qua­<lb/>drata, vt con&longs;tat. </s> </p> <p id="N171DB" type="main"> <s id="N171DD">Secundum experimentum e&longs;t in plano inclinato, in quo corpus graue <lb/>de&longs;cendit iuxta prædictam progre&longs;&longs;ionem, quod expre&longs;&longs;is verbis te&longs;tatur <lb/>Galileus à &longs;e fui&longs;&longs;e probatum &longs;æpiùs, nec vnquam à vero ne tantillùm <lb/>quidem aberra&longs;&longs;e. </s> <s id="N171E6"><!-- NEW -->&longs;ed in perpendiculari deor&longs;um eadem proportione <lb/>cre&longs;cit motus, quâ in plano inclinato; licèt in plano inclinato tardior &longs;it <lb/>motus, vt demon&longs;trabimus aliàs. </s> </p> <p id="N171EE" type="main"> <s id="N171F0"><!-- NEW -->Tertium experimentum petitur ex funependulis; </s> <s id="N171F4"><!-- NEW -->in quibus &longs;æpiùs <lb/>ob&longs;eruatum e&longs;t longitudinem funis, & con&longs;equenter arcum quadrantis <lb/>longioris funependuli e&longs;&longs;e ad longitudinem, &longs;eu quadrantem alterius <lb/>breuioris, vt quadratum temporis, quo perficitur vibratio maioris ad <lb/>quadratum temporis, quo perficitur vibratio minoris.v.g.&longs;it longitudo <lb/>funependuli maioris, CG minoris verò &longs;ubquadrupla CF; </s> <s id="N17202"><!-- NEW -->eleuetur vter­<lb/>que funis, cui pondus æquale &longs;it appen&longs;um v&longs;que ad horizontalem <lb/>CDE & alterum ex D; </s> <s id="N1720A"><!-- NEW -->alterum verò ex E demi&longs;&longs;um cadat deor&longs;um; haud <lb/>dubiè funependulum CE duplum temporis collocabit in decurrendo <lb/>quadrante EG, & funependulum ED &longs;ubduplum. </s> <s id="N17212"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i CD conficit <lb/>&longs;uam vibrationem DF vno &longs;ecundo, EG conficiet &longs;uam EG duobus, vt <lb/>centies ob&longs;eruatum e&longs;t; </s> <s id="N1721E"><!-- NEW -->&longs;ed EG e&longs;t quadruplus DF, vt patet; igitur EG <lb/>& DF &longs;unt vt quadrata temporum, quibus percurritur EG & DF &longs;ed vt <lb/>de&longs;cendit graue per DF & EG, ita de&longs;cendit per CF & CG, quippe <lb/>DF & EG habent rationem plani inclinati deor&longs;um. </s> </p> <p id="N17228" type="main"> <s id="N1722A"><!-- NEW -->Adde quod, vt &longs;e habet tempus, quo de&longs;cendit per totum quadrantem <lb/>DF, ad tempus, quo de&longs;cendit per totum quadrantem EG. &longs;ic &longs;e habet <lb/>tempus, quo de&longs;cendit per arcum DL &longs;ubduplum DF ad tempus, quo <lb/>de&longs;cendit per arcum EI &longs;ubduplum EG; </s> <s id="N17234"><!-- NEW -->item tempus, quo de&longs;cendit <pb pagenum="100" xlink:href="026/01/132.jpg"/>per arcum DM &longs;ubquadruplum DF.ad tempus, quo de&longs;cendit per arcum <lb/>EK &longs;ubquadruplum EG; </s> <s id="N1723F"><!-- NEW -->denique vt tempus, quo per minimum ar­<lb/>cum quadrantis DF, ad tempus, quo de&longs;cendit per alium proportiona­<lb/>lem, &longs;cilicet quadruplum in quadrante EG; </s> <s id="N17247"><!-- NEW -->atqui tam parui arcus po&longs;­<lb/>&longs;unt a&longs;&longs;umi, vt &longs;int ad in&longs;tar lineæ rectæ deor&longs;um tangentis &longs;cilicet in D <lb/>& in E; </s> <s id="N1724F"><!-- NEW -->igitur in his rectis de&longs;cendunt grauia iuxta progre&longs;&longs;ionem præ­<lb/>dictam; </s> <s id="N17255"><!-- NEW -->id e&longs;t, cum arcus minimus a&longs;&longs;umptus ab E, qui æquiualet rectæ, <lb/>&longs;it quadruplus arcus minimi a&longs;&longs;umpti à puncto D, tempus, quo percurri­<lb/>tur ille primus, e&longs;t ad tempus, quo percurritur hic &longs;ubquadruplus, vt tem­<lb/>pus, quo percurritur EG ad tempus, quo percurritur DF vt dictum e&longs;t; </s> <s id="N1725F"><!-- NEW --><lb/>&longs;ed tempus, quo percurritur EG e&longs;t duplum illius, quo percurritur DF; </s> <s id="N17264"><!-- NEW --><lb/>igitur tempus, quo percurritur minimus arcus a&longs;&longs;umptus ab E, & qui e&longs;t <lb/>ad in&longs;tar rectæ, e&longs;t duplum temporis quo percurritur minimus arcus a&longs;­<lb/>&longs;umptus à puncto D &longs;ubquadruplus prioris, & qui e&longs;t etiam ad in&longs;tar re­<lb/>ctæ; igitur &longs;patia &longs;unt vt temporum quadrata. </s> </p> <p id="N1726F" type="main"> <s id="N17271"><!-- NEW -->Quod autem tempus, quo percurritur EG &longs;it duplum illius, quo per­<lb/>curritur DF, patet experientiâ; </s> <s id="N17277"><!-- NEW -->nam &longs;i numerentur ducentæ vibrationes <lb/>funependuli CD; </s> <s id="N1727D"><!-- NEW -->eodem tempore numerabuntur centum vibrationes <lb/>maioris CE; </s> <s id="N17283"><!-- NEW -->igitur vibrationum minoris numerus e&longs;t duplus numeri vi­<lb/>brationum maioris, dum &longs;imul vibrantur; </s> <s id="N17289"><!-- NEW -->igitur eo tempore, quo fiunt <lb/>100.maioris, fient 200. minoris; nam omnes vibrationes eiu&longs;dem fune­<lb/>penduli &longs;unt æquò diuturnæ, licèt fiant per arcus inæquales eiu&longs;dem. </s> <s id="N17291"><lb/>quadrantis, vt &longs;æpè ob&longs;eruatum e&longs;t. </s> <s id="N17295">In his tribus poti&longs;&longs;imum experimen­<lb/>tis fundatur hæc hypothe&longs;is Galilei, quæ nec clariùs meo. </s> <s id="N1729A">iudicio, nec <lb/>&longs;inceriùs exponi po&longs;&longs;unt. </s> </p> <p id="N1729F" type="main"> <s id="N172A1"><!-- NEW -->Antequam rationes, quæ pro hac &longs;ententia facere videntur, propona­<lb/>mus, refellamu&longs;que; </s> <s id="N172A7"><!-- NEW -->o&longs;tendo primò quomodo cum his experimentis <lb/>&longs;tare po&longs;&longs;it no&longs;tra hypothe&longs;is; </s> <s id="N172AD"><!-- NEW -->igitur ex iis hypothe&longs;is Galilei rectè de­<lb/>duci non pote&longs;t: </s> <s id="N172B3"><!-- NEW -->quippe hæc e&longs;t certi&longs;&longs;ima regula, quam nemo Philo&longs;o­<lb/>phus negare au&longs;it: </s> <s id="N172B9"><!-- NEW -->Quotie&longs;cumque aliquod experimentum tale e&longs;t, vt <lb/>cum eo &longs;tare po&longs;&longs;int contrariæ hypothe&longs;es; </s> <s id="N172BF"><!-- NEW -->ex eo certè neutra deduci po­<lb/>te&longs;t; igitur ex propo&longs;itis experimentis &longs;uam hypothe&longs;im Galileus non <lb/>legitimè deducit, quod vt clari&longs;&longs;imè o&longs;tendam. </s> </p> <p id="N172C7" type="main"> <s id="N172C9"><!-- NEW -->Suppono, quando dicitur &longs;ecundum &longs;patium e&longs;&longs;e triplum primi &longs;up­<lb/>po&longs;itis æqualibus temporibus, non ita Geometricè, certaque, & acuratâ <lb/>a&longs;&longs;ertione hoc dici; </s> <s id="N172D1"><!-- NEW -->quin vel aliqua puncta in &longs;patiis, vel in&longs;tantia in <lb/>temporibus de&longs;int, vel &longs;uper&longs;int; </s> <s id="N172D7"><!-- NEW -->&longs;i enim quis diceret &longs;patium e&longs;&longs;e tri­<lb/>plum primi minus 100000. punctis, vel &longs;ecundum tempus e&longs;&longs;e maius <lb/>primo 100000. in&longs;tantibus; quis hanc, vel &longs;patij, vel temporis differen­<lb/>tiam &longs;en&longs;u percipiat? </s> <s id="N172E1"><!-- NEW -->cum tamen experimentum omne phy&longs;icum &longs;en&longs;ui <lb/>&longs;ube&longs;&longs;e po&longs;&longs;it; </s> <s id="N172E7"><!-- NEW -->nec e&longs;t quod aliquis dicat hoc idem toties ob&longs;eruatum <lb/>e&longs;&longs;e, tam multis locis temporibus, totque ac tantis etiam te&longs;tibus, vt mi­<lb/>nimè fraus aliqua, vel error &longs;ubrepere potuerit; nam cum parua &longs;it, & <lb/>in&longs;en&longs;ibilis tùm &longs;patiorum, tùm temporum differentia, maius vel minus <lb/>æquali tempus, pro æquali, maius.vel minus triplò &longs;patium pro triplo <pb pagenum="101" xlink:href="026/01/133.jpg"/>facilè accipi pote&longs;t, cum nullum di&longs;crimen &longs;en&longs;ibile e&longs;t. </s> </p> <p id="N172F8" type="main"> <s id="N172FA"><!-- NEW -->Adde quod non de&longs;unt viri graui&longs;&longs;imi qui dicant &longs;e vix ob&longs;eruare po­<lb/>tui&longs;&longs;e hanc &longs;patiorum progre&longs;&longs;ionem; </s> <s id="N17300"><!-- NEW -->plures appellare po&longs;&longs;em; </s> <s id="N17304"><!-- NEW -->vnus <lb/>Ga&longs;&longs;endus e&longs;t in&longs;tar omnium; </s> <s id="N1730A"><!-- NEW -->qui &longs;anè in ob&longs;eruando fuit acurati&longs;&longs;imus, <lb/>qui literis &longs;criptis, quas ego vidi, expre&longs;&longs;is verbis a&longs;&longs;erit progre&longs;&longs;ionem <lb/>hanc non e&longs;&longs;e omninò iuxta hos numeros 1.3.5.7. &longs;ed &longs;ingulis addendas <lb/>e&longs;&longs;e &longs;uas minutias, quas ip&longs;e habet; </s> <s id="N17314"><!-- NEW -->&longs;ed ego omitto, quia etiam &longs;ua incer­<lb/>titudine laborant; </s> <s id="N1731A"><!-- NEW -->igitur nullo experimento ad amu&longs;&longs;im concludes, <lb/>vel <expan abbr="æqualitat&etilde;">æqualitatem</expan> vel aliam accuratam tùm temporum tùm &longs;patiorum pro­<lb/>portionem: </s> <s id="N17326"><!-- NEW -->Equidem &longs;en&longs;u percipio practicam hanc e&longs;&longs;e maiorem pede; </s> <s id="N1732A"><!-- NEW --><lb/>at tot lineis vel <expan abbr="pũctis">punctis</expan> &longs;uperare ne Argus quidem certò, ac di&longs;tinctè cer­<lb/>neret: </s> <s id="N17335"><!-- NEW -->Sed efficaciter, meo iudicio, hanc Galilei hypothe&longs;im refello; </s> <s id="N17339"><!-- NEW -->&longs;int <lb/> 2.partes temporis æquales AE, EF, eæque &longs;en&longs;ibiles; </s> <s id="N1733F"><!-- NEW -->nec enim aliæ a&longs;­<lb/>&longs;umi po&longs;&longs;unt; </s> <s id="N17345"><!-- NEW -->&longs;intque minimæ omnium &longs;en&longs;ibilium; </s> <s id="N17349"><!-- NEW -->haud dubiè con&longs;tant <lb/>&longs;ingulæ infinitis ferè aliis in&longs;en&longs;ibilibus, vt patet; </s> <s id="N1734F"><!-- NEW -->igitur &longs;ic ratiocinatur <lb/>Galileus; </s> <s id="N17355"><!-- NEW -->in prima parte temporis AE corpus graue percurrit &longs;patium <lb/>GH, & in &longs;ecunda æquali EF percurrit &longs;patium HL triplum prioris; </s> <s id="N1735B"><!-- NEW --><lb/>igitur &longs;patia &longs;unt vt quadrata temporum, rectè; &longs;ed antequam vlterius <lb/>progrediar;</s> <s id="N17362"> Quæro vel à Galileo, vel à quolibet alto, vtrum &longs;patium <lb/>HL &longs;it omnino triplum? </s> <s id="N17367">& &longs;i aliquis contenderet dee&longs;&longs;e (1/1000000) GH <lb/>vtrum experimento præ&longs;enti conuinci po&longs;&longs;it? </s> <s id="N1736C"><!-- NEW -->nemo, vt puto, id a&longs;&longs;erere <lb/>au&longs;it; </s> <s id="N17372"><!-- NEW -->hoc po&longs;ito, a&longs;&longs;umptaque progre&longs;&longs;ione arithmetica <expan abbr="quã">quam</expan> no&longs;tra &longs;en­<lb/>tentia in &longs;patiis ad&longs;truit; </s> <s id="N1737C"><!-- NEW -->&longs;i prima parte temporis AE percurratur &longs;pa­<lb/>tium GH, &longs;ecunda EF. percurretur tantùm HK duplum GH; </s> <s id="N17382"><!-- NEW -->igitur <lb/>minus e&longs;t hoc &longs;patium vero &longs;patio 1/4. &longs;cilicet tota KL; </s> <s id="N17388"><!-- NEW -->res pror&longs;us de­<lb/>mon&longs;trata e&longs;&longs;et, &longs;i termini proportionis vnius e&longs;&longs;ent tantùm 2. id e&longs;t, &longs;i <lb/>progre&longs;&longs;io fieret in partibus temporis &longs;en&longs;ibilibus; </s> <s id="N17390"><!-- NEW -->at po&longs;ito quod &longs;int <lb/>plures termini, vt reuerâ &longs;unt; </s> <s id="N17396"><!-- NEW -->nam in totidem terminis fit progre&longs;&longs;io, in <lb/>quibus fit augmentum impetus, vel accelerationis acce&longs;&longs;io; </s> <s id="N1739C"><!-- NEW -->atqui hæc <lb/>fit in &longs;ingulis in&longs;tantibus, licèt finitis, igitur & progre&longs;&longs;io; </s> <s id="N173A2"><!-- NEW -->Quare duæ <lb/>partes temporis AE, EF diuidantur in 4. æquales AD; certè in duabus <lb/>primis percurretur &longs;patium. </s> <s id="N173AA">VQ æquale GH; igitur duabus vltimis per­<lb/>curretur QK, quæ &longs;it ad QV vt 7. ad 3. nam prima parte percurritur 1. <lb/>&longs;patium. </s> <s id="N173B1"><!-- NEW -->&longs;ecunda 2. igitur QV continet tria &longs;patia; </s> <s id="N173B5"><!-- NEW -->tertia verò 3. quarta <lb/>4.ergo hæ duæ vltimæ 7. &longs;ed QM e&longs;t dupla QV; </s> <s id="N173BB"><!-- NEW -->igitur continet 6. igi­<lb/>tur MK e&longs;t 1/3 VQ, vel KL; </s> <s id="N173C1"><!-- NEW -->igitur KM e&longs;t (1/12) GL; </s> <s id="N173C5"><!-- NEW -->igitur 12. L (1/10), vel <lb/>1/6, igitur VK e&longs;t ad GL vt 10.ad 12. igitur totum &longs;patium VK e&longs;t mi­<lb/>nus vero 1/6. Præterea 2. partes temporis AE EF diuidantur in 8. partes <lb/>æquales AE; </s> <s id="N173CF"><!-- NEW -->haud dubiè 4. primis percurretur &longs;patium XT æquale <lb/>GH, quod debet diuidi in 10. &longs;patia; </s> <s id="N173D5"><!-- NEW -->nam 4. terminis, &longs;eu temporibus <lb/>re&longs;pondent &longs;patia 10. quibus æqualia &longs;unt 40. in teta GL, cuius XT e&longs;t <lb/>(1/14), &longs;ed &longs;i in 4.primis acquiruntur 10. 4. vltimis EF acquiruntur 26.&longs;cili­<lb/>cet T 5; igitur tota X 5. e&longs;t 6. igitur e&longs;t ad GL vt 36. ad 40. &longs;eu 9. ad <lb/>10. igitur X 5. e&longs;t &longs;patium minus vero (1/10). </s> </p> <p id="N173E1" type="main"> <s id="N173E3"><!-- NEW -->Præterea diuidatur tempus AF in 16. partes æquales AB; </s> <s id="N173E7"><!-- NEW -->haud dubiè <pb pagenum="102" xlink:href="026/01/134.jpg"/>8 primis acquiritur &longs;patium YS æquale GH; quod debet diuidi in &longs;pa­<lb/>tiola 36, quæ re&longs;pondent 8. temporibus, &longs;eu terminis huius progre&longs;&longs;io­<lb/>nis, quibus æqualia &longs;unt 144. in GL, cuius YS e&longs;t 1/4, &longs;ed &longs;i in 8. primis <lb/>acquiruntur 36. in 8. vltimis acquirentur 100. igitur S 6. e&longs;t 100. igitur <lb/>Y6. e&longs;t 136. igitur e&longs;t ad GL vt 136. ad 144.&longs;eu 17.ad 18.igitur Y6.e&longs;t <lb/>&longs;patium totale minus vero (1/18). </s> </p> <p id="N173FA" type="main"> <s id="N173FC">Deinde diuidatur adhuc tempus AF in partes 32. æquales, 16. pri­<lb/>mis acquiritur ZR æquale GH, quod debet diuidi in &longs;patiola 136.quæ <lb/>re&longs;pondent 16. temporibus quibus æqualia &longs;unt 544. in tota GL, cuius <lb/>ZR e&longs;t 1/4 &longs;ed &longs;i in 16. primis temporibus acquiruntur 136. in vltimis <lb/>16. acquiruntur 392. igitur R 7. e&longs;t 392. & ZR 136. igitur Z 7.528. <lb/>igitur Z 7. e&longs;t ad GL, vt 528. ad 544. &longs;eu vt 33. ad 34. igitur Z 7 e&longs;t <lb/>&longs;patium minus verò (1/34) </s> </p> <p id="N1740B" type="main"> <s id="N1740D"><!-- NEW -->Denique &longs;i diuidatur tempus AF in partes 64.&longs;patium acqui&longs;itum erit <lb/>minus vero, a&longs;&longs;umpto &longs;cilicet tota HL (1/66), &longs;i diuidatur in 128. partes, erit <lb/>minus (1/130) &longs;i diuidatur in 256. partes, erit minus (1/258) &longs;ed temporis par­<lb/>tes 2.AE. EF minimè &longs;en&longs;ibilium diuidi po&longs;&longs;unt in infinita ferè in&longs;tan­<lb/>tia; &longs;int tantùm ex.g. </s> <s id="N17419">1000000. igitur &longs;patium tunc acqui&longs;itum erit mi­<lb/>nus &longs;uppo&longs;ito vero HL (1/1000002), quæ &longs;i de&longs;it tantùm &longs;patio KL vt &longs;it 1/4 <lb/>totius GL, quis hoc di&longs;cernat? </s> <s id="N17420"><!-- NEW -->igitur etiam &longs;uppo&longs;ita progre&longs;&longs;ione arith­<lb/>metica, quæ fiat in finitis in&longs;tantibus; </s> <s id="N17426"><!-- NEW -->&longs;i ob&longs;eruetur acurati&longs;&longs;imè &longs;patium, <lb/>quod percurritur in vna parte temporis &longs;en&longs;ibili v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;patium GH in <lb/>parte temporis AE; </s> <s id="N17432"><!-- NEW -->&longs;patium, quod acquiretur in tempore &longs;ecundo æqua­<lb/>li tàm propè accedet ad &longs;patium HL, id e&longs;t ad triplum prioris GH, vt <lb/>nullus mortalium di&longs;cernere po&longs;&longs;it; igitur cum hoc experimento tàm <lb/>pote&longs;t &longs;tare no&longs;tra hypothe&longs;is, quàm alia Galilei, igitur neutra ex eo tan­<lb/>tùm euinci pote&longs;t. </s> </p> <p id="N1743E" type="main"> <s id="N17440"><!-- NEW -->Hinc obiter ob&longs;erua progre&longs;&longs;ionem differentiarum; </s> <s id="N17444"><!-- NEW -->quippe &longs;i &longs;int <lb/>tantùm 2. partes temporis, differentia e&longs;t 1/4; </s> <s id="N1744A"><!-- NEW -->&longs;i 4.1/6 &longs;i 8. (1/10); &longs;i 16.(1/18); &longs;i 32. <lb/>(1/34); </s> <s id="N17450"><!-- NEW -->&longs;i 64.(1/66) nam primò denominator fractionis &longs;uperat tantùm binario <lb/>numerum partium temporis; &longs;ecundò differentiæ denominatorum &longs;unt <lb/>in progre&longs;&longs;ione geometrica dupla numerorum 2. 4. 8. 16. 32. 64. <lb/>128. &c. </s> </p> <p id="N1745A" type="main"> <s id="N1745C"><!-- NEW -->Eodem modo &longs;oluendum e&longs;t &longs;ecundum experimentum rotati globi in <lb/>plano decliui; </s> <s id="N17462"><!-- NEW -->præ&longs;ertim cum globus ab incur&longs;u a&longs;periorum partium <lb/>tùm globi, tùm plani &longs;altuatim de&longs;cendat; </s> <s id="N17468"><!-- NEW -->quod dubium e&longs;&longs;e non pote&longs;t, <lb/>& quò decliuius erit, faciliùs re&longs;iliet a plano, vt patet; &longs;ed de motu in <lb/>planis inclinatis fusè agemus infrà libro integro. </s> </p> <p id="N17470" type="main"> <s id="N17472"><!-- NEW -->Quod &longs;pectat ad tertium experimentum; </s> <s id="N17476"><!-- NEW -->multa in eo &longs;upponuntur <lb/>vel fal&longs;a, vel &longs;altem dubia: vel ea quæ cum no&longs;tra hypothe&longs;i optimè con­<lb/>ueniant. </s> <s id="N1747E"><!-- NEW -->Primum e&longs;t, quando dicuntur omnes vibrationes eiu&longs;dem fune­<lb/>penduli, &longs;iue maiores, &longs;iue minores e&longs;&longs;e æquediuturnæ, quod manife&longs;tis <lb/>experimentis repugnat; </s> <s id="N17486"><!-- NEW -->quippe vibratio maior plùs temporis; </s> <s id="N1748A"><!-- NEW -->minor ve­<lb/>rò minùs in &longs;uo de&longs;cen&longs;u ponit; </s> <s id="N17490"><!-- NEW -->dimittantur enim duo funependula æ­<lb/>qualia; </s> <s id="N17496"><!-- NEW -->alterum quidem ex altitudine 90.graduum, alterum ex altitudine <pb pagenum="103" xlink:href="026/01/135.jpg"/>10. vel 15.graduum; </s> <s id="N1749F"><!-- NEW -->ita vt &longs;imul vibrationes &longs;uas incipiant; </s> <s id="N174A3"><!-- NEW -->numerentur <lb/>vibrationes vtriu&longs;que, vbi 100. è minoribus numerat&ecedil; fuerint, numera­<lb/>buntur circiter 97. è maioribus, quod &longs;æpiùs ob&longs;eruaui te&longs;tibus etiam <lb/>adhibitis; </s> <s id="N174AD"><!-- NEW -->hoc ip&longs;um etiam ob&longs;eruarunt alij; </s> <s id="N174B1"><!-- NEW -->atque adeo ip&longs;e P.Mer&longs;en­<lb/>nus, qui L. 2. &longs;uæ ver&longs;ionis, Ar.17. Galileum arguit parùm acurati &longs;tu­<lb/>dij in his ob&longs;eruationibus adhibiti: </s> <s id="N174B9"><!-- NEW -->rationem huius effectus in libro de <lb/>funependulis explicabimus; </s> <s id="N174BF"><!-- NEW -->immò &longs;i omnes vibratìones maiores primæ <lb/>vibrationi 90. grad. <!-- REMOVE S-->e&longs;&longs;ent æquales, & aliæ minores alterius funependu­<lb/>li &longs;en&longs;un, vt &longs;it, minuerentur; </s> <s id="N174C9"><!-- NEW -->vix 90. maiores numerare po&longs;&longs;es, iam enu­<lb/>meratis 100. ex minoribus; </s> <s id="N174CF"><!-- NEW -->&longs;ed de his omnibus &longs;uo loco; </s> <s id="N174D3"><!-- NEW -->in vna tamen <lb/>vel altera vibratione vix aliquod di&longs;crimen ob&longs;eruatur; quod tamen ob­<lb/>&longs;eruari facilè po&longs;&longs;et in maioribus funependulis. </s> </p> <p id="N174DB" type="main"> <s id="N174DD"><!-- NEW -->Secundum, quod &longs;upponitur, e&longs;t quod longitudines funependulorum <lb/>&longs;int pror&longs;us, vt quadrata temporum, quibus vibrationes &longs;ingulorum <lb/>fiunt, v.g. <!-- REMOVE S-->funependulum longitudinis 4. pedum facere vnam vibratio­<lb/>nem eo tempore, quo funependulum longitudinis vnius pedis facit duas; </s> <s id="N174E9"><!-- NEW --><lb/>quod primò in multis vibrationibus non tàm accuratè ob&longs;eruatur; </s> <s id="N174EE"><!-- NEW --><expan abbr="&longs;ecū-dò">&longs;ecun­<lb/>dò</expan> licèt ob&longs;eruaretur &longs;en&longs;ibiliter, idem re&longs;ponderi debet, quod &longs;uprà in <lb/>&longs;ingulis vibrationibus e&longs;&longs;e tantùm di&longs;crimen; </s> <s id="N174F9"><!-- NEW -->quod etiam in multis &longs;en&longs;i­<lb/>bile non e&longs;t; &longs;i enim di&longs;crimen primarum vibrationem v.g.&longs;it (1/100000000) <lb/>certè vltimarum adhuc in&longs;en&longs;ibile erit. </s> </p> <p id="N17501" type="main"> <s id="N17503"><!-- NEW -->Tertium &longs;uppo&longs;itum fuit, minimum arcum minoris quadrantis a&longs;&longs;um­<lb/>ptum, & alium minoris quadrantis e&longs;&longs;e ad in&longs;tar perpendicularium; </s> <s id="N17509"><!-- NEW -->cùm <lb/>tamen diuer&longs;a &longs;it inclinatio minoris, & maioris quadrantis: </s> <s id="N1750F"><!-- NEW -->quippe <lb/>principium maioris accedit propiùs ad perpendicularem; </s> <s id="N17515"><!-- NEW -->facit enim <lb/>angulum contingentiæ minorem; </s> <s id="N1751B"><!-- NEW -->alia verò extremitas accedit propiùs <lb/>ad horizontalem propter rationem prædictam; </s> <s id="N17521"><!-- NEW -->hinc illa extremitas ma­<lb/>ioris, vnde e&longs;t initium motus, planum decliuius facit; altera verò minùs <lb/>decliue; &longs;ed hæc fusè pro&longs;equar &longs;uo loco. </s> </p> <p id="N17529" type="main"> <s id="N1752B"><!-- NEW -->Quartum, quod &longs;upponitur e&longs;t, accelerationem motus fieri in qua­<lb/>drante in ea ratione, in qua fit per plana chordarum inclinata, quod <lb/>etiam fal&longs;um e&longs;t; </s> <s id="N17533"><!-- NEW -->quia in eodem plano inclinato &longs;upponitur eadem <lb/>inclinatio; </s> <s id="N17539"><!-- NEW -->&longs;ecus in quadrante, cuius &longs;ingula puncta nouam faciunt in­<lb/>clinationem: </s> <s id="N1753F"><!-- NEW -->adde quod quarta pars quadrantis maioris EK non facit <lb/>eandem inclinationem, quam totus quadrans minor DF ip&longs;i EK æqua­<lb/>lis; quamquam hoc ip&longs;i vltrò concedent aduer&longs;arij. </s> </p> <p id="N17547" type="main"> <s id="N17549"><!-- NEW -->Præterea, &longs;it ita vt &longs;upponitur; </s> <s id="N1754D"><!-- NEW -->ita vt &longs;en&longs;ibiliter differentia huius <lb/>progre&longs;&longs;ionis percipi non po&longs;&longs;it, &longs;intque numeratæ omnes vibrationes <lb/>&longs;en&longs;ibiles dati funependuli ex altitudine 90, grad. <!-- REMOVE S-->demi&longs;&longs;i; </s> <s id="N17557"><!-- NEW -->quæ vix e&longs;&longs;e <lb/>po&longs;&longs;unt 1800; </s> <s id="N1755D"><!-- NEW -->&longs;int autem plures &longs;cilicet 2000. dicis confectas e&longs;&longs;e 2000 <lb/>minoris funependuli eo tempore, quo 1000. tantùm in quadruplo fune­<lb/>pendulo numerantur; </s> <s id="N17565"><!-- NEW -->annuo quidem, &longs;i res tantùm &longs;en&longs;ibiliter con&longs;ide­<lb/>retur; </s> <s id="N1756B"><!-- NEW -->&longs;in verò &longs;ecùs, id pernego; &longs;ed dico dee&longs;&longs;e v. <!-- REMOVE S-->g. <!-- REMOVE S-->1000000. puncta <lb/>&longs;patij, quæ di&longs;cerni non po&longs;&longs;unt, ita vt primæ vibrationi 1000. pun­<lb/>cta &longs;ecundæ, 2000. tertiæ 3000. &c. </s> <s id="N17577"><!-- NEW -->vltimæ verò, &longs;eu mille&longs;imæ <pb pagenum="104" xlink:href="026/01/136.jpg"/>1000000. quæ omnia &longs;unt in&longs;en&longs;ibilia, neque maiorem habent diffi­<lb/>cultatem, quàm in motu perpendiculari, de quo &longs;uprà; etiam conce&longs;&longs;is <lb/>vltrò omnibus experimétis propo&longs;itis. </s> <s id="N17584"><!-- NEW -->Igitur &longs;uppo&longs;itâ progre&longs;&longs;ione &longs;pa­<lb/>tiorum arithmetica in in&longs;tantibus, tàm propè accedit ad aliam, quàm <lb/>Galileus ponit, &longs;iue in perpendiculari deor&longs;um, &longs;iue in quadrante fune­<lb/>penduli; </s> <s id="N1758E"><!-- NEW -->a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus, vt differentia <lb/>di&longs;cernit non po&longs;&longs;it; </s> <s id="N17594"><!-- NEW -->immò nec duplum differentiæ, nec centuplum, nec <lb/>millecuplum; </s> <s id="N1759A"><!-- NEW -->&longs;ed de his &longs;atis quæ ex dictis &longs;uprà facilè intelligi po&longs;&longs;unt: <lb/>quare veniemus iam ad rationes. </s> </p> <p id="N175A0" type="main"> <s id="N175A2"><!-- NEW -->Prima ratio, quam affert Galileus e&longs;t; </s> <s id="N175A6"><!-- NEW -->quia cum natura in &longs;uis opera­<lb/>tionibus adhibeat &longs;implici&longs;&longs;ima media; </s> <s id="N175AC"><!-- NEW -->& cum acceleratio motus natu­<lb/>ralis non po&longs;&longs;it fieri iuxta faciliorem, vel &longs;impliciorem progre&longs;&longs;ionem, <lb/>quàm &longs;it-ea quæ fit per quadrata; </s> <s id="N175B4"><!-- NEW -->non e&longs;t dubium, quin iuxta illam pro­<lb/>gre&longs;&longs;io motus naturaliter accelerati fieri debeat; præ&longs;ertim cùm omni­<lb/>bus experimentis con&longs;entiat, & in ea omnia phænomena explicari <lb/>po&longs;&longs;int. </s> </p> <p id="N175BE" type="main"> <s id="N175C0">Re&longs;p. Primò progre&longs;&longs;ionem arithmeticam &longs;implicem iuxta hos nu­<lb/>meros 1.2.3.4. longè &longs;impliciorem e&longs;&longs;e alia quæ fit iuxta illos 1.3.5.7.vt <lb/>nemo non iudicabit. </s> <s id="N175C7"><!-- NEW -->Secundò <expan abbr="cũ">cum</expan> accidit duas hypothe&longs;es conuenire cum <lb/>omnibus experimentis &longs;eu phænomonis, debet e&longs;&longs;e aliqua ratio, cur ad­<lb/>hibeatur vna potiùs quàm alia; </s> <s id="N175D3"><!-- NEW -->&longs;ed nulla e&longs;t ratio, cur Galileus adhibeat <lb/>&longs;uam, vti videbimus; </s> <s id="N175D9"><!-- NEW -->nos verò ratione demon&longs;tratiuâ probamus no&longs;tram; </s> <s id="N175DD"><!-- NEW --><lb/>igitur no&longs;tra e&longs;t præferenda pro theorica rei veritate; quia verò alia in <lb/>temporibus &longs;en&longs;ibilibus proximè ad verum accedit eam adhibendam e&longs;&longs;e <lb/>decernemus infrà ad praxim, & communem i&longs;torum motuum men­<lb/>&longs;uram. </s> </p> <p id="N175E8" type="main"> <s id="N175EA">Secunda ratio e&longs;t; </s> <s id="N175ED"><!-- NEW -->quia, &longs;i accipiatur &longs;ubduplum maximæ, & minimæ <lb/>velocitatis; &longs;itque ex his qua&longs;i conflata velocitas motus æquabilis, hoc <lb/>motu æquabili æquali tempore pèrcurretur &longs;patium idem, quod antè <lb/>motu naturaliter accelerato v.g. <!-- REMOVE S-->&longs;int numeri datæ progre&longs;&longs;ionis 1.3.5.7. <lb/>9.11. certè &longs;umma terminorum &longs;eu totum &longs;patium erit 36. accipiatur <lb/>&longs;ubduplum primi 1/2 & &longs;exti 5. 1/2 habebitur velocitas vt 6. igitur cum <lb/>velocitate vt 6. æquali tempore percurretur &longs;patium 36. quod rectè de­<lb/>mon&longs;trauit Galileus. <!-- KEEP S--></s> </p> <p id="N17602" type="main"> <s id="N17604"><!-- NEW -->Re&longs;pondeo non minùs no&longs;tram hypothe&longs;im cum hoc ip&longs;o &longs;tare, quàm <lb/>&longs;tet hypothe&longs;is Galilei: </s> <s id="N1760A"><!-- NEW -->&longs;int enim 6. in&longs;tantia, & &longs;ingulis &longs;ua tribuantur <lb/>&longs;patiola more dicto 1 2 3 4 5 6. &longs;umma &longs;patiorum e&longs;t 21. a&longs;&longs;umatur &longs;ub­<lb/>duplum velocitatis primi in&longs;tantis 1/2, & &longs;ubduplum &longs;exti in&longs;tantis, &longs;cili­<lb/>cet 3. conflatum ex vtroque 3 1/3; </s> <s id="N17614"><!-- NEW -->ducatur in 6.id e&longs;t in numerum termi­<lb/>norum, vel in&longs;tantium; &longs;umma erit 21. igitur quod tribuit Galileus &longs;uæ <lb/>progre&longs;&longs;ioni, etiam no&longs;træ competit. </s> </p> <p id="N1761C" type="main"> <s id="N1761E"><!-- NEW -->Tertia ratio petitur ex mathe&longs;i &longs;it enim linea AE diui&longs;a in quatuor <lb/>partes æquales, quæ nobis repre&longs;entent 4. partes temporis æquales; </s> <s id="N17624"><!-- NEW --><lb/>haud dubiè, cùm acquirantur temporibus æqualibus æqualia velocitatis <lb/>momenta; </s> <s id="N1762B"><!-- NEW -->haud dubiè, inquam, his 4. temporibus AB, BC, CD, DE, ac-<pb pagenum="105" xlink:href="026/01/137.jpg"/>quirentur æquales velocitatis gradus; </s> <s id="N17634"><!-- NEW -->&longs;it autem BI, men&longs;ura velocitatis, <lb/>quam acquirit mobile cadens ex &longs;ua quiete in fine primæ partis tempo­<lb/>ris AB; </s> <s id="N1763C"><!-- NEW -->certè in fine &longs;ecundæ partis temporis BC acquiret velocitatem, <lb/>quæ coniuncta cum priore BI faciet duplam CH, & in fine tertiæ par­<lb/>tiæ CD triplam DG; </s> <s id="N17644"><!-- NEW -->denique in fine quartæ DE quadruplam EF; </s> <s id="N17648"><!-- NEW -->quip­<lb/>pe cum in parte BC remaneat tota velocitas B, & acquiratur æqualis; </s> <s id="N1764E"><!-- NEW --><lb/>certè in fine BC e&longs;t velocitas CH dupla illius quæ commen&longs;uratur BI. <lb/>&longs;imiliter in parte CD remanebit vtraque, & accedet altera; </s> <s id="N17655"><!-- NEW -->igitur e&longs;t ve­<lb/>locitas DG tripla BI, & EF e&longs;t quadrupla: Similiter ita &longs;e ratio habet <lb/>cuiu&longs;libet alterius partis inter AB ad aliam alterius partis inter BC, vt <lb/>lineæ ductæ parallelæ BICH, &c. </s> <s id="N1765F"><!-- NEW -->igitur cum &longs;patium acqui&longs;itum re&longs;­<lb/>pondeat exercitio huius velocitatis; </s> <s id="N17665"><!-- NEW -->&longs;itque in&longs;tanti B vt BI, & in&longs;tanti <lb/>C vt CH; </s> <s id="N1766B"><!-- NEW -->certè tempore AB e&longs;t vt triangulum AIB; </s> <s id="N1766F"><!-- NEW -->nam &longs;patium AIB <lb/>e&longs;t collectio omnium linearum, quæ duci po&longs;&longs;unt parallelæ in tempore <lb/>AB; </s> <s id="N17677"><!-- NEW -->idem dico de trapezo CBIH, qui e&longs;t triplus trianguli IBA; </s> <s id="N1767B"><!-- NEW -->& de <lb/>trapezo GDCH, qui e&longs;t quintuplus; </s> <s id="N17681"><!-- NEW -->igitur triangulum HCA e&longs;t qua­<lb/>druplum IBA; </s> <s id="N17687"><!-- NEW -->quia hæc triangula &longs;unt vt quadrata laterum; </s> <s id="N1768B"><!-- NEW -->igitur &longs;pa­<lb/>tium acqui&longs;itum temporibus AB, BC, e&longs;t ad &longs;patium acqui&longs;itum tempo­<lb/>re AB, vt triangulum HCB ad triangulum IBA; </s> <s id="N17693"><!-- NEW -->igitur vt quadratum <lb/>AB ad quadratum AC; </s> <s id="N17699"><!-- NEW -->igitur vt quadratum temporis AB ad quadra­<lb/>tum temporis AC; igitur &longs;patia diuer&longs;is temporibus decur&longs;a &longs;unt vt qua­<lb/>drata temporum, quibus &longs;ingula decurruntur. </s> </p> <p id="N176A1" type="main"> <s id="N176A3"><!-- NEW -->Hæc ratio ad &longs;peciem videtur e&longs;&longs;e demon&longs;tratiua, deficit tamen à ve­<lb/>ra demon&longs;tratione; </s> <s id="N176A9"><!-- NEW -->primo, quia &longs;upponit in&longs;tantia infinita, quæ multi <lb/>pa&longs;&longs;im negabunt in tempore; </s> <s id="N176AF"><!-- NEW -->immò aliquis vltrò demon&longs;trare tentaret <lb/>non e&longs;&longs;e infinita; </s> <s id="N176B5"><!-- NEW -->itaque ex &longs;uppo&longs;itione quod &longs;int tantùm finita in&longs;tan­<lb/>tia a&longs;&longs;umantur 4. æqualia AC, CD, DE, EF, certè cum in&longs;tans &longs;it to­<lb/>rum &longs;imul, velocitatem habet æquabilem &longs;ibi toti re&longs;pondentem; </s> <s id="N176BD"><!-- NEW -->igitur <lb/>in&longs;tanti AC re&longs;pondeat velocitas, cuius men&longs;ura &longs;it ABCG; </s> <s id="N176C3"><!-- NEW -->haud du­<lb/>biè in&longs;tanti CD re&longs;pondebit velocitas CH, &longs;cilicet dupla AB; </s> <s id="N176C9"><!-- NEW -->nam re­<lb/>manet primus velocitatis gradus acqui&longs;itus primo in&longs;tanti: </s> <s id="N176CF"><!-- NEW -->&longs;ed alter æ­<lb/>qualis acquiritur; </s> <s id="N176D5"><!-- NEW -->igitur e&longs;t duplus prioris; </s> <s id="N176D9"><!-- NEW -->igitur re&longs;pondet lineæ DK. <lb/>quæ tripla e&longs;t AB, & quarto lineæ FN, quæ e&longs;t quadrupla AB; </s> <s id="N176DF"><!-- NEW -->igitur <lb/>cre&longs;cit &longs;patium, vt rectangula CB, DH, EK, FM; </s> <s id="N176E5"><!-- NEW -->&longs;ed hæc cre&longs;cunt iuxta <lb/>progre&longs;&longs;ionem numerorum 1.2.3.4. nec aliter res e&longs;&longs;e pote&longs;t ex &longs;uppo&longs;i­<lb/>tione quod &longs;int in&longs;tantia finita; </s> <s id="N176ED"><!-- NEW -->quod alibi ex profe&longs;&longs;o tractamus: </s> <s id="N176F1"><!-- NEW -->quippe <lb/>illa quæ&longs;tio pertinet ad Metaphy&longs;icam, non verò ad phy&longs;icun; </s> <s id="N176F7"><!-- NEW -->nam vel <lb/>&longs;ingula aliquid addunt, vel nihil: aliquid addunt haud dubiè; </s> <s id="N176FD"><!-- NEW -->igitur con­<lb/>&longs;iderantur tantùm 4. in&longs;tantia prima AC, CD, DE, EF, in &longs;ua &longs;crie; </s> <s id="N17703"><!-- NEW -->certè <lb/>non po&longs;&longs;unt aliam progre&longs;&longs;ionem facere quàm eam, quæ e&longs;t iuxta hos <lb/>numeros 1.2.3.4.vnde non fit per triangula &longs;ed per rectangula minima; <lb/>igitur linea AF præcedentis figuræ non e&longs;t recta, &longs;ed denticulata, qualis <lb/>e&longs;&longs;et ABGHIKLMN, &longs;ed longè minoribus gradibus, &longs;eu denticulis. </s> <s id="N1770F"><lb/>Hinc quò rectangula CB, DH, &c. </s> <s id="N17713"><!-- NEW -->fient maiora in partibus &longs;cilicet tem­<lb/>poris &longs;en&longs;ibilibus, &longs;eruata &longs;cilicet in illis progre&longs;&longs;ione numerorum 1.2.3. <pb pagenum="106" xlink:href="026/01/138.jpg"/>4.progre&longs;&longs;io longiùs di&longs;cedet à vera; </s> <s id="N1771E"><!-- NEW -->vt &longs;uprà iam totius repetitum fuit: </s> <s id="N17722"><!-- NEW --><lb/>quippe hæc progre&longs;&longs;io in puris in&longs;tantibus fieri tantùm pote&longs;t, cum &longs;in­<lb/>gulis in&longs;tantibus noua fiat acce&longs;&longs;io velocitatis, in hoc enim e&longs;t error, <lb/>quòd in tota parte temporis AC ponatur æquabilis velocitas, eiu&longs;que <lb/>principium A, &longs;it æquale fini C; </s> <s id="N1772D"><!-- NEW -->nam AB, & GH &longs;unt æquales; </s> <s id="N17731"><!-- NEW -->cùm ta­<lb/>men &longs;it minor velocitas in A, quàm in C, ni&longs;i AC &longs;it tantùm <expan abbr="in&longs;tãs">in&longs;tans</expan>; </s> <s id="N1773B"><!-- NEW -->vnde <lb/>tota velocitas in hypothe&longs;i Galilei acqui&longs;ita in 4.partibus temporis a&longs;­<lb/>&longs;umptis e&longs;t, vt triangulum AFN; </s> <s id="N17743"><!-- NEW -->acqui&longs;ita verò in no&longs;tra hypothe&longs;i e&longs;t vt <lb/>&longs;umma rectangulorum CB, CI, EK, EN, quæ &longs;umma e&longs;t ad triangulum <lb/>AFN, vt 10, ad 8. vel vt 5.ad 4. igitur maior 1/4; nam prima pars tempo­<lb/>ris addit triangulum ABG, &longs;ecunda GHI. &c. </s> </p> <p id="N1774D" type="main"> <s id="N1774F"><!-- NEW -->Si tamen diuidantur i&longs;tæ partes temporis in minores v. <!-- REMOVE S-->g. <!-- REMOVE S-->in 8. tunc <lb/>&longs;umma rectangulorum erit tantùm maior 1/8; </s> <s id="N17759"><!-- NEW -->&longs;i in 16. (1/16) &longs;i in 32. (1/32); </s> <s id="N1775D"><!-- NEW -->&longs;i in <lb/>64.(11/64), cuius &longs;ehema hîc habes; &longs;int enim 3.partes temporis &longs;en&longs;ibiles A <lb/>CDFE, & &longs;patium vt triangulum AFN, &longs;patia verò acqui&longs;ita in &longs;ingulis <lb/>partibus, vt portiones trianguli prædicti, quæ ip&longs;is re&longs;pondent v. <!-- REMOVE S-->g. <!-- REMOVE S-->ac­<lb/>qui&longs;itum in prima parte ad acqui&longs;itum in &longs;ecunda tantùm, vt triangu­<lb/>lum ACG ad trapezum GCDI &c. </s> <s id="N1776F"><!-- NEW -->denique acqui&longs;itum in temporibus <lb/>inæqualibus, vt quadrata temporum v. <!-- REMOVE S-->g. <!-- REMOVE S-->acqui&longs;itum in prima parte ad <lb/>acqui&longs;itum in duabus, vt triangulum ACG ad triangulum ADI; </s> <s id="N1777B"><!-- NEW -->id e&longs;t <lb/>quadratum CA ad quadratum DA; </s> <s id="N17781"><!-- NEW -->in no&longs;tra verò hypothe&longs;i, &longs;i velocitas <lb/>in tota prima parte AC ponatur vt CG æquabiliter; </s> <s id="N17787"><!-- NEW -->haud dubiè &longs;patium <lb/>acqui&longs;itum in prædictis 4. temporibus erit, vt &longs;umma rectangulorum C <lb/>B, CI, EK, EN, quæ maior e&longs;t toto triangulo, AFN, 4. triangulis ABG, <lb/>GHI, IKL, LMN, ie e&longs;t 1/4 totius trianguli AFN; atque ita &longs;umma re­<lb/>ctangulorum continet 10. quadrata æqualia quadrato CB, & triangu­<lb/>lum AFN, continet. </s> <s id="N17795">tantùm 8. </s> </p> <p id="N17798" type="main"> <s id="N1779A"><!-- NEW -->Iam verò diuidantur 4. partes temporis AF, in 8. æquales; </s> <s id="N1779E"><!-- NEW -->in &longs;enten­<lb/>tia Galilei totum &longs;patium erit &longs;emper triangulum AFN, id e&longs;t vt &longs;ubdu­<lb/>plum quadrati &longs;ub AF; </s> <s id="N177A6"><!-- NEW -->quæ cùm &longs;it 8. quadratum erit 64.& &longs;ubduplum <lb/>quadrati 32. at verò &longs;umma rectangulorum e&longs;t 36. id e&longs;t continet 36. <lb/>quadrata æqualia quadrato XA; cùm tamen triangulum AFN, conti­<lb/>neat tantùm 32. igitur &longs;umma prædicta e&longs;t ad triangulum AFN, vt 36. <lb/>ad 32. id e&longs;t vt 9.ad 8. igitur &longs;umma e&longs;t maior triangulo 1/8, quæ omnia <lb/>con&longs;tant. </s> </p> <p id="N177B4" type="main"> <s id="N177B6"><!-- NEW -->Præterea diuidatur vlteriùs tempus AF in 16. æquales partes; </s> <s id="N177BA"><!-- NEW -->qua­<lb/>dratum 16. cum &longs;it 256. accipiatur &longs;ubduplum id e&longs;t 128. & erit trian­<lb/>gulum AFN, cui &longs;emper re&longs;pondet totum &longs;patium acqui&longs;itum in &longs;enten­<lb/>tia Galilei; </s> <s id="N177C4"><!-- NEW -->at verò &longs;umma rectangulorum erit 136. igitur &longs;umma e&longs;t ad <lb/>&longs;ummam vt 136.ad 128.id e&longs;t vt 17.ad 16. igitur e&longs;t maior &longs;umma trian­<lb/>gulo (1/16) atque ita deinceps; </s> <s id="N177CC"><!-- NEW -->&longs;i vlteriùs diuidas prædictum tempus in par­<lb/>tes minores: quot porrò erunt, antequam fiat tota re&longs;olutio in in&longs;tan­<lb/>tia, &longs;int enim v. <!-- REMOVE S-->g. <!-- REMOVE S-->in tempore AF in&longs;tantia 1000000. &longs;umma quæ re&longs;­<lb/>pondet no&longs;træ progre&longs;&longs;ioni, erit maior altera, quæ re&longs;pondet progre&longs;&longs;io­<lb/>ni Galilei (1/1000000) quis hoc percipiat? </s> </p> <pb pagenum="107" xlink:href="026/01/139.jpg"/> <p id="N177E0" type="main"> <s id="N177E2"><!-- NEW -->Si verò in no&longs;tra hypothe&longs;i &longs;patium, quod re&longs;pondet primæ parti tem­<lb/>poris AC &longs;it idem cum illo, quod re&longs;pondet eidem parti in &longs;ententia <lb/>Galilei, id e&longs;t æquale triangulo CAG, &longs;umma &longs;patiorum erit minor in <lb/>no&longs;tra hypothe&longs;i triangulo AFN &longs;ex triangulis æqualibus triangulo <lb/>ACG; igitur erit vt 10.ad 16. igitur minor 1/8. </s> <s id="N177EE"><!-- NEW -->&longs;i verò diuidantur in 8. <lb/>temporis partes, triangulum AFN continebit 64. triangula æqualia <lb/>AXQ: </s> <s id="N177F6"><!-- NEW -->at verò &longs;umma quæ re&longs;pondet no&longs;træ hypothe&longs;i 36.igitur minor <lb/>(7/16). denique &longs;i diuidantur in 16. partes, triangulum AFN continebit <lb/>256. triangula æqualia AYZ; at verò &longs;umma no&longs;tra 136. igitur minor <lb/>(15/52) &longs;ed nunquam erit minor 1/2. </s> </p> <p id="N17800" type="main"> <s id="N17802"><!-- NEW -->Ob&longs;eruabis obiter dictum e&longs;&longs;e &longs;uprà &longs;ummam rectangulorum CB CI <lb/>EK EN e&longs;&longs;e maiorem triangulo AFN, 2.quadratis æqualibus CB; </s> <s id="N17808"><!-- NEW -->&longs;i <lb/>verò diuidatur tempus in 8. partes, &longs;umma rectangulorum e&longs;t minor præ­<lb/>cedenti &longs;ummâ, toto quadrato æquali CB, id e&longs;t 4.quadratis æqualibus <lb/>XB, id e&longs;t 1/2 primæ differentiæ, quæ e&longs;t &longs;umma duorum quadratorum <lb/>æqualium CB; </s> <s id="N17814"><!-- NEW -->at &longs;i diuidatur in 16. partes, tempus AF, &longs;umma rectan­<lb/>gulorum e&longs;t minor præcedente 8. quadratis æqualibus QZ, vel &longs;ubdu­<lb/>plo quadrati CB, id e&longs;t 1/4 primæ differentiæ quæ e&longs;t &longs;umma duorum <lb/>quadratorum æqualium CB; </s> <s id="N1781E"><!-- NEW -->&longs;i 4. partes temporis diuidantur in 8. de­<lb/>trahitur 1/2 differentiæ, quæ e&longs;t inter &longs;ummam primam rectangulorum, <lb/>& triangulum AFN; </s> <s id="N17826"><!-- NEW -->&longs;i diuidantur in 16. detrahitur 1/4 eiu&longs;dem diffe­<lb/>rentiæ; </s> <s id="N1782C"><!-- NEW -->&longs;i diuidantur in 32. detrahitur 1/8, &longs;i in 64. (1/16); </s> <s id="N17830"><!-- NEW -->atque ita deinceps, <lb/>& nunquam hæ minutiæ &longs;ubtractæ in infinitum totam differentiam ex­<lb/>haurient; hinc minutiæ i&longs;tæ 1/2 1/4 1/8 (1/16) (1/32) (1/64) &c. </s> <s id="N17838"><!-- NEW -->in infinitum non fa­<lb/>ciunt vnum integrum; &longs;ed hæc &longs;unt facilia. </s> </p> <p id="N1783E" type="main"> <s id="N17840"><!-- NEW -->Quarta ratio, quam afferunt aliqui, e&longs;t; </s> <s id="N17844"><!-- NEW -->quia &longs;i cum eadem velocita­<lb/>te acqui&longs;ita in fine temporis dati &longs;ine augmento nouo moueatur mobi­<lb/>le; </s> <s id="N1784C"><!-- NEW -->haud dubiè acquiret duplum &longs;patium tempore æquali tempori dato; </s> <s id="N17850"><!-- NEW --><lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it triangulum AFE; </s> <s id="N17859"><!-- NEW -->&longs;itque velocitas acqui&longs;ita EF in 4. parti­<lb/>bus temporis AE, vt iam &longs;uprà dictum e&longs;t, ne cogar repetere: </s> <s id="N1785F"><!-- NEW -->certè &longs;i du­<lb/>catur velocitas EF in tempus AE, vel EL æquale; </s> <s id="N17865"><!-- NEW -->habebitur rectan­<lb/>gulum EK duplum trianguli AFE: </s> <s id="N1786B"><!-- NEW -->&longs;ed triangulum AFE e&longs;t &longs;umma <lb/>&longs;patiorum motus accelerati tempore AE, & rectangulum EK e&longs;t &longs;um­<lb/>ma &longs;patiorum motus æquabilis cum velocitate EF; igitur duplum e&longs;t <lb/>&longs;patium motus æquabilis, quod erat demon&longs;trandum. </s> <s id="N17875"><!-- NEW -->Præterea &longs;i diui­<lb/>datur velocitas EF, & eius &longs;ubdupla ducatur in tempus AE; habebitur <lb/>rectangulum æquale triangulo AFE, vt con&longs;tat. </s> <s id="N1787D"><!-- NEW -->Re&longs;pondeo facilè ex di­<lb/>ctis, hoc ip&longs;um etiam ex no&longs;tra hypothe&longs;i proxime &longs;equi; </s> <s id="N17883"><!-- NEW -->&longs;int enim duo <lb/>in&longs;tantia; </s> <s id="N17889"><!-- NEW -->haud dubie &longs;i non cre&longs;cit velocitas, &longs;ecundo in&longs;tanti æquale <lb/>&longs;patium percurretur; </s> <s id="N1788F"><!-- NEW -->&longs;i vero &longs;ecundo in&longs;tanti cre&longs;cat, percurrentur illo <lb/>motu 3.&longs;patia; </s> <s id="N17895"><!-- NEW -->& cùm velocitas <expan abbr="&longs;ecũdi">&longs;ecundi</expan> <expan abbr="in&longs;tãtis">in&longs;tantis</expan> &longs;it dupla velocitatis primi <lb/>in&longs;tantis, primo in&longs;tanti &longs;it 1.gradus v.g. <!-- REMOVE S-->&longs;ecundo erunt 2. gradus; </s> <s id="N178A5"><!-- NEW -->igi­<lb/>tur moueatur per duo in&longs;tantia motu æquabili veloci vt 2. percurrentur <lb/>4. &longs;patia; </s> <s id="N178AD"><!-- NEW -->igitur totum &longs;patium, quod percurritur motu veloci vt 2. per <lb/>2.in&longs;tantia e&longs;t ad totum &longs;patium, quod percurritur æquali tempore mo-<pb pagenum="108" xlink:href="026/01/140.jpg"/>tu naturaliter accelerato vt 4. ad 3. igitur continet illud 1. (11/3); </s> <s id="N178B8"><!-- NEW -->&longs;i verò <lb/>&longs;int 3. in&longs;tantis continet illud, 1/2; &longs;i 4. continet 1. 3/5, &longs;i 5. continet 1.2/3 <lb/>&longs;i 5. continet 1 2/3. &longs;i 6. continet 1 5/7. &longs;i 7. continet 1 3/4. &longs;i 8. continet <lb/>1 7/9. &longs;i 9. continet 1 (4/11). &longs;i 10. continet 1 9/5 &longs;ic quo plura erunt in&longs;tantia <lb/>accedet propiùs ad rationem duplam, nunquam tamen ad illam perue­<lb/>niet. </s> <s id="N178C6"><!-- NEW -->Ex dictis multa tumultuatim Corollaria congeri po&longs;&longs;unt; </s> </p> <p id="N178CA" type="main"> <s id="N178CC"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N178D9" type="main"> <s id="N178DB"><!-- NEW -->Etiam&longs;i non &longs;int partes infinitæ temporis; </s> <s id="N178DF"><!-- NEW -->in ordine tamen ad praxim <lb/>eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent infinitæ; quia licèt finitæ &longs;int, nume­<lb/>rari tamen non po&longs;&longs;unt. </s> </p> <p id="N178E7" type="main"> <s id="N178E9"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N178F6" type="main"> <s id="N178F8"><!-- NEW -->Etiam &longs;i non &longs;int infiniti tarditatis gradus, vt con&longs;tat ex dictis, &longs;ed fi­<lb/>niti; </s> <s id="N178FE"><!-- NEW -->in ordine tamen ad praxim eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent in­<lb/>finiti; quia non pote&longs;t di&longs;tingui primus, & minimus ab omnibus <lb/>aliis. </s> </p> <p id="N17906" type="main"> <s id="N17908"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N17915" type="main"> <s id="N17917"><!-- NEW -->Licèt hypothe&longs;is Galilei &longs;it fal&longs;a in hypothe&longs;i in&longs;tantium finitorum; </s> <s id="N1791B"><!-- NEW --><lb/>nam &longs;ingulis in&longs;tantibus noua fit velocitatis acce&longs;&longs;io; </s> <s id="N17920"><!-- NEW -->phy&longs;icè tamen lo­<lb/>quendo eodem modo &longs;e habet, ac &longs;i e&longs;&longs;et vera; </s> <s id="N17926"><!-- NEW -->quia cum non po&longs;&longs;it pro­<lb/>bari, ni&longs;i in partibus temporis &longs;en&longs;ibilibus; </s> <s id="N1792C"><!-- NEW -->certà, cùm quælibet pars <lb/>&longs;en&longs;ibilis innumera ferè in&longs;tantia contineat, in quibus fit progre&longs;&longs;io; </s> <s id="N17932"><!-- NEW --><lb/>differentia vtriu&longs;que &longs;en&longs;ibilis e&longs;&longs;e non pote&longs;t; </s> <s id="N17937"><!-- NEW -->igitur linea denticulata <lb/> eodem modo &longs;e habet phy&longs;icè, hoc e&longs;t &longs;en&longs;ibiliter, ac &longs;i e&longs;&longs;et recta; </s> <s id="N1793D"><!-- NEW -->&longs;ic­<lb/>que progre&longs;&longs;io arithmetica in multis terminis reducitur &longs;en&longs;ibiliter ad <lb/>Geometriam in paucioribus terminis; immò in communi illa &longs;ententia. </s> <s id="N17945"><!-- NEW --><lb/>in qua dicitur tempus con&longs;tare ex partibus actu infinitis, progre&longs;&longs;io Ga­<lb/>lilei tantùm locum habere pete&longs;t; </s> <s id="N1794C"><!-- NEW -->igitur hæc e&longs;to clauis huius difficul­<lb/>tatis; </s> <s id="N17952"><!-- NEW -->progre&longs;&longs;io &longs;implex principium phy&longs;icum habet, non experimen­<lb/>tum; </s> <s id="N17958"><!-- NEW -->progre&longs;&longs;io numerorum imparium experimentum non principium; </s> <s id="N1795C"><!-- NEW --><lb/>vtramque cum principio & experimento componimus; prima enim &longs;i. </s> <s id="N17961"><lb/>a&longs;&longs;umantur partes temporis &longs;en&longs;ibiles tran&longs;it in &longs;ecundam, &longs;ecunda in <lb/>primam, &longs;i vltima a&longs;&longs;umantur in&longs;tantia. </s> </p> <p id="N17967" type="main"> <s id="N17969"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N17976" type="main"> <s id="N17978">Cognito &longs;patio quod percurritur in data parte temporis &longs;en&longs;ibili, co­<lb/>gno&longs;ci pote&longs;t &longs;patium quod in duabus æqualibus vel 3.vel 4.&c.percurri <lb/>pote&longs;t.v.g. </s> <s id="N1797F"><!-- NEW -->multi probarunt &longs;æpiùs primo &longs;ecundo minuto corpus graue <lb/>percurrere 12. pedes; igitur duobus percurreret 48. accipe enim 9. 2. <lb/>id e&longs;t 4. & in 4. duces 12. vt habeas 48. 4. verò minutis percurret 192. <lb/>nam accipe 9. 4. id e&longs;t 16. & in 16. duces 12.vt habeat 192. res omninò <lb/>facilis. </s> </p> <p id="N1798B" type="main"> <s id="N1798D"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1799A" type="main"> <s id="N1799C"><!-- NEW -->Similiter cognito &longs;patio quod percurrit 4. &longs;ecundis minutis, cogno­<lb/>&longs;ces &longs;patium, quod percurret 2. vel 1. v.g. <!-- REMOVE S-->percurrit 4. &longs;ecundis 192. pe-<pb pagenum="109" xlink:href="026/01/141.jpg"/>des; </s> <s id="N179A9"><!-- NEW -->accipe 9.4. id e&longs;t 16. & per 16. diuide 192. quotíens dabit 12. pro <lb/>primo &longs;ecundo: accipe 9.2. id e&longs;t, 4. & diuide 192. per 4.quotiens dabit <lb/>48. pro duobus minutis, atque ita deinceps. </s> </p> <p id="N179B1" type="main"> <s id="N179B3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N179C0" type="main"> <s id="N179C2"><!-- NEW -->Similiter cognito tempore cogno&longs;ci pote&longs;t &longs;patium decur&longs;um; </s> <s id="N179C6"><!-- NEW -->quia <lb/>&longs;patia &longs;unt vt quadrata temporum; </s> <s id="N179CC"><!-- NEW -->vel cognito &longs;patio cogno&longs;ci pote&longs;t <lb/>tempus; quia tempora &longs;unt, vt radices &longs;patiorum, hæc elementa &longs;altem <lb/>Arithmetices de&longs;iderant. </s> </p> <p id="N179D4" type="main"> <s id="N179D6">Sed iam re&longs;tat, vt &longs;oluamus objectiones aliquas, quæ contra motus ac­<lb/>celerationem pugnare videntur. </s> </p> <p id="N179DB" type="main"> <s id="N179DD">Prima objectio e&longs;t; </s> <s id="N179E0"><!-- NEW -->&longs;i motus acceleratio fieret in in&longs;tantibus, &longs;ecundo <lb/>in&longs;tanti idem corpus e&longs;&longs;et in duobus locis adæquatis quod &longs;ic o&longs;tendo: </s> <s id="N179E6"><!-- NEW --><lb/> &longs;it &longs;patium AB quod percurrit corpus graue primo in&longs;tanti; </s> <s id="N179EB"><!-- NEW -->haud du­<lb/>biè AB, e&longs;t eius locus adæquatus; </s> <s id="N179F1"><!-- NEW -->&longs;ecundo in&longs;tanti percurrit BC duplum <lb/>AB; </s> <s id="N179F7"><!-- NEW -->igitur eodem in&longs;tanti re&longs;pondet loco BD, & DC, quorum vterque <lb/>e&longs;t æqualis AB; igitur &longs;ecundo in&longs;tanti e&longs;t in duobus locis, &longs;cilicet BD <lb/>& DC, quod dici non pote&longs;t. </s> </p> <p id="N179FF" type="main"> <s id="N17A01"><!-- NEW -->Hæc objectio impugnat omnem velocitatem; </s> <s id="N17A05"><!-- NEW -->hoc e&longs;t, non modò eam, <lb/>quæ motui naturaliter accelerato competit; </s> <s id="N17A0B"><!-- NEW -->verùm etiam illam, quæ <lb/>ine&longs;t motui violento; igitur vt re&longs;pondeam faciliùs; </s> <s id="N17A11"><!-- NEW -->&longs;uppono punctum <lb/>phy&longs;icum, mobile &longs;cilicet A; </s> <s id="N17A17"><!-- NEW -->aut &longs;i mauis Angelum coëxten&longs;um quadra­<lb/>to A; </s> <s id="N17A1D"><!-- NEW -->qui &longs;cilicet moueatur motu accelerato, & primo in&longs;tanti acquirat <lb/>locum immediatum æqualem priori, &longs;cilicet AB; </s> <s id="N17A23"><!-- NEW -->licèt enim po&longs;&longs;et ac­<lb/>quirere vibrationem participantem de priori; </s> <s id="N17A29"><!-- NEW -->quia tamen acquireret <lb/>tandem non participantem, id e&longs;t, quæ tota &longs;it extra illam, cui e&longs;t imme­<lb/>diata, qualis e&longs;t AB. &longs;uppono hîc acquiri vibrationem non participan­<lb/>tem de priori, id e&longs;t &longs;patium AB, æquale priori, in quo erat A, & pror­<lb/>&longs;us extra illud po&longs;itum licèt immediatum; </s> <s id="N17A35"><!-- NEW -->hoc po&longs;ito, primo in&longs;tanti pun­<lb/>ctum A acquirit AB tanquam locum adæquatum, vt certum e&longs;t: </s> <s id="N17A3B"><!-- NEW -->certum <lb/>e&longs;t etiam loca BC, CD, e&longs;&longs;e adæquata: </s> <s id="N17A41"><!-- NEW -->igitur &longs;imul, id e&longs;t eodem in­<lb/>&longs;tanti in vtroque e&longs;&longs;e non pote&longs;t; </s> <s id="N17A47"><!-- NEW -->nam in&longs;tans &longs;imul totum e&longs;t; </s> <s id="N17A4B"><!-- NEW -->igitur <lb/>&longs;ecundo in&longs;tanti non percurrit BC, &longs;ed &longs;ecundo tempore æquali primo; </s> <s id="N17A51"><!-- NEW --><lb/>hoc enim &longs;ecundum tempus con&longs;tat duobus in&longs;tantibus, quod &longs;imul <lb/>vtrumque re&longs;pondet primo: </s> <s id="N17A58"><!-- NEW -->quippe dari po&longs;&longs;unt in&longs;tantia phy&longs;ica; </s> <s id="N17A5C"><!-- NEW -->igitur <lb/>primum in&longs;tans quo percurritur AB e&longs;t æquale duobus aliis, quibus <lb/>percurruntur BD, & CD; vnde quando dixi primo in&longs;tanti acquiri &longs;pa­<lb/>tium duplum primi, idem e&longs;t, ac &longs;i dixi&longs;&longs;em &longs;ecundo tempore æquali pri­<lb/>mo, quod reuerà tempus con&longs;tat 2. in&longs;tantibus, quorum alterum re&longs;pon­<lb/>det &longs;patio BC, & alterum &longs;patio DC. <!-- KEEP S--></s> </p> <p id="N17A6B" type="main"> <s id="N17A6D">Secunda objectio; </s> <s id="N17A70"><!-- NEW -->Sed inquiet aliquis, igitur non e&longs;t continua acce­<lb/>leratio motus; nam in&longs;tans quo percurritur &longs;ecundum &longs;patium BD, cùm <lb/>&longs;it æquale in&longs;tanti quo percurritur tertium &longs;patium DC, in vtroque &longs;pa­<lb/>tio e&longs;t æquabilis motus. </s> <s id="N17A7A"><!-- NEW -->Re&longs;pondeo in&longs;tans quo percurritur &longs;ecundum <lb/>&longs;patium BD, e&longs;&longs;e maius in&longs;tanti, quo percurritur tertium &longs;patium DC; </s> <s id="N17A80"><!-- NEW --><lb/>tà tamen lege, vt vtrumque &longs;imul &longs;umptum &longs;it omninò equale in&longs;tanti, <pb pagenum="110" xlink:href="026/01/142.jpg"/>quo percurritur primum &longs;patíum AB; </s> <s id="N17A8A"><!-- NEW -->&longs;imiliter totum &longs;patium CG ita <lb/>percurritur tertio tempore, vt &longs;ingula &longs;patia CE. EI. FG. &longs;ingulis in­<lb/>&longs;tantibus percurrantur; </s> <s id="N17A92"><!-- NEW -->&longs;ed hæc tria in&longs;tantia &longs;imul &longs;umpta &longs;unt æqualia <lb/>primo in&longs;tanti, quo percurritur &longs;patium; licèt primum quo percurritur <lb/>CE &longs;it maius &longs;ecundo, quo percurritur EF, & hoc maius tertio, quo per­<lb/>curritur FG, atque ita deinceps. </s> </p> <p id="N17A9C" type="main"> <s id="N17A9E">Ob&longs;eruabis po&longs;&longs;e velocitatem motus explicari duobus modis. </s> <s id="N17AA1">Primò, <lb/>&longs;i a&longs;&longs;umantur tempora æqualia, & &longs;patia inæqualia in ea progre&longs;&longs;ione, <lb/>quam hactenus explicuimus. </s> <s id="N17AA8">Secundò &longs;i accipiantur &longs;patia æqualia & <lb/>tempora inæqualia, quod duobus modis fieri tantùm pote&longs;t. </s> <s id="N17AAD">Primò &longs;i ac­<lb/>cipiantur &longs;patia æqualia primo &longs;patio, quod percurritur primo in&longs;tanti. </s> <s id="N17AB2"><!-- NEW --><lb/>Secundò &longs;i accipiantur &longs;patia æqualia alteri &longs;patio, quod in parte tempo­<lb/>ris &longs;en&longs;ibili percurritur; </s> <s id="N17AB9"><!-- NEW -->in qua verò proportione tempora fiant &longs;emper <lb/>minora, 'dicemus infrà; </s> <s id="N17ABF"><!-- NEW -->nec dicas durum e&longs;&longs;e dicere in&longs;tans e&longs;&longs;e po&longs;&longs;e <lb/>minus in&longs;tanti; </s> <s id="N17AC5"><!-- NEW -->nam equidem fateor in&longs;tanti mathematico nihil e&longs;&longs;e <lb/>po&longs;&longs;e minus; </s> <s id="N17ACB"><!-- NEW -->&longs;ecus verò in&longs;tanti phy&longs;ico, quod e&longs;t diui&longs;ibile potentiâ, vt <lb/>dicemus aliàs; nomine in&longs;tantis phy&longs;ici intelligo durationem indiui&longs;i­<lb/>bilem, hoc e&longs;t, cuius entitas tota &longs;imul e&longs;t. </s> </p> <p id="N17AD3" type="main"> <s id="N17AD5">Tertia objectio. </s> <s id="N17AD8"><!-- NEW -->Sed inquies, igitur &longs;ecundo tempore æquali primo <lb/>acquiruntur 2.gradus velocitatis, vel impetus; </s> <s id="N17ADE"><!-- NEW -->igitur tria &longs;patia &longs;ecun­<lb/>do tempore percurruntur, quod e&longs;t contra hypothe&longs;im; </s> <s id="N17AE4"><!-- NEW -->quippe duo gra­<lb/>dus impetus accedunt primo, &longs;imiliter tertio tempore producentur tres <lb/>gradus impetus; </s> <s id="N17AEC"><!-- NEW -->qui &longs;i iungantur tribus præcedentibus, erunt 6. Igitur <lb/>percurrentur tertio tempore 6. &longs;patia, & quarto 10.quinto 15. quia &longs;in­<lb/>gulis in&longs;tantibus debet produci impetus; e&longs;t enim cau&longs;a nece&longs;&longs;aria ap­<lb/>plicata. </s> </p> <p id="N17AF6" type="main"> <s id="N17AF8"><!-- NEW -->Re&longs;pond&etail;o, equidem eo in&longs;tanti, quo percurritur &longs;patium BD, pro­<lb/>duci aliquid impetus, & aliquid eo in&longs;tanti, quo percurritur &longs;patium <lb/>DC; </s> <s id="N17B00"><!-- NEW -->ita vt tamen totus ille impetus, qui producitur his duobus in&longs;tan­<lb/>tibus, &longs;it æqualis illi, qui producitur primo in&longs;tanti, quo &longs;cilicet percurri­<lb/>tur &longs;patium AB; </s> <s id="N17B08"><!-- NEW -->quia duo illa in&longs;tantia &longs;imul &longs;umpta faciunt tempus <lb/>æquale primo in&longs;tanti; </s> <s id="N17B0E"><!-- NEW -->atqui temporibus æqualibus eadem cau&longs;a nece&longs;­<lb/>&longs;aria non impedita æqualem effectum producit per Ax.3.hinc vides &longs;in­<lb/>gulis in&longs;tantibus eadem proportione decre&longs;cere impetum in perfectio­<lb/>ne, qua tempus e&longs;t breuius, &longs;eu velocior motus; &longs;ed de hoc infrà. </s> </p> <p id="N17B18" type="main"> <s id="N17B1A">Quarta objectio; </s> <s id="N17B1D"><!-- NEW -->&longs;i impetus &longs;ingulis in&longs;tantibus cre&longs;ceret, vel intende­<lb/>retur, augeretur grauitatio: </s> <s id="N17B23"><!-- NEW -->quippe &longs;i grauitas primo in&longs;tanti producat <lb/>vnum gradum impetus; </s> <s id="N17B29"><!-- NEW -->&longs;ecundo æqualem producet, & tertio, atque ita <lb/>deinceps, quod e&longs;&longs;et ab&longs;urdum; alioqui minima atomus quodlibet cor­<lb/>pus graue adæquaret, quod e&longs;t ab&longs;urdum. </s> </p> <p id="N17B31" type="main"> <s id="N17B33"><!-- NEW -->Re&longs;pondeo nunquam impetum intendi, ni&longs;i &longs;it motus, qui e&longs;t illius fi­<lb/>nis; </s> <s id="N17B39"><!-- NEW -->alioquin fru&longs;tra e&longs;&longs;et per plura in&longs;tantia; </s> <s id="N17B3D"><!-- NEW -->igitur de&longs;trui deberet; </s> <s id="N17B41"><!-- NEW -->nec <lb/>dicas impetum naturalem etiam fru&longs;trà e&longs;&longs;e &longs;ine motu; </s> <s id="N17B47"><!-- NEW -->quia cum mo­<lb/>tus non &longs;it eius finis adæquatus; </s> <s id="N17B4D"><!-- NEW -->non mirum e&longs;t &longs;i po&longs;&longs;it e&longs;&longs;e &longs;ine motu; </s> <s id="N17B51"><!-- NEW --><lb/>atqui iam diximus &longs;uprà habere duos fines, quorum alterum &longs;emper ha-<pb pagenum="111" xlink:href="026/01/143.jpg"/>bet; </s> <s id="N17B5B"><!-- NEW -->primus e&longs;t grauitatio, &longs;eu ni&longs;us ver&longs;us centrum; &longs;ecundus motus <lb/>deor&longs;um; </s> <s id="N17B61"><!-- NEW -->cùm tamen impetus additivius motum tantùm pro fine habeat; <lb/>igitur &longs;i impeditur totus motus, non producitur hic impetus. </s> </p> <p id="N17B67" type="main"> <s id="N17B69">Quinta objectio; </s> <s id="N17B6C"><!-- NEW -->&longs;i impetum &longs;uum intendit corpus graue; </s> <s id="N17B70"><!-- NEW -->&longs;imiliter <lb/>Ignis diceretur intendere calorem; Sol lucem, &c. </s> <s id="N17B76"><!-- NEW -->Re&longs;pondeo primò de <lb/>luce &longs;ingularem e&longs;&longs;e rationem; </s> <s id="N17B7C"><!-- NEW -->quia &longs;cilicet con&longs;eruatur à cau&longs;a &longs;ua pri­<lb/>mo productiua; quidquid &longs;it; </s> <s id="N17B82"><!-- NEW -->&longs;i viderem effectum caloris, vel frigoris <lb/>perpetuò cre&longs;cere; </s> <s id="N17B88"><!-- NEW -->haud dubiè dicerem etiam cau&longs;as ip&longs;as intendi; </s> <s id="N17B8C"><!-- NEW -->atqui <lb/>hoc ip&longs;um video in motu naturali, qui effectus impetus e&longs;t; </s> <s id="N17B92"><!-- NEW -->adde quod <lb/>argumentum à pari debile e&longs;t; </s> <s id="N17B98"><!-- NEW -->cum enim &longs;int diuer&longs;i naturæ fines, diuer­<lb/>&longs;æ &longs;unt viæ quibus &longs;uos fines con&longs;equítur; </s> <s id="N17B9E"><!-- NEW -->denique contrarietas caloris, <lb/>& frigoris impedit fortè, ne vlterius vtraque qualitas intendatur, de qua <lb/>fusè &longs;uo loco; </s> <s id="N17BA6"><!-- NEW -->porrò dicemus Tomo &longs;exto calorem con&longs;eruari à cau&longs;a &longs;ua <lb/>primo productiua; quo po&longs;ito ce&longs;&longs;at difficultas; quod licèt alicui durum <lb/>videri po&longs;&longs;it, demon&longs;trabo tamen. </s> </p> <p id="N17BAE" type="main"> <s id="N17BB0">Sexta objectio; igitur &longs;i ex infinita di&longs;tantia lapis de&longs;cenderet, inten­<lb/>deret etiam &longs;uum motum. </s> <s id="N17BB5">Re&longs;pondeo primò, non po&longs;&longs;e dari infinitam il­<lb/>lam di&longs;tantiam. </s> <s id="N17BBA"><!-- NEW -->Secundò etiam&longs;i daretur lapis, ex ea non caderet; </s> <s id="N17BBE"><!-- NEW -->fru&longs;trà <lb/>enim e&longs;&longs;et ille motus: </s> <s id="N17BC4"><!-- NEW -->Tertiò, &longs;i daretur motus infinitus, haud dubiè e&longs;&longs;et <lb/>æquabilis; </s> <s id="N17BCA"><!-- NEW -->qualis e&longs;t motus circularis corporum cœle&longs;tium; </s> <s id="N17BCE"><!-- NEW -->at verò <lb/>motus naturalis deor&longs;um corporum grauium debet e&longs;&longs;e acceleratus ne <lb/>vel de&longs;cenderent tardiùs, &longs;i cum primo tantùm velocitatis gradu de&longs;cen­<lb/>derent; </s> <s id="N17BD8"><!-- NEW -->vel &longs;u&longs;tineri vix po&longs;&longs;ent, &longs;i impetum innatum intentiorem habe­<lb/>rent; vtrum verò &longs;emper intendatur, & ex quacumque altitudine cadat <lb/>corpus graue, videbimus infrà. </s> </p> <p id="N17BE0" type="main"> <s id="N17BE2">Ex dictis hactenus facilè refelluntur aliæ &longs;ententiæ de proportione <lb/>motus naturaliter accelerati. </s> </p> <p id="N17BE7" type="main"> <s id="N17BE9"><!-- NEW -->Et primò quidem illa, quæ vult fieri &longs;ecundum rationem &longs;inuum <lb/>ver&longs;orum, licèt initio tàm propè accedat ad proportionem Galilei, vt <lb/>di&longs;cerni &longs;en&longs;ibiliter ab ea non po&longs;&longs;it; </s> <s id="N17BF1"><!-- NEW -->quare tutò &longs;atis a&longs;&longs;umi po­<lb/>terit, &longs;i quando &longs;it opus illius loco, quod nos in explicandis motibus cœ­<lb/>le&longs;tibus præ&longs;tabimus; </s> <s id="N17BF9"><!-- NEW -->interim quia faciliùs explicatur in motu recto per <lb/>rationem quadratorum quàm &longs;inuum, illam retinebimus; </s> <s id="N17BFF"><!-- NEW -->præ&longs;ertim cùm <lb/>vtraque ad no&longs;tram reducatur; modò progre&longs;&longs;io fiat in in&longs;tantibus. </s> <s id="N17C05"><!-- NEW --><lb/>Secundò reiicitur &longs;ententia illorum qui volunt hanc progre&longs;&longs;ionem fie­<lb/>ri iuxta proportionem geometricam, quam vides in his numeris 1.2.4.8. <lb/>16. quæ licèt initio minùs recedat à vera, in progre&longs;&longs;u tamen multùm <lb/>aberrat, nec e&longs;t vlla ratio quæ pro illa faciat: </s> <s id="N17C10"><!-- NEW -->Et verò nulla in mentem <lb/>venire pote&longs;t; ni&longs;i fortè dicatur, cùm &longs;ecundo in&longs;tanti &longs;it dupla velocitas, <lb/>tertio <expan abbr="pon&etilde;dam">ponendam</expan> e&longs;&longs;e quadruplam, & 4°ree;. </s> <s id="N17C1C">octuplam; </s> <s id="N17C1F"><!-- NEW -->quia vt velocitas pri­<lb/>mi in&longs;tantis e&longs;t ad velocitatem &longs;ecundi, ita velocitas huius ad velocita­<lb/>tem tertij, & velocitas huius ad velocitatem quarti; </s> <s id="N17C27"><!-- NEW -->igitur &longs;equitur pro­<lb/>gre&longs;&longs;ionem rationis geometricæ duplæ; cur enim e&longs;&longs;et maior ratio pri­<lb/>mi in&longs;tantis ad &longs;ecundum quàm &longs;ecundi ad tertium tertij ad quartum? <lb/></s> <s id="N17C30">&c. </s> <s id="N17C33"><!-- NEW -->&longs;ed profectò vix vlla apparet rationis &longs;pecies, cùm nulla &longs;it cau&longs;a, <pb pagenum="112" xlink:href="026/01/144.jpg"/>quæ 3°ree; in&longs;tanti, & 4°ree; plùs agat <expan abbr="quã">quam</expan> primo, & &longs;ecundo; </s> <s id="N17C40"><!-- NEW -->igitur e&longs;t peculiaris <lb/>cau&longs;a huius inæqualitatis rationum; </s> <s id="N17C46"><!-- NEW -->quòd &longs;cilicet æqualibus temporibus <lb/>æqualia acquirantur velocitatis momenta; vt &longs;uprà demon&longs;trauimus; </s> <s id="N17C4C"><!-- NEW --><lb/>quippe id præ&longs;tari debet in explicandis inæqualitatibus motuum recto­<lb/>rum naturalium, quod præ&longs;tant A&longs;tronomi in explicanda inæqualitate <lb/>motuum cæle&longs;tium; qui &longs;emper æqualitatem aliquam &longs;upponunt, nec e&longs;t <lb/>quòd hanc &longs;ententiam nonnullis experimentis ictuum qui&longs;quam con­<lb/>firmet, in quibus multa fraus &longs;ube&longs;&longs;e pote&longs;t. </s> </p> <p id="N17C59" type="main"> <s id="N17C5B">Tertiò reiicitur illa quoque &longs;ententia, quæ proportionem lineæ &longs;ectæ <lb/>in mediam, & extremam rationem huic lineæ tribuit, quam ferè in his <lb/>numeris vides 1.2.3.5.8, 13. 21. 34. 55. quæ &longs;ub finem etiam longi&longs;&longs;imè <lb/>aberrat, vt videre e&longs;t, quare ii&longs;dem rationibus impugnatur, quibus iam <lb/>aliam impugnauimus. </s> </p> <p id="N17C66" type="main"> <s id="N17C68"><!-- NEW -->Scio e&longs;&longs;e alias multas rationes, quibus aliqui recentiores motus natu­<lb/>ralis accelerationem explicare nituntur, &longs;ed iam &longs;uprà &longs;atis &longs;uperque re­<lb/>iectæ fuerunt, vel profectò eæ &longs;unt, quæ ne quidem inter fabulo&longs;a poë­<lb/>tarum commenta locum aliquem habere po&longs;&longs;int: </s> <s id="N17C72"><!-- NEW -->Et verò ni&longs;i me ani­<lb/>mus fallit in re clari&longs;&longs;ima, rationem huius effectus ex communibus <lb/>principiis deductam cum ip&longs;is etiam experimentis con&longs;entire hactenus <lb/>ita demon&longs;trauimus, vt iam vix vllus dubitationi locus relinquatur; &longs;ed <lb/>interruptam Theorematum &longs;eriem tandem repetimus. </s> </p> <p id="N17C7E" type="main"> <s id="N17C80"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N17C8C" type="main"> <s id="N17C8E"><!-- NEW --><emph type="italics"/>Si accipiantur &longs;patia æqualia primo &longs;patio, quod vno in&longs;tanti percurritur, <lb/>in&longs;tantia &longs;unt inæqualia in motu natur aliter accelerato<emph.end type="italics"/>; </s> <s id="N17C99"><!-- NEW -->probatur, quia &longs;e­<lb/>cundum &longs;patium æquale primo percurritur motu velociore, quàm pri­<lb/>mo, & tertium quam &longs;ecundo: </s> <s id="N17CA1"><!-- NEW -->ergo minori tempore per Def.2.l.1. &longs;ed <lb/>primum &longs;patium conficitur vno in&longs;tanti; </s> <s id="N17CA7"><!-- NEW -->igitur &longs;ecundum vno in&longs;tanti, <lb/>&longs;ed minore; idem dico de tertio. </s> </p> <p id="N17CAD" type="main"> <s id="N17CAF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s> </p> <p id="N17CBB" type="main"> <s id="N17CBD"><!-- NEW --><emph type="italics"/>In ea proportione decre&longs;cunt hæc instantia,<emph.end type="italics"/> vt primum &longs;it maius &longs;ecundo, <lb/>&longs;ecundum tertio, tertium quarto, quartum quinto, quintum &longs;exto, <lb/>atque ita deinceps; ita vt &longs;ecundum & tertium &longs;imul &longs;umpta, item quar­<lb/>tum, quintum, &longs;extum, &longs;eptimum, item octauum, nonum, decimum, &longs;imul <lb/>&longs;umpta adæquent primum, hoc e&longs;t vt vnum, duo, tria, quatuor, quinque, <lb/>&longs;ex, &c. </s> <s id="N17CD0">faciant &longs;emper tempora æqualia, quia temporibus æqualibus æ­<lb/>qualia acquiruntur velocitatis momenta? </s> <s id="N17CD5"><!-- NEW -->igitur &longs;i primo in&longs;tanti per­<lb/>curritur vnum &longs;patium; </s> <s id="N17CDB"><!-- NEW -->&longs;ecundo tempore æquali percurruntur duo &longs;pa­<lb/>tia æqualia primo, & tertio, tria; atque deinceps; </s> <s id="N17CE1"><!-- NEW -->&longs;ed vt &longs;uprà dictum e&longs;t <lb/>in re&longs;pon&longs;. ad obiect. primam, vno, & <expan abbr="eod&etilde;">eodem</expan> in&longs;tanti non pote&longs;t idem cor­<lb/>pus percurrere duo &longs;patia, ne &longs;imul e&longs;&longs;et in duobus locis; </s> <s id="N17CED"><!-- NEW -->igitur &longs;ingula <lb/>&longs;patia re&longs;pondent &longs;ingulis in&longs;tantibus licèt minoribus; </s> <s id="N17CF3"><!-- NEW -->&longs;ed &longs;ecundo tem­<lb/>pore æquali primo in&longs;tanti percurruntur duo &longs;patia æqualia primo &longs;pa­<lb/>tio; </s> <s id="N17CFB"><!-- NEW -->igitur &longs;ecundum, & tertium in&longs;tans debent &longs;imul &longs;umpta adæquare <pb pagenum="113" xlink:href="026/01/145.jpg"/>primum, &longs;ed non &longs;unt æqualia, vt con&longs;tat; </s> <s id="N17D04"><!-- NEW -->alioquin duobus illis in&longs;tanti<lb/>bus motus e&longs;&longs;et æquabilis; igitur &longs;ecundum e&longs;t maius tertio, ita vt tamen <lb/>ex vtroque tempus fiat æquale primo in&longs;tanti. </s> </p> <p id="N17D0C" type="main"> <s id="N17D0E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s> </p> <p id="N17D1A" type="main"> <s id="N17D1C"><!-- NEW --><emph type="italics"/>Non decre&longs;cunt illa in&longs;tantia &longs;ecundum lineam &longs;extam in extremam & <lb/>mediam rationem propagatam; </s> <s id="N17D24"><!-- NEW -->ita vt primum &longs;it ad &longs;ecundum, vt &longs;ecundum <lb/>ad tertium, tertium ad quartum, quartum ad quintum at que ita deinceps<emph.end type="italics"/>; </s> <s id="N17D2D"><!-- NEW --><lb/>&longs;it enim aliqua &longs;eries numerorum, qui aliquo modo accedant ad prædi­<lb/>ctam proportionem 1.2.3.5.8.13.21.34.55. &longs;itque primum in&longs;tans vlti­<lb/>mus numerus 55. &longs;ecundum 34.tertium 21. atque ita deinceps: </s> <s id="N17D36"><!-- NEW -->Equidem <lb/>&longs;ecundum, & tertium adæquant primum; </s> <s id="N17D3C"><!-- NEW -->at verò quartum, quintum, <lb/>&longs;extum nullo modo adæquant; </s> <s id="N17D42"><!-- NEW -->immò ne quidem eius &longs;ubduplum, & <lb/>multò minus 3. alij addito primo: </s> <s id="N17D48"><!-- NEW -->immò &longs;i linea data duodecies propor­<lb/>tionaliter diuidatur, vltimum &longs;egmentum vix e&longs;&longs;et &longs;ubcentuplum primi, <lb/>vt con&longs;tat; igitur reiici debet hæc propo&longs;itio. </s> </p> <p id="N17D50" type="main"> <s id="N17D52"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s> </p> <p id="N17D5E" type="main"> <s id="N17D60"><!-- NEW --><emph type="italics"/>In&longs;tans primum non e&longs;t ad &longs;ecundum vt numerus ad numerum; </s> <s id="N17D66"><!-- NEW -->nec ad <lb/>tertium, quartum, quintum, &longs;extum, &c.<emph.end type="italics"/> probatur, quia nullus numerus <lb/>excogitari pote&longs;t quo de&longs;ignari po&longs;&longs;it quantitas, &longs;eu perfectio, &longs;eu va­<lb/>lor i&longs;torum in&longs;tantium; </s> <s id="N17D73"><!-- NEW -->&longs;it enim primum in&longs;tans &longs;ecundum &longs;it 3/5. tertium <lb/>2/5 quartum 4/9 quintum 2/9 &longs;extum 2/9. <!--neuer Satz-->Equidem &longs;ecundum, & tertium ad&etail;­<lb/>quant primum; </s> <s id="N17D7D"><!-- NEW -->adde quod non pote&longs;t amplius &longs;eries propagari per nu­<lb/>meros rationales; </s> <s id="N17D83"><!-- NEW -->&longs;it autem &longs;ecundum (6/11) 3. (5/11) cum tribus aliis 4/9 1/9 7/9; </s> <s id="N17D87"><!-- NEW --><lb/>equidem &longs;i reducantur hæ 5. minutiæ, re&longs;pondebunt his (54/99) (45/99) (44/99) (12/99) (26/99): </s> <s id="N17D8C"><!-- NEW --><lb/>igitur &longs;ecunda erit maior quarta; </s> <s id="N17D91"><!-- NEW -->at prima &longs;uperat &longs;ecundam (9/999) &longs;ecunda <lb/>tertiam (1/99) tertia quartam (11/99) quarta quintam (12/99). Cur porrò hæc inæqua­<lb/>litas, igitur numeri po&longs;&longs;unt a&longs;&longs;ignari; non po&longs;&longs;unt etiam poni in &longs;erie <lb/>geometrica &longs;ubdupla 1. 1/2 1/4 1/8 &c. </s> <s id="N17D9B">quia &longs;ecunda. </s> <s id="N17D9E"><!-- NEW -->& tertia non adæquant <lb/>primam idem dicendum e&longs;t potiori iure de tribus aliis; </s> <s id="N17DA4"><!-- NEW -->nec etiam in &longs;e­<lb/>rie arithmetica &longs;implici 1. 1/2 1/3 1/4 2/5 1/6; quia &longs;ecunda, & tertia &longs;unt mi­<lb/>nores prima 1/6, vt quarta, quinta, &longs;exta &longs;unt minores prima (26/74). </s> </p> <p id="N17DAC" type="main"> <s id="N17DAE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s> </p> <p id="N17DBA" type="main"> <s id="N17DBC"><!-- NEW --><emph type="italics"/>Datur aliquis &longs;eries numerorum irrationabilium, &longs;eu &longs;urdorum minorum, & <lb/>minorum<emph.end type="italics"/>; quorum primus ita &longs;uperet &longs;ecundum, &longs;ecundus tertium, <lb/>tertius quartum, &c. </s> <s id="N17DC9">vt &longs;ecundus, & tertius adæquent primum, item <lb/>quartus, quintus, &longs;extus. </s> <s id="N17DCE">item 4. alij, qui &longs;equuntur, item 5. item 6. &c. </s> <s id="N17DD1"><!-- NEW --><lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->pote&longs;t dari linea AG con&longs;tans tribus partibus æqualibus, &longs;cilicet <lb/>AB, BC, CG, & &longs;ecunda BC duabus BD maiore, & DC minore, & ter­<lb/>tia tribus prima CE minore ED, &longs;ed maiore EF, &longs;ecunda EF maiore F <lb/>G, atque ita deinceps; </s> <s id="N17DE0"><!-- NEW -->addi pote&longs;t quartum &longs;egmentum æquale AB; </s> <s id="N17DE4"><!-- NEW -->quod <lb/>&longs;ubdiuidetur in 4. partes, quarum prima &longs;it maior &longs;ecunda, & <expan abbr="h&etail;c">haec</expan> tertia <lb/>& hæc quarta, & omnes minores FG; </s> <s id="N17DF0"><!-- NEW -->ita autem &longs;uperant primæ &longs;equen­<lb/>tes, vt differentia primæ, & &longs;ecundæ &longs;it maior differentia &longs;ecundæ, & <pb pagenum="114" xlink:href="026/01/146.jpg"/>tertiæ, & hæc maior differentia tertiæ, & quartæ; atque ita deinceps, nec <lb/>aliter res e&longs;&longs;e pote&longs;t. </s> </p> <p id="N17DFD" type="main"> <s id="N17DFF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s> </p> <p id="N17E0B" type="main"> <s id="N17E0D"><!-- NEW --><emph type="italics"/>Hinc partes, quo fiunt minores, accedunt propiùs ad æqualitatem,<emph.end type="italics"/> v.g. <!-- REMOVE S-->BD, <lb/>& DC accedunt propiùs ad æqualitatem quàm AB, BD, & DC, CE, pro­<lb/>piùs quàm CD, DB, & CE, EF, quàm EC, CD, atque ita deinceps, vt patet; <lb/>hinc po&longs;t aliquot in&longs;tantia motus, æqualia ferè redduntur in&longs;tantia, vt <lb/>con&longs;tat. </s> </p> <p id="N17E20" type="main"> <s id="N17E22"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s> </p> <p id="N17E2E" type="main"> <s id="N17E30"><!-- NEW --><emph type="italics"/>Hinc qua proportione decre&longs;cunt instantia, decre&longs;cit etiam perfectio <lb/>impetus<emph.end type="italics"/>; </s> <s id="N17E3B"><!-- NEW -->quia temporibus æqualibus eadem cau&longs;a nece&longs;&longs;aria æqualem ef­<lb/>fectum producit per Ax. tertium igitur inæqualem inæqualibus, per Ax. <!-- REMOVE S--><lb/>13. num.4. igitur minorem minore tempore; </s> <s id="N17E44"><!-- NEW -->igitur minorem in eadem <lb/>proportione, in qua tempus e&longs;t; igitur qua proportione, &c. </s> </p> <p id="N17E4A" type="main"> <s id="N17E4C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s> </p> <p id="N17E58" type="main"> <s id="N17E5A"><!-- NEW --><emph type="italics"/>Hinc vides quâm &longs;it nece&longs;&longs;aria illa diuer&longs;a perfectio impetus, quam indi­<lb/>cauimus lib.<emph.end type="italics"/>1. hinc impetus productus &longs;ecundo, & tertio in&longs;tanti adæ­<lb/>quat impetum productum primo, quem etiam adæquat productus quar­<lb/>to, quinto, &longs;exto, item productus &longs;eptimo, octauo, nono; decimo, atque ita <lb/>deinceps; </s> <s id="N17E6B"><!-- NEW -->hinc e&longs;t eadem differentia impetuum, quæ <expan abbr="in&longs;tãtium">in&longs;tantium</expan>; </s> <s id="N17E73"><!-- NEW -->hinc &longs;in­<lb/>gulis &longs;patiis æqualibus primo &longs;patio, quod percurritur primo in&longs;tanti; </s> <s id="N17E79"><!-- NEW --><lb/>re&longs;pondent &longs;ingula in&longs;tantia, & &longs;ingulis in&longs;tantibus &longs;inguli, & &longs;ingulares <lb/>impetus; </s> <s id="N17E80"><!-- NEW -->hinc non e&longs;t quod primo in&longs;tanti dicantur produci plura pun­<lb/>cta impetus in eodem puncto corporis grauis; </s> <s id="N17E86"><!-- NEW -->&longs;ed vnicum tantùm pun­<lb/>ctum talis perfectionis &longs;cilicet phy&longs;icum; cur enim potius duo puncta, <lb/>quam tria? </s> <s id="N17E8E">&longs;ed quod vnum e&longs;t determinatum e&longs;t per Ax. 5. lib. 1. hinc <lb/>optima ratio cur potius tali in&longs;tanti producatur impetus talis perfectio­<lb/>nis quàm alterius? </s> <s id="N17E95"><!-- NEW -->quippe perfectio impetus &longs;equitur perfectionem in­<lb/>&longs;tantis quo producitur; </s> <s id="N17E9B"><!-- NEW -->hinc dicendum videtur omnia puncta impetus <lb/>e&longs;&longs;e diuer&longs;æ perfectionis, vel heterogenea; vt vulgò aiunt Philo&longs;ophi; </s> <s id="N17EA1"><!-- NEW --><lb/>cuius rationem demon&longs;tratiuam afferemus lib. &longs;equenti cum de motu <lb/>violento; </s> <s id="N17EA8"><!-- NEW -->hinc vides duplicem progre&longs;&longs;ionem; </s> <s id="N17EAC"><!-- NEW -->primam &longs;cilicet, qua ex <lb/>&longs;uppo&longs;itis temporibus æqualibus acquiruntur &longs;patia inæqualia, de qua <lb/>fusè &longs;uprà; </s> <s id="N17EB4"><!-- NEW -->in hac enim velocitas eadem proportione cum impetu cre&longs;­<lb/>cit, & cum ip&longs;o tempore; </s> <s id="N17EBA"><!-- NEW -->hoc e&longs;t tempore triplo e&longs;t tripla, quadruplo <lb/>quadrupla; </s> <s id="N17EC0"><!-- NEW -->item impetus in duplo tempore e&longs;t duplus, in triplo triplus; </s> <s id="N17EC4"><!-- NEW --><lb/>modò progre&longs;&longs;io fiat in temporibus primo in&longs;tanti æqualibus; </s> <s id="N17EC9"><!-- NEW -->&longs;ecunda <lb/>progre&longs;&longs;io e&longs;t qua ex &longs;uppo&longs;itis &longs;patiis æqualibus tempora fluunt inæ­<lb/>qualia, hoc e&longs;t minora & minora; </s> <s id="N17ED1"><!-- NEW -->quibus etiam re&longs;pondet impetus im­<lb/>perfectior in eadem proportione temporum; prima fit per differentias <lb/>æquales, & proportiones inæquales, &longs;ecunda verò per differentias inæ­<lb/>quales, & proportiones inæquales. </s> </p> <pb pagenum="115" xlink:href="026/01/147.jpg"/> <p id="N17EDF" type="main"> <s id="N17EE1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s> </p> <p id="N17EED" type="main"> <s id="N17EEF"><!-- NEW --><emph type="italics"/>Si a&longs;&longs;umantur &longs;patia &longs;en&longs;ibilia æqualia, tempora &longs;unt ferè in ratione &longs;ubdu­<lb/>plicata &longs;patiorum<emph.end type="italics"/>; </s> <s id="N17EFA"><!-- NEW -->crun enim &longs;patia &longs;int vt quadrata <expan abbr="t&etilde;porum">temporum</expan> &longs;en&longs;ibiliter; </s> <s id="N17F02"><!-- NEW --><lb/>certè tempora &longs;unt, vt radices i&longs;torum quadratorum, &longs;cilicet &longs;patiorum; </s> <s id="N17F07"><!-- NEW --><lb/>&longs;int enim quæcunque &longs;patia æqualia in linea AF; </s> <s id="N17F0C"><!-- NEW -->&longs;intque &longs;patia AC 4. <lb/>AE 16. radix quadr.4. e&longs;t 2.16. verò 4. igitur tempora &longs;unt vt 4.2.&longs;i ve­<lb/>rò accipiatur primum &longs;patium, quod vno tempore percurritur; </s> <s id="N17F14"><!-- NEW -->tempus <lb/>quo percurruntur duo &longs;patia æqualia primum e&longs;t v.2.quo percurruntur <lb/>tria v.3.quo percurruntur 4.&longs;patia, 2. atque ita deinceps; igitur in praxi <lb/>quæ tantùm fit in &longs;patiis &longs;en&longs;ibilibus hæc progre&longs;&longs;io adhibenda e&longs;t, il­<lb/>lamque deinceps, &longs;i quando opus e&longs;t, adhibebimus. </s> </p> <p id="N17F20" type="main"> <s id="N17F22"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s> </p> <p id="N17F2E" type="main"> <s id="N17F30"><!-- NEW --><emph type="italics"/>In vacuo &longs;i corpus graue de&longs;cenderet, prædictæ proportiones accurati&longs;&longs;imè <lb/>&longs;eruarentur<emph.end type="italics"/>; </s> <s id="N17F3B"><!-- NEW -->quia &longs;cilicet nullum e&longs;&longs;e impedimentum; </s> <s id="N17F3F"><!-- NEW -->at verò &longs;i aliquod <lb/>intercedit impedimentum; </s> <s id="N17F45"><!-- NEW -->haud dubiè non &longs;eruantur accuratè; e&longs;t autem <lb/>aliquod impedimentum in medio, quantumuis liberum e&longs;&longs;e videatur, <lb/>quæ omnia con&longs;tant. </s> </p> <p id="N17F4D" type="main"> <s id="N17F4F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s> </p> <p id="N17F5B" type="main"> <s id="N17F5D"><!-- NEW --><emph type="italics"/>Impetus naturalis addititius de&longs;truitur<emph.end type="italics"/>; patet experientiâ; </s> <s id="N17F66"><!-- NEW -->quippe pila <lb/>deor&longs;um cadens tandem quie&longs;cit, licèt à terra reflectatur ratione impe­<lb/>dimenti, ex quo re&longs;ultat duplex determinatio, ratione cuius idem im­<lb/>petus &longs;ibi aliquo modo redditur <expan abbr="cõtrarius">contrarius</expan>; </s> <s id="N17F74"><!-- NEW -->&longs;ed de his fusè in primo libro <lb/>à Th.148. ad finem v&longs;que libri: </s> <s id="N17F7A"><!-- NEW -->nam reuerâ duæ determinationes op­<lb/>po&longs;itæ pugnant pro rata per Ax. 15.l.1. & quotie&longs;cunque idem impetus <lb/>e&longs;t ad lineas oppo&longs;itas determinatus eodem modo &longs;e habet, ac &longs;i duplex <lb/>e&longs;&longs;et, & quilibet &longs;uæ &longs;ube&longs;&longs;et determinationi; </s> <s id="N17F84"><!-- NEW -->atqui &longs;i duplex e&longs;&longs;et oppo­<lb/>&longs;itus, pugnarent pro rata; </s> <s id="N17F8A"><!-- NEW -->igitur tàm pugnant duæ determinationes op­<lb/>po&longs;itæ in eodem impetu, quàm duo impetus ad oppo&longs;itas lineas deter­<lb/>minati; igitur impetus naturalis aduentitius de&longs;truitur, &c. </s> </p> <p id="N17F92" type="main"> <s id="N17F94"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s> </p> <p id="N17FA0" type="main"> <s id="N17FA2"><!-- NEW --><emph type="italics"/>Impetus naturalis innatus nunquam de&longs;truitur<emph.end type="italics"/>; </s> <s id="N17FAB"><!-- NEW -->Probatur, quia nihil e&longs;te <lb/>quod exigat eius de&longs;tructionem, quia &longs;cilicet nunquam e&longs;t fru&longs;trà; </s> <s id="N17FB1"><!-- NEW -->nam <lb/>vel habet motum deor&longs;um, vel grauitationis effectum, vel de&longs;truit impe­<lb/>tum extrin&longs;ecum in motu violento; igitur nunquam e&longs;t fru&longs;trà, cum &longs;em­<lb/>per habeat aliquem effectum. </s> </p> <p id="N17FBB" type="main"> <s id="N17FBD"><!-- NEW -->Dices lignum vi extrin&longs;eca in aqua immer&longs;um &longs;ua &longs;ponte a&longs;cendit; </s> <s id="N17FC1"><!-- NEW --><lb/>igitur ille gradus impetus grauitationis de&longs;truitur, & alius producitur; <lb/>hæc quæ&longs;tio ad præ&longs;ens in&longs;titutum non pertinet, &longs;ed ad librum de gra­<lb/>uitate, & leuitate. </s> <s id="N17FCA">Igitur breuiter re&longs;pondeo illum impetum nunquam <lb/>de&longs;trui, quandiu mobile grauitat, vel grauitatione &longs;ingulari, (&longs;ic corpus <lb/>grauitat in manum &longs;u&longs;tinentis,) vel grauitatione communi, (&longs;ic lignum <lb/>humori innatans grauitat, non quidem in aquam, at &longs;imul cum aqua;) <lb/>&longs;ed de grauitate, & grauitatione in Tomo de &longs;tatibus corporum &longs;en&longs;ibi-<pb pagenum="116" xlink:href="026/01/148.jpg"/>libus, in quo o&longs;tendemus ideo lignum &longs;ur&longs;um emergere, quia ab aqua <lb/>extenditur, & ideo corpora &longs;ur&longs;um ire, quia alia deor&longs;um eunt. </s> </p> <p id="N17FDC" type="main"> <s id="N17FDE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s> </p> <p id="N17FEA" type="main"> <s id="N17FEC"><!-- NEW --><emph type="italics"/>Quando lapis de&longs;cendit per medium aëra, impeditur aliquantulum eius <lb/>motus<emph.end type="italics"/>: </s> <s id="N17FF7"><!-- NEW -->Probatur primò experientiâ, quæ certa e&longs;t; </s> <s id="N17FFB"><!-- NEW -->tàm enim aër impe­<lb/>dit motum deor&longs;um, quàm &longs;ur&longs;um, vt videre e&longs;t in mobili leuiore &longs;eu ra­<lb/>riore, quod etiam flante vento ob&longs;eruare omnes po&longs;&longs;unt; </s> <s id="N18003"><!-- NEW -->quomodo ve­<lb/>rò impediat, dicemus aliàs; </s> <s id="N18009"><!-- NEW -->&longs;ecundò corpus immobile, in quod mobile <lb/>impingitur, motum illius impedit; </s> <s id="N1800F"><!-- NEW -->&longs;ed in diuer&longs;as partes aëris corpus <lb/>graue impingitur in de&longs;cen&longs;u; igitur aliquantulum impeditur eius <lb/>motus. </s> </p> <p id="N18017" type="main"> <s id="N18019"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s> </p> <p id="N18025" type="main"> <s id="N18027"><!-- NEW --><emph type="italics"/>Hinc motus naturalis deor&longs;um aliquantulum retardatur,<emph.end type="italics"/> quia nihil aliud <lb/>præ&longs;tare pote&longs;t huiu&longs;modi impedimentum, ni&longs;i aliquam retardationem; <lb/>igitur motus inde redditur tardior. </s> </p> <p id="N18034" type="main"> <s id="N18036"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s> </p> <p id="N18042" type="main"> <s id="N18044"><!-- NEW --><emph type="italics"/>Hinc etiam impetus producitur imperfectior<emph.end type="italics"/>; quia ex imperfectione ef­<lb/>fectus requiritur imperfectio cau&longs;æ per Ax. 13.l. </s> <s id="N1804F">1. & quâ proportione <lb/>e&longs;t tardior motus eâdem impetus e&longs;t imperfectior, per Ax. <!-- REMOVE S-->5. excipe ta­<lb/>men impetum innatum, qui &longs;emper habet eundem effectum grauitatio­<lb/>nis, vel &longs;ingularis, quâ grauitas cum ip&longs;o medio, &longs;i reuerâ medium gra­<lb/>uitat, de quo aliàs. </s> </p> <p id="N1805C" type="main"> <s id="N1805E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s> </p> <p id="N1806A" type="main"> <s id="N1806C"><!-- NEW --><emph type="italics"/>Quo medium den&longs;ius e&longs;t plus impedit motum deor&longs;um<emph.end type="italics"/>; </s> <s id="N18075"><!-- NEW -->Probatur, quia &longs;i <lb/>motum impedit; certè non totum; quis enim hoc dicat; </s> <s id="N1807B"><!-- NEW -->&longs;ed eæ dumta­<lb/>xat partes, quibus incubat corpus graue; </s> <s id="N18081"><!-- NEW -->igitur quò &longs;unt plures huiu&longs;­<lb/>modi partes, maius e&longs;t impedimentum; </s> <s id="N18087"><!-- NEW -->&longs;ed in medio den&longs;iori plures &longs;unt <lb/>cum minore exten&longs;ione; </s> <s id="N1808D"><!-- NEW -->hoc enim e&longs;t, quod voco den&longs;ius; igitur me­<lb/>dium den&longs;ius plùs impedit. </s> </p> <p id="N18093" type="main"> <s id="N18095"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s> </p> <p id="N180A1" type="main"> <s id="N180A3"><emph type="italics"/>Hinc tardiùs de&longs;cendit mobile per mediam aquam, quàm per medium <lb/>aëra,<emph.end type="italics"/> quia aqua e&longs;t den&longs;ior aëre. </s> </p> <p id="N180AD" type="main"> <s id="N180AF"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N180BB" type="main"> <s id="N180BD"><!-- NEW -->Ob&longs;erua e&longs;&longs;e aliqua corpora minus den&longs;a, quæ motum omninò im­<lb/>pediunt; </s> <s id="N180C3"><!-- NEW -->quippe certum e&longs;t aquam e&longs;&longs;e den&longs;iorem ligno; </s> <s id="N180C7"><!-- NEW -->atqui li­<lb/>gnum de&longs;cen&longs;um lapidis impedit, non verò aqua; </s> <s id="N180CD"><!-- NEW -->quia &longs;cilicet lignum <lb/>non e&longs;t medium, vt aqua; </s> <s id="N180D3"><!-- NEW -->vt enim aliquod corpus &longs;it medium, debet e&longs;&longs;e <lb/>liquidum, vt, aqua & alij liquores; vel &longs;pirabile vt aër, vapor, &c. </s> <s id="N180D9"><!-- NEW -->ratio <lb/>e&longs;t, quia partes ligni, vel alterius corporis durioris, ita &longs;unt inter &longs;e con­<lb/>junctæ, vel implicatæ, vt omnem tran&longs;itum intercludant, ni&longs;i corpus ip­<lb/>&longs;um graue valido ictu vel impetu &longs;ibi viam aperiat; </s> <s id="N180E3"><!-- NEW -->igitur vt corpus ali­<lb/>quod vice medij defungatur, debet in eo &longs;tatu e&longs;&longs;e, in quo eius partes <pb pagenum="117" xlink:href="026/01/149.jpg"/>modico ferè ni&longs;u &longs;eiungantur, & loco cedant; </s> <s id="N180EE"><!-- NEW -->&longs;ed de his &longs;tatibus cor­<lb/>porum fusè agemus Tomo 5. adde quod ad medium &longs;ufficit vacuum &longs;i <lb/>motus in vacuo e&longs;&longs;e pote&longs;t, de quo alibi; quod certè e&longs;t omnium me­<lb/>diorum optimum, cum nullo modo re&longs;i&longs;tar mobili. </s> </p> <p id="N180F8" type="main"> <s id="N180FA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s> </p> <p id="N18106" type="main"> <s id="N18108"><emph type="italics"/>Hinc producitur impetus imperfectior in medio den&longs;iore:<emph.end type="italics"/> quia in eo tar­<lb/>dior e&longs;t motus, ex cuius tarditate arguitur imperfectio impetus per Ax. <!-- REMOVE S--><lb/>13.num.4. </s> </p> <p id="N18115" type="main"> <s id="N18117"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N18123" type="main"> <s id="N18125"><!-- NEW -->Ob&longs;erua den&longs;itatem medij cogno&longs;ci ex eius grauitate; </s> <s id="N18129"><!-- NEW -->illud enim <lb/>den&longs;ius e&longs;t, quod e&longs;t grauius & vici&longs;&longs;im; </s> <s id="N1812F"><!-- NEW -->quod fusè explicabimus &longs;uo lo­<lb/>co; </s> <s id="N18135"><!-- NEW -->e&longs;t enim grauitas quædam <emph type="italics"/>den&longs;itas, vt ait<emph.end type="italics"/> Philo&longs;ophus <emph type="italics"/>tùm l.<emph.end type="italics"/>4.<emph type="italics"/>pb.c.<emph.end type="italics"/><lb/>9.<emph type="italics"/>t.<emph.end type="italics"/>85. & 86. <emph type="italics"/>den&longs;um & rarum,<emph.end type="italics"/> inquit, <emph type="italics"/>&longs;unt lationis efficientia,<emph.end type="italics"/> & paulò &longs;u­<lb/>periùs; </s> <s id="N18160"><!-- NEW --><emph type="italics"/>e&longs;t autem den&longs;um graue, rarum verò leue, & l.<emph.end type="italics"/>8.<emph type="italics"/>c.<emph.end type="italics"/>7.<emph type="italics"/>t.<emph.end type="italics"/>55. <emph type="italics"/>hæc habet, <lb/>graue & leue; molle & durum den&longs;itates quædam e&longs;&longs;e, & raritates videntur,<emph.end type="italics"/><lb/>quæ adnotare volui, vt vel inde con&longs;tet doctrinam hanc cum Peripate­<lb/>tica optimè con&longs;entire. </s> </p> <p id="N18180" type="main"> <s id="N18182"><!-- NEW -->Ob&longs;eruabis etiam hîc à me non di&longs;cuti, in quo con&longs;i&longs;tat den&longs;itas, vel <lb/>raritas, grauitas, vel leuitas; </s> <s id="N18188"><!-- NEW -->&longs;uppono tantùm graue illud e&longs;&longs;e, quod ten­<lb/>dit deor&longs;um; </s> <s id="N1818E"><!-- NEW -->leue illud, quod tendit &longs;ur&longs;um &longs;iue pellatur à grauiori, &longs;iue <lb/>non, den&longs;um verò e&longs;&longs;e id quod multùm materia habet &longs;ub parua exten­<lb/>&longs;ione, rarum è contrario; quorum omnium cau&longs;as, & rationes &longs;uo loco <lb/>explicabimus. </s> </p> <p id="N18198" type="main"> <s id="N1819A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s> </p> <p id="N181A6" type="main"> <s id="N181A8"><!-- NEW --><emph type="italics"/>Sub medium leuius corpus graue de&longs;cendit<emph.end type="italics"/>; </s> <s id="N181B1"><!-- NEW -->certa e&longs;t hypothe&longs;is, ni&longs;i for­<lb/>tè aliquando per accidens &longs;ecus accidat; </s> <s id="N181B7"><!-- NEW -->ratio porrò petitur ex ip&longs;a <lb/>grauitatis natura, quâ corpus graue tendit deor&longs;um; </s> <s id="N181BD"><!-- NEW -->nihil enim aliud <lb/>grauitas e&longs;t, quidquid tandem illa &longs;it; </s> <s id="N181C3"><!-- NEW -->quippe corpus graue de&longs;cendit, <lb/>quando medium liberum habet, idemque leuius, per quod de&longs;cendat; </s> <s id="N181C9"><!-- NEW --><lb/>quod certè &longs;i grauius e&longs;&longs;et, haud dubiè non de&longs;cenderet; </s> <s id="N181CE"><!-- NEW -->&longs;ic ferrum, & <lb/>&longs;axum plumbo liquato innatant; </s> <s id="N181D4"><!-- NEW -->cum tamen per mediam aquam de­<lb/>&longs;cendant; </s> <s id="N181DA"><!-- NEW -->fic lignum aquæ &longs;upernatat, quod per liberum aëra de&longs;cendit; </s> <s id="N181DE"><!-- NEW --><lb/>ratio e&longs;t, quia grauius de&longs;cendit &longs;ub medium leuius; </s> <s id="N181E3"><!-- NEW -->cur autem id fiat <lb/>fusè alibi explicabo; id tantùm obiter indico. </s> <s id="N181E9"><!-- NEW -->Omnis motus, qui fit à <lb/>principio intrin&longs;eco per lineam rectam propter locum e&longs;t, vt patet; quis <lb/>enim neget corpus graue ideo de&longs;cendere &longs;ub leuius, vt occupet aliquem <lb/>locum quo prius carebat, qui tamen illi connaturalis e&longs;t in hoc rerum <lb/>ordine? </s> <s id="N181F5"><!-- NEW -->cum à natura acceperit vim illam intrin&longs;ecam, quâ in eum lo­<lb/>cum &longs;e&longs;e recipere pote&longs;t; </s> <s id="N181FB"><!-- NEW -->quam certè vim intrin&longs;ecam nunquam à na­<lb/>tura rebus creatis in&longs;itam e&longs;&longs;e con&longs;tat, ni&longs;i ad eum finem con&longs;equendum, <lb/>cui à natura de&longs;tinantur; </s> <s id="N18203"><!-- NEW -->cur verò locus connaturalis corporis grauio­<lb/>ris &longs;it ille, in quo leuiori &longs;ube&longs;t, non diu hærebit animus, quin &longs;tatim ra­<lb/>tio affulgeat; </s> <s id="N1820B"><!-- NEW -->cum enim corpus, quod e&longs;t &longs;uprà, &longs;u&longs;tineatur ab eo quod e&longs;t <lb/>infrà; </s> <s id="N18211"><!-- NEW -->illud certè infra e&longs;&longs;e connaturalius e&longs;t, quod aptius e&longs;t ad &longs;u&longs;tinen-<pb pagenum="118" xlink:href="026/01/150.jpg"/>dum; </s> <s id="N1821A"><!-- NEW -->atqui den&longs;um aptius e&longs;t ad id munus, quia plures partes &longs;u&longs;tinentis <lb/>pauciores &longs;u&longs;tinent alterius leuioris, &longs;eu rarioris, vt con&longs;tat; </s> <s id="N18220"><!-- NEW -->v.g. <!-- REMOVE S-->certum <lb/>e&longs;t <expan abbr="cãdem">eandem</expan> aëris partem pluribus aquæ partibus re&longs;pondere; </s> <s id="N1822C"><!-- NEW -->&longs;ed de hoc <lb/>alias fusè; </s> <s id="N18232"><!-- NEW -->hæc interim &longs;ufficiat indica&longs;&longs;e, vt vel aliqua ratio affulgeat; </s> <s id="N18236"><!-- NEW --><lb/>cur &longs;cilicet corpus graue &longs;ub medium leuius &longs;ua &longs;ponte de&longs;cendat; </s> <s id="N1823B"><!-- NEW -->adde <lb/>quod cum omne corpus graue tendat deor&longs;um, tunc vnum infra aliud de­<lb/>&longs;cendit, cum &longs;unt plures partes pellentis, quàm pul&longs;i; denique per va­<lb/>cuum modicum &longs;ine vlla re&longs;i&longs;tentia de&longs;cenderet. </s> </p> <p id="N18245" type="main"> <s id="N18247"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s> </p> <p id="N18253" type="main"> <s id="N18255"><!-- NEW --><emph type="italics"/>Sub medium grauius corpus leuius minimè de&longs;cendit, &longs;ed huic inna­<lb/>tat<emph.end type="italics"/>; </s> <s id="N18260"><!-- NEW -->v.g. <!-- REMOVE S-->lignum aquæ, ferrum plumbo liquato; </s> <s id="N18266"><!-- NEW -->certa e&longs;t hypothe&longs;is: </s> <s id="N1826A"><!-- NEW -->ratio <lb/>e&longs;t, quia ideo de&longs;cendit graue &longs;ub medium, quia grauius &longs;eu den&longs;ius e&longs;t <lb/>medio; </s> <s id="N18272"><!-- NEW -->igitur, &longs;i den&longs;ius e&longs;t ip&longs;um medium, non de&longs;cendet; clarum e&longs;t; <lb/>cur verò a&longs;cendat &longs;upra medium. </s> <s id="N18278"><!-- NEW -->v.g. <!-- REMOVE S-->cur lignum aquæ immer&longs;um tan­<lb/>dem emergat hîc non di&longs;cutio, &longs;ed tantùm indico ab ip&longs;a aqua &longs;ur&longs;um <lb/>extendi; quanta verò parte lignum emergat, dicemus aliàs, cum de in­<lb/>natantibus humido. </s> </p> <p id="N18284" type="main"> <s id="N18286"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s> </p> <p id="N18292" type="main"> <s id="N18294"><!-- NEW --><emph type="italics"/>Sub medium æquè graue corpus non de&longs;cendit, nec etiam &longs;upra a&longs;cendit<emph.end type="italics"/>; </s> <s id="N1829D"><!-- NEW -->ra­<lb/>tio e&longs;t, quia ideo de&longs;cendit &longs;ub medium, quia medium leuius e&longs;t, ideo <lb/>a&longs;cendit &longs;upra, quia medium grauius e&longs;t; </s> <s id="N182A5"><!-- NEW -->igitur &longs;i nec &longs;it grauius nec <lb/>leuius, non e&longs;t quod a&longs;cendat vel de&longs;cendat; </s> <s id="N182AB"><!-- NEW -->nihil tamen illius &longs;upra <lb/>primam medij &longs;uperficiem extare poterit; alioqui e&longs;&longs;et leuius medio, <lb/>contra &longs;uppo&longs;itionem. </s> </p> <p id="N182B3" type="main"> <s id="N182B5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s> </p> <p id="N182C1" type="main"> <s id="N182C3"><!-- NEW --><emph type="italics"/>Aër &longs;uam grauitatem habet<emph.end type="italics"/>; </s> <s id="N182CC"><!-- NEW -->quod iam à nullo in dubium reuocari po­<lb/>te&longs;t; </s> <s id="N182D2"><!-- NEW -->nam &longs;i comprimatur intra vas æneum v.g. <!-- REMOVE S-->etiam minimæ cra&longs;&longs;itu­<lb/>dinis; </s> <s id="N182DA"><!-- NEW -->&longs;i deinde ponderetur, maius e&longs;t haud dubiè pondus, quo maior <lb/>e&longs;t aëris copia intru&longs;a; </s> <s id="N182E0"><!-- NEW -->atqui non modo triplum totius aëris, qui ante <lb/>compre&longs;&longs;ionem totam va&longs;is capacitatem occupabat intrudi pote&longs;t, vel <lb/>decuplum; </s> <s id="N182E8"><!-- NEW -->verùm etiam vigecuplum; </s> <s id="N182EC"><!-- NEW -->immò centuplum, & millecuplum <lb/>adhibita cochleâ, vel alio mechanico organo, & aucta va&longs;is cra&longs;&longs;itudine, <lb/>de quo aliàs: </s> <s id="N182F4"><!-- NEW -->quanta verò &longs;it grauitas aëris comparata cum grauitate <lb/>aquæ, cen&longs;et Galileus e&longs;&longs;e ferè vt 1. ad 400. Mer&longs;ennus verò vt 1. ad <lb/>1356. vel &longs;altem vt 1.ad 1300. Nos maiorem illà; </s> <s id="N182FC"><!-- NEW -->hâc vero minorem <lb/>e&longs;&longs;e ob&longs;eruauimus, de quo aliàs; </s> <s id="N18302"><!-- NEW -->nec enim e&longs;t præ&longs;entis in&longs;tituti, pro <lb/>quo &longs;ufficiat modò, aëri aliquam ine&longs;&longs;e grauitatem; </s> <s id="N18308"><!-- NEW -->nec dicas aëra le­<lb/>uem e&longs;&longs;e; </s> <s id="N1830E"><!-- NEW -->nam reuerâ leuis e&longs;t, &longs;i comparetur cum aqua; </s> <s id="N18312"><!-- NEW -->grauis autem &longs;i <lb/>comparetur cum a&longs;cendente halitu, vel fortè cum vacuo; </s> <s id="N18318"><!-- NEW -->nec e&longs;t quod <lb/>aliquis fortè metuat, ne &longs;i aër &longs;it grauis, ab eo tandem opprimatur, nam <lb/>etiam&longs;i aqua &longs;it grauis non tamen opprimit vrinatores, cuius rei veri&longs;&longs;i­<lb/>mam rationem &longs;uo loco afferemus; </s> <s id="N18322"><!-- NEW -->denique non e&longs;t quod aliqui &longs;atis <lb/>incautè re&longs;pondeant, ip&longs;um aëra non e&longs;&longs;e grauem, &longs;ed tantùm &longs;entiri ali­<lb/>quod pondus cra&longs;&longs;ioris vaporis immixti; </s> <s id="N1832A"><!-- NEW -->nam de alio aëre non affirmo <pb pagenum="119" xlink:href="026/01/151.jpg"/>grauem e&longs;&longs;e, ni&longs;i tantùm de illo, quem &longs;piramus, in quo ambulamus, qui <lb/>nos ambit: </s> <s id="N18335"><!-- NEW -->adde quod Ari&longs;toteles l.4. <emph type="italics"/>de Cœlo, c.<emph.end type="italics"/>5.<emph type="italics"/>t.<emph.end type="italics"/>36. tribuit aëri gra­<lb/>uitatem his verbis; <emph type="italics"/>quapropter<emph.end type="italics"/> inquit, <emph type="italics"/>aër, & aqua habent & leuitatem, & <lb/>grauitatem.<emph.end type="italics"/></s> </p> <p id="N18354" type="main"> <s id="N18356"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s> </p> <p id="N18362" type="main"> <s id="N18364"><!-- NEW --><emph type="italics"/>Medium eiu&longs;dem grauitatis cum dato corpore graui detrahit totam eius <lb/>grauitationem &longs;ingularem; </s> <s id="N1836C"><!-- NEW -->hoc e&longs;t corpus graue in medium æquè graue non <lb/>grauitat<emph.end type="italics"/>; </s> <s id="N18375"><!-- NEW -->quia &longs;i grauitaret de&longs;cenderet; </s> <s id="N18379"><!-- NEW -->&longs;ic pars aquæ in aliam partem <lb/>aquæ non grauitat, & &longs;i aqua ponderetur in aqua, nullius ponderis e&longs;t; </s> <s id="N1837F"><!-- NEW --><lb/>cum enim nulla &longs;it ratio cur vna &longs;it infrà potiùs, quàm alia, vna certè al­<lb/>terius locum non ambit; igitur caret grauitatione &longs;ingulari. </s> </p> <p id="N18386" type="main"> <s id="N18388"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s> </p> <p id="N18394" type="main"> <s id="N18396"><!-- NEW --><emph type="italics"/>Medium graue detrahit aliquid de &longs;ingulari grauitatione corporis grauio­<lb/>ris<emph.end type="italics"/>; </s> <s id="N183A1"><!-- NEW -->certa e&longs;t hypothe&longs;is; </s> <s id="N183A5"><!-- NEW -->nec enim plumbum e&longs;t eius ponderis &longs;ingula­<lb/>ris in aqua, cuius e&longs;t in aëre; dixi &longs;ingularis; </s> <s id="N183AB"><!-- NEW -->nam &longs;i plumbum & ip&longs;a <lb/>aqua &longs;imul appendantur, haud dubiè totum habebis pondus plumbi, & <lb/>totum pondus aquæ; </s> <s id="N183B3"><!-- NEW -->ratio verò huius effectus non e&longs;t huius loci; </s> <s id="N183B7"><!-- NEW -->quid­<lb/>quid &longs;it, &longs;i æqualis grauitas medij tollit totam æqualem alterius corpo­<lb/>ris; certè maiorem alterius corporis totam non tollit per Th. 80. &longs;ed <lb/>tantùm aliquid illius, quod quomodo fiat, dicemus Tomo quinto cum de <lb/>graui, & leui. </s> </p> <p id="N183C3" type="main"> <s id="N183C5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s> </p> <p id="N183D1" type="main"> <s id="N183D3"><!-- NEW --><emph type="italics"/>Medium graue detrahit eam partem grauitationis corporis grauioris, quæ <lb/>e&longs;t æqualis &longs;uæ grauitationi.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i medij grauitas e&longs;t &longs;ubdupla, detrahit <lb/>&longs;ubduplum grauitationis; </s> <s id="N183E4"><!-- NEW -->&longs;i &longs;ubdecupla, &longs;ubdecuplum, atque ita dein­<lb/>ceps; hoc iam olim &longs;uppo&longs;uit magnus Archim. <!-- KEEP S--></s> <s id="N183EB"><!-- NEW -->&longs;upponunt etiam reliqui <lb/>omnes, præ&longs;ertim recentior Galileus; </s> <s id="N183F1"><!-- NEW -->&longs;i enim æqualis &longs;uperat æqualem, <lb/>ergo inæqualis pro rata; &longs;cilicet &longs;ubdupla &longs;ubduplum &longs;ubtripla, &c. </s> <s id="N183F7"><!-- NEW -->Præ­<lb/>terea, cum detrahat aliquam partem grauitationis maioris per Th.85.nec <lb/>detrahat inæqualem maiorem, per Th.80.nec inæqualem minorem; cur <lb/>enim potius vnam minorem quam aliam? </s> <s id="N18401">certè æqualem tantùm <lb/>detrahere pote&longs;t, quod &longs;uo loco per Principium po&longs;itiuum demon&longs;tra­<lb/>bimus. </s> </p> <p id="N18408" type="main"> <s id="N1840A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s> </p> <p id="N18416" type="main"> <s id="N18418"><!-- NEW --><emph type="italics"/>Hinc ratio cur grauia de&longs;cendant tardius in aqua, quàm in aëre, & in <lb/>aëre, quàm in vacuo<emph.end type="italics"/>; </s> <s id="N18423"><!-- NEW -->hinc etiam maioris &longs;unt ponderis in aëre quam in <lb/>aqua; </s> <s id="N18429"><!-- NEW -->hinc &longs;i grauitas alicuius corporis &longs;it ad grauitatem aëris vt 100. <lb/>ad 1. haud dubiè decre&longs;cet eius pondus in aëre (1/100); </s> <s id="N1842F"><!-- NEW -->id e&longs;t, &longs;i penderet 100. <lb/>libras in vacuo, in aëre penderet 99. & eo tempore quo in vacuo decur­<lb/>reret 100. pa&longs;&longs;us, in aëre decurreret 99. &longs;i nulla &longs;it aliunde re&longs;i&longs;tentia, <lb/>qualis reuerâ e&longs;t, vt dicam infrà; </s> <s id="N18439"><!-- NEW -->&longs;imiliter &longs;i grauitas alicuius corporis <lb/>&longs;it ad grauitatem aquæ, vt 10. ad 1. decre&longs;cet eius pondus in aqua (1/10), & <lb/>eo tempore quo decurreret in vacuo 10. palmos &longs;patij, in aqua decurre <pb pagenum="120" xlink:href="026/01/152.jpg"/>ret tantùm 9. po&longs;ito quod non &longs;it aliud quod re&longs;i&longs;tat; </s> <s id="N18448"><!-- NEW -->quanta verò &longs;it <lb/>grauitas omnium corporum tùm duriorum, qualia &longs;unt metalla, tùm li­<lb/>quidorum, tùm &longs;pirabilium, dicemus &longs;uo loco; illorum tabulas habes <lb/>apud Gethaldum, & Mer&longs;ennum. </s> </p> <p id="N18452" type="main"> <s id="N18454"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s> </p> <p id="N18460" type="main"> <s id="N18462"><!-- NEW --><emph type="italics"/>Hinc, &longs;i nihil aliud de&longs;cen&longs;um corporum grauium impediret, cognito pen­<lb/>dere vtriu&longs;que, medij & corporis grauis, &longs;patio, quod in vno illorum conficit, <lb/>cogno&longs;ci po&longs;&longs;et &longs;patium, quod in alio conficeret æquali tempore<emph.end type="italics"/>, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;uppona­<lb/>mus grauitatem aquæ e&longs;&longs;e ad grauitatem aëris vt 400. ad 1. &longs;itque corpus, <lb/>cuius grauitas &longs;it dupla grauitatis aquæ; </s> <s id="N18477"><!-- NEW -->haud dubiè eo tempore, quo <lb/>conficit in aëre 799. &longs;patia, in aqua conf;iciet tantùm 400. quia in vacuo <lb/>conficeret 800. aër autem detrahit (1/800), & aqua 1/2, vt con&longs;tat ex dictis; </s> <s id="N1847F"><!-- NEW -->&longs;i­<lb/>militer cognitis &longs;patiis in vtroque medio confectis, & grauitate vtriu&longs;que <lb/>medij cogno&longs;ceretur grauitas corporis de&longs;cendentis; quia tamen e&longs;t alia <lb/>re&longs;i&longs;tentiæ ratio, hîc non hæreo. </s> </p> <p id="N18489" type="main"> <s id="N1848B"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N18497" type="main"> <s id="N18499"><!-- NEW -->Ob&longs;eruabis dictum e&longs;&longs;e hactenus; </s> <s id="N1849D"><!-- NEW -->&longs;i nihil aliud de&longs;cen&longs;um corporis <lb/>grauis impedit; </s> <s id="N184A3"><!-- NEW -->nam certè aliud e&longs;t, de quo infrà, ex cuius ignoratione <lb/>plures haud dubiè in inue&longs;tigandis grauitatum medij rationibus hallu­<lb/>cinarentur; </s> <s id="N184AB"><!-- NEW -->cum enim ob&longs;eruatum &longs;it globum plumbeum, cuius graui­<lb/>tas e&longs;t ferè dodecupla grauitatis aquæ, conficere in libero aëre 48. pedes <lb/>&longs;patij tempore duorum &longs;ecundorum, in aqua verò 12. pedes eodem tem­<lb/>pore; </s> <s id="N184B5"><!-- NEW -->certè in vacuo ip&longs;o moueretur tardiùs quàm in aëre; </s> <s id="N184B9"><!-- NEW -->quia eo tem­<lb/>pore, quo conficit in aqua 12.pedes in vacuo conficeret (13 1/21), &longs;i tantùm <lb/>detrahitur (1/12) grauitationis, & de&longs;cen&longs;us; </s> <s id="N184C1"><!-- NEW -->atqui in aëre eodem tempore <lb/>conficit 48. pedes; </s> <s id="N184C7"><!-- NEW -->igitur velociùs moueretur in aëre quàm in vacuo; </s> <s id="N184CB"><!-- NEW --><lb/>igitur e&longs;t aliquid aliud quod impedit motum; </s> <s id="N184D0"><!-- NEW -->vt enim optimè monet <lb/>Mer&longs;ennus, &longs;i grauitas aquæ &longs;it ad grauitatem aëris vt 400 ad 1.& graui­<lb/>tas plumbi ad grauitatem aquæ vt 12. ad 1.eadem grauitas plumbi e&longs;t ad <lb/>grauitatem aëris vt 4800. igitur &longs;i &longs;patium, quod decurrit plumbum in <lb/>vacuo diuidatur in 4800. partes, decurret in aëre 4799. partes; </s> <s id="N184DC"><!-- NEW -->in aqua <lb/>verò 4400. quod e&longs;t contra experientiam; </s> <s id="N184E2"><!-- NEW -->nam &longs;patium, quod decurrit <lb/>in aëre e&longs;t maius &longs;patio, quod decurrit in aqua 3/4; </s> <s id="N184E8"><!-- NEW -->quippe conficit 12. <lb/>pedes in aqua eodem tempore, quo in aëre conficit 48; </s> <s id="N184EE"><!-- NEW -->igitur in aqua <lb/>amittit 3/4 &longs;uæ grauitationis, & &longs;ui motus; igitur 3600. partes; </s> <s id="N184F4"><!-- NEW -->igitur <lb/>plumbi grauitas e&longs;&longs;et ad grauitatem aquæ vt 4.ad 3.& ad grauitatem aë­<lb/>ris vt 3600. ad 1. atqui vtrumque fal&longs;um e&longs;&longs;e con&longs;tat; </s> <s id="N184FC"><!-- NEW -->igitur e&longs;t aliquid <lb/>aliud, quod etiam impedit motum; nec ex motu diuer&longs;o per diuer&longs;a me­<lb/>dia cogno&longs;ci pote&longs;t eorum grauitas. </s> </p> <p id="N18504" type="main"> <s id="N18506"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s> </p> <p id="N18512" type="main"> <s id="N18514"><!-- NEW --><emph type="italics"/>Hinc potiori iure reiicies illorum &longs;ententiam, qui volunt impediri motum <lb/>corporis de&longs;cendentis per diuer&longs;a media pro diuer&longs;a ratione grauitatum vtriu&longs;­<lb/>que medy<emph.end type="italics"/>; quod certè fal&longs;um e&longs;t; </s> <s id="N18521"><!-- NEW -->nam aqua &longs;it ad grauitatem aëris vt <lb/>400. ad 1. deberet omne corpus de&longs;cendere velociùs in aëre quadrin-<pb pagenum="121" xlink:href="026/01/153.jpg"/>gente&longs;ies, quàm in aqua, quod fal&longs;um e&longs;t; cum aliquod corpus nullo mo­<lb/>do de&longs;cendat in aqua, quod de&longs;cendit in aëre, vt lignum. </s> </p> <p id="N1852E" type="main"> <s id="N18530"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s> </p> <p id="N1853C" type="main"> <s id="N1853E"><!-- NEW --><emph type="italics"/>Non pote&longs;t corpus graue per medium corporeum de&longs;cendere, ni&longs;i vel totum <lb/>medium loco cedat, vel aliquæ partes eiu&longs;dem medij,<emph.end type="italics"/> patet; quia vnum cor­<lb/>pus non pote&longs;t penetrari cum alio. </s> </p> <p id="N1854B" type="main"> <s id="N1854D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 91.<emph.end type="center"/></s> </p> <p id="N18559" type="main"> <s id="N1855B"><!-- NEW --><emph type="italics"/>Totum medium loco non cedit in de&longs;cen&longs;u grauium<emph.end type="italics"/>; </s> <s id="N18564"><!-- NEW -->patet etiam, tùm <lb/>quia ad mouendum totum medium exigua vis corporis grauis non &longs;uffi­<lb/>cit; </s> <s id="N1856C"><!-- NEW -->tùm quia tàm facilè per medium durum eiu&longs;dem grauitatis de&longs;cen­<lb/>deret; denique patet manife&longs;tâ experientiâ. </s> </p> <p id="N18572" type="main"> <s id="N18574"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s> </p> <p id="N18580" type="main"> <s id="N18582"><!-- NEW --><emph type="italics"/>Hinc aliqua tantùm partes medij loco cedunt<emph.end type="italics"/>; probatur, quia vel totum <lb/>medium, vel aliquæ eius partes, per Th.90.non primum per Th.91. igitur <lb/>&longs;ecundum, in his certè non e&longs;t vlla difficultas. </s> </p> <p id="N1858F" type="main"> <s id="N18591"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s> </p> <p id="N1859D" type="main"> <s id="N1859F"><!-- NEW --><emph type="italics"/>Non po&longs;&longs;unt illæ partes loco cedere &longs;ine motu; </s> <s id="N185A5"><!-- NEW -->nec moueri &longs;ine impetu, nec <lb/>habere impetum, ni&longs;i producatur in illis à cau&longs;a aliqua applicata; quæ certè <lb/>alia none&longs;t quàm impetus corporis de&longs;cendentis,<emph.end type="italics"/> vt con&longs;tat ex iis, quæ dixi­<lb/>mus primo lib. <!-- REMOVE S--><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s> </p> <p id="N185BD" type="main"> <s id="N185BF"><!-- NEW --><emph type="italics"/>Illæ partes, quæ loco cedunt de&longs;cendenti corpori graui, nece&longs;&longs;ariò ab aliis <lb/>&longs;eparantur, & &longs;uo appul&longs;u, vel impul&longs;u alias multas impellunt, ac &longs;eparant,<emph.end type="italics"/><lb/>atqui &longs;eparari non po&longs;&longs;unt ab aliis, ni&longs;i &longs;oluatur vnio, &longs;eu nexus, <lb/>quo cum aliis deuinciuntur; quidquid tandem &longs;it illa vnio, de qua <lb/>aliàs. </s> </p> <p id="N185CF" type="main"> <s id="N185D1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s> </p> <p id="N185DD" type="main"> <s id="N185DF"><!-- NEW --><emph type="italics"/>Hinc quò arctior e&longs;t ille nexus, difficilius &longs;oluitur<emph.end type="italics"/>; igitur maiore vi, vel <lb/>impetu opus e&longs;t, vt &longs;olui po&longs;&longs;it, vt con&longs;tat. </s> </p> <p id="N185EA" type="main"> <s id="N185EC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s> </p> <p id="N185F8" type="main"> <s id="N185FA"><!-- NEW --><emph type="italics"/>Hinc corpus grauius &longs;ustinetur à leuiore.<emph.end type="italics"/> v.g. <!-- REMOVE S-->plumbum à ligno propter <lb/>arctiorem nexum partium ligni, qui ab impetu plumbi quantumuis gra­<lb/>ui&longs;&longs;imi &longs;uperari non pote&longs;t; </s> <s id="N18609"><!-- NEW -->hinc corpus illud, medium tantùm appello <lb/>in quo po&longs;&longs;int corpora moueri, cuius nexus &longs;uperari pote&longs;t à corpore <lb/>grauiori in aliqua &longs;altem figura, vel &longs;itu; </s> <s id="N18611"><!-- NEW -->hinc corpora dura non po&longs;&longs;unt <lb/>e&longs;&longs;e medium; </s> <s id="N18617"><!-- NEW -->immò neque mollia, vt cera, argilla; &longs;ed vel liquida, vel <lb/>&longs;pirabilia. </s> </p> <p id="N1861D" type="main"> <s id="N1861F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s> </p> <p id="N1862B" type="main"> <s id="N1862D"><!-- NEW --><emph type="italics"/>Hinc ducitur euidens ratio, cur medium impediat motum &longs;i dumtaxat ha­<lb/>beat arctiorum partium implicationem & nexum<emph.end type="italics"/>; </s> <s id="N18638"><!-- NEW -->quia non modo partes <pb pagenum="122" xlink:href="026/01/154.jpg"/>medij amouendæ &longs;unt è &longs;uo loco; </s> <s id="N18641"><!-- NEW -->verùm etiam nexus ille partium &longs;ol­<lb/>uendus; igitur ex vtroque capite impeditur motus. </s> </p> <p id="N18647" type="main"> <s id="N18649"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s> </p> <p id="N18655" type="main"> <s id="N18657"><!-- NEW --><emph type="italics"/>Quo &longs;ubtiliores &longs;unt partes difficilius inter &longs;e implicari po&longs;&longs;unt &longs;eu ligari <lb/>quibu&longs;dam filamentis<emph.end type="italics"/>, con&longs;tat; </s> <s id="N18662"><!-- NEW -->igitur cum aëris partes &longs;int magis lubricæ, <lb/>quàm partes aquæ, & faciliùs per obuia quæque foramina irrepere po&longs;­<lb/>&longs;int, non po&longs;&longs;unt ita contineri; </s> <s id="N1866A"><!-- NEW -->&longs;ic videmus multùm aquæ hauriri, dum <lb/>arctioribus retibus attollitur; </s> <s id="N18670"><!-- NEW -->immò dum aquam manu &longs;tringimus, ali­<lb/>quam re&longs;i&longs;tentiam &longs;en&longs;u percipimus; quæ certè nulla e&longs;t, dum aëra &longs;trin­<lb/>gimus. </s> </p> <p id="N18678" type="main"> <s id="N1867A"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N18686" type="main"> <s id="N18688">Ob&longs;eruabis vnionem continuatiuam corporum aliquando po&longs;itam <lb/>e&longs;&longs;e in plexu, vel implicatione partium, vt videmus in fune, ligno, carne, <lb/>o&longs;&longs;ibus, &c. </s> <s id="N1868F">aliquando in vacui metu; </s> <s id="N18692"><!-- NEW -->&longs;ic aqua, vt &longs;uo va&longs;i adhæreat, <lb/>a&longs;cendit, vel &longs;ur&longs;um attollitur, ne detur vacuum; </s> <s id="N18698"><!-- NEW -->aliquando in coitione <lb/>quadam magnetica; </s> <s id="N1869E"><!-- NEW -->porrò hic plexus con&longs;tat ex infinitis ferè tenui&longs;&longs;i­<lb/>morum filamentorum voluminibus, vel aduncis &longs;iue hamatis partibus, <lb/>&longs;eu corpu&longs;culis: </s> <s id="N186A6"><!-- NEW -->Vtrum verò præter hæc requiratur alius vnionis mo­<lb/>dus, di&longs;cutiemus fusè Tomo 5. quidquid &longs;it; certum e&longs;t medium illud, <lb/>cuius partes arctiori maiorique nexu copulantur, longè difficiliùs per­<lb/>curri po&longs;&longs;e, &longs;eu perrumpi. </s> </p> <p id="N186B0" type="main"> <s id="N186B2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 99.<emph.end type="center"/></s> </p> <p id="N186BE" type="main"> <s id="N186C0"><!-- NEW --><emph type="italics"/>Hinc non modò aqua detrahit plumbo<emph.end type="italics"/> (1/22) <emph type="italics"/>&longs;ui motus, quod &longs;cilicet plumbi gra­<lb/>uitas &longs;it dedecupla grauitatis aquæ, verùm etiam propter re&longs;istentiam petitam <lb/>ex alio capite aliquid adhuc detrahere pote&longs;t<emph.end type="italics"/>; </s> <s id="N186D3"><!-- NEW -->&longs;cilicet quia partes aquæ non <lb/>po&longs;&longs;unt amoueri, ni&longs;i ab aliis &longs;eparentur; </s> <s id="N186D9"><!-- NEW -->atqui maiore vi opus e&longs;t ad­<lb/>&longs;oluendum &longs;trictiorem nexum; </s> <s id="N186DF"><!-- NEW -->immò licèt partes aquæ nullo penitus <lb/>nexu vniantur, &longs;ed tantùm vel vacui metu, vel alio modo, quod alibi ex­<lb/>plicabimus; </s> <s id="N186E7"><!-- NEW -->omninò detraherent adhuc plumbo (1/12) motus; </s> <s id="N186EB"><!-- NEW -->igitur, &longs;i <lb/>præter illud impedimentum, quod petitur à comparatione grauitatis <lb/>corporis mobilis cum grauitate medij, addatur aliud longè robu&longs;tius; <lb/>non mirum e&longs;t, &longs;i maior inde &longs;equatur effectus, id e&longs;t maior imminutio <lb/>motus, qui qua&longs;i frangitur ab impedimento. </s> </p> <p id="N186F7" type="main"> <s id="N186F9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s> </p> <p id="N18705" type="main"> <s id="N18707"><!-- NEW --><emph type="italics"/>Hinc petitur ratio illius experimenti, &longs;i verum e&longs;t, duobus &longs;ecundis per­<lb/>currere plumbeam pilam in aëre<emph.end type="italics"/> 48. <emph type="italics"/>&longs;patij pedes, in aqua verò<emph.end type="italics"/> 12. <emph type="italics"/>pedes<emph.end type="italics"/>; </s> <s id="N1871E"><!-- NEW -->hinc <lb/>tenui nexu partes aëris copulantur; </s> <s id="N18724"><!-- NEW -->partes verò aquæ firmiori; </s> <s id="N18728"><!-- NEW -->hinc aër <lb/>minùs re&longs;i&longs;tit etiam motibus violentis; </s> <s id="N1872E"><!-- NEW -->hinc vix pote&longs;t qui&longs;piam in aqua <lb/>currere propter maiorem aquæ re&longs;i&longs;tentiam; </s> <s id="N18734"><!-- NEW -->hinc pote&longs;t dici quota parte <lb/>firmior &longs;it nexus vnius corporis quàm alterius; </s> <s id="N1873A"><!-- NEW -->hinc non tantùm copu­<lb/>lantur partes metu vacui; </s> <s id="N18740"><!-- NEW -->alioquin æquè re&longs;i&longs;terent partes aëris, ac par­<lb/>tes aquæ ratione nexus; </s> <s id="N18746"><!-- NEW -->hinc videntur guttulæ illæ &longs;phericæ inuolui te­<lb/>nui qua&longs;i membranula, &longs;eu &longs;uperficie, cuius analogiam videmus in aqua <pb pagenum="123" xlink:href="026/01/155.jpg"/>feruente; </s> <s id="N18752"><!-- NEW -->in bullis, quæ ex guttis pluuiæ re&longs;ilientibus na&longs;ci videntur; </s> <s id="N18756"><!-- NEW -->in <lb/>bullis etiam illis &longs;aponariis, quas leui calamo pueri inter ludendum in­<lb/>flant; </s> <s id="N1875E"><!-- NEW -->hinc ex minimo ferè contactu guttula &longs;pargitur, ni&longs;i fortè cum <lb/>multo a&longs;per&longs;a puluere cru&longs;tam quamdam induit &longs;olidiorem; </s> <s id="N18764"><!-- NEW -->&longs;ic bullæ il­<lb/>læ ad minimum etiam contactum di&longs;&longs;ipantur; </s> <s id="N1876A"><!-- NEW -->hinc ip&longs;a &longs;uperficies <lb/>aquæ plus videtur re&longs;i&longs;tere quod multis experimentis comprobatur; </s> <s id="N18770"><!-- NEW -->&longs;ed <lb/>illo maximè, quo videmus findi à remo cum quodam qua&longs;i &longs;tridulo cre­<lb/>pitu re&longs;i&longs;tentiæ maioris te&longs;te; </s> <s id="N18778"><!-- NEW -->immò cum ab ip&longs;a naui qua&longs;i &longs;ulcatur, <lb/>idem &longs;tridor auditur, maximè in iis tractibus; </s> <s id="N1877E"><!-- NEW -->in quibus nullis fluctibus <lb/>agitata læuigati&longs;&longs;imam faciem præfert; </s> <s id="N18784"><!-- NEW -->habes analogiam in illa cru&longs;ta, <lb/>quæ concre&longs;cit in &longs;uperficie liquorum, &longs;ed præ&longs;ertim o&longs;&longs;arum: </s> <s id="N1878A"><!-- NEW -->adde quod <lb/>aër paulò compre&longs;&longs;ior vndique guttulam premens æquali ni&longs;u eam miri­<lb/>ficè tornat: </s> <s id="N18792"><!-- NEW -->hæc tantùm tumultuatim conge&longs;ta alibi fusè pertractabi­<lb/>mus, & ex &longs;implici&longs;&longs;imis principiis demon&longs;trabimus; plura hîc de graui­<lb/>tate crant dicenda, & de grauitatione, quæ tantùm indica&longs;&longs;e &longs;ufficiat, vt <lb/>deinde Tomo quinto fusè explicentur. </s> </p> <p id="N1879C" type="main"> <s id="N1879E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 101.<emph.end type="center"/></s> </p> <p id="N187AA" type="main"> <s id="N187AC"><!-- NEW --><emph type="italics"/>Non re&longs;istit medium propter compre&longs;&longs;ionem partium inferiorum, quas nullo <lb/>modo comprimi nece&longs;&longs;e e&longs;t, vel in&longs;en&longs;ibiliter<emph.end type="italics"/>; </s> <s id="N187B7"><!-- NEW -->cum enim tantus relinquatur <lb/>locus retrò, quantus acquiritur antè, nulla opus e&longs;t compre&longs;&longs;ione; </s> <s id="N187BD"><!-- NEW -->&longs;ed <lb/>partes à fronte pul&longs;æ factâ circuitione retror&longs;um eunt, non certè tramite <lb/>recto; </s> <s id="N187C5"><!-- NEW -->&longs;i enim frons ip&longs;ius lata &longs;it, haud dubiè partes pul&longs;æ alias pellunt, <lb/>& hæ vici&longs;&longs;im alias longo circuitu, vt patet experientia; nulla tamen, vel <lb/>modica fieri videtur compre&longs;&longs;io. </s> </p> <p id="N187CD" type="main"> <s id="N187CF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 102.<emph.end type="center"/></s> </p> <p id="N187DB" type="main"> <s id="N187DD"><!-- NEW --><emph type="italics"/>Hinc quo &longs;unt plures partes diuidendæ, quæ antè uniebantur, maior e&longs;t re&longs;i­<lb/>&longs;tentia<emph.end type="italics"/>; </s> <s id="N187E8"><!-- NEW -->igitur maiore vi opus e&longs;t, igitur maiore grauitate; </s> <s id="N187EC"><!-- NEW -->&longs;ed in medio <lb/>den&longs;iore ab eodem mobili plures &longs;eparantur quàm in rariore; </s> <s id="N187F2"><!-- NEW -->quia &longs;ci­<lb/>licet corpus den&longs;um plures habet &longs;ub minori exten&longs;ione, & rarum è con­<lb/>trario, vt videbimus &longs;uo loco; </s> <s id="N187FA"><!-- NEW -->igitur in medio den&longs;iore idem mobile ma­<lb/>jorem re&longs;i&longs;tentiam inuenit, quàm in rariore; </s> <s id="N18800"><!-- NEW -->licèt vtriu&longs;que partes <lb/>æquali nexu &longs;eu fibula copulentur; </s> <s id="N18806"><!-- NEW -->quia &longs;cilicet plures &longs;unt diuidendæ <lb/>in den&longs;iore; </s> <s id="N1880C"><!-- NEW -->quia plures &longs;cilicet in æquali &longs;patio occurrunt, quàm in ra­<lb/>riore; igitur maiore vi grauitatis opus e&longs;t. </s> </p> <p id="N18812" type="main"> <s id="N18814"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 103.<emph.end type="center"/></s> </p> <p id="N18820" type="main"> <s id="N18822"><!-- NEW --><emph type="italics"/>Hinc medium pote&longs;t comparari cum alio in<emph.end type="italics"/> 2. <emph type="italics"/>capitibus<emph.end type="italics"/>; </s> <s id="N18831"><!-- NEW -->Primum e&longs;t in <lb/>grauitate, vel den&longs;itate, nam reuerâ ex maiori den&longs;itate maiorem gra­<lb/>uitatem reducimus; </s> <s id="N18839"><!-- NEW -->Secundum e&longs;t in maiori, vel minori partium nexu, <lb/>ex quo 4. &longs;equuntur combinationes 2.mediorum; </s> <s id="N1883F"><!-- NEW -->nam vel &longs;unt eiu&longs;dem <lb/>grauitatis, & mollitiei; </s> <s id="N18845"><!-- NEW -->vel eiu&longs;dem grauitatis & diuer&longs;æ mollitiei; </s> <s id="N18849"><!-- NEW -->vel <lb/>eiu&longs;dem mollitiei, & diuer&longs;æ grauitatis; </s> <s id="N1884F"><!-- NEW -->vel diuer&longs;æ grauitatis, & eiu&longs;­<lb/>dem mollitiei; </s> <s id="N18855"><!-- NEW -->mollius autem illud appello, cuius partes laxiori nexu <lb/>copulantur; </s> <s id="N1885B"><!-- NEW -->porrò 4. i&longs;tæ combinationes &longs;upponunt <expan abbr="id&etilde;">idem</expan> mobile <expan abbr="invtroq;">in vtroque</expan> <lb/>medio; </s> <s id="N18869"><!-- NEW -->&longs;i &longs;it prima combinatio, motus e&longs;t æqualis in vtroque; </s> <s id="N1886D"><!-- NEW -->&longs;i &longs;ecunda <pb pagenum="124" xlink:href="026/01/156.jpg"/>maior e&longs;t in molliori; </s> <s id="N18876"><!-- NEW -->&longs;i tertia maior in grauiori; </s> <s id="N1887A"><!-- NEW -->&longs;i verò quarta &longs;ubdi­<lb/>uidi pote&longs;t in duas; </s> <s id="N18880"><!-- NEW -->nam vel grauius e&longs;t conjunctum cum maiori molli­<lb/>tie, vel leuius; </s> <s id="N18886"><!-- NEW -->&longs;i leuius, haud dubiè maior e&longs;t motus in leuiore; </s> <s id="N1888A"><!-- NEW -->&longs;i gra­<lb/>uius & mollities compen&longs;et grauitatem, id e&longs;t, &longs;i vt &longs;e habet grauitas gra­<lb/>uioris ad leuitatem leuioris; </s> <s id="N18892"><!-- NEW -->ita &longs;e habet mollities illius ad mollitiem <lb/>huius, æqualis e&longs;t in vtroque; &longs;i &longs;ecus, pro rata; </s> <s id="N18898"><!-- NEW -->hinc pote&longs;t e&longs;&longs;e æqualis <lb/>motus in grauiore & leuiore medio, & in æquè graui pote&longs;t e&longs;&longs;e maior <lb/>in grauiore; & minor; </s> <s id="N188A0"><!-- NEW -->maior quidem, &longs;i maior &longs;it ratio mollitiei gra­<lb/>uioris ad mollitiem leuioris, quàm grauitatis ad grauitatem; </s> <s id="N188A6"><!-- NEW -->minor ve­<lb/>rò, &longs;i maior &longs;it ratio grauitatis ad grauitatem, quàm mollitiei ad molli­<lb/>tiem; </s> <s id="N188AE"><!-- NEW -->æqualis denique &longs;i æqualis ratio; </s> <s id="N188B2"><!-- NEW -->& his regulis cuncta facilè ex­<lb/>plicari po&longs;&longs;unt; </s> <s id="N188B8"><!-- NEW -->hîc porrò &longs;uppono idem mobile, quod per vtrumque me­<lb/>dium de&longs;cendere po&longs;&longs;it, id e&longs;t, quod &longs;it vtroque grauius, medium autem <lb/>appello illud, per quod mobile grauius per &longs;e de&longs;cendit; dixi per &longs;e quia <lb/>nonnunquam accidit, vt vel ratione figuræ, vel alterius impedimenti non <lb/>de&longs;cendat. </s> </p> <p id="N188C4" type="main"> <s id="N188C6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 104.<emph.end type="center"/></s> </p> <p id="N188D2" type="main"> <s id="N188D4"><!-- NEW --><emph type="italics"/>Sunt tres combinationes mobilis cum medio<emph.end type="italics"/>; </s> <s id="N188DD"><!-- NEW -->prima, &longs;i &longs;it idem mobile <lb/>cum diuer&longs;is mediis; </s> <s id="N188E3"><!-- NEW -->&longs;ecunda, &longs;i idem medium cum diuer&longs;is mobilibus; </s> <s id="N188E7"><!-- NEW --><lb/>tertia &longs;i diuer&longs;a mobïlia cum diuer&longs;is mediis; </s> <s id="N188EC"><!-- NEW -->de primâ actum e&longs;t iam <lb/>&longs;uprà; &longs;ecunda &longs;ube&longs;t 4. combinationibus. </s> <s id="N188F2"><!-- NEW -->Prima &longs;i mobilia &longs;int eiu&longs;­<lb/>dem materiæ, &longs;ed diuer&longs;æ figuræ; Secunda eiu&longs;dem figuræ & diuer&longs;æ <lb/>materiæ. </s> <s id="N188FA"><!-- NEW -->Quarta diuer&longs;æ materiæ & figuræ; </s> <s id="N188FE"><!-- NEW -->&longs;i prima & &longs;ecunda, vel &longs;unt <lb/>figuræ æquales, vel inæquales; </s> <s id="N18904"><!-- NEW -->&longs;i primum &longs;unt eiu&longs;dem grauitatis; &longs;i &longs;e­<lb/>cundum diuer&longs;æ; </s> <s id="N1890A"><!-- NEW -->quippe figuræ &longs;imiles po&longs;&longs;unt e&longs;&longs;e æquales, vel inæ­<lb/>quales; </s> <s id="N18910"><!-- NEW -->& figuræ æquales po&longs;&longs;unt e&longs;&longs;e &longs;imiles, vel di&longs;&longs;imiles; </s> <s id="N18914"><!-- NEW -->&longs;i &longs;it tertia <lb/>combinatio, in qua &longs;int eiu&longs;dem figuræ, & diuer&longs;æ materiæ, diuer&longs;æ in­<lb/>quam in grauitate; </s> <s id="N1891C"><!-- NEW -->&longs;i figuræ &longs;unt æquales, &longs;emper e&longs;t diuer&longs;a grauitas; </s> <s id="N18920"><!-- NEW -->&longs;i <lb/>inæquales pote&longs;t e&longs;&longs;e vel eadem, vel tertia; </s> <s id="N18926"><!-- NEW -->in quarta combinatione di­<lb/>uer&longs;a compen&longs;atio fieri pote&longs;t; idem dicendum e&longs;t de tertia combinatio­<lb/>ne diuer&longs;orum mobilium, & mediorum, de quibus omnibus &longs;eor&longs;im iam <lb/>dicemus. </s> </p> <p id="N18930" type="main"> <s id="N18932"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 105.<emph.end type="center"/></s> </p> <p id="N1893E" type="main"> <s id="N18940"><emph type="italics"/>Si mobilia duo eiu&longs;dem materiæ, figuræ, & grauitatis in eodem medio de­<lb/>&longs;cendant, æquali motu feruntur<emph.end type="italics"/> dem. </s> <s id="N1894A"><!-- NEW -->vbi e&longs;t eadem proportio cau&longs;æ & re&longs;i­<lb/>&longs;tentiæ ibi e&longs;t idem effectus, per Ax. 5. &longs;ed in hoc ca&longs;u eadem e&longs;t illa pro­<lb/>portio; </s> <s id="N18952"><!-- NEW -->nam e&longs;t æqualis cau&longs;a, &longs;cilicet grauitas; </s> <s id="N18956"><!-- NEW -->idem medium æqualiter <lb/>vtrique re&longs;i&longs;tens, cum non plures medij partes re&longs;i&longs;tant vni, quam alteri; <lb/>igitur æqualis proportio. </s> </p> <p id="N1895E" type="main"> <s id="N18960"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 106.<emph.end type="center"/></s> </p> <p id="N1896C" type="main"> <s id="N1896E"><!-- NEW --><emph type="italics"/>Maior e&longs;t re&longs;istentia eiu&longs;dem medij ratione &longs;cilicet partium, cum plures <lb/>eius partes re&longs;istunt quàm cum pauciores<emph.end type="italics"/>; patet, quia maior effectus re­<lb/>&longs;pondet pluribus partibus cau&longs;æ per Ax.13.l.1. num.2. </s> </p> <pb pagenum="125" xlink:href="026/01/157.jpg"/> <p id="N1897F" type="main"> <s id="N18981"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 107.<emph.end type="center"/></s> </p> <p id="N1898D" type="main"> <s id="N1898F"><!-- NEW --><emph type="italics"/>Plures partes re&longs;istunt, quando plures pelluntur à mobili deor&longs;um<emph.end type="italics"/>; </s> <s id="N18998"><!-- NEW -->quip­<lb/>pe in tantum re&longs;i&longs;tunt, in quantum ab aliis &longs;eparantur; </s> <s id="N1899E"><!-- NEW -->atqui in tantum <lb/>&longs;eparantur, in quantum amouentur è &longs;uo loco; </s> <s id="N189A4"><!-- NEW -->&longs;ed ideo amouentur è <lb/>&longs;uo loco, in quantum pelluntur; </s> <s id="N189AA"><!-- NEW -->igitur cum plures pelluntur tunc plures <lb/>re&longs;i&longs;tunt; igitur tunc maior e&longs;t re&longs;i&longs;tentia. </s> </p> <p id="N189B0" type="main"> <s id="N189B2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 108.<emph.end type="center"/></s> </p> <p id="N189BE" type="main"> <s id="N189C0"><!-- NEW --><emph type="italics"/>Plures pelluntur à maiori &longs;uperficie, quàm à minori, quæ tendit deor&longs;um <lb/>parallela horizonti.<emph.end type="italics"/> v.g. <!-- REMOVE S-->à &longs;uperficie cubi maioris, quàm minoris; quippe <lb/>tot pelluntur quot re&longs;pondent primæ faciei, &longs;eu primo plano, quod e&longs;t in <lb/>fronte. </s> </p> <p id="N189D1" type="main"> <s id="N189D3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 109.<emph.end type="center"/></s> </p> <p id="N189DF" type="main"> <s id="N189E1"><emph type="italics"/>Si diuidatur cubus in cubos minores, ratio &longs;uperficierum erit duplicat a la­<lb/>terum, & ratio &longs;olidorum triplicata,<emph.end type="italics"/> con&longs;tat ex Geometria, &longs;it enim cubus </s> </p> <p id="N189EB" type="main"> <s id="N189ED"><!-- NEW --><arrow.to.target n="note2"/><lb/>GK, nam in gratiam eorum qui Geometriam ignorant hoc ip&longs;um ocu­<lb/>lis &longs;ubiiciendum e&longs;&longs;e videtur; diuidantur 6. eius facies in 4. quadrata <lb/>æqualia v. <!-- REMOVE S-->g. <!-- REMOVE S-->facies AI in quad. </s> <s id="N189FD"><!-- NEW -->AE. EC. EG. EI. idem fiat in aliis <lb/>5. faciebus, quarum duæ hîc tantum apparent; &longs;cilicet AK. KL; </s> <s id="N18A03"><!-- NEW -->&longs;ed <lb/>tribus aliis parallelis; </s> <s id="N18A09"><!-- NEW -->his tribus cædem diui&longs;iones re&longs;pondent; </s> <s id="N18A0D"><!-- NEW -->haud <lb/>dubiè erunt cubi minores, quorum latus &longs;it æquale AB, & quælibet fa­<lb/>cies æqualis quadrato AE, &longs;ed facies maior AI, e&longs;t quadrupla minoris <lb/>AE, ergo AI e&longs;t ad AE vt quadratum lateris AG ad quadratum lateris <lb/>AD; &longs;ed hæc e&longs;t ratio duplicata laterum 1. 2. 4. &longs;imiliter cubus maior <lb/>GK e&longs;t octuplum minoris DN, igitur vt cubus lateris AG ad cubum <lb/>lateris AD. &longs;ed hæc e&longs;t ratio triplicata. </s> <s id="N18A1D">1.2.4.8. </s> </p> <p id="N18A20" type="margin"> <s id="N18A22"><margin.target id="note2"/>a <emph type="italics"/>Fig.<emph.end type="italics"/>26 <lb/><emph type="italics"/>Tab.<emph.end type="italics"/>1.<!-- KEEP S--></s> </p> <p id="N18A35" type="main"> <s id="N18A37"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 110.<emph.end type="center"/></s> </p> <p id="N18A43" type="main"> <s id="N18A45"><!-- NEW --><emph type="italics"/>Hinc plùs minuitur &longs;olidum in diuer&longs;ione cubi quam facies, & plùs facies <lb/>quàm latus<emph.end type="italics"/>; </s> <s id="N18A50"><!-- NEW -->patet ex dictis, nam latus minoris cubi e&longs;t tantùm &longs;ubdu­<lb/>plum lateris maioris, & facies &longs;ubquadrupla; &longs;olidum verò &longs;ub­<lb/>octuplum. </s> </p> <p id="N18A58" type="main"> <s id="N18A5A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 111.<emph.end type="center"/></s> </p> <p id="N18A66" type="main"> <s id="N18A68"><!-- NEW --><emph type="italics"/>Hinc plùs minuitur grauitas, quàm re&longs;i&longs;tentia minoris cubi<emph.end type="italics"/>; </s> <s id="N18A71"><!-- NEW -->quia grauitas <lb/>re&longs;pondet &longs;olido, & re&longs;i&longs;tentia prim&etail; faciei; </s> <s id="N18A77"><!-- NEW -->re&longs;i&longs;tentia <expan abbr="inquā">inquam</expan> ratione par­<lb/>tium medij; </s> <s id="N18A81"><!-- NEW -->&longs;ed &longs;olidum plus minuitur quàm facies, vt dictum e&longs;t; </s> <s id="N18A85"><!-- NEW -->igitur <lb/>plus minuitur grauitas, quæ e&longs;t cau&longs;a virium quàm hæc re&longs;i&longs;tentia; ergo <lb/>decre&longs;cunt vires in maiore proportione quàm hæc re&longs;i&longs;tentia, quod be­<lb/>nè ob&longs;eruauit Galileus in dìalogis. </s> </p> <p id="N18A8F" type="main"> <s id="N18A91"><!-- NEW -->Hinc concludit Galileus duos cubos eiu&longs;dem materiæ, &longs;ed inæquales <lb/>de&longs;cendere inæquali motu; </s> <s id="N18A97"><!-- NEW -->maiorem &longs;cilicet velociùs minori; </s> <s id="N18A9B"><!-- NEW -->demon­<lb/>&longs;trare videtur, quia maior habet maiorem proportionem virium ad re­<lb/>&longs;i&longs;tentiam, quàm minor; igitur maiorem habet effectum per Ax. 5. igi­<lb/>tur maiorem, & velociorem motum. </s> </p> <p id="N18AA5" type="main"> <s id="N18AA7"><!-- NEW -->Scio non dee&longs;&longs;e multos viros doctos qui acriter in hanc &longs;ententiam <pb pagenum="126" xlink:href="026/01/158.jpg"/>in&longs;urgant: </s> <s id="N18AB0"><!-- NEW -->Obiicient fortè primò, experientiam e&longs;&longs;e contrariam; </s> <s id="N18AB4"><!-- NEW -->&longs;i enim <lb/>accipiantur duo cubi maior, & minor eiu&longs;dem materiæ, & dimittantur <lb/>ex eadem altitudine eodem pror&longs;us momento terram ferient; </s> <s id="N18ABC"><!-- NEW -->Re&longs;ponde­<lb/>ri pote&longs;t momentum illud &longs;en&longs;u percipi non po&longs;&longs;e; &longs;i enim dicam ma­<lb/>iorem tangere terram 1000. in&longs;tantibus ante minorem, an fortè &longs;en&longs;u <lb/>hoc percipies, vi&longs;u &longs;cilicet vel auditu? </s> <s id="N18AC6"><!-- NEW -->igitur in maxima altitudine hæc <lb/>&longs;patiorum inæqualitas, & temporum &longs;en&longs;u percipi po&longs;&longs;et, quæ in minori <lb/>&longs;ub &longs;en&longs;um non cadit: præterea accipe pulueris granulum eiu&longs;dem ma­<lb/>teriæ, tuncque etiam &longs;en&longs;ibilem motuum differentiam videbîs, atqui <lb/>e&longs;t eadem ratio de omni minore. </s> </p> <p id="N18AD2" type="main"> <s id="N18AD4"><!-- NEW -->Secundò obiicient, &longs;i &longs;uperponatur cubus minor maiori in &longs;uo motu <lb/>nunquam &longs;eparantur; igitur æquali motu de&longs;cendunt. </s> <s id="N18ADA"><!-- NEW -->Re&longs;p. videri po­<lb/>te&longs;t equidem æquali motu de&longs;cendere quia &longs;unt veluti partes eiu&longs;dem <lb/>corporis, & grauitant grauitatione communi, neque minor habet &longs;ingu­<lb/>larem re&longs;i&longs;tentiam &longs;uperandam; </s> <s id="N18AE4"><!-- NEW -->immò &longs;i &longs;uperponatur minor maiori, <lb/>vel maior minori, motus e&longs;t velocior quàm e&longs;&longs;et &longs;olius maioris; </s> <s id="N18AEA"><!-- NEW -->quia <lb/>cum non &longs;it maior re&longs;i&longs;tentia, maiores illi vires opponuntur; igitur fa­<lb/>ciliùs &longs;uperatur. </s> </p> <p id="N18AF2" type="main"> <s id="N18AF4">Tertiò obiicient; </s> <s id="N18AF7"><!-- NEW -->e&longs;t eadem &longs;pecie grauitas; </s> <s id="N18AFB"><!-- NEW -->igitur eadem grauitatio, <lb/>idemque motus deor&longs;um; </s> <s id="N18B01"><!-- NEW -->Re&longs;ponderi po&longs;&longs;et concedendo antecedens, <lb/>vnde in vacuo omnia grauia æquè velociter de&longs;cenderent, &longs;i in eo mo­<lb/>tus e&longs;&longs;et; at verò altera duarum cau&longs;arum eiu&longs;dem &longs;peciei, quæ habet mi­<lb/>norem proportionem actiuitatis ad re&longs;i&longs;tentiam, profectò minùs agit, <lb/>quod certum e&longs;t. </s> </p> <p id="N18B0D" type="main"> <s id="N18B0F"><!-- NEW -->Quartò obij:igitur motus po&longs;&longs;et e&longs;&longs;e velocior, & velocior in infini­<lb/>tum; </s> <s id="N18B15"><!-- NEW -->&longs;i enim maior cubus de&longs;cenderet velociùs; </s> <s id="N18B19"><!-- NEW -->igitur &longs;i detur maior ad­<lb/>huc velociùs, atque ita deinceps: </s> <s id="N18B1F"><!-- NEW -->Re&longs;p. inanem pror&longs;us e&longs;&longs;e difficulta­<lb/>tem; </s> <s id="N18B25"><!-- NEW -->quia cubus ille quantumuis maximus in vacuo de&longs;cendit velociùs <lb/>quàm in aliquo medio v.g.in aëre, igitur nunquam augmentum veloci­<lb/>tatis infinitum e&longs;t; quippe inter duos gradus velocitatis infiniti &longs;unt <lb/>po&longs;&longs;ibiles. </s> <s id="N18B2F"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it velocitas, quam habet in vacuo vt 2. illa verò quàm <lb/>habet in aëre vt 1. &longs;i cre&longs;cat velocitas iuxta has minutias &longs;ingulis in&longs;tan­<lb/>tibus 1/2 1/4 1/8 (1/16) (1/32), atque ita deinceps; quàm porrò multæ &longs;unt huiu&longs;modi <lb/>progre&longs;&longs;iones 1/3 1/6 (1/12) (1/24) &c. </s> <s id="N18B3D">igitur obiectiones illæ non euertunt Gali­<lb/>lei &longs;ententiam. </s> </p> <p id="N18B42" type="main"> <s id="N18B44"><!-- NEW -->Inde idem Galileus o&longs;tendere videtur cur atomi materiæ etiam gra­<lb/>ui&longs;&longs;imæ, &longs;eu granula pulueris motu tardi&longs;&longs;imo de&longs;cendant in aëre vel in <lb/>aqua; quia &longs;cilicet per illam diui&longs;ionem ita imminutæ &longs;unt vires graui­<lb/>tatis, vt iam re&longs;i&longs;tentiam medij &longs;uperare non po&longs;&longs;int. </s> </p> <p id="N18B4E" type="main"> <s id="N18B50"><!-- NEW -->Sed videtur e&longs;&longs;e graui&longs;&longs;ima difficultas, &longs;int enim duo cubi, maior B <lb/>F, minor GM, & vterque innatet medio liquido duplo grauiori; </s> <s id="N18B56"><!-- NEW -->certè ex­<lb/>tabit maior toto rectangulo CA æquali CF, & minor toto rectangulo <lb/>KH æquali KM; </s> <s id="N18B5E"><!-- NEW -->igitur e&longs;t eadem proportio grauitatis maioris ad re&longs;i­<lb/>&longs;tentiam medij in grauitatione, quæ e&longs;t minoris; igitur & in motu. </s> </p> <p id="N18B64" type="main"> <s id="N18B66"><!-- NEW -->Re&longs;ponderi pote&longs;t e&longs;&longs;e maximam di&longs;paritatem inter grauitationem, & <pb pagenum="127" xlink:href="026/01/159.jpg"/>motum; </s> <s id="N18B6F"><!-- NEW -->&longs;it enim cubus BD qui de&longs;cendat per totam AH; </s> <s id="N18B73"><!-- NEW -->haud dubiè <lb/>cum &longs;patium DI, contineat 3. cubos medij æquales DB, eos debet remo­<lb/>uere in &longs;uo de&longs;cen&longs;u; </s> <s id="N18B7B"><!-- NEW -->&longs;it autem cubus BG; </s> <s id="N18B7F"><!-- NEW -->haud dubiè, cum &longs;it eadem pro­<lb/>portio cubi AE ad cubum medij DM, quæ e&longs;t cubi BG ad cubum me­<lb/>dij FL, eodem tempore vterque cubum medij &longs;uppo&longs;iti è &longs;uo loco extru­<lb/>det; igitur eo tempore, quo AE expellet 3. DI, FL extrudet 3. EO, ergo <lb/>æquabili tempore inæquale &longs;patium percurrunt. </s> </p> <p id="N18B8B" type="main"> <s id="N18B8D"><!-- NEW -->Dices ergo &longs;patia &longs;unt vt latera: </s> <s id="N18B91"><!-- NEW -->Re&longs;ponderi pote&longs;t hoc reuerâ per &longs;e <lb/>e&longs;&longs;e debere; </s> <s id="N18B97"><!-- NEW -->&longs;ed quia cubus DM vt extrudatur, maiorem debet facere cir­<lb/>cuitionem, vt à fronte retrò eat, velociori motu extrudi debet; </s> <s id="N18B9D"><!-- NEW -->igitur vi­<lb/>res &longs;uas in eo con&longs;umit maiori ex parte cubus AE; hinc compen&longs;atio e&longs;&longs;e <lb/>videtur. </s> </p> <p id="N18BA5" type="main"> <s id="N18BA7">Vt &longs;olui po&longs;&longs;it præ&longs;ens difficultas, quæ cettè maxima e&longs;t, totam rem <lb/>i&longs;tam paulò fu&longs;iùs e&longs;&longs;e explicandam iudico. </s> <s id="N18BAC"><!-- NEW -->Primò itaque certum e&longs;t <lb/>partes medij, quæ prius in fronte erant, retroire; </s> <s id="N18BB2"><!-- NEW -->hoc ip&longs;um videmus in <lb/>naui quæ &longs;ulcat aquas, hoc ip&longs;um accidit in omni corpore natante etiam <lb/>immobili, quippe partes aquæ retinentur ab illa membranula, de qua &longs;u­<lb/>prà; </s> <s id="N18BBC"><!-- NEW -->&longs;ic enim &longs;æpè a&longs;&longs;urgunt, & intume&longs;cunt &longs;upra labra va&longs;is; </s> <s id="N18BC0"><!-- NEW -->cur verò <lb/>continui penè circulares limbi dilatentur: </s> <s id="N18BC6"><!-- NEW -->Re&longs;p. nullo flante vento <lb/>vix aliquem circulum huiu&longs;modi in &longs;uperficie aquæ apparere à fronte, <lb/>&longs;ed tantùm à tergo, & lateribus, qua&longs;i ad in&longs;tar pyramidis; &longs;ed de his aliàs <lb/>fusè. </s> </p> <p id="N18BD0" type="main"> <s id="N18BD2"><!-- NEW -->Secundò certum e&longs;t numerum partium, quas impellit maior cubus A <lb/>E; </s> <s id="N18BD8"><!-- NEW -->e&longs;&longs;e quadruplum numeri partium, quas impellit cubus BG: </s> <s id="N18BDC"><!-- NEW -->&longs;int autem <lb/>v.g.8. partes re&longs;i&longs;tentes cubo maiori, &longs;unt duæ re&longs;i&longs;tentes cubo minoris; <lb/>&longs;ed vires cubi maioris &longs;unt ad vires cubi minoris vt 8. ad 1. igitur vires <lb/>vt 8. &longs;uperabunt faciliùs re&longs;i&longs;tentiam vt 8. quam vires vt 1. re&longs;i&longs;tentiam <lb/>vt 2.vnde duplò velociùs moueretur, ni&longs;i aër duplò velociori motu amo­<lb/>uendus e&longs;&longs;et, quod vt clarius explicetur;</s> </p> <p id="N18BEA" type="main"> <s id="N18BEC"><!-- NEW -->Sit cubus maior AF octuplus cubi GI, vt iam dictum e&longs;t; </s> <s id="N18BF0"><!-- NEW -->haud <lb/>dubiè aër qui &longs;ub&longs;tat cubo AF e&longs;t quadruplus aëris, qui &longs;ub&longs;tat cubo GI, <lb/>vnde &longs;i vires cubi AF e&longs;&longs;ent quadruplæ virium cubi GI, e&longs;&longs;et æqualis <lb/>proportio in vtroque virium, & re&longs;i&longs;tentiæ; </s> <s id="N18BFA"><!-- NEW -->&longs;ed &longs;unt octuplæ; </s> <s id="N18BFE"><!-- NEW -->igitur faci­<lb/>liùs vincetur re&longs;i&longs;tentia; </s> <s id="N18C04"><!-- NEW -->igitur amouebitur aër faciliùs; &longs;it autem aër <lb/>expre&longs;&longs;us in globulis EFB, &c. </s> <s id="N18C0A"><!-- NEW -->cuius &longs;uperficies cum relinquatur retrò <lb/>ver&longs;us AB, & occupetur illa quæ e&longs;t in fronte EF; </s> <s id="N18C10"><!-- NEW -->haud dubiè partes <lb/>hinc inde diuiduntur in D, & &longs;egmentum NB tran&longs;it in locum relicti <lb/>loci BC, FN tran&longs;it in NB, & DF, in FN; </s> <s id="N18C18"><!-- NEW -->idem dico de &longs;egmentis oppo­<lb/>&longs;itis; </s> <s id="N18C1E"><!-- NEW -->idem pror&longs;us dico de minori globo; </s> <s id="N18C22"><!-- NEW -->nam MH tran&longs;it in HQ, & H <lb/>Q in QG, & QG in GL, idem dico de &longs;egmentis oppo&longs;itis; </s> <s id="N18C28"><!-- NEW -->igitur hæc <lb/>e&longs;t circuitio partium medij, quàm &longs;uprà indicauimus; hinc aër, qui amo­<lb/>uetur à corpore graui de&longs;cendente moueri debet nece&longs;&longs;ariò velociùs <lb/>quàm ip&longs;um corpus graue, quod de&longs;cendit. </s> </p> <p id="N18C32" type="main"> <s id="N18C34"><!-- NEW -->In hoc porrò ob&longs;erua &longs;egmentum MH moueri tardiùs quàm DF; </s> <s id="N18C38"><!-- NEW -->quia <lb/>conficit &longs;ubduplum &longs;patium, eo tempore, quo DF conficit duplum; </s> <s id="N18C3E"><!-- NEW --><pb pagenum="128" xlink:href="026/01/160.jpg"/>nam DF & FN &longs;unt duplæ MH & & HQ igitur dupla vi motrice opus <lb/>e&longs;t; </s> <s id="N18C48"><!-- NEW -->&longs;ed vires cubi AF &longs;unt ad vires cubi GI, vt 8. ad 1. partes verò aëris, <lb/>quas impellit AF, &longs;unt ad partes aëris, quas impellit GI, vt 4.ad 1. igitur <lb/>&longs;i partes aëris mouerentur æquali motu cum ip&longs;is cubis, à quibus mo­<lb/>uentur; </s> <s id="N18C52"><!-- NEW -->certè maior moueretur motu velociori; </s> <s id="N18C56"><!-- NEW -->vt autem moueantur par­<lb/>tes DF duplò velociore motu, quàm partes MH; </s> <s id="N18C5C"><!-- NEW -->debent vires, quæ mo­<lb/>nent DF, e&longs;&longs;e in ratione dupla ad illas, quæ mouent MH, id e&longs;t eo tem­<lb/>pore, quo vires vt 8.mouebunt mobile vt 4. motu vt 2. vires vt 1.moue­<lb/>bunt mobile vt 1. motu vt 1. licèt enim &longs;uperficies aëris EF moueatur <lb/>deor&longs;um; attamen ab alio aëere inferiore ita repertitur, vt &longs;ur&longs;um ver&longs;us <lb/>FN repellatur. </s> </p> <p id="N18C6A" type="main"> <s id="N18C6C"><!-- NEW -->Equidem tota &longs;uperficies aëris DF, cum pluribus partibus con&longs;tet, <lb/>non pote&longs;t &longs;imul tran&longs;ire in FN; </s> <s id="N18C72"><!-- NEW -->quia pars D antequam perueniat ad F <lb/>tran&longs;it per medium DF; igitur &longs;ucce&longs;&longs;iuè per mea ad illud &longs;patium DF, <lb/>quo tempore quie&longs;ceret globus AF, quod ridiculum e&longs;t. </s> </p> <p id="N18C7A" type="main"> <s id="N18C7C"><!-- NEW -->Quare fit nece&longs;&longs;ariò aliqua circuitio, & partium aëris commixtio, <lb/>&longs;eu conflictus; </s> <s id="N18C82"><!-- NEW -->ita vt retroeant pul&longs;æ prius & repercu&longs;&longs;æ; </s> <s id="N18C86"><!-- NEW -->non quidem <lb/>tramite recto, &longs;ed cum aliqua circuitione; </s> <s id="N18C8C"><!-- NEW -->quod certè facilè concipi po­<lb/>te&longs;t, quæ circuitio eò maior e&longs;t, quo latera cuborum &longs;unt maiora; ita­<lb/>que cum hæc &longs;atis fusè videantur e&longs;&longs;e explicata, &longs;it. </s> </p> <p id="N18C94" type="main"> <s id="N18C96"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 112.<emph.end type="center"/></s> </p> <p id="N18CA2" type="main"> <s id="N18CA4"><!-- NEW --><emph type="italics"/>Duo cubi eiu&longs;de<emph.end type="italics"/>m <emph type="italics"/>materiæ, & diuer&longs;æ grauitatis æquali motu per &longs;e de&longs;­<lb/>cendunt<emph.end type="italics"/>; </s> <s id="N18CB5"><!-- NEW -->probatur, quia licèt &longs;it maior proportio actiuitatis minus ad <lb/>&longs;uam re&longs;i&longs;tentiam, quàm alterius; </s> <s id="N18CBB"><!-- NEW -->illud tamen compen&longs;atur; </s> <s id="N18CBF"><!-- NEW -->eóque par­<lb/>tes aëris velociùs moueri debeant iuxta rationem laterum, vt patet ex <lb/>dictis; </s> <s id="N18CC7"><!-- NEW -->vnde nece&longs;&longs;ariò &longs;equitur motus æqualis in vtroque cubo; </s> <s id="N18CCB"><!-- NEW -->igitur <lb/>licèt maioris cubi vires habeant maiorem proportionem ad molem, <lb/>quæ præcipuum illius motus retardat; </s> <s id="N18CD3"><!-- NEW -->tum tamen aër, qui re&longs;i&longs;tit maiori <lb/>cubo debeat amoueri, vt dictum e&longs;t velociore motu quam aër, qui re&longs;i­<lb/>&longs;tit minori, &longs;itque eadem proportio re&longs;i&longs;tentiæ ratione motus minoris <lb/>ad maiorem, quæ e&longs;t ratione molis maioris ad minorem; </s> <s id="N18CDD"><!-- NEW -->certè ratio <lb/>compo&longs;ita vtriu&longs;què erit eadem in vtroque cubo; </s> <s id="N18CE3"><!-- NEW -->igitur æquè velociter <lb/>vterque de&longs;cendet: </s> <s id="N18CE9"><!-- NEW -->hinc &longs;atís facilè &longs;oluitur ratio Galilei, quam multi <lb/>parum cauti pro demon&longs;tratione venditarunt, ad aliam verò rationem, <lb/>quam ex minuto puluere ducere videtur, etiam facilè re&longs;ponderi pote&longs;t; </s> <s id="N18CF1"><!-- NEW --><lb/>ideo corpu&longs;cula illa diu fluitare in aëre, tùm quòd minimo ferè tenuis <lb/>auræ flatu agitentur; </s> <s id="N18CF8"><!-- NEW -->&longs;ic pulueris nubes medius ventus agit; </s> <s id="N18CFC"><!-- NEW -->quis enim <lb/>ne&longs;cit aëris partes agitari perpetuò; </s> <s id="N18D02"><!-- NEW -->immò & aquæ inter &longs;e mi&longs;ceri; </s> <s id="N18D06"><!-- NEW -->igi­<lb/>tur ab agitationis veluti impre&longs;&longs;ione fluitant illa corpu&longs;cula, cum mini­<lb/>mus ferè impetus extrin&longs;ecus illa commouere po&longs;&longs;it; </s> <s id="N18D0E"><!-- NEW -->tùm etiam quòd à <lb/>filamentis illis, quibus partes aëris implicantur facilè detineantur; ana­<lb/>logiam habes in lapillo, qui ab araneæ tela intercipitur. </s> </p> <pb pagenum="129" xlink:href="026/01/161.jpg"/> <p id="N18D1A" type="main"> <s id="N18D1C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 113.<emph.end type="center"/></s> </p> <p id="N18D28" type="main"> <s id="N18D2A"><!-- NEW --><emph type="italics"/>Duo globi eiu&longs;dem materiæ, & diuer&longs;æ diametri de&longs;cendunt etiam æquali <lb/>motu propter <expan abbr="eãdem">eandem</expan> rationem<emph.end type="italics"/>; </s> <s id="N18D39"><!-- NEW -->immò e&longs;t perfectior æqualitas in globis, <lb/>quàm in cubis; </s> <s id="N18D3F"><!-- NEW -->quia perfectior fit circuitio, vt con&longs;ideranti patebit; <lb/>hinc globus eiu&longs;dem materiæ, & grauitatis cum cubo de&longs;cendit velociùs <lb/>quia &longs;cilicet aër in de&longs;cen&longs;u globi faciliùs agitur retrò, vt con&longs;tat. </s> </p> <p id="N18D47" type="main"> <s id="N18D49"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 114.<emph.end type="center"/></s> </p> <p id="N18D55" type="main"> <s id="N18D57"><!-- NEW --><emph type="italics"/>Corpus vtrimque in mucronem de&longs;inens faciliùs adhuc de&longs;cendit, <lb/>quâm globus eiu&longs;dem materiæ<emph.end type="italics"/>; ratio e&longs;t; </s> <s id="N18D62"><!-- NEW -->quia breuiore circuitu partes re­<lb/>troeunt; </s> <s id="N18D68"><!-- NEW -->quippe tunc maxima e&longs;t facilitas in pellendo aëre, qui e&longs;t à fron­<lb/>te mobilis, cum velociùs moueri non debet ip&longs;o mobili; </s> <s id="N18D6E"><!-- NEW -->atqui hoc ip­<lb/>&longs;um e&longs;t quod accidit mobili vtrimque aucto; </s> <s id="N18D74"><!-- NEW -->nam linea curua DBA, <lb/>quam percurrit de&longs;criptum mobile, non e&longs;t multò longior; </s> <s id="N18D7A"><!-- NEW -->at verò in <lb/>quadrato &longs;uperiori AF maiori e&longs;t duplò; in circulo quidem minor dia­<lb/>meter &longs;emiperipheriæ, &longs;ed non duplò. </s> </p> <p id="N18D82" type="main"> <s id="N18D84"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 115.<emph.end type="center"/></s> </p> <p id="N18D90" type="main"> <s id="N18D92"><!-- NEW --><emph type="italics"/>Idem corpus diuer&longs;o motu de&longs;cendere pote&longs;t,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->parallipedum A, &longs;i re­<lb/>ctangulum BF &longs;it in fronte tardiùs de&longs;cendet, quàm &longs;i in fronte &longs;it re­<lb/>ctangulum CE, vel rectangulum FH; </s> <s id="N18DA3"><!-- NEW -->hinc tribus motibus diuer&longs;is de&longs;­<lb/>cendere pote&longs;t idem parallipedum, modò habeat &longs;emper alteram facie­<lb/>rum horizonti parallelam; </s> <s id="N18DAB"><!-- NEW -->hinc cylindrus eiu&longs;dem grauitatis de&longs;cendet <lb/>velociùs quàm parallelipedum, vt patet ex dictis; </s> <s id="N18DB1"><!-- NEW -->ex quibus facilè intel­<lb/>ligi pote&longs;t, quænam corpora faciliùs quàm alia de&longs;cendant; quippe illa <lb/>regula e&longs;t certi&longs;&longs;ima quàm &longs;uprà attulimus. </s> <s id="N18DB9"><!-- NEW -->Porrò ob&longs;eruabis omne <lb/>corpus difficiliùs pelli per lineam perpendicularem quàm per obliquam; </s> <s id="N18DBF"><!-- NEW --><lb/>hinc globus pellit tantùm vnicum punctum perpendiculariter; </s> <s id="N18DC4"><!-- NEW -->idem di­<lb/>co de cono; cylindrus verò vnam lineam, cubus integrum planum. </s> </p> <p id="N18DCA" type="main"> <s id="N18DCC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 116.<emph.end type="center"/></s> </p> <p id="N18DD8" type="main"> <s id="N18DDA"><!-- NEW --><emph type="italics"/>Hinc duo corpora eiu&longs;dem grauitatis, &longs;ed quorum alterum<emph.end type="italics"/> f<emph type="italics"/>aciem, quæ e&longs;t <lb/>in fronte, habet maiorem, inæquali motu de&longs;cendunt<emph.end type="italics"/>; patet ex dictis; quia in <lb/>vtroque &longs;unt æquales vires, &longs;ed diuer&longs;a re&longs;i&longs;tentia. </s> </p> <p id="N18DED" type="main"> <s id="N18DEF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 117.<emph.end type="center"/></s> </p> <p id="N18DFB" type="main"> <s id="N18DFD"><!-- NEW --><emph type="italics"/>Hinc tenues illæ &longs;uperficies corporum etiam materiæ graui&longs;&longs;imæ, vel in <lb/>aëre fluitant, vel aquis innatant<emph.end type="italics"/>; ratio e&longs;t, quia re&longs;i&longs;tentia &longs;uperat <lb/>vires. </s> </p> <p id="N18E0A" type="main"> <s id="N18E0C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N18E18" type="main"> <s id="N18E1A"><!-- NEW -->Ob&longs;eruabis primam &longs;uperficiem aquæ habere maiorem quamdam re­<lb/>&longs;i&longs;tentiam propter illam, qua&longs;i membranulam, de qua &longs;uprà; </s> <s id="N18E20"><!-- NEW -->vnde a&longs;&longs;ur­<lb/>git quiddam lymbus in margine bracteæ ferri, vel auri innatantis; vel <lb/>etiam globuli paulò grauioris aquâ, igitur vt immergatur corpus debet <lb/>e&longs;&longs;e grauius totâ illâ aquâ, quæ capacitatem illam non cauam occu­<lb/>paret. </s> </p> <pb pagenum="130" xlink:href="026/01/162.jpg"/> <p id="N18E30" type="main"> <s id="N18E32"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 118.<emph.end type="center"/></s> </p> <p id="N18E3E" type="main"> <s id="N18E40"><!-- NEW --><emph type="italics"/>Globi æquales diuer&longs;æ materiæ inæqualiter de&longs;cendunt<emph.end type="italics"/>; </s> <s id="N18E49"><!-- NEW -->quia &longs;cilicet alte­<lb/>rum e&longs;t grauius, quod &longs;uppono; </s> <s id="N18E4F"><!-- NEW -->igitur æqualis e&longs;t re&longs;i&longs;tentia, & vires <lb/>inæquales; </s> <s id="N18E55"><!-- NEW -->igitur non e&longs;t eadem proportio actiuitatis: & re&longs;i&longs;tentiæ; igi­<lb/>tur non e&longs;t æqualis motus per Ax.5. <!-- KEEP S--></s> </p> <p id="N18E5C" type="main"> <s id="N18E5E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 119.<emph.end type="center"/></s> </p> <p id="N18E6A" type="main"> <s id="N18E6C"><!-- NEW --><emph type="italics"/>Globi otiam inæquales diuer&longs;æ materiæ inæqualiter de&longs;cendunt<emph.end type="italics"/>; quod de­<lb/>mon&longs;tro; </s> <s id="N18E77"><!-- NEW -->quia globi eiu&longs;dem materiæ inæqualiter de&longs;cendunt per Th. <!-- REMOVE S--><lb/>113. &longs;ed duo globi æquales diuer&longs;æ materiæ de&longs;cendunt inæqualiter per <lb/>Th.118. igitur, & inæquales; quod dico de globis', dicatur de cubis, & <lb/>aliis figuris &longs;imilibus. </s> </p> <p id="N18E82" type="main"> <s id="N18E84"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 120.<emph.end type="center"/></s> </p> <p id="N18E90" type="main"> <s id="N18E92"><!-- NEW --><emph type="italics"/>Globus materiæ leuioris pote&longs;t de&longs;cendere velociori motu quam parallelipe­<lb/>dum grauioris<emph.end type="italics"/>; </s> <s id="N18E9D"><!-- NEW -->con&longs;tat experientia; ratio e&longs;t, quia cum globus ferreus de&longs;­<lb/>cendat velociùs, quàm ligneus per Th. 118. in data ratione, putà (1/100) <lb/>haud dubiè bractea ferri non modo (1/100) tardiùs de&longs;cendet, verùm etiam <lb/>(20/100) in quo non e&longs;t difficultas. </s> </p> <p id="N18EA7" type="main"> <s id="N18EA9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 121.<emph.end type="center"/></s> </p> <p id="N18EB5" type="main"> <s id="N18EB7"><!-- NEW --><emph type="italics"/>Hinc &longs;i mutetur figura po&longs;&longs;unt grauia diuer&longs;æ materiæ ita de&longs;cendere, vn <lb/>vel grauius, vel leuius, vel grauioris materiæ, vel leuioris velociùs de&longs;cendat<emph.end type="italics"/>; <lb/>vt con&longs;tat ex regulis præ&longs;criptis. </s> </p> <p id="N18EC4" type="main"> <s id="N18EC6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 122.<emph.end type="center"/></s> </p> <p id="N18ED2" type="main"> <s id="N18ED4"><!-- NEW --><emph type="italics"/>Globi æquales diuer&longs;æ materiæ,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->ligneus, & plumbeus de&longs;cendunt <lb/>inæqualiter iuxta proportionem grauitatis, & re&longs;i&longs;tentiæ medij compa­<lb/>ratæ cum vtroque, v.g. <!-- REMOVE S-->plumbo detrahitur (1/4800); ligno verò (8/300) v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i <lb/>grauitas ligni &longs;it ad grauitatem aëris vt 300.ad 1. & plumbi vt 4800. ad <lb/>1. &longs;it enim altitudo 33. pedum 4. digit. </s> <s id="N18EEF"><!-- NEW -->reducantur in digitos erunt 400. <lb/>in lineas 4800. igitur detrahetur vna linea &longs;patij plumbeo globo; </s> <s id="N18EF5"><!-- NEW -->ligneo <lb/>verò vnus digitus cum 4. lineis; &longs;ed quis hoc ob&longs;eruet? </s> </p> <p id="N18EFB" type="main"> <s id="N18EFD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 123.<emph.end type="center"/></s> </p> <p id="N18F09" type="main"> <s id="N18F0B"><!-- NEW --><emph type="italics"/>Corpus graue &longs;pongio&longs;um longè tardiùs de&longs;cendit<emph.end type="italics"/>; </s> <s id="N18F14"><!-- NEW -->quia aër in perexigua <lb/>illa foramina inten&longs;us frangitur, re&longs;ilit, ac proinde motum impedit; talis <lb/>e&longs;t medulla &longs;ambuci, &longs;pongia, &longs;tupa, &c. </s> <s id="N18F1C">immò a&longs;perum corpus tardiùs <lb/>de&longs;cendit, quòd &longs;cilicet aër ab a&longs;perioribus illis &longs;alebris re&longs;iliens mo­<lb/>tum retardet, hinc &longs;ibilus ille auditur &c. </s> </p> <p id="N18F23" type="main"> <s id="N18F25"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N18F31" type="main"> <s id="N18F33"><!-- NEW -->Ex his con&longs;tat quid dicendum &longs;it de motu corporum grauium in <lb/>medio, &longs;iue &longs;int eiu&longs;dem materiæ, & &longs;imilis figuræ, maioris vel minoris, <lb/>vel æqualis; </s> <s id="N18F3B"><!-- NEW -->tunc enim de&longs;cendunt æqualiter contra Galileum, &longs;iue <lb/>&longs;int diuer&longs;æ materiæ, & &longs;imilis figuræ, æqualis, vel inæqualis, <pb pagenum="131" xlink:href="026/01/163.jpg"/>tunc enim de&longs;cendunt inæqualiter, &longs;iue diuer&longs;æ materiæ & diuer&longs;æ fi­<lb/>guræ; </s> <s id="N18F48"><!-- NEW -->tunc enim de&longs;cendunt modò æqualiter, modò inæqualiter; </s> <s id="N18F4C"><!-- NEW -->æquali­<lb/>ter certè, cum figura compen&longs;at materiam; </s> <s id="N18F52"><!-- NEW -->cum verò non compen&longs;at, <lb/>inæqualiter pro rata; </s> <s id="N18F58"><!-- NEW -->denique &longs;i comparentur duo corpora cum diuer&longs;is <lb/>mediis; primo inuenienda e&longs;t proportio motuum vtriu&longs;que in eodem <lb/>tùm &longs;ingulorum in diuer&longs;is mediis, vt &longs;uprà dictum e&longs;t. </s> </p> <p id="N18F60" type="main"> <s id="N18F62"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 124.<emph.end type="center"/></s> </p> <p id="N18F6E" type="main"> <s id="N18F70"><!-- NEW --><emph type="italics"/>In modico vacuo omnia æquè velociter de&longs;cenderent<emph.end type="italics"/>: </s> <s id="N18F79"><!-- NEW -->Probatur, quia tota <lb/>diuer&longs;itas vel inæqualitas mediorum petitur à diuer&longs;a proportione acti­<lb/>uitatis cum re&longs;i&longs;tentia medij per Ax. 5. &longs;ed in vacuo nulla e&longs;t re&longs;i&longs;ten­<lb/>tia; </s> <s id="N18F83"><!-- NEW -->igitur nulla proportio; igitur nulla ratio motus inæqualis. </s> </p> <p id="N18F87" type="main"> <s id="N18F89"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 125.<emph.end type="center"/></s> </p> <p id="N18F95" type="main"> <s id="N18F97"><!-- NEW --><emph type="italics"/>In motu natur aliter accelerato deor&longs;um cre&longs;cit re&longs;istentia medij &longs;ingulis in­<lb/>&longs;tantibus<emph.end type="italics"/>: </s> <s id="N18FA2"><!-- NEW -->probatur, quia &longs;ingulis in&longs;tantibus plures partes medij &longs;unt <lb/>&longs;uperandæ; </s> <s id="N18FA8"><!-- NEW -->cre&longs;cunt enim &longs;patia, vt con&longs;tat ex dictis; igitur cre&longs;cit re&longs;i­<lb/>&longs;tentia &longs;ingulis in&longs;tantibus. </s> </p> <p id="N18FAE" type="main"> <s id="N18FB0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 126.<emph.end type="center"/></s> </p> <p id="N18FBC" type="main"> <s id="N18FBE"><!-- NEW --><emph type="italics"/>Cre&longs;cit re&longs;istentia iuxta rationem &longs;patiorum,<emph.end type="italics"/> probatur; </s> <s id="N18FC7"><!-- NEW -->quia cre&longs;cit iux­<lb/>ta rationem plurium partium medij, quæ temporibus æqualibus percur­<lb/>runtur; &longs;ed eæ cre&longs;cunt iuxta rationem &longs;patiorum, vt con&longs;tat. </s> </p> <p id="N18FCF" type="main"> <s id="N18FD1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 127.<emph.end type="center"/></s> </p> <p id="N18FDD" type="main"> <s id="N18FDF"><!-- NEW --><emph type="italics"/>Hinc cre&longs;cit re&longs;i&longs;tentia iuxta rationem velocitatum &longs;ingulis instantibus<emph.end type="italics"/>; </s> <s id="N18FE8"><!-- NEW --><lb/>quæ ratio &longs;equitur progre&longs;&longs;ionem arithmeticam &longs;implicem numerorum <lb/>1.2.3.4.5.6. ex &longs;uppo&longs;itione quòd tempus con&longs;tet ex partibus finitis actu; </s> <s id="N18FEF"><!-- NEW --><lb/>nam eodem modo cre&longs;cit velocitas, quo cre&longs;cunt numeri prædicti; </s> <s id="N18FF4"><!-- NEW -->&longs;ed <lb/>eodem modo cre&longs;cunt &longs;patia, &longs;i dumtaxat accipiantur in &longs;ingulis in&longs;tan­<lb/>tibus; </s> <s id="N18FFC"><!-- NEW -->re&longs;i&longs;tentia cre&longs;cit iuxta rationem &longs;patiorum; igitur iuxta ratio­<lb/>nem velocitatum. </s> </p> <p id="N19002" type="main"> <s id="N19004"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N19010" type="main"> <s id="N19012"><!-- NEW -->Ob&longs;eruabis, &longs;i tempus con&longs;tet ex infinitis actu partibus, ita vt &longs;ingu­<lb/>læ partes motus &longs;ingulis partibus temporis & infinitæ infinitis re&longs;pon­<lb/>deant; </s> <s id="N1901A"><!-- NEW -->non pote&longs;t e&longs;&longs;e alia progre&longs;&longs;io, in qua fiat acceleratio motus na­<lb/>turalis, quàm illa Galilei iuxta hos numeros 1. 3. 5. 7. vt con&longs;tat ex dictis <lb/>per illud Principium; </s> <s id="N19022"><!-- NEW --><emph type="italics"/>æqualibus temporibus æqualia acquiruntur velocita­<lb/>tis momenta<emph.end type="italics"/>; </s> <s id="N1902D"><!-- NEW -->&longs;i verò tempus con&longs;tat ex finitis in&longs;tantibus æqualibus, nul­<lb/>la datur progre&longs;&longs;io motus naturaliter accelerati; </s> <s id="N19033"><!-- NEW -->quia motus accelerari <lb/>non pote&longs;t; </s> <s id="N19039"><!-- NEW -->ne &longs;cilicet eodem in&longs;tanti mobile &longs;it in pluribus locis adæ­<lb/>quatis; denique &longs;i tempus con&longs;tat ex finitis in&longs;tantibus actu, & infinitis <lb/>potentiâ, non pote&longs;t e&longs;&longs;e alia progre&longs;&longs;io huius accelerationis, quam hæc <lb/>no&longs;tra iuxta numeros toties repetitos 1.2.3.4.5. attamen quia illa finita <lb/>in&longs;tantia &longs;unt ferè innumera in qualibet parte &longs;en&longs;ibili temporis, in <lb/>praxi &longs;ine &longs;en&longs;ibili errore in partibus temporis &longs;en&longs;ibilibus po&longs;&longs;umus <pb pagenum="132" xlink:href="026/01/164.jpg"/>adhibere priorem progre&longs;&longs;ionem Galilei, & in hoc cardine tota verri­<lb/>tur, meo iudicio, propo&longs;itæ quæ&longs;tionis difficultas. </s> </p> <p id="N1904E" type="main"> <s id="N19050"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 128.<emph.end type="center"/></s> </p> <p id="N1905C" type="main"> <s id="N1905E"><!-- NEW --><emph type="italics"/>Hinc cre&longs;cit re&longs;istentia iuxta rationem crementi impetus<emph.end type="italics"/>; cum enim cre­<lb/>&longs;cant impetus in ratione velocitatum, vt con&longs;tat, & cre&longs;cat re&longs;i&longs;tentia <lb/>medij in eadem ratione per Theor. <!-- REMOVE S-->127. cre&longs;cit etiam in ratione im­<lb/>petuum. </s> </p> <p id="N1906F" type="main"> <s id="N19071"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 129.<emph.end type="center"/></s> </p> <p id="N1907D" type="main"> <s id="N1907F"><!-- NEW --><emph type="italics"/>Hinc cre&longs;cit re&longs;istentia medij in eadem ratione, in qua cre&longs;cunt vires mobi­<lb/>lis<emph.end type="italics"/>; demon&longs;tr. </s> <s id="N1908A"><!-- NEW -->quia cre&longs;cunt vires, vt cre&longs;cit impetus; nam impetus e&longs;t <lb/>vis illa, quâ mobile &longs;uperat re&longs;i&longs;tentiam medij vt con&longs;tat, &longs;ed re&longs;i&longs;ten­<lb/>tia cre&longs;cit vt impetus per Th. 128. igitur cre&longs;cit in ratione virium. </s> </p> <p id="N19092" type="main"> <s id="N19094"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 130.<emph.end type="center"/></s> </p> <p id="N190A0" type="main"> <s id="N190A2"><!-- NEW --><emph type="italics"/>Si cre&longs;cit re&longs;i&longs;tentia in eadem ratione in qua cre&longs;cunt vires, non mutatur <lb/>progre&longs;&longs;io effectuum.<emph.end type="italics"/> v.g. <!-- REMOVE S-->primo in&longs;tanti impetus &longs;it vt 1.&longs;itque 1.&longs;patium, <lb/>in quo e&longs;t re&longs;i&longs;tentia, vt 1. Secundo in&longs;tanti &longs;it impetus vt 2. re&longs;i&longs;tentia in <lb/>2. &longs;patiis vt 2. haud dubiè &longs;i vno in&longs;tanti vnus gradus impetus &longs;uperat <lb/>re&longs;i&longs;tentiam vt 1. dum percurrit 1.&longs;patium; </s> <s id="N190B5"><!-- NEW -->certè 2. gradus impetus vno <lb/>in&longs;tanti &longs;uperabunt re&longs;i&longs;tentiam vt 2. dum conficit mobile 2. &longs;patia; at­<lb/>que ita deinceps. </s> </p> <p id="N190BD" type="main"> <s id="N190BF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 132.<emph.end type="center"/></s> </p> <p id="N190CB" type="main"> <s id="N190CD"><!-- NEW --><emph type="italics"/>Hinc certè concludo contra Galileum, & alios quo&longs;dam motum grauium <lb/>po&longs;t aliquod &longs;patium decur&longs;um ex naturaliter accelerato non fieri æquabilem,<emph.end type="italics"/><lb/>quia in tantum fieret æquabilis in quantum tanta e&longs;&longs;et re&longs;i&longs;tentia, vt no­<lb/>uam accelerationem impediret; </s> <s id="N190DB"><!-- NEW -->&longs;ed hæc ratio nulla e&longs;t; </s> <s id="N190DF"><!-- NEW -->quia in eadem <lb/>ratione cre&longs;cit re&longs;i&longs;tentia, in qua cre&longs;cunt vires per Th. 129. igitur non <lb/>mutatur progre&longs;&longs;io motuum per Th. 130. igitur nec acceleratio; </s> <s id="N190E7"><!-- NEW -->igitur <lb/>motus naturalis ex accelerato non fit æquabilis: Equidem, vt iam &longs;uprà <lb/>dictum e&longs;t, in minori &longs;emper ratione cre&longs;cit velocitas, itémque ip&longs;a re&longs;i­<lb/>&longs;tentia quod in omni progre&longs;&longs;ione arithmetica iuxta numeros 1.2.3.4.5. </s> </p> <p id="N190F1" type="main"> <s id="N190F3"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N190FF" type="main"> <s id="N19101"><!-- NEW -->Ob&longs;eruabis remitti à nobis motum leuium &longs;ur&longs;um in 5. Tomum, in cu­<lb/>ius tertio libro agemus de graui, & leui; quia ideo corpus a&longs;cendit, quia <lb/>ab alio de&longs;cendente truditur &longs;ur&longs;um. </s> </p> </chap> <chap id="N19109"> <pb pagenum="133" xlink:href="026/01/165.jpg"/> <figure id="id.026.01.165.1.jpg" xlink:href="026/01/165/1.jpg"/> <p id="N19113" type="head"> <s id="N19115"><emph type="center"/>LIBER TERTIVS, <lb/><emph type="italics"/>DE MOTV VIOLENTO <lb/>&longs;ur&longs;um Perpendiculariter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N19124" type="main"> <s id="N19126"><!-- NEW -->OMnis certè motus, qui e&longs;t à principio ex­<lb/>trin&longs;eco, violentus appellari pote&longs;t, attamen <lb/>hîc non ago de omni violento, &longs;ed dumta­<lb/>xat de illo, qui fit &longs;ursùm per lineam verticalem; </s> <s id="N19130"><!-- NEW -->quia <lb/>&longs;cilicet ex diametro opponitur motui naturali, qui <lb/>fit deorsùm perpendiculariter; igitur cum de hoc <lb/>ip&longs;o in &longs;ecundo Libro egerimus, de illo in hoc non <lb/>agemus. <lb/><gap desc="hr tag"/></s> </p> <p id="N1913F" type="main"> <s id="N19141"><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1914D" type="main"> <s id="N1914F"><emph type="italics"/>MOtus violentus e&longs;t, quo corpus graue mouetur &longs;ursùm per li­<lb/>neam verticalem à principio extrin&longs;eco mediatè, vel immediatè vt <lb/>plurimùm.<emph.end type="italics"/></s> </p> <p id="N1915A" type="main"> <s id="N1915C"><!-- NEW -->Dixi à principio extrin&longs;eco, &longs;iue conjuncto, vt cum manu attollo &longs;ur­<lb/>&longs;um corpus graue, &longs;iue non conjuncto, vt cum quis proiicit lapidem &longs;ur­<lb/>sùm, &longs;iue &longs;it verum principium effectiuum, vt cum impetus, quem poten­<lb/>tia motrix producit in manu, producit alium in mobili; </s> <s id="N19166"><!-- NEW -->&longs;iue non &longs;it <lb/>principium effectiuum, &longs;ed tantùm determinans, vt cum mobile quod <lb/>cadit deor&longs;um, &longs;ur&longs;um deinde repercutitur; </s> <s id="N1916E"><!-- NEW -->nec enim corpus repercu­<lb/>tiens producit impetum nouum, vt dicemus cum de motu reflexo; </s> <s id="N19174"><!-- NEW -->quin <lb/>potiùs producti partem de&longs;truit per accidens, & quidquid illius &longs;upere&longs;t, <lb/>ad nouam lineam determinat; quod quomodo fiat fusè &longs;uo loco expli­<lb/>cabimus, igitur licèt corpus reflectens &longs;it tantùm principium nouæ de­<lb/>terminationis, non verò alicuius impetus producti, dici pote&longs;t princi­<lb/>pium huius motus violenti. </s> </p> <p id="N19182" type="main"> <s id="N19184">Dixi vt plurimùm, nam &longs;i terra ducto per centrum foramine e&longs;&longs;et <lb/>peruia, haud dubiè lapis demi&longs;&longs;us versùs centrum iret motu naturaliter <pb pagenum="134" xlink:href="026/01/166.jpg"/>accelerato, tùm deinde propter impetus acqui&longs;iti vim, à centro versùs <lb/>oppo&longs;itum circumferentiæ punctum iret, motu certè violento, qui ta­<lb/>men ab extrin&longs;eco non e&longs;&longs;et. </s> </p> <p id="N19192" type="main"> <s id="N19194"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N191A1" type="main"> <s id="N191A3"><!-- NEW --><emph type="italics"/>Corpus graue projectum &longs;ur&longs;um tandem redit<emph.end type="italics"/>; Hæc hypothe&longs;is certa e&longs;t, <lb/>& nemo e&longs;t qui eam in dubium vocet. </s> </p> <p id="N191AE" type="main"> <s id="N191B0"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N191BD" type="main"> <s id="N191BF"><!-- NEW --><emph type="italics"/>Quidquid erat, & de&longs;init e&longs;&longs;e de&longs;truitur<emph.end type="italics"/>; Hoc Axioma certum e&longs;t, quip­<lb/>pe de&longs;trui hoc tantùm dicitur, quod de&longs;init e&longs;&longs;e. </s> </p> <p id="N191CA" type="main"> <s id="N191CC"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N191D9" type="main"> <s id="N191DB"><emph type="italics"/>Quidquid destruitur, ad exigentiam alicuius destruitur, &longs;altem totius na­<lb/>turæ.<emph.end type="italics"/></s> <s id="N191E4"> Hoc Axioma idem e&longs;t cum Axiom. <!-- REMOVE S-->14. l. <!-- REMOVE S-->1. n. </s> <s id="N191EB"><!-- NEW -->2. vnde alia expli­<lb/>catione minimè indiget; hoc ip&longs;um etiam demon&longs;traui in Th.147.149. <lb/>150,&c. </s> <s id="N191F3">l. <!-- REMOVE S-->1. <!-- KEEP S--></s> </p> <p id="N191F9" type="main"> <s id="N191FB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N19208" type="main"> <s id="N1920A"><!-- NEW --><emph type="italics"/>Datur motus violentus<emph.end type="italics"/>; demon&longs;tro; corpus proiicitur per lineam ver­<lb/>ticalem per hyp. </s> <s id="N19215"><!-- NEW -->1. &longs;ed hic motus e&longs;t à principio extrin&longs;eco, igitur e&longs;t <lb/>violentus per def.1. probatur minor; Primò, quia illud e&longs;t principium, <lb/>&longs;eu cau&longs;a motus, ex cuius applicatione &longs;emper &longs;equitur motus per Ax.11. <lb/>l. <!-- REMOVE S-->1.n. </s> <s id="N19221">1. &longs;ed ex applicatione potentiæ extrin&longs;ecæ v. <!-- REMOVE S-->g. <!-- REMOVE S-->arcus, manus, &c. </s> <s id="N19228"><!-- NEW --><lb/>ad lineam &longs;ur&longs;um &longs;emper &longs;equitur motus &longs;ur&longs;um; igitur e&longs;t illius cau&longs;a. </s> <s id="N1922D"><lb/>Secundò probatur, quia mobile projectum &longs;ursùm mouetur adhuc &longs;epa­<lb/>ratum à potentia motrice per hyp. </s> <s id="N19233">6.l.1. igitur potentia motrix impre&longs;­<lb/>&longs;it aliquid mobili, vi cuius deinde mouetur, igitur hic motus e&longs;t à prin­<lb/>cipio extrin&longs;eco. </s> </p> <p id="N1923A" type="main"> <s id="N1923C"><!-- NEW -->Diceret fortè aliquis produci hunc motum ab ip&longs;o mobili; &longs;ed con­<lb/>trà; </s> <s id="N19242"><!-- NEW -->igitur &longs;emper produceret, quod ab&longs;urdum e&longs;t: </s> <s id="N19246"><!-- NEW -->dicet, ad hoc vt pro­<lb/>ducat determinari debere ab aliquo, &longs;ed contrà; </s> <s id="N1924C"><!-- NEW -->illud à quo determina­<lb/>tur vel e&longs;t extrin&longs;ecum, vel intrin&longs;ecum, &longs;i primum, ergo hic motus e&longs;t <lb/>&longs;emper à principio extrin&longs;eco, quod &longs;atis e&longs;t e&longs;&longs;e determinans per def.1. <lb/>&longs;i verò e&longs;t intrin&longs;ecum; igitur &longs;emper e&longs;&longs;et hic motus, quamdiu e&longs;&longs;et <lb/>ip&longs;um mobile, quod e&longs;t contra hyp. </s> <s id="N19258">1. nam reuera non &longs;emper mo­<lb/>uetur. </s> </p> <p id="N1925D" type="main"> <s id="N1925F"><!-- NEW -->Diceret fortè alius excitari quædam corpu&longs;cula, à quibus mouetur <lb/>corpus graue &longs;ursùm; &longs;ed contrà; </s> <s id="N19265"><!-- NEW -->nam vel &longs;unt in ip&longs;o mobili illa cor­<lb/>pu&longs;cula, vel extra mobile; &longs;i primum; </s> <s id="N1926B"><!-- NEW -->igitur hic motus &longs;emper erit ab <lb/>extrin&longs;eco mediatè, cum ab extrin&longs;eco excitentur; </s> <s id="N19271"><!-- NEW -->&longs;ed hoc &longs;ufficit ad <lb/>hoc; vt motus dicatur violentus per def. </s> <s id="N19277"><!-- NEW -->1. &longs;i verò &longs;unt extra mobile; <lb/>igitur motus ille e&longs;t &longs;emper ab extrin&longs;eco, idque duplici nomine. </s> </p> <p id="N1927D" type="main"> <s id="N1927F"><!-- NEW -->Denique diceret alius ex &longs;uppo&longs;itione, quod terra moueatur non po&longs;­<lb/>&longs;e corpus graue proiici &longs;ursùm per lineam verticalem, ni&longs;i tantùm ad <lb/>&longs;peciem; </s> <s id="N19287"><!-- NEW -->vt &longs;i quis è naui mobili &longs;ur&longs;um proiiceret pilam rectà omni­<lb/>nò, quoad eius fieri po&longs;&longs;it; videbitur enim iis, qui vehuntur eadem naui <pb pagenum="135" xlink:href="026/01/167.jpg"/>&longs;ur&longs;um ferri per lineam verticalem, aliis verò in&longs;tantibus videbitur cla­<lb/>ri&longs;&longs;imè ferri per lineam nouam inclinatam. </s> </p> <p id="N19294" type="main"> <s id="N19296"><!-- NEW -->Re&longs;pondeo etiam admi&longs;&longs;a &longs;uppo&longs;itione dici à me motum illum &longs;ur­<lb/>&longs;um e&longs;&longs;e per lineam verticalem, quando eadem linea recta connectit <lb/>&longs;emper hæc tria puncta; </s> <s id="N1929E"><!-- NEW -->&longs;cilicet centrum terræ, idem punctum &longs;uperfi­<lb/>ciei terræ, & ip&longs;am pilam; </s> <s id="N192A4"><!-- NEW -->ad illud verò quod dicitur de naui, non diffi­<lb/>teor verum e&longs;&longs;e; &longs;ed dico non e&longs;&longs;e propriè motum violentum, de quo hîc <lb/>tantùm e&longs;t quæ&longs;tio, &longs;ed e&longs;&longs;e motum mixtum, de quo fusè &longs;uo loco. </s> <s id="N192AC"><!-- NEW -->Ob&longs;er­<lb/>uabis autem hîc me ab&longs;tinere à refellendis ab&longs;urdis illis &longs;uppo&longs;itioni­<lb/>bus, quibus præmi&longs;&longs;æ objectiones innituntur; nam, cui quæ&longs;o in men­<lb/>tem venire pote&longs;t ab ip&longs;a entitate corporis grauis produci motum in &longs;e? </s> <s id="N192B6"><lb/>quis credat produci frigus ab igne? </s> <s id="N192BA">calorem à niue? </s> <s id="N192BD">lucem à tenebris? </s> <s id="N192C0"><lb/>quæ porrò fabulæ, quæ commenta, quæ &longs;omnia excogitari po&longs;&longs;unt, quæ <lb/>non vile&longs;cant &longs;i cum his comparentur. </s> </p> <p id="N192C6" type="main"> <s id="N192C8"><!-- NEW -->Illa quoque corpu&longs;cula excitata leuiora &longs;unt, quàm vt aliquod præfe­<lb/>rant rationis momentum; cum mera &longs;int philo&longs;ophiæ ludibria. </s> </p> <p id="N192CE" type="main"> <s id="N192D0">Denique illa hypothe&longs;is de terræ motu nullis demon&longs;trationibus fir­<lb/>mata e&longs;t, vt videbimus &longs;uo loco. </s> </p> <p id="N192D5" type="main"> <s id="N192D7"><!-- NEW -->Vnum fortè e&longs;t, quod difficilius obiici pote&longs;t; </s> <s id="N192DB"><!-- NEW -->&longs;it enim linea vertica­<lb/>lis AC, &longs;itque globus in A æqualiter impul&longs;us per lineas AD & AB; </s> <s id="N192E1"><!-- NEW --><lb/>haud dubiè &longs;i anguli DAC, BAC &longs;int æquales: certè mobile feretur <lb/>per lineam verticalem AC, vt con&longs;tat ex dictis. </s> <s id="N192E8"><!-- NEW -->Re&longs;pondeo motum illum <lb/>e&longs;&longs;e violentum; e&longs;t enim à principio extrin&longs;eco, coque gemino, &longs;eu mix­<lb/>to, in quo non e&longs;t difficultas. </s> </p> <p id="N192F0" type="main"> <s id="N192F2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N192FF" type="main"> <s id="N19301"><!-- NEW --><emph type="italics"/>Motus violentus habet cau&longs;am<emph.end type="italics"/>; quia de nouo e&longs;t, & tandem de&longs;init per <lb/>hypoth. </s> <s id="N1930C">1. igitur habet cau&longs;am per Ax.8.l.1. </s> </p> <p id="N1930F" type="main"> <s id="N19311"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1931E" type="main"> <s id="N19320"><!-- NEW --><emph type="italics"/>I&longs;te motus &longs;upponit impetum<emph.end type="italics"/>; quia ni&longs;i e&longs;&longs;et impetus non e&longs;&longs;et natura­<lb/>liter motus per Th.18.l.1. </s> </p> <p id="N1932B" type="main"> <s id="N1932D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1933A" type="main"> <s id="N1933C"><!-- NEW --><emph type="italics"/>I&longs;te impetus debet e&longs;&longs;e in mobili projecto &longs;ur&longs;um<emph.end type="italics"/>; </s> <s id="N19345"><!-- NEW -->quia ibi e&longs;t cau&longs;a, vbi <lb/>e&longs;t effectus formalis, &longs;ed motus e&longs;t effectus formalis &longs;ecundarius impe­<lb/>tus per Th.15.l.1. igitur cum motus &longs;it in projecto &longs;ur&longs;um, in eo e&longs;t etiam <lb/>impetus: </s> <s id="N1934F"><!-- NEW -->præterea &longs;ecunda pars motus non ponitur à potentia motrice; <lb/>quia illa non e&longs;t applicata mobili cum ponitur noua pars motus, igitur <lb/>ab alia cau&longs;a applicata, &longs;ed nulla e&longs;t extrin&longs;eca, vt patet, nulla intrin&longs;eca <lb/>præter impetum. </s> </p> <p id="N19359" type="main"> <s id="N1935B"><!-- NEW -->Diceret aliquis ab aëre extrin&longs;ecùs ambiente mobile ip&longs;um propelli; </s> <s id="N1935F"><!-- NEW --><lb/>&longs;ed contra, nam aër, & omne aliud medium re&longs;i&longs;tit potiùs quàm iuuet, vt <lb/>demon&longs;trauimus l. <!-- REMOVE S-->&longs;ecundo Th. 1. Nec dicas fui&longs;&longs;e mentem Ari&longs;totelis, <lb/>cum nobiles Peripatetici contrâ &longs;entiant; </s> <s id="N1936A"><!-- NEW -->Albertus Magnus, Toletus, <lb/>Scaliger, Suarius, & recentiores; </s> <s id="N19370"><!-- NEW -->neque hoc negauit vnquam Ari&longs;tote-<pb pagenum="136" xlink:href="026/01/168.jpg"/>les, &longs;ed in hoc non multùm laboramus; nec dicas hinc &longs;equi motum <lb/>violentum e&longs;&longs;e à principio intrin&longs;eco contra def. </s> <s id="N1937B"><!-- NEW -->1. nam e&longs;t quidem à <lb/>principio intrin&longs;eco formali, non tamen à principio intrin&longs;eco mouen­<lb/>te vel agente; </s> <s id="N19383"><!-- NEW -->nec enim impetus e&longs;t cau&longs;a efficiens motus &longs;ui &longs;ubjecti; <lb/>&longs;ed cau&longs;a formalis vt &longs;æpè explicuimus. </s> </p> <p id="N19389" type="main"> <s id="N1938B"><!-- NEW -->Diceret fortè alius primam partem motus produci à potentiâ motri­<lb/>ce, &longs;ecundam verò ab entitate ip&longs;ius corporis; &longs;ed contrà; </s> <s id="N19391"><!-- NEW -->igitur corpus <lb/>e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; igitur &longs;emper produceret. </s> <s id="N19397"><!-- NEW -->Dices &longs;emper producere <lb/>&longs;i determinetur, &longs;ed contrà; à quo determinatur ad producendam &longs;ecun­<lb/>dam partem? </s> <s id="N1939F"><!-- NEW -->nihil e&longs;t enim applicatum, à quo determinari po&longs;&longs;it; </s> <s id="N193A3"><!-- NEW -->Dices <lb/>accepi&longs;&longs;e determinationem; &longs;ed contrà; quid e&longs;t illa determinatio? </s> <s id="N193A9"><!-- NEW --><lb/>Dices e&longs;&longs;e modum; </s> <s id="N193AE"><!-- NEW -->igitur permanentem; igitur e&longs;t cau&longs;a motus per Ax. <!-- REMOVE S--><lb/>1. l. <!-- REMOVE S-->1. n. </s> <s id="N193B7">1. igitur e&longs;t impetus per def. </s> <s id="N193BA"><!-- NEW -->3. l. <!-- REMOVE S-->1. Dices determinari à priori <lb/>parte motus; &longs;ed contrà primò, nam reuerâ non e&longs;t illa pars cum deter­<lb/>minatur corpus. </s> <s id="N193C4">Secundò, quid e&longs;t illa prima pars motus, ni&longs;i migratio è <lb/>loco in locum, quæ reuerâ à potentia motrice produci propriè non po­<lb/>te&longs;t per Th.2. l. <!-- REMOVE S-->1. &longs;ed de his iam fusè actum e&longs;t in toto ferè libro primo, <lb/>&longs;ed præ&longs;ertim in Th.6. </s> </p> <p id="N193CF" type="main"> <s id="N193D1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N193DE" type="main"> <s id="N193E0"><!-- NEW --><emph type="italics"/>Ille impetus e&longs;t vera qualitas Phy&longs;ica ab&longs;oluta<emph.end type="italics"/>; </s> <s id="N193E9"><!-- NEW -->hoc iam &longs;uprà demon­<lb/>&longs;tratum e&longs;t, &longs;cilicet phy&longs;icè; immò ex motu violento maximè probatur <lb/>dari impetum, & vix quidquam e&longs;t in rerum naturâ, quod clariùs euin­<lb/>cat aliquid de nouo produci. </s> </p> <p id="N193F3" type="main"> <s id="N193F5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N19402" type="main"> <s id="N19404"><!-- NEW --><emph type="italics"/>I&longs;te impetus producitur ab aliqua cau&longs;a<emph.end type="italics"/>; </s> <s id="N1940D"><!-- NEW -->Probatur, quia e&longs;t de nouo; </s> <s id="N19411"><!-- NEW -->igi­<lb/>tur non e&longs;t à &longs;e per Ax. 8. l. <!-- REMOVE S-->1. igitur e&longs;t ab alio; igitur ab aliqua <lb/>cau&longs;a. </s> </p> <p id="N1941B" type="main"> <s id="N1941D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N19429" type="main"> <s id="N1942B"><!-- NEW --><emph type="italics"/>Producitur ab aliqua cau&longs;a extrin&longs;eca<emph.end type="italics"/>; </s> <s id="N19434"><!-- NEW -->Probatur primò, quia aliquis <lb/>motus violentus e&longs;t à cau&longs;a extrin&longs;eca per def.1. Secundò, e&longs;t ab aliqua <lb/>cau&longs;a applicata, &longs;ed e&longs;t tantùm applicata potentia motrix; </s> <s id="N1943C"><!-- NEW -->igitur e&longs;t cau­<lb/>&longs;a, per Ax. 11. l. <!-- REMOVE S-->1. nec enim producitur hic impetus ab entitate corpo­<lb/>ris projecti, quod plu&longs;quàm certum e&longs;t ex dictis; hîc enim tantùm <lb/>e&longs;t quæ&longs;tio de illo motu, qui extrin&longs;ecùs aduenit, non vero de reflexo <lb/>&longs;ursùm, &c. </s> </p> <p id="N1944A" type="main"> <s id="N1944C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N19458" type="main"> <s id="N1945A"><!-- NEW --><emph type="italics"/>Producitur ab alio impetu<emph.end type="italics"/>; </s> <s id="N19463"><!-- NEW -->quia potentia motrix non agit ad extra ni&longs;i <lb/>per impetum productum in organo, vt patet; præterea &longs;i e&longs;t cau&longs;a vni­<lb/>uoca &longs;ufficiens applicata, non e&longs;t ponenda æquiuoca per Ax.11.l.1. adde <lb/>quod impetus producitur &longs;emper ad extra ab alio impetu per Th. 42. <lb/>l.1.nec in his hactenus propo&longs;itis vlla e&longs;t difficultas. </s> </p> <p id="N1946F" type="main"> <s id="N19471"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N1947D" type="main"> <s id="N1947F"><!-- NEW --><emph type="italics"/>Impetus impre&longs;&longs;us mobili &longs;ur&longs;um con&longs;eruatur per aliquod tempus<emph.end type="italics"/>; </s> <s id="N19488"><!-- NEW -->Probatur, <pb pagenum="137" xlink:href="026/01/169.jpg"/>quia mobile &longs;eparatum à potentia motrice adhuc mouetur per hyp.6.l.1, <lb/>igitur ille motus habet cau&longs;am, vt &longs;æpè dictum e&longs;t; </s> <s id="N19493"><!-- NEW -->non aliam, quàm im­<lb/>petum per Th.4. non productum de nouo, quippe nulla e&longs;t cau&longs;a mobili <lb/>applicata per Th. 7. & 8. igitur iam antè productam; igitur con&longs;er­<lb/>uatur. </s> </p> <p id="N1949D" type="main"> <s id="N1949F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N194AB" type="main"> <s id="N194AD"><!-- NEW --><emph type="italics"/>Con&longs;eruatur ab aliqua cau&longs;a extrin&longs;eca applicata<emph.end type="italics"/>; </s> <s id="N194B6"><!-- NEW -->vt patet ex dictis, non <lb/>ab aëre; </s> <s id="N194BC"><!-- NEW -->igitur à nullo corpore; </s> <s id="N194C0"><!-- NEW -->igitur ab alia causâ in&longs;en&longs;ibili; </s> <s id="N194C4"><!-- NEW -->igitur <lb/>illam e&longs;&longs;e oportet, & no&longs;&longs;e rerum omnium exigentias, & po&longs;&longs;e cuncta <lb/>producere; </s> <s id="N194CC"><!-- NEW -->quippe con&longs;eruatio e&longs;t repetita productio; </s> <s id="N194D0"><!-- NEW -->immò con&longs;erua­<lb/>re per actionem, per quam &longs;it res in tali loco, & tali tempore; </s> <s id="N194D6"><!-- NEW -->illa porrò <lb/>cau&longs;a in&longs;en&longs;ibilis incorporea, quæ vbique e&longs;t, & &longs;emper, Deus e&longs;t: Nec <lb/>puta po&longs;&longs;e exi&longs;tentiam cau&longs;æ primæ probari &longs;en&longs;ibiliori, vt &longs;ic loquar, <lb/>argumento, quàm eo, quod petitur ex motu projectorum, quorum motus <lb/>durat etiam&longs;i à potentia motrice mobile ip&longs;um &longs;it &longs;eparatum. </s> </p> <p id="N194E2" type="main"> <s id="N194E4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N194F0" type="main"> <s id="N194F2"><emph type="italics"/>Hinc multa colligi po&longs;&longs;unt.<emph.end type="italics"/></s> <s id="N194F9"> Primò, &longs;i nullus e&longs;&longs;et impetus extrin&longs;ecus, <lb/>vel acqui&longs;itus, nullus e&longs;&longs;et motus violentus, ni&longs;i tantùm motus reflexus <lb/>cadentium deorsùm. </s> <s id="N19500"><!-- NEW -->Secundò, &longs;i nullus e&longs;&longs;et Deus, nullus e&longs;&longs;et motus <lb/>violentus; immò nec vllus naturaliter acceleratus. </s> <s id="N19506">Tertiò, &longs;i impetus e&longs;­<lb/>&longs;et fluens vt motus, nullus e&longs;&longs;et motus violentus. </s> <s id="N1950B">Quartò, &longs;i &longs;ingulæ par­<lb/>tes motus produci debent ab aliquâ causâ efficiente, nullus etiam e&longs;&longs;et <lb/>motus violentus. </s> </p> <p id="N19512" type="main"> <s id="N19514"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N19520" type="main"> <s id="N19522"><!-- NEW --><emph type="italics"/>Vt &longs;it motus violentus debent produci plures partes impetus violenti <lb/>quàm &longs;int partes impetus naturalis<emph.end type="italics"/>; </s> <s id="N1952D"><!-- NEW -->Probatur, quia &longs;i e&longs;&longs;ent plures natura­<lb/>lis deorsùm, quàm &longs;int violenti &longs;ur&longs;um, corpus tenderet deor&longs;um; &longs;ed <lb/>tardiùs per Th.134.l.1. & &longs;i tot e&longs;&longs;ent vnius, quot alterius, mobile ip&longs;um <lb/>non moueretur per Th.133.l.1. </s> </p> <p id="N19537" type="main"> <s id="N19539"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N19545" type="main"> <s id="N19547"><!-- NEW --><emph type="italics"/>Motus violentus non e&longs;t acceleratus<emph.end type="italics"/>; probatur primò experientiâ, quæ <lb/>certa e&longs;t. </s> <s id="N19552"><!-- NEW -->Secundò, quia &longs;i &longs;emper cre&longs;ceret, numquam rediret mobile <lb/>contra hyp.1. nec enim ab vllo reflectitur; &longs;i enim reflecteretur ab aëre <lb/>inten&longs;us, multò magis remi&longs;&longs;us. </s> </p> <p id="N1955A" type="main"> <s id="N1955C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N19568" type="main"> <s id="N1956A"><emph type="italics"/>Hinc impetus in mobili &longs;ur&longs;um projecto non intenditur,<emph.end type="italics"/> quia non inten­<lb/>ditur effectus per Th.13. igitur nec cau&longs;a per Ax.2.l.2. </s> </p> <p id="N19574" type="main"> <s id="N19576"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N19582" type="main"> <s id="N19584"><!-- NEW --><emph type="italics"/>Motus violentus non e&longs;t æquabilis<emph.end type="italics"/>; </s> <s id="N1958D"><!-- NEW -->quia mobile tandem redit per hyp.1. <lb/>&longs;ed numquam rediret, &longs;i e&longs;&longs;et æquabilis; cur enim potiùs hoc in&longs;tanti <lb/>quàm alio? </s> <s id="N19595">cur ab hoc puncto &longs;patij potiùs, quàm ab alio? </s> </p> <pb pagenum="138" xlink:href="026/01/170.jpg"/> <p id="N1959C" type="main"> <s id="N1959E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N195AA" type="main"> <s id="N195AC"><!-- NEW --><emph type="italics"/>Hinc non con&longs;eruatur intactus impetus<emph.end type="italics"/>; </s> <s id="N195B5"><!-- NEW -->quia &longs;i e&longs;&longs;et intactus, e&longs;&longs;et &longs;em­<lb/>per æqualis; igitur haberet &longs;emper æqualem motum per Ax.3.l.2. igitur <lb/>motus e&longs;&longs;et æquabilis, contra Th.15. </s> </p> <p id="N195BD" type="main"> <s id="N195BF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N195CB" type="main"> <s id="N195CD"><!-- NEW --><emph type="italics"/>Hinc nece&longs;&longs;e e&longs;t aliquid impetus destrui<emph.end type="italics"/>; </s> <s id="N195D6"><!-- NEW -->cum enim non remaneat inta­<lb/>ctus, & æqualis; nec fiat maior per Th.14. certè fit minor, igitur detra­<lb/>ctione aliqua per Ax.1.l.2. </s> </p> <p id="N195DE" type="main"> <s id="N195E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N195EC" type="main"> <s id="N195EE"><!-- NEW --><emph type="italics"/>Singulis in&longs;tantibus aliquid de&longs;truitur impetus impre&longs;&longs;i<emph.end type="italics"/>; probatur quia <lb/>cur potiùs vno quam alio? </s> <s id="N195F9">quippe illa ratio, quæ probat de vno probat <lb/>de &longs;ingulis. </s> </p> <p id="N195FE" type="main"> <s id="N19600"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N1960C" type="main"> <s id="N1960E"><!-- NEW --><emph type="italics"/>Hinc nece&longs;&longs;ariè eadem vel aqualis cau&longs;a de&longs;tructionis debet e&longs;&longs;e applicata<emph.end type="italics"/>; <lb/>probatur, quia æqualis effectus æqualem cau&longs;am &longs;upponit, per Ax. <!-- REMOVE S--><lb/>3. l. <!-- REMOVE S-->2. <!-- KEEP S--></s> </p> <p id="N1961F" type="main"> <s id="N19621"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N1962D" type="main"> <s id="N1962F"><!-- NEW --><emph type="italics"/>Illa cau&longs;a non e&longs;t tantùm aër ambiens vt volunt aliqui<emph.end type="italics"/>; </s> <s id="N19638"><!-- NEW -->quia licèt re&longs;i­<lb/>&longs;tat motui, &longs;eu potius mobili, non tamen e&longs;t ea re&longs;i&longs;tentia, quæ po&longs;&longs;it <lb/>impetum tam citò de&longs;truere; </s> <s id="N19640"><!-- NEW -->probatur primò, quia &longs;i hoc e&longs;&longs;et, de&longs;true­<lb/>retur æquali tempore per omnem lineam &longs;ur&longs;um, quod e&longs;t contra expe­<lb/>rientiam, vt dicemus infrà; </s> <s id="N19648"><!-- NEW -->e&longs;&longs;et enim eadem cau&longs;a applicata; </s> <s id="N1964C"><!-- NEW -->igitur idem <lb/>& æqualis effectus; </s> <s id="N19652"><!-- NEW -->probatur &longs;ecundò, quia non de&longs;truit aër primum il­<lb/>lum gradum impetus naturalis acqui&longs;iti, vt con&longs;tat in motu deor&longs;um, qui <lb/>tamen e&longs;t imperfecti&longs;&longs;imus; igitur non e&longs;t &longs;ufficiens ad de&longs;truendum im­<lb/>petum violentum, ni&longs;i longo tempore. </s> <s id="N1965C"><!-- NEW -->Tertiò, globus &longs;ursùm projectus <lb/>a&longs;cendit, & deinde de&longs;cendit æquali tempore; </s> <s id="N19662"><!-- NEW -->igitur &longs;altem &longs;ingulis in­<lb/>&longs;tantibus de&longs;truitur vnus gradus impetus violenti æqualis primo gradui <lb/>innato; </s> <s id="N1966A"><!-- NEW -->atqui aër non pote&longs;t vno in&longs;tanti de&longs;truere impetum æqualem <lb/>primo innato; alioqui non intenderetur motus naturalis. </s> <s id="N19670"><!-- NEW -->Quartò, & hæc <lb/>e&longs;t ratio à priori, quotie&longs;cumque &longs;unt in eodem mobili duo impetus ad <lb/>oppo&longs;itas lineas determinati, pugnant pro rata, vt demon&longs;trauimus l.1. <lb/>Th. 149. 150. 152. & in toto Schol. <!-- REMOVE S-->& multis aliis pa&longs;&longs;im; atqui con&longs;er­<lb/>uatur &longs;emper impetus naturalis innatus per Sch. <!-- REMOVE S-->Th.152.n.6.l.1.per Th. <!-- REMOVE S--><lb/>9. & Schol.Th.14. & Th.73.l.2. </s> </p> <p id="N19683" type="main"> <s id="N19685"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N19691" type="main"> <s id="N19693"><!-- NEW --><emph type="italics"/>Illa cau&longs;a non e&longs;t entitas corporis mobilis, vel ip&longs;a grauitas, di&longs;tincta &longs;cili­<lb/>cet ab impetu innato &longs;i quæ e&longs;t de quæ alias,<emph.end type="italics"/> probatur, quia non e&longs;&longs;et potior <lb/>ratio cur vno in&longs;tanti de&longs;truerentur duo gradus impetus, quàm 3. 4. 5. <lb/>quippe grauitas exigeret de&longs;tructionem omnium: præterea omnis impe­<lb/>tus de&longs;truitur ne &longs;it fru&longs;trà per Schol, Th.152. & Th.162.l.1. denique &longs;i <pb pagenum="139" xlink:href="026/01/171.jpg"/>ade&longs;t contrarius impetus de&longs;tructiuus eo modo, quo explicuimus l. <!-- REMOVE S-->1. non <lb/>e&longs;t ponenda alia cau&longs;a de&longs;tructiua. </s> </p> <p id="N196AD" type="main"> <s id="N196AF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N196BB" type="main"> <s id="N196BD"><!-- NEW --><emph type="italics"/>Hinc nece&longs;&longs;e e&longs;t impetum violentum de&longs;trui ab impetu naturali innato<emph.end type="italics"/>; </s> <s id="N196C6"><!-- NEW -->pro­<lb/>batur, quia nulla e&longs;t cau&longs;a extrin&longs;eca de&longs;tructiua &longs;altem adæquatè per hT. <lb/>20.igitur e&longs;t intrin&longs;eca per Ax.4. l.2. &longs;ed intrin&longs;eca vel e&longs;t mobilis enti­<lb/>tas, vel grauitas, vel impetus innatus; </s> <s id="N196D0"><!-- NEW -->&longs;ed mobilis entitas non e&longs;t cau&longs;a <lb/>de&longs;tructiua; nec etiam ip&longs;a grauitas per Th.21. igitur impetus naturalis <lb/>innatus. </s> </p> <p id="N196D8" type="main"> <s id="N196DA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N196E6" type="main"> <s id="N196E8"><emph type="italics"/>Hinc vera ratio cur &longs;ingulis in&longs;tantibus aliquid de&longs;truatur,<emph.end type="italics"/> quia &longs;ingulis <lb/>in&longs;tantibus e&longs;t cau&longs;a de&longs;tructiua applicata, igitur &longs;ingulis in&longs;tantibus de­<lb/>&longs;truit per Ax. 12. l. <!-- REMOVE S-->1. <!-- KEEP S--></s> </p> <p id="N196F7" type="main"> <s id="N196F9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N19705" type="main"> <s id="N19707"><!-- NEW --><emph type="italics"/>Hinc etiam ratio cur &longs;ingulis instantibus, &longs;eu æqualibus temporibus æqua­<lb/>liter de&longs;truatur<emph.end type="italics"/>; </s> <s id="N19712"><!-- NEW -->quia &longs;ingulis in&longs;tantibus e&longs;t eadem cau&longs;a de&longs;tructiua ap­<lb/>plicata; igitur &longs;ingulis in&longs;tantibus æqualiter de&longs;truit per Ax.3.l.2.porrò <lb/>in tantum de&longs;truit in quantum efficit, vt aliquid &longs;it fru&longs;trà, vt fusè di­<lb/>ctum e&longs;t lib.1.vel in quantum exigit eius <expan abbr="de&longs;truction&etilde;">de&longs;tructionem</expan>, nam perinde e&longs;t. </s> </p> <p id="N19720" type="main"> <s id="N19722"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N1972E" type="main"> <s id="N19730"><!-- NEW --><emph type="italics"/>Hinc etiam petitur ratio, propter quam talis portio impetus violenti de­<lb/>&longs;truatur vne in&longs;tanti<emph.end type="italics"/>; quia &longs;cilicet contraria pugnant prorata per Ax.15. <lb/>& per Th.134.l.1. </s> </p> <p id="N1973D" type="main"> <s id="N1973F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N1974B" type="main"> <s id="N1974D"><!-- NEW --><emph type="italics"/>Hinc illa inuer&longs;a communis dicti, æqualibus temporibus æqualia de&longs;truun­<lb/>tur velocitatis momenta in motu violento<emph.end type="italics"/>; quippe eadem cau&longs;a eidem &longs;ub­<lb/>jecto applicata æqualibus temporibus æqualem effectum producit per <lb/>Ax.3.l.2. &longs;ed impetus innatus e&longs;t cau&longs;a de&longs;tructiua impetus violenti per <lb/>Th. 22. igitur æqualibus temporibus, &c. </s> </p> <p id="N1975E" type="main"> <s id="N19760"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N1976C" type="main"> <s id="N1976E"><!-- NEW --><emph type="italics"/>In eadem proportione retardatur motus violentus, in qua naturalis accele­<lb/>ratur<emph.end type="italics"/>: </s> <s id="N19779"><!-- NEW -->probatur quia &longs;ingulis in&longs;tantibus æqualibus acquiritur æqualis <lb/>gradus impetus, vt &longs;æpè dictum e&longs;t &longs;uprà; </s> <s id="N1977F"><!-- NEW -->atqui &longs;ingulis in&longs;tantibus de­<lb/>&longs;truitur vnus gradus impetus violenti per Th.24. &longs;ed ille gradus re&longs;pon­<lb/>det impetui innato per Th. 25. igitur æqualibus temporibus tantùm de­<lb/>&longs;truitur violenti, quantùm acquiritur naturalis; cum enim primo in­<lb/>&longs;tanti &longs;it impetus naturalis, & &longs;ecundo tempore æquali acquiratur æqua­<lb/>lis, item tertio, quarto, &c. </s> <s id="N1978D"><!-- NEW -->certè cum impetus innatus pugnet cum vio­<lb/>lento pro rata; </s> <s id="N19793"><!-- NEW -->nec &longs;it potior ratio cur maiorem portionem quàm mino­<lb/>rem de&longs;truat, æqualem certè de&longs;truit, itemque &longs;ecundo in&longs;tanti æqua­<lb/>lem, item tertio, quarto; igitur in eadem proportione decre&longs;cit violentus, <lb/>&longs;eu retardatur, in qua naturalis acceleratur. </s> </p> <pb pagenum="140" xlink:href="026/01/172.jpg"/> <p id="N197A1" type="main"> <s id="N197A3"><!-- NEW -->Hinc inuertenda e&longs;t progre&longs;&longs;ionis linea; </s> <s id="N197A7"><!-- NEW -->quippe linea AE repræ&longs;en­<lb/>tat nobis progre&longs;&longs;ionem motus accelerati, quæ fit in in&longs;tantibus, & li­<lb/>nea FK progre&longs;&longs;ionem motus, quæ fit in partibus temporis &longs;en&longs;ibilibus; </s> <s id="N197AF"><!-- NEW --><lb/>in illa primo in&longs;tanti decurritur primum &longs;patium AB, &longs;ecundo tempore <lb/>æquali BC, tertio CD, quarto DE: </s> <s id="N197B6"><!-- NEW -->in hac vero prima parte acquiritur <lb/>&longs;patium FG &longs;ecunda æquali primæ GH, tertia HI, quarta IK; </s> <s id="N197BC"><!-- NEW -->igitur &longs;i ac­<lb/>cipiatur linea AE, progrediendo ab A ver&longs;us E, vel linea FK progre­<lb/>diendo ab F ver&longs;us K habebitur progre&longs;&longs;io motus naturaliter accelerati; </s> <s id="N197C4"><!-- NEW --><lb/>&longs;i verò accipiatur EA, vel KF, progrediendo &longs;cilicet ab E ver&longs;us A, vel à <lb/>K ver&longs;us F, erit progre&longs;&longs;io motus violenti naturaliter retardati; </s> <s id="N197CB"><!-- NEW -->vt con­<lb/>&longs;tat ex præcedèntibus Theorematis; & quemadmodum progre&longs;&longs;io acce­<lb/>lerationis in in&longs;tantibus finitis fit iuxta &longs;eriem i&longs;torum numerorum 1.2. <lb/>3.4. in partibus verò temporis &longs;en&longs;ibilibus iuxta &longs;eriem i&longs;torum 1.3.5.7. <lb/>ita fit omninò progre&longs;&longs;io retardationis in in&longs;tantibus iuxta hos nume­<lb/>ros 4.3.2.1. in partibus temporis &longs;en&longs;ibilibus iuxta hos 7.5. 3. 1. <!-- KEEP S--></s> </p> <p id="N197DA" type="main"> <s id="N197DC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N197E8" type="main"> <s id="N197EA"><!-- NEW --><emph type="italics"/>Motus violentus durat tot in&longs;tantibus &longs;cilicet æquiualentibus quot &longs;unt ij <lb/>gradus impetus quibus violentus &longs;uperat innatum,<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;it vnus gradus im­<lb/>petus innati; </s> <s id="N197F9"><!-- NEW -->producantur 5. gradus violenti, quorum &longs;inguli &longs;int æqua­<lb/>les innato etiam <expan abbr="æquiual&etilde;ter">æquiualenter</expan>, motus durabit 4. in&longs;tantibus etiam æqui­<lb/>ualenter id e&longs;t 4. temporibus, quorum &longs;ingula erunt æqualia primo in­<lb/>&longs;tanti motus naturalis, probatur, cum &longs;ingulis in&longs;tantibus æqualibus de­<lb/>&longs;truatur vnus gradus; certè 4. in&longs;tantibus durat motus. </s> </p> <p id="N19809" type="main"> <s id="N1980B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N19817" type="main"> <s id="N19819"><!-- NEW --><emph type="italics"/>Si accipiantur &longs;patia æqualia in hac progre&longs;&longs;ione retardationis, e&longs;t inuer&longs;a <lb/>illius, quàm tribuimus &longs;uprà accelerationi, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus; </s> <s id="N19821"><!-- NEW --><lb/>tum &longs;i accipiantur &longs;patia æqualia prime &longs;patie quod decurritur prime in&longs;tan­<lb/>ti metus naturalis, tum &longs;i accipiantur &longs;patia æqualia date &longs;patie quod in par­<lb/>te temporis &longs;en&longs;ibili percurritur<emph.end type="italics"/>; </s> <s id="N1982D"><!-- NEW -->quippe quemadmodum in progre&longs;&longs;ione <lb/>accelerationis decre&longs;cunt tempora; </s> <s id="N19833"><!-- NEW -->&longs;ic in progre&longs;&longs;ione retardationis <lb/>cre&longs;cunt, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus; quare ne iam dicta hic re­<lb/>petam, con&longs;ule quæ diximus lib.2. de hac progre&longs;&longs;ione. </s> </p> <p id="N1983B" type="main"> <s id="N1983D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N19849" type="main"> <s id="N1984B"><!-- NEW --><emph type="italics"/>Hinc instantia initio huius metus &longs;unt minora &longs;icut initio motus naturalis <lb/>&longs;unt maiora; </s> <s id="N19853"><!-- NEW -->& &longs;ub finem in motu violente &longs;unt maiora, in naturali &longs;unt mi­<lb/>nora<emph.end type="italics"/>; </s> <s id="N1985C"><!-- NEW -->quia &longs;cilicet hic acceleratur, ille retardatur: </s> <s id="N19860"><!-- NEW -->igitur velo­<lb/>citas accelerati cre&longs;cit; </s> <s id="N19866"><!-- NEW -->igitur &longs;i accipiantur &longs;patia æqualia, decre&longs;cit tem­<lb/>pus; </s> <s id="N1986C"><!-- NEW -->at verò velocitas retardati decre&longs;cit, igitur a&longs;&longs;umptis &longs;patiis æquali­<lb/>bus, cre&longs;cit tempus; </s> <s id="N19872"><!-- NEW -->igitur &longs;i accipiatur &longs;patium, quod percurritur primo <lb/>in&longs;tanti huius motus, & deinde alia huic æqualia; </s> <s id="N19878"><!-- NEW -->haud dubiè, cum &longs;e­<lb/>cundo in&longs;tanti motus &longs;it tardior, &longs;itque a&longs;&longs;umptum æquale &longs;patium; haud <lb/>dubiè inquam in&longs;tans &longs;ecundum erit maius primo, & tertium &longs;ecundo, <lb/>atque ita deinceps. </s> </p> <pb pagenum="141" xlink:href="026/01/173.jpg"/> <p id="N19886" type="main"> <s id="N19888"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N19894" type="main"> <s id="N19896"><!-- NEW --><emph type="italics"/>Hinc primo in&longs;tanti motus violenti de&longs;truitur minor gradus impetus quàm <lb/>&longs;ecundo,<emph.end type="italics"/> quod demon&longs;tro; </s> <s id="N198A1"><!-- NEW -->quia eadem cau&longs;a breuiore tempore minùs agit <lb/>per Ax.3.l.2. & Ax. 13.l.1. num.4. igitur minùs impetus de&longs;truitur pri­<lb/>mo, quàm &longs;ecundo, & minùs &longs;ecundo quàm tertio, atque ita deinceps; <lb/>idem enim dici debet de cau&longs;a de&longs;tructiua, quod de productiua. </s> </p> <p id="N198AB" type="main"> <s id="N198AD">Dices, igitur idem impetus de&longs;truitur primo in&longs;tanti, quo e&longs;t, &longs;i de&longs;trui­<lb/>tur primo in&longs;tanti motus. </s> <s id="N198B2">Re&longs;pondeo negando; quia primo in&longs;tanti, quo <lb/>e&longs;t impetus, non e&longs;t motus per Th.34.l.1. <!-- KEEP S--></s> </p> <p id="N198B8" type="main"> <s id="N198BA"><!-- NEW -->Dices, igitur impetus ille e&longs;t fru&longs;trà, quia nullus effectus, &longs;eu motus <lb/>ex eo &longs;equitur; Re&longs;pondeo negando; nam omnes gradus impetus qui ei­<lb/>dem parti mobilis in&longs;unt, communi qua&longs;i actione, vel exigentia indi­<lb/>ui&longs;ibiliter exigunt motum. </s> </p> <p id="N198C4" type="main"> <s id="N198C6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N198D2" type="main"> <s id="N198D4"><!-- NEW --><emph type="italics"/>Hinc gradus omnes producti in eadem parte &longs;ubiecti &longs;unt inæquales in­<lb/>perfectione<emph.end type="italics"/>; </s> <s id="N198DF"><!-- NEW -->cum enim &longs;inguli &longs;ingulis in&longs;tantibus de&longs;truantur, vt dictum <lb/>e&longs;t; quippe e&longs;t tantùm vnus gradus impetus innati, & cum &longs;ingula in­<lb/>&longs;tantia &longs;int inæqualia, etiam &longs;inguli gradus illius impetus &longs;unt inæquales <lb/>in perfectione. </s> </p> <p id="N198E9" type="main"> <s id="N198EB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N198F7" type="main"> <s id="N198F9"><!-- NEW --><emph type="italics"/>Hinc redditur optima ratio, cur tot producantur potiùs quàm plures, quæ <lb/>alioquin minimè afferri pote&longs;t<emph.end type="italics"/>; </s> <s id="N19904"><!-- NEW -->immò, ni&longs;i hoc e&longs;&longs;et, nulla e&longs;&longs;et huiu&longs;modi <lb/>naturalis retardatio; nam producantur, &longs;i fieri pote&longs;t, omnes æquales, &longs;int­<lb/>que v.g.20. nunquid po&longs;&longs;unt e&longs;&longs;e 40. perfectionis &longs;ubduplæ, vel 10. du­<lb/>plæ, vel 5. quadruplæ &c. </s> <s id="N1990E">cur autem potiùs vnum dices quàm aliud? </s> <s id="N19911"><!-- NEW -->at <lb/>verò optimam inde reddo rationem quòd cum &longs;int omnes inæquales, cò <lb/>plures &longs;unt, quò maior e&longs;t ni&longs;us; pauciores verò, quò minor. </s> </p> <p id="N19919" type="main"> <s id="N1991B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N19927" type="main"> <s id="N19929"><emph type="italics"/>Hinc &longs;unt inæquales in eâdem proportione, in quæ in&longs;tantia &longs;unt inæqualia<emph.end type="italics"/><lb/>v. </s> <s id="N19932">g. <!-- REMOVE S-->quà proportione primum in&longs;tans e&longs;t minus &longs;ecundo, & &longs;ecundum <lb/>tertio, ita ille gradus impetus, qui de&longs;truitur primo in&longs;tanti, e&longs;t minor <lb/>vel imperfectior co, qui de&longs;truitur &longs;ecundo, & qui de&longs;truitur &longs;ecundo <lb/>imperfectior co, qui de&longs;truitur tertio, atque ita deinceps. </s> </p> <p id="N1993D" type="main"> <s id="N1993F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N1994B" type="main"> <s id="N1994D"><!-- NEW --><emph type="italics"/>Hinc perfecti&longs;&longs;imus omnium graduum ille e&longs;t qui de&longs;truitur vltimo in&longs;tan­<lb/>ti, de quo infrá<emph.end type="italics"/>; </s> <s id="N19958"><!-- NEW -->quod &longs;equitur ex dictis nece&longs;&longs;ariò: vtrùm verò ille &longs;it æ­<lb/>qualis omninò in perfectione impetui naturali innato, dicemus <lb/>infrà. </s> </p> <p id="N19960" type="main"> <s id="N19962"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1996E" type="main"> <s id="N19970"><!-- NEW -->Hic ob&longs;eruabis mirabilem &longs;anæ naturæ prouidentiam, quæ motus <lb/>omnes cum ip&longs;o naturali ita compo&longs;uit, vt &longs;it veluti regula omnium mo­<lb/>tuum, &longs;itque vnum qua&longs;i principium perfectionis totius impetus; </s> <s id="N19978"><!-- NEW -->tùm in <pb pagenum="142" xlink:href="026/01/174.jpg"/>motu naturali, in cuius progre&longs;&longs;ione producitur &longs;emper imperfectior, <lb/>tùm in violento, in cuius progre&longs;&longs;ione de&longs;truitur &longs;emper perfectior; </s> <s id="N19983"><!-- NEW --><lb/>producitur imperfectior ab eadem cau&longs;a in minoribus temporibus, & <lb/>de&longs;truitur perfectior ab eadem cau&longs;a in maioribus temporibus; </s> <s id="N1998A"><!-- NEW -->& cum <lb/>impetus innatus &longs;it cau&longs;a de&longs;tructiua impetus violenti, habet inæqualem <lb/>proportionem cum &longs;uo effectu pro temporibus inæqualibus; </s> <s id="N19992"><!-- NEW -->& cum <lb/>idem impetus innatus &longs;it qua&longs;i principium crementi, vel accelerationis, <lb/>&longs;icut e&longs;t principium retardationis; </s> <s id="N1999A"><!-- NEW -->certè pro inæqualitate temporum e&longs;t <lb/>diuer&longs;a proportio crementorum; quo nihil clarius in hac materia meo <lb/>iudicio dici pote&longs;t. </s> </p> <p id="N199A2" type="main"> <s id="N199A4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N199B0" type="main"> <s id="N199B2"><!-- NEW --><emph type="italics"/>Hinc finis motus naturalis omninò conuenit cum principio motus violenti; </s> <s id="N199B8"><!-- NEW --><lb/>& finis huius cum principio illius<emph.end type="italics"/>; quæcumque tandem progre&longs;&longs;io accipia­<lb/>tur; </s> <s id="N199C2"><!-- NEW -->&longs;iue temporum æqualium in &longs;patiis inæqualibus; &longs;iue &longs;patio­<lb/>rum æqualium in temporibus inæqualibus, &longs;iue a&longs;&longs;umantur in&longs;tan­<lb/>tia in progre&longs;&longs;ione arithmetica &longs;implici iuxta hos numeros 1.2.3.4. &longs;iue <lb/>a&longs;&longs;umantur temporis partes &longs;en&longs;ibiles in progre&longs;&longs;ione Galilei iuxta hos <lb/>numeros 1.3.5.7. quæ omnia ex dictis nece&longs;&longs;ariò con&longs;equuntur. </s> </p> <p id="N199CE" type="main"> <s id="N199D0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N199DC" type="main"> <s id="N199DE"><!-- NEW --><emph type="italics"/>Nec modò conuenit principium vnius cum alterius fine, & vici&longs;&longs;im, &longs;ed <lb/>etiam aliæ partes motus in di&longs;tantiis æqualibus<emph.end type="italics"/> &longs;it enim linea AG, quam <lb/>percurrit mobile demi&longs;&longs;um ex puncto A deor&longs;um motu naturaliter ac­<lb/>celerato, & moueatur per 6. in&longs;tantia, &longs;eu 6. tempora æqualia: </s> <s id="N199ED"><!-- NEW -->Primo <lb/>in&longs;tanti, quo percurrit &longs;patium AB; </s> <s id="N199F3"><!-- NEW -->haud dubiè, quando peruenit ad pun­<lb/>ctum G, habet 7. gradus impetus æquales, quia ante motum AB habebat <lb/>innatum; </s> <s id="N199FB"><!-- NEW -->&longs;ed in motu illo fluunt 6. tempora æqualia, vt dictum e&longs;t; </s> <s id="N199FF"><!-- NEW -->igitur <lb/>6. acquirit gradus impetus, quorum quidem vltimò acqui&longs;itus nullum <lb/>adhuc habuit motum; </s> <s id="N19A07"><!-- NEW -->&longs;ed haud dubiè haberet, &longs;i vlteriùs hic motus pro­<lb/>pagaretur: </s> <s id="N19A0D"><!-- NEW -->his po&longs;itis imprimantur mobili in O 7.gradus impetus æqua­<lb/>les prioribus &longs;ursùm motu violento, per lineam OH; </s> <s id="N19A13"><!-- NEW -->certè primo in&longs;tan­<lb/>ti motus, &longs;eu tempore æquali prioribus percurret ON, id e&longs;t 6. &longs;patiola; </s> <s id="N19A19"><!-- NEW --><lb/>quia licèt &longs;int 7.gradus; </s> <s id="N19A1E"><!-- NEW -->attamen impetus innatus corporis grauis detra­<lb/>hit vnum &longs;patium, &longs;imulque de&longs;truit vnum gradum, &longs;ecundo tempore <lb/>percurret NM 5. tertio ML 4. quarto LK 3. quinto KI 2. &longs;exto IH 1. <lb/>igitur primum violenti ON re&longs;pondet vltimo naturali FG &longs;eu &longs;ecun­<lb/>dum illius quinto huius, tertium illius quarto huius, quartum tertio, <lb/>quintum &longs;ecundo &longs;extum primo, & vici&longs;&longs;im; idem pror&longs;us in progre&longs;&longs;ione <lb/>Galilei accidit, a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus. </s> </p> <p id="N19A2E" type="main"> <s id="N19A30"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N19A3C" type="main"> <s id="N19A3E"><!-- NEW --><emph type="italics"/>Hinc ad eam altitudinem a&longs;cendit motu violento cum iis gradibus impe­<lb/>tus, quos habuit ab eadem altitudine decidens motu naturali<emph.end type="italics"/>; con&longs;tat ex <lb/>dictis. </s> </p> <pb pagenum="143" xlink:href="026/01/175.jpg"/> <p id="N19A4F" type="main"> <s id="N19A51"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N19A5D" type="main"> <s id="N19A5F"><!-- NEW --><emph type="italics"/>Hinc &longs;i motus violentus, & naturalis durent æqualibus temporibus, &longs;patia <lb/>vtriu&longs;que erunt æqualia<emph.end type="italics"/>; </s> <s id="N19A6A"><!-- NEW -->con&longs;tat etiam ex dictis v.g. <!-- REMOVE S-->corpus graue, motu <lb/>naturali in libero aëre tempore duorum &longs;ecundorum percurrit 48. pe­<lb/>des, igitur &longs;i moueatur &longs;ur&longs;um æquali tempore percurret 48. pedes per <lb/>&longs;e, dico per &longs;e; quippe ratione figuræ corporis &longs;ecus accidere pote&longs;t, vt <lb/>plurimùm etiam accedit ratione motus mixti ex motu centri recto, & <lb/>motu orbis circulari, de quo infrà. </s> </p> <p id="N19A7A" type="main"> <s id="N19A7C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N19A88" type="main"> <s id="N19A8A"><emph type="italics"/>Hinc, vt &longs;patia vtroque motu diuer&longs;a &longs;unt æqualia, ita tempora quibus de­<lb/>curruntur &longs;unt æqualia,<emph.end type="italics"/> & impetus acqui&longs;itus in fine naturalis cum in­<lb/>nato e&longs;t æqualis impetui producta in principio violenti. </s> </p> <p id="N19A96" type="main"> <s id="N19A98"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N19AA4" type="main"> <s id="N19AA6"><!-- NEW --><emph type="italics"/>Hinc tandiu durat de&longs;cen&longs;us mobilis proiecti &longs;ursùm motu violento, quan­<lb/>diu durat eiu&longs;dem a&longs;cen&longs;us, & tot habet gradus impetus in fine de&longs;cen&longs;us, <lb/>quot habet in principio a&longs;cen&longs;us<emph.end type="italics"/>; </s> <s id="N19AB3"><!-- NEW -->e&longs;t enim æquale &longs;patium; </s> <s id="N19AB7"><!-- NEW -->igitur æquale <lb/>tempus; igitur æqualis vtrobique impetus. </s> <s id="N19ABD"><!-- NEW -->Sed hîc duo obiici po&longs;&longs;unt, <lb/>primò &longs;agittam per lineam verticalem vibratam po&longs;ui&longs;&longs;e tantùm in a&longs;­<lb/>cen&longs;u 3. &longs;ecunda, in de&longs;cen&longs;u verò 5. vt &longs;æpiùs ob&longs;eruatum e&longs;t, te&longs;te Mer­<lb/>&longs;enno; </s> <s id="N19AC7"><!-- NEW -->&longs;ecundò, &longs;i eodem tempore corpus graue &longs;ursùm proiectum motu <lb/>violento a&longs;cenderet, quo deinde de&longs;cendit, in fine de&longs;cen&longs;us æqualis <lb/>e&longs;&longs;et ictus, &longs;eu percu&longs;&longs;io vtriu&longs;que; cum tamen illa &longs;it maior, quæ infli­<lb/>gitur motu violento, vt con&longs;tat multis experimentis. </s> </p> <p id="N19AD1" type="main"> <s id="N19AD3"><!-- NEW -->Re&longs;pondeo ad primum etiam te&longs;te Mer&longs;enno globum ferreum trium <lb/>aut 4. librarum &longs;ur&longs;um explo&longs;um è breuiore tormento &longs;ed latiore, æqua­<lb/>le tempus in a&longs;cen&longs;u, & in de&longs;cen&longs;u in&longs;ump&longs;i&longs;&longs;e; </s> <s id="N19ADB"><!-- NEW -->quod reuerâ &longs;ecùs acci­<lb/>dit &longs;agittæ, cuius differentia a&longs;cen&longs;us, & de&longs;cen&longs;us &longs;en&longs;u etiam percipi <lb/>pote&longs;t; </s> <s id="N19AE3"><!-- NEW -->tùm quia lignea materia multò leuior e&longs;t ferro, tùm quia leui&longs;&longs;i­<lb/>mæ illæ pennæ, quibus in&longs;truitur, motum retardant in de&longs;cen&longs;u; </s> <s id="N19AE9"><!-- NEW -->quod <lb/>maximè confirmatur ex eo quod pluma facilè anhelitu &longs;ur&longs;um pellatur <lb/>&longs;atis veloci motu, quæ deinde tardi&longs;&longs;imo &longs;ua &longs;ponte de&longs;cendit: </s> <s id="N19AF1"><!-- NEW -->præterea <lb/>mucro ferreus, quo &longs;agitta armatur, &longs;emper præire debet, cuius rei ratio­<lb/>nem afferemus infrà; </s> <s id="N19AF9"><!-- NEW -->igitur cum in a&longs;cen&longs;u præeat, vt præeat in de&longs;cen­<lb/>&longs;u, altera extremitas &longs;emicirculum &longs;uo motu facere debet, qui certè ad <lb/>naturalem motum pertinet, altera tamen extremitas, quæ mouetur mo­<lb/>tu contrario alterius motum retardat; ad &longs;ecundam obiectionem <lb/>re&longs;pondebo Th.44. </s> </p> <p id="N19B05" type="main"> <s id="N19B07"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N19B13" type="main"> <s id="N19B15"><!-- NEW --><emph type="italics"/>Si motus violentus e&longs;&longs;et æquabilis, &longs;patium e&longs;&longs;et ferè duplum illius, quod <lb/>percurritur motu naturaliter retardato, a&longs;&longs;umptis &longs;cilicet <expan abbr="t&etilde;poribus">temporibus</expan> æqualibus<emph.end type="italics"/>; </s> <s id="N19B24"><!-- NEW --><lb/>cum enim motu æquabili compo&longs;ito ex &longs;ubdupla velocitate maximæ, & <lb/>minimæ motus accelerati æquali tempore percurratur æquale &longs;patium, <lb/>&longs;ubduplum minimæ pro nihilo ferè habetur; </s> <s id="N19B2D"><!-- NEW -->igitur pote&longs;t tantùm a&longs;&longs;u-<pb pagenum="144" xlink:href="026/01/176.jpg"/>mi &longs;ubduplum maximæ; igitur velocitas motus &longs;it æqualis maximæ, haud <lb/>dubiè &longs;patium duplum percurretur. </s> </p> <p id="N19B38" type="main"> <s id="N19B3A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N19B46" type="main"> <s id="N19B48"><!-- NEW --><emph type="italics"/>Hinc benè à naturâ in&longs;titutum fuit impetum naturalem innatum &longs;emper <lb/>con&longs;eruari<emph.end type="italics"/>; </s> <s id="N19B53"><!-- NEW -->alioqui violentus e&longs;&longs;et æquabilis, igitur nunquam de&longs;ineret: <lb/>quantum ab&longs;urdum! quale incommodum &c. </s> </p> <p id="N19B59" type="main"> <s id="N19B5B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N19B67" type="main"> <s id="N19B69"><!-- NEW --><emph type="italics"/>Eadem e&longs;t ratio &longs;eu proportio ictuum, & percu&longs;&longs;ionum, quæ integrorum <lb/>&longs;patiorum quæ &longs;cilicet toto motu percurruntur in a&longs;cen&longs;u & de&longs;cen&longs;u,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->corpus graue cadens ex data altitudine 48 pedum æqualem ictum infli­<lb/>git in fine de&longs;cen&longs;us, & in principio a&longs;cen&longs;us, quo &longs;cilicet ad <expan abbr="eãdem">eandem</expan> <lb/>altitudinem a&longs;cenderet; </s> <s id="N19B81"><!-- NEW -->probatur, quia æqualis acquiritur impetus in <lb/>de&longs;cen&longs;u alteri, qui de&longs;truitur in a&longs;cen&longs;u, a&longs;&longs;umptis dumtaxat &longs;patiis illis <lb/>æqualibus; </s> <s id="N19B89"><!-- NEW -->igitur æqualis e&longs;t in fine de&longs;cen&longs;us, in quo e&longs;t totus acqui&longs;i­<lb/>tus, atque in principio a&longs;cen&longs;us, in quo nullus e&longs;t de&longs;tructus: </s> <s id="N19B8F"><!-- NEW -->ad id verò, <lb/>quod dicebatur &longs;uprà de &longs;agitta, cuius ictus maior e&longs;t initio a&longs;cen&longs;us, <lb/>quàm in fine de&longs;cen&longs;us non diffiteor; </s> <s id="N19B97"><!-- NEW -->quia materia &longs;agittæ, tùm lignea <lb/>tùm plumea motum &longs;atis &longs;uperque retardat, vt differentia ictuum &longs;en&longs;u <lb/>ip&longs;o percipi po&longs;&longs;it; quæ tamen nulla perciperetur in a&longs;cen&longs;u de&longs;cen&longs;u­<lb/>que globi ferrei. </s> </p> <p id="N19BA1" type="main"> <s id="N19BA3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N19BAF" type="main"> <s id="N19BB1"><!-- NEW --><emph type="italics"/>Hinc reiicies Galileum, & alios eius &longs;ectatores qui volunt impetum corpori <lb/>impre&longs;&longs;um de&longs;trui tantùm ab aëre<emph.end type="italics"/>; </s> <s id="N19BBC"><!-- NEW -->quod plu&longs;quàm fal&longs;um e&longs;&longs;e comper­<lb/>tum e&longs;t, vt demon&longs;trauimus &longs;uprà Th. 20. qua&longs;i verò non ad&longs;it aliqua <lb/>cau&longs;a nece&longs;&longs;aria de&longs;tructiua, &longs;cilicet impetus innatus; </s> <s id="N19BC4"><!-- NEW -->hinc etiam eum­<lb/>dem reiicies, qui vult numquam fieri po&longs;&longs;e, vt motu naturaliter accelera­<lb/>to tanta acquiratur velocitas, quanta imprimitur in motu violento; </s> <s id="N19BCC"><!-- NEW -->vult <lb/>enim motum acceleratum tran&longs;ire in æquabilem, cuius contrarium de­<lb/>mon&longs;trauimus &longs;uprà Th. 131, l. <!-- REMOVE S-->2. igitur cum cre&longs;cat &longs;emper velocitas, <lb/>nullus e&longs;t finitus gradus, quem tandem non a&longs;&longs;equatur; immò vt dictum <lb/>e&longs;t in præcedenti Th. a&longs;&longs;umptis æqualibus &longs;patiis, impetus, qui e&longs;t in <lb/>principio a&longs;cen&longs;us, æqualis e&longs;t cum eo, qui e&longs;t in fine de&longs;cen&longs;us. </s> </p> <p id="N19BDC" type="main"> <s id="N19BDE"><!-- NEW -->Diceret fortè aliquis cadentem globum ex alti&longs;&longs;imæ turris apice de­<lb/>clinare à perpendiculari antequam terram feriat, vt con&longs;tat ex multis <lb/>experimentis; </s> <s id="N19BE6"><!-- NEW -->igitur præualet tandem re&longs;i&longs;tentia aëris: </s> <s id="N19BEA"><!-- NEW -->&longs;ed re&longs;pondeo id <lb/>tantùm accidere propter currentem illac aëris tractum; alioquin non <lb/>e&longs;&longs;et potiùs ratio, cur in vnam partem declinaret, quàm in aliam. </s> </p> <p id="N19BF2" type="main"> <s id="N19BF4"><emph type="center"/><emph type="italics"/>Theoroma<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N19C00" type="main"> <s id="N19C02"><!-- NEW --><emph type="italics"/>Non e&longs;t eadem ratio ictuum, &longs;eu percu&longs;&longs;ionum, quæ e&longs;t &longs;egmentorum in­<lb/>tegri &longs;patij<emph.end type="italics"/>; </s> <s id="N19C0D"><!-- NEW -->v.g. <!-- REMOVE S-->in &longs;ubduplo &longs;patij &longs;egmento non e&longs;t &longs;ubduplus ictus, &longs;it <lb/> enim &longs;patium integrum motus vîolenti OH, & principium motus &longs;it <lb/>in O, finis in H; </s> <s id="N19C17"><!-- NEW -->accipiatur &longs;egmentum OM, quod e&longs;t qua&longs;i &longs;ubduplum O <lb/>H, ictus in M non e&longs;t profectò &longs;ubduplus ictus in O, &longs;ed tantùm in L, vt <pb pagenum="145" xlink:href="026/01/177.jpg"/>con&longs;tat ex dictis; igitur rationes ictuum non &longs;unt, vt rationes &longs;egmen­<lb/>torum integri &longs;patij. </s> </p> <p id="N19C24" type="main"> <s id="N19C26"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N19C32" type="main"> <s id="N19C34"><!-- NEW --><emph type="italics"/>Vt in praxi determinentur rationes ictuum<emph.end type="italics"/>; </s> <s id="N19C3D"><!-- NEW -->a&longs;&longs;umatur progre&longs;&longs;io Gali­<lb/>lei in AF, ita vt &longs;i prima parte temporis &longs;en&longs;ibili percurratur &longs;patium <lb/>FE 9 partium æqualium; </s> <s id="N19C45"><!-- NEW -->&longs;ecunda percurratur ED. 7. partium, tertia <lb/>DC 5. quarta CB 3; </s> <s id="N19C4B"><!-- NEW -->quinta BA 1. hoc po&longs;ito facilè erit determinare <lb/>rationes ictuum; </s> <s id="N19C51"><!-- NEW -->nam in de&longs;cen&longs;u ictus &longs;unt vt velocitates, & hæ vt tem­<lb/>pora; </s> <s id="N19C57"><!-- NEW -->igitur &longs;i AB percurritur in dato tempore, & AC in duobus prio­<lb/>ri æqualibus; </s> <s id="N19C5D"><!-- NEW -->certè ictus in de&longs;cen&longs;u AC e&longs;t duplus ictus in de&longs;cen&longs;u <lb/>AB; in AD triplus, &c. </s> <s id="N19C63">Igitur in a&longs;cen&longs;u ictus in F erit quintuplus, <lb/>ictus in E quadruplus in D triplus, &c. </s> <s id="N19C68">igitur ictus &longs;unt in ratione dupli­<lb/>cata &longs;patiorum facto &longs;patij initio à &longs;ummo puncto A. <!-- KEEP S--></s> </p> <p id="N19C6E" type="main"> <s id="N19C70"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N19C7C" type="main"> <s id="N19C7E"><!-- NEW --><emph type="italics"/>Hinc cognitis viribus, quibus corpus graue proijcitur ad datam altitudi­<lb/>nem, cogno&longs;ci po&longs;&longs;unt vires, quibus ad aliam quamcumque proijciatur<emph.end type="italics"/>; </s> <s id="N19C89"><!-- NEW -->v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->proiiciatur corpus graue ad altitudinem 48. pedum; </s> <s id="N19C92"><!-- NEW -->vires &longs;unt iis æqua­<lb/>les, quas acquirit in de&longs;cen&longs;u eiu&longs;dem altitudinis 48. pedum; </s> <s id="N19C98"><!-- NEW -->&longs;it alia di­<lb/>&longs;tantia 100. pedum; haud dubiè vires nece&longs;&longs;ariæ ad motum hunc violen­<lb/>tum &longs;unt æquales iis, quas acquireret in de&longs;cen&longs;u 100. pedum per Th. <!-- REMOVE S--><lb/>40. atqui ita &longs;e habent vires acqui&longs;itæ in de&longs;cen&longs;u 48. pedum ad vires <lb/>acqui&longs;itas in de&longs;cen&longs;u 100. vt v.g. <!-- REMOVE S-->48. ad v.g. <!-- REMOVE S-->100. id e&longs;t ferè vt 7. <lb/>ad 10. </s> </p> <p id="N19CAB" type="main"> <s id="N19CAD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N19CB9" type="main"> <s id="N19CBB"><!-- NEW --><emph type="italics"/>Cognitis etiam &longs;patiis cogno&longs;cetur tempus<emph.end type="italics"/>; </s> <s id="N19CC4"><!-- NEW -->&longs;it enim decur&longs;um idem &longs;pa­<lb/>tium 48. pedum motu violento &longs;ur&longs;um; </s> <s id="N19CCA"><!-- NEW -->idque v. <!-- REMOVE S-->g. <!-- REMOVE S-->tempore 2. &longs;ecundo­<lb/>rum, quod ferè cum experientia con&longs;entit; </s> <s id="N19CD4"><!-- NEW -->&longs;it aliud &longs;patium 100. tempus <lb/>primi motus e&longs;t ad tempus &longs;ecundi vt v. <!-- REMOVE S-->g. <!-- REMOVE S-->48. ad v. <!-- REMOVE S-->g. <!-- REMOVE S-->100. quia &longs;patia <lb/>&longs;unt vt quadrata temporum; </s> <s id="N19CE4"><!-- NEW -->igitur tempora vt radices 4. hinc vires &longs;unt <lb/>in ratione temporum; </s> <s id="N19CEA"><!-- NEW -->quia vt temporibus æqualibus acquiruntur æqua­<lb/>lia velocitatis momenta in motu naturali, ita & de&longs;truuntur æqualia in <lb/>motu violento, quæ omnia con&longs;tant; igitur ictus &longs;unt vt vires, vires vt <lb/>tempora, tempora denique, vt radices <expan abbr="q.">que</expan> &longs;patiorum. </s> </p> <p id="N19CF8" type="main"> <s id="N19CFA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 150.<emph.end type="center"/></s> </p> <p id="N19D06" type="main"> <s id="N19D08"><!-- NEW --><emph type="italics"/>In vltimo contactu motus violenti nullus e&longs;t ictus, v. <!-- REMOVE S-->g. <!-- REMOVE S-->mobile projectum <lb/>&longs;ur&longs;um<emph.end type="italics"/> <emph type="italics"/>per lineam<emph.end type="italics"/> FA <emph type="italics"/>nullam percu&longs;&longs;ionem infligeret in<emph.end type="italics"/> A; </s> <s id="N19D23"><!-- NEW -->probatur <lb/>quia non tendit vlteriùs; </s> <s id="N19D29"><!-- NEW -->igitur non impeditur eius motus à &longs;uperficie <lb/>corporis terminati ad punctum A; igitur nullum impetum in eo produ­<lb/>cit, qui tantùm producitur ad tollendum impedimentum per Th.44.l.1. <lb/>igitur nullum ictum infligit, qui tantùm infligitur per impetum, vt <lb/>con&longs;tat. </s> </p> <p id="N19D35" type="main"> <s id="N19D37"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N19D43" type="main"> <s id="N19D45"><emph type="italics"/>Ex his &longs;atis facilè comparari po&longs;&longs;unt rationes percu&longs;&longs;ionis,<emph.end type="italics"/> quæ infliguntur <pb pagenum="146" xlink:href="026/01/178.jpg"/>tùm ex ca&longs;u corporis grauis cadentis, tùm ex vi mallei impacti, tùm ex <lb/>impetu corporis projecti, tùm ex grauitatione corporis grauis incum­<lb/>bentis, quæ omnia hîc fu&longs;iùs e&longs;&longs;ent tractanda, ni&longs;i locum proprium infrà <lb/>&longs;ibi vendicarent. </s> </p> <p id="N19D58" type="main"> <s id="N19D5A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N19D66" type="main"> <s id="N19D68"><emph type="italics"/>Ad motum violentum non concurrit impetus innatus,<emph.end type="italics"/> probatur, quia im­<lb/>petus ad lineas oppo&longs;itas ex diametro determinati ad communem li­<lb/>neam determinari non po&longs;&longs;unt, cur enim potiùs dextror&longs;um quam &longs;ini­<lb/>stror&longs;um? </s> <s id="N19D76">igitur non concurrunt ad communem motum, ni&longs;i dicatur <lb/>impetus innatus valeo nomine concurrere ad violentum, quod eius li­<lb/>neam &longs;ingulis temporibus qua&longs;i ca&longs;tiget, vltróque, vel vlteriùs currentem <lb/>contineat. </s> </p> <p id="N19D7F" type="main"> <s id="N19D81"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N19D8D" type="main"> <s id="N19D8F"><!-- NEW --><emph type="italics"/>Hinc ad motum violentum impetus ab exteriore potentia mobili impre&longs;&longs;us <lb/>tantùm concurrit<emph.end type="italics"/>; </s> <s id="N19D9A"><!-- NEW -->patet, cum enim in mobili projecto &longs;ur&longs;um &longs;it tantùm <lb/>ille impetus præter innatum, nec innatus concurrat per Th. 52. illum <lb/>tantùm concurrere nece&longs;&longs;e e&longs;t: excipe &longs;emper impetum acqui&longs;itum, de <lb/>quo iam &longs;uprà. </s> </p> <p id="N19DA4" type="main"> <s id="N19DA6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N19DB2" type="main"> <s id="N19DB4"><!-- NEW --><emph type="italics"/>Primo instanti quo producitur impetus ille à potentia motrice in mobili, me­<lb/>diante &longs;cilicet impetu producto in organo proprio, non e&longs;t motus<emph.end type="italics"/>; probatur, <lb/>quia primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, per Th.34.l.1. <!-- KEEP S--></s> </p> <p id="N19DC2" type="main"> <s id="N19DC4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N19DD0" type="main"> <s id="N19DD2"><!-- NEW --><emph type="italics"/>Impetus productus in manu producit impetum in organo vel in mobili pri­<lb/>mo in&longs;tanti, quo e&longs;t<emph.end type="italics"/>; </s> <s id="N19DDD"><!-- NEW -->probatur, quia &longs;ecundo in&longs;tanti exigit motum &longs;ui &longs;ub­<lb/>jecti; </s> <s id="N19DE3"><!-- NEW -->igitur tolli etiam impedimentum; </s> <s id="N19DE7"><!-- NEW -->igitur per motum medij; </s> <s id="N19DEB"><!-- NEW -->igitur <lb/>priori in&longs;tanti in eodem mobili debet e&longs;&longs;e impetus; </s> <s id="N19DF1"><!-- NEW -->igitur produci ab <lb/>impetu organi; igitur & in organo ab impetu manus. </s> </p> <p id="N19DF7" type="main"> <s id="N19DF9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N19E05" type="main"> <s id="N19E07"><!-- NEW --><emph type="italics"/>Primo in&longs;tanti, quo producitur impetus in motu violento, nullus eius gra­<lb/>dus de&longs;truitur<emph.end type="italics"/>; probatur, quia alioquin &longs;imul eodem in&longs;tanti, quo e&longs;&longs;e in­<lb/>ciperet, e&longs;&longs;e de&longs;ineret, quod dici non pote&longs;t. </s> </p> <p id="N19E14" type="main"> <s id="N19E16"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N19E22" type="main"> <s id="N19E24"><!-- NEW --><emph type="italics"/>Impetus innatus impedit ne producatur tantus impetus in motu violento,<emph.end type="italics"/><lb/>probatur, quia certè tàm impedit primam productionem, quàm con&longs;er­<lb/>uationem, vt patet; </s> <s id="N19E30"><!-- NEW -->e&longs;t enim par vtrobique ratio; præterea agit in ip&longs;am <lb/>manum. </s> </p> <p id="N19E36" type="main"> <s id="N19E38"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s> </p> <p id="N19E44" type="main"> <s id="N19E46"><!-- NEW --><emph type="italics"/>Impetus violentus producitur minor, quàm produceretur vno dumtaxat gra­<lb/>du aquali ip&longs;i impetui innato<emph.end type="italics"/>; </s> <s id="N19E51"><!-- NEW -->quippe &longs;icut de&longs;truit &longs;ingulis in&longs;tantibus <lb/>æqualibus vnum gradum; </s> <s id="N19E57"><!-- NEW -->quia pugnat pro rata; </s> <s id="N19E5B"><!-- NEW -->ita pror&longs;us impedit, ne <pb pagenum="147" xlink:href="026/01/179.jpg"/>producatur vnus gradus &longs;ibi æqualis primo in&longs;tanti; cur enim duo po­<lb/>tiùs, quàm tres? </s> </p> <p id="N19E66" type="main"> <s id="N19E68"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N19E74" type="main"> <s id="N19E76"><!-- NEW --><emph type="italics"/>Secundo &longs;tatim in&longs;tanti de&longs;truit alterum gradum<emph.end type="italics"/>: </s> <s id="N19E7F"><!-- NEW -->quippe e&longs;t cau&longs;a ne­<lb/>ce&longs;&longs;aria; </s> <s id="N19E85"><!-- NEW -->igitur &longs;tatim primo in&longs;tanti exigit de&longs;tructionem; </s> <s id="N19E89"><!-- NEW -->non certè <lb/>pro primo in&longs;tanti per Th.56.igitur pro &longs;ecundo, atque ita pro aliis dein­<lb/>ceps; de&longs;truitur autem, ne &longs;it fru&longs;trà eo modo, quo diximus &longs;uprà. </s> </p> <p id="N19E91" type="main"> <s id="N19E93"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N19E9F" type="main"> <s id="N19EA1"><!-- NEW --><emph type="italics"/>Hinc optima ratio illius instituti naturæ, quo factum e&longs;t, vt impetus innatus <lb/>numquam destruatur<emph.end type="italics"/>; </s> <s id="N19EAC"><!-- NEW -->ne &longs;i aliquando de&longs;trueretur, nulla e&longs;&longs;et cau&longs;a de­<lb/>&longs;tructiua impetus violenti; ac proinde æquabilis e&longs;&longs;et, &longs;emperque dura­<lb/>ret, de&longs;tructiua inquam &longs;uo modo. </s> </p> <p id="N19EB4" type="main"> <s id="N19EB6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N19EC2" type="main"> <s id="N19EC4"><!-- NEW --><emph type="italics"/>Hinc corpus quod non grauitat, facilè proijcitur, vel impellitur<emph.end type="italics"/>: </s> <s id="N19ECD"><!-- NEW -->&longs;ic na­<lb/>uis aquis innatans, nubes in aëre liberatæ; halitus, atque adeo ip&longs;æ partes <lb/>aquæ, quas perexiguus lapillus in orbes penè innumeros agit, ne quid <lb/>dicam de partibus aëris, quæ tam citò & procul mouentur, vt con&longs;tat in <lb/>&longs;ono, motu &longs;cilicet ferè æquabili. </s> </p> <p id="N19ED9" type="main"> <s id="N19EDB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N19EE7" type="main"> <s id="N19EE9"><!-- NEW --><emph type="italics"/>Hinc etiam è contrario corpus grauius difficiliùs &longs;ur&longs;um proijcitur<emph.end type="italics"/>: </s> <s id="N19EF2"><!-- NEW -->tùm <lb/>quia plures partes impetus &longs;unt producendæ in &longs;ubjecto grauiore quod <lb/>pluribus partibus con&longs;tat, tùm impetus innatus maior e&longs;t, non quidem in <lb/>inten&longs;ione &longs;ed in exten&longs;ione, ac proinde impedit ne plures gradus pro­<lb/>ducantur; quippe maius impedimentum plus impedit, quis hoc neget? </s> </p> <p id="N19EFE" type="main"> <s id="N19F00"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s> </p> <p id="N19F0C" type="main"> <s id="N19F0E"><!-- NEW --><emph type="italics"/>Omnes partes impetus productæ in mobili primo instanti concurrunt ad <lb/>motum &longs;ecundi instantis<emph.end type="italics"/>; probatur, quia alioqui aliqua e&longs;&longs;et fru&longs;trà, quod <lb/>dici non debet. </s> </p> <p id="N19F1B" type="main"> <s id="N19F1D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s> </p> <p id="N19F29" type="main"> <s id="N19F2B"><!-- NEW --><emph type="italics"/>Concurrunt omnes illæ, quæ in&longs;unt eidem parti &longs;eu puncto mobilis <expan abbr="commun">communes</expan> <lb/>qua&longs;i actione vel exigentia<emph.end type="italics"/>; patet ex dictis de impetu, quia concurrunt ad <lb/>velocitatem, quæ e&longs;t indiui&longs;ibilis actu. </s> </p> <p id="N19F3C" type="main"> <s id="N19F3E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s> </p> <p id="N19F4A" type="main"> <s id="N19F4C"><!-- NEW --><emph type="italics"/>Non ponitur tamen totus motus &longs;ecundo instanti, quem exigunt primo; <emph.end type="italics"/><lb/>quia impetus innatus aliquid detrahit, cum exigat motum deor&longs;um per <lb/>lineam oppo&longs;itam, igitur imminuitur motus pro rata. </s> </p> <p id="N19F58" type="main"> <s id="N19F5A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s> </p> <p id="N19F66" type="main"> <s id="N19F68"><!-- NEW --><emph type="italics"/>Hinc ille gradus motus qui non ponitur &longs;ecundo instanti respondet gradus <lb/>impetus qui destruitur<emph.end type="italics"/>; cum vterque habeat <expan abbr="eãdem">eandem</expan> men&longs;uram, &longs;cilicet <lb/>impetum innatum. </s> </p> <pb pagenum="148" xlink:href="026/01/180.jpg"/> <p id="N19F7D" type="main"> <s id="N19F7F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s> </p> <p id="N19F8B" type="main"> <s id="N19F8D"><!-- NEW --><emph type="italics"/>Hinc effectus pete&longs;t e&longs;&longs;e eo instanti quo non existit eius cau&longs;a partialis<emph.end type="italics"/>; </s> <s id="N19F96"><!-- NEW -->v.g. <!-- REMOVE S--><lb/>motus qui ponitur &longs;ecundo in&longs;tanti non minùs exigitur ab eo gradu im­<lb/>petus qui de&longs;truitur &longs;ecundò in&longs;tanti, quàm ab aliis, non exigitur qui­<lb/>dem &longs;ecundo &longs;ed primo pro &longs;ecundo; </s> <s id="N19FA1"><!-- NEW -->vnde dixi cau&longs;am partialem, quia <lb/>etiam exigitur ab aliis gradibus impetus, qui non de&longs;truuntur exigentiâ <lb/>communi; </s> <s id="N19FA9"><!-- NEW -->quippe impetus non exigit ni&longs;i pro &longs;ecundo in&longs;tanti; </s> <s id="N19FAD"><!-- NEW -->nec vl­<lb/>lum ab&longs;urdum e&longs;t eo in&longs;tanti cau&longs;am exigentiæ non exi&longs;tere cum poni­<lb/>tur eius effectus, &longs;cilicet id quod exigebat priori in&longs;tanti quo erat; </s> <s id="N19FB5"><!-- NEW -->nul­<lb/>lus e&longs;t enim influxus huius cau&longs;æ; præ&longs;ertim cum non &longs;it cau&longs;a <lb/>totalis. </s> </p> <p id="N19FBD" type="main"> <s id="N19FBF"><!-- NEW -->Vnde cum effectus qui ponitur &longs;ecundo in&longs;tanti non re&longs;pondeat per­<lb/>fectioni cau&longs;æ totius propter impedimentum, aliquis gradus cau&longs;æ e&longs;&longs;et <lb/>fru&longs;trà; </s> <s id="N19FC7"><!-- NEW -->igitur eodem in&longs;tanti &longs;ecundo de&longs;trui debet, alioqui ni&longs;i de&longs;true­<lb/>retur &longs;ingulis in&longs;tantibus poneretur effectus non re&longs;pondens perfectioni <lb/>cau&longs;æ; </s> <s id="N19FCF"><!-- NEW -->immò numquam de&longs;trueretur totus motus violentus, vt con&longs;tat; </s> <s id="N19FD3"><!-- NEW --><lb/>itaque primo in&longs;tanti omnes gradus impetus qui &longs;unt exigunt motum <lb/>pro &longs;ecundo ne aliquis eo in&longs;tanti &longs;it fru&longs;trà &longs;i non exigeret, & &longs;ecundo <lb/>in&longs;tanti aliquis gradus impetus de&longs;truitur, ne &longs;it fru&longs;trà eodem in&longs;tanti <lb/>&longs;ecundo, cum &longs;cilicet non &longs;int tot gradus motus, quot &longs;unt gradus impe­<lb/>tus; atque ita deinceps tertio in&longs;tanti de&longs;truitur vnus gradus, vt iam &longs;u­<lb/>prà dictum e&longs;t. </s> </p> <p id="N19FE2" type="main"> <s id="N19FE4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s> </p> <p id="N19FF0" type="main"> <s id="N19FF2"><emph type="italics"/>Ideo de&longs;truitur potiùs vnus gradus impetus quàm alius &longs;ecundo in&longs;tanti, <lb/>tertioque, &c. </s> <s id="N19FF9"><!-- NEW -->quia talis e&longs;t perfectionis<emph.end type="italics"/>; </s> <s id="N1A000"><!-- NEW -->hoc iam &longs;uprà explicatum e&longs;t; quia <lb/>cum motus initio &longs;it velocior, in&longs;tantia &longs;unt minora, igitur minùs im­<lb/>petus in &longs;ingulis de&longs;truitur, pater ex dictis. </s> </p> <p id="N1A008" type="main"> <s id="N1A00A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s> </p> <p id="N1A016" type="main"> <s id="N1A018"><!-- NEW --><emph type="italics"/>Ille gradus impetus qui de&longs;truitur &longs;ecundo in&longs;tanti non concurrit ad motum <lb/>tertij in&longs;tantis<emph.end type="italics"/>; </s> <s id="N1A023"><!-- NEW -->quia non pote&longs;t concurrere ad motum ni&longs;i exigendo; </s> <s id="N1A027"><!-- NEW -->at­<lb/>qui exigere tantùm pote&longs;t, quando e&longs;t; </s> <s id="N1A02D"><!-- NEW -->quod enim non e&longs;t non exigit, <lb/>&longs;ed motus tertij in&longs;tantis exigitur &longs;ecundo; </s> <s id="N1A033"><!-- NEW -->&longs;ic enim tota res motus pro­<lb/>cedit vt impetus primo in&longs;tanti exigat motum pro &longs;ecundo; </s> <s id="N1A039"><!-- NEW -->& &longs;ecundo <lb/>pro tertio; </s> <s id="N1A03F"><!-- NEW -->& tertio pro quarto, atque ita deinceps; </s> <s id="N1A043"><!-- NEW -->igitur impetus ille <lb/>qui de&longs;truitur; &longs;ecundo in&longs;tanti non exigit motum pro tertio, & qui de­<lb/>&longs;truitur tertio non exigit pro quarto, atque ita deinceps. </s> </p> <p id="N1A04B" type="main"> <s id="N1A04D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s> </p> <p id="N1A059" type="main"> <s id="N1A05B"><!-- NEW --><emph type="italics"/>Hinc impetus innatus non concurrit ad motum violentum,<emph.end type="italics"/> vt dictum e&longs;t, <lb/>&longs;ed tantùm impedit, immediatè quidem, quia cum exigat motum deor­<lb/>sùm, facit vt non &longs;it tantus motus &longs;ur&longs;um; </s> <s id="N1A068"><!-- NEW -->mediatè verò, quia cum non <lb/>&longs;it tantus motus &longs;ursùm, quantus e&longs;&longs;et, haud dubiè non re&longs;pondet adæ­<lb/>quatè cau&longs;æ; </s> <s id="N1A070"><!-- NEW -->igitur aliquid cau&longs;æ fru&longs;trà e&longs;t; </s> <s id="N1A074"><!-- NEW -->igitur de&longs;trui debet; hinc <pb pagenum="149" xlink:href="026/01/181.jpg"/>de&longs;truitur etiam hic impetus per principium commune, ne aliquid &longs;it <lb/>fru&longs;trà. </s> </p> <p id="N1A07F" type="main"> <s id="N1A081"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s> </p> <p id="N1A08D" type="main"> <s id="N1A08F"><!-- NEW --><emph type="italics"/>Linea motus &longs;ur&longs;um determinatur à potentia motrice<emph.end type="italics"/>; </s> <s id="N1A098"><!-- NEW -->probatur, quia hæc <lb/>determinat impetum productum in manu vel in organo; </s> <s id="N1A09E"><!-- NEW -->hic verò im­<lb/>petum, quem producit in mobili &longs;ursùm projecto; patet, quia nulla e&longs;t <lb/>alia cau&longs;a applicata. </s> </p> <p id="N1A0A6" type="main"> <s id="N1A0A8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s> </p> <p id="N1A0B4" type="main"> <s id="N1A0B6"><!-- NEW --><emph type="italics"/>Tandem duo impetus violentus, &longs;cilicet, & innatus ad æqualitatem perue­<lb/>nirent, &longs;i vel vnus gradus violenti e&longs;&longs;et æqualis perfectionis cum innato<emph.end type="italics"/>; </s> <s id="N1A0C1"><!-- NEW -->cum <lb/>enim detrahatur &longs;emper pars aliquota alicuius totius, tandem perueni­<lb/>tur ad vltimam; </s> <s id="N1A0C9"><!-- NEW -->igitur &longs;int 100. gradus impetus violenti, quorum quili­<lb/>bet &longs;it æqualis impetui innato; </s> <s id="N1A0CF"><!-- NEW -->certè cum temporibus æqualibus æqua­<lb/>lis gradus impetus de&longs;truatur; </s> <s id="N1A0D5"><!-- NEW -->accipiatur illud tempus, in quo de&longs;trui­<lb/>tur vnus, haud dubiè 100. æqualibus temporibus de&longs;truentur omnes 100. <lb/>igitur 99. in&longs;tantibus de&longs;truentur 99. gradus; </s> <s id="N1A0DD"><!-- NEW -->igitur &longs;upere&longs;t vnus; igitur <lb/>duo illi impetus perueniunt tandem ad æqualitatem. </s> </p> <p id="N1A0E3" type="main"> <s id="N1A0E5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s> </p> <p id="N1A0F1" type="main"> <s id="N1A0F3"><!-- NEW --><emph type="italics"/>Vbi vterque perueni&longs;&longs;et ad æqualitatem, non e&longs;&longs;et potior ratio cur mobile mo­<lb/>ueretur &longs;ursùm quàm deor&longs;um in&longs;tanti &longs;equenti<emph.end type="italics"/>; </s> <s id="N1A0FE"><!-- NEW -->probatur, quia tàm gra­<lb/>dus impetus innati exigit motum deor&longs;um quàm gradus impetus vio­<lb/>lenti &longs;ursùm; igitur neuter habebit motum per Th.133.l. </s> <s id="N1A106">1. <!-- KEEP S--></s> </p> <p id="N1A10A" type="main"> <s id="N1A10C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s> </p> <p id="N1A118" type="main"> <s id="N1A11A"><!-- NEW --><emph type="italics"/>Hinc ip&longs;o in&longs;tanti, quo e&longs;&longs;et æqualitas, e&longs;&longs;et adhuc motus<emph.end type="italics"/>; </s> <s id="N1A123"><!-- NEW -->quia in&longs;tanti <lb/>immediatè antecedenti erant duo gradus impetus violenti, & vnus in­<lb/>nati; igitur duo illi præualent pro in&longs;tanti &longs;equenti, in quo e&longs;t æqua­<lb/>litas. </s> </p> <p id="N1A12D" type="main"> <s id="N1A12F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s> </p> <p id="N1A13B" type="main"> <s id="N1A13D"><!-- NEW --><emph type="italics"/>Itaque quie&longs;ceret mobile ip&longs;o &longs;tatim in&longs;tanti, quod in&longs;tanti æqualitatis &longs;uc­<lb/>cedit<emph.end type="italics"/>; patet, quia neuter impetus pro illo in&longs;tanti præualere po&longs;&longs;et per <lb/>Th. 73. </s> </p> <p id="N1A14A" type="main"> <s id="N1A14C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s> </p> <p id="N1A158" type="main"> <s id="N1A15A"><!-- NEW --><emph type="italics"/>Igitur in&longs;tanti quietis nullus e&longs;&longs;et ampliùs impetus violentus<emph.end type="italics"/>; </s> <s id="N1A163"><!-- NEW -->cum enim <lb/>&longs;ingulis in&longs;tantibus de&longs;truatur vnus gradus, v. <!-- REMOVE S-->g in&longs;tanti illo, quod &longs;e­<lb/>quitur po&longs;t in&longs;tans æqualitatis, de&longs;truitur ille gradus, qui &longs;upere&longs;t; </s> <s id="N1A16D"><!-- NEW -->nec <lb/>pote&longs;t vel plùs, vel minùs de&longs;trui; </s> <s id="N1A173"><!-- NEW -->pugnant enim pro rata; quod certè <lb/>cuiquam fortè paradoxor videbitur, &longs;cilicet nullum tune e&longs;&longs;e motum <lb/>propter pugnam, cum tamen nulla e&longs;t amplius pugna. </s> </p> <p id="N1A17B" type="main"> <s id="N1A17D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s> </p> <p id="N1A189" type="main"> <s id="N1A18B"><!-- NEW --><emph type="italics"/>Quies illa duraret tantùm vno in&longs;tanti,<emph.end type="italics"/> probatur, quia cum in&longs;tanti quie­<lb/>tis &longs;it tantùm impetus innatus per Th. 76. certè non impeditur quomi­<lb/>nus habeat motum pro in&longs;tanti &longs;equenti, quem reuerà exigit; </s> <s id="N1A198"><!-- NEW -->igitur pro <pb pagenum="150" xlink:href="026/01/182.jpg"/>in&longs;tanti &longs;equenti moueritur; </s> <s id="N1A1A1"><!-- NEW -->&longs;ed pro alio antecedente mouebatur; igi­<lb/>tur quies illa durat tantùm vno in&longs;tanti. </s> </p> <p id="N1A1A7" type="main"> <s id="N1A1A9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s> </p> <p id="N1A1B5" type="main"> <s id="N1A1B7"><!-- NEW --><emph type="italics"/>Quies illa non fit propter aliquam reflexionem, vt aliqui dicunt<emph.end type="italics"/>; </s> <s id="N1A1C0"><!-- NEW -->quia nul­<lb/>la pror&longs;us e&longs;t reflexio, vbi nullum e&longs;t reflectens; </s> <s id="N1A1C6"><!-- NEW -->atqui nullum e&longs;t refle­<lb/>ctens, vt patet, quia nullum e&longs;t corpus impediens motus propagationem; </s> <s id="N1A1CC"><!-- NEW --><lb/>licèt enim medium impediat, non tamen per modum reflectentis pro­<lb/>priè; </s> <s id="N1A1D3"><!-- NEW -->immo vt dicemus infrà in puncto reflexionis nulla datur quies; &longs;ed <lb/>motus reflexus &longs;ibi vendicat librum &longs;ingularem. </s> </p> <p id="N1A1D9" type="main"> <s id="N1A1DB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s> </p> <p id="N1A1E7" type="main"> <s id="N1A1E9"><emph type="italics"/>Hinc &longs;iue præce&longs;&longs;erit motus violentus, &longs;iue non, corpus graue eodem vel æ­<lb/>quali motu deor&longs;um cadit,<emph.end type="italics"/> quia nullus amplius remanet impetus violen­<lb/>tus in fine motus violenti, per Th.76. igitur &longs;olus impetus naturalis li­<lb/>bero motu deorsùm fertur. </s> </p> <p id="N1A1F7" type="main"> <s id="N1A1F9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s> </p> <p id="N1A205" type="main"> <s id="N1A207"><!-- NEW --><emph type="italics"/>Hinc reiicies aliquos apud Galileum, qui volunt ideo motum naturalem <lb/>accelerari, quia &longs;en&longs;im de&longs;truitur impetus violentus antè impre&longs;&longs;us,<emph.end type="italics"/> quod pe­<lb/>nitus ridiculum e&longs;t; quia lapis deci&longs;us è rupe etiam motu naturaliter <lb/>accelerato deor&longs;um cadit, licèt eò nunquam motu violento euectus <lb/>fuerit. </s> </p> <p id="N1A218" type="main"> <s id="N1A21A">Ob&longs;eruabis hanc hypothe&longs;im gradus impetus violenti æqualis perfe­<lb/>ctionis cum innato e&longs;&longs;e fal&longs;am. </s> <s id="N1A21F">Primò, quia commodius e&longs;t potentiæ <lb/>motrici producere imperfectiorem impetum, &longs;ic enim plures illius gra­<lb/>dus producere pote&longs;t. </s> <s id="N1A226"><!-- NEW -->Secundò, quia in reflexo &longs;ur&longs;um vltimus gradus <lb/>qui de&longs;truitur e&longs;t imperfectior innato, e&longs;t enim acqui&longs;itus; igitur in omni <lb/>alio motu &longs;ursùm. </s> <s id="N1A22E"><!-- NEW -->Tertiò, quia violentus e&longs;t cum innato in eadem &longs;ubie­<lb/>cti parte; &longs;ed idem &longs;ubiectum formas homogeneas non patitur, de quò <lb/>aliàs, hinc dicendum &longs;upere&longs;t non quie&longs;cere mobile in fine motus </s> </p> <p id="N1A236" type="main"> <s id="N1A238"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s> </p> <p id="N1A244" type="main"> <s id="N1A246"><!-- NEW --><emph type="italics"/>Corpus quod non grauitat proiicitur &longs;ur&longs;um motu æquabili per &longs;e<emph.end type="italics"/>; </s> <s id="N1A24F"><!-- NEW -->patet, quia <lb/>nihil e&longs;t quod de&longs;truat ip&longs;um impetum; </s> <s id="N1A255"><!-- NEW -->igitur &longs;emper moueretur, ni&longs;i <lb/>per accidens ab ip&longs;o medio eius motus retardaretur; </s> <s id="N1A25B"><!-- NEW -->vnde dixi <emph type="italics"/>per &longs;e,<emph.end type="italics"/><lb/>cum ratione medij retardetur; immò quò leuius e&longs;t, faciliùs à medio re­<lb/>tinetur, vide Th.61. </s> </p> <p id="N1A268" type="main"> <s id="N1A26A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s> </p> <p id="N1A276" type="main"> <s id="N1A278"><!-- NEW --><emph type="italics"/>Non cre&longs;cit impetus naturalis in motu violento &longs;ur&longs;um<emph.end type="italics"/>; probatur primò, <lb/>quia impetus naturalis aduentitius &longs;upponit motum deor&longs;um, ad cuius <lb/>inten&longs;ionem à natura fuit in&longs;titutus per re&longs;p. </s> <s id="N1A285">ad quartam obiect. </s> <s id="N1A288"><!-- NEW -->in di&longs;­<lb/>&longs;ert.l.2. adde quod tardiùs a&longs;cenderet, quàm de&longs;cenderet; </s> <s id="N1A28E"><!-- NEW -->deinde velo­<lb/>ciùs de&longs;cenderet po&longs;tmotum violentum corpus graue, quàm &longs;i nullo mo­<lb/>tu violento præuio demitteretur deor&longs;um, quæ omnia experimentis <pb pagenum="151" xlink:href="026/01/183.jpg"/><expan abbr="etiã">etiam</expan> vulgaribus repugnant; immò & cunctis ferè præmi&longs;&longs;is Theorematis. <!-- KEEP S--></s> </p> <p id="N1A29F" type="main"> <s id="N1A2A1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s> </p> <p id="N1A2AD" type="main"> <s id="N1A2AF"><!-- NEW --><emph type="italics"/>Motus violentus non tendit ad quietem per omnes tarditatis gradus, vt <lb/>pa&longs;&longs;im a&longs;&longs;erit Galileus<emph.end type="italics"/>; </s> <s id="N1A2BA"><!-- NEW -->Primò, quia non &longs;unt infinita in&longs;tantia, &longs;ed retarda­<lb/>tur tantùm &longs;ingulis in&longs;tantibus; </s> <s id="N1A2C0"><!-- NEW -->Secundò in medio den&longs;iore minùs du­<lb/>rat; </s> <s id="N1A2C6"><!-- NEW -->igitur non tran&longs;it per tot gradus tarditatis; </s> <s id="N1A2CA"><!-- NEW -->præterea in plano incli­<lb/>nato &longs;ur&longs;um în minore proportione retardatur motus, quod etiam in <lb/>plano horizontali certi&longs;&longs;imum e&longs;t; quorum omnium rationes &longs;uo loco <lb/>videbimus. </s> </p> <p id="N1A2D4" type="main"> <s id="N1A2D6"><!-- NEW -->Nec e&longs;t quod aliqui dicant infinito tribui non po&longs;&longs;e hæc prædicata <lb/>æqualitatis vel inæqualitatis, quod fal&longs;um e&longs;t, loquamur de infinito actu; </s> <s id="N1A2DC"><!-- NEW --><lb/>&longs;i enim e&longs;&longs;et numerus infinitus hominum, nunquid verum e&longs;&longs;et dicere <lb/>numerum oculorum e&longs;&longs;e maiorem numero hominum; </s> <s id="N1A2E3"><!-- NEW -->nec e&longs;t quod ali­<lb/>qui confugiant ad di&longs;iunctiones; </s> <s id="N1A2E9"><!-- NEW -->nos rem i&longs;tam &longs;uo loco fusè tractabi­<lb/>mus & demon&longs;trabimus, ni fallor, cum Ari&longs;totele, fieri non pò&longs;&longs;e vt &longs;it <lb/>aliquod creatum infinitum actu; </s> <s id="N1A2F1"><!-- NEW -->licèt vltrò concedamus plura e&longs;&longs;e infi­<lb/>nita potentiâ; & verò certum e&longs;t infinito potentiâ non ine&longs;&longs;e huiu&longs;modi <lb/>prædicata æqualitatis, vel inæqualitatis. </s> </p> <p id="N1A2F9" type="main"> <s id="N1A2FB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s> </p> <p id="N1A307" type="main"> <s id="N1A309"><!-- NEW --><emph type="italics"/>Immò &longs;i tran&longs;iret mobile &longs;ursùm proiectum per omnes tarditatis gradus, <lb/>nunquam profectò de&longs;cenderat<emph.end type="italics"/>; </s> <s id="N1A314"><!-- NEW -->quia cum &longs;ingulis in&longs;tantibus &longs;inguli gra­<lb/>dus re&longs;pondeant, & duo in&longs;tantia &longs;imul e&longs;&longs;e non po&longs;&longs;int; </s> <s id="N1A31A"><!-- NEW -->nunquam certè <lb/>verum e&longs;&longs;et dicere fluxi&longs;&longs;e infinita; </s> <s id="N1A320"><!-- NEW -->igitur nec mobile per infinitos tar­<lb/>ditatis gradus ad quietem perueni&longs;&longs;e; hoc Theorema &longs;upponit e&longs;&longs;e tan­<lb/>tùm finita in&longs;tantia. </s> </p> <p id="N1A328" type="main"> <s id="N1A32A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s> </p> <p id="N1A336" type="main"> <s id="N1A338"><!-- NEW --><emph type="italics"/>Re&longs;i&longs;tentia aëris est maior initio, quàm in fine motus violenti,<emph.end type="italics"/> vt con&longs;tat ex <lb/>dictis, quia initio motus e&longs;t velocior, igitur plures partes aëris æquali <lb/>tempore re&longs;i&longs;tunt; in fine verò è contrario. </s> </p> <p id="N1A345" type="main"> <s id="N1A347"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s> </p> <p id="N1A353" type="main"> <s id="N1A355"><!-- NEW --><emph type="italics"/>Hinc oppo&longs;ita e&longs;t omninò ratio re&longs;istentia, quæ &longs;equitur ex motu violento illi, <lb/>quæ cum naturali e&longs;t coniuncta,<emph.end type="italics"/> hæc enim initio minor, in fine maior, illa <lb/>verò initio maior, & in fine minor; hinc prima cre&longs;cit cam &longs;uo motu, <lb/>&longs;ecunda cum &longs;uo decre&longs;cit. </s> </p> <p id="N1A364" type="main"> <s id="N1A366"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s> </p> <p id="N1A372" type="main"> <s id="N1A374"><!-- NEW --><emph type="italics"/>Decre&longs;cit igitur impetus eadem proportione, qua decre&longs;cit re&longs;i&longs;tentia<emph.end type="italics"/>; vt pa­<lb/>tet ex dictis; igitur in toto motu eadem e&longs;t re&longs;i&longs;tentiæ proportio. </s> </p> <p id="N1A37F" type="main"> <s id="N1A381"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s> </p> <p id="N1A38D" type="main"> <s id="N1A38F"><!-- NEW --><emph type="italics"/>Variæ &longs;unt potentiæ motrices, à quibus mobile &longs;ur&longs;um proiici potest motu <lb/>violento,<emph.end type="italics"/> v.g. <!-- REMOVE S-->potentia motrix animantium, potentia motrix grauium mo­<lb/>bili &longs;cilicet &longs;ur&longs;um repercu&longs;&longs;o; potentia motrix, quæ &longs;equitur ex com­<lb/>pre&longs;&longs;ione & rarefactione corporum, &longs;ed de his omnibus aliàs. </s> </p> <pb pagenum="152" xlink:href="026/01/184.jpg"/> <p id="N1A3A4" type="main"> <s id="N1A3A6"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A3B2" type="main"> <s id="N1A3B4"><!-- NEW -->Ob&longs;eruabis primò &longs;i aliquando accidat, vt aliqui volunt ictum, qui <lb/>&longs;tatim initio motus violenti infligitur, non e&longs;&longs;e maximum, &longs;ed minorem <lb/>eo, qui po&longs;t aliquod confectum &longs;patium infligitur; </s> <s id="N1A3BC"><!-- NEW -->quod probant in pila <lb/>ex fi&longs;tula ænea &longs;ur&longs;um emi&longs;&longs;a, quæ <expan abbr="moior&etilde;">maiorem</expan> ictum infligit in data di&longs;tantia, <lb/>quod &longs;anè &longs;i verum e&longs;t, hæc vnica e&longs;t, &longs;eu ratio, &longs;eu cau&longs;a, quòd &longs;cilicet &longs;ur­<lb/>&longs;um pila pellatur ab igne, qui ab ore fi&longs;tulæ erumpens per aliquod &longs;pa­<lb/>tium à tergo vrget; igni enim innatum e&longs;t &longs;ur&longs;um euolare. </s> </p> <p id="N1A3CC" type="main"> <s id="N1A3CE">Ob&longs;eruabis &longs;ecundò, vix po&longs;&longs;e manu mobile &longs;ur&longs;um rectà proiici, quia <lb/>&longs;cilicet manus extremitas motu mixto mouetur ex duobus vel pluribus <lb/>circularibus, de quo infrà. </s> </p> <p id="N1A3D5" type="main"> <s id="N1A3D7"><!-- NEW -->Ob&longs;erua tertiò, non tantùm propter grauitationem con&longs;eruari impe­<lb/>tum naturalem innatum, &longs;ed etiam vt motui violento re&longs;i&longs;tat; at verò <lb/>non re&longs;i&longs;teret, ni&longs;i grauitaret. </s> </p> <p id="N1A3DF" type="main"> <s id="N1A3E1"><!-- NEW -->Ob&longs;erua quartò, reciprocas rationes motus naturalis & violenti; in <lb/>quibus mirabile pror&longs;us fuit naturæ in&longs;titutum, cum idem in vtroque il­<lb/>larum &longs;it principium. </s> </p> <p id="N1A3E9" type="main"> <s id="N1A3EB"><!-- NEW -->Ob&longs;erua quintò, finem motus violenti e&longs;&longs;e multiplicem, nullum ta­<lb/>men à natura in&longs;titutum; </s> <s id="N1A3F1"><!-- NEW -->quippe potentia motrix, quæ agit ex appetitu <lb/>elicito, (vt vulgò aiunt,) &longs;eu cum cognitione, finem &longs;ibi proponit ad libi­<lb/>tùm; </s> <s id="N1A3F9"><!-- NEW -->illa verò quæ vi compre&longs;&longs;ionis excitatur per accidens &longs;ur&longs;um agit <lb/>mobile potiùs, quàm per aliam lineam; repercu&longs;&longs;a &longs;ursùm videntur e&longs;&longs;e <lb/>magis iuxta in&longs;titutum naturæ. <lb/><figure id="id.026.01.184.1.jpg" xlink:href="026/01/184/1.jpg"/></s> </p> </chap> <chap id="N1A407"> <pb pagenum="153" xlink:href="026/01/185.jpg"/> <figure id="id.026.01.185.1.jpg" xlink:href="026/01/185/1.jpg"/> <p id="N1A411" type="head"> <s id="N1A413"><emph type="center"/>LIBER QVARTVS, <lb/><emph type="italics"/>DE MOTV MIXTO EX <lb/>duobus, vel pluribus rectis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N1A422" type="main"> <s id="N1A424"><!-- NEW -->MOTVM mixtum eum e&longs;&longs;e non dico, qui <lb/>ex pluribus aliis motibus componatur; <lb/>&longs;eu mi&longs;ceatur; </s> <s id="N1A42C"><!-- NEW -->nec enim plures motus <lb/>&longs;imul e&longs;&longs;e po&longs;&longs;unt in eodem mobili; </s> <s id="N1A432"><!-- NEW -->cùm <lb/>tantùm e&longs;&longs;e po&longs;&longs;it vno dumtaxat in&longs;tan­<lb/>ti vnica migratio ex loco in locum; </s> <s id="N1A43A"><!-- NEW -->nec plura loca <lb/>naturali virtute &longs;imul acquiri po&longs;&longs;unt; </s> <s id="N1A440"><!-- NEW -->Igitur nec &longs;i­<lb/>mul e&longs;&longs;e duo motus; </s> <s id="N1A446"><!-- NEW -->Itaque motus mixtus &longs;implex <lb/>e&longs;t, &longs;i con&longs;ideretur ratio, & linea motus; </s> <s id="N1A44C"><!-- NEW -->mixtus verò <lb/>dicitur, quod ex pluribus re&longs;ultet, qui reuerâ non <lb/>&longs;unt, &longs;ed cùm e&longs;&longs;e po&longs;&longs;int, qua&longs;i confluunt in tertium <lb/>motum communi &longs;umptu qua&longs;i de vtroque partici­<lb/>pantem, quod totum fit propter diuer&longs;os impetus, <lb/>vel <expan abbr="eũdem">eundem</expan> ad diuer&longs;as lineas determinatum, vt fusè <lb/>explicabimus infrà: Porrò in hoc Libro explicamus <lb/>tantùm motum mixtum, qui re&longs;ultat ex pluribus re­<lb/>ctis, vt titulus ip&longs;e præfert. <lb/><gap desc="hr tag"/></s> </p> <p id="N1A467" type="main"> <s id="N1A469"><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A475" type="main"> <s id="N1A477"><!-- NEW --><emph type="italics"/>MOtus mixtus e&longs;t, qui &longs;equitur ex multiplici impetu ad <expan abbr="eãdem">eandem</expan>, vel di­<lb/>uer&longs;as lineas determinato, vel eodem ad diuer&longs;as<emph.end type="italics"/>; </s> <s id="N1A486"><!-- NEW -->hæc definitio cla­<lb/>ra e&longs;t; </s> <s id="N1A48C"><!-- NEW -->ob&longs;eruabis tantùm ad motum mixtum &longs;ufficere duplicem impe-<pb pagenum="154" xlink:href="026/01/186.jpg"/>tum ad <expan abbr="eãdem">eandem</expan> lineam determinatam, deor&longs;um, v.g. <!-- REMOVE S-->in mobili proiecto; </s> <s id="N1A49B"><!-- NEW --><lb/>nec enim e&longs;t motus purè naturalis, nec etiam violentus, vt con&longs;tat; igi­<lb/>tur mixtus. </s> </p> <p id="N1A4A2" type="main"> <s id="N1A4A4"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A4B1" type="main"> <s id="N1A4B3"><!-- NEW --><emph type="italics"/>Cum proiicitur corpus per lineam horizontalem, vel inclinatum &longs;ur&longs;um, <lb/>vel deor&longs;um mobile percurrit lineam curuam<emph.end type="italics"/>; quod etiam pueri &longs;ciunt, qui <lb/>di&longs;co ludunt. </s> </p> <p id="N1A4C0" type="main"> <s id="N1A4C2"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A4CF" type="main"> <s id="N1A4D1"><!-- NEW --><emph type="italics"/>Globus etiam plumbeus è &longs;ummo malo malo mobilis nauis demi&longs;&longs;us per <lb/>lineam perpendicularem deor&longs;um minimè cadit, &longs;ed per curuam inclinatam<emph.end type="italics"/>: </s> <s id="N1A4DC"><!-- NEW --><lb/>hæc hypothe&longs;is mille &longs;altem nititur experimentis; </s> <s id="N1A4E1"><!-- NEW -->modò &longs;ufficiat quod <lb/>&longs;it; nam propter quid &longs;it, demon&longs;trabo. </s> </p> <p id="N1A4E7" type="main"> <s id="N1A4E9"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A4F6" type="main"> <s id="N1A4F8"><!-- NEW --><emph type="italics"/>Proiectum per horizontalem &longs;ub finem motus minùs ferit quàm initio, imò <lb/>& proiectum per inclinatam deor&longs;um<emph.end type="italics"/>; </s> <s id="N1A503"><!-- NEW -->hæc hypothe&longs;is centies probata fuit; <lb/>nec in dubium reuocari pote&longs;t. </s> </p> <p id="N1A509" type="main"> <s id="N1A50B"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A518" type="main"> <s id="N1A51A"><!-- NEW --><emph type="italics"/>Omnis impetus qui mobili ine&longs;t dum ip&longs;um mouetur, præ&longs;tat aliquid ad mo­<lb/>tum<emph.end type="italics"/>; </s> <s id="N1A525"><!-- NEW -->vel enim retardat, vt impetus innatus retardat violentum, vt &longs;uprà <lb/>diximus; vel ad motum vnà cum alio, vel &longs;olus concurrit. </s> <s id="N1A52B">Ax.2. </s> </p> <p id="N1A52E" type="main"> <s id="N1A530"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A53D" type="main"> <s id="N1A53F"><!-- NEW --><emph type="italics"/>Ille impetus qui alium retardat, haud dubiè retardat tantùm pro rata<emph.end type="italics"/>; <lb/>hoc etiam &longs;uprà demon&longs;trauimus, & qui de&longs;truitur, de&longs;truitur quoque <lb/>pro rata, ne &longs;it fru&longs;trà qui de&longs;truitur. </s> </p> <p id="N1A54C" type="main"> <s id="N1A54E"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A55B" type="main"> <s id="N1A55D"><!-- NEW --><emph type="italics"/>Ille impetus qui cum alio ad <expan abbr="eũdem">eundem</expan> motum concurrit, concurrit etiam pro <lb/>rata<emph.end type="italics"/>; hoc etiam &longs;uprà demon&longs;tratum e&longs;t, e&longs;t enim cau&longs;a nece&longs;&longs;aria, igitur <lb/>quantum pote&longs;t concurrit, igitur pro rata &longs;uæ virtutis. </s> </p> <p id="N1A56E" type="main"> <s id="N1A570"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A57D" type="main"> <s id="N1A57F"><!-- NEW --><emph type="italics"/>Licèt &longs;int plures impetus in eodem mobili, non &longs;unt tamen plures &longs;imul li­<lb/>neæ motus<emph.end type="italics"/>; ne mobile &longs;it &longs;imul in pluribus locis. </s> </p> <p id="N1A58A" type="main"> <s id="N1A58C"><emph type="center"/><emph type="italics"/>Po&longs;tulatum<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A599" type="main"> <s id="N1A59B"><emph type="italics"/>Liceat a&longs;&longs;umere quamlibet coniugationem motuum,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->vel duorum æ­<lb/>quabilium, vel alterius æquabilis, & alterius retardati, vel alterius æqua­<lb/>bilis, & alterius accelerati, vel alterius retardati, & alterius accelera­<lb/>ti, &c. </s> </p> <p id="N1A5AD" type="main"> <s id="N1A5AF"><emph type="center"/><emph type="italics"/>Po&longs;tulatum<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A5BC" type="main"> <s id="N1A5BE"><emph type="italics"/>Illa linea vocetur curua quæ con&longs;tat infinitis prope lateribus polygoni.<emph.end type="italics"/></s> </p> <p id="N1A5C5" type="main"> <s id="N1A5C7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A5D4" type="main"> <s id="N1A5D6"><!-- NEW --><emph type="italics"/>Motus mixtus ex duobus æquabilibus æqualibus e&longs;t rectus<emph.end type="italics"/>; &longs;it enim mo-<pb pagenum="155" xlink:href="026/01/187.jpg"/>bile in A, &longs;itque impetus per AB, & alter æqualis per AD, motus mixtus <lb/>fiet per AE, a&longs;&longs;umpta &longs;cilicet DE æquali, & parallela AB, quod probatur <lb/>per Th.137.l.1. <!-- KEEP S--></s> </p> <p id="N1A5E9" type="main"> <s id="N1A5EB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A5F8" type="main"> <s id="N1A5FA"><!-- NEW --><emph type="italics"/>Linea AE e&longs;t diagonalis quadrati, quotie&longs;cumque vterque impetus e&longs;t æ­<lb/>qualis, & lineæ determinationum decu&longs;&longs;antur ad angulos rectos<emph.end type="italics"/>; probatur per <lb/>idem Th.137. <!-- KEEP S--></s> </p> <p id="N1A608" type="main"> <s id="N1A60A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A617" type="main"> <s id="N1A619"><!-- NEW --><emph type="italics"/>Hinc de&longs;truitur aliquid impetus<emph.end type="italics"/>; </s> <s id="N1A622"><!-- NEW -->alioquin motus e&longs;&longs;et duplus cuiu&longs;li­<lb/>bet &longs;eor&longs;im &longs;umpti, quod fal&longs;um e&longs;t; </s> <s id="N1A628"><!-- NEW -->nam motus &longs;unt vt lineæ &longs;ed diago­<lb/>nalis quadrati non e&longs;t dupla lateris; hoc etiam probatur per Th. 141. <lb/>& 142.l.1. </s> </p> <p id="N1A630" type="main"> <s id="N1A632"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A63F" type="main"> <s id="N1A641"><!-- NEW --><emph type="italics"/>Motus mixtus ex duobus æquabilibus inæqualibus est etiam rectus<emph.end type="italics"/>; &longs;it <lb/>enim mobile in A eadem figura &longs;itque impetus per AC, & alter &longs;ubdu­<lb/>plus prioris per AD, motus fiet per AF ducta DF æquali, & parallela AC, <lb/>quod probatur per Th.137.l.1. <!-- KEEP S--></s> </p> <p id="N1A651" type="main"> <s id="N1A653"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A660" type="main"> <s id="N1A662"><!-- NEW --><emph type="italics"/>Linea AF e&longs;t diagonalis rectanguli, quotie&longs;cunque lineæ determinationum <lb/>decu&longs;&longs;antur ad angulos rectos<emph.end type="italics"/>; probatur per idem Th.137. <!-- KEEP S--></s> </p> <p id="N1A66E" type="main"> <s id="N1A670"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A67D" type="main"> <s id="N1A67F"><emph type="italics"/>Hinc de&longs;truitur aliquid impetus per Th.<emph.end type="italics"/>141. & 142.<emph type="italics"/>l.<emph.end type="italics"/>1. idque pro rata <lb/>ne aliquid &longs;it fru&longs;trà per Ax.2. & &longs;æpè iam probatum e&longs;t. </s> </p> <p id="N1A68F" type="main"> <s id="N1A691"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1A69D" type="main"> <s id="N1A69F"><!-- NEW --><emph type="italics"/>Hinc determinari pote&longs;t portio vtriu&longs;que impetus destructi,<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;i &longs;int æ­<lb/>quales, portio detracta vtrique æqualibus temporibus e&longs;t differentia <lb/>diagonalis & compo&longs;itæ ex DA, AB, quod clarum e&longs;t; &longs;i vero impetus <lb/>&longs;int inæquales, portio de&longs;tructa erit &longs;emper differentia diagonalis, v.g. <!-- REMOVE S--><lb/>AF & compo&longs;itæ ex AC.AD. </s> </p> <p id="N1A6B3" type="main"> <s id="N1A6B5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1A6C1" type="main"> <s id="N1A6C3"><!-- NEW --><emph type="italics"/>Aliquando impetus qui remanet in motu mixto est rationalis<emph.end type="italics"/>; </s> <s id="N1A6CC"><!-- NEW -->id e&longs;t habet <lb/>proportionem ad vtrumque, quæ appellari pote&longs;t, aliquando ad neutrum, <lb/><expan abbr="aliquãdo">aliquando</expan> ad alterutrum; </s> <s id="N1A6D7"><!-- NEW -->ad vtrumque v.g. <!-- REMOVE S-->&longs;i alter impetuum &longs;it 8.alter 6. <lb/>haud dubiè linea motus mixti erit 10. ad neutrum vt in diagonali qua­<lb/>drati, & in multis aliis; </s> <s id="N1A6E1"><!-- NEW -->ad alterum denique v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i alter &longs;it &longs;ubduplus la­<lb/>teris æquilateri; alter verò eiu&longs;dem perpendicularis; nam diagonalis, &longs;eu <lb/>linea motus mixti erit latus ip&longs;um æquilateri. </s> </p> <p id="N1A6ED" type="main"> <s id="N1A6EF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N1A6FB" type="main"> <s id="N1A6FD"><!-- NEW --><emph type="italics"/>Si lineæ determinationum decu&longs;&longs;entur ad angulum obtu&longs;um, &longs;intque æqua­<lb/>les impetus, linea motus mixti erit diagonalis Rhombi<emph.end type="italics"/>; </s> <s id="N1A708"><!-- NEW -->vt patet per Th.140. <lb/>l.1. pote&longs;t autem hæc diagonalis e&longs;&longs;e vel æqualis alteri laterum, vel ma-<pb pagenum="156" xlink:href="026/01/188.jpg"/>ior, vel minor; e&longs;t æqualis, quando angulus maior Rhombi e&longs;t 120. e&longs;t <lb/>minor cùm angulus minor e&longs;t 60. denique e&longs;t maior, cùm maior angu­<lb/>lus e&longs;t minor 120, quæ omnia con&longs;tant ex Geometria. <!-- KEEP S--></s> </p> <p id="N1A718" type="main"> <s id="N1A71A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N1A726" type="main"> <s id="N1A728"><!-- NEW --><emph type="italics"/>Si lineæ determinationum decu&longs;&longs;entur ad angulum acutum, & &longs;int æqua­<lb/>les impetus, linea motus mixti erit diagonalis Rhombi<emph.end type="italics"/>; quæ certè eò longior <lb/>erit, quò angulus erit acutior per Th. 139. l.1. porrò e&longs;t &longs;emper maior <lb/>lateribus &longs;eor&longs;im &longs;umptis. </s> </p> <p id="N1A737" type="main"> <s id="N1A739"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A745" type="main"> <s id="N1A747"><!-- NEW -->Ob&longs;erua in Rhombo e&longs;&longs;e duas diagonales inæquales, vt con&longs;tat; </s> <s id="N1A74B"><!-- NEW -->igi­<lb/>tur cùm lineæ determinationum decu&longs;&longs;antur ad angulum obtu&longs;um, linea <lb/>motus mixti &longs;emper e&longs;t diagonalis minor; cùm verò decu&longs;&longs;antur ad an­<lb/>gulum acutum, &longs;emper e&longs;t diagonalis maior. </s> </p> <p id="N1A755" type="main"> <s id="N1A757"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A764" type="main"> <s id="N1A766"><!-- NEW -->Hinc quò acutior e&longs;t angulus diagonalis accedit propiùs ad duplum <lb/>lateris, donec tandem vtraque linea coëat; tunc enim linea motus e&longs;t du­<lb/>pla lateris. </s> </p> <p id="N1A76E" type="main"> <s id="N1A770"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A77D" type="main"> <s id="N1A77F"><!-- NEW -->Hinc quoque quò angulus e&longs;t obtu&longs;ior diagonalis accedit propiùs ad <lb/>nullam, vt &longs;ic loquar, donec tandem vtraque linea concurrat in rectam, <lb/>tunc enim nulla e&longs;t diagonalis; igitur nulla linea motus. </s> </p> <p id="N1A787" type="main"> <s id="N1A789"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N1A795" type="main"> <s id="N1A797"><!-- NEW --><emph type="italics"/>Cum alter impetuum e&longs;t maior, linea motus e&longs;t diagonalis Rhomboidis, mi­<lb/>nor quidem &longs;i lineæ decu&longs;&longs;entur ad angulum obtu&longs;um; </s> <s id="N1A79F"><!-- NEW -->maior verò &longs;i decu&longs;&longs;en­<lb/>tur ad angulum acutum<emph.end type="italics"/>; vt patet ex dictis. </s> </p> <p id="N1A7A8" type="main"> <s id="N1A7AA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N1A7B6" type="main"> <s id="N1A7B8"><!-- NEW --><emph type="italics"/>Cum alter impetus in motu mixto est maior, linea motus mixti accedit <lb/>proprius ad lineam maioris; </s> <s id="N1A7C0"><!-- NEW -->hoc est facit angulum acutiorem cum illa<emph.end type="italics"/>; v.g. <!-- REMOVE S-->in <lb/>eadem figura &longs;it linea impetus maioris AC, & minoris AD, linea motus <lb/>mixti e&longs;t diagonalis AF, quæ accedit propiùs ad AC, quàm ad AD, id e&longs;t <lb/>facit angulum acutiorem cum AC, vt patet ex dictis. </s> </p> <p id="N1A7CF" type="main"> <s id="N1A7D1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N1A7DD" type="main"> <s id="N1A7DF"><!-- NEW --><emph type="italics"/>Cum verò impetus &longs;unt æquales, linea motus mixti facit angulum æqualem <lb/>cum linea vtriu&longs;que<emph.end type="italics"/>; vt AE in eadem figura quod etiam dici debet, licèt <lb/>lineæ determinationum decu&longs;&longs;entur ad angulum obtu&longs;um vel acutum, <lb/> vt AC, EG. IM. </s> </p> <p id="N1A7EE" type="main"> <s id="N1A7F0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N1A7FC" type="main"> <s id="N1A7FE"><!-- NEW --><emph type="italics"/>Non cre&longs;cit, vel decre&longs;cit in eadem ratione, in quæ vnus impetus &longs;uperat <lb/>alium<emph.end type="italics"/>; </s> <s id="N1A809"><!-- NEW -->cum enim impetus &longs;int vt lineæ, &longs;ub quibus fiunt rectangula vel <lb/>Rhomboides; v.g. <!-- REMOVE S-->impetus AC e&longs;t duplus impetus AD, &longs;ed angulus D <lb/>AF non e&longs;t duplus anguli FAC, vt con&longs;tat ex Geometria. <!-- KEEP S--></s> </p> <pb pagenum="157" xlink:href="026/01/189.jpg"/> <p id="N1A818" type="main"> <s id="N1A81A"><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A826" type="main"> <s id="N1A828"><!-- NEW -->Ob&longs;eruabis dari de facto hunc motum mixtum ex duobus æquabilibus <lb/>in rerum natura; </s> <s id="N1A82E"><!-- NEW -->talis e&longs;t motus nauis, quam geminus ventus impellit in <lb/>mari, vel nubis, imò aëris pars in medio aëre, atque adeo ip&longs;ius venti, <lb/>&longs;unt enim hi motus æquabiles per &longs;e; quippe retardantur &longs;olummodo <lb/>propter re&longs;i&longs;tentiam medij, non verò propter vllam grauitationem. </s> </p> <p id="N1A838" type="main"> <s id="N1A83A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N1A846" type="main"> <s id="N1A848"><!-- NEW --><emph type="italics"/>Motus mixtus ex duobus retardatis e&longs;t rectus<emph.end type="italics"/>; </s> <s id="N1A851"><!-- NEW -->&longs;it enim duplex impetus <lb/>per AE & AH æqualis; </s> <s id="N1A857"><!-- NEW -->ita vt in dato tempore percurrat &longs;eor&longs;im AE mo­<lb/>tu retardato; </s> <s id="N1A85D"><!-- NEW -->item AH iuxta proportionem Galilei; </s> <s id="N1A861"><!-- NEW -->certè eo tempore quo <lb/>percurreret AD in AE, & AI in AH percurrit AG motu mîxto per Th. <!-- REMOVE S--><lb/>5. Similiter eo tempore quo percurreret AE &longs;eor&longs;im, & AH, percurrit <lb/>AF per Th.5. Igitur hic motus mixtus e&longs;t rectus, dum &longs;it vterque retar­<lb/>datus iuxta <expan abbr="eãdem">eandem</expan> progre&longs;&longs;ionem; </s> <s id="N1A872"><!-- NEW -->&longs;imiliter &longs;i alter impetus impetus <lb/>&longs;it inæqualis, vt patet in &longs;equenti figura, &longs;it enim impetus per AE, & <lb/>alter minor per AH, certè ex AD, AI fit AG, & ex AE, AH fit AF, quam <lb/>rectam e&longs;&longs;e con&longs;tat ex Geometria; nec vlla e&longs;t difficultas, quæ ex &longs;upe­<lb/>rioribus Theorematis facilè &longs;olui non po&longs;&longs;it. </s> </p> <p id="N1A87E" type="main"> <s id="N1A880"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A88D" type="main"> <s id="N1A88F">Hinc linea motus mixti ex duobus retardatis &longs;iue æqualibus, &longs;iue <lb/>inæqualibus e&longs;t diagonalis parallelogrammatis &longs;ub lineis determina­<lb/>tionum. </s> </p> <p id="N1A896" type="main"> <s id="N1A898"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A8A4" type="main"> <s id="N1A8A6">Ob&longs;eruabis dari de facto hunc motum in rerum natura, &longs;i v. <!-- REMOVE S-->g. <!-- REMOVE S-->in pla­<lb/>no horizontali idem globus, vel &longs;imul gemino ictu impellatur, vel &longs;i iam <lb/>impul&longs;um mobile per nouam lineam impellatur. </s> </p> <p id="N1A8B1" type="main"> <s id="N1A8B3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N1A8BF" type="main"> <s id="N1A8C1"><!-- NEW --><emph type="italics"/>Motus mixtus ex duobus acceleratis uniformiter e&longs;t etiam rectus<emph.end type="italics"/>; </s> <s id="N1A8CA"><!-- NEW -->Proba­<lb/>tur, quia debet tantùm inuerti linea prioris &longs;cilicet mixti ex duobus re­<lb/>tardatis; </s> <s id="N1A8D2"><!-- NEW -->&longs;i enim à puncto F pellatur per FE, FH, motu accelerato, ita <lb/>primo, tempori re&longs;pondeat FM, FN, &longs;ecundo NH, ME; </s> <s id="N1A8D8"><!-- NEW -->haud dubiè li­<lb/>nea motus mixti erit FA; nam primò tempori re&longs;pondebit FG, & duo­<lb/>bus FA, vt con&longs;tat ex dictis, &longs;iue vterque impetus &longs;it æqualis, &longs;iue alter <lb/>maior altero. </s> </p> <p id="N1A8E2" type="main"> <s id="N1A8E4"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A8F1" type="main"> <s id="N1A8F3">Hinc etiam linea motus mixti ex duobus acceleratis e&longs;t diagonalis, <lb/>vt iam &longs;uprà dictum e&longs;t de omnibus aliis. </s> </p> <p id="N1A8F8" type="main"> <s id="N1A8FA"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A906" type="main"> <s id="N1A908"><!-- NEW -->Ob&longs;eruabis hunc motum dari in rerum natura &longs;altem in corporibus <lb/>&longs;ublunaribus; nec enim e&longs;t acceleratus ni&longs;i &longs;it motus naturalis, qui à <lb/>duplici impetu e&longs;&longs;e non pote&longs;t. </s> </p> <pb pagenum="158" xlink:href="026/01/190.jpg"/> <p id="N1A914" type="main"> <s id="N1A916"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N1A922" type="main"> <s id="N1A924"><!-- NEW --><emph type="italics"/>Si motus mixtus con&longs;tet ex æquabili, & accelerato naturaliter &longs;it per li­<lb/>neam curuam<emph.end type="italics"/>; </s> <s id="N1A92F"><!-- NEW -->&longs;it enim impetus per AF motu æquabili, & per AC motu <lb/>accelerato naturaliter, ita vt eo tempore quo percurritur &longs;eor&longs;im &longs;pa­<lb/>tium AB percurratur AD triplum; </s> <s id="N1A937"><!-- NEW -->certè ex vtroque primo tempore re­<lb/>&longs;ultat linea motus mixti AE, &longs;ecundo tempore EG, &longs;ed AEG non e&longs;t <lb/>recta; alioquin duo triangula ABE, ACG e&longs;&longs;ent proportionalia, quod <lb/>e&longs;t ab&longs;urdum. </s> </p> <p id="N1A941" type="main"> <s id="N1A943"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N1A94F" type="main"> <s id="N1A951"><!-- NEW --><emph type="italics"/>Hæc linea e&longs;t Parabola<emph.end type="italics"/>; </s> <s id="N1A95A"><!-- NEW -->quod ip&longs;e Galileus toties in&longs;inuauit, & quiuis <lb/>etiam rudior Geometra intelliget; in quo diutiùs non hæreo, præ&longs;ertim <lb/>cùm nullus &longs;it motus, qui con&longs;tet ex æquabili, & naturaliter accelerato, <lb/>vt demon&longs;trabimus infrà. </s> </p> <p id="N1A964" type="main"> <s id="N1A966"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N1A972" type="main"> <s id="N1A974"><!-- NEW --><emph type="italics"/>Si motus mixtus con&longs;tet ex æquabili & naturaliter retardato, fit per lineam <lb/>curuam<emph.end type="italics"/>; &longs;i enim eo <expan abbr="t&etilde;pore">tempore</expan> quo per NE &longs;ur&longs;um proiicitur corpus graue <lb/>& con&longs;equenter motu naturaliter retardato impellatur per NI motu <lb/>æquabili, diuidatur NI in 4. partes æquales v.g. <!-- REMOVE S-->ductis parallelis RD, <lb/>NE, PC, &c. </s> <s id="N1A98B"><!-- NEW -->a&longs;&longs;umatur NS vel RM, cui affigatur quilibet numerus impar; </s> <s id="N1A98F"><!-- NEW --><lb/>putà 7. itaque RM &longs;int 7. ducatur HM parallelæ IN, a&longs;&longs;umatur QL 5. <lb/>ducatur GL parallela, accipiatur VK 3. ducatur FK: </s> <s id="N1A996"><!-- NEW -->denique a&longs;&longs;umatur <lb/>FAI ducaturque AE parallela IN, & de&longs;cribatur per puncta AKLMN, <lb/>linea curua; </s> <s id="N1A99E"><!-- NEW -->hæc e&longs;t Parabola, vt con&longs;tat ex Geometria; </s> <s id="N1A9A2"><!-- NEW -->nam &longs;i BK e&longs;t 1. <lb/>CL erit 4. DM 9. EV 16. &longs;ed æquales &longs;unt AF.AG.AH.AI. prioribus vt <lb/>patet; </s> <s id="N1A9AA"><!-- NEW -->igitur &longs;agittæ &longs;unt vt quadrata <expan abbr="applicatarũ">applicatarum</expan>; </s> <s id="N1A9B2"><!-- NEW -->igitur hæc e&longs;t Parabola; <lb/>igitur curua, atqui motus mixtus prædictus fieret per hanc lineam, nam <lb/>eo tempore quo mobile e&longs;&longs;et in S, erit in M, concurrit enim vterque im­<lb/>petus pro rata, & eo tempore, quo e&longs;&longs;et in K erit in L, atque ita <lb/>deinceps. </s> </p> <p id="N1A9BE" type="main"> <s id="N1A9C0"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1A9CC" type="main"> <s id="N1A9CE"><!-- NEW -->Ob&longs;eruabis e&longs;&longs;e pror&longs;us inuer&longs;am prioris, quæ &longs;it ex motu æquabili, & <lb/>naturaliter accelerato; </s> <s id="N1A9D4"><!-- NEW -->&longs;i enim per AE &longs;it æquabilis & æqualis priori <lb/>per NI, & per AI &longs;it acceleratus, &longs;i quo tempore peruenit in B motu æ­<lb/>quabili perueniat in F motu accelerato; haud dubiè perueniet in K, mox <lb/>in L, &c. </s> <s id="N1A9DE"><!-- NEW -->quia eadem proportione, &longs;ed inuer&longs;a quâ retardatur, <lb/>acceleratur; </s> <s id="N1A9E4"><!-- NEW -->igitur &longs;i vltimo tempore retardati acquirit tantùm <lb/>YE; </s> <s id="N1A9EA"><!-- NEW -->primo tempore æquali &longs;cilicet accelerati acquiret AF, atque ita <lb/>deinceps &longs;i per NE &longs;it retardatus, & per NI æquabilis linea motus mixti <lb/>erit NLA; </s> <s id="N1A9F2"><!-- NEW -->&longs;i verò &longs;it per AI acceleratus, & per AE æquabilis æqualis <lb/>priori per NI, lineamosus mixti erit ALN eadem &longs;cilicet cum priori <lb/>mutatis tantùm terminis à quo, & ad quem; vtrùm verò in rerum natu­<lb/>ra &longs;it huiu&longs;modi motus videbimus infrà. </s> </p> <pb pagenum="159" xlink:href="026/01/191.jpg"/> <p id="N1AA00" type="main"> <s id="N1AA02"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N1AA0E" type="main"> <s id="N1AA10"><!-- NEW --><emph type="italics"/>Si con&longs;tet ex retardato & accelerato, vt fit in perpendiculari &longs;ur&longs;um, & <lb/>deor&longs;um motus mixtus, linea per quam fit e&longs;t curua,<emph.end type="italics"/> &longs;it enim retardatus <lb/>per AD, &longs;it acceleratus per AG, a&longs;&longs;umatur AB cum numero impari, putà <lb/>5.BC.3. CD.1. accipiatur AE.1. EF.3. ducantur parallelæ BK. CL. DI. <lb/>& aliæ EM. FH. GI. & per puncta AM. HI. ducatur linea curua, hæc e&longs;t <lb/>linea motus mixti ex retardato & accelerato; hæc porrò non e&longs;t Parabo­<lb/>la, vt con&longs;tat, quia quadratum AE non e&longs;t ad ad quadratum AF, vt qua­<lb/>dratum AB, vel EM ad quadratum FH, vel AC. <!-- KEEP S--></s> </p> <p id="N1AA28" type="main"> <s id="N1AA2A"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1AA36" type="main"> <s id="N1AA38"><!-- NEW -->Ob&longs;eruabis in fine huius motus amplitudinem, &longs;eu &longs;inum rectum li­<lb/>neæ &longs;cilicet GI, e&longs;&longs;e æqualem altitudini &longs;eu &longs;inui ver&longs;o, vel &longs;agittæ AG; </s> <s id="N1AA3E"><!-- NEW --><lb/>cùm enim motus naturaliter acceleratus in eadem proportione cre&longs;cat, <lb/>quod hic &longs;uppono, in qua retardatus decre&longs;cit; </s> <s id="N1AA45"><!-- NEW -->certè AG quæ e&longs;t linea <lb/>accelerati e&longs;t æqualis GI, quæ e&longs;t linea retardati: non tamen dicendum <lb/>e&longs;t lineam AI e&longs;&longs;e circulum, alioquin GH e&longs;&longs;et æqualis GI, &longs;ed GH e&longs;t, v. <!-- REMOVE S--><lb/>g. <!-- REMOVE S-->89. cum GI &longs;it radix quadr.81. e&longs;t enim 9. licèt GM &longs;it æqualis GH. <lb/>&longs;ed de his lineis infrà. </s> <s id="N1AA54">Vtrùm verò &longs;it aliquis motus huiu&longs;modi, videbi­<lb/>mus in &longs;equentibus Theorematis. <!-- KEEP S--></s> </p> <p id="N1AA5A" type="main"> <s id="N1AA5C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N1AA68" type="main"> <s id="N1AA6A"><!-- NEW --><emph type="italics"/>Quando corpus proiicitur per horizontalem in aëre libero, mouetur motu <lb/>mixto<emph.end type="italics"/>; </s> <s id="N1AA75"><!-- NEW -->probatur, quia &longs;unt duo impetus in eo corpore, &longs;cilicet innatus <lb/>deor&longs;um, & impre&longs;&longs;us per horizontalem, vt patet; igitur vterque aliquid <lb/>præ&longs;tat ad illum motum per Ax. 1. igitur e&longs;t motus mixtus per def. </s> <s id="N1AA7D">1. <!-- KEEP S--></s> </p> <p id="N1AA81" type="main"> <s id="N1AA83"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N1AA8F" type="main"> <s id="N1AA91"><emph type="italics"/>Ille motus non e&longs;t mixtus ex vtroque æquabili.<emph.end type="italics"/></s> <s id="N1AA98"> Demon&longs;tro; motus mixtus <lb/>ex vtroque æquabili e&longs;t rectus per Th.1.& 4. &longs;ed hic motus proiecti per <lb/>horizontalem non e&longs;t rectus per hyp.1. </s> </p> <p id="N1AA9F" type="main"> <s id="N1AAA1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N1AAAD" type="main"> <s id="N1AAAF"><!-- NEW --><emph type="italics"/>Ille motus non e&longs;t mixtus ex naturali æquabili & alio accelerato<emph.end type="italics"/>; patet, <lb/>quia nulla e&longs;t cau&longs;a, à qua violentus po&longs;&longs;it accelerari. </s> </p> <p id="N1AABA" type="main"> <s id="N1AABC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N1AAC8" type="main"> <s id="N1AACA"><!-- NEW --><emph type="italics"/>Non est mixtus ex naturali æquabili & violento retardato<emph.end type="italics"/>; </s> <s id="N1AAD3"><!-- NEW -->Primò, quia <lb/>cùm pro tata concurrant po&longs;t integrum quadrantem vix &longs;patium vnius <lb/>palmi confeci&longs;&longs;et in perpendiculari deor&longs;um per Th.59.l.2.quod tamen <lb/>e&longs;t contra experientiam.Secundò, quia ad aliquod tandem punctum per­<lb/>ueniretur, in quo mobile haberet tantùm impetum innatun; igitur nul­<lb/>lus e&longs;&longs;et ictus contra experientiam. </s> <s id="N1AAE1"><!-- NEW -->Tertiò, quia naturalis impetus in­<lb/>tenditur in plano inclinato; </s> <s id="N1AAE7"><!-- NEW -->igitur in motu per inclinatam, e&longs;t enim <lb/>motus deor&longs;um; igitur intenditur impetus naturalis, vt patet ex lib. 2. <lb/>igitur non e&longs;t mixtus. </s> </p> <pb pagenum="160" xlink:href="026/01/192.jpg"/> <p id="N1AAF3" type="main"> <s id="N1AAF5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N1AB01" type="main"> <s id="N1AB03"><!-- NEW --><emph type="italics"/>Motus ille non e&longs;t mixtus ex naturali retardator & violento æquabili, vel <lb/>accelerato<emph.end type="italics"/>; quia numquam de&longs;truitur impetus innatus, vt &longs;æpiùs dictum <lb/>e&longs;t &longs;uprà, tùm primo, tùm &longs;ecundo libro, nec in hoc e&longs;t vlla diffi­<lb/>cultas. </s> </p> <p id="N1AB12" type="main"> <s id="N1AB14"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N1AB20" type="main"> <s id="N1AB22"><!-- NEW --><emph type="italics"/>Non est mixtus ex naturali accelerato & violento æquabili<emph.end type="italics"/>; </s> <s id="N1AB2B"><!-- NEW -->demon&longs;tra­<lb/>tur, primò, quia &longs;ub finem motus e&longs;&longs;et maior impetus; </s> <s id="N1AB31"><!-- NEW -->quippè nihil de­<lb/>traheretur violento, &longs;ed multùm accederet naturali; igitur e&longs;&longs;et maior, <lb/>igitur e&longs;&longs;et maior ictus contra hyp. </s> <s id="N1AB39">3. &longs;ecundò, quotie&longs;cunque &longs;unt duo <lb/>impetus in eodem mobili ad diuer&longs;as lineas determinati, aliquid illo­<lb/>rum de&longs;truitur per Th.141.l.1.tertiò &longs;i e&longs;&longs;et vterque æquabilis, aliquid <lb/>de&longs;trueretur per Theorema 6. igitur potiori iure, &longs;i impetus naturalis <lb/>cre&longs;cat. </s> </p> <p id="N1AB44" type="main"> <s id="N1AB46"><!-- NEW -->Diceret fortè aliquis impetum de&longs;trui ab aëre, &longs;ed iam &longs;uprà re&longs;pon­<lb/>&longs;um e&longs;t modicum inde imminui; </s> <s id="N1AB4C"><!-- NEW -->nec enim vnquam aër in corpore graui <lb/>de&longs;truit tantùm impetus, quantùm producitur naturalis &longs;i &longs;it acceleratus; <lb/>alioquin motus deor&longs;um non cre&longs;ceret contra experientiam, & &longs;uprà in <lb/>toto ferè 2.lib. demon&longs;trauimus. </s> </p> <p id="N1AB56" type="main"> <s id="N1AB58"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N1AB64" type="main"> <s id="N1AB66"><!-- NEW --><emph type="italics"/>Hinc linea huius motus non e&longs;t Parabola<emph.end type="italics"/>; quia vt &longs;it Parabola, debet ille <lb/>motus con&longs;tare vel ex naturali æquabili, & violento retardato per Th. <!-- REMOVE S--><lb/>19. vel ex naturali accelerato & violento æquabili per Th. <!-- REMOVE S-->18. &longs;ed hic <lb/>motus neuter e&longs;t, non primum per Th. <!-- REMOVE S-->25. non &longs;ecundum per Theo­<lb/>rema 26. </s> </p> <p id="N1AB7C" type="main"> <s id="N1AB7E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N1AB8A" type="main"> <s id="N1AB8C"><!-- NEW --><emph type="italics"/>Hinc reiicies Galileum,<emph.end type="italics"/> qui in dialogis hæc &longs;emper &longs;uppo&longs;uit, &longs;ed nun­<lb/>quam probauit, nec probare vnquam potuit; </s> <s id="N1AB97"><!-- NEW -->hoc etiam &longs;upponunt <lb/>multi Galilei &longs;ectatores, qui cen&longs;ent impetum nunquam de&longs;trui ni&longs;i à <lb/>re&longs;i&longs;tentia medij; </s> <s id="N1AB9F"><!-- NEW -->&longs;ed quæro ab illis quodnam medium de&longs;truat partem <lb/>impetus in motu mixto; </s> <s id="N1ABA5"><!-- NEW -->nec enim linea motus mixti adæquat duas alias <lb/>ex quibus qua&longs;i re&longs;ultat; </s> <s id="N1ABAB"><!-- NEW -->certè hoc non pote&longs;t explicari cum infinitis fetè <lb/>aliis, ni&longs;i dicatur impetum de&longs;trui ab alio impetu, eo modo quo &longs;æpè <lb/>diximus, hoc e&longs;t ne &longs;it fru&longs;trà; igitur impetus violentus de&longs;truitur ab in­<lb/>nato, non tamen innatus à violento, vt &longs;æpiùs inculcauimus. </s> </p> <p id="N1ABB5" type="main"> <s id="N1ABB7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N1ABC3" type="main"> <s id="N1ABC5"><!-- NEW --><emph type="italics"/>Non e&longs;t mixtus ex naturali accelerato eo modo quo acceleratur deor&longs;um per <lb/>lineam perpendicularem & ex violento retardato<emph.end type="italics"/>: </s> <s id="N1ABD0"><!-- NEW -->Probatur, &longs;i ita e&longs;t, <expan abbr="tãtùm">tantùm</expan> <lb/>additur naturali, quantum detrahitur violento, imò plùs; </s> <s id="N1ABDA"><!-- NEW -->igitur &longs;emper <lb/>e&longs;t in eo mobili æqualis vel maior impetus; igitur æqualis e&longs;t &longs;emper, <lb/>vel maior ictus contra hyp. </s> <s id="N1ABE2"><!-- NEW -->3. adde quod non minùs impeditur ab im­<lb/>petu violento naturalis motus, quàm ab inclinato plano; </s> <s id="N1ABE8"><!-- NEW -->&longs;ed in plano <pb pagenum="161" xlink:href="026/01/193.jpg"/>inclinato non acceleratur motus cum eadem acce&longs;&longs;ione, qua &longs;cilicet in­<lb/>tenditur in perpendiculari deorsùm; </s> <s id="N1ABF3"><!-- NEW -->nec enim tam citò de&longs;cendit mobi­<lb/>le, quod certum e&longs;t, & in lib.de planis inclinatis demon&longs;trabo, cum tan­<lb/>tùm hîc &longs;upponam ad in&longs;tar phy&longs;icæ hypothe&longs;eos; adde quod idem mo­<lb/>bile proiectum per horizontalem in data di&longs;tantia minùs ferit, quàm pro­<lb/>iectum per inclinatam deor&longs;um. </s> </p> <p id="N1ABFF" type="main"> <s id="N1AC01"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N1AC0D" type="main"> <s id="N1AC0F"><!-- NEW --><emph type="italics"/>Itaque motus prædictus mixtus est ex violento retardato & naturali acce­<lb/>lerato, non eo quidem modo quo acceleratur in perpendiculari, &longs;ed eo quo acce­<lb/>leratur in plano inclinato, quod hic &longs;ingulis <expan abbr="in&longs;tãtibus">in&longs;tantibus</expan> mutatur<emph.end type="italics"/>; </s> <s id="N1AC20"><!-- NEW -->probatur pri­<lb/>mo, quia inductione facta non <expan abbr="cõ&longs;tat">con&longs;tat</expan> ex omnibus aliis; </s> <s id="N1AC2A"><!-- NEW -->&longs;unt enim tantùm <lb/>9 combinationes, quia &longs;unt tres differentiæ, &longs;cilicet æquabilibus, retarda­<lb/>tio, acceleratio; </s> <s id="N1AC32"><!-- NEW -->igitur &longs;i 3.ducantur in 3. &longs;unt 9. &longs;unt autem prima ex na­<lb/>turali, quem deinceps voco primum, æquabili & violento (quem voca­<lb/>bo &longs;ecundum) æquabili, &longs;ecunda ex prima æquabili & &longs;ecundo accelera­<lb/>to, tertia ex primo æquabili & &longs;ecundo retardato, quarta ex primo acce­<lb/>lerato & &longs;ecundo æquabili, quinta ex primo accelerato & &longs;ecundo acce­<lb/>lerato, &longs;exta ex primo accelerato & &longs;ecundo retardato, &longs;eptima ex primo <lb/>retardato & &longs;ecundo æquabili, octaua ex primo retardato & &longs;ecundo ac­<lb/>celerato, nona ex primo retardato, & &longs;ecundo retardato: non e&longs;t prima <lb/>per Th.22. non &longs;ecunda per Th. 21. non tertia per Th. 24. non quarta, <lb/>per Th.26. non quinta per T.2h.23. non &longs;exta per Th.29. eo modo quo <lb/>diximus, non &longs;eptima per Th. 25. non octaua per Th. 25. non denique <lb/>nona per Th.25. igitur debet e&longs;&longs;e alius motus, &longs;ed alius excogitari non <lb/>pote&longs;t præter illum quem adduxi. </s> <s id="N1AC4E"><!-- NEW -->Probatur &longs;ecundò, quia non minùs <lb/>impeditur ab impetu violento impetus naturalis acqui&longs;itus quàm à pla­<lb/>no inclinato vt iam dictum e&longs;t; </s> <s id="N1AC56"><!-- NEW -->igitur acceleratur quidem &longs;ed minùs; </s> <s id="N1AC5A"><!-- NEW -->nec <lb/>enim vterque e&longs;t æquabilis, nam linea e&longs;&longs;et recta per Th.4. & naturalis <lb/>cre&longs;cit quia de&longs;cendit deor&longs;um; præterea per Th.24. non pote&longs;t impetus <lb/>naturalis e&longs;&longs;e æquabilis, igitur non pote&longs;t violentus e&longs;&longs;e vel æquabilis, <lb/>vel acceleratus, igitur retardatus. </s> </p> <p id="N1AC66" type="main"> <s id="N1AC68"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N1AC74" type="main"> <s id="N1AC76"><!-- NEW --><emph type="italics"/>Motus naturalis acceleratus ex quo hic motus con&longs;tat acceleratur in alia <lb/>proportione quàm fit ea, in qua acceleratur, dum per idem planum inclina­<lb/>tum de&longs;cendit<emph.end type="italics"/>; </s> <s id="N1AC83"><!-- NEW -->probatur, quia &longs;ingulis in&longs;tantibus mutatur inclinatio pla­<lb/>ni &longs;eu lineæ; igitur &longs;ingulis in&longs;tantibus mutatur proportio accelera­<lb/>tionis. </s> </p> <p id="N1AC8B" type="main"> <s id="N1AC8D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N1AC99" type="main"> <s id="N1AC9B"><emph type="italics"/>Hinc perpetuò cre&longs;cit proportio accelerationis, quia &longs;emper cre&longs;cit inclina­<lb/>tio plani,<emph.end type="italics"/> vt patet, cùm enîm &longs;it linea curua per hyp. </s> <s id="N1ACA5">1. quo magis incur­<lb/>uatur, accedit propiùs ad perpendicularem, igitur motus magis accele­<lb/>ratur. </s> </p> <pb pagenum="162" xlink:href="026/01/194.jpg"/> <p id="N1ACB0" type="main"> <s id="N1ACB2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N1ACBE" type="main"> <s id="N1ACC0"><emph type="italics"/>Hinc ratio hypothe&longs;eos primæ,<emph.end type="italics"/> cùm enim con&longs;tet hic motus ex accelera­<lb/>to & retardato, eius linea e&longs;t curua per Th.20. non tamen e&longs;t Parabola, <lb/>vt con&longs;tat ex eodem Th.20. Vnde reiicies Galileum, qui vult lineam mo­<lb/>tus proiecti per horizontalem in aëre libero e&longs;&longs;e Parabolam. <!-- KEEP S--></s> </p> <p id="N1ACCF" type="main"> <s id="N1ACD1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N1ACDD" type="main"> <s id="N1ACDF"><!-- NEW --><emph type="italics"/>In hoc motu retardatur in maiori proportione violentus quàm acceleretur <lb/>natur alis<emph.end type="italics"/>; </s> <s id="N1ACEA"><!-- NEW -->probatur, non in minore, quia plùs impetus adderetur quàm de­<lb/>traheretur; igitur maior e&longs;&longs;et in fine motus quàm initio, igitur maior <lb/>ictus contra hyp.;. </s> <s id="N1ACF2">non in æquali, quia &longs;emper e&longs;&longs;et æqualis ictus con­<lb/>tra hyp.3.& contra Th.29. </s> </p> <p id="N1ACF7" type="main"> <s id="N1ACF9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N1AD05" type="main"> <s id="N1AD07"><!-- NEW --><emph type="italics"/>Hinc plùs detrahitur impetus quàm addatur,<emph.end type="italics"/> quia &longs;cilicet detrahitur <lb/>pro rata, vt dicemus infrà; at verò cùm acceleretur tantùm naturalis <lb/>iuxta rationem motus, & motus &longs;it iuxta rationem plani, minùs accele­<lb/>ratur quàm &longs;i caderet mobile perpendiculariter deor&longs;um. </s> </p> <p id="N1AD16" type="main"> <s id="N1AD18"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N1AD24" type="main"> <s id="N1AD26"><!-- NEW --><emph type="italics"/>Hinc ratio clara cur &longs;it minor ictus in &longs;ine huius motus<emph.end type="italics"/>; </s> <s id="N1AD2F"><!-- NEW -->quia &longs;cilicet e&longs;t <lb/>minùs impetus, quia plùs detractum e&longs;t quàm additum; </s> <s id="N1AD35"><!-- NEW -->nec e&longs;t quod <lb/>tribuant hanc retardationem medio; </s> <s id="N1AD3B"><!-- NEW -->quippe aër non plùs re&longs;i&longs;tit motui <lb/>violento quàm naturali; </s> <s id="N1AD41"><!-- NEW -->&longs;ed id quod detrahitur ab aëre corpori graui, v. <!-- REMOVE S--><lb/>g. <!-- REMOVE S-->pilæ plumbeæ e&longs;t in&longs;en&longs;ibile, vt fatentur omnes; igitur idem <expan abbr="dicen-dū">dicen­<lb/>dum</expan> e&longs;t de motu violento & mixto, hinc hoc ip&longs;um etiam fieret in vacuo. </s> </p> <p id="N1AD50" type="main"> <s id="N1AD52"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N1AD5E" type="main"> <s id="N1AD60"><!-- NEW --><emph type="italics"/>Impetus naturalis concurrit ad hunc motum<emph.end type="italics"/>; probatur, quia alioquin <lb/>e&longs;&longs;et rectus contra hyp. </s> <s id="N1AD6B">3. prætereà pote&longs;t concurrere; </s> <s id="N1AD6E"><!-- NEW -->nec enim &longs;unt li­<lb/>neæ determinationum oppo&longs;itæ; igitur concurrit per Th.137.l.1. <!-- KEEP S--></s> </p> <p id="N1AD75" type="main"> <s id="N1AD77"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N1AD83" type="main"> <s id="N1AD85"><!-- NEW --><emph type="italics"/>Si impetus naturalis non concurreret ad hunc motum, proiectum moueretur <lb/>per lineam horizontalem rectam, vt con&longs;tat, motu æquabili<emph.end type="italics"/>; po&longs;ito quod non <lb/>retardaretur in horizontali, eodem modo moueretur quo in verticali <lb/>&longs;ur&longs;um, quæ omnia con&longs;tant ex dictis &longs;uprà. </s> </p> <p id="N1AD94" type="main"> <s id="N1AD96"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N1ADA2" type="main"> <s id="N1ADA4"><!-- NEW --><emph type="italics"/>Patest vtrimque de&longs;cribi linea curua huius motus<emph.end type="italics"/>; </s> <s id="N1ADAD"><!-- NEW -->&longs;it enim mobile pro­<lb/>jectum ex E per horizontalem EI <expan abbr="eã">eam</expan> &longs;cilicet velocitate, quam acqui&longs;iui&longs;­<lb/>&longs;et motu naturaliter accelerato de&longs;cendendo ex A in E; </s> <s id="N1ADB9"><!-- NEW --><expan abbr="&longs;it&qacute;ue">&longs;itque</expan> AB &longs;pa­<lb/>tium acqui&longs;itum primo in&longs;tanti de&longs;cen&longs;us; BC duplum, CD triplum, &c. </s> <s id="N1ADC2"><!-- NEW --><lb/>iuxta progre&longs;&longs;ionem arithmeticam, &longs;it EI æqualis EA, diuidatur que eo­<lb/>dem modo in 4. &longs;patia vt diui&longs;a e&longs;t EA; </s> <s id="N1ADC9"><!-- NEW -->a&longs;&longs;umpta EO æqualis AB, ducan­<lb/>tur FN. GM. HL. IK. parallelæ EV; </s> <s id="N1ADCF"><!-- NEW -->a&longs;&longs;umatur OP æqualis OE, & PQ,<lb/>quæ &longs;it ad OE, vt OE ad hypothenu&longs;im &longs;eu planum inclinatum EN, a&longs;-<pb pagenum="163" xlink:href="026/01/195.jpg"/>&longs;inuatur QR æqualis OE, tum RS quæ &longs;it ad OE vt OQ ad planum incli­<lb/>natum NM; </s> <s id="N1ADDC"><!-- NEW -->denique a&longs;&longs;umatur ST æqualis OE, tum TV, quæ &longs;it ad OF, <lb/>vt QS ad inclinatam ML; ducantur ON. QM. SL. VK. parallelæ EI, <lb/>tùm per puncta E.N.M.L.X ducatur curua, hæc e&longs;t linea prædicti motus, <lb/>demon&longs;tratur. </s> </p> <p id="N1ADE6" type="main"> <s id="N1ADE8"><!-- NEW -->Impetus violentus percurrit EF eo tempore, quo naturalis percurrit <lb/>EO; </s> <s id="N1ADEE"><!-- NEW -->igitur linea motus mixti ex vtroque ducitur per punctum N, & licèt <lb/>videatur e&longs;&longs;e recta EN, &longs;cilicet diagonalis rectanguli OF, e&longs;t tamen cur­<lb/>ua, quia mobile non percurrit EF vno in&longs;tanti; </s> <s id="N1ADF6"><!-- NEW -->igitur nec EO, igitur <lb/>motu æqualiter accelerato percurrit EO; </s> <s id="N1ADFC"><!-- NEW -->igitur EN non e&longs;t recta per <lb/>Th.20. Præterea.Secundo tempore impetus innatus remanet; </s> <s id="N1AE02"><!-- NEW -->igitur per­<lb/>curratur OP cui addit ut PQ, quia impetus naturalis minùs cre&longs;cit, vt di­<lb/>ctum e&longs;t in Th.34. quippe cre&longs;cit iuxta rationem plani inclinati EN.ad <lb/>EO permutando, quæ &longs;it v.g. <!-- REMOVE S-->&longs;ubquadrupla; </s> <s id="N1AE0E"><!-- NEW -->igitur PQ e&longs;t &longs;ubquadrupla <lb/>EO; </s> <s id="N1AE14"><!-- NEW -->& cùm de&longs;trui &longs;upponatur vnus gradus violenti, v.g. <!-- REMOVE S-->&longs;uper&longs;unt tan­<lb/>tùm 3. quibus percurritur FG; igitur linea huius motus duci debet per <lb/>punctum M, idem dico de punctis L & K, igitur hæc e&longs;t linea motus <lb/>mixti, quàm &longs;cilicet corpus graue proiectum per horizontalem &longs;uo fluxu <lb/>de&longs;cribit, & cuius alias proprietates demon&longs;trabimus. </s> </p> <p id="N1AE22" type="main"> <s id="N1AE24"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N1AE30" type="main"> <s id="N1AE32"><emph type="italics"/>Hinc impetus naturalis in motu mixto cre&longs;cit &longs;emper in maiori proportione<emph.end type="italics"/><lb/>v.g. </s> <s id="N1AE3B">Oq.e&longs;t maior EO, & QS maior OQ atque ita deinceps. </s> </p> <p id="N1AE3E" type="main"> <s id="N1AE40"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N1AE4C" type="main"> <s id="N1AE4E"><!-- NEW --><emph type="italics"/>Impetus violentus hîc &longs;upponitur decre&longs;cere &longs;emper in eadem proportione<emph.end type="italics"/>; </s> <s id="N1AE57"><!-- NEW --><lb/>v.g. <!-- REMOVE S-->FG e&longs;t minor EF vno &longs;patio, GH minor EF vno &longs;patio; HI minor <lb/>GH vno &longs;patio, quæ omnia con&longs;tant. </s> <s id="N1AE60">Vtrùm verò id fiat, dicemus infrà, <lb/>& exempli gratia tantùm dictum e&longs;&longs;e volo. </s> </p> <p id="N1AE65" type="main"> <s id="N1AE67"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N1AE73" type="main"> <s id="N1AE75"><emph type="italics"/>Hinc quò maior e&longs;t impetus violentus in hoc motu, amplitudo huius linea <lb/>e&longs;t maior<emph.end type="italics"/> v.g. <!-- REMOVE S-->VK, quæ &longs;emper maior e&longs;t altitudine VE, vt enim e&longs;&longs;et æ­<lb/>qualis, impetus naturalis deberet cre&longs;cere in eadem proportione, in qua <lb/>decre&longs;cit violentus, vt dictum e&longs;t &longs;uprà. </s> </p> <p id="N1AE85" type="main"> <s id="N1AE87"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N1AE93" type="main"> <s id="N1AE95"><!-- NEW --><emph type="italics"/>Determinari po&longs;&longs;et hæc amplitudo, &longs;i decre&longs;cat violentus in EI, vt decre­<lb/>&longs;cit in verticali EA<emph.end type="italics"/>; </s> <s id="N1AEA0"><!-- NEW -->nam EI & EA &longs;unt æquales, &longs;ed EI & VK &longs;unt æqua­<lb/>les, AE verò e&longs;t linea, vel quam conficit mobile proiectum &longs;ur&longs;um cum <lb/>eodem, vel æquali impetu alteri quo proiicitur per horizontalem; &longs;eu <lb/>e&longs;t linea quam percurrit corpus graue deor&longs;um, dum acquirit æqualem <lb/>impetum alteri impre&longs;&longs;o eidem mobili per horizontalem EI. </s> </p> <p id="N1AEAC" type="main"> <s id="N1AEAE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N1AEBA" type="main"> <s id="N1AEBC"><!-- NEW --><emph type="italics"/>Hinc non pote&longs;t proijci in libero medio mobile graue per rectam horizonta­<lb/>lem<emph.end type="italics"/>; </s> <s id="N1AEC7"><!-- NEW -->quippe moueri non pote&longs;t ni&longs;i motu mixto ex naturali accelerato <pb pagenum="164" xlink:href="026/01/196.jpg"/>eo modo quo diximus, & violento retardato; </s> <s id="N1AED0"><!-- NEW -->igitur linea e&longs;t curua; dixi <lb/>in medio libero, cùm in plano duro horizontali per lineam rectam pro­<lb/>iici po&longs;&longs;it. </s> </p> <p id="N1AED8" type="main"> <s id="N1AEDA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N1AEE6" type="main"> <s id="N1AEE8"><!-- NEW --><emph type="italics"/>Hinc funis ten&longs;us, cuius &longs;cilicet vtraque extremitas immobiliter affixa e&longs;t, <lb/>nunquam e&longs;t rectus, &longs;ed inflectitur<emph.end type="italics"/>; </s> <s id="N1AEF3"><!-- NEW -->ratio e&longs;t, quia haud dubiè grauitat, igi­<lb/>tur incuruatur; vtrùm verò faciat Parabolam hæc linea curua, vt vult <lb/>Galileus, examinabimus in libro de lineis motus. </s> </p> <p id="N1AEFB" type="main"> <s id="N1AEFD"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1AF09" type="main"> <s id="N1AF0B"><!-- NEW -->Ob&longs;eruabis funem ten&longs;um &longs;emper incuruari, ni&longs;i fortè ex maxima tra­<lb/>ctione &longs;uam flexibilitatem amittat, cuius ope tantùm curuatur, imò ita <lb/>tendi pote&longs;t, vt ten&longs;ioni cedens frangatur: Equidem po&longs;ito quod vel in­<lb/>flecti po&longs;&longs;it, vel reduci, nece&longs;&longs;ariò inflectetur in medio, vt benè demon­<lb/>&longs;trat Galileus in dialogis, no&longs;que infrà ad potentiam vectis reducemus, <lb/>ne multiplicemus figuras. </s> </p> <p id="N1AF19" type="main"> <s id="N1AF1B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N1AF27" type="main"> <s id="N1AF29"><!-- NEW --><emph type="italics"/>Hinc ducitur optima ratio, cur proiectum per lineam horizontalem, v.g.pi­<lb/>la è tormento explo&longs;a, vel &longs;agitta arcu emi&longs;&longs;a per plura &longs;ecunda minuta mo­<lb/>ueatur in medio aëre antequam terram attingat<emph.end type="italics"/>; </s> <s id="N1AF36"><!-- NEW -->quod plu&longs;quàm mille ex­<lb/>perimentis comprobatum e&longs;t; </s> <s id="N1AF3C"><!-- NEW -->plura leges apud Mer&longs;ennum, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it tor­<lb/>mentum horizonti parallelum extans &longs;upra horizontem tribus pedibus; </s> <s id="N1AF46"><!-- NEW --><lb/>certum e&longs;t &longs;patium illud trium pedum confici à globo perpendiculariter <lb/>demi&longs;&longs;o tempore 30. tertiorum; </s> <s id="N1AF4D"><!-- NEW -->cùm tamen explo&longs;us per lineam hori­<lb/>zontalem terram tantùm attingat po&longs;t 4. &longs;ecunda, ide&longs;t 240. tertia; </s> <s id="N1AF53"><!-- NEW -->ita <lb/>Mer&longs;ennus l.2. de motu Prop. vltima, imò l. <!-- REMOVE S-->5. &longs;uæ ver&longs;ionis art.5. con­<lb/>tra Galileum o&longs;tendit glandem emi&longs;&longs;am è tormento minori conficere <lb/>75. exapedas, tempore vnius &longs;ecundi minuti in linea, quæ parùm decli­<lb/>nat ab horizontali; </s> <s id="N1AF61"><!-- NEW -->atqui tempore vnius &longs;ecundi minuti conficit 2.exa­<lb/>pedas in perpendiculari deor&longs;um; </s> <s id="N1AF67"><!-- NEW -->igitur deberet glans infrà &longs;copum de­<lb/>&longs;cendere notabiliter, id e&longs;t, toto 12. pedum interuallo, cùm tamen vix <lb/>tantillùm aberret à &longs;copo 1.Idem Mer&longs;ennus habet in Bali&longs;tica Prop.25. <lb/>globum è maiore tormento horizonti parallelo emi&longs;&longs;um in aëre tractu <lb/>continuo vola&longs;&longs;e toto tempore 8. &longs;ecundorum, antequam planum hori­<lb/>zontale attigi&longs;&longs;et, cùm tamen &longs;ex tantùm exapedis tormentum extaret <lb/>&longs;upra horizontem; </s> <s id="N1AF77"><!-- NEW -->alter globus ex alio tormento explo&longs;us 6. tantum &longs;e­<lb/>cunda in aëre con&longs;ump&longs;it; </s> <s id="N1AF7D"><!-- NEW -->imò bombardarum globi aliquando tota 14. <lb/>&longs;ecunda po&longs;uerunt; </s> <s id="N1AF83"><!-- NEW -->habet idem Mer&longs;ennus alia plura, quorum fides &longs;it <lb/>penes authores à quibus accepit; </s> <s id="N1AF89"><!-- NEW -->nam vt dicam quod res e&longs;t vix accu­<lb/>ratè minima illa tempora metiri po&longs;&longs;umus; </s> <s id="N1AF8F"><!-- NEW -->quidquid &longs;it, ex illis &longs;altem <lb/>euinco mobile projectum per horizontalem plùs temporis in&longs;umere in <lb/>&longs;uo fluxu, quam &longs;i ex eadem altitudine perpendiculariter demittatur; vt <lb/>vult Galileus; </s> <s id="N1AF99"><!-- NEW -->cuius ratio alia non e&longs;t ab ea, quàm &longs;uprà indicauimus, <lb/>quòd &longs;cilicet motus naturalis minùs cre&longs;cat in motu mixto quàm in na-<pb pagenum="165" xlink:href="026/01/197.jpg"/>turali, vt &longs;uprà demon&longs;trauimus; </s> <s id="N1AFA4"><!-- NEW -->imò &longs;i cre&longs;ceret vt vult Galileus, ictus; <lb/>haud dubiè e&longs;&longs;et maior in fine motus quàm initio, quod omninò expe­<lb/>rientiæ repugnat. </s> </p> <p id="N1AFAC" type="main"> <s id="N1AFAE"><!-- NEW -->Nec e&longs;t quod aliquis dicat glandem emi&longs;&longs;am per horizontalem tan­<lb/>tillùm a&longs;cendere; </s> <s id="N1AFB4"><!-- NEW -->vnde plus temporis in a&longs;cen&longs;u &longs;imul & de&longs;cen&longs;u col­<lb/>locatur, quàm in &longs;olo de&longs;cen&longs;u; </s> <s id="N1AFBA"><!-- NEW -->nam primò vix hoc aliquis &longs;ibi per&longs;ua­<lb/>&longs;erit, cùm experimento percipi non po&longs;&longs;it; </s> <s id="N1AFC0"><!-- NEW -->Secundò licèt verum e&longs;&longs;et, <lb/>non tamen e&longs;t tantus a&longs;cen&longs;us, quin adhuc plùs temporis ponat in a&longs;­<lb/>cen&longs;u, atqué in de&longs;cen&longs;u, quàm in alti&longs;&longs;ima perpendiculari quadruplæ al­<lb/>titudinis, vt con&longs;tat; </s> <s id="N1AFCA"><!-- NEW -->&longs;it enim horizontalis AF, di&longs;tans à plano hori­<lb/>zontali altitudine BA; </s> <s id="N1AFD0"><!-- NEW -->&longs;it tormentum directum per lineam AF, & glo­<lb/>bus percurrat lineam curuam AEF, idque &longs;patio 8.&longs;ecundorum minu­<lb/>torum; </s> <s id="N1AFD8"><!-- NEW -->&longs;itque DE 3. pedum; </s> <s id="N1AFDC"><!-- NEW -->certè eo tempore quo conficit AE, &longs;i in <lb/>perpendiculari conficiat ED, cum ED conficiat tempore 30‴; </s> <s id="N1AFE2"><!-- NEW -->haud <lb/>dubiè AE eodem tempore conficere deberet; </s> <s id="N1AFE8"><!-- NEW -->&longs;ed conficit AE tempore <lb/>4. &longs;ecundorum, vt con&longs;tat ex ip&longs;is multorum ob&longs;eruationibus; </s> <s id="N1AFEE"><!-- NEW -->igitur to­<lb/>tam AEF deberet percurrere tempore 1″, id e&longs;t eo tempore quo in per­<lb/>pendiculari deor&longs;um percurruntur 12. pedes; </s> <s id="N1AFF6"><!-- NEW -->denique &longs;i verum &longs;it glo­<lb/>bum a&longs;cendere tantillùm dum emittitur è tormento horizonti paralle­<lb/>lo; </s> <s id="N1AFFE"><!-- NEW -->crediderim id e&longs;&longs;e tùm ex aliqua repercu&longs;&longs;ione aëris, tùm eo quod à <lb/>flamma &longs;ur&longs;um a&longs;cendente &longs;ur&longs;um etiam aliquantulum inclinetur; </s> <s id="N1B004"><!-- NEW -->quod <lb/>verò &longs;pectat ad &longs;agittam, alia cau&longs;a non e&longs;t ni&longs;i modica aëris repercu&longs;&longs;io; </s> <s id="N1B00A"><!-- NEW --><lb/>e&longs;t enim leuior &longs;agittæ materia; &longs;ed de repercu&longs;&longs;ione fusè agemus <lb/>infrà. </s> </p> <p id="N1B011" type="main"> <s id="N1B013"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N1B01F" type="main"> <s id="N1B021"><!-- NEW --><emph type="italics"/>Motus projecti &longs;ur&longs;um per inclinatam e&longs;t mixtus<emph.end type="italics"/>; </s> <s id="N1B02A"><!-- NEW -->probatur, quia con&longs;tat <lb/>ex naturali, & violenti; qui cùm non &longs;int in oppo&longs;itis lineis, ad commu­<lb/>nem motum concurrunt, vt patet. </s> </p> <p id="N1B032" type="main"> <s id="N1B034"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N1B040" type="main"> <s id="N1B042"><!-- NEW --><emph type="italics"/>Non e&longs;t mixtus ex vtroque æquabili<emph.end type="italics"/>; quia linea e&longs;&longs;et recta per Th.1.&longs;ed <lb/>linea huius motus e&longs;t curua per hyp. </s> <s id="N1B04D">non pertinet etiam hic motus ad <lb/>&longs;ecundam combinationem de qua Th. 30. nec ad quintam, nec ad <lb/>octauam, nec ad nonam, de aliis videbimus infrà. </s> </p> <p id="N1B054" type="main"> <s id="N1B056"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N1B062" type="main"> <s id="N1B064"><!-- NEW --><emph type="italics"/>Non e&longs;t mixtus ex naturali accelerato, & violento æquabili<emph.end type="italics"/>; </s> <s id="N1B06D"><!-- NEW -->probatur, <lb/>quia in fine motus e&longs;&longs;et maior impetus, igitur e&longs;&longs;et maior ictus contra ex­<lb/>perientiam; </s> <s id="N1B075"><!-- NEW -->imò longè maior quàm &longs;i mobile proiiceretur per horizon­<lb/>talem, quia diutiùs durat ille motus; </s> <s id="N1B07B"><!-- NEW -->igitur plures gradus impetus na­<lb/>turalis acquiruntur; </s> <s id="N1B081"><!-- NEW -->igitur longè maior e&longs;t ictus; prætereà &longs;i impetus <lb/>naturalis de&longs;truit impetum &longs;ur&longs;um in verticali, cur non in inclinata? </s> <s id="N1B087"><!-- NEW -->nam <lb/>e&longs;t eadem omninò ratio; </s> <s id="N1B08D"><!-- NEW -->quippe ideò de&longs;truitur in verticali, quia cor­<lb/>pus graue &longs;ur&longs;um attollitur; </s> <s id="N1B093"><!-- NEW -->cùm tamen &longs;ua &longs;ponte deor&longs;um ferri debe­<lb/>ret; </s> <s id="N1B099"><!-- NEW -->&longs;ed non minùs, cùm per inclinatam &longs;ur&longs;um proiicitur, remouetur à <pb pagenum="166" xlink:href="026/01/198.jpg"/>&longs;uo centro, & &longs;ur&longs;um rapitur; </s> <s id="N1B0A2"><!-- NEW -->nec ob&longs;tat oppo&longs;itio lineæ verticalis &longs;ur­<lb/>&longs;um cum perpendiculari deor&longs;um; quia etiam per inclinatam deor&longs;um <lb/>fertur in plano inclinato, quæ opponitur ex diametro alteri inclinatæ <lb/>&longs;ur&longs;um. </s> </p> <p id="N1B0AC" type="main"> <s id="N1B0AE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s> </p> <p id="N1B0BA" type="main"> <s id="N1B0BC"><!-- NEW --><emph type="italics"/>Non e&longs;t mixtus in a&longs;cen&longs;u ex primo accelerato & &longs;ecundo retardato, acce­<lb/>lerato inquam eo modo quo acceleratur in perpendiculari deor&longs;um<emph.end type="italics"/>; </s> <s id="N1B0C7"><!-- NEW -->probatur <lb/>primò, quia motus ille e&longs;&longs;et &longs;emper æqualis, quia tantùm adderetur im­<lb/>petus quantùm detraheretur, igitur e&longs;&longs;et idem ictus in fine qui in princi­<lb/>pio; Secundò, quia tempora motuum e&longs;&longs;ent breuiora quàm par &longs;it con­<lb/>tra experientiam, vt patet ex Th.46. </s> </p> <p id="N1B0D3" type="main"> <s id="N1B0D5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N1B0E1" type="main"> <s id="N1B0E3"><!-- NEW --><emph type="italics"/>Non e&longs;t mixtus in a&longs;cen&longs;u ex violento retardato, & naturali accelerato, eo <lb/>modo quo diximus in Th.<emph.end type="italics"/> 30. probatur, quia cùm acceleretur iuxta ratio­<lb/>nem plani inclinati deor&longs;um, vt dictum e&longs;t, &longs;upra horizontalem; </s> <s id="N1B0F0"><!-- NEW -->nullum <lb/>e&longs;t ampliùs planum inclinatum deor&longs;um; </s> <s id="N1B0F6"><!-- NEW -->igitur nulla acceleratio, imò <lb/>impetus naturalis, vt iam &longs;uprà dictum e&longs;t cre&longs;cit tantùm vt motus deor­<lb/>&longs;um acceleretur; </s> <s id="N1B0FE"><!-- NEW -->&longs;ed nullus e&longs;t hîc motus deor&longs;um; </s> <s id="N1B102"><!-- NEW -->modicùm figuræ <lb/>rem ob oculos ponit; </s> <s id="N1B108"><!-- NEW -->motus in plano AB e&longs;t ad motum in AC vt <lb/>AC ad AB, & in AD, vt AD ad AB, & in AE, vt AE ad AB; </s> <s id="N1B10E"><!-- NEW -->igitur immi­<lb/>nuitur in infinitum; &longs;ed acceleratur in inclinata deor&longs;um iuxta hanc ra­<lb/>tionem, igitur nulla &longs;upere&longs;t ampliùs proportio, &longs;ecundum quam acce­<lb/>lerari po&longs;&longs;et in inclinata &longs;ur&longs;um. </s> </p> <p id="N1B118" type="main"> <s id="N1B11A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N1B126" type="main"> <s id="N1B128"><!-- NEW --><emph type="italics"/>Hic motus e&longs;t mixtus ex naturali æquabili, & violento retardato in a&longs;cen­<lb/>&longs;u<emph.end type="italics"/>; </s> <s id="N1B133"><!-- NEW -->probatur, quia nulla alia combinatio præter hanc &longs;upere&longs;t, quam <lb/>tertio loco &longs;uprà collocauimus in Th. 30. ratio à priori e&longs;t, quia natura­<lb/>lis innatus non retardatur; </s> <s id="N1B13B"><!-- NEW -->quia nunquam de&longs;truitur, nec acceleratur; </s> <s id="N1B13F"><!-- NEW --><lb/>quia &longs;ur&longs;um tendit mobile; </s> <s id="N1B144"><!-- NEW -->igitur &longs;upere&longs;t tantùm quod &longs;it æquabilis, <lb/>violentus verò non acceleratur, vt patet, quia nulla e&longs;t cau&longs;a: </s> <s id="N1B14A"><!-- NEW -->non e&longs;t <lb/>æquabilis, quia coniunctus e&longs;t cum cau&longs;a de&longs;tructiua; igitur e&longs;t re­<lb/>tardatus. </s> </p> <p id="N1B152" type="main"> <s id="N1B154"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N1B160" type="main"> <s id="N1B162"><!-- NEW --><emph type="italics"/>Hic motus e&longs;t mixtus in arcu de&longs;cen&longs;us ex naturali accelerato eo modo, quo <lb/>diximus &longs;uprà in Th.<emph.end type="italics"/> 30. <emph type="italics"/>& violento retardato<emph.end type="italics"/>; </s> <s id="N1B173"><!-- NEW -->probatur per idem Th.e&longs;t <lb/>enim par vtrique motui ratio; quippe hic perinde &longs;e habet, atque &longs;i mo­<lb/>bile per horizontalem proiiceretur, nam præuius motus <expan abbr="nequidquã">nequidquam</expan> facit. </s> </p> <p id="N1B17F" type="main"> <s id="N1B181"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N1B18D" type="main"> <s id="N1B18F"><!-- NEW --><emph type="italics"/>Arcus vterque constat linea curua<emph.end type="italics"/>; probatur per Th.19. non e&longs;t tamen <lb/>Parabola linea arcus de&longs;cen&longs;us per Th.20.& 27. </s> </p> <p id="N1B19A" type="main"> <s id="N1B19C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N1B1A8" type="main"> <s id="N1B1AA"><!-- NEW --><emph type="italics"/>Pote&longs;t hac linea vtcumque de&longs;cribi, &longs;uppo&longs;ita retardatione violenti in pro<emph.end type="italics"/>-<pb pagenum="167" xlink:href="026/01/199.jpg"/><emph type="italics"/>portione arithmetica &longs;implici<emph.end type="italics"/>; </s> <s id="N1B1BD"><!-- NEW -->&longs;it enim verticalis, AG horizontalis AN, <lb/>linea projectionis AD; </s> <s id="N1B1C3"><!-- NEW -->&longs;itque primum &longs;egmentum AD, quod &longs;cilicet <lb/>percurritur eo tempore quo in perpendiculari deor&longs;um percurritur DF, <lb/>id e&longs;t, v.g. <!-- REMOVE S-->&longs;exta eius pars, ducatur AFG, &longs;itque FG 5. partium, quarum <lb/>&longs;cilicet AD e&longs;t 6. a&longs;&longs;umatur GH æqualis DF, ducaturque FHI; </s> <s id="N1B1CF"><!-- NEW -->&longs;itque <lb/>HI 4. partium, a&longs;&longs;umatur IP æqualis GH, ducaturque HP; </s> <s id="N1B1D5"><!-- NEW -->accipiatur <lb/>PK 3. partium; </s> <s id="N1B1DB"><!-- NEW -->iam motus naturalis acceleratur eo modo quo &longs;uprà di­<lb/>ctum e&longs;t iuxta rationem inclinationis deor&longs;um; </s> <s id="N1B1E1"><!-- NEW -->itaque a&longs;&longs;umatur KL <lb/>paulo maior IP; &longs;imiliter ducatur PLM, &longs;itque LM duarum partium, <lb/>& MN paulò maior KL, tum &longs;it LNO, &longs;itque NO 1. partis, & OB ma­<lb/>ior MN, & ducatur curua per puncta A.F.H.P.L.N.B. & habebis <lb/>intentum. </s> </p> <p id="N1B1ED" type="main"> <s id="N1B1EF"><!-- NEW -->Porrò hæc linea non e&longs;t parabolica, vt con&longs;tat ex Geometria & plura <lb/>puncta habebis &longs;i minora &longs;patiola a&longs;&longs;umas; &longs;uppono enim DF e&longs;&longs;e tan­<lb/>tùm id &longs;patij quod primo in&longs;tanti in perpendiculari deor&longs;um à corpore <lb/>graui percurritur. </s> </p> <p id="N1B1F9" type="main"> <s id="N1B1FB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N1B207" type="main"> <s id="N1B209"><!-- NEW --><emph type="italics"/>Aliter hæc linea pote&longs;t de&longs;cribi &longs;uppo&longs;ita retardatione per numeros impa­<lb/>res; vt habes in fig.<emph.end type="italics"/> 46.T.1. in qua AC e&longs;t verticalis, AB horizontalis, <lb/>AD inclinata 9. partium, FG 7. HI 5. reliqua vt &longs;uprà dictum e&longs;t. </s> </p> <p id="N1B216" type="main"> <s id="N1B218"><!-- NEW -->Si verò linea inclinata recedat longiùs ab horizontali, & accedat pro­<lb/>piùs ad verticalem; vt habeantur puncta, transferantur eadem &longs;patia, & <lb/>habebis puncta, per quæ de&longs;cribes prædictam lineam. </s> </p> <p id="N1B220" type="main"> <s id="N1B222">Denique &longs;i inclinata accedat propiùs ad horizontalem, transferantur <lb/>&longs;imiliter &longs;patia vnius in alteram. </s> </p> <p id="N1B227" type="main"> <s id="N1B229">Ob&longs;eruabis autem crementa de&longs;cen&longs;us in GH. IB e&longs;&longs;e iuxta nume­<lb/>ros impares 1.3.5.7.&c. </s> <s id="N1B22E"><!-- NEW -->quandoquidem a&longs;&longs;umitur &longs;patium quod confi­<lb/>citur in tempore &longs;en&longs;ibili, habita tamen &longs;emper ratione accelerationis, <lb/>quæ fit in plano inclinato, vnde cre&longs;cit &longs;emper proportio acceleratio­<lb/>nis, vt &longs;uprà demon&longs;trauimus; quæ certè proportionum inæqualitas ef­<lb/>ficit, ne po&longs;&longs;int accuratè de&longs;cribi prædictæ lineæ, &longs;ed tantùm rudi Miner­<lb/>uâ, cum &longs;ingulis in&longs;tantibus mutetur proportio accelerationis. </s> </p> <p id="N1B23C" type="main"> <s id="N1B23E"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B24A" type="main"> <s id="N1B24C"><!-- NEW -->Ob&longs;eruabis nondum e&longs;&longs;e à nobis determinatam proportionem illam, <lb/>in qua de&longs;truitur impetus violentus in motu mixto, quæ tamen ex dictis <lb/>&longs;uprà pote&longs;t colligi; quippe de&longs;truitur pro rata, ide&longs;t qua proportione <lb/>linea motus mixti e&longs;t minor linea compo&longs;ita ex vtroque, &longs;it ergo. </s> </p> <p id="N1B256" type="main"> <s id="N1B258"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N1B264" type="main"> <s id="N1B266"><!-- NEW --><emph type="italics"/>Impetus violentus &longs;olus de&longs;truitur in arcu a&longs;cen&longs;us<emph.end type="italics"/>; </s> <s id="N1B26F"><!-- NEW -->probatur, quia natu­<lb/>ralis non cre&longs;cit, vt patet; con&longs;tat enim arcus a&longs;cen&longs;us ex naturali æqua­<lb/>bili, &longs;ed aliquis impetus decre&longs;cit, vt con&longs;tat ex dictis, igitur &longs;olus <lb/>violentus. </s> </p> <pb pagenum="168" xlink:href="026/01/200.jpg"/> <p id="N1B27D" type="main"> <s id="N1B27F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s> </p> <p id="N1B28B" type="main"> <s id="N1B28D"><!-- NEW --><emph type="italics"/>Impetus naturalis non decre&longs;cit etiam in arcu de&longs;cen&longs;us<emph.end type="italics"/>; probatur quia <lb/>cre&longs;cit, vt dictum e&longs;t &longs;uprà, igitur non decre&longs;cit. </s> </p> <p id="N1B298" type="main"> <s id="N1B29A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N1B2A6" type="main"> <s id="N1B2A8"><emph type="italics"/>De&longs;truitur impetus violentus pro rata. </s> <s id="N1B2AD">id e&longs;t, qua proportione e&longs;t frustrà;<emph.end type="italics"/><lb/>v.g. </s> <s id="N1B2B4"><!-- NEW -->&longs;it impetus per AD inclinatam &longs;ur&longs;um, & alius per AB perpendi­<lb/>cularem deor&longs;um; </s> <s id="N1B2BA"><!-- NEW -->haud dubiè motus erit per AC; </s> <s id="N1B2BE"><!-- NEW -->igitur concurrunt <lb/>ad motum AC motus AB & AD, vel potiùs impetus; </s> <s id="N1B2C4"><!-- NEW -->igitur debet de­<lb/>&longs;trui impetus in ea proportione, in qua AC e&longs;t minor AG, id e&longs;t com­<lb/>po&longs;ita ex AD, DC, quod impetus AB non po&longs;&longs;it de&longs;trui; </s> <s id="N1B2CC"><!-- NEW -->totum id <lb/>quod de&longs;truetur detrahetur impetui AD; </s> <s id="N1B2D2"><!-- NEW -->igitur a&longs;&longs;umatur DF &longs;cilicet <lb/>differentia AC, & AG; impetus de&longs;tructus ita &longs;e habet ad impetum <lb/>AD, vt DF ad AD, & ad re&longs;iduum impetum ex AD, vt DF ad FA, <lb/>quæ omnia con&longs;tant ex Th.7. &longs;it ergo AC fig. </s> <s id="N1B2DC"><!-- NEW -->49. perpendicularis &longs;ur­<lb/>&longs;um, AD inclinata, AB horizontalis; &longs;it impetus violentus re&longs;pondens <lb/>AD, & naturalis DG, ducatur AGK, ex AD detrahatur DF, id e&longs;t <lb/>differentia AG & compo&longs;itæ ex AD. DG, &longs;upere&longs;t AF, cui a&longs;&longs;umitur <lb/>æqualis GK, ex qua detrahitur KH, id e&longs;t differentia GL, & compo&longs;itæ <lb/>ex GK, KL, &longs;upere&longs;t GH, cui LO accipitur æqualis, cui detrahitur <lb/>OM, id e&longs;t differentia LP & compo&longs;itæ ex LO, OP, &longs;upere&longs;t ML, cui <lb/>æqualis accipitur PR, atque ita deinceps. </s> <s id="N1B2EE"><!-- NEW -->Porrò demon&longs;tratur de&longs;trui <lb/>impetum violentum iuxta hanc proportionem; </s> <s id="N1B2F4"><!-- NEW -->quia de&longs;truitur, qua <lb/>proportione e&longs;t fru&longs;trà, pro rata per Ax.2.& Th.7.&longs;ed totus impetus qui <lb/>concurrit ad &longs;ecundam lineam AG, e&longs;t compo&longs;itus ex AD, GD; </s> <s id="N1B2FC"><!-- NEW -->quia &longs;i <lb/>naturalis &longs;olus e&longs;&longs;et, percurreret &longs;patium æquale DG; </s> <s id="N1B302"><!-- NEW -->&longs;i verò &longs;olus e&longs;&longs;et <lb/>violentus percurreret &longs;patium æquale AD; </s> <s id="N1B308"><!-- NEW -->igitur vterque &longs;imul &longs;umptus <lb/>e&longs;t vt <expan abbr="cõpo&longs;ita">compo&longs;ita</expan>, ex AG. DG. igitur &longs;i ea proportione e&longs;t fru&longs;trà, qua motus <lb/>deficit, cùm AG &longs;it motus; </s> <s id="N1B314"><!-- NEW -->certè motus e&longs;t ad impetum, vt AG ad <expan abbr="compo-&longs;itã">compo­<lb/>&longs;itam</expan> ex AD. DG; </s> <s id="N1B31E"><!-- NEW -->igitur deficit motus tota DF quæ e&longs;t differentia AG & <lb/><expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> ex AD. DG; </s> <s id="N1B327"><!-- NEW -->igitur impetus e&longs;t fru&longs;trà in ratione DF; </s> <s id="N1B32B"><!-- NEW -->igitur de­<lb/>bet de&longs;trui in ratione DF; </s> <s id="N1B331"><!-- NEW -->&longs;ed impetus DG &longs;eu naturalis nihil de&longs;trui­<lb/>tur per Th.57. & 58. igitur ex violento AD de&longs;truitur DF; </s> <s id="N1B337"><!-- NEW -->igitur &longs;u­<lb/>pere&longs;t tantum AF vel æqualis GK; </s> <s id="N1B33D"><!-- NEW -->&longs;imiliter impetui GK & KL re­<lb/>&longs;pondet motus GL, &longs;ed GL e&longs;t minor compo&longs;ita ex GK & KL &longs;eg­<lb/>mento KH; </s> <s id="N1B345"><!-- NEW -->igitur e&longs;t fru&longs;trà impetus in ratione KH; </s> <s id="N1B349"><!-- NEW -->igitur de&longs;truitur <lb/>in eadem ratione KH, non ex naturali KL; </s> <s id="N1B34F"><!-- NEW -->igitur ex violento GK; <lb/>igitur &longs;upere&longs;t tantum GH, vel æqualis LO, in qua &longs;imiliter procedi­<lb/>tur. </s> <s id="N1B357">& &longs;upere&longs;t LM vel æqualis PR, atque ita deinceps. </s> </p> <p id="N1B35A" type="main"> <s id="N1B35C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B369" type="main"> <s id="N1B36B">Hinc de&longs;truitur impetus initio motus in maiori quantitate, quia <pb pagenum="169" xlink:href="026/01/201.jpg"/>DF. v. <!-- REMOVE S-->g. <!-- REMOVE S-->e&longs;t maxima omnium differentiarum. </s> </p> <p id="N1B377" type="main"> <s id="N1B379"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B386" type="main"> <s id="N1B388"><!-- NEW -->Hinc &longs;ub finem differentia lineæ motus v. <!-- REMOVE S-->g. <!-- REMOVE S-->TB &longs;emper e&longs;t maius <lb/>latus trianguli TXB; </s> <s id="N1B392"><!-- NEW -->idem dico de aliis; igitur differentia lineæ motus <lb/>& compo&longs;itæ ex duplici impetu e&longs;t &longs;emper minor & minor in in­<lb/>finitum. </s> </p> <p id="N1B39A" type="main"> <s id="N1B39C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B3A9" type="main"> <s id="N1B3AB">Po&longs;&longs;unt determinari à Geometria omnes anguli triangulorum ADG. <lb/>GKL. OLP. nam ADG e&longs;t æqualis CAD, at verò GKL æqualis <lb/>KGD, & hic duobus &longs;imul ADG & DAG, igitur determinari facilè <lb/>poterunt ex doctrina triangulorum. </s> </p> <p id="N1B3B4" type="main"> <s id="N1B3B6"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B3C3" type="main"> <s id="N1B3C5">Hinc etiam &longs;ciri poterit in quo puncto linea motus v.g. <!-- REMOVE S-->LP cum per­<lb/>pendiculari OP faciat angulum rectum, quod &longs;atis e&longs;t indica&longs;&longs;e, nam hic <lb/>Geometram non ago. </s> </p> <p id="N1B3CE" type="main"> <s id="N1B3D0"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B3DD" type="main"> <s id="N1B3DF">Hinc quoque &longs;ciri pote&longs;t maxima altitudo huius projectionis, quæ <lb/>&longs;cilicet in eo puncto e&longs;t, in quo linea motus cum perpendiculari deor­<lb/>&longs;um facit angulum rectum, v.g. <!-- REMOVE S-->in puncto P, &longs;i angulus LPO e&longs;t <lb/>rectus. </s> </p> <p id="N1B3EA" type="main"> <s id="N1B3EC"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B3F9" type="main"> <s id="N1B3FB"><!-- NEW -->Hinc pote&longs;t etiam &longs;ciri altitudo operâ triangulorum productorum <lb/>AG 2. GK 3. OLP. quod quiuis Geometra facilè intelliget; hîc quo­<lb/>que obiter ob&longs;erua vnum, quod &longs;æpè aliàs indicauimus, quanti videlicet <lb/>momenti &longs;it Geometria in rebus phy&longs;icis. </s> </p> <p id="N1B405" type="main"> <s id="N1B407"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1B413" type="main"> <s id="N1B415"><!-- NEW -->Hinc etiam colligo arcum a&longs;cen&longs;us maiorem e&longs;&longs;e arcu de&longs;cen&longs;us &longs;u­<lb/>pra idem planum horizontale AB; </s> <s id="N1B41B"><!-- NEW -->quia in arcu de&longs;cen&longs;us acceleratur <lb/>pro ratione diuer&longs;æ inclinationis impetus naturalis; </s> <s id="N1B421"><!-- NEW -->igitur lineam mo­<lb/>tus addunt propiùs ad perpendicularem, vt vides in TB; </s> <s id="N1B427"><!-- NEW -->igitur minùs <lb/>acquirit in horizontali; </s> <s id="N1B42D"><!-- NEW -->igitur minor amplitudo horizontalis &longs;ube&longs;t ar­<lb/>cui de&longs;cen&longs;us projectorum quàm arcui a&longs;cen&longs;us; dixi &longs;uprà idem pla­<lb/>num, quia arcus de&longs;cen&longs;us infra planum AB propagatur ferè in infi­<lb/>nitum. </s> </p> <p id="N1B437" type="main"> <s id="N1B439"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1B445" type="main"> <s id="N1B447"><!-- NEW -->Hinc reiicio Galileum qui nulla pror&longs;us fultus ratione phy&longs;ica vult <lb/>vtrumque e&longs;&longs;e æqualem, quod tamen omnibus experimentis repugnat, & <lb/>ip&longs;i etiam pueri, qui di&longs;co ludunt ob&longs;eruare po&longs;&longs;unt arcum de&longs;cen&longs;us &longs;ui <lb/>di&longs;ci e&longs;&longs;e longè minorem, nec e&longs;t quod ad &longs;uam Parabolam confugiat, <lb/>quæ duo fal&longs;a &longs;upponit principia, &longs;cilicet æquabilitatem motus violen­<lb/>ti, & accelerationem naturalis eo &longs;cilicet modo quo fieret in perpendi­<lb/>culari; at vtrumque fal&longs;um e&longs;&longs;e &longs;uprà demon&longs;trauimus, adde quod vt iam <pb pagenum="170" xlink:href="026/01/202.jpg"/>dixi in &longs;agitta emi&longs;&longs;a, projecto di&longs;co, &c. </s> <s id="N1B45C"><!-- NEW -->omnes ob&longs;eruare po&longs;&longs;unt ar­<lb/>cum a&longs;cen&longs;us maiorem e&longs;&longs;e arcu de&longs;cen&longs;us, quod etiam &longs;upponunt om­<lb/>nes, qui de re tormentaria &longs;crip&longs;erunt; præ&longs;ertim Vfanus tract. <!-- REMOVE S-->3. <lb/>c. <!-- REMOVE S-->13. </s> </p> <p id="N1B46A" type="main"> <s id="N1B46C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N1B478" type="main"> <s id="N1B47A">Hinc etiam colliges contra Vfanum globum è tormento emi&longs;&longs;um per <lb/>inclinatam &longs;ur&longs;um non ferri primò per lineam rectam, quia mouetur <lb/>motu mixto, qui rectus e&longs;&longs;e non pote&longs;t in hoc ca&longs;u per Th.54. </s> </p> <p id="N1B481" type="main"> <s id="N1B483"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N1B48F" type="main"> <s id="N1B491"><!-- NEW -->Motus mixtus arcus de&longs;cen&longs;us v&longs;que ad centrum terræ durare po&longs;&longs;et <lb/>&longs;i producerentur tot partes impetus quot &longs;unt in&longs;tantia illius motus; quia <lb/>cùm &longs;emper de&longs;truatur minor impetus, & minor in infinitum, po&longs;t ali­<lb/>quod &longs;patium de&longs;cen&longs;us tam parùm de&longs;truitur v&longs;que ad centrum terræ vt <lb/>non adæquet totus ille impetus primam partem primo in&longs;tanti de&longs;tru­<lb/>ctam, at tunc linea motus à perpendiculari deor&longs;um di&longs;tingui non <lb/>pote&longs;t. </s> </p> <p id="N1B4A1" type="main"> <s id="N1B4A3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N1B4AF" type="main"> <s id="N1B4B1"><!-- NEW -->Sed ne Geometriam omninò de&longs;picere videar, in circulo demon&longs;tro <lb/>proportiones omnes in quibus decre&longs;cit motus violentus per quamlibet <lb/>lineam inclinatam &longs;ur&longs;um, vel deor&longs;um; </s> <s id="N1B4B9"><!-- NEW -->&longs;it ergo circulus ADGQ cen­<lb/>tro B; </s> <s id="N1B4BF"><!-- NEW -->&longs;it motus violentus &longs;ur&longs;um BD coniunctus cum naturali BR, &longs;int­<lb/>que ex gr. <!-- REMOVE S-->BR. RQ æquales; </s> <s id="N1B4C7"><!-- NEW -->haud dubiè linea motus erit BC, quia na­<lb/>turalis BR pugnat pro rata per Th.134.l.1. eritque BC &longs;ubdupla BD; <lb/>igitur centro R. &longs;emidiametro RC de&longs;cribatur circulus CLPS, erit <lb/>æqualis priori, ducanturque ex centro B infinitæ lineæ BE. BF. BK. <lb/>BN, & vt res fit clarior, &longs;int omnes anguli DBE. EBF. FBG, &c. <lb/></s> <s id="N1B4D4">æquales &longs;cilicet grad. <!-- REMOVE S-->30. & ex punctis E.F.G.K.N.q. </s> <s id="N1B4D9"><!-- NEW -->ducantur lineæ <lb/>ad circunferentiam circuli CLPS. parallelæ DP.Dico omnes e&longs;&longs;e æqua­<lb/>les DC; </s> <s id="N1B4E1"><!-- NEW -->nam primò FH. GL. KM. QP &longs;unt æquales, vt patet: </s> <s id="N1B4E5"><!-- NEW -->deinde <lb/>CE & QO &longs;unt æquales; </s> <s id="N1B4EB"><!-- NEW -->igitur EV. OX, quod etiam certum e&longs;t; igi­<lb/>tur &longs;i &longs;upponatur idem motus violentus æqualis BD per omnes inclina­<lb/>tas BE. BF, &c. </s> <s id="N1B4F3">coniunctus naturali æquali BR; </s> <s id="N1B4F6"><!-- NEW -->primum &longs;patium erit <lb/>BC, &longs;ecundum BV, tertium BH, quartum BL, quintum BM, &longs;extum <lb/>BO<emph type="sub"/>2<emph.end type="sub"/> &longs;eptimum BP. quod certè mirabile e&longs;t; </s> <s id="N1B504"><!-- NEW -->nam ex BE. EV. fit BV per <lb/>Th.5. &longs;imiliter ex BF. FH. fit BH, ex BG. GL. fit BL; </s> <s id="N1B50A"><!-- NEW -->denique ex <lb/><expan abbr="Bq.">Bque</expan> QP fit BP; iam verò proportiones i&longs;tarum linearum ex Trigo­<lb/>nometria facilè intelligi po&longs;&longs;unt. </s> </p> <p id="N1B515" type="main"> <s id="N1B517"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N1B523" type="main"> <s id="N1B525"><!-- NEW --><emph type="italics"/>Iactus per horizontalem, & per verticalem nihil acquirit per &longs;e in eodem <lb/>plane horizontali, vnde incipit iactus<emph.end type="italics"/>; </s> <s id="N1B530"><!-- NEW -->probatur, quia verticalis iactus per <lb/><expan abbr="eãdem">eandem</expan> lineam redit; </s> <s id="N1B539"><!-- NEW -->horizontalis verò &longs;tatim de&longs;cendit; quia motus <pb pagenum="171" xlink:href="026/01/203.jpg"/>mixtus e&longs;t per Th.44. dixi per &longs;e, nam fortè per accidens fieri pote&longs;t, vt <lb/>iactus horizontalis habeat arcum a&longs;cen&longs;us, & de&longs;cen&longs;us. </s> </p> <p id="N1B544" type="main"> <s id="N1B546"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N1B552" type="main"> <s id="N1B554"><!-- NEW --><emph type="italics"/>Hinc quò iactus propiùs accedit ad horizontalem &longs;eu verticalem, minùs <lb/>acquirit in eodem plano horizontali, &longs;cilicet in eo à cuius extremitate inci­<lb/>pit iactus<emph.end type="italics"/>; </s> <s id="N1B561"><!-- NEW -->probatur, quia cùm iactus verticalis nihil pror&longs;us acqui­<lb/>rat in horizontali plano per Theorema 60. certè quò propiùs ad illum <lb/>iactus inclinatus accedet, minùs acquiret; idem dico de iactu hori­<lb/>zontali. </s> </p> <p id="N1B56B" type="main"> <s id="N1B56D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N1B579" type="main"> <s id="N1B57B"><!-- NEW --><emph type="italics"/>Hinc quò iactus longiùs recedit ab vtroque &longs;cilicet à verticali, & hori­<lb/>zontali, plùs acquiret in eodem plano horizontali<emph.end type="italics"/>; &longs;i enim quò plùs ac­<lb/>cedit ad vtrumque, minùs acquirit, igitur plùs acquirit, quò plùs re­<lb/>cedit. </s> </p> <p id="N1B58A" type="main"> <s id="N1B58C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s> </p> <p id="N1B598" type="main"> <s id="N1B59A"><!-- NEW --><emph type="italics"/>Hinc iactus medius &longs;eu per inclinatam qua cum verticali, vel horizontali <lb/>facit angulum<emph.end type="italics"/> 45.<emph type="italics"/>&longs;eu &longs;emirectum, e&longs;t omnium maximus, id e&longs;t plùs acqui­<lb/>rit in eodem plano horizontali, quàm reliqui omnes<emph.end type="italics"/>; </s> <s id="N1B5AD"><!-- NEW -->experientia certi&longs;&longs;ima <lb/>e&longs;t, ratio e&longs;t quia ab horizontali & verticali maximè omnium di&longs;tat; <lb/>igitur maximus e&longs;t per Theorema 62. nec e&longs;t vlla alia ratio geome­<lb/>trica. </s> </p> <p id="N1B5B7" type="main"> <s id="N1B5B9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s> </p> <p id="N1B5C5" type="main"> <s id="N1B5C7"><!-- NEW --><emph type="italics"/>Iactus qui æqualiter ab horizontali & verticali di&longs;tant, &longs;unt æquales<emph.end type="italics"/>; </s> <s id="N1B5D0"><!-- NEW --><lb/>probatur, quia qua proportione ad horizontalem &longs;eu verticalem acce­<lb/>dit iactus, in ea proportione minor e&longs;t; </s> <s id="N1B5D7"><!-- NEW -->igitur qui æqualiter acce­<lb/>dunt in proportione æquali, minores &longs;unt; </s> <s id="N1B5DD"><!-- NEW -->igitur æquales, quod mo­<lb/>dica figura ob oculos ponet; </s> <s id="N1B5E3"><!-- NEW -->&longs;it enim quadrans ABF, iactus verti­<lb/>calis AB, horizontalis AF, medius AD, hic maximus omnium <lb/>erit; at verò AC, & AE, qui ab AD æqualiter di&longs;tant, erunt æ­<lb/>quales. </s> </p> <p id="N1B5ED" type="main"> <s id="N1B5EF"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B5FB" type="main"> <s id="N1B5FD">Ob&longs;eruabis primò, omitti à me multa quæ &longs;uis Parabolis aliqui af­<lb/>fingunt, quæ nec experimentis, nec vllis rationibus con&longs;en­<lb/>tiunt. </s> </p> <p id="N1B604" type="main"> <s id="N1B606"><!-- NEW -->Secundò rationem i&longs;torum omnium Theorematum; </s> <s id="N1B60A"><!-- NEW -->quia quo iactus <lb/>ad verticalem propiùs accedit, maior quantitas impetus de&longs;truitur <lb/> v.g. <!-- REMOVE S-->in AD plùs quàm in GK; </s> <s id="N1B614"><!-- NEW -->igitur citò deficiunt vires huic iactui; </s> <s id="N1B618"><!-- NEW --><lb/>adde quod acquirit in verticali, quod alius acquirit in horizontali; </s> <s id="N1B61D"><!-- NEW -->at <pb pagenum="172" xlink:href="026/01/204.jpg"/>verò qui propiùs accedit ad horizontalem citò de&longs;cendit infra planum <lb/>horizontale, tùm quia propior e&longs;t, tum quia citò naturalis impetus <lb/>acceleratur; </s> <s id="N1B62A"><!-- NEW -->igitur plùs acquirit in perpendiculari deor&longs;um, quàm in <lb/>horizontali; quæ omnia ex certis principiis, non fictitiis dedu­<lb/>cuntur. </s> </p> <p id="N1B632" type="main"> <s id="N1B634"><!-- NEW -->Tertiò, ob&longs;eruabis talem e&longs;&longs;e hypothe&longs;im illam Paraboli&longs;tarum, de <lb/>qua &longs;uprà; </s> <s id="N1B63A"><!-- NEW -->&longs;it enim iactus verticalis EA; </s> <s id="N1B63E"><!-- NEW -->medius EB; </s> <s id="N1B642"><!-- NEW -->certè ex eorum <lb/>etiam principio eo tempore, quo motu æquabili percurreret mobile &longs;pa­<lb/>tium EA, motu naturaliter retardato percurreret &longs;patium EG &longs;ubdu­<lb/>plum; </s> <s id="N1B64C"><!-- NEW -->atqui percurrit EG eo tempore, quo idem percurreret GE motu <lb/>naturaliter accelerato; </s> <s id="N1B652"><!-- NEW -->&longs;ed percurret inclinatam EC eo tempore quo <lb/>percurret EA, &longs;cilicet motu æquabili; </s> <s id="N1B658"><!-- NEW -->&longs;unt enim æquales: Volunt autem <lb/>FE diuidi in 16. partes, & ED in 8. ducique parallelas HQ IP, &c. </s> <s id="N1B65E"><!-- NEW -->& ac­<lb/>cipi VR (1/16) FE, ita vt RQ &longs;it ad RH vt 9.ad 7. & PS (4/16) & NT (9/16), vel O <lb/>T (1/16) PS (4/16) PR (9/16); </s> <s id="N1B666"><!-- NEW -->igitur eo tempore, quo mobile e&longs;&longs;et in IX, erit in M; </s> <s id="N1B66A"><!-- NEW --><lb/>igitur motus naturalis acqui&longs;iuit XM, id e&longs;t 1/4 AE; </s> <s id="N1B66F"><!-- NEW -->igitur eo tempore quo <lb/>e&longs;&longs;et in B erit in D; </s> <s id="N1B675"><!-- NEW -->igitur motus naturalis acqui&longs;iuit BD quadruplum X <lb/>M; </s> <s id="N1B67B"><!-- NEW -->nam &longs;i vno tempore motu æquabili conficit EX, duobus conficit E <lb/>D & &longs;i motu naturaliter accelerato conficit vno tempore XM, duobus <lb/>conficit BD iuxta proportionem Galilei, in qua &longs;patia &longs;unt vt temporum <lb/>quadrata; </s> <s id="N1B685"><!-- NEW -->& quo tempore motu æquabili conficeret EA, vel EB naturali <lb/>conficeret GE vel CZ æqualem GE; ducatur igitur linea per puncta E. <lb/>RS, OM, hæc e&longs;t &longs;emiparabola cui &longs;i addas MZD, habebis totam ampli­<lb/>tudinem Parabolæ ED, hoc e&longs;t totum &longs;patium, quod acquirit in plano <lb/>horizontali ED iactus medius EB. <!-- KEEP S--></s> </p> <p id="N1B692" type="main"> <s id="N1B694"><!-- NEW -->Si verò &longs;it inclinata EY; </s> <s id="N1B698"><!-- NEW -->vt habeatur iuxta hanc hypothe&longs;im amplitu­<lb/>do horizontalis; </s> <s id="N1B69E"><!-- NEW -->fiat &longs;emicirculus centro G, &longs;emidiametro GE; </s> <s id="N1B6A2"><!-- NEW -->&longs;it per­<lb/>pendicularis YK, erit &longs;ubdupla amplitudo; </s> <s id="N1B6A8"><!-- NEW -->&longs;icut perpendicularis XL de­<lb/>finit &longs;ubduplam amplitudinem LE iactus EB; </s> <s id="N1B6AE"><!-- NEW -->&longs;imiliter YK definit &longs;ubdu­<lb/>plam amplitudinem iactus E 4.3. nam arcus YX e&longs;t æqualis arcui X 4. <lb/>igitur anguli YEC, CE. 3. &longs;unt æquales; hinc iactus &longs;unt æquales &longs;upra, & <lb/>infra grad.45. vt autem habeatur altitudo Parabolæ &longs;ubdupla XL e&longs;t al­<lb/>titudo Parabolæ iactus EC, &longs;ubdupla YX e&longs;t altitudo iactus EY, &longs;ubdu­<lb/>pla 4.K e&longs;t altitudo iactus E 3. <!-- KEEP S--></s> </p> <p id="N1B6BD" type="main"> <s id="N1B6BF"><!-- NEW -->Ex his facilè iuxta hypethe&longs;im tabulæ omnium iactuum, cuiu&longs;libet <lb/>eleuationis con&longs;trui po&longs;&longs;unt; </s> <s id="N1B6C5"><!-- NEW -->de quibus habes plura apud Galileum in <lb/>dialogis, & plurima apud Mer&longs;ennum in Bali&longs;tica; </s> <s id="N1B6CB"><!-- NEW -->quare ab illis ab&longs;ti­<lb/>neo: præ&longs;ertim cum &longs;it fal&longs;a illa hypothe&longs;is, eiu&longs;que &longs;ectatores vltrò fa­<lb/>teantur tabulas illas non parum à vero abe&longs;&longs;e, de quo vide Mer&longs;ennum <lb/>prop. 30. Bali&longs;t. <!-- KEEP S--></s> </p> <p id="N1B6D6" type="main"> <s id="N1B6D8"><!-- NEW -->Quartò, po&longs;&longs;unt iuxta no&longs;tram hypothe&longs;im tabulæ nouæ con&longs;trui, quod <lb/>& ego præ&longs;tarem, ni&longs;i pror&longs;us inutiles e&longs;&longs;ent; </s> <s id="N1B6DE"><!-- NEW -->quare prudenter omi&longs;&longs;as <lb/>e&longs;&longs;e prudentes omnes cen&longs;ebunt, cum hîc calculatorem non <expan abbr="agã">agam</expan>, &longs;ed phi­<lb/>lo&longs;ophum; </s> <s id="N1B6EA"><!-- NEW -->id certè tolerari potuit in analyticis, quæ &longs;ine calculationibus <lb/>intelligi non po&longs;&longs;unt; </s> <s id="N1B6F0"><!-- NEW -->&longs;ed minimè ferendum in Phy&longs;ica, quæ &longs;ucculen-<pb pagenum="173" xlink:href="026/01/205.jpg"/>tior e&longs;t, quàm vt numeris tantùm, <expan abbr="&longs;icci&longs;&qacute;ue">&longs;icci&longs;que</expan> calculis nutriatur; </s> <s id="N1B6FD"><!-- NEW -->adde quod <lb/>Praxis Theoricæ in his omninò præferenda e&longs;t; </s> <s id="N1B703"><!-- NEW -->quamquam huic etiam <lb/>parti dee&longs;&longs;e nolumus, &longs;ed in &longs;ingularem libellum omnes i&longs;tas tabulas & <lb/>alias huiu&longs;modi remittimus; cum hic tantùm rerum phy&longs;icarum cau&longs;as <lb/>explicemus. </s> </p> <p id="N1B70D" type="main"> <s id="N1B70F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s> </p> <p id="N1B71B" type="main"> <s id="N1B71D"><emph type="italics"/>Si accipiatur planum horizontale intra illud vnde incipit iactus haud du­<lb/>biè iactus omnium maximus erit horizontalis in vtraque hypothe&longs;i.<emph.end type="italics"/></s> <s id="N1B726"><!-- NEW --> Primo in <lb/>hypothe&longs;i Galilci, in qua Parabola GD figurâ &longs;uperiore habet maximum <lb/>omnium amplitudinem; </s> <s id="N1B72E"><!-- NEW -->licèt iactus per GX; </s> <s id="N1B732"><!-- NEW -->ex quo &longs;equitur, non ha­<lb/>beat impetum maiorem, quâm iactus per EY, vel EX; </s> <s id="N1B738"><!-- NEW -->in no&longs;tra verò, ia­<lb/>ctus per BG primo tempore plùs acquirit in horizontali BG, quàm ia­<lb/>ctus per BF; </s> <s id="N1B740"><!-- NEW -->igitur plùs etiam &longs;ecundo tempore; </s> <s id="N1B744"><!-- NEW -->nam BF acquirit tantùm <lb/>primo tempore BH, at verò BG acquirit RL; </s> <s id="N1B74A"><!-- NEW -->adde quod minùs perit ex <lb/>iactu BG; </s> <s id="N1B750"><!-- NEW -->quippe a&longs;&longs;umatur BL in B 2. & GL in 2. 3. detrahitur tantùm <lb/>G. 3.ex BG; </s> <s id="N1B756"><!-- NEW -->at verò a&longs;&longs;umatur BH in B 4. & FH in 4.5. detrahitur F 5.ex <lb/>BF; </s> <s id="N1B75C"><!-- NEW -->igitur plùs ex BF quàm ex BG; quæ omnia ex &longs;uperioribus regulis <lb/>iu&longs;ta no&longs;tram hypothe&longs;im præ&longs;criptis con&longs;equuntur. </s> </p> <p id="N1B762" type="main"> <s id="N1B764"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s> </p> <p id="N1B770" type="main"> <s id="N1B772"><!-- NEW --><emph type="italics"/>Immò probabile e&longs;t æquales fore iactus per inclinatas &longs;ur&longs;um, & deor&longs;um <lb/>æqualiter ab horizontali, vnde incipit iactus, distantes; </s> <s id="N1B77A"><!-- NEW -->æquales inquam in ali­<lb/>quo plano horizontali, inferiore<emph.end type="italics"/>; </s> <s id="N1B783"><!-- NEW -->&longs;i enim iactus fiat per BD eadem figura & <lb/>BP nihil acquiritur in horizontali, vt con&longs;tat; </s> <s id="N1B789"><!-- NEW -->&longs;i verò iactus &longs;it per BG <lb/>maximum &longs;patium acquirunt in horizontali plano inferiore; </s> <s id="N1B78F"><!-- NEW -->igitur qua <lb/>proportione propiùs accedent lineæ &longs;eu iactus ad BD, PP minùs acqui­<lb/>rent; </s> <s id="N1B797"><!-- NEW -->qua verò proportione propiùs accedent ad RG plùs acquirent; </s> <s id="N1B79B"><!-- NEW -->igi­<lb/>tur æqualiter plùs, & minùs hinc inde, &longs;i æqualiter hinc inde di&longs;tent; </s> <s id="N1B7A1"><!-- NEW -->im­<lb/>mò hoc ip&longs;um præ&longs;entibus oculis intueri licèt; </s> <s id="N1B7A7"><!-- NEW -->&longs;i enim iactus BF compa­<lb/>retur cum iactu BK; </s> <s id="N1B7AD"><!-- NEW -->certè BK acquirit RK, BF acquirit BH æqualem B <lb/>K; </s> <s id="N1B7B3"><!-- NEW -->&longs;ed BF & BK æqualiter di&longs;tant ab horizontali BG; </s> <s id="N1B7B7"><!-- NEW -->nam arcus GF, & <lb/>GK &longs;unt æquales, vt con&longs;tat: idem dico de iactu BE, & BX, qui acquirunt <lb/>æquale &longs;patium in horizontali æquale &longs;cilicet BZ. </s> </p> <p id="N1B7BF" type="main"> <s id="N1B7C1"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1B7CD" type="main"> <s id="N1B7CF"><!-- NEW -->Ob&longs;eruabis hoc omninò licèt mirum cuiquam fortè videatur, certè <lb/>in&longs;titutum e&longs;&longs;e à natura; </s> <s id="N1B7D5"><!-- NEW -->&longs;i enim comparentur omnes iactus &longs;uprà hori­<lb/>zontalem BG, haud dubiè cum duo extremi &longs;cilicet BD, & BG nihil <lb/>pror&longs;us acquirant, vt con&longs;tat ex dictis, iactus medius &longs;cilicet ad gradum <lb/>45.erit omnium maximus, quia æqualiter ab vtraque extremitate di&longs;tat, <lb/>vt demon&longs;trauimus &longs;uprà; </s> <s id="N1B7E1"><!-- NEW -->&longs;i verò comparentur omnes iactus, qui po&longs;­<lb/>&longs;unt fieri à centro B per totum &longs;emicirculum <expan abbr="DGq;">DGque</expan> certè cum duo ex­<lb/>tremi BD, BQ nihil pror&longs;us acquirant, vt con&longs;tat, iactus medius, &longs;cilicet <lb/>ad gradum 90.qui e&longs;t BG erit omnium maximus, quia æqualiter ab vtra-<pb pagenum="174" xlink:href="026/01/206.jpg"/>que di&longs;tat extremitate; &longs;imiliter quemadmodum iactus æqualiter à me­<lb/>dio iactu 45. di&longs;tantes æqualem amplitudinem acquirunt in horizontali <lb/>BG, ita qui æqualiter di&longs;tant à medio iactu 90.vel horizontali BG æqua­<lb/>lem amplitudinem acquirunt in aliquo plano horizontali, &longs;cilicet in eo <lb/>vnde vterque iactus de&longs;init in perpendicularem deor&longs;um. </s> </p> <p id="N1B7FC" type="main"> <s id="N1B7FE"><!-- NEW -->Ob&longs;eruabis &longs;ecundo, omnes perpendiculares deor&longs;um perinde accipi, <lb/>atque &longs;i e&longs;&longs;ent parallelæ propter in&longs;en&longs;ibilem differentium; </s> <s id="N1B804"><!-- NEW -->quod certè <lb/>ab omnibus admittitur; quomodo verò per diuer&longs;a plana deor&longs;um cor­<lb/>pus tendere po&longs;&longs;it, v&longs;que ad centrum terræ, Libro &longs;equenti explica­<lb/>bimus. </s> </p> <p id="N1B80E" type="main"> <s id="N1B810"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s> </p> <p id="N1B81C" type="main"> <s id="N1B81E"><!-- NEW --><emph type="italics"/>In iactu per inclinatam deor&longs;um dato tempore minùs detrahitur de impetu <lb/>violento, quàm in iactu per inclinatam &longs;ur&longs;um<emph.end type="italics"/> &longs;it enim circulus centro A <lb/>&longs;emidiametro AG; </s> <s id="N1B82B"><!-- NEW -->&longs;itque AG horizontalis, & AO perpendiculatis deor­<lb/>&longs;um; </s> <s id="N1B831"><!-- NEW -->&longs;it iactus per inclinatam &longs;ur&longs;um AD, &longs;itque impetus violentus vt A <lb/>D, & naturalis deor&longs;um vt DE; </s> <s id="N1B837"><!-- NEW -->linea motus erit DAE; </s> <s id="N1B83B"><!-- NEW -->igitur a&longs;&longs;umatur A <lb/>E in AC, & DE in CB, ex impetu AD detrahitur DB, vt con&longs;tat ex dictis <lb/>quia totius ille fru&longs;trà e&longs;t; </s> <s id="N1B843"><!-- NEW -->&longs;it autem inclinata deor&longs;um cum impetu vio­<lb/>lento æquali AI æqualis AD, &longs;itque naturalis deor&longs;um acceleratus pro <lb/>rata plani inclinati vt IL, linea motus erit AL; </s> <s id="N1B84B"><!-- NEW -->a&longs;&longs;umatur AK, vt AL, & <lb/>KH vt IL, detrahitur tantùm IH, &longs;ed IH e&longs;t minor DB; igitur tempore <lb/>&longs;equenti æquali impetus violentus inclinatæ &longs;ur&longs;um erit vt EF æqualis <lb/>AB inclinatæ deor&longs;um, vt LM, quæ maior e&longs;t EF, quia e&longs;t æqua­<lb/>lis AH. </s> </p> <p id="N1B857" type="main"> <s id="N1B859"><!-- NEW -->Ratio à priori e&longs;t, quia cum inclinata deor&longs;um faciat acutum angu­<lb/>lum cum perpendiculari deor&longs;um, cum quo obtu&longs;um facit inclinata &longs;ur­<lb/>&longs;um, maior e&longs;t in illa linea motus; </s> <s id="N1B861"><!-- NEW -->e&longs;t enim maior diagonalis, in hac ve­<lb/>rò minor, igitur in illa minùs impetus e&longs;t fru&longs;trà, in i&longs;ta verò plùs, igitur <lb/>minùs impetus in illa de&longs;truitur, plùs in i&longs;ta; quæ omnia con&longs;tant ex <lb/>Th. 110. & 139. & 140. l.1. habes etiam in qua proportione decre&longs;cat <lb/>impetus. </s> </p> <p id="N1B86D" type="main"> <s id="N1B86F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s> </p> <p id="N1B87B" type="main"> <s id="N1B87D"><!-- NEW --><emph type="italics"/>Hinc in iactu qui fit per inclinatam deor&longs;um minùs detrahitur,<emph.end type="italics"/> & in eo <lb/>qui fit per inclinationem &longs;ur&longs;um plùs detrahitur, in perpendiculari deor­<lb/>&longs;um nihil detrahitur, in perpendiculari &longs;ur&longs;um totus detrahitur qui po­<lb/>te&longs;t extrahi, id e&longs;t ex collectione vtriu&longs;que naturalis, & violenti dupli <lb/>naturalis in prima linea motus; hæc omnia &longs;equuntur ex dictis. </s> </p> <p id="N1B88E" type="main"> <s id="N1B890"><!-- NEW -->Obiici pote&longs;t vnum &longs;atis difficile; quia &longs;i in perpendiculari deor&longs;um <lb/>purà in AP nihil detrahitur impetus violenti, igitur cre&longs;cit &longs;emper vis <lb/>ictus, quod videtur e&longs;&longs;e contra experientiam. </s> </p> <p id="N1B898" type="main"> <s id="N1B89A"><!-- NEW -->Re&longs;p. me aliquando fui&longs;&longs;e in ea &longs;ententiâ, vt reuerâ exi&longs;timarem de­<lb/>cre&longs;cere impetum violentum in iactu perpendiculari deor&longs;um; </s> <s id="N1B8A0"><!-- NEW -->cum <lb/>etiam exi&longs;timarem decre&longs;cere vim ictus; </s> <s id="N1B8A6"><!-- NEW -->&longs;ed re melius con&longs;iderata, cum <lb/>nunquam id experiri potuerim; </s> <s id="N1B8AC"><!-- NEW -->nam &longs;emper &longs;entio vim ictus maiorem, <pb pagenum="175" xlink:href="026/01/207.jpg"/>cum deor&longs;um mobile proiicitur, quàm cum &longs;ua &longs;ponte ex eadem altitu­<lb/>dine de&longs;cendit; certè ni fallor cum ratio demon&longs;tratiua pro hac &longs;en­<lb/>tentia faciat, non dubitaui ampliùs priorem &longs;ententiam immutare. </s> </p> <p id="N1B8B9" type="main"> <s id="N1B8BB"><!-- NEW -->Porrò ratio, quæ pro hac &longs;ententia facit, remque ip&longs;am euincit, talis <lb/>e&longs;t; </s> <s id="N1B8C1"><!-- NEW -->certum e&longs;t impetum violentum de&longs;trui à naturali aliquando in ma­<lb/>iori, aliquando in minori proportione, vt con&longs;tat ex dictis; </s> <s id="N1B8C7"><!-- NEW -->illa autem, <lb/>&longs;eu maior, &longs;eu minor proportio aliam regulam non habet præter illam <lb/>quam toties inculcauimus, id e&longs;t impetum de&longs;trui pro rata, id e&longs;t qua <lb/>proportione e&longs;t fru&longs;trà, id e&longs;t qua proportione e&longs;t minor motus eo, qui <lb/>e&longs;&longs;et ab vtroque impetu &longs;i ad <expan abbr="eãdem">eandem</expan> lineam vterque determinatus e&longs;&longs;et <lb/>atqui cum proiicitur mobile deor&longs;um, vterque impetus ad <expan abbr="eãdem">eandem</expan> li­<lb/>neam e&longs;t determinatus; </s> <s id="N1B8DF"><!-- NEW -->igitur nihil motus dee&longs;t per Th.138.l.1. igitur <lb/>nihil impetus e&longs;t fru&longs;trà; igitur nihil impetus illius de&longs;truitur. </s> </p> <p id="N1B8E5" type="main"> <s id="N1B8E7"><!-- NEW -->Quod dictum e&longs;&longs;e velim non con&longs;iderata medij re&longs;i&longs;tentiâ, quæ certè <lb/>aliquid impetus de&longs;truit, quod tamen in&longs;en&longs;ibile e&longs;t in medio libero, pu­<lb/>tà in aëre; </s> <s id="N1B8EF"><!-- NEW -->&longs;i enim in&longs;en&longs;ibilis e&longs;t hæc re&longs;i&longs;tentia in motu naturali; </s> <s id="N1B8F3"><!-- NEW -->dum <lb/>mobile &longs;it eius &longs;oliditatis, quæ &longs;uperet facilè vim aëris; certè etiam in­<lb/>&longs;en&longs;ibilis e&longs;t in motu proiectorum, præ&longs;ertim in mediocri &longs;patio, e&longs;t <lb/>enim par vtrobique ratio. </s> </p> <p id="N1B8FD" type="main"> <s id="N1B8FF"><!-- NEW -->Equidem fateor in longi&longs;&longs;imo &longs;patio po&longs;&longs;e tandem de&longs;trui totum im­<lb/>petum violentum; </s> <s id="N1B905"><!-- NEW -->nam &longs;i aliquid in dato &longs;patio de&longs;truitur; </s> <s id="N1B909"><!-- NEW -->igitur in ma­<lb/>iore piùs de&longs;truitur; </s> <s id="N1B90F"><!-- NEW -->atque ita deinceps, donec tandem totus de&longs;tructus <lb/>&longs;it; at verò in iis altitudinibus, ex quibus corpus deor&longs;um proiicere po&longs;­<lb/>&longs;umus, vix quidquam facit prædicta re&longs;i&longs;tentia. </s> </p> <p id="N1B917" type="main"> <s id="N1B919">Nec e&longs;t quod aliquis dicat ab hac re&longs;i&longs;tentia non de&longs;trui impetum <lb/>naturalem in motu naturaliter accelerato, vt dictum e&longs;t in &longs;ecundo lib. <!-- KEEP S--></s> <s id="N1B91F"><lb/>Igitur nec de&longs;trui violentum; </s> <s id="N1B923"><!-- NEW -->nam qua proportione cre&longs;cit medij re&longs;i­<lb/>&longs;tentia, cre&longs;cunt vires impetus, qui perpetuò augetur; </s> <s id="N1B929"><!-- NEW -->vnde cum <lb/>remaneat &longs;emper eadem re&longs;i&longs;tentiæ proportio &longs;icut primo tempore mo­<lb/>tus impedit hæc re&longs;i&longs;tentia, ne tantillùm impetus producatur; </s> <s id="N1B931"><!-- NEW -->ita &longs;ecun­<lb/>do tempore impedit ne tantillùm æquale producatur; </s> <s id="N1B937"><!-- NEW -->igitur nihil pro­<lb/>ducti impetus ab illa de&longs;truitur propter augmentum continuum: </s> <s id="N1B93D"><!-- NEW -->at ve­<lb/>rò cum impetus violentus non intendatur; </s> <s id="N1B943"><!-- NEW -->certè &longs;i tantillùm illus perit, <lb/>primo vel &longs;ecundo in&longs;tanti motus, propter medij re&longs;i&longs;tentis, tantillùm <lb/>æquale &longs;ingulis temporibus æqualibus de&longs;truitur; igitur cum infinitus <lb/>non &longs;it po&longs;t longi&longs;&longs;imum &longs;patij tractum totus tandem de&longs;truetur vio­<lb/>lentus &longs;olo &longs;uper&longs;tite naturali. </s> </p> <p id="N1B94F" type="main"> <s id="N1B951"><!-- NEW -->Hinc fortè &longs;agitta ex notabili altitudine minùs ferit; </s> <s id="N1B955"><!-- NEW -->quia materia illa <lb/>lignea, & plumea, ex qua con&longs;tat, multùm ab aëre re&longs;i&longs;tente accipit de­<lb/>trimenti: </s> <s id="N1B95D"><!-- NEW -->adde quod licèt initio deor&longs;um rectà emittatur; </s> <s id="N1B961"><!-- NEW -->attamen mini­<lb/>mo aëris flatu declinat tantillùm obliqua; hæc verò obliquitas maximam <lb/>ictus vim infringit, & conflictus impetuum qua&longs;i ip&longs;um ictum di&longs;trahit, <lb/>quod facilè probabis, &longs;i modico ferè tactu cadentem perpendiculariter <lb/>&longs;agittam à &longs;uo tramite deturbes. </s> </p> <p id="N1B96D" type="main"> <s id="N1B96F">Dices, etiam in glande è tormento explo&longs;a hoc ip&longs;um cernitur </s> </p> <pb pagenum="176" xlink:href="026/01/208.jpg"/> <p id="N1B976" type="main"> <s id="N1B978"><!-- NEW -->Re&longs;p. e&longs;t minor vis ictus inflicti à glande deor&longs;um, quàm &longs;ur&longs;um vt <lb/>aliqui putant; </s> <s id="N1B97E"><!-- NEW -->id autem ex duplici capite procedere; </s> <s id="N1B982"><!-- NEW -->primum e&longs;t, cum fe­<lb/>ratur glans ab igne per aliquod tempus, non e&longs;t dubium, quin vis ignis <lb/>&longs;ur&longs;um maior &longs;it quàm deor&longs;um; </s> <s id="N1B98A"><!-- NEW -->cum &longs;ur&longs;um gemino qua&longs;i impetu fera­<lb/>tur, deor&longs;um verò impetu tantùm explo&longs;ionis; </s> <s id="N1B990"><!-- NEW -->&longs;ecundum e&longs;t, quia cum <lb/>glans iam deor&longs;um &longs;ua &longs;ponte de&longs;cendat, haud dubiè ab igne minus eò <lb/>impelli pote&longs;t, vt &longs;æpè diximus &longs;uprà; quidquid &longs;it, &longs;i proiiciatur deor&longs;um <lb/>globus plumbeus vel arcu, vel manu, ob&longs;eruabitur maiorem ab eo ictum <lb/>infligi, quàm &longs;i &longs;ua &longs;ponte de&longs;cenderet. </s> </p> <p id="N1B99C" type="main"> <s id="N1B99E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s> </p> <p id="N1B9AA" type="main"> <s id="N1B9AC"><!-- NEW --><emph type="italics"/>Si corpus moueatur deor&longs;um perpendiculariter motu mixto, eo tempore que <lb/>motu naturali acquireret illum impetum quem habet motu violento, acquirit <lb/>triplum illius &longs;patium<emph.end type="italics"/> v.g. <!-- REMOVE S-->in figura &longs;uperiore &longs;it linea perpendiculatis <lb/>deor&longs;um A E, in qua motu naturali dato tempore acquiratur AB, & &longs;e­<lb/>cundo tempore æquali BC; </s> <s id="N1B9BF"><!-- NEW -->&longs;itque impetus violentus vt AC: </s> <s id="N1B9C3"><!-- NEW -->Dico quod <lb/>æquali tempore prioribus acquireret AE triplum AC, quia motu ve­<lb/>loci vt AC acquirit CE eo tempore, quo motu veloci vt AB acquirit A <lb/>B, & veloci vt BC acquirit BC; </s> <s id="N1B9CD"><!-- NEW -->nam eo tempore, quo acquirit AB acqui­<lb/>rit CD, & eo tempore, quo acquirit BC acquirit DE; </s> <s id="N1B9D3"><!-- NEW -->ergo eo tempore, <lb/>quo acquirit AC acquirit CE; </s> <s id="N1B9D9"><!-- NEW -->ergo &longs;i iungatur motus naturalis violento, <lb/>eo tempore, quo motu naturali acquiretur tantùm AC, motu mixto ex <lb/>naturali & tali violento acquiretur AE, id e&longs;t triplum: </s> <s id="N1B9E1"><!-- NEW -->&longs;i verò moueatur <lb/>duobus temporibus, ita vt primò acquirat AC, & altero triplum AC, <lb/>&longs;itque coniunctus impetus violentus vt AC; certè duobus temporibus <lb/>acquiretur motu mixto octuplum AC, &longs;ed hæc &longs;unt facilia. </s> </p> <p id="N1B9EB" type="main"> <s id="N1B9ED"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s> </p> <p id="N1B9F9" type="main"> <s id="N1B9FB"><emph type="italics"/>Si corpus graue proiiciatur deor&longs;um per medium aëra, qui re&longs;i&longs;tat, cum <lb/>tandem de&longs;truatur impetus violentus, vbi totus de&longs;tructus e&longs;t, minor e&longs;t ictus <lb/>quàm e&longs;&longs;et. </s> <s id="N1BA04"><!-- NEW -->&longs;i corpus graue &longs;olo impetu natur ali eò de&longs;cendi&longs;&longs;et<emph.end type="italics"/>; </s> <s id="N1BA0B"><!-- NEW -->quod demon­<lb/>&longs;tro, &longs;it enim &longs;patium AD, quod percurrit motu mixto eo tempore, quo <lb/>motu naturali puro &longs;patium BC idem mobile percurreret, &longs;itque de&longs;tru­<lb/>ctus in puncto D totus impetus violentus; </s> <s id="N1BA15"><!-- NEW -->certè remanet tantùm natu­<lb/>ralis acqui&longs;itus eo tempore, quo mobile percurrit BC; </s> <s id="N1BA1B"><!-- NEW -->&longs;ed temporibus æ­<lb/>qualibus acquiruntur æqualia velocitatis momenta; </s> <s id="N1BA21"><!-- NEW -->igitur æqualis im­<lb/>petus; </s> <s id="N1BA27"><!-- NEW -->igitur in C tantùm ille impetus, qui e&longs;&longs;et in E vel in D; </s> <s id="N1BA2B"><!-- NEW -->&longs;ed dum <lb/>percurreret ED motu puro naturali, augetur impetus; </s> <s id="N1BA31"><!-- NEW -->igitur maior e&longs;&longs;et <lb/>impetus in D &longs;ub finem motus naturalis per AD, quam motus mixti per <lb/>eamdem AD; </s> <s id="N1BA39"><!-- NEW -->igitur maior ictus &longs;ub finem naturalis; igitur minus &longs;ub fi­<lb/>nem violenti. </s> </p> <p id="N1BA3F" type="main"> <s id="N1BA41"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s> </p> <p id="N1BA4D" type="main"> <s id="N1BA4F"><emph type="italics"/>Hinc paradoxon egregium; </s> <s id="N1BA54"><!-- NEW -->mobile proiectum in data di&longs;tantia minùs ferit <lb/>quàm &longs;ua &longs;ponte demi&longs;&longs;um<emph.end type="italics"/>; quod nece&longs;&longs;ariò &longs;equitur ex dictis. </s> </p> <p id="N1BA5D" type="main"> <s id="N1BA5F"><!-- NEW -->Ob&longs;eruabis &longs;crupulum adhuc fortè hærere, cur &longs;cilicet impetus <pb pagenum="177" xlink:href="026/01/209.jpg"/>violentus non de&longs;truatur à naturali, cuius &longs;cilicet iu&longs;tam impedit propa­<lb/>gationem; </s> <s id="N1BA6A"><!-- NEW -->&longs;ed profectò nullo modo impetus ille violentus impedit effe­<lb/>ctum impetus naturalis innati vel addititij; </s> <s id="N1BA70"><!-- NEW -->quia vterque totum &longs;uum ef­<lb/>fectum &longs;ortitur; </s> <s id="N1BA76"><!-- NEW -->quod autem &longs;pectat ad propagationem; certè ita propa­<lb/>gatur, vt temporibus æqualibus æqualis impetus accedat. </s> </p> <p id="N1BA7C" type="main"> <s id="N1BA7E">Dices, debes quidem nouus impetus accedere, &longs;ed non tali <lb/>modo. </s> </p> <p id="N1BA83" type="main"> <s id="N1BA85">Re&longs;p. non e&longs;&longs;e alium modum à natura in&longs;titutum, ni&longs;i vt temporibus <lb/>æqualibus æqualia velocitatis momenta acquirantur. </s> </p> <p id="N1BA8A" type="main"> <s id="N1BA8C">Dices præterea, fru&longs;trà accedit nouus impetus naturalis, cum iam ad­<lb/>&longs;it violentus, qui eius munere defungi pote&longs;t. </s> </p> <p id="N1BA91" type="main"> <s id="N1BA93"><!-- NEW -->Re&longs;p. cau&longs;am nece&longs;&longs;ariam nece&longs;&longs;ariò agere; igitur corpus graue perpe­<lb/>tuò in medio libero &longs;uum motum intendit. </s> </p> <p id="N1BA99" type="main"> <s id="N1BA9B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s> </p> <p id="N1BAA7" type="main"> <s id="N1BAA9"><!-- NEW --><emph type="italics"/>Pote&longs;t vtcumque delineari linea motus mixti per inclinatam deor&longs;um<emph.end type="italics"/> &longs;it <lb/>enim perpendicularis deor&longs;um AB &longs;it iactus per inclinatam AF; </s> <s id="N1BAB4"><!-- NEW -->&longs;itque <lb/>impetus violentus vt AE naturalis vt EC, linea motus erit AC; </s> <s id="N1BABA"><!-- NEW -->a&longs;&longs;umatur <lb/>AF æqualis AC, & DF æqualis EC, &longs;itque CH vt AD, & impetus natu­<lb/>ralis auctus vt HK, linea motus erit CK; </s> <s id="N1BAC2"><!-- NEW -->&longs;it CI æqualis DK, & IG æqua­<lb/>lis HK, & KL æqualis CG; </s> <s id="N1BAC8"><!-- NEW -->&longs;it que impetus naturalis &longs;ecundò auctus vt L <lb/>M; </s> <s id="N1BACE"><!-- NEW -->linea motus erit KM; igitur connectantur puncta AC, KM per lineam <lb/>curuam, hæc e&longs;t linea quæ&longs;ita, vt con&longs;tat ex dictis &longs;uprà. </s> </p> <p id="N1BAD4" type="main"> <s id="N1BAD6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s> </p> <p id="N1BAE2" type="main"> <s id="N1BAE4"><!-- NEW --><emph type="italics"/>Hinc pote&longs;t aliquo tempore tantùm impetus violenti de&longs;trui quantùm pro­<lb/>ducitur naturalis<emph.end type="italics"/>; igitur &longs;i non con&longs;ideres re&longs;i&longs;tentiam medij, tunc æqua­<lb/>lis e&longs;&longs;et ictus, & æquabilis motus. </s> </p> <p id="N1BAF1" type="main"> <s id="N1BAF3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s> </p> <p id="N1BAFF" type="main"> <s id="N1BB01"><!-- NEW --><emph type="italics"/>Quando mobile peruenit in M, & acqui&longs;iuit in perpendiculari deor&longs;um to­<lb/>tam altitudinem AR, non habet totum impetum naturalem, quem acquireret <lb/>motu naturali per totam AR, &longs;ed tantùm illum, quem acquireret in compo&longs;ita <lb/>ex &longs;egmentis NO, PB, QR<emph.end type="italics"/>; quia ad motum i&longs;tum deor&longs;um non tantùm <lb/>concurrit impetus naturalis, &longs;ed etiam violentus vt con&longs;tat. </s> </p> <p id="N1BB12" type="main"> <s id="N1BB14"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s> </p> <p id="N1BB20" type="main"> <s id="N1BB22"><emph type="italics"/>Hinc reiicies Galileum, & alios,<emph.end type="italics"/> qui volunt in linea motus AC ac­<lb/>quiri <expan abbr="tantũdem">tantundem</expan> impetus naturalis quantum in perpendiculari AB ac­<lb/>quireretur. </s> </p> <p id="N1BB32" type="main"> <s id="N1BB34"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s> </p> <p id="N1BB40" type="main"> <s id="N1BB42"><!-- NEW --><emph type="italics"/>In naui mobili &longs;i è &longs;ummo malo remittatur corpus graue, de&longs;cendit motu <emph.end type="italics"/><pb pagenum="178" xlink:href="026/01/210.jpg"/><emph type="italics"/>mixto<emph.end type="italics"/>; probatur, quia duplex impetus concurrit ad illum motum, &longs;cilicet <lb/>naturalis deor&longs;um, & horizontalis impre&longs;&longs;us à naui, vt con&longs;tat ex defini­<lb/>tione 1.hyp.2. & Ax.1. </s> </p> <p id="N1BB58" type="main"> <s id="N1BB5A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s> </p> <p id="N1BB66" type="main"> <s id="N1BB68"><!-- NEW --><emph type="italics"/>Ille motus e&longs;t mixtus ex naturali accelerato, & violento per horizontalem <lb/>retardato<emph.end type="italics"/>; quod eodem modo probatur, quo &longs;uprà probatum e&longs;t in mobi­<lb/>li proiecto per horizontalem Th.30. e&longs;t enim pror&longs;us eadem, cum à na­<lb/>ui reuera imprimatur impetus iis omnibus, quæ motu nauis fe­<lb/>runtur. </s> </p> <p id="N1BB79" type="main"> <s id="N1BB7B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s> </p> <p id="N1BB87" type="main"> <s id="N1BB89"><!-- NEW --><emph type="italics"/>Hinc reiicio omnes alias combinationes recepta &longs;exta; </s> <s id="N1BB8F"><!-- NEW -->immò &longs;extam <lb/>ip&longs;am ex parte<emph.end type="italics"/>; nec enim naturalis acceleratur in hoc motu in ea <lb/>proportione, in qua acceleratur per lineam perpendicularem deor­<lb/>&longs;um per Th. 29.&longs;ed iuxta rationem planorum inclinatorum per Theo­<lb/>rema 31. nec etiam violentus de&longs;truitur vniformiter, &longs;ed pro rata per <lb/>Th. 39. </s> </p> <p id="N1BBA0" type="main"> <s id="N1BBA2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s> </p> <p id="N1BBAE" type="main"> <s id="N1BBB0"><!-- NEW --><emph type="italics"/>Hinc initio plùs detrahitur violenti, & minùs additur naturalis, in <lb/>fine plùs additur naturalis & minùs detrahitur violenti<emph.end type="italics"/>; hinc minor e&longs;t <lb/>ictus in fine ni&longs;i malus nauis ad eam altitudinem a&longs;cenderet, ad quam <lb/>profectò nullus a&longs;cendit, quæ omnia con&longs;tant per Theorema 34. <lb/>35. 36. </s> </p> <p id="N1BBC1" type="main"> <s id="N1BBC3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s> </p> <p id="N1BBCF" type="main"> <s id="N1BBD1"><!-- NEW --><emph type="italics"/>Hinc ratio curuitatis huius lineæ, vel hypothe&longs;is &longs;ecundæ<emph.end type="italics"/>; </s> <s id="N1BBDA"><!-- NEW -->quæ tamen non <lb/>e&longs;t Parabola vt volunt aliqui; </s> <s id="N1BBE0"><!-- NEW -->hinc non eo tempore de&longs;cendit in nauim <lb/>prædictus globus, quo de&longs;cenderet per ip&longs;am perpendicularem motu <lb/>purè naturali ex eadem altitudine, &longs;ed maiore tempore; quia motu mix­<lb/>to non acceleratur iuxta proportionem motus naturalis puri per Th. <!-- REMOVE S--><lb/>77. quod confirmatur illis omnibus experimentis, quæ &longs;uprà adduxi <lb/>Th. <!-- REMOVE S-->46. </s> </p> <p id="N1BBF1" type="main"> <s id="N1BBF3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s> </p> <p id="N1BBFF" type="main"> <s id="N1BC01"><!-- NEW --><emph type="italics"/>Hinc &longs;i nauis moueretur eadem velocitate, qua funis arcus cum re­<lb/>dit, e&longs;&longs;etque aptata &longs;agitta, & directa horizontaliter in naui; </s> <s id="N1BC09"><!-- NEW -->haud <lb/>dubiè &longs;i po&longs;t aliquod tempus &longs;taret illicò immota nauis: </s> <s id="N1BC0F"><!-- NEW -->emitteretur &longs;a­<lb/>gita, non minore certè vi quàm ab ip&longs;o arcu<emph.end type="italics"/>; </s> <s id="N1BC18"><!-- NEW -->hinc etiam cum <lb/>nauis appellitur ad littus, &longs;i &longs;tatim &longs;ub&longs;i&longs;tat; </s> <s id="N1BC1E"><!-- NEW -->omnia quæ &longs;unt in <lb/>naui &longs;uccutiuntur & <expan abbr="pleriq;">plerique</expan> cadunt incauti in partem aduer&longs;am propter <pb pagenum="179" xlink:href="026/01/211.jpg"/>impetum à naui acceptum; ex quo certè experimento maximè confir­<lb/>matur hic impetus à naui impre&longs;&longs;us, per quem Galileus ex hypothe&longs;i mo­<lb/>tus æ&longs;tum maris explicat exemplo appul&longs;arum nauium ad littus, quæ <lb/>aquam vehunt. </s> </p> <p id="N1BC33" type="main"> <s id="N1BC35"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s> </p> <p id="N1BC41" type="main"> <s id="N1BC43"><!-- NEW --><emph type="italics"/>Hinc demi&longs;&longs;us globus plumbeus, vel alterius materiæ, quæ facilè vim aëris <lb/>infringat è &longs;ummo malo nauis ad imum ferè malum de&longs;cendit,<emph.end type="italics"/> hæc e&longs;t ex­<lb/>perientia à Galileo producta, non tamen adinuenta, à Ga&longs;&longs;endo do­<lb/>cti&longs;&longs;imè & eleganti&longs;&longs;imè explicata, ab omnibus Copernici &longs;ectatoribus <lb/>toties decantata, quæ vulgus ignobile ad admirationem adducit; </s> <s id="N1BC54"><!-- NEW -->imò <lb/>plures è Philo&longs;ophis fuere, qui eam in dubium adducerent, cum cam &longs;uis <lb/>principiis, ne dicam fortè &longs;omniis aduer&longs;ari putarent; certi&longs;&longs;imum tamen <lb/>e&longs;t illud experimentum centies, imò millies comprobatum, totis etiam <lb/>vrbibus &longs;pectantibus. </s> <s id="N1BC60"><!-- NEW -->Nec ratio huius experimenti adeo ab&longs;tru&longs;a e&longs;t, <lb/>vel recondita, quin à vulgari, ne dicam triobolari Philo&longs;opho &longs;tatim ex­<lb/>plicari po&longs;&longs;it; </s> <s id="N1BC68"><!-- NEW -->cum enim imprimatur à naui mobili impetus pendulo <lb/>globo per horizontalem, & alius ab ip&longs;a grauitate deor&longs;um per Th. 71. <lb/>certè mouetur globus demi&longs;&longs;us re&longs;ecto funiculo motu mixto ex hori­<lb/>zontali nauis, naturali corporis grauis; </s> <s id="N1BC72"><!-- NEW -->igitur per lineam curuam, quæ <lb/>ferè ad imum malum terminatur &longs;ed modicum figuræ adhibendum e&longs;t; </s> <s id="N1BC78"><!-- NEW --><lb/>&longs;it planum aquæ <expan abbr="horizõtale">horizontale</expan>, cui innatat nauis IH; </s> <s id="N1BC81"><!-- NEW -->&longs;it malus IA perpen­<lb/>dicularis altus 48. pedes; </s> <s id="N1BC87"><!-- NEW -->diuidatur in 4. partes æquales; </s> <s id="N1BC8B"><!-- NEW -->corpus graue <lb/>conficiat &longs;patium illud duobus &longs;ecundis, v.g.igitur AK vno &longs;ecundo; </s> <s id="N1BC91"><!-- NEW -->e&longs;t <lb/>autem VK 12. pedum; </s> <s id="N1BC97"><!-- NEW -->iam verò moueatur nauis per horizontalem IH, <lb/>vel AL maxima qua&longs;i velocitate qua triremis moueri pote&longs;t; </s> <s id="N1BC9D"><!-- NEW -->ita vt vna <lb/>hora faciat 16. milliaria Germanica, & 15′.4. milliaria, 3′ 800. pa&longs;&longs;us, <lb/>1′ 266. 1″ 4. pa&longs;&longs;us & (13/30); </s> <s id="N1BCA5"><!-- NEW -->&longs;upponamus 1″ conficere 18. pedes, &longs;itque AC <lb/>18. & AK vel CE 12. haud dubiè motu mixto faciet lineam AE, & &longs;e­<lb/>cundo tempore lineam EH, donec tandem cadat in punctum H nauis, <lb/>quò ferè peruenit punctum I; </s> <s id="N1BCAF"><!-- NEW -->nam eodem modo retardatur motus <lb/>nauis; </s> <s id="N1BCB5"><!-- NEW -->immò plùs quàm motus globi; </s> <s id="N1BCB9"><!-- NEW -->quod &longs;cilicet partes aquæ, quæ à <lb/>naui diuiduntur multum re&longs;i&longs;tant; </s> <s id="N1BCBF"><!-- NEW -->vnde fit compen&longs;atio; </s> <s id="N1BCC3"><!-- NEW -->nam initio <lb/>motus violentus, qua&longs;i &longs;ecum rapit motum naturalem initio tardi&longs;&longs;i­<lb/>mum; præ&longs;ertim cum non acceleretur, ni&longs;i iuxta rationem plani incli­<lb/>nati, vt &longs;uprà dictum e&longs;t, & in fine naturalis rapit violentum. </s> </p> <p id="N1BCCD" type="main"> <s id="N1BCCF"><!-- NEW -->Dixi ad imum ferè malum; </s> <s id="N1BCD3"><!-- NEW -->nam reuera aliquid dee&longs;t quod tamen in­<lb/>&longs;en&longs;ibile e&longs;t; </s> <s id="N1BCD9"><!-- NEW -->&longs;ed quia modico tempore globus de&longs;cendit; </s> <s id="N1BCDD"><!-- NEW -->&longs;it enim malus <lb/>108. pedum altitudinis, de&longs;cendit globus tempore 3″; </s> <s id="N1BCE3"><!-- NEW -->&longs;it 192.4; </s> <s id="N1BCE7"><!-- NEW -->&longs;it &longs;i <lb/>fieri pote&longs;t 432. de&longs;cendet 6″, &longs;ed nunquam accedit ad tantam altitudi­<lb/>nem, igitur duobus vel tribus &longs;ecundis de&longs;cendit; </s> <s id="N1BCEF"><!-- NEW -->igitur modico tem­<lb/>pore; </s> <s id="N1BCF5"><!-- NEW -->igitur violentus motus cen&longs;eri debet eo tempore æquabilis &longs;en&longs;i­<lb/>biliter; </s> <s id="N1BCFB"><!-- NEW -->& cum motus nauis nunquam &longs;it eiu&longs;dem velocitatis cum illa <lb/>quæ acquiritur tempore 2″ in de&longs;cen&longs;u, quia cum in de&longs;cen&longs;u acquiran­<lb/>tur, hoc dato tempore ferè 48. pedes &longs;patij; </s> <s id="N1BD03"><!-- NEW -->certè motu æquabili cuius <pb pagenum="180" xlink:href="026/01/212.jpg"/>e&longs;&longs;et eadem velocitas acquirerentur 96. &longs;ed vix acquirerentur 24.vt di­<lb/>ctum e&longs;t &longs;uprà; </s> <s id="N1BD0E"><!-- NEW -->igitur vix nauis percurrit in horizontali æqualem lineam <lb/>longitudini mali eo tempore, quo globus nauim attingit &longs;it enim <lb/>altitudo mali FA 48. pedum; </s> <s id="N1BD16"><!-- NEW -->&longs;it amplitudo &longs;patij horizontalis æqualis <lb/>FA; haud dubiè 1″ percurret AD, id e&longs;t 12.pedes ferè, quo tempore per­<lb/>currat FG. 24. pedes & 20″ percurret DF, & GI. &longs;i motus &longs;umatur vt <lb/>æquabilis, vel GH, &longs;i retardatur, igitur 1°ree;″ mobile percurrit &longs;egmentum <lb/>curuæ AE & 2°ree; EH. </s> </p> <p id="N1BD22" type="main"> <s id="N1BD24"><!-- NEW -->Et licèt videatur tantùm acquirere MI, quæ e&longs;t minor DF 15. per­<lb/>pendiculari deor&longs;um, acquirit totam EH, quæ non modo e&longs;t à motu na­<lb/>turali, verùm etiam à motu violento; </s> <s id="N1BD2C"><!-- NEW -->nec enim motu naturali dum mi­<lb/>&longs;cetur cum alio, tantùm acquiritur deor&longs;um, quantùm reuerâ acquiritur <lb/>motu naturali puro, vt &longs;uprà monuimus; </s> <s id="N1BD34"><!-- NEW -->quia tamen etiam deor&longs;um mo­<lb/>tus violentus deflectitur, etiam aliquid &longs;patij ratione violenti deor&longs;um <lb/>acquiritur; </s> <s id="N1BD3C"><!-- NEW -->&longs;i enim vbi peruenit in E vterque impetus intactus remane­<lb/>ret &longs;ine acce&longs;&longs;ione, &longs;ine imminutione; </s> <s id="N1BD42"><!-- NEW -->haud dubiè per <expan abbr="eãdem">eandem</expan> EM, quæ <lb/>&longs;it tangens huius curuæ AEH &longs;uum cur&longs;um pro&longs;equeretur; </s> <s id="N1BD4C"><!-- NEW -->igitur ac­<lb/>quireret deor&longs;um totam DN, vel EO propter impetum naturalem præ­<lb/>uium; &longs;i verò aliquid naturalis accedat, quid mirum &longs;i ratione illius ac­<lb/>quiratur MI, vel NF? </s> </p> <p id="N1BD56" type="main"> <s id="N1BD58">Dices non de&longs;cendit tam citò motu naturali accelerato, mixto cum <lb/>violento, quàm motu puro naturali. </s> </p> <p id="N1BD5D" type="main"> <s id="N1BD5F"><!-- NEW -->Re&longs;pondeo concedo; </s> <s id="N1BD63"><!-- NEW -->vnde nunquam ex A in H 2″ de&longs;cendit; </s> <s id="N1BD67"><!-- NEW -->&longs;ed <lb/>tardiùs, licèt FA &longs;it 48. ped. <!-- REMOVE S-->&longs;ed parùm abe&longs;t tùm propter minorem re&longs;i­<lb/>&longs;tentiam huius impetus violenti, qui facilè detorquetur, & con&longs;equen­<lb/>tur minùs illius perit, tùm quia etiam de&longs;truitur aliquid violenti; igitur <lb/>paulò plùs temporis collocat in GI, quàm in FG. </s> </p> <p id="N1BD75" type="main"> <s id="N1BD77"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1BD83" type="main"> <s id="N1BD85"><!-- NEW -->Ob&longs;eruabis primò, &longs;i nouus impetus accedat, non e&longs;&longs;e expectandum <lb/>hunc effectum; quippe nihil accipit à naui globus deinceps, vbi &longs;emel <lb/>re&longs;ecto fune ab ea qua&longs;i &longs;eparatur. </s> </p> <p id="N1BD8D" type="main"> <s id="N1BD8F">Secundò, &longs;i &longs;tatim &longs;i&longs;tat nauis demi&longs;&longs;o globo ad vnum malum nullo <lb/>modo de&longs;cendet, vt patet, &longs;ed antè. </s> </p> <p id="N1BD94" type="main"> <s id="N1BD96"><!-- NEW -->Tertiò, &longs;i demittatur globus dum &longs;i&longs;tit nauis, tùm deinde, vbi <lb/>demi&longs;&longs;us e&longs;t, impellatur nauis; non de&longs;cendet etiam ad radicem, &longs;ed <lb/>retrò. </s> </p> <p id="N1BD9E" type="main"> <s id="N1BDA0"><!-- NEW -->Quartò, motus nauis non e&longs;t æquabilis, quidquid dicat Galileus; </s> <s id="N1BDA4"><!-- NEW -->alio­<lb/>quin vna remorum impul&longs;ione opus e&longs;&longs;et, vt &longs;emper eodem motu moue­<lb/>retur, aut certè &longs;i continua remigatione impellatur; </s> <s id="N1BDAC"><!-- NEW -->cre&longs;ceret in infini­<lb/>tum velocitas motus, &longs;i nihil de priori, velocitate detraheretur; </s> <s id="N1BDB2"><!-- NEW -->retarda­<lb/>tur igitur ille nauis motus propter re&longs;i&longs;tentiam aquæ, cuius partes & im­<lb/>pellendæ & &longs;ulcandæ, &longs;eu diuidendæ &longs;unt; </s> <s id="N1BDBA"><!-- NEW -->hinc fiunt ro&longs;tratæ naues <lb/>vel cu&longs;pidatæ vt faciliùs aquam findere po&longs;&longs;int; </s> <s id="N1BDC0"><!-- NEW -->igitur ille motus nauis <lb/>non e&longs;t æquabilis; Idem pror&longs;us dicendum e&longs;t de impetu impre&longs;&longs;o in <pb pagenum="181" xlink:href="026/01/213.jpg"/>globo, cuius aliquæ partes de&longs;truuntur, ne &longs;int fru&longs;trà, quod &longs;uprà de pro­<lb/>jecto per horizontalem vel inclinatam luculenter demon&longs;trauimus. </s> </p> <p id="N1BDCD" type="main"> <s id="N1BDCF"><!-- NEW -->Quintò &longs;i demittatur ex alia naui proxima immobili perpendiculari­<lb/>ter omninò de&longs;cendet; </s> <s id="N1BDD5"><!-- NEW -->Vnde valde hallucinantur ij, qui exi&longs;timant hunc <lb/>motum e&longs;&longs;e ab aëre quem nauis commouet, quod fal&longs;i&longs;&longs;imum e&longs;t, quia <lb/>pertica ad in&longs;tar mali parùm aëris commouet; </s> <s id="N1BDDD"><!-- NEW -->adde quod aër retrò agi­<lb/>tur, vt patet in aqua; </s> <s id="N1BDE3"><!-- NEW -->præterea &longs;i è curru immobili demittatur globus eo <lb/>tempore, quo alius currus præteruolat, de&longs;cendit perpendiculariter; </s> <s id="N1BDE9"><!-- NEW -->&longs;i ve­<lb/>rò è curru mobili etiam in maiori di&longs;tantia porrecta &longs;cilicet maximè <lb/>extra currum demittente dextera; </s> <s id="N1BDF1"><!-- NEW -->globus ab ip&longs;o curru capietur; </s> <s id="N1BDF5"><!-- NEW -->hîc <lb/>etiam ob&longs;eruabis idem pror&longs;us accidere in curru mobili, quod in naui; </s> <s id="N1BDFB"><!-- NEW -->&longs;i <lb/>enim è fene&longs;tra currus mobilis demittas pilam, &longs;emper cadet ex aduer&longs;o; <lb/>idem dico de currente equo, cui in&longs;idens demittat globum, imò &longs;i locus <lb/>&longs;it planus & politus, pila per aliquod tempus currum, vel equitem in&longs;e­<lb/>quetur, quod qui&longs;que probare poterit, vt reuerâ centies probatum <lb/>fuit. </s> </p> <p id="N1BE09" type="main"> <s id="N1BE0B"><!-- NEW -->Sextò ad rationem Galilei, qui contendit motum circularem circa <lb/>centrum terræ e&longs;&longs;e æquabilem, quia &longs;cilicet mobile non recedit à centro: <lb/>leuis e&longs;t omninò ratio; </s> <s id="N1BE13"><!-- NEW -->quia globus in medio aëre motu mixto mouetur, <lb/>id e&longs;t habet impetum partim deor&longs;um, partim per tangentem, & nullo <lb/>modo per circularem, vt certum e&longs;t; </s> <s id="N1BE1B"><!-- NEW -->nec enim rotata alium impetum im­<lb/>primunt, igitur violentus e&longs;t; </s> <s id="N1BE21"><!-- NEW -->igitur de&longs;trui debet etiam iuxta commu­<lb/>nia principia: </s> <s id="N1BE27"><!-- NEW -->adde quod motus mixtus fit per Diagonalem quod etiam <lb/>ip&longs;e admittit; </s> <s id="N1BE2D"><!-- NEW -->igitur totus impetus æqualem motum non habet; </s> <s id="N1BE31"><!-- NEW -->nec enim <lb/>Diagonalis æqualis e&longs;t vnquam duobus lateribus; </s> <s id="N1BE37"><!-- NEW -->igitur aliquid illius <lb/>fru&longs;trà e&longs;t; </s> <s id="N1BE3D"><!-- NEW -->igitur de&longs;trui debet; </s> <s id="N1BE41"><!-- NEW -->præterea licèt motus circularis &longs;it peren­<lb/>nis circa centrum mundi; </s> <s id="N1BE47"><!-- NEW -->nam de illo tantùm e&longs;t quæ&longs;tio, hoc ip&longs;um <lb/>&longs;upponit primò motum illum e&longs;&longs;e &longs;implicem; </s> <s id="N1BE4D"><!-- NEW -->&longs;ecundò, nullam pror&longs;us <lb/>e&longs;&longs;e re&longs;i&longs;tentiam; </s> <s id="N1BE53"><!-- NEW -->atqui in hoc ca&longs;u vtrumque deficit; </s> <s id="N1BE57"><!-- NEW -->nam motus ille <lb/>circularis non e&longs;t &longs;implex &longs;ed mixtus, & obe&longs;t re&longs;i&longs;tentia aquæ, vt &longs;uprà <lb/><expan abbr="dictũ">dictum</expan> e&longs;t; ni&longs;i verò con&longs;ideres <expan abbr="de&longs;cendent&etilde;">de&longs;cendentem</expan> globum è &longs;ummo malo, quis <lb/>dicat e&longs;&longs;e circularem? </s> <s id="N1BE68"><!-- NEW -->adde quod nauis imprimit tantùm rectum per <lb/>tangentem, vt iam &longs;uprà dictum e&longs;t; </s> <s id="N1BE6E"><!-- NEW -->porrò ad illud, quod dicit non de­<lb/>&longs;trui motum circularem à naturali, cui non e&longs;t contrarius, cum non re­<lb/>moueat longiùs à centro; </s> <s id="N1BE76"><!-- NEW -->videtur omninò di&longs;&longs;imulare cau&longs;am impetus <lb/><expan abbr="de&longs;tructiuã">de&longs;tructiuam</expan>, quæ cettè in <expan abbr="cõtrarietate">contrarietate</expan> tantùm determinationis po&longs;ita e&longs;t, <lb/>vt &longs;uprà dictum e&longs;t; </s> <s id="N1BE85"><!-- NEW -->ex qua &longs;equitur aliquid impetus fru&longs;trà e&longs;&longs;e; </s> <s id="N1BE89"><!-- NEW -->ac pro­<lb/>inde de&longs;trui per Axioma illud toties decantatum, <emph type="italics"/>Quod frustrà e&longs;t, non e&longs;t<emph.end type="italics"/>: </s> <s id="N1BE95"><!-- NEW --><lb/>Præterea non video quomodo hanc rationem proponat magnus Gali­<lb/>leus, qui nullum alium impetum violentum de&longs;trui putat, nî&longs;i tantùm il­<lb/>lum, qui e&longs;t per lineam verticalem &longs;ur&longs;um; nam ex motu illo impre&longs;&longs;o <lb/>æquabili, & naturali accelerato &longs;uas Parabolas ad&longs;truit. </s> </p> <p id="N1BEA0" type="main"> <s id="N1BEA2"><!-- NEW -->Septimò, non e&longs;t tamen quod diffitear ingeniosè excogitatum ab eo <lb/>fui&longs;&longs;e, ideo globum è &longs;ummo malo demi&longs;&longs;um ad imum de&longs;cendere, quod <lb/>&longs;cilicet de&longs;cendat motu mixto ex naturali accelerato, & violento æqua-<pb pagenum="182" xlink:href="026/01/214.jpg"/>bili, quod vt breuiter ob oculos ponatur &longs;it malus nauis mobilis IA, <lb/>quæ eo tempore, quo corpus graue de&longs;cendit ab A in D motu naturali, <lb/>percurrit FG æquabili motu, & con&longs;equenter GI æqualem FG eo tem­<lb/>pore, quo idem corpus graue percurrit DF triplam AD; </s> <s id="N1BEB5"><!-- NEW -->igitur globus <lb/>demi&longs;&longs;us ex A &longs;uo motu de&longs;cribit Parabolam AEH; quod etiam accidet <lb/>a&longs;&longs;umpta quacunque altitudine mali vel quocunque &longs;patio confecto à <lb/>naui mobili eo tempore, quo corpus graue motu naturali accelerato <lb/>conficit &longs;patium æquale altitudini mali. </s> </p> <p id="N1BEC1" type="main"> <s id="N1BEC3"><!-- NEW -->Octauò, non e&longs;t tamen di&longs;&longs;imulandum, quod etiam non di&longs;&longs;imulauit <lb/>Mer&longs;ennus, talem non fore de&longs;cen&longs;um, &longs;i nauis v. <!-- REMOVE S-->g. <!-- REMOVE S-->eadem cum emi&longs;&longs;a <lb/>&longs;agitta, vel explo&longs;a è tormento glande velocitate moueretur; </s> <s id="N1BECF"><!-- NEW -->non quod <lb/>aër vel medium ob&longs;i&longs;tat, vt ip&longs;i dicunt; </s> <s id="N1BED5"><!-- NEW -->hoc enim iam &longs;uprà rejecimus; </s> <s id="N1BED9"><!-- NEW --><lb/>&longs;ed quod major impetus violentus efficiat, vt iam &longs;uprà dictum e&longs;t, ne in <lb/>tanta proportione naturalis acceleretur; </s> <s id="N1BEE0"><!-- NEW -->quod etiam &longs;uo boatu intonant <lb/>tormenta maiora, è quibus horizontaliter directis explo&longs;æ pilæ per plu­<lb/>ra &longs;ecunda in libero aëre moueantur, licèt os tormenti à plano horizon­<lb/>tis vix tribus pedibus ab&longs;it; </s> <s id="N1BEEA"><!-- NEW -->igitur non de&longs;cribunt &longs;uo motu Parabolas; </s> <s id="N1BEEE"><!-- NEW --><lb/>hinc &longs;ub finem minor e&longs;t ictus; hinc etiam fatetur idem Mer&longs;ennus &longs;e­<lb/>cundum &longs;patium horizontale confici tardiore motu quàm primum & <lb/>tertium quàm &longs;ecundum, atque ita deinceps. </s> </p> <p id="N1BEF7" type="main"> <s id="N1BEF9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s> </p> <p id="N1BF05" type="main"> <s id="N1BF07"><!-- NEW --><emph type="italics"/>Si corpus graue proiiciatur &longs;ur&longs;um perpendiculariter è naui mobili, &longs;unt tres <lb/>impetus qui concurrunt ad illum motum<emph.end type="italics"/> &longs;it enim nauis mobilis per hori­<lb/>zontalem LF, è qua &longs;ur&longs;um rectâ per lineam perpendicularem LA pro­<lb/>iiciatur corpus graue; </s> <s id="N1BF16"><!-- NEW -->huic certè ine&longs;t triplus impetus, &longs;cilicet duo vio­<lb/>lenti, alter per verticalem LA impre&longs;&longs;us à proiiciente; </s> <s id="N1BF1C"><!-- NEW -->alter per horizon­<lb/>talem LF impre&longs;&longs;us à naui; </s> <s id="N1BF22"><!-- NEW -->tertius denique naturalis per ip&longs;am perpen­<lb/>dicularem deor&longs;um LP; </s> <s id="N1BF28"><!-- NEW -->igitur tres i&longs;ti impetus &longs;uo modo concurrunt <lb/>ad motum per Ax.1.certè &longs;i ine&longs;&longs;ent tantùm duo impetus &longs;cilicet LA, & <lb/>LF, motus fieret per inclinatam rectam LC; </s> <s id="N1BF30"><!-- NEW -->vel &longs;i tantùm duo LP, & <lb/>LA fieret per ip&longs;am LA motus retardatus; </s> <s id="N1BF36"><!-- NEW -->vel &longs;i LF & LP fieret per <lb/>curuam deor&longs;um, vt con&longs;tat ex dictis; igitur per aliam lineam fieri de­<lb/>bet ad quam tres illi impetus concurrunt. </s> </p> <p id="N1BF3E" type="main"> <s id="N1BF40"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s> </p> <p id="N1BF4C" type="main"> <s id="N1BF4E"><emph type="italics"/>Tam pugnat impetus naturalis per LP cum verticali LA quando e&longs;t con­<lb/>junctus cum horizontali LF, quàm cum nullus e&longs;t horizontalis,<emph.end type="italics"/> probatur, <lb/>quia &longs;emper mobile deor&longs;um trahit, vt patet. </s> </p> <p id="N1BF5A" type="main"> <s id="N1BF5C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s> </p> <p id="N1BF68" type="main"> <s id="N1BF6A"><!-- NEW --><emph type="italics"/>Hinc naturalis e&longs;t æquabilis, & violentus &longs;ur&longs;um e&longs;t retardatus; </s> <s id="N1BF70"><!-- NEW -->horizon­<lb/>talis verò e&longs;t æquabilis &longs;altem æquiualenter<emph.end type="italics"/>; </s> <s id="N1BF79"><!-- NEW -->quia cum illo non pugnat ho­<lb/>rizontalis, in a&longs;cen&longs;u &longs;altem perinde &longs;e habet; </s> <s id="N1BF7F"><!-- NEW -->immò cum illo conuenit <lb/>ad de&longs;truendum violentum &longs;ur&longs;um, id e&longs;t ad deflectendum deor&longs;um <lb/>mobile vt con&longs;tat; </s> <s id="N1BF87"><!-- NEW -->igitur hic motus con&longs;tat ex naturali & horizontali <pb pagenum="183" xlink:href="026/01/215.jpg"/>æquabilibus, & violento retardato &longs;int enim tres impetus ab eodem <lb/>puncto E &longs;cilicet EF, ED, EA; </s> <s id="N1BF92"><!-- NEW -->ex EA ED fit mixtus EG, ex EA, <lb/>EF, violentus EB; denique ex mixto EG à naturali EF fit EC, quæ <lb/>omnia &longs;unt clara. </s> </p> <p id="N1BF9A" type="main"> <s id="N1BF9C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s> </p> <p id="N1BFA8" type="main"> <s id="N1BFAA"><emph type="italics"/>A&longs;cendit mobile ad <expan abbr="eãdem">eandem</expan> altitudinem hoc motu, ad quem a&longs;cenderet <lb/>&longs;ine horizontali<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;ine horizontali a&longs;cendit in B, cum horizontali <lb/>a&longs;cendit in C, &longs;ed DC, & EB &longs;unt eiu&longs;dem altitudinis. </s> </p> <p id="N1BFBE" type="main"> <s id="N1BFC0"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1BFCC" type="main"> <s id="N1BFCE"><!-- NEW -->Ob&longs;eruabis, licèt i&longs;te motus non fiat per lineam parabolicam, vt &longs;uprà <lb/>demon&longs;trauimus Th. 54. & reliquis; quia tamen &longs;en&longs;ibiliter proximè <lb/>accedit, deinceps vtemur Parabola vt in fig. </s> <s id="N1BFD6"><!-- NEW -->Th. 83. & horizontalem <lb/>motum accipiemus pro æquabili; </s> <s id="N1BFDC"><!-- NEW -->licèt omninò æquabilis non &longs;it; </s> <s id="N1BFE0"><!-- NEW -->ni&longs;i <lb/>tantùm æquiualenter; </s> <s id="N1BFE6"><!-- NEW -->dixi æquiualenter, quia eodem modo &longs;e habet hic <lb/>motus, ac &longs;i per inclinatam &longs;ur&longs;um LC impetu &longs;cilicet LC mobile pro­<lb/>iiceretur; </s> <s id="N1BFEE"><!-- NEW -->&longs;ed in hoc ca&longs;u de&longs;trueretur impetus ille per inclinatam &longs;im­<lb/>plex; </s> <s id="N1BFF4"><!-- NEW -->igitur & mixtus; </s> <s id="N1BFF8"><!-- NEW -->quia tamen ille qui remanet partim ex LA, par­<lb/>tim ex LF eodem modo ferè &longs;e habet ac &longs;i totus LF intactus maneret; <lb/>hinc dictum e&longs;t &longs;uprà æquiualenter e&longs;&longs;e æquabilem. </s> </p> <p id="N1C000" type="main"> <s id="N1C002"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s> </p> <p id="N1C00E" type="main"> <s id="N1C010"><!-- NEW --><emph type="italics"/>A&longs;cendit hoc motu ad &longs;ubduplam altitudinem illius, ad quam motu mixto <lb/>tantum ex verticali & horizontali &longs;ine naturali a&longs;cenderet<emph.end type="italics"/>; quippe a&longs;cende­<lb/>ret in C fig. </s> <s id="N1C01D"><!-- NEW -->Th.83. &longs;ine impetu naturali, &longs;ed FC & LA æquales &longs;unt; </s> <s id="N1C021"><!-- NEW --><lb/>atqui motu violento puro, ni&longs;i naturalis obe&longs;&longs;et, a&longs;cenderet in A; </s> <s id="N1C026"><!-- NEW -->at ve­<lb/>rò &longs;i obe&longs;t naturalis; </s> <s id="N1C02C"><!-- NEW -->a&longs;cendit tantùm motu violento in K, & mixto in <lb/>in D; </s> <s id="N1C032"><!-- NEW -->quia ex K in L motu naturali tot acquireret mobile gradus impe­<lb/>tus naturalis quot amittit in motu violento ab L in K; </s> <s id="N1C038"><!-- NEW -->&longs;ed cum in impe­<lb/>tu acqui&longs;ito à K in L motu æquabili a&longs;cenderet ab L in A, quæ e&longs;t dupla <lb/>LK vt o&longs;tendimus in &longs;ecundo libro; </s> <s id="N1C040"><!-- NEW -->&longs;ed motu mixto, & verticali, & ho­<lb/>rizontali a&longs;cenderet in C; </s> <s id="N1C046"><!-- NEW -->&longs;ed FD e&longs;t &longs;ubdupla FE; igitur motu mixto <lb/>a&longs;cendit ad &longs;ubduplam altitudinem, &c. </s> </p> <p id="N1C04C" type="main"> <s id="N1C04E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s> </p> <p id="N1C05A" type="main"> <s id="N1C05C"><!-- NEW --><emph type="italics"/>Mobile projectum è naui mobili, vbi ad &longs;ummam altitudinem peruenit mo­<lb/>tu mixto ex verticali retardato, horizontali æquabili, & naturali item æqua­<lb/>bili, de&longs;cendit etiam motu mixto ex horizontali retardato &longs;altem æquiualenter, <lb/>& naturali accelerato<emph.end type="italics"/>; </s> <s id="N1C06B"><!-- NEW -->dixi æquiualenter, quia vt dixi in Sch. <!-- REMOVE S-->Th.86. licèt <lb/>remaneat aliquid impetus verticalis qui in communem lineam abit cum <lb/>horizontali; </s> <s id="N1C075"><!-- NEW -->res tamen perinde &longs;e habet atque &longs;i totus verticalis de&longs;true­<lb/>retur, & totus horizontalis intactus permaneret; igitur de&longs;cen&longs;us fit mo­<lb/>tu mixto ex naturali accelerato & horizontali retardato per Th.30. quia <lb/>tamen modico illo tempore parùm retardatur, vt &longs;uprà monui, &longs;en&longs;ibili­<lb/>ter accipi pote&longs;t pro æquabili. </s> </p> <pb pagenum="184" xlink:href="026/01/216.jpg"/> <p id="N1C085" type="main"> <s id="N1C087"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s> </p> <p id="N1C093" type="main"> <s id="N1C095"><!-- NEW --><emph type="italics"/>Hinc &longs;en&longs;ibiliter ex a&longs;cen&longs;u & de&longs;cen&longs;u fit<emph.end type="italics"/> <emph type="italics"/>integra Parabola<emph.end type="italics"/>; </s> <s id="N1C0A4"><!-- NEW -->nam pro­<lb/>iiciatur ex L in A, eo tempore, quo nauis mouetur ex L in F, certè &longs;i <lb/>tempus illud diuidatur bifariam prima parte mobile percurret LI tri­<lb/>plam IK in verticali, & LM &longs;ubduplam LF in horizontali; </s> <s id="N1C0AE"><!-- NEW -->igitur erit <lb/>in G; </s> <s id="N1C0B4"><!-- NEW -->&longs;ecunda verò parte temporis in verticali percurrit IK, & MF in <lb/>horizontali; </s> <s id="N1C0BA"><!-- NEW -->igitur erit in D; </s> <s id="N1C0BE"><!-- NEW -->præterea &longs;i accipiantur duæ aliæ partes tem­<lb/>poris æquales; </s> <s id="N1C0C4"><!-- NEW -->prima in perpendiculari deor&longs;um percurret DE æqua­<lb/>lem LK, & in horizontali DO; </s> <s id="N1C0CA"><!-- NEW -->igitur erit in N; </s> <s id="N1C0CE"><!-- NEW -->&longs;ecunda vero in per­<lb/>pendiculari percurret NQ triplam NO, & NR in horizontali; igitur <lb/>erit in S; </s> <s id="N1C0D6"><!-- NEW -->&longs;ed hæc e&longs;t Parabola; </s> <s id="N1C0DA"><!-- NEW -->nam vt &longs;e habent quadrata applicatarum <lb/>v.g. <!-- REMOVE S-->EG, FL, ita &longs;agittæ DE, DF; dixi &longs;en&longs;ibiliter, nam vt &longs;uprà mo­<lb/>nui e&longs;t alia linea, quæ tamen proximè accedit ad Parabolam. <!-- KEEP S--></s> </p> <p id="N1C0E5" type="main"> <s id="N1C0E7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s> </p> <p id="N1C0F3" type="main"> <s id="N1C0F5"><!-- NEW --><emph type="italics"/>Hinc ferè recedit mobile in idem punctum nauis, è quo &longs;ur&longs;um proiectum <lb/>e&longs;t<emph.end type="italics"/>; </s> <s id="N1C100"><!-- NEW -->dixi ferè, quia non e&longs;t omninò Parabola; immò &longs;upponitur motus <lb/>horizontalis tùm nauis tùm mobilis omninò æquabilis, à quo tamen <lb/>tantillùm deficit, &longs;ed in tam breui tempore non e&longs;t &longs;en&longs;ibile. </s> </p> <p id="N1C108" type="main"> <s id="N1C10A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 91.<emph.end type="center"/></s> </p> <p id="N1C116" type="main"> <s id="N1C118"><!-- NEW --><emph type="italics"/>Hinc quantùm initio detrahit horizontali verticalis inten&longs;ior, & &longs;ub finem <lb/>remittit, tantùm initio remittit horizontali naturalis tardior, & &longs;ub finem ve­<lb/>locior detrahit<emph.end type="italics"/>; </s> <s id="N1C125"><!-- NEW -->&longs;ic in a&longs;cen&longs;u linea curua LD, initio parùm recedit à ver­<lb/>ticali LK, & multùm &longs;ub finem; in de&longs;cen&longs;u verò curua DS accedit <lb/>propiùs ad horizontalem DT, à qua multùm recedit &longs;ub finem. </s> </p> <p id="N1C12D" type="main"> <s id="N1C12F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s> </p> <p id="N1C13B" type="main"> <s id="N1C13D"><!-- NEW --><emph type="italics"/>Hinc eadem, quâ mobilis proijcitur &longs;ur&longs;um è naui mobili, recipitur manu<emph.end type="italics"/>; <lb/>probata centies experientia; idem dico de &longs;agitta, arcu emi&longs;&longs;a, glande <lb/>tormento explo&longs;a, &c. </s> <s id="N1C14A"><!-- NEW -->&longs;ic dum demittis manu in eadem naui aliquod <lb/>graue deor&longs;um, eadem &longs;emper à te di&longs;tantia cadit; &longs;ic in rhodis currenti­<lb/>bus poma odorifera, &longs;ur&longs;um modica vi projecta eadem &longs;emper excipiun­<lb/>tur manu, perinde atque &longs;i currus ip&longs;e &longs;taret. </s> <s id="N1C154"><!-- NEW -->Ita pror&longs;us &longs;e res habet <lb/>dum in&longs;idens equo etiam pernici&longs;&longs;imè currenti ludis huiu&longs;modi moti­<lb/>bus; </s> <s id="N1C15C"><!-- NEW -->quorum nullum pror&longs;us di&longs;crimen ob&longs;eruabis in naui, &longs;iue &longs;tet &longs;iue <lb/>moueatur &longs;olito cur&longs;u; </s> <s id="N1C162"><!-- NEW -->&longs;i enim eadem velocitate, qua vel emi&longs;&longs;a &longs;agitta, <lb/>vel glans explo&longs;a moueretur; haud dubiè maximum di&longs;crimen inter­<lb/>cederet. </s> </p> <p id="N1C16A" type="main"> <s id="N1C16C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s> </p> <p id="N1C178" type="main"> <s id="N1C17A"><!-- NEW --><emph type="italics"/>Hinc &longs;i pilam projectam è naui mobili continuo intuitu pro&longs;equaris &longs;ur&longs;um <lb/>rectà ferri iudicabis<emph.end type="italics"/>; </s> <s id="N1C185"><!-- NEW -->quippe cum perpetuò mutes perpendicularem pro­<lb/>pter motum nauis, in eadem &longs;emper e&longs;&longs;e putas, in qua pila &longs;emper <lb/>occurrat; </s> <s id="N1C18D"><!-- NEW -->licèt reuerâ qui &longs;unt in naui immobili rem aliter e&longs;&longs;e <pb pagenum="185" xlink:href="026/01/217.jpg"/>iudicent; </s> <s id="N1C196"><!-- NEW -->quippe vident pilam &longs;uo motu de&longs;cribere curuam non &longs;imi­<lb/>lem illi, quam di&longs;cus per lineam inclinatam &longs;ur&longs;um proiectus &longs;uo mo­<lb/>tu de&longs;criberet; neque mirum e&longs;t, cum &longs;int eædem vtriu&longs;que rationes, cum <lb/>hac tantum differentia, quòd inclinata di&longs;ci &longs;it motus &longs;implicis, inclina­<lb/>ta verò pilæ a&longs;cendentis &longs;it motus mixti ex horizontali & verticali, æ­<lb/>quabili quidem in a&longs;cen&longs;us accelerato in de&longs;cen&longs;u. </s> </p> <p id="N1C1A4" type="main"> <s id="N1C1A6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s> </p> <p id="N1C1B2" type="main"> <s id="N1C1B4"><emph type="italics"/>Ex his vides non valere vulgarem rationem, quæ vulgò affertur contra mo­<lb/>tum terræ, &longs;equi &longs;cilicet ex eo lapidem proiectum &longs;ur&longs;um per verticalem longo <lb/>interuallo ver&longs;us occa&longs;um retrò de&longs;cen&longs;urum,<emph.end type="italics"/> quod tamen etiam ex motu <lb/>terræ &longs;uppo&longs;ito non &longs;equeretur, cum non &longs;equatur ex motu nauis. </s> </p> <p id="N1C1C2" type="main"> <s id="N1C1C4"><!-- NEW -->Igitur alia ratione impugnari debet hypothe&longs;is illa, quæ terræ motum <lb/>de&longs;truit; </s> <s id="N1C1CA"><!-- NEW -->quod certè &longs;i à me fieri po&longs;&longs;it, in tractatu de corporibus cœle&longs;ti­<lb/>bus, vel de nouo &longs;y&longs;temate aliquando præ&longs;tabimus; </s> <s id="N1C1D0"><!-- NEW -->non tamen e&longs;t quod <lb/>hîc di&longs;&longs;imulem aliquorum agendi methodum, qui ex hoc phœnome­<lb/>no con&longs;tanter a&longs;&longs;erunt terram moueri; </s> <s id="N1C1D8"><!-- NEW -->nam primò, &longs;equeretur tantùm <lb/>moueri circa centrum id e&longs;t motu orbis, non verò motu centri; quæ e&longs;t <lb/>hypothe&longs;is Origani. </s> <s id="N1C1E0"><!-- NEW -->Secundò ex quiete terræ hoc idem phœnomenon <lb/>&longs;equitur; </s> <s id="N1C1E6"><!-- NEW -->quippe, &longs;i terra quie&longs;cit, eadem manu cadentem excipio lapi­<lb/>dem, quæ &longs;ur&longs;um rectà proiicit; </s> <s id="N1C1EC"><!-- NEW -->igitur quemadmodum ex hoc non infero <lb/>terræ quietem, &longs;ed aliunde; </s> <s id="N1C1F2"><!-- NEW -->ita neque ex hoc inferri pote&longs;t terræ motus; </s> <s id="N1C1F6"><!-- NEW --><lb/>cum enim duplex hypothe&longs;is eodem phœnomeno &longs;tare pote&longs;t, neutra ex <lb/>eo euincitur; igitur &longs;icuti fateor ex hoc phœnomeno minimè demon­<lb/>&longs;trari terræ quietem ita & tu fateri debes ex eo minimè ad&longs;trui po&longs;&longs;e <lb/>terræ motum. </s> </p> <p id="N1C201" type="main"> <s id="N1C203"><!-- NEW -->Adde quod, haud dubiè &longs;i terra quie&longs;cit citiùs proiectus lapis &longs;ur&longs;um <lb/>de&longs;cendit, quàm &longs;i mouetur; </s> <s id="N1C209"><!-- NEW -->nec enim vt dictum e&longs;t &longs;uprà proiecta velo­<lb/>ci&longs;&longs;imo motu per horizontalem de&longs;cendunt eo tempore, quo ex eadem <lb/>altitudine motu purè naturali de&longs;cenderent; </s> <s id="N1C211"><!-- NEW -->quod multis euincitur ex­<lb/>perimentis, vt vidimus in Th.46. atqui punctum terræ &longs;ub æquatore ve­<lb/>loci&longs;&longs;imè moueretur, quod vno temporis &longs;ecundo conficeret 1250.pedes <lb/>geometricos &longs;i 5. pedes geometrici tribuantur pa&longs;&longs;ui, 4000. pa&longs;&longs;us leucæ <lb/>germanicæ, 15. leucæ germanicæ gradui Æquatoris, toti demum Æqua­<lb/>tori 360. gradus; </s> <s id="N1C21F"><!-- NEW -->cum autem iactus medius tormenti validi&longs;&longs;imi &longs;it <lb/>15000. pedum, duretque 30″ temporis; </s> <s id="N1C225"><!-- NEW -->certè 30″ temporis con&longs;icit pun­<lb/>ctum æquatoris 37500. pedes; </s> <s id="N1C22B"><!-- NEW -->igitur mouetur velociùs explo&longs;a glande; </s> <s id="N1C22F"><!-- NEW --><lb/>igitur &longs;i hæc velocitas glandis impedit, ne tàm citò deor&longs;um cadat, ma­<lb/>jor velocitas motus terræ potiori iure illud ip&longs;um impediet; igitur &longs;i <lb/>terra quie&longs;cit, globus &longs;ur&longs;um proiectus velociùs recidet in terram, et&longs;i <lb/>terra moueatur tardiùs. </s> </p> <p id="N1C23A" type="main"> <s id="N1C23C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C248" type="main"> <s id="N1C24A"><!-- NEW -->Ob&longs;eruabis duos tantùm motus in naui mobili fui&longs;&longs;e hactenus explica­<lb/>tos; </s> <s id="N1C250"><!-- NEW -->primus e&longs;t, quo demittitur plumbea pila è &longs;ummo mali; </s> <s id="N1C254"><!-- NEW -->&longs;ecundus e&longs;t, <lb/>quo ex <expan abbr="sũmo">summo</expan> malo, vel ex alio nauis mobilis puncto proiicitur <expan abbr="&longs;ursũ">&longs;ursum</expan> cor-<pb pagenum="186" xlink:href="026/01/218.jpg"/>pus graue per lineam verticalem; </s> <s id="N1C267"><!-- NEW -->&longs;unt autem plures alij motus, tot &longs;cili­<lb/>cet, quot po&longs;&longs;unt duci lineæ è &longs;ummo malo in orbem quoquo ver&longs;um; <lb/>quarum hæ &longs;unt præcipuæ. </s> <s id="N1C26F"><!-- NEW -->&longs;it apex mali B; </s> <s id="N1C273"><!-- NEW -->circa quem de&longs;cribatur cir­<lb/>culus ACDE, &longs;itque primò circulus ille verticalis parallelus &longs;cilicet li­<lb/>neæ directionis nauis BA, quæ &longs;it v. <!-- REMOVE S-->g. <!-- REMOVE S-->ver&longs;us Boream; </s> <s id="N1C27F"><!-- NEW -->primò habes li­<lb/>neam verticalem &longs;ur&longs;um BE; </s> <s id="N1C285"><!-- NEW -->&longs;ecundò perpendicularem deor&longs;um BC; </s> <s id="N1C289"><!-- NEW --><lb/>tertiò lineam directionis ver&longs;us Boream BA; </s> <s id="N1C28E"><!-- NEW -->quartò illi oppo&longs;itum <lb/>ver&longs;us Au&longs;trum BD; tùm voluatur circulus circa axem immobilem AD <lb/>per quadrantem integrum, dum &longs;cilicet BE &longs;it ad Ortum, quæ e&longs;t quinta <lb/>linea, & BC ip&longs;i oppo&longs;ita ad Occa&longs;um, quæ e&longs;t &longs;exta. </s> <s id="N1C298">Igitur habes 6. li­<lb/>neas; </s> <s id="N1C29D"><!-- NEW -->&longs;cilicet &longs;ur&longs;um, deor&longs;um, ver&longs;us Boream & Au&longs;trum, ver&longs;us Ortum, <lb/>& Occa&longs;um; linea quæ tendit deor&longs;um pote&longs;t dupliciter con&longs;iderari, vel <lb/>enim demittitur &longs;ua &longs;ponte, vel proiicitur. </s> </p> <p id="N1C2A5" type="main"> <s id="N1C2A7"><!-- NEW -->Iam verò inter <expan abbr="Boreã">Boream</expan>, & Occa&longs;um habes lineas triplicis generis, primò <lb/>horizonti parallelas, quæ vt con&longs;iderentur; </s> <s id="N1C2B1"><!-- NEW -->cen&longs;eatur prædictus circulus <lb/>parallelus horizonti, ita vt ex centro B ducantur ad <expan abbr="circumferentiã">circumferentiam</expan> tot <lb/>lineæ, quot &longs;unt puncta in circumferentia; </s> <s id="N1C2BD"><!-- NEW -->&longs;ecundò inclinatas &longs;ur&longs;um & <lb/>inclinatas deor&longs;um; </s> <s id="N1C2C3"><!-- NEW -->&longs;imiliter inter Occa&longs;um & Au&longs;trum, inter Au&longs;trum <lb/>& Ortum, inter Ortum & Boream; porrò exprimes omnes lineas, &longs;i api­<lb/>cem mali fingas centrum globi, &longs;eu &longs;i in circulo prædicto verticali à <lb/>centro B ad circumferentiam ducantur tot lineæ quot po&longs;&longs;unt duci, <lb/>tuncque circa axem EC immobilem voluatur circulus, &c. </s> <s id="N1C2CF">his po&longs;i­<lb/>tis &longs;it. </s> </p> <p id="N1C2D4" type="main"> <s id="N1C2D6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s> </p> <p id="N1C2E2" type="main"> <s id="N1C2E4"><!-- NEW --><emph type="italics"/>Si proijciatur globus deor&longs;um à &longs;ummo malo, de&longs;cendet ferè ad imum ma­<lb/>lum<emph.end type="italics"/>; </s> <s id="N1C2EF"><!-- NEW -->probatur, quia de&longs;cendet quidem velociùs quàm &longs;i motu naturali <lb/>de&longs;cenderet vt con&longs;tat per Th. 69. &longs;ed profectò nihil acquiret in hori­<lb/>zontali globus, quod non acquirat nauis; </s> <s id="N1C2F7"><!-- NEW -->igitur imùm ferè malum attin­<lb/>git &longs;ed opus e&longs;t aliqua figurâ; </s> <s id="N1C2FD"><!-- NEW -->&longs;it enim apex mali A, de&longs;cendatque pri­<lb/>mò ex A &longs;ua &longs;ponte in H; </s> <s id="N1C303"><!-- NEW -->haud dubiè &longs;i eo tempore, quo motu na­<lb/>turali conficit AD, mixto deor&longs;um conficit AF, eo tempore cadet in G <lb/>ex A &longs;i hic impetus deor&longs;um adueniat; </s> <s id="N1C30B"><!-- NEW -->&longs;ed res e&longs;t clara; </s> <s id="N1C30F"><!-- NEW -->hæc porrò figura <lb/>non e&longs;t Parabola, licèt &longs;it curua; </s> <s id="N1C315"><!-- NEW -->con&longs;tat autem hîc motus ex naturali <lb/>accelerato, ex impre&longs;&longs;o deor&longs;um æquabili per &longs;e, & horizontali &longs;en&longs;i­<lb/>biliter æquabili; pote&longs;t autem de&longs;ignari hæc linea motus ex &longs;uprà <lb/>dictis. </s> </p> <p id="N1C31F" type="main"> <s id="N1C321"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s> </p> <p id="N1C32D" type="main"> <s id="N1C32F"><!-- NEW --><emph type="italics"/>Si in circulo verticali prædicto proijciatur per lineam horizontalem ver­<lb/>&longs;us Boream, mouebitur globus motu mixto ex duplici horizontali per <expan abbr="eãdem">eandem</expan> <lb/>lineam ferè æquabili; </s> <s id="N1C33D"><!-- NEW -->id e&longs;t &longs;en&longs;ibiliter, licèt geometricè loquendo retardetur, <lb/>& naturali accelerato<emph.end type="italics"/>; </s> <s id="N1C346"><!-- NEW -->&longs;it perpendicularis deor&longs;um AH, <expan abbr="horizōtalis">horizontalis</expan> AC, <lb/>quam conficiat eo tempore, quo conficit AH motu naturali, motu mixto <lb/>perueniet in K; </s> <s id="N1C352"><!-- NEW -->&longs;i verò duplicetur horizontalis, ita vt eo tempore quo <lb/>conficit AH, conficiat AD, motu mixto perueniet in L; </s> <s id="N1C358"><!-- NEW -->hæc autem curua <pb pagenum="187" xlink:href="026/01/219.jpg"/>HL accedit ad Parabolam licèt non &longs;it vera Parabola; quia quando ia­<lb/>ctus horizontalis e&longs;t veloci&longs;&longs;imus, qualis in arce, vel in tormentis belli­<lb/>cis, eodem tempore mobile non decidit in terram, quo de&longs;cenderet mo­<lb/>tu purè naturali ex eadem altitudine. </s> </p> <p id="N1C367" type="main"> <s id="N1C369"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s> </p> <p id="N1C375" type="main"> <s id="N1C377"><!-- NEW --><emph type="italics"/>Hinc, &longs;i motus nauis e&longs;&longs;et æqualis motui &longs;agittæ, motus ex vtroque mixtus <lb/>duplam amplitudinem in plano hòrizontali acquireret, v.g. <!-- REMOVE S-->&longs;i<emph.end type="italics"/> tantùm &longs;agitta <lb/>emi&longs;&longs;a arcu extra nauim ex A perueniret in K, in naui mobili perueniret <lb/>in L; </s> <s id="N1C388"><!-- NEW -->&longs;i verò nauis, vt reuerâ fit, tardiùs moueatur, &longs;agitta è naui emi&longs;&longs;a <lb/>ver&longs;us Boream &longs;cilicet acquiret pro rata, id e&longs;t &longs;i nauis motus &longs;it tantùm <lb/>&longs;ubduplus perueniret in M; &longs;i &longs;ubquadruplus in N &c. </s> </p> <p id="N1C390" type="main"> <s id="N1C392"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s> </p> <p id="N1C39E" type="main"> <s id="N1C3A0"><!-- NEW --><emph type="italics"/>Hinc tormentum bellicum quod e&longs;t in prora directum ad <expan abbr="eãdem">eandem</expan> lineam, <lb/>quam &longs;uo motu conficit nauis maiorem iactum habebit, non tamen &longs;en&longs;ibiliter<emph.end type="italics"/>; </s> <s id="N1C3AF"><!-- NEW --><lb/>quia motus nauis parum addit; </s> <s id="N1C3B4"><!-- NEW -->ob&longs;eruabis tamen non videri maiorem <lb/>quàm &longs;i nauis quie&longs;ceret, quia eo tempore, quo &longs;agitta ex A peruenit in <lb/>L, nauis ex H peruenit in K; </s> <s id="N1C3BC"><!-- NEW -->igitur videtur &longs;emper e&longs;&longs;e idem iactus, &longs;iue <lb/>moueatur nauis &longs;iue non, quia e&longs;t &longs;emper eadem di&longs;tantia nauis, & ter­<lb/>mini iactus; cum nauis id totum acquirat &longs;patij, quod motui &longs;agittæ <lb/>accedit. </s> </p> <p id="N1C3C6" type="main"> <s id="N1C3C8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 99.<emph.end type="center"/></s> </p> <p id="N1C3D4" type="main"> <s id="N1C3D6"><!-- NEW --><emph type="italics"/>Hinc vt quis maiore ni&longs;u lapidem v. <!-- REMOVE S-->g. <!-- REMOVE S-->proijciat, tùm longiore tempore <lb/>brachium rotat, tùm præuio cur&longs;u impetum auget,<emph.end type="italics"/> quia non tantùm impe­<lb/>tus brachij imprimitur mobili, &longs;ed etiam impetus totius corporis; </s> <s id="N1C3E7"><!-- NEW -->hinc <lb/>etiam &longs;i præmittatur cur&longs;us longiore &longs;altu in plano horizontali maius <lb/>&longs;patium traiicitur; quæ omnia ex ii&longs;dem principiis manife&longs;tè &longs;e­<lb/>quuntur. </s> </p> <p id="N1C3F3" type="main"> <s id="N1C3F5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s> </p> <p id="N1C401" type="main"> <s id="N1C403"><!-- NEW --><emph type="italics"/>Si verò per oppo&longs;itam lineam ver&longs;us Au&longs;trum proijcitur mobile, mouebitur <lb/>motu mixto ex duobus horizontalibus ad oppo&longs;itas lineas, & ex naturali ac­<lb/>celerato<emph.end type="italics"/>; </s> <s id="N1C410"><!-- NEW -->&longs;it proiectio per AB, ita vt mobilè perueniat in L ni&longs;i impedia­<lb/>tur; </s> <s id="N1C416"><!-- NEW -->certè &longs;i nauis motu &longs;ubduplo in oppo&longs;itam partem feratur, peruenit <lb/>tantùm in K, quæ omnia con&longs;tant ex dictis; </s> <s id="N1C41C"><!-- NEW -->nam impetus oppo&longs;iti pu­<lb/>gnant pro rata, vt &longs;æpè diximus; </s> <s id="N1C422"><!-- NEW -->videbitur tamen e&longs;&longs;e æqualis iactus; </s> <s id="N1C426"><!-- NEW -->&longs;i <lb/>enim eo tempore, quo &longs;agitta peruenit in K, nauis fertur in oppo&longs;itam <lb/>partem &longs;patio æquali KL, haud dubiè di&longs;tantia &longs;emper erit æqualis; tan­<lb/>tùm enim recedit ver&longs;us Boream nauis, quantùm &longs;agitta à puncto L ad <lb/>punctum K reducitur. </s> </p> <p id="N1C432" type="main"> <s id="N1C434"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 101.<emph.end type="center"/></s> </p> <p id="N1C440" type="main"> <s id="N1C442"><!-- NEW --><emph type="italics"/>Si motus nauis e&longs;&longs;et æqualis motui &longs;agittæ v. <!-- REMOVE S-->g.<emph.end type="italics"/> <emph type="italics"/>&longs;i nauis ferretur per <lb/>lineam GC &longs;eu TA ver&longs;us Boream, & &longs;agitta è &longs;ummo malo emitteretur <lb/>per lineam TO ver&longs;us Au&longs;trum, de&longs;cenderet per lineam T.G. nec quidquam<emph.end type="italics"/><pb pagenum="188" xlink:href="026/01/220.jpg"/><emph type="italics"/>acquireret in horizontali<emph.end type="italics"/>; </s> <s id="N1C460"><!-- NEW -->quod probatur per Th. 133. l.1. &longs;ic globus tor­<lb/>menti etiam ne latum quidem vnguem pertran&longs;iret in horizontali, vide­<lb/>tur tamen &longs;emper e&longs;&longs;e idem iactus; </s> <s id="N1C468"><!-- NEW -->nam eo tempore, quo &longs;agitta caderet <lb/>à T in G, nauis e&longs;&longs;et in C, atqui CG & GM &longs;unt a&longs;&longs;umptæ æquales; hinc <lb/>potiùs arcus e&longs;&longs;et emi&longs;&longs;us quàm &longs;agitta, & tormentum explo&longs;um quàm <lb/>globus. </s> </p> <p id="N1C472" type="main"> <s id="N1C474"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C480" type="main"> <s id="N1C482"><!-- NEW -->Ob&longs;eruabis, &longs;i nauis motus &longs;it ad motum &longs;agittæ v. <!-- REMOVE S-->g. <!-- REMOVE S-->in ratione &longs;ub­<lb/>dupla, &longs;cilicet vt FG, vel LM ad GM peruenit in L per Parabolam TL; </s> <s id="N1C48C"><!-- NEW -->&longs;t <lb/>vt EG vel KM ad GL peruenit in K per Parabolam TK; &longs;i vt DG vel I <lb/>M ad GM peruenitin I per Parabolam TI, &c. </s> <s id="N1C494"><!-- NEW -->vnde vides Parabolas <lb/>i&longs;tas &longs;emper in infinitum contrahi, donec tandem in rectam TG de&longs;i­<lb/>nant vbi motus nauis e&longs;t æqualis motui &longs;agittæ: Parabolas dixi &longs;en&longs;ibi­<lb/>liter, &longs;cilicet eo modo, quo &longs;uprà. </s> </p> <p id="N1C49E" type="main"> <s id="N1C4A0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 102.<emph.end type="center"/></s> </p> <p id="N1C4AC" type="main"> <s id="N1C4AE"><emph type="italics"/>Si verò motus nauis e&longs;&longs;et maior motu &longs;agittæ, &longs;agitta fèrretur in <expan abbr="eãdem">eandem</expan> <lb/>partem in quam fertur nauis per &longs;patium æquale differentia illorum motuum,<emph.end type="italics"/><lb/>v.g. </s> <s id="N1C4BD"><!-- NEW -->&longs;i nauis moueatur per GM & &longs;agitta per TA, &longs;itque motus nauis ad <lb/>motum &longs;agittæ, vt GM, ad IM; eo tempore quo nauis attinget M, &longs;agitta <lb/>cadet in I, & &longs;i motus &longs;it vt GM ad KM cadet in K vel vt GM ad GL <lb/>cadet in L. per Parabolas, quæ omnia con&longs;tant ex dictis, & ex Theore­<lb/>mate per 134. l.1. </s> </p> <p id="N1C4C9" type="main"> <s id="N1C4CB"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C4D8" type="main"> <s id="N1C4DA"><!-- NEW --><emph type="italics"/>Ex illa hypothe&longs;i &longs;equitur egregium paradoxon &longs;cilicet &longs;agittam retorqueri <lb/>in &longs;agittarium<emph.end type="italics"/>; </s> <s id="N1C4E5"><!-- NEW -->&longs;it enim motus nauis ad motum &longs;agittæ vt GM ad LM; </s> <s id="N1C4E9"><!-- NEW --><lb/>haud dubiè per Th. &longs;uperius eo tempore, quo nauis peruenit ad M &longs;a­<lb/>gitta attinget punctum L, & eo tempore quo nauis e&longs;&longs;et in L &longs;agitta e&longs;­<lb/>&longs;et in puncto Y, &longs;i cum nauis peruenit in L illicò &longs;i&longs;tat &longs;agitta, cadet in <lb/>ip&longs;am nauim; </s> <s id="N1C4F4"><!-- NEW -->nam cadet in L quod clarum e&longs;t: </s> <s id="N1C4F8"><!-- NEW -->dixi &longs;i nauis &longs;i&longs;tat po&longs;t <lb/>emi&longs;&longs;am &longs;agittam, &longs;i enim nauis &longs;emper moueatur, æquabilis &longs;emper e&longs;&longs;e <lb/>videbitur &longs;agittæ iactus, &longs;i enim è naui immobili emi&longs;&longs;a fui&longs;&longs;et prædicta <lb/>&longs;agitta per horizontalem TO, acqui&longs;iui&longs;&longs;et &longs;patium vel amplitudinem G <lb/>L; </s> <s id="N1C504"><!-- NEW -->&longs;ed videtur confeci&longs;&longs;e ML, cum nauis mouetur; atqui ML e&longs;t æqualis <lb/>LG, quid clarius? </s> </p> <p id="N1C50A" type="main"> <s id="N1C50C"><!-- NEW -->Hinc &longs;i quis in naui currat per lineam directionis id e&longs;t ver&longs;us eain <lb/>partem, in quam mouetur nauis, curret velociùs; </s> <s id="N1C512"><!-- NEW -->immò &longs;i ambulet, ingen­<lb/>tes faciet pa&longs;&longs;us &longs;eu &longs;altus v.g.&longs;i nauis conficit &longs;patium GM eo tempore <lb/>quo aliquis &longs;altat ex G in H; </s> <s id="N1C51A"><!-- NEW -->haud dubiè amplitudo eius &longs;altus erit com­<lb/>po&longs;ita ex tota GM & GH; </s> <s id="N1C520"><!-- NEW -->&longs;i verò in partem oppo&longs;itam ver&longs;us C currat: </s> <s id="N1C524"><!-- NEW --><lb/>vel currit velociùs, vel tardiùs, vel æquali motu: </s> <s id="N1C529"><!-- NEW -->&longs;i primum, aliquid &longs;patij <lb/>acquiret ver&longs;us C æqualis &longs;cilicet <expan abbr="differ&etilde;tiæ">differentiæ</expan> motuum; </s> <s id="N1C533"><!-- NEW -->&longs;i <expan abbr="&longs;ecundũ">&longs;ecundum</expan>, recedet <lb/>ver&longs;us M &longs;patio æquali eidem differentiæ; &longs;i tertium, nec accedet, nec re­<lb/>cedet, &longs;ed totis viribus currens &longs;eu tentans currere in eodem &longs;emper lo-<pb pagenum="189" xlink:href="026/01/221.jpg"/>co &longs;tabit, vel &longs;i &longs;it rotatus globus in tabulato nauis mouebitur motu or­<lb/>bis circa centrum immobile. </s> </p> <p id="N1C546" type="main"> <s id="N1C548"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 103.<emph.end type="center"/></s> </p> <p id="N1C554" type="main"> <s id="N1C556"><!-- NEW --><emph type="italics"/>Si proiiciatur mobile per lineam inclinatam deor&longs;um, quæ &longs;it hypothenu&longs;is <lb/>trianguli orthogonij, cuius ba&longs;is &longs;it horizontalis & perpendiculum &longs;patium,<emph.end type="italics"/><lb/>quod percurritur motu naturali æquali tempore, idque in naui mobili <lb/>in eam <expan abbr="part&etilde;">partem</expan>, ver&longs;us quam mouetur nauis, erit motus mixtus ex naturali <lb/>accelerato & inclinato mixto ex horizontali & alio inclinato &longs;it enim <lb/>horizontalis AD, perpendicularis AMK, &longs;it AM &longs;patium quod percurri­<lb/>tur in perpendiculari motu purè naturali, eo tempore, quo percurritur <lb/>AC &longs;ubdupla AD, &longs;itque AM &longs;ubdupla AC, & &longs;ecundo tempore æquali <lb/>percurratur in horizontali CD, & in perpendiculari MK tripla AM; </s> <s id="N1C572"><!-- NEW --><lb/>erit motus mixtus per lineam parabolicam ANH; </s> <s id="N1C577"><!-- NEW -->nam &longs;uppono hori­<lb/>zontalem æquabilem, cùm parùm ab eo ab&longs;it, vt &longs;upradictum e&longs;t; præ&longs;er­<lb/>tim cum &longs;en&longs;ibiliter hæc linea &longs;it parabolica. </s> </p> <p id="N1C57F" type="main"> <s id="N1C581"><!-- NEW -->Iam verò in eadem naui proiiciatur mobile per inclinatam AP, quæ <lb/>&longs;it diagonalis quadrati AP, & impetus perinclinatam AP &longs;it ad impetum <lb/>per horizontalem AC, vt AP ad AC; </s> <s id="N1C589"><!-- NEW -->ducatur LPF parallela MN, & CF <lb/>parallela AP; </s> <s id="N1C58F"><!-- NEW -->denique diagonalis AF: </s> <s id="N1C593"><!-- NEW -->haud dubiè ML e&longs;t æqualis AM, vt <lb/>patet; </s> <s id="N1C599"><!-- NEW -->& &longs;i motus e&longs;&longs;et tantum mixtus ex AC & AP fieret per diagona­<lb/>lem AF, quam mobile eodem tempore percurreret quo vel AC vel AP; </s> <s id="N1C59F"><!-- NEW --><lb/>igitur &longs;i dum percurrit AF percurrit AM, motu naturali, certè dum per­<lb/>currit AN &longs;ubdupla AF, percurret tantùm &longs;ubquadruplam AM; </s> <s id="N1C5A6"><!-- NEW -->a&longs;&longs;uma­<lb/>tur ergo NO æqualis AS, & FG æqualis AM; <expan abbr="ducaturq;">ducaturque</expan> curua AOG, hæc <lb/>e&longs;t linea qu&ecedil;&longs;ita. </s> </p> <p id="N1C5B2" type="main"> <s id="N1C5B4"><!-- NEW -->Itaque idem dicendum e&longs;t de his inclinatis, quod de aliis &longs;uprà di­<lb/>ctum e&longs;t Th.72. ni&longs;i quod accipitur inclinata mixta ex horizontali & da­<lb/>ta inclinata, v.g. <!-- REMOVE S-->ANF ex AC & AP; </s> <s id="N1C5BE"><!-- NEW -->hæc autem linea non e&longs;t Parabolica, <lb/>quia quadratum MN, vel VO e&longs;t ad quadratum RG vt 1.ad 4.at verò &longs;a­<lb/>gitta AV e&longs;t ad &longs;agittam AP, vt 5.ad 12.porrò hæc linea &longs;ecat Parabolam <lb/>vt patet; &longs;i verò accipiatur inclinatata AI, mixta inclinata erit AH igitur <lb/>a&longs;&longs;umatur HX æqualis AM, & PZ æqualis AS ducetur linea huius mo­<lb/>tus per AZX. quænam verò &longs;int hç lineæ, dicemus aliàs Tomo &longs;equenti. </s> </p> <p id="N1C5CC" type="main"> <s id="N1C5CE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 104.<emph.end type="center"/></s> </p> <p id="N1C5DA" type="main"> <s id="N1C5DC"><!-- NEW --><emph type="italics"/>Si proiiciatur per inclinatam &longs;ur&longs;um in eam partem, in quam mouetur nauis, <lb/>erit etiam mixtus ex naturali, & inclinato ex horizontali, & data inclinata<emph.end type="italics"/>; <lb/>vnde idem pror&longs;us <expan abbr="dic&etilde;duin">dicendum</expan> e&longs;t de mixta inclinata, quod de &longs;implici in­<lb/>clinata, de qua multa &longs;uprà dicta &longs;unt à Th.47. &longs;uppo&longs;ito tamen motu na­<lb/>turali accelerato, ad quem proximè accedit propter mutationem perpe­<lb/>tuam lineæ. </s> <s id="N1C5F3"><!-- NEW -->&longs;it enim inclinata &longs;ur&longs;um AB, quæ percurratur motu <lb/>æquabili eo tempore, quo horizontalis AE, vel quo motu naturali LA; </s> <s id="N1C5F9"><!-- NEW --><lb/>diuidatur AE bifariam in D; </s> <s id="N1C5FE"><!-- NEW -->ducatur DG, tùm DC, AC, hæc e&longs;t linea mo­<lb/>tus mixti ex inclinata AG, & horizontali AD; </s> <s id="N1C604"><!-- NEW -->&longs;equitur deinde Parabola; </s> <s id="N1C608"><!-- NEW --><lb/>nam &longs;i eo tempore quo percurritur AD, percurritur AG, & LM vel FA; </s> <s id="N1C60D"><!-- NEW --><pb pagenum="190" xlink:href="026/01/222.jpg"/>certè eodem percurritur AC, igitur &longs;ubduplo tempore <expan abbr="percurr&etilde;tur">percurrentur</expan> AN; </s> <s id="N1C619"><!-- NEW --><lb/>igitur FO, quæ e&longs;t &longs;ubquadrupla FA; </s> <s id="N1C61E"><!-- NEW -->igitur a&longs;&longs;umatur NH æqualis FO, & <lb/>CK æqualis FA, & ducatur curua per puncta AHK; hæc e&longs;t &longs;emiparabo­<lb/>la, nam KI e&longs;t ad KE vt quadratum IH ad quadratum EA. </s> </p> <p id="N1C626" type="main"> <s id="N1C628"><!-- NEW -->Vnde vides omnes inclinatas &longs;ur&longs;um v&longs;que ab horizontali DB ad <lb/>verticalem DA inclu&longs;iuè e&longs;&longs;e Parabolas; omnes verò inclinatas ab ea­<lb/>dem horizontali DB ad perpendicularem DC inclu&longs;iuè non e&longs;&longs;e Para­<lb/>bolas, &longs;ed propiùs accedere ad rectam, vnde aliquis &longs;u&longs;picari po&longs;&longs;et e&longs;&longs;e <lb/>Hyperbolas. </s> </p> <p id="N1C634" type="main"> <s id="N1C636"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 105.<emph.end type="center"/></s> </p> <p id="N1C642" type="main"> <s id="N1C644"><!-- NEW --><emph type="italics"/>Si proijciatur mobile per inclinatam &longs;ur&longs;um vel deor&longs;um in partem oppo&longs;i­<lb/>tam directionis nauis,<emph.end type="italics"/> <emph type="italics"/>&longs;cilicet per diagonales de&longs;cendit & a&longs;cendit per li­<lb/>neam rectam, &longs;ur&longs;um vel deor&longs;um, v.g.<emph.end type="italics"/> &longs;it horizontalis KL, inclinata <lb/>deor&longs;um KB, mixta erit KL; </s> <s id="N1C659"><!-- NEW -->&longs;it etiam inclinata KL, & horizontalis <lb/>CH; </s> <s id="N1C65F"><!-- NEW -->mixta erit KH, cui addatur in eadem KF portio &longs;patij, quod motu <lb/>naturali percurritur; idem dico de aliis inclinatis. </s> </p> <p id="N1C665" type="main"> <s id="N1C667"><!-- NEW -->Præterea &longs;it horizontalis VX, inclinata <expan abbr="&longs;ursũ">&longs;ursum</expan> VN; </s> <s id="N1C66F"><!-- NEW -->mixta erit VY; </s> <s id="N1C673"><!-- NEW -->&longs;ic <lb/>ex VOVX fiet VS detracta &longs;cilicet portioni &longs;patij, quod detrahitur à <lb/>motu naturali; &longs;i verò &longs;it vel major motus horizontalis, vel minor eo, <lb/>quem a&longs;&longs;ump&longs;imus, non percurrit mobile lineam rectam &longs;ed vel Para­<lb/>bolam &longs;i &longs;ur&longs;um proiiciatur, vel &longs;i deor&longs;um aliam nouam, quam ad Hy­<lb/>perbolam accedere &longs;uprà diximus. </s> </p> <p id="N1C681" type="main"> <s id="N1C683"><!-- NEW -->Hinc certè, quod mirabile dictu e&longs;t, &longs;i è puncto nauis V &longs;ur&longs;um per <lb/>inclinatam VO proiiciatur, &longs;tatimque po&longs;t proiectionem &longs;i&longs;tat nauis, in <lb/>ip&longs;am nauim de&longs;cendet mobile; </s> <s id="N1C68B"><!-- NEW -->atque ita ex his habeo omnes motus cir­<lb/>culi verticalis paralleli lineæ directionis; </s> <s id="N1C691"><!-- NEW -->quare &longs;upere&longs;t vt explicemus <lb/>alios motus; ac primò quidem per circulum horizontalem, cuius habeo <lb/>quoque duas lineas, &longs;cilicet communes &longs;ectiones horizontalis & prio­<lb/>ris verticalis, id e&longs;t lineam directionis ver&longs;us Boream, & oppo&longs;itam ver­<lb/>&longs;us Au&longs;trum. <!-- KEEP S--></s> </p> <p id="N1C69E" type="main"> <s id="N1C6A0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 106.<emph.end type="center"/></s> </p> <p id="N1C6AC" type="main"> <s id="N1C6AE"><!-- NEW --><emph type="italics"/>Si proijciatur mobile per horizontalem ver&longs;us Ortum è naui mobili, <lb/>monebitur motu mixto ex duplici horizontali, & naturali deor&longs;um<emph.end type="italics"/>, &longs;it <lb/>enim horizontalis ver&longs;us Boream AC, & alia horizontalis AH ver&longs;us <lb/>ortum in eodem plano horizontali; </s> <s id="N1C6BD"><!-- NEW -->certè ex vtraque fit mixta AK, quæ <lb/>&longs;i percurratur æquali tempore cum AC, & eius &longs;ubdupla cum AB, AC <lb/>verò æquali tempore cum AF; </s> <s id="N1C6C5"><!-- NEW -->quamquàm &longs;uppono iam e&longs;&longs;e perpendi­<lb/>cularem deor&longs;um AB; </s> <s id="N1C6CB"><!-- NEW -->denique cum AG &longs;ubquadrupla AF a&longs;&longs;umatur <lb/>ED æqualis AG perpendiculariter ducta in AD, & KL æqualis AF <lb/>parallela ED, & per puncta AEL ducatur curua, hæc e&longs;t linea motus <lb/>quæ&longs;ita; </s> <s id="N1C6D5"><!-- NEW -->voluatur autem triangulum AKL, donec &longs;it parallelum circulo <lb/>verticali vel alteri, ACO erit in proprio &longs;itu; </s> <s id="N1C6DB"><!-- NEW -->vnde eo tempore, quo e&longs;­<lb/>&longs;et in DE punctum nauis A e&longs;&longs;et in B, & eo, quo e&longs;&longs;et in KL, punctum A <lb/>e&longs;&longs;et in C; hoc e&longs;t &longs;ingula puncta AK, è regione AC ductis parallelis <pb pagenum="191" xlink:href="026/01/223.jpg"/>BD, CK, ac proinde nauis & mobile &longs;emper e&longs;&longs;ent è regione in linea <lb/>ver&longs;us ortum. </s> </p> <p id="N1C6EA" type="main"> <s id="N1C6EC"><!-- NEW -->Hinc &longs;i ex A dirigas <expan abbr="&longs;agittã">&longs;agittam</expan> in H feris punctum K, quam artem probè <lb/>no&longs;&longs;e debent rei tormentariæ præfecti; </s> <s id="N1C6F6"><!-- NEW -->quippe &longs;agitta aberrabit à &longs;copo <lb/>ver&longs;us Boream declinans toto eo &longs;patio, quod conficit nauis eodem tem­<lb/>pore, quo mouetur &longs;agitta; ita pror&longs;us &longs;i moueatur H ver&longs;us K, vt attin­<lb/>gas ex puncto immobili A debes dirigere ictum in K, &longs;i quo tempore <lb/>&longs;agitta conficit AK &longs;copus H percurrit HK.Idem pror&longs;us dicendum e&longs;t <lb/>de iaculatione per lineam oppo&longs;itam ver&longs;us occa&longs;um. </s> </p> <p id="N1C704" type="main"> <s id="N1C706"><!-- NEW -->Si verò proiiciatur mobile per lineam inter Boream, & Ortum, linea <lb/>motus erit Parabola cuius Tangens erit mixta ex horizontali ver&longs;us <lb/>Boream, & declinante ver&longs;us Ortum, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it horizontalis ver&longs;us Boream <lb/>AF, quam hactenus a&longs;&longs;ump&longs;i pro linea directionis; </s> <s id="N1C714"><!-- NEW -->&longs;it linea ver&longs;us <lb/>Ortum AC; </s> <s id="N1C71A"><!-- NEW -->&longs;it declinans ver&longs;us Boream AL; </s> <s id="N1C71E"><!-- NEW -->&longs;itque impetus AL, ad <lb/>AE vt AL ad AE, quod hactenus &longs;uppo&longs;ui; </s> <s id="N1C724"><!-- NEW -->&longs;it LG æqualis AE, AG <lb/>e&longs;t mixta ex AE, AL; </s> <s id="N1C72A"><!-- NEW -->a&longs;&longs;umatur KI, & GH vt iam diximus; fiatque <lb/>Parabola AIH, quæ circa axem AE ita voluatur, vt &longs;it perpendicularis <lb/>plano horizontali LF. </s> </p> <p id="N1C732" type="main"> <s id="N1C734">Idem dico de omni alia declinante vel à Borea ad Ortum, vel ad Oc­<lb/>ca&longs;um. </s> </p> <p id="N1C739" type="main"> <s id="N1C73B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 107.<emph.end type="center"/></s> </p> <p id="N1C747" type="main"> <s id="N1C749"><!-- NEW --><emph type="italics"/>Si mobile proiiciatur per declinantem ab Austro ad Ortum, cuius impetus <lb/>&longs;it vt linea; </s> <s id="N1C751"><!-- NEW -->conficit lineam parabolicam, cuius tangens vel amplitudo e&longs;t re­<lb/>sta ad Ortum<emph.end type="italics"/>; </s> <s id="N1C75A"><!-- NEW -->&longs;it enim NF ad Boream, NA ad Au&longs;trum, NI ad Or­<lb/>tum, ND ad Occa&longs;um; </s> <s id="N1C760"><!-- NEW -->&longs;it NL declinans ab au&longs;tro ad Ortum, &longs;itque im­<lb/>petus per NL ad impetum per NF, vt NL ad NF; </s> <s id="N1C766"><!-- NEW -->mixta ex NF NL <lb/>e&longs;t HK; </s> <s id="N1C76C"><!-- NEW -->&longs;it autem KH æqualis &longs;patio, quod conficitur motu naturali eo <lb/>tempore, quo percurritur NF, &longs;it KI æqualis NK, & IG quadrupla KH; <lb/>Parabola NHG e&longs;t linea motus quæ&longs;ita dum voluatur NIG circa axem <lb/>NI, dum IG pendeat perpendicularitur ex plano horizontali ON. </s> </p> <p id="N1C776" type="main"> <s id="N1C778">Idem fiet, &longs;i proiiciatur per declinantem NB ab Au&longs;tro &longs;cilicet ad <lb/>Occa&longs;um. <!-- KEEP S--></s> </p> <p id="N1C77E" type="main"> <s id="N1C780"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 108.<emph.end type="center"/></s> </p> <p id="N1C78C" type="main"> <s id="N1C78E"><!-- NEW --><emph type="italics"/>Si mobile proiiciatur per inclinantem &longs;ur&longs;um in circulo verticali, cuius &longs;e­<lb/>ctio cum horizontali tendit ad Ortum, conficit lineam parabolicam, cuius am­<lb/>plitudo e&longs;t mixta ex horizontali ver&longs;us Boream, & horizontali ver&longs;us Ortum,<emph.end type="italics"/><lb/> &longs;it linea ver&longs;us Boream AB, ver&longs;us Ortum AK, mixta ex vtraque AF, <lb/>linea inclinata &longs;ur&longs;um AP, Parabola AMN, quæ vertatur circa A do­<lb/>nec incubet AFG, denique AFG circa FA voluatur, donec incubet <lb/>perpendiculariter plano; porrò perinde e&longs;t, &longs;iue proiiciatur per inclina­<lb/>tam &longs;ur&longs;um ver&longs;us Ortum, &longs;iue ver&longs;us Occa&longs;um. <!-- KEEP S--></s> </p> <p id="N1C7A5" type="main"> <s id="N1C7A7"><!-- NEW -->Si verò proiiciatur per inclinatam deor&longs;um ver&longs;us Ortum, de&longs;cribit <lb/>lineam, quæ non e&longs;t Parabola, &longs;ed propiùs accedit ad Hyperbolam, cuius <pb pagenum="192" xlink:href="026/01/224.jpg"/>tangens e&longs;t mixta ex inclinata deor&longs;um ex horizontali ver&longs;us Boream, <lb/> &longs;it enim AC ver&longs;us Boream, AB ver&longs;us Ortum, AD inclinata deor­<lb/>&longs;um &longs;ub horizontali AB, AG quæ e&longs;t in eodem plano cum AD DG, <lb/>mixta ex AD, & AC; </s> <s id="N1C7B8"><!-- NEW -->a&longs;&longs;umatur EF æqualis &longs;patio, quod conficitur <lb/>motu naturali eo tempore, quo conficitur AE, & GH æqualis &longs;patio, <lb/>quod conficitur motu naturali eo tempore, quo percurritur AG; </s> <s id="N1C7C0"><!-- NEW -->duca­<lb/>tur curua AFH, cuius &longs;itus vt habeatur &longs;it AB ver&longs;us Ortum, ex qua <lb/>pendeat perpendiculariter deor&longs;um triangulum ABH, tùm circa axem <lb/>AD voluatur triangulum ADH, donec HD &longs;it parallela horizonti; </s> <s id="N1C7CA"><!-- NEW -->tùm <lb/>circa axem AG voluatur triangulum AGH, dum GH &longs;it perpendicu­<lb/>laris deor&longs;um, tunc enim linea motus AFH habebit proprium &longs;itum; <lb/>idem fiet &longs;i proiiciatur per inclinatam deor&longs;um ver&longs;us Occa&longs;um. <!-- KEEP S--></s> </p> <p id="N1C7D5" type="main"> <s id="N1C7D7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 109.<emph.end type="center"/></s> </p> <p id="N1C7E3" type="main"> <s id="N1C7E5"><!-- NEW --><emph type="italics"/>Si proijciatur per inclinatam &longs;ur&longs;um, & declinantem ad Ortum, linea mo­<lb/>tus erit Parabola, cuius amplitudo erit mixta ex declinante horizontali, & <lb/>horizontali ver&longs;us Boream,<emph.end type="italics"/> &longs;it enim horizontalis ver&longs;us Boream AK, <lb/>horizontalis ver&longs;us Ortum AR, declinans à Borea in Ortum AD, mixta <lb/>ex AD, AK &longs;it AI, &longs;itque Rhomboides AE parallelus horizonti; </s> <s id="N1C7F6"><!-- NEW -->&longs;it <lb/>EG perpendicularis &longs;ur&longs;um, &longs;it HD parallela GE; differentia &longs;patij, <lb/>quod acquiritur motu naturali eo tempore, quo percurritur AI, & FC, <lb/>quæ &longs;it &longs;ubdupla EG. </s> <s id="N1C800"><!-- NEW -->Dico lineam motus AHF e&longs;&longs;e parabolicam, quæ <lb/>omnia con&longs;tant ex dictis; </s> <s id="N1C806"><!-- NEW -->idemque dictum e&longs;to de omni alia inclinata <lb/>&longs;ur&longs;um &longs;imul, & declinante, &longs;eu ver&longs;us Ortum &longs;eu ver&longs;us Occa&longs;um; </s> <s id="N1C80C"><!-- NEW -->porrò <lb/>triangulum AEG incubat <expan abbr="perp&etilde;diculariter">perpendiculariter</expan> plano horizontali ADEK; </s> <s id="N1C816"><!-- NEW --><lb/>&longs;i verò proiiciatur per inclinatam deor&longs;um voluatur AKE, dum KO <lb/>&longs;it perpendicularis deor&longs;um; </s> <s id="N1C81D"><!-- NEW -->&longs;it planum RK horizontale, voluatur <lb/>AKE circa A, ita vt KO &longs;it &longs;emper perpendicularis deor&longs;um, donec <lb/>AE &longs;ecet planum RK in AD &longs;int IO. & EA vt EF, GH in &longs;uperio­<lb/>re figura, & per puncta AOM ducatur curua; hæc e&longs;t linea motus <lb/>quæ&longs;ita. </s> </p> <p id="N1C829" type="main"> <s id="N1C82B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 110.<emph.end type="center"/></s> </p> <p id="N1C837" type="main"> <s id="N1C839"><!-- NEW --><emph type="italics"/>Si proiiciatur per declinantem ab Austro ad Ortum & inclinatam &longs;ur&longs;um, <lb/>de&longs;cribet Parabolam, cuius amplitudo erit mixta ex horizontali ver&longs;us Bo­<lb/>ream & declinante horizontali ab Au&longs;tro ad Ortum<emph.end type="italics"/> &longs;it AF horizontalis <lb/>ver&longs;us Boream, AG ver&longs;us Ortum, AI declinans ab Au&longs;tro ad Ortum, <lb/>AG mixta ex AF AI AL inclinata, ANK Parabola; </s> <s id="N1C84A"><!-- NEW -->&longs;it enim planum <lb/>FI horizontale cui triangulum ALI incubet perpendiculariter in &longs;e­<lb/>ctione AG, reliqua &longs;unt facilia; </s> <s id="N1C852"><!-- NEW -->idem dico de inclinata &longs;ur&longs;um &longs;imul, & <lb/>declinante ab Au&longs;tro ad Occa&longs;um; </s> <s id="N1C858"><!-- NEW -->&longs;i verò &longs;it inclinata deor&longs;um, &longs;it pla­<lb/>num ACB horizontale, AB &longs;it declinans, AC &longs;it mixta ex AB & ho­<lb/>rizontali ver&longs;us Boream AF; &longs;it AD inclinata deor&longs;um, fiatque cur­<lb/>ua AQE more &longs;olito, ita vt triangulum ACE perpendiculariter <lb/>deor&longs;um pendeat ex plano horizontali ACB, reliqua &longs;unt facilia. </s> </p> <pb pagenum="193" xlink:href="026/01/225.jpg"/> <p id="N1C868" type="main"> <s id="N1C86A"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C876" type="main"> <s id="N1C878"><!-- NEW -->Ob&longs;eruabis a&longs;&longs;umptam e&longs;&longs;e à me hactenus Parabolam, licèt accurate <lb/>non &longs;int parabolicæ lineæ, quia proximè ad Parabolas accedunt; <lb/>certè Phy&longs;icè loquendo & &longs;en&longs;ibiliter pro Parabolis a&longs;&longs;umi po&longs;&longs;e ni­<lb/>hil vetat. </s> </p> <p id="N1C882" type="main"> <s id="N1C884"><emph type="center"/><emph type="italics"/>Corollaria.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N1C88F" type="main"> <s id="N1C891">Ex his colligis mirabilium motuum rationem. </s> <s id="N1C894">Primò mobile proje­<lb/>ctum per lineam declinantem ab Ortu ferri po&longs;&longs;e rectà ad Ortum. </s> </p> <p id="N1C899" type="main"> <s id="N1C89B">Secundò projectum per inclinatam deor&longs;um, ferri po&longs;&longs;e per ip&longs;am <lb/>perpendicularem deor&longs;um. </s> </p> <p id="N1C8A0" type="main"> <s id="N1C8A2">Tertiò projectum per inclinatam &longs;ur&longs;um, ferri po&longs;&longs;e per verti­<lb/>calem. </s> </p> <p id="N1C8A7" type="main"> <s id="N1C8A9">Quartò, rationem à priori habes, cur &longs;i ex equo vel &longs;puas, vel ali­<lb/>quid demittas deor&longs;um, rectà perpendiculariter non cadat, &longs;ed &longs;emper <lb/>è regione, quod maximè videre e&longs;t cum purgatur nauis mobilis, eiecta <lb/>&longs;cilicet aquâ, quæ &longs;emper nauim in&longs;equi videtur, imò & cum quis pe­<lb/>dem effert in naui hunc motum quoque ob&longs;eruat. </s> </p> <p id="N1C8B4" type="main"> <s id="N1C8B6"><!-- NEW -->Quintò non erit etiam iniucundum inde elicere quomodo in maiore <lb/>naui, di&longs;co ludere vel pila quis po&longs;&longs;it, licèt nauis motus nullo modo lu­<lb/>dum impediat; quæ omnia ex iis, quæ diximus nece&longs;&longs;ariò con&longs;equuntur, <lb/>& quæ manife&longs;tum probat experimentum. </s> </p> <p id="N1C8C0" type="main"> <s id="N1C8C2"><!-- NEW -->Sextò, inde etiam eruuntur rationes motuum mixtorum ex pluribus <lb/>motibus v.g.4.5.6.7.&c.in infinitum &longs;iue in eodem plano, &longs;iue in diuer­<lb/>&longs;is; </s> <s id="N1C8CA"><!-- NEW -->In diuer&longs;is vt hactenus explicuimus; </s> <s id="N1C8CE"><!-- NEW -->in eodem vero &longs;iv.g.per BC, <lb/>BE, BA &longs;imul imprimantur impetus eidem mobili qui &longs;int vt ip&longs;æ li­<lb/>neæ; </s> <s id="N1C8D6"><!-- NEW -->primò fiat ex BA BC mixta BD, & ex BD BE, mixta BF, vel ex <lb/>BE BC mixta BG, & ex BG BA mixta BF, vel ex BE BA mixta <lb/>BH, & ex BH BC mixta BF; </s> <s id="N1C8DE"><!-- NEW -->vides &longs;emper e&longs;&longs;e <expan abbr="cãdem">eandem</expan> vltimam <lb/>mixtam in diuer&longs;is planis; iam o&longs;tendimus e&longs;&longs;e plures &longs;uprà in naui <lb/>mobili v.g. <!-- REMOVE S-->per planum verticale, horizontale, & inclinatum. </s> </p> <p id="N1C8EC" type="main"> <s id="N1C8EE"><!-- NEW -->Septimò, &longs;i in naui mobili curreret equus, vel currus, e&longs;&longs;et motus mix­<lb/>tus ex quatuor aliis, & &longs;i terra moueretur in naui mobili e&longs;&longs;ent quatuor <lb/>motus, &longs;i ex ea aliquod mobile proiiceretur; inuenitur autem linea mix­<lb/>ta in diuer&longs;is planis per quamdam planorum circuitionem, de qua <lb/>&longs;uprà. </s> </p> <p id="N1C8FA" type="main"> <s id="N1C8FC"><!-- NEW -->Octauò, po&longs;&longs;et facilè in eodem plano motus mixtus conflari ex qua­<lb/>tuor aliis vel etiam pluribus, &longs;int enim quatuor in eodem plano AD <lb/>AE. AF. AH. ex AD AE fit AB, ex AB, A fi fit AC, ex AC AH <lb/>fit AG, quæ e&longs;t longior AC, & AC longior AB: po&longs;&longs;es etiam compo­<lb/>nere ex AH AF, atque ita deinceps eodem ordine, & &longs;emper vltima <lb/>linea erit AG, quod certè mirabile e&longs;t, & à Geometris demon&longs;trari <lb/>pote&longs;t. </s> </p> <p id="N1C90C" type="main"> <s id="N1C90E"><!-- NEW -->Nonò, ex his motibus mixtis educi po&longs;&longs;unt rationes multorum effe-<pb pagenum="194" xlink:href="026/01/226.jpg"/>ctuum naturalium, qui ob&longs;eruantur in rebus naturalibus, quales &longs;unt v.g. <!-- REMOVE S--><lb/>nubium, vaporum, ventorumque motus, qui &longs;æpè turbinatim procellas <lb/>agunt, quorum turbinum ratio referri non debet, vt videbimus &longs;uo loco, <lb/>in repercu&longs;&longs;ionem aliquam, quæ fiat à concauis montibus, qui longi&longs;&longs;i­<lb/>mo interuallo &longs;æpiùs ab&longs;unt; </s> <s id="N1C920"><!-- NEW -->&longs;ed potiùs petenda e&longs;t ab ip&longs;a mixti motus <lb/>naturâ; </s> <s id="N1C926"><!-- NEW -->quippè rara materies venti facilè recipit omnem impetum; </s> <s id="N1C92A"><!-- NEW -->ita­<lb/>que ex prægnantibus &longs;æpè nubibus conferta tenui&longs;&longs;imorum halituum <lb/>examina fractis qua&longs;i carceribus quacumque linea erumpunt; <lb/>hinc infiniti propemodum motus, hinc turbines illi, &c. </s> <s id="N1C934"><lb/>atque hæc de motu mixto ex pluribus <lb/>rectis &longs;int &longs;atis. <lb/><figure id="id.026.01.226.1.jpg" xlink:href="026/01/226/1.jpg"/></s> </p> </chap> <chap id="N1C940"> <pb pagenum="195" xlink:href="026/01/227.jpg"/> <figure id="id.026.01.227.1.jpg" xlink:href="026/01/227/1.jpg"/> <p id="N1C94A" type="head"> <s id="N1C94C"><emph type="center"/>LIBER QVINTVS, <lb/><emph type="italics"/>DE MOTV IN DIVERSIS <lb/>Planis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N1C95B" type="main"> <s id="N1C95D"><!-- NEW -->HACTENVS con&longs;iderauimus motum in libe­<lb/>ro medio; iam verò con&longs;iderabimus in planis <lb/>durioribus, in quibus mobilè feratur vel &longs;ua <lb/>&longs;ponte vel ab extrin&longs;eco impul&longs;um. <lb/><gap desc="hr tag"/></s> </p> <p id="N1C96A" type="main"> <s id="N1C96C"><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C978" type="main"> <s id="N1C97A"><!-- NEW --><emph type="italics"/>PLanum inclinatum e&longs;t corpus durum læuigati&longs;&longs;imum, in quo mobile quod­<lb/>piam moueri po&longs;&longs;it, quod nec &longs;it verticale &longs;ur&longs;um, nec perpendiculare deor­<lb/>&longs;um,<emph.end type="italics"/> non addo, nec horizonti parallelum; quia planum rectilineum hori­<lb/>zontale e&longs;t etiam decliue, vt &longs;uo loco videbimus. </s> </p> <p id="N1C989" type="main"> <s id="N1C98B"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C998" type="main"> <s id="N1C99A"><emph type="italics"/>Corpus graue per planum inclinatum de&longs;cendit, & quidem velociùs per illud <lb/>planum, quod minùs recedit à perpendiculari, tardiùs verò per illud, quod plùs <lb/>recedit.<emph.end type="italics"/></s> </p> <p id="N1C9A5" type="main"> <s id="N1C9A7"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C9B4" type="main"> <s id="N1C9B6"><emph type="italics"/>Corpus graue in plano inclinato minùs grauitat, id e&longs;t faciliùs &longs;ustinetur, & <lb/>tardiore motu de&longs;cendit, quàm in perpendiculari deor&longs;um.<emph.end type="italics"/></s> </p> <p id="N1C9BF" type="main"> <s id="N1C9C1">Vtraque hypothe&longs;is certa e&longs;t, & de vtraque &longs;upponimus tantùm, quòd <lb/>&longs;it, nam demon&longs;trabimus infrà propter quid &longs;it. </s> </p> <p id="N1C9C6" type="main"> <s id="N1C9C8"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1C9D5" type="main"> <s id="N1C9D7"><!-- NEW --><emph type="italics"/>Corpus graue ideò tantùm mouetur &longs;ua &longs;ponte, vt deor&longs;um tendat<emph.end type="italics"/>: </s> <s id="N1C9E0"><!-- NEW -->hoc <lb/>Axioma con&longs;tat ex iis, quæ fusè demon&longs;traui &longs;ecundò lib. adde quod, <lb/>deor&longs;um tendere, & corpus graue &longs;ua &longs;ponte moueri idem pror&longs;us &longs;onare <lb/>videntur; </s> <s id="N1C9EA"><!-- NEW -->nec enim loquor de potentiâ motrice animantium, vel de alia <lb/>quacumque magneticâ, &longs;ed de potentiâ motrice grauium; </s> <s id="N1C9F0"><!-- NEW -->graue autem <lb/>illud appello, quod in medio rariore po&longs;itum deor&longs;um tendit, ni&longs;i impe­<lb/>diatur, denique hîc &longs;uppono dari motum naturalem grauium deor&longs;um <pb pagenum="196" xlink:href="026/01/228.jpg"/>quod demon&longs;tratum e&longs;t &longs;ecundo lib. & verò &longs;i tibi adhuc non fiat &longs;atis, <lb/>probetur hoc Axioma per hypothe&longs;im primam; nam reuerâ &longs;uppono <lb/>quòd omnibus experimentis comprobatur, &longs;cilicet corpus graue per pla­<lb/>num Inclinatum deor&longs;um &longs;ua &longs;ponte de&longs;cendere, non verò a&longs;cendere ni&longs;i <lb/>propter aliquam reflexionem. </s> </p> <p id="N1CA05" type="main"> <s id="N1CA07"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CA14" type="main"> <s id="N1CA16"><!-- NEW --><emph type="italics"/>Motus, qui impeditur, imminuitur, idque pro rata, & vici&longs;&longs;im impeditur <lb/>qui imminuitur<emph.end type="italics"/>; cur enim imminueretur &longs;eu retardaretur, &longs;i nullum &longs;it <lb/>impedimentum? </s> </p> <p id="N1CA23" type="main"> <s id="N1CA25"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CA32" type="main"> <s id="N1CA34"><!-- NEW --><emph type="italics"/>Omne quod impedit motum, debet e&longs;&longs;e applicatum mobili vel per &longs;e, vel <lb/>per &longs;uam virtutem<emph.end type="italics"/>; hoc Axioma etiam certum e&longs;t. </s> </p> <p id="N1CA3F" type="main"> <s id="N1CA41"><emph type="center"/><emph type="italics"/>Po&longs;tulatum.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CA4D" type="main"> <s id="N1CA4F"><!-- NEW --><emph type="italics"/>Liceat accipere in perpendiculari deor&longs;um, parallelas, cum &longs;cilicet a&longs;&longs;umi­<lb/>tur modica altitudo<emph.end type="italics"/>; licèt enim non &longs;int parallel&etail;, quia tamen in&longs;en&longs;ibili <lb/>interuallo ad &longs;e&longs;e inuicem accedunt, pro parallelis accipiuntur. </s> </p> <p id="N1CA5C" type="main"> <s id="N1CA5E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CA6B" type="main"> <s id="N1CA6D"><!-- NEW --><emph type="italics"/>Impeditur motus corporis in plano inclinato<emph.end type="italics"/>; certum e&longs;t quod impedia­<lb/>tur, quia tardiore motu de&longs;cendit mobile per hyp. </s> <s id="N1CA78">2. igitur impeditur <lb/>per Axio.2. </s> </p> <p id="N1CA7D" type="main"> <s id="N1CA7F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CA8C" type="main"> <s id="N1CA8E"><!-- NEW --><emph type="italics"/>Ideo impeditur, quia impeditur linea ad quam determinatus e&longs;t impetus <lb/>innatus<emph.end type="italics"/>; cum &longs;it determinatus ad lineam perpendicularem deor&longs;um per <lb/>Ax.1. cur enim potiùs ad vnam lineam quàm ad aliam? </s> <s id="N1CA9B"><!-- NEW -->atqui id tan­<lb/>tùm planum inclinatum efficit, vel impedit, ne deor&longs;um rectà tendere <lb/>po&longs;&longs;it; igitur ex eo tantùm capite impedit. </s> </p> <p id="N1CAA3" type="main"> <s id="N1CAA5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CAB2" type="main"> <s id="N1CAB4"><!-- NEW --><emph type="italics"/>Non totus impeditur motus in plano inclinato<emph.end type="italics"/>; </s> <s id="N1CABD"><!-- NEW -->quia &longs;i totus impediretur, <lb/>nullus e&longs;&longs;et omninò motus &longs;uper eodem plano, &longs;ed per planum inclina­<lb/>tum mobile deor&longs;um mouetur per hyp.1.igitur totus motus non impedi­<lb/>tur; hinc ratio à priori primæ hypothe&longs;eos. </s> </p> <p id="N1CAC7" type="main"> <s id="N1CAC9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CAD6" type="main"> <s id="N1CAD8"><!-- NEW --><emph type="italics"/>In ea proportione minùs mouetur, in quæ plùs impeditur<emph.end type="italics"/>; </s> <s id="N1CAE1"><!-- NEW -->probatur per <lb/>Axioma 2.cum enim motus imminuatur, quia impeditur per idem Axio­<lb/>ma; </s> <s id="N1CAE9"><!-- NEW -->certè quò plùs impeditur, plùs imminuitur; &longs;ed quò plùs imminui­<lb/>tur, minor e&longs;t, ergo quò plùs impeditur, minor e&longs;t. </s> </p> <p id="N1CAEF" type="main"> <s id="N1CAF1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CAFE" type="main"> <s id="N1CB00"><!-- NEW --><emph type="italics"/>Eò plùs impeditur motus, quò maius &longs;patium conficiendum e&longs;t ad ac­<lb/>quirendam <expan abbr="eãdem">eandem</expan> altitudinem, &longs;eu di&longs;tantiam à centro, illo &longs;patio, <lb/>quod conficitur in perpendiculari deor&longs;um<emph.end type="italics"/>; hoc Theor. <!-- REMOVE S-->vt clariùs <lb/>demon&longs;tretur, aliquid figuræ tribuendum e&longs;t. </s> <s id="N1CB15"><!-- NEW -->&longs;it perpendicularis deor-<pb pagenum="197" xlink:href="026/01/229.jpg"/>&longs;um, AB, &longs;it planum inclinatum AE duplum AB; </s> <s id="N1CB1E"><!-- NEW -->certè vbi mobile ex A <lb/>peruenit in E per planum AE, di&longs;tat æquè à centro, ac &longs;i e&longs;&longs;et in B; </s> <s id="N1CB24"><!-- NEW -->&longs;up­<lb/>pono enim perpendiculares omnes deor&longs;um e&longs;&longs;e parallelas per po&longs;tula­<lb/>tum; </s> <s id="N1CB2C"><!-- NEW -->igitur non acce&longs;&longs;it propiùs ad centrum confecto &longs;patio AE, quàm <lb/>confecto AB; </s> <s id="N1CB32"><!-- NEW -->igitur impeditur in plano AE in ea proportione, in qua <lb/>AB e&longs;t minor AE, nam haud dubiè AE e&longs;t maior AB, &longs;it autem dupla v.g. <!-- REMOVE S--><lb/>igitur impeditur non quidem totus motus &longs;ed &longs;ubduplus; </s> <s id="N1CB3B"><!-- NEW -->in plano verò <lb/>AD impeditur iuxta cam proportionem in qua AB e&longs;t minor AD, nec <lb/>enim aliunde pote&longs;t impediri, cum &longs;cilicet impediatur tantùm, quia im­<lb/>peditur linea ad quam ab ip&longs;a natura determinatus e&longs;t per Th.2. v. <!-- REMOVE S-->g.li­<lb/>nea deor&longs;um AB; </s> <s id="N1CB49"><!-- NEW -->quippè lineæ comparantur inter &longs;e v.g. <!-- REMOVE S-->AE cum AB, <lb/>nam impedimentum lineæ AE in eo tantùm po&longs;itum e&longs;t, quòd difficiliùs <lb/>per illam quàm per AB ad <expan abbr="c&etilde;trum">centrum</expan> feratur mobile, quod certum e&longs;t, cum <lb/>imperimentum petatur a difficultate; </s> <s id="N1CB59"><!-- NEW -->atqui difficultas motus, qui fit per <lb/>lineam AE in eo tantùm e&longs;t, quòd &longs;it maius &longs;patium conficiendum, igi­<lb/>tur quò maius &longs;patium e&longs;t, maior difficultas e&longs;t; igitur quò maior linea <lb/>e&longs;t, maius impedimentum e&longs;t. </s> </p> <p id="N1CB63" type="main"> <s id="N1CB65"><!-- NEW -->Adde quod vel impedimenti proportio petitur ab angulis vel à Tan­<lb/>gentibus, vel à &longs;ecantibus; </s> <s id="N1CB6B"><!-- NEW -->nihil enim aliud ade&longs;&longs;e pote&longs;t; </s> <s id="N1CB6F"><!-- NEW -->igitur per Ax. <!-- REMOVE S--><lb/>3. pote&longs;t tantùm impediri ab his; </s> <s id="N1CB76"><!-- NEW -->&longs;ed proportio impedimenti non pote&longs;t <lb/>e&longs;&longs;e ab angulis; </s> <s id="N1CB7C"><!-- NEW -->quod probatur primò, quia &longs;i ego quæram à te in qua <lb/>proportione motus per AE e&longs;t tardior motu per AB; </s> <s id="N1CB82"><!-- NEW -->dices in ea, in qua <lb/>angulus EAB e&longs;t maior nullo angulo, quod e&longs;t ridiculum: </s> <s id="N1CB88"><!-- NEW -->Equidem di­<lb/>ceres motum per AD e&longs;&longs;e velociorem motu per AE in ea proportione, <lb/>in qua angulus EAB e&longs;t maior angulo BAD, quod tamen fal&longs;um e&longs;t; </s> <s id="N1CB90"><!-- NEW -->e&longs;&longs;et <lb/>enim ferè duplò maior, quod repugnat <expan abbr="experim&etilde;tis">experimentis</expan> omnibus; </s> <s id="N1CB9A"><!-- NEW -->at &longs;i <expan abbr="accipiã">accipiam</expan> <lb/>angulum BA, qui &longs;it tantùm vnius gradus &longs;eu minuti, &longs;itque EAB angu­<lb/>lus 60. grad. <!-- REMOVE S-->&longs;i velocitas motus per AI e&longs;&longs;et ad velocitatem motus per <lb/>AE vt angulus EAB ad angulum BAI, motus per AI e&longs;&longs;et &longs;exagecuplò <lb/>velocior, quàm per AE, quod e&longs;t ab&longs;urdum: Diceret fortè aliquis in to­<lb/>to angulo 90. GAB di&longs;tribui huius impedimenti motum v.g. <!-- REMOVE S-->&longs;i angulus <lb/>BAI &longs;it 1.grad. </s> <s id="N1CBB2"><!-- NEW -->motus per AI amittit tantùm (1/90) &longs;ui motus; &longs;i angulus D <lb/>AB circiter 40.grad. </s> <s id="N1CBB8"><!-- NEW -->motus per AD amittit tantùm (40/90), & per AE (60/90); cum <lb/>&longs;it angulus BAE 60. grad. <!-- REMOVE S-->igitur motus per AB e&longs;t ad motum per AE <lb/>vt 3.ad 1. quod omnibus experimentis repugnat. </s> </p> <p id="N1CBC2" type="main"> <s id="N1CBC4">Secundò probatur, quia &longs;i fiat inclinata proximè accedens ad AG v. <!-- REMOVE S--><lb/>g.4′.& a&longs;&longs;umatur alia accedens 3′. </s> <s id="N1CBCA">differentia anguli erit tantùm 2′. </s> <s id="N1CBCD">cum <lb/>tamen differentia longitudinis plani &longs;eu &longs;ecantis huius, & illius, &longs;it ma­<lb/>xima, vt con&longs;tat ex canone &longs;inuum, igitur non imminueretur motus in <lb/>plano inclinato ratione impedimenti contra Th.4. quis enim neget e&longs;&longs;e <lb/>maximum impedimentum motus tantum &longs;patium, quod <expan abbr="conficiendũ">conficiendum</expan> e&longs;t. </s> </p> <p id="N1CBDC" type="main"> <s id="N1CBDE"><!-- NEW -->Tertiò, omnia experimenta con&longs;entiunt huic Theoremati, & repu­<lb/>gnant huic propo&longs;itioni quæ petitur ab angulis; </s> <s id="N1CBE4"><!-- NEW -->adde quod angulus ni­<lb/>hil pror&longs;us facit ad motum, &longs;ed linea &longs;eu &longs;patium; denique hoc ip&longs;um e&longs;t <lb/>quod ab omnibus Mechanicis vulgò &longs;upponitur perinde qua&longs;i prima <pb pagenum="198" xlink:href="026/01/230.jpg"/>notio, quæ tamen aliquâ demon&longs;tratione indiget. </s> </p> <p id="N1CBF1" type="main"> <s id="N1CBF3"><!-- NEW -->Equidem explicari pote&longs;t hæc demon&longs;tratio operâ libræ; </s> <s id="N1CBF7"><!-- NEW -->&longs;it enim <lb/>libra CG cuius centrum immobile e&longs;t A; </s> <s id="N1CBFD"><!-- NEW -->&longs;it autem diameter libræ CG, <lb/>pondus in C &longs;e habet ad pondus in D, tran&longs;lata &longs;cilicet diametro in DH <lb/>vt CA, ad BA; </s> <s id="N1CC05"><!-- NEW -->igitur pondus in D grauitaret minùs in planum inclina­<lb/>tum DA, quàm in horizontali CAI; </s> <s id="N1CC0B"><!-- NEW -->nam pondus in D idem præ&longs;tat, quod <lb/>præ&longs;taret appen&longs;um in D fune DE; </s> <s id="N1CC11"><!-- NEW -->igitur grauitatio in C e&longs;t ad grauita­<lb/>tionem in D, vt CA, vel DA ad BA; </s> <s id="N1CC17"><!-- NEW -->&longs;ed quâ proportione decre&longs;cit graui­<lb/>tatio in planum, cre&longs;cit motus in plano inclinato, quia minùs impeditur <lb/>per Th.4. igitur in perpendiculari ea nulla e&longs;t gtauitatio in planum; </s> <s id="N1CC1F"><!-- NEW -->nec <lb/>impeditur vllo modo motus, igitur ab E ver&longs;us C ita impeditur motus, vt <lb/>AC ver&longs;us C impeditur grauitatio in planum, &longs;ed impeditur grauitatio <lb/>in D v.g. <!-- REMOVE S-->in ratione totius CA ad EA, vel DA ad DI; igitur impeditur <lb/>motus in eadem proportione v.g. <!-- REMOVE S-->in plano DA ad DB vel AI, igitur in <lb/>ratione plani inclinati ad perpendicularem. </s> </p> <p id="N1CC31" type="main"> <s id="N1CC33"><!-- NEW -->Hæc omnia veri&longs;&longs;ima &longs;unt; </s> <s id="N1CC37"><!-- NEW -->&longs;upere&longs;t tamen vt &longs;ciatur ratio phy&longs;ica cur <lb/>pondus in D æquiualeat ponderi in B quod &longs;upponunt quidem omnes <lb/>Mechanici, & omnibus experimentis congruit: </s> <s id="N1CC3F"><!-- NEW -->Equidem pondus pendu­<lb/>lum ex D fune DB, vel longiore, e&longs;t eiu&longs;dem momenti, cuius e&longs;t affixum <lb/>in D, ita vt linea directionis, quæ ducitur ab eius centro re&longs;pondeat fu­<lb/>ni DB; </s> <s id="N1CC49"><!-- NEW -->vnde rectè concluditur ab Archimede idem pondus affixum bra­<lb/>chio BA eiu&longs;dem e&longs;&longs;e momenti cum pendulo DB, vel affixo puncto D, <lb/>quod certè veri&longs;&longs;umum e&longs;t, nondum tamen rationem phy&longs;icam video; </s> <s id="N1CC51"><!-- NEW --><lb/>verum quidem e&longs;t idem pondus pendulum fune DB minoris e&longs;&longs;e <lb/>momenti, quàm &longs;i e&longs;&longs;et affixum puncto C; </s> <s id="N1CC58"><!-- NEW -->nam &longs;uppono CG e&longs;&longs;e libram <lb/>in &longs;itu horizontali; </s> <s id="N1CC5E"><!-- NEW -->tum quia pondus illud DB trahit deor&longs;um extremum <lb/>libræ D per arcum DC longo circuitu, maximè declinante à &longs;ua linea <lb/>directionis DB; </s> <s id="N1CC66"><!-- NEW -->tùm quia ex hoc &longs;equitur nece&longs;&longs;ariò pondus B deflecti <lb/>à &longs;ua perpendiculari curua linea; </s> <s id="N1CC6C"><!-- NEW -->tùm quia linea DA, quæ rigida &longs;uppo­<lb/>nitur, re&longs;i&longs;tit motui DB & patet; in qua verò proportione, dictum e&longs;t <lb/>certè hactenus, &longs;ed phy&longs;icè non demon&longs;tratum. </s> </p> <p id="N1CC74" type="main"> <s id="N1CC76"><!-- NEW -->Pater Mer&longs;ennus multis locis ex docti&longs;&longs;imo Roberuallo demon&longs;trat <lb/>rem i&longs;tam ingenio&longs;i&longs;&longs;imè; </s> <s id="N1CC7C"><!-- NEW -->&longs;it enim circulus centro R; </s> <s id="N1CC80"><!-- NEW -->&longs;int vectes æqua­<lb/>les BF horizonti, DN perpendiculari paralleli; </s> <s id="N1CC86"><!-- NEW -->tùm CL, FO, æqualiter <lb/>inclinati, ducantur CO EL; </s> <s id="N1CC8C"><!-- NEW -->haud dubiè &longs;i pondera C & L &longs;int æqualia <lb/>erit æquilibrium; </s> <s id="N1CC92"><!-- NEW -->quod certum e&longs;t, & demon&longs;trabimus cum de libra; </s> <s id="N1CC96"><!-- NEW -->e&longs;t <lb/>enim quarta propo&longs;itio Vbaldi de libra; </s> <s id="N1CC9C"><!-- NEW -->&longs;ed pondus in O pendulum &longs;ci­<lb/>licet filo CO e&longs;t eiu&longs;dem momenti, cuius e&longs;t pondus in P; </s> <s id="N1CCA2"><!-- NEW -->igitur pon­<lb/>dus in P æquale ponderi O &longs;u&longs;tineret pondus ML, &longs;ed pondus in P <lb/>e&longs;t ad pondus in B vel in F, ad hoc, vt &longs;it æquilibrium, RF ad R <lb/>P; </s> <s id="N1CCAC"><!-- NEW -->igitur pondus in A vel in R, quod erit ad pondus in L, vt P ad R <lb/>L, &longs;u&longs;tinebit pondus in L; </s> <s id="N1CCB2"><!-- NEW -->&longs;ed &longs;i applicetur potentia in C quæ trahat per <lb/>tangentem CT, faciet idem momentum quod faceret in B trahens per <lb/>tangentem BA; </s> <s id="N1CCBA"><!-- NEW -->at vicem illius potentiæ gerit pondus B vel A, quod gra­<lb/>uitat per BA; </s> <s id="N1CCC0"><!-- NEW -->igitur potentia applicata C per CT, æqualis ponderi A <pb pagenum="199" xlink:href="026/01/231.jpg"/>retineret pondus in L; </s> <s id="N1CCC9"><!-- NEW -->ducatur autem KLG Tangens parallela CT; </s> <s id="N1CCCD"><!-- NEW -->certè <lb/>eadem potentia in L per LG retinebit pondus in L; </s> <s id="N1CCD3"><!-- NEW -->quæ idem retine­<lb/>ret applicata in C per CT; </s> <s id="N1CCD9"><!-- NEW -->cum enim RC & RL &longs;int æquales &longs;i &longs;int ap­<lb/>plicatæ duæ potentiæ æquales in C quidem per CT, & in L per LG; </s> <s id="N1CCDF"><!-- NEW --><lb/>haud dubiè erit perfectum æquilibrium; </s> <s id="N1CCE4"><!-- NEW -->igitur &longs;i pondus A pendeat in <lb/>H fune LGH, retinebit pondus L in plano inclinato GLK; </s> <s id="N1CCEA"><!-- NEW -->e&longs;t autem <lb/>pondus H ad pondus LN SR ad RL; </s> <s id="N1CCF0"><!-- NEW -->&longs;ed triangula RSL, & GKI <lb/>&longs;unt proportionalia; </s> <s id="N1CCF6"><!-- NEW -->igitur pondus in H e&longs;t ad pondus L, vt GI ad G <lb/>K; </s> <s id="N1CCFC"><!-- NEW -->igitur &longs;i vires, quæ retinent pondus in plano inclinato GK &longs;unt ad vi­<lb/>res, quæ retinent pondus in perpendiculari GI, vt GI ad GK; igitur im­<lb/>petus &longs;eu motus mobilis in plano GK e&longs;t ad impetum, &longs;eu motum eiu&longs;­<lb/>dem in perpendiculo GI, vt GI ad GK. </s> </p> <p id="N1CD06" type="main"> <s id="N1CD08"><!-- NEW -->Hæc omnia veri&longs;&longs;ima &longs;unt, &longs;emper tamen de&longs;iderari videtur ratio phy­<lb/>&longs;ica, cur idem pondus pendulum ex C in O, &longs;it eiu&longs;dem momenti cum <lb/>pondere affixo puncto P, &longs;eu brachio libræ horizontalis PS. quod certè <lb/>Mechanica Axiomatis, vel hypothe&longs;eos loco iure a&longs;&longs;umere pote&longs;t; </s> <s id="N1CD12"><!-- NEW -->at ve­<lb/>rò phy&longs;ica non &longs;atis habet de re cogno&longs;cere quod &longs;it, ni&longs;i &longs;ciat propter <lb/>quid &longs;it; igitur nos aliquam afferre conabimur. </s> <s id="N1CD1A">Suppono tantùm tunc <lb/>e&longs;&longs;e æquilibrium perfectum duorum ponderum æqualium cum <expan abbr="vtrimq;">vtrimque</expan> <lb/>æqualia illa pondera ita &longs;unt appen&longs;a, vt linea directionis vnius æqua­<lb/>lis &longs;it lineæ directionis alterius, cur enim alterum præualeret &longs;i &longs;int æ­<lb/>qualia? </s> <s id="N1CD29">hoc po&longs;ito. </s> </p> <p id="N1CD2C" type="main"> <s id="N1CD2E"><!-- NEW -->Dico pondus affixum P æquale ponderi L facere æquilibrium; cum <lb/>enim linea directionis &longs;it PO, &longs;i de&longs;cenderet liberè per PO. </s> <s id="N1CD34"><!-- NEW -->L eodem <lb/>tempore attolleretur per LS, quod certè applicatis planis SL PO facilè <lb/>fieri po&longs;&longs;et; </s> <s id="N1CD3C"><!-- NEW -->&longs;ed eodem modo P grauitat, quo &longs;i de&longs;cenderet per PO; </s> <s id="N1CD40"><!-- NEW -->e&longs;t <lb/>enim eius linea directionis; </s> <s id="N1CD46"><!-- NEW -->atqui tunc faceret æquilibrium, quod o&longs;ten­<lb/>do; </s> <s id="N1CD4C"><!-- NEW -->æquale &longs;patium conficeret L, per LS a&longs;cendendo, quod P per PO <lb/>de&longs;cendendo; </s> <s id="N1CD52"><!-- NEW -->igitur &longs;i attolleret L in S, &longs;imiliter pondus L æquale P in S <lb/>attolleret pondus P ex O in P, igitur neutrum præualere pote&longs;t; &longs;ed quia <lb/>hæc fu&longs;iùs explicabimus cum de libra, nunc tantùm indica&longs;&longs;e &longs;ufficiat. </s> </p> <p id="N1CD5A" type="main"> <s id="N1CD5C"><!-- NEW -->Supere&longs;t vt breuiter o&longs;tendamus accipi non po&longs;&longs;e hanc proportio­<lb/>nem imminutionis motus in plano inclinato à Tangente BE tùm <lb/>quia; </s> <s id="N1CD64"><!-- NEW -->iam à &longs;ecante accipi o&longs;tendimus, tùm quia &longs;it Tangens BD æqualis <lb/>&longs;umi toti &longs;eu perpendiculari AB; </s> <s id="N1CD6A"><!-- NEW -->&longs;equeretur motum per AD æqualem <lb/>e&longs;&longs;e motui per AB; </s> <s id="N1CD70"><!-- NEW -->Equidem in maxima di&longs;tantia accedit Tangens ad <lb/>&longs;ecantem; igitur eò plùs impeditur motus, quò maius &longs;patium conficien­<lb/>dum e&longs;t, &c. </s> </p> <p id="N1CD78" type="main"> <s id="N1CD7A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CD87" type="main"> <s id="N1CD89"><emph type="italics"/>Ex hoc &longs;equitur nece&longs;&longs;ariò motum in plano inclinato e&longs;&longs;e ad motum in per­<lb/>pendiculari, vt ip&longs;a perpendicularis ad ip&longs;um planum inclinatum,<emph.end type="italics"/> v.g. <!-- REMOVE S-->velo­<lb/>citas motus per AE e&longs;t ad velocitatem motus per AB, vt ip&longs;a AB e&longs;t <lb/>ad ip&longs;am AE, &longs;it enim AE dupla AB, velocitas per AB e&longs;t dupla veloci­<lb/>tatis per AE. <!-- KEEP S--></s> </p> <pb pagenum="200" xlink:href="026/01/232.jpg"/> <p id="N1CDA0" type="main"> <s id="N1CDA2"><!-- NEW -->Ob&longs;erua quæ&longs;o, cum dico motum in plano inclinato e&longs;&longs;e ad motum <lb/>in perpendiculo, vt ip&longs;æ lineæ permutando, ita intelligendum e&longs;&longs;e, vt <lb/>vel a&longs;&longs;umatur motus in &longs;ingulis in&longs;tantibus, ita vt eo in&longs;tanti, quo datum <lb/>&longs;patium in inclinata acquiritur, acquiratur duplum in perpendiculo; </s> <s id="N1CDAC"><!-- NEW -->quo <lb/>po&longs;ito valet certè tantùm illa proportio ratione motus æquabilis, &longs;i &longs;er­<lb/>uari debet; nam perinde &longs;e habet phy&longs;icè, atque &longs;i e&longs;&longs;et, vt iam fusè ex­<lb/>plicatum e&longs;t lib.2. in re &longs;imili. </s> </p> <p id="N1CDB6" type="main"> <s id="N1CDB8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1CDC4" type="main"> <s id="N1CDC6"><!-- NEW --><emph type="italics"/>Hinc de&longs;cendit mobile per &longs;e in plano inclinato<emph.end type="italics"/>; </s> <s id="N1CDCF"><!-- NEW -->ratio e&longs;t, quia totus mo­<lb/>tus non impeditur, cum &longs;it eadem proportio, quæ e&longs;t perpendicularis <lb/>ad inclinatam; dixi per &longs;e, nam per accidens in plano &longs;cabro tantillùm <lb/>inclinato mobile de&longs;cendit, adde quod corpus graue tamdiu mouetur <lb/>quandiu accedere pote&longs;t ad centrum terræ. </s> </p> <p id="N1CDDB" type="main"> <s id="N1CDDD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1CDE9" type="main"> <s id="N1CDEB"><emph type="italics"/>Motus in infinitum imminui pote&longs;t,<emph.end type="italics"/> probatur, quia proportio perpen­<lb/>dicularis ad inclinatam pote&longs;t e&longs;&longs;e minor in infinitum, quia inclinata <lb/>pote&longs;t e&longs;&longs;e longior, & in infinitum. </s> </p> <p id="N1CDF7" type="main"> <s id="N1CDF9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N1CE05" type="main"> <s id="N1CE07"><!-- NEW --><emph type="italics"/>Ex his vera redditur ratio cur in plano inclinato ad angulum BG motus &longs;it <lb/>&longs;ubduplus illius qui fit in perpendiculari<emph.end type="italics"/>; v.g. <!-- REMOVE S-->&longs;it angulus BAE 60. certè <lb/>AE e&longs;t dupla AB, &longs;ed motus in AB e&longs;t ad motum in AE vt AE ad AB <lb/>per Th.6. igitur e&longs;t duplus. </s> </p> <p id="N1CE18" type="main"> <s id="N1CE1A"><!-- NEW -->Ex his reiicies quoque Cardanum, & alios quo&longs;dam, qui diuer&longs;am <lb/>proportionem motuum in planis inclinatis deducunt ex diuer&longs;is angu­<lb/>lis inclinationis; iuxta quam proportionem motus in AE e&longs;&longs;et &longs;ubtri­<lb/>plus in AB contra experimentum. </s> </p> <p id="N1CE24" type="main"> <s id="N1CE26"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N1CE32" type="main"> <s id="N1CE34"><!-- NEW --><emph type="italics"/>Motus acceleratur in plano inclinato<emph.end type="italics"/>; </s> <s id="N1CE3D"><!-- NEW -->experientia clari&longs;&longs;ima e&longs;t, ratio <lb/>eadem cum illa, quam adduximus lib.3. cum de motu naturali, quia &longs;ci­<lb/>licet prior impetus con&longs;eruatur, & acquiritur nouus, Imò acceleratur <lb/>iuxta <expan abbr="eãdem">eandem</expan> proportionem, vel no&longs;tram &longs;ingulis in&longs;tantibus, vel Gali­<lb/>lei in partibus temporum &longs;en&longs;ibilibus; vnde a&longs;&longs;umemus deinceps i&longs;tam <lb/>Galilei proportionem, quia &longs;cilicet partes temporis &longs;en&longs;ibiles tantùm <lb/>a&longs;&longs;umere po&longs;&longs;umus. </s> </p> <p id="N1CE51" type="main"> <s id="N1CE53"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N1CE5F" type="main"> <s id="N1CE61"><!-- NEW --><emph type="italics"/>In plano inclinato e&longs;t idem impetus innatus qui est in perpendiculari,<emph.end type="italics"/> &longs;ed <lb/>in hac habet totum &longs;uum motum, non verò in illa, quia impeditur, ni&longs;i <lb/>enim totus e&longs;&longs;et, non grauitaret corpus illud in planum inclinatum; </s> <s id="N1CE6E"><!-- NEW --><lb/>quippe &longs;uas omnes vires impetus ille exereret circa motum; </s> <s id="N1CE73"><!-- NEW -->igitur ali­<lb/>quid illarum exerit circa motum aliquid circa planum, in quod ex parte <lb/>grauitat; igitur idem e&longs;t impetus innatus, adde quod ille e&longs;t in&longs;epa­<lb/>rabilis. </s> </p> <pb pagenum="201" xlink:href="026/01/233.jpg"/> <p id="N1CE81" type="main"> <s id="N1CE83"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N1CE8F" type="main"> <s id="N1CE91"><!-- NEW --><emph type="italics"/>Impetus naturalis aduentitius productus à corpore graui in plano inclinato <lb/>e&longs;t minor eo, qui producitur in perpendiculari<emph.end type="italics"/>; </s> <s id="N1CE9C"><!-- NEW -->probatur, quia e&longs;t minor <lb/>motus, igitur minor impetus, vt &longs;æpè diximus; </s> <s id="N1CEA2"><!-- NEW -->&longs;ecundò (hæc e&longs;t ratio <lb/>à priori;) quia cum ideo producatur impetus i&longs;te aduentitius, vt motus <lb/>acceleretur; </s> <s id="N1CEAA"><!-- NEW -->certè debet re&longs;pondere motui, qui competit impetui innati; </s> <s id="N1CEAE"><!-- NEW --><lb/>&longs;i enim nullum habet motum, nullus accedit de nouo impetus, è con­<lb/>tra verò &longs;i e&longs;t motus, &longs;ed maior, &longs;i maior e&longs;t motus, & minor &longs;i e&longs;t minor; <lb/>quia hic impetus tantùm e&longs;t propter motum. </s> </p> <p id="N1CEB7" type="main"> <s id="N1CEB9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N1CEC5" type="main"> <s id="N1CEC7"><!-- NEW --><emph type="italics"/>Impetus qui producitur in acceleratione motus habet totum motum quem <lb/>exigit (præ&longs;cindendo à re&longs;i&longs;tentia medij)<emph.end type="italics"/>; </s> <s id="N1CED2"><!-- NEW -->nec enim per illum mobile graui­<lb/>tat in planum; </s> <s id="N1CED8"><!-- NEW -->alioquin cre&longs;ceret &longs;emper grauitatio; </s> <s id="N1CEDC"><!-- NEW -->igitur totus exerce­<lb/>tur circa motum; </s> <s id="N1CEE2"><!-- NEW -->ratio e&longs;t quia hic impetus addititius non e&longs;t in&longs;titutus <lb/>propter grauitationem, &longs;ed tantùm propter motum: adde quod ad om­<lb/>nem lineam determinari pote&longs;t, &longs;ecùs verò naturalis &longs;altem om­<lb/>ninò. </s> </p> <p id="N1CEEC" type="main"> <s id="N1CEEE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N1CEFA" type="main"> <s id="N1CEFC"><!-- NEW --><emph type="italics"/>Imminuitur motu illo grauitatio corporis in planum<emph.end type="italics"/>; ratio e&longs;t primò; </s> <s id="N1CF05"><!-- NEW -->quia <lb/>quò velociùs mouetur in plano, breuiori tempore &longs;ingulis partibus in­<lb/>cumbit: </s> <s id="N1CF0D"><!-- NEW -->&longs;ecundò quia motu illo accelerato qua&longs;i di&longs;trahitur mobile ab <lb/>illa linea grauitationis in planum; hinc mobile celeri motu moueretur <lb/>in plano illo inclinato, quod eiu&longs;dem &longs;ub&longs;i&longs;tentis grauitationi & ponde­<lb/>ri vltrò cederet. </s> </p> <p id="N1CF17" type="main"> <s id="N1CF19"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N1CF25" type="main"> <s id="N1CF27"><!-- NEW --><emph type="italics"/>Impetus innatus ex &longs;e e&longs;t &longs;emper determinatus ad lineam perpendicularem <lb/>deor&longs;um<emph.end type="italics"/>; </s> <s id="N1CF32"><!-- NEW -->quia grauitas tendit ad commune centrum, vt videbimus tra­<lb/>ctatu &longs;equenti; </s> <s id="N1CF38"><!-- NEW -->tamen ratione plani qua&longs;i detorquetur ad lineam plani <lb/>ad quam tamen omninò non determinatur, alioquin non grauitaret in <lb/>planum: </s> <s id="N1CF40"><!-- NEW -->vnde dixi, detorquetur &longs;eu qua&longs;i diuiditur, perinde qua&longs;i e&longs;&longs;et <lb/>duplex impetus, quorum alter per lineam perpendicularem deor&longs;um <lb/>e&longs;&longs;et determinatus, in quo non e&longs;t difficultas; impetus tamen aduenti­<lb/>tius determinatur omninò ad lineam plani. </s> </p> <p id="N1CF4A" type="main"> <s id="N1CF4C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1CF58" type="main"> <s id="N1CF5A"><!-- NEW -->Dubitari pote&longs;t an grauitatio in planum inclinatum &longs;it vt re&longs;iduum <lb/>plani, cui detrahitur perpendiculum v.g. <!-- REMOVE S-->&longs;it planum inclinatum CD ad <lb/>angulum ACD 60. potentia quæ &longs;u&longs;tinet pondus B per EB e&longs;t ad præ­<lb/>dictum pondus vt CA ad CD; </s> <s id="N1CF66"><!-- NEW -->detrahitur CA ex CD, &longs;upere&longs;t FD æqua­<lb/>lis &longs;cilicet CA; </s> <s id="N1CF6C"><!-- NEW -->an fortè grauitatio ponderis B in planum inclinatum C <lb/>D e&longs;t ad grauitationem eiu&longs;dem in planum horizontale; </s> <s id="N1CF72"><!-- NEW -->quæ e&longs;t graui­<lb/>tatio tota, id e&longs;t nihil imminuta vt DF ad DC; </s> <s id="N1CF78"><!-- NEW -->attollatur enim totum <lb/>triangulum CAD in eadem &longs;itu altera manu, & altera filo EB paralle-<pb pagenum="202" xlink:href="026/01/234.jpg"/>lo CF, retineatur pondus B ne &longs;cilicet deor&longs;um cadat; </s> <s id="N1CF83"><!-- NEW -->tùm &longs;ubtrahatur <lb/>pondus trianguli CAD; nunquid fortè altera manus &longs;u&longs;tinebit tantùm <lb/>&longs;ubduplum ponderis B? & altera &longs;ubduplum? </s> <s id="N1CF8B"><!-- NEW -->igitur vt habeatur quod <lb/>&longs;u&longs;tinet &longs;uppo&longs;ita dextra v.g. <!-- REMOVE S-->debet &longs;ub&longs;trahi, quod &longs;u&longs;tinet &longs;ini&longs;tra, &longs;ed <lb/>quod &longs;u&longs;tinet &longs;ini&longs;tra, e&longs;t vt ip&longs;a potentia, id e&longs;t vt CA ad CD; igitur <lb/>tota CD repræ&longs;entat totum pondus, &longs;egmentum CF partem ponderis <lb/>quæ competit potentiæ E, FD verò partem quæ &longs;u&longs;tinetur à pla­<lb/>no CF. <!-- KEEP S--></s> </p> <p id="N1CF9C" type="main"> <s id="N1CF9E"><!-- NEW -->Hinc facilè po&longs;&longs;et determinari quota pars ponderis incubet plano,<lb/>&longs;it enim planum inclinatum AC, perpendiculum AB, accipiatur AB <lb/>æqualis AB, &longs;itque AC tripla AB, duæ tertiæ ponderis incubant plano <lb/>&longs;i verò &longs;it horizontale planum, totum pondus grauitat in illud; </s> <s id="N1CFA8"><!-- NEW -->nulla e&longs;t <lb/>enim perpendicularis, &longs;i &longs;it perpendiculare planum, nihil pror&longs;us gra­<lb/>uitat; </s> <s id="N1CFB0"><!-- NEW -->quia nulla e&longs;t inclinata, & quò propiùs accedit planum inclina­<lb/>tum ad horizontalem plùs grauitat pondus in illud, minùs verò; quò <lb/>propiùs accedit ad perpendicularem. </s> </p> <p id="N1CFB8" type="main"> <s id="N1CFBA"><!-- NEW -->Hinc e&longs;&longs;et oppo&longs;ita ratio grauitationis, & motus, in plano inclinato; </s> <s id="N1CFBE"><!-- NEW --><lb/>nam quò plùs e&longs;t grauitationis minùs e&longs;t motus, quò plùs motus, minùs <lb/>grauitationis; </s> <s id="N1CFC5"><!-- NEW -->quando verò planum inclinatum e&longs;t duplum perpendicu­<lb/>culi vt planum CFD, tunc <expan abbr="tantũdem">tantundem</expan> detrahitur de grauitatione in <lb/>planum quantùm de motu in eodem plano; </s> <s id="N1CFD1"><!-- NEW -->ide&longs;t vtrique &longs;ubduplum, <lb/>&longs;i verò vt in plano ADC perpendiculum e&longs;t &longs;ubtriplum plani, detrahun­<lb/>tur de motu 2/3 & de grauitatione 1/3, idem dico de aliis, quæ certè omnia <lb/>ex veris principiis phy&longs;icis con&longs;equi videntur, quò enim plus grauitat <lb/>mobile in planum, plùs &longs;u&longs;tinetur; </s> <s id="N1CFDD"><!-- NEW -->quò plùs &longs;u&longs;tinetur, plùs impeditur il­<lb/>lius motus; </s> <s id="N1CFE3"><!-- NEW -->&longs;ed hoc repugnat communi Mechanicorum &longs;ententiæ, qui <lb/>cen&longs;ent grauitationem in planum inclinatum e&longs;&longs;e ad grauitationem in <lb/>horizontale, vt Tangens e&longs;t ad &longs;ecantem, quæ &longs;it linea plani inclinati, <lb/>v.g. <!-- REMOVE S-->vt AB ad CD, quod certè omnes &longs;upponunt, &longs;ed minimè <expan abbr="demon-&longs;trãt">demon­<lb/>&longs;trant</expan>, &longs;i quid video &longs;altem phy&longs;icè; </s> <s id="N1CFF5"><!-- NEW -->nec enim illud nemon&longs;trant propriè ex <lb/>eo quòd pondus in extremitate libræ affixum habeat diuer&longs;a momenta <lb/>iuxta rationem Tangentium ad &longs;ecantes, v.g. <!-- REMOVE S-->in &longs;ecunda figura Th.5. <lb/>pondus in D e&longs;t ad pondus in C vt BA ad DA, quod veri&longs;&longs;imum e&longs;t, & <lb/>&longs;uprà demon&longs;trauimus; </s> <s id="N1D003"><!-- NEW -->quippe hoc pertinet ad rationem momenti, non <lb/>verò grauitationis in planum; </s> <s id="N1D009"><!-- NEW -->adde quod affixum e&longs;t pondus vecti; </s> <s id="N1D00D"><!-- NEW -->igi­<lb/>tur vectis &longs;u&longs;tinet totum illius pondus; </s> <s id="N1D013"><!-- NEW -->vtrùm verò &longs;i pondus in plano <lb/>inclinato veluti in vecte moueatur pondus quo grauitat in planum &longs;it <lb/>ad pondus quo grauitat in horizontali vt Tangens ad &longs;ecantem, certè <lb/>non demon&longs;trant; </s> <s id="N1D01D"><!-- NEW -->attamen ita res pror&longs;us &longs;e habet; quare fit. </s> </p> <p id="N1D021" type="main"> <s id="N1D023"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N1D02F" type="main"> <s id="N1D031"><emph type="italics"/>Grauitatio ponderis in planum inclinatum e&longs;t ad grauitationem eiu&longs;dem <lb/>in planum horizontale, vt Tangens, vel horizontalis ad &longs;ecantem, vel incli­<lb/>natam,<emph.end type="italics"/> quod demon&longs;tro. </s> <s id="N1D03D"><!-- NEW -->Primò &longs;it planum inclinatum GD, pondus in-<pb pagenum="203" xlink:href="026/01/235.jpg"/>cubans F; </s> <s id="N1D046"><!-- NEW -->dico grauitationem ponderis F in inclinatam GD e&longs;&longs;e ad gra­<lb/>uitationem in horizontalem CD vt CD ad GD; </s> <s id="N1D04C"><!-- NEW -->quia pondus F pellit <lb/>planum per lineam FE &longs;eu GB Tangentem; </s> <s id="N1D052"><!-- NEW -->quia determinari non po­<lb/>te&longs;t &longs;eu percu&longs;&longs;io, &longs;eu impre&longs;&longs;io ex alio capite quàm ex linea ducta à <lb/>centro grauitatis perpendiculariter in planum, vt demon&longs;trauimus <lb/>in Th. 120. l. <!-- REMOVE S-->1. atqui libræ extremitas G initio de&longs;cendit per Tangen­<lb/>tem GB, id e&longs;t per minimum arcum, qui ferè concurrit cum Tangente; </s> <s id="N1D060"><!-- NEW --><lb/>&longs;ed ideò de&longs;cendit in AB, quia pellitur deor&longs;um à pondere; </s> <s id="N1D065"><!-- NEW -->igitur men­<lb/>&longs;ura grauitationis e&longs;t de&longs;cen&longs;us libræ, &longs;ed libra faciliùs de&longs;cendit ex A <lb/>deor&longs;um quàm ex G in proportione AD ad CD vel GD ad CD; </s> <s id="N1D06D"><!-- NEW -->igitur <lb/>grauitatio ponderis in A e&longs;t ad grauitationem eiu&longs;dem in G, vt GD ad <lb/>CD; quia rationes cau&longs;arum &longs;unt eædem cum rationibus effectuum. </s> </p> <p id="N1D075" type="main"> <s id="N1D077"><!-- NEW -->Præterea &longs;it planum inclinatum GD, &longs;it IF parallela GD; </s> <s id="N1D07B"><!-- NEW -->&longs;int IK, I <lb/>M & quadrans KFR; </s> <s id="N1D081"><!-- NEW -->punctum I &longs;it centrum libræ immobile; </s> <s id="N1D085"><!-- NEW -->certè &longs;i &longs;it <lb/>alterum brachium libræ æquale IF in&longs;tructum æquali pondere F, erit æ­<lb/>quilibrium; &longs;ed pondus illud in F e&longs;t ad idem in R, vt IM ad IF, &longs;eu vt <lb/>CD ad GD, quod erat dem. </s> </p> <p id="N1D08F" type="main"> <s id="N1D091"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1D09D" type="main"> <s id="N1D09F"><!-- NEW -->Ob&longs;eruabis po&longs;&longs;e facilè ex dictis explicari diuer&longs;as potentias applica­<lb/>tas ponderi F in eodem plano GD, primò &longs;i accipiatur IHF parallela <lb/>GH cum centro immobili I pondus retinebitur, &longs;i potentia in I &longs;it ad <lb/>globum vt GC ad GD, vt demon&longs;tratum e&longs;t; &longs;i verò pellat potentia per <lb/>lineam IF, globus de&longs;cendet, vt patet. </s> </p> <p id="N1D0AB" type="main"> <s id="N1D0AD"><!-- NEW -->Hinc &longs;ecundò &longs;u&longs;tinens MF totum pondus F &longs;u&longs;tinet, patet, quia &longs;i­<lb/>ue planum inclinatum pondus ip&longs;um tangat, &longs;iue perpendiculare, totum <lb/>&longs;u&longs;tinet pondus; &longs;ub&longs;tracto enim plano pondus immobile manet, adde <lb/>quod non pote&longs;t pondus F &longs;u&longs;tineri in brachio IM, ni&longs;i æquale pondus <lb/>ex æquali brachio oppo&longs;ito pendeat. </s> </p> <p id="N1D0B9" type="main"> <s id="N1D0BB"><!-- NEW -->Tertiò ex puncto T lineâ TFE non pote&longs;t &longs;u&longs;tineri pondus licèt po­<lb/>tentia in T e&longs;&longs;et infinita, quia ex TE de&longs;cendet in TV, patet; idem <lb/>dico de omnibus aliis lineis ductis ab F ad aliquod punctum inter <lb/>TM. </s> </p> <p id="N1D0C5" type="main"> <s id="N1D0C7"><!-- NEW -->Quartò ex puncto X linea XF &longs;u&longs;tinebitur pondus dum potentia ap­<lb/>plicetur in X, maior quidem potentia applicata in I, &longs;ed minor applica­<lb/>ta in M; </s> <s id="N1D0CF"><!-- NEW -->nam potentia M e&longs;t ad potentiam I vt IF ad MF; </s> <s id="N1D0D3"><!-- NEW -->igitur poten­<lb/>tia X e&longs;t ad potentiam M vt MF ad XF; ad potentiam verò I vt IF <lb/>ad XF. <!-- KEEP S--></s> </p> <p id="N1D0DC" type="main"> <s id="N1D0DE">Quintò, cùm triangula IF M.HF 4. &longs;int proportionalia, potentia M <lb/>e&longs;t ad potentiam I vt HF ad 4. F. <!-- KEEP S--></s> </p> <p id="N1D0E4" type="main"> <s id="N1D0E6"><!-- NEW -->Sextò, &longs;i applicetur potentia, vel in T pellendo per lineam TFE, quæ <lb/>cadit perpendiculariter in planum GD, vel &longs;i applicetur in A per lineam <lb/>AE trahendo, non poterit retineri globus, quæcunque tandem poten­<lb/>tia applicetur; </s> <s id="N1D0F0"><!-- NEW -->quia &longs;emper per GD globus rotari poterit nullo cor­<lb/>pore impediente; </s> <s id="N1D0F6"><!-- NEW -->&longs;uppono enim tùm planum tùm globum e&longs;&longs;e perfectè <pb pagenum="204" xlink:href="026/01/236.jpg"/>politum, quod tamen nobis dee&longs;&longs;e certum e&longs;t ad experimentum, &longs;uppo­<lb/>no nullam e&longs;&longs;e partium compre&longs;&longs;ionem, qua vna pars in aliam qua&longs;i pe­<lb/>netret; </s> <s id="N1D103"><!-- NEW -->&longs;i enim totus locus datur ad de&longs;cen&longs;um; </s> <s id="N1D107"><!-- NEW -->certè non e&longs;t vlla ratio <lb/>propter quam non de&longs;cendat; </s> <s id="N1D10D"><!-- NEW -->nec dicas affigi plano GD ab ip&longs;a vi ex­<lb/>teriùs affigente; </s> <s id="N1D113"><!-- NEW -->quia nullo modo impeditur motus, per datam lineam, <lb/>ni&longs;i vel aliquod corpus opponatur, vel alius impetus detrahat ab eadem <lb/>linea; atqui nihil horum prorsùs e&longs;t in hoc ca&longs;u. </s> </p> <p id="N1D11B" type="main"> <s id="N1D11D"><!-- NEW -->Si potentia applicetur in N per lineam NF, maior e&longs;&longs;e debet quàm in <lb/>I, &longs;ed minor quàm in A; </s> <s id="N1D123"><!-- NEW -->e&longs;t autem ad potentiam in I vt IF ad NF; </s> <s id="N1D127"><!-- NEW --><lb/>quippe re&longs;i&longs;tit planum GD huic potentiæ in N, non tamen re&longs;i&longs;tit in I; </s> <s id="N1D12C"><!-- NEW --><lb/>igitur illa maior e&longs;&longs;e debet, quod autem potentia in N &longs;it ad potentiam <lb/>in I, vt IF ad NF (po&longs;ito &longs;cilicet quod vtraque pondus E &longs;u&longs;tineat) plùs <lb/>quàm certum e&longs;t; </s> <s id="N1D135"><!-- NEW -->quia cùm pondus po&longs;&longs;it tantùm moueri per EG &longs;eu per <lb/>lineam FI potentia NF trahit per FN; </s> <s id="N1D13B"><!-- NEW -->igitur potentia in N &longs;u&longs;tinens <lb/>pondus F e&longs;t ad potentiam in I &longs;u&longs;tinentem idem pondus, vt IF ad NF; <lb/>&longs;imiliter potentia in K &longs;u&longs;tinens idem pondus F e&longs;t ad potentiam in I vt <lb/>IF ad ZF, nam IZ e&longs;t perpendicularis in KF, donec tandem potentia <lb/>&longs;it in A applicata per AF in quam IF cadit perpendiculariter, igitur po­<lb/>tentia in A debet e&longs;&longs;e infinita. </s> </p> <p id="N1D149" type="main"> <s id="N1D14B"><!-- NEW -->Octauò, &longs;i pellatur pondus F per omnes lineas contentas &longs;ini&longs;tror&longs;um <lb/>inter FT & FA deor&longs;um faciliùs cadet; </s> <s id="N1D151"><!-- NEW -->&longs;i verò trahatur per lineas con­<lb/>tentas inter TF & FA dextror&longs;um, etiam deor&longs;um cadit; </s> <s id="N1D157"><!-- NEW -->quia perinde <lb/>e&longs;t &longs;iue trahatur per lineam IF, &longs;iue pellatur æquali ni&longs;u per lineam VF <lb/>quæ concurrit cum FI; </s> <s id="N1D15F"><!-- NEW -->& perinde e&longs;t &longs;iue pellatur per IF, &longs;iue trahatur <lb/>per FV; idem dictum &longs;it de omnibus aliis lineis, quæ per centrum F <lb/>hinc inde ducuntur. </s> </p> <p id="N1D167" type="main"> <s id="N1D169"><!-- NEW -->Vnum e&longs;t, quod de&longs;iderari videtur ex quo reliqua ferè omnia depen­<lb/>dent, quomodo &longs;cilicet potentia in N trahens per FN &longs;it ad potentiam <lb/>in I trahentem per FI vt FI e&longs;t ad FN, quod &longs;ic breuiter demon&longs;tro: </s> <s id="N1D171"><!-- NEW --><lb/> &longs;it horizontalis BD, & triangulum ECD; ex centro D ducatur arcus <lb/>BE, qui &longs;it v.g. <!-- REMOVE S-->30.grad. </s> <s id="N1D17A"><!-- NEW -->vt CE &longs;it &longs;ubdupla ED; </s> <s id="N1D17E"><!-- NEW -->certè potentia in B <lb/>e&longs;t ad potentiam in E per EC vt BD, vel ED ad CD; </s> <s id="N1D184"><!-- NEW -->&longs;ed potentia in E <lb/>per EA Tangentem e&longs;t æqualis potentiæ in B; </s> <s id="N1D18A"><!-- NEW -->&longs;it autem planum EA, & <lb/>connectatur AC; </s> <s id="N1D190"><!-- NEW -->triangula AEC & ECD &longs;unt proportionalia; </s> <s id="N1D194"><!-- NEW -->igitur <lb/>&longs;it AC verticalis, EC horizontalis, & AE inclinata; </s> <s id="N1D19A"><!-- NEW -->&longs;it potentia in A <lb/>per AE trahens pondus E; </s> <s id="N1D1A0"><!-- NEW -->&longs;it potentia C trahens per CE; </s> <s id="N1D1A4"><!-- NEW -->dico quod <lb/>impeditur tractio toto angulo AEC, &longs;icut ante impediebatur grauitatio <lb/>toto angulo AEC; </s> <s id="N1D1AC"><!-- NEW -->igitur vtrobique e&longs;t æquale impedimentum; </s> <s id="N1D1B0"><!-- NEW -->&longs;ed in <lb/>primo ca&longs;u ratione impedimenti ita &longs;e habet potentia in E per EA ad <lb/>potentiam in E per EC, vt ED ad CD, vel vt EA ad EC; igitur in &longs;e­<lb/>cundo in quo e&longs;t idem impedimentum potentia in A per EA e&longs;t ad po­<lb/>tentiam in C per EC, vt ip&longs;a inclinata AE ad EC. <!-- KEEP S--></s> </p> <p id="N1D1BD" type="main"> <s id="N1D1BF"><!-- NEW -->Nonò denique ob&longs;eruabis, egregium e&longs;&longs;e apud Mer&longs;ennum tractatum <lb/>authore docti&longs;&longs;imo Roberuallo &longs;uper hac tota re, in quo certè Geome-<pb pagenum="205" xlink:href="026/01/237.jpg"/>tria nihil de&longs;iderare pote&longs;t; </s> <s id="N1D1CA"><!-- NEW -->licèt phy&longs;ica fortè aliquid de&longs;iderare po&longs;&longs;it; <lb/>adde quod implicatior illa figura infinitis ferè contexta lineis, quam ha­<lb/>bet, equidem erudito Geometræ faciet &longs;atis, non tamen rudiori Tyroni, <lb/>qui vix in hoc labyrintho tutum &longs;e e&longs;&longs;e putabit. </s> </p> <p id="N1D1D4" type="main"> <s id="N1D1D6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N1D1E2" type="main"> <s id="N1D1E4"><!-- NEW --><emph type="italics"/>Si globus incumbat<emph.end type="italics"/> <emph type="italics"/>plano inclinato rotatur nece&longs;&longs;ariò deor&longs;um<emph.end type="italics"/>; </s> <s id="N1D1F3"><!-- NEW -->&longs;it enim <lb/>globus F in plano ED; </s> <s id="N1D1F9"><!-- NEW -->ducatur FH perpendicularis deor&longs;um; </s> <s id="N1D1FD"><!-- NEW -->hæc e&longs;t <lb/>linea directionis centri grauitatis, vt con&longs;tat; </s> <s id="N1D203"><!-- NEW -->igitur cùm non &longs;u&longs;tinea­<lb/>tur in prædicta linea, nec enim terminatur ad punctum contactus G, cer­<lb/>tè debet rotari; </s> <s id="N1D20B"><!-- NEW -->adde quod non e&longs;t in æquilibrio, vt patet, ratio autem <lb/>inæqualitatis e&longs;t vt GF ad FN, nec vlla e&longs;t difficultas; igitur duplici <lb/>qua&longs;i motu de&longs;cendet in prædicto plano ille globus, &longs;cilicet motu centri <lb/>propter inclinationem plani, & motu orbis, tùm quia non e&longs;t in æqui­<lb/>librio, tùm quia in linea directionis FH non &longs;u&longs;tinetur à plano. </s> </p> <p id="N1D217" type="main"> <s id="N1D219"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N1D225" type="main"> <s id="N1D227"><!-- NEW --><emph type="italics"/>Si corpus aliquod incumbat<emph.end type="italics"/> <emph type="italics"/>plano inclinato, &longs;ique linea directionis <lb/>centri grauitatis &longs;ecet ip&longs;um planum intra ba&longs;im corpus repit quidem in <lb/>prædicto plano &longs;ed non rotatur, &longs;i verò cadat extra ba&longs;im rotatur, non repit<emph.end type="italics"/>; </s> <s id="N1D23A"><!-- NEW --><lb/>&longs;it enim planum inclinatum BC, cui incubet cubus DL, cuius cen­<lb/>trum grauitatis &longs;it I; </s> <s id="N1D241"><!-- NEW -->ducatur RG perpendicularis deor&longs;um per cen­<lb/>trum grauitatis I cadit in punctum G intra ba&longs;im BG; </s> <s id="N1D247"><!-- NEW -->igitur non ro­<lb/>tabitur, &longs;ed repet; </s> <s id="N1D24D"><!-- NEW -->quia &longs;i &longs;u&longs;tinetur in G remoto &longs;en&longs;im plano BC; <lb/>haud dubiè portio GD non præponderat portioni GL, vt patet ex <lb/>libra. </s> </p> <p id="N1D255" type="main"> <s id="N1D257"><!-- NEW -->Sit quoque parallelipedum EK, centrum grauitatis N, perpendicu­<lb/>laris ducta per centrum HNM cadit intra ba&longs;im; </s> <s id="N1D25D"><!-- NEW -->igitur non rotabi­<lb/>tur, quia &longs;ubmoto plano BC non &longs;u&longs;tinetur quidem in M, &longs;ed minimè <lb/>inclinabitur dextror&longs;um; igitur non rotabitur. </s> <s id="N1D265"><!-- NEW -->Si verò cadat extra ba­<lb/>&longs;im haud dubiè rotabitur, &longs;it enim planum inclinatum AC, cui in­<lb/>cumbat parallelipedum FN, cuius centrum grauitatis &longs;it L; </s> <s id="N1D26D"><!-- NEW -->ducatur L <lb/>perpendicularis, cadit in E extra ba&longs;im FD; </s> <s id="N1D273"><!-- NEW -->certè latus DN inclinabi­<lb/>tur deor&longs;um; igitur rotabitur, quia eodem modo &longs;e habet, quo &longs;e ha­<lb/>beret, &longs;i &longs;ubmoto plano &longs;u&longs;tineretur in linea DX, &longs;ed trapezus DX <lb/>PN triangulo FXD præponderat per regulas libræ, de quibus &longs;uo <lb/>loco. </s> </p> <p id="N1D27F" type="main"> <s id="N1D281"><!-- NEW -->Ob&longs;eruabis autem primò &longs;ciri po&longs;&longs;e data plani inclinatione & ba&longs;i <lb/>parallelipedi maximam illius altitudinem, qua po&longs;ita non rotetur; </s> <s id="N1D287"><!-- NEW --><lb/>&longs;ecus verò po&longs;ita quacunque alia maiore; </s> <s id="N1D28C"><!-- NEW -->&longs;it enim planum AC, ba­<lb/>&longs;is parallelipedi FD; </s> <s id="N1D292"><!-- NEW -->erigantur FO, DN perpendiculares in <pb pagenum="206" xlink:href="026/01/238.jpg"/>AC; </s> <s id="N1D29B"><!-- NEW -->tùm erigatur perpendicularis DX parallela AB; </s> <s id="N1D29F"><!-- NEW -->connectantur R <lb/>M: dico FX e&longs;&longs;e maximam altitudinem, vt con&longs;tat ex dictis. </s> </p> <p id="N1D2A5" type="main"> <s id="N1D2A7"><!-- NEW -->Secundò, quotie&longs;cunque rectangulum, ita e&longs;t &longs;itum, vt eius <lb/>diagonalis &longs;it perpendicularis; </s> <s id="N1D2AD"><!-- NEW -->dico e&longs;&longs;e in perfecto æquilibrio; </s> <s id="N1D2B1"><!-- NEW --><lb/>&longs;it enim rectangulum BE, cuius diagonalis BE perpendicula­<lb/>riter cadit in horizontalem AC; </s> <s id="N1D2B8"><!-- NEW -->certè erit in æqualibrio; </s> <s id="N1D2BC"><!-- NEW -->&longs;it enim <lb/>diui&longs;um per lineam BE ita vt FH vel KI &longs;it libra quæ &longs;u&longs;tineatur in ful­<lb/>cro BG; &longs;itque totum pondus trianguli BED appen&longs;um brachio GH, <lb/>& aliud BET appen&longs;um brachio æquali GF, erit perfectum æquili­<lb/>brium per regulas libræ, &longs;ed duo triangula eodem modo &longs;e habent <lb/>conjuncta, quo &longs;e haberent &longs;eparata & appen&longs;a, vt patet. </s> </p> <p id="N1D2CA" type="main"> <s id="N1D2CC">Tertiò, omnia rectangula proportionalia in eodem æquilibrio rema­<lb/>nerent v.g. <!-- REMOVE S-->rectangulum BG cum rectangulo BE, idem dico de Rhom­<lb/>bo, Rhomboide, &c. </s> </p> <p id="N1D2D5" type="main"> <s id="N1D2D7">Quartò, inde etiam cogno&longs;citur in qua proportione minuatur pondus. </s> <s id="N1D2DA"><!-- NEW --><lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it enim cylindrus AE horizontalis, &longs;u&longs;tineaturque in A immo­<lb/>biliter, itemque in E; </s> <s id="N1D2E5"><!-- NEW -->certè qui &longs;u&longs;tinet in E æqualiter &longs;u&longs;tinet; </s> <s id="N1D2E9"><!-- NEW -->at verò <lb/>&longs;i attollatur in AD; </s> <s id="N1D2EF"><!-- NEW -->certè potentia quæ in D &longs;u&longs;tinet, e&longs;t ad potentiam <lb/>quæ &longs;u&longs;tinet in E, vt AF ad AE, quia pondus grauitaret in D & in E in <lb/>eadem ratione per Th. 16. &longs;ed potentia &longs;u&longs;tinens adæquat ponderis ra­<lb/>tionem, &longs;u&longs;tinens inquam, per DH; </s> <s id="N1D2F9"><!-- NEW -->nam reuerà &longs;u&longs;tinens per DF æqua­<lb/>lis e&longs;&longs;e debet potentiæ in E: </s> <s id="N1D2FF"><!-- NEW -->idem dico &longs;i attollatur in AP, nam potentia <lb/>trahens in P, per CP, e&longs;t ad potentiam in E, vt QA ad AP, vel AE; <lb/>igitur pondus in D e&longs;t ad pondus in P vt FA ad QA. </s> </p> <p id="N1D307" type="main"> <s id="N1D309">Quintò, hinc &longs;i duo ferant parallelipedum in &longs;itu inclinato v.g.vt AD, <lb/>ferunt inæqualiter, &longs;cilicet in ratione AD FA, itemque &longs;i ferant in &longs;itu <lb/>inclinato AP, vel AC, donec tandem AE attollatur in B, nihil amplius <lb/>&longs;u&longs;tinet potentia in B, & potentia in A totum &longs;u&longs;tinet. </s> </p> <p id="N1D312" type="main"> <s id="N1D314"><!-- NEW -->Sextò, hinc cùm attollitur cylindrus continuò minùs &longs;entitur pondus <lb/>& faciliùs attollitur; &longs;ic qui attollunt pontes illos ver&longs;atiles, initio maxi­<lb/>mo ni&longs;u, & modico &longs;ub finem trahunt. </s> </p> <p id="N1D31C" type="main"> <s id="N1D31E"><!-- NEW -->Septimò ob&longs;eruabis, &longs;i circa centrum immobile A attollatur cylindrus <lb/>AE fune BE, potentia po&longs;ita in B, vel fune EO, potentia po&longs;ita in O; </s> <s id="N1D324"><!-- NEW --><lb/>hæc deber e&longs;&longs;e minor quàm po&longs;ita in B, vt autem cogno&longs;catur propor­<lb/>tio, fiat angulus PAE æqualis angulo OEB; </s> <s id="N1D32B"><!-- NEW -->ducatur PQ; </s> <s id="N1D32F"><!-- NEW -->dico poten­<lb/>tiam in O e&longs;&longs;e ad potentiam B, vt AQ ad AP, quia &longs;i anguli OEB & <lb/>PAQ &longs;unt æquales etiam anguli APQ & AEB &longs;unt æquales; igitur <lb/>perinde e&longs;t &longs;iue trahatur PA circa A per lineam PQ, &longs;iue trahatur EA <lb/>circa A per lineam EB. <!-- KEEP S--></s> <s id="N1D33C">Idem dictum &longs;it de aliis lincis. </s> </p> <p id="N1D33F" type="main"> <s id="N1D341"><!-- NEW -->Octauò &longs;i attollendum &longs;it rectangulum non quidem circa axem; </s> <s id="N1D345"><!-- NEW -->&longs;ed <lb/>circa angulum immobilem, etiam decre&longs;cit proportio ponderis, &longs;it enim <lb/>v.g. <expan abbr="quadratũ">quadratum</expan> ACFD, &longs;itque AD horizontalis, AI perpendicularis, duca­<lb/>tur diagonalis AF, attollatur circa punctum A, ita vt transferatur in AG, <lb/>ducatur GB perpendicularis: </s> <s id="N1D355"><!-- NEW -->dico potentiam in G e&longs;&longs;e ad potentiam in <lb/>in A, vt AB ad AD; quippe res eodem modo &longs;e habet, ac &longs;i AF a&longs;cenderet <pb pagenum="207" xlink:href="026/01/239.jpg"/>per arcum FM, donec vbi AF traducta &longs;it in AM, tunc enim nulla erit <lb/>potentia in M propter æquilibrium. </s> </p> <p id="N1D362" type="main"> <s id="N1D364"><!-- NEW -->Nonò, hinc initio decre&longs;cit in maiori proportione ratione præpon­<lb/>derantiæ; </s> <s id="N1D36A"><!-- NEW -->quia po&longs;ita ba&longs;i KN, angulus KAN e&longs;t omnium maximus; at <lb/>verò decre&longs;cit in minori proportione initio ratione &longs;egmenti horizon­<lb/>talis AD, in quam cadit perpendicularis. </s> </p> <p id="N1D372" type="main"> <s id="N1D374"><!-- NEW -->Decimò, &longs;i &longs;it rectangulum oblongum horizontale vt AE diffici­<lb/>liùs attolletur; </s> <s id="N1D37A"><!-- NEW -->quia quadratum AF figuræ prioris debet tantùm attolli <lb/>per arcum FM, vt &longs;tatuatur in æquilibro; </s> <s id="N1D380"><!-- NEW -->at verò rectangulum AE fi­<lb/>guræ huius attolli debet per arcum EC longè maiorem; </s> <s id="N1D386"><!-- NEW -->igitur difficiliùs: <lb/>porrò potentia in D e&longs;t ad potentiam in F vt AG ad AF, vt con&longs;tat ex <lb/>dictis. </s> </p> <p id="N1D38E" type="main"> <s id="N1D390"><!-- NEW -->Vndecimò, denique, &longs;i &longs;it rectangulum oblongum, &longs;ed verticale vt <lb/>HK longè faciliùs attolletur, quia diagonalis HK debet tantùm percur­<lb/>rere arcum KM vt &longs;tatuatur in æquilibrio; </s> <s id="N1D398"><!-- NEW -->igitur minorem, igitur longè <lb/>faciliùs; porrò hæc omnia omnibus experimentis con&longs;entiunt, & ex <lb/>principiis facillimis demon&longs;trantur. </s> <s id="N1D3A0">Hæc paulò fu&longs;iùs pro&longs;equutus &longs;um, <lb/>quia pertinent ad rationem plani inclinati. </s> </p> <p id="N1D3A5" type="main"> <s id="N1D3A7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N1D3B3" type="main"> <s id="N1D3B5"><!-- NEW --><emph type="italics"/>In plano inclinato acceleratur motus in eadem proportione qua acceleratur <lb/>in perpendiculari<emph.end type="italics"/>; </s> <s id="N1D3C0"><!-- NEW -->&longs;it enim planum inclinatum AC, perpendicularis A <lb/>E, in qua primo tempore &longs;en&longs;ibili percurrat AD; </s> <s id="N1D3C6"><!-- NEW -->&longs;ecundò DE; </s> <s id="N1D3CA"><!-- NEW -->certè dato <lb/>etiam tempore licèt maiore percurret AB; </s> <s id="N1D3D0"><!-- NEW -->igitur alio æquali percurret <lb/>CB; </s> <s id="N1D3D6"><!-- NEW -->nam vt &longs;e habet AE ad AG; </s> <s id="N1D3DA"><!-- NEW -->ita &longs;e habet AD ad AB, & DE ad BC; <lb/>quæ omnia &longs;unt certa. </s> </p> <p id="N1D3E0" type="main"> <s id="N1D3E2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N1D3EE" type="main"> <s id="N1D3F0"><!-- NEW --><emph type="italics"/>Hinc æqualis ine&longs;t velocitas mobili decur&longs;a AC, inclinata & decur&longs;a AE <lb/>perpendiculari,<emph.end type="italics"/> probatur, motus per AC e&longs;t ad motum per AE, vt AE, ad <lb/>AC per Th.6.igitur motus per AC e&longs;t tardior; </s> <s id="N1D3FD"><!-- NEW -->&longs;ed motu tardiore minùs <lb/>&longs;patium conficitur æquali tempore in ca proportione, in qua motus e&longs;t <lb/>tardior; </s> <s id="N1D405"><!-- NEW -->&longs;ed proportio velocitatis e&longs;t vt AC ad AE: </s> <s id="N1D409"><!-- NEW -->atqui quâ propor­<lb/>tione motus e&longs;t tardior alio, maius &longs;patium decurri debet, vt motu acce­<lb/>lerato per minora crementa acquiratur velocitas alteri æqualis; </s> <s id="N1D411"><!-- NEW -->igitur <lb/>eò &longs;patium debet e&longs;&longs;e maius, quò motus erit tardior; </s> <s id="N1D417"><!-- NEW -->igitur debet percur­<lb/>ri AC in inclinata, & AE in perpendiculari, vt &longs;it æqualis velocitas; </s> <s id="N1D41D"><!-- NEW --><lb/>&longs;it autem v.g. <!-- REMOVE S-->AC dupla AE, certè motus per AC e&longs;t &longs;ubduplus motus <lb/>pes AE; </s> <s id="N1D426"><!-- NEW -->ducatur EB perpendicularis, certè AB e&longs;t &longs;ubdupla AE; </s> <s id="N1D42A"><!-- NEW -->igitur <lb/>eo tempore, quo percurret AE, percurret tantùm AB &longs;ubduplum &longs;cili­<lb/>cet motu &longs;ubduplo; </s> <s id="N1D432"><!-- NEW -->igitur tempore æquali BC triplam AB; </s> <s id="N1D436"><!-- NEW -->&longs;ed tem­<lb/>poribus æqualibus acquiruntur æqualia velocitatis momenta; </s> <s id="N1D43C"><!-- NEW -->igitur ve­<lb/>locitas in C e&longs;t dupla illius, quæ erat in B; </s> <s id="N1D442"><!-- NEW -->&longs;ed quæ e&longs;t in E e&longs;t dupla il­<lb/>lius, quæ e&longs;t in B; igitur quæ e&longs;t in E e&longs;t æqualis illi, quæ e&longs;t in C. <!-- KEEP S--></s> <s id="N1D449"><!-- NEW -->Adde <lb/>quod in ea proportione in qua motus e&longs;t tardior, &longs;patium e&longs;t maius, vt <lb/>æqualis velocitas acquiratur; </s> <s id="N1D451"><!-- NEW -->igitur &longs;i quælibet pars &longs;patij motum auget <pb pagenum="208" xlink:href="026/01/240.jpg"/>minùs quidem qua proportione motus e&longs;t tardior, & &longs;i &longs;patium AC ma­<lb/>jus e&longs;t &longs;patio AE in ca proportione in qua motus per AE e&longs;t velocior; </s> <s id="N1D45C"><!-- NEW --><lb/>pauciores partes &longs;patij AE augent motum, &longs;ed plùs &longs;ingulæ, & plures <lb/>&longs;patij AC augent motum, &longs;ed minùs &longs;ingulæ; </s> <s id="N1D463"><!-- NEW -->&longs;ed cum &longs;int plures in ea­<lb/>dem proportione, in qua minùs augent; certè plures quarum &longs;ingulæ mi­<lb/>nùs augent, &longs;imul &longs;umptæ æqualiter augent, v.g. <!-- REMOVE S-->&longs;int AC 4. partes, & AE <lb/>2. &longs;ingulæ AE augeant motum vt 4. & &longs;ingulæ AC vt 2. quia in ca pro­<lb/>portione minùs augent in qua 2. &longs;unt ad 4. certè 2. &longs;imul &longs;umptæ augent <lb/>motum vt 8. & 4. &longs;imul &longs;umptæ etiam vt 8. quæ dicta &longs;unt in gratiam <lb/>Geometrarum, &longs;ed meliùs adhuc ex dictis patebit. </s> </p> <p id="N1D475" type="main"> <s id="N1D477"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N1D483" type="main"> <s id="N1D485"><!-- NEW --><emph type="italics"/>Hinc aqualis e&longs;&longs;et ictus ab eodem mobili po&longs;t motum per AE. AF. AC. <lb/>AG.<emph.end type="italics"/> quia e&longs;&longs;et acqui&longs;itus æqualis impetus; igitur e&longs;&longs;et æqualis ictus, <lb/>quod certè mirabile e&longs;t. </s> </p> <p id="N1D492" type="main"> <s id="N1D494"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N1D4A0" type="main"> <s id="N1D4A2"><!-- NEW --><emph type="italics"/>Hinc pote&longs;t determinari &longs;patij quæcunque petita proportio ad &longs;patium da­<lb/>tum<emph.end type="italics"/>; </s> <s id="N1D4AD"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it ictus inflictus à mobili decur&longs;a perpendiculari AE: </s> <s id="N1D4B5"><!-- NEW -->vis æ­<lb/>qualem ictum &longs;ed confecto &longs;patio duplo; </s> <s id="N1D4BB"><!-- NEW -->accipe AC duplam AE: vis æ­<lb/>qualem ictum &longs;ed confecto &longs;patio triplo, accipe AG triplam AE. <!-- KEEP S--></s> </p> <p id="N1D4C2" type="main"> <s id="N1D4C4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N1D4D0" type="main"> <s id="N1D4D2"><!-- NEW --><emph type="italics"/>Tempora quibus percurruntur &longs;patia planorum &longs;unt vt planorum longitu­<lb/>dines,<emph.end type="italics"/> v.g.tempus quo percurritur planum inclinatum AC e&longs;t ad tempus <lb/>quo percurritur perpendicularis AE, vt AC ad AE; </s> <s id="N1D4DF"><!-- NEW -->probatur, cùm enim <lb/>mobile in C & in E habeat æqualem impetum &longs;eu velocitatem per Th. <!-- REMOVE S--><lb/>20. certè cùm motus in AC &longs;it &longs;ubduplus v.g. <!-- REMOVE S-->motus in AE, e&longs;t enim <lb/>vt AE ad AC per Th.6. igitur cum &longs;ubduplo motu æquali tempore ac­<lb/>quiritur &longs;ubduplus impetus; </s> <s id="N1D4EE"><!-- NEW -->igitur tempore duplo æqualis impetus; </s> <s id="N1D4F2"><!-- NEW -->at­<lb/>qui tempus motus per AC e&longs;t ad tempus motus per AE vt AC ad AE, <lb/>ide&longs;t duplum; </s> <s id="N1D4FA"><!-- NEW -->adde quod &longs;i æqualis impetus e&longs;t in C & in E; </s> <s id="N1D4FE"><!-- NEW -->igitur æqua­<lb/>lis in D & in B, &longs;ed AB e&longs;t ad BC vt AD ad DE; </s> <s id="N1D504"><!-- NEW -->igitur &longs;i cre&longs;cit impe­<lb/>tus per partes &longs;ubduplas in AC, nece&longs;&longs;ariò cre&longs;cit per partes duplas in <lb/>&longs;patio, atque in tempore; </s> <s id="N1D50C"><!-- NEW -->cùm enim motus &longs;it &longs;ubduplus, tarditas e&longs;t &longs;ub­<lb/>dupla; </s> <s id="N1D512"><!-- NEW -->igitur acquiritur in AC &longs;patium AB &longs;ubduplum AE eo tempore, <lb/>quo percurritur AE, &longs;i enim accipiantur æqualia tempora, &longs;patia &longs;unt vt <lb/>motus; </s> <s id="N1D51A"><!-- NEW -->&longs;ed motus per AC e&longs;t &longs;ubduplus; </s> <s id="N1D51E"><!-- NEW -->igitur &longs;patium AB e&longs;t &longs;ubdu­<lb/>plum AE; </s> <s id="N1D524"><!-- NEW -->&longs;ed tempore æquali conficit BC triplum AB, igitur tota AC <lb/>e&longs;t dupla AE; </s> <s id="N1D52A"><!-- NEW -->&longs;ed percurritur tempore duplo; </s> <s id="N1D52E"><!-- NEW -->igitur tempora &longs;unt vt <lb/><expan abbr="lõgitudines">longitudines</expan> planorum; </s> <s id="N1D537"><!-- NEW -->&longs;ed clariùs, & breuiùs illud demon&longs;tro; </s> <s id="N1D53B"><!-- NEW -->In ea pro­<lb/>portione erit maius tempus per AC quàm per AE, in qua minor e&longs;t <lb/>motus per AC quàm per AE; </s> <s id="N1D543"><!-- NEW -->&longs;i enim motus per AF e&longs;&longs;et ad motum per <lb/>AE vt AF ad AE, certè æquali tempore AF & AE percurrerentur; </s> <s id="N1D549"><!-- NEW -->igitur <lb/>qua proportione motus per AF e&longs;t minor, tempus e&longs;t maius; </s> <s id="N1D54F"><!-- NEW --><expan abbr="tantũdem">tantundem</expan> <lb/>enim additur tempori, quantum detrahitur motui; igitur tempora &longs;unt <pb pagenum="209" xlink:href="026/01/241.jpg"/>vt lineæ. </s> <s id="N1D55D">Hinc acquiritur velocitas æqualis, vt dictum e&longs;t Th. 20. quia <lb/>&longs;i tantùm addit tempus per AF &longs;upra tempus per AE, quantum addit <lb/>motus per AE &longs;upra motum per AF, haud dubiè e&longs;t æqualitas. </s> </p> <p id="N1D564" type="main"> <s id="N1D566"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N1D572" type="main"> <s id="N1D574"><emph type="italics"/>Hinc pote&longs;t determinari longitudo plani, quæ dato tempore percurratur,<emph.end type="italics"/> v. <!-- REMOVE S--><lb/>g. <!-- REMOVE S-->perpendicularis 3. pedum percurritur 30‴. </s> <s id="N1D581">igitur &longs;i a&longs;&longs;umas planum <lb/>inclinatum 6. pedum, percurretur 1″. </s> <s id="N1D586">&longs;i 12. 2′. </s> <s id="N1D589">&longs;i 24. 4″. </s> <s id="N1D58C">atque ita dein­<lb/>ceps; </s> <s id="N1D591"><!-- NEW -->hinc po&longs;&longs;et dari planum inclinatum quod tantùm 100. annis per­<lb/>curretur, &longs;cilicet &longs;i longitudo plani a&longs;&longs;umpti &longs;it æque multiplex longitu­<lb/>dinis 12. pedum atque 100. anni vnius &longs;ecundi; quod facilè e&longs;t, imò da­<lb/>to plano cuiu&longs;cunque longitudinis, pote&longs;t dari tempus quodcunque quo <lb/>percurratur, de quo infrà. </s> </p> <p id="N1D59D" type="main"> <s id="N1D59F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N1D5AB" type="main"> <s id="N1D5AD"><!-- NEW --><emph type="italics"/>Determinari pote&longs;t quantum &longs;patium conficiat mobile in plano inclinato; <lb/>dum conficit perpendicularem<emph.end type="italics"/>; </s> <s id="N1D5B8"><!-- NEW -->&longs;it enim perpendiculum AE, inclinata AC; </s> <s id="N1D5BC"><!-- NEW --><lb/>ducatus, EB perpendicularis in AC; </s> <s id="N1D5C1"><!-- NEW -->dico quod eodem tempore percur­<lb/>ret AE & AB, quod demon&longs;tro; </s> <s id="N1D5C7"><!-- NEW -->quia triangula EAB, EAC &longs;unt pro­<lb/>portionalia: </s> <s id="N1D5CD"><!-- NEW -->igitur AB e&longs;t ad AE vt AE ad AC; </s> <s id="N1D5D1"><!-- NEW -->igitur motus in AB <lb/>e&longs;t ad motum in DE vt AB ad AE; </s> <s id="N1D5D7"><!-- NEW -->igitur &longs;i tempora a&longs;&longs;umantur æqua­<lb/>lia &longs;patia erunt vt motus, vt patet, id e&longs;t motu &longs;ubduplo acquiritur &longs;pa­<lb/>tium &longs;ubduplum: </s> <s id="N1D5DF"><!-- NEW -->nec alia e&longs;&longs;e pote&longs;t regula tarditatis, igitur &longs;patia <lb/>erunt vt AB ad AE, id e&longs;t in ratione motuum; </s> <s id="N1D5E5"><!-- NEW -->licèt enim motus veloci­<lb/>tas cre&longs;cat, attamen &longs;i accipiatur velocitas compo&longs;ita ex &longs;ubdupla maxi­<lb/>mæ & minimæ, percurretur AE motu æquabili æquali tempore; &longs;ed <lb/>compo&longs;ita ex &longs;ubdupla maximæ & minimæ per AB habet <expan abbr="eãdem">eandem</expan> ra­<lb/>tionem ad priorem compo&longs;itam, quàm motus per AB ad motum per AE. <lb/>& hic quam habet AB ad AE. <!-- KEEP S--></s> <s id="N1D5F8">Sed hæc &longs;unt clara. </s> </p> <p id="N1D5FB" type="main"> <s id="N1D5FD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N1D609" type="main"> <s id="N1D60B"><!-- NEW --><emph type="italics"/>Hinc æquali tempore de&longs;cendit per inclinatam BE,<emph.end type="italics"/> &longs;it enim inclinata <lb/>AG, perpendicularis AE; &longs;it quoque FC perpendicularis in AG, & FD, <lb/>in CF. <!-- KEEP S--></s> <s id="N1D619">Dico quòd eo tempore, quo conficit CD perpendicularem <lb/>conficit CF inclinatam per Th.24. e&longs;t enim DF perpendicularis in IC. <lb/>&longs;icut FC in AG, &longs;ed CD e&longs;t æqualis AF, vt patet. </s> </p> <p id="N1D620" type="main"> <s id="N1D622"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N1D62E" type="main"> <s id="N1D630"><!-- NEW --><emph type="italics"/>Hinc cognito &longs;patio quod percurritur in plano inclinato, cogno&longs;citur &longs;pa­<lb/>tium quod conficeretur tempore æquali in perpendiculari,<emph.end type="italics"/> &longs;it enim tempus <lb/>quo percurritur AC; ducatur ex C perpendicularis CF. <!-- KEEP S--></s> <s id="N1D63E"><!-- NEW -->Dico confici AF <lb/>in perpendiculari eo tempore, quo percurritur AC: </s> <s id="N1D644"><!-- NEW -->vel &longs;it inclinata C <lb/>F, ducatur ex F perpendicularis FD; percurretur CD eo tempore, quo <lb/>percurritur CF, quæ probantur per Th.24.& 25. </s> </p> <pb pagenum="210" xlink:href="026/01/242.jpg"/> <p id="N1D650" type="main"> <s id="N1D652"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N1D65E" type="main"> <s id="N1D660"><!-- NEW --><emph type="italics"/>Hinc per omnes chordas in&longs;criptas circulo ad alteram extremitatem, <lb/>diametri perpendicularis terminatas de&longs;cendit mobile æquali tempore<emph.end type="italics"/>; </s> <s id="N1D66B"><!-- NEW -->a &longs;it <lb/>enim circulus centro B; </s> <s id="N1D671"><!-- NEW -->&longs;it diameter AE perpendicularis deor&longs;um; </s> <s id="N1D675"><!-- NEW -->du­<lb/>catur AC inclinata, tùm CE; </s> <s id="N1D67B"><!-- NEW -->de&longs;cendat haud dubiè æquali tempore <lb/>per AC.CE.AE. per Th.24.25.26. idem dico de omnibus aliis AD.D <lb/>E. AG.GE.AF.FE; </s> <s id="N1D683"><!-- NEW -->e&longs;t enim eadem omnibus ratio; hinc non pote&longs;t da­<lb/>ri planum tam paruæ longitudinis, quo non po&longs;&longs;it dari minus, quod dato <lb/>tempore percurratur. </s> <s id="N1D68B"><!-- NEW -->Hæc e&longs;t illa propo&longs;itio toties à Galileo enuncia­<lb/>ta; </s> <s id="N1D691"><!-- NEW -->cum enim motus per BE &longs;it ad motum per GE vt GE ad BE, & tem­<lb/>pus per BE ad tempus per GE vt BE ad GE; </s> <s id="N1D697"><!-- NEW -->cumque &longs;it vt BE ad GE <lb/>rita GE ad AE; </s> <s id="N1D69D"><!-- NEW -->certè motus per AE e&longs;t ad motum per GE vt AE ad G <lb/>E; </s> <s id="N1D6A3"><!-- NEW -->igitur tantùm addit AE &longs;upra GE ratione &longs;patij, quantum ratione <lb/>motus: igitur tempore æquali per AE. & GE fiet motus, idem dico de <lb/>aliis chordis. </s> </p> <p id="N1D6AB" type="main"> <s id="N1D6AD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N1D6B9" type="main"> <s id="N1D6BB"><!-- NEW --><emph type="italics"/>Hinc datis duabus inclinatis æqualibus pote&longs;t determinari ratio tempo­<lb/>rum, in quibus percurruntur<emph.end type="italics"/>; </s> <s id="N1D6C6"><!-- NEW -->&longs;int enim AG.AH æquales, &longs;ed diuer&longs;æ incli­<lb/>nationis; haud dubiè cum æquali tempore AG. AF percurrantur per <lb/>Th. 27. tempora quibus percurruntur AGAH erunt vt tempora quibus <lb/>percurruntur AF AH, & hæc vt tempora quibus percurruntur AE. <!-- REMOVE S-->A <lb/>K, & hæc vt radices quadratæ illorum &longs;patiorum AE. AK, cum autem <lb/>&longs;patia &longs;int vt quadrata temporum, vel in duplicata ratione, &longs;i inter AE <lb/>& AK &longs;it media proportionalis AN. v. <!-- REMOVE S-->g. <!-- REMOVE S-->tempus quo percurretur AE <lb/>erit ad tempus, quo percurretur AK vt AE ad AN, vel AN ad AK. </s> </p> <p id="N1D6DE" type="main"> <s id="N1D6E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N1D6EC" type="main"> <s id="N1D6EE"><emph type="italics"/>Hinc cognito tempore quo percurritur data portio linea cogno&longs;ci potest <lb/>tempus, quo percurritur aliud &longs;patium vel alia portio,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->cogno&longs;co tem­<lb/>pus quo percurritur AK, & volo cogno&longs;cere tempus quo percurritur K <lb/>E, con&longs;equenti motu ex AK, &longs;cio tempus quo percurritur &longs;ola AE, quod <lb/>e&longs;t ad tempus quo percurritur AK vt AE ad AN per Th. 28. igitur <lb/>tempus quo percurritur KE con&longs;equenti motu ex AK e&longs;t ad tempus, <lb/>quo percurritur AK vt EN ad NA, vel vt NK, ad NA. </s> </p> <p id="N1D706" type="main"> <s id="N1D708"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N1D714" type="main"> <s id="N1D716"><!-- NEW --><emph type="italics"/>Hinc in planis inæqualibus tùm in longitudine, tùns in inclinatione, <lb/>pote&longs;t &longs;ciri ratio temporum, quibus percurruntur<emph.end type="italics"/>; </s> <s id="N1D721"><!-- NEW -->&longs;int enim AC AR duo pla­<lb/>na; </s> <s id="N1D727"><!-- NEW -->&longs;it autem AE perpendicularis indefinita; </s> <s id="N1D72B"><!-- NEW -->diuidatur AC bifariam <lb/>in V ducta perpendiculari VB; </s> <s id="N1D731"><!-- NEW -->ex B fiat circulus, &longs;ecabit puncta <lb/>ACE; </s> <s id="N1D737"><!-- NEW -->&longs;ecat etiam AR; </s> <s id="N1D73B"><!-- NEW -->in D igitur AC, & AD percurruntur æquali <lb/>tempore per Th. 27. &longs;imiliter fiat circulus ART eodem modos certè A <lb/>R & AT percurruntur æqualibus temporibus per Th. 27. igitur tempus, <lb/>quo per curritur AR, vel AD e&longs;t ad tempus, quo percurritur AR vt <lb/>tempus, quo percurritur AE ad tempus, quo percurritur AT; </s> <s id="N1D747"><!-- NEW -->&longs;ed hæc <pb pagenum="211" xlink:href="026/01/243.jpg"/>&longs;unt vt radices AEAT, id e&longs;t tempus quo percurritur AE e&longs;t ad tem­<lb/>pus, quo percurritur AT, vt AE ad mediam proportionalem inter AE <lb/>AT, vel vt AD ad mediam proportionalem inter AD AR; quippe AD <lb/>e&longs;t ad AR vt AE ad AT. </s> </p> <p id="N1D756" type="main"> <s id="N1D758"><!-- NEW -->Galileus verò demon&longs;trat rationem i&longs;torum temporum e&longs;&longs;e compo&longs;i­<lb/>tam ex ratione longitudinem planorum & ex ratione &longs;ubduplicata al­<lb/>titudinum eorumdem permutatim accepta: pro quo ob&longs;erua à Galileo <lb/>rationem duplicatam appellari duplam, & &longs;ubduplicatam appellari &longs;ub­<lb/>duplam. </s> </p> <p id="N1D764" type="main"> <s id="N1D766"><!-- NEW -->Ob&longs;eruabis denique plurima ex his colligi po&longs;&longs;e præ&longs;ertim ex Th. 27. <lb/>quæ quia &longs;unt purè geometrica, certè phy&longs;ic&etail; minimè competunt; aliqua <lb/>tamen omittere non po&longs;&longs;um. </s> </p> <p id="N1D76E" type="main"> <s id="N1D770"><!-- NEW -->Primò, &longs;i &longs;int duo plana inæqualia ad angulum rectum, qui &longs;u&longs;tinea­<lb/>tur ab horizontali, determinari po&longs;&longs;unt tempora de&longs;cen&longs;uum &longs;it enim <lb/>triangulum orthogonium ABE, ita vt AE &longs;it horizontalis; </s> <s id="N1D778"><!-- NEW -->ducatur B <lb/>G indefinita perpendicularis in ba&longs;im AE; </s> <s id="N1D77E"><!-- NEW -->tùm FA perpendicularis in <lb/>AB; </s> <s id="N1D784"><!-- NEW -->tùm FC perpendicularis in BE; </s> <s id="N1D788"><!-- NEW -->tùm denique GE in BE; </s> <s id="N1D78C"><!-- NEW -->dico BA <lb/>BFBC percurri temporibus æqualibus, item BE, BG, EG, etiam æqua­<lb/>libus; </s> <s id="N1D794"><!-- NEW -->igitur tempus, quo percurritur BA e&longs;t ad tempus quo percurri­<lb/>tur BE, vt tempus, quo percurritur BF ad tempus quo percurritur BG; <lb/>hæc porrò &longs;unt in &longs;ubduplicata ratione BFBG vel BC, & BE. <!-- KEEP S--></s> </p> <p id="N1D79D" type="main"> <s id="N1D79F"><!-- NEW -->Secundò, &longs;i planum &longs;u&longs;tinens angulum rectum non &longs;it parallelum <lb/>horizonti 6. res &longs;imiliter determinari poterit; </s> <s id="N1D7A5"><!-- NEW -->&longs;it enim triangulum or­<lb/>thogonium ABC ex B, ducatur perpendicularis deor&longs;um indefinitè BF, <lb/>tùm EA in AB, tùm DC in CB, tùm EH parallela DC, tùm GC in A <lb/>C; </s> <s id="N1D7AF"><!-- NEW -->denique AG parallela BF; dico quod BABEHE AE percurren­<lb/>tur æqualibus temporibus item BCCDBD. </s> </p> <p id="N1D7B5" type="main"> <s id="N1D7B7"><!-- NEW -->Tertiò, &longs;iue de&longs;cendat ex B in C per lineam perpendicularem BC, <lb/>&longs;iue ex A per inclinatam AC, eodem modo de&longs;cendet &longs;iue per CD, &longs;iue <lb/>per CE; ratio e&longs;t clara, quia acquirit æqualem velocitatem &longs;iue ex A &longs;i­<lb/>ue ex B de&longs;cendat pet Th. 20. erit autem tempus per CE ad tempus per <lb/>CD, vt CE ad CD per Th.23.& motus per CE ad motum per CD, vt <lb/>CD ad CE per Th.6. po&longs;ito initio motus in C. <!-- KEEP S--></s> </p> <p id="N1D7C6" type="main"> <s id="N1D7C8"><!-- NEW -->Quartò, præuio motu ex A vel ex B ad C pote&longs;t inueniri inclinata, <lb/>per quam mobile pergat moueri motu &longs;cilicet naturaliter accelerato, ita <lb/>vt æquali tempore illam conficiat; </s> <s id="N1D7D0"><!-- NEW -->&longs;i enim BC conficiet dato tempore; </s> <s id="N1D7D4"><!-- NEW --><lb/>igitur CF triplum CB conficiet tempore æquali; </s> <s id="N1D7D9"><!-- NEW -->&longs;it autem planum ho­<lb/>rizontale EDK ad quod ex C ducendum &longs;it planum inclinatum, quod <lb/>eodem tempore percurratur, quo CF, diuidatur CF bifariam in H, & ex <lb/>puncto H fiat arcus CK, ducaturque CK: </s> <s id="N1D7E3"><!-- NEW -->Dico CF & CK æquali tem­<lb/>pore confici per Th. 27. modò ex quiete C procedat motus: </s> <s id="N1D7E9"><!-- NEW -->&longs;imiliter a&longs;­<lb/>&longs;umi pote&longs;t alia horizontalis LM ducto arcu LF ex centro H; </s> <s id="N1D7EF"><!-- NEW -->nam CL <lb/>& CF æquali tempore percurruntur; </s> <s id="N1D7F5"><!-- NEW -->&longs;i verò præ&longs;upponatur motus præ­<lb/>uius ex A vel ex B, haud dubiè CK breuiori tempore percurretur, quàm <lb/>CF, idem dico de CL; </s> <s id="N1D7FD"><!-- NEW -->alioqui CE & CI eodem præuio motu &longs;uppo <pb pagenum="212" xlink:href="026/01/244.jpg"/>&longs;ito æquali tempore percurrerentur, quod fal&longs;um e&longs;t; </s> <s id="N1D806"><!-- NEW -->nam &longs;it AC ad A <lb/>N vt AN ad AE; </s> <s id="N1D80C"><!-- NEW -->&longs;itque BC ad BO vt BO ad BI; </s> <s id="N1D810"><!-- NEW -->certè tempus, quo <lb/>percurritur BC e&longs;t ad tempus, quo percurritur CI vt CB ad CO, & <lb/>tempus quo percurritur BC e&longs;t ad tempus quo percurritur CE vt BC ad <lb/>CN; </s> <s id="N1D81A"><!-- NEW -->&longs;ed CN e&longs;t minor quàm CO, vt con&longs;tat ex Geometria, quod bre­<lb/>uiter in tironum <expan abbr="gratiã">gratiam</expan> in terminis rationabilibus o&longs;tendo, &longs;it planum <lb/>inclinatum AE 9. &longs;itque AE id e&longs;t 9. ad AD. 6. vt AD ad AC 4. ex <lb/>centro C a&longs;&longs;umpta CH 3. ducatur arcus HB & ex A ad prædictum ar­<lb/>cum Tangens AB, tùm ex BC G indefinitè & ex E, EG perpendicularis <lb/>in EA; </s> <s id="N1D82C"><!-- NEW -->haud dubiè triangula CGE, CAB &longs;unt proportionalia; </s> <s id="N1D830"><!-- NEW -->igitur vt <lb/>CB;.ad CA. 4.ita CE 5. ad CG 6. 2/3; </s> <s id="N1D836"><!-- NEW -->igitur tota BG e&longs;t 9. 2/3; &longs;itque B <lb/>G ad BF, vt BF ad DC, quod vt fiat BG 9. 2/3 in BC 3. productum erit <lb/>29. igitur BF e&longs;t Rad. </s> <s id="N1D83E">quad. </s> <s id="N1D841"><!-- NEW -->29.igitur e&longs;t maior 5. &longs;ed &longs;i e&longs;&longs;et maior 5. C <lb/>M & CD e&longs;&longs;ent æquales; </s> <s id="N1D847"><!-- NEW -->igitur CF e&longs;t maior CD; </s> <s id="N1D84B"><!-- NEW -->e&longs;t enim BF ferè 3. <lb/>1/2 paulò minùs: </s> <s id="N1D851"><!-- NEW -->vt autem reperiatur linea inclinata, quæ percurratur æ­<lb/>quali tempore cum BC &longs;uppo&longs;ito præuio motu per BC, a&longs;&longs;umatur CK <lb/>æqualis CB id e&longs;t 3.partium, <expan abbr="fiat&qacute;ue">fiatque</expan> vt AC ad AK, ita AK ad AN; </s> <s id="N1D85D"><!-- NEW -->haud <lb/>dubiè percurret CN æquali tempore, quo BC; </s> <s id="N1D863"><!-- NEW -->vt verò habeatur pun­<lb/>ctum in horizontali, &longs;it AF perpendicularis bifariam diui&longs;a in K, &longs;it K <lb/>F diui&longs;a in 4. partes æquales, quibus addatur FP 1/4 KFEK V dupla FA, <lb/>& producatur in X; </s> <s id="N1D86D"><!-- NEW -->ita vt EX &longs;it 1/4 EK: </s> <s id="N1D871"><!-- NEW -->dico quod præuio motu ex A in <lb/>K, & deinde deflexo per KX conficietur KX æquali tempore cum AK; </s> <s id="N1D877"><!-- NEW --><lb/>&longs;i enim caderet mobile ex V primo tempore percurreret VL, id e&longs;t 1/4 V <lb/>K eo tempore, quo percurreret AK per Th.6. igitur &longs;ecundo tempore <lb/>æquali LK, id e&longs;t 3/4 VK; </s> <s id="N1D880"><!-- NEW -->igitur tertio tempore æquali KX 5/4 VK; nam eo­<lb/>dem modo &longs;e habet in k &longs;iue de&longs;cendat ex V, &longs;iue ex A per Th.20. </s> </p> <p id="N1D886" type="main"> <s id="N1D888"><!-- NEW -->Porrò vt habeatur in horizontali FS; </s> <s id="N1D88C"><!-- NEW -->&longs;it FR æqualis KF; </s> <s id="N1D890"><!-- NEW -->&longs;it FT æ­<lb/>qualis KR; </s> <s id="N1D896"><!-- NEW -->&longs;it arcus TS ex k: </s> <s id="N1D89A"><!-- NEW -->Dico quod ks e&longs;t linea quæ&longs;ita; </s> <s id="N1D89E"><!-- NEW -->nam &longs;i &longs;it <lb/>vt BS ad BZ, ita BZ ad BK, kz erit æqualis KF, vel AK; </s> <s id="N1D8A4"><!-- NEW -->&longs;ed tempus <lb/>quo percurritur AK e&longs;t ad tempus quo percurritur Dk vt BK ad AK <lb/>per Th.23.& ad tempus, quo percurritur BS, vt Bk ad BZ, & ad tem­<lb/>pus quo percurritur ks vt Bk ad kz; ergo Ak & ks percurruntur æ­<lb/>quali tempore, &longs;i kz &longs;it æqualis KF, quod &longs;ic breuiter demon&longs;tro, cùm <lb/>figura apud Galileum de&longs;ideretur. </s> <s id="N1D8B2"><!-- NEW -->&longs;int AFFE æquales; </s> <s id="N1D8B6"><!-- NEW -->ducatur AE <lb/>quæ transferatur iu FG, &longs;itque GI æqualis AG, &longs;ic tota AG mihi repræ­<lb/>&longs;entat totam BS &longs;uperioris figuræ, vt con&longs;tat; </s> <s id="N1D8BE"><!-- NEW -->&longs;it autem AG ad AH vt A <lb/>H ad AI: </s> <s id="N1D8C4"><!-- NEW -->Dico GH e&longs;&longs;e æqualem AF; </s> <s id="N1D8C8"><!-- NEW -->&longs;it enim quadratum HD mediæ <lb/>proportionalis: </s> <s id="N1D8CE"><!-- NEW -->Dico e&longs;&longs;e æquale rectangulo IC, dùm AC &longs;it æqualis A <lb/>G; </s> <s id="N1D8D4"><!-- NEW -->igitur quadratum PR cuius latus e&longs;t æquale FG, &longs;eu AE continet <lb/>duo quadrata RDSN; </s> <s id="N1D8DA"><!-- NEW -->ergo GH e&longs;t æqualis VN; igitur GH quod erat <lb/>demon&longs;trandum. </s> </p> <p id="N1D8E0" type="main"> <s id="N1D8E2"><!-- NEW -->Quintò, hinc nunquam ks vel kx pote&longs;t e&longs;&longs;e tripla Ak donec tan­<lb/>dem perueniatur ad perpendiculum kH; </s> <s id="N1D8E8"><!-- NEW -->nam &longs;ecundo tempore percur­<lb/>ritur kH triplum Ak, &longs;i primo percurritur Ak; </s> <s id="N1D8EE"><!-- NEW -->nunquam etiam ks vel <lb/>vlla alia inclinata pote&longs;t e&longs;&longs;e dupla tantùm Ak; </s> <s id="N1D8F4"><!-- NEW -->&longs;ed &longs;emper e&longs;t maior, do-<pb pagenum="213" xlink:href="026/01/245.jpg"/>nec tandem perueniat ad horizontalem KY, quæ e&longs;t dupla AK, quia in <lb/>horizontali non acceleratur motus; </s> <s id="N1D8FF"><!-- NEW -->igitur cum impetu acqui&longs;ito in de&longs;­<lb/>cen&longs;u AK, conficiet motu æquabili KY duplum AK per Th.42.l.3. po&longs;ito <lb/>quòd non de&longs;truatur; atque ex his &longs;atis facilè intelligentur, quæcumque <lb/>habes apud Galileum in dialog.3.à propo&longs;itione 3.ad 23. </s> </p> <p id="N1D909" type="main"> <s id="N1D90B">Sextò non probat Galileus, &longs;ed tantùm &longs;upponit mobile ad <expan abbr="eãdem">eandem</expan> <expan abbr="alti-tudin&etilde;">alti­<lb/>tudinem</expan> a&longs;cendere po&longs;&longs;e motu reflexo ex qua de&longs;cendit, quod examinabi­<lb/>mus lib. <!-- REMOVE S--><expan abbr="&longs;equ&etilde;ti">&longs;equenti</expan>, hinc non laborabimus in <expan abbr="examinãdis">examinandis</expan> prop. 24.25.26.27. </s> </p> <p id="N1D923" type="main"> <s id="N1D925"><!-- NEW -->Septimò, cognito tempore, quo percurrit mobile perpendiculum EC <lb/>quod &longs;it diameter circuli; </s> <s id="N1D92B"><!-- NEW -->&longs;ciri pote&longs;t quo tempore percurrat duas chor­<lb/>das &longs;imul EGGC; </s> <s id="N1D931"><!-- NEW -->&longs;it enim Tangens EF, &longs;itque vt FG ad FD, ita FD ad <lb/>FC; </s> <s id="N1D937"><!-- NEW -->cum EG & EC de&longs;cendat æquali tempore per Th.27. cum in G &longs;it <lb/>idem motus, &longs;iue ex E, &longs;iue ex F de&longs;cendat per Th.20. certè &longs;i de&longs;cendit <lb/>per EG dato tempore, quod &longs;it vt EG, de&longs;cendit per GC tempore, quod <lb/>e&longs;t vt GD; igitur tempus, quo de&longs;cendit per EC e&longs;t ad tempus, quo de&longs;­<lb/>cendit per EGC, vt EG ad EGD. </s> </p> <p id="N1D943" type="main"> <s id="N1D945"><!-- NEW -->Ob&longs;eruabis autem GF e&longs;&longs;e ad EF vt EF ad FC; </s> <s id="N1D949"><!-- NEW -->igitur FD e&longs;t media <lb/>inter FC GF, & e&longs;t æqualis FE, igitur anguli FDE.FED æquales; </s> <s id="N1D94F"><!-- NEW -->&longs;ed FD <lb/>E e&longs;t æqualis duobus DCE.DEC, & FEG, e&longs;t æqualis DCE; igitur duo G <lb/>DE DEC &longs;unt æquales. </s> </p> <p id="N1D957" type="main"> <s id="N1D959"><!-- NEW -->Octauò, &longs;i accipiantur æquales horizontalis, & perpendicularis, v.g. <!-- REMOVE S--><lb/>BA AC, ducaturque BC: </s> <s id="N1D960"><!-- NEW -->Dico nullum duci po&longs;&longs;e planum inclinatum à <lb/>puncto B ad perpendiculum AEM, quod breuiori tempore percurratur, <lb/>quàm BC, nec intra angulum vt BR, nec extra vt BM; </s> <s id="N1D968"><!-- NEW -->&longs;it enim vt BC ad <lb/>BI ita BI ad BH, e&longs;t autem BI æqualis BA, igitur &longs;i BA, &longs;it 4.BC e&longs;t v.g. <!-- REMOVE S--><lb/>32. & BH radix q.8.igitur HI e&longs;t ferè I paulò plùs; igitur cum BH percur­<lb/>ratur æquali tempore cum AC, e&longs;t tempus, quo percurritur BH ad tem­<lb/>pus quo percurritur HC vt BH ad HI. <!-- KEEP S--></s> </p> <p id="N1D976" type="main"> <s id="N1D978"><!-- NEW -->Sit autem BR dupla AR, &longs;itque perpendicularis AK in BR; </s> <s id="N1D97C"><!-- NEW -->certè KR <lb/>e&longs;t &longs;ubquadrupla BR; </s> <s id="N1D982"><!-- NEW -->igitur percurritur BL æqualis KR eo tempore quo <lb/>percurritur AR; </s> <s id="N1D988"><!-- NEW -->igitur BL &longs;it ad BV vt BV ad BR; </s> <s id="N1D98C"><!-- NEW -->igitur temporibus æ­<lb/>qualibus percurruntur BL LR; </s> <s id="N1D992"><!-- NEW -->igitur &longs;i tempus quo percurritur BL &longs;it vt <lb/>BH, tempus quo percurretur LR erit etiam vt BH; </s> <s id="N1D998"><!-- NEW -->igitur totum tempus <lb/>quo percurritur tota BR erit vt tota BE, &longs;ed tempus quo percurritur tota <lb/>BC e&longs;t tantum vt BI qu&etail; e&longs;t minor BC; </s> <s id="N1D9A0"><!-- NEW -->igitur BC breuiori tempore per­<lb/>curritur quàm BR; &longs;it <expan abbr="etiã">etiam</expan> vt BP ad BX ita BX ad BM, &longs;i BO e&longs;t 4. OP 2. <lb/>certè BP e&longs;t rad.q. </s> <s id="N1D9AC"><!-- NEW -->12.id e&longs;t ferè 3.1/2 paulò minùs, BM verò e&longs;t dupla BA <lb/>vel BO; </s> <s id="N1D9B2"><!-- NEW -->igitur e&longs;t 8. ducatur ergo 8. in 4. 1/3 productum erit 28. cuius radix <lb/>e&longs;t ferè 5.1/3 paulò minùs; </s> <s id="N1D9B8"><!-- NEW -->igitur BX e&longs;t 5.1/3 paulò minùs; </s> <s id="N1D9BC"><!-- NEW -->cum autem BH <lb/>&longs;it 2.q.8.e&longs;t ferè 2.5/6, paulò minùs; </s> <s id="N1D9C2"><!-- NEW -->igitur &longs;it vt BP 3.1/2 ad BX 5.1/3, ita BH <lb/>2.5/6 ad aliam; </s> <s id="N1D9C8"><!-- NEW -->certè erit 144. id e&longs;t 4.(26/63), licèt minùs acceptum &longs;it; </s> <s id="N1D9CC"><!-- NEW -->igitur <lb/>126.e&longs;t maior BI, quæ e&longs;t tantùm 4; igitur BE breuiori tempore percur­<lb/>ritur, quàm BM. </s> </p> <p id="N1D9D4" type="main"> <s id="N1D9D6"><!-- NEW -->Nonò, per duas chordas quadrantis de&longs;cendit breuiori tempore mo­<lb/>bile, quàm per alteram tantùm inferiorem &longs;cilicet &longs;it enim tantùm <pb pagenum="214" xlink:href="026/01/246.jpg"/>quadrans ABG in quo &longs;int duæ chordæ GC, CB: </s> <s id="N1D9E1"><!-- NEW -->Dico quòd per vtram­<lb/>que ex G breuiori tempore de&longs;cendit, quàm per inferiorem CB; </s> <s id="N1D9E7"><!-- NEW -->quia <lb/>per CB, & GB æquali tempore de&longs;cendit per Th.27.&longs;ed per GCB bre­<lb/>uiori tempore de&longs;cendit, quàm per GB; </s> <s id="N1D9EF"><!-- NEW -->&longs;it enim GD perpendicularis <lb/>parallela AB; </s> <s id="N1D9F5"><!-- NEW -->&longs;it ED perpendicularis in CG, & per 3. puncta GCD <lb/>ducatur circulus: </s> <s id="N1D9FB"><!-- NEW -->his po&longs;itis, GH & GC eodem tempore percurrentur, <lb/>& in C idem erit motus, &longs;iue ex G per GE, &longs;iue ex E per EC de&longs;cen­<lb/>dat mobile per Th.27.& 20. &longs;it autem EB ad EK vt EK ad EC, &longs;itque <lb/>BE v.g, dupla BE vel BA: </s> <s id="N1DA05"><!-- NEW -->dico EK e&longs;&longs;e æqualem BG; </s> <s id="N1DA09"><!-- NEW -->e&longs;t autem BH <lb/>maior BC vel AB, vel HG minor CK; </s> <s id="N1DA0F"><!-- NEW -->&longs;it etiam GH ad GI, ita GI <lb/>ad GB: </s> <s id="N1DA15"><!-- NEW -->dico tempus, quo de&longs;cendit per GCB e&longs;&longs;e ad tempus quo de­<lb/>&longs;cendit per GB vt GCK ad compo&longs;itam ex GC, HI; </s> <s id="N1DA1B"><!-- NEW -->&longs;ed hæc e&longs;t ma­<lb/>ior illa, vt patet ex Geometria, & analytica; </s> <s id="N1DA21"><!-- NEW -->igitur breuiori tempore de­<lb/>&longs;cendit per GCB, quàm per GB; &longs;ed de hoc aliàs. </s> </p> <p id="N1DA27" type="main"> <s id="N1DA29"><!-- NEW -->Sit enim EB 8. dupla &longs;cilicet AB; </s> <s id="N1DA2D"><!-- NEW -->&longs;it autem EE &longs;ubdupla EB ad <lb/>EK vt EK ad EB; </s> <s id="N1DA33"><!-- NEW -->a&longs;&longs;umatur GE, &longs;itque tempus, quo continetur GC. <lb/>vt GC, & quo conficitur BC vt CK; </s> <s id="N1DA39"><!-- NEW -->igitur quo conficitur GCB vt <lb/>GCK: </s> <s id="N1DA3F"><!-- NEW -->&longs;imiliter &longs;it &longs;ecunda linea GB, &longs;itque tempus, quo percurritur <lb/>GH vt GC, vel NO æqualis GC, &longs;itque vt GH ad GN, ita GN ad <lb/>GB certè &longs;i GH decurratur tempore GH, AB decurretur tempore <lb/>HN; </s> <s id="N1DA49"><!-- NEW -->&longs;ed HN maior e&longs;t MB, vel CG, vt con&longs;tat ex analytica; </s> <s id="N1DA4D"><!-- NEW -->adde quod <lb/>in figura prima &longs;it GI ad GM vt GM ad GB; </s> <s id="N1DA53"><!-- NEW -->certè &longs;i tempore GI <lb/>percurratur GI, percurretur GB tempore GM; </s> <s id="N1DA59"><!-- NEW -->e&longs;t autem GM æqua­<lb/>lis AB, vel EC; </s> <s id="N1DA5F"><!-- NEW -->&longs;imiliter &longs;it EC ad EK vt EK ad EB, &longs;i percurratur <lb/>EC tempore EC, percurretur EB tempore EK; </s> <s id="N1DA65"><!-- NEW -->&longs;ed GC percurretur <lb/>tempore GC &longs;ed GCK minor e&longs;t GIM; </s> <s id="N1DA6B"><!-- NEW -->&longs;it enim GM. 4. EK R. <expan abbr="q.">que</expan> <lb/>32. id e&longs;t, 5 7/8 paulò minùs, quibus &longs;i &longs;ubtrahas CE 4. & &longs;ub&longs;tituas CG <lb/>2. paulò plùs habebis 3 7/8; igitur GCK minor e&longs;t GIM. </s> <s id="N1DA77"><!-- NEW -->Ex his habes <lb/>omnes Galilei propo&longs;itiones de motu in planis inclinatis numero 38. in <lb/>quo &longs;tudio, vt verum fatear, maximam &longs;ibi laudem peperit; </s> <s id="N1DA7F"><!-- NEW -->in quo ta­<lb/>men opere duo de&longs;iderari videntur, <expan abbr="alterũ">alterum</expan> à Philo&longs;ophis, quod ita phy&longs;i­<lb/>cæ partes omnes neglexerit, vt ferè vni Geometriæ &longs;atisfaceret; alterum <lb/>ab Geometris quod Geometriam equidem accuratè tractarit. </s> <s id="N1DA8D"><!-- NEW -->Sed minùs <lb/>ad captum Tyronum: atque hæc de his &longs;int &longs;atis, vt tandem no&longs;trorum <lb/>Theorematum &longs;eriem interruptam repetamus. </s> </p> <p id="N1DA95" type="main"> <s id="N1DA97"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N1DAA3" type="main"> <s id="N1DAA5"><!-- NEW --><emph type="italics"/>Ex dictis &longs;equitur pondus centum librarum po&longs;&longs;e habere tantùm grauitatio­<lb/>nem vnius libræ<emph.end type="italics"/>; </s> <s id="N1DAB0"><!-- NEW -->&longs;it enim planum inclinatum centuplum horizontalis, id <lb/>e&longs;t, &longs;ecans centupla Tangentis; haud dubiè grauitatio in prædictum pla­<lb/>num erit tantùm &longs;ubcentupla per Th.16. <!-- KEEP S--></s> </p> <p id="N1DAB9" type="main"> <s id="N1DABB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N1DAC7" type="main"> <s id="N1DAC9"><!-- NEW --><emph type="italics"/>Ex duobus ferentibus idem parallelipedum in &longs;itu inclinato pote&longs;t alter fer­<lb/>re tantùm vnam libram, licèt pendat centum libras<emph.end type="italics"/>; </s> <s id="N1DAD4"><!-- NEW -->&longs;it enim ita inclina-<pb pagenum="215" xlink:href="026/01/247.jpg"/>tum, vt linea inclinationis &longs;it centupla horizontalis oppo&longs;itæ; certè qui <lb/>&longs;u&longs;tinet in altera extremitate eleuata (1/100) tantùm &longs;u&longs;tinet ponderis par­<lb/>tem per Th. 18. alius verò &longs;u&longs;tinet in altera extremitate, quæ deor&longs;um <lb/>e&longs;t (93/100). </s> </p> <p id="N1DAE3" type="main"> <s id="N1DAE5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N1DAF1" type="main"> <s id="N1DAF3"><!-- NEW --><emph type="italics"/>Qui pote&longs;t tantùm datum pondus &longs;ur&longs;um attollere per lineam verticalem, <lb/>centuplum per inclinatum planum ad <expan abbr="eãdem">eandem</expan> altitudinem attollet<emph.end type="italics"/>; </s> <s id="N1DB02"><!-- NEW -->&longs;i enim &longs;it <lb/>inclinata ad perpendiculum in ratione centupla; haud dubiè qui attollit <lb/>datum pondus per ip&longs;um perpendiculum &longs;ine viribus auctis per inclina­<lb/>tum planum, pondus centuplò maius attollet, quia potentia per inclina­<lb/>tam e&longs;t ad potentiam per ip&longs;um perpendiculum vel altitudo ad inclina­<lb/>tam per Theor. <!-- REMOVE S-->6. igitur &longs;i æqualis vtrobique applicetur potentia, pon­<lb/>dus centuplò maius attollet per inclinatam, &longs;eu pellendo, &longs;eu tra­<lb/>hendo. </s> </p> <p id="N1DB16" type="main"> <s id="N1DB18"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N1DB24" type="main"> <s id="N1DB26"><emph type="italics"/>Hinc ratio plani inclinati demon&longs;trat<emph.end type="italics"/> <emph type="italics"/>cochleæ vires.<emph.end type="italics"/> v.g. <!-- REMOVE S-->pellitur &longs;ur&longs;um <lb/>per DE inclinatam faciliùs quàm verticalem DH in ratione DE ad <lb/>DH, quæ &longs;i e&longs;t tripla, eadem potentia quæ datum pondus attollit per <lb/>DH, triplò maius attollet per DE, vel &longs;i attollat per DA verticalem, <lb/>triplò maius attollet per &longs;piras vel Helices DE EC, CF, &c. </s> <s id="N1DB3E">v&longs;que ad <lb/>A; hinc quò Helix erit inclinatior, potentia maius pondus illius operâ <lb/>attollet. </s> </p> <p id="N1DB45" type="main"> <s id="N1DB47"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N1DB53" type="main"> <s id="N1DB55"><!-- NEW --><emph type="italics"/>Hinc clarè vides compen&longs;ari longitudinem motus, &longs;patij vel temporis, pon­<lb/>deris acce&longs;&longs;ione,<emph.end type="italics"/> v.g. <!-- REMOVE S-->triplò maius pondus attollitur per DE quàm per <lb/>DH; </s> <s id="N1DB64"><!-- NEW -->quia &longs;patium DE e&longs;t triplum DH; igitur motus triplus, &longs;cilicet in <lb/>duratione, (loquor enim de motu æquabili quo &longs;ur&longs;um corpus, vel tra­<lb/>hitur, vel continuò pellitur.) </s> </p> <p id="N1DB6C" type="main"> <s id="N1DB6E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N1DB7A" type="main"> <s id="N1DB7C"><!-- NEW --><emph type="italics"/>Hinc nullus mons e&longs;&longs;e pote&longs;t quantumuis arduus, ad cuius apicem via faci­<lb/>li in modum cochleæ &longs;trata pertingi non po&longs;&longs;it<emph.end type="italics"/>; & quò plures erunt &longs;piræ, eo <lb/>facilior erit & minùs decliuis via. </s> </p> <p id="N1DB89" type="main"> <s id="N1DB8B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N1DB97" type="main"> <s id="N1DB99"><!-- NEW --><emph type="italics"/>Quando de&longs;cendit mobile per multas &longs;piras, &longs;eu volutas, pote&longs;t determinari <lb/>altitudo perpendicularis, ex qua eodem tempore de&longs;cenderet<emph.end type="italics"/>; </s> <s id="N1DBA4"><!-- NEW -->&longs;it enim &longs;pira <lb/>&longs;eu cochlea AFCHD, & perpendiculum AD; </s> <s id="N1DBAA"><!-- NEW -->certè eodem tempore <lb/>de&longs;cendit per AFC, quo de&longs;cenderet per AG duplam AF; </s> <s id="N1DBB0"><!-- NEW -->&longs;ed eo tem­<lb/>pore, quo de&longs;cendit per AF inclinatam, conficit AD per Th.27. quæ e&longs;t <lb/>ad AF vt AF ad BA; </s> <s id="N1DBB8"><!-- NEW -->&longs;it autem dupla: </s> <s id="N1DBBC"><!-- NEW -->&longs;imiliter eodem tempore conficit <lb/>AFG vel AFG, quo conficit AE duplam AG; denique eo tempore, <lb/>quo conficit AF CHD, vel AGD, conficit duplam AE. <!-- KEEP S--></s> </p> <pb pagenum="216" xlink:href="026/01/248.jpg"/> <p id="N1DBC9" type="main"> <s id="N1DBCB">Sic etiam eo tempore, quo in perpendiculo conficit AD conficit &longs;ub­<lb/>duplam &longs;cilicet AF, &longs;ed hæc &longs;unt clara. </s> </p> <p id="N1DBD0" type="main"> <s id="N1DBD2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N1DBDE" type="main"> <s id="N1DBE0"><!-- NEW --><emph type="italics"/>Quando proiicitur mobile per planum inclinatum &longs;ur&longs;um in ea proportione <lb/>proiicitur longiùs, quò inclinata ip&longs;a longior e&longs;t perpendiculari.<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;i proii­<lb/>citur per BA in verticali, illa eadem <expan abbr="pot&etilde;tia">potentia</expan> quæ proiicit in A ex B, pro­<lb/>iiciet <expan abbr="quoq;">quoque</expan> ex F in A, ex M in A, atque ita deinceps ex &longs;ingulis punctis <lb/>horizontalis BM; </s> <s id="N1DBFB"><!-- NEW -->ratio e&longs;t, quia in ea proportione de&longs;truitur impetus <lb/>per BA, in qua motus per AB de&longs;cendit; </s> <s id="N1DC01"><!-- NEW -->nam impetus innatus deor­<lb/>&longs;um qua&longs;i trahit mobile graue; </s> <s id="N1DC07"><!-- NEW -->impetus verò impre&longs;&longs;us &longs;ur&longs;um attollit; </s> <s id="N1DC0B"><!-- NEW --><lb/>igitur pugnant pro rata, vt &longs;æpè diximus in tertio libro, & alibi: </s> <s id="N1DC10"><!-- NEW -->&longs;imiliter <lb/>in inclinata FA impetus innatus qua&longs;i reducit mobile deor&longs;um dum <lb/>impre&longs;&longs;us violentus &longs;ur&longs;um promouet; </s> <s id="N1DC18"><!-- NEW -->igitur &longs;i impetus innatus per AB, <lb/>& per AT æqualem vim haberet, haud dubiè æquale &longs;patium contine­<lb/>ret mobile projectum per BA & FA; </s> <s id="N1DC20"><!-- NEW -->nam eadem potentia cum æquali <lb/>re&longs;i&longs;tentia idem præ&longs;tat & inæqualiter de&longs;cendit per AB AF, & motus <lb/>per AF e&longs;t ad motum per AB, vt AB ad AF. v.g. <!-- REMOVE S-->&longs;ubduplus; </s> <s id="N1DC2A"><!-- NEW -->igitur re­<lb/>&longs;i&longs;tentia per BA erit dupla re&longs;i&longs;tentiæ per FA; </s> <s id="N1DC30"><!-- NEW -->igitur &longs;patium per FA <lb/>erit duplum; </s> <s id="N1DC36"><!-- NEW -->igitur ex F a&longs;cendet in A, quo cum eo impetu ex B a&longs;cendet <lb/>in A, &longs;uppo&longs;ita eadem potentia; </s> <s id="N1DC3C"><!-- NEW -->idem etiam dicendum de aliis punctis <lb/>horizontalis BM: </s> <s id="N1DC42"><!-- NEW -->præterea ille impetus &longs;ufficit ad motum &longs;ur&longs;um per <lb/>FA, qui accipitur in de&longs;cen&longs;u AF, vt con&longs;tat ex dictis; </s> <s id="N1DC48"><!-- NEW -->itemque &longs;ufficit <lb/>ad motum &longs;ur&longs;um per BA qui acquiritur in de&longs;cen&longs;u AB; &longs;ed æqualis ve­<lb/>locitas, vel impetus acquiritur in vtroque de&longs;cen&longs;u AB AF per Th. 20. <lb/>igitur idem impetus &longs;ufficit ad de&longs;cen&longs;um BA FA. </s> </p> <p id="N1DC52" type="main"> <s id="N1DC54"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N1DC60" type="main"> <s id="N1DC62"><!-- NEW --><emph type="italics"/>Hinc dicendum e&longs;t impetum naturalem per inclinatam FA vel MA non <lb/>&longs;ur&longs;um intendi, &longs;eu cre&longs;cere<emph.end type="italics"/>; </s> <s id="N1DC6D"><!-- NEW -->alioqui ex A mobile de&longs;cenderet citiùs in F, <lb/>po&longs;tquàm ex F proiectum e&longs;&longs;et in A, quàm &longs;i tantùm ex A in F demit­<lb/>teretur, quod e&longs;t contra experientiam; adde quòd impetus naturalis &longs;ur­<lb/>&longs;um non cre&longs;cit, vt iam &longs;æpè dictum e&longs;t. </s> </p> <p id="N1DC77" type="main"> <s id="N1DC79"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N1DC85" type="main"> <s id="N1DC87"><!-- NEW --><emph type="italics"/>Destruitur aliquid impetus impre&longs;&longs;i in mobili per planum inclinatum.<emph.end type="italics"/><lb/>Probatur, quia tandem quie&longs;cit mobile; </s> <s id="N1DC91"><!-- NEW -->igitur ce&longs;&longs;at motus; </s> <s id="N1DC95"><!-- NEW -->igitur & im­<lb/>petus: </s> <s id="N1DC9B"><!-- NEW -->nec dicas id fieri ab aëre, vel plani &longs;cabritie; </s> <s id="N1DC9F"><!-- NEW -->nam, &longs;i hoc e&longs;&longs;et, <lb/>æquale &longs;patium conficeret in FA & LA; </s> <s id="N1DCA5"><!-- NEW -->quippe æqualis portio plani <lb/>æqualiter re&longs;i&longs;tit; Idem dico de aëre; igitur de&longs;truitur impetus impre&longs;­<lb/>&longs;us ab impetu naturali. </s> </p> <p id="N1DCAD" type="main"> <s id="N1DCAF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N1DCBB" type="main"> <s id="N1DCBD"><!-- NEW --><emph type="italics"/>Destruitur tantùm pro rata, hoc e&longs;t in ratione, quam habet perpendiculum <lb/>ad inclinatam.<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;it perpendiculum FCA; </s> <s id="N1DCCA"><!-- NEW -->haud dubiè &longs;i non de&longs;true­<lb/>retur motus &longs;ur&longs;um cum eo gradu impetus, quo ex F a&longs;cendit in C motu <lb/>retardato, a&longs;cenderet in A motu æquabili, & eodem tempore; </s> <s id="N1DCD2"><!-- NEW -->igitur eo <pb pagenum="217" xlink:href="026/01/249.jpg"/>tempore de&longs;truitur totus impetus; </s> <s id="N1DCDB"><!-- NEW -->&longs;i verò proiiciatur per LC; </s> <s id="N1DCDF"><!-- NEW -->certè im­<lb/>petus totus non de&longs;truitur per LC, eo tempore, quo ex F a&longs;cenderet in <lb/>C, &longs;ed pro rata, id e&longs;t in ratione FC ad LC, quæ &longs;it &longs;ubdupla v.g. <!-- REMOVE S-->igitur <lb/>impetus de&longs;truitur tantùm &longs;ubduplus; </s> <s id="N1DCEB"><!-- NEW -->igitur eo tempore, quo ex F a&longs;cen­<lb/>dit in C, ex L a&longs;cendet in K, ita vt LM æquali FC addatur MK æqua­<lb/>lis EB; e&longs;t autem EB &longs;ubdupla CA vel EF. </s> <s id="N1DCF3"><!-- NEW -->Similiter &longs;it perpendicu­<lb/>lum FG, & inclinata HF tripla FG; </s> <s id="N1DCF9"><!-- NEW -->a&longs;&longs;umatur FC æqualis FG, item­<lb/>que HO æqualis GF; </s> <s id="N1DCFF"><!-- NEW -->certè eo tempore, quo perpendiculari detrahitur <lb/>totus impetus, detrahitur tantùm &longs;ubtriplum per inclinatam HF; </s> <s id="N1DD05"><!-- NEW -->igitur <lb/>a&longs;&longs;umatur ER &longs;ubtripla EF; </s> <s id="N1DD0B"><!-- NEW -->& addatur OP æqualis FR: </s> <s id="N1DD0F"><!-- NEW -->dico quod eo <lb/>tempore, quo ex G a&longs;cendit in F, ex H a&longs;cendit in P; </s> <s id="N1DD15"><!-- NEW -->quippe a&longs;cenderet <lb/>in O, &longs;i eo tempore totus impetus de&longs;trueretur, & in S &longs;i nullus; </s> <s id="N1DD1B"><!-- NEW -->igitur <lb/>in P, &longs;i &longs;ubtriplus tantùm de&longs;truatur, de&longs;truitur porrò &longs;ubtriplus, quia vis <lb/>impetus innati per FH e&longs;t tantùm &longs;ubtripla eiu&longs;dem per FG; </s> <s id="N1DD23"><!-- NEW -->atqui de­<lb/>&longs;truitur tantùm ab impetu innato, quæ omnia certi&longs;&longs;imè con&longs;tant; Ex <lb/>quo habes tempora e&longs;&longs;e vt lineas. </s> </p> <p id="N1DD2B" type="main"> <s id="N1DD2D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N1DD39" type="main"> <s id="N1DD3B"><!-- NEW --><emph type="italics"/>Hinc pote&longs;t dici quo tempore conficiatur tota inclinata &longs;ur&longs;um &longs;cilicet eo <lb/>tempore quo inclinata deor&longs;um percurritur.<emph.end type="italics"/> v.g, CL dupla CF percurritur <lb/>tempore duplo illius, quo percurritur CF; </s> <s id="N1DD48"><!-- NEW -->igitur mobile proiectum ex <lb/>L in C percurrit LC eodem tempore a&longs;cendendo, quo percurrit EL de­<lb/>&longs;cendendo; &longs;ed percurrit EL de&longs;cendendo eodem tempore, quo percur­<lb/>rit perpendicularem quadruplam CF, vt &longs;uprà diximus. </s> </p> <p id="N1DD52" type="main"> <s id="N1DD54"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N1DD60" type="main"> <s id="N1DD62"><!-- NEW --><emph type="italics"/>Hinc nunquam in inclinata &longs;ur&longs;um proiectum mobile acquirit duplum &longs;pa­<lb/>tium illius quod acquirit idem proiectum in verticali &longs;ur&longs;um,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->ex H pro­<lb/>iectum nunquam acquiret in HF duplum &longs;patium GF, po&longs;ito quòd ex <lb/>G proiiciatur tantùm in F dato tempore, &longs;itque eadem potentia per HF. <lb/>Probatur, quia &longs;emper de&longs;truitur aliquid impetus iuxta proportionem <lb/>FG ad FH per Th.40. &longs;ed &longs;i nullus de&longs;truitur impetus, duplum &longs;patium <lb/>conficit; </s> <s id="N1DD7B"><!-- NEW -->igitur &longs;i aliquid de&longs;truitur, duplum &longs;patium non conficitur: po­<lb/>te&longs;t tamen propiùs in infinitum ad duplum accedere. </s> </p> <p id="N1DD81" type="main"> <s id="N1DD83"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N1DD8F" type="main"> <s id="N1DD91"><!-- NEW --><emph type="italics"/>Hinc erecta perpendiculari<emph.end type="italics"/> FC, <emph type="italics"/>ductaque horizontali<emph.end type="italics"/> FL, <emph type="italics"/>productaque <lb/>in infinitum, &longs;i ex quolibet illius puncto eleuetur planum inclinatum termina­<lb/>tum ad<emph.end type="italics"/> C, <emph type="italics"/>eadem potentia que ex<emph.end type="italics"/> F <emph type="italics"/>in<emph.end type="italics"/> C <emph type="italics"/>mobile proiiciet, etiam ex quolibet <lb/>puncto de&longs;ignato in horizontali proiiciet in<emph.end type="italics"/> C <emph type="italics"/>per planum inclinatum<emph.end type="italics"/>; quod <lb/>probatur per Th. 38. </s> </p> <p id="N1DDC6" type="main"> <s id="N1DDC8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N1DDD4" type="main"> <s id="N1DDD6"><!-- NEW --><emph type="italics"/>Ex his etiam probatur proiici ex<emph.end type="italics"/> L <emph type="italics"/>in<emph.end type="italics"/> C <emph type="italics"/>ab ea potentia, quæ ex<emph.end type="italics"/> F <emph type="italics"/>proiicit in<emph.end type="italics"/><lb/>C; </s> <s id="N1DDF2"><!-- NEW -->cum enim primo tempore proiiciat ex L in K (&longs;uppono enim LC <lb/>e&longs;&longs;e quadruplam KC) certè &longs;ecundo conficit tantùm KC; </s> <s id="N1DDF8"><!-- NEW -->e&longs;t enim mo­<lb/>tus violentus &longs;ur&longs;um retardatus inuer&longs;us motus deor&longs;um accelerati; </s> <s id="N1DDFE"><!-- NEW -->at-<pb pagenum="218" xlink:href="026/01/250.jpg"/>qui motu naturaliter accelerato &longs;i primo tempore conficit KC, &longs;ecun­<lb/>do conficit KL triplum CK; igitur &longs;i motu retardato primo tempore <lb/>conficit LK, &longs;ecundo conficit KC &longs;ubtriplum LK. <!-- KEEP S--></s> </p> <p id="N1DE0C" type="main"> <s id="N1DE0E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N1DE1A" type="main"> <s id="N1DE1C"><!-- NEW --><emph type="italics"/>Si proiiciatur in horizontali motus per &longs;e e&longs;t æqualis in &longs;patio modico<emph.end type="italics"/>: </s> <s id="N1DE25"><!-- NEW -->Pro­<lb/>batur, quia in nulla proportione de&longs;truitur, vt patet; </s> <s id="N1DE2B"><!-- NEW -->dixi per &longs;e, quia re­<lb/>uera nullum e&longs;t planum perfectè l&etail;uigatum, nec etiam mobile: </s> <s id="N1DE31"><!-- NEW -->vnde cum <lb/>a&longs;peritas plani re&longs;i&longs;tat, inde maximè motus retardatur; dixi in &longs;patio <lb/>modico, nam planum horizontale rectilineum longius, e&longs;t planum incli­<lb/>natum, de quo infrà, vnde vt motus &longs;it æqualis, debet proiici in &longs;uperfi­<lb/>cie curua æqualiter di&longs;tante à centro mundi. </s> </p> <p id="N1DE3D" type="main"> <s id="N1DE3F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N1DE4B" type="main"> <s id="N1DE4D"><!-- NEW --><emph type="italics"/>Si proiiciatur mobile deor&longs;um per inclinatum planum, mouetur velociùs<emph.end type="italics"/> B; <lb/>certum e&longs;t, & acquirit maius &longs;patium &longs;ingulis temporibus iuxta ratio­<lb/>nem impetus accepti. </s> <s id="N1DE5A"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it planum ABE, in quo primo dato tem­<lb/>pore mobile acquirat AB, &longs;itque impetus impre&longs;&longs;us æqualis împetui, <lb/>quem acquirit dum percurrit &longs;patium AB; </s> <s id="N1DE64"><!-- NEW -->haud dubiè primo tempore <lb/>ratione vtriu&longs;que impetus percurrit AC, &longs;cilicet, duo &longs;patia; </s> <s id="N1DE6A"><!-- NEW -->&longs;ecundo <lb/>CD, id e&longs;t 4. &longs;patia; </s> <s id="N1DE70"><!-- NEW -->tertio DE, id e&longs;t 6. &longs;patia; atque ita deinceps: vn­<lb/>de vides proportionem arithmeticam, quæ na&longs;citur ex acce&longs;&longs;ione quan­<lb/>tumuis modica noui impetus. </s> </p> <p id="N1DE78" type="main"> <s id="N1DE7A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N1DE86" type="main"> <s id="N1DE88"><!-- NEW --><emph type="italics"/>In plano inclinato non de&longs;truitur impetus impre&longs;&longs;us, quia non e&longs;t frustrà<emph.end type="italics"/>; <lb/>igitur non de&longs;truitur per Sch. <!-- REMOVE S-->Th.152.lib.1. &longs;ic diximus in Theoremate <lb/>68. l.4. in proiecto deor&longs;um per lineam perpendicularem deor&longs;um non <lb/>de&longs;trui quidquam impetus impre&longs;&longs;i, licèt de&longs;truatur in proiecto per in­<lb/>clinatam deor&longs;um in libero medio, vt diximus in Th.67. lib.4. vide Th. <!-- REMOVE S--><lb/>68.lib.4. </s> </p> <p id="N1DE9E" type="main"> <s id="N1DEA0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N1DEAC" type="main"> <s id="N1DEAE"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari quantus impetus imprimi debeat mobili per planum in­<lb/>clinatum, vt æquali velocitate moueatur quo mouetur in perpendiculari &longs;uæ <lb/>&longs;ponte,<emph.end type="italics"/> hoc e&longs;t vt æquali tempore æquale &longs;patium vtrimque acquiratur, <lb/>a&longs;&longs;umpto &longs;cilicet &longs;patio totali, quod toti motui competit, non verò eius <lb/>tantùm parte; debet enim a&longs;&longs;umi impetus iuxta proportionem differen­<lb/>tiæ &longs;patij, quod acquiritur in perpendiculari, & alterius &longs;patij, quod ac­<lb/>quiritur in perpendiculari, & alterius &longs;patij, quod acquiritur in inclina­<lb/>ta. </s> <s id="N1DEC5"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it planum inclinatum AH, perpendiculum verò AE; </s> <s id="N1DECB"><!-- NEW -->ducatur <lb/>EB perpendicularis in AH, mobile percurrit AB in inclinata eo tem­<lb/>pore, quo percurrit AE in perpendiculo; </s> <s id="N1DED3"><!-- NEW -->a&longs;&longs;umatur AC æqualis AE; </s> <s id="N1DED7"><!-- NEW --><lb/>&longs;i imprimatur impetus, qui &longs;it ad acqui&longs;itum in &longs;patio AB vt BC ad AB: </s> <s id="N1DEDC"><!-- NEW --><lb/>dico quod mobile eodem tempore percurret AE, & AC, vt con&longs;tat; </s> <s id="N1DEE1"><!-- NEW --><lb/>quia impetus in C e&longs;t æqualis impetui in E; </s> <s id="N1DEE6"><!-- NEW -->vt verò percurrat in incli­<lb/>nata AH æquale &longs;patium AG, æquali tempore, quo percurrit AG; </s> <s id="N1DEEC"><!-- NEW -->a&longs;-<pb pagenum="219" xlink:href="026/01/251.jpg"/>&longs;umatur AF æqualis AH, addaturque impetus, qui &longs;it ad acqui&longs;itum in <lb/>H, vt GF ad FA, vel AH, & habebitur intentum: </s> <s id="N1DEF7"><!-- NEW -->dixi totum &longs;patium re­<lb/>&longs;pondens &longs;cilicet toti motui; </s> <s id="N1DEFD"><!-- NEW -->alioqui &longs;i pars tantùm accipiatur tùm &longs;pa­<lb/>tij, tùm motus, res procul dubio &longs;ecus accidet; &longs;it enim impetus impre&longs;­<lb/>&longs;us vt BC ad AB. <!-- KEEP S--></s> <s id="N1DF06"><!-- NEW -->Equidem primò tempore, quo in perpendiculari con­<lb/>citur AE, conficitur AC æqualis; </s> <s id="N1DF0C"><!-- NEW -->at verò &longs;ecundo, quo conficitur EG <lb/>triplum AE in perpendiculari, conficitur CI quadruplum AC, vel <lb/>AE; </s> <s id="N1DF14"><!-- NEW -->igitur non &longs;unt æqualia &longs;patia; &longs;ed hæc &longs;unt &longs;atis facilia. </s> </p> <p id="N1DF18" type="main"> <s id="N1DF1A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s> </p> <p id="N1DF26" type="main"> <s id="N1DF28"><!-- NEW --><emph type="italics"/>Si planum horizontale &longs;it perfectè læuigatum in vne tantùm illius puncto &longs;i­<lb/>&longs;tere pote&longs;t mobile graue<emph.end type="italics"/>; </s> <s id="N1DF33"><!-- NEW -->&longs;it enim globus terræ centro A &longs;emidiametro <lb/>AE; </s> <s id="N1DF39"><!-- NEW -->&longs;itque planum horizontale FEGN læuigati&longs;&longs;imum: dico quòd in <lb/>puncto contactus E quie&longs;cet mobile. </s> <s id="N1DF3F"><!-- NEW -->Probatur, quia ex omni alio puncto <lb/>mobile pote&longs;t de&longs;cendere; </s> <s id="N1DF45"><!-- NEW -->&longs;it enim in G. v.g. <!-- REMOVE S-->haud dubiè GA maior e&longs;t <lb/>AE; </s> <s id="N1DF4D"><!-- NEW -->igitur GE planum e&longs;t inclinatum, id e&longs;t, E propiùs accedet ad cen­<lb/>trum terræ A; &longs;ed per planum inclinatum mobile de&longs;cendit per hyp. </s> <s id="N1DF53"><!-- NEW -->1. <lb/>idem dico de omni alio plani puncto, excepto puncto E, ex quo non <lb/>pote&longs;t moueri, ni&longs;i a&longs;cendat, id e&longs;t à centro A recedat; igitur in eo <lb/>quie&longs;cet. </s> </p> <p id="N1DF5D" type="main"> <s id="N1DF5F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N1DF6B" type="main"> <s id="N1DF6D"><emph type="italics"/>Hinc in men&longs;a lauigati&longs;&longs;ima globus vel eburneus, vel cry&longs;tallinus vix vn­<lb/>quam &longs;istit, ni&longs;i in eius centro,<emph.end type="italics"/> quod multis experimentis comprobatum <lb/>e&longs;t, & ratio luce meridianâ clarior à rudioribus etiam primo &longs;tatim ob­<lb/>tutu cernitur. </s> </p> <p id="N1DF7B" type="main"> <s id="N1DF7D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N1DF89" type="main"> <s id="N1DF8B"><!-- NEW --><emph type="italics"/>Hinc ridiculum &longs;eu joculare paradoxon, quo &longs;cilicet dici pote&longs;t duorum alter <lb/>in eodem plano a&longs;cendere, alter de&longs;cendere, licèt in <expan abbr="eãdem">eandem</expan> cœli plagam con­<lb/>uer&longs;i ambulent<emph.end type="italics"/>; </s> <s id="N1DF9C"><!-- NEW -->&longs;i enim alter ex G in E; </s> <s id="N1DFA0"><!-- NEW -->alter verò ex E in F tenderet; </s> <s id="N1DFA4"><!-- NEW -->hic <lb/>certè a&longs;cenderet, quia recederet à terræ centro A; </s> <s id="N1DFAA"><!-- NEW -->ille verò de&longs;cende­<lb/>ret, quia ad centrum accederet; & &longs;i in partes oppo&longs;itas ambulent, in <lb/>hoc eodem plano vterque &longs;imul a&longs;cendere, vel &longs;imul de&longs;cendere pote&longs;t. </s> </p> <p id="N1DFB2" type="main"> <s id="N1DFB4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N1DFC0" type="main"> <s id="N1DFC2"><!-- NEW --><emph type="italics"/>E&longs;t etiam aliud paradoxon, &longs;cilicet in eodem puncto E duo plana eadem li­<lb/>neâ contenta hinc inde a&longs;cendere; </s> <s id="N1DFCA"><!-- NEW -->vel duos montes alti&longs;&longs;imos in eadem recta <lb/>linea contineri; </s> <s id="N1DFD0"><!-- NEW -->vel mediam vallem, & gemines montes linea recti&longs;&longs;ima &longs;imul <lb/>connecti<emph.end type="italics"/>; hæc porrò &longs;unt &longs;atis facilia, & vix &longs;upra vulgi captum. </s> </p> <p id="N1DFD9" type="main"> <s id="N1DFDB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N1DFE7" type="main"> <s id="N1DFE9"><!-- NEW --><emph type="italics"/>Adde aliud paradoxon &longs;cilicet idem mobile per duo plana parallela inæ­<lb/>quali motu de&longs;cendere.<emph.end type="italics"/> v.g. <!-- REMOVE S-->per plana XFB, VEA, nam VEA e&longs;t per­<lb/>pendiculum; at verò XFB e&longs;t horizontale, vt clarum e&longs;t. </s> </p> <p id="N1DFF8" type="main"> <s id="N1DFFA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N1E006" type="main"> <s id="N1E008"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari motus proportio cuiu&longs;libet puncti a&longs;&longs;ignati in plano EN<emph.end type="italics"/>; </s> <s id="N1E011"><!-- NEW --><pb pagenum="220" xlink:href="026/01/252.jpg"/>&longs;it enim punctum G; </s> <s id="N1E019"><!-- NEW -->ducatur à centro A recta AGH; </s> <s id="N1E01D"><!-- NEW -->haud dubiè e&longs;t per­<lb/>pendicularis; </s> <s id="N1E023"><!-- NEW -->ducatur IGK &longs;ecans GH; ad angulos rectos; </s> <s id="N1E027"><!-- NEW -->hæc e&longs;t ho­<lb/>rizontalis, quæ ad hanc perpendicularem pertinet; </s> <s id="N1E02D"><!-- NEW -->ducatur HI parallela <lb/>EG; </s> <s id="N1E033"><!-- NEW -->hæc e&longs;t inclinata, vt patet ex dictis; immò per ip&longs;am deff. </s> <s id="N1E037">1. &longs;ed mo­<lb/>tus in inclinata e&longs;t vt ip&longs;um perpendiculum ad inclinatam per Th. 6. <lb/>igitur motus per HI in ip&longs;o puncto H, vel per GE in ip&longs;o puncto G e&longs;t <lb/>ad motum per HG, vt HG ad HI. <!-- KEEP S--></s> </p> <p id="N1E041" type="main"> <s id="N1E043"><!-- NEW -->Aliter ducatur HZ perpendicularis IH; </s> <s id="N1E047"><!-- NEW -->dico motum in G vel ex G <lb/>initio e&longs;&longs;e ad motum per VE vel GL vt GH ad GZ; &longs;unt enim duo <lb/>triangula IGH, ZGH proportionalia. </s> </p> <p id="N1E04F" type="main"> <s id="N1E051"><!-- NEW -->Aliter ducatur LK parallela GG; </s> <s id="N1E055"><!-- NEW -->triangula GKL, GHI &longs;unt propor­<lb/>tionalia; igitur motus per GE e&longs;t ad motum per HG, vt LG ad LK. <!-- KEEP S--></s> </p> <p id="N1E05C" type="main"> <s id="N1E05E"><!-- NEW -->Aliter ducatur QL, triangula QLA, LGK &longs;unt proportionalia; </s> <s id="N1E062"><!-- NEW -->igi­<lb/>tur motus per GE e&longs;t ad motum per HG vt QL ad AL; igitur vt &longs;inus <lb/>rectus anguli QAL ad totum. </s> <s id="N1E06A">Idem dico de puncto O, & omnibus alia <lb/>in quibus e&longs;t eadem praxis. </s> </p> <p id="N1E06F" type="main"> <s id="N1E071"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N1E07D" type="main"> <s id="N1E07F"><!-- NEW --><emph type="italics"/>In &longs;ingulis punctis plani EN e&longs;t diuer&longs;us motus<emph.end type="italics"/>; </s> <s id="N1E088"><!-- NEW -->nam in puncto E nullus <lb/>e&longs;t motus per Th. 50.atqui in puncto G e&longs;t motus; </s> <s id="N1E08E"><!-- NEW -->idem dico de puncto <lb/>O, atqui in puncto O e&longs;t maior motus, quàm in G, &longs;cilicet initio, id e&longs;t <lb/>velocior incipit motus in O, quàm in G; </s> <s id="N1E096"><!-- NEW -->probatur quia in G e&longs;t ad mo­<lb/>tum maximum qui fit in perpendiculari vt QL ad LA, & in puncto O <lb/>vt YP ad PA, &longs;ed YP e&longs;t maior QL, vt con&longs;tat; </s> <s id="N1E09E"><!-- NEW -->igitur initio e&longs;t maior <lb/>motus in O quàm in G; igitur quâ proportione horizontalis EN erit <lb/>longior, puncta, quæ longiùs di&longs;tabunt, habebunt rationem plani ma­<lb/>gis inclinati. </s> </p> <p id="N1E0A8" type="main"> <s id="N1E0AA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N1E0B6" type="main"> <s id="N1E0B8"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari grauitatio in &longs;ingulis punctis plani EN<emph.end type="italics"/>; </s> <s id="N1E0C1"><!-- NEW -->cum enim <lb/>grauitatio in plano inclinato &longs;it ad grauitationem in horizontali vt <lb/>Tangens ad &longs;ecantem, vel vt horizontalis, in quam &longs;cilicet cadit perpen­<lb/>lum ad inclinatam per Th. 16. &longs;it punctum, G grauitatio in eo puncto <lb/>e&longs;t ad grauitationem in puncto E, vt QA ad AL, & in puncto O ve YA <lb/>ad AP: idem dico de aliis punctis. </s> </p> <p id="N1E0CF" type="main"> <s id="N1E0D1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s> </p> <p id="N1E0DD" type="main"> <s id="N1E0DF"><!-- NEW --><emph type="italics"/>Hinc eò minor e&longs;t grauitatio, quò maior e&longs;t di&longs;tantia ab E<emph.end type="italics"/>; </s> <s id="N1E0E8"><!-- NEW -->atque ita ab E <lb/>ver&longs;us N cre&longs;cit motus, & decre&longs;cit grauitatio; at verò ab N ver&longs;us B <lb/>cre&longs;cit grauitatio, & decre&longs;cit motus. </s> </p> <p id="N1E0F0" type="main"> <s id="N1E0F2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N1E0FE" type="main"> <s id="N1E100"><emph type="italics"/>Globus ab O ver&longs;us E rotatus &longs;emper acceleraret &longs;uum motum.<emph.end type="italics"/></s> <s id="N1E107"><!-- NEW --> Demon­<lb/>&longs;tro, quia impetus productus in O con&longs;eruaretur etiam in G, & nouus <lb/>produceretur, igitur acceleraret &longs;uum motum; </s> <s id="N1E10F"><!-- NEW -->&longs;uppono enim planum E <lb/>N e&longs;&longs;e læuigati&longs;&longs;imum; </s> <s id="N1E115"><!-- NEW -->igitur nihil e&longs;&longs;et, à quo de&longs;trueretur: </s> <s id="N1E119"><!-- NEW -->adde quòd <pb pagenum="221" xlink:href="026/01/253.jpg"/>&longs;emper haberet &longs;uum effectum; </s> <s id="N1E122"><!-- NEW -->igitur non e&longs;&longs;et fru&longs;trà; igitur per Schol. <!-- REMOVE S--><lb/>Th.152.l.1. </s> </p> <p id="N1E129" type="main"> <s id="N1E12B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N1E137" type="main"> <s id="N1E139"><!-- NEW --><emph type="italics"/>Ille motus acceleratur per partes inæquales<emph.end type="italics"/>; </s> <s id="N1E142"><!-- NEW -->quia &longs;cilicet motus additus <lb/>in O minor e&longs;&longs;et quàm in N, & in G quàm in O per Th. 56. igitur per <lb/>partes inæquales acceleraretur, immò pote&longs;t determinari proportio cre­<lb/>menti motus in &longs;ingulis; </s> <s id="N1E14C"><!-- NEW -->cum enim in O &longs;it vt YP, in QL. in Yvt T <foreign lang="greek">d</foreign><lb/>ad AC; certè cre&longs;cit in proportione &longs;inuum rectorum ad &longs;inum totum. </s> </p> <p id="N1E155" type="main"> <s id="N1E157"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N1E163" type="main"> <s id="N1E165"><!-- NEW --><emph type="italics"/>Mobile de&longs;cendens ex O in E tran&longs;it per tot plana inclinata diuer&longs;a, quot <lb/>&longs;unt puncta in tota EO vt con&longs;tat, vel potiùs quot po&longs;&longs;unt duci Tangentes di­<lb/>uer&longs;æ in toto arcu PE<emph.end type="italics"/>; quippe Tangens puncti P e&longs;&longs;et parallela IG, idem <lb/>dico de omnibus aliis punctis arcus PE. </s> </p> <p id="N1E174" type="main"> <s id="N1E176"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N1E182" type="main"> <s id="N1E184"><!-- NEW --><emph type="italics"/>Motus funependuli in quolibet puncto arcus, per quem de&longs;cendit, e&longs;t ad mo­<lb/>tum in perpendiculari, vt &longs;inus re&longs;idui arcus ad &longs;emidiametrum<emph.end type="italics"/>; </s> <s id="N1E18F"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it fune­<lb/>pendulum AD in perpendiculari, quod vibrari po&longs;&longs;it circa punctum im­<lb/>mobile A, eleuetur in A<foreign lang="greek">b</foreign>, ducatur Tangens <foreign lang="greek">b</foreign> V motus funependiculi in <lb/>puncto <foreign lang="greek">b</foreign> &longs;cilicet initio, idem e&longs;t, qui e&longs;&longs;et in plano inclinato <foreign lang="greek">b</foreign>V vt patet, <lb/>atqui motus in inclinato plano <foreign lang="greek">b</foreign> V e&longs;t ad motum in <expan abbr="perp&etilde;diculari">perpendiculari</expan> vt <foreign lang="greek">a</foreign> V. <lb/>ad <foreign lang="greek">b</foreign> V, &longs;ed <foreign lang="greek">a</foreign>V e&longs;t ad <foreign lang="greek">b</foreign>V vt <foreign lang="greek">ab</foreign> ad A<foreign lang="greek">b</foreign>, &longs;unt enim triangula proportionalia; <lb/>igitur motus initio &longs;cilicet in puncto arcus putà B e&longs;t ad motum in per­<lb/>pendiculari etiam initio con&longs;ideratum, vt &longs;inus rectus re&longs;idui arcus, putà <lb/><foreign lang="greek">b</foreign> D ad &longs;emidiametrum, vel &longs;inum totum, id e&longs;t <foreign lang="greek">a b</foreign> ad A <foreign lang="greek">b</foreign>, idem dico de <lb/>omnibus aliis punctis. </s> </p> <p id="N1E1E2" type="main"> <s id="N1E1E4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s> </p> <p id="N1E1F0" type="main"> <s id="N1E1F2"><!-- NEW --><emph type="italics"/>Hinc proportio accelerationis motus in de&longs;cen&longs;u funependuli &longs;eu incremen­<lb/>ti in &longs;ingulis punctis additi e&longs;t in proportione huiu&longs;modi &longs;inuum minorum &longs;em­<lb/>per & minorum<emph.end type="italics"/>; v.g. <!-- REMOVE S-->motus in puncto B e&longs;t vt BA &longs;emidiameter in <foreign lang="greek">t</foreign> vt <foreign lang="greek">t</foreign><lb/><foreign lang="greek">m</foreign> in <foreign lang="greek">b</foreign> vt <foreign lang="greek">b a</foreign>, id e&longs;t licèt maior &longs;it motus in <foreign lang="greek">t</foreign> quàm in B, cum &longs;cilicet <lb/>de&longs;cendit ex B in <foreign lang="greek">t</foreign>, vt illa portio crementi quæ in ip&longs;o puncto <foreign lang="greek">t</foreign> addi­<lb/>tur e&longs;t ad primam in B vt <foreign lang="greek">t m</foreign> ad BA. </s> </p> <p id="N1E229" type="main"> <s id="N1E22B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s> </p> <p id="N1E237" type="main"> <s id="N1E239"><!-- NEW --><emph type="italics"/>Hinc velocitas acqui&longs;ita in arcu BT e&longs;t ad acqui&longs;itam in arcu B <foreign lang="greek">b</foreign>, vt <lb/>omnes &longs;inus eiu&longs;dem arcus B <foreign lang="greek">t</foreign> ad omnes &longs;inus arcus B <foreign lang="greek">b</foreign>, & hæc ad acqui&longs;i­<lb/>tum in toto quadrante BD, vt hi ad omnes &longs;inus quadrantis<emph.end type="italics"/>; </s> <s id="N1E252"><!-- NEW -->&longs;imiliter pote&longs;t <lb/>comparari acqui&longs;ita tantùm in arcu BT, cum acqui&longs;ita in arcu <foreign lang="greek">t b</foreign> vel <foreign lang="greek">b</foreign><lb/>D, quod probatur; quia motus, qui re&longs;pondet &longs;ingulis punctis arcus initio <lb/>e&longs;t in proportione &longs;inuum &longs;eu tran&longs;uer&longs;arum BA, <foreign lang="greek">t m, b a</foreign>, &c. </s> <s id="N1E267"><!-- NEW -->igitur &longs;i <lb/>à &longs;ingulis punctis arcus quadrantis in rectam lineam compo&longs;iti duce­<lb/>rentur; </s> <s id="N1E26F"><!-- NEW -->haùd dubiè prædictam aream qua&longs;i occupabunt; igitur acqui&longs;ita <lb/>in vno puncto e&longs;t ad acqui&longs;itam in alio puncto vt linea tran&longs;uer&longs;a ad <pb pagenum="222" xlink:href="026/01/254.jpg"/>tran&longs;uer&longs;am v. <!-- REMOVE S-->g. <!-- REMOVE S-->acqui&longs;ita in &longs;olo puncto <foreign lang="greek">t</foreign> nulla habita ratione &longs;upe­<lb/>riorum ad acqui&longs;itam in &longs;olo puncto <foreign lang="greek">b</foreign> vt <foreign lang="greek">tm</foreign> ad <foreign lang="greek">ba</foreign> ita acqui&longs;ita in arcu <lb/>B <foreign lang="greek">t</foreign> e&longs;t ad acqui&longs;itam in arcu <foreign lang="greek">t b</foreign>, vt area &longs;inuum B <foreign lang="greek">t a</foreign>, ad aream &longs;inum <lb/>arcus <foreign lang="greek">t b. </foreign></s> </p> <p id="N1E2A3" type="main"> <s id="N1E2A5"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1E2B1" type="main"> <s id="N1E2B3"><!-- NEW -->Ob&longs;eruabis prædicta ita intelligenda e&longs;&longs;e, vt a&longs;&longs;umantur arcus exten&longs;i <lb/>in lineam rectam, ne &longs;cilicet &longs;inus plùs æquo contrahantur, &longs;eu potius <lb/>aliquo modo compenetrentur; </s> <s id="N1E2BB"><!-- NEW -->&longs;emper enim accidet trapezus mixtus, v. <!-- REMOVE S--><lb/>g. <!-- REMOVE S-->&longs;it trapezus A <foreign lang="greek">t</foreign> a&longs;&longs;umatur recta æqualia arcui B <foreign lang="greek">t</foreign> & duæ rectæ æqua­<lb/>les duabus BA <foreign lang="greek">t m</foreign>, quarta erit curua; igitur erit trapezus mixtus, quæ cer­<lb/>tè cautio adhibenda e&longs;t, alioquin fal&longs;um e&longs;&longs;et &longs;uperius Theorema, &longs;ed de <lb/>funependulis infrà. </s> </p> <p id="N1E2D6" type="main"> <s id="N1E2D8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s> </p> <p id="N1E2E4" type="main"> <s id="N1E2E6"><!-- NEW --><emph type="italics"/>In plano horizontali E O motus incrementa in diuer&longs;is punctis habent <lb/><expan abbr="eãdem">eandem</expan> proportionem quam habent in motu funependuli per arcum &longs;uum<emph.end type="italics"/> v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->fit planum EO ducatur AP O, motus in O e&longs;t ad motum in perpendicu­<lb/>lari vt PX ad AE, &longs;it funependulum AP cuius centrum; </s> <s id="N1E2FB"><!-- NEW -->cui affixa e&longs;t im­<lb/>mobiliter extremitas funis, &longs;it A & punctum quietis &longs;it E, motus illius in <lb/>puncto P e&longs;t ad motum in puncto C vt PX ad AB: </s> <s id="N1E303"><!-- NEW -->&longs;imiliter motus in G <lb/>puncto plani e&longs;t ad motum in perpendiculari vt LQ ad AE per Th.55. <lb/><expan abbr="item&qacute;ue">itemque</expan> &longs;it funependulum in L, motus in L e&longs;t ad motum in C vt LQ <lb/>ad AE, idem dico de punctis T & Y & omnibus aliis; igitur crementa <lb/>motus tùm in motu tùm in arcu &longs;unt in eadem proportione. </s> </p> <p id="N1E312" type="main"> <s id="N1E314"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s> </p> <p id="N1E320" type="main"> <s id="N1E322"><emph type="italics"/>Determinari pote&longs;t velocitas acqui&longs;ita in de&longs;cen&longs;u OE,<emph.end type="italics"/> e&longs;t enim vt trian­<lb/>gulum <expan abbr="mixtũ">mixtum</expan> cuius alterum latus rectum &longs;it ad OE, alterum ad angulos <lb/>rectos PX, tertium curua connectens &longs;inus rectos infra PX ver&longs;us vt E <lb/>vides in figura EO 4. e&longs;t autem hæc velocitas ad velocitatem acqui&longs;i­<lb/>tam in perpendiculari æquali OE vt prædictum triangulum EO 4. ad <lb/>rectangulum &longs;ub OEA. </s> </p> <p id="N1E338" type="main"> <s id="N1E33A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s> </p> <p id="N1E346" type="main"> <s id="N1E348"><!-- NEW --><emph type="italics"/>Non de&longs;cendit mobile per per OE & GE æquali tempore vt patet,<emph.end type="italics"/> quia <lb/>hæc Tangens EO pote&longs;t e&longs;&longs;e longior in infinitum; &longs;ed has proportiones <lb/>demon&longs;trabimus Tom, &longs;equenti, quia multam Geometriam de&longs;ide­<lb/>rant. </s> </p> <p id="N1E357" type="main"> <s id="N1E359"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s> </p> <p id="N1E365" type="main"> <s id="N1E367"><!-- NEW --><emph type="italics"/>Omne planum quod ad aliquod punctum circumferentiæ globi terre&longs;tris <lb/>terminatur, & productum vlterius non &longs;ecat centrum pote&longs;t plænum inclina­<lb/>tum e&longs;&longs;e,<emph.end type="italics"/> v.g. <!-- REMOVE S-->in planum LD vel YD, immò nullum e&longs;t planum quod non <lb/>&longs;it horizontale, id e&longs;t quod non cadat perpendiculariter in aliquem ra­<lb/>dium vel in aliquod perpendiculum v.g. <!-- REMOVE S-->LD e&longs;t horizontalis quia ca-<pb pagenum="223" xlink:href="026/01/255.jpg"/>dit perpendiculariter in perpendiculum AD, idem dico de plano YD, <lb/>cuius perpendiculum vt inueniatur, ex centro A adducatur perpendicu­<lb/>laris in YD: </s> <s id="N1E385"><!-- NEW -->hinc non pote&longs;t de&longs;cendere corpus ad centrum terræ per <lb/>planum inclinatum rectilineum quia linea recta quæ ducitur ad cen­<lb/>trum e&longs;t perpendiculum; igitur non e&longs;t planum inclinatum. </s> </p> <p id="N1E38D" type="main"> <s id="N1E38F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s> </p> <p id="N1E39B" type="main"> <s id="N1E39D"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari motus duorum planorum inclinatorum quorum idem <lb/>est perpendiculum,<emph.end type="italics"/> &longs;it enim arcus terræ GFC centro A; </s> <s id="N1E3A8"><!-- NEW -->&longs;int duo plana <lb/>FK GFL quorum idem e&longs;t perpendiculum LA; </s> <s id="N1E3AE"><!-- NEW -->motus in K per KF initio <lb/>e&longs;t ad motum per K vt DC ad DCA; </s> <s id="N1E3B4"><!-- NEW -->ducatur autem AH perpendicula­<lb/>ris in GL, & centro A ducatur arcus HE, ducaturque vel HO perpendi­<lb/>cularis in AL vel CP in AH; </s> <s id="N1E3BC"><!-- NEW -->dico motum in L e&longs;&longs;e vt PC ad CA: &longs;ed <lb/>hæc &longs;unt facilia. </s> </p> <p id="N1E3C2" type="main"> <s id="N1E3C4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s> </p> <p id="N1E3D0" type="main"> <s id="N1E3D2"><!-- NEW --><emph type="italics"/>Nullus gradus impetus de&longs;truitur in de&longs;cen&longs;u KF vel MF per &longs;e<emph.end type="italics"/>; </s> <s id="N1E3DB"><!-- NEW -->quia nihil <lb/>e&longs;t à quo de&longs;truatur, dixi per &longs;e; nam per accidens aliquid de&longs;trui pote&longs;t <lb/>tùm ratione plani &longs;cabri tùm etiam ratione aëris. </s> </p> <p id="N1E3E3" type="main"> <s id="N1E3E5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s> </p> <p id="N1E3F1" type="main"> <s id="N1E3F3"><!-- NEW --><emph type="italics"/>Omnes gradus acqui&longs;iti in de&longs;cen&longs;u concurrunt ad de&longs;cen&longs;um præter vnum <lb/>&longs;cilicet præter acqui&longs;itum vltimo instanti de&longs;cen&longs;us<emph.end type="italics"/>; quia impetus non con­<lb/>currit ad motum primo in&longs;tanti quo e&longs;t, per Th. 34. lib.1. de omnibus <lb/>aliis certum e&longs;t quod concurrant, quia non impediuntur, igitur concur­<lb/>runt per Ax.12. lib.1. <!-- KEEP S--></s> </p> <p id="N1E405" type="main"> <s id="N1E407"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s> </p> <p id="N1E413" type="main"> <s id="N1E415"><!-- NEW --><emph type="italics"/>Omnes gradus impetus qui concurrunt ad de&longs;cen&longs;um, concurrunt ad a&longs;cen­<lb/>&longs;um præter vnum<emph.end type="italics"/>; </s> <s id="N1E420"><!-- NEW -->probatur, quia &longs;i omnes concurrerent, maior e&longs;&longs;et a&longs;­<lb/>cen&longs;us de&longs;cen&longs;u quod e&longs;t ab&longs;urdum: adde quod impetus innatus ad li­<lb/>neam &longs;ur&longs;um determinari non pote&longs;t per Th.12. &longs;ed impetus innatus <lb/>concurrit ad de&longs;cen&longs;um, vt patet. </s> </p> <p id="N1E42A" type="main"> <s id="N1E42C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s> </p> <p id="N1E438" type="main"> <s id="N1E43A"><!-- NEW --><emph type="italics"/>Hinc tot concurrunt ad a&longs;cen&longs;um quot ad de&longs;cen&longs;um<emph.end type="italics"/>; </s> <s id="N1E443"><!-- NEW -->nam ad a&longs;cen&longs;um <lb/>omnes præter vltimum, ad de&longs;cen&longs;um omnes præter primum; igitur tot <lb/>concurrunt ad a&longs;cen&longs;um, quot ad de&longs;cen&longs;um. </s> </p> <p id="N1E44B" type="main"> <s id="N1E44D">Dices, primo in&longs;tanti a&longs;cen&longs;us aliquis gradus de&longs;truitur. </s> <s id="N1E450"><!-- NEW -->Re&longs;ponderet <lb/>aliquis, tran&longs;eat antecedens, quia cùm in&longs;tanti vltimo de&longs;cen&longs;us omnes <lb/>gradus præter innatum exigant motus pro &longs;equenti in&longs;tanti, quod e&longs;t pri­<lb/>mum in&longs;tans a&longs;cen&longs;us; certè tot concurrunt ad primum in&longs;tans a&longs;cen&longs;us, <lb/>quot ad vltimum de&longs;cen&longs;us, licèt aliquis gradus de&longs;truatur pro primo in­<lb/>&longs;tanti a&longs;cen&longs;us. </s> <s id="N1E45E"><!-- NEW -->Re&longs;ponderet alius, cùm primo in&longs;tanti a&longs;cen&longs;us gradus <lb/>ille qui vltimo de&longs;cen&longs;us productus e&longs;t concurrat ad motum, igitur illo <lb/>in&longs;tanti fru&longs;trà non e&longs;&longs;e, igitur non debere de&longs;trui, cùm eo tantùm no­<lb/>mine de&longs;truatur impetus; </s> <s id="N1E468"><!-- NEW -->igitur primo in&longs;tanti a&longs;cen&longs;us non de&longs;trui <pb pagenum="224" xlink:href="026/01/256.jpg"/>vllum <expan abbr="gradũ">gradum</expan> impetus, quia &longs;cilicet impetus innatus in omnibus in&longs;tan­<lb/>tibus præcedentibus habuit motum <expan abbr="deorsũ">deorsum</expan>; </s> <s id="N1E47B"><!-- NEW -->igitur nullo <expan abbr="in&longs;tãti">in&longs;tanti</expan> præteri­<lb/>to exigebat motum oppo&longs;itum: adde quod vltimo in&longs;tanti de&longs;cen&longs;us quo <lb/>mobile ponitur in F impetus naturalis non exigit ampliùs motum, cur <lb/>enim potius ver&longs;us M quàm ver&longs;us N, igitur primo tantùm in&longs;tanti a&longs;­<lb/>cen&longs;us quo mobile fertur ver&longs;us N, impetus naturalis exigit mobile re­<lb/>dire in F. <!-- KEEP S--></s> </p> <p id="N1E48E" type="main"> <s id="N1E490"><!-- NEW -->Dices, &longs;i primo in&longs;tanti a&longs;cen&longs;us nullus gradus impetus de&longs;truitur; igi­<lb/>tur nec &longs;ecundo neque tertio, non e&longs;t enim potior ratio pro vno quàm <lb/>pro altero. </s> <s id="N1E498"><!-- NEW -->Re&longs;ponderet negando, nam ideo, vt iam indicaui, primo <expan abbr="in&longs;tã-ti">in&longs;tan­<lb/>ti</expan> a&longs;cen&longs;us nullus gradus de&longs;truitur, quia in&longs;tanti immediatè <expan abbr="anteced&etilde;ti">antecedenti</expan>, <lb/>quod erat vltimum de&longs;cen&longs;us, impetus innatus non exigebat quidquam <lb/>ampliùs, igitur nullus gradus e&longs;t fru&longs;trà, igitur nullus de&longs;truitur, at verò <lb/>in&longs;tanti a&longs;cen&longs;us impetus innatus exigit pro &longs;equente, quod e&longs;t &longs;ecun­<lb/>dum a&longs;cen&longs;us mobile redire in F, igitur ex illa pugna &longs;ecundi in&longs;tantis <lb/>de&longs;truitur aliquid impetus; </s> <s id="N1E4B0"><!-- NEW -->&longs;ed profectò primo a&longs;cen&longs;us de&longs;truitur ali­<lb/>quid impetus, quia aliquid motus remittitur, propter impetum inna­<lb/>tum; </s> <s id="N1E4B8"><!-- NEW -->igitur aliquis impetus e&longs;t fru&longs;trà: </s> <s id="N1E4BC"><!-- NEW -->non tamen hoc facit, quin omnes <lb/>gradus in de&longs;cen&longs;u acqui&longs;iti concurrant ad a&longs;cen&longs;um; igitur tot concur­<lb/>runt ad a&longs;cen&longs;um, quot ad de&longs;cen&longs;um, cum hac tamen differentia, quod <lb/>impetus innatus, qui concurrit ad de&longs;cen&longs;um, non ad a&longs;cen&longs;um &longs;it longè <lb/>velocior vltimo in&longs;tanti motus acqui&longs;ito, qui concurrit ad de&longs;cen&longs;um, <lb/>non ad a&longs;cen&longs;um, </s> </p> <p id="N1E4CA" type="main"> <s id="N1E4CC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s> </p> <p id="N1E4D8" type="main"> <s id="N1E4DA"><!-- NEW --><emph type="italics"/>Hinc in ea proportione cre&longs;cit impetus in de&longs;cen&longs;u, qua decre&longs;cit in a&longs;cen&longs;u, <lb/>& in eadem cre&longs;cit, & decre&longs;cit motus in eadem cre&longs;cunt, & decre&longs;cunt &longs;pa­<lb/>tia,<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;int &longs;ex in&longs;tantia de&longs;cen&longs;us iuxta proportionem &longs;cilicet in&longs;tan­<lb/>tium, in qua res i&longs;ta faciliùs explicatur: </s> <s id="N1E4EB"><!-- NEW -->primo in&longs;tanti motus &longs;unt duo <lb/>gradus impetus, quorum alter tantùm concurrit, &longs;cilicet qui præextitit; </s> <s id="N1E4F1"><!-- NEW --><lb/>qui enim producitur primo illo in&longs;tanti, non concurrit ad illum motum <lb/>per Th. 34. lib. 1. igitur primo in&longs;tanti &longs;unt duo gradus impetus, vnus <lb/>gradus motus, & vnum &longs;patium; </s> <s id="N1E4FA"><!-- NEW -->&longs;ecundo verò in&longs;tanti &longs;unt tres gradus <lb/>impetus quorum vnus non concurrit, 2. gradus motus, 2.&longs;patia, atque ita <lb/>deinceps; donec tandem &longs;exto eo vltimo in&longs;tanti de&longs;cen&longs;us &longs;int 7. gra­<lb/>dus impetus, quorum vnus non concurrit, 6. gradus motus, & 6. <lb/>&longs;patia. </s> </p> <p id="N1E506" type="main"> <s id="N1E508"><!-- NEW -->Similiter primo in&longs;tanti a&longs;cen&longs;us &longs;unt 7. gradus impetus, quorum <lb/>vnus non concurrit &longs;cilicet innatus, 6. gradus motus, 6. &longs;patia; &longs;ecundo <lb/>6.gradus impetus, quorum vnus non concurrit &longs;cilicet innatus, 5.gradus <lb/>motus, 5.&longs;patia, atque ita deinceps. </s> </p> <p id="N1E512" type="main"> <s id="N1E514"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N1E520" type="main"> <s id="N1E522"><!-- NEW --><emph type="italics"/>Hinc æqualia ferè vtrimque &longs;unt &longs;patia de&longs;cen&longs;us &longs;cilicet, & a&longs;cen&longs;us<emph.end type="italics"/>; v.g. <!-- REMOVE S--><lb/>MF æquale FN, quia e&longs;t &longs;umma eorumdem terminorum per Th. 74. <lb/>igitur ex F mobile a&longs;cendit ad altitudinem FN æqualem altitudini FM, <pb pagenum="225" xlink:href="026/01/257.jpg"/>ex qua priùs de&longs;cenderat dixi ferè, quia cum innatus &longs;it perfectior vlti­<lb/>mo acqui&longs;ito paulò plùs &longs;patij acquiritur in de&longs;cen&longs;u, quàm in a&longs;cen&longs;u, <lb/>&longs;ed minimum e&longs;t &longs;en&longs;ibile. </s> </p> <p id="N1E539" type="main"> <s id="N1E53B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s> </p> <p id="N1E547" type="main"> <s id="N1E549"><emph type="italics"/>Hinc æqualibus temporibus a&longs;cendit ferè ab F in N, & de&longs;cendit ex M <lb/>in F,<emph.end type="italics"/> quia numerus terminorum æqualis e&longs;t numero in&longs;tantium. </s> </p> <p id="N1E553" type="main"> <s id="N1E555"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s> </p> <p id="N1E561" type="main"> <s id="N1E563"><!-- NEW --><emph type="italics"/>Hinc motum haberet ferè perpetuum ab M in F ab F in N, ab N ite­<lb/>rum in F, &c.<emph.end type="italics"/> &longs;i enim de&longs;cendens ex M in F a&longs;cendit ad æqualem altitu­<lb/>dinem FN, ita & de&longs;cendens ex N in F a&longs;cendet ad æqualem altitudi­<lb/>nem FM, atque ita deinceps; </s> <s id="N1E572"><!-- NEW -->igitur motus erit ferè perpetuus; </s> <s id="N1E576"><!-- NEW -->&longs;ed pro­<lb/>fectò nullum e&longs;t corpus tàm læuigatum, quod motum non impediat: dixi <lb/>ferè, quia de&longs;cen&longs;us tantillùm &longs;uperat a&longs;cen&longs;um, &longs;ed vix intra mille an­<lb/>nos &longs;en&longs;u id percipi po&longs;&longs;et. </s> </p> <p id="N1E580" type="main"> <s id="N1E582"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s> </p> <p id="N1E58E" type="main"> <s id="N1E590"><!-- NEW --><emph type="italics"/>Hinc &longs;i terrestris globus e&longs;&longs;et perforatus in perpendiculo FAI, &longs;i ex puncto <lb/>F demitteretur globus plumbeus per FAI de&longs;cenderet ex F in A, tum ex <lb/>Aa&longs;cenderet in I æquali ferè tempore<emph.end type="italics"/>; </s> <s id="N1E59D"><!-- NEW -->quod nece&longs;&longs;ariò &longs;equitur ex dictis; </s> <s id="N1E5A1"><!-- NEW --><lb/>quia omnes gradus qui concurrent ad a&longs;cen&longs;um, etiam concurrerent ad <lb/>de&longs;cen&longs;um, præter vnum, &longs;cilicet vltimo in&longs;tanti de&longs;cen&longs;us acqui&longs;itum; </s> <s id="N1E5A8"><!-- NEW --><lb/>& omnes, qui concurrerent ad de&longs;cen&longs;um, concurrerent etiam ad a&longs;cen­<lb/>&longs;um præter vnum, &longs;cilicet primum vel innatum; </s> <s id="N1E5AF"><!-- NEW -->igitur æquale &longs;patium <lb/>æquali tempore percurreretur; </s> <s id="N1E5B5"><!-- NEW -->quod certè dictum &longs;it ab&longs;trahendo à re­<lb/>&longs;i&longs;tentia aëris, quæ fortè modica e&longs;&longs;et; </s> <s id="N1E5BB"><!-- NEW -->Ex hac perpetua vibrationum &longs;e­<lb/>rie aliquando explicabimus cau&longs;as phy&longs;icas apogæi & perigæi Solis, & <lb/>aliorum planetarum; adhibe <expan abbr="cãdem">eandem</expan> cautionem, de qua &longs;uprà. </s> </p> <p id="N1E5C7" type="main"> <s id="N1E5C9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s> </p> <p id="N1E5D5" type="main"> <s id="N1E5D7"><emph type="italics"/>Si duo plana inclinata faciunt angulum e&longs;t ferè æqualis a&longs;cen&longs;us de&longs;cen&longs;ui.<emph.end type="italics"/><lb/>v. </s> <s id="N1E5E0"><!-- NEW -->g. <!-- REMOVE S-->de&longs;cendat per LF dico quod a&longs;cendet per FR ad altitudinem ferè <lb/>æqualem LF, quia licèt in angulo illo LFR &longs;it noua determinatio ad <lb/>nouam lineam motus, id e&longs;t qua&longs;i reflexio; </s> <s id="N1E5EA"><!-- NEW -->nihil e&longs;t tamen quod de&longs;truat <lb/>impetum; nam in reflexione &longs;eu noua determinatione non perit aliquid <lb/>impetus nece&longs;&longs;ariò vt lib. &longs;equenti demon&longs;trabimus. </s> </p> <p id="N1E5F2" type="main"> <s id="N1E5F4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s> </p> <p id="N1E600" type="main"> <s id="N1E602"><emph type="italics"/>E&longs;t tamen alia ratio de motu funependuli quâ euincemus a&longs;cen&longs;um e&longs;&longs;e mi­<lb/>norem de&longs;cen&longs;u,<emph.end type="italics"/> de qua infrà. </s> </p> <p id="N1E60C" type="main"> <s id="N1E60E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s> </p> <p id="N1E61A" type="main"> <s id="N1E61C"><!-- NEW --><emph type="italics"/>Initio a&longs;cen&longs;us per FN de&longs;truuntur gradus impetus producti &longs;ub finem de­<lb/>&longs;cen&longs;us, & &longs;ub finem a&longs;cen&longs;us destruuntur producti initio de&longs;cen&longs;us:<emph.end type="italics"/> ratio e&longs;t <lb/>clara, quia producti &longs;ub finem de&longs;cen&longs;us &longs;unt imperfectiores, cùm plùs <lb/>recedant à perpendiculari, per Th. 55. &longs;imiliter initio a&longs;cen&longs;us longiùs <lb/>recedit linea à verticali; </s> <s id="N1E62D"><!-- NEW -->igitur minùs de&longs;truetur impetus, vt &longs;æpè incul-<pb pagenum="226" xlink:href="026/01/258.jpg"/>cauimus; nam idem de&longs;truitur in dato puncto a&longs;cen&longs;us, qui producere­<lb/>tur in eodem puncto de&longs;cen&longs;us. </s> </p> <p id="N1E638" type="main"> <s id="N1E63A">Dices, gradus productus vltimo in&longs;tanti de&longs;cen&longs;us non de&longs;truitur pri­<lb/>mo a&longs;cen&longs;us. </s> <s id="N1E63F"><!-- NEW -->Re&longs;pondeo de&longs;trui; </s> <s id="N1E643"><!-- NEW -->hinc eadem cau&longs;a idem de&longs;truit primo <lb/>in&longs;tanti a&longs;cen&longs;us quod produxit vltimo in&longs;tanti de&longs;cen&longs;us; de&longs;truit in­<lb/>quam mediatè. </s> </p> <p id="N1E64B" type="main"> <s id="N1E64D"><!-- NEW -->Hîc ob&longs;eruabis &longs;ingulare di&longs;crimen, quod intercedit inter cau&longs;am <lb/>producentem, & exigentem; </s> <s id="N1E653"><!-- NEW -->nam producens verè agit, exigens verò tan­<lb/>tùm exigit; </s> <s id="N1E659"><!-- NEW -->illa con&longs;equitur effectum eo in&longs;tanti quo agit; </s> <s id="N1E65D"><!-- NEW -->hæc verò non <lb/>habet effectum eo in&longs;tanti, quo exigit, &longs;ed pro &longs;equenti; </s> <s id="N1E663"><!-- NEW -->e&longs;t tamen cau&longs;a <lb/>eo in&longs;tanti, quo exigit, non certè agens, &longs;ed exigens: </s> <s id="N1E669"><!-- NEW -->exemplum habes <lb/>in impetu, qui non habet motum eo in&longs;tanti quo exigit, &longs;ed tantùm &longs;e­<lb/>quenti pro quo exigit; </s> <s id="N1E671"><!-- NEW -->igitur e&longs;t cau&longs;a motus antequàm &longs;it motus, non <lb/>agens &longs;ed exigens; at verò cum impetus alium impetum producit e&longs;t <lb/>tantùm cau&longs;a illius cum agit. </s> </p> <p id="N1E679" type="main"> <s id="N1E67B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s> </p> <p id="N1E687" type="main"> <s id="N1E689"><!-- NEW --><emph type="italics"/>Vltimo in&longs;tanti a&longs;cen&longs;us &longs;unt duo gradus impetus, &longs;cilicet productus primo <lb/>in&longs;tanti de&longs;cen&longs;us cum innato<emph.end type="italics"/>; </s> <s id="N1E694"><!-- NEW -->igitur in&longs;tanti &longs;equenti erit motus, id e&longs;t, <lb/>de&longs;cen&longs;us, quia præualet innatus qui perfectior e&longs;t, vt con&longs;tat ex dictis; </s> <s id="N1E69A"><!-- NEW --><lb/>igitur nullum erit in&longs;tans quietis; quæ omnia explicari debent eodem <lb/>modo, quo iam explicuimus in motu violento, lib.3. e&longs;t enim eadem ra­<lb/>tio, &c. </s> <s id="N1E6A3">quæ omitto ne multa hîc repetere cogar. </s> </p> <p id="N1E6A6" type="main"> <s id="N1E6A8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s> </p> <p id="N1E6B4" type="main"> <s id="N1E6B6"><!-- NEW --><emph type="italics"/>Ictus e&longs;&longs;ent ferè æquales in &longs;egmentis æqualibus a&longs;cen&longs;us & de&longs;cen&longs;us,<emph.end type="italics"/> quia <lb/>motus e&longs;&longs;et æqualis in illis; igitur ictus æquales, quod facilè e&longs;t. </s> </p> <p id="N1E6C1" type="main"> <s id="N1E6C3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s> </p> <p id="N1E6CF" type="main"> <s id="N1E6D1"><!-- NEW --><emph type="italics"/>In planis eiu&longs;dem inclinationis idem corpus graue e&longs;t eiu&longs;dem ponderis<emph.end type="italics"/> v. <!-- REMOVE S--><lb/>g. <!-- REMOVE S-->&longs;int plana FE. GD. HO eiu&longs;dem inclinationis cum communi &longs;ci­<lb/>licet perpendiculo ODEA; </s> <s id="N1E6E1"><!-- NEW -->certè pondus corporis in O e&longs;t ad pondus <lb/>eiu&longs;dem in H vt AH ad AO per Th.57. & pondus corporis eiu&longs;dem in <lb/>D e&longs;t ad pondus eiu&longs;dem in G vt AG ad AD, & in E vt AF ad AE; </s> <s id="N1E6E9"><!-- NEW --><lb/>&longs;ed AF e&longs;t ad AE vt AG ad AD, vt AH ad AO; &longs;unt enim triangula <lb/>proportionalia. </s> </p> <p id="N1E6F0" type="main"> <s id="N1E6F2">Hinc reiice quorumdam recentiorum &longs;ententiam, qui volunt corpus, <lb/>quod propiùs ad centrum terræ accedit, e&longs;&longs;e minùs graue, & grauius quod <lb/>longiùs à centro recedit, quod de grauitate corporis ab&longs;olutè &longs;umpti nul­<lb/>latenus dici pote&longs;t vt con&longs;tat, vtrum verò &longs;i cum alio in eadem libra &longs;ta­<lb/>tuatur hinc inde, videbimus &longs;uo loco. </s> </p> <p id="N1E6FD" type="main"> <s id="N1E6FF"><!-- NEW -->Diceret fortè aliquis in ip&longs;o centro &longs;poliari &longs;ua tota grauitate; </s> <s id="N1E703"><!-- NEW -->igitur <lb/>quo propiùs accedit ad centrum maiori grauitatis portione multatur; </s> <s id="N1E709"><!-- NEW -->&longs;ed <lb/>nego con&longs;equentiam; </s> <s id="N1E70F"><!-- NEW -->nec enim &longs;equitur priuari parte grauitatis dum <lb/>abe&longs;t à centro, licèt tota priuetur cum e&longs;t in centro &longs;ed de hac quæ&longs;tione <lb/>plura aliàs; nec enim huius loci e&longs;t. </s> </p> <pb pagenum="227" xlink:href="026/01/259.jpg"/> <p id="N1E71B" type="main"> <s id="N1E71D"><!-- NEW -->Sed ne hoc fortè excidat &longs;i Globus CGLH de&longs;cendat ex A ad cen­<lb/>trum mundi &longs;eu grauium E, quæri pote&longs;t vtrum omnes partes mouean­<lb/>tur &longs;ua &longs;ponte ver&longs;us L etiam illæ quæ vltra centrum E proce&longs;&longs;erunt, &longs;eu <lb/>quod idem e&longs;t, vtrum globus CGLH, cuius centrum E e&longs;t coniun­<lb/>ctum cum centro grauium E tran&longs;latus in IFKB eiu&longs;dem &longs;it ponderis, <lb/>cuius e&longs;&longs;et in A. v.g. <!-- REMOVE S-->Re&longs;p. primò globum prædictum, cuius centrum e&longs;t in E, nullius e&longs;&longs;e <lb/>ponderis, vt con&longs;tat; nec enim potiùs in vnam partem, quàm in aliam <lb/>inclinat. </s> </p> <p id="N1E731" type="main"> <s id="N1E733"><!-- NEW -->Re&longs;pondeo &longs;ecundò globum <expan abbr="eũdem">eundem</expan>, cuius centrum e&longs;t D ex­<lb/>tra centrum grauium E grauitare, quia inclinat ver&longs;us E.R e&longs;pondeo ter­<lb/>tiò non æqualiter grauitare, &longs;iue &longs;it in D, &longs;iue &longs;it in A; </s> <s id="N1E73F"><!-- NEW -->quia grauitat per <lb/>&longs;uam entitatem quatenus coniuncta e&longs;t cum inclinatione; </s> <s id="N1E745"><!-- NEW -->&longs;ed non e&longs;t ea­<lb/>dem entitas in A quæ in D cum eadem inclinatione, igitur nec eadem <lb/>grauitas; </s> <s id="N1E74D"><!-- NEW -->non enim grauitat inde &longs;ecundum totam &longs;uam entitatem; <lb/>quia &longs;cilicet &longs;ectio MFNE non pote&longs;t ampliùs grauitare infrà E, quan­<lb/>doquidem E e&longs;t locus infimus. </s> </p> <p id="N1E755" type="main"> <s id="N1E757">Dices grauitare grauitatione communi. </s> <s id="N1E75A"><!-- NEW -->Re&longs;pondeo ad extra conce­<lb/>do, &longs;cilicet ad producendum impetum in corpore quod impedit motum, <lb/>&longs;ecus verò grauitatione intrin&longs;ecâ; vnde &longs;i &longs;u&longs;tineretur globus in F non <lb/>&longs;u&longs;tineretur totus, &longs;ed fortè detraheretur de toto pondere, primò &longs;ectio <lb/>MFNE, quæ non grauitat ver&longs;us F & altera æqualis quæ ab ea &longs;u&longs;tine­<lb/>retur. </s> <s id="N1E768"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;i &longs;ectio OCPD immediatè incumberet &longs;ectioni MFNE, <lb/>ita vt corda OP iungeretur cordæ MN; </s> <s id="N1E770"><!-- NEW -->certè vtraque con&longs;i&longs;teret; dixi <lb/>fortè, quia non e&longs;t ita certum, vt videbimus alias. </s> <s id="N1E776"><!-- NEW -->Dices igitur &longs;i globus <lb/>ille e&longs;&longs;et in centro, minima vi adhibita amoueretur; </s> <s id="N1E77C"><!-- NEW -->igitur idem timen­<lb/>dum e&longs;&longs;et de toto terre&longs;tri globo; </s> <s id="N1E782"><!-- NEW -->&longs;ed noli timere quæ&longs;o tàm facilè terræ <lb/>motum; </s> <s id="N1E788"><!-- NEW -->immò &longs;i globus ille &longs;emel occuparet centrum E., cum non tan­<lb/>tum hemi&longs;pherium GLH contra nitatur GCH; </s> <s id="N1E78E"><!-- NEW -->verùm etiam CGL, <lb/>CHL, & infinita alia; </s> <s id="N1E794"><!-- NEW -->certè vt moueatur vbi &longs;emel centrum E occupat, <lb/>debent tot ferè produci gradus impetus, quot produci deberent vt mo­<lb/>ueretur extra centrum, vt probabimus cum de grauitate &longs;cilicet in tra­<lb/>ctatu &longs;equenti phy&longs;icæ &longs;ingulari: Interim dicendum e&longs;t &longs;ingulas partes <lb/>huius globi &longs;eor&longs;im grauitare, cum centrum occupat, excepto illo puncto <lb/>quod in centro e&longs;t. </s> </p> <p id="N1E7A2" type="main"> <s id="N1E7A4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s> </p> <p id="N1E7B0" type="main"> <s id="N1E7B2"><!-- NEW --><emph type="italics"/>Pote&longs;t, corpus graue de&longs;cendere ad centrum terræ per planum conuexum <lb/>quadrantis,<emph.end type="italics"/> &longs;it enim globus terræ GBCK, centrum A; de&longs;cribatur ex <lb/>K &longs;emidiametro KA quadrans KLA. </s> <s id="N1E7BF">Dico quòd corpus graue de&longs;cen­<lb/>det per conuexum arcum LVA, non tamen per concauum. </s> <s id="N1E7C4"><!-- NEW -->Probatur <lb/>prima pars, quia à puncto L per arcum LVA &longs;emper accedit propiùs ad <lb/>centrum A; </s> <s id="N1E7CC"><!-- NEW -->igitur per illam de&longs;cendet, quia nulla e&longs;t alia linea minor <lb/>dextror&longs;um; </s> <s id="N1E7D2"><!-- NEW -->&longs;i enim e&longs;&longs;et aliqua, e&longs;&longs;et LCA; </s> <s id="N1E7D6"><!-- NEW -->quia po&longs;&longs;unt tantùm duci <lb/>duæ illæ rectæ breui&longs;&longs;imæ, quæ terminentur ad puncta LC vt patet; </s> <s id="N1E7DC"><!-- NEW -->&longs;ed <lb/>LCA e&longs;t maior arcu LVA: </s> <s id="N1E7E2"><!-- NEW -->Probatur &longs;ecunda pars, quia ab L in A in-<pb pagenum="228" xlink:href="026/01/260.jpg"/>tror&longs;um pote&longs;t duci linea LA breuior arcu LVA; igitur per concauum <lb/>LVA non de&longs;cenderet mobile. </s> </p> <p id="N1E7ED" type="main"> <s id="N1E7EF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s> </p> <p id="N1E7FB" type="main"> <s id="N1E7FD"><!-- NEW --><emph type="italics"/>Motus puncti L initio e&longs;&longs;et minor motu puncti V initio; </s> <s id="N1E803"><!-- NEW -->id e&longs;t po&longs;ito quod <lb/>demittatur ex V ver&longs;us A<emph.end type="italics"/>; </s> <s id="N1E80C"><!-- NEW -->demon&longs;tro, quia eodem modo &longs;e habet in L, <lb/>atque &longs;i e&longs;&longs;et in puncto L <expan abbr="Tãgentis">Tangentis</expan> LC, vt pater; </s> <s id="N1E816"><!-- NEW -->&longs;ed motus per LC ini­<lb/>tio e&longs;t ad motum per LA vt ND ad NA vel vt LC ad LA per Th.55. <lb/>at verò motus in V vel in F initio per FE <expan abbr="Tãgentem">Tangentem</expan> e&longs;t ad motum per­<lb/>pendiculi FA vt FE ad FA; </s> <s id="N1E824"><!-- NEW -->&longs;ed e&longs;t maior ratio FE ad FA, quàm LE <lb/>ad LA, vt con&longs;tat; igitur motus initio in V e&longs;t minor quàm in L <lb/>initio. </s> </p> <p id="N1E82C" type="main"> <s id="N1E82E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s> </p> <p id="N1E83A" type="main"> <s id="N1E83C"><!-- NEW --><emph type="italics"/>Hinc e&longs;t inuer&longs;a ratio motus funependuli vulgaris & plani inclinati recti,<emph.end type="italics"/><lb/>in quibus motus &longs;upremi puncti e&longs;t maior motu cuiu&longs;libet alterius pun­<lb/>cti, vnde inciperet motus, cum tamen hic &longs;it minor: porrò po&longs;&longs;et e&longs;&longs;e <lb/>funependulum KLA dum vel LVA e&longs;&longs;et orbis durus quem media di­<lb/>uideret rima qua&longs;i ecliptica globi penduli ex K fune exten&longs;o, & per ri­<lb/>mam incerto KL, vel quod faciliùs e&longs;&longs;et &longs;i KL e&longs;&longs;et pri&longs;ma durum, quod <lb/>circa K immobile moueri &longs;eu volui po&longs;&longs;et. </s> </p> <p id="N1E850" type="main"> <s id="N1E852"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s> </p> <p id="N1E85E" type="main"> <s id="N1E860"><emph type="italics"/>Alia via facilior occurrit, quæ mihi videtur non e&longs;&longs;e omittenda qua propor­<lb/>tiones illæ diuer&longs;i motus demonstrari po&longs;&longs;ent,<emph.end type="italics"/> &longs;it. </s> <s id="N1E86A"><!-- NEW -->v.g. <!-- REMOVE S-->punctum L; </s> <s id="N1E870"><!-- NEW -->a&longs;&longs;umatur <lb/>arcus LQ æqualis arcui LA; </s> <s id="N1E876"><!-- NEW -->ducatur recta AQ, in quam ducatur LK <lb/>perpendicularis: </s> <s id="N1E87C"><!-- NEW -->dico motum in L per arcum LVA initio e&longs;&longs;e ad motum <lb/>per LA vt KA ad LA: </s> <s id="N1E882"><!-- NEW -->&longs;imiliter &longs;it punctum V; </s> <s id="N1E886"><!-- NEW -->a&longs;&longs;umatur VL æqualis <lb/>arcui VA; </s> <s id="N1E88C"><!-- NEW -->& in hanc perpendicularis VX.dico motum in V per arcum <lb/>VA e&longs;&longs;e ad motum per ip&longs;um perpendiculum VA vt XA ad rectam <lb/>VA; </s> <s id="N1E894"><!-- NEW -->idem dico de omnibus aliis: </s> <s id="N1E898"><!-- NEW -->Ratio e&longs;t, quia Tangens, quæ ducere­<lb/>tur in V e&longs;&longs;et parallela AX; igitur triangula vtrimque e&longs;&longs;ent æqualia. </s> <s id="N1E89E"><!-- NEW --><lb/>v.g. <!-- REMOVE S-->FEA & FYA: item motus in P e&longs;t ad motum per ip&longs;um perpen­<lb/>diculum, vt Tangens PM ad PA, vt con&longs;tat ex dictis. </s> </p> <p id="N1E8A7" type="main"> <s id="N1E8A9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s> </p> <p id="N1E8B5" type="main"> <s id="N1E8B7"><!-- NEW --><emph type="italics"/>Hinc totus motus per LA perpendiculum e&longs;t ad totum motum per arcum <lb/>LVA, vt omnes chordæ ductæ ab A ad omnia puncta quadrantis AVL <lb/>&longs;imul &longs;umptæ ad totidem &longs;ubduplas chordarum ductarum ab A ad alterna <lb/>puncta totius &longs;emicirculi ALQ vel ad totidem <expan abbr="Tãgentes">Tangentes</expan> &longs;imul &longs;umptas<emph.end type="italics"/>: </s> <s id="N1E8CA"><!-- NEW -->cum <lb/>enim motus in L per arcum LVA &longs;it ad motum in L por ip&longs;um perpen­<lb/>diculum LA vt &longs;ubdupla AQ ad LA, & motus in V per arcum in A <lb/>&longs;it ad motum in V per rectam VA, vt &longs;ubdupla chordæ AL ad rectam <lb/>VA, atque ita deinceps per Th.88. certè omnia antecedentis &longs;imul &longs;um­<lb/>pta habent illam rationem ad omnia con&longs;equentia &longs;imul &longs;umpta, vt con­<lb/>&longs;tat; igitur totus motus, &c. </s> </p> <pb pagenum="229" xlink:href="026/01/261.jpg"/> <p id="N1E8DE" type="main"> <s id="N1E8E0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s> </p> <p id="N1E8EC" type="main"> <s id="N1E8EE"><!-- NEW --><emph type="italics"/>Globus de&longs;cendens B per conuexum arcum LVA in quo A e&longs;t centrum <lb/>terræ a&longs;cenderet denuò per quadrantem oppo&longs;itum AFS<emph.end type="italics"/>; </s> <s id="N1E8F9"><!-- NEW -->patet, quia totus <lb/>impetus non de&longs;trueretur in centro A, qui &longs;cilicet e&longs;&longs;et inten&longs;ior pro­<lb/>pter accelerationem de&longs;cen&longs;us, quàm vt in momento de&longs;truatur; quod <lb/>probatur ex aliis funependulis, & reflexis. </s> </p> <p id="N1E903" type="main"> <s id="N1E905"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 91.<emph.end type="center"/></s> </p> <p id="N1E911" type="main"> <s id="N1E913"><!-- NEW --><emph type="italics"/>Non a&longs;cenderet per totum arcum AFS<emph.end type="italics"/>; </s> <s id="N1E91C"><!-- NEW -->hoc Theorema probabitur cum <lb/>de motu funependuli, e&longs;t enim eadem pro vtroque ratio; quæ in eo po­<lb/>&longs;ita e&longs;t, quòd in a&longs;cen&longs;u aliquid impetus de&longs;truatur. </s> </p> <p id="N1E924" type="main"> <s id="N1E926"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s> </p> <p id="N1E932" type="main"> <s id="N1E934"><!-- NEW --><emph type="italics"/>Velociùs de&longs;cenderet per arcum maiorem LVA quam per minorem XA; </s> <s id="N1E93A"><!-- NEW --><lb/>velociùs, inquam, pro rata<emph.end type="italics"/>; </s> <s id="N1E942"><!-- NEW -->nam arcum XA citiùs percurreret; </s> <s id="N1E946"><!-- NEW -->ratio e&longs;t, <lb/>quia modicus XA e&longs;t magis curuus, vt patet; </s> <s id="N1E94C"><!-- NEW -->igitur determinatio­<lb/>nis mutatio maior e&longs;t: adde quod maior arcus accedit propiùs ad <lb/>rectam. </s> </p> <p id="N1E954" type="main"> <s id="N1E956"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s> </p> <p id="N1E962" type="main"> <s id="N1E964"><!-- NEW --><emph type="italics"/>Non modo per quadrantem circuli de&longs;cendere pote&longs;t in centrum terræ, &longs;ed <lb/>etiam per &longs;emicirculum<emph.end type="italics"/>; </s> <s id="N1E96F"><!-- NEW -->vt videre e&longs;t in eadem figura, nam &longs;i globus &longs;ta­<lb/>tueretur iuxta Quantulùm, &longs;cilicet, extra perpendiculum AQ dextror­<lb/>&longs;um, v.g. <!-- REMOVE S-->versùs P; </s> <s id="N1E979"><!-- NEW -->certè de&longs;cenderet v&longs;que ad A per conuexum &longs;emicir­<lb/>culi QLA; per conuexum, inquam, non per concauum, vt dictum e&longs;t <lb/>de quadrante LVA. </s> <s id="N1E981"><!-- NEW -->Ratio e&longs;t, quia accederet &longs;emper propiùs ad cen­<lb/>trum A; </s> <s id="N1E987"><!-- NEW -->igitur e&longs;&longs;et planum inclinatum per Th. 2. igitur per illud de­<lb/>&longs;cenderet, nec vlla e&longs;&longs;et difficultas; </s> <s id="N1E98D"><!-- NEW -->quod autem accedat &longs;emper propiùs <lb/>ad A per &longs;emicirculum QLA, certum e&longs;t; </s> <s id="N1E993"><!-- NEW -->quia PA minor e&longs;t QA; nam <lb/>diameter e&longs;t maxima &longs;ubten&longs;arum in circulo. </s> <s id="N1E999"><!-- NEW -->Immò per alium &longs;emi­<lb/>circulum ASQ a&longs;cenderet denuóque de&longs;cenderet repetitis pluribus vi­<lb/>brationibus; nunquam tamen a&longs;cenderet v&longs;que ad punctum Q propter <lb/>tamdem rationem, quam in Theoremate 92. adduximus. </s> </p> <p id="N1E9A3" type="main"> <s id="N1E9A5">Ob&longs;eruabis præterea non tantùm corpus graue po&longs;&longs;e de&longs;cendere per <lb/>&longs;emicirculum, qui &longs;ecet centrum mundi A, &longs;ed etiam per plures alios. </s> <s id="N1E9AA"><lb/>v.g. <!-- REMOVE S-->per &longs;emicirculum ROB, quia &longs;cilicet ab R ver&longs;us BO & ab O <lb/>ver&longs;us B &longs;emper de&longs;cendit, a&longs;cenditque propiùs ad A, cùm nulla linea in­<lb/>ter AOB duci po&longs;&longs;it ad punctum A, quæ non &longs;it maior BA, vt <lb/>con&longs;tat. </s> </p> <p id="N1E9B6" type="main"> <s id="N1E9B8"><!-- NEW -->Vt autem habeas i&longs;tos circulos; accipe centrum &longs;uprà A ver&longs;us K, mo­<lb/>do radius &longs;eu &longs;emidiameter de&longs;cendat infrà A. v.g. <!-- REMOVE S-->IB vel KB, &c. </s> </p> <p id="N1E9C0" type="main"> <s id="N1E9C2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s> </p> <p id="N1E9CE" type="main"> <s id="N1E9D0"><!-- NEW --><emph type="italics"/>Hinc pote&longs;t aliquis dimidium globum terre&longs;trem percurrere, licèt &longs;emper <lb/>de&longs;cendat<emph.end type="italics"/>; </s> <s id="N1E9DB"><!-- NEW -->vt&longs;i conficiat &longs;emicirculum ROB, & licet &longs;emper a&longs;cendat, <pb pagenum="230" xlink:href="026/01/262.jpg"/>vt &longs;i conficiat &longs;emicirculum BIIR; hæc ita clara &longs;unt, vt oculis tantùm <lb/>indigeant. </s> </p> <p id="N1E9E6" type="main"> <s id="N1E9E8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N1E9F4" type="main"> <s id="N1E9F6"><!-- NEW --><emph type="italics"/>Hinc pote&longs;t e&longs;&longs;e mons per quem aliquis a&longs;cendat, licèt &longs;ub planum horizon­<lb/>tale de&longs;cendat.<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;it Tangens in puncto B; </s> <s id="N1EA03"><!-- NEW -->haud dubiè qui ex B ver&longs;us <lb/>H procederet per arcum BH, haud dubiè a&longs;cenderet, quia recederet <lb/>&longs;emper à centro mundi A; </s> <s id="N1EA0B"><!-- NEW -->de&longs;cenderet tamen infra Tangentem in B; </s> <s id="N1EA0F"><!-- NEW -->igi­<lb/>tur mons e&longs;&longs;et infra horizontale planum; montem enim appello tractum <lb/>arduum, in quo dum aliquis ambulat, a&longs;cendit, hoc e&longs;t recedit à terræ <lb/>centro. </s> </p> <p id="N1EA19" type="main"> <s id="N1EA1B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s> </p> <p id="N1EA27" type="main"> <s id="N1EA29"><!-- NEW --><emph type="italics"/>Diuer&longs;æ e&longs;&longs;ent rationes motus in de&longs;cen&longs;u per &longs;emicirculum QLA<emph.end type="italics"/>; </s> <s id="N1EA32"><!-- NEW -->&longs;cilicet <lb/>in iis punctis, quæ propiùs accedunt ad A motus e&longs;&longs;et velocior initio <lb/>&longs;cilicet; </s> <s id="N1EA3A"><!-- NEW -->pote&longs;t autem haberi hæc proportio ductis Tangentibus, vt &longs;æpè <lb/>iam dixi; </s> <s id="N1EA40"><!-- NEW -->at verò in &longs;emicirculo ROB in puncto T e&longs;&longs;et veloci&longs;&longs;imus mo­<lb/>tus initio, quia angulus ITA e&longs;t maximus eorum omnium, qui po&longs;&longs;unt <lb/>fieri ductis duabus rectis ab A & I coëuntibus in &longs;emicirculo ROB, igi­<lb/>tur & illi oppo&longs;itus; </s> <s id="N1EA4A"><!-- NEW -->igitur perpendiculum AT accedit propiùs ad Tan­<lb/>gentem; </s> <s id="N1EA50"><!-- NEW -->igitur planum inclinatius e&longs;t; </s> <s id="N1EA54"><!-- NEW -->igitur in puncto T e&longs;t velocior mo­<lb/>tus initio quàm in aliis; igitur acceleratur motus ab R in T per cre­<lb/>menta &longs;emper maiora, & ab ip&longs;o T ad B per crementa minora. </s> </p> <p id="N1EA5C" type="main"> <s id="N1EA5E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s> </p> <p id="N1EA6A" type="main"> <s id="N1EA6C"><!-- NEW --><emph type="italics"/>Pote&longs;t de&longs;cendere corpus graue v.g. <!-- REMOVE S-->globus v&longs;que ad centrum terræ per He­<lb/>licem<emph.end type="italics"/>; </s> <s id="N1EA79"><!-- NEW -->&longs;it enim globus terræ AEQO, centrum K; </s> <s id="N1EA7D"><!-- NEW -->diuidatur QK in 4. <lb/>partes æquales QR.RP.PS.SK; </s> <s id="N1EA83"><!-- NEW -->a&longs;&longs;umatur EH æqualis QR, & AC æqua­<lb/>lis QP, & OM æqualis QS; </s> <s id="N1EA89"><!-- NEW -->tùm per &longs;ignata puncta de&longs;cribatur helix Q <lb/>HCZMK: </s> <s id="N1EA8F"><!-- NEW -->dico quod per eius conuexum globus de&longs;cenderet ex Q, ad <lb/>centrum terræ; </s> <s id="N1EA95"><!-- NEW -->quia &longs;emper accedit propiùs ad centrum; </s> <s id="N1EA99"><!-- NEW -->immò per plura <lb/>volumina de&longs;cendere pote&longs;t; &longs;it enim QK diui&longs;a in 8. partes æquales Q <lb/>TTR, &c. </s> <s id="N1EAA1"><!-- NEW -->tùm a&longs;&longs;umatur EF æqualis QT, AB æqualis QR, ON æqualis <lb/>QV tùm QR in ip&longs;a QK, & æqualis QY, ED, a qualis QS, & OL æqualis <lb/>QX; & per puncta a&longs;&longs;ignata de&longs;cribatur Helix QFBNPIDLK, per cam <lb/>de&longs;cenderet globus ad centrum terræ K po&longs;t duas circumuolutiones. </s> </p> <p id="N1EAAB" type="main"> <s id="N1EAAD"><!-- NEW -->Per aliam quoque &longs;piralem compo&longs;itam ex &longs;emicirculis de&longs;cendere <lb/>pote&longs;t ad centrum terræ B; </s> <s id="N1EAB3"><!-- NEW -->&longs;it enim centrum terræ F & globus terræ A <lb/>CMD; </s> <s id="N1EAB9"><!-- NEW -->accipiantur duo puncta hinc inde HK ad libitum; </s> <s id="N1EABD"><!-- NEW -->tunc ex H <lb/>fiat &longs;emicirculus MB; </s> <s id="N1EAC3"><!-- NEW -->haud dubiè globus po&longs;itus in M de&longs;cendet in B per <lb/>conuexum &longs;emicirculi in B; </s> <s id="N1EAC9"><!-- NEW -->quia B inter omnia illius puncta accedit pro­<lb/>ximè ad F; </s> <s id="N1EACF"><!-- NEW -->tùm ex K ducatur &longs;emicirculus BI; </s> <s id="N1EAD3"><!-- NEW -->certè ex B de&longs;cenderet in I <lb/>propter <expan abbr="eãdem">eandem</expan> rationem, tùm ex H de&longs;cribatur &longs;emicirculus IF; </s> <s id="N1EADD"><!-- NEW -->certè <lb/>ex I de&longs;cendet in F, quæ omnia patent ex dictis; </s> <s id="N1EAE3"><!-- NEW -->po&longs;&longs;unt autem multipli­<lb/>cari i&longs;tæ &longs;piræ in infinitum: Hinc licèt globus &longs;ingulis horis 100000. leu­<lb/>cas conficeret in de&longs;cen&longs;u, non tamen attingeret centrum ni&longs;i po&longs;t 1000. <lb/>annos, immò plures &longs;ecundùm numerum &longs;pirarum. </s> </p> <pb pagenum="231" xlink:href="026/01/263.jpg"/> <p id="N1EAF1" type="main"> <s id="N1EAF3"><!-- NEW -->Denique pote&longs;t de&longs;cendere per plura plana inclinata AKLMNO <lb/>PQRST, &longs;iue ducantur perpendiculariter, &longs;cilicet AK in BC, KL in B <lb/>D, atque ita deinceps; </s> <s id="N1EAFB"><!-- NEW -->&longs;iue non perpendiculariter, modò DL &longs;it maior C <lb/>K, EM maior DL, at que ita deinceps; attamen vltimum planum TB non <lb/>erit inclinatum, &longs;ed perpendiculum, vt patet. </s> </p> <p id="N1EB03" type="main"> <s id="N1EB05"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s> </p> <p id="N1EB11" type="main"> <s id="N1EB13"><!-- NEW --><emph type="italics"/>Po&longs;&longs;unt e&longs;&longs;e infinita plana inter orbem terræ, & horizontale per quæ globus <lb/>&longs;eu corpus graue non de&longs;cendet<emph.end type="italics"/>; </s> <s id="N1EB1E"><!-- NEW -->&longs;it enim centrum terræ C, ex quo de&longs;cri­<lb/>batur arcus QMH ducta diametro MCA in M; </s> <s id="N1EB24"><!-- NEW -->ducatur Tangens NM <lb/>L; </s> <s id="N1EB2A"><!-- NEW -->hæc erit horizontale planum, vt con&longs;tat; </s> <s id="N1EB2E"><!-- NEW -->tùm ex aliquo puncto infra C <lb/>putà ex A de&longs;cribatur arcus SMK; </s> <s id="N1EB34"><!-- NEW -->cercè &longs;i ponatur globus in M non <lb/>de&longs;cendet per arcum MG, quia potiùs a&longs;cenderet; </s> <s id="N1EB3A"><!-- NEW -->immò &longs;i ponatur <lb/>in T de&longs;cendet in M, immò faciliùs pelleretur corpus graue per arcum <lb/>MT, quàm per horizontalem MN, vt patet; </s> <s id="N1EB42"><!-- NEW -->igitur potentia illa, quæ per <lb/>horizontalem pellit non e&longs;t omnium minima, quæ per arcum MQ pel­<lb/>lit; quia in eo nullo modo globus a&longs;cendit, &longs;ed &longs;emper à centro C æqui­<lb/>di&longs;tat. </s> <s id="N1EB4C"><!-- NEW -->Si verò a&longs;&longs;umas quæcumque centra &longs;upra B putà D, & E, & ducas <lb/>arcus TMGPOMF; </s> <s id="N1EB52"><!-- NEW -->certè globus de&longs;cendet per MO, & MP, vt manife­<lb/>&longs;tum e&longs;t ex dictis, & hoc fortè ludicrum cuiquam videbitur; </s> <s id="N1EB58"><!-- NEW -->&longs;i enim col­<lb/>locetur globus in T, de&longs;cendit ver&longs;us M; </s> <s id="N1EB5E"><!-- NEW -->&longs;i verò in Y de&longs;cendet ver&longs;us <lb/>P; </s> <s id="N1EB64"><!-- NEW -->licèt V & T non di&longs;tét pollice; </s> <s id="N1EB68"><!-- NEW -->po&longs;&longs;unt enim accipi minima illa &longs;patia <lb/>ver&longs;us M, vbi e&longs;t angulus contingentiæ; </s> <s id="N1EB6E"><!-- NEW -->nulla tamen pote&longs;t duci recta ab <lb/>M infra MN, per quam globus non de&longs;cendat velociùs initio, quàm per <lb/>vllum arcum, &longs;iue MP, &longs;iue MO, &longs;iue quemcumque alium quamtumuis <lb/>maximè incuruatum vel inclinatum; </s> <s id="N1EB78"><!-- NEW -->quia &longs;cilicet recta illa ducta ex M <lb/>infra MN &longs;ecat omnes illos arcus, vt patet; </s> <s id="N1EB7E"><!-- NEW -->igitur initio facit planum <lb/>inclinatius: dixi initio, quia deinde in arcu multùm inuale&longs;cit motus, <lb/>cum &longs;emper deficiat in recta, vt diximus abundè &longs;uprà. </s> </p> <p id="N1EB86" type="main"> <s id="N1EB88"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 99.<emph.end type="center"/></s> </p> <p id="N1EB94" type="main"> <s id="N1EB96"><!-- NEW --><emph type="italics"/>Si quadrans ita di&longs;tet à centro mundi, vt tùm alter eius radius, tùm Tan­<lb/>gens ip&longs;i parallela cen&longs;eantur perpendiculares, globus de&longs;cendet ex eius vertice <lb/>per arcum<emph.end type="italics"/>: </s> <s id="N1EBA3"><!-- NEW -->Sit enim quadrans ATE erectus &longs;upra horizontem, ita vt <lb/>AE &longs;it horizontalis, & tùm TA, tùm 3. A perpendiculares; </s> <s id="N1EBA9"><!-- NEW -->certè de&longs;cen­<lb/>det globus per eius conuexum VBA in eadem proportione, in qua de&longs;­<lb/>cerdit per &longs;emicirculum, de quo &longs;uprà; </s> <s id="N1EBB1"><!-- NEW -->Igitur motus per quadrantem T <lb/>BE e&longs;t ad motum per ip&longs;um perpendiculum in eadem ratione, in qua e&longs;t <lb/>ad motum per &longs;emicirculum; </s> <s id="N1EBB9"><!-- NEW -->quippe motus in T nullus e&longs;t per arcum TE; </s> <s id="N1EBBD"><!-- NEW --><lb/>5.verò motus per arcum 5.E, initio &longs;cilicet, vt &longs;æpè dictum e&longs;t, e&longs;t ad mo­<lb/>tum per ip&longs;am perpendicularem vt A 7.ad A 5.in 4.vt A 7.ad A 4. in B <lb/>vt A <foreign lang="greek">d</foreign> ad AB, in D vt AH ad AD in X vt AF ad AX, in E, vt AE ad A <lb/>E; </s> <s id="N1EBCC"><!-- NEW -->vides autem tran&longs;ire motum hunc ferè per omnes gradus tarditatis: </s> <s id="N1EBD0"><!-- NEW -->di­<lb/>co ferè, quia reuerâ non tran&longs;it per omnes; quippe &longs;i fieret maior qua­<lb/>drans tangens i&longs;tum in T, motus e&longs;&longs;et iuxta initium præ&longs;ertim tar­<lb/>dior. </s> </p> <pb pagenum="232" xlink:href="026/01/264.jpg"/> <p id="N1EBDE" type="main"> <s id="N1EBE0"><!-- NEW -->Ob&longs;erua&longs;ti iam vt puto motum per Arcum TBE e&longs;&longs;e inuer&longs;um vul­<lb/>garis funependuli; </s> <s id="N1EBE6"><!-- NEW -->quippe in illo motuum incrementa initio &longs;unt mino­<lb/>ra, & &longs;emper cre&longs;cunt; at verò in hoc initio &longs;unt maiora, & &longs;emper de­<lb/>cre&longs;cunt. </s> </p> <p id="N1EBEE" type="main"> <s id="N1EBF0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s> </p> <p id="N1EBFC" type="main"> <s id="N1EBFE"><!-- NEW --><emph type="italics"/>Po&longs;&longs;unt determinari vires, quæ &longs;u&longs;tinere po&longs;&longs;unt datum pondus collocatu&mtail;<emph.end type="italics"/><lb/><emph type="italics"/>in arcu erecto ATE<emph.end type="italics"/>: </s> <s id="N1EC0D"><!-- NEW -->quippe ad &longs;u&longs;tinendum pondus in T nullæ vires <lb/>requiruntur, ad &longs;u&longs;tinendum in E æqualis potentia ponderi requiritur; </s> <s id="N1EC13"><!-- NEW --><lb/>at verò potentia, quæ &longs;u&longs;tinet in 5. &longs;e habet ad æqualem vt A 7.ad AE, <lb/>in 4.vt A Z.ad AE, in B vt A<foreign lang="greek">d</foreign> ad AE, in D vt AH ad AE, in X vt AF ad <lb/>AE; </s> <s id="N1EC20"><!-- NEW -->denique in E vt AE ad AE; ratio e&longs;t, quia potentia debet e&longs;&longs;e pro­<lb/>portionata momento ponderis, &longs;eu motus, &longs;ed motus in B.v.g.per BE e&longs;t <lb/>ad motum qui fit per perpendicularem vt A<foreign lang="greek">d</foreign> ad AB vel AE, igitur po­<lb/>tentia quæ impedit hunc motum, id e&longs;t quæ &longs;u&longs;tinet pondus in B e&longs;t ad <lb/>illam quæ &longs;u&longs;tinet in E vt A <foreign lang="greek">d</foreign> ad AE. <!-- KEEP S--></s> </p> <p id="N1EC35" type="main"> <s id="N1EC37"><!-- NEW -->Debet autem &longs;u&longs;tineri pondus vel per Tangentem ductam ad punctum <lb/>B vel ip&longs;i parallelam in certo dumtaxat funiculo, vt fit in trochleis; vnde <lb/>&longs;i &longs;emicirculus A 2.E &longs;it trochlea, & pondus pendeat ex E, <expan abbr="adhibeaturq;">adhibeaturque</expan> <lb/>potentia trahens in A, debet e&longs;&longs;e æqualis ponderi, &longs;ed de trochleis fusè <lb/>lib. 11. </s> </p> <p id="N1EC47" type="main"> <s id="N1EC49"><!-- NEW -->Hinc etiam facilè determinari pote&longs;t quomodo de&longs;truatur impetus, <lb/>&longs;i proiiciatur globus per arcum EBT &longs;ur&longs;um; </s> <s id="N1EC4F"><!-- NEW -->nam in eadem proportione <lb/>de&longs;truetur in a&longs;cendendo, qua acceleratur de&longs;cendendo; </s> <s id="N1EC55"><!-- NEW -->neque e&longs;t hîc <lb/>&longs;ingularis difficultas; </s> <s id="N1EC5B"><!-- NEW -->quemadmodum enim in de&longs;cen&longs;u &longs;emper accele­<lb/>ratur per incrementa inæqualia iuxta rationem explicatam; </s> <s id="N1EC61"><!-- NEW -->ita in a&longs;cen­<lb/>&longs;u &longs;emper retardatur per detractiones inæquales; </s> <s id="N1EC67"><!-- NEW -->in de&longs;cen&longs;u quidem per <lb/>incrementa initio minora, & maiora &longs;ub finem; in a&longs;cen&longs;u è contrario <lb/>per detractiones initio maiores &longs;ub finem minores. </s> </p> <p id="N1EC6F" type="main"> <s id="N1EC71"><!-- NEW -->Hinc denique determinari pote&longs;t quantùm corpus grauitet in toto <lb/>arcu TBE; </s> <s id="N1EC77"><!-- NEW -->in E nihil grauitat, in T totum grauitat; igitur grauitatio in <lb/>T, &longs;eu tota e&longs;t ad grauitationem in E, vt TA ad nihil, in 5. verò vt AT <lb/>ad AT, in 4. vt AT ad AA, in B vt AT ad AS, atque ita deinceps, quæ <lb/>con&longs;tant ex dictis. </s> </p> <p id="N1EC81" type="main"> <s id="N1EC83">In&longs;uper ob&longs;erua corpus graue incumbens arcui TBE, per varias lineas <lb/>po&longs;&longs;e pelli, vel trahi, de quibus idem pror&longs;us dicendum e&longs;t, quod dictum <lb/>e&longs;t in Th.5. & Sch.Th.16. </s> </p> <p id="N1EC8A" type="main"> <s id="N1EC8C"><!-- NEW -->Adde quod omi&longs;imus, &longs;ed facilè ex dictis lib. 1. intelligi pote&longs;t, im­<lb/>petum qui producitur in acceleratione motus per planum inclinatum <lb/>e&longs;&longs;e imperfectiorem ex duplici capite; primò ratione minoris temporis, <lb/>quo producitur ex ratione maioris vel minoris inclinationis, &longs;eu longi­<lb/>tudinis. </s> <s id="N1EC98"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it planum inclinatum AC; </s> <s id="N1EC9E"><!-- NEW -->certè cum po&longs;t motum per A <lb/>E, & per AB &longs;it æqualis ictus vel impetus; </s> <s id="N1ECA4"><!-- NEW -->& cùm tempus quo de&longs;cendit <lb/>per AE &longs;it duplum temporis, quo de&longs;cendit per AB; </s> <s id="N1ECAA"><!-- NEW -->certè &longs;ingulis in&longs;tan­<lb/>tibus, quibus durat motus per AC, producitur impetus &longs;ubduplus tan-<pb pagenum="233" xlink:href="026/01/265.jpg"/>tùm in perfectione illius, qui producitur per AB; &longs;i enim æqualis perfe­<lb/>ctionis; </s> <s id="N1ECB7"><!-- NEW -->igitur impetus po&longs;t de&longs;cen&longs;um per AC e&longs;&longs;et duplus illius qui ha­<lb/>betur in B po&longs;t de&longs;cen&longs;um per AB; </s> <s id="N1ECBD"><!-- NEW -->&longs;i autem e&longs;&longs;et minor &longs;ubduplo; </s> <s id="N1ECC1"><!-- NEW -->igitur <lb/>in C, vel impetus e&longs;&longs;et minor quam in B contra hypothe&longs;im; </s> <s id="N1ECC7"><!-- NEW -->igitur debet <lb/>&longs;ubduplus; </s> <s id="N1ECCD"><!-- NEW -->igitur duplò plures &longs;unt gradus impetus in C quàm in B, cùm <lb/>&longs;cilicet &longs;inguli gradus impetus in B æquiualeant duobus impetus in A: <lb/>his adde aliqua breuia Corollaria, quæ qui&longs;que ex dictis facilè colligere <lb/>poterit. </s> </p> <p id="N1ECD7" type="main"> <s id="N1ECD9"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1ECE6" type="main"> <s id="N1ECE8"><!-- NEW -->Ex his primò vides perfectam analogiam impetus in omni motu, qui <lb/>reuera explicari non pote&longs;t, ni&longs;i detur impetus alio imperfectior: </s> <s id="N1ECEE"><!-- NEW -->Porrò <lb/>multa hîc de&longs;iderantur, quæ ad motum in planis inclinatis pertinent, que <lb/>in Tomum &longs;equentem remittimus; quia potiori iure ad Mathematicam <lb/>&longs;pectant, quàm ad Phy&longs;icam. </s> </p> <p id="N1ECF8" type="main"> <s id="N1ECFA"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1ED07" type="main"> <s id="N1ED09"><!-- NEW -->Secundò, impetus po&longs;&longs;e in infinitum decre&longs;cere perfectionem quod <lb/>primò con&longs;tat ex eo, quòd infra horizontalem po&longs;&longs;int duci lineæ minùs <lb/>& minùs inclinatæ: &longs;ecundò ex eo, quòd po&longs;&longs;int inter quamlibet inclina­<lb/>tam deor&longs;um rectam, & &longs;uperficiem orbis terræ de&longs;cribi infiniti orbes, <lb/>quorum centrum &longs;it &longs;upra centrum terræ, quorum arcus initio faciunt <lb/>minorem, & minorem inclinationem. </s> </p> <p id="N1ED17" type="main"> <s id="N1ED19"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1ED26" type="main"> <s id="N1ED28">Tertiò, hinc colliges impetum qui producitur in primo puncto de&longs;­<lb/>cen&longs;us illorum arcuum e&longs;&longs;e pror&longs;us alogum cum illo, qui producitur in <lb/>primo puncto de&longs;cen&longs;us cuiu&longs;libet rectæ inclinatæ, & illum qui à pro­<lb/>ximo puncto ver&longs;us punctum contactus in Tangente producitur <lb/>e&longs;&longs;e etiam alogum cum illo, qui in proximo puncto ver&longs;us idem pun­<lb/>ctum contactus producitur in circumferentia circuli, cuius centrum &longs;it <lb/>infra centrum terræ, id e&longs;t cuius radius &longs;it longior radio orbis terræ, </s> </p> <p id="N1ED37" type="main"> <s id="N1ED39"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1ED46" type="main"> <s id="N1ED48">Quartò, quid mirabilius quam ad idem punctum contactus po&longs;&longs;e du­<lb/>ci infinitos circulos quorum arcus omnes in ea&longs;dem partes incuruan­<lb/>tur, licèt &longs;int infiniti? </s> <s id="N1ED4F"><!-- NEW -->quia &longs;umpto termino in eodem puncto contactus <lb/>omninò a&longs;cendant &longs;cilicet ij, qui maiores &longs;unt orbe terræ, & infiniti, qui <lb/>de&longs;cendunt, ij &longs;cilicet qui minores &longs;unt; & vnicus tantùm medius, qui <lb/>nec a&longs;cendat nec de&longs;cendat, qui e&longs;t orbis terræ. </s> </p> <p id="N1ED59" type="main"> <s id="N1ED5B"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1ED68" type="main"> <s id="N1ED6A"><!-- NEW -->Quintò, non po&longs;&longs;e faciliùs globum moueri, quàm in &longs;uperficie terræ, <lb/>&longs;i probè læuigata e&longs;&longs;et; </s> <s id="N1ED70"><!-- NEW -->nullum enim e&longs;t planum &longs;upra &longs;iue rectum, &longs;iue <lb/>curuum, quod non a&longs;cendat; </s> <s id="N1ED76"><!-- NEW -->nullum infrà quod non de&longs;cendat: hinc mo­<lb/>tus e&longs;&longs;et æquabilis. </s> </p> <pb pagenum="234" xlink:href="026/01/266.jpg"/> <p id="N1ED80" type="main"> <s id="N1ED82"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1ED8F" type="main"> <s id="N1ED91"><!-- NEW -->Sextò, cum globus rotatur in plano inclinato mouetur motu mixto, <lb/>&longs;cilicet ex motu orbis & centri, <expan abbr="mouetur&qacute;ue">moueturque</expan> velociùs quàm cubus eiu&longs;­<lb/>dem ponderis; </s> <s id="N1ED9D"><!-- NEW -->quia pauciores partes plani fricantur à globo; </s> <s id="N1EDA1"><!-- NEW -->&longs;ed hæc ra­<lb/>tio non valet, ni&longs;i &longs;upponatur planum non e&longs;&longs;e perfectè læuigatum; </s> <s id="N1EDA7"><!-- NEW -->igi­<lb/>tur e&longs;t alia ratio: an quia cubus mouetur motu centri? </s> <s id="N1EDAD"><!-- NEW -->globus verò motu <lb/>centri & orbis; </s> <s id="N1EDB3"><!-- NEW -->&longs;ed motus orbis iuuat motum centri; </s> <s id="N1EDB7"><!-- NEW -->&longs;ed hæc ratio nulla <lb/>e&longs;t, quia <expan abbr="tantũdem">tantundem</expan> pars &longs;uperior globi addit motui centri quantùm <lb/>inferior detrahit; </s> <s id="N1EDC3"><!-- NEW -->igitur alia ratio e&longs;t, &longs;cilicet non tantùm globum de&longs;­<lb/>cendere in plano inclinato per grauitatem ab&longs;olutam, &longs;ed etiam per re&longs;­<lb/>pectiuam, <expan abbr="e&longs;t&qacute;ue">e&longs;tque</expan> veluti potentia Mechanica admota, &longs;cilicet vectis, cu­<lb/>jus qua&longs;i vicem gerit &longs;emidiameter circuli: </s> <s id="N1EDD1"><!-- NEW -->porrò vectis centrum e&longs;t <lb/>punctum contactus; </s> <s id="N1EDD7"><!-- NEW -->dixi &longs;emidiametrum, non verò diametrum; </s> <s id="N1EDDB"><!-- NEW -->quia to­<lb/>tum pondus globi non e&longs;t appen&longs;um extremæ diametro, &longs;ed extremæ &longs;e­<lb/>midiametro in hoc ca&longs;u; illa autem extremitas e&longs;t centrum grauitatis <lb/>globi. </s> </p> <p id="N1EDE5" type="main"> <s id="N1EDE7"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1EDF3" type="main"> <s id="N1EDF5"><!-- NEW -->Septimò, hinc etiam apparet analogia impetus imperfectioris, qui pro­<lb/>ducitur ver&longs;us centrum vectis, & illius, qui producitur in mobili per <lb/>planum inclinatum; </s> <s id="N1EDFD"><!-- NEW -->nam ideo e&longs;t imperfectior, qui producitur ver&longs;us <lb/>centrum vectis, quia temporibus æqualibus partes mobiles vectis, quæ <lb/>&longs;unt ver&longs;us centrum acquirunt &longs;patia inæqualia &longs;cilicet, minora, & mi­<lb/>nora in infinitum; </s> <s id="N1EE07"><!-- NEW -->ita pror&longs;us in planis inclinatis cum acquirantur tem­<lb/>poribus æqualibus &longs;patia inæqualia; </s> <s id="N1EE0D"><!-- NEW -->minora certè in longioribus, &longs;up­<lb/>po&longs;ita dumtaxat eadem perpendiculi altitudine debet produci impetus <lb/>imperfectior; nam ex imperfectione effectus id e&longs;t motus, benè colligitur <lb/>imperfectio cau&longs;æ id e&longs;t impetus. </s> </p> <p id="N1EE17" type="main"> <s id="N1EE19"><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1EE25" type="main"> <s id="N1EE27"><!-- NEW -->Octauò denique, mirabile e&longs;t, quî fieri po&longs;&longs;it, vt eadem potentia quæ <lb/>totas &longs;uas vires exerens globum proiicit per lineam verticalem ad al­<lb/>titudinem vnius pollicis, id e&longs;t quæ proiicere tantùm pote&longs;t per &longs;patium <lb/>digitale, per omnes tamen inclinatas, quæ ad extremitatem huius per­<lb/>pendiculi duci po&longs;&longs;unt, cuiu&longs;cunque &longs;int longitudinis, non auctis viri­<lb/>bus proiiciat; quis hoc crederet? </s> <s id="N1EE35">ni&longs;i manife&longs;ta cogeret demon&longs;tratio, <lb/>quam habes in Th.20.27. &c. </s> </p> </chap> <chap id="N1EE3A"> <pb pagenum="235" xlink:href="026/01/267.jpg"/> <figure id="id.026.01.267.1.jpg" xlink:href="026/01/267/1.jpg"/> <p id="N1EE44" type="head"> <s id="N1EE46"><emph type="center"/>LIBER SEXTVS, <lb/><emph type="italics"/>DE MOTV REFLEXO.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N1EE53" type="main"> <s id="N1EE55"><!-- NEW -->DE motu reflexo agendum e&longs;&longs;e videtur hoc <lb/>loco; præmittendu&longs;que e&longs;t motui circula­<lb/>ri, qui fortè &longs;ine motu reflexo nunquam fit, <lb/>vt dicemus infrà. <lb/><gap desc="hr tag"/></s> </p> <p id="N1EE62" type="main"> <s id="N1EE64"><emph type="center"/><emph type="italics"/>DEPINITIO 1.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EE70" type="main"> <s id="N1EE72"><emph type="italics"/>MOtus reflexus e&longs;t reditus mobilis ratione corporis impedientis primam <lb/>lineam motus.<emph.end type="italics"/></s> </p> <p id="N1EE7B" type="main"> <s id="N1EE7D"><!-- NEW -->Hæc definitio e&longs;t clara; </s> <s id="N1EE81"><!-- NEW -->dicitur reditus, quia reuerâ mobile, quod re­<lb/>percutitur, &longs;eu reflectitur, qua&longs;i redit, &longs;eu retrò agitur; </s> <s id="N1EE87"><!-- NEW -->&longs;iue id fiat per <lb/>eandem lineam, quâ appul&longs;um fuit; &longs;iue per aliam: </s> <s id="N1EE8D"><!-- NEW -->&longs;ic pila in murum <lb/>impacta reflecti dicitur, ita vt eius linea frangatur in ip&longs;a muri &longs;uperfi­<lb/>cie, quod duobus tantùm modis fieri pote&longs;t: primò &longs;ine angulo, vt cum <lb/>redit mobile per eandem lineam, per quam priùs acce&longs;&longs;erat, &longs;icque linea <lb/>reflexionis opponi videtur ex diametro lineæ incidentiæ. </s> <s id="N1EE99">Secundò cum <lb/>angulo, quòd &longs;cilicet in puncto reflexionis linea reflexionis cum linea <lb/>incidentiæ faciat angulum. </s> </p> <p id="N1EEA0" type="main"> <s id="N1EEA2"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EEAF" type="main"> <s id="N1EEB1"><emph type="italics"/>Corpus reflectens e&longs;t, quod motum liberum alterius corporis impacti non <lb/>permittit vlteriùs per eandem lineam propagari, &longs;ed illius lineam frangit, & <lb/>inflectit,<emph.end type="italics"/> &c. </s> <s id="N1EEBD">huius corporis conditiones in &longs;equentibus Theorematis <lb/>definiemus. </s> </p> <p id="N1EEC2" type="main"> <s id="N1EEC4"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EED1" type="main"> <s id="N1EED3"><emph type="italics"/>Punctum reflexionis e&longs;t punctum illud plani reflectentis, in quo linea refle­<lb/>xionis, & linea incidentiæ coëunt.<emph.end type="italics"/></s> </p> <p id="N1EEDC" type="main"> <s id="N1EEDE"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EEEB" type="main"> <s id="N1EEED"><emph type="italics"/>Linea incidentiæ e&longs;t illa linea motus. </s> <s id="N1EEF2">per quam mobile ante reflexionem ap­<lb/>pellitur ad planum reflectens.<emph.end type="italics"/></s> </p> <pb pagenum="236" xlink:href="026/01/268.jpg"/> <p id="N1EEFD" type="main"> <s id="N1EEFF"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EF0C" type="main"> <s id="N1EF0E"><!-- NEW --><emph type="italics"/>Linea reflexionis e&longs;t illa linea motus, per quam mobile po&longs;t reflexionem re­<lb/>cedit à plano inclinato<emph.end type="italics"/>; hinc vides punctum reflexionis e&longs;&longs;e terminum ad <lb/>quem illius lineæ, & terminum à quo huius. </s> </p> <p id="N1EF1B" type="main"> <s id="N1EF1D"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EF2A" type="main"> <s id="N1EF2C"><emph type="italics"/>Angulus incidentiæ e&longs;t, quem facit cum plano reflectente linea inci­<lb/>dentiæ.<emph.end type="italics"/></s> </p> <p id="N1EF35" type="main"> <s id="N1EF37"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1EF43" type="main"> <s id="N1EF45"><emph type="italics"/>Angulus reflexionis e&longs;t, quem facit linea reflexionis cum eodem plano.<emph.end type="italics"/></s> </p> <p id="N1EF4C" type="main"> <s id="N1EF4E"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1EF5A" type="main"> <s id="N1EF5C"><!-- NEW --><emph type="italics"/>Cathetus e&longs;t linea perpendiculariter cadens in planum reflectens ducta ab <lb/>aliquo puncto linea incidentia<emph.end type="italics"/>; </s> <s id="N1EF67"><!-- NEW -->& tunc dicitur Cathetus incidentiæ; </s> <s id="N1EF6B"><!-- NEW -->vel <lb/>ab aliquo lineæ reflexionis, & tunc dicitur Cathetus reflexionis; hæc <lb/>omnia &longs;unt facilia, quæ in gratiam Tyronum breuiter in figura <lb/>propono. </s> </p> <p id="N1EF75" type="main"> <s id="N1EF77"><!-- NEW -->Sit FB linea plani reflectentis; </s> <s id="N1EF7B"><!-- NEW -->&longs;it D punctum reflexionis; &longs;it AD <lb/>linea incidentiæ, DH linea reflexionis, AB Cathetus incidentiæ, HF <lb/>Cathetus reflexionis, ADB angulus incidentiæ, EDF oppo&longs;itus, <lb/>HDF angulus reflexionis, CDB oppo&longs;itus, ADH angulus aperturæ <lb/>vel pyramidis reflexionis, EDC oppo&longs;itus, ADE angulus &longs;upplementi <lb/>anguli incidentiæ, HDG angulus complementi anguli reflexionis, re­<lb/>ctangulum BH &longs;uperficies reflexionis, BF &longs;ectio plani reflectentis, & <lb/>prædictæ &longs;uperficiei. </s> </p> <p id="N1EF8D" type="main"> <s id="N1EF8F"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EF9C" type="main"> <s id="N1EF9E"><emph type="italics"/>Aliquod corpus in aliud cum impetu impaction reflectitur,<emph.end type="italics"/> hæc hypothe­<lb/>&longs;is certa e&longs;t. </s> </p> <p id="N1EFA8" type="main"> <s id="N1EFAA"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EFB7" type="main"> <s id="N1EFB9"><!-- NEW --><emph type="italics"/>Corpus reflexum in aliud impactum aliquando illud mouet<emph.end type="italics"/>; &longs;ic pila ab <lb/>aliquo corpore reflexa in aliam incidens mouet illam. </s> </p> <p id="N1EFC4" type="main"> <s id="N1EFC6"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EFD3" type="main"> <s id="N1EFD5"><!-- NEW --><emph type="italics"/>Quo motus directus, &longs;cilicet qui &longs;is per lineam incidentia, e&longs;t maior, maior <lb/>e&longs;t quoque motus reflexus<emph.end type="italics"/>; &longs;i enim maiore vi pila appellitur in parietem <lb/>maiore vi etiam retorquctur. </s> </p> <p id="N1EFE2" type="main"> <s id="N1EFE4"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1EFF1" type="main"> <s id="N1EFF3"><!-- NEW --><emph type="italics"/>Idem impetus ad plures lineas determinari pere&longs;t &longs;eor&longs;um<emph.end type="italics"/>; </s> <s id="N1EFFC"><!-- NEW -->hoc Axima <lb/>certum e&longs;t; probatum e&longs;t in libro 1. Th.113.114. &c. </s> <s id="N1F002">dixi &longs;eor&longs;im, nam <lb/>plures &longs;imul lineas habere non pote&longs;t per Th.115.l.1. </s> </p> <p id="N1F007" type="main"> <s id="N1F009"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F016" type="main"> <s id="N1F018"><!-- NEW --><emph type="italics"/>Vbi e&longs;t effectus, ibi e&longs;t cau&longs;a, effectus inquam formalis,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->vbi e&longs;t album, <lb/>ibi e&longs;t id, quod exigit motum, &longs;eu præ&longs;tat illum motum in mobili; </s> <s id="N1F027"><!-- NEW -->id e&longs;t <pb pagenum="237" xlink:href="026/01/269.jpg"/>impetus: quippe omnis motus e&longs;t ab impetu, quod &longs;æpiùs in toto libro <lb/>primo demon&longs;tratum e&longs;t. </s> </p> <p id="N1F032" type="main"> <s id="N1F034"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F041" type="main"> <s id="N1F043"><emph type="italics"/>Impetus destruitur tantùm ne &longs;it frustra per Sch. <!-- REMOVE S-->Theor.<emph.end type="italics"/>152.<emph type="italics"/>& alia multa <lb/>libro primò,<emph.end type="italics"/> &longs;i enim impetus &longs;uum po&longs;&longs;et habere effectum reuerâ non de­<lb/>&longs;trueretur. </s> </p> <p id="N1F057" type="main"> <s id="N1F059"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F066" type="main"> <s id="N1F068"><!-- NEW --><emph type="italics"/>Tunc dici non pote&longs;t tota cau&longs;a destructa (cau&longs;a inquam formalis) cum <lb/>tuus effectus non e&longs;t de&longs;tructus<emph.end type="italics"/>; &longs;eu tunc non debet dici de&longs;tructus totus <lb/>impetus cum totus motus non e&longs;t de&longs;tructus. </s> </p> <p id="N1F075" type="main"> <s id="N1F077"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F084" type="main"> <s id="N1F086"><!-- NEW --><emph type="italics"/>Datur motus reflexus<emph.end type="italics"/>; </s> <s id="N1F08F"><!-- NEW -->nemo dubitat: </s> <s id="N1F093"><!-- NEW -->quippe aliquod corpus in aliud <lb/>impactum reflectitur per Ax. primum &longs;ed &longs;i corpus reflectitur e&longs;t motus <lb/>reflexus; </s> <s id="N1F09B"><!-- NEW -->igitur certum e&longs;t de motu reflexo quod &longs;it; infrà verò videbi­<lb/>mus propter quid &longs;it. </s> </p> <p id="N1F0A1" type="main"> <s id="N1F0A3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F0B0" type="main"> <s id="N1F0B2"><!-- NEW --><emph type="italics"/>In motu reflexo e&longs;t impetus<emph.end type="italics"/>; probatur, quia vbi e&longs;t motus, ibi e&longs;t impe­<lb/>tus per Axioma 2. <!-- KEEP S--></s> </p> <p id="N1F0BE" type="main"> <s id="N1F0C0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F0CD" type="main"> <s id="N1F0CF"><!-- NEW --><emph type="italics"/>Hinc cau&longs;a motus reflexi e&longs;t impetus qui ine&longs;t corpori reflexo<emph.end type="italics"/>; </s> <s id="N1F0D8"><!-- NEW -->nec enim e&longs;t <lb/>quidquam aliud applicatum cum mobile &longs;eparatum tùm à corpore refle­<lb/>ctente, tùm à manu proiicientis etiam moueatur; </s> <s id="N1F0E0"><!-- NEW -->igitur nihil extrin&longs;e­<lb/>cum pote&longs;t e&longs;&longs;e cau&longs;a huius motus; </s> <s id="N1F0E6"><!-- NEW -->igitur aliquod intrin&longs;ecum, voco <lb/>impetum; </s> <s id="N1F0EC"><!-- NEW -->hîc diutiùs non hæreo, quia &longs;imile argumentum habes in ter­<lb/>tio libro, in quo fusè probaui requiri impetum ad motum violentum, <lb/>atqui nullus motus reflexus e&longs;t naturalis; igitur violentus vel mixtus, <lb/>igitur requirit nece&longs;&longs;ariò impetum. </s> </p> <p id="N1F0F6" type="main"> <s id="N1F0F8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F105" type="main"> <s id="N1F107"><!-- NEW --><emph type="italics"/>Ille impetus vel producitur nouus, vel con&longs;eruatur prauius<emph.end type="italics"/>; clarum e&longs;t, <lb/>nec aliud excogitari pote&longs;t. </s> </p> <p id="N1F112" type="main"> <s id="N1F114"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F121" type="main"> <s id="N1F123"><!-- NEW --><emph type="italics"/>Ille impetus non producitur à corpore reflectente<emph.end type="italics"/>: </s> <s id="N1F12C"><!-- NEW -->probatur primò, quia <lb/>omnis impetus producitur ad extra ab alio impetu per Theor. <!-- REMOVE S-->42. lib.1. <lb/>Secundò probatur, quia corpus reflectens &longs;emper produceret impetum <lb/>in alio corpore applicato; </s> <s id="N1F138"><!-- NEW -->e&longs;&longs;et enim cau&longs;a nece&longs;&longs;aria; </s> <s id="N1F13C"><!-- NEW -->igitur nece&longs;&longs;ariò <lb/>ageret per Ax.12. lib.1. nec e&longs;t quod dicas agere tantùm po&longs;ita tali con­<lb/>ditione: </s> <s id="N1F144"><!-- NEW -->hoc e&longs;t po&longs;ito motu præuio, quod &longs;atis ridiculum e&longs;t, vt iam <lb/>aliàs monui; </s> <s id="N1F14A"><!-- NEW -->quia conditio nihil aliud præ&longs;tat in cau&longs;a quàm applicatio­<lb/>nem &longs;ubiecti apti, in quo agat, & &longs;ubtractionem omnis impedimenti; </s> <s id="N1F150"><!-- NEW --><lb/>atqui cum proximè pila parieti adhæret, e&longs;t omninò applicata, & abe&longs;t <lb/>omne impedimentum: </s> <s id="N1F157"><!-- NEW -->præterea &longs;i corpus reflectens ageret; </s> <s id="N1F15B"><!-- NEW -->haud dubiè <pb pagenum="238" xlink:href="026/01/270.jpg"/>&longs;i maius e&longs;t maiorem impetum produceret; </s> <s id="N1F164"><!-- NEW -->nec enim agit tantùm pars, <lb/>quæ tangitur; </s> <s id="N1F16A"><!-- NEW -->alioqui globus qui tangit tantùm in puncto minimè re­<lb/>flecteretur; quid enim punctum agere pote&longs;t? </s> <s id="N1F170"><!-- NEW -->Igitur &longs;i tantùm agit, quo <lb/>maius e&longs;t plùs agit; quæ omnia &longs;unt perab&longs;urda; Igitur non producitur <lb/>ille impetus à corpore reflectente. </s> <s id="N1F178">Vide Th. 40.lib.1.&c. </s> </p> <p id="N1F17B" type="main"> <s id="N1F17D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F18A" type="main"> <s id="N1F18C"><!-- NEW --><emph type="italics"/>Non producitur ab vllo alio extrin&longs;eco<emph.end type="italics"/>; </s> <s id="N1F195"><!-- NEW -->non ab aëre, qui motui ob&longs;i­<lb/>&longs;tit; </s> <s id="N1F19B"><!-- NEW -->&longs;ed nihil e&longs;t aliud extrin&longs;ecum applicatum; </s> <s id="N1F19F"><!-- NEW -->Igitur non producitur <lb/>ab vlla cau&longs;a extrin&longs;eca: </s> <s id="N1F1A5"><!-- NEW -->adde &longs;i vis rationem euidenti&longs;&longs;imam, quæ Theo­<lb/>rema &longs;uperius mirificè confirmat; </s> <s id="N1F1AB"><!-- NEW -->quia &longs;cilicet maximè applicatur mo­<lb/>bile corpori reflectenti per lineam perpendicularem; </s> <s id="N1F1B1"><!-- NEW -->igitur per illam <lb/>maximè deberet agere: </s> <s id="N1F1B7"><!-- NEW -->quippè per lineam obliquam qua&longs;i tantùm allam­<lb/>bitur corpus reflectens; </s> <s id="N1F1BD"><!-- NEW -->atqui linea reflexionis perpendicularis minima <lb/>e&longs;t omnium quamuis per accidens, vt con&longs;tat experientiâ, & nos infrà <lb/>demon&longs;trabimus; </s> <s id="N1F1C5"><!-- NEW -->cùm tamen deberet e&longs;&longs;e maxima; </s> <s id="N1F1C9"><!-- NEW -->igitur impetus non <lb/>producitur in mobili reflexo, nec ab ip&longs;o corpore reflectente, nec ab vllo <lb/>alio extrin&longs;eco; quia nihil pror&longs;us aliud applicatum e&longs;t, à quo produci <lb/>po&longs;&longs;it. </s> <s id="N1F1D3"><!-- NEW -->Re&longs;pondent aliqui produci à generante; &longs;ed quodnam e&longs;t illud <lb/>generans? </s> <s id="N1F1D9"><!-- NEW -->non cau&longs;a &longs;ecunda, vt patet; an verò prima? </s> <s id="N1F1DD">&longs;ed quis dicat <lb/>moueri tantùm à Deo pilam à muro repercu&longs;&longs;am? </s> <s id="N1F1E2">&longs;ed quidquid moue­<lb/>tur, inquies, ab alio mouetur, vt vult Philo&longs;ophus. <!-- KEEP S--></s> <s id="N1F1E8"><!-- NEW -->Re&longs;pondeo mediatè <lb/>&longs;cilicet, vel immediatè; </s> <s id="N1F1EE"><!-- NEW -->quippe illa pila à &longs;e ip&longs;a non mouetur, &longs;ed ab <lb/>impul&longs;ore mediante, &longs;cilicet, impetu impre&longs;&longs;o; &longs;ed hæc alibi iam indi­<lb/>cauimus. </s> </p> <p id="N1F1F6" type="main"> <s id="N1F1F8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N1F204" type="main"> <s id="N1F206"><!-- NEW --><emph type="italics"/>Non producitur ille impetus ab ip&longs;o mobili,<emph.end type="italics"/> vt con&longs;tat nec enim exigit <lb/>moueri illo motu; </s> <s id="N1F211"><!-- NEW -->adde quod e&longs;t cau&longs;a nece&longs;&longs;aria; </s> <s id="N1F215"><!-- NEW -->igitur nulla e&longs;&longs;et ra­<lb/>tio, cur modò maiorem, modò minorem effectum, hoc e&longs;t impetum pro­<lb/>duceret; quod tamen accidit; &longs;ed hæc &longs;unt facilia. </s> </p> <p id="N1F21D" type="main"> <s id="N1F21F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N1F22B" type="main"> <s id="N1F22D"><!-- NEW --><emph type="italics"/>Non producitur nouus impetus in reflectione pura:<emph.end type="italics"/> probatur, quia produ­<lb/>ceretur ab aliqua cau&longs;a: </s> <s id="N1F238"><!-- NEW -->illa autem e&longs;&longs;et vel extrin&longs;eca, vel intrin&longs;eca; </s> <s id="N1F23C"><!-- NEW --><lb/>non producitur ab vlla causâ extrin&longs;ecà per Theor.6.nec ab vlla intrin­<lb/>&longs;ecâ per Th.7. igitur à nulla; </s> <s id="N1F243"><!-- NEW -->igitur nullus producitur; </s> <s id="N1F247"><!-- NEW -->dixi in reflexio­<lb/>ne purâ, quia præter reflexionem fieri pote&longs;t, vt corpus reflectens mobi­<lb/>le impellat; </s> <s id="N1F24F"><!-- NEW -->vt cum duo globi mutuò colliduntur, vel vt &longs;it aliqua com­<lb/>pre&longs;&longs;io, quâ po&longs;itâ nouus impetus producetur; </s> <s id="N1F255"><!-- NEW -->non e&longs;t tamen quòd ali­<lb/>quis dicat motum reflexum e&longs;&longs;e tantùm à compre&longs;&longs;ione; </s> <s id="N1F25B"><!-- NEW -->quia quò corpus <lb/>durius e&longs;t; </s> <s id="N1F261"><!-- NEW -->& minùs redit, meliùs reflectitur; &longs;ic marmor à marmore fa­<lb/>cilè reflectitur. </s> </p> <p id="N1F267" type="main"> <s id="N1F269"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N1F275" type="main"> <s id="N1F277"><!-- NEW --><emph type="italics"/>Hinc impetus ille, qui e&longs;t cau&longs;a motus reflexi, e&longs;t idem cum præuio con&longs;er<emph.end type="italics"/>-<pb pagenum="239" xlink:href="026/01/271.jpg"/><emph type="italics"/>uato<emph.end type="italics"/>; quia vel e&longs;t productus de nouo, vel præuius, per Th. 4. non pri­<lb/>mum per Th.8.igitur e&longs;t præuius. </s> </p> <p id="N1F28C" type="main"> <s id="N1F28E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N1F29A" type="main"> <s id="N1F29C"><!-- NEW --><emph type="italics"/>Hinc potentia motrix, quæ priùs impegit mobile in corpus reflectens e&longs;t cau­<lb/>&longs;a huius motus reflexi<emph.end type="italics"/>; </s> <s id="N1F2A7"><!-- NEW -->quia &longs;cilicet e&longs;t cau&longs;a impetus, vi cuius mobile <lb/>mouetur etiam motu reflexo; hinc qui ludit pilá, verè dicitur cau&longs;a re­<lb/>flexionis pilæ, cau&longs;a inquam, &longs;ed mouens. </s> </p> <p id="N1F2AF" type="main"> <s id="N1F2B1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N1F2BD" type="main"> <s id="N1F2BF"><!-- NEW --><emph type="italics"/>Corpus reflectens dici pote&longs;t aliquo modo cau&longs;a reflexionis, id e&longs;t, cau&longs;a no­<lb/>uæ determinationis lineæ motus<emph.end type="italics"/>; ni&longs;i enim occurreret paries. </s> <s id="N1F2CA"><!-- NEW -->v.g. <!-- REMOVE S-->non re­<lb/>flecteretur pila; quamquam dici debet potiùs occa&longs;io, immò impedi­<lb/>mentum prioris lineæ, ex quo nece&longs;&longs;ariò &longs;equitur noua linea, ve dicam <lb/>infrà. </s> </p> <p id="N1F2D6" type="main"> <s id="N1F2D8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N1F2E4" type="main"> <s id="N1F2E6"><!-- NEW --><emph type="italics"/>Hinc habetur veri&longs;&longs;ima cau&longs;a reflexionis<emph.end type="italics"/>; </s> <s id="N1F2EF"><!-- NEW -->cum enim impetus non con­<lb/>&longs;eruetur à cau&longs;a primò producente, vt &longs;æpè dictum e&longs;t &longs;uprà, nec de&longs;trui <lb/>po&longs;&longs;it &longs;altem totus à corpore reflectente; </s> <s id="N1F2F7"><!-- NEW -->certè debet &longs;uum motum vlte­<lb/>riùs propagare; </s> <s id="N1F2FD"><!-- NEW -->igitur per aliquam lineam; quomodo verò determine­<lb/>tur linea reflexionis, dicemus infrà. </s> </p> <p id="N1F303" type="main"> <s id="N1F305"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N1F311" type="main"> <s id="N1F313"><emph type="italics"/>Hinc non destruitur totus impetus in puncto reflexionis.<emph.end type="italics"/></s> <s id="N1F31A"> Probatur primò, <lb/>quia motus reflexus e&longs;t ab impetu per Th. 3. &longs;ed non producitur nouus <lb/>impetus per Theorema 8. igitur e&longs;t impetus, qui erat ante reflexionem <lb/>per Th.9. igitur non de&longs;truitur totus, &longs;altem per &longs;e, in puncto reflexio­<lb/>nis. </s> <s id="N1F325">Probatur &longs;ecundò à priori; </s> <s id="N1F328"><!-- NEW -->quia nunquam de&longs;truitur impetus, ni&longs;i <lb/>quando e&longs;t fru&longs;tra per Ax.3.&longs;ed corpus reflectens non facit, vt &longs;it fru&longs;trà, <lb/>quia non impedit omnem lineam motus; </s> <s id="N1F330"><!-- NEW -->igitur &longs;i ad aliquam determi­<lb/>nari pote&longs;t, impetus non erit fru&longs;trà: ad quam autem determinari de­<lb/>beat, dicemus infrà. </s> </p> <p id="N1F338" type="main"> <s id="N1F33A"><!-- NEW -->Dixi, non de&longs;truitur totus impetus; </s> <s id="N1F33E"><!-- NEW -->quia fortè aliqua pars illius de­<lb/>&longs;truitur in reflexione vt demon&longs;trabo, &longs;cilicet per accidens: dixi præterea <lb/>per &longs;e, quia per accidens pote&longs;t accidere vt totus impetus de&longs;truatur pro­<lb/>pter mollitiem vel corporis reflexi, vel propter aliam cau&longs;am, de quo <lb/>aliàs. </s> </p> <p id="N1F34A" type="main"> <s id="N1F34C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N1F358" type="main"> <s id="N1F35A"><!-- NEW --><emph type="italics"/>Ex hoc etiam habetur impetum non e&longs;&longs;e &longs;ucce&longs;&longs;iuum &longs;ed qualitatem perma­<lb/>nentem eamque durare, licèt à cau&longs;a primò producente non con&longs;eruetur &longs;ed ab <lb/>alia<emph.end type="italics"/>; vt iam alias demon&longs;trauimus. </s> </p> <p id="N1F367" type="main"> <s id="N1F369"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N1F375" type="main"> <s id="N1F377"><!-- NEW -->In omni reflexione determinatur noua linea motus; </s> <s id="N1F37B"><!-- NEW -->clarum e&longs;t, quia <lb/>non e&longs;t motus &longs;ine linea determinata, vt patet; </s> <s id="N1F381"><!-- NEW -->&longs;ed non remanet prior <pb pagenum="240" xlink:href="026/01/272.jpg"/>linea; </s> <s id="N1F38A"><!-- NEW -->igitur e&longs;t noua, igitur illa determinatur; cur enim potiùs, quàm <lb/>alia, ni&longs;i determinaretur vna. </s> </p> <p id="N1F390" type="main"> <s id="N1F392"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N1F39E" type="main"> <s id="N1F3A0"><!-- NEW --><emph type="italics"/>Non determinatur à puncto contactus <expan abbr="tamũm">tantum</expan><emph.end type="italics"/>; </s> <s id="N1F3AC"><!-- NEW -->quia ab eodem puncto <lb/>plures lineæ reflexionis procedere po&longs;&longs;unt; </s> <s id="N1F3B2"><!-- NEW -->non à linea incidentiæ tan­<lb/>tùm; </s> <s id="N1F3B8"><!-- NEW -->quia &longs;i tantillùm inclinetur planum eadem linea incidentiæ pote&longs;t <lb/>habere diuer&longs;as lineas reflexionis; </s> <s id="N1F3BE"><!-- NEW -->non determinatur <expan abbr="deniq;">denique</expan> ab ip&longs;o plano <lb/>inclinato quod diuer&longs;as lineas reflectit; </s> <s id="N1F3C8"><!-- NEW -->non determinatur, inquam, ab <lb/>his omnibus &longs;eor&longs;im &longs;umptis, vt patet, &longs;ed ab omnibus coniunctim: </s> <s id="N1F3CE"><!-- NEW --><lb/>quippe ab his determinatur linea motus, ex quibus po&longs;itis, & applicatis <lb/>nece&longs;&longs;ariò &longs;equitur; </s> <s id="N1F3D5"><!-- NEW -->&longs;ed ex applicatione i&longs;torum omnium &longs;eor&longs;im non &longs;e­<lb/>quitur talis linea; </s> <s id="N1F3DB"><!-- NEW -->quæ tamen &longs;equitur ex applicatione omnium coniun­<lb/>ctim, vt patet; igitur ab his coniunctim &longs;umptis determinatur linea. </s> </p> <p id="N1F3E1" type="main"> <s id="N1F3E3"><!-- NEW -->Dices, linea incidentiæ non e&longs;t ampliùs, quando linea reflexionis <lb/>determinatur; igitur non pote&longs;t illam determinare. </s> <s id="N1F3E9"><!-- NEW -->Re&longs;pondeo deter­<lb/>minationem in eo e&longs;&longs;e po&longs;itam tantùm, quòd impetus po&longs;ito tali angulo <lb/>incidentiæ non po&longs;&longs;it aliam inire lineam, præter illam vnicam; </s> <s id="N1F3F1"><!-- NEW -->cùm enim <lb/>impetus ex &longs;e &longs;it indifferens ad omnes lineas, eo ip&longs;o determinatur ad <lb/>vnam, quo impeditur ne per alias motus propagetur; </s> <s id="N1F3F9"><!-- NEW -->atqui angulus inci­<lb/>dentiæ non modò dicit lineam incidentiæ, &longs;ed lineam plani, atque adeo <lb/>apicem anguli qui e&longs;t in puncto contactus; igitur po&longs;ito illo angulo <lb/>incidentiæ impetus determinatur ad lineam reflexionis. </s> </p> <p id="N1F403" type="main"> <s id="N1F405"><!-- NEW -->Porrò quod impediatur omnis alia linea, patet ex eo, quod primo ip&longs;a <lb/>linea incidentiæ impeditur ne vlteriùs producatur ab impenetrabilita­<lb/>te; & duritie plani reflectentis; immò & omnes aliæ impediuntur, quæ <lb/>per ip&longs;um planum duci po&longs;&longs;unt. </s> </p> <p id="N1F40F" type="main"> <s id="N1F411">Secundò, quod &longs;pectat ad alias, quæ citra planum reflectens à pun­<lb/>cto contactus duci quoque po&longs;&longs;unt, omnes præter vnam impediuntur, <lb/>quæ &longs;cilicet facit angulum cum plano æqualem angulo incidentiæ, vt <lb/>demon&longs;trabimus infrà. </s> </p> <p id="N1F41A" type="main"> <s id="N1F41C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N1F428" type="main"> <s id="N1F42A"><!-- NEW --><emph type="italics"/>Ideo determinatur impetus ad omnem lineam, quia impeditur prior linea<emph.end type="italics"/>; <lb/>clarum e&longs;t; ni&longs;i enim impediretur prior; </s> <s id="N1F435"><!-- NEW -->certè non determinaretur ad <lb/>nouam, quod certum e&longs;t: </s> <s id="N1F43B"><!-- NEW -->adde quod planum reflectens perinde &longs;e habet, <lb/>que &longs;i mobile impelleret cum eo impetus gradu, quem ip&longs;um mobile <lb/>iam habet; </s> <s id="N1F443"><!-- NEW -->impelleret autem per lineam perpendicularem in puncto <lb/>contactus erectam; &longs;ed propter priorem determinationem fit noua linea <lb/>mixta, de qua infrà. </s> </p> <p id="N1F44B" type="main"> <s id="N1F44D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N1F459" type="main"> <s id="N1F45B"><!-- NEW --><emph type="italics"/>Corpus reflectens impedit motum<emph.end type="italics"/>; </s> <s id="N1F464"><!-- NEW -->quia e&longs;t impenetrabile, durum, den­<lb/>&longs;um; &longs;ed de his infrà, quando con&longs;iderabimus impedimenta ratione <lb/>materiæ. </s> </p> <pb pagenum="241" xlink:href="026/01/273.jpg"/> <p id="N1F470" type="main"> <s id="N1F472"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N1F47E" type="main"> <s id="N1F480"><emph type="italics"/>Corpus reflectens plùs, vel minùs impedit motum ratione diuer&longs;æ appul&longs;io­<lb/>nis:<emph.end type="italics"/> probatur, quia motus reflexus aliquando e&longs;t maior, aliquando e&longs;t <lb/>minor, de quo infrà. </s> </p> <p id="N1F48C" type="main"> <s id="N1F48E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N1F49A" type="main"> <s id="N1F49C"><!-- NEW --><emph type="italics"/>Si corpus reflectens impingeretur in mobile, cui nullus prius ine&longs;&longs;et impetus, <lb/>punctum contactus determinaret lineam motus<emph.end type="italics"/>; vt demon&longs;trauimus lib.10. <lb/><expan abbr="moueret&qacute;ue">moueretque</expan> globum, v.g. <!-- REMOVE S-->per lineam perpendicularem ductam à puncto <lb/>contactus per centrum globi per Th.120.& 121. lib.1. <!-- KEEP S--></s> </p> <p id="N1F4B1" type="main"> <s id="N1F4B3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N1F4BF" type="main"> <s id="N1F4C1"><!-- NEW --><emph type="italics"/>Quò maiorem ictum infligit mobile per lineam incidentiæ corpori refle­<lb/>ctenti, e&longs;t maius impedimentum<emph.end type="italics"/>; </s> <s id="N1F4CC"><!-- NEW -->cum enim impetus agat tantùm ad extra, <lb/>vt tollat impedimentum; </s> <s id="N1F4D2"><!-- NEW -->certè quò maior e&longs;t ictus, plùs agit impetus; </s> <s id="N1F4D6"><!-- NEW --><lb/>igitur quò maior e&longs;t ictus, e&longs;t maius impedimentum, & vici&longs;&longs;im quò <lb/>maius e&longs;t impedimentum e&longs;t maior ictus; & contrà, quò minor e&longs;t ictus, <lb/>e&longs;t minus impedimentum, & vici&longs;&longs;im &longs;uppo&longs;ita &longs;cilicet eadem potentiâ <lb/>impellente, vt demon&longs;tratum e&longs;t libro primo. </s> </p> <p id="N1F4E1" type="main"> <s id="N1F4E3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N1F4EF" type="main"> <s id="N1F4F1"><!-- NEW --><emph type="italics"/>Quando linea incidentiæ cadit perpendiculariter in planum reflectens e&longs;t <lb/>maximum impedimentum<emph.end type="italics"/>; quia &longs;cilicet e&longs;t maximus ictus, vt probauimus <lb/>lib.1. <!-- KEEP S--></s> </p> <p id="N1F4FF" type="main"> <s id="N1F501"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N1F50D" type="main"> <s id="N1F50F"><emph type="italics"/>Quò linea incidentiæ cadit obliquiùs in <expan abbr="planũ">planum</expan>, e&longs;t minùs <expan abbr="impedimentũ">impedimentum</expan>,<emph.end type="italics"/> quia <lb/>e&longs;t minor ictus. </s> <s id="N1F521">v.g.in fig. </s> <s id="N1F524"><!-- NEW -->Definitione.8. ictus per lineam GD e&longs;t ad <lb/>ictum per lineam AD, vt AD ad AB; </s> <s id="N1F52A"><!-- NEW -->nec in his immoror, quæ lib.1. <lb/>& aliis &longs;ufficienter demon&longs;trata &longs;unt; </s> <s id="N1F530"><!-- NEW -->præ&longs;ertim cum de planis inclina­<lb/>tis; </s> <s id="N1F536"><!-- NEW -->nam perinde &longs;e habet inflictus ictus, atque grauitatio in ip&longs;um pla­<lb/>num; </s> <s id="N1F53C"><!-- NEW -->e&longs;t enim grauitatio in planum inclinatum, vt &longs;uprà fusè dictum e&longs;t <lb/>in Th.16. lib.5.ad grauitationem in horizontale, vt Tangens horizonta­<lb/>les ad &longs;ecantem, id e&longs;t, vt AB ad AD; </s> <s id="N1F544"><!-- NEW -->nam BD e&longs;t qua&longs;i perpendicu­<lb/>laris; igitur ictus &longs;unt, vt &longs;inus anguli incidentiæ ad &longs;inum totum. </s> <s id="N1F54A"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S--><lb/>vt AB, ad AD hinc per lineam, AD, e&longs;t minùs impedimentum quàm <lb/>per GD immò eadem e&longs;t ratio impedimentorum & ictuum; igitur im­<lb/>pedimentum in linea, GD e&longs;t ad impedimentum per lineam, AD, vt <lb/>AD, ad AB. <!-- KEEP S--></s> </p> <p id="N1F55A" type="main"> <s id="N1F55C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N1F568" type="main"> <s id="N1F56A"><!-- NEW --><emph type="italics"/>Hinc plùs, vel minùs determinat nouam lineam motus planum reflectens<emph.end type="italics"/>; </s> <s id="N1F573"><!-- NEW --><lb/>cum enim ideo determinetur impetus ad nouam lineam, quia impeditur <lb/>prior per Theorema 17. certè in eadem proportione determinatur ad <lb/>nouam, in qua impeditur prior; </s> <s id="N1F57C"><!-- NEW -->&longs;ed plùs vel minùs impeditur per Th. <!-- REMOVE S--><lb/>23. igitur plùs vel minùs determinatur impetus; </s> <s id="N1F583"><!-- NEW -->igitur plùs vel minùs <lb/>determinat planum reflectens: porrò planum BD, determinat mobile <pb pagenum="242" xlink:href="026/01/274.jpg"/>quod reflectit per lineam DG, & ni&longs;i e&longs;&longs;et alia determinatio per DG <lb/>reflecteretur mobile, vt reuerâ fit, cum linea incidentiæ e&longs;t perpen­<lb/>dicularis. </s> </p> <p id="N1F592" type="main"> <s id="N1F594"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N1F5A0" type="main"> <s id="N1F5A2"><!-- NEW --><emph type="italics"/>Hinc planum reflectens maximè determinat impetum ad nouam lineam <lb/>cum linea incidentiæ e&longs;t perpendicularis<emph.end type="italics"/>; </s> <s id="N1F5AD"><!-- NEW -->quia tunc e&longs;t maximum impedi­<lb/>mentum per Th.22.igitur maximè determinat per Th.24. & contrà, quò <lb/>linea incidentiæ e&longs;t obliquior, minor e&longs;t determinatio ad lineam no­<lb/>uam; igitur hæc tria &longs;unt in eadem proportione, &longs;cilicet ictus, impedi­<lb/>mentum, determinatio noua. </s> </p> <p id="N1F5B9" type="main"> <s id="N1F5BB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N1F5C7" type="main"> <s id="N1F5C9"><!-- NEW --><emph type="italics"/>Maxima determinatio, quâ planum reflectens po&longs;&longs;it impetum, mobili im­<lb/>pre&longs;&longs;um, qua&longs;i retorquere, e&longs;t illa, quæ fit per lineam perpendicularem.<emph.end type="italics"/> v.g.per <lb/>DG; </s> <s id="N1F5D6"><!-- NEW -->&longs;i enim planum ip&longs;um mobile impelleret à puncto contactus D; <lb/>certè impelleret tantùm per lineam perpendicularem, &longs;eu per lineam <lb/>ductam à puncto D per centrum globi, &longs;i v. <!-- REMOVE S-->g. <!-- REMOVE S-->e&longs;&longs;et globus, vt demon­<lb/>&longs;trauimus in primo lib.1. Igitur maxima determinatio, quæ po&longs;&longs;it inferri <lb/>à plano e&longs;t in ip&longs;a perpendiculari. </s> </p> <p id="N1F5E6" type="main"> <s id="N1F5E8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N1F5F4" type="main"> <s id="N1F5F6"><!-- NEW --><emph type="italics"/>Hinc, &longs;i linea incidentiæ e&longs;t perpendicularis GD, linea quoque reflexionis <lb/>e&longs;t eadem DG<emph.end type="italics"/>; </s> <s id="N1F601"><!-- NEW -->quia huic e&longs;t maximum impedimentum, quia &longs;cilicet e&longs;t <lb/>maximus ictus; igitur maxima determinatio per Th. 25. &longs;ed maxima e&longs;t <lb/>illa, quâ mobile per ip&longs;am perpendicularem DG à puncto contactus D <lb/>retorquetur per Th.26. Igitur &longs;i linea incidentiæ, &c. </s> <s id="N1F60B">quod erat proban­<lb/>dum. </s> <s id="N1F610"><!-- NEW -->Probatur præterea, quia &longs;i linea incidentiæ e&longs;t perpendicularis <lb/>GD, non e&longs;t potior ratio, cur linea reflexionis inclinet dextror&longs;um ver­<lb/>&longs;us A, quàm &longs;ini&longs;tror&longs;um ver&longs;us H; igitur debet e&longs;&longs;e perpendicu­<lb/>laris. </s> </p> <p id="N1F61A" type="main"> <s id="N1F61C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N1F628" type="main"> <s id="N1F62A"><!-- NEW --><emph type="italics"/>Si linea incidentiæ cadat obliquè in planum, linea reflexionis non erit per­<lb/>pendicularis<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it linea incidentia AD, linea reflexionis non e&longs;t per­<lb/>pendicularis DG; quia tunc non e&longs;t maximus ictus, nec maximum im­<lb/>pedimentum per Th.23.igitur nec maxima determinatio per Theor.24. <lb/>igitur non fit per ip&longs;am perpendicularem DG per Th. 26. </s> </p> <p id="N1F63F" type="main"> <s id="N1F641"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N1F64D" type="main"> <s id="N1F64F"><!-- NEW --><emph type="italics"/>Hinc linea reflexionis, quæ &longs;equitur lineam incidentiæ obliquè cadentem in <lb/>planum non tantùm determinatur à plane reflectente &longs;ed participat aliquid de <lb/>priori determinatione.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it linea incidentiæ AD, linea reflexionis <lb/>DH; </s> <s id="N1F662"><!-- NEW -->non tantùm determinatur hæc linea à plano FB, alioqui e&longs;&longs;et DG, <lb/>nec e&longs;t eadem cum prima; alioqui e&longs;&longs;et DE, &longs;ed partim determinatur à <lb/>plano FB per DG partimque reti nec aliquid primæ determinationis, & <lb/>ex vtraque fit DH, vt con&longs;tat, quia quò linea incidentiæ e&longs;t obliquior, <lb/>planum minùs determinat per Th. 25. </s> </p> <pb pagenum="243" xlink:href="026/01/275.jpg"/> <p id="N1F672" type="main"> <s id="N1F674"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N1F680" type="main"> <s id="N1F682"><!-- NEW --><emph type="italics"/>Hinc quâ proportione planum minùs confert ad nouam determinationem, <lb/>plùs remanet prioris determinationis; </s> <s id="N1F68A"><!-- NEW -->quò verò plùs illud confert, huius minùs <lb/>restat<emph.end type="italics"/>; </s> <s id="N1F693"><!-- NEW -->hinc, cum planum totam confert <expan abbr="nouã">nouam</expan> <expan abbr="determination&etilde;">determinationem</expan> vt in per­<lb/>pendiculari DD, nihil prioris remanet; </s> <s id="N1F6A1"><!-- NEW -->hinc &longs;i linea incidentiæ &longs;it pa­<lb/>rallela plano BF nulla fiet noua determinatio, tota priore intacta; </s> <s id="N1F6A7"><!-- NEW -->&longs;i ve­<lb/>rò &longs;it perpendicularis GD, tota determinatio e&longs;t noua, & nihil prioris <lb/>remanet; &longs;i demum lineæ incidentiæ &longs;int aliæ, confert vtrumque ad no­<lb/>uam determinationem pro rata. </s> </p> <p id="N1F6B1" type="main"> <s id="N1F6B3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N1F6BF" type="main"> <s id="N1F6C1"><!-- NEW --><emph type="italics"/>Si pellatur mobile per AD in planum FB, determinatio lineæ reflexionis <lb/>erit qua&longs;i mixta &longs;inistror&longs;um<emph.end type="italics"/>; </s> <s id="N1F6CC"><!-- NEW -->&longs;i enim ex D propagaretur motus in E rectè <lb/>&longs;ini&longs;tror&longs;um acquireret DF in linea BF, vt patet; </s> <s id="N1F6D2"><!-- NEW -->igitur &longs;i &longs;it linea inci­<lb/>dentiæ AD, noua determinatio per DH con&longs;tabit partim ex eo, quòd <lb/>planum reflectens confert partim ex eo, quod remanet prioris determi­<lb/>nationis, quod re&longs;pondet DF, & ex eo quod confert planum FB, quod <lb/>re&longs;pondet DP; </s> <s id="N1F6DE"><!-- NEW -->quia ictus per AD e&longs;t ad ictum per GD, vt PD ad DP <lb/>vel DG; </s> <s id="N1F6E4"><!-- NEW -->&longs;ed e&longs;t eadem ratio impedimenti eademque determinationis <lb/>per Theoremata &longs;uperiora; atqui ex DPDF fit DHGO. igitur deter­<lb/>minatio lineæ reflexæ e&longs;t mixta, quod erat probandum. </s> </p> <p id="N1F6EC" type="main"> <s id="N1F6EE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N1F6FA" type="main"> <s id="N1F6FC"><!-- NEW --><emph type="italics"/>Hinc decre&longs;cit determinatio, quam confert planum iuxta rationem &longs;inuum <lb/>ver&longs;orum in<emph.end type="italics"/> GD. v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i &longs;it linea incidentiæ AD; </s> <s id="N1F70B"><!-- NEW -->ducatur APH paral­<lb/>lela FB, determinatio quam confert planum, decre&longs;cit &longs;inu ver&longs;o PG; </s> <s id="N1F711"><!-- NEW -->&longs;i <lb/>verò &longs;it linea incidentiæ ID, decre&longs;cit &longs;inu ver&longs;o LG; atque ita dein­<lb/>ceps; at verò cre&longs;cit portio prioris determinationis lineæ incidentiæ <lb/>iuxta rationem &longs;inuum rectorum in DB v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i &longs;it linea incidentiæ AD, <lb/>cre&longs;cit &longs;inu recto AP æquali BD &longs;i &longs;it IL cre&longs;cit &longs;inu recto IL vel RD. </s> </p> <p id="N1F721" type="main"> <s id="N1F723"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N1F72F" type="main"> <s id="N1F731"><!-- NEW --><emph type="italics"/>Hinc angulus reflexionis e&longs;t æqualis angulo incidentiæ, & hoc e&longs;t principium <lb/>po&longs;itiuum huius æqualitatis angulorum.<emph.end type="italics"/> &longs;it enim linea incidentiæ AD, du­<lb/>catur APH, AB, HF; </s> <s id="N1F73E"><!-- NEW -->certè DF & DB &longs;unt æquales APPH; </s> <s id="N1F742"><!-- NEW -->item­<lb/>que ABPDHF &longs;unt æquales; </s> <s id="N1F748"><!-- NEW -->atqui determinatio lineæ reflexionis <lb/>e&longs;t mixta ex DFH; </s> <s id="N1F74E"><!-- NEW -->igitur erit DH; </s> <s id="N1F752"><!-- NEW -->&longs;ed triangula DFH, DAB &longs;unt <lb/>æqualia & anguli HDFADB &longs;unt æquales: </s> <s id="N1F758"><!-- NEW -->&longs;imiliter &longs;it linea inciden­<lb/>tiæ ID, ducatur IN parallela AHIRNM; </s> <s id="N1F75E"><!-- NEW -->certè duo anguli IDR, <lb/>NDM &longs;unt æquales; </s> <s id="N1F764"><!-- NEW -->idem dico de omnibus aliis lineis incidentiæ, & <lb/>hæc e&longs;t vera ratio po&longs;itiua à priori, de qua plura infrà; </s> <s id="N1F76A"><!-- NEW -->non dee&longs;t etiam <lb/>negatiua, quia &longs;cilicet po&longs;ita linea incidentiæ AD cùm &longs;ini&longs;tror&longs;um &longs;int <lb/>infiniti anguli inæquales angulo incidentiæ; </s> <s id="N1F772"><!-- NEW -->non e&longs;t potior ratio, cur <lb/>per vnum fiat quàm per alium, & cum &longs;it tantùm vnus æqualis HDM in <lb/>eodem &longs;cilicet plano; </s> <s id="N1F77A"><!-- NEW -->certè per illum fieri debet; </s> <s id="N1F77E"><!-- NEW -->quippe quod vnum <lb/>e&longs;t, determinatum e&longs;t, vt &longs;æpè diximus aliàs; </s> <s id="N1F784"><!-- NEW -->nec e&longs;t quòd aliqui delica-<pb pagenum="244" xlink:href="026/01/276.jpg"/>tioris &longs;thomachi rationem hanc negatiuam, cum tanta nau&longs;ea re&longs;puant, <lb/>cum optima &longs;it; </s> <s id="N1F78F"><!-- NEW -->nec vlli fallaciæ &longs;ubiiciatur, non tamen &longs;olitariam e&longs;&longs;e <lb/>oportuit; quippe effectus po&longs;itiuus per principium po&longs;itiuum ad &longs;uam <lb/>cau&longs;am reducendus e&longs;t. </s> </p> <p id="N1F797" type="main"> <s id="N1F799"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N1F7A5" type="main"> <s id="N1F7A7"><!-- NEW --><emph type="italics"/>Hinc vides e&longs;&longs;e &longs;emper quatuor angulos æquales,<emph.end type="italics"/> &longs;cilicet, angulum inci­<lb/>dentiæ, angulum reflexionis & duos his oppo&longs;itos; allos verò quatuor <lb/>etiam inter &longs;e æquales, &longs;cilicet duos angulos complementi & duos his <lb/>oppo&longs;itos. </s> </p> <p id="N1F7B6" type="main"> <s id="N1F7B8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N1F7C4" type="main"> <s id="N1F7C6"><!-- NEW --><emph type="italics"/>Hinc quoque reiicies illos, qui nolunt in reflexione impetum produci in mo­<lb/>bili à plano reflectente<emph.end type="italics"/>; quod reuerâ, &longs;i fieret nulla e&longs;&longs;et ratio æqualitatis <lb/>angulorum incidentiæ, & reflexionis, reiicies quoque aliquos apud Mer­<lb/>&longs;ennum in phænom. </s> <s id="N1F7D5"><!-- NEW -->Balli&longs;t. <!-- REMOVE S-->prop. 24. qui ponunt duo qualitatum gene­<lb/>ra, quarum aliæ mobile firmiter affigant plano, aliæ à plano remoueant, <lb/>quod plu&longs;quàm ridiculum e&longs;t; </s> <s id="N1F7DF"><!-- NEW -->itemque alios ibidem, qui nolunt circa <lb/>punctum reflexionis ab impre&longs;&longs;ione mobilis fo&longs;&longs;ulam fieri, &longs;ed non &longs;ine <lb/>compre&longs;&longs;ione, cuius deinde vi repellitur idem mobile; </s> <s id="N1F7E7"><!-- NEW -->&longs;ed in duro mar­<lb/>more nullum omninò apparet ve&longs;tigium huius fo&longs;&longs;ulæ, adde quod &longs;i hoc <lb/>e&longs;&longs;et, &longs;emper reflexio fieret per ip&longs;am perpendicularem; </s> <s id="N1F7EF"><!-- NEW -->quod vero perti­<lb/>net ad illas qualitates magneticas, quarum aliæ retinent, aliæ repellunt <lb/>mobile, pœnitus in hoc ca&longs;u in&longs;ul&longs;æ &longs;unt; </s> <s id="N1F7F7"><!-- NEW -->alioqui etiam &longs;ine motu præ­<lb/>uio repellerent: vtrum verò in magnete admittendæ &longs;int, fusè di&longs;puta­<lb/>bimus &longs;uo loco. </s> </p> <p id="N1F7FF" type="main"> <s id="N1F801"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N1F80D" type="main"> <s id="N1F80F"><emph type="italics"/>Ex hac angulorum æqualitate tùm Captotrica infinita ferè Theoremata de­<lb/>monstrat in radiis vi&longs;ilibus, in &longs;peculis v&longs;toriis, tùm Echometria in reflexione <lb/>&longs;onorum.<emph.end type="italics"/></s> <s id="N1F81A"><!-- NEW --> Et verò noua Catoptrica pote&longs;t e&longs;&longs;e in motu, quæ eadem pror­<lb/>&longs;us demon&longs;trabit, tùm in &longs;peculis parabolicis, à quibus omnia mi&longs;&longs;ilia <lb/>projecta per parallelas axi Parabolæ in idem punctum reflectentur; </s> <s id="N1F822"><!-- NEW -->vel <lb/>Ellipticis, à quibus omnia mi&longs;&longs;ilia projecta à dato puncto per omnes li­<lb/>neas ad idem punctum reflectentur; </s> <s id="N1F82A"><!-- NEW -->vel Hyperbolicis, à quibus mi&longs;&longs;ilia <lb/>projecta per plures lineas ad idem punctum ad aliud punctum omnes re­<lb/>flectuntur; </s> <s id="N1F832"><!-- NEW -->vel Sphæricis concauis, à quibus mi&longs;&longs;ilia projecta per plures <lb/>lineas decu&longs;&longs;atas in eodem puncto ad idem punctum reflectuntur; vel <lb/>Sphæricis conuexis, à quibus mi&longs;&longs;ile proiectum à quolibet puncto dato <lb/>ad quodlibet aliud datum reflectitur. </s> <s id="N1F83C"><!-- NEW -->Ratio e&longs;t, quia in circulo &longs;unt om­<lb/>nia plana; </s> <s id="N1F842"><!-- NEW -->quælibet enim Tangens planum e&longs;t; &longs;iue denique in Cylin­<lb/>dricis, Conicis, &c. </s> <s id="N1F848"><!-- NEW -->quæ omnia ex principiis Catoptricis demon&longs;trari <lb/>po&longs;&longs;unt: </s> <s id="N1F84E"><!-- NEW -->adde &longs;i vis in hac Catoptrica ver&longs;atos e&longs;&longs;e debere, qui pilâ lu­<lb/>dunt, quos nunquam falleret ictus, &longs;i hanc rationem angulorum non mo­<lb/>dò perfectè callerent, verùm etiam ad praxim reducerent: immò po&longs;&longs;et <lb/>e&longs;&longs;e aliqua portio muri talis figuræ, vt &longs;emper inde reflexa pila per da­<lb/>tum cuniculum rectà traiiceretur. </s> </p> <pb pagenum="245" xlink:href="026/01/277.jpg"/> <p id="N1F85E" type="main"> <s id="N1F860"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N1F86C" type="main"> <s id="N1F86E"><!-- NEW --><emph type="italics"/>In reflexione destruitur aliquid impetus, &longs;i talis &longs;it vtriu&longs;que determina­<lb/>tionis pugna, vt aliquid impetus &longs;it frustrà<emph.end type="italics"/>; </s> <s id="N1F879"><!-- NEW -->vt con&longs;tat ex his, quæ diximus <lb/>libro primo; </s> <s id="N1F87F"><!-- NEW -->con&longs;tat autem in reflexione e&longs;&longs;e determinationum pugnam <lb/>per Th 31. & 32. pugnat enim &longs;uo modo prior determinatio per GD <lb/>cum &longs;ecunda oppo&longs;ita per DG; igitur aliquid impetus de&longs;truitur, &longs;i ex <lb/>tali pugna aliquid &longs;it fru&longs;trà. </s> <s id="N1F889"><!-- NEW -->Ob&longs;eruabis autem eundem impetum in eo­<lb/>dem mobili cum duplici determinatione perinde &longs;e habere in ordine <lb/>ad nouam, vt patet, lineam, atque &longs;i e&longs;&longs;ent duo impetus in ratione deter­<lb/>minationum: vtrùm autem aliquid impetus &longs;it fru&longs;trà per &longs;e, determina­<lb/>bimus infrà. </s> </p> <p id="N1F895" type="main"> <s id="N1F897"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N1F8A3" type="main"> <s id="N1F8A5"><!-- NEW --><emph type="italics"/>Si totus impetus destrueretur nulla e&longs;&longs;et reflexio<emph.end type="italics"/>; </s> <s id="N1F8AE"><!-- NEW -->quod maximè e&longs;&longs;et ab­<lb/>&longs;urdum & incommodum toti naturæ; </s> <s id="N1F8B4"><!-- NEW -->&longs;i verò nullus impetus de&longs;truere­<lb/>tur, &longs;eu per &longs;e, &longs;eu per accidens, daretur motus perpetuus; </s> <s id="N1F8BA"><!-- NEW -->quippe mo­<lb/>bile ad eandem altitudinem a&longs;cenderet po&longs;t reflexionem, iterumque de­<lb/>&longs;cendens ad <expan abbr="eãdem">eandem</expan> a&longs;cenderet atque ita deinceps; igitur motus e&longs;&longs;et <lb/>perpetuus, & nunquam corpus illud quie&longs;ceret, quod e&longs;t contra in&longs;titu­<lb/>tum naturæ. </s> </p> <p id="N1F8CA" type="main"> <s id="N1F8CC"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1F8D8" type="main"> <s id="N1F8DA">Ob&longs;erua primò ex hypothe&longs;i certa haberi, dari motum reflexum, ex <lb/>qua colligo totum impetum non de&longs;trui. </s> <s id="N1F8DF">Secundò ex hypothe&longs;i certa <lb/>haberi, motum reflexum e&longs;&longs;e minorem directo vlteriùs propagato, vt <lb/>con&longs;tat experientiâ, ex qua colligo aliquam portionem impetus de&longs;trui, <lb/>&longs;altem per accidens propter compre&longs;&longs;ionem, & alli&longs;ionem partium. </s> </p> <p id="N1F8E8" type="main"> <s id="N1F8EA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N1F8F6" type="main"> <s id="N1F8F8"><!-- NEW --><emph type="italics"/>Maior e&longs;t determinatio, quæ confertur à plano mobili per lineam perpendi­<lb/>cularem incidenti, quàm prior, quæ inerat mobili<emph.end type="italics"/>; </s> <s id="N1F903"><!-- NEW -->probatur, quia nec e&longs;t <lb/>minor, nec æqualis, non minor; </s> <s id="N1F909"><!-- NEW -->alioquin prior vinceret; </s> <s id="N1F90D"><!-- NEW -->non æqualis, <lb/>quia neutra præualeret; </s> <s id="N1F913"><!-- NEW -->igitur e&longs;t maior; </s> <s id="N1F917"><!-- NEW -->&longs;i vtraque determinatio e&longs;&longs;et <lb/>aqualis totus impetus de&longs;trui deberet; </s> <s id="N1F91D"><!-- NEW -->igitur eadem e&longs;t proportio impe­<lb/>tus remanentis, quæ e&longs;t mixtæ determinationis ex priori, & noua; </s> <s id="N1F923"><!-- NEW -->nul­<lb/>lus enim impetus e&longs;&longs;e pote&longs;t &longs;ine determinatione; </s> <s id="N1F929"><!-- NEW -->igitur &longs;i tota perit de­<lb/>terminatio, totus etiam perit impetus, qui illi re&longs;pondet; </s> <s id="N1F92F"><!-- NEW -->& &longs;i remanet <lb/>aliquid determinationis mixtæ, aliquid etiam impetus remanet, qui e&longs;t <lb/>ad priorem impetum, vt hæc determinatio re&longs;idua ad priorem determi­<lb/>nationem; quantum verò remaneat prioris impetus, dicam infrà. </s> </p> <p id="N1F939" type="main"> <s id="N1F93B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N1F947" type="main"> <s id="N1F949"><!-- NEW --><emph type="italics"/>Determinatio per DG à plano e&longs;t dupla determinationis prioris per lineam <lb/>incidentiæ GD<emph.end type="italics"/>; quod &longs;ic demon&longs;tro; </s> <s id="N1F954"><!-- NEW -->&longs;it linea incidentiæ ID, linea re­<lb/>flexionis erit DN, &longs;cilicet ad angulos æquales, per Th. 33. &longs;it autem an­<lb/>gulus NDM 30. graduum, & NDG 60. ducatur NO parallela GD; </s> <s id="N1F95C"><!-- NEW --><pb pagenum="246" xlink:href="026/01/278.jpg"/>tùm ID producatur in O, denique ducatur NG: </s> <s id="N1F964"><!-- NEW -->prima determinatio <lb/>lineæ incidentiæ ID, e&longs;t per DO, determinatio plani e&longs;t per DG; </s> <s id="N1F96A"><!-- NEW -->&longs;ed <lb/>DO e&longs;t æqualis DG; </s> <s id="N1F970"><!-- NEW -->nam DON, DNG &longs;unt æquilatera æqualia; </s> <s id="N1F974"><!-- NEW --><lb/>hinc determinatio mixta e&longs;t per DN, diuidens angulum GDO bifa­<lb/>riam; </s> <s id="N1F97B"><!-- NEW -->igitur &longs;i &longs;it linea incidentiæ ID & angulus ID B. 30. graduum, <lb/>æqualis e&longs;t determinatio plani determinationi prioris lineæ; </s> <s id="N1F981"><!-- NEW -->hinc angu­<lb/>lus diuiditur æqualiter bifariam; </s> <s id="N1F987"><!-- NEW -->&longs;it verò linea incidentiæ AD produ­<lb/>cta v&longs;que ad E, linea reflexionis DH; </s> <s id="N1F98D"><!-- NEW -->ducatur HE; </s> <s id="N1F991"><!-- NEW -->a&longs;&longs;umatur DT <lb/>æqualis EH: </s> <s id="N1F997"><!-- NEW -->dico determinationem plani e&longs;&longs;e ad determinationem <lb/>prioris lineæ AD vel DE, vt DT ad DE; </s> <s id="N1F99D"><!-- NEW -->cum enim determinatio mix­<lb/>ta &longs;it per DH; </s> <s id="N1F9A3"><!-- NEW -->certè DH accedit propiùs ADDG, quàm ad DE; </s> <s id="N1F9A7"><!-- NEW -->igi­<lb/>tur determinatio per DG e&longs;t ad determinationem, per DE vt DT <lb/>æqualis HE ad DE; nam perinde &longs;e habent, atque &longs;i e&longs;&longs;ent duo impe­<lb/>tus determinati ad duas lineas, de quibus hoc ip&longs;um demon&longs;trauimus <lb/>tùm libro 1. Th.137. 138. 139. &c. </s> <s id="N1F9B3"><!-- NEW -->tùm lib.4. à Th. 1. ad Th.14.quippe <lb/>linea determinationis mixtæ e&longs;t diagonalis, vt &longs;æpè probauimus: </s> <s id="N1F9B9"><!-- NEW -->deinde <lb/>&longs;it linea incidentiæ per KD; </s> <s id="N1F9BF"><!-- NEW -->&longs;it DX linea reflexionis; </s> <s id="N1F9C3"><!-- NEW -->&longs;it XQ, ip&longs;ique <lb/>æqualis DZ, dico determinationem per DG e&longs;&longs;e ad determinationem <lb/>per DQ vt DZ ad DQ, &longs;ed XQ e&longs;t minor GS, vt con&longs;tat; </s> <s id="N1F9CB"><!-- NEW -->igitur quò <lb/>linea incidentiæ accedit propiùs ad perpendicularem GD, determinatio <lb/>plani e&longs;t maior, e&longs;tque vt chordæ NO, HE, <expan abbr="Xq;">Xque</expan> igitur &longs;i tandem li­<lb/>nea incidentiæ &longs;it perpendicularis GD, determinatio plani e&longs;t ad deter­<lb/>minationem lineæ incidentiæ, vt DY æqualis GS ad DG: </s> <s id="N1F9DB"><!-- NEW -->&longs;ed cum ex <lb/>Th.4. multa lux reliquis con&longs;equentibus immò & antecedentibus afful­<lb/>gere po&longs;&longs;it, paulò fu&longs;iùs explicandum, & demon&longs;trandum e&longs;&longs;e videtur: </s> <s id="N1F9E3"><!-- NEW --><lb/>itaque duobus modis, primò ex hypothe&longs;i anguli reflexionis æqualis an­<lb/>gulo incidentiæ, quod iam reuerâ præ&longs;titum e&longs;t; &longs;ed cum ex hoc Theo­<lb/>remate prædicta æqualitas angulorum reflexionis tanquam per princi­<lb/>pium immediatum po&longs;itiuum demon&longs;trari po&longs;&longs;it, ne &longs;it aliqua circuli <lb/>&longs;pecies, quo determinatio noua dupla prioris po&longs;ita linea incidentiæ <lb/>perpendiculari per æqualitatem anguli reflexionis, & hæc æqualitas per <lb/>illam eandem determinationem duplam demon&longs;tretur, aliam viam inire <lb/>oportet, vnde intima totius reflexionis principia eruantur, quod vt <lb/>fiat. </s> </p> <p id="N1F9F8" type="main"> <s id="N1F9FA"><!-- NEW -->Primò certum e&longs;t, corpus reflectens in perpendiculari, (quæ e&longs;t cum <lb/>linea incidentiæ terminata ad punctum contactus ducitur per centrum <lb/>grauitatis globi reflexi) certum e&longs;t inquam corpus reflectens in prædi­<lb/>cta linea aliquando cedere, aliquando non cedere; </s> <s id="N1FA04"><!-- NEW -->cedere autem dici­<lb/>tur cùm vel amouetur à corpore impacto, vel &longs;altem concutitur: <lb/>tunc autem nullo modo cedere dicitur, cum ab ictu nullo modo mo­<lb/>uetur. </s> </p> <p id="N1FA0E" type="main"> <s id="N1FA10"><!-- NEW -->Secundò, ce&longs;&longs;io, & re&longs;i&longs;tentia ita po&longs;&longs;unt comparari, vt vel ce&longs;&longs;io &longs;it <lb/>æqualis re&longs;i&longs;tentiæ, vel ce&longs;&longs;io &longs;ine re&longs;i&longs;tentia, vel re&longs;i&longs;tentia &longs;ine ce&longs;&longs;ione: </s> <s id="N1FA16"><!-- NEW --><lb/>porrò tunc e&longs;t ce&longs;&longs;io tota, cum nulla e&longs;t re&longs;i&longs;tentia, quod tantum accide­<lb/>ret, &longs;i corpus moueretur in vacuo; </s> <s id="N1FA1D"><!-- NEW -->quippe nullum e&longs;t medium quamtum-<pb pagenum="247" xlink:href="026/01/279.jpg"/>uis rarum, & tenue, quod aliquantulum non re&longs;i&longs;tat, vt clarum e&longs;t; </s> <s id="N1FA26"><!-- NEW -->tunc <lb/>quoque e&longs;t re&longs;i&longs;tentia &longs;ine ce&longs;&longs;ione, &longs;eu tota re&longs;i&longs;tentia, cum ip&longs;um cor­<lb/>pus re&longs;i&longs;tens nullo modo cedit; </s> <s id="N1FA2E"><!-- NEW -->id e&longs;t nullo modo mouetur ab ictu; </s> <s id="N1FA32"><!-- NEW -->neque <lb/>enim excogitari pote&longs;t maior re&longs;i&longs;tentia; </s> <s id="N1FA38"><!-- NEW -->denique tunc e&longs;t æqualis ce&longs;­<lb/>&longs;io re&longs;i&longs;tentiæ, cum ip&longs;um corpus, in quod aliud impingitur (vocetur re­<lb/>flectens) tantùm cedit quantum re&longs;i&longs;tit; </s> <s id="N1FA40"><!-- NEW -->cedit autem per motum; </s> <s id="N1FA44"><!-- NEW -->igitur <lb/>&longs;i reflectenti imprimitur æqualis motus ab impacto reflectens æqualiter <lb/>cedit, & re&longs;i&longs;tit, &longs;i minor minùs cedit, & plùs re&longs;i&longs;tit, &longs;i nullus nullo mo­<lb/>do cedit, &longs;ed tantùm re&longs;i&longs;tit; &longs;i maior plùs cedit, & minùs re&longs;i&longs;tit, &longs;cili­<lb/>cet in infinitum, donec tandem in vacuo &longs;it tantum ce&longs;&longs;io, nulla re&longs;i­<lb/>&longs;tentia. </s> </p> <p id="N1FA52" type="main"> <s id="N1FA54"><!-- NEW -->Tertiò, tunc impactum motum æqualem imprimit reflectenti, cum <lb/>impactum æquale e&longs;t reflectenti, tùm mole, tùm pondere v.g. <!-- REMOVE S-->globus A <lb/>impactus in globum B eiu&longs;dem materiæ, & diametri, modo nullus fiat <lb/>attritus partium, &longs;eu compre&longs;&longs;io, &longs;itque linea directionis connectens <lb/>centra per punctum contactus, quod in primo libro iam demon&longs;tratum <lb/>e&longs;t; </s> <s id="N1FA64"><!-- NEW -->cum enim totus impetus globi A agat, & quantum pote&longs;t; </s> <s id="N1FA68"><!-- NEW -->certè pro­<lb/>ducit æqualem; </s> <s id="N1FA6E"><!-- NEW -->nec enim aliunde determinari pote&longs;t æqualitas effectus <lb/>quàm ab æqualitate cau&longs;æ po&longs;itis ii&longs;dem circum&longs;tantiis, & cum impetus <lb/>in B impre&longs;&longs;us di&longs;tribuatur tot partibus quot producens æqualis in A, <lb/>vterque impetus e&longs;t æquè inten&longs;us; </s> <s id="N1FA78"><!-- NEW -->igitur æquè velox motus per &longs;e; </s> <s id="N1FA7C"><!-- NEW -->cum <lb/>per accidens aliquando &longs;ecus accidat; </s> <s id="N1FA82"><!-- NEW -->&longs;i verò reflectens &longs;it minor, idem <lb/>impetus paucioribus partibus di&longs;tribuitur; </s> <s id="N1FA88"><!-- NEW -->igitur inten&longs;ior e&longs;t; </s> <s id="N1FA8C"><!-- NEW -->igitur <lb/>velocior motus, &longs;ecus verò cum maior e&longs;t, donec tandem tanta &longs;it moles, <lb/>vt plura &longs;int puncta in reflectente, quàm &longs;int in impacto puncta impe­<lb/>tus; tunc enim nullus imprimitur impetus, vt con&longs;tat ex dictis lib. 1. <!-- KEEP S--></s> </p> <p id="N1FA97" type="main"> <s id="N1FA99"><!-- NEW -->Quartò, quod autem &longs;it æqualis re&longs;i&longs;tentia, & ce&longs;&longs;io globi B æqualis <lb/>globo A etiam certum e&longs;t; </s> <s id="N1FA9F"><!-- NEW -->tùm quia, &longs;i æqualiter mouetur, æqualiter ce­<lb/>dit, vt iam dixi &longs;i æqualiter cedit, æqualiter re&longs;i&longs;tit; </s> <s id="N1FAA5"><!-- NEW -->nam quâ proportio­<lb/>ne minùs cedit, plùs re&longs;i&longs;tit; </s> <s id="N1FAAB"><!-- NEW -->igitur qua proportione ce&longs;&longs;io augetur, re&longs;i­<lb/>&longs;tentia imminuitur: præterea cum re&longs;i&longs;tat per &longs;uam entitatem impene­<lb/>trabilem, duram &c. </s> <s id="N1FAB3"><!-- NEW -->certè &longs;i e&longs;t æqualis entitas, e&longs;t æqualis re&longs;i&longs;tentia; </s> <s id="N1FAB7"><!-- NEW --><lb/>quod etiam videmus in corporibus immer&longs;is eiu&longs;dem grauitatis cum <lb/>medio, ita vt tot &longs;int partes impellentes, quot impul&longs;æ; </s> <s id="N1FABE"><!-- NEW -->denique illud <lb/>experimentum quo videmus globum A impactum in B æqualem per li­<lb/>neam connectentem centra immobilem &longs;i&longs;tere, rem i&longs;tam euincit; </s> <s id="N1FAC6"><!-- NEW -->nam <lb/>ideo &longs;i&longs;tit, quia e&longs;t æqualis determinatio noua priori; </s> <s id="N1FACC"><!-- NEW -->nam vt &longs;e habet <lb/>re&longs;i&longs;tentia reflectentis, ita &longs;e habet noua determinatio, quam &longs;uo modo <lb/>confert impacto, vt &longs;uprà demon&longs;tratum e&longs;t: </s> <s id="N1FAD4"><!-- NEW -->& cùm &longs;int ad lineas op­<lb/>po&longs;itas ex diametro hæ duæ determinationes, neutra præualere pote&longs;t; <lb/>igitur nece&longs;&longs;e e&longs;t &longs;i&longs;tere globum impactum. </s> </p> <p id="N1FADC" type="main"> <s id="N1FADE"><!-- NEW -->Quintò, certum e&longs;t determinationem nouam e&longs;&longs;e iuxta proportionem <lb/>re&longs;i&longs;tentiæ, & hanc iuxta proportionem minoris ce&longs;&longs;ionis; </s> <s id="N1FAE4"><!-- NEW -->vnde cum <lb/>nulla e&longs;t re&longs;i&longs;tentia, &longs;ed tantùm ce&longs;sio, nulla pror&longs;us e&longs;t noua determina­<lb/>tio igitur à termino nullius re&longs;i&longs;tentiæ, & totius ce&longs;sionis ad terminum <pb pagenum="248" xlink:href="026/01/280.jpg"/>æqualis ce&longs;&longs;ionis, & re&longs;i&longs;tentiæ, acquiritur tantùm noua determinatio <lb/>æqualis priori: </s> <s id="N1FAF3"><!-- NEW -->&longs;imiliter à termino nullius ce&longs;&longs;ionis, & totius re&longs;i&longs;tentiæ <lb/>ad terminum æqualis re&longs;i&longs;tentiæ, & ce&longs;&longs;ionis, acquiritur tantùm æqualis <lb/>ce&longs;&longs;io; </s> <s id="N1FAFB"><!-- NEW -->&longs;ed qua proportione cre&longs;cit ce&longs;&longs;io, imminuitur re&longs;i&longs;tentia, & vi­<lb/>ci&longs;sim; </s> <s id="N1FB01"><!-- NEW -->igitur cum æqualis ce&longs;sio, & re&longs;i&longs;tentia &longs;int in communi medio; </s> <s id="N1FB05"><!-- NEW --><lb/>tantùm enim e&longs;t ab æquali re&longs;i&longs;tentia & æquali ce&longs;sione ad totam ce&longs;­<lb/>&longs;ionem, & nullam re&longs;i&longs;tentiam, quantùm e&longs;t ab æquali re&longs;i&longs;tentia & ce&longs;­<lb/>&longs;ione æquali ad totam re&longs;i&longs;tentiam, & nullam ce&longs;sionem; </s> <s id="N1FB0E"><!-- NEW -->& cum à nulla <lb/>re&longs;i&longs;tentia ad æqualem acquiritur noua determinatio æqualis priori; </s> <s id="N1FB14"><!-- NEW -->cer­<lb/>tè ab æquali ad totam acquiretur <expan abbr="tantũdem">tantundem</expan> determinationis nouæ; igi­<lb/>tur tunc erit dupla prioris, quod erat demon&longs;trandum. </s> </p> <p id="N1FB20" type="main"> <s id="N1FB22"><!-- NEW -->Sextò, præterea globus A impactus &longs;ine acce&longs;sione noui impetus non <lb/>pote&longs;t velociùs moueri, quàm antè moueretur; </s> <s id="N1FB28"><!-- NEW -->&longs;ed per reflexionem non <lb/>acquirit maiorem impetum, vt con&longs;tat; </s> <s id="N1FB2E"><!-- NEW -->igitur velociùs, quàm antè non <lb/>mouetur; </s> <s id="N1FB34"><!-- NEW -->igitur &longs;i con&longs;ideretur globus A impactus; </s> <s id="N1FB38"><!-- NEW -->&longs;i e&longs;t æqualis re&longs;i­<lb/>&longs;tentia, nullo modo mouetur; </s> <s id="N1FB3E"><!-- NEW -->&longs;i e&longs;t maior re&longs;i&longs;tentia, &longs;ed non tota; </s> <s id="N1FB42"><!-- NEW -->mo­<lb/>uetur quidem motu reflexo; </s> <s id="N1FB48"><!-- NEW -->&longs;ed inæquali priori, &longs;i adhuc maior moue­<lb/>tur etiam, &longs;ed velociore motu, donec tandem in tota re&longs;i&longs;tentia toto <lb/>priore motu moueatur per &longs;e, vt dicemus paulò pò&longs;t; </s> <s id="N1FB50"><!-- NEW -->&longs;i verò &longs;it minor <lb/>re&longs;i&longs;tentia ce&longs;sione, mouetur quidem per eandem lineam, &longs;ed tardiore <lb/>motu, &longs;i adhuc minor mouetur quoque, &longs;ed velociore motu, donec tan­<lb/>dem in nulla re&longs;i&longs;tentia &longs;it totus prior motus; </s> <s id="N1FB5A"><!-- NEW -->&longs;i verò con&longs;ideretur glo­<lb/>bus reflectens, &longs;i e&longs;t æqualis re&longs;i&longs;tentia mouetur æquali motu; &longs;i maior <lb/>minore; &longs;i tota nullo; </s> <s id="N1FB62"><!-- NEW -->&longs;i vero &longs;it minor re&longs;i&longs;tentia mouetur motu velo­<lb/>ciore, atque ita deinceps; &longs;i nulla qua&longs;i infinito: </s> <s id="N1FB68"><!-- NEW -->dico qua&longs;i, quia &longs;i va­<lb/>cuum moueri po&longs;&longs;et per impo&longs;sibile, certè cum non re&longs;i&longs;tat, infinitè ce­<lb/>deret; igitur infinito motu qua&longs;i moueretur. </s> </p> <p id="N1FB70" type="main"> <s id="N1FB72"><!-- NEW -->Septimò, vnde vides ab illo communi medio ver&longs;us vtrumque extre­<lb/>mum cre&longs;cere &longs;emper motum globi impacti; </s> <s id="N1FB78"><!-- NEW -->donec tandem in vtroque <lb/>extremo æquali motu moueatur, quo iam priùs mouebatur in linea inci­<lb/>dentiæ; </s> <s id="N1FB80"><!-- NEW -->at verò globi reflectentis ver&longs;us extremum nullius ce&longs;sionis im­<lb/>minui motum, donec tandem in illo extremo nullus &longs;it; </s> <s id="N1FB86"><!-- NEW -->cre&longs;cere vero <lb/>ver&longs;us aliud extremum, donec tandem in illo infinitus &longs;it, eo modo, quo <lb/>diximus, id e&longs;t infinita ce&longs;sio, quam accipio ad in&longs;tar motus infinitæ ve­<lb/>locitatis; quemadmodum accipi pote&longs;t nulla ce&longs;sio, &longs;eu tota re&longs;i&longs;tentia <lb/>ad in&longs;tar motus infinitæ tarditatis. </s> </p> <p id="N1FB92" type="main"> <s id="N1FB94"><!-- NEW -->Octauò, globus impactus imprimit &longs;emper æqualem impetum refle­<lb/>ctenti, qui pro diuer&longs;a huius mole diuer&longs;um modum præ&longs;tat; </s> <s id="N1FB9A"><!-- NEW -->&longs;i refle­<lb/>ctens æqualis e&longs;t æqualem, &longs;i maior minorem, &longs;i minor maiorem; </s> <s id="N1FBA0"><!-- NEW -->quippe <lb/>idem impetus in paucioribus partibus facit maiorem motum, in totidem <lb/>æqualem, in pluribus minorem, donec tandem &longs;i plures &longs;int partes &longs;ub­<lb/>jecti quàm partes impetus, nullus &longs;it motus; igitur nullus impetus, vt <lb/>con&longs;tat ex his, quæ diximus lib.1. <!-- KEEP S--></s> </p> <p id="N1FBAD" type="main"> <s id="N1FBAF"><!-- NEW -->Nonò, hinc motus reflexus in perpendiculari minor e&longs;t ea parte mo­<lb/>tus, quæ reflectenti imprimitur; </s> <s id="N1FBB5"><!-- NEW -->vel enim imprimitur motus æqualis, <pb pagenum="249" xlink:href="026/01/281.jpg"/>vel inæqualis, &longs;i æqualis, certè toto motu multatur globus impactus; </s> <s id="N1FBBE"><!-- NEW -->&longs;i <lb/>inæqualis, vel minor, vel maior; </s> <s id="N1FBC4"><!-- NEW -->&longs;i minor, certè e&longs;t aliquis motus refle­<lb/>xus æqualis priori minùs ea parte, quæ reflectenti imprimitur, donec <lb/>tandem nullus imprimatur motus; </s> <s id="N1FBCC"><!-- NEW -->tunc enim reflexus e&longs;t priori æqua­<lb/>lis; &longs;i verò maior imprimitur, fortè nullus e&longs;t reflexus po&longs;ito &longs;cilicet ra­<lb/>dio incidentiæ perpendiculari, minor tamen erit idem motus globi im­<lb/>pacti vlteriùs per eandem lineam propagati. </s> <s id="N1FBD6"><!-- NEW -->v.g.&longs;i &longs;it duplus detrahitur <lb/>priori motui 1/2, &longs;i triplus 1/3, &longs;i quadruplus 1/4, atque ita deinceps; &longs;i de­<lb/>nique infinities velocior ex &longs;uppo&longs;itione impo&longs;sibili detrahitur aliquid, <lb/>quod habet ad priorem motum proportionem minoris inæqualitatis in­<lb/>finitam. </s> </p> <p id="N1FBE2" type="main"> <s id="N1FBE4"><!-- NEW -->Decimò, ex his rectè concludi pote&longs;t non produci infinita puncta im­<lb/>petus, nec e&longs;&longs;e infinitas partes &longs;ubjecti actu; </s> <s id="N1FBEA"><!-- NEW -->alioqui punctum mouere­<lb/>tur motu infinito, qui repugnat: </s> <s id="N1FBF0"><!-- NEW -->præterea nullum e&longs;&longs;et corpus quamtum­<lb/>nis magnum, cui modico ictu non imprimatur impetus, &longs;i impetus con­<lb/>flat infinitis partibus; </s> <s id="N1FBF8"><!-- NEW -->quare in vtraque progre&longs;sione &longs;i&longs;tendum e&longs;t; <lb/>primò in nulla ce&longs;sione & tota re&longs;i&longs;tentia, cum &longs;cilicet plura &longs;unt pun­<lb/>cta &longs;ubjecti, quàm impetus. </s> <s id="N1FC00"><!-- NEW -->Secundò cum reflectens tantùm con&longs;tat <lb/>vnico puncto, in quo &longs;cilicet impetus finitus impre&longs;&longs;us præ&longs;tat veloci&longs;­<lb/>&longs;imum motum quem præ&longs;tare pote&longs;t; </s> <s id="N1FC08"><!-- NEW -->licèt enim dato quocunque motu <lb/>po&longs;sit dari velocior, non tamen cum dato impetu finito determinato &longs;i­<lb/>ne acce&longs;sione alterius; &longs;ed iam interruptam no&longs;trorum Theorematum &longs;e­<lb/>riem pro&longs;equamur. </s> </p> <p id="N1FC12" type="main"> <s id="N1FC14"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N1FC20" type="main"> <s id="N1FC22"><!-- NEW --><emph type="italics"/>Determinatio noua cuiu&longs;libet alterius anguli incidentiæ obliqui, vel acuti, <lb/>e&longs;t ad priorem, vt duplum &longs;inus recti eiu&longs;dem anguli ad &longs;inum totum.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <lb/> &longs;it radius incidentiæ AD in <expan abbr="planũ">planum</expan> immobile BDF: </s> <s id="N1FC35"><!-- NEW -->dico nouam de­<lb/>terminationem e&longs;&longs;e ad priorem, vt duplum AB, id e&longs;t BC ad DA. De­<lb/>mon&longs;tro; </s> <s id="N1FC3D"><!-- NEW -->cum enim ictus per AD obliquam &longs;it ad ictum per AB per­<lb/>pendicularem, vt AB ad AD, vt con&longs;tat ex dictis, tùm &longs;upra, tùm in lib. <lb/>de planis inclinatis; </s> <s id="N1FC45"><!-- NEW -->ictus enim habent eam proportionem, quam ha­<lb/>bent grauitationes; </s> <s id="N1FC4B"><!-- NEW -->&longs;ed grauitatio in inclinatam AD e&longs;t ad grauitatio­<lb/>nem in horizontalem DB, vt DB ad DA; </s> <s id="N1FC51"><!-- NEW -->igitur ictus inflictus plano <lb/>DB per inclinatam AD e&longs;t ad inflictum per ip&longs;am perpendicularem <lb/>GD vt PR æqualem AB ad DA; </s> <s id="N1FC59"><!-- NEW -->nam ictus in planum AD per GD <lb/>idem e&longs;t cum ictu in DB per AD: </s> <s id="N1FC5F"><!-- NEW -->&longs;imiliter &longs;it incidens KD, &longs;itque an­<lb/>gulus IDR æqualis KDG, ictus in ID per GD e&longs;t æqualis ictui in <lb/>DR per KD; </s> <s id="N1FC67"><!-- NEW -->&longs;unt enim GDI, KDR æquales; </s> <s id="N1FC6B"><!-- NEW -->&longs;ed ictus in ID e&longs;t, vt <lb/>grauitatio in eandem ID; </s> <s id="N1FC71"><!-- NEW -->hæc autem in inclinatam DI, ad aliam in <lb/>horizontalem DR vt DR ad DI; </s> <s id="N1FC77"><!-- NEW -->igitur ictus in DI per GD e&longs;t ad <lb/>ictum in DR per GD, vt DR vel LI ad ID; </s> <s id="N1FC7D"><!-- NEW -->&longs;ed K <foreign lang="greek">b</foreign> e&longs;t æqualis IL; </s> <s id="N1FC85"><!-- NEW --><lb/>nam arcus KG & IR &longs;unt æquales; </s> <s id="N1FC8A"><!-- NEW -->igitur ictus per GD in DR e&longs;t ad <lb/>ictum in DR per KD e&longs;t vt DK ad K <foreign lang="greek">b</foreign>; &longs;ed impedimentum e&longs;t vt ictus. </s> <s id="N1FC94"><!-- NEW --><lb/>re&longs;i&longs;tentia vt impedimentum, determinatio noua, vt re&longs;i&longs;tentia; </s> <s id="N1FC99"><!-- NEW -->igitur <pb pagenum="250" xlink:href="026/01/282.jpg"/>determinatio noua in linea incidentiæ GD e&longs;t ad nouam in linea inci­<lb/>dentiæ KD, vt GD vel KD ad K <foreign lang="greek">b</foreign>, & in linea incidentiæ AD vt AD <lb/>ad AB; </s> <s id="N1FCAA"><!-- NEW -->igitur vt &longs;inus totus ad &longs;inum rectum dati anguli incidentiæ; </s> <s id="N1FCAE"><!-- NEW -->&longs;ed <lb/>in linea incidentiæ perpendiculari GD, determinatio noua e&longs;t ad pri o­<lb/>rem in ratione dupla; </s> <s id="N1FCB6"><!-- NEW -->igitur vt G <foreign lang="greek">d</foreign> ad GD; </s> <s id="N1FCBE"><!-- NEW -->ergo noua per KD e&longs;t <lb/>ad nouam per DG, vt K <foreign lang="greek">q</foreign>, ad G <foreign lang="greek">d</foreign>; </s> <s id="N1FCCC"><!-- NEW -->nam vt e&longs;t K <foreign lang="greek">b</foreign> ad GD ita K <foreign lang="greek">q</foreign> ad <lb/>G <foreign lang="greek">d</foreign>; </s> <s id="N1FCDE"><!-- NEW -->ergo noua per KD e&longs;t ad priorem vt K <foreign lang="greek">q</foreign> ad KD, & noua per <lb/>AD, vt AC ad AD, atque ita deinceps; ergo determinatio noua per <lb/>lineam incidentiæ obliquam e&longs;t ad priorem, vt duplum &longs;inus recti an­<lb/>guli incidentiæ ad &longs;inum totum, quod erat demon&longs;trandum. </s> </p> <p id="N1FCEC" type="main"> <s id="N1FCEE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N1FCFA" type="main"> <s id="N1FCFC"><!-- NEW --><emph type="italics"/>Hinc in ip&longs;o angulo<emph.end type="italics"/> 60. <emph type="italics"/>determinatio noua e&longs;t æqualis priori, id e&longs;t in an­<lb/>gulo incidentiæ<emph.end type="italics"/> 30. &longs;it enim prædictus angulus IDR; </s> <s id="N1FD0D"><!-- NEW -->certè RI e&longs;t &longs;ubdu­<lb/>pla ID, vt con&longs;tat; </s> <s id="N1FD13"><!-- NEW -->&longs;ed determinatio noua per ID e&longs;t ad priorem, vt <lb/>dupla IR ad ID; ergo vt æqualis ad æqualem. </s> </p> <p id="N1FD19" type="main"> <s id="N1FD1B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N1FD27" type="main"> <s id="N1FD29"><emph type="italics"/>Hinc &longs;upra angulum<emph.end type="italics"/> 30.<emph type="italics"/>v&longs;que ad<emph.end type="italics"/> 90. <emph type="italics"/>noua determinatio e&longs;t maior priore,<emph.end type="italics"/><lb/>donec tandem in ip&longs;a GD vel in ip&longs;o angulo GDR 90. &longs;it dupla prio­<lb/>ris, infrà verò angulum 30. e&longs;t minor priore, donec tandem in ip&longs;a &longs;e­<lb/>ctione plani FDB nulla &longs;it noua. </s> </p> <p id="N1FD42" type="main"> <s id="N1FD44"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N1FD50" type="main"> <s id="N1FD52"><!-- NEW --><emph type="italics"/>Ex his demonstratur acurati&longs;&longs;imè æqualitas anguli reflexionis cum &longs;uo an­<lb/>gulo incidentiæ<emph.end type="italics"/>; </s> <s id="N1FD5D"><!-- NEW -->&longs;it enim linea incidentiæ KD v. <!-- REMOVE S-->g. <!-- REMOVE S-->determinatio noua <lb/>per DG e&longs;t ad priorem per DQ, vt K <foreign lang="greek">q</foreign> vel XQ æqualis ad DQ; igi­<lb/>tur vt DZ æqualis QX ad DX; </s> <s id="N1FD71"><!-- NEW -->&longs;ed quotie&longs;cumque &longs;unt duæ determi­<lb/>nationes, fit mixta per diagonalem Parallelo grammatis; </s> <s id="N1FD77"><!-- NEW -->&longs;ed QZ e&longs;t pa­<lb/>rallelogramma, & DX diagonalis; </s> <s id="N1FD7D"><!-- NEW -->igitur determinatio mixta ex vtra­<lb/>que e&longs;t per DX; </s> <s id="N1FD83"><!-- NEW -->&longs;ed angulus XDG e&longs;t æqualis KDG, vt patet, nam <lb/>XDG e&longs;t æqualis DXQ, & hic DQX, & hic QD <foreign lang="greek">d</foreign>, & hic QDK; </s> <s id="N1FD8D"><!-- NEW --><lb/>igitur KDR, qui e&longs;t angulus incidentiæ e&longs;t æqualis angulo XDF, qui <lb/>e&longs;t angulus reflexionis: idem dico de omni alio. </s> </p> <p id="N1FD94" type="main"> <s id="N1FD96">Ob&longs;erua&longs;ti iam ni fallor primò determinationes nouas e&longs;&longs;e vt chor­<lb/>das arcus &longs;ubdupli incidentiæ. </s> <s id="N1FD9B">Secundò planum reflectens qua&longs;i repelle­<lb/>re omnes ictus per DG, id e&longs;t per lineam, quæ à puncto contactus duci­<lb/>tur per centrum grauitatis, vt demon&longs;tratum e&longs;t lib.1. Th.120.121. </s> </p> <p id="N1FDA2" type="main"> <s id="N1FDA4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N1FDB0" type="main"> <s id="N1FDB2"><!-- NEW --><emph type="italics"/>Nullus impetus de&longs;truitur per &longs;e in pura reflexione<emph.end type="italics"/>; </s> <s id="N1FDBB"><!-- NEW -->nam per accidens vt <lb/>plurimùm de&longs;truitur, vt dicemus infrà: </s> <s id="N1FDC1"><!-- NEW -->dixi in pura reflexione; </s> <s id="N1FDC5"><!-- NEW -->quia cum <lb/>fit aliqua compre&longs;&longs;io, vel repellitur corpus impactus ni&longs;u po&longs;itiuo, etiam <lb/>de&longs;truitur impetus; </s> <s id="N1FDCD"><!-- NEW -->demon&longs;tratur Th. quia nihil impetus e&longs;t fru&longs;trà; </s> <s id="N1FDD1"><!-- NEW --><lb/>igitur nihil de&longs;truitur: </s> <s id="N1FDD6"><!-- NEW -->con&longs;equentia patet ex dictis; probatur antece­<lb/>dens, quia linea determinationis mixtæ e&longs;t &longs;emper æqualis lineæ prioris <lb/>determinationis, &longs;i remoto obice fui&longs;&longs;et propagata. </s> <s id="N1FDDE"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it linea inciden-<pb pagenum="251" xlink:href="026/01/283.jpg"/>tiæ AD, quæ vlteriùs producta &longs;ine reflexione &longs;it, vt DE; </s> <s id="N1FDE9"><!-- NEW -->certè deter­<lb/>minatio, &longs;eu motus e&longs;t vt DE, vt patet: </s> <s id="N1FDEF"><!-- NEW -->iam reflectatur in D à plano <lb/>BF; </s> <s id="N1FDF5"><!-- NEW -->noua determinatio per DG e&longs;t ad priorem, vt DT æqualis HE ad <lb/>DE; </s> <s id="N1FDFB"><!-- NEW -->igitur determinatio mixta per DH e&longs;t vt DH, &longs;ed DH e&longs;t æqua­<lb/>lis DE; </s> <s id="N1FE01"><!-- NEW -->igitur determinatio mixta e&longs;t æqualis priori; </s> <s id="N1FE05"><!-- NEW -->igitur nihil im­<lb/>petus e&longs;t fru&longs;trà; </s> <s id="N1FE0B"><!-- NEW -->igitur nihil illius de&longs;truitur, quod erat demon&longs;trandum: </s> <s id="N1FE0F"><!-- NEW --><lb/>Idem demon&longs;trari pote&longs;t in quacunque lineâ; in perpendiculo verò <lb/>GD; </s> <s id="N1FE16"><!-- NEW -->cùm noua per DG &longs;it dupla prioris per D <foreign lang="greek">d</foreign>, id e&longs;t, vt DY æqua­<lb/>lis GD, ad DA; certè mixta erit DG æqualis DA. <!-- KEEP S--></s> </p> <p id="N1FE21" type="main"> <s id="N1FE23"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N1FE2F" type="main"> <s id="N1FE31"><!-- NEW --><emph type="italics"/>Hinc omnes lineæ reflexæ per &longs;e &longs;unt æquales,<emph.end type="italics"/> quia &longs;unt &longs;emidiametri eiu&longs;­<lb/>dem circuli; </s> <s id="N1FE3C"><!-- NEW -->dico per &longs;e; </s> <s id="N1FE40"><!-- NEW -->nam per accidens &longs;ecùs accidit; </s> <s id="N1FE44"><!-- NEW -->hinc malè di­<lb/>citur reflexam perpendicularem e&longs;&longs;e omnium reflexarum breui&longs;&longs;imam <lb/>per &longs;e; quod licèt ita e&longs;&longs;e videatur, illud reuerâ e&longs;t per accidens. </s> </p> <p id="N1FE4C" type="main"> <s id="N1FE4E">Obiiceret fortè aliquis <expan abbr="pilã">pilam</expan> reflexam nunquam ad eam a&longs;cendere <expan abbr="&longs;ubli-mitat&etilde;">&longs;ubli­<lb/>mitatem</expan> ex qua priùs demi&longs;&longs;a fuerat. </s> <s id="N1FE5B"><!-- NEW -->Re&longs;p. hoc <expan abbr="veri&longs;&longs;imũ">veri&longs;&longs;imum</expan> e&longs;&longs;e &longs;ed per acci­<lb/>dens hoc ita fieri certum e&longs;t propter diui&longs;ionem, attritum, compre&longs;&longs;io­<lb/>nem, ce&longs;&longs;ionemque partium; </s> <s id="N1FE67"><!-- NEW -->vnde pila eò altiùs a&longs;cendit, quò durior, & <lb/>leuigatior e&longs;t illa materia, ex qua con&longs;tat, planumque ip&longs;um leuigatius, <lb/>durius & ad libellam acuratius ita compo&longs;itum, vt &longs;it omninò horizonti <lb/>parallelum: </s> <s id="N1FE71"><!-- NEW -->adde quod planum debet e&longs;&longs;e pror&longs;us immobile; &longs;i enim mo­<lb/>bile &longs;it, multus impetus de&longs;truitur. </s> </p> <p id="N1FE77" type="main"> <s id="N1FE79"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N1FE85" type="main"> <s id="N1FE87"><!-- NEW --><emph type="italics"/>Hinc licèt non po&longs;&longs;it e&longs;&longs;e motus mixtus ex duplici impetu ad diuer&longs;as lineas <lb/>determinato, ni&longs;i aliquid impetus destruatur, vt constat ex dictis; </s> <s id="N1FE8F"><!-- NEW -->pote&longs;t ta­<lb/>men e&longs;&longs;e linea motus qua&longs;i mixta ex duabus cum eodem &longs;cilicet impetu licèt <lb/>nihil impetus destruatur; e&longs;t enim maximum di&longs;crimen vtriu&longs;que, vt <lb/>patet.<emph.end type="italics"/></s> </p> <p id="N1FE9B" type="main"> <s id="N1FE9D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N1FEA9" type="main"> <s id="N1FEAB"><!-- NEW --><emph type="italics"/>Ideo perpendicularis reflexa e&longs;t reflexarum minima, non quidem per &longs;e, <lb/>&longs;ed per accidens<emph.end type="italics"/>; </s> <s id="N1FEB6"><!-- NEW -->quia cum perpendicularis maximum ictum infligat, fit <lb/>maior compre&longs;&longs;io partium, attritus, diui&longs;io; ex quibus nece&longs;&longs;ariò &longs;equi­<lb/>tur plùs impetus de&longs;trui. </s> </p> <p id="N1FEBE" type="main"> <s id="N1FEC0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N1FECC" type="main"> <s id="N1FECE"><!-- NEW --><emph type="italics"/>Motus reflexus non e&longs;t mixtus ex motu plani pellentis & alio<emph.end type="italics"/>; </s> <s id="N1FED7"><!-- NEW -->quia reue­<lb/>rà planum nullum imprimit impetum, quod etiam ex dictis nece&longs;&longs;ariò <lb/>&longs;equitur; </s> <s id="N1FEDF"><!-- NEW -->&longs;ed e&longs;t veluti occa&longs;io, ex qua re&longs;ultat noua determinatio mix­<lb/>ta, ratione &longs;cilicet impedimenti, eo modo, quo diximus; &longs;i enim pla­<lb/>num ip&longs;um nouum impetum imprimeret mobili, non e&longs;&longs;et pura reflexio. </s> <s id="N1FEE7"><lb/>de qua modo agimus, &longs;ed alia, de qua infrà. </s> </p> <p id="N1FEEB" type="main"> <s id="N1FEED"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s> </p> <p id="N1FEF9" type="main"> <s id="N1FEFB"><!-- NEW --><emph type="italics"/>Non datur quies vlla in puncto reflexionis<emph.end type="italics"/>; </s> <s id="N1FF04"><!-- NEW -->appello puram reflexionem, <pb pagenum="252" xlink:href="026/01/284.jpg"/>in qua nullus &longs;it attritus nec <expan abbr="cõpre&longs;&longs;io">compre&longs;&longs;io</expan>, vel in mobili impacto, vel in pla­<lb/>no reflectente; prob. </s> <s id="N1FF13"><!-- NEW -->quia mobile vno tantùm in&longs;tanti tangit <expan abbr="planũ">planum</expan>; </s> <s id="N1FF1B"><!-- NEW -->igitur <lb/>nullo in&longs;tanti quie&longs;cit; </s> <s id="N1FF21"><!-- NEW -->antecedens certum e&longs;t, quia eo in&longs;tanti, quo primò <lb/>tangit, habet <expan abbr="impetũ">impetum</expan>; </s> <s id="N1FF2B"><!-- NEW -->nec enim de&longs;truitur totus per Th.38.igitur in&longs;tanti <lb/>&longs;equenti habebit &longs;uum effectum, ergo motum; </s> <s id="N1FF31"><!-- NEW -->ergo vno tantùm in&longs;tanti <lb/>tangit; </s> <s id="N1FF37"><!-- NEW -->nec dicas impetum illum impediri; </s> <s id="N1FF3B"><!-- NEW -->nam ideo impediretur motus <lb/>pro &longs;equenti in&longs;tanti, quia tangitur planum primo in&longs;tanti; </s> <s id="N1FF41"><!-- NEW -->igitur &longs;imi­<lb/>liter, non moueretur tertio in&longs;tanti, quia priori, id e&longs;t &longs;ecundo planum <lb/>tangeretur; idem dico de quarto, quinto &c. </s> <s id="N1FF49"><!-- NEW -->ergo mobile omninò quie­<lb/>&longs;ceret, nec reflecteretur, quod e&longs;t contra Th.1.igitur vno tantùm in&longs;tanti <lb/>tangit mobile planum, quod erat antecedens propo&longs;itum: Iam verò pro­<lb/>batur con&longs;equentia; </s> <s id="N1FF53"><!-- NEW -->&longs;i quie&longs;cit in puncto reflexionis mobile; </s> <s id="N1FF57"><!-- NEW -->igitur eo <lb/>in&longs;tanti, quo tangit illud punctum; </s> <s id="N1FF5D"><!-- NEW -->&longs;ed eo in&longs;tanti non quie&longs;cit, quo reue­<lb/>râ mouetur; </s> <s id="N1FF63"><!-- NEW -->atqui eo in&longs;tanti quo tangit reuerâ mouetur; quia moueri, e&longs;t <lb/>nouum locum primò acquirere per def.1. l.1. </s> </p> <p id="N1FF69" type="main"> <s id="N1FF6B">Obiicies, primo in&longs;tanti contactus mobile tangit planum quie&longs;cens, <lb/>ergo non mouetur. </s> <s id="N1FF70">Re&longs;pondeo negando <expan abbr="con&longs;e&qtilde;uens">con&longs;equens</expan>, nam reuerâ pote&longs;t <lb/>mobile in plano immobili moueri. </s> </p> <p id="N1FF79" type="main"> <s id="N1FF7B"><!-- NEW -->Obiicies &longs;ecundò, mobile in puncto non mouetur; igitur in puncto <lb/>reflexionis non mouetur. </s> <s id="N1FF81">Re&longs;pondeo primò negando antecedens; qui <lb/>enim admittunt puncta phy&longs;ica, dicent acquiri po&longs;&longs;e motu punctum phy­<lb/>&longs;icum &longs;patij. </s> <s id="N1FF88">Re&longs;pondeo &longs;ecundò eandem e&longs;&longs;e difficultatem pro motu &longs;e­<lb/>quentis in&longs;tantis, quidquid &longs;it, &longs;iue dentur puncta &longs;iue non, cuius di&longs;cu&longs;­<lb/>&longs;io pertinet ad Metaphy&longs;icam, ne.no negabit motum reuerâ e&longs;&longs;e, cum pri­<lb/>mo nouus locus acquiritur, in quo non e&longs;t difficultas. </s> </p> <p id="N1FF91" type="main"> <s id="N1FF93"><!-- NEW -->Obiicies tertiò, in puncto nulla e&longs;t &longs;ucce&longs;&longs;io; igitur neque motus. </s> <s id="N1FF97"><!-- NEW --><lb/>Re&longs;pondeo primò, nulla e&longs;t &longs;ucce&longs;sio actu, concedo, potentia, nego; </s> <s id="N1FF9C"><!-- NEW -->Re­<lb/>&longs;pondeo &longs;ecundò, concedo antecedens, di&longs;tinguo con&longs;equens; </s> <s id="N1FFA2"><!-- NEW -->nullus e&longs;t <lb/>motus &longs;ucce&longs;siuus, concedo; in&longs;tantaneus, nego. </s> </p> <p id="N1FFA8" type="main"> <s id="N1FFAA">Obiicies quartò, nullus datur motus in&longs;tantaneus. </s> <s id="N1FFAD"><!-- NEW -->Re&longs;pondeo, nullus <lb/>datur in&longs;tantaneus actu nego, potentiâ concedo; quia quocunque dato <lb/>motu pote&longs;t dari minor. </s> </p> <p id="N1FFB5" type="main"> <s id="N1FFB7">Obiicies quintò, igitur motus in eo puncto non pote&longs;t e&longs;&longs;e tardior, & <lb/>velocior. </s> <s id="N1FFBC">Re&longs;pondeo primo negando; nam vno motu in&longs;tantaneo actu <lb/>pote&longs;t dari velocior, vel tardior, quæ omnia facilè in Metaphy&longs;icis expli­<lb/>cantur, & demon&longs;trantur, ex quibus certè res i&longs;ta phy&longs;ica minimè de­<lb/>pendet. </s> </p> <p id="N1FFC5" type="main"> <s id="N1FFC7">Obiicies &longs;extò, authoritatem Ari&longs;totelis. <!-- KEEP S--></s> <s id="N1FFCB"><!-- NEW -->Re&longs;pondeo Ari&longs;totelem in­<lb/>telligendum e&longs;&longs;e de corpore projecto &longs;ur&longs;um motu violento, quod ante­<lb/>quam de&longs;cendat vno in&longs;tanti quie&longs;cit; quod etiam demon&longs;traui lib. 3.Im­<lb/>mò plerique &longs;unt inter Peripateticos qui tenent in puncto reflexionis <lb/>non dari quietem, in hoc &longs;cilicet reflexionis genere, de quo hîc agimus, <lb/>qui fusè hanc quæ&longs;tionem di&longs;cutiunt, nos breuiore methodo v&longs;i rem <lb/>ip&longs;am, ni fallor ex no&longs;tris principiis demon&longs;trauimus. </s> </p> <pb pagenum="253" xlink:href="026/01/285.jpg"/> <p id="N1FFDF" type="main"> <s id="N1FFE1"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N1FFED" type="main"> <s id="N1FFEF">Ob&longs;erua primò, &longs;i planum reflectens cedit, vel mobile ip&longs;um, rem <lb/>aliter e&longs;&longs;e explicandam. </s> </p> <p id="N1FFF4" type="main"> <s id="N1FFF6"><!-- NEW -->Secundò tribus modis <expan abbr="planũ">planum</expan> cedere, primò per <expan abbr="diui&longs;ion&etilde;">diui&longs;ionem</expan> partium &longs;i fran<lb/>gantur; 2°ree; per diui&longs;ionem &longs;ine fractione propriè &longs;umpta, &longs;ed <expan abbr="cũ">cum</expan> ce&longs;sione. </s> </p> <p id="N20008" type="main"> <s id="N2000A">Tertiò, &longs;ine diui&longs;ione, &longs;ed non &longs;ine compre&longs;sione. </s> </p> <p id="N2000D" type="main"> <s id="N2000F"><!-- NEW --><expan abbr="Ex&etilde;plum">Exemplum</expan> primi generis habes in charta, &longs;eu vitro, quæ <expan abbr="dũ">dum</expan> reflectit fran­<lb/>gitur: </s> <s id="N2001C"><!-- NEW --><expan abbr="exemplũ">exemplum</expan> &longs;ecundi in cera molli, vel pingui terrâ; </s> <s id="N20023"><!-- NEW -->tertii <expan abbr="deniq;">denique</expan> in <expan abbr="m&etilde;-brana">men­<lb/>brana</expan> ten&longs;a, vel fune ten&longs;o: </s> <s id="N20031"><!-- NEW -->&longs;imiliter mobile ip&longs;um tribus modis cedere <lb/>pote&longs;t 1°ree; <expan abbr="cũ">cum</expan> diui&longs;ione partium, & fractione, &longs;ic <expan abbr="dũ">dum</expan> <expan abbr="vitrũ">vitrum</expan> à marmore refle­<lb/>ctitur in mille partes abit.2°ree; &longs;ine fractione, &longs;ed non &longs;ine depre&longs;sione; &longs;ic <lb/>plumbum deprimitur in corpus durum impactum, aut cera mollis. </s> <s id="N20047">3°ree; &longs;ine <lb/>diui&longs;ione, &longs;ed <expan abbr="nõ">non</expan> &longs;ine aliqua compre&longs;sione, &longs;ic ve&longs;icca inflata reflectitur. </s> </p> <p id="N20050" type="main"> <s id="N20052">Itaque duo &longs;unt planorum genera. </s> <s id="N20055">Primum e&longs;t eorum, quæ non cedunt <lb/>præ duritie. </s> <s id="N2005A"><!-- NEW -->Secundum eorum, quæ cedunt vel per fractionem, vel per de­<lb/>pre&longs;sionem, vel per compre&longs;sionem: </s> <s id="N20060"><!-- NEW -->per fractionem dupliciter; </s> <s id="N20064"><!-- NEW -->primò &longs;i <lb/>alterantur tantùm aliquæ partes minutiores, vt fit in molliori lapide; </s> <s id="N2006A"><!-- NEW --><lb/>Secundò &longs;i per fractionem corpus diuidatur in partes notabiles, vt fit in <lb/>vitro, glacie; adde totidem genera mobilium. </s> </p> <p id="N20071" type="main"> <s id="N20073"><!-- NEW -->Ob&longs;erua tertiò e&longs;&longs;e tres alias combinationes; </s> <s id="N20077"><!-- NEW -->vel enim mobile reflecti­<lb/>tur à mobili, &longs;ed non pellitur à plano, & hæc e&longs;t pura reflexio; vel pellitur <lb/>à plano &longs;ine motu præuio, vel &longs;imul reflectitur, & pellitur à plano, quod <lb/>&longs;imul mouetur. </s> <s id="N20081">Ob&longs;erua 4°ree; <expan abbr="cũ">cum</expan> mouetur corpus reflectens à mobili im­<lb/>pacto tres e&longs;&longs;e quoque <expan abbr="cõbinationes">combinationes</expan>, vel enim cum mouetur corpus refle­<lb/>ctens, reflectitur, &longs;eu retroagitur mobile impactum, vel <expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>, &longs;eu quie­<lb/>&longs;cit, vel non retroagitur, &longs;ed idem iter pro&longs;equitur. </s> <s id="N20096"><!-- NEW -->Ob&longs;erua 5°ree; <expan abbr="cū">cum</expan> &longs;int <lb/>quinque veluti &longs;tatus corporis reflectentis; </s> <s id="N200A0"><!-- NEW -->nam vel e&longs;t molle, vel pre&longs;si­<lb/>bile, vel durum vel fragile, vel friabile, & totidem &longs;tatus mobilis, e&longs;&longs;e 25. <lb/>combinationes, vt patet ex regula combinationum, in quo non e&longs;t diffi­<lb/>cultas; igitur deinceps con&longs;iderabo reflexionem ratione potiùs materiæ <lb/>corporis, tùm reflexi, tùm reflectentis, &longs;it ergo. </s> </p> <p id="N200AC" type="main"> <s id="N200AE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N200BA" type="main"> <s id="N200BC"><!-- NEW --><emph type="italics"/>De&longs;truitur impetus in reflexione ex multis capitibus<emph.end type="italics"/>: primò, ratione diuer­<lb/>&longs;æ determinationis, &longs;i talis e&longs;t vt aliquid impetus &longs;it fru&longs;trà, &longs;uppo&longs;ita <lb/>etiam perfecta duritie mobilis, & plani & figura apta. </s> <s id="N200C9"><!-- NEW -->Secundò, ratione <lb/>diui&longs;ionis partium vel plani, vel mobilis, vel vtriu&longs;que; </s> <s id="N200CF"><!-- NEW -->&longs;i enim alteran­<lb/>tur partes, fit qua&longs;i fo&longs;&longs;ula, quam &longs;en&longs;im &longs;ubit mobile, cumque &longs;ingulis <lb/>in&longs;tantibus &longs;it noua difficultas &longs;uperanda, &longs;emper inde imminuitur impe­<lb/>tus: </s> <s id="N200D9"><!-- NEW -->adde quod minor e&longs;t determinatio plani quod cadit; </s> <s id="N200DD"><!-- NEW -->igitur minor <lb/>e&longs;t motus reflexus; </s> <s id="N200E3"><!-- NEW -->igitur plùs impetus e&longs;t fru&longs;trà; </s> <s id="N200E7"><!-- NEW -->igitur plùs de&longs;truitur; </s> <s id="N200EB"><!-- NEW --><lb/>&longs;i autem planum vel ip&longs;um mobile propter fragilitatem in partes di&longs;si­<lb/>liat, etiam de&longs;truitur aliquid impetus; Tertio ratione impre&longs;sionis; <lb/>Quarto ratione compre&longs;sionis; Quintò ratione repul&longs;ionis; Sextò ra­<lb/>tione liberioris ce&longs;sionis; &longs;ed hæc omnia minutiùs videntur e&longs;&longs;e ex­<lb/>plicanda. </s> </p> <pb pagenum="254" xlink:href="026/01/286.jpg"/> <p id="N200FC" type="main"> <s id="N200FE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N2010A" type="main"> <s id="N2010C"><!-- NEW --><emph type="italics"/>De&longs;truitur impetus cum &longs;cilicet mobili impacto in planum atteruntur par­<lb/>tes vel plani, vel mobilis, vel vtriu&longs;que,<emph.end type="italics"/> &longs;ic cum &longs;axum alliditur molliori la­<lb/>pidi, prima &longs;uperficies re&longs;i&longs;tit quidem; </s> <s id="N20119"><!-- NEW -->at certè minùs quàm par &longs;it, vt <lb/>&longs;i&longs;tat mobile; </s> <s id="N2011F"><!-- NEW -->de&longs;truitur tamen aliquid impetus, quia impeditur tantil­<lb/>lùm &longs;altem prima illa determinatio; </s> <s id="N20125"><!-- NEW -->Secunda &longs;uperficies re&longs;i&longs;tit etiam in <lb/>maiori &longs;cilicet proportione, tùm quia impetus eua&longs;it infirmior ex primo <lb/>qua&longs;i conflictu, tùm quia paulò durior e&longs;t &longs;ecunda &longs;uperficies quàm pri­<lb/>ma, quod &longs;cilicet aliquæ partes qua&longs;i intrudantur in vacuitates interce­<lb/>ptas; </s> <s id="N20131"><!-- NEW -->&longs;ic pila lignea multis ictibus confu&longs;a durior e&longs;t; </s> <s id="N20135"><!-- NEW -->denique tertia &longs;u­<lb/>perficies re&longs;i&longs;tit in maiori proportione quàm &longs;ecunda & quarta quàm <lb/>tertia; </s> <s id="N2013D"><!-- NEW -->atque ita deinceps, donec tandem, vel totus impetus vincatur, vel <lb/>determinatio prior &longs;uperetur: </s> <s id="N20143"><!-- NEW -->hinc &longs;i alterantur partes plani tantùm, mi­<lb/>nùs impetus de&longs;truetur, quàm &longs;i atterantur partes mobilis; quia impetus <lb/>partium mobilis attritarum totus de&longs;init, nec vllam vim ampliùs facit, <lb/>quod potiori iure dicendum e&longs;t, &longs;i atterantur partes vtriu&longs;que. </s> </p> <p id="N2014D" type="main"> <s id="N2014F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N2015B" type="main"> <s id="N2015D"><!-- NEW --><emph type="italics"/>Hinc pluribus licèt in&longs;tantibus mobile tangat planum, non tamen vllo quie­<lb/>&longs;cit<emph.end type="italics"/>; alioqui &longs;emper quie&longs;ceret per Th.50. </s> </p> <p id="N20168" type="main"> <s id="N2016A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N20176" type="main"> <s id="N20178"><!-- NEW --><emph type="italics"/>Hinc cum atteruntur partes plani ab impactione mobilis, minor e&longs;t reflexio<emph.end type="italics"/>; </s> <s id="N20181"><!-- NEW --><lb/>quia minor e&longs;t cau&longs;a, &longs;cilicet impetus, quæ minor e&longs;t adhuc &longs;i atterantur <lb/>partes mobilis, & minor adhuc, &longs;i partes vtriu&longs;que; quæ omnia con&longs;tant <lb/>ex dictis. </s> </p> <p id="N2018A" type="main"> <s id="N2018C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N20198" type="main"> <s id="N2019A"><!-- NEW --><emph type="italics"/>Cum re&longs;iliunt partes mobilis, destruitur impetus pen &longs;e, quia &longs;cilicet illa di­<lb/>ui&longs;io, vel &longs;olutio continuitatis &longs;eu plexus re&longs;i&longs;tit<emph.end type="italics"/>; </s> <s id="N201A5"><!-- NEW -->igitur impedit, &longs;ed omne im­<lb/>pedimentum detrahit aliquid impetus: </s> <s id="N201AB"><!-- NEW -->dixi per &longs;e, nam per accidens fieri <lb/>pote&longs;t, vt aliqua particula re&longs;iliens maiore cum impetu moueatur, vt pa­<lb/>tet aliquando experientiâ; quia præter priorem impetum, qui cum aliis <lb/>partibus illi communis erat, additur alius propter nouam alli&longs;ionem, &longs;eu, <lb/>quod mirabilius e&longs;t, cum aliqua particula ex maiore ma&longs;sâ diuellitur, im­<lb/>petus totius mobilis qua&longs;i migrat in particulam illam, perinde qua&longs;i ab <lb/>eo emitteretur, id e&longs;t cum antè totum mobile veloci&longs;&longs;imo motu ferretur, <lb/>particula auul&longs;a, eodem deinde mouetur. </s> </p> <p id="N201BD" type="main"> <s id="N201BF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N201CB" type="main"> <s id="N201CD"><!-- NEW --><emph type="italics"/>Porrò re&longs;iliunt particulæ mobilis per omnes ferè lineas, quæ determinantur <lb/>per accidens à forma vel &longs;ectione diui&longs;ionis<emph.end type="italics"/>; </s> <s id="N201D8"><!-- NEW -->quæ enim dextror&longs;um &longs;eparan­<lb/>tur, dextror&longs;um eunt; </s> <s id="N201DE"><!-- NEW -->atque ita in omnem partem &longs;ine alia regula; </s> <s id="N201E2"><!-- NEW -->cur <lb/>verò ab ictu diuellantur partes, non e&longs;t huius loci di&longs;cutere; </s> <s id="N201E8"><!-- NEW -->&longs;ic enim <lb/>qua&longs;i finditur &longs;axum ex colli&longs;ione; </s> <s id="N201EE"><!-- NEW -->tùm quia ex illo omnium partium <lb/>&longs;uccu&longs;&longs;u &longs;oluitur illarum nexus; </s> <s id="N201F4"><!-- NEW -->tùm quia intruduntur aliquæ partes, <pb pagenum="255" xlink:href="026/01/287.jpg"/>qua&longs;i ad in&longs;tar cunei, quæ aliàs diuidunt; </s> <s id="N201FD"><!-- NEW -->tùm denique, quia e&longs;t aliqua <lb/>compre&longs;&longs;io, cuius vires certè maximæ &longs;unt, vt dicemus alibi: </s> <s id="N20203"><!-- NEW -->Exemplum <lb/>habes tùm in corpore duro, quale e&longs;t vitrum, cuius modicam laminam &longs;i <lb/>duriori pauimento impingas, hinc inde mille particulæ tumultuatim re­<lb/>&longs;ilient; tùm in corpore liquido, vt in aqua, quæ etiam ad corpus durum <lb/>alli&longs;a in mille guttulas di&longs;pergitur, quia eius partes facilè &longs;eparantur. </s> </p> <p id="N2020F" type="main"> <s id="N20211"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N2021D" type="main"> <s id="N2021F"><!-- NEW --><emph type="italics"/>Si vel mobile e&longs;t mollius, vel ip&longs;um planum, vel vtrumque, ita vt non atte­<lb/>rantur partes, &longs;ed tantùm citra compre&longs;&longs;ionem cedant, de&longs;truitur etiam multus <lb/>impetus<emph.end type="italics"/>; </s> <s id="N2022C"><!-- NEW -->&longs;it enim v.g.pila ex molliori cera, haud dubiè ex impactione non <lb/>comprimitur quidem, &longs;ed deprimitur, nec amplius figuram &longs;phæræ, &longs;ed <lb/>portionis habet: </s> <s id="N20234"><!-- NEW -->in qua reuerâ depre&longs;&longs;ione multus e&longs;t conflictus, nec &longs;uf­<lb/>ficienter prima &longs;uperficies re&longs;i&longs;tit, licèt aliquid impetus de&longs;truat, nec <lb/>etiam &longs;ecunda, nec tertia, quæ tamen re&longs;i&longs;tunt &longs;emper in maiori propor­<lb/>tione; </s> <s id="N2023E"><!-- NEW -->donec tandem vel totus ictus qua&longs;i extinguatur, vel determinatio <lb/>prior &longs;uperetur; </s> <s id="N20244"><!-- NEW -->ex quo &longs;equitur reflexio, &longs;ed minor: </s> <s id="N20248"><!-- NEW -->porrò minor refle­<lb/>xio re&longs;ultat ex mollitie mobilis, quam plani, cæteris paribus, & minor <lb/>adhuc ex mollitie vtriu&longs;que; in quo verò con&longs;i&longs;tat mollities corpo­<lb/>rum, & quomodo deprimantur &longs;ine compre&longs;&longs;ione, explicabimus tra­<lb/>ctatu &longs;equenti. </s> </p> <p id="N20254" type="main"> <s id="N20256"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s> </p> <p id="N20262" type="main"> <s id="N20264"><!-- NEW --><emph type="italics"/>Hinc plumbum ad reflexionem minùs aptum e&longs;t,<emph.end type="italics"/> quia &longs;cilicet eius partes <lb/>difficiliùs auelluntur, & à maiore ictu, qui ex grauitate maiore re&longs;ultat, <lb/>faciliùs deprimuntur; </s> <s id="N20271"><!-- NEW -->hinc cum in molliorem terram pila alliditur, qua&longs;i <lb/>emoritur eius &longs;altus; </s> <s id="N20277"><!-- NEW -->hinc, &longs;i grauior ictus e&longs;t, qualis e&longs;t maioris vel mi­<lb/>noris pilæ è tormento explo&longs;æ, & mollior terra, qualis e&longs;t illa quâ vulgò <lb/>aggeres munitionum farciuntur, pila terram ip&longs;am facto foramine pene­<lb/>trat, cùm facilè cedat materia; nec inde amplius re&longs;ultat, cuius rei ratio <lb/>e&longs;t clari&longs;&longs;ima quia &longs;en&longs;im extinguitur impetus, nec angu&longs;tiæ foraminis <lb/>reditum patiuntur. </s> </p> <p id="N20285" type="main"> <s id="N20287"><!-- NEW -->Hinc multâ lanâ muniuntur latera nauium contra maiora tormenta; </s> <s id="N2028B"><!-- NEW --><lb/>quippe globi vis &longs;en&longs;im emoritur in lana, quia &longs;inguli pili re&longs;i&longs;tunt; & <lb/>quia facilè cedunt difficiliùs diuiduntur, &longs;ed fallenti illa ce&longs;&longs;ione ictum <lb/>quoque fallunt, in quo non e&longs;t difficultas. </s> </p> <p id="N20294" type="main"> <s id="N20296"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N202A2" type="main"> <s id="N202A4"><!-- NEW --><emph type="italics"/>Quando fit aliqua compre&longs;&longs;io, distribuitur etiam impetus<emph.end type="italics"/>; </s> <s id="N202AD"><!-- NEW -->e&longs;t enim con­<lb/>flictus, & pugna partium inter &longs;e; </s> <s id="N202B3"><!-- NEW -->&longs;it enim ve&longs;icca in pauimentum alli­<lb/>&longs;a, partes anticæ aëris, quo ve&longs;icca inflatur, comprimunt, & qua&longs;i po&longs;ti­<lb/>cas repellunt, à quibus mutuò retruduntur; </s> <s id="N202BB"><!-- NEW -->vides pugnam; </s> <s id="N202BF"><!-- NEW -->igitur de­<lb/>&longs;truitur impetus: </s> <s id="N202C5"><!-- NEW -->&longs;ed re&longs;tituitur &longs;tatim à potentia motrice media, quâ <lb/>&longs;cilicet corpus omne compre&longs;&longs;um plùs æquo, vt &longs;e&longs;e in pri&longs;tinum exten­<lb/>&longs;ionis &longs;tatum re&longs;tituat, producit in &longs;e impetum: porrò de hac potentiâ <pb pagenum="254" xlink:href="026/01/288.jpg"/>agemus fusè tractatu &longs;equenti lib.2. porrò vel comprimitur tantum mo­<lb/>bile, vel tantùm ip&longs;um planum, vel &longs;imul vtrumque. </s> </p> <p id="N202D4" type="main"> <s id="N202D6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N202E2" type="main"> <s id="N202E4"><!-- NEW --><emph type="italics"/>Ex hac compre&longs;&longs;ione &longs;equitur aliqua reflexio<emph.end type="italics"/>; </s> <s id="N202ED"><!-- NEW -->&longs;iue tantùm mobile com­<lb/>primatur, vt ve&longs;icca inflata vel pila; </s> <s id="N202F3"><!-- NEW -->quippe præter reflexionem puram, <lb/>id e&longs;t præter priorem impetum, qui tamen ex parte de&longs;truitur, fit acce&longs;&longs;io <lb/>noui impetus; </s> <s id="N202FB"><!-- NEW -->igitur maior e&longs;t motus qui reuerâ impetus maior e&longs;t, quò <lb/>maior e&longs;t compre&longs;&longs;io, quæ maior e&longs;t, quò maior e&longs;t ictus; </s> <s id="N20301"><!-- NEW -->hinc maximè <lb/>apta e&longs;t ad reflexionem pila, & ve&longs;icca; </s> <s id="N20307"><!-- NEW -->&longs;i tamen excipias mobile duri&longs;­<lb/>&longs;imum in planum duri&longs;&longs;imum impactum; </s> <s id="N2030D"><!-- NEW -->tunc enim maxima e&longs;t reflexio, <lb/>experientiâ te&longs;te; </s> <s id="N20313"><!-- NEW -->&longs;i verò planum ip&longs;um comprimatur, ex illa quoque <lb/>compre&longs;&longs;ione &longs;equitur noui impetus acce&longs;&longs;io: </s> <s id="N20319"><!-- NEW -->Exemplum habes in fune <lb/>ten&longs;o, vel in membrana timpani bellici, in qua pi&longs;a tam facilè &longs;ub&longs;ultant; </s> <s id="N2031F"><!-- NEW --><lb/>emoritur tamen ferè totus prior impetus propter ce&longs;&longs;ionem plani; </s> <s id="N20324"><!-- NEW -->& ni&longs;i <lb/>nouus accederet, haud dubiè vel nulla penitus vei minima fieret refle­<lb/>xio; </s> <s id="N2032C"><!-- NEW -->denique fieri pote&longs;t compre&longs;&longs;io tùm in mobili, tùm in plano v.g. <!-- REMOVE S-->&longs;i <lb/>ve&longs;icca inflata repercutiatur à membrana tympani maximè ten&longs;a, in hoc <lb/>ca&longs;u maxima fit noui impetus acce&longs;&longs;io ex duplici compre&longs;&longs;ione; </s> <s id="N20336"><!-- NEW -->&longs;ed ma­<lb/>xima fit etiam prioris impetus imminutio ex duplici etiam capite, nem­<lb/>pè ex compre&longs;&longs;ione, eaque duplici, & noua determinatione; &longs;ed hæc &longs;unt <lb/>facilia. </s> </p> <p id="N20340" type="main"> <s id="N20342"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N2034E" type="main"> <s id="N20350"><emph type="italics"/>Si corpus in aliud impactum repellatur per productionem impetus. </s> <s id="N20355"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;i <lb/>duo globi mutuò impellantur, de&longs;truitur etiam impetus ex hoc capite,<emph.end type="italics"/> vt patet <lb/>experientia: </s> <s id="N20362"><!-- NEW -->immò &longs;i globus in æqualem globum impingatur de&longs;truitur <lb/>totus impetus prior; </s> <s id="N20368"><!-- NEW -->vt dictum e&longs;t alibi, de quo etiam infrà: </s> <s id="N2036C"><!-- NEW -->Ratio huius <lb/>Theorematis e&longs;t, quia aliqua impetus portio e&longs;t fru&longs;trà; </s> <s id="N20372"><!-- NEW -->quia non pote&longs;t <lb/>habere &longs;uum effectum; igitur de&longs;trui debet. </s> </p> <p id="N20378" type="main"> <s id="N2037A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N20386" type="main"> <s id="N20388"><!-- NEW --><emph type="italics"/>Si globus in alium æqualem impingitur, ita vt punctum contactus, & cen­<lb/>trum vtriu&longs;que &longs;int in eadem linea, multa <expan abbr="&longs;equũtur">&longs;equuntur</expan> phænomena, quæ iam atti­<lb/>gimus lib.<emph.end type="italics"/>1.<emph type="italics"/>à Th.<emph.end type="italics"/>60.Primò, æqualis impetus in globo, in quem impactus <lb/>e&longs;t, producitur per Th.60.lib.1. Secundò, æqualis e&longs;t determinatio noua <lb/>priori; </s> <s id="N203A3"><!-- NEW -->probatur per Th.127.lib.1. Tertiò, de&longs;truitur totus impetus prior <lb/>per Th.128. hinc quie&longs;cit globus impactus; </s> <s id="N203A9"><!-- NEW -->cuius rei non pote&longs;t e&longs;&longs;e alia <lb/>cau&longs;a; </s> <s id="N203AF"><!-- NEW -->nec enim dicas de&longs;trui totum impetum illum (vt reuerâ totus de­<lb/>&longs;truitur) ratione re&longs;i&longs;tentiæ, quæ minor e&longs;t, quàm e&longs;&longs;et, &longs;i in parietem il­<lb/>lideretur; </s> <s id="N203B7"><!-- NEW -->igitur tota ratio, cur de&longs;truatur totus impetus, duci tantùm <lb/>pote&longs;t ex eo, quod &longs;it fru&longs;trà; </s> <s id="N203BD"><!-- NEW -->e&longs;t autem fru&longs;trà, quia cum prior deter­<lb/>minatio ferat globum impactùm per eandem lineam, & noua per oppo­<lb/>&longs;itam; </s> <s id="N203C5"><!-- NEW -->vtraque certè æqualis e&longs;t; </s> <s id="N203C9"><!-- NEW -->igitur neutra præualet; </s> <s id="N203CD"><!-- NEW -->igitur globus <lb/>con&longs;i&longs;tit; </s> <s id="N203D3"><!-- NEW -->&longs;i quis enim diceret non e&longs;&longs;e æquales; </s> <s id="N203D7"><!-- NEW -->igitur altera maior e&longs;t; </s> <s id="N203DB"><!-- NEW --><lb/>igitur debet præualere; </s> <s id="N203E0"><!-- NEW -->igitur &longs;i prior e&longs;t, debet vlteriùs propagari motus <pb pagenum="255" xlink:href="026/01/289.jpg"/>in eadem linea; </s> <s id="N203E9"><!-- NEW -->&longs;i noua, igitur debet tantillùm reflecti; igitur cum nec <lb/>vlteriùs producatur motus, nec retrò agatur mobile, vtraque determi­<lb/>natio nece&longs;&longs;ariò æqualis e&longs;t. </s> <s id="N203F1"><!-- NEW -->Quænam verò &longs;it huius æqualitatis ratio à <lb/>priori, difficilè dictu e&longs;t; </s> <s id="N203F7"><!-- NEW -->dico tamen petendam e&longs;&longs;e ab æqualitate glo­<lb/>borum; </s> <s id="N203FD"><!-- NEW -->cum enim determinatio noua &longs;it duplò maior à plano immobili <lb/>& duro; </s> <s id="N20403"><!-- NEW -->certè à plano mobili minor e&longs;t, vt con&longs;tat, quia cedit; </s> <s id="N20407"><!-- NEW -->igitur <lb/>quâ proportione plùs, vel minùs cedit, e&longs;t minor dupla; </s> <s id="N2040D"><!-- NEW -->&longs;ed maior glo­<lb/>bus minùs cedit, quàm æqualis; </s> <s id="N20413"><!-- NEW -->quia ce&longs;&longs;io e&longs;t minor impul&longs;ione; </s> <s id="N20417"><!-- NEW -->igitur <lb/>quando ce&longs;&longs;io e&longs;t æqualis impul&longs;ioni, æquales &longs;unt determinationes; </s> <s id="N2041D"><!-- NEW -->at­<lb/>qui cum producitur æqualis impetus, & imprimitur æqualis motus, <lb/>æqualis e&longs;t ce&longs;&longs;iò impul&longs;ioni, id e&longs;t æquè cedit, ac impellitur; cum tamen, <lb/>&longs;i maior &longs;it globus, non æquè citò cedat, quia tardior motus imprimitur, <lb/>& hæc e&longs;t, ni fallor, vera ratio huius æqualitatis determinationum, & <lb/>hæc vera cau&longs;a quietis globi impacti, de qua iam &longs;uprà Th. 40. </s> </p> <p id="N2042B" type="main"> <s id="N2042D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s> </p> <p id="N20439" type="main"> <s id="N2043B"><!-- NEW --><emph type="italics"/>Cum verò globus impellitur in globum æqualem per lineam obliquam, num­<lb/>quam quie&longs;cit<emph.end type="italics"/>; </s> <s id="N20446"><!-- NEW -->quod demon&longs;tratur, quia &longs;emper e&longs;t determinatio mixta; </s> <s id="N2044A"><!-- NEW --><lb/>quod vt meliùs intelligatur, opus e&longs;t nouâ figurâ &longs;it ergo punctum con­<lb/>tactus duorum globorum B, & ip&longs;a CBN &longs;it Tangens communis, &longs;eu <lb/>&longs;ectio plani, quæ gerit vicem plani reflectentis; </s> <s id="N20453"><!-- NEW -->fit autem primò linea <lb/>incidentiæ connectens centra FBA; </s> <s id="N20459"><!-- NEW -->nulla fit in ea reflexio per Th. 61. <lb/>quia &longs;cilicet determinatio noua per lineam BF e&longs;t æqualis priori per <lb/>FB; </s> <s id="N20461"><!-- NEW -->&longs;it EB linea incidentiæ faciens angulum EBC cum Tangente <lb/>NC; </s> <s id="N20467"><!-- NEW -->determinatio noua e&longs;t ad determinationem priorem vt BG vel <lb/>ER ad BE, & &longs;i &longs;it linea incidentiæ DB vt BH, vel SD ad BD; </s> <s id="N2046D"><!-- NEW -->deni­<lb/>que &longs;i &longs;it BV vt TV ad BV, donec tandem linea incidentiæ &longs;it CB, quâ <lb/>po&longs;itâ nulla e&longs;t determinatio noua; </s> <s id="N20475"><!-- NEW -->vides e&longs;&longs;e eandem viam proportio­<lb/>num quæ fuit &longs;uprà; </s> <s id="N2047B"><!-- NEW -->licèt non &longs;it futura eadem angulorum reflexionis <lb/>proportio, quia determinationum nouarum rationes non &longs;unt eædem; <lb/>producatur enim EBL DBM &c. </s> <s id="N20483"><!-- NEW -->determinatio prior per EB e&longs;t ad <lb/>nouam per BF, vt BE ad BG; </s> <s id="N20489"><!-- NEW -->igitur ducantur EP PL; </s> <s id="N2048D"><!-- NEW -->a&longs;&longs;umatur LI <lb/>æqualis BG, & GI, BL æqualis BE; </s> <s id="N20493"><!-- NEW -->denique ducatur BI: dico BI e&longs;&longs;e <lb/>lineam reflexionis &longs;eu determinationem mixtam ex BG BL per Th. <!-- REMOVE S--><lb/>137.lib.1.&c. <!-- KEEP S--></s> <s id="N2049D"><!-- NEW -->Similiter &longs;i &longs;it linea incidentiæ DBN, ducanturque DO. <lb/>OM, & a&longs;&longs;umatur MK æqualis BH, vel SD, dico lineam BK e&longs;&longs;e de­<lb/>terminationem mixtam ex BH BM, ex quibus etiam longitudo omnium <lb/>reflexarum facilè determinari pote&longs;t; quippe longitudo e&longs;t vt linea de­<lb/>terminationis mixtæ. </s> <s id="N204A9"><!-- NEW -->v.g. <!-- REMOVE S-->BI, BK; </s> <s id="N204AF"><!-- NEW -->demon&longs;tratur autem hæc determi­<lb/>nationum progre&longs;&longs;io, quia determinatio per EB e&longs;t ad determinationem <lb/>per FB vt ictus per EB ad ictum per FB, vt iam &longs;æpè dictum e&longs;t; </s> <s id="N204B7"><!-- NEW -->&longs;ed <lb/>ictus per EB in CN e&longs;t ad ictum per FB vt ER ad FB vel EB, id e&longs;t, vt <lb/>&longs;inus rectus anguli incidentiæ ad &longs;inum totum; </s> <s id="N204BF"><!-- NEW -->&longs;ed determinatio noua <lb/>in perpendiculo FB e&longs;t ad priorem, vt FB ad BF per Th.62. igitur noua <lb/>determinatio per EB e&longs;t ad priorem vt ER &longs;eu &longs;inus rectus anguli EBC <pb pagenum="256" xlink:href="026/01/290.jpg"/>ad &longs;inum totum EB, & per DB vt DS ad DB: idem dico de aliis. </s> </p> <p id="N204CC" type="main"> <s id="N204CE">Hinc colligo primò, omnes determinationes nouas in hypothe&longs;i glo­<lb/>borum æqualium e&longs;&longs;e &longs;ubduplas in ei&longs;dem angulis priorum determina­<lb/>tionum in hypothe&longs;i corporis reflectentis immobilis. </s> </p> <p id="N204D5" type="main"> <s id="N204D7">Colligo &longs;ecundò, omnes reflexiones fieri nece&longs;&longs;ariò per eandem li­<lb/>neam, quæ &longs;cilicet e&longs;t Tangens puncti contactus globi reflectentis, quod <lb/>valdè mirificum e&longs;t, & facilè ob&longs;eruabunt, qui Tudicula minore ludunt. </s> <s id="N204DE"><!-- NEW --><lb/>Colligo &longs;exto, cum angulus incidentiæ e&longs;t 60. lineam reflexam e&longs;&longs;e &longs;ub­<lb/>duplam directæ quæ vlteriùs produceretur; infrà verò &longs;exto e&longs;&longs;e maio­<lb/>rem, &longs;uprà verò e&longs;&longs;e minorem, e&longs;t autem longitudo lineæ &longs;inus comple­<lb/>menti anguli incidentiæ. </s> <s id="N204E9">v.g. <!-- REMOVE S-->&longs;i linea incidentiæ &longs;it EB e&longs;t EG, &longs;i DB <lb/>e&longs;t DH, &longs;i VB e&longs;t VX. </s> </p> <p id="N204F0" type="main"> <s id="N204F2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s> </p> <p id="N204FE" type="main"> <s id="N20500"><!-- NEW --><emph type="italics"/>Si globus minor in maiorem impingatur, qui ab eo tamen moueatur per li­<lb/>neam connectentem centra vtriu&longs;que impactus, reflectitur<emph.end type="italics"/>; </s> <s id="N2050B"><!-- NEW -->ratio e&longs;t, quia ma­<lb/>ior globus e&longs;t maius impedimentum, vt iam diximus Th. 131.lib.1.id <lb/>e&longs;t, vt clariùs hic explicetur, quæ ibidem tantùm obiter indicauimus, <lb/>noua determinatio maior e&longs;t priore, quia ce&longs;sio e&longs;t minor impul&longs;ione; &longs;it <lb/>autem. </s> <s id="N20517"><!-- NEW -->v.g. <!-- REMOVE S-->globus reflectens duplus impacto; </s> <s id="N2051D"><!-- NEW -->igitur motus e&longs;t &longs;ubduplus, <lb/>quia &longs;cilicet impetus di&longs;tribuitur pluribus partibus &longs;ubjecti; </s> <s id="N20523"><!-- NEW -->igitur &longs;in­<lb/>gulæ minùs habent; </s> <s id="N20529"><!-- NEW -->igitur impetus e&longs;t remi&longs;sior; </s> <s id="N2052D"><!-- NEW -->igitur motus tardior; </s> <s id="N20531"><!-- NEW --><lb/>igitur ce&longs;sio minor &longs;ubduplo; </s> <s id="N20536"><!-- NEW -->igitur determinatio noua e&longs;t maior æqua­<lb/>li 1/2 hinc debet nece&longs;&longs;ariò reflecti, quia quotie&longs;cunque ad lineas op­<lb/>po&longs;itas ex diametro determinatur impetus, maior determinatio præua­<lb/>let pro rata per Th.134.lib.1. nam perinde &longs;e habet, atque &longs;i e&longs;&longs;et duplex <lb/>impetus; </s> <s id="N20542"><!-- NEW -->quanta porrò e&longs;&longs;e debeat linea reflexa, determinari pote&longs;t; </s> <s id="N20546"><!-- NEW -->&longs;i <lb/>enim determinatio noua e&longs;&longs;et &longs;olilaria mobile cum eo impetu, quem ha­<lb/>bet <expan abbr="cõficeret">conficeret</expan> v.g. <!-- REMOVE S-->BA vel BF; </s> <s id="N20554"><!-- NEW -->diuidatur BF in duas partes æquales in <foreign lang="greek">u</foreign>, <lb/>determinatio noua e&longs;t ad priorem vt 3. ad 2. a&longs;&longs;umatur F<foreign lang="greek">b</foreign> æqualis B<foreign lang="greek">u</foreign>; </s> <s id="N20566"><!-- NEW --><lb/>igitur propter determinationem priorem oppo&longs;itam &longs;cilicet BA detra­<lb/>hi debent duæ partes toti B<foreign lang="greek">b</foreign> &longs;cilicet <foreign lang="greek">bu</foreign> æqualis BA; </s> <s id="N20575"><!-- NEW -->igitur linea re­<lb/>flexa erit B<foreign lang="greek">u</foreign> dupla totius BF; </s> <s id="N2057F"><!-- NEW -->&longs;it etiam globus reflectens, qui mouetur <lb/>ab impacto, quadruplus, determinatio noua erit ad priorem vt 7. ad 4. <lb/>fit B<foreign lang="greek">d</foreign> ad BA vt 7. ad 4. ex B<foreign lang="greek">d</foreign> detrahatur DH æqualis BA, &longs;upere&longs;t <lb/>HB id e&longs;t 3/4 totius BF; non pote&longs;t autem e&longs;&longs;e maior determinatio no­<lb/>ua priore quàm in ratione dupla, vt diximus &longs;uprà. </s> <s id="N20593"><!-- NEW -->Ratio e&longs;t, quia eò mi­<lb/>nor e&longs;t determinatio noua, quò maior e&longs;t motus impre&longs;&longs;us globo maiori <lb/>reflectenti; </s> <s id="N2059B"><!-- NEW -->igitur tantum detrahitur duplæ, quantum additur motus; </s> <s id="N2059F"><!-- NEW -->&longs;i <lb/>motus e&longs;t æqualis, detrahitur duplæ æqualis priori; </s> <s id="N205A5"><!-- NEW -->igitur &longs;upere&longs;t æqua­<lb/>lis; </s> <s id="N205AB"><!-- NEW -->&longs;i motus e&longs;t &longs;ubduplus, detrahitur duplæ &longs;ubdupla prioris; </s> <s id="N205AF"><!-- NEW -->igitur &longs;u­<lb/>pere&longs;t 1/2 &longs;i &longs;ubquadruplus detrahitur duplæ &longs;ubquadrupla prioris, igitur <lb/>&longs;upere&longs;t 1 3/4 &longs;i &longs;it duplus motus, determinatio noua e&longs;t &longs;ubdupla; </s> <s id="N205B7"><!-- NEW -->igitur <lb/>priori detrahitur 1/2 de quo infrà; </s> <s id="N205BD"><!-- NEW -->quod autem &longs;pectat ad longitudi­<lb/>nes linearum non e&longs;t difficultas; quippe determinatio minor detrahi <lb/>deber maiori. </s> </p> <pb pagenum="257" xlink:href="026/01/291.jpg"/> <p id="N205C9" type="main"> <s id="N205CB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s> </p> <p id="N205D7" type="main"> <s id="N205D9"><!-- NEW --><emph type="italics"/>Si globus minor in maiorem impingatur per lineam obliquam incidentiæ, <lb/>&longs;emper reflectitur<emph.end type="italics"/>; </s> <s id="N205E4"><!-- NEW -->quippè &longs;it determinatio mixta ex priore, & noua, quæ <lb/>determinari pote&longs;t, &longs;i aliquid à nouæ figuræ de&longs;cribatur; </s> <s id="N205EA"><!-- NEW -->&longs;it circulus <lb/>FQCD; </s> <s id="N205F0"><!-- NEW -->&longs;int diametri QD, FC; </s> <s id="N205F4"><!-- NEW -->&longs;it AI dupla AF, &longs;itque determi­<lb/>natio prior vt FA, &longs;i &longs;ecunda &longs;it vt AI, erit dupla prioris; </s> <s id="N205FA"><!-- NEW -->igitur corpus <lb/>reflectens erit immobile; </s> <s id="N20600"><!-- NEW -->igitur &longs;i linea incidentiæ &longs;it EA, reflexa erit <lb/>AT, ita vt anguli TAF, EAF &longs;int æquales; </s> <s id="N20606"><!-- NEW -->&longs;i autem determinatio no­<lb/>ua &longs;it ad priorem vt AH ad AF, id e&longs;t, v.g. <!-- REMOVE S-->vt 3. ad 2. po&longs;itâ &longs;cilicet li­<lb/>neâ incidentiæ perpendiculari FA in planum reflectens QD, quod certè <lb/>mouebitur per Th. 64. aliter procedendum e&longs;t vt inueniatur linea re­<lb/>flexa re&longs;pondens lineæ incidentiæ obliquæ; </s> <s id="N20614"><!-- NEW -->diuidatur FAMK ita vt <lb/>KN &longs;it ad AF vt 3.ad 2. ac proinde AH &longs;it diui&longs;a bifariam in K; </s> <s id="N2061A"><!-- NEW -->de­<lb/>&longs;cribatur circulus KMNR, &longs;it linea quælibet incidentiæ obliqua EA; </s> <s id="N20620"><!-- NEW --><lb/>producatur in B; </s> <s id="N20625"><!-- NEW -->ducantur OX BT parallelæ AH; </s> <s id="N20629"><!-- NEW -->a&longs;&longs;umatur AG æqua­<lb/>lis OX, & GS æqualis AB; </s> <s id="N2062F"><!-- NEW -->certè BS erit æqualis OX vel AG; </s> <s id="N20633"><!-- NEW -->duca­<lb/>tur AS, hæc erit reflexa quæ&longs;ita: </s> <s id="N20639"><!-- NEW -->idem dico de omnibus aliis lineis in­<lb/>cidentiæ; demon&longs;tratur eodem modo quo &longs;uprà in Th. 30. 31. 32. quæ <lb/>con&longs;ule, ne hic repetere cogar. </s> </p> <p id="N20641" type="main"> <s id="N20643"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s> </p> <p id="N2064F" type="main"> <s id="N20651"><!-- NEW --><emph type="italics"/>Si globus maior impingatur in minorem, per lineam incidentiæ connecten­<lb/>tem centra nullo modo reflectitur &longs;ed per eandem lineam primum motum pro­<lb/>pagat licèt tardiùs per Th.<emph.end type="italics"/>132. lib.1. in qua verò proportione retardetur <lb/>motus non ita facilè dictu e&longs;t; dici tamen pote&longs;t & explicari in fig. </s> <s id="N20660"><!-- NEW -->Th. <!-- REMOVE S--><lb/>63. &longs;i enim globi &longs;unt æquales, ce&longs;&longs;io æqualis e&longs;t impul&longs;ioni; </s> <s id="N20667"><!-- NEW -->&longs;i globus <lb/>impactus &longs;it maior, ce&longs;&longs;io e&longs;t maior impul&longs;ione, vt con&longs;tat; </s> <s id="N2066D"><!-- NEW -->igitur, &longs;i globus <lb/>e&longs;t ad globum vt FB ad FB; </s> <s id="N20673"><!-- NEW -->determinatio noua erit ad priorem vt FB <lb/>ad FB; </s> <s id="N20679"><!-- NEW -->igitur quie&longs;cet globus impactus per Th. 62. &longs;i verò globus impa­<lb/>ctus &longs;it ad alium vt EB ad ER; </s> <s id="N2067F"><!-- NEW -->determinatio noua erit ad priorem, vt <lb/>BG ad BF; </s> <s id="N20685"><!-- NEW -->igitur motus retardatus globi impacti e&longs;t ad non retardatum <lb/>vt FG ad FB; </s> <s id="N2068B"><!-- NEW -->quod &longs;i globus impactus e&longs;t ad alium vt DB ad DS, deter­<lb/>minatio noua e&longs;t ad priorem vt BH ad BF; </s> <s id="N20691"><!-- NEW -->&longs;i &longs;it vt TV, ad VB, deter­<lb/>minatio noua erit ad priorem vt BX ad BF, donec tandem nullus &longs;it <lb/>globus re&longs;i&longs;tens; neque res aliter e&longs;&longs;e pote&longs;t. </s> </p> <p id="N20699" type="main"> <s id="N2069B"><!-- NEW -->Hinc vides duos terminos oppo&longs;itos, qui &longs;unt, nulla re&longs;i&longs;tentia, & infi­<lb/>nita re&longs;i&longs;tentia; </s> <s id="N206A1"><!-- NEW -->nulla e&longs;t re&longs;i&longs;tentia, cum globus impactus in nullum in­<lb/>cidit, &longs;ed e&longs;t veluti infinita ce&longs;&longs;io; </s> <s id="N206A7"><!-- NEW -->cum verò globus in corpus immobile <lb/>impingitur, e&longs;t veluti infinita re&longs;i&longs;tentia ratione huius motus; </s> <s id="N206AD"><!-- NEW -->cum verò <lb/>globus in alium globum, quem mouet, impingitur, &longs;i vterque æqualis e&longs;t; </s> <s id="N206B3"><!-- NEW --><lb/>e&longs;t etiam æqualis ce&longs;&longs;io re&longs;i&longs;tentiæ; </s> <s id="N206B8"><!-- NEW -->igitur globus impactus quie&longs;cit, & <lb/>hoc e&longs;t iu&longs;tum medium extremorum prædictorum, id e&longs;t, inter nullam <lb/>ce&longs;&longs;ionem, & infinitam ce&longs;&longs;ionem; </s> <s id="N206C0"><!-- NEW -->media e&longs;t æqualis ce&longs;&longs;io; </s> <s id="N206C4"><!-- NEW -->& inter nul­<lb/>lam re&longs;i&longs;tentiam & infinitam re&longs;i&longs;tentiam media e&longs;t æqualis re&longs;i&longs;tentia; </s> <s id="N206CA"><!-- NEW --><pb pagenum="258" xlink:href="026/01/292.jpg"/>re&longs;i&longs;tentia autem con&longs;ideratur in globo impacto, cuius re&longs;i&longs;titur motui; </s> <s id="N206D2"><!-- NEW --><lb/>ce&longs;&longs;io verò in alio, qui motui cedit; </s> <s id="N206D7"><!-- NEW -->appello autem infinitam re&longs;i&longs;ten­<lb/>tiam cui nulla re&longs;pondet ce&longs;&longs;io; </s> <s id="N206DD"><!-- NEW -->nihil enim aliud præ&longs;taret infinita; </s> <s id="N206E1"><!-- NEW -->por­<lb/>rò cum nulla e&longs;t ce&longs;&longs;io, determinatio noua e&longs;t dupla prioris, vt demon­<lb/>&longs;tratum e&longs;t &longs;uprà; </s> <s id="N206E9"><!-- NEW -->igitur nihil prioris remanet; </s> <s id="N206ED"><!-- NEW -->cum verò nulla e&longs;t re&longs;i­<lb/>&longs;tentia, tota prior remanet, & nulla e&longs;t noua: </s> <s id="N206F3"><!-- NEW -->denique cum ce&longs;&longs;io æqua­<lb/>lis e&longs;t re&longs;i&longs;tentiæ, tantùm remanet prioris quantùm e&longs;t nouæ; </s> <s id="N206F9"><!-- NEW -->igitur <lb/>vtraque æqualis e&longs;t: Vnde vides, ni fallor, perfectam analogiam, &c. </s> <s id="N206FF">Ob­<lb/>&longs;erua&longs;ti ni fallor, quod in hac re poti&longs;&longs;imum e&longs;t. </s> <s id="N20704"><!-- NEW -->Primò, tunc e&longs;&longs;e infini­<lb/>tam re&longs;i&longs;tentiam, cum nulla e&longs;t ce&longs;&longs;io: vt in corpore reflectente pror&longs;us <lb/>immobili. </s> <s id="N2070C">Secundò, tunc e&longs;&longs;e infinitam ce&longs;&longs;ionem, cum nulla e&longs;t re&longs;i­<lb/>&longs;tentia vt in vacuo. </s> <s id="N20711"><!-- NEW -->Tertiò, æqualitatem ce&longs;&longs;ionis, & re&longs;i&longs;tentiæ æquali­<lb/>ter ab vtroque di&longs;tare; tantùm enim e&longs;t inter æqualitatem illam, & in­<lb/>finitam ce&longs;&longs;ionem quantum inter eandem æqualitatem, & infinitam re­<lb/>&longs;i&longs;tentiam. </s> <s id="N2071B">Quartò ab infinita ce&longs;&longs;ione ad æqualitatem accedere nouam <lb/>determinationem æqualem priori. </s> <s id="N20720"><!-- NEW -->Quintò, ab eadem æqualitate ad in­<lb/>finitam re&longs;i&longs;tentiam <expan abbr="tantũdem">tantundem</expan> accedere, ac proinde nouam determi­<lb/>nationem e&longs;&longs;e duplam prioris; ex quo etiam probatur æqualitas angulo­<lb/>rum incidentiæ, & reflexionis. </s> </p> <p id="N2072E" type="main"> <s id="N20730"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s> </p> <p id="N2073C" type="main"> <s id="N2073E"><!-- NEW --><emph type="italics"/>Si globus maior impingatur in minorem per lineam obliquam &longs;emper re­<lb/>flectitur, licèt aliquando in&longs;en&longs;ibiliter, quia fit determinatio mixta ex noua & <lb/>priore, cuius proportio determinari pote&longs;t<emph.end type="italics"/>; &longs;it enim determinatio noua ad <lb/>priorem in linea incidentiæ perpendiculari vt C<foreign lang="greek">d</foreign> ad CA fig. </s> <s id="N20751"><!-- NEW -->Th. 65. <lb/> vel vt AZ ad AF, &longs;it linea incidentiæ obliqua EA producta in B; </s> <s id="N20757"><!-- NEW --><lb/>certè &longs;i determinatio noua per lineam incidentiæ obliquam EA e&longs;t ad <lb/>priorem, vt AZ ad AF; </s> <s id="N2075E"><!-- NEW -->&longs;umatur B<foreign lang="greek">u</foreign> æqualis AY; </s> <s id="N20766"><!-- NEW -->ducantur Y<foreign lang="greek">u</foreign> A<foreign lang="greek">u</foreign><lb/>dico A<foreign lang="greek">u</foreign> e&longs;&longs;e lineam reflexionis, quia e&longs;t mixta ex AY & AB, vt con­<lb/>&longs;tat ex dictis; Idem dico de aliis incidentiæ. </s> </p> <p id="N20779" type="main"> <s id="N2077B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s> </p> <p id="N20787" type="main"> <s id="N20789"><!-- NEW --><emph type="italics"/>Si globus in æqualem globum impingatur, qui æquali impetu in eum etiam <lb/>impingitur per lineam connectentem centra<emph.end type="italics"/>; </s> <s id="N20794"><!-- NEW -->vterque retro agitur æquali <lb/>pœnitus motu, quo &longs;uam lineam vlteriùs propaga&longs;&longs;et, &longs;i in alterum glo­<lb/>bum non incidi&longs;&longs;et per Th.137.lib.1.&longs;i autem inæquali impetu mouean­<lb/>tur, non e&longs;t determinatum &longs;uprà; pote&longs;t autem &longs;it determinari, fig. </s> <s id="N2079E"><!-- NEW -->1. <lb/>Tab.1.&longs;it globus A impactus in alium B motu vt 4. eodem tempore, quo <lb/>globus B impingitur in A motu vt 2. certè globus B retrò agetur motu vt <lb/>4. quippè &longs;iue moueatur æquali motu, &longs;iue minori, &longs;iue etiam quie&longs;cat, <lb/>&longs;emper æquali motu à globo A impelletur; quod certè mirabile e&longs;t; pri­<lb/>mum con&longs;tat per Th. 135.lib. tertium con&longs;tat per Theor.128.lib.1. </s> <s id="N207AC"><!-- NEW -->Igi­<lb/>tur &longs;ecundum con&longs;tat, &longs;i enim impellitur motu vt 4.dum in contrariam <lb/>partem mouetur vt 4. multò magis &longs;i tantùm mouetur vt 2. & &longs;i tantùm <lb/>impellitur motu vt 4. dum quie&longs;cit multò magis motu vt 4. dum in <pb pagenum="259" xlink:href="026/01/293.jpg"/>contrariam partem mouetur motu vt 2. at verò globus A non retro age­<lb/>tur: </s> <s id="N207BD"><!-- NEW -->motu vt 4. &longs;ed tantùm motu vt 2. vt patet; </s> <s id="N207C1"><!-- NEW -->quippe omninò con&longs;i&longs;teret, <lb/>&longs;i globus B nullum præuium impetum habui&longs;&longs;et; &longs;i verò habui&longs;&longs;et mo­<lb/>tum vt 4. tùm etiam A retroageretur motu vt 4. igitur motu vt duo, &longs;i <lb/>B impre&longs;&longs;it impetum vt duo. </s> </p> <p id="N207CB" type="main"> <s id="N207CD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s> </p> <p id="N207D9" type="main"> <s id="N207DB"><!-- NEW --><emph type="italics"/>Si globus A inæqualem globum impingatur per lineam obliquam, ita vt al­<lb/>ter in alterum impetu mutuo impingatur, determinari pote&longs;t motus vtriu&longs;que <lb/>vterque reflectetur<emph.end type="italics"/>; </s> <s id="N207E8"><!-- NEW -->certum e&longs;t, fit enim determinatio mixta ex noua, & <lb/>priore; </s> <s id="N207EE"><!-- NEW -->igitur e&longs;t motus, quod duobus modis fieri pote&longs;t; </s> <s id="N207F2"><!-- NEW -->primò &longs;i æqua­<lb/>lis vtriu&longs;que &longs;it motus, &longs;it linea incidentiæ EB producta in L fig.Th.63. <lb/> per quam globus A ab E proiicitur in globum B; </s> <s id="N207FA"><!-- NEW -->e&longs;tque LB linea in­<lb/>cidentiæ, per quam globus proiicitur in globum A, ita vt punctum con­<lb/>tactus &longs;it B, & linea connectens centra ABF; </s> <s id="N20802"><!-- NEW -->&longs;i globus B con&longs;i&longs;teret in <lb/>puncto B globus A reflecteretur per lineam BI, vt demon&longs;tratum e&longs;t in <lb/>Theoremate 63. quia determinatio prior e&longs;t, vt BL, noua vt BG; </s> <s id="N2080A"><!-- NEW -->igitur <lb/>ex vtraque fit BI; </s> <s id="N20810"><!-- NEW -->at verò &longs;i globus B imprimat impetum in globo A <lb/>æqualem quidem, &longs;i linea incidentiæ e&longs;&longs;et perpendicularis, minorem ta­<lb/>men, quia e&longs;t obliqua qui e&longs;t ad æqualem vt BG ad BF; </s> <s id="N20818"><!-- NEW -->certè determina­<lb/>tio noua e&longs;t dupla BG; </s> <s id="N2081E"><!-- NEW -->quippe ratione reflexionis e&longs;t vt BG, ratione <lb/>impul&longs;ionis vt BG; </s> <s id="N20824"><!-- NEW -->igitur compo&longs;ita ex vtraque vt B<foreign lang="greek">d</foreign> dupla BG; </s> <s id="N2082C"><!-- NEW -->a&longs;&longs;u­<lb/>matur LP æqualis; </s> <s id="N20832"><!-- NEW -->haud dubiè B<foreign lang="greek">d</foreign>, & P<foreign lang="greek">d</foreign> BL; certè determinatio mix­<lb/>ta ex B<foreign lang="greek">d</foreign>, BL erit BP, quæ erit linea reflexionis. </s> <s id="N20844"><!-- NEW -->Hinc egregium Corol­<lb/>larium deduco quod &longs;cilicet reflectatur globus A per angulos æquales, <lb/>quotie&longs;cunque globo æquali impetu contranitente repellitur; </s> <s id="N2084C"><!-- NEW -->quippe <lb/>angulus PBF e&longs;t æqualis angulo EBF: alterum etiam deduco, omnes li­<lb/>neas reflexionis ad quo&longs;cunque angulos &longs;iue rectos, &longs;iue obliquos dum <lb/>vterque globus mutuo impetu ab æquali potentia in &longs;e&longs;e inuicem impin­<lb/>guntur, e&longs;&longs;e æquales, quod certè mirabile e&longs;t. </s> <s id="N20858">Secundò, &longs;i non &longs;it æqualis <lb/>vtriu&longs;que motus, &longs;ed motus globi DB &longs;it ad motum globi A vt AZ ad <lb/>AF fig. </s> <s id="N2085F"><!-- NEW -->Th.65. res ferè eodem modo determinari pote&longs;t; </s> <s id="N20863"><!-- NEW -->quippè mo­<lb/>tus impre&longs;&longs;us à globo B per lineam perpendicularem e&longs;t ad motum im­<lb/>pre&longs;&longs;um A per inclinatam EA vt AZ ad AY; &longs;it autem linea inci­<lb/>dentiæ DB fig. </s> <s id="N2086D">Th. 63. eiu&longs;dem incidentiæ cum EA fig. </s> <s id="N20870"><!-- NEW -->Th. 65. igitur <lb/>globus A incidat per DB, & globus B per MB, ita vt punctum conta­<lb/>ctus &longs;it B, & linea connectens centra FA; determinatio noua ratione in­<lb/>cidentiæ e&longs;t vt BH, cui addatur HF æqualis AY fig. </s> <s id="N2087A"><!-- NEW -->alterius ratione <lb/>motus impre&longs;&longs;i à globo B; </s> <s id="N20880"><!-- NEW -->tota determinatio erit BF; </s> <s id="N20884"><!-- NEW -->a&longs;&longs;umatur MT <lb/>æqualis BF: dico nouam lineam quæ&longs;itam e&longs;&longs;e B<foreign lang="greek">q</foreign> mixtam &longs;cilicet ex <lb/>BF BM, quod probatur vt &longs;uprà. </s> </p> <p id="N20890" type="main"> <s id="N20892"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s> </p> <p id="N2089E" type="main"> <s id="N208A0"><!-- NEW --><emph type="italics"/>Si duo globi inæquales inuicem impingantur per lineam connectentem cen­<lb/>tra diuer&longs;imodè <expan abbr="po&longs;sũt">po&longs;sunt</expan> reflecti<emph.end type="italics"/>; </s> <s id="N208AF"><!-- NEW -->Primò, &longs;i motus vtriu&longs;que e&longs;t æqualis, minor <lb/>globus retroagetur; </s> <s id="N208B5"><!-- NEW -->accipit enim totum impetum maioris globi, id e&longs;t, <pb pagenum="260" xlink:href="026/01/294.jpg"/>impetum æqualem; </s> <s id="N208BE"><!-- NEW -->igitur retro agitur velociore motu in eadem propor­<lb/>tione qua alter globus maior e&longs;t altero, v.g. <!-- REMOVE S-->&longs;i maior e&longs;t duplus, retroa­<lb/>getur motu duplo illius, quo &longs;uum iter pro&longs;equeretur, ni&longs;i maior globus <lb/>occurreret; </s> <s id="N208CA"><!-- NEW -->at verò globus maior duplus &longs;cilicet alterius non retroage­<lb/>tur; </s> <s id="N208D0"><!-- NEW -->quippè &longs;i minor globus con&longs;i&longs;teret in puncto contactus, maior glo­<lb/>bus &longs;uum iter pro&longs;equeretur motu &longs;ubduplo; </s> <s id="N208D6"><!-- NEW -->quippe determinatio noua <lb/>e&longs;&longs;et &longs;ubdupla prioris, vt patet ex Th.66. &longs;ed accipit etiam impetum &longs;ub­<lb/>duplum illius, quem habet, igitur determinatio noua e&longs;t compo&longs;ita ex <lb/>duabus &longs;ubduplis; </s> <s id="N208E0"><!-- NEW -->igitur e&longs;t æqualis priori; </s> <s id="N208E4"><!-- NEW -->igitur <expan abbr="nõ">non</expan> retroagetur, &longs;ed con­<lb/>&longs;i&longs;tet &longs;i duplus e&longs;t; &longs;i verò maior duplo &longs;uum iter pro&longs;equetur &longs;ed minore <lb/>motu pro rata, &longs;i minor duplo retroagetur. </s> <s id="N208F0"><!-- NEW -->Hinc egregium effatum, &longs;i duo <lb/>globi in &longs;e &longs;e inuicem allidantur æquali motu, &longs;i maior duplus e&longs;t, con&longs;i­<lb/>&longs;tet ad punctum contactus; </s> <s id="N208F8"><!-- NEW -->&longs;i maior duplo &longs;uum iter pro&longs;equetur; &longs;i mi­<lb/>nor reflectetur; </s> <s id="N208FE"><!-- NEW -->quod &longs;i motu inæquali mouentur, vel maior mouetur <lb/>maiori motu, vel minor; </s> <s id="N20904"><!-- NEW -->&longs;i maior, minor retroagetur, maior verò vel re­<lb/>troagetur, vel con&longs;i&longs;tet, vel eadem via mouebitur; </s> <s id="N2090A"><!-- NEW -->retroagetur quidem, &longs;i <lb/>noua determinatio compo&longs;ita &longs;cilicet ex impetu impre&longs;&longs;o à minore glo­<lb/>bo, & determinatione reflexionis quam conferet globus minor, etiam&longs;i <lb/>quie&longs;ceret; </s> <s id="N20914"><!-- NEW -->&longs;i noua inquam determinatio &longs;it maior priore; </s> <s id="N20918"><!-- NEW -->con&longs;i&longs;tet verò, <lb/>&longs;i fit æqualis; </s> <s id="N2091E"><!-- NEW -->&longs;uum denique iter pro&longs;equetur, &longs;i &longs;it minor: quæ omnia ex <lb/>dictis facilè determinari po&longs;&longs;unt. </s> </p> <p id="N20924" type="main"> <s id="N20926"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s> </p> <p id="N20932" type="main"> <s id="N20934"><!-- NEW --><emph type="italics"/>Si verò duo globi inæquales in &longs;e&longs;e inuicem impingantur per lineas obliquas, <lb/>&longs;unt quoque tres combinationes<emph.end type="italics"/>; </s> <s id="N2093F"><!-- NEW -->vel enim vterque impingitur motu æquali, <lb/>vel maior globus maiore motu, vel minor; vt autem habeatur linea, &longs;eu <lb/>determinatio cuiu&longs;libet globi, &longs;upponi debet primò linea incidentiæ al­<lb/>terius v.g. <!-- REMOVE S-->maioris. </s> <s id="N2094B"><expan abbr="Secũdò">Secundò</expan> &longs;upponi debet minor quie&longs;cere. </s> <s id="N20951"><!-- NEW -->Tertiò, inue­<lb/>niri noua determinatio, quæ confertur maiori à minore quie&longs;cente, quæ <lb/>facilè inueniri pote&longs;t cognita determinatione noua, quam conferret &longs;i <lb/>linea incidentiæ e&longs;&longs;et perpendicularis; Quartò, debet inueniri determi­<lb/>natio noua quæ confertur à minore maiori ratione impetus, quæ facilè <lb/>inueniri pote&longs;t cognita determinatione huius impetus per lineam per­<lb/>pendicularem. </s> <s id="N20961">Quintò, debet componi determinatio noua ex vtraque. </s> <s id="N20964"><!-- NEW --><lb/>Sextò denique, ex his habebitur determinatio mixta ex hac compo&longs;ita, & <lb/>linea incidentiæ producta, quod facilè ex dictis intelligitur; &longs;imiliter, vt <lb/>habeatur reflexo minoris, debent eadem præ&longs;upponi in maiore. </s> </p> <p id="N2096D" type="main"> <s id="N2096F"><!-- NEW -->Obiiceret hic &longs;ortè aliquis mirari &longs;e quamobrem duo globi æquales <lb/>in &longs;e&longs;e inuicem æquali motu impinguntur vterque retroagatur, cùm po­<lb/>tiùs vterque con&longs;i&longs;tere deberet: quemadmodum quie&longs;cit globus cui im­<lb/>primuntur duo impetus contrarij, hoc e&longs;t ad lineas oppo&longs;itas determi­<lb/>nati. </s> <s id="N2097B"><!-- NEW -->Re&longs;pondeo cum eodem in&longs;tanti eidem globo duplex ille impetus <lb/>imprimitur, non videri vllam rationem, cur alter præualeat; </s> <s id="N20981"><!-- NEW -->at verò vbi <lb/>iam impetus e&longs;t productus, pote&longs;t ad aliam lineam determinari, vt patet; </s> <s id="N20987"><!-- NEW --><lb/>igitur ratione determinationis nouæ, quæ e&longs;t æqualis priori de&longs;truitur; </s> <s id="N2098C"><!-- NEW --><pb pagenum="261" xlink:href="026/01/295.jpg"/>igitur, &longs;i nihil aliud e&longs;&longs;et, globus quie&longs;ceret; at verò ratione impetus <lb/>noui producti ab alio globo, vel eius impetu, retroagitur. </s> </p> <p id="N20996" type="main"> <s id="N20998"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s> </p> <p id="N209A4" type="main"> <s id="N209A6"><!-- NEW --><emph type="italics"/>Pote&longs;t globus retroagi, licèt in aliud corpus non incidat<emph.end type="italics"/>: hoc e&longs;t vulgare, <lb/>mirificum tamen experimentum, &longs;it enim globus ECBL incubans <lb/>plano horizontali MLG, in quem de&longs;cendat planum, quod ni&longs;i globi <lb/>re&longs;i&longs;teret materies, re&longs;ecaret &longs;ectionem DHE. </s> <s id="N209B5"><!-- NEW -->Dico quod ab i&longs;to ictu <lb/>globus determinabitur ad duos motus, alterum centri K ver&longs;us A, alte­<lb/>rum orbis puncti D &longs;cilicet, vel C ver&longs;us E, ita vt initio motus centri <lb/>præualeat ver&longs;us A, qui citò de&longs;truitur propter affrictum partium plani; </s> <s id="N209BF"><!-- NEW --><lb/>vnde remanet tantùm motus orbis, quo &longs;cilicet globus rotatur ver&longs;us F; </s> <s id="N209C4"><!-- NEW --><lb/>nec e&longs;t alia ratio huius experimenti, in quo habetur quædam reflexio &longs;i­<lb/>ne corpore reflectente: pro quo ob&longs;erua fore vt experimentum meliùs <lb/>&longs;uccedat, &longs;i cadat ictus propiùs ad punctum C, quia diutiùs voluitur <lb/>orbis. </s> </p> <p id="N209CF" type="main"> <s id="N209D1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s> </p> <p id="N209DD" type="main"> <s id="N209DF"><!-- NEW --><emph type="italics"/>Hinc etiam ratio euidenti&longs;&longs;ima alterius experimenti, quod valdè familiare <lb/>e&longs;t iis, qui breuioribus globulis ludunt<emph.end type="italics"/>; </s> <s id="N209EA"><!-- NEW -->&longs;i enim ita proiiciatur per medium <lb/>aëra globulus, vt eius hemi&longs;phærium &longs;uperiùs moueatur contrario motu <lb/>motui centri, vel vt A&longs;tronomi loquuntur in Antecedentia, vbi globulus <lb/>terræ planum attingit, vel illico con&longs;i&longs;tit, vel retroagitur, ni&longs;i aliqua <lb/>portio plani inæqualis aliò reflectat; </s> <s id="N209F6"><!-- NEW -->cuius rei ratio e&longs;t duplex ille mo­<lb/>tus, quorum &longs;i determinatio æqualis e&longs;t, con&longs;i&longs;tit globus; </s> <s id="N209FC"><!-- NEW -->&longs;i verò determi­<lb/>natio motus orbis &longs;it maior, quod &longs;emper accidit in breuiore ictu; certè <lb/>cum præualeat, globum retroire nece&longs;&longs;e e&longs;t. </s> </p> <p id="N20A04" type="main"> <s id="N20A06"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s> </p> <p id="N20A12" type="main"> <s id="N20A14"><!-- NEW --><emph type="italics"/>Globulus eburneus in alium impactus con&longs;istit quidem &longs;i centrum respicias<emph.end type="italics"/>; </s> <s id="N20A1D"><!-- NEW --><lb/>at verò &longs;æpè accidit globulum circa centrum &longs;uum immobile motu cir­<lb/>culari & horizontali ad in&longs;tar vorticis conuolui; </s> <s id="N20A24"><!-- NEW -->cuius effectus ratio e&longs;t, <lb/>quia cùm prior impetus ideo tantùm de&longs;truatur, quia e&longs;t fru&longs;trà, & fru­<lb/>&longs;trà e&longs;t, quia æqualis e&longs;t determinatio vtraque per lineas oppo&longs;itas, de­<lb/>terminatio inquam motus centri; </s> <s id="N20A2E"><!-- NEW -->&longs;i tamen globi deficiat æquilibrium, vt <lb/>&longs;emper reuerâ tantillùm deficit, in partem illam globus voluitur, vt vide­<lb/>mus in corpore oblongo, cuius dum vna extremitas pellitur circa cen­<lb/>trum aliquod voluitur; </s> <s id="N20A38"><!-- NEW -->&longs;ed de motu circulari infrà; &longs;ed tanti&longs;per &longs;phæ­<lb/>ri&longs;terium ingredi placuit, vt alios effectus motus reflexi demon­<lb/>&longs;tremus. </s> </p> <p id="N20A40" type="main"> <s id="N20A42"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s> </p> <p id="N20A4E" type="main"> <s id="N20A50"><!-- NEW --><emph type="italics"/>Cum pila coniicitur in parietem ad latus, re&longs;ilit in pauimentum, vnde ite­<lb/>rum repercutitur fallente &longs;altu<emph.end type="italics"/>; </s> <s id="N20A5B"><!-- NEW -->ratio e&longs;t clara, quia quadruplici qua&longs;i <lb/>motu mouetur pila in vltimo &longs;altu; </s> <s id="N20A61"><!-- NEW -->Primus e&longs;t motus centri bis reflexus; </s> <s id="N20A65"><!-- NEW --><pb pagenum="262" xlink:href="026/01/296.jpg"/>Secundus primus motus orbis, quo &longs;cilicet primum in parietem illi&longs;a e&longs;t, <lb/>Tertius motus orbus mixtus, quo ex pariete re&longs;i&longs;tit; </s> <s id="N20A6F"><!-- NEW -->Quartus denique <lb/>motus orbis, quo mouetur po&longs;t quàm à pauimento repercu&longs;&longs;a e&longs;t, exem­<lb/>plum habes in pila rotata per planum horizontale, quæ obliquè in aduer­<lb/>&longs;um planum impingitur; </s> <s id="N20A79"><!-- NEW -->&longs;tatim enim ob&longs;eruas nouum motum orbis mix­<lb/>tum ex priori & nouo, in quo e&longs;t quidem maxima difficultas; &longs;ed de his <lb/>motibus mixtis agemus infrà lib. 9. </s> </p> <p id="N20A81" type="main"> <s id="N20A83"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s> </p> <p id="N20A8F" type="main"> <s id="N20A91"><!-- NEW --><emph type="italics"/>Cum pila emittitur rotato &longs;ur&longs;um pilari reticulo &longs;altus vt plurimùm fallit, <lb/>&longs;ecus verò &longs;i emittatur reticulo deor&longs;um acto<emph.end type="italics"/>; </s> <s id="N20A9C"><!-- NEW -->ratio e&longs;t, quia in primo ca&longs;u <lb/>motus orbis pilæ e&longs;t contrarius motui centri, vt patet; inde fraus &longs;altus, <lb/>&longs;ecus verò in &longs;ecundo ca&longs;u. </s> </p> <p id="N20AA4" type="main"> <s id="N20AA6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s> </p> <p id="N20AB2" type="main"> <s id="N20AB4"><!-- NEW --><emph type="italics"/>Cum pila veloci&longs;&longs;imè ita emittitur, vt linea incidentiæ faciat angulum acu­<lb/>ti&longs;&longs;imum cum pauimento, nullus ferè e&longs;t &longs;altus<emph.end type="italics"/>; </s> <s id="N20ABF"><!-- NEW -->quia cum parùm valeat vis <lb/>reflexiua ad angulum acuti&longs;&longs;imum; </s> <s id="N20AC5"><!-- NEW -->quia prior determinatio ferè præua­<lb/>let, & remanet tota, non quidem intacta, &longs;ed vix &longs;aucia; </s> <s id="N20ACB"><!-- NEW -->determinatio <lb/>motus orbis, qui promouet motum centri, iuuat priorem determina­<lb/>tionem motus centri; igitur vel nullus, vel modicus, i&longs;que celerrimus <lb/>fit &longs;altus. </s> </p> <p id="N20AD5" type="main"> <s id="N20AD7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s> </p> <p id="N20AE3" type="main"> <s id="N20AE5"><emph type="italics"/>Cum pila cadit obliqua linea in pauimentum non longo à pariete interuallo, <lb/>in quem linea &longs;ur&longs;um inclinata po&longs;t &longs;altum &longs;tatim impingitur longè altiùs <lb/>a&longs;cendit pilæ &longs;altus,<emph.end type="italics"/> ratio petitur à noua reflexione, quod facilè e&longs;t. </s> </p> <p id="N20AF1" type="main"> <s id="N20AF3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s> </p> <p id="N20AFF" type="main"> <s id="N20B01"><!-- NEW --><emph type="italics"/>Cum pila obliquè cadit in iuncturam parietis & pauimenti, non reflectitur, <lb/>& tunc maximè fallit &longs;altus<emph.end type="italics"/>; </s> <s id="N20B0C"><!-- NEW -->ratio e&longs;t, quia e&longs;t duplex punctum conta­<lb/>ctus; </s> <s id="N20B12"><!-- NEW -->igitur determinationum nouarum conflictus; </s> <s id="N20B16"><!-- NEW -->quippè paries ver&longs;us <lb/>pauimentum; </s> <s id="N20B1C"><!-- NEW -->hoc verò ver&longs;us parietem repellit; igitur tantùm &longs;upere&longs;t, <lb/>vt in pauimento rotetur &longs;ine &longs;altu, quod accidit ad omnem angulum in­<lb/>cidentiæ obliquum, vt patet experientiâ, cuius ratio communis e&longs;t. </s> </p> <p id="N20B24" type="main"> <s id="N20B26"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s> </p> <p id="N20B32" type="main"> <s id="N20B34"><!-- NEW --><emph type="italics"/>Cum leniore affrictu pilæ funis perstringitur vel, vt aiunt, crispatur, &longs;altus <lb/>etiam ludentis manum frustratur<emph.end type="italics"/>; quia motus orbis mutatur in illo funis <lb/>incu&longs;&longs;u, vt patet. </s> </p> <p id="N20B41" type="main"> <s id="N20B43"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s> </p> <p id="N20B4F" type="main"> <s id="N20B51"><!-- NEW --><emph type="italics"/>Denique, cum reticulo motus orbis is a intorquetur, vt vel circulo horizon­<lb/>tali, vel alteri inclinato &longs;it parallelus, &longs;altus pilæ fallaciæ &longs;ube&longs;t<emph.end type="italics"/>; </s> <s id="N20B5C"><!-- NEW -->quippe à <lb/>priori determinatione motus orbis tuebatur; </s> <s id="N20B62"><!-- NEW -->omitto inæqualitatem pa­<lb/>uimenti, quæ &longs;altum pilæ &longs;æpi&longs;&longs;imè à &longs;ua linea detorquet; &longs;ed fortè &longs;atis <lb/>lu&longs;um e&longs;t. </s> </p> <pb pagenum="263" xlink:href="026/01/297.jpg"/> <p id="N20B6E" type="main"> <s id="N20B70"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s> </p> <p id="N20B7C" type="main"> <s id="N20B7E"><!-- NEW --><emph type="italics"/>Cum planus lapis per lineam incidentiæ valdè obliquam in &longs;uperficiem <lb/>aquæ proijcitur, qua&longs;i repit lapis in ip&longs;a &longs;uperficie &longs;eu plurimo &longs;altu di&longs;currit<emph.end type="italics"/>; </s> <s id="N20B89"><!-- NEW --><lb/>quia &longs;cilicet modica re&longs;i&longs;tentia &longs;ufficit ad reflexionem, cum angulus in­<lb/>cidentiæ e&longs;t obliquior, vt con&longs;tat ex dictis; </s> <s id="N20B90"><!-- NEW -->vt tamen longiorem tractum <lb/>percurrat lapis, ita proiiciendus e&longs;t, vt eius horizonti planior &longs;uperficies <lb/>&longs;it parallela; </s> <s id="N20B98"><!-- NEW -->immò tantillùm portio anthica attollatur: cur autem, & <lb/>quomodo re&longs;i&longs;tat &longs;uperficies aquæ, dicemus &longs;uo loco. </s> </p> <p id="N20B9E" type="main"> <s id="N20BA0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s> </p> <p id="N20BAC" type="main"> <s id="N20BAE"><!-- NEW --><emph type="italics"/>Immò &longs;æpiùs accidit maiorum tormentorum pilas ab aqua reflecti aliquo­<lb/>ties, vt multis experimentis comprobatum e&longs;t<emph.end type="italics"/>; </s> <s id="N20BB9"><!-- NEW -->nec enim ab interiore maris <lb/>fundo reflecti po&longs;&longs;unt, &longs;ed lineam incidentiæ valdè obliquam e&longs;&longs;e nece&longs;­<lb/>&longs;e e&longs;t; habes egregium experimentum apud Mercennum in phœn. </s> <s id="N20BC1"><!-- NEW --><lb/>Balli&longs;t propo&longs;itione 25. ab illu&longs;tri viro petro Petito ob&longs;eruatum, quo <lb/>duntaxat a&longs;&longs;erit pilam è tormento ferreo 10 pedes longo, & horizontali <lb/>parallelo emi&longs;&longs;am, quinquies à &longs;uperficie Oceani reflexam fui&longs;&longs;e; &longs;ed de <lb/>hoc paulò pò&longs;t. </s> </p> <p id="N20BCC" type="main"> <s id="N20BCE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s> </p> <p id="N20BDA" type="main"> <s id="N20BDC"><!-- NEW --><emph type="italics"/>Addo vnum, quod &longs;æpiùs ob&longs;eruatum e&longs;t in illo iactu planorum lapidum, <lb/>quòd &longs;cilicet &longs;ub finem iactus qua&longs;i in orbem dextror&longs;um reflectantur<emph.end type="italics"/>; </s> <s id="N20BE7"><!-- NEW -->cuius <lb/>ratio manife&longs;ta e&longs;t motus orbis horizontali parallelus, qui præter motum <lb/>centri lapidi impre&longs;&longs;us e&longs;t; </s> <s id="N20BEF"><!-- NEW -->quia faciliùs de&longs;truitur motus centri, quàm <lb/>motus orbis; </s> <s id="N20BF5"><!-- NEW -->vnde &longs;ub finem hic illum in &longs;uas partes trahit, dextror&longs;um <lb/>&longs;cilicet, &longs;i dextra proiiciatur lapis; </s> <s id="N20BFB"><!-- NEW -->quia duobus primis digitis po&longs;terior <lb/>lapidis portio &longs;ini&longs;tror&longs;um inflectitur; igitur anterior dextror&longs;um, in <lb/>quo non e&longs;t difficultas. </s> </p> <p id="N20C03" type="main"> <s id="N20C05"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s> </p> <p id="N20C11" type="main"> <s id="N20C13"><!-- NEW --><emph type="italics"/>Cum proiicitur globus in aquam per lineam incidentiæ obliquam, &longs;i non re­<lb/>flectitur ab ip&longs;a &longs;uperficie aquæ; </s> <s id="N20C1B"><!-- NEW -->incuruatur eius linea producta per mediam <lb/>aquam,<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;it vas ABD G, &longs;olidum aquæ va&longs;e contentum CBDF; </s> <s id="N20C26"><!-- NEW -->li­<lb/>nea obliqua incidentiæ globi projecti IH, producta HD: </s> <s id="N20C2C"><!-- NEW -->dico quod <lb/>frangetur in H, & qua&longs;i refringetur in HE; </s> <s id="N20C32"><!-- NEW -->experientia certi&longs;&longs;ima e&longs;t; </s> <s id="N20C36"><!-- NEW --><lb/>ratio verò e&longs;t, quia cùm vis reflexiua puncti H &longs;it aliqua, hoc e&longs;t, cùm &longs;it <lb/>aliquid determinationis nouæ, quæ haud dubiè minor e&longs;t priore, debet <lb/>nece&longs;&longs;ariò mutari linea; </s> <s id="N20C3F"><!-- NEW -->quod autem &longs;it aliquid determinationis nouæ <lb/>in H, patet ex eo quod angulus incidentiæ &longs;it valdè obliquus, reflectitur <lb/>globus; igitur in altero angulo incidentiæ debet e&longs;&longs;e aliquid nouæ de­<lb/>terminationis. </s> <s id="N20C49"><!-- NEW -->Secundò, quia plùs re&longs;i&longs;tit aqua, quàm aër; </s> <s id="N20C4D"><!-- NEW -->igitur fran­<lb/>gitur prior determinatio, & hæc e&longs;t vera ratio huius effectus, quem ali­<lb/>qui ob&longs;eruarunt; </s> <s id="N20C55"><!-- NEW -->Et fortè dici po&longs;&longs;et refractio motus, quæ pror&longs;us e&longs;t <lb/>contraria refractioni luminis; </s> <s id="N20C5B"><!-- NEW -->quippe refractio luminis talis e&longs;t, vt radius <lb/>primo medio raro in den&longs;um incidens incuruetur ad perpendicularem, <lb/>cum tamen linea motus obliquè incidens è medio raro in den&longs;um incur-<pb pagenum="264" xlink:href="026/01/298.jpg"/>uetur à perpendiculari: </s> <s id="N20C68"><!-- NEW -->An fortè etiam ex hoc phænomeno duci pote&longs;t <lb/>vera men&longs;ura, &longs;eu regula refractionum, quod ingenio&longs;i&longs;&longs;imè excogitauit <lb/>vir illu&longs;tris Renatus De&longs;cartes in &longs;ua Dioptrica; </s> <s id="N20C70"><!-- NEW -->&longs;ed di&longs;crimen maximum <lb/>e&longs;t, quòd luminis diffu&longs;io &longs;eu propagatio nullum dicat motum localem, <lb/>vt &longs;uo loco demon&longs;trabimus; </s> <s id="N20C78"><!-- NEW -->quippe lumen qualitas e&longs;t, vt impetus; quod <lb/>tamen ad rem præ&longs;entem nihil pror&longs;us facit. </s> </p> <p id="N20C7E" type="main"> <s id="N20C80"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s> </p> <p id="N20C8C" type="main"> <s id="N20C8E"><!-- NEW --><emph type="italics"/>Linea refractionis motus non e&longs;t recta (&longs;ic eam deinceps appellabimus.)<emph.end type="italics"/><lb/><expan abbr="Cũ">Cum</expan> enim ideo deflectat à recta HD, quia <expan abbr="planũ">planum</expan> in H re&longs;i&longs;tit motui globi; <lb/>igitur etiam in K deflectet à recta KE, quia etiam medium in K re&longs;i&longs;tit. </s> </p> <p id="N20CA1" type="main"> <s id="N20CA3"><!-- NEW -->Ob&longs;eruabis tamen primò, vix hoc di&longs;cerni po&longs;&longs;e, ni&longs;i &longs;it maxima vis <lb/>motus; </s> <s id="N20CA9"><!-- NEW -->quippe grauitas corporis defert corpus deor&longs;um; vnde vis illa <lb/>grauitationis impedit, ne corpus reflectat &longs;eu re&longs;iliat &longs;ur&longs;um Secundò, &longs;i <lb/>corpus in aquam projectum &longs;it leuius aqua, non modò hæc refractio &longs;en­<lb/>&longs;ibilis e&longs;t, verùm etiam illa perpetua refractionum &longs;eries, quia aqua &longs;em­<lb/>per attollit &longs;ur&longs;um corpus leuius. </s> <s id="N20CB5">Tertiò, in corpore oblongo hoc expe­<lb/>rimentum maximè probatur, quia plures partes aquæ &longs;imul reflectunt. </s> </p> <p id="N20CBA" type="main"> <s id="N20CBC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s> </p> <p id="N20CC8" type="main"> <s id="N20CCA"><emph type="italics"/>Linea motus refracti non e&longs;t recta,<emph.end type="italics"/> prob. </s> <s id="N20CD2">quia cum in &longs;ingulis punctis <lb/>aquæ ferè mutetur, curuam e&longs;&longs;e nece&longs;&longs;e e&longs;t. </s> </p> <p id="N20CD7" type="main"> <s id="N20CD9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s> </p> <p id="N20CE5" type="main"> <s id="N20CE7"><!-- NEW --><emph type="italics"/>Hinc optima ratio ducitur, cur globus ex tormento excu&longs;&longs;us ad angulum <lb/>incidentiæ valdè acutum &longs;uperficiem aquæ penetret<emph.end type="italics"/>; </s> <s id="N20CF2"><!-- NEW -->ex qua denuò emergit <lb/>qua&longs;i per arcum primum deor&longs;um; </s> <s id="N20CF8"><!-- NEW -->tùm demum &longs;ur&longs;um inflexum immò <lb/>plures accidunt huiu&longs;modi repetitæ emer&longs;iones: </s> <s id="N20CFE"><!-- NEW -->hinc valdè falluntur, <lb/>qui credunt ab ip&longs;o fundo maris globum repercuti; quod plu&longs;quàm ri­<lb/>diculum e&longs;t; hoc quoque <expan abbr="experimentũ">experimentum</expan> in projectis &longs;axis &longs;æpiùs ob&longs;eruaui. </s> </p> <p id="N20D0A" type="main"> <s id="N20D0C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s> </p> <p id="N20D18" type="main"> <s id="N20D1A"><!-- NEW --><emph type="italics"/>Hinc cum &longs;axa planiora &longs;unt in medio aëre &longs;imile ob&longs;eruari pote&longs;t experi­<lb/>mentum<emph.end type="italics"/>; </s> <s id="N20D25"><!-- NEW -->nam po&longs;t aliquem de&longs;cen&longs;um iterum a&longs;cendit &longs;axum; nec e&longs;t <lb/>quod aliquis vento flanti cau&longs;am huius effectus tribuat, qui &longs;emper acci­<lb/>dit etiam valdè &longs;ereno cœlo. </s> </p> <p id="N20D2D" type="main"> <s id="N20D2F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s> </p> <p id="N20D3B" type="main"> <s id="N20D3D"><!-- NEW --><emph type="italics"/>Hinc cau&longs;a euidens illius a&longs;cen&longs;us &longs;agittæ quamtumuis per lineam horizon­<lb/>ti parallelam emitatur<emph.end type="italics"/>; </s> <s id="N20D48"><!-- NEW -->quippè ab aëre inferiori qua&longs;i repercutitur, ali­<lb/>quid &longs;imile coniicio in glandibus ex tormento explo&longs;is; </s> <s id="N20D4E"><!-- NEW -->e&longs;t enim aliquis <lb/>quamuis in&longs;en&longs;ibilis a&longs;cen&longs;us; </s> <s id="N20D54"><!-- NEW -->hinc fortè ratio, cur in &longs;copum lineas di­<lb/>rectionis horizonti parallelæ re&longs;pondentem globus incidat, cùm infra <lb/>&longs;copum cadere deberet, vt reuerâ fit in notabili di&longs;tantia propter mo­<lb/>tum mixtum; </s> <s id="N20D5E"><!-- NEW -->exemplum huius effectus clari&longs;&longs;imum video in illis auicu­<lb/>lis, quæ per &longs;altus, vel arcus huiu&longs;modi volant; primò enim de&longs;cendere <lb/>videntur, &longs;ed vix a&longs;cendunt. </s> </p> <pb pagenum="265" xlink:href="026/01/299.jpg"/> <p id="N20D6A" type="main"> <s id="N20D6C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 91.<emph.end type="center"/></s> </p> <p id="N20D78" type="main"> <s id="N20D7A"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari proportio anguli huius refractionis motus, &longs;i cogno&longs;catur <lb/>re&longs;i&longs;tentia, qua medium re&longs;istit perpendiculari<emph.end type="italics"/>; </s> <s id="N20D85"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i globus plumbeus ex <lb/>aëre perpendiculariter cadat in &longs;uperficiem aquæ, haud dubiè ip&longs;am <lb/>aquam &longs;ubit, &longs;ed minore motu; </s> <s id="N20D91"><!-- NEW -->quippe frangitur ab ip&longs;a den&longs;itate aquæ <lb/>vis primi impetus, quo &longs;cilicet per liberiorem aëra priùs ferebatur: </s> <s id="N20D97"><!-- NEW -->vnde <lb/>&longs;i habeatur proportio re&longs;i&longs;tentiæ aquæ po&longs;ita linea incidentiæ perpendi­<lb/>culari, non e&longs;t dubium, quin habeatur etiam re&longs;i&longs;tentia po&longs;ita linea in­<lb/>cidentiæ obliqua; nam eodem modo hoc determinandum e&longs;t, quo &longs;uprà <lb/>determinatum fuit Th. 66. 67. v. <!-- REMOVE S-->g. <!-- REMOVE S-->in fig. </s> <s id="N20DA7"><!-- NEW -->Th. 65. determinatio noua <lb/>po&longs;ita perpendiculari &longs;it ad priorem vt AZ ad AF, ita vt per mediam <lb/>aquam conficiat tantùm &longs;patium A<foreign lang="greek">d</foreign> v. <!-- REMOVE S-->g. <!-- REMOVE S-->eo tempore, quo in libero aë­<lb/>re conficit AC; </s> <s id="N20DB9"><!-- NEW -->certè &longs;i linea incidentiæ &longs;it inclinata EA, determinatio <lb/>noua erit ad priorem, vt AY ad AE, vel AB; </s> <s id="N20DBF"><!-- NEW -->igitur fiet mixta ex AY <lb/>AB, &longs;cilicet A<foreign lang="greek">u</foreign>; </s> <s id="N20DC9"><!-- NEW -->non tamen eo tempore conficiet A<foreign lang="greek">u</foreign>, quo conficiet <lb/>A<foreign lang="greek">d</foreign>; </s> <s id="N20DD7"><!-- NEW -->quia &longs;cilicet omnes partes aquæ re&longs;i&longs;tunt, vt con&longs;tat; </s> <s id="N20DDB"><!-- NEW -->igitur con­<lb/>ficietur A <foreign lang="greek">q</foreign> æqualis A<foreign lang="greek">d</foreign>; quæ porrò &longs;it proportio re&longs;i&longs;tentiæ, quæ mobi­<lb/>le retardat in aqua, & re&longs;i&longs;tentiæ, quæ idem retardat in aëre determina­<lb/>ri non pote&longs;t, ni&longs;i primò cogno&longs;catur proportio grauitatis vtriu&longs;que. </s> <s id="N20DEB"><!-- NEW --><lb/>Secundò, ni&longs;i &longs;ciatur in quo po&longs;ita &longs;it hæc re&longs;i&longs;tentia: Tertiò, ni&longs;i per­<lb/>&longs;pectum &longs;it, an maiore nexu partes aquæ inter &longs;e copulentur, an mino­<lb/>re, vel æquali, de quo alias. </s> <s id="N20DF4">Equidem P. <!-- REMOVE S-->Mer&longs;ennus lib.1.a.15. &longs;uæ ver­<lb/>&longs;ionis a&longs;&longs;erit corpus graue per mediam aquam conficere 12. pedes &longs;patij <lb/>eo <expan abbr="t&etilde;pore">tempore</expan>, quo 48. percurrit in aëre, id e&longs;t, tempore duorum &longs;ecundorum. </s> </p> <p id="N20E01" type="main"> <s id="N20E03">Ob&longs;eruabis autem hîc tantùm con&longs;ideratam fui&longs;&longs;e lineam A<foreign lang="greek">q</foreign> rectam <lb/>&longs;ine noua determinatione, quæ &longs;cilicet in&longs;en&longs;ibilis e&longs;t, quando linea in­<lb/>cidentiæ non e&longs;t tam obliqua, nec impetus tantarum virium. </s> <s id="N20E0E"><!-- NEW -->Denique <lb/>ob&longs;eruabis cognito vno angulo motus refracti ad datum angulum inci­<lb/>dentiæ cogno&longs;ci facilè quemlibet alium, qui alteri angulo incidentiæ re­<lb/>&longs;pondeat, vt patet ex dictis: </s> <s id="N20E18"><!-- NEW -->Vtrum verò anguli refractionum motus ex <lb/>aëre in aquam &longs;int iidem cum angulis refractionum luminis ex aqua in <lb/>aëra, examinabimus alibi: </s> <s id="N20E20"><!-- NEW -->hæc interim &longs;ufficiant de motu refracto; quem <lb/>tamen adhuc reflexum e&longs;&longs;e contendo, immò nulla e&longs;t refractio in motu, <lb/>quæ non &longs;it reflexio, & nulla reflexio in lumine, quæ non &longs;it refractio, de <lb/>quo fusè alibi. </s> </p> <p id="N20E2A" type="main"> <s id="N20E2C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s> </p> <p id="N20E38" type="main"> <s id="N20E3A"><!-- NEW --><emph type="italics"/>Aqua, quæ cadit in planum durum re&longs;ilit in mille partes quoquo ver&longs;um<emph.end type="italics"/>; </s> <s id="N20E43"><!-- NEW --><lb/>non certè, quòd partes inferiores pellantur à &longs;uperioribus, vt volunt ali­<lb/>qui; </s> <s id="N20E4A"><!-- NEW -->&longs;ed quòd facilè &longs;eparentur partes aquæ; </s> <s id="N20E4E"><!-- NEW -->vnde non mirum e&longs;t, &longs;i vel <lb/>modico impetu di&longs;pergantur; </s> <s id="N20E54"><!-- NEW -->quippe, vt corpus aliquod reflectatur in­<lb/>tegrum, id e&longs;t &longs;ine partium di&longs;per&longs;ione, debet re&longs;i&longs;tentia vnionis partium <lb/>e&longs;&longs;e maior tota vi impetus ad nouam lineam determinati; </s> <s id="N20E5C"><!-- NEW -->cur verò po­<lb/>tiùs vna guttula dextror&longs;um repercutiatur, quàm &longs;ini&longs;tror&longs;um; </s> <s id="N20E62"><!-- NEW -->certè alia <lb/>ratio e&longs;&longs;e non pote&longs;t, ni&longs;i primò diuer&longs;a figura tùm aquæ impactæ, tùm <pb pagenum="266" xlink:href="026/01/300.jpg"/>plani reflectentis; Secundò aër re&longs;iliens; </s> <s id="N20E6D"><!-- NEW -->Tertiò &longs;ectio ip&longs;a, vt &longs;ic lo­<lb/>quar, diui&longs;ionis, &longs;eu conflictus aliarum partium: idem, cæteris paribus, de <lb/>lapide, cuius mille particulæ re&longs;iliunt. </s> </p> <p id="N20E75" type="main"> <s id="N20E77"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s> </p> <p id="N20E83" type="main"> <s id="N20E85"><!-- NEW --><emph type="italics"/>Globus reflectens, qui ab ictu alterius mouetur, non mouetur ip&longs;o instanti con­<lb/>tactus<emph.end type="italics"/>; prob. </s> <s id="N20E90"><!-- NEW -->quia eo primum in&longs;tanti ab alio globo accipit impetum; &longs;ed <lb/>primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, vt demon&longs;tratum e&longs;t lib. <!-- REMOVE S--><lb/>1.igitur globus reflectens, &c. </s> <s id="N20E99">mouetur tamen. </s> <s id="N20E9C">Secundò in&longs;tans; vnde <lb/>vno tantùm in&longs;tanti contactus e&longs;t. </s> </p> <p id="N20EA1" type="main"> <s id="N20EA3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s> </p> <p id="N20EAF" type="main"> <s id="N20EB1"><!-- NEW --><emph type="italics"/>Hinc colligo produci illum impetum ip&longs;o in&longs;tanti contactus<emph.end type="italics"/>; </s> <s id="N20EBA"><!-- NEW -->alioqui in&longs;tan­<lb/>ti &longs;equenti non e&longs;&longs;et motus; </s> <s id="N20EC0"><!-- NEW -->immò daretur quies in puncto reflexionis; </s> <s id="N20EC4"><!-- NEW --><lb/>quippe, &longs;i tantùm &longs;ecundo in&longs;tanti produceretur, fieret contactus in duo­<lb/>bus in&longs;tantibus; igitur e&longs;&longs;et quies. </s> </p> <p id="N20ECB" type="main"> <s id="N20ECD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s> </p> <p id="N20ED9" type="main"> <s id="N20EDB"><!-- NEW --><emph type="italics"/>Figura corporis impacti variare pote&longs;t reflexionem<emph.end type="italics"/>; &longs;i enim corpus impa­<lb/>ctum &longs;it parallelipedum v. <!-- REMOVE S-->g. <!-- REMOVE S-->multiplex e&longs;&longs;e pote&longs;t reflexionis variatio <lb/>pro diuer&longs;o appul&longs;u, vt con&longs;ideranti patebit. </s> </p> <p id="N20EEC" type="main"> <s id="N20EEE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s> </p> <p id="N20EFA" type="main"> <s id="N20EFC"><!-- NEW --><emph type="italics"/>Si impetus e&longs;&longs;et tantùm determinatus ad vnam lineam; </s> <s id="N20F02"><!-- NEW -->nulla daretur re­<lb/>flexio<emph.end type="italics"/>; patet, quia nulla daretur cau&longs;a reflexionis, quæ tantùm e&longs;t impe­<lb/>tus prior ad nouam lineam determinatus ratio plani oppo&longs;iti. </s> </p> <p id="N20F0D" type="main"> <s id="N20F0F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s> </p> <p id="N20F1B" type="main"> <s id="N20F1D"><!-- NEW --><emph type="italics"/>Quò angulus incidentiæ e&longs;t obliquior, faciliùs fit reflexio<emph.end type="italics"/>; </s> <s id="N20F26"><!-- NEW -->quia minor por­<lb/>tio impetus de&longs;truitur quamuis per accidens; </s> <s id="N20F2C"><!-- NEW -->igitur motus propagatur <lb/>faciliùs; adde quod noua determinatio minùs recedit à priori. </s> </p> <p id="N20F32" type="main"> <s id="N20F34"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N20F40" type="main"> <s id="N20F42"><!-- NEW -->Primò ob&longs;eruabis cau&longs;æ reflexionis e&longs;&longs;e multiplices; </s> <s id="N20F46"><!-- NEW -->&longs;cilicet planum <lb/>reflectens, priorem impetum permanentem, nouam determinationem: </s> <s id="N20F4C"><!-- NEW -->in <lb/>plano verò reflectente con&longs;iderantur impenetrabilitas, durities, & im­<lb/>mobilitas: </s> <s id="N20F54"><!-- NEW -->in priore impetu con&longs;ideratur capacitas ad nouam lineam <lb/>motus, & &longs;ufficiens inten&longs;io ad hoc, vt aliquid impetus ab ictu vel con­<lb/>tactu remaneat; </s> <s id="N20F5C"><!-- NEW -->denique noua determinatio, &longs;i radius incidentiæ &longs;it <lb/>perpendicularis, debet e&longs;&longs;e maior priore; </s> <s id="N20F62"><!-- NEW -->alioqui nulla erit reflexio; &longs;i <lb/>verò linea incidentiæ &longs;it obliqua, pote&longs;t e&longs;&longs;e maior, vel minor, vel <lb/>æqualis. </s> </p> <p id="N20F6A" type="main"> <s id="N20F6C"><!-- NEW -->Secundò ob&longs;eruabis veri&longs;&longs;imam cau&longs;am reflexionis po&longs;itam e&longs;&longs;e in de­<lb/>terminatione noua, ratione cuius pote&longs;t e&longs;&longs;e motus; </s> <s id="N20F72"><!-- NEW -->igitur impetus non <lb/>e&longs;t fru&longs;trà; igitur non debet de&longs;trui &longs;ecundùm illam portionem, quæ <lb/>non e&longs;t fru&longs;trà. </s> </p> <pb pagenum="267" xlink:href="026/01/301.jpg"/> <p id="N20F7E" type="main"> <s id="N20F80"><!-- NEW -->Tertiò, quod &longs;pectat ad æqualitatem anguli reflexionis, & anguli in­<lb/>cidentiæ, non e&longs;t alia huius æqualitatis ratio præter illam, quam attuli­<lb/>mus; </s> <s id="N20F88"><!-- NEW -->nec e&longs;t quod aliqui aliam rationem commini&longs;cantur, cuius prin­<lb/>cipia the&longs;im ip&longs;am &longs;upponunt; </s> <s id="N20F8E"><!-- NEW -->nam primò &longs;upponunt omnem virtutem <lb/>quantumuis impeditam eniti maximè quantum pote&longs;t, vt producat ef­<lb/>fectum &longs;ecundùm inten&longs;ionem agentis; </s> <s id="N20F96"><!-- NEW -->cùm fortè Geometra admitte­<lb/>ret hoc principium &longs;ine alia probatione: an fortè virtus ip&longs;a cogno&longs;cit <lb/>intentionem, agentis, id e&longs;t impetus potentiæ motricis? </s> <s id="N20F9E">numquid impe­<lb/>tus ip&longs;e determinari debet ab ip&longs;a potentia motrice? </s> <s id="N20FA3">numquid e&longs;t deter­<lb/>minatio noua à plano reflectente? </s> <s id="N20FA8">an fortè potentia motrix intendit <lb/>motum per aliam lineam, quàm per lineam incidentiæ? </s> <s id="N20FAD"><!-- NEW -->cum ip&longs;a linea <lb/>reflexionis &longs;emper accidat præter intentionem potentiæ motricis natu­<lb/>ralis; denique licèt hoc totum verum e&longs;&longs;et, vnde probatur po&longs;&longs;e impe­<lb/>tum ad angulum reflexionis æqualem &longs;e ip&longs;um determinare? </s> <s id="N20FB7"><!-- NEW -->Secundò, <lb/>&longs;upponunt impetum e&longs;&longs;e indifferentem ad diuer&longs;as lineas, quod &longs;anè ve­<lb/>rum e&longs;t; </s> <s id="N20FBF"><!-- NEW -->probare tamen deberent, & di&longs;cernere impetum innatum ab <lb/>omni aliò, at, e&longs;to id verum &longs;it; cur potiùs determinatur ad lineam quæ <lb/>faciat angulum æqualem, quàm inæqualem angulo incidentiæ? </s> <s id="N20FC7">ex hoc <lb/>enim principio non probatur hæc æqualitas. </s> </p> <p id="N20FCC" type="main"> <s id="N20FCE"><!-- NEW -->Tertiò, &longs;upponunt dextra fieri &longs;ini&longs;tra in reflexione, & transferri an­<lb/>gulos, idque in eodem plano; </s> <s id="N20FD4"><!-- NEW -->benè e&longs;t; </s> <s id="N20FD8"><!-- NEW -->rem factam &longs;upponunt, quam <lb/>nemo negat; </s> <s id="N20FDE"><!-- NEW -->&longs;ed propter quid fiat demon&longs;trandum e&longs;&longs;et; &longs;i enim quæ­<lb/>ram, cur in eodem plano &longs;int radius incidentiæ. </s> <s id="N20FE4">radius reflexus, & &longs;e­<lb/>ctio communis plani reflectentis? </s> <s id="N20FE9">non video quonam modo demon­<lb/>&longs;trent. </s> <s id="N20FEE"><!-- NEW -->Dicent fortè, quia ita fit in lumine; </s> <s id="N20FF2"><!-- NEW -->belle! ob&longs;curum per ob&longs;cu­<lb/>rius; </s> <s id="N20FF8"><!-- NEW -->quippe ratio reflexionis clarior e&longs;t in motu, quàm in flumine, vt <lb/>&longs;uo loco videbimus; </s> <s id="N20FFE"><!-- NEW -->igitur negari po&longs;&longs;et de lumine, licèt verum &longs;it, do­<lb/>nec &longs;it demon&longs;tratum; immò quamuis probatum e&longs;&longs;et de lumine, quis <lb/>vnquam deduxit à pari argumentum demon&longs;tratiuum? </s> <s id="N21006"><!-- NEW -->Dicent non e&longs;&longs;e <lb/>potiùs rationem, cur fiat per vnum planum ex aliis infinitis, quàm per <lb/>aliud; </s> <s id="N2100E"><!-- NEW -->benè e&longs;t, iam vtuntur illa negatiua ratione, quam paulò antè re­<lb/>&longs;puebant, licèt optima &longs;it, nec quidquam in contrarium afferunt; </s> <s id="N21014"><!-- NEW -->at &longs;o­<lb/>litariam e&longs;&longs;e non oportet; quippe vt iam &longs;uprà monui, effectus po­<lb/>&longs;itiuus per principium po&longs;itiuum ad &longs;uam cau&longs;am reducendus e&longs;t. </s> </p> <p id="N2101C" type="main"> <s id="N2101E"><!-- NEW -->Denique dicent hanc e&longs;&longs;e demon&longs;trationem <emph type="italics"/>Aristotelis in Problematis <lb/>&longs;ect.<emph.end type="italics"/>17.<emph type="italics"/>Probl.<emph.end type="italics"/>13. quod vt palam fiat, textum ip&longs;um de&longs;cribo, <emph type="italics"/>quamobrem,<emph.end type="italics"/><lb/>inquit, <emph type="italics"/>corpora, quæ feruntur, vbi alicubi occurrerunt, re&longs;ilire in partem con­<lb/>trariam &longs;olent, nec ni&longs;i ad &longs;imiles angulos, an quod non &longs;olum eo feruntur im­<lb/>petu, quo pro &longs;ua parte ip&longs;a fieri apti&longs;&longs;ima &longs;unt, verùm etiam illo, qui à mittente <lb/>profici&longs;citur; </s> <s id="N21040"><!-- NEW -->&longs;uus igitur ce&longs;&longs;at cuique impetus, cum &longs;uum ad locum peruene­<lb/>rint, omnia namque requie&longs;cere &longs;olent vbi in eam &longs;edem &longs;e&longs;e contulerunt, quam <lb/>&longs;uapte naturâ de&longs;iderant; </s> <s id="N21048"><!-- NEW -->&longs;ed externo illo, quem habent, impetu nece&longs;&longs;itas ori­<lb/>tur amplius mouendi; </s> <s id="N2104E"><!-- NEW -->quod cùm in partem priorem effici neque at, quia re pro­<lb/>hibetur objecta, vel in latus, vel in rectum agi nece&longs;&longs;e e&longs;t; </s> <s id="N21054"><!-- NEW -->omnia autem in an­<lb/>gulos re&longs;iliunt &longs;imiles, quoniam eodem ferri cogantur, quò motus ducat; </s> <s id="N2105A"><!-- NEW -->quem<emph.end type="italics"/><pb pagenum="268" xlink:href="026/01/302.jpg"/><emph type="italics"/>is dedit, qui mi&longs;erit; </s> <s id="N21067"><!-- NEW -->eo autem vt angulo, vel acuto, vel recto ferantur omninò <lb/>incidit; vt igitur in &longs;peculis extremum lineæ rectæ, &c. </s> <s id="N2106D">itaque feruntur, &c. </s> <s id="N21070"><lb/>cum angulo tanto retorqueantur, quanto vertex con&longs;titerit,<emph.end type="italics"/> &c Sed quæ&longs;o, quis <lb/>vmquam agno&longs;cet demon&longs;trationem in mera comparatione præ&longs;ertim <lb/>in problematis quorum rationes Ari&longs;toteles, vel alter, vt aliqui volunt, <lb/>illorum auctor dubitanter tantùm proponit? </s> <s id="N2107D"><!-- NEW -->Igitur vix au&longs;im a&longs;&longs;erere ab <lb/>Ari&longs;totele hoc ip&longs;um fui&longs;&longs;e demon&longs;tratum; </s> <s id="N21083"><!-- NEW -->&longs;ed aliam demon&longs;trationem <lb/>aggrediuntur, pro qua &longs;upponunt primò determinationem e&longs;&longs;e formam, <lb/>&longs;eu formalitatem, &longs;eu connotationem; </s> <s id="N2108B"><!-- NEW -->quam parùm hæc phy&longs;icam &longs;a­<lb/>piunt, & demon&longs;trationem olent! Secundò, vnumquodque per &longs;e deter­<lb/>minare ad aliud, ad quod e&longs;t determinatum, & determinationem fieri <lb/>per id, quod e&longs;t maximè determinatum; </s> <s id="N21095"><!-- NEW -->quia propter quod vnumquod­<lb/>que tale e&longs;t, & illud magis; </s> <s id="N2109B"><!-- NEW -->quam debile fulcrum! Tertiò &longs;upponunt, <lb/>principium determinans effectum &longs;ecundum genus, & &longs;peciem &longs;imilem <lb/>&longs;ibi reddere in vtroque, etiam Logicè; </s> <s id="N210A3"><!-- NEW -->Quartò, &longs;upponunt ex duobus <lb/>indeterminatis po&longs;&longs;e fieri determinatum; quid inde? </s> <s id="N210A9"><!-- NEW -->Quintò, &longs;uppo­<lb/>nunt angulum reflexionis determinari ab angulo incidentiæ; &longs;ed hæc e&longs;t <lb/>the&longs;is. </s> <s id="N210B1"><!-- NEW -->Ex his principiis primò concludunt reflexionem fieri per angulos <lb/>æquales, idque in eodem plano; </s> <s id="N210B7"><!-- NEW -->&longs;cio quidem de re quod &longs;it, &longs;ed non vi­<lb/>deo demon&longs;trari propter quid &longs;it ex his principiis, vt con&longs;ideranti pate­<lb/>bit; </s> <s id="N210BF"><!-- NEW -->nec e&longs;t quod vlteriùs in iis refutandis immoremur; </s> <s id="N210C3"><!-- NEW -->præ&longs;ertim cùm <lb/>rem hanc acurati&longs;&longs;imè demon&longs;trauerimus &longs;uprà; </s> <s id="N210C9"><!-- NEW -->&longs;ed antequam ab hoc <lb/>motu reflexo di&longs;cedam, alia demon&longs;tratio reiicienda e&longs;t, quæ &longs;ic propo­<lb/>nitur &longs;it planum reflectens immobile, MR, &longs;it linea incidentiæ KD; </s> <s id="N210D1"><!-- NEW --><lb/>hæc e&longs;t, vt aiunt, determinatio mixta ex duabus K<foreign lang="greek">b</foreign>, K<foreign lang="greek">q</foreign>: </s> <s id="N210DE"><!-- NEW -->hoc po&longs;ito, li­<lb/>nea reflexa erit DX, mixta &longs;cilicet ex D<foreign lang="greek">q</foreign> D<foreign lang="greek">u</foreign>; </s> <s id="N210EC"><!-- NEW -->&longs;ed profectò non video, <lb/>nec &longs;entio vim huius determinationis; </s> <s id="N210F2"><!-- NEW -->primò enim nego motum per <lb/>KD e&longs;&longs;e mixtum; </s> <s id="N210F8"><!-- NEW -->e&longs;t enim tantùm vnicum principium determinationis; </s> <s id="N210FC"><!-- NEW --><lb/>igitur vna tantùm e&longs;t determinatio; </s> <s id="N21101"><!-- NEW -->nam primò hæc eadem linea KD <lb/>po&longs;&longs;et e&longs;&longs;e mixta ex pluribus aliis; </s> <s id="N21107"><!-- NEW -->quippè po&longs;&longs;unt e&longs;&longs;e infinita Paralle­<lb/>logrammata, quibus hæc diagonalis KD communis e&longs;&longs;e po&longs;&longs;it; cur au­<lb/>tem potiùs erit diagonalis vnius quàm alterius. </s> <s id="N2110F">Secundò, &longs;i cadat deor­<lb/>&longs;um corpus graue impingaturque in planum inclinatum, nunquid e&longs;t <lb/>motus &longs;implex, & purus naturalis? </s> <s id="N21116">quis e&longs;t qui hoc neget, &longs;i terminos <lb/>ip&longs;os capiat? </s> <s id="N2111B"><!-- NEW -->&longs;ed dicunt, &longs;i proiiciatur mobile per inclinatam in planum <lb/>horizontale, e&longs;t motus mixtus ex naturali accelerato, & impre&longs;&longs;o; </s> <s id="N21121"><!-- NEW -->equi­<lb/>dem hic motus mixtus e&longs;t, &longs;ed tota linea curua; </s> <s id="N21127"><!-- NEW -->quæ non e&longs;t parabolica, <lb/>vt con&longs;tat ex dictis &longs;uprà lib.4.non facit lineam directionis, &longs;ed vltimum <lb/>illius &longs;egmentum, &longs;eu vltima Tangens, quæ tanquam recta a&longs;&longs;umitur: <lb/>præterea quis vmquam lineam incidentiæ a&longs;&longs;ump&longs;it ni&longs;i rectum? </s> <s id="N21131"><!-- NEW -->igitur <lb/>licèt linea incidentiæ po&longs;&longs;it e&longs;&longs;e mixta ex duabus aliis, quod negari non <lb/>pote&longs;t; </s> <s id="N21139"><!-- NEW -->pote&longs;t tamen e&longs;&longs;e &longs;implex, quod nemo etiam negabit; </s> <s id="N2113D"><!-- NEW -->igitur hoc <lb/>ip&longs;um nihil facit ad hanc incidentiæ lineam; </s> <s id="N21143"><!-- NEW -->igitur illud primum an­<lb/>tecedens e&longs;t fal&longs;um, in quo habetur lineam incidentiæ e&longs;&longs;e mixtam; </s> <s id="N21149"><!-- NEW -->quia <lb/>cùm debeat e&longs;&longs;e vniuer&longs;ale, vt &longs;cilicet vniuer&longs;aliter concludat; </s> <s id="N2114F"><!-- NEW -->certè, &longs;i <pb pagenum="269" xlink:href="026/01/303.jpg"/>vniuer&longs;ale e&longs;t, fal&longs;um e&longs;&longs;e con&longs;tat; addunt aliqui e&longs;&longs;e mixtam æquiualen­<lb/>ter. </s> <s id="N2115A"><!-- NEW -->Tertiò, cum &longs;it eadem potentia motrix applicata, tùm in K, tùm in <lb/>A; </s> <s id="N21160"><!-- NEW -->certè debet e&longs;&longs;e idem impetus; </s> <s id="N21164"><!-- NEW -->cum autem duæ lineæ K <foreign lang="greek">q</foreign> K <foreign lang="greek">b</foreign> repræ­<lb/>&longs;entent duos impetus, qui concurrunt ad motum mixtum per KD (nam <lb/>hoc ip&longs;i dicunt) certè duo ABAP &longs;imul &longs;umpti æquales e&longs;&longs;e deberent <lb/>duobus K <foreign lang="greek">q</foreign> K <foreign lang="greek">b</foreign>, quod fal&longs;um e&longs;t; quia KD &longs;it 4. &longs;itque angulus GDK <lb/>30.grad. </s> <s id="N21180">K <foreign lang="greek">q</foreign> e&longs;t 2. igitur collecta <foreign lang="greek">q</foreign> K <foreign lang="greek">b</foreign> e&longs;t 6. & eius quadratum 36. at <lb/>verò quadratum AB e&longs;t 18. ergo quadratum collectæ ex ABAP e&longs;t <lb/>32. igitur illa maior e&longs;t. </s> </p> <p id="N21193" type="main"> <s id="N21195"><!-- NEW -->Sed iam ad aliam propo&longs;itionem venio, in qua dicitur linea reflexio­<lb/>nis DX e&longs;&longs;e mixta ex D <foreign lang="greek">q</foreign> D <foreign lang="greek">u</foreign> quod fal&longs;um e&longs;t; </s> <s id="N211A3"><!-- NEW -->nam primò hoc dicis, <lb/>hoc proba po&longs;itiuo argumento: </s> <s id="N211A9"><!-- NEW -->Dices, quia non pote&longs;t aliter explicari <lb/>æqualitas anguli reflexionis; </s> <s id="N211AF"><!-- NEW -->bellè! nego antecedens; nam licèt nondum <lb/>verus illius modus explicatus non e&longs;&longs;et, proba tuum e&longs;&longs;e verum. </s> <s id="N211B5"><!-- NEW -->Secundò <lb/>vel aliquid prioris determinationis manet, vel nihil; </s> <s id="N211BB"><!-- NEW -->non primum, vt ip&longs;i <lb/>volunt; </s> <s id="N211C1"><!-- NEW -->alioqui DX e&longs;&longs;et mixta ex tribus &longs;cilicet DQ, D <foreign lang="greek">q</foreign>, D <foreign lang="greek">u</foreign>, quod <lb/>ab&longs;urdum e&longs;t; </s> <s id="N211CF"><!-- NEW -->quod &longs;i nihil remaneat prioris determinationis; </s> <s id="N211D3"><!-- NEW -->ergo ni­<lb/>hil prioris impetus, quod etiam concedunt; </s> <s id="N211D9"><!-- NEW -->igitur producitur nouus, &longs;ci­<lb/>licet propter compre&longs;&longs;ionem aëris, corporis reflexi, & reflectentis; </s> <s id="N211DF"><!-- NEW -->&longs;ed <lb/>profectò, licèt hoc totum verum e&longs;&longs;et, cùm illa compre&longs;&longs;io fieret in linea <lb/>quæ per centrum globi producitur, &longs;cilicet à puncto contactus, &longs;cilicet <lb/>in linea DG; </s> <s id="N211E9"><!-- NEW -->certè per illam fieret repercu&longs;&longs;io; </s> <s id="N211ED"><!-- NEW -->Tertiò tunc maxima e&longs;t <lb/>percu&longs;&longs;io, cum linea incidentiæ e&longs;t perpendicularis; </s> <s id="N211F3"><!-- NEW -->igitur tunc e&longs;&longs;e de­<lb/>bet maxima vis compre&longs;&longs;ionis; </s> <s id="N211F9"><!-- NEW -->igitur maxima vis repercu&longs;&longs;ionis, &longs;ed e&longs;t <lb/>tantùm vt DG; at verò, &longs;i linea incidentiæ &longs;it AD, vis repercu&longs;&longs;ionis <lb/>erit, vt collecta ex DFDP quæ maior e&longs;t priore. </s> <s id="N21201">Quartò, cur DX erit <lb/>potiùs mixta ex duabus D <foreign lang="greek">q</foreign>, D <foreign lang="greek">u</foreign>, quàm ex duabus aliis? </s> <s id="N2120E"><!-- NEW -->Quintò, perinde <lb/>&longs;e habet planum reflectens, atque &longs;i globum ip&longs;um pelleret, cùm nihil de­<lb/>terminationis prioris remaneat, vt ip&longs;i volunt, &longs;ed pelleret per ip&longs;am <lb/>DG. Sextò, proba argumento po&longs;itiuo e&longs;&longs;e mixtam DX ex D <foreign lang="greek">u</foreign>, D <foreign lang="greek">q</foreign>; nam <lb/>hoc reuerâ fingis &longs;ine ratione. </s> <s id="N21222">Septimò, præterea &longs;i corpus e&longs;&longs;et duri&longs;&longs;i­<lb/>mum minùs reflecti po&longs;&longs;et à plano duri&longs;&longs;imo, &longs;i nulla fieret compre&longs;&longs;io. </s> <s id="N21227"><!-- NEW --><lb/>Octauò proba mihi impetum priorem de&longs;trui per &longs;e; </s> <s id="N2122C"><!-- NEW -->nam cùm &longs;it indif­<lb/>ferens ad omnes lineas, nunquam de&longs;truitur, ni&longs;i &longs;it fru&longs;trà; </s> <s id="N21232"><!-- NEW -->hic autem <lb/>fru&longs;trà non e&longs;t: </s> <s id="N21238"><!-- NEW -->Itaque manife&longs;tum efficitur, non modò ex his principiis <lb/>non demon&longs;trari æqualitatem anguli reflexionis, &longs;ed ne argumento qui­<lb/>dem probabili comprobari; quia tamen in no&longs;tra demon&longs;tratione multa <lb/>&longs;unt, quæ ip&longs;is non probantur, breuiter recen&longs;eo. </s> </p> <p id="N21242" type="main"> <s id="N21244">Suppono primò, planum reflectens e&longs;&longs;e principium nouæ determina­<lb/>tionis, quod nemo inficiebitur. </s> <s id="N21249">Secundò, e&longs;&longs;e tantùm principium vnius <lb/>determinationis quia vnum principium e&longs;t. </s> <s id="N2124E">Tertiò, per quamcunque li­<lb/>neam incidat globus in punctum D plani &longs;cilicet immobilis, e&longs;t &longs;emper <lb/>idem punctum contactus & eadem <expan abbr="Tãgens">Tangens</expan>. <!-- KEEP S--></s> <s id="N2125A">Quartò, à puncto contactus <lb/>globi duci tantùm po&longs;&longs;e vnicam lineam ad centrum. </s> <s id="N2125F"><!-- NEW -->Quintò, cum deter­<lb/>minationis terminus à quo &longs;it illud punctum contactus, per illam tan-<pb pagenum="270" xlink:href="026/01/304.jpg"/>tum lineam fieri pote&longs;t; </s> <s id="N2126A"><!-- NEW -->nam perinde &longs;e habet globus ille, atque &longs;i re­<lb/>pelleretur à plano; </s> <s id="N21270"><!-- NEW -->nec alia e&longs;&longs;e pote&longs;t linea directionis globi, vt fusè <lb/>probauimus, cum de impetu; </s> <s id="N21276"><!-- NEW -->nec in hoc e&longs;t vlla difficultas, quia cen­<lb/>trum grauitatis dirigit lineam motus; hoc po&longs;ito. </s> </p> <p id="N2127C" type="main"> <s id="N2127E"><!-- NEW -->Si nulla e&longs;&longs;et determinatio præter hanc, haud dubiè globus per DG <lb/>moueretur, vt reuerâ &longs;it cum linea incidentiæ e&longs;t perpendicularis; </s> <s id="N21284"><!-- NEW -->quia <lb/>duæ lineæ oppo&longs;itæ non faciunt determinationem mixtam; </s> <s id="N2128A"><!-- NEW -->&longs;ecus verò <lb/>omnes alias; </s> <s id="N21290"><!-- NEW -->cum igitur globus prædictus reflectatur per DX, illud &longs;it <lb/>nece&longs;&longs;ariò per determinationem mixtam, quod etiam fatentur omnes: </s> <s id="N21296"><!-- NEW --><lb/>mixta e&longs;&longs;e non pote&longs;t ni&longs;i ex duabus &longs;it, vnica tantùm à plano reflecten­<lb/>te e&longs;t, &longs;cilicet per DG; </s> <s id="N2129D"><!-- NEW -->igitur altera e&longs;&longs;e debet, eáque prior per KDQ; </s> <s id="N212A1"><!-- NEW --><lb/>cùm enim prior determinatio &longs;upponatur, vt KD vel vt DQ: e&longs;t enim <lb/>&longs;emper eadem, & cùm noua &longs;it per DG, po&longs;ita diagonali DX, quis non <lb/>videt e&longs;&longs;e mixtam ex DQ & DZ æquali QX? nam perinde &longs;e habet <lb/>globus in D, atque &longs;i pelleretur hinc per DQ, hinc per DZ, ita vt impe­<lb/>tus e&longs;&longs;ent vt lineæ DZ DQ. </s> </p> <p id="N212AE" type="main"> <s id="N212B0"><!-- NEW -->Ex his concludo determinationem nouam e&longs;&longs;e ad priorem po&longs;itâ li­<lb/>neâ incidentiæ KD, vt DZ vel QX ad DQ po&longs;itâ verò lineâ inciden­<lb/>tiæ AD, vt EH ad DE; </s> <s id="N212B8"><!-- NEW -->denique in perpendiculari GD, vt <foreign lang="greek">d</foreign> G ad DG, <lb/>id e&longs;t, in ratione dupla; </s> <s id="N212C2"><!-- NEW -->& nemo e&longs;t meo iudicio, qui rem i&longs;tam attentè <lb/>con&longs;iderans non concedat vltrò de re quod &longs;it, ex hypothe&longs;i æqualitatis <lb/>angulorum reflexionis cum aliis incidentiæ; vt autem demon&longs;tretur <lb/>propter quid &longs;it, aliud principium adhibendum e&longs;t, quod fusè præ&longs;titi­<lb/>mus &longs;uprà. </s> <s id="N212CE"><!-- NEW -->Sed obiiciunt i&longs;tam determinationem nouam quæ fit à plano <lb/>e&longs;&longs;e fictitiam, & chymericam; </s> <s id="N212D4"><!-- NEW -->&longs;ed meo iudicio chymeram facit, qui rem <lb/>tam claram non capit; </s> <s id="N212DA"><!-- NEW -->cum enim non negent nouam determinationem <lb/>e&longs;&longs;e in motu reflexo, nam impetus e&longs;t indifferens, vt &longs;uprà probatum e&longs;t <lb/>abundè, & ex motu funependuli euincitur; </s> <s id="N212E2"><!-- NEW -->certè &longs;i noua e&longs;t, à plano e&longs;t: </s> <s id="N212E6"><!-- NEW --><lb/>&longs;ed à plano e&longs;t per ip&longs;am perpendicularem vt demon&longs;tratum e&longs;t &longs;uprà; <lb/>igitur hæc noua determinatio fictitia non e&longs;t. </s> </p> <p id="N212ED" type="main"> <s id="N212EF"><!-- NEW -->Sed dicunt ab eodem plano e&longs;&longs;e non po&longs;&longs;e determinationem inæqua­<lb/>lem; quia idem principium eundem effectum habet. </s> <s id="N212F5">Re&longs;p. negando ante­<lb/>cedens; </s> <s id="N212FA"><!-- NEW -->cùm enim pro diuer&longs;a re&longs;i&longs;tentia diuer&longs;a &longs;it determinatio, & <lb/>cùm planum prædictum modò plùs, modò minùs re&longs;i&longs;tat; quid mirum &longs;i <lb/>diuer&longs;a &longs;it etiam determinatio? </s> </p> <p id="N21302" type="main"> <s id="N21304">In&longs;tant, lineam determinationis eiu&longs;dem impetus e&longs;&longs;e &longs;emper æqua­<lb/>lem. </s> <s id="N21309">Re&longs;p. negando; </s> <s id="N2130C"><!-- NEW -->quia idem impetus ad duas lineas pote&longs;t determi­<lb/>nari &longs;imul, quæ faciant determinationem mixtam; vnde licèt idem im­<lb/>petus habeat eandem lineam &longs;patij, non tamen eandem lineam determi­<lb/>nationis. </s> <s id="N21316"><!-- NEW -->v.g. <!-- REMOVE S-->quando dico determinationem nouam in perpendiculari <lb/>e&longs;&longs;e ad priorem vt DY ad DG; </s> <s id="N2131E"><!-- NEW -->non dico propterea DY e&longs;&longs;e lineam &longs;pa­<lb/>tij; &longs;ed cùm duæ determinationes comparantur, a&longs;&longs;umi po&longs;&longs;unt lineæ, <lb/>quæ de&longs;ignent proportionem &longs;eu rationem determinationum, quid fa­<lb/>cilius? </s> </p> <p id="N21328" type="main"> <s id="N2132A"><!-- NEW -->Quæres, quid &longs;it illa determinatio: facilis quæ&longs;tio. </s> <s id="N2132E"><!-- NEW -->Re&longs;p. e&longs;&longs;e ip&longs;um <pb pagenum="271" xlink:href="026/01/305.jpg"/>impetum cum habitudine actuali ad talem vel talem lineam; </s> <s id="N21337"><!-- NEW -->quod au­<lb/>tem po&longs;&longs;it e&longs;&longs;e plùs vel minùs determinatus ad vnam, quàm ad aliam, du­<lb/>bium e&longs;&longs;e non pote&longs;t, nec in dubium reuocari, & benè di&longs;tinguitur li­<lb/>nea quanta in ratione determinationis, & quanta in ratione &longs;patij: </s> <s id="N21341"><!-- NEW -->immò <lb/>hoc ip&longs;i &longs;upponunt; nam &longs;i KD e&longs;t mixta ex K <foreign lang="greek">b</foreign> & K <foreign lang="greek">q</foreign>, quis non vi­<lb/>det e&longs;&longs;e eundem impetum cum determinatione duplici inæquali? </s> <s id="N21351">præ­<lb/>terea, quis neget globum impactum perpendiculariter in alium æqua­<lb/>lem quie&longs;cere? </s> <s id="N21358"><!-- NEW -->cur verò quie&longs;cit, ni&longs;i quia impetus e&longs;t fru&longs;trà; <lb/>cur autem e&longs;t fru&longs;trà, ni&longs;i quia cum determinatio <lb/>noua &longs;it æqualis priori? </s> <s id="N21360">&longs;ed de <lb/>his &longs;atis. <lb/><figure id="id.026.01.305.1.jpg" xlink:href="026/01/305/1.jpg"/></s> </p> </chap> <chap id="N2136B"> <pb pagenum="272" xlink:href="026/01/306.jpg"/> <figure id="id.026.01.306.1.jpg" xlink:href="026/01/306/1.jpg"/> <p id="N21375" type="head"> <s id="N21377"><emph type="center"/>LIBER SEPTIMVS, <lb/><emph type="italics"/>DE MOTV CIRCVLARI.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N21385" type="main"> <s id="N21387">CVM in natura minimè de&longs;ideretur motus cir­<lb/>cularis, eius affectiones breuiter in hoc libro <lb/>demon&longs;trantur. <lb/><gap desc="hr tag"/></s> </p> <p id="N21391" type="main"> <s id="N21393"><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2139F" type="main"> <s id="N213A1"><emph type="italics"/>MOtus circularis e&longs;t, cuius linea æqualiter in omnibus &longs;uis punctis à com­<lb/>muni centro distat.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i punctum in periphæria circuli moue­<lb/>retur. </s> </p> <p id="N213B1" type="main"> <s id="N213B3"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N213C0" type="main"> <s id="N213C2"><emph type="italics"/>Radius motus e&longs;t linea recta ducta ab illo communi centro ad periphæ­<lb/>riam.<emph.end type="italics"/></s> </p> <p id="N213CB" type="main"> <s id="N213CD"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N213DA" type="main"> <s id="N213DC"><emph type="italics"/>Arcus e&longs;t pars periphæria maior, vel minor.<emph.end type="italics"/></s> </p> <p id="N213E3" type="main"> <s id="N213E5"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N213F2" type="main"> <s id="N213F4"><!-- NEW --><emph type="italics"/>Tangens e&longs;t linea, quæ tangit periphæriam in vnico puncto, quam tamen <lb/>non &longs;ecat<emph.end type="italics"/>; hæc omnia clara &longs;unt, immò vulgaria. </s> </p> <p id="N213FF" type="main"> <s id="N21401"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2140E" type="main"> <s id="N21410"><!-- NEW --><emph type="italics"/>Si dum rota vertitur imponatur eius &longs;umma &longs;uperficiei aliquod mobile, <lb/>proijcitur à rota, &longs;eu potiùs amouetur<emph.end type="italics"/>; res clara e&longs;t in molari lapide, in <lb/>funda, &c. </s> </p> <p id="N2141D" type="main"> <s id="N2141F"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2142C" type="main"> <s id="N2142E"><!-- NEW --><emph type="italics"/>Illa mouentur æqualiter, quæ temporibus æqualibus aqualia &longs;patia percur­<lb/>runt; inæqualiter verò qua inæqualia; qua maiora, celeriùs; tardiùs, qua <lb/>minora.<emph.end type="italics"/></s> </p> <p id="N2143A" type="main"> <s id="N2143C"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N21449" type="main"> <s id="N2144B"><emph type="italics"/>Qua &longs;imul incipiunt moueri, & de&longs;inunt, aquali tempore mouentur.<emph.end type="italics"/></s> </p> <pb pagenum="273" xlink:href="026/01/307.jpg"/> <p id="N21456" type="main"> <s id="N21458"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N21465" type="main"> <s id="N21467"><emph type="italics"/>Datur motus circularis.<emph.end type="italics"/></s> <s id="N2146E"><!-- NEW --> Probatur infinitis ferè experimentis; primò in <lb/>librâ cuius brachia motu tantùm circulari de&longs;cendunt. </s> <s id="N21474"><!-- NEW -->Secundò in ve­<lb/>cte, qui etiam mouetur circulari motu; </s> <s id="N2147A"><!-- NEW -->Tertiò in turbine, rota molari, <lb/>liquore contento intra vas &longs;phæricum; Quartò in funependulo vibrato. </s> <s id="N21480"><!-- NEW --><lb/>Probatur &longs;ecundò; </s> <s id="N21485"><!-- NEW -->quia pote&longs;t imprimi impetus vtrique extremitati ci­<lb/>lindri in partes oppo&longs;itas, &longs;it enim cilindrus, vel parallelipedum LC, <lb/>cuius extremitati imprimatur impetus, per lineam CP, itemque extre­<lb/>mitati L æqualis per lineam LG oppo&longs;itam CP. Dico, quod mouebitur <lb/>circulariter circa centrum K, ita vt extremitas L conficiat arcum LB & <lb/>C arcum CE; </s> <s id="N21493"><!-- NEW -->nec enim C moueri pote&longs;t per CP neque L per LM; </s> <s id="N21497"><!-- NEW --><lb/>quippe cùm &longs;it æqualis impetus, neutra extremitas præualere pote&longs;t: </s> <s id="N2149C"><!-- NEW -->non <lb/>vtraque, quia MP e&longs;t maior LC; </s> <s id="N214A2"><!-- NEW -->nec dici pote&longs;t neutram moueri, cum <lb/>moueri po&longs;&longs;it L per arcum LT, & C per arcum CS; </s> <s id="N214A8"><!-- NEW -->quippe impetus <lb/>e&longs;t indifferens ad omnem lineam; & hæc e&longs;t ratio à priori circularis <lb/>motus de qua fusè infrà. </s> </p> <p id="N214B0" type="main"> <s id="N214B2"><!-- NEW -->Ob&longs;eruabis motum circularem ab iis negari, qui ex punctis mathema­<lb/>ticis continuum componunt; </s> <s id="N214B8"><!-- NEW -->quia ex eo &longs;equeretur non po&longs;&longs;e dari mo­<lb/>tum continuum velociorem, vel tardiorem, quod ridiculum e&longs;t; </s> <s id="N214BE"><!-- NEW -->&longs;i enim <lb/>punctum Q æquali tempore moueatur cum puncto C certè arcus QR <lb/>quem percurrit eo tempore, quo C percurrit arcum CS, e&longs;&longs;et æqualis <lb/>arcui CS, quod e&longs;t ab&longs;urdum; </s> <s id="N214C8"><!-- NEW -->quod certè ne admittere cogantur, mo­<lb/>tum circularem negant, quod æquè ab&longs;urdum e&longs;t; </s> <s id="N214CE"><!-- NEW -->præ&longs;ertim eum ad vi­<lb/>tandum motum circularem infinita quoque ab&longs;urda deglutiant, ma­<lb/>nife&longs;tis experimentis contradicant, oculos ip&longs;os intuentium præ&longs;tigiis <lb/>illudi a&longs;&longs;erant, ferreum vectem dum mouetur in mille partes diffringi <lb/>etiam iurent; &longs;ed hæc omitto. </s> </p> <p id="N214DA" type="main"> <s id="N214DC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N214E9" type="main"> <s id="N214EB"><!-- NEW --><emph type="italics"/>Ni&longs;i impediretur impetus determinatio per lineam rectam, non daretur mo­<lb/>tus circularis &longs;altem in &longs;ublunaribus.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->ni&longs;i impediretur determinatio <lb/>impetus, qui ine&longs;t puncto L per lineam LM; </s> <s id="N214FC"><!-- NEW -->haud dubiè non mouere­<lb/>tur per arcum LB, &longs;ed per rectam LM; igitur ille motus non e&longs;&longs;et cir­<lb/>cularis. </s> </p> <p id="N21504" type="main"> <s id="N21506"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N21513" type="main"> <s id="N21515"><!-- NEW --><emph type="italics"/>Hinc motus circularis oritur ex recto impedito in &longs;ingulis punctis<emph.end type="italics"/>: </s> <s id="N2151E"><!-- NEW -->dixi in <lb/>&longs;ingulis punctis; </s> <s id="N21524"><!-- NEW -->quia licèt in puncto L impediretur, non tamen in &longs;e­<lb/>quenti; </s> <s id="N2152A"><!-- NEW -->e&longs;&longs;et quidem noua linea determinationis, non tamen curua; &longs;i <lb/>tamen in &longs;ingulis punctis impediatur æquali &longs;emper radio, haud dubiè <lb/>e&longs;t circularis. </s> </p> <p id="N21532" type="main"> <s id="N21534">Ob&longs;eruabis dictum e&longs;&longs;e &longs;upra in &longs;ublunaribus quia corpora cœle&longs;tia <lb/>mouentur motu circulari non habita vlla ratione motus recti, de quo <lb/>&longs;uo loco. </s> </p> <pb pagenum="274" xlink:href="026/01/308.jpg"/> <p id="N2153F" type="main"> <s id="N21541"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2154E" type="main"> <s id="N21550"><!-- NEW --><emph type="italics"/>Hinc &longs;ingulis instantibus punctum dum mouetur circa centrum<emph.end type="italics"/> K <emph type="italics"/>deter­<lb/>minatur ad nouam lineam<emph.end type="italics"/>; </s> <s id="N21561"><!-- NEW -->quia &longs;cilicet &longs;ingulis in&longs;tantibus impeditur; </s> <s id="N21565"><!-- NEW --><lb/>igitur &longs;ingulis in&longs;tantibus nouam determinationem accipit; e&longs;t enim ea­<lb/>dem ratio pro &longs;ecundo in&longs;tanti, quæ e&longs;t pro primo, itemque pro tertio, <lb/>quarto, &c. </s> </p> <p id="N2156E" type="main"> <s id="N21570"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2157D" type="main"> <s id="N2157F"><!-- NEW --><emph type="italics"/>Hinc tot &longs;unt determinationes &longs;ingulis in&longs;tantibus re&longs;pondentes, quot &longs;unt <lb/>Tangentes in circulo<emph.end type="italics"/>; </s> <s id="N2158A"><!-- NEW -->quippè in &longs;ingulis punctis determinatur ad Tan­<lb/>gentem; </s> <s id="N21590"><!-- NEW -->&longs;ed impeditur denuò pro &longs;equenti in&longs;tanti; </s> <s id="N21594"><!-- NEW -->igitur ad nouam <lb/>Tangentem determinatur; </s> <s id="N2159A"><!-- NEW -->e&longs;t autem hæc veri&longs;&longs;ima motus circularis ra­<lb/>tio; </s> <s id="N215A0"><!-- NEW -->quod &longs;cilicet cum &longs;ingulis in&longs;tantibus æqualiter impediatur motus <lb/>rectus; </s> <s id="N215A6"><!-- NEW -->quia altera mobilis extremitas accedere non pote&longs;t, &longs;ingulis quo­<lb/>que in&longs;tantibus ad nouam Tangentem determinatur æquali &longs;emper ra­<lb/>dio; vnde nece&longs;&longs;ariò &longs;equitur motus circularis. </s> </p> <p id="N215AE" type="main"> <s id="N215B0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N215BD" type="main"> <s id="N215BF"><!-- NEW --><emph type="italics"/>Hinc reiicies aliquem recentiorem, qui vult motum circularem e&longs;&longs;e mixtum <lb/>ex duobus rectis, quorum alter &longs;it vt &longs;inus recti, alter verò vt &longs;inus ver&longs;i,<emph.end type="italics"/> &longs;it <lb/>enim quadrans KCE; &longs;it impetus per EK, & per EO, vel duplex, vel <lb/>idem determinatus ad duas i&longs;tas lineas, ita vt determinatio per EK &longs;it <lb/>ad determinationem EO, vt &longs;inus ver&longs;i ad rectos. </s> <s id="N215D0"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->a&longs;&longs;umpto arcu <lb/>EM, vt EN ad NM; certè hoc po&longs;ito debet moueri punctum E per li­<lb/>neam circularem EMC. </s> <s id="N215DC"><!-- NEW -->Equidem &longs;i e&longs;&longs;et duplex impetus, vel vnus tan­<lb/>tùm cum duplici illa determinatione, ex eo &longs;equeretur motus circularis <lb/>mixtus ex duobus rectis; </s> <s id="N215E4"><!-- NEW -->&longs;icut rectus pote&longs;t ex duobus circularibus ori­<lb/>ri, vt dicemus aliàs; </s> <s id="N215EA"><!-- NEW -->non tamen inde &longs;equitur omnem motum circula­<lb/>rem e&longs;&longs;e mixtum ex duobus rectis, quod nemo non videt: </s> <s id="N215F0"><!-- NEW -->quippe po&longs;ito <lb/>quòd radius KE &longs;it affixus immobiliter centro K, licèt pellatur tantùm, <lb/>per Tangentem EO etiam cum valido impetu, nihilo tamen minus mo­<lb/>tu circulari mouebitur: </s> <s id="N215FA"><!-- NEW -->Adde quod difficile e&longs;&longs;et duos impetus ita attem­<lb/>perare, vt cre&longs;ceret vnus in ratione &longs;inuum ver&longs;orum, & alter in ratione <lb/>&longs;inuum rectorum; </s> <s id="N21602"><!-- NEW -->nec enim motus illi recti, ex quibus circularis qua&longs;i <lb/>na&longs;ceretur, æquales e&longs;&longs;e po&longs;&longs;unt; </s> <s id="N21608"><!-- NEW -->igitur &longs;ufficit vnius impetus ad vnam <lb/>tantùm lineam primo in&longs;tanti determinatus v.g. <!-- REMOVE S-->ad Tangentem EO, qui <lb/>ratione impedimenti in K &longs;uum effectum habere non pote&longs;t, &longs;ed reduci­<lb/>tur continuò ver&longs;us K æquali &longs;emper di&longs;tantia; </s> <s id="N21614"><!-- NEW -->ex quo &longs;equitur nece&longs;&longs;a­<lb/>riò motus circularis, &longs;cilicet ex illa qua&longs;i funis adductione; </s> <s id="N2161A"><!-- NEW -->&longs;i enim ex <lb/>puncto K laxaretur habena &longs;egmentis æqualibus; </s> <s id="N21620"><!-- NEW -->differentiæ &longs;inus totius <lb/>& &longs;ecantis v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;egmento VO in arcu EP; certè E moueretur per <lb/>rectam EO. </s> </p> <p id="N2162C" type="main"> <s id="N2162E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N2163A" type="main"> <s id="N2163C"><!-- NEW --><emph type="italics"/>Hinc optimè intelligitur ratio hypothe&longs;eos primæ<emph.end type="italics"/>; </s> <s id="N21645"><!-- NEW -->&longs;i enim punctum E &longs;epara-<pb pagenum="275" xlink:href="026/01/309.jpg"/>retur à recta EK eo in&longs;tanti, quo imprimitur impetus; </s> <s id="N2164E"><!-- NEW -->haud dubiè per <lb/>rectam EO moueretur; </s> <s id="N21654"><!-- NEW -->quia &longs;cilicet impetus puncti E determinatus e&longs;t <lb/>in puncto E ad motum per Tangentem EO; </s> <s id="N2165A"><!-- NEW -->& &longs;i nullum e&longs;&longs;et impedi­<lb/>mentum per rectam EO, moueretur; </s> <s id="N21660"><!-- NEW -->atqui &longs;i &longs;eparetur punctum E, ce&longs;­<lb/>&longs;at impedimentum, vt patet; </s> <s id="N21666"><!-- NEW -->nec enim amplius retinetur ex puncto K; </s> <s id="N2166A"><!-- NEW --><lb/>igitur ce&longs;&longs;at ratio motus circularis; </s> <s id="N2166F"><!-- NEW -->igitur motu recto per rectam EO <lb/>mouebitur; </s> <s id="N21675"><!-- NEW -->&longs;ic lapis impo&longs;itus rotæ dum maximo cum impetu vertitur, <lb/>per Tangentem proiicitur; </s> <s id="N2167B"><!-- NEW -->&longs;ic gutta aquæ, quæ cadit in volubilem tro­<lb/>chum etiam di&longs;pergitur; </s> <s id="N21681"><!-- NEW -->&longs;ic rota ip&longs;a, cuius aliqua pars præ nimia vi <lb/>motus diffringitur, illam qua&longs;i proiicit per rectam; </s> <s id="N21687"><!-- NEW -->hinc ratio vnica <lb/>proiectionis quæ fit operâ fundarum; </s> <s id="N2168D"><!-- NEW -->&longs;it enim funda KE vel KL, quæ <lb/>moueatur per arcum LE; </s> <s id="N21693"><!-- NEW -->certè, &longs;i lapis demittatur in puncto E, lapis <lb/>proiicietur per rectam LO; </s> <s id="N21699"><!-- NEW -->nec enim ad aliam lineam lapis, dum e&longs;t in <lb/>puncto E, e&longs;t determinatus, ni&longs;i ad Tangentem EO, ad quam dumtaxat <lb/>impetus puncti EA e&longs;t determinatus; in hoc igitur Fundibularij tan­<lb/>tùm in&longs;i&longs;tit indu&longs;tria, quâ &longs;cilicet &longs;axum in funda rotatum &longs;copum cui <lb/>de&longs;tinatur, attingat, vt illam Tangentem inueniat quæ à prædicto &longs;copo <lb/>in circulum, quem &longs;uo motu de&longs;cribit, funda ducitur. </s> <s id="N216A7"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it radius fun­<lb/>dæ KL hypomoclium K, circulus quem de&longs;cribit funda LEC; </s> <s id="N216AF"><!-- NEW -->&longs;it &longs;co­<lb/>pus O, ducatur tangens EO; </s> <s id="N216B5"><!-- NEW -->certè, &longs;i vbi funda peruenit in E, dimit­<lb/>tat lapidem, prædictum &longs;copum non illicò feriet; </s> <s id="N216BB"><!-- NEW -->hinc etiam ratio, cur in <lb/>naui dum motu recto mouetur facilè con&longs;i&longs;tamus; cum tamen (quod in <lb/>longioribus illis nauiculis facilè contingere pote&longs;t) &longs;i circa centrum <lb/>&longs;uum nauis vertatur, quod accidit cum vtraque extremitas in partes op­<lb/>po&longs;itas, vel remo, vel pertica pellitur, nec in ca con&longs;i&longs;tamus. </s> </p> <p id="N216C7" type="main"> <s id="N216C9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N216D5" type="main"> <s id="N216D7"><!-- NEW --><emph type="italics"/>Si rota plana in circulo horizontali voluatur, &longs;itque pondus plano rotæ incu­<lb/>bans, in eo producetur impetus<emph.end type="italics"/>; vt certum e&longs;t; </s> <s id="N216E2"><!-- NEW -->an verò pondus retroagi de­<lb/>beat, præ&longs;ertim &longs;i &longs;it globus, vel aqua; </s> <s id="N216E8"><!-- NEW -->an verò per Tangentem proiici, <lb/>dubium e&longs;&longs;e pote&longs;t; </s> <s id="N216EE"><!-- NEW -->videntur enim pro vtraque hypothe&longs;i facere expe­<lb/>rientiæ; </s> <s id="N216F4"><!-- NEW -->pro prima quidem, &longs;i rotetur rota concaua &longs;eu &longs;cutella plena <lb/>aqua; </s> <s id="N216FA"><!-- NEW -->aqua enim in partem contrariam volui videbitur; &, &longs;i plano <lb/>quod in circulo horizontali voluitur imponatur globus leuigati&longs;&longs;imus, <lb/>certè in partem oppo&longs;itam ibit. </s> <s id="N21702"><!-- NEW -->Secundæ hypothe&longs;i alia videntur fauere <lb/>experimenta; </s> <s id="N21708"><!-- NEW -->&longs;i enim trochus volubilis, vel aqua, vel puluere a&longs;perga­<lb/>tur, &longs;tatim aqua re&longs;ilit per Tangentem, idem dico de puluere, &longs;i funda in <lb/>circulo horizontali voluatur, lapis demi&longs;&longs;us per Tangentem ibit: &longs;ed <lb/>hæc omnia, quæ ad proiectiones pertinent, licèt illæ &longs;equantur ex motu <lb/>circulari, examinabimus & demon&longs;trabimus lib. 10. cum de proiectis. </s> </p> <p id="N21714" type="main"> <s id="N21716"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N21722" type="main"> <s id="N21724"><!-- NEW --><emph type="italics"/>Cau&longs;a motus circularis e&longs;t ea, quæ cum tali impedimento coniuncta e&longs;t<emph.end type="italics"/>; </s> <s id="N2172D"><!-- NEW -->ex <lb/>quo accidit diametrum mobilis in aliquo &longs;ui puncto retineri immobi­<lb/>lem; &longs;unt autem varij modi huius applicationis. </s> <s id="N21735">Primus e&longs;t ille, quem <lb/>indicauimus &longs;uprà Th.1.cum &longs;cilicet vtraque extremitas cylindri æquali <pb pagenum="276" xlink:href="026/01/310.jpg"/>impetu in partes oppo&longs;itas pellitur. </s> <s id="N2173F">v.g. <!-- REMOVE S-->C per CP, L per LG. Secundus<lb/>e&longs;t, cum affigitur altera extremitas. </s> <s id="N21746">v.g. <!-- REMOVE S-->punctum K affigitur, ita vt tamen <lb/>propter flexibilitatem radij KL, idem radius moueri po&longs;&longs;it circa cen­<lb/>trum K, vt videmus in funependulis. </s> <s id="N2174F"><!-- NEW -->Tertius e&longs;t, &longs;i diameter fulcro K <lb/>in&longs;eratur, vt in obelis ferri, vel magnetica acu: huc reuoca rotas omnes, <lb/>quæ in circulo horizontali, & verticali voluuntur. </s> <s id="N21757">Quartus, &longs;i cum ali­<lb/>qua explo&longs;ione digitorum motus imprimatur, vel globo, vel trocho, vel <lb/>iis cubis, quibus in&longs;cripti numeri po&longs;t girationem &longs;ortem indicant. </s> <s id="N2175E"><!-- NEW --><lb/>Quintus, &longs;i cum flagello trochus agatur; </s> <s id="N21763"><!-- NEW -->cum enim implicetur flagel­<lb/>lum trocho, vbi retrahitur, in gyros agitur trochus; </s> <s id="N21769"><!-- NEW -->huc reuoca funem <lb/>illum plicatilem, quibus armatus ferro trochus voluitur: </s> <s id="N2176F"><!-- NEW -->adde his refle­<lb/>xionem variam ex qua &longs;æpè oritur hæc turbinatio; </s> <s id="N21775"><!-- NEW -->tùm etiam figuram <lb/>va&longs;is; </s> <s id="N2177B"><!-- NEW -->&longs;ic aqua intra vas &longs;phæricum voluitur; </s> <s id="N2177F"><!-- NEW -->&longs;ic in vorticibus voluitur <lb/>aqua propter præruptum de&longs;cen&longs;um aluei; </s> <s id="N21785"><!-- NEW -->&longs;ic etiam turbinatim de&longs;cen­<lb/>dit aqua per tubum infundibuli; cætera omitto, quæ ex his facilè intel­<lb/>ligi po&longs;&longs;unt. </s> </p> <p id="N2178D" type="main"> <s id="N2178F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N2179B" type="main"> <s id="N2179D"><!-- NEW --><emph type="italics"/>Datur impetus in motu circulari<emph.end type="italics"/>; </s> <s id="N217A6"><!-- NEW -->probatur facilè, quia etiam ab&longs;ente <lb/>potentia motrice durat motus; </s> <s id="N217AC"><!-- NEW -->igitur ade&longs;&longs;e debet illius cau&longs;a; igitur <lb/>impetus, clarum e&longs;t; </s> <s id="N217B2"><!-- NEW -->debet autem e&longs;&longs;e hic impetus ita determinatus, vt <lb/>determinatio vnius puncti impediat determinationem alteriùs; &longs;ed aliam <lb/>permittat, alioqui de&longs;trueretur totus impetus, & hæc vici&longs;&longs;im illam. </s> </p> <p id="N217BA" type="main"> <s id="N217BC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N217C8" type="main"> <s id="N217CA"><!-- NEW --><emph type="italics"/>Subjectum huius impetus e&longs;t omne mobile<emph.end type="italics"/>; </s> <s id="N217D3"><!-- NEW -->non e&longs;t difficultas pro mobili <lb/>corporeo, quod pluribus partibus con&longs;tat; </s> <s id="N217D9"><!-- NEW -->quippe impetus vnius partis <lb/>pote&longs;t impedire impetum alterius; </s> <s id="N217DF"><!-- NEW -->at difficilius e&longs;t dictu, an punctum, <lb/>&longs;i detur, moueri po&longs;&longs;it circulariter: de puncto phy&longs;ico loquor? </s> <s id="N217E5"><!-- NEW -->cui cer­<lb/>tè non repugnat motus circularis; quippè licèt careat partibus actu, non <lb/>tamen caret partibus potentiâ. </s> <s id="N217ED"><!-- NEW -->Dices, non mutat locum; </s> <s id="N217F1"><!-- NEW -->igitur non mo­<lb/>uetur: </s> <s id="N217F7"><!-- NEW -->antecedens con&longs;tare videtur, quia &longs;emper remanet in eodem loco: </s> <s id="N217FB"><!-- NEW --><lb/>con&longs;equentia etiam videtur e&longs;&longs;e clara per Def.1. lib. 1. Re&longs;pondeo pri­<lb/>mò mutare locum re&longs;pectiuum; </s> <s id="N21802"><!-- NEW -->quippe licèt punctum phy&longs;icum non ha­<lb/>beat partes, habet tamen facies; </s> <s id="N21808"><!-- NEW -->vnde facies conuertuntur per motum <lb/>circularem; </s> <s id="N2180E"><!-- NEW -->igitur non habent ampliùs eundem re&longs;pectum; igitur nec <lb/>eundem locum re&longs;pectiuum. </s> <s id="N21814"><!-- NEW -->Re&longs;pondeo &longs;ecundò, punctum phy&longs;icum ha­<lb/>bere partes potentiâ, non actu; </s> <s id="N2181A"><!-- NEW -->vnde mutat locum, dum voluitur; </s> <s id="N2181E"><!-- NEW -->quia <lb/>quælibet pars potentiâ diuer&longs;æ parti &longs;patij potentiâ re&longs;pondet; </s> <s id="N21824"><!-- NEW -->&longs;ed hîc <lb/>non di&longs;cutio quæ&longs;tionem illam, an dentur puncta phy&longs;ica; </s> <s id="N2182A"><!-- NEW -->&longs;ed tantùm <lb/>a&longs;&longs;ero, ex &longs;uppo&longs;itione quòd detur punctum phy&longs;icum moueri po&longs;&longs;e mo­<lb/>tu circulari: </s> <s id="N21832"><!-- NEW -->Idem de Angelo dici pote&longs;t, non tamen de puncto mathe­<lb/>matico, cuius motus concipi non pote&longs;t; </s> <s id="N21838"><!-- NEW -->vnde optimè negat Ari&longs;toteles <lb/>punctum mathematicum moueri po&longs;&longs;e; </s> <s id="N2183E"><!-- NEW -->immò nos aliquando repugnare <lb/>dari punctum mathematicum o&longs;tendemus; igitur ex dictis patet, omne <pb pagenum="277" xlink:href="026/01/311.jpg"/>mobile, quod &longs;cilicet moueri pote&longs;t motu recto, motu circulari etiam <lb/>moueri po&longs;&longs;e. </s> </p> <p id="N2184B" type="main"> <s id="N2184D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N21859" type="main"> <s id="N2185B"><!-- NEW --><emph type="italics"/>Finis huius motus varius e&longs;t in naturâ, & multiplex v&longs;us<emph.end type="italics"/>; primò enim <lb/>ex motu circulari fit, vt impetus qui e&longs;t ad omnem lineam indifferens <lb/>habeat &longs;uum effectum, cum omnes lineæ impediuntur præter vnam, & <lb/>hoc e&longs;t vera ratio à priori huius motus. </s> <s id="N2186A"><!-- NEW -->Secundò nulla libratio, &longs;eu vi­<lb/>bratio e&longs;&longs;e po&longs;&longs;et, ni&longs;i motus circularis e&longs;&longs;et; hinc nullus libræ v&longs;us, ve­<lb/>ctis, trochleæ, aliorumque organorum mechanicorum quorum opera <lb/>inutilis e&longs;&longs;et &longs;ine motu circulari. </s> <s id="N21874"><!-- NEW -->Tertiò, omitto gyros, & &longs;piras, turbi­<lb/>num, rotarum, lapidum molarium, immò & &longs;yderum orbitas, fundarum <lb/>librationes; </s> <s id="N2187C"><!-- NEW -->immò & ip&longs;orum brachiorum; digitorum, tybiarum v&longs;um; <lb/>immò au&longs;im dicere motum circularem non minùs toti naturæ vtilem <lb/>e&longs;&longs;e, quàm rectum. </s> </p> <p id="N21884" type="main"> <s id="N21886"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N21892" type="main"> <s id="N21894"><!-- NEW --><emph type="italics"/>Motus circularis pote&longs;t appellari &longs;implex<emph.end type="italics"/>; </s> <s id="N2189D"><!-- NEW -->quia ex pluribus mixtus non <lb/>e&longs;t omnis motus circularis, licèt aliquis motus circularis po&longs;&longs;it e&longs;&longs;e mixtus <lb/>ex duobus rectis, vt dictum e&longs;t &longs;uprà; </s> <s id="N218A5"><!-- NEW -->non minùs quàm rectus pote&longs;t e&longs;&longs;e <lb/>mixtus ex duobus circularibus; </s> <s id="N218AB"><!-- NEW -->non e&longs;t tamen propterea dicendum om­<lb/>nem circularem e&longs;&longs;e mixtum; </s> <s id="N218B1"><!-- NEW -->cum &longs;cilicet in mobili, quod circulari mo­<lb/>tu mouetur, non fit duplex impetus; quis autem dicat motum funepen­<lb/>duli &longs;ur&longs;um vibrati e&longs;&longs;e mixtum? </s> <s id="N218B9"><!-- NEW -->equidem in &longs;ublunaribus nullus e&longs;t mo­<lb/>tus circularis qui ex multiplici determinatione non con&longs;tet, vt dictum <lb/>e&longs;t &longs;uprà; </s> <s id="N218C1"><!-- NEW -->Vnde fortè vel eo nomine mixtus dici po&longs;&longs;et, &longs;ed propter ean­<lb/>dem rationem motus reflexus mixtus dici po&longs;&longs;et; </s> <s id="N218C7"><!-- NEW -->quidquid &longs;it, dum rem <lb/>intelligas, loquere vt voles; </s> <s id="N218CD"><!-- NEW -->dixi in &longs;ublunaribus, quia corpora cœle&longs;tia <lb/>ita &longs;unt à natura in&longs;tituta, vt circulari motu rotari po&longs;tulent; de quo &longs;uo <lb/>loco: </s> <s id="N218D5"><!-- NEW -->Et verò hæc legitima videtur e&longs;&longs;e Ari&longs;totelis &longs;ententia, qui motum <lb/>naturalem rectum grauibus, & leuibus tribuit, circularem verò cœle&longs;ti­<lb/>bus; </s> <s id="N218DD"><!-- NEW -->ex quo etiam motu tanquam ex natiua proprietate quintam cœlo­<lb/>rum e&longs;&longs;entiam concludit; denique nulla videtur e&longs;&longs;e repugnantia, nul­<lb/>lumque ab&longs;urdum, &longs;i motus circularis alicui corpori competat. </s> <s id="N218E5"><!-- NEW -->Vtrum <lb/>verò motus circularis dici po&longs;&longs;it naturalis, dubium e&longs;&longs;e non pote&longs;t, pro <lb/>cœle&longs;tibus illis corporibus, &longs;i à principio intrin&longs;eco rotantur; </s> <s id="N218ED"><!-- NEW -->pro &longs;ub­<lb/>lunaribus aliquod fortè dubium e&longs;&longs;et; &longs;ed quæ&longs;o te cum funependulum <lb/>&longs;ua &longs;ponte vibratum de&longs;cendit, quo nomine motum illum appellas? </s> <s id="N218F5">Nun­<lb/>quid e&longs;t à principio intrin&longs;eco? </s> <s id="N218FA">cur igitur naturalem appellare detrectas? </s> <s id="N218FD"><lb/>rem intelligis, loquere vt voles. </s> </p> <p id="N21901" type="main"> <s id="N21903"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N2190F" type="main"> <s id="N21911"><emph type="italics"/>Omnia puncta eiu&longs;dem circuli mouentur æquali motu.<emph.end type="italics"/></s> <s id="N21918"><!-- NEW --> Probatur quia <lb/>æqualibus temporibus æquales arcus percurrunt, vt con&longs;tat; igitur mo­<lb/>uentur æquali motu, id e&longs;t æquè velociter per Axioma 1. <!-- KEEP S--></s> </p> <pb pagenum="278" xlink:href="026/01/312.jpg"/> <p id="N21925" type="main"> <s id="N21927"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N21933" type="main"> <s id="N21935"><!-- NEW --><emph type="italics"/>Puncta diuer&longs;orum circulorum mouentur inæquali motu<emph.end type="italics"/>; </s> <s id="N2193E"><!-- NEW -->quia tempori­<lb/>bus æqualibus inæquales percurrunt arcus; </s> <s id="N21944"><!-- NEW -->igitur inæquali motu per <lb/>Axio. <!-- REMOVE S-->1. v.g. <!-- REMOVE S-->puncta L & C quæ di&longs;tant æqualiter à centro K, mouentur <lb/>æquali motu, quia æquali tempore conficiunt æquales arcus CS, LT; at <lb/>verò puncta CQ inæquali motu mouentur, quia æquali tempore arcus <lb/>inæquales percurrunt, &longs;cilicet CS, QX. </s> </p> <p id="N21954" type="main"> <s id="N21956"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N21962" type="main"> <s id="N21964"><!-- NEW --><emph type="italics"/>Hinc puncta, quæ accedunt propiùs ad centrum mouentur tardiùs, quæ lon­<lb/>giùs recedunt, mouentur velociùs.<emph.end type="italics"/> v.g. <!-- REMOVE S-->C velociùs, quia conficit arcum ma­<lb/>iorem; </s> <s id="N21973"><!-- NEW -->CSQ tardiùs, quia æquali tempore conficit arcum minorem <lb/>QR &longs;unt autem arcus &longs;imiles, vt radij, id e&longs;t QR e&longs;t ad CS, vt radius <lb/>KQ ad QC, &longs;ed motus &longs;unt vt arcus; igitur motus, vt radij, vel di&longs;tantiæ <lb/>à centro communi. </s> </p> <p id="N2197D" type="main"> <s id="N2197F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N2198B" type="main"> <s id="N2198D"><!-- NEW --><emph type="italics"/>Ex his constat impetum, qui præstat motum circularem distribui in mobili <lb/>vniformiter, id e&longs;t æqualem in eodem circulo, vel in distantia æquali, & dif­<lb/>formiter, id e&longs;t inæqualem in diuer&longs;is circulis, vel in diuer&longs;a distantia<emph.end type="italics"/>; </s> <s id="N2199A"><!-- NEW -->quia <lb/>ex inæqualitate motus cogno&longs;ci tantùm pote&longs;t inæqualitas impetus; </s> <s id="N219A0"><!-- NEW -->fit <lb/>autem hæc diffu&longs;io, &longs;eu propagatio in ratione longitudinum v. <!-- REMOVE S-->g. <!-- REMOVE S-->impe­<lb/>tus in Q e&longs;t ad impetum in C, vt longitudo KQ ad KC, vt con&longs;tat ex <lb/>dictis; </s> <s id="N219AE"><!-- NEW -->accipio autem omnes partes impetus, quæ &longs;unt in Q, & compa­<lb/>ro omnes illas cum omnibus illis, quæ in&longs;unt puncto C; </s> <s id="N219B4"><!-- NEW -->nam certum e&longs;t <lb/>ex his quæ fusè diximus lib.1.non produci plures partes impetus in C, <expan abbr="quã">quam</expan> <lb/>in <expan abbr="q;">que</expan> &longs;ed perfectiorem impetum produci in C, quàm in Q: </s> <s id="N219C4"><!-- NEW -->recole quæ <lb/>diximus lib.1. à Th. 99. ad Th.112. in quibus habes totam propagatio­<lb/>nem impetus determinati ad motum circularem; </s> <s id="N219CC"><!-- NEW -->&longs;iue applicetur po­<lb/>tentia centro, id e&longs;t iuxta centrum; &longs;iue circumferentiæ. </s> </p> <p id="N219D2" type="main"> <s id="N219D4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N219E0" type="main"> <s id="N219E2"><!-- NEW --><emph type="italics"/>Motus puncti C non e&longs;t velocior motu puncti Q ratione temporis, &longs;ed &longs;patij<emph.end type="italics"/>; </s> <s id="N219EB"><!-- NEW --><lb/>quia vtrumque mouetur &longs;emper æquali tempore, quia &longs;unt in eodem ra­<lb/>dio; </s> <s id="N219F2"><!-- NEW -->recole etiam, quæ diximus alibi, &longs;cilicet lib. 2. in comparatione <lb/>motuum, vel a&longs;&longs;umi po&longs;&longs;e &longs;patia æqualia cum temporibus inæqualibus, <lb/>vel tempora æqualia cum &longs;patiis inæqualibus; </s> <s id="N219FA"><!-- NEW -->atqui in motu circulari <lb/>cum omnes partes eiu&longs;dem mobilis &longs;imul moueantur, id e&longs;t &longs;imul inci­<lb/>piant, & de&longs;inant moueri; </s> <s id="N21A02"><!-- NEW -->certè æquali tempore mouentur; </s> <s id="N21A06"><!-- NEW -->&longs;ed motus <lb/>e&longs;t inæqualis; igitur non ratione temporis, quod æquale e&longs;t, &longs;ed <lb/>&longs;patij. </s> </p> <p id="N21A0E" type="main"> <s id="N21A10">Hic fortè aliquis de&longs;ideraret &longs;olutionem illius argumenti, quod vul­<lb/>gò ducitur ex motu circulari contra puncta phy&longs;ica, quod &longs;ic breuiter <lb/>proponi pote&longs;t. </s> <s id="N21A17"><!-- NEW -->Sit punctum Q, quod acquirat punctum &longs;patij ver&longs;us R <lb/>vno in&longs;tanti; </s> <s id="N21A1D"><!-- NEW -->certe punctum C, quod mouetur ver&longs;us S, acquiret eodem <pb pagenum="279" xlink:href="026/01/313.jpg"/>illo in&longs;tanti plu&longs;quam punctum &longs;patij; </s> <s id="N21A26"><!-- NEW -->igitur eodem in&longs;tanti erit in <lb/>duobus loris, quod e&longs;t ab&longs;urdum; </s> <s id="N21A2C"><!-- NEW -->nec pote&longs;t dici punctum C moueri <lb/>duobus in&longs;tantibus, &longs;ed minoribus, quæ &longs;cilicet re&longs;pondeant in&longs;tanti, quo <lb/>mouetur punctum <expan abbr="q;">que</expan> quia &longs;i po&longs;t primum in&longs;tans C &longs;i&longs;teret, Q mouere­<lb/>tur adhuc, quod e&longs;t ab&longs;urdum; nam &longs;imul incipit, & de&longs;init moueri, <lb/>cum puncto C. <!-- KEEP S--></s> <s id="N21A3D"><!-- NEW -->Equidem non pote&longs;t explicari maior velocitas motus C <lb/>per in&longs;tantia minora, vt patet; igitur per &longs;patia maiora. </s> <s id="N21A43"><!-- NEW -->Itaque re&longs;pon­<lb/>deo &longs;i C & Q mouentur in eodem radio conjunctim non po&longs;&longs;e pun­<lb/>ctum K acquirere punctum &longs;patij nullo modo participans cum priori, <lb/>&longs;ed participans; </s> <s id="N21A4D"><!-- NEW -->licèt enim punctum &longs;patij careat partibus actu, habet <lb/>tamen partes potentia, vt explicabimus fusè &longs;uo loco; </s> <s id="N21A53"><!-- NEW -->&longs;unt enim vbica­<lb/>tiones communicantes, & non communicantes, quod explico in Ange­<lb/>lo &longs;it enim Angelus coëxten&longs;us quadrato FC, (quam hypothe&longs;im <lb/>nemo negabit;) &longs;it alius æqualis exten&longs;ionis coëxten&longs;us quadrato HE, <lb/>qui con&longs;i&longs;tat dum primus Angelus mouetur; </s> <s id="N21A5F"><!-- NEW -->certè ita moueri pote&longs;t, vt <lb/>primo in&longs;tanti occupet &longs;patium CK, & coëxtendatur alteri Angelo, vt <lb/>certum e&longs;t; </s> <s id="N21A67"><!-- NEW -->quippè vnico in&longs;tanti locum &longs;ibi adæquatum occupare po­<lb/>te&longs;t; </s> <s id="N21A6D"><!-- NEW -->vel ita moueri pote&longs;t, vt primo in&longs;tanti occupet &longs;patium GD, & <lb/>coëxtendatur quidem alteri Angelo &longs;ed inadæquatè: </s> <s id="N21A73"><!-- NEW -->his po&longs;itis, &longs;patium <lb/>HE comparatum cum &longs;patio FC e&longs;t non communicans; </s> <s id="N21A79"><!-- NEW -->&longs;patium verò <lb/>GD communicans, tum cum HE, tum cum HA, po&longs;&longs;unt autem dari <lb/>huiu&longs;modi &longs;patia in infinitum plùs vel minùs participantia v. <!-- REMOVE S-->g. <!-- REMOVE S-->LM <lb/>plus participat de AC quam BD, & BD plu&longs;quam NO; </s> <s id="N21A87"><!-- NEW -->igitur non <lb/>e&longs;t dubium quin Angelus moueatur eo tardiùs, &longs;uppo&longs;ito æquali tempo­<lb/>re, quo acquirit &longs;patium plùs participans de priore; </s> <s id="N21A8F"><!-- NEW -->vnde quando vno <lb/>in&longs;tanti acquirit &longs;patium non communicans HE, non pote&longs;t velociùs <lb/>moueri illo in&longs;tanti, vel æquali; </s> <s id="N21A97"><!-- NEW -->nec pote&longs;t motus e&longs;&longs;e velocior ratione <lb/>&longs;patij, licèt po&longs;&longs;it e&longs;&longs;e ratione temporis; quia &longs;patium HE acquirere po­<lb/>te&longs;t minore in&longs;tanti. </s> <s id="N21A9F"><!-- NEW -->Quod dicitur de Angelo, dicatur de puncto phy&longs;i­<lb/>co; cuius exten&longs;io e&longs;t quidem indiui&longs;ibilis actu vt exten&longs;io Angeli diui­<lb/>&longs;ibilis tamen potentia in infinitum. </s> </p> <p id="N21AA7" type="main"> <s id="N21AA9"><!-- NEW -->His po&longs;itis, motus extremitatis radij dirigit motum aliorum puncto­<lb/>rum ver&longs;us centrum; &longs;ed punctum extremitatis radij non pote&longs;t <lb/>dato in&longs;tanti moueri velociùs quàm &longs;i punctum &longs;patij non communi­<lb/>cans acquirat, quo po&longs;ito nullum aliud punctum radij acquirit eodem <lb/>in&longs;tanti &longs;patium non communicans. </s> </p> <p id="N21AB5" type="main"> <s id="N21AB7"><!-- NEW -->Dices, ponamus punctum extremitatis facta acce&longs;&longs;ione noui &longs;egmenti <lb/>moueri eadem velocitate, quâ priùs mouebatur, cum terminabat radium; </s> <s id="N21ABD"><!-- NEW --><lb/>igitur acquirit punctum &longs;patij non participans; igitur extremitas noua <lb/>illo in&longs;tanti acquirit plu&longs;quam punctum. </s> <s id="N21AC4"><!-- NEW -->Re&longs;pondeo, &longs;i addatur extremi­<lb/>tas noua facta &longs;cilicet acce&longs;&longs;ione noui &longs;egmenti, po&longs;ito quod punctum <lb/>prioris extremitatis moueatur æquè velociter ac priùs; </s> <s id="N21ACC"><!-- NEW -->certè noua ex­<lb/>tremitas velociùs mouebitur priore, vt con&longs;tat; </s> <s id="N21AD2"><!-- NEW -->igitur in&longs;tanti minore <lb/>acquiret &longs;patium non communicans; igitur hoc in&longs;tanti minore prior <lb/>extremitas acquirit &longs;patium communicans. </s> <s id="N21ADA"><!-- NEW -->Ex his vides velocitatem <pb pagenum="280" xlink:href="026/01/314.jpg"/>motus circularis ratione eiu&longs;dem radij, vel mobilis explicari per &longs;patia <lb/>magis, vel minùs communicantia; </s> <s id="N21AE5"><!-- NEW -->at verò velocitatem motus recti per <lb/>in&longs;tantia maiora, & minora: </s> <s id="N21AEB"><!-- NEW -->Sed hæc fusè in Metaphy&longs;ica explicabimus; </s> <s id="N21AEF"><!-- NEW --><lb/>neque hîc contendimus dari vel puncta, vel in&longs;tantia; </s> <s id="N21AF4"><!-- NEW -->&longs;ed tantùm po&longs;ito <lb/>quod dentur, ita &longs;olui po&longs;&longs;e argumentum illud, quod vulgò ducitur ex <lb/>motu circulari, quo reuerâ puncta Mathematica non tamen phy&longs;ica pro­<lb/>fligantur: </s> <s id="N21AFE"><!-- NEW -->&longs;imiliter &longs;olues argumentum illud vix triobolare, quo dicuntur <lb/>e&longs;&longs;e tot puncta in minore circulo, quot in maiore, eo quod iidem radij <lb/>vtrumque &longs;ecent, quia &longs;i duo radij ad duo puncta immediata maioris <lb/>terminentur, penetrantur inadæquatè in &longs;ectione minoris circuli; &longs;ed <lb/>de hoc aliàs. </s> </p> <p id="N21B0A" type="main"> <s id="N21B0C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N21B18" type="main"> <s id="N21B1A"><!-- NEW --><emph type="italics"/>Motus circularis pote&longs;t e&longs;&longs;e velocior, & tardior in infinitum<emph.end type="italics"/>; </s> <s id="N21B23"><!-- NEW -->quia quocun­<lb/>que dato radio pote&longs;t dari maior, & minor; </s> <s id="N21B29"><!-- NEW -->immò pote&longs;t compen&longs;ari <lb/>motus; </s> <s id="N21B2F"><!-- NEW -->&longs;it enim radius EC diui&longs;us bifariam in H; </s> <s id="N21B33"><!-- NEW -->certè &longs;i moueatur <lb/>EC circa centrum E; </s> <s id="N21B39"><!-- NEW -->C mouebitur duplo velociùs quàm H, quia arcus <lb/>CN e&longs;t duplus HT; </s> <s id="N21B3F"><!-- NEW -->&longs;i tamen &longs;it radius AH; </s> <s id="N21B43"><!-- NEW -->certè &longs;i pote&longs;t moueri <lb/>æquè velociter, &longs;i enim a&longs;&longs;umatur H <foreign lang="greek">m</foreign> æqualis HT, & percurrat H <foreign lang="greek">m</foreign><lb/>eo tempore, quo alter radius EC percurrit CN, motus erit æqualis; </s> <s id="N21B52"><!-- NEW -->quia <lb/>arcus CN & H <foreign lang="greek">m</foreign> &longs;unt æquales, vt con&longs;tat: </s> <s id="N21B5C"><!-- NEW -->pote&longs;t etiam vectis longio­<lb/>ris extremitas moueri motu æquali cum extremitate minoris; </s> <s id="N21B62"><!-- NEW -->&longs;i enim <lb/>H extremitas HE percurrit H <foreign lang="greek">m</foreign>, & a&longs;&longs;umatur vectis duplus EC, diuida­<lb/>tur H <foreign lang="greek">m</foreign> bifariam in T ducaturque ETN; </s> <s id="N21B72"><!-- NEW -->certè &longs;i C conficiat CN co­<lb/>dem tempore, vtraque extremitas C & H æquè velociter mouebitur; </s> <s id="N21B78"><!-- NEW -->&longs;i <lb/>autem duplicetur adhuc longitudo radij, diuidatur HT bifariam in X, <lb/>ducaturque linea, atque ita deinceps; quæ omnia &longs;unt trita. </s> </p> <p id="N21B80" type="main"> <s id="N21B82"><!-- NEW -->Ex his habes principium motus tardioris, & velocioris in infinitum; </s> <s id="N21B86"><!-- NEW -->&longs;i <lb/>enim punctum H &longs;emper æquali tempore conficiat arcum H <foreign lang="greek">m</foreign>; </s> <s id="N21B90"><!-- NEW -->certè <lb/>punctum C conficiet arcum C <foreign lang="greek">b</foreign> duplum prioris; </s> <s id="N21B9A"><!-- NEW -->quia EC e&longs;t dupla <lb/>EH; </s> <s id="N21BA0"><!-- NEW -->&longs;i verò accipiatur tripla, conficiet triplum, atque ita deinceps; </s> <s id="N21BA4"><!-- NEW -->&longs;ed <lb/>pote&longs;t vectis e&longs;&longs;e longior, & longior in infinitum; </s> <s id="N21BAA"><!-- NEW -->igitur motus velo­<lb/>cior, & velocior; </s> <s id="N21BB0"><!-- NEW -->&longs;i verò punctum C conficiat tantùm arcum CN æqua­<lb/>lem H <foreign lang="greek">m</foreign>; haud dubiè punctum H mouebitur duplò tardiùs, & &longs;i acci­<lb/>piatur vectis duplus CE, cuius extremitas percurrat arcum æqualem <lb/>CN, punctum H mouebitur quadruplò tardiùs, atque ita deinceps. </s> </p> <p id="N21BBE" type="main"> <s id="N21BC0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N21BCC" type="main"> <s id="N21BCE"><emph type="italics"/>Motus circularis non e&longs;t naturaliter acceleratus.<emph.end type="italics"/></s> <s id="N21BD5"><!-- NEW --> Probatur, quia in infi­<lb/>nitum intenderetur, quod e&longs;&longs;et ab&longs;urdum in natura; </s> <s id="N21BDB"><!-- NEW -->caret enim termino: </s> <s id="N21BDF"><!-- NEW --><lb/>non e&longs;t difficultas pro motu circulari violento quo v.g. <!-- REMOVE S-->vertitur rota in <lb/>circulo verticali, vel mixto, quo &longs;cilicet lapis &longs;phæricus ita de&longs;cendit, vt <lb/>circa &longs;uum centrum etiam voluatur, vel indifferenti, quo recta vertitur <lb/>in circulo horizontali; </s> <s id="N21BEC"><!-- NEW -->quia nullum e&longs;t principium accelerationis i&longs;to­<lb/>rum motuum; </s> <s id="N21BF2"><!-- NEW -->igitur e&longs;t tantùm difficultas pro naturali circulari, quo <pb pagenum="281" xlink:href="026/01/315.jpg"/>fortè &longs;ydera rotantur; qui tamen non e&longs;t acceleratus per &longs;e, propter ra­<lb/>tionem prædictam. </s> </p> <p id="N21BFD" type="main"> <s id="N21BFF">Obiiceret fortè aliquis; </s> <s id="N21C02"><!-- NEW -->eadem ratio quæ probat motum naturalem <lb/>deor&longs;um accelerari, eadem probat circularem naturalem etiam intendi: <lb/>quippè &longs;emper ade&longs;t principium intrin&longs;ecum applicatum. </s> <s id="N21C0A"><!-- NEW -->Re&longs;pondeo <lb/>negandam e&longs;&longs;e paritatem; </s> <s id="N21C10"><!-- NEW -->quia naturalis motus grauium non accelera­<lb/>tur fru&longs;trà; </s> <s id="N21C16"><!-- NEW -->Nunquam enim recedit à &longs;uo fine; </s> <s id="N21C1A"><!-- NEW -->at verò, &longs;i motus circula­<lb/>ris &longs;yderum acceleraretur, tandem abiret in infinitum, quod reuerâ e&longs;&longs;et <lb/>contra finem à natura in&longs;titutum; quippè carerent &longs;uo fine, & v&longs;u corpo­<lb/>ra cœle&longs;tia, &longs;i longè celeriori motu rotarentur. </s> </p> <p id="N21C24" type="main"> <s id="N21C26">Obiiceret alius, motus circularis naturalis non acceleraretur, igitur <lb/>tardi&longs;&longs;imus e&longs;&longs;et, qualis reuerâ motus naturalis grauium deor&longs;um, quod <lb/>e&longs;t contra experientiam. </s> <s id="N21C2D"><!-- NEW -->Re&longs;pondeo, vel determinatum impetus gradum, <lb/>eumque valdè intentum produxi&longs;&longs;e iuxta in&longs;titutum &longs;uæ naturæ, vel per <lb/>aliquot minuta &longs;e&longs;e moui&longs;&longs;e motu recto naturaliter accelerato; &longs;ed de <lb/>hoc motu &longs;yderum agemus fusè aliquando, cum de cau&longs;is corporum cœ­<lb/>le&longs;tium. </s> </p> <p id="N21C39" type="main"> <s id="N21C3B"><!-- NEW -->Obiicies de&longs;cen&longs;um funependuli, qui e&longs;t naturaliter acceleratus; &longs;ed <lb/>profectò ille motus e&longs;t tantùm per accidens circularis. </s> </p> <p id="N21C41" type="main"> <s id="N21C43"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N21C4F" type="main"> <s id="N21C51"><!-- NEW -->Ob&longs;eruabis ex dictis &longs;atis con&longs;tare, quàm temerè mirentur aliqui tan­<lb/>tam motuum cœle&longs;tium celeritatem, cum motus circularis velocitas in <lb/>infinitum augeri po&longs;&longs;it: Ob&longs;eruabis præterea, &longs;i fortè motus rectus corpo­<lb/>rum cœle&longs;tium præce&longs;&longs;it per aliquot minuta, motum illum, qui deinde <lb/>&longs;ucce&longs;&longs;it, non e&longs;&longs;e perfectè circularem, &longs;ed mixtum, quem aliquando ex­<lb/>plicabimus, & ex eo cau&longs;as Apogæi, Perigæi, declinationis, &c. </s> <s id="N21C5F">omné&longs;­<lb/>que anomalias deducemus &longs;uo loco. </s> </p> <p id="N21C64" type="main"> <s id="N21C66"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N21C72" type="main"> <s id="N21C74"><!-- NEW --><emph type="italics"/>Rota circulo verticali parallela circa axem mobilis addito minimo im­<lb/>petu per &longs;e moueri pote&longs;t<emph.end type="italics"/>; </s> <s id="N21C7F"><!-- NEW -->&longs;it enim ABCD plano verticali parallela circa <lb/>centrum E volubilis; </s> <s id="N21C85"><!-- NEW -->&longs;itque in perfecto æquilibrio, & accedat minima <lb/>vis impetus in A v.g. <!-- REMOVE S-->haud dubiè punctum E de&longs;cendet deor&longs;um, alio­<lb/>quin maneret æquilibrium, & non maneret: dixi per &longs;e; </s> <s id="N21C8F"><!-- NEW -->nam cùm non <lb/>po&longs;&longs;it volui circa centrum E, ni&longs;i vel cum mobili axe duobus hinc inde <lb/>lunatis fulcris &longs;u&longs;tentato, vel facto foramine circa axem immobilem, vel <lb/>circa geminos apices conicos immi&longs;&longs;os iu&longs;tis apothecis in plano rotæ <lb/>excauatis, quales videmus in acu magnetica; atqui non pote&longs;t volui rota <lb/>&longs;iue primo, &longs;iue &longs;ecundo, &longs;iue tertio modo voluatur &longs;ine multa compre&longs;­<lb/>&longs;ione partium, id e&longs;t, &longs;ine aliquo affrictu, in quo multæ particulæ vnius <lb/>plani cum particulis alterius qua&longs;i pectinatim commi&longs;&longs;æ, motum & im­<lb/>petunt &longs;i&longs;tunt. </s> </p> <p id="N21CA3" type="main"> <s id="N21CA5"><emph type="center"/><emph type="italics"/>Theorèma<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N21CB1" type="main"> <s id="N21CB3"><emph type="italics"/>Rota minor in eodem &longs;itu de quo &longs;uprà æquè facilè moueri pote&longs;t, ac maior<emph.end type="italics"/><pb pagenum="282" xlink:href="026/01/316.jpg"/><emph type="italics"/>per &longs;e.<emph.end type="italics"/></s> <s id="N21CC3"><!-- NEW --> Probatur primò, quia vtraque minimo impetu moueri pote&longs;t per <lb/>Th. 21. Secundò, quia addita minima vi impetus in F, & minima in A <lb/>tàm facilè maior rota de&longs;cendit, quàm minor, quia æqualiter tollitur <lb/>æquilibrium vtriu&longs;que: dixi per &longs;e, quia maior rota propter maius pon­<lb/>dus maiore affrictu motum impedit. </s> </p> <p id="N21CCF" type="main"> <s id="N21CD1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N21CDD" type="main"> <s id="N21CDF"><!-- NEW --><emph type="italics"/>Pote&longs;t vis aliqua applicata rotæ in A v.g. <!-- REMOVE S-->rotam mouere in eodem &longs;itu ver­<lb/>ticali; licèt nullum impetum producat.<emph.end type="italics"/></s> <s id="N21CEB"><!-- NEW --> Probatur, quia vis minima pote&longs;t <lb/>deprimere rotam ABCD. v.g. <!-- REMOVE S-->per Th.21. &longs;ed vis minima non pote&longs;t <lb/>producere impetum in qualibet rota, vt patet; </s> <s id="N21CF5"><!-- NEW -->nec enim producere po­<lb/>te&longs;t, ni&longs;i in tota rota producat per Th.33. lib. primo; &longs;ed vis minima im­<lb/>petus tot partes impetus, producere non pote&longs;t, quot e&longs;&longs;ent nece&longs;&longs;ariæ, vt <lb/>omnibus partibus rotæ di&longs;tribuerentur. </s> </p> <p id="N21CFF" type="main"> <s id="N21D01"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N21D0D" type="main"> <s id="N21D0F"><emph type="italics"/>Hinc egregium paradoxum; </s> <s id="N21D14"><!-- NEW -->pote&longs;t aliquid mouere rotam, & non agere in <lb/>rotam<emph.end type="italics"/>; </s> <s id="N21D1D"><!-- NEW -->quia vis mouens non pote&longs;t in rotam agere, ni&longs;i impetum in ea <lb/>producat, vt patet; </s> <s id="N21D23"><!-- NEW -->&longs;ed pote&longs;t illa vis rotam mouere licèt impetum in ea <lb/>non producat per Th.23. igitur mouere, & non agere: </s> <s id="N21D29"><!-- NEW -->quod quomodo <lb/>fiat facilè explicari pote&longs;t; quippè illa vis ponderis. </s> <s id="N21D2F"><!-- NEW -->v.g. <!-- REMOVE S-->quæ accedit pun­<lb/>cto A cum toto pondere &longs;emicirculi BA DE, grauitatione communi <lb/>præualet grauitationi alterius &longs;emicirculi rotæ BC DE; </s> <s id="N21D39"><!-- NEW -->quia &longs;cilicet <lb/>maior e&longs;t; &longs;ic pondus vnius &longs;crupuli &longs;uperpo&longs;itum ingenti rupi non pro­<lb/>ducit in rupe impetum, &longs;ed &longs;i fortè appendatur rupes, &longs;imul cum illa gra­<lb/>uitat, quod facilè concipi pote&longs;t. </s> </p> <p id="N21D43" type="main"> <s id="N21D45"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N21D51" type="main"> <s id="N21D53"><!-- NEW --><emph type="italics"/>Cum de&longs;cendit deor&longs;um &longs;emicirculus BA DE, attollitur &longs;ur&longs;um &longs;emicir­<lb/>culus oppo&longs;itus<emph.end type="italics"/>; </s> <s id="N21D5E"><!-- NEW -->quia &longs;cilicet impetus illius producit in i&longs;to alium impe­<lb/>tum; </s> <s id="N21D64"><!-- NEW -->nec enim corpus graue a&longs;cendit &longs;ur&longs;um &longs;ua &longs;ponte in medio leuio­<lb/>re; igitur ab extrin&longs;eco; </s> <s id="N21D6A"><!-- NEW -->&longs;ed nulla e&longs;t alia cau&longs;a applicata præter impe­<lb/>tum &longs;emicirculi de&longs;cendentis; </s> <s id="N21D70"><!-- NEW -->igitur ab eo producitur hic impetus, <lb/>i&longs;que omninò æqualis; quia &longs;cilicet vterque mouetur motu æquali. </s> </p> <p id="N21D76" type="main"> <s id="N21D78"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N21D84" type="main"> <s id="N21D86"><!-- NEW --><emph type="italics"/>Hinc impetus deor&longs;um producere pote&longs;t impetum &longs;ur&longs;um<emph.end type="italics"/>; quippe <lb/>ad aliam lineam determinare non pote&longs;t, quod valdè paradoxum e&longs;t. </s> </p> <p id="N21D91" type="main"> <s id="N21D93"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N21D9F" type="main"> <s id="N21DA1"><!-- NEW --><emph type="italics"/>Hinc impetus vnius partis mobilis continui pote&longs;t impetum &longs;imilem produ­<lb/>cere in alia parte eiu&longs;dem mobilis<emph.end type="italics"/>; vt patet ex dictis, quod tantùm locum <lb/>habet in motu circulari. </s> <s id="N21DAE">Diceret aliquis, igitur in motu recto etiam lo­<lb/>cum habebit. </s> <s id="N21DB3"><!-- NEW -->Re&longs;pondeo negando, alioqui minima potentia quodlibet <lb/>pondus motu recto moueret etiam nullo adhibito mechanico organo; </s> <s id="N21DB9"><!-- NEW --><lb/>quia modo produceretur tantulus impetus in aliqua parte, hic produce­<lb/>ret alium, & hic alium, immò vterque &longs;ecundo in&longs;tanti alium produce-<pb pagenum="283" xlink:href="026/01/317.jpg"/>ret: </s> <s id="N21DC5"><!-- NEW -->e&longs;&longs;et enim cau&longs;a nece&longs;&longs;aria; </s> <s id="N21DC9"><!-- NEW -->&longs;ed hoc e&longs;t ab&longs;urdum: ratio verò di&longs;pa­<lb/>ritatis e&longs;t, quia mobile, quod motu circulari voluitur circa centrum, <lb/>quod e&longs;t in ip&longs;o mobili duplicis mobilis vicem gerit, quorum vnum im­<lb/>pedit motum alterius, nec moueri po&longs;&longs;unt, ni&longs;i motibus oppo&longs;itis. </s> </p> <p id="N21DD4" type="main"> <s id="N21DD6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N21DE2" type="main"> <s id="N21DE4"><!-- NEW --><emph type="italics"/>Si applicetur pondus in<emph.end type="italics"/> K, <emph type="italics"/>minus erit illius<emph type="sup"/>a<emph.end type="sup"/> momentum, quàm in A, erit­<lb/>que ad momentum in A, vt LE ad AE<emph.end type="italics"/>; </s> <s id="N21DFB"><!-- NEW -->quod &longs;æpiùs iam &longs;uprà dictum <lb/>e&longs;t; </s> <s id="N21E01"><!-- NEW -->præ&longs;ertim lib.4. Inde tamen egregium deduco paradoxum, &longs;cilicet <lb/>minimam vim &longs;ufficere ad deprimendum &longs;emicirculum BA DE &longs;iue &longs;it <lb/>applicata in A &longs;iue in K; faciliùs tamen id præ&longs;tare in C, quàm in K, <lb/>id e&longs;t velociore motu. </s> </p> <p id="N21E0B" type="main"> <s id="N21E0D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N21E19" type="main"> <s id="N21E1B"><!-- NEW --><emph type="italics"/>Potentia in C applicata etiam minima per lineam CN, mouebit &longs;emicir­<lb/>culum DE BE &longs;ur&longs;um<emph.end type="italics"/>; vt patet; </s> <s id="N21E26"><!-- NEW -->nullum tamen producet impetum, &longs;i <lb/>minima &longs;it; </s> <s id="N21E2C"><!-- NEW -->ratio e&longs;t, quia eodem modo &longs;e habet, ac &longs;i detraheret partem <lb/>ponderis &longs;emicirculi DC BE, qua detracta non e&longs;t ampliùs æquili­<lb/>brium; </s> <s id="N21E34"><!-- NEW -->igitur oppo&longs;itus &longs;emicirculus BA DE præualere debet; </s> <s id="N21E38"><!-- NEW -->vnde <lb/>ideo a&longs;cendit ille, quia de&longs;cendit i&longs;te; </s> <s id="N21E3E"><!-- NEW -->qui ideo de&longs;cendit, quia vel de­<lb/>trahitur aliquid de momento alterius, vel impeditur; </s> <s id="N21E44"><!-- NEW -->atqui impedire <lb/>tantùm pote&longs;t, vel per productionem impetus, vel per applicationem po­<lb/>tentiæ per CN, quæ actione communi cum toto impetu &longs;emicirculi <lb/>BA DE iuuat eius de&longs;cen&longs;um; </s> <s id="N21E4E"><!-- NEW -->nam perinde &longs;e habet potentia, &longs;iue &longs;it, <lb/>applicata in A per lineam AO &longs;iue in C per CN: quod certè manife­<lb/>&longs;tum e&longs;t. </s> </p> <p id="N21E56" type="main"> <s id="N21E58"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N21E64" type="main"> <s id="N21E66"><!-- NEW --><emph type="italics"/>Hinc etiam habes duo paradoxa<emph.end type="italics"/>; </s> <s id="N21E6F"><!-- NEW -->primum e&longs;t, potentiam immediatè <lb/>concurrere ad motum &longs;emicirculi, cui non e&longs;t applicata, & mediatè tan­<lb/>tùm ad motum illius, cui applicata e&longs;t; nam potentia applicata in C per <lb/>CN concurrit immediatè ad motum A deor&longs;um, & &longs;imul cum A ad mo­<lb/>tum Cur&longs;um. </s> <s id="N21E7B">Secundum e&longs;t, &longs;olam negationem e&longs;&longs;e cau&longs;am motus, &longs;ci­<lb/>licet detractionem partis momenti, quod clarum e&longs;t. </s> </p> <p id="N21E80" type="main"> <s id="N21E82"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N21E8E" type="main"> <s id="N21E90"><emph type="italics"/>Hinc etiam alia deduco paradoxa.<emph.end type="italics"/></s> <s id="N21E97"> Primum e&longs;t, faciliùs &longs;u&longs;tineri maius <lb/>pondus, quàm minus. </s> <s id="N21E9C">Secundum plùs addi ponderis, quò plùs detrahi­<lb/>tur. </s> <s id="N21EA1"><!-- NEW -->Tertium plùs detrahi, quò plùs additur, v.g. <!-- REMOVE S-->&longs;i detrahatur aliqua por­<lb/>tio ex &longs;emicirculo BC DE, &longs;emicirculus rotæ oppo&longs;itus de&longs;cendet, ni&longs;i <lb/>&longs;it potentia in CA, qua &longs;u&longs;tineatur; </s> <s id="N21EAB"><!-- NEW -->& quò maior portio detrahetur po­<lb/>tentiæ, maius pondus incumbet; quò minor, minus. </s> <s id="N21EB1">Sed hæc clara <lb/>&longs;unt. </s> </p> <p id="N21EB6" type="main"> <s id="N21EB8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N21EC4" type="main"> <s id="N21EC6"><emph type="italics"/>Impetus productus in rota con&longs;eruatur aliquamdiu.<emph.end type="italics"/></s> <s id="N21ECD"><!-- NEW --> Duplex impetus con­<lb/>&longs;iderari pote&longs;t in rota; </s> <s id="N21ED3"><!-- NEW -->primus e&longs;t productus ad intra accedente, &longs;cilicet <lb/>minima vi ponderis alteri &longs;emicirculo, putâ puncto A, qua po&longs;ita tolla-<pb pagenum="284" xlink:href="026/01/318.jpg"/>tur æquilibrium, quo &longs;ublato &longs;ua &longs;ponte mouetur rota; </s> <s id="N21EDE"><!-- NEW -->hic autem impe­<lb/>tus primò durat in toto de&longs;cen&longs;u quadrantis AD; </s> <s id="N21EE4"><!-- NEW -->immò acceleratur tan­<lb/>tillùm motus, licèt longè minùs, quàm in funependulo propter re&longs;i&longs;ten­<lb/>tiam &longs;emicirculi oppo&longs;iti contranitentis; </s> <s id="N21EEC"><!-- NEW -->vbi verò A peruenit in D, <lb/>non acceleratur ampliùs motus, &longs;ed tantillùm a&longs;cendit ver&longs;us C &, dein­<lb/>de de&longs;cendit, tandemque quie&longs;cit in D paucis confectis vibrationibus; </s> <s id="N21EF4"><!-- NEW --><lb/>&longs;ed de hoc cur&longs;u, & recur&longs;u agemus fusè lib. &longs;equenti; </s> <s id="N21EF9"><!-- NEW -->alter impetus e&longs;t <lb/>productus ab extrin&longs;eco, applicata &longs;cilicet valida potentiá, qui rotam <lb/>agit velociore motu, vt patet, cùm præter impetum ad intra &longs;it etiam im­<lb/>petus productus ab extrin&longs;eca cau&longs;a; </s> <s id="N21F03"><!-- NEW -->igitur maior e&longs;t impetus; igitur <lb/>maior motus: </s> <s id="N21F09"><!-- NEW -->porrò hic impetus aliquandiu con&longs;eruatur, vt patet expe­<lb/>rientiâ; nec e&longs;t vlla cau&longs;a &longs;ufficiens applicata, à qua tam citò de­<lb/>&longs;truatur. </s> </p> <p id="N21F11" type="main"> <s id="N21F13"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N21F1F" type="main"> <s id="N21F21"><!-- NEW --><emph type="italics"/>Quando voluitur rota ab applicata valida potentia in A. v.g. <!-- REMOVE S-->per AO, <lb/>non modo producitur impetus in &longs;emicirculo BA DE, &longs;ed etiam in oppo&longs;ito<emph.end type="italics"/>; <lb/>cùm vtrique mediatè vel immediatè &longs;it applicata &longs;ufficienter, exemplo <lb/>vectis. </s> </p> <p id="N21F32" type="main"> <s id="N21F34"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N21F40" type="main"> <s id="N21F42"><emph type="italics"/>Non destruitur per &longs;e impetus productus in rota ab extrin&longs;eco.<emph.end type="italics"/></s> <s id="N21F49"><!-- NEW --> Probatur, <lb/>quia licèt &longs;ingulis in&longs;tantibus mutetur eius determinatio, vt con&longs;tat ex <lb/>dictis; </s> <s id="N21F51"><!-- NEW -->nam per &longs;e impetus in hoc motu e&longs;t determinatus ad lineam re­<lb/>ctam; </s> <s id="N21F57"><!-- NEW -->nullus tamen impetus e&longs;t fru&longs;trà: </s> <s id="N21F5B"><!-- NEW -->quippè illud &longs;patium acquiritur <lb/>in linea curua, quod in recta percurreretur &longs;i nullum e&longs;&longs;et impedimen­<lb/>tum; </s> <s id="N21F63"><!-- NEW -->quemadmodum enim in reflexione, quæ fit à plano immobili, nul­<lb/>lus de&longs;truitur impetus; </s> <s id="N21F69"><!-- NEW -->ita nullus hîc de&longs;truitur; tàm enim centrum il­<lb/>lud immobile ad &longs;e qua&longs;i mobile trahit, quàm planum immobile ad &longs;e re­<lb/>pellit. </s> </p> <p id="N21F71" type="main"> <s id="N21F73">Quæreret fortè aliquis, vtrum in &longs;emicirculo a&longs;cendente impetus de­<lb/>&longs;truatur ab impetu naturali grauitationis. </s> <s id="N21F78"><!-- NEW -->Re&longs;pondeo negando, quia <lb/>nunquam a&longs;cendit C, ni&longs;i de&longs;cendat A; </s> <s id="N21F7E"><!-- NEW -->nunquam verò de&longs;cendit A, ni&longs;i <lb/>&longs;it maior vis in A quam in C, quod certum e&longs;t; </s> <s id="N21F84"><!-- NEW -->igitur grauitatio C impe­<lb/>dit quidem, ne &longs;it tantus motus in A, nunquam tamen impedit totum <lb/>motum, cum maius e&longs;t momentum in A; </s> <s id="N21F8C"><!-- NEW -->quod &longs;i æquale &longs;it vtrinque mo­<lb/>mentum; certè totus motus vtrinque impeditur, & hæc e&longs;t vera ratio <lb/>æquilibrij, de quo aliàs. </s> </p> <p id="N21F94" type="main"> <s id="N21F96"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N21FA2" type="main"> <s id="N21FA4"><!-- NEW --><emph type="italics"/>Hinc &longs;i nullus &longs;it partium affrictus, e&longs;&longs;et motus ille perpetuus<emph.end type="italics"/>; quia nul­<lb/>lus de&longs;truitur impetus per Th. 34. igitur ille motus e&longs;&longs;et perpetuus. </s> </p> <p id="N21FAF" type="main"> <s id="N21FB1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N21FBD" type="main"> <s id="N21FBF"><!-- NEW --><emph type="italics"/>In maiore rota e&longs;t maior affrictus partium, & impetus citiùs destruitur.<emph.end type="italics"/><lb/>Secunda pars &longs;equitur ex prima; hæc autem ex maiore ponderis grauita­<lb/>tione, vel in axem, vel in &longs;ubjectum planum. </s> </p> <pb pagenum="285" xlink:href="026/01/319.jpg"/> <p id="N21FCF" type="main"> <s id="N21FD1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N21FDD" type="main"> <s id="N21FDF"><!-- NEW --><emph type="italics"/>Licèt impetus non destruatur in motu rotæ, & impediatur determinatio <lb/>prima, vt patet; </s> <s id="N21FE7"><!-- NEW -->attamen impedimentum non pote&longs;t minus excogitari<emph.end type="italics"/>; </s> <s id="N21FEE"><!-- NEW -->cùm <lb/>nulla po&longs;&longs;it duci linea recta declinans ab AO, per quam noua determi­<lb/>natio fieri po&longs;&longs;it; </s> <s id="N21FF6"><!-- NEW -->fit enim ratione anguli contingentiæ; </s> <s id="N21FFA"><!-- NEW -->igitur determi­<lb/>natio noua proximè accedit ad priorem; igitur e&longs;t minimum impedi­<lb/>mentum. </s> </p> <p id="N22002" type="main"> <s id="N22004"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N22010" type="main"> <s id="N22012"><!-- NEW --><emph type="italics"/>Hinc in maiori rota minus e&longs;t impedimentum<emph.end type="italics"/>; </s> <s id="N2201B"><!-- NEW -->quia &longs;cilicet minor e&longs;t <lb/>angulus contingentiæ; </s> <s id="N22021"><!-- NEW -->maius verò in minori rota: </s> <s id="N22025"><!-- NEW -->porrò minor rota à <lb/>maiore &longs;eparata citiùs &longs;uos gyros ab&longs;oluit; </s> <s id="N2202B"><!-- NEW -->quia &longs;unt minores, (&longs;uppono <lb/>æqualem impetum in extremo orbe rotæ vtriu&longs;que productum,) idque <lb/>pro rata; &longs;i enim minor &longs;it &longs;ubdupla maioris, maior vnum tantum gyrum <lb/>aget eo tempore, quo minor duos percurret. </s> </p> <p id="N22035" type="main"> <s id="N22037"><!-- NEW -->Ob&longs;erua primò, pondus applicatum in A non modò producere impe­<lb/>tum in toto radio AE; </s> <s id="N2203D"><!-- NEW -->&longs;ed etiam in toto radio oppo&longs;ito EC; </s> <s id="N22041"><!-- NEW -->ratio e&longs;t, <lb/>quia &longs;i impetus radij AE producit impetum in radio EC; </s> <s id="N22047"><!-- NEW -->certè pondus <lb/>additum radio AE cen&longs;etur pars eiu&longs;dem radij; </s> <s id="N2204D"><!-- NEW -->igitur impetus illius <lb/>ponderis immediatè producit impetum in radio EC; </s> <s id="N22053"><!-- NEW -->quia impedit hic <lb/>radius oppo&longs;itus motum alterius AE; </s> <s id="N22059"><!-- NEW -->igitur, vt tollat impedimentum, <lb/>producit AE impetum in EC; </s> <s id="N2205F"><!-- NEW -->&longs;i autem produceretur tantùm impetus in <lb/>EC ab impetu radij AE; </s> <s id="N22065"><!-- NEW -->igitur, vel aliquid impetus e&longs;&longs;et fru&longs;trà, vel <lb/>nunquam radius minor po&longs;&longs;et attollere maiorem, quacunque accedente <lb/>potentia; </s> <s id="N2206D"><!-- NEW -->&longs;it enim radius FE, in quo producatur quilibet impetus, &longs;it­<lb/>que radius oppo&longs;itus maior duplo EC; </s> <s id="N22073"><!-- NEW -->certè &longs;i impetus radij FE produ­<lb/>cit impetum in radio EC, vel producit æqualem, vel minorem, maiorem <lb/>enim producere non pote&longs;t; &longs;i minorem, vel æqualem; </s> <s id="N2207B"><!-- NEW -->igitur remi&longs;&longs;io­<lb/>rem, quia pluribus partibus &longs;ubjecti di&longs;tribuitur; </s> <s id="N22081"><!-- NEW -->igitur vel motus e&longs;&longs;et <lb/>remi&longs;&longs;ior radij EC quàm radij FE, quod dici non pote&longs;t; </s> <s id="N22087"><!-- NEW -->vel aliquid <lb/>impetus radij FE e&longs;&longs;et fru&longs;trà, quod etiam dici non pote&longs;t; itaque poten­<lb/>tia applicata in F, mediante &longs;cilicet organo, quodcumque tandem illud <lb/>&longs;it.v.g. </s> <s id="N22091"><!-- NEW -->pugno, producit impetum in ip&longs;o organo, impetus verò organi, <lb/>&longs;eu pugni producit impetum primò in toto radio FE, tùm in toto radio <lb/>EC, id e&longs;t totus impetus tùm pugni, tùm radij FC, &longs;cilicet innatus pro­<lb/>ducit impetum in alio radio EC; </s> <s id="N2209B"><!-- NEW -->nec enim producitur tantùm ab impe­<lb/>tu radij propter rationem &longs;uprà allatam, cùm &longs;it maior impetus in radio <lb/>EC quàm in radio FE; </s> <s id="N220A3"><!-- NEW -->nec tantùm ab impetu pugni, vel organi admo­<lb/>ti; </s> <s id="N220A9"><!-- NEW -->quia etiam&longs;i nullus accederet nouus impetus radio AE, &longs;ed tantùm <lb/>minimum pondus; </s> <s id="N220AF"><!-- NEW -->haud dubiè attolleret radium EC: </s> <s id="N220B3"><!-- NEW -->Adde quod ra­<lb/>dius EC impedit motum radij FE; </s> <s id="N220B9"><!-- NEW -->igitur ab impetu huius producitur <lb/>etiam in illo impetus; igitur tùm ab impetu pugni, vel organi, tùm ab <lb/>impetu radij FE producitur impetus in radio EC. <!-- KEEP S--></s> </p> <pb pagenum="286" xlink:href="026/01/320.jpg"/> <p id="N220C7" type="main"> <s id="N220C9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N220D5" type="main"> <s id="N220D7"><!-- NEW --><emph type="italics"/>Hæc inæqualis distributio impetus e&longs;t veri&longs;&longs;ima cau&longs;a girationis illius, quam <lb/>videmus in cylindro projecto per vibrationem &longs;iue brachium &longs;ur&longs;um &longs;iue deor­<lb/>&longs;um vibretur<emph.end type="italics"/>; </s> <s id="N220E4"><!-- NEW -->quod ab omnibus facilè ob&longs;eruari pote&longs;t &longs;it enim cylin­<lb/>drus ED libratus per arcum AD, &longs;tatimque demittatur; </s> <s id="N220EA"><!-- NEW -->vbi attigit <lb/>punctum D, e&longs;t quidem determinatus ad Tangentem DP, & punctum I <lb/>ad Tangentem IR; </s> <s id="N220F2"><!-- NEW -->quia tamen e&longs;t minor impetus in I, quàm in D, & <lb/>minor adhuc in E; </s> <s id="N220F8"><!-- NEW -->certè D debet moueri velociùs quàm I, & I quam E; </s> <s id="N220FC"><!-- NEW --><lb/>igitur motu recto moueri non pote&longs;t prædictus cylindrus ED; </s> <s id="N22101"><!-- NEW -->moueri <lb/>motu recto, id e&longs;t in &longs;itu parallelo ED; </s> <s id="N22107"><!-- NEW -->igitur extremitas D gyros aget, <lb/>quia retinetur ab aliis punctis, quorum tardior e&longs;t motus; </s> <s id="N2210D"><!-- NEW -->&longs;ed hîc erit <lb/>motus mixtus, de quo in lib.9.agemus, & totam rem i&longs;tam fusè explica­<lb/>bimus; </s> <s id="N22115"><!-- NEW -->hîc tantùm &longs;ufficiat dixi&longs;&longs;e cau&longs;am legitimam illius circuitionis <lb/>e&longs;&longs;e tantùm inæqualem illam di&longs;tributionem impetus in cylindro ED; <lb/>a&longs;&longs;ignauimus autem ibidem lineam, quam &longs;uo motu de&longs;cribit extremitas <lb/>D, & centrum, circa quod &longs;uos gyros agit. </s> </p> <p id="N2211F" type="main"> <s id="N22121"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N2212D" type="main"> <s id="N2212F"><!-- NEW --><emph type="italics"/>Diu durat motus impre&longs;&longs;us rotæ in circulo verticali, &longs;i vel modicus &longs;it par­<lb/>tium affrictus<emph.end type="italics"/>; </s> <s id="N2213A"><!-- NEW -->Probatur, quia cùm non de&longs;truatur impetus aliunde, quàm <lb/>ab affrictu, dicendum e&longs;t minimum etiam &longs;ingulis in&longs;tantibus de&longs;trui <lb/>impetum; </s> <s id="N22142"><!-- NEW -->igitur diu durat impetus; </s> <s id="N22146"><!-- NEW -->igitur diu durat motus: nec e&longs;t alia <lb/>ratio vulgaris illius experimenti, quo videmus perforatam acum circa <lb/>cylindrum leuigati&longs;&longs;imum diu rotari. </s> </p> <p id="N2214E" type="main"> <s id="N22150"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N2215C" type="main"> <s id="N2215E"><!-- NEW --><emph type="italics"/>Cum rota voluitur in circulo horizontali, non pote&longs;t moueri applicata mini­<lb/>ma potentia<emph.end type="italics"/>; </s> <s id="N22169"><!-- NEW -->Probatur, quia nullo modo rotatur ad intra, id e&longs;t non pro­<lb/>ducit in &longs;e impetum, vt patet; </s> <s id="N2216F"><!-- NEW -->igitur debet produci impetus in illa à po­<lb/>tentia applicata; igitur tot partes impetus, quot &longs;unt &longs;altem in tota rota, <lb/>cum &longs;ingulæ partes moueantur. </s> </p> <p id="N22177" type="main"> <s id="N22179"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N22185" type="main"> <s id="N22187"><!-- NEW --><emph type="italics"/>Hinc difficiliùs mouetur in circulo horizontali quàm in verticali<emph.end type="italics"/>; patet, <lb/>quia in hoc à minima potentia applicata pote&longs;t moueri per Th.21. &longs;ecus <lb/>verò in illo per Th.41. igitur in horizontali difficiliùs moueri pote&longs;t, <lb/>quàm in verticali. </s> <s id="N22196">Ob&longs;eruabis autem tribus modis volui po&longs;&longs;e huiu&longs;modi <lb/>rotam. </s> <s id="N2219B">Primò &longs;i in plano horizontali leuigati&longs;&longs;imo voluatur. </s> <s id="N2219E">Secundò, &longs;i <lb/>circa cylindrum immobilem, qui aperto foramini in&longs;eritur. </s> <s id="N221A3"><!-- NEW -->Tertiò, &longs;i <lb/>vno concauo vnius axis ducatur per centrum rotæ, in&longs;eratur vnus &longs;oli­<lb/>dus, quo fulcitus orbis con&longs;i&longs;tat in æquilibrio, difficiliùs voluitur primo <lb/>modo rota propter affrictum plurimarum partium; &longs;ecundo faciliùs, &longs;ed <lb/>longè faciliùs tertio &longs;ic autem voluitur acus magnetica. </s> </p> <p id="N221AF" type="main"> <s id="N221B1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N221BD" type="main"> <s id="N221BF"><!-- NEW --><emph type="italics"/>Potentia applicata talis e&longs;&longs;e debet, vt po&longs;&longs;it imprimere impetum toti rota<emph.end type="italics"/>; </s> <s id="N221C8"><!-- NEW --><pb pagenum="287" xlink:href="026/01/321.jpg"/>cum enim non po&longs;&longs;it moueri vna pars rotæ &longs;ine alia; </s> <s id="N221D0"><!-- NEW -->certè, vel impetus <lb/>imprimitur omnibus, vel nulli per Th.37. lib.1.præ&longs;ertim cùm totus im­<lb/>petus, qui rotæ imprimitur, &longs;it ab extrin&longs;eco; nec enim accidit huic rotæ, <lb/>quod alteri, quæ &longs;itum verticalem habet, cuius &longs;emicirculus, cui admoue­<lb/>tur potentia per lineam deor&longs;um motu naturali ex parte deor&longs;um fertur, <lb/>vt &longs;upra explicatum e&longs;t. </s> <s id="N221DE"><!-- NEW -->Hinc totus impetus in rota horizontali produ­<lb/>citur ab extrin&longs;eco; hinc ab ea tantùm potentia volui pote&longs;t, quæ tot <lb/>partes impetus pote&longs;t producere, quot &longs;unt nece&longs;&longs;ariæ, vt omnibus parti­<lb/>bus plani illius circularis di&longs;tribuantur, iuxta propagationem, quæ motui <lb/>circulari competit. </s> </p> <p id="N221EA" type="main"> <s id="N221EC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N221F8" type="main"> <s id="N221FA"><!-- NEW --><emph type="italics"/>Hinc in vtroque &longs;emicirculo plani producitur impetus ab ip&longs;a potentia ap­<lb/>plicata, non vero ab impetu producto in altero &longs;emicirculo producitur impetus <lb/>in alio,<emph.end type="italics"/> vt con&longs;tat ex dictis; </s> <s id="N22207"><!-- NEW -->&longs;it enim rota horizonti parallela ABCD, & <lb/>applicetur potentia in A per AO, non pote&longs;t produci impetus in radio <lb/>AE, ni&longs;i tollatur impedimentum; </s> <s id="N2220F"><!-- NEW -->impedit autem radius EC eo primo <lb/>in&longs;tanti; </s> <s id="N22215"><!-- NEW -->igitur debet &longs;imul tolli impedimentum, & produci impetus in <lb/>AE; </s> <s id="N2221B"><!-- NEW -->&longs;ed non pote&longs;t tolli impedimentum, ni&longs;i per impetum; </s> <s id="N2221F"><!-- NEW -->igitur non <lb/>modò producitur impetus in AE, &longs;ed etiam in EC; </s> <s id="N22225"><!-- NEW -->atqui impetus in <lb/>EC non producitur ab impetu producto in EA; </s> <s id="N2222B"><!-- NEW -->applicetur enim poten­<lb/>tia in F; </s> <s id="N22231"><!-- NEW -->certè minùs impetus producetur in FE, quàm in EC, vt con­<lb/>&longs;tat; </s> <s id="N22237"><!-- NEW -->igitur impetus in EC producitur ab ip&longs;a potentia applicata in A, <lb/>vel in F; &longs;i verò rota &longs;it verticalis, ab eadem potentia, & impetu innato <lb/>radij AE. vel &longs;emicirculi DA BE. <!-- KEEP S--></s> </p> <p id="N22240" type="main"> <s id="N22242"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N2224E" type="main"> <s id="N22250"><!-- NEW --><emph type="italics"/>Hinc faciliùs mouetur rota motu illo circulari, quàm recto<emph.end type="italics"/>; </s> <s id="N22259"><!-- NEW -->quia &longs;it dia­<lb/>meter AC, vt moueatur motu recto per &longs;e debet produci impetus eiu&longs;­<lb/>dem perfectionis in omnibus partibus AC, vt con&longs;tat ex dictis lib. 1. &longs;i <lb/>enim motus omnium partium e&longs;t æqualis; </s> <s id="N22263"><!-- NEW -->igitur & impetus, at verò, vt <lb/>moueatur motu circulari in plano horizontali facto &longs;cilicet circulo <lb/>ABCD, & admota potentia in A; </s> <s id="N2226D"><!-- NEW -->certè impetus qui producitur in A, <lb/>& in C, e&longs;t minor impetu producto in F, & in H; </s> <s id="N22273"><!-- NEW -->igitur &longs;i producatur <lb/>in A impetus eiu&longs;dem perfectionis ad motum circularem cum eo, qui <lb/>produceretur admotum rectum; </s> <s id="N2227B"><!-- NEW -->haud dubiè totus impetus productus in <lb/>AC ad motum rectum e&longs;t perfectior toto impetu producto ad circula­<lb/>rem; igitur difficiliùs ille, hic faciliùs producitur. </s> </p> <p id="N22283" type="main"> <s id="N22285"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N22291" type="main"> <s id="N22293"><!-- NEW --><emph type="italics"/>Si applicetur potentia in F, difficiliùs mouebit rotam, quàm &longs;i applicetur in <lb/>A<emph.end type="italics"/>; </s> <s id="N2229E"><!-- NEW -->ratio clara e&longs;t, quia producet in F impetum eiu&longs;dem perfectionis, <lb/>quem produceret in A, vt certum e&longs;t; </s> <s id="N222A4"><!-- NEW -->igitur maior erit impetus in to­<lb/>ta AC; </s> <s id="N222AA"><!-- NEW -->igitur difficiliùs mouebitur rota: adde quod longitudo vectis <lb/>iuuat motum EC. <!-- KEEP S--></s> </p> <pb pagenum="288" xlink:href="026/01/322.jpg"/> <p id="N222B5" type="main"> <s id="N222B7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s> </p> <p id="N222C3" type="main"> <s id="N222C5"><!-- NEW --><emph type="italics"/>Facilè cogno&longs;citur, in qua proportione potentia applicata puncte A faciliùs <lb/>vertat rotam, quàm applicata puncto F in circulo &longs;cilicet horizontali<emph.end type="italics"/>; </s> <s id="N222D0"><!-- NEW -->&longs;it enim <lb/>&longs;olus vectis FC, cuius centrum &longs;it E; </s> <s id="N222D6"><!-- NEW -->certè &longs;i vertatur in circulo hori­<lb/>zontali, potentia applicata extremitati C faciliùs ver&longs;abit, quàm appli­<lb/>cata puncto F, iuxta proportionem CE ad EF, vel ad HE; igitur po­<lb/>tentia applicata puncto H, vectis CF e&longs;t eiu&longs;dem momenti, cuius e&longs;t ea­<lb/>dem applicata puncto F, quia æqualem pror&longs;us effectum, &longs;cilicet impe­<lb/>tum, debet producere in vecte CF, vt moueatur in circulo horizontali <lb/>circa centrum E. <!-- KEEP S--></s> <s id="N222E7"><!-- NEW -->Probatur vlteriùs, quia motus, æquabiles &longs;cilicet, &longs;unt <lb/>vt &longs;patia, impetus vt motus, vires vt impetus; </s> <s id="N222ED"><!-- NEW -->igitur applicata potentiæ <lb/>in C producat impetum in vecte CF, vt vertatur in plano horizontali, & <lb/>C eo motu acquirat CS &longs;egmentum CE &longs;ectorem CES; </s> <s id="N222F5"><!-- NEW -->&longs;egmentum <lb/>verò FE &longs;ectorem FEV; </s> <s id="N222FB"><!-- NEW -->applicetur autem eadem potentia in F, vt ver­<lb/>tatur, idem vectis FC, & producatur in F impetus æqualis impetui an­<lb/>tè producto in C; </s> <s id="N22303"><!-- NEW -->haud dubiè punctum F percurret arcum FG eo tem­<lb/>pore, quo C priore motu percurrebat CS, vt patet; </s> <s id="N22309"><!-- NEW -->quia arcus CS e&longs;t <lb/>æqualis quadranti FG; igitur &longs;egmentum FE quadrantem FEG, & &longs;eg­<lb/>mentum EC quadrantem CED. </s> </p> <p id="N22311" type="main"> <s id="N22313"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s> </p> <p id="N2231F" type="main"> <s id="N22321"><!-- NEW --><emph type="italics"/>Ex his determinantur omnes aliæ proportiones<emph.end type="italics"/>; </s> <s id="N2232A"><!-- NEW -->&longs;i enim fit vectis AC <lb/>(quem &longs;uppono æqualem in omnibus &longs;uis partibus & volubilem circa <lb/>centrum E in plano horizontali) & applicetur potentia in puncto A, in <lb/>quo producat minimum impetum, quem pote&longs;t immediatè producere ex <lb/>hypothe&longs;i toties repetita, ita vt dato tempore percurrat A arcum AK, &longs;i <lb/>&longs;it vectis AH, & applicetur potentia in A, mouebit faciliùs, quàm AC <lb/>iuxta proportionem 8/5; </s> <s id="N2233A"><!-- NEW -->nam in vecte AC &longs;patium e&longs;t compo&longs;itum ex <lb/>gemino &longs;ectore AEK, CES, & in vecte AH &longs;patium e&longs;t compo&longs;itum <lb/>ex &longs;ectore AEK & ZEH, qui &longs;ubquadruplus e&longs;t AEK; </s> <s id="N22342"><!-- NEW -->igitur hoc &longs;pa­<lb/>tium totum confectum hoc vltimo motu e&longs;t ad prius &longs;patium vt 5. ad 8. <lb/>igitur & motus; </s> <s id="N2234A"><!-- NEW -->igitur & impetus; </s> <s id="N2234E"><!-- NEW -->&longs;ed quò minor e&longs;t impetus, e&longs;t maior <lb/>facilitas; igitur facilitas vltimi motus e&longs;t ad facilitatem primi, vt 8. ad 5. <lb/>idem dico, &longs;i applicetur potentia in H. <!-- KEEP S--></s> </p> <p id="N22357" type="main"> <s id="N22359"><!-- NEW -->Si verò retento &longs;emper eodem vecte AC applicetur potentia tùm in <lb/>A, tùm in F, facilitas motus potentiæ applicatæ in A e&longs;t ad facilitatem <lb/>motus potentiæ applicatæ in F, vt AE ad FE, vel vt AB ad AK, vel <lb/>vt AEB ad AEK, quæ omnia con&longs;tant ex dictis; igitur applicata in F <lb/>in vecte AC e&longs;t ad applicatam in F in vecte FE vt 5. ad 8. &longs;ed hæc &longs;unt <lb/>&longs;atis clara, nec vlteriore explicatione indigent. </s> </p> <p id="N22367" type="main"> <s id="N22369"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s> </p> <p id="N22375" type="main"> <s id="N22377"><!-- NEW --><emph type="italics"/>Hinc quò propiùs ad centrum applicatur potentia, eò maior e&longs;t difficultas <lb/>motus<emph.end type="italics"/>; </s> <s id="N22382"><!-- NEW -->igitur &longs;i applicetur ip&longs;i centro mathematicè con&longs;iderato e&longs;t infi­<lb/>nita difficultas; </s> <s id="N22388"><!-- NEW -->igitur nulla potentia &longs;uperare po&longs;&longs;et hanc difficultatem; </s> <s id="N2238C"><!-- NEW --><pb pagenum="289" xlink:href="026/01/323.jpg"/>hinc vt artifices &longs;uas ver&longs;ent rotas faciliùs, vel maximè curuum manu­<lb/>brium adhibent, vel affixo ver&longs;us circumferentiam in plano rotæ clauo <lb/>rotam agunt in orbes; quæ omnia clarè &longs;equuntur ex dictis. </s> </p> <p id="N22398" type="main"> <s id="N2239A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s> </p> <p id="N223A6" type="main"> <s id="N223A8"><!-- NEW --><emph type="italics"/>Minor rota faciliùs vertitur in circulo horizontali; </s> <s id="N223AE"><!-- NEW -->quàm maior.<emph.end type="italics"/> v. <!-- REMOVE S-->g.ro­<lb/>ta FGHI, quàm AB CD; </s> <s id="N223B9"><!-- NEW -->quia &longs;cilicet producitur minùs impetus in <lb/>minore, quàm in maiore, vt patet; </s> <s id="N223BF"><!-- NEW -->&longs;unt enim pauciores partes in mino­<lb/>re, plures in maiore; </s> <s id="N223C5"><!-- NEW -->mouetur autem faciliùs minor, quàm maior iuxta <lb/>rationem diametrorum, permutando; </s> <s id="N223CB"><!-- NEW -->Probatur, quia producatur impe­<lb/>tus in A maioris rotæ, ita vt dato tempore conficiat AK; </s> <s id="N223D1"><!-- NEW -->tùm æqualis <lb/>impetus in F minoris rotæ; </s> <s id="N223D7"><!-- NEW -->certè eodem tempore conficiet punctum F <lb/>arcum FG æqualem AK; </s> <s id="N223DD"><!-- NEW -->&longs;ed quadrans FEG e&longs;t ad &longs;ectorem AEK, vt <lb/>FE ad AE, vt con&longs;tat; </s> <s id="N223E3"><!-- NEW -->igitur facilitas motus minoris rotæ e&longs;t ad facili­<lb/>tatem motus maioris, vt FE ad AE; </s> <s id="N223E9"><!-- NEW -->igitur & impetus; &longs;ed quò minor <lb/>e&longs;t impetus, e&longs;t maior facilitas, &c. </s> </p> <p id="N223EF" type="main"> <s id="N223F1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s> </p> <p id="N223FD" type="main"> <s id="N223FF"><!-- NEW --><emph type="italics"/>Hinc tantæ molis po&longs;&longs;et e&longs;&longs;e rota in &longs;itu horizontali, vt à potentia etiam ve­<lb/>geta minimè verti po&longs;&longs;es,<emph.end type="italics"/> vt clarum e&longs;t; </s> <s id="N2240A"><!-- NEW -->neque hîc vllo modo con&longs;idero <lb/>re&longs;i&longs;tentiam, quæ petitur à compre&longs;&longs;ione, & affrictu partium, qui haud <lb/>dubiè maior e&longs;t in maiore rota; </s> <s id="N22412"><!-- NEW -->&longs;ed tantùm con&longs;idero re&longs;i&longs;tentiam ne­<lb/>gatiuam, hoc e&longs;t eam, quæ tantùm petitur à maiore numero partium ro­<lb/>tæ; </s> <s id="N2241A"><!-- NEW -->quò enim &longs;unt plures &longs;ubjecti partes, plures etiam partes impetus de­<lb/>&longs;iderantur, vt &longs;æpè dictum e&longs;t; igitur maior potentia. </s> </p> <p id="N22420" type="main"> <s id="N22422"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s> </p> <p id="N2242E" type="main"> <s id="N22430"><!-- NEW --><emph type="italics"/>Destruitur impetus productus in hac rotæ horizontali, &longs;ed &longs;en&longs;im &longs;ine &longs;en&longs;u <lb/>propter affrictum,<emph.end type="italics"/> vt &longs;uprà dictum e&longs;t: </s> <s id="N2243B"><!-- NEW -->hinc e&longs;&longs;et motus perpetuus, &longs;i nul­<lb/>lus e&longs;&longs;et affrictus; </s> <s id="N22441"><!-- NEW -->minùs impetus de&longs;truitur in maiore rota, quàm in mi­<lb/>nore: hinc gyrus minoris citiùs peragitur, & de&longs;init minor citiùs <lb/>moueri. </s> </p> <p id="N22449" type="main"> <s id="N2244B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s> </p> <p id="N22457" type="main"> <s id="N22459"><!-- NEW --><emph type="italics"/>Minor rota citiùs &longs;uum gyrum ab&longs;oluit, quàm maior,<emph.end type="italics"/> vt dictum e&longs;t &longs;uprà, <lb/>&longs;iue &longs;it in &longs;itu verticali, &longs;iue in &longs;itu horizontali; </s> <s id="N22464"><!-- NEW -->&longs;ed non e&longs;t determinata <lb/>proportio, quàm hîc de&longs;ideramus; dico enim tempora motuum e&longs;&longs;e, vt <lb/>radios. </s> <s id="N2246C"><!-- NEW -->v.g.tempus, quo rota minor FGHI &longs;uum gyrum ab&longs;oluit, e&longs;&longs;e ad <lb/>tempus, quo maior ABCD &longs;uum perficit, vt e&longs;t radius FE ad radium <lb/>AE, quod demon&longs;tro; </s> <s id="N22474"><!-- NEW -->quia &longs;it impetus æqualis impre&longs;&longs;us puncto A ma­<lb/>ioris rotæ puncto F minoris, ita vt A & F moueantur æquali motu; </s> <s id="N2247A"><!-- NEW -->mi­<lb/>nor rota conficit duos orbes eo tempore, quo maior vnum conficit, vt <lb/>con&longs;tat ex dictis; quia &longs;uppono. </s> <s id="N22482"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->circulum minoris e&longs;&longs;e &longs;ubduplum; <lb/></s> <s id="N22483"><!-- NEW -->igitur tempus, quo peragitur maior e&longs;t ad tempus, quo peragitur minor <lb/>in ratione dupla; </s> <s id="N22484"><!-- NEW -->igitur vt radius AE ad radium FE, quod erat demon­<lb/>&longs;trandum. </s> </p> <pb pagenum="290" xlink:href="026/01/324.jpg"/> <p id="N22494" type="main"> <s id="N22496"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s> </p> <p id="N224A2" type="main"> <s id="N224A4"><!-- NEW --><emph type="italics"/>Hinc &longs;i tantùm habeatur ratio vectis, maior difficiliùs ver&longs;atur in plano <lb/>horizontali, quàm minor.<emph.end type="italics"/> v.g. <!-- REMOVE S-->AE circa centrum E quam FE, producto <lb/>&longs;cilicet æquali motu in extremitate vtriu&longs;que A & F; </s> <s id="N224B3"><!-- NEW -->&longs;i enim A dato <lb/>tempore percurrit AK; </s> <s id="N224B9"><!-- NEW -->certè F percurret FG; </s> <s id="N224BD"><!-- NEW -->&longs;ed quadrans FEG e&longs;t <lb/>&longs;ubduplus &longs;ectoris AEK, vt con&longs;tat; </s> <s id="N224C3"><!-- NEW -->igitur faciliùs vertitur FE, quàm <lb/>AE in proportione AE, ad FE: </s> <s id="N224C9"><!-- NEW -->&longs;i tamen non con&longs;ideretur pondus &longs;eu <lb/>re&longs;i&longs;tentia vectis, haud dubiè &longs;i pondus &longs;it in Q, faciliùs mouebitur ope­<lb/>ra maioris vectis AE, quàm minoris FE; </s> <s id="N224D1"><!-- NEW -->quia opera maioris mouetur <lb/>motu vt QT; </s> <s id="N224D7"><!-- NEW -->operâ verò minoris motu vt QY, igitur difficiliùs opera <lb/>minoris in proportione QY ad QT; </s> <s id="N224DD"><!-- NEW -->denique &longs;i pondus &longs;it in F maioris <lb/>vectis, & in <foreign lang="greek">d</foreign> minoris, &longs;itque AE ad AF, vt FE ad F <foreign lang="greek">d</foreign>, æquale erit <lb/>momentum vtriu&longs;que vectis ad mouendum pondus; </s> <s id="N224ED"><!-- NEW -->quia arcus FV erit <lb/>æqualis arcui <foreign lang="greek">d</foreign> Y; </s> <s id="N224F7"><!-- NEW -->hîc autem nullomodo con&longs;ideratur vectis re&longs;i&longs;ten­<lb/>tia; </s> <s id="N224FD"><!-- NEW -->&longs;i verò producatur <expan abbr="tantũdem">tantundem</expan> impetus in toto vecte AE quamtum <lb/>in FE; </s> <s id="N22507"><!-- NEW -->certè pro rata &longs;ingulæ partes FE duplum habent; </s> <s id="N2250B"><!-- NEW -->igitur tempo­<lb/>ra gyrorum erunt in ratione duplicata radiorum; </s> <s id="N22511"><!-- NEW -->quia cum F habeat du­<lb/>plum impetum A, certè de&longs;cribit orbem integrum eo tempore, quo A <lb/>quadrantem; </s> <s id="N22519"><!-- NEW -->ergo F 4. orbes, dum A vnicum: &longs;ed hæc &longs;unt facilia. </s> </p> <p id="N2251D" type="main"> <s id="N2251F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s> </p> <p id="N2252B" type="main"> <s id="N2252D"><!-- NEW --><emph type="italics"/>Si vectis BH ita pellatur in B in plano horizontali, in quo liberè moueri <lb/>po&longs;&longs;it<emph.end type="italics"/> <emph type="italics"/>v.g. <!-- REMOVE S-->dum aquæ &longs;upernatat, nulli centro immobili affixus, &longs;it que aqualis <lb/>den&longs;itatis in omnibus &longs;uis partibus; mouebitur circa aliquod centrum, etiam&longs;i <lb/>nulli centro affigatur.<emph.end type="italics"/></s> <s id="N22543"><!-- NEW --> Probatur, quia punctum B velociùs mouebitur, quàm <lb/>A vel H, vt patet experientiâ: </s> <s id="N22549"><!-- NEW -->ratio e&longs;t, quia minùs impetus producitur <lb/>in toto cylindro BH, applicata potentia in B, quàm in A, quod e&longs;t cen­<lb/>trum grauitatis cylindri BA, vt iam o&longs;tendimus Th. 68. 69. BB; </s> <s id="N22551"><!-- NEW -->porrò <lb/>ratio à priori e&longs;t, quia cùm impetus producatur tantùm ad extra, vt tol­<lb/>latur impedimentum motus, vt fusè o&longs;tendimus lib. 1. certè in tantùm <lb/>amouetur impedimentum, in quantum amouetur corpus impediens mo­<lb/>tum alterius; </s> <s id="N2255D"><!-- NEW -->atqui amoueri tantùm pote&longs;t per motum; </s> <s id="N22561"><!-- NEW -->igitur eo motu <lb/>amouetur, quo faciliùs amoueri pote&longs;t, & minore &longs;umptu, vt ita dicam, <lb/>id e&longs;t minore impetu: </s> <s id="N22569"><!-- NEW -->porrò cum potentia &longs;it determinata ad producen­<lb/>dum tabem impetum, immediatè &longs;cilicet, id e&longs;t, in ea parte, cui immedia­<lb/>tè admouetur; </s> <s id="N22571"><!-- NEW -->alioqui &longs;i po&longs;&longs;et minorem, & minorem in infinitum pro­<lb/>ducere po&longs;&longs;et etiam immediatè &longs;ine operâ organi mechanici quodlibet <lb/>pondus mouere, quod e&longs;t ab&longs;urdum, de quo iam &longs;uprà; </s> <s id="N22579"><!-- NEW -->&longs;it igitur potentia <lb/>applicata in A, &longs;cilicet in centro grauitatis cylindri BH; </s> <s id="N2257F"><!-- NEW -->certè producit <lb/>maximum impetum, quem pote&longs;t producere in cylindro BH (&longs;uppono <lb/>enim e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam, & producere perfecti&longs;&longs;imum impetum, <lb/>quem producere po&longs;&longs;it) producit inquam maximum ratione numeri; </s> <s id="N22589"><!-- NEW --><lb/>cùm in toto cylindro BH producat impetum eiu&longs;dem perfectionis; </s> <s id="N2258E"><!-- NEW -->igi­<lb/>tur mouetur motu recto; </s> <s id="N22594"><!-- NEW -->igitur æquali in omnibus partibus; </s> <s id="N22598"><!-- NEW -->igitur æqua­<lb/>lis e&longs;t impetus in omnibus partibus, id e&longs;t, æquè inten&longs;us; </s> <s id="N2259E"><!-- NEW -->&longs;it autem po-<pb pagenum="291" xlink:href="026/01/325.jpg"/>tentia applicata in B, ita vt in puncto B producatur impetus eiu&longs;dem <lb/>perfectionis, de quo &longs;uprà: </s> <s id="N225A9"><!-- NEW -->&longs;i mouetur motu circulari circa aliquod cen­<lb/>trum v. <!-- REMOVE S-->g. <!-- REMOVE S-->circa centrum H, & punctum B conficiat arcum BD æqua­<lb/>lem rectæ b I, vel BL quam æquali tempore B vel A antè percurrebant <lb/>motu recto; </s> <s id="N225B7"><!-- NEW -->certè totus cylindrus BH acquiret tantùm &longs;patium BHD <lb/>motu circulari circa centrum H; </s> <s id="N225BD"><!-- NEW -->&longs;ed motu recto acqui&longs;iuit &longs;patium re­<lb/>ctanguli BK, quod maius e&longs;t, vt patet; </s> <s id="N225C3"><!-- NEW -->igitur motus circularis circa H <lb/>cylindri BH e&longs;t ad rectum, vt &longs;ector BHD ad rectangulum BK; </s> <s id="N225C9"><!-- NEW -->igitur <lb/>facilitas motus circularis e&longs;t ad facilitatem motus recti præ&longs;entis, vt re­<lb/>ctangulum BK ad &longs;ectorem BHD; </s> <s id="N225D1"><!-- NEW -->quænam verò &longs;it hæc proportio pa­<lb/>tet ex Cyclometria, &longs;uppo&longs;itâ ratione Archimedis periphæriæ ad diame­<lb/>trum; </s> <s id="N225D9"><!-- NEW -->igitur cum cylindrus impul&longs;us in B faciliùs moueri po&longs;&longs;it motu <lb/>circulari, quàm recto, vt con&longs;tat ex dictis; </s> <s id="N225DF"><!-- NEW -->& cùm eo motu moueatur, <lb/>quo faciliùs moueri pote&longs;t; </s> <s id="N225E5"><!-- NEW -->modò po&longs;&longs;it ad illum determinari, non mirum <lb/>e&longs;t &longs;i eo moueatur, & minor impetus producatur in eodem cylindro <lb/>BH; debet autem e&longs;&longs;e aliquod centrum huius motus, quod determina­<lb/>bimus paulò pò&longs;t, po&longs;tquam breuiter exilem quamdam objectionem de <lb/>impetu refutauerimus. </s> </p> <p id="N225F1" type="main"> <s id="N225F3"><!-- NEW -->Itaque obiiciunt aliqui, impetum non produci ad extra ab impetu; </s> <s id="N225F7"><!-- NEW --><lb/>quia &longs;cilicet impetus habet iam effectum &longs;cilicet motum; </s> <s id="N225FC"><!-- NEW -->igitur aliud <lb/>munus non e&longs;t illi imponendum; </s> <s id="N22602"><!-- NEW -->igitur non producit alium effectum; <lb/>igitur non e&longs;t cau&longs;a impetus. </s> </p> <p id="N22608" type="main"> <s id="N2260A"><!-- NEW -->Re&longs;pondeo primò, calor e&longs;t cau&longs;a rarefactionis; </s> <s id="N2260E"><!-- NEW -->igitur non producit <lb/>alium calorem, quia habet iam vnum effectum; &longs;i tuum argumentum <lb/>concludit, meum quoque concludet. </s> <s id="N22616"><!-- NEW -->Re&longs;pondeo &longs;ecundò, anima produ­<lb/>cit vi&longs;ionem, ergo auditionem producere non pote&longs;t, cùm iam habeat <lb/>vnum effectum: </s> <s id="N2261E"><!-- NEW -->Dices, eandem cau&longs;am po&longs;&longs;e habere plures effectus; cur <lb/>igitur negas de impetu? </s> </p> <p id="N22624" type="main"> <s id="N22626"><!-- NEW -->Re&longs;pondeo tertiò directè, motum e&longs;&longs;e effectum impetus ad intra, quem <lb/>præ&longs;tat in &longs;uo &longs;ubjecto; </s> <s id="N2262C"><!-- NEW -->igitur e&longs;t effectus formalis &longs;ecundarius; </s> <s id="N22630"><!-- NEW -->nec <lb/>alius e&longs;&longs;e pote&longs;t, vt lib.1. demon&longs;trauimus; </s> <s id="N22636"><!-- NEW -->at verò impetus e&longs;t effectus <lb/>alterius impetus ad extra; </s> <s id="N2263C"><!-- NEW -->igitur impetus e&longs;t cau&longs;a efficiens impetus, id­<lb/>que ad extra & cau&longs;a formalis, vel exigitiua motus ad intra; </s> <s id="N22642"><!-- NEW -->&longs;icut calor <lb/>e&longs;t cau&longs;a formalis, vel exigitiua rarefactionis ad intra, cau&longs;a verò effi­<lb/>ciens alterius caloris ad extra; </s> <s id="N2264A"><!-- NEW -->& verò nullo argumento probabis calo­<lb/>rem à calore produci, quo ego non probem impetum ab impetu produ­<lb/>ci; </s> <s id="N22652"><!-- NEW -->igitur impetus e&longs;t cau&longs;a alterius impetus; </s> <s id="N22656"><!-- NEW -->quia phy&longs;icè loquendo il­<lb/>lud vocamus cau&longs;am, ex cuius applicatione &longs;equitur nece&longs;&longs;ariò effectus; </s> <s id="N2265C"><!-- NEW --><lb/>atqui applicato corpore &longs;olo &longs;ine impetu nullus impetus producitur ad <lb/>extra, vt patet; </s> <s id="N22663"><!-- NEW -->applicato verò cum impetu, producitur &longs;tatim alius im­<lb/>petus; </s> <s id="N22669"><!-- NEW -->igitur ip&longs;e impetus e&longs;t cau&longs;a: </s> <s id="N2266D"><!-- NEW -->nec dicas requiri, vt conditionem; </s> <s id="N22671"><!-- NEW --><lb/>quia primò, nullum e&longs;&longs;et munus huius conditionis; nec enim applica­<lb/>ret cau&longs;am &longs;ubjecto, nec remoueret vllum impedimentum. </s> <s id="N22678">Secundò di­<lb/>cam &longs;imiliter calorem e&longs;&longs;e conditionem. </s> <s id="N2267D">Tertiò, dicerem etiam e&longs;&longs;e con­<lb/>ditionem ad motum. </s> <s id="N22682"><!-- NEW -->Quartò, quis dicat corpus graue producere impe-<pb pagenum="292" xlink:href="026/01/326.jpg"/>tum &longs;ur&longs;um immediatè per &longs;e; &longs;ed hæc omittamus, quæ leuia &longs;unt, præ­<lb/>&longs;ertim cùm demon&longs;trauerimus luculenter lib.1.impetum produci ab im­<lb/>petu, vt &longs;cilicet tollatur impedimentum. </s> </p> <p id="N2268F" type="main"> <s id="N22691"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s> </p> <p id="N2269D" type="main"> <s id="N2269F"><emph type="italics"/>Quando pellitur cylindrus innatans<emph.end type="italics"/> <emph type="italics"/>in puncto L non vertitur circa cen­<lb/>trum A.<emph.end type="italics"/><!-- KEEP S--></s> <s id="N226AF"><!-- NEW --> Probatur, quia vertatur circa centrum A. v.g. <!-- REMOVE S-->& percurrat B <lb/>arcum BC, & totus cylindrus duos &longs;ectores BAC, GAH; </s> <s id="N226B7"><!-- NEW -->&longs;it autem <lb/>BC &longs;ubduplus quadrantis BE, & duo &longs;ectores prædicti æquales qua­<lb/>dranti BAE; </s> <s id="N226BF"><!-- NEW -->hoc po&longs;ito, &longs;patium totius cylindri erit, vt quadrans; </s> <s id="N226C3"><!-- NEW -->igi­<lb/>tur motus; igitur impetus: </s> <s id="N226C9"><!-- NEW -->iam verò vertatur circa centrum H, ita vt B <lb/>percurrat arcum BD æqualem BC (erit autem BD &longs;ubquadruplus qua­<lb/>drantis BF;) igitur totus cylindrus circa centrum H percurret &longs;patium <lb/>&longs;ectoris BHD æqualis quadranti BAE; </s> <s id="N226D3"><!-- NEW -->igitur motus circa centrum H <lb/>e&longs;t æqualis motui circa centrum A; </s> <s id="N226D9"><!-- NEW -->igitur e&longs;t eadem difficultas motus; <lb/>igitur non vertitur potiùs circa centrum A, quàm circa centrum H. <!-- KEEP S--></s> </p> <p id="N226E0" type="main"> <s id="N226E2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s> </p> <p id="N226EE" type="main"> <s id="N226F0"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari centrum, circa quod vertitur cylindrus BH innatans <lb/>humido, modo &longs;upponatur æqualis den&longs;itatis, & cra&longs;&longs;itudinis<emph.end type="italics"/>; </s> <s id="N226FB"><!-- NEW -->diuidatur enim <lb/>AH bifariam in M: </s> <s id="N22701"><!-- NEW -->Dico vertiginem futuram circa centrum M, quod <lb/>demon&longs;tro; </s> <s id="N22707"><!-- NEW -->quia vertatur circa M, & extremitas B moueatur æquali <lb/>motu, quo priùs moueri &longs;upponebatur circa A, vel circa H; </s> <s id="N2270D"><!-- NEW -->certè cùm <lb/>arcus BR &longs;it ad arcum BE vt BM ad BA, id e&longs;t vt 3. ad 2. erit BN <lb/>&longs;ubtripla BR, cùm &longs;it æqualis BC &longs;ubdupla BE; </s> <s id="N22715"><!-- NEW -->totum autem &longs;patium <lb/>confectum hoc motu erit conflatum ex &longs;ectoribus BMN, & HMO, vt <lb/>patet: </s> <s id="N2271D"><!-- NEW -->porrò &longs;ector BMN e&longs;t &longs;ubtriplus quadrantis BMR, qui quadrans <lb/>e&longs;t ad priorem BAE, vt 9. ad 4. id e&longs;t, vt quadratum 3. ad quadratum 2. <lb/>vt con&longs;tat; </s> <s id="N22725"><!-- NEW -->igitur conflatum ex &longs;ectore BMN, & &longs;ectore HMO e&longs;t ad <lb/>quadrantem BAE, vel conflatum ex geminis &longs;ectoribus BAC, HAG <lb/>vt 3 1/3 ad 4. &longs;i autem accipiatur centrum, vel inter MA, vel MH, maius <lb/>erit &longs;patium, vt con&longs;tat ex Geometria; </s> <s id="N2272F"><!-- NEW -->igitur circa centrum M e&longs;t mini­<lb/>mum &longs;patium; </s> <s id="N22735"><!-- NEW -->igitur minimus motus; </s> <s id="N22739"><!-- NEW -->igitur minimus impetus; igitur <lb/>maxima facilitas; igitur &longs;i pellatur in B, vertetur circa M, quod hactenus <lb/>non explicatum modò ab aliquo, quod &longs;ciam, verùm etiam ne propo&longs;itum <lb/>quidem fuit. </s> </p> <p id="N22743" type="main"> <s id="N22745"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s> </p> <p id="N22751" type="main"> <s id="N22753"><!-- NEW --><emph type="italics"/>Hinc facilè dictu e&longs;t, cur naues ita impul&longs;æ ab altera extremitate circa al­<lb/>teram extremitatem non vertantur,<emph.end type="italics"/> vt patet experientiâ; </s> <s id="N2275E"><!-- NEW -->quia hæc tendit <lb/>in partem oppo&longs;itam; </s> <s id="N22764"><!-- NEW -->nec etiam circa centrum grauitatis nauis, quod <lb/>etiam manife&longs;tis experientiis confirmatur, cùm &longs;cilicet impul&longs;a extremi­<lb/>tas maiorem arcum de&longs;cribat, &longs;ed circa medium centrum inter vtrum­<lb/>que, ex quo principio tota remigationis ratio pendet: </s> <s id="N2276E"><!-- NEW -->immò & guber­<lb/>naculi, quod puppi affigitur, vt con&longs;ideranti patebit, quod &longs;ufficiat indi­<lb/>ca&longs;&longs;e; </s> <s id="N22776"><!-- NEW -->&longs;i verò pellatur idem cylindrus in T. v.g. <!-- REMOVE S-->mouebitur circa cen-<pb pagenum="293" xlink:href="026/01/327.jpg"/>trum, quod e&longs;t inter MH, licèt propiùs accedat ad M, quàm ad H, vt <lb/>con&longs;tat ex calculatione; </s> <s id="N22783"><!-- NEW -->e&longs;t autem aliquod punctum inter TA, ex quo &longs;i <lb/>pellatur, mouebitur circa punctum H; </s> <s id="N22789"><!-- NEW -->&longs;i verò a&longs;&longs;umantur alia puncta <lb/>ver&longs;us A, ex quibus pellatur, centra motus, erunt extra BH, ac proinde <lb/>extremitas B pul&longs;a ex B mouetur per arcum BN; </s> <s id="N22791"><!-- NEW -->pul&longs;a ex A per rectam <lb/>AL; pul&longs;a denique ex punctis, quæ &longs;unt inter BA, per arcus maiorum <lb/>circulorum, eò &longs;anè maiorum, quò propiùs punctum, ex quo pellitur, ac­<lb/>cedit ad A. <!-- KEEP S--></s> </p> <p id="N2279C" type="main"> <s id="N2279E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s> </p> <p id="N227AA" type="main"> <s id="N227AC"><!-- NEW --><emph type="italics"/>Si pellatur nauis, vel cylindrus BH in puncto T, difficiliùs mouebitur, etiam <lb/>ex &longs;uppo&longs;itione, quòd circa centrum M moueatur<emph.end type="italics"/>; </s> <s id="N227B7"><!-- NEW -->quod eodem modo de­<lb/>mon&longs;tratur, quo &longs;uprà; </s> <s id="N227BD"><!-- NEW -->accipiatur TZ æqualis BC; </s> <s id="N227C1"><!-- NEW -->&longs;it autem BT æqua­<lb/>lis TA; </s> <s id="N227C7"><!-- NEW -->certè arcus TS erit æqualis arcui BE; </s> <s id="N227CB"><!-- NEW -->igitur &longs;ector VMB erit <lb/>&longs;ubduplus quadrantis BMR: </s> <s id="N227D1"><!-- NEW -->&longs;imiliter &longs;ector HMX erit &longs;ubduplus qua­<lb/>drantis HMP; </s> <s id="N227D7"><!-- NEW -->igitur motus erit, vt aggregatum ex his duobus &longs;ectori­<lb/>bus; </s> <s id="N227DD"><!-- NEW -->&longs;ed cum applicatur potentia in B, motus e&longs;t vt aggregatum ex duo­<lb/>bus &longs;ectoribus BMN, HNO; </s> <s id="N227E3"><!-- NEW -->&longs;it autem quadrans BMR, vt 9. & qua­<lb/>drans HMP vt 1. igitur cum applicatur potentia in B, motus e&longs;t ad mo­<lb/>tum cum applicatur in T vt 3 1/3 ad 5. igitur & impetus; igitur facilitas <lb/>primi motus e&longs;t ad facilitatem &longs;ecundi, vt 5. ad 3 1/3 igitur in T diffici­<lb/>liùs pellitur, quàm in B. </s> </p> <p id="N227EF" type="main"> <s id="N227F1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s> </p> <p id="N227FD" type="main"> <s id="N227FF"><!-- NEW --><emph type="italics"/>Hinc maxima difficultas e&longs;t ad minimam, vt rectangulum BK ad aggre­<lb/>tum ex duobus &longs;ectoribus BMN & HMO, id e&longs;t vt<emph.end type="italics"/> 6. 2/7 ad 2. (13/21): </s> <s id="N2280A"><!-- NEW -->hinc <lb/>nauis, quæ pellitur è lateris puncto, quod re&longs;pondet centro A, difficiliùs <lb/>longè mouetur; &longs;uppono enim nauim e&longs;&longs;e eiu&longs;dem latitudinis, & den&longs;i­<lb/>tatis, nec &longs;abulo adhærere. </s> </p> <p id="N22814" type="main"> <s id="N22816"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s> </p> <p id="N22822" type="main"> <s id="N22824"><!-- NEW --><emph type="italics"/>Si &longs;uperponatur corpus plano rotæ, quæ voluitur in circulo horizontali, pro­<lb/>iicietur per Tangentem extremam.<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;it rota ABCD horizontali pa­<lb/>rallela quæ vertatur ab A ver&longs;us B celeri motu, &longs;itque planum eius le­<lb/>uigati&longs;&longs;imum; </s> <s id="N22835"><!-- NEW -->imponatur globus etiam leuigati&longs;&longs;imus puncto A: </s> <s id="N22839"><!-- NEW -->dico <lb/>quod proiicietur per Tangentem AF, quia impetus, qui in illo impri­<lb/>mitur in puncto F e&longs;t determinatus ad Tangentem A <foreign lang="greek">q</foreign>; </s> <s id="N22845"><!-- NEW -->&longs;ed non impe­<lb/>ditur, quominus habeat &longs;uum motum; </s> <s id="N2284B"><!-- NEW -->nec enim globus prædictus ita <lb/>affigitur plano rotæ, quin liberè &longs;eor&longs;im moueri po&longs;&longs;it: </s> <s id="N22851"><!-- NEW -->dixi per Tangen­<lb/>tem extremam, quia &longs;i imponatur globus puncto F; </s> <s id="N22857"><!-- NEW -->certè non impelle­<lb/>tur per Tangentem F <foreign lang="greek">u</foreign>, vt patebit ex &longs;equenti propo&longs;itione; quod à nul­<lb/>lo hactenus, quod &longs;ciam, ob&longs;eruatum fuit. </s> </p> <p id="N22863" type="main"> <s id="N22865"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s> </p> <p id="N22871" type="main"> <s id="N22873"><emph type="italics"/>Si imponatur globus puncto F plani horizontalis rotæ ABCD, non proii­<lb/>cietur per Tangentem F<emph.end type="italics"/> <foreign lang="greek">u</foreign> quod primò manife&longs;tis experimentis comproba­<lb/>tum e&longs;t. </s> <s id="N22883"><!-- NEW -->Secundò probatur, quia dum globus his punctis, in quibus re-<pb pagenum="294" xlink:href="026/01/328.jpg"/>cta F <foreign lang="greek">u</foreign> &longs;ecat alios maiores circulos concentricos, ab his punctis nouum <lb/>impetum accipit, ratione cuius debet mutare lineam, quod certum e&longs;t; </s> <s id="N22892"><!-- NEW --><lb/>cum autem circuli maiores rotæ moueantur velociùs, quàm FGH, po­<lb/>tiori iure mutari debet determinatio currentis globi in prædicto plano; </s> <s id="N22899"><!-- NEW --><lb/>quænam verò &longs;it hæc linea motus, difficilè dictu e&longs;t; dicemus tamen <lb/>Tomo &longs;equenti, cum de lineis motus. </s> </p> <p id="N228A0" type="main"> <s id="N228A2"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N228AE" type="main"> <s id="N228B0"><!-- NEW -->Ob&longs;eruabis primò, &longs;i &longs;it rota ABCD verticali circulo parallela, proii­<lb/>ci corpus ab eius periphæria per lineam minùs di&longs;tantem ab ip&longs;a peri­<lb/>phæria, quò maior e&longs;t circulus; </s> <s id="N228B8"><!-- NEW -->quia &longs;cilicet tunc angulus contingentiæ <lb/>e&longs;t maior; </s> <s id="N228BE"><!-- NEW -->hinc &longs;i terra moueretur (licèt reuerâ, quie&longs;cat) non e&longs;&longs;et pe­<lb/>riculum, ne proiicerentur lapides per Tangentem, quæ vix di&longs;taret per <lb/>longum &longs;patij tractum ab ip&longs;o arcu terræ, vt ob&longs;eruat Galileus, & res <lb/>ip&longs;a facilis e&longs;t; vnde miror nonnullos Philo&longs;ophos, alioquin docti&longs;&longs;i­<lb/>mos, id argumenti contra motum terræ áttuli&longs;&longs;e, cuius nulla penitus <lb/>vis e&longs;t, vt nonnemo in elementis Geometricis etiam mediocriter tinctus <lb/>facilè demon&longs;trabit. </s> </p> <p id="N228CE" type="main"> <s id="N228D0"><!-- NEW -->Ob&longs;erua &longs;ecundò, ex his peti rationes projectionis fundæ, quæ in quo­<lb/>cunque circulo &longs;uos gyros habet; e&longs;t enim eadem ratio. </s> </p> <p id="N228D6" type="main"> <s id="N228D8"><!-- NEW -->Ob&longs;erua tertiò, cum aliquod corpus incubat plano, quod motu recto <lb/>mouetur, numquam ab eo &longs;eparari, quamdiu planum ip&longs;um æquabili mo­<lb/>tu mouetur; </s> <s id="N228E0"><!-- NEW -->quià non mutatur determinatio impetus împre&longs;&longs;i corpori <lb/>incubanti; & cùm æqualis &longs;it impetus tùm in plano, tùm in globo. </s> <s id="N228E6"><!-- NEW -->v.g. <lb/><!-- REMOVE S-->&longs;uperimpo&longs;ito, vtrumque æquali motu nece&longs;&longs;ario mouetur; </s> <s id="N228ED"><!-- NEW -->igitur &longs;ine <lb/>projectione; </s> <s id="N228F3"><!-- NEW -->&longs;ic dum nauis recto cur&longs;u mouetur &longs;ecundo flumine, omnia <lb/>quæ naui in&longs;unt, æqualiter cum ip&longs;a naui mouentur; at verò &longs;i planum <lb/>mouetur motu circulari, mutatur determinatio &longs;ingulis in&longs;tantibus, vnde <lb/>&longs;equitur projectio, vt dictum e&longs;t &longs;uprà. </s> </p> <p id="N228FD" type="main"> <s id="N228FF">Ob&longs;erua quartò, globum impo&longs;itum rotæ ABCD initio tardiùs, tùm <lb/>deinde velociùs moueri, quò &longs;cilicet plùs recedit à centro E, quia à pun­<lb/>ctis plani, in quibus rotatur, & quæ maiore motu vertuntur, maiorem <lb/>quoque impetus vim accipit. </s> </p> <p id="N22908" type="main"> <s id="N2290A"><!-- NEW -->Ob&longs;erua quintò, globum in plano ABCD per lineam FVB rotatum <lb/>moueri velociùs ip&longs;is punctis plani, in quibus rotatur, excepto primo <lb/>in&longs;tanti motus; </s> <s id="N22912"><!-- NEW -->quia accipit à &longs;ingulis punctis æqualem impetum ip&longs;i <lb/>impetui, qui ip&longs;is ine&longs;t; qui cum priori conjunctus diagonalem facit, vt <lb/>&longs;uprà dictum e&longs;t, cum de motu mixto & lib. 1. cum de determinatione <lb/>motus. </s> </p> <p id="N2291C" type="main"> <s id="N2291E"><!-- NEW -->Ob&longs;eruabis &longs;extò, moueri motu accelerato maiori & maiori, quod <lb/>certè mirum e&longs;t; </s> <s id="N22924"><!-- NEW -->cum tamen rota in cuius plano horizontali rotatur, <lb/>motu æquali moueatur; </s> <s id="N2292A"><!-- NEW -->maximè autem cre&longs;cit ille motus, quia priorem <lb/>&longs;emper impetum &longs;eruat, cui nouus &longs;emper accedit, exceptis paucis <lb/>gradibus, qui ob conflictum determinationum, & impetuum excidunt; <pb pagenum="295" xlink:href="026/01/329.jpg"/>quia quotie&longs;cunque nouus impetus ad nouam lineam determinatus ac­<lb/>cedit priori, non e&longs;t dubium, quin de&longs;truatur aliquid impetus, quia ali­<lb/>quid fru&longs;trà e&longs;t, vt lib. 1. demon&longs;tratum e&longs;t. </s> </p> <p id="N2293B" type="main"> <s id="N2293D"><!-- NEW -->Ob&longs;erua &longs;eptimò, aliud mirabilius, &longs;cilicet impetum po&longs;&longs;e produci in <lb/>eo mobili, cui iam ine&longs;t maior impetus, quàm in&longs;it alteri, à quo nouus <lb/>imprimitur; quod certè nunquam fieri pote&longs;t, cum nouus impetus ad <lb/>eandem lineam e&longs;t determinatus, ad quam prior impetus, qui mobili <lb/>ine&longs;t, iam determinatus e&longs;t. </s> </p> <p id="N22949" type="main"> <s id="N2294B">Ob&longs;eruabis octauò; </s> <s id="N2294E"><!-- NEW -->quotie&longs;cunque planum, quod mouetur motu re­<lb/>cto, vel de&longs;init illicò moueri, vel tardiùs mouetur, tunc globus incubans <lb/>mouetur vlteriùs, & qua&longs;i proiicitur; </s> <s id="N22956"><!-- NEW -->hoc ip&longs;um vidimus in naui: </s> <s id="N2295A"><!-- NEW -->ratio <lb/>clara e&longs;t; quia prior impetus in globo productus, qui manet intactus, <lb/>&longs;uum effectum habet. </s> </p> <p id="N22962" type="main"> <s id="N22964">Ob&longs;eruabis nonò, &longs;i terra moueretur ex hypothe&longs;i Copernici, quæ <lb/>tamen fal&longs;i&longs;&longs;ima e&longs;t, idem Parallelus terre&longs;tris globi inæquali motu mo­<lb/>ueretur. </s> <s id="N2296B"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->idem punctum Æquatoris, dum Soli directè re&longs;pondet de <lb/>meridie tardiore motu; </s> <s id="N22975"><!-- NEW -->oppo&longs;itum verò de media nocte velociùs moue­<lb/>retur; </s> <s id="N2297B"><!-- NEW -->ex qua tamen inæqualitate motus aliqui malè &longs;u&longs;picantur æ&longs;tum <lb/>maris oriri; </s> <s id="N22981"><!-- NEW -->quippe licèt fortè aliquis æ&longs;tus maris ex illa hypothe&longs;i &longs;e­<lb/>queretur, longè tamen diuer&longs;us ab eo, qui nunc e&longs;t; nam primò, iis omni­<lb/>bus qui eidem Meridiano &longs;ub&longs;unt eodem tempore accideret æ&longs;tus &longs;cili­<lb/>cet de meridie. </s> <s id="N2298B"><!-- NEW -->Secundò his, qui propiùs accedunt ad polos longè minor <lb/>æ&longs;tus e&longs;&longs;et; vtrumque autem fal&longs;um e&longs;&longs;e con&longs;tat. </s> <s id="N22991"><!-- NEW -->Tertiò, eadem &longs;emper <lb/>hora in &longs;ingulis punctis eiu&longs;dem Paralleli &longs;eor&longs;im ferueret æ&longs;tus; &longs;ed de <lb/>his aliàs plura. </s> </p> <p id="N22999" type="main"> <s id="N2299B"><!-- NEW -->Ob&longs;eruabis decimò, quò diutius potentia motrix manet applicata, ac­<lb/>cedente continenter maiore ni&longs;u, maior quoque impetus producitur in <lb/>rota, quod clarum e&longs;t; vnde diutiùs deinde rota ver&longs;atur. </s> </p> <p id="N229A3" type="main"> <s id="N229A5"><!-- NEW -->Ob&longs;eruabis vndecimò, trochum in gyros actum ita aliquando ver&longs;ari, <lb/>vt &longs;tare pror&longs;us immobilis videatur; quia ferreum fulcrum, cui ligneus <lb/>conus innititur vel excauato &longs;ibi foramine excurrere vltrà non pote&longs;t, <lb/>vel motu centri penitus quie&longs;cente &longs;upere&longs;t tantùm motus orbis. </s> </p> <p id="N229AF" type="main"> <s id="N229B1"><!-- NEW -->Ob&longs;eruabis duodecimò, antequam quie&longs;cat trochus, inclinata verti­<lb/>gine per aliquod tempus ver&longs;ari, moxque, vbi decidit, in plano ip&longs;o ad <lb/>in&longs;tar globi adhuc rotari; </s> <s id="N229B9"><!-- NEW -->&longs;ed quia hæc pertinent ad motum mixtum ex <lb/>circularibus in libro 9. remitto: & verò multa &longs;unt in hoc trochi motu, <lb/>quæ&longs;i attentè con&longs;iderentur, maximam admirationem mouere po&longs;&longs;int. </s> </p> <p id="N229C1" type="main"> <s id="N229C3"><!-- NEW -->Ob&longs;eruabis decimotertiò, &longs;i ferrum, quo trochus armatur, ita e&longs;&longs;et <lb/>infixum vt reuerâ centrum grauitatis cum puncto contactus plani con­<lb/>necteret; </s> <s id="N229CB"><!-- NEW -->nulla e&longs;&longs;et inclinata vertigo, antequam impetus extinguere­<lb/>tur; cur enim potiùs in vnam partem, quàm in aliam. </s> </p> <p id="N229D1" type="main"> <s id="N229D3"><!-- NEW -->Ob&longs;eruabis decimoquarto aquam in vorticibus facilè circulari motu <lb/>conuolui, & aëra, vel halitum in turbinibus; </s> <s id="N229D9"><!-- NEW -->quia &longs;cilicet vel nullus, vel <lb/>modicus e&longs;t obex: idem dico de nube, fumo, acu magnetica, trocho, vel <lb/>&longs;phæra læuigata in plano leuigato. </s> </p> <pb pagenum="296" xlink:href="026/01/330.jpg"/> <p id="N229E5" type="main"> <s id="N229E7"><!-- NEW -->Ob&longs;eruabis decimoquintò, &longs;i in eadem parte plani diu vertatur Tro­<lb/>chus, qua&longs;i excauat &longs;ibi foramen; </s> <s id="N229ED"><!-- NEW -->arrodit enim plani partes &longs;uis denti­<lb/>culis; etiam pelitum ferrum: </s> <s id="N229F3"><!-- NEW -->inde etiam impetum de&longs;trui certum e&longs;t; <lb/>nec enim &longs;ine re&longs;i&longs;tentia id fieri pote&longs;t. </s> </p> <p id="N229F9" type="main"> <s id="N229FB">Ob&longs;eruabis decimo&longs;extò, impetum eundem habere po&longs;&longs;e motum cir­<lb/>cularem, & rectum in &longs;ublunaribus, & per accidens determinari tantùm <lb/>ad motum circularem, ratione &longs;cilicet impedimenti, vt con&longs;tat ex dictis. </s> </p> <p id="N22A02" type="main"> <s id="N22A04"><!-- NEW -->Ob&longs;eruabis decimo&longs;eptimò, motum rectum accelerari, &longs;ed diu non <lb/>durare; </s> <s id="N22A0A"><!-- NEW -->retardari verò violentum, ac æquè diu durare; </s> <s id="N22A0E"><!-- NEW -->circularem <lb/>verò non accelerari, &longs;ed minùs retardari, atque adeo <lb/>longè diutiùs durare; quia tantùm per accidens <lb/>retardatur, &longs;ed de his <lb/>&longs;atis. <lb/><figure id="id.026.01.330.1.jpg" xlink:href="026/01/330/1.jpg"/></s> </p> </chap> <chap id="N22A20"> <pb pagenum="297" xlink:href="026/01/331.jpg"/> <figure id="id.026.01.331.1.jpg" xlink:href="026/01/331/1.jpg"/> <p id="N22A2A" type="head"> <s id="N22A2C"><emph type="center"/>LIBER OCTAVVS, <lb/><emph type="italics"/>DE MOTV FVNEPENDVLORVM.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22A3A" type="main"> <s id="N22A3C"><!-- NEW -->NIHIL inuenio apud antiquos, quod ad <lb/>hoc genus motus pertineat; </s> <s id="N22A42"><!-- NEW -->&longs;unt tamen <lb/>plerique recentiores qui fusè de illo di­<lb/>&longs;putarunt, quorum haud dubiè princi­<lb/>pem locum obtinet Galileus, qui &longs;anè <lb/>mirabiles aliquas huius motus affectiones explicat <lb/>tùm in gemino Sy&longs;themate; tùm in Dialogis, cui ac­<lb/>cedunt Balianus Mercennus, & nonnulli alij. </s> </p> <p id="N22A52" type="main"> <s id="N22A54"><!-- NEW -->Ego verò in hoc libro omnium vibrationum cau­<lb/>&longs;as inquiram, quæ &longs;unt duplicis generis: </s> <s id="N22A5A"><!-- NEW -->Primum e&longs;t <lb/>earum, quibus vibrata hinc inde funependula agun­<lb/>tur, quæ titulum huic libro fecerunt; &longs;unt autem tres <lb/>funependulorum &longs;pecies. </s> <s id="N22A64">Prima e&longs;t eorum, quæ in al­<lb/>tera extremitate fune appen&longs;a vibrantur in circulo <lb/>verticali. </s> <s id="N22A6B">Secunda e&longs;t eorum, quæ ab altera etiam ex­<lb/>tremitate appen&longs;a fune priùs obtorto in circulo ho­<lb/>rizontali &longs;uos agunt gyros. </s> <s id="N22A72">Tertia e&longs;t chordarum, <lb/>quarum vtraque extremitas clauo immobili affigi­<lb/>tur. </s> <s id="N22A79"><!-- NEW -->Secundum genus vibrationum e&longs;t earum, quibus <lb/>aguntur grauia cum à &longs;uo centro grauitatis remouen­<lb/>tur, vt &longs;e&longs;e reducant, quarum &longs;unt duæ &longs;pecies; prima <lb/>e&longs;t earum, quibus vibratur in circulo verticali corpus <lb/>aliquod circa alteram extremitatem, vt campana. </s> <s id="N22A85"><lb/>Secunda e&longs;t earum, quibus vibrantur grauia circa <pb pagenum="298" xlink:href="026/01/332.jpg"/>punctum proximum &longs;uo centro grauitatis, &longs;ic v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->trabs trabi &longs;uperimpo&longs;ita libratur, & vibratur. <lb/><gap desc="hr tag"/></s> </p> <p id="N22A96" type="main"> <s id="N22A98"><emph type="center"/><emph type="italics"/>DEFINITIO I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22AA4" type="main"> <s id="N22AA6"><!-- NEW --><emph type="italics"/>VIbratio funependuli primæ &longs;peciei e&longs;t motus circularis, quo a&longs;cendit, & <lb/>de&longs;cendit funependulum<emph.end type="italics"/>; </s> <s id="N22AB1"><!-- NEW -->&longs;unt autem aliæ æquales, aliæ inæquales: </s> <s id="N22AB5"><!-- NEW --><lb/>æquales &longs;unt, quæ &longs;unt eiu&longs;dem radij, inæquales è contrario: </s> <s id="N22ABA"><!-- NEW -->aliæ &longs;imi­<lb/>les, quæ &longs;imiles arcus complectuntur; di&longs;&longs;imiles è contrario: </s> <s id="N22AC0"><!-- NEW -->aliæ æquè <lb/>diuturnæ, quæ temporibus æqualibus perficiuntur: </s> <s id="N22AC6"><!-- NEW -->aliæ integræ, quarum <lb/>de&longs;cen&longs;us integrum quadrantem comprehendit; non integræ è contra­<lb/>rio; </s> <s id="N22ACE"><!-- NEW -->portio vetò vibrationis e&longs;t arcus; &longs;ed hæc omnia in propo&longs;ito. </s> <s id="N22AD2"><!-- NEW -->Sche­<lb/>mate explicamus; </s> <s id="N22AD8"><!-- NEW -->&longs;it enim plumbeus globus E appen&longs;us fune EA ex <lb/>puncto A immobili, AE e&longs;t radius, vel longitudo funependuli E, NEC <lb/>e&longs;t vibratio integra, LER non integra, LE portio vibrationis NEC, <lb/>NL & MF portiones &longs;imiles, MDB, NEC vibrationes inæquales: ex <lb/>his reliqua facilè intelligi poterunt. </s> </p> <p id="N22AE4" type="main"> <s id="N22AE6"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22AF3" type="main"> <s id="N22AF5"><emph type="italics"/>Momentum e&longs;t exce&longs;&longs;us virtutis mouentis &longs;upra re&longs;istentiam alterius.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->&longs;int brachia vectis inæqualia, momentum e&longs;t in longiore ea vis, qua de­<lb/>&longs;cendens deor&longs;um &longs;ur&longs;um attollit minus &longs;eu breuius. </s> </p> <p id="N22B04" type="main"> <s id="N22B06"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22B13" type="main"> <s id="N22B15"><!-- NEW --><emph type="italics"/>Ten&longs;io e&longs;t vis allata ab extrin&longs;eco corpore, qua augetur eius exten&longs;io<emph.end type="italics"/>; </s> <s id="N22B1E"><!-- NEW -->res <lb/>e&longs;t clara in ten&longs;o fune, quomodocunque id fiat, quod hîc non di&longs;cutio; <lb/>compre&longs;&longs;io verò e&longs;t vis illata ab extrin&longs;eco corpori, qua contrahitur eius <lb/>exten&longs;io v.g. <!-- REMOVE S-->in intorto fune. </s> </p> <p id="N22B2A" type="main"> <s id="N22B2C"><!-- NEW -->Ob&longs;eruabis autem ad ten&longs;ionem, & compre&longs;&longs;ionem requiri, vt &longs;ubla­<lb/>ta illa vi extrin&longs;eca, vel impedimento admoto corpus ten&longs;um, vel com­<lb/>pre&longs;&longs;um ad pri&longs;tinam exten&longs;ionem &longs;e&longs;e reducat; neque di&longs;puto de mo­<lb/>do, quo id fieri po&longs;&longs;it, qui alterius loci e&longs;t. </s> </p> <p id="N22B38" type="main"> <s id="N22B3A"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22B47" type="main"> <s id="N22B49"><!-- NEW --><emph type="italics"/>Corpus graue funependulum à &longs;uæ quiete, vel è &longs;uo centro grauitatis remo­<lb/>tum de&longs;cendit &longs;uâ &longs;ponte, iterumque a&longs;cendit, id e&longs;t vibratur<emph.end type="italics"/>; cer­<lb/>tum e&longs;t. </s> </p> <p id="N22B56" type="main"> <s id="N22B58"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22B65" type="main"> <s id="N22B67"><!-- NEW --><emph type="italics"/>Funependula longiora maiore tempore &longs;uam vibrationam conficiunt, bre­<lb/>uiora minore<emph.end type="italics"/>; quod etiam certum e&longs;t. </s> </p> <p id="N22B72" type="main"> <s id="N22B74"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22B81" type="main"> <s id="N22B83"><emph type="italics"/>Motus naturalis e&longs;t acceleratus in tempore &longs;en&longs;ibili in proportione nume­<lb/>rorum<emph.end type="italics"/> 1.3.5.7. <emph type="italics"/>&c.<emph.end type="italics"/> quod multis explicatum e&longs;t lib. 2. &longs;i verò acceleratio <pb pagenum="299" xlink:href="026/01/333.jpg"/>a&longs;&longs;umatur in &longs;ingulis in&longs;tantibus finitis, e&longs;t iuxta &longs;eriem &longs;implicem nu­<lb/>merorum 1. 2. 3. 4. &c. </s> </p> <p id="N22B9A" type="main"> <s id="N22B9C"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22BA9" type="main"> <s id="N22BAB"><!-- NEW --><emph type="italics"/>Motus in plano inclinato e&longs;t ad motum in perpendiculari, vt perpendicula­<lb/>ris ad inclinatam<emph.end type="italics"/>; </s> <s id="N22BB6"><!-- NEW -->quod etiam lib.5.fusè explicatum e&longs;t; e&longs;t autem &longs;em­<lb/>per in plano inclinato motus prioris grauis. </s> </p> <p id="N22BBC" type="main"> <s id="N22BBE"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22BCB" type="main"> <s id="N22BCD"><!-- NEW --><emph type="italics"/>In quadrante incubante perpendiculariter plano horizontali, tot &longs;unt di­<lb/>uer&longs;a plana inclinata, quot &longs;unt puncta, &longs;eu Tangentes<emph.end type="italics"/>; hoc etiam certum <lb/>e&longs;t, & angulus contingentiæ maior e&longs;t in minore circulo, minor in <lb/>maiore. </s> </p> <p id="N22BDC" type="main"> <s id="N22BDE"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22BEB" type="main"> <s id="N22BED"><!-- NEW --><emph type="italics"/>Nullus arcus circuli e&longs;t vt linea recta, nec &longs;ine errore accipi pote&longs;t vt recta,<emph.end type="italics"/><lb/>contrariam hypothe&longs;im aliqui &longs;upponunt, quam tamen fal&longs;am e&longs;&longs;e &longs;ciunt; </s> <s id="N22BF7"><!-- NEW --><lb/>licèt enim quoad &longs;en&longs;um error &longs;ube&longs;&longs;e non po&longs;&longs;it; </s> <s id="N22BFC"><!-- NEW -->attamen repugnat <lb/>Geometriæ: </s> <s id="N22C02"><!-- NEW -->hinc &longs;uppo&longs;itio no&longs;tra Geometricè vera e&longs;t; &longs;ed de hoc in­<lb/>frà fusè. </s> </p> <p id="N22C08" type="main"> <s id="N22C0A"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22C17" type="main"> <s id="N22C19"><!-- NEW --><emph type="italics"/>Tamdiu durat motus, quandiu durat impetus; hic autem tandiu durat, <lb/>quamdiu non e&longs;t frustrà.<emph.end type="italics"/></s> </p> <p id="N22C23" type="main"> <s id="N22C25"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22C32" type="main"> <s id="N22C34"><emph type="italics"/>Noua determinatio impotus cum priore facit mixtum &longs;i determinatio mixta <lb/>facit nouam lineam.<emph.end type="italics"/></s> </p> <p id="N22C3D" type="main"> <s id="N22C3F"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22C4C" type="main"> <s id="N22C4E"><emph type="italics"/>Quotie&longs;cunque fit mixta determinatio per acce&longs;&longs;ionem noni impetus, de­<lb/>&longs;truitur aliquid impetus prioris, patet.<emph.end type="italics"/></s> </p> <p id="N22C57" type="main"> <s id="N22C59"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22C66" type="main"> <s id="N22C68"><emph type="italics"/>Impetus innatus non concurrit ad motum &longs;ur&longs;um.<emph.end type="italics"/></s> </p> <p id="N22C6F" type="main"> <s id="N22C71"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22C7E" type="main"> <s id="N22C80"><!-- NEW --><emph type="italics"/>In inclinata minùs destruitur impetus dato tempore, quàm in perpendicu­<lb/>lari &longs;ur&longs;um, plùs verò destruitur, quò propiùs accedit ad verticalem<emph.end type="italics"/>; hæc <lb/>omnia quæ loco Axiomatum hîc propo&longs;ui, in &longs;uperioribus libris, præ­<lb/>&longs;ertim in Quinto abundè demon&longs;traui. </s> </p> <p id="N22C8F" type="main"> <s id="N22C91"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22C9E" type="main"> <s id="N22CA0"><!-- NEW --><emph type="italics"/>Funependulum de&longs;cendit motu accelerato<emph.end type="italics"/>; </s> <s id="N22CA9"><!-- NEW -->experientia certa e&longs;t, eius <lb/>ratio e&longs;t eadem cum ea, quam attuli lib.2. de motu naturali, vt eius ac­<lb/>celerationem demon&longs;trarem; </s> <s id="N22CB1"><!-- NEW -->&longs;cilicet impetus nouus &longs;ingulis in&longs;tantibus <lb/>producitur, cùm &longs;it &longs;emper eadem cau&longs;a applicata; </s> <s id="N22CB7"><!-- NEW -->corpus enim graue <lb/>&longs;ua &longs;ponte de&longs;cendit; </s> <s id="N22CBD"><!-- NEW -->quod autem impetui priori accedat, patet; </s> <s id="N22CC1"><!-- NEW -->nec <lb/>enim de&longs;truitur &longs;altem totus alioqui fru&longs;trà produceretur, contra Axio­<lb/>ma primum, adde quòd in plano inclinato deor&longs;um graue de&longs;cendit motu <pb pagenum="300" xlink:href="026/01/334.jpg"/>naturaliter accelerato; </s> <s id="N22CCE"><!-- NEW -->igitur in arcu NLE. v. <!-- REMOVE S-->g. <!-- REMOVE S-->qui habet rationem <lb/>plani inclinati in omnibus &longs;uis punctis per hypothe&longs;im 5. Præterea ictus <lb/>e&longs;t maior, quò maior e&longs;t arcus vibrationîs; </s> <s id="N22CDA"><!-- NEW -->igitur impetus maior; </s> <s id="N22CDE"><!-- NEW -->igitur <lb/>cre&longs;cit impetus; </s> <s id="N22CE4"><!-- NEW -->igitur motus e&longs;t acceleratus; </s> <s id="N22CE8"><!-- NEW -->deinde maior vibratio, & <lb/>minor eiu&longs;dem penduli fiunt ferè temporibus æqualibus; </s> <s id="N22CEE"><!-- NEW -->igitur nece&longs;&longs;a­<lb/>riò acceleratur motus: </s> <s id="N22CF4"><!-- NEW -->Denique probatur euidenter non de&longs;trui totum <lb/>priorem impetum; </s> <s id="N22CFA"><!-- NEW -->quia &longs;cilicet idem e&longs;t impedimentum, &longs;i quod e&longs;t ad <lb/>productionem noui, quod e&longs;t ad con&longs;eruationem prioris; </s> <s id="N22D00"><!-- NEW -->&longs;ed illud im­<lb/>pedimentum, id e&longs;t inclinatio plani, non impedit productionem noui, <lb/>licèt minoris, vt videbimus paulò pò&longs;t; </s> <s id="N22D08"><!-- NEW -->quia &longs;cilicet in omni plano in­<lb/>clinato corpus graue mouetur per hypoth.4. igitur non impedit con&longs;er­<lb/>uationem prioris, &longs;altem totam, licèt fortè aliquid de&longs;trueretur, de quo <lb/>paulò pò&longs;t; </s> <s id="N22D12"><!-- NEW -->igitur acceleratur nece&longs;&longs;ariò ille motus: </s> <s id="N22D16"><!-- NEW -->Et hæc e&longs;t ratio à <lb/>priori huius effectus, quòd &longs;cilicet plùs addatur impetus, quàm tollatur; </s> <s id="N22D1C"><!-- NEW --><lb/>igitur remanet maior; </s> <s id="N22D21"><!-- NEW -->igitur velocior motus; in qua verò ratione minùs <lb/>de&longs;truatur quàm producatur, vel nouus &longs;it minor priore, dicemus <lb/>infrà. </s> </p> <p id="N22D29" type="main"> <s id="N22D2B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22D38" type="main"> <s id="N22D3A"><emph type="italics"/>In motu funependuli decre&longs;cunt &longs;emper incrementa motus.<emph.end type="italics"/></s> <s id="N22D41"> Probatur faci­<lb/>lè; </s> <s id="N22D46"><!-- NEW -->quia cùm in &longs;ingulis punctis de&longs;cen&longs;us arcus NE mutetur ratio plani <lb/>inclinati diuer&longs;a ab ea, quæ e&longs;t in puncto <expan abbr="q;">que</expan> &longs;unt enim vt Tangentes; </s> <s id="N22D50"><!-- NEW --><lb/>certè Tangentes punctorum, quæ propiùs accedunt ad N, accedunt <lb/>etiam propiùs ad perpendicularem deor&longs;um, à qua longiùs recedunt <lb/>Tangentes, quæ accedunt propiùs ad E, vt con&longs;tat; </s> <s id="N22D59"><!-- NEW -->at qui motus in planis, <lb/>quæ accedunt propiùs ad horizontalem, minor e&longs;t; </s> <s id="N22D5F"><!-- NEW -->igitur incrementa <lb/>motus quæ in de&longs;cen&longs;u NE accedunt, minora &longs;unt ver&longs;us E, maiora ver­<lb/>&longs;us N; igitur decre&longs;cunt, quod erat demon&longs;trandum. </s> </p> <p id="N22D67" type="main"> <s id="N22D69">Ob&longs;eruabis iam demon&longs;tratum lib.5. Th.62.63. hæc incrementa e&longs;&longs;e, <lb/>vt &longs;inus arcus re&longs;idui, quæ tu con&longs;ule, ne hic repetere cogar. </s> </p> <p id="N22D6E" type="main"> <s id="N22D70"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22D7D" type="main"> <s id="N22D7F"><!-- NEW --><emph type="italics"/>Hinc &longs;emper cre&longs;cit motus funependuli in de&longs;cen&longs;u arcus NE, &longs;ed minori­<lb/>bus &longs;en&longs;im incrementis<emph.end type="italics"/>; </s> <s id="N22D8A"><!-- NEW -->quod etiam aliàs ob&longs;eruatum e&longs;t; </s> <s id="N22D8E"><!-- NEW -->vnde nece&longs;&longs;ariò <lb/>concludo minùs accelerari in quadrante NE, quàm in perpendiculari <lb/>NS, quod demon&longs;tratum e&longs;t, & minus &longs;patium percurri in arcu NE <lb/>æquali &longs;cilicet tempore, quàm in perpendiculari NS, quod nece&longs;&longs;arium <lb/>e&longs;t: </s> <s id="N22D9A"><!-- NEW -->Nec e&longs;t quod aliquis &longs;ua experimenta opponat, &longs;cilicet quadrantem <lb/>NE percurri tempore vnius &longs;ecundi, &longs;i radius AE &longs;it tripedalis, cùm <lb/>alioqui perpendiculum AE graue corpus percurrat eodem tempore, <lb/>quorum alterum, vel potiùs vtrumque fal&longs;um e&longs;&longs;e nece&longs;&longs;e e&longs;t; </s> <s id="N22DA4"><!-- NEW -->nam primò <lb/>quadrans NE e&longs;t maior radio AE; </s> <s id="N22DAA"><!-- NEW -->igitur percurrit citiùs AE quàm <lb/>NE: &longs;ecundò, minora &longs;unt motus incrementa in quadrante, quia &longs;in­<lb/>gula puncta illius habent rationem plani inclinati, quis autem tam ac­<lb/>curatè in tripedali <expan abbr="p&etilde;dulo">pendulo</expan> iu&longs;tum tempus ob&longs;eruare po&longs;&longs;it? </s> <s id="N22DB8">nec accuratæ <pb pagenum="301" xlink:href="026/01/335.jpg"/>illæ ob&longs;eruationes e&longs;&longs;e po&longs;&longs;unt, quæ &longs;en&longs;ibiles non &longs;unt, &longs;iue aures con­<lb/>&longs;ulas, quæ &longs;onum excipiunt, &longs;iue oculos, qui motum ip&longs;um ob&longs;eruant. </s> <s id="N22DC2"><!-- NEW --><lb/>Tertiò, &longs;i oculos con&longs;ulis; num ip&longs;i potiùs vident motum vibrati pendu­<lb/>li e&longs;&longs;e tardiorem, quàm demi&longs;&longs;i per lineam perpendicularem? </s> <s id="N22DC9">nec alius <lb/>nodus hic &longs;oluendus e&longs;t, nec aër &longs;en&longs;ibiliter pilæ plumbeæ re&longs;i&longs;tit, nec <lb/>minùs re&longs;i&longs;tit motui circulari quàm recto. </s> <s id="N22DD0"><!-- NEW -->Denique compertum e&longs;t à me <lb/>in longiore pendulo motum in arcu e&longs;&longs;e tardiorem, quàm in perpendi­<lb/>culo: </s> <s id="N22DD8"><!-- NEW -->nodus ob&longs;eruationis facilis e&longs;t, nam adhibui AE planum durum <lb/>re&longs;pondens accuratè perpendiculari, cui aliud planum E <foreign lang="greek">b</foreign> ad angulos <lb/>rectos affixum erat <expan abbr="re&longs;põdens">re&longs;pondens</expan> Tangenti; </s> <s id="N22DE8"><!-- NEW -->tùm demi&longs;&longs;o ex A globulo plum­<lb/>beo &longs;imulque alio æquali pendulo &longs;cilicet circa A ex N per NE; </s> <s id="N22DEE"><!-- NEW -->ex quo <lb/>accidit citiùs auditum e&longs;&longs;e ictum globi cadentis perpendiculariter, quàm <lb/>vibrati per arcum NE: quis autem hoc non videat, &longs;iue &longs;en&longs;um ip&longs;um, <lb/>&longs;iue rationem con&longs;ulat? </s> <s id="N22DF8">fuit meum pendulum 12. pedes longum. </s> </p> <p id="N22DFB" type="main"> <s id="N22DFD">Quæreret aliquis primò quanta fuerit differentia temporum Secundò, <lb/>quanto tempore globus pendulus ex N in E peruenerit. </s> <s id="N22E02"><!-- NEW -->Re&longs;pondeo inu­<lb/>tilem e&longs;&longs;e quæ&longs;tionem; </s> <s id="N22E08"><!-- NEW -->nec enim minimas illas temporum differentias <lb/>&longs;en&longs;u metiri po&longs;&longs;umus; </s> <s id="N22E0E"><!-- NEW -->&longs;i enim affirmarem cum nonnullis corpus graue <lb/>per medium liberum 12. &longs;patij pedes conficere vno temporis &longs;ecundo; </s> <s id="N22E14"><!-- NEW --><lb/>certè &longs;i quis contenderet vel dee&longs;&longs;e, vel &longs;upere&longs;&longs;e 1000. in&longs;tantia; quonam <lb/>argumento, vel experimento contrarium euincere po&longs;&longs;em? </s> <s id="N22E1B">quod certè <lb/>dictum e&longs;&longs;e velim, vt vel inde o&longs;tendatur in ca&longs;&longs;um laborare eos, qui <lb/>hanc &longs;cientiam his tantùm experimentis confirmant, quæ circa in&longs;en&longs;i­<lb/>bilia ver&longs;antur. </s> <s id="N22E24"><!-- NEW -->Equidem magnifacio in rebus phy&longs;icis experimentum, <lb/>&longs;ine quo nulla hypothe&longs;is e&longs;&longs;e pote&longs;t; </s> <s id="N22E2A"><!-- NEW -->at modo &longs;en&longs;ibile &longs;it, alioqui cer­<lb/>tum e&longs;&longs;e non pote&longs;t; </s> <s id="N22E30"><!-- NEW -->&longs;i autem &longs;en&longs;ibile e&longs;t, omnibus commune e&longs;&longs;e debet, <lb/>sum &longs;en&longs;us applicent; </s> <s id="N22E36"><!-- NEW -->igitur nunquam vir prudens &longs;e&longs;e accinget ad in­<lb/>dagandam rationem alicuius experimenti, quod certum e&longs;&longs;e non pote&longs;t: </s> <s id="N22E3C"><!-- NEW --><lb/>vnde &longs;i quis omnes ob&longs;eruationes, tùm à Plinio, tùm à Cardano, tùm à <lb/>Fraca&longs;torio, tùm à Porta, tùm ab aliis propo&longs;itas ad principia phy&longs;ica re­<lb/>ducere velit, per me &longs;tat, non contradico; numquam tamen illa mihi <lb/>mens erit, cui &longs;atis e&longs;t rationes, & cau&longs;as phy&longs;icas illorum tantùm expe­<lb/>rimentorum explicare, quæ mihi certa &longs;unt, &longs;untque omnibus commu­<lb/>nia, vel e&longs;&longs;e po&longs;&longs;unt. </s> </p> <p id="N22E4B" type="main"> <s id="N22E4D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22E5A" type="main"> <s id="N22E5C"><emph type="italics"/>In motu funependuli &longs;ingulis instantibus e&longs;t noua determinatio motus.<emph.end type="italics"/></s> <s id="N22E63"><!-- NEW --> Pro­<lb/>batur, quia &longs;ingulis in&longs;tantibus e&longs;t qua&longs;i nouum planum; </s> <s id="N22E69"><!-- NEW -->tot &longs;unt enim <lb/>plana in quadrante NE, quot Tangentes, & tot Tangentes quot pun­<lb/>cta, tot denique puncta, quot in&longs;tantia; </s> <s id="N22E71"><!-- NEW -->atqui in &longs;ingulis nouis planis <lb/>mutatur determinatio; </s> <s id="N22E77"><!-- NEW -->igitur in &longs;ingulis punctis; igitur in &longs;ingulis in­<lb/>&longs;tantibus. </s> </p> <p id="N22E7D" type="main"> <s id="N22E7F"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22E8B" type="main"> <s id="N22E8D">Ob&longs;eruabis e&longs;&longs;e aliqua Lemmata præmittenda antequam proportio­<lb/>nes motus per arcum NE demon&longs;trentur. </s> </p> <pb pagenum="302" xlink:href="026/01/336.jpg"/> <p id="N22E96" type="main"> <s id="N22E98"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22EA5" type="main"> <s id="N22EA7"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari tempus, quo percurruntur duo &longs;patia æqualia motu na­<lb/>turaliter accelerato inæquali.<emph.end type="italics"/> &longs;it v.g. <!-- REMOVE S-->tempus AF; </s> <s id="N22EB4"><!-- NEW -->&longs;it velocitas EF ac­<lb/>qui&longs;ita tempore AF motu &longs;cilicet naturaliter accelerato minore; </s> <s id="N22EBA"><!-- NEW -->&longs;it <lb/>etiam velocitas FD acqui&longs;ita alio motu maiore eodem tempore AF; </s> <s id="N22EC0"><!-- NEW --><lb/>haud dubiè &longs;patium acqui&longs;itum primo motu erit ad acqui&longs;itum &longs;ecundo, <lb/>æquali &longs;cilicet tempore, vt triangulum EAF ad triangulum DAF, vt <lb/>con&longs;tat ex dictis lib.2. in controuer&longs;ia; </s> <s id="N22EC9"><!-- NEW -->&longs;patium verò acqui&longs;itum tempo­<lb/>re AF primo motu, &longs;cilicet minore, idque v.g. <!-- REMOVE S-->in ratione &longs;ubdupla erit <lb/>ad &longs;patium acqui&longs;itum &longs;ecundo motu maiore tempore &longs;ubduplo AI, vt <lb/>triangulum EAF ad triangulum BAI, &longs;ed BAI, e&longs;t &longs;ubduplum EAF, <lb/>id e&longs;t, vt FA ad IA, vt patet: </s> <s id="N22ED7"><!-- NEW -->vt autem inueniantur tempora, quæ re­<lb/>&longs;pondent &longs;patiis inæqualibus; </s> <s id="N22EDD"><!-- NEW -->&longs;it AH media proportionalis inter AI & <lb/>AF; </s> <s id="N22EE3"><!-- NEW -->haud dubiè triangulum CHA e&longs;t &longs;ubduplum DAF; </s> <s id="N22EE7"><!-- NEW -->igitur æquale <lb/>EAF; </s> <s id="N22EED"><!-- NEW -->igitur velocitas acqui&longs;ita tempore AF &longs;it FE, motu &longs;cilicet mi­<lb/>nore; </s> <s id="N22EF3"><!-- NEW -->acqui&longs;ita verò tempore AH motu maiore &longs;it HC; </s> <s id="N22EF7"><!-- NEW -->certè &longs;patia <lb/>erunt vt CHA & DAF: </s> <s id="N22EFD"><!-- NEW -->&longs;ed hæc &longs;unt æqualia; igitur motu maiore con­<lb/>ficitur æquale &longs;patium tempore AH & motu minore tempore AF. <!-- KEEP S--></s> </p> <p id="N22F04" type="main"> <s id="N22F06"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22F13" type="main"> <s id="N22F15"><!-- NEW --><emph type="italics"/>Si accipiantur tempora æqualia cum motibus inæqualibus, &longs;patia &longs;unt vt <lb/>ba&longs;es triangulorum<emph.end type="italics"/>; </s> <s id="N22F20"><!-- NEW -->&longs;it enim tempus AI, quo motu maiore acquiratur ve­<lb/>locitas IB, & minore IK; </s> <s id="N22F26"><!-- NEW -->certè &longs;patia &longs;unt vt triangula BAI, KAI; </s> <s id="N22F2A"><!-- NEW --><lb/>&longs;ed hæc &longs;unt vt ba&longs;es BI, KI, immò &longs;unt vt rectangula BA KA; nec <lb/>in his e&longs;t quidquam difficultatis. </s> </p> <p id="N22F31" type="main"> <s id="N22F33"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22F40" type="main"> <s id="N22F42"><!-- NEW -->Po&longs;&longs;unt determinari vel &longs;patia inæqualia temporibus æqualibus, vel <lb/>tempora inæqualia &longs;patiis æqualibus in chordis eiu&longs;dem quadrantis, & <lb/>in perpendiculari, &longs;it tempus DI; </s> <s id="N22F4A"><!-- NEW -->&longs;it motus per ip&longs;am perpendicula­<lb/>rem AP, vel DI; </s> <s id="N22F50"><!-- NEW -->&longs;it motus etiam per chordam inclinatam DP; </s> <s id="N22F54"><!-- NEW -->velo­<lb/>citas primi e&longs;t ad velocitatem &longs;ecundi in tempore DI, vt DP ad DI, <lb/>vel vt AK ad &longs;inum VK, vel vt IP ad NP, vel vt quadratum IA ad <lb/>rectangulum NA; </s> <s id="N22F5E"><!-- NEW -->&longs;ed &longs;patia &longs;unt vt velocitates &longs;uppo&longs;itis temporibus <lb/>æqualibus; </s> <s id="N22F64"><!-- NEW -->igitur &longs;patium, quod percurritur in ip&longs;a perpendiculari e&longs;t <lb/>ad &longs;patium, quod percurritur in inclinata DP temporibus æqualibus, vt <lb/>quadratum IA ad rectangulum NA, vel vt DP ad DI, vel vt DT ad <lb/>DP, quæ omnia con&longs;tant; </s> <s id="N22F6E"><!-- NEW -->&longs;it autem motus in inclinata FP; certè &longs;pa­<lb/>tium acqui&longs;itum in perpendiculari e&longs;t ad &longs;patium acqui&longs;itum in FP, vt <lb/>QZP ad ZI, vel FP ad FY, vel AP ad PR, vel AL ad LX, vel PI <lb/>ad PM, vel vt quadratum IA, ad rectangulum MA, vel vt F <foreign lang="greek">d</foreign> ad PF, <lb/>&longs;ed F <foreign lang="greek">d</foreign> e&longs;t æqualis DT, quia cum DP & FP percurrantur temporibus <lb/>æqualibus, <expan abbr="&longs;i&qacute;ue">&longs;ique</expan> eo tempore quo percurritur DP, percurritur DT, & <lb/>eo quo percurritur FP, percurritur. </s> <s id="N22F8A">F <foreign lang="greek">d</foreign>; certè DT & F <foreign lang="greek">d</foreign> &longs;unt <lb/>quales. </s> </p> <pb pagenum="303" xlink:href="026/01/337.jpg"/> <p id="N22F9B" type="main"> <s id="N22F9D">Idem dico de omnibus aliis chordis, quarum motus, & velocitates, <lb/>&longs;patia temporibus æqualibus acqui&longs;ita &longs;unt ad motus, velocitates, &longs;patia <lb/>acqui&longs;ita in perpendiculari, vt ip&longs;arum longitudines ad DT, vel duplam <lb/>DI, vel vt earum &longs;ubduplæ &longs;eu &longs;inus recti &longs;ubdupli &longs;ui arcus ad &longs;inum to­<lb/>tum DI, vel vt rectangula &longs;ub illis &longs;inubus comprehen&longs;a, & &longs;inu toto <lb/>ad quadratum &longs;inus totius. </s> </p> <p id="N22FAA" type="main"> <s id="N22FAC"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N22FB9" type="main"> <s id="N22FBB"><!-- NEW --><emph type="italics"/>Si &longs;int duæ quantitates in data ratione, & aliæ duæ in data, &longs;ed minore; </s> <s id="N22FC1"><!-- NEW -->&longs;i &longs;it <lb/>media proportionalis inter duas primas & media inter duas posteriores, &longs;itque <lb/>data noua quantitas ad aliam, vt prima priorum quantitatum ad primam <lb/>mediam proportionalem, &longs;it denique eadem quantitas noua ad aliam vt prima <lb/>po&longs;teriorum quantitatum ad &longs;ecundam mediam proportionalem, certè erit mi­<lb/>nor ratio noua quantitatis ad &longs;ecundam que&longs;itam, quàm ad primam<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it <lb/>DE prima quantitas, & LK &longs;ecunda; </s> <s id="N22FD8"><!-- NEW -->&longs;it KR tertia, VZ quarta; </s> <s id="N22FDC"><!-- NEW -->&longs;itque <lb/>prima ad &longs;ecundam, vt 4. ad 9. & tertia ad quartam, vt 3. ad 12. certè e&longs;t <lb/>minor ratio tertiæ ad quartam, quàm primæ ad &longs;ecundam; </s> <s id="N22FE4"><!-- NEW -->inter primam <lb/>& &longs;ecundam &longs;it media proportionalis AC æqualis FH, id e&longs;t <foreign lang="greek">s</foreign>, & &longs;it <lb/>quinta quantitas; </s> <s id="N22FF0"><!-- NEW -->&longs;it etiam alia inter tertiam & quartam; </s> <s id="N22FF4"><!-- NEW -->&longs;it TS æqualis <lb/>VY, &longs;cilicet <foreign lang="greek">s</foreign>; &longs;itque &longs;exta quantitas, & vt prima ad &longs;ecundam, ita <lb/>&longs;eptima quantitas v. <!-- REMOVE S-->g. <!-- REMOVE S-->DE ad octauam AC, &longs;itque vt tertia quantitas <lb/>VX vel QR ad &longs;extam VY, vel TS, ita eadem &longs;eptima DE ad nonam <lb/>AC. <!-- KEEP S--></s> <s id="N23009">Dico e&longs;&longs;e minorem ratione &longs;eptimæ DE ad nonam AT, quàm <lb/>eiu&longs;dem &longs;eptimæ DE ad octauam AC, quia AB vel DE e&longs;t ad AC vt <lb/>2. ad 3. & ad X, vt a. </s> <s id="N23010">ad 4. quæ omnia con&longs;tant ex Geometria. <!-- KEEP S--></s> </p> <p id="N23014" type="main"> <s id="N23016"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23023" type="main"> <s id="N23025"><!-- NEW --><emph type="italics"/>Si &longs;int<emph.end type="italics"/> <emph type="italics"/>duæ chordæ in quadrante EIB, & producatur BI v&longs;que ad G; </s> <s id="N23031"><!-- NEW -->&longs;it­<lb/>que EM perpendicularis, in quam cadat IH, quæ cum EI faciat angulum <lb/>rectum; </s> <s id="N23039"><!-- NEW -->ex eodem puncto H ducatur HQ perpendicularis in EB: </s> <s id="N2303D"><!-- NEW -->dico mino­<lb/>rem e&longs;&longs;e proportionem EQ ad EB, quàm GI ad GB<emph.end type="italics"/>; </s> <s id="N23046"><!-- NEW -->&longs;it enim IP paral­<lb/>lela EG, vt EP e&longs;t ad EB, &longs;ic GI ad GB; </s> <s id="N2304C"><!-- NEW -->igitur EQ habet minorem <lb/>proportionem ad EB, quam GI ad GB; </s> <s id="N23052"><!-- NEW -->&longs;imiliter &longs;int chordæ EIL, <lb/>EL; </s> <s id="N23058"><!-- NEW -->ducatur HK perpendicularis in EL: </s> <s id="N2305C"><!-- NEW -->dico EK habere minorem <lb/>rationem ad EL, quàm FI ad FL; </s> <s id="N23062"><!-- NEW -->nam vt EO e&longs;t ad EL, ita FI ad FL; </s> <s id="N23066"><!-- NEW --><lb/>igitur minor e&longs;t ratio EK ad EL, quàm FI ad FL; </s> <s id="N2306B"><!-- NEW -->Idem dico de om­<lb/>nibus aliis chordis: </s> </p> <p id="N23071" type="main"> <s id="N23073"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23080" type="main"> <s id="N23082"><!-- NEW --><emph type="italics"/>Cognite tempore, quo percurritur &longs;egmentum, lineæ cogno&longs;ci pote&longs;t tempus, que <lb/>aliud &longs;egmentum percurretur motu &longs;cilicet propagate<emph.end type="italics"/>; </s> <s id="N2308D"><!-- NEW -->&longs;it v. <!-- REMOVE S-->g. <!-- REMOVE S-->perpendicu­<lb/>laris deor&longs;um DI; </s> <s id="N23097"><!-- NEW -->&longs;it primum &longs;egmentum DG decur&longs;um tempore AB; </s> <s id="N2309B"><!-- NEW --><lb/>&longs;it vt DC ad DH, ita DH ad DI; </s> <s id="N230A0"><!-- NEW --><expan abbr="&longs;it&qacute;ue">&longs;itque</expan> vt DG ad DH, ita tempus <lb/>AB ad AC; dico quod &longs;ecundum &longs;egmentum percurretur tempore BC <lb/>po&longs;t primum decur&longs;um, patet ex dictis lib.2. & 5. <!-- KEEP S--></s> </p> <pb pagenum="304" xlink:href="026/01/338.jpg"/> <p id="N230B0" type="main"> <s id="N230B2"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N230BE" type="main"> <s id="N230C0"><!-- NEW --><emph type="italics"/>Cognito tempore, quo percurritur chorda cuiu&longs;libet arcus, cogno&longs;ci pote&longs;t <lb/>quantum &longs;paty eodem tempore percurratur in <expan abbr="perp&etilde;diculari">perpendiculari</expan> & in alia chorda<emph.end type="italics"/>; </s> <s id="N230CF"><!-- NEW --><lb/> &longs;it chorda EL; </s> <s id="N230D4"><!-- NEW -->fiat angulus rectus ELM, itemque MDE: </s> <s id="N230D8"><!-- NEW -->dico quod <lb/>eodem tempore percurretur EL EM ED; </s> <s id="N230DE"><!-- NEW -->&longs;imiliter fiat angulus re­<lb/>ctus EIH, itemque HKE, HQE: dico quod eodem tempore percur­<lb/>rentur EI, EH, EK,EQ. idem dico de omnibus aliis chordis, quæ <lb/>omnia con&longs;tant ex his quæ diximus lib.2. & 5. <!-- KEEP S--></s> </p> <p id="N230ED" type="main"> <s id="N230EF"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N230FB" type="main"> <s id="N230FD"><!-- NEW --><emph type="italics"/>Due chorda ELB citiùs percurruntur quàm &longs;ola EB; </s> <s id="N23103"><!-- NEW -->itemque due EIB, <lb/>quàm EB<emph.end type="italics"/>; </s> <s id="N2310C"><!-- NEW -->quia eodem tempore percurruntur EI, <expan abbr="Eq;">Eque</expan> & IB eodem <lb/>tempore percurritur &longs;iue à G incipiat motus &longs;iue ab E; </s> <s id="N23116"><!-- NEW -->nam ab æquali <lb/>altitudine æqualis acquiritur impetus, &longs;ed minor e&longs;t proportio EQ ad <lb/>EB, quam GI ad GB per Lemma quintum; </s> <s id="N2311E"><!-- NEW -->igitur &longs;i &longs;it media propor­<lb/>tionalis inter GI, GB, & &longs;ecunda inter EQEB, &longs;itque vt GI ad pri­<lb/>mam proportionalem; </s> <s id="N23126"><!-- NEW -->ita tempus, quo percurritur EI ad aliud X, & vt <lb/>EQ ad &longs;ecundam proportionalem, ita idem tempus, quo percurritur EI, <lb/>vel EQ ad aliud Z; </s> <s id="N2312E"><!-- NEW -->certè tempus Z e&longs;t maius tempore X per Lemma <lb/>4. &longs;ed EQB percurritur tempore Z, & EIB tempore X; </s> <s id="N23134"><!-- NEW -->EQ verò, & <lb/>EI tempore æquali per Lemma 7. igitur duæ EIB citiùs percurruntur, <lb/>quàm EB; </s> <s id="N2313C"><!-- NEW -->idem dico de aliis: hoc ip&longs;um etiam demon&longs;trauit Galil. <!-- REMOVE S-->in <lb/>dialogis. </s> </p> <p id="N23144" type="main"> <s id="N23146"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N23152" type="main"> <s id="N23154"><!-- NEW --><emph type="italics"/>Tres chordæ faciliùs percurruntur, quàm duæ<emph.end type="italics"/>; </s> <s id="N2315D"><!-- NEW -->&longs;int enim tres EILB; </s> <s id="N23161"><!-- NEW --><lb/>&longs;int duæ ELB. Primò, duæ EIL citiùs percurruntur quàm EL, quia <lb/>IL eodem tempore percurritur, &longs;iue initium motus ducatur ab F, &longs;iue ab <lb/>E; </s> <s id="N2316A"><!-- NEW -->& minor e&longs;t ratio EK ad EL, quàm FI ad FL per Lem.5.EI, & EK <lb/>æquè citò percurruntur per Lem. <!-- REMOVE S-->7. igitur &longs;it vt FI ad mediam propor­<lb/>tionalem inter FI & FL; </s> <s id="N23174"><!-- NEW -->ita tempus Z ad tempus X, & vt EK ad me­<lb/>diam proportionalem inter EK EL, ita tempus Z ad tempus Y; </s> <s id="N2317A"><!-- NEW -->certè <lb/>tempus Y erit maius tempore X per Lem. <!-- REMOVE S-->8. igitur citiùs percurrentur <lb/>duæ EIL, quàm EL; </s> <s id="N23184"><!-- NEW -->&longs;ed &longs;i eodem tempore percurrerentur duæ EIL <lb/>cum EL; </s> <s id="N2318A"><!-- NEW -->certè LB æquali tempore percurreretur, quia e&longs;t idem impetus <lb/>in L, &longs;iue ab E per EL, &longs;iue ab F per FL incipiat motus, vt con&longs;tat, & e&longs;t <lb/>idem in I, &longs;iue ab E, &longs;iue ab F incipiat; </s> <s id="N23192"><!-- NEW -->igitur idem in L &longs;iue ab E per <lb/>EIL, &longs;iue ab F per FL, &longs;iue ab E per EL; </s> <s id="N23198"><!-- NEW -->igitur LB æquali tempore <lb/>percurretur, &longs;iue motus &longs;it ab E per ELB, &longs;iue ab E per EI, LB, po&longs;ito <lb/>quòd EIL & EL æquali tempore percurrantur; </s> <s id="N231A0"><!-- NEW -->&longs;ed EIL percurrun­<lb/>tur citiùs quàm EL; </s> <s id="N231A6"><!-- NEW -->igitur citiùs EILB, quàm ELB; </s> <s id="N231AA"><!-- NEW -->igitur cùm ELB <lb/>percurrantur citiùs, quàm EB, & EILB, quàm ELB; </s> <s id="N231B0"><!-- NEW -->certè EILB per­<lb/>curruntur citiùs, quàm EB: Eodem modo demon&longs;trabitur 4. chordas ci­<lb/>tiùs percurri, quàm 3. 5. quàm 4. atque ita deinceps. </s> </p> <pb pagenum="305" xlink:href="026/01/339.jpg"/> <p id="N231BC" type="main"> <s id="N231BE"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N231CA" type="main"> <s id="N231CC"><!-- NEW --><emph type="italics"/>Velocitas acqui&longs;ita in duabus chordis EIB e&longs;t æqualis acqui&longs;itæ in EB<emph.end type="italics"/>; </s> <s id="N231D5"><!-- NEW --><lb/>quia acqui&longs;ita in EI e&longs;t æqualis acqui&longs;itæ in GI; </s> <s id="N231DA"><!-- NEW -->&longs;unt enim eiu&longs;dem al­<lb/>titudinis; </s> <s id="N231E0"><!-- NEW -->igitur acqui&longs;ita in EIB æqualis acqui&longs;itæ in GB: </s> <s id="N231E4"><!-- NEW -->&longs;ed acqui­<lb/>&longs;ita in GB e&longs;t æqualis acqui&longs;itæ in EIB; </s> <s id="N231EA"><!-- NEW -->igitur acqui&longs;ita in EB e&longs;t æqua­<lb/>lis acqui&longs;itæ in EIB, itemque acqui&longs;ita in ELB acqui&longs;itæ in EB: </s> <s id="N231F0"><!-- NEW -->immò <lb/>acqui&longs;ita in tribus EILB e&longs;t æqualis acqui&longs;itæ in EB; </s> <s id="N231F6"><!-- NEW -->quia acqui&longs;ita in <lb/>EIL e&longs;t æqualis acqui&longs;itæ in EL; </s> <s id="N231FC"><!-- NEW -->igitur acqui&longs;ita in EILB æqualis <lb/>acqui&longs;itæ in ELB: </s> <s id="N23202"><!-- NEW -->&longs;ed acqui&longs;ita in ELB e&longs;t æqualis acqui&longs;itæ in EB; igi­<lb/>tur acqui&longs;ita in EB æqualis acqui&longs;itæ in EILB idem dico de 5. chordis, <lb/>6.7. atque ita deinceps. </s> </p> <p id="N2320B" type="main"> <s id="N2320D"><!-- NEW -->Quod certè mirabile e&longs;t, & qua&longs;i paradoxon; </s> <s id="N23211"><!-- NEW -->præ&longs;ertim cùm duplici <lb/>motu acquiratur æqualis velocitas in &longs;patiis inæqualibus, quorum mauis <lb/>citiùs percurritur; </s> <s id="N23219"><!-- NEW -->Equidem in AB, EB acquiritur æqualis velocitas, <lb/>vel impetus, &longs;ed breuius &longs;patium, &longs;cilicet AB citius percurritur; </s> <s id="N2321F"><!-- NEW -->at verò <lb/>in EB, & ELB acquiritur æqualis velocitas; </s> <s id="N23225"><!-- NEW -->licèt &longs;patium longius ELB <lb/>percurratur citiùs, quàm EB; &longs;imiliter EILB velociùs, quam ELB & EB. <!-- KEEP S--></s> </p> <p id="N2322C" type="main"> <s id="N2322E"><!-- NEW -->Hinc &longs;uprà velocitas acqui&longs;ita in perpendiculari &longs;eu radio quadrantis <lb/>non e&longs;t ad velocitatem acqui&longs;itam in toto arcu quadrantis vt quadratum <lb/>&longs;ub radio ad ip&longs;um quadrantem, quia &longs;cilicet velocitas acqui&longs;ita per ar­<lb/>cum ELB e&longs;t æqualis acqui&longs;itæ per omnes chordas facto initio motus <lb/>ab E; &longs;ed velocitas acqui&longs;ita in 6. chordis. </s> <s id="N2323A">v. <!-- REMOVE S-->g. <!-- REMOVE S-->e&longs;t æqualis acqui&longs;itæ in <lb/>5. 4. 3. 2. 1. igitur velocitas acqui&longs;ita in EB e&longs;t æqualis acqui&longs;itæ in ar­<lb/>cu ELB, & in ip&longs;a perpendiculari ER. </s> </p> <p id="N23245" type="main"> <s id="N23247"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N23253" type="main"> <s id="N23255"><!-- NEW --><emph type="italics"/>Hinc Lemma vniuer&longs;ali&longs;&longs;imum &longs;tatuitur, &longs;cilicet ab eodem puncto altitudi­<lb/>nîs ad <expan abbr="eãdem">eandem</expan> horizontalem, vel ab eadem horizontali ad idem punctum <lb/>deor&longs;um, vel ab eadem horizontali ad aliam horizontalem aquales acquiri <lb/>velocitates, &longs;iue plures &longs;int lineæ, &longs;ine vnica, &longs;iue &longs;implices, &longs;iue compo&longs;itæ, &longs;iue <lb/>recta, &longs;iue curua<emph.end type="italics"/>; quæ omnia ex Lemmate decimo manife&longs;ta redduntur. </s> </p> <p id="N2326A" type="main"> <s id="N2326C"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N23278" type="main"> <s id="N2327A"><!-- NEW --><emph type="italics"/>Velocitas acqui&longs;ita in toto arcu quadrantis ELB non debet a&longs;&longs;umi in area <lb/>tota quadrantis AEB, &longs;ed in linea recta æquali toti arcui ELB, ductis &longs;ci­<lb/>licet lineis rectis tran&longs;uer&longs;is, qua &longs;int ip&longs;is &longs;inubus rectis æquales, cuius con&longs;tru­<lb/>ctionis<emph.end type="italics"/>; </s> <s id="N23289"><!-- NEW -->&longs;it enim linea AN æqualis arcui quadrantis, & NT radio; </s> <s id="N2328D"><!-- NEW -->igi­<lb/>tur totum triangulum mixtum ex rectis AN, NT, & curua TQH, e&longs;t <lb/>velocitas acqui&longs;ita in toto arcu quadrantis; &longs;it autem A <foreign lang="greek">s</foreign> æqualis lateri <lb/>quadrati in&longs;cripti qua e&longs;t ad AN proximè vt 10. ad 11. e&longs;t enim AB ra­<lb/>dix quad. </s> <s id="N2329D">98. &longs;itque AE &longs;inus rectus quad. </s> <s id="N232A0"><!-- NEW -->45. certè rectangulum NE <lb/>e&longs;t velocitas acqui&longs;ita in chorda A <foreign lang="greek">s</foreign>, &longs;ed hæc e&longs;t æqualis acqui&longs;itæ in <lb/>toto arcu quadrantis AN; </s> <s id="N232AC"><!-- NEW -->igitur rectangulum NE e&longs;t æquale triangulo <lb/>mixto NTOA, denique velocitas acqui&longs;ita in radio A 4. æquali AF, <lb/>e&longs;t vt quadratum 4 F, &longs;ed quadratum 4. F e&longs;t æquale rectangulo BE, vt <lb/>con&longs;tat, nam A <foreign lang="greek">s</foreign> e&longs;t dupla AE; </s> <s id="N232BA"><!-- NEW -->igitur rectangulum e&longs;t &longs;ubduplum qua-<pb pagenum="306" xlink:href="026/01/340.jpg"/>drati &longs;ub A <foreign lang="greek">s</foreign>, &longs;ed quadratum &longs;ub A <foreign lang="greek">s</foreign> e&longs;t duplum quadrati 4 F; </s> <s id="N232CB"><!-- NEW -->igitur <lb/>quadratum 4 F e&longs;t æquale rectangulo <foreign lang="greek">s</foreign> E; igitur & triangulo mixto <lb/>NTQA. </s> </p> <p id="N232D7" type="main"> <s id="N232D9"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N232E5" type="main"> <s id="N232E7"><!-- NEW -->Inde Corollarium cyclometricum deduci pote&longs;t, &longs;cilicet proportio, <lb/>quam habet triangulum mixtum NTQA ad quadrantem, cuius arcus <lb/>æqualis e&longs;t rectæ AN, & radius rectæ AF. v.g. <!-- REMOVE S-->ad quadrantem AFL, <lb/>vel INT, vel LAC; </s> <s id="N232F3"><!-- NEW -->porrò triangulum prædictum e&longs;t maius quadrante <lb/>&longs;ectione ex curua TQA, & rectâ AT; </s> <s id="N232F9"><!-- NEW -->aut certè qui inuenerit triangu­<lb/>lum mixtum KLQ æquale mixto FQ <foreign lang="greek">d</foreign>, habebit rectangulum KF æqua­<lb/>le quadranti AFL; </s> <s id="N23305"><!-- NEW -->& vt res i&longs;ta promoueatur à Geometris: </s> <s id="N23309"><!-- NEW -->dico qua­<lb/>dratum &longs;ub radio e&longs;&longs;e ad &longs;emicirculum, vt triangulum mixtum NTQA <lb/>ad rectangulum NF; </s> <s id="N23311"><!-- NEW -->porrò mixtum FTQA con&longs;tat ex omnibus &longs;inu­<lb/>bus ver&longs;is collectis; </s> <s id="N23317"><!-- NEW -->illud verò ex omnibus &longs;inubus rectis; vt autem in­<lb/>ueniatur illud collectum, accipi debet motus qui cre&longs;cat &longs;ecundum pro­<lb/>portionem &longs;inuum ver&longs;orum v.g. <!-- REMOVE S-->in linea FT, velocitas puncti F e&longs;t vt <lb/>FA, in <foreign lang="greek">q</foreign>, vt <foreign lang="greek">q</foreign> O in <foreign lang="greek">b</foreign>, vt <foreign lang="greek">b</foreign> P, &c. </s> </p> <p id="N23333" type="main"> <s id="N23335"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23341" type="main"> <s id="N23343">Ob&longs;eruabis autem primò lineas tran&longs;uer&longs;as FA, <foreign lang="greek">q</foreign> O, <foreign lang="greek">b</foreign> P, <foreign lang="greek">d</foreign> Q, CR, <lb/>&c. </s> <s id="N23354">e&longs;&longs;e æquales lineis CB <foreign lang="greek">m u</foreign> ZT <foreign lang="greek">w</foreign> S, <foreign lang="greek">u</foreign> R, &c. </s> <s id="N23363">quia BC figura <lb/> quam vocemus <expan abbr="figurã">figuram</expan> primam, e&longs;t æqualis AF, fig. </s> <s id="N2336C">quam vocemus <lb/>&longs;ecundam. </s> <s id="N23371"><!-- NEW -->O <foreign lang="greek">q</foreign> &longs;ecundæ e&longs;t æqualis H <foreign lang="greek">q</foreign>, minùs HO; </s> <s id="N2337D"><!-- NEW -->&longs;ed HO &longs;ecundæ <lb/>e&longs;t æqualis QM primæ, vel BD; </s> <s id="N23383"><!-- NEW -->igitur O <foreign lang="greek">q</foreign> &longs;ecundæ e&longs;t æqualis DC <lb/>primæ; </s> <s id="N2338D"><!-- NEW -->&longs;ed DC e&longs;t æqualis VA, quia VD e&longs;t quadratum, &longs;ed V <foreign lang="greek">m</foreign> e&longs;t <lb/>æqualis VA; </s> <s id="N23397"><!-- NEW -->igitur DC; </s> <s id="N2339B"><!-- NEW -->igitur O <foreign lang="greek">q</foreign> &longs;ecundæ: </s> <s id="N233A3"><!-- NEW -->præterea IP &longs;ecundæ e&longs;t <lb/>æqualis AD, quæ e&longs;t &longs;ubdupla AF; </s> <s id="N233A9"><!-- NEW -->igitur æqualis P <foreign lang="greek">b</foreign>; </s> <s id="N233B1"><!-- NEW -->&longs;ed IP e&longs;t æqua­<lb/>lis BT primæ; </s> <s id="N233B7"><!-- NEW -->igitur BT, cui etiam e&longs;t æqualis TZ; </s> <s id="N233BB"><!-- NEW -->igitur TZ æqualis <lb/>P <foreign lang="greek">b</foreign> &longs;ecundæ: </s> <s id="N233C5"><!-- NEW -->idem dico de aliis tran&longs;uer&longs;is: immò demon&longs;trabimus tom. <lb/></s> <s id="N233CA"><!-- NEW --><expan abbr="&longs;eq.">&longs;eque</expan> quadratricem quadrantis, cuius radius &longs;it NA terminari ad punctum <lb/>T, ita vt NT &longs;it ba&longs;is quadratricis, & NA latus; non tamen propterea <lb/>hæc linea &longs;inuum e&longs;t quadratrix, vt demon&longs;trabimus. </s> </p> <p id="N233D5" type="main"> <s id="N233D7"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N233E3" type="main"> <s id="N233E5"><!-- NEW --><emph type="italics"/>Diuer&longs;æ chordæ acquirunt diuer&longs;am velocitatem pro diuer&longs;a ratione &longs;inuum <lb/>ver&longs;orum &longs;uorum arcuum.<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->velocitas acqui&longs;ita in chorda AM e&longs;t <lb/>æqualis acqui&longs;itæ in &longs;inu ver&longs;o AQ, & acqui&longs;ita in chorda AL æqualis <lb/>acqui&longs;itæ in &longs;inu ver&longs;o AR, atque ita deinceps; donec acqui&longs;ita in AC <lb/>&longs;it æqualis acqui&longs;itæ in &longs;inu toto AB. <!-- KEEP S--></s> </p> <p id="N233FB" type="main"> <s id="N233FD">Itaque in chorda quæ ducitur ab A, velocitas cre&longs;cit vt in &longs;inu ver&longs;o <lb/>eiu&longs;dem.v.g. </s> <s id="N23402">in AM, AL, AK; in chorda verò, quæ ducitur ab aliquo <lb/>puncto arcus AC v&longs;que ad C, cre&longs;cit vt in &longs;inu recto. </s> <s id="N23407"><!-- NEW -->v.g. <!-- REMOVE S-->velocitas ac­<lb/>qui&longs;ita in chorda LC e&longs;t æqualis acqui&longs;itæ in perpendiculari LE, quæ <lb/>e&longs;t &longs;inus rectus arcus LC; item acquiritur æqualis velocitas in duabus <lb/>at que in vna, dum &longs;cilicet communes terminos habeant. </s> <s id="N23413"><!-- NEW -->v.g. <!-- REMOVE S-->in duabus <pb pagenum="307" xlink:href="026/01/341.jpg"/>AKC acquiritur æqualis acqui&longs;itæ in AC; </s> <s id="N2341E"><!-- NEW -->nam in AK, A <foreign lang="greek">w</foreign> acquiritur <lb/>æqualis; </s> <s id="N23426"><!-- NEW -->tùm etiam in KC, <foreign lang="greek">w</foreign> C; Item in tribus acquiritur æqualis ac­<lb/>qui&longs;itæ in duabus, atque ita deinceps. </s> </p> <p id="N23430" type="main"> <s id="N23432">Præterea velocitas acqui&longs;ita in chordis mediis.v.g. </s> <s id="N23435"><!-- NEW -->in chorda LI e&longs;t <lb/>æqualis acqui&longs;itæ in LZ, vel RT, vel in &longs;inu toto AB, minùs &longs;inu ver&longs;o <lb/>arcus LA, & &longs;inu recto arcus IC; &longs;ed hæc &longs;unt &longs;atis facilia. </s> </p> <p id="N2343D" type="main"> <s id="N2343F"><!-- NEW -->Idem dico de chordis arcus quadrantis funependuli AEB figura Lem­<lb/>ma.4. v. <!-- REMOVE S-->g. <!-- REMOVE S-->de chorda IB, in qua velocitas acqui&longs;ita e&longs;t æqualis acqui­<lb/>&longs;itæ in RB, vel in duabus ILB, vel in tribus 4. 5. atque ita deinceps: <lb/>hinc etiam vides in quadrante EB acquiri æqualem velocitatem, &longs;iue <lb/>EA &longs;it perpendicularis deor&longs;um, &longs;iue AB. <!-- KEEP S--></s> </p> <p id="N23450" type="main"> <s id="N23452"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N2345E" type="main"> <s id="N23460"><!-- NEW --><emph type="italics"/>Citiùs de&longs;cendet corpus per duas EIB, quàm per IB<emph.end type="italics"/>; </s> <s id="N23469"><!-- NEW -->quia de&longs;cen&longs;us e&longs;t <lb/>æquè diuturnus per EB, & IB; &longs;ed citiùs de&longs;cendit per EIB, quàm per <lb/>EB, vt iam &longs;uprà dictum e&longs;t in Lem. <!-- REMOVE S-->8. igitur citiùs per EIB, quàm <lb/>per IB. </s> </p> <p id="N23475" type="main"> <s id="N23477"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N23483" type="main"> <s id="N23485"><!-- NEW --><emph type="italics"/>Citiùs de&longs;cendet per<emph.end type="italics"/> <emph type="italics"/>duas chordas BHF, quàm per duas BGF, à quiete<emph.end type="italics"/><lb/>B; </s> <s id="N23495"><!-- NEW -->&longs;int enim duæ BHF, &longs;itque BH. v.g. <!-- REMOVE S-->chorda arcus 30.grad.&longs;c.5 1764. <lb/>earum partium, quarum &longs;inus totus e&longs;t 100000. &longs;it Tangens BE; </s> <s id="N2349D"><!-- NEW -->&longs;it HD <lb/>perpendicularis in BH, & HT in BD; </s> <s id="N234A3"><!-- NEW -->certè HT e&longs;t media proportio­<lb/>nalis inter DT, & TB; </s> <s id="N234A9"><!-- NEW -->e&longs;tque differentia &longs;inus totius, & &longs;inus OH 60. <lb/>grad. <!-- REMOVE S-->e&longs;t autem OH 86603. igitur HT 13397. quadretur HT, produ­<lb/>ctum diuidatur per BT 50000. quotiens dabit TD 3589. quæ &longs;i adda­<lb/>tur BT, habebitur tota BD 53589. quadretur BD; </s> <s id="N234B5"><!-- NEW -->a&longs;&longs;umatur &longs;ubduplum <lb/>quadrati, ex quo extrahatur radix; </s> <s id="N234BB"><!-- NEW -->habebitur KD, vel BK 37893. &longs;it <lb/>autem LF 200000. ad 141422. æqualem BF, ita BF ad LH 100000. <lb/>certè tempus per LH e&longs;t ad tempus per BH, vt LH ad BH; </s> <s id="N234C3"><!-- NEW -->&longs;ed tempus <lb/>per LH e&longs;t ad tempus per LF, vt LH ad 141422.igitur tempus per BH <lb/>e&longs;t ad tempus per HF facto initio motus ex L, vt BH 51764. ad 41422. <lb/>igitur ad tempus per BHF, vt 51764.ad 93186. porrò BH & BK æqua­<lb/>li tempore percurruntur; </s> <s id="N234CF"><!-- NEW -->igitur tempus per BK e&longs;t BH, id e&longs;t 51764. <lb/>cùm autem &longs;patia in eadem linea &longs;int in ratione duplicata temporum; <lb/>certè &longs;patium BK acqui&longs;itum tempore 51764.e&longs;t ad &longs;patium acqui&longs;itum <lb/>in BF tempore 93186. vt quadratum 51764. ad quadratum 93186.id e&longs;t, <lb/>vt 2679511696.ad 8676630576.vnde factâ regulâ trium habeo &longs;patium <lb/>decur&longs;um in BF 122702. tempore 93186. &longs;ed tota BF e&longs;t 141422. igitur <lb/>citiùs percurruntur duæ BHF, quàm BF. </s> </p> <p id="N234DF" type="main"> <s id="N234E1"><!-- NEW -->Præterea &longs;int duæ BGF, BG e&longs;t 100000.&longs;it perpendicularis G 4 cùm <lb/>angulus GB 4.&longs;it grad.30. erit vt 5 G ad GB, ita BG ad B 4. igitur B 4. <lb/>erit 115469. &longs;it 4.3.perpendicularis in BF, quadratum B 4. e&longs;t duplum <lb/>quadrati B 3.igitur B 3. erit 81655. iam verò FN e&longs;t &longs;ecans grad.75. &longs;ci­<lb/>licet 386370.igitur GN e&longs;t 334606. detracta &longs;cilicet FG æquali BH; </s> <s id="N234ED"><!-- NEW -->&longs;it <lb/>autem NG ad 359557. vt hæc ad NF; </s> <s id="N234F3"><!-- NEW -->certè tempus per BG e&longs;t ad tem-<pb pagenum="308" xlink:href="026/01/342.jpg"/>pus per NG, vt BG ad NG, & ad tempus per GF, vt BG ad 24951. & <lb/>ad tempus per BGF, vt BG id e&longs;t, 100000. ad 124951. porrò tempus <lb/>per B 3. e&longs;t BG; </s> <s id="N23500"><!-- NEW -->ergo vt quadratum temporis per BG ad quadratum <lb/>temporis per BGF, &longs;cilicet vt 10000000000. ad 1561475241. ita B 3. <lb/>&longs;cilicet 81655. ad aliam, hæc erit 123496. igitur in BF, quæ e&longs;t partium <lb/>141422. percurruntur partes 123496. eo tempore, quo percurruntur <lb/>BGF; </s> <s id="N2350C"><!-- NEW -->at verò eo tempore, quo percurruntur BHF; </s> <s id="N23510"><!-- NEW -->percurruntur in <lb/>BF 122702. igitur pauciores; </s> <s id="N23516"><!-- NEW -->igitur minore tempore; igitur duæ BHF <lb/>percurruntur minore tempore, quàm duæ BGF, quod erat demon­<lb/>&longs;trandum. </s> </p> <p id="N2351E" type="main"> <s id="N23520"><!-- NEW -->Similiter de&longs;cendet citiùs per duas BHF, quàm per duas BZF: </s> <s id="N23524"><!-- NEW -->immò <lb/>quod mirabile e&longs;t, patetque ex analytica, citiùs per duas BGF, quàm per <lb/>duas BZF; </s> <s id="N2352C"><!-- NEW -->(&longs;uppono enim BZ e&longs;&longs;e arcum grad. <!-- REMOVE S-->45.) &longs;it enim Z <foreign lang="greek">u</foreign> per­<lb/>pendicularis, itemque Z <foreign lang="greek">d, d</foreign> B e&longs;t æqualis BR. igitur 70711. Z <foreign lang="greek">d</foreign> e&longs;t <lb/>29289. igitur <foreign lang="greek">d u</foreign> 1223. igitur B <foreign lang="greek">u</foreign> 71924. igitur B <foreign lang="greek">b</foreign> 51858. iam tempus <lb/>per BZ e&longs;t ad tempus per YZ vt BZ ad YZ. id e&longs;t, vt 76536. ad 184777. <lb/>&longs;it autem vt AYF 261313. ad aliam 219737.ita hæc ad YZ; </s> <s id="N23552"><!-- NEW -->certè tem­<lb/>pus per BZ e&longs;t ad tempus per BZF, vt BZ ad 111496. igitur B <foreign lang="greek">b</foreign> fit <lb/>tempore BZ; ergo vt quadratum BZ ad quadratum 111496. id e&longs;t, vt <lb/>4857759296. ad 12431358016. ita &longs;it B <foreign lang="greek">b</foreign>, id e&longs;t 51858.ad 132708.igitur <lb/>eo tempore, quo percurruntur BZF, percurruntur in BF 132708.earum <lb/>partium, quarum BF e&longs;t 141422. &longs;ed pauciores percurruntur eo tempo­<lb/>re, quo fit de&longs;cen&longs;us per BHF, vel BGF. </s> </p> <p id="N2356A" type="main"> <s id="N2356C"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N23578" type="main"> <s id="N2357A"><emph type="italics"/>Citiùs percurruntur duæ inferiores.v.g. </s> <s id="N2357F"><!-- NEW -->HGF, quàm duæ BHF<emph.end type="italics"/>; </s> <s id="N23586"><!-- NEW -->e&longs;t enim <lb/>PF &longs;ubdupla &longs;ecantis NF; </s> <s id="N2358C"><!-- NEW -->igitur 193185. FG e&longs;t 51764. GP 141421. <lb/>&longs;it autem PG ad 165285.vt hæc ad PF; </s> <s id="N23592"><!-- NEW -->certè tempus per HG e&longs;t ad <lb/>tempus per PG, vt HG ad PG; </s> <s id="N23598"><!-- NEW -->igitur tempus per HG e&longs;t ad tempus <lb/>per HGF, vt 51764. ad 75628. &longs;ed BX e&longs;t æqualis, eiu&longs;demque incli­<lb/>nationis cum HG; </s> <s id="N235A0"><!-- NEW -->igitur tempus, quo percurritur BX e&longs;t BX. vel HG; </s> <s id="N235A4"><!-- NEW --><lb/>&longs;it autem vt BX ad 75628. ita hæc ad aliam 111092. igitur eo tempore, <lb/>quo percurruntur HGF, percurruntur in BF 111092. minor BF; igitur <lb/>citiùs percurruntur HGF quàm BHF, vel BZF, &c. </s> <s id="N235AD">igitur duæ infe­<lb/>riores citiùs, quàm duæ &longs;uperiores. </s> </p> <p id="N235B2" type="main"> <s id="N235B4"><!-- NEW -->Ex his manife&longs;tum e&longs;t, quænam &longs;int qua&longs;i termini progre&longs;&longs;ionis in a&longs;­<lb/>&longs;umptis duabus chordis; &longs;i enim diuidatur arcus BF in 6.arcus æquales, <lb/>BF tardi&longs;&longs;imè, BHF veloci&longs;&longs;imè, &c. </s> <s id="N235BC">po&longs;t BHF, BGF, tùm &longs;ingulæ ab <lb/>H ver&longs;us Z & ver&longs;us V re&longs;pondent &longs;ingulæ immediatè AG ver&longs;us Z, & <lb/>ver&longs;us <foreign lang="greek">q. </foreign></s> </p> <p id="N235C6" type="main"> <s id="N235C8"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N235D4" type="main"> <s id="N235D6"><!-- NEW --><emph type="italics"/>Si &longs;int duo pendula inæqualia, tempora de&longs;cen&longs;uum per chordas &longs;imiles, <lb/>&longs;unt in ratione &longs;ubduplicat a earumdem; </s> <s id="N235DE"><!-- NEW -->hæ verò &longs;unt vt radij<emph.end type="italics"/>; </s> <s id="N235E5"><!-- NEW -->&longs;it enim qua­<lb/>drans A <foreign lang="greek">a r</foreign>, cuius radius A <foreign lang="greek">a</foreign> &longs;it &longs;ubquadruplus radij AB; </s> <s id="N235F3"><!-- NEW -->&longs;int chordæ <lb/>&longs;imiles <foreign lang="greek">a r</foreign>, BF; </s> <s id="N235FD"><!-- NEW -->hæc e&longs;t quadrupla illius; </s> <s id="N23601"><!-- NEW -->igitur cum &longs;it eadem vtriu&longs;-<pb pagenum="309" xlink:href="026/01/343.jpg"/>que inclinatio; </s> <s id="N2360A"><!-- NEW -->eo tempore, quo percurretur tota <foreign lang="greek">a r</foreign> percurretur tan­<lb/>tùm quarta pars BF; </s> <s id="N23614"><!-- NEW -->igitur &longs;uper&longs;unt 1/4 BF; </s> <s id="N23618"><!-- NEW -->&longs;ed &longs;ecundo tempore &longs;en­<lb/>&longs;ibili æquali primo percurritur &longs;patium triplum &longs;patij primi temporis; </s> <s id="N2361E"><!-- NEW --><lb/>igitur tota BF percurritur tempore duplo, & <foreign lang="greek">a r</foreign> &longs;ubduplo; </s> <s id="N23627"><!-- NEW -->igitur tem­<lb/>pora &longs;unt vt radices 1. & 4. igitur in ratione &longs;ubduplicata; </s> <s id="N2362D"><!-- NEW -->præterea &longs;int <lb/>chordæ <foreign lang="greek">a</foreign> X <foreign lang="greek">r</foreign>, & aliæ duæ BZF &longs;imiles prioribus; certè &longs;i prima mino­<lb/>ris quadrantis <foreign lang="greek">a</foreign> X percurratur vno tempore. </s> <s id="N23641"><!-- NEW -->Prima maioris BF, percur­<lb/>ritur duobus temporibus; </s> <s id="N23647"><!-- NEW -->&longs;ed in eadem proportione percurrentur duæ <lb/>X <foreign lang="greek">b</foreign> ZF, vt patet; </s> <s id="N23651"><!-- NEW -->quia vt e&longs;t <foreign lang="greek">w</foreign> X ad X <foreign lang="greek">r</foreign>, ita XZ ad ZF: idem pror&longs;us di­<lb/>co, &longs;i accipiantur tres chordæ, 4.5.6. &c. </s> <s id="N2365F">in vtroque arcu. </s> </p> <p id="N23662" type="main"> <s id="N23664"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23671" type="main"> <s id="N23673"><emph type="italics"/>Vibratio minor eiu&longs;dem, vel æqualis funependuli breuiore tempore percurri­<lb/>tur.<emph.end type="italics"/></s> <s id="N2367C"><!-- NEW --> Probatur quia percurruntur citiùs duæ chordæ inferiores HGF, <lb/>quàm duæ &longs;uperiores quæcunque per Lem. <!-- REMOVE S-->16. immò & tres inferiores, <lb/>quàm tres &longs;uperiores, atque ita deinceps; igitur totus arcus inferior <lb/>HGF, qui con&longs;tat ex his chordis minoribus &longs;emper, & minoribus per­<lb/>curretur citiùs, quàm &longs;uperior, & maior.v.g. </s> <s id="N2368A">BHF. </s> </p> <p id="N2368D" type="main"> <s id="N2368F"><!-- NEW -->Adde quod, multis con&longs;tat experimentis minorem vibrationem citiùs <lb/>peragi, quod plu&longs;quam centies à me probatum e&longs;t; </s> <s id="N23695"><!-- NEW -->&longs;i enim &longs;imul demit­<lb/>tantur duo funependula æqualia; </s> <s id="N2369B"><!-- NEW -->alterum quidem è &longs;ummo quadrantis <lb/>puncto, alterum ex decimo, vel decimoquinto altitudinis gradu, appo&longs;ito <lb/>in puncto quietis aliquo &longs;onoro corpore; </s> <s id="N236A3"><!-- NEW -->haud dubiè ictum, qui &longs;equitur <lb/>ex minori vibratione, priùs audies; </s> <s id="N236A9"><!-- NEW -->tùm &longs;tatim alium; </s> <s id="N236AD"><!-- NEW -->immò &longs;i numeren­<lb/>tur vibrationes vtriu&longs;que eodem tempore plures minoris, maioris verò <lb/>pauciores numerabuntur; </s> <s id="N236B5"><!-- NEW -->&longs;æpiùs numeraui 11.minores eo tantùm tem­<lb/>pore, quo alter, qui mecum erat 10. maiores numerabat, & 40. circiter <lb/>minores dum alter 37.maiores recen&longs;eret; </s> <s id="N236BD"><!-- NEW -->& certè &longs;i vibratio vtraque <lb/>maior &longs;cilicet, & minor per <expan abbr="eũdem">eundem</expan> arcum recurreret, centum minores <lb/>eo ferè tempore agerentur, quo 90.maiores; licèt enim vtraque decre&longs;­<lb/>cat, maior tamen decre&longs;cit in maiore proportione, quàm minor, cuius <lb/>rei rationem afferemus infrà. </s> </p> <p id="N236CD" type="main"> <s id="N236CF">Nec e&longs;t quod aliquis cum Galileo, Baliano, & aliis opponat, omnes <lb/>vibrationes, &longs;iue maiores &longs;int, &longs;iue minores e&longs;&longs;e æquè diuturnas, idque <lb/>manife&longs;tis con&longs;tare experimentis, quibus ego alia certi&longs;&longs;ima experimen­<lb/>ta oppono, quibus etiam vltrò a&longs;&longs;entitur P. Mer&longs;ennus, Galileo alioqui <lb/>addicti&longs;&longs;imus, in ver&longs;ione eiu&longs;dem Galilei lib. 1. art. </s> <s id="N236DA"><!-- NEW -->18. & verò docti <lb/>omnes Galileo &longs;unt addicti&longs;simi; </s> <s id="N236E0"><!-- NEW -->in qua verò proportione minor vibra­<lb/>tio breuiore tempore peragatur, quàm major, difficilè dictu e&longs;t, & vix <lb/>determinari pote&longs;t, ni&longs;i fortè dicatur in ea proportione arcum HF citiùs <lb/>percurri, quàm arcum BHF, in qua duæ chordæ HGF citiùs percur­<lb/>runtur, quàm duæ BZF; &longs;ed de his fusè aliàs. </s> </p> <p id="N236EC" type="main"> <s id="N236EE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N236FB" type="main"> <s id="N236FD"><!-- NEW --><emph type="italics"/>Velocitates acqui&longs;ita in vibrationibus inæqualibus &longs;unt vt altitudines<emph.end type="italics"/>; </s> <s id="N23706"><!-- NEW -->&longs;int <lb/>enim vibrationes duæ BF, HF; </s> <s id="N2370C"><!-- NEW -->dico velocitatem acqui&longs;itam in de&longs;cen-<pb pagenum="310" xlink:href="026/01/344.jpg"/>&longs;u BF e&longs;&longs;e ad acqui&longs;itam in de&longs;cen&longs;u HP, vt vecta AF ad rectam OF, <lb/>quod facilè probatur; </s> <s id="N23717"><!-- NEW -->quia ex B in F æqualis acquiritur velocitas &longs;iue <lb/>per rectam BF <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> mobile, &longs;iue per duas BHF, &longs;iue per tres BHGF, <lb/>&longs;iue per totum quadrantem BHF; </s> <s id="N23723"><!-- NEW -->&longs;ed æqualis e&longs;t acqui&longs;ita per BF ac­<lb/>qui&longs;itæ per AF, vel BE; </s> <s id="N23729"><!-- NEW -->quæ omnia con&longs;tant per Lemm.10.& 11.&longs;imili­<lb/>ter acqui&longs;ita in recta HF e&longs;t æqualis acqui&longs;itæ in recta OF in duabus <lb/>HGF; </s> <s id="N23731"><!-- NEW -->immò & in arcu HZF; </s> <s id="N23735"><!-- NEW -->igitur acqui&longs;ita in arcu BHF e&longs;t ad <lb/>acqui&longs;itam in arcu HZF, vt acqui&longs;ita in AF ad acqui&longs;itam in OF; </s> <s id="N2373B"><!-- NEW -->&longs;ed <lb/>illa e&longs;t ad hanc vt AF ad OF, vt con&longs;tat; igitur &longs;unt vt altitudines, quod <lb/>erat probandum. </s> </p> <p id="N23743" type="main"> <s id="N23745"><!-- NEW -->Hinc non &longs;unt vt chordæ, neque vt arcus; </s> <s id="N23749"><!-- NEW -->hinc acqui&longs;ita in arcu <lb/>BHF e&longs;t dupla acqui&longs;itæ in arcu HZF; </s> <s id="N2374F"><!-- NEW -->cùm tamen arcus BF non &longs;it <lb/>duplus; &longs;ed &longs;e&longs;quialter arcus HZF. </s> </p> <p id="N23755" type="main"> <s id="N23757"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N23763" type="main"> <s id="N23765"><!-- NEW --><emph type="italics"/>Hinc &longs;unt diuer&longs;i ictus inæqualium vibrationum in eadem altitudinum ra­<lb/>tione<emph.end type="italics"/>; </s> <s id="N23770"><!-- NEW -->quia eadem e&longs;t ratio ictuum, quæ velocitatum acqui&longs;itarum in <lb/>puncto percu&longs;sionis; </s> <s id="N23776"><!-- NEW -->&longs;ed ratio velocitatum e&longs;t eadem quæ altitudinum, <lb/>&longs;eu perpendicularium per Th.7. igitur eadem ratio ictuum, quæ altitu­<lb/>dinum; </s> <s id="N2377E"><!-- NEW -->&longs;ed inæqualium vibrationum eiu&longs;dem funependuli diuer&longs;æ &longs;unt <lb/>altitudines; igitur diuer&longs;i ictus, quod erat demon&longs;trandum. </s> </p> <p id="N23784" type="main"> <s id="N23786"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N23792" type="main"> <s id="N23794"><!-- NEW --><emph type="italics"/>In diuer&longs;is funependulis &longs;imilium vibrationum velocitates &longs;unt vt chordæ<emph.end type="italics"/>; </s> <s id="N2379D"><!-- NEW --><lb/>&longs;int enim duo funependula inæqualis A <foreign lang="greek">r</foreign>, AF; </s> <s id="N237A6"><!-- NEW -->certè &longs;it vibratio maio­<lb/>ris BF, & minoris vibratio &longs;imilis <foreign lang="greek">a r</foreign>, velocitas vibrationis BF e&longs;t vt al­<lb/>titudo AF & minoris <foreign lang="greek">a r</foreign>, vt altitudo A <foreign lang="greek">r</foreign>; </s> <s id="N237BA"><!-- NEW -->&longs;ed vt AF e&longs;t ad A <foreign lang="greek">r</foreign>, ita BF <lb/>ad <foreign lang="greek">a r</foreign>; </s> <s id="N237C8"><!-- NEW -->&longs;unt enim triangula proportionalia; </s> <s id="N237CC"><!-- NEW -->idem dico de aliis.v.g ZF <lb/>& X <foreign lang="greek">r</foreign>, iu quo non e&longs;t difficultas: hinc percu&longs;siones vtriu&longs;que erunt etiam <lb/>vt chordæ, quia &longs;unt vt altitudines. </s> </p> <p id="N237D8" type="main"> <s id="N237DA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N237E6" type="main"> <s id="N237E8"><!-- NEW --><emph type="italics"/>Tempora, quibus peraguntur vibrationes &longs;imiles funependulorum inæqua­<lb/>lium &longs;unt ferè in ratione &longs;ubduplicata longitudinum, &longs;eu radiorum<emph.end type="italics"/>: </s> <s id="N237F3"><!-- NEW -->Probatur, <lb/>quia tempora de&longs;cen&longs;uum per chordas &longs;imiles &longs;unt in ratione &longs;ubdupli­<lb/>cata earumdem chordarum, &longs;iue &longs;int 2.&longs;iue &longs;int tres, & per Lemma 17. <lb/>&longs;ed &longs;i accipiantur plures chordæ, tandem habebitur arcus; </s> <s id="N237FD"><!-- NEW -->igitur vibra­<lb/>tio per arcum e&longs;t veluti de&longs;cen&longs;us per infinitas ferè chordas æquales; </s> <s id="N23803"><!-- NEW -->&longs;ed <lb/>tempora horum de&longs;cen&longs;uum &longs;unt in ratione &longs;ubduplicata chordarum; </s> <s id="N23809"><!-- NEW -->& <lb/>hæc e&longs;t eadem ratio cum &longs;ubduplicata radiorum; igitur tempora vibra­<lb/>tionum &longs;imilium &longs;unt ferè in ratione &longs;ubduplicata radiorum. </s> </p> <p id="N23811" type="main"> <s id="N23813"><!-- NEW -->Ob&longs;eruabis rem <expan abbr="i&longs;tã">i&longs;tam</expan> accuratè, & analyticè di&longs;cuti po&longs;&longs;e, &longs;it enim qua­<lb/>drans ADH maioris vibrationis, & quadrans CED minoris; </s> <s id="N2381D"><!-- NEW -->&longs;itque <lb/>CD &longs;ubquadrupla AD, & arcus DE &longs;ubquadruplus DKH; </s> <s id="N23823"><!-- NEW -->a&longs;&longs;umatur <lb/>DN &longs;ubquadruplus DH; </s> <s id="N23829"><!-- NEW -->&longs;itque DN æqualis DE; </s> <s id="N2382D"><!-- NEW -->certè eo tempore, <pb pagenum="311" xlink:href="026/01/345.jpg"/>quo percurretur DE, percurretur plu&longs;quam DN; </s> <s id="N23836"><!-- NEW -->quippe DN e&longs;t minùs <lb/>inclinatus, quàm DE: </s> <s id="N2383C"><!-- NEW -->porrò recta NH eodem deinde tempore percur­<lb/>retur, &longs;iue ducatur initium motus AD per arcum DN, &longs;iue AD per re­<lb/>ctam DN, &longs;iue ab O per rectam ON; quia in N e&longs;t æqualis velocitas <lb/>per Lemm. </s> <s id="N23846"><!-- NEW -->11. igitur tempus, quo percurritur recta NH, facto initio <lb/>motus ex D per rectam, vel arcum DN, e&longs;t ad tempus, quo percurritur <lb/>DN, vt 42466.ad DN, id e&longs;t ad 390181. &longs;it enim vt ON ad 111347. <lb/>ita hæc ad OH 179995. detrahatur ON ex 111347.&longs;upere&longs;t 42466.igi­<lb/>tur eo tempore, quo percurritur DE, percurritur plu&longs;quam DN; </s> <s id="N23852"><!-- NEW -->per­<lb/>curritur tamen minùs, quàm DL; </s> <s id="N23858"><!-- NEW -->quia tempus, quo percurritur DL e&longs;t <lb/>ad tempus quo percurritur LH facto initio motus in D, vt DL 51764. <lb/>ad 41422. igitur eo tempore, quo percurritur DE; percurritur minùs <lb/>quàm DL. </s> </p> <p id="N23862" type="main"> <s id="N23864"><!-- NEW -->Adde quod rectæ DE, DM, æquali tempore percurruntur; </s> <s id="N23868"><!-- NEW -->&longs;ed DM <lb/>breuiore tempore percurritur, quàm arcus DL, immò arcus DE citiùs <lb/>peragitur, quàm recta DE; </s> <s id="N23870"><!-- NEW -->igitur citiùs quàm arcus DL; </s> <s id="N23874"><!-- NEW -->&longs;i verò acci­<lb/>piatur arcus DR; </s> <s id="N2387A"><!-- NEW -->certè tempus per arcum DE e&longs;t paulò minus tempo­<lb/>re per arcum DR; quia tempus, quo percurritur DR e&longs;t ad tempus, quo <lb/>percurretur RH, facto initio motus in D, vt 45444.ad 41705.&longs;ed vtrum­<lb/>que tempus debet e&longs;&longs;e æquale, vt &longs;cilicet arcus in DH æquali tempore <lb/>cum arcu DE percurratur. </s> </p> <p id="N23886" type="main"> <s id="N23888"><!-- NEW -->Ob&longs;eruabis præterea, vt inueniatur arcus quadrantis DH, cuius tem­<lb/>pus &longs;it &longs;ubduplum ip&longs;ius quadrantis, vel æquale tempori per arcum DE, <lb/>a&longs;&longs;umendum e&longs;&longs;e punctum in arcu DH, puta N; </s> <s id="N23890"><!-- NEW -->per quod &longs;i ducatur <lb/>HNO, &longs;itque vt ON ad OV, ita OV ad OH, ip&longs;a NV erit æqualis <lb/>ip&longs;i ND; </s> <s id="N23898"><!-- NEW -->quippè tempus per DN e&longs;t ad tempus per ON, vt ip&longs;a DN ad <lb/>ON; </s> <s id="N2389E"><!-- NEW -->&longs;ed tempus per ON e&longs;t ad tempus per NH, vt ON ad NV; </s> <s id="N238A2"><!-- NEW -->igi­<lb/>tur tempus per DN e&longs;t ad tempus per NH, vt DN ad NV; </s> <s id="N238A8"><!-- NEW -->igitur DN, <lb/>& NH facto initio motus à D fiunt tempore æquali; </s> <s id="N238AE"><!-- NEW -->&longs;ed vt tempus per <lb/>rectam DN ad tempus per rectam NH; </s> <s id="N238B4"><!-- NEW -->ita tempus per duas DXN ad <lb/>tempus per duas NZH; </s> <s id="N238BA"><!-- NEW -->ita tempus per 4. æquales in&longs;criptas arcui DN <lb/>ad tempus per 4.æquales in&longs;criptas arcui NZH, atque ita deinceps; igi­<lb/>tur ita tempus per arcum DN ad tempus per arcum NZH. </s> </p> <p id="N238C2" type="main"> <s id="N238C4"><!-- NEW -->Quomodo verò po&longs;&longs;it inueniri punctum N, viderint Geometræ; </s> <s id="N238C8"><!-- NEW -->nec <lb/>enim phy&longs;ici e&longs;t in&longs;tituti; habetur autem ex analytica, &longs;i excipiatur ar­<lb/>cus DN. 24. gra. </s> <s id="N238D0">20′. </s> <s id="N238D3">circiter; &longs;itque HO &longs;ecans anguli AHO grad.57. <lb/>10′. </s> <s id="N238D8"><!-- NEW -->&longs;itque ON, ad OV vt OV ad OH, ip&longs;a NV erit proximè æqualis <lb/>ip&longs;i ND: igitur DN. & NH æqualibus temporibus percurrentur. </s> <s id="N238DE">Simili­<lb/>ter opera eiu&longs;dem analyticæ habebitur arcus, qui peragitur in DZH eo <lb/>tempore, quo arcus DNF percurritur, po&longs;&longs;untque hæc omnia in cano­<lb/>nes redigi. </s> </p> <p id="N238E7" type="main"> <s id="N238E9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N238F5" type="main"> <s id="N238F7"><emph type="italics"/>In diuer&longs;is punctis arcus diuer&longs;us impetus producitur.<emph.end type="italics"/></s> <s id="N238FE"> Prob. </s> <s id="N23901"><!-- NEW -->&longs;it enim <lb/>pendulum fune ex centro immobili A; </s> <s id="N23907"><!-- NEW -->&longs;itque AO horizontalis, AD <pb pagenum="312" xlink:href="026/01/346.jpg"/>perpendicularis; </s> <s id="N23910"><!-- NEW -->haud dubiè producit maiorem impetum in O, quàm in <lb/>LH quippè in D nullo modo grauitat in &longs;uppo&longs;itam manum, in H mi­<lb/>nùs grauitat, in O maximè; &longs;ed qua proportione plùs, vel minùs graui­<lb/>tat, producit maiorem vel minorem impetum, vt patet. </s> </p> <p id="N2391A" type="main"> <s id="N2391C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N23928" type="main"> <s id="N2392A"><emph type="italics"/>Impetus, quem producit in H, e&longs;t ad impetum, quem producit in O, vt HC <lb/>ad DA vel OA.<emph.end type="italics"/></s> <s id="N23933"><!-- NEW --> Probatur, quia grauitatio in H e&longs;t ad grauitationem in <lb/>O, vt CH ad DA, vt demon&longs;tratum e&longs;t &longs;uprà lib. de motu in planis in­<lb/>clinatis; </s> <s id="N2393B"><!-- NEW -->ratio e&longs;t, quia in ea proportione maior e&longs;t, vel minor grauita­<lb/>tio, in qua plùs vel minùs impeditur; </s> <s id="N23941"><!-- NEW -->atqui in O non impeditur; </s> <s id="N23945"><!-- NEW -->quia li­<lb/>nea determinationis ad motum e&longs;t eadem cum linea grauitationis; </s> <s id="N2394B"><!-- NEW -->quip­<lb/>pè globus O grauitat per <expan abbr="Oq;">Oque</expan> &longs;ed OQ e&longs;t Tangens puncti O; </s> <s id="N23955"><!-- NEW -->igitur e&longs;t <lb/>linea determinationis in puncto O; </s> <s id="N2395B"><!-- NEW -->igitur linea determinationis in pun­<lb/>cto O e&longs;t eadem cum linea grauitationis; at verò in H linea grauitatio­<lb/>nis e&longs;t HG, & determinationis HF diuer&longs;a à priore, &longs;ed de his iam plu­<lb/>ra aliàs. </s> </p> <p id="N23965" type="main"> <s id="N23967"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23973" type="main"> <s id="N23975">Ob&longs;eruabis globum prædictum in H diuer&longs;imode po&longs;&longs;e &longs;u&longs;tineri. </s> <s id="N23978">Pri­<lb/>mò, per Tangentem HI. <!-- KEEP S--></s> <s id="N2397E">Secundò applicata potentia in F per FH. Tertiò, <lb/>per horizontalem HV tracto &longs;cilicet fune. </s> <s id="N23983">Quartò, per HK. Quintò, per <lb/>GH. </s> <s id="N23988"><!-- NEW -->Sextò denique in aliis punctis intermediis applicari pote&longs;t poten­<lb/>tia; </s> <s id="N2398E"><!-- NEW -->&longs;i primo modo, & &longs;ecundo potentia &longs;u&longs;tinens pondus in H e&longs;t ad <lb/>&longs;u&longs;tinentem in D ex A vel in O ex Q vt HC ad DA vel HA; </s> <s id="N23994"><!-- NEW -->ad &longs;u&longs;ti­<lb/>nentem verò ex A in H, vt CH ad CA, &longs;i tertio per HV potentia ap­<lb/>plicata in V e&longs;t ad applicatam in A, dum vtraque &longs;imul agat vt HC ad <lb/>HA; </s> <s id="N2399E"><!-- NEW -->&longs;i quarto modo applicata in K æqualis e&longs;t applicatæ in A, itemque <lb/>applicata in Y per YH, vel in O per OH, po&longs;ita HZ æquali HA; </s> <s id="N239A4"><!-- NEW -->&longs;i <lb/>quinto modo applicata in G per GHS &longs;u&longs;tinet totum pondus, itemque <lb/>applicata in S per SH; &longs;i denique &longs;exto modo, pro rata. </s> </p> <p id="N239AC" type="main"> <s id="N239AE"><!-- NEW -->Ob&longs;eruabis &longs;ecundò rem omninò &longs;citu digni&longs;&longs;imam, e&longs;&longs;e duas tantùm <lb/>lineas, quibus applicata potentia totum pondus &longs;u&longs;tinet, &longs;cilicet GH, HS, <lb/>e&longs;&longs;e quoque duas quibus applicata potentia pondus pendulum &longs;u&longs;tinens <lb/>in dato puncto puta H, habet minimam rationem, quæ haberi po&longs;&longs;it ad <lb/>potentiam applicatam in A per AH; &longs;unt autem illæ CH, HV, quæ e&longs;t <lb/>ip&longs;a horizontalis. </s> </p> <p id="N239BC" type="main"> <s id="N239BE"><!-- NEW -->Ob&longs;eruabis tertiò, applicatam in puncto C per CH e&longs;&longs;e minimam <lb/>earum omnium, quæ cum alia applicata in A per HA pendulum pondus <lb/>&longs;u&longs;tinere po&longs;&longs;it; </s> <s id="N239C6"><!-- NEW -->aliàs verò hinc inde applicatas e&longs;&longs;e maiores, v.g. <!-- REMOVE S-->applica­<lb/>tam in E per EH e&longs;&longs;e ad applicatam in A per HA, vt EH ad HA; </s> <s id="N239CE"><!-- NEW -->appli­<lb/>catam verò in Z e&longs;&longs;e ad <expan abbr="eãdem">eandem</expan> vt ZH ad HA; applicatam in T vt <lb/>TH ad HA, &c. </s> <s id="N239DA"><!-- NEW -->&longs;unt autem 4.æquales exceptis maxima, quæ totum pon­<lb/>dus &longs;u&longs;tinet per lineas HS GH, & minimâ, quæ cum applicata in A mi­<lb/>nimis viribus &longs;u&longs;tinet, per lineas CH HV; </s> <s id="N239E2"><!-- NEW -->&longs;i verò a&longs;&longs;umantur quæcum­<lb/>que aliæ lineæ, &longs;unt 4. æquales v.g. <!-- REMOVE S-->accipiatur EH, &longs;it HB ip&longs;i æqualis <pb pagenum="313" xlink:href="026/01/347.jpg"/>producta per H ad X; </s> <s id="N239EF"><!-- NEW -->erunt haud dubiè 4.lineæ, quibus eadem applica­<lb/>ta potentia cum altera in A &longs;u&longs;tinebit pondus, &longs;cilicet HE & oppo&longs;ita <lb/>HI, HB cum oppo&longs;ita HX, &longs;uppono enim HB e&longs;&longs;e æqualem HE, & BH <lb/>pellere ver&longs;us H: quæ omnia certè ob&longs;erua&longs;&longs;e non piget, præ&longs;ertim cùm <lb/>tota res i&longs;ta iucunda iuxta, atque vtilis e&longs;&longs;e videatur. </s> </p> <p id="N239FB" type="main"> <s id="N239FD"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23A0A" type="main"> <s id="N23A0C"><!-- NEW -->Colligo primò ex his determinationem impetus producti in puncto <lb/>O e&longs;&longs;e omninò &longs;implicem à propria &longs;cilicet ponderis penduli grauitatio­<lb/>ne, nec quidquam facere potentiam applicatam in A; </s> <s id="N23A14"><!-- NEW -->quippe impetus <lb/>determinatur ad Tangentem OQ, quæ e&longs;t eadem cum linea grauitatio­<lb/>nis; vnde reuerâ &longs;u&longs;tinetur totum pondus in O. <!-- KEEP S--></s> </p> <p id="N23A1D" type="main"> <s id="N23A1F"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23A2C" type="main"> <s id="N23A2E"><!-- NEW -->Secundò, &longs;i pondus &longs;it in D, e&longs;t determinatio mixta vtraque æqualis, <lb/>nam neque potentia retinens in A e&longs;t maior potentia grauitationis in­<lb/>clinantis deor&longs;um; </s> <s id="N23A36"><!-- NEW -->alioquin &longs;i maior e&longs;&longs;et, præualeret; </s> <s id="N23A3A"><!-- NEW -->igitur mobile fer­<lb/>retur ver&longs;us A; </s> <s id="N23A40"><!-- NEW -->cùm tamen quie&longs;cat in D, nec etiam maior e&longs;t potentia <lb/>grauitationis; </s> <s id="N23A46"><!-- NEW -->alioqui pondus ferretur deor&longs;um, nec dicas nullam e&longs;&longs;e <lb/>potentiam applicatam in A; </s> <s id="N23A4C"><!-- NEW -->nam reuerâ, &longs;i quis ex puncto A &longs;u&longs;tinet <lb/>pendulum pondus, maximè defatigatur, & maximè agit eius potentia mo­<lb/>trix; quomodo verò &longs;u&longs;tineantur pondera, dicemus lib. 10. </s> </p> <p id="N23A54" type="main"> <s id="N23A56"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23A63" type="main"> <s id="N23A65"><!-- NEW -->Tertiò, &longs;i pondus &longs;it in H vel in L e&longs;t determinatio mixta ex duabus <lb/>inæqualibus, ita vt determinatio potentiæ, quæ e&longs;t applicata in A &longs;it mi­<lb/>nor determinatione, quæ e&longs;t à grauitatione ponderis; </s> <s id="N23A6D"><!-- NEW -->&longs;it enim pondus in <lb/>H, &longs;itque determinatio altera per lineam HA, altera per lineam HG; </s> <s id="N23A73"><!-- NEW -->&longs;i <lb/>vtraque æqualis e&longs;t, linea determinationis mixtæ non e&longs;&longs;et Tangens HF; </s> <s id="N23A79"><!-- NEW --><lb/>nec enim angulus AHG diuidit æqualiter bifariam ip&longs;am HF; atqui <lb/>cum vtraque determinatio e&longs;t æqualis, po&longs;ita quod vtraque linea faciat <lb/>angulum, linea nouæ determinationis facit angulum vtrimque æqualem, <lb/>vt demon&longs;trauimus &longs;uprà. </s> </p> <p id="N23A85" type="main"> <s id="N23A87"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23A94" type="main"> <s id="N23A96"><!-- NEW -->Quartò hinc colligo, determinationem, quæ e&longs;t à potentia applicata <lb/>in A cre&longs;cere continuè ab O ad D, ita vt in O &longs;it nulla, in D &longs;it maxima, <lb/>id e&longs;t æqualis alteri determinationi propriæ grauitationis; </s> <s id="N23A9E"><!-- NEW -->in reliquis ve­<lb/>rò punctis prima e&longs;t ad &longs;ecundam, vt &longs;inus rectus &longs;uperioris arcus ad &longs;i­<lb/>num totum, v.g.&longs;i pondus &longs;it in L, determinatio grauitationis e&longs;t ad aliam <lb/>vt LA ad LR, &longs;i &longs;it in H vt HA ad HS, &longs;i &longs;it in O vt OA ad nihil; </s> <s id="N23AA8"><!-- NEW -->&longs;i <lb/>&longs;it in D vt DA ad DA; idem dico de omnibus aliis punctis inter­<lb/>mediis. </s> </p> <p id="N23AB0" type="main"> <s id="N23AB2"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23ABF" type="main"> <s id="N23AC1">Quintò colligo, impetum grauitationis productum in &longs;ingulis pun­<lb/>ctis e&longs;&longs;e ad impetum productum in O, id e&longs;t ad maximum, qui po&longs;&longs;it <pb pagenum="314" xlink:href="026/01/348.jpg"/>produci </s> <s id="N23ACB"><!-- NEW -->vno in&longs;tanti ab ip&longs;o corpore grani, vt &longs;inum rectum arcus infe­<lb/>rioris ad &longs;inum totum; </s> <s id="N23AD1"><!-- NEW -->&longs;it enim pondus in L, impetus productus in L <lb/>e&longs;t ad productum in O, vt &longs;inus BL ad LA; </s> <s id="N23AD7"><!-- NEW -->&longs;it in H, vt &longs;inus HC ad <lb/>HA; </s> <s id="N23ADD"><!-- NEW -->&longs;it in O vt OA ad OA, &longs;it in D vt nihil ad DA: </s> <s id="N23AE1"><!-- NEW -->hinc vides con­<lb/>trarias vices impetus producti in &longs;ingulis punctis, & determinationis, <lb/>quæ e&longs;t à potentia applicata in A; </s> <s id="N23AE9"><!-- NEW -->quippè ille continuò imminuitur ab <lb/>O ad D; </s> <s id="N23AEF"><!-- NEW -->hæc verò continuo cre&longs;cit; </s> <s id="N23AF3"><!-- NEW -->ille totus e&longs;t in O nullus in D; </s> <s id="N23AF7"><!-- NEW -->hæc <lb/>tota in D, nulla in O; </s> <s id="N23AFD"><!-- NEW -->ille e&longs;t ad totum, vt &longs;inus arcus inferioris ad &longs;i­<lb/>num totum; hæc verò e&longs;t ad totam, &longs;eu maximam, vt &longs;inus arcus &longs;uperio­<lb/>ris ad &longs;inum totum. </s> </p> <p id="N23B05" type="main"> <s id="N23B07"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23B14" type="main"> <s id="N23B16"><!-- NEW -->Sextò, hinc colligo rationem à priori huius imminutionis impetus; </s> <s id="N23B1A"><!-- NEW --><lb/>cum enim impetus de&longs;truatur ne &longs;it fru&longs;trà; </s> <s id="N23B1F"><!-- NEW -->certè propter <expan abbr="eãdem">eandem</expan> ratio­<lb/>nem non producitur, ne &longs;cilicet &longs;it fru&longs;trà; </s> <s id="N23B29"><!-- NEW -->cùm enim impetus &longs;it vt mo­<lb/>tus, &longs;it mobile in L cum duplici determinatione alteram per lineam LA <lb/>alteram L <foreign lang="greek">d</foreign>; </s> <s id="N23B35"><!-- NEW -->&longs;it autem hæc ad illam vt LA ad LR, vel vt L <foreign lang="greek">d</foreign> æqualis <lb/>LA ad L <foreign lang="greek">b</foreign> æqualem LR, &longs;itque arcus LO grad. <!-- REMOVE S-->30. LR e&longs;t &longs;ubdupla <lb/>LA; </s> <s id="N23B47"><!-- NEW -->&longs;it <foreign lang="greek">b u</foreign> æqualis L <foreign lang="greek">d</foreign>, ip&longs;ique parallela, & <foreign lang="greek">u d</foreign> æqualis L <foreign lang="greek">b</foreign> & paralle­<lb/>la; </s> <s id="N23B5D"><!-- NEW -->certè hoc po&longs;ito, motus erit per L <foreign lang="greek">u</foreign>, &longs;cilicet per diagonalem, vt &longs;æ­<lb/>piùs &longs;uprà demon&longs;trauimus; </s> <s id="N23B67"><!-- NEW -->igitur &longs;i tantùm e&longs;&longs;et determinatio L <foreign lang="greek">d</foreign> mo­<lb/>tus e&longs;&longs;et L <foreign lang="greek">d</foreign>; </s> <s id="N23B75"><!-- NEW -->&longs;i verò conjungatur determinatio L <foreign lang="greek">b</foreign>, motus erit L <foreign lang="greek">u</foreign>; </s> <s id="N23B81"><!-- NEW -->&longs;ed <lb/>impetus e&longs;t vt motus; </s> <s id="N23B87"><!-- NEW -->igitur impetus L <foreign lang="greek">d</foreign>, cum vtraque determinatione <lb/>conjunctus non haberet totum &longs;uum effectum, id e&longs;t motum L <foreign lang="greek">d</foreign>; </s> <s id="N23B95"><!-- NEW -->igitur <lb/>aliquid illius e&longs;t fru&longs;trà; </s> <s id="N23B9B"><!-- NEW -->igitur producitur tantùm impetus vt L <foreign lang="greek">u</foreign>; </s> <s id="N23BA3"><!-- NEW -->&longs;ed <lb/>vt L <foreign lang="greek">u</foreign> ad L <foreign lang="greek">d</foreign>, ita LB ad LA; nam triangula L <foreign lang="greek">u d</foreign>, & BLA &longs;unt æqua­<lb/>lia, & æquiangula, vt patet. </s> </p> <p id="N23BB7" type="main"> <s id="N23BB9"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N23BC5" type="main"> <s id="N23BC7"><!-- NEW -->Septimò colligo, &longs;ingulis in&longs;tantibus mutari determinationem quæ e&longs;t <lb/>ab A, & con&longs;equenter determinationem mixtam, ip&longs;amque acce&longs;&longs;ionem <lb/>impetus noui: </s> <s id="N23BCF"><!-- NEW -->hinc etiam rectè explicatur, in quo po&longs;itum &longs;it illud impe­<lb/>dimentum ratione cuius corpus rectà deor&longs;um non tendit; quippè in <lb/>eo tantùm po&longs;itum e&longs;t, quod &longs;it noua determinatio, idem dico de re&longs;i­<lb/>&longs;tentia. </s> </p> <p id="N23BD9" type="main"> <s id="N23BDB">Ob&longs;eruabis autem idem præ&longs;tare funem affixum in A ratione conti­<lb/>nuitatis, & vnionis &longs;uarum partium, quod præ&longs;taret potentia in A fune <lb/>ip&longs;o trahens, vt con&longs;tat, &longs;eu pondus contranitens ex rotula appen&longs;um. </s> </p> <p id="N23BE2" type="main"> <s id="N23BE4"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N23BF0" type="main"> <s id="N23BF2"><!-- NEW -->Octauò colligo, cre&longs;cere impedimentum ab O in D in ratione &longs;i­<lb/>nuum ver&longs;orum arcus &longs;uperioris; </s> <s id="N23BF8"><!-- NEW -->cùm enim in L v. <!-- REMOVE S-->g. <!-- REMOVE S-->motus &longs;it ad mo­<lb/>tum liberum in O vt L <foreign lang="greek">u</foreign> ad L <foreign lang="greek">d</foreign> vel vt LB ad LA, impeditur motus vt <lb/>RO; </s> <s id="N23C0C"><!-- NEW -->nam motus, vel impetus in L e&longs;t minor impetu in O, differentia <lb/>vtriu&longs;que RO, &longs;ed RO e&longs;t &longs;inus ver&longs;us arcus OL; idem dico de <lb/>reliquis. </s> </p> <pb pagenum="315" xlink:href="026/01/349.jpg"/> <p id="N23C18" type="main"> <s id="N23C1A"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N23C26" type="main"> <s id="N23C28"><!-- NEW -->Nonò colligo hoc impedimentum facere quidem, ne tantus impetus <lb/>nouus accidat, non tamen facere vt productus antè pereat; </s> <s id="N23C2E"><!-- NEW -->quippe ni­<lb/>hil impetus antè producti de&longs;truitur per &longs;e; </s> <s id="N23C34"><!-- NEW -->licèt determinatio noua per <lb/>Tangentem nouam accedat in &longs;ingulis punctis; </s> <s id="N23C3A"><!-- NEW -->nihil tamen impetus e&longs;t <lb/>fru&longs;trà; </s> <s id="N23C40"><!-- NEW -->vt in reflexione dictum e&longs;t, adde quod determinatio prior, nihil <lb/>pror&longs;us confert; </s> <s id="N23C46"><!-- NEW -->quia tota impeditun à potentia retinente in A immo­<lb/>biliter; dixi per &longs;e, quia per accidens propter aliquam ten&longs;ionem chor­<lb/>dæ pote&longs;t aliquid de&longs;trui, quæ ten&longs;io e&longs;t pror&longs;us per accidens. </s> </p> <p id="N23C4E" type="main"> <s id="N23C50"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N23C5C" type="main"> <s id="N23C5E"><!-- NEW -->Decimò colligo inde reddi rationem à priori, cur ille motus vibra­<lb/>tionis funependuli &longs;it acceleratus; </s> <s id="N23C64"><!-- NEW -->quia impetus additur &longs;ingulis in&longs;tan­<lb/>tibus, & nihil de&longs;truitur; </s> <s id="N23C6A"><!-- NEW -->immò &longs;i de&longs;trueretur iuxta rationem prædicti <lb/>impedimenti, & pondus e&longs;&longs;et in H, cùm ratio impedimenti &longs;it SO, & <lb/>ratio noui impetus CH æqualis SO; </s> <s id="N23C72"><!-- NEW -->haud dubiè in H <expan abbr="tantũdem">tantundem</expan> pro­<lb/>duceretur impetus, quantum de&longs;trueretur; igitur nullum &longs;entiretur pon­<lb/>dus in H, quod ab&longs;urdum e&longs;t. </s> </p> <p id="N23C7E" type="main"> <s id="N23C80"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N23C8C" type="main"> <s id="N23C8E"><!-- NEW --><emph type="italics"/>Velocitates acqui&longs;itæ in funependulis inæqualibus &longs;unt vt altitudines<emph.end type="italics"/>; &longs;it <lb/>enim in figura. </s> <s id="N23C99"><!-- NEW -->Th. 10. Funependulum maius AH, minus GH; </s> <s id="N23C9D"><!-- NEW -->&longs;it vi­<lb/>bratio minoris FYH; </s> <s id="N23CA3"><!-- NEW -->&longs;it vibratio maioris DKH: </s> <s id="N23CA7"><!-- NEW -->dico velocitatem <lb/>acqui&longs;itam in prima vibratione e&longs;&longs;e ad acqui&longs;itam in &longs;ecunda, vt AH ad <lb/>GH; </s> <s id="N23CAF"><!-- NEW -->&longs;i verò vibratio maioris &longs;it tantùm LKH; </s> <s id="N23CB3"><!-- NEW -->dico e&longs;&longs;e æqualem ve­<lb/>locitatem vtriu&longs;que, quæ omnia patent ex dictis: hinc &longs;eruari po&longs;&longs;unt <lb/>quæ cumque proportiones ictuum inflictorum à malleis, vel &longs;imul, vel <lb/>&longs;ucce&longs;&longs;iue, &c. </s> </p> <p id="N23CBD" type="main"> <s id="N23CBF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N23CCB" type="main"> <s id="N23CCD"><!-- NEW --><emph type="italics"/>Ex dictis po&longs;&longs;unt multa determinari, &longs;eu cogno&longs;ci primo cognito numero vi­<lb/>brationum funependulorum inæqualium, quæ eodem tempore peraguntur, co­<lb/>gno&longs;ci po&longs;&longs;unt altitudines, &longs;eu longitudines funium<emph.end type="italics"/>; </s> <s id="N23CDA"><!-- NEW -->&longs;unt enim longitudines, <lb/>vt quadrati numerorum permutando; </s> <s id="N23CE0"><!-- NEW -->&longs;int enim duo funependula A, & <lb/>B, & numerentur vibrationes 5. penduli A & 7. penduli B æquali tem­<lb/>pore; a&longs;&longs;umantur quadrati vtriu&longs;que 25. & 49. certè longitudo penduli <lb/>A, erit ad longitudinem penduli B vt 49. ad 25. Secundò, ex cognita <lb/>minima longitudine cogno&longs;citur maxima v.g.&longs;it funependulum tripeda­<lb/>le, cuius integra vibratio tempore vnius &longs;ecundi minuti peragitur, vt <lb/>aliqui volunt (quod tantùm exempli gratia a&longs;&longs;umptum &longs;it) numerentur. </s> <s id="N23CF0"><!-- NEW --><lb/>v.g. <!-- REMOVE S-->10. vibrationes huius tripedalis funependuli eo tempore, quo duæ <lb/>æntùm vibrationes alterius maioris numerantur; &longs;int quadrati 100. & <lb/>4. certè longitudo maioris e&longs;t ad longitudinem maioris vt 4. ad 100.igi­<lb/>tur &longs;i 4. dant 100. quid dabunt 3. habeo 75. igitur longitudo maioris <lb/>funependuli e&longs;t 75. pedum. </s> <s id="N23CFF"><!-- NEW -->Tertiò, pote&longs;t cogno&longs;ci altitudo putei quan­<lb/>tumuis alti&longs;&longs;imi, vel alterius loci editi, ex quo demittitur corpus graue; </s> <s id="N23D05"><!-- NEW --><pb pagenum="316" xlink:href="026/01/350.jpg"/>&longs;i enim toto eo tempore, quo corpus graue cadit, numerentur 6. vibratio­<lb/>nes tripedalis funependuli; </s> <s id="N23D0F"><!-- NEW -->haud dubiè motus ille durauit &longs;ex minutis <lb/>&longs;ecundis; igitur &longs;i præcogno&longs;catur quantum &longs;patij percurratur deor&longs;um, <lb/>dum fluit vnum &longs;ecundum minutum, quod &longs;it.v.g. </s> <s id="N23D17">&longs;patium pedum 18 6/7 <lb/>hoc po&longs;ito, quadrentur tempora &longs;cilicet, & 6. habeo 1. & 36. iam facio <lb/>regulam trium, &longs;i 1.dat. </s> <s id="N23D1E">18 6/7 quid dabunt 36. & habeo 619. pedes mi­<lb/>nus 1/2. </s> </p> <p id="N23D25" type="main"> <s id="N23D27"><!-- NEW -->Ob&longs;eruabis autem dictum fui&longs;&longs;e à me &longs;uprà funependulum tripedale <lb/>peragere &longs;uam integram vibrationem tempore vnius &longs;ecundi minuti; </s> <s id="N23D2D"><!-- NEW --><lb/>quod certè, vt ait eruditus Mer&longs;ennus, &longs;æpiùs ob&longs;eruatum e&longs;t; </s> <s id="N23D32"><!-- NEW -->hæc autem <lb/>e&longs;t ob&longs;eruatio Mer&longs;enni, quam habet in Bali&longs;t. <!-- REMOVE S-->prop.15.eamque &longs;æpiùs, <lb/>vt ip&longs;e ait, iteratam: </s> <s id="N23D3C"><!-- NEW -->Itaque dicit tripedalis &longs;ili &longs;patio quadrantis horæ, <lb/>nongentas vibrationes fui&longs;&longs;e numeratas, &longs;ed in quadrante horæ &longs;unt 15. <lb/>minuta prima; </s> <s id="N23D44"><!-- NEW -->igitur nongenta &longs;ecunda; igitur cum &longs;ingulæ vibrationes <lb/>æquali tempore peragantur &longs;ingulis &longs;ecundis minutis re&longs;pondent. </s> </p> <p id="N23D4A" type="main"> <s id="N23D4C"><!-- NEW -->Inde in&longs;ignem difficultatem educit idem auctor; </s> <s id="N23D50"><!-- NEW -->cum enim in per­<lb/>pendiculari deor&longs;um percurrantur 12. pedes tempore vnius &longs;ecundi mi­<lb/>nuti, & 48. tempore duorum &longs;ecundorum, quod multis ob&longs;eruationibus <lb/>comprobatum e&longs;t; </s> <s id="N23D5A"><!-- NEW -->certè tempore &longs;emi&longs;ecundi minuti 3. tantùm pedes <lb/>confici nece&longs;&longs;e e&longs;t; </s> <s id="N23D60"><!-- NEW -->igitur eo tempore, quo radius tripedalis percurritur, <lb/>totus etiam percurritur quadrantis arcus, qui e&longs;t 4 3/7; </s> <s id="N23D66"><!-- NEW -->igitur maior e&longs;t <lb/>motus in arcu, quàm in perpendiculari, quod dici non pote&longs;t; cùm ne <lb/>æqualis quidem &longs;it. </s> </p> <p id="N23D6E" type="main"> <s id="N23D70"><!-- NEW -->Ad &longs;oluendum hunc nodum &longs;upponendum e&longs;t vibrationes minores <lb/>citiùs peragi, quàm maiores; </s> <s id="N23D76"><!-- NEW -->quod etiam ibidem ob&longs;eruat idem auctor; </s> <s id="N23D7A"><!-- NEW --><lb/>igitur non e&longs;t dubium, quin longè plures vibrationes fiant, quàm fierent <lb/>&longs;i omnes e&longs;&longs;ent æquales arcui quadrantis; </s> <s id="N23D81"><!-- NEW -->&longs;i enim numeres minores dum <lb/>alius numerat maiores; </s> <s id="N23D87"><!-- NEW -->cum numerabis 10. ille vix 9. habebit, & &longs;i <lb/>omnes maiores e&longs;&longs;ent æquales primæ integræ, dum habes 9. vix haberet <lb/>8. itaque non re&longs;pondent &longs;ingulæ vibrationes æquales primæ integræ <lb/>&longs;ingulis &longs;ecundis minutis; &longs;ed ferè &longs;ingulis plùs 16. vel 17. minutis <lb/>tertiis. </s> </p> <p id="N23D93" type="main"> <s id="N23D95"><!-- NEW -->Quare eo tempore, quo percurritur arcus quadrantis funependuli tri­<lb/>pedalis non percurruntur in perpendiculo 6. pedes; </s> <s id="N23D9B"><!-- NEW -->quia in perpendi­<lb/>culo percurruntur 6. pedes eo tempore, quo diagonalis quadrati, &longs;eu latus <lb/>quadrati in&longs;cripti percurritur; </s> <s id="N23DA3"><!-- NEW -->v.g. <!-- REMOVE S-->in figura Lem.3.percurruntur DT <lb/>dupla radij ID, eo tempore, quo percurritur DP; </s> <s id="N23DAB"><!-- NEW -->&longs;ed DP percurritur <lb/>tardiùs, quàm arcus DKP; </s> <s id="N23DB1"><!-- NEW -->igitur DKP citiùs quàm DT; </s> <s id="N23DB5"><!-- NEW -->igitur non <lb/>percurritur &longs;patium 6. pedum in perpendiculo eo tempore, quo percur­<lb/>ritur arcus quadrantis DKP, cuius radius ID &longs;it tripedalis; </s> <s id="N23DBD"><!-- NEW -->præterea <lb/>non percurruntur tantùm in perpendiculo eodem tempore pedes &longs;patij <lb/>4 5/7, vel vndecim, &longs;i radius con&longs;tat 7. pedibus, vt voluit idem auctor l. <!-- REMOVE S-->2. <lb/>de cau&longs;is &longs;onorum Prop. 27. Cor. <!-- REMOVE S-->3. quia &longs;i radius habet 3. arcus <lb/>quadrantis habet 4 5/7. &longs;i radius habet 7. arcus quadrantis habet 11. <lb/>&longs;ed eodem tempore conficitur maius &longs;patium in perpendiculo, quàm in <pb pagenum="317" xlink:href="026/01/351.jpg"/>arcu, cuius ratio con&longs;tat clari&longs;&longs;imè ex dictis, quia dum mobile mouea­<lb/>tur in perpendiculo &longs;ingulis in&longs;tantibus nouum impetum æqualem pri­<lb/>mo producit, in arcu verò minorem; </s> <s id="N23DD8"><!-- NEW -->igitur minor e&longs;t motus; </s> <s id="N23DDC"><!-- NEW -->igitur mi­<lb/>nus &longs;patium eodem tempore percurritur in arcu, & maius in perpendi­<lb/>culo; </s> <s id="N23DE4"><!-- NEW -->igitur non percurruntur 11. tantùm in perpendiculo eo tempore <lb/>quo 11. percurruntur in arcu; quantum verò &longs;patium in perpendiculo <lb/>percurratur eo tempore, quo arcus quadrantis dati conficitur, determi­<lb/>nabimus infrà. </s> </p> <p id="N23DEE" type="main"> <s id="N23DF0"><!-- NEW -->Denique ob&longs;eruabis, ex hoc etiam po&longs;&longs;e concludi omnes vibrationes <lb/>eiu&longs;dem funependuli non e&longs;&longs;e æquè diuturnas; </s> <s id="N23DF6"><!-- NEW -->nam reuerà &longs;i æquè diu­<lb/>turnæ e&longs;&longs;ent, & nongentæ numeratæ e&longs;&longs;ent &longs;patio 15. minutorum; </s> <s id="N23DFC"><!-- NEW -->haud <lb/>dubiè &longs;ingulæ &longs;ingulis &longs;ecundis minutis re&longs;ponderent; igitur eo tempore, <lb/>quo tres &longs;patij pedes decurrerentur in perpendiculo, in quadrantis arcu <lb/>4. 3/7 conficerentur, quod fieri non pote&longs;t. </s> </p> <p id="N23E06" type="main"> <s id="N23E08"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N23E14" type="main"> <s id="N23E16"><!-- NEW --><emph type="italics"/>In a&longs;cen&longs;u vibrationis funependuli de&longs;truitur impetus<emph.end type="italics"/>; patet, quia de&longs;init <lb/>motus; </s> <s id="N23E21"><!-- NEW -->igitur & impetus, ne &longs;it fru&longs;trà; </s> <s id="N23E25"><!-- NEW -->præterea applicatum e&longs;t princi­<lb/>pium de&longs;tructionis impetus; </s> <s id="N23E2B"><!-- NEW -->igitur de&longs;truitur; antecedens ex dicendis <lb/>infra clari&longs;&longs;imum euadet. </s> </p> <p id="N23E31" type="main"> <s id="N23E33"><!-- NEW -->De&longs;truitur autem impetus propter impetum innatum, qui &longs;ingulis in­<lb/>&longs;tantibus contranititur; </s> <s id="N23E39"><!-- NEW -->quemadmodum enim in motu violento &longs;ur&longs;um <lb/>ideo de&longs;truitur impetus ab innato, quia hic e&longs;t determinatus ad lineam <lb/>deor&longs;um; </s> <s id="N23E41"><!-- NEW -->ille verò &longs;ur&longs;um, ex quo determinatio mixta oritur; </s> <s id="N23E45"><!-- NEW -->vnde ali­<lb/>quid impetus de&longs;truitur, ne &longs;it fru&longs;trà; idem pror&longs;us dicendum e&longs;t in a&longs;­<lb/>cen&longs;u per arcum. </s> </p> <p id="N23E4D" type="main"> <s id="N23E4F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N23E5B" type="main"> <s id="N23E5D"><!-- NEW --><emph type="italics"/>Singulis in&longs;tantibus inæqualiter de&longs;truitur impetus in a&longs;cen&longs;u illo vibratio­<lb/>nis<emph.end type="italics"/>; prob. </s> <s id="N23E6A"><!-- NEW -->quia &longs;ingulis in&longs;tantibus mutatur determinatio, id e&longs;t ratio <lb/>plani inclinati; </s> <s id="N23E70"><!-- NEW -->nam quodlibet punctum arcus, vt &longs;æpè dictum e&longs;t, facit <lb/>planum inclinatum diuer&longs;um; </s> <s id="N23E76"><!-- NEW -->igitur lineæ vtriu&longs;que determinationis <lb/>faciunt diuer&longs;um angulum; </s> <s id="N23E7C"><!-- NEW -->igitur determinatio noua mixta diuer&longs;a e&longs;t; <lb/>igitur plùs vel minùs impetus de&longs;truitur, quia plùs vel minùs e&longs;t fru&longs;trà, <lb/>quod ex dicendis patebit. </s> </p> <p id="N23E84" type="main"> <s id="N23E86"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N23E92" type="main"> <s id="N23E94"><!-- NEW --><emph type="italics"/>De&longs;truitur impetus in &longs;ingulis punctis iuxta rationem &longs;inuum rectorum ar­<lb/>cuum inferiorum<emph.end type="italics"/> v.g. <!-- REMOVE S-->&longs;it arcus a&longs;cen&longs;us DIO, &longs;itque mobile pendulum in <lb/>H; </s> <s id="N23EA3"><!-- NEW -->impetus qui de&longs;truitur in H, e&longs;t ad impetum qui de&longs;truitur in per­<lb/>pendiculari &longs;ur&longs;um (&longs;uppo&longs;ito &longs;cilicet <expan abbr="t&etilde;pore">tempore</expan>) vt &longs;inus HC ad &longs;inum HA; </s> <s id="N23EAD"><!-- NEW --><lb/>nam de&longs;truitur in ea ratione, iuxta quam de&longs;trueretur in plano inclinato <lb/>EH; </s> <s id="N23EB4"><!-- NEW -->&longs;ed in planis inclinatis iuxta prædictam rationem impetum de&longs;trui <lb/>demon&longs;tratum e&longs;t &longs;uo loco; </s> <s id="N23EBA"><!-- NEW -->adde quod impetus innatus determinat mo­<lb/>bile ad lineam deor&longs;um HG, alius verò ad lineam HM; </s> <s id="N23EC0"><!-- NEW -->atqui &longs;i e&longs;&longs;ent <lb/>duo gradus impetus, quorum alter e&longs;&longs;et determinatus per HM, alter per <pb pagenum="318" xlink:href="026/01/352.jpg"/>HGV, motus fieret per HX, &longs;ed HX e&longs;t æqualis HM; </s> <s id="N23ECC"><!-- NEW -->igitur de&longs;truitur <lb/>&longs;ubduplus impetus, quia e&longs;t fru&longs;trà; </s> <s id="N23ED2"><!-- NEW -->&longs;ed HC e&longs;t &longs;ubdupla HA: </s> <s id="N23ED6"><!-- NEW -->præterea <lb/>impetus innatus retrahit mobile per HE minùs, quàm AD iuxta eam <lb/>proportionem, in qua motus per HE e&longs;t minor quàm motus per AD; </s> <s id="N23EDE"><!-- NEW -->&longs;ed <lb/>motus per HE e&longs;t ad motum per AD vt HE ad AE, vel vt HC ad HA; </s> <s id="N23EE4"><!-- NEW --><lb/>igitur illa vis, quæ retrahit mobile per HE e&longs;t ad eam, qua retrahitur <lb/>per AD vt HC ad HA; </s> <s id="N23EEB"><!-- NEW -->&longs;ed in eadem proportione de&longs;truitur impetus, <lb/>quo mobile fertur &longs;ur&longs;um, in qua retrahitur deor&longs;um; </s> <s id="N23EF1"><!-- NEW -->igitur impetus de­<lb/>&longs;tructus in H e&longs;t ad de&longs;tructum in perpendiculo vt HC ad HA; ergo <lb/>vt &longs;inus rectus arcus inferioris e&longs;t ad &longs;inum totum. </s> </p> <p id="N23EF9" type="main"> <s id="N23EFB"><!-- NEW -->Dictum e&longs;t eodem tempore; </s> <s id="N23EFF"><!-- NEW -->nam minori tempore minùs impetus de­<lb/>&longs;truitur, plùs verò maiori; </s> <s id="N23F05"><!-- NEW -->vnde quando comparatur impetus de&longs;tructus <lb/>in plano inclinato &longs;ur&longs;um cum de&longs;tructo in verticali, &longs;emper intelligi­<lb/>tur vtrumque de&longs;trui eodem tempore; </s> <s id="N23F0D"><!-- NEW -->alioquin vitio&longs;a e&longs;&longs;et proportio, <lb/>& comparatio; idem dico de impetu producto, quod de de&longs;tructo. </s> </p> <p id="N23F13" type="main"> <s id="N23F15"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23F21" type="main"> <s id="N23F23">Inde colliges in eadem proportione minùs impetus de&longs;trui in a&longs;cen&longs;u <lb/>per planum inclinatum, quâ minùs producitur in de&longs;cen&longs;u. </s> </p> <p id="N23F28" type="main"> <s id="N23F2A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N23F36" type="main"> <s id="N23F38"><emph type="italics"/>Totus impetus qui concurrit ad de&longs;cen&longs;um funependuli, non concurrit ad <lb/>a&longs;cen&longs;um,<emph.end type="italics"/> prob. </s> <s id="N23F42"><!-- NEW -->quia impetus innatus non concurrit ad a&longs;cen&longs;um, vt <lb/>con&longs;tat ex dictis alibi; &longs;ed hic concurrit ad de&longs;cen&longs;um. </s> </p> <p id="N23F48" type="main"> <s id="N23F4A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N23F56" type="main"> <s id="N23F58"><!-- NEW --><emph type="italics"/>Aliquis etiam gradus impetus concurrit ad a&longs;cen&longs;um, qui non concurrit <lb/>ad de&longs;cen&longs;um,<emph.end type="italics"/> probatur, quia vltimo in&longs;tanti de&longs;cen&longs;us aliquid impetus <lb/>noui producitur quantumuis minimi, quia &longs;ingulis in&longs;tantibus motus <lb/>deor&longs;um aliquid impetus accedit; </s> <s id="N23F67"><!-- NEW -->&longs;ed ille impetus non concurrit ad mo­<lb/>tum deor&longs;um; </s> <s id="N23F6D"><!-- NEW -->quia cum primo illo in&longs;tanti, quo e&longs;t, non concurrat ad <lb/>motum, cumque illud in&longs;tans &longs;it vltimum motus deor&longs;um; </s> <s id="N23F73"><!-- NEW -->certè ad mo­<lb/>tum deor&longs;um non concurrit, &longs;ed ad motum &longs;ur&longs;um concurrit, nam pri­<lb/>mo in&longs;tanti, quo e&longs;t, exigit motum pro &longs;equenti; e&longs;t autem &longs;equens <lb/>in&longs;tans primum a&longs;cen&longs;us. </s> </p> <p id="N23F7D" type="main"> <s id="N23F7F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N23F8B" type="main"> <s id="N23F8D"><!-- NEW --><emph type="italics"/>A&longs;cen&longs;us funependuli non e&longs;t æqualis de&longs;cen&longs;ui:<emph.end type="italics"/> patet experientiâ; ratio <lb/>e&longs;t manife&longs;ta; </s> <s id="N23F98"><!-- NEW -->quia impetus innatus non concurrit ad a&longs;cen&longs;um, licèt ad <lb/>de&longs;cen&longs;um concurrat; </s> <s id="N23F9E"><!-- NEW -->nec dicas impetus gradum vltimum non concur­<lb/>rere etiam ad de&longs;cen&longs;um, licèt concurrat ad a&longs;cen&longs;um; </s> <s id="N23FA4"><!-- NEW -->nec enim e&longs;t pa­<lb/>ritas; </s> <s id="N23FAA"><!-- NEW -->quia impetus innatus, &longs;eu primus gradus e&longs;t perfecti&longs;&longs;imus omnium <lb/>productorum; </s> <s id="N23FB0"><!-- NEW -->vltimus verò imperfecti&longs;&longs;imus, tùm quia producitur mi­<lb/>nori tempore, tùm quia producitur in plano inclinati&longs;&longs;imo; igitur &longs;i <lb/>comparetur cum primo, pro nullo ferè haberi deber impetus. </s> </p> <pb pagenum="319" xlink:href="026/01/353.jpg"/> <p id="N23FBC" type="main"> <s id="N23FBE"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N23FCA" type="main"> <s id="N23FCC"><!-- NEW -->Hinc manife&longs;ta ratio, cur funependulum po&longs;t vibrationem de&longs;cen&longs;us <lb/>non perueniat in a&longs;cen&longs;u ad tantam altitudinem; </s> <s id="N23FD2"><!-- NEW -->nec e&longs;t quod aliqui di­<lb/>cant aëra interceptum efficere, ne ad æqualem altitudinem a&longs;cendat, <lb/>cùm aër non minùs re&longs;i&longs;tat de&longs;cen&longs;ui, quàm a&longs;cen&longs;ui; quod quomodo <lb/>fiat, iam alibi explicuimus. </s> </p> <p id="N23FDC" type="main"> <s id="N23FDE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N23FEA" type="main"> <s id="N23FEC"><!-- NEW --><emph type="italics"/>Maioris vibrationis a&longs;cen&longs;us imminuitur in maiori proportione, quàm mi­<lb/>noris<emph.end type="italics"/>; </s> <s id="N23FF7"><!-- NEW -->certa experientia, cuius ratio e&longs;t, quia in arcu &longs;uperiore plùs im­<lb/>petus de&longs;truitur, in inferiore minùs; </s> <s id="N23FFD"><!-- NEW -->igitur plùs &longs;patij detrahitur maiori <lb/>vibrationi, quàm minori, &longs;cilicet in a&longs;cen&longs;u; </s> <s id="N24003"><!-- NEW -->hæc ratio demon&longs;tratiua e&longs;t, <lb/>quia quò minùs impetus de&longs;truitur &longs;ingulis in&longs;tantibus, plùs &longs;patij ac­<lb/>quiritur, vt con&longs;tat ex planis inclinatis; </s> <s id="N2400B"><!-- NEW -->&longs;it enim in eadem figura pla­<lb/>num inclinatum DO, & verticale DA; </s> <s id="N24011"><!-- NEW -->imprimatur impetus mobili ex D, <lb/>certè cum eodem impetu a&longs;cendet per DA & per DO, vt demon&longs;traui­<lb/>mus cum de planis inclinatis; </s> <s id="N24019"><!-- NEW -->igitur &longs;ingulis in&longs;tantibus minùs impetus <lb/>in DO de&longs;truitur, quàm in DA; </s> <s id="N2401F"><!-- NEW -->vnde maius &longs;patium conficitur; </s> <s id="N24023"><!-- NEW -->e&longs;t enim <lb/>DO maior DA: </s> <s id="N24029"><!-- NEW -->ita pror&longs;us accidit in arcu a&longs;cen&longs;us funependuli; </s> <s id="N2402D"><!-- NEW -->&longs;it enim <lb/>arcus a&longs;cen&longs;us DH æqualis arcui de&longs;cen&longs;us oppo&longs;iti; </s> <s id="N24033"><!-- NEW -->certè tantillùm im­<lb/>petus de&longs;truetur; </s> <s id="N24039"><!-- NEW -->igitur arcus a&longs;cen&longs;us ferè accedet ad A; </s> <s id="N2403D"><!-- NEW -->&longs;i vetò arcus <lb/>de&longs;cen&longs;us &longs;it æqualis DL, plùs impetus de&longs;truetur in a&longs;cen&longs;u; igitur ar­<lb/>cus a&longs;cen&longs;us habebit minorem proportionem ad DL, quàm prior ad DH, <lb/>& hæc e&longs;t veri&longs;&longs;ima ratio luculenti&longs;&longs;imi experimenti, quod ferè omnibus <lb/>notum e&longs;t. </s> </p> <p id="N24049" type="main"> <s id="N2404B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N24057" type="main"> <s id="N24059"><!-- NEW --><emph type="italics"/>Si proijciatur mobile per ip&longs;um perpendiculum DA cum eo impetu, quo <lb/>ex D feratur in A motu naturaliter retardato; </s> <s id="N24061"><!-- NEW -->certè cum eodem impetu fere­<lb/>tur in O per DO, & per arcum DLO:<emph.end type="italics"/> probatur quia ex A in D, vel ex O <lb/>in D &longs;iue per chordam OD, &longs;iue per arcum OHD æqualis impetus ac­<lb/>quiritur per Lemma 11. &longs;ed cum eodem impetu, quo ex A fertur in D. <lb/>vel ex O in D motu naturaliter accelerato, ex D ferri pote&longs;t in A vel in <lb/>O: </s> <s id="N24072"><!-- NEW -->dixi cum eodem impetu, ita vt tot gradus impetus concurrant ad a&longs;­<lb/>cen&longs;um, quot ad de&longs;cen&longs;um; </s> <s id="N24078"><!-- NEW -->&longs;i enim aliquis gradus concurrens ad de&longs;­<lb/>cen&longs;um, non concurreret ad a&longs;cen&longs;um; </s> <s id="N2407E"><!-- NEW -->haud dubiè non perueniret mo­<lb/>bile ad <expan abbr="eãdem">eandem</expan> altitudinem; quod autem æquale &longs;patium re&longs;pondeat <lb/>a&longs;cen&longs;ui, & de&longs;cen&longs;ui &longs;uppo&longs;ito æquali impetu, iam demon&longs;tratum e&longs;t &longs;u­<lb/>prà l. <!-- REMOVE S-->3. & 5. &longs;ed iam examinandæ &longs;unt proportiones huius de&longs;tructio­<lb/>nis impetus in maioribus, & minoribus vibrationibus. </s> </p> <p id="N24090" type="main"> <s id="N24092"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N2409E" type="main"> <s id="N240A0"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari in qua parte arcus de&longs;inat motus &longs;ur&longs;um in a&longs;cen&longs;u <lb/>vibrationis, &longs;i cogno&longs;catur ad quam altitudinem ferretur mobile per ip&longs;um <lb/>perpendiculum<emph.end type="italics"/>; </s> <s id="N240AD"><!-- NEW -->fit cum punctum infimum D, &longs;itque in pendule ille impe­<lb/>tus, haud dubiè per arcum ferretur in <foreign lang="greek">a</foreign>, ducatur <foreign lang="greek">a</foreign>Q parallela AO; </s> <s id="N240B7"><!-- NEW -->haud <pb pagenum="320" xlink:href="026/01/354.jpg"/>dubiè per arcum feretur in Q & per chordam DO perueniet in <foreign lang="greek">q</foreign>; </s> <s id="N240C4"><!-- NEW -->&longs;i ve­<lb/>rò illo impetu ferri tantùm po&longs;&longs;it in B per per DA, fertur in 4.per DO, <lb/>& in L per arcum; </s> <s id="N240CC"><!-- NEW -->denique &longs;i ferri tantùm po&longs;&longs;it illo impetu per DA in <lb/>G, feretur in 3 per DO, & in H per arcum; </s> <s id="N240D2"><!-- NEW -->quæ omnia con&longs;tant ex Th. <!-- REMOVE S--><lb/>20. quia cum eodem impetu a&longs;cendit mobile ad <expan abbr="eãdem">eandem</expan> altitudinem <lb/>&longs;iue per ip&longs;um perpendiculum, &longs;iue per chordas, &longs;iue per arcus; ex hoc <lb/>confirmatur maximè Th.10. quia &longs;i diuidatur perpendiculum in partes <lb/>æquales ductis parallelis AO, arcus ita diuidetur, vt &longs;uperior arcus &longs;it <lb/>minor. </s> <s id="N240E5"><!-- NEW -->v.g. <!-- REMOVE S-->diuidatur DA in B æqualiter bifariam; </s> <s id="N240EB"><!-- NEW -->ducatur BL parallela <lb/>AO, non diuidit arcum OD bifariam, cùm arcus OL &longs;it &longs;ubtriplus arcus <lb/>OD; </s> <s id="N240F3"><!-- NEW -->igitur cùm eo tantùm impetu, quo in perpendiculo acquireretur in <lb/>a&longs;cen&longs;u DB &longs;ubduplum DA, in arcu acquiretur DL, quæ e&longs;t 2/3 totius D­<lb/>O; igitur minores vibrationes minùs imminuuntur in a&longs;cen&longs;u, quàm <lb/>maiores. </s> </p> <p id="N240FD" type="main"> <s id="N240FF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N2410B" type="main"> <s id="N2410D"><!-- NEW --><emph type="italics"/>Hinc tam facilè vibratur funependulum per minimum arcum, v. <!-- REMOVE S-->g. <!-- REMOVE S-->cum <lb/>primo impetu, quo a&longs;cenderet ex D in C vel in<emph.end type="italics"/> 3. <emph type="italics"/>a&longs;cendit in H<emph.end type="italics"/>; quia &longs;cilicet <lb/>cum eo impetu, quo minimum ferè &longs;patium acquirit in perpendiculo, <lb/>notabile &longs;atis &longs;patium decurrit in arcu. </s> </p> <p id="N24126" type="main"> <s id="N24128"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N24134" type="main"> <s id="N24136"><!-- NEW --><emph type="italics"/>Hinc tamdiu durant minimæ illa vibrationes; </s> <s id="N2413C"><!-- NEW -->quia &longs;ingulæ minima por­<lb/>tione imminuuntur, & maiores è contrariò tam citò decurtantur<emph.end type="italics"/>; </s> <s id="N24145"><!-- NEW -->cuius re&longs; <lb/>non e&longs;t alia ratio præter eam, quam &longs;uprà adduximus, quæ rem ip&longs;am <lb/>euincit; e&longs;t tamen in&longs;ignis difficultas, quam paulò po&longs;t di&longs;cutiemus in <lb/>&longs;equenti Schol. <!-- REMOVE S--><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N2415A" type="main"> <s id="N2415C"><!-- NEW --><emph type="italics"/>Hinc ratio, cur minimo &longs;erè cur&longs;u funependulum etiam graui&longs;&longs;imum modi­<lb/>ca libratione vibretur<emph.end type="italics"/>; </s> <s id="N24168"><!-- NEW -->immò, quod fortè alicui mirum videretur, ip&longs;o an­<lb/>helitu graui&longs;&longs;ima pondera moueri po&longs;&longs;unt, quod quiuis facilè probare <lb/>poterit; </s> <s id="N24170"><!-- NEW -->pro quo diligenter ob&longs;eruandum e&longs;t, vt eo dumtaxat ordine an­<lb/>helitus repetatur, quo vibrationes fiunt, ita vt iam euntem molem à <lb/>tergo impellat; vnde accidet, vt repetito tandem anhelitu maiore motu <lb/>funependulum vibretur. </s> </p> <p id="N2417A" type="main"> <s id="N2417C"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24188" type="main"> <s id="N2418A"><!-- NEW -->Ob&longs;eruabis primò maximam occurrere difficultatem contra ea, quæ <lb/>hactenus demon&longs;trauimus; &longs;it enim quadrans AIE, &longs;itque EA diui&longs;a <lb/>in 4. partes æquales. </s> <s id="N24192"><!-- NEW -->v.g. <!-- REMOVE S-->ex A cadat corpus graue in E, & ex E a&longs;cen­<lb/>dat denuò per EA eâ lege, vt omnes gradus impetus acqui&longs;iti in de&longs;cen­<lb/>&longs;u concurrant ad a&longs;cen&longs;um, excepto primo gradu impetus innati; </s> <s id="N2419C"><!-- NEW -->certè <lb/>non a&longs;cendet in A, vt con&longs;tat ex dictis; </s> <s id="N241A2"><!-- NEW -->igitur a&longs;cendat in B, & ex B ite­<lb/>rum de&longs;cendat in E, redeatque ver&longs;us A; </s> <s id="N241A8"><!-- NEW -->haud dubiè perueniet tantùm <lb/>in C; </s> <s id="N241AE"><!-- NEW -->ita vt tantum detrahatur &longs;patij in hoc &longs;ecundo a&longs;cen&longs;u, quantum <lb/>detractum e&longs;t in primo: idem dico de tertio, quarto, &c. </s> <s id="N241B4"><!-- NEW -->ducantur BH, <pb pagenum="321" xlink:href="026/01/355.jpg"/>CG, D F parallelæ AI; </s> <s id="N241BD"><!-- NEW -->cum &longs;patium eo modo decidatur ex area EI, quo <lb/>ex perpendiculo EA maiori vibrationi detrahitur IH, &longs;ecundæ minori <lb/>HG, tertiæ GF, quartæ FE; igitur plùs detrahitur minoribus, quàm <lb/>maioribus. </s> </p> <p id="N241C7" type="main"> <s id="N241C9"><!-- NEW -->Re&longs;pondeo, maius &longs;patium percurri &longs;ur&longs;um maiore tempore, quàm <lb/>minus; </s> <s id="N241CF"><!-- NEW -->&longs;it enim EA con&longs;tans 36. &longs;patia iuxta nouam progre&longs;&longs;ionem <lb/>arithmeticam, &longs;intque 8. gradus impetus acqui&longs;iti in de&longs;cen&longs;u AE con­<lb/>iuncti cum innato: </s> <s id="N241D7"><!-- NEW -->primo in&longs;tanti, &longs;eu tempore percurrentur tantùm 7. <lb/>&longs;patia, <expan abbr="de&longs;truetur&qacute;ue">de&longs;trueturque</expan> vnus gradus impetus, &longs;ecundo 6. <expan abbr="de&longs;truetur&qacute;ue">de&longs;trueturque</expan> al­<lb/>ter gradus impetus; denique tertio 5. quatto 4. &c. </s> <s id="N241E7"><!-- NEW -->igitur 28. &longs;patia 7. <lb/>in&longs;tantibus; igitur non perueniet in A mobile, &longs;ed conficiet &longs;patium, <lb/>quod erit ad EA, vt 28. ad 36. porrò &longs;i cadat ex 28. acquiret 7. gradus <lb/>impetus præter innatum, quorum ope &longs;ecundo a&longs;cendet ad 21. tertio ad <lb/>15. quartò ad 10. quinto ad 6. &longs;extò ad 3. &longs;eptimo ad 1. igitur &longs;patium <lb/>quod amittit in a&longs;cen&longs;u continet 8. in &longs;ecundo 7. in tertio 6. in quarto <lb/>5. in quinto 4. in &longs;exto 3. in &longs;eptimo 2. igitur e&longs;t maxima inæqualitas, <lb/>quæ pari modo explicari pote&longs;t in progre&longs;&longs;ione Galilei. <!-- KEEP S--></s> </p> <p id="N241FA" type="main"> <s id="N241FC">Secundò, obijci pote&longs;t: </s> <s id="N241FF"><!-- NEW -->amitti tantùm &longs;patij &longs;ingulis temporibus, <lb/>quantum acquiritur primo tempore, vel in&longs;tanti, cum impetu innato: &longs;ed <lb/>cum primo ille velocitatis gradu vix intra multos annos conficeretur <lb/>modicum &longs;patium. </s> <s id="N2420A"><!-- NEW -->Re&longs;pondeo, &longs;i con&longs;ideretur tantùm illud &longs;patium, <lb/>quod acquiritur primo tempore cum impetu non impedito; </s> <s id="N24210"><!-- NEW -->haud dubiè <lb/>in&longs;en&longs;ibile e&longs;t, & licèt infinitus ferè repetatur illud idem &longs;patium; </s> <s id="N24216"><!-- NEW -->haud <lb/>dubiè in&longs;en&longs;ibile manet: vnde &longs;i a&longs;cen&longs;us fiat in 10000. in&longs;tantibus, to­<lb/>ties accipi debet illud ip&longs;um &longs;patium, ex quo modicum tantùm re&longs;ultat, <lb/>quod minuitur in &longs;ecundo a&longs;cen&longs;u, itemque in tertio, quarto, &c. </s> </p> <p id="N24220" type="main"> <s id="N24222"><!-- NEW -->Vnde ten&longs;io funis, ex quo pendet corpus graue con&longs;ideranda e&longs;t, qui <lb/>cum propter impetum de&longs;cen&longs;us mox dilatetur, & tendatur, mox contra­<lb/>hatur, tùm in a&longs;cen&longs;u, tùm in de&longs;cen&longs;u; </s> <s id="N2422A"><!-- NEW -->certè multùm impetus de&longs;truitur, <lb/>quod autem tendatur maximè in de&longs;cen&longs;u prædictus funis, con&longs;tat <lb/>multis experimentis &longs;i minor e&longs;t; nam reuerâ; &longs;i maior, e&longs;&longs;et multum re­<lb/>tardaret motum tùm aëris re&longs;i&longs;tentia, quæ etiam aliquid facit, licèt totus <lb/>hic effectus ab illa pendere non po&longs;&longs;it, vt aliqui volunt, tùm etiam partes <lb/>funis propiùs ad centrum accedentes, quæ citiùs de&longs;cendunt, &c. </s> </p> <p id="N24238" type="main"> <s id="N2423A"><!-- NEW -->Tertiò, &longs;unt tres determinationes in a&longs;cen&longs;u; </s> <s id="N2423E"><!-- NEW -->prima e&longs;t impetus pro­<lb/>ducti in de&longs;cen&longs;u determinati ad Tangentem; &longs;ecunda funis per &longs;uam li­<lb/>neam qua&longs;i retrahentis pendulum. </s> <s id="N24246"><!-- NEW -->tertia ip&longs;ius impetus innati qua&longs;i tra­<lb/>hentis deor&longs;um idem pondus; atqui ex pugna trium determinationum in <lb/>eodem mobili de&longs;truitur multùm impetus, vt patet ex dictis alibi. </s> </p> <p id="N2424E" type="main"> <s id="N24250"><!-- NEW -->Quartò, cum eo impetu, cuius ope non po&longs;&longs;et corpus a&longs;cendere per <lb/>ip&longs;um perpendiculum EA, a&longs;cendit adhuc per arcum EI; </s> <s id="N24256"><!-- NEW -->licèt enim cum <lb/>co impetu, quo fertur in F po&longs;&longs;it fieri in D, &longs;ed tardiori motu; </s> <s id="N2425C"><!-- NEW -->attamen <lb/>quia impetus qui pendulo ine&longs;t, e&longs;t determinatus ad talem gradum ve­<lb/>locitatis, quo certè per ip&longs;am ED ferri non pote&longs;t; </s> <s id="N24264"><!-- NEW -->quod etiam euincitur <lb/>ex organis mechanicis, & planis inclinatis; </s> <s id="N2426A"><!-- NEW -->nam reuerà moueret aliquis <pb pagenum="322" xlink:href="026/01/356.jpg"/>per planum tantillùm inclinatum maximam corporis molem, quam per <lb/>aliud planum inclinatius, & accedens propiùs ad verticale minimè mo­<lb/>uere po&longs;&longs;et; </s> <s id="N24277"><!-- NEW -->cuius effectus alia ratio non e&longs;t, ni&longs;i quod impetus, qui im­<lb/>primitur mobili ad talem gradum velocitatis &longs;it determinatus; atqui in <lb/>perpendiculo eo motu moueri non pote&longs;t, vt con&longs;tat, &longs;ed in plano lon­<lb/>giore. </s> </p> <p id="N24281" type="main"> <s id="N24283"><!-- NEW -->Quintò, hinc vera ratio, cur in &longs;uperiore arcu de&longs;truatur citò impe­<lb/>tus; </s> <s id="N24289"><!-- NEW -->tardiùs verò in in inferiore; </s> <s id="N2428D"><!-- NEW -->quia, cùm Tangens cuiu&longs;libet puncti ar­<lb/>cus &longs;it eius planum, & hæc in arcu &longs;uperiore accedat propiùs ad perpen­<lb/>diculum; non mirum e&longs;t, &longs;i cum eo impetu per arcum &longs;uperiorem mo­<lb/>ueri non po&longs;&longs;it mobile cò non a&longs;cendat, cuius tantùm ope per inferio­<lb/>rem arcum a&longs;cendit. </s> </p> <p id="N24299" type="main"> <s id="N2429B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N242A7" type="main"> <s id="N242A9"><!-- NEW --><emph type="italics"/>Omnes vibrationes numerari non po&longs;&longs;unt<emph.end type="italics"/>; </s> <s id="N242B2"><!-- NEW -->certum e&longs;t, cùm &longs;int infinitæ <lb/>ferè in&longs;en&longs;ibiles, nec pote&longs;t &longs;en&longs;u di&longs;cerni, quantus &longs;it arcus minimæ vi­<lb/>brationis; </s> <s id="N242BC"><!-- NEW -->&longs;i tamen de&longs;trueretur tantùm impetus in a&longs;cen&longs;u ab impetu <lb/>innato, nec ten&longs;io funis, triplex determinatio, re&longs;i&longs;tentia aëris, & diuer&longs;æ <lb/>partes funis, quarum minores vibrationes impediuntur quidquam fa­<lb/>cerent, cognita differentia prima vibrationis & &longs;ecunda, fortè cogno&longs;ci <lb/>po&longs;&longs;et numerus vibrationum cognito principio progre&longs;&longs;ionis; </s> <s id="N242C8"><!-- NEW -->quantus <lb/>verò &longs;it numerus vibrationum, quæ incipiunt à maiore, & quantus illa­<lb/>rum quæ incipiunt à minore etiam incertum e&longs;t, v.g. <!-- REMOVE S-->&longs;i funependulum <lb/>AD demittatur ex O, & deinde ex L; </s> <s id="N242D4"><!-- NEW -->certum e&longs;t quidem e&longs;&longs;e plures vi­<lb/>brationes cum demittitur ex O toto eo vibrationum numero, quæ re­<lb/>cen&longs;entur, donec perueniatur ad illam vibrationem, cuius a&longs;cen&longs;us con­<lb/>&longs;tat arcu DL; </s> <s id="N242DE"><!-- NEW -->nam deinceps æqualis erit numerus earum, quæ con&longs;e­<lb/>quentur, & earum, quarum prima demittitur ex L, vt patet; </s> <s id="N242E4"><!-- NEW -->quot verò <lb/>præcedant vibrationes antequam perueniatur ad illam, cuius a&longs;cen&longs;us <lb/>e&longs;t arcus DL; equidem aliqua ob&longs;eruatione affixo &longs;cilicet maiore qua­<lb/>drante parieti iu &longs;uos gradus di&longs;tributo cogno&longs;ci pote&longs;t, &longs;ed nunquam <lb/>&longs;atis acurata. </s> </p> <p id="N242F0" type="main"> <s id="N242F2">Itaque certum e&longs;t primò accelerari motum in de&longs;cen&longs;u, &c retardari <lb/>in a&longs;cen&longs;u. </s> </p> <p id="N242F7" type="main"> <s id="N242F9"><!-- NEW -->Secundò, certum e&longs;t impetum nouum accedere in de&longs;cen&longs;u in &longs;ingu­<lb/>lis punctis arcus iuxta rationem &longs;inus recti arcus inferioris; in a&longs;cen&longs;u <lb/>verò imminui acqui&longs;itum impetum in eadem ratione, omi&longs;&longs;a ea parte <lb/>impetus, quæ de&longs;truitur tùm in a&longs;cen&longs;u tùm in de&longs;cen&longs;u propter ten&longs;io­<lb/>nem funis, & re&longs;i&longs;tentiam aëris. </s> </p> <p id="N24305" type="main"> <s id="N24307"><!-- NEW -->Tertiò, certum e&longs;t, primum gradum impetus &longs;cilicet innatum concur­<lb/>rere ad de&longs;cen&longs;um; </s> <s id="N2430D"><!-- NEW -->&longs;ecus verò ad a&longs;cen&longs;um, & contra vltimum gradum <lb/>concurrere ad a&longs;cen&longs;um, &longs;ecus ad de&longs;cen&longs;um; &longs;ed hic vltimus gradus mi­<lb/>nimus e&longs;t, & pro nihilo reputandus. </s> </p> <p id="N24315" type="main"> <s id="N24317">Quartò, certum e&longs;t a&longs;cen&longs;um minorem e&longs;&longs;e de&longs;cen&longs;u, nec funependu­<lb/>lum ad <expan abbr="cãdem">eandem</expan>, vnde dimi&longs;&longs;um e&longs;t priùs, a&longs;cendere altitudinem. </s> </p> <pb pagenum="323" xlink:href="026/01/357.jpg"/> <p id="N24324" type="main"> <s id="N24326">Quintò, certum e&longs;t arcum de&longs;cen&longs;us maioris vibrationis habere ma­<lb/>iorem proportionem ad arcum a&longs;cen&longs;us, qui &longs;equitur, quàm habeat ar­<lb/>cus de&longs;cen&longs;us minoris vibrationis ad &longs;uum a&longs;cen&longs;um. </s> </p> <p id="N2432D" type="main"> <s id="N2432F"><!-- NEW -->Sextò, certum e&longs;t non tantùm imminui arcum a&longs;cen&longs;us ab aëre ob&longs;i­<lb/>&longs;tente &longs;ed maximè ab impetu innato retrahente deor&longs;um funependulum; </s> <s id="N24335"><!-- NEW --><lb/>tùm etiam maximè ab ip&longs;a ten&longs;ione funis, tùm ab ip&longs;o fune adducente <lb/>pondus; tùm denique à diuer&longs;is partibus funis, quæ dum ab aliis retinen­<lb/>tur, qua&longs;i cum illis pugnant, ex qua pugna &longs;equitur aliqua clades in <lb/>motu. </s> </p> <p id="N24340" type="main"> <s id="N24342"><!-- NEW -->Septimò, certum e&longs;t acquiri æqualem impetum ex eadem altitudine <lb/>in motu deor&longs;um, &longs;iue per arcum, &longs;iue per chordam, &longs;iue per ip&longs;um per­<lb/>pendiculum inæquali tamen tempore; &longs;imiliter de&longs;trui æqualem im­<lb/>petum in a&longs;cen&longs;u, qui ad <expan abbr="eãdem">eandem</expan> altitudinem pertingit. </s> </p> <p id="N24350" type="main"> <s id="N24352">Octauò, certum e&longs;t eo tempore, quo de&longs;cendit mobile per arcum in <lb/>ip&longs;o perpendiculo acquirere &longs;patium maius ip&longs;o arcu, minus tamen du­<lb/>plo radij. </s> </p> <p id="N24359" type="main"> <s id="N2435B"><!-- NEW -->Nonò, certum e&longs;t, non accelerari motum per arcum in de&longs;cen&longs;u iuxta <lb/>proportionem numerorum 1.3.5.7. vt volunt aliqui; </s> <s id="N24361"><!-- NEW -->quia hæc accele­<lb/>ratio ex ip&longs;o Galileo &longs;upponit principium illud, æqualibus temporibus <lb/>acquiruntur æqualis velocitatis momenta, &longs;ed inæqualia acquiruntur in <lb/>arcu; vt patet ex dictis; </s> <s id="N2436B"><!-- NEW -->multò minùs intenditur iuxta proportionem <lb/>arcuum qui &longs;ecantur à lineis ductis parallelis horizontali ab iis punctis <lb/>perpendiculi, in quibus &longs;ecatur iuxta hos numeros 1. 3. 5. 7. certum e&longs;t <lb/>etiam non retardari iuxta <expan abbr="eãdem">eandem</expan> proportionem in a&longs;cen&longs;u: quippe in <lb/>hoc in eadem proportione retardatur, qua in illo acceleratur. </s> </p> <p id="N2437B" type="main"> <s id="N2437D"><!-- NEW -->Decimò, certum e&longs;t omnes vibrationes non po&longs;&longs;e numerari cuiu&longs;cum­<lb/>que longitudinis &longs;it ip&longs;um funependulum: immò hoc valdè e&longs;&longs;et inutile, <lb/>vt inutile e&longs;t no&longs;&longs;e numerum granorum arenæ maris. </s> </p> <p id="N24385" type="main"> <s id="N24387">Vndecimò, <expan abbr="certũ">certum</expan> e&longs;t vibrationes minores citiùs ab&longs;olui, quàm maiores. </s> </p> <p id="N2438E" type="main"> <s id="N24390"><!-- NEW -->Duodecimò, certum e&longs;t tempora vibrationum funependulorum inæ­<lb/>qualium e&longs;&longs;e ferè, vt radices longitudinum, & longitudines, vt quadrata <lb/>temporum dixi: ferè, nec enim omninò res ita &longs;e habet. </s> </p> <p id="N24398" type="main"> <s id="N2439A"><!-- NEW -->Con&longs;tat ex iis quæ &longs;uprà diximus, ea omnia, quæ hactenus enumerata <lb/>&longs;unt, certa e&longs;&longs;e cum aliis plurimis &longs;uprà recen&longs;itis; &longs;unt etiam aliqua in­<lb/>nota. </s> <s id="N243A2"><!-- NEW -->Primò incertum fuit hactenus, in qua proportione temporum per­<lb/>curratur arcus: </s> <s id="N243A8"><!-- NEW -->Equidem certum e&longs;t in qua proportione velocitas cre&longs;cit, <lb/>vt &longs;uprà demon&longs;tratum e&longs;t; incertum, quænam &longs;it progre&longs;&longs;io &longs;patiorum <lb/>&longs;eu proportio motus in &longs;patio arcus, dato &longs;cilicet tempore &longs;en&longs;ibili. </s> </p> <p id="N243B0" type="main"> <s id="N243B2"><!-- NEW -->Ob&longs;eruo tamen, &longs;i con&longs;ideretur hic motus in in&longs;tantibus, demitta­<lb/>túrque funependulum è &longs;ummo arcu, &longs;patium quod acquiritur primo in­<lb/>&longs;tanti e&longs;t ad &longs;patium, quod acquiritur &longs;ecundo, vt &longs;inus totus ad colle­<lb/>ctum ex &longs;inu toto & &longs;inu recto immediato arcus inferioris, qui proximè <lb/>accedit ad totum; e&longs;t autem ad &longs;patium, quod acquiritur tertio in&longs;tan­<lb/>ti, vt &longs;inus totus ad collectum ex &longs;inu toto & duobus &longs;inubus rectis im­<lb/>mediatis, atque ita deinceps. </s> </p> <pb pagenum="324" xlink:href="026/01/358.jpg"/> <p id="N243C6" type="main"> <s id="N243C8"><!-- NEW -->Ob&longs;eruo &longs;ecundò iuxta progre&longs;&longs;ionem Galilei, &longs;i a&longs;&longs;umatur pars tem­<lb/>poris &longs;en&longs;ibilis, in qua percurratur &longs;patium &longs;uperius in arcu, non po&longs;&longs;e <lb/>cogno&longs;ci quanto tempore percurratur reliquus arcus; </s> <s id="N243D0"><!-- NEW -->&longs;it enim trian­<lb/>gulum mixtum ABE, quale iam expre&longs;&longs;imus, &longs;itque primus arcus dato <lb/>tempore decur&longs;us ad reliquum vt AD ad DE; </s> <s id="N243D8"><!-- NEW -->ducatur DC, &longs;itque v.g. <!-- REMOVE S--><lb/>trapezus DCBA ad triangulum ABE vt 2. ad.7.dico velocitatem, quæ <lb/>acquiritur in arcu AD, e&longs;&longs;e ad illam, quæ acquiritur in AE vt 2.ad 7. & <lb/>ad illam, quæ acquiritur in DE, vt 2.ad 5.&longs;ed in hoc motu tempora non <lb/>&longs;unt vt velocitates; </s> <s id="N243E5"><!-- NEW -->quia temporibus æqualibus non acquiruntur æqua­<lb/>les velocitatis gradus; </s> <s id="N243EB"><!-- NEW -->igitur nec &longs;patia vt temporum, &longs;eu velocitatum <lb/>quadrata; igitur incertum e&longs;t hactenus, in qua proportione temporum <lb/>percurrantur duo arcus dati in quadrante, & quæ proportio &longs;patiorum <lb/>re&longs;pondeat temporibus datis. </s> </p> <p id="N243F5" type="main"> <s id="N243F7">Secundò, incertum etiam hactenus in qua proportione percurratur <lb/>velociùs arcus, quàm chorda, & tardiùs, quàm radius in perpendiculo, <lb/>& quantum &longs;patium in eodem perpendiculo percurratur eo tempore, <lb/>quo totus arcus quadrantis peragitur. </s> </p> <p id="N24400" type="main"> <s id="N24402"><!-- NEW -->Tertiò incertum e&longs;t, in qua proportione minor vibratio citiùs peraga­<lb/>tur, quàm maior; </s> <s id="N24408"><!-- NEW -->licèt cogno&longs;ci po&longs;&longs;it in qua proportione peragantur ci­<lb/>tiùs duæ chordæ in&longs;criptæ arcui minori, quàm duæ in&longs;criptæ arcui maio­<lb/>ri; & licèt certum &longs;it omnes chordas &longs;eor&longs;im &longs;umptas æqualibus tem­<lb/>poribus decurri, & citiùs decurri duas eidem arcui in&longs;criptas, quàm &longs;o­<lb/>lam inferiorem. </s> </p> <p id="N24414" type="main"> <s id="N24416"><!-- NEW -->Quartò incertum e&longs;t, in qua proportione imminuatur impetus tùm in <lb/>de&longs;cen&longs;u, tùm in a&longs;cen&longs;u, tùm propter re&longs;i&longs;tentiam aëris, tùm propter ten­<lb/>&longs;ionem chordæ, tùm ratione triplicis determinationis in &longs;ingulis pun­<lb/>ctis arcus; </s> <s id="N24420"><!-- NEW -->licèt certum &longs;it quantum &longs;ingulis in&longs;tantibus detrahatur im­<lb/>petus in a&longs;cen&longs;u ab impetu innato retrahente pendulum deor&longs;um; incer­<lb/>tum e&longs;t tamen, quantus &longs;it ille impetus innatus. </s> </p> <p id="N24428" type="main"> <s id="N2442A"><!-- NEW -->Quintò, incertum e&longs;t in qua proportione a&longs;cen&longs;us primæ vibrationis <lb/>&longs;it minor de&longs;cen&longs;u eiu&longs;dem; </s> <s id="N24430"><!-- NEW -->incertum etiam, in qua proportione a&longs;cen­<lb/>&longs;us &longs;ecundæ &longs;it minor a&longs;cen&longs;u prime; </s> <s id="N24436"><!-- NEW -->incertum denique, in qua proportio­<lb/>ne plùs imminuatur a&longs;cen&longs;us maiorum vibrationum, quàm minorum; li­<lb/>cèt certum &longs;it plùs imminui. </s> </p> <p id="N2443E" type="main"> <s id="N24440"><!-- NEW -->Sextò, incertum e&longs;t, quot peragantur vibrationes dati funependulis <lb/>item quantus &longs;it arcus vltimæ vibrationis; </s> <s id="N24446"><!-- NEW -->item in qua proportione ma­<lb/>ior &longs;it numerus vibrationum, quarum prima maior e&longs;t numero vibratio­<lb/>num, quarum prima minor e&longs;t; denique quot intercipiantur vibratio­<lb/>nes in differentia data duorum arcuum. </s> </p> <p id="N24450" type="main"> <s id="N24452"><!-- NEW -->Hæc, quæ hactenus propo&longs;uimus in 6. vltimis capitibus, &longs;unt omninò <lb/>incerta, ita vt neque &longs;en&longs;u percipi po&longs;&longs;int, neque fuerit hactenus vllum <lb/>principium, per quod po&longs;sint demon&longs;trari; ni&longs;i fortè primum caput ex­<lb/>cipias, de quo infrà. </s> </p> <p id="N2445C" type="main"> <s id="N2445E">Primò dubium e&longs;t an numerus vibrationum funependuli maioris &longs;it <lb/>maior numero vibrationum funependuli minoris, po&longs;ito quòd primam <pb pagenum="325" xlink:href="026/01/359.jpg"/>vtriu&longs;que vibratio &longs;it &longs;imilis. </s> </p> <p id="N24468" type="main"> <s id="N2446A"><!-- NEW -->Secundò dubium e&longs;t, an numerus vibrationum funependuli longio­<lb/>ris &longs;it æqualis numero vibrationum alterius minoris, po&longs;ito quòd prima <lb/>vtriu&longs;que ab eadem altitudine demittatur; vel po&longs;ito quòd arcus primæ <lb/>maioris funependuli &longs;it æqualis arcui primæ minoris. </s> </p> <p id="N24474" type="main"> <s id="N24476"><!-- NEW -->Tertiò dubium e&longs;t, in qua proportione pendula materiæ grauiores <lb/>&longs;uas vibrationes citiùs peragant, quàm pendula materiæ leuioris; </s> <s id="N2447C"><!-- NEW -->item­<lb/>que dubium, quanto tempore citiùs extinguantur vibrationes penduli <lb/>materiæ leuioris, quàm grauioris: </s> <s id="N24484"><!-- NEW -->licèt certum &longs;it citiùs ab&longs;olui vibra­<lb/>tiones funependuli materiæ leuioris, quàm grauioris; hæc &longs;unt dubia, <lb/>quæ breuiter di&longs;cutiemus in &longs;equentibus Theorematis. <!-- KEEP S--></s> </p> <p id="N2448D" type="main"> <s id="N2448F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N2449B" type="main"> <s id="N2449D"><!-- NEW --><emph type="italics"/>Funependula longiora diutiùs vibrantur, quàm breuiora, &longs;i prima vtriu&longs;­<lb/>que vibratio &longs;it &longs;imilis<emph.end type="italics"/>; </s> <s id="N244A8"><!-- NEW -->experientia manife&longs;ta e&longs;t; </s> <s id="N244AC"><!-- NEW -->ratio etiam euidens, <lb/>quia vt &longs;e habent &longs;ingulæ vibrationes minoris ad &longs;ingulas maioris; </s> <s id="N244B2"><!-- NEW -->ita <lb/>omnes minoris &longs;e habent ad omnes maioris, vt patet; </s> <s id="N244B8"><!-- NEW -->&longs;ed &longs;ingulæ maio­<lb/>ris diutiùs durant, quàm &longs;ingulæ minoris; </s> <s id="N244BE"><!-- NEW -->igitur omnes maioris diutiùs <lb/>durant, quàm omnes minoris; igitur funependula longiora diutiùs vi­<lb/>brantur, &c. </s> </p> <p id="N244C6" type="main"> <s id="N244C8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N244D4" type="main"> <s id="N244D6"><!-- NEW --><emph type="italics"/>Tot &longs;unt vibrationes maioris funependuli per &longs;e, quot &longs;unt minoris, po&longs;ito <lb/>quod vtriu&longs;que vibratio prima &longs;it &longs;imilis<emph.end type="italics"/>; </s> <s id="N244E1"><!-- NEW -->demon&longs;tratur, &longs;it enim fune­<lb/>pendulum maius AG, & minus AO &longs;ubquadruplum &longs;cilicet AG, demit­<lb/>tatur AG ex AD, & AO ex AB, impetus acqui&longs;itus in G per DG e&longs;t <lb/>æqualis acqui&longs;ito in perpendiculo AG; </s> <s id="N244EB"><!-- NEW -->& impetus acqui&longs;itus in O per <lb/>BO e&longs;t æqualis acqui&longs;ito in perpendiculo per AO; </s> <s id="N244F1"><!-- NEW -->&longs;ed acqui&longs;itus in per­<lb/>pendiculo AG e&longs;t duplus acqui&longs;iti in perpendiculo AO, vt con&longs;tat; </s> <s id="N244F7"><!-- NEW -->&longs;unt <lb/>enim velocitates, vel impetus acqui&longs;iti in ratione &longs;ubduplicata &longs;patio­<lb/>rum; </s> <s id="N244FF"><!-- NEW -->præterea impetus, qui de&longs;truitur in a&longs;cen&longs;u GK, e&longs;t æqualis acqui&longs;i­<lb/>to in de&longs;cen&longs;u DG, excepto primo gradu; </s> <s id="N24505"><!-- NEW -->itemque de&longs;tructus in a&longs;cen&longs;u <lb/>OM æqualis acqui&longs;ito in de&longs;cen&longs;u BO; </s> <s id="N2450B"><!-- NEW -->igitur de&longs;tructus in a&longs;cen&longs;u GK <lb/>e&longs;t duplus de&longs;tructi in a&longs;cen&longs;u OM; </s> <s id="N24511"><!-- NEW -->itaque po&longs;t de&longs;cen&longs;um BO a&longs;cendat <lb/>funependulum in N, ita vt a&longs;cen&longs;us ON &longs;it minor de&longs;cen&longs;u arcu NM: </s> <s id="N24517"><!-- NEW --><lb/>quia &longs;cilicet ad a&longs;cen&longs;um non concurrit impetus innatus: </s> <s id="N2451C"><!-- NEW -->dico quòd po&longs;t <lb/>de&longs;cen&longs;um DG a&longs;cendet tantùm in H; </s> <s id="N24522"><!-- NEW -->ita vt a&longs;cen&longs;us GH &longs;it minor de&longs;­<lb/>cen&longs;u toto arcu HK quadruplo MN: </s> <s id="N24528"><!-- NEW -->porrò tempus a&longs;cen&longs;us per GK <lb/>e&longs;t duplum a&longs;cen&longs;us per OM; </s> <s id="N2452E"><!-- NEW -->& &longs;i concurreret impetus innatus, a&longs;cen­<lb/>&longs;us e&longs;&longs;et æqualis de&longs;cen&longs;ui per &longs;e; </s> <s id="N24534"><!-- NEW -->igitur perueniret in K; </s> <s id="N24538"><!-- NEW -->igitur &longs;i non <lb/>concurrat vno tempore dee&longs;t &longs;patium NM, vel IK, id e&longs;t toto eo tem­<lb/>pore, quo a&longs;cendit pendulum AO; </s> <s id="N24540"><!-- NEW -->impetus innatus cum aliis concur­<lb/>rens ad a&longs;cen&longs;um promoueret mobile toto &longs;patio NM, quod dee&longs;t tan­<lb/>tùm defectu illius concur&longs;us; </s> <s id="N24548"><!-- NEW -->igitur, &longs;i æquali tempore non concurrat ad <lb/>a&longs;cen&longs;um GK; </s> <s id="N2454E"><!-- NEW -->certè ex a&longs;cen&longs;u detrahetur tantùm IK æqualis v.g. <!-- REMOVE S-->MN; </s> <s id="N24554"><!-- NEW --><lb/>&longs;i verò duobus temporibus æqualibus non concurrat; </s> <s id="N24559"><!-- NEW -->certè ex a&longs;cen&longs;u <pb pagenum="326" xlink:href="026/01/360.jpg"/>detrahetur HK quadruplum MN; </s> <s id="N24562"><!-- NEW -->nam &longs;icut idem impetus concurrens <lb/>duobus temporibus addit quadruplum &longs;patium propter motum accele­<lb/>ratum; </s> <s id="N2456A"><!-- NEW -->ita &longs;i non concurrat duobus temporibus, deerit &longs;patium qua­<lb/>druplum illius quod dee&longs;&longs;et, &longs;i tantùm vno tempore non concurreret; </s> <s id="N24570"><!-- NEW --><lb/>igitur a&longs;cen&longs;us maioris funependuli erit OH; </s> <s id="N24575"><!-- NEW -->igitur OH, ON erunt vi­<lb/>brationes &longs;imiles; </s> <s id="N2457B"><!-- NEW -->igitur &longs;i de&longs;cendat AG ex H, & AO ex N, vibrationes <lb/>a&longs;cen&longs;us &longs;ecundi erunt adhuc &longs;imiles propter <expan abbr="cãdem">eandem</expan> rationem; </s> <s id="N24585"><!-- NEW -->igitur <lb/>& vibrationes tertij a&longs;cen&longs;us, quarti, quinti, atque ita deinceps; </s> <s id="N2458B"><!-- NEW -->igitur tot <lb/>erunt vibrationes maioris, quot minoris per &longs;e, &longs;i prima vtriu&longs;que vi­<lb/>bratio &longs;it &longs;imilis: dixi per &longs;e; nam per accidens ratione funis ferè &longs;emper <lb/>accidit mutari i&longs;tum ordinem vibrationum. </s> </p> <p id="N24595" type="main"> <s id="N24597"><!-- NEW -->Præterea, cùm impetus, quo pendulum maius a&longs;cendit per GK, <lb/>&longs;it duplus illius, quo minus a&longs;cendit per OM, cùm in &longs;ingulis punctis <lb/>a&longs;cen&longs;us OM, & &longs;ingulis a&longs;cen&longs;us GK de&longs;truatur impetus; </s> <s id="N2459F"><!-- NEW -->cum GK &longs;it <lb/>quadruplum OM; </s> <s id="N245A5"><!-- NEW -->certè in &longs;ingulis punctis GK impetus de&longs;truitur &longs;ub­<lb/>duplus illius, qui in &longs;ingulis punctis OM de&longs;truitur; &longs;i enim æqualis; </s> <s id="N245AB"><!-- NEW -->igi­<lb/>tur impetus per GK e&longs;&longs;et quadruplus impetus per OM; </s> <s id="N245B1"><!-- NEW -->&longs;i minor &longs;ubdu­<lb/>plo v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;ubquadruplus; </s> <s id="N245BB"><!-- NEW -->igitur impetus per GK e&longs;&longs;et æqualis impetui <lb/>per OM; </s> <s id="N245C1"><!-- NEW -->&longs;ed e&longs;t tantum duplus; </s> <s id="N245C5"><!-- NEW -->igitur &longs;ubduplus de&longs;truitur in &longs;ingulis <lb/>punctis, igitur in æquali punctorum GK numero, &longs;ubduplus tantùm im­<lb/>petus de&longs;trueretur; </s> <s id="N245CD"><!-- NEW -->in duplo punctorum numero, æqualis, in quadruplo <lb/>punctorum numero, duplus; </s> <s id="N245D3"><!-- NEW -->de&longs;truitur autem in &longs;ingulis punctis GK <lb/>&longs;ubduplus; quia &longs;ubduplum tantùm tempus re&longs;pondet &longs;ingulis punctis G <lb/>K illius temporis, quod re&longs;pondet &longs;ingulis punctis OM. </s> </p> <p id="N245DB" type="main"> <s id="N245DD"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N245EA" type="main"> <s id="N245EC">Primò colligo, &longs;olutionem primi dubij propo&longs;iti &longs;uprà, ita vt non iam <lb/>dubium, at certum omninò &longs;uper&longs;it. </s> </p> <p id="N245F1" type="main"> <s id="N245F3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24600" type="main"> <s id="N24602"><!-- NEW -->Secundò colligo, &longs;olutionem &longs;ecundi dubij, &longs;i enim funependulum P <lb/>G demittatur ex PR & AG ex AT; </s> <s id="N24608"><!-- NEW -->haud dubiè plures erunt vibrationes <lb/>penduli PG, quàm AG; </s> <s id="N2460E"><!-- NEW -->quia tot e&longs;&longs;ent AG, quot PG, &longs;i AG demittere­<lb/>tur ex AD; </s> <s id="N24614"><!-- NEW -->&longs;ed plures &longs;unt vibrationes funependuli AG demi&longs;&longs;i ex AD, <lb/>quàm eiu&longs;dem ex AT; </s> <s id="N2461A"><!-- NEW -->ergo plures funependuli PG demi&longs;&longs;i ex PR, <lb/>quàm AG demi&longs;&longs;i ex AT; vnde &longs;oluitur prima pars dubij &longs;ecundi, </s> </p> <p id="N24620" type="main"> <s id="N24622"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2462F" type="main"> <s id="N24631"><!-- NEW -->Tertiò colligo, &longs;olutionem &longs;ecundæ partis eiu&longs;dem dubij; </s> <s id="N24635"><!-- NEW -->&longs;i enim A <lb/>O demittatur ex AB, & AG ex AS; </s> <s id="N2463B"><!-- NEW -->ita arcus GS &longs;it æqualis arcui OB; </s> <s id="N2463F"><!-- NEW --><lb/>certè erunt plures vibrationes AO, quàm AG, vt patet ex dictis; quod <lb/>&longs;pectat ad tertium dubium, illud ip&longs;um &longs;oluemus paulò pò&longs;t. </s> </p> <p id="N24646" type="main"> <s id="N24648"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24655" type="main"> <s id="N24657"><!-- NEW -->Quartò &longs;i demittatur funependulum AG ex AV, vt de&longs;cendat in A <lb/>G, &longs;itque clauus horizonti parallelus in P, non a&longs;cendet &longs;egmentum PG <lb/>in PR, vt vult Galileus; </s> <s id="N2465F"><!-- NEW -->quia AV non a&longs;cenderet in AT, quod ip&longs;e &longs;up-<pb pagenum="327" xlink:href="026/01/361.jpg"/>ponit; </s> <s id="N24668"><!-- NEW -->atqui &longs;uprà demon&longs;trauimus a&longs;cen&longs;um minorem e&longs;&longs;e de&longs;cen&longs;u, non <lb/>tantùm propter re&longs;i&longs;tentiam aëris, vt vult ip&longs;e Galileus; &longs;ed propter prin­<lb/>cipium intrin&longs;ecum de&longs;tructiuum impetus acqui&longs;iti in de&longs;cen&longs;u, de quo <lb/>&longs;uprà. </s> </p> <p id="N24672" type="main"> <s id="N24674"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24681" type="main"> <s id="N24683"><!-- NEW -->Quintò equidem, &longs;i AG demittatur ex AR, affixo clauo in P, non <lb/>modò &longs;egmentum PG a&longs;cendet in PR, verùm etiam altiùs a&longs;cendet ver­<lb/>&longs;us A; immò gyri plures erunt, &longs;i clauus affigatur propiùs ad punctum <lb/>G, qui certè gyri quò minores erunt, eò citiùs conficientur. </s> </p> <p id="N2468D" type="main"> <s id="N2468F"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2469C" type="main"> <s id="N2469E"><!-- NEW -->Pote&longs;t determinari numerus i&longs;torum gyrorum; </s> <s id="N246A2"><!-- NEW -->&longs;it enim primò clauus <lb/>in P, &longs;int que APG æquales; </s> <s id="N246A8"><!-- NEW -->&longs;i AG po&longs;t de&longs;cen&longs;um KG a&longs;cenderet in <lb/>AD; </s> <s id="N246AE"><!-- NEW -->certè &longs;egmentum PG a&longs;cenderet per &longs;emicircumferentiam GR <lb/>A, in PA, ratio e&longs;t, quia pendulum G de&longs;cendent ex K; </s> <s id="N246B4"><!-- NEW -->&longs;ed ex hypothe&longs;i <lb/>Galilei &longs;i de&longs;cenderet ex Y a&longs;cenderet in R; </s> <s id="N246BA"><!-- NEW -->igitur cùm a&longs;cendet cum illo <lb/>impetu acqui&longs;ito in de&longs;cen&longs;u KG, a&longs;cenderet in D ex hypothe&longs;i Galilei; <lb/>&longs;ed arcus GD e&longs;t æqualis GRA. </s> </p> <p id="N246C2" type="main"> <s id="N246C4"><!-- NEW -->Ob&longs;eruabis primò iuxta no&longs;tram hypothe&longs;im, qua diximus pendulum <lb/>AG po&longs;t de&longs;cen&longs;um per KG non a&longs;cendere in D, vix po&longs;&longs;e cogno&longs;ci <lb/>affixo clauo, ad quod punctum circuli GRA ex G pendulum peruentu­<lb/>rum &longs;it; </s> <s id="N246CE"><!-- NEW -->&longs;i enim per GD a&longs;cendat in F, & a&longs;&longs;umatur AZ æqualis DF, non <lb/>dee&longs;&longs;ent fortè, qui exi&longs;timarent arcum a&longs;cen&longs;us per GRA e&longs;&longs;e GRZ <lb/>æqualem GF; </s> <s id="N246D6"><!-- NEW -->&longs;ed plùs impetus de&longs;truitur in arcu GRZ, quàm in arcu <lb/>GF, vt patet ex dictis; </s> <s id="N246DC"><!-- NEW -->nam nullum e&longs;t punctum in arcu GF, in quo plùs <lb/>impetus de&longs;truatur, quàm in alio dato arcus GRZ; </s> <s id="N246E2"><!-- NEW -->cùm tamen &longs;int ali­<lb/>qua puncta in arcu GRZ, in quibus plùs impetus de&longs;truitur, quàm in <lb/>arcu GF, v.g. <!-- REMOVE S-->in puncto R; </s> <s id="N246EC"><!-- NEW -->itaque ducatur FD parallela AD: </s> <s id="N246F0"><!-- NEW -->dico quòd <lb/>perueniet pendulum in <foreign lang="greek">d</foreign>; </s> <s id="N246FA"><!-- NEW -->quippe cum eodem impetu ad <expan abbr="eãdem">eandem</expan> altitu­<lb/>dinem a&longs;cenditur; quæ omnia certa &longs;unt. </s> </p> <p id="N24704" type="main"> <s id="N24706"><!-- NEW -->Ob&longs;eruabis &longs;ecundò, &longs;i affigatur clauus in <foreign lang="greek">q</foreign>, &longs;intque P<foreign lang="greek">q</foreign>G æquales, ex <lb/>hypothe&longs;i Galilei; </s> <s id="N24714"><!-- NEW -->pendulum G primò ex G perueniet in P, cum eo &longs;cili­<lb/>cet impetu, quo perueniet in T; </s> <s id="N2471A"><!-- NEW -->tùm deinde ex P per <foreign lang="greek">b</foreign> redit in G aucto <lb/>&longs;cilicet impetu in de&longs;cen&longs;u P <foreign lang="greek">b</foreign> G, & ex G iterum a&longs;cendit in P; atque ita <lb/>deinceps; quippe gyri perennes e&longs;&longs;ent, ni&longs;i tandem totum filum circa <lb/>clauum conuolueretur. </s> </p> <p id="N2472C" type="main"> <s id="N2472E"><!-- NEW -->Ob&longs;eruabis præterea, aliquid &longs;imile contingere, cum pondus filo pen­<lb/>dulum in gyros, agimus circa mobilem digitum, v.g. <!-- REMOVE S-->quippe vltimi gyri <lb/>citiùs ab&longs;oluuntur; </s> <s id="N24738"><!-- NEW -->quia &longs;cilicet breuiores &longs;unt, &longs;ed hæc &longs;unt facilia; </s> <s id="N2473C"><!-- NEW -->ob­<lb/>&longs;eruabis tamen cum voluitur filum illud circa digitum pendulum, non <lb/>moueri motu circulari, &longs;ed &longs;pirali; vnde cùm motus mixtus &longs;it, in librum <lb/>&longs;equentem reiicimus. </s> </p> <p id="N24746" type="main"> <s id="N24748"><!-- NEW -->Ob&longs;eruabis deinde, cum pendulum AG de&longs;cendit ex V in G, & prop­<lb/>ter clauum, à quo retinetur, filum a&longs;cendit in R, a&longs;cen&longs;um GR ferri bre­<lb/>uiore tempore, quàm a&longs;cen&longs;um GT; </s> <s id="N24750"><!-- NEW -->quia a&longs;cen&longs;us GT & GD æquali &longs;e-<pb pagenum="328" xlink:href="026/01/362.jpg"/>rè tempore peraguntur; </s> <s id="N24759"><!-- NEW -->&longs;unt enim vibrationes eiu&longs;dem funependuli; </s> <s id="N2475D"><!-- NEW --><lb/>quippe licèt minor vibratio minore tempore fiat; </s> <s id="N24762"><!-- NEW -->illud tamen &longs;en&longs;u di&longs;­<lb/>cerni non pote&longs;t, ni&longs;i in &longs;erie multarum vibrationum; </s> <s id="N24768"><!-- NEW -->atqui GR perfici­<lb/>tur æquali tempore, &longs;iue pendulum de&longs;cendat ex V; &longs;iue ex Y; </s> <s id="N2476E"><!-- NEW -->acquiritur <lb/>enim æqualis impetus vtroque modo; </s> <s id="N24774"><!-- NEW -->&longs;ed a&longs;cen&longs;us GR fieret æquali <lb/>tempore cum de&longs;cen&longs;u YG; </s> <s id="N2477A"><!-- NEW -->hic verò breuiore, quàm VG, vt patet; &longs;unt <lb/>enim numeri vibrationum, vt radices longitudinum. </s> </p> <p id="N24780" type="main"> <s id="N24782"><!-- NEW -->Ob&longs;eruabis denique po&longs;&longs;e funependulum, PG &longs;olidum demitti ex A, <lb/>&longs;i tantillùm inclinctur; fed de hoc funependulorum genere agemus <lb/>infrà. </s> </p> <p id="N2478A" type="main"> <s id="N2478C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N24798" type="main"> <s id="N2479A"><!-- NEW --><emph type="italics"/>Funependulum in fine a&longs;cen&longs;us non quie&longs;cit vno in&longs;tanti<emph.end type="italics"/>; </s> <s id="N247A3"><!-- NEW -->quia numquam <lb/>ad perfectam æqualitatem peruenitur; quod eodem modo probatur, <lb/>quo &longs;uprà l. <!-- REMOVE S-->3. e&longs;t enim par ratio. </s> </p> <p id="N247AD" type="main"> <s id="N247AF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N247BB" type="main"> <s id="N247BD"><!-- NEW --><emph type="italics"/>Figura penduli multum facit ad motum vibrationis<emph.end type="italics"/>: </s> <s id="N247C6"><!-- NEW -->&longs;phærica omnium <lb/>ferè apti&longs;&longs;ima e&longs;t præter Conchoidem, & eam, quæ con&longs;taret ex duobus <lb/>conis in communi ba&longs;i coniunctis, vel in gemina pyramide; ratio con&longs;tat <lb/>ex cis, quæ diximus de motu naturali. </s> </p> <p id="N247D0" type="main"> <s id="N247D2"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N247DE" type="main"> <s id="N247E0"><!-- NEW --><emph type="italics"/>Funis multùm etiam facit<emph.end type="italics"/>; </s> <s id="N247E9"><!-- NEW -->omnium optimus e&longs;t tenui&longs;&longs;imus, qui &longs;ci­<lb/>licet faciliùs aëra &longs;ecat; </s> <s id="N247EF"><!-- NEW -->nec enim dubium e&longs;t, quin huic diui&longs;ioni re&longs;i&longs;tat <lb/>aër, cuius re&longs;i&longs;tentiæ analogiam videmus in aqua, quam funis oblongus <lb/>non ni&longs;i cum &longs;en&longs;ibili re&longs;i&longs;tentia diuidit, vt videre e&longs;t in iis funibus, qui­<lb/>bus ab equis naues trahuntur; </s> <s id="N247F9"><!-- NEW -->aliqui adhibent ductum auri filum; </s> <s id="N247FD"><!-- NEW -->&longs;ed <lb/>vnum præ&longs;ertim ob&longs;eruandum e&longs;t, &longs;cilicet ne præ nimia tenuitate maio­<lb/>ris fortè vi ponderis vlterius ducatur, vel dilatetur; </s> <s id="N24805"><!-- NEW -->vtrumque enim mo­<lb/>tum vibrationis retardat: </s> <s id="N2480B"><!-- NEW -->immò pendulum ip&longs;um non de&longs;criberet &longs;emi­<lb/>circulum; an verò &longs;emiellyp&longs;im vt volunt aliqui, definiemus &longs;uo loco, <lb/>cum de lineis motus. </s> </p> <p id="N24813" type="main"> <s id="N24815"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N24821" type="main"> <s id="N24823"><!-- NEW --><emph type="italics"/>Pondus funependuli multùm facit ad vibrationis motum<emph.end type="italics"/>; </s> <s id="N2482C"><!-- NEW -->&longs;i enim granu­<lb/>lum plumbeum appendatur, vix &longs;uperabit re&longs;i&longs;tentiam funis, qui vt vi­<lb/>bretur, optimè ten&longs;us e&longs;&longs;e debet; atqui notabili pondere tendi tantùm <lb/>pote&longs;t. </s> </p> <p id="N24836" type="main"> <s id="N24838"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N24844" type="main"> <s id="N24846"><!-- NEW --><emph type="italics"/>Materia funependuli multùm etiam facit ad vibrationis motum &longs;uppo&longs;ita <lb/>&longs;cilicet eadem figura<emph.end type="italics"/>; </s> <s id="N24851"><!-- NEW -->quippe tam leuis e&longs;&longs;e po&longs;&longs;et materia, vt nec aëris <lb/>vim nec funis re&longs;i&longs;tentiam &longs;uperaret: </s> <s id="N24857"><!-- NEW -->hinc globus &longs;ubereus vel è &longs;ambu­<lb/>cea medulla con&longs;tans, tardiùs de&longs;cendit, quàm plumbeus; </s> <s id="N2485D"><!-- NEW -->habes apud <lb/>Mer&longs;ennum has proportiones; </s> <s id="N24863"><!-- NEW -->globus plumbeus pendulus fune pedum <lb/>3. 1/2 è &longs;ummo quadrantis arcu demi&longs;&longs;us a&longs;cendit per arcum oppo&longs;itum <pb pagenum="329" xlink:href="026/01/363.jpg"/>æqualem minus vno digito; </s> <s id="N2486E"><!-- NEW -->&longs;ubereus verò minus 4/9 arcus quadrantis; </s> <s id="N24872"><!-- NEW --><expan abbr="&longs;ã-buceus">&longs;am­<lb/>buceus</expan> minus 6/7 cereus minus tribus digitis; </s> <s id="N2487B"><!-- NEW -->addit præterea <expan abbr="plumbeũ">plumbeum</expan> in <lb/>perpendiculo conficere 48. pedes tempore duorum &longs;ecundorum, cereum <lb/>paulò maiore tempore; </s> <s id="N24887"><!-- NEW -->quod tamen percipi non pote&longs;t; </s> <s id="N2488B"><!-- NEW -->&longs;ubereum in eo­<lb/>dem &longs;patio percurrendo ponere tria &longs;ecunda medullarum 5. ve&longs;icam pi&longs;­<lb/>cis inflatam 8. &longs;ed hæc accuratè ob&longs;eruari non po&longs;&longs;unt; </s> <s id="N24893"><!-- NEW -->&longs;i enim dicam <lb/>&longs;upere&longs;&longs;e, vel dee&longs;&longs;e aliquid, vel &longs;patij, vel temporis, quod tamen &longs;en&longs;u <lb/>minimè percipiatur; quis e&longs;t qui contrarium probare po&longs;&longs;it. </s> </p> <p id="N2489B" type="main"> <s id="N2489D"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N248A9" type="main"> <s id="N248AB">Ob&longs;eruabis primò non e&longs;&longs;e omittendum, quod habet Galileus in dia­<lb/>logis, & facilè ex dictis colligi pote&longs;t, &longs;cilicet pendula diuer&longs;æ longitu­<lb/>dini sita po&longs;&longs;e componi, vt vnum vnicam vibrationem efficiat, dum aliud <lb/>percurrit 2. vel 3. &c. </s> <s id="N248B4"><!-- NEW -->atque ita haberi po&longs;&longs;e quemdam oculorum qua&longs;i <lb/>concentum non &longs;onorum &longs;ed motuum, v.g. <!-- REMOVE S-->&longs;i &longs;it alter funis longus 4. pe­<lb/>des; </s> <s id="N248BE"><!-- NEW -->alter verò vnum, pendulum ex illo duas percurret; </s> <s id="N248C2"><!-- NEW -->quia numeri <lb/>vibrationum &longs;unt, vt tempora; </s> <s id="N248C8"><!-- NEW -->hæc verò &longs;ubduplicata longitudinum; </s> <s id="N248CC"><!-- NEW -->hîc <lb/>autem vides quadam &longs;peciem diapa&longs;on, cuius proportio in his numeris <lb/>po&longs;ita e&longs;t 1/2; </s> <s id="N248D4"><!-- NEW -->&longs;i vero aliud funependulum &longs;it longum 9. pedes, conficiet <lb/>& alterum vnum hoc eodem tempore tres vibrationes; </s> <s id="N248DA"><!-- NEW -->&longs;i &longs;it aliud, 16. <lb/>pedes longum, & alterum vnum; </s> <s id="N248E0"><!-- NEW -->hec eodem tempore conficiet 4. vibra­<lb/>tiones, atque ita deinceps poteris habere quamlibet proportionem in <lb/>numeris vibrationum ex ip&longs;a combinationum regula; &longs;ed profectò non <lb/>magnam voluptatem ex hac qua&longs;i oculorum mu&longs;icâ percipies, &longs;altem <lb/>ego modicam percipere potui. </s> </p> <p id="N248EC" type="main"> <s id="N248EE"><!-- NEW -->Ob&longs;eruabis &longs;ecundò, hactenus actum e&longs;&longs;e à nobis de primo funepen­<lb/>dulorum genere &longs;atis longa tractatione; iam ergo &longs;upere&longs;t, vt de aliis <lb/>agamus. </s> </p> <p id="N248F6" type="main"> <s id="N248F8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N24904" type="main"> <s id="N24906"><!-- NEW --><emph type="italics"/>Pondus pendulum contorto fune gyros agit reciprocos in plano horizontali<emph.end type="italics"/>; </s> <s id="N2490F"><!-- NEW --><lb/>ratio petitur tantùm ex compre&longs;&longs;ione intorti funis, qui dum &longs;e &longs;e redu­<lb/>cit ad pri&longs;tinum &longs;tatum, pendulum pondus in gyros agit; </s> <s id="N24916"><!-- NEW -->cum verò acce­<lb/>leretur motus, & nouus &longs;emper accedat impetus, pendulum ip&longs;um funi <lb/>etiam pri&longs;tino &longs;tatui re&longs;tituto qua&longs;i primam gratiam refert, cùm impe­<lb/>tum in eum refundat; </s> <s id="N24920"><!-- NEW -->&longs;i enim funis &longs;olus ade&longs;&longs;et nullo pendulo pondere <lb/>ten&longs;us; </s> <s id="N24926"><!-- NEW -->haud dubiè &longs;tatim quie&longs;ceret, vbi &longs;ublata e&longs;&longs;et compre&longs;&longs;io; </s> <s id="N2492A"><!-- NEW -->at verò <lb/>quia impetus ponderi pendulo impre&longs;&longs;us adhuc durat funem ip&longs;um in <lb/>contrariam partem intorquet; donec tandem po&longs;t multos gyros repeti­<lb/>tos pendulum pondus quie&longs;cat. </s> </p> <p id="N24934" type="main"> <s id="N24936"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24942" type="main"> <s id="N24944"><!-- NEW -->Ob&longs;eruabis plures e&longs;&longs;e huius funependuli motus affectiones, quæ certè <lb/>demon&longs;trari po&longs;&longs;unt; quia tamen cau&longs;a huius, qui &longs;equitur ex compre&longs;­<lb/>&longs;ione e&longs;t noua potentia motrix, quàm mediam vocamus, cuius mirifica <lb/>vis vix cogno&longs;ci pote&longs;t, ni&longs;i probè cogno&longs;catur ratio den&longs;i, rari, &c. </s> <s id="N2494E">tra-<pb pagenum="330" xlink:href="026/01/364.jpg"/>ctationem hanc in alium Tomum reiicimus, in quo fusè agemus de om­<lb/>nibus affectionibus huius potentiæ. </s> </p> <p id="N24958" type="main"> <s id="N2495A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N24966" type="main"> <s id="N24968"><!-- NEW --><emph type="italics"/>Corpus oblongum flexibile in altera extremitate immobiliter affixum, &longs;i in­<lb/>curuetur non modò reducit &longs;e&longs;e ad pri&longs;tinum &longs;tatum, verùm etiam multas <lb/>tremulas vibrationes hinc inde facit<emph.end type="italics"/>; </s> <s id="N24975"><!-- NEW -->quarum cau&longs;a e&longs;t motus acceleratus <lb/>eiu&longs;dem potentiæ motricis mediæ; has quoque vibrationes remitti­<lb/>mus. </s> </p> <p id="N2497D" type="main"> <s id="N2497F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s> </p> <p id="N2498B" type="main"> <s id="N2498D"><!-- NEW --><emph type="italics"/>Funis ten&longs;us in vtraque extremitate affixus, &longs;i pul&longs;etur infinitas ferè tremu­<lb/>la&longs;que vibrationes hinc inde peragit<emph.end type="italics"/>; &longs;unt etiam mirabiles harum vibra­<lb/>tionum affectiones, quas multis Theorematis in eodem volumine pro­<lb/>&longs;equemur. </s> </p> <p id="N2499C" type="main"> <s id="N2499E">Diceret aliquis; </s> <s id="N249A1"><!-- NEW -->igitur in hoc tractatu omnia, quæ &longs;pectant ad motum <lb/>non habentur; Re&longs;pondeo, tractatum hunc e&longs;&longs;e poti&longs;&longs;imum in&longs;titutum <lb/>ad demon&longs;trandas omnes affectiones tùm motus grauium, tùm motus <lb/>impre&longs;&longs;i à principio extrin&longs;eco, intactis pror&longs;us iis motibus, qui &longs;unt vel <lb/>à potentia motrice animantium, in ip&longs;is dumtaxat animantibus, de qui­<lb/>bus agemus &longs;uo loco, quales &longs;unt progredi, currere, volare, notare, repe­<lb/>re, &c. </s> <s id="N249B1"><!-- NEW -->vel à leuitate corporum, &longs;i fortè aliquis motus e&longs;t à leuitate, <lb/>quod hîc non di&longs;cutio, &longs;ed remitto in librum de graui & leui; </s> <s id="N249B7"><!-- NEW -->vel de­<lb/>nique ab illa potentiâ mediâ, cui omnes motus ten&longs;orum; compre&longs;&longs;orum, <lb/>arcuum; reique tormentariæ tùm hydraulicæ, pneumaticæ, &c. </s> <s id="N249BF">tribue­<lb/>mus: </s> <s id="N249C4"><!-- NEW -->de his certè motibus in hoc tractatu non agemus; </s> <s id="N249C8"><!-- NEW -->quia cùm non <lb/>po&longs;&longs;int demon&longs;trari illorum affectiones, ni&longs;i cogno&longs;cantur illorum cau­<lb/>&longs;æ; </s> <s id="N249D0"><!-- NEW -->neque hæ cogno&longs;ci po&longs;&longs;int, ni&longs;i multa alia cogno&longs;cantur, vt certi&longs;&longs;i­<lb/>mum e&longs;t; minùs prudenter factum e&longs;&longs;et, &longs;i de iis hoc loco di&longs;putatio <lb/>in&longs;titueretur. </s> </p> <p id="N249D8" type="main"> <s id="N249DA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s> </p> <p id="N249E6" type="main"> <s id="N249E8"><!-- NEW --><emph type="italics"/>E&longs;t aliud corporis libratilis genus<emph.end type="italics"/> &longs;i &longs;it v. <!-- REMOVE S-->g. <!-- REMOVE S-->corpus oblongum, grane, <lb/>& &longs;olidum AF in &longs;itu horizontali innixum plano verticali EBCD; </s> <s id="N249F7"><!-- NEW -->&longs;i <lb/>enim extremitas F attollatur per arcum FG circa centrum B; </s> <s id="N249FD"><!-- NEW -->haud du­<lb/>biè altera A deprimetur per arcum AI circa idem centrum B; </s> <s id="N24A03"><!-- NEW -->at &longs;tatim <lb/>G de&longs;cendet motu naturaliter accelerato in F, & propter acqui&longs;itum in <lb/>de&longs;cen&longs;u, de&longs;cendet infra horizontalem GF per arcum FH, circa cen­<lb/>trum C, & I a&longs;cendet in A, tùm in K, &longs;ed K &longs;tatim de&longs;cendet, atque ita <lb/>deinceps; </s> <s id="N24A0F"><!-- NEW -->donec tandem po&longs;t multas vibrationes quie&longs;cat AF in &longs;itu <lb/>horizontali; </s> <s id="N24A15"><!-- NEW -->porrò G de&longs;cendit, quia GI non e&longs;t in æquilibrio, cùm <lb/>centrum grauitatis &longs;it in M; igitur BG, quæ e&longs;t longior BI, de&longs;cen­<lb/>det. </s> </p> <p id="N24A1D" type="main"> <s id="N24A1F"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24A2C" type="main"> <s id="N24A2E">Primò colligo, motum accelerari in de&longs;cen&longs;u GF, quia impetus acqui­<lb/>&longs;itus in G remanet adhuc in Q, & nouus acquiritur, vt &longs;æpe dictum e&longs;t. </s> </p> <pb pagenum="331" xlink:href="026/01/365.jpg"/> <p id="N24A37" type="main"> <s id="N24A39"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24A46" type="main"> <s id="N24A48"><!-- NEW -->Secundò, impetum acqui&longs;itum in G e&longs;&longs;e minorem acqui&longs;ito in <expan abbr="q;">que</expan> & <lb/>acqui&longs;itum in Q minorem acqui&longs;ito in F; quia momentum in G e&longs;t ad <lb/>momentum in E, vt OB ad FB, vt &longs;uprà dictum e&longs;t multis locis. </s> </p> <p id="N24A54" type="main"> <s id="N24A56"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24A63" type="main"> <s id="N24A65"><!-- NEW -->Tertiò colligo, e&longs;&longs;e inuer&longs;as rationes accelerationis in funependulo, & <lb/>in priori, quod vibratur in plano verticali: quippe in i&longs;to impetus ac­<lb/>qui&longs;itus in &longs;uperiore arcu e&longs;t maior acqui&longs;ito in inferiore, &longs;ecus in illo. </s> </p> <p id="N24A6D" type="main"> <s id="N24A6F"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24A7C" type="main"> <s id="N24A7E">Quartò colligo, i&longs;tas vibrationes non e&longs;&longs;e perpetuas, quia &longs;ecunda e&longs;t <lb/>minor prima, & tertia minor &longs;ecunda, atque ita deinceps propter ratio­<lb/>nem, quam attulimus &longs;uprà. </s> </p> <p id="N24A85" type="main"> <s id="N24A87"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24A94" type="main"> <s id="N24A96"><!-- NEW -->Quintò colligo, vibrationes minores fieri citiùs, quàm maiores, v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->QF quàm GF, quod multis con&longs;tat experimentis, & ratio e&longs;t manife&longs;ta; </s> <s id="N24A9F"><!-- NEW --><lb/>quia QF &longs;it æqualis QG; certè QF accedit propiùs ad perpendicularem <lb/>quàm GQ; </s> <s id="N24A9G"> igitur cùm &longs;it æqualis, breuiore tempore percurretur, quod <lb/>clari&longs;&longs;imum e&longs;t. </s> </p> <p id="N24AAC" type="main"> <s id="N24AAE"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24ABB" type="main"> <s id="N24ABD"><!-- NEW -->Sextò colligo, ea&longs;dem vel &longs;imiles &longs;equi &longs;i AF &longs;u&longs;pendatur ex LN; e&longs;t <lb/>enim pror&longs;us eadem ratio. </s> </p> <p id="N24AC3" type="main"> <s id="N24AC5"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N24AD1" type="main"> <s id="N24AD3"><!-- NEW -->Septimò colligo, alia corpora etiam cubica, vel alterius figuræ plano <lb/>horizontali v. <!-- REMOVE S-->g. <!-- REMOVE S-->ip&longs;i &longs;olo incubantia, &longs;i tantillùm è &longs;uo &longs;itu remouean­<lb/>tur per &longs;imiles vibrationes &longs;e&longs;e in illum re&longs;tituere; immò ex minima <lb/>percu&longs;&longs;ione multis huiu&longs;modi vibrationibus percu&longs;&longs;um corpus contre­<lb/>mi&longs;cit. </s> </p> <p id="N24AE3" type="main"> <s id="N24AE5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s> </p> <p id="N24AF1" type="main"> <s id="N24AF3"><!-- NEW --><emph type="italics"/>Si corpus &longs;olidum pendulum circa punctum immobile ita voluatur, vt ex <lb/>verticali &longs;itu amoueatur; </s> <s id="N24AFB"><!-- NEW -->haud dubiè de&longs;cendet, a&longs;cendetque per vibrationes <lb/>repetitas<emph.end type="italics"/>; </s> <s id="N24B04"><!-- NEW -->& hoc e&longs;t vltimum vibrationum genus, quarum eadem e&longs;t <lb/>pror&longs;us ratio, & cau&longs;a, quam &longs;uperioribus tribuimus, iis &longs;cilicet, quæ in <lb/>plano verticali à pendulo pondere de&longs;cribuntur; </s> <s id="N24B0C"><!-- NEW -->nam in vtroque genere <lb/>vibrationum primò acceleratur motus; </s> <s id="N24B12"><!-- NEW -->&longs;ecundò plùs initio, minùs ad fi­<lb/>nem vibrationis, tertiò non &longs;unt perpetuæ vibrationes; </s> <s id="N24B18"><!-- NEW -->quartò ad a&longs;cen­<lb/>&longs;um non concurrit impetus innatus; quintò, impetus de&longs;truitur cum ma­<lb/>iore proportione in maiore vibratione, quàm in minore, &c. </s> <s id="N24B20">quæ vtri­<lb/>que generi &longs;unt communia. </s> </p> <p id="N24B25" type="main"> <s id="N24B27"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s> </p> <p id="N24B33" type="main"> <s id="N24B35"><!-- NEW --><emph type="italics"/>Funependulum, & corpus oblongum eiu&longs;dem longitudinis non de&longs;cendunt <lb/>equè velociter, &longs;i ex eadem altitudine demi&longs;&longs;a circa <expan abbr="centrũ">centrum</expan> immobile vibrentur<emph.end type="italics"/>; </s> <s id="N24B44"><!-- NEW --><pb pagenum="332" xlink:href="026/01/366.jpg"/> &longs;it enim corpus oblongum AB vibratum circa centrum immobile A <lb/>per arcum BC, &longs;itque pendulum pondus C fune CA, demi&longs;&longs;um, & vi­<lb/>bratum per arcum BC; </s> <s id="N24B50"><!-- NEW -->certè tardiùs funependulum hoc arcum BC per­<lb/>curret, quàm corpus oblongum, quod multis experimentis comprobatum <lb/>e&longs;t; </s> <s id="N24B58"><!-- NEW -->ratio e&longs;t, quia in pondere funependulo &longs;olum pondus E cen&longs;eri de­<lb/>bet cau&longs;a motus; </s> <s id="N24B5E"><!-- NEW -->quippe, licèt funis aliquid conferat; </s> <s id="N24B62"><!-- NEW -->quia tamen tam <lb/>exilis e&longs;&longs;e pote&longs;t, vt vix quidquam addat póderis, pro nihilo computatur; </s> <s id="N24B68"><!-- NEW --><lb/>igitur totus motus e&longs;t ab ip&longs;o pondere pendulo; at verò in corpore ob­<lb/>longo AB, quod &longs;it v. <!-- REMOVE S-->g. <!-- REMOVE S-->parallelipedum, vel cylindricum, non tantùm e&longs;t <lb/>motus à puncto B, verùm etiam à punctis FE, &c. </s> <s id="N24B75">cum enim punctum <lb/>F, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i &longs;eor&longs;im &longs;umatur, percurrat arcum FG citiùs quàm punctum B <lb/>&longs;eor&longs;im arcum BC, certè punctum F, qua&longs;i deor&longs;um rapit punctum B igi­<lb/>tur totum corpus AB citiùs ab&longs;oluit &longs;uam vibrationem, quàm funepen­<lb/>dulum, quod erat probandum. </s> </p> <p id="N24B84" type="main"> <s id="N24B86"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s> </p> <p id="N24B92" type="main"> <s id="N24B94"><!-- NEW --><emph type="italics"/>Vt &longs;u&longs;tineatur corpus oblongum AB, faciliùs &longs;u&longs;tinetur in B, quàm in P, <lb/>& in F, quàm in E, & in E quàm in H,<emph.end type="italics"/> atque ita deinceps (&longs;uppono autem, <lb/>quòd po&longs;&longs;it volui circa centrum A) ratio clara e&longs;t ex vecte, de quo &longs;uo <lb/>loco; immò licèt AB penderet tantùm vnam vnciam, po&longs;&longs;et aliquod <lb/>a&longs;&longs;ignari punctum iuxta A, in quo ab homine robu&longs;ti&longs;&longs;imo &longs;u&longs;tineri non <lb/>po&longs;&longs;et in &longs;itu horizontali AB. <!-- KEEP S--></s> </p> <p id="N24BA8" type="main"> <s id="N24BAA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s> </p> <p id="N24BB6" type="main"> <s id="N24BB8"><!-- NEW --><emph type="italics"/>Si de&longs;cendat cylindrus AB in AC circa centrum A, & occurrat in AC <lb/>alteri corpori, ictum maximum infliget ex puncte F, &longs;i AF e&longs;t media pro­<lb/>portionalis inter AE, AB, & habeatur tantum ratio impetus ab&longs;olutè &longs;umpti <emph.end type="italics"/>; </s> <s id="N24BC5"><!-- NEW --><lb/>hoc fuit iucundi&longs;&longs;imum Theorema, quod in lib. 1. demon&longs;trauimus; </s> <s id="N24BCA"><!-- NEW -->ne­<lb/>que hîc repeto; </s> <s id="N24BD0"><!-- NEW -->vnum tantùm addo valdè paradoxon in punctum G e&longs;&longs;e <lb/>maximum ictum, non tamen maximam vim, &longs;cilicet ad mouendum; </s> <s id="N24BD6"><!-- NEW --><lb/>nam in D maior erit vis, quàm in G, & in I, quàm in D; erit tamen mi­<lb/>nor motus, &longs;eu minor impre&longs;&longs;io. </s> </p> <p id="N24BDD" type="main"> <s id="N24BDF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s> </p> <p id="N24BEB" type="main"> <s id="N24BED"><!-- NEW --><emph type="italics"/>In maiori proportione de&longs;truitur impetus in a&longs;cen&longs;u vibrationis eiu&longs;dem <lb/>corporis oblongi, quam in a&longs;cen&longs;it vibrationis funependuli<emph.end type="italics"/>; </s> <s id="N24BFA"><!-- NEW -->con&longs;tat certè cla­<lb/>ri&longs;&longs;imis experimentis; </s> <s id="N24C00"><!-- NEW -->ratio e&longs;t, quia plures partes impetus innati re&longs;i­<lb/>&longs;tunt; quippè impetus innatus funis tam paruus e&longs;t, vt pro nullo ha­<lb/>beatur. </s> </p> <p id="N24C08" type="main"> <s id="N24C0A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s> </p> <p id="N24C16" type="main"> <s id="N24C18"><!-- NEW --><emph type="italics"/>Hinc &longs;unt pauciores vibrationes corporis oblongi, quàm funependuli,<emph.end type="italics"/> cum <lb/>&longs;inguli a&longs;cen&longs;us plùs impetus de&longs;truant in vibrationibus corporis ob­<lb/>longi, quàm funependuli: </s> <s id="N24C2B"><!-- NEW -->Hinc citiùs quie&longs;cit corpus oblongum vibra­<lb/>tum, quàm funependulum; </s> <s id="N24C31"><!-- NEW -->licèt vtrumque ex eadem altitudine demitta­<lb/>tur; quod etiam multis experimentis comprobatur, & ratio patet ex <lb/>dictis. </s> </p> <pb pagenum="333" xlink:href="026/01/367.jpg"/> <p id="N24C3D" type="main"> <s id="N24C3F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s> </p> <p id="N24C4B" type="main"> <s id="N24C4D"><!-- NEW --><emph type="italics"/>Vibrationes minores corporis oblongi citiùs peraguntur, quàm minores<emph.end type="italics"/>; ex­<lb/>perientia certa e&longs;t, ratio verò eadem cum ea, quam explicuimus &longs;uprà <lb/>in funependulis. </s> </p> <p id="N24C5A" type="main"> <s id="N24C5C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s> </p> <p id="N24C68" type="main"> <s id="N24C6A"><!-- NEW --><emph type="italics"/>Minùs producitur impetus in E, v.g. <!-- REMOVE S-->corporis oblongi, &longs;cilicet in de&longs;cen&longs;u, <lb/>quàm &longs;i AE &longs;eparata e&longs;&longs;et ab AB<emph.end type="italics"/>; </s> <s id="N24C77"><!-- NEW -->patet, plùs tamen producitur, quàm &longs;i <lb/>E deferretur à B, vt accidit in funependulis; </s> <s id="N24C7D"><!-- NEW -->prima pars e&longs;t certa; </s> <s id="N24C81"><!-- NEW -->quia <lb/>corpus oblongum AE perficit citiùs &longs;uam vibrationem, quàm AB; &longs;ecun­<lb/>da etiam probatur, quia alioqui vibratio corporis oblongi, & vibratio <lb/>funependuli eiu&longs;dem longitudinis æquali tempore percurreretur. </s> </p> <p id="N24C8B" type="main"> <s id="N24C8D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s> </p> <p id="N24C99" type="main"> <s id="N24C9B"><!-- NEW --><emph type="italics"/>Si punctum H e&longs;&longs;et nodus longè grauior reliquo AB, extremitas B percur­<lb/>reret citius arcum BC, quàm ip&longs;um perpendiculum<emph.end type="italics"/>; </s> <s id="N24CA8"><!-- NEW -->quia &longs;cilicet impetus <lb/>nodi A &longs;eg mentum HB &longs;ecum abriperet; </s> <s id="N24CAE"><!-- NEW -->&longs;ed eo tempore, quo percurri­<lb/>tur arcus HI, non percurritur, perpendiculum æquale arcui BC, vt pa­<lb/>tet; </s> <s id="N24CB6"><!-- NEW -->immò po&longs;&longs;et ita componi corpus oblongum, vt punctum B tùm in <lb/>perpendiculo, tùm in arcu BC, æquè citò moueretur; multa haud <lb/>dubiè dicenda &longs;uper&longs;unt de hoc pendulorum genere, quæ <lb/>remittimus in appendicem, quam huic Tomo <lb/>&longs;ubnectimus. <lb/><figure id="id.026.01.367.1.jpg" xlink:href="026/01/367/1.jpg"/></s> </p> </chap> <chap id="N24CC8"> <pb pagenum="334" xlink:href="026/01/368.jpg"/> <figure id="id.026.01.368.1.jpg" xlink:href="026/01/368/1.jpg"/> <p id="N24CD2" type="head"> <s id="N24CD4"><emph type="center"/>LIBER SECVNDVS, <lb/><emph type="italics"/>DE MOTV NATVRALI.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N24CE1" type="head"> <s id="N24CE3"><emph type="center"/>LIBER NONVS, <lb/><emph type="italics"/>DE MOTV MIXTO EX RECTO, ET <lb/>Circulari, vel ex pluribus Circularibus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N24CF2" type="main"> <s id="N24CF4"><!-- NEW -->MOTVS mixtus e&longs;&longs;e pote&longs;t vel ex recto, <lb/>& circulari, vel ex duobus rectis, & <lb/>circulari, vel ex duobus circularibus, & <lb/>recto, vel ex pluribus circularibus, at­<lb/>que ita deinceps: de iis acturus &longs;um in <lb/>hoc libro, reiectis tamen lineis i&longs;torum motuum in <lb/>Tomum &longs;equentem. <lb/><gap desc="hr tag"/></s> </p> <p id="N24D07" type="main"> <s id="N24D09"><emph type="center"/><emph type="italics"/>DEFINITIO<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24D16" type="main"> <s id="N24D18"><emph type="italics"/>MOtus mixtus ex circulari & recto ille e&longs;t, ad quem concurrit duplex <lb/>impetus, quorum vnus &longs;it determinatus ad motum rectum, & alius <lb/>ad circularem, vel vnus tantum impetus, ad cremam, & rectam lineam &longs;im <lb/>modo determinatus.<emph.end type="italics"/></s> </p> <p id="N24D25" type="main"> <s id="N24D27"><!-- NEW -->Hunc modum explicabimus infrà in Theorematis; </s> <s id="N24D2B"><!-- NEW -->interea definitio, <lb/>&longs;atis clara e&longs;t mihi videtur: exemplum habes in rota, quæ in recto plano <lb/>voluitur. </s> </p> <p id="N24D33" type="main"> <s id="N24D35"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24D42" type="main"> <s id="N24D44"><!-- NEW --><emph type="italics"/>Motus mixtus ex duobus circularibus e&longs;t, ad quem concurvit impetus, vel <lb/>vnicus, vel duplex ad duas lineas circulares determinatus<emph.end type="italics"/>; </s> <s id="N24D4F"><!-- NEW -->&longs;imiliter de­<lb/>finiri pote&longs;t mixtus in duobus, & circulari; duobus circularibus & recto, <lb/>pluribus circularibus. </s> </p> <p id="N24D57" type="main"> <s id="N24D59"><!-- NEW -->Sed quæ&longs;o, cum audis motum mixtum ex duobus, caue credas, duos <lb/>motus ine&longs;&longs;e eidem mobili; quod certè fieri non pote&longs;t, &longs;ed tantùm plu­<lb/>res impetus, vel vnicum ad diuer&longs;as lineas determinatum. </s> </p> <pb pagenum="335" xlink:href="026/01/369.jpg"/> <p id="N24D65" type="main"> <s id="N24D67"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24D74" type="main"> <s id="N24D76"><emph type="italics"/>Illa partes mouentur velociùs, quæ tempore aquali maius &longs;patium acquirunt <lb/>tardiùs verò, que minus &longs;patium, clari&longs;&longs;imum e&longs;t, nec maiori indiget expli­<lb/>catione.<emph.end type="italics"/></s> </p> <p id="N24D82" type="main"> <s id="N24D84"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24D91" type="main"> <s id="N24D93"><!-- NEW --><emph type="italics"/>Cum vtraque determinatio motus ad <expan abbr="eãdem">eandem</expan> partem &longs;pectat, acquiritur <lb/>maius &longs;patium; </s> <s id="N24D9F"><!-- NEW -->tum verò ad diuer&longs;as partes minus, at que ita prorata<emph.end type="italics"/>; hoc <lb/>etiam Axioma certum e&longs;t. </s> </p> <p id="N24DA8" type="main"> <s id="N24DAA"><emph type="center"/><emph type="italics"/>Hypothe&longs;is.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24DB6" type="main"> <s id="N24DB8"><!-- NEW --><emph type="italics"/>Rotæ circa idem centrum mobilis &longs;emicirculi oppo&longs;iti in partes contrarias <lb/>feruntur, motu &longs;cilicet orbis per arcus &longs;cilicet æquales<emph.end type="italics"/>; </s> <s id="N24DC7"><!-- NEW -->nam anguli oppo&longs;iti <lb/>æquales &longs;unt; &longs;ed arcus &longs;unt vt anguli. </s> </p> <p id="N24DCD" type="main"> <s id="N24DCF"><emph type="center"/><emph type="italics"/>Po&longs;tulatum.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24DDB" type="main"> <s id="N24DDD"><emph type="italics"/>Liceat rotare orbem in plana &longs;uperficie, in conuexa, in concaua, in æquali. </s> <s id="N24DE4"><lb/>inæquali, ita vt motus orbis conueniat cum motu centri, vel ab eo diuer&longs;us &longs;it.<emph.end type="italics"/></s> </p> <p id="N24DEA" type="main"> <s id="N24DEC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24DF9" type="main"> <s id="N24DFB"><!-- NEW --><emph type="italics"/>Rota, quæ mouetur in &longs;uperficie plana, mouetur motu mixto ex recto centri <lb/>& circulari orbis<emph.end type="italics"/>; </s> <s id="N24E0A"><!-- NEW -->&longs;it enim AQLZ incubans plano AD in quo rotatur, <lb/>&longs;itque AD recta æqualis arcui <expan abbr="Aq;">Aque</expan> certè po&longs;ito quod motus orbis &longs;it æ­<lb/>qualis motui centri, id e&longs;t po&longs;ito quod æqualibus temporibus &longs;egmentum <lb/>plani percurratur motu centri v.g. <!-- REMOVE S-->QE vel AD æquale arcui, qui circa <lb/>centrum O conuoluitur motu orbis, v.g. <!-- REMOVE S-->arcui AQ, quodlibet punctum <lb/>peripheriæ rotæ mouebitur motu mixto ex recto, & circulari v. <!-- REMOVE S-->g. <!-- REMOVE S-->pun­<lb/>ctum L motu centri fertur ver&longs;us V & motu orbis ver&longs;us Q; &longs;i enim <lb/>e&longs;&longs;et tantum motus centri ver&longs;us E, omnes partes mouerentur motu recto <lb/>v.g. <!-- REMOVE S-->L per rectam LV, A per rectam AD; </s> <s id="N24E30"><!-- NEW -->&longs;i verò e&longs;&longs;et tantùm motus <lb/>orbis, omnes partes mouerentur tantùm motu circulari v. <!-- REMOVE S-->g. <!-- REMOVE S-->L, per ar­<lb/>cum LZ; A per arcum AZ; </s> <s id="N24E3C"><!-- NEW -->at cum &longs;imul &longs;it vterque motus, id e&longs;t vtraque <lb/>determinatio, certè vtraque confert de &longs;uo; igitur e&longs;t motus mixtus. </s> </p> <p id="N24E42" type="main"> <s id="N24E44"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24E51" type="main"> <s id="N24E53"><!-- NEW --><emph type="italics"/>Vnicum tantùm punctum rotæ mouetur metu recto, &longs;cilicet centrum, cætera <lb/>per lineam curuam<emph.end type="italics"/>; </s> <s id="N24E60"><!-- NEW -->de centro con&longs;tat, quia cùm &longs;emper æqualiter di&longs;ter <lb/>à planis AD & LV, &longs;cilicet eodem radio OL, ON; </s> <s id="N24E66"><!-- NEW -->certè percurrit OE <lb/>parallelam vtrique; &longs;ed parallela vtrique e&longs;t recta, punctum verò L mo­<lb/>uetur per lineam curuam, vt con&longs;tabit ex illius de&longs;criptione, quàm tra­<lb/>demus infrà. </s> </p> <p id="N24E72" type="main"> <s id="N24E74"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24E81" type="main"> <s id="N24E83"><!-- NEW --><emph type="italics"/>Si diuidatur arcus LQ in tres arcus aquales & planum AD in tres par­<lb/>tes æquales, pote&longs;t a&longs;&longs;ignari punctum, in quo &longs;it L decur&longs;o prime arcu LK<emph.end type="italics"/>; </s> <s id="N24E92"><!-- NEW -->&longs;i <lb/>enim e&longs;&longs;et tantùm <expan abbr="co&etail;tri">centri</expan>, e&longs;&longs;et in <foreign lang="greek">m</foreign>, &longs;i motus orbis e&longs;&longs;et in K; </s> <s id="N24E9C"><!-- NEW -->igitur <lb/>&longs;it recta MI parallela LV, &longs;itque KI æqualis AB, vel L <foreign lang="greek">m</foreign>; </s> <s id="N24EA6"><!-- NEW -->haud dubiè erit <pb pagenum="336" xlink:href="026/01/370.jpg"/>in I; </s> <s id="N24EAF"><!-- NEW -->nec enim de&longs;cendet infra MI, vt con&longs;tat: </s> <s id="N24EB3"><!-- NEW -->&longs;ic motus orbis dat LK, <lb/>vel MK motus centri L <foreign lang="greek">m</foreign> vel KI; </s> <s id="N24EBD"><!-- NEW -->igitur vterque &longs;imul LI vel KI: </s> <s id="N24EC1"><!-- NEW -->&longs;i­<lb/>militer decur&longs;o arcu KH, punctum rotæ L erit in G; </s> <s id="N24EC7"><!-- NEW -->nam motus orbis <lb/>dat LH, vel NH, vel motus centri AC vel LV; igitur &longs;i a&longs;&longs;umatur HG <lb/>æqualis LV, vterque motus dabit LIG. </s> </p> <p id="N24ECF" type="main"> <s id="N24ED1"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24EDE" type="main"> <s id="N24EE0"><!-- NEW -->Hinc colligo, de&longs;criptionem lineæ, quam &longs;uo motu &longs;eu flux^u de&longs;cri­<lb/>bit punctum L, cuius infinita puncta a&longs;&longs;ignari po&longs;&longs;unt, &longs;i enim diuidatur <lb/>planum æquale arcui LQ in tot partes, in quot diuiditur arcus LQ, & <lb/>cuilibet &longs;inui recto arcus a&longs;&longs;umpti addatur &longs;egmentum plani con&longs;tans <lb/>tot partibus, quot partibus arcus aliis arcubus v.g.&longs;inui MK, KI æqua­<lb/>lis L <foreign lang="greek">m</foreign>, &longs;inui NH, LV, denique &longs;inui toti OQ tota LY, habebuntur &longs;in­<lb/>gula puncta huius lineæ L, I, G, F quam rotatilem appellamus; quæ certè <lb/>eò acuratiùs de&longs;cribetur, quò plura eius puncta &longs;ignabuntur, id e&longs;t quò <lb/>diuidetur arcus LQ in plures arcus, & planum LV in plures partes. </s> </p> <p id="N24EFA" type="main"> <s id="N24EFC"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24F09" type="main"> <s id="N24F0B"><!-- NEW -->Linea quoque rotatilis puncti A de&longs;cribi pote&longs;t diui&longs;o arcu AZ in <lb/>tres arcus, & plano AD in 3. pattes; </s> <s id="N24F11"><!-- NEW -->&longs;int enim &longs;inus TX, Y <foreign lang="greek">p</foreign> &longs;itque TS <lb/>æqualis AB, YR æqualis AG, & ZP æqualis AD; </s> <s id="N24F1B"><!-- NEW -->certè de&longs;cribetur hæc <lb/>linea per puncta ASRP à quo plura puncta &longs;ignabuntur, eò accuratiùs <lb/>de&longs;cribetur, quæ omnia con&longs;tant ex dictis; </s> <s id="N24F23"><!-- NEW -->nam motus orbis dat AT vel <lb/>XT motus centri AB; igitur TS. </s> </p> <p id="N24F29" type="main"> <s id="N24F2B"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24F38" type="main"> <s id="N24F3A"><!-- NEW -->Hinc vides punctum L oppo&longs;itum puncto contactus ita moueri, vt <lb/>motus orbis addatur motui centri; punctum verò A ita mouetur, vt mo­<lb/>tus orbis detrahatur motui centri. </s> </p> <p id="N24F42" type="main"> <s id="N24F44"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24F51" type="main"> <s id="N24F53"><!-- NEW -->Hinc etiam de&longs;cribi pote&longs;t linea, quam de&longs;cribit quodlibet punctum <lb/>interioris circuli v.g. <!-- REMOVE S-->punctum E; </s> <s id="N24F5B"><!-- NEW -->de&longs;cribatur enim arcus quadrátis & 2. <lb/>diuidatur in 3. arcus æquales, ducanturque per puncta &longs;ignata 3.4. rectæ <lb/>parallelæ OE, a&longs;&longs;umatur 3. 5. æqualis L <foreign lang="greek">m</foreign> & 4, 6, æqualis LV; denique <lb/>2.7. æqualis LV, <expan abbr="connectantur&qacute;ue">connectanturque</expan> puncta &longs;ignata per lineam nouam, E <lb/>5.6.7. hæc e&longs;t linea quam de&longs;cribit &longs;uo motu mixto punctum C, quæ <lb/>con&longs;tat ex dictis. </s> </p> <p id="N24F71" type="main"> <s id="N24F73"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24F80" type="main"> <s id="N24F82"><!-- NEW -->Aliter de&longs;cribi pote&longs;t hæc linea rotatilis; </s> <s id="N24F86"><!-- NEW -->&longs;it enim AD diui&longs;a v.g. <!-- REMOVE S-->in <lb/>tres partes æquales, <expan abbr="item&qacute;ue">itemque</expan> OE ex punctis <foreign lang="greek">r</foreign> Q, de&longs;cribantur circuli <lb/>æquales rotæ, <expan abbr="a&longs;&longs;umantur&qacute;ue">a&longs;&longs;umanturque</expan> arcus BS æqualis LK & arcus CR æqua­<lb/>lis LK, & habebis puncta SR: </s> <s id="N24F9E"><!-- NEW -->&longs;imiliter a&longs;&longs;umatur arcus <foreign lang="greek">m</foreign> I æqualis LK <lb/>& alter V.G. æqualis LH, & habebis puncta IG, idem fiet pro aliis pun­<lb/>ctis; hinc vides rotatiles de&longs;cribi po&longs;&longs;e per &longs;inus, & per arcus. </s> </p> <pb pagenum="337" xlink:href="026/01/371.jpg"/> <p id="N24FAE" type="main"> <s id="N24FB0"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N24FBD" type="main"> <s id="N24FBF"><!-- NEW -->Collige punctum L in arcu de&longs;cen&longs;us LQ ita moueri, vt motus orbis <lb/>addat &longs;inus rectos motui centri v.g. <!-- REMOVE S-->motus orbis LK addit &longs;inum rectum <lb/>MK; punctum vero oppo&longs;itum A ita mouetur in arcu AZ, vt motus or­<lb/>bis detrahat &longs;inus rectos motui centri v. <!-- REMOVE S-->g. <!-- REMOVE S-->motus orbis AT detrahit <lb/>&longs;inum XT, punctum Z ita vt a&longs;cendit per arcum ZL, vt motus orbis <lb/>addat motui centri &longs;inus ver&longs;os v. <!-- REMOVE S-->g. <!-- REMOVE S-->motus orbis arcus ZQ addit &longs;inum <lb/>ver&longs;um Z 11. denique punctum oppo&longs;itum Q ita de&longs;cendit per arcum <lb/>QA vt motus orbis detrahat motui &longs;inus ver&longs;os v. <!-- REMOVE S-->g. <!-- REMOVE S-->motus orbis arcus <lb/>QT detrahit &longs;inum ver&longs;um Q 13. hinc vides quàm benè conueniant, <lb/>&longs;ingulæ quadrantes rotæ cuius rei ratio clari&longs;&longs;ima e&longs;t. </s> </p> <p id="N24FE3" type="main"> <s id="N24FE5"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N24FF1" type="main"> <s id="N24FF3">Hinc punctum Z in a&longs;cen&longs;u Z, 10.grad. </s> <s id="N24FF6"><!-- NEW -->60. tantùm addit motui cen­<lb/>tri, quantum L in de&longs;cen&longs;u L, 10.grad.30. a&longs;cen&longs;us verò 10. L grad. <!-- REMOVE S-->30. <lb/>tantum addet quantum a&longs;cen&longs;us 10, Q grad. <!-- REMOVE S-->denique &longs;i accipiatur primus <lb/>arcus a&longs;cen&longs;us addit &longs;inum ver&longs;um, &longs;i vltimus, rectum; at verò primus <lb/>de&longs;cen&longs;us in &longs;emicirculo dumtaxat &longs;uperiore addit &longs;inum rectum, vlti­<lb/>mus ver&longs;um, quæ omnia certi&longs;&longs;imè con&longs;tant. </s> </p> <p id="N25008" type="main"> <s id="N2500A"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25016" type="main"> <s id="N25018">Ob&longs;eruabis hanc e&longs;&longs;e liueam rotatilem, quàm à multis annis cum in­<lb/>finitis ferè rotatilium &longs;peciebus & proprietatibus no&longs;ter Philo&longs;ophus in­<lb/>uenit, de quibus &longs;equenti Tomo. <!-- KEEP S--></s> </p> <p id="N25020" type="main"> <s id="N25022"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2502F" type="main"> <s id="N25031"><!-- NEW --><emph type="italics"/>Omnia puncta rotæ AQLZ, quæ rotatur in plano, mouentur inæquali mo­<lb/>tu<emph.end type="italics"/>; </s> <s id="N2503C"><!-- NEW -->de duobus oppo&longs;itis LA con&longs;tat manife&longs;tè, quia æquali tempore <lb/>L acquirit maius &longs;patium, quàm A, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;patium LI eo tempo­<lb/>re quo A acquirit &longs;patium AS: </s> <s id="N25048"><!-- NEW -->de duobus QZ etiam con&longs;tat; </s> <s id="N2504C"><!-- NEW -->nam <lb/>Z ita mouetur ver&longs;us L, vt motus orbis addat &longs;inum ver&longs;um motui centri <lb/>Q verò ita mouetur, vt detrahat <expan abbr="eũdem">eundem</expan> &longs;inum; </s> <s id="N25058"><!-- NEW -->igitur Z mouetur velo­<lb/>ciùs, quàm <expan abbr="q;">que</expan> de duobus K & 10. certum e&longs;t, nam 10. plùs addit a&longs;cen­<lb/>dendo quàm K de&longs;cendendo æquali tempore; </s> <s id="N25064"><!-- NEW -->nam 10. in arcu 10. L ad­<lb/>dit motui centri 10. M, & K in de&longs;cen&longs;u KH addit addit tantùm 14. H; </s> <s id="N2506A"><!-- NEW --><lb/>&longs;ed hæc e&longs;t minor.10. M, vt con&longs;tat toto &longs;inu ver&longs;o arcus HQ; & licèt <lb/>punctum 10. in a&longs;cen&longs;u eodem tempore addat 10. M quo punctum L <lb/>in de&longs;cen&longs;u addit MK æqualem; </s> <s id="N25077"><!-- NEW -->non tamen propterea mouentur æquè <lb/>velociter; </s> <s id="N2507D"><!-- NEW -->quia punctum L initio mouetur velociùs, & &longs;ub finem tardiùs; </s> <s id="N25081"><!-- NEW --><lb/>at verò punctum 10. initio mouetur tardiùs; vnde quocunque arcu a&longs;­<lb/>&longs;umpto inter 10. L, & alio æquali inter LK, punctum L mouebitur <lb/>velociùs initio. </s> </p> <p id="N2508A" type="main"> <s id="N2508C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25099" type="main"> <s id="N2509B"><!-- NEW -->Hinc colligo, punctum L omnium veloci&longs;&longs;imè moueri initio & pun­<lb/>ctum A omnium tardi&longs;&longs;imè; ratio e&longs;t quia puncto L motus orbis addit <pb pagenum="338" xlink:href="026/01/372.jpg"/>totum id quod pote&longs;t addere, po&longs;ito quod &longs;it æqualis motui centri, & pun­<lb/>cto A detrahit totum id, quod pote&longs;t detrahere. </s> </p> <p id="N250A8" type="main"> <s id="N250AA"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N250B7" type="main"> <s id="N250B9">Colligo &longs;ecundò, duo puncta eodem tempore &longs;patia æqualia po&longs;&longs;e ac­<lb/>quire, licèt vtrumque mobile inæquali motu moueatur. </s> </p> <p id="N250BE" type="main"> <s id="N250C0"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N250CD" type="main"> <s id="N250CF"><!-- NEW -->Tertiò, &longs;i a&longs;&longs;umatur punctum <foreign lang="greek">b</foreign> grad.45, illud ip&longs;um e&longs;&longs;e, quod maxi­<lb/>mum omnium &longs;patium conficit eo tempore, quo reuoluitur quadrans, <lb/>id e&longs;t eo tempore, quo percurrit lineam L, I, G, F; nam percurrit &longs;egmen­<lb/>tum rotatilis, cuius chorda e&longs;t <foreign lang="greek">d b</foreign>, &longs;eu percurrit duplam &longs;egmenti L 15. <lb/>atqui dupla L 15. e&longs;t maior LF, vt con&longs;tat. </s> </p> <p id="N250E3" type="main"> <s id="N250E5"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N250F2" type="main"> <s id="N250F4"><!-- NEW -->Centrum O mouetur velociùs, quàm punctum contactus A, vt certum <lb/>e&longs;t; nam eo tempore quo centrum conficit OP æqualem AB, punctum A <lb/>conficit tantùm AS. </s> </p> <p id="N250FC" type="main"> <s id="N250FE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2510B" type="main"> <s id="N2510D"><!-- NEW --><emph type="italics"/>Punctum A non regreditur, &longs;ed tantillùm accedit dextror&longs;um<emph.end type="italics"/>; </s> <s id="N25116"><!-- NEW -->ratio e&longs;t, <lb/>quia dextror&longs;um acquirit AB, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;ini&longs;tror&longs;um verò acquirit XT æ­<lb/>qualem arcui AV; </s> <s id="N25122"><!-- NEW -->&longs;ed arcus e&longs;t mâior &longs;uo &longs;inu; </s> <s id="N25126"><!-- NEW -->igitur plùs acquirit dex­<lb/>tror&longs;um, quàm &longs;ini&longs;tror&longs;um; igitur non regreditur, nec etiam remanet in <lb/>linea AO. </s> </p> <p id="N2512E" type="main"> <s id="N25130"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2513D" type="main"> <s id="N2513F"><!-- NEW --><emph type="italics"/>Omnia puncta inter OQ mouentur tardiùs, quàm centrum O<emph.end type="italics"/>; </s> <s id="N25148"><!-- NEW -->&longs;it enim <lb/>punctum P v.g. <!-- REMOVE S-->certè perueniet in 7. ita vt OPR 7. &longs;int æquales; </s> <s id="N25150"><!-- NEW -->&longs;ed P <lb/>7. e&longs;t minor OE, licèt P 7. tantillùm incuruetur; </s> <s id="N25156"><!-- NEW -->è contrario verò nullum <lb/>e&longs;t punctum inter GZ, quod non moueatur velociùs, quàm O, vt patet; </s> <s id="N2515C"><!-- NEW --><lb/>hinc Z mouetur veloci&longs;&longs;imè omnium punctorum diametri ZQ, Q verò <lb/>tardi&longs;&longs;imè; </s> <s id="N25163"><!-- NEW -->O denique medio qua&longs;i motu inter vtrumque; </s> <s id="N25167"><!-- NEW -->tardiùs qui­<lb/>dem cæteris inter ZO, velociùs tamen aliis, quæ &longs;unt inter OQ; immò <lb/>omnia puncta radiorum OA, OQ, quæ di&longs;tant æqualiter ab O eo tem­<lb/>pore, quo centrum O percurrit totam OE, acquirunt æqualia &longs;patia, <lb/>itemque æqualia, quæ &longs;unt in radiis OL, OZ, licèt prioribus maiora: <lb/>&longs;imiliter motus aliarum partium, quæ &longs;unt intra circulum, <expan abbr="earum&qacute;ue">earumque</expan> <lb/>&longs;patia, dato tempore cogno&longs;ci po&longs;&longs;unt, & ex dictis facilè intelliguntur. </s> </p> <p id="N2517F" type="main"> <s id="N25181"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2518E" type="main"> <s id="N25190"><!-- NEW -->Hînc collige vulgi &longs;en&longs;um; nam plerique omnes exi&longs;timant tùm om­<lb/>nes partes peripheriæ rotæ moueri æquè velociter, tùm nullam e&longs;&longs;e par­<lb/>tem intra circulum vel arcum, quæ non moueatur tardiùs, tùm partibus <lb/>peripheriæ, tum ip&longs;o centro. </s> </p> <p id="N2519A" type="main"> <s id="N2519C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N251A9" type="main"> <s id="N251AB"><!-- NEW -->Colligo &longs;ecundò, fi fiat quadrans A, 18. 16. vt e&longs;t arcus 18.16.ad rectá <pb pagenum="339" xlink:href="026/01/373.jpg"/>18.A.ita rectam 18. A e&longs;&longs;e ad LA; </s> <s id="N251B5"><!-- NEW -->quia A 16. e&longs;t æqualis &longs;emicirculo L <lb/>QA, & hic arcui quadrantis L. 19. &longs;ed vt 16.18.ad 18.A vel L 19. æqua­<lb/>lem, ita L 19. ad LA; igitur A e&longs;t media proportionalis inter LA, & ar­<lb/>cum 18. 16. &longs;ed de hoc aliàs. </s> </p> <p id="N251BF" type="main"> <s id="N251C1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N251CD" type="main"> <s id="N251CF"><!-- NEW --><emph type="italics"/>Punctum L mouetur velociùs, & velociùs in infinitum puncto A<emph.end type="italics"/>; </s> <s id="N251D8"><!-- NEW -->a&longs;&longs;umatur <lb/>enim motus puncti L per vnicum gradum quadrantis LQ addatur &longs;inus <lb/>rectus vnius grad. <!-- REMOVE S-->1745. ip&longs;i gradui, &longs;cilicet 1746. eritque &longs;patium con­<lb/>&longs;ectum 3491. paulò plùs; </s> <s id="N251E8"><!-- NEW -->detrahatur autem gradus ex &longs;inu &longs;upere&longs;t I, &longs;it­<lb/>que &longs;inus ver&longs;us vnius gradus 15. certè erit &longs;patium decur&longs;um ab A da­<lb/>to illo tempore paulò plùs; </s> <s id="N251F0"><!-- NEW -->&longs;ed velocitates motuum æquàli tempore &longs;unt <lb/>vt &longs;patia; </s> <s id="N251F6"><!-- NEW -->igitur velocitas motus puncti L e&longs;t ad velocitatem motus pun­<lb/>cti A, vt 3491.ad 15.id e&longs;t vt 232.ad I; atqui &longs;i accipiatur in orbe &longs;patium <lb/>minus vno gradu, erit adhuc maior proportio motus puncti L ad motum <lb/>puncti A. <!-- KEEP S--></s> </p> <p id="N25201" type="main"> <s id="N25203"><!-- NEW -->Immò, &longs;i ponas &longs;inum totum partium 1000000. & a&longs;&longs;umat motum L, <lb/>& A per vnum minutum arcus erit 2910, & eius &longs;inus rectus 2908.ver­<lb/>&longs;us verò; igitur motus A erit vt 2. motus L 5818. igitur motus L ad mo­<lb/>tum A per vnum minutum quadrantis, vt 2909. ad I, <expan abbr="atq;">atque</expan> ita in infinitú. </s> </p> <p id="N25211" type="main"> <s id="N25213"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N2521F" type="main"> <s id="N25221"><!-- NEW --><emph type="italics"/>Minor rota inclu&longs;a maiori ita mouetur, vt &longs;it maior in illa motus centri, <lb/>quàm motus orbis<emph.end type="italics"/>; </s> <s id="N2522C"><!-- NEW -->&longs;it enim minor rota P <foreign lang="greek">p</foreign>; </s> <s id="N25234"><!-- NEW -->haud dubiè centrum O acqui­<lb/>ret &longs;patium OE duplò maius arcu P <foreign lang="greek">w</foreign> eo tempore, quo motus orbis per­<lb/>curret <expan abbr="eũdem">eundem</expan> arcum P <foreign lang="greek">w</foreign>; an verò &longs;ingula puncta quadrantis P <foreign lang="greek">w</foreign> re&longs;­<lb/>pondeant &longs;ingulis punctis plani <foreign lang="greek">w q</foreign>, vel &longs;ingula duobus, vulgaris diffi­<lb/>cultas e&longs;t, quæ ab Ari&longs;totelica rota &longs;ibi nomen fecit, quam hîc breuiter <lb/>di&longs;cutimus. <lb/><gap desc="hr tag"/></s> </p> <p id="N2525A" type="main"> <s id="N2525C"><emph type="center"/>DIGRESSIO<emph.end type="center"/></s> </p> <p id="N25263" type="main"> <s id="N25265"><emph type="center"/><emph type="italics"/>De Rota Ari&longs;totelica.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25271" type="main"> <s id="N25273">ARi&longs;toteles hanc difficultatem habet, quæ&longs;t. </s> <s id="N25276"><!-- NEW -->24. Mechanicorum, <expan abbr="quã">quam</expan> <lb/>etiam explicat Blancanus, <expan abbr="proponit&qacute;">proponitque</expan>; </s> <s id="N25284"><!-- NEW -->Mer&longs;ennus in præfatione &longs;uæ <lb/>ver&longs;ionis <expan abbr="mechanicarũ">mechanicarum</expan> Galilei; nos illam hoc loco breuiter di&longs;cutiemus. </s> </p> <p id="N2528E" type="main"> <s id="N25290">1. Tribus modis pote&longs;t moueri rota in plano 1°ree;. </s> <s id="N25293">ita vt motus centri <lb/>motui orbis &longs;it æqualis, id e&longs;t vt centrum percurrat lineam rectam æqua­<lb/>lem arcui orbis, qui <expan abbr="eod&etilde;">eodem</expan> <expan abbr="t&etilde;pore">tempore</expan> conuertitur. </s> <s id="N252A2">2°ree;. </s> <s id="N252A5">ita vt motus orbis &longs;it mi­<lb/>nor motu centri, id e&longs;t vt centrum percurrat lineam rectam <expan abbr="maior&etilde;">maiorem</expan> arcu, <lb/>qui <expan abbr="eod&etilde;">eodem</expan> <expan abbr="t&etilde;pore">tempore</expan> conuoluitur. </s> <s id="N252B8">3°ree;. </s> <s id="N252BB">ita vt motus centri &longs;it minor motu orbis. </s> </p> <p id="N252BE" type="main"> <s id="N252C0">2. Primum motus modum di&longs;cu&longs;&longs;imus in &longs;uperioribus Theorematis, <lb/>2. verò, & 3. di&longs;cutiemus hoc loco. </s> <s id="N252C5">&longs;it ergo in præ&longs;enti fig. </s> <s id="N252C8"><!-- NEW -->rota incubans <lb/>plano CN in puncto C centro A, radio AC, quæ <expan abbr="aliã">aliam</expan> includat <expan abbr="conc&etilde;tricã">concentricam</expan> <pb pagenum="340" xlink:href="026/01/374.jpg"/>radio AB; </s> <s id="N252DB"><!-- NEW -->&longs;itque v.g. <!-- REMOVE S-->AB &longs;ubdupla AC; </s> <s id="N252E1"><!-- NEW -->&longs;it planum CE æquale arcui C <lb/>H; </s> <s id="N252E7"><!-- NEW -->ita vt in decur&longs;u &longs;ingula puncta CH re&longs;pondeant &longs;ingulis CE; </s> <s id="N252EB"><!-- NEW -->cùm <lb/>autem rapiatur rota ABD à maiore; </s> <s id="N252F1"><!-- NEW -->haud dubiè punctum D peruenit <lb/>in F, cum punctum A peruenit in G; id e&longs;t radius AD conuenit <lb/>cum GF. </s> </p> <p id="N252F9" type="main"> <s id="N252FB"><!-- NEW -->3. Porrò caput difficultatis poti&longs;&longs;imum in eo po&longs;itum e&longs;t, quod BF &longs;it <lb/>dupla arcus BD; </s> <s id="N25301"><!-- NEW -->igitur vel &longs;ingula puncta arcus BD re&longs;pondent in de­<lb/>cur&longs;u &longs;ingulis BF, vel &longs;ingula BD re&longs;pondent duobus BF, vel alterna <lb/>puncta BF &longs;altuatim remanent penitus intacta; </s> <s id="N25309"><!-- NEW -->&longs;ed nihil horum dici <lb/>po&longs;&longs;e videtur: </s> <s id="N2530F"><!-- NEW -->non primum.quia alioquin tot e&longs;&longs;ent puncta in arcu DB, <lb/>quot in plano BF æquali arcui CH, igitur &longs;ubduplus arcus e&longs;&longs;et æqualis <lb/>duplo, quòd dici non pote&longs;t; </s> <s id="N25317"><!-- NEW -->licèt aliqui vltrò concedant, quod ego mi­<lb/>nimè concedere, nedum concipere po&longs;&longs;um; </s> <s id="N2531D"><!-- NEW -->non pote&longs;t etiam dici quod <lb/>&longs;ingula puncta arcus DB re&longs;pondeant duobus punctis plani BF; </s> <s id="N25323"><!-- NEW -->cùm <lb/>enim puncta D, & C &longs;int in eodem radio AC; </s> <s id="N25329"><!-- NEW -->certè &longs;i punctum tangit in <lb/>motu punctum plani proximè &longs;equens dextror&longs;um; </s> <s id="N2532F"><!-- NEW -->igitur AB cadit per­<lb/>pendiculariter in BF; </s> <s id="N25335"><!-- NEW -->igitur & AC in CE; </s> <s id="N25339"><!-- NEW -->igitur punctum C tangit <lb/>etiam in motu punctum proximè &longs;equens plani CE; </s> <s id="N2533F"><!-- NEW -->igitur planum CE <lb/>e&longs;&longs;et duplum arcus CH, &longs;ed e&longs;t æquale per con&longs;tructionem; </s> <s id="N25345"><!-- NEW -->nec e&longs;t quod <lb/>aliqui prouocent ad experimentum, quod nullum e&longs;t; quippe quod certæ <lb/>& geometricæ demon&longs;trationi repugnaret. </s> </p> <p id="N2534D" type="main"> <s id="N2534F"><!-- NEW -->4. Non pote&longs;t etiam dici, quòd alterna puncta plani BF qua&longs;i &longs;altua­<lb/>tim remaneant intacta; </s> <s id="N25355"><!-- NEW -->nam eo tempore, quo aliquod punctum plani C <lb/>E re&longs;pondens puncto intacto plani BF tangitur; </s> <s id="N2535B"><!-- NEW -->haud dubiè aliquod pun­<lb/>ctum arcus BD tangit planum BF; </s> <s id="N25361"><!-- NEW -->alioquin centrum A de&longs;cenderet &longs;u­<lb/>pra lineam AG; </s> <s id="N25367"><!-- NEW -->igitur maior rota non incubaret plano CE contra hy­<lb/>pothe&longs;im; </s> <s id="N2536D"><!-- NEW -->igitur quolibet in&longs;tanti aliquod punctum arcus BD tangit <lb/>planum BF; </s> <s id="N25373"><!-- NEW -->igitur nullum punctum plani BF intactum e&longs;t; </s> <s id="N25377"><!-- NEW -->quippe om­<lb/>ne punctum contactus plani CE, & maioris circuli re&longs;pondet puncto <lb/>contactus oppo&longs;ito plani BF, & minoris circuli; igitur non remanent al­<lb/>terna puncta plani BF qua&longs;i &longs;altuatim intacta. </s> </p> <p id="N25381" type="main"> <s id="N25383"><!-- NEW -->5. Hinc reiicies infinita illa vacuola Galilei; </s> <s id="N25387"><!-- NEW -->&longs;i enim in linea BM re­<lb/>manent infinita puncta intacta, non verò in CN; </s> <s id="N2538D"><!-- NEW -->certè vbi punctum, <lb/>quod immediatè &longs;equitur C tangitur, & fit punctum contactus, vel nul­<lb/>lum punctum in BF tangitur vel aliquod; &longs;i <expan abbr="primũ">primum</expan>; </s> <s id="N25399"><!-- NEW -->igitur radius minoris <lb/>rotæ imminuitur, quod e&longs;t ab&longs;urdum: &longs;i &longs;ecundum; </s> <s id="N2539F"><!-- NEW -->igitur nullum vacuo­<lb/>lum intercipitur, quod e&longs;t contra Galileum; </s> <s id="N253A5"><!-- NEW -->quod verò &longs;pectat ad poli­<lb/>gona concentrica determinabimus paulò pò&longs;t; </s> <s id="N253AB"><!-- NEW -->&longs;int enim duo poligona <lb/>concentrica centro D, quorum maius ita voluatur, vt AI re&longs;pondeat AF, <lb/>id e&longs;t circa centrum A; </s> <s id="N253B3"><!-- NEW -->certè M mouebitur per arcum MI, D per arcum <lb/>DE, B per arcum BM; </s> <s id="N253B9"><!-- NEW -->igitur &longs;ingula puncta mouebuntur motu &longs;implicis <lb/>circulari <expan abbr="co&qacute;ue">coque</expan> velociùs, quò recedent longiùs ab A: hinc punctum B <lb/>mouebitur omnium veloci&longs;&longs;imè, quia longi&longs;&longs;imè di&longs;tat à puncto A. <!-- KEEP S--></s> </p> <p id="N253C6" type="main"> <s id="N253C8"><!-- NEW -->6. Si verò minus poligonum dirigat motum, qui primò fiat ciat cen­<lb/>trum D; </s> <s id="N253CE"><!-- NEW -->haud dubiè punctum A mouebitur per arcum AV, per <expan abbr="qu&etilde;">quem</expan> retro-<pb pagenum="341" xlink:href="026/01/375.jpg"/>agetur; </s> <s id="N253DB"><!-- NEW -->igitur &longs;i maius poligonum dirigat motum, relinquentur plura <lb/>&longs;egmenta in plano CH intacta æqualia DE; &longs;i verò minus dirigat latera <lb/>maioris poligoni, aliquid &longs;emper de priori &longs;patio in plano BF qua&longs;i re­<lb/>petent per regre&longs;&longs;um. </s> </p> <p id="N253E5" type="main"> <s id="N253E7"><!-- NEW -->7. Hinc tamen malè concludit Galileus &longs;imile quid accidere in mo­<lb/>tu circulorum concentricorum; </s> <s id="N253ED"><!-- NEW -->e&longs;t enim maxima di&longs;paritas: Primò, quia <lb/>centrum A circuli in priori figurâ nunquam recedit à linea AL, alio­<lb/>qui radij circuli eiu&longs;dem e&longs;&longs;ent inæquales, cùm tamen M poligoni a&longs;cen­<lb/>dat &longs;upra MI. Secundò, quia nullum punctum peripheriæ circuli quie&longs;­<lb/>cit. </s> <s id="N253F9">Tertiò, quia omnia puncta circuli mouentur motu mixto ex recto, <lb/>& circulari, excepto centro, cùm tamen omnia puncta poligoni motu <lb/>circulari moueantur, excepto puncto contactus, quod quie&longs;cit. </s> </p> <p id="N25402" type="main"> <s id="N25404"><!-- NEW -->8. Et ne omittam aliud, quod miraculi loco e&longs;t apud <expan abbr="eũdem">eundem</expan> <expan abbr="Galileã">Galileam</expan>, <lb/>quo &longs;cilicet primum illud &longs;uum effectum confirmare concendit, &longs;cilicet <lb/>punctum dici po&longs;&longs;e æquale lineæ &longs;it enim &longs;emicirculus ABMC, rectan­<lb/>gulum BN, triangulum ALN, recta KD parallela BC, denique AI circa <lb/>axem AM; </s> <s id="N2541A"><!-- NEW -->voluantur hæc tria; </s> <s id="N2541E"><!-- NEW -->certè rectangulum relinquit cylindrum, <lb/>triangulum, conum, & &longs;emicirculus hemi&longs;phærium; </s> <s id="N25424"><!-- NEW -->&longs;it autem idem pla­<lb/>num KD parallelum BC &longs;ecans hæc tria; </s> <s id="N2542A"><!-- NEW -->haud dubiè &longs;ectio coni HF <lb/>erit circulus, i&longs;que æqualis plano contento duobus circulis parallelis, <lb/>quorum maior habeat diametrum KD, & minor IE, quod breuiter de­<lb/>mon&longs;tratur; </s> <s id="N25434"><!-- NEW -->quia quando IA e&longs;t æquale quadratis IGA, led BA e&longs;t æ­<lb/>qualis AI, & BC æqualis KD dupla AI; </s> <s id="N2543A"><!-- NEW -->igitur quadratum KD e&longs;t qua­<lb/>druplum quadrati KG, vel IA; </s> <s id="N25440"><!-- NEW -->igitur continet quatuor quadrata AI, & <lb/>AI quatuor AG, vel HG; </s> <s id="N25446"><!-- NEW -->igitur continet quadratum IE, & HF; </s> <s id="N2544A"><!-- NEW -->&longs;ed cir­<lb/>culi &longs;unt vt quadrata diametrorum; </s> <s id="N25450"><!-- NEW -->igitur circulus diametri KD conti­<lb/>net circulos diametri IE, & HF; </s> <s id="N25456"><!-- NEW -->igitur, &longs;i ex circulo diametri CD de­<lb/>trahatur circulus diametri IE, &longs;upere&longs;t corona illa, cuius latitudo e&longs;t IK, <lb/>& ED, de qua &longs;uprà; igitur æqualis e&longs;t circulo diametri AF. <!-- KEEP S--></s> </p> <p id="N2545F" type="main"> <s id="N25461"><!-- NEW -->9. Hinc concludit Galileus punctum apicis coni A e&longs;&longs;e æquale cir­<lb/>culo diametri BC; </s> <s id="N25467"><!-- NEW -->quod certè non mihi videtur &longs;equi; </s> <s id="N2546B"><!-- NEW -->cùm &longs;emper aga­<lb/>tur de ba&longs;i coni, quæ non e&longs;t punctum, & licèt conus HF A &longs;it æqualis <lb/>&longs;olido KIB in orbem &longs;cilicet ducto, detracto dumtaxat hemi&longs;phærio ex <lb/>cylindro, quod tamen non demon&longs;trat Galileus, &longs;ed demon&longs;trarum &longs;up­<lb/>ponit à Luca Valerio; </s> <s id="N25477"><!-- NEW -->nunquam pao&longs;ectò perueniet ad punctum mathe­<lb/>maticum; </s> <s id="N2547D"><!-- NEW -->quippe &longs;emper habebit conum æqualem alteri &longs;olido; &longs;i verò <lb/>quis admittat puncta phy&longs;ica, concedi po&longs;&longs;et vltrò punctum phy&longs;icum <lb/>conicum æquale e&longs;&longs;e alteri &longs;olido maximè dilatato propter angulum <lb/>contingentiæ KBI in quo non videtur e&longs;&longs;e difficultas. </s> </p> <p id="N25488" type="main"> <s id="N2548A"><!-- NEW -->10. Quod autem conus HAF &longs;it æqualis prædicto &longs;olido, quod Ga­<lb/>lileus vocat &longs;calprum orbiculare, breuiter demon&longs;tro; </s> <s id="N25490"><!-- NEW -->quia cum ba&longs;is HF <lb/>&longs;it æqualis KI, ED, id e&longs;t coronæ, itemque &longs;ingulæ ba&longs;es &longs;upra HF v&longs;que <lb/>adverticem A; </s> <s id="N25498"><!-- NEW -->certè totum HFA conflatum ex omnibus ba&longs;ibus e&longs;t æ­<lb/>quale toti &longs;olido &longs;eu &longs;calpro conflato ex omnibus coronis; hæc obiter <lb/>attigi&longs;&longs;e volui, ne fortè di&longs;&longs;imulatum à nobis e&longs;&longs;e qui&longs;quam exi&longs;timaret, <pb pagenum="342" xlink:href="026/01/376.jpg"/>&longs;ed iam hoc poti&longs;&longs;imum &longs;upere&longs;t, vt difficultatem propo&longs;itam de rota <lb/>Ari&longs;totelica breuiter &longs;oluamus, </s> </p> <p id="N254A9" type="main"> <s id="N254AB"><!-- NEW -->11. Certum e&longs;t primò in hypothe&longs;i, quæ componit continuum ex <lb/>punctis mathematicis vix po&longs;&longs;e explicari, &longs;iue dicantur e&longs;&longs;e infinita, vt <lb/>vult Galileus, &longs;iue finita vt alij volunt; </s> <s id="N254B3"><!-- NEW -->quia nec idem punctum minoris <lb/>rotæ pluribus &longs;ui plani re&longs;pondet, nec &longs;ingula &longs;ingulis re&longs;pondent, nec <lb/>etiam fiunt illi &longs;altus intactis finitis, vel infinitis vacuolis; immò talis e&longs;t <lb/>motus circularis natura, vt minimè concipi, nedum explicari po&longs;&longs;it iuxta <lb/>hypothe&longs;im punctorum mathematicorum. </s> </p> <p id="N254BF" type="main"> <s id="N254C1"><!-- NEW -->12. Certum e&longs;t &longs;ecundò, vix etiam explicari po&longs;&longs;e iuxta hypothe&longs;im <lb/>partium proportionalium infinitarum actu; </s> <s id="N254C7"><!-- NEW -->quia contactus ip&longs;e globi, & <lb/>plani tam ob&longs;curè in hac hypothe&longs;i explicatur, vt etiam authores ip&longs;i, <lb/>qui huic &longs;ententiæ patrocinantur, vltrò a&longs;&longs;erant in&longs;eparabilem e&longs;&longs;e diffi­<lb/>cultatem; </s> <s id="N254D1"><!-- NEW -->quod enim dicunt contactum fieri in parte indeterminata, <lb/>ne&longs;cio an aliquis &longs;i non blandiens capere po&longs;&longs;it: nunquid enim contactus <lb/>e&longs;t determinatus qui realis e&longs;t, & &longs;ingularis, id e&longs;t hic & non alius? </s> <s id="N254DB">nun­<lb/>quid e&longs;t aliquid, quod tangit ab omni, eo quod tangit, di&longs;tinctum? </s> <s id="N254E0"><!-- NEW -->quip­<lb/>pe tangere, & non tangere &longs;unt prædicata contradictoria; &longs;ed de his fusè <lb/>in Metaphy&longs;ica. </s> </p> <p id="N254E8" type="main"> <s id="N254EA"><!-- NEW -->13. Adde quod, licèt contactus globi in plano explicari po&longs;&longs;et, &longs;upe­<lb/>re&longs;&longs;et tamen eadem difficultas; nam cùm nulla &longs;it pars, &longs;iue indetermina­<lb/>ta, &longs;iue determinata in plano BF, quæ &longs;it intacta, & cum eadem pars <lb/>arcus BD non re&longs;pondeat pluribus partibus plani BF, & cùm &longs;ingu­<lb/>læ partes arcus &longs;ingulis partibus non re&longs;pondeant (quæ omnia <lb/>con&longs;tant ex dictis) profectò eadem e&longs;t difficultas iuxta hypothe&longs;im par­<lb/>tium proportionalium infinitarum actu, quæ e&longs;t iuxta hypothe&longs;im pun­<lb/>ctorum mathematicorum finitorum, vel infinitorum. </s> </p> <p id="N254FE" type="main"> <s id="N25500">14. His po&longs;itis, &longs;upere&longs;t tantùm vt &longs;oluatur hæc difficultas iuxta hy­<lb/>pothe&longs;im punctorum phy&longs;icorum, vel partium diui&longs;ibilium in infini­<lb/>tum potentiâ, cuius principia & difficultates in Metaphy&longs;ica di&longs;cu­<lb/>tiemus. </s> </p> <p id="N25509" type="main"> <s id="N2550B"><!-- NEW -->Dico ergo &longs;atis facilè iuxta hanc hypothe&longs;im explicari, & &longs;olui po&longs;&longs;e <lb/>nodum rotæ Ari&longs;totelicæ: </s> <s id="N25511"><!-- NEW -->quippe punctum phy&longs;icum curuum tangit <lb/>punctum phy&longs;icum planum, &longs;ed non adæquatè; </s> <s id="N25517"><!-- NEW -->quippè nullum curuum <lb/>adæquari pote&longs;t plano, &longs;eu cum plano conuenire, quod nemo Geometra <lb/>negare poterit: </s> <s id="N2551F"><!-- NEW -->quippe duæ quantitates po&longs;&longs;unt duobus modis con&longs;ide­<lb/>rari: Primò in ordine ad æqualitatem, vel inæqualitatem. </s> <s id="N25525"><!-- NEW -->Secundò, in <lb/>ordine ad commen&longs;urationem, vel conuenientiam, vel <expan abbr="incommen&longs;ura-bilitat&etilde;">incommen&longs;ura­<lb/>bilitatem</expan>; </s> <s id="N25531"><!-- NEW -->&longs;i primo modo, vna quantitas, vel dicitur alteri æqualis, vel inæ­<lb/>qualis; </s> <s id="N25537"><!-- NEW -->&longs;i inæqualis, vel maior, vel minor; </s> <s id="N2553B"><!-- NEW -->&longs;i maior vel minor, dicitur <lb/>rationalis, vel irrationalis &longs;eu aloga; &longs;ed hæc &longs;unt vulgaria, paulò ob&longs;cu­<lb/>riora, quæ &longs;equuntur. </s> </p> <p id="N25543" type="main"> <s id="N25545"><!-- NEW -->15. Si enim &longs;ecundo modo con&longs;iderentur, vel po&longs;&longs;unt commen&longs;urari, <lb/>vel non po&longs;&longs;unt; </s> <s id="N2554B"><!-- NEW -->&longs;i primum, &longs;unt nece&longs;&longs;ariò æquales; </s> <s id="N2554F"><!-- NEW -->&longs;i inæquales illæ &longs;unt <lb/>vel alogæ eædem quæ &longs;uprà, &longs;ic diagonalis <expan abbr="cõparata">comparata</expan> cum latere quadrati <pb pagenum="343" xlink:href="026/01/377.jpg"/>e&longs;t aloga, hoc e&longs;t ita inæqualis, vt nulla &longs;it vtrique pars aliquota commu­<lb/>munis; </s> <s id="N25560"><!-- NEW -->alogæ quidem in ordine ad commen&longs;urationem, non tamen in <lb/>ordines ad partes aliquotas; </s> <s id="N25566"><!-- NEW -->&longs;ic maior arcus comparatus cum linea recta <lb/>&longs;ubdupla non e&longs;t alogus primo modo &longs;ed <expan abbr="&longs;ecũdo">&longs;ecundo</expan>, id e&longs;t illa linea, quæ e&longs;t <lb/>&longs;ubdupla arcus, non pote&longs;t conuenire cum arcu toto, nec cum aliqua <lb/>eius parte; </s> <s id="N25574"><!-- NEW -->&longs;i verò &longs;int æquales, po&longs;&longs;unt etiam dici alogæ in ordine ad <lb/>commen&longs;urationem, &longs;i nullo modo conuenire po&longs;&longs;unt quamtumuis diui­<lb/>dantur; </s> <s id="N2557C"><!-- NEW -->&longs;ic angulus, quem faciunt duæ circumferentiæ, pote&longs;t quidem e&longs;&longs;e <lb/>&etail;qualis angulo dato rectilineo; </s> <s id="N25582"><!-- NEW -->nunquam tamen cum eo conuenire po­<lb/>te&longs;t; </s> <s id="N25588"><!-- NEW -->&longs;ic arcus æqualis rectæ, &longs;ic denique punctum curuum æquale puncto <lb/>plano; </s> <s id="N2558E"><!-- NEW -->licèt enim totum punctum tangatur ab alîo puncto, non tamen <lb/>adæquatè, quia exten&longs;io vnius e&longs;t aloga cum exten&longs;ione alterius; </s> <s id="N25594"><!-- NEW -->analo­<lb/>giam habes in duobus Angelis; </s> <s id="N2559A"><!-- NEW -->quorum vnus figuram &longs;phæricam <expan abbr="pedal&etilde;">pedalem</expan> <lb/>induat, alter cubicam, & alter alterum tangat; </s> <s id="N255A4"><!-- NEW -->nam reuerâ totus Angelus <lb/>tangitur, quia caret partibus, non tamen adæquatè, vt certum e&longs;t; </s> <s id="N255AA"><!-- NEW -->immò <lb/>po&longs;&longs;et Angelus cuius e&longs;t figura &longs;phærica, ita duobus aliis, quorum e&longs;&longs;et <lb/>figura cubica adhærere, vt <expan abbr="vtriq;">vtrique</expan> inadæquatè adhæreret v.g. <!-- REMOVE S-->Angelus A <lb/>punctis BC ita vt ip&longs;um punctum contactus e&longs;&longs;et in ip&longs;a qua&longs;i commi&longs;­<lb/>&longs;ura: </s> <s id="N255BC"><!-- NEW -->immò pote&longs;t Angelus, cuius e&longs;t figura &longs;phærica habere diuer&longs;os con­<lb/>tactus inadæquatos in tota facie Angeli, cuius e&longs;t figura cubica v.g. <!-- REMOVE S-->An­<lb/>gelus A vel in D vel in E, vel in F; </s> <s id="N255C6"><!-- NEW -->immò &longs;unt infiniti potentia huiu&longs;modi <lb/>inadæquatè diuer&longs;i; </s> <s id="N255CC"><!-- NEW -->denique Angelus A pote&longs;t longo tempore in &longs;uper­<lb/>ficie v.g. <!-- REMOVE S-->Angeli C &longs;ucce&longs;&longs;iuè moueri, acquirendo &longs;cilicet nouos conta­<lb/>ctus inadæquatos; </s> <s id="N255D6"><!-- NEW -->vocetur autem contactus E centralis, &longs;eu medius; con­<lb/>tactus verò B extremus. </s> </p> <p id="N255DC" type="main"> <s id="N255DE">16. Nec A e&longs;t; </s> <s id="N255E1"><!-- NEW -->quòd aliqui ne&longs;cio quas partes viruales in angelo ex­<lb/>ten&longs;o agno&longs;cant, quæ certè à me concipi non po&longs;&longs;unt; </s> <s id="N255E7"><!-- NEW -->ni&longs;i fortè aliquid <lb/>extrin&longs;ecum &longs;onent, &longs;cilicet Angelum exten&longs;um multis &longs;imul partibus <lb/>alicuius corporis coextendi po&longs;&longs;e; </s> <s id="N255EF"><!-- NEW -->vnde fit &longs;ingulis inadæquatè coexten­<lb/>di; quod nemo negabit; </s> <s id="N255F5"><!-- NEW -->&longs;ed ne dici moremur in hac materia, quam hîc <lb/>ex profe&longs;&longs;o non tractamus; </s> <s id="N255FB"><!-- NEW -->cettum e&longs;t iuxta hanc hypothe&longs;im punctorum <lb/>phy&longs;icorum facilè explicari motum rotæ Ari&longs;totelicæ: </s> <s id="N25601"><!-- NEW -->quippe dum pun­<lb/>ctum quod proximè accedit ad C in arcu CH incubat puncto plani C <lb/>E, quòd immediatè &longs;equitur C, idque centrali contactu punctum, quod <lb/>proximè &longs;equitur B in arcu BD, quem &longs;ubduplum CH &longs;uppono, tangit <lb/>punctum, quod &longs;equitur immediatè B in plano BF contactu extremo, id <lb/>e&longs;t commi&longs;&longs;ura puncti B & alterius contactu medio, tangit <expan abbr="punctũ">punctum</expan> plani <lb/>quod probatur; </s> <s id="N25615"><!-- NEW -->quia punctum, quod immediatè &longs;equitur B in arcu BDC <lb/>quod vocabimus deinceps &longs;ecundum, tangit contactu tertium punctum <lb/>plani BF eo in&longs;tanti, quo tertium punctum arcus CH tangit contactu <lb/>medio tertium plani CE; igitur eo in&longs;tanti, quo &longs;ecundum CH tangit <lb/>contactu medio &longs;ecundum CE, &longs;ecundum BD tangit contactu extremo <lb/>primum BF, extremo inquam ratione puncti arcus, non ratione puncti <lb/>plani. </s> </p> <p id="N25625" type="main"> <s id="N25627"><!-- NEW -->17. Si verò e&longs;&longs;et maior rota, eîu&longs;que contactus e&longs;&longs;et inter BC, e&longs;&longs;ent <pb pagenum="344" xlink:href="026/01/378.jpg"/>alij contactus inadæquati, vt facilè intelligi pote&longs;t ex dictis, pote&longs;t au­<lb/>tem fieri, vt dixi, vt &longs;int plures contactus inadæquati etiam arcus CH, <lb/>ni&longs;i veloci&longs;&longs;imè moueatur ratione loci, id e&longs;t ni&longs;i punctum phy&longs;icum <lb/>mobile acquirat &longs;ingulis in&longs;tantibus punctum loci immediatum non <lb/>participans de priori; </s> <s id="N25638"><!-- NEW -->quod certè pote&longs;t acquirere duplici motu, &longs;cilicet <lb/>vel recto vel mixto ex recto, & circulari; nec e&longs;t enim dubium, quin An­<lb/>gelus v. <!-- REMOVE S-->g. <!-- REMOVE S-->inducta figura &longs;phærica non po&longs;&longs;it volui circa &longs;e ip&longs;um velo­<lb/>ciùs, & velociùs in infinitum. </s> </p> <p id="N25646" type="main"> <s id="N25648"><!-- NEW -->18. V. g.. angelus A pote&longs;t circa centrum mathematicum, id e&longs;t <lb/>imaginatum B immobile agi in orbem tardiùs, & tardiùs quidem, &longs;i <expan abbr="vnũ">vnum</expan> <lb/>orbem faciat pluribus, & pluribus in&longs;tantibus; velociùs verò, &longs;i pauciori­<lb/>bus; </s> <s id="N25656"><!-- NEW -->quot verò in&longs;tantibus vnum integrum orbem peragat, &longs;i tempus <lb/>con&longs;tet finitis in&longs;tantibus; </s> <s id="N2565C"><!-- NEW -->exi&longs;timo primò, po&longs;&longs;e pluribus, & pluribus pe­<lb/>ragere quia tardiùs, & tardiùs in infinitum moueri pote&longs;t; </s> <s id="N25662"><!-- NEW -->&longs;ecundò pau­<lb/>cioribus, & paucioribus, donec tandem vno in&longs;tanti conficiat integrum <lb/>orbem; </s> <s id="N2566A"><!-- NEW -->vt autem moueatur adhuc velociùs in infinitum; aget quidem &longs;in­<lb/>gulos orbes &longs;ingulis in&longs;tantibus, &longs;ed minoribus, &longs;eu breuioribus. </s> </p> <p id="N25670" type="main"> <s id="N25672"><!-- NEW -->19. Ob&longs;eruabis Angelum A po&longs;&longs;e tribus modis moueri; </s> <s id="N25676"><!-- NEW -->primò circa <lb/>centrum B immobile, vt iam dictum e&longs;t, idque velociùs, & tardiùs in in­<lb/>finitum, & hic motus e&longs;t perfectè circularis: </s> <s id="N2567E"><!-- NEW -->Secundò motu recto &longs;impli­<lb/>ci per lineas BE, IH, idque etiam tardiùs, & velociùs; </s> <s id="N25684"><!-- NEW -->tardiùs quidem, &longs;i <lb/>plura ponat in&longs;tantia, vt centrum B re&longs;pondeat E, vel totus circulus A <lb/>toti F; </s> <s id="N2568C"><!-- NEW -->velociùs uerò &longs;i pauciora donec tandem vno in&longs;tanti circulus A <lb/>re&longs;pondeat F adæquatè, id e&longs;t acquirat locum immediatum non partici­<lb/>pantem, quod adhuc fiet velociùs, & velociùs in infinitum; quia pote&longs;t id <lb/>fieri per in&longs;tantia breuiora, & breuiora. </s> </p> <p id="N25696" type="main"> <s id="N25698"><!-- NEW -->20. Tertiò pote&longs;t moueri motu mixto ex duobus præcedentibus, ita <lb/>vt qua&longs;i rotetur in plano IH, quod tribus modis fieri pote&longs;t: </s> <s id="N25699"><!-- NEW -->primo &longs;i D <lb/>punctum &longs;cilicet <expan abbr="re&longs;põderet">re&longs;ponderet</expan> H; </s> <s id="N256A4"><!-- NEW -->&longs;ecundo, &longs;i aliud punctum inter DI tertio; </s> <s id="N256A8"><!-- NEW --><lb/>&longs;i aliquod inter DCI, primo pote&longs;t fieri, vel &longs;ucce&longs;&longs;iuè per contactus <lb/>inadæquatos, vel in in&longs;tanti, idem dico de &longs;ecundo, & tertio, donec <lb/>tandem eo motu tran&longs;eat in F, ita vt punctum F re&longs;pondeat H & circa B <lb/>totum orbem confecerit; &longs;ed de his plura cum de Angelis. <!-- KEEP S--></s> </p> <p id="N256B4" type="main"> <s id="N256B6"><!-- NEW -->21. Porrò punctum B eo in&longs;tanti, quo &longs;ecundum CH tangit conta­<lb/>ctu medio, &longs;ecundum CE tangit extremo &longs;ecundum BF; </s> <s id="N256BC"><!-- NEW -->igitur &longs;imul <lb/>cum alio id e&longs;t cum &longs;ecundo BD; </s> <s id="N256C2"><!-- NEW -->&longs;i verò accipiatur quodlibet aliud pun­<lb/>ctum inter RC; </s> <s id="N256C8"><!-- NEW -->illud certè non tangit vllo modo ad primum BF eo in­<lb/>&longs;tanti, quo &longs;ecundum CH tangit contactu medio &longs;ecundum CE; </s> <s id="N256CE"><!-- NEW -->&longs;i ta­<lb/>men accipiatur aliquod punctum inter BA v.g. <!-- REMOVE S-->R; certè punctum R tan­<lb/>git &longs;olum &longs;ecundum RV, &longs;ed contactu, qui nec e&longs;t extremus, nec medius, <lb/>&longs;ed inter vtrumque, eo &longs;cilicet in&longs;tanti, quo &longs;ecundum CH tangit con­<lb/>tactu medio primum CE. </s> </p> <p id="N256DC" type="main"> <s id="N256DE"><!-- NEW -->22. Ex his facilè intellegi pote&longs;t hic motus; quic &longs;cilicet idem punctum <lb/>rotæ minoris pote&longs;t re&longs;pondere diuer&longs;is punctis &longs;ui plani, &longs;ed diuer&longs;o <lb/>contactu, quod facilè explicatur, tùm per analogiam motus angelici, tùm <pb pagenum="345" xlink:href="026/01/379.jpg"/>per analogiam partium curuarum rotæ exten&longs;arum. </s> <s id="N256ED">Vnde ex &longs;uperiori­<lb/>bus re&longs;pon&longs;ionibus, duæ &longs;i rectè explicentur &longs;oluunt hunc nodum. </s> <s id="N256F2"><!-- NEW -->Tertia <lb/>verò omninò fal&longs;a e&longs;t; </s> <s id="N256F8"><!-- NEW -->nam primùm dici pote&longs;t fieri aliquos &longs;altus con­<lb/>tactuum inadæquatorum; </s> <s id="N256FE"><!-- NEW -->quia &longs;cilicet punctum &longs;ecundum BD tangit &longs;e­<lb/>cundum BF contactu quidem extremo in puncto arcus, &longs;ed medio in <lb/>puncto plani; </s> <s id="N25706"><!-- NEW -->igitur plures contactus inadæquati inter extremum & me­<lb/>dium qua&longs;i omittuntur per &longs;altus; nullum e&longs;t tamen in&longs;tans, quod ali­<lb/>quo punctum plani non tangatur aliquo contactu, ab aliquo puncto ar­<lb/>cus, vel etiam à duobus in ip&longs;a commi&longs;&longs;ura, quæ commi&longs;&longs;ura ad in&longs;tar <lb/>puncti mathematici imaginarij concipi pote&longs;t. </s> </p> <p id="N25712" type="main"> <s id="N25714"><!-- NEW -->23. Secundò dici pote&longs;t, quod idem punctum arcus BD tangat duo <lb/>puncta plani BF &longs;ed diuer&longs;o contactu; nec enim duo puncta plani tan­<lb/>guntur ab eodem puncto arcus contactu medio in ip&longs;o puncto arcus. </s> <s id="N2571C"><lb/>Tertiò denique dici non pote&longs;t &longs;ingula puncta BD &longs;ingulis punctis B <lb/>F re&longs;pondere, vt con&longs;tat ex dictis, atque ita ex iis, quæ hactenus diximus <lb/>&longs;ufficienter explicatus e&longs;t &longs;ecundus modus motus rotæ in plano. </s> </p> <p id="N25724" type="main"> <s id="N25726"><!-- NEW -->Quod verò &longs;pectat ad tertium; </s> <s id="N2572A"><!-- NEW -->&longs;i minor globus centro G in eadem <lb/>figura moueatur, vt motus orbis &longs;it æqualis motui centri v.g. <!-- REMOVE S-->ex G mo­<lb/>ueatur in I, ex K perueniat in M, &longs;itque FM vel GI æqualis arcus FK, <lb/>& rota minor GF &longs;ecum rapiat maiorem GE; </s> <s id="N25736"><!-- NEW -->haud dubiè motus orbis <lb/>maioris rotæ e&longs;t maior motu centri, vt patet; quippe eo tempore, quo re­<lb/>uoluitur arcus quadrantis, & centrum acquirit tantùm GI &longs;ubduplum <lb/>eiu&longs;dem arcus. </s> </p> <p id="N25740" type="main"> <s id="N25742"><!-- NEW -->24. E&longs;t autem in hoc motu eadem difficultas; </s> <s id="N25746"><!-- NEW -->nam vel &longs;ingula pun­<lb/>cta EI re&longs;pondent &longs;ingulis EN, vel duæ EI re&longs;pondent eidem EN vel <lb/>alterna EI non tangunt per &longs;altus; </s> <s id="N2574E"><!-- NEW -->atqui nihil horum dici po&longs;&longs;e videtur: </s> <s id="N25752"><!-- NEW --><lb/>non primum, quia &longs;unt plura puncta EI quam EN: </s> <s id="N25757"><!-- NEW -->non &longs;ecundum, <lb/>quia &longs;i duo puncta EI tangerent idem EN; </s> <s id="N2575D"><!-- NEW -->igitur duo FK tangerent <lb/>idem FM quod fal&longs;um e&longs;t, non denique tertium; </s> <s id="N25765"><!-- NEW -->quia &longs;i punctum &longs;ecun­<lb/>dum FK tangat contactu tantum extremo primum FK, ita vt &longs;it conta­<lb/>ctus extremus in vtroque id e&longs;t in &longs;ecundo plani, & in &longs;ecundo arcus; <lb/>haud dubiè &longs;ecundus EI tangit &longs;ecundum EN contactu medio in pun­<lb/>cto arcus & extremo in puncto plani </s> </p> <p id="N25771" type="main"> <s id="N25773"><!-- NEW -->25. Itaque hic motus explicari debet per diuer&longs;os contactas inadæ­<lb/>quatos; non pote&longs;t tamen fieri, quin minor rota &longs;uum motum componat <lb/>cum motu maioris, vt explicauimus abundè, cum de motu circulari, v.g. <!-- REMOVE S--><lb/>non pote&longs;t minor rota ita moueri, vt acquirat quodlibet eius punctum <lb/>locum immediatè non participantem vno in&longs;tanti, &longs;i ex eo &longs;equatur aliud <lb/>punctum, vel eiu&longs;dem rotæ, vel alterius coniunctæ moueri velociùs, vt <lb/>con&longs;tat ex dictis. </s> </p> <p id="N25784" type="main"> <s id="N25786"><!-- NEW -->26. Vides autem primò, motum maioris rotæ accedere propiùs ad cir­<lb/>cularem, cum mouetur hoc &longs;ecundo motus genere; </s> <s id="N2578C"><!-- NEW -->quia &longs;cilicet motus <lb/><expan abbr="c&etilde;tri">centri</expan> &longs;i <expan abbr="cõparetur">comparetur</expan> cum motu orbis maioris rotæ, minor e&longs;t; </s> <s id="N25799"><!-- NEW -->&longs;i enim nullus <lb/>e&longs;&longs;et motus centri, &longs;ed tantùm motus orbis, e&longs;&longs;et motus perfectè circula­<lb/>ris; </s> <s id="N257A1"><!-- NEW -->igitur quo minor e&longs;t motus centri, & maior motus orbis, accedit ille <pb pagenum="346" xlink:href="026/01/380.jpg"/>motus propiùs ad circularem, & è contrario quò maior e&longs;t motus centri, <lb/>vt accidit in &longs;ecundo genere motus, accedit propiùs ad motum rectum; <lb/>cum verò alter alteri æqualis e&longs;t motus mixtus, quem medium appellare <lb/>po&longs;&longs;umus. </s> </p> <p id="N257B0" type="main"> <s id="N257B2"><!-- NEW -->27. Aliqua puncta maioris rotæ; </s> <s id="N257B6"><!-- NEW -->cuius motus à minori dirigitur re­<lb/>troëunt, &longs;cilicet, quæ accedunt propiùs ad punctum contactus E, v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->ip&longs;um E vbi centrum rotæ e&longs;t in KI regreditur in O: </s> <s id="N257C1"><!-- NEW -->immò regredi vi­<lb/>detur v&longs;que ad X, id e&longs;t, donec &longs;ecus lineam BM; </s> <s id="N257C7"><!-- NEW -->igitur cum arcus ZE <lb/>M, &longs;it &longs;ubduplus arcus ZIM, vt con&longs;tat, & cùm motus centri &longs;it &longs;ubduplus <lb/>motus orbis, etiam arcus, qui regreditur, e&longs;t &longs;ubduplus illius, qui non re­<lb/>greditur; &longs;ed <expan abbr="motũ">motum</expan> centri &longs;equitur. </s> <s id="N257D5"><!-- NEW -->Tertiò, &longs;i ducas multas parallelas AL, <lb/>quæ diuidant YE in arcus æquales, habebis puncta lineæ motus v.g. <!-- REMOVE S-->&longs;it E <lb/>V &longs;ubduplus EY &longs;it, VO &longs;ubdupla EN, &longs;it EZ 2/3 XY; </s> <s id="N257DF"><!-- NEW -->&longs;it IX 2/3 EN; deni­<lb/>que ip&longs;a YP æqualis EN. </s> </p> <p id="N257E5" type="main"> <s id="N257E7"><!-- NEW -->28. Quartò, aliquod punctum nec progreditur, nec regreditur vno <lb/>in&longs;tanti, eo &longs;cilicet; </s> <s id="N257ED"><!-- NEW -->quo tantum detrahit motus orbis, quantum addit <lb/>motus centri, <expan abbr="pote&longs;t&qacute;ue">pote&longs;tque</expan> determinari punctum illud; </s> <s id="N257F7"><!-- NEW -->imò & proportiones <lb/>motus cuiu&longs;libet puncti; &longs;ed hæc ex po&longs;itis principiis facilè colligitur <lb/>operâ analytices. </s> </p> <p id="N257FF" type="main"> <s id="N25801"><!-- NEW -->Quintò punctum E mouetur velociùs, cum dirigitur motus â minori <lb/>rota, quàm punctum C, cum dirigitur motus à maiori; </s> <s id="N25807"><!-- NEW -->quia motus orbis <lb/>multùm illud retroagit: </s> <s id="N2580D"><!-- NEW -->immò non mouetur tardi&longs;&longs;imè omnium; </s> <s id="N25811"><!-- NEW -->&longs;ed pun­<lb/>ctum illud, quod nec progreditur, nec regreditur, &longs;ed modicùm vel a&longs;cen­<lb/>dit vel de&longs;cendit; &longs;unt autem duo huiu&longs;modi puncta, alterum in arcu I <lb/>E, alterum in YE. <!-- KEEP S--></s> </p> <p id="N2581C" type="main"> <s id="N2581E"><!-- NEW -->29. Sextò denique ex his principis benè èxplicatur quomodo maior <lb/>vel minor rota, cuius motus ab alia minore dirigitur, moueri pote&longs;t; </s> <s id="N25824"><!-- NEW -->nec <lb/>e&longs;t quod in his diutiùs immoremur, vt tandem interruptam no&longs;tro­<lb/>rum Theorematum &longs;eriem repetamus, &longs;unt enim plures alij motus mixti <lb/>non tantùm ex recto, & circulari, &longs;ed ex duobus & pluribus circularibus; <lb/>quorum omnium rationes ni&longs;i me veritas ip&longs;a fallit (quæ tamen falle­<lb/>re non pote&longs;t) ad &longs;ua principiæ phy&longs;ica reducemus. </s> </p> <p id="N25833" type="main"> <s id="N25835"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N25841" type="main"> <s id="N25843"><!-- NEW --><emph type="italics"/>Globus, qui de&longs;cendit deor&longs;um in plano inclinato, mouetur motu mix­<lb/>to ex recto centri, & circulari orbis<emph.end type="italics"/>; </s> <s id="N25850"><!-- NEW -->patet ex dictis, cum more rotæ <lb/>moueatur, &longs;ic etiam mouetur globus deor&longs;um demi&longs;&longs;us cum aliqua in­<lb/>clinatione; </s> <s id="N25858"><!-- NEW -->cuius certè nulla pars a&longs;cendit, &longs;en regreditur; </s> <s id="N2585C"><!-- NEW -->e&longs;t enim <lb/>eadem illius ratio; </s> <s id="N25862"><!-- NEW -->cur autem moueatur ille motu mixto, & non <lb/>recto &longs;implici: </s> <s id="N25868"><!-- NEW -->ratio e&longs;t, quia propter primam illam inclinationem <lb/>tollitur eius æquilibrium; </s> <s id="N2586E"><!-- NEW -->cùm enim globus perfectus in aëre vibratus, <lb/>&longs;i nulla ad&longs;it inclinatio, &longs;it in perfecto æquilibrio, certè, &longs;i vel modica in­<lb/>clinatio accedat vel in C vel in D tolletur æquilibrium, quia illa incli­<lb/>natio <expan abbr="id&etilde;">idem</expan> præ&longs;tat quod pondus nouum <expan abbr="additũ">additum</expan>; porrò huius inclinationis: <pb pagenum="347" xlink:href="026/01/381.jpg"/>ratio ex eo petitur primò, quòd prius globus demittatur per planum <lb/>inclinatum, &longs;iue cadat ex ip&longs;a manu, &longs;iue ex alio plano v.g. <!-- REMOVE S-->ex recto vel <lb/>alio plano decliui. </s> <s id="N2588B"><!-- NEW -->Secundò ex eo, quòd priùs moueatur altera extremi­<lb/>tas putà C, quàm D; </s> <s id="N25891"><!-- NEW -->igitur acquirit C plùs impetus motu naturaliter ac­<lb/>celerato; </s> <s id="N25897"><!-- NEW -->igitur retinetur à puncto; </s> <s id="N2589B"><!-- NEW -->quòd licèt deinde moueatur, tardiùs <lb/>tamen mouetur; </s> <s id="N258A1"><!-- NEW -->igitur C vbi ad imum de&longs;cendit iterum videtur a&longs;cen­<lb/>dere tùm propter determinationem nouam; </s> <s id="N258A7"><!-- NEW -->tùm quia ab oppo&longs;ito pun­<lb/>cto de&longs;cendente qua&longs;i attollitur: </s> <s id="N258AD"><!-- NEW -->non dixi a&longs;cendere, &longs;ed tantùm videri <lb/>a&longs;cendere, quia reuerâ non a&longs;cendit; </s> <s id="N258B3"><!-- NEW -->alioquin aliquod punctum regrede­<lb/>retur, quod fal&longs;um e&longs;t; nec enim pote&longs;t a&longs;cendere, ni&longs;i regrediatur, vt <lb/>con&longs;tat. </s> </p> <p id="N258BB" type="main"> <s id="N258BD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N258C9" type="main"> <s id="N258CB"><!-- NEW --><emph type="italics"/>Hinc non de&longs;truitur ille impetus ab impetu innato, vt fit in funependulis<emph.end type="italics"/>; </s> <s id="N258D4"><!-- NEW --><lb/>quia &longs;cilicet de&longs;truitur tantùm ab innato in a&longs;cen&longs;u; </s> <s id="N258D9"><!-- NEW -->&longs;ed nullum pun­<lb/>ctum globi a&longs;cendit, vt dictum e&longs;t, quod vt meliùs intelligatur, &longs;it in fi­<lb/>gura Th. 1. globus centro O; </s> <s id="N258E1"><!-- NEW -->&longs;itque OF perpendicularis deor&longs;um, quæ <lb/>percurritur ab eodem centro O motu centri; </s> <s id="N258E7"><!-- NEW -->&longs;itque motus orbis ab L <lb/>in <expan abbr="q;">que</expan> intelligatur autem planium AI 6; </s> <s id="N258F1"><!-- NEW -->certè punctum A, quod perinde <lb/>&longs;e habet, atque &longs;i e&longs;&longs;et punctum contactus, de&longs;cribit lineam ARP ergo <lb/>non a&longs;cendit; igitur non de&longs;truitur impetus productus ab impetu in­<lb/>nato. </s> </p> <p id="N258FB" type="main"> <s id="N258FD"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25909" type="main"> <s id="N2590B">Ob&longs;eruabis 1°ree;. </s> <s id="N2590E"><!-- NEW -->mirificam e&longs;&longs;e impetus propagationem in hoc motu; <lb/>quippe omnes partes mouentur inæquali motu, licèt moueantur à prin­<lb/>cipio intrin&longs;eco. </s> </p> <p id="N25916" type="main"> <s id="N25918">1. Non tantum accelerari motum centri, &longs;ed etiam motum orbis, vt <lb/>patet experientiâ in globo de&longs;cendente per decliue planum. </s> </p> <p id="N2591F" type="main"> <s id="N25921"><!-- NEW -->3. Si globus non de&longs;cendat in plano declini &longs;ed in libero aëre po&longs;t <lb/>primam librationem motus orbis non cre&longs;cit; </s> <s id="N25929"><!-- NEW -->quia omnes partes ten­<lb/>dere po&longs;&longs;unt deor&longs;um, nec ab vllo obice impediuntur; non e&longs;t autem <lb/>par ratio pro motu in plano decliui, vt patet. </s> </p> <p id="N25931" type="main"> <s id="N25933"><!-- NEW -->4. Hinc motus orbis &longs;en&longs;im dece&longs;cit, &longs;ed omninò in&longs;en&longs;ibiliter; </s> <s id="N25937"><!-- NEW --><lb/>quia non de&longs;truitur ab impetu innato, vt iam dictum e&longs;t; </s> <s id="N2593C"><!-- NEW -->nec enim &longs;ic <lb/>motus circularis e&longs;t contrarius motui recto; </s> <s id="N25942"><!-- NEW -->quippe modò centrum <lb/>grauitatis globi feratur motu recto, hoc &longs;atis e&longs;&longs;e videtur, &longs;iue partes mo­<lb/>tu circulari ferantur: circa idem centrum, &longs;iue omnes motu recto per <lb/>lineas parallelas ferantur:</s> <s id="N25943"><!-- NEW -->ratio à priori e&longs;t, quia in tantum vnus impe­<lb/>tus de&longs;truit alium in eadem parte mobilis, in quantum impeditur ab eo <lb/>eius motus deor&longs;um totius globi nullo modo impeditur ab illo motu <lb/>circulari, quia globus æquè citò de&longs;cendit vno, atque alio motu, vt con­<lb/>&longs;tat mille experientiæ. </s> </p> <p id="N25957" type="main"> <s id="N25959"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N25965" type="main"> <s id="N25967"><!-- NEW --><emph type="italics"/>Si corporis grauis altera extremitas &longs;it grauior demittaturque in eo &longs;itu,<emph.end type="italics"/><pb pagenum="348" xlink:href="026/01/382.jpg"/><emph type="italics"/>in quo &longs;it parallelum horizonti; </s> <s id="N25976"><!-- NEW -->haud dubiè extremitas grauior præit motu <lb/>mixto<emph.end type="italics"/>; </s> <s id="N2597F"><!-- NEW -->quia &longs;cilicet qua&longs;i ab aliâ leuiore retinetur, exemplum habes in <lb/>&longs;agittâ ferro armatâ, & in fune ex quo plumbum pendet; ratio euiden­<lb/>ti&longs;&longs;ima e&longs;t; </s> <s id="N25987"><!-- NEW -->quia illa extremitas faciliùs medij re&longs;i&longs;tentiam &longs;uperat, igitur <lb/>præire debet; </s> <s id="N2598D"><!-- NEW -->igitur motu mixto; </s> <s id="N25991"><!-- NEW -->illa tamen tardiùs de&longs;cendit, quàm <lb/>de&longs;cenderet, &longs;i à leuiore e&longs;&longs;et &longs;eparata; </s> <s id="N25997"><!-- NEW -->leuior verò velociùs, quàm &longs;i e&longs;­<lb/>&longs;et &longs;olitaria; quod autem non &longs;it alia ratio, patet poti&longs;&longs;imum ex eo, quòd <lb/>plumbum ita demi&longs;&longs;um, vt funis præeat, tandem funem a&longs;&longs;equitur, & tan­<lb/>dem à tergo relinquit. </s> </p> <p id="N259A1" type="main"> <s id="N259A3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N259B0" type="main"> <s id="N259B2">Hinc petenda e&longs;t vera ratio illius phœnomeni, quod iam &longs;uprà l. <!-- REMOVE S-->3. <lb/>indicauimus, &longs;cilicet &longs;agittam plùs temporis ponere in de&longs;cen&longs;u, quàm <lb/>in a&longs;cen&longs;u minoremque infligere ictum, quàm leuius lignum, & multò <lb/>leuior penna cu&longs;pidis ferreæ motum retardat. </s> </p> <p id="N259BD" type="main"> <s id="N259BF"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N259CC" type="main"> <s id="N259CE"><!-- NEW -->Si altera extremitas &longs;agittæ plumis in&longs;truatur, licèt proijciatur motu <lb/>violento &longs;ur&longs;um extremitas ferro armata præit plumis à tergo relictis; </s> <s id="N259D4"><!-- NEW --><lb/>ratio e&longs;t, quia aër fortiùs re&longs;i&longs;tit pluuis, quàm ferro, vel ligno; igitur ca­<lb/>rum motum retardat. </s> </p> <p id="N259DB" type="main"> <s id="N259DD"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N259EA" type="main"> <s id="N259EC"><!-- NEW -->Hinc &longs;agitta pennis atton&longs;is fertur in incertum, & &longs;copum fallit, cui <lb/>fuerat de&longs;tinata; </s> <s id="N259F2"><!-- NEW -->quia licèt lignum minore vi polleat, quàm ferrum; </s> <s id="N259F6"><!-- NEW -->vix <lb/>tamen &longs;en&longs;ibilis e&longs;t differentia; </s> <s id="N259FC"><!-- NEW -->adde quod minima deflexio, vel decli­<lb/>natio ad retrò agendum ferrum &longs;ufficit; corpus enim facilè mouetur mo­<lb/>tu mixto ex recto, & circulari. </s> </p> <p id="N25A04" type="main"> <s id="N25A06"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25A13" type="main"> <s id="N25A15"><!-- NEW -->Hinc ratio illius iaculi breui cu&longs;pide armati, cuius altera extremitas <lb/>decu&longs;&longs;atim fi&longs;&longs;a cra&longs;&longs;iore charta paululùm expan&longs;a munitur, qu&etail; deflexio­<lb/>nem impedit; </s> <s id="N25A1D"><!-- NEW -->cuius rei analogiam habes in nauis gubernaculo; </s> <s id="N25A21"><!-- NEW -->e&longs;t enim <lb/>ad in&longs;tar quadruplicis claui motum dirigentis; </s> <s id="N25A27"><!-- NEW -->quîppe inclinari non <lb/>pote&longs;t, ni&longs;i multum aëris pellant alæ illæ chartaceæ: In &longs;agitta aliquid <lb/>&longs;imile habes. </s> </p> <p id="N25A2F" type="main"> <s id="N25A31"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25A3E" type="main"> <s id="N25A40"><!-- NEW -->Hinc &longs;i euibretur iaculum illud per horizontalem v.g. <!-- REMOVE S-->circa pro­<lb/>prium axem conuoluitur; </s> <s id="N25A48"><!-- NEW -->quia aër tenues illas tran&longs;uerberat alas, ex <lb/>qua aëris vel colli&longs;ione, vel appul&longs;u, vel qua&longs;i reflexione facilè &longs;equitur <lb/>circularis motus, qui nullatenus impedit rectum, vt iam dixi &longs;uprà; </s> <s id="N25A50"><!-- NEW -->&longs;ed <lb/>cum eo motum mixtum componit, de quo paulò pò&longs;t; nunc tantùm &longs;uf­<lb/>ficiat attigi&longs;&longs;e veri&longs;&longs;imam rationem illorum gyrorum. </s> </p> <p id="N25A58" type="main"> <s id="N25A5A"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25A67" type="main"> <s id="N25A69"><!-- NEW -->Simile phœnomenum habes in illis volatilibus calamis, qui multis <lb/>copiam ludi faciunt; </s> <s id="N25A6F"><!-- NEW -->nam primò tignea illa, vel o&longs;&longs;ea theca, cui com-<pb pagenum="349" xlink:href="026/01/383.jpg"/>mittuntur plumæ, plumas ip&longs;as præit propter rationem prædictam; </s> <s id="N25A78"><!-- NEW -->nam <lb/>aëra faciliùs diuidit; </s> <s id="N25A7E"><!-- NEW -->&longs;ecundò vertiginem illam habet, de qua &longs;uprà; </s> <s id="N25A82"><!-- NEW -->quia <lb/>aër qua&longs;i reuerberat, <expan abbr="torquetq;">torquetque</expan> plumas; de hoc motu paulò pò&longs;t agemus. </s> </p> <p id="N25A8D" type="main"> <s id="N25A8F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N25A9B" type="main"> <s id="N25A9D"><!-- NEW --><emph type="italics"/>Cum Cylindrus ita dimittitur, vt altera extremitas motu circulari praeat, <lb/>remanente initio aliquo centro immobili, de&longs;cendit motu mixto ex recto & <lb/>circulari<emph.end type="italics"/>; </s> <s id="N25AAA"><!-- NEW -->vt con&longs;tat ex iis, quæ diximus de globo deor&longs;um cadente hoc <lb/>genere motus; &longs;unt tamen hîc multa ob&longs;eruanda. </s> <s id="N25AB0">Primò omnes partes <lb/>globi initio moueri, &longs;ed inæqualiter, cùm tamen aliqua pars cylindri non <lb/>moueatur. </s> <s id="N25AB7"><!-- NEW -->Sit enim cylindrus AC ita innixus B, vt liberè moueri po&longs;&longs;it; </s> <s id="N25ABB"><!-- NEW --><lb/>haud dubiè, cùm non &longs;it æquilibrium, &longs;egmentum BC præualebit; </s> <s id="N25AC0"><!-- NEW -->igitur <lb/>circa centrum B extremitas C de&longs;cendet per arcum CD, & A per arcum <lb/>AE; donec tandem punctum B moueatur per rectam BF, &longs;eu per aliam <lb/>proximè accedentem, &longs;i. </s> <s id="N25ACA"><!-- NEW -->tantillùm à plano BF repellatur; </s> <s id="N25ACE"><!-- NEW -->punctum verò <lb/>C motu mixto ex recto deor&longs;um, & circulari circa B; </s> <s id="N25AD4"><!-- NEW -->ea tamen lege, vt <lb/>motus orbis nullo modo acceleretur, &longs;ed tantùm motus centri; igitur <lb/>hic motus con&longs;tat ex motu centri accelerato, & motu orbis qua&longs;i æqua­<lb/>bili, cuius linea de&longs;cribi pote&longs;t, vt videbimus l. <!-- REMOVE S-->12. dixi, ferè æquabilem, <lb/>quia aliquid de&longs;truitur &longs;ingulis in&longs;tantibus ratione nouæ determinatio­<lb/>nis, vt diximus &longs;uprà cum de motu circulari, &longs;ed parùm pro nihilo repu­<lb/>tatur. </s> </p> <p id="N25AE6" type="main"> <s id="N25AE8"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N25AF4" type="main"> <s id="N25AF6">Ob&longs;erua 1°ree;. </s> <s id="N25AF9">e&longs;&longs;e plures huius motus mixti &longs;pecies. </s> <s id="N25AFC">Primò e&longs;t mixtus <lb/>ex motu centri & motu orbis æquali. </s> <s id="N25B01">Secundo ex 1°ree;. </s> <s id="N25B04">maiore & 2°ree;. </s> <s id="N25B07">mi­<lb/>nore. </s> <s id="N25B0C">Tertiò ex 1°ree;. </s> <s id="N25B0F">minore & 2°ree;. </s> <s id="N25B12">maiore. </s> <s id="N25B15">Quartò ex 1°ree;. </s> <s id="N25B18">accelerato 2°ree;. <lb/></s> <s id="N25B1C">æquabili Quintò ex 1°ree;. </s> <s id="N25B1F">accelerato 2°ree;. </s> <s id="N25B22">retardato. </s> <s id="N25B25">Sextò ex vtroque retar­<lb/>dato. </s> <s id="N25B2A">Septimò ex vtroque accelerato. </s> <s id="N25B2D">Octauò ex 1°ree;. </s> <s id="N25B30">æquabili 2°ree;. </s> <s id="N25B33">accele­<lb/>rato.Nono ex 1°ree;. </s> <s id="N25B38">retardato 2°ree;. </s> <s id="N25B3B">accelerato. </s> <s id="N25B3E">Decimò ex 1°ree;. </s> <s id="N25B41">æquabili 2°ree;. </s> <s id="N25B44">ac­<lb/>celerato.Vndecimò ex 1°ree;. </s> <s id="N25B49">æquabili 2°ree;. </s> <s id="N25B4C">retardato &c. </s> <s id="N25B4F"><!-- NEW -->nec enim hîc dee&longs;t <lb/>maxima motuum &longs;ylua, quorum tamen, quia e&longs;t eadem ratio, nimis acu­<lb/>ratam di&longs;tributionem omittimus, non facilè haberi pote&longs;t; </s> <s id="N25B57"><!-- NEW -->cùm enim <lb/>&longs;int tres termini, &longs;cilicet æquabilis, retardatus, acceleratus, erunt 9. <lb/>combinationes; </s> <s id="N25B5F"><!-- NEW -->& cùm &longs;ingulæ tres differentias habeant; nam vel mo­<lb/>tus orbis e&longs;t æqualis motui centri, vel maior, vel minor, ducantur 9.in 3. <lb/>& erunt 27. </s> </p> <p id="N25B67" type="main"> <s id="N25B69"><!-- NEW -->Ob&longs;erua &longs;ecundò centrum motus po&longs;&longs;e vel propiùs accedere ad A <lb/>v.g.&longs;i e&longs;&longs;et in G, vel ad C v.g. <!-- REMOVE S-->&longs;i e&longs;&longs;et Z. &longs;i primum, maior e&longs;t motus orbis, <lb/>id e&longs;t velocior, licèt pauciores circuitus fiant; </s> <s id="N25B73"><!-- NEW -->quia extremitas C ma­<lb/>iorem arcum de&longs;cribens plùs temporis in de&longs;cen&longs;u ponit; </s> <s id="N25B79"><!-- NEW -->igitur maio­<lb/>rem velocitatem acquirit; &longs;i verò &longs;ecundum, è contrario. </s> </p> <p id="N25B7F" type="main"> <s id="N25B81"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N25B8D" type="main"> <s id="N25B8F"><!-- NEW --><emph type="italics"/>Cum cylindrus proijcitur &longs;ur&longs;um it a vt aliquod punctum rectà feratur, cir­<lb/>ca quod voluitur cylindrus, </s> <s id="N25B98"><!-- NEW -->est motus mixtus ex recto centri, & circulari orbis,<emph.end type="italics"/><pb pagenum="350" xlink:href="026/01/384.jpg"/>pro quo non e&longs;t noua difficultas; nam e&longs;t pror&longs;us eadem ratio, ni&longs;i <lb/>quod primò debet priùs imprimi motus rectus omnibus partibus erecto <lb/>cylindro tùm vbi &longs;eparatur à manu circulariis. </s> <s id="N25BA8">Secundò centrum pote&longs;t <lb/>accedere propiùs ad &longs;ummam extremitatem vel ad imam. </s> <s id="N25BAD">Tertiò, a&longs;cendit <lb/>eò altiùs cylindrus, quò centrum motus orbis accedit propiùs ad &longs;um­<lb/>mam extremitatem. </s> <s id="N25BB4"><!-- NEW -->Quartò, pote&longs;t extremitas ima impelli duobus mo­<lb/>dis: </s> <s id="N25BBA"><!-- NEW -->primò &longs;i retrò agitur, &longs;ecundò &longs;i antè; </s> <s id="N25BBE"><!-- NEW -->&longs;ed quia hæc omnia perti­<lb/>nent ad diuer&longs;os oblongæ ha&longs;tæ motus iucundaque militaris illius exer­<lb/>citationis phœnomena, quorum omnium rationem in &longs;ingulari Theo­<lb/>remate afferemus; eò totam rem i&longs;tam remittimus. </s> </p> <p id="N25BC8" type="main"> <s id="N25BCA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N25BD6" type="main"> <s id="N25BD8"><!-- NEW --><emph type="italics"/>Quando globus, &longs;eu rota voluitur in &longs;uperficie curua immobili, omnes eius <lb/>partes mouentur motu mixto ex duobus circularibus, &longs;cilicet ex motu circula­<lb/>ri centri, & circulari orbis,<emph.end type="italics"/> e&longs;t enim motus centri circularis &longs;i voluatur <lb/>globus in orbe, hoc e&longs;t in &longs;uperficie curua; </s> <s id="N25BE7"><!-- NEW -->porrò hæc &longs;uperficies vel e&longs;t <lb/>conuexa, vel concaua, vel e&longs;t circuli maioris, vel minoris; </s> <s id="N25BED"><!-- NEW -->itemque &longs;i con­<lb/>caua vel e&longs;t circuli æqualis, vel maioris, vel minoris; igitur &longs;unt 6. nouæ <lb/>combinationes, quæ &longs;i ducantur in 27. habebis 162. &longs;ed quia, &longs;i e&longs;t con­<lb/>caua minoris, vel æqualis, non pote&longs;t globus in ea rotari. </s> <s id="N25BF7"><!-- NEW -->Hinc &longs;unt tan­<lb/>tùm 4. legitimæ combinationes nouæ, quæ &longs;i ducantur in 27, habebis <lb/>108; &longs;ed iam &longs;eor&longs;im rem i&longs;tam con&longs;ideremus. </s> </p> <p id="N25BFF" type="main"> <s id="N25C01"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N25C0D" type="main"> <s id="N25C0F"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena rotæ, quæ circa æqualem rotam immo­<lb/>bilem it a rotatur, vt arcus mobilis, & immobilis decur&longs;i &longs;int æquales.<emph.end type="italics"/></s> <s id="N25C18"><!-- NEW --> Sit rota <lb/>immobilis centro L, radio AB; &longs;it alia centro C æqualis priori, quæ ita <lb/>moueatur, vt &longs;inguli arcus BE re&longs;pondeant &longs;ingulis arcubus BT, & pun­<lb/>ctum E tangat in T, D in X, F in D. <!-- KEEP S--></s> <s id="N25C23">Primò centrum mouetur motu cir­<lb/>culari, de&longs;cribitque circulum radio AC, &longs;cilicet duplum circuli immobi­<lb/>lis ABX. </s> <s id="N25C2A"><!-- NEW -->Secundò motus centri e&longs;t duplò maior motu orbis, id e&longs;t eo <lb/>tempore, quo in &longs;uperficie conuexa decur&longs;us e&longs;t arcus BT, centrum C <lb/>confecit arcum CV duplum; cuius phœnomeni ratio clara e&longs;t, quia &longs;ci­<lb/>licet centrum C di&longs;tat &longs;emper ab A toto radio AC duplo AB. <!-- KEEP S--></s> </p> <p id="N25C35" type="main"> <s id="N25C37"><!-- NEW -->Tertiò pote&longs;t de&longs;cribi linea, quam punctum B &longs;uo fluxu de&longs;cribit; <lb/>ducatur &longs;emicirculus CVT; diuidatur in 12. partes æquales ductis radiis <lb/>AC, AL, AV &c.qui &longs;ecant circulum ABX in punctis YZ <foreign lang="greek">dg</foreign> &c. </s> <s id="N25C43"><!-- NEW -->tùm <lb/>ex punctis, quæ terminant ductos radios in &longs;emicirculo CVT de&longs;cri­<lb/>bantur circuli radio CB; haud dubiè tangent hi circuli circulum ABX <lb/>in punctis YZ <foreign lang="greek">dg</foreign> &c. </s> <s id="N25C51">denique accipiatur arcus YG æqualis YB, tùm <lb/>ZH æqualis ZB, tùm <foreign lang="greek">d</foreign> I æqualis <foreign lang="greek">d</foreign> B, atque ita deinceps, & per puncta <lb/>BGHIK. &c. </s> <s id="N25C60"><!-- NEW -->ducantur curua BGLMOQS, atque idem fiat &longs;ini­<lb/>&longs;tror&longs;um, & habebitur linea, quam &longs;uo fluxu de&longs;cribit punctum B; </s> <s id="N25C66"><!-- NEW -->quod <lb/>breuiter demon&longs;tratur, quia quando centrum C e&longs;t in L, decurrit arcum <lb/>CL &longs;ubduplum CV; </s> <s id="N25C6E"><!-- NEW -->igitur tangit in <foreign lang="greek">d</foreign>; </s> <s id="N25C76"><!-- NEW -->igitur decurrit B <foreign lang="greek">d</foreign> &longs;ubduplum <lb/>BT; </s> <s id="N25C80"><!-- NEW -->igitur circa centrum C motu orbis conuer&longs;us e&longs;t arcus &longs;ubduplus <pb pagenum="351" xlink:href="026/01/385.jpg"/>BE e&longs;t æqualis <foreign lang="greek">d</foreign> B; </s> <s id="N25C8D"><!-- NEW -->&longs;ed <foreign lang="greek">d</foreign> I e&longs;t æqualis <foreign lang="greek">d</foreign> B; igïtur punctum circuli mo­<lb/>bilis e&longs;t in I, idem pror&longs;us demon&longs;trabitur de aliis punctis. </s> </p> <p id="N25C9B" type="main"> <s id="N25C9D"><!-- NEW -->Quartò, hinc triangula curuilinea BYG, BZH, B <foreign lang="greek">d</foreign> I &longs;unt I&longs;o&longs;celia; </s> <s id="N25CA5"><!-- NEW --><lb/>ip&longs;um vero BVK e&longs;t æquilaterum quia AK e&longs;t Tangens, vt con&longs;tat; </s> <s id="N25CAA"><!-- NEW --><lb/>immò &longs;inguli circuli debent tangere &longs;uum radium, vt patet; porrò miri­<lb/>fica e&longs;t huius lineæ figura, quæ &longs;ectionem cordis exhibet, quam ideo <lb/>deinceps lineam cordis appellabimus, cuius &longs;unt in&longs;ignes omninò pro­<lb/>prietates, quas &longs;uo loco demon&longs;trabimus. </s> </p> <p id="N25CB5" type="main"> <s id="N25CB7"><!-- NEW -->Quintò, punctum B initio tardi&longs;&longs;imè mouetur cum eo tempore, quo <lb/>decurrit BG punctum oppo&longs;itum D decurrat D6; </s> <s id="N25CBD"><!-- NEW -->ratio e&longs;t, quia motus <lb/>centri defert D in I, cui motus orbis cum motu centri con&longs;entiens ad­<lb/>dit P6, cùm tamen motus orbis puncti B &longs;it contrarius motui centri; </s> <s id="N25CC5"><!-- NEW --><lb/>adde quod motus centri circa centrum A tribuit maiorem motum <lb/>puncto D, quàm B iuxta proportionem radiorum; igitur cùm DA <lb/>&longs;it tripla BA, motus centri D e&longs;t triplus motus centri B, igitur duplici <lb/>nomine motus puncti B e&longs;t tardior. </s> <s id="N25CD0">Primò, quia motus orbis <lb/>tantùm addit D, quantum detrahit B. Secundò, quia motus centri addit <lb/>D motum triplum illius, quem addit B. </s> </p> <p id="N25CD7" type="main"> <s id="N25CD9"><!-- NEW -->Sextò po&longs;&longs;unt haberi per <expan abbr="analyticã">analyticam</expan> proportiones arcuum lineæ motus, <lb/>quos B &etail;qualibus <expan abbr="t&etilde;poribus">temporibus</expan> percurrit v.g.BG, GH, HI, IK, KL, LM, <expan abbr="deniq;">denique</expan> <lb/>vltimus RS æqualis D6; </s> <s id="N25CED"><!-- NEW -->indico breuiter huius proportionem, cum BGDP <lb/>e&longs;t tripla BY, & P6; </s> <s id="N25CF3"><!-- NEW -->e&longs;t quadrupla; </s> <s id="N25CF7"><!-- NEW -->igitur ferè æqualis BV, &longs;i ducantur <lb/>duæ rectæ YB, YG angulus rectilineus GYB e&longs;t æqualis YAB, id e&longs;t <lb/>15 grad.igitur ita &longs;e habet arcus BG ad BY vt recta BY ad BA, id e&longs;t ferè, <lb/>vt 1.ad 4.paulò minùs; </s> <s id="N25D01"><!-- NEW -->&longs;ed D6 e&longs;t quadruplus BY; </s> <s id="N25D05"><!-- NEW -->igitur BG e&longs;t ad D6 <lb/>vt 1. ad 16.paulò minus; </s> <s id="N25D0B"><!-- NEW -->&longs;ed eo maior erit proportio motus D, quo a&longs;­<lb/>&longs;umetur minor arcus; </s> <s id="N25D11"><!-- NEW -->vt autem habeatur proportio a&longs;&longs;umpto arcu in­<lb/>tegro quadrantis e&longs;t vt M S ad MB; porrò e&longs;t ferè eadem proportio <lb/>motuum punctorum appo&longs;itorum rotæ mobilis, &longs;iue rotetur in plano re­<lb/>ctilineæ, &longs;iue in &longs;uperficie curua. </s> </p> <p id="N25D1B" type="main"> <s id="N25D1D"><!-- NEW -->Septimò, puncta B & E de tempore, quo percurritur arcus quadran­<lb/>tis percurrunt &longs;patia æqualia: </s> <s id="N25D23"><!-- NEW -->hinc ET, BM &longs;unt æquales; </s> <s id="N25D27"><!-- NEW -->immò <lb/>&longs;i ducantur rectæ BEMTB, erit ET perfectum quadratum vt con&longs;tat, <lb/>cuius diagonalis erit BM; </s> <s id="N25D2F"><!-- NEW -->igitur æqualis BX, quæ omnia con&longs;tant ex <lb/>ip&longs;is elementis; porrò punctum B veloci&longs;&longs;imè omnium mouetur, vt pa­<lb/>tet ex dictis. </s> </p> <p id="N25D37" type="main"> <s id="N25D39"><!-- NEW -->Octauò, quodlibet punctum circuli mobilis BEDF &longs;uo motu de­<lb/>&longs;cribit arcum lineæ cordis, vt certum e&longs;t, qui in mille punctis decu&longs;­<lb/>&longs;antur cum linea puncti, quam de&longs;cribit punctum B v.g. <!-- REMOVE S-->linea puncti D <lb/>decu&longs;&longs;atur cum linea puncti B in <expan abbr="q.">que</expan> quippe D q, S q &longs;unt æquales, linea <lb/>puncti E cum linea puncti B in L; denique de&longs;cribi pote&longs;t hæc linea <lb/>BKMN &c. </s> <s id="N25D4D"><!-- NEW -->ductis radiis ex centro ad libitum &longs;ine vllo diui&longs;ionis <lb/>ordine v.g. <!-- REMOVE S-->ducatur A <foreign lang="greek">d</foreign>; </s> <s id="N25D59"><!-- NEW -->L nulla habita diui&longs;ionis ratione; </s> <s id="N25D5D"><!-- NEW -->ex L de&longs;cri­<lb/>batur arcus radio L <foreign lang="greek">d</foreign>; </s> <s id="N25D67"><!-- NEW -->a&longs;&longs;umantur <foreign lang="greek">d</foreign> I, <foreign lang="greek">d</foreign> B æquales, per I; </s> <s id="N25D73"><!-- NEW -->haud dubiè <lb/>ducetur linea; idem dico de aliis punctis. </s> </p> <pb pagenum="352" xlink:href="026/01/386.jpg"/> <p id="N25D7D" type="main"> <s id="N25D7F"><!-- NEW -->Nonò, &longs;i a&longs;&longs;umatur quodlibet punctum intra rotam v.g. <!-- REMOVE S-->punctum <lb/>X perueniet in A eo tempore, quo B erit in M, vt patet; </s> <s id="N25D87"><!-- NEW -->hinc moue­<lb/>bitur per lineam motus mixti, qui accedit propiùs ad circularem; <lb/>quemadmodum enim cum rota mouetur in plano rectilineo, punctum <lb/>illius, quod accedit propiùs ad centrum mouetur eo motu, qui accedit <lb/>propiùs ad motum centri, id e&longs;t ad motum rectum. </s> <s id="N25D93"><!-- NEW -->Similiter punctum, <lb/>quod accedit propiùs ad Q in hac rota mouetur eo motu, qui accedit <lb/>propiùs ad motum centri C, id e&longs;t ad motum circularem; igitur hic mo­<lb/>tus puncti X plùs participat de motu centri, quàm de motu orbis, qui <lb/>&longs;cilicet in eo minimus e&longs;t. </s> </p> <p id="N25D9F" type="main"> <s id="N25DA1"><!-- NEW -->Decimò, hinc &longs;i motus minoris rotæ radio CX dirigatur à motu ma­<lb/>ioris radio CB; </s> <s id="N25DA7"><!-- NEW -->hæc quidem ita mouetur vt &longs;ingula puncta BE re­<lb/>&longs;pondeant &longs;ingulis BT, non tamen &longs;ingula XY &longs;ingulis XB; </s> <s id="N25DAD"><!-- NEW -->&longs;ed hic <lb/>etiam accer&longs;endi &longs;unt contactus illi inadæquati extremi plùs, minu&longs;ue, <lb/>de quibus &longs;uprà; e&longs;t enim pror&longs;us eadem difficultas, quam &longs;uprà di&longs;cu&longs;­<lb/>&longs;imus &longs;uo titulo rotæ Ari&longs;totelicæ, quam hîc tantùm indica&longs;&longs;e &longs;ufficiat, <lb/>cùm ex prædictis principiis omninò &longs;oluatur. </s> </p> <p id="N25DB9" type="main"> <s id="N25DBB"><!-- NEW -->Vndecimò &longs;imiliter, &longs;i minor rota motum maioris dirigat; </s> <s id="N25DBF"><!-- NEW -->haud du­<lb/>biè maioris idem punctum pluribus punctis &longs;uperficiei curuæ, cui in­<lb/>cumbit inadæquato dumtaxat contactu re&longs;pondebit, eritque diuer&longs;a li­<lb/>nea huius motus, & aliqua puncta retroagentur; </s> <s id="N25DC9"><!-- NEW -->quod quomodo fiat, <lb/>iam &longs;uprà explicuimus; quod verò &longs;pectat ad proprietates i&longs;tarum linea­<lb/>rum, in &longs;ingularem tractatum cas remittimus. </s> </p> <p id="N25DD1" type="main"> <s id="N25DD3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N25DDF" type="main"> <s id="N25DE1"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena, quæ in &longs;uperficie curua circuli <lb/>maioris rotatur<emph.end type="italics"/>; </s> <s id="N25DEC"><!-- NEW -->&longs;it enim &longs;uperficies curua BF radius AB, &longs;itque rota <lb/>radio NB, cuius peripheria e&longs;t æqualis BF; </s> <s id="N25DF2"><!-- NEW -->igitur M tanget C, O tan­<lb/>get D, & B tandem tanget F; igitur mouetur hæc rota motu mixto ex <lb/>duobus circularibus. </s> </p> <p id="N25DFA" type="main"> <s id="N25DFC">Primò, &longs;ignari po&longs;&longs;unt omnia puncta huius lineæ v. <!-- REMOVE S-->g. <!-- REMOVE S-->MIHF <lb/>per quæ ducenda e&longs;t linea curua, cuius etiam affectiones aliàs demon­<lb/>&longs;trabimus. </s> </p> <p id="N25E07" type="main"> <s id="N25E09"><!-- NEW -->Secundò, punctum B mouetur initio tardi&longs;&longs;imè, O veloci&longs;&longs;imè; </s> <s id="N25E0D"><!-- NEW -->ratio­<lb/>nem iam bis attulimus; </s> <s id="N25E13"><!-- NEW -->quia &longs;cilicet maior e&longs;t motus, cum motus centri <lb/>conuenit cum motu orbis; minor verò è contrario. </s> </p> <p id="N25E19" type="main"> <s id="N25E1B"><!-- NEW -->Tertiò, motus huius rotæ accedit propiùs ad motum rotæ in plano <lb/>rectilineo, quàm motus rotæ &longs;uperioris; quia BF, quæ e&longs;t &longs;uperficies ma­<lb/>ioris circuli, accedit propiùs ad lineam rectam. </s> </p> <p id="N25E23" type="main"> <s id="N25E25"><!-- NEW -->Quartò, &longs;i &longs;it minor rota radio NR cuius motus dirigatur à motu <lb/>maioris radio NB, de&longs;cribit lineam, quæ accedit propiùs ad lineam <lb/>rectam RSTVX, &longs;eu potiùs ad motum centri, quod mouetur motu <lb/>circulari per arcum NG, à quo non recedit, vt patet: </s> <s id="N25E2F"><!-- NEW -->porrò minor <lb/>rota percurrit maiorem &longs;uperficiem &longs;ua peripheria, quod etiam expli-<pb pagenum="353" xlink:href="026/01/387.jpg"/>candum e&longs;t per contactus inadæquatos; tunc enim motus centri longè <lb/>&longs;uperat motum orbis. </s> </p> <p id="N25E3C" type="main"> <s id="N25E3E"><!-- NEW -->Quintò, &longs;i vera e&longs;&longs;et hypothe&longs;is Copernici, terra moueretur hoc vlti­<lb/>mo motu mixto ex motu centri, & motu orbis; </s> <s id="N25E44"><!-- NEW -->vnde omnia puncta <lb/>eiu&longs;dem circuli paralleli mouerentur inæquali motui tardi&longs;&longs;imo qui­<lb/>dem punctum contactus hoc e&longs;t meridiano re&longs;pondens, veloci&longs;&longs;imo ve­<lb/>rò ip&longs;i oppo&longs;itum, &longs;cilicet de media nocte: porrò in hoc motu motus <lb/>centri e&longs;&longs;et ferè maior motu orbis iuxta communem de diametro ma­<lb/>gni orbis &longs;ententiam. </s> </p> <p id="N25E52" type="main"> <s id="N25E54">Sextò, &longs;i motus maioris rotæ dirigatur à minore res eodem modo <lb/>explicanda e&longs;t, quo explicuimus illam per <expan abbr="cõtactus">contactus</expan> diuer&longs;os inadæquatos <lb/>tùm Th. 15. num. </s> <s id="N25E5F"><!-- NEW -->11. tùm in digre&longs;&longs;ione multis locis: </s> <s id="N25E63"><!-- NEW -->porrò po&longs;&longs;unt e&longs;&longs;e <lb/>diuer&longs;æ proportiones circuli mobilis, & immobilis; qui &longs;i maximus e&longs;t, <lb/>minimus illius arcus accipi pote&longs;t pro linea recta. </s> </p> <p id="N25E6B" type="main"> <s id="N25E6D"><!-- NEW -->Septimò, pote&longs;t ita rota moueri, vt pars &longs;uperior retrò agatur, id e&longs;t, <lb/>vt motus orbis &longs;it oppo&longs;itus motui <expan abbr="cétri">centri</expan> v.g.&longs;i punctum N moueatur qui­<lb/>dem dextror&longs;um motu centri, O verò &longs;ini&longs;tror&longs;um motu orbis; </s> <s id="N25E75"><!-- NEW -->&longs;ed tunc <lb/>punctum B mouebitur dextror&longs;um motu orbis, &longs;ed e&longs;t noua difficultas: </s> <s id="N25E7B"><!-- NEW --><lb/>quippe ex hac hypothe&longs;i punctum O de&longs;criberet &longs;uo motu lineam &longs;imi­<lb/>lem, & æqualem lineæ rotatili BMIHF; punctum verò B moueretur <lb/>iuxta hanc hypothe&longs;in eo modo, quo mouetur punctum O iuxta prio­<lb/>rem. </s> <s id="N25E86"><!-- NEW -->Sic autem moueri dicuntur quidam Epicycli ab A&longs;tronomis, quo­<lb/>rum centrum mouetur in con&longs;equentia, hoc e&longs;t &longs;ecundum &longs;eriem <lb/>&longs;ignorum; </s> <s id="N25E8E"><!-- NEW -->&longs;ummum verò punctum, &longs;eu &longs;tella apogæa retrò agitur, &longs;eu <lb/>in partem aduer&longs;am contendit, vel vt vocant, in præcedentia: </s> <s id="N25E94"><!-- NEW -->tales <lb/>vulgò ponuntur Solis Epicycli & Lunæ; vnde obiter colligo, quàm &longs;it <lb/>nece&longs;&longs;aria A&longs;tronomis hæc de motu mixto &longs;ententia, vt &longs;ua phœnome­<lb/>na ad &longs;uas cau&longs;as phy&longs;icas reducant. </s> </p> <p id="N25E9E" type="main"> <s id="N25EA0">Octauò denique, po&longs;&longs;unt e&longs;&longs;e diuer&longs;æ lineæ huius motus pro diuer&longs;a <lb/>circulorum proportione, quarum figuras, de&longs;criptiones, affectiones &longs;uo <lb/>loco demon&longs;trabimus, & nouos latices tum Geometris, tùm Phy&longs;icis <lb/>aperiemus, ex quibus vbertim fluit infinitarum ferè demon&longs;trationum <lb/>materia. </s> </p> <p id="N25EAD" type="main"> <s id="N25EAF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N25EBB" type="main"> <s id="N25EBD"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt cuncta phœnomena rotæ maioris mobilis circa minore&mtail; <lb/>immobilem<emph.end type="italics"/>; &longs;it enim rota minor centro A, cui incubet maior rota cen­<lb/>tro K, radio KB duplo BA, roteturque circa &longs;uperficiem BDFTH <lb/>punctum 5 re&longs;pondebit F & Q po&longs;t decur&longs;am &longs;uperficiem puncto B, <lb/>eritque motus mixtus. </s> </p> <p id="N25ECE" type="main"> <s id="N25ED0"><!-- NEW -->Primò, centrum K mouebitur motu circulari, quia &longs;emper æqualiter <lb/>di&longs;tat à puncto A; igitur de&longs;cribit circulum, cuius radius e&longs;t KA. </s> </p> <p id="N25ED6" type="main"> <s id="N25ED8"><!-- NEW -->Secundò, pote&longs;t facilè de&longs;cribi linea motus puncti B v.g. <!-- REMOVE S-->diuidatur <lb/>enim BDFH in 8 arcus æquales, & B5 in 4; tùm per puncta <pb pagenum="354" xlink:href="026/01/388.jpg"/>CDE &c. </s> <s id="N25EE5"><!-- NEW -->de&longs;cribantur circuli radio KB; </s> <s id="N25EE9"><!-- NEW -->& a&longs;&longs;umatur CR æqualis <lb/>B 2; tùm DL æqualis B 3, tùm EM æqualis B 4, tùm FN æqualis B 5, <lb/>atque ita deinceps, vt per puncta &longs;ignata de&longs;cribatur linea curua <lb/>BRLMNOPRQ, hæc e&longs;t linea huius motus. </s> </p> <p id="N25EF3" type="main"> <s id="N25EF5"><!-- NEW -->Tertiò, omnia puncta mouentur inæqualiter, B quidem tardi&longs;&longs;imè, <lb/>Q veloci&longs;&longs;imè; </s> <s id="N25EFB"><!-- NEW -->nam eo tempore, quò B conficit BR, modicum illud <lb/>&longs;patium IQ decuerit QS, cuius proportio ex analy&longs;i cogno&longs;ci pote&longs;t; </s> <s id="N25F01"><!-- NEW --><lb/>idem dico de motu aliorum punctorum; e&longs;t etiam eadem ratio huius <lb/>inæqualitatis, de qua &longs;uprâ, cuius omnes proportiones a&longs;&longs;ignari po&longs;­<lb/>&longs;unt. </s> </p> <p id="N25F0A" type="main"> <s id="N25F0C"><!-- NEW -->Quartò ob&longs;erua, figuram huius lineæ, quæ accedere videtur ad &longs;pi­<lb/>ralem: præterea linea puncti B, &longs;cilicet BRLMNOPRQ, &longs;ecat li­<lb/>neam puncti Q in 8 mirabili implicatione, cuius interior portio exhibet <lb/>&longs;ectionem cordis &longs;cilicet BRLMN 8 XY <foreign lang="greek">d</foreign> B. </s> </p> <p id="N25F1A" type="main"> <s id="N25F1C"><!-- NEW -->Quintò, deinde pro diuer&longs;a proportione rotarum maioris, &longs;cilicet & <lb/>minoris rotæ, &longs;unt diuer&longs;æ lineæ, & motus mixti diuer&longs;i; immò po&longs;&longs;et <lb/>rota immobilis, circa quam alia rotatur, tam parua e&longs;&longs;e, vt linea tantùm <lb/>po&longs;t multas gyrationes perfici po&longs;&longs;et. </s> </p> <p id="N25F26" type="main"> <s id="N25F28"><!-- NEW -->Sextò, po&longs;&longs;unt etiam determinari lineæ aliorum punctorum intra <lb/>rotam mobilem v, g.puncti T; </s> <s id="N25F2E"><!-- NEW -->quod vt fiat, &longs;emper e&longs;t a&longs;&longs;umendus ra­<lb/>dius KB, qui &longs;cilicet, dum K e&longs;t in <foreign lang="greek">m</foreign>, incubat <foreign lang="greek">m</foreign> R, dum e&longs;t in M incubat <lb/>ML, dum e&longs;t in <foreign lang="greek">q</foreign> re&longs;pondet <foreign lang="greek">q</foreign> M; </s> <s id="N25F46"><!-- NEW -->denique dum e&longs;t in 9 re&longs;pondet <lb/>9 N; itaque a&longs;&longs;umantur <foreign lang="greek">m</foreign> 3, M <foreign lang="greek">w, q</foreign> 7, 9 <foreign lang="greek">b</foreign> æquales K, & ducatur per <lb/>&longs;ignata puncta linea curua T3 <foreign lang="greek">p</foreign> 7 <foreign lang="greek">b</foreign>, hæc e&longs;t linea motus mixti pun­<lb/>cti T. </s> </p> <p id="N25F64" type="main"> <s id="N25F66"><!-- NEW -->Septimò, quando motus minoris rotæ radio KT dirigitur à motu <lb/>maioris radio KB, rotatur illa in &longs;uperficie circuli radio AT, &longs;ed ita <lb/>quadratus TV qua&longs;i repat per contactus inadæquatos in &longs;emicirculo <lb/>T 11 10; </s> <s id="N25F70"><!-- NEW -->porrò in hoc ca&longs;u maxima e&longs;&longs;et difficultas rotæ Ari&longs;totelicæ; <lb/>denique, quando maior dirigitur à minori, quadrans B5 qua&longs;i contra­<lb/>hitur in arcu minore BC, quæ contractio explicatur per contractus in­<lb/>adæquatos, vt iam &longs;æpè diximus in aliis motibus. </s> </p> <p id="N25F7A" type="main"> <s id="N25F7C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N25F88" type="main"> <s id="N25F8A"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena rotæ mobilis in &longs;uperficie concaua <lb/>maioris circuli<emph.end type="italics"/>; dixi maioris circuli; </s> <s id="N25F95"><!-- NEW -->quia in &longs;uperficie concaua mi­<lb/>noris, vel æqualis moueri non pote&longs;t, vt con&longs;tat; </s> <s id="N25F9B"><!-- NEW -->&longs;it ergo fig.4. rota <lb/>mobilis radio PC; </s> <s id="N25FA1"><!-- NEW -->&longs;it &longs;uperficies concaua circuli dupli prioris in <lb/>peripheria CGK; </s> <s id="N25FA7"><!-- NEW -->diuidatur CGK in 8 arcus æquales; haud <lb/>dubiè tota &longs;uperficies rotæ mobilis &longs;ucce&longs;&longs;iuè percurret totam <lb/>&longs;uperficiem concauam CGK, cùm illa &longs;it huic æqualis, hoc po­<lb/>&longs;ito. </s> </p> <p id="N25FB1" type="main"> <s id="N25FB3"><!-- NEW -->Primò, punctum C percurret rectam CAK, nec vnquam ab <lb/>ea di&longs;cedet, & centrum P percurret &longs;emicirculum PQN; </s> <s id="N25FB9"><!-- NEW -->quippe <pb pagenum="355" xlink:href="026/01/389.jpg"/>&longs;emper æqualem &longs;eruabit di&longs;tantiam à &longs;uperficie concaua CGK; </s> <s id="N25FC2"><!-- NEW -->&longs;ed illa <lb/>e&longs;t PC; </s> <s id="N25FC8"><!-- NEW -->igitur &longs;emper di&longs;tabit æqualiter à centro A; igitur de&longs;cribit &longs;e­<lb/>micirculum PQN. </s> </p> <p id="N25FCE" type="main"> <s id="N25FD0"><!-- NEW -->Secundò, quod &longs;pectat ad primum; </s> <s id="N25FD4"><!-- NEW -->certè punctum A rotæ mobilis <lb/>tanget ip&longs;um G; </s> <s id="N25FDA"><!-- NEW -->e&longs;t enim quadrans CG æqualis &longs;emicirculo CA, &longs;ed <lb/>cum A tanget G, C erit in A; </s> <s id="N25FE0"><!-- NEW -->denique C tanget K; </s> <s id="N25FE4"><!-- NEW -->igitur C percurret <lb/>rectam CAK; </s> <s id="N25FEA"><!-- NEW -->porrò facilè o&longs;tendetur punctum C moueri per alia pun­<lb/>cta v.g.per punctum T; </s> <s id="N25FF0"><!-- NEW -->nam punctum 9.tanget E; </s> <s id="N25FF4"><!-- NEW -->igitur TY e&longs;t tangens <lb/>igitur AY & YE; </s> <s id="N25FFA"><!-- NEW -->igitur ET, TA &longs;unt æquales, vt con&longs;tat; igitur C duce­<lb/>tur per. </s> <s id="N26000">T; </s> <s id="N26003"><!-- NEW -->præterea C 4. DV &longs;unt arcus æquales, quia angulus CAD e&longs;t <lb/>&longs;ubduplus CP 4. vel YTD, vt con&longs;tat; </s> <s id="N26009"><!-- NEW -->igitur arcus DV e&longs;t æqualis C 4. <lb/>igitur C ducitur per V: idem o&longs;tendetur pro aliis punctis. </s> </p> <p id="N2600F" type="main"> <s id="N26011">Tertiò, hinc pote&longs;t determinari longitudo di&longs;tantiarum CV, VT, &c. </s> <s id="N26014"><!-- NEW --><lb/>nam AE e&longs;t chorda arcus 135. id e&longs;t, e&longs;t dupla &longs;inus grad. <!-- REMOVE S-->67. 1/2 AT e&longs;t <lb/>chorda arcus 90. id e&longs;t latus quadrati in&longs;cripti: denique RA e&longs;t chorda <lb/>arcus 45. id e&longs;t dupla &longs;inus 22. 1/2 hinc vides quàm acuratè recta AC &longs;e­<lb/>cet omnes arcus DV, ET, &c.ita vt &longs;int æquales aliis arcubus maioris cir­<lb/>culi, &longs;cilicet DC, DV, EC, ET, PR, PC, &c. </s> </p> <p id="N26023" type="main"> <s id="N26025">Quartò, hinc vides punctum C initio tardi&longs;&longs;imè moueri, & continuè <lb/>&longs;uum motum accelerare, donec perueniat in A, quem ab A in K retar­<lb/>dat in eadem proportione, in qua AC in A accelerat, CV e&longs;t ferè &longs;ubtri­<lb/>pla VT, &longs;cilicet 15224. ad 43354.TR e&longs;t ad CT vt 64886.ad 58578. vt <lb/>con&longs;tat ex tabulis &longs;inuum. </s> </p> <p id="N26030" type="main"> <s id="N26032"><!-- NEW -->Quintò, non modò punctum C rotæ mobilis mouetur motu recto, <lb/>verùm etiam alia puncta circumferentiæ eiu&longs;dem rotæ; </s> <s id="N26038"><!-- NEW -->e&longs;t enim par om­<lb/>nium ratio v.g. <!-- REMOVE S-->punctum 2. mouetur per rectam 3.A punctum 4.per re­<lb/>ctam DA. punctum 9.per rectam EA; quod certè mirabile videtur, & <lb/>primo intuitu vix credi po&longs;&longs;et. </s> </p> <p id="N26044" type="main"> <s id="N26046"><!-- NEW -->Sextò, &longs;i a&longs;&longs;umatur aliud punctum intra rotam de&longs;cribi poterit facilè <lb/>linea illius motus; </s> <s id="N2604C"><!-- NEW -->&longs;it v.g. <!-- REMOVE S-->punctum 6. ducantur rectæ TYYTZR; nam <lb/>radius PR migrat in TV, YTZRQA, &longs;umantur TV, YT, Z<foreign lang="greek">d</foreign>, QX æ­<lb/>quales P6.& per &longs;ignata puncta de&longs;cribatur curua 6. T<foreign lang="greek">d</foreign>X, hæc e&longs;t linea <lb/>motus puncti 6. cuius motus initio e&longs;t tardior, &longs;ub finem velocior. </s> </p> <p id="N26060" type="main"> <s id="N26062"><!-- NEW -->Septimò, hinc pote&longs;t dirigi motus minoris à motu maioris, & vici&longs;&longs;im, <lb/>quod explicandum e&longs;t eodem pror&longs;us modo, quo iam &longs;æpè explicatum <lb/>e&longs;t per diuer&longs;os &longs;cilicet contactus inadæquatos, pro quo tantùm ob&longs;erua, <lb/>&longs;i minor dirigatur à maiore, puncta minoris dextror&longs;um mouentur <lb/>tùm &longs;ini&longs;trorum; contra verò &longs;i maior dirigatur à minore, puncta maio­<lb/>ris mouentur &longs;ini&longs;tror&longs;um, tùm dextror&longs;um, quæ omnia ex dictis facilè <lb/>intelligi po&longs;&longs;unt, & explicari. </s> </p> <p id="N26072" type="main"> <s id="N26074"><!-- NEW -->Octauò, præterea puncta radij RC a&longs;&longs;umpta, quæ propiùs ad extre­<lb/>mitatem C accedunt, de&longs;cribunt lineam, quæ propiùs accedit ad rectam; </s> <s id="N2607A"><!-- NEW --><lb/>quæ verò accedunt propiùs ad centrum P, de&longs;cribunt lineam magis cur­<lb/>uam; </s> <s id="N26081"><!-- NEW -->idem de punctis in radio PA; </s> <s id="N26085"><!-- NEW -->nam e&longs;t eadem ratio, quæ omnia ex <lb/>dictis con&longs;tant; </s> <s id="N2608B"><!-- NEW -->an fortè cùm punctum C de&longs;cribat rectam, punctum P <pb pagenum="356" xlink:href="026/01/390.jpg"/>circulum, & quæ propiùs accedunt ad C minùs curuam, quæ propiùs <lb/>ad P magis curuam; &longs;ed tractatu &longs;equenti omnes i&longs;tas lineas explica­<lb/>bimus. </s> </p> <p id="N26098" type="main"> <s id="N2609A"><!-- NEW -->Nonò, &longs;i &longs;uperficies &longs;it minoris circuli quàm dupli; </s> <s id="N2609E"><!-- NEW -->certè punctum C, <lb/>v.g. <!-- REMOVE S-->non de&longs;cribet rectum CK, &longs;ed aliam curuam &longs;ini&longs;tror&longs;um; &longs;i verò <lb/>&longs;it maioris circuli quàm dupli, de&longs;cribet aliam curuam dextror&longs;um, quæ <lb/>omnia con&longs;tant ex dictis. </s> </p> <p id="N260AA" type="main"> <s id="N260AC"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N260B8" type="main"> <s id="N260BA"><!-- NEW -->Non videntur omittenda aliqua Corollaria Cyclomètrica, quæ ex di­<lb/>ctis &longs;ua &longs;ponte na&longs;ci videntur; </s> <s id="N260C0"><!-- NEW -->nam primò &longs;emicirculus AQG e&longs;t æqua­<lb/>lis triangulo mixto ex arcubus GC, & GA, & recta AC; quia quadrans <lb/>AGC e&longs;t æqualis circulo A9.CB, vt patet. </s> </p> <p id="N260C8" type="main"> <s id="N260CA"><!-- NEW -->Secundò, omnes radij eodem modo &longs;ecantur à circulo v.g. <!-- REMOVE S-->AC, AD. <lb/>AE: &longs;unt enim CVE <foreign lang="greek">w</foreign>, D4.æquales, item C 3. T, VE9. &c. </s> </p> <p id="N260D6" type="main"> <s id="N260D8">Tertiò, omnes arcus intercepti inter radios &longs;unt æquales v.g. <!-- REMOVE S-->DY, C 4. <lb/>T4. E4. GF, F9.9 <foreign lang="greek">d. </foreign><!-- KEEP S--></s> <s id="N260E3">&c. </s> </p> <p id="N260E6" type="main"> <s id="N260E8"><!-- NEW -->Quartò, præterea arcus à puncto contactus maioris, & dupli circuli <lb/>v&longs;que ad quemlibet radium &longs;unt æquales, v.g. <!-- REMOVE S-->G9, A & GC, G9. <foreign lang="greek">d</foreign> GD, <lb/>G9. & GE, GF, & GC, tùm FR, & FC, F <foreign lang="greek">b</foreign>, & FD, F <foreign lang="greek">w</foreign> & FE, tùm ET, <lb/>& EC, E 4. & ED; denique DV, DC. <!-- KEEP S--></s> </p> <p id="N26101" type="main"> <s id="N26103"><!-- NEW -->Quintò, triangula illa mixta ex duplici arcu æquali maioris, & minoris <lb/>circuli, & altero latere recto, &longs;unt æqualia &longs;ectionibus minoribus circuli, <lb/>quarum arcus æquales &longs;unt prioribus minoris circuli, &longs;ic triangulum <lb/>mixtum ex arcubus GC, G9. A, & recta AE e&longs;t æquale &longs;emicirculo G9. <lb/>A; </s> <s id="N2610F"><!-- NEW -->mixtun verò ex arcubus FC, FR, & recta RC, e&longs;t æquale &longs;ectioni VA <lb/>vel E9. A, mixtum ex arcubus ET, EC & recta, æquale e&longs;t &longs;ectioni TA <lb/>vel 9. <foreign lang="greek">d</foreign> A; denique mixtum ex arcu DC, DV, & recta CV e&longs;t æquale <lb/>&longs;ectioni RA. </s> </p> <p id="N2611D" type="main"> <s id="N2611F"><!-- NEW -->Sextò &longs;ubtracto ex prædictis triangulis alio triangulo mixto per da­<lb/>tum radium quemcumque, &longs;ubtrahitur portio æqualis ex &longs;emicirculo <lb/>minore, & re&longs;iduum æquale e&longs;t re&longs;iduo v.g.ex triangulo mixto G9. AC <lb/>G ducto radio AF, detrahitur triangulum mixtum GF <foreign lang="greek">r</foreign>, ex &longs;emicir­<lb/>culo A9. C, detrahitur portio æqualis 7. A; </s> <s id="N2612F"><!-- NEW -->igitur re&longs;iduum &longs;emicirculi <lb/>e&longs;t æquale re&longs;iduo trianguli mixti; </s> <s id="N26135"><!-- NEW -->deinde ducto radio AC detrahitur <lb/>triangulo mixto prædicto aliud mixtum minus GE9. ex &longs;emicirculo A <lb/>9. C detrahitur portio A9. æqualis detracto; igitur Trapezus re&longs;iduus, E <lb/>9. A 7. E, e&longs;t æqualis triangulo mixto CA9. C. idem dico de aliis. </s> </p> <p id="N2613F" type="main"> <s id="N26141"><!-- NEW -->Septimò, cùm &longs;ector AFG &longs;it æqualis quadranti AP9. &longs;ectio ACZ, <lb/>e&longs;t maior quadrante prædicto triangulo mixto GCF vel &longs;ectiore 7. A; </s> <s id="N26147"><!-- NEW --><lb/>atqui &longs;ectio ACZ habet arcum 135. & A 7. arcum 90. igitur &longs;ectio ar­<lb/>cus 135. e&longs;t æqualis quadranti plus &longs;ectione arcus; </s> <s id="N2614E"><!-- NEW -->igitur triangulum A <lb/>7.4.A e&longs;t æquale quadranti; triangulum verò mixtum GCA e&longs;t æquale <lb/>quadranti, minùs prædicta &longs;ectione arcus 90. </s> </p> <p id="N26156" type="main"> <s id="N26158"><!-- NEW -->Octauò, hinc triangulum mixtum ex arcubus A 7.9. GG & recta AG <pb pagenum="357" xlink:href="026/01/391.jpg"/>e&longs;t æquale quadrato radij AQ; idem dico de mixto ex arcubus AT9. 9. <lb/>C, & recta AC; </s> <s id="N26167"><!-- NEW -->hinc vtrumque &longs;imul &longs;umptum detracta &longs;cilicet duplici <lb/>portione A 7.9. TA e&longs;t æquale quadrato in&longs;cripto, & duplex illa &longs;ectio <lb/>figura ouali e&longs;t æqualis triangulo mixto ex tribus arcubus G9. 9. C, C <lb/>G; quod facilè geometricè demon&longs;tratur; </s> <s id="N26171"><!-- NEW -->&longs;it enim circulus centro B; </s> <s id="N26175"><!-- NEW --><lb/>&longs;int duæ diametri, GE, AC, quibus in 4. quadrantes diuidatur circulus; </s> <s id="N2617A"><!-- NEW --><lb/>tùm a&longs;&longs;umatur arcus GF, æqualis FC, & CD; </s> <s id="N2617F"><!-- NEW --><expan abbr="ducãtur">ducantur</expan> rectæ AD, AF, GF, <lb/>IF: </s> <s id="N26188"><!-- NEW -->dico triangulum mixtum ex rectis AF, FG, & arcu GA, e&longs;&longs;e æquale <lb/>quadranti, quod demon&longs;tro; </s> <s id="N2618E"><!-- NEW -->triangula KAL, KFG &longs;unt æquiangula, quia <lb/>anguli K vtrinque &longs;unt æquales: </s> <s id="N26194"><!-- NEW -->&longs;ed DAF, & AFG, &longs;u&longs;tinent æquales ar­<lb/>cus; </s> <s id="N2619A"><!-- NEW -->igitur &longs;unt æquales; </s> <s id="N2619E"><!-- NEW -->igitur &longs;unt proportionalia; igitur vt quadr. </s> <s id="N261A2">BA ad <lb/>quadr. </s> <s id="N261A7">IF: &longs;ed quadr. </s> <s id="N261AA">BF e&longs;t duplum quadr. </s> <s id="N261AD">IF; </s> <s id="N261B0"><!-- NEW -->igitur & BA e&longs;t duplum; </s> <s id="N261B4"><!-- NEW --><lb/>igitur KAL duplum KFG; </s> <s id="N261B9"><!-- NEW -->igitur BAK æquale; </s> <s id="N261BD"><!-- NEW -->igitur tantum additur, <lb/>quantum tollitur; igitur prædictum triangulum e&longs;t æquale quadranti. </s> </p> <p id="N261C3" type="main"> <s id="N261C5"><!-- NEW -->Nonò præterea, Trapezus FC9. AEF e&longs;t æqualis triangulo mixto ex <lb/>arcubus ABC, TAR, & recta RC; </s> <s id="N261CB"><!-- NEW -->Trapezus verò E9. TA, CE æqualis <lb/>mixto triangulo ex arcubus ABCAT, & recta TC; </s> <s id="N261D1"><!-- NEW -->Trapezus verò D<foreign lang="greek">m</foreign>A <lb/>CD e&longs;t æqualis mixto ex arcubus ABC, AV, & recta VC; </s> <s id="N261DB"><!-- NEW -->hinc lulu­<lb/>la DCBAVD e&longs;t æqualis &longs;ectori ACD; </s> <s id="N261E1"><!-- NEW -->igitur quadranti P9. C: </s> <s id="N261E5"><!-- NEW -->hinc <lb/>altera lulula AT 4. ECBA e&longs;t dupla prioris; </s> <s id="N261EB"><!-- NEW -->igitur æqualis &longs;emicircu­<lb/>lo AC, vel &longs;ectori AEC: hinc tota figura ex AC, CE, & recto CA, e&longs;t <lb/>æqualis circulo A9. CB. <!-- KEEP S--></s> </p> <p id="N261F4" type="main"> <s id="N261F6"><!-- NEW -->Decimò, Trapezus E <foreign lang="greek">w b</foreign> RCE e&longs;t æqualis quadranti P9. C: </s> <s id="N261FE"><!-- NEW -->hinc &longs;i <lb/>detrahatur ex prædicto Trapezo triangulum mixtum E 4. TCE, illa <lb/>figura E <foreign lang="greek">w b</foreign> RT 4. E e&longs;t æqualis triangulo rectilineo AP9. &longs;imiliter <lb/>aliæ figuræ T 4. DVT, R <foreign lang="greek">b</foreign> 4. TRA <foreign lang="greek">m b</foreign> RA, A <foreign lang="greek">m</foreign> 9. <foreign lang="greek">r</foreign> F <foreign lang="greek">w</foreign> RA; item 9. <lb/><foreign lang="greek">r</foreign> F <foreign lang="greek">p r</foreign>, &c. </s> </p> <p id="N26229" type="main"> <s id="N2622B"><!-- NEW -->Vndecimò, &longs;ector ACE diuiditur in duas partes æquales ab arcu R <lb/><foreign lang="greek">w</foreign>; </s> <s id="N26234"><!-- NEW -->item &longs;ector ADF ab arcu <foreign lang="greek">m r</foreign>; </s> <s id="N2623C"><!-- NEW -->item totus quadrans AGC ab arcu A <lb/>9. G; </s> <s id="N26242"><!-- NEW -->denique illa figura E <foreign lang="greek">w</foreign> RTE e&longs;t æqualis Trapezo D <foreign lang="greek">b</foreign> RVD; </s> <s id="N2624E"><!-- NEW -->igi­<lb/>tur Trapezus æqualis rectilineo A9.P, itemque Trapezus T9.ECT <lb/>æqualis quadranti P9. C; </s> <s id="N26256"><!-- NEW -->igitur Trapezo E <foreign lang="greek">p</foreign> RCE; </s> <s id="N2625E"><!-- NEW -->igitur triangulum <lb/>mixtum <foreign lang="greek">b</foreign> 9. <foreign lang="greek">w b</foreign> æquale mixto T <foreign lang="greek">b</foreign> R; </s> <s id="N26270"><!-- NEW -->&longs;ed de his &longs;atis, quæ tantùm indi­<lb/>ca&longs;&longs;e &longs;ufficiat; omitto enim infinita alia, de quibus in Cyclometria. </s> </p> <p id="N26276" type="main"> <s id="N26278"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N26284" type="main"> <s id="N26286"><emph type="italics"/>Si ita moueatur cylindrus per quamcunque lineam, vt eius axis moueatur <lb/>motu recto, totu&longs;que cylindrus circa axem motu circulari moueatur, motus <lb/>mixtus e&longs;t, cuius diuer&longs;a &longs;unt phœnomena.<emph.end type="italics"/></s> </p> <p id="N26291" type="main"> <s id="N26293"><!-- NEW -->Primò, axis mouetur tantùm motu recto; </s> <s id="N26297"><!-- NEW -->aliæ verò partes motu mixto <lb/> &longs;it enim cylindrus CH, cuius axis &longs;it AB, circa quem moueatur cylin­<lb/>drus motu circulari, & qui per <expan abbr="eãdem">eandem</expan> lineam AB indefinitè produ­<lb/>ctam mouetur; certè punctum C, v.g. <!-- REMOVE S-->mouetur motu mixto ex motu cen­<lb/>tri A, vel axis AB, & motus orbis. </s> </p> <p id="N262A9" type="main"> <s id="N262AB"><!-- NEW -->Secundò, punctum C mouetur motu &longs;piræ; nam &longs;i tantùm motu orbis <pb pagenum="358" xlink:href="026/01/392.jpg"/>moueretur, decur&longs;o &longs;emicirculo peruenire in L, F, N, &c. </s> <s id="N262B4"><!-- NEW -->igitur &longs;i eo <lb/>tempore, quo C decurrit motu centri, &longs;emicirculum CD; </s> <s id="N262BA"><!-- NEW -->punctum axis <lb/>A decurrit AK; </s> <s id="N262C0"><!-- NEW -->haud dubiè punctum C erit in E, tùm in F, tùm in G, tùm <lb/>in T; &longs;ed hic motus &longs;piralis e&longs;t, vt con&longs;tat. </s> </p> <p id="N262C6" type="main"> <s id="N262C8"><!-- NEW -->Tertiò, omnia puncta peripheriæ CD mouentur æquali motu; quia <lb/>&longs;cilicet æqualem motum centri, & orbis participant. </s> </p> <p id="N262CE" type="main"> <s id="N262D0"><!-- NEW -->Quartò, &longs;i motus centri vel axis &longs;it minor, frequentiores &longs;unt Helices <lb/>v.g. <!-- REMOVE S-->&longs;i eo tempore, quo C decurrit &longs;emicirculum CD, A decurreret tan­<lb/>tùm AR, C perueniret tantùm in Q, mox in I, atque ita deinceps moue­<lb/>retur per frequentiores &longs;piras; &longs;i verò motus axis &longs;it maior, &longs;piræ erunt <lb/>rariores, vt patet, v.g. <!-- REMOVE S-->&longs;i eo tempore, quo C motu centri decurrit &longs;emi­<lb/>circulum CD, punctum A decurrit AL, punctum C decurret &longs;piram C <lb/>M, mox MT, &c. </s> </p> <p id="N262E4" type="main"> <s id="N262E6"><!-- NEW -->Quintò, areæ circuli CAD mouebuntur motu &longs;pirali, excepto centro <lb/>A, minores tamen &longs;piras conficeret, &longs;cilicet circa cylindrum cuius minor <lb/>e&longs;t ba&longs;is, vt patet; </s> <s id="N262EE"><!-- NEW -->vnde minore motu mouentur, quàm C vel D; </s> <s id="N262F2"><!-- NEW -->igitur <lb/>axis AB tardi&longs;&longs;imo motu mouentur; </s> <s id="N262F8"><!-- NEW -->partes verò &longs;uperficiei cylindri <lb/>veloci&longs;&longs;imè; aliarum verò partium, quæ accedunt propiùs ad periphæ­<lb/>riam, velociùs. </s> <s id="N26300"><!-- NEW -->quæ propiùs ad centrum, tardiùs: </s> <s id="N26304"><!-- NEW -->hoc motu mouentur alæ <lb/>auium; </s> <s id="N2630A"><!-- NEW -->quæ directo volatu tendunt per lineam rectam, vt grues; nam <lb/>quælibet pars alæ motum axis habet, & orbis. </s> </p> <p id="N26310" type="main"> <s id="N26312"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N2631E" type="main"> <s id="N26320"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena calami volatilis<emph.end type="italics"/> &longs;it enim calamus <lb/>&longs;eu cylindrus DA, in altera extremitate D ita excauatus, vt duæ pennæ <lb/>BD, CE in&longs;eri po&longs;&longs;int eo ferè modo, quo vides. </s> </p> <p id="N2632C" type="main"> <s id="N2632E"><!-- NEW -->Primò, mouetur axis FA motu recto; reliquæ verò partes motu mix­<lb/>to ex recto axis, & circulari orbis eo modo, quo diximus de cylindro <lb/>in &longs;uperiore Theoremate. <!-- KEEP S--></s> </p> <p id="N26337" type="main"> <s id="N26339"><!-- NEW -->Secundò, &longs;emper calamus DA præit, &longs;cilicet ip&longs;a ba&longs;is A, & &longs;equun­<lb/>tur pennæ; </s> <s id="N2633F"><!-- NEW -->ratio e&longs;t, quia pennîs re&longs;i&longs;tit fortiùs aër, vt pater; </s> <s id="N26343"><!-- NEW -->igitur earum <lb/>vim faciliùs &longs;uperat; </s> <s id="N26349"><!-- NEW -->hinc &longs;emper retinentur à tergo, nec alia ratio e&longs;&longs;e <lb/>pote&longs;t; </s> <s id="N2634F"><!-- NEW -->præ&longs;ertim cùm pennæ ita &longs;int compo&longs;itæ propter diuaricationem, <lb/>vt multum aëra verberent; </s> <s id="N26355"><!-- NEW -->quod autem pennis maximè re&longs;i&longs;tat aër, patet <lb/>ex auium volatu; </s> <s id="N2635B"><!-- NEW -->imò ex ip&longs;o plumarum de&longs;cen&longs;u; </s> <s id="N2635F"><!-- NEW -->hinc pennæ illæ, qui­<lb/>bus ornantur equitum pilei, &longs;emper à tergo &longs;equuntur currentem equi­<lb/>tem; </s> <s id="N26367"><!-- NEW -->idem dico de fa&longs;ciis illis tran&longs;uer&longs;ariis, quibus iunguntur equites; <lb/>idem de militaribus &longs;ignis, &longs;eu vexillis. </s> </p> <p id="N2636F" type="main"> <s id="N26371">Tertiò, hinc ratio motus recti calami, quia, cùm &longs;emper præeat, <lb/><expan abbr="eũdem">eundem</expan> &longs;itum &longs;eruat, penna&longs;que ip&longs;as qua&longs;i reluctantes trahit, &longs;untque <lb/>ip&longs;æ ad in&longs;tar claui, qui puppim regit. </s> </p> <p id="N2637B" type="main"> <s id="N2637D"><!-- NEW -->Quartò, cum plumæ ita deuaricatæ qua&longs;i à reflante aëra pellantur &longs;e­<lb/>quitur nece&longs;&longs;ario motus orbis circa axem calami DA; </s> <s id="N26383"><!-- NEW -->quippe hîc motus <lb/>facilis e&longs;t; </s> <s id="N26389"><!-- NEW -->&longs;ic enim voluitur vectis &longs;eu cylindrus, quotie&longs;cumque ab altera <pb pagenum="359" xlink:href="026/01/393.jpg"/>tremitate pellitur; </s> <s id="N26392"><!-- NEW -->igitur cum pellantur D & C; quid mirum &longs;i totus ca­<lb/>lamus cum ip&longs;is pennis conuertatur. </s> </p> <p id="N26398" type="main"> <s id="N2639A"><!-- NEW -->Quintò, hinc motus calami e&longs;t mixtus ex recto axis, & circulari or­<lb/>bis; </s> <s id="N263A0"><!-- NEW -->igitur &longs;piralis e&longs;t; </s> <s id="N263A4"><!-- NEW -->&longs;piræ autem maiores &longs;unt, vel minores pro diuer&longs;a <lb/>di&longs;tantia partium ab axe AF, qui debet cen&longs;eri productus v&longs;que ad G; </s> <s id="N263AA"><!-- NEW --><lb/>nam partes, quæ longiùs di&longs;tant ab axe, maiores &longs;piras decurrunt; aliæ <lb/>verò minores; porrò &longs;piræ ip&longs;æ eò frequentiores &longs;unt, quò motus orbis <lb/>velocior e&longs;t, & contrà rariores, quò tardior. </s> </p> <p id="N263B3" type="main"> <s id="N263B5"><!-- NEW -->Sextò, &longs;i &longs;it tantùm vnica penna, calamus non mouetur hoc motu; </s> <s id="N263B9"><!-- NEW --><lb/>quia vix aër verberatur; </s> <s id="N263BE"><!-- NEW -->adde quod in eam partem, quæ caret penna im­<lb/>pul&longs;us nece&longs;&longs;ariò inclinatur; idem accidit cum altera penna fracta e&longs;t, <lb/>vel minus aptè diuaricata. </s> </p> <p id="N263C6" type="main"> <s id="N263C8"><!-- NEW -->Septimò, in cam partem conuertitur, &longs;eu &longs;piras agit, in quam pennæ <lb/>ip&longs;æ detorquentur; </s> <s id="N263CE"><!-- NEW -->alioquin non e&longs;&longs;et, cur potiùs in vnam, quàm in aliam <lb/>&longs;uos agerent orbes; </s> <s id="N263D4"><!-- NEW -->igitur ita diuaricantur pennæ, vt earum plana &longs;ibi in­<lb/>uicem &longs;int obliqua; </s> <s id="N263DA"><!-- NEW -->cuius rei ratio prædicta clari&longs;&longs;ima cùm &longs;it; non e&longs;t <lb/>quod amplius de hac re laboremus. </s> </p> <p id="N263E0" type="main"> <s id="N263E2"><!-- NEW -->Octauò, &longs;i pennæ di&longs;tractiones &longs;unt, & maximè diuaricatæ; </s> <s id="N263E6"><!-- NEW -->motus <lb/>axis e&longs;t tardior; ratio e&longs;t, quia in eo &longs;tatu multum aëra pellunt, &longs;eu venti­<lb/>lant, à quo retinentur. </s> </p> <p id="N263EE" type="main"> <s id="N263F0"><!-- NEW -->Nonò, &longs;i di&longs;tractiores &longs;unt, motus orbis e&longs;t etiam tardior, &longs;untque <lb/>rariores &longs;piræ; </s> <s id="N263F6"><!-- NEW -->ratio e&longs;t eadem, quia cùm motus orbis e&longs;t maior, etiam <lb/>plùs aëris vertigo illa &longs;ecum abripit; </s> <s id="N263FC"><!-- NEW -->hinc maior e&longs;t re&longs;i&longs;tentia; vnde <lb/>ob&longs;eruabis, vt motus orbis minùs impediatur, ita pennas e&longs;&longs;e componen­<lb/>das, vt aëra &longs;ua qua&longs;i acie cæ&longs;im diuidant, ne &longs;i pellant tota &longs;ua &longs;uperficie, <lb/>maior &longs;it re&longs;i&longs;tentia. </s> </p> <p id="N26406" type="main"> <s id="N26408">Decimò, &longs;i demum plùs æquo &longs;int diuaricatæ, ita vt angulum obtu&longs;i&longs;­<lb/>&longs;imum faciant, ce&longs;&longs;at omninò motus orbis propter maiorem re&longs;i&longs;tentiam, <lb/>quæ vertiginem illam impedit. </s> </p> <p id="N2640F" type="main"> <s id="N26411"><!-- NEW -->Vndecimò, ita pennæ aptari debent, vt &longs;en&longs;im inflexæ à radice DE <lb/>ver&longs;us apices BC afflatum aëris diuer&longs;um excipiant, & di&longs;&longs;imilem: vnde <lb/>accidit, vt partes ip&longs;æ, quæ retardantur, & maiore vi pollent in vertigi­<lb/>nem agantur, in eam &longs;cilicet partem, in quam aliqua inclinatio conducit <lb/>&longs;ic globus retentus à corpore oppo&longs;ito in orbem agitur propter rationem <lb/>prædictam, ne ille impetus &longs;it fru&longs;trà, qui adhuc &longs;upere&longs;t. </s> <s id="N2641F"><!-- NEW -->Hinc vides <lb/>motum orbis non imprimi calamo à pennis, &longs;ed pennis à calamo; </s> <s id="N26425"><!-- NEW -->qui <lb/>cùm ab illis retardetur, ne aliquid impetus &longs;it fru&longs;trà, &longs;upplet motu cir­<lb/>culari, quod recto difficiliori propter re&longs;i&longs;tentiam orbis con&longs;equi non <lb/>pote&longs;t; </s> <s id="N2642F"><!-- NEW -->determinatur quidem motus circularis in talem partem ab ip&longs;a <lb/>pennarum deflexione; </s> <s id="N26435"><!-- NEW -->non tamen imprimitur: </s> <s id="N26439"><!-- NEW -->hinc &longs;i fortè in via pen­<lb/>næ ex &longs;ua theca decidant, calamus ip&longs;e &longs;ine nouo impul&longs;u longiùs &longs;pa­<lb/>tium conficito; tribuit enim motui recto non impedito, quod circulari, <lb/>vel &longs;pirali, &longs;i pennæ ade&longs;&longs;ent tribueret. </s> </p> <p id="N26443" type="main"> <s id="N26445"><!-- NEW -->Duodecimò, &longs;i pennæ contractiores &longs;unt, & angulum acutiorem fa­<lb/>ciant, calamus velociùs mouetur motu axis; </s> <s id="N2644B"><!-- NEW -->ratio e&longs;t, quia re&longs;i&longs;tentia mi-<pb pagenum="360" xlink:href="026/01/394.jpg"/>nùs retardat; </s> <s id="N26454"><!-- NEW -->&longs;unt enim pauciores partes, quæ valde obliquè cadunt: </s> <s id="N26458"><!-- NEW --><lb/>hinc minor e&longs;t appul&longs;us, quod clarum e&longs;t; hinc, vt calamus velociùs per­<lb/>gat, con&longs;tringuntur pennæ. </s> </p> <p id="N2645F" type="main"> <s id="N26461"><!-- NEW -->Decimotertiò, &longs;i contractiores &longs;unt, & rectè compo&longs;itæ, cum illa &longs;cili­<lb/>cet inflexione, <expan abbr="eoq;">eoque</expan> &longs;itu, de quo n.11.non modò velocior erit motus axis, <lb/>&longs;ed etiam motus orbis; </s> <s id="N2646D"><!-- NEW -->ratio e&longs;t, quia minor orbis citiùs perficitur: </s> <s id="N26471"><!-- NEW -->adde <lb/>quod minus aëris huic motui re&longs;i&longs;tit; </s> <s id="N26477"><!-- NEW -->vnde vides ita e&longs;&longs;e aptandas pen­<lb/>nas, vt re&longs;i&longs;tentia aëris inæqualis cau&longs;et illam vertiginem, quæ tamen <lb/>tanta e&longs;&longs;e non debet; alioquin ip&longs;um motum orbis omninò impediret, <lb/>vt diximus n. </s> <s id="N26481">10. </s> </p> <p id="N26484" type="main"> <s id="N26486"><!-- NEW -->Decimoquartò, denique &longs;i plùs æquo contractæ &longs;unt, e&longs;&longs;et motus or­<lb/>bis; quippe modica e&longs;t aëris re&longs;i&longs;tentia, quæ ad motum illum non &longs;ufficit, <lb/>licèt &longs;emper &longs;int aliqui gyri, &longs;ed rariores. </s> </p> <p id="N2648E" type="main"> <s id="N26490"><!-- NEW -->Decimoquintò, tres aliquando, aliquando duæ in&longs;eruntur pennæ; </s> <s id="N26494"><!-- NEW -->e&longs;t <lb/>enim eadem vertiginis cau&longs;a, imò quatuor in&longs;eri po&longs;&longs;ent; &longs;unt enim qua&longs;i <lb/>totidem claui, qui dirigunt illum motum. </s> </p> <p id="N2649C" type="main"> <s id="N2649E"><!-- NEW -->Decimo&longs;extò, &longs;i pennæ delicatioribus pilis tenera lanugine ve&longs;tian­<lb/>tur, tardiùs mouetur calamus vtroque motu; quia vix aëra penetrare po&longs;­<lb/>&longs;unt delicatiores molliore&longs;que pili. </s> </p> <p id="N264A6" type="main"> <s id="N264A8"><!-- NEW -->Decimo&longs;eptimò, &longs;i proiicitur &longs;ur&longs;um, de&longs;cendatque deor&longs;um rectà, e&longs;t <lb/>motus mixtus ex recto & circulari; </s> <s id="N264AE"><!-- NEW -->&longs;i verò proiiciatur per horizontalem, <lb/>vel inclinatam, e&longs;t motus mixtus ex duobus rectis & circulari, vt con­<lb/>&longs;tat; </s> <s id="N264B6"><!-- NEW -->ex quo motu fit linea mixta ex Parabola & Helice; </s> <s id="N264BA"><!-- NEW -->&longs;it enim cylin­<lb/>drus CH, cuius motus &longs;piralis &longs;it CEFGT mixtus ex recto CT, & cir­<lb/>culari orbis CD; &longs;it etiam mixtus LTQ ex accelerato LM, & æquabili <lb/>MQ certè &longs;i addatur LQ circulus &longs;eu &longs;pira CEF, &c. </s> <s id="N264C7">&longs;itque RC æqua­<lb/>lis IE, & VT æqualis NG, habebitur &longs;pira mixta LCSVQ </s> </p> <p id="N264D0" type="main"> <s id="N264D2"><!-- NEW -->Decimooctauò, &longs;i pennæ latiores &longs;unt, &longs;eu maiorem habent &longs;uperfi­<lb/>ciem, minùs aptæ &longs;unt ad vtrumque motum, &longs;cilicet axis, & centri; quia <lb/>aër plùs æquo re&longs;i&longs;tit, nam plures illius pelluntur partes. </s> </p> <p id="N264DA" type="main"> <s id="N264DC"><!-- NEW -->Decimononò, &longs;i verò contractiores &longs;unt, etiam minùs aptæ videntur: <lb/>quippe aëra facilè diuidunt. </s> </p> <p id="N264E2" type="main"> <s id="N264E4"><!-- NEW -->Vige&longs;imò, &longs;i breuiores, certi&longs;&longs;imus e&longs;t motus orbis; quia minor circu­<lb/>lus citiùs perficitur. </s> </p> <p id="N264EA" type="main"> <s id="N264EC"><!-- NEW -->Vige&longs;imoprimò, &longs;i longiores, è contrario: adde quod ab axis leuioris <lb/>motu, dirigi vix po&longs;&longs;unt. </s> </p> <p id="N264F2" type="main"> <s id="N264F4"><!-- NEW -->Vige&longs;imo&longs;ecundò, &longs;i altera pennarum &longs;it fracta, e&longs;&longs;et motus orbis; quia <lb/>&longs;egmentum fractum aliarum partium motum non &longs;equitur, vt patet. </s> </p> <p id="N264FA" type="main"> <s id="N264FC"><!-- NEW -->Vige&longs;imotertiò, &longs;i calamus &longs;it leuior, ineptus e&longs;t; </s> <s id="N26500"><!-- NEW -->quia re&longs;i&longs;tentiam <lb/>pennarum non &longs;uperat; quippe contra reflantis aëris vim, calami præua­<lb/>lens impetus leuiores pennas &longs;ecum abripere debet. </s> </p> <p id="N26508" type="main"> <s id="N2650A"><!-- NEW -->Vige&longs;imoquartò, &longs;i longior &longs;it calamus, minùs aptus e&longs;t; quia &longs;cilicet <lb/>plures partes impetus quæ in&longs;unt grauiori calamo nullo negotio re&longs;i­<lb/>&longs;tentiam aëris, & retardationem pennarum &longs;uperant. </s> </p> <p id="N26512" type="main"> <s id="N26514"><!-- NEW -->Vige&longs;imoquintò, &longs;i longior &longs;it calamus, minùs aptus e&longs;t; </s> <s id="N26518"><!-- NEW -->tùm quia gra-<pb pagenum="361" xlink:href="026/01/395.jpg"/>uior e&longs;t, tùm quia difficiliùs conuertitur, vt &longs;emper præeat; e&longs;t enim ma­<lb/>ior re&longs;i&longs;tentia ad conuertendum longius corpus, vt patet. </s> </p> <p id="N26523" type="main"> <s id="N26525"><!-- NEW -->Vige&longs;imo&longs;extò, &longs;i breuior & leuior, ineptus e&longs;t propter rationem alla­<lb/>tam; </s> <s id="N2652B"><!-- NEW -->nam &longs;i breui&longs;&longs;imus &longs;it, eius tamen grauitatis, quæ &longs;ufficiat ad &longs;upe­<lb/>randam aëris vim, apti&longs;&longs;imus cen&longs;eri debet: hinc aliquando globulus per­<lb/>foratus calami vicem gerit. </s> </p> <p id="N26533" type="main"> <s id="N26535"><!-- NEW -->Vige&longs;imo&longs;eptimò, extremitas calami, quæ præit, debet e&longs;&longs;e paulò maior, <lb/>& qua&longs;i nodo armata, vt &longs;cilicet faciliùs præire po&longs;&longs;it, ne alia extremitas <lb/>qua&longs;i reluctetur; igitur ad in&longs;tar clauæ calamus componi debet. </s> </p> <p id="N2653D" type="main"> <s id="N2653F"><!-- NEW -->Vige&longs;imooctauò, in vacuo nulla pror&longs;us e&longs;&longs;et vertigo huius volatilis <lb/>calami; </s> <s id="N26545"><!-- NEW -->quia nulla e&longs;&longs;et aëris re&longs;i&longs;tentia; &longs;ed de his &longs;atis. </s> </p> <p id="N26549" type="main"> <s id="N2654B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N26557" type="main"> <s id="N26559"><!-- NEW --><emph type="italics"/>Cuncta phœnomena teli &longs;eu iaculi volatilis explicari po&longs;&longs;unt<emph.end type="italics"/>: </s> <s id="N26562"><!-- NEW -->huius teli <lb/>figuram habes rudiore manu adumbratam; hîc habes. </s> <s id="N26568">cu&longs;pis e&longs;t C, du­<lb/>plex clauus &longs;eu quadruplex BGDABE, ex aliqua leuiore materia <lb/>con&longs;tans v.g. <!-- REMOVE S-->ex charta duplicata, vel pennis, hoc po&longs;ito. </s> </p> <p id="N26571" type="main"> <s id="N26573"><!-- NEW -->Primò, cu&longs;pis C po&longs;t eiaculationem &longs;emper præit; </s> <s id="N26577"><!-- NEW -->ratio e&longs;t, quia <lb/>alæ illæ leuiores à tergo &longs;equuntur; minùs enim aëris vim frangere <lb/>queunt. </s> </p> <p id="N2657F" type="main"> <s id="N26581"><!-- NEW -->Secundò, in eo &longs;tatu &longs;emper remanet iaculum; </s> <s id="N26585"><!-- NEW -->quia non pote&longs;t &longs;ur&longs;um <lb/>attolli extremitas B, nec deor&longs;um deprimi; </s> <s id="N2658B"><!-- NEW -->quia ala ABE impedit; </s> <s id="N2658F"><!-- NEW -->nec <lb/>etiam dextror&longs;um, vel &longs;ini&longs;tror&longs;um inclinari; </s> <s id="N26595"><!-- NEW -->quia ala BGD prohibet; </s> <s id="N26599"><!-- NEW --><lb/>igitur &longs;i nec &longs;ur&longs;um, neque deor&longs;um, nec &longs;ini&longs;tror&longs;um, nec extror&longs;um <lb/>inclinari pote&longs;t; haud dubiè in eodem &longs;itu remanebit. </s> </p> <p id="N265A0" type="main"> <s id="N265A2"><!-- NEW -->Tertiò, citi&longs;&longs;imo motu fertur hoc iaculi genus; </s> <s id="N265A6"><!-- NEW -->quia nihil prohibet; </s> <s id="N265AA"><!-- NEW --><lb/>quippe aër facilè diuiditur ab ip&longs;o iaculo CB; </s> <s id="N265AF"><!-- NEW -->tùm deinde ab ip&longs;is alis <lb/>cæ&longs;im qua&longs;i &longs;ecatur acie dumtaxat, nunquam &longs;uperficie oppo&longs;ita; adde <lb/>quod, aër facilè fluit per 4. illas cauitates BGFE, DGFA, &c. </s> <s id="N265B7"><!-- NEW -->&longs;emper <lb/>enim aëri opponitur acies anguli; &longs;ed hæc &longs;unt facilia. </s> </p> <p id="N265BD" type="main"> <s id="N265BF"><!-- NEW -->Quartò, non agitur in vertiginem hoc iaculum; </s> <s id="N265C3"><!-- NEW -->quia &longs;cilicet non e&longs;t <lb/>tanta aëris re&longs;i&longs;tentia, quantam e&longs;&longs;e oportet; </s> <s id="N265C9"><!-- NEW -->adde quod nulla e&longs;t alarum <lb/>inflexio, quæ faciat inæqualem re&longs;i&longs;tentiam, vt in calamo volatili; </s> <s id="N265CF"><!-- NEW -->igitur <lb/>e&longs;t tantùm motus axis; vbi tamen vibratur per horizontalem, vel incli­<lb/>natam, mouetur motu mixto ex duobus rectis, de quo iam aliàs. </s> </p> <p id="N265D7" type="main"> <s id="N265D9"><!-- NEW -->Quintò, huc reuoca &longs;agittas, quæ tribus in&longs;tructæ pennis <expan abbr="eũdem">eundem</expan> <lb/>&longs;emper retinent &longs;itum in motu, vt ferrum &longs;eu mucro præeat; vnde vides <lb/>eumdem &longs;emper &longs;equi effectum, &longs;iue tres &longs;int alæ, &longs;iue quatuor. </s> </p> <p id="N265E5" type="main"> <s id="N265E7"><!-- NEW -->Sextò, huc reuoca minima illa &longs;picula &longs;picâ in&longs;tructa, quæ per tubum <lb/>pneumaticum pueri flatu eiaculantur; </s> <s id="N265ED"><!-- NEW -->nam cu&longs;pis &longs;emper præit, quia <lb/>motus alterius extremitatis leuiore &longs;pica retardatur; &longs;ed hæc &longs;unt fa­<lb/>cilia. </s> </p> <p id="N265F5" type="main"> <s id="N265F7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N26603" type="main"> <s id="N26605"><!-- NEW --><emph type="italics"/>Explicatur etiam motus illius, qua&longs;i velaris moletrinæ, qua pueri curren­<lb/>tes &longs;æpi&longs;&longs;imè ludum<emph.end type="italics"/>; cuius figuram hîc habes; nam eo tempore, <pb pagenum="362" xlink:href="026/01/396.jpg"/>quo DA fertur per ip&longs;um D, BC cum &longs;uis velus vertitur circa DA. <!-- KEEP S--></s> </p> <p id="N26616" type="main"> <s id="N26618">Primò, hinc e&longs;t motus mixtus, & recto axis DA & circulari CB. <!-- KEEP S--></s> </p> <p id="N2661C" type="main"> <s id="N2661E"><!-- NEW -->Secundò, hinc e&longs;t motus perfectè &longs;piralis, nec enim differt à motu cy­<lb/>lindri; de quo &longs;uprà. </s> </p> <p id="N26624" type="main"> <s id="N26626">Tertiò, &longs;piræ &longs;unt frequentiores, quò motus e&longs;t velocior motu centri <lb/>A, maiores è contrario. </s> </p> <p id="N2662B" type="main"> <s id="N2662D"><!-- NEW -->Quartò, debet con&longs;tare debet CB ex leui&longs;&longs;ima materia; alioquin non <lb/>mouebitur motu orbis. </s> </p> <p id="N26633" type="main"> <s id="N26635"><!-- NEW -->Quintò, debet facilè po&longs;&longs;e moueri circa A; alioquin vis illa reflantis <lb/>aëris, quæ CB motum circularem imprimit, non &longs;ufficeret. </s> </p> <p id="N2663B" type="main"> <s id="N2663D"><!-- NEW -->Sextò, ideo BC mouetur circa A; </s> <s id="N26641"><!-- NEW -->quia cum vela C & B polleant mul­<lb/>tum aëra, maior e&longs;t re&longs;i&longs;tentia; </s> <s id="N26647"><!-- NEW -->hinc propter modicam inclinationem <lb/>axis DA aër in &longs;uperficies C & B obliquè incidens illas impellit; &longs;ed <lb/>quia axis BA re&longs;i&longs;tit nece&longs;&longs;ariò circa A, motu circulari cientur. </s> </p> <p id="N2664F" type="main"> <s id="N26651"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N2665D" type="main"> <s id="N2665F"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt omnes motus ponderis, &longs;eu plumei à tergo valuarum fu­<lb/>nependuli, cuius vi valuæ ip&longs;a claudantur,<emph.end type="italics"/> v.g.&longs;it fores AE quarum va­<lb/>rum e&longs;t AF; &longs;it funis CDG, cuius extremitas immobiliter affixa &longs;it C, <lb/>pondus appen&longs;um &longs;it G, cuius vi &longs;eu motu fores ip&longs;æ clauduntur. </s> </p> <p id="N2666E" type="main"> <s id="N26670"><!-- NEW -->Primò, certum e&longs;t pondus G non moueri motu recto; quia cum ip­<lb/>&longs;o rectangulo AE mouetur circa axem immobilem AB. <!-- KEEP S--></s> </p> <p id="N26677" type="main"> <s id="N26679">Secundò, certum e&longs;t non moueri motu purè circulari, qui mouetur <lb/>per lineam GD. </s> </p> <p id="N2667E" type="main"> <s id="N26680"><!-- NEW -->Tertiò, certum e&longs;t rectangulum A moueri motu purè circulari, vt pa­<lb/>tet; ita vt DE &longs;uo motu de&longs;cribat cylindrum, cuius radius &longs;eu &longs;emidia­<lb/>meter ba&longs;is e&longs;t BE. <!-- KEEP S--></s> </p> <p id="N26689" type="main"> <s id="N2668B">Quartò, certum e&longs;t, quodlibet punctum huius rectanguli de&longs;­<lb/>cribere circulum, maiorem &longs;cilicet vel minorem pro diuer&longs;a di&longs;tan­<lb/>tia ab axe AB, v. <!-- REMOVE S-->g. <!-- REMOVE S-->punctum D de&longs;cribit circulum, cuius radius <lb/>e&longs;t DA, punctum verò I de&longs;cribit circulum, cuius radius e&longs;t HI. <!-- KEEP S--></s> </p> <p id="N26699" type="main"> <s id="N2669B">Quintò, certum e&longs;t pondus G moueri motu mixto ex circulari forium. <lb/></s> <s id="N2669F">& recto deor&longs;um. </s> </p> <p id="N266A2" type="main"> <s id="N266A4"><!-- NEW -->Sextò, habes &longs;chema huius motus in cylindro A quem de&longs;cribunt <lb/>fores &longs;uo motu, &longs;i enim A moueatur per &longs;emicirculum AB, & rectam A <lb/>C; </s> <s id="N266AC"><!-- NEW -->haud dubiè mouebitur per AD; igitur hic motus e&longs;t &longs;piralis, nec e&longs;t <lb/>alia difficultas. </s> </p> <p id="N266B2" type="main"> <s id="N266B4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N266C0" type="main"> <s id="N266C2"><!-- NEW --><emph type="italics"/>Quando voluitur funis circa cylindrum, vel axem, mouetur motu <lb/>&longs;pirali, &longs;ed diuer&longs;o à prioribus<emph.end type="italics"/>; </s> <s id="N266CF"><!-- NEW -->&longs;unt enim veræ &longs;piræ ad in&longs;tar &longs;apien­<lb/>tia in diuer&longs;a volumina contorti; </s> <s id="N266D5"><!-- NEW -->&longs;ic funis circa digitum &longs;æpè <lb/>rotatur.; </s> <s id="N266DB"><!-- NEW -->e&longs;t enim motus mixtus ex diuer&longs;is circularibus: </s> <s id="N266E1"><!-- NEW -->quippè <pb pagenum="363" xlink:href="026/01/397.jpg"/>in &longs;ingulis punctis e&longs;t diuer&longs;a determinatio ad nouum circulum, quia <lb/>e&longs;t nouus radius, quia continuò radius huius vertiginis imminuitur; <lb/>porrò duobus modis pote&longs;t funis circa axem vel cylindrum conuolui. </s> <s id="N266EE"><!-- NEW --><lb/>Primò, &longs;i &longs;emper circa <expan abbr="eũdem">eundem</expan> cylindri circulum voluatur; </s> <s id="N266F7"><!-- NEW -->tunc autem <lb/>facit veras &longs;piras, vt vides in A. Secundò, &longs;i circa diuer&longs;os eiu&longs;dem axis <lb/>circulos, vel potius diuer&longs;a eiu&longs;dem axis puncta voluatur, & hic e&longs;t mo­<lb/>tus &longs;piralis conicus, vt vides in cono FDE; </s> <s id="N26701"><!-- NEW -->idem e&longs;&longs;et motus &longs;i conus <lb/>circa axem volueretur &longs;imulque aliquod punctum peripheriæ ba&longs;is coni <lb/>rectà ab ip&longs;a peripheria ad verticem coni tenderet; </s> <s id="N26709"><!-- NEW -->&longs;i enim totus conus <lb/>moueatur motu axis recto, quodlibet punctum &longs;uperficiei coni mouetur <lb/>motu &longs;pirali cylindrico, excepto dumtaxat ip&longs;o vertice; hoc denique <lb/>motu mouerentur &longs;ingula puncta baculi ED, qui in conum rotaretur à <lb/>vertice E eo tempore, quo rotans ip&longs;e per rectam EG moueretur. </s> </p> <p id="N26715" type="main"> <s id="N26717"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N26723" type="main"> <s id="N26725"><!-- NEW --><emph type="italics"/>Similiter po&longs;&longs;unt explicari motus &longs;pirales &longs;phærici, quos habes in<emph.end type="italics"/>; </s> <s id="N2672E"><!-- NEW -->hic au­<lb/>tem motus duplex e&longs;t; </s> <s id="N26734"><!-- NEW -->primus mixtus ex recto per axem KL, quo totus <lb/>globus mouetur, & ex circulari circa axem KL, qui reuerâ e&longs;t &longs;piralis <lb/>cylindricus; </s> <s id="N2673C"><!-- NEW -->&longs;ecundus mixtus ex duobus circularibus, &longs;cilicet ex circulari <lb/>circa axem KL, & circulari per arcum IL, v.g. <!-- REMOVE S-->&longs;i punctum eo tempore <lb/>voluatur circa axem KL per arcum IO, quo fertur per arcum IL vnde <lb/>habes in hac figura tres motus &longs;pirales, quorum &longs;inguli con&longs;tant ex circu­<lb/>lari circa axem KL; </s> <s id="N2674A"><!-- NEW -->&longs;ed deinde con&longs;tant &longs;inguli ex &longs;ingulis motibus di­<lb/>uer&longs;is, &longs;cilicet &longs;piralis cylindricus ex motu puncti I v.g. <!-- REMOVE S-->per rectam IN <lb/>parallelam KL; </s> <s id="N26754"><!-- NEW -->&longs;piralis conicus per rectam IL, & &longs;piralis &longs;phæricus <lb/>per arcum IPL; </s> <s id="N2675A"><!-- NEW -->hinc duo primi con&longs;tant ex circulari, & recto; certius <lb/>verò ex duobus circularibus. </s> </p> <p id="N26760" type="main"> <s id="N26762"><!-- NEW -->Denique pote&longs;t e&longs;&longs;e &longs;piralis concoidicus qualem vides in i&longs;que du­<lb/>plex; </s> <s id="N26768"><!-- NEW -->primò &longs;i vertatur conois circa axem SV; </s> <s id="N2676C"><!-- NEW -->&longs;ecundò, &longs;i vertatur circa <lb/>axem XZ: </s> <s id="N26772"><!-- NEW -->quippe hoc modo &longs;piræ erunt maiores; </s> <s id="N26776"><!-- NEW -->&longs;unt quoque &longs;inguli <lb/>triplicis generis; </s> <s id="N2677C"><!-- NEW -->e&longs;t enim vel parabolicus, vel ellipticus, vel hyperboli­<lb/>cus; porrò, qui dicunt motus cœle&longs;tes e&longs;&longs;e &longs;pirales, viderint an &longs;int cy­<lb/>lindrici vel &longs;phærici, vel conici, vel elliptici &c. </s> <s id="N26784">omitto &longs;piralem in pla­<lb/>no, mixtum &longs;cilicet ex circulari & recto, cuius &longs;chema habes Th.24. tùm <lb/>L 5. Th.79. de quo etiam aliàs, cum de lineis motus. </s> </p> <p id="N2678B" type="main"> <s id="N2678D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N26799" type="main"> <s id="N2679B"><!-- NEW --><emph type="italics"/>Cum taleola &longs;upra planum rectilineum ita repit, vt etiam circa propriu&mtail; <lb/>centrum voluatur, est motus mixtus ex recto & circulari<emph.end type="italics"/>; </s> <s id="N267A6"><!-- NEW -->neque hic motus <lb/>diuer&longs;us e&longs;t à motu rotæ in plano, &longs;it enim taleola centro A, circa quod <lb/>vertitur dum centrum A repit motu recto per rectam AD, perinde &longs;e <lb/>habet, atque &longs;i rota in plano BE vel CF rotaretur; </s> <s id="N267B0"><!-- NEW -->immò pote&longs;t tabella <lb/>GK ita moueri, vt eius centrum A moueatur per AD, dum reliquæ par­<lb/>tes circa centrum A voluuntur; </s> <s id="N267B8"><!-- NEW -->tunc enim punctum H eodem motu <lb/>moueretur, quo alia puncta peripheriæ huius rotæ; </s> <s id="N267BE"><!-- NEW -->punctum verò I eo <lb/>modo quo I in radio BA, dum rota mouetur, quod &longs;uprà fusè explicui-<pb pagenum="364" xlink:href="026/01/398.jpg"/>mus; denique ita moueri pote&longs;t taleola, vt primò B moueatur motu or­<lb/>bis ver&longs;us. </s> <s id="N267CB">Secundò, ver&longs;us K; Tertiò, vt motus centri &longs;it maior vel minor <lb/>motu orbis. </s> <s id="N267D0">Quartò, vt &longs;it æqualis. </s> </p> <p id="N267D3" type="main"> <s id="N267D5"><!-- NEW -->Denique, ne omittam motum illum, quo clauis &longs;eu planum &longs;olidum <lb/>in læuigata men&longs;a mouetur, dico mixtum e&longs;&longs;e ex recto alicuius centri & <lb/>circularis orbis; </s> <s id="N267DD"><!-- NEW -->&longs;it enim v.g.baculus AD, qui ita repat in plano læui­<lb/>gato vt altera eius extremitas fortiùs impellatur, mouebitur motu mixto <lb/>ex circulari circa centrum C per Th.55.l.7. & recto orbis circa C; </s> <s id="N267E5"><!-- NEW -->de­<lb/>&longs;cribent autem duæ extremitates A & D lineas rotatiles diuer&longs;as; hic au­<lb/>tem motus diuer&longs;us erit pro diuer&longs;a coniugatione motus orbis, & mo­<lb/>tus centri, cùm hic po&longs;&longs;it e&longs;&longs;e vel maior, vel minor motu orbis, vel <lb/>æqualis, </s> </p> <p id="N267F1" type="main"> <s id="N267F3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N267FF" type="main"> <s id="N26801"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena motus globi.<emph.end type="italics"/></s> </p> <p id="N26808" type="main"> <s id="N2680A"><!-- NEW -->Primò, ita globus rotatur aliquando in plano, vt motus orbis de&longs;cri­<lb/>bat circulos perpendiculariter incubantes plano; </s> <s id="N26810"><!-- NEW -->&longs;ic vulgò proijcitur <lb/>globus, nec differt hic motus à motu rotæ in plano; e&longs;t enim mixtus ex <lb/>recto centri & circulari orbis. </s> </p> <p id="N26818" type="main"> <s id="N2681A"><!-- NEW -->Secundò, ita rotatur aliquandò, vt &longs;it &longs;emper idem punctum contactus, <lb/>& motus orbis de&longs;cribat circulos parallelos plano in quo rotatur; non <lb/>differt etiam hic motus à motu rotæ, quæ in plano verticali rotaretur. </s> </p> <p id="N26822" type="main"> <s id="N26824"><!-- NEW -->Tertiò, ita rotatur, vt motus orbis de&longs;cribat circulos inclinatos plùs, <lb/>vel minùs; </s> <s id="N2682A"><!-- NEW -->non differt autem hic motus à motu rotæ, quæ in plano in­<lb/>clinato rotaretur; mutatur autem continue punctum contactus in 1°ree;. <lb/></s> <s id="N26831">& 3°ree;. </s> <s id="N26834">motu. </s> </p> <p id="N26837" type="main"> <s id="N26839">Porrò, &longs;æpiùs ob&longs;eruabis i&longs;tos motus globi in aqua, in qua &longs;cilicet fa­<lb/>cilè circa centrum voluitur per quodcunque planum. </s> </p> <p id="N2683E" type="main"> <s id="N26840"><!-- NEW -->Quartò, ita mouetur vt con&longs;tet hic motus ex duobus qua&longs;i circulari­<lb/>bus, & ex recto; </s> <s id="N26846"><!-- NEW -->quando &longs;cilicet inflectitur ita motus centri, vt mouea­<lb/>tur centrum per lineam curuam; </s> <s id="N2684C"><!-- NEW -->dixi curuam; non verò circularem; </s> <s id="N26850"><!-- NEW --><lb/>quia non habet centrum motus purè circularem, &longs;ed mixtum ex <lb/>recto & circulari; </s> <s id="N26857"><!-- NEW -->exemplum habes clari&longs;&longs;imum in illo deflexu <lb/>globi, qui valdè familiaris e&longs;t iis, qui trunculorum ludum exercent; <lb/>quippe tantillùm detorquetur circa horizontalem, ex qua declinatione <lb/>&longs;equitur motus mixtus ex tribus, &longs;cilicet ex motu orbis in circulo hori­<lb/>zontali, ex motu orbis in verticali, & motu centri recto. </s> </p> <p id="N26863" type="main"> <s id="N26865"><!-- NEW -->Quintò, ita proijcitur globus aliquandò, vt motus centri &longs;it contrarius <lb/>motui orbis; tunc autem vel &longs;i&longs;tit globus, vel etiam redit, cum motus or­<lb/>bis inten&longs;ior e&longs;t, de quo iam &longs;uprà. </s> </p> <p id="N2686D" type="main"> <s id="N2686F">Sextò, cum proijcitur &longs;ur&longs;um per lineam perpendicularem, ita vt non <lb/>modò motus centri, verùm etiam motus orbis imprimatur, mouetur mo­<lb/>tu mixto ex recto centri & circulari orbis, nec differt hic motus à motu <lb/>rotæ in plano recto, idem dico de de&longs;cen&longs;u & de iactu circuli ferrei vel <lb/>lignei. </s> </p> <pb pagenum="365" xlink:href="026/01/399.jpg"/> <p id="N2687E" type="main"> <s id="N26880"><!-- NEW -->Septimò, cum proijcitur globus per inclinatam, mouetur motu mixto <lb/>ex tribus &longs;cilicet ex recto violento centri, ex naturali deor&longs;um & ex cir­<lb/>culari orbis, e&longs;tque idem motus, qui e&longs;&longs;et, &longs;i globus rotaretur in plano <lb/>curuo ferè parabolico; </s> <s id="N2688A"><!-- NEW -->quippe centrum de&longs;cribit hanc lineam; &longs;ed linea <lb/>centri e&longs;t &longs;emper parallela plano, in quo rotatur globus. </s> </p> <p id="N26890" type="main"> <s id="N26892"><!-- NEW -->Octauò, cum rotatur globus in plano decliui per lineam inclinatam <lb/>mouetur motu mixto ex tribus, &longs;cilicet ex duobus rectis centri, & circu­<lb/>lari orbis; </s> <s id="N2689A"><!-- NEW -->hic motus &longs;imilis e&longs;t priori; </s> <s id="N2689E"><!-- NEW -->quippe centrum de&longs;cribit ferè Pa­<lb/>rabolam; hinc facilis methodus de&longs;cribendæ Parabolæ ex iactu globuli <lb/>atramento tincti, quam etiam tradit Galileus. <!-- KEEP S--></s> </p> <p id="N268A7" type="main"> <s id="N268A9"><!-- NEW -->Nonò, &longs;i globi alterum hemi&longs;phærium &longs;it grauius, cum rotatur in recto <lb/>plano, deflectit in cam partem quam &longs;pectat hemi&longs;phærium grauius; </s> <s id="N268AF"><!-- NEW --><lb/>imò deinde detorquetur in oppo&longs;itam, e&longs;tque motus mixtus ex duobus <lb/>circularibus, altero &longs;cilicet librationis, altero gyri rotatilis, & recto cen­<lb/>tri; </s> <s id="N268B8"><!-- NEW -->porrò mouetur centrum motu curuo qui aliquando accedit propiùs <lb/>ad circularem; </s> <s id="N268BE"><!-- NEW -->huc etiam reuoca motum parop&longs;idis rotulæ, quæ in mul­<lb/>tos agitur gyros & &longs;piras; quia præualet portio grauior, eóque detorquet <lb/>centrum motus. </s> </p> <p id="N268C6" type="main"> <s id="N268C8"><!-- NEW -->Decimò, hinc quod iucundum e&longs;&longs;et, &longs;i huiu&longs;modi globum in datum <lb/>&longs;copum proijceres; </s> <s id="N268CE"><!-- NEW -->haud dubiè alium feriret; </s> <s id="N268D2"><!-- NEW -->igitur vt &longs;copum &longs;ignatum <lb/>tangas, aliò collimare debes; </s> <s id="N268D8"><!-- NEW -->porrò linea huius motus eadem e&longs;t, quæ <lb/>e&longs;&longs;et, &longs;i globus rotaretur in linea parallela lineæ, quam de&longs;cribit cen­<lb/>trum; </s> <s id="N268E1"><!-- NEW -->quæ vel e&longs;t &longs;pira, vel circulus, vel alia curua, iuxta diuer&longs;am con­<lb/>iugationem motum; illa autem facilè haberi pote&longs;t ex dictis &longs;uprà. </s> </p> <p id="N268E7" type="main"> <s id="N268E9">Vndecimo, &longs;i in plano recto ita rotetur cylindrus, vt &longs;inguli circuli <lb/>paralleli ba&longs;i rotentur æqualiter, &longs;inguli circuli mouentur motu mixto <lb/>ex recto centri, & circulari orbis, e&longs;tque hic motus &longs;imilis motui rotæ <lb/>in plano recto, de quo &longs;uprà. </s> </p> <p id="N268F2" type="main"> <s id="N268F4"><!-- NEW -->Duodecimò, &longs;i verò ita rotetur, vt altera eius extremitas velociore <lb/>motu feratur, e&longs;t alius motus mixtus ex curuo axis & circulari orbis, <lb/>dixi curuum axis; quia non e&longs;t nece&longs;&longs;ariò circularis. </s> </p> <p id="N268FC" type="main"> <s id="N268FE"><!-- NEW -->Decimotertiò, cum rotatur conus, mouetur motu mixto ex curuo axis <lb/>& circulari orbis, hic motus &longs;atis communis e&longs;t; eius porrò ratio e&longs;t; </s> <s id="N26904"><!-- NEW --><lb/>quia cùm &longs;inguli circuli &longs;uperficiei coni ita rotentur, vt motus orbis &longs;u <lb/>æqualis motui centri; certè cùm &longs;int omnes inæquales, &longs;patium decur­<lb/>runt. </s> <s id="N2690D"><!-- NEW -->Hinc vertex retrò relinquitur à ba&longs;i; </s> <s id="N26911"><!-- NEW -->hinc ba&longs;is nece&longs;&longs;ariò retor­<lb/>quetur; </s> <s id="N26917"><!-- NEW -->dixi autem curuum axis; </s> <s id="N2691B"><!-- NEW -->quippe centrum ba&longs;is non mouetur <lb/>motu purè circulari; nam tantillùm verticem promouet, quia motus <lb/>eius centri maximè iuuatur à motu eius orbis, qui longè maior e&longs;t. </s> </p> <p id="N26923" type="main"> <s id="N26925"><!-- NEW -->Decimoquartò, huc demum reuoca gyros illarum pyxidum, quarum <lb/>margines oppo&longs;iti &longs;unt circuli inæquales; quippe &longs;unt veluti fru&longs;ta co­<lb/>ni, cuius angulus verticis e&longs;t valde acutus. </s> </p> <p id="N2692D" type="main"> <s id="N2692F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N2693B" type="main"> <s id="N2693D"><!-- NEW --><emph type="italics"/>Morus di&longs;ci facilè explicari potest;<emph.end type="italics"/>; </s> <s id="N26946"><!-- NEW -->e&longs;t enim planum circulare, cuius <pb pagenum="366" xlink:href="026/01/400.jpg"/>centrum de&longs;cribit ferè Parabolam; </s> <s id="N2694F"><!-- NEW -->vnde eius motus e&longs;t mixtus ex para­<lb/>bolico centri, & circulari orbis in circulo horizontali; </s> <s id="N26955"><!-- NEW -->igitur motus cen­<lb/>tri con&longs;tat ex duobus rectis, &longs;cilicet ex violento, & naturali deor&longs;um; <lb/>porrò e&longs;t idem motus qui e&longs;&longs;et, &longs;i circulus verticali parallelus rotaretur <lb/>in linea parabolica de&longs;cripta in plano horizontali. </s> </p> <p id="N2695F" type="main"> <s id="N26961"><!-- NEW -->Ob&longs;eruo autem primò motum orbis di&longs;ci e&longs;&longs;e po&longs;&longs;e maiorem motu <lb/>centri, vel minorem, vel ip&longs;i æqualem; quod quomodo fieri po&longs;&longs;it, fusè <lb/>&longs;uprà explicuimus. </s> </p> <p id="N26969" type="main"> <s id="N2696B"><!-- NEW -->Secundò, &longs;i altera eius portio &longs;it grauior motus orbis, non e&longs;t idem <lb/>cum centro di&longs;ci, vt patet; præualet enim portio grauior, &longs;ed propiùs <lb/>accedit ad portionem grauiorem. </s> </p> <p id="N26973" type="main"> <s id="N26975"><!-- NEW -->Tertiò, hinc cùm di&longs;cus cadit in terram, re&longs;itit altera eius portio, &longs;ci­<lb/>licet leuior; </s> <s id="N2697B"><!-- NEW -->quia cùm de&longs;cribat maiorem circulum orbis, maiorem im­<lb/>petum habet; hinc conuertitur di&longs;cus. </s> </p> <p id="N26981" type="main"> <s id="N26983"><!-- NEW -->Quartò, imprimitur motus orbis in ip&longs;o iactu; </s> <s id="N26987"><!-- NEW -->quia &longs;cilicet vna pars <lb/>mouetur, antequam alia di&longs;cedat è manu proijcientis; vnde &longs;equitur <lb/>nece&longs;&longs;ariò motus orbis. </s> </p> <p id="N2698F" type="main"> <s id="N26991"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N2699D" type="main"> <s id="N2699F"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena longioris<emph.end type="italics"/> <emph type="italics"/>ha&longs;tæ vel &longs;ari&longs;&longs;æ.<emph.end type="italics"/></s> </p> <p id="N269AC" type="main"> <s id="N269AE"><!-- NEW -->Primò, &longs;it ha&longs;ta in plano horizontali BG; &longs;i motu &longs;implici attollatur <lb/>extremitas B, mouebitur per arcum BA circa centrum G. <!-- KEEP S--></s> </p> <p id="N269B5" type="main"> <s id="N269B7"><!-- NEW -->Secundò, &longs;i non modò attollatur, &longs;ed euibretur cum aliquo vi&longs;u, ele­<lb/>uata &longs;cilicet tantillùm extremitate G, mouebitur vtraque extremitas; <lb/>non certè circa F, &longs;ed circa E, vel D, ita vt GE &longs;it 1/4 AG per Th.55.l.7. </s> </p> <p id="N269BF" type="main"> <s id="N269C1"><!-- NEW -->Tertiò, &longs;i extremitas G non adducatur &longs;ed B per aliquam Tangentem <lb/>arcus BA euibretur pro diuer&longs;a Tangente diuer&longs;us erit motus, &longs;i v.g.per <lb/>Tangentem BH punctum D a&longs;&longs;urget per DE, igitur G redibit in C, B <lb/>verò &longs;patium compo&longs;itum ex tota v. <!-- REMOVE S-->g. <!-- REMOVE S-->& eius &longs;ubdupla BC; </s> <s id="N269CF"><!-- NEW -->e&longs;t autem <lb/>hic motus mixtus ex recto centri D & circulari orbis; </s> <s id="N269D5"><!-- NEW -->&longs;i verò extremi­<lb/>tas B euibretur per Tangentem HL & D, vel E per EK; haud dubiè ex­<lb/>tremitas G minùs retroagetur, & acquiret dextror&longs;um maius &longs;patium. </s> </p> <p id="N269DD" type="main"> <s id="N269DF"><!-- NEW -->Quartò, &longs;i nullo modo adducatur centrum D, vel extremitas G; </s> <s id="N269E3"><!-- NEW -->nun­<lb/>quam G ad manum ludentis perueniet, id e&longs;t nunquam perueniet in B; <lb/>vnde manife&longs;tè patet hunc motum circularem non fieri circa C. <!-- KEEP S--></s> </p> <p id="N269EC" type="main"> <s id="N269EE"><!-- NEW -->Quintò, &longs;i ita euibretur ha&longs;ta, vt tantillùm adducatur centrum motus <lb/>circularis, &longs;cilicet D; </s> <s id="N269F4"><!-- NEW -->haud dubiè altera extremitas G cadere poterit in <lb/>B, id e&longs;t peruenire ad manum ludentis; </s> <s id="N269FA"><!-- NEW -->&longs;i verò plùs æquo adducatur, <lb/>manum ludentis fallet, &longs;eu præteribit; </s> <s id="N26A00"><!-- NEW -->&longs;i denique minùs adducatur, por­<lb/>rigi manum oportet, vt extremitates G excipiat: porrò hic motus e&longs;t <lb/>mixtus ex tribus, &longs;cilicet ex duobus rectis centri, & circulari orbis. </s> </p> <p id="N26A08" type="main"> <s id="N26A0A"><!-- NEW -->Sextò, ita poterit adduci centrum D, & &longs;imul euibrari B, vt ha&longs;tæ me­<lb/>dium C facto &longs;emicircuitu in dextram erectam cadat, quadretque ad in­<lb/>&longs;tar iaculi mi&longs;&longs;ilis, cuius mucro deor&longs;um vergens prædæ plagam inten­<lb/>tat; hoc ludi genus o&longs;tentationem Hi&longs;panicam vulgò vocant. </s> </p> <pb pagenum="367" xlink:href="026/01/401.jpg"/> <p id="N26A18" type="main"> <s id="N26A1A"><!-- NEW -->Septimò, erigitur ha&longs;ta, &longs;i extremitas G tantillùm eleuata cum altera <lb/>oppo&longs;ita B, tùm &longs;tatim B deprimatur; </s> <s id="N26A20"><!-- NEW -->vnde accidit ip&longs;am G noua acce&longs;­<lb/>&longs;ione impetus &longs;ur&longs;um promoueri; </s> <s id="N26A26"><!-- NEW -->quippe &longs;i deprimatur B circa aliquod <lb/>centrum, attollitur G; </s> <s id="N26A2C"><!-- NEW -->adde aliquam refluxionem ip&longs;ius G, quæ valdè <lb/>initio remouetur à manu, vt cum deinde adducitur, maiorem faciat ar­<lb/>cum; </s> <s id="N26A34"><!-- NEW -->igitur maiore tempore; </s> <s id="N26A38"><!-- NEW -->igitur &longs;en&longs;im ab ip&longs;a manu maior in illam <lb/>deriuatur impetus; denique vt deinde maiore quoque arcu extremitas B <lb/>deprimatur, remoueaturque, & con&longs;equenter oppo&longs;ita G magis attolla­<lb/>tur, & accedat. </s> </p> <p id="N26A42" type="main"> <s id="N26A44">Octauò, duobus aliis modis erigitur ha&longs;ta è &longs;itu horizontali. </s> </p> <p id="N26A47" type="main"> <s id="N26A49"><!-- NEW -->Primò, conuer&longs;o intror&longs;um brachio; </s> <s id="N26A4D"><!-- NEW -->eleuatur enim extremitas G. <lb/>& deprimitur illicò B; </s> <s id="N26A53"><!-- NEW -->vnde minore conatu deinde attollitur; </s> <s id="N26A57"><!-- NEW -->minus e&longs;t <lb/>enim momentum vectis; </s> <s id="N26A5D"><!-- NEW -->&longs;it enim vectis in &longs;itu horizontali LN, &longs;itque <lb/>eius momentum vt LN; </s> <s id="N26A63"><!-- NEW -->certè &longs;i attollatur in LO, eius momentum erit <lb/>tantùm vt LM; </s> <s id="N26A69"><!-- NEW -->hinc facilè eleuatur pertica po&longs;t aliquam inclinationem <lb/>&longs;ur&longs;um; &longs;ecundus modus, cum torquetur extrin&longs;ecus brachium, pro quo <lb/>e&longs;t eadem pror&longs;us ratio. </s> </p> <p id="N26A71" type="main"> <s id="N26A73">Nonò, erigitur adhuc duobus modis ha&longs;ta. </s> </p> <p id="N26A76" type="main"> <s id="N26A78">Primò, intorto extrin&longs;ecus brachio, detortoque. </s> <s id="N26A7B"><!-- NEW -->Secundò, contorte <lb/>intror&longs;um reductóque traiecto &longs;ub ha&longs;tam capite; e&longs;t autem eadem ra­<lb/>tio, quæ &longs;uprà. </s> </p> <p id="N26A83" type="main"> <s id="N26A85"><!-- NEW -->Decimò, cum erecta ha&longs;ta &longs;ur&longs;um ita proijcitur, vt po&longs;t circuitum pot <lb/>medium truncum excipiatur, mouetur motu mixto ex recto centri, & <lb/>circulari orbis; quod duobus modis fieri pote&longs;t. </s> <s id="N26A8D"><!-- NEW -->Primò, &longs;i extremitas quæ <lb/>tenetur manu, retrò agatur, vbi priùs &longs;ur&longs;um tota ha&longs;ta impul&longs;a e&longs;t; quip­<lb/>pe ex eo duplici motu centri, & orbis &longs;equetur conuer&longs;io ha&longs;tæ, & is <lb/>de&longs;cen&longs;us in quo commodè per medium truncum excipi po&longs;&longs;it. </s> <s id="N26A97">Secundò. </s> <s id="N26A9A"><lb/>hoc eodem motu mouebitur, eritque &longs;imile phœnomenum, &longs;i extremitas, <lb/>quæ tenetur manu impul&longs;a primò &longs;ur&longs;um cum tota ha&longs;ta, tùm deinde <lb/>antè pellatur, ita vt extremitas oppo&longs;ita retrò agatur. </s> </p> <p id="N26AA2" type="main"> <s id="N26AA4"><!-- NEW -->Vndecimò, motus orbis pote&longs;t aliquando e&longs;&longs;e maior, aliquando minor, <lb/>pro diuer&longs;o &longs;cilicet impul&longs;u: </s> <s id="N26AAA"><!-- NEW -->idem dico de motu centri; </s> <s id="N26AAE"><!-- NEW -->imò po&longs;&longs;et <lb/>e&longs;&longs;e tantus motus centri, vt conuer&longs;io ha&longs;tæ perfici non po&longs;&longs;et; </s> <s id="N26AB4"><!-- NEW -->e&longs;t au­<lb/>tem motus centri velocior initio in a&longs;cen&longs;u, & tardior in fine; & contrà <lb/>tardior initio de&longs;cen&longs;us, & in fine velocior, vt con&longs;tat ex dictis l.2. & 3. <!-- KEEP S--></s> </p> <p id="N26ABD" type="main"> <s id="N26ABF"><!-- NEW -->Duodecimò, cum motus centri modicus e&longs;t, parùm a&longs;&longs;urgit ha&longs;ta, & <lb/>licèt morus orbis &longs;it maximus vix integram conuer&longs;ionem perficere <lb/>pote&longs;t; cum verò motus centri maximus e&longs;t, & motus orbis modicus, <lb/>etiam &longs;uam conuer&longs;ionem non perficit, &longs;ed altiùs a&longs;&longs;urgit mucro. </s> </p> <p id="N26AC9" type="main"> <s id="N26ACB"><!-- NEW -->Decimotertiò, centrum motus orbis non videtur e&longs;&longs;e aliud ab ip&longs;is <lb/>3/4 ver&longs;us mucronem, vt iam &longs;æpe indicauimus: porrò ni&longs;i hoc centrum <lb/>motus orbis retroagatur tantillùm, id e&longs;t 1/4 longitudinis ha&longs;tæ, non po­<lb/>terit excipi per medium truncum, ni&longs;i maius producatur. </s> </p> <p id="N26AD5" type="main"> <s id="N26AD7"><!-- NEW -->Decimoquartò, pote&longs;t centrum orbis, vel plùs æquo retrò agi, vel ante <lb/>pelli, vt con&longs;tat; </s> <s id="N26ADD"><!-- NEW -->vnde tota ferè indu&longs;tria po&longs;ita e&longs;t in temperando illius <pb pagenum="368" xlink:href="026/01/402.jpg"/>motu recto; </s> <s id="N26AE6"><!-- NEW -->denique non e&longs;t omittendum etiam ha&longs;tam eratam &longs;olo <lb/>nixam &longs;ur&longs;um intorto pugno ita proijci po&longs;&longs;e, vt po&longs;t circuitum excipia­<lb/>tur, nec e&longs;t noua difficultas; communicatur enim primò motus centri <lb/>rectus, tùm motus orbis, immò, &longs;i &longs;it breuior, etiam geminos circuitus <lb/>facit, antequam iu&longs;ta manu excipiatur. </s> </p> <p id="N26AF2" type="main"> <s id="N26AF4">Decimoquintò, extremitas, quæ manu tenetur velociùs deinde moue­<lb/>tur. </s> <s id="N26AF9">Primò, patet experientia. </s> <s id="N26AFC">Secundò, maius &longs;patium conficit; </s> <s id="N26AFF"><!-- NEW -->ratio e&longs;t, <lb/>quia mouetur circa centrum maiore &longs;emidiametro, quas con&longs;tat 1/4 totius <lb/>ha&longs;tæ, quod vt faciliùs videatur, &longs;it ha&longs;ta AE, quæ pellatur &longs;ur&longs;um mo­<lb/>tu recto CE, &longs;itque motus orbis circa centrum C; </s> <s id="N26B09"><!-- NEW -->vbi verò C peruenit <lb/>in D, A peruenit in L, & D in I; </s> <s id="N26B0F"><!-- NEW -->vbi verò C peruenit in E, A peruenit in <lb/>G & D rediit in D; </s> <s id="N26B15"><!-- NEW -->vides quanta &longs;it differentia motus; </s> <s id="N26B19"><!-- NEW -->nam eo tempore, <lb/>quo A decurrit &longs;patium AKL, D decurrit tantùm DHI; </s> <s id="N26B1F"><!-- NEW -->quænam por­<lb/>rò &longs;it hæc figura; </s> <s id="N26B25"><!-- NEW -->certè &longs;i non e&longs;t Ellip&longs;is, propiùs ad illam accedit: <lb/>idem dico de de&longs;cen&longs;u ha&longs;tæ, quod dictum e&longs;t de a&longs;cen&longs;u. </s> </p> <p id="N26B2B" type="main"> <s id="N26B2D"><!-- NEW -->Decimo&longs;extò, duobus aliis modis pote&longs;t ha&longs;ta in aëre <expan abbr="cõuerti">conuerti</expan>; </s> <s id="N26B35"><!-- NEW -->primò, &longs;i <lb/>mucro agatur retrò, vtraque manu admota alteri extremitati: </s> <s id="N26B3B"><!-- NEW -->hic autem <lb/>modus differt à prioribus, quod in illis motus centri rectus præcedat <lb/>motum orbis; in hoc verò vterque &longs;imul incipiat. </s> <s id="N26B43"><!-- NEW -->Secundò, &longs;i primò in <lb/>humeris liberetur ha&longs;ta, tùm &longs;ur&longs;um euibretur; &longs;ed hæc &longs;unt facilia. </s> </p> <p id="N26B49" type="main"> <s id="N26B4B"><!-- NEW -->Decimo&longs;eptimò, ad ha&longs;tam reuocabis baculum rotatum ab altera ex­<lb/>tremitate; &longs;it enim baculus AE rotatus circa extremitatem A, tùm &longs;ta­<lb/>tim demi&longs;&longs;us. </s> <s id="N26B53"><!-- NEW -->Primò, E po&longs;t &longs;emicirculum peruenit in A. Secundò, E im­<lb/>primitur maior impetus, vt patet: hinc tertiò mouetur velocius. </s> <s id="N26B59"><!-- NEW -->Quartò, <lb/>A non de&longs;cendit infra AE, po&longs;t quam demi&longs;&longs;us e&longs;t baculus, vt pater ex­<lb/>perientiâ; ratio e&longs;t, quia E per tangentem EL determinata impedit, ne <lb/>A deor&longs;um tendat. </s> <s id="N26B63"><!-- NEW -->Quintò, E per arcum EG non mouetur; </s> <s id="N26B67"><!-- NEW -->alioquin A <lb/>e&longs;&longs;et immobilis: </s> <s id="N26B6D"><!-- NEW -->præterea F. non mouetur motu circulari, ni&longs;i retineatur <lb/>in A; </s> <s id="N26B73"><!-- NEW -->&longs;ed non retinetur; igitur non mouetur per EG. Sextò, non moue­<lb/>tur quoque per rectam EF, quia retinetur E ab A, & reliquis partibus, <lb/>quæ minùs habent impetus. </s> <s id="N26B7B"><!-- NEW -->Septimò, mouetur E per lineam curuam, quæ <lb/>accedit ad ellip&longs;im, &longs;cilicet per EHA; </s> <s id="N26B81"><!-- NEW -->A verò a&longs;&longs;urgit &longs;upra AE; </s> <s id="N26B85"><!-- NEW -->ratio <lb/>huius motus petitur ex eo quod, neque per EF, neque per arcum EG <lb/>mouetur extremitas E; igitur per curuam de vtraque participan­<lb/>tem. </s> </p> <p id="N26B8F" type="main"> <s id="N26B91"><!-- NEW -->Decimooctauò, cum ita proijcitur baculus, vt altera extremitas citíùs <lb/>moueatur quàm alia, &longs;equitur motus mixtus ex recto centri, & circulari <lb/>orbis; </s> <s id="N26B99"><!-- NEW -->quia &longs;cilicet illa pars, quæ maiorem impetum habet, qua&longs;i retrò <lb/>agitur ab alia, quæ minorem habet, non quidem motu purè circulari; <lb/>alioqui omninò retineretur ab alia extremitate, &longs;ed alio mixto, quia non <lb/>omninò retinetur. </s> </p> <p id="N26BA3" type="main"> <s id="N26BA5"><!-- NEW -->Decimononò, hinc pote&longs;t ita temperari motus ille orbis, vt tantùm <lb/>&longs;emicircuitum in toto cur&longs;u impleat, cum &longs;cilicet partes omnes æquali <lb/>ferè cum impetu mouentur; </s> <s id="N26BAD"><!-- NEW -->&longs;i enim æqualitas e&longs;t in motu omnium <lb/>partium, mouentur omnes motu recto; </s> <s id="N26BB3"><!-- NEW -->&longs;i verò motus &longs;ingularum &longs;unt <pb pagenum="369" xlink:href="026/01/403.jpg"/>vt radij, motus e&longs;t purè circularis; &longs;i verò e&longs;t alia inæqualitas, erit <lb/>mixtus, qui magis accedet ad circularem, quò maior erit inæqualitas, & <lb/>magis ad rectum, quò minor erit. </s> </p> <p id="N26BC0" type="main"> <s id="N26BC2"><!-- NEW -->Vige&longs;imò, hinc qui ludunt trunculis illis lu&longs;oriis inuer&longs;o tamen mo­<lb/>re, quod &longs;æpè hic fit, quo &longs;cilicet non globus in trunculos, &longs;ed trunculi <lb/>in globum proijciantur, arripiunt trunculum ip&longs;um per medium trun­<lb/>cum, vt &longs;cilicet æqualem impetum &longs;ingulis partibus imprimant; vnde <lb/>&longs;equitur motus rectus, & ex motu recto vniformis trunculi ca&longs;us, ne &longs;i <lb/>altera extremitas ante aliam &longs;olum tangat, &longs;tatim re&longs;iliat alia per ali­<lb/>quot gyros, & à &longs;copo di&longs;cedat. </s> </p> <p id="N26BD2" type="main"> <s id="N26BD4">Vige&longs;imoprimò, mouetur baculus proiectus eo modo, de quo num. </s> <s id="N26BD7"><!-- NEW -->18. <lb/>circa aliquod centrum, quod tribus quartis tribuimus ver&longs;us eam ex­<lb/>tremitatem, quæ vltimò à manu dimittitur; quippe faciliùs circa hoc <lb/>centrum mouetur, de quo alibi, vnde e&longs;t motus mixtus ex recto centri, <lb/>ex recto naturali, & ex circulari orbis, quæ omnia ex dictis &longs;atis intelli­<lb/>guntur. </s> </p> <p id="N26BE5" type="main"> <s id="N26BE7">Vige&longs;imo&longs;ecundò, non &longs;unt omittenda aliquot phœnomena, quæ in <lb/>trunculorum ludo ferè &longs;emper occurrunt. </s> <s id="N26BEC">1°ree;. </s> <s id="N26BEF">&longs;i iuxta verticem tangan­<lb/>tur faciliùs decutiuntur, quia maior e&longs;t vectis, 2°ree;. </s> <s id="N26BF4">minùs deflectit glo­<lb/>bus à &longs;uo tramite, &longs;i per &longs;ummos vertices decutiat, quia minùs re&longs;i&longs;tunt. </s> <s id="N26BF9"><lb/>3°ree;. </s> <s id="N26BFD">hinc, &longs;i etiam per imum pedem directo ictu verberentur, plùs re&longs;i­<lb/>&longs;tunt, quia minor e&longs;t vectis, 4°ree;. </s> <s id="N26C02">hinc &longs;tatim à recta via globus deflectit, <lb/>5°ree;. </s> <s id="N26C07">&longs;i obliquè globus feriat trunculum, qua&longs;i lambendo, parùm declinat <lb/>à &longs;uo cur&longs;u, quia minima e&longs;t re&longs;i&longs;tentia, quía obliquus ictus minimus <lb/>e&longs;t, vt con&longs;tat ex dictis &longs;æpiùs in &longs;uperioribus libris. </s> <s id="N26C10"><!-- NEW -->6. cum &longs;ic obliquè <lb/>decutitur trunculus, hic decu&longs;&longs;us deinde alios decutit; </s> <s id="N26C16"><!-- NEW -->quia ex obliquo <lb/>ictu cra&longs;&longs;ioris pedis agitur in vertiginem circa verticem ad in&longs;tar coni, <lb/>de quo &longs;uprà; & cum maiorem gyrum de&longs;cribit, vix vnquam accidit, vt <lb/>in &longs;atis frequenti &longs;ylua in alium trunculum non incidat, quem etiam <lb/>decutit. </s> <s id="N26C22">7°ree;. </s> <s id="N26C25">aliqui tradunt artem, qua nouem trunculi decuti po&longs;&longs;unt, <lb/>quod multis modis præ&longs;tari pote&longs;t, &longs;ed ad rem præ&longs;entem non &longs;pe­<lb/>ctat. </s> </p> <p id="N26C2C" type="main"> <s id="N26C2E"><!-- NEW -->Vige&longs;imotertiò, e&longs;t etiam aliud ludi genus, quo pueri ru&longs;ticani ludunt; </s> <s id="N26C32"><!-- NEW --><lb/>e&longs;t autem minimum parallelipedum gemino mucrone hinc inde in&longs;tru­<lb/>ctum, vel cuius vtraque extremitas e&longs;t emarginata, vel ad in&longs;tar fu&longs;i in <lb/>apicem coni, hinc inde de&longs;inens; &longs;i enim baculo ro&longs;trum illud ferias, <lb/>&longs;tatim a&longs;&longs;urgit. </s> <s id="N26C3D"><!-- NEW -->Sit enim primò parallelipedum emarginatum AD in­<lb/>cubans &longs;olo EC; </s> <s id="N26C43"><!-- NEW -->&longs;i ro&longs;trum A baculo percutiatur, deprimitur A circa <lb/>centrum E, & attollitur D maiore quidem arcu; igitur maiore impe­<lb/>tu, qui quia non retinetur omninò non mouetur circulari motu D, <lb/>&longs;ed curuo mixto circa centrum E, quod ab extremitate D tantillùm <lb/>eleuatur. </s> <s id="N26C4F">Secundò, ex hoc phœnomeno manife&longs;tè confirmatur, quod <lb/>diximus &longs;uprà de baculo num. </s> <s id="N26C54">17. quod &longs;cilicet a&longs;&longs;urgat extremitas illa, <lb/>quæ manu tenetur &longs;upra horizontalem. </s> <s id="N26C59"><!-- NEW -->Tertiò, idem pror&longs;us accidet <pb pagenum="370" xlink:href="026/01/404.jpg"/> &longs;i &longs;upra planum horizontale BA v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it cylindrus CB extans aliqua <lb/>&longs;ui parte putà FC; </s> <s id="N26C68"><!-- NEW -->&longs;i percutiatur baculo ED in C, a&longs;&longs;urget propter <lb/><expan abbr="eãdem">eandem</expan> rationem motu mixto; </s> <s id="N26C71"><!-- NEW -->nam primò circa centrum F deprimi­<lb/>tur C, & attollitur B; </s> <s id="N26C77"><!-- NEW -->B quidem velociore motu, vt patet; igitur &longs;ecum <lb/>attollit extremitatem oppo&longs;itam C motu mixto propter rationem iam <lb/>&longs;uprà allatam. </s> </p> <p id="N26C7F" type="main"> <s id="N26C81">Vige&longs;imoquartò, AB &longs;i baculus in aëre libratus perpendiculariter. </s> <s id="N26C84"><!-- NEW --><lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->percutiatur altero baculo ED. Primò, in centro grauitatis C <lb/>baculi AB, mouebitur AB motu recto; </s> <s id="N26C8F"><!-- NEW -->ratio e&longs;t, quia omnes partes mo­<lb/>uentur æqualiter; igitur motu recto. </s> <s id="N26C95"><!-- NEW -->Secundò, tunc erit maximus <lb/>iactus, &longs;i ED percutiat C, ita vt EC media proportionalis inter ED, <lb/>& eius &longs;ubduplam EG; </s> <s id="N26C9D"><!-- NEW -->quia ED producit maximum impetum & to­<lb/>tum; e&longs;t enim C centrum grauitatis impetus totius ED, & centrum gra­<lb/>uitatis corporis impedientis AB. Tertiò, hinc &longs;i ED feriat in puncto G, <lb/>non erit tantus iactus licèt AB proijciatur motu recto. </s> <s id="N26CA7"><!-- NEW -->Quartò, &longs;i <lb/>percutiatur in F, non mouebitur motu recto, vt con&longs;tat experientiâ; </s> <s id="N26CAD"><!-- NEW --><lb/>quippe maior impetus producetur in extremitate B, quàm in A; </s> <s id="N26CB2"><!-- NEW -->igitur <lb/>non mouebitur motu recto, &longs;ed mixto circa centrum mobile H. Quintò, <lb/>non producetur totus impetus, qui pote&longs;t produci ab ip&longs;o ED; </s> <s id="N26CBA"><!-- NEW -->quia <lb/>non impedietur totus, vt patet: quippe extremitas B faciliùs cedit. </s> <s id="N26CC0"><!-- NEW --><lb/>Sextò, quo punctum ictus accedet propiùs ad extremitatem B, minor <lb/>erit motus centri, <expan abbr="maior&qacute;ue">maiorque</expan> motus circularis, & con&longs;equenter minor <lb/>iactus, & contrà, quò punctum ictus accedet propiùs ad centrum C. <lb/>Septimò, &longs;unt 6. ictuum combinationes in hoc ca&longs;u; </s> <s id="N26CCF"><!-- NEW -->nam vel ictus <lb/>cadet in centrum grauitatis C baculi AB vel extra; &longs;i primum, tribus <lb/>modis id fieri pote&longs;t. </s> <s id="N26CD7"><!-- NEW -->Primò, &longs;i centrum grauitatis impetus baculi ED <lb/>feriat &longs;cilicet ip&longs;um C. Secundò, &longs;i aliud punctum inter CD putà K. <lb/>Tertiò, &longs;i aliquod inter CE putà G; </s> <s id="N26CDF"><!-- NEW -->&longs;i verò &longs;ecundum ii&longs;dem tribus mo­<lb/>dis fieri pote&longs;t, &longs;ed de his &longs;atis; &longs;upere&longs;t tantùm, ni fallor, vt ea phœno­<lb/>mena, quæ in tudiaria gladiatura ob&longs;eruari po&longs;&longs;unt, eorumque cau&longs;as <lb/>explicemus, &longs;ed illud præ&longs;tabimus in lib. &longs;equenti. </s> </p> <p id="N26CE9" type="main"> <s id="N26CEB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N26CF7" type="main"> <s id="N26CF9"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena turbinis &longs;en trochi circumacti<emph.end type="italics"/>: </s> <s id="N26D02"><!-- NEW -->Tur­<lb/>binum puerilium duo &longs;unt genera: primum e&longs;t eorum, qui ferro mu­<lb/>niuntur, qui certè diuer&longs;æ &longs;unt figuræ, &longs;phæricæ, conicæ &c. </s> <s id="N26D0A">communi­<lb/>ter tamen fiunt iuxta figuram cordis, vt vides in A. <!-- KEEP S--></s> <s id="N26D10">Secundum e&longs;t eo­<lb/>rum, qui ferro carent, quorum &longs;unt etiam diuer&longs;æ figuræ, communior e&longs;t <lb/>conois, vt vides in</s> </p> <p id="N26D17" type="main"> <s id="N26D19"><!-- NEW -->Primò, circumagitur vel &longs;cutica vt B, vel funiculo intorto vt A: <lb/>vtriu&longs;que ratio eadem e&longs;t; cùm enim circumuolutus funiculus reduci­<lb/>tur, haud dubiè trochum ip&longs;um in orbem agit. </s> </p> <p id="N26D21" type="main"> <s id="N26D23">Secundò, cum mouetur trochus circa axem CD immobilem, e&longs;t mo­<lb/>tus purè circularis. </s> </p> <pb pagenum="371" xlink:href="026/01/405.jpg"/> <p id="N26D2C" type="main"> <s id="N26D2E"><!-- NEW -->Tertiò, cùm mouetur circa axem mobilem motu recto, e&longs;t motus mix­<lb/>tus ex recto & circulari &longs;imilis motui rotæ; cum verò mouetur axis in <lb/>orbem, mouetur motu mixto ex duobus circularibus, & hic e&longs;t motus <lb/>veri&longs;&longs;imus turbinationis. </s> </p> <p id="N26D38" type="main"> <s id="N26D3A"><!-- NEW -->Quartò, cau&longs;a motus orbis e&longs;t prima reductio &longs;cuticæ, &longs;eu funiculi, <lb/>quæ circumagit turbinem; </s> <s id="N26D40"><!-- NEW -->cau&longs;a verò motus axis e&longs;t extremitas funicu­<lb/>li, vel &longs;cuticæ, quæ trochum aliquo modo, vel adducit, vel qua&longs;i explodit, <lb/>vel expellit; </s> <s id="N26D48"><!-- NEW -->adducit quidem funiculus, cuius altera extremitas etiam <lb/>adducitur; </s> <s id="N26D4E"><!-- NEW -->expellitur verò trochus, cum verbere adigitur: &longs;ed de his <lb/>paulò pò&longs;t. </s> </p> <p id="N26D54" type="main"> <s id="N26D56"><!-- NEW -->Quintò, ideò trochus mouetur motu orbis, &longs;eu motu circulari, quia <lb/>impetus contrarii &longs;imul imprimuntur, v.g.in fig.B imprimitur impetus E <lb/>per artum EHF, & F per arcum FGE: </s> <s id="N26D5E"><!-- NEW -->vndè &longs;equitur nece&longs;&longs;ariò motus <lb/>circularis; hinc digitis in contrarias partes explo&longs;is turbo in orbem <lb/>agitur. </s> </p> <p id="N26D66" type="main"> <s id="N26D68"><!-- NEW -->Sextò, diu durat i&longs;te motus circularis turbinis, quia non de&longs;truitur <lb/>ab impetu contrario grauitationis, vt iam diximus alibi, &longs;ed tantùm ab <lb/>affrictu ad planum illud, in quo vertitur, & à noua determinatione, quæ <lb/>&longs;ingulis in&longs;tantibus ponitur, quæ pro nihilo ferè haberi debet; hinc quò <lb/>vertex turbinis, politior e&longs;t, & planum in quo &longs;uos gyros agit, læuiga­<lb/>tius, diutiùs durat eius motus. </s> </p> <p id="N26D76" type="main"> <s id="N26D78"><!-- NEW -->Septimò, aliquando dormire dicitur turbo cum celerrimè mouetur, <lb/>defixo &longs;cilicet axe in eodem loco, & &longs;itu, ratio petitur ex eo quòd ver­<lb/>tex certè componitur cum ip&longs;o plano factâ &longs;ibi veluti in&longs;en&longs;ibili apo­<lb/>theca &longs;eu fo&longs;&longs;ula, cuius tenuis margo impedit motum centri; igitur mo­<lb/>tus orbis vnicus e&longs;t, igitur maior. </s> </p> <p id="N26D84" type="main"> <s id="N26D86"><!-- NEW -->Octauò, verbere adigitur trochus, <expan abbr="ip&longs;i&qacute;ue">ip&longs;ique</expan> imprimitur primò motus <lb/>orbis, quia lora illa &longs;cuticæ trocho aduoluta, vbi deinde explicantur, tro­<lb/>chum ip&longs;um circumagunt: </s> <s id="N26D92"><!-- NEW -->&longs;ecundò motus centri, quia eadem lora ad in­<lb/>&longs;tar fundæ qua&longs;i trochum explodunt; </s> <s id="N26D98"><!-- NEW -->&longs;ic plerumque accidit adhiberi lora, <lb/>vt longiùs ligneus orbis proijciatur; </s> <s id="N26D9E"><!-- NEW -->quippe dum explicantur lora, du­<lb/>plex ille motus nece&longs;&longs;ariò imprimitur; </s> <s id="N26DA4"><!-- NEW -->primus quidem, quia explicari <lb/>non po&longs;&longs;unt, ni&longs;i trochus circumagatur; </s> <s id="N26DAA"><!-- NEW -->&longs;ecundus verò, quia explicari <lb/>lora non po&longs;&longs;unt ni&longs;i in aliquam partem ferantur, & trochum ip&longs;um tra­<lb/>hant, vel &longs;altem impellant; </s> <s id="N26DB2"><!-- NEW -->adde quod diutiùs manet potentia applicata; <lb/>hinc maior effectus, analogiam habes in funda. </s> </p> <p id="N26DB8" type="main"> <s id="N26DBA"><!-- NEW -->Nonò, quando turbo ferro in&longs;tructus, cui funiculus aduolutus e&longs;t, re­<lb/>tentâ alterà funiculi extremitate, & explicato eodem funiculo circum­<lb/>agitur; </s> <s id="N26DC2"><!-- NEW -->haud dubiè maiore vi pollet hic motus, <expan abbr="durat&qacute;ue">duratque</expan> diù, tùm quie <lb/>funiculus e&longs;t, longior, tùm quia maiore ni&longs;u qua&longs;i euibratur, tùm quia <lb/>diù manet potentia applicata; </s> <s id="N26DCE"><!-- NEW -->porrò duobus modis explicatur funiculus; </s> <s id="N26DD2"><!-- NEW --><lb/>primò enim adducitur, &longs;eu retrahitur, ex quo accidit, vt motus centri <lb/>determinetur in <expan abbr="eãdem">eandem</expan> partem; </s> <s id="N26DDD"><!-- NEW -->&longs;ecundò non adducitur, &longs;ed tantùm <lb/>altera extremitas retinetur; vnde fit, vt motus centri nullus ferè &longs;it. </s> </p> <p id="N26DE3" type="main"> <s id="N26DE5"><!-- NEW -->Decimò, motus centri circularis in cam &longs;emper e&longs;t partem, in quam <pb pagenum="372" xlink:href="026/01/406.jpg"/>exterior turbinis portio motu orbis conuoluitur; </s> <s id="N26DEE"><!-- NEW -->v.g. <!-- REMOVE S-->turbo B mouetur <lb/>motu orbis per arcum EHF; </s> <s id="N26DF6"><!-- NEW -->igitur motu circulari centri vel axis moue­<lb/>bitur per DK, &longs;i &longs;upponatur erectus perpendiculariter in plano LDK; </s> <s id="N26DFC"><!-- NEW --><lb/>ratio e&longs;t, quia circularis axis determinatur à circulari orbis; igitur vter­<lb/>que fit in <expan abbr="eãdem">eandem</expan> partem. </s> </p> <p id="N26E07" type="main"> <s id="N26E09"><!-- NEW -->Vndecimò, diuer&longs;a &longs;cabrities plani in quo circumagitur turbo mul­<lb/>tùm immutat turbinationis modum; tunc enim vel diuer&longs;a plani incli­<lb/>natî ratio, vel diuer&longs;æ qua&longs;i fo&longs;&longs;ulæ, vel in&longs;en&longs;ibiles &longs;copuli turbinem eò <lb/>&longs;æpe adigunt, quo impre&longs;&longs;i motus indoles minimè ferret. </s> <s id="N26E13"><!-- NEW --><lb/>Duodecimò, licèt imprimatur motus rectus axi per adductionem, vel <lb/>emi&longs;&longs;ionem funiculi, non tamen mouetur axis motu recto; quia hic mo­<lb/>tus rectus ab ip&longs;o motu orbis immutatur, ita vt ex vtroque motus fiat <lb/>mixtus, ip&longs;eque adeò axis motu qua&longs;i &longs;pirali, reliquæ verò partes inæ­<lb/>quali motu circumagantur. </s> </p> <p id="N26E20" type="main"> <s id="N26E22"><!-- NEW -->Decimotertiò, quando axis mouetur motu circulari, pote&longs;t e&longs;&longs;e circu­<lb/>lus, quem de&longs;cribit maior vel minor; </s> <s id="N26E28"><!-- NEW -->&longs;i maior e&longs;t, i&longs;que duplus circuli <lb/>ba&longs;is trochi &longs;ingula puncta ba&longs;is de&longs;cribunt lineam cordis, dum motus <lb/>orbis, & axis æquali numero circulorum con&longs;tent; </s> <s id="N26E30"><!-- NEW -->&longs;i verò axis de&longs;cribit <lb/>circulum æqualem ba&longs;i, <expan abbr="&longs;it&qacute;ue">&longs;itque</expan> numerus circulorum <expan abbr="vtriu&longs;&qacute;ue">vtriu&longs;que</expan> motus æ­<lb/>qualis, de&longs;cribit quodlibet punctum periphæriæ ba&longs;is lineam nouam, <lb/>cuius &longs;chema hic habes, &longs;it enim circulus, quem de&longs;cribit punctum <lb/>axis, quod e&longs;t centrum ba&longs;is &longs;upremæ trochi, <expan abbr="AHKq;">AHKque</expan> &longs;itque ba&longs;is ip&longs;a <lb/>circulus EDBC; </s> <s id="N26E4A"><!-- NEW -->hoc po&longs;ito moueatur centrum A per circulum AHK <lb/>Q, cum erit in G, erit in F, cum in H erit in D, cum in D, erit in L; &c. </s> <s id="N26E50"><lb/>igitur punctum periphæriæ ba&longs;is E de&longs;cribit &longs;uo motu lineam curuam <lb/>EFADLMPCAE, quæ &longs;uas habet proprietates, de quibus &longs;uo loco. </s> </p> <p id="N26E56" type="main"> <s id="N26E58"><!-- NEW -->Decimoquartò, ob&longs;eruas, ni&longs;i fallor, mirabilem huius motus analo­<lb/>giam; </s> <s id="N26E5E"><!-- NEW -->&longs;it enim centrum circuli, qui circa alium immobilem conuertitur, <lb/>decurrat circulum duplò maiorem, de&longs;cribit lineam cordis, de qua &longs;uprà, <lb/>&longs;i maiorem duplo (eâ tamen lege vt centrum, & orbis æquali tempore <lb/>&longs;uum circulum decurrant) de&longs;cribitur linea, quæ accedit propiùs ad cir­<lb/>culum; </s> <s id="N26E6A"><!-- NEW -->&longs;i verò circulus centri &longs;it æqualis circulo orbis, habes lineam in <lb/>&longs;uperiore &longs;chemate, quæ geminum <expan abbr="circulũ">circulum</expan> imperfectum præfert, qui eò <lb/>propiùs ad &longs;e inuicem <expan abbr="accedũt">accedunt</expan>, quo circulus centri minor e&longs;t; </s> <s id="N26E7A"><!-- NEW -->cùm enim <lb/>nullus e&longs;t omninò <expan abbr="c&etilde;tri">centri</expan> circulus, tunc ambo circuli imperfecti in vnum <lb/><expan abbr="perfectũ">perfectum</expan> coëunt; &longs;i verò circulus centri &longs;it minor duplò, &longs;ed maior æquali, <lb/>minor erit &longs;uperior illa figura EFA, &c. </s> <s id="N26E8B">donec tandem vbi circulus cen­<lb/>tri e&longs;t duplus circuli orbis vnica tantùm figura de&longs;cribatur, &longs;cilicet linea <lb/>cordis. </s> <s id="N26E92"><!-- NEW -->Sed de his omnibus fusè &longs;uo loco; &longs;unt enim mirificæ harum <lb/>linearum proprietates. </s> </p> <p id="N26E98" type="main"> <s id="N26E9A"><!-- NEW -->Decimoquintò, &longs;altitat initio proiectus turbo; </s> <s id="N26E9E"><!-- NEW -->ratio e&longs;t, quia motus <lb/>centri maior e&longs;t; </s> <s id="N26EA4"><!-- NEW -->igitur ob maiorem affrictum &longs;æpiùs re&longs;ilit; quod pro­<lb/>fectò non accideret, &longs;i planum læuigati&longs;&longs;imum e&longs;&longs;et, & ferreus mucro <lb/>politi&longs;&longs;imus hinc &longs;tatim primus ille ardor deferue&longs;cit, & miliùs turbi­<lb/>natur. </s> </p> <pb pagenum="373" xlink:href="026/01/407.jpg"/> <p id="N26EB2" type="main"> <s id="N26EB4"><!-- NEW -->Decimo&longs;extò, antequam quie&longs;cat turbo, inclinatur, &longs;uo&longs;que orbes agit <lb/>inclinato qua&longs;i corpore, & obliquo axe; </s> <s id="N26EBA"><!-- NEW -->ratio e&longs;t, quia vel axis &longs;eu ferreus <lb/>mucro tantillùm abe&longs;t à grauitatis centro, vel aliquis plani &longs;copulus, vel <lb/>decliuis plaga turbinem ip&longs;um inclinat; agit tamen adhuc aliquot obli­<lb/>quos gyros propter vim prioris impetus, quæ &longs;en&longs;im à grauitatione tur­<lb/>binis frangitur, & tandem omninò &longs;uperatur. </s> </p> <p id="N26EC6" type="main"> <s id="N26EC8"><!-- NEW -->Decimo&longs;eptimò, hinc, vbi terrarum tangit depre&longs;&longs;us turbo, ad in&longs;tar <lb/>rotæ deindæ rotatur; ratio e&longs;t, quia multus adhuc remanet impetus ad <lb/>motum orbis determinatus, qui vbi tangitur, &longs;olum trochum ip&longs;um cum <lb/>centro ad in&longs;tar rotæ præcipitem agit. </s> </p> <p id="N26ED4" type="main"> <s id="N26ED6"><!-- NEW -->Decimooctauò, hinc vides naturam maximè gaudere motu recto qui <lb/>paulò ante turbini erecto minimè concedebatur; cur enim in vnam po­<lb/>tiùs partem, quàm in aliam? </s> <s id="N26EDE"><!-- NEW -->at verò lap&longs;o iacentique facilè permittitur; <lb/>nam in plano motus orbis rotæ facilè determinat motum rectum <lb/>centri. </s> </p> <p id="N26EE6" type="main"> <s id="N26EE8"><!-- NEW -->Decimononò, ad turbinem reuoco cubum illum, &longs;uis numeris vel <lb/>characteribus in&longs;tructum, & duobus hinc inde in &longs;uprema, & ima facie, <lb/>qua&longs;i paxillis, vel communi axe munitum, cuius figuram hîc habes; vol­<lb/>uitur enim hic cubus circa &longs;uum axem, neque e&longs;t noua difficultas. </s> </p> <p id="N26EF2" type="main"> <s id="N26EF4">Vige&longs;imò, huc etiam reuoca fu&longs;um, qui dum turbinatim ver&longs;atur, di­<lb/>uer&longs;is etiam motibus moueri pote&longs;t &longs;ur&longs;um, deor&longs;um, dextror&longs;um, &longs;ini­<lb/>&longs;tror&longs;um, ïta vt in eo mira motuum varietas ob&longs;eruari po&longs;&longs;it. </s> </p> <p id="N26EFB" type="main"> <s id="N26EFD"><!-- NEW -->Vige&longs;imoprimò, reuocabis quoque motum parop&longs;idis, dum digito <lb/>qua&longs;i flagellatur; e&longs;t enim quoddam turbinationis genus, cuius ratio <lb/>facilis e&longs;t, & con&longs;tat ex dictis. </s> </p> <p id="N26F05" type="main"> <s id="N26F07"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N26F13" type="main"> <s id="N26F15"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt phœnomena<emph.end type="italics"/> <emph type="italics"/>motus Excentricorum<emph.end type="italics"/>; </s> <s id="N26F24"><!-- NEW -->&longs;it circulus ALK <lb/>M centro E; </s> <s id="N26F2A"><!-- NEW -->&longs;it alius excentricus ACOD centro B, circa quod mouea­<lb/>tur punctum A v.g. <!-- REMOVE S-->motu orbis; </s> <s id="N26F32"><!-- NEW -->Primò, nulla erit inæqualitàs motus, &longs;ed <lb/>tantùm videbitur e&longs;&longs;e; </s> <s id="N26F38"><!-- NEW -->nam <expan abbr="punctũ">punctum</expan> A, in quo &longs;it a&longs;trum po&longs;t decur&longs;um <lb/>quadrantem; </s> <s id="N26F42"><!-- NEW -->videbitur in N; </s> <s id="N26F46"><!-- NEW -->igitur videbitur tantùm confeci&longs;&longs;e arcum A <lb/>N minorem quadrante; </s> <s id="N26F4C"><!-- NEW -->hinc motus ab A ad C indicabitur tardior; </s> <s id="N26F50"><!-- NEW -->at ve­<lb/>AC ad O videbitur velocior; </s> <s id="N26F56"><!-- NEW -->quia credetur confeci&longs;&longs;e arcum maiorem <lb/>NK, æquali &longs;cilicet tempore, quo AN; </s> <s id="N26F5C"><!-- NEW -->hinc ab A ad C, id e&longs;t ab apogæo <lb/>dicitur e&longs;&longs;e tardior; vel ocior verò AC ad I, id e&longs;t ad perigæum, &longs;ed hæc <lb/>&longs;unt facilia, & communia, per quæ explicantur anomaliæ, & inæquali­<lb/>tates &longs;impliciores motuum cæle&longs;tium. </s> </p> <p id="N26F66" type="main"> <s id="N26F68"><!-- NEW -->Secundò, &longs;i voluatur circulus radio AE circa centrum E, nec &longs;it vllus <lb/>motus circa centrum B; haud dubiè omnes partes excentrici ADOC <lb/>mouebuntur motu circulari &longs;ed inæquali, vt patet. </s> </p> <p id="N26F70" type="main"> <s id="N26F72"><!-- NEW -->Tertiò, &longs;i &longs;it motus circularis circa vtrumque centrum; certè centrum <lb/>B circumagetur per circellum BGHF, punctum verò A excentrici <lb/>de&longs;cribet hanc lineam APIQBSIRA, vt con&longs;tat ex dictis Th. 30. <lb/>num. </s> <s id="N26F7C">30. </s> </p> <pb pagenum="374" xlink:href="026/01/408.jpg"/> <p id="N26F83" type="main"> <s id="N26F85">Quartò, hine in &longs;ingulis circuitionibus videretur facere duas, & pe­<lb/>rigæum videretur ver&longs;us eam partem, ver&longs;us quam videretur apogæum. </s> </p> <p id="N26F8A" type="main"> <s id="N26F8C"><!-- NEW -->Quintò, centrum B po&longs;&longs;et moueri per circellum minorem BGHF, <lb/>vel per alium, cuius centrum e&longs;&longs;et inter BE; per hos autem circellos <lb/>explicant A&longs;tronomi diuer&longs;as excentricitatis mutationes. </s> </p> <p id="N26F94" type="main"> <s id="N26F96">Sextò, moueretur punctum A inæqualiter, v.g. <!-- REMOVE S-->eo tempore, quo per­<lb/>currit AP, percurrit tantùm SI, vt con&longs;tat ex dictis &longs;uprà. </s> </p> <p id="N26F9D" type="main"> <s id="N26F9F"><!-- NEW -->Septimò, po&longs;&longs;unt etiam determinari illi arcus, qui tardiùs licèt de­<lb/>cur&longs;i, velociùs tamen decurri viderentur; </s> <s id="N26FA5"><!-- NEW -->nam in A videretur moueri <lb/>tardi&longs;&longs;imè; at verò veloci&longs;&longs;imè in B. </s> </p> <p id="N26FAB" type="main"> <s id="N26FAD"><!-- NEW -->Octauò, po&longs;&longs;unt plures excentrici &longs;imul componi cum pluribus etiam <lb/>concentricis; </s> <s id="N26FB3"><!-- NEW -->&longs;ed de iis fusè in A&longs;tronomia; hîc tantum &longs;ufficiat indi­<lb/>ca&longs;&longs;e, & qua&longs;i reduxi&longs;&longs;e ad principia motuum mixtorum. </s> </p> <p id="N26FB9" type="main"> <s id="N26FBB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N26FC7" type="main"> <s id="N26FC9"><emph type="italics"/>Po&longs;&longs;unt explicari omnia phœnomena<emph.end type="italics"/> <emph type="italics"/>Epiciclorum.<emph.end type="italics"/></s> <s id="N26FD6"><!-- NEW --> Primò &longs;it circulus H <lb/>BCK centro A, &longs;it epicyclus LIQG, centro G; </s> <s id="N26FDC"><!-- NEW -->a&longs;&longs;umatur quodlibet <lb/>eius punctum, putà G, quod moueatur motu mixto id e&longs;t, motu centri, <lb/>& motu orbis: po&longs;&longs;unt a&longs;&longs;ignari omnia puncta lineæ huius motus, om­<lb/>nes velocitatis proportiones, &c. </s> </p> <p id="N26FE6" type="main"> <s id="N26FE8">Secundò, &longs;i H moueatur ver&longs;us K, & G ver&longs;us Q de&longs;cribet &longs;peciem <lb/>lineæ cordis GZMNE. </s> </p> <p id="N26FED" type="main"> <s id="N26FEF">Tertiò, G mouebitur velociùs, in G quam in N, E, &c. </s> <s id="N26FF2">tardi&longs;&longs;imè in <lb/>perigæo E, veloci&longs;&longs;imè in Apogæo G. <!-- KEEP S--></s> </p> <p id="N26FF8" type="main"> <s id="N26FFA">Quartò, temporibus æqualibus diuer&longs;os arcus de&longs;cribit, &longs;cilicet ar­<lb/>cum compræhen&longs;um angulo HAN, NAC. </s> </p> <p id="N26FFF" type="main"> <s id="N27001"><!-- NEW -->Quintò, &longs;i G moueatur ver&longs;us L & H ver&longs;us K, tardi&longs;&longs;imus motus <lb/>erît in apogæo G, veloci&longs;&longs;imus in perigæo E; nam eo tempore, quo à pe­<lb/>rigæo conficit arcum compræhen&longs;um angulo CAM, conficit ab apo­<lb/>gæo arcum compræhen&longs;um angulo MAH. </s> </p> <p id="N2700B" type="main"> <s id="N2700D">Sextò, &longs;i motus epicycli &longs;it inæqualis motui centri, diuer&longs;a erit linea <lb/>huîus motus mixti, diuer&longs;æ motuum, & velocitatum proportiones. </s> </p> <p id="N27012" type="main"> <s id="N27014"><!-- NEW -->Septimò, &longs;i &longs;int duo Epicycli, erit etiam diuer&longs;a linea, & diuer&longs;a mo­<lb/>tuum proportio; pote&longs;t autem accidere, vt vel vterque in <expan abbr="eãdem">eandem</expan> par­<lb/>tem, vel in diuer&longs;as tendant. </s> </p> <p id="N27020" type="main"> <s id="N27022"><!-- NEW -->Octauò, pote&longs;t etiam Epicyclus rotari in excentrico, in quo ca&longs;u di­<lb/>uer&longs;us erit motus, diuer&longs;a linea; quæ omnia facilè ex dictis con&longs;tant, de <lb/>quibus fusè agemus &longs;uo loco. </s> </p> <p id="N2702A" type="main"> <s id="N2702C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N27038" type="main"> <s id="N2703A"><!-- NEW --><emph type="italics"/>Si rota moueatur in circulo parallelo illi plane, cui incubat perpendicula­<lb/>riter eodem ferè motu moneri videtur, quo turbo, de quo &longs;uprà<emph.end type="italics"/>; </s> <s id="N27045"><!-- NEW -->a&longs;&longs;umatur <lb/>enim figura prima Th. 15. in qua &longs;it circulus immobilis in plano hori­<lb/>zontali BTXD, & erigatur rota BEDF, ita vt &longs;it parallela circulo <lb/>verticali, tangatque priorem circulum in B, cuius deinde periphæriam <lb/>&longs;en&longs;im percurrat; </s> <s id="N27051"><!-- NEW -->haud dubiè punctum B de&longs;cribet &longs;uo motu lineam, quæ <pb pagenum="375" xlink:href="026/01/409.jpg"/>pote&longs;t declinari; </s> <s id="N2705A"><!-- NEW -->&longs;it enim circulus immobilis BDFC, mobilis FEG, <lb/>punctum F po&longs;t decur&longs;um quadrantem FD extat &longs;upra planum hori­<lb/>zontis tota ID erecta; </s> <s id="N27062"><!-- NEW -->po&longs;t decur&longs;um verò &longs;emicirculum tota BK <lb/>erecta æquali BF, vt con&longs;tat; </s> <s id="N27068"><!-- NEW -->igitur vertatur FBK, circa FB, donec incu­<lb/>bet perpendiculariter plano horizontali in BF; </s> <s id="N2706E"><!-- NEW -->tùm circa FK, ita ere­<lb/>ctam vertatur planum, donec incubet DI, erecta in I, fiet planum, in quo <lb/>de&longs;cribetur linea huius motus; </s> <s id="N27076"><!-- NEW -->a&longs;&longs;umatur autem DH æqualis AI; </s> <s id="N2707A"><!-- NEW -->dico <lb/>quod ducetur per FHK: </s> <s id="N27080"><!-- NEW -->&longs;imiliter inuenientur alia puncta, quod &longs;uffi­<lb/>ciat indica&longs;&longs;e; </s> <s id="N27086"><!-- NEW -->e&longs;t autem hic motus maximè inæqualis propter ratio­<lb/>nem, de qua &longs;uprà: </s> <s id="N2708C"><!-- NEW -->&longs;ed de his &longs;atis; </s> <s id="N27090"><!-- NEW -->immò certum e&longs;t punctum F &longs;uo <lb/>motu prædicto de&longs;cribere perfectum circulum duplum circuli rota­<lb/>ti, cuius centrum e&longs;t D erectum in A, nam DH, DF, DK &longs;unt æqua­<lb/>les; </s> <s id="N2709A"><!-- NEW -->&longs;i enim circulus tangat in M, punctum F erectum toto arcu FM, <lb/>re&longs;pondebit perpendiculariter puncto O, ita vt OM &longs;it æqualis PB, vel <lb/>HS, vel AN; erigatur autem OR, donec incubet perpendiculariter, <lb/>extat &longs;uper AD erecta in A tota QR, ita OQ &longs;it æqualis AD. <!-- KEEP S--></s> <s id="N270A5">Sed <lb/>quad. </s> <s id="N270AA"><!-- NEW -->AO e&longs;t æquale quadratis AM, MO; igitur &longs;it quad. </s> <s id="N270AE"><!-- NEW -->AM qua­<lb/>dratum MO erit 8. igitur quadratum A 24. &longs;ed extat &longs;uper MO, QR, <lb/>æqualis OM; </s> <s id="N270B6"><!-- NEW -->igitur &longs;i à D erecto ducantur duæ rectæ, altera ad Q, altera <lb/>ad R, lineæ OR erectæ; </s> <s id="N270BC"><!-- NEW -->certè DQ erit æqualis AO; </s> <s id="N270C0"><!-- NEW -->e&longs;t enim ip&longs;i pa­<lb/>rallela; </s> <s id="N270C6"><!-- NEW -->tùm fiet triangulum ortogon ex tribus DQ, QR, DR; igitur <lb/>quadr. </s> <s id="N270CC"><!-- NEW -->DR e&longs;t æquale duobus DQ, QR, &longs;ed DQ e&longs;t æqualis A <lb/>O; igitur quadr. </s> <s id="N270D2"><!-- NEW -->DQ e&longs;t 24. QR e&longs;t æqualis OM; igitur quadr. </s> <s id="N270D6">QR <lb/>e&longs;t 8. igitur quadratum DR e&longs;t 32. &longs;ed quadr. </s> <s id="N270DB"><!-- NEW -->DF e&longs;t 32. po&longs;ito <lb/>quadrato AF 16.igitur DR erit æqualis DF; igitur circu­<lb/>lus duplus, &c. </s> <s id="N270E3">quod erat demon­<lb/>&longs;trandum. <lb/><figure id="id.026.01.409.1.jpg" xlink:href="026/01/409/1.jpg"/></s> </p> </chap> <chap id="N270EE"> <pb pagenum="376" xlink:href="026/01/410.jpg"/> <figure id="id.026.01.410.1.jpg" xlink:href="026/01/410/1.jpg"/> <p id="N270F8" type="head"> <s id="N270FA"><emph type="center"/>LIBER DECIMVS, <lb/><emph type="italics"/>DE DIVERSIS MOTIONVM, VEL <lb/>imprimendi motus rationibus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N27109" type="main"> <s id="N2710B"><!-- NEW -->HACTENVS explicauimus naturam cau­<lb/>&longs;æ formalis motus, ide&longs;t impetus in <lb/>libro primo: proprietates motus natu­<lb/>ralis &longs;ecundo: tertio violenti affectio­<lb/>nes; </s> <s id="N27117"><!-- NEW -->quarto mixti ex pluribus rectis: </s> <s id="N2711B"><!-- NEW --><lb/>quinto motum in diuer&longs;is planis con&longs;iderauimus; <lb/>&longs;exto reflexum; &longs;eptimo circularem; octauo fune­<lb/>pendulorum vibrationes; nono mixtum ex circulari, <lb/>quæ omnia &longs;pectant, vel ad cau&longs;am formalem, vel <lb/>ad principium intrin&longs;ecum, vel ad modum etiam <lb/>intrin&longs;ecum, vel ad &longs;patium, &c. </s> <s id="N2712A"><!-- NEW -->iam verò con&longs;i­<lb/>deramus diuer&longs;os modos, quibus impetus imprimi <lb/>pote&longs;t; pote&longs;t enim mobile proijci, pelli, trahi, percuti, <lb/>premi, &longs;u&longs;tineri, tornari, &c. </s> <s id="N27134">de quibus omnibus iam <lb/>nobis, hoc decimo libro agendum videtur, vt dein­<lb/>de vndecimo de organis motus, & duodecimo de li­<lb/>neis tandem agamus. <lb/><gap desc="hr tag"/></s> </p> <p id="N27140" type="main"> <s id="N27142"><emph type="center"/><emph type="italics"/>DEFINITIO I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2714E" type="main"> <s id="N27150"><!-- NEW --><emph type="italics"/>IMpre&longs;&longs;io e&longs;t productio impetus in exteriore mobili, vel ni&longs;us ad illam.<emph.end type="italics"/><lb/>Explicatione multa non indiget hæc definitio; </s> <s id="N2715A"><!-- NEW -->dicitur productio <lb/>impetus, quia reuerâ quando proijcitur lapis, in eum deriuatur aliquid <pb pagenum="377" xlink:href="026/01/411.jpg"/>ab ip&longs;o proijciente mediatè, vel immediatè, cuius vi deinde mouetur; </s> <s id="N27165"><!-- NEW -->at­<lb/>qui vnus impetus illud ip&longs;um præ&longs;tare pote&longs;t, vr con&longs;tat ex dictis, toto, <lb/>lib. 1. additum e&longs;t, vel ni&longs;us ad illam, vt producitur impetus in omni <lb/>pul&longs;ione, nec in omni percu&longs;&longs;ione; </s> <s id="N2716F"><!-- NEW -->cum enim quis pellit ingentem rupem <lb/>&longs;eu percutit pugno; nullum certè producit impetum, ni&longs;i aliqua pars <lb/>auolet, quæ omnia con&longs;tant ex dictis l.1. </s> </p> <p id="N27177" type="main"> <s id="N27179"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N27186" type="main"> <s id="N27188"><emph type="italics"/>Re&longs;i&longs;tentia mobilis e&longs;t illa ratio, que mobili ine&longs;t, cuius vi vel motum omnem <lb/>ip&longs;um mobile ab applicata potentia renuit vel tardiorem tantum permittit.<emph.end type="italics"/></s> </p> <p id="N27191" type="main"> <s id="N27193"><!-- NEW -->Quid verò &longs;it illa ratio, & in quo po&longs;ita &longs;it explicabimus infrà; </s> <s id="N27197"><!-- NEW -->nihil <lb/>enim aliud nomine re&longs;i&longs;tentiæ intelligi pote&longs;t, quàm id, quo mobile re­<lb/>&longs;i&longs;tit motui; </s> <s id="N2719F"><!-- NEW -->re&longs;i&longs;tere autem motui, e&longs;t vel totum impedire motum vel <lb/>eius partem, per quid autem re&longs;i&longs;tat, & propter quid dicemus infrà: &longs;atis <lb/>e&longs;t dixi&longs;&longs;e, quid &longs;it re&longs;i&longs;tere & re&longs;i&longs;tentia. </s> </p> <p id="N271A7" type="main"> <s id="N271A9"><emph type="center"/><emph type="italics"/>Hypothe&longs;is.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N271B5" type="main"> <s id="N271B7"><!-- NEW -->Lapis 20. librarum difficiliùs proijcitur, vel &longs;u&longs;tinetur ab eadem po­<lb/>tentiâ, quàm lapis vnius libræ; hypothe&longs;is certa e&longs;t. </s> </p> <p id="N271BD" type="main"> <s id="N271BF">Axiomata nulla præmittemus cum Theoremata lib. 1. demon&longs;trata <lb/>&longs;ufficiant. </s> </p> <p id="N271C4" type="main"> <s id="N271C6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N271D3" type="main"> <s id="N271D5"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena &longs;u&longs;tentationis.<emph.end type="italics"/></s> </p> <p id="N271DC" type="main"> <s id="N271DE"><!-- NEW -->Primò, vt manus &longs;u&longs;tineat pondus in &longs;itu horizontali producit in &longs;e <lb/>impetum; </s> <s id="N271E4"><!-- NEW -->quia, cùm brachium libero motu librari po&longs;&longs;it, &longs;uo pondere <lb/>de&longs;cenderet, ni&longs;i aliquod re&longs;i&longs;teret; </s> <s id="N271EA"><!-- NEW -->&longs;ed ip&longs;um brachium non re&longs;i&longs;tit; </s> <s id="N271EE"><!-- NEW -->igi­<lb/>tur aliquid quod brachio ine&longs;t; igitur impetus. </s> </p> <p id="N271F4" type="main"> <s id="N271F6"><!-- NEW -->Secundò, impetus, quem ip&longs;a potentia motrix in brachio producit, <lb/>non e&longs;t maior impetu grauitationis ip&longs;ius brachij; </s> <s id="N271FC"><!-- NEW -->quia alioquin præua­<lb/>leret; igitur brachium a&longs;cenderet, contra hypothe&longs;im. </s> </p> <p id="N27202" type="main"> <s id="N27204"><!-- NEW -->Tertiò, ille impetus non e&longs;t etiam minor; </s> <s id="N27208"><!-- NEW -->quia alioqui impetus gra­<lb/>uitationis præualeret; igitur brachium de&longs;cenderet, contra hypo­<lb/>the&longs;im. </s> </p> <p id="N27210" type="main"> <s id="N27212">Quartò, hinc &longs;equitur e&longs;&longs;e æqualem, cùm &longs;it per n.1.nec &longs;it maior per <lb/>2.nec minor per 3. &longs;equitur nece&longs;&longs;ariò e&longs;&longs;e æqualem. </s> </p> <p id="N27217" type="main"> <s id="N27219"><!-- NEW -->Quintò, &longs;ingulis in&longs;tantibus impetus productus priore in&longs;tanti de­<lb/>&longs;truitur; probatur, quia quotie&longs;cumque ad lineas oppo&longs;itas ex diame­<lb/>tro determinantur duo impetus æquales, de&longs;truuntur, &longs;i de&longs;trui po&longs;&longs;unt <lb/>per Theorema 123.lib.1. at verò impetus innatus de&longs;trui non pote&longs;t, per <lb/>Theorema 77. libro 2. igitur de&longs;truitur productus à potentia mo­<lb/>trice. </s> </p> <p id="N27227" type="main"> <s id="N27229"><!-- NEW -->Sextò, propter molliores partes organi, v. <!-- REMOVE S-->g. <!-- REMOVE S-->mu&longs;culorum, neruo­<lb/>rum, impetus naturalis aliquem &longs;emper effectum &longs;ortitur, com­<lb/>pre&longs;&longs;ionis, diui&longs;ionis, ten&longs;ionis: </s> <s id="N27235"><!-- NEW -->ratio e&longs;t, quia anima non produ-<pb pagenum="378" xlink:href="026/01/412.jpg"/>cit impetum in omnibus immediatè; vt patet; </s> <s id="N2723E"><!-- NEW -->alioquin etiam re&longs;ectis <lb/>neruis brachij po&longs;&longs;et brachium moueri; </s> <s id="N27244"><!-- NEW -->igitur illæ partes, quæ tan­<lb/>tùm habent impetum grauitationis deor&longs;um, qua&longs;i pugnant cum <lb/>aliis, quæ impetum grauitationis habent impetum ab ima; </s> <s id="N2724C"><!-- NEW -->quemad­<lb/>modum enim, cum aliquod pondus humeris incubat, vel manui; &longs;en­<lb/>tio ponderis vim, cuius effectus rationem afferemus paulò pò&longs;t, ita <lb/>pror&longs;us partes, quæ immediatè ab anima impetum non accipiunt, alias <lb/>deprimunt. </s> </p> <p id="N27258" type="main"> <s id="N2725A"><!-- NEW -->Septimò, &longs;ingulis in&longs;tantibus anima producit impetum in organo; <lb/>quia &longs;ingulis de&longs;truitur per num. </s> <s id="N27260"><!-- NEW -->5. igitur cùm re&longs;i&longs;tat continuò graui­<lb/>tationi, tùm ip&longs;ius organi, tùm partium coniunctarum cum organo, <lb/>&longs;iue &longs;int animatæ, &longs;iue inanimes, debet ade&longs;&longs;e cau&longs;a huius re&longs;i&longs;tentiæ; <lb/>igitur nouus impetus, cùm prior de&longs;truatur. </s> </p> <p id="N2726A" type="main"> <s id="N2726C"><!-- NEW -->Octauò, impetus productus in organo, quod mouetur, produ­<lb/>cit impetum in aliis partibus cum ip&longs;o organo coniunctis; pro­<lb/>batur; </s> <s id="N27274"><!-- NEW -->cum enim &longs;ingulæ partes mouentur, &longs;ingulæ habent impe­<lb/>tum, &longs;ed &longs;ingulæ impetum ab anima non habent immediatè, vt <lb/>con&longs;tat; </s> <s id="N2727C"><!-- NEW -->igitur aliquæ partes habent impetum ab impetu ip&longs;ius <lb/>organi: </s> <s id="N27282"><!-- NEW -->&longs;ecundò eodem pror&longs;us modo moueo vnguem, quo lapil­<lb/>lum; </s> <s id="N27288"><!-- NEW -->&longs;ed lapillus, quem moueo manu, non accipiet impetum imme­<lb/>diatè ab anima, &longs;ed ab organo, vel potiùs ab impetu organi; igitur nec <lb/>vnguis, nec aliæ partes, quæ non &longs;unt organum motus, licèt cum eo <lb/>coniunctæ &longs;int. </s> </p> <p id="N27292" type="main"> <s id="N27294"><!-- NEW -->Nonò, cum verò organum non mouetur.v.g.manus quantumuis ex­<lb/>ten&longs;a, vel erecta, non producit impetum in aliis partibus coniun­<lb/>ctis, licèt animatis; probatur primò, fru&longs;trà produceretur, cùm <lb/>impediri po&longs;&longs;it earum motus deor&longs;um &longs;ine impetu, alioquin men&longs;a, <lb/>quæ &longs;u&longs;tinet pondus, produceret in eo impetum, quod e&longs;t ridicu­<lb/>lum. </s> <s id="N272A2"><!-- NEW -->Secundò, quia &longs;i impetus organi producit impetum in partibus <lb/>vnitis, quo eas qua&longs;i reducit &longs;ur&longs;um; </s> <s id="N272A8"><!-- NEW -->igitur impetus grauitationis <lb/>partium vnitarum producit etiam impetum deor&longs;um in organo; <lb/>immò daretur proce&longs;&longs;us in infinitum, de quo paulò pò&longs;t. </s> </p> <p id="N272B0" type="main"> <s id="N272B2"><!-- NEW -->Decimò, cum manus &longs;u&longs;tinet aliquod pondus immobiliter, non <lb/>producit in eo impetum; </s> <s id="N272B8"><!-- NEW -->Primò, quia, &longs;i non producitur impe­<lb/>tus in alijs partibus vnitis, licèt animatis, multò minùs in alijs; </s> <s id="N272BE"><!-- NEW --><lb/>Secundò, quia eodem modo &longs;u&longs;tinetur pondus à manu, quo ab alio <lb/>corpore inanimo, v. <!-- REMOVE S-->g. <!-- REMOVE S-->à men&longs;a; </s> <s id="N272C9"><!-- NEW -->&longs;ed hæc non producit impetum in <lb/>pondere, quod &longs;u&longs;tinet, vt dicam paulò pò&longs;t; </s> <s id="N272CF"><!-- NEW -->Tertiò, quia fru&longs;trà pro­<lb/>duceretur; </s> <s id="N272D5"><!-- NEW -->quia modò manus &longs;u&longs;tinens &longs;tet immobilis; haud dubiè etiam <lb/>&longs;ublato omni extrin&longs;eco impetu à pondere adhuc &longs;u&longs;tinebitur. </s> </p> <p id="N272DB" type="main"> <s id="N272DD">Dices; </s> <s id="N272E0"><!-- NEW -->igitur fru&longs;trà produceretur impetus in manu; </s> <s id="N272E4"><!-- NEW -->Re&longs;p. negando <lb/>quia ni&longs;i potentia motrix produceret impetum in manu, ab ip&longs;o pon­<lb/>dere deprimeretur; igitur non e&longs;t fru&longs;trà omninò ille impetus. </s> </p> <p id="N272EC" type="main"> <s id="N272EE"><!-- NEW -->Dices, non habet motum; </s> <s id="N272F2"><!-- NEW -->igitur e&longs;t fru&longs;trà; </s> <s id="N272F6"><!-- NEW -->Re&longs;p. omnem impetum <lb/>non e&longs;&longs;e fru&longs;trà, licèt careat motu, vt patet in ip&longs;o impetu innato, <pb pagenum="379" xlink:href="026/01/413.jpg"/>cuius duplex e&longs;t effectum; </s> <s id="N27301"><!-- NEW -->&longs;cilicet grauitatio, & motus, vt aliàs iam in­<lb/>dicauimus; </s> <s id="N27307"><!-- NEW -->&longs;imiliter impetus productus à potentia motrice, in &longs;uo or­<lb/>gano habere pote&longs;t duplicem effectum; </s> <s id="N2730D"><!-- NEW -->primus e&longs;t motus; </s> <s id="N27311"><!-- NEW -->&longs;ecundus e&longs;t <lb/>ni&longs;us &longs;eu conatus oppo&longs;itus extrin&longs;eco motui; </s> <s id="N27317"><!-- NEW -->quemadmodum enim in­<lb/>natus &longs;emper habet motum, ni&longs;i impediatur ab alio corpore, ita & im­<lb/>petus organi potentiæ motricis, nec e&longs;t magna difficultas; immò cla­<lb/>ri&longs;&longs;ima vtriu&longs;que potentiæ analogia. </s> </p> <p id="N27321" type="main"> <s id="N27323"><!-- NEW -->Vndecimò, hinc benè explicatur, quomodo defatigetur ten&longs;um bra­<lb/>&longs;iue coniunctum &longs;iue coniunctum; </s> <s id="N27329"><!-- NEW -->&longs;it cum extrin&longs;eco <expan abbr="põdere">pondere</expan>, &longs;iue <expan abbr="cũ">cum</expan> pro­<lb/>pria tantùm grauitate; </s> <s id="N27337"><!-- NEW -->quia partes aliquæ tendunt deor&longs;um, aliæ verò &longs;ur­<lb/>&longs;um; </s> <s id="N2733D"><!-- NEW -->hinc &longs;emper fit aliqua ten&longs;io; igitur aliqua diui&longs;io; </s> <s id="N27341"><!-- NEW -->igitur dolor, &longs;ic <lb/>enim tenditur funis à <expan abbr="põdere">pondere</expan> pendulo, pondus verò <expan abbr="incubãs">incubans</expan> tùm aliquas <lb/>partes premit, tùm alias maximè di&longs;trahit, in quo non e&longs;t difficultas; </s> <s id="N27351"><!-- NEW -->&longs;i <lb/>autem manus incubet men&longs;æ, v. <!-- REMOVE S-->g. <!-- REMOVE S-->& pondus manui fit tantùm com­<lb/>pre&longs;&longs;io partium, quæ pro mollitie facilè cedunt & &longs;eparantur; </s> <s id="N2735D"><!-- NEW -->igitur <lb/>pondus producit impetum in manu & neruis; alioquin nulla e&longs;&longs;et ten­<lb/>&longs;io, neque compre&longs;&longs;io. </s> </p> <p id="N27365" type="main"> <s id="N27367"><!-- NEW -->Duodecimò, hinc benè colligo non produci impetum à potentia mo­<lb/>trice in toto organo; </s> <s id="N2736D"><!-- NEW -->quia &longs;i hoc e&longs;&longs;et, omnes partes &longs;tarent immobili­<lb/>ter; </s> <s id="N27373"><!-- NEW -->e&longs;&longs;et enim hic impetus æqualis impetui grauitationis, tùm organi, <lb/>tùm ponderis; </s> <s id="N27379"><!-- NEW -->tùm aliarum partium, cum organo coniunctarum; </s> <s id="N2737D"><!-- NEW -->igitur <lb/>nulla e&longs;&longs;et defatigatio; quia tam facilè anima produceret impetum, <lb/>2°ree;.in&longs;tanti, 3°ree;, 4°ree;. </s> <s id="N27385">&c. </s> <s id="N27388">quàm 1°ree;; </s> <s id="N2738B"><!-- NEW -->&longs;ed nulla e&longs;t defatigatio pro 1°ree;; </s> <s id="N2738F"><!-- NEW -->igitur <lb/>nulla e&longs;&longs;et in reliquis, quod tamen e&longs;t contra hypothe&longs;im; </s> <s id="N27395"><!-- NEW -->immò po&longs;&longs;e­<lb/>mus liberè moueri per medium aëra; cum enim 1°ree;. </s> <s id="N2739B"><!-- NEW -->in&longs;tanti po&longs;&longs;emus <lb/>producere impetum maiorem impetu grauitationis, vt patet; </s> <s id="N273A1"><!-- NEW -->certè non <lb/>de&longs;trueretur totus, 2°ree; in&longs;tanti; igitur cum 2°ree;. </s> <s id="N273A7">in&longs;tanti po&longs;&longs;et æqualis <lb/>1°ree;. </s> <s id="N273AC"><!-- NEW -->impetus produci; </s> <s id="N273B0"><!-- NEW -->&longs;emper intenderetur; </s> <s id="N273B4"><!-- NEW -->igitur facilè moueremur, <lb/>quod ab&longs;urdum e&longs;t; </s> <s id="N273BA"><!-- NEW -->igitur po&longs;&longs;umus quidem &longs;altu &longs;ur&longs;um totum cor­<lb/>pus attollere; </s> <s id="N273C0"><!-- NEW -->at cùm in omnibus partibus potentia motrix non pro­<lb/>ducat impetum immediatè; </s> <s id="N273C6"><!-- NEW -->certè deor&longs;um tendunt, motu naturaliter <lb/>accelerato, vnde tandem organum ip&longs;um deor&longs;um &longs;ecum trahunt; &longs;ed <lb/>de his aliàs plura, cum de potentia progre&longs;&longs;iua. </s> </p> <p id="N273CE" type="main"> <s id="N273D0"><!-- NEW -->Decimotertiò, quando pondus &longs;u&longs;tinetur à plano immobili, v. <!-- REMOVE S-->g. <!-- REMOVE S-->à <lb/>men&longs;a, non producitur in eo impetus &longs;ur&longs;um à men&longs;a; quia impetus <lb/>producitur tantùm ad extra ab alio impetu, per Th.42. l.1. &longs;ed nullus e&longs;t <lb/>impetus &longs;ur&longs;um in men&longs;a, vt patet. </s> </p> <p id="N273DE" type="main"> <s id="N273E0"><!-- NEW -->Decimoquartò, pondus non producit impetum in ip&longs;a men&longs;a, ni&longs;i vel <lb/>tota men&longs;a, vel aliquæ eius partes moueantur, vel comprimantur, vel <lb/>dilatentur; </s> <s id="N273E8"><!-- NEW -->quod reuera ferè &longs;emper accidit; </s> <s id="N273EC"><!-- NEW -->quia cum &longs;it perpetuum <lb/>corporum effluuium, multæ partes &longs;eparantur vi ponderis, quæ ab iis <lb/>corpu&longs;culis, quæ auolarunt, continebantur; </s> <s id="N273F4"><!-- NEW -->&longs;ic tandem po&longs;t multos an­<lb/>nos trabs lignea incubanti ponderi cedit; </s> <s id="N273FA"><!-- NEW -->&longs;ic lapis &longs;en&longs;im terram de­<lb/>primit, &longs;ic globus plumbeus diutiùs molæ incubans, &longs;ibi qua&longs;i fo&longs;&longs;ulam <lb/>fingit, depre&longs;&longs;is duntaxat mollioribus partibus; </s> <s id="N27402"><!-- NEW -->quod certè fit vel in-<pb pagenum="380" xlink:href="026/01/414.jpg"/>&longs;en&longs;ibili motu vel per &longs;eparationem aliquarum partium; </s> <s id="N2740B"><!-- NEW -->cum enim da­<lb/>to quocumque motu, dari po&longs;&longs;it tardior; certè pote&longs;t e&longs;&longs;e continuus <lb/>motus, quo per centum annos, vix latus vnguis acquiratur, quod nemo <lb/>Philo&longs;ophus mirabitur, qui naturam motus circularis probè intelle­<lb/>xerit. </s> </p> <p id="N27417" type="main"> <s id="N27419"><!-- NEW -->Decimoquintò, brachium omninò explicatum difficiliùs &longs;u&longs;tinet <lb/>pondus, quam contractum; </s> <s id="N2741F"><!-- NEW -->quia maius e&longs;t explicati momentum, vt pa­<lb/>tet; e&longs;t enim qua&longs;i longior vectis circa extremum humerum rotatus. </s> </p> <p id="N27425" type="main"> <s id="N27427">Obijceret aliquis, contra ea quæ diximus num. </s> <s id="N2742A"><!-- NEW -->14. &longs;it globulus libram <lb/>pendens incubans men&longs;æ 99. librarum; </s> <s id="N27430"><!-- NEW -->haud dubiè qui men&longs;am pon­<lb/>derat, centum librarum pondus &longs;u&longs;tinet; igitur globulus producit in <lb/>men&longs;a impetum. </s> <s id="N27438"><!-- NEW -->Re&longs;p. neg. <!-- REMOVE S--><expan abbr="con&longs;eq.">con&longs;eque</expan> nam ideò &longs;entitur pondus 100. li­<lb/>brarum; quia vtrumque pondus grauitatione communi in &longs;uppo&longs;itam <lb/>grauitat manum. </s> </p> <p id="N27445" type="main"> <s id="N27447"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N27454" type="main"> <s id="N27456"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena detentionis.<emph.end type="italics"/></s> </p> <p id="N2745D" type="main"> <s id="N2745F"><!-- NEW -->Primò, aliquis detinetur, &longs;imul, & &longs;u&longs;tinetur; </s> <s id="N27463"><!-- NEW -->&longs;it globum pendulum <lb/>fune, cuius altera extremitas manu tenetur immobili; </s> <s id="N27469"><!-- NEW -->nullus autem <lb/>producitur impetus in ip&longs;o globo, quo &longs;ur&longs;um, qua&longs;i attollatur; </s> <s id="N2746F"><!-- NEW -->quod <lb/>probatur, ii&longs;dem omninò rationibus, quibus probauimus in &longs;uperiori <lb/>Theo. <!-- REMOVE S-->de &longs;u&longs;tentatione; </s> <s id="N27479"><!-- NEW -->ip&longs;a tamen chorda, &longs;i vel brachio, vel digito cir­<lb/>cumuoluatur, &longs;ua vbique inurit ve&longs;tigia; </s> <s id="N2747F"><!-- NEW -->premit enim molliorem car­<lb/>nem, & neruos; huic aliqua diui&longs;io; hinc dolor: nec in hoc &longs;ingularis <lb/>e&longs;t difficultas. </s> </p> <p id="N27487" type="main"> <s id="N27489"><!-- NEW -->Secundò, retinetur aliquod mobile, per quamlibet lineam, vel fune, <lb/>vel vnco, vel manu, v.g. <!-- REMOVE S-->auolans auis filo, indomitus equus fræno, di&longs;ce­<lb/>dens homo pallio vel manu; </s> <s id="N27493"><!-- NEW -->hoc po&longs;ito, non producitur impetus à reti­<lb/>nente in mobili retento per &longs;e; </s> <s id="N27499"><!-- NEW -->quia perinde &longs;e habet, atque &longs;i rupes im­<lb/>mobilis retineret annulo ferreo, vel vnco; </s> <s id="N2749F"><!-- NEW -->&longs;ed rupes non producit im­<lb/>petum in eo corpore, quod retinet, dixi per &longs;e; </s> <s id="N274A5"><!-- NEW -->nam &longs;i partes aliquæ <lb/>&longs;eparari po&longs;&longs;int vel dilatari; haud dubiè producitur in iis impetus. </s> </p> <p id="N274AB" type="main"> <s id="N274AD"><!-- NEW -->Tertiò, hinc &longs;i duo retineant &longs;e &longs;e inuicem vel fune, vel annulo, vel <lb/>cylindro, multus impetus producitur ab vtroque in altero; </s> <s id="N274B3"><!-- NEW -->quippe ten­<lb/>duntur nerui & mu&longs;culi, ex qua ten&longs;ione multæ partes &longs;eparantur; </s> <s id="N274B9"><!-- NEW -->hinc <lb/>dolor & defatigatio; </s> <s id="N274BF"><!-- NEW -->igitur producitur impetus, quod certè clari&longs;&longs;imè <lb/>&longs;equitur ex no&longs;tris principiis; </s> <s id="N274C5"><!-- NEW -->cum enim potentia motrix alicui mobili <lb/>applicatur, quod &longs;imul totum mouere non pote&longs;t propter re&longs;i&longs;tentiam <lb/>vel ip&longs;ius molis, vel impetus contrarij; </s> <s id="N274CD"><!-- NEW -->&longs;i fortè aliqua pars amoueri po­<lb/>te&longs;t, & &longs;eparari ab aliis in eam potentia applicata &longs;uas vires exerit; quo­<lb/>modo verò rumpatur funis, vtrimque tractus, dicemus paulò pò&longs;t, cum <lb/>de tractione. </s> </p> <p id="N274D7" type="main"> <s id="N274D9"><!-- NEW -->Quartò, retinetur aliquod mobile immobiliter in plano decliui, id­<lb/>que duobus modus; primò, qua&longs;i trahendo: </s> <s id="N274DF"><!-- NEW -->&longs;ecundò, qua&longs;i pellendo, nul­<lb/>lus impetus producitur per &longs;e in mobili retento à retinente; </s> <s id="N274E5"><!-- NEW -->quod pro-<pb pagenum="381" xlink:href="026/01/415.jpg"/>batur eodem modo, quo &longs;uprà; </s> <s id="N274EE"><!-- NEW -->per accidens autem producitur propter <lb/><expan abbr="eãdem">eandem</expan> rationem vnde &longs;uprà; </s> <s id="N274F7"><!-- NEW -->&longs;uppono autem nullo modo vel trahi <lb/>&longs;ur&longs;um, vel pelli vtrimque: porrò retinetur ab æquali potentia, quod <lb/>iam alibi demon&longs;trauimus lib.5. in quo etiam fusè explicuimus diuer­<lb/>&longs;as lineas, quibus potentia applicari pote&longs;t. </s> </p> <p id="N27501" type="main"> <s id="N27503"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N27510" type="main"> <s id="N27512"><emph type="italics"/>Hinc facilè explicantur omnia phœnomena lationis.<emph.end type="italics"/></s> </p> <p id="N27519" type="main"> <s id="N2751B"><!-- NEW -->Primò, lationem appello illam impre&longs;&longs;ionem, qua potentia motrix <lb/>aliquid &longs;uo organo, mediatè vel immediatè coniunctum &longs;ecum vna de­<lb/>fert; </s> <s id="N27523"><!-- NEW -->&longs;ic dum quis ambulat, pileum etiam, quo caput tegitur, mouet; &longs;ic <lb/>equus rapit, nauis vehit nautam, currus aurigam defert. </s> </p> <p id="N27529" type="main"> <s id="N2752B"><!-- NEW -->Secundò, imprimitur impetus in vtroque; probatur facilè; quia <lb/>vtrumque mouetur cum eo tamen di&longs;crimine, quod lator in &longs;e producit <lb/>impetum, qui in mobili delato alium producit. </s> </p> <p id="N27533" type="main"> <s id="N27535"><!-- NEW -->Tertiò, impetus latoris æqualis e&longs;t impetui delato, quia vtrique ine&longs;t <lb/>æqualis motus; igitur æqualis impetus. </s> </p> <p id="N2753B" type="main"> <s id="N2753D"><!-- NEW -->Quartò, hinc cùm nauis imprimat impetum iis omnibus, quæ vehit <lb/>æqualem &longs;uo, non e&longs;t mirum &longs;i motus qui ob&longs;eruantur è naui mobili <lb/>tùm in proiectis, tùm in demi&longs;&longs;is, tùm in di&longs;per&longs;is, &longs;imiles omninò iis <lb/>appareant, qui ob&longs;eruantur è naui immobili, licèt omninò &longs;int di&longs;&longs;imi­<lb/>les; quæ omnia fusè explicui l.4. </s> </p> <p id="N27549" type="main"> <s id="N2754B"><!-- NEW -->Quintò, hinc quæ vehuntur naui non &longs;eparantur ab ip&longs;a naui, quia <lb/>æquali motu feruntur, ni&longs;i nauis illicò &longs;i&longs;tat; </s> <s id="N27551"><!-- NEW -->quia impetus prior, non &longs;ta­<lb/>tim de&longs;truitur, quod iam explicuimus alibi; </s> <s id="N27557"><!-- NEW -->immò &longs;e&longs;e aliquando &longs;ub­<lb/>trahit equiti; </s> <s id="N2755D"><!-- NEW -->quia, &longs;cilicet, demi&longs;&longs;o vel inflexo tantillùm dor&longs;o, perni­<lb/>citer &longs;e&longs;e eripit; </s> <s id="N27563"><!-- NEW -->idem accidit globo, quem in plano horizontali læui­<lb/>gato &longs;u&longs;tines; </s> <s id="N27569"><!-- NEW -->&longs;i enim illicò demittas orbem velociter ductum, vel &longs;ta­<lb/>tim ducas reduca&longs;que; haud dubiè globus in eo plano mouebitur. </s> </p> <p id="N2756F" type="main"> <s id="N27571"><!-- NEW -->Sextò, quædam humeris & collo, quædam capite, alia manu feruntur, <lb/>etiam liquida va&longs;e contenta; </s> <s id="N27577"><!-- NEW -->vas autem ip&longs;um effunditur, &longs;i motus ali­<lb/>qua notabili morula interrumpatur; </s> <s id="N2757D"><!-- NEW -->cùm enim &longs;uperficies aquæ v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->in eam partem adhuc moueatur, in quam priùs erat denominata; certe <lb/>&longs;i maior e&longs;t motus, effunditur aqua. </s> </p> <p id="N27588" type="main"> <s id="N2758A"><!-- NEW -->Septimò, hinc e&longs;t aliquod artificium, quo ita po&longs;&longs;int in plano hori­<lb/>zontali verticali manubrio in&longs;tructo deferri orbes pleni liquore, vt ni­<lb/>hil penitus effundatur; </s> <s id="N27592"><!-- NEW -->&longs;i enim ita temperetur brachij motus, vt &longs;it con­<lb/>tinuus & æquabilis, non modò nihil effundetur; </s> <s id="N27598"><!-- NEW -->verùm etiam, ne ip&longs;a <lb/>quidem &longs;uperficies liquoris mutabitur; </s> <s id="N2759E"><!-- NEW -->vt autem &longs;it continuus ille bra­<lb/>chij motus, & æquabilis; </s> <s id="N275A4"><!-- NEW -->debet ita porrigi brachium, &longs;eu componi cum <lb/>inæquali reliqui corporis motu, vt eo aliquando tardior, aliquando ve­<lb/>locior &longs;it; porrò hæc inæqualitas motus progre&longs;&longs;iui procedit ex duplici <lb/>illo qua&longs;i gemini crucis arcu, geminoque vtriu&longs;que centro, &longs;ed de hoc <lb/>alibi. </s> </p> <p id="N275B0" type="main"> <s id="N275B2"><!-- NEW -->Octauò, hinc quò velociùs corpus progredietur minoribu&longs;que, licèt, <pb pagenum="382" xlink:href="026/01/416.jpg"/>frequentioribus pa&longs;&longs;ibus, brachij motus accedit propiùs ad æquabilem; </s> <s id="N275BB"><!-- NEW --><lb/>igitur minùs mutatur &longs;uperficies liquoris va&longs;e contenti; </s> <s id="N275C0"><!-- NEW -->hinc in naui, <lb/>quæ veloci&longs;&longs;imo motu fertur, ne tremit quidem &longs;uperficies aquæ, quam <lb/>repo&longs;itam quis habet in va&longs;e; denique quò &longs;uperficies concaua orbis <lb/>&longs;eu va&longs;is e&longs;t maioris circuli faciliùs effunditur liquor, quia planum e&longs;t <lb/>minus decliue, & minus recedit ab horizontali, & contrà &longs;i e&longs;t minoris <lb/>&longs;phæræ &longs;eu circuli, hinc fortè tantus e&longs;t maris æ&longs;tus in Oceano, & mo­<lb/>dicus valdè in Mediterraneo, &longs;ed de his alibi. </s> </p> <p id="N275D0" type="main"> <s id="N275D2">Nonò, his adde amphoras illas aqua, vel lacte ad &longs;ummum v&longs;que <lb/>marginem repletas, quas ru&longs;ticanæ fœminæ è &longs;ummo capite ita portant, <lb/>vt nihil penitus effundatur, quia &longs;cilicet ten&longs;o collo ambulant, vt capi­<lb/>tis motus ad æquabilem propius accedat. </s> </p> <p id="N275DB" type="main"> <s id="N275DD"><!-- NEW -->Decimò, non e&longs;t omittendum ille orbis gyrus cum &longs;cypho pleno; <lb/>quod vt melius intelligatur. </s> <s id="N275E3"><!-- NEW -->Sit orbis AFEG pendulus filo FA; </s> <s id="N275E7"><!-- NEW -->&longs;it <lb/>&longs;cyphus EDC plenus aqua vel alio liquore, puncto circuli E in&longs;idens, <lb/>tùm rotetur orbis circa centrum F; </s> <s id="N275EF"><!-- NEW -->haud dubiè, ne gutta quidem aquæ <lb/>effundetur; </s> <s id="N275F5"><!-- NEW -->ratio e&longs;t, cùm E &longs;it &longs;emper punctum oppo&longs;itum centro, mo­<lb/>tus F & &longs;cyphus motu illo circulari maximè pellatur, prematurque ver­<lb/>&longs;us E, aqua ip&longs;a etiam ver&longs;us E recipit impetum ver&longs;us fundum &longs;cyphi; </s> <s id="N275FD"><!-- NEW --><lb/>qui cùm &longs;it inten&longs;ior natiuo propriæ grauitationis aquæ, non e&longs;t mirum <lb/>&longs;i præualeat, & nihil penitus effundatur in gyro, præ&longs;ertim cùm partes <lb/>omnes aquæ moueantur eo motu, quo in primo &longs;itu omninò relinquun­<lb/>tur; </s> <s id="N27608"><!-- NEW -->adde quod licèt impetus innatus tantillùm obe&longs;&longs;et, impeditur ta­<lb/>men ab illa vligine, quæ cum aqua commixta e&longs;t, de qua iam &longs;uprà; </s> <s id="N2760E"><!-- NEW --><lb/>quod autem &longs;cyphus impellatur ver&longs;us E, patet clari&longs;&longs;imè in funda, in <lb/>qua lapis circumagitur, &longs;ed de funda infrà, cum de proiectione; tunc <lb/>enim rem i&longs;tam demon&longs;trabimus. </s> </p> <p id="N27617" type="main"> <s id="N27619"><!-- NEW -->Vndecimò, vt feratur cylindrus humeris commodiùs in &longs;itu e&longs;&longs;e de­<lb/><arrow.to.target n="note3"/><lb/>bet, vt &longs;uprà horizontalem eleuetur ad angulum 45. grad. <!-- REMOVE S-->&longs;it enim 60. <lb/>grad &longs;itque cylindrus AF, cuius centrum grauitatis C incubans puncto <lb/>humeri C, tunc humerus &longs;u&longs;tinet totum pondus ab&longs;olutum cylindri, <lb/>& manus nihil: </s> <s id="N2762B"><!-- NEW -->&longs;i verò manu erectum &longs;u&longs;tineatur in DG; haud du­<lb/>biè manus totum &longs;u&longs;tinet pondus ab&longs;olutum, humerus nihil, &longs;i &longs;u&longs;ti­<lb/>neatur KCI in C, vel NCL in C, maius pondus &longs;u&longs;tinebitur propter <lb/>rationem vectis de quo in lib. &longs;equenti. </s> <s id="N27635"><!-- NEW -->Denique, &longs;i &longs;u&longs;tineatur in HCE <lb/>ad angulum HCA, 60. grad. <!-- REMOVE S-->humerus &longs;u&longs;tinet vt BH, manus vt EI; </s> <s id="N2763D"><!-- NEW --><lb/>ergo non di&longs;tribuitur pondus æqualiter humero & manui; igitur com­<lb/>modiùs fieri pote&longs;t, &longs;i æqualiter di&longs;tribuitur, quod vt fiat debet e&longs;&longs;e ad <lb/>eleuationem anguli 45. &longs;ed hæc pertinent ad libram, & vectem de quibus <lb/>agemus infrà, etiam &longs;upra lib.5. &longs;æpiùs indicauimus. </s> </p> <p id="N27648" type="margin"> <s id="N2764A"><margin.target id="note3"/>b <emph type="italics"/>Fig.<emph.end type="italics"/>28 <lb/><emph type="italics"/>Tab.<emph.end type="italics"/>4.</s> </p> <p id="N2765C" type="main"> <s id="N2765E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2766B" type="main"> <s id="N2766D"><emph type="italics"/>Aliquod mobile graue dimittitur deor&longs;um multis modis.<emph.end type="italics"/></s> </p> <p id="N27674" type="main"> <s id="N27676">Primò, per lineam perpendicularem, & tunc e&longs;t motus purè natura­<lb/>lis, &longs;imulque omnes partes mobilis dimittuntur. </s> </p> <pb pagenum="383" xlink:href="026/01/417.jpg"/> <p id="N2767F" type="main"> <s id="N27681">Secundò, per planum inclinatum tuncque &longs;i globus e&longs;t, rotatur, quia <lb/>tollitur æquilibrium. </s> </p> <p id="N27686" type="main"> <s id="N27688">Tertiò, ita dimittitur globus, vt primò per manum qua&longs;i decliuem ca­<lb/>dat, tuncque &longs;imiliter rotatur propter <expan abbr="eãdem">eandem</expan> rationem. </s> </p> <p id="N27691" type="main"> <s id="N27693">Quartò, dimittitur funependulum, & tunc de&longs;cendit per arcum. </s> </p> <p id="N27696" type="main"> <s id="N27698">Quintò, dimittitur cylindrus, cuius altera extremitas nititur &longs;olo, & <lb/>tunc de&longs;cendit etiam per arcum. </s> </p> <p id="N2769D" type="main"> <s id="N2769F">Sextò, dimittitur baculus; </s> <s id="N276A2"><!-- NEW -->&longs;ed inæqualiter, ita vt altera eius extremitas <lb/>cadat, antequam alia dimittatur, & tunc etiam circumagitur baculus; &longs;ed <lb/>hæc &longs;unt facilis. </s> </p> <p id="N276AA" type="main"> <s id="N276AC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N276B9" type="main"> <s id="N276BB"><emph type="italics"/>Aliquotâ mobile proiectum excipitur manu multis modis.<emph.end type="italics"/></s> </p> <p id="N276C2" type="main"> <s id="N276C4">Primò, firma & fixa manu, in quam cadit eodem modo, quo caderet <lb/>in parietem, vt patet. </s> </p> <p id="N276C9" type="main"> <s id="N276CB">Secundò, manu repellente, tunque e&longs;t maior ictus. </s> </p> <p id="N276CE" type="main"> <s id="N276D0"><!-- NEW -->Tertiò, manu &longs;en&longs;im &longs;ub&longs;idente, vt fallat ictum; </s> <s id="N276D4"><!-- NEW -->&longs;ic lapidem &longs;ur&longs;um <lb/>proiectum cadentem ita excipimus manu, immò & maiorem globum, <lb/>vt vix vllum ictum &longs;entiamus; </s> <s id="N276DC"><!-- NEW -->quod vt fiat, manus retroagi debet, non <lb/>quidem pari velocitate cum globo, &longs;ed paulò tardiore motu, vt &longs;cilicet <lb/>modicum impetum imprimat globus; </s> <s id="N276E4"><!-- NEW -->&longs;i enim manus pari velocitate <lb/>moueretur, nullum pror&longs;us impetum imprimeret globus; </s> <s id="N276EA"><!-- NEW -->&longs;i verò non <lb/>moueretur, &longs;ed omninò manus quie&longs;ceret, maximum ictum exceptus <lb/>globus infligeret; </s> <s id="N276F2"><!-- NEW -->&longs;i verò moueatur &longs;ed paulò tardius aliquid impetus <lb/>imprimetur &longs;ingulis in&longs;tantibus, donec tandem totus ictus extingua­<lb/>tur; </s> <s id="N276FA"><!-- NEW -->adde quod mollities manus ad extinguendum ictum poti&longs;&longs;imum <lb/>confert; analogiam habes in lana, quæ tormentorum vim penitus <lb/>eneruat. </s> </p> <p id="N27702" type="main"> <s id="N27704">Quartò, vt longiùs repellatur pila, &longs;ecundus modus adhiberi debet <lb/>eritque motus mixtus ex directo & reflexo. </s> </p> <p id="N27709" type="main"> <s id="N2770B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N27718" type="main"> <s id="N2771A"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena tractionis.<emph.end type="italics"/></s> </p> <p id="N27721" type="main"> <s id="N27723"><!-- NEW -->Primò, trahitur mobile per productionem impetus; </s> <s id="N27727"><!-- NEW -->nec enim po­<lb/>tentia motrix, quæ reuerâ cau&longs;a e&longs;t tractionis, quidquam aliud produce­<lb/>re pote&longs;t; </s> <s id="N2772F"><!-- NEW -->præterea quod trahitur, verè mouetur; igitur per impetum, <lb/>&longs;ic differt tractio à mera detentione, de qua &longs;uprà. </s> </p> <p id="N27735" type="main"> <s id="N27737"><!-- NEW -->Secundò, hinc tractio e&longs;t actio potentiæ motricis, qua mobile ip&longs;um <lb/>propiùs accedit ad motorem; </s> <s id="N2773D"><!-- NEW -->nam motor ad &longs;e trahit mobile; </s> <s id="N27741"><!-- NEW -->igitur <lb/>mobile accedit ad motorem: </s> <s id="N27747"><!-- NEW -->quod tantùm dictum &longs;it de tractione di­<lb/>recta; nam per reflexam, ip&longs;e motor ad mobile accedit, de qua <lb/>infrà. </s> </p> <p id="N2774F" type="main"> <s id="N27751"><!-- NEW -->Tertiò, quando trahitur aliquod mobile, impetus producitur in om­<lb/>nibus illius partibus; </s> <s id="N27757"><!-- NEW -->probatur, quia omnes mouentur; igitur omnes <lb/>recipiunt impetum. </s> <s id="N2775D">Secundò, quia &longs;i tantùm in vna produci impetum <pb pagenum="384" xlink:href="026/01/418.jpg"/>oporteret, vt reliquæ etiam mouerentur à quacumque potentia quodli­<lb/>bet mobile trahi po&longs;&longs;et, quod e&longs;t ab&longs;urdum. </s> </p> <p id="N27767" type="main"> <s id="N27769">Dices, alias partes re&longs;i&longs;tere. </s> <s id="N2776C"><!-- NEW -->Re&longs;p. igitur vt moueantur, &longs;uperari debet <lb/>illarum re&longs;i&longs;tentia; </s> <s id="N27772"><!-- NEW -->igitur per aliquid de nouo proctum; </s> <s id="N27776"><!-- NEW -->igitur per <lb/>impetum: </s> <s id="N2777C"><!-- NEW -->immò non producitur in vna, ni&longs;i producatur in aliis; </s> <s id="N27780"><!-- NEW --><lb/>alioquin fru&longs;trà e&longs;&longs;et ille impetus, cui nullus effectus re&longs;pon­<lb/>deret; </s> <s id="N27787"><!-- NEW -->igitur &longs;i de&longs;truitur, quando fru&longs;trà e&longs;&longs;et, &longs;i con&longs;eruaretur; </s> <s id="N2778B"><!-- NEW -->ita <lb/>etiam non producitur quando fru&longs;trà e&longs;&longs;et, &longs;i produceretur; e&longs;t enim <lb/>par vtrimque ratio. </s> </p> <p id="N27793" type="main"> <s id="N27795"><!-- NEW -->Quartò, hinc licèt trahatur ingens rupes, non propterea mouetur, <lb/>quia non pote&longs;t impetus produci in omnibus illius partibus ab applica­<lb/>ta potentia; igitr in nulla per Th.33.l.1. </s> </p> <p id="N2779D" type="main"> <s id="N2779F">Dices, e&longs;t cau&longs;a nece&longs;&longs;aria applicata. </s> <s id="N277A2">Re&longs;p. e&longs;&longs;e quidem applicatam, &longs;ed <lb/>e&longs;&longs;e impeditam propter maximam rupis re&longs;i&longs;tentiam, quam debiliores <lb/>potentiæ vires &longs;uperare non po&longs;&longs;unt. </s> </p> <p id="N277A9" type="main"> <s id="N277AB"><!-- NEW -->Quintò, hinc vna pars tracta non &longs;equitur aliam vltrò; </s> <s id="N277AF"><!-- NEW -->&longs;i enim vltrò <lb/>&longs;equeretur minima potentia, &longs;ufficeret ad trahendum maximum pondus; <lb/>præterea &longs;ingulæ partes mouentur per impetum. </s> </p> <p id="N277B7" type="main"> <s id="N277B9">Diceret aliquis, impetus productus in vna parte producit impetum <lb/>in alia. </s> <s id="N277BE">Re&longs;p. negando; alioquin minima potentia quodlibet pondus <lb/>moueret contra experientiam. </s> </p> <p id="N277C3" type="main"> <s id="N277C5"><!-- NEW -->Dices, impetus vnius corporis producit impetum in alio, à quo eius <lb/>motus impeditur; igitur impetus vnius partis producit impetum in <lb/>alia, à qua eius motus impeditur. </s> <s id="N277CD"><!-- NEW -->Re&longs;p. impetum, qui reuerâ alicui <lb/>corpori ine&longs;t, hoc ip&longs;um præ&longs;tare; </s> <s id="N277D3"><!-- NEW -->at impetus non producitur in vna <lb/>parte mobilis, ni&longs;i &longs;imul in aliis producatur; </s> <s id="N277D9"><!-- NEW -->vel enim producitur in <lb/>omnibus, vel in nulla; </s> <s id="N277DF"><!-- NEW -->hinc colliges quantum ab&longs;urdum &longs;equeretur, <lb/>ni&longs;i hoc e&longs;&longs;et; </s> <s id="N277E5"><!-- NEW -->quia perpetua e&longs;&longs;et impetus productio, & minimus im­<lb/>petus totam ip&longs;am terram moueret; </s> <s id="N277EB"><!-- NEW -->vide quæ diximus &longs;uper ea re toto <lb/>lib.1. nec enim totus impetus motoris producit totum &longs;uum effectum <lb/>in vnico puncto mobilis, quod ridiculum dictu e&longs;t; </s> <s id="N277F3"><!-- NEW -->alioquin produ­<lb/>ceretur impetus inten&longs;i&longs;&longs;imus; </s> <s id="N277F9"><!-- NEW -->igitur in pluribus; igitur in omnibus, <lb/>quæ &longs;imul moueri debent, vel in multa. </s> </p> <p id="N277FF" type="main"> <s id="N27801">Diceret aliquis; </s> <s id="N27804"><!-- NEW -->quando mouetur corpus equi, mouetur etiam ani­<lb/>ma; </s> <s id="N2780A"><!-- NEW -->igitur &longs;ine impetu; </s> <s id="N2780E"><!-- NEW -->igitur per impetum corporis; </s> <s id="N27812"><!-- NEW -->igitur nomine <lb/>tantùm vnionis; </s> <s id="N27818"><!-- NEW -->igitur pars corporis alteri vnita etiam &longs;ine impetu, <lb/>&longs;cilicet per impetum alterius moueri pote&longs;t: hanc difficultatem iam <lb/>&longs;oluimus &longs;uprà l.1.Th.38.Cor.12. </s> </p> <p id="N27820" type="main"> <s id="N27822"><!-- NEW -->Sextò, producitur impetus æqualis in omnibus partibus, quod trahi­<lb/>tur motu recto; </s> <s id="N27828"><!-- NEW -->quia &longs;cilicet motus e&longs;t æqualis; igitur & impetus. </s> </p> <p id="N2782C" type="main"> <s id="N2782E">Septimò, funis trahi pote&longs;t diuer&longs;imodè. </s> <s id="N27831"><!-- NEW -->Primò, &longs;i altera eius extre­<lb/>mitas annulo, &longs;eu clauo immobili affixa &longs;it; </s> <s id="N27837"><!-- NEW -->alteri verò applicetur po­<lb/>tentia, vel pondus; &longs;iue &longs;it in &longs;itu horizontali, &longs;iue in verticali. </s> <s id="N2783D">Secundò, <lb/>&longs;i vtrique extremitati applicetur pondus vel alia potentia motrix. </s> <s id="N27842"><!-- NEW --><lb/>Tertiò, &longs;i vtraque extremitas clauo immobiliter affigatur in &longs;itu hori-<pb pagenum="385" xlink:href="026/01/419.jpg"/>zontali, admoueaturque pondus, &longs;eu potentia alicui chordæ puncto <lb/>deor&longs;um trahens: </s> <s id="N2784E"><!-- NEW -->denique &longs;i ponticulo maximè attollatur, & tendatur <lb/>chorda po&longs;ita in priori &longs;itu; </s> <s id="N27854"><!-- NEW -->&longs;i primò, rumpetur chorda per &longs;e in ea ex­<lb/>tremitate, quæ immobiliter clauo affigitur; </s> <s id="N2785A"><!-- NEW -->&longs;i tertio & quarto in ea <lb/>parte, in qua vel deprimitur, vel attollitur: dixi per &longs;e, quia per acci­<lb/>dens &longs;ecus accidit, vt reuerâ &longs;æpè fit, vel propter inflexionem nodi, vel <lb/>aliquas partes debiliores, vel pre&longs;&longs;ionem maiorem cum ten&longs;ione con­<lb/>iunctam &c. </s> <s id="N27866"><!-- NEW -->&longs;ed quia hæc phœnomena pertinent partim ad ten&longs;ionem, <lb/>& compre&longs;&longs;ionem, partim ad re&longs;i&longs;tentiam corporum, de quibus agemus <lb/>Tomo &longs;equenti; </s> <s id="N2786E"><!-- NEW -->certè hoc loco demon&longs;trari non po&longs;&longs;unt; </s> <s id="N27872"><!-- NEW -->igitur &longs;atis <lb/>e&longs;t modò indica&longs;&longs;e huius demon&longs;trationis locum, qui talis e&longs;t: inter il­<lb/>las duas partes fieri debet diui&longs;io chordæ, quarum vna reuerâ trahitur, <lb/>alia verò non mouetur, vel quarum vtraque mouetur &longs;ed in partes op­<lb/>po&longs;itas, quod nemo negabit. </s> <s id="N2787E"><!-- NEW -->Et hoc principio hæc omnia, demon&longs;trari <lb/>po&longs;&longs;unt; </s> <s id="N27884"><!-- NEW -->&longs;ed de his omnibus &longs;uo loco fusè agemus; hæc enim vberri­<lb/>mam demon&longs;trationum &longs;egetem dabunt, præ&longs;ertim &longs;i comparentur inter <lb/>&longs;e omnes chordarum affectiones, v.g. <!-- REMOVE S-->materia, figura, pondus, longitudo, <lb/>cra&longs;&longs;ities, &longs;itus, diuer&longs;a potentiæ applicatio. </s> </p> <p id="N27890" type="main"> <s id="N27892"><!-- NEW -->Octauò, quando corpus trahitur fune, quò funis e&longs;t longior per &longs;e, <lb/>difficiliùs trahitur; </s> <s id="N27898"><!-- NEW -->ratio e&longs;t, quia funis tantæ longitudinis e&longs;&longs;e pote&longs;t, <lb/>vt ne ip&longs;e quidem &longs;ine pondere trahi po&longs;&longs;it; </s> <s id="N2789E"><!-- NEW -->igitur quâ proportione <lb/>erit breuior dum applicari po&longs;&longs;it potentia, faciliùs trahet, dixi per &longs;e; </s> <s id="N278A4"><!-- NEW --><lb/>quia funis longior, cuius plures partes &longs;unt, maiorem patitur ten&longs;io­<lb/>nem; </s> <s id="N278AB"><!-- NEW -->hinc vt partes &longs;e&longs;e reducant corpus ip&longs;um adducunt; </s> <s id="N278AF"><!-- NEW -->adde quod, <lb/>quò aliquod corpus magis tenditur, maioris impetus e&longs;t capax, quia <lb/>priori remanenti qui non e&longs;t fru&longs;trà, quia &longs;uum effectum habet, &longs;ecun­<lb/>dus accedit à &longs;ecundo ni&longs;u, igitur, quando dico corpus trahi faciliùs <lb/>breuiori fine, nullam habeo rationem ten&longs;ionis; quæ certè facere po­<lb/>te&longs;t, dum funis non &longs;it tantæ longitudinis, vt corpus faciliùs trahatur <lb/>propter illa duo capita, quæ indicauimus. </s> </p> <p id="N278BF" type="main"> <s id="N278C1">Nonò, hinc vno fune faciliùs trahitur corpus, quàm duobus. </s> <s id="N278C4"><!-- NEW -->Primò, <lb/>quia pluribus partibus funis di&longs;tribuitur impetus; </s> <s id="N278CA"><!-- NEW -->igitur eò minus &longs;in­<lb/>gulæ habent, quò plures &longs;unt; &longs;ecundò, quia cum vnus e&longs;t funis, e&longs;t <lb/>maior ten&longs;io, quæ iuuat corporis tracti motum. </s> <s id="N278D2"><!-- NEW -->Tertiò, quia &longs;i &longs;unt duo <lb/>funis vel diuer&longs;is partibus corporis tracti affliguntur, vel vni, &longs;i pri­<lb/>mum; </s> <s id="N278DA"><!-- NEW -->igitur &longs;unt duæ lineæ directionis, ex quibus fit altera mixta; </s> <s id="N278DE"><!-- NEW --><lb/>&longs;ed nunquam mi&longs;centur duæ determinationes &longs;ine aliqua iactura, quan­<lb/>do e&longs;t duplex impetus, vt fusè &longs;atis demon&longs;tratum e&longs;t &longs;uprà, &longs;i &longs;ecun­<lb/>dum etiam &longs;unt duæ, vt patet; </s> <s id="N278E7"><!-- NEW -->igitur eadem valet ratio; </s> <s id="N278EB"><!-- NEW -->cum verò &longs;unt <lb/>plures funes, minùs impetus &longs;ingulis di&longs;tribuitur; hinc plura fila te­<lb/>nui&longs;&longs;ima &longs;u&longs;tinere po&longs;&longs;unt ingens pondus. </s> </p> <p id="N278F3" type="main"> <s id="N278F5"><!-- NEW -->Decimò, hinc facilè colligi pote&longs;t, quid dicendum &longs;it de pluribus equis <lb/>trahentibus currum; </s> <s id="N278FB"><!-- NEW -->qui certè ad currum iungi non po&longs;&longs;unt, ni&longs;i &longs;int <lb/>plures funes, qui tamen in communem &longs;eu funem &longs;eu temonem de&longs;i­<lb/>nunt; &longs;it autem pondus A, linea directionis GE. </s> <s id="N27904">Si &longs;it tantùm vnus <pb pagenum="386" xlink:href="026/01/420.jpg"/>equus, vel trahet duobus funibus BECE, vel vnico GE, addito axe <lb/>DF, & duobus funibus DHFH. </s> <s id="N2790E"><!-- NEW -->Hoc &longs;ecundo modo faciliùs trahet; <lb/>quia impetus meliùs deriuatur in pondus A per lineam EG, quæ per <lb/>centrum grauitatis ducitur. </s> </p> <p id="N27916" type="main"> <s id="N27918"><!-- NEW -->Ob&longs;eruabis autem, &longs;i cylindrus quo trahitur quodlibet pondus per <lb/>lineam AB; </s> <s id="N2791E"><!-- NEW -->trahatur per duas CFDF, tùm æqualibus viribus per duas <lb/>CHGD, haud dubiè hoc &longs;ecundo modo faciliùs trahetur, vt con&longs;tat, <lb/>& faciliùs per duas CFDF, quàm per duas CEDE; </s> <s id="N27926"><!-- NEW -->&longs;uppono autem <lb/>ita trahi CF, vt æqualiter trahatur per DF; </s> <s id="N2792C"><!-- NEW -->alioqui axis volueretur <lb/>circa B, in quo non e&longs;t difficultas: </s> <s id="N27932"><!-- NEW -->hoc po&longs;ito, dico po&longs;&longs;e a&longs;&longs;ignari dif­<lb/>ferentiam i&longs;torum motuum; </s> <s id="N27938"><!-- NEW -->a&longs;&longs;umatur enim punctum D, quod trahi­<lb/>tur per DF & per DI parallelam CF æqualiter vtrimque; </s> <s id="N2793E"><!-- NEW -->certè mo­<lb/>uebitur per DGL; &longs;i autem trahatur CD per duas CHDG æqualibus <lb/>viribus ab eadem potentia faciliùs trahetur iuxta rationem DF ad DG, <lb/>vel DFL ad DE, vt con&longs;tat ex dictis l. <!-- REMOVE S-->4. de motu mixto tùm etiam l.1. </s> </p> <p id="N2794A" type="main"> <s id="N2794C"><!-- NEW -->Vndecimò, &longs;i autem iungantur duo equi ad trahendum pondus A <lb/>axe DF, & fune EG; </s> <s id="N27952"><!-- NEW -->&longs;i æqualiter trahant, quod tamen vix accidere po­<lb/>te&longs;t, licèt differentia &longs;it pror&longs;us in&longs;en&longs;ibilis; </s> <s id="N27958"><!-- NEW -->&longs;i autem inæqualiter tra­<lb/>hant, perit aliquid impetus vtriu&longs;que, vt patet; </s> <s id="N2795E"><!-- NEW -->nam eo tempore, quo <lb/>D, cui maior vis ine&longs;t v.g. <!-- REMOVE S-->progreditur, F regreditur; </s> <s id="N27966"><!-- NEW -->igitur meo iudi­<lb/>cio, ne pereat quidquam impetus, ita debent collocari equi, vt pondus <lb/>&longs;it A, funis communis BC, primus axis DE, primus equus F trahens <lb/>funibus FDFE, tùm &longs;ecundus axis GH coniunctus cum primo funibus <lb/>GDHE, &longs;ecundus equus I trahens funibus IG, IH, atque ita deinceps: </s> <s id="N27972"><!-- NEW --><lb/>hoc po&longs;ito totus impetus productus à primo equo F <expan abbr="cõmunicatur">communicatur</expan> primo <lb/>axi DE; </s> <s id="N2797D"><!-- NEW -->præterea totus impetus productus à &longs;ecundo equo I communi­<lb/>catur &longs;ecundo axi GH, & ex hoc primo DE; </s> <s id="N27983"><!-- NEW -->igitur DE recipit totum im­<lb/>petum ab vtroque equo productum; </s> <s id="N27989"><!-- NEW -->qui certè inten&longs;i&longs;&longs;imus e&longs;&longs;et, ni&longs;i axis <lb/>DE coniunctus e&longs;&longs;et cum pondere A; </s> <s id="N2798F"><!-- NEW -->igitur totus impetus ab vtroque <lb/>equo productus toti ponderi di&longs;tribuitur, ni&longs;i fortè maius &longs;it <expan abbr="põdus">pondus</expan>; </s> <s id="N27999"><!-- NEW -->tunc <lb/>enim tertius equus M accedere deberet; igitur nihil pror&longs;us perit impetus. </s> </p> <p id="N2799F" type="main"> <s id="N279A1"><!-- NEW -->Duodecimò, vterque equus producit impetum in pondere A actione <lb/>communi; probatur, quia, &longs;i qui&longs;que &longs;ingularem impetum produceret, <lb/>qui toti ponderi di&longs;tribui non po&longs;&longs;et, cur potiùs his partibus quam aliis? </s> <s id="N279A9"><lb/>igitur cùm omnibus di&longs;tribuatur; </s> <s id="N279AD"><!-- NEW -->certè ab vtroque &longs;imul producitur; </s> <s id="N279B1"><!-- NEW --><lb/>nec enim alter equus trahit tantùm alteram partem ponderis; </s> <s id="N279B6"><!-- NEW -->quæ enim <lb/>a&longs;&longs;ignari pote&longs;t, &longs;ed &longs;inguli totum pondus, &longs;ed coniunctim, id e&longs;t quæli­<lb/>bet pars ponderis ab vtroque trahitur, &longs;ed non &longs;ola, totum pondus ab <lb/>altero trahitur, &longs;ed non &longs;olo; </s> <s id="N279C0"><!-- NEW -->equidem equus F non producit impetum in <lb/>funibus DGI, nec in axe GH, nec equus I in funibus DFE, quia nullo <lb/>modo impediunt motum, vnde equus I, vt æqualiter cum æquo F trahat <lb/>pondus A, debet paulò maiore ni&longs;u trahere; qui certè determinari pote&longs;t; </s> <s id="N279CA"><!-- NEW --><lb/>&longs;uppono enim primò vtrumque F, I totis viribus eniti: </s> <s id="N279CF"><!-- NEW -->&longs;ecundò equum I <lb/>non minùs conferre ad motum ponderis A, quàm equum F 3. funes DG, <lb/>EH & axem GH e&longs;&longs;e (1/1000) ponderis A; certè hoc po&longs;ito equus I e&longs;t fortior <lb/>equo F (1/1000). </s> </p> <pb pagenum="387" xlink:href="026/01/421.jpg"/> <p id="N279DD" type="main"> <s id="N279DF"><!-- NEW -->Decimotertiò, currus initio difficiliùs trahitur; </s> <s id="N279E3"><!-- NEW -->ratio e&longs;t, quia nullus <lb/>impetus ine&longs;t initio, qui vbi &longs;emel productus primo in&longs;tanti; </s> <s id="N279E9"><!-- NEW -->nec totus <lb/>de&longs;truatur &longs;ecundo; </s> <s id="N279EF"><!-- NEW -->nec enim totus fru&longs;trà e&longs;t; </s> <s id="N279F3"><!-- NEW -->habet enim aliquem effe­<lb/>ctum, id e&longs;t motum; </s> <s id="N279F9"><!-- NEW -->augetur per acce&longs;&longs;ionem noui impetus &longs;ecundo in­<lb/>&longs;tanti producti; idem dico de tertio, quarto, quinto, &c. </s> <s id="N279FF"><!-- NEW -->donec tandem <lb/>po&longs;t aliquod tempus motu <expan abbr="æ&qacute;uabili">æquabili</expan> procedat currus; </s> <s id="N27A09"><!-- NEW -->quia &longs;cilicet quan­<lb/>tum de&longs;truitur &longs;ingulis in&longs;tantibus, <expan abbr="tantũdem">tantundem</expan> ferè producitur, &longs;ed mi­<lb/>nùs profectò, quàm initio; igitur faciliùs; </s> <s id="N27A15"><!-- NEW -->igitur initio difficiliùs; </s> <s id="N27A19"><!-- NEW -->hinc <lb/>equi totis neruis enituntur initio, præ&longs;ertim in plano arduo; </s> <s id="N27A1F"><!-- NEW -->at vbi cur­<lb/>rus primum impetum accepit, longè faciliùs deinde propagatur; </s> <s id="N27A25"><!-- NEW -->hinc &longs;i <lb/>rumpatur funis, quo trahitur currus præcipiti equorum cur&longs;u, currus <lb/>ip&longs;e deinde per aliquod tempus adhuc rotatur; </s> <s id="N27A2D"><!-- NEW -->igitur prior impetus du­<lb/>rat adhuc; nec enim nouus producitur. </s> </p> <p id="N27A33" type="main"> <s id="N27A35"><!-- NEW -->Decimoquartò, &longs;i dum quis trahit toto ni&longs;u magnum aliquod pondus, <lb/>funis rumpatur, pronus corruit; </s> <s id="N27A3B"><!-- NEW -->ratio e&longs;t, quia totum <expan abbr="impetũ">impetum</expan> in &longs;e produ­<lb/>cit, quem in &longs;e &longs;imul & pondere integro fune &longs;eruato produxi&longs;&longs;et; </s> <s id="N27A45"><!-- NEW -->hinc <lb/>dum duo in partes aduer&longs;as cylindrum, vel funem trahunt, &longs;i dimittat <lb/>vnus &longs;upinus, alter proruit; </s> <s id="N27A4D"><!-- NEW -->quæ omnia ex no&longs;tris principijs luce clariora <lb/>redduntur; </s> <s id="N27A53"><!-- NEW -->non e&longs;t tamen, quod aliquis exi&longs;timet huius phœnomeni ra­<lb/>tionem tantùm à priori impetu con&longs;eruato e&longs;&longs;e; </s> <s id="N27A59"><!-- NEW -->qui certè minor erat in <lb/>trahente, quàm vt hunc effectum præ&longs;tare po&longs;&longs;it, cùm toti ponderi di­<lb/>&longs;tribuatur; igitur poti&longs;&longs;ima ratio duci debet ab impetu nouo producto, <lb/>quî cùm in auul&longs;um pondus tran&longs;ire non po&longs;&longs;it, totus in ip&longs;o trahente <lb/>qua&longs;i &longs;ub&longs;i&longs;tit. </s> </p> <p id="N27A65" type="main"> <s id="N27A67"><!-- NEW -->Decimoquintò, vt quis fortius trahat firmo pede, & crure intento, &longs;o­<lb/>lum ip&longs;um aduer&longs;o ni&longs;u premit; </s> <s id="N27A6D"><!-- NEW -->ratio in promptu e&longs;t, quia dum manu <lb/>trahit corporis truncum lumborum vi, & o&longs;&longs;ium contractorum explica­<lb/>tione &longs;ur&longs;um attollit; </s> <s id="N27A75"><!-- NEW -->igitur nouus impetus ponderi tracto accedit; </s> <s id="N27A79"><!-- NEW -->hinc <lb/>pede, vel genu in partem aduer&longs;am contranititur, qui trahit; </s> <s id="N27A7F"><!-- NEW -->nam que­<lb/>madmodum gemino brachio fortiùs trahimus, quàm vno; ita pror&longs;us, <lb/>cum brachiorum vis iuuatur à lumbis, cruribus, &c. </s> <s id="N27A87">haud dubiè vali­<lb/>dior e&longs;t. </s> </p> <p id="N27A8C" type="main"> <s id="N27A8E"><!-- NEW -->Decimo&longs;extò, cum faciliùs amoueri pote&longs;t, quod pellimus pede, vel <lb/>genu, quàm quod trahimus manu, vel vnco, illud ip&longs;um mouetur; </s> <s id="N27A94"><!-- NEW -->hinc <lb/>vnco, &longs;i quis annulum apprehen&longs;um trahat quantumuis immobilem, & <lb/>pede firmo nauim pellat in aduer&longs;am partem; </s> <s id="N27A9C"><!-- NEW -->haud dubiè, quia faciliùs <lb/>moueri pote&longs;t nauis quàm annulus, ver&longs;us annulum ibit; </s> <s id="N27AA2"><!-- NEW -->&longs;ed ne diuer­<lb/>&longs;as impre&longs;&longs;ionum rationes, quæ in motu nauis vulgò apparent di&longs;traha­<lb/>mus; hoc loco breuiter omnes congerendas e&longs;&longs;e putaui. </s> <s id="N27AAA">Primò ad lit­<lb/>tus tendit cum trahitur vnco annullus immobilis, vt iam dictum e&longs;t. </s> <s id="N27AAF"><!-- NEW -->Se­<lb/>cundò, &longs;i pellitur, vel fundum aquæ, vel aliud corpus immobile longio­<lb/>ri ligno, & pede pellatur ip&longs;a nauis in aduer&longs;am partem, in cam ibit <lb/>propter <expan abbr="eãdem">eandem</expan> rationem; Tertiò &longs;i pellatur aqua remis fixo etiam pe­<lb/>de vel crure contranitente in aduer&longs;am partem, idem &longs;equetur effectus. </s> <s id="N27ABF"><lb/>Quartò, hinc quò remus latior e&longs;t, & longior erit, maior erit effectus, <pb pagenum="388" xlink:href="026/01/422.jpg"/>modò &longs;uppetant vires. </s> <s id="N27AC8"><!-- NEW -->Quintò, hinc latioris claui inflexione vertitur <lb/>nauis; </s> <s id="N27ACE"><!-- NEW -->Sextò, inflata ventis &longs;ecundis vela nauem agunt; </s> <s id="N27AD2"><!-- NEW -->ratio clari&longs;&longs;ima <lb/>e&longs;t, quia non po&longs;&longs;unt vela impelli, ni&longs;i alia nauis, cui &longs;unt coniuncta mo­<lb/>ueatur; &longs;ed de re nautica agemus fusè &longs;uo loco, atque adeo de tota re <lb/>hydraulica. </s> </p> <p id="N27ADC" type="main"> <s id="N27ADE">Decimo&longs;eptimò, denique ex dictis multa corollaria con&longs;equi po&longs;&longs;unt. </s> <s id="N27AE1"><lb/>Certum e&longs;t. </s> <s id="N27AE7">1°ree;. </s> <s id="N27AEA">pars tracta non &longs;equitur trahentem &longs;ua &longs;ponte 2.°ree;. </s> <s id="N27AED">re&longs;i&longs;tit <lb/>alteri trahenti, 3°ree;. </s> <s id="N27AF2">non producit impetum pars trahens in tracta. </s> <s id="N27AF5">4°ree;. </s> <s id="N27AF8">non <lb/>trahitur immediatè, & aliæ mediatè, &longs;ed omnes &longs;imul immediatè. </s> <s id="N27AFD">5°ree;. </s> <s id="N27B00">nul­<lb/>lus impetus productus in corpore tracto impeditur. </s> <s id="N27B05">6°ree;. </s> <s id="N27B08">impetus primæ <lb/>partis non producit impetum in aliis. </s> <s id="N27B0D">7°ree;. </s> <s id="N27B10">quando dico tauri trahunt iu­<lb/>gum producunt impetum actione communi. </s> <s id="N27B15">8°ree;. </s> <s id="N27B18"><!-- NEW -->rota faciliùs trahitur, <lb/>quàm cubus; quia pauciores partes plani re&longs;i&longs;tunt. </s> <s id="N27B1E">9°ree;. </s> <s id="N27B21">quando fracto <lb/>fune trahens pronus corruit, non tantùm hic ca&longs;us procedit à priore <lb/>impetu, &longs;ed maximè à nouo. </s> <s id="N27B28">1°ree;. </s> <s id="N27B2B">extremitas funis fracti re&longs;ilit propter <lb/>præcedentem ten&longs;ionem. </s> <s id="N27B30">11°ree;. </s> <s id="N27B33">hinc cum di&longs;cerpitur charta vel tela edi­<lb/>tur &longs;onus &longs;tridulus, qui prouenit à motu extremorum filorum quæ re&longs;i­<lb/>liunt. </s> <s id="N27B3A">12°ree;. </s> <s id="N27B3D">immò cum baculus frangitur, aliqua &longs;egmenta maxima vi eui­<lb/>brantur, &longs;entiturque in manu qua&longs;i formicans dolor, propter illas tre­<lb/>mulas &longs;uccu&longs;&longs;iones. </s> <s id="N27B44">13°ree;. </s> <s id="N27B47"><!-- NEW -->cum trahitur cylindrus vtrimque in aduer&longs;as <lb/>partes à duobus contranitentibus æqualium virium, &longs;i minimè inflecti <lb/>po&longs;&longs;it, ille præualebit, cuius vtraque manus propiùs ad medium cylin­<lb/>drum accedit; </s> <s id="N27B51"><!-- NEW -->&longs;ecùs verò, &longs;i inflectatur; e&longs;t enim ad in&longs;tar gemini vectis. </s> <s id="N27B55"><lb/>14°ree;. </s> <s id="N27B59"><!-- NEW -->cum trahitur cylindrus æqualiter vtrimque, qui neque flecti, ne­<lb/>que tendi po&longs;&longs;it; </s> <s id="N27B5F"><!-- NEW -->haud dubiè nullum impetum habet, quia e&longs;&longs;et fru&longs;trà, <lb/>15. de&longs;truitur impetus in tractione, ne &longs;it fru&longs;trà: ex his reliqua facilè <lb/>intelligentur. </s> </p> <p id="N27B67" type="main"> <s id="N27B69"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N27B75" type="main"> <s id="N27B77"><emph type="italics"/>Explicari po&longs;&longs;unt omnia, quæ pertinent ad impul&longs;um.<emph.end type="italics"/></s> </p> <p id="N27B7E" type="main"> <s id="N27B80"><!-- NEW -->Primò, impul&longs;us duplicis e&longs;t generis: </s> <s id="N27B84"><!-- NEW -->primus e&longs;t coniunctus cum per­<lb/>cu&longs;&longs;ione, &longs;ic tudicula impul&longs;us globus emittitur: &longs;ecundus &longs;ine percu&longs;&longs;io­<lb/>ne; </s> <s id="N27B8C"><!-- NEW -->& hic duplex e&longs;t: </s> <s id="N27B90"><!-- NEW -->Primus, quo mobile impul&longs;um &longs;eparatur ab impel­<lb/>lente: </s> <s id="N27B96"><!-- NEW -->Secundus, quo non &longs;eparatur, &longs;ed ip&longs;i continuò adhæret; </s> <s id="N27B9A"><!-- NEW -->quia <lb/>continuo impul&longs;u mouetur; de hoc tantùm vltimo impul&longs;u agitur in <lb/>hoc Th. <!-- REMOVE S-->Secundò, ex dictis de tractatione colligi po&longs;&longs;unt ea, quæ dici debent <lb/>de impul&longs;u, quatenus nulli percu&longs;&longs;ioni nec emi&longs;&longs;ioni coniunctus e&longs;t. </s> </p> <p id="N27BA6" type="main"> <s id="N27BA8"><lb/>1°ree;. </s> <s id="N27BAC"><!-- NEW -->impellens producit impetum in &longs;e ip&longs;e; 2°ree;. </s> <s id="N27BB0">impetus impellentis pro­<lb/>ducit impetum in corpore. </s> <s id="N27BB5">3°ree;. </s> <s id="N27BB8">&longs;ingulis in&longs;tantibus de&longs;truitur aliquid <lb/>impetus impellentis, & impul&longs;i. </s> <s id="N27BBD">4°ree;. </s> <s id="N27BC0">initio difficiliùs mobile mouetur <lb/>impul&longs;u. </s> <s id="N27BC5">5°ree;. </s> <s id="N27BC8">po&longs;t primum motum tùm deinde faciliùs mouetur corpus <lb/>impul&longs;um, nec tanto ni&longs;u potentiæ opus e&longs;t. </s> <s id="N27BCD">6°ree;. </s> <s id="N27BD0">cum æquali motu mo­<lb/>uetur impul&longs;um tantùm impetus producitur, quantùm de&longs;truitur. </s> <s id="N27BD5">7°ree;. </s> <s id="N27BD8"><lb/>cum pellitur rupes immobilis, nullus in ea producitur impetus, ni&longs;i <pb pagenum="389" xlink:href="026/01/423.jpg"/>fortè aliqua pars &longs;eparetur, vel comprimatur. </s> <s id="N27BE1">8°ree;. </s> <s id="N27BE4"><!-- NEW -->producitur tamen <lb/>împetus in organo; probatur ex ni&longs;u; immò & compre&longs;&longs;ione molliorum <lb/>partium. </s> <s id="N27BEC">9°ree;. </s> <s id="N27BEF"><!-- NEW -->quando duo &longs;e&longs;e mutuò, & æquali ni&longs;u pellunt, vterque in &longs;e <lb/>ip&longs;o, & in alio producit impetum; </s> <s id="N27BF5"><!-- NEW -->in &longs;e quidem, quia maximè euitetur, <lb/>& defatigatur potentia motrix; in alio verò, in quo fit aliqua partium <lb/>compre&longs;&longs;io, quæ &longs;ine impetu <expan abbr="nũquam">nunquam</expan> fit. </s> <s id="N27C01">10°ree;. </s> <s id="N27C04"><!-- NEW -->&longs;i os pelleret os, &longs;eu corpus <lb/>durum aliud durum, natiua vi di&longs;tincta à grauitatione, in neutro pro­<lb/>duceretur impetus; </s> <s id="N27C0C"><!-- NEW -->quia e&longs;&longs;et fru&longs;trà: vide quæ diximus &longs;uprà de tra­<lb/>ctione. </s> <s id="N27C12">11°ree;. </s> <s id="N27C15"><!-- NEW -->pellens etiam firmo pede &longs;olum, in aduer&longs;am partem pellit, <lb/>&longs;eu premit; rationem iam attulimus &longs;uprà. </s> <s id="N27C1B">12°ree;. </s> <s id="N27C1E">&longs;i dum reluctantem alium <lb/>& contranitentem pellis, &longs;e&longs;e illicò cedens eripiat, pronus in terram <lb/>corrues. </s> <s id="N27C25"><!-- NEW -->13°ree;.&longs;í plures idem pondus pellant, actione communi impetum <lb/>producunt; hæc, & alia multa ex dictis de tractione facilè per eadem <lb/>principia demon&longs;trantur. </s> </p> <p id="N27C2D" type="main"> <s id="N27C2F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N27C3B" type="main"> <s id="N27C3D"><emph type="italics"/>Attolli aliquid pote&longs;t & eleuari,<emph.end type="italics"/> 1°ree;. </s> <s id="N27C45"><!-- NEW -->&longs;i producatur impetus maior impe­<lb/>tu grauitationis; ratio clara e&longs;t, quia fortior præualet. </s> <s id="N27C4B">2°ree;. </s> <s id="N27C4E"><!-- NEW -->de&longs;truitur &longs;e­<lb/>cundo in&longs;tanti aliquid impetus producti; quia e&longs;t fru&longs;trà propter <lb/>impetum natiuum. </s> <s id="N27C56">3°ree;. </s> <s id="N27C59"><!-- NEW -->&longs;i tantùm producatur impetus &longs;ingulis in­<lb/>&longs;tantibus, quantum de&longs;truitur, motus erit æquabilis, &longs;i plùs, acceleratus, <lb/>&longs;i minùs, retardatus, patet ex dictis.4°ree;.pondus attollitur initio difficiliùs <lb/>propter rationem prædictam; minùs enim produci debet impetus &longs;ecun­<lb/>do in&longs;tanti, quàm primò. </s> <s id="N27C65">5°ree;. </s> <s id="N27C68">&longs;ub funem tamen valdè laborat potentia <lb/>propter <expan abbr="compre&longs;&longs;ion&etilde;">compre&longs;&longs;ionem</expan>, & ten&longs;ionem partium, de qua &longs;uprà.6°ree;. </s> <s id="N27C71"><!-- NEW -->difficiliùs <lb/>attollitur ingens pondus, quàm modicum; ratio clara e&longs;t, quia plures <lb/>partes impetus imprimi debent maiori, cui plures in&longs;unt, quàm minori. </s> <s id="N27C79"><lb/>7°ree;. </s> <s id="N27C7D">facilius attollitur per planum inclinatum, quàm per lineam vertica­<lb/>lem deor&longs;um, rationem iam attulimus l. <!-- REMOVE S-->5. 8°ree;. </s> <s id="N27C84">hinc etiam organo me­<lb/>chanico faciliùs attollitur pondus, de quo lib. 11. 9°ree;. </s> <s id="N27C89"><!-- NEW -->licèt grauitas non <lb/>re&longs;i&longs;teret, corpus maius difficilius attolleretur, quàm minus; quia plures <lb/>partes impetus illius motus de&longs;ideraret, quàm huius, &longs;ed maior impetus <lb/>difficiliùs imprimitur, quàm minor. </s> </p> <p id="N27C93" type="main"> <s id="N27C95"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N27CA1" type="main"> <s id="N27CA3"><!-- NEW --><emph type="italics"/>Corpus<emph.end type="italics"/> 1°ree;. <emph type="italics"/>deprimitur per impetum infra medium grauius, v. <!-- REMOVE S-->g. <!-- REMOVE S-->lignu&mtail; <lb/>infra aquam<emph.end type="italics"/>; ratio clara e&longs;t. </s> <s id="N27CB8">2°ree;. </s> <s id="N27CBB">deprimitur, vel trahendo, vel impellen­<lb/>do, vel calcando. </s> <s id="N27CC0">3°ree;. </s> <s id="N27CC3">trahimus &longs;æpè deor&longs;um, vt corpus attollatur &longs;ur­<lb/>&longs;um, vt in trochleis. </s> <s id="N27CC8">4°ree;. </s> <s id="N27CCB"><!-- NEW -->quò corpus maius e&longs;t, & leuius difficiliùs depri­<lb/>mitur infra medium grauius, quia non pote&longs;t deprimi ni&longs;i plures medij <lb/>grauiores partes attollantur, vt clarum e&longs;t; exemplum habes in nauibus, <lb/>5°ree;. </s> <s id="N27CD5">deprimimus aliquando corpora per ten&longs;ionem, vt ramos arborum, <lb/>&longs;eu per librationem, vt campanarum funes, &longs;eu extremos vectes. </s> <s id="N27CDA">6°ree;. </s> <s id="N27CDD"><lb/>clauus deprimitur, vel palus tribus modis. </s> <s id="N27CE1">1°ree;. </s> <s id="N27CE4">percu&longs;&longs;ione; 2°ree;. </s> <s id="N27CE7">ia­<lb/>ctu &longs;eu eiaculatione. </s> <s id="N27CEC">3°ree;. </s> <s id="N27CEF">impul&longs;ione; de hac iam &longs;uprà actum e&longs;t, de dua­<lb/>bus primis paulò pò&longs;t agetur, &longs;ed hæc &longs;unt facilia, & faciles cau&longs;æ. </s> </p> <pb pagenum="390" xlink:href="026/01/424.jpg"/> <p id="N27CF8" type="main"> <s id="N27CFA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N27D06" type="main"> <s id="N27D08"><emph type="italics"/>Omnes gyrationum modi explicari, & demon&longs;trari po&longs;&longs;unt.<emph.end type="italics"/></s> </p> <p id="N27D0F" type="main"> <s id="N27D11"><!-- NEW -->Primò, vertitur baculus manu primo circa proprium axem, vt <expan abbr="veru">verum</expan>, <lb/>quia inflectitur eodem modo manus & inferior brachij portio: </s> <s id="N27D17"><!-- NEW -->&longs;ecundo <lb/>circa alteram extremitatem quæ manu tenetur: tertio circa quodlibet <lb/>aliud punctum, ratio petitur tùm à tali brachij motu, tùm ab eo modo, <lb/>quo baculus tenetur, </s> </p> <p id="N27D21" type="main"> <s id="N27D23"><!-- NEW -->Secundò, circumagitur funis vel funda; </s> <s id="N27D27"><!-- NEW -->quia producitur maior im­<lb/>petus in extremitate remota circa centrum immobile; hinc circulus; </s> <s id="N27D2D"><!-- NEW --><lb/>hinc quia extremitatis illius motus determinatur &longs;emper ad Tangentem, <lb/>tenditur funis; &longs;ed de funda infrà, cum de proiectione. </s> </p> <p id="N27D34" type="main"> <s id="N27D36"><!-- NEW -->Tertiò, multos alios gyros facimus, manu, brachio, collo, pede, toto <lb/>denique corporis trunco; </s> <s id="N27D3C"><!-- NEW -->quot enim habemus articulos, tot motus cir­<lb/>cularis habemus centra; </s> <s id="N27D42"><!-- NEW -->hinc &longs;uæ apothecæ caput o&longs;&longs;is tam aptè in&longs;e­<lb/>ritur, vt circa illam facilè moueatur; </s> <s id="N27D48"><!-- NEW -->exemplum habes in oculo, dum <lb/>infra &longs;uam thecam voluitur; </s> <s id="N27D4E"><!-- NEW -->&longs;ed de tota corporis fabrica, quatenus con­<lb/>ducit ad motum, &longs;uo loco agemus; nec enim hi motus ad hunc tracta­<lb/>tum pertinent. </s> </p> <p id="N27D56" type="main"> <s id="N27D58"><!-- NEW -->Quartò, hinc reuoca deflexionem illam iacti globi, de qua &longs;uprà, quæ <lb/>familiaris e&longs;t trunculorum ludo, item gyros globi, quem, vel inter duas <lb/>volas circumagis, vel inter volam, & aliud planum, qui partim ad impul­<lb/>&longs;um, partim ad tractum pertinent; &longs;ed neque hæc &longs;unt difficilia. </s> </p> <p id="N27D62" type="main"> <s id="N27D64"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N27D70" type="main"> <s id="N27D72"><!-- NEW -->Ob&longs;eruabis vix po&longs;&longs;e vno Theoremate compræhendi omnia phœno­<lb/>mena percu&longs;&longs;ionis, cuius &longs;unt tria veluti prima genera, &longs;cilicet ictus, ca­<lb/>&longs;us, iactus: ictum appello illam percu&longs;&longs;ionem, quæ infligitur pugno, ma­<lb/>nu, calce, cornu, vel quolibet organo, cum potentia motrice coniuncto, <lb/>v.g. <!-- REMOVE S-->fu&longs;te, &longs;axo, flagello, &c. </s> <s id="N27D80"><!-- NEW -->ca&longs;us e&longs;t percu&longs;&longs;io à corpore graui deor&longs;um <lb/>cadente inflicta; iactus denique e&longs;t percu&longs;&longs;io, quæ aliquam emi&longs;&longs;ionem, <lb/>&longs;eu vibrationem &longs;upponit, lapidis, pilæ, &c. </s> <s id="N27D88">itaque vt omnia percu&longs;&longs;io­<lb/>nis phœnomena di&longs;tinctiùs explicemus, &longs;ingulis Theorematis &longs;ingulos <lb/>percu&longs;&longs;ionis modos explicabimus. </s> </p> <p id="N27D8F" type="main"> <s id="N27D91"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N27D9D" type="main"> <s id="N27D9F"><emph type="italics"/>Explicantur omnia phœnomena percu&longs;&longs;ionis, quæ infligitur manu, pugno, <lb/>brachio, calce, cornu.<emph.end type="italics"/></s> </p> <p id="N27DA8" type="main"> <s id="N27DAA"><!-- NEW -->Primò, pugnus infligit ictum diuer&longs;o motu; primò, motu recto; </s> <s id="N27DAE"><!-- NEW -->&longs;it <lb/>enim humerus AB, caput cubiti B, os cubiti BF, fiat arcus AC, & KI, <lb/>ita vt ABK &longs;it æqualis ABF; </s> <s id="N27DB6"><!-- NEW -->certè ACI erit totum brachium ten&longs;um, <lb/>caput B nunquam recedit ab arcu BC, nec extremitas F à recta FI; </s> <s id="N27DBC"><!-- NEW -->vbi <lb/>autem F peruenit in G; </s> <s id="N27DC2"><!-- NEW -->a&longs;&longs;umatur GE æqualis FB: </s> <s id="N27DC6"><!-- NEW -->vbi verò F peruenit <lb/>in H; </s> <s id="N27DCC"><!-- NEW -->a&longs;&longs;umatur HD æqualis FB, & habebitur proportio motus extre­<lb/>mitatis F & capitis B; vides motum rectum FI mixtum ex duobus cir­<lb/>cularibus circa centrum immobile A, & mobile B. </s> </p> <p id="N27DD4" type="main"> <s id="N27DD6"><!-- NEW -->Secundò, po&longs;&longs;et moueri per omnem lineam, v. <!-- REMOVE S-->g. <!-- REMOVE S-->FN, FM, <pb pagenum="391" xlink:href="026/01/425.jpg"/>immò, & per lineam perpendicularem &longs;ur&longs;um, vel deor&longs;um, & quò <lb/>plùs contrahetur brachium, motus rectus per horizontalem erit maior; </s> <s id="N27DE5"><!-- NEW --><lb/>&longs;it enim angulus cubiti ABO, ita vt BF &longs;it in BO; </s> <s id="N27DEA"><!-- NEW -->certè extremitas <lb/>O percurret motu recto totam OP; </s> <s id="N27DF0"><!-- NEW -->& &longs;i omninò contrahatur brachium, <lb/>ita vt F &longs;it in A, percurret extremitas A totam rectam AP; </s> <s id="N27DF6"><!-- NEW -->tunc au­<lb/>tem ictus e&longs;t fortior, cum linea motus recti e&longs;t maior; quippe &longs;ingulis in­<lb/>&longs;tantibus nouus impetus accedit. </s> </p> <p id="N27DFE" type="main"> <s id="N27E00"><!-- NEW -->Tertiò, pote&longs;t inueniri maximum &longs;patium quod pote&longs;t confici ab ex­<lb/>tremitate brachij motu recto; </s> <s id="N27E06"><!-- NEW -->&longs;it enim centrum humeri immobile A, <lb/>&longs;it AC os humeri, CD cubiti, &longs;it AD perpendicularis deor&longs;um; </s> <s id="N27E0C"><!-- NEW -->&longs;it <lb/>angulus BAC maximæ deflexionis, qua os humeri po&longs;&longs;it retrò agi; </s> <s id="N27E12"><!-- NEW -->&longs;it <lb/>CGK, item DFO, &longs;it BG recta, BH æqualis CD; </s> <s id="N27E18"><!-- NEW -->ducatur EHL <lb/>perpendicularis &longs;ur&longs;um, &longs;itque CEOS cubiti: dico EL e&longs;&longs;e maximum <lb/>&longs;patium, &c. </s> <s id="N27E20"><!-- NEW -->cùm enim caput cubiti C po&longs;&longs;it tantùm retroagi in B; </s> <s id="N27E24"><!-- NEW -->certè <lb/>non pote&longs;t extremitas D, in quocumque loco &longs;it, circuli DFO &longs;ecare <lb/>BG, in puncto quod propiùs accedat ad centrum A quàm H; </s> <s id="N27E2C"><!-- NEW -->&longs;ed om­<lb/>nium linearum, quæ po&longs;&longs;unt duci per H &longs;ur&longs;um perpendiculariter, ma­<lb/>xima e&longs;t EL; </s> <s id="N27E34"><!-- NEW -->immò EL e&longs;t omnium maxima, quæ duci po&longs;&longs;unt po&longs;ita <lb/>extremitate inter DE; </s> <s id="N27E3A"><!-- NEW -->vt autem habeatur omnium maxima; </s> <s id="N27E3E"><!-- NEW -->&longs;it punctum <lb/>K &longs;ur&longs;um, ad quod tantùm nodus, &longs;eu caput cubiti C peruenire pote&longs;t; </s> <s id="N27E44"><!-- NEW --><lb/>a&longs;&longs;umatur KO æqualis CD, ex centro B fiat arcus AH, tùm ex O ad <lb/>arcum AH; </s> <s id="N27E4B"><!-- NEW -->ducatur Tangens OQF; </s> <s id="N27E4F"><!-- NEW -->certum e&longs;t e&longs;&longs;e maximam lineam; <lb/>quia accedit propiùs ad centrum A, vt con&longs;tat. </s> </p> <p id="N27E55" type="main"> <s id="N27E57">Quartò, pote&longs;t pugnus ferire motu perfectè circulari, idque duobus <lb/>modis. </s> <s id="N27E5C">Primò, &longs;i brachium exten&longs;um AD circa centrum moueatur per <lb/>arcum DFO figura prima. </s> <s id="N27E61"><!-- NEW -->Secundò, &longs;i moueatur caput cubiti; </s> <s id="N27E65"><!-- NEW -->&longs;it enim <lb/> os humeri AB, & cubiti BC caput cubiti B; </s> <s id="N27E6B"><!-- NEW -->ex A fiat arcus BEL; </s> <s id="N27E6F"><!-- NEW --><lb/>tùm ex aliquo puncto &longs;uprà A, putà ex N radio NC fiat arcus CI; </s> <s id="N27E74"><!-- NEW -->tùm <lb/>a&longs;&longs;umpta AK æquali AB fiat arcus KH &longs;ecans priorem in I; </s> <s id="N27E7A"><!-- NEW -->certè extre­<lb/>mitas C moueri poterit per arcum CI, donec brachium extentum &longs;it <lb/>in AI, quod non e&longs;t difficile; hîc porrò vides motum circularem ex <lb/>duobus alijs circularibus mixtum. </s> </p> <p id="N27E84" type="main"> <s id="N27E86">Quintò, moueri per quamcumque aliam lineam curuam, ellipticam, <lb/>parabolicam &c. </s> <s id="N27E8B"><!-- NEW -->immò per infinitas alias nouas; </s> <s id="N27E8F"><!-- NEW -->vides nouam FDC, <lb/>quæ vt fiat cubitus IF e&longs;t &longs;emper &longs;ibi ip&longs;i parallelus; </s> <s id="N27E95"><!-- NEW -->quod vt fiat, caput <lb/>I & extremitas F debent moueri æquali motu; </s> <s id="N27E9B"><!-- NEW -->&longs;unt enim CBLDEK <lb/>FI æquales & parallelæ: </s> <s id="N27EA1"><!-- NEW -->ex quo fit hanc curuam e&longs;&longs;e &longs;peciem nouæ <lb/>Conchoidis, de qua aliàs; mouetur autem initio tardiùs, & &longs;ub <lb/>finem velociùs, non quidem proprio motu circa centrum I, &longs;ed motu <lb/>mixto. </s> </p> <p id="N27EAB" type="main"> <s id="N27EAD"><!-- NEW -->Sextò, e&longs;t maximus ictus inflictus à pugno, qui mouetur motu re­<lb/>cto per longiorem lineam, quæ accedit propiùs ad lineam brachij dein­<lb/>de extenti; </s> <s id="N27EB5"><!-- NEW -->quò enim e&longs;t longior linea producitur &longs;en&longs;un maior impe­<lb/>tus; </s> <s id="N27EBB"><!-- NEW -->e&longs;t enim motus naturaliter acceleratus, cùm &longs;it applicata con­<lb/>tinuô potentia motrix: </s> <s id="N27EC1"><!-- NEW -->præterea ictus e&longs;t magis directus, &longs;i linea <pb pagenum="392" xlink:href="026/01/426.jpg"/>motus propiùs accedit ad lineam brachij extenti: hinc quò plus cotra­<lb/>hitur brachium ad infligendum ictum e&longs;t validior ictus, quia e&longs;t lon­<lb/>gior linea & magis directa, quod natura ip&longs;a docuit pueros pugnis con­<lb/>tendentes. </s> </p> <p id="N27ED0" type="main"> <s id="N27ED2"><!-- NEW -->Septimò, auer&longs;a manu impingitur validior colaphus, quàm aduer&longs;a; </s> <s id="N27ED6"><!-- NEW --><lb/>quia mouetur manus per arcum paulò maiorem &longs;emicirculo; </s> <s id="N27EDB"><!-- NEW -->in quo <lb/>motus continuò cre&longs;cit; at verò &longs;i aduer&longs;â; </s> <s id="N27EE1"><!-- NEW -->non validus e&longs;t ictus; </s> <s id="N27EE5"><!-- NEW -->pri­<lb/>mò quia quando auer&longs;a infligitur, & e&longs;t motus circa duplex centrum, <lb/>vterque circularis in <expan abbr="eãdem">eandem</expan> partem tendit; </s> <s id="N27EF1"><!-- NEW -->igitur maior e&longs;t; </s> <s id="N27EF5"><!-- NEW -->&longs;ecus <lb/>accidit cum aduer&longs;à: </s> <s id="N27EFB"><!-- NEW -->Secundò, non tam extendi pote&longs;t brachium impa­<lb/>ctum intror&longs;um, quàm in aduer&longs;am partem; igitur minor e&longs;t arcus, <lb/>vel os humeri &longs;i&longs;titur, atque ita ex parte extinguitur ictus. </s> <s id="N27F03"><!-- NEW -->Tertiò <lb/>manus auer&longs;a durior e&longs;t, quàm aduer&longs;a; </s> <s id="N27F09"><!-- NEW -->e&longs;t enim vola mollior; </s> <s id="N27F0D"><!-- NEW -->hæc <lb/>verò mollities extinguit vim ictus, vt &longs;æpè demon&longs;trauimus: de rota­<lb/>tione brachij, quæ maximè vim auget, dicemus infrà, cum de Tudicu­<lb/>la, clauâ, baculo, de lineis verò dicemus lib.12. </s> </p> <p id="N27F17" type="main"> <s id="N27F19"><!-- NEW -->Octauò, qui longioribus brachijs in&longs;tructi &longs;unt, maiores ictus <lb/>infligunt; </s> <s id="N27F1F"><!-- NEW -->patet, quia maiorem de&longs;cribunt arcum; </s> <s id="N27F23"><!-- NEW -->igitur velociore <lb/>motu rotatur pugnus; </s> <s id="N27F29"><!-- NEW -->cum tamen motu circulari mouetur brachium; <lb/>certum e&longs;t maiorem ictum minimè infligi ab extremitate, vt con&longs;tat <lb/>ex dictis de baculo lib.1. Th.73. ni&longs;i fortè ratione contracti pugni, quod <lb/>iam ibidem indicauimus. </s> </p> <p id="N27F33" type="main"> <s id="N27F35"><!-- NEW -->Nonò, cum deor&longs;um impingitur pugnus, cre&longs;cit ictus propter acce&longs;­<lb/>&longs;ionem motus naturalis accelerati; </s> <s id="N27F3B"><!-- NEW -->e&longs;t enim corpus graue; </s> <s id="N27F3F"><!-- NEW -->cum &longs;ur&longs;um, <lb/>è contrario imminuitur motus: in qua verò proportione, dicemus in­<lb/>frà cum de malleo. </s> </p> <p id="N27F47" type="main"> <s id="N27F49">Decimò, aliquando rotatur brachium, antequam infligatur ictus, <lb/>vel intror&longs;um, vel in partem oppo&longs;itam, præ&longs;ertim vt longiùs ia­<lb/>ciatur lapis, vt pila reticulo, vel auer&longs;o, vel aduer&longs;o procul <lb/>emittatur, &c. </s> <s id="N27F52">ratio e&longs;t, quia continuò augetur motus, vt iam di­<lb/>ctum e&longs;t. </s> </p> <p id="N27F57" type="main"> <s id="N27F59"><!-- NEW -->Vndecimò, breuiter indico ictum inflictum ab ip&longs;o cubiti capi­<lb/>te retrò acto, &longs;atis grauem e&longs;&longs;e; </s> <s id="N27F5F"><!-- NEW -->tùm quia durior e&longs;t ille nodus; tùm <lb/>quia ad eius motum non modò &longs;uperius brachij &longs;egmentum, verùm <lb/>etiam inferius concurrit. </s> </p> <p id="N27F67" type="main"> <s id="N27F69"><!-- NEW -->Duodecimò, infligitur etiam grauis ictus calce, cuius e&longs;t eadem <lb/>ratio, quæ &longs;uprà; e&longs;t enim duplex centrum, duplex motus, &c. </s> <s id="N27F6F">Ob­<lb/>&longs;eruabis tamen. </s> <s id="N27F74">Primò ictum maiorem infligi, &longs;i crura longiora &longs;unt. </s> <s id="N27F77"><!-- NEW --><lb/>Secundò aduer&longs;o calce quam auer&longs;o; e&longs;t enim oppo&longs;ita brachiorum <lb/>ratio, cùm genu aduer&longs;um &longs;it, & auersum cubiti caput. </s> <s id="N27F7E"><!-- NEW -->Tertiò, equi <lb/>è contrario calcem fortiùs retroagunt, quia tibiæ po&longs;terioris ge­<lb/>nu auer&longs;um e&longs;t; </s> <s id="N27F86"><!-- NEW -->adde quoque ictum ab ip&longs;o genu inflictum; </s> <s id="N27F8A"><!-- NEW -->de <lb/>quo idem dicendum e&longs;t, quod de ictu à nodo cubiti inflicto iam <lb/>diximus; quippe in eo tantùm differunt, quòd habeant contrarios <lb/>&longs;itus. </s> </p> <pb pagenum="393" xlink:href="026/01/427.jpg"/> <p id="N27F98" type="main"> <s id="N27F9A"><!-- NEW -->Decimotertiò, explo&longs;ione inten&longs;i digiti talitrum imprimitur, cuius <lb/>&longs;unt tres modi; primus e&longs;t, cum vngue medij, vel alterius digiti pul&longs;o <lb/>tanti&longs;per molliore &longs;ummi pollicis apice, inten&longs;us deinde digitus eo­<lb/>dem vngue talitrum impingit. </s> <s id="N27FA4">Secundum e&longs;t, cum retento &longs;ummo di­<lb/>gito ab aliquo molliori corpore &longs;tatim dimittitur. </s> <s id="N27FA9"><!-- NEW -->Tertium e&longs;t, cum <lb/>mollior medij digiti, & pollicis apex po&longs;t aliquam pre&longs;&longs;ionem, non <lb/>&longs;ine aliquo &longs;trepitu exploditur; </s> <s id="N27FB1"><!-- NEW -->ratio primi e&longs;t, quia dum vnguis mol­<lb/>liorem &longs;ub&longs;tantiam premit, auget impetum potentia motrix in illa <lb/>mora, neruu&longs;que maximè intenditur; </s> <s id="N27FB9"><!-- NEW -->igitur maior e&longs;t ictus; </s> <s id="N27FBD"><!-- NEW -->eadem <lb/>ratio valet pro &longs;ecundo, & tertio modo; </s> <s id="N27FC3"><!-- NEW -->&longs;trepitus ille oritur à colli­<lb/>&longs;ione, vel compre&longs;&longs;ione: </s> <s id="N27FC9"><!-- NEW -->immò &longs;i nulla fieret compre&longs;&longs;io aut certè <lb/>&longs;i nulla cederet mollior materia, non e&longs;&longs;et maior ictus; adde quod <lb/>non tantùm augetur impetus à potentia motrice diutiùs agente, &longs;ed <lb/>etiam ratione compre&longs;&longs;ionis noua &longs;it impetus acce&longs;&longs;io, vt patet in <lb/>arcu. </s> </p> <p id="N27FD5" type="main"> <s id="N27FD7">Decimoquartò, denique quod &longs;pectat ad cornu facilè explicari pote&longs;t <lb/>quomodo ab irato tauro intendatur, vno &longs;cilicet durioris capitis motu, <lb/>atque adeò totius corporis. </s> </p> <p id="N27FDE" type="main"> <s id="N27FE0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N27FEC" type="main"> <s id="N27FEE"><emph type="italics"/>Explicari po&longs;&longs;unt omnes ictus, qui infliguntur impacto &longs;cilicet &longs;axo, fu&longs;te, <lb/>flagello, & alio quouis organo, cui &longs;emper potentia motrix coniuncta e&longs;t, nec <lb/>ab ea &longs;eparatur,<emph.end type="italics"/> excepto dumtaxat omni malleorum genere, gladiorum, <lb/>&c. </s> <s id="N27FFC"><!-- NEW -->primò manus in&longs;tructa &longs;axo grauiorem ictum infligit; </s> <s id="N28000"><!-- NEW -->tùm quia <lb/>multus impetus imprimitur graui &longs;axo; </s> <s id="N28006"><!-- NEW -->tùm quia durior e&longs;t materia; </s> <s id="N2800A"><!-- NEW --><lb/>igitur nihil cedit: </s> <s id="N2800F"><!-- NEW -->porrò maior e&longs;t ictus, &longs;i deor&longs;um intendatur, cùm <lb/>accedat impetus grauitatis ip&longs;ius &longs;axi: adde ferream manicam, quæ prop­<lb/>ter <expan abbr="eãdem">eandem</expan> rationem petentem colaphum infringit. </s> </p> <p id="N2801B" type="main"> <s id="N2801D"><!-- NEW -->Secundò, fu&longs;tis impingi pote&longs;t duobus modis; </s> <s id="N28021"><!-- NEW -->primò motu recto, <lb/>cùm &longs;cilicet porrecto brachio extremitas fu&longs;tis &longs;copum attingit; </s> <s id="N28027"><!-- NEW -->&longs;ecundò <lb/>motu circulari rotato &longs;cilicet brachio: </s> <s id="N2802D"><!-- NEW -->primo modo infligitur ictus pun­<lb/>ctim, vt vulgò dicunt: &longs;ecundo qua&longs;i cæ&longs;im, vterque &longs;ua phœnomena <lb/>habet. </s> </p> <p id="N28035" type="main"> <s id="N28037"><!-- NEW -->Tertiò, cum punctim impingitur fu&longs;tis, quò hic maior e&longs;t, maiorem <lb/>incutit ictum; </s> <s id="N2803D"><!-- NEW -->præ&longs;ertim, &longs;i gemina manu intenditur; </s> <s id="N28041"><!-- NEW -->quia &longs;cilicet ma­<lb/>jor impetus imprimitur; </s> <s id="N28047"><!-- NEW -->huc reuoca &longs;ari&longs;&longs;æ graui&longs;&longs;imum ictum, quo <lb/>ferrea lorica perfodi pote&longs;t; </s> <s id="N2804D"><!-- NEW -->quia &longs;cilicet maior impetus imprimitur in­<lb/>tentis priùs, & vibratis brachijs; </s> <s id="N28053"><!-- NEW -->multùm enim confert, tum illa bra­<lb/>chiorum, atque adeò totius &longs;ari&longs;&longs;æ vibratio; </s> <s id="N28059"><!-- NEW -->tùm etiam neruorum ten­<lb/>&longs;io, vt videmus in arcu; </s> <s id="N2805F"><!-- NEW -->&longs;ed hoc iam &longs;uprà explicuimus; huc etiam <lb/>reuoca cra&longs;&longs;iorem illum vectem, quo fores ip&longs;i pul&longs;ati perrum­<lb/>puntur. </s> </p> <p id="N28067" type="main"> <s id="N28069"><!-- NEW -->Quartò, longitudo &longs;ari&longs;&longs;æ compen&longs;ari pote&longs;t cra&longs;&longs;itie; </s> <s id="N2806D"><!-- NEW -->&longs;it enim <lb/>&longs;ari&longs;&longs;a 12. pedes alta pendens 12. libras; </s> <s id="N28073"><!-- NEW -->&longs;it alia 6. pedes alta pendens <pb pagenum="394" xlink:href="026/01/428.jpg"/>12. libras vtraque æquali ni&longs;u, & modo ab eadem potentia impacta æ­<lb/>qualem ictum infligit; </s> <s id="N2807E"><!-- NEW -->probatur quia tantumdem impetus imprimitur <lb/>vni, quantum alteri; </s> <s id="N28084"><!-- NEW -->nam aëris re&longs;i&longs;tentia vix quidquam facit; </s> <s id="N28088"><!-- NEW -->licèt pau­<lb/>lò plùs re&longs;i&longs;tat aër breuiori, cuius ba&longs;is latior e&longs;t in ratione dupla, quàm <lb/>longiori; hinc cra&longs;&longs;iori fu&longs;te licèt breuiore maximus ictus infringitur, <lb/>vt patet experientiâ. </s> </p> <p id="N28092" type="main"> <s id="N28094">Diceret aliquis hæc repugnare omnibus experimentis, quibus &longs;cili­<lb/>cet clari&longs;&longs;imè con&longs;tat minorem e&longs;&longs;e breuiorum &longs;ari&longs;&longs;arum vim. </s> </p> <p id="N28099" type="main"> <s id="N2809B">Re&longs;p. hoc ip&longs;um accidere; quia breuiores &longs;ari&longs;&longs;æ, quas habemus, vel <lb/>exiliores &longs;unt longioribus, vel &longs;altem non cra&longs;&longs;iores, cùm tamen cra&longs;&longs;io­<lb/>res e&longs;&longs;e oporteat in eadem ratione, in qua illæ longiores &longs;unt vt æqualis <lb/>&longs;it ictus. </s> </p> <p id="N280A4" type="main"> <s id="N280A6"><!-- NEW -->Quintò, cur verò maior fu&longs;tis maiorem impetum à brachiorum vi <lb/>recipiat; </s> <s id="N280AC"><!-- NEW -->ratio e&longs;t, primò quia maiori vtrumque brachium admouetur: </s> <s id="N280B0"><!-- NEW --><lb/>&longs;ecundò, quia vibratur antequam intendatur; </s> <s id="N280B5"><!-- NEW -->atqui ex ea vibratione <lb/>multus impetus accedit, vt patet ex vibrato ariete: </s> <s id="N280BB"><!-- NEW -->tertiò, quia maior <lb/>fu&longs;tis tardiùs mouetur, vt con&longs;tat; </s> <s id="N280C1"><!-- NEW -->igitur plùs impetus in eo producit <lb/>potentia motrix, quæ &longs;ingulis in&longs;tantibus toto ni&longs;u fu&longs;tem impellit; </s> <s id="N280C7"><!-- NEW -->& <lb/>hæc e&longs;t vera ratio à priori: </s> <s id="N280CD"><!-- NEW -->quartò, adde quod pondus maioris fu&longs;tis <lb/>qua&longs;i neruos extendit; </s> <s id="N280D3"><!-- NEW -->atqui ten&longs;i nerui fortiores &longs;unt; </s> <s id="N280D7"><!-- NEW -->in qua verò <lb/>proportione &longs;it maior ictus, dicemus numero &longs;equenti; e&longs;t enim res <lb/>&longs;citu digni&longs;&longs;ima. </s> </p> <p id="N280DF" type="main"> <s id="N280E1"><!-- NEW -->Sextò, determinari pote&longs;t proportio ictuum maioris, & minoris <lb/>fu&longs;tis, cum vterque punctim impingitur ab eadem potentiâ per eam­<lb/>dem lineam æquali ni&longs;u; </s> <s id="N280E9"><!-- NEW -->&longs;it fu&longs;tis minor H duarum librarum; </s> <s id="N280ED"><!-- NEW -->&longs;it <lb/>maior I 8. librarum; </s> <s id="N280F3"><!-- NEW -->&longs;it datum tempus L, quo I &longs;uam lineam K <lb/>motu accelerato &longs;patium conficit: </s> <s id="N280F9"><!-- NEW -->dico H eodem tempore L con­<lb/>ficere tantùm &longs;patium prioris &longs;ubquadruplum; </s> <s id="N280FF"><!-- NEW -->igitur duplo tem­<lb/>pore conficit &longs;patium K: </s> <s id="N28105"><!-- NEW -->&longs;ed æqualibus temporibus acquiruntur <lb/>æqualia velocitatis momenta motu accelerato; </s> <s id="N2810B"><!-- NEW -->igitur vbi H confi­<lb/>cit &longs;patium K, habet &longs;ubduplam velocitatem illius, quam habet I <lb/>confecto eodem &longs;patio K; </s> <s id="N28113"><!-- NEW -->&longs;ed moles H e&longs;t quadrupla molis I; </s> <s id="N28117"><!-- NEW -->igi­<lb/>tur impetus H e&longs;t duplus impetu I; </s> <s id="N2811D"><!-- NEW -->igitur duplò maior ictus: </s> <s id="N28121"><!-- NEW -->quod <lb/>vt clariùs videatur, in &longs;chemate hoc ip&longs;um demon&longs;tro, producitur <lb/>æqualis impetus eodem tempore in H & in I; </s> <s id="N28129"><!-- NEW -->e&longs;t enim eadem poten­<lb/>tia, idem ni&longs;us, &longs;ed di&longs;tribuitur in H numero partium quadru­<lb/>plo numeri partium I; </s> <s id="N28131"><!-- NEW -->igitur velocitas, vel inten&longs;io impetus H e&longs;t <lb/>&longs;ubquadrupla; </s> <s id="N28137"><!-- NEW -->igitur &longs;i I tempore L percurrit AG; </s> <s id="N2813B"><!-- NEW -->certè H eodem <lb/>tempore percurrit AB &longs;ubquadruplam AG; </s> <s id="N28141"><!-- NEW -->igitur duplo tempore <lb/>AC æqualem AG; </s> <s id="N28147"><!-- NEW -->&longs;ed H decur&longs;a AC, habet &longs;ubuplam veloci­<lb/>tatem I, decur&longs;a AG; </s> <s id="N2814D"><!-- NEW -->quia decur&longs;a AF habet æqualem: </s> <s id="N28151"><!-- NEW -->&longs;ed AF e&longs;t <lb/>quadrupla AC; igitur decur&longs;a AC habet &longs;ubduplam, &c. </s> <s id="N28157"><!-- NEW -->&longs;ed ra­<lb/>tione molis habet H quadruplum impetus; igitur ratione vtriu&longs;que <lb/>duplum. </s> </p> <pb pagenum="395" xlink:href="026/01/429.jpg"/> <p id="N28163" type="main"> <s id="N28165">Ob&longs;eruabis autem primò ratione ponderis H, quod &longs;u&longs;tińetur, aliquid <lb/>impetus detrahendum e&longs;&longs;e. </s> <s id="N2816A">Secundò, vt accuratè procedatur vtrumque <lb/>fu&longs;tem funependulum e&longs;&longs;e po&longs;&longs;e. </s> <s id="N2816F"><!-- NEW -->Tertiò, ictus e&longs;&longs;e vt impetus; impetus <lb/>verò in ratione &longs;ubduplicata ponderum, hoc e&longs;t, vt radices quadratas. </s> <s id="N28175"><!-- NEW --><lb/>v.g. <!-- REMOVE S-->fu&longs;tis maior pendit 36. libras, minor 4; </s> <s id="N2817C"><!-- NEW -->ictus maioris e&longs;t ad ictum <lb/>minoris vt 6. ad 2. Quartò, denique plures partes percuti à maiore <lb/>fu&longs;te, cuius ba&longs;is latior e&longs;t, nec tam facilè comprimi, nec ip&longs;um fu&longs;tem <lb/>incuruari; ac proinde minùs ictui detrahi, &longs;ed de his &longs;atis. </s> </p> <p id="N28186" type="main"> <s id="N28188">Septimò, &longs;i fu&longs;tis cæ&longs;im impingatur, maiorem ictum infligit. </s> <s id="N2818B">Primò, <lb/>non circa extremitatem &longs;ed circa 2/3, vt demon&longs;trabimus infrà. </s> <s id="N28190"><!-- NEW -->Secundò, <lb/>quò maior e&longs;t arcus fu&longs;tis e&longs;t maior ictus; </s> <s id="N28196"><!-- NEW -->ratio patet ex dictis; cùm &longs;it <lb/>motus acceleratus. </s> <s id="N2819C">Tertiò, pote&longs;t hic motus totum implere orbem, &longs;iue <lb/>fieri auer&longs;a, &longs;iue aduer&longs;a manu. </s> <s id="N281A1">Quartò, auer&longs;a manu impactus fu&longs;tis ma­<lb/>iorem ictum infligit, quia brachium hoc modo intentum maiore vi <lb/>pollet, vt dictum e&longs;t &longs;uprà. </s> <s id="N281A8"><!-- NEW -->Quintò, hinc &longs;æpè ita inflecti &longs;eu tornari po­<lb/>te&longs;t brachium, vt de&longs;cribat arcum minoris circuli, &longs;ed maiorem, &longs;eu po­<lb/>tius lineam &longs;piralem, in qua de&longs;cribenda diutiùs moratur; </s> <s id="N281B0"><!-- NEW -->hinc motus fit <lb/>maior, quia e&longs;t acceleratus; igitur maior ictus. </s> <s id="N281B6">Sextò, &longs;i fu&longs;tis deor&longs;um <lb/>feratur motu circulari, impetus naturalis accedit impre&longs;&longs;o. </s> <s id="N281BB">Septimò, &longs;i <lb/>vtraque manu intendatur fu&longs;tis, maior erit ictus, vt con&longs;tat ex dictis. </s> <s id="N281C0"><lb/>Octauò denique, quod dictum e&longs;t de fu&longs;te impacto cæ&longs;im, dici debet <lb/>en&longs;e. </s> </p> <p id="N281C6" type="main"> <s id="N281C8">Octauò, aliquando fu&longs;tis inflectitur; </s> <s id="N281CB"><!-- NEW -->quia flexibilis e&longs;t; </s> <s id="N281CF"><!-- NEW -->cum &longs;cilicet <lb/>motu circulari, &longs;eu cæ&longs;im diuerberat, &longs;eu flagellat; </s> <s id="N281D5"><!-- NEW -->&longs;it enim fu&longs;tis CA, <lb/>qui rotetur circa centrum C; </s> <s id="N281DB"><!-- NEW -->certè vbi B peruenerit in E, A perueniet <lb/>in H; </s> <s id="N281E1"><!-- NEW -->igitur inflexus e&longs;t fu&longs;tis HEC, vel GFC; ratio e&longs;t, quia cùm po­<lb/>tentia applicata in C agat toto ni&longs;u. </s> <s id="N281E7"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i &longs;egmentum CB &longs;eiunctum <lb/>e&longs;&longs;et à &longs;egmento BA; </s> <s id="N281F1"><!-- NEW -->haud dubiè punctum B perueniet citiùs in F, quàm <lb/>&longs;i vtrumque &longs;egmentum coniunctum e&longs;&longs;et, vt notum e&longs;t; </s> <s id="N281F7"><!-- NEW -->quia maior im­<lb/>petus imprimitur B &longs;eiuncta; </s> <s id="N281FD"><!-- NEW -->atqui licèt CB &longs;it coniunctum BA, ab eo <lb/>tamen facilè, non quidem omninò &longs;eiungi, &longs;ed deflecti, dimoueri pote&longs;t <lb/>propter flexibilitatem materiæ; </s> <s id="N28205"><!-- NEW -->igitur B relinquet à tergo BA; </s> <s id="N2820B"><!-- NEW -->igitur <lb/>fu&longs;tis inflectetur, & hæc e&longs;t vera ratio huius phœnomeni: hinc virgulæ <lb/>&longs;ucco & humore plenæ, nerui bubuli latiores, canones, funiculi, lora, <lb/>en&longs;es, manubria Tudiculæ maioris, & alia huiu&longs;modi propter rationem <lb/>prædictam inflectuntur. </s> </p> <p id="N28217" type="main"> <s id="N28219"><!-- NEW -->Nonò, extremitas fu&longs;tis inflexi, cum deinde redit, maiorem ictum in­<lb/>fligit: </s> <s id="N2821F"><!-- NEW -->ratio e&longs;t, v.g. <!-- REMOVE S-->A vbi attingit D po&longs;t inflexionem; </s> <s id="N28225"><!-- NEW -->quia maiorem <lb/>impetum habet; </s> <s id="N2822B"><!-- NEW -->nam præter impre&longs;&longs;um à potentia applicata in C, acce­<lb/>dit alius ab ip&longs;a inflexione, cuius rationem afferemus tractatu &longs;equenti, <lb/>cum de compre&longs;&longs;ione, & ten&longs;ione corporum; </s> <s id="N28233"><!-- NEW -->e&longs;t enim quædam potentia <lb/>media inter potentiam grauitationis, & potentiam animatorum, quam <lb/>proinde mediam appellabimus; </s> <s id="N2823B"><!-- NEW -->quâ &longs;cilicet corpora &longs;e&longs;e re&longs;tituunt pri­<lb/>&longs;tinæ exten&longs;ioni, cuius mirificos effectus habemus in arcu chordis pul-<pb pagenum="396" xlink:href="026/01/430.jpg"/>&longs;atis, va&longs;is pneumaticis, & hydraulicis, denique in tota re tormentaria; <lb/>hinc primò Tudiculæ maioris manubrium inflexum multùm auget ip&longs;am <lb/>vim ictus, de quo infrà. </s> <s id="N2824A">Secundò, neruus bubulus, primò inflexus, tùm <lb/>&longs;tatim rediens &longs;capulas malè afficit. </s> <s id="N2824F">Tertiò, flexibiles virgæ tran&longs;uer&longs;as <lb/>plagas cum tanto dolore infligunt inu&longs;tis vibicibus. </s> <s id="N28254">Quartò, idem dico <lb/>de regula illa latiore, qua remigiorum præ&longs;ides, remiges tardos ca&longs;ti­<lb/>gant &c. </s> </p> <p id="N2825B" type="main"> <s id="N2825D"><!-- NEW -->Decimò, non videtur omittendum flagelli phœnomenum; </s> <s id="N28261"><!-- NEW -->e&longs;t autem <lb/>duplex flagellorum genus; </s> <s id="N28267"><!-- NEW -->primum illorum e&longs;t, quibus aurigæ &longs;uos <lb/>equos agunt; </s> <s id="N2826D"><!-- NEW -->&longs;ecundum eorum, quibus &longs;eges in area teritur; </s> <s id="N28271"><!-- NEW -->quod &longs;pe­<lb/>ctat ad primum, vel loris vel funiculis con&longs;tat; </s> <s id="N28277"><!-- NEW -->acris verò e&longs;t ictus, quem <lb/>inurit eius præ&longs;ertim extremitas; </s> <s id="N2827D"><!-- NEW -->ratio e&longs;t, quia cùm partes funis, quæ <lb/>propius ad manubrium accedunt, citiùs moueantur, & alias ponè relin <lb/>quant, i&longs;tæ deinde in &longs;uo motu plùs temporis ponunt; </s> <s id="N28285"><!-- NEW -->igitur, cùm &longs;it <lb/>motus acceleratus, maiorem induunt impetum, maioremque imprimunt: </s> <s id="N2828B"><!-- NEW --><lb/>adde quòd, continuò arcum minoris circuli extremitas ip&longs;a de&longs;cribit, <lb/>quæ vltimò tantum applicatur: </s> <s id="N28292"><!-- NEW -->hinc nouus accelerationis modus, vt <lb/>clari&longs;&longs;imè videtur in funiculo circa digitum, cui aduoluitur in gyros <lb/>acto: </s> <s id="N2829A"><!-- NEW -->Quod &longs;pectat ad flagellum frumentarium, mouetur motu mixto <lb/>ex duobus circularibus; </s> <s id="N282A0"><!-- NEW -->con&longs;tat enim de gemino fu&longs;te, quorum alter <lb/>circa alterius extremitatem rotatur; </s> <s id="N282A6"><!-- NEW -->hic verò circa centrum humeri: </s> <s id="N282AA"><!-- NEW --><lb/>porrò extremus fu&longs;tis facit integrum circulum, vnde maximum ictum <lb/>infligit, quem &longs;cilicet præce&longs;&longs;it longior motus; </s> <s id="N282B1"><!-- NEW -->adde quod qua&longs;i à tergo <lb/>relinquitur extremus fu&longs;tis ab altero; </s> <s id="N282B7"><!-- NEW -->igitur diutiùs potentia maner ap­<lb/>plicata; </s> <s id="N282BD"><!-- NEW -->igitur maiorem impetum producit, ex quo &longs;equitur maior ictus; </s> <s id="N282C1"><!-- NEW --><lb/>porrò vt vltima extremitas extremi fu&longs;tis qua&longs;i retroagitur; </s> <s id="N282C6"><!-- NEW -->quod &longs;cilicet <lb/>eius centrum antè producatur, &longs;eu porrigatur; </s> <s id="N282CC"><!-- NEW -->cùm enim attollitur fla­<lb/>gellum illud plicatile; haud dubiè extremitas deor&longs;um tendit proprio <lb/>pondere, & producto in aduer&longs;am partem eius centro, vel altera extre­<lb/>mitate, quid mirum &longs;i perficit circulum? </s> <s id="N282D6">eius lineam de&longs;cribemus l.12. </s> </p> <p id="N282D9" type="main"> <s id="N282DB"><!-- NEW -->Vndecimò, &longs;ed aliquam huius phœnomeni adumbrationem iuuerit <lb/>exhibere; </s> <s id="N282E1"><!-- NEW -->&longs;it flagellum plicatile DAB, &longs;itque AB &longs;olum areæ horizon­<lb/>ti parallelum; </s> <s id="N282E7"><!-- NEW -->porrò &longs;it AB extremus fu&longs;tis, qui voluitur circa cen­<lb/>trum A; </s> <s id="N282ED"><!-- NEW -->DA verò &longs;it primus fu&longs;tis ad in&longs;tar manubrij volubilis circa <lb/>centrum D; </s> <s id="N282F3"><!-- NEW -->&longs;it autem circellus DO, EF, & brachium LMD, cuius <lb/>contractione dum erigitur flagellum, extremitas B de&longs;cribit &longs;ecirculum <lb/>DOE, & A curuam AXG in a&longs;cen&longs;u, in de&longs;cen&longs;u GTA; </s> <s id="N282FB"><!-- NEW -->B verò in <lb/>a&longs;cen&longs;u curuam BECK, in de&longs;cen&longs;u denique curuam KRB: </s> <s id="N28301"><!-- NEW -->itaque <lb/>motus extremitas D mouetur motu circulari; </s> <s id="N28307"><!-- NEW -->A verò motu mixto ex <lb/>circulari duplici, &longs;cilicet punctorum A & D; </s> <s id="N2830D"><!-- NEW -->D quidem per circellum <lb/>DFEO; </s> <s id="N28313"><!-- NEW -->A verò per arcum AC, denique B motu mixto ex tribus cir­<lb/>cularibus D &longs;cilicet in circello DFEO, A in arcu AC, B denique in <lb/>circulo ABS; </s> <s id="N2831B"><!-- NEW -->igitur B mouetur integro circulo circa A, A circa D per <lb/>arcum AC, & D circa Y integro etiam circulo; </s> <s id="N28321"><!-- NEW -->vbi verò A e&longs;t in G, & <lb/>D in E, B e&longs;t in H; </s> <s id="N28327"><!-- NEW -->mouetur autem B velociùs quàm A, tùm in a&longs;cen&longs;u, <pb pagenum="397" xlink:href="026/01/431.jpg"/>tùm de&longs;cen&longs;u; </s> <s id="N28330"><!-- NEW -->quia tota GH eodem in&longs;tanti cadit in AB; quippe H <lb/>participat motum A per GA, & motum D per ED, quod clari&longs;&longs;imum <lb/>e&longs;t. </s> </p> <p id="N28338" type="main"> <s id="N2833A"><!-- NEW -->Duodecimò, maior e&longs;t ictus, &longs;i initio de&longs;cen&longs;us fu&longs;tis AB tantillùm <lb/>retrò inclinet, vt GH; </s> <s id="N28340"><!-- NEW -->quia B ab H in B plùs temporis ponit, quàm à <lb/>Q, vt patet; </s> <s id="N28346"><!-- NEW -->igitur diutiùs potentia manet applicata; </s> <s id="N2834A"><!-- NEW -->igitur maiorem <lb/>impetum producit; </s> <s id="N28350"><!-- NEW -->igitur maior e&longs;t ictus; </s> <s id="N28354"><!-- NEW -->debet autem in eo &longs;itu e&longs;&longs;e, <lb/>in quo motus A in G ita temperetur cum motu B in H, vt eodem mo­<lb/>mento vtrumque feriat planum AB; </s> <s id="N2835C"><!-- NEW -->&longs;i enim vel A attingat antè B, vel <lb/>B antè A, minor e&longs;t ictus, vt con&longs;tat; </s> <s id="N28362"><!-- NEW -->quia totus motus &longs;imul non im­<lb/>peditur; </s> <s id="N28368"><!-- NEW -->pote&longs;t autem cogno&longs;ci ille &longs;itus vel illa inclinatio cognita pro­<lb/>portione motus circularis circa D, & circa A; </s> <s id="N2836E"><!-- NEW -->immò ni&longs;i retineatur <lb/>DA; </s> <s id="N28374"><!-- NEW -->haud dubiè A tanget &longs;olum AB ex G, antequam B de&longs;cendat in B <lb/>ex H; </s> <s id="N2837A"><!-- NEW -->igitur attemperandus e&longs;t motus fu&longs;tis DA; </s> <s id="N2837E"><!-- NEW -->præterea pondus in <lb/>de&longs;cen&longs;u auget ictum, deinde B de&longs;cendit deor&longs;um motu orbis & motu <lb/>centri: </s> <s id="N28386"><!-- NEW -->præterea B pote&longs;t in a&longs;cen&longs;u maiorem arcum &longs;ui orbis decurre­<lb/>re, quàm in de&longs;cen&longs;u, vel æqualem: denique maior e&longs;t ictus quando po­<lb/>tentia toto ni&longs;u euidente fu&longs;tis AB plùs temporis ante ictum in &longs;uo mo­<lb/>tu in&longs;umit. </s> </p> <p id="N28390" type="main"> <s id="N28392"><!-- NEW -->Decimotertiò, e&longs;t etiam aliud flagelli genus pluribus catenulis ferreis <lb/>in&longs;tructi, ex quibus &longs;ingulis &longs;inguli ferrei globi aliquando &longs;piculis, & <lb/>clauis armati pendent, quorum graui&longs;&longs;imus e&longs;t ictus propter rationes <lb/>prædictas; </s> <s id="N2839C"><!-- NEW -->præ&longs;ertim cùm catenula, &longs;eu funiculus, faciliùs adduci, & in­<lb/>flecti po&longs;&longs;it, quàm extremus ille fu&longs;tis, de quo &longs;uprà; </s> <s id="N283A2"><!-- NEW -->neque dee&longs;t ar­<lb/>tificium; quo quis hoc armorum genere vtens etiam contra plures &longs;e&longs;e <lb/>tueri po&longs;&longs;it. </s> </p> <p id="N283AA" type="main"> <s id="N283AC">Decimoquartò, denique vulgare e&longs;t phœnomenum illud funiculi, &longs;en <lb/>flagelli, quo &longs;cilicet initio remouetur manubrij extremitas, mox &longs;tatim <lb/>adducitur, ex qua productione, & adductione per vndantem funem <lb/>propagatur impetus v&longs;que ad eiu&longs;dem extremitatem nodo vt plurimùm <lb/>ad&longs;trictam. </s> <s id="N283B7"><!-- NEW -->Hinc primò &longs;trepitus ille aurigis familiari&longs;&longs;imus; </s> <s id="N283BB"><!-- NEW -->quippe <lb/>maxima fit aëris colli&longs;io in extremo fune; immo, & partium ten&longs;io, &longs;eu <lb/>di&longs;tractio propter motus illos contrarios productionis. </s> <s id="N283C3">Secundò, hinc <lb/>di&longs;trahitur funis, & qua&longs;i laceratur, di&longs;tractis &longs;cilicet tenui&longs;&longs;imis illis <lb/>filamentis, ex quibus con&longs;tat. </s> <s id="N283CA">Tertiò, hinc &longs;tringitur illa extremitas no­<lb/>do, tùm vt acrior &longs;it ictus, tùm vt filamenta illa nodo illo contineantur. </s> <s id="N283CF"><!-- NEW --><lb/>Quartò, duplex e&longs;t motus illius funis propter flexibilitatem; </s> <s id="N283D4"><!-- NEW -->hinc illæ <lb/>vndæ &longs;eu &longs;piræ; </s> <s id="N283DA"><!-- NEW -->nam remouetur caput funis, quod deinde &longs;equuntur <lb/>aliæ partes per &longs;inuo&longs;os flexus; &longs;ed mox vbi adducitur idem caput, maios <lb/>impetus producitur in aliis partibus. </s> <s id="N283E2"><!-- NEW -->Quintò, currentes vndæ &longs;eu flexus <lb/>adductionis, quæ fit maiore impetu, quàm productio, tandem in <lb/>primos flexus &longs;inuatos ab ip&longs;a productione incurrunt: hinc augetur <lb/>impetus, & motus extremitatis. </s> <s id="N283EC"><!-- NEW -->Sextò, adde quod licèt &longs;it tantùm, vel <lb/>productio, vel adductio flagelli, &longs;unt iidem &longs;erè effectus, &longs;ed minimè <lb/>æquales, quia augetur continuò motus flexuum; </s> <s id="N283F4"><!-- NEW -->tùm quia funis ver&longs;us <pb pagenum="398" xlink:href="026/01/432.jpg"/>extremitatem &longs;en&longs;im imminuitur; </s> <s id="N283FD"><!-- NEW -->tùm quia minor e&longs;t radius illius mo­<lb/>tus, quia circulari incipit: hinc extremitas funis veloci&longs;&longs;imè tandem <lb/>mouetur, & impacta acuti&longs;&longs;imum ictum incutit. </s> <s id="N28405"><!-- NEW -->Septimò, ob&longs;erua pro­<lb/>pter illam inflexionem motum diutiùs per&longs;euerare; </s> <s id="N2840B"><!-- NEW -->igitur potentia <lb/>manet diutiùs applicata; </s> <s id="N28411"><!-- NEW -->igitur maiorem effectum producit, vnde re­<lb/>uocare pote&longs;t: hunc effectum ad illud phænomenum baculi flexibilis, <lb/>de quo &longs;uprà. </s> <s id="N28419">Octauò, hinc pueri &longs;trophiolis prædicto modo inflexis <lb/>inter &longs;e contendunt, pro quo e&longs;t eadem ratio. </s> <s id="N2841E"><!-- NEW -->Nonò, hinc vt excutiatur <lb/>puluis ex pannis, eodem modo &longs;uccutiuntur; </s> <s id="N28424"><!-- NEW -->tùm propter ten&longs;ionem <lb/>filorum, quæ pulueri liberiores meatus aperit; </s> <s id="N2842A"><!-- NEW -->tùm propter vibrationes <lb/>quæ puluerem abigunt: </s> <s id="N28430"><!-- NEW -->immò flexibus aduer&longs;is tapetes ita &longs;uccutiun­<lb/>tur, vt flexus hinc inde currentes qua&longs;i tumentes fluctus, &longs;ibi inuicem <lb/>occurrant in medio tapete, & allidantur; </s> <s id="N28438"><!-- NEW -->hinc &longs;equitur ten&longs;io; </s> <s id="N2843C"><!-- NEW -->hinc <lb/>vibratio, pulueris excu&longs;&longs;io, hinc etiam &longs;trepitus; denique clari&longs;&longs;imè vi­<lb/>dentur flexus illi volubiles in exten&longs;a mappa, quorum ratio patet ex <lb/>dictis. </s> </p> <p id="N28446" type="main"> <s id="N28448"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N28454" type="main"> <s id="N28456"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt omnia percu&longs;&longs;ionum phœnomena, quæ fiunt opera mallei,<emph.end type="italics"/><lb/>hîc con&longs;ideratur malleus qua&longs;i incu&longs;&longs;us circulari motu, qui nullo mo­<lb/>do coniunctus &longs;it cum motu naturali deor&longs;um, quod tamen infrà ex­<lb/>plicabimus; hoc po&longs;ito. </s> </p> <p id="N28464" type="main"> <s id="N28466"><!-- NEW -->Primò, quò maior e&longs;t malleus eodem arcu impactus & manubrio, <lb/>maior e&longs;t ictus, quia tardiùs mouetur; </s> <s id="N2846C"><!-- NEW -->igitur potentia manet diutiùs <lb/>applicata; igitur maior e&longs;t ictus, vt con&longs;tat ex dictis. </s> </p> <p id="N28472" type="main"> <s id="N28474"><!-- NEW -->Secundò, hinc ex hac hypothe&longs;i ictus &longs;unt in ratione &longs;ubduplicata <lb/>ponderum malleorum; con&longs;tat etiam, po&longs;ita &longs;cilicet eadem longitudine <lb/>manubrij. </s> </p> <p id="N2847C" type="main"> <s id="N2847E">Tertiò, maior incutitur ictus non quidem circa extremitatem <lb/>ba&longs;is mallei, nec circa medium, &longs;ed circa mediam proportionalem <lb/>inter diametrum ba&longs;is, & &longs;ubduplum, patet per Th. 73. l. <!-- REMOVE S-->1. Co­<lb/>rol. <!-- REMOVE S-->4. <!-- KEEP S--></s> </p> <p id="N2848C" type="main"> <s id="N2848E"><!-- NEW -->Quartò, &longs;i &longs;it longius manubrium mallei, maiorem ictum infliget; </s> <s id="N28492"><!-- NEW --><lb/>quia tardius maiorem arcum decurrit, quàm minorem; </s> <s id="N28497"><!-- NEW -->igitur potentia <lb/>manet diutiùs applicata; </s> <s id="N2849D"><!-- NEW -->igitur maiorem effectum producit; </s> <s id="N284A1"><!-- NEW -->quod au­<lb/>tem tardiùs &longs;uum arcum perficiat maior radius, patet experientia ma­<lb/>ioris perticæ & breuioris fu&longs;tis; cuius ratio e&longs;t, quia idem impetus ma­<lb/>iori moli impre&longs;&longs;us remi&longs;&longs;ior e&longs;t, quia &longs;cilicet pluribus partibus di&longs;tri­<lb/>buitur. </s> </p> <p id="N284AD" type="main"> <s id="N284AF"><!-- NEW -->Quintò, velocitates extremitatum, po&longs;ita diuer&longs;a longitudine manu­<lb/>brij, &longs;unt vt ip&longs;æ longitudines permutando: </s> <s id="N284B5"><!-- NEW -->probatur, quia cùm &longs;it mo­<lb/>tus acceleratus, &longs;patia &longs;unt vt quadrata temporum; &longs;ed velocitates <lb/>&longs;unt vt tempora, & tempora &longs;unt in ratione &longs;ubduplicata &longs;patiorum. </s> <s id="N284BD"><!-- NEW --><lb/>id e&longs;t, vt diametri quadratorum, id e&longs;t, vt longitudines, &longs;it enim lon­<lb/>gitudo AB, quæ dato tempore H decurrat &longs;patium ABF, potentia <pb pagenum="399" xlink:href="026/01/433.jpg"/>&longs;cilicet toto ni&longs;u applicata, &longs;it etiam longitudo AC dupla AB: </s> <s id="N284C9"><!-- NEW -->dico <lb/>quod eodem tempore H acquiret æquale &longs;patium &longs;cilicet CAD; </s> <s id="N284CF"><!-- NEW -->igitur <lb/>CAD e&longs;t 1/4 CAG, quia e&longs;t æquale BAF; </s> <s id="N284D5"><!-- NEW -->igitur CD e&longs;t 1/4 CG, &longs;ed <lb/>CG e&longs;t duplus BF; </s> <s id="N284DB"><!-- NEW -->igitur CD e&longs;t &longs;ubduplus BF; </s> <s id="N284DF"><!-- NEW -->igitur velocitas ex­<lb/>tremitatis C in CA e&longs;t &longs;ubdupla velocitatis B in BA: </s> <s id="N284E5"><!-- NEW -->adde quod AC cùm <lb/>numerus partium AC &longs;it duplus numeri partium AB, & cùm in eadem <lb/>proportione di&longs;tribuatur impetus AC, & AB; certè partes maioris &longs;i <lb/>comparentur cum partibus proportionalibus minoris, &longs;ubduplam tan­<lb/>tùm habebunt portionem. </s> </p> <p id="N284F1" type="main"> <s id="N284F3"><!-- NEW -->Sextò, ictus inflicti à malleis, quorum manubria diuer&longs;am longitu­<lb/>dinem habent, &longs;uppo&longs;ito eodem angulo, &longs;unt vt longitudines; </s> <s id="N284F9"><!-- NEW -->&longs;i enim <lb/>eo tempore, quo AB facit &longs;patium BAF, AC facit CAD; </s> <s id="N284FF"><!-- NEW -->certè æquali <lb/>tempore AC faciet DAG, vt con&longs;tat ex natura motus accelerati; </s> <s id="N28505"><!-- NEW --><lb/>igitur acquirit <expan abbr="tantũdem">tantundem</expan> impetus; </s> <s id="N2850E"><!-- NEW -->&longs;ed eo tempore, quo AC decurrit <lb/>CAD, acquirit æqualem impetum AB dum percurrit BAF, vt patet ex <lb/>dictis; </s> <s id="N28516"><!-- NEW -->igitur AC decur&longs;o CAG habet duplum impetum AB decur&longs;o <lb/>BAF; </s> <s id="N2851C"><!-- NEW -->igitur dupla e&longs;t vis ictus; </s> <s id="N28520"><!-- NEW -->igitur ictus &longs;unt in ratione &longs;ubdupli­<lb/>cata CAG, BAF; igitur vt ACAB. </s> </p> <p id="N28526" type="main"> <s id="N28528"><!-- NEW -->Septimò, diceret aliquis velocitatem C decur&longs;o CD, e&longs;&longs;e &longs;ubduplam <lb/>velocitatis B decur&longs;o BF; </s> <s id="N2852E"><!-- NEW -->&longs;ed velocitas C, decur&longs;o CG, e&longs;t dupla velo­<lb/>citatis eiu&longs;dem C decur&longs;o CD; </s> <s id="N28534"><!-- NEW -->igitur velocitas C, decur&longs;o CG, e&longs;t <lb/>æqualis velocitati B, decur&longs;o BF; igitur æqualis ictus. </s> <s id="N2853A"><!-- NEW -->Re&longs;p. conce&longs;&longs;a <lb/>primâ con&longs;equentiâ, vltimâ verò negatâ; </s> <s id="N28540"><!-- NEW -->quia non tantùm impetus <lb/>puncti C incutit ictum &longs;ed totius CA, qui cen&longs;etur e&longs;&longs;e collectus in <lb/>malleo in quo e&longs;t qua&longs;i centrum huius impetus, vt iam explicuimus <lb/>aliàs; &longs;ed velocitas totius CA confecto CAD e&longs;t æqualis velocitati <lb/>totius BA confecto BAF, cuius velocitas CA confecto CAG e&longs;t dupla, <lb/>vt iam probatum e&longs;t. </s> </p> <p id="N2854E" type="main"> <s id="N28550"><!-- NEW -->Octauò, hinc ictus CA confecto CAD e&longs;t æqualis ictui AB con­<lb/>fecto BAF, & ictus CA confecto CI duplo CD e&longs;t ad ictum CA con­<lb/>fecto CD, vt radix CA ad radicem CI: </s> <s id="N28558"><!-- NEW -->hinc vides hunc motum con­<lb/>uenire in eo cum recto, quòd &longs;cilicet ictus inflictus motu recto à mi­<lb/>nori mole, &longs;it ad ictum maioris, &longs;uppo&longs;ita linea motus æquali in ratio­<lb/>ne &longs;ubduplicata ponderum; quòd dicitur etiam de motu circulari duo­<lb/>rum fu&longs;tium inæqualium, quorum ictus &longs;unt in ratione &longs;ubduplicata <lb/>longitudinum, a&longs;&longs;umptis duntaxat arcubus æqualibus ab extremitate <lb/>vtriu&longs;que decur&longs;is. </s> </p> <p id="N28568" type="main"> <s id="N2856A"><!-- NEW -->Nonò, cum mallei &longs;unt diuer&longs;i ponderis, & longitudinis, facilè co­<lb/>gno&longs;ci poterit proportio ictuum; </s> <s id="N28570"><!-- NEW -->e&longs;t enim compo&longs;ita ex ratione lon­<lb/>gitudinum & &longs;ubduplicata ponderum v.g. <!-- REMOVE S-->&longs;it malleus A, cuius longitu­<lb/>do &longs;it 2. pondus 4. &longs;it malleus B cuius longitudo &longs;it pondus; </s> <s id="N2857A"><!-- NEW -->rectè ra­<lb/>tio longitudinum e&longs;t 2/3, & &longs;ubduplicata ponderum e&longs;t 2/3; </s> <s id="N28580"><!-- NEW -->ducatur vna <lb/>in aliam, vt euadat compo&longs;ita &longs;cilicet 4/1 vel longitudo A &longs;it I, & B 2; </s> <s id="N28586"><!-- NEW --><lb/>habebitur ratio &longs;ubduplicata ponderum 2/1, & ratio longitudinum 3/2; </s> <s id="N2858D"><!-- NEW --><lb/>ducatur vna in aliam, habebitur ratio compo&longs;ita 2/2; </s> <s id="N28594"><!-- NEW -->igitur &longs;unt æqua-<pb pagenum="400" xlink:href="026/01/434.jpg"/>les, quæ omnia facilè intelliguntur ex dictis; </s> <s id="N2859D"><!-- NEW -->itaque habes 4. combina­<lb/>tiones duorum malleorum; </s> <s id="N285A3"><!-- NEW -->vel enim e&longs;t idem pondus vtrique, & ea­<lb/>dem longitudo, vel idem pondus, &longs;ed diuer&longs;a longitudo, vel eadem lon­<lb/>gitudo & diuer&longs;um pondus, vel diuer&longs;um pondus & diuer&longs;a longitudo; <lb/>&longs;i verò e&longs;t diuer&longs;a longitudo &longs;imul, & diuer&longs;um pondus, vel eidem ine&longs;t <lb/>maius pondus, & maior longitudo, vel maior longitudo, & minus pon­<lb/>dus, & contrà alteri minor longitudo, & minus pondus, vel maius pon­<lb/>dus, & minor longitudo, quorum omnium proportiones &longs;unt determi­<lb/>natæ. </s> </p> <p id="N285B5" type="main"> <s id="N285B7"><!-- NEW -->Decimò, quod &longs;pectat ad cra&longs;&longs;itudinem manubrij, illa haud dubiè <lb/>auget aliquando vim ictus, aliquando imminuit; </s> <s id="N285BD"><!-- NEW -->auget quidem, cum <lb/>malleus centrum impetus occupat eo modo, quo explicuimus l. <!-- REMOVE S-->1.Th.73. <lb/>Corol.4. imminuit verò cum ab eo centro recedit, vt manife&longs;tum e&longs;t ex <lb/>dictis ibidem, cum infligitur ictus eo mallei puncto, in quo non e&longs;t <lb/>prædictum centrum, formicat manus infligentis, vt patet experientiâ; </s> <s id="N285CB"><!-- NEW --><lb/>quippe extremitas illa manubrij, quæ manu tenetur, vel attollitur, vel <lb/>deprimitur; </s> <s id="N285D2"><!-- NEW -->attollitur quidem, &longs;i punctum contactus, vel ictus e&longs;t inter <lb/>prædictum centrum & manum; </s> <s id="N285D8"><!-- NEW -->& è contrario deprimitur, &longs;i centrum <lb/>ip&longs;um &longs;it inter punctum contactus & manum; </s> <s id="N285DE"><!-- NEW -->& quia manus im­<lb/>pedit, ne vel attollatur, vel deprimatur, impetus in illam qua­<lb/>&longs;i refunditur; </s> <s id="N285E6"><!-- NEW -->hinc illa formicatio non &longs;ine maximo &longs;æpiùs do­<lb/>loris &longs;en&longs;u; </s> <s id="N285EC"><!-- NEW -->denique ob&longs;erua nouem e&longs;&longs;e combinationes, &longs;i con­<lb/>&longs;iderentur in malleo longitudo, & latitudo manubrij cum ip&longs;o <lb/>pondere; quippe &longs;i 3. ducantur in 3. erunt 9. &longs;ed hæc &longs;unt fa­<lb/>cilia. </s> </p> <p id="N285F6" type="main"> <s id="N285F8"><!-- NEW -->Vndecimò, &longs;i malleus impingatur deor&longs;um cre&longs;cit ictus propter mo­<lb/>tum naturaliter acceleratum, additum &longs;cilicet extrin&longs;ecùs impre&longs;&longs;o; </s> <s id="N285FE"><!-- NEW --><lb/>&longs;i enim mallei cadunt ex eadem altitudine, &longs;untque eiu&longs;dem ponderis, <lb/>ictus æquales e&longs;&longs;e nece&longs;&longs;e e&longs;t; </s> <s id="N28605"><!-- NEW -->&longs;i verò &longs;unt eiu&longs;dem ponderis, & cadunt <lb/>ex diuer&longs;a altitudine impetus acqui&longs;iti motu naturali, &longs;unt in ratione <lb/>&longs;ubduplicata altitudinum; </s> <s id="N2860D"><!-- NEW -->&longs;i verò &longs;unt diuer&longs;i ponderis, & cadunt ex <lb/>diuer&longs;a altitudine, &longs;unt in ratione compo&longs;ita aliquomodo ex vtraque; </s> <s id="N28613"><!-- NEW --><lb/>dico aliquo modo, quia non e&longs;t omninò propria compo&longs;itio rationum; </s> <s id="N28618"><!-- NEW --><lb/>pote&longs;t tamen facilè proportio ictuum inueniri, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it malleus A, & <lb/>malleus B, ictus A ratione impetus impre&longs;&longs;i extrin&longs;ecus &longs;it vt 8, ratione <lb/>ca&longs;us &longs;it vt 2; </s> <s id="N28625"><!-- NEW -->at verò ictus B ratione impetus impre&longs;&longs;i &longs;it vt 6, ratione <lb/>ca&longs;us vt 3: </s> <s id="N2862B"><!-- NEW -->addantur 8, & 2 erunt 10; </s> <s id="N2862F"><!-- NEW -->adduntur 6, & 3 erunt 9; </s> <s id="N28633"><!-- NEW -->igitur <lb/>ictus &longs;unt in ratione (10/9), vt con&longs;tat: </s> <s id="N28639"><!-- NEW -->porrò quemadmodum nouus im­<lb/>petus accedit ratione motus naturalis, cum malleus impingitur deor­<lb/>&longs;um, ita aliquid impetus de&longs;truitur cum malleus impingitur &longs;ur&longs;um, vt <lb/>patet; </s> <s id="N28643"><!-- NEW -->denique, quia &longs;unt 5 termini, quos re&longs;picit ictus, &longs;cilicet pondus <lb/>mallei, longitudo manubrij, cra&longs;&longs;itudo arcus extremitatis, & linea &longs;ur­<lb/>&longs;um vel deor&longs;um, ita &longs;unt 25. combinationes ictuum; &longs;ed hoc fa­<lb/>cile e&longs;t. </s> </p> <pb pagenum="401" xlink:href="026/01/435.jpg"/> <p id="N28651" type="main"> <s id="N28653">Duodecimò, ictus eiu&longs;dem mallei per diuer&longs;os arcus &longs;unt in ra­<lb/>tione &longs;ubduplicata arcuum. </s> <s id="N28658"><!-- NEW -->v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it malleus AC arcus CD, tùm <lb/>arcus CG: </s> <s id="N28662"><!-- NEW -->dico ictus per vtrumque arcum e&longs;&longs;e in ratione &longs;ubdu­<lb/>plicata arcuum CD, EG, id e&longs;t in ratione 2/3, vt con&longs;tat ex dictis; <lb/>pote&longs;t etiam facilè inueniri proportio, &longs;i &longs;it diuer&longs;a longitudo, vel <lb/>diuer&longs;um pondus &c. </s> <s id="N2866C"><!-- NEW -->hinc ratio manife&longs;ta, cur per minimum ictum <lb/>nullus ferè &longs;it ictus: &longs;ed hæc ex dicendis infrà de ca&longs;u clari&longs;&longs;imè intel­<lb/>ligentur. </s> </p> <p id="N28674" type="main"> <s id="N28676">Decimotertiò, claua reduci debet ad malleum. </s> <s id="N28679"><!-- NEW -->Primò, deter­<lb/>minari pote&longs;t, ex quo puncto maiorem ictum infligit, quando mo­<lb/>uetur motu recto; </s> <s id="N28681"><!-- NEW -->&longs;it enim centrum grauitatis clauæ I, in quo &longs;i <lb/>&longs;u&longs;tineatur, &longs;tabit in æquilibrio; </s> <s id="N28687"><!-- NEW -->ducatur FIE, maiorem ictum <lb/>infliget ex puncto E, quia <expan abbr="tantũdem">tantundem</expan> e&longs;t impetus in &longs;egmento <lb/>FEK quantum in &longs;egmento FEA; </s> <s id="N28693"><!-- NEW -->igitur totus impeditur impe­<lb/>tus; igitur maximus erit ictus &longs;i infligat ictum motu circulari circa <lb/>aliud e&longs;t centrum percu&longs;&longs;ionis, de quo infrà. </s> <s id="N2869B"><!-- NEW -->Tertiò, hoc percu&longs;&longs;io­<lb/>nis organum validum ictum infligit propter illam extremam cra&longs;­<lb/>&longs;itudinem, e&longs;t enim quoddam mallei genus, & valdè periculo&longs;um; </s> <s id="N286A3"><!-- NEW --><lb/>præ&longs;ertim &longs;i ferreis clauis armetur; </s> <s id="N286A8"><!-- NEW -->hinc vulgò tribuitur Herculi tan­<lb/>quam in&longs;igne fortitudinis &longs;ymbolum; porrò tàm altè clauum infigit <lb/>&longs;ibi coniunctum, quam infigeret, &longs;i claua ip&longs;a erectum, & qua&longs;i expe­<lb/>ctantem ictum feriret. </s> </p> <p id="N286B2" type="main"> <s id="N286B4">Decimoquartò, Tudicula maior reuocatur ad malleum. </s> <s id="N286B7"><!-- NEW -->Primò <lb/>faciunt ad ictum longitudo manubrij, flexibilitas, inæqualitas, mal­<lb/>lei pondus, durities materiæ, arcus motus, vegetæ potentiæ vires; </s> <s id="N286BF"><!-- NEW --><lb/>omitto ea, quæ cum malleo habet communia, quorum ratio ex <lb/>dictis con&longs;tare pote&longs;t; igitur non videntur e&longs;&longs;e repetenda. </s> <s id="N286C6">Secundò, <lb/>flexibilitas manubrij auget vim ictus, tùm quia potentia diutiùs <lb/>manet applicata, cùm aliquo tempore in ip&longs;a vibratione malleus à <lb/>tergo relinquatur, tùm quia potentia illa media, de qua &longs;upra, &longs;uum <lb/>impetum, impetui alterius adiungit. </s> <s id="N286D1"><!-- NEW -->Tertiò, ita manubrium fa­<lb/>bricatur, vt continua imminutione ver&longs;us malleum decre&longs;cat, quod <lb/>multum facit ad ictum, quia hæc inæqualitas inflexioni re&longs;i&longs;tit ver­<lb/>&longs;us caput manubrij; </s> <s id="N286DB"><!-- NEW -->igitur initio inflectitur manubrium, non pro­<lb/>cul à malleo, tùm deinde aucto impetu in partibus remotioribus, <lb/>quæ difficiliùs inflectuntur; </s> <s id="N286E3"><!-- NEW -->igitur inæqualiter partes illæ redeunt, <lb/>atque &longs;e&longs;e pri&longs;tino &longs;tatui re&longs;tituunt; </s> <s id="N286E9"><!-- NEW -->atqui ex illa inæqualitate diu­<lb/>tiùs durat motus; </s> <s id="N286EF"><!-- NEW -->igitur inde maior euadit: </s> <s id="N286F3"><!-- NEW -->&longs;imile quid videmus in <lb/>arcu, cuius medium cra&longs;&longs;ius e&longs;t: adde quod &longs;i æqualis &longs;it cra&longs;&longs;itudo, <lb/>incipit inflexio ver&longs;us illam extremitatem, quæ propiùs accedit ad <lb/>manum, longiùs recedit à malleo, vt patet experientiâ, in fune, <lb/>virgâ &c. </s> <s id="N286FF"><!-- NEW -->&longs;ed de arcu, ten&longs;ione, compre&longs;&longs;ione fusè agemus <lb/>tractatu &longs;ingulari: </s> <s id="N28705"><!-- NEW -->hæc tantum obiter indica&longs;&longs;e &longs;ufficiat. <pb pagenum="402" xlink:href="026/01/436.jpg"/>Quartò, maximus e&longs;t ictus, cum malleus eo in&longs;tanti attingit pilam, quo <lb/>manubrium e&longs;t rectum; </s> <s id="N28710"><!-- NEW -->tunc enim e&longs;t modum vibrationis &longs;eu reditus; <lb/>igitur maximus impetus. </s> <s id="N28716"><!-- NEW -->Quintò, &longs;i altera extremitas mallei, quæ glo­<lb/>bum attingit, &longs;it obliqua, globum ip&longs;um attollit propter punctum con­<lb/>tactus; quod certè clarum e&longs;t. </s> <s id="N2871E"><!-- NEW -->Sextò, durities mallei multùm facit ad <lb/>ictum; </s> <s id="N28724"><!-- NEW -->&longs;i enim cedat lignum, imminuitur impetus, vt patet; hinc ar­<lb/>millâ, vel annulo ferreo armatur vtraque ba&longs;is mallei, vt firmior eua­<lb/>dat. </s> <s id="N2872C"><!-- NEW -->Septimò, globi ratio multa habenda e&longs;t, cui infligitur ictus; </s> <s id="N28730"><!-- NEW -->quippe <lb/>&longs;i leuior e&longs;t ab aëre ambiente impeditur, & retinetur; </s> <s id="N28736"><!-- NEW -->&longs;i verò mollior <lb/>minor ictus infligitur, quia cedit materies; </s> <s id="N2873C"><!-- NEW -->hinc pilæ è duriore buxo <lb/>tornantur; </s> <s id="N28742"><!-- NEW -->hinc etiam tunduntur pilæ malleo, vt materies den&longs;ior <lb/>euadat, impleanturque infinita ferè vacuola aëre plena, quæ pilam le­<lb/>uiorem reddunt; &longs;ed hæc ad emi&longs;&longs;ionem, & proiectionem pertinent, <lb/>de quibus infrà. </s> <s id="N2874C"><!-- NEW -->Octauò, vt recta via procedat pila debet in id punctum <lb/>malleus infligi, ex quo ducta per centrum pilæ linea, & deinde produ­<lb/>cta concurrat cum ip&longs;a linea directionis; nec enim aliter determinari <lb/>pote&longs;t linea motus globi per Th... l.1. hinc manubrium debet &longs;emper <lb/>facere angulos rectos cum linea directionis. </s> <s id="N28758"><!-- NEW -->Nonò, ad ictum inflictum <lb/>à maiori Tudicula tres potentiæ motrices concurrunt, &longs;cilicet ip&longs;a po­<lb/>tentia impellentis, potentia motus deor&longs;um, & ip&longs;a media; </s> <s id="N28760"><!-- NEW -->igitur hæc <lb/>ars in eo præ&longs;ertim po&longs;ita e&longs;t, quod hæ potentiæ ita temperentur, &longs;eu <lb/>componantur, vt vna non ob&longs;it alteri, & &longs;ingulæ pro viribus agat: ex <lb/>his alia facilè intelligentur. </s> </p> <p id="N2876A" type="main"> <s id="N2876C"><!-- NEW -->Decimoquintò, &longs;upere&longs;t familiaris ille &longs;oni effectus, quem mal­<lb/>leus cadens in incudem edit, quem tamen hîc non di&longs;cutiemus; quia <lb/>naturam & affectiones &longs;onorum alio Tomo de qualitatibus &longs;en&longs;ibilibus <lb/>libro &longs;ingulari fusè explicabimus. </s> </p> <p id="N28776" type="main"> <s id="N28778"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N28784" type="main"> <s id="N28786"><!-- NEW --><emph type="italics"/>Ex dictis explicaris po&longs;&longs;um omnia phœnomena, quæ ob&longs;eruantur in ludo <lb/>rudis gladiatoriæ<emph.end type="italics"/>; </s> <s id="N28792"><!-- NEW -->Primo, tria &longs;unt in hac arte, ad quæ reliqua facilè re­<lb/>ducuntur; </s> <s id="N28798"><!-- NEW -->primum e&longs;t declinatio; &longs;ecundum petitio; tertium confla­<lb/>tum ex vtroque. </s> <s id="N2879E">Secundò, pote&longs;t declinati, vel auerti ictus, &longs;eu petitio <lb/>duobus modis. </s> </p> <p id="N287A3" type="main"> <s id="N287A5">Primò, &longs;i declinatio cum aliqua impactione coniungatur. </s> </p> <p id="N287A8" type="main"> <s id="N287AA">Secundò, &longs;i tantùm cum mera re&longs;i&longs;tentia, vel &longs;implici impul­<lb/>&longs;ione. <lb/><arrow.to.target n="note4"/></s> </p> <p id="N287B3" type="margin"> <s id="N287B5"><margin.target id="note4"/>a <emph type="italics"/>Fig.<emph.end type="italics"/> 17 <lb/><emph type="italics"/>Tab.<emph.end type="italics"/> 5.<!-- KEEP S--></s> </p> <p id="N287C8" type="main"> <s id="N287CA"><!-- NEW -->Tertiò, vtriu&longs;que modi &longs;unt 4. combinationes; </s> <s id="N287CE"><!-- NEW -->&longs;iue enim duo gladij <lb/>AC, DF, capulares pilæ AD; </s> <s id="N287D4"><!-- NEW -->&longs;it autem gladius AC declinans petitio­<lb/>nem alterius DF; id certè quatuor modis præ&longs;tare pote&longs;t. </s> <s id="N287DA">Primò, &longs;i <lb/>punctum contactus ad mucronem vtriu&longs;que propiùs accedat. </s> <s id="N287DF"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;i <lb/>vterque &longs;it in &longs;itu ACDF. Secundò, &longs;i propiùs accedat ad capulum <lb/>vtriu&longs;que, talis e&longs;t &longs;itus DFCH. Tertiò, &longs;i accedat propiùs ad mu­<lb/>cronem gladij petentis DF, & propiùs ad capulum declinantis. <pb pagenum="403" xlink:href="026/01/437.jpg"/>Quartò, è contrario &longs;i accedat propiùs ad capulum petentis DF, & pro­<lb/>piùs ad mucronem declinantis; addi pote&longs;t quinta combinatio, cum <lb/>&longs;cilicet contactus e&longs;t in medio vtriu&longs;que. </s> </p> <p id="N287F6" type="main"> <s id="N287F8"><!-- NEW -->Quartò, &longs;i &longs;it mera impul&longs;io &longs;ine percu&longs;&longs;ione, vel impactione, maxi­<lb/>ma vis e&longs;t declinationis, cum punctum contactus accedit propiùs ad ca­<lb/>pulum declinantis, & ad mucronem petentis iuxta tertiam combinatio­<lb/>nem, & &longs;itum DFPE, & punctum contactus in B; </s> <s id="N28802"><!-- NEW -->ratio e&longs;t, cum verta­<lb/>tur PE circa P applicatæ potentiæ in P, maius e&longs;t momentum in B <lb/>quàm in alio puncto ver&longs;us E, vt patet; </s> <s id="N2880A"><!-- NEW -->quippe B mouetur minore motu; </s> <s id="N2880E"><!-- NEW --><lb/>igitur faciliùs; </s> <s id="N28813"><!-- NEW -->præterea FD mouetur circa D; igitur in B faciliùs pelli­<lb/>tur, quàm in vllo puncto ver&longs;us D ratione vectis. </s> </p> <p id="N28819" type="main"> <s id="N2881B"><!-- NEW -->Quintò, cum punctum contactus accedit propiùs ad capulum peten­<lb/>tis, & ad mucronem impellentis, minima vis e&longs;t declinationis, &longs;cilicet <lb/>iuxta quartam combinationem, & &longs;itum DFRG: ratio e&longs;t, quia minor <lb/>e&longs;t vis potentiæ applicatæ in R, & maior re&longs;i&longs;tentia applicatæ in D, vt <lb/>patet ex dictis. </s> </p> <p id="N28827" type="main"> <s id="N28829"><!-- NEW -->Sextò, cum punctum contactus accedit propiùs ad capulum vtrïu&longs;que <lb/>iuxta &longs;ecundam combinationem, & &longs;itum DFSH, tunc e&longs;t maxima vis <lb/>declinantis, & maxima re&longs;i&longs;tentia petentis; </s> <s id="N28831"><!-- NEW -->vnde vna compen&longs;atur ab <lb/>alia; </s> <s id="N28837"><!-- NEW -->cum verò punctum contactus accedit propiùs ad mucronem vtriu&longs;­<lb/>que, minima e&longs;t vis impellentis, & minima re&longs;i&longs;tentia impul&longs;i iuxta pri­<lb/>mam combinationem, & &longs;itum DFAC; ratio patet ex dictis. </s> </p> <p id="N2883F" type="main"> <s id="N28841"><!-- NEW -->Septimò, hinc tam facilè declinatur ictus gladij DF, &longs;iue fiat iuxta <lb/>primam combinationem, &longs;iue iuxta &longs;ecundam, quia licèt &longs;it minima vis <lb/>in prima; </s> <s id="N28849"><!-- NEW -->e&longs;t etiam minima re&longs;i&longs;tentia; </s> <s id="N2884D"><!-- NEW -->& licèt &longs;it maxima re&longs;i&longs;tentia <lb/>in &longs;ecunda, e&longs;t etiam maxima vis; </s> <s id="N28853"><!-- NEW -->igitur vna compen&longs;at aliam, vt patet; </s> <s id="N28857"><!-- NEW --><lb/>immò iuxta &longs;itum DFQK, po&longs;ito puncto contactus in L, & iuxta om­<lb/>nem alium &longs;itum, in quo punctum contactus æqualiter di&longs;tat à mucro­<lb/>ne vtriu&longs;que, vis declinantis æqualis e&longs;t; e&longs;t enim æqualis ratio virium, <lb/>& re&longs;i&longs;tentiæ, vt con&longs;tat, po&longs;ita vtriu&longs;que longitudine. </s> </p> <p id="N28862" type="main"> <s id="N28864"><!-- NEW -->Octauò, &longs;i verò impul&longs;io, vel declinatio fiat cum impactione, tribus <lb/>modis id fieri pote&longs;t; </s> <s id="N2886A"><!-- NEW -->primo, motu circulari circa pilam capularem A: </s> <s id="N2886E"><!-- NEW --><lb/>&longs;ecundo, motu circulari circa centrum di&longs;tans 3/4 à capulò, tertio, motu <lb/>recto ducto &longs;cilicet gladio dextror&longs;um, vel &longs;ini&longs;tror&longs;um horizonti pa­<lb/>rallelo; primus modus pe&longs;&longs;imus e&longs;t, quia totum corpus, defectum manet. </s> <s id="N28877"><lb/>Tertius proximè ad priorem accedit propter <expan abbr="eãdem">eandem</expan> rationem. </s> <s id="N2887F">Secun­<lb/>dus optimus omnium, & communis e&longs;t, quia &longs;emper gladius tegit <lb/>corpus. </s> </p> <p id="N28886" type="main"> <s id="N28888"><!-- NEW -->Nonò, &longs;i primo modo declinatur ictus repul&longs;o petentis gladio maxi­<lb/>ma vis erit; &longs;i punctum contactus fiat circa 2/3 de quo infrà, quod verò <lb/>&longs;pectat ad gladium, qui repellitur, eò faciliùs repellitur, quò punctum <lb/>contactus propiùs ad eius mucronem accedet. </s> <s id="N28892"><!-- NEW -->Si tertio modo, & gla­<lb/>dius &longs;olus ita libraretur maxima vis e&longs;&longs;et circa centrum eius grauitatis; </s> <s id="N28898"><!-- NEW --><lb/>in hoc enim puncto maximum ictum infligunt, quæ motu recto mo­<lb/>uentur; quia verò totum &longs;egmentum brachij, quod inter manum, & <pb pagenum="404" xlink:href="026/01/438.jpg"/>caput cubiti intercipitur, mouetur &longs;imul cum gladio motu recto, circa <lb/>capulum erit maxima vis, cùm propiùs accedat ad centrum grauitatis <lb/>totius conflati ex illo &longs;egmento brachij, & gladio. </s> </p> <p id="N288A8" type="main"> <s id="N288AA">Decimò, denique &longs;i &longs;ecundo modo declinetur ictus, idem dicendum <lb/>e&longs;t quod de motu circulari dictum, mutato dumtaxat centro, v.g. <!-- REMOVE S-->&longs;it gla­<lb/>dius declinantis RG, &longs;itque IG 1/4 totius RG circa I &longs;it motus circula­<lb/>tis, centrum percu&longs;&longs;ionis erit circa 2/3 IG, vel IR. </s> </p> <p id="N288B5" type="main"> <s id="N288B7"><!-- NEW -->Vndecimò, vix tamen ita acuratè hoc &longs;ecundo modo declinatur ictus, <lb/>quin tertius etiam cum &longs;ecundo coniunctus &longs;it, vt patet experientiâ; </s> <s id="N288BD"><!-- NEW --><lb/>rotatur autem manus declinantis vt illo qua&longs;i gyro maiorem impetum <lb/>acquirat, de quo iam &longs;uprà: immò ni&longs;i tertius modus cum &longs;ecundo e&longs;&longs;et <lb/>coniunctus, non po&longs;&longs;et delinari ictus, &longs;i contactus gladiorum fieret in <lb/>centro illius motus, vt patet. </s> </p> <p id="N288C8" type="main"> <s id="N288CA"><!-- NEW -->Duodecimò, quò longior e&longs;t gladius declinantis, cum iuxta mucro­<lb/>nem fit contactus &longs;ine impactione e&longs;t vis debilior, quàm e&longs;&longs;et in breuio­<lb/>re, patet ex vecte; &longs;i verò &longs;it impactio iuxta &longs;ecundum. </s> <s id="N288D2"><!-- NEW -->n.10. vis maior <lb/>e&longs;t cum gladius longior e&longs;t; </s> <s id="N288D8"><!-- NEW -->e&longs;t enim maior motus; </s> <s id="N288DC"><!-- NEW -->igitur maior ictus li­<lb/>cèt tardior; </s> <s id="N288E2"><!-- NEW -->hinc longiore gladio equidem fortiùs auertitur ictus quàm <lb/>breuiore, &longs;ed tardiùs; breuiore verò citiùs quàm longiore, &longs;ed debiliùs, <lb/>vt patet ex dictis. </s> </p> <p id="N288EA" type="main"> <s id="N288EC">Decimotertiò, longior gladius &longs;u&longs;tinetur facilè opera capularis pilæ, <lb/>quæ momentum longitudinis gladij &longs;upplet, vt con&longs;tat ex &longs;tatera, cuius <lb/>proportiones videbimus lib.&longs;eq. </s> <s id="N288F3">quippe &longs;i pila faciat æquipendium, cum <lb/>lamella manus &longs;u&longs;tinet tantùm pondus ab&longs;olutum &longs;ine momento, &c. </s> </p> <p id="N288F8" type="main"> <s id="N288FA"><!-- NEW -->Decimoquartò, hinc gladius, qui in mucronem ita de&longs;init, vt ea por­<lb/>tio, quæ ad capulum propiùs accedit, &longs;it cra&longs;&longs;ior, faciliùs &longs;u&longs;tineri pote&longs;t, <lb/>licèt &longs;it eiu&longs;dem ponderis cum alio; quia &longs;cilicet non e&longs;t tantum mo­<lb/>mentum. </s> </p> <p id="N28904" type="main"> <s id="N28906"><!-- NEW -->Decimoquintò, mucro intentatus per lineam rectam horizonti pa­<lb/>rallelus difficiliùs excipitur, & auertitur; </s> <s id="N2890C"><!-- NEW -->certa e&longs;t experientia, cuius <lb/>ratio in promptu e&longs;t, quia vel gladius declinantis e&longs;t horizonti paralle­<lb/>lus, vel non parallelus: &longs;i primum; </s> <s id="N28914"><!-- NEW -->igitur vix excipere pote&longs;t, quia cum <lb/>alia non decu&longs;&longs;atur; &longs;i verò &longs;ecundum; </s> <s id="N2891A"><!-- NEW -->plùs æquo demitti capulum opor­<lb/>tet; </s> <s id="N28920"><!-- NEW -->hinc non modò manus debilior e&longs;t; </s> <s id="N28924"><!-- NEW -->verùm etiam corpus detegitur: <lb/>adde quod ictus validior e&longs;t per lineam perpendicularem. </s> </p> <p id="N2892A" type="main"> <s id="N2892C"><!-- NEW -->Decimo&longs;extò, hinc ita debet extremitas manus per horizontalem <lb/>porrigi & brachium contractum explicari, vt maiorem lineam rectam <lb/>de&longs;cribat; </s> <s id="N28934"><!-- NEW -->acquiritur enim maior impetus in maiori &longs;patio, quod per­<lb/>curritur motu accelerato, vt con&longs;tat ex dictis, &longs;ed quò brachium con­<lb/>tractius e&longs;t, cò maiorem lineam eius extremitas motu recto decurrit: <lb/>adde quod impre&longs;&longs;io totius corporis, quod in <expan abbr="eãdem">eandem</expan> partem agitur, <lb/>multùm auget vim brachij mucronem in aduer&longs;um pectus inten­<lb/>tantis. </s> </p> <p id="N28946" type="main"> <s id="N28948">Decimo&longs;eptimò, &longs;i longior e&longs;t gladius impetus, hæc videntur e&longs;&longs;e <lb/>commoda. </s> <s id="N2894D"><!-- NEW -->Primò, eius mucro longiùs producitur, & procul attingit. <pb pagenum="405" xlink:href="026/01/439.jpg"/>Secundò maiorem ictum infligit, vt iam &longs;upra dictum e&longs;t de &longs;ari&longs;&longs;a, mo­<lb/>dò in eadem ratione aucta &longs;it cra&longs;&longs;itudo; non de&longs;unt tamen incommo­<lb/>da. </s> <s id="N2895A">Primò ratione vectis maius e&longs;t illius pondus. </s> <s id="N2895D">Secundò faciliùs de­<lb/>clinatur ictus propter <expan abbr="eãdem">eandem</expan> rationem. </s> <s id="N28966">Tertiò, &longs;i tantillùm deflecte­<lb/>tur, corpus omninò detegit propter maiorem cum, &longs;unt enim arcus <lb/>vt radij, vel longitudines. </s> <s id="N2896D">Quartò, hinc pugiles faciliùs decu&longs;&longs;atis gla­<lb/>dijs &longs;e&longs;e mutuò præhendunt, & luctâ decernunt. </s> </p> <p id="N28972" type="main"> <s id="N28974"><!-- NEW -->Decimooctauò, ni&longs;i per lineam horizontali parallelam mucro in&longs;en­<lb/>tetur, minor e&longs;t vis ictus, quia obliquè cadit; </s> <s id="N2897A"><!-- NEW -->igitur debilior e&longs;t: </s> <s id="N2897E"><!-- NEW -->&longs;i porrò <lb/>extante brachio mucro intenditur; haud dubiè ictus obliquus erit, cùm <lb/>circa extremum humerum brachium libretur. </s> </p> <p id="N28986" type="main"> <s id="N28988">Decimononò, cum auertitur, &longs;eu repellitur impetus gladius, ferro <lb/>directo id fieri debet, &longs;cilicet iuxta &longs;ecundum modum n. </s> <s id="N2898D"><!-- NEW -->10. alioquin <lb/>ferrum læuigatum in alio læuigato facilè decurrit, &longs;i obliquè in ip&longs;um <lb/>cadat; </s> <s id="N28995"><!-- NEW -->porrò ex hac repercu&longs;&longs;ione mucro impetens mouetur motu mixto, <lb/>dextror&longs;um &longs;cilicet vel &longs;ini&longs;tror&longs;um declinante: hinc qui impetit id po­<lb/>ti&longs;&longs;imum curare debet, vt eius ferrum ferro alterius obliquè accidat. </s> </p> <p id="N2899D" type="main"> <s id="N2899F"><!-- NEW -->Vige&longs;imò, eodem ni&longs;u pote&longs;t quis ictum aduer&longs;arij declinare, ip&longs;ique <lb/>adeo ictum infligere, quod gladiatoribus valde familiare e&longs;t; </s> <s id="N289A5"><!-- NEW -->hinc autem <lb/>&longs;ingulari motu mouetur manus, mixto &longs;cilicet ex recto, & circulari; cir­<lb/>culari quidem iuxta &longs;ecundum modum traditum n. </s> <s id="N289AD">10. recto verò iuxta <lb/>modum traditum n.15. quod certè &longs;i expeditè, & accuratè fiat, imparatus <lb/>ho&longs;tis intercipitur, vt vix ictum excipere po&longs;&longs;it. </s> </p> <p id="N289B4" type="main"> <s id="N289B6"><!-- NEW -->Vige&longs;imoprimò, ita ho&longs;tis gladio impeti debet, vt corpus impetentis <lb/>tectum remaneat: omitto alia, quæ ad hanc artem pertinent v.g corporis <lb/>&longs;itum, gladiorum temperaturam, cochleam gladij, &c. </s> <s id="N289BE"><!-- NEW -->quæ cùm ad mo­<lb/>tum minimè &longs;pectent, huius loci e&longs;&longs;e non po&longs;&longs;unt: </s> <s id="N289C4"><!-- NEW -->omitto etiam illos <lb/>ictus, qui cæ&longs;im infliguntur, quia ex dictis de baculo &longs;uprà facilè intelli­<lb/>gi po&longs;&longs;unt; denique omitto varios illos gladij breuioris latiori&longs;que gyros, <lb/>quibus &longs;e&longs;e qua&longs;i, vt vulgò aiunt, induit qui contra plures &longs;e&longs;e tuetur. </s> </p> <p id="N289CE" type="main"> <s id="N289D0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N289DC" type="main"> <s id="N289DE"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena percu&longs;&longs;ionis, quæ infligitur à <lb/>corpore graui deor&longs;um &longs;ua &longs;ponte cadente motu naturaliter accele­<lb/>rato.<emph.end type="italics"/></s> </p> <p id="N289E9" type="main"> <s id="N289EB"><!-- NEW -->Primò, corpus graue cadens ex maiore altitudine fortiùs ferit: ratio <lb/>e&longs;t; </s> <s id="N289F1"><!-- NEW -->quia de&longs;cendit motu naturaliter accelerato; </s> <s id="N289F5"><!-- NEW -->igitur maiorem acqui­<lb/>rit impetum; </s> <s id="N289FB"><!-- NEW -->igitur maiorem impetum ad extra producit; igitur maio­<lb/>rem ictum infligit. </s> </p> <p id="N28A01" type="main"> <s id="N28A03"><!-- NEW -->Secundò, &longs;unt 4. combinationes grauium; </s> <s id="N28A07"><!-- NEW -->vel enim e&longs;t idem pondus <lb/>e&longs;t altitudo; </s> <s id="N28A0D"><!-- NEW -->vel idem pondus, diuer&longs;a altitudo; </s> <s id="N28A11"><!-- NEW -->vel eadem altitudo di­<lb/>uer&longs;um pondus; </s> <s id="N28A17"><!-- NEW -->vel diuer&longs;um pondus & diuer&longs;a altitudo; addi pote&longs;t <lb/>diuer&longs;us incidentiæ angulus, immò diuer&longs;a figura corporis cadentis, quæ <lb/>omnia infrà demon&longs;trabimus. </s> </p> <p id="N28A1F" type="main"> <s id="N28A21"><!-- NEW -->Tertiò, &longs;i &longs;it æquale pondus, & æqualis altitudo &longs;uppo&longs;ito ca&longs;u <pb pagenum="406" xlink:href="026/01/440.jpg"/>perpendiculari æquales &longs;unt ictus, patet; quia eadem cau&longs;a <expan abbr="eũdem">eundem</expan> ha<lb/>bet effectum. </s> </p> <p id="N28A31" type="main"> <s id="N28A33"><!-- NEW -->Quartò, &longs;i &longs;it æquale pondus, & inæqualis altitudo, ictus &longs;unt in ra­<lb/>tione &longs;ubduplicata altitudinum v.g. <!-- REMOVE S-->&longs;it altitudo 4. cubitorum, & altera <lb/>tantum cubitalis; </s> <s id="N28A3D"><!-- NEW -->certè cùm acquirantur æqualibus temporibus æqua­<lb/>lia velocitatis momenta, velocitates acqui&longs;itæ &longs;unt vt tempora, impetus <lb/>vt velocitates, ictus vt impetus; </s> <s id="N28A45"><!-- NEW -->&longs;ed tempora &longs;unt in ratione &longs;ubdupli­<lb/>cata &longs;patiorum vel altitudinem; </s> <s id="N28A4B"><!-- NEW -->igitur & ictus; igitur ictus inflictus à <lb/>corpore cadente ex altitudine 4. cubitorum e&longs;t duplus ictus eiu&longs;dem <lb/>corporis cadentis ex altitudine cubitali. </s> </p> <p id="N28A53" type="main"> <s id="N28A55"><!-- NEW -->Quintò, &longs;i &longs;it æqualis altitudo, & diuer&longs;um pondus, ictus per &longs;e &longs;unt <lb/>vt pondera: </s> <s id="N28A5B"><!-- NEW -->probatur facilè, quia e&longs;t duplus impetus in corpora duplo, <lb/>non quidem ratione inten&longs;ionis, &longs;ed ratione exten&longs;ionis, vt patet: dixi <lb/>per &longs;e, quia diuer&longs;a ratio re&longs;i&longs;tentiæ medij hanc proportionem mutare <lb/>pote&longs;t. </s> </p> <p id="N28A65" type="main"> <s id="N28A67"><!-- NEW -->Sextò, &longs;i &longs;int infinita in&longs;tantia, e&longs;t infinita proportio inter actum in­<lb/>flictum à corpore cadente, & vim grauitationis eiu&longs;dem; </s> <s id="N28A6D"><!-- NEW -->quia dato quo­<lb/>cunque tempore po&longs;&longs;et dari minus, & minus; igitur dato quocunque <lb/>ictu po&longs;&longs;et dari minor, & minor in infinitum, quod ex illa hypothe&longs;i <lb/>nece&longs;&longs;ariò con&longs;equitur. </s> </p> <p id="N28A77" type="main"> <s id="N28A79"><!-- NEW -->Septimò, immò &longs;i &longs;int infinita in&longs;tantia, &longs;ique infinita proportio in­<lb/>ter ictum inflictum à corpore cadente, & vim grauitationis eiu&longs;dem, e&longs;t <lb/>etiam infinita proportio inter <expan abbr="eumd&etilde;">eumdem</expan> ictum, & vim grauitationis cuiu&longs;­<lb/>libet alterius corporis quantumuis immen&longs;i, inter duas grauitationes <lb/>duorum corporum datur proportio, vt con&longs;tat; </s> <s id="N28A89"><!-- NEW -->&longs;unt enim vt pondera; <lb/>igitur &longs;i nullam habet proportionem cum ictu corporis grauis cadentis, <lb/>nullam etiam habebit altera, vt patet ex elementis. </s> </p> <p id="N28A91" type="main"> <s id="N28A93"><!-- NEW -->Octauò, hinc negamus e&longs;&longs;e infinita illa in&longs;tantia; </s> <s id="N28A97"><!-- NEW -->quia ex illa hypothe­<lb/>&longs;i hoc ab&longs;urdum nece&longs;&longs;ariò &longs;equitur, quod experimento repugnat; quis <lb/>enim neget maiorem e&longs;&longs;e vim 100000. librarum ferri in modicum cy­<lb/>lindrum plumbi incubantis, quàm modici granuli in <expan abbr="eũdem">eundem</expan> cylin­<lb/>drum ex altitudine lineæ cadentis. </s> </p> <p id="N28AA7" type="main"> <s id="N28AA9">Nonò, &longs;i altitudo &longs;it diuer&longs;a, & pondus diuer&longs;um, ictus &longs;unt in ratione <lb/>compo&longs;ita ex ratione ponderum, & &longs;ubduplicata altitudinum, patet ex <lb/>dictis. </s> </p> <p id="N28AB0" type="main"> <s id="N28AB2">Decimò, &longs;i &longs;int infinita in&longs;tantia dato ictu cuiu&longs;libet corporis caden­<lb/>tis ex quacunque altitudine, non pote&longs;t dari vlla corporis moles, qua <lb/>&longs;uo pondere id præ&longs;tat, quod illud præ&longs;titit &longs;uo ca&longs;u. </s> <s id="N28ABB">Probatur ex n. </s> <s id="N28ABE"><!-- NEW -->7. <lb/>hinc fru&longs;trà proponitur hæc quæ&longs;tio ab ijs, qui agno&longs;cunt infinitos tar­<lb/>ditatis gradus, per quos propagatur motus; nam reuerâ ex hac hypothe&longs;i <lb/>e&longs;t infinita proportio inter ictum, & vim grauitationis. </s> </p> <p id="N28AC8" type="main"> <s id="N28ACA"><!-- NEW -->Vndecimò, &longs;i tamen ponantur finita in&longs;tantia; </s> <s id="N28ACE"><!-- NEW -->haud dubiè hæc pro­<lb/>po&longs;itio non e&longs;t infinita; </s> <s id="N28AD4"><!-- NEW -->&longs;it enim quodlibet corpus cadens ex quacun­<lb/>que data altitudine per 100. in&longs;tantia, &longs;eu partes temporis æquales pri­<lb/>mo in&longs;tanti quo mouetur; </s> <s id="N28ADC"><!-- NEW -->haud dubiè ictus ab eo inflictus cadendo e&longs;t <pb pagenum="407" xlink:href="026/01/441.jpg"/>ad vim grauitationis eiu&longs;dem vt 1001. ad 1. cùm enim &longs;ingulis in&longs;tan­<lb/>tibus æqualibus acquirantur æqualia velocitatis momenta, &longs;eu æqualis <lb/>impetus; </s> <s id="N28AE9"><!-- NEW -->certè 1000. in&longs;tantibus, quibus mouetur acqui&longs;iuit 1000. gra­<lb/>dus impetus æquales primo, quem habebat in prima grauitatione; </s> <s id="N28AEF"><!-- NEW -->& <lb/>qui fuit cau&longs;a motus primi in&longs;tantis; </s> <s id="N28AF5"><!-- NEW -->igitur &longs;i hic addatur 1000. erunt <lb/>1001. hinc &longs;i corpus moueatur tantùm vno in&longs;tanti, ictus erit duplus <lb/>tantùm grauitationis: &longs;uppono autem nullam e&longs;&longs;e medij re&longs;i&longs;tentiam, <lb/>ictumque infligi per lineam directam. </s> </p> <p id="N28AFF" type="main"> <s id="N28B01">Duodecimò, hinc, &longs;i a&longs;&longs;umatur corpus, cuius pondus &longs;it ad pondus <lb/>corporis prædicti vt 1001. ad 1. idem erit effectus eius grauitationis, & <lb/>illius ictus vno in&longs;tanti. </s> <s id="N28B08"><!-- NEW -->Probatur manife&longs;tè, quia, quæ habent <expan abbr="eũdem">eundem</expan> <lb/>rationem ad aliud tertium; </s> <s id="N28B12"><!-- NEW -->&longs;unt æqualia; dixi vno in&longs;tanti; nam reuerâ <lb/>corpus graue, quod primo in&longs;tanti imprimit aliquid impetus primo in­<lb/>&longs;tanti, illum auget, &longs;ecundo, tertio, &c. </s> <s id="N28B1A">quod maximè ob&longs;eruandum e&longs;t; <lb/>alioqui maxima erit hallucinatio. </s> </p> <p id="N28B1F" type="main"> <s id="N28B21"><!-- NEW -->Decimotertiò, hinc non pote&longs;t determinari proportio corporis ca­<lb/>dentis, & grauitantis, ni&longs;i ex hypothe&longs;i; </s> <s id="N28B27"><!-- NEW -->quia nemo &longs;cit quot fluxerint <lb/>in&longs;tantia in dato motu; </s> <s id="N28B2D"><!-- NEW -->quoad reuerâ &longs;ciri po&longs;&longs;et &longs;i po&longs;&longs;et aliqua arte in­<lb/>ueniri corpus, cuius grauitatio haberet <expan abbr="effectũ">effectum</expan>, quem habet alterius ictus, <lb/>quod nec &longs;ciri pote&longs;t per depre&longs;&longs;um cylindrum cereum vel <expan abbr="plumbeũ">plumbeum</expan>, vel <lb/>alterius mollioris materiæ, quia æqualis depre&longs;&longs;io accuratè cogno&longs;ci non <lb/>pote&longs;t; &longs;i quis enim diceret dee&longs;&longs;e, vel &longs;upere&longs;&longs;e 1000. &longs;uperficies, quà <lb/>ratione conuinci po&longs;&longs;et? </s> <s id="N28B43"><!-- NEW -->non pote&longs;t etiam &longs;ciri operâ libræ, in cuius al­<lb/>terum brachium cadat mobile, quia &longs;unt ferè infiniti motus in&longs;en&longs;ibiles, <lb/>vt con&longs;ideranti patebit; igitur proportio hæc tantùm, determinari pote&longs;t <lb/>ex hypothe&longs;i data, vt clari&longs;&longs;imè con&longs;tat ex dictis. </s> </p> <p id="N28B4D" type="main"> <s id="N28B4F"><!-- NEW -->Decimoquartò, hinc maxima e&longs;t proportio inter ictum, & grauita­<lb/>tionem; </s> <s id="N28B55"><!-- NEW -->cùm modicus malleoli ca&longs;us eum effectum præ&longs;tet, quem in­<lb/>gens corporis moles &longs;ua grauitatione præ&longs;tare non po&longs;&longs;et; </s> <s id="N28B5B"><!-- NEW -->non e&longs;t tamen <lb/>infinita proportio, quia pote&longs;t tanta e&longs;&longs;e moles grauitatis, & tam par­<lb/>uum corporis cadentis pondus, vt illa præualeat, vt con&longs;tat experientiâ, <lb/>quæ nobis euidenti&longs;&longs;imam &longs;uggerit rationem; </s> <s id="N28B65"><!-- NEW -->quia reiicimus infinitos <lb/>illos tarditatis gradus, quos a&longs;&longs;ump&longs;it Galilæus ad probandam &longs;uam <lb/>hypothe&longs;im de motu accelerato, & infinita eiu&longs;dem & aliorum multo­<lb/>rum in&longs;tantia, de quibus alibi in Metaphy&longs;icâ; </s> <s id="N28B6F"><!-- NEW -->e&longs;t tamen maxima illa <lb/>proportio, vt dixi; </s> <s id="N28B75"><!-- NEW -->quia perexigua temporis pars infinitis ferè in&longs;tanti­<lb/>bus con&longs;tat; </s> <s id="N28B7B"><!-- NEW -->quorum certè numerum recen&longs;ere po&longs;&longs;emus, &longs;i quis mo­<lb/>dum inueniat, quo po&longs;&longs;it ab&longs;olutè adæquare grauitationis dati corporis <lb/>effectum cum effectu ictus alterius cadentis: quod meo iudicio non <lb/>modo geometricè, verùm etiam mechanicè, &longs;altem accuratè fieri non <lb/>pote&longs;t. </s> </p> <p id="N28B87" type="main"> <s id="N28B89">Decimoquintò, nec illud, quod habet Dominus Hobs apud Mer&longs;en­<lb/>num, in phœnom. </s> <s id="N28B8E">Mech. </s> <s id="N28B91">pr. <!-- REMOVE S-->25. videtur &longs;atisfacere. </s> <s id="N28B96">Primè, quia &longs;up­<lb/>ponit primum illum conatum cylindri AB, & puncti phy&longs;ici A'C, <lb/>&longs;ed non tradit modum, quo po&longs;&longs;it cogno&longs;ci. </s> <s id="N28B9D"><!-- NEW -->Secundò, quia dicit cona-<pb pagenum="408" xlink:href="026/01/442.jpg"/>tum primum puncti AC, & totius axis AB, quamdiu de&longs;cendit vterque, <lb/>e&longs;&longs;e æqualem; </s> <s id="N28BA8"><!-- NEW -->quod tamen dici non pote&longs;t, quia conatus &longs;ingulorum <lb/>punctorum &longs;eor&longs;im &longs;unt æquales; </s> <s id="N28BAE"><!-- NEW -->&longs;ed conatus omnium coniunctim e&longs;t <lb/>maior conatu &longs;ingulorum; </s> <s id="N28BB4"><!-- NEW -->nam &longs;ingula habent &longs;uum impetum; verum <lb/>e&longs;t quidem moueri motu æquali, quia &longs;ingula æquali impetu mouentur. </s> <s id="N28BBA"><!-- NEW --><lb/>Tertiò, quia vult po&longs;ito cylindro &longs;upra ba&longs;im 4. illam immediatè premi <lb/>à puncto EB, hoc verò punctum à puncto DE, & hoc ab CD, & hoc ab <lb/>AC; </s> <s id="N28BC3"><!-- NEW -->quod tamen dici non pote&longs;t; quis enim dicat granulum &longs;uperpo&longs;i­<lb/>tum rupi in illam grauitare? </s> <s id="N28BC9">Equidem cum illa grauitat grauitatione <lb/>communi, vt dictum e&longs;t &longs;uprà, non tamen in illam. </s> <s id="N28BCE"><!-- NEW -->Quartò, quia dicit <lb/>pumum B cum conatu totius cylinèri incubantis eo tempore, quo pun­<lb/>ctum AC conficeret AC, conficere AB, quod repugnat progre&longs;&longs;ioni <lb/>Galilei, quam &longs;equitur ip&longs;e; </s> <s id="N28BD8"><!-- NEW -->quia conatus &longs;unt, vt velocitates; </s> <s id="N28BDC"><!-- NEW -->hæ verò <lb/>vt tempora; &longs;ed &longs;patia in ratione duplicata temporum. </s> </p> <p id="N28BE2" type="main"> <s id="N28BE4"><!-- NEW -->Denique non video, quomodo ex his etiam datis demon&longs;tret pro­<lb/>portionem quæ&longs;itam percu&longs;&longs;ionis, & grauitationis; </s> <s id="N28BEA"><!-- NEW -->igitur non e&longs;t con&longs;u­<lb/>lendum &longs;patium, &longs;ed tempus eo modo, quo diximus; </s> <s id="N28BF0"><!-- NEW -->&longs;i enim punctum <lb/>moueatur per 1000. in&longs;tantia, acquiret mille puncta impetus; </s> <s id="N28BF6"><!-- NEW -->igitur ha­<lb/>bebit 1001. igitur &longs;i a&longs;&longs;umatur corpus, quod con&longs;tet 1001. punctis habe­<lb/>bit 1001. puncta impetus, id e&longs;t &longs;ingula in &longs;ingulis; quæ cum omnia gra­<lb/>uitent grauitatione communi, æqualis e&longs;t priori effectus. </s> </p> <p id="N28C00" type="main"> <s id="N28C02"><!-- NEW -->Decimo&longs;extò, hinc vides, quàm &longs;it difficilis, vel potiùs impo&longs;&longs;ibilis <lb/>huius proportionis inuentio, ex cuius cognitione tempus re&longs;oluitur in <lb/>&longs;ua in&longs;tantia, immò & quantitas in &longs;ua puncta: primum quidem; </s> <s id="N28C0A"><!-- NEW -->&longs;it enim <lb/>data moles, cuius grauitatio æqualis e&longs;t ictui alterius cadentis dato <lb/>tempore; haud dubiè tot &longs;unt in&longs;tantia in toto illo tempore, quoties <lb/>pondus cadens continetur in grauitante, vt patet ex dictis. </s> </p> <p id="N28C14" type="main"> <s id="N28C16">Decimo&longs;eptimò, pote&longs;t a&longs;&longs;umi perexigua pars temporis pro in&longs;tanti <lb/>phy&longs;ico, nec tam &longs;en&longs;ibilis erit error, & modicum &longs;patium pro puncto <lb/>phy&longs;ico, vt deinde mechanicè procedatur ad indagandam hanc propor­<lb/>tionem percu&longs;&longs;ionis, & grauitationis. </s> </p> <p id="N28C1F" type="main"> <s id="N28C21">Decimooctauò, pote&longs;t explicari quomodo defigatur palus ab ictu <lb/>corporis deor&longs;um cadentis. </s> <s id="N28C26">Primò enim, ideò defigitur, quia materia <lb/>mollior cedit non &longs;ine aliqua compre&longs;&longs;ione. </s> <s id="N28C2B"><!-- NEW -->Secundò, hinc in mucro­<lb/>nem de&longs;inere debet, vt faciliùs penetret, quod ad cuneum reducemus <lb/>alibi: idem dico de &longs;ecuri, gladio, en&longs;e, &c. </s> <s id="N28C33"><!-- NEW -->Tertiò, initio faciliùs <lb/>defigitur, con&longs;tat experientiâ; ratio e&longs;t, quia plures partes deinde com­<lb/>primuntur propter longitudinem, & cra&longs;&longs;itudinem pali &longs;eu claui. </s> <s id="N28C3B"><!-- NEW -->Quar­<lb/>tò, hinc minùs defigitur &longs;ecundo ictu, quàm primo; </s> <s id="N28C41"><!-- NEW -->igitur maiore ni&longs;u <lb/>opus e&longs;t: </s> <s id="N28C47"><!-- NEW -->in qua verò proportione difficilè dictu e&longs;t; inueniri tamen po­<lb/>te&longs;t de qua numero &longs;equenti. </s> <s id="N28C4D">Quintò, pote&longs;t etiam dici vel po&longs;ito &longs;e­<lb/>cundô ictu æquali primo quantum defigat &longs;upra primum, vel po&longs;ita de­<lb/>fixione illa, qua defigitur &longs;ecundo ictu æquali primæ, quam proportio­<lb/>nem habeant ictus. </s> <s id="N28C56">Tertiò, po&longs;ito vtroque inæquali, quæ &longs;it etiam vtriu&longs;­<lb/>que proportio. </s> </p> <pb pagenum="409" xlink:href="026/01/443.jpg"/> <p id="N28C5F" type="main"> <s id="N28C61">Decimononò, &longs;i æqualis &longs;it &longs;ecundus ictus. </s> <s id="N28C64"><!-- NEW -->Primò, pote&longs;t determina­<lb/>ri proportio iuxta quam defigitur palus, quod vt melius explicetur, &longs;it <lb/>cuneus BE, cuius &longs;olidum facilè demon&longs;tratur; </s> <s id="N28C6C"><!-- NEW -->e&longs;t enim &longs;ubduplum pa­<lb/>rallelipedi, cuius ba&longs;is &longs;it quadratum AC, & altitudo RE; </s> <s id="N28C72"><!-- NEW -->&longs;i enim trian­<lb/>gulum ADE ducatur in latus AB vel EF habebitur &longs;olidum cunci, vt <lb/>con&longs;tat, vnde cunei eiu&longs;dem latitudinis &longs;unt, vt triangula, v.g. <!-- REMOVE S-->cuneus A <lb/>F ad eumdem YF; </s> <s id="N28C7E"><!-- NEW -->vt triangulum ADE ad triangulum YHE: </s> <s id="N28C82"><!-- NEW -->hoc po­<lb/>&longs;ito &longs;it triangulum MKN æqualis ADF, & primo ictu tota EI vel N <lb/>Z &longs;ecundo ictu defigitur, non quidem æquali altitudine, &longs;ed æquali &longs;oli­<lb/>do; </s> <s id="N28C8C"><!-- NEW -->cùm autem triangulum XZN &longs;it &longs;ubquadruplum trianguli QON <lb/>&longs;it media proportionalis N inter NZNO, triangulum N <foreign lang="greek">b</foreign> Y e&longs;t du­<lb/>plum NZX; </s> <s id="N28C98"><!-- NEW -->igitur &longs;ecundo ictu defigetur N <foreign lang="greek">b</foreign>: </s> <s id="N28CA0"><!-- NEW -->&longs;imiliter &longs;i vt NZ ad N <lb/><foreign lang="greek">b</foreign>, ita N <foreign lang="greek">b</foreign> ad N. Tertio, ita defigetur NT, & quarto NO dupla NI: ra­<lb/>tio e&longs;t, quia æquales ictus æquales habent effectus. </s> </p> <p id="N28CAF" type="main"> <s id="N28CB1"><!-- NEW -->Vige&longs;imò, &longs;i æquales accipiantur altitudines &longs;ingulis ictibus, ictus <lb/>&longs;unt in ratione duplicata altitudinum, &longs;uppo&longs;itâ prædicta hypothe&longs;i cunei <lb/>v.g.&longs;i dato ictu defigatur NZ, & altero NO, &longs;ecundus e&longs;t ictus quadruplus <lb/>primi; </s> <s id="N28CBB"><!-- NEW -->&longs;i verò tertio ictu defigatur N<foreign lang="greek">q</foreign> tripla NZ, ictus e&longs;t ad primum <lb/>in ratione 9/1. &longs;i denique dato ictu defigatur NM, ictus e&longs;t ad primum <lb/>in ratione 36/3, vt patet ex dictis; &longs;i verò primo ictu defigatur NZ, &longs;ecundo <lb/>ZO, tertio O <foreign lang="greek">q</foreign>, quarto <foreign lang="greek">q</foreign> M, ictus &longs;unt, vt numeri impares 1. <lb/>3. 7. 9. </s> </p> <p id="N28CD5" type="main"> <s id="N28CD7"><!-- NEW -->Vige&longs;imoprimò, hinc &longs;i dentur duo ictus, & eorum proportio deter­<lb/>minari, vt pote&longs;t proportio altitudinum, quæ defiguntur, quæ &longs;unt in <lb/>ratione &longs;ubduplicata ictuum, &longs;uppo&longs;ito cuneo: </s> <s id="N28CDF"><!-- NEW -->&longs;imiliter, &longs;i dentur alti­<lb/>tudines, carumque proportio, determinari pote&longs;t proportio ictum; </s> <s id="N28CE5"><!-- NEW -->&longs;unt <lb/>enim in ratione duplicata, vt patet ex dictis; porrò vtrumque pote&longs;t <lb/>con&longs;iderari duobus modis. </s> <s id="N28CED">Primò, coniunctim, &longs;i &longs;ecundus ictus &longs;ucce­<lb/>dat primo, & eius altitudinem augeat. </s> <s id="N28CF4">Secundò, &longs;i &longs;eor&longs;im vterque <lb/>con&longs;ideretur, &c. </s> </p> <p id="N28CF9" type="main"> <s id="N28CFB"><!-- NEW -->Vige&longs;imo&longs;ecundò, in clauis, vel conis altitudines &longs;unt in ratione <lb/>&longs;ubtriplicata <expan abbr="ictuũ">ictuum</expan>, & ictus in ratione triplicata altitudinum defixarum, <lb/>quòd manife&longs;tum e&longs;t ex Geometria; </s> <s id="N28D07"><!-- NEW -->&longs;it enim conus BAF, qui defigatur <lb/>vno ictu; </s> <s id="N28D0D"><!-- NEW -->&longs;itque alter ictus, quo defigatur tantùm FD &longs;ubdupla FA: </s> <s id="N28D11"><!-- NEW --><lb/>cùm ictus &longs;int vt defixa &longs;olida; </s> <s id="N28D16"><!-- NEW -->certè conus FD e&longs;t ad conum FA in <lb/>ratione triplicata, id e&longs;t vt cubus FD ad cubum FA, id e&longs;t vt 1. ad 8. <lb/>quæ omnia con&longs;tant: </s> <s id="N28D1E"><!-- NEW -->idem dico de pyramide, quod de cono: hinc vi­<lb/>detur differentia ictuum, quibus defigitur cuneus, & conus, </s> </p> <p id="N28D24" type="main"> <s id="N28D26"><!-- NEW -->Vige&longs;imotertiò, pote&longs;t explicari quomodo deprimatur cylindrus con­<lb/>&longs;tans ex molliori materia; </s> <s id="N28D2C"><!-- NEW -->nam primò deprimitur prima &longs;uperficies <lb/>cylindri, & extenditur; quia cùm materia. </s> <s id="N28D32">&longs;it mollior, prematurque a <lb/>duobus corporibus duris vtrinque, &longs;cilicet ab vtraque ba&longs;i, cedit & di­<lb/>latatur propter humorem in cauitatibus contentum. </s> <s id="N28D39">Secundò, aliquan­<lb/>do totus cylindrus deprimitur &longs;eruatà &longs;emper cylindri licet cra&longs;&longs;io­<lb/>ris figurâ, quod vt fiat, molli&longs;&longs;imam materiam e&longs;&longs;e nece&longs;&longs;e e&longs;t. </s> <s id="N28D40"><!-- NEW -->Ter-<pb pagenum="410" xlink:href="026/01/444.jpg"/>tiò, aliquando primæ tantùm &longs;uperficies extenduntur, vt videmus in <lb/>capite, &longs;eu ba&longs;i cuneorum; quia materies durior multùm re&longs;i&longs;tit. </s> <s id="N28D4B"><!-- NEW -->Quartò, <lb/>limbus ba&longs;is dilatatæ contrahitur deinde, &longs;eu retorquetur deor&longs;um; </s> <s id="N28D51"><!-- NEW -->quia <lb/>cùm interiores circuli dilatentur, deberet facere limbus ille maiorem <lb/>circulum; quod cùm fieri non po&longs;&longs;it, contrahitur &longs;eu incuruatur deor­<lb/>&longs;um, quod facilè &longs;ine figura intelligi pote&longs;t. </s> <s id="N28D5B"><!-- NEW -->Quintò, pote&longs;t deter­<lb/>minari proportio ictuum, quibus deprimuntur cylindri; </s> <s id="N28D61"><!-- NEW -->&longs;i enim &longs;up­<lb/>ponatur eadem altitudo, &longs;eu linea depre&longs;&longs;ionis, & diuer&longs;a cra&longs;&longs;i­<lb/>tudo cylindrorum ictus, erunt vt ba&longs;es; </s> <s id="N28D69"><!-- NEW -->nam quò plures partes de­<lb/>primendæ &longs;unt, maiore ictu opus e&longs;t, &longs;i opponatur eadem cra&longs;&longs;itudo <lb/>vtriu&longs;que cylindri &longs;ed diuer&longs;a depre&longs;&longs;ionis linea vel altitudo, ictus <lb/>erunt vt altitudines; </s> <s id="N28D73"><!-- NEW -->&longs;i vtraque &longs;upponitur diuer&longs;a, ictus erunt in ra­<lb/>tione compo&longs;ita ex ratione ba&longs;ium, & altitudinum; quæ omnia con&longs;tant <lb/>ex dictis. </s> </p> <p id="N28D7B" type="main"> <s id="N28D7D">Ob&longs;eruabis tamen cre&longs;cere re&longs;i&longs;tentiam ex duplici capite. </s> <s id="N28D80"><!-- NEW -->Primò, <lb/>ex eo quod aliquæ vacuitates occupentur à partibus depre&longs;&longs;is, ac proin­<lb/>de cylindrus induretur; &longs;ic intus durior euadit &longs;ub malleo, & & pila <lb/>lignea &longs;ub ictibus. </s> <s id="N28D8A"><!-- NEW -->Secundò, latiorem illam &longs;uperficiem impedire di­<lb/>latationem aliarum partium: </s> <s id="N28D90"><!-- NEW -->hinc variè di&longs;cerpitur eius limbus, vt <lb/>videre e&longs;t in cuneo ferreo: </s> <s id="N28D96"><!-- NEW -->atqui in explicandis &longs;uprà ictuum propor­<lb/>tionibus, hoc geminum re&longs;i&longs;tentiæ caput nullo modo con&longs;iderauimus: </s> <s id="N28D9C"><!-- NEW --><lb/>&longs;extò, quærunt aliqui dato ictu, quo deprimitur cylindrus data alti­<lb/>tudine, quantum pondus e&longs;&longs;e debeat, quod &longs;ua grauitatione eum­<lb/>dem præ&longs;tet effectum; &longs;ed profectò id nemo vnquam determinauit, <lb/>ni&longs;i primò inueniat pondus, cuius ca&longs;u prædictus cylindrus eodem <lb/>modo deprimatur. </s> <s id="N28DA9"><!-- NEW -->Secundò, ni&longs;i &longs;ciat quot in&longs;tantibus de&longs;cendat, vt <lb/>patet ex his quæ diximus &longs;uprà; vt autem comparetur ictus inflictus <lb/>à brachio cum ictu inflicto à pondere cadente, debet con&longs;uli diuer&longs;a <lb/>depre&longs;&longs;io, vel defixio. </s> </p> <p id="N28DB3" type="main"> <s id="N28DB5"><!-- NEW -->Vige&longs;imoqnartò, corpus cadens in planum horizontale per lineam <lb/>perpendicularem, maximum ictum infligit: </s> <s id="N28DBB"><!-- NEW -->maiorem, cum cadit in pla­<lb/>num decliue, quod manife&longs;tum e&longs;t; </s> <s id="N28DC1"><!-- NEW -->pote&longs;t autem determinari propor­<lb/>tio ictuum ratione planorum; </s> <s id="N28DC7"><!-- NEW -->&longs;it enim perpendicularis KN cadens in <lb/>planum horizontale AD, erit maximus ictus; </s> <s id="N28DCD"><!-- NEW -->&longs;it vt AD; </s> <s id="N28DD1"><!-- NEW -->fiat quadrans <lb/>ADG: </s> <s id="N28DD7"><!-- NEW -->&longs;it planum decliue AE, in quod cadit KM; </s> <s id="N28DDB"><!-- NEW -->ducatur EC vel <lb/>EI; </s> <s id="N28DE1"><!-- NEW -->primus ictus e&longs;t ad &longs;ecundum, vt AD ad AC vel IE; </s> <s id="N28DE5"><!-- NEW -->&longs;it aliud <lb/>planum decliue AF, in quod cadit KN; </s> <s id="N28DEB"><!-- NEW -->ducantur FBFH, primus e&longs;t <lb/>ad tertium, vt AD ad AB; patet ex dictis &longs;uprà, cum de planis in­<lb/>clinatis. </s> </p> <p id="N28DF3" type="main"> <s id="N28DF5"><!-- NEW -->Vige&longs;imoquintò, &longs;i verò cadat corpus graue in globum, a&longs;&longs;umenda e&longs;t <lb/>Tangens puncti contactus v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;it globus centro A &longs;it corpus cadens <lb/>per FD; </s> <s id="N28E01"><!-- NEW -->&longs;it punctum contactus D; </s> <s id="N28E05"><!-- NEW -->&longs;it Tangens CE; </s> <s id="N28E09"><!-- NEW -->idem e&longs;t ictus, <lb/>qui e&longs;&longs;et, &longs;i corpus graue caderet in planum inclinatum CE; </s> <s id="N28E0F"><!-- NEW -->&longs;i verò <lb/>globus cadat in aliud corpus v. <!-- REMOVE S-->g. <!-- REMOVE S-->globus A in corpus HG <lb/>per lineam RG; </s> <s id="N28E1B"><!-- NEW -->ducatur AG, tùm GS, ictus in G e&longs;t ad ictum <pb pagenum="411" xlink:href="026/01/445.jpg"/>in L vt SA ad AL: denique &longs;i globus cadat in globum, id pote&longs;t fieri <lb/>duobus modis. </s> <s id="N28E26">Primò, &longs;i L cadat in X, id e&longs;t linea directionis ducatur <lb/>per centrum vtriu&longs;que, & tunc maximus ictus. </s> <s id="N28E2B"><!-- NEW -->Secundò, &longs;i &longs;ecus v.g. <!-- REMOVE S-->&longs;i <lb/>globus A cadat in globum O, &longs;itque punctum contactus in M; &longs;ic autem <lb/>ictus e&longs;t ad priorem in compo&longs;ita ex OYZA ad compo&longs;itam ex MO <lb/>MA vel vt chorda MY, &longs;eu MP ad diametrum LB, quæ omnia patent <lb/>ex dictis. </s> </p> <p id="N28E39" type="main"> <s id="N28E3B"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N28E47" type="main"> <s id="N28E49"><!-- NEW -->Ob&longs;erua &longs;upere&longs;&longs;e tertium modum percu&longs;&longs;ionis, qui fit emi&longs;&longs;ione; cum <lb/>autem emi&longs;&longs;io tribus modis fieri po&longs;&longs;it.1°ree;. </s> <s id="N28E4F">&longs;implici impul&longs;ione &longs;ine ictu, <lb/>& proiectione. </s> <s id="N28E54">2°ree;. Percu&longs;&longs;ione. </s> <s id="N28E57"><!-- NEW -->3°ree;. Proiectione, cui adde eiaculationem, <lb/>vel euibrationem; de his tribus &longs;equentibus Theorematis agendum e&longs;t. </s> </p> <p id="N28E5D" type="main"> <s id="N28E5F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N28E6B" type="main"> <s id="N28E6D"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena emi&longs;&longs;ionis, quæ fit primo modo, &longs;cilicet <lb/>per meram impul&longs;ionem.<emph.end type="italics"/></s> </p> <p id="N28E76" type="main"> <s id="N28E78"><!-- NEW -->Primò, emittitur vt plurimùm globus, &longs;eu pila Tudiculâ dumtaxat <lb/>minori; vix enim e&longs;&longs;e pote&longs;t alius emi&longs;&longs;ionis modus, qui ad hunc facilè <lb/>non reuocetur. </s> </p> <p id="N28E80" type="main"> <s id="N28E82"><!-- NEW -->Secundò, imprimitur impetus Tudiculæ &longs;imul, & globo, quia <expan abbr="vtrumq;">vtrumque</expan> <lb/>motum brachij impedit; hoc etiam demon&longs;trauimus lib.1. <!-- KEEP S--></s> </p> <p id="N28E8D" type="main"> <s id="N28E8F"><!-- NEW -->Tertiò, quò maior e&longs;t Tudicula, tardiùs mouetur, vt patet: </s> <s id="N28E93"><!-- NEW -->hinc po­<lb/>tentia manet diutiùs applicata; </s> <s id="N28E99"><!-- NEW -->non tamen propterea globus velociùs <lb/>mouetur, vt patet, quia &longs;ingulis in&longs;tantibus minùs in eo producitur; </s> <s id="N28E9F"><!-- NEW -->e&longs;t <lb/>enim qua&longs;i pars Tudiculæ; &longs;ecus tamen accidit, &longs;i Tudicula verberet <lb/>pilam, de quo infrà. </s> </p> <p id="N28EA7" type="main"> <s id="N28EA9"><!-- NEW -->Quartò, &longs;i Tudicula &longs;it longior, longiùs emittitur pila; </s> <s id="N28EAD"><!-- NEW -->ratio e&longs;t, quia <lb/>diutiùs manet potentia applicata pilæ; </s> <s id="N28EB3"><!-- NEW -->quippe magis contrahitur bra­<lb/>chium: hinc longiùs porrigitur, vt clarum e&longs;t. </s> </p> <p id="N28EB9" type="main"> <s id="N28EBB"><!-- NEW -->Quintò, &longs;i maior &longs;it Tudicula, & pila emittatur verberatione, longiùs <lb/>emittitur; </s> <s id="N28EC1"><!-- NEW -->ratio e&longs;t, quia maior impetus imprimitur Tudiculæ à potentia <lb/>diutiùs applicata; diutiùs autem applicatur maiori, quia tardiùs moue­<lb/>tur, vt &longs;uprà diximus. </s> </p> <p id="N28EC9" type="main"> <s id="N28ECB"><!-- NEW -->Sextò, pila emi&longs;&longs;a veloci&longs;&longs;imè mouetur eo in&longs;tanti, quo vltimo tan­<lb/>gitur à Tudicula; quia deinceps nihil pror&longs;us impetus accedit, ac proin­<lb/>de continuò &longs;en&longs;im de&longs;truitur ab eo in&longs;tanti. </s> </p> <p id="N28ED3" type="main"> <s id="N28ED5"><!-- NEW -->Septimò, nunquam mouetur pila emi&longs;&longs;a velociùs ip&longs;a Tudiculâ, cum <lb/>&longs;cilicet emi&longs;&longs;io fit per meram impul&longs;ionem; </s> <s id="N28EDB"><!-- NEW -->quia &longs;cilicet vltimo in&longs;tanti, <lb/>contactus veloci&longs;&longs;imè mouetur pila; </s> <s id="N28EE1"><!-- NEW -->&longs;ed eo in&longs;tanti æquè velociter mo­<lb/>uetur Tudicula, vt con&longs;tat: porrò ideo emittitur pila, quia retinetur Tu­<lb/>dicula, ne longiùs recedat. </s> </p> <p id="N28EE9" type="main"> <s id="N28EEB"><!-- NEW -->Octauò, cum verò emittitur pila per verberationem; </s> <s id="N28EEF"><!-- NEW -->haud dubiè, &longs;i <lb/>pila leuior e&longs;t Tudicula, mouetur deinde velociùs; </s> <s id="N28EF5"><!-- NEW -->&longs;ecus verò, &longs;i grauior <lb/>e&longs;t & æquè velocior, &longs;i æqualis e&longs;t grauitatis; </s> <s id="N28EFB"><!-- NEW -->patet ex dictis de impetu; <pb pagenum="412" xlink:href="026/01/446.jpg"/>hinc vides emi&longs;&longs;ionem cæteris paribus maiorem e&longs;&longs;e per verberationem, <lb/>quàm per meram impul&longs;ionem. </s> </p> <p id="N28F06" type="main"> <s id="N28F08"><!-- NEW -->Nonò, pila grauior emi&longs;&longs;a eodem ni&longs;u potentiæ grauiorem ictum in­<lb/>fligit occurrenti globo, quia &longs;cilicet plùs habet impetus; </s> <s id="N28F0E"><!-- NEW -->nam diutiùs <lb/>potentia fuit applicata: adde quod, &longs;i tardiore motu mouetur propter <lb/>maiorem molem, diutiùs pila intacta manet applicata, de quo infrà. </s> </p> <p id="N28F16" type="main"> <s id="N28F18"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N28F24" type="main"> <s id="N28F26">Ob&longs;eruabis e&longs;&longs;e plura alia phœnomena in ludo minoris Tudiculæ <lb/>v.g. <!-- REMOVE S-->1°ree;.quod &longs;pectat ad proportionem ictuum ratione puncti contactus, <lb/>de qua idem dicendum e&longs;t, quod &longs;uprà dictum e&longs;t Th. 15. num. </s> <s id="N28F2F">25. <lb/>2°ree;.quod &longs;pectat ad lineam motus, per quam pila impacta impellit aliam, <lb/>de qua lib.1. Th.50. 51. 52.& alibi pa&longs;&longs;im. </s> <s id="N28F36">3°ree;. </s> <s id="N28F39">quod &longs;pectat ad reflexio­<lb/>nem, de qua fusè lib.6. à Th.62. ad 75. </s> </p> <p id="N28F3E" type="main"> <s id="N28F40"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N28F4C" type="main"> <s id="N28F4E"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena emi&longs;&longs;ionum, quæ fiunt cum percu&longs;&longs;ione.<emph.end type="italics"/></s> </p> <p id="N28F55" type="main"> <s id="N28F57"><!-- NEW -->Primò, &longs;it percu&longs;&longs;io minoris Tudiculæ v.g. <!-- REMOVE S-->eo maior e&longs;t, quò Tudi­<lb/>cula maior e&longs;t; rationem iam attulimus &longs;uprà num.5.Th.16. </s> </p> <p id="N28F5F" type="main"> <s id="N28F61"><!-- NEW -->Secundò, quo Tudicula longior e&longs;t, maior ictus, & emi&longs;&longs;io; quia <lb/>&longs;cilicet diutiùs potentia manet applicata, quia brachium longiùs extens <lb/>pote&longs;t, vt diximus numero 4. Th.16. <!-- KEEP S--></s> </p> <p id="N28F6A" type="main"> <s id="N28F6C">Tertiò, quod &longs;pectat ad &longs;ecundum ictum, idem pror&longs;us dicendum e&longs;t <lb/>quod dictum e&longs;t Theoremate &longs;uperiore num.9. </s> </p> <p id="N28F71" type="main"> <s id="N28F73"><!-- NEW -->Quartò, quod &longs;pectat ad Tudiculam maiorem, iam &longs;uprà explicuimus <lb/>cuncta illius phœnomena, cum de malleo: certum e&longs;t enim primò ma­<lb/>iorem à maiore ictum infligi, cæteris partibus, quàm à minore propter <lb/>prædictum rationem. </s> <s id="N28F7D">Secundò, certum e&longs;t longitudinem manubrij fle­<lb/>xibilitatem, inæqualitatem, materiem, duritiem mallei, æqualitatem ba&longs;is <lb/>&c. </s> <s id="N28F84">multùm conferre ad maiorem cùm ictus. </s> <s id="N28F87">Tertiò certum e&longs;t mino­<lb/>rem globum, in quem impingitur Tudicula, citiùs moueri, inaiorem tar­<lb/>diùs, cæteris paribus. </s> <s id="N28F8E"><!-- NEW -->Quartò, globus maior in alium impactus Tudiculâ <lb/>maiorem ictum infligit, vt con&longs;tat experientiâ; ratî; </s> <s id="N28F94"><!-- NEW -->e&longs;t, quia tardiùs <lb/>mouetur; </s> <s id="N28F9A"><!-- NEW -->igitur diutiùs applicatur: Equidem globus proiectus in alium <lb/>fortiorem ictum infligit ex duplici capite, vt dicam infrà. </s> <s id="N28FA0">1°ree;. </s> <s id="N28FA3"><!-- NEW -->Quia ma­<lb/>iorem impetum à potentia diutiùs applicata.2°ree;.Quia diutiùs applicatur <lb/>globo in quem impingitur; at verò quando impingitur Tudiculâ maiore, <lb/>ex duplici quoque capite cre&longs;cit ictus.1°ree;.quia globus globo diutiùs ma­<lb/>net applicatus, cùm tardior motus dicat plùs temporis. </s> <s id="N28FAF">2°ree;. </s> <s id="N28FB2"><!-- NEW -->quia malleus <lb/>tardiorem motum imprimis globo; </s> <s id="N28FB8"><!-- NEW -->igitur diutiùs manet applicatus: e&longs;t <lb/>enim hæc abta lex agentium, vt longiore tempore maior effectus produ­<lb/>catur, minor verò minore, reliqua ex dictis facilè intelligentur. </s> </p> <p id="N28FC2" type="main"> <s id="N28FC4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N28FD0" type="main"> <s id="N28FD2"><emph type="italics"/>Explicari po&longs;&longs;unt omnia phœnomena emi&longs;&longs;ionum, quæ fiunt per iactum.<emph.end type="italics"/></s> </p> <p id="N28FD9" type="main"> <s id="N28FDB"><!-- NEW -->Primò, Iactus duobus modis fieri pote&longs;t: primò brachio: </s> <s id="N28FDF"><!-- NEW -->&longs;ecundò, <lb/>aliquo organo; </s> <s id="N28FE5"><!-- NEW -->e&longs;t autem multiplex organi genus, de quo infrà; omitto <pb pagenum="413" xlink:href="026/01/447.jpg"/>enim iactum illum, qui fit pede mini&longs;tro, cuius eadem e&longs;t ratio, quæ <lb/>brachij. </s> </p> <p id="N28FF0" type="main"> <s id="N28FF2"><!-- NEW -->Secundò, iactu lapidis maioris, maior ictus infligitur; </s> <s id="N28FF6"><!-- NEW -->ratio e&longs;t, quia <lb/>diutiùs manet lapis applicatus potentiæ, ip&longs;ique adeo corpori, in quod <lb/>impingitur; </s> <s id="N28FFE"><!-- NEW -->vtrumque certè, quia tardiùs mouetur, ergo tardiùs &longs;epara­<lb/>tur à manu; ergo etiam in&longs;tans contactus maius e&longs;t. </s> </p> <p id="N29004" type="main"> <s id="N29006"><!-- NEW -->Tertiò, hinc proportio ictuum &longs;atis facilè ex dictis &longs;uprà determinari <lb/>pote&longs;t; </s> <s id="N2900C"><!-- NEW -->&longs;i enim habeatur tantùm ratio impetus maioris, qui imprimitur <lb/>&longs;axo ab ip&longs;a potentia, ictus &longs;unt in ratione &longs;ubduplicata ponderum, id <lb/>e&longs;t, vt tempora, quibus &longs;axum adhæret manui; </s> <s id="N29014"><!-- NEW -->&longs;i verò habeatur ratio <lb/>contactus, ictus &longs;unt vt motus permutando, &longs;uppo&longs;ito æquali impetu; </s> <s id="N2901A"><!-- NEW --><lb/>igitur, &longs;i habeatur ratio vtriu&longs;que, ictus &longs;unt in ratione compo&longs;ita ex ra­<lb/>tione &longs;ubduplicata ponderum, & ratione permutata motuum v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;int <lb/>&longs;axa AB &longs;it A 4.librarum, B vnius; </s> <s id="N29027"><!-- NEW -->ratio &longs;ubduplicata e&longs;t 2/1 motus A e&longs;t <lb/>vt velocitas; </s> <s id="N2902D"><!-- NEW -->igitur e&longs;t ad motum B, vt 1/2. permutetur, erit 2/1 componatur <lb/>vtraque ratio, eritque ratio 4/1; </s> <s id="N29033"><!-- NEW -->igitur ictus lapidis &longs;unt vt pondera; quæ <lb/>omnia con&longs;tant ex dictis &longs;uprà. </s> </p> <p id="N29039" type="main"> <s id="N2903B"><!-- NEW -->Quartò, leui&longs;&longs;imi lapides vix iaciuntur ad modicam di&longs;tantiam v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->granula &longs;abuli; ratio e&longs;t, 1°ree;. </s> <s id="N29044">quia accipiunt minùs impetus, quia citiùs <lb/>&longs;eparantur à iaciente manu, vt patet. </s> <s id="N29049">2°ree;. </s> <s id="N2904C"><!-- NEW -->quia mouetur initio velociùs in <lb/>aëre; </s> <s id="N29053"><!-- NEW -->igitur &longs;ingulis in&longs;tantibus plùs impetus de&longs;truitur, vt con&longs;tat; nam <lb/>in maiori &longs;patio aëris e&longs;t maior re&longs;i&longs;tentia. </s> <s id="N29059"><!-- NEW -->3°ree;.quia cùm aër perpetuo <lb/>motu agitetur, vt certum e&longs;t, in leuiori corpore impetum imprimit; igi­<lb/>tur aliam &longs;i&longs;tit vel deflectit. </s> <s id="N29061">4°ree;.quia manu non pote&longs;t rectè prehendi ia­<lb/>ciendus lapillus &c. </s> </p> <p id="N29066" type="main"> <s id="N29068"><!-- NEW -->Quintò, grauior lapis ad modicam tantùm di&longs;tantiam iacitur; ratio <lb/>e&longs;t 1°ree;.quia producitur remi&longs;&longs;ior impetus, cùm &longs;cilicet pluribus partibus <lb/>&longs;ubiecti di&longs;tribuatur. </s> <s id="N29070">2°ree;.quia impetus grauitationis citiùs de&longs;truit impe­<lb/>tum extrin&longs;ecus aduenientem. </s> </p> <p id="N29075" type="main"> <s id="N29077"><!-- NEW -->Sextò, figura corporis iacti multùm confert ad iactum, quia ratione <lb/>figuræ pote&longs;t aër plùs, vel minùs re&longs;i&longs;tere: </s> <s id="N2907D"><!-- NEW -->hinc figura circularis depre&longs;­<lb/>&longs;ior apti&longs;&longs;ima e&longs;t ad iactum; </s> <s id="N29083"><!-- NEW -->quia minor e&longs;t aëris re&longs;i&longs;tentia, qualis e&longs;t <lb/>figura lenticularis: </s> <s id="N29089"><!-- NEW -->hinc &longs;cabri corporis, qualis e&longs;t tophus, iactus e&longs;t <lb/>difficilior; </s> <s id="N2908F"><!-- NEW -->quia &longs;cilicet aër &longs;alebris illis, vel a&longs;peritatibus interceptus <lb/>magis re&longs;i&longs;tit: hinc &longs;ibilus propter colli&longs;ionem aëris &c. </s> </p> <p id="N29095" type="main"> <s id="N29097">Septimò, iacitur lapis multis modis 1°ree;. </s> <s id="N2909A">rotato infrà brachio extento: </s> <s id="N2909D"><!-- NEW --><lb/>&longs;ic vulgò iaciuntur grauiora &longs;axa; </s> <s id="N290A2"><!-- NEW -->ad iactum autem conferunt vires po­<lb/>tentiæ, brachium longiùs, longior arcus, Tangens, per quam emittitur di­<lb/>mi&longs;&longs;um &longs;axum, quæ debet facere cum horizontali angulum grad.45. ma­<lb/>nus &longs;imul explicata; </s> <s id="N290AC"><!-- NEW -->&longs;i enim vna pars ante aliam dimittatur, retinetur <lb/>iactus, vt vulgò dicitur, figura, & moles lapidis; </s> <s id="N290B2"><!-- NEW -->&longs;i enim maior e&longs;t, non <lb/>procul emittitur præuia brachij gyratio, quia impetus augetur: denique <lb/>impre&longs;&longs;us toti corpori impetus, quæ omnia mirificè maiorem iactum ef­<lb/>ficiunt, vt con&longs;tat ex dictis &longs;uprà. </s> <s id="N290BD"><!-- NEW -->2°ree;.iacitur lapis rotato quidem deor&longs;um <lb/>brachio, &longs;ed non &longs;iue aliqua eiu&longs;dem brachij contractione, & aliquot <pb pagenum="414" xlink:href="026/01/448.jpg"/>gyris: &longs;ic vulgò iaciuntur &longs;axa minora, tuncque præ&longs;ertim contentis ner­<lb/>uis toti corpori impetus accedit, qui deinde ad augendam iactum in <lb/>ip&longs;um brachium qua&longs;i refunditur.3°ree;. </s> <s id="N290CC">iacitur lapis negligenti qua&longs;i ni&longs;u, <lb/>&longs;eu reiectione circumacta manu horizonti parallela, & contracto tan­<lb/>tillùm brachio. </s> <s id="N290D3">4°ree;. </s> <s id="N290D6">additur aliquando deflexio vel declinatio iactui <lb/>præ&longs;ertim in ludo trunculorum, præ&longs;ertim cùm trunculorum lineæ ad­<lb/>uer&longs;æ omninò & directæ iacienti re&longs;pondens. </s> <s id="N290DD"><!-- NEW -->5°ree;.denique, iacitur &longs;axum <lb/>rotato &longs;upra brachio implicatis gyris, qui reuerâ iactus augetur ex ii&longs;­<lb/>dem omninò capitibus; de quibus iam &longs;uprà, quorum omnium cau&longs;æ & <lb/>rationes parent manife&longs;tæ ex dictis. </s> </p> <p id="N290E7" type="main"> <s id="N290E9"><!-- NEW -->Octauò, corporis iacti impetus de&longs;truitur &longs;en&longs;im, tùm ab impetu nati­<lb/>uo ab occur&longs;u aliorum corporum; </s> <s id="N290EF"><!-- NEW -->hinc in plano a&longs;periore citiùs rota­<lb/>tus globus &longs;i&longs;tit; quæ certè omnia &longs;unt facilia. </s> </p> <p id="N290F5" type="main"> <s id="N290F7"><!-- NEW -->Nonò, eiaculatio e&longs;t iactus &longs;eu vibratio alicuius mi&longs;&longs;ilis oblongi, qua­<lb/>le e&longs;t iaculum vel telum, pro qua non e&longs;t difficultas; </s> <s id="N290FD"><!-- NEW -->fit enim porrecto <lb/>antè per &longs;uperiorem arcum brachio; infligetur autem maior ictus, cum <lb/>1°ree;. </s> <s id="N29105"><!-- NEW -->iaculum e&longs;t maius, propter eandem rationem quam &longs;uprà attulimus <lb/>pro &longs;ari&longs;&longs;a.2°ree;.cum directus e&longs;t ictus; </s> <s id="N2910B"><!-- NEW -->pote&longs;t autem e&longs;&longs;e obliquus, vel quia <lb/>in planum cadit obliquè, licèt non declinet telum à &longs;ua linea, vel quia à <lb/>&longs;ua linea declinat, quæ cadit alioquin perpendiculariter in planum, vel <lb/>denique ex vtroque capite: omitto alia capita, quæ maiorem vim ictui <lb/>conciliant, de quibus &longs;uprà num.7. 3°ree;. </s> <s id="N29117"><!-- NEW -->multùm facit ad maiorem ictum <lb/>concitatus in eam partem equus, in quam vibratur telum; hinc equites <lb/>antiquioris militiæ telis & iaculis pugnabant. </s> </p> <p id="N2911F" type="main"> <s id="N29121">Decimò, iactus fieri pote&longs;t multiplici organo ejaculatorio, 1°ree;. </s> <s id="N29124">&longs;ypho­<lb/>ne, 2°ree;.fi&longs;tula tormentaris, 3°ree;.arcu, 4°ree;.funda, 5°ree;. </s> <s id="N29129"><!-- NEW -->reticulo pilari vel cla­<lb/>uula denique infinita e&longs;t ferè organorum huiu&longs;modi &longs;uppellex; </s> <s id="N2912F"><!-- NEW -->omitto <lb/>motus omnes rei tormentariæ, balli&longs;ticæ, hydraulicæ, & pneumaticæ, de <lb/>quibus fusè Tomo &longs;equenti; </s> <s id="N29137"><!-- NEW -->quod &longs;pectat ad &longs;yphonem, quo aquam vel <lb/>globulos ejaculari &longs;olemus, non e&longs;t dubium quin illa ejaculatio &longs;it effe­<lb/>ctus compre&longs;&longs;ionis, de qua etiam, Tomo &longs;equenti; igitur &longs;uper&longs;unt tan­<lb/>tùm duo prædictorum organorum genera, &longs;cilicet funda & pilaris cla­<lb/>uula. </s> </p> <p id="N29143" type="main"> <s id="N29145">Vndecimò, funda vulgare e&longs;t organum iactus, cuius phœnomena fa­<lb/>cilè explicari po&longs;&longs;unt.1°ree;. </s> <s id="N2914A"><!-- NEW -->rotatur vt maiorem impetum acquirat ad mo­<lb/>tus reticulo lapis, 2°ree;.quò longior e&longs;t funda, longiùs lapis abigitur, quia <lb/>diutiùs manet applicatus, cùm maiorem arcum decurrat, 3°ree;.lapis in reti­<lb/>culo fundæ retinetur; quia cùm per Tangentem lineam &longs;ingulis in&longs;tanti­<lb/>bus determinetur, vt con&longs;tat ex dictis &longs;uprà, impeditur & retinetur à re­<lb/>ticulo, per quod Tangens illa duci tantùm pote&longs;t, e&longs;t eadem ratio, quæ <lb/>orbis rotati, de quo Th.3.num.10. 4°ree;. </s> <s id="N2915A">hinc demi&longs;&longs;o altero fundæ funi­<lb/>culo lapis iacitur, quia nihil e&longs;t à quo retineri ampliùs queat. </s> <s id="N2915F">5°ree;. </s> <s id="N29162"><!-- NEW -->quò <lb/>maior e&longs;t lapis cæteris paribus, tardiùs rotatur funda, at maior impetus <lb/>lapidi imprimitur; quia diutiùs manet applicatus. </s> <s id="N2916A">6°ree;. </s> <s id="N2916D"><!-- NEW -->tenditur conti­<lb/>nuò rota, quantumuis rotetur; quia &longs;cilicet non quidem à pondere <pb pagenum="415" xlink:href="026/01/449.jpg"/>lapidis, &longs;ed ab eius impetu ad Tangentem determinato eò trahitur. </s> <s id="N29178"><lb/>Septimò, quod autem ad Tangentem continuò determinetur linea mo­<lb/>tus, patet ex dictis, cum de motu circulari. </s> <s id="N29180">Octauò, longi&longs;&longs;imus erit ia­<lb/>ctus, &longs;i Tangens, ad quam motus lapidis determinatur, eo in&longs;tanti, quo <lb/>demittitur faciat angulum 45. grad. <!-- REMOVE S-->cum horizontali. </s> <s id="N29189"><!-- NEW -->Nonò, vt rectè <lb/>collimetur, &longs;eu dirigatur lapis ad propo&longs;itum &longs;copum, egregium artifi­<lb/>cium e&longs;&longs;e pote&longs;t; quod totum in eo po&longs;itum e&longs;t, vt inueniatur illa Tan­<lb/>gens, quæ ducitur ad &longs;copum. </s> <s id="N29193"><!-- NEW -->Decimò, ad fundam reuocari pote&longs;t, li­<lb/>nea illa fi&longs;&longs;i baculi furca, cui &longs;i lapis in&longs;eratur, facilè deinde emittitur; </s> <s id="N29199"><!-- NEW --><lb/>&longs;it enim linea furca AB; </s> <s id="N2919E"><!-- NEW -->&longs;it lapis in&longs;ertus B, &longs;i rotetur maximo ni&longs;u furca <lb/>AB circa centrum A, vel circa centrum humeri; </s> <s id="N291A4"><!-- NEW -->haud dubiè lapis B <lb/>cum aliquo impetu di&longs;cedet: ratio e&longs;t, quia cùm &longs;tatim retineatur furca <lb/>impre&longs;&longs;a priùs maxima impetus vi, tùm lapidi tùm furcæ, &longs;uperat vis <lb/>illa impetus, quæ lapidi ine&longs;t, modicam illam &longs;trictionem fi&longs;&longs;æ rimæ, <lb/>nec e&longs;t alia difficultas. </s> </p> <p id="N291B0" type="main"> <s id="N291B2">Vndecimò, ad fundam reuocabis vibrationes arietis, Tudiculæ, æris <lb/>campani, & omnium funependulorum, quas &longs;uis vibrationibus aliquod <lb/>corpus eiaculantur, vel ictum infligunt. </s> </p> <p id="N291B9" type="main"> <s id="N291BB">Duodecimò, claua pilaris, &longs;eu reticulum notum e&longs;t omnibus or­<lb/>ganum, cuius phœnomena clari&longs;&longs;ima &longs;unt. </s> <s id="N291C0">Primò, reticulo longiùs <lb/>emittitur pila, quàm clauiculâ, propter ten&longs;ionem & reditum chordarum. </s> <s id="N291C5"><lb/>Secundò, quò longiùs e&longs;t clauulæ manubrium, longiùs abigitur pila. </s> <s id="N291C9"><lb/>Tertiò, vt &longs;u&longs;tineatur ictus breui manubrio, reticulo opus e&longs;t. </s> <s id="N291CD">Quartò, <lb/>auer&longs;a manu impacto reticulo, pila longiùs emittitur. </s> <s id="N291D2">Quintò, quò &longs;unt <lb/>ten&longs;iores chordæ reticuli, maior e&longs;t ictus. </s> <s id="N291D7"><!-- NEW -->Sextò, hinc recens reticulum <lb/>veteri, & iam attrito præferri debet; hinc ille chordarum &longs;onus. </s> <s id="N291DD"><!-- NEW -->Septimò <lb/>pote&longs;t a&longs;&longs;ignari clauulæ locus, in quo &longs;i fiat percu&longs;&longs;io, fit maximus ictus, <lb/>&longs;it enim clauula AE, cuius centrum grauitatis &longs;it C; </s> <s id="N291E5"><!-- NEW -->haud dubiè, &longs;i mo­<lb/>ueatur motu recto, maximum ictum infliget in C; </s> <s id="N291EB"><!-- NEW -->&longs;i verò motu circu­<lb/>lari circa E e&longs;t aliud centrum percu&longs;&longs;ionis, de quo infrà; </s> <s id="N291F1"><!-- NEW -->&longs;i tamen reticu­<lb/>lum propter ten&longs;ionem chordarum, quæ maximum addit momentum in <lb/>centro reticuli, erit ferè maximus ictus in linea AD, &longs;iue &longs;it reticulum, <lb/>&longs;iue &longs;it clauula, debet fieri contactus; alioqui &longs;i in F, v.g. <!-- REMOVE S-->fieret declina­<lb/>ret planum clauulæ, vt patet. </s> <s id="N291FF"><!-- NEW -->Nonò, cra&longs;&longs;itudo clauulæ multùm facit ad <lb/>augendam vim ictus; e&longs;t enim eadem pror&longs;us ratio, quæ mallei. </s> <s id="N29205"><!-- NEW -->Decimò, <lb/>firmitas, & qua&longs;i ten&longs;io carpi multùm facit ad ictum; </s> <s id="N2920B"><!-- NEW -->præ&longs;ertim cùm pila <lb/>retorquetur; quia &longs;cilicet ratione vectis ferè circa extremitatem manu­<lb/>brij pellitur clauula ab immi&longs;&longs;a pilâ. </s> <s id="N29213"><!-- NEW -->Vndecimò, vt &longs;it maior ictus, ali­<lb/>quo tempore reticulum comitatur pilam, adhæretque à tergo: </s> <s id="N29219"><!-- NEW -->ratio e&longs;t, <lb/>quia potentia manet diutiùs applicata: vide alia, quæ pertinent ad de­<lb/>flexionem pilæ, & reflexionem lib.6. de motu reflexo à Th.75. ad 81. </s> </p> <p id="N29221" type="main"> <s id="N29223"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N2922F" type="main"> <s id="N29231"><emph type="italics"/>Aliæ &longs;unt plurimæ motionum &longs;pecies, quas in hoc Theoremate exponi­<lb/>mus.<emph.end type="italics"/></s> </p> <pb pagenum="416" xlink:href="026/01/450.jpg"/> <p id="N2923E" type="main"> <s id="N29240"><!-- NEW -->Primò, occurrit pre&longs;&longs;io, & dilatatio: </s> <s id="N29244"><!-- NEW -->premitur corpus ab impetu <lb/>impre&longs;&longs;o à circumferentia ad centrum; </s> <s id="N2924A"><!-- NEW -->&longs;ic premitur aër, & aqua intra <lb/>vas; </s> <s id="N29250"><!-- NEW -->dilatatur verò per impetum à centro ad circumferentiam; &longs;ed mira­<lb/>biles &longs;unt pre&longs;&longs;ionis & dilatationis effectus, qui propterea librum &longs;ingu­<lb/>larem de&longs;iderant. </s> </p> <p id="N29258" type="main"> <s id="N2925A">Secundò, intru&longs;io, & extru&longs;io: </s> <s id="N2925D"><!-- NEW -->illa e&longs;t impul&longs;io intror&longs;um; </s> <s id="N29261"><!-- NEW -->hæc verò <lb/>extror&longs;um: </s> <s id="N29267"><!-- NEW -->vtraque fit vt plurimùm cum pre&longs;&longs;ione; </s> <s id="N2926B"><!-- NEW -->&longs;ic defigîtur clauus; <lb/>vi mallei; </s> <s id="N29271"><!-- NEW -->&longs;ic excluditur alius: </s> <s id="N29275"><!-- NEW -->ad intru&longs;ionem & extru&longs;ionem reuocari <lb/>pote&longs;t ductus auri, vel argenti, vel alterius ductilis materiæ; &longs;ed hunc <lb/>rei ductilis &longs;tatum Tomo quinto explicabimus cum alijs corporeum &longs;ta­<lb/>tibus. </s> </p> <p id="N2927F" type="main"> <s id="N29281"><!-- NEW -->Tertiò, di&longs;po&longs;itio fit per eiaculationem, vel minimarum partium, quæ <lb/>&longs;imul omnes vno iactu demittuntur manu; </s> <s id="N29287"><!-- NEW -->&longs;it plura grana tritici vel <lb/>arenæ iaciuntur, vel alicuius corporis, cuius partes &longs;eparantur in ip&longs;o <lb/>iactu; cur verò vna per hanc lineam, alia per aliam feratur, determi­<lb/>natur vel à concur&longs;u cum alia parte, vel à &longs;itu, quem &longs;ingulæ in iacien­<lb/>tis manu habebant priùs, vel ab ordine, quo &longs;ingulæ proce&longs;&longs;erunt. </s> </p> <p id="N29293" type="main"> <s id="N29295"><!-- NEW -->Quartò, adductio ad tractionem reuocari pote&longs;t; </s> <s id="N29299"><!-- NEW -->&longs;unt tamen plures <lb/>illius modi; </s> <s id="N2929F"><!-- NEW -->vel enim per meram tractionem; </s> <s id="N292A3"><!-- NEW -->&longs;ic adducitur clauus, vel <lb/>truncus, vel per circuitionem &longs;implicem, &longs;ic adducitur rotati baculi ex­<lb/>tremitas; vel per circuitionem mixtam: </s> <s id="N292AB"><!-- NEW -->&longs;ic adducitur extremitas funis <lb/>flagelli; vel cum aliquo iactu; </s> <s id="N292B1"><!-- NEW -->&longs;ic adducitur pulmentum vt in va&longs;e <lb/>optimè commi&longs;ceatur v.g. <!-- REMOVE S-->&longs;ic coqui adducunt frixum & inuertunt, por­<lb/>recto tantillùm, tùm deinde rotato &longs;artaginis manubrio: </s> <s id="N292BB"><!-- NEW -->&longs;i enim e&longs;&longs;et <lb/>vera rotatio, frixum per Tangentem erit; at verò propter motum rectum <lb/>po&longs;t inuer&longs;ionem ab ip&longs;a &longs;artagine minimè recedit. </s> </p> <p id="N292C3" type="main"> <s id="N292C5"><!-- NEW -->Quintò, ventilatio e&longs;t motio, quâ frumentum excernitur vanno; </s> <s id="N292C9"><!-- NEW -->van­<lb/>nus circuli e&longs;t vulgare &longs;atis frumentarium organum duabus an&longs;is in&longs;tru­<lb/>ctum, quibus vibratur tùm in aduer&longs;am partem, vt ip&longs;o &longs;uccu&longs;&longs;u paleæ, <lb/>ari&longs;tæ, & aliæ fe&longs;tucæ auolent; </s> <s id="N292D3"><!-- NEW -->tùm dextror&longs;um &longs;ini&longs;tror&longs;umque libratur <lb/>vt leuior materia extet; </s> <s id="N292D9"><!-- NEW -->triticum enim grauius e&longs;t; </s> <s id="N292DD"><!-- NEW -->igitur deor&longs;um ten­<lb/>dit; palea verò &longs;ur&longs;um; </s> <s id="N292E3"><!-- NEW -->ideo verò attollitur, &longs;ub&longs;ultatque triticum in van­<lb/>no, quia po&longs;t impre&longs;&longs;um impetum per vibrationem &longs;ur&longs;um, manus ip&longs;a <lb/>deor&longs;um cum aliquo impetu truditur, in quo non e&longs;t difficultas, alio <lb/>verò motu qua&longs;i recto repit frumentum in vanni aluo, quia per addu­<lb/>ctionem vanni impul&longs;æ priùs &longs;ini&longs;tror&longs;um frumentum in eam partem <lb/>adhuc propter priorem impetum fertur; &longs;ic cum nauis illicò &longs;i&longs;tit in <lb/>potu, qui &longs;unt in ea & portum a&longs;piciunt, proni cadunt, de quo iam <lb/>&longs;uprà. </s> </p> <p id="N292F5" type="main"> <s id="N292F7"><!-- NEW -->Sextò, remigatio fit pellendo, trahendoque, de qua iam &longs;uprà Th. 6. <lb/>16.longior & latior remus maiorem vim aquæ impellit; </s> <s id="N292FD"><!-- NEW -->difficiliùs taman <lb/>mouetur, quò maior e&longs;t illius portio à centro motus ver&longs;us manum re­<lb/>migantis, faciliùs mouetur propter rationem vectis; </s> <s id="N29305"><!-- NEW -->faciliùs mouetur, &longs;i <lb/>aduer&longs;o flumine feratur nauis: </s> <s id="N2930B"><!-- NEW -->ratio e&longs;t, quia aqua pul&longs;a ver&longs;us eam <lb/>partem, in quam fluit minùs re&longs;i&longs;tit, quando eundem remum tractant, <pb pagenum="417" xlink:href="026/01/451.jpg"/>ille plus confert, qui ad extremitatem propiùs accedit; ratio clara e&longs;t: </s> <s id="N29316"><!-- NEW --><lb/>&longs;ed de re nautica aliàs; vide interim locum citatum. </s> </p> <p id="N2931B" type="main"> <s id="N2931D"><!-- NEW -->Septimò, tritus fit, cum ab impacto aliquo duriore corpore malleo, <lb/>v.g. <!-- REMOVE S-->vel pilo aliud teritur, quod &longs;cilicet impetus partibus illis impre&longs;&longs;is <lb/>&longs;uperet vim implicationis, vel vnionis partium; </s> <s id="N29327"><!-- NEW -->e&longs;t etiam eadem ratio <lb/>fracturæ eadem ten&longs;ionis, vel inflexionis; per quid verò corpus ip&longs;um <lb/>&longs;it vel friabile, vel fragile, vel flexibile, fusè explicamus Tomo <lb/>quinto. </s> </p> <p id="N29331" type="main"> <s id="N29333"><!-- NEW -->Octauò, &longs;uccu&longs;&longs;us e&longs;t impetus impre&longs;&longs;us repetito frequenti ni&longs;u; </s> <s id="N29337"><!-- NEW -->&longs;ic <lb/>vulgò &longs;uccutiuntur arbores, vt fructus maturi cadant; </s> <s id="N2933D"><!-- NEW -->excuti verò ali­<lb/>quid dicitur, cum impetus vi ab alio &longs;eparatur; </s> <s id="N29343"><!-- NEW -->&longs;ic excuti dicuntur den­<lb/>tes; &longs;ic excutitur malleo marmoris fragmentum, &c. </s> <s id="N29349"><!-- NEW -->in quo non e&longs;t <lb/>difficultas; </s> <s id="N2934F"><!-- NEW -->nam quoties maior e&longs;t vis impetus, quàm implicationis par­<lb/>tium, vel vnionis, tunc aliqua pars auolat ab ictu: </s> <s id="N29355"><!-- NEW -->denique ca&longs;us alicuius <lb/>corporis facilè intelligi pote&longs;t; </s> <s id="N2935B"><!-- NEW -->periculo&longs;ior e&longs;t altioris hominis, quàm <lb/>pu&longs;illi: </s> <s id="N29361"><!-- NEW -->hinc animalcula cadentia vix quidquam detrimenti à <lb/>ca&longs;u accipiunt: </s> <s id="N29367"><!-- NEW -->præterea ictus grauior e&longs;t, &longs;i quis cadat in eam partem, <lb/>ver&longs;us quam &longs;ummo ni&longs;u fertur; </s> <s id="N2936D"><!-- NEW -->quia impetus grauitatis augetur ab alio <lb/>impre&longs;&longs;o: </s> <s id="N29373"><!-- NEW -->deinde pars illa corporis, quæ ca&longs;u altitudine multùm auget <lb/>vel imminuit grauitatem ictus, vt certum e&longs;t; </s> <s id="N29379"><!-- NEW -->immò corpus illud, cui <lb/>alliditur: </s> <s id="N2937F"><!-- NEW -->hinc caput in marmor impactum graui&longs;&longs;imum ictum refert: </s> <s id="N29383"><!-- NEW --><lb/>hinc tybiæ, vel brachij os ita impingitur ca&longs;u, vt frangatur, vel propter <lb/>rationem vectis, vel propter inæqualitatem corporis, in quod impingi­<lb/>tur; </s> <s id="N2938C"><!-- NEW -->hinc franguntur o&longs;&longs;a facilè modico ictu, &longs;i vtrimque &longs;u&longs;tineantur; </s> <s id="N29390"><!-- NEW --><lb/>in medio vero ab&longs;it fulcrum: &longs;ed hæc pertinent ad re&longs;i&longs;tentiam corporum, <lb/>de qua Tomo &longs;equenti, </s> </p> <p id="N29397" type="main"> <s id="N29399"><!-- NEW -->Nonò, explo&longs;io fit, cum aliquid emittitur, vel cum aliquo &longs;trepitu, <lb/>vt glans è fi&longs;tula, vel per continuam pre&longs;&longs;ionem digitorum; </s> <s id="N2939F"><!-- NEW -->&longs;ic nucleus <lb/>cera&longs;i vulgo exploditur à pueris: </s> <s id="N293A5"><!-- NEW -->Ratio e&longs;t, quia propter vliginem nu­<lb/>clei recenter extracti digiti in eius &longs;uperficie conuexa facilè repunt; </s> <s id="N293AD"><!-- NEW -->hinc <lb/>aucto &longs;emper impetu, & nouo etiam addito ex porrecto brachio pro­<lb/>cul exploditur: &longs;ic omnia lubrica è manibus facilè elabuntur, vt &longs;æpè <lb/>pi&longs;ces, &c. </s> </p> <p id="N293B7" type="main"> <s id="N293B9"><!-- NEW -->Decimò, re&longs;i&longs;tentia corporum procedit tum ex impenetrabilitate <lb/>tùm ex duritie, tùm ex den&longs;itate; </s> <s id="N293BF"><!-- NEW -->nos verò hos &longs;tatus alio Tomo expli­<lb/>cabimus; </s> <s id="N293C5"><!-- NEW -->e&longs;t autem duplex re&longs;i&longs;tentia; </s> <s id="N293C9"><!-- NEW -->prima e&longs;t formalis, quæ in eo <lb/>po&longs;ita e&longs;t, quod non corpus impediat motum alterius, non per aliquid <lb/>contrarium, quod in eo producat, &longs;ed vel per &longs;uam impenetrabilitatem, <lb/>vel per &longs;uam duritiem, vel per &longs;uam molem; </s> <s id="N293D3"><!-- NEW -->nam inde oritur noua de­<lb/>terminatio, vt alibi explicuimus, vel denique per &longs;uam grauitationem, <lb/>&c, &longs;ecunda e&longs;t actiua, vt cum imum corpus imprimit alteri impetum; <lb/>&longs;ed hæc facilè ex dictis intelligi po&longs;&longs;unt. </s> </p> <p id="N293DD" type="main"> <s id="N293DF">Vndecimò, omitto varias motiones corporis humani. </s> <s id="N293E2"><!-- NEW -->Primò, motum <lb/>progre&longs;&longs;iuum &longs;iue fiat cur&longs;u, &longs;iue lentiore gradu: quippè tùm coxæ <lb/>mouentur motu mixto ex duobus circularibus. </s> <s id="N293EA">& crura ex tribus. </s> <s id="N293ED">Se-<pb pagenum="418" xlink:href="026/01/452.jpg"/>cundò, &longs;altum. </s> <s id="N293F5">Tertiò, luctum. </s> <s id="N293F8">Quartò, chorum, &longs;eu numero&longs;am &longs;alta­<lb/>tionem. </s> <s id="N293FD"><!-- NEW -->Quintò denique aliorum animalium motus, qui reuerâ huius <lb/>loci e&longs;&longs;e non po&longs;&longs;unt; nam perfectam mu&longs;culorum, atque adeo totius <lb/>fabricæ corporis humani cognitionem &longs;upponunt, quam trademus &longs;uo <lb/>loco, cum de homine, addemu&longs;que alios motus v.g. <!-- REMOVE S-->re&longs;pirationis, &longs;ter­<lb/>nutationis, tu&longs;&longs;is, &longs;ingultus, o&longs;citationis, ri&longs;us, fletus, fi&longs;toles, & dia&longs;to­<lb/>les, &c. </s> <s id="N2940D"><!-- NEW -->quorum omnium veri&longs;&longs;imas cau&longs;as afferemus; </s> <s id="N29411"><!-- NEW -->omitto etiam <lb/>cau&longs;as phy&longs;icas motuum cœle&longs;tium, quas certè, ni&longs;i me veritas fallit, <lb/>Tomo &longs;equenti demon&longs;trabimus per &longs;implici&longs;&longs;ima principia, cum aliquo <lb/>&longs;altem rei a&longs;tronomicæ incremento: denique omitto alios motus, qui <lb/>certæ materiæ affiguntur v.g.æ&longs;tus maris, libræ motus, fluuiorum fluxus, <lb/>ventorum vis, fluminis ira, magnetis virtus, & electri, &c. </s> <s id="N2941F">de quibus <lb/>&longs;uo loco: quippe hoc loco con&longs;ideramus tantùm motiones, quatenus <lb/>certæ materiæ copulantur. </s> </p> <p id="N29426" type="main"> <s id="N29428"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N29434" type="main"> <s id="N29436"><!-- NEW --><emph type="italics"/>Explicari po&longs;&longs;unt &longs;ingulares aquarum motus,<emph.end type="italics"/> quod tantum hîc breuiter <lb/>præ&longs;tabimus: </s> <s id="N29441"><!-- NEW -->itaque primò, aqua fluit cum plano decliui, quod liquo­<lb/>ris proprium e&longs;t; </s> <s id="N29447"><!-- NEW -->ideo verò fluit, quia cum vna pars alteri extare non <lb/>po&longs;&longs;it; nec enim leuior e&longs;t, deor&longs;um fluit, de quo aliàs fusè. </s> </p> <p id="N2944D" type="main"> <s id="N2944F"><!-- NEW -->Secundò, &longs;tillatim cadit, quia &longs;cilicet colligitur in &longs;phærulas, quæ <lb/>tandem proprio pondere deor&longs;um eunt; cur verò in &longs;phærulas torne­<lb/>tur, veri&longs;&longs;imam rationem dabimus &longs;uo loco. </s> </p> <p id="N29457" type="main"> <s id="N29459">Tertiò, &longs;tillicidium facilè re&longs;i&longs;tit, quia &longs;cilicet aquæ partes, quæ tan­<lb/>tùm modico glutine continentur, diuelluntur facilè, & repercu&longs;&longs;u illo, <lb/>præ&longs;ertim &longs;i à corpore duriore fiat, in omnem partem eunt. </s> </p> <p id="N29460" type="main"> <s id="N29462"><!-- NEW -->Quartò, a&longs;per&longs;io aquæ valdè familiaris e&longs;t, quod &longs;cilicet vi iàctus mi­<lb/>nutim emittatur aqua, in quo non e&longs;t vlla difficultas; nam aqua facilè <lb/>diuiditur. </s> </p> <p id="N2946A" type="main"> <s id="N2946C"><!-- NEW -->Quintò, aqua diluit facilè tùm alios liquores; quia facilè mi&longs;cetur <lb/>tùm corpora &longs;pongio&longs;a, quorum poros, & cauitates facilè &longs;ubit. </s> </p> <p id="N29472" type="main"> <s id="N29474">Sextò, abluit corpora, quibus &longs;cilicet facilè adhæret, & denique cum <lb/>&longs;ordibus exprimitur. </s> </p> <p id="N29479" type="main"> <s id="N2947B">Septimò, aqua fluit, quæ &longs;cilicet in minuti&longs;&longs;imas particulas di&longs;tincta <lb/>&longs;en&longs;im lique&longs;cente vapore in terram cadit. </s> </p> <p id="N29480" type="main"> <s id="N29482"><!-- NEW -->Octauò, infunditur ex vno &longs;cilicet va&longs;e in aliud; </s> <s id="N29486"><!-- NEW -->affunditur, &longs;ubiectis <lb/>&longs;cilicet manibus; effunditur, &longs;cilicet ex &longs;uo va&longs;e. </s> </p> <p id="N2948C" type="main"> <s id="N2948E"><!-- NEW -->Nonò, exundat &longs;æpiùs v. <!-- REMOVE S-->g. <!-- REMOVE S-->fluuius alueo; </s> <s id="N29496"><!-- NEW -->&longs;ic palus etiam & mare <lb/>re&longs;tagnant propter nimiam aquarum copiam: hinc &longs;æpè terram in­<lb/>undat. </s> </p> <p id="N2949E" type="main"> <s id="N294A0"><!-- NEW -->Decimò, libratur &longs;æpiùs in &longs;uo va&longs;e v.g. <!-- REMOVE S-->in latiore cratere; </s> <s id="N294A6"><!-- NEW -->nam facilè <lb/>a&longs;cendit per planum modicè inclinatum, reditque per diuer&longs;as vices; fa­<lb/>ciliùs tamen in latiori, quàm in angu&longs;tiore calice. </s> </p> <p id="N294AE" type="main"> <s id="N294B0"><!-- NEW -->Vndecimò, fluctuat, cum &longs;cilicet eius &longs;uperficies agitatur ventorum <lb/>vi; e&longs;t enim aqua corpus facilè mobile. </s> </p> <pb pagenum="419" xlink:href="026/01/453.jpg"/> <p id="N294BA" type="main"> <s id="N294BC">Duodecimò, cri&longs;patur, cum &longs;cilicet vel leuior e&longs;t afflatus, vel tremu­<lb/>lo &longs;uccutitur motu vas illud, in quo continetur. </s> </p> <p id="N294C1" type="main"> <s id="N294C3"><!-- NEW -->Decimotertiò, in circulos agitur, cum aliquod corpus immergitur <lb/>quia <expan abbr="tantũdem">tantundem</expan> aquæ attollitur &longs;en&longs;im; </s> <s id="N294CD"><!-- NEW -->quod quia extare non pote&longs;t, in <lb/>orbem &longs;uperficiei reliquæ coextenditur: hinc continuò illius circuli, <lb/>tantillùm extantis decre&longs;cit tumor. </s> </p> <p id="N294D5" type="main"> <s id="N294D7"><!-- NEW -->Decimoquartò, facilè mi&longs;cetur cum aqua; quia facilè partes aquæ mi­<lb/>nimo &longs;cilicet impetu diuiduntur. </s> </p> <p id="N294DD" type="main"> <s id="N294DF"><!-- NEW -->Decimoquintò, feruet aqua calore; quia &longs;cilicet partes calidiores <lb/>in vaporem conuer&longs;æ retentæ in bullis &longs;ur&longs;um eas attollunt in &longs;pu­<lb/>mam. </s> </p> <p id="N294E7" type="main"> <s id="N294E9"><!-- NEW -->Decimo&longs;extò, &longs;altitat aqua, cum &longs;cilicet aluei fundum e&longs;t paulò a&longs;pe­<lb/>rius: ratio clari&longs;&longs;ima e&longs;t, quia à &longs;axis occurrentibus repercutitur. </s> </p> <p id="N294EF" type="main"> <s id="N294F1">Decimo&longs;eptimò, agit verticem &longs;æpius, cum &longs;cilicet tractu re&longs;pondet <lb/>profundiori, vel cum repellitur à littore, remo, &c. </s> </p> <p id="N294F6" type="main"> <s id="N294F8"><!-- NEW -->Decimooctauò, agitatur facilè &longs;eu baculo, &longs;eu libratione va&longs;is: </s> <s id="N294FC"><!-- NEW -->&longs;ed <lb/>hæc tantùm breuiter indica&longs;&longs;e &longs;ufficiat, quæ alibi &longs;uis locis fusè omninò <lb/>explicabimus: atque hæc de diuer&longs;is motionibus &longs;int &longs;atis. <lb/><figure id="id.026.01.453.1.jpg" xlink:href="026/01/453/1.jpg"/></s> </p> <pb pagenum="420" xlink:href="026/01/454.jpg"/> <figure id="id.026.01.454.1.jpg" xlink:href="026/01/454/1.jpg"/> <p id="N29513" type="main"> <s id="N29515"><emph type="center"/>APPENDIX PRIMA <lb/>PHYSICOMATHEMATICA,<emph.end type="center"/></s> </p> <p id="N2951E" type="main"> <s id="N29520"><emph type="center"/><emph type="italics"/>De centro percu&longs;sionis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2952B" type="main"> <s id="N2952D"><!-- NEW -->DE duplici centro hactenus actum e&longs;t, <lb/>magnitudinis, &longs;cilicet, & grauitatis; <lb/>præ&longs;ertim de hoc vltimo: </s> <s id="N29535"><!-- NEW -->in quo certè <lb/>opere non &longs;ine maxima laude præ­<lb/>&longs;tanti&longs;&longs;imi Mathematici de&longs;udarunt, <lb/>&longs;cilicet Archimedes, Commandinus, Lucas Vale­<lb/>rius, Steuinus, Guldinus, Galileus paucis: </s> <s id="N29541"><!-- NEW -->&longs;ed du­<lb/>plex aliud centrum con&longs;iderari pote&longs;t; </s> <s id="N29547"><!-- NEW -->primum di­<lb/>citur centrum impre&longs;&longs;ionis: vtrumque pror&longs;us inta­<lb/>ctum aliis doctâ paucarum licèt propo&longs;itionum co­<lb/>ronâ, vel peripheria in hac appendice corona­<lb/>mus. <lb/><gap desc="hr tag"/></s> </p> <p id="N29556" type="main"> <s id="N29558"><emph type="center"/><emph type="italics"/>DEFINITIO I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29564" type="main"> <s id="N29566"><emph type="italics"/>CEntrum grauitatis e&longs;t punctum, quod omnia grauitatis momenta æqua­<lb/>liter dirimit.<emph.end type="italics"/></s> </p> <p id="N2956F" type="main"> <s id="N29571"><!-- NEW -->Clara e&longs;t definitio; centrum enim grauitatis e&longs;t illud punctum, ex <lb/>quo pendulum corpus per quamlibet lineam &longs;eruat æquilibrium. </s> </p> <p id="N29577" type="main"> <s id="N29579"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29586" type="main"> <s id="N29588"><emph type="italics"/>Centrum impre&longs;&longs;ionis e&longs;t illud, per quod, &longs;i ducatur planum vtrimque, di­<lb/>rimit æqualem impetum.<emph.end type="italics"/></s> </p> <p id="N29591" type="main"> <s id="N29593"><!-- NEW -->Hæc etiam clara e&longs;t; </s> <s id="N29597"><!-- NEW -->con&longs;ideratur autem impetus non modò ratione <pb pagenum="421" xlink:href="026/01/455.jpg"/>inten&longs;ionis verùm etiam exten&longs;ionis; debet etiam accipi punctum illud <lb/>in linea motus. </s> </p> <p id="N295A2" type="main"> <s id="N295A4"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N295B1" type="main"> <s id="N295B3"><emph type="italics"/>Centrum percu&longs;&longs;ionis e&longs;t punctum illud corporis impacti in quo &longs;i contactus <lb/>fiat, maximus ictus infligitur.<emph.end type="italics"/></s> </p> <p id="N295BC" type="main"> <s id="N295BE"><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N295CB" type="main"> <s id="N295CD"><emph type="italics"/>Linea directionis e&longs;t linea motus centri grauitatis.<emph.end type="italics"/></s> </p> <p id="N295D4" type="main"> <s id="N295D6"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N295E3" type="main"> <s id="N295E5"><emph type="italics"/>Centrum grauitatis dirigit linea motus aliorum punctorum.<emph.end type="italics"/></s> </p> <p id="N295EC" type="main"> <s id="N295EE"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N295FB" type="main"> <s id="N295FD"><emph type="italics"/>Si percu&longs;&longs;io ita fiat, vt totus impetus corporis impacti impediatur maxi­<lb/>ma e&longs;t.<emph.end type="italics"/></s> </p> <p id="N29606" type="main"> <s id="N29608"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29615" type="main"> <s id="N29617"><emph type="italics"/>Momenta &longs;unt, vt di&longs;tantiæ.<emph.end type="italics"/></s> </p> <p id="N2961E" type="main"> <s id="N29620"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2962D" type="main"> <s id="N2962F"><emph type="italics"/>Omnes partes corporis, quod mouetur motu recto, mouentur æqua­<lb/>liter.<emph.end type="italics"/></s> </p> <p id="N29638" type="main"> <s id="N2963A"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29647" type="main"> <s id="N29649"><emph type="italics"/>Corpus graue &longs;u&longs;tinetur in æquilibrio, cum &longs;u&longs;tinetur in linea dire­<lb/>ctionis.<emph.end type="italics"/></s> </p> <p id="N29652" type="main"> <s id="N29654"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29661" type="main"> <s id="N29663"><emph type="italics"/>Centrum percu&longs;&longs;ionis e&longs;t in illa linea, quæ dirimit vtrimque momenta, tùm <lb/>ratione impetus, tùm ratione di&longs;tantiæ.<emph.end type="italics"/></s> </p> <p id="N2966C" type="main"> <s id="N2966E"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N2967A" type="main"> <s id="N2967C"><!-- NEW --><emph type="italics"/>Si pondera inæqualia &longs;unt in æquilibrio, di&longs;tantiæ &longs;unt, vt pondera per­<lb/>mutando; vel collectio di&longs;tantiarum e&longs;t ad maiorem, vt collectio ponderum ad <lb/>alterum pondus, quod maius est, &c.<emph.end type="italics"/></s> </p> <p id="N29688" type="main"> <s id="N2968A"><emph type="center"/><emph type="italics"/>Po&longs;itiones<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N29696" type="main"> <s id="N29698"><emph type="italics"/>Maximus ictus infligitur in linea directionis, per &longs;e,<emph.end type="italics"/> vt con&longs;tat ex <lb/>po&longs;.5.6.2. </s> </p> <p id="N296A2" type="main"> <s id="N296A4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N296B1" type="main"> <s id="N296B3"><emph type="italics"/>Centrum percu&longs;&longs;ionis lineæ mobilis motu recto e&longs;t idem cum centro graui­<lb/>tatis eiu&longs;dem.<emph.end type="italics"/></s> </p> <p id="N296BC" type="main"> <s id="N296BE"><!-- NEW -->Sit enim linea AC, horizonti parallela, v.g. <!-- REMOVE S-->quæ cadat perpendi­<lb/>culariter; </s> <s id="N296C6"><!-- NEW -->&longs;it eius centrum grauitatis B, quod &longs;cilicet vtrimque æqua­<lb/>liter di&longs;tat ab AC; </s> <s id="N296CC"><!-- NEW -->centrum percu&longs;&longs;ionis e&longs;t in B. Probatur; </s> <s id="N296D0"><!-- NEW -->quia cùm <lb/>in B impediatur totus impetus; </s> <s id="N296D6"><!-- NEW -->quippe neutrum &longs;egmentum præualere <lb/>pote&longs;t; e&longs;t enim vtrimque æqualis impetus, per po&longs;it. </s> <s id="N296DC">3. 4. certè maxi­<lb/>ma percu&longs;&longs;io e&longs;t in B, per po&longs;it.2. igitur e&longs;t centrum percu&longs;&longs;ionis per <pb pagenum="422" xlink:href="026/01/456.jpg"/>def.5. igitur centrum percu&longs;&longs;ionis e&longs;t idem cum centro grauitatis, quod <lb/>erat dem. </s> </p> <p id="N296E8" type="main"> <s id="N296EA"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N296F7" type="main"> <s id="N296F9">Hinc quatuor centra concurrunt in idem punctum, &longs;cilicet magni­<lb/>tudinis, grauitatis, impre&longs;&longs;ionis, & percu&longs;&longs;ionis. </s> </p> <p id="N296FE" type="main"> <s id="N29700"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2970D" type="main"> <s id="N2970F">Idem pror&longs;us dicendum e&longs;t de Rectangulo, Parallelogrammate, Cir­<lb/>culo, Ellip&longs;i, Cylindro, Pri&longs;mate, Parallelipedo, Sphæra, &c. </s> <s id="N29714">in quibus <lb/>po&longs;ito motu recto, hæc quatuor centra in eodem plano, immò & linea <lb/>reperiuntur. </s> </p> <p id="N2971B" type="main"> <s id="N2971D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2972A" type="main"> <s id="N2972C"><emph type="italics"/>Si planum triangulare cadat motu recto deor&longs;um, v.g. <!-- REMOVE S-->horizonti paralle­<lb/>lum, centrum percu&longs;&longs;ionis e&longs;t idem cum centro grauitatis eiu&longs;dem.<emph.end type="italics"/></s> </p> <p id="N29737" type="main"> <s id="N29739"><!-- NEW -->Sit enim triangulare planum FBH, cuius centrum grauitatis &longs;it I: </s> <s id="N2973D"><!-- NEW --><lb/>dico e&longs;&longs;e centrum percu&longs;&longs;ionis; </s> <s id="N29742"><!-- NEW -->quia, cùm &longs;it æqualis motus, & impetus <lb/>omnium partium plani, &longs;i &longs;u&longs;tineatur in I, &longs;tat in æquilibrio, per def.1. <lb/>igitur totus impetus impeditur; igitur e&longs;t maxima percu&longs;&longs;io, per <lb/>Po&longs;. <!-- REMOVE S-->2. <!-- KEEP S--></s> </p> <p id="N2974F" type="main"> <s id="N29751"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2975D" type="main"> <s id="N2975F"><!-- NEW -->Ob&longs;eruabis punctum I po&longs;&longs;e haberi duobus modis; </s> <s id="N29763"><!-- NEW -->Primò, &longs;i ducatur <lb/>FC diuidens æqualiter HB; </s> <s id="N29769"><!-- NEW -->diuidit etiam æqualiter GA, & omnes alias <lb/>parallelas HB; </s> <s id="N2976F"><!-- NEW -->igitur in FC e&longs;t centrum grauitatis: </s> <s id="N29773"><!-- NEW -->&longs;imiliter ducatur <lb/>HD diuidens æqualiter FB, centrum grauitatis erit etiam in HD; </s> <s id="N29779"><!-- NEW -->igi­<lb/>tur in communi puncto I. Secundò, ita diuidatur FH in G, vt FG &longs;it <lb/>dupla GH, ducaturque GA: </s> <s id="N29781"><!-- NEW -->&longs;imiliter ducatur KE diuidens HB eodem <lb/>modo, punctum communis &longs;ectionis I e&longs;t centrum grauitatis; </s> <s id="N29787"><!-- NEW -->quippe <lb/>duo triangula DIC, FIH &longs;unt proportionalia; </s> <s id="N2978D"><!-- NEW -->igitur vt DC ad FH, <lb/>ita DI ad IH, &longs;ed DC e&longs;t &longs;ubdupla FH; </s> <s id="N29793"><!-- NEW -->igitur DI &longs;ubdupla IH: </s> <s id="N29797"><!-- NEW -->&longs;imi­<lb/>liter IC &longs;ubdupla IF; </s> <s id="N2979D"><!-- NEW -->igitur GH &longs;ubdupla GF; igitur inuentum e&longs;t <lb/>centrum grauitatis, quod erat faciendum. </s> </p> <p id="N297A3" type="main"> <s id="N297A5"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N297B2" type="main"> <s id="N297B4"><!-- NEW --><emph type="italics"/>Si planum triangulare cadat parallelum lineæ verticali,<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->in &longs;itu FH <lb/>B, ita vt FH &longs;it parallela horizonti, centrum percu&longs;&longs;ionis e&longs;t in G; </s> <s id="N297C3"><!-- NEW -->cùm <lb/>enim GA ducatur per centrum grauitatis I, &longs;itque parallela HB, e&longs;t <lb/>linea directionis, per def.4. igitur &longs;i &longs;u&longs;tineatur in G, &longs;tabit in æquili­<lb/>brio, per p.5. igitur totus impetus impeditur, vt patet; igitur e&longs;t maxi­<lb/>ma percu&longs;&longs;io per p. </s> <s id="N297CF">2. igitur centrum percu&longs;&longs;ionis e&longs;t G, quod erat de­<lb/>mon&longs;t. </s> </p> <p id="N297D4" type="main"> <s id="N297D6"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N297E3" type="main"> <s id="N297E5"><!-- NEW -->Hinc corpus &longs;olidum ex multis huiu&longs;modi triangulis æqualibus qua&longs;i <lb/>conflatum, idem pror&longs;us percu&longs;&longs;ionis centrum habet; &longs;iue cadat lineæ <lb/>verticali parallelum, &longs;iue ip&longs;i verticali. </s> </p> <pb pagenum="423" xlink:href="026/01/457.jpg"/> <p id="N297F1" type="main"> <s id="N297F3"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29800" type="main"> <s id="N29802"><!-- NEW -->Hinc etiam ad Mechanicam reduci pote&longs;t inuentio praxis prædictæ; </s> <s id="N29806"><!-- NEW --><lb/>&longs;it enim triangulum AGD; </s> <s id="N2980B"><!-- NEW -->diuidatur AD in tres partes in BC; </s> <s id="N2980F"><!-- NEW -->du­<lb/>cantur BI, CH, parallelæ DG, itemque IE, HF parallelæ AD; </s> <s id="N29815"><!-- NEW -->&longs;u&longs;ti­<lb/>neaturque prædictum planum erectum in C, &longs;tabit in æquilibrio; </s> <s id="N2981B"><!-- NEW -->cùm <lb/>enim momenta ponderum æqualium &longs;int vt di&longs;tantiæ, rectangulo CE <lb/>re&longs;pondet æquale, & æquedi&longs;tans CI, itemque trianguli EHK, æquale <lb/>& æquedi&longs;tans IKD, triangulo demum GHE, triangulum &longs;ubduplum <lb/>AIB, cuius momentum adæquat momentum alterius dupli GHB; quia <lb/>di&longs;tantia e&longs;t dupla. </s> </p> <p id="N29829" type="main"> <s id="N2982B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29838" type="main"> <s id="N2983A"><!-- NEW --><emph type="italics"/>Si Pyramis, cuius axis &longs;it parallela horizonti, cadat deor&longs;um; </s> <s id="N29840"><!-- NEW -->centrum <lb/>percu&longs;&longs;ionis e&longs;t in linea derectionis, quæ &longs;cilicet ducetur deor&longs;um à centro gra­<lb/>tatis,<emph.end type="italics"/> quod eodem modo demon&longs;tratur, quo &longs;uprà; </s> <s id="N2984B"><!-- NEW -->e&longs;t autem centrum <lb/>grauitatis illud punctum, quod ita axem diuidit, vt &longs;egmentum ver&longs;us <lb/>ba&longs;im &longs;it &longs;ubtriplum alterius ver&longs;us verticem, quod multi hactenus de­<lb/>mon&longs;trarunt, &longs;cilicet Commandinus, Valerius, Steuinus, Galileus; &longs;it <lb/>enim conus ENI, &longs;it axis AI diui&longs;us in 4. partes æquales BCD, pa­<lb/>rallelus horizonti, &longs;u&longs;tineatur in M, &longs;tabit in æquilibrio. </s> </p> <p id="N29859" type="main"> <s id="N2985B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29868" type="main"> <s id="N2986A"><!-- NEW --><emph type="italics"/>Si quodlibet aliud planum, vel corpus, deor&longs;um cadat, motu recto, cen­<lb/>trum percu&longs;&longs;ionis e&longs;t in linea directionis<emph.end type="italics"/>; </s> <s id="N29875"><!-- NEW -->quod eodem modo probatur, quo <lb/>&longs;uprà: </s> <s id="N2987B"><!-- NEW -->quodnam verò &longs;it centrum grauitatis omnium corporum, plano­<lb/>rum, figurarum, hîc non di&longs;putamus; con&longs;ulantur authores citati, quibus <lb/>addatur La Faille, qui egregiè centrum grauitatis partium circuli, & <lb/>Eclip&longs;is demon&longs;trauit. </s> </p> <p id="N29885" type="main"> <s id="N29887"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29894" type="main"> <s id="N29896"><!-- NEW --><emph type="italics"/>Si linea circa centrum immobile mobilis, voluatur, centrum percu&longs;&longs;ionis <lb/>non e&longs;t centrum grauitatis<emph.end type="italics"/>; </s> <s id="N298A1"><!-- NEW -->&longs;it enim linea AD, quæ voluatur circa cen­<lb/>trum A; </s> <s id="N298A7"><!-- NEW -->diuidatur bifariam in G, punctum G e&longs;t centrum grauitatis: vt <lb/>con&longs;tat; </s> <s id="N298AD"><!-- NEW -->non tamen e&longs;t centrum percu&longs;&longs;ionis, quia in &longs;egmento GD e&longs;t <lb/>quidem æquale momentum ratione di&longs;tantiæ, &longs;ed maius ratione impe­<lb/>tus; quippe GD mouetur velociùs, quàm GA vt certum e&longs;t. </s> </p> <p id="N298B5" type="main"> <s id="N298B7"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N298C3" type="main"> <s id="N298C5"><!-- NEW --><emph type="italics"/>In hac eadem hypothe&longs;i centrum percu&longs;&longs;ionis non e&longs;t idem cum centro im­<lb/>pre&longs;&longs;ionis<emph.end type="italics"/>; </s> <s id="N298D0"><!-- NEW -->diuidatur enim AD in M, ita vt AM, &longs;it media propor­<lb/>tionalis inter AG, & AD; </s> <s id="N298D6"><!-- NEW -->certè M e&longs;t centrum impre&longs;&longs;ionis, vt de­<lb/>mon&longs;tratum e&longs;t lib. 1.non tamen e&longs;t centrum percu&longs;&longs;ionis; </s> <s id="N298DC"><!-- NEW -->quia &longs;eg­<lb/>mentum MA habet quidem æqualem impetum cum &longs;egmento MD; </s> <s id="N298E2"><!-- NEW -->ha­<lb/>bet tamen maius momentum, quia maiorem habet di&longs;tantiam; igitur <lb/>non erit æquilibrium in M. </s> </p> <pb pagenum="424" xlink:href="026/01/458.jpg"/> <p id="N298EE" type="main"> <s id="N298F0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N298FC" type="main"> <s id="N298FE"><!-- NEW --><emph type="italics"/>Si diuidatur AD in tres partes æquales, &longs;it que ID<emph.end type="italics"/> 1/3 <emph type="italics"/>centrum percu&longs;&longs;io­<lb/>nis erit in I<emph.end type="italics"/>; </s> <s id="N2990F"><!-- NEW -->demon&longs;tratur, quia impetus puncti G e&longs;t ad impetum pun­<lb/>cti D; </s> <s id="N29915"><!-- NEW -->vt arcus EG, ad arcum BD; </s> <s id="N29919"><!-- NEW -->&longs;it autem DC æqualis DB; </s> <s id="N2991D"><!-- NEW -->ducatur <lb/>AC, triangulum ACD erit æquale &longs;ectori ADB, vt con&longs;tat; impetus in <lb/>D erit, vt recta DC, & in I, vt recta IH, & in G, vt recta GF, &c. </s> <s id="N29925">igi­<lb/>tur perinde &longs;e habet impetus, qui ine&longs;t puncto D, atque &longs;i incubaret ip&longs;i <lb/>D.DC, & I, IH, & G, GF, &c. </s> <s id="N2992C"><!-- NEW -->atqui &longs;i hoc e&longs;&longs;et, centrum grauitatis <lb/>e&longs;&longs;et in I, vt patet ex dictis; ibique e&longs;&longs;et percu&longs;&longs;ionis, per Th. 3. igitur <lb/>I e&longs;t centrum percu&longs;&longs;ionis. </s> </p> <p id="N29934" type="main"> <s id="N29936"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29942" type="main"> <s id="N29944">Colligo primò, ex dictis in hac hypothe&longs;i tria centra &longs;eparari. </s> </p> <p id="N29947" type="main"> <s id="N29949">Secundò &longs;i nullum e&longs;&longs;et momentum ratione di&longs;tantiæ, centrum per­<lb/>cu&longs;&longs;ionis idem e&longs;&longs;et cum centro impre&longs;&longs;ionis. </s> </p> <p id="N2994E" type="main"> <s id="N29950"><!-- NEW -->Tertiò, centrum percu&longs;&longs;ionis lineæ circa alteram extremitatem mo­<lb/>bilis; </s> <s id="N29956"><!-- NEW -->idem e&longs;&longs;e cum centro percu&longs;&longs;ionis trianguli, &longs;eu plani triangula­<lb/>ris; de quo &longs;uprà. </s> </p> <p id="N2995C" type="main"> <s id="N2995E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N2996A" type="main"> <s id="N2996C"><!-- NEW --><emph type="italics"/>Si rotetur planum rectangulum circa alterum laterum centrum percu&longs;&longs;ionis <lb/>e&longs;t in linea, quæ diuidit rectangulum æqualiter, & cadit perpendiculariter <lb/>in axem, circa quem rotatur<emph.end type="italics"/>; </s> <s id="N29979"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it rectangulum CF, rotatum circa C <lb/>A; </s> <s id="N29981"><!-- NEW -->&longs;it BG, dirimens æqualiter CA & HF, centrum grauitatis e&longs;t in <lb/>BG; quia e&longs;t æquale momentum in BF & BH, tùm ratione impetus, <lb/>tùm ratione di&longs;tantiæ, vt pater per p.6. </s> </p> <p id="N29989" type="main"> <s id="N2998B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s> </p> <p id="N29997" type="main"> <s id="N29999"><!-- NEW --><emph type="italics"/>Si BG diuidatur in tres partes æquales B, D, I, G, rotetur que circa CA, <lb/>vt dictum e&longs;t &longs;uprà, centrum percu&longs;&longs;ionis e&longs;t in I<emph.end type="italics"/>; </s> <s id="N299A4"><!-- NEW -->quia &longs;i volueretur &longs;ola <lb/>AF, e&longs;&longs;et in E, &longs;i &longs;ola CH, e&longs;&longs;et in K, &longs;i &longs;ola BG, e&longs;&longs;et in I, per Th. 8. <lb/>igitur centra percu&longs;&longs;ionis omnium &longs;unt in linea EK; &longs;ed lineæ EK, cuius <lb/>&longs;ingula puncta mouentur æquali motu, centrum percu&longs;&longs;ionis e&longs;t in I, per <lb/>Th.1. igitur centrum percu&longs;&longs;ionis totius CF acti circum CA, e&longs;t in I, <lb/>quod erat demon&longs;tr. </s> </p> <p id="N299B2" type="main"> <s id="N299B4"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N299C0" type="main"> <s id="N299C2">Primò, &longs;i rotetur circa CH, eodem modo inuenietur centrum per­<lb/>cu&longs;&longs;ionis, &longs;cilicet N ita vt NO &longs;it 1/3 MO. </s> </p> <p id="N299C7" type="main"> <s id="N299C9"><!-- NEW -->Secundò, &longs;i rotetur circa OM rectangulum CF; </s> <s id="N299CD"><!-- NEW -->diuidatur in tres <lb/>partes æquales, &longs;itque PG 1/3 NG, centrum percu&longs;&longs;ionis e&longs;t P; </s> <s id="N299D3"><!-- NEW -->e&longs;t enim <lb/>eadem ratio, quæ &longs;uprà; </s> <s id="N299D9"><!-- NEW -->nec e&longs;t minor ictus, quàm in I; </s> <s id="N299DD"><!-- NEW -->rotato &longs;cilicet <lb/>rectangulo circa CA; quia e&longs;t æqualis impetus. </s> </p> <p id="N299E3" type="main"> <s id="N299E5">Tertiò, &longs;i rotetur circa BR, in quam AH cadit perpendiculariter, e&longs;t <lb/>alia ratio, de qua infrà. </s> </p> <pb pagenum="425" xlink:href="026/01/459.jpg"/> <p id="N299EE" type="main"> <s id="N299F0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s> </p> <p id="N299FC" type="main"> <s id="N299FE"><!-- NEW --><emph type="italics"/>Si<emph.end type="italics"/> <emph type="italics"/>triangulum BIG voluatur circa CA, in quam BH cadit perpendi­<lb/>culariter, &longs;itque BH axis per centrum grauitatis ductus, diui&longs;u&longs;que in<emph.end type="italics"/> 4. <lb/><emph type="italics"/>partes æquales B.F.E.D.H. centrum percu&longs;&longs;ionis e&longs;t in D<emph.end type="italics"/>; quod facilè de­<lb/>mon&longs;tratur; </s> <s id="N29A18"><!-- NEW -->nam IG in i&longs;to motu de&longs;cribit &longs;uperficiem cylindri, & <lb/>triangulum GBI de&longs;cribit, vt &longs;ic loquar, &longs;ectorem cylindri; </s> <s id="N29A1E"><!-- NEW -->igitur im­<lb/>petus in IG e&longs;t ad impetum in NM, vt &longs;uperficies curua terminata in I <lb/>G, ad &longs;uperficiem terminatam in NM, &longs;ub eodem &longs;cilicet angulo; </s> <s id="N29A26"><!-- NEW -->vel vt <lb/>ba&longs;is pyramidis IG, ad ba&longs;im NM; igitur perinde &longs;e habet IG, ac &longs;i <lb/>incumberet prædicta ba&longs;is, itemque NM, &c. </s> <s id="N29A2E"><!-- NEW -->igitur ac &longs;i e&longs;&longs;et &longs;olida <lb/>pyramis quadrilatera; &longs;ed pyramidis centrum grauitatis e&longs;t D, per <lb/>Theorema 4. <!-- KEEP S--></s> </p> <p id="N29A37" type="main"> <s id="N29A39"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s> </p> <p id="N29A45" type="main"> <s id="N29A47"><!-- NEW --><emph type="italics"/>Si idem triangulum GIB voluatur circa IG, centrum percu&longs;&longs;ionis e&longs;t in <lb/>E, quod diuidit HB bifariam æqualiter<emph.end type="italics"/>; </s> <s id="N29A52"><!-- NEW -->quod vt demon&longs;tretur, perinde <lb/>&longs;e habet triangulum BGI circumactum, atque &longs;i &longs;ingulis partibus in­<lb/>cumberent perpendiculares, quæ e&longs;&longs;ent vt earumdem partium motus; </s> <s id="N29A5A"><!-- NEW --><lb/>&longs;it autem triangulum BAC æquale priori; </s> <s id="N29A5F"><!-- NEW -->ba&longs;is cunei ABHKDC; </s> <s id="N29A63"><!-- NEW --><lb/>ducatur planum DBA, quod dirimat cuneum in duo &longs;olida, &longs;cilicet in <lb/>pyramidem ABHKD, & &longs;olidum ABDC; </s> <s id="N29A6A"><!-- NEW -->pyramis continet 2/3 totius <lb/>cunei, vt con&longs;tat; </s> <s id="N29A70"><!-- NEW -->e&longs;t enim prædictus cuneus &longs;ubduplus pri&longs;matis, cuius <lb/>ba&longs;is &longs;it HA, & altitudo ID; </s> <s id="N29A76"><!-- NEW -->cuius pyramis prædicta continet 1/3; </s> <s id="N29A7A"><!-- NEW -->igitur <lb/>&longs;i pri&longs;ma &longs;it vt 6. pyramis erit vt 2. & cuneus vt 3. igitur pyramis conti­<lb/>net 2/3 cunci; </s> <s id="N29A82"><!-- NEW -->igitur alterum &longs;olidum ABDC e&longs;t 1/3 cunei; </s> <s id="N29A86"><!-- NEW -->cunei cen­<lb/>trum grauitatis idem e&longs;t, quod trianguli HKD, per Corol. <!-- REMOVE S-->1. Th.3.igi­<lb/>tur e&longs;t in linea directionis MF.ita vt IM &longs;it 1/3 totius ID, per Th 3. py­<lb/>ramidis verò centrum grauitatis e&longs;t in linea NG, ita vt IN &longs;it 1/4 totius <lb/>ID, per Th.4. igitur &longs;i e&longs;t NM ad ML, vt &longs;olidum ABDC ad pyra­<lb/>midem AHD, id e&longs;t vt 1.ad 2. certè NI, & NL erunt æquales; </s> <s id="N29A96"><!-- NEW -->&longs;ed IN <lb/>e&longs;t 1/4 totius ID; igitur IL 1/2 ergo L dirimit æqualiter ID, quod erat <lb/>demon&longs;tr. </s> <s id="N29A9E">&longs;it ID 12.IN 3.IM 4. IL 6. <!-- KEEP S--></s> </p> <p id="N29AA2" type="main"> <s id="N29AA4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s> </p> <p id="N29AB0" type="main"> <s id="N29AB2"><!-- NEW --><emph type="italics"/>Si voluatur &longs;ector circa axem parallelum &longs;ubten&longs;æ, determinari pote&longs;t cen­<lb/>trum percu&longs;&longs;ionis, dato centro grauitatis &longs;ectoris, quod tantum hactenus in­<lb/>uentum e&longs;t ex &longs;uppo&longs;ita circuli quadratura<emph.end type="italics"/>: </s> <s id="N29ABF"><!-- NEW -->&longs;it enim &longs;ector AKHM, &longs;ub­<lb/>ten&longs;a KM; </s> <s id="N29AC5"><!-- NEW -->diuidatur AI in tres partes æquales ADFI, item AH, in <lb/>tres æquales AEGH, centrum grauitatis &longs;ectoris non e&longs;t in F, quod e&longs;t <lb/>centrum grauitatis trianguli AMK, &longs;ed propiùs accedit ad H; </s> <s id="N29ACD"><!-- NEW -->nec <lb/>etiam e&longs;t in G, quod e&longs;t centrum grauitatis trianguli ALN, &longs;ed propiùs <lb/>accedit ad A; </s> <s id="N29AD5"><!-- NEW -->ergo e&longs;t inter FG, v.g. <!-- REMOVE S-->in R, ita vt AH &longs;it ad AR vt arcus <lb/>MHK ad 2/3 &longs;ubten&longs;æ MK; </s> <s id="N29ADD"><!-- NEW -->id e&longs;t ad MP; </s> <s id="N29AE1"><!-- NEW -->vt demon&longs;trat La Faille Prop. <!-- REMOVE S--><lb/>34. pote&longs;t etiam haberi centrum grauitatis &longs;egmenti circuli; </s> <s id="N29AE8"><!-- NEW -->&longs;it enim <lb/>&longs;egmentum FCHI cuius centrum &longs;it B; </s> <s id="N29AF0"><!-- NEW -->&longs;int BC. BI. BH. diuidens æ-<pb pagenum="426" xlink:href="026/01/460.jpg"/>qualiter CI; </s> <s id="N29AF9"><!-- NEW -->&longs;itque D centrum grauitatis trianguli BCI; </s> <s id="N29AFD"><!-- NEW -->&longs;it E centrum <lb/>grauitatis &longs;ectoris BCHI, &longs;itque vt &longs;ectio FCHI, ad triangulum BEI, <lb/>ita DE ad EG, vel vt &longs;ectio ad &longs;ectorem, ita DE ad DG; G e&longs;t centrum <lb/>grauitatis &longs;ectionis, per p.7. </s> </p> <p id="N29B07" type="main"> <s id="N29B09"><!-- NEW -->His po&longs;itis voluatur &longs;ector AKHM, circa axem CB, perinde &longs;e ha­<lb/>bet circumactus, atque &longs;i &longs;ingulis partibus incumberent rectæ, quæ e&longs;&longs;ent <lb/>vt motus earumdem pretium, vt con&longs;tat ex dictis; </s> <s id="N29B11"><!-- NEW -->igitur &longs;it &longs;ector AEF <lb/>D, æqualis priori, perinde &longs;e habet, atque &longs;olidum AEFDCB, quod <lb/>&longs;cilicet con&longs;tat ex pyramide AEDCB, & &longs;egmento cylindri EFDCB; </s> <s id="N29B19"><!-- NEW --><lb/>pyramidis centrum grauitatis &longs;it I, ita vt IG &longs;it 1/4 GA, &longs;it M centrum <lb/>grauitatis &longs;egmenti &longs;olidi, &longs;eu potiùs &longs;it terminus perpendicularis deor­<lb/>&longs;um, quæ ducatur per centrum grauitatis eiu&longs;dem &longs;olidi; </s> <s id="N29B22"><!-- NEW -->diuidatur IM <lb/>in N, ita vt IN &longs;it ad NM, vt &longs;egmentum cylindri GEFDCB, ad <lb/>pyramidem AEDCB; certè N e&longs;t centrum grauitatis &longs;olidi AEFDCHB, <lb/>per p.7. igitur N e&longs;t centrum percu&longs;&longs;ionis &longs;ectoris circumacti. </s> </p> <p id="N29B2C" type="main"> <s id="N29B2E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s> </p> <p id="N29B3A" type="main"> <s id="N29B3C"><!-- NEW --><emph type="italics"/>Si<emph.end type="italics"/> <emph type="italics"/>&longs;ector AKHM voluatur circa Tangentem NHL, determinari <lb/>pote&longs;t centrum percu&longs;&longs;ionis eodem modo<emph.end type="italics"/>; </s> <s id="N29B4D"><!-- NEW -->nam a&longs;&longs;umi pote&longs;t cuneus, vt &longs;uprà, <lb/>cuius ba&longs;is &longs;it &longs;egmentum cylindri; </s> <s id="N29B53"><!-- NEW -->tùm pyramis cum eadem ba&longs;i; </s> <s id="N29B57"><!-- NEW -->tùm in­<lb/>ueniri centrum grauitatis vtriu&longs;que; </s> <s id="N29B5D"><!-- NEW -->tùm detracta pyramide ex cuneo, <lb/>haberi re&longs;iduum &longs;olidum, cuius centrum grauitatis inuenietur, iuxta pr&etail;­<lb/>dictam praxim; quippe hoc erit centrum percu&longs;&longs;ionis quæ&longs;itum. </s> </p> <p id="N29B65" type="main"> <s id="N29B67"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s> </p> <p id="N29B73" type="main"> <s id="N29B75"><!-- NEW --><emph type="italics"/>Si voluatur<emph.end type="italics"/> <emph type="italics"/>triangulum FBH circa FM, in quam cadit HF perpen­<lb/>diculariter: </s> <s id="N29B83"><!-- NEW -->&longs;i a&longs;&longs;umatur NH<emph.end type="italics"/> 1/4 <emph type="italics"/>FI, ducaturque NP parallela HB, &longs;e­<lb/>cans FC in O, dico punctum O e&longs;&longs;e centrum percu&longs;&longs;ionis<emph.end type="italics"/>; quod eodem modo <lb/>probatur quo &longs;uprà Th.11. <!-- KEEP S--></s> </p> <p id="N29B95" type="main"> <s id="N29B97"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s> </p> <p id="N29BA3" type="main"> <s id="N29BA5"><!-- NEW --><emph type="italics"/>Si voluatur quodlibet triangulum circa angulum rectum, determinari pe­<lb/>test centrum percu&longs;&longs;ionis<emph.end type="italics"/>; </s> <s id="N29BB0"><!-- NEW -->&longs;it enim triangulum ABC; </s> <s id="N29BB4"><!-- NEW -->ducatur quælibet <lb/>linea Tangens angulum, v.g. <!-- REMOVE S-->DBE, circa quam voluatur triangulum, du­<lb/>cantur AE, CD perpendiculares AD; </s> <s id="N29BBE"><!-- NEW -->aliæ duæ ip&longs;is æquales AFCG, <lb/>perpendicularis in AC; </s> <s id="N29BC4"><!-- NEW -->tùm FG connectantur; </s> <s id="N29BC8"><!-- NEW -->eleueturque Trapezus <lb/>AG, donec AF, CG incubent perpendiculariter plano ABC; </s> <s id="N29BCE"><!-- NEW -->denique <lb/>à B ducantur rectæ ad omnia puncta Trapezi erecti, habebitur pyramis, <lb/>cuius centrum grauitatis, dabit centrum percu&longs;&longs;ionis quæ&longs;itum, per Th. <!-- REMOVE S--><lb/>11. quod vt fiat, inueniatur centrum grauitatis Trapezi AG, modo di­<lb/>cto, ducta &longs;cilicet FC, a&longs;&longs;umptoque I centro grauitatis trianguli FGC <lb/>& L centro grauitatis trianguli FAC; </s> <s id="N29BDD"><!-- NEW -->&longs;i enim ducatur LI, &longs;itque LI <lb/>ad LP, vt Trapezium AG, ad triangulum FGC; </s> <s id="N29BE3"><!-- NEW -->certè P e&longs;t centrum <lb/>grauitatis Trapezij per p.7. tùm ex P erecto ducatur recta ad B, hæc erit <lb/>axis pyramidis; </s> <s id="N29BEB"><!-- NEW -->porrò &longs;i ducatur perpendicularis PO; </s> <s id="N29BEF"><!-- NEW -->tùm BO habebi-<pb pagenum="427" xlink:href="026/01/461.jpg"/>tur orthogonium POB; denique a&longs;&longs;umatur OR 1/4 totius OB, R erit <lb/>centrum percu&longs;&longs;ionis trianguli ACB per Th. 11. </s> </p> <p id="N29BFA" type="main"> <s id="N29BFC"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29C08" type="main"> <s id="N29C0A"><!-- NEW -->Hinc colligo quid dicendum &longs;it de rectangulo ita rotato, vt diagona­<lb/>lis cadat perpendiculariter in axem, circa quem rotatur; </s> <s id="N29C10"><!-- NEW -->&longs;it enim re­<lb/>ctangulum CF, cuius diagonalis AIA, axis circa quem voluitur BR, in­<lb/>ueniantur centra percu&longs;&longs;ionis vtriu&longs;que trianguli &longs;eor&longs;im AFH, ACH, <lb/>rotati circa axem BR per Th. 16. connectantur rectâ, in hac erit cen­<lb/>trum percu&longs;&longs;ionis totius rectanguli; </s> <s id="N29C1C"><!-- NEW -->cù di&longs;tantiæ à centro communi <lb/>&longs;int vt pyramides permutando per p.7. vt con&longs;tat ex dictis; ex quibus <lb/>etiam &longs;atis intelligetur quid de alijs planis, tùm regularibus, tùm irre­<lb/>gularibus dicendum &longs;it, cù &longs;cilicet po&longs;&longs;int in triangula diuidi. </s> </p> <p id="N29C26" type="main"> <s id="N29C28"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s> </p> <p id="N29C34" type="main"> <s id="N29C36"><!-- NEW --><emph type="italics"/>Si voluatur triangulare planum parallelum circulo, in quo voluitur, deter­<lb/>minari pote&longs;t eius centrum percu&longs;&longs;ionis<emph.end type="italics"/>; </s> <s id="N29C41"><!-- NEW -->&longs;it enim triangulum AFH, quod <lb/>ita voluatur, vt extremitas H de&longs;cribat arcum HS, & F arcum FR; </s> <s id="N29C47"><!-- NEW -->certè <lb/>F mouetur velociùs quàm H iuxta rationem AF ad AH; </s> <s id="N29C4D"><!-- NEW -->&longs;it ergo FM æ­<lb/>qualis FA, & HN æqualis HA; </s> <s id="N29C53"><!-- NEW -->ducatur MN, erigatur Trapezus FN, <lb/>donec incubet plano AFH, & cen&longs;eantur ductæ ab A rectæ ad puncta <lb/>MN erecta; </s> <s id="N29C5B"><!-- NEW -->habebitur pyramis; </s> <s id="N29C5F"><!-- NEW -->&longs;it autem centrum grauitatis L, Trapezij <lb/>FN, &longs;itque LG perpendicularis in FH, ducatur AG, a&longs;&longs;umaturque DG <lb/>1/4 AG; </s> <s id="N29C67"><!-- NEW -->haud dubiè D e&longs;t centrum grauitatis huius; </s> <s id="N29C6B"><!-- NEW -->&longs;it linea directionis <lb/>DC; </s> <s id="N29C71"><!-- NEW -->quippe punctum D mouetur per Tangentem: </s> <s id="N29C75"><!-- NEW -->quod etiam de alijs <lb/>punctis dictum e&longs;to; </s> <s id="N29C7B"><!-- NEW -->e&longs;t enim hæc ratio motus circularis; igitur maximus <lb/>ictus erit in C per p. </s> <s id="N29C81">8. igitur C e&longs;t centrum percu&longs;&longs;ionis. </s> </p> <p id="N29C84" type="main"> <s id="N29C86"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29C93" type="main"> <s id="N29C95">Collige perinde &longs;e habere motum puncti F, atque &longs;i ip&longs;i incumberet <lb/>linea FM, & puncto H, HN. </s> </p> <p id="N29C9A" type="main"> <s id="N29C9C"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29CA9" type="main"> <s id="N29CAB"><!-- NEW -->Præterea centrum percu&longs;&longs;ionis aliquando e&longs;&longs;e extra rectam AH, cum <lb/>&longs;cilicet angulus circa, quem voluitur e&longs;t minùs acutus, &longs;it enim trian­<lb/>gulum AGL quod voluatur circa A, &longs;itque centrum grauitatis Trapezij <lb/>E, de quo &longs;uprà; </s> <s id="N29CB5"><!-- NEW -->ducantur EC, AC, &longs;it CB 1/4 AC, ducatur linea dire­<lb/>ctionis BI; vides I e&longs;&longs;e extra AL. <!-- KEEP S--></s> </p> <p id="N29CBC" type="main"> <s id="N29CBE"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29CCB" type="main"> <s id="N29CCD"><!-- NEW -->Præterea o&longs;tendi po&longs;&longs;e longè faciliùs totam rem i&longs;tam; </s> <s id="N29CD1"><!-- NEW -->&longs;it enim tri­<lb/>angulum ABD; </s> <s id="N29CD7"><!-- NEW -->ducatur HBG æqualis BA, perpendicularis in BD; </s> <s id="N29CDB"><!-- NEW --><lb/>diuidatur AD bifariam æqualiter in L; </s> <s id="N29CE0"><!-- NEW -->a&longs;&longs;umatur DE æqualis DL, <lb/>rùm ducantur HL, GE; </s> <s id="N29CE6"><!-- NEW -->inueniatur centrum grauitatis C, Trapezij H <lb/>LEG; </s> <s id="N29CEC"><!-- NEW -->ducatur AC, cuius KC &longs;it 1/4 ducatur KD perpendicularis in <lb/>AC, punctum D e&longs;t centrum percu&longs;&longs;ionis; </s> <s id="N29CF2"><!-- NEW -->quippe &longs;i vertatur Trapezus <lb/>HE, circa axem BD, donec AD cadat in illum perpendiculariter, &longs;it-<pb pagenum="428" xlink:href="026/01/462.jpg"/>que &longs;ectio communis BD; certè habebitur ba&longs;is pyramidis, cuius axis <lb/>erit AC, quæ omnia con&longs;tant. </s> </p> <p id="N29CFF" type="main"> <s id="N29D01"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s> </p> <p id="N29D0D" type="main"> <s id="N29D0F"><!-- NEW --><emph type="italics"/>Determinari pote&longs;t centrum percu&longs;&longs;ionis in latere orthogonij &longs;ubten&longs;o angulo <lb/>recto<emph.end type="italics"/>; </s> <s id="N29D1A"><!-- NEW -->&longs;it enim AGB, latu&longs;que &longs;ubten&longs;um angulo recto AB, &longs;it Trape­<lb/>zus KD, eo modo quo diximus, cuius centrum grauitatis &longs;it H, ducatur <lb/>AH, a&longs;&longs;umatur IH 1/4: </s> <s id="N29D22"><!-- NEW -->AH, ducatur IM perpendicularis in AH: dico <lb/>punctum M e&longs;&longs;e centrum percu&longs;&longs;ionis, quod demon&longs;tratur per Theo­<lb/>rema 17. </s> </p> <p id="N29D2A" type="main"> <s id="N29D2C"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s> </p> <p id="N29D38" type="main"> <s id="N29D3A"><!-- NEW --><emph type="italics"/>Si voluatur triangulum prædictum, circa angulum rectum, determinari <lb/>pote&longs;t<emph.end type="italics"/> <emph type="italics"/>centrum percu&longs;&longs;ionis<emph.end type="italics"/>; </s> <s id="N29D4B"><!-- NEW -->&longs;it enim triangulum ABH, quod voluatur <lb/>circa centrum B; </s> <s id="N29D51"><!-- NEW -->motus puncti A e&longs;t ad motum H, vt BA, ad BH; </s> <s id="N29D55"><!-- NEW -->&longs;it ergo <lb/>Trapezus MG, cuius latus ML &longs;it æquale AB, & GI æquale BH; </s> <s id="N29D5B"><!-- NEW -->erit <lb/>pyramis, eo modo, quo diximus &longs;uprà; </s> <s id="N29D61"><!-- NEW -->&longs;it autem D centrum grauitatis <lb/>ba&longs;is, &longs;eu Trapezij, & AD axis; </s> <s id="N29D67"><!-- NEW -->&longs;it KD 1/4 BD; </s> <s id="N29D6B"><!-- NEW -->&longs;it denique KE perpen­<lb/>dicularis in DB: dico punctum E e&longs;&longs;e centrum percu&longs;&longs;ionis, quod co­<lb/>dem modo demon&longs;tratur, quo &longs;uprà. </s> </p> <p id="N29D73" type="main"> <s id="N29D75"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29D81" type="main"> <s id="N29D83">Hinc colligo primò, de omni triangulo idem pror&longs;us dicendum e&longs;&longs;e, <lb/>e&longs;t enim eadem ratio, vt con&longs;ideranti patebit. </s> </p> <p id="N29D88" type="main"> <s id="N29D8A"><!-- NEW -->Secundò, &longs;i voluatur circa punctum aliquod lateris, po&longs;&longs;e determinari <lb/>centrum percu&longs;&longs;ionis; </s> <s id="N29D90"><!-- NEW -->&longs;it enim triangulum ABC; </s> <s id="N29D94"><!-- NEW -->a&longs;&longs;umatur punctum <lb/>M, circa quod voluatur mode prædicto, motus puncti C, e&longs;t ad motum <lb/>puncti A, vt MC, vel DX, ad MA, vel PO; </s> <s id="N29D9C"><!-- NEW -->hinc Trapezus DPOX, id e&longs;t <lb/>ba&longs;is pyramidis, cuius axis e&longs;t MG, & centrum grauitatis K: </s> <s id="N29DA4"><!-- NEW -->&longs;imiliter <lb/>habetur Trapezus DRNX; </s> <s id="N29DAA"><!-- NEW -->id e&longs;t ba&longs;is alterius pyramidis, cuius axis e&longs;t <lb/>MV, & centrum grauitatis H; </s> <s id="N29DB0"><!-- NEW -->fiat autem vt vtraque pyramis ad eam, cuius <lb/>axis e&longs;t MG, ita tota HK, ad HI; </s> <s id="N29DB6"><!-- NEW -->dico I e&longs;&longs;e centrum commune graui­<lb/>tatis; </s> <s id="N29DBC"><!-- NEW -->ducatur IL perpendicularis in IM; dico L e&longs;&longs;e centrum percu&longs;­<lb/>&longs;ionis quæ&longs;itum. </s> </p> <p id="N29DC2" type="main"> <s id="N29DC4">Tertiò, &longs;i voluatur circa aliud punctum, res eodem modo &longs;uc­<lb/>cedet. </s> </p> <p id="N29DC9" type="main"> <s id="N29DCB"><!-- NEW -->Quartò, &longs;i &longs;it &longs;olidum ad in&longs;tar cunei, con&longs;tans &longs;cilicet ex multis pla­<lb/>nis triangularibus, quæ probè inter &longs;e conueniant; idem etiam accidet, <lb/>quæ omnia ex &longs;uprà dictis clari&longs;&longs;ima efficiuntur. </s> </p> <p id="N29DD3" type="main"> <s id="N29DD5">Quintò, &longs;i &longs;it triangulum EAD, fig. </s> <s id="N29DD8"><!-- NEW -->quod ita voluatur circa centrum <lb/>A, vt latus AE, modò accedat ad CB, modò recedat; </s> <s id="N29DDE"><!-- NEW -->&longs;itque ita diui&longs;a AS <lb/>in R, vt RS &longs;it 1/4 AS, &longs;i ducatur RN, centrum percu&longs;&longs;ionis erit in N, <lb/>quia R e&longs;t centrum grauitatis geminæ pyramidis; </s> <s id="N29DE6"><!-- NEW -->igitur RN linea di­<lb/>rectionis in&longs;tanti percu&longs;&longs;ionis; &longs;i verò producatur AS in G, &longs;intque I & <lb/>M centra grauitatis pyramidum ducanturque IF, MF perpendiculares <lb/>in AI. AM, centrum percu&longs;&longs;ionis erit F, vt con&longs;tat ex dictis. </s> </p> <pb pagenum="429" xlink:href="026/01/463.jpg"/> <p id="N29DF4" type="main"> <s id="N29DF6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s> </p> <p id="N29E02" type="main"> <s id="N29E04"><!-- NEW --><emph type="italics"/>Sectoris minoris quadrante determinari pote&longs;t centrum percu&longs;&longs;ionis, cum <lb/>&longs;cilicet voluitur in plano, cui eiu&longs;dem planum e&longs;t parallelum<emph.end type="italics"/>; </s> <s id="N29E0F"><!-- NEW -->&longs;it enim <lb/>quadrans BAI; </s> <s id="N29E15"><!-- NEW -->ducatur BI, &longs;it pyramis cuius ba&longs;is &longs;it &longs;ectio cylindri, <lb/>erectis, &longs;cilicet perpendicularibus tran&longs;uer&longs;is &longs;upra arcum BTI, eo <lb/>modo, quo &longs;uprà iam &longs;æpè diximus; </s> <s id="N29E1D"><!-- NEW -->v.g. <!-- REMOVE S-->ducta &longs;it Tangens ZT, diui&longs;a bi­<lb/>fariam in C, puncto &longs;cilicet contactus, quæ tandiu voluatur circa CA, <lb/>dum &longs;ecet arcum ad angulos rectos: </s> <s id="N29E27"><!-- NEW -->idem fiat in alijs punctis arcus; </s> <s id="N29E2B"><!-- NEW -->de­<lb/>nique ad extremitates Tangentium ducantur vtrimque à centro A rectæ, <lb/>& habebitur prædicta pyramis mixta, cuius centrum grauitatis inuen­<lb/>tum dabit centrum percu&longs;&longs;ionis; </s> <s id="N29E35"><!-- NEW -->quod vt meliùs oculo &longs;ubijciatur, &longs;it <lb/>triangulum ZTA, voluatur circa CA, donec eius planum &longs;ecet ad an­<lb/>gulos iectos planum quadrantis BAI; </s> <s id="N29E3D"><!-- NEW -->tùm in eo &longs;itu voluatur axis AC <lb/>per totum arcum BI, & habebitur &longs;olidum quæ&longs;itum, cuius centrum gra­<lb/>uitatis ita pote&longs;t inueniri; </s> <s id="N29E45"><!-- NEW -->ducatur BI, tùm AC diuidens BI bifariam <lb/>in E, centrum grauitatis e&longs;t in AC; </s> <s id="N29E4D"><!-- NEW -->a&longs;&longs;umatur GE 1/4 totius AE; </s> <s id="N29E51"><!-- NEW -->certè G <lb/>e&longs;t centrum grauitatis pyramidis ABEI; </s> <s id="N29E57"><!-- NEW -->&longs;it autem D centrum grauitatis <lb/>reliqui &longs;olidi BEIC, &longs;itque vt hoc &longs;olidum ad pyramidem ABEI, ita <lb/>GF ad FD: dico F e&longs;&longs;e centrum grauitatis per p. </s> <s id="N29E5F">7. ducatur FK perpen­<lb/>dicularis in AC, K e&longs;t centrum percu&longs;&longs;ionis per Th.17. <!-- KEEP S--></s> </p> <p id="N29E65" type="main"> <s id="N29E67"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29E73" type="main"> <s id="N29E75">Colligo primò; </s> <s id="N29E78"><!-- NEW -->prædictam pyramidem mixtam e&longs;&longs; 2/3 &longs;ectoris cylindrj; </s> <s id="N29E7C"><!-- NEW --><lb/>&longs;it enim triangulum ACZ erectum, atque îta voluatur per totam pe­<lb/>ripheram IBPVI. fiet &longs;olidum cauum, cuius cauitas erit conus, cuius <lb/>altitudo erit CZ, & ba&longs;is orbis BPVI; </s> <s id="N29E85"><!-- NEW -->&longs;ed hic conus e&longs;t 1/3 cylindri, &longs;ub <lb/>eadem ba&longs;i, & altitudine; </s> <s id="N29E8B"><!-- NEW -->igitur &longs;olidum, quod &longs;upere&longs;t, continet 2/3 cy­<lb/>lindri &longs;ub altitudine CZ, & ba&longs;i BPVI; </s> <s id="N29E91"><!-- NEW -->&longs;ed cauum BAI de quo &longs;uprà <lb/>e&longs;t 1/3 totius; igitur reliquum continet 2/3 &longs;ectoris cylindri BA, &longs;ub alti­<lb/>tudine CT. </s> </p> <p id="N29E99" type="main"> <s id="N29E9B"><!-- NEW -->Secundò colligo, &longs;i a&longs;&longs;umatur &longs;emicirculus PBI momentum quadran­<lb/>tis PBA, æquale e&longs;&longs;e momento quadrantis IA <foreign lang="greek">b</foreign>, vt con&longs;tat; nam I, per <lb/>IM, idem præ&longs;tat quod P, per PQ, & S per SR, idem quod L, <lb/>per LV, &c. </s> </p> <p id="N29EA9" type="main"> <s id="N29EAB"><!-- NEW -->Tertiò, &longs;i voluatur tantùm triangulum ABI, ducaturque GX per­<lb/>pendicularis in AC punctum X erit centrum percu&longs;&longs;ionis; quid mirum <lb/>igitur, &longs;i addito &longs;egmento BCIE, &longs;it in K? </s> </p> <p id="N29EB3" type="main"> <s id="N29EB5">Quartò, &longs;i quadrans AI <foreign lang="greek">b</foreign> trahat deor&longs;um adducto filo ex K, certè in <lb/>K erit centrum percu&longs;&longs;ionis, vt con&longs;tat. </s> </p> <p id="N29EBE" type="main"> <s id="N29EC0">Quintò, &longs;i vterque quadrans BI <foreign lang="greek">b</foreign> A &longs;imul cadat, centrum percu&longs;&longs;io­<lb/>nis erit in K, &longs;ed duplò maior ictus. </s> </p> <p id="N29EC9" type="main"> <s id="N29ECB">Sexto, &longs;i &longs;emicirculus APBI cadar, centrum etiam percu&longs;&longs;ionis erit <lb/>in K, quia quadrans PBA æquiualet quadranti A <foreign lang="greek">b</foreign> I. <!-- KEEP S--></s> </p> <p id="N29ED5" type="main"> <s id="N29ED7"><!-- NEW -->Septimò, &longs;i a&longs;&longs;umatur &longs;ector maior quadrante, &longs;ed minor &longs;emicirculo, <lb/>v.g. <!-- REMOVE S-->ASBI, &longs;it BAC æqualis BAS; </s> <s id="N29EDF"><!-- NEW -->inueniatur centrum grauitatis BA <pb pagenum="430" xlink:href="026/01/464.jpg"/>C eodem modo, quo inuentum e&longs;t centrum F quadrantís rotati: </s> <s id="N29EEC"><!-- NEW -->&longs;imili­<lb/>ter inueniatur centrum grauitatis TAI rotati; </s> <s id="N29EF2"><!-- NEW -->connectantur rectâ hæc <lb/>duo centra inuenta, &longs;itque vt duplum BAC ad CAI, ita &longs;egmentum <lb/>connectentïs centra, quod terminatur in centro CAI ad aliud &longs;egmen­<lb/>tum; punctum diuidens &longs;egmenta erit centrum grauitatis quæ&longs;itum, à <lb/>quo &longs;i ducatur perpendicularis, eo modo, quo diximus, hæc dabit cen­<lb/>trum percu&longs;&longs;ionis. </s> </p> <p id="N29F00" type="main"> <s id="N29F02"><!-- NEW -->Octauò, &longs;i a&longs;&longs;umatur &longs;ector maior &longs;emicirculo, v.g. <!-- REMOVE S-->AVBL, eodem <lb/>modo procedendum e&longs;t; quippe PAV æquiualet CAB, & IAL æquiua­<lb/>let CAI, & BAP æquiualet BAI, nec e&longs;t noua difficultas. </s> </p> <p id="N29F0C" type="main"> <s id="N29F0E">Nonò, hinc &longs;i circulus integer circa centrum voluatur, centrum per­<lb/>cu&longs;&longs;ionis erit in K, &longs;ed ictu quadruplo ictus inflicti à quadrante. </s> </p> <p id="N29F13" type="main"> <s id="N29F15"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s> </p> <p id="N29F21" type="main"> <s id="N29F23"><!-- NEW --><emph type="italics"/>Si rotetur circulus circa punctum circumferentia vel circa Tangentem, <lb/>determinari pote&longs;t centrum percu&longs;&longs;ionis<emph.end type="italics"/>; </s> <s id="N29F2E"><!-- NEW -->&longs;it enim centro B, ANCP, rota­<lb/>tus circa TA, in quam diameter AC cadit perpendiculariter; </s> <s id="N29F34"><!-- NEW -->a&longs;&longs;umatur <lb/>RC 1/3 AC: </s> <s id="N29F3A"><!-- NEW -->dico R e&longs;&longs;e centrum percu&longs;&longs;ionis; quia motus C e&longs;t ad mo­<lb/>tum R, vt CF ad RH, & ad motum B, vt CF ad BL, &c. </s> <s id="N29F40"><!-- NEW -->igitur perinde <lb/>&longs;e habet planum ANCP, atque &longs;i &longs;emicylindrus ACF ip&longs;i incubaret, <lb/>vt patet, &longs;ed centrum grauitatis huius &longs;olidi e&longs;t X in quo CL & FB de­<lb/>cu&longs;&longs;antur; </s> <s id="N29F4A"><!-- NEW -->&longs;ed vt demon&longs;tratum e&longs;t &longs;uprà, &longs;i ducatur HXR, RC e&longs;t 2/3 <lb/>totius AC; igitur R e&longs;t centrum percu&longs;&longs;ionis. </s> </p> <p id="N29F50" type="main"> <s id="N29F52"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N29F5E" type="main"> <s id="N29F60"><!-- NEW -->Primò colligo, &longs;i &longs;egmentum circuli voluatur: </s> <s id="N29F64"><!-- NEW -->&longs;imiliter haberi pote&longs;t <lb/>centrum percu&longs;&longs;ionis, inuento &longs;cilicet centro grauitatis ba&longs;is vtriu&longs;que <lb/>v.g. <!-- REMOVE S-->&longs;i &longs;egmentum OAQ voluatur circa TA, inueniri debet centrum <lb/>grauitatis eiu&longs;dem & ad illud à puncto H recta ducenda; </s> <s id="N29F70"><!-- NEW -->itemque in­<lb/>ueniendum e&longs;t centrum grauitatis &longs;egmenti Ellip&longs;eos HAI, & ad illud <lb/>à puncto R ducenda recta; nam vtriu&longs;que decu&longs;&longs;ationis punctum dabit <lb/>centrum grauitatis huius &longs;olidi, ex qua &longs;i ducatur perpendicularis in AR, <lb/>extremitas dabit centrum percu&longs;&longs;ionis. </s> </p> <p id="N29F7D" type="main"> <s id="N29F7F">Secundò, &longs;i voluatur circulus CNAH circa PN, habebitur centrum <lb/>percu&longs;&longs;ionis eodem modo, inuentis &longs;cilicet centris grauitatis &longs;emicir­<lb/>culi PNC, & &longs;emiellip&longs;eos, cuius altera &longs;emidiameter &longs;it BF, altera BP, <lb/>vt con&longs;tat ex dictis, </s> </p> <p id="N29F88" type="main"> <s id="N29F8A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s> </p> <p id="N29F96" type="main"> <s id="N29F98"><!-- NEW --><emph type="italics"/>Si voluatur circulus circa punctum circumferentia in circulo parallelo &longs;uo <lb/>plano, determinari pote&longs;t centrum percu&longs;&longs;ionis, quod di&longs;tat <emph.end type="italics"/>2/3 <emph type="italics"/>diametri à cen­<lb/>tro motus<emph.end type="italics"/>; </s> <s id="N29FB1"><!-- NEW -->&longs;it enim circulus ACFG, centro B, qui voluatur circa cen­<lb/>trum A; </s> <s id="N29FB7"><!-- NEW -->motus puncti F e&longs;t ad motum puncti B, vt recta AF ad rectam <lb/>AD, & ad motum puncti C, vt AF ad AC; </s> <s id="N29FBD"><!-- NEW -->idem dico de alis punctis; </s> <s id="N29FC1"><!-- NEW --><lb/>&longs;it EH æqualis AF, diui&longs;a bifariam in F, quæ tandiu voluatur, donec <pb pagenum="431" xlink:href="026/01/465.jpg"/>&longs;ecet arcum CFG ad angulos rectos; idem pror&longs;us fiat in aliis punctis <lb/>peripheriæ, a&longs;&longs;umptis &longs;cilicet lineis æqualibus &longs;ubten&longs;is arcuum, v.g. <!-- REMOVE S-->in <lb/>puncto D, a&longs;&longs;umpta linea æquali AD, in puncto C, a&longs;&longs;umpta æquali AC, <lb/>&c. </s> <s id="N29FD3"><!-- NEW -->hoc po&longs;ito habetur &longs;olidum, quod facilè vocauerim Elliptico cylin­<lb/>dricum, cuius con&longs;tructio talis e&longs;t, &longs;it cylindrus RI, cuius diameter <lb/>ba&longs;is &longs;it KI, æqualis diametro AF circuli prioris; </s> <s id="N29FDB"><!-- NEW -->&longs;it etiam altitudo KR, <lb/>æqualis prædictæ diametro KI, &longs;it KR diui&longs;a bifariam in L, &longs;itque pla­<lb/>num IL &longs;ecans cylindrum, itemque alterum LP, vtraque &longs;ectio Ellip&longs;is <lb/>e&longs;t, vt patet; </s> <s id="N29FE5"><!-- NEW -->ac proinde habetur &longs;olidum quæ&longs;itum LIP con&longs;tans gemi­<lb/>na ba&longs;i LI. & LP Elliptica, & reliqua circumferentià cylindricâ, cuius <lb/>centrum grauitatis e&longs;t in N, id e&longs;t in puncto decu&longs;&longs;ationis rectarum PM, <lb/>IS, quæ diuidunt ILPL bifariam æqualiter, e&longs;t autem NO 1/3 totius <lb/>LO, per Sch. <!-- REMOVE S-->Th.2. hoc po&longs;ito &longs;it XF 1/3 totius AF: dico e&longs;&longs;e centrum <lb/>percu&longs;&longs;ionis quæ&longs;itum circuli ACFG rotati circa A, quia perinde &longs;e <lb/>habet, atque &longs;i puncto X incubaret prædictum &longs;olidum ellipticocylindri­<lb/>cum, cuius X e&longs;&longs;et centrum grauitatis. </s> </p> <p id="N29FF9" type="main"> <s id="N29FFB"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A007" type="main"> <s id="N2A009">Ob&longs;eruabis primò, in plano ACFG, vt punctum X &longs;it centrum per­<lb/>cu&longs;&longs;ionis, incidendam e&longs;&longs;e &longs;triam quamdam, &longs;eu rimam, quæ termi­<lb/>netur in X. <!-- KEEP S--></s> </p> <p id="N2A011" type="main"> <s id="N2A013">Secundò, idem e&longs;&longs;e centrum percu&longs;&longs;ionis rectæ AF, quæ voluitur <lb/>circa A, &longs;iue &longs;it &longs;implex linea, &longs;iue diameter circuli. </s> </p> <p id="N2A018" type="main"> <s id="N2A01A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s> </p> <p id="N2A026" type="main"> <s id="N2A028"><!-- NEW --><emph type="italics"/>Si voluatur rectangulum parallelum orbi in quo voluitur determinari<emph.end type="italics"/> <emph type="italics"/>po­<lb/>test centrum percu&longs;&longs;ionis<emph.end type="italics"/>; </s> <s id="N2A039"><!-- NEW -->&longs;it enim rectangulum AD, quod voluatur circa <lb/>centrum A, eo modo, quo dictum e&longs;t &longs;it ducta AD, inueniatur centrum <lb/>I, trianguli ABD; </s> <s id="N2A041"><!-- NEW -->itemque centrum H, trianguli ADF, per Th. 17. <lb/>tùm ducta IH, diuidatur bifariam in K; </s> <s id="N2A047"><!-- NEW -->ducatur AK, tùm GK perpen­<lb/>dicularis in AK: dico G e&longs;&longs;e centrum percu&longs;&longs;ionis, per po&longs;.7.& Theo­<lb/>rema 17. </s> </p> <p id="N2A04F" type="main"> <s id="N2A051"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A05D" type="main"> <s id="N2A05F"><!-- NEW -->Colligo ex his facilè po&longs;&longs;e determinari centrum percu&longs;&longs;ionis in alijs <lb/>figuris planis; quia diuidi po&longs;&longs;unt in plura triangula. </s> </p> <p id="N2A065" type="main"> <s id="N2A067"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s> </p> <p id="N2A073" type="main"> <s id="N2A075"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari centrum percu&longs;&longs;ionis &longs;olidi<emph.end type="italics"/> <emph type="italics"/>trium facierum ABDE<emph.end type="italics"/>; </s> <s id="N2A084"><!-- NEW --><lb/>vt demon&longs;tretur centrum percu&longs;&longs;ionis pyramidis, & pri&longs;matis, præmitti <lb/>debuit hoc &longs;olidum; </s> <s id="N2A08B"><!-- NEW -->&longs;it enim &longs;olidum priori &longs;imile, A.M. G.C. motus <lb/>puncti M, e&longs;t ad motum puncti G, vt recta BM ad rectam BG; </s> <s id="N2A091"><!-- NEW -->igitur &longs;it <lb/>NK ad OH, vt BM ad BG; </s> <s id="N2A097"><!-- NEW -->certè perinde &longs;e habet punctum M, atque <lb/>&longs;i NMK incubaret, non quidem per MG, &longs;ed per lineam perpendicu­<lb/>larem ductam in BM, vt patet ex dictis: </s> <s id="N2A09F"><!-- NEW -->idem dico de puncto G, quod <lb/>perinde &longs;e habet, atque &longs;i incubaret OGH; </s> <s id="N2A0A5"><!-- NEW -->itaque inuenire oportet <lb/>centrum grauitatis &longs;olidi ACHKNOA, quod vt fiat, a&longs;&longs;umatur IP <pb pagenum="432" xlink:href="026/01/466.jpg"/>æqualis AC; </s> <s id="N2A0B0"><!-- NEW -->ducantur AP, CI centrum grauitatis &longs;olidi ACIKNP <lb/>re&longs;pondet per lineam directionis puncto E, ita vt EG &longs;it 1/3 GB per Co­<lb/>roll.1. Th.3.&longs;i autem a&longs;&longs;umatur FG 1/4 totius BG, &longs;itque linea QFX, <lb/>& ex puncto F &longs;u&longs;tineatur vtraque pyramis AOPN, & CIHK, erit <lb/>perfectum æquilibrium per Th. 4. igitur &longs;it FE ad ED, vt &longs;olidum <lb/>ACHKNO ad vtramque pyramidem AOPN, CIHK, certè pun­<lb/>ctum D erit centrum grauitatis &longs;olidi ACHKNO, per p.7. a&longs;&longs;umatur <lb/>GL æqualis GD; </s> <s id="N2A0C2"><!-- NEW -->ducatur BL, hæc e&longs;t axis vt patet, modò GM &longs;it æqua­<lb/>lis GB; </s> <s id="N2A0C8"><!-- NEW -->&longs;i enim inæqualis e&longs;t, &longs;it GL ad GM, vt GD ad GB: </s> <s id="N2A0CC"><!-- NEW -->præterea <lb/>ducatur DR parallela GM; </s> <s id="N2A0D2"><!-- NEW -->denique ducatur perpendicularis FR in B <lb/>L; </s> <s id="N2A0D8"><!-- NEW -->dico F e&longs;&longs;e centrum percu&longs;&longs;ionis, vt patet ex dictis &longs;uprà, præ&longs;ertim in <lb/>Th. 17. & alibi pa&longs;&longs;im, ne toties eadem repetere cogar ad nau&longs;eam; <lb/>quamquam enim hæc &longs;atis noua &longs;unt, illa tamen indicanda potiùs, quàm <lb/>fusè tractanda e&longs;&longs;e putaui. </s> </p> <p id="N2A0E2" type="main"> <s id="N2A0E4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s> </p> <p id="N2A0F0" type="main"> <s id="N2A0F2"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari centrum percu&longs;&longs;ionis pyramidis, cum voluitur circa <lb/>verticem<emph.end type="italics"/>; </s> <s id="N2A0FD"><!-- NEW -->&longs;it enim &longs;olidum, de quo &longs;uprà ABCGM, fitque aliud &longs;oli­<lb/>dum ABCHKMNOG, cuius axis &longs;it BL & centrum grauitatis R, <lb/>hoc ip&longs;um e&longs;t centrum percu&longs;&longs;ionis &longs;olidi ABCGM, ducta &longs;cilicet RF, <lb/>per Th.24. iam verò &longs;i ex &longs;olido ACIKNP, detrahatur prædictum <lb/>&longs;olidum ABCGM, &longs;upere&longs;t vtrimque integra pyramis, &longs;cilicet CMK <lb/>IG, & AMNPG, cuius axis communis erit eadem BL, vt patet; </s> <s id="N2A10B"><!-- NEW -->itaque <lb/>a&longs;&longs;umatur LY 1/4 LB, Y re&longs;pondebit centrum percu&longs;&longs;ionis &longs;olidi ACIK <lb/>NP per Corol.4. Th.19. igitur &longs;it vt vtraque pyramis ANPG, & AK <lb/>IG, ad reliquum &longs;olidum ABCGM, ita RY, ad YZ; </s> <s id="N2A115"><!-- NEW -->dico Z e&longs;&longs;e cen­<lb/>trum percu&longs;&longs;ionis vtriu&longs;que pyramidis, ductâ &longs;cilicet perpendiculari <lb/>Z <foreign lang="greek">d</foreign>, vt con&longs;tat ex dictis; quare in axe pyramidis a&longs;&longs;umatur æqualis BZ, <lb/>& habebitur intentum. </s> </p> <p id="N2A123" type="main"> <s id="N2A125"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A131" type="main"> <s id="N2A133"><!-- NEW -->Ob&longs;eruabis primò, &longs;olidum integrum AKNPI e&longs;&longs;e &longs;ubduplum pri&longs;­<lb/>matis eiu&longs;dem altitudinis & ba&longs;is NI; pyramidem verò CMI e&longs;&longs;e 1/6 <lb/>eiu&longs;dem pri&longs;matis, ergo vtramque æqualem 1/3 igitur &longs;olidum ABCGM <lb/>1/6. igitur æquale alteri pyramidum, igitur RY duplam e&longs;&longs;e YZ. </s> </p> <p id="N2A13E" type="main"> <s id="N2A140">Secundò, ob&longs;eruabis punctum Z dici po&longs;&longs;e centrum percu&longs;&longs;ionis in­<lb/>terius, à quo deinde &longs;i ducatur recta Z <foreign lang="greek">d</foreign> perpendicularis in BL, termi­<lb/>nabitur in <foreign lang="greek">d</foreign>, quod dici pote&longs;t centrum percu&longs;&longs;ionis exterius. </s> </p> <p id="N2A14F" type="main"> <s id="N2A151">Tertiò, ob&longs;eruabis, centrum percu&longs;&longs;ionis exterius aliquando e&longs;&longs;e in <lb/>ip&longs;a facie, &longs;eu linea BG, cum &longs;cilicet angulus MPG e&longs;t valdè acutus, <lb/>aliquando e&longs;&longs;e extra &longs;uperficiem corporis, v. <!-- REMOVE S-->g. <!-- REMOVE S-->in <foreign lang="greek">d</foreign>, cum &longs;cilicet an­<lb/>gulus MBG e&longs;t obtu&longs;ior, quod iam &longs;uprà ob&longs;eruatum e&longs;t, cum de trian­<lb/>gulo Cor.2. Th.17. <!-- KEEP S--></s> </p> <pb pagenum="433" xlink:href="026/01/467.jpg"/> <p id="N2A169" type="main"> <s id="N2A16B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s> </p> <p id="N2A177" type="main"> <s id="N2A179"><!-- NEW --><emph type="italics"/>Pote&longs;t determinari centrum percu&longs;&longs;ionis parallelipedi<emph.end type="italics"/>; </s> <s id="N2A182"><!-- NEW -->&longs;it enim paralle­<lb/>lipedum MF quod voluatur circa MK; </s> <s id="N2A188"><!-- NEW -->&longs;it rectangulum LE &longs;ecans bifa­<lb/>riam æqualiter parallelipedum; </s> <s id="N2A18E"><!-- NEW -->centrum percu&longs;&longs;ionis erit in plano re­<lb/>ctanguli LE; </s> <s id="N2A194"><!-- NEW -->ducatur LE, diagonalis; </s> <s id="N2A198"><!-- NEW -->inueniatur centrum percu&longs;&longs;ionis <lb/>rectanguli LE, per Th.23. &longs;itque N, v.g. <!-- REMOVE S-->ducatur NO, dico O e&longs;&longs;e cen­<lb/>trum percu&longs;&longs;ionis quæ&longs;itum, &longs;cilicet exterius, vt patet ex dictis; </s> <s id="N2A1A2"><!-- NEW -->pote&longs;t <lb/>etiam determinari, &longs;i voluatur circa AC, vel circa PR, nam perinde <lb/>&longs;e habet prædictum parallelipedum, atque ip&longs;um rectangulum; hoc verò <lb/>atque ip&longs;um triangulum, in quo nulla pror&longs;us e&longs;t difficultas. </s> </p> <p id="N2A1AC" type="main"> <s id="N2A1AE"><!-- NEW -->Pote&longs;t etiam determinari centrum percu&longs;&longs;ionis cunei, id e&longs;t &longs;emipa­<lb/>rallelipedi, &longs;iue circa MK, &longs;ine circa IG voluatur; quæ omnia pa­<lb/>tent ex dictis. </s> </p> <p id="N2A1B6" type="main"> <s id="N2A1B8"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s> </p> <p id="N2A1C4" type="main"> <s id="N2A1C6"><!-- NEW --><emph type="italics"/>Determinari<emph.end type="italics"/> <emph type="italics"/>pote&longs;t centrum percu&longs;&longs;ionis &longs;olidi ABDE, &longs;i voluatur circa <lb/>axem IDH<emph.end type="italics"/>; </s> <s id="N2A1D7"><!-- NEW -->nam motus puncti C e&longs;t ad motum puncti E, vt DC ad <lb/>DE, vel vt BN æqualis DC ad LK æqualem ED; </s> <s id="N2A1DD"><!-- NEW -->mouentur enim AC <lb/>B æquali motu; </s> <s id="N2A1E3"><!-- NEW -->itaque perinde &longs;e habet prædictum &longs;olidum in ordine <lb/>ad percu&longs;&longs;ionem, atque &longs;i e&longs;&longs;et &longs;olidum BMKLD; </s> <s id="N2A1E9"><!-- NEW -->id e&longs;t duplex pyra­<lb/>mis, &longs;cilicet DNMKL, & DMNBA, quarum centra grauitatis &longs;int <lb/>PQ, & commune vtriu&longs;que &longs;it R iuxtam modum &longs;uprà po&longs;itum; </s> <s id="N2A1F1"><!-- NEW -->duca­<lb/>tur SR perpendicularis in RD: dico S e&longs;&longs;e centrum percu&longs;&longs;ionis exte­<lb/>rius quæ&longs;itum, quod eodem modo probatur, quo &longs;uprà. </s> </p> <p id="N2A1F9" type="main"> <s id="N2A1FB"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A207" type="main"> <s id="N2A209"><!-- NEW -->Primò colligo inde, vbi &longs;it centrum percu&longs;&longs;ionis cylindri, &longs;iue volua­<lb/>tur circa Tangentem ba&longs;is, &longs;iue circa diametrum eiu&longs;dem; nam idem de <lb/>cylindro dicendum e&longs;t, quod de parallelipedo dictum e&longs;t Th.26. </s> </p> <p id="N2A211" type="main"> <s id="N2A213"><!-- NEW -->Secundò colligo, centrum percu&longs;&longs;ionis coni; quippe vt &longs;e habet pyra­<lb/>mis ad pri&longs;ma, ita &longs;e habet conus ad cylindrum. </s> </p> <p id="N2A219" type="main"> <s id="N2A21B">Tertiò, colligo centrum percu&longs;&longs;ionis Pyramidis quando voluitur cir­<lb/>ca latus ba&longs;is per Th.27. </s> </p> <p id="N2A220" type="main"> <s id="N2A222"><!-- NEW -->Quartò, colligo centrum percu&longs;&longs;ionis cylindri; cum voluitur circa <lb/>Tangentem parallelum axi per Th.22. <!-- KEEP S--></s> </p> <p id="N2A229" type="main"> <s id="N2A22B"><!-- NEW -->Quintò, colligo centrum grauitatis pri&longs;matis, &longs;iue voluatur circa la­<lb/>tus ba&longs;is; </s> <s id="N2A231"><!-- NEW -->tunc enim idem pror&longs;us dicendum e&longs;t, quod de parallelipedo; </s> <s id="N2A235"><!-- NEW --><lb/>&longs;iue circa lineam parallelam axi; tunc enim centrum percu&longs;&longs;ionis co­<lb/>gno&longs;citur ex centro percu&longs;&longs;ionis ba&longs;is cognito, &longs;i voluatur in circulo &longs;uo <lb/>plano parallelo per Cor. <!-- REMOVE S-->Th.22. <!-- KEEP S--></s> </p> <p id="N2A241" type="main"> <s id="N2A243">Sextò denique, colligo centrum percu&longs;&longs;ionis cuiu&longs;libet alterius <lb/>&longs;olidi, planis rectilineis contenti, quod &longs;cilicet in pyramides diui­<lb/>di pote&longs;t. </s> </p> <pb pagenum="434" xlink:href="026/01/468.jpg"/> <p id="N2A24E" type="main"> <s id="N2A250"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A25C" type="main"> <s id="N2A25E"><!-- NEW -->Ob&longs;eruabis non dee&longs;&longs;e fortè aliquos, quibus centrum grauitatis Py­<lb/>ramidos difficile inuentu videatur; </s> <s id="N2A264"><!-- NEW -->quare in eorum gratiam facilem de­<lb/>mon&longs;trationem &longs;ubijcio; </s> <s id="N2A26A"><!-- NEW -->&longs;it enim pyramis EFBA, cuius ba&longs;is &longs;it trian­<lb/>gularis EFB; </s> <s id="N2A270"><!-- NEW -->ducatur EC diuidens bifariam FB, &longs;itque DC 1/3 totius <lb/>EC, centrum grauitatis ba&longs;is EFB e&longs;t D, per Sch.Th.2. ducatur AD, id <lb/>e&longs;t axis pyramidos, per communem definitionem; </s> <s id="N2A278"><!-- NEW -->quippe axis e&longs;t recta <lb/>ducta à vertice ad centrum grauitatis ba&longs;is oppo&longs;itæ; </s> <s id="N2A27E"><!-- NEW -->ducatur AC, diui­<lb/>dens BF bifariam æqualiter; </s> <s id="N2A284"><!-- NEW -->a&longs;&longs;umatur GC, 1/3 AC, ducatur EG, hæc <lb/>e&longs;t axis, vt patet ex dictis; </s> <s id="N2A28A"><!-- NEW -->a&longs;&longs;umatur autem triangulum AEC, &longs;itque HO <lb/>K maioris claritatis gratia, &longs;intque gemini axes HL, OI, centrum py­<lb/>ramis e&longs;t in OI & in HL; igitur in M; </s> <s id="N2A292"><!-- NEW -->&longs;ed ML e&longs;t 1/4 totius LH, quod <lb/>&longs;ic demon&longs;tro; </s> <s id="N2A298"><!-- NEW -->triangula PIM, OLM &longs;unt æquiangula; </s> <s id="N2A29C"><!-- NEW -->igitur propor­<lb/>tionalia; </s> <s id="N2A2A2"><!-- NEW -->itemque duo HIN, & HKO; </s> <s id="N2A2A6"><!-- NEW -->igitur vt HK ad KO, ita HI ad <lb/>IN; </s> <s id="N2A2AC"><!-- NEW -->&longs;ed HI continet 2/4 HK, per hypothe&longs;im; </s> <s id="N2A2B0"><!-- NEW -->igitur IN continet 2/3 KO; </s> <s id="N2A2B4"><!-- NEW --><lb/>igitur IN e&longs;t æqualis LO; </s> <s id="N2A2B9"><!-- NEW -->igitur vt IP e&longs;t ad LO, ita PM ad ML; &longs;ed <lb/>PI e&longs;t ad LO vt 2. 2/3 ad 8. id e&longs;t vt 3. ad 9. nam &longs;it OK 12. IN æqualis <lb/>LO e&longs;t 8.igitur PM e&longs;t ad ML, vt 3. ad 9. vel vt 1. ad 3. igitur &longs;it HL <lb/>12. PL erit 4. igitur PM 1. ML 3. igitur ML e&longs;t 1/4 LH, quod erat <lb/>demon&longs;trandum. </s> </p> <p id="N2A2C5" type="main"> <s id="N2A2C7"><!-- NEW -->Si verò pyramidos ba&longs;is &longs;it quadrilatera, vel polygona, diuidi pote&longs;t in <lb/>plures, quarum ba&longs;is &longs;it trilatera; quare in omni pyramide facilè de­<lb/>mon&longs;tratur centrum grauitatis ita dirimere axem, vt &longs;egmentum ver&longs;us <lb/>ba&longs;im &longs;it 1/4 totius. </s> </p> <p id="N2A2D1" type="main"> <s id="N2A2D3"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s> </p> <p id="N2A2DF" type="main"> <s id="N2A2E1"><!-- NEW --><emph type="italics"/>Determinari pote&longs;t centrum percu&longs;&longs;ionis coni mixti, cuius ba&longs;is &longs;it portio <lb/>&longs;uperficiei &longs;phæræ, cuius centrum &longs;it in apice coni<emph.end type="italics"/>; </s> <s id="N2A2EC"><!-- NEW -->quia vt &longs;e habet triangu­<lb/>lum I&longs;o&longs;celes ad conum, ita &longs;e habet &longs;ector &longs;ub eodem angulo ad prædi­<lb/>ctum conum mixtum, vt patet; </s> <s id="N2A2F4"><!-- NEW -->quia vt conus ille rectus formatur a trian­<lb/>gulo circa &longs;uum axem circumacto, ita & mixtus formatur à &longs;ectore circa <lb/>&longs;uum axem circumuoluto; </s> <s id="N2A2FC"><!-- NEW -->igitur vt &longs;e habet di&longs;tantia inter centrum vel <lb/>apicem trianguli, circa quem voluitur, & centrum percu&longs;&longs;ionis eiu&longs;dem <lb/>ad di&longs;tantiam inter eo&longs;dem terminos in cono recto, ita &longs;e habet di&longs;tan­<lb/>tia inter eo&longs;dem terminos in &longs;ectore, ad di&longs;tantiam inter eo&longs;dem termi­<lb/>nos in prædicto cono mixto; </s> <s id="N2A308"><!-- NEW -->&longs;ed cogno&longs;cuntur ex dictis &longs;uprà tres pri­<lb/>mi termini huius proportionis; igitur cogno&longs;ci pote&longs;t quartus, igitur <lb/>determinari centrum percu&longs;&longs;ionis, quod erat demon&longs;trandum. </s> </p> <p id="N2A310" type="main"> <s id="N2A312"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A31E" type="main"> <s id="N2A320">Colligo primò, ex his facilè cogno&longs;ci po&longs;&longs;e centrum percu&longs;&longs;ionis &longs;e­<lb/>ctoris &longs;phæræ, nam vt &longs;e habet conus rectus ad pyramidem, ita &longs;e habes <lb/>prædictus conus mixtus ad &longs;ectorem, &longs;ub eodem &longs;cilicet angulo. </s> </p> <p id="N2A327" type="main"> <s id="N2A329"><!-- NEW -->Colligo &longs;ecundò, etiam po&longs;&longs;e cogno&longs;ci centrum percu&longs;&longs;ionis eiu&longs;dem <lb/>&longs;ectoris circumacti, non tantùm circa centrum &longs;phæræ, &longs;ed circa radium; </s> <s id="N2A32F"><!-- NEW --><pb pagenum="435" xlink:href="026/01/469.jpg"/>immò gemini &longs;ectoris coniuncti, &longs;eu quartæ partis &longs;phæræ, ex quo etiam <lb/>&longs;equitur determinatio centri grauitatis Hemi&longs;phærij, atque adeo totius <lb/>&longs;phæræ; quæ omnia pendent ex dictis &longs;uprà. </s> </p> <p id="N2A33B" type="main"> <s id="N2A33D"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A349" type="main"> <s id="N2A34B">Ob&longs;eruabis &longs;upere&longs;&longs;e innumeras ferè corporum rationes, v.g.&longs;phæram <lb/>ex dato puncto &longs;uperficiei libratam, tùm elliptica &longs;olida, parabolica, hy­<lb/>perbolica, &c. </s> <s id="N2A352"><!-- NEW -->quorum centra percu&longs;&longs;ionis determinari po&longs;&longs;unt; &longs;ed ab­<lb/>&longs;tineo, tùm quia cum multam mathe&longs;im de&longs;iderent, vix habent aliquem <lb/>in phy&longs;ica locum, tùm quia plura excerpere non potui, ex innumeris pe­<lb/>nè, quæ apud &longs;e no&longs;ter Philo&longs;ophus habet. </s> </p> <p id="N2A35C" type="main"> <s id="N2A35E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s> </p> <p id="N2A36A" type="main"> <s id="N2A36C"><!-- NEW --><emph type="italics"/>Determinari pote&longs;t centrum impre&longs;&longs;ionis, tùm in linea, tùm in plano, tù&mtail; <lb/>in &longs;olido quæ circumaguntur<emph.end type="italics"/>; quia pote&longs;t diuidi bifariam, tùm planum illud <lb/>&longs;i &longs;it linea, tùm &longs;olidum, &longs;i planum vel &longs;olidum, vt patet per def.2. </s> </p> <p id="N2A379" type="main"> <s id="N2A37B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s> </p> <p id="N2A387" type="main"> <s id="N2A389"><!-- NEW --><emph type="italics"/>Si linea rigida libretur circa alteram extremitatem immobilem a&longs;&longs;uma­<lb/>turque funependulum, cuius longitudo contineat<emph.end type="italics"/> 2/3 <emph type="italics"/>prædictæ lineæ, vibrationes <lb/>vtriu&longs;que erunt æquediuturnæ<emph.end type="italics"/>; quod demon&longs;tratur; </s> <s id="N2A39C"><!-- NEW -->quia centrum percu&longs;­<lb/>&longs;ionis prædictæ lineæ di&longs;tat 2/3 ab altera extremitate immobili per Th.8. <lb/>atqui centrum percu&longs;&longs;ionis in hoc motu circulari dirigit motum aliorum <lb/>punctorum; </s> <s id="N2A3A6"><!-- NEW -->quia defungitur munere centri grauitatis, vt patet ex dictis; </s> <s id="N2A3AA"><!-- NEW --><lb/>nec enim alterum &longs;egmentorum præualet; </s> <s id="N2A3AF"><!-- NEW -->&longs;ed totus motus impeditur, <lb/>per po&longs;.2. igitur perinde &longs;e habet atque &longs;i totum pondus, vel totam vim <lb/>collectam haberet; </s> <s id="N2A3B7"><!-- NEW -->&longs;ed in hoc ca&longs;u e&longs;&longs;et ad in&longs;tar funependuli, in quo <lb/>non habetur vlla ratio fili, &longs;ed ponderis appen&longs;i; igitur eius vibratio e&longs;t <lb/>æquediuturna cum vibratione prædicti funependuli quod erat demon­<lb/>&longs;trandum. </s> </p> <p id="N2A3C1" type="main"> <s id="N2A3C3"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A3CF" type="main"> <s id="N2A3D1"><!-- NEW -->Ob&longs;eruabis, ex hoc vno certi&longs;&longs;imo principio egregium experimentum <lb/>mirificè comprobari; nempè &longs;æpiùs compertum e&longs;t innumeris ferè expe­<lb/>rimentis, tùm ab erudito Mer&longs;enno, tùm à no&longs;tro Philo&longs;opho longitu­<lb/>dinem funependuli i&longs;ochroni cum cylindro continere 2/3 cylindri. </s> </p> <p id="N2A3DB" type="main"> <s id="N2A3DD"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s> </p> <p id="N2A3E9" type="main"> <s id="N2A3EB"><!-- NEW --><emph type="italics"/>Si voluatur planum rectangulum circa alterum laterum, funependulum <lb/>i&longs;ochronum continet duas tertias<emph.end type="italics"/>; probatur eodem modo; nam perinde &longs;e <lb/>habet illud planum, atque &longs;i multæ lineæ parallelæ &longs;imul volueren­<lb/>tur. </s> </p> <p id="N2A3FA" type="main"> <s id="N2A3FC"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s> </p> <p id="N2A408" type="main"> <s id="N2A40A"><!-- NEW --><emph type="italics"/>Si voluatur planum triangulare circa angulum, eo modo quo diximus in <lb/>Th.<emph.end type="italics"/>11. <emph type="italics"/>funependulum i&longs;ochronum continet<emph.end type="italics"/> 3/4 <emph type="italics"/>axis prædicti trianguli<emph.end type="italics"/>; quia in <lb/>1/4 e&longs;t centrum percu&longs;&longs;ionis per Th. 11. </s> </p> <pb pagenum="436" xlink:href="026/01/470.jpg"/> <p id="N2A428" type="main"> <s id="N2A42A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s> </p> <p id="N2A436" type="main"> <s id="N2A438"><!-- NEW --><emph type="italics"/>Si voluatur prædictum planum circa ba&longs;im eo modo, quo dictum e&longs;t Th.<emph.end type="italics"/>12. <lb/><emph type="italics"/>funependulum i&longs;ochronum continet<emph.end type="italics"/> 1/2 <emph type="italics"/>eiu&longs;dem axis<emph.end type="italics"/>; quod eodem modo de­<lb/>mon&longs;tratur per Th.12. </s> </p> <p id="N2A450" type="main"> <s id="N2A452"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A45E" type="main"> <s id="N2A460">Colligo primò, cuilibet &longs;ectori funependulum i&longs;ochronum po&longs;&longs;e a&longs;&longs;i­<lb/>gnari, quia cuiu&longs;libet &longs;ectoris, qui voluitur circa angulum, eo modo <lb/>quo diximus Th.13. centrum percu&longs;&longs;ionis determinatum e&longs;t. </s> </p> <p id="N2A467" type="main"> <s id="N2A469">Colligo &longs;ecundò, &longs;i rotetur planum circulare, eo modo quo diximus <lb/>Th.21. funependuli i&longs;ochroni longitudinem continere 2/3 diametri eiu&longs;­<lb/>dem circuli, quia ibi e&longs;t centrum percu&longs;&longs;ionis eiu&longs;dem circuli, per <lb/>Th. 21. </s> </p> <p id="N2A472" type="main"> <s id="N2A474">Colligo tertiò, &longs;i rotetur planum circulare circa diametrum, etiam <lb/>po&longs;&longs;e determinari ex centro percu&longs;&longs;ionis inuento, longitudinem fune­<lb/>penduli i&longs;ochroni, vt patet ex dictis. </s> </p> <p id="N2A47B" type="main"> <s id="N2A47D"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s> </p> <p id="N2A489" type="main"> <s id="N2A48B"><!-- NEW --><emph type="italics"/>Quando voluitur planum triangulare parallelum plano in quo voluitur, <lb/>determinari pote&longs;t longitudo funependuli i&longs;ochroni<emph.end type="italics"/>; &longs;it enim AFH, cuius <lb/>centrum extrin&longs;ecum percu&longs;&longs;ionis fit C, longitudo funependuli i&longs;ochro­<lb/>ni erit AC, quod eodem modo demon&longs;tratur. </s> </p> <p id="N2A49A" type="main"> <s id="N2A49C"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A4A8" type="main"> <s id="N2A4AA"><!-- NEW -->Colligo primò, etiam determinari po&longs;&longs;e, quando ita voluitur vt latus <lb/>in quo fit percu&longs;&longs;io &longs;u&longs;tineat angulum rectum, v.g. <!-- REMOVE S-->triangulum AGB <lb/>circumactum circa A, habet centrum percu&longs;&longs;ionis in M; igitur AM e&longs;t <lb/>longitudo funependuli i&longs;ochroni. </s> </p> <p id="N2A4B6" type="main"> <s id="N2A4B8"><!-- NEW -->Secundò, &longs;i voluatur circa angulum rectum; </s> <s id="N2A4BC"><!-- NEW -->v.g. <!-- REMOVE S-->triangulum ABH <lb/>circa B, centrum percu&longs;&longs;ionis e&longs;t in E; igitur BE e&longs;t longitudo funepen­<lb/>duli i&longs;ochroni. </s> </p> <p id="N2A4C6" type="main"> <s id="N2A4C8"><!-- NEW -->Tertiò, aliquando longitudo prædicta e&longs;t minor latere, in quo fit <lb/>percu&longs;&longs;io, vt patet in exemplis adductis; </s> <s id="N2A4CE"><!-- NEW -->aliquando e&longs;t æqualis, vt in <lb/>triangulo ABD volutum circa A, nam centrum percu&longs;&longs;ionis e&longs;t D; </s> <s id="N2A4D4"><!-- NEW -->igi­<lb/>tur longitudo funependuli i&longs;ochroni e&longs;t AD; </s> <s id="N2A4DA"><!-- NEW -->aliquando e&longs;t maior, vt <lb/>videre e&longs;t in triangulo ALG, quod voluitur circa A; nam longitudo fu­<lb/>nependuli i&longs;ochroni e&longs;t AI, quæ e&longs;t maior AL. <!-- KEEP S--></s> </p> <p id="N2A4E3" type="main"> <s id="N2A4E5">Quartò, &longs;i coniungantur duo triangula v.g. <!-- REMOVE S-->EAS. ADS. voluan­<lb/>turque &longs;imul circa A, eo modo quo diximus &longs;cilicet parallela plano, in <lb/>quo voluuntur, longitudo i&longs;ochroni funependuli erit AF, po&longs;ito quòd <lb/>F &longs;it centrum percu&longs;&longs;ionis, vt dictum e&longs;t &longs;uprà Corol. <!-- REMOVE S-->5. Th.19. </s> </p> <p id="N2A4F2" type="main"> <s id="N2A4F4">Quintò, hinc vides rationem egregij experimenti, quod &longs;æpè Doctus <lb/>Mer&longs;ennus propo&longs;uit, &longs;cilicet longitudinem funependuli i&longs;ochroni e&longs;&longs;e <lb/>ferè quadruplam perpendicularis ductæ in ba&longs;im trianguli I&longs;o&longs;celis, li­<lb/>brati circa angulum verticis 150.grad. </s> <s id="N2A4FD">quod certè ad veritatem tam pro­<lb/>pè accedit ex geometrica calculatione, vt nullum pror&longs;us di&longs;crimen <pb pagenum="Tabula sexta" xlink:href="026/01/471.jpg"/><pb xlink:href="026/01/473.jpg"/><pb pagenum="437" xlink:href="026/01/473.jpg"/>e&longs;&longs;e videatur, methodus huius calculationis facilis e&longs;t, & à mediocri <lb/>Logi&longs;ta haberi pote&longs;t. </s> </p> <p id="N2A509" type="main"> <s id="N2A50B"><!-- NEW -->Sextò, hinc etiam habetur longitudo funependuli i&longs;ochroni, &longs;i vol­<lb/>uatur planum circulare parallelum plano, in quo voluitur, continet <lb/>enim 2/3 diametri circuli, qui voluitur; vt patet ex Th. 22. idem dico de <lb/>quolibet &longs;ectore, qui eodem modo voluatur. </s> </p> <p id="N2A515" type="main"> <s id="N2A517"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s> </p> <p id="N2A523" type="main"> <s id="N2A525"><!-- NEW --><emph type="italics"/>Si voluatur pyramis circa verticem, determinari pote&longs;t longitudo funepen­<lb/>duli i&longs;ochroni, idem dico de parallelipedo, pri&longs;mate, cono, cylindro, &c.<emph.end type="italics"/> per <lb/>Th.25. 26. & Corollaria; </s> <s id="N2A532"><!-- NEW -->quia inuento centro percu&longs;&longs;ionis extrin&longs;eco, <lb/>habetur prædicta longitudo; idem dico de cono mixto, &longs;ectore &longs;olido, <lb/>&c. </s> <s id="N2A53A">per Th.28. & Coroll. <!-- KEEP S--></s> </p> <p id="N2A53E" type="main"> <s id="N2A540"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A54C" type="main"> <s id="N2A54E">Hinc colligo primò ex dato centro percu&longs;&longs;ionis extrin&longs;eco, dari &longs;tatim <lb/>longitudinem funependuli i&longs;ochroni, & vici&longs;&longs;im. </s> </p> <p id="N2A553" type="main"> <s id="N2A555">Secundò, data quacunque longitudine funependuli i&longs;ochroni, v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->tripla perpendicularis, cadentis in ba&longs;im trianguli i&longs;o&longs;celis, dari po&longs;&longs;e <lb/>triangulum, cuius libratio &longs;it æquediuturna, &longs;ed hæc breuiter indica&longs;&longs;e <lb/>&longs;ufficiat. <lb/><figure id="id.026.01.473.1.jpg" xlink:href="026/01/473/1.jpg"/></s> </p> <pb pagenum="438" xlink:href="026/01/474.jpg"/> <figure id="id.026.01.474.1.jpg" xlink:href="026/01/474/1.jpg"/> <p id="N2A570" type="main"> <s id="N2A572"><emph type="center"/>APPENDIX SECVNDA.<emph.end type="center"/></s> </p> <p id="N2A579" type="main"> <s id="N2A57B"><emph type="center"/><emph type="italics"/>DE PRINCIPIO PHYSICOSTATICO, <lb/>ad mouenda ingentia pondera.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2A588" type="main"> <s id="N2A58A"><!-- NEW -->DVo &longs;unt in Statica, quæ demon&longs;trationem de&longs;idera­<lb/>re po&longs;&longs;unt; Primum e&longs;t, quod &longs;pectat ad proportio­<lb/>nes potentiarum, ponderum, re&longs;i&longs;tentiæ, motuum, <lb/>temporum, di&longs;tantiarum, &c. </s> <s id="N2A594">Secundum pertinet <lb/>ad cau&longs;as Phy&longs;icas huiu&longs;modi effectuum, qui cùm &longs;int <lb/>naturales, & &longs;en&longs;ibiles, &longs;ua cau&longs;a carere non po&longs;&longs;unt. </s> <s id="N2A59B"><lb/>Primum &longs;anè quod ad Mathe&longs;im attinet egregiè præ­<lb/>&longs;titerunt hactenus docti&longs;&longs;imi viri Vbaldus, Steuinus, Galileus, &c. </s> <s id="N2A5A1"><!-- NEW --><lb/>ita vt nihil amplius de&longs;iderari po&longs;&longs;it; </s> <s id="N2A5A6"><!-- NEW -->Secundum tamen quod iuris phy­<lb/>&longs;ici e&longs;t, vix, ac ne vix quidem delibatum inuenio; quare ad huius libri <lb/>calcem principium Phy&longs;ico&longs;taticum breuiter explicandum &longs;u&longs;cipio, per <lb/>quod duntaxat illi omnes mirifici effectus ad &longs;uas cau&longs;as reducantur, <lb/>quod ni&longs;i fallor huic tractatui dee&longs;&longs;e videtur. <lb/><gap desc="hr tag"/></s> </p> <p id="N2A5B5" type="main"> <s id="N2A5B7"><emph type="center"/><emph type="italics"/>AXIOMA<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A5C4" type="main"> <s id="N2A5C6"><emph type="italics"/>AB eadem potentiâ faciliùs producitur in eodem mobili minor motus, <lb/>quam maior.<emph.end type="italics"/></s> </p> <p id="N2A5CF" type="main"> <s id="N2A5D1"><!-- NEW -->Hoc Axioma manife&longs;tum redditur ex ijs, quæ pa&longs;&longs;im habentur in lib. <!-- REMOVE S--><lb/>1. de impetu; </s> <s id="N2A5D8"><!-- NEW -->quippe motus ex duplici tantùm capite minor e&longs;&longs;e pote&longs;t; </s> <s id="N2A5DC"><!-- NEW --><lb/>primò, ex eo quòd &longs;ingulis partibus mobilis pauciores partes impetus <lb/>in&longs;int; </s> <s id="N2A5E3"><!-- NEW -->&longs;ecundò ex eo quòd imperfectior impetus mobili imprimatur; </s> <s id="N2A5E7"><!-- NEW --><lb/>atqui ex vtroque capite faciliùs producit ut minor motus; quia faciliùs <lb/>imprimitur minor, vel imperfectior impetus, nempe minore ni&longs;u agit <lb/>potentia. </s> </p> <p id="N2A5F0" type="main"> <s id="N2A5F2"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A5FF" type="main"> <s id="N2A601"><emph type="italics"/>Quò maiore tempore datum &longs;patium percurritur, eò minor e&longs;t motus, id e&longs;t <lb/>tardior, vt patet ex dictis l.<emph.end type="italics"/>1. </s> </p> <pb pagenum="439" xlink:href="026/01/475.jpg"/> <p id="N2A60F" type="main"> <s id="N2A611"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A61E" type="main"> <s id="N2A620"><!-- NEW --><emph type="italics"/>Quò minus &longs;patium decurritur dato tempore minor, & tardior e&longs;t motus<emph.end type="italics"/>; <lb/>hoc etiam con&longs;tat ex eadem dem. </s> </p> <p id="N2A62B" type="main"> <s id="N2A62D"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A63A" type="main"> <s id="N2A63C"><emph type="italics"/>Maiore tempore potentia applicata &longs;i &longs;emper agit, plus agit.<emph.end type="italics"/></s> <s id="N2A643"> Quid clarius? </s> </p> <p id="N2A646" type="main"> <s id="N2A648"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A655" type="main"> <s id="N2A657"><!-- NEW --><emph type="italics"/>Pondus alteri æquale illud mouere tantum non pote&longs;t motu æquali<emph.end type="italics"/>; </s> <s id="N2A660"><!-- NEW -->cur <lb/>enim pondus A mouebit B potiùs quàm B. A: quod certum e&longs;t. </s> </p> <p id="N2A666" type="main"> <s id="N2A668"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A675" type="main"> <s id="N2A677"><!-- NEW --><emph type="italics"/>Pondus alteri æquale mouere pote&longs;t illud motu minore<emph.end type="italics"/>; </s> <s id="N2A680"><!-- NEW -->quia cùm æquali <lb/>mouere tantùm non po&longs;&longs;it, & cùm po&longs;&longs;it faciliùs minore, quàm maiore; <lb/>certè minore mouere pote&longs;t. </s> </p> <p id="N2A68A" type="main"> <s id="N2A68C"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N2A698" type="main"> <s id="N2A69A"><emph type="italics"/>Pondus minus pote&longs;t mouere maius motu minore, &longs;i maior &longs;it proportio mo­<lb/>tuum, quàm ponderum,<emph.end type="italics"/> v.g. <!-- REMOVE S-->pondus duarum librarum quod mouetur <lb/>motu vt 3.pote&longs;t mouere pondus 4.librarum motu vt 1.vt patet ex dictis. </s> </p> <p id="N2A6A8" type="main"> <s id="N2A6AA"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N2A6B6" type="main"> <s id="N2A6B8"><!-- NEW --><emph type="italics"/>Eò faciliùs mouetur pondus per inclinatam, quàm per ip&longs;um perpendicu­<lb/>lum, quò inclinata maior e&longs;t perpendiculo<emph.end type="italics"/>; vt patet ex ijs, quæ dicta &longs;unt l.5. <lb/>de planis inclinatis. </s> </p> <p id="N2A6C5" type="main"> <s id="N2A6C7"><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N2A6D3" type="main"> <s id="N2A6D5"><emph type="italics"/>Pondus maius mouet tantùm minus motu maiore, cum e&longs;t maior proportio <lb/>ponderum quàm motuum,<emph.end type="italics"/> vt patet. </s> </p> <p id="N2A6DF" type="main"> <s id="N2A6E1"><emph type="center"/><emph type="italics"/>Problema vniuer&longs;ali&longs;&longs;imum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2A6EC" type="main"> <s id="N2A6EE"><emph type="italics"/>Mouere quodcumque pondus à qualibet applicata potentia moueatur motu <lb/>minore, ita vt &longs;it maior proportio motuum, quàm ponderum,<emph.end type="italics"/> per Ax. 7. </s> </p> <p id="N2A6F8" type="main"> <s id="N2A6FA"><emph type="center"/><emph type="italics"/>Coroll. <!-- REMOVE S-->vniuer&longs;ali&longs;&longs;imum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2A707" type="main"> <s id="N2A709"><!-- NEW -->Hinc colligo, in eo tantùm po&longs;itam e&longs;&longs;e indu&longs;triam, qua po&longs;&longs;int <lb/>pondera moueri, vt minore, & minore motu moueantur; igitur, qua <lb/>proportione imminues motum, eâdem maius pondus mouebis. </s> </p> <p id="N2A711" type="main"> <s id="N2A713"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A720" type="main"> <s id="N2A722"><emph type="italics"/>Æqualia pondera æquali vtrimque brachio libræ appen&longs;a &longs;unt in æquilibrio<emph.end type="italics"/><lb/>per Ax.5. <!-- KEEP S--></s> </p> <p id="N2A72C" type="main"> <s id="N2A72E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A73B" type="main"> <s id="N2A73D"><!-- NEW --><emph type="italics"/>In æqualia pondera inæquali brachio librata faciunt æquilibrium &longs;i &longs;it ea­<lb/>dem proportio brachiorum quæ ponderum permutando<emph.end type="italics"/>; </s> <s id="N2A748"><!-- NEW -->quia e&longs;t eadem pro­<lb/>portio motuum, quæ brachiorum, vt patet; igitur &longs;unt in æquilibrio nec <lb/>enim minus pondus attolli pote&longs;t à maiori per Ax.9.nec maius à mino­<lb/>re per Ax.7. igitur &longs;unt in æquilibrio. </s> </p> <pb pagenum="440" xlink:href="026/01/476.jpg"/> <p id="N2A756" type="main"> <s id="N2A758"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A764" type="main"> <s id="N2A766"><!-- NEW -->Hinc collige omnes rationes, quæ &longs;pectant ad libram; </s> <s id="N2A76A"><!-- NEW -->hinc vulgare <lb/>illud dictum mechanicum: Si pondera &longs;int vt di&longs;tantiæ, &longs;unt in æqui­<lb/>librio. </s> </p> <p id="N2A772" type="main"> <s id="N2A774">Hinc coniugari po&longs;&longs;unt infinitis modis pondera, & di&longs;tantiæ, quorum <lb/>omnium rationes compo&longs;itæ ob&longs;eruari debent. </s> </p> <p id="N2A779" type="main"> <s id="N2A77B">Hinc etiam obliqua libra, & inclinata, &longs;i &longs;upponantur brachia adin­<lb/>&longs;tar lineæ indiui&longs;ibilis facit æquilibrium. </s> </p> <p id="N2A780" type="main"> <s id="N2A782"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A78F" type="main"> <s id="N2A791"><!-- NEW --><emph type="italics"/>Ideo facilè ingens pondus attollitur vecte, quia mouetur motu minore iux­<lb/>ta <expan abbr="eãdem">eandem</expan> rationem, de quo &longs;uprà<emph.end type="italics"/>; </s> <s id="N2A7A0"><!-- NEW -->cùm enim &longs;upponatur in vecte pun­<lb/>ctum immobile, quod certo nititur fulcro; </s> <s id="N2A7A6"><!-- NEW -->nece&longs;&longs;e e&longs;t vtrimque moueri <lb/>&longs;egmenta vectis motu circulari, <expan abbr="eo&qacute;ue">eoque</expan> inæquali; </s> <s id="N2A7B0"><!-- NEW -->quia &longs;unt inæqualia; </s> <s id="N2A7B4"><!-- NEW -->igi­<lb/>tur altero minore; & hæc e&longs;t prima ratio imminuendi motus. </s> </p> <p id="N2A7BA" type="main"> <s id="N2A7BC"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A7C8" type="main"> <s id="N2A7CA"><!-- NEW -->Hinc datum quodcunque pondus attollitur vecte; hinc quò &longs;egmen­<lb/>tum, quod à fulcro porrigitur ver&longs;us pondus quod attollitur e&longs;t breuius, <lb/>eò maius pondus attolli pote&longs;t. </s> </p> <p id="N2A7D2" type="main"> <s id="N2A7D4">Hinc vectis per Tangentem &longs;emper attolli debet, vt maiorem præ&longs;tet <lb/>effectum, vt con&longs;tat ex ijs, quæ diximus l.4. </s> </p> <p id="N2A7D9" type="main"> <s id="N2A7DB"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A7E8" type="main"> <s id="N2A7EA"><!-- NEW --><emph type="italics"/>Ideo facilè attollitur ingens pondus trochlea, quia mouetur motu minor&etail;, <lb/>vt manife&longs;tum e&longs;t<emph.end type="italics"/>; </s> <s id="N2A7F5"><!-- NEW -->e&longs;t autem minor motus in ea proportione, in qua lon­<lb/>gitudo funis adducti &longs;uperat altitudinem &longs;patij decur&longs;i à pondere, quod <lb/>attollitur; mirabile &longs;anè inuentum, &longs;i quod aliud. </s> </p> <p id="N2A7FD" type="main"> <s id="N2A7FF"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A80B" type="main"> <s id="N2A80D"><!-- NEW -->Hinc, &longs;i funis adducatur deor&longs;um, vnica rotula non iuuat potentiam; </s> <s id="N2A811"><!-- NEW --><lb/>quia longitudo funis adducti e&longs;t æqualis altitudini &longs;patij decur&longs;i à pon­<lb/>dere; </s> <s id="N2A818"><!-- NEW -->&longs;i verò adducatur &longs;ur&longs;um vnica rotula duplicat potentiam; </s> <s id="N2A81C"><!-- NEW -->quia lon­<lb/>gitudo prædicta funis adducti e&longs;t dupla prædictæ altitudinis; </s> <s id="N2A822"><!-- NEW -->igitur mo­<lb/>tus ponderis a&longs;cendentis e&longs;t &longs;ubduplus; </s> <s id="N2A828"><!-- NEW -->igitur duplum pondus eadem po­<lb/>tentia attollet, vel idem pondus &longs;ubdupla per Ax. 1. &longs;i verò &longs;int duæ ro­<lb/>tulæ adducaturque deor&longs;um, duplum etiam pondus attollet eadem po­<lb/>tentia; </s> <s id="N2A832"><!-- NEW -->quia longitudo funis adducti e&longs;t dupla altitudinis; ex his reliqua <lb/>de trochlea facilè intelligentur, </s> </p> <p id="N2A838" type="main"> <s id="N2A83A"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A846" type="main"> <s id="N2A848"><!-- NEW -->Equidem demon&longs;trari pote&longs;t aliter à debili potentia &longs;u&longs;tineri po&longs;&longs;e <lb/>ingens pondus operâ trochleæ; </s> <s id="N2A84E"><!-- NEW -->quia &longs;cilicet pluribus di&longs;tribuitur &longs;u&longs;ti­<lb/>nendi munus, vt clarum e&longs;t; </s> <s id="N2A854"><!-- NEW -->quod verò &longs;pectat ad motum, vnum tantùm <lb/>e&longs;t illius principium, &longs;cilicet potentia, quæ trahit; licèt enim clauus, cui <lb/>affigitur altera extremitas funis po&longs;&longs;it &longs;u&longs;tinere, non tamen mouere. </s> </p> <p id="N2A85C" type="main"> <s id="N2A85E"><!-- NEW -->Hinc demum ratio, cur &longs;i multiplicentur funes, & orbiculi ingens-<pb pagenum="441" xlink:href="026/01/477.jpg"/>etiam pondus perexiguis fu&longs;ciculis &longs;u&longs;tineri po&longs;&longs;it; </s> <s id="N2A867"><!-- NEW -->quia pluribus di&longs;tri­<lb/>buitur: hinc, &longs;i plura e&longs;&longs;ent araneæ fila, maximum &longs;axum &longs;u&longs;tinere po&longs;&longs;ent. </s> </p> <p id="N2A86D" type="main"> <s id="N2A86F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A87C" type="main"> <s id="N2A87E"><!-- NEW --><emph type="italics"/>Ideo mouetur ingens pondus operâ axis, vel &longs;uculæ; quia &longs;cilicet imminuitur <lb/>matus,<emph.end type="italics"/> vt clarum e&longs;t. </s> </p> <p id="N2A889" type="main"> <s id="N2A88B"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A897" type="main"> <s id="N2A899"><!-- NEW -->Hinc, quò minor e&longs;t diameter axis, maius pondus attollitur &longs;eu mo­<lb/>uetur; </s> <s id="N2A89F"><!-- NEW -->quia cùm circulorum peripheriæ &longs;int vt &longs;emidiametri, quò minor <lb/>e&longs;t diameter axis cui aduoluitur funis ductarius, e&longs;t minor motus; </s> <s id="N2A8A7"><!-- NEW -->igi­<lb/>tur maius pondus attollitur; </s> <s id="N2A8AD"><!-- NEW -->igitur &longs;i longitudo vectis &longs;it dupla &longs;emidia­<lb/>metri &longs;uculæ, duplum pondus attollitur; &longs;i tripla, triplum, &c. </s> </p> <p id="N2A8B3" type="main"> <s id="N2A8B5">Huc reuoca terebraś, & manubria, &c. </s> </p> <p id="N2A8B8" type="main"> <s id="N2A8BA"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A8C7" type="main"> <s id="N2A8C9"><!-- NEW --><emph type="italics"/>Ideo cochlea mouet ingens pondus<emph.end type="italics"/>; quia imminuit motum, vt videre e&longs;t <lb/>in torcularibus, in quibus Helicis opera ingens pri&longs;ma attollitur. </s> </p> <p id="N2A8D4" type="main"> <s id="N2A8D6"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A8E2" type="main"> <s id="N2A8E4"><!-- NEW -->Hinc quò &longs;unt plures Helices, & decliuiores motus rectus e&longs;t minor; <lb/>hinc faciliùs attollitur pondus; &longs;i enim longitudo &longs;piræ e&longs;t decupla axis, <lb/>potentia decuplum pondus attollet. </s> </p> <p id="N2A8ED" type="main"> <s id="N2A8EF"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s> </p> <p id="N2A8FB" type="main"> <s id="N2A8FD"><emph type="italics"/>Ideò tantæ &longs;unt cunei vires, quia motum imminuit.<emph.end type="italics"/></s> </p> <p id="N2A904" type="main"> <s id="N2A906"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A912" type="main"> <s id="N2A914"><!-- NEW -->Hinc quò angulus cunei e&longs;t acutior, maius pondus attollitur eius ope­<lb/>râ; hinc proportiones omnes demon&longs;trari po&longs;&longs;unt, hinc cuneus ad angu­<lb/>lum 45. & &longs;uprà non iuuat potentiam, &longs;ecus infrà, ad cuneum reuoca <lb/>clauos & gladios. </s> </p> <p id="N2A920" type="main"> <s id="N2A922"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s> </p> <p id="N2A92E" type="main"> <s id="N2A930"><!-- NEW --><emph type="italics"/>Ideo rotis denticulatis mouetur ingens pondus<emph.end type="italics"/>; quia imminuitur motus, <lb/>vt clarum e&longs;t. </s> </p> <p id="N2A93B" type="main"> <s id="N2A93D"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A949" type="main"> <s id="N2A94B">Ob&longs;eruabis huius organi operâ imminui po&longs;&longs;e motum in infinitum, <lb/>atque ad eo maius &longs;emper pondus, & maius in infinitum attolli po&longs;&longs;e. </s> </p> <p id="N2A950" type="main"> <s id="N2A952"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A95E" type="main"> <s id="N2A960">Ex his facilè colliges ad mouenda pondera in eo tantùm po&longs;itam e&longs;&longs;e <lb/>indu&longs;triam, vt motus imminuatur, & vnicum illud e&longs;&longs;e principium phy­<lb/>&longs;icomechanicum. </s> </p> <p id="N2A967" type="main"> <s id="N2A969"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s> </p> <p id="N2A975" type="main"> <s id="N2A977"><emph type="italics"/>Vt pondus attollatur adhiberi pote&longs;t alia indu&longs;tria &longs;cilicet plani inclinati, in <lb/>quo faciliùs pondus attollitur, quàm in verticali,<emph.end type="italics"/> de quo iam &longs;uprà in lib. 5.<!-- REMOVE S--><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A98D" type="main"> <s id="N2A98F"><!-- NEW -->Ob&longs;eruabis autem, organum mechanicum adhiberi po&longs;&longs;e ad mouen-<pb pagenum="442" xlink:href="026/01/478.jpg"/>dum pondus per omne planum, in plano horizontali facillimè ingens <lb/>pondus moueri pote&longs;t; præ&longs;ertim &longs;i plani &longs;cabrities non impediat motum. </s> </p> <p id="N2A99A" type="main"> <s id="N2A99C">Hinc modico organo ingentem nauim facilè mouebat Archimedes, <lb/>quam &longs;ine organo tota ciuitas non mouere poterat. </s> </p> <p id="N2A9A1" type="main"> <s id="N2A9A3">Quæres, quot &longs;int potentiæ mechanicæ? </s> <s id="N2A9A6"><!-- NEW -->Re&longs;p. quinque hactenus <lb/>numeratas e&longs;&longs;e, quæ &longs;unt, vectis, trochlea, axis, cuneus, cochlea; addi <lb/>po&longs;&longs;unt rotæ denticulatæ. </s> </p> <figure id="id.026.01.478.1.jpg" xlink:href="026/01/478/1.jpg"/> <p id="N2A9B3" type="main"> <s id="N2A9B5"><emph type="center"/>APPENDIX TERTIA.<emph.end type="center"/><!-- KEEP S--></s> </p> <p id="N2A9BD" type="main"> <s id="N2A9BF"><emph type="center"/><emph type="italics"/>DE PRINCIPIO PHYSICO­<lb/>mechanico impre&longs;sionis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2A9CC" type="main"> <s id="N2A9CE"><!-- NEW -->NON ago hîc de impre&longs;&longs;ione, quæ fit operâ pulueris tormen­<lb/>tarij, vel nerui ten&longs;i, vel aëris compre&longs;&longs;i; nec enim e&longs;t huius­<lb/>loci, &longs;ed de illâ, quæ fit operâ alterius potentiæ motricis. </s> </p> <p id="N2A9D6" type="main"> <s id="N2A9D8"><!-- NEW -->Iniactu duo tantùm con&longs;iderari debent: </s> <s id="N2A9DC"><!-- NEW -->Primum e&longs;t po­<lb/>tentia, &longs;ecundum linea directionis, quod &longs;pectat ad primum, <lb/>commune e&longs;t iactui & percu&longs;&longs;ioni; de &longs;ecundo iam &longs;uprà dictum e&longs;t lib.4. <lb/>vbi diximus maximum iactum fieri ad angulum &longs;emirectum. </s> </p> <p id="N2A9E6" type="main"> <s id="N2A9E8"><emph type="center"/><emph type="italics"/>Principium vniuer&longs;ali&longs;&longs;imum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2A9F3" type="main"> <s id="N2A9F5"><!-- NEW --><emph type="italics"/>Quò diutius potentia manet applicata maior e&longs;t impre&longs;&longs;io<emph.end type="italics"/>; veritas huius <lb/>axiomatis certi&longs;&longs;ima e&longs;t, & con&longs;tat ex Ax.13. l.1.n.4. ad hoc autem reuo­<lb/>cari po&longs;&longs;unt omnia organa, quæ potentia motrix adhibet ad motum im­<lb/>primendum. </s> </p> <p id="N2AA04" type="main"> <s id="N2AA06"><emph type="center"/><emph type="italics"/>Corollaria.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2AA11" type="main"> <s id="N2AA13"><!-- NEW -->1. Hinc diu rotatum brachium maiorem ictum infligit; </s> <s id="N2AA17"><!-- NEW -->hinc rotatum <lb/>pendulum fune plumbum forti&longs;&longs;imè ferit; </s> <s id="N2AA1D"><!-- NEW -->hinc fundæ iactus potentior; <lb/>hinc longior funda longiorem iactum præ&longs;tat, &c. </s> </p> <p id="N2AA23" type="main"> <s id="N2AA25"><!-- NEW -->2. Hinc pertica longior, quæ diu vibratur propter maiorem arcum <lb/>validum ictum incutit; adde fu&longs;tem, flagellum, <expan abbr="lõgum">longum</expan> mallei manubrium. </s> </p> <p id="N2AA2F" type="main"> <s id="N2AA31"><!-- NEW -->3. Hinc corpus diu cadens deor&longs;um grauius ferit; hinc aries ille, <lb/>cuius ca&longs;us pali figuntur. </s> </p> <p id="N2AA37" type="main"> <s id="N2AA39">4. Hinc maius &longs;axum, vel grauior &longs;udes maiorem ictum infligit. </s> </p> <p id="N2AA3C" type="main"> <s id="N2AA3E"><!-- NEW -->5. Hinc trochus ductario funiculo vibratus celerrimè agitur; </s> <s id="N2AA42"><!-- NEW -->hinc <lb/>etiam plani orbes explicata, & exporrecta zona procul abiguntur; quia <lb/>&longs;cilicet potentia diu manet applicata. </s> </p> <p id="N2AA4A" type="main"> <s id="N2AA4C">6. Hinc antiquus aries diu vibratus, ita verberabat muros, vt &longs;tatim <lb/>di&longs;ijceret propter eandem rationem. </s> </p> <p id="N2AA51" type="main"> <s id="N2AA53"><!-- NEW -->7. Hinc demum antiquæ illæ machinæ, quarum opera ingentia &longs;axa <lb/>iaciebantur; hæc & innumera propemodum alia ex eodem principio <lb/>con&longs;equuntur. </s> </p> <pb pagenum="443" xlink:href="026/01/479.jpg"/> <figure id="id.026.01.479.1.jpg" xlink:href="026/01/479/1.jpg"/> <p id="N2AA64" type="main"> <s id="N2AA66"><emph type="center"/>APPENDIX QVARTA.<emph.end type="center"/></s> </p> <p id="N2AA6D" type="main"> <s id="N2AA6F"><emph type="center"/><emph type="italics"/>DE PRINCIPIO PHYSICO <lb/>Rationis duplicatæ Phy&longs;icæ.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2AA7C" type="main"> <s id="N2AA7E">VIx credi pote&longs;t quam multis effectibus naturalibus hæc <lb/>duplicata ratio affigatur, aliquos cur&longs;im indicabo vt ve­<lb/>rum germanumque illius principium &longs;tatuatur. </s> </p> <p id="N2AA85" type="main"> <s id="N2AA87"><!-- NEW -->1. In motu recto naturaliter accelerato, decur&longs;a &longs;patia <lb/>&longs;unt in ratione duplicata temporum, id e&longs;t vt temporum <lb/>quadrata; dixi in motu recto, tùm eo, qui fit deor&longs;um in perpendiculo, <lb/>tùm eo, qui fit in plano inclinato. </s> </p> <p id="N2AA91" type="main"> <s id="N2AA93">2. Si iaciantur lapides inæqualis ponderis à potentia toto ni&longs;u agente <lb/>& eodem arcu, lapides &longs;unt in ratione duplicata inflictorum ictuum. </s> </p> <p id="N2AA98" type="main"> <s id="N2AA9A">3. Si impingantur &longs;udes inæquales eodem brachiorum arcu, pondera <lb/>&longs;unt in ratione duplicata ictuum. </s> </p> <p id="N2AA9F" type="main"> <s id="N2AAA1">4. Si malleus impingatur diuer&longs;o arcu ab eadem potentia, arcus &longs;unt <lb/>in ratione duplicata ictuum. </s> </p> <p id="N2AAA6" type="main"> <s id="N2AAA8">5. Si ex tubis erectis eiu&longs;dem cauitatis æqualique foramine fluat aqua, <lb/>longitudines tuborum &longs;unt in ratione duplicata quantitatum aquæ, quæ <lb/>ex tubis æquali tempore fluunt. </s> </p> <p id="N2AAAF" type="main"> <s id="N2AAB1">6. Similiter &longs;i ex &longs;iphonibus fluat aqua æquali foramine, longitudines <lb/>&longs;iphonum &longs;unt in ratione duplicata quantitatum aquæ, &c. </s> <s id="N2AAB6">vt &longs;uprà. </s> </p> <p id="N2AAB9" type="main"> <s id="N2AABB">7. Si chordæ ten&longs;æ eiu&longs;dem longitudinis appendantur inæqualia pon­<lb/>dera, hæc &longs;unt in ratione duplicata &longs;onorum in ratione acuti & grauis. </s> </p> <p id="N2AAC0" type="main"> <s id="N2AAC2">8. Si chordæ ten&longs;æ &longs;int eiu&longs;dem longitudinis & diuer&longs;æ cra&longs;&longs;itiei, ba­<lb/>&longs;es &longs;unt in ratione duplicata &longs;onorum permutando. </s> </p> <p id="N2AAC7" type="main"> <s id="N2AAC9">9. Lumen ita propagatur vt lumina propagata &longs;ub eodem angulo, & <lb/>cono &longs;int in ratione duplicata di&longs;tantiarum permutando. </s> </p> <p id="N2AACE" type="main"> <s id="N2AAD0">10. Idem dico pror&longs;us de propagatione &longs;onorum, immò au&longs;im dicere <lb/>toti rei &longs;onorum familiari&longs;&longs;imam e&longs;&longs;e hanc rationem duplicatam. </s> </p> <p id="N2AAD5" type="main"> <s id="N2AAD7"><!-- NEW -->11. In funependulis res e&longs;t clari&longs;&longs;ima; nam longitudines &longs;unt in ratio­<lb/>ne duplicata temporum quibus vibrationes perficiuntur. </s> </p> <p id="N2AADD" type="main"> <s id="N2AADF">12. Non e&longs;t omittendum quod in humana voce ob&longs;eruatur pro ratio­<lb/>ne grauis & acuti, &longs;cilicet ni&longs;us e&longs;&longs;e in ratione duplicata <expan abbr="&longs;onorũ">&longs;onorum</expan>. </s> <s id="N2AAE8">Omitto <lb/>infinita ferè alia quæ huic rationi duplicatæ &longs;ub&longs;unt, &longs;ed iam principia <lb/>phy&longs;ica his effectibus quibus ine&longs;t hæc ratio duplicata, tribuamus. </s> </p> <p id="N2AAEF" type="main"> <s id="N2AAF1"><!-- NEW -->Primum caput & vndecimum hoc principio nituntur, eadem cau&longs;a <lb/>æquali tempore æqualem effectum producit vnde illud; corpus graue <lb/>æqualibus temporibus æqualia acquirit velocitatis momenta, de quo lib. <!-- REMOVE S--><lb/>2. Ex hoc principio demon&longs;trauimus in partibus temporis &longs;en&longs;ibilibus <lb/>&longs;patia e&longs;&longs;e temporum quadrata. </s> </p> <pb pagenum="444" xlink:href="026/01/480.jpg"/> <p id="N2AB02" type="main"> <s id="N2AB04">Secundum & tertium hoc principio nituntur, motus impre&longs;&longs;i diuer&longs;is <lb/>corporibus ab eadem potentia æquali tempore &longs;unt vt corpora permu­<lb/>tando v.g.motus impre&longs;&longs;us corpori vnius libræ e&longs;t ad motum impre&longs;&longs;um <lb/>corpori quatuor librarum vt 4.ad 1.æquali &longs;cilicet tempore quod clarum <lb/>e&longs;t, igitur graue 4.librarum decurrit tantùm quartam partem arcus, igitur <lb/>&longs;ecundo tempore æquali decurrit tres alias partes, vide qu&etail; diximus l.10. </s> </p> <p id="N2AB11" type="main"> <s id="N2AB13">Quartum nititur hoc principio &longs;patia &longs;unt quadrata temporum, ve­<lb/>locitates &longs;unt vt tempora, ictus vt velocitates. </s> </p> <p id="N2AB18" type="main"> <s id="N2AB1A"><!-- NEW -->Quintum, &longs;extum, &longs;eptimum habent hoc commune principium: </s> <s id="N2AB1E"><!-- NEW -->eadem <lb/>e&longs;t proportio effectuum quæ cau&longs;arum; </s> <s id="N2AB24"><!-- NEW -->quippe cau&longs;a quæ aquam excu­<lb/>dit e&longs;t pondus &longs;uperimpo&longs;itum, igitur cum imprimat motum pluribus <lb/>partibus, velociorem imprimit &longs;ingulis, igitur ex duplici capite cre&longs;cit <lb/>effectus, &longs;cilicet ex maiore <expan abbr="quãtitate">quantitate</expan> aquæ & ex velociore motu; &longs;it enim <lb/>v.g.maior tubus quadruplus alterius cau&longs;a e&longs;t quadrupla, igitur duplam <lb/>quantitatem aquæ extrudet æquali tempore, quia duplo velociore motu. </s> <s id="N2AB36"><!-- NEW --><lb/>nam extrudere æqualem quantitatem duplo velociore motu e&longs;t effectus <lb/>duplus; igitur duplam quantitatem extrudere duplo velociore motu e&longs;t <lb/>effectus quadruplus, igitur e&longs;t eadem proportio cau&longs;&ecedil; quæ effectus. </s> <s id="N2AB3F">De &longs;i­<lb/>phone idem dictum e&longs;to, præ&longs;tat enim <expan abbr="eũdem">eundem</expan> effectum trahendo, quem <lb/>tubus aquæ pellendo, denique vnica vibratio chordæ ten&longs;æ duplo velo­<lb/>cior e&longs;t effectus duplus, igitur duæ duplo velociores effectus quadruplus. </s> </p> <p id="N2AB4C" type="main"> <s id="N2AB4E"><!-- NEW -->Octauum habet idem principium, nam chordæ eiu&longs;dem longitudinis <lb/>&longs;unt vt ba&longs;es, &longs;it vna quadrupla alterius v. <!-- REMOVE S-->g. <!-- REMOVE S-->appendatur vtrique æquale <lb/>pondus, ten&longs;io maioris e&longs;t &longs;ubquadrupla; </s> <s id="N2AB5A"><!-- NEW -->igitur &longs;i huic appendatur pon­<lb/>dus quadruplum &longs;onum edet duplo acutiorem; igitur ba&longs;es &longs;unt vt qua­<lb/>drata &longs;onorum. </s> </p> <p id="N2AB62" type="main"> <s id="N2AB64"><!-- NEW -->Nonum, & decimum nituntur hoc principio, lumen minus e&longs;t in ea <lb/>proportione in qua plus di&longs;trahitur; igitur lumina &longs;unt vt ba&longs;es permu­<lb/>tando, &longs;ed ba&longs;es &longs;unt in ratione duplicata di&longs;tantiarum, idem dico de &longs;ono. </s> </p> <p id="N2AB6C" type="main"> <s id="N2AB6E"><!-- NEW -->Duodecimum denique idem principium habet cum &longs;eptimo: </s> <s id="N2AB72"><!-- NEW -->vis enim <lb/>illa &longs;eu ni&longs;us quo adducitur arteria æquiualet ponderi; &longs;ed de his &longs;atis. </s> </p> <p id="N2AB78" type="main"> <s id="N2AB7A"><emph type="center"/><emph type="italics"/>Schol. <!-- REMOVE S-->quod pertinet ad reflexionem.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2AB87" type="main"> <s id="N2AB89"><!-- NEW -->Ob&longs;erua&longs;ti in Th.8.l.6.quo&longs;dam nolui&longs;&longs;e impetum in reflexione pro­<lb/>duci propter compre&longs;&longs;ionem, vel corporis reflexi, vel reflectentis, vel <lb/>vtriu&longs;que, quod certè fieri non pote&longs;t, alioquin &longs;it globus reflexus; </s> <s id="N2AB91"><!-- NEW -->certè <lb/>comprimitur nece&longs;&longs;ariò à puncto contactus ver&longs;us centrum quod certum <lb/>e&longs;t; </s> <s id="N2AB99"><!-- NEW -->igitur redit nece&longs;&longs;ariò per lineam ductam à puncto contactus per <lb/>idem centrum quod fal&longs;um e&longs;t vt patet; igitur e&longs;t alia cau&longs;a huius motus <lb/>&longs;cilicet præuius impetus. </s> </p> <p id="N2ABA1" type="main"> <s id="N2ABA3">Quidam etiam volunt hunc impetum produci ab ip&longs;o corpore re­<lb/>flectente quod tamen ab&longs;urdum e&longs;t, alioquin per <expan abbr="eãdem">eandem</expan> lineam ductam <lb/>à puncto contactus per centrum globi fieret reflexio, &longs;ic enim globus <lb/>tantùm impelli pote&longs;t, vt demon&longs;tratum e&longs;t lib.1. &longs;ed de his fatis. </s> </p> <pb pagenum="445" xlink:href="026/01/481.jpg"/> <p id="N2ABB4" type="main"> <s id="N2ABB6"><emph type="center"/><emph type="italics"/>Schol. <!-- REMOVE S-->pag.<emph.end type="italics"/> 217. <emph type="italics"/>num.<emph.end type="italics"/>8.<emph.end type="center"/></s> </p> <p id="N2ABCA" type="main"> <s id="N2ABCC"><!-- NEW -->Ob&longs;eruabis primò, fœdatam e&longs;&longs;e pulcherrimam demon&longs;trationem quæ <lb/>habetur loco citato innumeris propemodum mendis, qua &longs;cilicet pro­<lb/>batur omnium inclinatarum, quæ ab eodem horizontalis puncto ad <lb/>idem perpendiculum ducuntur, cam quæ e&longs;t ad angulum 45. grad. <!-- REMOVE S-->bre­<lb/>ui&longs;&longs;imo tempore decurri; </s> <s id="N2ABDA"><!-- NEW -->&longs;it enim Fig.49. Tab.2. in qua &longs;it EC diui&longs;a <lb/>bifariam in A, ex quo ducatur circulus radio AC, &longs;it AB perpendicula­<lb/>ris in AC; </s> <s id="N2ABE4"><!-- NEW -->ducantur BC.BR.BM. dico BC breuiore tempore quàm B <lb/>R, BM, percurri, quod breuiter demon&longs;tro: </s> <s id="N2ABEA"><!-- NEW -->ducatur AH perpendicula­<lb/>ris in BC, &longs;itque vt BH ad BI, ita BI ad BC; </s> <s id="N2ABF0"><!-- NEW -->certè BH & AC æquali <lb/>tempore percurruntur; &longs;it autem tempus quo percurritur BH, vel AC <lb/>vt. </s> <s id="N2ABF8">BH; </s> <s id="N2ABFB"><!-- NEW -->haud dubiè tempus quo percurretur BC erit vt BI, e&longs;t autem B <lb/>I æqualis AC,, quæ e&longs;t media proportionalis inter BC & BH, vt con­<lb/>&longs;tat; </s> <s id="N2AC03"><!-- NEW -->&longs;it autem BR dupla AR, & angulus ABR 30. grad. <!-- REMOVE S-->ducatur BY <lb/>perpendicularis in BR, certè RY e&longs;t dupla BR, &longs;unt enim triangula RB <lb/>A, RBY proportionalia; </s> <s id="N2AC0D"><!-- NEW -->igitur BR & YR perpendicularis eodem tem­<lb/>pore percurruntur; </s> <s id="N2AC13"><!-- NEW -->&longs;ed YR e&longs;t maior EC, nam EC e&longs;t dupla AB, & R <lb/>Y dupla RB, quæ e&longs;t maior AB, ergo YR maiore tempore percurritur <lb/>quam CE, igitur BR quam BC, &longs;imiliter ducatur BM ad angulum ABM <lb/>60. grad. <!-- REMOVE S-->&longs;it QB perpendicularis in BM; </s> <s id="N2AC1F"><!-- NEW -->igitur QM e&longs;t dupla QB, <lb/>igitur maior EC; </s> <s id="N2AC25"><!-- NEW -->igitur maiore tempore percurritur; </s> <s id="N2AC29"><!-- NEW -->&longs;ed BM & QM <lb/>æquali tempore decurruntur; igitur BM maiore tempore, quam BC <lb/>quod erat demon&longs;trandum. </s> </p> <p id="N2AC31" type="main"> <s id="N2AC33"><!-- NEW -->Ob&longs;eruabis &longs;ecundò BM & BR æquali tempore decurri, vnde quod <lb/>&longs;anè mirificum e&longs;t, &longs;i pariter vtrimque cre&longs;cat, & decre&longs;cat angulus in <lb/>puncto B, &longs;upra & infra BC, æquali tempore percurrentur duo plana in­<lb/>clinata; v.g.angulus RBA detrahit angulo ABC angulum CBR 15.grad. <lb/></s> <s id="N2AC3E">& angulus ABM addit angulum CBM 15.grad. </s> <s id="N2AC41">motus per BR & B <lb/>M fient æqualibus temporibus, vt con&longs;tat ex dictis. </s> </p> <p id="N2AC46" type="main"> <s id="N2AC48"><!-- NEW -->Ob&longs;eruabis tertiò rationem à priori inde e&longs;&longs;e ducendam; </s> <s id="N2AC4C"><!-- NEW -->quod cum <lb/>perpendiculum &longs;eu diagonalis quæ &longs;u&longs;tinet angulum rectum &longs;it regula <lb/>temporis quo decurritur omnis inclinata, diagonalis quadrati &longs;it om­<lb/>nium aliarum minima in rectangulis quorum minus latus &longs;it maius &longs;e­<lb/>midiagonali quadrati, in eodem &longs;cilicet perpendiculo; </s> <s id="N2AC58"><!-- NEW -->v.g. <!-- REMOVE S-->&longs;it diagona­<lb/>lis EC, &longs;int latera quadrati EBC, ducatur infra BA quælibet recta, v.g. <!-- REMOVE S--><lb/>BR, & in BR ducatur perpendicularis BY, certè YR e&longs;t maior EC, <lb/>quia vt e&longs;t RA ad AB, ita AB ad AY, igitur AB e&longs;t media proportionalis <lb/>communis; </s> <s id="N2AC67"><!-- NEW -->&longs;ed collectum ex extremis inæqualibus, e&longs;t &longs;emper maius <lb/>collecto ex æqualibus, po&longs;ita &longs;cilicet eadem media proportionali; </s> <s id="N2AC6D"><!-- NEW -->&longs;i enim <lb/>&longs;unt æqualia, media proportionalis e&longs;t &longs;emidiameter circuli cuius dia­<lb/>meter e&longs;t æqualis collecto; </s> <s id="N2AC75"><!-- NEW -->&longs;i verò &longs;unt inæqualia, media proportiona­<lb/>lis e&longs;t &longs;unicorda circuli, cuius diameter e&longs;t æqualis collecto; igitur col­<lb/>lectum i&longs;tud e&longs;t maius priore, &longs;ed hæc &longs;unt &longs;atis clara. </s> </p> <p id="N2AC7D" type="main"> <s id="N2AC7F">Quod &longs;pectat ad demon&longs;trationem num. </s> <s id="N2AC82">9. ibidem po&longs;itam, & peni-<pb pagenum="446" xlink:href="026/01/482.jpg"/>tus mendis fædatam, duces &longs;pongiam v&longs;que ad lineam 22. pag.214. vbi <lb/>legis hæc verba, adde quod præ&longs;ertim, cùm illam alibi, &longs;cilicet lib. 8. de­<lb/>mon&longs;tremus. </s> </p> <p id="N2AC8E" type="main"> <s id="N2AC90"><!-- NEW -->Cæterum vnum ob&longs;eruabis in Fig. <!-- REMOVE S-->1.Tab.4. &longs;i diuidatur BE bifariam <lb/>æqualiter in T ducaturque FTG, fore vt mobile citiùs decurrat BTF <lb/>facto initio motus in B, quam chordam BF: </s> <s id="N2AC9A"><!-- NEW -->cum enim FG &longs;it dupla FT, <lb/>&longs;it media proportionalis inter GT, GF; haud dubiè quadratum illius erit <lb/>duplum quadr. </s> <s id="N2ACA2">TF, & &longs;ubduplum quadr.BF, igitur &longs;it EG 4.ET 2FT erit <lb/>Rad. </s> <s id="N2ACA7"><expan abbr="q.">que</expan> 20. igitur FG rad. </s> <s id="N2ACAD"><expan abbr="q.">que</expan> 80. igitur media proportionalis (quæ &longs;it, <lb/>v.g. <!-- REMOVE S-->G <foreign lang="greek">m</foreign>) rad. </s> <s id="N2ACBB"><expan abbr="q.">que</expan> 40. igitur &longs;i &longs;ubtrahatur GT, id e&longs;t rad. </s> <s id="N2ACC1"><!-- NEW -->q.20. id e&longs;t 4. <lb/>1/2 paulò minùs, &longs;ed plùs quàm 4. 1/3 ex G <foreign lang="greek">m</foreign>; id e&longs;t ex rad. </s> <s id="N2ACCB"><!-- NEW -->q.40. id e&longs;t 6. <lb/>1/3 paulò minùs &longs;upere&longs;t <foreign lang="greek">tm</foreign>, quæ minor e&longs;t 2. &longs;ed &longs;i tempore BT, per­<lb/>curritur BT, æquali tempore percurretur tripla BT; </s> <s id="N2ACD7"><!-- NEW -->igitur tempus quo <lb/>percurritur dupla BE, e&longs;t vt BE; </s> <s id="N2ACDD"><!-- NEW -->&longs;ed tempus quo percurritur BTF e&longs;t vt <lb/>BT <foreign lang="greek">m</foreign>; </s> <s id="N2ACE7"><!-- NEW -->atqui T <foreign lang="greek">m</foreign> e&longs;t minor TE; </s> <s id="N2ACEF"><!-- NEW -->id e&longs;t 2. igitur breuiore tempore percur­<lb/>ritur BTF, quam dupla DE; </s> <s id="N2ACF5"><!-- NEW -->&longs;ed quo tempore percurritur dupla BE, <lb/>etiam percurritur BF; </s> <s id="N2ACFB"><!-- NEW -->igitur BTF breuiore tempore percurritur quam <lb/>BF; </s> <s id="N2AD01"><!-- NEW -->vt autem &longs;cias quantum percurritur in perpendiculari, quo tempore <lb/>percurritur BTF, &longs;it FE 100000. erit FT 111800. igitur G <foreign lang="greek">m</foreign> 151657. <lb/>igitur &longs;i vt BT 50000. ad BT <foreign lang="greek">m</foreign>, id e&longs;t ad 89857. ita BT <foreign lang="greek">m</foreign> ad aliam, hæc <lb/>erit 161485. hoc &longs;patium decurretur in perpendiculari, vides quam &longs;it <lb/>minor dupla BE, id e&longs;t 200000. Si autem accipis Fig.1. Tab.3. BZE &longs;it <lb/>GP 100000.GZ 42265.&longs;it etiam vt EZ ad EY ita EY ad CB; GZ erit <lb/>87757. igitur acquiretur in perpendiculari 182253.eo tempore quo per­<lb/>curretur GZB, facto initio motus à G, &longs;ed hæc e&longs;t minor dupla GP, id <lb/>e&longs;t 200000. accedit tamen propiùs quam &longs;uperior, igitur longiore tem­<lb/>pore decurit duas GZB huius figuræ quam duas BTF &longs;uperioris fig. </s> </p> <p id="N2AD23" type="main"> <s id="N2AD25"><!-- NEW -->Denique in Fig. <!-- REMOVE S-->32. Tab. <!-- REMOVE S-->3.&longs;it BY ita vt angulus BYA &longs;it grad.15.&longs;itque <lb/>v.g. <!-- REMOVE S-->vt YZ, ad YL, ita YL ad YB; </s> <s id="N2AD31"><!-- NEW -->iuxta canonem &longs;inuum BY erit 386370. <lb/>YL 330171. ZL 47739. EZ 73205. ELZ 120944. igitur acquiretur in <lb/>perpendiculari 199814. quo tempore decurretur EZB; vides quàm pro­<lb/>ximè accedat ad duplam EM id e&longs;t ad 200000. </s> </p> <p id="N2AD3D" type="main"> <s id="N2AD3F"><!-- NEW -->Denique &longs;i percurrat EMB, &longs;cilicet EM motu accelerato, tum MB <lb/>æquabili; </s> <s id="N2AD45"><!-- NEW -->certè MB percurret &longs;ubduplo tempore illius, quo percurrit E <lb/>M, vt con&longs;tat; igitur &longs;it EM tempus quo percurrit EM v. <!-- REMOVE S-->g. <!-- REMOVE S-->2.percurret <lb/>EMB tempore EMS &longs;cilicet 3. &longs;ed &longs;i percurrat EM tempore EM, du­<lb/>plam decurrit tempore EB, &longs;ed EB e&longs;t minor EMS, e&longs;t enim rad. </s> <s id="N2AD53">quadr. </s> <s id="N2AD56"><lb/>8. igitur EB decurritur citiùs quàm EMB, &longs;ed de his &longs;atis. <lb/><gap desc="hr tag"/></s> </p> <p id="N2AD5D" type="main"> <s id="N2AD5F"><emph type="center"/><emph type="italics"/>ERRATA.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p id="N2AD6A" type="main"> <s id="N2AD6C"><emph type="italics"/>Pag.<emph.end type="italics"/> 10. <emph type="italics"/>lin. 4<emph.end type="italics"/> magnete. <emph type="italics"/>p.13 l.vlt.<emph.end type="italics"/>non decre&longs;cit <emph type="italics"/>p.<emph.end type="italics"/>17.<emph type="italics"/>Th.<emph.end type="italics"/> 10.<emph type="italics"/>l.<emph.end type="italics"/> 2. non exigeret.<emph type="italics"/>p.<emph.end type="italics"/>20. <lb/><emph type="italics"/>l .ult.<emph.end type="italics"/> in &longs;e ip&longs;o. <emph type="italics"/>p.21.t.26.l.2.<emph.end type="italics"/> non pote&longs;t. <emph type="italics"/>p.<emph.end type="italics"/>24.<emph type="italics"/>t.<emph.end type="italics"/>32.<emph type="italics"/>l.<emph.end type="italics"/>5. duabus. <emph type="italics"/>p.<emph.end type="italics"/>25.<emph type="italics"/>t.<emph.end type="italics"/> 33. <emph type="italics"/>l.<emph.end type="italics"/> 15.tertiò <lb/>probatur. <emph type="italics"/>Ca&longs;tiga ibidem multas interpunctiones p.<emph.end type="italics"/>28.<emph type="italics"/>l.<emph.end type="italics"/> 1. maioris. <emph type="italics"/>p .<emph.end type="italics"/>31 <emph type="italics"/>l.<emph.end type="italics"/>3. Ax. 12. <lb/><emph type="italics"/>l.<emph.end type="italics"/>8 primo <emph type="italics"/>l.9. &longs;ecundo l.35.<emph.end type="italics"/> cum tu. <emph type="italics"/>p.<emph.end type="italics"/>33.<emph type="italics"/>l.<emph.end type="italics"/> 1. motus.<emph type="italics"/>p.<emph.end type="italics"/> 35. min 5s. <emph type="italics"/>t.<emph.end type="italics"/> 51.& 52. fig.2. <lb/><emph type="italics"/>t.<emph.end type="italics"/> 55.<emph type="italics"/>l.<emph.end type="italics"/>2. immobilis A. <emph type="italics"/>p.<emph.end type="italics"/>36. fig.2. <emph type="italics"/>p.<emph.end type="italics"/>49.<emph type="italics"/>t.<emph.end type="italics"/>86.<emph type="italics"/>l.<emph.end type="italics"/>3.lib.2.<emph type="italics"/>p.<emph.end type="italics"/>54.<emph type="italics"/>l.<emph.end type="italics"/>1. Th. 81.<emph type="italics"/>p.<emph.end type="italics"/>25.<emph type="italics"/>l.<emph.end type="italics"/>17. in EL. <pb xlink:href="026/01/483.jpg"/><emph type="italics"/>l.<emph.end type="italics"/>38.AB ad GB, id e&longs;t vt 1.ad 5.<emph type="italics"/>p.<emph.end type="italics"/>66.<emph type="italics"/>t.<emph.end type="italics"/>137.<emph type="italics"/>l.<emph.end type="italics"/>4. AD & AB.<emph type="italics"/>t.<emph.end type="italics"/>738.<emph type="italics"/>l.<emph.end type="italics"/>5. tota AC. <emph type="italics"/>t.<emph.end type="italics"/>140<lb/>fig. </s> <s id="N2AE60">15.tab.1. <emph type="italics"/>p.<emph.end type="italics"/>80.<emph type="italics"/>l.<emph.end type="italics"/> 3. idem e&longs;&longs;et, <emph type="italics"/>p.<emph.end type="italics"/>83.<emph type="italics"/>l.<emph.end type="italics"/>20. non e&longs;t.<emph type="italics"/>p.<emph.end type="italics"/>88.<emph type="italics"/>l.<emph.end type="italics"/>4. &longs;ecundo erunt, <emph type="italics"/>p.<emph.end type="italics"/>89. <emph type="italics"/>in <lb/>Sch.l.<emph.end type="italics"/>5. 1.&longs;patium, <emph type="italics"/>l.<emph.end type="italics"/> 7, <emph type="italics"/>ca&longs;tiga interpunctionem, p.<emph.end type="italics"/>90, <emph type="italics"/>t.<emph.end type="italics"/>41.<emph type="italics"/>l.<emph.end type="italics"/>3. terminus &longs;it 1.<emph type="italics"/>t.<emph.end type="italics"/>43. <emph type="italics"/>lege <lb/>ter<emph.end type="italics"/> rad.q. <emph type="italics"/>p.<emph.end type="italics"/>91 <emph type="italics"/>l.<emph.end type="italics"/>5. <emph type="italics"/>dele hac verba<emph.end type="italics"/> quàm &longs;patij quod, &c. </s> <s id="N2AECD">v&longs;que ad quàm, <emph type="italics"/>p.92.l.<emph.end type="italics"/> 15. <lb/>& 17. <emph type="italics"/>ca&longs;tiga interpunctiones p.<emph.end type="italics"/> 101. <emph type="italics"/>l.<emph.end type="italics"/> 10. perticam, <emph type="italics"/>l.<emph.end type="italics"/>26. proportionis primæ. <emph type="italics"/>l.<emph.end type="italics"/>39. <lb/>æquales AC.<emph type="italics"/>l.<emph.end type="italics"/>42. 1/4 &longs;ed, <emph type="italics"/>p.<emph.end type="italics"/>102. <emph type="italics"/>l.<emph.end type="italics"/>17. minimæ, <emph type="italics"/>p.<emph.end type="italics"/>104.<emph type="italics"/>l.<emph.end type="italics"/>4.acceditur. <emph type="italics"/>l.<emph.end type="italics"/>7.di&longs;cerni.<emph type="italics"/>p.<emph.end type="italics"/>105. <lb/><emph type="italics"/>l.<emph.end type="italics"/> 6, BI, <emph type="italics"/>l.<emph.end type="italics"/>32 igitur tertio. <emph type="italics"/>l.<emph.end type="italics"/>33. FM, <emph type="italics"/>p.<emph.end type="italics"/>106.<emph type="italics"/>l.<emph.end type="italics"/>1. toties, <emph type="italics"/>l.<emph.end type="italics"/>8. & 10. AFM, <emph type="italics"/>p.<emph.end type="italics"/>108.<emph type="italics"/>l.<emph.end type="italics"/>27.in­<lb/>&longs;tantia illud 1. 1/2 <emph type="italics"/>l.<emph.end type="italics"/>4. &longs;i 9. continet 1. 4/5 &longs;i 10. 1. (9/12) <emph type="italics"/>Coroll.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>4. <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>2.<emph type="italics"/>l.<emph.end type="italics"/>6.q.4. <emph type="italics"/>p.<emph.end type="italics"/> 109.<emph type="italics"/>l.<emph.end type="italics"/>1. <lb/>q.4. <emph type="italics"/>l.<emph.end type="italics"/>2. q.2. <emph type="italics"/>Cor.<emph.end type="italics"/>6. <emph type="italics"/>l.<emph.end type="italics"/>20. & 22. vbicationem, <emph type="italics"/>l.<emph.end type="italics"/>30. phy&longs;ica minora. <emph type="italics"/>l.<emph.end type="italics"/>32. &longs;ecundo in­<lb/>&longs;tanti, <emph type="italics"/>p.<emph.end type="italics"/> 113.<emph type="italics"/>t.<emph.end type="italics"/>64.<emph type="italics"/>l.<emph.end type="italics"/>1. &longs;ectam, <emph type="italics"/>t.<emph.end type="italics"/>65.<emph type="italics"/>l.<emph.end type="italics"/>4.primum in&longs;tans. </s> <s id="N2AFC3"><!-- NEW -->1.<emph type="italics"/>l.<emph.end type="italics"/>7.tertium. (5/11) <emph type="italics"/>t.<emph.end type="italics"/>66.<emph type="italics"/>l.<emph.end type="italics"/>1.aliqua <lb/><emph type="italics"/>l.<emph.end type="italics"/>7. minore CD.<emph type="italics"/>p.<emph.end type="italics"/>115. <emph type="italics"/>t.<emph.end type="italics"/>70.<emph type="italics"/>l.<emph.end type="italics"/>7.primo e&longs;t rad.q.2. <emph type="italics"/>l.<emph.end type="italics"/>8. tria rad.q.3.<emph type="italics"/>Th.<emph.end type="italics"/>71-<emph type="italics"/>l.<emph.end type="italics"/>2.nullum <lb/>e&longs;&longs;et. <emph type="italics"/>p.<emph.end type="italics"/>116.<emph type="italics"/>t.<emph.end type="italics"/>76. <emph type="italics"/>l.<emph.end type="italics"/>5.vel communis qua grauitat, <emph type="italics"/>l.<emph.end type="italics"/>6. de quo aliàs, vel &longs;ingularis, <emph type="italics"/>p,<emph.end type="italics"/> 117. <lb/><emph type="italics"/>in Sch.l.<emph.end type="italics"/>12. materiæ, <emph type="italics"/>p.<emph.end type="italics"/>118.<emph type="italics"/>t.<emph.end type="italics"/>81. <emph type="italics"/>l.<emph.end type="italics"/>7. extrudi, <emph type="italics"/>p.<emph.end type="italics"/>123. <emph type="italics"/>t.<emph.end type="italics"/> 103.<emph type="italics"/>l.<emph.end type="italics"/>6. vel diuer&longs;æ grauitatis, <lb/>& mollitiei, <emph type="italics"/>p,<emph.end type="italics"/> 124. <emph type="italics"/>l.<emph.end type="italics"/>4. grauioris, <emph type="italics"/>t.<emph.end type="italics"/>104. <emph type="italics"/>l<emph.end type="italics"/> 5. &longs;ecunda eiu&longs;dem materiæ, & figuræ ter­<lb/>tia.<emph type="italics"/>l.<emph.end type="italics"/>12. vel eadem vel diuer&longs;a <emph type="italics"/>p.<emph.end type="italics"/>125.<emph type="italics"/>t.<emph.end type="italics"/>109.<emph type="italics"/>L.B.K.L. t.<emph.end type="italics"/>110.<emph type="italics"/>l.<emph.end type="italics"/>1. diui&longs;ione, <emph type="italics"/>p.<emph.end type="italics"/>127.<emph type="italics"/>l.<emph.end type="italics"/>25. <lb/>cubo minori, <emph type="italics"/>p.<emph.end type="italics"/>128.<emph type="italics"/>l.<emph.end type="italics"/>7.mouent, <emph type="italics"/>l.<emph.end type="italics"/>10. aëre repellitur. <emph type="italics"/>l.<emph.end type="italics"/> 14. permeat, <emph type="italics"/>t.<emph.end type="italics"/>112. <emph type="italics"/>l<emph.end type="italics"/> 2. actiui­<lb/>tatis vnius.<emph type="italics"/>l.<emph.end type="italics"/>7. motum retardat; cum.<emph type="italics"/>l.<emph.end type="italics"/>16. modicus ventus.<emph type="italics"/>p.<emph.end type="italics"/>129. <emph type="italics"/>t.<emph.end type="italics"/>114.<emph type="italics"/>l.<emph.end type="italics"/>5.acuto. <emph type="italics"/>l.<emph.end type="italics"/><lb/>6. mobile, <emph type="italics"/>l.<emph.end type="italics"/>7.maior e&longs;t.<emph type="italics"/>l.<emph.end type="italics"/>8. &longs;emiperipheriæ, <emph type="italics"/>l.vlt.<emph.end type="italics"/> illam cauam, <emph type="italics"/>p.<emph.end type="italics"/>130.<emph type="italics"/>l.<emph.end type="italics"/>2.alter grauior <lb/><emph type="italics"/>t.<emph.end type="italics"/>123.<emph type="italics"/>l.<emph.end type="italics"/>2. intru&longs;us, <emph type="italics"/>p.<emph.end type="italics"/>133.<emph type="italics"/>l.<emph.end type="italics"/>7. in hoc agemus, <emph type="italics"/>p.<emph.end type="italics"/>13.<emph type="italics"/>l.<emph.end type="italics"/>1. ad&longs;tantibus, <emph type="italics"/>p.<emph.end type="italics"/>137. <emph type="italics"/>l.<emph.end type="italics"/>4. produ­<lb/>ctum. <emph type="italics"/>p<emph.end type="italics"/> 143.<emph type="italics"/>l.<emph.end type="italics"/>7. accidit <emph type="italics"/>l.<emph.end type="italics"/>12. producto.<emph type="italics"/>p.<emph.end type="italics"/>145. <emph type="italics"/>habes.<emph.end type="italics"/> v.g. <!-- REMOVE S-->pro R, Q, & radices 4. pro <lb/><expan abbr="q.">que</expan> <emph type="italics"/>& alibi pa&longs;&longs;im<emph.end type="italics"/> 9.<emph type="italics"/>pro Q, t.<emph.end type="italics"/>47. <emph type="italics"/>l.<emph.end type="italics"/>9. &longs;ubduplicata. <emph type="italics"/>p.<emph.end type="italics"/>151. <emph type="italics"/>l.<emph.end type="italics"/>11. &longs;i loquamur. <emph type="italics"/>l.<emph.end type="italics"/>14. di­<lb/>&longs;tinctiones, <emph type="italics"/>l.<emph.end type="italics"/>21. de&longs;cenderet. <emph type="italics"/>p.<emph.end type="italics"/>154.<emph type="italics"/>l.<emph.end type="italics"/>1. determinatum, <emph type="italics"/>l.<emph.end type="italics"/>5. inclinatam &longs;ur&longs;um.<emph type="italics"/>p.<emph.end type="italics"/>256. <lb/><emph type="italics"/>t.<emph.end type="italics"/>13, <emph type="italics"/>l.<emph.end type="italics"/>4.IM.&longs;eq.fig.pro fig.37. lege 13. <emph type="italics"/>p.<emph.end type="italics"/>157.<emph type="italics"/>l,<emph.end type="italics"/> 3.partis.<emph type="italics"/>l<emph.end type="italics"/>28. ita vt, <emph type="italics"/>l.<emph.end type="italics"/> 37. non dati.<emph type="italics"/>p.<emph.end type="italics"/><lb/>158.<emph type="italics"/>t.<emph.end type="italics"/> 19.<emph type="italics"/>l.<emph.end type="italics"/> 6. parallela.<emph type="italics"/>p.<emph.end type="italics"/>161.l.12. æquabilitas. <emph type="italics"/>l.<emph.end type="italics"/>15. primo æquabibi <emph type="italics"/>p.<emph.end type="italics"/> 162.<emph type="italics"/>t.<emph.end type="italics"/>39. <emph type="italics"/>l.<emph.end type="italics"/>1. <lb/>vtcumque, <emph type="italics"/>l.<emph.end type="italics"/>6. EO æquali.<emph type="italics"/>p.<emph.end type="italics"/>165.<emph type="italics"/>t.<emph.end type="italics"/>42.<emph type="italics"/>l,<emph.end type="italics"/> 3. violento.<emph type="italics"/>p.<emph.end type="italics"/>167.fig.47.<emph type="italics"/>Th.<emph.end type="italics"/>57. <emph type="italics"/>l.<emph.end type="italics"/>2. decre&longs;cit. <lb/><emph type="italics"/>p.<emph.end type="italics"/>173.<emph type="italics"/>c.<emph.end type="italics"/> 1.<emph type="italics"/>l.<emph.end type="italics"/>4. linea motus accedit, <emph type="italics"/>p.<emph.end type="italics"/>172. <emph type="italics"/>t.<emph.end type="italics"/>2.<emph type="italics"/>l.<emph.end type="italics"/>15. QR (2/16) in X.<emph type="italics"/>l.<emph.end type="italics"/> 19.EB.<emph type="italics"/>l.<emph.end type="italics"/>31.EYEZ. <lb/><emph type="italics"/>p.<emph.end type="italics"/>137.<emph type="italics"/>l.<emph.end type="italics"/>8. infra.<emph type="italics"/>l.<emph.end type="italics"/>10 maximam <emph type="italics"/>t.<emph.end type="italics"/> 66 <emph type="italics"/>l.<emph.end type="italics"/>7. BG. l. <!-- REMOVE S-->12. æqualem RK. <emph type="italics"/>p.<emph.end type="italics"/> 174. <emph type="italics"/>l.<emph.end type="italics"/>7. diffe­<lb/>rentiam, <emph type="italics"/>l.<emph.end type="italics"/> 9. tendere, centrum, <emph type="italics"/>l.<emph.end type="italics"/>16.erit AE, <emph type="italics"/>l.<emph.end type="italics"/>18.totus ille, <emph type="italics"/>t.<emph.end type="italics"/>62 <emph type="italics"/>l.<emph.end type="italics"/>2. inclinatiorem, <emph type="italics"/>l.<emph.end type="italics"/>4. <lb/>detrahi, <emph type="italics"/>p.<emph.end type="italics"/>175.<emph type="italics"/>l.<emph.end type="italics"/>35. re&longs;i&longs;tentiam, <emph type="italics"/>p.<emph.end type="italics"/>176.<emph type="italics"/>t.<emph.end type="italics"/>70.fig. </s> <s id="N2B2C1">54.<emph type="italics"/>l.<emph.end type="italics"/>9.in E &longs;ed.<emph type="italics"/>p.<emph.end type="italics"/>177.<emph type="italics"/>l<emph.end type="italics"/>7.debet. <emph type="italics"/>t.<emph.end type="italics"/>72. <lb/>tab.2.<emph type="italics"/>l.<emph.end type="italics"/> 5. æqualis CR.<emph type="italics"/>l.vlt.<emph.end type="italics"/> demittatur, <emph type="italics"/>p.<emph.end type="italics"/>178.<emph type="italics"/>t.<emph.end type="italics"/>77.<emph type="italics"/>l.<emph.end type="italics"/> 3.eadem ratio.<emph type="italics"/>t.<emph.end type="italics"/>78.<emph type="italics"/>l.<emph.end type="italics"/>1.excepta. <lb/><emph type="italics"/>t.<emph.end type="italics"/>80.<emph type="italics"/>l.<emph.end type="italics"/>4.motus mixtus, <emph type="italics"/>p.<emph.end type="italics"/> 179.<emph type="italics"/>l.<emph.end type="italics"/>2. motus terræ, <emph type="italics"/>l<emph.end type="italics"/> 24. AK. tab.2.<emph type="italics"/>l.<emph.end type="italics"/>27.AD.<emph type="italics"/>l.<emph.end type="italics"/>28. DE <emph type="italics"/>p.<emph.end type="italics"/><lb/>180.<emph type="italics"/>l.<emph.end type="italics"/>7. 20 .<emph type="italics"/>l.<emph.end type="italics"/> 33. imum malum, <emph type="italics"/>p.<emph.end type="italics"/>18 1.<emph type="italics"/>l.<emph.end type="italics"/>11.rapietur.<emph type="italics"/>l.<emph.end type="italics"/>32.&longs;i verò.<emph type="italics"/>p.<emph.end type="italics"/>182.<emph type="italics"/>l.<emph.end type="italics"/>2.FA, <emph type="italics"/>p.<emph.end type="italics"/>183.<emph type="italics"/>l<emph.end type="italics"/> 3. <lb/>mixtus EB denique, <emph type="italics"/>l<emph.end type="italics"/> 6. ad quam.<emph type="italics"/>l.<emph.end type="italics"/>27. cum impetu, <emph type="italics"/>l.<emph.end type="italics"/>29. ex verticali.<emph type="italics"/>p.<emph.end type="italics"/>184.<emph type="italics"/>l.<emph.end type="italics"/>6.parte. <lb/><emph type="italics"/>l.<emph.end type="italics"/>9. æqualem IK, <emph type="italics"/>l.<emph.end type="italics"/>15. recidit.<emph type="italics"/>l.<emph.end type="italics"/>26. mobile, <emph type="italics"/>l.<emph.end type="italics"/>29. rhedis. <emph type="italics"/>p.<emph.end type="italics"/>185.<emph type="italics"/>l.<emph.end type="italics"/>2. motu non a&longs;&longs;imi­<lb/>lem.<emph type="italics"/>p.<emph.end type="italics"/>186. <emph type="italics"/>l.<emph.end type="italics"/>8. oppo&longs;itam, <emph type="italics"/>p.<emph.end type="italics"/>187.<emph type="italics"/>l<emph.end type="italics"/> 2. arcu <emph type="italics"/>p.<emph.end type="italics"/>188. <emph type="italics"/>l.<emph.end type="italics"/>10. ad GM, <emph type="italics"/>l.<emph.end type="italics"/>28. puncto Z, <emph type="italics"/>p.<emph.end type="italics"/>189. <lb/><emph type="italics"/>l.<emph.end type="italics"/>24. &longs;ubduplam, <emph type="italics"/>l.<emph.end type="italics"/>31.&longs;agittam AR.<emph type="italics"/>p.<emph.end type="italics"/>190.<emph type="italics"/>l.<emph.end type="italics"/>14. erit KI inclinata KC, <emph type="italics"/>l.<emph.end type="italics"/>37.quam &longs;up­<lb/>pono.<emph type="italics"/>l.<emph.end type="italics"/>38. ca&longs;t.interpunct.<emph type="italics"/>p.<emph.end type="italics"/>191.<emph type="italics"/>t.<emph.end type="italics"/>107.<emph type="italics"/>l.<emph.end type="italics"/>6.e&longs;t AH.<emph type="italics"/>p.<emph.end type="italics"/>92.<emph type="italics"/>t.<emph.end type="italics"/>109.<emph type="italics"/>l.<emph.end type="italics"/>5. &longs;it AE.<emph type="italics"/>l.<emph.end type="italics"/>6.&longs;it HN, <lb/><emph type="italics"/>l.<emph.end type="italics"/>7. AO & FG.<emph type="italics"/>l.<emph.end type="italics"/>15. & EM.<emph type="italics"/>l.<emph.end type="italics"/>16. AM, ca&longs;t.interp.<emph type="italics"/>t.<emph.end type="italics"/>110.<emph type="italics"/>l.<emph.end type="italics"/>5.<emph type="italics"/>p.<emph.end type="italics"/>193.<emph type="italics"/>n.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>5. è naui. <emph type="italics"/>n.<emph.end type="italics"/>8. <lb/><emph type="italics"/>l,<emph.end type="italics"/> 3. ex ABAF, <emph type="italics"/>p.<emph.end type="italics"/>197.<emph type="italics"/>l.<emph.end type="italics"/>38. tantum I, <emph type="italics"/>l.<emph.end type="italics"/>28. BAI.<emph type="italics"/>p.<emph.end type="italics"/>198.<emph type="italics"/>l.<emph.end type="italics"/>6. CA. nam.<emph type="italics"/>l.<emph.end type="italics"/>7.fune DB.<emph type="italics"/>l.<emph.end type="italics"/>10. <lb/>EA.<emph type="italics"/>l.<emph.end type="italics"/> 12.AC ver&longs;us E.<emph type="italics"/>l.<emph.end type="italics"/> 13.ad BA.<emph type="italics"/>l.<emph.end type="italics"/> 34. EO, <emph type="italics"/>l.<emph.end type="italics"/>40. vt RF, <emph type="italics"/>l.<emph.end type="italics"/>41. vel in B vt PR.<emph type="italics"/>p.<emph.end type="italics"/>199. <lb/><emph type="italics"/>l.<emph.end type="italics"/>7. LM.vt SR.<emph type="italics"/>l<emph.end type="italics"/> 35.&longs;inui.<emph type="italics"/>p.<emph.end type="italics"/>200.<emph type="italics"/>t.<emph.end type="italics"/>70.<emph type="italics"/>l.<emph.end type="italics"/>4.non de&longs;cendit.<emph type="italics"/>t.<emph.end type="italics"/> 9.<emph type="italics"/>l.<emph.end type="italics"/>1. BAE, <emph type="italics"/>t.<emph.end type="italics"/>10. <emph type="italics"/>l.<emph.end type="italics"/>2. lib.2. <emph type="italics"/>p.<emph.end type="italics"/><lb/>201.<emph type="italics"/>l.<emph.end type="italics"/>7. innato, <emph type="italics"/>l. <!-- REMOVE S-->vlt.<emph.end type="italics"/> eodem. <emph type="italics"/>in Sch.<emph.end type="italics"/>fig.26.tab. <!-- REMOVE S-->1. <emph type="italics"/>p.<emph.end type="italics"/>202.<emph type="italics"/>l.<emph.end type="italics"/>2.AD.fig.27, <emph type="italics"/>l.<emph.end type="italics"/> 30. vt AD. <lb/>Th. 16.Fig. </s> <s id="N2B52F">31. Tab.2.<emph type="italics"/>p.<emph.end type="italics"/>203.<emph type="italics"/>l.<emph.end type="italics"/>8. in A.<emph type="italics"/>l.<emph.end type="italics"/>21. GD.<emph type="italics"/>p.<emph.end type="italics"/>205 <emph type="italics"/>t.<emph.end type="italics"/>18.<emph type="italics"/>l.<emph.end type="italics"/>15.ducatur LE.<emph type="italics"/>l.<emph.end type="italics"/>6.DG. <emph type="italics"/>l. <!-- REMOVE S--><lb/>vlt.<emph.end type="italics"/> FP.DN, <emph type="italics"/>p.<emph.end type="italics"/>206. <emph type="italics"/>n.<emph.end type="italics"/>8.<emph type="italics"/>l.<emph.end type="italics"/>3.AIFD, <emph type="italics"/>l.<emph.end type="italics"/>4. in AG.<emph type="italics"/>p.<emph.end type="italics"/>207.<emph type="italics"/>t.<emph.end type="italics"/>19.<emph type="italics"/>habes<emph.end type="italics"/> L pro G.<emph type="italics"/>p.<emph.end type="italics"/>209.<emph type="italics"/>t.<emph.end type="italics"/>25. <lb/><emph type="italics"/>l.<emph.end type="italics"/>3. ducatur, <emph type="italics"/>t.<emph.end type="italics"/>26.<emph type="italics"/>l.<emph.end type="italics"/>2. AF.<emph type="italics"/>p.<emph.end type="italics"/>210.<emph type="italics"/>l.<emph.end type="italics"/>4.de&longs;cendet fig.42.tab.2. <emph type="italics"/>t.<emph.end type="italics"/>28.loco B.lege X.<emph type="italics"/>t.<emph.end type="italics"/> 30.<emph type="italics"/>l.<emph.end type="italics"/>7. <lb/>ad KA.<emph type="italics"/>t,<emph.end type="italics"/> 30. <emph type="italics"/>l.<emph.end type="italics"/>8.petcurritur A.D.<emph type="italics"/>p.<emph.end type="italics"/>211.<emph type="italics"/>l.<emph.end type="italics"/>6. longitudinum, <emph type="italics"/>p.<emph.end type="italics"/>212.<emph type="italics"/>l.<emph.end type="italics"/>12. ad BC ducatur <lb/>BG. </s> <s id="N2B5F6">Si non e&longs;&longs;et maior 5. CF, <emph type="italics"/>l.<emph.end type="italics"/> 14. CF ferè 2. 1/2 <emph type="italics"/>l.<emph.end type="italics"/> 30, BKAK, <emph type="italics"/>p.<emph.end type="italics"/>213.<emph type="italics"/>l.<emph.end type="italics"/>41.&longs;it rad. </s> <s id="N2B611"><lb/>q.8.<emph type="italics"/>l.<emph.end type="italics"/>20.GED.num. <!-- REMOVE S-->8, & 9.&longs;catent mendis tu ca&longs;tigabis iuxta Sch. <!-- REMOVE S-->vltimæ appendicis. <lb/><emph type="italics"/>p.<emph.end type="italics"/>215. <emph type="italics"/>t.<emph.end type="italics"/>37.<emph type="italics"/>l.<emph.end type="italics"/>7. vel AFC. <emph type="italics"/>p.<emph.end type="italics"/>216. <emph type="italics"/>t.<emph.end type="italics"/>38.<emph type="italics"/>l.<emph.end type="italics"/>11. conficeret per AF. <emph type="italics"/>l. <!-- REMOVE S-->vlt.<emph.end type="italics"/> a&longs;cen&longs;um. </s> <s id="N2B64C">Th.40. <lb/><emph type="italics"/>l.<emph.end type="italics"/>4. MA <emph type="italics"/>t.<emph.end type="italics"/>41.fig.3.tab, 3.<emph type="italics"/>p.<emph.end type="italics"/>217. <emph type="italics"/>l.<emph.end type="italics"/>6.21.22. E.pro C.<emph type="italics"/>p.<emph.end type="italics"/>218.<emph type="italics"/>t.<emph.end type="italics"/>47.<emph type="italics"/>l.<emph.end type="italics"/>4. &longs;ubduplus impetus <lb/><emph type="italics"/>t.<emph.end type="italics"/>49. <emph type="italics"/>l.<emph.end type="italics"/>11. vt &longs;ubdupla BC <emph type="italics"/>l.<emph.end type="italics"/>13. <emph type="italics"/>dele<emph.end type="italics"/> a, quia v&longs;que vt verò, <emph type="italics"/>p.<emph.end type="italics"/>219. <emph type="italics"/>l.<emph.end type="italics"/>2. vt &longs;ubdupla GF <lb/><emph type="italics"/>l.<emph.end type="italics"/>5. vt &longs;ubdupla BC.<emph type="italics"/>l.<emph.end type="italics"/>7. quadruplum AB.<emph type="italics"/>p.<emph.end type="italics"/>220.<emph type="italics"/>l.<emph.end type="italics"/> 8.perpendicularis GH.<emph type="italics"/>l.<emph.end type="italics"/>11.paral­<lb/>lela EG.<emph type="italics"/>t.<emph.end type="italics"/> 56. habes Y lege & <emph type="italics"/>t.<emph.end type="italics"/>58. <emph type="italics"/>l.<emph.end type="italics"/>2. ver&longs;us E, <emph type="italics"/>p.<emph.end type="italics"/>221.<emph type="italics"/>t.<emph.end type="italics"/>60, Y pro & <emph type="italics"/>t.<emph.end type="italics"/>62. V pro <foreign lang="greek">g</foreign>, <lb/><emph type="italics"/>l.<emph.end type="italics"/>8.puta <foreign lang="greek">b.</foreign><emph type="italics"/>t.<emph.end type="italics"/>64. T pro <foreign lang="greek">t</foreign> <emph type="italics"/>p.<emph.end type="italics"/>222.<emph type="italics"/>l.<emph.end type="italics"/>9. æqualis.<emph type="italics"/>t.<emph.end type="italics"/>65. X pro & <emph type="italics"/>l.<emph.end type="italics"/>10.in plano.<emph type="italics"/>t.<emph.end type="italics"/>66 P & <emph type="italics"/>t<emph.end type="italics"/>68. <lb/><emph type="italics"/>l.<emph.end type="italics"/>3.vt planum fig.7, tab. </s> <s id="N2B727"><!-- NEW -->3. <emph type="italics"/>p.<emph.end type="italics"/>223. <emph type="italics"/>l.<emph.end type="italics"/>11.per KA vt DC ad CA, <emph type="italics"/>l.<emph.end type="italics"/>13. EPPEEA, <emph type="italics"/>l.<emph.end type="italics"/>37. <lb/>enotum, <emph type="italics"/>p.<emph.end type="italics"/>225.<emph type="italics"/>l.<emph.end type="italics"/>3. non e&longs;t <emph type="italics"/>p.<emph.end type="italics"/>228.<emph type="italics"/>t.<emph.end type="italics"/>86.<emph type="italics"/>l.<emph.end type="italics"/>6. LC.<emph type="italics"/>l.<emph.end type="italics"/>7. maior <emph type="italics"/>t.<emph.end type="italics"/>87.<emph type="italics"/>l.<emph.end type="italics"/>6. in&longs;erte.<emph type="italics"/>t,<emph.end type="italics"/> 89 <emph type="italics"/>t.<emph.end type="italics"/>8.an-<pb xlink:href="026/01/484.jpg"/>tecedentia.<emph type="italics"/>p.<emph.end type="italics"/>219.<emph type="italics"/>t.<emph.end type="italics"/>93. <emph type="italics"/>l.<emph.end type="italics"/> 17. accedit.<emph type="italics"/>p.<emph.end type="italics"/>230.<emph type="italics"/>t.<emph.end type="italics"/>97.<emph type="italics"/>l.<emph.end type="italics"/> 90. tum QP. & EI.æqualia QYA <lb/>D.<emph type="italics"/>p.<emph.end type="italics"/>231.<emph type="italics"/>t.<emph.end type="italics"/>98.<emph type="italics"/>l.<emph.end type="italics"/>6, MK.<emph type="italics"/>l.<emph.end type="italics"/>11. &longs;upra C.<emph type="italics"/>l.<emph.end type="italics"/>12. arcus MGP.<emph type="italics"/>l.<emph.end type="italics"/>14.&longs;i verò in V.<emph type="italics"/>t.<emph.end type="italics"/>99.<emph type="italics"/>l.<emph.end type="italics"/>11.in 4. <lb/>vt AZ.<emph type="italics"/>l.<emph.end type="italics"/>4. 3 E.<emph type="italics"/>l.<emph.end type="italics"/>5. TBE <emph type="italics"/>p.<emph.end type="italics"/>232.<emph type="italics"/>t.<emph.end type="italics"/>100.<emph type="italics"/>l.<emph.end type="italics"/>12. in&longs;erto.<emph type="italics"/>l.<emph.end type="italics"/>33. & ratione. <emph type="italics"/>l.<emph.end type="italics"/>13. EQE.<emph type="italics"/>l.<emph.end type="italics"/>27. <lb/>ad AT ad A <foreign lang="greek"><expan abbr="q.">que</expan></foreign><emph type="italics"/>l.<emph.end type="italics"/>36. motum per AC.<emph type="italics"/>l.<emph.end type="italics"/>37. per AC.<emph type="italics"/>p.<emph.end type="italics"/>233.<emph type="italics"/>l.<emph.end type="italics"/>3.e&longs;&longs;et. <emph type="italics"/>l.<emph.end type="italics"/>4.debet e&longs;&longs;e <emph type="italics"/>co.<emph.end type="italics"/>4. <lb/><emph type="italics"/>l<emph.end type="italics"/> 5 de&longs;cendant.<emph type="italics"/>p.<emph.end type="italics"/>235.<emph type="italics"/>l.<emph.end type="italics"/>20. ADG.<emph type="italics"/>l.<emph.end type="italics"/>39. vbi e&longs;t motus.<emph type="italics"/>p.<emph.end type="italics"/>238.<emph type="italics"/>l.<emph.end type="italics"/>3. totum agit. <emph type="italics"/>p.<emph.end type="italics"/>240.<emph type="italics"/>t.<emph.end type="italics"/><lb/>17.<emph type="italics"/>l.<emph.end type="italics"/>4. atque, <emph type="italics"/>p.<emph.end type="italics"/>241.<emph type="italics"/>t.<emph.end type="italics"/>20.<emph type="italics"/>l.<emph.end type="italics"/>2. lib. 1.<emph type="italics"/>t.<emph.end type="italics"/>23.<emph type="italics"/>l.<emph.end type="italics"/>8. horizontalis.<emph type="italics"/>l.<emph.end type="italics"/>13. GD ad AB. <emph type="italics"/>p.<emph.end type="italics"/>243.<emph type="italics"/>l.<emph.end type="italics"/>5. D <lb/>G. <emph type="italics"/>l.<emph.end type="italics"/>17. ad DA.<emph type="italics"/>l.<emph.end type="italics"/>19. dele GO, <emph type="italics"/>p.<emph.end type="italics"/>244.<emph type="italics"/>t.<emph.end type="italics"/>33.<emph type="italics"/>l.<emph.end type="italics"/>6. volunt.<emph type="italics"/>p.<emph.end type="italics"/>246.<emph type="italics"/>l.<emph.end type="italics"/>19.& 23. G <foreign lang="greek">d.</foreign><emph type="italics"/>l.<emph.end type="italics"/>24. Th. <!-- REMOVE S--><lb/>40.<emph type="italics"/>l.<emph.end type="italics"/>42. idque duobus.<emph type="italics"/>p.<emph.end type="italics"/>248.<emph type="italics"/>l.<emph.end type="italics"/>38. motum.<emph type="italics"/>p.<emph.end type="italics"/>249:<emph type="italics"/>t.<emph.end type="italics"/>41.<emph type="italics"/>l.<emph.end type="italics"/> 11. PD æqualis, <emph type="italics"/>p.<emph.end type="italics"/>250.<emph type="italics"/>t.<emph.end type="italics"/>44.<emph type="italics"/>l.<emph.end type="italics"/>8. <lb/>& hic GDK.<emph type="italics"/>p.<emph.end type="italics"/>251.<emph type="italics"/>l.<emph.end type="italics"/>9. G <foreign lang="greek">d.</foreign><emph type="italics"/>p.<emph.end type="italics"/>252.<emph type="italics"/>l.<emph.end type="italics"/>4. quie&longs;cit vt vult; &longs;ed rem demon&longs;traui.<emph type="italics"/>p.<emph.end type="italics"/>253. <lb/><emph type="italics"/>l.<emph.end type="italics"/>7. quod dum.<emph type="italics"/>l.<emph.end type="italics"/>17.& 36.atterantur.<emph type="italics"/>l.<emph.end type="italics"/>39.cedit.<emph type="italics"/>p.<emph.end type="italics"/> 254.<emph type="italics"/>l.<emph.end type="italics"/> 13.atterantur, <emph type="italics"/>p.<emph.end type="italics"/>253. <emph type="italics"/>t.<emph.end type="italics"/>59.<emph type="italics"/>l.<emph.end type="italics"/> 1. <lb/>de&longs;truitur.<emph type="italics"/>p.<emph.end type="italics"/>254.<emph type="italics"/>t.<emph.end type="italics"/>62.<emph type="italics"/>l.<emph.end type="italics"/>12. oppo&longs;itam.<emph type="italics"/>p.<emph.end type="italics"/>255.<emph type="italics"/>l.<emph.end type="italics"/>34. DBM. <emph type="italics"/>p.<emph.end type="italics"/>266.<emph type="italics"/>l.<emph.end type="italics"/>9. verò 60.<emph type="italics"/>t.<emph.end type="italics"/>64. <emph type="italics"/>l.<emph.end type="italics"/><lb/>19. &longs;ubdupla habent &longs;æpius V.pro <foreign lang="greek">g.</foreign><emph type="italics"/>l.<emph.end type="italics"/>21.detrahatur <foreign lang="greek">d</foreign> H.<emph type="italics"/>l.<emph.end type="italics"/>28. 1 1/2 <emph type="italics"/>p.<emph.end type="italics"/>257..<emph type="italics"/>l.<emph.end type="italics"/>12.FAN <lb/>C. fig.23. tab. </s> <s id="N2B9BC"><!-- NEW -->3. <emph type="italics"/>p.<emph.end type="italics"/>258.<emph type="italics"/>t.<emph.end type="italics"/>68.<emph type="italics"/>l.<emph.end type="italics"/> 3 autem &longs;ic <emph type="italics"/>l.<emph.end type="italics"/>10. Th. 135. lib. 1.<emph type="italics"/>t.<emph.end type="italics"/> 67. <emph type="italics"/>habes &longs;æpius<emph.end type="italics"/> <foreign lang="greek">n</foreign><lb/>pro <foreign lang="greek">g.</foreign><emph type="italics"/>p.<emph.end type="italics"/>259.<emph type="italics"/>l.<emph.end type="italics"/>14. globus B. <emph type="italics"/>l.<emph.end type="italics"/>31. globi B. <emph type="italics"/>l.<emph.end type="italics"/>29. a&longs;&longs;umatur M <foreign lang="greek">q</foreign>, <emph type="italics"/>p.<emph.end type="italics"/> 262. <emph type="italics"/>l.<emph.end type="italics"/>2. re&longs;ilit. <emph type="italics"/>p.<emph.end type="italics"/><lb/>264. Th.90.<emph type="italics"/>l.<emph.end type="italics"/>6. lineæ.<emph type="italics"/>l.<emph.end type="italics"/>9. &longs;ed mox.<emph type="italics"/>p.<emph.end type="italics"/> 265. <foreign lang="greek">u</foreign> pro <foreign lang="greek">g</foreign> <emph type="italics"/>p.<emph.end type="italics"/>266. <emph type="italics"/>t.<emph.end type="italics"/>93. in&longs;tanti. <emph type="italics"/>t.<emph.end type="italics"/>97.<emph type="italics"/>in Sch. <!-- REMOVE S--><lb/>l.<emph.end type="italics"/>1. cau&longs;as multiplices.<emph type="italics"/>p.<emph.end type="italics"/>267.<emph type="italics"/>l.<emph.end type="italics"/>6. an fortè.<emph type="italics"/>l.<emph.end type="italics"/>26. lumine.<emph type="italics"/>l.<emph.end type="italics"/>39: fori.<emph type="italics"/>p.<emph.end type="italics"/>268, <emph type="italics"/>l.<emph.end type="italics"/>40. rectam. <lb/><emph type="italics"/>p.<emph.end type="italics"/>269.<emph type="italics"/>l,<emph.end type="italics"/> 7. e&longs;t minor 3 1/2 & eius quadr.minus 31.<emph type="italics"/>l.<emph.end type="italics"/>8. e&longs;t 8.<emph type="italics"/>l<emph.end type="italics"/> 9. igitur hæc. <emph type="italics"/>l.<emph.end type="italics"/>14. <emph type="italics"/>dele<emph.end type="italics"/><lb/>non <emph type="italics"/>in hac pa.& &longs;up. </s> <s id="N2BA9B">legs <foreign lang="greek">g</foreign> pro n. </s> <s id="N2BAA2">p.<emph.end type="italics"/>270. <emph type="italics"/>l.<emph.end type="italics"/>8.aliæ. <emph type="italics"/>p.<emph.end type="italics"/>273.<emph type="italics"/>l.<emph.end type="italics"/>9. lineam LM. <emph type="italics"/>p.<emph.end type="italics"/>274.<emph type="italics"/>t.<emph.end type="italics"/>6.<emph type="italics"/>l.<emph.end type="italics"/><lb/>17.vnus <emph type="italics"/>p.<emph.end type="italics"/>275.<emph type="italics"/>l.<emph.end type="italics"/>13.<emph type="italics"/>dele.<emph.end type="italics"/>A, <emph type="italics"/>l.<emph.end type="italics"/>21.<emph type="italics"/>dele<emph.end type="italics"/> non, <emph type="italics"/>l.<emph.end type="italics"/>25. vix in.<emph type="italics"/>p.<emph.end type="italics"/>276.<emph type="italics"/>l.<emph.end type="italics"/>1.LM.<emph type="italics"/>p.<emph.end type="italics"/>278.<emph type="italics"/>t.<emph.end type="italics"/>15.<emph type="italics"/>l.<emph.end type="italics"/>7. QR. <lb/><emph type="italics"/>p.<emph.end type="italics"/>279.<emph type="italics"/>l.<emph.end type="italics"/>2.locis.<emph type="italics"/>l.<emph.end type="italics"/>9, <expan abbr="q.">que</expan><emph type="italics"/>p.<emph.end type="italics"/>280.<emph type="italics"/>t.<emph.end type="italics"/> 19. <emph type="italics"/>lege<emph.end type="italics"/> L pro T.<emph type="italics"/>p.<emph.end type="italics"/>281.<emph type="italics"/>l.<emph.end type="italics"/>11.&longs;i motus.<emph type="italics"/>l.<emph.end type="italics"/> 14.inten&longs;um.<emph type="italics"/>t.<emph.end type="italics"/>21. <lb/>A.<emph type="italics"/>p,<emph.end type="italics"/> 283. <emph type="italics"/>t.<emph.end type="italics"/>29.<emph type="italics"/>l.<emph.end type="italics"/>2. DC.<emph type="italics"/>t.<emph.end type="italics"/>30.<emph type="italics"/>l.<emph.end type="italics"/>5. C &longs;ur&longs;um.<emph type="italics"/>p.<emph.end type="italics"/>284.<emph type="italics"/>t.<emph.end type="italics"/>34.<emph type="italics"/>l.<emph.end type="italics"/>8. à &longs;e. <emph type="italics"/>p.<emph.end type="italics"/> 286. <emph type="italics"/>t.<emph.end type="italics"/> 42.<emph type="italics"/>l.<emph.end type="italics"/>7. cono <lb/><emph type="italics"/>l.<emph.end type="italics"/>4. cuius axis, conus, <emph type="italics"/>p.<emph.end type="italics"/>287.<emph type="italics"/>t.<emph.end type="italics"/>45.<emph type="italics"/>l.<emph.end type="italics"/>7.maior, <emph type="italics"/>p.<emph.end type="italics"/>288.<emph type="italics"/>t.<emph.end type="italics"/>48.<emph type="italics"/>l.<emph.end type="italics"/>18.FC.<emph type="italics"/>p.<emph.end type="italics"/>289.<emph type="italics"/>t.<emph.end type="italics"/>50.<emph type="italics"/>l.<emph.end type="italics"/> 10.ad AE <lb/>permutando, <emph type="italics"/>p.<emph.end type="italics"/>292.<emph type="italics"/>t.<emph.end type="italics"/>57, <emph type="italics"/>l.<emph.end type="italics"/>7. &longs;ubduplæ, <emph type="italics"/>p.<emph.end type="italics"/>293.<emph type="italics"/>t.<emph.end type="italics"/>61.<emph type="italics"/>l.<emph.end type="italics"/>5. A <foreign lang="greek">q</foreign>, <emph type="italics"/>l.<emph.end type="italics"/>6, puncto A, <emph type="italics"/>ibidem lege<emph.end type="italics"/><lb/>Y <emph type="italics"/>pro<emph.end type="italics"/> V.<emph type="italics"/>p.<emph.end type="italics"/>298.<emph type="italics"/>def,<emph.end type="italics"/> 9.<emph type="italics"/>l.<emph.end type="italics"/>1. corpori, <emph type="italics"/>l.<emph.end type="italics"/>6. à moto, <emph type="italics"/>p.<emph.end type="italics"/>299. <emph type="italics"/>l.<emph.end type="italics"/>6. corporis, <emph type="italics"/>l.<emph.end type="italics"/>22. mixtam, <emph type="italics"/>p.<emph.end type="italics"/>300. <lb/><emph type="italics"/>t.<emph.end type="italics"/>2.<emph type="italics"/>l<emph.end type="italics"/> 3. L, <emph type="italics"/>p.<emph.end type="italics"/>131.<emph type="italics"/>l.<emph.end type="italics"/>8. motus, <emph type="italics"/>p.<emph.end type="italics"/>302. <emph type="italics"/>Lem.<emph.end type="italics"/>1, <emph type="italics"/>l.<emph.end type="italics"/>12. æqualibus, <emph type="italics"/>Lem.<emph.end type="italics"/>3. <emph type="italics"/>l.<emph.end type="italics"/> 13. <emph type="italics"/>dele<emph.end type="italics"/> Q, <emph type="italics"/>l.<emph.end type="italics"/>18. <lb/>æquales, <emph type="italics"/>p.<emph.end type="italics"/> 303. <emph type="italics"/>Lem.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>7. &longs;it QR, <emph type="italics"/>l.<emph.end type="italics"/>12. ad quintam, <emph type="italics"/>l<emph.end type="italics"/> 15. Ax.rationem, <emph type="italics"/>l.<emph.end type="italics"/>17. Ax.<emph type="italics"/>Lem.<emph.end type="italics"/><lb/>6.<emph type="italics"/>l.<emph.end type="italics"/>4. <emph type="italics"/>in DG, p.<emph.end type="italics"/>303. <emph type="italics"/>Lem.<emph.end type="italics"/> 10.<emph type="italics"/>l.<emph.end type="italics"/>12. maius, <emph type="italics"/>Lem.<emph.end type="italics"/>12.<emph type="italics"/>l.<emph.end type="italics"/> 4. <emph type="italics"/>dele<emph.end type="italics"/> cuius con&longs;tructionis <emph type="italics"/>l.<emph.end type="italics"/>5. <lb/>TQA, <emph type="italics"/>l.<emph.end type="italics"/>7. quæ AB, <emph type="italics"/>l.<emph.end type="italics"/>8. quad.45.<emph type="italics"/>l.<emph.end type="italics"/>12. BE, <emph type="italics"/>p.<emph.end type="italics"/>306 <emph type="italics"/>in Sch l,<emph.end type="italics"/> 2. <foreign lang="greek">m a</foreign>, YR, <emph type="italics"/>p.<emph.end type="italics"/>307.<emph type="italics"/>Lem<emph.end type="italics"/> 15. <lb/><emph type="italics"/>l.<emph.end type="italics"/>23.ad BG, B 4, <emph type="italics"/>p.<emph.end type="italics"/>308. <foreign lang="greek">u</foreign> <emph type="italics"/>pro <foreign lang="greek">g</foreign> pa&longs;&longs;im, l.<emph.end type="italics"/> 17. vt YZF, <emph type="italics"/>Lem.<emph.end type="italics"/> 16. <emph type="italics"/>l.<emph.end type="italics"/>11. quinam, <emph type="italics"/>p<emph.end type="italics"/> 307. <lb/><emph type="italics"/>l.<emph.end type="italics"/>9. <foreign lang="greek">a</foreign> ad BZ, <emph type="italics"/>p.<emph.end type="italics"/>310.<emph type="italics"/>l.<emph.end type="italics"/>1, recta, <emph type="italics"/>t.<emph.end type="italics"/>8, <emph type="italics"/>l,<emph.end type="italics"/> 2. inæqualia, <emph type="italics"/>l.<emph.end type="italics"/>6. in quo, <emph type="italics"/>p.<emph.end type="italics"/>311.<emph type="italics"/>l.<emph.end type="italics"/>36. 34.grad.<emph type="italics"/>p.<emph.end type="italics"/> 313. <lb/><emph type="italics"/>Cor.<emph.end type="italics"/>3.<emph type="italics"/>l.<emph.end type="italics"/>6.angulum ip&longs;a.<emph type="italics"/>p.<emph.end type="italics"/>316. <emph type="italics"/>l<emph.end type="italics"/> 36. percurritur, <emph type="italics"/>p.<emph.end type="italics"/>317.<emph type="italics"/>t.<emph.end type="italics"/>16. fig.3. Tab.4.<emph type="italics"/>l.<emph.end type="italics"/>4, tempore <lb/>æquali, <emph type="italics"/>p.<emph.end type="italics"/>319.<emph type="italics"/>t.<emph.end type="italics"/> 20.<emph type="italics"/>l.<emph.end type="italics"/>13. ad H <emph type="italics"/>t.<emph.end type="italics"/>22.<emph type="italics"/>l.<emph.end type="italics"/> 3.enim, <emph type="italics"/>l,<emph.end type="italics"/> 4. impetus quo a&longs;cendat in <foreign lang="greek">w</foreign> dele hæc <lb/>verba haud dubiè per arcum ferretur in <foreign lang="greek">w</foreign> <emph type="italics"/>p.<emph.end type="italics"/>320.<emph type="italics"/>l.<emph.end type="italics"/> 1. perueniet in <foreign lang="greek">q</foreign> <emph type="italics"/>l.<emph.end type="italics"/>4. C fertur, <emph type="italics"/>t,<emph.end type="italics"/> 23. <lb/><emph type="italics"/>l<emph.end type="italics"/> 1, ni&longs;it, <emph type="italics"/>p.<emph.end type="italics"/> 321 <emph type="italics"/>l.<emph.end type="italics"/>6, &longs;patiis, <emph type="italics"/>l.<emph.end type="italics"/>15.primo a&longs;cen&longs;u, <emph type="italics"/>l<emph.end type="italics"/> 34. ferri, <emph type="italics"/>p.<emph.end type="italics"/> 322.<emph type="italics"/>t.<emph.end type="italics"/>26.<emph type="italics"/>l.<emph.end type="italics"/>6.primæ, &longs;ecun­<lb/>dæ, <emph type="italics"/>p<emph.end type="italics"/> 323.<emph type="italics"/>l.<emph.end type="italics"/> 34. ignota, <emph type="italics"/>p.<emph.end type="italics"/> 324.<emph type="italics"/>l vlt.<emph.end type="italics"/> prima, <emph type="italics"/>p.<emph.end type="italics"/>326. <emph type="italics"/>cor.<emph.end type="italics"/>3.<emph type="italics"/>l.<emph.end type="italics"/>2. ita vt, <emph type="italics"/>p.<emph.end type="italics"/>327.<emph type="italics"/>cor.<emph.end type="italics"/> 5 <emph type="italics"/>l.<emph.end type="italics"/>4. de­<lb/>&longs;cenderet, <emph type="italics"/>l<emph.end type="italics"/> 33.ferri, <emph type="italics"/>p.<emph.end type="italics"/>329 <emph type="italics"/>l.<emph.end type="italics"/>5. medullaceum, <emph type="italics"/>l.<emph.end type="italics"/> 17.quamdam, <emph type="italics"/>l<emph.end type="italics"/> 18. <emph type="italics"/>dele<emph.end type="italics"/> conficiet, <emph type="italics"/>l<emph.end type="italics"/> 19. <lb/>conficiet tres, <emph type="italics"/>l.<emph.end type="italics"/>41. huius motus <emph type="italics"/>p.<emph.end type="italics"/> 331. <emph type="italics"/>cor.<emph.end type="italics"/>2.l.3. in F. <emph type="italics"/>cor.<emph.end type="italics"/>3. <emph type="italics"/>l<emph.end type="italics"/> 1 in hoc & <emph type="italics"/>cor.<emph.end type="italics"/> 5.<emph type="italics"/>l.<emph.end type="italics"/>3. <lb/>quia enim, <emph type="italics"/>cor.<emph.end type="italics"/>6.<emph type="italics"/>l.<emph.end type="italics"/>1. &longs;u&longs;pendatur, <emph type="italics"/>p.<emph.end type="italics"/>332.<emph type="italics"/>l.<emph.end type="italics"/>3. pondus, C. <emph type="italics"/>t.<emph.end type="italics"/>41. <emph type="italics"/>l.<emph.end type="italics"/> 5. puncto, <emph type="italics"/>p.<emph.end type="italics"/> 333. <emph type="italics"/>l.<emph.end type="italics"/> 2. <lb/>quam maiores, <emph type="italics"/>p.<emph.end type="italics"/>334. <emph type="italics"/>def.<emph.end type="italics"/> 1 <emph type="italics"/>l.<emph.end type="italics"/>3. curuam, <emph type="italics"/>def<emph.end type="italics"/> 2.<emph type="italics"/>l.<emph.end type="italics"/> 3. ex duobus rectis & <emph type="italics"/>p<emph.end type="italics"/> 335 <emph type="italics"/>t.<emph.end type="italics"/> 1.<emph type="italics"/>l<emph.end type="italics"/> 12. <lb/>LQA. <emph type="italics"/>p.<emph.end type="italics"/> 336 <emph type="italics"/>l<emph.end type="italics"/> 2. vel MI, <emph type="italics"/>l.<emph.end type="italics"/> 4. & motus <emph type="italics"/>cor.<emph.end type="italics"/>1 <emph type="italics"/>l<emph.end type="italics"/> 6, L <foreign lang="greek">g</foreign>, <emph type="italics"/>cor.<emph.end type="italics"/> 2. <emph type="italics"/>l<emph.end type="italics"/> 3. AC, & <emph type="italics"/>l<emph.end type="italics"/> 4.& quo <lb/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>2 <emph type="italics"/>p<emph.end type="italics"/> 2 <emph type="italics"/>cor<emph.end type="italics"/> 5.<emph type="italics"/>l<emph.end type="italics"/> 4. LH, & <emph type="italics"/>p.<emph.end type="italics"/>337. <emph type="italics"/>cor.<emph.end type="italics"/>6.<emph type="italics"/>l<emph.end type="italics"/> 5.<emph type="italics"/>dele,<emph.end type="italics"/> vt <emph type="italics"/>l<emph.end type="italics"/> 6.Z, 9, <emph type="italics"/>l.<emph.end type="italics"/> 10. &longs;inguli. <emph type="italics"/>cor.<emph.end type="italics"/>7 <emph type="italics"/>l.<emph.end type="italics"/><lb/>3.9.grad <emph type="italics"/>p.<emph.end type="italics"/>338. <emph type="italics"/>t.<emph.end type="italics"/> 5 <emph type="italics"/>l.<emph.end type="italics"/> 2. AB, æqualem arcui AV. <emph type="italics"/>l.<emph.end type="italics"/> 3. æqualem XV, id e&longs;t arcum <lb/>&longs;ious AV, &longs;ed, <emph type="italics"/>t.<emph.end type="italics"/>6.<emph type="italics"/>l<emph.end type="italics"/> 2 OPDL 4. OZP, <emph type="italics"/>p.<emph.end type="italics"/>339.<emph type="italics"/>l.<emph.end type="italics"/>4. A. 18. <emph type="italics"/>t.<emph.end type="italics"/>7.<emph type="italics"/>l.<emph.end type="italics"/>4.&longs;inus ex gradu, <emph type="italics"/>l.<emph.end type="italics"/> 11. <lb/>a&longs;&longs;umas, <emph type="italics"/>l<emph.end type="italics"/> 13. vero 1.<emph type="italics"/>t.<emph.end type="italics"/>8. <emph type="italics"/>l<emph.end type="italics"/> 1. rota, <emph type="italics"/>p.<emph.end type="italics"/>340.<emph type="italics"/>l.<emph.end type="italics"/>15. puncta B, punctum B, <emph type="italics"/>l.<emph.end type="italics"/>25. a&longs;cenderet. <emph type="italics"/>l.<emph.end type="italics"/><lb/>39 centro M, <emph type="italics"/>l.<emph.end type="italics"/>41. &longs;implici, <emph type="italics"/>l.<emph.end type="italics"/>42. punctum P, <emph type="italics"/>l.<emph.end type="italics"/>44 circa centrum, <emph type="italics"/>p.<emph.end type="italics"/>341.<emph type="italics"/>n.<emph.end type="italics"/>8.<emph type="italics"/>l<emph.end type="italics"/> 2.illum <lb/>fig.10 tab.4, lib.10 quadrat IA, <emph type="italics"/>l.<emph.end type="italics"/>15. KD, <emph type="italics"/>l.<emph.end type="italics"/>16. HF, <emph type="italics"/>n<emph.end type="italics"/> 9.<emph type="italics"/>l.<emph.end type="italics"/>6. profecto, <emph type="italics"/>p.<emph.end type="italics"/>342. <emph type="italics"/>n.<emph.end type="italics"/>12 <emph type="italics"/>l.<emph.end type="italics"/>4. <lb/>in&longs;uperabilem <emph type="italics"/>l<emph.end type="italics"/> 6.&longs;ibi non, <emph type="italics"/>l.<emph.end type="italics"/>8. non tangit, <emph type="italics"/>p.<emph.end type="italics"/>343 <emph type="italics"/>l.<emph.end type="italics"/>21 huiu&longs;modi contactus, <emph type="italics"/>l.<emph.end type="italics"/>25.DN, <lb/><emph type="italics"/>l.<emph.end type="italics"/>5. ne diu, <emph type="italics"/>l.<emph.end type="italics"/>13. arcu BD, <emph type="italics"/>l<emph.end type="italics"/> 14. contactu medio, <emph type="italics"/>p.<emph.end type="italics"/>344. <emph type="italics"/>dele<emph.end type="italics"/> non n 18.fig 12. tab.4. <emph type="italics"/>l<emph.end type="italics"/> 2. <lb/>imaginarium, <emph type="italics"/>n<emph.end type="italics"/> 20. DC primum, <emph type="italics"/>p.<emph.end type="italics"/>345 <emph type="italics"/>l.<emph.end type="italics"/>8. quod n.24 <emph type="italics"/>l<emph.end type="italics"/> 2. duo, <emph type="italics"/>p.<emph.end type="italics"/>346.n.27. <emph type="italics"/>l<emph.end type="italics"/> 3.in K, <lb/><emph type="italics"/>l<emph.end type="italics"/> 4. &longs;ecet, <emph type="italics"/>l.<emph.end type="italics"/>9. <emph type="italics"/>lege <foreign lang="greek">g</foreign> pro<emph.end type="italics"/> V, &longs;it ZX n.28.<emph type="italics"/>l.<emph.end type="italics"/>4 colliguntur, <emph type="italics"/>p<emph.end type="italics"/> 347.<emph type="italics"/>l<emph.end type="italics"/> 5. puncto D. <emph type="italics"/>in Sch.l. </s> <s id="N2C0F1"><!-- NEW --><lb/>vlt<emph.end type="italics"/> experientia, <emph type="italics"/>p<emph.end type="italics"/> 348.<emph type="italics"/>l, vlt<emph.end type="italics"/> lignea, <emph type="italics"/>p.<emph.end type="italics"/>349 <emph type="italics"/>l.<emph.end type="italics"/>9. nam, <emph type="italics"/>p.<emph.end type="italics"/>350. <emph type="italics"/>t.<emph.end type="italics"/>15. <emph type="italics"/>l.<emph.end type="italics"/> 3. centro A, lege <foreign lang="greek">t</foreign><lb/><emph type="italics"/>pro<emph.end type="italics"/> T, ter, <emph type="italics"/>p.<emph.end type="italics"/>351. <emph type="italics"/>l.<emph.end type="italics"/>1. qui e&longs;t, <emph type="italics"/>n.<emph.end type="italics"/> 5.<emph type="italics"/>l.<emph.end type="italics"/>3 in P, <emph type="italics"/>n.<emph.end type="italics"/>6.<emph type="italics"/>l<emph.end type="italics"/> 3. BGDP, <emph type="italics"/>l.<emph.end type="italics"/>4.p.6. igitur BD e&longs;t qua­<lb/>drupla BV, <emph type="italics"/>l.<emph.end type="italics"/>11. oppo&longs;itorum, <emph type="italics"/>l<emph.end type="italics"/> 12. rectilineo, <emph type="italics"/>lege<emph.end type="italics"/> <foreign lang="greek">t</foreign> pro T bis, <emph type="italics"/>p.<emph.end type="italics"/>352 <emph type="italics"/>n<emph.end type="italics"/> 9 & 10 <emph type="italics"/>pa&longs;­<lb/>&longs;im lege<emph.end type="italics"/> <foreign lang="greek">r</foreign> pro X <emph type="italics"/>n<emph.end type="italics"/> 9.<emph type="italics"/>l.<emph.end type="italics"/>7. ad C, <foreign lang="greek">m</foreign> 10.<emph type="italics"/>l.<emph.end type="italics"/>3. BT non &longs;ingula <foreign lang="greek">r a</foreign> &longs;ingulis <foreign lang="greek">r</foreign> B <emph type="italics"/>t.<emph.end type="italics"/>16.<emph type="italics"/>l.<emph.end type="italics"/>1. rotæ, <lb/>quæ <emph type="italics"/>p.<emph.end type="italics"/>353. <emph type="italics"/>n<emph.end type="italics"/> 5.<emph type="italics"/>l<emph.end type="italics"/> 3. motu, <emph type="italics"/>l<emph.end type="italics"/> 6. triplo maior, <emph type="italics"/>t<emph.end type="italics"/> 17.<emph type="italics"/>l<emph.end type="italics"/> 3.<emph type="italics"/>dele<emph.end type="italics"/> T, <emph type="italics"/>p.<emph.end type="italics"/>354 <emph type="italics"/>n<emph.end type="italics"/> 3 <emph type="italics"/>l.<emph.end type="italics"/>2. configit BG,. <lb/>I 3 <emph type="italics"/>dele<emph.end type="italics"/> I, <emph type="italics"/>n,<emph.end type="italics"/> 6.<emph type="italics"/>l<emph.end type="italics"/> 5.KT, <emph type="italics"/>n.<emph.end type="italics"/>7.<emph type="italics"/>l.<emph.end type="italics"/>3. vt quadrans <emph type="italics"/>l<emph.end type="italics"/> 6. contactus, <emph type="italics"/>t.<emph.end type="italics"/>13. <emph type="italics"/>l.<emph.end type="italics"/>3. <emph type="italics"/>dele<emph.end type="italics"/> 4 <emph type="italics"/>p,<emph.end type="italics"/> 355 <emph type="italics"/>n.<emph.end type="italics"/>2 <emph type="italics"/>l,<emph.end type="italics"/><pb xlink:href="026/01/485.jpg"/>8. VTD, <emph type="italics"/>n.<emph.end type="italics"/>3.<emph type="italics"/>l.<emph.end type="italics"/> nam AV, <emph type="italics"/>n.<emph.end type="italics"/>4. AC, <emph type="italics"/>n.<emph.end type="italics"/> 6 <emph type="italics"/>l.<emph.end type="italics"/>2. TVY, <emph type="italics"/>l.<emph.end type="italics"/>3. radius PCTV &longs;umantur <foreign lang="greek">t g</foreign> Y <lb/>YT: <emph type="italics"/>l.<emph.end type="italics"/>4.6 T <foreign lang="greek">d</foreign>, <emph type="italics"/>n.<emph.end type="italics"/>8.<emph type="italics"/>l.<emph.end type="italics"/>1.PC, <emph type="italics"/>l.<emph.end type="italics"/>5.igitur cum, <emph type="italics"/>p.<emph.end type="italics"/>356.<emph type="italics"/>l.<emph.end type="italics"/>5.rectam in Coroll.ita peccatum e&longs;t <lb/>vt errata ca&longs;tigati vix po&longs;&longs;int <emph type="italics"/>p.<emph.end type="italics"/>358.<emph type="italics"/>n.<emph.end type="italics"/> 5 <emph type="italics"/>l.<emph.end type="italics"/>1. partes areæ, <emph type="italics"/>l<emph.end type="italics"/> 2. conficient, <emph type="italics"/>l.<emph.end type="italics"/>4. mouetur, <lb/><emph type="italics"/>t.<emph.end type="italics"/>20.<emph type="italics"/>n.<emph.end type="italics"/>3. <emph type="italics"/>l.<emph.end type="italics"/>8, cinguntur, <emph type="italics"/>p.<emph.end type="italics"/>359.<emph type="italics"/>l.<emph.end type="italics"/>1. B & C, <emph type="italics"/>n,<emph.end type="italics"/> 11.<emph type="italics"/>l.<emph.end type="italics"/>9. aëris, <emph type="italics"/>p,<emph.end type="italics"/> 360 <emph type="italics"/>n.<emph.end type="italics"/> 14.<emph type="italics"/>l<emph.end type="italics"/> 1.ce&longs;&longs;at motus, <lb/><emph type="italics"/>n.<emph.end type="italics"/> 17. tab. </s> <s id="N2C2F3"><!-- NEW -->5.<emph type="italics"/>n.<emph.end type="italics"/>20. citi&longs;&longs;imus, <emph type="italics"/>n.<emph.end type="italics"/>22.<emph type="italics"/>l<emph.end type="italics"/> 1 ce&longs;&longs;at motus, <emph type="italics"/>n.<emph.end type="italics"/>24.<emph type="italics"/>l.<emph.end type="italics"/>1. &longs;i grauior, <emph type="italics"/>p,<emph.end type="italics"/> 361.<emph type="italics"/>t.<emph.end type="italics"/>21.<emph type="italics"/>n.<emph.end type="italics"/>2. <lb/><emph type="italics"/>l.<emph.end type="italics"/>4. nec dextror&longs;um, <emph type="italics"/>p,<emph.end type="italics"/> 362.<emph type="italics"/>l.<emph.end type="italics"/>1. ip&longs;am DA, velis, <emph type="italics"/>l<emph.end type="italics"/> 2. ex recto, <emph type="italics"/>l<emph.end type="italics"/> 5. motus orbis, <emph type="italics"/>l<emph.end type="italics"/> 11. <lb/>pollant, <emph type="italics"/>t.<emph.end type="italics"/>23.<emph type="italics"/>l<emph.end type="italics"/> 1. plumbi, <emph type="italics"/>l.<emph.end type="italics"/>2. &longs;int, <emph type="italics"/>l.<emph.end type="italics"/>7. quia, <emph type="italics"/>l<emph.end type="italics"/> 9. <foreign lang="greek">a</foreign>, <emph type="italics"/>n.<emph.end type="italics"/> 6.<emph type="italics"/>l.<emph.end type="italics"/>1. adde, <emph type="italics"/>t.<emph.end type="italics"/>25. <emph type="italics"/>l.<emph.end type="italics"/>2. &longs;erpentis, <emph type="italics"/>p.<emph.end type="italics"/><lb/>363.<emph type="italics"/>t.<emph.end type="italics"/>25 <emph type="italics"/>l.<emph.end type="italics"/>13. conoidicus, <emph type="italics"/>p.<emph.end type="italics"/>364.<emph type="italics"/>l.<emph.end type="italics"/>2. ver&longs;us G, <emph type="italics"/>t.<emph.end type="italics"/>27.<emph type="italics"/>n.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>4. motum, <emph type="italics"/>p.<emph.end type="italics"/>365.<emph type="italics"/>n<emph.end type="italics"/> 9.<emph type="italics"/>l<emph.end type="italics"/> 6.rota­<lb/>tæ, <emph type="italics"/>p.<emph.end type="italics"/> 366.<emph type="italics"/>l,<emph.end type="italics"/> 12. re&longs;ilit., <emph type="italics"/>t.<emph.end type="italics"/>29 <emph type="italics"/>l.<emph.end type="italics"/>4. ni&longs;u, <emph type="italics"/>l.<emph.end type="italics"/>10.faciet vero, <emph type="italics"/>l.<emph.end type="italics"/> 14.AI, <emph type="italics"/>l.<emph.end type="italics"/>23.extremitatem.<emph type="italics"/>p.<emph.end type="italics"/>367. <lb/><emph type="italics"/>n.<emph.end type="italics"/>13. <emph type="italics"/>l<emph.end type="italics"/> 4.manus, <emph type="italics"/>p.<emph.end type="italics"/> 368.<emph type="italics"/>l.<emph.end type="italics"/>1. erectam, <emph type="italics"/>n.<emph.end type="italics"/>10.<emph type="italics"/>l.<emph.end type="italics"/>3. quæ, <emph type="italics"/>n.<emph.end type="italics"/>1.<emph type="italics"/>l<emph.end type="italics"/>6 5. libretur, <emph type="italics"/>n.<emph.end type="italics"/> 17.<emph type="italics"/>l.<emph.end type="italics"/> 6. EL, <emph type="italics"/>p.<emph.end type="italics"/><lb/>369.<emph type="italics"/>l,<emph.end type="italics"/> 1. qua, <emph type="italics"/>p.<emph.end type="italics"/>370 <emph type="italics"/>n<emph.end type="italics"/> 24.<emph type="italics"/>l.<emph.end type="italics"/>24. rudiaria <emph type="italics"/>lege pa&longs;&longs;im<emph.end type="italics"/> G <emph type="italics"/>pro<emph.end type="italics"/> C, <emph type="italics"/>t<emph.end type="italics"/> 30.<emph type="italics"/>l<emph.end type="italics"/> 7. vt C, <emph type="italics"/>p.<emph.end type="italics"/>371.<emph type="italics"/>l.<emph.end type="italics"/>6. <lb/>qui, <emph type="italics"/>p.<emph.end type="italics"/>372 <emph type="italics"/>n.<emph.end type="italics"/>13.<emph type="italics"/>l.<emph.end type="italics"/>10. GE cum in I, erit in L, <emph type="italics"/>n.<emph.end type="italics"/>15.<emph type="italics"/>l.<emph.end type="italics"/>4. mitius, <emph type="italics"/>p.<emph.end type="italics"/>373.<emph type="italics"/>l.<emph.end type="italics"/>7.terram <emph type="italics"/>p.<emph.end type="italics"/>374.<emph type="italics"/>t.<emph.end type="italics"/><lb/>33.fig.13 tab.4.<emph type="italics"/>p.<emph.end type="italics"/>375. <emph type="italics"/>lege<emph.end type="italics"/> Q <emph type="italics"/>pro<emph.end type="italics"/> K <emph type="italics"/>pa&longs;&longs;im<emph.end type="italics"/> LB erectæ, <emph type="italics"/>l<emph.end type="italics"/> 1.delineari fig.8.tab.5.<emph type="italics"/>l.<emph.end type="italics"/>16 ita <lb/>vt, <emph type="italics"/>l<emph.end type="italics"/> 17.quadratum AM 16.<emph type="italics"/>l<emph.end type="italics"/> 17. quadratum AO, <emph type="italics"/>p<emph.end type="italics"/> 377.<emph type="italics"/>l<emph.end type="italics"/> 3. nec producitur, <emph type="italics"/>t.<emph.end type="italics"/>1 <emph type="italics"/>l<emph.end type="italics"/> 4.ali­<lb/>quid, <emph type="italics"/>p<emph.end type="italics"/> 378.<emph type="italics"/>l.<emph.end type="italics"/>4 anima, <emph type="italics"/>p<emph.end type="italics"/> 379.<emph type="italics"/>l.<emph.end type="italics"/>1. effectus, <emph type="italics"/>l.<emph.end type="italics"/>8.brachium, <emph type="italics"/>l penult.<emph.end type="italics"/> volæ, <emph type="italics"/>p.<emph.end type="italics"/>380.<emph type="italics"/>t.<emph.end type="italics"/>2 <emph type="italics"/>l.<emph.end type="italics"/>2.ali­<lb/>quid &longs;ic globus pendulus, <emph type="italics"/>p.<emph.end type="italics"/>381.<emph type="italics"/>t.<emph.end type="italics"/>3 <emph type="italics"/>l.<emph.end type="italics"/>5. æquitem capiti, <emph type="italics"/>l<emph.end type="italics"/> 18. imo equus, <emph type="italics"/>l.<emph.end type="italics"/>26 determi­<lb/>nata, <emph type="italics"/>l<emph.end type="italics"/> 36. cruris, <emph type="italics"/>p.<emph.end type="italics"/> 392.<emph type="italics"/>n.<emph.end type="italics"/>10.fig 28.<emph type="italics"/>l<emph.end type="italics"/> 21. omittendus, <emph type="italics"/>n.<emph.end type="italics"/>11.fig.27.<emph type="italics"/>l.<emph.end type="italics"/>9 vt BC <emph type="italics"/>p<emph.end type="italics"/> 383.<emph type="italics"/>l.<emph.end type="italics"/>10. <lb/>facilia, <emph type="italics"/>p.<emph.end type="italics"/>384.<emph type="italics"/>l.<emph.end type="italics"/>4. productum, <emph type="italics"/>p.<emph.end type="italics"/> 385 n.8.<emph type="italics"/>l<emph.end type="italics"/> 10, fune; </s> <s id="N2C5A7"><!-- NEW --><emph type="italics"/>n.<emph.end type="italics"/>9.<emph type="italics"/>l<emph.end type="italics"/> 5. funes, <emph type="italics"/>p.<emph.end type="italics"/>386 <emph type="italics"/>l.<emph.end type="italics"/>15. ad DL, <lb/><emph type="italics"/>n.<emph.end type="italics"/>11, fig.31.<emph type="italics"/>l<emph.end type="italics"/> 7.fig.30 <emph type="italics"/>p.<emph.end type="italics"/>388.<emph type="italics"/>l.<emph.end type="italics"/>3.etiam nauis, <emph type="italics"/>l.<emph.end type="italics"/>11. duo tauri, <emph type="italics"/>t.<emph.end type="italics"/>7.<emph type="italics"/>l.<emph.end type="italics"/>10. &longs;e ip&longs;o, <emph type="italics"/>l<emph.end type="italics"/> 11. corpore <lb/>impul&longs;o, <emph type="italics"/>p.<emph.end type="italics"/>389.<emph type="italics"/>t<emph.end type="italics"/> 8 <emph type="italics"/>l.<emph.end type="italics"/>8. finem, <emph type="italics"/>p.<emph.end type="italics"/>390.<emph type="italics"/>t.<emph.end type="italics"/>11.<emph type="italics"/>l<emph.end type="italics"/> 4, arcus BC, <emph type="italics"/>p.<emph.end type="italics"/>392.<emph type="italics"/>n.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>6.ABC, <emph type="italics"/>p.<emph.end type="italics"/>194.<emph type="italics"/>n.<emph.end type="italics"/>6. <lb/><emph type="italics"/>l<emph.end type="italics"/> 8. conficit, <emph type="italics"/>l.<emph.end type="italics"/>18. &longs;ubduplam, <emph type="italics"/>p.<emph.end type="italics"/>395.<emph type="italics"/>n.<emph.end type="italics"/>8. <emph type="italics"/>l.<emph.end type="italics"/>8. &longs;e iuncto, <emph type="italics"/>p<emph.end type="italics"/> 396.<emph type="italics"/>n.<emph.end type="italics"/>21.<emph type="italics"/>l.<emph.end type="italics"/>6. de <emph type="italics"/>p.<emph.end type="italics"/>399. <emph type="italics"/>l<emph.end type="italics"/> 9. <lb/>proportionem, <emph type="italics"/>n.<emph.end type="italics"/>3 <emph type="italics"/>l.<emph.end type="italics"/>3. vt radix CD, <emph type="italics"/>n.<emph.end type="italics"/>9.<emph type="italics"/>l.<emph.end type="italics"/>4. &longs;it 1 pondus 2. certè, <emph type="italics"/>p<emph.end type="italics"/> 401.<emph type="italics"/>l.<emph.end type="italics"/>6. arcum, <emph type="italics"/>l.<emph.end type="italics"/><lb/>15. circa K, <emph type="italics"/>p.<emph.end type="italics"/>402.<emph type="italics"/>l<emph.end type="italics"/> 2. medium, <emph type="italics"/>l<emph.end type="italics"/> 22. agant, <emph type="italics"/>t.<emph.end type="italics"/>14.<emph type="italics"/>l.<emph.end type="italics"/>9. &longs;int, in hoc Th. affige literas af­<lb/>firas mucroni gladij ipfi capulari pilæ, & vici&longs;&longs;im, <emph type="italics"/>p.<emph.end type="italics"/>403.<emph type="italics"/>n.<emph.end type="italics"/>7.<emph type="italics"/>l.<emph.end type="italics"/>8.æquali vtriu&longs;que, <emph type="italics"/>n.<emph.end type="italics"/> 8. <lb/><emph type="italics"/>l.<emph.end type="italics"/>5. detectum, <emph type="italics"/>p<emph.end type="italics"/> 404 <emph type="italics"/>n.<emph.end type="italics"/>13.<emph type="italics"/>l<emph.end type="italics"/> 3. æquipondium, <emph type="italics"/>n.<emph.end type="italics"/> 15.<emph type="italics"/>l.<emph.end type="italics"/>5. alio, <emph type="italics"/>p.<emph.end type="italics"/>405 <emph type="italics"/>n.<emph.end type="italics"/>18.<emph type="italics"/>l.<emph.end type="italics"/>1. intentetur, <lb/><emph type="italics"/>l.<emph.end type="italics"/>3.extento, <emph type="italics"/>n<emph.end type="italics"/> 19.<emph type="italics"/>l<emph.end type="italics"/> 1. impetens gladius, <emph type="italics"/>t.<emph.end type="italics"/>23 <emph type="italics"/>n.<emph.end type="italics"/>2.<emph type="italics"/>l.<emph.end type="italics"/>2. & eadem altitudo, <emph type="italics"/>p.<emph.end type="italics"/>406. <emph type="italics"/>n.<emph.end type="italics"/>5.<emph type="italics"/>l.<emph.end type="italics"/>2. <lb/>corpore <emph type="italics"/>n.<emph.end type="italics"/> 6 <emph type="italics"/>l.<emph.end type="italics"/>1. ictum, <emph type="italics"/>n.<emph.end type="italics"/>10.<emph type="italics"/>l.<emph.end type="italics"/>2. quæ, <emph type="italics"/>n.<emph.end type="italics"/>11.<emph type="italics"/>l.<emph.end type="italics"/>2. proportio, <emph type="italics"/>l.<emph.end type="italics"/>3: 1000.<emph type="italics"/>p.<emph.end type="italics"/>407.<emph type="italics"/>l.<emph.end type="italics"/>4.gradus, <emph type="italics"/>n.<emph.end type="italics"/><lb/>12.<emph type="italics"/>l.<emph.end type="italics"/>3. <expan abbr="eãdem">eandem</expan>, <emph type="italics"/>p.<emph.end type="italics"/>409.<emph type="italics"/>n.<emph.end type="italics"/>19.fig.20.<emph type="italics"/>l.<emph.end type="italics"/>11. P <foreign lang="greek">n</foreign> N <foreign lang="greek">b g.</foreign><emph type="italics"/>n.<emph.end type="italics"/>22. fig. </s> <s id="N2C7AF">16.<emph type="italics"/>p.<emph.end type="italics"/>410. <emph type="italics"/>n.<emph.end type="italics"/>24.<emph type="italics"/>l.<emph.end type="italics"/>2. mino­<lb/>rem, <emph type="italics"/>lege<emph.end type="italics"/> N, <emph type="italics"/>pro<emph.end type="italics"/> F, <emph type="italics"/>p.<emph.end type="italics"/>411.<emph type="italics"/>l.<emph.end type="italics"/>5. vt <emph type="italics"/>l.<emph.end type="italics"/>6. vt chorda MV, <emph type="italics"/>l.vlt.<emph.end type="italics"/>velociter, <emph type="italics"/>p.<emph.end type="italics"/>402. <emph type="italics"/>t.<emph.end type="italics"/>17.<emph type="italics"/>l.<emph.end type="italics"/>5.ex­<lb/>tendi, <emph type="italics"/>l.<emph.end type="italics"/>12. prædictam, <emph type="italics"/>l.<emph.end type="italics"/>24. imprimit, <emph type="italics"/>l.<emph.end type="italics"/>25. certa.<emph type="italics"/>p.<emph.end type="italics"/>413.<emph type="italics"/>n.<emph.end type="italics"/>4.<emph type="italics"/>l.<emph.end type="italics"/>3. mouentur, <emph type="italics"/>l.<emph.end type="italics"/>7. alium, <lb/><emph type="italics"/>p.<emph.end type="italics"/>414.<emph type="italics"/>l.<emph.end type="italics"/>2. augendum, <emph type="italics"/>n.<emph.end type="italics"/>10.<emph type="italics"/>l.<emph.end type="italics"/>2. tormentaria, <emph type="italics"/>l.<emph.end type="italics"/>3.<emph type="italics"/>n.<emph.end type="italics"/>11. reticulo in fig. </s> <s id="N2C84D">24. tab. </s> <s id="N2C850">5. adhibe <lb/>H &longs;ub G tum L &longs;ub Z, in fig.22. adhibe omnes literas in quadrante AGD, in fig. </s> <s id="N2C855">25. <lb/>tab. </s> <s id="N2C85A">lege C inter BA, <emph type="italics"/>p.<emph.end type="italics"/>415.<emph type="italics"/>l.<emph.end type="italics"/>12. fig.37. tab. </s> <s id="N2C869"><!-- NEW -->3.<emph type="italics"/>n.<emph.end type="italics"/>11.<emph type="italics"/>l.<emph.end type="italics"/>2. quæ, <emph type="italics"/>n.<emph.end type="italics"/> 12.<emph type="italics"/>l.<emph.end type="italics"/> 12. tamen &longs;it <emph type="italics"/>l<emph.end type="italics"/> 14. <lb/>octauo, in <emph type="italics"/>p.<emph.end type="italics"/>416.<emph type="italics"/>n.<emph.end type="italics"/>2.<emph type="italics"/>l.<emph.end type="italics"/>5. corporum, <emph type="italics"/>n.<emph.end type="italics"/>3.<emph type="italics"/>l.<emph.end type="italics"/>1. di&longs;per&longs;io, <emph type="italics"/>l.<emph.end type="italics"/>2. &longs;ic <emph type="italics"/>n.<emph.end type="italics"/>3.<emph type="italics"/>l.<emph.end type="italics"/>2. vannus autem; <emph type="italics"/>l.<emph.end type="italics"/><lb/>12.portu, <emph type="italics"/>n.<emph.end type="italics"/>6.<emph type="italics"/>l.<emph.end type="italics"/>6. quando duo, <emph type="italics"/>p.<emph.end type="italics"/>417.<emph type="italics"/>l.<emph.end type="italics"/>4.impre&longs;&longs;us, <emph type="italics"/>n.<emph.end type="italics"/>8.<emph type="italics"/>l.<emph.end type="italics"/> 11. cadit, <emph type="italics"/>n.<emph.end type="italics"/>9.<emph type="italics"/>l.<emph.end type="italics"/>6.&longs;apo.<emph type="italics"/>n.<emph.end type="italics"/>10.<emph type="italics"/>l.<emph.end type="italics"/>4. <lb/>vnum corpus, <emph type="italics"/>l.<emph.end type="italics"/>8. vnum corpus <emph type="italics"/>p.<emph.end type="italics"/>418.<emph type="italics"/>l.<emph.end type="italics"/>9.luctam, <emph type="italics"/>l.<emph.end type="italics"/> 12. fulminis, <emph type="italics"/>t.<emph.end type="italics"/> 20. <emph type="italics"/>l.<emph.end type="italics"/>2. in plano, <lb/><emph type="italics"/>l.<emph.end type="italics"/>8.re&longs;ilit, <emph type="italics"/>l.<emph.end type="italics"/> 18. pluit, <emph type="italics"/>p.<emph.end type="italics"/> 419. <emph type="italics"/>l.<emph.end type="italics"/> 14. vorticem, <emph type="italics"/>p.<emph.end type="italics"/> 421. <emph type="italics"/>l.<emph.end type="italics"/> 9. lineas, <emph type="italics"/>lege pa&longs;&longs;im<emph.end type="italics"/> po&longs;itio <lb/>propo&longs;itiones, <emph type="italics"/>p.<emph.end type="italics"/>423.<emph type="italics"/>l.<emph.end type="italics"/>7.triangulo, <emph type="italics"/>l.<emph.end type="italics"/>8. IKD, <emph type="italics"/>p.<emph.end type="italics"/>426.<emph type="italics"/>t.<emph.end type="italics"/>15.rig.12. <emph type="italics"/>t.<emph.end type="italics"/>16. <emph type="italics"/>l.<emph.end type="italics"/>5. perpendi­<lb/>culares,. 427.<emph type="italics"/>l.<emph.end type="italics"/>6. AH, <emph type="italics"/>cor.<emph.end type="italics"/>2.fig.14.<emph type="italics"/>p.<emph.end type="italics"/>430<emph type="italics"/>l.<emph.end type="italics"/> 32. CNAP, <emph type="italics"/>t.<emph.end type="italics"/>22.<emph type="italics"/>l.<emph.end type="italics"/>4. D vt. </s> </p> <p id="N2C9A5" type="main"> <s id="N2C9A7"><emph type="center"/><emph type="italics"/>FINIS,<emph.end type="italics"/><emph.end type="center"/></s> </p> </chap> <pb xlink:href="026/01/486.jpg"/> </body> <back> <section> <pb xlink:href="026/01/487.jpg"/> <figure id="id.026.01.487.1.jpg" xlink:href="026/01/487/1.jpg"/> <p id="N2C9BF" type="head"> <s id="N2C9C1"> TABVLA I </s> </p> <pb xlink:href="026/01/488.jpg"/> <figure id="id.026.01.488.1.jpg" xlink:href="026/01/488/1.jpg"/> <p id="N2C9CC" type="head"> <s id="N2C9CE"> TABVLA 2 </s> </p> <pb xlink:href="026/01/489.jpg"/> <figure id="id.026.01.489.1.jpg" xlink:href="026/01/489/1.jpg"/> <p id="N2C9D9" type="head"> <s id="N2C9DB"> TABVLA TERTIA </s> </p> <pb xlink:href="026/01/490.jpg"/> <figure id="id.026.01.490.1.jpg" xlink:href="026/01/490/1.jpg"/> <p id="N2C9E6" type="head"> <s id="N2C9E8"> TABVLA QVARTA </s> </p> <pb xlink:href="026/01/491.jpg"/> <figure id="id.026.01.491.1.jpg" xlink:href="026/01/491/1.jpg"/> <p id="N2C9F3" type="head"> <s id="N2C9F5"> TABVLA QVINTA </s> </p> </section> </back> </text> </archimedes>